Abstract:
Method for calculating or optimizing a spectacle lens, including
       specifying at least one surface for the spectacle lens to be calculated or optimized;   determining the course of a main ray through at least one visual point of the at least one surface;   determining a first primary set and a second primary set of coefficients of the local aberration of a local wavefront;   specifying at least one function which assigns a second secondary set of coefficients to a second primary set of coefficients, said second secondary set of coefficients defining the higher-order aberration of a propagated wavefront;   determining a higher-order aberration of a local wavefront propagated starting from the at least one visual point along the main ray depending on at least the second primary set of coefficients on the basis of the specified function; and   calculating or optimizing the at least one surface of the spectacle lens based on the determined higher-order aberration of the propagated local wavefront.

Description:
BACKGROUND 
     The present invention relates to a method, a device, and a corresponding computer program product for calculating (optimizing) and producing a spectacle lens taking into consideration higher-order aberrations of both the eye and the spectacle lens. 
     For the production or optimization of spectacle lenses, in particular of individual spectacle lenses, each spectacle lens is manufactured such that the best possible correction of a refractive error of the respective eye of the spectacles wearer is obtained for each desired direction of sight or each desired object point. In general, a spectacle lens is said to be fully corrective for a given direction of sight if the values sphere, cylinder, and axis of the wavefront upon passing the vertex sphere match with the values for sphere, cylinder, and axis of the prescription for the eye having the visual defect. In the refraction determination for an eye of a spectacles wearer, dioptric values (particularly sphere, cylinder, cylinder axis) for a far (usually infinite) distance and optionally (for multifocal or progressive lenses) an addition for a near distance (e.g. according to DIN 58208) are determined. In this way, the prescription (in particular sphere, cylinder, cylinder axis, and optionally addition) that is sent to a spectacles manufacturer is stipulated. In modern spectacle lenses, object distances deviating from the standard, which are used in the refraction determination, can be indicated additionally. 
     However, a full correction for all directions of sight at the same time is normally not possible. Therefore, the spectacle lenses are manufactured such that they achieve a good correction of visual defects of the eye and only small aberrations in the main regions of use, especially in central visual regions, while larger aberrations are permitted in peripheral regions. 
     In order to be able to manufacture a spectacle lens in this way, the spectacle lens surfaces or at least one of the spectacle lens surfaces is first of all calculated such that the desired distribution of the unavoidable aberrations is effected thereby. This calculation and optimization is usually performed by means of an iterative variation method by minimization of a target function. As a target function, particularly a function F having the following functional connection with the spherical power S, the magnitude of the cylindrical power Z, and the axis of the cylinder a (also referred to as “SZA” combination) is taken into account and minimized: 
     
       
         
           
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     In the target function F, at the evaluation points i of the spectacle lens, at least the actual refractive deficits of the spherical power S Δ,i  and the cylindrical power Z Δ,i  as well as target values for the refractive deficits of the spherical power S Δ,i,target  and the cylindrical power Z Δ,i,target  are taken into consideration. 
     It was found in DE 103 13 275 that it is advantageous to not indicate the target values as absolute values of the properties to be optimized, but as their deviation from the prescription, i.e. as the required misadjustment. This has the advantage that the target values are independent of the prescription (Sph V ,Zyl V ,Axis V ,Pr V ,B V ) and that the target values do not have to be changed for every individual prescription. Thus, as “actual” values of the properties to be optimized, not absolute values of these optical properties are taken into account in the target function, but the deviations from the prescription. This has the advantage that the target values can be specified independent of the prescription and do not have to be changed for every individual prescription. 
     The respective refractive deficits at the respective evaluation points are preferably taken into consideration with weighting factors g i,SΔ  and g i,ZΔ . Here, the target values for the refractive deficit of the spherical power S Δ,i,target  and/or the cylindrical power Z Δ,i,target , particularly together with the weighting factor g i,SΔ  and g i,ZΔ , form the so-called spectacle lens design. In addition, particularly further residues, especially further parameters to be optimized, such as coma and/or spherical aberration and/or prism and/or magnification and/or anamorphic distortion, etc., can be taken into consideration, which is particularly implied by the expression “+ . . . ”. 
     In some cases, this can contribute to a clear improvement particularly of an individual adjustment of a spectacle lens if in the optimization of the spectacle lens not only aberrations up to the second order (sphere, magnitude of astigmatism, and cylinder axis), but also higher-order aberrations (e.g. coma, trefoil, spherical aberration) are taken into consideration. 
     It is known from the prior art to determine the shape of a wavefront for optical elements and particularly spectacle lenses that are delimited by at least two refractive boundary surfaces. For example, this can be done by means of a numerical calculation of a sufficient number of neighboring rays, along with a subsequent fit of the wavefront data by Zernike polynomials. Another approach is based on local wavefront tracing in the refraction (cf. WO 2008/089999 A1). Here, only one single ray (the main ray) per visual point is calculated, accompanied by the derivatives of the vertex depth of the wavefront according to the transversal coordinates (perpendicular to the main ray). These derivatives can be formed up to a specific order, wherein the second derivatives describe the local curvature properties of the wavefront (such as refractive power, astigmatism) and the higher derivatives are connected with the higher-order aberrations. 
     In the tracing of light through a spectacle lens, the local derivatives of the wavefront are calculated at a suitable position in the course of the ray in order to compare them with desired values obtained from the refraction of the spectacle lens wearer. This position can be the vertex sphere, for example. In this respect, it is assumed that a spherical wavefront starts at the object point and propagates up to the first spectacle lens surface. There, the wavefront is refracted and subsequently propagates to the second spectacle lens surface, where it is refracted again. If further surfaces exist, the alternation of propagation and refraction will be continued until the last boundary surface has been passed. The last propagation takes place from this last boundary surface to the vertex sphere. 
     WO 2008/089999 A1 discloses the laws of refraction at refractive surfaces not only for aberrations or optical properties of second order, but also for higher orders. If a wavefront with local derivatives known up to a specific order is obliquely incident on a boundary surface, the vertex depth of which can itself be described by known local derivatives up to the same order, then the local derivatives of the outgoing wavefront can be calculated up to the same order with the calculation methods according to WO 2008/089999 A1. Such a calculation, particularly up to the second order, is very helpful for assessing the image formation properties or optical properties of a spectacle lens in the wearing position. Specifically, such a calculation is of great importance if a spectacle lens is to be optimized in the wearing position over all visual points. 
     Even if the process of refraction can be described and calculated very efficiently therewith, the consideration of higher-order aberrations remains very expensive nevertheless, since especially the required iterative ray tracing for the propagation of the wavefronts involves great computing effort. 
     SUMMARY 
     It is the object of the invention to provide an improved method for calculating or optimizing a spectacle lens, preferably a progressive spectacle lens, wherein the spectacle lens is adapted to the individual needs of the spectacles wearer in an improved way. This object is solved by a computer-implemented method, a device, a computer program product, and a storage medium with the features indicated in the independent claims. Preferred embodiments are subject of the dependent claims. 
     According to a first aspect, the invention provides a computer-implemented method for calculating or optimizing a spectacle lens. First of all, the method comprises specifying at least one surface for the spectacle lens to be calculated or optimized. This surface particularly serves as a starting surface for the further individual optimization. Preferably, the surface is the back surface of the spectacle lens. Preferably, a corresponding starting surface is specified both for the front surface and for the back surface of the spectacle lens. In a preferred embodiment, only one surface is iteratively varied or optimized during the optimization process. The other surface of the spectacle lens may be a simple spherical or rotationally symmetric aspherical surface. However, it is also possible to optimize both surfaces. 
     Starting from the at least one specified surface, the method comprises determining the course of a main ray through at least one visual point (i) of the at least one surface. Particularly preferably, the course of the main ray is determined starting from an object point through the spectacle lens up to a reference surface or assessment surface or comparative surface, in particular the vertex sphere. 
     At the at least one visual point (i), a first primary set (s io ) and a second primary set (e iok ) of coefficients of the local aberration of a local wavefront going out from the at least one visual point (i) are determined in a vicinity of the main ray, wherein the first primary set of coefficients (s io ) specifies the spherical and astigmatic aberration of the outgoing local wavefront and the second primary set of coefficients (e iok ) specifies at least one further higher-order aberration (k&gt;2) of the outgoing local wavefront. The outgoing wavefront (or later also referred to as the original wavefront) is the result of the refraction of light on the at least one surface. If the one surface is the front surface, the outgoing wavefront preferably reflects the local refraction properties of the surface in addition to the object distance (and the thus involved vergence of the wavefront before hitting said surface). The process of refraction could be determined particularly by analogy with the description in WO 2008/089999 A1 or in J. Opt. Soc. Am. A 27, 218-237. 
     Now, the invention suggests specifying a function e pk =ƒ(e ok ) which assigns a second secondary set of coefficients (e pk ) to a second primary set of coefficients (e ok ), said second secondary set of coefficients specifying the at least one higher-order aberration of a propagated wavefront. Specifically, by means of this function, it is very easily possible to determine the propagation of a wavefront on the basis of coefficients of the original wavefront. 
     Thus, the method according to the invention comprises determining at least one higher-order aberration of a local wavefront propagated starting from the at least one visual point (i) along the main ray depending on at least the second primary set of coefficients (e iok ) on the basis of the specified function e pk =ƒ(e ok ). Specifically, to this end, a first and a second secondary set of coefficients of the propagated wavefront are determined. While the spherical and astigmatic curvature of the wavefront, i.e. the first secondary set of coefficients, are determined particularly on the basis of the first primary set of coefficients basically in a known manner, the invention suggests a very efficient method for determining the second secondary set of coefficients as well without being dependent on complex ray tracing like in conventional methods. 
     For the benefit of a clear representation of aspects of the invention, the following terminology for the designation of the coefficients will be used within the scope of this description. Here, the coefficients or sets of coefficients used or determined for describing the outgoing (original) local wavefront will be referred to as “primary” coefficients or “primary” sets of coefficients. Accordingly, the coefficients or sets of coefficients used or determined for describing the propagated local wavefront will be referred to as “secondary” coefficients or “secondary” sets of coefficients. Moreover, the coefficients or sets of coefficients used or determined for describing the spherical and astigmatic aberration of the respective local wavefront will be referred to as “first” coefficients or “first” sets of coefficients, while the coefficients or sets of coefficients used or determined for describing the aberrations, i.e. higher-order aberrations, deviating from the spherical and astigmatic aberrations will be referred to as “second” coefficients or “second” sets of coefficients. 
     After the refraction-induced influence of the spectacle lens on the wavefront was very efficiently determined for higher-order aberrations (as described in e.g. WO 2008/089999 A1 or J. Opt. Soc. Am. A 27, 218-237), the method according to the invention now comprises calculating or optimizing the at least one surface of the spectacle lens taking into consideration the determined aberration or higher-order aberrations of the propagated local wavefront. Preferably, a comparison of the influence on the wavefront, caused by the spectacle lens to be optimized, with the required prescription is performed at the vertex sphere taking higher-order aberrations into consideration. Depending on the deviation, at least one of the surfaces of the spectacle lens is varied and the process is repeated correspondingly until the spectacle lens comes sufficiently close to the required power. 
     In this way, the invention allows a very fast and precise consideration of higher-order aberrations with clearly reduced computing effort. Thus, an improvement of the individual adjustment of a spectacle lens can be achieved without substantially increasing the computing effort. 
     Preferably, specifying at least one function e pk =ƒ(e ok ) comprises specifying a linear function e pk =B k (e ok +r k ), or the function e pk =ƒ(e ok ) is specified as a linear function e pk =B k (e ok +r k ) in which the proportionality term B k  depends on the first primary set of coefficients (s io ), but not on the second primary set of coefficients (e ok ). Preferably, the remainder term r k  does also not depend on the second primary set of coefficients (e ok ). Particularly preferably, it holds that r k =0, i.e. e pk =B k e ok . These embodiments result in a very efficient optimization particularly for the consideration of third-order aberrations (k=3). 
     For a particularly efficient consideration of third-order aberrations (e.g. coma), it is preferred that determining a first primary set of coefficients comprise determining a power vector 
                 s   o     =     (           S   oxx               S   axy               S   oyy           )       ,         
wherein determining a second primary set of coefficients comprises determining a coma vector
 
                 e     o   ⁢           ⁢   3       =     (           E   oxxx               E   oxxy               E   oxyy               E   oyyy           )       ,         
and wherein the function
 
               e     p   ⁢           ⁢   3       =         B   3     ⁢     e     o   ⁢           ⁢   3         =         γ   3     ⁡     (           β   y     -   3             3   ⁢     β   y     -   2       ⁢     d   n     ⁢     S   xy             3   ⁢         β   y     -   1       ⁡     (       d   n     ⁢     S   xy       )       2               (       d   n     ⁢     S   xy       )     3                 β   y     -   2       ⁢     d   n     ⁢     S   xy               β   y     -   1       ⁡     (       1   γ     +     3   ⁢       (       d   n     ⁢     S   xy       )     2         )                 2   γ     ⁢     d   n     ⁢     S   xy       +     3   ⁢       (       d   n     ⁢     S   xy       )     3                   β   x     -   1       ⁡     (       d   n     ⁢     S   xy       )       2                   β   y     -   1       ⁡     (       d   n     ⁢     S   xy       )       2               2   γ     ⁢     d   n     ⁢     S   xy       +     3   ⁢       (       d   n     ⁢     S   xy       )     3                 β   x     -   1       ⁡     (       1   γ     +     3   ⁢       (       d   n     ⁢     S   xy       )     2         )               β   x     -   2       ⁢     d   n     ⁢     S   xy                   (       d   n     ⁢     S   xy       )     3           3   ⁢         β   x     -   1       ⁡     (       d   n     ⁢     S   xy       )       2             3   ⁢     β   x     -   2       ⁢     d   n     ⁢     S   xy             β   y     -   3             )       ⁢     e     o   ⁢           ⁢   3                           with   ⁢           ⁢   γ     =     1       1   -       d   n     ⁢     S   oxx       -       (       d   n     ⁢     S   oxy       )     2     -       d   n     ⁢     S   oyy       +         (     d   n     )     2     ⁢     S   oxx     ⁢     S   oyy         )         ,       β   x     =         1     1   -       d   n     ⁢     S   oxx           ⁢           ⁢   and   ⁢           ⁢     β   y       =     1     1   -       d   n     ⁢     S   oyy                     
is specified as at least one function e p3 =ƒ(e o3 ).
 
     In a preferred embodiment, determining the second primary set of coefficients (e iok ) comprises determining at least a plurality of primary subsets (e iok , for k=3, 4, . . . ) of the second primary set of coefficients (e iok ). Here, specifying at least one function e pk =ƒ(e ok ) preferably comprises specifying a linear function e pk =B k (e ok +r k ) for each subset (i.e. for each k=3, 4, . . . ) such that for each of the linear functions (i.e. for each k=3, 4, . . . ) the proportionality term B k  depends on the first primary set of coefficients (s io ), but not on the second primary set of coefficients (e ok ). Preferably, the remainder term r k     0    of a linear function for the subset of the order k 0  does also not depend on the coefficients of the respective subset k 0 , but at most on coefficients of the preceding subset or the preceding subsets (i.e. the coefficients of the order k&lt;k 0 ). To this end, the subsets (e iok , for k=3, 4, . . . ) of the second primary set of coefficients (e iok ) are preferably determined such that they form a series of consecutive subsets in a specific or specifiable sequence of ascending orders. In particular, each subset represents aberrations of the corresponding order, which corresponds e.g. to the order (k=3, 4, . . . ) of associated Zernike polynomials of the respective aberrations. 
     In a particularly preferred embodiment, determining the second primary set of coefficients (e iok ) comprises determining at least one first primary subset (e io3 ) and one second primary subset (e io4 ) of the second primary set of coefficients (e iok ). Here, specifying at least one function e pk =ƒ(e ok ) preferably comprises specifying a first linear function e p3 =B 3 (e o3 +r 3 ) in which the proportionality term B 3  depends on the first primary set of coefficients (s io ), but not on the second primary set of coefficients (e ok ). Preferably, the remainder term r 3  does also not depend on the second primary set of coefficients (e ok ). Particularly preferably, it even holds that r 3 =0, i.e. e p3 =B 3 e o3 . In addition, specifying at least one function e pk =ƒ(e ok ) in this embodiment preferably comprises specifying a second linear function e p4 =B 4 (e o4 +r 4 ) in which the proportionality term B 4  depends on the first primary set of coefficients (s io ), but not on the second primary set of coefficients (e ok ). Preferably, the remainder term r 4  does not depend on the second primary subset (e io4 ) of the second primary set of coefficients (e ok ). This results in a very efficient optimization particularly for the consideration of fourth-order aberrations (k=4). Particularly preferably, the remainder term r 4  of the second linear function depends on the first primary subset (e io3 ) of the second primary set of coefficients (e ok ). 
     For a particularly efficient consideration of fourth-order aberrations (e.g. spherical aberration), it is preferred that determining a first primary set of coefficients comprise determining a power vector 
                 s   o     =     (           S   oxx               S   axy               S   oyy           )       ,         
wherein determining a second primary set of coefficients comprises determining a coma vector
 
               e     o   ⁢           ⁢   3       =     (           E   oxxx               E   oxxy               E   oxyy               E   oyyy           )           
and determining a spherical aberration vector
 
                 e     o   ⁢           ⁢   4       =     (           E   oxxxx               E   oxxxy               E   oxxyy               E   oxyyy               E   oyyyy           )       ,         
and
 
wherein the function
 
               e     p   ⁢           ⁢   4       =       (           β   x   4         …                   …       0           ⋮           β   x   3     ⁢     β   y   1                                   ⋮                                       β   x   2     ⁢     β   y   2                                       ⋮                                   β   x   1     ⁢     β   y   3           ⋮           0                               …         β   y   4           )     ⁢     (       e     o   ⁢           ⁢   4       +       d   n     ⁢     (           3   ⁢     (         β   x     ⁢     E   oxxx   2       +       β   y     ⁢     E   oxxy   2       -       S   oxx   4       n   2         )                 3   ⁢           ⁢       E   oxxy     ⁡     (         β   x     ⁢     E   oxxx       +       β   y     ⁢     E   oxyy         )                       β   x     ⁡     (       2   ⁢           ⁢     E   oxxy   2       +       E   oxxx     ⁢     E   oxyy         )       +       β   y     ⁡     (       2   ⁢           ⁢     E   oxyy   2       +       E   oxxy     ⁢     E   oyyy         )       -       (         S   oxx     ⁢     S   oyy       n     )     2                 3   ⁢           ⁢       E   oxyy     ⁡     (         β   x     ⁢     E   oxxy       +       β   y     ⁢     E   oyyy         )                   3   ⁢     (         β   x     ⁢     E   oxyy   2       +       β   y     ⁢     E   oyyy   2       -       S   oyy   4       n   2         )             )         )                           ⁢       with   ⁢           ⁢     β   x       =         1     1   -       d   n     ⁢     S   oxx           ⁢           ⁢   and   ⁢           ⁢     β   y       =     1     1   -       d   n     ⁢     S   oyy                     
is specified as at least one function e pk =ƒ(e ok ). Here, the coefficients of the coma vector particularly form the first primary subset, and the coefficients of the spherical aberration vector particularly form the second primary subset of the second primary set of coefficients.
 
     Preferably, the method further comprises determining an angle α between a first plane of refraction of the main ray at a first surface of the spectacle lens and a second plane of refraction of the main ray at a second surface of the spectacle lens, wherein determining a higher-order aberration comprises
         determining a second secondary set (e ipk ) of coefficients of the local aberration of the propagated wavefront; and   determining a transformed second secondary set ({tilde over (e)} ipk ) of coefficients depending on the determined angle α, particularly by applying a rotation matrix R k (α). Thus, for each refraction step, a Cartesian coordinate system can preferably be selected such that an axis of a Cartesian coordinate system is perpendicular to the plane of refraction and remains unchanged upon transition through the refractive surface, which results in an easy calculation of the refraction process.       

     Preferably, the method further comprises collecting prescription or refraction data V, wherein the prescription data comprises data with respect to the spherical power Sph V , the magnitude of the astigmatism Zyl V , the astigmatism axis Axis V , as well as at least one further predetermined higher-order refraction HOA V . 
     Preferably, the method comprises minimizing a target function in which values of higher-order aberrations are explicitly taken into consideration, e.g. in the form of 
     
       
         
           
             
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         R actual (i) actual refractive error (difference spectacle lens and eye) at the i th  evaluation point 
         R target (i) required refractive error at the i th  evaluation point 
         G R,i  weighting of the refractive error at the i th  evaluation point 
         A actual (i) actual astigmatic difference (difference spectacle lens and eye) at the i th  evaluation point 
         A target (i) required astigmatic difference at the i th  evaluation point 
         G A,i  weighting of the astigmatic difference at the i th  evaluation point
 
and additionally
 
         C actual  (i) actual difference of the coma of the spectacle lens and the eye at the i th  evaluation point 
         C target (i) required difference of the coma at the i th  evaluation point 
         G C,i  weighting of the coma at the i th  evaluation point 
         S actual  (i) actual difference of the spherical aberration of the spectacle lens and the eye at the i th  evaluation point 
         S target (i) required difference of the spherical aberration at the i th  evaluation point 
         G S,i  weighting of the spherical aberration at the i th  evaluation point. 
       
    
     If the target function is extended, the degree of overdetermination will increase, whereby the stability of the optimization method can deteriorate. Therefore, it is preferred that the target function is not extended, i.e. that the higher-order aberrations are not explicitly taken into account in the target function, but that both the higher-order aberrations of the eye and of the spectacle lens are taken into account in the respective spherocylindrical values. In this case, the method preferably comprises minimizing a target function in which values of higher-order aberrations are implicitly taken into consideration over transformed values of the refractive error and the astigmatic difference, e.g. in the form of 
     
       
         
           
             
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         R actual,t (i) transformed refractive error (difference spectacle lens and eye) at the i th  evaluation point 
         R target (i) required refractive error at the i th  evaluation point 
         G R,i  weighting of the refractive error at the i th  evaluation point 
         A actual,t (i) transformed astigmatic difference (difference spectacle lens and eye) at the i th  evaluation point 
         A target (i) required astigmatic difference at the i th  evaluation point 
       
    
     G A,i  weighting of the astigmatic difference at the i th  evaluation point. 
     In a preferred embodiment, the method comprises determining a transformed first secondary set of coefficients from the first secondary set of coefficients and the second secondary set of coefficients. Specifically, a transformed first secondary coefficient is determined for every first secondary coefficient of the set of first secondary coefficients in dependence on at least one coefficient of the second secondary set of coefficients, i.e. in dependence on at least one higher-order aberration. The transformed first secondary coefficients then specifically define the transformed spherocylindrical refraction. 
     For specifying the way in which the transformed spherocylindrical refraction (in minus cylinder notation) depends on the higher-order aberrations in preferred embodiments, preferably the following function are provided: 
     
       
         
           
             
               sph 
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                   4 
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                   r 
                   0 
                   2 
                 
               
               ⁢ 
               
                 ( 
                 
                   u 
                   + 
                   
                     
                       
                         2 
                       
                       ⁢ 
                       
                         
                           
                             v 
                             2 
                           
                           + 
                           
                             w 
                             2 
                           
                         
                       
                     
                     2 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               zyl 
               ⁡ 
               
                 ( 
                 
                   v 
                   , 
                   
                     w 
                     ; 
                     
                       r 
                       0 
                     
                   
                 
                 ) 
               
             
             = 
             
               
                 - 
                 
                   
                     4 
                     ⁢ 
                     
                       3 
                     
                   
                   
                     r 
                     0 
                     2 
                   
                 
               
               ⁢ 
               
                 2 
               
               ⁢ 
               
                 
                   
                     v 
                     2 
                   
                   + 
                   
                     w 
                     2 
                   
                 
               
             
           
         
       
       
         
           
             
               a 
               ⁡ 
               
                 ( 
                 
                   v 
                   , 
                   
                     w 
                     ; 
                     
                       r 
                       0 
                     
                   
                 
                 ) 
               
             
             = 
             
               
                 1 
                 2 
               
               ⁢ 
               
                 arctan 
                 ⁡ 
                 
                   ( 
                   
                     w 
                     , 
                     v 
                   
                   ) 
                 
               
             
           
         
       
       
         
           with 
         
       
       
         
           
             
               arctan 
               ⁡ 
               
                 ( 
                 
                   x 
                   , 
                   y 
                 
                 ) 
               
             
             := 
             
               { 
               
                 
                   
                     
                       
                         arctan 
                         ⁡ 
                         
                           ( 
                           
                             y 
                             / 
                             x 
                           
                           ) 
                         
                       
                       , 
                     
                   
                   
                     
                       x 
                       &gt; 
                       0 
                     
                   
                 
                 
                   
                     
                       
                         
                           arctan 
                           ⁡ 
                           
                             ( 
                             
                               y 
                               / 
                               x 
                             
                             ) 
                           
                         
                         + 
                         π 
                       
                       , 
                     
                   
                   
                     
                       
                         x 
                         &lt; 
                         0 
                       
                       , 
                       
                         y 
                         &gt; 
                         0 
                       
                     
                   
                 
                 
                   
                     
                       π 
                       , 
                     
                   
                   
                     
                       
                         x 
                         &lt; 
                         0 
                       
                       , 
                       
                         y 
                         = 
                         0 
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           arctan 
                           ⁡ 
                           
                             ( 
                             
                               y 
                               / 
                               x 
                             
                             ) 
                           
                         
                         - 
                         π 
                       
                       , 
                     
                   
                   
                     
                       
                         x 
                         &lt; 
                         0 
                       
                       , 
                       
                         y 
                         &lt; 
                         0 
                       
                     
                   
                 
               
             
           
         
       
     
     In an embodiment in which the higher-order aberrations are at least not explicitly used for the transformation of the spherocylindrical values, a transformed refraction is preferably determined by:
 
 Sph ( c   2   0   ,c   2   −2   ,c   2   2   ,{c   n   m   };r   0 )= sph ( c   2   0   ,c   2   −2   ,c   2   2   ;r   0 )
 
 Zyl ( c   2   0   ,c   2   −2   ,c   2   2   ,{c   n   m   };r   0 )= zyl ( c   2   0   ,c   2   −2   ,c   2   2   ;r   0 )
 
 A ( c   2   0   ,c   2   −2   ,c   2   2   ,{c   n   m   };r   0 )= sph ( c   2   0   ,c   2   −2   ,c   2   2   ;r   0 ),
 
where c 2   0 , c 2   −2 ,c 2   2  represent second-order Zernike coefficients and c n   m  with n≧3 higher-order Zernike coefficients of the wavefront in the notation of the OSA standard, cf. for example Jason Porter et al. “Adaptive Optics for Vision Science”, Wiley (2006), p. 522. The parameter r 0  represents the pupil radius. To this end, the method preferably comprises detecting a pupil radius r 0 . Moreover, the method preferably comprises determining second-order Zernike coefficients (c 2   0 ,c 2   −2 ,c 2   2 ).
 
     A preferred embodiment taking into consideration the higher-order aberrations in the determination of the transformed spherocylindrical refraction, i.e. in a transformation of the spherocylindrical values in dependence on the second secondary coefficients, is determined by:
 
 Sph ( c   2   0   ,c   2   −2   ,c   2   2   ,{c   n   m })= sph ( c   2   0   ,c   2   −2 +√{square root over ( 5/3)} c   4   −2   ,c   2   2 +√{square root over ( 5/3)} c   4   2 )
 
 Zyl ( c   2   0   ,c   2   −2   ,c   2   2   ,{c   n   m })= zyl ( c   2   −2 +√{square root over ( 5/3)} c   4   −2   ,c   2   2 +√{square root over ( 5/3)} c   4   2 )
 
 A ( c   2   0   ,c   2   −2   ,c   2   2   ,{c   n   m })= a ( c   2   −2 +√{square root over ( 5/3)} c   4   −2   ,c   2   2 +√{square root over ( 5/3)} c   4   2 )
 
     A further alternative preferred embodiment is determined by:
 
 Sph ( c   2   0   ,c   2   −2   ,c   2   2   ,{c   n   m })= sph ( c   2   0 +√{square root over ( 5/3)} c   4   −2   ,c   2   2 +√{square root over ( 5/3)} c   4   2   ,c   2   2 +√{square root over ( 5/3)} c   4   2 )
 
 Zyl ( c   2   0   ,c   2   −2   ,c   2   2   ,{c   n   m })= zyl ( c   2   −2 +√{square root over ( 5/3)} c   4   −2   ,c   2   2 +√{square root over ( 5/3)} c   4   2 )
 
 A ( c   2   0   ,c   2   −2   ,c   2   2   ,{c   n   m })= a ( c   2   −2 +√{square root over ( 5/3)} c   4   −2   ,c   2   2 +√{square root over ( 5/3)} c   4   2 )
 
     A further preferred embodiment is determined by:
 
 Sph ( c   2   0   ,c   2   −2   ,c   2   2   ,{c   n   m })= sph ( c   2   0 +2√{square root over (15)} c   4   0   ,c   2   −2 +3√{square root over (15)} c   4   −2   ,c   2   2 +3√{square root over (15)} c   4   2 )
 
 Zyl ( c   2   0   ,c   2   −2   ,c   2   2   ,{c   n   m })= zyl ( c   2   −2 +3√{square root over (15)} c   4   −2   ,c   2   2 +3√{square root over (15)} c   4   2 )
 
 A ( c   2   0   ,c   2   −2   ,c   2   2   ,{c   n   m })= a ( c   2   −2 +3√{square root over (15)} c   4   −2   ,c   2   2 +3√{square root over (15)} c   4   2 )
 
     Slightly more general, a preferred embodiment is determined by:
 
 Sph ( c   2   0   ,c   2   −2   ,c   2   2   {c   n   m })= sph ( c   2   0   +Δc   2   0   ,c   2   −2   +Δc   2   −2   ,c   2   2   Δc   2   2 )
 
 Zyl ( c   2   0   ,c   2   −2   ,c   2   2   {c   n   m })= zyl ( c   2   −2   +Δc   2   −2   ,c   2   2   Δc   2   2 )
 
 A ( c   2   0   ,c   2   −2   ,c   2   2   {c   n   m })= a ( c   2   −2   +Δc   2   −2   ,c   2   2   Δc   2   2 )
 
in which the changes Δc 2   0 , Δc 2   −2 , Δc 2   2  are functions (particularly continuous functions) of the higher-order Zernike coefficients c n   m  of the wavefront, where n≧3.
 
     Preferably, collecting prescription data comprises collecting first prescription data for a first object distance and second prescription data for a second object distance. Specifically, the influence of the object distance on the pupil size can be taken into consideration individually. In this way, a changed pupil size mainly influences the higher-order aberrations. 
     Preferably, the method further comprises:
         specifying an object distance model A1(x, y), where A1 designates the object distance and (x,y) a visual point or visual spot of the spectacle lens in a specified or specifiable direction of sight;   specifying a function r 0 =g (A1), which describes the dependence of a pupil size r 0  on the object distance A1;   determining a pupil size for the at least one main ray ( 10 ) on the basis of the object distance model A1(x, y) and the specified function r 0 =g(A1).       

     Preferably, the spectacle lens to be optimized is a progressive spectacle lens. 
     In a further aspect, the invention provides a device for calculating and optimizing a spectacle lens, comprising:
         a surface model database for specifying at least one surface for the spectacle lens to be calculated or optimized;   main ray determining means for determining the course of a main ray ( 10 ) through at least one visual point (i) of said at least one surface;   primary coefficient determining means for determining a first primary set (s io ) and a second primary set (e iok ) of coefficients of the local aberration of a local wavefront ( 18 ) going out from the at least one visual point (i) in a vicinity of the main ray ( 10 ), wherein the first primary set of coefficients (s io ) specifies the spherical and astigmatic aberration of the outgoing local wavefront and the second primary set of coefficients (e iok ) specifies a further higher-order aberration (k&gt;2) of the outgoing local wavefront;   a propagation model database for specifying at least one function e pk =ƒ(e ok ), which assigns a second secondary set of coefficients (e pk )) to a second primary set of coefficients (e ok ), said second secondary set of coefficients specifying the at least one higher-order aberration of a propagated wavefront;   secondary coefficient determining means for determining a higher-order aberration of a local wavefront propagated starting from the at least one visual point (i) along the main ray depending on at least the second primary set of coefficients (e iok ) on the basis of the specified function e pk =ƒ(e ok ); and   calculating or optimizing means adapted to calculate or optimize the at least one surface of the spectacle lens taking into consideration the determined higher-order aberration of the propagated local wavefront.       

     Further, the invention provides a computer program product adapted, when loaded and executed on a computer, to perform a method for calculating or optimizing a spectacle lens according to the present invention, particularly in a preferred embodiment thereof. 
     Moreover, the invention provides a storage medium with a computer program stored thereon, the computer program being adapted, when loaded and executed on a computer, to perform a method for calculating or optimizing a spectacle lens according to the present invention, particularly in a preferred embodiment thereof. 
     In addition, the invention provides a method for producing a spectacle lens, comprising:
         calculating or optimizing a spectacle lens according to the method for calculating or optimizing a spectacle lens according to the present invention, particularly in a preferred embodiment thereof;   manufacturing the thus calculated or optimized spectacle lens.       

     Moreover, the invention provides a device for producing a spectacle lens, comprising:
         calculating and optimizing means adapted to calculate or optimize a spectacle lens according to the present invention, particularly in a preferred embodiment thereof;   machining means adapted to finish the spectacle lens.       

     Further, the invention provides a use of a spectacle lens, produced according to the preferred production method, in a predetermined average or individual wearing position of the spectacle lens in front of the eyes of a specific spectacles wearer, for correcting a visual defect of the spectacles wearer. 
    
    
     
       DRAWINGS 
       Preferred embodiments of the invention will be described by way of example in the following with reference to the accompanying drawings, which show: 
         FIG. 1  a schematic illustration of the physiological and physical model of a spectacle lens along with a ray course in a predetermined wearing position; 
         FIG. 2  a schematic illustration of a coordinate system with an original wavefront and a propagated wavefront; 
         FIG. 3  a schematic illustration of a spherical wavefront with a vergence distance s o  at a distance d from a propagated wavefront with the vergence distance s p ; 
         FIG. 4  a schematic illustration of the process of propagation of a wavefront; 
         FIG. 5  a schematic illustration of local coordinate systems of a refractive surface, an incoming and an outgoing wavefront; and 
         FIG. 6  a flow chart for illustrating a method for optimizing an individual spectacle lens according to an embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  shows a schematic illustration of the physiological and physical model of a spectacle lens in a predetermined wearing position along with an exemplary ray course, on which an individual spectacle lens calculation or optimization according to a preferred embodiment of the invention is based. 
     Here, preferably only one single ray (the main ray  10 ) is calculated per visual point of the spectacle lens, but further also the derivatives of the vertex depths of the wavefront according to the transversal coordinates (perpendicular to the main ray). These derivatives are taken into consideration up to the desired orders, wherein the second derivatives describe the local curvature properties of the wavefront and the higher derivatives are related to the higher-order aberrations. 
     In the tracing of light through the spectacle lens, the local derivatives of the wavefronts are ultimately determined at a suitable position in the ray course in order to compare them with the required values of the refraction of the spectacles wearer there. In a preferred embodiment, this position is for example the vertex sphere or the entrance pupil of the eye  12 . To this end, it is assumed that a spherical wavefront originates at an object point and propagates to the first spectacle lens surface  14 . There, it is refracted and subsequently it propagates (ST 2 ) up to the second spectacle lens surface  16 , where it is refracted again. If further surfaces to be considered exist, the alternation of propagation and refraction is continued until the last boundary surface has been passed, and the last propagation (ST 4 ) then takes place from this last boundary surface to the vertex sphere (or the entrance pupil of the eye). 
     In the following, the propagation of the wavefront according to a preferred embodiment of the present invention will be described in more detail. These statements can e.g. be applied to the propagation of the wavefront between the two spectacle lens surfaces and/or to the propagation of the wavefront from the rear spectacle lens surface to the vertex sphere. 
     As illustrated in  FIG. 2 , preferably a Cartesian coordinate system (with an x axis, a y axis, and a z axis) is defined, the origin of which being at the intersection point of the main ray  10  with the original wavefront  18  for a predetermined main ray  10 . The z axis preferably points in the direction of the main ray  10 . The directions of the x axis and the y axis are preferably selected to be perpendicular to the z axis and perpendicular to each other such that the coordinate system is right-handed. If the original wavefront is assumed to be a wavefront at a refractive surface, i.e. a surface of the spectacle lens, the x axis and/or the y axis is preferably selected to be parallel to the surface or surface tangent in the penetration point of the main ray. In another preferred embodiment, the x axis and the y axis are selected to be parallel to the main curvatures of the original wavefront  18 . 
     Preferably, a description of the wavefront according to
 
 w ( x,y )=( x,y,w ( x,y ))  (1)
 
is assumed, where the value w(x,y) is represented by
 
                     w   ⁡     (     x   ,   y     )       =       ∑     k   =   0     ∞     ⁢           ⁢       ∑     m   =   0     k     ⁢           ⁢         a     m   ,     k   -   m             m   !     ⁢       (     k   -   m     )     !         ⁢     x   m     ⁢     y     k   -   m                     (   2   )               
by means of the coefficients
 
     
       
         
           
             
               
                 
                   
                     a 
                     
                       m 
                       , 
                       
                         k 
                         - 
                         m 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ∂ 
                           k 
                         
                         
                           
                             ∂ 
                             
                               x 
                               m 
                             
                           
                           ⁢ 
                           
                             ∂ 
                             
                               y 
                               
                                 k 
                                 - 
                                 m 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         w 
                         ⁡ 
                         
                           ( 
                           
                             x 
                             , 
                             y 
                           
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       | 
                       
                         
                           x 
                           = 
                           0 
                         
                         , 
                         
                           y 
                           = 
                           0 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Thus, the connection between the coefficients a k     x     ,k     y    and the local aberrations E k     x     ,k     y    can be described by:
 
 E   k     x     ,k     y     =na   k     x     ,k     y     E   2,0   =S   xx   na   In,2,0   E   1,1   =S   xy   =na   1,1   E   0,2   =S   yy   =na   0,2   E   3,0   =na   3,0  
 
     For aberrations up to the second order, the propagation of a spherical wavefront with the vergence S o =n/s o  of the original wavefront in a surrounding around a main ray can preferably be expressed in a known manner by the propagation equation 
                     S   p     =       1     1   -       d   n     ⁢     S   o           ⁢     S   o               (   4   )               
where S p =n/s p  designates the vergence of the propagated wavefront. As illustrated in  FIG. 3 , s o  and s p  designate the vertex distance of the original wavefront  18  and the propagated wavefront  20 , respectively, (distance along the main ray  10  from the wavefront to the image point  22 ). n designates the refractive index and d the propagation distance.
 
     By an extension to three dimensions, the spherocylindrical form of the wavefront can be represented as follows. First of all, the curvatures 1/s o  and 1/s p  are identified with the second derivatives of the vertex depths of the original wavefront  18  and the propagated wavefront  20 , respectively. In the three-dimensional representation, the two derivatives w o   (2,0) =∂ 2 w o /∂x 2 , w o   (1,1) =∂ 2 w o /∂x∂y, w o   (0,2) =∂ 2 w o /∂y 2  the original wavefront  18  and correspondingly for the propagated wavefront  20  are respectively summarized in form of a vergence matrix: 
                       S   o     =       (           S   oxx           S   oxy               S   oxy           S   oyy           )     =     n   ⁡     (           w   o     (     2   ,   0     )             w   o     (     1   ,   1     )                 w   o     (     1   ,   1     )             w   o     (     0   ,   2     )             )           ,     
     ⁢       S   p     =       (           S   pxx           S   pxy               S   pxy           S   pyy           )     =     n   ⁡     (           w   p     (     2   ,   0     )             w   p     (     1   ,   1     )                 w   p     (     1   ,   1     )             w   p     (     0   ,   2     )             )                   (   5   )               
According to
 
                       S   oxx     =       (     Sph   +     Cyl   2       )     -       Cyl   2     ⁢   cos   ⁢           ⁢   2   ⁢   α         ⁢     
     ⁢       S   oxy     =       -     Cyl   2       ⁢   sin   ⁢           ⁢   2   ⁢   α       ⁢     
     ⁢       S   oyy     =       (     Sph   +     Cyl   2       )     +       Cyl   2     ⁢   cos   ⁢           ⁢   2   ⁢   α                 (   6   )               
(and analogously for the propagated wavefront) the components of the respective vergence matrix are connected with the known parameters of spherical power Sph, the magnitude Cyl of cylindrical power, and the cylinder axis a of the cylindrical power. By means of the representation in form of the vergence matrix, by analogy with equation (4), the propagation of an astigmatic wavefront can be represented via the generalized propagation equation
 
                     S   p     =       1     1   -       d   n     ⁢     S   o           ⁢     S   o               (   7   )               
with the identity matrix
 
             1   =       (         1       0           0       1         )     .           
Equivalent to this representation in form of the vergence matrix,
 
                       s   o     =       (           S   oxx               S   oxy               S   oyy           )     =     n   ⁡     (           w   o     (     2   ,   0     )                 w   o     (     1   ,   1     )                 w   o     (     0   ,   2     )             )           ,     
     ⁢       s   p     =       (           S   pxx               S   pxy               S   pyy           )     =     n   ⁡     (           w   p     (     2   ,   0     )                 w   p     (     1   ,   1     )                 w   p     (     0   ,   2     )             )                   (   8   )               
are introduced as power vectors in the three-dimensional vector space for the original wavefront  18  and the propagated wavefront  20 .
 
     Now, for consideration of higher-order aberrations in the propagation of the wavefront, corresponding vectors e k  of the dimension k+1 are introduced: 
     
       
         
           
             
               
                 
                   
                     
                       e 
                       ok 
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               
                                 E 
                                 
                                   ox 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   … 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   xx 
                                 
                               
                             
                           
                           
                             
                               
                                 E 
                                 
                                   ox 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   … 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   xy 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 E 
                                 
                                   oy 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   … 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   yy 
                                 
                               
                             
                           
                         
                         ) 
                       
                       := 
                       
                         n 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 
                                   w 
                                   o 
                                   
                                     ( 
                                     
                                       k 
                                       , 
                                       0 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   w 
                                   o 
                                   
                                     ( 
                                     
                                       
                                         k 
                                         - 
                                         1 
                                       
                                       , 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 ⋮ 
                               
                             
                             
                               
                                 
                                   w 
                                   o 
                                   
                                     ( 
                                     
                                       0 
                                       , 
                                       k 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       e 
                       pk 
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               
                                 E 
                                 
                                   px 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   … 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   xx 
                                 
                               
                             
                           
                           
                             
                               
                                 E 
                                 
                                   px 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   … 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   xy 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 E 
                                 
                                   py 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   … 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   yy 
                                 
                               
                             
                           
                         
                         ) 
                       
                       := 
                       
                         n 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 
                                   w 
                                   o 
                                   
                                     ( 
                                     
                                       k 
                                       , 
                                       0 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   w 
                                   o 
                                   
                                     ( 
                                     
                                       
                                         k 
                                         - 
                                         1 
                                       
                                       , 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 ⋮ 
                               
                             
                             
                               
                                 
                                   w 
                                   o 
                                   
                                     ( 
                                     
                                       0 
                                       , 
                                       k 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     For further consideration, at first only a two-dimensional representation will be described for reasons of simplification. Here, some point on the original wavefront (r=o) or the propagated wavefront (r=p) is described by 
                       w   r     ⁡     (   y   )       =     (         y               w   r     ⁡     (   y   )             )             (   10   )               
where w r (y) is described by:
 
     
       
         
           
             
               
                 
                   
                     
                       w 
                       r 
                     
                     ⁡ 
                     
                       ( 
                       y 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         0 
                       
                       ∞ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           a 
                           
                             r 
                             , 
                             k 
                           
                         
                         
                           k 
                           ! 
                         
                       
                       ⁢ 
                       
                         y 
                         k 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     The coefficients a o,k  of the original wavefront  18  correspond to the derivatives of the wavefront with y=0: 
     
       
         
           
             
               
                 
                   
                     a 
                     
                       o 
                       , 
                       k 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             ∂ 
                             k 
                           
                           
                             ∂ 
                             
                               y 
                               k 
                             
                           
                         
                         ⁢ 
                         
                           
                             w 
                             o 
                           
                           ⁡ 
                           
                             ( 
                             y 
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         | 
                         
                           y 
                           = 
                           0 
                         
                       
                     
                     = 
                     
                       
                         w 
                         o 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       ⁡ 
                       
                         ( 
                         0 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     In two dimensions, the vergence matrix S o  in equation (5) is reduced to a scalar E o,k =nw o   (k) =na o,k . For second or third-order aberrations, e.g. S o =E o,2 =nw o   (2) =na o,2 , E o,3 =nw o   (k) =na o,3 , etc. result. The same applies to the propagated wavefront  20 . 
     Here, it is to be noted that any wavefront at the intersection point with the main ray  10  is not inclined with respect to the z axis. Since the z axis points along the direction of the main ray  10 , it is perpendicular to the original and propagated wavefronts in the intersection points of the main ray  10  with the wavefronts  18 ,  20 . Moreover, since the origin of the coordinate system is at the original wavefront  18 , it holds for the coefficients that: a o,0 =0, a o,1 =0, a p,0 =d, and a p,1 =0 
     In two dimensions, the normal vector n w (y) for a wavefront w(y) results from n w (y)=(−w (1) (y),1) T /√{square root over (1+w (1) (y) 2 )}, Where w (1) =∂w/∂y. For Reasons of a simplified notation, first of all v≡w (1)  and the following function is introduced: 
     
       
         
           
             
               
                 
                   
                     n 
                     ⁡ 
                     
                       ( 
                       v 
                       ) 
                     
                   
                   := 
                   
                     
                       1 
                       
                         
                           1 
                           + 
                           
                             v 
                             2 
                           
                         
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               - 
                               v 
                             
                           
                         
                         
                           
                             1 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     As derivatives n (i) (0)≡∂ i /∂v i  n(v)| v=0  of this function there result: 
     
       
         
           
             
               
                 
                   
                     
                       n 
                       ⁡ 
                       
                         ( 
                         0 
                         ) 
                       
                     
                     := 
                     
                       ( 
                       
                         
                           
                             0 
                           
                         
                         
                           
                             1 
                           
                         
                       
                       ) 
                     
                   
                   , 
                   
                     
                       
                         n 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       ⁡ 
                       
                         ( 
                         0 
                         ) 
                       
                     
                     := 
                     
                       ( 
                       
                         
                           
                             
                               - 
                               1 
                             
                           
                         
                         
                           
                             0 
                           
                         
                       
                       ) 
                     
                   
                   , 
                   
                     
                       
                         n 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       ⁡ 
                       
                         ( 
                         0 
                         ) 
                       
                     
                     := 
                     
                       ( 
                       
                         
                           
                             0 
                           
                         
                         
                           
                             
                               - 
                               1 
                             
                           
                         
                       
                       ) 
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         n 
                         
                           ( 
                           3 
                           ) 
                         
                       
                       ⁡ 
                       
                         ( 
                         0 
                         ) 
                       
                     
                     := 
                     
                       ( 
                       
                         
                           
                             3 
                           
                         
                         
                           
                             0 
                           
                         
                       
                       ) 
                     
                   
                   , 
                   
                     
                       
                         n 
                         
                           ( 
                           4 
                           ) 
                         
                       
                       ⁡ 
                       
                         ( 
                         0 
                         ) 
                       
                     
                     := 
                     
                       ( 
                       
                         
                           
                             0 
                           
                         
                         
                           
                             9 
                           
                         
                       
                       ) 
                     
                   
                   , 
                   
                     etc 
                     . 
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     The normal vector, which is perpendicular to both the original wavefront  18  and the propagated wavefront  20 , can be designated uniformly with n w . Thus, for the first derivative of the normal vector there is determined: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           ∂ 
                           y 
                         
                       
                       ⁢ 
                       
                         
                           n 
                           w 
                         
                         ⁡ 
                         
                           ( 
                           y 
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       | 
                       
                         y 
                         = 
                         0 
                       
                     
                     ⁢ 
                     
                       ≡ 
                       
                         
                           n 
                           w 
                           
                             ( 
                             1 
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           n 
                           
                             ( 
                             1 
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           w 
                           o 
                           
                             ( 
                             2 
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               
                                 - 
                                 1 
                               
                             
                           
                           
                             
                               0 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           w 
                           o 
                           
                             ( 
                             2 
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     The same applies to the higher derivatives. 
     With the local aberrations of the original wavefront  18 , the corresponding coefficients a k  and, equivalent thereto, the derivatives of the wavefront are directly defined as well. Subsequently, the propagated wavefront  20  is determined therefrom particularly by determining its derivatives or coefficients a k  for all orders 2≦k≦k 0  up to the desired value k 0 , and thus the values of the local aberrations of the propagated wavefront  20  are determined. 
     As a starting point, the following situation with respect to  FIG. 4  will be considered in an illustrative way. While the main ray  10  and the coordinate system are fixed, a neighboring ray  24  scans the original wavefront  18  ({w o }) and strikes it in a section y o ≠0. From there, it propagates further to the propagated wavefront  20  ({w p }). As illustrated in  FIG. 4 , y o  designates the projection of the intersection point of the neighboring ray  24  with the original wavefront {w o } to the y axis, while analogously the projection of the intersection point with the propagated wavefront {w p } to the y axis is designated with y p . 
     The vector w o =w o (y o ) (cf. equation (10)) points to the intersection point of the neighboring ray  24  with the original wavefront  18 , and the optical path difference (OPD) with respect to the propagated wavefront  20  is designated with τ. Accordingly, the vector from the original wavefront  18  to the propagated surface  20  is represented by τ/n n w . Thus, it results for the vector to the corresponding point of the propagated wavefront: w p =w o +τ/n n w . As a basic equation there is introduced: 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             
                               y 
                               o 
                             
                           
                         
                         
                           
                             
                               
                                 w 
                                 o 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   y 
                                   o 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                     + 
                     
                       
                         τ 
                         n 
                       
                       ⁢ 
                       
                         n 
                         w 
                       
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             y 
                             p 
                           
                         
                       
                       
                         
                           
                             
                               w 
                               p 
                             
                             ⁡ 
                             
                               ( 
                               
                                 y 
                                 p 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Now, from this equation, the desired relations are derived order by order. Here, y p  is preferably used as a free variable, on which y o  depends in turn. For solving the equation, first of all the vector 
                     p   ⁡     (     y   p     )       =     (             y   o     ⁡     (     y   p     )                   w   p     ⁡     (     y   p     )             )             (   17   )               
on the boundary condition
 
               p   ⁡     (   0   )       =     (         0             τ   n           )           
can be introduced. Based on this, the following function is introduced for the further consideration:
 
                     f   ⁡     (     p   ,     y   p       )       =     (             y   o     +       τ   n     ⁢       n     w   ,   y       ⁡     (       w   o     (   1   )       ⁡     (     y   o     )       )         -     y   p                     w   o     ⁡     (     y   o     )       +       τ   n     ⁢       n     w   ,   z       ⁡     (       w   o     (   1   )       ⁡     (     y   o     )       )         -     w   p             )             (   18   )               
where (p 1 ,p 2 )=(y o ,w p ) are the components of p. Now, if p=p(y p ), the equation (16) can be represented in a compact form by:
 
 f ( p ( y   p ) y   p )=0  (19)
 
     The derivatives of this function according to y p  are preferably expressed by the following system of differential equations: 
                           ∑     j   =   1     2     ⁢           ⁢         ∂     f   i         ∂     p   j         ⁢       p   j     (   1   )       ⁡     (     y   p     )           +       ∂     f   i         ∂     y   p           =   0     ,     
     ⁢     i   =   1.2             (   20   )               
where the matrix with the elements A ij :=∂ƒ i /∂p j  is referred to as a Jacobi matrix A. The Jacobi matrix A thus reads
 
     
       
         
           
             
               
                 
                   
                     A 
                     ⁢ 
                     
                       : 
                     
                     = 
                     
                       ( 
                       
                         
                           
                             
                               
                                 ∂ 
                                 
                                   f 
                                   1 
                                 
                               
                               
                                 ∂ 
                                 
                                   y 
                                   o 
                                 
                               
                             
                           
                           
                             
                               
                                 ∂ 
                                 
                                   f 
                                   1 
                                 
                               
                               
                                 ∂ 
                                 
                                   w 
                                   p 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ∂ 
                                 
                                   f 
                                   2 
                                 
                               
                               
                                 ∂ 
                                 
                                   y 
                                   o 
                                 
                               
                             
                           
                           
                             
                               
                                 ∂ 
                                 
                                   f 
                                   2 
                                 
                               
                               
                                 ∂ 
                                 
                                   w 
                                   p 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             1 
                             + 
                             
                               
                                 τ 
                                 n 
                               
                               ⁢ 
                               
                                 n 
                                 
                                   w 
                                   , 
                                   y 
                                 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 w 
                                 o 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           
                             
                               w 
                               o 
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                             + 
                             
                               
                                 τ 
                                 n 
                               
                               ⁢ 
                               
                                 n 
                                 
                                   w 
                                   , 
                                   z 
                                 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 w 
                                 o 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             - 
                             1 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     The terms appearing in this equation are to be understood as w o   (1) ≡w o   (1) (y o ) w o   (2) ≡w o   (2) (y o ), n w,y ≡n w,y (w o   (1) (y o )), n w,y   (1) ≡n w,y   (1) (w o   (1) (y o )), etc., where y o ,w p  are in turn themselves functions of y p . 
     The derivative vector ∂ƒ i /∂y p  can be summarized as 
     
       
         
           
             
               
                 
                   
                     b 
                     ⁢ 
                     
                       : 
                     
                     = 
                     
                       
                         ∂ 
                         f 
                       
                       
                         ∂ 
                         
                           y 
                           p 
                         
                       
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           1 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     Thus, the above differential equation system can be represented as:
 
 A ( p ( y   p )) p   (1) ( y   p )= b   (23)
 
     Formally, this equation is solved by:
 
 p   (1) ( y   p )= A ( p ( y   p )) −1   b   (24)
 
with the boundary condition
 
               p   ⁡     (   0   )       =       (         0             τ   n           )     .           
Based on this, the equation system for higher-order aberrations is preferably solved recursively as follows:
 
                         p     (   1   )       ⁡     (   0   )       =       A     -   1       ⁢   b       ⁢     
     ⁢         p     (   2   )       ⁡     (   0   )       =         (     A     -   1       )       (   1   )       ⁢   b       ⁢     
     ⁢   …   ⁢     
     ⁢           p     (   k   )       ⁡     (   0   )       =         (     A     -   1       )       (     k   -   1     )       ⁢   b       ,             (   25   )               
with the abbreviatory designations A −1 =A(p(0)) −1 =A(0) −1 
 
     
       
         
           
             
               
                 ( 
                 
                   A 
                   
                     - 
                     1 
                   
                 
                 ) 
               
               
                 ( 
                 1 
                 ) 
               
             
             = 
             
               
                 
                   ⅆ 
                   
                       
                   
                 
                 
                   ⅆ 
                   
                     y 
                     p 
                   
                 
               
               ⁢ 
               
                 
                   A 
                   ⁡ 
                   
                     ( 
                     
                       p 
                       ⁡ 
                       
                         ( 
                         
                           y 
                           p 
                         
                         ) 
                       
                     
                     ) 
                   
                 
                 
                   - 
                   1 
                 
               
               ⁢ 
               
                 
                   
                     
                        
                       
                         
                           y 
                           p 
                         
                         = 
                         0 
                       
                     
                     ⁢ 
                     
                       , 
                       … 
                       ⁢ 
                       
                           
                       
                       , 
                       
                         
                           
                             ( 
                             
                               A 
                               
                                 - 
                                 1 
                               
                             
                             ) 
                           
                           
                             ( 
                             
                               k 
                               - 
                               1 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               ⅆ 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             
                               ⅆ 
                               
                                 y 
                                 p 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
                               A 
                               ⁡ 
                               
                                 ( 
                                 
                                   p 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       y 
                                       p 
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                      
                   
                   
                     
                       y 
                       p 
                     
                     = 
                     0 
                   
                 
                 . 
               
             
           
         
       
     
     In an alternative approach, it is suggested performing the recursion on the basis of equation (23) instead of equation (24). The first (k−1) derivatives of equation (23) yield: 
                           ⁢                 Ap     (   1   )       ⁡     (   0   )       =   b           (   a   )                     A     (   1   )       ⁢       p     (   1   )       ⁡     (   0   )         +       Ap     (   2   )       ⁡     (   0   )         =   0           (   b   )                     A     (   2   )       ⁢       p     (   1   )       ⁡     (   0   )         +     2   ⁢     A     (   1   )       ⁢       p     (   2   )       ⁡     (   0   )         +       Ap     (   3   )       ⁡     (   0   )         =   0           (   c   )             …                               ∑     j   =   1     k     ⁢           ⁢       (           k   -   1               j   -   1           )     ⁢           ⁢     A     (     k   -   j     )       ⁢       p     (   j   )       ⁡     (   0   )           =   0     ,     k   ≥   2             (   d   )           ⁢     
     ⁢           ⁢   where   ⁢     
     ⁢           ⁢       A   =       A   ⁡     (     p   ⁡     (   0   )       )       =     A   ⁡     (   0   )           ,     
     ⁢       A     (   1   )       =       ⅆ     ⅆ     y   p         ⁢     A   ⁡     (     p   ⁡     (     y   p     )       )       ⁢                y   p     =   0       ⁢     ,   …   ⁢           ,       A     (     k   -   j     )       =         ⅆ     k   -   j         ⅆ     y   p     k   -   j           ⁢     A   ⁡     (     p   ⁡     (     y   p     )       )                      y   p     =   0                       (   26   )               
designate the total derivatives of the function. Formally, these equations are solved by:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               p 
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                           = 
                           
                             
                               A 
                               
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             b 
                           
                         
                         , 
                       
                     
                     
                       
                         k 
                         = 
                         1 
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               p 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                           = 
                           
                             
                               - 
                               
                                 A 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   j 
                                   = 
                                   1 
                                 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           k 
                                           - 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         
                                           j 
                                           - 
                                           1 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   A 
                                   
                                     ( 
                                     
                                       k 
                                       - 
                                       j 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     p 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     0 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         , 
                       
                     
                     
                       
                         k 
                         ≥ 
                         2. 
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     In order to obtain A(0) −1 , preferably equation (21) is evaluated for p=0 and equation (14) is applied. This yields: 
                     A   ⁡     (   0   )       =         (           1   +       τ   n     ⁢     w   o     (   2   )               0           0         -   1           )     ⇒       A   ⁡     (   0   )         -   1         =     (           1     1   -       τ   n     ⁢     w   o     (   2   )                 0           0         -   1           )               (   28   )               
from which it results for p (1) (0):
 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       
                         ( 
                         1 
                         ) 
                       
                     
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         A 
                         
                           - 
                           1 
                         
                       
                       ⁢ 
                       b 
                     
                     = 
                     
                       ( 
                       
                         
                           
                             
                               1 
                               
                                 1 
                                 - 
                                 
                                   
                                     τ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       2 
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             0 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
     In turn, this means 
                 y   o     (   1   )       ⁡     (   0   )       =         1     1   -       τ   n     ⁢     w   o     (   2   )             ⁢           ⁢   and   ⁢           ⁢       w   p     (   1   )       ⁡     (   0   )         =   0.           
For orders k≧2, preferably equation (27) is applied. The derivatives
 
                 A     (   1   )       =         ⅆ     ⅆ     y   p         ⁢     A   ⁡     (     p   ⁡     (     y   p     )       )         ⁢     ❘       y   p     =   0           ,         
etc. are preferably determined from equation (21) and preferably equation (14) is applied again. Thus, it results in the second order:
 
                     w   p     (   2   )       =       1     1   -       τ   n     ⁢     w   o     (   2   )             ⁢     w   o     (   2   )                 (   30   )               
which basically corresponds to the above-described propagation equation. The higher orders can analogously be expressed by:
 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             w 
                             p 
                             
                               ( 
                               3 
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   
                                     1 
                                     - 
                                     
                                       
                                         τ 
                                         n 
                                       
                                       ⁢ 
                                       
                                         w 
                                         o 
                                         
                                           ( 
                                           2 
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               3 
                             
                             ⁢ 
                             
                               w 
                               o 
                               
                                 ( 
                                 3 
                                 ) 
                               
                             
                           
                         
                         ⁢ 
                         
                           
 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             w 
                             p 
                             
                               ( 
                               4 
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   
                                     1 
                                     - 
                                     
                                       
                                         τ 
                                         n 
                                       
                                       ⁢ 
                                       
                                         w 
                                         o 
                                         
                                           ( 
                                           2 
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               4 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   w 
                                   o 
                                   
                                     ( 
                                     4 
                                     ) 
                                   
                                 
                                 + 
                                 
                                   3 
                                   ⁢ 
                                   
                                     τ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           1 
                                           
                                             1 
                                             - 
                                             
                                               
                                                 τ 
                                                 n 
                                               
                                               ⁢ 
                                               
                                                 w 
                                                 o 
                                                 
                                                   ( 
                                                   2 
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           w 
                                           o 
                                           
                                             
                                               ( 
                                               3 
                                               ) 
                                             
                                             2 
                                           
                                         
                                       
                                       - 
                                       
                                         w 
                                         o 
                                         
                                           
                                             ( 
                                             2 
                                             ) 
                                           
                                           4 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                           
 
                         
                         ⁢ 
                         
                           w 
                           p 
                           
                             ( 
                             5 
                             ) 
                           
                         
                       
                       = 
                       
                         
                           
                             ( 
                             
                               1 
                               
                                 1 
                                 - 
                                 
                                   
                                     τ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       2 
                                       ) 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           5 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               w 
                               o 
                               
                                 ( 
                                 5 
                                 ) 
                               
                             
                             + 
                             
                               5 
                               ⁢ 
                               
                                 1 
                                 
                                   1 
                                   - 
                                   
                                     
                                       τ 
                                       n 
                                     
                                     ⁢ 
                                     
                                       w 
                                       o 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                 
                               
                               ⁢ 
                               
                                 τ 
                                 n 
                               
                               ⁢ 
                               
                                 
                                   w 
                                   o 
                                   
                                     ( 
                                     3 
                                     ) 
                                   
                                 
                                 ( 
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       w 
                                       o 
                                       
                                         ( 
                                         4 
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     3 
                                     ⁢ 
                                     
                                       1 
                                       
                                         1 
                                         - 
                                         
                                           
                                             τ 
                                             n 
                                           
                                           ⁢ 
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               2 
                                               ) 
                                             
                                           
                                           ⁢ 
                                           n 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       τ 
                                       n 
                                     
                                     ⁢ 
                                     
                                       w 
                                       o 
                                       
                                         
                                           ( 
                                           3 
                                           ) 
                                         
                                         2 
                                       
                                     
                                   
                                   - 
                                   
                                     6 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       w 
                                       o 
                                       
                                         
                                           ( 
                                           2 
                                           ) 
                                         
                                         3 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     … 
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
     Equation (31) correspondingly applies to the derivatives and the coefficients a o,k  and a p,k  due to equations (10) to (12). Now, if one replaces d=τ/n and 
               β   =     1     1   -       d   n     ⁢     S   o             ,         
the local aberrations can be expressed as follows:
 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           S 
                           p 
                         
                         = 
                         
                           β 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             S 
                             o 
                           
                         
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           E 
                           
                             p 
                             , 
                             3 
                           
                         
                         = 
                         
                           
                             β 
                             3 
                           
                           ⁢ 
                           
                             E 
                             
                               o 
                               , 
                               3 
                             
                           
                         
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           E 
                           
                             p 
                             , 
                             4 
                           
                         
                         = 
                         
                           
                             β 
                             4 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 E 
                                 
                                   o 
                                   , 
                                   4 
                                 
                               
                               + 
                               
                                 3 
                                 ⁢ 
                                 
                                   d 
                                   n 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       β 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         E 
                                         
                                           o 
                                           , 
                                           3 
                                         
                                         2 
                                       
                                     
                                     - 
                                     
                                       
                                         S 
                                         o 
                                         4 
                                       
                                       
                                         n 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           E 
                           
                             p 
                             , 
                             5 
                           
                         
                         = 
                         
                           
                             β 
                             5 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 E 
                                 
                                   o 
                                   , 
                                   5 
                                 
                               
                               + 
                               
                                 5 
                                 ⁢ 
                                 β 
                                 ⁢ 
                                 
                                   d 
                                   n 
                                 
                                 ⁢ 
                                 
                                   
                                     E 
                                     
                                       o 
                                       , 
                                       3 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         2 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           E 
                                           
                                             o 
                                             , 
                                             4 
                                           
                                         
                                       
                                       + 
                                       
                                         3 
                                         ⁢ 
                                         β 
                                         ⁢ 
                                         
                                           d 
                                           n 
                                         
                                         ⁢ 
                                         
                                           E 
                                           
                                             o 
                                             , 
                                             3 
                                           
                                           2 
                                         
                                       
                                       - 
                                       
                                         6 
                                         ⁢ 
                                         
                                           
                                             S 
                                             o 
                                             3 
                                           
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                         E 
                         
                           p 
                           , 
                           6 
                         
                       
                     
                     = 
                     
                       
                         β 
                         6 
                       
                       ( 
                       
                         
                           E 
                           
                             o 
                             , 
                             6 
                           
                         
                         + 
                         
                           5 
                           ⁢ 
                           β 
                           ⁢ 
                           
                             d 
                             n 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 3 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   E 
                                   
                                     o 
                                     , 
                                     3 
                                   
                                 
                                 ⁢ 
                                 
                                   E 
                                   
                                     o 
                                     , 
                                     5 
                                   
                                 
                               
                               + 
                               
                                 21 
                                 ⁢ 
                                 β 
                                 ⁢ 
                                 
                                   d 
                                   n 
                                 
                                 ⁢ 
                                 
                                   E 
                                   
                                     o 
                                     , 
                                     3 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   E 
                                   
                                     o 
                                     , 
                                     4 
                                   
                                 
                               
                               - 
                               
                                 12 
                                 ⁢ 
                                 
                                   
                                     
                                       S 
                                       o 
                                       3 
                                     
                                     ⁢ 
                                     
                                       E 
                                       
                                         o 
                                         , 
                                         4 
                                       
                                     
                                   
                                   
                                     n 
                                     2 
                                   
                                 
                               
                               + 
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   E 
                                   
                                     o 
                                     , 
                                     4 
                                   
                                 
                               
                               - 
                               
                                 9 
                                 ⁢ 
                                 β 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   S 
                                   0 
                                   2 
                                 
                                 ⁢ 
                                 
                                   E 
                                   
                                     o 
                                     , 
                                     3 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   
                                     3 
                                     + 
                                     
                                       4 
                                       ⁢ 
                                       
                                         d 
                                         n 
                                       
                                       ⁢ 
                                       
                                         S 
                                         o 
                                       
                                     
                                   
                                   
                                     n 
                                     2 
                                   
                                 
                               
                               + 
                               
                                 21 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       β 
                                       ⁢ 
                                       
                                         d 
                                         n 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   E 
                                   
                                     o 
                                     , 
                                     3 
                                   
                                   4 
                                 
                               
                               + 
                               
                                 9 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   S 
                                   o 
                                   6 
                                 
                                 ⁢ 
                                 
                                   
                                     3 
                                     + 
                                     
                                       
                                         d 
                                         n 
                                       
                                       ⁢ 
                                       
                                         S 
                                         o 
                                       
                                     
                                   
                                   
                                     n 
                                     4 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     For 2&lt;k≦6, this is preferably represented by
 
 E   p,k =β k ( E   o,k   +R   k )  (33)
 
in a generalized way, where in R k  all wavefront derivatives E o,j  of the lower orders (j&lt;k) are expressed in form of local aberrations.
 
     Even if a three-dimensional representation is more complex, it can basically be established by analogy with the two-dimensional representation. Therefore, for the fully three-dimensional representation, only a few essential additional considerations will be described in the following. 
     Preferably, the original wavefront can be expressed by the 3D vector 
     
       
         
           
             
               
                 
                   
                     
                       w 
                       o 
                     
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           x 
                         
                       
                       
                         
                           y 
                         
                       
                       
                         
                           
                             
                               w 
                               o 
                             
                             ⁡ 
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     where w o  (x, y) is determined according to equation (2), and the relationship between the coefficients and the derivatives is determined according to equation (3). The connection between the coefficients and the local aberrations results from a multiplication of the coefficient by the refractive index. Preferably, by analogy with equation (13), formal vectors are introduced: 
                     n   ⁡     (     u   ,   v     )       :=       1       1   +     u   2     +     v   2           ⁢     (           -   u               -   v             1         )               (   35   )               
so that the normal vectors with respect to a surface w(x,y):=(x,y,w(x,y)) T  are determined by:
 
     
       
         
           
             
               
                 
                   w 
                   
                     ( 
                     
                       1 
                       , 
                       0 
                     
                     ) 
                   
                 
                 × 
                 
                   w 
                   
                     ( 
                     
                       0 
                       , 
                       1 
                     
                     ) 
                   
                 
               
               
                  
                 
                   
                     w 
                     
                       ( 
                       
                         1 
                         , 
                         0 
                       
                       ) 
                     
                   
                   × 
                   
                     w 
                     
                       ( 
                       
                         0 
                         , 
                         1 
                       
                       ) 
                     
                   
                 
                  
               
             
             = 
             
               
                 
                   1 
                   
                     
                       1 
                       + 
                       
                         w 
                         
                           
                             
                               ( 
                               
                                 1 
                                 , 
                                 0 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             w 
                             
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   1 
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                     
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       
                         
                           - 
                           
                             w 
                             
                               ( 
                               
                                 1 
                                 , 
                                 0 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           - 
                           
                             w 
                             
                               ( 
                               
                                 0 
                                 , 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         1 
                       
                     
                   
                   ) 
                 
               
               = 
               
                 
                   n 
                   ⁡ 
                   
                     ( 
                     
                       
                         w 
                         
                           ( 
                           
                             1 
                             , 
                             0 
                           
                           ) 
                         
                       
                       , 
                       
                         w 
                         
                           ( 
                           
                             0 
                             , 
                             1 
                           
                           ) 
                         
                       
                     
                     ) 
                   
                 
                 = 
                 
                   n 
                   ⁡ 
                   
                     ( 
                     
                       ∇ 
                       w 
                     
                     ) 
                   
                 
               
             
           
         
       
     
     In the intersection point, it thus results n w (0,0)=(0,0,1) T , and the derivatives according to equation (14) are preferably determined from equation (35). 
     As the basis for the consideration of a connection between the original and propagated wavefronts, preferably substantially equation (16) is used, with the difference that now x and y components are considered at the same time. As a vector of unknown functions, there is preferably determined: 
                     p   ⁡     (       x   p     ,     y   p       )       =     (             x   o     ⁡     (       x   p     ,     y   p       )                   y   o     ⁡     (       x   p     ,     y   p       )                   w   p     ⁡     (       x   p     ,     y   p       )             )             (   36   )               
and by analogy with equation (16), there is preferably used for the three-dimensional consideration:
 
 f ( p ( x   p   ,y   p ) x   p   ,y   p )=0  (37)
 
where f is analogous to equation (18).
 
     An importance difference compared to the two-dimensional consideration is that in the three-dimensional case two arguments exist, with respect to which the derivatives are taken into account. Thus, already in the first order, two equations are considered:
 
 A ( p ( x   p   ,y   p )) p   (1,0) ( x   p   ,y   p )= b   x  
 
 A ( p ( x   p   ,y   p )) p   (0,1) ( x   p   ,y   p )= b   y   (38)
 
where the inhomogeneity is described by the column vectors:
 
     
       
         
           
             
               
                 
                   
                     
                       b 
                       x 
                     
                     = 
                     
                       
                         - 
                         
                           
                             ∂ 
                             f 
                           
                           
                             ∂ 
                             
                               x 
                               p 
                             
                           
                         
                       
                       = 
                       
                         
                           ( 
                           
                             
                               
                                 1 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ) 
                         
                         T 
                       
                     
                   
                   , 
                   
                     
                       b 
                       y 
                     
                     = 
                     
                       
                         - 
                         
                           
                             ∂ 
                             f 
                           
                           
                             ∂ 
                             
                               y 
                               p 
                             
                           
                         
                       
                       = 
                       
                         
                           ( 
                           
                             
                               
                                 0 
                               
                               
                                 1 
                               
                               
                                 0 
                               
                             
                           
                           ) 
                         
                         T 
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     The Jacobi matrix A(p(x 2 ,y 2 )) with the elements A ij :=∂ƒ i /∂p j  is the same for both equations and analogous to equation (21), but now in the size 3×3. 
     
       
         
           
             
               
                 
                   
                     A 
                     ⁡ 
                     
                       ( 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           
                             
                               x 
                               p 
                             
                             , 
                             
                               y 
                               p 
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             
                               
                                 
                                   1 
                                   + 
                                   
                                     
                                       τ 
                                       n 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             n 
                                             
                                               w 
                                               , 
                                               x 
                                             
                                             
                                               ( 
                                               
                                                 0 
                                                 , 
                                                 1 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 1 
                                                 , 
                                                 1 
                                               
                                               ) 
                                             
                                           
                                         
                                         + 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       n 
                                       
                                         w 
                                         , 
                                         x 
                                       
                                       
                                         ( 
                                         
                                           1 
                                           , 
                                           0 
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       w 
                                       o 
                                       
                                         ( 
                                         
                                           2 
                                           , 
                                           0 
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               τ 
                               n 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     n 
                                     
                                       w 
                                       , 
                                       x 
                                     
                                     
                                       ( 
                                       
                                         0 
                                         , 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         0 
                                         , 
                                         2 
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     n 
                                     
                                       w 
                                       , 
                                       x 
                                     
                                     
                                       ( 
                                       
                                         1 
                                         , 
                                         0 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         1 
                                         , 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           
                             
                               τ 
                               n 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     n 
                                     
                                       w 
                                       , 
                                       y 
                                     
                                     
                                       ( 
                                       
                                         0 
                                         , 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         1 
                                         , 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     n 
                                     
                                       w 
                                       , 
                                       y 
                                     
                                     
                                       ( 
                                       
                                         1 
                                         , 
                                         0 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         2 
                                         , 
                                         0 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             1 
                             + 
                             
                               
                                 τ 
                                 n 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                       n 
                                       
                                         w 
                                         , 
                                         y 
                                       
                                       
                                         ( 
                                         
                                           0 
                                           , 
                                           1 
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       w 
                                       o 
                                       
                                         ( 
                                         
                                           0 
                                           , 
                                           2 
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       n 
                                       
                                         w 
                                         , 
                                         y 
                                       
                                       
                                         ( 
                                         
                                           1 
                                           , 
                                           0 
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       w 
                                       o 
                                       
                                         ( 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         1 
                                         , 
                                         0 
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     τ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           n 
                                           
                                             w 
                                             , 
                                             z 
                                           
                                           
                                             ( 
                                             
                                               0 
                                               , 
                                               1 
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               1 
                                               , 
                                               1 
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           n 
                                           
                                             w 
                                             , 
                                             z 
                                           
                                           
                                             ( 
                                             
                                               1 
                                               , 
                                               0 
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               2 
                                               , 
                                               0 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         0 
                                         , 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     τ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           n 
                                           
                                             w 
                                             , 
                                             z 
                                           
                                           
                                             ( 
                                             
                                               0 
                                               , 
                                               1 
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               0 
                                               , 
                                               2 
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           n 
                                           
                                             w 
                                             , 
                                             z 
                                           
                                           
                                             ( 
                                             
                                               1 
                                               , 
                                               0 
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               1 
                                               , 
                                               1 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             - 
                             1 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
     The direct solutions by analogy with equation (25) are now determined by 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           p 
                           
                             ( 
                             
                               1 
                               , 
                               0 
                             
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             0 
                             , 
                             0 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           A 
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           b 
                           x 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           p 
                           
                             ( 
                             
                               0 
                               , 
                               1 
                             
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             0 
                             , 
                             0 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           A 
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           b 
                           y 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           p 
                           
                             ( 
                             
                               2 
                               , 
                               0 
                             
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             0 
                             , 
                             0 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           
                             ( 
                             
                               A 
                               
                                 - 
                                 1 
                               
                             
                             ) 
                           
                           
                             ( 
                             
                               1 
                               , 
                               0 
                             
                             ) 
                           
                         
                         | 
                         
                           b 
                           x 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           p 
                           
                             ( 
                             
                               1 
                               , 
                               1 
                             
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             0 
                             , 
                             0 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           
                             
                               ( 
                               
                                 A 
                                 
                                   - 
                                   1 
                                 
                               
                               ) 
                             
                             
                               ( 
                               
                                 0 
                                 , 
                                 1 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             b 
                             x 
                           
                         
                         = 
                         
                           
                             
                               ( 
                               
                                 A 
                                 
                                   - 
                                   1 
                                 
                               
                               ) 
                             
                             
                               ( 
                               
                                 1 
                                 , 
                                 0 
                               
                               ) 
                             
                           
                           | 
                           
                             b 
                             y 
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           p 
                           
                             ( 
                             
                               0 
                               , 
                               2 
                             
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             0 
                             , 
                             0 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           
                             ( 
                             
                               A 
                               
                                 - 
                                 1 
                               
                             
                             ) 
                           
                           
                             ( 
                             
                               0 
                               , 
                               1 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           b 
                           y 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     … 
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         
                           p 
                           
                             ( 
                             
                               
                                 k 
                                 x 
                               
                               , 
                               
                                 k 
                                 y 
                               
                             
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             0 
                             , 
                             0 
                           
                           ) 
                         
                       
                       = 
                       
                         { 
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             ( 
                                             
                                               A 
                                               
                                                 - 
                                                 1 
                                               
                                             
                                             ) 
                                           
                                           
                                             ( 
                                             
                                               
                                                 
                                                   k 
                                                   x 
                                                 
                                                 - 
                                                 1 
                                               
                                               , 
                                               0 
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           b 
                                           x 
                                         
                                       
                                       , 
                                     
                                   
                                   
                                     
                                       
                                         
                                           k 
                                           x 
                                         
                                         ≠ 
                                         0 
                                       
                                       , 
                                       
                                         
                                           k 
                                           y 
                                         
                                         = 
                                         0 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               ( 
                                               
                                                 A 
                                                 
                                                   - 
                                                   1 
                                                 
                                               
                                               ) 
                                             
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     k 
                                                     x 
                                                   
                                                   - 
                                                   1 
                                                 
                                                 , 
                                                 
                                                   k 
                                                   y 
                                                 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             b 
                                             x 
                                           
                                         
                                         = 
                                         
                                           
                                             
                                               ( 
                                               
                                                 A 
                                                 
                                                   - 
                                                   1 
                                                 
                                               
                                               ) 
                                             
                                             
                                               ( 
                                               
                                                 
                                                   k 
                                                   x 
                                                 
                                                 , 
                                                 
                                                   
                                                     k 
                                                     y 
                                                   
                                                   - 
                                                   1 
                                                 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             b 
                                             y 
                                           
                                         
                                       
                                       , 
                                     
                                   
                                   
                                     
                                       
                                         
                                           k 
                                           x 
                                         
                                         ≠ 
                                         0 
                                       
                                       , 
                                       
                                         
                                           k 
                                           y 
                                         
                                         ≠ 
                                         0 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           
                                             ( 
                                             
                                               A 
                                               
                                                 - 
                                                 1 
                                               
                                             
                                             ) 
                                           
                                           
                                             ( 
                                             
                                               0 
                                               , 
                                               
                                                 
                                                   k 
                                                   y 
                                                 
                                                 - 
                                                 1 
                                               
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           b 
                                           y 
                                         
                                       
                                       , 
                                     
                                   
                                   
                                     
                                       
                                         
                                           k 
                                           x 
                                         
                                         = 
                                         0 
                                       
                                       , 
                                       
                                         
                                           k 
                                           y 
                                         
                                         ≠ 
                                         0 
                                       
                                     
                                   
                                 
                               
                               ⁢ 
                               
                                 
 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               where 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 A 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             = 
                             
                               
                                 
                                   A 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       p 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           0 
                                           , 
                                           0 
                                         
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   - 
                                   1 
                                 
                               
                               = 
                               
                                 
                                   A 
                                   ⁡ 
                                   
                                     ( 
                                     0 
                                     ) 
                                   
                                 
                                 
                                   - 
                                   1 
                                 
                               
                             
                           
                           , 
                           
                             
 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 ( 
                                 
                                   A 
                                   
                                     - 
                                     1 
                                   
                                 
                                 ) 
                               
                               
                                 ( 
                                 
                                   1 
                                   , 
                                   0 
                                 
                                 ) 
                               
                             
                             = 
                             
                               
                                 
                                   ⅆ 
                                   
                                     ⅆ 
                                     
                                       x 
                                       p 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     A 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         p 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               x 
                                               p 
                                             
                                             , 
                                             
                                               y 
                                               p 
                                             
                                           
                                           ) 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                               
                               ⁢ 
                               
                                 | 
                                 
                                   
                                     
                                       x 
                                       p 
                                     
                                     = 
                                     0 
                                   
                                   , 
                                   
                                     
                                       y 
                                       p 
                                     
                                     = 
                                     0 
                                   
                                 
                               
                             
                           
                           , 
                           
                             
 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 ( 
                                 
                                   A 
                                   
                                     - 
                                     1 
                                   
                                 
                                 ) 
                               
                               
                                 ( 
                                 
                                   
                                     k 
                                     x 
                                   
                                   , 
                                   
                                     k 
                                     y 
                                   
                                 
                                 ) 
                               
                             
                             = 
                             
                               
                                 
                                   
                                     ⅆ 
                                     
                                       k 
                                       x 
                                     
                                   
                                   
                                     ⅆ 
                                     
                                       x 
                                       p 
                                       
                                         k 
                                         x 
                                       
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     ⅆ 
                                     
                                       k 
                                       y 
                                     
                                   
                                   
                                     ⅆ 
                                     
                                       x 
                                       p 
                                       
                                         k 
                                         y 
                                       
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     A 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         p 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               x 
                                               p 
                                             
                                             , 
                                             
                                               y 
                                               p 
                                             
                                           
                                           ) 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                               
                               ⁢ 
                               
                                 | 
                                 
                                   
                                     
                                       x 
                                       p 
                                     
                                     = 
                                     0 
                                   
                                   , 
                                   
                                     
                                       y 
                                       p 
                                     
                                     = 
                                     0 
                                   
                                 
                               
                             
                           
                           , 
                           
                             etc 
                             .. 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
     By analogy with equations (28) and (29), it results for the three-dimensional consideration: 
                       A   ⁡     (   0   )       =         (           1   -       τ   n     ⁢     w   o     (     2   ,   0     )                   -     τ   n       ⁢     w   o     (     1   ,   1     )             0               -     τ   n       ⁢     w   o     (     1   ,   1     )               1   -       τ   n     ⁢     w   o     (     0   ,   2     )               0           0       0         -   1           )     ⇒       A   ⁡     (   0   )         -   1         =     (           γ   ⁡     (           1   -       τ   n     ⁢     w   o     (     0   ,   2     )                   τ   n     ⁢     w   o     (     1   ,   1     )                     τ   n     ⁢     w   o     (     1   ,   1     )               1   -       τ   n     ⁢     w   o     (     2   ,   0     )                 )                             0           0                       0                                                       0                           -   1           )         ⁢     
     ⁢           ⁢   with   ⁢     
     ⁢     γ   =         -   1       det   ⁡     (     A   ⁡     (   0   )       )         =     1       1   -       τ   n     ⁢     w   o     (     2   ,   0     )         -       (       τ   n     ⁢     w   o     (     1   ,   1     )         )     2     -       τ   n     ⁢     w   o     (     0   ,   2     )         +         (     τ   n     )     2     ⁢     w   o     (     2   ,   0     )       ⁢     w   o     (     0   ,   2     )           )                   (   42   )               
and after application of equations (39) and (41), the solutions
 
     
       
         
           
             
               
                 
                   
                     
                       
                         p 
                         
                           ( 
                           
                             1 
                             , 
                             0 
                           
                           ) 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           0 
                           , 
                           0 
                         
                         ) 
                       
                     
                     = 
                     
                       γ 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 n 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     
                                       τ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         w 
                                         o 
                                         
                                           ( 
                                           
                                             0 
                                             , 
                                             2 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 n 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 τ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   w 
                                   o 
                                   
                                     ( 
                                     
                                       1 
                                       , 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               0 
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         p 
                         
                           ( 
                           
                             0 
                             , 
                             1 
                           
                           ) 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           0 
                           , 
                           0 
                         
                         ) 
                       
                     
                     = 
                     
                       γ 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 n 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 τ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   w 
                                   o 
                                   
                                     ( 
                                     
                                       1 
                                       , 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 n 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     
                                       τ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         w 
                                         o 
                                         
                                           ( 
                                           
                                             2 
                                             , 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               0 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
     
     After further application of equations (39) and (41), it results in the second order
 
 w   p   (2,0) =γ(τ/ n ( w   o   (1,1) ) 2 +(1 −τ/n w   o   (0,2) ) w   o   (2,0) )
 
 w   p   (1,1)   =γw   o   (1,1)  
 
 w   p   (0,2) =γ(τ/ n ( w   o   (1,1) ) 2 +(1 −τ/n w   o   (2,0) ) w   o   (0,2) )  (44)
 
     In a preferred embodiment, the coordinate axes for determination of the propagation are selected or determined such that the x axis and the y axis coincide with the directions of the main curvatures of the original wavefront. It thereby holds that w o   (1,1) =0, and the equations (44) are simplified as 
     
       
         
           
             
               
                 
                   
                     
                       w 
                       p 
                       
                         ( 
                         
                           2 
                           , 
                           0 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         1 
                         
                           1 
                           - 
                           
                             
                               τ 
                               n 
                             
                             ⁢ 
                             
                               w 
                               o 
                               
                                 ( 
                                 
                                   2 
                                   , 
                                   0 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         w 
                         o 
                         
                           ( 
                           
                             2 
                             , 
                             0 
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       w 
                       p 
                       
                         ( 
                         
                           1 
                           , 
                           1 
                         
                         ) 
                       
                     
                     = 
                     0 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       w 
                       p 
                       
                         ( 
                         
                           0 
                           , 
                           2 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         1 
                         
                           1 
                           - 
                           
                             
                               τ 
                               n 
                             
                             ⁢ 
                             
                               w 
                               o 
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   2 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         w 
                         o 
                         
                           ( 
                           
                             0 
                             , 
                             2 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
     In a corresponding way, the equations in the third order are preferably expressed as follows: 
     
       
         
           
             
               
                 
                   
                     
                       w 
                       p 
                       
                         ( 
                         
                           3 
                           , 
                           0 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         γ 
                         3 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     
                                       τ 
                                       n 
                                     
                                     ⁢ 
                                     
                                       w 
                                       o 
                                       
                                         ( 
                                         
                                           0 
                                           , 
                                           2 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               3 
                             
                             ⁢ 
                             
                               w 
                               o 
                               
                                 ( 
                                 
                                   3 
                                   , 
                                   0 
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               τ 
                               n 
                             
                             ⁢ 
                             
                               
                                 w 
                                 o 
                                 
                                   ( 
                                   
                                     1 
                                     , 
                                     1 
                                   
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     3 
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           1 
                                           - 
                                           
                                             
                                               τ 
                                               n 
                                             
                                             ⁢ 
                                             
                                               w 
                                               o 
                                               
                                                 ( 
                                                 
                                                   0 
                                                   , 
                                                   2 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       w 
                                       o 
                                       
                                         ( 
                                         
                                           2 
                                           , 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       τ 
                                       n 
                                     
                                     ⁢ 
                                     
                                       
                                         w 
                                         o 
                                         
                                           ( 
                                           
                                             1 
                                             , 
                                             1 
                                           
                                           ) 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             τ 
                                             n 
                                           
                                           ⁢ 
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 0 
                                                 , 
                                                 3 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 1 
                                                 , 
                                                 1 
                                               
                                               ) 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     3 
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           1 
                                           - 
                                           
                                             
                                               τ 
                                               n 
                                             
                                             ⁢ 
                                             
                                               w 
                                               o 
                                               
                                                 ( 
                                                 
                                                   0 
                                                   , 
                                                   2 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       w 
                                       o 
                                       
                                         ( 
                                         
                                           2 
                                           , 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       w 
                       p 
                       
                         ( 
                         
                           2 
                           , 
                           1 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         γ 
                         3 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             w 
                             o 
                             
                               ( 
                               
                                 2 
                                 , 
                                 1 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               τ 
                               n 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         1 
                                         , 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         2 
                                         ⁢ 
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               1 
                                               , 
                                               2 
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         w 
                                         o 
                                         
                                           ( 
                                           
                                             3 
                                             , 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         2 
                                         ⁢ 
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               0 
                                               , 
                                               2 
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         w 
                                         o 
                                         
                                           ( 
                                           
                                             2 
                                             , 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         2 
                                         , 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 ( 
                                 
                                   τ 
                                   n 
                                 
                                 ) 
                               
                               2 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         2 
                                         , 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       
                                         ( 
                                         
                                           0 
                                           , 
                                           2 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                                 - 
                                 
                                   2 
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               1 
                                               , 
                                               1 
                                             
                                             ) 
                                           
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               w 
                                               o 
                                               
                                                 ( 
                                                 
                                                   1 
                                                   , 
                                                   2 
                                                 
                                                 ) 
                                               
                                             
                                             + 
                                             
                                               w 
                                               o 
                                               
                                                 ( 
                                                 
                                                   3 
                                                   , 
                                                   0 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       - 
                                       
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               2 
                                               , 
                                               0 
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               2 
                                               , 
                                               1 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         0 
                                         , 
                                         2 
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         0 
                                         , 
                                         3 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       
                                         ( 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       w 
                                       o 
                                       
                                         ( 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                         ) 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 1 
                                                 , 
                                                 1 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 2 
                                                 , 
                                                 1 
                                               
                                               ) 
                                             
                                           
                                         
                                         - 
                                         
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 1 
                                                 , 
                                                 2 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 2 
                                                 , 
                                                 0 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 ( 
                                 
                                   τ 
                                   n 
                                 
                                 ) 
                               
                               3 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         1 
                                         , 
                                         2 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       
                                         ( 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                         ) 
                                       
                                       3 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               0 
                                               , 
                                               3 
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               2 
                                               , 
                                               0 
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         2 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               0 
                                               , 
                                               2 
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               2 
                                               , 
                                               1 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       
                                         ( 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       w 
                                       o 
                                       
                                         ( 
                                         
                                           0 
                                           , 
                                           2 
                                         
                                         ) 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           2 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 1 
                                                 , 
                                                 2 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 2 
                                                 , 
                                                 0 
                                               
                                               ) 
                                             
                                           
                                         
                                         + 
                                         
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 0 
                                                 , 
                                                 2 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 3 
                                                 , 
                                                 0 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         1 
                                         , 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     w 
                                     o 
                                     
                                       
                                         ( 
                                         
                                           0 
                                           , 
                                           2 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         2 
                                         , 
                                         0 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         2 
                                         , 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       w 
                       p 
                       
                         ( 
                         
                           1 
                           , 
                           2 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         γ 
                         3 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             w 
                             o 
                             
                               ( 
                               
                                 1 
                                 , 
                                 2 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               τ 
                               n 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         1 
                                         , 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         2 
                                         ⁢ 
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               2 
                                               , 
                                               1 
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         w 
                                         o 
                                         
                                           ( 
                                           
                                             0 
                                             , 
                                             3 
                                           
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         2 
                                         ⁢ 
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               2 
                                               , 
                                               0 
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         w 
                                         o 
                                         
                                           ( 
                                           
                                             0 
                                             , 
                                             2 
                                           
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         1 
                                         , 
                                         2 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 ( 
                                 
                                   τ 
                                   n 
                                 
                                 ) 
                               
                               2 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         1 
                                         , 
                                         2 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       
                                         ( 
                                         
                                           2 
                                           , 
                                           0 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                                 - 
                                 
                                   2 
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               1 
                                               , 
                                               1 
                                             
                                             ) 
                                           
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               w 
                                               o 
                                               
                                                 ( 
                                                 
                                                   2 
                                                   , 
                                                   1 
                                                 
                                                 ) 
                                               
                                             
                                             + 
                                             
                                               w 
                                               o 
                                               
                                                 ( 
                                                 
                                                   0 
                                                   , 
                                                   3 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       - 
                                       
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               0 
                                               , 
                                               2 
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               1 
                                               , 
                                               2 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         2 
                                         , 
                                         0 
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         3 
                                         , 
                                         0 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       
                                         ( 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       w 
                                       o 
                                       
                                         ( 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                         ) 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 1 
                                                 , 
                                                 1 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 1 
                                                 , 
                                                 2 
                                               
                                               ) 
                                             
                                           
                                         
                                         - 
                                         
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 2 
                                                 , 
                                                 1 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 0 
                                                 , 
                                                 2 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 ( 
                                 
                                   τ 
                                   n 
                                 
                                 ) 
                               
                               3 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         2 
                                         , 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       
                                         ( 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                         ) 
                                       
                                       3 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               3 
                                               , 
                                               0 
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               0 
                                               , 
                                               2 
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         2 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               2 
                                               , 
                                               0 
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           w 
                                           o 
                                           
                                             ( 
                                             
                                               1 
                                               , 
                                               2 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       
                                         ( 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       w 
                                       o 
                                       
                                         ( 
                                         
                                           2 
                                           , 
                                           0 
                                         
                                         ) 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           2 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 2 
                                                 , 
                                                 1 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 0 
                                                 , 
                                                 2 
                                               
                                               ) 
                                             
                                           
                                         
                                         + 
                                         
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 2 
                                                 , 
                                                 0 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 0 
                                                 , 
                                                 3 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         1 
                                         , 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     w 
                                     o 
                                     
                                       
                                         ( 
                                         
                                           2 
                                           , 
                                           0 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         0 
                                         , 
                                         2 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     w 
                                     o 
                                     
                                       ( 
                                       
                                         1 
                                         , 
                                         2 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       w 
                       p 
                       
                         ( 
                         
                           0 
                           , 
                           3 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         γ 
                         3 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     
                                       τ 
                                       n 
                                     
                                     ⁢ 
                                     
                                       w 
                                       o 
                                       
                                         ( 
                                         
                                           2 
                                           , 
                                           0 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               3 
                             
                             ⁢ 
                             
                               w 
                               o 
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   3 
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               τ 
                               n 
                             
                             ⁢ 
                             
                               
                                 w 
                                 o 
                                 
                                   ( 
                                   
                                     1 
                                     , 
                                     1 
                                   
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     3 
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           1 
                                           - 
                                           
                                             
                                               τ 
                                               n 
                                             
                                             ⁢ 
                                             
                                               w 
                                               o 
                                               
                                                 ( 
                                                 
                                                   2 
                                                   , 
                                                   0 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       w 
                                       o 
                                       
                                         ( 
                                         
                                           1 
                                           , 
                                           2 
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       τ 
                                       n 
                                     
                                     ⁢ 
                                     
                                       
                                         w 
                                         o 
                                         
                                           ( 
                                           
                                             1 
                                             , 
                                             1 
                                           
                                           ) 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             τ 
                                             n 
                                           
                                           ⁢ 
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 3 
                                                 , 
                                                 0 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             w 
                                             o 
                                             
                                               ( 
                                               
                                                 1 
                                                 , 
                                                 1 
                                               
                                               ) 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     3 
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           1 
                                           - 
                                           
                                             
                                               τ 
                                               n 
                                             
                                             ⁢ 
                                             
                                               w 
                                               o 
                                               
                                                 ( 
                                                 
                                                   2 
                                                   , 
                                                   0 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       w 
                                       o 
                                       
                                         ( 
                                         
                                           2 
                                           , 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
           
         
       
     
     Now, if one replaces 
               d   =         τ   n     ⁢           ⁢   and   ⁢           ⁢   γ     =     1       1   -       d   n     ⁢     S   oxx       -       (       d   n     ⁢     S   oxy       )     2     -       d   n     ⁢     S   oyy       +         (     d   n     )     2     ⁢     S   oxx     ⁢     S   oyy         )           ,         
the propagation of the wavefront in the second order in the form of the local aberrations can be expressed as follows:
 
     
       
         
           
             
               
                 
                   
                     s 
                     p 
                   
                   = 
                   
                     γ 
                     ( 
                     
                       
                         s 
                         o 
                       
                       + 
                       
                         
                           d 
                           n 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     S 
                                     oxy 
                                     2 
                                   
                                   - 
                                   
                                     
                                       S 
                                       oxx 
                                     
                                     ⁢ 
                                     
                                       S 
                                       oyy 
                                     
                                   
                                 
                               
                             
                             
                               
                                 0 
                               
                             
                             
                               
                                 
                                   
                                     S 
                                     oxy 
                                     2 
                                   
                                   - 
                                   
                                     
                                       S 
                                       oxx 
                                     
                                     ⁢ 
                                     
                                       S 
                                       oyy 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
     Moreover, if one replaces 
                 β   x     =         1     1   -       d   n     ⁢     S   xx           ⁢           ⁢   and   ⁢           ⁢     β   y       =     1     1   -       d   n     ⁢     S   yy               ,         
the propagation of the wavefront in the third order can be described by:
 
     
       
         
           
             
               
                 
                   
                     e 
                     
                       p 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       3 
                     
                   
                   = 
                   
                     
                       
                         γ 
                         3 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 β 
                                 y 
                                 
                                   - 
                                   3 
                                 
                               
                             
                             
                               
                                 3 
                                 ⁢ 
                                 
                                   β 
                                   y 
                                   
                                     - 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   d 
                                   n 
                                 
                                 ⁢ 
                                 
                                   S 
                                   xy 
                                 
                               
                             
                             
                               
                                 3 
                                 ⁢ 
                                 
                                   
                                     
                                       β 
                                       y 
                                       
                                         - 
                                         1 
                                       
                                     
                                     ( 
                                     
                                       
                                         d 
                                         n 
                                       
                                       ⁢ 
                                       
                                         S 
                                         xy 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   ( 
                                   
                                     
                                       d 
                                       n 
                                     
                                     ⁢ 
                                     
                                       S 
                                       xy 
                                     
                                   
                                   ) 
                                 
                                 3 
                               
                             
                           
                           
                             
                               
                                 
                                   β 
                                   y 
                                   
                                     - 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   d 
                                   n 
                                 
                                 ⁢ 
                                 
                                   S 
                                   xy 
                                 
                               
                             
                             
                               
                                 
                                   β 
                                   y 
                                   
                                     - 
                                     1 
                                   
                                 
                                 ( 
                                 
                                   
                                     1 
                                     γ 
                                   
                                   + 
                                   
                                     3 
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           
                                             d 
                                             n 
                                           
                                           ⁢ 
                                           
                                             S 
                                             xy 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                             
                               
                                 
                                   
                                     2 
                                     γ 
                                   
                                   ⁢ 
                                   
                                     d 
                                     n 
                                   
                                   ⁢ 
                                   
                                     S 
                                     xy 
                                   
                                 
                                 + 
                                 
                                   3 
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         
                                           d 
                                           n 
                                         
                                         ⁢ 
                                         
                                           S 
                                           xy 
                                         
                                       
                                       ) 
                                     
                                     3 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     β 
                                     x 
                                     
                                       - 
                                       1 
                                     
                                   
                                   ( 
                                   
                                     
                                       d 
                                       n 
                                     
                                     ⁢ 
                                     
                                       S 
                                       xy 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     β 
                                     y 
                                     
                                       - 
                                       1 
                                     
                                   
                                   ( 
                                   
                                     
                                       d 
                                       n 
                                     
                                     ⁢ 
                                     
                                       S 
                                       xy 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             
                               
                                 
                                   
                                     2 
                                     γ 
                                   
                                   ⁢ 
                                   
                                     d 
                                     n 
                                   
                                   ⁢ 
                                   
                                     S 
                                     xy 
                                   
                                 
                                 + 
                                 
                                   3 
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         
                                           d 
                                           n 
                                         
                                         ⁢ 
                                         
                                           S 
                                           xy 
                                         
                                       
                                       ) 
                                     
                                     3 
                                   
                                 
                               
                             
                             
                               
                                 
                                   β 
                                   x 
                                   
                                     - 
                                     1 
                                   
                                 
                                 ( 
                                 
                                   
                                     1 
                                     γ 
                                   
                                   + 
                                   
                                     3 
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           
                                             d 
                                             n 
                                           
                                           ⁢ 
                                           
                                             S 
                                             xy 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                             
                               
                                 
                                   β 
                                   x 
                                   
                                     - 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   d 
                                   n 
                                 
                                 ⁢ 
                                 
                                   S 
                                   xy 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       d 
                                       n 
                                     
                                     ⁢ 
                                     
                                       S 
                                       xy 
                                     
                                   
                                   ) 
                                 
                                 3 
                               
                             
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   
                                     
                                       β 
                                       x 
                                       
                                         - 
                                         1 
                                       
                                     
                                     ( 
                                     
                                       
                                         d 
                                         n 
                                       
                                       ⁢ 
                                       
                                         S 
                                         xy 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                             
                               
                                 3 
                                 ⁢ 
                                 
                                   β 
                                   x 
                                   
                                     - 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   d 
                                   n 
                                 
                                 ⁢ 
                                 
                                   S 
                                   xy 
                                 
                               
                             
                             
                               
                                 β 
                                 x 
                                 
                                   - 
                                   3 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       e 
                       
                         o 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         3 
                       
                     
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
           
         
       
     
     In a preferred embodiment, the coordinate axes for determination of the propagation are selected or determined such that the x axis and the y axis coincide with the directions of the main curvatures of the original wavefront. Thereby, the equations (47) and (48) are simplified as 
     
       
         
           
             
               
                 
                   
                     s 
                     p 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               β 
                               x 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               β 
                               y 
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       s 
                       o 
                     
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
             
               
                 
                   
                     e 
                     
                       p 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       3 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               β 
                               x 
                               3 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               
                                 β 
                                 x 
                                 2 
                               
                               ⁢ 
                               
                                 β 
                                 y 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               
                                 β 
                                 x 
                               
                               ⁢ 
                               
                                 β 
                                 y 
                                 2 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               β 
                               y 
                               3 
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       e 
                       
                         o 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         3 
                       
                     
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
           
         
       
     
     The propagation of fourth-order aberrations can be determined in a comparatively simply way by: 
     
       
         
           
             
               
                 
                   
                     e 
                     
                       p 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       4 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               β 
                               x 
                               4 
                             
                           
                           
                             … 
                           
                           
                             
                                 
                             
                           
                           
                             … 
                           
                           
                             0 
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             
                               
                                 β 
                                 x 
                                 3 
                               
                               ⁢ 
                               
                                 β 
                                 y 
                                 1 
                               
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             ⋮ 
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                               
                                 β 
                                 x 
                                 2 
                               
                               ⁢ 
                               
                                 β 
                                 y 
                                 2 
                               
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                               
                                 β 
                                 x 
                                 1 
                               
                               ⁢ 
                               
                                 β 
                                 y 
                                 3 
                               
                             
                           
                           
                             ⋮ 
                           
                         
                         
                           
                             0 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             … 
                           
                           
                             
                               β 
                               y 
                               4 
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           e 
                           
                             o 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             4 
                           
                         
                         + 
                         
                           
                             d 
                             n 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     3 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             β 
                                             x 
                                           
                                           ⁢ 
                                           
                                             E 
                                             oxxx 
                                             2 
                                           
                                         
                                         + 
                                         
                                           
                                             β 
                                             y 
                                           
                                           ⁢ 
                                           
                                             E 
                                             oxxy 
                                             2 
                                           
                                         
                                         - 
                                         
                                           
                                             S 
                                             oxx 
                                             4 
                                           
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     3 
                                     ⁢ 
                                     
                                       
                                         E 
                                         oxxy 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             
                                               β 
                                               x 
                                             
                                             ⁢ 
                                             
                                               E 
                                               oxxx 
                                             
                                           
                                           + 
                                           
                                             
                                               β 
                                               y 
                                             
                                             ⁢ 
                                             
                                               E 
                                               oxyy 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         β 
                                         x 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             2 
                                             ⁢ 
                                             
                                               E 
                                               oxxy 
                                               2 
                                             
                                           
                                           + 
                                           
                                             
                                               E 
                                               oxxx 
                                             
                                             ⁢ 
                                             
                                               E 
                                               oxyy 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       
                                         β 
                                         y 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             2 
                                             ⁢ 
                                             
                                               E 
                                               oxyy 
                                               2 
                                             
                                           
                                           + 
                                           
                                             
                                               E 
                                               oxxy 
                                             
                                             ⁢ 
                                             
                                               E 
                                               oyyy 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             S 
                                             oxx 
                                           
                                           ⁢ 
                                           
                                             S 
                                             oyy 
                                           
                                         
                                         n 
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     3 
                                     ⁢ 
                                     
                                       
                                         E 
                                         oxyy 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             
                                               β 
                                               x 
                                             
                                             ⁢ 
                                             
                                               E 
                                               oxxy 
                                             
                                           
                                           + 
                                           
                                             
                                               β 
                                               y 
                                             
                                             ⁢ 
                                             
                                               E 
                                               oyyy 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     3 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             β 
                                             x 
                                           
                                           ⁢ 
                                           
                                             E 
                                             oxyy 
                                             2 
                                           
                                         
                                         + 
                                         
                                           
                                             β 
                                             y 
                                           
                                           ⁢ 
                                           
                                             E 
                                             oyyy 
                                             2 
                                           
                                         
                                         - 
                                         
                                           
                                             S 
                                             oyy 
                                             4 
                                           
                                           
                                             n 
                                             2 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
           
         
       
     
     For 2&lt;k≦4, this is preferably generalized by 
                     e   pk     =       B   k     ⁡     (       e   ok     +     r   k       )               (   52   )                 with   ⁢           ⁢     B   k       =     (           β   x   k         …                   …       0           ⋮           β   x     k   -   1       ⁢     β   y   1                                   ⋮                                   ⋱                                   ⋮                                   β   x   1     ⁢     β   y     k   -   1             ⋮           0                               …         β   y   k           )             (   53   )               
where r k  represents a vector in which by analogy with R k  in equation (33) all remainder terms R k     x     ,k     y    are included.
 
     In a preferred embodiment it holds:
 
 s   p   =T   (2) ( {circumflex over (R)} ) T   (2) ({circumflex over (β)}){tilde over ({circumflex over ( s )})} p   S ( {circumflex over (R)} )
 
 e   p3   =T   (3) ( {circumflex over (R)} ) T   (3) ({circumflex over (β)}){tilde over ({circumflex over ( e )})} p3   S ( {circumflex over (R)} )
 
 e   p4   =T   (4) ( {circumflex over (R)} ) T   (4) ({circumflex over (β)}){tilde over ({circumflex over ( e )})} p4   S ( {circumflex over (R)} )
 
etc.  (54)
 
where s p , e p3 , e p4 , . . . apply in every coordinate system and where
 
                     ⁢       β   ^     =     (             β   ^     xx         0           0           β   ^     yy           )                           ⁢   with                       ⁢         β   ^     xx     =       (     1   -       τ   n     ⁢       w   ^       (     2   ,   0     )           )       -   1                             ⁢         β   ^     yy     =       (     1   -       τ   n     ⁢       w   ^       (     0   ,   2     )           )       -   1                       (             w   ^       (     2   ,   0     )                   w   ^       (     1   ,   1     )                   w   ^       (     0   ,   2     )             )     =         1   2     ⁢     (       w     (     2   ,   0     )       +     w     (     0   ,   2     )         )     ⁢     (         1           0           1         )       +       1   2     ⁢     (       w     (     2   ,   0     )       -     w     (     0   ,   2     )         )     ⁢       1   +       (       2   ⁢     w     (     1   ,   1     )             w     (     2   ,   0     )       -     w     (     0   ,   2     )           )     2         ⁢     (         1           0             -   1           )               
is an auxiliary matrix, which can be referred to back to the matrix
 
             β   =       (           β   xx           β   xy               β   xy           β   yy           )     =       (     1   -       τ   n     ⁢     (           w   o     (     2   ,   0     )             w   o     (     1   ,   1     )                 w   o     (     1   ,   1     )             w   o     (     0   ,   2     )             )         )       -   1                   by               β   ^     =       R   ^     ⁢           ⁢   β   ⁢           ⁢       R   ^       -   1                   where               R   ^     =     (           cos   ⁢           ⁢   φ             -   sin     ⁢           ⁢   φ               sin   ⁢           ⁢   φ           cos   ⁢           ⁢   φ           )           
is a rotation matrix, which transforms from the special system in which the x axis and the y axis coincide with the directions of the main curvatures of the original wavefront into the general system. Here,
 
     
       
         
           
             φ 
             = 
             
               
                 1 
                 2 
               
               ⁢ 
               arc 
               ⁢ 
               
                   
               
               ⁢ 
               tan 
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   2 
                   ⁢ 
                   
                     β 
                     xy 
                   
                 
                 
                   
                     β 
                     yy 
                   
                   - 
                   
                     β 
                     xx 
                   
                 
               
             
           
         
       
     
     In equation (54), the matrix 
               S   ⁡     (     R   ^     )       ⁢     :     =     (           R   ^         0           0       1         )           
is used, and, further, T (1) , T (2) , T (3) , T (4)  in equation (54) are matrix-like functions which assign the matrices
 
                     ⁢         T     (   1   )       ⁡     (   X   )       =     (         a       c           b       d         )                           ⁢         T     (   2   )       ⁡     (   X   )       =     (           a   2           2   ⁢   ac           c   2             ab         ad   +   bc         cd             b   2           2   ⁢   bd           d   2           )                           ⁢         T     (   3   )       ⁡     (   X   )       =     (           a   3           3   ⁢     a   2     ⁢   c           3   ⁢     ac   2             c   3                 a   2     ⁢   b           a   ⁡     (     ad   +     2   ⁢   bc       )             c   ⁡     (       2   ⁢   ad     +   bc     )               c   2     ⁢   d               ab   2           b   ⁡     (       2   ⁢   ad     +   bc     )             d   ⁡     (     ad   +     2   ⁢   bc       )             cd   2               b   3           3   ⁢     b   2     ⁢   d           3   ⁢     bd   2             d   3           )                       T     (   4   )       ⁡     (   X   )       =     (           a   4           4   ⁢     a   3     ⁢   c           6   ⁢     a   2     ⁢     c   2             4   ⁢     ac   3             c   4                 a   3     ⁢   b             a   2     ⁡     (       3   ⁢   bc     +   ad     )             3   ⁢     ac   ⁡     (     bc   +   ad     )                 c   2     ⁡     (     bc   +     3   ⁢   ad       )               c   3     ⁢   d                 a   2     ⁢     b   2             2   ⁢     ab   ⁡     (     bc   +   ad     )                   b   2     ⁢     c   2       +     4   ⁢   abcd     +       a   2     ⁢     d   2               2   ⁢     cd   ⁡     (     bc   +   ad     )                 c   2     ⁢     d   2                 ab   3             b   2     ⁡     (     bc   +     3   ⁢   ad       )             3   ⁢     bd   ⁡     (     bc   +   ad     )                 d   2     ⁡     (       3   ⁢   bc     +   ad     )             cd   3               b   4           4   ⁢     b   3     ⁢   d           6   ⁢     b   2     ⁢     d   2             4   ⁢     bd   3             d   4           )           
to a predetermined matrix
 
     
       
         
           
             x 
             = 
             
               ( 
               
                 
                   
                     a 
                   
                   
                     b 
                   
                 
                 
                   
                     c 
                   
                   
                     d 
                   
                 
               
               ) 
             
           
         
       
     
     For even higher orders, the matrices T (n)  can be defined with n&gt;4. Finally, as expressions for solutions on which the solutions for the propagated wavefronts can be formed by the transformation in equation (54), there are predetermined for the order n=2 
                     s   ~     ^     p     =       (               w   ~     ^     p     (     2   ,   0     )                     w   ~     ^     p     (     1   ,   1     )                     w   ~     ^     p     (     0   ,   2     )             )     =       (               β   ^     11     -   1       ⁢       w   ^     o     (     2   ,   0     )                       0                 β   ^     22     -   1       ⁢       w   ^     o     (     0   ,   2     )                     )     =       (             w   ^       (     2   ,   0     )               0               w   ^     o     (     0   ,   2     )             )     -       τ   n     ⁢     (             w   ^     o       (     2   ,   0     )     2               0               w   ^     o       (     0   ,   2     )     2             )               ,         
for the order n=3
 
                   e   ~     ^       p   ⁢           ⁢   3       =       (               w   ~     ^     p     (     3   ,   0     )                     w   ~     ^     p     (     2   ,   1     )                     w   ~     ^     p     (     1   ,   2     )                     w   ~     ^     p     (     0   ,   3     )             )     =     (             w   ^     o     (     3   ,   0     )                   w   ^     o     (     2   ,   1     )                   w   ^     o     (     1   ,   2     )                   w   ^     o     (     0   ,   3     )             )             
and for the order n=4
 
     
       
         
           
             
               
                 
                   e 
                   ~ 
                 
                 ^ 
               
               
                 p 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 4 
               
             
             = 
             
               
                 ( 
                 
                   
                     
                       
                         
                           
                             w 
                             ~ 
                           
                           ^ 
                         
                         p 
                         
                           ( 
                           
                             4 
                             , 
                             0 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             w 
                             ~ 
                           
                           ^ 
                         
                         p 
                         
                           ( 
                           
                             3 
                             , 
                             1 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             w 
                             ~ 
                           
                           ^ 
                         
                         p 
                         
                           ( 
                           
                             2 
                             , 
                             2 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             w 
                             ~ 
                           
                           ^ 
                         
                         p 
                         
                           ( 
                           
                             1 
                             , 
                             3 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             w 
                             ~ 
                           
                           ^ 
                         
                         p 
                         
                           ( 
                           
                             0 
                             , 
                             4 
                           
                           ) 
                         
                       
                     
                   
                 
                 ) 
               
               = 
               
                 
                   ( 
                   
                     
                       
                         
                           
                             w 
                             ^ 
                           
                           o 
                           
                             ( 
                             
                               4 
                               , 
                               0 
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             w 
                             ^ 
                           
                           o 
                           
                             ( 
                             
                               3 
                               , 
                               1 
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             w 
                             ^ 
                           
                           o 
                           
                             ( 
                             
                               2 
                               , 
                               2 
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             w 
                             ^ 
                           
                           o 
                           
                             ( 
                             
                               1 
                               , 
                               3 
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             w 
                             ^ 
                           
                           o 
                           
                             ( 
                             
                               0 
                               , 
                               4 
                             
                             ) 
                           
                         
                       
                     
                   
                   ) 
                 
                 + 
                 
                   
                     τ 
                     n 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           
                             3 
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     
                                       β 
                                       ^ 
                                     
                                     11 
                                   
                                   ⁢ 
                                   
                                     
                                       w 
                                       ^ 
                                     
                                     o 
                                     
                                       
                                         ( 
                                         
                                           3 
                                           , 
                                           0 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       β 
                                       ^ 
                                     
                                     22 
                                   
                                   ⁢ 
                                   
                                     
                                       w 
                                       ^ 
                                     
                                     o 
                                     
                                       
                                         ( 
                                         
                                           2 
                                           , 
                                           1 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     w 
                                     ^ 
                                   
                                   o 
                                   
                                     
                                       ( 
                                       
                                         2 
                                         , 
                                         0 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             3 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 
                                   w 
                                   ^ 
                                 
                                 o 
                                 
                                   ( 
                                   
                                     2 
                                     , 
                                     1 
                                   
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         β 
                                         ^ 
                                       
                                       11 
                                     
                                     ⁢ 
                                     
                                       
                                         w 
                                         ^ 
                                       
                                       o 
                                       
                                         ( 
                                         
                                           3 
                                           , 
                                           0 
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         β 
                                         ^ 
                                       
                                       22 
                                     
                                     ⁢ 
                                     
                                       
                                         w 
                                         ^ 
                                       
                                       o 
                                       
                                         ( 
                                         
                                           1 
                                           , 
                                           2 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         w 
                                         ^ 
                                       
                                       o 
                                       
                                         ( 
                                         
                                           1 
                                           , 
                                           2 
                                         
                                         ) 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               β 
                                               ^ 
                                             
                                             11 
                                           
                                           ⁢ 
                                           
                                             
                                               w 
                                               ^ 
                                             
                                             o 
                                             
                                               ( 
                                               
                                                 3 
                                                 , 
                                                 0 
                                               
                                               ) 
                                             
                                           
                                         
                                         + 
                                         
                                           2 
                                           ⁢ 
                                           
                                             
                                               β 
                                               ^ 
                                             
                                             22 
                                           
                                           ⁢ 
                                           
                                             
                                               w 
                                               ^ 
                                             
                                             o 
                                             
                                               ( 
                                               
                                                 1 
                                                 , 
                                                 2 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         w 
                                         ^ 
                                       
                                       o 
                                       
                                         ( 
                                         
                                           2 
                                           , 
                                           1 
                                         
                                         ) 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           2 
                                           ⁢ 
                                           
                                             
                                               β 
                                               ^ 
                                             
                                             11 
                                           
                                           ⁢ 
                                           
                                             
                                               w 
                                               ^ 
                                             
                                             o 
                                             
                                               ( 
                                               
                                                 2 
                                                 , 
                                                 1 
                                               
                                               ) 
                                             
                                           
                                         
                                         + 
                                         
                                           
                                             
                                               β 
                                               ^ 
                                             
                                             22 
                                           
                                           ⁢ 
                                           
                                             
                                               w 
                                               ^ 
                                             
                                             o 
                                             
                                               ( 
                                               
                                                 0 
                                                 , 
                                                 3 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       w 
                                       ^ 
                                     
                                     o 
                                     
                                       
                                         ( 
                                         
                                           2 
                                           , 
                                           0 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       w 
                                       ^ 
                                     
                                     o 
                                     
                                       
                                         ( 
                                         
                                           0 
                                           , 
                                           2 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             3 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 
                                   w 
                                   ^ 
                                 
                                 o 
                                 
                                   ( 
                                   
                                     1 
                                     , 
                                     2 
                                   
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         β 
                                         ^ 
                                       
                                       11 
                                     
                                     ⁢ 
                                     
                                       
                                         w 
                                         ^ 
                                       
                                       o 
                                       
                                         ( 
                                         
                                           2 
                                           , 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         β 
                                         ^ 
                                       
                                       22 
                                     
                                     ⁢ 
                                     
                                       
                                         w 
                                         ^ 
                                       
                                       o 
                                       
                                         ( 
                                         
                                           0 
                                           , 
                                           3 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             3 
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     
                                       β 
                                       ^ 
                                     
                                     11 
                                   
                                   ⁢ 
                                   
                                     
                                       w 
                                       ^ 
                                     
                                     o 
                                     
                                       
                                         ( 
                                         
                                           1 
                                           , 
                                           2 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       β 
                                       ^ 
                                     
                                     22 
                                   
                                   ⁢ 
                                   
                                     
                                       w 
                                       ^ 
                                     
                                     o 
                                     
                                       
                                         ( 
                                         
                                           0 
                                           , 
                                           3 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     w 
                                     ^ 
                                   
                                   o 
                                   
                                     
                                       ( 
                                       
                                         0 
                                         , 
                                         2 
                                       
                                       ) 
                                     
                                     4 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     In the following, it will be shown how the aberrations of a spectacle lens are considered in the optimization thereof in a preferable way by the wavefronts being described in different coordinate systems that are rotated relative to each other. As described with respect to  FIG. 1  and  FIG. 2 , the coordinate systems are preferably defined by the intersection points of the main ray  10  with the refractive surface  14 ,  16 , by the refractive surface, and by the direction of the main ray  10 . In order to describe an incoming wavefront, the refractive surface itself, and the outgoing wavefront for the process of refraction on the refractive surface, preferably three different local Cartesian coordinate systems (x,y,z), ( x , y , z ), and (x′,y′,z′) are used. The origin of all these coordinate systems preferably coincides with the intersection point of the main ray  10  with the refractive surface. While the systems have the normal direction to the plane of refraction (i.e. the plane in which the incoming and the outgoing main ray are located) as the common axis x=x′= x , the z axis points along the incoming main ray, the z′ axis along the outgoing main ray, and the  z  axis along the normal of the refractive surface. The orientations of the axis, axis, and axis are preferably selected such that each system is right-handed (cf.  FIG. 5 ) 
     At the transition between the coordinate systems, all vector quantities v depend on each other via the following relations
 
 v=R (ε)   v ,v′=R (ε)   v     (55)
 
where R designates the rotations about the common x axis and is defined by the three-dimensional rotation matrix
 
     
       
         
           
             
               
                 
                   
                     R 
                     ⁡ 
                     
                       ( 
                       ɛ 
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ɛ 
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ɛ 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ɛ 
                           
                         
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ɛ 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   56 
                   ) 
                 
               
             
           
         
       
     
     In case of a rotation of the coordinate system by the angle α about the z axis, the coordinate transformation is described by 
                               x   ~     =       x   ⁢           ⁢   cos   ⁢           ⁢   α     -     y   ⁢           ⁢   sin   ⁢           ⁢   α                     y   ~     =       x   ⁢           ⁢   sin   ⁢           ⁢   α     +     y   ⁢           ⁢   cos   ⁢           ⁢   α               ⁢           ⁢   or   ⁢           ⁢     (           x   ~               y   ~           )       =       R   ⁡     (   α   )       ⁢     (         x           y         )               (   57   )               
with the rotation matrix
 
     
       
         
           
             
               
                 
                   
                     R 
                     ⁡ 
                     
                       ( 
                       α 
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                         
                       
                       
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   58 
                   ) 
                 
               
             
           
         
       
     
     Thus, the wavefront {tilde over (w)} in the rotated coordinate system {tilde over (x)}, {tilde over (y)} is described by
 
 {tilde over (w)} ( {tilde over (x)},{tilde over (y)} )= w ( x ( {tilde over (x)},{tilde over (y)} ), y ( {tilde over (x)},{tilde over (y)} ))  (59)
 
     If one derives the wavefront {tilde over (w)} according to {tilde over (x)},{tilde over (y)}, one obtains the new coefficients ã m,k−m  relative to the coefficients a m,k−m . 
     
       
         
           
             
               
                 
                   
                     
                       a 
                       ~ 
                     
                     
                       m 
                       , 
                       
                         k 
                         - 
                         m 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ∂ 
                           k 
                         
                         
                           
                             ∂ 
                             
                               
                                 x 
                                 ~ 
                               
                               m 
                             
                           
                           ⁢ 
                           
                             ∂ 
                             
                               
                                 y 
                                 ~ 
                               
                               
                                 k 
                                 - 
                                 m 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         w 
                         ⁡ 
                         
                           ( 
                           
                             
                               x 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     ~ 
                                   
                                   , 
                                   
                                     y 
                                     ~ 
                                   
                                 
                                 ) 
                               
                             
                             , 
                             
                               y 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     ~ 
                                   
                                   , 
                                   
                                     y 
                                     ~ 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       | 
                       
                         
                           
                             x 
                             ~ 
                           
                           = 
                           0 
                         
                         , 
                         
                           
                             y 
                             ~ 
                           
                           = 
                           0 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   60 
                   ) 
                 
               
             
           
         
       
     
     In the second order, the aberrations are preferably represented by the vector 
     
       
         
           
             
               
                 
                   s 
                   = 
                   
                     ( 
                     
                       
                         
                           
                             S 
                             xx 
                           
                         
                       
                       
                         
                           
                             S 
                             xy 
                           
                         
                       
                       
                         
                           
                             S 
                             yy 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   61 
                   ) 
                 
               
             
           
         
       
     
     If the coordinate system is rotated by the angle α, the new aberrations {tilde over (s)} of second order (in the rotated coordinate system ({tilde over (x)},{tilde over (y)})) are calculated via 
     
       
         
           
             
               
                 
                   
                     
                       s 
                       ~ 
                     
                     = 
                     
                       
                         
                           R 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           α 
                           ) 
                         
                       
                       ⁢ 
                       s 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   with 
                 
               
               
                 
                   ( 
                   62 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       R 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       α 
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             
                               cos 
                               2 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             
                               sin 
                               2 
                             
                             ⁢ 
                             α 
                           
                         
                       
                       
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             
                               
                                 cos 
                                 2 
                               
                               ⁢ 
                               α 
                             
                             - 
                             
                               
                                 sin 
                                 2 
                               
                               ⁢ 
                               α 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                         
                       
                       
                         
                           
                             
                               sin 
                               2 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             
                               cos 
                               2 
                             
                             ⁢ 
                             α 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   63 
                   ) 
                 
               
             
           
         
       
     
     For higher orders of the aberrations, the dependency of the new coefficients a m,k−m  on the old coefficients a m,k−m  is preferably expressed by 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             00 
                           
                         
                       
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             01 
                           
                         
                       
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             10 
                           
                         
                       
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             02 
                           
                         
                       
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             11 
                           
                         
                       
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             20 
                           
                         
                       
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             03 
                           
                         
                       
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             12 
                           
                         
                       
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             21 
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       
                         R 
                         Pot 
                       
                       ⁡ 
                       
                         ( 
                         
                           N 
                           , 
                           α 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               a 
                               00 
                             
                           
                         
                         
                           
                             
                               a 
                               01 
                             
                           
                         
                         
                           
                             
                               a 
                               10 
                             
                           
                         
                         
                           
                             
                               a 
                               02 
                             
                           
                         
                         
                           
                             
                               a 
                               11 
                             
                           
                         
                         
                           
                             
                               a 
                               20 
                             
                           
                         
                         
                           
                             
                               a 
                               03 
                             
                           
                         
                         
                           
                             
                               a 
                               12 
                             
                           
                         
                         
                           
                             
                               a 
                               21 
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   64 
                   ) 
                 
               
             
           
         
       
     
     The resulting rotation matrix has the block structure, which shows that the coefficients a m,k−m  of the order k only depend on coefficients a m,k−m  of the same order k. The rotation matrix for the first 15 coefficients (N=15) up to the order (k=4) thus reads 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       Pot 
                     
                     ⁡ 
                     
                       ( 
                       
                         15 
                         , 
                         α 
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           
                               
                           
                         
                         
                           … 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             
                               R 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               α 
                               ) 
                             
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                             
                               R 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               α 
                               ) 
                             
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                             
                               R 
                               3 
                             
                             ⁡ 
                             
                               ( 
                               α 
                               ) 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           … 
                         
                         
                           
                               
                           
                         
                         
                           0 
                         
                         
                           
                             
                               R 
                               4 
                             
                             ⁡ 
                             
                               ( 
                               α 
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   65 
                   ) 
                 
               
             
           
         
       
     
     The matrix elements of the block structures R k (α) of the first order (k=1) yield the known rotation matrix 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       α 
                       ) 
                     
                   
                   = 
                   
                     
                       R 
                       ⁡ 
                       
                         ( 
                         α 
                         ) 
                       
                     
                     = 
                     
                       ( 
                       
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                             
                           
                         
                         
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   66 
                   ) 
                 
               
             
           
         
       
     
     In the second order (k=2), the rotation matrix reads 
                       R   2     ⁡     (   α   )       =     (             cos   2     ⁢   α             -   2     ⁢           ⁢   cos   ⁢           ⁢   α   ⁢           ⁢   sin   ⁢           ⁢   α             sin   2     ⁢   α               cos   ⁢           ⁢   α   ⁢           ⁢   sin   ⁢           ⁢   α               cos   2     ⁢   α     -       sin   2     ⁢   α               -   cos     ⁢           ⁢   α   ⁢           ⁢   sin   ⁢           ⁢   α                 sin   2     ⁢   α           2   ⁢           ⁢   cos   ⁢           ⁢   α   ⁢           ⁢   sin   ⁢           ⁢   α             cos   2     ⁢   α           )             (   67   )               
in the third order (k=3)
 
                       R   3     ⁡     (   α   )       =     (             cos   3     ⁢   α             -   3     ⁢           ⁢     cos   2     ⁢   α   ⁢           ⁢   sin   ⁢           ⁢   α           3   ⁢           ⁢   cos   ⁢           ⁢   α   ⁢           ⁢     sin   2     ⁢   α             sin   3     ⁢   α                 cos   2     ⁢   α   ⁢           ⁢   sin   ⁢           ⁢   α               cos   3     ⁢   α     -     2   ⁢           ⁢   cos   ⁢           ⁢     αsin   2     ⁢   α                 sin   3     ⁢   α     -     2   ⁢           ⁢     cos   2     ⁢   α   ⁢           ⁢   sin   ⁢           ⁢   α             cos   ⁢           ⁢   α   ⁢           ⁢     sin   2     ⁢   α               cos   ⁢           ⁢   α   ⁢           ⁢     sin   2     ⁢   α           -     (         sin   3     ⁢   α     -     2   ⁢           ⁢     cos   2     ⁢   α   ⁢           ⁢   sin   ⁢           ⁢   α       )                 cos   3     ⁢   α     -     2   ⁢           ⁢   cos   ⁢           ⁢     αsin   2     ⁢   α               cos   2     ⁢   α   ⁢           ⁢   sin   ⁢           ⁢   α                 sin   3     ⁢   α           3   ⁢           ⁢   cos   ⁢           ⁢   α   ⁢           ⁢     sin   2     ⁢   α           3   ⁢           ⁢     cos   2     ⁢   α   ⁢           ⁢   sin   ⁢           ⁢   α             cos   3     ⁢   α           )             (   68   )               
and in the fourth order (k=4)
 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       4 
                     
                     ⁡ 
                     
                       ( 
                       α 
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             
                               cos 
                               4 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             
                               - 
                               
                                 cos 
                                 3 
                               
                             
                             ⁢ 
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             
                               cos 
                               2 
                             
                             ⁢ 
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sin 
                               2 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sin 
                               3 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             
                               sin 
                               4 
                             
                             ⁢ 
                             α 
                           
                         
                       
                       
                         
                           
                             4 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               cos 
                               3 
                             
                             ⁢ 
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             
                               
                                 cos 
                                 4 
                               
                               ⁢ 
                               α 
                             
                             - 
                             
                               3 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 cos 
                                 2 
                               
                               ⁢ 
                               
                                 sin 
                                 2 
                               
                               ⁢ 
                               α 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         α 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           sin 
                                           3 
                                         
                                         ⁢ 
                                         α 
                                       
                                       - 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       cos 
                                       3 
                                     
                                     ⁢ 
                                     α 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     ( 
                                     
                                       
                                         
                                           sin 
                                           4 
                                         
                                         ⁢ 
                                         α 
                                       
                                       - 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     3 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       cos 
                                       2 
                                     
                                     ⁢ 
                                     
                                       sin 
                                       2 
                                     
                                     ⁢ 
                                     α 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               4 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sin 
                               3 
                             
                             ⁢ 
                             α 
                           
                         
                       
                       
                         
                           
                             6 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               cos 
                               2 
                             
                             ⁢ 
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sin 
                               2 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           cos 
                                           3 
                                         
                                         ⁢ 
                                         α 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         α 
                                       
                                       - 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       sin 
                                       3 
                                     
                                     ⁢ 
                                     α 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 4 
                               
                               ⁢ 
                               α 
                             
                             - 
                             
                               4 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 cos 
                                 2 
                               
                               ⁢ 
                               α 
                             
                             + 
                             
                               
                                 sin 
                                 4 
                               
                               ⁢ 
                               α 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   3 
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         α 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           sin 
                                           3 
                                         
                                         ⁢ 
                                         α 
                                       
                                       - 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       cos 
                                       3 
                                     
                                     ⁢ 
                                     α 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             6 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               cos 
                               2 
                             
                             ⁢ 
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sin 
                               2 
                             
                             ⁢ 
                             α 
                           
                         
                       
                       
                         
                           
                             4 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sin 
                               3 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             - 
                             
                               ( 
                               
                                 
                                   
                                     sin 
                                     4 
                                   
                                   ⁢ 
                                   α 
                                 
                                 - 
                                 
                                   3 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     cos 
                                     2 
                                   
                                   ⁢ 
                                   
                                     sin 
                                     2 
                                   
                                   ⁢ 
                                   α 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     - 
                                     2 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         α 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           sin 
                                           3 
                                         
                                         ⁢ 
                                         α 
                                       
                                       - 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       cos 
                                       3 
                                     
                                     ⁢ 
                                     α 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     
                                       cos 
                                       4 
                                     
                                     ⁢ 
                                     α 
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   3 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     cos 
                                     2 
                                   
                                   ⁢ 
                                   
                                     sin 
                                     2 
                                   
                                   ⁢ 
                                   α 
                                 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               4 
                             
                             ⁢ 
                             
                               cos 
                               3 
                             
                             ⁢ 
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                         
                       
                       
                         
                           
                             
                               sin 
                               4 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sin 
                               3 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             
                               cos 
                               2 
                             
                             ⁢ 
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sin 
                               2 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             
                               cos 
                               3 
                             
                             ⁢ 
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                         
                         
                           
                             
                               cos 
                               4 
                             
                             ⁢ 
                             α 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   69 
                   ) 
                 
               
             
           
         
       
     
     The equations (66) to (69) show that the block matrix elements e i,j (α) of the respective rotation matrix R k (α) have the symmetry e i,j (α)=e k+2−i,k+2−j (−α). With c=cos α, s=sin α, the block matrices can be simplified to read 
     
       
         
           
             
               
                 
                   
                     
                       
                         R 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         α 
                         ) 
                       
                     
                     = 
                     
                       ( 
                       
                         
                           
                             
                               
                                 c 
                                 k 
                               
                               ⁢ 
                               
                                 s 
                                 0 
                               
                             
                           
                           
                             * 
                           
                         
                         
                           
                             
                               
                                 c 
                                 0 
                               
                               ⁢ 
                               
                                 s 
                                 k 
                               
                             
                           
                           
                             * 
                           
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         R 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         α 
                         ) 
                       
                     
                     = 
                     
                       ( 
                       
                         
                           
                             
                               
                                 c 
                                 k 
                               
                               ⁢ 
                               
                                 s 
                                 0 
                               
                             
                           
                           
                             * 
                           
                           
                             * 
                           
                         
                         
                           
                             
                               
                                 kc 
                                 1 
                               
                               ⁢ 
                               
                                 s 
                                 1 
                               
                             
                           
                           
                             
                               
                                 c 
                                 2 
                               
                               - 
                               
                                 s 
                                 2 
                               
                             
                           
                           
                             * 
                           
                         
                         
                           
                             
                               
                                 c 
                                 0 
                               
                               ⁢ 
                               
                                 s 
                                 k 
                               
                             
                           
                           
                             
                               
                                 c 
                                 1 
                               
                               ⁢ 
                               
                                 s 
                                 1 
                               
                             
                           
                           
                             * 
                           
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         R 
                         3 
                       
                       ⁡ 
                       
                         ( 
                         α 
                         ) 
                       
                     
                     = 
                     
                       ( 
                       
                         
                           
                             
                               
                                 c 
                                 k 
                               
                               ⁢ 
                               
                                 s 
                                 0 
                               
                             
                           
                           
                             * 
                           
                           
                             * 
                           
                           
                             * 
                           
                         
                         
                           
                             
                               
                                 kc 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 s 
                                 1 
                               
                             
                           
                           
                             
                               
                                 c 
                                 k 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   c 
                                   1 
                                 
                                 ⁢ 
                                 
                                   s 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                               
                             
                           
                           
                             * 
                           
                           
                             * 
                           
                         
                         
                           
                             
                               
                                 kc 
                                 1 
                               
                               ⁢ 
                               
                                 s 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   s 
                                   k 
                                 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   c 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   s 
                                   1 
                                 
                               
                             
                           
                           
                             * 
                           
                           
                             * 
                           
                         
                         
                           
                             
                               
                                 c 
                                 0 
                               
                               ⁢ 
                               
                                 s 
                                 k 
                               
                             
                           
                           
                             
                               
                                 c 
                                 1 
                               
                               ⁢ 
                               
                                 s 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 c 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 s 
                                 1 
                               
                             
                           
                           
                             * 
                           
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         R 
                         4 
                       
                       ⁡ 
                       
                         ( 
                         α 
                         ) 
                       
                     
                     = 
                     
                       ( 
                       
                         
                           
                             
                               
                                 c 
                                 k 
                               
                               ⁢ 
                               
                                 s 
                                 0 
                               
                             
                           
                           
                             * 
                           
                           
                             * 
                           
                           
                             * 
                           
                           
                             * 
                           
                         
                         
                           
                             
                               
                                 kc 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 s 
                                 1 
                               
                             
                           
                           
                             
                               
                                 c 
                                 k 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                                 ⁢ 
                                 
                                   s 
                                   2 
                                 
                               
                             
                           
                           
                             * 
                           
                           
                             * 
                           
                           
                             * 
                           
                         
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 ( 
                                 
                                   k 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 c 
                                 2 
                               
                               ⁢ 
                               
                                 s 
                                 2 
                               
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   k 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                       c 
                                       
                                         k 
                                         - 
                                         1 
                                       
                                     
                                     ⁢ 
                                     
                                       s 
                                       1 
                                     
                                   
                                   - 
                                   
                                     
                                       c 
                                       1 
                                     
                                     ⁢ 
                                     
                                       s 
                                       
                                         k 
                                         - 
                                         1 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 c 
                                 k 
                               
                               - 
                               
                                 4 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                                 ⁢ 
                                 
                                   s 
                                   2 
                                 
                               
                               + 
                               
                                 s 
                                 k 
                               
                             
                           
                           
                             * 
                           
                           
                             * 
                           
                         
                         
                           
                             
                               
                                 kc 
                                 1 
                               
                               ⁢ 
                               
                                 s 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   s 
                                   k 
                                 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                                 ⁢ 
                                 
                                   s 
                                   2 
                                 
                               
                             
                           
                           
                             
                               2 
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                       c 
                                       
                                         k 
                                         - 
                                         1 
                                       
                                     
                                     ⁢ 
                                     
                                       s 
                                       1 
                                     
                                   
                                   - 
                                   
                                     
                                       c 
                                       1 
                                     
                                     ⁢ 
                                     
                                       s 
                                       
                                         k 
                                         - 
                                         1 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             * 
                           
                           
                             * 
                           
                         
                         
                           
                             
                               
                                 c 
                                 0 
                               
                               ⁢ 
                               
                                 s 
                                 k 
                               
                             
                           
                           
                             
                               
                                 c 
                                 1 
                               
                               ⁢ 
                               
                                 s 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 c 
                                 2 
                               
                               ⁢ 
                               
                                 s 
                                 2 
                               
                             
                           
                           
                             
                               
                                 c 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 s 
                                 1 
                               
                             
                           
                           
                             * 
                           
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         R 
                         5 
                       
                       ⁡ 
                       
                         ( 
                         α 
                         ) 
                       
                     
                     = 
                     
                       ( 
                       
                         
                           
                             
                               
                                 c 
                                 k 
                               
                               ⁢ 
                               
                                 s 
                                 0 
                               
                             
                           
                           
                             * 
                           
                           
                             * 
                           
                           
                             * 
                           
                           
                             * 
                           
                           
                             * 
                           
                         
                         
                           
                             
                               
                                 kc 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 s 
                                 1 
                               
                             
                           
                           
                             
                               
                                 c 
                                 k 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   c 
                                   
                                     k 
                                     - 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   s 
                                   2 
                                 
                               
                             
                           
                           
                             * 
                           
                           
                             * 
                           
                           
                             * 
                           
                           
                             * 
                           
                         
                         
                           
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 kc 
                                 
                                   k 
                                   - 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 s 
                                 2 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   c 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   s 
                                   1 
                                 
                               
                               - 
                               
                                 2 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     k 
                                     - 
                                     2 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                                 ⁢ 
                                 
                                   s 
                                   
                                     k 
                                     - 
                                     2 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 c 
                                 k 
                               
                               - 
                               
                                 6 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   c 
                                   
                                     k 
                                     - 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   s 
                                   2 
                                 
                               
                               + 
                               
                                 3 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   c 
                                   1 
                                 
                                 ⁢ 
                                 
                                   s 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                               
                             
                           
                           
                             * 
                           
                           
                             * 
                           
                           
                             * 
                           
                         
                         
                           
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 kc 
                                 2 
                               
                               ⁢ 
                               
                                 s 
                                 
                                   k 
                                   - 
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       
                                         - 
                                         
                                           ( 
                                           
                                             k 
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         c 
                                         1 
                                       
                                       ⁢ 
                                       
                                         s 
                                         
                                           k 
                                           - 
                                           1 
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       2 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           k 
                                           - 
                                           2 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         c 
                                         
                                           k 
                                           - 
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         s 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 s 
                                 k 
                               
                               - 
                               
                                 6 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                                 ⁢ 
                                 
                                   s 
                                   
                                     k 
                                     - 
                                     2 
                                   
                                 
                               
                               + 
                               
                                 3 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   c 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   s 
                                   1 
                                 
                               
                             
                           
                           
                             * 
                           
                           
                             * 
                           
                           
                             * 
                           
                         
                         
                           
                             
                               
                                 kc 
                                 1 
                               
                               ⁢ 
                               
                                 s 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   s 
                                   k 
                                 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                                 ⁢ 
                                 
                                   s 
                                   
                                     k 
                                     - 
                                     2 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   - 
                                   2 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   c 
                                   1 
                                 
                                 ⁢ 
                                 
                                   s 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                               
                               + 
                               
                                 3 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   c 
                                   
                                     k 
                                     - 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   s 
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   c 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   s 
                                   1 
                                 
                               
                               - 
                               
                                 3 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                                 ⁢ 
                                 
                                   s 
                                   
                                     k 
                                     - 
                                     2 
                                   
                                 
                               
                             
                           
                           
                             * 
                           
                           
                             * 
                           
                         
                         
                           
                             
                               
                                 c 
                                 0 
                               
                               ⁢ 
                               
                                 s 
                                 k 
                               
                             
                           
                           
                             
                               
                                 c 
                                 1 
                               
                               ⁢ 
                               
                                 s 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 c 
                                 2 
                               
                               ⁢ 
                               
                                 s 
                                 
                                   k 
                                   - 
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 c 
                                 
                                   k 
                                   - 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 s 
                                 2 
                               
                             
                           
                           
                             
                               
                                 c 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 s 
                                 1 
                               
                             
                           
                           
                             * 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   70 
                   ) 
                 
               
             
           
         
       
     
     In a preferred embodiment, the aberrations are described in the form of Zernike polynomials. In this case, the rotation is performed in the space of the Zernike polynomials. The wavefront is preferably spanned by the Zernike polynomials in polar coordinates: 
     
       
         
           
             
               
                 
                   
                     
                       
                         Z 
                         
                           0 
                           , 
                           0 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           ρ 
                           , 
                           φ 
                         
                         ) 
                       
                     
                     = 
                     1 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         Z 
                         
                           1 
                           , 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           ρ 
                           , 
                           φ 
                         
                         ) 
                       
                     
                     = 
                     
                       2 
                       ⁢ 
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       φ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         Z 
                         
                           1 
                           , 
                           
                             - 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           ρ 
                           , 
                           φ 
                         
                         ) 
                       
                     
                     = 
                     
                       2 
                       ⁢ 
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       φ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         Z 
                         
                           2 
                           , 
                           0 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           ρ 
                           , 
                           φ 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         3 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                               ρ 
                               2 
                             
                           
                           - 
                           1 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         Z 
                         
                           2 
                           , 
                           2 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           ρ 
                           , 
                           φ 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         6 
                       
                       ⁢ 
                       
                         ρ 
                         2 
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                       ⁢ 
                       φ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         Z 
                         
                           2 
                           , 
                           
                             - 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           ρ 
                           , 
                           φ 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         6 
                       
                       ⁢ 
                       
                         ρ 
                         2 
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                       ⁢ 
                       φ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   ⋮ 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   with 
                 
               
               
                 
                   ( 
                   71 
                   ) 
                 
               
             
             
               
                 
                   
                     W 
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         0 
                       
                       ∞ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ∑ 
                         m 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           c 
                           
                             k 
                             , 
                             m 
                           
                         
                         ⁢ 
                         
                           Z 
                           
                             k 
                             , 
                             m 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   72 
                   ) 
                 
               
             
           
         
       
     
     The Zernike coefficients corresponding to a wavefront w(x, y) are preferably determined via the integral 
                     c   k   m     =       1     π   ⁢           ⁢     r   0   2         ⁢         ∫             ∫     pupil     ⁢       Z   k   m     ⁡     (       x     r   0       ,     y     r   0         )       ⁢     w   ⁡     (     x   ,   y     )       ⁢     ⅆ   x     ⁢           ⁢     ⅆ   y               (   73   )               
where r:=√{square root over (x 2 +y 2 )}, x=ρ cos φ, y=ρ sin φ, and r 0  the pupil size.
 
     In the preferred representation by means of Zernike polynomials in polar coordinates, the rotation for the Zernike coefficients is very simple. The vector of Zernike coefficients is transformed by the rotation 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             00 
                           
                         
                       
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             01 
                           
                         
                       
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             10 
                           
                         
                       
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             02 
                           
                         
                       
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             11 
                           
                         
                       
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             20 
                           
                         
                       
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             03 
                           
                         
                       
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             12 
                           
                         
                       
                       
                         
                           
                             
                               a 
                               ~ 
                             
                             21 
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       
                         R 
                         Pot 
                       
                       ⁡ 
                       
                         ( 
                         
                           N 
                           , 
                           α 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               a 
                               00 
                             
                           
                         
                         
                           
                             
                               a 
                               01 
                             
                           
                         
                         
                           
                             
                               a 
                               10 
                             
                           
                         
                         
                           
                             
                               a 
                               02 
                             
                           
                         
                         
                           
                             
                               a 
                               11 
                             
                           
                         
                         
                           
                             
                               a 
                               20 
                             
                           
                         
                         
                           
                             
                               a 
                               03 
                             
                           
                         
                         
                           
                             
                               a 
                               12 
                             
                           
                         
                         
                           
                             
                               a 
                               21 
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   74 
                   ) 
                 
               
             
           
         
       
     
     In a block matrix representation, the rotation matrix is directly based on the elementary rotation matrix of equation (57). For N=15, the rotation matrix has the form: 
     
       
                 
         
             
             
         
      
     
     For illustration purposes, every block belonging to the same radial order is framed. 
     If the wavefront is represented via a series as in equations (70) and (71), a series representation, i.e. a linear combination of the coefficients a m,k−m  results for the integral of equation (72) as well. If the coefficients c k   m  or a m,k−m  are summed as vectors up to a specific order k, a transition matrix T(N) between the Zernike subspace and the Taylor series subspace of the order k can be indicated by 
                     (           c     0   ,   0                 c     1   ,   1                 c     1   ,     -   1                   c     2   ,   0                 c     2   ,   2                 c     2   ,     -   2                   c     3   ,   1                 c     3   ,     -   1                   c     3   ,   3               ⋮         )     =         T   ⁡     (   N   )       ⁢     (         E             E   x               E   y               E   xx               E   xy               E   yy               E   xxx               E   xxy             ⋮             E     yy   ⁢           ⁢   …   ⁢           ⁢   y             )       =     n   ⁢           ⁢     T   ⁡     (   N   )       ⁢     (           a   00               a   01               a   10               a   02               a   11               a   20               a   03               a   12               a   21             ⋮         )                 (   76   )               
with T(N)=Z(N)D(N), where e.g. for N=9
 
                     D   ⁡     (   9   )       =     (         1       …                                                                   …       0           0         r   0                                                                                 ⋮           ⋮                     r   0                                                                                                                           r   0   2                                                                                                                           r   0   2                                                                                                                           r   0   2                                                                                                                           r   0   3                                     ⋮                                                                                 r   0   3         ⋮           0       …                                                       …       0         r   0   3           )             (   77   )               
designates a matrix that indicates the correct power of the pupil radius. The basic transformation matrix Z(N) is determined by Zernike expansion of the power series. Preferably, the following representation is provided for the transformation matrix for N=15:
 
     
       
         
           
             
               
                 Z 
                 
                   - 
                   1 
                 
               
               ⁡ 
               
                 ( 
                 15 
                 ) 
               
             
             = 
             
               ( 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       - 
                       
                         3 
                       
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     * 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     2 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       
                         - 
                         4 
                       
                       ⁢ 
                       
                         2 
                       
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     0 
                   
                   
                     2 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       
                         - 
                         4 
                       
                       ⁢ 
                       
                         2 
                       
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     ⋮ 
                   
                   
                     0 
                   
                   
                     
                       4 
                       ⁢ 
                       
                         3 
                       
                     
                   
                   
                     
                       2 
                       ⁢ 
                       
                         6 
                       
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     * 
                   
                   
                     0 
                   
                   
                     * 
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     ⋮ 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       2 
                       ⁢ 
                       
                         6 
                       
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     * 
                   
                   
                     0 
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                       4 
                       ⁢ 
                       
                         3 
                       
                     
                   
                   
                     
                       
                         - 
                         2 
                       
                       ⁢ 
                       
                         6 
                       
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     * 
                   
                   
                     0 
                   
                   
                     * 
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       36 
                       ⁢ 
                       
                         2 
                       
                     
                   
                   
                     0 
                   
                   
                     
                       12 
                       ⁢ 
                       
                         2 
                       
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     0 
                   
                   
                     
                       12 
                       ⁢ 
                       
                         2 
                       
                     
                   
                   
                     0 
                   
                   
                     
                       12 
                       ⁢ 
                       
                         2 
                       
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                       12 
                       ⁢ 
                       
                         2 
                       
                     
                   
                   
                     0 
                   
                   
                     
                       
                         - 
                         12 
                       
                       ⁢ 
                       
                         2 
                       
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     0 
                   
                   
                     
                       36 
                       ⁢ 
                       
                         2 
                       
                     
                   
                   
                     0 
                   
                   
                     
                       
                         - 
                         12 
                       
                       ⁢ 
                       
                         2 
                       
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     ⋮ 
                   
                   
                     0 
                   
                   
                     ⋮ 
                   
                   
                     0 
                   
                   
                     * 
                   
                   
                     0 
                   
                   
                     * 
                   
                   
                     0 
                   
                   
                     * 
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     ⋮ 
                   
                   
                     
                         
                     
                   
                   
                     ⋮ 
                   
                   
                     0 
                   
                   
                     * 
                   
                   
                     0 
                   
                   
                     * 
                   
                   
                     0 
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     * 
                   
                   
                     0 
                   
                   
                     * 
                   
                   
                     0 
                   
                   
                     * 
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     0 
                   
                   
                     * 
                   
                   
                     0 
                   
                   
                     * 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     * 
                   
                   
                     0 
                   
                   
                     * 
                   
                   
                     0 
                   
                   
                     * 
                   
                 
               
               ) 
             
           
         
       
     
     In this equation as well, the blocks belonging to the same radial order are framed for purposes of illustration. It can be seen that non-disappearing elements also exist outside the diagonal blocks. However, they do not influence the rotation matrix R Pot (N, α). 
     In order to determine the rotation matrix R Pot (N, α), R Zernike (N, α) is transformed to the coefficient system of the power series development with equation (76): 
     
       
         
           
             
               ( 
               
                 
                   
                     
                       
                         a 
                         ~ 
                       
                       00 
                     
                   
                 
                 
                   
                     
                       
                         a 
                         ~ 
                       
                       01 
                     
                   
                 
                 
                   
                     
                       
                         a 
                         ~ 
                       
                       10 
                     
                   
                 
                 
                   
                     
                       
                         a 
                         ~ 
                       
                       02 
                     
                   
                 
                 
                   
                     
                       
                         a 
                         ~ 
                       
                       11 
                     
                   
                 
                 
                   
                     
                       
                         a 
                         ~ 
                       
                       20 
                     
                   
                 
                 
                   
                     
                       
                         a 
                         ~ 
                       
                       03 
                     
                   
                 
                 
                   
                     
                       
                         a 
                         ~ 
                       
                       12 
                     
                   
                 
                 
                   
                     
                       
                         a 
                         ~ 
                       
                       21 
                     
                   
                 
                 
                   
                     ⋮ 
                   
                 
               
               ) 
             
             = 
             
               
                 
                   
                     T 
                     
                       - 
                       1 
                     
                   
                   ⁡ 
                   
                     ( 
                     N 
                     ) 
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       
                         
                           
                             c 
                             ~ 
                           
                           
                             0 
                             , 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           
                             c 
                             ~ 
                           
                           
                             1 
                             , 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           
                             c 
                             ~ 
                           
                           
                             1 
                             , 
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             c 
                             ~ 
                           
                           
                             2 
                             , 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           
                             c 
                             ~ 
                           
                           
                             2 
                             , 
                             2 
                           
                         
                       
                     
                     
                       
                         
                           
                             c 
                             ~ 
                           
                           
                             2 
                             , 
                             
                               - 
                               2 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             c 
                             ~ 
                           
                           
                             3 
                             , 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           
                             c 
                             ~ 
                           
                           
                             3 
                             , 
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             c 
                             ~ 
                           
                           
                             3 
                             , 
                             3 
                           
                         
                       
                     
                     
                       
                         ⋮ 
                       
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     
                       T 
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       N 
                       ) 
                     
                   
                   ⁢ 
                   
                     
                       R 
                       Zernike 
                     
                     ⁡ 
                     
                       ( 
                       
                         N 
                         , 
                         α 
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           
                             c 
                             
                               0 
                               , 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             c 
                             
                               1 
                               , 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             c 
                             
                               1 
                               , 
                               
                                 - 
                                 1 
                               
                             
                           
                         
                       
                       
                         
                           
                             c 
                             
                               2 
                               , 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             c 
                             
                               2 
                               , 
                               2 
                             
                           
                         
                       
                       
                         
                           
                             c 
                             
                               2 
                               , 
                               
                                 - 
                                 2 
                               
                             
                           
                         
                       
                       
                         
                           
                             c 
                             
                               3 
                               , 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             c 
                             
                               3 
                               , 
                               
                                 - 
                                 1 
                               
                             
                           
                         
                       
                       
                         
                           
                             c 
                             
                               3 
                               , 
                               3 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     
                       T 
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       N 
                       ) 
                     
                   
                   ⁢ 
                   
                     
                       R 
                       Zernike 
                     
                     ⁡ 
                     
                       ( 
                       
                         N 
                         , 
                         α 
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     T 
                     ⁡ 
                     
                       ( 
                       N 
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           
                             a 
                             00 
                           
                         
                       
                       
                         
                           
                             a 
                             01 
                           
                         
                       
                       
                         
                           
                             a 
                             10 
                           
                         
                       
                       
                         
                           
                             a 
                             02 
                           
                         
                       
                       
                         
                           
                             a 
                             11 
                           
                         
                       
                       
                         
                           
                             a 
                             20 
                           
                         
                       
                       
                         
                           
                             a 
                             03 
                           
                         
                       
                       
                         
                           
                             a 
                             12 
                           
                         
                       
                       
                         
                           
                             a 
                             21 
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     From this, it follows that
 
 R   Pot ( N ,α)= T   −1 ( N ) R   Zernike ( N ,α) T ( N )  (78)
 
with a block structure of the form
 
                       R   Pot     ⁡     (   15   )       =     (         1       0                   …       0           0           R   1     ⁡     (   α   )                                   ⋮                                       R   2     ⁡     (   α   )                                       ⋮                                   R   3     ⁡     (   α   )           0           0       …                   0           R   4     ⁡     (   α   )             )             (   79   )               
wherein the block matrices are identical with those of equation (70).
 
       FIG. 6  illustrates an exemplary method for individually optimizing a spectacle lens taking higher-order aberrations (HOA) of both the eye and the spectacle lens into consideration. In a step ST 12 , not only the local aberrations of 2 nd  order (S′ xx , S′ xy , S′ yy ) but also the aberrations of a higher order (K′ xxx , K′ xxy , K′ xyy  etc.) at the vertex sphere are calculated on the basis of wavefront tracing (ST 10 ). 
     From these, from the local aberrations, the values for sphere, cylinder, and cylinder axis (sph, zyl, A) of the spectacle lens are calculated with the help of Zernike polynomials and/or other suitable metrics, preferably taking the pupil diameter or pupil radius into consideration. Preferably, in a step S 14 , first of all Zernike coefficients (c 2   0 , c 2   2 , c 2   −2 , . . . ) are determined. Since now also the higher-order local aberrations are known, it is possible to calculate the ideal sph, zyl, A values of the spectacle lens for a finite pupil opening, which preferably correspond to the above-described transformed values. Both the connection between the local aberrations (S′ xx , S′ xy , S′ yy , K′ xxx , K′ xxy , K′ xyy , . . . ) and the Zernike coefficients (c 2   0 , c 2   2 , c 2   −2 , . . . ), as it is particularly referred to in step ST 14 , and the connection between the Zernike coefficients (c 2   0 , c 2   2 , c 2   −2 , . . . ) and the values for sphere (Sph), cylinder (Zyl bzw. Cyl), and cylinder axis (A or α) are provided as functional connections c 2   0 , c 2   2 , c 2   −2 , . . . )=f(r, ′ xx , S′ xy , S′ yy , K′ xxx , K′ xxy , K′ xyy , . . . ) and Sph, Zyl, A=f(r, c 2   0 , c 2   2 , c 2   −2 , . . . ) in a step ST 18 , particularly taking the pupil radius r into consideration. 
     Now, it is preferred that the pupil size r be specified to be variable for every visual point. It is particularly preferred that the pupil size be specified as a function of the object distance, which in turn represents a function of the visual point. This can be based e.g. on the near reflex, so that with near objects the assumed pupil diameter decreases. 
     Preferably, in the refraction determination (ST 20 ), not only the values for sphere, cylinder, and cylinder axis, particularly for distance and near vision, are determined subjectively, but additionally the higher-order aberrations (c 2   0 , c 2   2 , c 2   −2 , . . . ) are determined with an aberrometer. In a step ST 22 , the subjective and objective refraction data are combined particularly considering object distance, direction of sight, and pupil diameter. Thus, it is possible to calculate ideal (transformed) prescription values (sph, zyl, A) particularly for different pupil diameters depending on the visual point with suitable metrics. It is particularly preferred that the ideal prescriptions be calculated once and then be deposited as a function of the object distance. Moreover, it is preferred that e.g. with the aberrometer also the individual pupil diameter be determined under photopic (small pupil) and mesopic (large pupil) conditions. Otherwise, standard values from literature have to be used. Subsequently, the spherocylindrical values of the spectacle lens (SL) can be combined with those of the eye (ST 24 ) in a known way (combination SL/eye K: K(Ref,Ast)=SL(Sph,Zyl,Axis)−eye(Sph,Zyl,axis). The target function (ST 26 ), in which particularly the target values S(Ref, Ast) provided in a step ST 28  are taken into account, preferably remains unchanged. The differences between the combination values K and the target values S determined in step ST 24  are particularly taken into account therein: K(Ref,Ast)−S(Ref,Ast). 
     LIST OF REFERENCE NUMERALS 
     
         
           10  main ray 
           12  eye 
           14  first refractive surface (front surface of the spectacle lens) 
           16  second refractive surface (back surface of the spectacle lens) 
           18  original wavefront 
           20  propagated wavefront 
           22  image point 
           24  neighboring ray 
         ST 2 , ST 4  propagation (and optionally rotation)