# Patent

## Title
Method of optimizing an objective with fluoride crystal lenses, and objective with fluoride crystal lenses

## Abstract

A numerical optimizing method serves to reduce harmful effects caused by intrinsic birefringence in lenses of a fluoride crystal material of cubic crystal structure in an objective, particularly a projection objective for a microlithography system. Under the optimizing method, an optimizing function which takes at least one birefringence-related image aberration into account is minimized. The birefringence-related image aberration is determined from a calculation for a light ray passing through the fluoride crystal lenses. To the extent that the birefringence-related image aberration is a function of parameters of the light ray, it depends only on geometric parameters of the light ray. The numerical optimizing method is used to produce objectives in which an optical retardation as well as an asymmetry of the optical retardation are corrected. The lenses are arranged in homogeneous groups, where each homogeneous group is corrected for the optical retardation asymmetry.

## Background
CROSS-REFERENCE TO RELATED APPLICATION
This application is a continuation of International Patent Application Serial No. PCT/EP 2003/009167 filed Aug. 19, 2003, in the German language and published as WO 2004/023172 A1 on Mar. 18, 2004, which, in turn, claims the priority of the German patent applications
102 41 102.6, filed Sep. 3, 2002,
102 43 080.2, filed Sep. 16, 2002,
102 53 355.5, filed Nov. 14, 2002,
103 04 116.8, filed Jan. 31, 2003,
all of which are hereby incorporated herein by reference in their entirety.


BACKGROUND OF THE INVENTION
The present invention relates to a numerical optimizing method for the determination of the optical data of an objective, and it further relates to an objective that is made in accordance with the method.
Numerical optimizing methods of this kind have long been known. The textbook “Synthese optischer Systeme” (Synthesis of Optical Systems) by H. Haferkorn and W. Richter (1984; VEB Deutscher Verlag der Wissenschaften: DDR-1080 Berlin) discusses the correction of optical systems in chapter 4. The first step is to find an optimizing function which takes a multitude of image aberrations into account such as, e.g., Seidel's aberrations, transverse aberrations, or wave aberrations. The individual image aberrations can be entered into the optimizing function with individual weight factors. In addition, boundary constraints such as focal lengths, or design constraints, can also be taken into account. The optimizing function depends on the degrees of freedom of the optical system, such as for example lens radii, asphericity parameters, lens thicknesses, distances between lenses, or indices of refraction. The purpose of the numerical optimizing method is to minimize the optimizing function and to thereby determine the optical data of the optical system. A variety of methods are used for the automatic correction, such as for example the correction methods with a linear approximation of the optimizing function, or the correction methods with a quadratic approximation of the optimizing function.
The aforementioned textbook is mentioned only as an example. The fundamentals of numerical optimizing methods are also described in the English-language literature related to the field of optical design.
A widely used computer program to run numerical optimizing processes is available under the trade name “Code V®” by Optical Research Associates (ORA®), Pasadena, Calif. (USA). With the Code V® program, an optimizing function can be defined which takes a variety of image aberrations into account. With a start-up system as a point of departure, the optical data of an objective can subsequently be determined with local and global optimizing procedures.
The computer program ZEMAX® of Focus Software, Inc., Tucson, Ariz., likewise provides the capability to determine the optical data of an objective through a numerical optimizing process.
It became known in May 2001, based on measurements that had been made, that calcium fluoride, in spite of having a cubic crystal structure, exhibits the characteristics of intrinsic birefringence. The measurement results supporting this discovery have been published in November 2001 in the article “Intrinsic birefringence in calcium fluoride and barium fluoride” by J. Burnett et al. (Physical Review B, Volume 64 (2001), pages 241102-1 to 241102-4).
To discuss the subject of intrinsic birefringence, it is essential to use an unambiguous notation of the crystallographic directions. Therefore, a system of notations is introduced below to identify crystallographic directions, crystallographic planes, and lenses whose lens axes are oriented in certain crystallographic directions.
The indices for the crystallographic directions will hereinafter be bracketed between the symbols “<” and “>”, and the indices for the crystallographic planes will be bracketed between the symbols “{” and “}”. The crystallographic directions are perpendicular to the correspondingly indexed crystallographic planes. For example, the crystallographic direction <100> is perpendicular to the crystallographic plane {100}. Crystals with a cubic lattice structure, which includes the fluoride crystals that are of interest in the present context, have the principal crystallographic directions <110>, < 110>, <1 10>, < 1 10>, <101>, <10 1>, < 101>, < 10 1>, <011>, <0 11>, <01 1>, <0 1 1>, <111>, < 1 1 1>, < 1 11>, < 11 1>, <1 1 1>, < 111>, <1 11>, <11 1>, <100>, <010>, <001>, < 100>, <0 10>, and <00 1>.
Because of the symmetries of cubic crystals, the principal crystallographic directions <100>, <010>, <001>, < 100>, <0 10>, and <00 1> are equivalent to each other. Therefore, those crystallographic directions that are oriented along one of the principal directions <100>, <010>, <001>, < 100>, <0 10>, and <00 1> will hereinafter be identified by the prefix “(100)-”, and crystallographic planes that are perpendicular to these directions will also be identified by the same prefix “(100)-”. Lenses whose lens axes are parallel to one of these principal crystallographic directions are likewise given the prefix “(100)-”.
Furthermore, the principal directions <110>, < 110>, <1 10>, < 1 10>, <101>, <10 1>, < 101>, < 10 1>, <011>, <0 11>, <01 1>, and <0 1 1> are likewise equivalent to each other. Therefore, those crystallographic directions that are oriented along one of the latter group of principal directions will hereinafter be identified by the prefix “(110)-”, and crystallographic planes that are perpendicular to these directions will also be identified by the same prefix “(110)-”. Lenses whose lens axes are parallel to one of these principal crystallographic directions are likewise given the prefix “(110)-”.
Finally, the principal directions <111>, < 1 1 1>, < 1 11>, < 11 1>, <1 1 1>, < 11 1>, <1 11>, and <11 1> are also equivalent to each other. Therefore, those crystallographic directions that are oriented along one of the latter group of principal directions will hereinafter be identified by the prefix “(111)-”, and crystallographic planes that are perpendicular to these directions will also be identified by the same prefix “(111)-”. Lenses whose lens axes are parallel to one of these principal crystallographic directions are likewise given the prefix “(111)-”.
Any statements made hereinafter in regard to one of the aforementioned principal crystallographic directions should be understood to be equally applicable to the equivalent principal crystallographic directions.
Consistent with the above, the crystallographic (111)-directions are not equivalent to the crystallographic (100)-directions or the crystallographic (110)-directions. Likewise, the crystallographic (100)-directions are not equivalent to the crystallographic (110)-directions.
According to the article in Physical Review B which was mentioned above, the intrinsic birefringence is strongly dependent on the material orientation of the fluoride crystal lens and on the direction of the light ray. It reaches its maximum in a light ray traveling through a lens in the crystallographic <110>-direction. The measurements presented in the article show that rays propagating in the (110)-direction of a calcium fluoride crystal are subject to a birefringence of (−11.8±0.4) nm/cm at a wavelength of λ=156.1 nm, of (−3.6±0.2) nm/cm at a wavelength of λ=193.09 nm, and of (−0.55±0.07) nm/cm at a wavelength of λ=253.65 nm. On the other hand, if the light propagation is oriented in the <100> direction or in the <111> direction of the crystal, no intrinsic birefringence occurs in calcium fluoride, as is also predicted by theory. Thus, the intrinsic birefringence has a strong directional dependence and increases significantly for shorter wavelengths.
The directional dependence of the intrinsic birefringence in a fluoride crystal with a cubic crystal structure is shown in the published article. “The trouble with calcium fluoride” by J. Burnett et al. (spie's oemagazine, March 2002, pp. 23-25 and FIG. 4). The intrinsic birefringence of a light ray depends in this case on the aperture angle as well as on the azimuth angle of a light ray. As is made evident in FIG. 4, the intrinsic birefringence has a fourfold azimuthal symmetry if the lens axis is oriented in the crystallographic (100)-direction, a threefold azimuthal symmetry if the lens axis is oriented in the crystallographic (111)-direction, and a twofold azimuthal symmetry if the lens axis is oriented in the crystallographic (110)-direction. By rotating two fluoride crystal lenses relative to each other about their lens axes, it is possible to reduce the detrimental influence of the intrinsic birefringence. An angle of rotation of 45° is proposed for two lenses whose lens axes are oriented in the crystallographic (100)-direction, an angle of rotation of 60° for two lenses whose lens axes are oriented in the crystallographic (111)-direction, and an angle of rotation of 90° for two lenses whose lens axes are oriented in the crystallographic (110)-direction. By simultaneously using pairs of (100)-lenses, (111)-lenses, and (110)-lenses with these respective angles of rotation, it is possible to reduce the optical path difference between two mutually orthogonal states of polarization. Furthermore, using calcium fluoride lenses and barium fluoride lenses in combination likewise results in a compensation of the detrimental influence of the intrinsic birefringence because, according to FIG. 2 of the same article, the respective birefringence effects for comparable crystallographic directions in barium fluoride and calcium fluoride have opposite signs.
As described in the articles by John Burnett et al., the detrimental influence of intrinsic birefringence manifests itself most of all in objectives that are used in the deep ultraviolet range (λ<200 nm), such as for example lithography projection objectives for applications in 157 nm-lithography.
Projection objectives and microlithography projections systems of this type have been disclosed, e.g., in the Patent Application Publication WO 01/50171 A1 (U.S. Ser. No. 10/177,580), which has the same assignee as the present application, and the references cited therein. The examples of embodiments presented in that patent application are purely refractive as well as catadioptric projection objectives with numerical aperture values of 0.8 and 0.9 at working wavelengths of 193 nm as well as 157 nm. The material used for the lenses is calcium fluoride.
The not pre-published patent application PCT/EP 02/05050, which has the same assignee as the present application, gives a description of different compensation methods to reduce the detrimental influence of the intrinsic birefringence, e.g., in the objectives that are presented as examples in WO 01/50171 A1 (U.S. Ser. No. 10/177,580). Among others, the solutions disclosed therein include the parallel use of (100)-lenses with (111)-lenses or (110)-lenses of the same fluoride crystal material as well as the use of compensation coatings. The disclosure of WO 01/50171 A1 (U.S. Ser. No. 10/177,580) is hereby incorporated in its entirety in the present application.
The not pre-published patent application DE 101 33 841.4 (U.S. Ser. No. 10/199,503), which has the same assignee as the present application, proposes the concept of using lenses of two different crystalline materials in parallel in order to reduce the harmful influence of intrinsic birefringence. Calcium fluoride and barium fluoride are suggested as a suitable pair of materials. The disclosure of DE 101 33 841.4 (U.S. Ser. No. 10/199,503) is hereby incorporated in its entirety in the present application.
The concept of rotating the orientation of lens elements in order to compensate for the effects of birefringence is also described in the not pre-published patent application DE 101 23 725.1, (PCT/EP 02/04900), whose content is hereby incorporated by reference in the present application.
However, the proposed methods are limited in their capability, or work only for a small number of suitable lenses, to provide a complete compensation of the aberrations caused by intrinsic birefringence. While it is known in theory, that two (100)-lenses rotated by 45° or two (111)-lenses rotated by 60°, relative to each other will reduce the harmful effects of intrinsic birefringence, this prediction is met in the ideal sense only if the lenses are adjoining planar-parallel plates. Transferring this concept to objectives with a multitude of individual lenses with different lens thickness and different surface parameters of the lens surfaces presents itself as a serious problem to the optical designer. For one, as the number of fluoride crystal lenses gets larger, the lens materials, the orientations of the lens axes and the angles of rotation have to be determined for a larger number of lenses. As a further problem, it is normally necessary to go through a comprehensive calculation of the polarization along specific light rays in order to determine the result of the compensation. However, calculating the polarization optics along the entire paths of representative rays is a complex undertaking.
OBJECT OF THE INVENTION
The task set for the present invention is therefore to propose a simple, effective, and also time-saving method for reducing the harmful influence of intrinsic birefringence in objectives with fluoride crystal lenses.
A further task set for the invention is to specify objectives that are distinguished by an excellent degree of correction in regard to the harmful influence of birefringence.
Finally, notwithstanding their superior birefringence correction, the objectives with fluoride crystal lenses need to lend themselves to a simple manufacturing process.
SUMMARY OF THE INVENTION
The foregoing objective is met by a numerical optimizing method, by an objective produced according to the method, an objective distinguished by an excellent degree of correction in regard to the harmful influence of birefringence, and an objective which combines a superior birefringence correction with a design that lends itself to a simple manufacturing process, and further by a microlithography projection system as well as a method for producing semiconductor elements by means of the microlithography projection system.
According to the invention, the harmful influence of intrinsic birefringence is reduced already in the phase of determining the optical data of an objective with fluoride crystal lenses, i.e., in the numerical optimizing process. The optical data of an objective that are considered in this process include the arrangement and dimensions of the optical elements as well as information regarding the lens materials being used and their orientation. As long as the arrangement and the dimensions of the optical elements, i.e., quantities such as the surface parameters of the lens surfaces, lens thicknesses and distances between the lenses of the objective, have not yet been selected and, consequently, remain available as degrees of freedom for the optimization, a significantly better range of possibilities remains open for correcting the aberrations caused by intrinsic birefringence. While it is possible to perform the optimization after the optical design of the objective has been decided upon, the only degrees of freedom remaining available at that point are the orientations of the lens axes and the angles of rotation. According to the invention, at least one birefringence-related image aberration is taken into account in an optimizing function that is to be minimized in accordance with the numerical optimizing method. This birefringence-related image aberration is determined from a calculation for a light ray passing through the fluoride crystal lenses. As the birefringence-related image aberration should serve to describe the reduction of the harmful influence of intrinsic birefringence, one would expect the birefringence-related image aberration to depend on polarization-optics parameters of the ray, such as for example the state of polarization or the optical path difference between two mutually orthogonal states of polarization which occurs in a ray passing through the fluoride crystal lenses. According to the invention, the birefringence-related image aberration, insofar as it depends on the parameters of the ray, is a function of only the geometric parameters of the light ray. The term “geometric parameters” refers for example to ray angles and ray path lengths, i.e., quantities that can be determined through geometric means. In comparison to the polarization-optics parameters, the determination of the geometric parameters is less complicated and can be accomplished at a lower cost. The geometric parameters are obtained almost immediately in the course of the calculation for a light ray. It is therefore surprising that a reduction of the harmful influence of intrinsic birefringence can be achieved by optimizing the birefringence-related image aberration which, insofar as is depends on parameters of the ray, is a function of only the geometric parameters of the ray.
It is advantageous to use an outer aperture ray to determine the birefringence-related image aberration. The term “outer aperture ray” in this context refers to a ray whose relative ray aperture lies between 0.5 and 1.0. The relative ray aperture is defined as the ratio between a ray aperture and the numeric aperture in the image plane of the objective, with the ray aperture in the image plane being defined as the sine of the angle between the outer aperture ray and the principal ray, where the principal ray and the outer aperture ray originate from the same object point in an object plane. If the object point lies on the optical axis, the principal ray runs along the optical axis. In this case, a relative ray aperture of, e.g., 1.0 means that the outer aperture ray is represented by the marginal ray of the light bundle originating from this object point. As geometric parameters for an outer aperture ray, at least one ray path OPL and an aperture angle θL are determined in each fluoride crystal lens. As a favorable circumstance, the birefringence-related image aberration in this case depends only on the ray path lengths OPL and the aperture angles θL in the fluoride crystal lenses. The term “ray path length OPL” means the geometric path length that a ray travels inside a lens. The aperture angle θL indicates the angle between a ray and the lens axis of the respective lens.
The reason why an outer aperture ray is used to determine the harmful influence of intrinsic birefringence is that an outer aperture ray normally has large aperture angles inside the lenses. The outer aperture rays are affected by the birefringent lenses especially in a case where the intrinsic birefringence—either in the individual fluoride crystal lenses or in a combination of lenses that are rotated relative to each other about the lens axis—causes an optical path difference between two mutually orthogonal states of polarization that increases with larger aperture angles.
In the calculation of the birefringence-related image aberrations, it is advantageous to determine the relative ray aperture of the outer aperture ray in such a way that the optical path difference for two mutually orthogonal states of polarization reaches its maximum when the fluoride crystal lenses are not rotated relative to each other. In determining the relative ray aperture, one begins by calculating the optical path differences for two mutually orthogonal states of polarization for relative ray apertures between 0.5 and 1.0. The relative ray aperture value for which the optical path difference of the outer aperture ray reaches its maximum is subsequently used for the further calculation of the birefringence-related image aberrations.
If the lens axis of a fluoride crystal lens is oriented in the crystallographic (111)-direction, it is advantageous if the birefringence-related image aberration also depends on the azimuth angle αL of the outer aperture ray. Due to the threefold azimuthal symmetry of the birefringence distribution Δn(θL,αL) of (111)-lenses, the harmful influence of intrinsic birefringence on a ray passing through two (111)-lenses of equal lens-axis orientation can already be reduced if the azimuth angle αL of the ray in one of the (111)-lenses differs by 180° from the other (111)-lens. This applies for example to a ray which runs in the meridian plane and which diverges from the lens axis in one of the (111)-lenses and converges with the lens axis in the other (111)-lens.
In the present context, the azimuth angle outside of a lens is defined as the angle between the projection of the ray into a plane that extends perpendicular to the optical axis of the objective and a fixed reference direction that is tied to the objective and runs perpendicular to the optical axis of the objective. The reference direction can, for example, coincide with the y-direction in the object plane or the image plane, if the z-direction coincides with the direction of the optical axis. The azimuth angle inside of a lens is defined as the angle αL between the projection of the ray into a plane that extends perpendicular to the lens axis and a reference direction that is tied to the lens and runs perpendicular to the lens axis. The reference direction of a lens is, for example, parallel to a direction that is obtained by projecting the crystallographic (110)-direction into a plane whose normal vector points in the crystallographic (100)- or (111)-direction, i.e., in that crystallographic direction which is aligned with the lens axis. The reference directions of the fluoride crystal lenses whose lens axes are oriented in the same principal crystallographic direction are thus tied in a consistent manner to the crystallographic structure.
The lens axis is defined, e.g., by the symmetry axis of a rotationally symmetric lens. If the lens does not have a symmetry axis, the lens axis may be defined as the centerline of an incident bundle of light rays, or as a line that runs in the median ray direction of all light rays in the lens. The lens axis of a planar-parallel plate is perpendicular to the surface planes of the lens. The lens axis is considered as aligned with a given crystallographic direction, if the angular deviation between the lens axis and the crystallographic direction is smaller than ±10°. The angular deviation should preferably be less than ±5°.
The lenses may be, e.g., refractive or diffractive lenses as well as correction plates with free-form corrective surfaces. Planar-parallel plates are likewise considered as lenses if they are arranged in the light path of the objective. However, elements with an explicit retarding function such as quarter-lambda plates or half-lambda plates are not considered to be lenses. They influence the state of polarization of all rays in approximately the same way and are therefore not taken into account in the calculation of the birefringence-related image aberrations.
Particularly in cases where different fluoride crystals of cubic structure, for example calcium fluoride and barium fluoride, are used as lens materials in an objective, it is of advantage if the birefringence-related image aberration depends on material factors that are characteristic for the lens materials being used. If, on the other hand, only one fluoride crystal material is used, the material factor has the same value in all of the fluoride crystal lenses and can therefore be considered to be a constant.
The optical path difference for two mutually orthogonal states of polarization is likewise influenced by the orientation of the lens axis of a fluoride crystal lens, i.e., the direction of the lens axis within the crystal. It is therefore advantageous if the birefringence-related image aberration is a function of orientation factors of the fluoride crystal lenses. If, on the other hand, the lens axes of the lenses in an objective or a lens group that is to be optimized are oriented in the same crystallographic direction, the orientation factor can be considered as a constant. In the process of reducing the harmful influence of intrinsic birefringence, it is advantageous if the birefringence-related image aberration is a function of ray factors SPL which are specific to each lens and are defined as

SPL=OPL·sin2 θL·(7·cos2 θL−1)  (1)

For aperture angles up to 40°, the following represents a good approximation of the ray factor:

SPL=(θL,OPL)≈OPL·(9/7)·sin2(2.17·θL)  (2)

Besides the functional relationships expressed in the equations (1) and (2) between the ray factor SPL and the quantities θL and OPL, one could also use a polynomial series or other representation of the functional relationship as a best approximation.
In addition to its dependence on the aperture angle θL, the intrinsic birefringence of fluoride crystals with a cubic crystal structure also depends on the azimuth angle αL of a light ray. Through an appropriately selected rotation of lenses whose lens axes run in the same principal crystallographic direction, one can achieve that the optical path difference for two mutually orthogonal states of polarization in light rays passing through these lenses will be nearly independent of the azimuth angle αL at least for one aperture angle θL. According to the invention, these lenses are assigned an effective amount of birefringence which depends only on the aperture angle θL of a light ray. The effective birefringence distribution is in this case proportionate to sin2 θL·(7·cos2 θL−1). For a ray passing through a lens with this effective birefringence distribution, the optical path difference for two mutually orthogonal states of polarization is proportionate to the value for the ray factor SPL which is obtained from equation (1) or (2). Consequently, it is advantageous if the birefringence-related image aberration is a function of the ray factors SPL of the fluoride crystal lenses.
In order to be able on the basis of the ray factors SPL to systematically minimize the optical path difference for two mutually orthogonal states of polarization occurring in an outer aperture ray in a fluoride crystal lens, it is advantageous if the birefringence-related image aberration is a function of birefringence factors BFL of the fluoride crystal lenses, where the individual birefringence factor

BFL=MAL·DIL·SPL  (1)

is defined as the product of a material factor MAL, an orientation factor DIL and the ray factor SPL which is determined by equations (1) and (2).

The material factor MAL for a fluoride crystal lens indicates the intrinsic birefringence for a ray that runs in the crystallographic (110)-direction. Consequently, the material factor depends on the wavelength. The material factor MAL for a working wavelength of 157 nm in calcium fluoride is (−11.8±0.4)nm/cm, and for barium fluoride it is (+33±3)nm/cm.
The orientation factor DIL for a fluoride crystal lens depends on the crystallographic direction in which the lens axis of the respective fluoride crystal lens is oriented. For a fluoride crystal lens whose lens axis is oriented in the crystallographic (100)-direction, DIL equals −1⁄2, and for a fluoride crystal lens whose lens axis runs in the crystallographic (111)-direction, DIL equals +1⁄3. Finally, for a fluoride crystal lens whose lens axis is oriented in the crystallographic (110)-direction, DIL equals +1⁄8.
The birefringence factor BFL can in effect be interpreted as an optical path difference that occurs in the outer aperture ray in the fluoride crystal lens.
Depending on the boundary conditions, the material factor or the orientation factor can also be set as constants.
The harmful influence of intrinsic birefringence caused by a group of fluoride crystal lenses can be substantially reduced if the birefringence-related image aberration for the group of lenses is defined as an optical retardation ORLG and the amount of the optical retardation ORLG is minimized by the optimizing method. The optical retardation




  
    
      
        
          OR
          LG
        
        =
        
          
            ∑
            
              L
              ⁢
              
                
              
              ⁢
              in
              ⁢
              
                
              
              ⁢
              LG
            
          
          ⁢
          
            BF
            L
          
        
      
    
    
      
        (
        4
        )
      
    
  




is defined as the sum of the birefringence factors BFL of the fluoride crystal lenses of the group.

Since the ray factor SPL by definition can take on only positive values, a reduction can be achieved for example if the fluoride crystal lenses of the group consist of different fluoride crystals whose material factors MAL have different signs, or if the lens axes of the fluoride crystal lenses of the group run in different crystallographic directions, so that the associated orientation factors DIL have different signs. While the degrees of freedom represented by the material factor MAL and the orientation factor DIL can take on only discrete values in the optimization process, the ray-specific factor SPL can be used for a fine correction by influencing the lens data. As the ray factor SPL has a linear dependency on the ray path length OPL, the magnitude of the ray factor SPL of a lens can be influenced directly through a change in the lens thickness. This is a favorable circumstance for the reason that the classic image aberration such as for example the distortion and the spherical image aberrations depend primarily on the surface parameters of the lens surfaces and only secondarily on the lens thicknesses. Thus, a sufficient number of degrees of freedom remain available in the simultaneous correction of birefringence-related image aberrations and classic image aberrations.
The optical retardation




  
    OR
    LG
  
  =
  
    
      ∑
      
        L
        ⁢
        
          
        
        ⁢
        in
        ⁢
        
          
        
        ⁢
        LG
      
    
    ⁢
    
      BF
      L
    
  




can in effect be interpreted as an optical path difference that occurs in the outer aperture ray in the group of fluoride crystal lenses. Minimizing this image aberration creates the requisite conditions for reducing the harmful influence of intrinsic birefringence by rotating the fluoride crystal lenses in relation to each other.

If an arbitrary constant is applied as a scale factor to the optical retardation ORLG, the foregoing interpretation of the optical retardation ORLG is equivalent to the definition expressed through equation (4).
In the practice of the numerical optimizing method for reducing the harmful influence of intrinsic birefringence in a group of fluoride crystal lenses, an advantageous approach is to start from an objective with known optical data and, as a first step, to assign to each of the fluoride crystal lenses a material factor MAL and a refractive index nL associated with the respective material. Next, an orientation factor DIL is assigned to the fluoride crystal lenses of the group. As a next step, the optimizing function is minimized with regard to the optical retardation ORLG of the group of fluoride crystal lenses.
If the optical retardation ORLG of the group of fluoride crystal lenses is larger than a given threshold, the foregoing steps are repeated, assigning different material factors MAL and/or orientation factors DIL to the fluoride crystal lenses.
Since under this approach, the material factors MAL and/or the orientation factors DIL are assigned to the fluoride crystal lenses, the quantities that remain available as degrees of freedom are the surface parameters of the lens surfaces, the lens thicknesses and the distances between the lenses.
It is advantageous to define an optical retardation asymmetry ORASLG as a further birefringence-related image aberration for a group of fluoride crystal lenses, as follows:




  
    
      
        
          ORAS
          LG
        
        =
        
          
          
            
              ∑
              
                L
                ⁢
                
                  
                
                ⁢
                in
                ⁢
                
                  
                
                ⁢
                G
              
            
            ⁢
            
              (
              
                
                  DS
                  L
                
                ·
                
                  AP
                  L
                
                ·
                
                  BF
                  L
                
              
              )
            
          
          
        
      
    
    
      
        (
        5
        )
      
    
  




The optical retardation asymmetry ORASLG is defined as the absolute amount of the sum of the respective products of a complex rotary position factor DSL with an azimuth factor APL and the birefringence factor BFL as defined in equation (3) for each of the fluoride crystal lenses of the group.

The angular position factor DSL=exp(i·SFDI—L·γL) is defined as a phase factor which depends on a symmetry factor SFDI—L and on an angle of rotation γL.
The symmetry factor SFDI—L equals 4 for (100)-lenses, 3 for (111)-lenses, and 2 for (110)-lenses. The symmetry factor indicates the angular periodicity which occurs in the azimuthal symmetry of the birefringence distribution Δn(θL,αL) and which is dependent on the orientation of the lens axis.
The angle of rotation γL indicates the angle between the reference direction of the fluoride crystal lens which has been rotated about the lens axis and a reference direction of the group. The reference direction of the group is the same for all lenses to the group. It coincides for example with the direction of the x-axis if the optical axis of the objective coincides with the z-axis.
For (100)-lenses, the azimuth factor APL equals +1, independent of the azimuth angle αL. In (111)-lenses, the azimuth factor APL depends on the azimuth angle αL of the outer aperture ray. The azimuth factor APL equals +1 for 0°≦αL<180°, and it equals −1 for 180°≦αL<360°. This takes into account the threefold azimuthal symmetry of the birefringence distribution Δn(θL,αL) of (111)-lenses which has been discussed above, so that the respective optical path differences for two mutually orthogonal states of polarization that occur in two rays whose azimuth angles differ by 180° have opposite signs and different amounts.
For (110)-lenses, the azimuth factor APL equals +1, independent of the azimuth angle αL.
If a group of fluoride crystal lenses has only a small asymmetry ORASLG of the optical retardation, the optical path difference between two mutually orthogonal states of polarization in an outer aperture ray depends almost exclusively on the aperture angle θL of the outer aperture ray.
The available degrees of freedom include primarily the angles of rotation γL as well as the ray factors SPL which can be influenced by way of the lens data.
If all or some of the angles of rotation γL of the fluoride crystal lenses are prescribed quantities, the number of degrees of freedom available for the optimization is reduced accordingly.
If one limits the choice of angles of rotation γL for a group of (100)-lenses to 0° and 45° or for a group of (111)-lenses to 0° and 60°, the angular position factor DSL can take on the values of ±1.
If one limits the choice of angles of rotation γL for a group of (110)-lenses to 0°, 45°, 90° and 135°, the angular position factor DSL can take on the values of ±1 or ±i.
A group of fluoride crystal lenses whose lens axes are all oriented in the same crystallographic direction and for which the asymmetry ORASHG of the optical retardation is minimized with the numerical optimizing method will hereinafter also be referred to as a homogeneous group. Image aberrations that relate to a special, i.e., homogenous, lens group, will hereinafter be identified by the subscript index “HG” instead of “LG”. Thus, there are homogeneous groups with (100)-lenses, with (111)-lenses, or with (110)-lenses. It is advantageous if the fluoride crystal lenses of a homogeneous group also consist of the same lens material.
It is further advantageous if the homogeneous groups have lenses in adjacent positions.
Following is a description of an advantageous procedure for determining the optical data of the fluoride crystal lenses of a homogeneous group in which the optical path difference in an outer aperture ray for two mutually orthogonal states of polarization depends almost exclusively on the aperture angle of the ray. The fluoride crystal lenses of the homogeneous group have the same orientation factors DIL.
Under the procedure, a respective material factor MAL and angle of rotation γL are assigned to each of the fluoride crystal lenses, and the optimizing function is minimized by taking into account the asymmetry ORASHG of the optical retardation in the homogeneous group of fluoride crystal lenses. If the asymmetry ORASHG of the optical retardation in the homogeneous group of fluoride crystal lenses exceeds a given threshold, a different set of material factors MAL and angles of rotation γL are assigned to the fluoride crystal lenses of the homogeneous group. The degrees of freedom which are available for the minimizing procedure are the surface parameters of the lens surfaces, the lens thicknesses and the distances between the lenses.
If the fluoride crystal lenses of the homogeneous group consist of the same lens material and thus have the same material factor MAL, there remains only a respective angle of rotation γL to be assigned to each of the fluoride crystal lenses.
Under a particularly preferred embodiment of the inventive method, the angles of rotation γL are used as additional degrees of freedom in the optimization, i.e., they are not assigned to the fluoride crystal lenses. This has the advantage that only small modifications of the surface parameters of the lens surfaces, the lens thicknesses and the lens distances are required so that the classical image aberrations such as the Seidel aberrations, which are independent of the angles of rotation γL of the fluoride crystal lenses, will be less affected by the optimization.
The best situation is achieved if the optimization can be performed by varying only the angles of rotation γL.
In an advantageous embodiment of the inventive method, the two birefringence-related imaging aberrations described above, i.e., the optical retardation ORLG and the asymmetry ORASHG of the optical retardation are minimized simultaneously for a group of fluoride crystal lenses that includes at least two homogeneous groups of fluoride crystal lenses. In this simultaneous minimization, the optical retardation ORLG is minimized for the overall group, and the asymmetry ORASHG is minimized for each homogeneous group.
Following are the steps of an advantageous procedure which can be applied when using the numerical optimizing method to reduce the optical retardation ORLG for a group of fluoride crystal lenses that includes at least two homogeneous groups of fluoride crystal lenses and to simultaneously reduce the asymmetry ORASHG of each of the homogeneous groups in an objective with known optical data:

    
    
        Assigning material factors MAL and refractive indices nL (if the lenses have different refractive indices nL) to the fluoride crystal lenses of the group.
        Assigning orientation factors DIL to the fluoride crystal lenses of the group.
        Allocating lenses of the same orientation factor DIL to one or more homogeneous groups.
        Assigning an angle of rotation γL to the fluoride crystal lenses of the individual homogeneous groups.
        Minimizing the optimizing function with regard to the optical retardation ORLG of the group and the respective asymmetry ORASHG of the optical retardation of each of the homogeneous groups.
        Repeating the foregoing steps, if the optical retardation ORLG of the group of fluoride crystal lenses exceeds a given threshold and if the respective asymmetries ORASHG of the optical retardation of the homogeneous groups exceed a further given threshold.
    
    


The surface parameters of the lens surfaces, the lens thicknesses, and the distances between the lenses are available as degrees of freedom in the minimization.
In a preferred version of the foregoing method, the step of assigning angles of rotation γL can be omitted, and the angles of rotation can be used as degrees of freedom.
The method can be used to particular advantage for lens groups that contain only fluoride crystal lenses of one and the same fluoride crystal, such as for example calcium fluoride, in which case the harmful effects of intrinsic birefringence can be reduced to a significant extent already by determining the orientation factors DIL and the angles of rotation γL. Under the optimizing procedure described above, significantly more degrees of freedom are available for the optimization, because at the outset the full range of angles of rotation γL are available rather than only those angles that are associated with the lens symmetries, i.e., 45° for (100)-lenses, 60° for (111)-lenses, and 90° for (110)-lenses.
Thus, the optimizing method of the foregoing description offers a way to design objectives where the classical image aberrations as well as the birefringence-related image aberrations are optimized already in the phase of determining of the optical data.
The objectives are distinguished by the fact that they produce a distortion of less than 5 nm within an image field of more than 15 mm diameter. The term “distortion” in this case encompasses the geometric distortion of the principal rays as well as the coma-induced distortion.
The mean deviation of the wave front from a perfect sphere can be optimized simultaneously for each image point, so that it is smaller than 10 mλ (thousandths of a wavelength) in terms of the working wavelength.
The objectives designed according to the numerical optimizing method are distinguished by the fact that the optical retardation ORLG for all fluoride crystal lenses that are arranged between an object plane and an image plane is smaller that 5%, in particular smaller than 2%, and preferably smaller than 1% of a working wavelength λ0.
If the objective includes polarization-selective elements which lead to an uncoupling between the individual partial systems of the objective with regard to their polarization characteristics, it is advantageous if the optical retardation ORLG of all fluoride crystal lenses in the partial system arranged in the light path directly ahead of the image plane is smaller that 25%, in particular smaller than 10%, and preferably smaller than 5% of a working wavelength λ0. Polarization-selective elements are, e.g., polarization filters, but they also include polarization-optics elements such as beam-splitting layers, which are used for example between the beam-splitter prisms of a beam-splitter cube in catadioptric projection objectives. In objectives with polarization-selective elements, the correction of the optical retardation for the partial system arranged ahead of the image plane is more difficult than for example in a refractive projection objective, because the partial system has fewer lenses than the refractive projection objective.
It is advantageous, if the optimization is directed not only to an outer aperture ray, but to a plurality of aperture rays with different relative ray apertures. For example, the optical retardation ORLG should be minimized for an outer aperture ray with a relative ray aperture between 0.75 and 1.0 and for an outer aperture ray with a relative ray aperture between 0.5 and 0.75. As a result, the distribution function of the optical path differences for two mutually orthogonal states of polarization in a ray bundle that originates from an object point shows reduced values for different rays independent of their relative ray apertures.
If the optimization is performed with regard to a plurality of outer aperture rays with different relative ray apertures, it is further of advantage if the calculation of the birefringence-related image aberrations is performed for an outer aperture ray whose relative ray aperture is determined so that the optical path difference for two mutually orthogonal states of polarization in this outer aperture ray takes on a maximum when the fluoride crystal lenses are not rotated relative to each other. In addition to an outer aperture ray defined by this condition, it is further an advantageous practice to also apply the optimization to an outer aperture ray with a relative ray aperture between 0.9 and 1.0. In order to keep the computing effort within limits, it is advantageous to perform the calculation especially for these two particular outer aperture rays.
If the values for the optical retardation of an objective or of a partial system are below the aforementioned threshold, a nearly complete reduction of the harmful effects of intrinsic birefringence can be achieved by further measures such a rotating the lenses relative to each other.
The proof for a successful compensation by rotating the fluoride crystal lenses relative to each other lies in a low value for the asymmetry ORASLG of the optical retardation, which is calculated for all fluoride crystal lenses of an objective or for the partial system arranged ahead of the image plane. A favorable value for the asymmetry ORASLG of the optical retardation is less than 10%, in particular less than 5%, and with special preference less than 2% of a working wavelength λ0.
When evaluating the asymmetry ORASLG of the optical retardation, it is likewise advisable to consider a plurality of different outer aperture rays with different relative ray aperture values.
It is advantageous to correct the asymmetry ORASHG of the optical retardation not only for all fluoride crystal lenses of an objective or a partial system, but also for individual homogeneous groups with small numbers of lenses.
Following is a discussion of the advantages of homogeneous groups with at least three fluoride crystal lenses. The asymmetry ORASHG of the optical retardation for all fluoride crystal lenses of this homogeneous group should be smaller than 5%, in particular smaller than 2% of a working wavelength λ0.
While an effective birefringence distribution Δeff(θL) that is independent of the azimuth angle can be achieved in two (111)-lenses or two (100)-lenses only if the lenses are rotated relative to each other by 60° or 45°, respectively, and if the two lenses have the same birefringence factors BFL, an arrangement of three lenses offers significantly more possibilities, since the angles of rotation γL are available as additional degrees of freedom for the optimization. This has particular bearing on the case where the inequality




  
    
      
        
          
          
            
              ∑
              
                L
                ⁢
                
                  
                
                ⁢
                in
                ⁢
                
                  
                
                ⁢
                HG
              
            
            ⁢
            
              (
              
                
                  c
                  L
                
                ·
                
                  BF
                  L
                
              
              )
            
          
          
        
        >
        0.0
      
    
    
      
        (
        6
        )
      
    
  




and in particular





  
    
      
        
          
          
            
              ∑
              
                L
                ⁢
                
                  
                
                ⁢
                in
                ⁢
                
                  
                
                ⁢
                HG
              
            
            ⁢
            
              (
              
                
                  c
                  L
                
                ·
                
                  BF
                  L
                
              
              )
            
          
          
        
        >
        
          2
          ⁢
          %
          ⁢
          
            
          
          ⁢
          of
          ⁢
          
            
          
          ⁢
          a
          ⁢
          
            
          
          ⁢
          working
          ⁢
          
            
          
          ⁢
          wavelength
          ⁢
          
            
          
          ⁢
          
            λ
            0
          
        
      
    
    
      
        (
        7
        )
      
    
  




is met for any arbitrary set of coefficients cL=±1. Especially in this case, the asymmetry ORASHG of the optical retardation can only be minimized it the angular position factor DSL=exp(i·SFDI—L·γL) assumes complex values. The optimizing method of the foregoing description allows angles of rotation γL to be calculated for the fluoride crystal lenses of the homogeneous group.

If the homogeneous group has exactly three lenses, the angles of rotation can be calculated directly from the ray factors SPL, the material factors MAL, the symmetry factors SFDI—L, and the azimuth factors APL of the three lenses, in accordance with the following equations:

γ1=0°±5°,  (8)





  
    
      
        
          γ
          2
        
        =
        
          
            
              1
              
                SF
                
                  DI_
                  ⁢
                  2
                
              
            
            ⁢
            
              arccos
              (
              
                
                  
                    
                      
                        
                          
                            (
                            
                              
                                AP
                                3
                              
                              ·
                              
                                SP
                                3
                              
                              ·
                              
                                MA
                                3
                              
                            
                            )
                          
                          2
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          
                            (
                            
                              
                                AP
                                2
                              
                              ·
                              
                                SP
                                2
                              
                              ·
                              
                                MA
                                2
                              
                            
                            )
                          
                          2
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          (
                          
                            
                              AP
                              1
                            
                            ·
                            
                              SP
                              1
                            
                            ·
                            
                              MA
                              1
                            
                          
                          )
                        
                        2
                      
                    
                  
                
                
                  2
                  ·
                  
                    (
                    
                      
                        AP
                        1
                      
                      ·
                      
                        SP
                        1
                      
                      ·
                      
                        MA
                        1
                      
                    
                    )
                  
                  ·
                  
                    (
                    
                      
                        AP
                        2
                      
                      ·
                      
                        SP
                        2
                      
                      ·
                      
                        MA
                        2
                      
                    
                    )
                  
                
              
              )
            
          
          ±
          
            5
            ⁢
            °
          
        
      
    
    
      
        (
        9
        )
      
    
  
  
    
      
        
          γ
          3
        
        =
        
          
            
              1
              
                SF
                
                  DI_
                  ⁢
                  3
                
              
            
            ⁢
            
              (
            
            ⁢
            
              arctan
              (
              
                
                  
                    (
                    
                      
                        AP
                        2
                      
                      ·
                      
                        SP
                        2
                      
                      ·
                      
                        MA
                        2
                      
                    
                    )
                  
                  ·
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          SF
                          
                            DI_
                            ⁢
                            2
                          
                        
                        ·
                        
                          γ
                          2
                        
                      
                      )
                    
                  
                
                
                  
                    
                      
                        
                          (
                          
                            
                              AP
                              1
                            
                            ·
                            
                              SP
                              1
                            
                            ·
                            
                              MA
                              1
                            
                          
                          )
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          (
                          
                            
                              AP
                              2
                            
                            ·
                            
                              SP
                              2
                            
                            ·
                            
                              MA
                              2
                            
                          
                          )
                        
                        ·
                      
                    
                  
                  
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              SF
                              
                                DI_
                                ⁢
                                2
                              
                            
                            ·
                            
                              γ
                              2
                            
                          
                          )
                        
                      
                    
                  
                
              
              )
            
          
          +
          
            
              180
              ⁢
              °
              ⁢
              
                )
              
            
            ±
            
              5
              ⁢
              °
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




It is advantageous if the fluoride crystal lenses of a homogeneous group are of the same fluoride crystal material, i.e. for example calcium fluoride or barium fluoride or another optically transparent fluoride crystal with a cubic crystal structure.
It is further advantageous if the fluoride crystal lenses of a homogeneous group are arranged in adjacent positions. This is a favorable arrangement because if there are other optical elements arranged between the fluoride crystal lenses of the homogeneous group, these other elements can change the state of polarization of a ray in such a manner that a ray passing through the homogeneous group can still be subject to an azimuth-angle-dependent path difference for two mutually orthogonal states of polarization, even if the asymmetry ORASHG of the optical retardation for all fluoride crystal lenses of this homogeneous group equals zero.
The concept of forming a homogeneous group with at least three fluoride crystal lenses is particularly advantageous in the case where at least one fluoride crystal lens of the homogeneous group has a birefringence factor BFL of more than 3% of a working wavelength λ0. If in this case one were to form a homogeneous group of two fluoride crystal lenses, the second of the fluoride crystal lenses would have to have the same birefringence factor BFL. If one allowed for example only the angles of rotation 0° and 60° for (111)-lenses, or 0° and 45° for (100)-lenses, the number of fluoride crystals in the homogeneous group would have be increased until




  
    
    
      
        ∑
        
          L
          ⁢
          
            
          
          ⁢
          in
          ⁢
          
            
          
          ⁢
          HG
        
      
      ⁢
      
        (
        
          
            c
            L
          
          ·
          
            BF
            L
          
        
        )
      
    
    
  
  =
  0




for an arbitrary combination of coefficients cL=±1. Thus for example in a homogeneous group of three lenses, the sum of the birefringence factor BFL for two lenses would have to equal the birefringence factor BFL for the fluoride crystal lens with the largest birefringence factor BFL.

It is advantageous if, besides the homogeneous group with at least three lenses, the objective has at least one further homogeneous group with at least two fluoride crystal lenses with an asymmetry ORASHG of the optical retardation of less than 10%, preferably less than 5%, and with particular preference less than 2% of a working wavelength λ0.
In order to allow a simultaneous minimization of the optical retardation ORLG, it is advantageous if the products MAL·DIL for the fluoride crystal lenses of at least two different homogeneous groups have different signs.
The larger the number of homogeneous groups in an objective or in a partial system, the more degrees of freedom remain available to perform an adjustment, because after the asymmetry ORASHG of the optical retardation has been corrected, the homogeneous groups can be positioned at arbitrary angles relative to each other. It is advantageous to form at least four homogeneous groups.
With particular preference, all fluoride crystal lenses of an objective, or of a partial system that is arranged between a polarization-selective optical element and an image plane, are assigned to one of the homogeneous groups. It is advantageous if the optical retardation ORLG of the fluoride crystal lenses of the objective or of the partial system is smaller than 25%, in particular smaller than 10%, and with special preference smaller than 5% of a working wavelength λ0.
The harmful influence of birefringence, in particular of the intrinsic birefringence of (100)- or (111)-lenses, becomes most noticeable if the light rays have large aperture angles inside the lenses. This is the case for objectives with an image-side numerical aperture of more than 0.7, in particular more than 0.8.
The magnitude of the intrinsic birefringence increases noticeably with a smaller working wavelength. For example in comparison to a wavelength of 248 nm, the intrinsic birefringence increases more than sixfold at a wavelength of 193 nm, and more than twentyfold at a wavelength of 157 nm. The invention can therefore be used to particular advantage if the working wavelength λ0 is smaller than 200 nm, and in particular if it is smaller than 160 nm.
Objectives designed according to the inventive method can be used advantageously in microlithography projection systems which include a light source, an illumination system, a mask-positioning system, a mask with a structure, a projection objective, an object-positioning system, and a light-sensitive substrate.
A microlithography projection system of this type serves to manufacture micro-structured semiconductor components.



BRIEF DESCRIPTION OF THE DRAWINGS
The invention will be explained in more detail hereinafter with reference to the drawings, wherein
FIG. 1 represents the lens section of a refractive projection objective according to the first embodiment;
FIG. 2 represents a coordinate system to illustrate the definition of the aperture angle and the azimuth angle;
FIGS. 3A-E illustrate the birefringence distribution for (100)-lenses in different representations, as well as the effective birefringence distribution for a homogeneous group of (100)-lenses;
FIGS. 4A-E illustrate the birefringence distribution for (111)-lenses in different representations, as well as the effective birefringence distribution for a homogeneous group of (111)-lenses;
FIGS. 5A-E illustrate the birefringence distribution for (110)-lenses in different representations, as well as the effective birefringence distribution for a homogeneous group of (110)-lenses;
FIG. 6 represents the lens section of a refractive projection objective according to the second embodiment;
FIG. 7 represents the lens section of a refractive projection objective according to the third embodiment;
FIG. 8 represents the lens section of a catadioptric projection objective according to the fourth embodiment; and
FIG. 9 schematically represents a microlithography projection system.



DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS
FIG. 1 represents in the form of a lens section a first embodiment of an objective 1 whose optical data were determined by the numerical optimizing method in accordance with the invention. The objective 1 is a refractive projection objective for a microlithography projection system designed for the working wavelength of 157 nm. The optical data for this objective are listed in Table 1. The lenses L101 to L130 of this objective consist of calcium fluoride crystal. The image-side numerical aperture of the objective is 0.9. The object field, which is centered on the optical axis OA has a diameter of 92.4 mm. Located on the optical axis OA is the central object point 3, while the object point 5 is located at the border of the object field. The imaging scale factor of the objective 1 between the object plane OB and the image plane IM is 0.25. The imaging performance of the objective 1 is corrected to such a high degree that for all image points of the object field the mean deviation of the wave front from an ideal spherical wave is smaller than 6 mλ in relation to a working wavelength of 157 nm, if only those aberrations are taken into account that are caused by the lens geometry. The principal-ray-induced and coma-induced distortion is smaller than 3 nm for all image points.
A practical example from the patent application WO 01/50171 (U.S. Ser. No. 10/177,580), owned by the same assignee as the present application, was chosen as the starting system for the optimization of the objective 1. The starting system is shown in FIG. 7 of WO 01/50171. The optical data for the starting system are listed in Table 6 of WO 01/50171. A more detailed description of the design and function of the starting system, i.e., of the objective to be optimized, may be found in the patent application WO 01/50171. The working wavelength, imaging scale ratio, size of the image field and numerical aperture of the starting system all have the same values as in the objective 1. All lenses of the objective to be optimized consist of calcium fluoride crystal material. Particularly in high-performance objectives of this kind, it is a requirement that the harmful effects of intrinsic birefringence be reduced as much as possible.
The invention specifies that in the optimizing function a plurality of birefringence-related image aberrations be taken into account, in addition to the image aberrations such as for example the geometric distortion, also referred to as principal-ray-induced distortion, the coma-induced distortion, wave-front aberrations, telecentrism, and boundary conditions such as image field size and imaging scale ratio.
In practice, the birefringence-related image aberrations are determined by running the complete calculations for individual rays.
In the first embodiment, the outer aperture ray 7 was chosen for the mathematical analysis. The outer aperture ray 7 originates from an object point 3 and its ray height in the aperture stop plane is equal to the radius of the aperture stop AS. The relative ray aperture in the image plane IM equals 1.0.
The birefringence-related image aberrations, to the extent that they depend on parameters of the aperture ray 7, are only a function of geometric parameters, i.e., of optical path lengths OPL, aperture angles θL and azimuth angles αL of the outer aperture ray 7 in the lenses L101 to L130.
FIG. 2 serves to visualize the definition of the aperture angle θL and azimuth angle αL of a ray 201 in a lens. The drawing shows the local x-y-z-coordinate system of a lens, where the z-axis is parallel to the direction of the lens axis. The ray 201 is directed at an aperture angle θL to the lens axis. The azimuth angle αL of the ray 201 is obtained by projecting the ray direction into a plane whose normal vector points in the direction of the lens axis and by measuring the angle of the projected ray direction 203 relative to a reference direction of the lens, in this case represented by the x-axis. The reference direction is fixed in the lens, and when the lens is rotated about the lens axis, the reference direction rotates along with the lens. In lenses whose lens axes are oriented in the crystallographic (100)- or (111)-direction, the reference direction in the present examples is oriented in the same direction as the projection of the crystallographic (110)-direction into the aforementioned plane whose normal vector points in the direction of the lens axis. The local x-axis does not need to have the same direction as the projection of the crystallographic (110)-direction. The essential point is that the local x-axis has a defined orientation in relation to the crystallographic orientation and that the relationship between the local x-axis and the crystal orientation is the same for all lenses with an equivalent crystal structure.
The birefringence-related image aberrations that are taken into account in the optimizing function are in all cases dependent on the ray factor SPL according to equation (1).
The functional dependency of the ray factor SPL on the aperture angle θL, i.e., the proportionality SPL∝sin2 θL·(7·cos2 θL−1), is the same as the functional dependency of the effective birefringence distribution Δn(θL) on the aperture angle θL of a ray in a fluoride crystal lens. FIGS. 3 to 5 support the discussion which follows below of the effective birefringence distribution Δneff(θL) of a calcium fluoride lens that is used according to the invention and whose lens axis is oriented, respectively, in the crystallographic (100)-, (111)-, or (110)-direction.
FIGS. 3A and 3B illustrate the birefringence distribution Δn(αL,θL) for (100)-lenses of calcium fluoride.
The contour lines in FIG. 3A indicate the amount of the intrinsic birefringence in nm/cm as a function of the aperture angle θL and the azimuth angle αL. The representations of the angles in FIGS. 3A to 3D are correlated to the local pupil coordinate system through the equations θL2=x2+y2 and αL=arctan(y/x).
Each line in FIG. 3B represents the amount and direction of the birefringence for a ray direction that is defined by the aperture angle θL and the azimuth angle αL. The length of each line is in proportion to the amount of the birefringence as represented by the difference between the principal axes of the elliptical section, while the direction of each line indicates the orientation of the longer of the two principal axes of the elliptical section. The elliptical section is obtained as the intersection between the index ellipsoid for a light ray in the direction (θL,αL) and a plane that contains the center point of the ellipsoid and is orthogonal to the light ray direction.
The four-fold azimuthal symmetry of the birefringence distribution of (100)-lenses is made evident by FIGS. 3A and 3B. The intrinsic birefringence has its maxima at the azimuth angles 0°, 90°, 180°, and 270°.
If the (100)-lenses that are combined into a homogeneous group are set at appropriately rotated position relative to each other, one can achieve a condition where the optical path difference for two mutually orthogonal states of polarization which occurs in a light ray is almost exclusively a function of the aperture angle θL of the light ray in relation to the optical axis. As an approximation, the lenses of this homogeneous group can be assigned an effective birefringence distribution Δneff(θL) which is a function of the aperture angle θL alone. This approximation becomes an exact representation in the case where the (100)-lenses are planar-parallel plates. With real lenses, the approximation can be achieved to a satisfactory degree at least for certain aperture angles θL.
FIGS. 3C and 3D represent the magnitude and direction of the effective birefringence for different ray directions in a homogeneous group of (100)-lenses.
FIG. 3E represents a radial profile of the effective birefringence distribution. The profile curve can be described by the equation:

Δneff(θL)=MAL·DIL·sin2 θL·(7·cos2 θL−1),  (11)

wherein the material factor MAL equals −11 nm/cm and the orientation factor DIL equals −0.5. For aperture angles θL<40°, the preceding equation can be approximated by

Δneff(θL)≈MAL·DIL·(9/7)·sin2(2.17·θL)  (12)

FIGS. 4A and 4B illustrate the birefringence distribution Δn(θL,αL) for (111)-lenses of calcium fluoride. The form of representation is analogous to FIGS. 3A and 3B. The three-fold azimuthal symmetry of the birefringence distribution of (111)-lenses is made evident by FIGS. 4A and 4B. The intrinsic birefringence has its maxima at the azimuth angles 0°, 120°, and 240°.
By rotating the (111)-lenses in relation to each other about the lens axes, it is likewise possible to produce an effective birefringence distribution Δneff(θL) which depends on the aperture angle θL alone.
FIGS. 4C and 4D represent the magnitude and direction of the effective birefringence for different ray directions in a homogeneous group of (111)-lenses.
FIG. 4E represents a radial profile of the effective birefringence distribution. The profile curve can likewise be described by equation (11), wherein the material factor MAL equals −11 nm/cm and the orientation factor DIL equals +1⁄3.
FIGS. 5A and 5B illustrate the birefringence distribution Δn(θL,αL) for (110)-lenses of calcium fluoride. The form of representation is analogous to FIGS. 3A and 3B. The two-fold azimuthal symmetry of the birefringence distribution of (110)-lenses is made evident by FIGS. 5A and 5B. The intrinsic birefringence has its maxima at the azimuth angles 0° and 180°.
By rotating the (110)-lenses in relation to each other about the lens axes, it is likewise possible to produce an effective birefringence distribution Δneff(θL) which depends on the aperture angle θL alone. However, in order to achieve this result to an ideal degree, at least four suitable (110)-lenses are required. But even with only two suitable and mutually rotated (110)-lenses, the effective birefringence distribution shows a dependency on the azimuth angle αL only at larger aperture angles θL.
FIGS. 5C and 5D represent the magnitude and direction of the effective birefringence for different ray directions in a homogeneous group of (110)-lenses.
FIG. 5E represents a radial profile of the effective birefringence distribution. The profile curve can likewise be described by equation (11), wherein the material factor MAL equals −11 nm/cm and the orientation factor DIL equals +1⁄8.
In a ray passing through a fluoride crystal lens that is used according to the invention, the effective optical path difference between two mutually orthogonal states of polarization is obtained by multiplying the ray path length OPL with the effective birefringence value Δneff for the aperture angle θL of the ray. The effective optical path difference is thus proportionate to the ray factor SPL according to equation (1).
In the first embodiment, the harmful influence of intrinsic birefringence is reduced by combining two mutually rotated (100)-lenses of calcium fluoride with mutually rotated (111)-lenses of calcium fluoride. All of the lenses L101 to L130 in this example have the same material parameter MAL.
The first step in the optimizing method consists of assigning orientation factors DIL to the lenses L101 to L130 of the given starting system, in other words, designating the lenses whose lens axes are to be oriented in the crystallographic (100)-direction and those whose lens axes are to be oriented in the crystallographic (111)-direction. The assignment of orientation factors is made in such a way that there are always at least two adjacent lenses with the same orientation factor DIL. The arrangement and the dimensions of the lenses of the objective remain unchanged for the time being.
In a second step of the optimizing method, the (100)-lenses are assigned to a first homogeneous group and the (111)-lenses are assigned to a second homogeneous group.
In a third step of the optimizing method, angles of rotation γL are assigned to the respective fluoride crystal lenses of the first homogeneous group and the second homogeneous group. In the first embodiment, the angles of rotation assigned to the individual (100)-lenses are either 0° or 45°, and the angles of rotation assigned to the individual (111)-lenses are either 0° or 60°.
In a fourth step of the optimizing method, the optical retardation




  
    OR
    LG
  
  =
  
    
      ∑
      
        L
        ⁢
        
          
        
        ⁢
        in
        ⁢
        
          
        
        ⁢
        LG
      
    
    ⁢
    
      BF
      L
    
  




is calculated for the lenses L101 to L130. The calculation is based on an outer aperture ray 7.

In a fifth step of the optimizing method, the respective asymmetries of the retardation



  
    
      
        ORAS
        HGI
      
      =
      
        
          
          
            
              ∑
              
                L
                ⁢
                
                  
                
                ⁢
                in
                ⁢
                
                  
                
                ⁢
                HGI
              
            
            ⁢
            
              (
              
                
                  MA
                  L
                
                ·
                
                  DI
                  L
                
                ·
                
                  DS
                  L
                
                ·
                
                  AP
                  L
                
                ·
                
                  SP
                  L
                
              
              )
            
          
          
        
        ⁢
        
          
        
        ⁢
        and
      
    
    ⁢
    
      
    
  




  
    
      ORAS
      HGII
    
    =
    
      
      
        
          ∑
          
            L
            ⁢
            
              
            
            ⁢
            in
            ⁢
            
              
            
            ⁢
            HGII
          
        
        ⁢
        
          (
          
            
              MA
              L
            
            ·
            
              DI
              L
            
            ·
            
              DS
              L
            
            ·
            
              AP
              L
            
            ·
            
              SP
              L
            
          
          )
        
      
      
    
  



are calculated for the lenses of the first homogeneous group and second homogeneous group.

The sixth step of the optimizing method consists of a test whether ORLG<1⁄4·λ0, wherein λ0 represents the working wavelength of the objective, as well as additional tests whether ORASLGI<1⁄4·λ0 and ORASLGII<1⁄4·λ0. If these threshold values are exceeded, the optimizing procedure is repeated with other orientation factors DIL for the lenses L101 to L130.
The different assignment possibilities in the foregoing steps are evaluated through a numerical optimizing program. Although the numerical optimizing program may not necessarily find the optimal solution, it will yield a solution in which the foregoing criteria are below the threshold values. A closely related problem in the mathematical literature is known as “the problem of the traveling salesman” where the shortest possible route has to be found to visit a given set of cities on a geographical map.
The optimization may be accomplished by one of the following methods, which are known from the literature under the terms:

    
    
        1. Monte Carlo Search,
        2. Simulated Annealing
        3. Threshold Accepting
        4. Simulated annealing with reheating
        5. Genetic algorithm

Should these methods fail to deliver a solution within a finite time period, the problem may also be solved by raising the threshold values.

    
    


In order to obtain a better starting system for the fine optimization, it is also possible to lower the threshold values to 1/10·λ0.
In the case of the first embodiment, a solution is found for the following orientations of the lens axes:

    
    
        first homogeneous group with (100)-lenses: L108, L109, L129, L130; and
        second homogeneous group with (111)-lenses: L101 to L107, L110 to L128.
    
    


In a seventh step of the method, the degrees of freedom represented by the arrangement and dimensions of the lenses of the objective are used to achieve a further optimization of the birefringence-related image aberrations ORLG, ORASHGI and ORASHGII. This step can be performed for example with the aforementioned optimizing programs “Code V®” or “ZEMAX®”, where the image aberrations ORLG, ORASHGI and ORASHGII are treated as user defined image aberrations in the optimizing function.
In an eighth step of the optimizing method, the first homogeneous group and the second homogeneous group are each split up into further homogeneous groups. In other words, further homogeneous groups are formed, each with at least two lenses. The purpose of splitting the first and second homogeneous groups into further homogeneous groups is to obtain an arrangement where a homogeneous group is made up only of neighboring lenses. Furthermore, increasing the number of homogeneous groups has the advantage that the angles of rotation of the homogeneous groups in relation to each other can be used as degrees of freedom in the adjustment process of the objective. The arrangement and dimensions of the optical elements of the lenses of the objective remain again unchanged at the outset.
As a ninth step in the optimizing method, the asymmetry




  
    ORAS
    
      HG
      n
    
  
  =
  
    
    
      
        ∑
        
          L
          ⁢
          
            
          
          ⁢
          aus
          ⁢
          
            
          
          ⁢
          
            HG
            n
          
        
      
      ⁢
      
        (
        
          
            MA
            L
          
          ·
          
            DI
            L
          
          ·
          
            DS
            L
          
          ·
          
            AP
            L
          
          ·
          
            SP
            L
          
        
        )
      
    
    
  




of the optical retardation is calculated for the lenses of the n-th homogeneous group.

The tenth step of the optimizing method consists of a test whether ORASHGn<1⁄4·λ0 for all of the homogeneous groups. If these threshold values are exceeded, the optimizing procedure is repeated from the eight step, splitting the first and second homogeneous groups differently into further homogeneous groups.
After a way of splitting up the objective into homogeneous groups has been determined, an eleventh step of the method follows in which the degrees of freedom represented by the arrangement and dimensions of the lenses of the objective are again used to achieve a further optimization of the birefringence-related image aberrations ORLG and the individual asymmetries ORASHGn of the optical retardation. This step can be performed for example with the aforementioned optimizing programs “Code V®” or “ZEMAX®”, where the image aberrations ORLG and ORASHGn for the individual homogeneous groups are treated as user defined image aberrations in the optimizing function.
The optimizing method can also be abbreviated with a modified second step in which the (100)-lenses are assigned to one or more homogeneous groups and the (111)-lenses are likewise assigned to one or more homogeneous groups. Two purposes are pursued in this step: For one, to allocate only neighboring lenses to homogeneous groups, and second, to increase the number of homogeneous groups. Steps 8 to 11 are dropped under this abbreviated version of the optimizing method.
In summary, with a given starting system as a point of departure, a solution was found for the objective 1 wherein the harmful effects of the intrinsic birefringence of the fluoride crystal lenses cause almost no polarization-dependent reduction of the imaging performance of the objective. This was achieved by orienting the lens axes of the lenses L108, L109, L129 and L130 in the crystallographic (100)-direction and orienting the lens axes of the remaining lenses in the crystallographic (111)-direction, and by appropriately adapting the arrangement and dimensions of the lenses through the optimizing method according to the invention.
The essential characterizing data of the objective 1 according to the invention are presented in Table 2. The quantities that depend on ray parameters have been calculated for the outer aperture ray 7 with the relative ray aperture of 1.0. The columns of Table 2 contain the following data:

    
    
        aperture angle θL in [°],
        ray path length OPL in [mm],
        ray factor SPL in [mm],
        orientation factor DIL,
        material factor MAL in [nm/cm],
        birefringence factor BFL in [nm],
        angle of rotation γL [°],
        symmetry factor SFDI—L,
        azimuth angle αL in [°], as well as
        azimuth parameter APL.
    
    


The reference direction of a lens is parallel to a direction that is defined by projecting the crystallographic (110)-direction into a plane whose normal vector points in the crystallographic (100)- or (111)-direction, depending on the direction of the lens axis.

















TABLE 2






θL
OPL
SPL

MAL
BFL
γL

αL



Lens
[°]
[mm]
[mm]
DIL
[nm/mm]
[nm]
[°]
SFDI—L
[°]
APL

























L101
8.15
15.92
1.87
0.33
−1.1
−0.69
60
3
60
1


L102
8.68
8.72
1.16
0.33
−1.1
−0.43
60
3
60
1


L103
7.74
10.75
1.14
0.33
−1.1
−0.42
60
3
60
1


L104
10.74
7.48
1.50
0.33
−1.1
−0.55
60
3
60
1


L105
9.42
6.87
1.07
0.33
−1.1
−0.39
60
3
60
1


L106
10.36
8.98
1.68
0.33
−1.1
−0.61
60
3
60
1


L107
21.80
12.83
8.91
0.33
−1.1
−3.27
0
3
0
1


L108
25.49
20.40
17.77
−0.5
−1.1
9.77
0
4
0
1


L109
16.51
39.96
17.53
−0.5
−1.1
9.64
45
4
45
1


L110
12.43
14.78
3.89
0.33
−1.1
−1.43
0
3
0
1


L111
2.35
25.63
0.26
0.33
−1.1
−0.09
0
3
0
1


L112
2.30
30.95
0.30
0.33
−1.1
−0.11
0
3
0
1


L113
18.47
30.77
16.36
0.33
−1.1
−6.00
0
3
180
−1


L114
18.84
22.13
12.17
0.33
−1.1
−4.46
60
3
240
−1


L115
14.22
9.98
3.36
0.33
−1.1
−1.23
0
3
180
−1


L116
1.35
30.01
0.10
0.33
−1.1
−0.04
0
3
180
−1


L117
26.48
32.58
29.85
0.33
−1.1
−10.94
0
3
0
1


L118
33.64
14.03
16.58
0.33
−1.1
−6.08
60
3
60
1


L119
26.60
7.07
6.52
0.33
−1.1
−2.39
60
3
60
1


L120
19.37
6.02
3.47
0.33
−1.1
−1.27
60
3
60
1


L121
6.66
6.55
0.52
0.33
−1.1
−0.19
60
3
60
1


L122
10.40
8.76
1.65
0.33
−1.1
−0.60
60
3
240
−1


L123
12.05
9.01
2.24
0.33
−1.1
−0.82
0
3
180
−1


L124
0.04
17.43
0.00
0.33
−1.1
−0.00
0
3
0
1


L125
5.50
15.89
0.87
0.33
−1.1
−0.32
0
3
0
1


L126
24.13
9.23
7.46
0.33
−1.1
−2.73
0
3
180
−1


L127
35.54
8.14
10.00
0.33
−1.1
−3.67
0
3
180
−1


L128
39.30
13.40
17.16
0.33
−1.1
−6.29
60
3
240
−1


L129
35.27
26.67
32.59
−0.5
−1.1
17.93
45
4
45
1


L130
35.27
26.45
32.33
−0.5
−1.1
17.78
0
4
0
1


Sum





0.10









The total optical retardation





  
    OR
    LG
  
  =
  
    
      ∑
      
        L101
        ⁢
        
          
        
        ⁢
        to
        ⁢
        
          
        
        ⁢
        L130
      
    
    ⁢
    
      BF
      L
    
  




of the lenses is ±0.1 nm, thus smaller than 1% of the working wavelength. The total for the asymmetry of the optical retardation





  
    ORAS
    LG
  
  =
  
    
    
      
        ∑
        
          L101
          ⁢
          
            
          
          ⁢
          bis
          ⁢
          
            
          
          ⁢
          L130
        
      
      ⁢
      
        (
        
          
            DS
            L
          
          ·
          
            AP
            L
          
          ·
          
            BF
            L
          
        
        )
      
    
    
  




is 1.96 nm, thus smaller than 2% of the working wavelength.

The lenses of the objective 1 are assigned to the five homogeneous groups of (111)-lenses HG1, HG3, HG4, HG5 and HG6 and to the two homogeneous groups HG2 and HG7 of (100)-lenses.
The lenses are assigned to the homogeneous groups as shown in Table 3.










TABLE 3





Homogeneous

Crystallographic orientation



group
Lenses
of the lens axis
ORAS [nm]







HG1
L101–L107
(111)
0.49


HG2
L108–L109
(100)
0.24


HG3
L110–L114
(111)
0.26


HG4
L115–L120
(111)
0.18


HG5
L121–L125
(111)
0.24


HG6
L126–L128
(111)
0.30


HG7
L129–L130
(100)
0.26









The asymmetry





  
    ORAS
    
      HG
      n
    
  
  =
  
    
    
      
        ∑
        
          L
          ⁢
          
            
          
          ⁢
          in
          ⁢
          
            
          
          ⁢
          
            HG
            n
          
        
      
      ⁢
      
        (
        
          
            MA
            L
          
          ·
          
            DI
            L
          
          ·
          
            DS
            L
          
          ·
          
            AP
            L
          
          ·
          
            SP
            L
          
        
        )
      
    
    
  




of the optical retardation in all groups is smaller than 1% of the working wavelength.

In each of the homogeneous groups HG2, HG3, HG4, HG6 and HG7 there is one lens whose birefringence factor BFL is larger that 3% of the working wavelength λ0=157 nm.
FIG. 6 illustrates a second embodiment in which the invention was put into practice in an objective 601. The central object point 603 is located on the optical axis OA. One difference to the first embodiment is that the optimization was performed by considering not only an outer aperture ray but two outer aperture rays. The birefringence-related image aberrations were optimized not only for the outer aperture ray 607 which has a relative ray aperture of 1.0 in the image plane IM, but also for the outer aperture ray 609 which has a relative ray aperture of 0.7 in the image plane IM.
The optical data of the objective 601 are listed in Table 4.
Table 5 presents the essential data that characterize the objective 601 in accordance with the invention. The quantities that depend on the ray parameters were calculated for the outer aperture ray 607 with the relative ray aperture 1.0 and the outer aperture ray with the relative ray aperture 0.7.

















TABLE 5









θL
OPL
SPL
BFL
θL
OPL
SPL
BFL




[°]
[mm]
[mm]
[nm]
[°]
[mm]
[mm]
[nm]

















MAL
Outer aperture ray 607
Outer aperture ray 609
γL

αL



Lens
DIL
[nm/mm]
Rel. ray aperture = 1.0
Rel. ray aperture = 0.7
[°]
SFDI—L
[°]
APL
























L601
0.33
−1.1
8.11
12.55
1.47
−0.54
5.67
12.56
0.73
−0.27
60
3
60
1


L602
0.33
−1.1
8.67
6.20
0.82
−0.30
6.06
6.24
0.41
−0.15
60
3
60
1


L603
0.33
−1.1
7.57
13.27
1.35
−0.50
5.30
13.58
0.69
−0.25
60
3
60
1


L604
0.33
−1.1
10.82
5.33
1.08
−0.40
7.56
5.50
0.56
−0.21
60
3
60
1


L605
0.33
−1.1
9.58
9.69
1.56
−0.57
6.75
8.89
0.73
−0.27
60
3
60
1


L606
0.33
−1.1
10.49
10.69
2.04
−0.75
7.39
8.71
0.85
−0.31
60
3
60
1


L607
0.33
−1.1
23.00
14.26
10.73
−3.94
15.93
9.39
3.87
−1.42
0
3
0
1


L608
−0.5
−1.1
27.16
21.22
20.07
11.04
18.31
26.19
13.72
7.54
0
4
0
1


L609
−0.5
−1.1
17.60
39.44
19.33
10.63
11.78
49.59
11.80
6.49
45
4
45
1


L610
0.33
−1.1
11.62
17.22
3.99
−1.46
7.77
19.79
2.12
−0.78
0
3
0
1


L611
0.33
−1.1
1.73
19.81
0.11
−0.04
1.12
25.23
0.06
−0.02
0
3
0
1


L612
0.33
−1.1
1.56
26.17
0.12
−0.04
1.00
33.62
0.06
−0.02
0
3
0
1


L613
0.33
−1.1
19.82
33.61
20.07
−7.36
13.08
47.57
13.74
−5.04
0
3
180
−1


L614
0.33
−1.1
20.20
25.32
15.59
−5.72
13.57
18.67
5.77
−2.12
0
3
180
−1


L615
0.33
−1.1
17.87
9.51
4.78
−1.75
11.97
7.13
1.75
−0.64
0
3
180
−1


L616
0.33
−1.1
3.53
29.20
0.66
−0.24
2.37
15.90
0.16
−0.06
60
3
240
−1


L617
0.33
−1.1
21.73
45.63
31.52
−11.56
14.12
32.49
10.80
−3.96
0
3
0
1


L618
0.33
−1.1
28.95
6.27
6.40
−2.35
18.21
11.72
6.09
−2.23
60
3
60
1


L619
0.33
−1.1
24.50
17.73
14.63
−5.36
15.13
24.17
9.09
−3.33
60
3
60
1


L620
0.33
−1.1
16.99
32.55
15.01
−5.50
10.03
40.58
7.12
−2.61
0
3
0
1


L621
0.33
−1.1
5.39
17.15
0.90
−0.33
2.68
35.09
0.46
−0.17
0
3
0
1


L622
0.33
−1.1
10.49
5.05
0.97
−0.35
7.07
21.98
1.96
−0.72
0
3
180
−1


L623
0.33
−1.1
11.47
5.19
1.18
−0.43
7.68
20.67
2.17
−0.80
0
3
180
−1


L624
0.33
−1.1
0.86
9.56
0.01
0.00
3.03
6.30
0.11
−0.04
0
3
0
1


L625
0.33
−1.1
6.84
12.84
1.08
−0.39
1.35
13.54
0.05
−0.02
0
3
0
1


L626
0.33
−1.1
22.16
5.73
4.08
−1.50
14.54
12.23
4.28
−1.57
0
3
180
−1


L627
0.33
−1.1
29.43
5.59
5.82
−2.13
19.01
9.31
5.19
−1.90
0
3
180
−1


L628
0.33
−1.1
39.04
14.56
18.62
−6.83
24.92
33.63
28.40
−10.41
60
3
240
−1


L629
−0.5
−1.1
35.27
30.56
37.35
20.54
23.84
27.28
21.64
11.90
45
4
45
1


L630
−0.5
−1.1
35.27
29.38
35.91
19.75
23.84
26.23
20.81
11.44
0
4
0
1


Sum





1.62



−1.93









The optical retardation




  
    OR
    LG
  
  =
  
    
      ∑
      
        L601
        ⁢
        
          
        
        ⁢
        to
        ⁢
        
          
        
        ⁢
        L630
      
    
    ⁢
    
      BF
      L
    
  




of all of the lenses is 1.6 nm for the outer aperture ray 607 and −1.9 nm for the outer aperture ray 609, thus smaller than 2% of the working wavelength.

The asymmetry of the optical retardation




  
    ORAS
    LG
  
  =
  
    
    
      
        ∑
        
          L601
          ⁢
          
            
          
          ⁢
          to
          ⁢
          
            
          
          ⁢
          L630
        
      
      ⁢
      
        (
        
          
            DS
            L
          
          ·
          
            AP
            L
          
          ·
          
            BF
            L
          
        
        )
      
    
    
  




for all of the lenses is 6.1 nm for the outer aperture ray 607 and 3.5 nm for the outer aperture ray 609, thus smaller than 4% of the working wavelength.

The lenses of the objective 601 are assigned to the two homogeneous groups HG61 and HG63 with (111)-lenses and the two homogeneous groups HG62 and HG64 with (100)-lenses.
Table 6 lists the assignments of the lenses to the homogeneous groups as well as the asymmetries of the optical retardation for the two outer aperture rays 607 and 609.











TABLE 6







Crystallographic




Homogeneous

orientation


group
Lenses
of the lens axis
ORAS [nm]







HG61
L601–L607
(111)
2.41
0.09


HG62
L608–L609
(100)
0.74
1.92


HG63
L610–L628
(111)
1.49
0.70


HG64
L629–L630
(100)
1.44
0.83









The asymmetry




  
    ORAS
    
      HG
      n
    
  
  =
  
    
    
      
        ∑
        
          L
          ⁢
          
            
          
          ⁢
          in
          ⁢
          
            
          
          ⁢
          
            HG
            n
          
        
      
      ⁢
      
        (
        
          
            MA
            L
          
          ·
          
            DI
            L
          
          ·
          
            DS
            L
          
          ·
          
            AP
            L
          
          ·
          
            SP
            L
          
        
        )
      
    
    
  




of the optical retardation for the outer aperture ray 607 as well as for the outer aperture ray 609 in all homogeneous groups is smaller than 2% of the working wavelength.

In each of the homogeneous groups HG62, HG63 and HG64, there is one lens whose birefringence factor BFL is larger than 3% of the working wavelength λ0=157 nm.
FIG. 7 illustrates a third embodiment in which the invention was put into practice in an objective 701. In this third embodiment, the harmful effects of intrinsic birefringence are reduced by using a combination of mutually rotated lenses of barium fluoride and mutually rotated lenses of calcium fluoride. The lens axes of the lens L701 to L730 are oriented in the crystallographic (111)-direction. Thus the orientation factor is the same for all of the lenses.
The optical data for the objective 701 are listed in Table 7. The image-side numerical aperture of the objective 701 is 0.9. The object field which is centered on the optical axis OA has a diameter of 92.4 mm. The central object point 703 is located on the optical axis OA, and a further object point 705 is located at the periphery of the object field. The imaging scale ratio of the objective 701 between the object plane OB and the image plane IM is 0.25. The imaging performance of the objective 701 is corrected to such a high degree that the deviation from the wave front of an ideal spherical wave for all image points of the image field is less than ten thousandths of a wavelength in relation to a working wavelength of 157 nm, if only those aberrations are considered that are caused by the lens geometry. The principle-ray-induced and coma-induced distortion for all image points is smaller than 3 nm.
The starting system for the optimization of the objective 701 is the same as in the first embodiment, i.e., the objective illustrated in FIG. 7 of WO 01/50171 (U.S. Ser. No. 10/177,580).
With a given starting system as a point of departure, the lenses L701 to L730 are in this case assigned material factors MAL rather than orientation factors DIL. In other words, the decision is made as to which of the lenses are to consist of barium fluoride and which are to consist of calcium fluoride. In this case, too, the assignment is made in such a way that in all cases at least two neighboring lenses have the same material factor MAL. Because a change of the material factor MAL also entails a change of the refractive index, a first step in the optimizing process serves to optimize the image aberrations caused by the change in the refractive index. Following this optimization, the arrangement and dimensions of the lenses of the objective remain at first unchanged.
In a second optimizing step, the barium fluoride lenses are assigned to a first homogeneous group and the calcium fluoride lenses are assigned to a second homogeneous group.
In a third step of the optimizing process, angles of rotation γL are assigned to the fluoride crystal lenses of the first homogeneous group and the second homogeneous group. Given that in this third embodiment the lens axes of the barium fluoride lenses as well as of the calcium fluoride lenses are oriented in the crystallographic (111)-direction, the individual lenses are assigned an angle of rotation of either 0° or 60°. If the lens axes were pointing in the crystallographic (100)-direction, the angle to assign to the lenses would be either 0° or 45°.
In a fourth step of the optimizing method, the optical retardation




  
    OR
    LG
  
  =
  
    
      ∑
      
        L701
        ⁢
        
          
        
        ⁢
        to
        ⁢
        
          
        
        ⁢
        L730
      
    
    ⁢
    
      BF
      L
    
  




is calculated for the lenses L701 to L730. The calculation is based on an outer aperture ray 707.

In a fifth step of the optimizing method, the respective asymmetries of the retardation



  
    
      ORAS
      HGI
    
    =
    
      
        
        
          
            ∑
            
              L
              ⁢
              
                
              
              ⁢
              in
              ⁢
              
                
              
              ⁢
              HGI
            
          
          ⁢
          
            (
            
              
                MA
                L
              
              ·
              
                DI
                L
              
              ·
              
                DS
                L
              
              ·
              
                AP
                L
              
              ·
              
                SP
                L
              
            
            )
          
        
        
      
      ⁢
      
        
      
      ⁢
      and
    
  




  
    
      ORAS
      HGII
    
    =
    
      
      
        
          ∑
          
            L
            ⁢
            
              
            
            ⁢
            in
            ⁢
            
              
            
            ⁢
            HGII
          
        
        ⁢
        
          (
          
            
              MA
              L
            
            ·
            
              DI
              L
            
            ·
            
              DS
              L
            
            ·
            
              AP
              L
            
            ·
            
              SP
              L
            
          
          )
        
      
      
    
  



are calculated for the lenses of the first homogeneous group and the second homogeneous group.

The sixth step of the optimizing method consists of a test whether ORLG<1⁄4·λ0, wherein λ0 represents the working wavelength of the objective, as well as additional tests whether ORASLGI<1⁄4·λ0 and ORASLGII<1⁄4·λ0. If these threshold values are exceeded, the optimizing procedure is repeated with other material factors MAL for the lenses L701 to L730.
The different assignment possibilities are evaluated through the numerical optimizing procedures that have already been discussed above in the context of the first embodiment.
Should these methods fail to deliver a solution within a finite time period, the problem may also be solved by raising the threshold values.
In order to obtain a better starting system for the fine optimization, it is also possible to lower the threshold values to 1/10·λ0.
In the case of the third embodiment, a solution is found for the following assignment of materials to the lenses:

    
    
        a first homogeneous group with barium fluoride lenses: L729, L730; and
        a second homogeneous group with calcium fluoride lenses: L701 to L728.
    
    


In a seventh step of the method, the degrees of freedom represented by the arrangement and dimensions of the lenses of the objective are used to achieve a further optimization of the birefringence-related image aberrations ORLG, ORASHGI and ORASHGII.
As described for the first embodiment, it is advantageous to define further homogeneous groups either in subsequent further optimizing steps or already in the second step of the optimizing method. As the first homogeneous group in the third embodiment consists of only two lenses, it cannot be split up further. However, the second homogeneous group can be subdivided into further homogeneous groups.
In summary, with a given starting system as a point of departure, a solution was found for the objective 701 wherein the harmful influence of the intrinsic birefringence of the fluoride crystal lenses is significantly reduced. This was achieved by changing the material in the lenses L729 and L730 of the starting system from calcium fluoride to barium fluoride and by appropriately adapting the arrangement and dimensions of the lenses through the optimizing method according to the invention.
The essential characterizing data of the objective 701 according to the invention are presented in Table 8. The quantities that depend on ray parameters have been calculated for the outer aperture ray 707 with the relative ray aperture of 1.0.

















TABLE 8






θL
OPL
SPL

MAL
BFL
γL

αL



Lens
[°]
[mm]
[mm]
DIL
[nm/mm]
[nm]
[°]
SFDI—L
[°]
APL

























L701
8.18
13.81
1.64
0.33
−1.1
−0.60
60
3
60
1.00


L702
8.79
6.72
0.91
0.33
−1.1
−0.34
60
3
60
1.00


L703
7.94
11.78
1.32
0.33
−1.1
−0.48
60
3
60
1.00


L704
10.84
7.99
1.63
0.33
−1.1
−0.60
60
3
60
1.00


L705
9.56
6.05
0.97
0.33
−1.1
−0.36
60
3
60
1.00


L706
10.34
8.86
1.65
0.33
−1.1
−0.60
60
3
60
1.00


L707
21.32
12.28
8.24
0.33
−1.1
−3.02
0
3
0
1.00


L708
24.76
20.99
17.57
0.33
−1.1
−6.38
0
3
0
1.00


L709
16.32
40.74
17.51
0.33
−1.1
−6.36
60
3
60
1.00


L710
11.24
10.74
2.34
0.33
−1.1
−0.86
60
3
60
1.00


L711
2.37
18.43
0.19
0.33
−1.1
−0.07
60
3
60
1.00


L712
2.31
31.79
0.31
0.33
−1.1
−0.11
60
3
60
1.00


L713
17.43
27.01
13.02
0.33
−1.1
−4.77
0
3
180
−1.00


L714
17.86
31.29
15.73
0.33
−1.1
−5.77
60
3
240
−1.00


L715
12.14
7.30
1.84
0.33
−1.1
−0.67
60
3
240
−1.00


L716
0.79
29.31
0.03
0.33
−1.1
−0.01
0
3
180
−1.00


L717
26.13
30.59
27.54
0.33
−1.1
−10.10
0
3
0
1.00


L718
33.53
14.96
17.64
0.33
−1.1
−6.47
60
3
60
1.00


L719
25.64
8.90
7.82
0.33
−1.1
−2.87
60
3
60
1.00


L720
18.57
7.41
3.98
0.33
−1.1
−1.46
60
3
60
1.00


L721
7.05
5.65
0.50
0.33
−1.1
−0.18
60
3
60
1.00


L722
10.96
6.76
1.40
0.33
−1.1
−0.51
60
3
240
−1.00


L723
12.62
6.07
1.64
0.33
−1.1
−0.60
0
3
180
−1.00


L724
0.29
19.14
0.00
0.33
−1.1
0.00
0
3
0
1.00


L725
4.50
19.05
0.70
0.33
−1.1
−0.26
0
3
0
1.00


L726
24.01
6.97
5.59
0.33
−1.1
−2.05
0
3
180
−1.00


L727
35.82
8.07
9.96
0.33
−1.1
−3.65
0
3
180
−1.00


L728
39.78
12.06
15.48
0.33
−1.1
−5.68
60
3
240
−1.00


L729
32.92
25.44
29.55
0.33
3.3
32.50
60
3
240
−1.00


L730
32.92
25.42
29.53
0.33
3.3
32.48
0
3
180
−1.00


Sum





0.16









The total optical retardation





  
    OR
    LG
  
  =
  
    
      ∑
      
        L701
        ⁢
        
          
        
        ⁢
        to
        ⁢
        
          
        
        ⁢
        L730
      
    
    ⁢
    
      BF
      L
    
  




of the lenses is +0.16 nm, thus smaller than 1% of the working wavelength. The total for the asymmetry of the optical retardation





  
    ORAS
    LG
  
  =
  
    
    
      
        ∑
        
          L701
          ⁢
          
            
          
          ⁢
          to
          ⁢
          
            
          
          ⁢
          L730
        
      
      ⁢
      
        (
        
          
            DS
            L
          
          ·
          
            AP
            L
          
          ·
          
            BF
            L
          
        
        )
      
    
    
  




is 0.53 nm, thus smaller than 1% of the working wavelength.

The lenses of the objective 701 are assigned to the six homogeneous groups HG71, HG72, HG73, HG74, HG75 and HG76 with calcium fluoride lenses and to the homogeneous group HG77 with barium fluoride lenses.
The lenses are assigned to the homogeneous groups as shown in Table 9.










TABLE 9





Homogeneous

Crystallographic orientation



group
Lenses
of the lens axis
ORAS [nm]







HG71
L701–L707
calcium fluoride
0.12


HG72
L708–L709
calcium fluoride
0.05


HG73
L710–L714
calcium fluoride
0.13


HG74
L715–L720
calcium fluoride
0.09


HG75
L721–L725
calcium fluoride
0.04


HG76
L726–L728
calcium fluoride
0.07


HG77
L729–L730
barium fluoride
0.02









The asymmetry





  
    ORAS
    
      HG
      n
    
  
  =
  
    
    
      
        ∑
        
          L
          ⁢
          
            
          
          ⁢
          in
          ⁢
          
            
          
          ⁢
          
            HG
            n
          
        
      
      ⁢
      
        (
        
          
            MA
            L
          
          ·
          
            DI
            L
          
          ·
          
            DS
            L
          
          ·
          
            AP
            L
          
          ·
          
            SP
            L
          
        
        )
      
    
    
  




of the optical retardation in all groups is smaller than 1% of the working wavelength.

In each of the homogeneous groups HG72, HG73, HG74, HG76 and HG77 there is one lens whose birefringence factor BFL is larger than 3% of the working wavelength λ0=157 nm.
FIG. 8 illustrates a fourth embodiment in which the invention was put into practice in an objective 801. The objective 801 is a catadioptric projection objective for a microlithography projection system. The optical data for the objective 801 are listed in Table 10. The image-side numerical aperture of the objective 801 is 0.9. The object field which is centered on the optical axis OA has a diameter of 106.4 mm. The central object point 803 is located on the optical axis OA, and a further object point 805 is located at the periphery of the object field. The imaging scale ratio of the objective 801 between the object plane OB and the image plane IM is 0.25. The imaging performance of the objective 801 is corrected to such a high degree that the deviation from the wave front of an ideal spherical wave for all object points of the object field is less than six thousandths of a wavelength in relation to a working wavelength of 157 nm, if only those aberrations are considered that are caused by the geometry of the optical components. The principle-ray-induced and coma-induced distortion for all image points is smaller than 3 nm.
Next to the object plane OB, the objective 801 has a first partial system 811 which contains the exclusively refractive optical components L801 and PE1, a beam-splitter device BSC, a second catadioptric partial system 813 with a concave mirror CM and a plurality of refractive optical components L802 to L804 and PE2, as well as a third partial system 815 with a beam-deflecting element MI and refractive optical components L805 to L820 and PE3.
The first partial system 811 includes the quarter-lambda plate PE1 as well as the lens L801.
The beam-splitter device BSC is configured as a beam-splitter cube composed of two prisms P1 and P2 with triangular cross-sectional shapes. A polarization-selective beam-splitting layer PSE which is interposed between the prisms is configured as a so-called S-P layer. This means that, ideally, the beam-splitting layer PSE reflects 100% of the electric field component that oscillates in the perpendicular direction to the plane of incidence of the light (S-component) while it transmits 100% of the electric field component that oscillates parallel to the plane of incidence of the light (P-component). Beam-splitting layers PSE of the S-P type which have been realized in practice come respectably close to these ideal values.
The quarter-lambda plate PE1 in the first partial system 811 has the purpose to change the circular polarization of the light that originates from the object, so that when the light meets the beam-splitting layer PSE, it has S-polarization as is necessary for reflection.
The light that was reflected by the beam-splitting layer PSE passes through the catadioptric partial system 813 and meets the beam-splitting layer a second time. A further quarter-lambda plate PE2 which is arranged in the catadioptric partial system 813 has the effect that the light meets the beam-splitting layer PSE with the required P-polarization so that it is transmitted by the beam-splitting layer PSE.
The light is reflected on a direction-changing mirror MI and passes through the lenses L805 to L820 and a further quarter-lambda plate PE3 to the image plane IM.
All refractive optical components of the objective 801 consist of calcium fluoride. Given that the objective has a working wavelength of 157 nm, the harmful influence of intrinsic birefringence needs to be reduced. Due to the configuration of the objective 801 as a catadioptric objective with a polarization-selective beam-splitting layer PSE, the partial systems 811, 813 and 815 are uncoupled from each other in regard to their polarizing properties. After it has been reflected or transmitted at the beam-splitting layer PSE, the light has a defined state of polarization, i.e., S-polarization in the case of reflection and P-polarization in the case of transmission. Other polarization components of the light falling on the beam-splitting layer are automatically filtered out. Light which is not S-polarized after passing through the first partial system 811 is absorbed or transmitted, while light which is not P-polarized after passing through the second catadioptric partial system 813 is absorbed or reflected.
It is therefore not possible to form homogeneous groups of lenses which are arranged in the light path ahead of and after the beam-splitting layer PSE. Each of the partial systems 811, 813 and 815 therefore has to be optimized by itself in regard to the birefringence-related image aberrations.
The lens axis of the lens L801 in the first partial system 811 is arranged parallel to the crystallographic (100)-direction.
In the second catadioptric partial system, the lens axis of the lens L802 is parallel to the crystallographic (100)-direction, while the lens axes of the lenses L803 and L804 are oriented parallel to the crystallographic (110)-direction. The lenses L803 and L804 form a homogeneous group HG85, wherein the lens L803 is set at an angle of rotation γL803=0° and the lens L804 is set at an angle of rotation γL804=90° relative to a reference direction that is fixed in the homogeneous group.
The most stringent requirements on the birefringence correction have to be imposed in the partial system 815, because the partial system 815 is arranged immediately ahead of the image plane IM and the harmful effect of the intrinsic birefringence of the lenses L805 to L820 directly reduces the imaging performance.
Following is a description of how the birefringence-related image aberrations are corrected in the third partial system 815 which is arranged between the polarization-selective optical element BSC and the image plane IM.
The reduction of the harmful effects of birefringence in the third partial system 815 is achieved by forming the homogeneous group HG84 of (100)-lenses and the homogeneous groups HG81 to HG83 of (111)-lenses. The lens axes of the lenses L805 to L817 are oriented in the crystallographic (111)-direction, and the lens axes of the lenses L818 to L820 are oriented in the crystallographic (100)-direction.
The essential characterizing data of the third partial system 815 according to the invention are presented in Table 11. The quantities that depend on ray parameters have been calculated for the outer aperture ray 807 with the relative ray aperture of 1.0.

















TABLE 11






θL
OPL
SPL

MAL
BFL
γL

αL



Lens
[°]
[mm]
[mm]
DIL
[nm/mm]
[nm]
[°]
SFDI—L
[°]
APL

























L805
7.94
16.32
1.83
0.33
−1.1
−0.67
0
3
0
1


L806
7.38
19.87
1.93
0.33
−1.1
−0.71
36.1
3
36.1
1


L807
6.49
29.15
2.20
0.33
−1.1
−0.81
78.8
3
78.8
1


L808
3.75
35.57
0.91
0.33
−1.1
−0.33
0
3
180
−1


L809
4.84
17.67
0.75
0.33
−1.1
−0.28
33.1
3
213.1
−1


L810
2.76
20.37
0.28
0.33
−1.1
−0.10
33.1
3
213.1
−1


L811
7.45
12.72
1.26
0.33
−1.1
−0.46
78
3
258
−1


PE3
1.98
10.01
0.07
0.33
−1.1
−0.03
80.2
3
260.2
−1


L812
4.80
30.66
1.28
0.33
−1.1
−0.47
0
3
180
−1


L813
0.66
5.83
0.00
0.33
−1.1
0.00
0
3
180
−1


L814
12.90
28.23
7.96
0.33
−1.1
−2.92
0
3
180
−1


L815
2.88
8.90
0.13
0.33
−1.1
−0.05
0
3
180
−1


L816
31.04
16.87
18.56
0.33
−1.1
−6.81
28.5
3
208.5
−1


L817
31.01
19.50
21.44
0.33
−1.1
−7.86
79.9
3
259.9
−1


L818
35.15
13.27
16.18
−0.5
−1.1
8.90
0
4
180
−1


L819
34.69
11.22
13.57
−0.5
−1.1
7.46
34.7
4
214.7
−1


L820
35.28
8.82
10.78
−0.5
−1.1
5.93
59.1
4
239.1
−1


Sum





0.81









The total optical retardation




  
    OR
    LG
  
  =
  
    
      ∑
      
        L805
        ⁢
        
          
        
        ⁢
        to
        ⁢
        
          
        
        ⁢
        L820
      
    
    ⁢
    
      BF
      L
    
  




for all of the lenses of the partial system 815 is +0.81 nm, thus smaller than 1% of the working wavelength.

The total for the asymmetry of the optical retardation




  
    ORAS
    LG
  
  =
  
    
    
      
        ∑
        
          L805
          ⁢
          
            
          
          ⁢
          to
          ⁢
          
            
          
          ⁢
          L820
        
      
      ⁢
      
        (
        
          
            DS
            L
          
          ·
          
            AP
            L
          
          ·
          
            BF
            L
          
        
        )
      
    
    
  




for all of the lenses of the partial system 815 is 0.06 nm, thus smaller than 1% of the working wavelength.

The lenses of the partial system 815 are assigned to the three homogeneous groups HG81, HG82 and HG93 with (111)-lenses and to the homogeneous group HG84 with (100)-lenses. The lenses are assigned to the homogeneous groups as shown in Table 12.










TABLE 12





Homogeneous

Crystallographic orientation



group
Lenses
of the lens axis
ORAS [nm]







HG81
L805–L807
(111)
0.002


HG82
L808–L811
(111)
0.003


HG83
L812–L817
(111)
0.012


HG84
L818–L820
(100)
0.043









The asymmetry




  
    ORAS
    
      HG
      n
    
  
  =
  
    
    
      
        ∑
        
          L
          ⁢
          
            
          
          ⁢
          in
          ⁢
          
            
          
          ⁢
          
            HG
            n
          
        
      
      ⁢
      
        (
        
          
            MA
            L
          
          ·
          
            DI
            L
          
          ·
          
            DS
            L
          
          ·
          
            AP
            L
          
          ·
          
            SP
            L
          
        
        )
      
    
    
  




of the optical retardation in all groups is smaller than 1% of the working wavelength.

In each of the homogeneous groups HG83 and HG84 there is one lens whose birefringence factor BFL is larger than 3% of the working wavelength λ0=157 nm.
In contrast to the first three examples where the (100)-lenses have angles of rotation of either γL=0° or γL=45° and the (111)-lenses have angles of rotation of either γL=0° or γL=60°, the fourth embodiment takes advantage of the fact that the asymmetry of the optical retardation ORAS of a homogeneous group can also be minimized through a specifically targeted adaptation of the angles of rotation γL.
The homogeneous group H84 will serve as an example to explain the concept of a targeted adaptation. The homogeneous group HG84 has the three lenses L818, L819 and L820.
The aim is for an asymmetry of the optical retardation




  
    ORAS
    
      HG
      84
    
  
  =
  
    
      
      
        
          ∑
          
            L
            ⁢
            
              
            
            ⁢
            in
            ⁢
            
              
            
            ⁢
            
              HG
              84
            
          
        
        ⁢
        
          (
          
            
              DS
              L
            
            ·
            
              AP
              L
            
            ·
            
              SP
              L
            
          
          )
        
      
      
    
    ⁢
    
      =
      !
    
    ⁢
    0.
  




As the lenses are (100)-lenses, the aperture factor APL equals +1 and the symmetry factor SFDI—L equals +4, independent of the azimuth angle αL of the outer aperture ray 807. Consequently, the preceding equation can be stated as:

ORASHG84=|exp(i·4·γL818)·SPL818+exp(i·4·γL819)·SPL819+exp(i·4·γ830)·SLL820|0

With the angle γL818=0° for the lens L818, one obtains for the two other lenses:




  
    
      
        γ
        L819
      
      =
      
        
          
            1
            4
          
          ·
          
            arccos
            ⁡
            
              (
              
                
                  
                    SP
                    L820
                    2
                  
                  -
                  
                    SP
                    L819
                    2
                  
                  -
                  
                    SP
                    L818
                    2
                  
                
                
                  2
                  ·
                  
                    SP
                    L818
                  
                  ·
                  
                    SP
                    L819
                  
                
              
              )
            
          
        
        =
        
          35
          ⁢
          °
        
      
    
    ,
    and
  




  
    
      γ
      L820
    
    =
    
      
        
          1
          4
        
        ·
        
          (
          
            
              arctan
              ⁡
              
                (
                
                  
                    
                      SP
                      L819
                    
                    ·
                    
                      sin
                      ⁡
                      
                        (
                        
                          4
                          ·
                          
                            γ
                            L819
                          
                        
                        )
                      
                    
                  
                  
                    
                      SP
                      L818
                    
                    +
                    
                      
                        SP
                        L819
                      
                      ·
                      
                        cos
                        ⁡
                        
                          (
                          
                            4
                            ·
                            
                              γ
                              L819
                            
                          
                          )
                        
                      
                    
                  
                
                )
              
            
            +
            
              180
              ⁢
              °
            
          
          )
        
      
      =
      
        59
        ⁢
        
          °
          .
        
      
    
  



On the other hand, it is impossible to form a homogeneous group with the lenses L818, L819 and L820 if the available angles of rotation are limited to 0° and 45°. This is a consequence of the following:




  
    
    
      
        ∑
        
          L
          ⁢
          
            
          
          ⁢
          in
          ⁢
          
            
          
          ⁢
          HG84
        
      
      ⁢
      
        (
        
          
            c
            L
          
          ·
          
            BF
            L
          
        
        )
      
    
    
  
  >
  0




for any combination of the coefficients cL=±1.













TABLE 13







cL818
cL819
cL820




  
    
      
        
          
          
            
              ∑
              
                L
                ⁢
                
                  
                
                ⁢
                in
                ⁢
                
                  
                
                ⁢
                HG84
              
            
            ⁢
            
              (
              
                
                  c
                  L
                
                ·
                
                  BF
                  L
                
              
              )
            
          
          
        
      
    
    
      
        
          [
          nm
          ]
        
      
    
  
   

























1
1
1
22.29



1
1
−1
10.43



1
−1
1
7.37



−1
1
1
4.49



1
−1
−1
4.49



−1
1
−1
7.37



−1
−1
1
10.43



−1
−1
−1
22.29










If the angles of rotation are limited to 0° and 45°, the minimum value that the asymmetry of the retardation can take is ORASLG=4.5 nm, which is larger than 2% of the working wavelength λ0.
The principal configuration of a microlithography projection system is illustrated in FIG. 9. The projection system 901 includes a light source 903, an illumination system 905, a mask 907 carrying a structure, a projection objective 909, and a substrate 911 that is to be exposed to the projection. The illumination system 905 collects the light of the light source 903—the latter being a KrF— or ArF laser depending on the working wavelength—and illuminates the mask 907. The light source 903 and the illumination system 905 provide a light distribution with a degree of homogeneity as required by the exposure process and a required lighting level at the entry pupil of the objective 909. The mask 907 is held in the light path by means of a mask holder 913. Masks 907 of the type used in microlithography have a structure with details in the micrometer-to-nanometer range. As an alternative to a so-called reticle, one could also use a program-controlled micro-mirror array or LCD array as a structure-carrying mask. An image of the mask 907 or of a partial surface range of the mask 907 is projected by the projection objective 909 onto the substrate 911 which is held in position by a substrate holder 915. Examples of actual designs of the projection objective 909 are shown in FIGS. 1, 6, 7 and 8. The substrate 911 is typically a silicon wafer which carries a light-sensitive coating, the so-called resist.
The minimal size of the structural details that can be optically resolved depends on the wavelength λ of the light that is used for the illumination and also on the image-side numerical aperture of the projection objective 909. The ultimately achievable resolution of the projection system 901 increases with shorter wavelengths λ of the light source 903 and with larger values of the image-side numerical aperture of the projection objective 909. With the objectives presented as examples in FIGS. 1, 6, 7 and 8, detail resolutions smaller than 150 nm can be achieved. It is therefore necessary to also minimize effects like the intrinsic birefringence. By providing a numerical optimizing method which not only corrects the classical image aberrations but also birefringence-related image aberrations already in the design phase, the invention has been successful in noticeably reducing the harmful effects of intrinsic birefringence particularly in projection objectives with a large numerical aperture on the image side.













TABLE 1









REFR. INDEX
1⁄2 FREE



SURFACE
RADII
THICKNESSES
MATERIAL
AT 157.63 nm
DIAMETER
LENS





















0
0.000000000
27.171000000
N2V157
1.00031429
46.200



1
0.000000000
0.619657764
N2V157
1.00031429
52.483


2
903.368459132AS
15.945973242
CAF2V157
1.55929034
53.245
L101


3
−226.803393202
8.774729563
N2V157
1.00031429
53.866


4
−163.038011767
8.803852757
CAF2V157
1.55929034
53.909
L102


5
−132.829604839
13.660124238
N2V157
1.00031429
54.704


6
−334.294132209
11.146309280
CAF2V157
1.55929034
53.671
L103


7
−155.654931629
16.012958308
N2V157
1.00031429
53.830


8
−73.621096309
7.898503828
CAF2V157
1.55929034
53.365
L104


9
−68.557324695AS
3.194701133
N2V157
1.00031429
54.820


10
−87.319562124AS
5.516496382
CAF2V157
1.55929034
52.130
L105


11
−223.001598894
4.902719287
N2V157
1.00031429
53.190


12
−156.862827901
5.476915177
CAF2V157
1.55929034
53.157
L106


13
155.322797951
34.113165766
N2V157
1.00031429
56.119


14
−92.560458488
5.019131834
CAF2V157
1.55929034
57.747
L107


15
3978.960899432AS
19.748553672
N2V157
1.00031429
73.712


16
−431.856884247
28.595253978
CAF2V157
1.55929034
86.500
L108


17
−141.788622554
0.750000000
N2V157
1.00031429
90.287


18
−4023.545513230AS
53.682797071
CAF2V157
1.55929034
113.819
L109


19
−198.268499421
1.506249020
N2V157
1.00031429
120.392


20
−391.620232439
21.637577380
CAF2V157
1.55929034
126.623
L110


21
−230.721564146
1.705183768
N2V157
1.00031429
127.866


22
727.797022829
27.725011822
CAF2V157
1.55929034
134.901
L111


23
1275.767906513
8.151260102
N2V157
1.00031429
135.253


24
1243.135878600
48.278536826
CAF2V157
1.55929034
135.967
L112


25
−284.760530410
2.485310208
N2V157
1.00031429
136.058


26
137.566103636
57.283374883
CAF2V157
1.55929034
107.395
L113


27
−5301.229434367AS
0.750000000
N2V157
1.00031429
103.559


28
2397.432274715
6.797107660
CAF2V157
1.55929034
101.025
L114


29
148.270665583
15.404593426
N2V157
1.00031429
83.008


30
274.187324420
6.600101122
CAF2V157
1.55929034
81.651
L115


31
182.196527770
32.159139227
N2V157
1.00031429
76.539


32
−221.531762691
5.000000000
CAF2V157
1.55929034
75.610
L116


33
101.456537540AS
57.213912935
N2V157
1.00031429
70.207


34
−106.788792156
5.626935721
CAF2V157
1.55929034
71.292
L117


35
1650.121069070
20.646988678
N2V157
1.00031429
87.893


36
−414.529202691
26.233453762
CAF2V157
1.55929034
95.039
L118


37
−204.791231248
0.751194338
N2V157
1.00031429
101.638


38
−652.140559964
25.729214428
CAF2V157
1.55929034
113.777
L119


39
−232.560945671
0.757093778
N2V157
1.00031429
116.017


40
−797.504020233
23.278583475
CAF2V157
1.55929034
125.215
L120


41
−295.378918809
0.750000000
N2V157
1.00031429
126.699


42
788.903405379
39.945207546
CAF2V157
1.55929034
136.621
L121


43
−429.239474965
11.617330294
N2V157
1.00031429
137.091


44
0.000000000
−7.643947086
N2V157
1.00031429
135.209


45
295.350327713AS
20.800474007
CAF2V157
1.55929034
136.602
L122


46
438.672121361
0.904568214
N2V157
1.00031429
135.313


47
352.144820344
48.858165030
CAF2V157
1.55929034
134.930
L123


48
−622.347578944
32.531844631
N2V157
1.00031429
133.556


49
−224.679062354
12.453352583
CAF2V157
1.55929034
131.448
L124


50
−251.422789558AS
16.098277935
N2V157
1.00031429
132.009


51
−194.298972775AS
15.943567731
CAF2V157
1.55929034
130.929
L125


52
−198.881082726
0.874791605
N2V157
1.00031429
133.010


53
204.783706668
19.816707893
CAF2V157
1.55929034
104.687
L126


54
329.789425103
0.940362287
N2V157
1.00031429
101.615


55
110.493410262
24.685776450
CAF2V157
1.55929034
87.597
L127


56
170.824363408
0.818849688
N2V157
1.00031429
84.672


57
118.591490343
31.724780430
CAF2V157
1.55929034
78.184
L128


58
919.618517702AS
7.153083120
N2V157
1.00031429
72.743


59
0.000000000
21.772860633
CAF2V157
1.55929034
66.406
L129


60
0.000000000
3.000000000
N2V157
1.00031429
50.979


61
0.000000000
21.595848862
CAF2V157
1.55929034
44.742
L130


62
0.000000000
8.604422398
N2V157
1.00031429
29.441


63
0.000000000
0.000009908

1.00000000
11.551









Formula for Aspherical Surfaces:




  
    z
    =
    
      
        
          
            1
            R
          
          ⁢
          
            h
            2
          
        
        
          1
          +
          
            
              1
              -
              
                
                  (
                  
                    1
                    +
                    K
                  
                  )
                
                ⁢
                
                  
                    (
                    
                      1
                      R
                    
                    )
                  
                  2
                
                ⁢
                
                  h
                  2
                
              
            
          
        
      
      +
      
        
          ∑
          
            k
            =
            1
          
        
        ⁢
        
          
            c
            k
          
          ⁢
          
            h
            
              
                2
                ⁢
                k
              
              +
              2
            
          
        
      
    
  




  
    
      
        
          z
          ⁢
          
            :
          
          ⁢
          
            
          
          ⁢
          sagittal
          ⁢
          
            
          
          ⁢
          height
        
      
      
        
          h
          ⁢
          
            :
          
          ⁢
          
            
          
          ⁢
          height
        
      
      
        
          R
          ⁢
          
            :
          
          ⁢
          
            
          
          ⁢
          radius
        
      
    
    
      
        
          k
          ⁢
          
            :
          
          ⁢
          
            
          
          ⁢
          conical
          ⁢
          
            
          
          ⁢
          constant
        
      
      
        
          
            c
            k
          
          ⁢
          
            :
          
          ⁢
          
            
          
          ⁢
          aspheric
          ⁢
          
            
          
          ⁢
          constants
        
      
      
        
          
        
      
    
  



Aspherical Constants:















SURFACE NO. 2
SURFACE NO. 9
SURFACE NO. 10















K
  0.0000
K
−1.3312
K
−1.1416


C1
  1.32190816e−007
C1
−4.03409633e−007
C1
  1.33648424e−007


C2
  1.18917598e−012
C2
  2.25345866e−011
C2
  1.59901344e−011


C3
  9.89822795e−017
C3
−2.16996631e−014
C3
−1.67348850e−014


C4
  2.42293194e−019
C4
  4.31434675e−018
C4
  3.68207274e−018


C5
−1.33402482e−022
C5
−7.88859611e−022
C5
−5.08498969e−022


C6
  3.58561252e−026
C6
  7.38357552e−026
C6
  9.85142914e−027


C7
−4.52438519e−030
C7
−6.76351468e−030
C7
  2.52500963e−030












SURFACE NO. 15
SURFACE NO. 18
SURFACE NO. 27















K
  0.0000
K
  0.0000
K
  0.0000


C1
  1.34648983e−007
C1
  6.83914897e−009
C1
  2.24804464e−008


C2
−2.19881265e−011
C2
−4.90773882e−013
C2
−4.42085557e−013


C3
  1.21097573e−015
C3
−7.12121614e−018
C3
−4.09221540e−017


C4
  4.34372877e−020
C4
−4.01052617e−022
C4
  4.34358300e−021


C5
−2.32460561e−023
C5
  2.86719325e−026
C5
−3.23260451e−025


C6
  2.73872082e−027
C6
  3.52889830e−031
C6
  1.32073344e−029


C7
−1.29386402e−031
C7
−4.92826156e−035
C7
−2.22573939e−034












SURFACE NO. 33
SURFACE NO. 45
SURFACE NO. 50















K
  0.0000
K
  0.0432
K
  0.0000


C1
−2.84926847e−008
C1
  7.18091870e−010
C1
  3.00490068e−010


C2
−1.11911567e−011
C2
−2.07564469e−014
C2
−2.29783717e−014


C3
−1.98673848e−016
C3
−6.90886458e−020
C3
−3.67108730e−019


C4
−1.64721030e−020
C4
  1.21545976e−024
C4
  1.62639442e−022


C5
−5.16269613e−024
C5
  1.92967617e−027
C5
−8.48026147e−027


C6
  9.67341069e−028
C6
−5.98854019e−032
C6
  2.19056910e−031


C7
−9.95326868e−032
C7
  2.74615721e−036
C7
−5.19843579e−036














SURFACE NO. 51
SURFACE NO. 58

















K
  0.0000
K
  0.0000



C1
−3.97920075e−010
C1
  4.41074602e−008



C2
  5.69350322e−014
C2
  1.63744260e−012



C3
  3.61452787e−018
C3
−5.56126851e−016



C4
−3.91454103e−023
C4
  6.15209417e−020



C5
−6.08597698e−027
C5
−4.00572791e−024



C6
  1.43804179e−031
C6
  1.31913491e−028



C7
−2.06445881e−036
C7
−7.75532664e−034























TABLE 4









REFR. INDEX
1⁄2 FREE



SURFACE
RADII
THICKNESSES
MATERIAL
AT 157.63 nm
DIAMETER
LENS





















0
0.000000000
27.171000000
N2V157
1.00031429
46.200



1
0.000000000
1.833271786
N2V157
1.00031429
52.489


2
783.892924679AS
12.563837338
CAF2V157
1.55928774
53.615
L601


3
−336.178598782
13.483853547
N2V157
1.00031429
54.059


4
−218.390059183
6.276663903
CAF2V157
1.55928774
54.534
L602


5
−171.176002650
16.023795053
N2V157
1.00031429
55.025


6
−823.461800230
13.872747800
CAF2V157
1.55928774
55.001
L603


7
−170.998932676
19.262379607
N2V157
1.00031429
55.155


8
−77.592079873
5.663422495
CAF2V157
1.55928774
54.129
L604


9
−73.368538462AS
0.750000000
N2V157
1.00031429
55.071


10
−90.155314860AS
8.095946137
CAF2V157
1.55928774
53.391
L605


11
−264.588184825
4.749021909
N2V157
1.00031429
55.096


12
−182.203588417
6.916852485
CAF2V157
1.55928774
55.160
L606


13
157.890718729
35.996521212
N2V157
1.00031429
58.722


14
−90.120717591
5.000000000
CAF2V157
1.55928774
59.783
L607


15
6089.901973469AS
19.262044202
N2V157
1.00031429
79.096


16
−411.048664379
30.748361980
CAF2V157
1.55928774
91.374
L608


17
−140.007509177
1.359795266
N2V157
1.00031429
94.872


18
−2454.195838756AS
58.056302567
CAF2V157
1.55928774
124.448
L609


19
−175.631024720
0.750000000
N2V157
1.00031429
127.844


20
−410.392028061
22.012182507
CAF2V157
1.55928774
136.898
L610


21
−289.170948659
0.750000000
N2V157
1.00031429
139.795


22
589.226039742
29.718069927
CAF2V157
1.55928774
149.822
L611


23
−1764.716438457
0.750000000
N2V157
1.00031429
149.885


24
−2091.165786491
39.679372288
CAF2V157
1.55928774
149.845
L612


25
−293.921040686
0.750000000
N2V157
1.00031429
149.889


26
141.074690739
59.683732997
CAF2V157
1.55928774
112.927
L613


27
1804.605848698AS
0.750000000
N2V157
1.00031429
107.252


28
808.544300607
12.982088851
CAF2V157
1.55928774
104.923
L614


29
160.839676617
9.023109975
N2V157
1.00031429
85.208


30
210.991861097
5.000000000
CAF2V157
1.55928774
83.898
L615


31
141.292536926
36.911717304
N2V157
1.00031429
77.686


32
−257.654928360
5.000000000
CAF2V157
1.55928774
76.651
L616


33
95.325935587AS
50.330906414
N2V157
1.00031429
69.151


34
−125.531518779
20.880133925
CAF2V157
1.55928774
69.569
L617


35
561.430837955
17.110748098
N2V157
1.00031429
87.312


36
−691.613901731
16.894726714
CAF2V157
1.55928774
90.907
L618


37
−238.199336305
0.750000000
N2V157
1.00031429
93.001


38
−554.106915935
29.955101210
CAF2V157
1.55928774
99.262
L619


39
−244.266487405
2.599916083
N2V157
1.00031429
105.597


40
−1623.632930866
46.979341235
CAF2V157
1.55928774
114.510
L620


41
−375.299861939
2.616280056
N2V157
1.00031429
122.751


42
602.812130427
47.262141123
CAF2V157
1.55928774
131.456
L621


43
−552.389590238
−2.631399003
N2V157
1.00031429
132.469


44
0.000000000
3.381399003
N2V157
1.00031429
131.890


45
270.247424481AS
32.690154914
CAF2V157
1.55928774
134.441
L622


46
1298.609020888
0.750000000
N2V157
1.00031429
133.808


47
882.110993726
29.603096927
CAF2V157
1.55928774
133.074
L623


48
−574.232533851
30.690878416
N2V157
1.00031429
132.370


49
−217.978483622
5.000000000
CAF2V157
1.55928774
131.415
L624


50
−240.865295878AS
16.295961279
N2V157
1.00031429
131.876


51
−186.443056187AS
13.165321161
CAF2V157
1.55928774
131.160
L625


52
−190.144960734
0.750000000
N2V157
1.00031429
133.154


53
255.761927404
15.864536301
CAF2V157
1.55928774
105.450
L626


54
500.272020192
0.750000000
N2V157
1.00031429
103.884


55
170.920660735
11.503993632
CAF2V157
1.55928774
94.653
L627


56
212.512004177
0.750000000
N2V157
1.00031429
92.402


57
96.902427286
43.472669787
CAF2V157
1.55928774
79.622
L628


58
1009.084489782AS
6.705862408
N2V157
1.00031429
74.819


59
0.000000000
24.951415814
CAF2V157
1.55928774
69.056
L629


60
0.000000000
3.000000000
N2V157
1.00031429
51.384


61
0.000000000
23.991545946
CAF2V157
1.55928774
45.154
L630


62
0.000000000
7.999939498
N2V157
1.00031429
28.161


63
0.000000000
0.000060502

1.00000000
11.552









Formula for Aspherical Surfaces:




  
    z
    =
    
      
        
          
            1
            R
          
          ⁢
          
            h
            2
          
        
        
          1
          +
          
            
              1
              -
              
                
                  (
                  
                    1
                    +
                    K
                  
                  )
                
                ⁢
                
                  
                    (
                    
                      1
                      R
                    
                    )
                  
                  2
                
                ⁢
                
                  h
                  2
                
              
            
          
        
      
      +
      
        
          ∑
          
            k
            =
            1
          
        
        ⁢
        
          
            c
            k
          
          ⁢
          
            h
            
              
                2
                ⁢
                k
              
              +
              2
            
          
        
      
    
  




  
    
      
        
          z
          ⁢
          
            :
          
          ⁢
          
            
          
          ⁢
          sagittal
          ⁢
          
            
          
          ⁢
          height
        
      
      
        
          h
          ⁢
          
            :
          
          ⁢
          
            
          
          ⁢
          height
        
      
      
        
          R
          ⁢
          
            :
          
          ⁢
          
            
          
          ⁢
          radius
        
      
    
    
      
        
          k
          ⁢
          
            :
          
          ⁢
          
            
          
          ⁢
          conical
          ⁢
          
            
          
          ⁢
          constant
        
      
      
        
          
            c
            k
          
          ⁢
          
            :
          
          ⁢
          
            
          
          ⁢
          aspheric
          ⁢
          
            
          
          ⁢
          constants
        
      
      
        
          
        
      
    
  



Aspherical Constants:















SURFACE NO. 2
SURFACE NO. 9
SURFACE NO. 10















K
  0.0000
K
−1.3746
K
−0.9533


C1
  1.42412806e−007
C1
−3.91462584e−007
C1
  1.01442979e−007


C2
  3.73474543e−013
C2
  2.46544593e−011
C2
  1.12967913e−011


C3
  2.54196922e−016
C3
−2.04805283e−014
C3
−1.60528229e−014


C4
  2.41381680e−020
C4
  4.41946133e−018
C4
  3.86881987e−018


C5
−3.53116720e−023
C5
−7.73696956e−022
C5
−5.92286133e−022


C6
  1.00285257e−026
C6
  5.61040821e−026
C6
−8.49115295e−027


C7
−1.43816000e−030
C7
−1.94057822e−031
C7
  1.37474277e−029












SURFACE NO. 15
SURFACE NO. 18
SURFACE NO. 27















K
  0.0000
K
  0.0000
K
  0.0000


C1
  1.35596315e−007
C1
  6.95701413e−009
C1
  2.24525214e−008


C2
−2.17108845e−011
C2
−5.35684131e−013
C2
−2.28054131e−013


C3
  1.10173478e−015
C3
−3.79501980e−018
C3
−2.89309559e−017


C4
  5.71322716e−020
C4
−3.23719916e−022
C4
  3.37316727e−021


C5
−2.18850275e−023
C5
  2.35866841e−026
C5
−3.33307440e−025


C6
  2.43291639e−027
C6
  5.05141936e−031
C6
  1.85894962e−029


C7
−1.08759646e−031
C7
−4.96033470e−035
C7
−3.99953115e−034












SURFACE NO. 33
SURFACE NO. 45
SURFACE NO. 50















K
  0.0000
K
−0.0114
K
  0.0000


C1
−2.49252058e−008
C1
  3.23760736e−010
C1
  1.35832451e−010


C2
−1.22122561e−011
C2
−5.68267268e−014
C2
−2.31285858e−014


C3
−5.59522075e−016
C3
−9.97094762e−019
C3
−3.70634585e−019


C4
−7.71755056e−020
C4
  1.41644139e−023
C4
  1.51653490e−022


C5
  5.15952342e−024
C5
  2.51891961e−027
C5
−5.78380400e−027


C6
−3.42583935e−028
C6
−1.67433921e−031
C6
  3.18449388e−031


C7
−1.14822731e−031
C7
  8.82789504e−036
C7
−1.23237976e−035














SURFACE NO. 51
SURFACE NO. 58

















K
  0.0000
K
  0.0000



C1
−2.57897816e−010
C1
  3.81430115e−008



C2
  6.04192858e−014
C2
  1.17273705e−012



C3
  4.22250731e−018
C3
−4.86017937e−016



C4
−2.38756301e−023
C4
  7.29496704e−020



C5
−6.77890210e−027
C5
−8.78403273e−024



C6
  1.05133445e−031
C6
  7.31555552e−028



C7
−1.84957630e−036
C7
−2.78651979e−032























TABLE 7









REFR. INDEX
1⁄2 FREE



SURFACE
RADII
THICKNESSES
MATERIAL
AT 157.63 nm
DIAMETER
LENS





















0
0.000000000
27.171000000
N2V157
1.00031429
46.200



1
0.000000000
0.000016882
N2V157
1.00031429
52.483


2
1100.049969943AS
13.835460517
CAF2V157
1.55929034
53.033
L701


3
−233.539657189
10.675240307
N2V157
1.00031429
53.534


4
−160.949399690
6.811295343
CAF2V157
1.55929034
53.674
L702


5
−128.254251442
12.843572407
N2V157
1.00031429
54.221


6
−288.287586108
12.129185515
CAF2V157
1.55929034
53.355
L703


7
−151.697146334
15.111929473
N2V157
1.00031429
53.645


8
−73.535662368
8.406531732
CAF2V157
1.55929034
53.333
L704


9
−68.884753499AS
3.691428280
N2V157
1.00031429
54.952


10
−87.410675872AS
5.073239609
CAF2V157
1.55929034
52.419
L705


11
−163.383192938
4.364422126
N2V157
1.00031429
53.327


12
−130.774825158
5.213204736
CAF2V157
1.55929034
53.205
L706


13
157.646556769
32.604386473
N2V157
1.00031429
56.407


14
−93.798275929
5.000000000
CAF2V157
1.55929034
57.566
L707


15
3368.901715215AS
19.800420879
N2V157
1.00031429
73.098


16
−454.955938310
28.589092202
CAF2V157
1.55929034
86.378
L708


17
−145.377556116
1.198146349
N2V157
1.00031429
90.432


18
−3181.938740937AS
55.253140410
CAF2V157
1.55929034
113.328
L709


19
−180.727077136
2.634998921
N2V157
1.00031429
118.959


20
−417.348137567
14.876175782
CAF2V157
1.55929034
125.828
L710


21
−280.965238275
5.601765271
N2V157
1.00031429
126.912


22
613.501207577
24.948281153
CAF2V157
1.55929034
134.461
L711


23
−4575.596777639
19.096198360
N2V157
1.00031429
134.559


24
−5577.600844823
44.584300088
CAF2V157
1.55929034
134.610
L712


25
−286.639122553
6.672038356
N2V157
1.00031429
134.739


26
140.838600122
52.023094126
CAF2V157
1.55929034
106.139
L713


27
−3009.300734493AS
0.750000000
N2V157
1.00031429
103.867


28
3005.251076354
15.763320827
CAF2V157
1.55929034
101.364
L714


29
139.656974557
17.052540516
N2V157
1.00031429
79.293


30
322.637994941
5.000000000
CAF2V157
1.55929034
78.279
L715


31
226.114107896
28.004102435
N2V157
1.00031429
74.948


32
−214.030514154
5.000000000
CAF2V157
1.55929034
74.063
L716


33
101.747356704AS
52.823748660
N2V157
1.00031429
69.305


34
−108.078160969
5.134531894
CAF2V157
1.55929034
69.902
L717


35
1340.722540577
20.639215645
N2V157
1.00031429
85.380


36
−403.094336268
27.036189958
CAF2V157
1.55929034
92.392
L718


37
−198.324297859
1.185943036
N2V157
1.00031429
99.441


38
−805.161246913
26.726916281
CAF2V157
1.55929034
112.885
L719


39
−252.128101241
1.573121915
N2V157
1.00031429
115.824


40
−1018.605718732
24.715209484
CAF2V157
1.55929034
125.021
L720


41
−317.737201000
2.188933650
N2V157
1.00031429
126.852


42
911.036061215
39.030703244
CAF2V157
1.55929034
136.017
L721


43
−397.514153561
8.829809473
N2V157
1.00031429
136.445


44
0.000000000
−8.059959988
N2V157
1.00031429
134.458


45
283.117650723AS
19.554556520
CAF2V157
1.55929034
135.792
L722


46
427.297926263
0.762082509
N2V157
1.00031429
134.749


47
343.801566914
45.281345111
CAF2V157
1.55929034
134.302
L723


48
−674.258075736
30.848389732
N2V157
1.00031429
133.402


49
−229.146438889
14.057444137
CAF2V157
1.55929034
132.464
L724


50
−256.824756474AS
16.154638811
N2V157
1.00031429
132.944


51
−201.324104340AS
19.005515015
CAF2V157
1.55929034
131.210
L725


52
−206.399552372
2.818955115
N2V157
1.00031429
133.337


53
209.927060432
17.433578142
CAF2V157
1.55929034
104.967
L726


54
342.745670850
1.853424300
N2V157
1.00031429
102.777


55
109.865402846
24.686242054
CAF2V157
1.55929034
87.784
L727


56
167.188686248
1.175052020
N2V157
1.00031429
84.827


57
114.740922142
31.059347722
CAF2V157
1.55929034
77.782
L728


58
774.185357516AS
6.852711919
N2V157
1.00031429
73.137


59
0.000000000
21.358665091
BAF2V157
1.65671094
68.663
L729


60
0.000000000
4.000000000
N2V157
1.00031429
54.812


61
0.000000000
21.341978310
BAF2V157
1.65671094
46.497
L730


62
0.000000000
10.153177731
N2V157
1.00031429
32.657


63
0.000000000
0.000003439

1.00000000
11.552









Formula for Aspherical Surfaces:




  
    z
    =
    
      
        
          
            1
            R
          
          ⁢
          
            h
            2
          
        
        
          1
          +
          
            
              1
              -
              
                
                  (
                  
                    1
                    +
                    K
                  
                  )
                
                ⁢
                
                  
                    (
                    
                      1
                      R
                    
                    )
                  
                  2
                
                ⁢
                
                  h
                  2
                
              
            
          
        
      
      +
      
        
          ∑
          
            k
            =
            1
          
        
        ⁢
        
          
            c
            k
          
          ⁢
          
            h
            
              
                2
                ⁢
                k
              
              +
              2
            
          
        
      
    
  




  
    
      
        
          z
          ⁢
          
            :
          
          ⁢
          
            
          
          ⁢
          sagittal
          ⁢
          
            
          
          ⁢
          height
        
      
      
        
          h
          ⁢
          
            :
          
          ⁢
          
            
          
          ⁢
          height
        
      
      
        
          R
          ⁢
          
            :
          
          ⁢
          
            
          
          ⁢
          radius
        
      
    
    
      
        
          k
          ⁢
          
            :
          
          ⁢
          
            
          
          ⁢
          conical
          ⁢
          
            
          
          ⁢
          constant
        
      
      
        
          
            c
            k
          
          ⁢
          
            :
          
          ⁢
          
            
          
          ⁢
          aspheric
          ⁢
          
            
          
          ⁢
          constants
        
      
      
        
          
        
      
    
  



Aspherical Constants:















SURFACE NO. 2
SURFACE NO. 9
SURFACE NO. 10















K
  0.0000
K
−1.3248
K
−1.1638


C1
  1.33150089e−007
C1
−4.04298090e−007
C1
  1.35760045e−007


C2
  1.13763804e−012
C2
  2.23884711e−011
C2
  1.74560325e−011


C3
  1.85005495e−016
C3
−2.13768099e−014
C3
−1.69784908e−014


C4
  1.89547123e−019
C4
  4.32945447e−018
C4
  3.92332145e−018


C5
−9.08656138e−023
C5
−7.67218945e−022
C5
−5.24149627e−022


C6
  2.18841677e−026
C6
  6.43225430e−026
C6
  1.80008094e−026


C7
−2.61083456e−030
C7
−5.59790324e−030
C7
  2.11070207e−031












SURFACE NO. 15
SURFACE NO. 18
SURFACE NO. 27















K
  0.0000
K
  0.0000
K
  0.0000


C1
  1.33720710e−007
C1
  6.96593187e−009
C1
  2.26412640e−008


C2
−2.21640015e−011
C2
−4.82835354e−013
C2
−3.47526105e−013


C3
  1.24670453e−015
C3
−5.25586943e−018
C3
−3.48221622e−017


C4
  4.68847161e−020
C4
−2.62033681e−022
C4
  4.13095973e−021


C5
−2.27314972e−023
C5
  2.45848299e−026
C5
−3.11442082e−025


C6
  2.64456225e−027
C6
  1.39224945e−031
C6
  1.49413133e−029


C7
−1.33093602e−031
C7
−4.77826351e−035
C7
−3.33941866e−034












SURFACE NO. 33
SURFACE NO. 45
SURFACE NO. 50















K
  0.0000
K
  0.0487
K
  0.0000


C1
−3.12818340e−008
C1
  8.27632088e−010
C1
  3.00931833e−010


C2
−1.18120980e−011
C2
−3.11431326e−014
C2
−2.08657685e−014


C3
−1.57343342e−016
C3
−9.63321269e−020
C3
  2.09222892e−020


C4
−2.85090644e−020
C4
  2.68685704e−024
C4
  1.34320349e−022


C5
−1.86083655e−024
C5
  1.77884350e−027
C5
−9.00430590e−027


C6
  2.75973692e−028
C6
−4.64279136e−032
C6
  2.84671618e−031


C7
−5.35892988e−032
C7
  2.28573552e−036
C7
−7.19510437e−036














SURFACE NO. 51
SURFACE NO. 58

















K
  0.0000
K
  0.0000



C1
−3.98465484e−010
C1
  4.37724949e−008



C2
  5.51545340e−014
C2
  1.43469905e−012



C3
  3.21278977e−018
C3
−4.92932980e−016



C4
−1.77115729e−023
C4
  5.42353817e−020



C5
−5.93690297e−027
C5
−3.34429725e−024



C6
  9.63915129e−032
C6
  7.19285978e−029



C7
−1.69407739e−036
C7
  2.37322857e−033























TABLE 10









REFR. INDEX
1⁄2 FREE



SURFACE
RADII
THICKNESSES
MATERIAL
AT 157.63 nm
DIAMETER
LENS





















0
0.000000000
38.000000000

1.00000000
53.151



1
0.000000000
0.000000000

1.00000000
61.364


2
0.000000000
10.000000000
CAF2HL
1.55840983
61.364
PE1


3
0.000000000
1.300000002

1.00000000
62.732


4
231.856805366
18.451814820
CAF2HL
1.55840983
65.024
L801


5
−67509.737986018
50.437190129

1.00000000
65.268


6
0.000000000
48.000000000
CAF2HL
1.55840983
67.346
P1


7
0.000000000
0.000000000
CAF2HL
1.55840983
68.613
P1


8
0.000000000
57.000000000
CAF2HL
1.55840983
68.613
P1


9
0.000000000
100.905000471

1.00000000
70.119


10
−313.089957435
10.000000000
CAF2HL
1.55840983
73.909
L802


11
−1163.721587893AS
24.741906635

1.00000000
75.819


12
0.000000000
10.000000000
CAF2HL
1.55840983
79.952
PE2


13
0.000000000
107.838657624

1.00000000
80.940


14
−230.899575687
15.000000000
CAF2HL
1.55840983
94.528
L803


15
−539.422130350
32.081825253

1.00000000
101.113


16
−196.716565694
15.000000000
CAF2HL
1.55840983
104.520
L804


17
−562.409459022AS
0.000000000

1.00000000
117.228


18
0.000000000
46.300616489

1.00000000
124.128


19
0.000000000
0.000000000

−1.00000000
156.068
REFL


20
247.087595697
46.300616489

1.00000000
129.897
REFL


21
0.000000000
0.000000000

1.00000000
124.128


22
562.409459022AS
15.000000000
CAF2HL
1.55840983
117.300
L804


23
196.716565694
32.081825253

1.00000000
104.670


24
539.422130350
15.000000000
CAF2HL
1.55840983
101.344
L803


25
230.899575687
107.838657624

1.00000000
94.823


26
0.000000000
10.000000000
CAF2HL
1.55840983
81.789
PE2


27
0.000000000
24.741906635

1.00000000
80.839


28
1163.721587893AS
10.000000000
CAF2HL
1.55840983
76.861
L802


29
313.089957435
100.905000471

1.00000000
74.991


30
0.000000000
105.000000000
CAF2HL
1.55840983
71.949
P1 + P2


31
0.000000000
36.835344000

1.00000000
69.716


32
0.000000000
0.000000000

1.00000000
68.495


33
0.000000000
50.000000000

1.00000000
68.495


34
213.488486974
16.533569876
CAF2HL
1.55840983
66.486
L805


35
587.420501402
25.000000008

1.00000000
65.178


36
0.000000000
53.534353960

1.00000000
61.944


37
721.619844096AS
19.668137773
CAF2HL
1.55840983
68.895
L806


38
439.459785709
92.812246323

1.00000000
71.582


39
466.205048699
30.401957151
CAF2HL
1.55840983
99.911
L807


40
−643.543871149AS
1.300000004

1.00000000
101.061


41
353.787925634
37.860558838
CAF2HL
1.55840983
102.798
L808


42
−496.630877032
103.286912892

1.00000000
102.044


43
−330.182777990
12.500000000
CAF2HL
1.55840983
78.570
L809


44
307.592249925AS
139.746687714

1.00000000
76.966


45
167.227205765
15.000000000
CAF2HL
1.55840983
90.214
L810


46
129.347785505
38.119964661

1.00000000
86.513


47
348.724479670
33.786518873
CAF2HL
1.55840983
90.897
L811


48
−322.472540992
1.308803146

1.00000000
91.396


49
0.000000000
10.000000000
CAF2HL
1.55840983
90.322
PE3


50
0.000000000
1.300000001

1.00000000
89.848


51
229.767680473AS
15.000000000
CAF2HL
1.55840983
88.591
L812


52
130.158437409
37.918545597

1.00000000
83.578


53
182.557769223
38.781957950
CAF2HL
1.55840983
91.097
L813


54
−1723.770535736
2.298748296

1.00000000
91.468


55
172.070308994AS
20.000000000
CAF2HL
1.55840983
92.034
L814


56
137.116503243
26.094678574

1.00000000
87.328


57
267.312915741
46.319407231
CAF2HL
1.55840983
89.247
L815


58
−218.724324573
9.877811439

1.00000000
89.123


59
94.463410833
35.776202355
CAF2HL
1.55840983
71.060
L816


60
221.924345592
2.207032267

1.00000000
63.953


61
216.844481669
23.001210787
CAF2HL
1.55840983
62.520
L817


62
685.044556108
2.652159904

1.00000000
53.800


63
148.223416916AS
14.400000000
CAF2HL
1.55840983
46.812
L818


64
321.611439620
2.000000000

1.00000000
40.243


65
349.600819610
10.800000000
CAF2HL
1.55840983
38.412
L819


66
−1602.590366624
1.300000000

1.00000000
33.175


67
0.000000000
7.200000000
CAF2HL
1.55840983
30.577
L820


68
0.000000000
8.000000000

1.00000000
25.931


69
0.000000000
0.000000000

1.00000000
13.288









Formula for Aspherical Surfaces:




  
    z
    =
    
      
        
          
            1
            R
          
          ⁢
          
            h
            2
          
        
        
          1
          +
          
            
              1
              -
              
                
                  (
                  
                    1
                    +
                    K
                  
                  )
                
                ⁢
                
                  
                    (
                    
                      1
                      R
                    
                    )
                  
                  2
                
                ⁢
                
                  h
                  2
                
              
            
          
        
      
      +
      
        
          ∑
          
            k
            =
            1
          
        
        ⁢
        
          
            C
            k
          
          ⁢
          
            h
            
              
                2
                ⁢
                k
              
              +
              2
            
          
        
      
    
  




  
    
      
        
          z
          :
          
            sagittal
            ⁢
            
              
            
            ⁢
            height
          
        
      
      
        
          h
          :
          height
        
      
      
        
          R
          :
          radius
        
      
    
    
      
        
          k
          :
          
            conical
            ⁢
            
              
            
            ⁢
            constant
          
        
      
      
        
          
            c
            k
          
          :
          
            aspheric
            ⁢
            
              
            
            ⁢
            constants
          
        
      
      
        
          
        
      
    
  



Aspherical Constants:















SURFACE NO. 11
SURFACE NO. 17
SURFACE NO. 22















K
  0.0000
K
  0.0000
K
  0.0000


C1
  1.89276799e−008
C1
−9.87667098e−009
C1
  9.87667098e−009


C2
−2.13543901e−013
C2
  1.55529229e−013
C2
−1.55529229e−013


C3
−1.11902161e−017
C3
−3.40988793e−019
C3
  3.40988793e−019


C4
−6.59518395e−022
C4
  7.86247628e−023
C4
−7.86247628e−023


C5
−1.30641213e−025
C5
  4.44084385e−027
C5
−4.44084385e−027












SURFACE NO. 28
SURFACE NO. 37
SURFACE NO. 40















K
  0.0000
K
  0.0000
K
  0.0000


C1
−1.89276799e−008
C1
−2.93715143e−008
C1
  2.54066895e−009


C2
  2.13543901e−013
C2
−2.77826255e−013
C2
  1.03623324e−013


C3
  1.11902161e−017
C3
  1.23346186e−018
C3
  3.44744026e−018


C4
  6.59518395e−022
C4
  1.23771524e−022
C4
  1.79842868e−023


C5
  1.30641213e−025
C5
−6.60110415e−027
C5
−3.76706748e−028












SURFACE NO. 44
SURFACE NO. 51
SURFACE NO. 55















K
  0.0000
K
  0.0000
K
  0.0000


C1
  2.02333456e−008
C1
−3.50173472e−008
C1
−3.47775090e−008


C2
−1.66963894e−013
C2
−8.25470279e−013
C2
  6.64588433e−013


C3
−2.27086237e−017
C3
  7.73468257e−017
C3
  4.15885969e−017


C4
−9.37730881e−022
C4
  3.00183277e−021
C4
−4.05856715e−021


C5
  9.33708378e−027
C5
  4.58314788e−025
C5
−2.48686531e−025













SURFACE NO. 63















K
  0.0000



C1
−1.25682736e−007



C2
−2.94013947e−011



C3
−9.23361808e−016



C4
−6.75007296e−019



C5
  1.59347903e−022

## Claims

1. A numerical optimizing method for determining optical data of an objective, wherein said optical data are determined so as to reduce harmful effects caused by intrinsic birefringence in lenses of the objective which comprise a fluoride crystal material of cubic crystal structure, said method comprising:
selecting an optimizing function which is to be minimized by the numerical optimizing method, wherein said optimizing function takes into account at least one birefringence-related image aberration which is determined from a calculation for a light ray passing through said fluoride crystal lenses, and wherein to the extent that said birefringence-related image aberration is a function of parameters of the light ray, said birefringence-related image aberration depends only on geometric parameters of the light ray; and
minimizing said optimizing function by varying said optical data.

2. The method of claim 1, wherein the light ray is an outer aperture ray which in each of the fluoride crystal lenses has a respective ray path length OPL and a respective aperture angle θL, and wherein the birefringence-related image aberration depends on the respective ray path lengths OPL and the respective aperture angles θL in the fluoride crystal lenses.

3. The method of claim 1, wherein the light ray is an outer aperture ray which in each of the fluoride crystal lenses has a respective azimuth angle αL, and wherein the birefringence-related image aberration depends on the respective azimuth angles αL in the fluoride crystal lenses.

4. The method of claim 1, wherein the birefringence-related image aberration depends on respective material factors which are characteristic for the fluoride crystal material in each of the fluoride crystal lenses.

5. The method of claim 1, wherein the birefringence-related image aberration depends on respective orientation factors which are characteristic for a crystallographic orientation of the fluoride crystal material in each of the fluoride crystal lenses.

6. The method of claim 2, wherein the birefringence-related image aberration depends on respective ray factors SPL of each of the fluoride crystal lenses, wherein the ray factor SPL of a fluoride crystal lens is defined as SPL=OPL·sin2 θL·(719 cos2 θL−1).

7. The method of claim 6, wherein the birefringence-related image aberration depends on respective birefringence factors BFL of each of the fluoride crystal lenses, wherein
the birefringence factor BFL=MAL·DIL·SPL is defined as the product of a material factor MAL, an orientation factor DIL and the ray factor SPL,
the material factor MAL for a fluoride crystal lens indicates the intrinsic birefringence of the lens material for a ray that propagates in the crystallographic (110)-direction,
the fluoride crystal lenses have lens axes, and
for a fluoride crystal lens whose lens axis is oriented in the crystallographic (100)-direction, the orientation factor DIL equals −1⁄2, for a fluoride crystal lens whose lens axis runs in the crystallographic (111)-direction, DIL equals +1⁄3, and for a fluoride crystal lens whose lens axis is oriented in the crystallographic (110)-direction, DIL equals +1⁄8.

8. The method of claim 7, wherein said at least one birefringence-related image aberration comprises an optical retardation ORLG of a group of fluoride crystal lenses, and wherein the optical retardation




  
    OR
    LG
  
  =
  
    
      ∑
      
        L
        ⁢
        
          
        
        ⁢
        in
        ⁢
        
          
        
        ⁢
        LG
      
    
    ⁢
    
      BF
      L
    
  




is defined as the sum of the birefringence factors BFL of the fluoride crystal lenses of the group.

9. The method of claim 7, wherein the birefringence-related image aberration comprises an optical retardation asymmetry ORASLG of a group of fluoride crystal lenses, and wherein




  
    ORAS
    LG
  
  =
  
    
    
      
        ∑
        
          L
          ⁢
          
            
          
          ⁢
          in
          ⁢
          
            
          
          ⁢
          LG
        
      
      ⁢
      
        (
        
          
            DS
            L
          
          ·
          
            AP
            L
          
          ·
          
            BF
            L
          
        
        )
      
    
    
  




is defined as the absolute amount of the sum of the respective products of a complex rotary position factor DSL with an azimuth factor APL and the birefringence factor BFL for each of the fluoride crystal lenses of the group,
the angular position factor DSL=exp(i·SFDI—L·γL) is defined as a phase factor which depends on a symmetry factor SFDI—L and on an angle of rotation γL,
for fluoride crystal lenses whose lens axis is oriented in the crystallographic (100)-direction, the symmetry factor SFDI—L equals 4 and the azimuth factor APL equals +1,
for fluoride crystal lenses whose lens axis is oriented in the crystallographic (111)-direction, the symmetry factor SFDI—L equals 3 and the azimuth factor APL depends on a respective azimuth angle αL of an outer aperture ray in the respective fluoride crystal lens, where APL=+1 for 0°≦αL<180°, and APL=−1 for 180°≦αL<360°,
for fluoride crystal lenses whose lens axis is oriented in the crystallographic (110)-direction, the symmetry factor SFDI—L equals 2 and the azimuth factor APL equals +1, and
the angle of rotation γL is defined as the angle between a reference direction of the fluoride crystal lens which has been rotated about the lens axis and a reference direction of the group.

10. The method of claim 9, wherein
the fluoride crystal lenses of a group are at the same time assigned to homogeneous groups,
the fluoride crystal lenses within each homogeneous group have the same material factor MAL and the same orientation factor DIL,
the respective products MAL·DIL for fluoride crystal lenses from at least two different homogeneous groups have opposite signs, and
the optimizing function is minimized by taking into account an optical retardation ORLG of the group and the respective asymmetries ORASHG of the homogeneous groups,
wherein the optical retardation





  
    OR
    LG
  
  =
  
    
      ∑
      
        L
        ⁢
        
          
        
        ⁢
        in
        ⁢
        
          
        
        ⁢
        LG
      
    
    ⁢
    
      BF
      L
    
  



 is defined as the sum of the birefringence factors BFL of the fluoride crystal lenses of the group.

11. An objective for a microlithography projection system with lenses of fluoride crystal material with a cubic crystal structure, wherein
each fluoride crystal lens has a lens axis that is oriented approximately parallel to one of the crystallographic (100)-direction, the crystallographic (111)-direction, and the crystallographic (110)-direction,
if the objective includes no polarization-selective optical element (PSE), all of the fluoride crystal lenses that are arranged between an object plane (OB) and an image plane (IM) form a group of fluoride crystal lenses,
if the objective includes a polarization-selective optical element (PSE), all of the fluoride crystal lenses that are arranged between the polarization-selective optical element (PSE) and the image plane (IM) form a group of fluoride crystal lenses,
an optical retardation ORLG of the group of fluoride crystal lenses is smaller than 5% of a working wavelength λ0,
the optical retardation ORLG of a group of fluoride crystal lenses is defined as the sum of birefringence factors BFL of the fluoride crystal lenses of the group,
the birefringence factor BFL of a fluoride crystal lens is defined as BFL=MAL·DIL·SPL,
the material factor MAL indicates the intrinsic birefringence of the lens material for a ray that propagates in the crystallographic (110)-direction,
the orientation factor DIL has a value of −1⁄2 for a fluoride crystal lens whose lens axis is oriented in the crystallographic (100)-direction, +1⁄3 for a fluoride crystal lens whose lens axis is oriented in the crystallographic (111)-direction, and +1⁄8 for a fluoride crystal lens whose lens axis is oriented in the crystallographic (110)-direction,
the ray factor SPL is defined as
 SPL=OPL·sin2 θL·(7·cos2 θL−1) and depends on a respective aperture angle θL and ray path length OPL of an outer aperture ray in each of the fluoride crystal lenses.

12. The objective of claim 11, wherein said optical retardation ORLG of the group of fluoride crystal lenses is smaller than 2% of the working wavelength λ0.

13. The objective of claim 12, wherein said optical retardation ORLG of the group of fluoride crystal lenses is smaller than 1% of the working wavelength λ0.

14. The objective of claim 11, wherein said optical retardation ORLG of the group of fluoride crystal lenses for an outer aperture ray with a relative ray aperture between 0.75 and 1.0 as well as for an outer aperture ray with a relative ray aperture between 0.5 and 0.75 is smaller than 10% of a working wavelength λ0, said relative ray aperture being the ratio between a ray aperture and the numerical aperture in an image plane (IM).

15. The objective of claim 14, wherein said optical retardation ORLG of the group of fluoride crystal lenses for an outer aperture ray with a relative ray aperture between 0.75 and 1.0 as well as for an outer aperture ray with a relative ray aperture between 0.5 and 0.75 is smaller than 5% of a working wavelength λ0.

16. An objective for a microlithography projection system with lenses of fluoride crystal material with a cubic crystal structure, wherein
each fluoride crystal lens has a lens axis that is oriented approximately parallel to one of the crystallographic (100)-direction, the crystallographic (111)-direction, and the crystallographic (110)-direction,
if the objective includes no polarization-selective optical element (PSE), all of the fluoride crystal lenses that are arranged between an object plane (OB) and an image plane (IM) form a group of fluoride crystal lenses,
if the objective includes a polarization-selective optical element (PSE), all of the fluoride crystal lenses that are arranged between the polarization-selective optical element (PSE) and the image plane (IM) form a group of fluoride crystal lenses,
an optical retardation asymmetry ORASLG of the group of fluoride crystal lenses is smaller than 10% of a working wavelength λ0,
the optical retardation asymmetry





  
    ORAS
    LG
  
  =
  
    
    
      
        ∑
        
          L
          ⁢
          
            
          
          ⁢
          in
          ⁢
          
            
          
          ⁢
          LG
        
      
      ⁢
      
        (
        
          
            DS
            L
          
          ·
          
            AP
            L
          
          ·
          
            BF
            L
          
        
        )
      
    
    
  



 is defined as the absolute amount of the sum of the respective products of a complex rotary position factor DSL with an azimuth factor APL and the birefringence factor BFL for each of the fluoride crystal lenses of the group,
the birefringence factor BFL of a fluoride crystal lens is defined as BFL=MAL·DIL·SPL,
the material factor MAL indicates the intrinsic birefringence of the lens material for a ray that is oriented in the crystallographic (110)-direction,
the orientation factor DIL has a value of −1⁄2 for a fluoride crystal lens whose lens axis is oriented in the crystallographic (100)-direction, +1⁄3 for a fluoride crystal lens whose lens axis is oriented in the crystallographic (111)-direction, and +1⁄8 for a fluoride crystal lens whose lens axis is oriented in the crystallographic (110)-direction,
the ray factor SPL is defined as
 SPL=OPL·sin2 θL·(7·cos2 θL−1) and depends on a respective aperture angle θL and ray path length OPL of an outer aperture ray in each of the fluoride crystal lenses,
the angular position factor DSL=exp(i·SFDI—L·γL) is defined as a phase factor which depends on a symmetry factor SFDI—L and on an angle of rotation γL,
for fluoride crystal lenses whose lens axis is oriented in the crystallographic (100)-direction, the symmetry factor SFDI—L equals 4 and the azimuth factor APL equals +1,
for fluoride crystal lenses whose lens axis is oriented in the crystallographic (111)-direction, the symmetry factor SFDI—L equals 3 and the azimuth factor APL depends on a respective azimuth angle αL of an outer aperture ray in the respective fluoride crystal lens, where APL=+1 for 0°≦αL<180°, and APL=−1 for 180°≦αL<360°,
for fluoride crystal lenses whose lens axis is oriented in the crystallographic (110)-direction, the symmetry factor SFDI—L equals 2 and the azimuth factor APL equals +1, and
the angle of rotation γL is defined as the angle between a reference direction of the fluoride crystal lens which has been rotated about the lens axis and a reference direction of the group.

17. The objective of claim 16, wherein the optical retardation asymmetry ORASLG of the group of fluoride crystal lenses is smaller than 5% of a working wavelength λ0.

18. The objective of claim 16, wherein the absolute amount of the optical retardation asymmetry ORASLG of the group of fluoride crystal lenses for an outer aperture ray with a relative ray aperture between 0.75 and 1.0 as well as for an outer aperture ray with a relative ray aperture between 0.5 and 0.75 is smaller than 15% of a working wavelength λ0, said relative ray aperture being the ratio between a ray aperture and the numerical aperture in an image plane (IM).

19. The objective of claim 18, wherein the absolute amount of the optical retardation asymmetry ORASLG of the group of fluoride crystal lenses for an outer aperture ray with a relative ray aperture between 0.75 and 1.0 as well as for an outer aperture ray with a relative ray aperture between 0.5 and 0.75 is smaller than 7% of a working wavelength λ0.