Patent Publication Number: US-11656385-B2

Title: Freeform Fresnel surface, method for producing a freeform Fresnel surface and method for creating a construction dataset

Description:
PRIORITY 
     This application claims the benefit of German Patent Application No. 10 2016 112 099.5, filed on Jul. 1, 2016, and German Patent Application No. 10 2016 115 937.9, filed on Aug. 26, 2016, both of which are hereby incorporated herein by reference in their entirety. 
     FIELD 
     The present invention relates to a freeform Fresnel surface and to a method for producing a freeform Fresnel surface by way of machining a base body. The invention also relates to a method for creating a construction dataset for the production of a freeform Fresnel surface and to a spectacle lens. 
     BACKGROUND 
     What is referred to as a freeform Fresnel surface is a freeform surface which is described by a polynomial and which is formed by multiple offsets by a preferably constant absolute value in the direction of its z-axis and by the corresponding cut at a carrier surface and also at a preferably parallel or preferably concentric surface. A freeform Fresnel surface  100  is illustrated by way of example in  FIG.  1   . The figure shows the individual Fresnel facets  102  with their respective Fresnel segments surfaces  104  and trailing edges  106  and the intersecting lines  108  with the carrier surface. The carrier surface can be plane or curved. Their origin coincides with the origin of the polynomial. The origin of the preferably parallel or preferably concentric surface is offset by a specific absolute value in the z-direction and creates the structure depth of the Fresnel segment surface. 
     DE 10 2009 010 537 A1 and DE 10 2009 010 538 A1 include examples of freeform Fresnel surfaces, wherein the latter document also describes a method for producing a freeform Fresnel surface. Freeform Fresnel surfaces are used for example in smart glasses or similar apparatuses, as are described for example in U.S. Pat. No. 5,369,415, in U.S. Pat. No. 6,204,974 B1, in US 2010/0260455 A1 or WO 2006/013565 A1. 
     The production of the freeform Fresnel surface (either as an original part or as an injection mold for example for industrial production) is effected by the infeed of a cutting tool in the direction of the z-axis up to the z-value given for the respective x-y-position by the mathematical description. 
     SUMMARY 
     An object herein is providing an advantageous method for producing freeform Fresnel surfaces. A second object is providing a method for creating an advantageous construction data set for producing a freeform Fresnel surface. A third objective is to provide an advantageous freeform Fresnel surface, and it is a fourth object to provide an advantageous spectacle lens. 
     In the method for producing a freeform Fresnel surface having a number of Fresnel facets that each have a Fresnel segment surface and a trailing edge, the freeform Fresnel surface is produced by way of machining a base body on the basis of construction data for the freeform Fresnel surface. The construction data for the freeform Fresnel surface can be based on:
         a carrier surface of the freeform Fresnel surface, which is defined in a coordinate system with an x-direction, a y-direction and a z-direction;   a family of freeform surfaces which are staggered in the z-direction of the coordinate system and inclined with respect to the x-x-plane of the carrier surface, wherein each of the freeform surfaces defines in each case the Fresnel segment surface of a Fresnel facet;   a family of intersection curves along which the freeform surfaces of the family of freeform surfaces intersect the carrier surface;   circular cylinder surfaces or cone surfaces whose circular cylinder axes or cone axes are perpendicular to the x-y-plane of the coordinate system, wherein each circular cylinder surface or cone surface intersects in each case a specific freeform surface from the family of freeform surfaces at the height of the carrier surface and additionally cuts the freeform surface arranged in each case in the z-direction directly below the specific freeform surface, with the result that the intersection with the specific freeform surface defines an upper intersection line and the intersection with the freeform surface that is arranged in the z-direction immediately below it defines a lower intersection line. The region of the circular cylinder surface or cone surface extending between the upper intersection line and the lower intersection line then defines the trailing edge of the Fresnel facet belonging to the selected freeform surface.       

     Producing the freeform Fresnel surface is then effected by way of forming the Fresnel segment surfaces and the trailing edges of the Fresnel facets by circular-path-shaped machining of the base body based on the construction data, wherein material is removed from the base body until the Fresnel segment surface and the trailing edge are exposed for every Fresnel facet. 
     For the production of the surface by machining it is advantageous if the movement of the tool used for the production can follow a circular path that is projected onto the x-y-plane. In the prior art, this is not possible in the region of the border of the Fresnel facets and in particular in the case of the trailing edges because the projection of the spatial intersection curves that define the shape onto the x-y-plane does not produce an arc. In the method according to the invention, however, the projection of the peripheries of the Fresnel facets onto the x-y-plane represents circular paths due to the use of the circular cylinder surfaces or cone surfaces for creating the construction data. Hereby, the production of the Fresnel segment surfaces and of the trailing edges using a tool that is guided on concentric circular paths becomes possible. In other words, in circular-path-shaped machining of the base body, the movement of the used tool can follow a circular path that is projected onto the x-y-plane, which simplifies the control of the machine tool for machining the freeform Fresnel surface. The infeed values of the used tool in the z-direction that are required for the circular-path-shaped machining of the base body can be converted into cylinder coordinates taking into account the position of the circular cylinder axis or of the cone axis. 
     Within the framework of the method it is advantageous when the origin of the coordinate system for the description of the freeform Fresnel surface is defined such that it is located centrally in the carrier surface, with the result that a compensation plane which is disposed through the freeform Fresnel surface is perpendicular to the z-axis of the system. 
     It is furthermore advantageous when the shape of the circular cylinder surfaces or cone surfaces is constructed such that the upper intersection lines are best approximated in each case to the shape of the intersection curve of the respective specific freeform surface with the carrier surface. This can be achieved in the case of circular cylinder surfaces by way of a suitable selection of the radius of the circular cylinder and in the case of cone surfaces by way of a suitable selection of the cone angle and of the distance of the cone tip from the x-y-plane. A good approximation of the upper intersection line to the shape of the intersection curve of the respective specific freeform surface with the carrier surface can be attained if a selected circular cylinder surface or cone surface is constructed such that it extends through the intersection curve end points of a selected intersection curve of the family of intersection curves and a point on the selected intersection curve that is located between the two intersection curve end points of said intersection curve. It is advantageous here if the point between the two intersection curve end points of the selected intersection curve is located at least approximately in the center between the two intersection curve end points. It is additionally advantageous if the selected intersection curve is an intersection curve that is located centrally in the family of intersection curves. The circular cylinder axes or cone axes of all other circular cylinder surfaces or cone surfaces are then selected such that they coincide with the circular cylinder axis or cone axis of the selected circular cylinder surface or cone surface. In addition, for all freeform surfaces, the circular cylinder surface or cone surface that is assigned to a specific freeform surface extends through a point on the intersection curve of said respective specific freeform surface with the carrier surface. Here, too, it is advantageous if the point on the intersection curve of the specific freeform surface with the carrier surface is located at least approximately in the center between the two intersection curve end points of the respective intersection curve. 
     A method for creating a construction data set for the production of a freeform Fresnel surface having a number of Fresnel facets that have in each case a Fresnel segment surface and a trailing edge is also provided. These methods can comprise the following steps:
         defining a carrier surface of the freeform Fresnel surface, which is defined in a coordinate system with an x-direction, a y-direction and a z-direction;   defining a family of freeform surfaces which are staggered in the z-direction of the coordinate system and extend at an angle greater than 0 and less than 90 degrees relative to the carrier surface, wherein each of the freeform surfaces defines in each case the Fresnel segment surface of a Fresnel facet;   ascertaining a family of intersection curves along which the freeform surfaces of the family of freeform surfaces intersect the carrier surface;   ascertaining circular cylinder surfaces or cone surfaces whose circular cylinder axes or cone axes are perpendicular to the x-y-plane of the coordinate system, wherein the circular cylinder surfaces or cone surfaces intersect in each case a specific freeform surface from the family of freeform surfaces at the height of the carrier surface and additionally cuts the freeform surface arranged in each case in the z-direction directly below the specific freeform surface, with the result that the intersection with the specific freeform surface defines an upper intersection line and the intersection with the freeform surface that is arranged in the z-direction immediately below it defines a lower intersection line. The region of the circular cylinder surface or cone surface extending between the upper intersection line and the lower intersection line then defines the trailing edge of the Fresnel facet belonging to the selected freeform surface.       

     Creating the construction data set typically also comprises ascertaining cylinder coordinates for forming the Fresnel segment surfaces and trailing edges of the Fresnel facets by way of circular-path-shaped machining of the base body using a machining tool, wherein the required infeed values for the tool in the z-direction are ascertained taking into account the position of the circular cylinder axis or the cone axis. 
     Within the framework of the method for creating a construction data set, it is advantageous if the origin of the coordinate system for the description of the freeform Fresnel surface is defined such that it is located centrally in the carrier surface. 
     It is furthermore advantageous when the shape of the circular cylinder surfaces or cone surfaces is constructed such that the upper intersection lines are best approximated in each case to the shape of the intersection curve of the respective specific freeform surface with the carrier surface. This can be done for example by way of a selected circular cylinder surface or cone surface being constructed by being arranged such that it extends through the intersection curve end points of a selected intersection curve of the family of intersection curves and a point on the selected intersection curve that is located between the two intersection curve end points of said intersection curve. It is advantageous here if the point between the two intersection curve end points of the selected intersection curve is disposed at least approximately in the center between the two intersection curve end points. It is additionally advantageous if an intersection curve that is located centrally in the family of intersection curves is selected as the selected intersection curve. The circular cylinder axes or cone axes of all other circular cylinder surfaces or cone surfaces are then arranged such that they coincide with the circular cylinder axis or cone axis of the selected circular cylinder surface or cone surface. In addition, the circular cylinder surface or cone surface that is assigned to a specific freeform surface is arranged such that it extends through a point on the intersection curve of the specific freeform surface with the carrier surface. Here, too, it is advantageous if the point on the intersection curve of the specific freeform surface with the carrier surface is located at least approximately in the center between the two intersection curve end points of the respective intersection curve. 
     Both in the method for producing a freeform Fresnel surface and in the method for creating a construction data set, it is possible, if the circular cylinder surfaces or cone surfaces are circular cylinder surfaces, for cone surfaces to be disposed through the Fresnel segment surfaces after the construction of the Fresnel segment surfaces and the trailing edges using the circular cylinder surfaces, with the associated cones being dimensioned with respect to their cone angle and the position of their tips above the x-y-plane such that they continuously intersect in each case two adjacent Fresnel segment surfaces and with the section of the cone surface located between the intersection lines of a cone surface with the adjacent Fresnel segment surfaces forming the trailing edge between the adjacent Fresnel segment surfaces. In this refinement of the method, the trailing edges are not parallel with respect to the z-direction, which simplifies the production of the trailing edges by cutting. If the circular cylinder surfaces or cone surfaces are cone surfaces, this development is not necessary. 
     Further provided is a computer program product, having software components for performing one of the methods provided herein. 
     Furthermore provided herein is a freeform Fresnel surface having a number of Fresnel facets that have in each case a Fresnel segment surface and a trailing edge and having a structure depth, which is defined in a coordinate system with an x-direction, a y-direction and a z-direction along the z-direction. In the freeform Fresnel surface according to the invention, the Fresnel segment surfaces and the trailing edge are bounded in each case by lines the projection of which onto the x-y-plane of the coordinate system are circle line sections in the x-y-plane, which is advantageous for the machining during the production of the freeform Fresnel surface, because the movement of the tool used for the production can follow a circular path that is projected onto the x-y-plane. 
     A spectacle lens can be equipped with a freeform Fresnel surface as provided herein. The use of such a freeform Fresnel surface is advantageous in the spectacle lens with respect to the production of said surface because the movement of the tool used for the production of the freeform Fresnel surface can follow a circular path that is projected onto the x-y-plane. 
     Further features, properties and advantages of the present invention will become apparent from the following description of exemplary embodiments with reference to the accompanying figures. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    shows an example of a freeform Fresnel surface according to the prior art. 
         FIG.  2    shows a step in a method for producing a freeform Fresnel surface according to the prior art. 
         FIG.  3    shows a further step in the method for producing a freeform Fresnel surface according to the prior art. 
         FIG.  4    shows a first step of an exemplary embodiment for the method according to the invention for creating a construction data set for the production of a freeform Fresnel surface. 
         FIG.  5    shows a further step of the exemplary embodiment of the method according to the invention. 
         FIG.  6    shows yet a further step of the exemplary embodiment of the method according to the invention. 
         FIG.  7    shows yet a further step of the exemplary embodiment of the method according to the invention. 
         FIG.  8    shows cone surfaces used to generate construction data for inclined trailing edges of the Fresnel facets of a freeform Fresnel surface. 
         FIG.  9    shows a first illustration for explaining the dimensioning of the cone surfaces from  FIG.  8   . 
         FIG.  10    shows a second illustration for explaining the dimensioning of the cone surfaces from  FIG.  8   . 
         FIG.  11    shows a freeform Fresnel surface with inclined trailing edges. 
         FIG.  12    shows a spectacle lens having a freeform Fresnel surface. 
     
    
    
     While the invention is amenable to various modifications and alternative forms, specifics thereof have been shown by way of example in the drawings and will be described in detail. It should be understood, however, that the intention is not to limit the invention to the particular example embodiments described. On the contrary, the invention is to cover all modifications, equivalents, and alternatives falling within the scope of the invention as defined by the appended claims. 
     DETAILED DESCRIPTION 
     In the following descriptions, the present invention will be explained with reference to various exemplary embodiments. Nevertheless, these embodiments are not intended to limit the present invention to any specific example, environment, application, or particular implementation described herein. Therefore, descriptions of these example embodiments are only provided for purpose of illustration rather than to limit the present invention. 
     Before discussing an exemplary embodiment of the invention, an example for producing a freeform Fresnel surface as per a method according to the prior art will be explained with reference to  FIGS.  2  and  3   . 
     As is illustrated in  FIG.  2   , in the method according to the prior art, the origin of the freeform Fresnel surface lies outside the used surface (surface aperture). The curvatures of carrier surface  110  and concentric surface  112  are illustrated by equatorial lines  114 ,  116 . The distance h between the two surfaces  110 ,  112  in the direction of the z-axis creates the structure depth and thus the width of the individual Fresnel segment surfaces  104 . 
     In mathematical terms, the freeform Fresnel surface is described as follows: 
     
       
         
           
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     Here, the index “F” designates the variables of the freeform Fresnel surface, while the index “T” designates the variables of the carrier surface  110 . The letter h designates the structure depth of the Fresnel segment surface, c the curvature, and k the conicity of the respective surface. A designates the coefficients of the freeform surface. 
       FIG.  2    shows the construction of the freeform Fresnel surface in accordance with the mathematical description. It illustrates freeform surfaces  118  which are staggered in the z-direction and the carrier surface  110  with the concentric surface  112 . Each of the freeform surfaces  118  produces in each case a spatial intersection curve  120 ,  122  with the carrier surface  110  and the concentric surface  112 , with the shape of said intersection curve  120 ,  122  depending on the shape of the freeform surface  118  and the curvature of the carrier surface and the concentric surface. (The uppermost and lowermost freeform surfaces intersect in each case only the carrier surface and concentric surface, respectively). The intersection curves ( 120 ,  122 ) produced form the respectively outer and inner boundary of the individual Fresnel segment surfaces  104 .  FIG.  3    illustrates the Fresnel segment surfaces  104  produced by cutting the freeform surfaces  118 . 
     Up to the state illustrated in  FIG.  3   , the constructed surface corresponds to the mathematical description. A real surface produced in a piece of material, e.g., by way of machining, however, also has the trailing edges  106  between the Fresnel segment surfaces, which can be seen in  FIG.  1   . The trailing edges  106  can be inclined relative to the x-y-plane (e.g., for technological reasons). They follow, e.g., the intersection curves on the concentric surface  112  and bound the Fresnel segment surfaces before they intersect the carrier surface  110 . 
     An exemplary embodiment for the method according to the invention for creating a construction data set for producing a freeform Fresnel surface will be described below with reference to  FIGS.  4  to  7   . 
     In the first step of the method, the origin 0 of the freeform Fresnel surface is located approximately centrally in the used surface aperture, with the result that a compensation plane (x-y-plane) that is disposed through the freeform Fresnel surface is perpendicular with respect to the z-axis of the system. This is shown in  FIG.  4   , which not only shows the origin 0 but also the carrier surface  2 , the freeform surfaces  4  and the concentric surface  6 . 
     Within the framework of the method according to the invention, for a family of freeform surfaces  4  which are staggered in the z-direction and inclined relative to the x-y-plane of the carrier surface such that they intersect the carrier surface  2 , the intersection curves  8  with the carrier surface  2  are ascertained, as in the prior art. As a result, a family of intersection curves  8 , which each represent the intersection of a freeform surface  4  with the carrier surface  2 , are obtained, wherein each freeform surface  4  defines a later Fresnel segment surface  10  (see  FIG.  7   ). 
     In the next step, illustrated graphically in  FIG.  5   , a circular cylinder surface  18 , whose circular cylinder axis A is perpendicular to the x-y-plane, is placed through the end points  14   a ,  14   b  and a center point  16  of a selected (central, in the present exemplary embodiment) section curve  8 . The position of the circular cylinder axis A with reference to the origin 0 of the freeform Fresnel surface is described by the coordinates x z  and y z . The position of the circular cylinder axis A is illustrated in the figure in shortened form due to a lack of space. 
     Due to the described type of positioning of the circular cylinder surface  18 , the latter intersects the freeform surface  4  belonging to the intersection curve  8  in the region of the carrier surface  2  and forms an intersection line  20  which is best approximated to the intersection curve  8  (see  FIG.  7   ), which will be referred to as upper intersection line  20  below. The projection of the upper intersection line  20  onto the x-y-plane then represents a section from a circle line which is best approximated to the projections of the intersection curves  8  onto the x-y-plane. Moreover, the circular cylinder surface  18  also intersects the freeform surface that is located in the family of freeform surfaces  4  directly below the freeform surface  4  producing the intersection curve  8 . The intersection line  22  of the circular cylinder surface  18  with said freeform surface will be referred to as lower intersection line  22  below. The projection of the lower intersection line onto the x-y-plane also represents a section from a circle line. 
     In a next step, which is graphically illustrated in  FIG.  6   , for each intersection curve  8  of a freeform surface  4  with the carrier surface  2 , a circular cylinder surface  18 ′ is disposed through a center point on the respective intersection curve  8 , wherein the axis of said circular cylinder surface  18 ′ coincides with the previously produced circular cylinder axis A. As a result, a set of concentric circular cylinder surfaces  18 ,  18 ′ is obtained, of which each intersects a specific freeform surface from the family of freeform surfaces  4  in the region of the carrier surface  2 , wherein in each case an upper intersection line  20  is formed. Likewise, a lower intersection line  22  is formed for the intersection of the respective circular cylinder surface  18 ′ with the freeform surface that is located under the specific freeform surface. Both the upper intersection lines  20  and the lower intersection lines  22  represent, in the projection onto the x-y-plane, in each case sections of circle lines, wherein the projection of an upper intersection line  20  onto the x-y-plane is best approximated to the projection of the intersection curve  8  of the corresponding freeform surface  4  with the carrier surface  2  onto the x-y-plane. 
     The upper intersection lines  20  and the lower intersection lines  22  within the framework of the invention define the outer and inner boundaries of the individual Fresnel segment surfaces  10  rather than the intersection curves of a freeform surface  4  with the carrier surface  2  and the concentric surface  6 . Here, a circular cylinder surface  18 ,  18 ′ is at the same time an outer boundary of a preceding Fresnel segment surface  10  and inner boundary of a subsequent Fresnel segment surface  10 , which means that the surface section of the corresponding freeform surface  4  which is located between the upper intersection line  20  of a circular cylinder surface  18 ,  18 ′ and the lower intersection line  22  of the circular cylinder surface  18 ,  18 ′ adjacent to the former forms the Fresnel segment surface  10  of a Fresnel facet  12 . The surface section of a circular cylinder surface  18 ,  18 ′ located between the upper intersection line  20  and the lower intersection line  22  then forms the trailing edge of the Fresnel facet  12 . 
     As a consequence of the described construction of the freeform Fresnel surface, the bounding edges of the Fresnel segment surfaces  10  are no longer located exactly on the carrier surface  2  and the concentric surface  6 , but are bounded by circular path segments, viewed in the projection onto the x-y-plane. During machining of a base body for producing the freeform Fresnel surface, the movement of the tool used can therefore follow a circular path projected onto the x-y-plane. The movements of the tool used that are required for machining the base body in the form of a circular path can be given in the form of polar coordinates, wherein the infeed values of the tool in the z-direction can be expressed by the z-component of the cylinder coordinates. 
     A freeform Fresnel surface produced in accordance with the described exemplary embodiment for the method according to the invention would have trailing edges that are perpendicular to the x-y-plane. However, it is frequently advantageous both in technological and functional terms if the trailing edges are inclined relative to the x-y-plane. This means, together with the objective of guiding the machining tool along circular paths, a requirement for embodying the trailing edges in the form of a section of a cone surface results. Here, the axes of the cones should preferably coincide with the circular cylinder axes described in the first exemplary embodiment. 
       FIG.  8    illustrates the freeform Fresnel surface produced using the previously described method together with a set of coaxial cone surfaces  24 . The associated cones are dimensioned with respect to their cone angle and the position of their tips above the x-y-plane such that they continuously intersect in each case two adjacent Fresnel segment surfaces  10 ,  10 ′ (see  FIGS.  9  and  10   ). Non-continuous intersection lines would create undefined gaps in the finished freeform Fresnel surface. 
     Expedient dimensioning of the cone surfaces  24  is achieved when the intersection curve  26  of a cone surface  24  at the second one of the adjacent Fresnel segment surfaces  10 ,  10 ′, that is to say the Fresnel segment surface  10 ′, is tangent to the lower intersection line  22 . In  FIGS.  9  and  10   , this dimensioning is illustrated using the example of a cone surface  24  and the two Fresnel segment surfaces  10 ,  10 ′ it intersects.  FIG.  9    here shows a cone surface and the two Fresnel segment surfaces  10 ,  10 ′ it intersects in a perspective view.  FIG.  10    shows the same arrangement with a view in the direction of the y-axis. The figure shows the intersection lines, produced by the cone surface  24 , on the Fresnel segment surfaces  10 ,  10 ′. The portion of the segments that is located in the z-direction above the intersection line of the first segment  10  and the portion of the segments that is located in the z-direction below the intersection line of the second segment  10 ′ is cut off by the cone surface. 
     The complete freeform Fresnel surface is formed by in each case alternating cutting of the Fresnel segment surfaces  10  and the cone surfaces  24 , wherein the remaining sections of the cone surfaces  24  form the trailing edges  28  of the Fresnel facets ( 12 ). 
     For the production of the complete freeform Fresnel surface by way of circular-path-shaped movements of the tool, the respective infeed values z w  of the tool in the z-direction, along which the structure depth of the freeform Fresnel surface is defined, is converted into polar coordinates of the machine tool taking into account the position of the cone axis (x z ;y z ) of Cartesian coordinates of the mathematical description of the freeform Fresnel surface.  FIG.  11    illustrates the finished freeform Fresnel surface and the relationships of the coordinates. The conversion is effected in accordance with the following equations:
 
 z=z   w   =f ( x   F   ;y   F )= f (α w   ;r   w )
 
 a   w =arctan [ y   z   −y   F )/( x   Z   −x   F )]
 
 r   w =√{square root over (( x   Z   −x   F ) 2 +( y   Z   −y   F ) 2 )}
 
     The index “F” designates the variables of the freeform Fresnel surface, while the index “W” designates the variables of the tool or the machine. The index Z designates the position coordinates of the cone axis. 
     Freeform Fresnel surfaces, as they are described in the present application, are used for example in the field of smart glasses. Smart glasses generally comprise, inter alia, at least one spectacle lens through which an imaging beam path originating from a display device is guided under multiple reflection at the spectacle lens surfaces to an output coupling structure, from which it is output-coupled from the spectacle lens in the direction of the eye of a user. An example of such a spectacle lens  100  is represented in  FIG.  12   . As shown in  FIG.  12   , the spectacle lens  200  may have a peripheral thickening region  206 , in which the thickness of the spectacle lens  200  is increased in order to achieve optimum guidance of the imaging beam path  202  to the output coupling structure  204 . The imaging beam path  202  is coupled into the spectacle lens  200  with the aid of a prism  208 , which serves as an input coupling device. By way of the prism  208 , which is occasionally also referred to as a tube, the imaging beam path  202  coming from a display (not shown) is coupled into the spectacle lens  200  at such an angle that it can reach the output coupling structure  204  under multiple reflections at the outer surface  201  and at the inner surface  203  of the spectacle lens  200 . The output coupling structure  204  in the present example is in the form of a freeform Fresnel surface which was produced in accordance with the described method. For this reason, the Fresnel segment surfaces and the trailing edges of the freeform Fresnel surfaces are bounded in each case by lines the projection of which onto the x-y-plane of the coordinate system are circle line sections in the x-y-plane. 
     The present invention has been described in detail on the basis of an exemplary embodiment for explanatory purposes. It goes without saying, however, that the invention is not intended to be restricted to the present exemplary embodiment. Rather, a person skilled in the art will recognize that it is possible to deviate from the exemplary embodiments within the scope of the invention as is defined in the claims. For example, there is the possibility to use, instead of the circular cylinder surfaces, cone surfaces from the start, wherein the construction of the upper and lower intersection lines of the cone surfaces with the freeform surfaces from the family of freeform surfaces can be constructed as was described for the circular cylinder surfaces. Therefore, the invention is intended to be restricted only by the appended claims.