Method of calculation for detailed waverfront calculation for spectacle lenses with a diffractive grating

Abstract

Method for assessing an optical property of k.sup.th order of an optical element at an evaluation point. The optical element has a boundary surface formed of a refractive base surface and a phase-modifying optical element. The method includes determining the properties of a wavefront in the local surrounding of the evaluation point by means of a local wavefront tracing, and determining the optical property at the evaluation point based on the properties of the wavefront in the local surrounding of the evaluation point, wherein the local wavefront tracing has a local wavefront tracing upon passage through the boundary surface, and the local wavefront tracing upon passage through the boundary surface is performed according to the equation for the local wavefront tracing through the refractive base surface, the equation being supplemented by an additive additional term PK.sup.(k).

Claims

1. A computer-implemented method for assessing at least one optical property of k.sup.th order, wherein k≧3 and is an integer, of an optical element at at least one evaluation point, wherein the optical element comprises at least one boundary surface formed of a refractive base surface and at least one phase-modifying optical element, and manufacturing the optical element, the method comprising: determining the properties of a wavefront in the local surrounding of the at least one evaluation point by a local wavefront tracing; and determining the at least one optical property at the at least one evaluation point on the basis of the properties of the wavefront in the local surrounding of the at least one evaluation point, wherein the local wavefront tracing comprises a local wavefront tracing upon passage through the at least one boundary surface, and the local wavefront tracing upon passage through the at least one boundary surface is performed according to an equation for the local wavefront tracing through the refractive base surface, said equation being supplemented by an additive additional term PK.sup.(k), and wherein the additive additional term PK.sup.(k) depends on the local derivatives of k.sup.th order of a phase function Ψ(x,y), assigned to the phase-modifying optical element, with respect to the coordinates tangentially to the base surface, wherein the phase function Ψ(x,y) represents the optical path length difference introduced by the at least one phase-modifying optical element as a function of the coordinates (x,y) tangentially to the base surface; and manufacturing the optical element based on the local wavefront tracing.

2. The method according to claim 1, wherein determining the at least one optical property comprises determining the course of a main ray passing through the evaluation point.

3. The method according to claim 1, further comprising obtaining surface data of the at least one refractive base surface and of the phase-modifying optical element.

4. The method according to claim 3, wherein the data of the phase-modifying optical element comprises data relating to the course of a grating lines of the phase-modifying optical element, and wherein the phase function ψ(x,y) is determined on the basis of the obtained data relating to the course of the grating lines of the phase-modifying optical element such that isolines of the phase function or curves with ψ(x,y)=c, where c is an integer constant and c≧0, are parallel to the grating lines of the phase-modifying optical element.

5. The method according to claim 4, wherein determining the phase function comprises: measuring a grating projection in a predetermined plane (x.sup.0,y.sup.0); determining a phase function ψ.sup.0(x.sup.0,y.sup.0) in a global coordinate system (x.sup.0,y.sup.0,z.sup.0) by regarding the grating lines projected in the plane (x.sup.0,y.sup.0) as curves ψ.sup.0(x.sup.0,y.sup.0)=c, where c is an integer constant and c≧0, wherein for all points lying between the grating lines, the values of the phase function are obtained by interpolation; transforming the determined phase function ψ.sup.0(x.sup.0,y.sup.0) into a local coordinate system (x,y,z), the z axis of which is perpendicular to the base surface in the evaluation point:
ψ(x,y)=ψ.sup.0(x.sup.0(x.sup.0,y.sup.0),y.sup.0(x,y)) specifying a wavelength and a diffraction order; determining the phase function Ψ(x,y) according to the equation Ψ(x,y)=Ψ(x,y;λ,m)=mλ.Math.ψ(x,y), wherein the global coordinate system (x.sup.0,y.sup.0,z.sup.0) is a coordinate system serving to describe the base surface by its vertex depth; and the local coordinate system x,y,z is a local coordinate system serving to describe the passage of a main ray through a boundary surface, wherein (x,y,z)=(0,0,0) holds at a penetration point of the main ray with the boundary surface, and wherein the z axis is perpendicular to the base surface at the penetration point.

6. The method according to claim 1, further comprising comparing the optical property determined at the at least one evaluation point to a predetermined target value for the at least one optical property at the at least one evaluation point.

7. The method according to claim 1, wherein the at least one optical property is a coma and/or a spherical aberration and/or a trefoil.

8. The method according to claim 1, wherein the at least one optical property is assessed at a plurality of evaluation points, wherein the plurality of evaluation points comprises at least 1000 evaluation points.

9. The method according to claim 1, wherein the optical element is a spectacle lens.

10. The method according to claim 9, wherein determining at least one optical property of the spectacle lens is performed taking a predetermined average or individual wearing position of the spectacle lens and/or a predetermined average or individual object distance model into consideration.

11. The method according to claim 1, wherein the phase-modifying optical element is a diffraction grating.

12. A non-transitory computer program product adapted, when loaded and executed on a computer, to perform a method for assessing at least one optical property of an optical element according claim 1.

13. A storage medium with a non-transitory computer program stored thereon, wherein the computer program is adapted, when loaded and executed on a computer, to perform a method for assessing at least one optical property of an optical element according to claim 1.

14. A device for assessing at least one optical property of an optical element at at least one evaluation point, comprising a computer adapted to perform a method for assessing at least one optical property of an optical element according to claim 1.

15. A method for manufacturing an optical element comprising at least one phase-modifying optical element arranged on a refractive base surface, comprising: specifying an optical element to be optimized; optimizing the specified optical element so as to reduce the deviation of the value of at least one optical property of the optical element from a target value at at least one evaluation point, wherein the at least one optical property is assessed according to the method for assessing at least one optical property of the optical element according to claim 1.

16. The method for manufacturing an optical element according to claim 15, wherein optimizing or modifying the specified optical element comprises calculating a target function, which depends on the at least one optical property.

17. The method for manufacturing an optical element according to claim 15, wherein optimizing or modifying the specified optical element comprises varying the refractive base surface and/or the phase-modifying optical element.

18. A device for manufacturing an optical element, the optical element being a spectacle lens, with at least one phase-modifying optical element, comprising a specifier adapted to specify an optical element to be optimized; an optimizer adapted to perform an optimization or modification of the specified optical element so as to reduce the deviation of the value of at least one optical property of the optical element from a target value at at least one evaluation point, wherein the at least one optical property is assessed according to the method for assessing at least one optical property of the optical element according to claim 1.

19. The method according to claim 1, wherein the at least one optical property is assessed at a plurality of evaluation points, wherein the plurality of evaluation points comprises at least 2000 evaluation points.

20. The method according to claim 1, wherein the at least one optical property is assessed at a plurality of evaluation points, wherein the plurality of evaluation points comprises at least 5000 evaluation points.

21. The method according to claim 1, wherein the at least one optical property is assessed at a plurality of evaluation points, wherein the plurality of evaluation points comprises at least 10000 evaluation points.

Description

DRAWINGS

(1) Further objects, features, and advantages of the present invention will become apparent from the following description of exemplary embodiments with reference to the drawings, which show:

(2) FIG. 1 a schematic illustration of the ray course for a ray tracing and wavefront tracing for an optical system;

(3) FIG. 2 a schematic illustration of the ray course for a ray tracing and wavefront tracing for a spectacle lens;

(4) FIG. 3 the ray course in the case of a plane boundary surface with a diffraction grating;

(5) FIG. 3A the mutual position of the coordinate system of the incident wavefront and the coordinate system of the base surface;

(6) FIG. 4 grating lines of a simple periodic diffraction grating on a plane boundary surface;

(7) FIG. 5 grating lines of a diffraction grating on a plane boundary surface;

(8) FIG. 6 grating lines of a diffraction grating on a curved boundary surface;

(9) FIG. 7 a flow diagram illustrating the transition between the vector field d(x.sup.0,y.sup.0) and the phase function ψ.sup.0(x.sup.0,y.sup.0);

(10) FIG. 8 the longitudinal chromatic aberration (FIG. 8A) and the lateral chromatic aberration (FIG. 8B) of a lens according to first comparative example, which has been optimized according to a monochromatic target function;

(11) FIG. 9 the longitudinal chromatic aberration (FIG. 9A) and the lateral chromatic aberration (FIG. 9B) of lenses optimized according to a polychromatic target function of a first type (first example) and of a second type (second example), respectively;

(12) FIG. 10A the prescription power, the refractive power, the diffractive power, and the total power in the lens center as a function of the wavelength of the lens according to the first comparative example;

(13) FIG. 10B the prescription power, the refractive power, the diffractive power, and the total power in the lens center as a function of the wavelength for two lenses according to the first and second examples;

(14) FIG. 11 the longitudinal chromatic aberration (FIG. 11A) and the lateral chromatic aberration (FIG. 11B) of a lens according to a second comparative example, which has been optimized according to a monochromatic target function;

(15) FIG. 12 the longitudinal chromatic aberration (FIG. 12A) and the lateral chromatic aberration (FIG. 12B) of a lens according to a third example, which has been optimized according to a polychromatic target function of a first type;

(16) FIG. 13 the longitudinal chromatic aberration (FIG. 13A) and the lateral chromatic aberration (FIG. 13B) of a lens according to a fourth example, which has been optimized according to a polychromatic target function of a second type;

(17) FIG. 14A the prescription power, the refractive power, the diffractive power, and the total power in the lens center as a function of the wavelength of the lens according to the second comparative example;

(18) FIG. 14B the prescription power, the refractive power, the diffractive power, and the total power in the lens center as a function of the wavelength of the lens according to the third example;

(19) FIG. 14C the prescription power, the refractive power, the diffractive power, and the total power in the lens center as a function of the wavelength of the lens according to the fourth example;

(20) FIG. 15A the refractive error at the wavelength λ.sub.C, the refractive error at the wavelength λ.sub.F, and the longitudinal chromatic aberration in the lens according to the second comparative example;

(21) FIG. 15B the refractive error at the wavelength λ.sub.C, the refractive error at the wavelength λ.sub.F, and the longitudinal chromatic aberration in the lens according to the third example;

(22) FIG. 15C the refractive error at the wavelength λ.sub.C, the refractive error at the wavelength λ.sub.F, and the longitudinal chromatic aberration in the lens according to the fourth example;

(23) FIGS. 16A-C the color fringe of a spectacle lens for different viewing angles a) 0° b) 10° c) 20° d) 30°, wherein

(24) FIG. 16A shows a non-corrected image formation through a monochromatically optimized spectacle lens;

(25) FIG. 16B shows an image formation through a spectacle lens with corrected color fringe, without an aspherical follow-up optimization of the refractive surfaces;

(26) FIG. 16C shows an image formation through a spectacle lens optimized by means of a simultaneous color fringe optimization and an aspherical follow-up optimization;

(27) FIG. 17 an exemplary color fringe correction of a non-corrected single-vision lens, wherein FIG. 17A shows the color fringe of a non-corrected single-vision lens with an Abbe number of 58.5, FIG. 17B shows the color fringe of a single-vision lens with an Abbe number of 40.5, and FIG. 17C shows the color fringe of a color fringe-corrected single-vision lens with an effective Abbe number of approximately 130;

(28) FIG. 18 an exemplary color fringe correction of a progressive spectacle lens, wherein FIG. 18A shows the astigmatism in the wearing position of the spectacle lens, FIG. 18B shows the diffractive phase or form of the grating lines of the diffraction grating, FIG. 18C shows the color fringe of the non-corrected spectacle lens, and FIG. 18D shows the color fringe of the corrected spectacle lens;

DETAILED DESCRIPTION

(29) The following example relates to an optical element, wherein the phase-modifying optical element is a diffraction grating. Moreover, the optical element is a spectacle lens. However, all explanations analogously apply to arbitrary optical elements as well, which have a phase-modifying optical element instead of a diffraction grating.

(30) The passage of light through an arbitrary optical element, which can comprise several optical components, or through an arbitrary optical system 100 can be described on the basis of ray tracing and wavefront tracing, as is schematically shown in FIG. 1 and FIG. 12. Here, the object of ray tracing is to calculate, for a specified optical element/system 100, the outgoing ray 108 going out of the element/system at the exit surface 106 from an incident ray 102 existing up to the entrance surface 104 of the optical element/system. The object of wavefront tracing is to calculate, for a specified optical element/system 100, the outgoing wavefront 112 at the exit surface 106 of the element/system 100 from an incident wavefront 110 existing at the entrance surface 104 of the optical element/system 100. The object of local wavefront tracing is to calculate, for a specified optical element/system 100, the local properties of the outgoing wavefront 112 at the exit surface 112 of the element/system 100 from the local properties of an incident wavefront 112 existing at the entrance surface 104 of the optical element 100.

(31) An optical element or optical system can be comprised of an arbitrary sequence of thin boundary surfaces, homogeneous materials, or inhomogeneous materials. In particular, an optical element (such as a spectacle lens, as shown in FIG. 2) can be comprised of the sequence of a curved refractive boundary surface 104 from air into a denser homogeneous material 114, the passage 116 through the material, and a second curved refractive boundary surface 106 back into air. FIG. 2 schematically shows the ray tracing and the wavefront tracing through such an optical element.

(32) To improve optical elements or optical systems, in particular spectacle lenses, it may be advantageous to additionally introduce optical components into the ray path, which are based on other physical effects than a mere refraction at a curved boundary surface. For example, it has been suggested that diffractive optical elements be used, to which diffraction effects are of importance. In particular, such an element can be phase-delaying or phase-modulating, in fact such that by adding the element, the change of the optical path length depends on the penetration point of the ray.

(33) However, methods allowing performing a precise tracing of the properties (in particular the image formation properties) of optical elements, which also comprise diffractive optical components, in particular taking a predetermined wearing position of the optical element into consideration, have not been known so far.

(34) An extended wavefront tracing for the case of complex optical elements, which have at least one diffraction grating, will be described in detail in the following. The method can also be considered an extension of the wavefront tracing through purely refractive boundary surfaces, which is known from document WO 2008/089999A1, for the case of complex optical systems with at least one diffraction grating.

(35) The most elementary description of a second-order wavefront tracing through a purely refractive boundary surface is known from the prior art (cf. e.g. the textbook “Optik und Technik der Brille” of H. Diepes and R. Blendowske, 2002 Optische Fachveröffentlichung GmbH, Heidelberg, pages 485 ff.) as the so-called BAD equation, or also vergence equation. In the case of a rotationally symmetric refractive surface and with a vertical, or perpendicular, incidence of rays, the BAD equation (vergence equation) reads as follow:
B=A+D.  (101)

(36) Here, A is the vergence (i.e. a measure for the local wavefront curvature) of the incident wavefront, B is the vergence of the outgoing wavefront, and D is the surface power of the refractive surface.

(37) If the requirements for rotational symmetry are not met, the BAD equation will be written vectorially for purely refractive surfaces according to the prior art (cf. e.g. the above-cited textbook “Optik und Technik der Brille”), i.e.
B=A+D  (102)

(38) Here, A is the power vector form of the vergence matrix of the incident wavefront, B is the power vector form of the vergence matrix of the outgoing wavefront, and D is the power vector form of the surface power matrix of the refractive surface. Instead of a power vector, linear combinations of power vector components, which have been combined to form a new vector, can be used as well. In particular, the normal curvatures and the torsion, i.e. the second derivatives with respect to transverse coordinates and the mixed derivative, can be used.

(39) Instead of the symbols A and A, B and B, as well as D and D, the following symbols are often used:

(40) A and A.fwdarw.S and S

(41) B and B.fwdarw.S′ and S′

(42) D and D.fwdarw.F and F.

(43) Accordingly, the equations (101) and (102) then read S′=S+F and S′=S+F.

(44) If the incidence of rays is not perpendicular, further modifications of the BAD equation can be introduced for purely refractive boundary surfaces according to the prior art, with which the wavefront passage can again be described precisely (cf. e.g. G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, and D. Uttenweiler, “Derivation of the refraction equations for higher-order aberrations of local wavefronts at oblique incidence”, JOSA. A, vol. 27, no. 2, February 2010, pages 218-237). These modifications are the Coddington equation and the generalized Coddington equation for second and higher-order aberrations. In particular, the publication of G. Esser et al. describes the power vector form of the generalized Coddington equation for second and higher-order aberrations.

(45) The equations (in power vector form) known from the prior art for second and higher-order wavefront tracings in the case of a passage through a purely refractive boundary surface are summarized in table 1:

(46) TABLE-US-00001 TABLE 1 vertical incidence + vertical incidence without no rotational symmetry rotational symmetry symmetry 2.sup.nd S′.sup.(2) = S.sup.(2) + F.sup.(2) S′.sup.(2) = S.sup.(2) + F.sup.(2) T′.sub.2 S′.sup.(2) = order alternative notation: alternative notation: T.sub.2S.sup.(2) + vF.sup.(2) S′ = S + F S′ = S + F B = A + D B = A + D 3.sup.rd S′.sup.(3) = S.sup.(3) + F.sup.(3) + S′.sup.(3) = S.sup.(3) + F.sup.(3) + T′.sub.3 S′.sup.(3) = order R.sub.3(S.sup.(2), F.sup.(2)) R.sub.3(S.sup.(2), F.sup.(2)) T.sub.3S.sup.(3) + vF.sup.(3) + Q.sub.3(S.sup.(2), F.sup.(2)) 4.sup.th S′.sup.(4) = S.sup.(4) + F.sup.(4) + S′.sup.(4) = S.sup.(4) + F.sup.(4) + T′.sub.4 S′.sup.(4) = order R.sub.4(S.sup.(3), F.sup.(3), S.sup.(2), F.sup.(2) R.sub.4(S.sup.(3), F.sup.(3), S.sup.(2), F.sup.(2)) T.sub.4S.sup.(4) + vF.sup.(4) + Q.sub.4(S.sup.(3), F.sup.(3), S.sup.(2), F.sup.(2)) etc.

(47) Tables 1A to 1C include explanations on the above-listed equations for the second order (table 1A), for the third order (table 1B), and for the fourth order (table 1C).

(48) TABLE-US-00002 TABLE 1A S.sup.(2) = nW.sub.In.sup.(2) (0) vergence of the incident wavefront: refractive index on the incidence side multiplied by the second derivative of the vertex depth of the incident wavefront S′.sup.(2) = n′W′.sub.Out.sup.(2) (0) vergence of the outgoing wavefront: refractive index on the emergence side multiplied by the second derivative of the vertex depth of the outgoing wavefront F.sup.(2) = (n′ − n)S.sup.(2) (0) surface power of the refractive surface: refractive index difference multiplied by the second derivative of the vertex depth of the refractive surface S and A vergence of the incident wavefront S′ and B vergence of the outgoing wavefront F and D surface power of the refractive surface $S^{\left(2\right)}=n\left(\begin{array}{c}{W}_{\mathrm{In}}^{\left(2,0\right)}\left(0,0\right)\\ W_{\mathrm{In}}^{\left(1,1\right)}\left(0,0\right)\\ W_{\mathrm{In}}^{\left(0,2\right)}\left(0,0\right)\end{array}\right)$ refractive index on the incidence side multiplied by the power vector of the second derivative of the vertex depth of the incident wavefront $S^{\prime \left(2\right)}=n\left(\begin{array}{c}{W}_{\mathrm{Out}}^{\prime \left(2,0\right)}\left(0,0\right)\\ W_{\mathrm{Out}}^{\prime \left(1,1\right)}\left(0,0\right)\\ W_{\mathrm{Out}}^{\prime \left(0,2\right)}\left(0,0\right)\end{array}\right)$ refractive index on the emergence side multiplied by the power vector of the second derivative of the vertex depth of the outgoing wavefront 0$F^{\left(2\right)}=\left(n^{\prime }-n\right)\left(\begin{array}{c}{\stackrel{\_}{S}}^{\left(2,0\right)}\left(0,0\right)\\ {\stackrel{\_}{S}}^{\left(1,1\right)}\left(0,0\right)\\ {\stackrel{\_}{S}}^{\left(0,2\right)}\left(0,0\right)\end{array}\right)$ refractive index difference multiplied by the power vector of the second derivative of the vertex depth of the refractive surface S and A power vector of the vergence of the incident wavefront S′ and B power vector of the vergence of the outgoing wavefront F and D power vector of the surface power of the refractive surface T.sub.2 incidence angle-dependent matrix for correction of oblique incidence T′.sub.2 emergence angle-dependent matrix for correction of oblique incidence v = n′ cos φ′ − n cos φ factor for correction of obliqe incidence

(49) TABLE-US-00003 TABLE 1B S.sup.(3) = nW.sub.In.sup.(3) (0) refractive index on the incidence side multiplied by the third derivative of the vertex depth of the incident wavefront S′.sup.(3) = n′W′.sub.Out.sup.(3) (0) refractive index on the emergence side multiplied by the third derivative of the vertex depth of the outgoing wavefront F.sup.(3) = (n′ − n)S.sup.(3) (0) refractive index difference multiplied by the third derivative of the vertex depth of the refractive surface $S^{\left(3\right)}=n\left(\begin{array}{c}{W}_{\mathrm{In}}^{\left(3,0\right)}\left(0,0\right)\\ W_{\mathrm{In}}^{\left(2,1\right)}\left(0,0\right)\\ W_{\mathrm{In}}^{\left(1,2\right)}\left(0,0\right)\\ W_{\mathrm{In}}^{\left(0,3\right)}\left(0,0\right)\end{array}\right)$ refractive index on the incidence side multiplied by the power vector of the third derivative of the vertex depth of the incident wavefront $S^{\prime \left(3\right)}=n\left(\begin{array}{c}{W}_{\mathrm{Out}}^{\prime \left(3,0\right)}\left(0,0\right)\\ W_{\mathrm{Out}}^{\prime \left(2,1\right)}\left(0,0\right)\\ W_{\mathrm{Out}}^{\prime \left(1,2\right)}\left(0,0\right)\\ W_{\mathrm{Out}}^{\prime \left(0,3\right)}\left(0,0\right)\end{array}\right)$ refractive index on the emergence side multiplied by the power vector of the third derivative of the vertex depth of the outgoing wavefront $F^{\left(3\right)}=\left(n^{\prime }-n\right)\left(\begin{array}{c}{\stackrel{\_}{S}}^{\left(3,0\right)}\left(0,0\right)\\ {\stackrel{\_}{S}}^{\left(2,1\right)}\left(0,0\right)\\ {\stackrel{\_}{S}}^{\left(1,2\right)}\left(0,0\right)\\ {\stackrel{\_}{S}}^{\left(0,3\right)}\left(0,0\right)\end{array}\right)$ refractive index difference multiplied by the power vector of the third derivative of the vertex depth of the refractive surface T.sub.3 incidence angle-dependent matrix for correction of oblique incidence T′.sub.3 emergence angle-dependent matrix for correction of oblique incidence v = n′ cos φ′ − n cos φ factor for correction of oblique incidence R.sub.3 (S.sup.(2), F.sup.(2)) additional term that only depends on the lower-order (here 2.sup.nd order) variables R.sub.3 (S.sup.(2), F.sup.(2)) vectorial additional term that only depends on the lower- order (here 2.sup.nd order) variables Q.sub.3 (S.sup.(2), F.sup.(2)) vectorial additional term that considers the oblique incidence and only depends on the lower-order (here 2.sup.nd order) variables

(50) TABLE-US-00004 TABLE 1C R.sub.4(S.sup.(3), F.sup.(3), S.sup.(2), F.sup.(2)) additional term that only depends on the lower-order (here 2.sup.nd and 3.sup.rd order) variables R.sub.4(S.sup.(3), F.sup.(3), S.sup.(2), F.sup.(2)) vectorial additional term that only depends on the lower-order (here 2.sup.nd and 3.sup.rd order) variables Q.sub.4(S.sup.(3), F.sup.(3), S.sup.(2), F.sup.(2)) vectorial additional term that considers the oblique incidence and only depends on the lower-order (here 2.sup.nd and 3.sup.rd order) variables

(51) The form of the additional terms R.sub.3(S.sup.(2),F.sup.(2)), R.sub.3(S.sup.(2),F.sup.(2)), Q.sub.3(S.sup.(2),F.sup.(2)) is further described in the publication [G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, and D. Uttenweiler, “Derivation of the refraction equations for higher-order aberrations of local wavefronts at oblique incidence”, JOSA. A, vol. 27, no. 2, February 2010, pages 218-237]. These terms disappear as soon as the lower-order terms S.sup.(2),F.sup.(2) and S.sup.(2),F.sup.(2) disappear or are equal to zero.

(52) The further explanations on the 4.sup.th order, etc., are analogous to the explanations on the 2.sup.nd and 3.sup.rd orders.

(53) Surprisingly, it has been shown that the equations describing the second and higher-order wavefront tracing by ordinary refractive boundary surfaces can be modified in a comparatively simple way, so that also diffractive optical elements or diffraction gratings can be taken into account. Here, either the passage of light through an isolated diffraction grating or the passage through a directly consecutive combination of a diffraction grating or a refractive boundary surface can be described.

(54) Moreover, it has turned out that a generally vectorial variable PK.sup.(k), k=2, 3, 4, . . . , hereinafter referred to as a phase curvature (for k=2) or as phase derivatives (for k>2), can be assigned to any arbitrary, non-rotationally symmetric diffraction grating even under arbitrary, oblique-angled ray incidence conditions, so that the corresponding BAD equation and the higher-order equations for refractive surfaces substantially only have to extended additively by the additional term PK.sup.(k), k=2, 3, 4, . . . in order to correctly describe the vergences for the wavefront passage in the presence of a non-rotationally symmetric diffraction grating.

(55) In other words, however asymmetric a situation may be for which the wavefront tracing could be described precisely for purely refractive surfaces, it is sufficient to extend the corresponding equation substantially only by an additive additional term PK in order to also describe the diffraction grating correctly.

(56) Further, it has been found that the vergence of the outgoing wavefront is independent of the order in which the refractive surface and the diffraction grating succeed one another.

(57) In the case of a second-order wavefront tracing (i.e. for k=2) for a rotationally symmetric diffraction grating and with a vertical, or perpendicular, incidence of rays, equation (101) is extended additively by the additional term PK.sup.(2) in order to correctly describe the vergences for the wavefront passage also in the presence of a diffraction grating:
B=A+D+PK.sup.(2).  (103)

(58) However, equation (101) is only valid exactly for the case that the ray impinges vertically, or perpendicularly, on the refractive surface and that the incident wavefront and the refractive surface are rotationally symmetric. But equation (101) is still a good approximation also for deviations from these prerequisites. Analogously, equation (103) is a good approximation in the presence of a diffraction grating.

(59) In the case of a second-order wavefront tracing (i.e. for k=2) for a non-rotationally symmetric diffraction grating and with a vertical, or perpendicular, incidence of rays, the diffraction grating can be assigned a vectorial variable PK.sup.(2), so that the corresponding vectorial BAD equation only has to be extended additively by the vectorial additional term PK.sup.(2) in order to correctly describe the vergences for the wavefront passage in the presence of a non-rotationally symmetric diffraction grating. In this case, it holds that
B=A+D+PK.sup.(2).  (104)

(60) As will be explained in detail in the following, a generally vectorial variable PK.sup.(k) can be assigned to any arbitrary, non-rotationally symmetric diffraction grating even under arbitrary, oblique-angled ray incidence conditions, so that the corresponding general BAD equation and the higher-order equations for the wavefront tracing through refractive surfaces substantially only have to be extended additively by the additional term PK.sup.(k) (PK.sup.(2), PK.sup.(3) PK.sup.(4), . . . ) in order to correctly describe the vergences for the wavefront passage in the presence of a non-rotationally symmetric diffraction grating.

(61) Moreover, is has been shown that the components of the additive additional term PK.sup.(k) can be described by the vector of the second or higher-order derivatives of a phase function Ψ(x,y) with respect to the coordinates x,y tangentially to the refractive surface (base surface). It holds for the additive second-order additional term that:

(62) $\begin{array}{cc}{{\mathrm{PK}}^{\left(2\right)}=-\left(\begin{array}{c}{\Psi }^{\left(2,0\right)}\left(0,0\right)\\ {\Psi }^{\left(1,1\right)}\left(0,0\right)\\ {\Psi }^{\left(0,2\right)}\left(0,0\right)\end{array}\right)=-\left(\begin{array}{c}{\partial }^2\Psi \left(\stackrel{\_}{x},\stackrel{\_}{y}\right)/\partial {\stackrel{\_}{x}}^2\\ {\partial }^2\Psi \left(\stackrel{\_}{x},\stackrel{\_}{y}\right)/\partial \stackrel{\_}{x}\partial \stackrel{\_}{y}\\ {\partial }^2\Psi \left(\stackrel{\_}{x},\stackrel{\_}{y}\right)/\partial {\stackrel{\_}{y}}^2\end{array}\right).Math.}_{\left(\stackrel{\_}{x},\stackrel{\_}{y}\right)=\left(0,0\right)}& \left(105\right)\end{array}$

(63) For this reason, the additive second-order additional term PK.sup.(2) is also referred to as a “phase curvature”.

(64) The phase function Ψ(x,y) represents the optical path length difference (optical path difference or OPD), which is introduced by the diffraction grating, as a function of the coordinates x,y tangentially to the refractive surface. The description of a diffraction grating by the phase function Ψ(x,y) allows determining the additive additional term PK.sup.(k) (PK.sup.(2), PK.sup.(3), PK.sup.(4), . . . ) constructively. Put differently, it is suggested that a phase function Ψ(x,y) be used to describe an arbitrary diffraction grating, the additive additional term PK.sup.(k) (PK.sup.(2), PK.sup.(3), PK.sup.(4), . . . ) being determined by the vector of the second and higher-order derivatives of the phase function with respect to the coordinates x,y tangentially to the refractive surface (cf. equation (105) for k=2).

(65) Table 2 summarizes the equations and wavefront equations (in power vector form) for the wavefront tracing in the case of a passage through a refractive boundary surface (base surface), to which a phase-modifying optical element is applied in addition. Tables 2A and 2B include explanations on the 2.sup.nd order (table 2A) and 3.sup.rd order (table 2B) equations listed in table 1.

(66) TABLE-US-00005 TABLE 2 vertical incidence + vertical incidence without no rotational symmetry rotational symmetry symmetry 2nd order S′.sup.(2) = S.sup.(2) + F.sup.(2) − Ψ.sup.(2) S′.sup.(2) = S.sup.(2) + F.sup.(2) − Ψ.sup.(2) T′.sub.2 S′.sup.(2) = alternative notation: alternative notation: T.sub.2S.sup.(2) + vF.sup.(2) − Ψ.sup.(2) S′ = S + F + PK.sup.(2) S′ = S + F + PK.sup.(2) alternative notation: B = A + D + PK.sup.(2) B = A + D + PK.sup.(2) T′.sub.2 S′.sup.(2) = T.sub.2S.sup.(2) + vF.sup.(2) + PK.sup.(2) 3rd order S′.sup.(3) = S.sup.(3) + F.sup.(3) − Ψ.sup.(3) + S′.sup.(3) = S.sup.(3) + F.sup.(3) − Ψ.sup.(3) + T′.sub.3 S′.sup.(3) = T.sub.3S.sup.(3) + R.sub.3(S.sup.(2), F.sup.(2)) R.sub.3(S.sup.(2), F.sup.(2)) vF.sup.(3) − Ψ.sup.(3) + alternative notation: alternative notation: Q.sub.3(S.sup.(2), F.sup.(2)) PK.sup.(3) = −Ψ.sup.(3) PK.sup.(3) = −Ψ.sup.(3) alternative notation: PK.sup.(3) = −Ψ.sup.(3) 4th order S′.sup.(4) = S.sup.(4) + F.sup.(4) − Ψ.sup.(4) + S′.sup.(4) = S.sup.(4) + F.sup.(4) − Ψ.sup.(3) + T′.sub.4 S′.sup.(4) = T.sub.4S.sup.(4) + R.sub.4(S.sup.(3), F.sup.(3), S.sup.(2), F.sup.(2)) R.sub.4(S.sup.(3), F.sup.(3), S.sup.(2), F.sup.(2)) vF.sup.(4) − Ψ.sup.(4) + alternative notation: alternative notation: Q.sub.4(S.sup.(3), F.sup.(3), S.sup.(2), F.sup.(2)) PK.sup.(4) = −Ψ.sup.(4) PK.sup.(4) = −Ψ.sup.(4) alternative notation PK.sup.(4) = −Ψ.sup.(4) etc.

(67) TABLE-US-00006 TABLE 2A Ψ.sup.(2) = Ψ.sup.(2) (0) second derivative of the phase function PK.sup.(2) = −Ψ.sup.(2) (0) phase curvature, i.e. negative second derivative of the phase function ${\Psi }^{\left(2\right)}=\left(\begin{array}{c}{\Psi }^{\left(2,0\right)}\left(0,0\right)\\ {\Psi }^{\left(1,1\right)}\left(0,0\right)\\ {\Psi }^{\left(0,2\right)}\left(0,0\right)\end{array}\right)$ power vector of the second derivatives of the phase function PK.sup.(2) = −Ψ.sup.(2) (0, 0) power vector of the phase curvature, i.e. the negative second derivatives of the phase function

(68) TABLE-US-00007 TABLE 2B Ψ.sup.(3) = Ψ.sup.(3) (0) third derivative of the phase function PK.sup.(3) = −Ψ.sup.(3) (0) additional term according to the invention, determined by negative third derivative of the phase function ${\Psi }^{\left(3\right)}=\left(\begin{array}{c}{\Psi }^{\left(3,0\right)}\left(0,0\right)\\ {\Psi }^{\left(2,1\right)}\left(0,0\right)\\ {\Psi }^{\left(1,2\right)}\left(0,0\right)\\ {\Psi }^{\left(0,3\right)}\left(0,0\right)\end{array}\right)$ power vector of the third derivatives of the phase function PK.sup.(3) = −Ψ.sup.(3) (0, 0) vectorial additional term according to the invention, determined by the power vector of the negative third derivatives of the phase function

(69) The equations for the 4th order and all higher orders are made up analogously.

(70) The coordinate system x,y,z is a local coordinate system, which serves to describe the passage of a ray through a boundary surface, wherein it holds at the penetration point of the main ray with the boundary surface that (x,y,z)=(0,0,0), and wherein the z axis is perpendicular to the base surface. A special possible choice of such a local system is to require that the incident ray be e.g. in the x-z plane or in the y-z plane. In general, however, this condition does not necessarily have to be satisfied. The use of local coordinate systems is known from WO 2008/089999 A1, for example, and is used in second-order wavefront tracing through surfaces without diffraction gratings. Typically, use is made of as many local systems as main rays are to be calculated. Further, a global coordinate system can be used.

(71) Moreover, it has been found that a connection can be established between the grating lines of the diffractive optical element or the diffraction grating and the phase function Ψ(x,y), which is based on the fact that the grating lines lie on curves with Ψ(x,y)=const. The determination of the phase function Ψ(x,y) on the basis of parameters of the diffraction grating will be described in detail in the following.

(72) Moreover, it has been found that different from a refraction at a purely refractive boundary surface (such as described in WO 2008 089999 A1), the incident main ray, the outgoing main ray, and the normal vector of the refractive surface generally will not have to be in one and the same plane any more if a diffraction grating is present.

(73) In the simplest case of a refractive, homogeneous boundary surface between two media with the refractive indices n and n′ without diffractive optical elements or diffraction gratings, the ray deviation is described by the law of refraction n′ sin φ′−n sin φ=0, where φ is the angle of incidence and φ′ is the angle of emergence.

(74) FIG. 3 schematically illustrates the ray path upon diffraction on a boundary surface 120 with a diffraction grating 122 in a special simple case in which it is possible to describe the ray deviation by a closed law. This law can be understood to be a modification of this law of refraction. This case is characterized in that the refractive surface 120 is a plane and that the diffraction grating 122 is an irregularity of this surface 120, which is translation-invariant perpendicular to the plane of incidence 124 and equidistant with period d in the direction of the plane of refraction (the plane of the refractive surface 120) (cf. FIG. 3). The irregularity can be a blazed grating, a rectangular grating, an alternation of translucent and opaque zones, or any other deviation from the homogeneous, plane, translucent, refractive surface. In this simple case, an incident monochromatic ray 102, which belongs to light with the wavelength λ, is split into many individual rays 108-m, which belong to the different diffraction order m, m= . . . , −2, −1, 0, 1, 2, . . . , by diffraction. The diffraction order m can be selected arbitrarily, but fixedly, and the situation for the ray pertaining to the diffraction order m can be described in an isolated manner, i.e. irrespective of the possible other diffraction orders, in the following. For a ray pertaining to the diffraction order m, the modified law of refraction applies n′ sin φ′−n sin φ=mλ/d, where n and n′ designate the refractive index of the material in front of and behind the boundary surface 120, φ is the angle of incidence, and φ′ is the angle of emergence.

(75) For every more general case, e.g. for rays being incident obliquely to the grating lines, for a non-equidistant grating and/or for a grating with curved grating lines and/or for a curved surface, no comprehensive laws on ray deviation and wavefront calculation have been known so far. In order to be able to calculate or optimize an optical element with arbitrary, in particular aspherical surfaces and at least one diffraction grating in the wearing position taking the second-order aberrations (e.g. refractive power and astigmatism) and optionally higher-order aberrations (e.g. coma or spherical aberration) into consideration, it is advantageous to also be able to perform exact wavefront tracing also for the general case.

(76) In the following, the principles of ray and wavefront tracing in the general case of an optical element or an optical system (e.g. a spectacle lens) with a diffraction grating will be described in more detail.

(77) Coordinate Systems

(78) First of all, variables capable of describing a boundary surface including at least one diffraction grating as generally as possible will be introduced. To this end, by analogy with the case of purely refractive surfaces, two types of coordinates or coordinate systems are used in principle.

(79) One type is global coordinates x.sup.0,y.sup.0,z.sup.0, which serve to describe the base surface (i.e. the purely refractive surface without the diffraction grating by its vertex depth z.sup.0(x.sup.0,y.sup.0). Here, the possibly existing vertex depth of the diffraction grating is not added to the vertex depth of the base surface. Instead, the diffraction grating itself is described by a separate property h(x.sup.0,y.sup.0). Here, h(x.sup.0,y.sup.0) can play the role of an additional vertex depth, so that the real (microscopic) physical vertex depth of the base surface is determined by z.sub.m.sup.0(x.sup.0,y.sup.0)=z.sup.0(x.sup.0,y.sup.0)+h(x.sup.0,y.sup.0). However, it is possible for h(x.sup.0,y.sup.0) to play the role of a transmission property or another property of the POE.

(80) The other type of coordinates is—as described above—local coordinates x,y,z, which serve to describe the passage of a ray through the boundary surface such that (x,y,z)=(0,0,0) applies at the penetration point and that the z axis is perpendicular to the base surface there. A special possible choice of such a local system is to require that the incident ray be e.g. in the x-z plane or in the y-z plane, for example. In general, however, this condition does not necessarily have to be satisfied. The use of local systems is known from WO 2008/089999 A1, for example, and is used in second-order wavefront tracing through surfaces without a diffraction grating. Typically, use is made of only a global coordinate system, but of as many local systems as main rays are to be calculated.

(81) FIG. 3A illustrates the position of the coordinate system x,y,z of the incident wavefront with respect to the coordinate system x,y,z of the refractive surface (base surface), expressed by the angles φ.sub.x, φ.sub.y, φ, and χ.

(82) For the sake of simplicity, FIG. 3A only shows the coordinate system of the incident wavefront and the coordinate system of the refractive surface. The coordinate system x′,y′,z′ of the outgoing wavefront can be specified by analogy with the coordinate system of the incident wavefront. Moreover, reference is made to FIG. 1 of the publication [G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, and D. Uttenweiler, “Derivation of the refraction equations for higher-order aberrations of local wavefronts at oblique incidence”, JOSA. A, vol. 27, no. 2, February 2010, pages 218-237], which shows a two-dimensional representation of the corresponding mutual position for all three coordinate systems.

(83) Description of a Diffraction Grating by the Phase Function Ψ(x,y)

(84) The phase function Ψ(x,y) represents the optical path length difference (optical path difference or OPD), which is introduced by the diffraction grating, as a function of the coordinates x,y tangentially to the refractive surface (base surface). The phase function Ψ(x,y) can be determined by means of the grating lines. Conversely, with a predetermined phase function, it is possible to determine the grating lines of the corresponding diffraction grating.

(85) In the simplest case of a constant, equidistant diffraction grating 122 on a plane surface 120 (cf. e.g. FIG. 3 and FIG. 4), which is described in global coordinates by z.sup.0(x.sup.0,y.sup.0)=a.sub.xx.sup.0+a.sub.yy.sup.0+t, it is possible to differentiate between a single-periodic and a double-periodic grating.

(86) In a single-periodic grating, a period vector d.sub.1 exists, so that
h(r.sup.0+d.sub.1)=h(r.sup.0)  (106a)
holds for all points r.sup.0=(x.sup.0,y.sup.0). Moreover, there exists a direction with translation invariance, i.e. a vector v with
h(r.sup.0+αv)=h(r.sup.0)  (106b)
for all α.

(87) In such a case, the grating lines 112a face toward v, whereas d.sub.1 does not necessarily have to describe the distance between the grating lines 122a, since d.sub.1 does not necessarily have to be perpendicular to v. In such a case, the vector d.sub.1 can be replaced by the vector with the components

(88) $d=\left(\begin{array}{c}{d}_x\\ d_y\end{array}\right),$
which is defined by
d=d.sub.1−(d.sub.1.Math.v)v  (107)

(89) This vector d is perpendicular to v and it further holds that

(90) $\begin{array}{cc}\begin{array}{c}h\left({\stackrel{\_}{r}}^0+d\right)=h\left({\stackrel{\_}{r}}^0+d_1-\left(d_1.Math.v\right)v\right)\\ =h\left({\stackrel{\_}{r}}^0+d_1-\alpha v\right)\\ =h\left({\stackrel{\_}{r}}^0+d_1\right)\\ =h\left({\stackrel{\_}{r}}^0\right)\end{array}& \left(108\right)\end{array}$
so that the vector d is also a period vector. However, in contrast to d.sub.1, the vector d also indicates the distance between two grating lines (cf. FIG. 4). The grating period is determined by the amount d=|d|.

(91) In a single-periodic grating 122, as shown in FIG. 4, for example, two further period vectors of practical importance exist in addition. They depend on the coordinate system and are determined by the vectors

(92) $\begin{array}{cc}{\delta }_x=\left(\begin{array}{c}{\delta }_x\\ 0\end{array}\right),{\delta }_y=\left(\begin{array}{c}0\\ {\delta }_y\end{array}\right),& \left(109\right)\end{array}$
which face toward the coordinate axes (cf. FIG. 4). The connection between d and the vectors δ.sub.x,δ.sub.y is determined by:

(93) 0$\begin{array}{cc}{\delta }_x=\frac{{.Math.d.Math.}^2}{d_x},{\delta }_y=\frac{{.Math.d.Math.}^2}{d_y},\mathrm{and}& \left(110a\right)\\ d_x=\frac{{\delta }_x{\delta }_y^2}{{\delta }_x^2+{\delta }_y^2},d_y=\frac{{\delta }_y{\delta }_x^2}{{\delta }_x^2+{\delta }_y^2}& \left(110b\right)\end{array}$

(94) In a double-periodic grating 122, two period vectors d.sub.1,d.sub.2 with
h(r.sup.0+d.sub.1)=h(r.sub.S)
h(r.sup.0+d.sub.2)=h(r.sub.S).  (111)
exist. Double-periodic within this scope means that there is no translation invariance in any direction, i.e. there is no vector v with h(r.sup.0+αv)=h(r.sub.S) for all α.

(95) From an inspection of the wave optics, one can say that a plane monochromatic wave of the wavelength λ, which is incident on a single- or double-periodic grating in an oblique manner, leads to a direction-dependent intensity distribution on the side of emergence due to interference. This distribution can be represented as a product of two direction-dependent factors, wherein said one factor (the form factor) is only determined by the form of the diffraction grating within the grating period, and the second factor (the grating or diffraction factor) is determined only by the grating periodicity. The second factor takes on a maximum in each of such directions in which the path difference between two points of the wave field on the boundary surface, which are displaced by one grating period, is an integer multiple of the wavelength of the light on the side of emergence.

(96) If, in the image of the geometric ray optics, the incident wave field is assigned the directional vector

(97) $\begin{array}{cc}N=\left(\begin{array}{c}\sin {\phi }_x\\ \sin {\phi }_y\\ \sqrt{1-{\sin }^2{\phi }_x-{\sin }^2{\phi }_y}\end{array}\right)& \left(112a\right)\end{array}$
and, on the side of emergence of each direction in which a maximum of the grating factor exists, a directional vector of the form

(98) $\begin{array}{cc}{N}^{\prime }=\left(\begin{array}{c}\sin {\phi }_x^{\prime }\\ \sin {\phi }_y^{\prime }\\ \sqrt{1-{\sin }^2{\phi }_x^{\prime }-{\sin }^2{\phi }_y^{\prime }}\end{array}\right)& \left(112b\right)\end{array}$
then the rays will be described by the laws

(99) $\begin{array}{cc}{n}^{\prime }\sin {\phi }_x^{\prime }-n\sin {\phi }_x=\frac{m_x\lambda }{{\delta }_x}{n}^{\prime }\sin {\phi }_y^{\prime }-n\sin {\phi }_y=\frac{m_y\lambda }{{\delta }_y}& \left(113\right)\end{array}$
where m.sub.x= . . . , −3, −2, −1, 0, 1, 2, 3, . . . and m.sub.y= . . . , −3, −2, −1, 0, 1, 2, 3, . . . are integers. The laws (113) can be considered to be the extended laws of refraction in the presence of a diffraction grating. In particular, the case m.sub.x=0, m.sub.y=0, i.e. the zeroth diffraction order, describes the situation without diffraction elements.

(100) In a double-periodic diffraction element, all integers m.sub.x,m.sub.y can be found independent of each other. In a single-periodic grating, only diffraction orders m.sub.x=σ.Math.m.sub.y of the same amount can be found, where σ=+1 applies to the case that the grating lines decrease for increasing values of x.sup.0 (such as in FIG. 4, ∂y.sup.0/∂x.sup.0<0) and σ=−1 applies in the case of increasing grating lines (∂y.sup.0/∂x.sup.0>0).

(101) In the following, single-periodic diffraction gratings (m:=m.sub.x=σ.Math.m.sub.y) will be discussed. However, all calculations can be modified accordingly for the case of double-periodic diffraction gratings.

(102) The equation (113) with m:=m.sub.x=σ.Math.m.sub.y on the right side can be interpreted such that two rays, which are refracted at two neighboring grating lines, have a non-vanishing path difference, i.e. a phase difference proportional to m and proportional to λ. Thus, there is the possibility of characterizing the course of the grating lines, namely on the one hand by grating properties that can be measured (e.g. with a microscope) and are based on the vector d, and on the other hand by the abstract property of introducing an additional location-dependent path difference into the ray path. In the second case, the course of the grating lines is determined by the difference between the values of a phase function Ψ.sup.0(x.sup.0,y.sup.0;λ,m), which in addition to the coordinates x.sup.0, y.sup.0 also depends on the wavelength λ and on the diffraction order m. Since this phase function is in any case proportional to λ and m these factors can be split off. Instead of the phase function Ψ.sup.0(x.sup.0,y.sup.0;λ,m), the phase function ψ.sup.0(x.sup.0,y.sup.0) can be used, where
Ψ.sup.0(x.sup.0,y.sup.0;λ,m)=mλ.Math.ψ.sup.0(x.sup.0,y.sup.0).  (114)

(103) FIG. 5 shows grating lines of a diffraction grating 122 on a plane boundary surface 120, and FIG. 6 shows grating lines of a diffraction grating 122 on a curved boundary surface 122.

(104) Except for the simplest case of a constant, equidistant grating on a plane surface, the grating lines extend in a different direction at each point of an optical element in the general case, as is shown in FIG. 5 and FIG. 6, for example. Moreover, their distance is generally different at each point (cf. e.g. FIG. 5). Strict periodicity is not present any more in principle. Consequently, the period vector d cannot be defined any more. Therefore, it is suggested replacing the period vector d by a coordinate-dependent function d(x.sup.0,y.sup.0) or, put differently, by a vector field) d(x.sup.0,y.sup.0) defined as the tangential vector field with respect to the trajectories 126, which are orthogonal to the grating lines 122a.

(105) In addition, in the general case of a curved base surface 120, as shown in FIG. 6, it has be to taken into account that the grating 122 is specified in global coordinates x.sup.0, y.sup.0 on the one hand, but local properties are relevant for an effect on the ray passage on the other hand, such as the local grating distance the grating 122 has along the tilted axes of a local coordinate system x,y.

(106) Instead of d(x.sup.0,y.sup.0), the effect of the grating 122 can be described by the phase function ψ(x.sup.0,y.sup.0) in this general case as well.

(107) The phase function of a diffraction grating ψ.sup.0(x.sup.0,y.sup.0) is more suitable for wavefront tracing than the vector field d(x.sup.0,y.sup.0), but it cannot be measured directly. In order to perform a wavefront tracing based on the phase function) ψ.sup.0(x.sup.0,y.sup.0), a method for determining the transition between the functions d(x.sup.0,y.sup.0) and ψ.sup.0(x.sup.0,y.sup.0) in both directions (i.e. d(x.sup.0,y.sup.0)←ψ.sup.0(x.sup.0,y.sup.0)) is proposed. The flow diagram shown in FIG. 7 illustrates the transition between the vector field d(x.sup.0,y.sup.0) and the phase function ψ.sup.0(x.sup.0,y.sup.0).

(108) In particular, in a predetermined grating, which can be known e.g. by measuring (cf. block 130 in FIG. 7) the microscope image of a grating projection or by a projection of another measurable property of the grating (e.g. a transmission property), the phase function ψ.sup.0(x.sup.0,y.sup.0) can be obtained in the global coordinate system (cf. block 132 in FIG. 7) by counting the grating lines and interpreting them as curves ψ.sup.0(x.sup.0,y.sup.0)=const. For the curves, the values ψ.sup.0(x.sup.0,y.sup.0)=0, ψ.sup.0(x.sup.0,y.sup.0)=1, ψ.sup.0(x.sup.0,y.sup.0)=2, etc., are assumed successively (cf. FIG. 7). For all points (x.sup.0,y.sup.0) not on but between the grating lines, the values of the phase function can be determined by suitable interpolation. Conversely, if the phase function ψ.sup.0(x.sup.0,y.sup.0) is known, the grating lines can be determined by calculating the curves ψ.sup.0(x.sup.0,y.sup.0)=const. with ψ.sup.0(x.sup.0,y.sup.0)=0, ψ.sup.0(x.sup.0,y.sup.0)=1, ψ.sup.0(x.sup.0,y.sup.0)=2, etc.

(109) After a local coordinate system (x,y,z) has been set, the phase function relevant for local ray tracing is the function
ψ(x,y)=ψ.sup.0(x.sup.0(x,y),y.sup.0(x,y)),  (115)
(cf. block 134 in FIG. 7), where the connections x.sup.0(x,y), y.sup.0(x,y) result from the transformation from the global coordinate system to the local coordinate system (for the respective penetration point). By setting (cf. block 136 in FIG. 7)
Ψ(x,y;λ,m)=mλ.Math.ψ(x,y)  (116)
analogously to equation (114), the local phase function can be obtained taking the diffraction order and the wavelength into account.
Wavefront Tracing in the Case of an Optical Element/System Comprising at Least One Diffraction Grating
First-Order Properties (Ray Deviation)

(110) For the tracing of rays described by the vectors N,N′ in the local system (cf. equations (112a) and (112b)), the wavefront tracing yields the laws for the ray deviation

(111) $\begin{array}{cc}{n}^{\prime }\sin {\phi }_x^{\prime }-n\sin {\phi }_x=\frac{\partial }{\partial x}\Psi \left(\stackrel{\_}{x},\stackrel{\_}{y};\lambda ,m\right)n^{\prime }\sin {\phi }_y^{\prime }-n\sin {\phi }_y=\frac{\partial }{\partial y}\Psi \left(\stackrel{\_}{x},\stackrel{\_}{y};\lambda ,m\right).& \left(117\right)\end{array}$

Example 1

(112) In the simplest case of a constant equidistant grating on a plane surface, which corresponds e.g. to FIG. 4, the phase function in the global system is determined by

(113) $\begin{array}{cc}{\psi }^0\left({\stackrel{\_}{x}}^0,{\stackrel{\_}{y}}^0\right)=\frac{{\stackrel{\_}{x}}^0}{{\delta }_x}+\frac{{\stackrel{\_}{y}}^0}{{\delta }_y}+{\psi }_0& \left(118\right)\end{array}$
where ψ.sub.0 is a constant. Since the base surface is plane, the local system can be selected identically with the global system, so that ψ(x,y)=ψ.sup.0(x,y). Since in this case it holds that

(114) $\begin{array}{cc}\frac{\partial }{\partial \stackrel{\_}{x}}\Psi \left(\stackrel{\_}{x},\stackrel{\_}{y};\lambda ,m\right)=\frac{m\lambda }{{\delta }_x}\frac{\partial }{\partial \stackrel{\_}{y}}\Psi \left(\stackrel{\_}{x},\stackrel{\_}{y};\lambda ,m\right)=\frac{m\lambda }{{\delta }_y}& \left(119\right)\end{array}$
equation (117) leads exactly to the special case of equation (113).

Example 2

(115) If, for an arbitrary grating, the local system x,y,z is selected at the penetration point such that the incident ray is in the y-z plane, then φ.sub.x=0. If the phase function in this local system is determined by Ψ(x,y;λ,m), then the laws for ray deviation according to equation (117) will read

(116) $\begin{array}{cc}{n}^{\prime }\sin {\phi }_x^{\prime }=\frac{\partial }{\partial \stackrel{\_}{x}}\Psi \left(\stackrel{\_}{x},\stackrel{\_}{y};\lambda ,m\right)n^{\prime }\sin {\phi }_y^{\prime }-n\sin {\phi }_y=\frac{\partial }{\partial \stackrel{\_}{y}}\Psi \left(\stackrel{\_}{x},\stackrel{\_}{y};\lambda ,m\right).& \left(120\right)\end{array}$

(117) If δΨ(x,y;λ,m)/≢x≠0, then φ′.sub.x#0. This means that—if the grating lines are not perpendicular to the plane of incidence—a ray deviation to the side will take place and the plane of emergence will not coincide with the plane of incidence (other than in the case of a mere refraction). If, conversely, ≢Ψ(x,y;λ,m)/≢x=0, then φ′.sub.x=0 and the ray deviation will take place only in the y-z plane.

(118) Second-Order Properties (Curvature Properties of the Wavefront)

(119) In order to describe wavefront properties, it is suggested that the ray tracing of a main ray passing through an evaluation point of the optical element be performed first. Thus, the main ray differs from possible neighboring rays that pass off the evaluation point. In the exemplary case of a spectacle lens, a main ray is particularly a light ray that, starting from the object point, passes through the center of the entrance pupil. Upon eye movements, the entrance pupil coincides with the ocular center of rotation, and not necessarily with the physical pupil of the eye. The angles φ.sub.x, φ.sub.y, φ′.sub.x, φ′.sub.y and thus the vectors N,N′ in equations (112a) and (112b) are known after this step.

(120) In addition, it is suggested that except for a local coordinate system, which serves to describe the base surface and in which also the incident and the outgoing rays are described, yet further coordinate systems be introduced as well.

(121) The coordinate system (x,y,z) serves to describe the incident wavefront and is directed such that the z axis is directed in the direction of light along the incident ray direction N and that the origin (x,y,z)=(0,0,0) coincides with the point (x,y,z)=(0,0,0).

(122) The coordinate system (x′,y′,z′) serves to describe the outgoing wavefront and is directed such that the z′ axis is directed in the direction of light along the outgoing ray direction N′ and that the origin (x′,y′,z′)=(0,0,0) also coincides with the point (x,y,z)=(0,0,0).

(123) The coordinates of a spatial vector can be described either by the variable v=(v.sub.x,v.sub.y,v.sub.z) in the coordinate system (x,y,z), by the variable v′=(v′.sub.x,v′.sub.y,v′.sub.z) in the coordinate system (x′,y′,z′), or by the variable v=(v.sub.x,v.sub.y,v.sub.z) in the coordinate system (x,y,z). The mutual position of the coordinate systems depends on the rays and is only set except for the degrees of freedom, which corresponds to a rotation of the system (x,y,z) about the incident ray and to a second independent rotation of the system (x′,y′,z′) about the outgoing ray.

(124) Preferably, the mutual position of the coordinate systems is set by

(125) $\begin{array}{cc}v=R.Math.\stackrel{\_}{v}R=R_z\left(\chi \right)R_x\left(\phi \right)R_z\left(-\chi \right)v^{\prime }=R^{\prime }.Math.\stackrel{\_}{v}{R}^{\prime }=R_z\left({\chi }^{\prime }\right)R_x\left({\phi }^{\prime }\right)R_z\left(-{\chi }^{\prime }\right)\mathrm{where}& \left(121\right)\\ R_x\left(\mathrm{.Math.}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \mathrm{.Math.}& -\sin \mathrm{.Math.}\\ 0& \sin \mathrm{.Math.}& \cos \mathrm{.Math.}\end{array}\right),R_y\left(\mathrm{.Math.}\right)=\left(\begin{array}{ccc}\cos \mathrm{.Math.}& 0& \sin \mathrm{.Math.}\\ 0& 1& 0\\ -\sin \mathrm{.Math.}& 0& \cos \mathrm{.Math.}\end{array}\right),R_z\left(\mathrm{.Math.}\right)=\left(\begin{array}{ccc}\cos \mathrm{.Math.}& -\sin \mathrm{.Math.}& 0\\ \sin \mathrm{.Math.}& \cos \mathrm{.Math.}& 0\\ 0& 0& 1\end{array}\right).& \left(122\right)\end{array}$

(126) The auxiliary angles φ, φ′, χ, χ′ in equation (121) must be expressed by the variables φ.sub.x, φ.sub.y, φ′.sub.x, φ′.sub.y known before the wavefront tracing. The matrices R, R′ are constructed such that φ, φ′ are the angles of incidence and emergence with respect to the surface normal, and it holds that

(127) $\begin{array}{cc}\cos \phi =\sqrt{1-{\sin }^2{\phi }_x-{\sin }^2{\phi }_y}\sin \phi =\sqrt{{\sin }^2{\phi }_x+{\sin }^2{\phi }_y},\cos {\phi }^{\prime }=\sqrt{1-{\sin }^2{\phi }_x^{\prime }-{\sin }^2{\phi }_y^{\prime }}\sin {\phi }^{\prime }=\sqrt{{\sin }^2{\phi }_x^{\prime }+{\sin }^2{\phi }_y^{\prime }},\mathrm{and}& \left(123\right)\\ \sin \chi =\frac{-\sin {\phi }_x}{\sqrt{{\sin }^2{\phi }_x+{\sin }^2{\phi }_y}}\cos \chi =\frac{\sin {\phi }_y}{\sqrt{{\sin }^2{\phi }_x+{\sin }^2{\phi }_y}},\sin {\chi }^{\prime }=\frac{-\sin {\phi }_x^{\prime }}{\sqrt{{\sin }^2{\phi }_x^{\prime }+{\sin }^2{\phi }_y^{\prime }}}\cos {\chi }^{\prime }=\frac{\sin {\phi }_y^{\prime }}{\sqrt{{\sin }^2{\phi }_x^{\prime }+{\sin }^2{\phi }_y^{\prime }}}.& \left(124\right)\end{array}$

(128) From the above equations (123) and (124), it follows that

(129) 0$\begin{array}{cc}\tan \chi =-\frac{\sin {\phi }_x}{\sin {\phi }_y},\tan {\chi }^{\prime }=-\frac{\sin {\phi }_x^{\prime }}{\sin {\phi }_y^{\prime }}.& \left(126\right)\end{array}$

(130) If the incident wavefront in the local coordinate system (x,y,z) is determined by W.sub.In(x,y), the refractive base surface in the system (x,y,z) will be determined by S(x,y), and the sought-for outgoing wavefront in the system (x′,y′,z′) is determined by W′.sub.Out(x′,y′) then the following formulae (127) and (128) will describe the dependence of the second local derivatives of the wavefronts (i.e. the incident and outgoing wavefronts), of the base surface, and of the phase Ψ(x,y;λ,m).

(131) $\begin{array}{cc}{n}^{\prime }{R}_2\left({\chi }^{\prime }\right)C_2^{\prime }{R}_2\left(-{\chi }^{\prime }\right)\left(\begin{array}{c}{W}_{\mathrm{Out}}^{\prime \left(2,0\right)}\left(0,0\right)\\ W_{\mathrm{Out}}^{\prime \left(1,1\right)}\left(0,0\right)\\ W_{\mathrm{Out}}^{\prime \left(0,2\right)}\left(0,0\right)\end{array}\right)-{\mathrm{nR}}_2\left(\chi \right)C_2{R}_2\left(-\chi \right)\left(\begin{array}{c}{W}_{\mathrm{In}}^{\left(2,0\right)}\left(0,0\right)\\ W_{\mathrm{In}}^{\left(1,1\right)}\left(0,0\right)\\ W_{\mathrm{In}}^{\left(0,2\right)}\left(0,0\right)\end{array}\right)==v\left(\begin{array}{c}{\stackrel{\_}{S}}^{\left(2,0\right)}\left(0,0\right)\\ {\stackrel{\_}{S}}^{\left(1,1\right)}\left(0,0\right)\\ {\stackrel{\_}{S}}^{\left(0,2\right)}\left(0,0\right)\end{array}\right)-\left(\begin{array}{c}{\Psi }^{\left(2,0\right)}\left(0,0\right)\\ {\Psi }^{\left(1,1\right)}\left(0,0\right)\\ {\Psi }^{\left(0,2\right)}\left(0,0\right)\end{array}\right),\mathrm{where}& \left(127\right)\\ \begin{array}{c}v=n^{\prime }\cos {\phi }^{\prime }-n\cos \phi \\ =n^{\prime }\sqrt{1-{\sin }^2{\phi }_x^{\prime }-{\sin }^2{\phi }_y^{\prime }}-n\sqrt{1-{\sin }^2{\phi }_x-{\sin }^2{\phi }_y}\end{array}& \left(128\right)\end{array}$
is satisfied. The phase Ψ(x,y;λ,m) is the phase defined in equation (116).

(132) The individual terms of equation (127) correspond to the terms of the BAD equation B−A=D+PK.sup.(2).

(133) In equation (127), the superscript symbols represent derivatives. It holds for an arbitrary function h(x,y) that:
h.sup.(k−m,m)(0,0):=≢.sup.k/≢x.sup.k−m≢y.sup.mh(x,y)|.sub.x=0,y=0.  (129)

(134) The function h(x,y) in the formula (129) optionally plays the role of the functions W.sub.In(x,y), W′.sub.Out(x′,y′), S(x,y) and Ψ(x,y;λ,m), wherein in the case of Ψ(x,y;λ,m), the derivatives refer to x,y. The matrices C.sub.2 and C′.sub.2 are defined as in the purely refractive case (cf. e.g. G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa and D. Uttenweiler, “Derivation of the refraction equations for higher-order aberrations of local wavefronts at oblique incidence”, J. Opt. Soc. Am. A/Vol. 27, No. 2/February 2010):

(135) $\begin{array}{cc}{C}_2=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \phi & 0\\ 0& 0& {\cos }^2\phi \end{array}\right),C_2^{\prime }=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos {\phi }^{\prime }& 0\\ 0& 0& {\cos }^2{\phi }^{\prime }\end{array}\right).& \left(130\right)\end{array}$

(136) In addition, the matrix R.sub.2(χ) is taken into account in equation (127), which describes the rotation of the wavefront. If, generally, a wavefront is determined by the function w(x,y), it will be described in rotated coordinates

(137) $\begin{array}{cc}\left(\begin{array}{c}\tilde{x}\\ \tilde{y}\end{array}\right)=\mathrm{Rot}\left(\alpha \right)\left(\begin{array}{c}x\\ y\end{array}\right)\mathrm{with}\mathrm{Rot}\left(\alpha \right)=\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right)& \left(132\right)\end{array}$
by the transformed function
{tilde over (w)}({tilde over (x)},{tilde over (y)})=w(x({tilde over (x)},{tilde over (y)}),y({tilde over (x)},{tilde over (y)}))  (133)

(138) The k.sup.th-order derivative

(139) $\frac{{\partial }^k}{\partial {\tilde{x}}^m\partial {\tilde{y}}^{k-m}}\tilde{w}\left(\tilde{x},\tilde{y}\right)$
with respect to the rotated coordinates can be expressed as a linear combination of the derivatives

(140) $\frac{{\partial }^k}{\partial x^l\partial y^{k-l}}w\left(x,y\right)$
with respect to the original coordinates. The (k+1)×(k+1) matrix R.sub.k(χ) describes the transition between the k.sup.th-order derivative in the coordinate system (x,y) and the k.sup.th-order derivative in the coordinate system ({tilde over (x)},{tilde over (y)}).

(141) $\begin{array}{cc}\frac{{\partial }^k}{\partial {\tilde{x}}^m\partial {\tilde{y}}^{k-m}}\tilde{w}\left(\tilde{x},\tilde{y}\right)=\underset{l=0}{\overset{k}{.Math.}}{\left(R_k\left(\chi \right)\right)}_{\left(m+1\right),\left(l+1\right)}\frac{{\partial }^k}{\partial x^l\partial y^{k-l}}w\left(x,y\right),m=0,\mathrm{.Math.}k,l=0,\mathrm{.Math.},k& \left(134\right)\end{array}$

(142) It explicitly holds for the first three orders that

(143) $\begin{array}{cc}{R}_1\left(\alpha \right)=\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right)=\mathrm{Rot}\left(\alpha \right)R_2\left(\alpha \right)=\left(\begin{array}{ccc}{\cos }^2\alpha & -2\cos \alpha \sin \alpha & {\sin }^2\alpha \\ \cos \alpha \sin \alpha & {\cos }^2\alpha -{\sin }^2\alpha & -\cos \alpha \sin \alpha \\ {\sin }^2\alpha & 2\cos \alpha \sin \alpha & {\cos }^2\alpha \end{array}\right)R_3\left(\alpha \right)=\left(\begin{array}{cccc}{\cos }^3\alpha & -3{\cos }^2\alpha \sin \alpha & 3\cos \alpha {\sin }^2\alpha & -{\sin }^3\alpha \\ {\cos }^2\alpha \sin \alpha & {\cos }^3\alpha -2\cos \alpha {\sin }^2\alpha & {\sin }^3\alpha -2{\cos }^2\alpha \sin \alpha & \cos \alpha {\sin }^2\alpha \\ \cos \alpha {\sin }^2\alpha & -\left({\sin }^3\alpha -2{\cos }^2\alpha \sin \alpha \right)& {\cos }^3\alpha -2\cos \alpha {\sin }^2\alpha & -{\cos }^2\alpha \sin \alpha \\ {\sin }^3\alpha & 3\cos \alpha {\sin }^2\alpha & 3{\cos }^2\alpha \sin \alpha & {\cos }^3\alpha \end{array}\right)& \left(135\right)\end{array}$

(144) In the following, some special cases of optical systems, in particular spectacle lenses with POE, will be discussed. In the case of a perpendicular incidence of light, it holds that:
v=n′−n;
R.sub.2=1;
C.sub.2=1;
C′.sub.2=1.

(145) It yields for equation (127):

(146) $\begin{array}{cc}{n}^{\prime }\left(\begin{array}{c}{W}_{\mathrm{Out}}^{\prime \left(2,0\right)}\left(0,0\right)\\ W_{\mathrm{Out}}^{\prime \left(1,1\right)}\left(0,0\right)\\ W_{\mathrm{Out}}^{\prime \left(0,2\right)}\left(0,0\right)\end{array}\right)-n\left(\begin{array}{c}{W}_{\mathrm{In}}^{\left(2,0\right)}\left(0,0\right)\\ W_{\mathrm{In}}^{\left(1,1\right)}\left(0,0\right)\\ W_{\mathrm{In}}^{\left(0,2\right)}\left(0,0\right)\end{array}\right)=\left(n^{\prime }-n\right)\left(\begin{array}{c}{\stackrel{\_}{S}}^{\left(2,0\right)}\left(0,0\right)\\ {\stackrel{\_}{S}}^{\left(1,1\right)}\left(0,0\right)\\ {\stackrel{\_}{S}}^{\left(0,2\right)}\left(0,0\right)\end{array}\right)-\left(\begin{array}{c}{\Psi }^{\left(2,0\right)}\left(0,0\right)\\ {\Psi }^{\left(1,1\right)}\left(0,0\right)\\ {\Psi }^{\left(0,2\right)}\left(0,0\right)\end{array}\right)& \left(127a\right)\end{array}$

(147) With a one-dimensional problem (meridional plane), instead of mixed derivatives with respect to x,y, only derivatives with respect to one coordinate (e.g. y) occur. Moreover, it holds that
R.sub.2=R.sub.2=1;
C.sub.2=C.sub.2=cos.sup.2φ;
C′.sub.2=C′.sub.2=cos.sup.2φ′.

(148) Consequently, equation (127) can be written as
n′ cos.sup.2φ′W′.sub.Out.sup.(2)(0)−n cos.sup.2φW.sub.In.sup.(2)(0)=vS.sup.(2)(0)−Ψ.sup.(2)(0)  (127b)

(149) With a perpendicular incidence of light and a one-dimensional problem, instead of mixed derivatives with respect to x,y, only derivatives with respect to one coordinate (e.g. y) occur. Moreover, it holds that
v=n′−n
R.sub.2=R.sub.2=1
C.sub.2=C.sub.2=1
C′.sub.2=C′.sub.2=1

(150) Consequently, equation (127) can be written as
n′W.sub.Out.sup.(2)(0)−nW.sub.In.sup.(2)(0)=(n′−n)S.sup.(2)(0)−Ψ.sup.(2)(0)  (127c)
Higher-Order Properties of the Wavefront (3.sup.rd, 4.sup.th, . . . -Order Properties)

(151) By analogy with equation (127), it holds for 3.sup.rd-order derivatives that

(152) $\begin{array}{cc}{n}^{\prime }{R}_3\left({\chi }^{\prime }\right)C_3^{\prime }{R}_3\left(-{\chi }^{\prime }\right)\left(\begin{array}{c}{W}_{\mathrm{Out}}^{\prime \left(3,0\right)}\left(0,0\right)\\ W_{\mathrm{Out}}^{\prime \left(2,1\right)}\left(0,0\right)\\ W_{\mathrm{Out}}^{\prime \left(1,2\right)}\left(0,0\right)\\ W_{\mathrm{Out}}^{\prime \left(0,3\right)}\left(0,0\right)\end{array}\right)-{\mathrm{nR}}_3\left(\chi \right)C_3{R}_3\left(-\chi \right)\left(\begin{array}{c}{W}_{\mathrm{In}}^{\left(3,0\right)}\left(0,0\right)\\ W_{\mathrm{In}}^{\left(2,1\right)}\left(0,0\right)\\ W_{\mathrm{In}}^{\left(1,2\right)}\left(0,0\right)\\ W_{\mathrm{In}}^{\left(0,3\right)}\left(0,0\right)\end{array}\right)==v\left(\begin{array}{c}{\stackrel{\_}{S}}^{\left(3,0\right)}\left(0,0\right)\\ {\stackrel{\_}{S}}^{\left(2,1\right)}\left(0,0\right)\\ {\stackrel{\_}{S}}^{\left(1,2\right)}\left(0,0\right)\\ {\stackrel{\_}{S}}^{\left(0,3\right)}\left(0,0\right)\end{array}\right)-\left(\begin{array}{c}{\Psi }^{\left(3,0\right)}\left(0,0\right)\\ {\Psi }^{\left(2,1\right)}\left(0,0\right)\\ {\Psi }^{\left(1,2\right)}\left(0,0\right)\\ {\Psi }^{\left(0,3\right)}\left(0,0\right)\end{array}\right)+Q_3\left(S^{\left(2\right)},F^{\left(2\right)}\right)& \left(136\right)\end{array}$
wherein analogously to equation (130), it holds that:

(153) 0$\begin{array}{cc}{C}_3=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \phi & 0& 0\\ 0& 0& {\cos }^2\phi & 0\\ 0& 0& 0& {\cos }^3\phi \end{array}\right),C_3^{\prime }=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\phi }^{\prime }& 0& 0\\ 0& 0& {\cos }^2{\phi }^{\prime }& 0\\ 0& 0& 0& {\cos }^3{\phi }^{\prime }\end{array}\right).& \left(137\right)\end{array}$

(154) Equations for even higher orders k=4, 5, . . . can be formed analogously.

(155) With the above-described procedure, it is possible to describe an optical system (e.g. a spectacle lens) with at least one diffraction grating in an exact manner. In particular, it is possible to describe the second and higher-order properties of the wavefront exiting the optical system in an exact manner. On the basis of the second and higher-order properties of the wavefront, the aberrations of the optical element (e.g. of the spectacle lens) can be determined in a manner per se known. In this respect, special reference is made to document WO 2008/089999 A1, to the article by W. Becken et al. “Brillengläser im Sport: Optimierung der Abbildungseigenschaften unter physiologischen Aspekten”, Z. Med. Phys., 17 (2007), pages 56-66, or to the article by G. Esser “Derivation of the refraction equations for higher-order aberrations of local wavefronts at oblique incidence”, JOSA A, vol. 27, no. 2, February 2010, pages 218-237. These documents are explicitly referred to with regard to the technical terminology used as well as the symbols used in equations (121) to (137) and the connection of sphere, cylinder, axis (SZA values) to wavefront properties. Thus, the corresponding explanations constitute an integral part of disclosure of the present application.

(156) As explained above, the method for assessing at least one optical property at at least one evaluation point of an optical element can be part of an optimization and a production process. In the following examples, the optical element is a spectacle lens and the at least one phase-modifying optical element is a diffraction grating. However, the optical element can also be an arbitrary optical element, such as a lens, a lens system, etc. The phase-modifying optical element can also be an arbitrary phase-delaying or phase-modulating element. The optical properties of the optical element (astigmatism and refractive error) are calculated or assessed by means of a method for assessing the at least one optical property according to one of the above-described preferred examples.

(157) Different procedures for performing a preferred optimization method according to the invention exist.

(158) Iterative Method

(159) In this method, at least one of the refractive surfaces of the spectacle lens, which contribute to the refraction, and the at least one diffraction grating are successively optimized or modified.

(160) In a first step, a monochromatic optimization of at least one of the surfaces (hereinafter referred to as lens surfaces), which contribute to the refraction, of a predetermined spectacle lens not having a diffraction grating is performed. In particular, at least one of the lens surfaces is modified and optimized until the refractive error and/or the astigmatic error is/are minimized. The monochromatic optimization is performed such that the refractive power in at least one reference point of the spectacle lens takes on the prescription value S.sub.prescription (S.sub.ref,0(λ.sub.d)=S.sub.prescription). The prescription value is the value that is determined for a spectacles wearer e.g. by refraction determination and that is required for correcting a visual defect of the spectacles wearer. The reference point can be the distance reference point, the centration or fitting point, the optical or geometric center of the spectacle lens, or any other suitable point on the spectacle lens. The monochromatic optimization can be performed for an arbitrary, suitable wavelength; preferably, the monochromatic optimization is performed for the wavelength λ.sub.d, which is also taken into account in the definition of the Abbe number. The spectacle lens is preferably optimized in the wearing position. Methods for the monochromatic optimization of spectacle lenses are known from the prior art (e.g. WO 2008/089999 A1).

(161) In a third step, a diffraction grating is added with lens surfaces being maintained. The diffraction grating is determined or calculated so as to optimally correct the color fringe of the spectacle lens optimized in the preceding step. However, by adding the diffraction grating, a refractive error is introduced.

(162) In a fourth step, with the grating being maintained, at least one of the lens surfaces contributing to the refraction can be optimized again until the refractive error, which has been introduced by the grating, is compensated for. In a next step, the grating is adjusted in order to compensate for the color fringe that formed due to the surface modification in the fourth step.

(163) Since for ordinary Abbe numbers the refractive power of a grating, which is to compensate for the color fringe of a spectacle lens, is in the order of 5% to 10% of the refractive power of the spectacle lens, the modifications in the iterative method become increasingly less, so that the method generally converges and can be discontinued after a suitably selected step.

(164) In this method, both the color fringe and the refractive error take on their conceivable minima only approximately, since the method is discontinued after a finite number of steps.

(165) Provision of Refractive Power

(166) As explained above, a diffraction grating also exhibits refractive power. The actual refractive power of a spectacle lens having a diffraction grating can be represented as the sum of the refractive power of the spectacle lens formed by purely refractive surfaces and of the refractive power of the diffraction grating. In particular, the actual refractive power S.sub.0(λ.sub.d) of a color fringe-corrected lens with a refractive power S.sub.ref,0(λ.sub.d) is determined by

(167) $\begin{array}{cc}{S}_0\left({\lambda }_d\right)=S_{\mathrm{ref},0}\left({\lambda }_d\right)\left(1-\frac{1}{v_d}.Math.\frac{{\lambda }_d}{{\lambda }_F-{\lambda }_C}\right),& \left(138\right)\end{array}$
where v.sub.d=(n.sub.d−1)/(n.sub.F−n.sub.C) is the (alternative) Abbe number of the optical material of the spectacle lens, and n.sub.d, n.sub.F and n.sub.C are the refractive indices of the optical material at the wavelengths λ.sub.d, λ.sub.F, and λ.sub.C. Preferably, λ.sub.d=587.562 nm, λ.sub.F=486.134 nm, and λ.sub.C=656.281 nm.

(168) Instead of optimizing at least one of the lens surfaces in an iterative process such that the refractive power of the spectacle lens in at least one reference point takes on the prescription value S.sub.prescription (S.sub.ref,0(λ.sub.d)=S.sub.prescription), the modification of the refractive power can be provided by a predetermined diffraction grating. This means that the optimization of the at least one lens surface in a first step is categorically organized such that the later total refractive power of the spectacle lens with the grating in at least one reference point takes on the prescription value, so that S.sub.0(λ.sub.d)=S.sub.prescription is satisfied. Consequently, the refractive, monochromatic optimization of the lens surfaces is performed such that for the refractive power of the spectacle lens in the predetermined reference point the condition

(169) $\begin{array}{cc}{S}_{\mathrm{ref},0}\left({\lambda }_d\right)=\frac{S_{\mathrm{prescription}}}{\left(1-\frac{1}{v_d}.Math.\frac{{\lambda }_d}{{\lambda }_F-{\lambda }_C}\right)}& \left(139\right)\end{array}$
is satisfied.

(170) In a second step, a diffraction grating is calculated and introduced such that it compensates for the color fringe of the spectacle lens of the first step. The thus added refractive error of the spectacle lens is exactly such that the total refractive power of the lens is S.sub.0(λ.sub.d)=S.sub.prescription.

(171) The method has the advantage that the color fringe is always optimally compensated for (within the possibilities of the grating). The refractive error, however, is only corrected approximately, since after introducing the diffraction grating, the lens surfaces of the spectacle lens cannot again be optimized in the wearing position.

(172) Provision of Diffractive Power

(173) Instead of determining the required grating at a later point by means of provision of the refractive power in the above-described method, the grating can categorically, i.e. right in a first step, be determined as the grating to compensate for the color fringe of a lens with refractive power S.sub.ref,0(λ.sub.d) (cf. equation (139)). In a second step, the refractive error of the spectacle lens is optimized, wherein with the grating being maintained, an optimization of at least one of the refractive lens surfaces is performed. Preferably, the optimization of at least one of the refractive lens surfaces is performed taking the wavefront tracing with the presence of a diffraction grating into consideration.

(174) The method according to the third example has the advantage that the refractive error is minimized in the wearing position in the best possible way, since the wavefront optimization in the wearing position is the last step to be performed. In this step, all other variables, such as the grating, are already known. A disadvantage might be that the color fringe might not be compensated for in the best possible way, since the grating is determined generally and not depending on the current surfaces in advance.

(175) Simultaneous Optimization of Surfaces and Grating

(176) With regard to a simultaneous optimization of the refractive portions and the diffractive portions of a lens, a preferred embodiment suggests minimizing both the refractive aberrations and the chromatic aberrations or the color fringe by minimizing a target function.

(177) It is known from WO 2008/089999 A1 to perform a monochromatic optimization of a spectacle lens by minimizing the following monochromatic target function:

(178) $\begin{array}{cc}{F}_{\mathrm{monochrom}}=\underset{i}{.Math.}{g}_Z\left(i\right){\left(Z_{\Delta }\left(i\right)-Z_{\Delta ,\mathrm{target}}\left(i\right)\right)}^2+g_S\left(i\right){\left(S_{\Delta }\left(i\right)-S_{\Delta ,\mathrm{target}}\left(i\right)\right)}^2,& \left(140\right)\end{array}$
where S.sub.Δ and Z.sub.Δ are the refractive error of the spherical equivalent or the amount of the astigmatic deviation, S.sub.Δ,target, Z.sub.Δ,target the corresponding target values, and g.sub.Z(i) and g.sub.S(i) the respective weightings.

(179) The image formation properties are evaluated at a predetermined wavelength. The sum over the index i goes over different evaluation points of the spectacle lens. A degree of freedom in the minimization of the target function in equation (140) is usually a vertex depth of at least one refractive surface, which is described by a function z(x,y). The degree of freedom in the optimization can e.g. be the vertex depth z.sub.1(x,y) of the front surface or the vertex depth z.sub.2(x,y) of the back surface of a spectacle lens. It is also possible that both the vertex depth of the front surface and that of the back surface are degrees of freedom in the optimization, as may be the case in a double progressive spectacle lens. The monochromatic target function can be a monocular or a binocular target function. A monochromatic binocular target function is described e.g. in WO 2008 089999 A1 or in the article of W. Becken, et al. “Brillengläser im Sport: Optimierung der Abbildungseigenschaften unter physiologischen Aspekten”, Z. Med. Phys., 17 (2007), pages 56-66. Reference is made to these documents with respect to the technical terms used, and in particular the symbols used in equation (140), as well as to the connection of sphere, cylinder, axis (SZA values) with wavefront properties. Thus, the corresponding explanations constitute an integral part of the disclosure or the present application.

(180) According to one embodiment of the invention, it is suggest that the monochromatic target function be expanded to take the wavelength dependency of the lens with a grating into consideration. In particular, the following three types of target functions are suggested:

(181) $\begin{array}{cc}\mathrm{Type}1)F_1=\underset{\lambda }{.Math.}{F}_{\mathrm{monochrom}}\left(\lambda \right)& \left(141\right)\\ \mathrm{Type}2)F_2=F_{\mathrm{monochrom}}\left({\lambda }_0\right)+\underset{i}{.Math.}{g}_{\mathrm{FLF}}\left(i\right)\times {f\left(S_{\mathrm{SK}}\left(i,{\lambda }_2\right)-S_{\mathrm{SK}}\left(i,{\lambda }_1\right)\right)}^2& \left(142\right)\\ \mathrm{Type}3)F_3=F_{\mathrm{monochrom}}\left({\lambda }_0\right)+\underset{i}{.Math.}{g}_{\mathrm{FQF}}\left(i\right)\times {g\left({\mathrm{\Delta \phi }}_{\mathrm{SK}}\left(i,{\lambda }_2,{\lambda }_1\right)\right)}^2.& \left(143\right)\end{array}$

(182) An example of a target function of type 1 is the target function

(183) $\begin{array}{cc}{F}_1=\underset{i,\lambda }{.Math.}{g}_Z\left(i,\lambda \right){\left(Z_{\Delta }\left(i,\lambda \right)-Z_{\Delta ,\mathrm{target}}\left(i,\lambda \right)\right)}^2+g_s\left(i,\lambda \right){\left(S_{\Delta }\left(i,\lambda \right)-S_{\Delta ,\mathrm{target}}\left(i,\lambda \right)\right)}^2.& \left(141a\right)\end{array}$

(184) An example of a target function of type 2 is the target function

(185) $\begin{array}{cc}{F}_2=\underset{i}{.Math.}\left(g_Z\left(i\right){\left(Z_{\Delta }\left(i,{\lambda }_0\right)-Z_{\Delta ,\mathrm{target}}\left(i,{\lambda }_0\right)\right)}^2+g_s\left(i\right){\left(S_{\Delta }\left(i,{\lambda }_0\right)-S_{\Delta ,\mathrm{target}}\left(i,{\lambda }_0\right)\right)}^2+g_{\mathrm{FLF}}\left(i\right)\times {f\left(S_{\mathrm{SK}}\left(i,{\lambda }_2\right)-S_{\mathrm{SK}}\left(i,{\lambda }_1\right)\right)}^2\right).& \left(142a\right)\end{array}$

(186) An example of a target function of type 3 is the target function

(187) $\begin{array}{cc}{F}_3=\underset{i}{.Math.}\left(g_Z\left(i\right){\left(Z_{\Delta }\left(i,{\lambda }_0\right)-Z_{\Delta ,\mathrm{target}}\left(i,{\lambda }_0\right)\right)}^2+g_s\left(i\right){\left(S_{\Delta }\left(i,{\lambda }_0\right)-S_{\Delta ,\mathrm{target}}\left(i,{\lambda }_0\right)\right)}^2+g_{\mathrm{FQF}}\left(i\right)\times {g\left({\mathrm{\Delta \phi }}_{\mathrm{SK}}\left(i,{\lambda }_2,{\lambda }_1\right)\right)}^2\right).& \left(143a\right)\end{array}$

(188) In the above equations (141a) to (142a):

(189) Z.sub.Δ(i,λ) is the actual value of the astigmatic error or the amount of the astigmatic deviation at the i.sup.th evaluation point of the spectacle lens for the wavelength λ;

(190) Z.sub.Δ,target(i,λ) is the target value of the astigmatic error or the amount of the astigmatic deviation at the i.sup.th evaluation point of the spectacle lens for the wavelength λ;

(191) S.sub.Δ(i,λ) is the actual value of the refractive error or the deviation of the spherical equivalent at the i.sup.th evaluation point of the spectacle lens for the wavelength λ;

(192) S.sub.Δ,target(i,λ) is the target value of the refractive error or the deviation of the spherical equivalent at the i.sup.th evaluation point of the spectacle lens for the wavelength λ;

(193) g.sub.Z(i,λ) is the weighting of the astigmatic error/of the amount of the astigmatic deviation at the i.sup.th evaluation point of the spectacle lens for the wavelength λ;

(194) g.sub.S(i,λ) is the weighting of the refractive error or the deviation of the spherical equivalent at the i.sup.th evaluation point of the spectacle lens for the wavelength λ;

(195) g.sub.FLF(i) is the weighting of the longitudinal chromatic aberration at the i.sup.th evaluation point of the spectacle lens;

(196) g.sub.FQF(i) is the weighting of the lateral chromatic aberration at the i.sup.th evaluation point of the spectacle lens;

(197) S.sub.SK(i,λ.sub.1) is the vergence matrix of the wavefront at the vertex sphere for the wavelength λ.sub.1 and for the i.sup.th evaluation point;

(198) S.sub.SK(i,λ.sub.2) is the vergence matrix of the wavefront at the vertex sphere for the wavelength λ.sub.2 and for the i.sup.th evaluation point; and

(199) Δφ.sub.SK(i,λ.sub.2,λ.sub.1) is the angle between the object-side main rays for two different wavelengths λ.sub.1 and λ.sub.2.

(200) In the target function of type 1, the common monochromatic target function is understood to be a function of the wavelength, i.e. F.sub.monochrom=F.sub.monochrom(λ). Subsequently, this target function is repeatedly evaluated for several wavelengths and summed via a predetermined set of wavelengths. The set of wavelengths comprises at least two wavelengths, e.g. λ.sub.1=λ.sub.F=486,134 nm and λ.sub.2=λ.sub.C=656,281 nm. In addition to the index i, the sum also goes over the wavelength λ correspondingly.

(201) The target function of type 2 is particularly obtained since a common monochromatic target function is evaluated at an operating wavelength λ.sub.0 and since another term is added to this function, which depends on the difference S.sub.SK(i,λ.sub.2)−S.sub.SK(i,λ.sub.1) of the vergence matrices S.sub.SK for two different wavelengths λ.sub.1, λ.sub.2. The type of dependence ƒ(S.sub.SK(i,λ.sub.2)−S.sub.SK(i,λ.sub.1)) can be selected differently. In particular, ƒ(S.sub.SK(i,λ.sub.2)−S.sub.SK(i,λ.sub.1)) can be the dioptric distance between the vergence matrices or the difference of the spherical equivalents. In the latter case, the term ƒ(S.sub.SK(i,λ.sub.2)−S.sub.SK(i,λ.sub.1)) represents a penalty term for the longitudinal chromatic aberration.

(202) The target function of type 3 is obtained by analogy with the target function of type 2, with the difference that the additional term g(Δφ.sub.SK(i,λ.sub.2,λ.sub.1)) is a penalty term for the lateral chromatic aberration determined by the angle Δφ.sub.SK(i,λ.sub.2,λ.sub.1) between the object-side main rays at the i.sup.th evaluation point. Here, g is a suitable function, e.g. the identity, a trigonometric function, or any other function.

(203) Irrespective of the type of target function (target function of type 1, type 2, or type 3), a function describing the diffraction grating is a degree of freedom of the optimization. In addition, the vertex depth z(x,y) of at least one of the surfaces of the spectacle lens is a degree of freedom in the optimization. The vertex depth can be described parametrically. A suitable representation of the vertex depth is a polynomial representation or a representation by means of splines, for example.

(204) Preferably, the function describing the diffraction grating is a phase function Ψ(x,y). The phase function Ψ(x,y) represents the optical path length difference (optical path difference or OPD), which is introduced by the diffraction grating, as a function of the coordinates x,y of a suitably selected coordinate system. Preferably, the phase function Ψ(x,y) is plotted in a local coordinate system (x,y,z), where x,y are the coordinates tangentially to the refracting surface. The description of a diffraction grating by a phase function will be described in detail in the following.

(205) The phase function can preferably be described parametrically, with the number of parameters being equal to the number of degrees of freedom of the phase function. Such a description is particularly suitable in the case that the phase function is to have certain symmetries. Generally, the phase function can be described by splines, like a free-form surface, wherein in the optimization, optionally up to several thousands of spline coefficients are available for variation then.

(206) The calculation of the wearing properties in equations (141) to (143) is performed in the presence of the diffraction grating, described by the current phase function Ψ(x,y). The refractive index relevant for the calculations is determined by its value n(λ) at the respective wavelength λ. The determined grating can generally be asymmetric, since the function Ψ(x,y) is freely optimized and does not or does not have to include any symmetries a priori.

(207) Instead of only one or two refractive surfaces, a higher number of refractive surfaces can be taken into consideration in the equations (141) to (143). For example, several refractive surfaces are present with compound systems. In equations (141) to (143), a higher number (two or more) of gratings can be taken into consideration as well. The several gratings can each be arranged on the boundary surfaces of a compound system, for example.

(208) The target function can also comprise terms that depend on the higher-order aberrations, such a coma, trefoil, spherical aberration. Moreover, instead of the monocular target function in equation (140), a binocular target function can be used as well. A monochromatic binocular target function is described e.g. in WO 2008/089999 A1 or in the publications of W. Becken, et al. “Brillengläser im Sport: Optimierung der Abbildungseigenschaften unter physiologischen Aspekten”, Z. Med. Phys., 17 (2007), pages 56-66.

(209) By introducing a target function in the method for optimizing a spectacle lens with at least one diffraction grating, the problem of meeting the refraction requirements on the spectacle lens and achieving the reduction of chromatic aberrations in the best possible way can be solved to a large extent. In particular, by means of the target function, a compromise between all requirements for a plurality of viewing directions can be sought. The target function can be based on common monochromatic target functions of the prior art; wherein according to a preferred example, it is suggested that additional terms be introduced, which directly include the lateral chromatic aberrations (type 3) and/or directly the longitudinal chromatic aberrations (type 2). Alternatively, the chromatic aberrations can be corrected indirectly by evaluating a monochromatic target function for several different wavelengths and summing it over all wavelengths (type 1). In order to meet the increasing requirements, it is suggested that additional degrees of freedom (parameters) be introduced and varied, namely the degrees of freedom of at least one diffraction grating, which is preferably taken into account in the target function via a modified wavefront tracing.

(210) It depends on the performance of the optimization steps how well the quality of the above-described iterative method, of the method including provision of refractive power, and of the method including provision of diffractive power comes near the quality of the method including a simultaneous optimization of the lens surfaces and of the grating by means of a target function. For example, if also a target function of type 1 is used in the iterative method, the quality of the iterative method including a sequential optimization of the lens surfaces and of the grating can come close to the quality of the method including a simultaneous optimization of the lens surfaces and of the grating by means of a target function, depending on the length of the iteration. An advantage of the iterative method including a sequential optimization of the lens surfaces and of the grating can be that also simpler target functions of the prior art can be used, as long as the wearing position optimization of the surface is performed in the last step. Moreover, if a complete compensation of the color fringe is not intended, but a certain residual color fringe error is admissible, then a relatively general selection of the grating will usually be sufficient, so that it does not have to be determined by optimization or by a target function. The same applies to the method including provision of diffractive power, since also in this case the last step is the wearing position optimization of the surface.

(211) In contrast, the method including provision of refractive power is mainly suitable for the optimization of single-vision lenses with gratings, since due to the grating being added at a later point, the ray path changes such that the lens in the wearing position does generally not meet the vision needs fully any more. However, in single-vision lenses, existing symmetries can be advantageously used such that due to the provision of right power in the optical center, sufficient quality can also be expected in the periphery.

EMBODIMENTS

(212) The following embodiments relate to rotationally symmetric single-vision lenses with a diffraction grating, wherein the astigmatism is disregarded and only one-dimensional problems are considered for the sake of simplicity. The single-vision lenses have a prescription power of S.sub.prescription+6.0 dpt and are made of an optical material having a refractive index of n.sub.d=1.597 with an Abbe number of v.sub.d=42.0. All spectacle lenses according to the examples of the invention are optimized according to the method for simultaneously optimizing lens surfaces and gratings with a target function. The optimization methods specifically used in embodiment 1 and embodiment 2 differ in some optimization objectives though.

(213) Other than most of the optimization problems, embodiment 1 is not an overdetermined problem and can therefore be solved exactly. It leads to a lens fully corrected within the scope of the model assumptions. Embodiment 2 is selected such that also within the scope of the one-dimensional treatment without astigmatism, a compromise for several viewing directions has to be found, which depends on the type of target function. This situation corresponds to the general case of a spectacle lens with a diffraction grating. For example, in progressive lenses, already without taking chromatic aberrations into account, the Minkwitz theorem causes the optimization problem to be overdetermined and therefore a compromise has to be found.

(214) It can be seen in both embodiments that the introduced grating can significantly improve the chromatic aberrations of the lens and at the same time also optimize the refractive error of the lens. Of course, the method is not limited to one-dimensional problems, but can also be applied to the tracing of a spectacle lens in the wearing position. In this more general case, an oblique-angled ray incidence is taken into consideration in the wavefront tracing.

(215) In the following, the optimization of a rotationally symmetric spectacle lens by means of a target function will be explained in more detail. As explained above, the astigmatism of the spectacle lens will be neglected for the sake of simplicity.

(216) An exemplary target function of type 1 in this case is a target function of the form:

(217) $\begin{array}{cc}{F}_1=\underset{i,\lambda }{\Sigma }{g}_S\left(i,\lambda \right){\left(S_{\Delta }\left(i,\lambda \right)-S_{\Delta ,\mathrm{target}}\left(i,\lambda \right)\right)}^2.& \left(141b\right)\end{array}$

(218) An exemplary target function of type 2 in this case is the target function of the form:

(219) $\begin{array}{cc}{F}_2=\underset{i}{\Sigma }{g}_S\left(i\right){\left(S_{\Delta }\left(i,{\lambda }_0\right)-S_{\Delta ,\mathrm{target}}\left(i,{\lambda }_0\right)\right)}^2+g_{\mathrm{FLF}}\left(i\right)\times {f\left(S_{\mathrm{SK}}\left(i,{\lambda }_2\right)-S_{\mathrm{SK}}\left(i,{\lambda }_1\right)\right)}^2.& \left(142b\right)\end{array}$

(220) An exemplary target function of type 3 in this case is the target function of the form:

(221) 0$\begin{array}{cc}{F}_3=\underset{i}{\Sigma }{g}_S\left(i\right)(S_{\Delta }\left(i,{\lambda }_0\right)-{S_{\Delta ,\mathrm{target}}\left(i,{\lambda }_0\right)}^2+g_{\mathrm{FQF}}\left(i\right)\times {f\left({\mathrm{\Delta \phi }}_{\mathrm{SK}}\left(i,{\lambda }_2,{\lambda }_1\right)\right)}^2.& \left(143b\right)\end{array}$

(222) The calculation of the values taken into account in equations (141b) to (143b) can be performed by means of wavefront tracing. Other than the wavefront tracing in the above-mentioned article of W. Becken, et, al. “Brillengläser im Sport: Optimierung der Abbildungseigenschaften unter physiologischen Aspekten”, Z. Med. Phys., 17 (2007), pages 56-66, the generalized Coddington equation described in the article, for the purely refractive case, is further extended by an additional term for the diffractive phase Ψ. The extension of the Coddington equation for an optical element with at least one diffraction grating will be described in detail in the following.

(223) In the case of a vertical, or perpendicular, light incidence on the individual boundary surface i(i=1,2) for the then one-dimensional problem, it holds for the wavefront tracing that
n′.sub.iW′.sub.Out,i.sup.(2)(0)−n.sub.iW.sub.In,i.sup.(2)(0)=(n′.sub.i−n.sub.i)S.sub.i.sup.(2)(0)−Ψ.sub.i.sup.(2)(0).  (144)

(224) In equation (144):

(225) n′.sub.i is the refractive index of the material behind the i.sup.th surface;

(226) n.sub.i is the refractive index of the material in front of the i.sup.th surface;

(227) S.sub.i is the refractive power of the i.sup.th surface;

(228) W′.sub.Out,i.sup.(2)(0) is the curvature of the outgoing wavefront at the i.sup.th surface;

(229) W.sub.In,i.sup.(2)(0) is the curvature of the incident wavefront at the i.sup.th surface;

(230) Ψ.sub.i.sup.(2)(0) is the phase of the t.sup.th diffraction grating.

(231) As can be seen from equation (144), the refractive portion of the refractive power is determined by (n.sub.i′−n.sub.i)S.sub.i.sup.(2)(0), and the diffractive portion of the refractive power is determined by −Ψ.sub.i.sup.(2)(0). In the present example, two refractive surfaces S.sub.i and S.sub.2 and two diffraction gratings Ψ.sub.1 and Ψ.sub.2, which limit a spectacle lens with a refractive index n in air, are contemplated. In this case, n.sub.1=1, n′.sub.1=n, n.sub.2=n, n′.sub.2=1.

(232) The change of the wavefront in the propagation through the spectacle lens and from the spectacle lens to the vertex sphere can be neglected. If the incident waveform is plane (i.e. W.sub.In,1.sup.(2)(0)=0), then the curvature W′.sub.Out,2.sup.(2)(0) of the outgoing wavefront can be equated directly with the refractive power of the lens. Accordingly, it holds for the refractive and diffractive portions of the refractive power that:

(233) $\begin{array}{cc}S\left(\stackrel{\_}{x},\lambda \right)=S_{\mathrm{ref}}\left(\stackrel{\_}{x},\lambda \right)+S_{\mathrm{diff}}\left(\stackrel{\_}{x},\lambda \right),\mathrm{where}& \left(145\right)\\ \begin{array}{c}{S}_{\mathrm{ref}}\left(\stackrel{\_}{x},\lambda \right)=\left(n\left(\lambda \right)-1\right){\stackrel{\_}{S}}^{\left(2\right)}\left(\stackrel{\_}{x}\right)\\ =\left(n\left(\lambda \right)-1\right)\left({\stackrel{\_}{S}}_1^{\left(2\right)}\left(\stackrel{\_}{x}\right)+{\stackrel{\_}{S}}_2^{\left(2\right)}\left(\stackrel{\_}{x}\right)\right)\\ S_{\mathrm{diff}}\left(\stackrel{\_}{x},\lambda \right)=-{\Psi }^{\left(2\right)}\left(\stackrel{\_}{x},\lambda \right)\\ =-\left({\Psi }_1^{\left(2\right)}\left(\stackrel{\_}{x},\lambda \right)+{\Psi }_2^{\left(2\right)}\left(\stackrel{\_}{x},\lambda \right)\right).\end{array}& \left(146\right)\end{array}$

(234) If one assumes that n(λ) depends linearly on the wavelength, it will hold that

(235) $\begin{array}{cc}\begin{array}{c}n\left(\lambda \right)-1=\left(n_0-1\right)+\frac{\partial n}{\partial \lambda }\left(\lambda -{\lambda }_0\right)\\ =\left(n_0-1\right)+\frac{n_C-n_F}{{\lambda }_C-{\lambda }_F}\left(\lambda -{\lambda }_0\right)\\ =\left(n_0-1\right)\left(1+\frac{n_C-n_F}{n_0-1}\left(\frac{\lambda -{\lambda }_0}{{\lambda }_C-{\lambda }_F}\right)\right).\end{array}.& \left(147\right)\end{array}$

(236) In the above equation, λ.sub.F=486.1 nm and λ.sub.C=656.3 nm. The wavelength λ.sub.0 in the expression n.sub.0=n(λ.sub.0) is a suitably selected wavelength.

(237) Preferably, λ.sub.0=λ.sub.d=587.1 nm is the wavelength in which the Abbe number

(238) ${\nu }_d=-\frac{n_d-1}{n_C-n_F}$
is defined, so that

(239) $\begin{array}{cc}n\left(\lambda \right)-1=\left(n_d-1\right)\left(1-\frac{1}{{\nu }_d}\left(\frac{\lambda -{\lambda }_d}{{\lambda }_C-{\lambda }_F}\right)\right)& \left(147a\right)\end{array}$

(240) The phase Ψ is proportional to the wavelength and reads in a spatial dimension:
Ψ(x;λ,m)mλ.Math.ψ(x).  (148)

(241) The total refractive power can be expressed by the sought functions S.sup.(2)(x) of the surface curvature and ψ.sup.(2)(x) of the phase curvature:

(242) $\begin{array}{cc}S\left(\stackrel{\_}{x},\lambda \right)=\left(n_d-1\right)\left(1-\frac{1}{{\nu }_d}\left(\frac{\lambda -{\lambda }_d}{{\lambda }_C-{\lambda }_F}\right)\right){\stackrel{\_}{S}}^{\left(2\right)}\left(\stackrel{\_}{x}\right)-m\lambda .Math.{\psi }^{\left(2\right)}\left(\stackrel{\_}{x}\right).& \left(149\right)\end{array}$

(243) In the following examples, the target function of type 1 will be evaluated for 2 wavelengths. For the weighting function, it holds that g.sub.S(i,λ)=1, g.sub.FLF(i)=1 and g.sub.FQF(i)=1. The target objectives for the refractive error are set to S.sub.Δ,target(i,λ)=0. Here, the variable S.sub.Δ(i,λ) is the difference of the existing refractive power S(x,λ) and the required prescription power of the lens S.sub.prescription.

(244) Under these conditions, the target functions of different types have the respective following formulae:

(245) $\begin{array}{cc}\begin{array}{cc}\mathrm{Type}1)& F_1={\underset{i}{\Sigma }\left(S\left({\stackrel{\_}{x}}_i,{\lambda }_1\right)-S_{\mathrm{prescription}}\right)}^2+{\underset{i}{\Sigma }\left(S\left({\stackrel{\_}{x}}_i,{\lambda }_2\right)-S_{\mathrm{prescription}}\right)}^2\\ \mathrm{Type}2)& F_2={\underset{i}{\Sigma }\left(S\left({\stackrel{\_}{x}}_i,{\lambda }_0\right)-S_{\mathrm{prescription}}\right)}^2+{f\left(S_{\mathrm{SK}}\left(i,{\lambda }_2\right)-S_{\mathrm{SK}}\left(i,{\lambda }_1\right)\right)}^2\\ \mathrm{Type}3)& F_3={\underset{i}{\Sigma }\left(S\left({\stackrel{\_}{x}}_i,{\lambda }_0\right)-S_{\mathrm{prescription}}\right)}^2+{g\left({\mathrm{\Delta \phi }}_{\mathrm{SK}}\left(i,{\lambda }_2,{\lambda }_1\right)\right)}^2.\end{array}& \left(150\right)\end{array}$