Population of an eye model using measurement data in order to optimize spectacle lenses

Abstract

A method, a device, and a corresponding computer program product for calculating (optimizing) and producing a spectacle lens with the aid of a semi-personalized eye model. In one approach, the method includes providing personalized refraction data of at least one eye of the spectacles wearer; establishing a personalized eye model in which at least the parameters: shape of an anterior corneal surface of a model eye; a cornea-lens distance; parameters of the lens of the model eye; and lens-retina distance are established using personalized measured values for the eye of the spectacles wearer, and/or using standard values, and/or using the provided personalized refraction data, such that the model eye has the provided personalized refraction data, wherein at least the establishment of the lens-retina distance takes place via calculation.

Claims

1. A computer-implemented method for determining personalized parameters of at least one eye of a spectacles wearer, the method comprising: providing personalized refraction data of the at least one eye of the spectacles wearer; and establishing a personalized eye model in which at least: a shape of an anterior corneal surface of a model eye; a cornea to eye-lens distance; parameters of the eye-lens of the model eye; and an eye-lens to retina distance, are established degrees of freedom, which are established using (i) personalized measured values for the eye of the spectacles wearer, (ii) standard values, or (iii) the provided personalized refraction data, wherein the personalized eye model is established at least partly based on the provided personalized refraction data such that the model eye has the provided personalized refraction data, and wherein at least the eye-lens to retina distance is established via calculation based on the provided personalized refraction data and one or more of the established degrees of freedom of the personalized eye model other than the eye-lens to retina distance.

2. The method according to claim 1, wherein the establishing the shape of the anterior corneal surface of the eye takes place using personalized measurements at least in part along the principal section of the cornea of the at least one eye.

3. The method according to claim 1, wherein the establishing the shape of the anterior corneal surface of the eye takes place using personalized measurements of the corneal topography of the at least one eye.

4. The method according to claim 1, wherein the establishing the cornea to eye-lens distance takes place using personalized measured values for the cornea to eye-lens distance.

5. The method according to claim 1, wherein the establishing the parameters of the eye-lens of the model eye comprises an establishment of the following parameters: shape of the anterior eye-lens surface; eye-lens thickness; and shape of the posterior eye-lens surface.

6. The method according to claim 5, wherein the establishing the eye-lens thickness and of the shape of the posterior eye-lens surface takes place using predetermined standard values, and the establishment of the shape of the anterior eye-lens surface comprises: providing standard values for a mean curvature of the anterior eye-lens surface; and calculating the shape of the anterior eye-lens surface at least partially based on the provided personalized refraction data.

7. The method according to claim 5, wherein the establishing the shape of the anterior eye-lens surface comprises: providing a personalized measured value of a curvature in a normal section of the anterior eye-lens surface.

8. The method according to claim 7, wherein the establishing the eye-lens thickness and of the shape of the posterior eye-lens surface takes place using standard values, and the establishment of the anterior eye-lens surface comprises: providing a personalized measured value of a curvature in a normal section of the anterior eye-lens surface; and calculating the shape of the anterior eye-lens surface at least partially based on the provided personalized refraction data.

9. The method according to claim 1, wherein the establishing the parameters of the eye-lens of the model eye includes establishing an optical effect of the eye-lens.

10. The method according to claim 1, further comprising: displaying the calculated eye-lens to retina distance.

11. The method according to claim 1, further comprising: determining an eye length of the model eye, at least partially based on the calculated eye-lens to retina distance; and displaying the determined eye length.

12. A computer-implemented method, comprising: a method for determining personalized parameters of the at least one eye of the spectacles wearer according to claim 1; predetermining a shape of a first surface and a shape of second surface for the spectacles lens to be calculated or optimized; determining the path of a principal ray through at least one visual point (i) of at least one surface of the spectacles lens into the model eye, which surface is to be calculated or optimized; evaluating an aberration of a wavefront at an evaluation surface, said wavefront resulting along the primary ray from a spherical wavefront striking the first surface of the spectacles lens, in comparison to a wavefront converging at a point on the retina of the personalized eye model; and iteratively varying the at least one surface to be calculated or optimized, until the evaluated aberration corresponds to a predetermined target aberration.

13. The method according to claim 12, wherein the evaluation surface is situated between the eye-lens and the retina of the model eye.

14. The method according to claim 12, wherein the evaluation surface is situated at the exit pupil of the model eye.

15. A system for determining personalized parameters of at least one eye of a spectacles wearer, comprising: a data interface configured to provide personalized refraction data of the at least one eye of the spectacles wearer; and a modeler configured to establish a personalized eye model which is established on at least: a shape of an anterior corneal surface of a model eye; a cornea to eye-lens distance; parameters of the eye-lens of the model eye; and an eye-lens to retina distance, are established degrees of freedom, which are established using (i) personalized measured values for the eye of the spectacles wearer, (ii) standard values, or (iii) the provided personalized refraction data, wherein the personalized eye model is established at least partly based on the provided personalized refraction data such that the model eye has the provided personalized refraction data, and wherein at least the eye-lens to retina distance is established via calculation based on the provided personalized refraction data and one or more of the established degrees of freedom of the personalized eye model other than the eye-lens to retina distance.

16. The system according to claim 15, wherein the modeler is configured to determine an eye length of the model eye at least partially based on the calculated eye-lens to retina distance.

17. The system according to claim 16, further comprising: a display configured to display the calculated eye-lens to retina distance or the determined eye length.

18. The system according to claim 15, wherein the personalized refraction data is measured by an aberrometer or the shape of the anterior corneal surface is measured by a topograph.

19. The system according to claim 15, further comprising: a surface model database configured to predetermine a shape of a first surface and a shape of a second surface for the spectacles lens to be calculated or optimized; a principal ray determiner configured to determine the path of a principal ray through at least one visual point (i) of at least one surface of the spectacles lens into the model eye, which surface is to be calculated or optimized; an evaluator configured to evaluate an aberration of a wavefront at an evaluation surface, said wavefront resulting along the principal ray from a spherical wavefront striking the first surface of the spectacles lens, in comparison to a wavefront converging at a point on the retina of the personalized eye model; and an optimizer configured to iteratively vary the at least one surface of the spectacles lens, said surface to be calculated or optimized, until the evaluated aberration corresponds to a predetermined target aberration.

20. A non-transitory computer program product having program code that is designed to implement a method for determining personalized parameters of at least one eye of a spectacles wearer according to claim 1, when loaded and executed on a computer.

21. A method for producing a spectacles lens, comprising: calculating or optimizing a spectacles lens according to the method of claim 12; and manufacturing the spectacles lens so calculated or optimized.

22. A system for producing a spectacles lens, comprising: a calculator or optimizer configured to calculate or optimize a spectacles lens according to the method of claim 12; and a machine configured to machine the spectacles lens according to the result of the calculation or optimization.

Description

BRIEF DESCRIPTION OF THE FIGURE

(1) Preferred embodiments of the invention are explained by way of example in the following, at least in part with reference to the accompanying drawing. Thereby shown is:

(2) FIG. 1 a schematic depiction of the physiological and physical model of a spectacles lens and of an eye, together with a ray path, in a predetermined usage position.

DETAILED DESCRIPTION

(3) First Approach

(4) Insofar as is not explicitly noted otherwise, initial details regarding exemplary and preferred implementations of the first approach of the invention are described in the following paragraphs:

(5) FIG. 1 shows a schematic depiction of the physiological and physical model of a spectacles lens and of an eye in a predetermined usage position, together with an exemplary ray path which forms the basis of a personalized spectacles lens calculation or optimization according to a preferred embodiment of the invention.

(6) Preferably, only a single ray (the principal ray 10, which preferably travels through the eye's center of rotation Z′) is hereby calculated per visual point of the spectacles lens, but moreover also accompanying the derivatives of the rises of the wavefront according to the transversal (orthogonal to the principal ray) coordinates. These derivatives are considered up to the desired order, wherein the second derivatives describe the local curvature properties of the wavefront, and the higher derivatives coincide with the higher-order aberrations.

(7) Given the calculation of light through the spectacles lens, up to the eye 12, according to the personalized prepared eye model, the local derivatives of the wavefronts are determined in the end effect at a suitable position in the ray path in order to compare them there with a reference wavefront which converges at a point on the retina of the eye 12. In particular, the two wavefronts (meaning the wavefront coming from the spectacles lens and the reference wavefront) are compared with one another at an evaluation surface.

(8) What is thereby meant by “position” is thereby not simply a defined value of the z-coordinate (in the light direction), but rather such a coordinate value in combination with the specification of all surfaces through which refraction has taken place before reaching the evaluation surface. In a preferred embodiment, refraction occurs through all refracting surfaces, including the posterior lens surface. In this instance, a spherical wavefront whose center of curvature lies on the retina of the eye 12 preferably serves as a reference wavefront.

(9) Particularly preferably, as of this last refraction propagation does not continue, so that the radius of curvature of this reference wavefront corresponds directly to the distance between posterior lens surface and retina. In a moreover preferred embodiment, propagation does continue after the last refraction, and in fact preferably up to the exit pupil AP of the eye 12. For example, this is situated at a distance d.sub.AR=d.sub.LR.sup.(b)=d.sub.LR−d.sub.LR.sup.(a)>d.sub.LR in front of the retina, and therefore even in front of the posterior lens surface, so that in this instance the propagation is a back-propagation (the terms d.sub.LR.sup.(a), d.sub.LR.sup.(b) are described further below in the enumeration of steps 1-6). In this instance as well, the reference wavefront is spherical with center of curvature on the retina, but has curvature radius 1/d.sub.AR.

(10) In this regard, it is assumed that a spherical wavefront w.sub.0 emanates from the object point and propagates up to the first spectacles lens surface 14. There it is refracted and subsequently propagates up to the second spectacles lens surface 16, wherein it is refracted again. The wavefront w.sub.g1 exiting from the spectacles lens subsequently propagates along the principal ray in the direction of the eye 12 (propagated wavefront w.sub.g2) until it strikes the cornea 18, where it is again refracted (wavefront w.sub.c). After a further propagation within the anterior chamber depth up to the eye lens 20, the wavefront is also refracted again by the eye lens 20, whereby the resulting wavefront w.sub.e is created at the posterior surface of the eye lens 20 or at the exit pupil of the eye, for example. This is compared with the spherical reference wavefront w.sub.s, and for all visual points the deviations are evaluated in the objective function (preferably with corresponding weightings for the individual visual points).

(11) The ametropia is thus no longer described only by a thin sphero-cylindrical lens, as this was typical in many conventional methods; rather, the corneal topography, the eye lens, the distances in the eye, and the deformation of the wavefront (including the lower-order aberrations—thus sphere, cylinder, and axis length—as well as preferably also including the higher-order aberrations) in the eye are preferably directly considered. In the eye model according to the invention, the vitreous body length d.sub.LR is thereby calculated in a personalized manner.

(12) An aberrometer measurement preferably delivers the personalized wavefront deformations of the real, ametropic eye for far and near (deviations, no absolute refractive powers), and the personalized mesopic and photopic pupil diameters. A personalized real anterior corneal surface that generally makes up nearly 75% of the total refractive power of the eye is preferably obtained from a measurement of the corneal topography (areal measurement of the anterior corneal surface). In a preferred embodiment, it is not necessary to measure the posterior corneal surface. Due to the small refractive index difference relative to the aqueous humor, and due to the small cornea thickness, it is preferably described in good approximation not by a separate refractive surface, but rather by an adaptation of the refractive index of the cornea.

(13) In general, in this specification bold-face lowercase letters designate vectors, and bold-face capital letters designate matrices, for example the (2×2) vergence matrices or refractive power matrices

(14) $S=\left(\begin{array}{cc}{S}_{\mathrm{xx}}& S_{xy}\\ S_{xy}& S_{yy}\end{array}\right),C=\left(\begin{array}{cc}{C}_{\mathrm{xx}}& C_{xy}\\ C_{xy}& C_{yy}\end{array}\right),L=\left(\begin{array}{cc}{L}_{\mathrm{xx}}& L_{xy}\\ L_{xy}& L_{yy}\end{array}\right),1=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),$
and cursive letters such as d designate scalar values.

(15) Furthermore, bold-face cursive capital letters should designate wavefronts or surfaces as a whole. For example, S is thus the vergence matrix of the identically named wavefront S; aside from the 2nd-order aberrations that are encompassed in S, S also includes the entirety of all higher-order aberrations (HOA) of the wavefront. Mathematically, S stands for the set of all parameters that are necessary in order to describe a wavefront (sufficiently precisely) with regard to a given coordinate system. S preferably stands for a set of Zernike coefficients having a pupil radius, or a set of coefficients of a Taylor series. S particularly preferably stands for the set from a vergence matrix S to describe the 2nd-order wavefront properties, and a set of Zernike coefficients (with a pupil radius) that serves to describe all remaining wavefront properties except for the 2nd order, or a set of coefficients according to a Taylor decomposition. Analogous statements apply to surfaces instead of wavefronts.

(16) Among other things, the following data may in principle be measured directly: the wavefront S.sub.M which is generated by the laser spot on the retina and the passage through the eye (from aberrometric measurement) shape of the anterior corneal surface C (via corneal topography) distance between cornea and anterior lens surface d.sub.CL (via pachymetry). This variable may also be determined indirectly via the measurement of the distance between the cornea and the iris; correction values may thereby be applied, if applicable. Such corrections may be the distance between the anterior lens surface and the iris, from known eye models (for example literature values). curvature of the anterior lens surface in a direction L.sub.1xx (via pachymetry). Without limitation of the generality, the x-plane may thereby be defined such that this section lies in the x-plane. The coordinate system is thus defined so that this plane lies obliquely; the derivative must be expanded by the functions of the corresponding angle. It is not required that it thereby be a principal section. For example, it may be the section in the horizontal plane.

(17) Furthermore—depending on the embodiment—the following data may either be measured or learned from the literature: thickness of the lens d.sub.L curvature of the posterior lens surface in the same direction as the anterior lens surface L.sub.2,xx (via pachymetry)

(18) Therefore, there are the following possibilities for the posterior lens surface: measurement of L.sub.2,xx (L.sub.2,M) and assumption of a rotational symmetry L.sub.2,xx=L.sub.2,yy=L.sub.2=L.sub.2,M and L.sub.2,xy=L.sub.2,yx=0 taking L.sub.2,xx from the literature (L.sub.2,Lit), and assumption of a rotational symmetry L.sub.2,xx=L.sub.2,yy=L.sub.2=L.sub.2,M and L.sub.2,xy=L.sub.2,yx=0 taking the complete (asymmetrical) shape L.sub.2 from the literature (L.sub.2,Lit) measurement of L.sub.2,xx (L.sub.2,M), and assumption of a cylinder or an otherwise specified asymmetry a.sub.Lit from the literature L.sub.2,xx=L.sub.2,M and L.sub.2,xy=L.sub.2,yx=ƒ(L.sub.2,xx,a.sub.Lit) as well as L.sub.2,yy=g(L.sub.2,xx,a.sub.Lit)

(19) The following data may be learned from the literature: refractive indices n.sub.CL of cornea and anterior chamber depth, as well as of the aqueous humor n.sub.LR and that of the lens n.sub.L

(20) In particular, the distance d.sub.LR between posterior lens surface and retina, as well as the components L.sub.1,yy and L.sub.1,xy=L.sub.1,yx of the anterior lens surface, therefore remain as unknown parameters. To simplify the formalism, the former may also be written as a vergence matrix D.sub.LR=D.sub.LR.Math.1 with D.sub.LR=n.sub.LR/d.sub.LR. Furthermore, the variable τ is generally used, which is defined as τ=d/n (wherein the corresponding index as is used for d and τ is always to be used for the refractive index as n, for example as τ.sub.LR=d.sub.LR/n.sub.LR, τ.sub.CL=d.sub.CL/n.sub.CL).

(21) In a preferred embodiment in which the lens is described via an anterior surface and a posterior surface, the modeling of the passage of the wavefront through the eye model used according to the invention, thus after the passage through the surfaces of the spectacles lens, may be described as follows, wherein the transformations of the vergence matrices are explicitly indicated: 1. Refraction of the wavefront S with the vergence matrix S at the cornea C with the surface refractive power matrix C, relative to the wavefront S′.sub.C with vergence matrix S′.sub.C=S+C 2. Propagation by the anterior chamber depth d.sub.CL (distance between cornea and anterior lens surface) relative to the wavefront S.sub.L1 with vergence matrix S.sub.L1=S′.sub.C/(1−τ.sub.CL.Math.S′)

(22) $S_{L1}=\frac{S_C^{\prime }}{\left(1-{\tau }_{CL}.Math.S_C^{\prime }\right)}$ 3. Refraction at the anterior lens surface L.sub.1 with the surface refractive power matrix L.sub.1 relative to the wavefront S′.sub.L1 with the vergence matrix S′.sub.L1=S.sub.L1+L.sub.1 4. Propagation by the lens thickness d.sub.L relative to the wavefront S.sub.L2 with vergence matrix S.sub.L2=S′.sub.L1/(1−τ.sub.L.Math.S′.sub.L1) 5. Refraction at the posterior lens surface L.sub.2 with the surface refractive power matrix L.sub.2 relative to the wavefront S′.sub.L2 with the vergence matrix S′.sub.L2=S.sub.L2+L.sub.2 6. Propagation by the distance between lens and retina d.sub.LR relative to the wavefront S.sub.R with the vergence matrix S.sub.R=S′.sub.L2/(1−τ.sub.LR.Math.S′.sub.L2)

(23) Each of the steps 2, 4, 6 in which propagation takes place over the distances τ.sub.CL, τ.sub.CL, or τ.sub.CL may thereby be divided up into two partial propagations 2a,b), 4a,b), or 6a,b) according to the following scheme, which for step 6a,b) explicitly reads: 6a. Propagation by the distance d.sub.LR.sup.(a) between lens and intermediate plane relative to the wavefront S.sub.LR with the vergence matrix S.sub.LR=S′.sub.L2/(1−τ.sub.LR.sup.(a)S′.sub.L2) 6b. Propagation by the distance d.sub.LR.sup.(b) between intermediate plane and retina relative to the wavefront S.sub.R with the vergence matrix S.sub.R=S.sub.LR/(1−τ.sub.LR.sup.(b)S.sub.LR)

(24) τ.sub.LR.sup.(a)=d.sub.LR.sup.(a)/n.sub.LR.sup.(a) and τ.sub.LR.sup.(b)=d.sub.LR.sup.(b)/n.sub.LR.sup.(b) may thereby be positive or negative, wherein n.sub.LR.sup.(a)=n.sub.LR.sup.(b)=n.sub.LR and τ.sub.LR.sup.(a)+τ.sub.LR.sup.(b)=τ.sub.LR should always be true. In each instance, step 6a and step 6b can be combined again via S.sub.R=S′.sub.L2/(1−(τ.sub.LR.sup.(a)+τ.sub.LR.sup.(b))S′.sub.L2)=S′.sub.L2/(1−τ.sub.LRS′.sub.L2). However, the division into step 6a and step 6b offers advantages, and the intermediate plane may preferably be placed in the plane of the exit pupil AP, which preferably is situated in front of the posterior lens surface. In this instance, τ.sub.LR.sup.(a)<0 and τ.sub.LR.sup.(b)>0.

(25) The division of steps 2, 4 may also take place analogous to the division of step 6 into 6a,b).

(26) For the selection of the evaluation surface of the wavefront, it is thus not only the absolute position in relation to the z-coordinate (in the light direction) but also the number of surfaces through which refraction has already taken place up to the evaluation surface. One and the same plane may thus be traversed repeatedly. For example, the plane of the AP (which normally is situated between the anterior lens surface and the posterior lens surface) is formally traversed by the light for the first time after a virtual step 4a, in which propagation takes place from the anterior lens surface by the length τ.sub.L.sup.(a)>0. The same plane is reached for the second time after step 6a if, after refraction by the posterior lens surface, propagation takes place again back to the AP plane, meaning that τ.sub.LR.sup.(a)=−τ.sub.L+τ.sub.L.sup.(a)=−τ.sub.L.sup.(b)<0, which is equivalent to τ.sub.LR.sup.(a)=τ.sub.LR−τ.sub.LR.sup.(b)<0. Given the wavefronts S.sub.AP, which relate in the text to the AP, what should preferably always be meant (if not explicitly noted otherwise) is the wavefront S.sub.AP=S.sub.LR, which is the result of step 6a.

(27) These steps 1 through 6 are referred to repeatedly in the further course of the specification. They describe a preferred correlation between the vergence matrix S of a wavefront S at the cornea and the vergence matrices of all intermediate wavefronts arising therefrom at the refractive intermediate surfaces of the eye, in particular the vergence matrix S′.sub.L2 of a wavefront S′.sub.L2 after the eye lens (or even of a wavefront S.sub.R at the retina). These correlations may be used both to calculate parameters (for example d.sub.LR or L.sub.1) that are not known a priori, and thus to populate the model with values in either a personalized or generic manner, and in order to simulate the propagation of the wavefront in the eye with then populated models to optimize spectacles lenses.

(28) In a preferred embodiment, the surfaces and wavefronts are treated up to the second order, for which a representation by vergence matrices is sufficient. Another preferred embodiment described still later takes into account and also utilizes higher orders of aberrations.

(29) In a preferred embodiment, in a second-order description the eye model has twelve parameters as degrees of freedom of the model that need to be populated. These preferably include the three degrees of freedom of the surface refractive power matrix C of the cornea C; the respective three degrees of freedom of the surface refractive power matrices L.sub.1 and L.sub.2 for the anterior lens surface or posterior lens surface; and respectively one for the length parameters of anterior chamber depth d.sub.CL, lens thickness d.sub.L, and the vitreous body length d.sub.LR.

(30) Populations of these parameters may in principle take place in a plurality of ways: i) directly, thus personalized measurement of a parameter ii) a priori given value of a parameter, for example as a literature value or from an estimate, for example due to the presence of a measured value for another variable that correlates with the parameter to be determined in a known manner using a preceding population analysis iii) calculation from consistency conditions, for example compatibility with a known refraction

(31) The total number df.sub.2 of second-order degrees of freedom of the eye model (df stands for “degree of freedom”, the index “2” stands for 2nd-order) is thus composed of
df.sub.2=df.sub.2(i)+df.sub.2(ii)+df.sub.2(iii)

(32) For example, if direct measured values are present for all twelve model parameters, then df.sub.2(i)=12, df.sub.2(ii)=0 and df.sub.2(iii)=0, which for the sake of simplicity is expressed in the following by the notation df.sub.2=12+0+0. In such an instance, the object refraction of the appertaining eye is also established, so that an objective refraction determination would no longer need to be additionally implemented.

(33) However, a central aspect of the invention directly relates to the goal of not needing to directly measure all parameters. It is thus in particular markedly simpler to measure, or objectively and/or subjectively determine, the refraction of the appertaining eye than to measure all parameters of the model eye in a personalized manner. At least one refraction, thus measurement data regarding the wavefront S.sub.M of the eye up to the 2nd order that correspond to the data of the vergence matrix S.sub.M, is thus preferably present. Given a population of the eye model purely on the basis of objectively measured data, these values may be taken from aberrometric measurements or autorefractometric measurements, or according to (ii) may be populated by data provided otherwise. A consideration of more subjective methods (i.e. subjective refraction), be it as a replacement for the objective measurement of the refraction or via the combination of both results, is further described later. The three conditions of the agreement with the three independent parameters of the vergence matrix S.sub.M therefore allow three parameters of the eye model to be derived, which in the notation introduced above corresponds to df.sub.2(iii)=3.

(34) In instances in which not all model parameters are accessible to direct measurements, or these measurements would be very costly, the invention thus utilizes the possibility of reasonably populating the missing parameters. For example, if direct measured values are present for at most nine model parameters (df.sub.2(i)≤9), then the cited conditions of the refraction may be used in order to calculate three of the model parameters (df.sub.2 (iii)=3). In the event that df.sub.2(i)=9 applies exactly, all twelve model parameters are then determined unambiguously via the measurements and the calculation, and (df.sub.2(ii)=0) applies. By contrast, if df.sub.2(i)<9, then df.sub.2(ii)=9−df.sub.2(i)>0, meaning that the model is underdetermined in the sense that df.sub.2(ii) parameters need to be established a priori.

(35) With the provision of a personalized refraction, thus measurement data regarding the wavefront S.sub.M of the eye, in particular up to the second order, the necessary data of the vergence matrix S.sub.M are present. According to a conventional method described in WO 2013/104548 A1, in particular the parameters {C, d.sub.CL, S.sub.M} are measured. By contrast, among other things the two length parameters d.sub.L and d.sub.LR (or D.sub.LR) are conventionally established a priori (for example via literature values or estimation). In WO 2013/104548 A1, in particular a differentiation is made between the two instances in which either L.sub.2 is established a priori and L.sub.L is calculated therefrom, or vice versa. The cited disclosure document discloses Equation (4) or Equation (5) as a calculation rule in this regard. For both instances, df.sub.2=4+5+3 applies.

(36) In the terminology of the aforementioned steps 1 through 6, the adaptation of L.sub.1 to the measurements in particular occurs in that, on the one hand, the measured vergence matrix S.sub.M is calculated through the likewise measured matrix C by means of the steps 1, 2, and propagated up to the object-side side of the anterior lens surface. On the other hand, a spherical wave is calculated from back to front from an imaginary point light source on the retina by means of the steps 6, 5, 4, run through in reverse, in that this spherical wave is refracted at the previously established surface refractive power matrix L.sub.2 of the posterior lens surface, and the wavefront that is then obtained propagates from the posterior lens surface up to the image-side side of the anterior lens surface. The difference of the vergence matrices S.sub.L1 and S′.sub.L1 that are determined in this manner, which difference must be present on the object side or image side of the anterior lens surface, must have been produced by the matrix L.sub.1, because in the aberrometric measurement the measured wavefront arises from a wavefront that emanates from a point on the retina and therefore, due to the reversibility of the ray paths, is identical to that incident wavefront (S=S.sub.M) that converges on this point of the retina. This leads to Equation (4) in the cited disclosure document:

(37) $\begin{array}{cc}{L}_1\left(D_{\mathrm{LR}}\right)=\frac{D_{\mathrm{LR}}.Math.1-L_2}{1+{\tau }_L.Math.\left(D_{\mathrm{LR}}.Math.1-L_2\right)}-\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}\left(S_M+C\right)}& \left(1a\right)\end{array}$

(38) The other instance in the cited disclosure document relates to the adaptation of the matrix L.sub.2 to the measurements after the matrix L.sub.1 has been established. A difference now exists merely in that: the measured wavefront S.sub.M is subjected to the steps 1, 2, 3, 4, and the assumed wavefront from the point light source is only subjected to step 6; and in that the missing step that is to take place for adaptation of the posterior lens surface L.sub.2 is now step 5, corresponding to Equation (5) of the cited disclosure document:

(39) $\begin{array}{cc}{L}_2=D_{\mathrm{LR}}-\left(\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}\left(S_M+C\right)}+L_1\right){\left(1-{\tau }_L\left(\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}\left(S_M+C\right)}+L_1\right)\right)}^{-1}& \left(1b\right)\end{array}$

(40) The central idea of the invention is to calculate at least the length parameter d.sub.LR (or D.sub.LR) from other measured data and a priori assumptions regarding other degrees of freedom, and not to assume it a priori as is conventional. Within the scope of the present invention, it has turned out that this therefore brought about a noteworthy improvement of the personalized adaptation at comparably low cost, because the wavefront calculation turned out to be very sensitively dependent on this length parameter. This means that, according to the invention, it is advantageous if at least the length parameter d.sub.LR, which belongs to the df.sub.2 (iii)=3 parameters, that is calculated. This parameter is in particular poorly accessible to a direct measurement; it varies strongly between different test subjects, and these variations comparably strongly influence the imaging of the eye.

(41) The data of the vergence matrix S.sub.M, and particularly preferably also the data regarding C from personalized measurements, are preferably available. In a further preferred aspect that is preferably also taken into account in the following embodiments, a spherical posterior surface, meaning a posterior surface without astigmatic components, is assumed given an assumption of data regarding the posterior lens surface.

(42) In a preferred embodiment of the invention, measurement data up to the second order that corresponding to the data of the surface refractive power matrix C are thus present with regard to the cornea C. Although these values may be learned from topographical measurements, the latter are not necessary. Rather, topometric measurements are sufficient. This situation corresponds to the instance df.sub.2=3+6+3, wherein in particular the anterior chamber depth d.sub.u, is one of the six parameters that are to be established a priori.

(43) Insofar as no further personalized measurements are performed, a situation with df.sub.2=3+6+3 is present. In order to be able to uniquely determine d.sub.LR, six parameters from {L.sub.1, L.sub.2, d.sub.L, d.sub.CL} must thus be populated via assumptions or literature values. The remaining two result from the calculation in addition to d.sub.LR. In a preferred embodiment, the parameters of the posterior lens surface, the mean curvature of the anterior lens surface, and the two length parameters d.sub.L and d.sub.CL are populated a priori (as predetermined standard values).

(44) In an instance that is particularly important to the invention, the anterior chamber depth d.sub.CL is thus additionally the distance between the cornea and the anterior lens surface, known for example from pachymetric or OCT measurements. The measured parameters therefore include {C, d.sub.CL, S.sub.M}. This situation corresponds to the instance of df.sub.2=4+5+3. Afterward the problem is still mathematically underdetermined; five parameters must thus be established a priori from {L.sub.1, L.sub.2, d.sub.L} via assumptions or literature values. In a preferred embodiment, the parameters are the posterior lens surface, the mean curvature of the anterior lens surface, and the lens thickness. The precise way of calculating for this instance is presented in more detail further below.

(45) Solely for the precision of the personalized adaptation, it is advantageous to be able to populate as many parameters as possible with personalized measurements. In a preferred embodiment, for this purpose the lens curvature is additionally provided in a normal section on the basis of a personalized measurement. A situation according to df.sub.2=5+4+3 then thereby results, and it is sufficient to establish four parameters from {L.sub.1yy, α.sub.L1, L.sub.2, d.sub.L} a priori. Here as well, in a preferred embodiment these are again the parameters of posterior lens surface and the lens thickness. The precise calculation is again described in more detail further below.

(46) In particular as an alternative to the normal step of the anterior lens surface, and particularly preferably in addition to the anterior chamber depth, the lens thickness may also be provided from a personalized measurement. The necessity to populate these parameters with model data or estimated parameters thereby disappears (df.sub.2=5+4+3). Otherwise, the statements as already made above apply. This embodiment is particularly advantageous if a pachymeter is used whose measurement depth allows the detection of the posterior lens surface, but not a sufficiently certain determination of the lens curvatures.

(47) In addition to the anterior chamber depth and a normal section of the anterior lens surface, in a preferred embodiment one additional parameter (for example measurement in two normal sections) or two additional parameters (measurement of both principal sections and the axis position) of the anterior lens surface are recorded via a personalized measurement. This additional information may in particular be utilized in two ways: Abandonment of a priori assumptions: one or two of the assumptions that were otherwise made a priori may be abandoned and be determined via calculation. In this instance, the situations df.sub.2=6+3+3 or df.sub.2=7+2+3 result. In the first instance, the mean curvature of the posterior surface (given assumption of an astigmatism-free posterior surface) may be determined, and in the second instance the surface astigmatism (including axis position) may be determined for a given mean curvature. Alternatively, in both instances the lens thickness may be determined from the measurements. However, such a procedure generally requires a certain caution, since noisy measurement data may easily lead to a “runaway” of the enabled parameters. The model may thereby as a whole become markedly worse instead of better. One possibility to prevent this is to predetermine anatomically reasonable limit values for these parameters, and to limit the variation of the parameters to this range. Of course, these limits may also be predetermined depending on the measured values. Reduction of the measurement uncertainty: if, by contrast, the same a priori assumptions continue to be made (preferably thus {L.sub.2, d.sub.L}), the situations df.sub.2=6+4+3 or df.sub.2=7+4+3 are present; the system is thus mathematically overdetermined. Instead of a simple analytical determination of D.sub.LR according to the subsequent embodiments, D.sub.LR (and possibly the still missing parameters from L.sub.1) is determined (“fit”) so that the distance between the L.sub.1 resulting from the equations and the measured L.sub.1 (or the measured L.sub.1, supplemented by the missing parameters) is minimal. A reduction of the measurement uncertainty may—obviously—be achieved via this procedure.

(48) In a further preferred implementation, the anterior chamber depth, two or three parameters of the anterior lens surface, and the lens thickness are measured in a personalized manner. The calculation of the remaining variables thereby takes place analogously, wherein the a priori assumption of the lens thickness may be replaced by the corresponding measurement.

(49) In a further preferred implementation, personalized measurements of the anterior chamber depth, at least one parameter of the anterior lens surface, the lens thickness, and at least one parameter of the posterior lens surface are provided. This is hereby an expansion of the aforementioned instances. The respective additionally measured parameters may take place analogous to the step-by-step expansions of the above segments. These instances are particularly advantageous if the aforementioned pachymetry units that measure in one plane, two planes, or over the entire surface are accordingly extended in terms of measurement depth, and are so precise that the curvature data can be sufficiently precisely determined.

(50) In the following it is shown, using a few examples, how the calculation of individual parameters may take place from the remaining measured parameters or parameters established a priori, and using the personalized refraction data.

(51) For example, in preferred embodiments, a measurement of the curvature of a lens surface is available in a normal section. Since the posterior surface cannot be measured in practice without the anterior surface also being measured, and the measurement of the anterior surface preferably occurs, the equations for the instances of a curvature of the anterior lens surface that is known in a normal section are specified in the following. If, instead of a normal section of the anterior lens surface, a normal section of the posterior lens surface is present (for example corresponding measurements, model assumptions), one must analogously proceed with Equation (1b). Without limiting the generality, the coordinate system is placed so that the normal section travels in the x-direction. In a next step, the matrix equation (1a) is then evaluated in the given normal section and solved for D.sub.LR, and this solution is subsequently used again in Equation (1a) for the complete specification of L.sub.1.

(52) If the xx-component of L.sub.1(D.sub.LR) from Equation (1) is set equal to the measured value L.sub.1,xx, for this matrix element a quadratic equation in D.sub.LR is obtained whose positive solution corresponds to the distance between posterior lens surface and retina:

(53) $\begin{array}{cc}{D}_{LR}=\frac{-b+\sqrt{b^2-4c}}{2a}& \left(2\right)\end{array}$

(54) It thereby applies that:
a=τ.sub.L(1+τ.sub.LA)
b=1−τ.sub.L(tr(L.sub.2)−AB)
c=A−L.sub.2,xx+τ.sub.L det L.sub.2(1+τ.sub.LA)−τ.sub.LA tr(L.sub.2)=A−L.sub.2,xx+a det L.sub.2−τ.sub.LA tr(L.sub.2)  (2a)
with
A=−S.sub.M,L1,xx−L.sub.1,xx
B=2−τtr(L.sub.2)
det(L.sub.2)=L.sub.2,xxL.sub.2,yy−L.sub.2,xy.sup.2
tr(L.sub.2)=L.sub.2,xx+L.sub.2,yy  (2b)
and

(55) 0$\begin{array}{cc}{S}_{M,L1,\mathrm{xx}}=\frac{{\tau }_{\mathrm{CL}}{S}_{M,C,\mathrm{xy}}^{″}+S_{M,C,\mathrm{xx}}^{\prime }.Math.\left(1-{\tau }_{\mathrm{CL}}{S}_{M,C,\mathrm{yy}}^{\prime }\right)}{-{\tau }_{\mathrm{CL}}^2{S}_{M,C,\mathrm{xy}}^{\prime 2}+\left(1-{\tau }_{\mathrm{CL}}{S}_{M,C,\mathrm{xx}}^{\prime }\right).Math.\left(1-{\tau }_{\mathrm{CL}}{S}_{M,C,\mathrm{yy}}^{\prime }\right)}{S}_{M,C,\mathrm{xx}}^{\prime }=S_{M,\mathrm{xx}}+C_{\mathrm{xx}}\left(\mathrm{xy}\mathrm{und}\mathrm{yy}\mathrm{analog}\right)& \left(2c\right)\end{array}$

(56) In the event of a symmetrical posterior lens surface (L.sub.2=L.sub.2,xx.Math.1), this simplifies to

(57) $\begin{array}{cc}{D}_{LR}=L_{2,\mathrm{xx}}+\frac{L_{1,\mathrm{xx}}+S_{M,L1,\mathrm{xx}}}{1-{\tau }_L.Math.\left(L_{1,\mathrm{xx}}+S_{M,L1,\mathrm{xx}}\right)}& \left(3\right)\end{array}$

(58) with S.sub.M,L1,xx from Equation (2c).

(59) In both instances, it is therefore possible to calculate the anterior lens surface L.sub.1 in that the respectively obtained D.sub.LR in Equation (1a) is used:

(60) $\begin{array}{cc}{L}_1=\frac{D_{\mathrm{LR}}-L_2}{1+{\tau }_L.Math.\left(D_{\mathrm{LR}}-L_2\right)}-\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}\left(S_M+C\right)}& \left(4\right)\end{array}$

(61) The result is naturally symmetrical (L.sub.1,xy=L.sub.1,yx), and for the component L.sub.1,xx reproduces the value used in (2b) or (3).

(62) In some preferred embodiments, a personalized measurement or a specification of a mean curvature of a lens surface is provided. For example, this situation is present when the mean curvature of the anterior lens surface may be measured, or no measurements at the lens surfaces may be performed and the mean curvature of a lens surface is assumed (for example taken from the literature). As was just now described, here the method for the anterior lens surface is described and can be analogously transferred to the posterior lens surface.

(63) In this instance of a given mean sphere L.sub.1,ms of the anterior lens surface, the free parameters are the cylinder L.sub.1,cyl and the axis length α.sub.L1. With L.sub.1,diff=L.sub.1,cyl/2, L.sub.1 becomes

(64) $\begin{array}{cc}{L}_1=\left(\begin{array}{cc}{L}_{1,ms}-L_{1,\mathrm{diff}}.Math.\cos 2{\alpha }_{L1}& -L_{1,\mathrm{diff}}.Math.\sin 2{\alpha }_{L1}\\ -L_{1,\mathrm{diff}}.Math.\sin 2{\alpha }_{L1}& L_{1,ms}+L_{1,\mathrm{diff}}.Math.\cos 2{\alpha }_{L1}\end{array}\right)& \left(5\right)\end{array}$

(65) One again proceeds from Equation (1a). If the expressions for L.sub.1 from Equations (5) and (1a) are now equated, an equation system is obtained that is made up of three equations (the two non-diagonal elements are identical) and the three unknowns L.sub.1,diff, α.sub.L1 and D.sub.LR. This has the physically relevant solution

(66) $\begin{array}{cc}{D}_{\mathrm{LR}}=\frac{-\stackrel{\_}{b}+\sqrt{{\stackrel{\_}{b}}^2-4\stackrel{\_}{\mathrm{ac}}}}{2\stackrel{\_}{a}}{L}_{1\mathrm{diff}}=\pm \sqrt{{\sigma }^2+{\gamma }^2}{\alpha }_{L1}=\frac{1}{2}\arctan \left(\pm \gamma ,\pm \sigma \right)+\frac{\pi }{2}\mathrm{with}& \left(6\right)\end{array}$$\begin{array}{cc}\stackrel{\_}{a}={\tau }_L\left(1+{\tau }_L\stackrel{\_}{A}\right)\stackrel{\_}{b}=1-{\tau }_L\left(\mathrm{tr}\left(L_2\right)-\stackrel{\_}{A}B\right)\stackrel{\_}{c}=\frac{1}{4}\left(\stackrel{\_}{A}{B}^2-B\mathrm{tr}\left(L_2\right)-\stackrel{\_}{a}{\mathrm{Ast}\left(L_2\right)}^2\right)\mathrm{and}\stackrel{\_}{A}={\stackrel{\_}{S}}_{M,L1}-{\stackrel{\_}{L}}_{1,\mathrm{meas}}\mathrm{Ast}\left(L_2\right)=\sqrt{{\mathrm{tr}\left(L_2\right)}^2-4{\mathrm{detL}}_2}\gamma =\frac{\begin{array}{c}2\left(-1+\sqrt{{\stackrel{\_}{b}}^2-4\stackrel{\_}{ac}}\right)\left(L_{2,\mathrm{xy}}-L_{2,\mathrm{vy}}\right)+\\ {\tau }_L^2{\mathrm{Ast}\left(L_2\right)}^2\left(S_{M,L1,\mathrm{xx}}-S_{M,L1,\mathrm{yy}}\right)\end{array}}{2{\tau }_L^2{\mathrm{Ast}\left(L_2\right)}^2}\sigma =\frac{2\left(-1+\sqrt{{\stackrel{\_}{b}}^2-4\stackrel{\_}{ac}}\right)L_{2,\mathrm{xy}}+{\tau }_L^2{\mathrm{Ast}\left(L_2\right)}^2{S}_{M,L1,\mathrm{xy}}}{2{\tau }_L^2{\mathrm{Ast}\left(L_2\right)}^2}& \left(6a\right)\end{array}$

(67) This can also be simplified for the instance of a rotationally symmetrical posterior lens surface:

(68) $\begin{array}{cc}{D}_{\mathrm{LR}}=L_2+\frac{{\stackrel{\_}{L}}_{1,\mathrm{mess}}+{\stackrel{\_}{S}}_{M,L1}}{1-{\tau }_L.Math.\left({\stackrel{\_}{L}}_{1,\mathrm{mess}}+{\stackrel{\_}{S}}_{M,L1}\right)}{L}_1=\left({\stackrel{\_}{L}}_{1,\mathrm{mess}}+{\stackrel{\_}{S}}_{M,L1}\right).Math.1-\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}\left(S_M+C\right)}\mathrm{wherein}{\stackrel{\_}{L}}_{1,\mathrm{meas}}=\frac{D_{\mathrm{LR}}-L_2}{1+{\tau }_L.Math.\left(D_{\mathrm{LR}}-L_2\right)}-{\stackrel{\_}{S}}_{M,L1}\mathrm{with}{\stackrel{\_}{S}}_{M,L1}=\frac{S_{M,L1,\mathrm{xx}}+S_{M,L1,\mathrm{yy}}}{2}& \left(7\right)\end{array}$

(69) The individual elements of the eye model can therefore be entirely calculated.

(70) Aside from a principal section with given angle position, or the mean curvature, the given (i.e. measured or assumed) variables may also be other parameters such as the thickest principal section, the thinnest principal section, the cylinder, and the axis position. In these instances, the procedure is analogous to the illustrated instances.

(71) Since the HOAs of the eye have also be now been taken into account in the optimization of spectacles lenses, it is advantageous to also consider the HOAs of the cornea or of the lens in the population of the eye model. Given the selection of HOAs for the lens, it generally applies that HOAs that may also represent the refractive index curve within the lens may be associated with the anterior lens surface or posterior lens surface.

(72) The previously depicted formalism is preferably expanded, in particular with regard to the cited steps 1 through 6, to the co-treatment of the HOAs, in that the calculation methods from the publications by G. Esser et al.: “Derivation of the refraction equations for higher order aberrations of local wavefronts at oblique incidence”, JOSA A, Vol. 27, No. 2 (2010), and by G. Esser et al.: “Derivation of the propagation equations for higher order aberrations of local wavefronts”, JOSA A, Vol. 28, No. 11 (2011), are applied aside from the formulas for the vergence matrices that are explicitly specified in steps 1 through 6.

(73) In general, the procedure with regard to the enumeration of degrees of freedom is executed in a manner very similar to as above. If, aside from data regarding 2nd-order errors, data about their HOAs are present (either from measurements or from reasonable assumptions) with regard to refractive surface C of the cornea and regarding the outgoing wavefront S.sub.M, the wavefront S.sub.L1 may also be determined computationally with accordingly many HOAs. This applies independently of the form in which the HOA presents itself. However, the Taylor series is particularly preferred, because in this form the statement exactly applies: if HOA coefficients up to the n-th order are present with regard to two surfaces C and S.sub.M, the corresponding HOA coefficients for S.sub.L1 can then also be computationally determined therefrom up to the n-th order. Furthermore, the Zernike basis is preferred, because here as well a similar statement applies. However, this is exact only when all Zernike coefficients with an order >n vanish.

(74) An order n is preferably established (in advance), up to which all participating surfaces and wavefronts should be treated. Independently of the presentation of the HOAs, aside from the three components for the 2nd-order errors, the wavefronts or surfaces then additionally possess N components for the HOAs, wherein N depends on n and, inter alia, on the presentation form of the HOAs (in the Taylor decomposition and Zernike decomposition, N=(n+1)(n+2)/2-6 applies).

(75) The adaptation condition using a measured wavefront, for example S.sub.M,L1, then accordingly no longer possesses only the three components described above, but rather a maximum of N+3 components in total. These are then accordingly accompanied by 3 (N+3)+3=3N+12 parameters (namely the three length parameters d.sub.CL, d.sub.L and d.sub.LR (or D.sub.LR), as well as respectively N+3 components of the cornea C and the lens surfaces L.sub.1 and L.sub.2). This means that

(76) $\begin{array}{c}{\mathrm{df}}_n={\mathrm{df}}_n\left(i\right)+{\mathrm{df}}_n\left(\mathrm{ii}\right)+{\mathrm{df}}_n\left(\mathrm{iii}\right)\\ =3N+12\end{array}$
applies, with df.sub.n(iii)=N+3. If the anterior chamber depth d.sub.CL and the cornea C are preferably measured again, df.sub.n(i)=N+4 applies, and consequently df.sub.n(ii)=N+5, corresponding to the situation of df.sub.n=(N+4)+(N+5)+(N+3).

(77) The further procedure may be implemented in a manner very analogous to as described above.

(78) Given the measurement device forming the basis of the procedure described here, the HOAs of the mapping of the eye on the retina may be detected in transmission with the aberrometry unit. The HOAs of the cornea surface may also be measured in reflection with the same device, via the topography unit. Both the exiting wavefront S.sub.M and the refractive surface C of the cornea, including the HOAs up to a defined order n, are therefore available. The wavefront S.sub.M supplies df.sub.n(iii)=N+3 conditions for parameter calculation. Aside from the cornea C, if it is again preferred to also measure the anterior chamber depth d.sub.CL, df.sub.n(i)=N+4 applies, and consequently df.sub.n(ii)=N+5, corresponding to the situation of df.sub.n=(N+4)+(N+5)+(N+3).

(79) In a preferred embodiment of the invention, in the population of the model the HOAs of the lens may now be selected so that, given the propagation of a wavefront emanating from a point on the retina according to steps 1 through 6, the measured wavefront arises in the reverse order.

(80) According to the invention, however, it is proposed that at least the length parameter d.sub.LR is measured neither a priori nor in a personalized manner, but rather is calculated using the personalized refraction data and the data otherwise established (in advance). For this purpose, at least one measured value or an assumption is provided in particular for one of the degrees of freedom of lens surfaces L.sub.1 or L.sub.2. for example, if this is a measured value for the curvature of L.sub.1 in a normal section, then d.sub.LR (or D.sub.LR) may be determined therefrom via calculation.

(81) If the specification in the vergence matrices relates to the local curvature (this corresponds to the specification of the HOAs as coefficients of a Taylor decomposition), for this purpose D.sub.LR and the missing parameters of the lens are first determined as has already been described above. Following this, the HOAs of the lens may be constructed step by step, starting from the second order to n-th order, with the formalism from G. Esser et al.: “Derivation of the refraction equations for higher order aberrations of local wavefronts at oblique incidence”, JOSA A, Vol. 27, No. 2 (2010), and from G. Esser et al.: “Derivation of the propagation equations for higher order aberrations of local wavefronts”, JOSA A, Vol. 28, No. 11 (2011).

(82) By contrast, if the mean curvature over a defined pupil is used—which is the case in the presentation according to Zernike, for example—the degree of freedom D.sub.LR is likewise established. In this formalism, an iterative procedure would be necessary due to the dependencies. However, this can be avoided via a conversion between the two notations before the beginning of the calculation.

(83) Even if neither a topograph nor an aberrometer is used, thus even if no personalized measurement data regarding HOAs are present, model-based assumptions about the HOAs of the cornea, the lens, or the eye may nevertheless be made and be used in the population of the eye model. The assumed values may thereby also be selected using corresponding models depending on measured data (for example refraction values, results of the topometry or autorefractometer measurement). Examples for the precise calculation have already been described further above, wherein the corresponding assumptions occur instead of the measured values for the HOAs. This also applies again in particular to spherical aberrations, and since this is markedly different than zero, averaged across the population. This may thereby be chosen independently of the measured data, or depending on measured data (for example refraction values, results of the topometry or autorefractometer measurement) and be associated with the cornea, one of the two lens surfaces, or combinations.

(84) Due to the great importance of subjective refraction, it is advantageous that the results of such a subjective eyeglass determination can at least partially enter into the population of the model for the optimization. Subjective refraction data are preferably provided in the form of sphere, cylinder, and axis position. For the sake of simplicity, the description of the procedure is orientated toward this notation, with sph, cyl, and α for the values of sphere, cylinder, and axis position.

(85) If HOAs are not considered, the process may continue as follows:

(86) If only the values of the subjective refraction enter into the optimization, the measurement of the wavefront S.sub.M by an aberrometer or an autorefractometer may be omitted, and instead the matrix S.sub.M may be constructed from the subjective values:

(87) $S_M=\left(\begin{array}{cc}\left(\mathrm{sph}+\frac{1}{2}.Math.\mathrm{cyl}\right)-\frac{1}{2}.Math.\mathrm{cyl}.Math.\cos \left(2a\right)& -\frac{1}{2}.Math.\mathrm{cyl}.Math.\sin \left(2a\right)\\ -\frac{1}{2}.Math.\mathrm{cyl}.Math.\sin \left(2a\right)& \left(\mathrm{sph}+\frac{1}{2}.Math.\mathrm{cyl}\right)+\frac{1}{2}.Math.\mathrm{cyl}.Math.\cos \left(2a\right))\end{array}\right)$

(88) However, the results of the subjective refraction are preferably combined with those of the aberrometric or autorefractometric measurement. For this purpose, an optimized refraction is determined on the basis of both data sets, for example according to a method described in DE 10 2007 032 564 A1. This is described by the values sph.sub.opt, cyl.sub.opt and a.sub.opt. Analogous to the preceding section, S.sub.M is obtained as

(89) $S_M=\left(\begin{array}{cc}\left({\mathrm{sph}}_{\mathrm{opt}}+\frac{1}{2}.Math.{\mathrm{cyl}}_{\mathrm{opt}}\right)-\frac{1}{2}.Math.{\mathrm{cyl}}_{\mathrm{opt}}.Math.\cos \left(2a_{\mathrm{opt}}\right)& -\frac{1}{2}.Math.{\mathrm{cyl}}_{\mathrm{opt}}.Math.\sin \left(2a_{\mathrm{opt}}\right)\\ -\frac{1}{2}.Math.{\mathrm{cyl}}_{\mathrm{opt}}.Math.\sin \left(2a_{\mathrm{opt}}\right)& \left({\mathrm{sph}}_{\mathrm{opt}}+\frac{1}{2}.Math.{\mathrm{cyl}}_{\mathrm{opt}}\right)+\frac{1}{2}.Math.{\mathrm{cyl}}_{\mathrm{opt}}.Math.\cos \left(2a_{\mathrm{opt}}\right)\end{array}\right)$

(90) According to DE 10 2007 032 564 A1, not all values of the subjective refraction or objective measurement need to enter into the optimized refraction values. For example, in the event of a determination of the optimized refraction values for near, or in the event of anticipated instrument myopia, the use of the objectively measured sphere or of the objectively measured defocus term may be omitted.

(91) Even given the incorporation of subjective refraction data, the HOAs may also be taken into account again in the population of the model. For this purpose, given use of the subjective refraction values it is necessary to have these enter into the data set in a consistent manner. To simplify the presentation, in the following a formalism is chosen on the basis of Zernike coefficients, wherein in principle a different basis may also be used.

(92) In the following, the correlation between a set of Zernike coefficients for representation of the wavefronts (c.sub.nm), with r.sub.0 as a radius of the wavefront, and refraction values (sph, cyl, a), is initially considered. The radius r.sub.0 is preferably either measured or is established on the basis of model assumptions. Given use of the RMS metric, for example, the bijective correlation results as

(93) $\left(\begin{array}{c}{c}_{2‐2}\\ c_{2,0}\\ c_{2+2}\end{array}\right)=g_{RMS}\left(\mathrm{sph},cyl,a\right)=\frac{r_0^2}{2\sqrt{6}}\left(\begin{array}{c}\frac{1}{2}.Math.\mathrm{cyl}.Math.\sin \left(2a\right)\\ -\frac{1}{\sqrt{2}}.Math.\left(\mathrm{sph}+\frac{1}{2}cyl\right)\\ \frac{1}{2}.Math.\mathrm{cyl}.Math.\cos \left(2a\right)\end{array}\right)\iff \left(\begin{array}{c}sph\\ cyl\\ a\end{array}\right)=f_{R\mathrm{MS}}\left(c_{2,-2},c_{2,0,}{c}_{2+2}\right)=\left(\begin{array}{c}-\frac{4\sqrt{3}}{r_0^2}.Math.\left(c_{2,0}-\frac{1}{\sqrt{2}}.Math.\sqrt{c_{2,-2}^2+c_{2,+2}^2}\right)\\ -\frac{4\sqrt{6}}{r_0^2}.Math.\sqrt{c_{2,-2}^2+c_{2,+2}^2}\\ \frac{1}{2}.Math.\arctan \left(c_{2,+2,}{c}_{2,-2}\right)+\frac{\pi }{2}\end{array}\right)$

(94) However, this is to be understood only as an example of a metric of the general form

(95) 0$\begin{array}{cc}\left(\begin{array}{c}sph\\ cyl\\ a\end{array}\right)=f_0\left(c_{2,-2},c_{2,0},c_{2,+2}\right)\iff \left(\begin{array}{c}{c}_{2,-2}\\ c_{2,0}\\ c_{2,+2}\end{array}\right)=g_0\left(\mathrm{sph},\mathrm{cyl},a\right).& \left(8\right)\end{array}$

(96) Moreover, there are correlations in which HOAs also enter into the refraction values. This mapping is then always surjective for the calculation of the refraction values, but no longer bijective, meaning that the complete set of all Zernike coefficients of all mapping errors cannot be unambiguously reproduced from the refraction values. However, the coefficients of the lower-order mapping errors can also be unambiguously determined here again if the coefficients for the HOAs are predetermined:

(97) $\begin{array}{cc}\left(\begin{array}{c}\mathrm{sph}\\ \mathrm{cyl}\\ a\end{array}\right)=f_1\left(c_{2,-2},c_{2,0},c_{2,+2},c_{i,j}\right)\iff \left(\begin{array}{c}{c}_{2,-2}\\ c_{2,0}\\ c_{2,+2}\end{array}\right)=g_1\left(\mathrm{sph},\mathrm{cyl},a,c_{i,j}\right)\left(i>2\right)& \left(9\right)\end{array}$

(98) Naturally, analogous calculations and derivation are also possible in other notations, for example with the local derivatives of the wavefronts that are used in the publications by G. Esser et al.: “Derivation of the refraction equations for higher order aberrations of local wavefronts at oblique incidence”, JOSA A, Vol. 27, No. 2 (2010), and by G. Esser et al.: “Derivation of the propagation equations for higher order aberrations of local wavefronts”, JOSA A, Vol. 28, No. 11 (2011). If autorefractometric measurement with data regarding HOAs are present, these data or portions of these data may be used, together with the subjective refraction values, in order to determine a set of optimized refraction data, for example according to DE 10 2007 032 564 A1. The simultaneous use of both subjective refraction data and of the measured data is thereby not necessary. The variables, which in this section are referred to in the following as optimized refraction values (sph.sub.opt, cyl.sub.opt and a.sub.opt), may thus also be directly adopted from the subjective refraction determination without the use of objective measurement variables.

(99) In principle, not all values of the subjective refraction or of the objective measurement need to enter into the optimized refraction values. For example, in the event of a determination of the optimized refraction values for near, or in the event of anticipated instrument myopia, the use of the objectively measured sphere or of the objectively measured defocus term may be omitted.

(100) A wavefront (preferably represented by the Zernike coefficients o.sub.i,j) that corresponds to these optimized values is then determined on the basis of the optimized refraction values. This wavefront is then used instead of the measured exiting wavefront described above. Given use of a metric according to Equation (8), the 2nd-order coefficients of this wavefront may be calculated according to Equation (8) from the optimized refraction values, and the higher-order coefficients may be directly adopted from the objective measurement of the exiting wavefront represented by the coefficient m.sub.i,j:

(101) $\begin{array}{cc}\left(\begin{array}{c}{o}_{2,-2}\\ o_{2,0}\\ o_{2,+2}\end{array}\right)=g_0\left({\mathrm{sph}}_{\mathrm{opt}},{\mathrm{cyl}}_{\mathrm{opt}},a_{\mathrm{opt}}\right)o_{i,j}=m_{i,j}\left(i>2\right)& \end{array}$

(102) By contrast, given use of a metric according to Equation (9), the second-order coefficients of the wavefront (o.sub.i,j) are not only dependent on the optimized refraction, but rather are by contrast to be chosen so that

(103) $\left(\begin{array}{c}{\mathrm{sph}}_{\mathrm{opt}}\\ {\mathrm{cyl}}_{\mathrm{opt}}\\ a_{\mathrm{opt}}\end{array}\right)=f_1\left(o_{2,-2},o_{2,0},o_{2,+2},c_{i,j}\right)\left(i>2\right)$
applies, and therefore additionally directly depend on the higher-order coefficients of the measured exiting wavefront (m.sub.i,j):

(104) $\left(\begin{array}{c}{o}_{2,-2}\\ o_{2,0}\\ o_{2,+2}\end{array}\right)=g_1\left(sp{h}_{\mathrm{opt}},\mathrm{cy}{l}_{\mathrm{opt}},a_{\mathrm{opt}},m_{i,j}\right)o_{ij}=m_{ij}$

(105) The evaluation of the aberrations during the calculation or optimization method may be performed at different locations in the ray path, meaning that the evaluation surface may be provided at different positions. Instead of taking place at the retina or at the posterior lens surface, an evaluation of the imaging wavefront may also already take place at a surface that is situated further forward in the model eye. For this purpose, within the model eye a reference wavefront R is defined that is then used in the lens optimization, for example. This reference wavefront thereby has the property that, given further propagation through the eye up to the retina, it leads to a point image. The reference wavefront may accordingly be determined, via back-propagation of a wavefront that converges at a point on the retina, from the retina up to the position of the reference wavefront. Since the measured wavefront S.sub.M is precisely the wavefront that emanates from a point light source on the retina, this may instead also be propagated inside the eye up to the position of the reference wavefront.

(106) Considered mathematically, both procedures are equivalent and lead to the same formulas for the reference wavefront. In the following, to derive the corresponding reference wavefronts the path is respectively chosen that manages with fewer propagation steps and enables a simpler representation. In the following, for example, only the treatment of the components of the defocus and astigmatism is described. However, an expansion to HOAs and the use of subjective refraction is likewise possible and advantageous.

(107) Given the consideration of HOAs, these may take place analogous to the calculation of the HOAs according to the embodiments in the following, via refraction (G. Esser et al.: “Derivation of the refraction equations for higher order aberrations of local wavefronts at oblique incidence”, JOSA A, Vol. 27, No. 2 (2010)) and propagation (G. Esser et al.: “Derivation of the propagation equations for higher order aberrations of local wavefronts”, JOSA A, Vol. 28, No. 11 (2011)).

(108) Since the wavefront propagation is a non-linear process, a spectacles lens optimization that evaluates an imaging wavefront via comparison with a reference wavefront generally leads to different results depending on at which surface within the eye this comparison occurs.

(109) In a preferred embodiment, only the ultimate step (in particular step 6b) is omitted, thus the propagation from the AP to the retina. The incident wavefront is thus only simulated up to the AP after the refraction at the posterior lens surface (thus calculation of S.sub.AP according to the aforementioned step 6a), and there is compared with a reference wavefront R.sub.AP. This is thereby characterized in that in that, given the propagation to the retina, a point image results there. According to the above statement, the vergence matrix of this wavefront is precisely

(110) $R_{AP}=D_{AP}=D_{LR}^{\left(b\right)}=\frac{1}{{\tau }_{LR}^{\left(b\right)}}1=\frac{1}{{\tau }_{\mathrm{LR}}-{\tau }_{LR}^{\left(a\right)}}1=\frac{1}{1/D_{\mathrm{LR}}-d_{\mathrm{LR}}^{\left(a\right)}/n_{LR}}1$
with the D.sub.LR determined from Equation (2) or (3), as well as the negative (accommodation-dependent) value d.sub.LR.sup.(a)<0, whose absolute value describes the distance between the posterior lens surface and the AP.

(111) In a furthermore preferred embodiment, the penultimate step, thus the propagation from the posterior lens surface to the retina as a whole, is moreover omitted. The incident wavefront is thus simulated only up to after the refraction at the posterior lens surface (thus calculation of S′.sub.L2 according to the aforementioned step 5), and there is compared with a reference wavefront R′.sub.L2. This is thereby characterized in that, given the propagation to the retina, it yields a point image there. According to the above statement, the vergence matrix of this wavefront is precisely
R′.sub.L2=D′.sub.L2=D.sub.LR.Math.1
with the D.sub.LR determined from Equation (2) or (3).

(112) A further simplification results if the comparison is placed before the refraction by the posterior lens surface. In this instance, the incident wavefront must be simulated, thus calculated, only up to S.sub.L2 according to the above step 4. For this purpose, analogous to S′.sub.L2, a reference wavefront R.sub.L2 is defined that, after the refraction at the posterior lens surface and the propagation to the retina, yields a point image there. This is determined as
R.sub.L2=R′.sub.L2−L.sub.2=D.sub.LR.Math.1−L.sub.2
with the D.sub.LR determined from Equation (2) or (3) and the L.sub.2 known from the literature or from measurements.

(113) In the event of a rotationally symmetrical posterior lens surface, this simplifies to
R.sub.L2=(D.sub.LR−L.sub.2,xx).Math.1

(114) In particular insofar as the lens thickness is likewise taken from the literature, in a further preferred embodiment it is suggested to omit the propagation through the lens as a next step, and to execute the comparison after the refraction by the anterior lens surface. In continuation of the above statement, for this purpose a reference wavefront R′.sub.L1 is preferably used that arises from R.sub.L2 via backward propagation by the lens thickness and has the following vergence matrix:
R′.sub.L1=R.sub.L2/(1+τ.sub.LR.sub.L2)
with the D.sub.LR determined from Equation (2) or (3), and the τ.sub.L=d.sub.L/n.sub.L known from the literature or from measurements, as well as the vergence matrix R.sub.L2 determined from Equation (6) or (7).

(115) In the event of a rotationally symmetrical posterior lens surface, this simplifies to

(116) $R_{L1}^{\prime }=\frac{D_{\mathrm{LR}}-L_{2,\mathrm{xx}}}{1+{\tau }_L.Math.\left(D_{\mathrm{LR}}-L_{2,\mathrm{xx}}\right)}.Math.1$

(117) As given the above model, here it also applies that, even if the consideration occurs before the last steps and—depending on notation—the variable D.sub.LR does not explicitly appear, this variable nevertheless at least implicitly appears together with d.sub.L and L.sub.2, since they together control the distribution of the effect L.sub.1 in the anterior lens surface.

(118) Yet another simplifications results if the comparison is placed before the refraction by the anterior lens surface. In this instance, the incident wavefront only needs to be simulated up to S.sub.L1 according to step 2. For this purpose, analogous to R′.sub.L1 a reference wavefront R.sub.L1 is defined that, after the refraction at the anterior lens surface and the further steps, converges to a point on the retina. This may be calculated either via the refraction of R′.sub.L1 at L.sub.1, or be determined directly from the refraction of the measured wavefront S.sub.M at the cornea C and a subsequent propagation by d.sub.CL. In both instances, one obtains

(119) $R_{L1}=\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}.Math.\left(S_M+C\right)}$

(120) The variables D.sub.LR, d.sub.L and L.sub.2 no longer enter therein; it is thus sufficient to know S.sub.M, C and d.sub.CL.

(121) One embodiment, in which the comparison is performed after the refraction at the cornea, is linked with relatively little computational cost. In this instance, only S.sub.M and C are still considered:
R′.sub.C=S.sub.M+C

(122) An additional, very efficient possibility is the positioning of the evaluation surface at the exit pupil of the model eye. This preferably lies before the posterior lens surface.

(123) The eye model and the population of the same may be expanded as follows:

(124) In principle, the eye model may differ between cornea and anterior chamber. For this purpose, a posterior corneal surface C.sub.2 is introduced at a distance d.sub.C after the anterior corneal surface anterior surface C.sub.1 (formerly C), and two different refraction indices n.sub.C or n.sub.CL are specified for cornea and anterior chamber. The first step stated above (refraction of the wavefront S at the cornea C into wavefront S′.sub.C with vergence matrix S′.sub.C=S+C) is also replaced by the following three steps: 1a: refraction of the wavefront S at the anterior corneal surface C.sub.1 into wavefront S′.sub.C1 with the vergence matrix S′C.sub.1=S+C.sub.1 1b: propagation by the thickness of the cornea d.sub.C to the wavefront S.sub.C2 with the vergence matrix S.sub.C2=S′.sub.C1/(1−τ.sub.CS′.sub.C1) 1c: refraction at the posterior corneal surface C.sub.2 into wavefront S′.sub.C2 with the vergence matrix S′C.sub.2=S.sub.C2+C.sub.2
wherein

(125) ${\tau }_c=\frac{d_c}{n_c}.$

(126) Analogous to the other values, here the values for d.sub.C and C.sub.2 may also be respectively measured, taken from the literature, or derived. As an example, a few possibilities for C.sub.2 are described here:

(127) In the event that no measurement at the posterior corneal surface is present, the shape of the posterior corneal surface may be taken from known eye models. Alternatively, in this instance the posterior corneal surface may also be derived from the measured shape of the anterior corneal surface. For this purpose, it is suggested to assume either a uniform corneal thickness (defined, for example, as emanating “in the direction of the rise” or “in a radial direction from a ‘center of corneal curvature’”). The thickness may thereby either be learned from a measurement, be derived from this, or be learned from the literature. Furthermore, local properties may also be transferred only in part to the posterior surface.

(128) If only a principal section of the posterior corneal surface is measured, this information may be used in order to reconstruct the entire posterior surface. For example, this may occur via the preparation of a function of the thickness or rise of the posterior corneal surface from the radius or the thickness of the rise of the anterior surface.

(129) In most such instances, the anterior and posterior corneal surfaces are thereby known in the same normal section (meaning here in the x-direction).

(130) The fact that the human eye is a non-centered optical system may thereby allow that the optical elements are arranged offset and/or tilted relative to a central axis.

(131) This may relate to the individual elements as a whole (i.e. cornea and lens), or to all refractive surfaces individually (anterior corneal surface, possibly posterior corneal surface, anterior lens surface, and posterior lens surface). The corresponding parameters are respectively, for example, two lateral coordinates of the displacement of the center of the element or of the surface from the central axis, and two tilt angles. Alternatively, first-order Zernike coefficients (tip/tilt) may also be used.

(132) The relevant variable that is affected by the change with regard to a centered system is the principal ray that forms the basis of the invention for all calculations, and which corresponds to the centered systems of the optical axis that have been dealt with up to now. In the general instance, the principal ray is that ray that emanates from the retina as a center of the measurement wavefront (preferably the site of the fovea) and passes through the middle of the entrance pupil. What is different than in the centered system, in which this ray coincides at suitable coordinates with the global z-axis of the eye model, is that the ray is now straight only in segments, from interface to interface, and also strikes off-center and at defined angles of incidence at each interface. The path of the principal ray, the positions of the penetration points, and the respective angles of incidence must be determined before calculation of the wavefronts (in the second order or higher order).

(133) If the changes of the individual elements relative to a centered system are small, the principal ray may be approximately determined via the following affine equations. These correspond to an affine, expanded form of the linear optics in relation to a global coordinate system. Each propagation of a ray with lateral coordinate r and direction angle α relative to the global z-axis by a length d is thereby mapped, via the 2×2 transfer matrix equation

(134) $\begin{array}{cc}\left(\begin{array}{c}{r}^{\prime }\\ {\alpha }^{\prime }\end{array}\right)=\left(\begin{array}{cc}1& d\\ 0& 1\end{array}\right)\left(\begin{array}{c}r\\ \alpha \end{array}\right),& \left(10a\right)\end{array}$
to the propagated ray with lateral coordinate r′ and direction angle α′. By contrast, the refraction is described by the expanded 2×2 transfer matrix equation

(135) 0$\begin{array}{cc}\left(\begin{array}{c}{r}^{\prime }\\ {\alpha }^{\prime }\end{array}\right)=\left(\begin{array}{cc}1& 0\\ \left(\frac{n}{n^{\prime }}-1\right)\rho & \frac{n}{n^{\prime }}\end{array}\right)\left(\begin{array}{c}r\\ \alpha \end{array}\right)+\left(\begin{array}{c}\Delta r\\ \Delta \alpha \end{array}\right).& \left(10b\right)\end{array}$
ρ is thereby the curvature of the refractive surface, and n, n′ are the refraction indices before and after the refraction. Δr and Δα are additionally correction portions of the ray parameters that materialize due to the lateral displacement and the tilting of the refractive interface, and may be determined from the tilt parameters and displacement parameters of the surface, for instance with Prentice's Rule. In the event of cylindrical surfaces, the 4×4 transfer matrix equations are to be used accordingly.

(136) If the approximation described in Equations (10a) and (10b) is not sufficient, the principal ray, meaning all penetration points through the surfaces, may be numerically determined. In both instances, the principal ray determination has the effect that all propagation distances, the coordinates of the penetration points, and the angles of incidence and emergence, ε, ε′ are determined at each interface. In the event of the affine equations, ε, ε′ result from α, α′, and the surface normals that can be determined from r, the decentration, and the dioptric effect according to Prentice's Rule at the penetration point. In the general instance, ε, ε′ result from the numerical principal ray calculation and the surface normals at the penetration point r. The latter may be calculated instead of the penetration point r, for example via derivation of the surface representation (for example Taylor representation or Zernike representation around the point r=0, or B-splines).

(137) In the event of the affine equations, the surface refractive power matrix C is constant and given by the respective refractive element. In the event of numerical calculation, C results at the penetration point via the local second derivatives in relation to a local coordinate system.

(138) With the angles of incidence and emergence ε, ε′ that are calculated in such a manner, and possibly the newly determined surface refractive power matrix C, the calculation methods of the invention as described in the following may also be applied to decentered systems:

(139) In the second order, instead of the vergence equation in matrix form S′.sub.C=S+C, the generalized Coddington equation occurs
Cos(ε′)S′.sub.C Cos(ε′)=Cos(ε)S Cos(ε)+vC  (11)
with

(140) $\begin{array}{cc}v=\frac{n^{\prime }\cos {\epsilon }^{\prime }-n\cos \epsilon }{n^{\prime }-n}\mathrm{Cos}\left(\epsilon \right)=\left(\begin{array}{cc}1& 0\\ 0& \cos \left(\epsilon \right)\end{array}\right)\mathrm{und}\mathrm{Cos}\left({\epsilon }^{\prime }\right)=\left(\begin{array}{cc}1& 0\\ 0& \cos \left({\epsilon }^{\prime }\right)\end{array}\right)& \left(11a\right)\end{array}$

(141) Instead of the propagation equation S′=S/(1−τS) with τ=d/n, the matrix equation
S′=S/(1−τ.sub.α,r.Math.S)mitτ.sub.α,r=d.sub.α,r/n  (12)
occurs. d.sub.α,r thereby designates the actual spatial distance between the penetration point of the successive surfaces.

(142) If HOAs should be considered as well, instead of Equations (11) and (12), for refraction and propagation the corresponding expanded equations for the respective orders are to be used from publications by G. Esser et al.: “Derivation of the refraction equations for higher order aberrations of local wavefronts at oblique incidence”, JOSA A, Vol. 27, No. 2 (2010), and by G. Esser et al.: “Derivation of the propagation equations for higher order aberrations of local wavefronts”, JOSA A, Vol. 28, No. 11 (2011), and for this purpose the coefficients of the Taylor expansion of the refractive surface are to be determined as described (ibid.) in the coordinate system of the ray incidence.

(143) Furthermore, a diaphragm—likewise also displaced or tilted—may be introduced in order to take into account the vignetting by the iris.

(144) Second Approach

(145) Insofar as is not explicitly noted otherwise, details regarding exemplary and preferred implementations of the second approach of the invention are now described in the following paragraphs:

(146) FIG. 1 shows a schematic depiction of the physiological and physical model of a spectacles lens and of an eye in a predetermined usage position, together with an exemplary ray path which forms the basis of a personalized spectacles lens calculation or optimization according to a preferred embodiment of the invention.

(147) Preferably, only a single ray (the principal ray 10, which preferably travels through the eye's center of rotation Z′) is hereby calculated per visual point of the spectacles lens, but moreover also accompanying the derivatives of the rises of the wavefront according to the transversal (orthogonal to the principal ray) coordinates. These derivatives are considered up to the desired order, wherein the second derivatives describe the local curvature properties of the wavefront, and the higher derivatives coincide with the higher-order aberrations.

(148) Given the calculation of light through the spectacles lens, up to the eye 12, according to the personalized prepared eye model, the local derivatives of the wavefronts are determined in the end effect at a suitable position in the ray path in order to compare them there with a reference wavefront which converges at a point on the retina of the eye 12. In particular, the two wavefronts (meaning the wavefront coming from the spectacles lens and the reference wavefront) are compared with one another at an evaluation surface.

(149) What is thereby meant by “position” is thereby not simply a defined value of the z-coordinate (in the light direction), but rather such a coordinate value in combination with the specification of all surfaces through which refraction has taken place before reaching the evaluation surface. In a preferred embodiment, refraction occurs through all refracting surfaces, including the posterior lens surface. In this instance, a spherical wavefront whose center of curvature lies on the retina of the eye 12 preferably serves as a reference wavefront.

(150) Particularly preferably, as of this last refraction propagation does not continue, so that the radius of curvature of this reference wavefront corresponds directly to the distance between posterior lens surface and retina. In an alternative possibility, propagation does continue after the last refraction, and in fact preferably up to the exit pupil AP of the eye 12. For example, this is situated at a distance d.sub.AR=d.sub.LR.sup.(b)=d.sub.LR−d.sub.LR.sup.(a)>d.sub.LR in front of the retina, and therefore even in front of the posterior lens surface, so that in this instance the propagation is a back-propagation (the terms d.sub.LR.sup.(a), d.sub.LR.sup.(b) are described further below in the enumeration of steps 1-6). In this instance as well, the reference wavefront is spherical with center of curvature on the retina, but has curvature radius 1/d.sub.AR.

(151) In this regard, it is assumed that a spherical wavefront w.sub.0 emanates from the object point and propagates up to the first spectacles lens surface 14. There it is refracted and subsequently propagates up to the second spectacles lens surface 16, where it is refracted again. The wavefront w.sub.g1 exiting from the spectacles lens subsequently propagates along the principal ray in the direction of the eye 12 (propagated wavefront w.sub.g2) until it strikes the cornea 18, where it is again refracted (wavefront w.sub.c). After a further propagation within the anterior chamber depth up to the eye lens 20, the wavefront is also refracted again by the eye lens 20, whereby the resulting wavefront w.sub.e is created at the posterior surface of the eye lens 20 or at the exit pupil of the eye, for example. This is compared with the spherical reference wavefront w.sub.s, and for all visual points the deviations are evaluated in the objective function (preferably with corresponding weightings for the individual visual points).

(152) The ametropia is thus no longer described only by a thin sphero-cylindrical lens, as this was typical in many conventional methods; rather, the corneal topography, the eye lens, the distances in the eye, and the deformation of the wavefront (including the lower-order aberrations—thus sphere, cylinder, and axis length—as well as preferably also including the higher-order aberrations) in the eye are preferably directly considered.

(153) An aberrometer measurement preferably delivers the personalized wavefront deformations of the real, ametropic eye for far and near (deviations, no absolute refractive powers), and the personalized mesopic and photopic pupil diameters. A personalized real anterior corneal surface that generally makes up nearly 75% of the total refractive power of the eye is preferably obtained from a measurement of the corneal topography (areal measurement of the anterior corneal surface). In a preferred embodiment, it is not necessary to measure the posterior corneal surface. Due to the small refractive index difference relative to the aqueous humor, and due to the small cornea thickness, it is preferably described in good approximation not by a separate refractive surface, but rather by an adaptation of the refractive index of the cornea.

(154) In general, in this specification bold-face lowercase letters designate vectors, and bold-face capital letters designate matrices, for example the (2×2) vergence matrices or refractive index matrices

(155) $S=\left(\begin{array}{cc}{S}_{\mathrm{xx}}& S_{xy}\\ S_{xy}& S_{yy}\end{array}\right),C=\left(\begin{array}{cc}{C}_{\mathrm{xx}}& C_{xy}\\ C_{xy}& C_{yy}\end{array}\right),L=\left(\begin{array}{cc}{L}_{\mathrm{xx}}& L_{xy}\\ L_{xy}& L_{yy}\end{array}\right),1=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),$
and cursive letters such as d designate scalar values.

(156) Furthermore, bold-face cursive capital letters should designate wavefronts or surfaces as a whole. For example, S is thus the vergence matrix of the identically named wavefront S; aside from the 2nd-order aberrations that are encompassed in S, S also includes the entirety of all higher-order aberrations (HOAs) of the wavefront. Mathematically, S stands for the set of all parameters that are necessary in order to describe a wavefront (sufficiently precisely) with regard to a given coordinate system. S preferably stands for a set of Zernike coefficients having a pupil radius, or a set of coefficients of a Taylor series. S particularly preferably stands for the set from a vergence matrix S to describe the 2nd-order wavefront properties, and a set of Zernike coefficients (with a pupil radius) that serves to describe all remaining wavefront properties except for the 2nd order, or a set of coefficients according to a Taylor decomposition. Analogous statements apply to surfaces instead of wavefronts.

(157) Among other things, the following data may in principle be measured directly: the wavefront S.sub.M, which is generated by the laser spot on the retina and the passage through the eye (from aberrometric measurement) shape of the anterior corneal surface C (via corneal topography) distance between cornea and anterior lens surface d.sub.CL (via pachymetry). This variable may also be determined indirectly via the measurement of the distance between the cornea and the iris; correction values may thereby be applied, if applicable. Such corrections may be the distance between the anterior lens surface and the iris, from known eye models (for example literature values). curvature of the anterior lens surface in a direction L.sub.1xx (via pachymetry). Without limitation of the generality, the x-plane may thereby be defined such that this section lies in the x-plane. The coordinate system is thus defined so that this plane lies obliquely; the derivative must be expanded by the functions of the corresponding angle. It is not required that it thereby be a principal section. For example, it may be the section in the horizontal plane.

(158) Furthermore—depending on the embodiment—the following data may either be measured or learned from the literature: thickness of the lens d.sub.L curvature of the posterior lens surface in the same direction as the anterior lens surface L.sub.2,xx (via pachymetry)

(159) Therefore, there are the following possibilities for the posterior lens surface: measurement of L.sub.2,xx (L.sub.2,M) and assumption of a rotational symmetry L.sub.2,xx=L.sub.2,yy=L.sub.2=L.sub.2,M and L.sub.2,xy=L.sub.2,yx=0 taking L.sub.2,xx from the literature (L.sub.2,Lit), and assumption of a rotational symmetry L.sub.2,xx=L.sub.2,yy=L.sub.2=L.sub.2,M and L.sub.2,xy=L.sub.2,yx=0 taking the complete (asymmetrical) shape L.sub.2 from the literature (L.sub.2,Lit) measurement of L.sub.2,xx (L.sub.2,M), and assumption of a cylinder or an otherwise specified asymmetry a.sub.Lit from the literature L.sub.2,xx=L.sub.2,M and L.sub.2,xy=L.sub.2,yx=f(L.sub.2,xx,a.sub.Lit) as well as L.sub.2,yy=g(L.sub.2,xx,a.sub.Lit)

(160) The following data may be learned from the literature: refractive indices n.sub.CL of cornea and anterior chamber depth, as well as of the aqueous humor n.sub.LR and that of the lens n.sub.L

(161) In particular, the distance d.sub.LR between posterior lens surface and retina, as well as the components L.sub.1,yy and L.sub.1,xy=L.sub.1,yx of the anterior lens surface, therefore remain as unknown parameters. To simplify the formalism, the former may also be written as a vergence matrix D.sub.LR=D.sub.LR.Math.1 with D.sub.LR=n.sub.LR/d.sub.LR. Furthermore, the variable z is generally used, which is defined as τ=d/n (wherein the corresponding index as is used for d and τ is always to be used for the refractive index as n, for example as τ.sub.LR=d.sub.LR/n.sub.LR, τ.sub.CL=d.sub.CL/n.sub.CL).

(162) In a preferred embodiment in which the lens is described via an anterior surface and a posterior surface, the modeling of the passage of the wavefront through the eye model used according to the invention, thus after the passage through the surfaces of the spectacles lens, may be described as follows, wherein the transformations of the vergence matrices are explicitly indicated: 7. Refraction of the wavefront S with the vergence matrix S at the cornea C with the surface refractive power matrix C, relative to the wavefront S′.sub.C with vergence matrix S′.sub.C=S+C 8. Propagation by the anterior chamber depth d.sub.CL (distance between cornea and anterior lens surface) relative to the wavefront S.sub.L1 with vergence matrix S.sub.L1=S′.sub.C/(1−τ.sub.CL.Math.S′)

(163) $S_{L1}=\frac{S_C^{\prime }}{\left(1-{\tau }_{CL}.Math.S_C^{\prime }\right)}$ 9. Refraction at the anterior lens surface L.sub.1 with the surface refractive power matrix L.sub.1 relative to the wavefront S′.sub.L1 with the vergence matrix S′.sub.L1=S.sub.L1+L.sub.1 10. Propagation by the lens thickness d.sub.L relative to the wavefront S.sub.L2 with vergence matrix S.sub.L2=S′.sub.L1/(1−τ.sub.L.Math.S′.sub.L1) 11. Refraction at the posterior lens surface L.sub.2 with the surface refractive power matrix L.sub.2 relative to the wavefront S′.sub.L2 with the vergence matrix S′.sub.L2=S.sub.L2+L.sub.2 12. Propagation by the distance between lens and retina d.sub.LR relative to the wavefront S.sub.R with the vergence matrix S.sub.R=S′.sub.L2/(1−τ.sub.LR.Math.S′.sub.L2)

(164) Each of the steps 2, 4, 6 in which propagation takes place over the distances τ.sub.CL, τ.sub.CL, or τ.sub.CL may thereby be divided up into two partial propagations 2a,b), 4a,b), or 6a,b) according to the following scheme, which for step 6a,b) explicitly reads: 6a. Propagation by the distance d.sub.LR.sup.(a) between lens and intermediate plane relative to the wavefront S.sub.LR with the vergence matrix S.sub.LR=S′.sub.L2/(1−τ.sub.LR.sup.(a)S′.sub.L2) 6b. Propagation by the distance d.sub.LR.sup.(b) between intermediate plane and retina relative to the wavefront S.sub.R with the vergence matrix S.sub.R=S.sub.LR/(1−τ.sub.LR.sup.(b)S.sub.LR)

(165) τ.sub.LR.sup.(a)=d.sub.LR.sup.(a)/n.sub.LR.sup.(a) and τ.sub.LR.sup.(b)=d.sub.LR.sup.(b)/n.sub.LR.sup.(b) may thereby be positive or negative, wherein n.sub.LR.sup.(a)=n.sub.LR.sup.(b)=n.sub.LR and τ.sub.LR.sup.(a)+τ.sub.LR.sup.(b)=τ.sub.LR should always be true. In each instance, step 6a and step 6b can be combined again via S.sub.R=S′.sub.L2/(1−(τ.sub.LR.sup.(a)+τ.sub.LR.sup.(b))S′.sub.L2)=S′.sub.L2/(1−τ.sub.LRS′.sub.L2). However, the division into step 6a and step 6b offers advantages, and the intermediate plane may preferably be placed in the plane of the exit pupil AP, which preferably is situated in front of the posterior lens surface. In this instance, τ.sub.LR.sup.(a)<0 and τ.sub.LR.sup.(b)>0.

(166) The division of steps 2, 4 may also take place analogous to the division of step 6 into 6a,b).

(167) For the selection of the evaluation surface of the wavefront, it is thus not only the absolute position in relation to the z-coordinate (in the light direction) but also the number of surfaces through which refraction has already taken place up to the evaluation surface. One and the same plane may thus be traversed repeatedly. For example, the plane of the AP (which normally is situated between the anterior lens surface and the posterior lens surface) is formally traversed by the light for the first time after a virtual step 4a, in which propagation takes place from the anterior lens surface by the length τ.sub.L.sup.(a)>0. The same plane is reached for the second time after step 6a if, after refraction by the posterior lens surface, propagation takes place again back to the AP plane, meaning that τ.sub.LR.sup.(a)=−τ.sub.L+τ.sub.L.sup.(a)=−τ.sub.L.sup.(b)<0, which is equivalent to τ.sub.LR.sup.(a)=τ.sub.LR−τ.sub.LR.sup.(b)<0. Given the wavefronts S.sub.AP, which relate in the text to the AP, what should preferably always be meant (if not explicitly noted otherwise) is the wavefront S.sub.AP=S.sub.LR, which is the result of step 6a.

(168) These steps 1 through 6 are referred to repeatedly in the further course of the specification. They describe a preferred correlation between the vergence matrix S of a wavefront S at the cornea and the vergence matrices of all intermediate wavefronts arising therefrom at the refractive intermediate surfaces of the eye, in particular the vergence matrix S′.sub.L2 of a wavefront S′.sub.L2 after the eye lens (or even of a wavefront S.sub.R at the retina). These correlations may be used both to calculate parameters (for example d.sub.LR or L.sub.1) that are not known a priori, and thus to populate the model with values in either a personalized or generic manner, and in order to simulate the propagation of the wavefront in the eye with then populated models to optimize spectacles lenses.

(169) Before the procedure according to the invention of the consideration of higher-order aberrations (meaning higher than second order, in particular in Taylor or Zernike decomposition of the aberrations) is discussed, for the sake of simplicity in the following an example of a principle of the formalism should be described using a description of the surfaces and wavefronts up to the second order, for which a representation by vergence matrices is sufficient. As is subsequently presented, this formalism may be used analogous to that for the implementation of the invention under consideration of higher orders of aberrations.

(170) In a preferred embodiment, in a second-order description the eye model has twelve parameters as degrees of freedom of the model that need to be populated. These preferably include the three degrees of freedom of the surface refractive power matrix C of the cornea C; the respective three degrees of freedom of the surface refractive power matrices L.sub.1 and L.sub.2 for the anterior lens surface or posterior lens surface; and respectively one for the length parameters of anterior chamber depth d.sub.CL, lens thickness d.sub.L, and the vitreous body length d.sub.LR.

(171) Populations of these parameters may in principle take place in a plurality of ways: iv) directly, thus personalized measurement of a parameter v) a priori given value of a parameter, for example as a literature value or from an estimate, for example due to the presence of a measured value for another variable that correlates with the parameter to be determined in a known manner using a preceding population analysis vi) calculation from consistency conditions, for example compatibility with a known refraction

(172) The total number df.sub.2 of second-order degrees of freedom of the eye model (df stands for “degree of freedom”, the index “2” stands for 2nd-order) is thus composed of
df.sub.2=df.sub.2(i)+df.sub.2(ii)+df.sub.2(iii)

(173) For example, if direct measured values are present for all twelve model parameters, then df.sub.2(i)=12, df.sub.2(ii)=0, and df.sub.2(iii)=0, which for the sake of simplicity is expressed in the following by the notation df.sub.2=12+0+0. In such an instance, the object refraction of the appertaining eye is also established, so that an objective refraction determination would no longer need to be additionally implemented.

(174) For the implementation of the present invention, it is not necessary to directly measure all parameters. Under the circumstances, it is thus simpler to measure, or objectively and/or subjectively determine, the refraction of the appertaining eye than to measure all parameters of the model eye in a personalized manner. At least one refraction, thus measurement data regarding the wavefront S.sub.M of the eye up to the 2nd order that correspond to the data of the vergence matrix S.sub.M, is thus preferably present. Given a population of the eye model purely on the basis of objectively measured data, these values may be taken from autorefractometric measurements, for example, or according to (ii) may be populated by data provided otherwise. The three conditions of the agreement with the three independent parameters of the vergence matrix S.sub.M therefore allow three parameters of the eye model to be derived, which in the notation introduced above corresponds to df.sub.2(iii)=3.

(175) In instances in which not all model parameters are accessible to direct measurements, or these measurements would be very costly, it is thus possible to reasonably populate the missing parameters. For example, if direct measured values are present for at most nine model parameters (df.sub.2(i)≤9), then the cited conditions of the refraction may be used in order to calculate three of the model parameters (df.sub.2(iii)=3). In the event that df.sub.2(i)=9 applies exactly, all twelve model parameters are then determined unambiguously via the measurements and the calculation, and (df.sub.2(ii)=0) applies. By contrast, if df.sub.2(i)<9, then df.sub.2(ii)=9−df.sub.2(i)>0, meaning that the model is underdetermined in the sense that df.sub.2 (ii) parameters need to be established a priori.

(176) With the provision of a personalized refraction, thus measurement data regarding the wavefront S.sub.M of the eye, in particular up to the second order, the necessary data of the vergence matrix S.sub.M are present. According to a conventional method described in WO 2013/104548 A1, in particular the parameters {C, d.sub.CL, S.sub.M} are measured. By contrast, among other things the two length parameters d.sub.L and d.sub.LR (or D.sub.LR) are conventionally established a priori (for example via literature values or estimation). In WO 2013/104548 A1, in particular a differentiation is made between the two instances in which either L.sub.2 is established a priori and L.sub.L is calculated therefrom, or vice versa. The cited disclosure document discloses Equation (4) or Equation (5) as a calculation rule in this regard. For both instances, df.sub.2=4+5+3 applies.

(177) In the terminology of the aforementioned steps 1 through 6, the adaptation of L.sub.1 to the measurements in particular occurs in that, on the one hand, the measured vergence matrix S.sub.M is calculated through the likewise measured matrix C by means of the steps 1, 2, and propagated up to the object-side side of the anterior lens surface. On the other hand, a spherical wave is calculated from back to front from an imaginary point light source on the retina by means of the steps 6, 5, 4, run through in reverse, in that this spherical wave is refracted at the previously established surface refractive power matrix L.sub.2 of the posterior lens surface, and the wavefront that is then obtained propagates from the posterior lens surface up to the image-side side of the anterior lens surface. The difference of the vergence matrices S.sub.L1 and S′.sub.L1 that are determined in this manner, which difference must be present on the object side or image side of the anterior lens surface, must have been produced by the matrix L.sub.1, because in the aberrometric measurement the measured wavefront arises from a wavefront that emanates from a point on the retina and therefore, due to the reversibility of the ray paths, is identical to that incident wavefront (S=S.sub.M) that converges on this point of the retina. This leads to Equation (4) in the cited disclosure document:

(178) $\begin{array}{cc}{L}_1\left(D_{\mathrm{LR}}\right)=\frac{D_{\mathrm{LR}}.Math.1-L_2}{1+{\tau }_L.Math.\left(D_{\mathrm{LR}}.Math.1-L_2\right)}-\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}\left(S_M+C\right)}& \left(1a\right)\end{array}$

(179) The other instance in the cited disclosure document relates to the adaptation of the matrix L.sub.2 to the measurements after the matrix L.sub.1 has been established. A difference now exists merely in that: the measured wavefront S.sub.M is subjected to the steps 1, 2, 3, 4, and the assumed wavefront from the point light source is only subjected to step 6; and in that the missing step that is to take place for adaptation of the posterior lens surface L.sub.2 is now step 5, corresponding to Equation (5) of the cited disclosure document:

(180) $\begin{array}{cc}{L}_2=D_{\mathrm{LR}}-\left(\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}(S_M+C}+L_1\right){\left(1-{\tau }_L\left(\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}\left(S_M+C\right)}+L_1\right)\right)}^{-1}& \left(1b\right)\end{array}$

(181) In a preferred implementation of the invention, at least one of the length parameters d.sub.L and d.sub.LR (or D.sub.LR) is calculated from other measured data and a priori assumptions regarding other degrees of freedom, and in particular is not assumed a priori.

(182) The data of the vergence matrix S.sub.M, and particularly preferably also the data regarding C from personalized measurements, are preferably available. In a further preferred embodiment, a spherical posterior surface, meaning a posterior surface without astigmatic components, is assumed given an assumption of data regarding the posterior lens surface.

(183) In a preferred embodiment of the invention, measurement data up to the second order that corresponding to the data of the surface refractive power matrix C are thus present with regard to the cornea C. Although these values may be learned from topographical measurements, the latter are not necessary. Rather, topometric measurements are sufficient. This situation corresponds to the instance df.sub.2=3+6+3, wherein in particular the anterior chamber depth d.sub.CL is one of the six parameters that are to be established a priori.

(184) Insofar as no further personalized measurements are performed, a situation with df.sub.2=3+6+3 is present. In order to be able to unambiguously determine d.sub.LR, six parameters from {L.sub.1, L.sub.2, d.sub.L, d.sub.CL} must thus be populated via assumptions or literature values. The remaining two result from the calculation in addition to d.sub.LR. In a preferred embodiment, the parameters of the posterior lens surface, the mean curvature of the anterior lens surface, and the two length parameters d.sub.L and d.sub.CL are populated a priori (as predetermined standard values).

(185) In a preferred implementation, the anterior chamber depth d.sub.CL is thus additionally the distance between the cornea and the anterior lens surface, known for example from pachymetric or OCT measurements. The measured parameters therefore include {C, d.sub.CL, S.sub.M}. This situation corresponds to the instance of df.sub.2=4+5+3. Afterward the problem is still mathematically underdetermined; five parameters must thus be established a priori from {L.sub.1, L.sub.2, d.sub.L} via assumptions or literature values. In a preferred embodiment, the parameters are hereby the posterior lens surface, the mean curvature of the anterior lens surface, and the lens thickness. The precise way of calculating for this instance is presented in more detail further below.

(186) Solely for the precision of the personalized adaptation, it is advantageous to be able to populate as many parameters as possible with personalized measurements. In a preferred embodiment, for this purpose the lens curvature is additionally provided in a normal section on the basis of a personalized measurement. A situation according to df.sub.2=5+4+3 then thereby results, and it is sufficient to establish four parameters from {L.sub.1yy, α.sub.L1, L.sub.2, d.sub.L} a priori. Here as well, in a preferred embodiment these are again the parameters of posterior lens surface and the lens thickness. The precise calculation is again described in more detail further below.

(187) In particular as an alternative to the normal step of the anterior lens surface, and particularly preferably in addition to the anterior chamber depth, the lens thickness may also be provided from a personalized measurement. The necessity to populate these parameters with model data or estimated parameters thereby disappears ((df.sub.2=5+4+3)). Otherwise, the statements as already made above apply. This embodiment is particularly advantageous if a pachymeter is used whose measurement depth allows the detection of the posterior lens surface, but not a sufficiently certain determination of the lens curvatures.

(188) In addition to the anterior chamber depth and a normal section of the anterior lens surface, in a preferred embodiment one additional parameter (for example measurement in two normal sections) or two additional parameters (measurement of both principal sections and the axis position) of the anterior lens surface are recorded via a personalized measurement. This additional information may in particular be utilized in two ways: Abandonment of a priori assumptions: one or two of the assumptions that were otherwise made a priori may be abandoned and be determined via calculation. In this instance, the situations df.sub.2=6+3+3 or df.sub.2=7+2+3 result. In the first instance, the mean curvature of the posterior surface (given assumption of an astigmatism-free posterior surface) may be determined, and in the second instance the surface astigmatism (including axis position) may be determined for a given mean curvature. Alternatively, in both instances the lens thickness may be determined from the measurements. However, such a procedure generally requires a certain caution, since noisy measurement data may easily lead to a “runaway” of the enabled parameters. The model may thereby as a whole become markedly worse instead of better. One possibility to prevent this is to predetermine anatomically reasonable limit values for these parameters, and to limit the variation of the parameters to this range. Of course, these limits may also be predetermined depending on the measured values. Reduction of the measurement uncertainty: if, by contrast, the same a priori assumptions continue to be made (preferably thus {L.sub.2, d.sub.L}), the situations df.sub.2=6+4+3 or df.sub.2=7+4+3 are present; the system is thus mathematically overdetermined. Instead of a simple analytical determination of D.sub.LR according to the subsequent embodiments, D.sub.LR (and possibly the still missing parameters from L.sub.1) is determined (“fit”) so that the distance between the L.sub.1 resulting from the equations and the measured L.sub.1 (or the measured L.sub.1, supplemented by the missing parameters) is minimal. A reduction of the measurement uncertainty may—obviously—be achieved via this procedure.

(189) In a further preferred implementation, the anterior chamber depth, two or three parameters of the anterior lens surface, and the lens thickness are measured in a personalized manner. The calculation of the remaining variables thereby takes place analogously, wherein the a priori assumption of the lens thickness may be replaced by the corresponding measurement.

(190) In a further preferred implementation, personalized measurements of the anterior chamber depth, at least one parameter of the anterior lens surface, the lens thickness, and at least one parameter of the posterior lens surface are provided. This is hereby an expansion of the aforementioned instances. The respective additionally measured parameters may take place analogous to the step-by-step expansions of the above segments. These instances are particularly advantageous if the aforementioned pachymetry units that measure in one plane, two planes, or over the entire surface are accordingly extended in terms of measurement depth, and are so precise that the curvature data can be sufficiently precisely determined.

(191) In the following it is shown, using a few examples, how the calculation of individual parameters may take place from the remaining measured parameters or parameters established a priori, and using the personalized refraction data.

(192) For example, in preferred embodiments, a measurement of the curvature of a lens surface is available in a normal section. Since the posterior surface cannot be measured in practice without the anterior surface also being measured, and the measurement of the anterior surface preferably occurs, the equations for the instances of a curvature of the anterior lens surface that is known in a normal section are specified in the following. If, instead of a normal section of the anterior lens surface, a normal section of the posterior lens surface is present (for example corresponding measurements, model assumptions), one must analogously proceed with Equation (1b). Without limiting the generality, the coordinate system is placed so that the normal section travels in the x-direction. In a next step, the matrix equation (1a) is then evaluated in the given normal section and solved for D.sub.LR, and this solution is subsequently used again in Equation (1a) for the complete specification of L.sub.1.

(193) If the xx-component of L.sub.1(D.sub.LR) from Equation (1) is set equal to the measured value L.sub.1,xx, for this matrix element a quadratic equation in D.sub.LR is obtained whose positive solution corresponds to the distance between posterior lens surface and retina:

(194) $\begin{array}{cc}{D}_{LR}=\frac{-b+\sqrt{b^2-4c}}{2a}& \left(2\right)\end{array}$

(195) It thereby applies that:
a=τ.sub.L(1+τ.sub.LA)
b=1−τ.sub.L(tr(L.sub.2)−AB)
c=A−L.sub.2,xx+τ.sub.L det L.sub.2(1+τ.sub.LA)−τ.sub.LA tr(L.sub.2)=A−L.sub.2,xx+a det L.sub.2−τ.sub.LA tr(L.sub.2)  (2a)
with
A=−S.sub.M,L1,xx−L.sub.1,xx
B=2−τtr(L.sub.2)
det(L.sub.2)=L.sub.2,xxL.sub.2,yy−L.sub.2,xy.sup.2
tr(L.sub.2)=L.sub.2,xx+L.sub.2,yy  (2b)
and

(196) $\begin{array}{cc}{S}_{M,L1,\mathrm{xx}}=\frac{{\tau }_{\mathrm{CL}}{S}_{M,C,\mathrm{xy}}^{\prime 2}+S_{M,C,\mathrm{xx}}^{\prime }.Math.\left(1-{\tau }_{\mathrm{CL}}{S}_{M,C,\mathrm{yy}}^{\prime }\right)}{-{\tau }_{\mathrm{CL}}^2{S}_{M,C,\mathrm{xy}}^{\prime 2}+\left(1-{\tau }_{\mathrm{CL}}{S}_{M,C,\mathrm{xx}}^{\prime }\right).Math.\left(1-{\tau }_{\mathrm{CL}}{S}_{M,C,\mathrm{yy}}^{″}\right)}{S}_{M,C,\mathrm{xx}}^{\prime }=S_{M,\mathrm{xx}}+C_{\mathrm{xx}}\left(\mathrm{xy}\mathrm{und}\mathrm{yy}\mathrm{analog}\right)& \left(2c\right)\end{array}$

(197) In the event of a symmetrical posterior lens surface (L.sub.2=L.sub.2,xx.Math.1), this simplifies to

(198) $\begin{array}{cc}{D}_{LR}=L_{2,\mathrm{xx}}+\frac{L_{1,\mathrm{xx}}+S_{M,\mathrm{L1},\mathrm{xx}}}{1-{\tau }_L.Math.\left(L_1+S_{M,L1,\mathrm{xx}}\right)}& \left(3\right)\end{array}$
with S.sub.M,L1,xx from Equation (2c).

(199) In both instances, it is therefore possible to calculate the anterior lens surface L.sub.1 in that the respectively obtained D.sub.LR in Equation (1a) is used:

(200) $\begin{array}{cc}{L}_1=\frac{D_{\mathrm{LR}}-L_2}{1+{\tau }_L.Math.\left(D_{\mathrm{LR}}-L_2\right)}-\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}\left(S_M+C\right)}& \left(4\right)\end{array}$

(201) The result is naturally symmetrical (L.sub.1,xy=L.sub.1,yx), and for the component L.sub.1,xx reproduces the value used in (2b) or (3).

(202) In some preferred embodiments, a personalized measurement or a specification of a mean curvature of a lens surface is provided. For example, this situation is present when the mean curvature of the anterior lens surface may be measured, or no measurements at the lens surfaces may be performed and the mean curvature of a lens surface is assumed (for example taken from the literature). As was just now described, here the method for the anterior lens surface is described and can be analogously transferred to the posterior lens surface.

(203) In this instance of a given mean sphere L.sub.1,ms of the anterior lens surface, the free parameters are the cylinder L.sub.1,cyl and the axis length α.sub.L1. With L.sub.1,diff=L.sub.1,cyl/2, L.sub.1 becomes

(204) 0$\begin{array}{cc}{L}_1=\left(\begin{array}{cc}{L}_{1,ms}-L_{1,\mathrm{diff}}.Math.\cos 2{\alpha }_{L1}& -L_{1,\mathrm{diff}}.Math.\sin 2{\alpha }_{L1}\\ -L_{1,\mathrm{diff}}.Math.\sin 2{\alpha }_{L1}& L_{1,ms}+L_{1,\mathrm{diff}}.Math.\cos 2{\alpha }_{L1}\end{array}\right)& \left(5\right)\end{array}$

(205) One again proceeds from Equation (1a). If the expressions for L.sub.1 from Equations (5) and (1a) are now equated, an equation system is obtained that is made up of three equations (the two non-diagonal elements are identical) and the three unknowns L.sub.1,diff, α.sub.L1 and D.sub.LR. This has the physically relevant solution

(206) $\begin{array}{cc}{D}_{\mathrm{LR}}=\frac{-\stackrel{\_}{b}+\sqrt{{\stackrel{\_}{b}}^2-4\stackrel{\_}{ac}}}{2\stackrel{\_}{a}}{L}_{1\mathrm{diff}}=\pm \sqrt{{\sigma }^2+{\gamma }^2}{\alpha }_{L1}=\frac{1}{2}\arctan \left(\pm \gamma ,\pm \sigma \right)+\frac{\pi }{2}\mathrm{with}& \left(6\right)\end{array}$$\begin{array}{cc}\stackrel{\_}{a}={\tau }_L\left(1+{\tau }_L\stackrel{\_}{A}\right)\stackrel{\_}{b}=1-{\tau }_L\left(\mathrm{tr}\left(L_2\right)-\stackrel{\_}{A}B\right)\stackrel{\_}{c}=\frac{1}{4}\left(\stackrel{\_}{A}{B}^2-B\mathrm{tr}\left(L_2\right)-\stackrel{\_}{a}{\mathrm{Ast}\left(L_2\right)}^2\right)\mathrm{and}\stackrel{\_}{A}={\stackrel{\_}{S}}_{M,L1}-{\stackrel{\_}{L}}_{1,\mathrm{mess}}\mathrm{Ast}\left(L_2\right)=\sqrt{{\mathrm{tr}\left(L_2\right)}^2-4\det L_2}\gamma =\frac{\begin{array}{c}2\left(-1+\sqrt{{\stackrel{\_}{b}}^2-4\stackrel{\_}{ac}}\right)\left(L_{2,\mathrm{xx}}-L_{2,\mathrm{xy}}\right)+\\ {\tau }_L^2{\mathrm{Ast}\left(L_2\right)}^2\left(S_{M,L1,\mathrm{xx}}-S_{M,L1,\mathrm{yy}}\right)\end{array}}{2{\tau }_L^2{\mathrm{Ast}\left(L_2\right)}^2}\sigma =\frac{2\left(-1+\sqrt{{\stackrel{\_}{b}}^2-4\stackrel{\_}{ac}}\right)L_{2,\mathrm{xy}}+{\tau }_L^2{\mathrm{Ast}\left(L_2\right)}^2{S}_{M,L1,\mathrm{xy}}}{2{\tau }_L^2{\mathrm{Ast}\left(L_2\right)}^2}& \left(6a\right)\end{array}$

(207) This can also be simplified for the instance of a rotationally symmetrical posterior lens surface:

(208) $\begin{array}{cc}{D}_{\mathrm{LR}}=L_2+\frac{{\stackrel{\_}{L}}_{1,\mathrm{mess}}+{\stackrel{\_}{S}}_{M,L1}}{1-{\tau }_L.Math.\left({\stackrel{\_}{L}}_{1,\mathrm{mess}}+{\stackrel{\_}{S}}_{M,L1}\right)}{L}_1=\left({\stackrel{\_}{L}}_{1,\mathrm{mess}}+{\stackrel{\_}{S}}_{M,L1}\right).Math.1-\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}\left(S_M+C\right)}\mathrm{wherein}{\stackrel{\_}{L}}_{1,\mathrm{meas}}=\frac{D_{\mathrm{LR}}-L_2}{1+{\tau }_L.Math.\left(D_{\mathrm{LR}}-L_2\right)}-{\stackrel{\_}{S}}_{M,L1}\mathrm{with}{\stackrel{\_}{S}}_{M,L1}=\frac{S_{M,L1,\mathrm{xx}}+S_{M,L1,\mathrm{yy}}}{2}& \left(7\right)\end{array}$

(209) The individual elements of the eye model can therefore be entirely calculated.

(210) Aside from a principal section with given angle position, or the mean curvature, the given (i.e. measured or assumed) variables may also be other parameters such as the thickest principal section, the thinnest principal section, the cylinder, and the axis position. In these instances, the procedure is analogous to the illustrated instances.

(211) Since the HOAs of the eye have also be now been taken into account in the optimization of spectacles lenses, it is advantageous to also consider the HOAs of the cornea or of the lens in the population of the eye model. Given the selection of HOAs for the lens, it generally applies that HOAs that may also represent the refractive index curve within the lens may be associated with the anterior lens surface or posterior lens surface.

(212) The previously depicted formalism is preferably expanded, in particular with regard to the cited steps 1 through 6, to the co-treatment of the HOAs, in that the calculation methods from the publications by G. Esser et al.: “Derivation of the refraction equations for higher order aberrations of local wavefronts at oblique incidence”, JOSA A, Vol. 27, No. 2 (2010), and by G. Esser et al.: “Derivation of the propagation equations for higher order aberrations of local wavefronts”, JOSA A, Vol. 28, No. 11 (2011), are applied aside from the formulas for the vergence matrices that are explicitly specified in steps 1 through 6.

(213) In general, the procedure with regard to the enumeration of degrees of freedom is executed in a manner very similar to as above. If, aside from data regarding 2nd-order errors, data about their HOAs are present (either from measurements or from reasonable assumptions) with regard to refractive surface C of the cornea and regarding the outgoing wavefront S.sub.M, the wavefront S.sub.L1 may also be determined computationally with accordingly many HOAs. This applies independently of the form in which the HOAs present themselves. However, the Taylor series is particularly preferred, because in this form the statement exactly applies: if HOA coefficients up to the n-th order are present with regard to two surfaces C and S.sub.M, the corresponding HOA coefficients for S.sub.L1 can then also be computationally determined therefrom up to the n-th order. Furthermore, the Zernike basis is preferred, because here as well a similar statement applies. However, this is exact only when all Zernike coefficients with an order >n vanish.

(214) An order n is preferably established (in advance), up to which all participating surfaces and wavefronts should be treated. Independently of the presentation of the HOAs, aside from the three components for the 2nd-order errors, the wavefronts or surfaces then additionally possess N components for the HOAs, wherein N depends on n and, inter alia, on the presentation form of the HOAs (in the Taylor decomposition and Zernike decomposition, N=(n+1)(n+2)/2-6 applies).

(215) The adaptation condition using a measured wavefront, for example S.sub.M,L1, then also accordingly no longer possesses only the three components described above, but rather a maximum of N+3 components in total. These are then accordingly accompanied by 3 (N+3)+3=3N+12 parameters (namely the three length parameters d.sub.CL, d.sub.L and d.sub.LR (or D.sub.LR), as well as respectively N+3 components of the cornea C and the lens surfaces L.sub.1 and L.sub.2). This means that

(216) $\begin{array}{c}{\mathrm{df}}_n={\mathrm{df}}_n\left(i\right)+{\mathrm{df}}_n\left(\mathrm{ii}\right)+{\mathrm{df}}_n\left(\mathrm{iii}\right)\\ =3N+12\end{array}$
applies, with df.sub.n(iii)=N+3. If the anterior chamber depth d.sub.CL and the cornea C are preferably measured again, df.sub.n(i)=N+4 applies, and consequently df.sub.n(ii)=N+5, corresponding to the situation of df.sub.n=(N+4)+(N+5)+(N+3).

(217) The further procedure may be implemented in a manner very analogous to as described above.

(218) In principle, the HOAs of the mapping of the eye onto the retina may be detected in transmission via suitable measurement devices with an aberrometry unit. On the other hand, the HOAs of the cornea surface may be measured in reflection by a topography unit. Both data of the exiting wavefront S.sub.M and a description of the refracting surface C of the cornea, including the HOAs up to a defined order n, are therefore available.

(219) In the event of a measurement of the S.sub.M for HOAs as well, this supplies df.sub.n(iii)=N+3 conditions for parameter calculation. If it is again preferred to also measure the d.sub.CL in addition to the cornea C, df.sub.n(i)=N+4 applies, and consequently df.sub.n(ii)=N+5, corresponding to the situation df.sub.n=(N+4)+(N+5)+(N+3).

(220) In such an instance, in the population of the model, the HOAs of the lens may be selected so that the measured wavefront is created given the propagation of a wavefront emanating from a point of the retina according to steps 1 through 6, in reverse order. If the parameters of the eye model are then populated, the propagation of this wavefront, emanating from a point of the retina up to the evaluation surface (according to at least one of the steps 1 through 6, in reverse order) may lead to the reference wavefront, which then is used for a comparison with the wavefront emanating from an object.

(221) In principle, in the adaptation of L.sub.1 the method may proceed analogous to the method described above with reference to WO 2013/104548 A1, wherein the two length parameters d.sub.L and d.sub.LR (or D.sub.LR) are established a priori. The single difference is now that the anterior lens surface L.sub.1, including its N HOA parameters up to the n-th order, may be adapted to the measurements, corresponding to df.sub.n(iii)=N+3. The posterior lens surface L.sub.2, which is unknown due to a lack of measured values, is preferably established in advance (for example via literature values regarding the average eye of the general population), including the N HOA parameters up to the n-th order, corresponding to df.sub.n(ii)=N+5. This occurs in particular in that, on the one hand, the measured wavefront S.sub.M is calculated through the likewise measured cornea C by means of steps 1, 2, and propagates up to the object-side side of the anterior lens surface L.sub.1. On the other hand, a spherical wave is calculated from back to front, by means of the steps 6, 5, 4 run through backward, from an imaginary point light source on the retina, in that this spherical wave refracts at the pre-established posterior lens surface L.sub.2, and the wavefront that is then obtained propagates from the posterior lens surface up to the image-side side of the anterior lens surface L.sub.1. The two wavefronts S.sub.L1 and S′.sub.L1 that are so determined, which are situated on the object side or, respectively, image side of the anterior lens surface, generally possess both lower-order aberrations and HOAs; however, their values differ between the two wavefronts. Since the two wavefronts occur in one and the same measurement ray path, and therefore must coincide beyond the still absent step 3, the refractive anterior lens surface L.sub.1 may be concluded unambiguously from this difference up to the n-th order, and in fact via the calculation methods known from G. Esser et al.: “Derivation of the refraction equations for higher order aberrations of local wavefronts at oblique incidence”, JOSA A, Vol. 27, No. 2 (2010), and from G. Esser et al.: “Derivation of the propagation equations for higher order aberrations of local wavefronts”, JOSA A, Vol. 28, No. 11 (2011), for example.

(222) On the other hand, in the adaptation of L.sub.2 it is also possible to proceed analogous to the method described above with regard to WO 2013/104548 A1, wherein again the two length parameters d.sub.L and d.sub.LR (or D.sub.LR) are established a priori. The posterior lens surface L.sub.2, including its HOAs up to the n-th order, is now adapted to the measurements after the anterior lens surface L.sub.1 has been established. A difference with regard to the adaptation of L.sub.1 in particular exists in that the measured wavefront S.sub.M is subjected to steps 1, 2, 3, 4, and the assumed wavefront from the point light source is only subjected to step 6, and in that the missing step that is to take place to adapt the posterior lens surface L.sub.2 is now step 5.

(223) For calculation, the formalism described in G. Esser et al.: “Derivation of the refraction equations for higher order aberrations of local wavefronts at oblique incidence”, JOSA A, Vol. 27, No. 2 (2010), and in G. Esser et al.: “Derivation of the propagation equations for higher order aberrations of local wavefronts”, JOSA A, Vol. 28, No. 11 (2011), for example, is thereby used for the refraction steps or propagation steps. In particular, it is reasonable to work from the lowest-order aberrations to the highest-order aberrations of interest (typically sixth).

(224) To use the aforesaid formalism, it is advantageous to describe the wavefronts or surfaces via the local derivation of the rise in the direction of the planes orthogonal to the direction of the propagation. Every surface or wavefront that is not present in this form is preferably initially brought into this form. For example, this may occur via transformation from a Zernike representation to the representation via local derivatives, or via a preceding fit of a rise representation. A suitable technical form of presentation of surfaces via Taylor coefficients is described in WO 2013/104548 A1, for example.

(225) Naturally, the deviations (including the second-order aberrations may also be distributed among the anterior lens surface and posterior lens surface, analogous to the above procedure.

(226) In a preferred embodiment, it is proposed that at least one of the length parameters d.sub.L and d.sub.LR is neither predetermined a priori nor measured in a personalized manner, but rather is calculated using the personalized refraction data and the other (pre-)established data. For this purpose, at least one measured value or an assumption is provided for one of the degrees of freedom of the lens surfaces L.sub.1 or L.sub.2. For example, if this is a measured value for the curvature of L.sub.1 in a normal section, then in particular d.sub.LR (or D.sub.LR) may be determined therefrom via calculation.

(227) If the specification in the vergence matrices refers to the local curvature (thus corresponds to the specification of the HOAs as coefficients of a Taylor decomposition), for this purpose D.sub.LR and the missing parameters of the lens are first determined as has already been described above. Following this, the HOAs of the lens may then be constructed step by step, starting from the second to n-th order, with the formalism from G. Esser et al.: “Derivation of the refraction equations for higher order aberrations of local wavefronts at oblique incidence”, JOSA A, Vol. 27, No. 2 (2010), and by G. Esser et al.: “Derivation of the propagation equations for higher order aberrations of local wavefronts”, JOSA A, Vol. 28, No. 11 (2011).

(228) By contrast to this, if the mean curvature over a defined pupil is used, which is the case given the representation according to Zernike, the degree of freedom D.sub.LR is likewise established. In this formalism, an iterative procedure would be necessary due to the dependencies. However, this can be avoided via a conversion between the two notations before the beginning of the calculation.

(229) In principle, the HOAs of the mapping of the eye onto the retina may be detected in transformation via suitable measurement devices having an aberrometry unit. However, such aberrometry units for the detection of HOAs are quite expensive and are not available to every optometrist. However, it is often possible to measure the HOAs of the corneal surface in reflection at less cost via a topography unit. Therefore, although no data of the exiting wavefront S.sub.M are available, at least a description of the refracting surface C of the cornea is available, including the HOAs up to a defined order n.

(230) The invention offers the possibility to use the personalized eye model if, although personalized measurements regarding the HOAs of the cornea are present, no personalized measurements of the HOAs of the eye are present. In a preferred implementation, aside from the cornea C, the anterior chamber depth d.sub.CL is thereby also measured, meaning that df.sub.n(i)=N+4 applies. Given use of an autorefractometer (meaning no measurement of the HOAs) instead of an aberrometer (also in combination with a subjective refraction), or the sole use of a subjective refraction without use of an aberrometer or autorefractometer, although the vergence matrix S.sub.M of the LOAs is known, no personalized information about the HOAs of the (measurement ray path) wavefront S.sub.M of the entire eye is present. This means that, exactly as in the instance without HOAs, instead of df.sub.n=N+3 calculation conditions only df.sub.n(iii)=3 calculation conditions are present. If it is desired to completely populate the model up to the n-th order, instead of df.sub.n(ii)=N+5 parameters df.sub.n(ii)=2N+5 parameters are preferably accordingly established a priori. The instance is thereby preferably considered again that both d.sub.L and d.sub.LR belong among the parameters established a priori. The model can therefore be populated in different ways with the additional parameters and be used for the calculation and optimization of a spectacles lens.

(231) In particular, this instance can be treated just as described above given the presence of measured HOAs of the eye, if assumptions are made about the HOAs of the eye. One example of this is values determined or model-based using a test subject collective. A remaining spherical aberration is thereby preferably assumed, since it is known—in particular from T. O. Salmon and C. van de Pol: Normal-eye Zernike coefficients and root-mean-square wavefront errors, J Cataract Refract Surg, Vol. 32, Pages 2064-2074 (2006), and from J. Porter et al.: Monochromatic aberrations of the human eye in a large population, JOSA A. Vol. 18, No. 8 (2001)—that this differs markedly from zero on average across the population. The calculation of the HOAs of the lens then takes place very analogously to the procedure described above, with the single difference that the HOA values for S.sub.M are not learned from a personalized measurement but rather are based on the aforementioned assumptions.

(232) Alternatively, if suitable assumptions are made about the HOAs of the lens, meaning that the HOAs of both lens surfaces L.sub.1 and L.sub.2 are established a priori, the HOAs of the wavefront S.sub.M up to the n-th order may take place with, for example, the algorithms from G. Esser et al.: “Derivation of the refraction equations for higher order aberrations of local wavefronts at oblique incidence”, JOSA A, Vol. 27, No. 2 (2010), and from G. Esser et al.: “Derivation of the propagation equations for higher order aberrations of local wavefronts”, JOSA A, Vol. 28, No. 11 (2011), in that steps 6, 5, 4, 3, 2, 1 are traversed in reverse from the retina to the cornea. In particular, d.sub.L and d.sub.LR established a priori also thereby enter into the calculation of S.sub.M.

(233) For the LOAs of the lens surfaces, no a priori establishments are made that exceed the above statements, since the LOAs of the wavefront S.sub.M are present, for example as a measured vergence matrix S.sub.M, from the subjective refraction, the autorefractor measurement, or a combination thereof.

(234) A preferred embodiment instance is hereby that the HOAs of the lens surfaces are set equal to zero in the basis that is used. This assumption is particularly preferably made in relation to the Taylor basis. This assumption is furthermore preferred in relation to the Zernike basis. Although the HOAs of S.sub.M are not a basis for a direct mapping of the HOAs of C, because the participating propagations in each instance also introduce HOAs, the advantage of vanishing HOAs of the lens surfaces exists in the reduction of the computation cost due to numerous vanishing terms.

(235) Alternatively, model-based values for the HOAs of the lens surfaces may also be selected. This applies in particular to spherical aberrations, in particular since it is known—from T. O. Salmon and C. van de Pol: Normal-eye Zernike coefficients and root-mean-square wavefront errors, J Cataract Refract Surg, Vol. 32, Pages 2064-2074 (2006), and from J. Porter et al.: Monochromatic aberrations of the human eye in a large population, JOSA A, Vol. 18, No. 8 (2001)—that the spherical aberration of the lens is on average markedly different than zero across the population. These may thereby be selected independently of the measured data, or depending on measured data (for example refraction values, spherical aberration of the cornea).

(236) Even if neither a topograph nor an aberrometer is used, thus no personalized measurement data of the HOAs are present, model-based assumptions about the HOAs of the cornea, the lens, or the eye may nevertheless be made and be used in the population of the eye model. The assumed values may thereby also be selected using corresponding models, depending on measured data (for example refraction values, results of the topometry measurement or autorefractometer measurement). Examples of the precise calculation have already been described further above, wherein the corresponding assumptions apply instead of the measured values for the HOAs. This also applies again in particular to spherical aberrations, since these are on average markedly different from zero across the population. This may thereby be chosen independently of the measured data, or depending on measured data (for example refraction values, results of the topometry measurement or autorefractometer measurement) and be associated with the cornea, one of the two lens surfaces, or combinations.

(237) The present invention offers the possibility of concluding S.sub.M via measurements of or assumptions about L.sub.1 and L.sub.2. Reasonable values for the HOAs of S.sub.M are thus obtained without aberrometric measurements. For this purpose, precise knowledge about the length parameters d.sub.L and d.sub.LR (or D.sub.LR) also do not need to be present, so that the formalism can be used even without the calculation of d.sub.LR that is described in Section 3. In contrast to the second-order errors of the wavefront S.sub.M, the HOAs of S.sub.M namely depend only so weakly on the length parameters d.sub.L and d.sub.LR (or D.sub.LR) that the selection of the values for d.sub.L and d.sub.LR that are to be established a priori—within the scope of the physiologically reasonable range—for adaptation of the HOAs of S.sub.M has only a small influence, and consequently standard parameters may also be used.

(238) One application of this method is that spectacles lens optimizations under consideration of the HOAs of the eye, such as the DNEye optimization, may be performed even without personalized aberrometric measurements (for example on the basis of topography measurements).

(239) The evaluation of the aberrations during the calculation method or optimization method may be performed at different locations in the ray path, meaning that the evaluation surface may be provided at different positions. Instead of taking place at the retina or at the posterior lens surface, an evaluation of the imaging wavefront may also be already take place at a surface situated further forward in the model eye. For this purpose, within the model eye a reference wavefront R is defined that is then used in the lens optimization, for example. This reference wavefront thereby has the property that it leads to a point image given further propagation through the eye, up to the retina. Accordingly, the reference wavefront may be determined via back-propagation of a wavefront, which wavefront converges at a point on the retina, from the retina up to the position of the reference wavefront. For example, since the measured wavefront S.sub.M is precisely the wavefront that emanates from a point light source on the retina, this may also instead be propagated inside the eye, up to the position of the reference wavefront.

(240) Considered mathematically, both procedures are equivalent and lead to the same formulas for the reference wavefront. In the following, to derive the corresponding reference wavefronts the respective way is chosen that manages with fewer propagation steps and enables a simpler representation. In the following, only the treatment of the components of the defocus and astigmatism is described by way of example. However, an extension to HOAs and the use of the subjective refraction is likewise possible and advantageous.

(241) Given the consideration of HOAs, analogous to the calculation of the HOAs according to the embodiments below, this may take place via refraction (G. Esser et al.: “Derivation of the refraction equations for higher order aberrations of local wavefronts at oblique incidence”, JOSA A, Vol. 27, No. 2 (2010)) and propagation (G. Esser et al.: “Derivation of the propagation equations for higher order aberrations of local wavefronts”, JOSA A, Vol. 28, No. 11 (2011)).

(242) Since the wavefront propagation is a non-linear process, a spectacles lens optimization that evaluates an imaging wavefront via comparison with a reference wavefront generally leads to different results depending on at which surface within the eye this comparison occurs.

(243) In a preferred embodiment, only the ultimate step (in particular step 6b) is omitted, thus the propagation from the AP to the retina. After the refraction at the posterior lens surface, the incident wavefront is thus simulated only up to the AP (thus calculation of S.sub.AP according to the aforementioned step 6a), and there is compared with a reference wavefront R.sub.AP. This is characterized in that, given the propagation to the retina, it yields a point image there. According to the above statement, the vergence matrix of this wavefront is

(244) $R_{AP}=D_{AP}=D_{LR}^{\left(b\right)}=\frac{1}{{\tau }_{LR}^{\left(b\right)}}1=\frac{1}{{\tau }_{\mathrm{LR}}-{\tau }_{LR}^{\left(a\right)}}1=\frac{1}{1/D_{IR}-d_{IR}^{\left(a\right)}/n_{LR}}1$
with the D.sub.LR determined from Equation (2) or (3), as well as the negative (accommodation-dependent) value d.sub.LR.sup.(a)<0 whose absolute magnitude describes the distance between the posterior lens surface and the AP.

(245) In a furthermore preferred embodiment, the penultimate step is moreover omitted, overall thus the propagation from the posterior lens surface to the retina. The incident wavefront is thus only simulated up to after the refraction at the posterior lens surface (thus calculation of S′.sub.L2 according to the aforementioned step 5), and there is compared with a reference wavefront R′.sub.L2. This is characterized in that, given the propagation to the retina, it yields a point image there. According to the above statement, the vergence matrix of this wavefront is
R′.sub.L2=D′.sub.L2=D.sub.LR.Math.1
with the D.sub.LR determined from Equation (2) or (3).

(246) A further simplification results if the comparison is placed before the refraction by the posterior lens surface. In this instance, the incident wavefront is simulated, thus calculated, only up to S.sub.L2 according to the above step 4. For this purpose, analogous to S′.sub.L2, a reference wavefront R.sub.L2 is defined that, after the refraction at the posterior lens surface and the propagation to the retina, yields a point image there. This is determined as
R.sub.L2=R′.sub.L2−L.sub.2=D.sub.LR.Math.1−L.sub.2
with the D.sub.LR determined from Equation (2) or (3), and the L.sub.2 known from the literature or from measurements.

(247) In the event of a rotationally symmetrical posterior lens surface, this simplifies to
R.sub.L2=(D.sub.LR−L.sub.2,xx).Math.1

(248) In particular insofar as the lens thickness is likewise learned from the literature, in a further preferred embodiment it is suggested as a next simplification step to omit the propagation through the lens and to execute the comparison after the refraction through the anterior lens surface. In a continuation of the above statement, for this purpose a reference wavefront R′.sub.L1 is preferably used that is created from R.sub.L2 via backward propagation by the lens thickness, and possesses the following vergence matrix:
R′.sub.L1=R.sub.L2/(1+τ.sub.LR.sub.L2)
with the D.sub.LR determined from Equation (2) or (3) and the τ.sub.L=d.sub.L/n.sub.L, known from the literature or from measurements, as well as the vergence matrix R.sub.L2 determined from Equation (6) or (7).

(249) In the event of a rotationally symmetrical posterior lens surface, this simplifies to

(250) $R_{L1}^{\prime }=\frac{D_{\mathrm{LR}}-L_{2,\mathrm{xx}}}{1+{\tau }_L.Math.\left(D_{\mathrm{LR}}-L_{2,\mathrm{xx}}\right)}.Math.1$

(251) As in the above models, it also applies here that, even if the consideration occurs before the last steps and—depending on notation—the variable D.sub.LR does not explicitly occur, this variable is nevertheless at least implicitly incurred together with d.sub.L and L.sub.2, since they together control the distribution of the effect L.sub.1 in the anterior lens surface.

(252) Yet another simplification results if the comparison is placed before the refraction by the anterior lens surface. In this instance, the incident wavefront needs to be simplified only up to S.sub.L1 according to step 2. For this purpose, analogous to R′.sub.L1, a reference wavefront R.sub.L1 is defined that converges at a point on the retina after the refraction at the anterior lens surface and the additional steps. This may either be calculated via the refraction of R′.sub.L1 at L.sub.1, or be determined directly from the refraction of the measured wavefront S.sub.M at the cornea C and a subsequent propagation by d.sub.CL. In both instances,

(253) $R_{L1}=\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}.Math.\left(S_M+C\right)}$
is obtained. The variables D.sub.LR, d.sub.L and L.sub.2 now no longer enter into it; it is thus sufficient to know S.sub.M, C and d.sub.CL.

(254) An embodiment in which the comparison is implemented after the refraction at the cornea is linked with relatively low computation cost. In this instance, only S.sub.M and C are still considered:
R′.sub.C=S.sub.M+C

(255) An additional, very efficient possibility is the positioning of the evaluation surface at the exit pupil of the model eye. This is preferably situated before the posterior lens surface.

(256) Additional Aspects

(257) Insofar as is not explicitly noted otherwise, aspects that are relevant to both the first and the second approach of the invention are described in the following paragraphs:

(258) In particular, in the following commercially available devices are cited in summary, again by way of example, with which devices parameter measurements that are necessary or preferred for the invention may be implemented. All devices listed here are, for example, also described in M. Kaschke et al., “Optical Devices in Ophthalmology and Optometry”, Wiley-VCH (2014): Shape of the anterior corneal surface: The shape of the anterior corneal surface may be determined with keratographs (for example Placido-Disk Keratograph ATLAS 9000 from Zeiss, Small-Target Keratograph E300 from Medmont, and Placido Disk unit of the Galilei G2 from Ziemer). In the instances in which only the curvatures are determined and used, the use of keratometers is also possible (for example manual Helmholtz-Littmann keratometer from Zeiss, manual Javal-Schiötz keratometer from Haag-Streit, and automatic electro-optical keratometry unit of the IOL Master from Zeiss). Shape of the anterior lens surface and posterior lens surface: The shape of the lens surfaces may be measured in a section or three-dimensionally with Scheimpflug cameras (for example Pantacam by Oculus, SL-45 by Topcon, and Galilei G2 by Ziemer), and OCTs (for example IOL Master of 500 by Zeiss, SL-OCT by Heidelberg, and Visante OCT by Zeiss). Distance between the described surfaces: Distances between the three cited surfaces may be measured both with some of the aforementioned Scheimpflug cameras and OCTs, and with the Lenstar LS900 from Haag-Streit. Some of these devices might also, in fact, be used in order to measure the distance between these surfaces and the retina. However, such measurements are often very costly, and may be directly avoided within the scope of the present invention. For this purpose, refer for example to R. B. Rabbetts, “Bennett & Rabbetts' Clinical Visual Optics”, Butterworth Heinemann Elsevier Health Sciences (2007). Refraction indices of the participating media: A citation of devices with which the refraction indices of the participating may be measured may be omitted here, since these values are preferably taken from the literature. For this purpose, refer for example to R. B. Rabbetts, “Bennett & Rabbetts' Clinical Visual Optics”, Butterworth Heinemann Elsevier Health Sciences (2007). Higher-order or lower-order aberrations of the eye: aberrations of the eye may be measured with aberrometers (for example iProfiler from Zeiss and KR-1W from Topcon based on Schack-Hartmann sensors, as well as OPD-Scan 111 from Nidek based on dynamic skiascopy). Given a consideration of lower-order aberrations, the use of autorefractometers (for example RM-8900 from Topcon and KW-2000 from Kowa) is sufficient.

REFERENCE LIST

(259) 10 principal ray 12 eye 14 first surface of the spectacles lens (anterior surface) 16 second surface of the spectacles lens (posterior surface) 18 anterior corneal surface 20 eye lens