Method for calculating and optimizing an eyeglass lens taking into consideration higher-order imaging errors

Abstract

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

Claims

1. A computer-implemented method for designing a spectacle lens and manufacturing the designed spectacle lens, comprising: specifying at least one surface for the spectacle lens to be designed; determining the course of a main ray through at least one visual point (i) of the at least one surface; determining a spherical and astigmatic aberration outgoing local wavefront set of coefficients (s.sub.io) and a higher-order aberration outgoing local wavefront set of coefficients (e.sub.iok) of the local aberration of a local wavefront going out from the at least one visual point (i) in a surrounding of the main ray; specifying at least one function f(e.sub.ok) which assigns a higher-order aberration propagated local wavefront set of coefficients (e.sub.pk) to a higher-order aberration outgoing local wavefront set of coefficients (e.sub.ok), wherein e.sub.pk=B.sub.k(e.sub.ok+r.sub.k), and the proportionality term B.sub.k depends on the spherical and astigmatic aberration outgoing local wavefront set of coefficients (s.sub.io), but not on the higher-order aberration outgoing local wavefront set of coefficients (e.sub.ok), and r.sub.k is a remainder term; determining a higher-order aberration of a local wavefront propagated starting from the at least one visual point (i) along the main ray depending on at least the higher-order aberration outgoing local wavefront set of coefficients (e.sub.iok) on the basis of the specified function f(e.sub.ok); designing the at least one surface of the spectacle lens based on the determined higher-order aberration of the propagated local wavefront; and manufacturing the designed spectacle lens.

2. The method according to claim 1, wherein determining a spherical and astigmatic aberration outgoing local wavefront set of coefficients comprises determining a power vector $s_{o} = (\begin{matrix} S_{oxx} \\ S_{oxy} \\ S_{oyy} \end{matrix}),$ wherein determining a higher-order aberration outgoing local wavefront set of coefficients comprises determining a coma vector $e_{o 3} = (\begin{matrix} E_{oxxx} \\ E_{oxxy} \\ E_{oxyy} \\ E_{oyyy} \end{matrix}),$ and wherein the function $e_{p 3} = γ^{3} (\begin{matrix} β_{y}^{- 3} & 3 β_{7}^{- 2} \frac{d}{n} S_{xy} & 3 {β_{y}^{- 1} (\frac{d}{n} S_{xy})}^{2} & {(\frac{d}{n} S_{xy})}^{3} \\ β_{y}^{- 2} \frac{d}{n} S_{xy} & β_{y}^{- 1} (\frac{1}{γ} + 3 {(\frac{d}{n} S_{xy})}^{2}) & \frac{2}{γ} \frac{d}{n} S_{xy} + 3 {(\frac{d}{n} S_{xy})}^{3} & {β_{x}^{- 1} (\frac{d}{n} S_{xy})}^{2} \\ {β_{y}^{- 1} (\frac{d}{n} S_{xy})}^{2} & \frac{2}{γ} \frac{d}{n} S_{xy} + 3 {(\frac{d}{n} S_{xy})}^{3} & β_{x}^{- 1} (\frac{1}{γ} + 3 {(\frac{d}{n} S_{xy})}^{2}) & β_{x}^{- 2} \frac{d}{n} S_{xy} \\ {(\frac{d}{n} S_{xy})}^{3} & 3 {β_{x}^{- 1} (\frac{d}{n} S_{xy})}^{2} & 3 β_{x}^{- 2} \frac{d}{n} S_{xy} & β_{x}^{- 3} \end{matrix}) e_{o 3}$ $with γ = \frac{1}{1 - \frac{d}{n} S_{oxx} - {(\frac{d}{n} S_{oxy})}^{2} - \frac{d}{n} S_{oyy} + {(\frac{d}{n})}^{2} S_{oxx} S_{oyy})}, β_{x} = \frac{1}{1 - \frac{d}{n} S_{oxx}} and β_{y} = \frac{1}{1 - \frac{d}{n} S_{oyy}}$ is specified as at least one function f(e.sub.o3).

3. The method according to claim 1, wherein determining a spherical and astigmatic aberration outgoing local wavefront set of coefficients comprises determining a power vector $s_{o} = (\begin{matrix} S_{oxx} \\ S_{oxy} \\ S_{oyy} \end{matrix}),$ wherein determining a higher-order aberration outgoing local wavefront set of coefficients comprises determining a coma vector $e_{o 3} = (\begin{matrix} E_{oxxx} \\ E_{oxxy} \\ E_{oxyy} \\ E_{oyyy} \end{matrix})$ and determining a spherical aberration vector $e_{o 4} = (\begin{matrix} E_{oxxxx} \\ E_{oxxxy} \\ E_{oxxyy} \\ E_{oxyyy} \\ E_{oyyyy} \end{matrix}),$ and wherein the function $e_{p 4} = (\begin{matrix} β_{x}^{4} & .Math. & .Math. & 0 \\ .Math. & β_{x}^{3} β_{y}^{1} & .Math. \\ β_{x}^{2} β_{y}^{2} \\ .Math. & β_{x}^{1} β_{y}^{3} & .Math. \\ 0 & .Math. & β_{y}^{4} \end{matrix}) (e_{o 4} + \frac{d}{n} (\begin{matrix} 3 (β_{x} E_{oxxx}^{2} + β_{y} E_{oxxy}^{2} - \frac{S_{oxx}^{4}}{n^{2}}) \\ 3 E_{oxxy} (β_{x} E_{oxxx} + β_{y} E_{oxyy}) \\ \begin{matrix} β_{x} (2 E_{oxxy}^{2} + E_{oxxx} E_{oxyy}) + β_{y} (2 E_{oxyy}^{2} + E_{oxxy} E_{oyyy}) - \\ {(\frac{S_{oxx} S_{oyy}}{n})}^{2} \end{matrix} \\ 3 E_{oxyy} (β_{x} E_{oxxy} + β_{y} E_{oyyy}) \\ 3 (β_{x} E_{oxyy}^{2} + β_{y} E_{oyyy}^{2} - \frac{S_{oyy}^{4}}{n^{2}}) \end{matrix}))$ $with β_{x} = \frac{1}{1 - \frac{d}{n} S_{oxx}} and β_{y} = \frac{1}{1 - \frac{d}{n} S_{oyy}}$ is specified as at least one function f(e.sub.ok).

4. The method according to claim 1, further comprising determining an angle α between a first plane of refraction of the main ray at a first surface of the spectacle lens and a second plane of refraction of the main ray at a second surface of the spectacle lens, wherein determining a higher-order aberration comprises: determining a higher-order aberration propagated local wavefront set of coefficients (e.sub.ipk) of the local aberration of the propagated wavefront; and determining a transformed higher-order aberration propagated local wavefront set of coefficients ({tilde over (e)}.sub.ipk) depending on the determined angle α.

5. The method according to claim 1, which further comprises collecting prescription data V, wherein the prescription data comprises data with respect to spherical power Sph.sub.V, magnitude of the astigmatism Zyl.sub.V, astigmatism axis Axis.sub.V, as well as at least one further higher-order refraction HOA.sub.V.

6. The method according to claim 5, wherein collecting prescription data comprises collecting first prescription data for a first object distance and second prescription data for a second object distance.

7. The method according to claim 1, further comprising: specifying an object distance model A1(x, y), where A1 designates the object distance and (x, y) a visual point or visual spot of the spectacle lens in a specified or specifiable direction of sight; specifying a function r.sub.0=g(A1), which describes the dependence of a pupil size r.sub.0 on the object distance A1; determining a pupil size for the at least one main ray on the basis of the object distance model A1(x, y) and the specified function r.sub.0=g(A1).

8. The method according to claim 1, wherein the spectacle lens is a progressive spectacle lens.

9. A computer program product stored on a non-transitory computer readable medium and adapted, when loaded and executed on a computer, to perform a method for designing a spectacle lens according to claim 1.

10. A method for producing a spectacle lens, comprising: designing a spectacle lens according to the method for designing a spectacle lens according to claim 1; and manufacturing the designed spectacle lens.

11. A device for producing a spectacle lens, comprising: a calculator configured to calculate the spectacle lens according to a method for designing a spectacle lens according to claim 1; and the device configured to produce the spectacle lens.

12. A device for designing a spectacle lens and manufacturing the designed spectacle lens, comprising: a surface model database configured to specify at least one surface for the spectacle lens to be designed; a main ray determiner configured to determine the course of a main ray through at least one visual point (i) of said at least one surface; a primary coefficient determiner configured to determine a spherical and astigmatic aberration outgoing local wavefront set of coefficients (s.sub.io) and a higher-order aberration outgoing local wavefront setoff coefficients (e.sub.iok) of the local aberration of a local wavefront going out from the at least one visual point (i) in a surrounding of the main ray; a propagation model database configured to specify at least one function f(e.sub.ok), which assigns a higher-order aberration propagated local wavefront set of coefficients (e.sub.pk) to a higher-order aberration outgoing local wavefront set of coefficients (e.sub.ok), wherein e.sub.pk=B.sub.k(e.sub.ok+r.sub.k), and the proportionality term B.sub.k depends on the spherical and astigmatic aberration outgoing local wavefront set of coefficients (s.sub.io), but not on the higher-order aberration outgoing local wavefront set of coefficients (e.sub.ok), and r.sub.k is a remainder term; a secondary coefficient determiner configured to determine a higher-order aberration of a local wavefront propagated starting from the at least one visual point (i) along the main ray depending on at least the higher-order aberration outgoing local wavefront set of coefficients (e.sub.iok) on the basis of the specified function f(e.sub.ok); a calculator or optimizer configured to calculate or optimize the at least one surface of the spectacle lens based on the determined higher-order aberration of the propagated local wavefront; and manufacturing the designed spectacle lens.

Description

DRAWINGS

(1) Preferred embodiments of the invention will be described by way of example in the following with reference to the accompanying drawings, which show:

(2) FIG. 1 a schematic illustration of the physiological and physical model of a spectacle lens along with a ray course in a predetermined wearing position;

(3) FIG. 2 a schematic illustration of a coordinate system with an original wavefront and a propagated wavefront;

(4) FIG. 3 a schematic illustration of a spherical wavefront with a vergence distance s.sub.o at a distance d from a propagated wavefront with the vergence distance s.sub.p;

(5) FIG. 4 a schematic illustration of the process of propagation of a wavefront;

(6) FIG. 5 a schematic illustration of local coordinate systems of a refractive surface, an incoming and an outgoing wavefront; and

(7) FIG. 6 a flow chart for illustrating a method for optimizing an individual spectacle lens according to an embodiment of the invention.

DETAILED DESCRIPTION

(8) FIG. 1 shows a schematic illustration of the physiological and physical model of a spectacle lens in a predetermined wearing position along with an exemplary ray course, on which an individual spectacle lens calculation or optimization according to a preferred embodiment of the invention is based.

(9) Here, preferably only one single ray (the main ray 10) is calculated per visual point of the spectacle lens, but further also the derivatives of the vertex depths of the wavefront according to the transversal coordinates (perpendicular to the main ray). These derivatives are taken into consideration up to the desired orders, wherein the second derivatives describe the local curvature properties of the wavefront and the higher derivatives are related to the higher-order aberrations.

(10) In the tracing of light through the spectacle lens, the local derivatives of the wavefronts are ultimately determined at a suitable position in the ray course in order to compare them with the required values of the refraction of the spectacles wearer there. In a preferred embodiment, this position is for example the vertex sphere or the entrance pupil of the eye 12. To this end, it is assumed that a spherical wavefront originates at an object point and propagates to the first spectacle lens surface 14. There, it is refracted and subsequently it propagates (ST2) up to the second spectacle lens surface 16, where it is refracted again. If further surfaces to be considered exist, the alternation of propagation and refraction is continued until the last boundary surface has been passed, and the last propagation (ST4) then takes place from this last boundary surface to the vertex sphere (or the entrance pupil of the eye).

(11) In the following, the propagation of the wavefront according to a preferred embodiment of the present invention will be described in more detail. These statements can e.g. be applied to the propagation of the wavefront between the two spectacle lens surfaces and/or to the propagation of the wavefront from the rear spectacle lens surface to the vertex sphere.

(12) As illustrated in FIG. 2, preferably a Cartesian coordinate system (with an x axis, a y axis, and a z axis) is defined, the origin of which being at the intersection point of the main ray 10 with the original wavefront 18 for a predetermined main ray 10. The z axis preferably points in the direction of the main ray 10. The directions of the x axis and the y axis are preferably selected to be perpendicular to the z axis and perpendicular to each other such that the coordinate system is right-handed. If the original wavefront is assumed to be a wavefront at a refractive surface, i.e. a surface of the spectacle lens, the x axis and/or the y axis is preferably selected to be parallel to the surface or surface tangent in the penetration point of the main ray. In another preferred embodiment, the x axis and the y axis are selected to be parallel to the main curvatures of the original wavefront 18.

(13) Preferably, a description of the wavefront according to
w(x,y)=(x,y,w(x,y)) (1)
is assumed, where the value w(x,y) is represented by

(14) $\begin{matrix} w (x, y) = {.Math.}_{k = 0}^{\infty} {.Math.}_{m = 0}^{k} \frac{a_{m, k - m}}{m! (k - m)!} x^{m} y^{k - m} & (2) \end{matrix}$
by means of the coefficients

(15) $\begin{matrix} a_{m, k - m} = \frac{\partial^{k}}{\partial x^{m} \partial y^{k - m}} w (x, y) |_{x = 0, y = 0} . & (3) \end{matrix}$

(16) Thus, the connection between the coefficients a.sub.k.sub.x.sub.,k.sub.y and the local aberrations E.sub.k.sub.x.sub.,k.sub.y can be described by:
E.sub.k.sub.x.sub.,k.sub.y=na.sub.k.sub.x.sub.,k.sub.yE.sub.2,0=S.sub.xxna.sub.In,2,0E.sub.1,1=S.sub.xy=na.sub.1,1E.sub.0,2=S.sub.yy=na.sub.0,2E.sub.3,0=na.sub.3,0

(17) For aberrations up to the second order, the propagation of a spherical wavefront with the vergence S.sub.o=n/s.sub.o of the original wavefront in a surrounding around a main ray can preferably be expressed in a known manner by the propagation equation

(18) $\begin{matrix} S_{p} = \frac{1}{1 - \frac{d}{n} S_{o}} S_{o} & (4) \end{matrix}$
where S.sub.p=n/s.sub.p designates the vergence of the propagated wavefront. As illustrated in FIG. 3, s.sub.o and s.sub.p designate the vertex distance of the original wavefront 18 and the propagated wavefront 20, respectively, (distance along the main ray 10 from the wavefront to the image point 22). n designates the refractive index and d the propagation distance.

(19) By an extension to three dimensions, the spherocylindrical form of the wavefront can be represented as follows. First of all, the curvatures 1/s.sub.o and 1/s.sub.p are identified with the second derivatives of the vertex depths of the original wavefront 18 and the propagated wavefront 20, respectively. In the three-dimensional representation, the two derivatives w.sub.o.sup.(2,0)=∂.sup.2w.sub.o/∂x.sup.2, w.sub.o.sup.(1,1)=∂.sup.2w.sub.o/∂x∂y, w.sub.o.sup.(0,2)=∂.sup.2w.sub.o/∂y.sup.2 the original wavefront 18 and correspondingly for the propagated wavefront 20 are respectively summarized in form of a vergence matrix:

(20) $\begin{matrix} S_{o} = (\begin{matrix} S_{oxx} & S_{oxy} \\ S_{oxy} & S_{oyy} \end{matrix}) = n (\begin{matrix} w_{o}^{(2, 0)} & w_{o}^{(1, 1)} \\ w_{o}^{(1, 1)} & w_{o}^{(0, 2)} \end{matrix}), S_{p} = (\begin{matrix} S_{pxx} & S_{pxy} \\ S_{pxy} & S_{pyy} \end{matrix}) = n (\begin{matrix} w_{p}^{(2, 0)} & w_{p}^{(1, 1)} \\ w_{p}^{(1, 1)} & w_{p}^{(0, 2)} \end{matrix}) & (5) \end{matrix}$
According to

(21) $\begin{matrix} S_{oxx} = (Sph + \frac{Cyl}{2}) - \frac{Cyl}{2} \cos 2 α S_{oxy} = - \frac{Cyl}{2} \sin 2 α S_{oyy} = (Sph + \frac{Cyl}{2}) + \frac{Cyl}{2} \cos 2 α & (6) \end{matrix}$
(and analogously for the propagated wavefront) the components of the respective vergence matrix are connected with the known parameters of spherical power Sph, the magnitude Cyl of cylindrical power, and the cylinder axis a of the cylindrical power. By means of the representation in form of the vergence matrix, by analogy with equation (4), the propagation of an astigmatic wavefront can be represented via the generalized propagation equation

(22) $\begin{matrix} S_{p} = \frac{1}{1 - \frac{d}{n} S_{o}} S_{o} & (7) \end{matrix}$
with the identity matrix

(23) $1 = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) .$
Equivalent to this representation in form of the vergence matrix,

(24) $\begin{matrix} s_{o} = (\begin{matrix} S_{oxx} \\ S_{oxy} \\ S_{oyy} \end{matrix}) = n (\begin{matrix} w_{o}^{(2, 0)} \\ w_{o}^{(1, 1)} \\ w_{o}^{(0, 2)} \end{matrix}), s_{p} = (\begin{matrix} S_{pxx} \\ S_{pxy} \\ S_{pyy} \end{matrix}) = n (\begin{matrix} w_{p}^{(2, 0)} \\ w_{p}^{(1, 1)} \\ w_{p}^{(0, 2)} \end{matrix}) & (8) \end{matrix}$
are introduced as power vectors in the three-dimensional vector space for the original wavefront 18 and the propagated wavefront 20.

(25) Now, for consideration of higher-order aberrations in the propagation of the wavefront, corresponding vectors e.sub.k of the dimension k+1 are introduced:

(26) 0 $\begin{matrix} e_{ok} = (\begin{matrix} E_{ox .Math. xx} \\ E_{ox .Math. xy} \\ .Math. \\ E_{oy .Math. yy} \end{matrix}) := n (\begin{matrix} w_{o}^{(k, 0)} \\ w_{o}^{(k - 1, 1)} \\ .Math. \\ w_{o}^{(0, k)} \end{matrix}), e_{pk} = (\begin{matrix} E_{px .Math. xx} \\ E_{px .Math. xy} \\ .Math. \\ E_{py .Math. yy} \end{matrix}) := n (\begin{matrix} w_{o}^{(k, 0)} \\ w_{o}^{(k - 1, 1)} \\ .Math. \\ w_{o}^{(0, k)} \end{matrix}) & (9) \end{matrix}$

(27) For further consideration, at first only a two-dimensional representation will be described for reasons of simplification. Here, some point on the original wavefront (r=o) or the propagated wavefront (r=p) is described by

(28) $\begin{matrix} w_{r} (y) = (\begin{matrix} y \\ w_{r} (y) \end{matrix}) & (10) \end{matrix}$
where w.sub.r(y) is described by:

(29) $\begin{matrix} w_{r} (y) = {.Math.}_{k = 0}^{\infty} \frac{a_{r, k}}{k!} y^{k} & (11) \end{matrix}$

(30) The coefficients a.sub.o,k of the original wavefront 18 correspond to the derivatives of the wavefront with y=0:

(31) $\begin{matrix} a_{o, k} = \frac{\partial^{k}}{\partial y^{k}} w_{o} (y) |_{y = 0} = w_{o}^{(k)} (0) & (12) \end{matrix}$

(32) In two dimensions, the vergence matrix S.sub.o in equation (5) is reduced to a scalar E.sub.o,k=nw.sub.o.sup.(k)=na.sub.o,k. For second or third-order aberrations, e.g. S.sub.o=E.sub.o,2=nw.sub.o.sup.(2)=na.sub.o,2, E.sub.o,3=nw.sub.o.sup.(k)=na.sub.o,3, etc. result. The same applies to the propagated wavefront 20.

(33) Here, it is to be noted that any wavefront at the intersection point with the main ray 10 is not inclined with respect to the z axis. Since the z axis points along the direction of the main ray 10, it is perpendicular to the original and propagated wavefronts in the intersection points of the main ray 10 with the wavefronts 18, 20. Moreover, since the origin of the coordinate system is at the original wavefront 18, it holds for the coefficients that: a.sub.o,0=0, a.sub.o,1=0, a.sub.p,0=d, and a.sub.p,1=0

(34) In two dimensions, the normal vector n.sub.w(y) for a wavefront w(y) results from n.sub.w(y)=(−w.sup.(1)(y),1).sup.T/√{square root over (1+w.sup.(1)(y).sup.2)}, Where w.sup.(1)=∂w/∂y. For Reasons of a simplified notation, first of all v≡w.sup.(1) and the following function is introduced:

(35) $\begin{matrix} n (v) := \frac{1}{\sqrt{1 + v^{2}}} (\begin{matrix} - v \\ 1 \end{matrix}) & (13) \end{matrix}$

(36) As derivatives n.sup.(i)(0)≡∂.sup.i/∂v.sup.i n(v)|.sub.v=0 of this function there result:

(37) $\begin{matrix} n (0) := (\begin{matrix} 0 \\ 1 \end{matrix}), n^{(1)} (0) := (\begin{matrix} - 1 \\ 0 \end{matrix}), n^{(2)} (0) := (\begin{matrix} 0 \\ - 1 \end{matrix}), n^{(3)} (0) := (\begin{matrix} 3 \\ 0 \end{matrix}), n^{(4)} (0) := (\begin{matrix} 0 \\ 9 \end{matrix}), etc . & (14) \end{matrix}$

(38) The normal vector, which is perpendicular to both the original wavefront 18 and the propagated wavefront 20, can be designated uniformly with n.sub.w. Thus, for the first derivative of the normal vector there is determined:

(39) $\begin{matrix} \frac{\partial}{\partial y} n_{w} (y) |_{y = 0} \equiv n_{w}^{(1)} (0) = n^{(1)} (0) w_{o}^{(2)} (0) = (\begin{matrix} - 1 \\ 0 \end{matrix}) w_{o}^{(2)} (0) & (15) \end{matrix}$

(40) The same applies to the higher derivatives.

(41) With the local aberrations of the original wavefront 18, the corresponding coefficients a.sub.k and, equivalent thereto, the derivatives of the wavefront are directly defined as well. Subsequently, the propagated wavefront 20 is determined therefrom particularly by determining its derivatives or coefficients a.sub.k for all orders 2≦k≦k.sub.0 up to the desired value k.sub.0, and thus the values of the local aberrations of the propagated wavefront 20 are determined.

(42) As a starting point, the following situation with respect to FIG. 4 will be considered in an illustrative way. While the main ray 10 and the coordinate system are fixed, a neighboring ray 24 scans the original wavefront 18 ({w.sub.o}) and strikes it in a section y.sub.o≠0. From there, it propagates further to the propagated wavefront 20 ({w.sub.p}). As illustrated in FIG. 4, y.sub.o designates the projection of the intersection point of the neighboring ray 24 with the original wavefront {w.sub.o} to the y axis, while analogously the projection of the intersection point with the propagated wavefront {w.sub.p} to the y axis is designated with y.sub.p.

(43) The vector w.sub.o=w.sub.o(y.sub.o) (cf. equation (10)) points to the intersection point of the neighboring ray 24 with the original wavefront 18, and the optical path difference (OPD) with respect to the propagated wavefront 20 is designated with τ. Accordingly, the vector from the original wavefront 18 to the propagated surface 20 is represented by τ/n n.sub.w. Thus, it results for the vector to the corresponding point of the propagated wavefront: w.sub.p=w.sub.o+τ/n n.sub.w. As a basic equation there is introduced:

(44) $\begin{matrix} (\begin{matrix} y_{o} \\ w_{o} (y_{o}) \end{matrix}) + \frac{τ}{n} n_{w} = (\begin{matrix} y_{p} \\ w_{p} (y_{p}) \end{matrix}) & (16) \end{matrix}$

(45) Now, from this equation, the desired relations are derived order by order. Here, y.sub.p is preferably used as a free variable, on which y.sub.o depends in turn. For solving the equation, first of all the vector

(46) $\begin{matrix} p (y_{p}) = (\begin{matrix} y_{o} (y_{p}) \\ w_{p} (y_{p}) \end{matrix}) & (17) \end{matrix}$
on the boundary condition

(47) $p (0) = (\begin{matrix} 0 \\ \frac{τ}{n} \end{matrix})$
can be introduced. Based on this, the following function is introduced for the further consideration:

(48) 0 $\begin{matrix} f (p, y_{p}) = (\begin{matrix} y_{o} + \frac{τ}{n} n_{w, y} (w_{o}^{(1)} (y_{o})) - y_{p} \\ w_{o} (y_{o}) + \frac{τ}{n} n_{w, z} (w_{o}^{(1)} (y_{o})) - w_{p} \end{matrix}) & (18) \end{matrix}$
where (p.sub.1,p.sub.2)=(y.sub.o,w.sub.p) are the components of p. Now, if p=p(y.sub.p), the equation (16) can be represented in a compact form by:
f(p(y.sub.p)y.sub.p)=0 (19)

(49) The derivatives of this function according to y.sub.p are preferably expressed by the following system of differential equations:

(50) $\begin{matrix} {.Math.}_{j = 1}^{2} \frac{\partial f_{i}}{\partial p_{j}} p_{j}^{(1)} (y_{p}) + \frac{\partial f_{i}}{\partial y_{p}} = 0, i = 1.2 & (20) \end{matrix}$
where the matrix with the elements A.sub.ij:=∂ƒ.sub.i/∂p.sub.j is referred to as a Jacobi matrix A. The Jacobi matrix A thus reads

(51) $\begin{matrix} A : = (\begin{matrix} \frac{\partial f_{1}}{\partial y_{o}} & \frac{\partial f_{1}}{\partial w_{p}} \\ \frac{\partial f_{2}}{\partial y_{o}} & \frac{\partial f_{2}}{\partial w_{p}} \end{matrix}) = (\begin{matrix} 1 + \frac{τ}{n} n_{w, y}^{(1)} w_{o}^{(2)} & 0 \\ w_{o}^{(1)} + \frac{τ}{n} n_{w, z}^{(1)} w_{o}^{(2)} & - 1 \end{matrix}) & (21) \end{matrix}$

(52) The terms appearing in this equation are to be understood as w.sub.o.sup.(1)≡w.sub.o.sup.(1)(y.sub.o) w.sub.o.sup.(2)≡w.sub.o.sup.(2)(y.sub.o), n.sub.w,y≡n.sub.w,y(w.sub.o.sup.(1)(y.sub.o)), n.sub.w,y.sup.(1)≡n.sub.w,y.sup.(1)(w.sub.o.sup.(1)(y.sub.o)), etc., where y.sub.o,w.sub.p are in turn themselves functions of y.sub.p.

(53) The derivative vector ∂ƒ.sub.i/∂y.sub.p can be summarized as

(54) $\begin{matrix} b : = \frac{\partial f}{\partial y_{p}} = (\begin{matrix} 1 \\ 0 \end{matrix}) & (22) \end{matrix}$

(55) Thus, the above differential equation system can be represented as:
A(p(y.sub.p))p.sup.(1)(y.sub.p)=b (23)

(56) Formally, this equation is solved by:
p.sup.(1)(y.sub.p)=A(p(y.sub.p)).sup.−1b (24)
with the boundary condition

(57) $p (0) = (\begin{matrix} 0 \\ \frac{τ}{n} \end{matrix}) .$
Based on this, the equation system for higher-order aberrations is preferably solved recursively as follows:

(58) $\begin{matrix} p^{(1)} (0) = A^{- 1} b p^{(2)} (0) = {(A^{- 1})}^{(1)} b .Math. p^{(k)} (0) = {(A^{- 1})}^{(k - 1)} b, & (25) \end{matrix}$
with the abbreviatory designations A.sup.−1=A(p(0)).sup.−1=A(0).sup.−1

(59) ${(A^{- 1})}^{(1)} = \frac{ⅆ}{ⅆ y_{p}} {A (p (y_{p}))}^{- 1} {{.Math.}_{y_{p} = 0}, .Math., {(A^{- 1})}^{(k - 1)} = \frac{ⅆ^{k - 1}}{ⅆ y_{p}^{k - 1}} {A (p (y_{p}))}^{- 1} .Math.}_{y_{p} = 0} .$

(60) In an alternative approach, it is suggested performing the recursion on the basis of equation (23) instead of equation (24). The first (k−1) derivatives of equation (23) yield:

(61) $\begin{matrix} \begin{matrix} {Ap}^{(1)} (0) = b & (a) \\ A^{(1)} p^{(1)} (0) + {Ap}^{(2)} (0) = 0 & (b) \\ A^{(2)} p^{(1)} (0) + 2 A^{(1)} p^{(2)} (0) + {Ap}^{(3)} (0) = 0 & (c) \\ .Math. \\ {.Math.}_{j = 1}^{k} (\begin{matrix} k - 1 \\ j - 1 \end{matrix}) A^{(k - j)} p^{(j)} (0) = 0, k \geq 2 & (d) \end{matrix} where A = A (p (0)) = A (0), A^{(1)} = \frac{ⅆ}{ⅆ y_{p}} A (p (y_{p})) {{.Math.}_{y_{p} = 0}, .Math., A^{(k - j)} = \frac{ⅆ^{k - j}}{ⅆ y_{p}^{k - j}} A (p (y_{p})) .Math.}_{y_{p} = 0} & (26) \end{matrix}$
designate the total derivatives of the function. Formally, these equations are solved by:

(62) $\begin{matrix} \begin{matrix} p^{(1)} (0) = A^{- 1} b, & k = 1 \\ p^{(k)} (0) = - A^{- 1} {.Math.}_{j = 1}^{k - 1} (\begin{matrix} k - 1 \\ j - 1 \end{matrix}) A^{(k - j)} p^{(j)} (0), & k \geq 2. \end{matrix} & (27) \end{matrix}$

(63) In order to obtain A(0).sup.−1, preferably equation (21) is evaluated for p=0 and equation (14) is applied. This yields:

(64) $\begin{matrix} A (0) = (\begin{matrix} 1 + \frac{τ}{n} w_{o}^{(2)} & 0 \\ 0 & - 1 \end{matrix}) .Math. {A (0)}^{- 1} = (\begin{matrix} \frac{1}{1 - \frac{τ}{n} w_{o}^{(2)}} & 0 \\ 0 & - 1 \end{matrix}) & (28) \end{matrix}$
from which it results for p.sup.(1)(0):

(65) 0 $\begin{matrix} p^{(1)} (0) = A^{- 1} b = (\begin{matrix} \frac{1}{1 - \frac{τ}{n} w_{o}^{(2)}} \\ 0 \end{matrix}) & (29) \end{matrix}$

(66) In turn, this means

(67) $y_{o}^{(1)} (0) = \frac{1}{1 - \frac{τ}{n} w_{o}^{(2)}} and w_{p}^{(1)} (0) = 0.$
For orders k≧2, preferably equation (27) is applied. The derivatives

(68) $A^{(1)} = \frac{ⅆ}{ⅆ y_{p}} A (p (y_{p})) {.Math.}_{y_{p} = 0},$
etc. are preferably determined from equation (21) and preferably equation (14) is applied again. Thus, it results in the second order:

(69) $\begin{matrix} w_{p}^{(2)} = \frac{1}{1 - \frac{τ}{n} w_{o}^{(2)}} w_{o}^{(2)} & (30) \end{matrix}$
which basically corresponds to the above-described propagation equation. The higher orders can analogously be expressed by:

(70) $\begin{matrix} w_{p}^{(3)} = {(\frac{1}{1 - \frac{τ}{n} w_{o}^{(2)}})}^{3} w_{o}^{(3)} w_{p}^{(4)} = {(\frac{1}{1 - \frac{τ}{n} w_{o}^{(2)}})}^{4} (w_{o}^{(4)} + 3 \frac{τ}{n} (\frac{1}{1 - \frac{τ}{n} w_{o}^{(2)}} w_{o}^{{(3)}^{2}} - w_{o}^{{(2)}^{4}})) w_{p}^{(5)} = {(\frac{1}{1 - \frac{τ}{n} w_{o}^{(2)}})}^{5} (w_{o}^{(5)} + 5 \frac{1}{1 - \frac{τ}{n} w_{o}^{(2)}} \frac{τ}{n} w_{o}^{(3)} (2 w_{o}^{(4)} + 3 \frac{1}{1 - \frac{τ}{n} w_{o}^{(2)} n} \frac{τ}{n} w_{o}^{{(3)}^{2}} - 6 w_{o}^{{(2)}^{3}})) .Math. & (31) \end{matrix}$

(71) Equation (31) correspondingly applies to the derivatives and the coefficients a.sub.o,k and a.sub.p,k due to equations (10) to (12). Now, if one replaces d=τ/n and

(72) $β = \frac{1}{1 - \frac{d}{n} S_{o}},$
the local aberrations can be expressed as follows:

(73) $\begin{matrix} S_{p} = β S_{o} E_{p, 3} = β^{3} E_{o, 3} E_{p, 4} = β^{4} (E_{o, 4} + 3 \frac{d}{n} (β E_{o, 3}^{2} - \frac{S_{o}^{4}}{n^{2}})) E_{p, 5} = β^{5} (E_{o, 5} + 5 β \frac{d}{n} E_{o, 3} (2 E_{o, 4} + 3 β \frac{d}{n} E_{o, 3}^{2} - 6 \frac{S_{o}^{3}}{n^{2}})) E_{p, 6} = β^{6} (E_{o, 6} + 5 β \frac{d}{n} (3 E_{o, 3} E_{o, 5} + 21 β \frac{d}{n} E_{o, 3}^{2} E_{o, 4} - 12 \frac{S_{o}^{3} E_{o, 4}}{n^{2}} + 2 E_{o, 4} - 9 β S_{0}^{2} E_{o, 3}^{2} \frac{3 + 4 \frac{d}{n} S_{o}}{n^{2}} + 21 {(β \frac{d}{n})}^{2} E_{o, 3}^{4} + 9 S_{o}^{6} \frac{3 + \frac{d}{n} S_{o}}{n^{4}})) & (32) \end{matrix}$

(74) For 2<k≦6, this is preferably represented by
E.sub.p,k=β.sup.k(E.sub.o,k+R.sub.k) (33)
in a generalized way, where in R.sub.k all wavefront derivatives E.sub.o,j of the lower orders (j<k) are expressed in form of local aberrations.

(75) Even if a three-dimensional representation is more complex, it can basically be established by analogy with the two-dimensional representation. Therefore, for the fully three-dimensional representation, only a few essential additional considerations will be described in the following.

(76) Preferably, the original wavefront can be expressed by the 3D vector

(77) $\begin{matrix} w_{o} (x, y) = (\begin{matrix} x \\ y \\ w_{o} (x, y) \end{matrix}) & (34) \end{matrix}$

(78) where w.sub.o (x, y) is determined according to equation (2), and the relationship between the coefficients and the derivatives is determined according to equation (3). The connection between the coefficients and the local aberrations results from a multiplication of the coefficient by the refractive index. Preferably, by analogy with equation (13), formal vectors are introduced:

(79) $\begin{matrix} n (u, v) := \frac{1}{\sqrt{1 + u^{2} + v^{2}}} (\begin{matrix} - u \\ - v \\ 1 \end{matrix}) & (35) \end{matrix}$
so that the normal vectors with respect to a surface w(x,y):=(x,y,w(x,y)).sup.T are determined by:

(80) $\frac{w^{(1, 0)} \times w^{(0, 1)}}{.Math. w^{(1, 0)} \times w^{(0, 1)} .Math.} = \frac{1}{\sqrt{1 + w^{{(1, 0)}^{2} + w^{{(0, 1)}^{2}}}}} (\begin{matrix} - w^{(1, 0)} \\ - w^{(0, 1)} \\ 1 \end{matrix}) = n (w^{(1, 0)}, w^{(0, 1)}) = n (\nabla w)$

(81) In the intersection point, it thus results n.sub.w(0,0)=(0,0,1).sup.T, and the derivatives according to equation (14) are preferably determined from equation (35).

(82) As the basis for the consideration of a connection between the original and propagated wavefronts, preferably substantially equation (16) is used, with the difference that now x and y components are considered at the same time. As a vector of unknown functions, there is preferably determined:

(83) 0 $\begin{matrix} p (x_{p}, y_{p}) = (\begin{matrix} x_{o} (x_{p}, y_{p}) \\ y_{o} (x_{p}, y_{p}) \\ w_{p} (x_{p}, y_{p}) \end{matrix}) & (36) \end{matrix}$
and by analogy with equation (16), there is preferably used for the three-dimensional consideration:
f(p(x.sub.p,y.sub.p)x.sub.p,y.sub.p)=0 (37)
where f is analogous to equation (18).

(84) An importance difference compared to the two-dimensional consideration is that in the three-dimensional case two arguments exist, with respect to which the derivatives are taken into account. Thus, already in the first order, two equations are considered:
A(p(x.sub.p,y.sub.p))p.sup.(1,0)(x.sub.p,y.sub.p)=b.sub.x
A(p(x.sub.p,y.sub.p))p.sup.(0,1)(x.sub.p,y.sub.p)=b.sub.y (38)
where the inhomogeneity is described by the column vectors:

(85) $\begin{matrix} b_{x} = - \frac{\partial f}{\partial x_{p}} = {(\begin{matrix} 1 & 0 & 0 \end{matrix})}^{T}, b_{y} = - \frac{\partial f}{\partial y_{p}} = {(\begin{matrix} 0 & 1 & 0 \end{matrix})}^{T} & (39) \end{matrix}$

(86) The Jacobi matrix A(p(x.sub.2,y.sub.2)) with the elements A.sub.ij:=∂ƒ.sub.i/∂p.sub.j is the same for both equations and analogous to equation (21), but now in the size 3×3.

(87) $\begin{matrix} A (p (x_{p}, y_{p})) = (\begin{matrix} \begin{matrix} 1 + \frac{τ}{n} (n_{w, x}^{(0, 1)} w_{o}^{(1, 1)} + \\ n_{w, x}^{(1, 0)} w_{o}^{(2, 0)}) \end{matrix} & \frac{τ}{n} (n_{w, x}^{(0, 1)} w_{o}^{(0, 2)} + n_{w, x}^{(1, 0)} w_{o}^{(1, 1)}) & 0 \\ \frac{τ}{n} (n_{w, y}^{(0, 1)} w_{o}^{(1, 1)} + n_{w, y}^{(1, 0)} w_{o}^{(2, 0)}) & 1 + \frac{τ}{n} (n_{w, y}^{(0, 1)} w_{o}^{(0, 2)} + n_{w, y}^{(1, 0)} w_{o}^{(1, 1)}) & 0 \\ \begin{matrix} w_{o}^{(1, 0)} + \\ \frac{τ}{n} (n_{w, z}^{(0, 1)} w_{o}^{(1, 1)} + n_{w, z}^{(1, 0)} w_{o}^{(2, 0)}) \end{matrix} & \begin{matrix} w_{o}^{(0, 1)} + \\ \frac{τ}{n} (n_{w, z}^{(0, 1)} w_{o}^{(0, 2)} + n_{w, z}^{(1, 0)} w_{o}^{(1, 1)}) \end{matrix} & - 1 \end{matrix}) & (40) \end{matrix}$

(88) The direct solutions by analogy with equation (25) are now determined by

(89) $\begin{matrix} p^{(1, 0)} (0, 0) = A^{- 1} b_{x} p^{(0, 1)} (0, 0) = A^{- 1} b_{y} p^{(2, 0)} (0, 0) = {(A^{- 1})}^{(1, 0)} | b_{x} p^{(1, 1)} (0, 0) = {(A^{- 1})}^{(0, 1)} b_{x} = {(A^{- 1})}^{(1, 0)} | b_{y} p^{(0, 2)} (0, 0) = {(A^{- 1})}^{(0, 1)} b_{y} .Math. p^{(k_{x}, k_{y})} (0, 0) = {\begin{matrix} {(A^{- 1})}^{(k_{x} - 1, 0)} b_{x}, & k_{x} \neq 0, k_{y} = 0 \\ {(A^{- 1})}^{(k_{x} - 1, k_{y})} b_{x} = {(A^{- 1})}^{(k_{x}, k_{y} - 1)} b_{y}, & k_{x} \neq 0, k_{y} \neq 0 \\ {(A^{- 1})}^{(0, k_{y} - 1)} b_{y}, & k_{x} = 0, k_{y} \neq 0 \end{matrix} where A^{- 1} = {A (p (0, 0))}^{- 1} = {A (0)}^{- 1}, {(A^{- 1})}^{(1, 0)} = \frac{ⅆ}{ⅆ x_{p}} {A (p (x_{p}, y_{p}))}^{- 1} |_{x_{p} = 0, y_{p} = 0}, {(A^{- 1})}^{(k_{x}, k_{y})} = \frac{ⅆ^{k_{x}}}{ⅆ x_{p}^{k_{x}}} \frac{ⅆ^{k_{y}}}{ⅆ x_{p}^{k_{y}}} {A (p (x_{p}, y_{p}))}^{- 1} |_{x_{p} = 0, y_{p} = 0}, etc .. & (41) \end{matrix}$

(90) By analogy with equations (28) and (29), it results for the three-dimensional consideration:

(91) $\begin{matrix} A (0) = (\begin{matrix} 1 - \frac{τ}{n} w_{o}^{(2, 0)} & - \frac{τ}{n} w_{o}^{(1, 1)} & 0 \\ - \frac{τ}{n} w_{o}^{(1, 1)} & 1 - \frac{τ}{n} w_{o}^{(0, 2)} & 0 \\ 0 & 0 & - 1 \end{matrix}) .Math. {A (0)}^{- 1} = (\begin{matrix} γ (\begin{matrix} 1 - \frac{τ}{n} w_{o}^{(0, 2)} & \frac{τ}{n} w_{o}^{(1, 1)} \\ \frac{τ}{n} w_{o}^{(1, 1)} & 1 - \frac{τ}{n} w_{o}^{(2, 0)} \end{matrix}) & \begin{matrix} 0 \\ 0 \end{matrix} \\ \begin{matrix} 0 & 0 \end{matrix} & - 1 \end{matrix}) with γ = \frac{- 1}{\det (A (0))} = \frac{1}{1 - \frac{τ}{n} w_{o}^{(2, 0)} - {(\frac{τ}{n} w_{o}^{(1, 1)})}^{2} - \frac{τ}{n} w_{o}^{(0, 2)} + {(\frac{τ}{n})}^{2} w_{o}^{(2, 0)} w_{o}^{(0, 2)})} & (42) \end{matrix}$
and after application of equations (39) and (41), the solutions

(92) $\begin{matrix} p^{(1, 0)} (0, 0) = γ (\begin{matrix} n (n - τ w_{o}^{(0, 2)}) \\ n τ w_{o}^{(1, 1)} \\ 0 \end{matrix}), p^{(0, 1)} (0, 0) = γ (\begin{matrix} n τ w_{o}^{(1, 1)} \\ n (n - τ w_{o}^{(2, 0)}) \\ 0 \end{matrix}) & (43) \end{matrix}$

(93) After further application of equations (39) and (41), it results in the second order
w.sub.p.sup.(2,0)=γ(τ/n(w.sub.o.sup.(1,1)).sup.2+(1−τ/n w.sub.o.sup.(0,2))w.sub.o.sup.(2,0))
w.sub.p.sup.(1,1)=γw.sub.o.sup.(1,1)
w.sub.p.sup.(0,2)=γ(τ/n(w.sub.o.sup.(1,1)).sup.2+(1−τ/n w.sub.o.sup.(2,0))w.sub.o.sup.(0,2)) (44)

(94) In a preferred embodiment, the coordinate axes for determination of the propagation are selected or determined such that the x axis and the y axis coincide with the directions of the main curvatures of the original wavefront. It thereby holds that w.sub.o.sup.(1,1)=0, and the equations (44) are simplified as

(95) $\begin{matrix} w_{p}^{(2, 0)} = \frac{1}{1 - \frac{τ}{n} w_{o}^{(2, 0)}} w_{o}^{(2, 0)} w_{p}^{(1, 1)} = 0 w_{p}^{(0, 2)} = \frac{1}{1 - \frac{τ}{n} w_{o}^{(0, 2)}} w_{o}^{(0, 2)} & (45) \end{matrix}$

(96) In a corresponding way, the equations in the third order are preferably expressed as follows:

(97) $\begin{matrix} w_{p}^{(3, 0)} = γ^{3} ({(1 - \frac{τ}{n} w_{o}^{(0, 2)})}^{3} w_{o}^{(3, 0)} + \frac{τ}{n} w_{o}^{(1, 1)} (3 {(1 - \frac{τ}{n} w_{o}^{(0, 2)})}^{2} w_{o}^{(2, 1)} + \frac{τ}{n} w_{o}^{(1, 1)} (\frac{τ}{n} w_{o}^{(0, 3)} w_{o}^{(1, 1)}) + 3 {(1 - \frac{τ}{n} w_{o}^{(0, 2)})}^{2} w_{o}^{(2, 1)})) w_{p}^{(2, 1)} = γ^{3} (w_{o}^{(2, 1)} + \frac{τ}{n} (w_{o}^{(1, 1)} (2 w_{o}^{(1, 2)} + w_{o}^{(3, 0)}) - (2 w_{o}^{(0, 2)} + w_{o}^{(2, 0)}) w_{o}^{(2, 1)}) + {(\frac{τ}{n})}^{2} (w_{o}^{(2, 1)} w_{o}^{{(0, 2)}^{2}} - 2 (w_{o}^{(1, 1)} (w_{o}^{(1, 2)} + w_{o}^{(3, 0)}) - w_{o}^{(2, 0)} w_{o}^{(2, 1)}) w_{o}^{(0, 2)} + w_{o}^{(0, 3)} w_{o}^{{(1, 1)}^{2}} + 2 w_{o}^{(1, 1)} (w_{o}^{(1, 1)} w_{o}^{(2, 1)} - w_{o}^{(1, 2)} w_{o}^{(2, 0)})) + {(\frac{τ}{n})}^{3} (w_{o}^{(1, 2)} w_{o}^{{(1, 1)}^{3}} - (w_{o}^{(0, 3)} w_{o}^{(2, 0)} + 2 w_{o}^{(0, 2)} w_{o}^{(2, 1)}) w_{o}^{{(1, 1)}^{2}} + w_{o}^{(0, 2)} (2 w_{o}^{(1, 2)} w_{o}^{(2, 0)} + w_{o}^{(0, 2)} w_{o}^{(3, 0)}) w_{o}^{(1, 1)} - w_{o}^{{(0, 2)}^{2}} w_{o}^{(2, 0)} w_{o}^{(2, 1)})) w_{p}^{(1, 2)} = γ^{3} (w_{o}^{(1, 2)} + \frac{τ}{n} (w_{o}^{(1, 1)} (2 w_{o}^{(2, 1)} + w_{o}^{(0, 3)}) - (2 w_{o}^{(2, 0)} + w_{o}^{(0, 2)}) w_{o}^{(1, 2)}) + {(\frac{τ}{n})}^{2} (w_{o}^{(1, 2)} w_{o}^{{(2, 0)}^{2}} - 2 (w_{o}^{(1, 1)} (w_{o}^{(2, 1)} + w_{o}^{(0, 3)}) - w_{o}^{(0, 2)} w_{o}^{(1, 2)}) w_{o}^{(2, 0)} + w_{o}^{(3, 0)} w_{o}^{{(1, 1)}^{2}} + 2 w_{o}^{(1, 1)} (w_{o}^{(1, 1)} w_{o}^{(1, 2)} - w_{o}^{(2, 1)} w_{o}^{(0, 2)})) + {(\frac{τ}{n})}^{3} (w_{o}^{(2, 1)} w_{o}^{{(1, 1)}^{3}} - (w_{o}^{(3, 0)} w_{o}^{(0, 2)} + 2 w_{o}^{(2, 0)} w_{o}^{(1, 2)}) w_{o}^{{(1, 1)}^{2}} + w_{o}^{(2, 0)} (2 w_{o}^{(2, 1)} w_{o}^{(0, 2)} + w_{o}^{(2, 0)} w_{o}^{(0, 3)}) w_{o}^{(1, 1)} - w_{o}^{{(2, 0)}^{2}} w_{o}^{(0, 2)} w_{o}^{(1, 2)})) w_{p}^{(0, 3)} = γ^{3} ({(1 - \frac{τ}{n} w_{o}^{(2, 0)})}^{3} w_{o}^{(0, 3)} + \frac{τ}{n} w_{o}^{(1, 1)} (3 {(1 - \frac{τ}{n} w_{o}^{(2, 0)})}^{2} w_{o}^{(1, 2)} + \frac{τ}{n} w_{o}^{(1, 1)} (\frac{τ}{n} w_{o}^{(3, 0)} w_{o}^{(1, 1)}) + 3 {(1 - \frac{τ}{n} w_{o}^{(2, 0)})}^{2} w_{o}^{(2, 1)})) & (46) \end{matrix}$

(98) Now, if one replaces

(99) $d = \frac{τ}{n} and γ = \frac{1}{1 - \frac{d}{n} S_{oxx} - {(\frac{d}{n} S_{oxy})}^{2} - \frac{d}{n} S_{oyy} + {(\frac{d}{n})}^{2} S_{oxx} S_{oyy})},$
the propagation of the wavefront in the second order in the form of the local aberrations can be expressed as follows:

(100) $\begin{matrix} s_{p} = γ (s_{o} + \frac{d}{n} (\begin{matrix} S_{oxy}^{2} - S_{oxx} S_{oyy} \\ 0 \\ S_{oxy}^{2} - S_{oxx} S_{oyy} \end{matrix})) & (47) \end{matrix}$

(101) Moreover, if one replaces

(102) 0 $β_{x} = \frac{1}{1 - \frac{d}{n} S_{xx}} and β_{y} = \frac{1}{1 - \frac{d}{n} S_{yy}},$
the propagation of the wavefront in the third order can be described by:

(103) $\begin{matrix} e_{p 3} = γ^{3} (\begin{matrix} β_{y}^{- 3} & 3 β_{y}^{- 2} \frac{d}{n} S_{xy} & 3 {β_{y}^{- 1} (\frac{d}{n} S_{xy})}^{2} & {(\frac{d}{n} S_{xy})}^{3} \\ β_{y}^{- 2} \frac{d}{n} S_{xy} & β_{y}^{- 1} (\frac{1}{γ} + 3 {(\frac{d}{n} S_{xy})}^{2}) & \frac{2}{γ} \frac{d}{n} S_{xy} + 3 {(\frac{d}{n} S_{xy})}^{3} & {β_{x}^{- 1} (\frac{d}{n} S_{xy})}^{2} \\ {β_{y}^{- 1} (\frac{d}{n} S_{xy})}^{2} & \frac{2}{γ} \frac{d}{n} S_{xy} + 3 {(\frac{d}{n} S_{xy})}^{3} & β_{x}^{- 1} (\frac{1}{γ} + 3 {(\frac{d}{n} S_{xy})}^{2}) & β_{x}^{- 2} \frac{d}{n} S_{xy} \\ {(\frac{d}{n} S_{xy})}^{3} & 2 {β_{x}^{- 1} (\frac{d}{n} S_{xy})}^{2} & 3 β_{x}^{- 2} \frac{d}{n} S_{xy} & β_{x}^{- 3} \end{matrix}) e_{o 3} & (48) \end{matrix}$

(104) In a preferred embodiment, the coordinate axes for determination of the propagation are selected or determined such that the x axis and the y axis coincide with the directions of the main curvatures of the original wavefront. Thereby, the equations (47) and (48) are simplified as

(105) $\begin{matrix} s_{p} = (\begin{matrix} β_{x} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & β_{y} \end{matrix}) s_{o} & (49) \\ e_{p 3} = (\begin{matrix} β_{x}^{3} & 0 & 0 & 0 \\ 0 & β_{x}^{2} β_{y} & 0 & 0 \\ 0 & 0 & β_{x} β_{y}^{2} & 0 \\ 0 & 0 & 0 & β_{y}^{3} \end{matrix}) e_{o 3} & (50) \end{matrix}$

(106) The propagation of fourth-order aberrations can be determined in a comparatively simply way by:

(107) $\begin{matrix} e_{p 4} = (\begin{matrix} β_{x}^{4} & .Math. & .Math. & 0 \\ .Math. & β_{x}^{3} β_{y}^{1} & .Math. \\ β_{x}^{2} β_{y}^{2} \\ .Math. & β_{x}^{1} β_{y}^{3} & .Math. \\ 0 & .Math. & β_{y}^{4} \end{matrix}) (e_{o 4} + \frac{d}{n} (\begin{matrix} 3 (β_{x} E_{oxxx}^{2} + β_{y} E_{oxxy}^{2} - \frac{S_{oxx}^{4}}{n^{2}}) \\ 3 E_{oxxy} (β_{x} E_{oxxx} + β_{y} E_{oxyy}) \\ β_{x} (2 E_{oxxy}^{2} + E_{oxxx} E_{oxyy}) + β_{y} (2 E_{oxyy}^{2} + E_{oxxy} E_{oyyy}) - \\ {(\frac{S_{oxx} S_{oyy}}{n})}^{2} \\ 3 E_{oxyy} (β_{x} E_{oxxy} + β_{y} E_{oyyy}) \\ 3 (β_{x} E_{oxyy}^{2} + β_{y} E_{oyyy}^{2} - \frac{S_{oyy}^{4}}{n^{2}}) \end{matrix})) & (51) \end{matrix}$

(108) For 2<k≦4, this is preferably generalized by

(109) $\begin{matrix} e_{pk} = B_{k} (e_{ok} + r_{k}) & (52) \\ with B_{k} = (\begin{matrix} β_{x}^{k} & .Math. & .Math. & 0 \\ .Math. & β_{x}^{k - 1} β_{y}^{1} & .Math. \\ ⋱ \\ .Math. & β_{x}^{1} β_{y}^{k - 1} & .Math. \\ 0 & .Math. & β_{y}^{k} \end{matrix}) & (53) \end{matrix}$
where r.sub.k represents a vector in which by analogy with R.sub.k in equation (33) all remainder terms R.sub.k.sub.x.sub.,k.sub.y are included.

(110) In a preferred embodiment it holds:
s.sub.p=T.sup.(2)({circumflex over (R)})T.sup.(2)({circumflex over (β)}){tilde over ({circumflex over (s)})}.sub.pS({circumflex over (R)})
e.sub.p3=T.sup.(3)({circumflex over (R)})T.sup.(3)({circumflex over (β)}){tilde over ({circumflex over (e)})}.sub.p3S({circumflex over (R)})
e.sub.p4=T.sup.(4)({circumflex over (R)})T.sup.(4)({circumflex over (β)}){tilde over ({circumflex over (e)})}.sub.p4S({circumflex over (R)})
etc. (54)
where s.sub.p, e.sub.p3, e.sub.p4, . . . apply in every coordinate system and where

(111) $\hat{β} = (\begin{matrix} {\hat{β}}_{xx} & 0 \\ 0 & {\hat{β}}_{yy} \end{matrix})$ $with$ ${\hat{β}}_{xx} = {(1 - \frac{τ}{n} {\hat{w}}^{(2, 0)})}^{- 1}$ ${\hat{β}}_{yy} = {(1 - \frac{τ}{n} {\hat{w}}^{(0, 2)})}^{- 1}$ $(\begin{matrix} {\hat{w}}^{(2, 0)} \\ {\hat{w}}^{(1, 1)} \\ {\hat{w}}^{(0, 2)} \end{matrix}) = \frac{1}{2} (w^{(2, 0)} + w^{(0, 2)}) (\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}) + \frac{1}{2} (w^{(2, 0)} - w^{(0, 2)}) \sqrt{1 + {(\frac{2 w^{(1, 1)}}{w^{(2, 0)} - w^{(0, 2)}})}^{2}} (\begin{matrix} 1 \\ 0 \\ - 1 \end{matrix})$
is an auxiliary matrix, which can be referred to back to the matrix

(112) $β = (\begin{matrix} β_{xx} & β_{xy} \\ β_{xy} & β_{yy} \end{matrix}) = {(1 - \frac{τ}{n} (\begin{matrix} w_{o}^{(2, 0)} & w_{o}^{(1, 1)} \\ w_{o}^{(1, 1)} & w_{o}^{(0, 2)} \end{matrix}))}^{- 1}$ $by$ $\hat{β} = \hat{R} β {\hat{R}}^{- 1}$ $where$ $\hat{R} = (\begin{matrix} \cos φ & - \sin φ \\ \sin φ & \cos φ \end{matrix})$
is a rotation matrix, which transforms from the special system in which the x axis and the y axis coincide with the directions of the main curvatures of the original wavefront into the general system. Here,

(113) $φ = \frac{1}{2} arc \tan \frac{2 β_{xy}}{β_{yy} - β_{xx}}$

(114) In equation (54), the matrix

(115) $S (\hat{R}) : = (\begin{matrix} \hat{R} & 0 \\ 0 & 1 \end{matrix})$
is used, and, further, T.sup.(1), T.sup.(2), T.sup.(3), T.sup.(4) in equation (54) are matrix-like functions which assign the matrices

(116) $T^{(1)} (X) = (\begin{matrix} a & c \\ b & d \end{matrix})$ $T^{(2)} (X) = (\begin{matrix} a^{2} & 2 ac & c^{2} \\ ab & ad + bc & cd \\ b^{2} & 2 bd & d^{2} \end{matrix})$ $T^{(3)} (X) = (\begin{matrix} a^{3} & 3 a^{2} c & 3 {ac}^{2} & c^{3} \\ a^{2} b & a (ad + 2 bc) & c (2 ad + bc) & c^{2} d \\ {ab}^{2} & b (2 ad + bc) & d (ad + 2 bc) & {cd}^{2} \\ b^{3} & 3 b^{2} d & 3 {bd}^{2} & d^{3} \end{matrix})$ $T^{(4)} (X) = (\begin{matrix} a^{4} & 4 a^{3} c & 6 a^{2} c^{2} & 4 {ac}^{3} & c^{4} \\ a^{3} b & a^{2} (3 bc + ad) & 3 ac (bc + ad) & c^{2} (bc + 3 ad) & c^{3} d \\ a^{2} b^{2} & 2 ab (bc + ad) & b^{2} c^{2} + 4 abcd + a^{2} d^{2} & 2 cd (bc + ad) & c^{2} d^{2} \\ {ab}^{3} & b^{2} (bc + 3 ad) & 3 bd (bc + ad) & d^{2} (3 bc + ad) & {cd}^{3} \\ b^{4} & 4 b^{3} d & 6 b^{2} d^{2} & 4 {bd}^{3} & d^{4} \end{matrix})$
to a predetermined matrix

(117) 0 $x = (\begin{matrix} a & b \\ c & d \end{matrix})$

(118) For even higher orders, the matrices T.sup.(n) can be defined with n>4. Finally, as expressions for solutions on which the solutions for the propagated wavefronts can be formed by the transformation in equation (54), there are predetermined for the order n=2

(119) ${\hat{\tilde{s}}}_{p} = (\begin{matrix} {\hat{\tilde{w}}}_{p}^{(2, 0)} \\ {\hat{\tilde{w}}}_{p}^{(1, 1)} \\ {\hat{\tilde{w}}}_{p}^{(0, 2)} \end{matrix}) = (\begin{matrix} {\hat{β}}_{11}^{- 1} {\hat{w}}_{o}^{(2, 0)} \\ \begin{matrix} 0 \\ {\hat{β}}_{22}^{- 1} {\hat{w}}_{o}^{(0, 2)} \end{matrix} \end{matrix}) = (\begin{matrix} {\hat{w}}^{(2, 0)} \\ 0 \\ {\hat{w}}_{o}^{(0, 2)} \end{matrix}) - \frac{τ}{n} (\begin{matrix} {\hat{w}}_{o}^{{(2, 0)}^{2}} \\ 0 \\ {\hat{w}}_{o}^{{(0, 2)}^{2}} \end{matrix}),$
for the order n=3

(120) ${\hat{\tilde{e}}}_{p 3} = (\begin{matrix} {\hat{\tilde{w}}}_{p}^{(3, 0)} \\ {\hat{\tilde{w}}}_{p}^{(2, 1)} \\ {\hat{\tilde{w}}}_{p}^{(1, 2)} \\ {\hat{\tilde{w}}}_{p}^{(0, 3)} \end{matrix}) = (\begin{matrix} {\hat{w}}_{o}^{(3, 0)} \\ {\hat{w}}_{o}^{(2, 1)} \\ {\hat{w}}_{o}^{(1, 2)} \\ {\hat{w}}_{o}^{(0, 3)} \end{matrix})$
and for the order n=4

(121) ${\hat{\tilde{e}}}_{p 4} = (\begin{matrix} {\hat{\tilde{w}}}_{p}^{(4, 0)} \\ {\hat{\tilde{w}}}_{p}^{(3, 1)} \\ {\hat{\tilde{w}}}_{p}^{(2, 2)} \\ {\hat{\tilde{w}}}_{p}^{(1, 3)} \\ {\hat{\tilde{w}}}_{p}^{(0, 4)} \end{matrix}) = (\begin{matrix} {\hat{w}}_{o}^{(4, 0)} \\ {\hat{w}}_{o}^{(3, 1)} \\ {\hat{w}}_{o}^{(2, 2)} \\ {\hat{w}}_{o}^{(1, 3)} \\ {\hat{w}}_{o}^{(0, 4)} \end{matrix}) + \frac{τ}{n} (\begin{matrix} 3 ({\hat{β}}_{11} {\hat{w}}_{o}^{{(3, 0)}^{2}} + {\hat{β}}_{22} {\hat{w}}_{o}^{{(2, 1)}^{2}} - {\hat{w}}_{o}^{{(2, 0)}^{4}}) \\ 3 {\hat{w}}_{o}^{(2, 1)} ({\hat{β}}_{11} {\hat{w}}_{o}^{(3, 0)} + {\hat{β}}_{22} {\hat{w}}_{o}^{(1, 2)}) \\ \begin{matrix} {\hat{w}}_{o}^{(1, 2)} ({\hat{β}}_{11} {\hat{w}}_{o}^{(3, 0)} + 2 {\hat{β}}_{22} {\hat{w}}_{o}^{(1, 2)}) + {\hat{w}}_{o}^{(2, 1)} (2 {\hat{β}}_{11} {\hat{w}}_{o}^{(2, 1)} + {\hat{β}}_{22} {\hat{w}}_{o}^{(0, 3)}) - \\ {\hat{w}}_{o}^{{(2, 0)}^{2}} {\hat{w}}_{o}^{{(0, 2)}^{2}} \end{matrix} \\ 3 {\hat{w}}_{o}^{(1, 2)} ({\hat{β}}_{11} {\hat{w}}_{o}^{(2, 1)} + {\hat{β}}_{22} {\hat{w}}_{o}^{(0, 3)}) \\ 3 ({\hat{β}}_{11} {\hat{w}}_{o}^{{(1, 2)}^{2}} + {\hat{β}}_{22} {\hat{w}}_{o}^{{(0, 3)}^{2}} - {\hat{w}}_{o}^{{(0, 2)}^{4}}) \end{matrix})$

(122) In the following, it will be shown how the aberrations of a spectacle lens are considered in the optimization thereof in a preferable way by the wavefronts being described in different coordinate systems that are rotated relative to each other. As described with respect to FIG. 1 and FIG. 2, the coordinate systems are preferably defined by the intersection points of the main ray 10 with the refractive surface 14, 16, by the refractive surface, and by the direction of the main ray 10. In order to describe an incoming wavefront, the refractive surface itself, and the outgoing wavefront for the process of refraction on the refractive surface, preferably three different local Cartesian coordinate systems (x,y,z), (x,y,z), and (x′,y′,z′) are used. The origin of all these coordinate systems preferably coincides with the intersection point of the main ray 10 with the refractive surface. While the systems have the normal direction to the plane of refraction (i.e. the plane in which the incoming and the outgoing main ray are located) as the common axis x=x′=x, the z axis points along the incoming main ray, the z′ axis along the outgoing main ray, and the z axis along the normal of the refractive surface. The orientations of the axis, axis, and axis are preferably selected such that each system is right-handed (cf. FIG. 5)

(123) At the transition between the coordinate systems, all vector quantities v depend on each other via the following relations
v=R(ε)v,v′=R(ε)v (55)
where R designates the rotations about the common x axis and is defined by the three-dimensional rotation matrix

(124) $\begin{matrix} R (.Math.) = (\begin{matrix} 1 & 0 & 0 \\ 0 & \cos .Math. & - \sin .Math. \\ 0 & \sin .Math. & \cos .Math. \end{matrix}) & (56) \end{matrix}$

(125) In case of a rotation of the coordinate system by the angle α about the z axis, the coordinate transformation is described by

(126) $\begin{matrix} \begin{matrix} \tilde{x} = x \cos α - y \sin α \\ \tilde{y} = x \sin α + y \cos α \end{matrix} or (\begin{matrix} \tilde{x} \\ \tilde{y} \end{matrix}) = R (α) (\begin{matrix} x \\ y \end{matrix}) & (57) \end{matrix}$
with the rotation matrix

(127) $\begin{matrix} R (α) = (\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}) & (58) \end{matrix}$

(128) Thus, the wavefront {tilde over (w)} in the rotated coordinate system {tilde over (x)}, {tilde over (y)} is described by
{tilde over (w)}({tilde over (x)},{tilde over (y)})=w(x({tilde over (x)},{tilde over (y)}),y({tilde over (x)},{tilde over (y)})) (59)

(129) If one derives the wavefront {tilde over (w)} according to {tilde over (x)},{tilde over (y)}, one obtains the new coefficients ã.sub.m,k−m relative to the coefficients a.sub.m,k−m.

(130) $\begin{matrix} {\tilde{a}}_{m, k - m} = \frac{\partial^{k}}{\partial {\tilde{x}}^{m} \partial {\tilde{y}}^{k - m}} w (x (\tilde{x}, \tilde{y}), y (\tilde{x}, \tilde{y})) |_{\tilde{x} = 0, \tilde{y} = 0} & (60) \end{matrix}$

(131) In the second order, the aberrations are preferably represented by the vector

(132) $\begin{matrix} s = (\begin{matrix} S_{xx} \\ S_{xy} \\ S_{yy} \end{matrix}) & (61) \end{matrix}$

(133) If the coordinate system is rotated by the angle α, the new aberrations {tilde over (s)} of second order (in the rotated coordinate system ({tilde over (x)},{tilde over (y)})) are calculated via

(134) $\begin{matrix} \tilde{s} = R_{2} (α) s with & (62) \\ R_{2} (α) = (\begin{matrix} \cos^{2} α & - 2 \cos α \sin α & \sin^{2} α \\ \cos α \sin α & \cos^{2} α - \sin^{2} α & - \cos α \sin α \\ \sin^{2} α & 2 \cos α \sin α & \cos^{2} α \end{matrix}) & (63) \end{matrix}$

(135) For higher orders of the aberrations, the dependency of the new coefficients a.sub.m,k−m on the old coefficients a.sub.m,k−m is preferably expressed by

(136) 0 $\begin{matrix} (\begin{matrix} {\tilde{a}}_{00} \\ {\tilde{a}}_{01} \\ {\tilde{a}}_{10} \\ {\tilde{a}}_{02} \\ {\tilde{a}}_{11} \\ {\tilde{a}}_{20} \\ {\tilde{a}}_{03} \\ {\tilde{a}}_{12} \\ {\tilde{a}}_{21} \\ .Math. \end{matrix}) = R_{Pot} (N, α) (\begin{matrix} a_{00} \\ a_{01} \\ a_{10} \\ a_{02} \\ a_{11} \\ a_{20} \\ a_{03} \\ a_{12} \\ a_{21} \\ .Math. \end{matrix}) & (64) \end{matrix}$

(137) The resulting rotation matrix has the block structure, which shows that the coefficients a.sub.m,k−m of the order k only depend on coefficients a.sub.m,k−m of the same order k. The rotation matrix for the first 15 coefficients (N=15) up to the order (k=4) thus reads

(138) $\begin{matrix} R_{Pot} (15, α) = (\begin{matrix} 1 & 0 & .Math. & 0 \\ 0 & R_{1} (α) & .Math. \\ R_{2} (α) \\ .Math. & R_{3} (α) & 0 \\ 0 & .Math. & 0 & R_{4} (α) \end{matrix}) & (65) \end{matrix}$

(139) The matrix elements of the block structures R.sub.k(α) of the first order (k=1) yield the known rotation matrix

(140) $\begin{matrix} R_{1} (α) = R (α) = (\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}) & (66) \end{matrix}$

(141) In the second order (k=2), the rotation matrix reads

(142) $\begin{matrix} R_{2} (α) = (\begin{matrix} \cos^{2} α & - 2 \cos α \sin α & \sin^{2} α \\ \cos α \sin α & \cos^{2} α - \sin^{2} α & - \cos α \sin α \\ \sin^{2} α & 2 \cos α \sin α & \cos^{2} α \end{matrix}) & (67) \end{matrix}$
in the third order (k=3)

(143) $\begin{matrix} R_{3} (α) = (\begin{matrix} \cos^{3} α & - 3 \cos^{2} α \sin α & 3 \cos α \sin^{2} α & \sin^{3} α \\ \cos^{2} α \sin α & \cos^{3} α - 2 \cos {αsin}^{2} α & \sin^{3} α - 2 \cos^{2} α \sin α & \cos α \sin^{2} α \\ \cos α \sin^{2} α & - (\sin^{3} α - 2 \cos^{2} α \sin α) & \cos^{3} α - 2 \cos {αsin}^{2} α & \cos^{2} α \sin α \\ \sin^{3} α & 3 \cos α \sin^{2} α & 3 \cos^{2} α \sin α & \cos^{3} α \end{matrix}) & (68) \end{matrix}$
and in the fourth order (k=4)

(144) $\begin{matrix} R_{4} (α) = (\begin{matrix} \cos^{4} α & - \cos^{3} α \sin α & \cos^{2} α \sin^{2} α & - \cos α \sin^{3} α & \sin^{4} α \\ 4 \cos^{3} α \sin α & \cos^{4} α - 3 \cos^{2} \sin^{2} α & \begin{matrix} 2 (\cos α \sin^{3} α - \\ \cos^{3} α \sin α) \end{matrix} & \begin{matrix} - (\sin^{4} α - \\ 3 \cos^{2} \sin^{2} α) \end{matrix} & - 4 \cos α \sin^{3} α \\ 6 \cos^{2} α \sin^{2} α & \begin{matrix} 2 (\cos^{3} α \sin α - \\ \cos α \sin^{3} α) \end{matrix} & \cos^{4} α - 4 \cos^{2} α + \sin^{4} α & \begin{matrix} 3 (\cos α \sin^{3} α - \\ \cos^{3} α \sin α) \end{matrix} & 6 \cos^{2} α \sin^{2} α \\ 4 \cos α \sin^{3} α & - (\sin^{4} α - 3 \cos^{2} \sin^{2} α) & \begin{matrix} - 2 (\cos α \sin^{3} α - \\ \cos^{3} α \sin α) \end{matrix} & \begin{matrix} \cos^{4} α - \\ 3 \cos^{2} \sin^{2} α \end{matrix} & - 4 \cos^{3} α \sin α \\ \sin^{4} α & \cos α \sin^{3} α & \cos^{2} α \sin^{2} α & \cos^{3} α \sin α & \cos^{4} α \end{matrix}) & (69) \end{matrix}$

(145) The equations (66) to (69) show that the block matrix elements e.sub.i,j(α) of the respective rotation matrix R.sub.k(α) have the symmetry e.sub.i,j(α)=e.sub.k+2−i,k+2−j(−α). With c=cos α, s=sin α, the block matrices can be simplified to read

(146) $\begin{matrix} R_{1} (α) = (\begin{matrix} c^{k} s^{0} & * \\ c^{0} s^{k} & * \end{matrix}) R_{2} (α) = (\begin{matrix} c^{k} s^{0} & * & * \\ {kc}^{1} s^{1} & c^{2} - s^{2} & * \\ c^{0} s^{k} & c^{1} s^{1} & * \end{matrix}) R_{3} (α) = (\begin{matrix} c^{k} s^{0} & * & * & * \\ {kc}^{k - 1} s^{1} & c^{k} - (k - 1) c^{1} s^{k - 1} & * & * \\ {kc}^{1} s^{k - 1} & - s^{k} + (k - 1) c^{k - 1} s^{1} & * & * \\ c^{0} s^{k} & c^{1} s^{k - 1} & c^{k - 1} s^{1} & * \end{matrix}) R_{4} (α) = (\begin{matrix} c^{k} s^{0} & * & * & * & * \\ {kc}^{k - 1} s^{1} & c^{k} - (k - 1) c^{2} s^{2} & * & * & * \\ 2 (k - 1) c^{2} s^{2} & (k - 1) (c^{k - 1} s^{1} - c^{1} s^{k - 1}) & c^{k} - 4 c^{2} s^{2} + s^{k} & * & * \\ {kc}^{1} s^{k - 1} & - s^{k} + (k - 1) c^{2} s^{2} & 2 (c^{k - 1} s^{1} - c^{1} s^{k - 1}) & * & * \\ c^{0} s^{k} & c^{1} s^{k - 1} & c^{2} s^{2} & c^{k - 1} s^{1} & * \end{matrix}) R_{5} (α) = (\begin{matrix} c^{k} s^{0} & * & * & * & * & * \\ {kc}^{k - 1} s^{1} & c^{k} - (k - 1) c^{k - 2} s^{2} & * & * & * & * \\ 2 {kc}^{k - 2} s^{2} & (k - 1) c^{k - 1} s^{1} - 2 (k - 2) c^{2} s^{k - 2} & c^{k} - 6 c^{k - 2} s^{2} + 3 c^{1} s^{k - 1} & * & * & * \\ 2 {kc}^{2} s^{k - 2} & \begin{matrix} - (k - 1) c^{1} s^{k - 1} + \\ 2 (k - 2) c^{k - 2} s^{2}) \end{matrix} & s^{k} - 6 c^{2} s^{k - 2} + 3 c^{k - 1} s^{1} & * & * & * \\ {kc}^{1} s^{k - 1} & - s^{k} + (k - 1) c^{2} s^{k - 2} & - 2 c^{1} s^{k - 1} + 3 c^{k - 2} s^{2} & 2 c^{k - 1} s^{1} - 3 c^{2} s^{k - 2} & * & * \\ c^{0} s^{k} & c^{1} s^{k - 1} & c^{2} s^{k - 2} & c^{k - 2} s^{2} & c^{k - 1} s^{1} & * \end{matrix}) & (70) \end{matrix}$

(147) In a preferred embodiment, the aberrations are described in the form of Zernike polynomials. In this case, the rotation is performed in the space of the Zernike polynomials. The wavefront is preferably spanned by the Zernike polynomials in polar coordinates:

(148) $\begin{matrix} Z_{0, 0} (ρ, φ) = 1 Z_{1, 1} (ρ, φ) = 2 ρ \cos φ Z_{1, - 1} (ρ, φ) = 2 ρ \sin φ Z_{2, 0} (ρ, φ) = \sqrt{3} (2 ρ^{2} - 1) Z_{2, 2} (ρ, φ) = \sqrt{6} ρ^{2} \cos 2 φ Z_{2, - 2} (ρ, φ) = \sqrt{6} ρ^{2} \cos 2 φ .Math. with & (71) \\ W (x, y) = {.Math.}_{k = 0}^{\infty} \underset{m}{.Math.} c_{k, m} Z_{k, m} & (72) \end{matrix}$

(149) The Zernike coefficients corresponding to a wavefront w(x, y) are preferably determined via the integral

(150) $\begin{matrix} c_{k}^{m} = \frac{1}{π r_{0}^{2}} \underset{pupil}{\int^{} \int} Z_{k}^{m} (\frac{x}{r_{0}}, \frac{y}{r_{0}}) w (x, y) ⅆ x ⅆ y & (73) \end{matrix}$
where r:=√{square root over (x.sup.2+y.sup.2)}, x=ρ cos φ, y=ρ sin φ, and r.sub.0 the pupil size.

(151) In the preferred representation by means of Zernike polynomials in polar coordinates, the rotation for the Zernike coefficients is very simple. The vector of Zernike coefficients is transformed by the rotation

(152) $\begin{matrix} (\begin{matrix} {\tilde{a}}_{00} \\ {\tilde{a}}_{01} \\ {\tilde{a}}_{10} \\ {\tilde{a}}_{02} \\ {\tilde{a}}_{11} \\ {\tilde{a}}_{20} \\ {\tilde{a}}_{03} \\ {\tilde{a}}_{12} \\ {\tilde{a}}_{21} \\ .Math. \end{matrix}) = R_{Pot} (N, α) (\begin{matrix} a_{00} \\ a_{01} \\ a_{10} \\ a_{02} \\ a_{11} \\ a_{20} \\ a_{03} \\ a_{12} \\ a_{21} \\ .Math. \end{matrix}) & (74) \end{matrix}$

(153) In a block matrix representation, the rotation matrix is directly based on the elementary rotation matrix of equation (57). For N=15, the rotation matrix has the form:

(154) ##STR00001##

(155) For illustration purposes, every block belonging to the same radial order is framed.

(156) If the wavefront is represented via a series as in equations (70) and (71), a series representation, i.e. a linear combination of the coefficients a.sub.m,k−m results for the integral of equation (72) as well. If the coefficients c.sub.k.sup.m or a.sub.m,k−m are summed as vectors up to a specific order k, a transition matrix T(N) between the Zernike subspace and the Taylor series subspace of the order k can be indicated by

(157) 0 $\begin{matrix} (\begin{matrix} c_{0, 0} \\ c_{1, 1} \\ c_{1, - 1} \\ c_{2, 0} \\ c_{2, 2} \\ c_{2, - 2} \\ c_{3, 1} \\ c_{3, - 1} \\ c_{3, 3} \\ .Math. \end{matrix}) = T (N) (\begin{matrix} E \\ E_{x} \\ E_{y} \\ E_{xx} \\ E_{xy} \\ E_{yy} \\ E_{xxx} \\ E_{xxy} \\ .Math. \\ E_{yy .Math. y} \end{matrix}) = n T (N) (\begin{matrix} a_{00} \\ a_{01} \\ a_{10} \\ a_{02} \\ a_{11} \\ a_{20} \\ a_{03} \\ a_{12} \\ a_{21} \\ .Math. \end{matrix}) & (76) \end{matrix}$
with T(N)=Z(N)D(N), where e.g. for N=9

(158) $\begin{matrix} D (9) = (\begin{matrix} 1 & .Math. & .Math. & 0 \\ 0 & r_{0} & .Math. \\ .Math. & r_{0} \\ r_{0}^{2} \\ r_{0}^{2} \\ r_{0}^{2} \\ r_{0}^{3} \\ .Math. & r_{0}^{3} & .Math. \\ 0 & .Math. & .Math. & 0 & r_{0}^{3} \end{matrix}) & (77) \end{matrix}$
designates a matrix that indicates the correct power of the pupil radius. The basic transformation matrix Z(N) is determined by Zernike expansion of the power series. Preferably, the following representation is provided for the transformation matrix for N=15:

(159) $Z^{- 1} (15) = (\begin{matrix} 1 & 0 & 0 & - \sqrt{3} & 0 & 0 & 0 & 0 & 0 & 0 & * & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 & - 4 \sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & .Math. & .Math. \\ .Math. & 0 & 2 & 0 & 0 & 0 & 0 & - 4 \sqrt{2} & 0 & 0 & 0 & 0 & 0 \\ .Math. & 0 & 4 \sqrt{3} & 2 \sqrt{6} & 0 & 0 & 0 & 0 & 0 & * & 0 & * \\ .Math. & 0 & 0 & 2 \sqrt{6} & 0 & 0 & 0 & 0 & 0 & * & 0 \\ 4 \sqrt{3} & - 2 \sqrt{6} & 0 & 0 & 0 & 0 & 0 & * & 0 & * \\ 0 & 0 & 0 & 36 \sqrt{2} & 0 & 12 \sqrt{2} & 0 & 0 & 0 & 0 \\ .Math. & .Math. & .Math. & 0 & 12 \sqrt{2} & 0 & 12 \sqrt{2} & 0 & 0 & 0 \\ 12 \sqrt{2} & 0 & - 12 \sqrt{2} & 0 & 0 & 0 & 0 & .Math. & .Math. \\ 0 & 36 \sqrt{2} & 0 & - 12 \sqrt{2} & 0 & 0 & 0 & 0 & 0 \\ .Math. & 0 & .Math. & 0 & * & 0 & * & 0 & * \\ .Math. & .Math. & 0 & * & 0 & * & 0 \\ * & 0 & * & 0 & * \\ .Math. & .Math. & .Math. & .Math. & .Math. & .Math. & .Math. & .Math. & .Math. & .Math. & 0 & * & 0 & * & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & * & 0 & * & 0 & * \end{matrix})$

(160) In this equation as well, the blocks belonging to the same radial order are framed for purposes of illustration. It can be seen that non-disappearing elements also exist outside the diagonal blocks. However, they do not influence the rotation matrix R.sub.Pot(N, α).

(161) In order to determine the rotation matrix R.sub.Pot(N, α), R.sub.Zernike(N, α) is transformed to the coefficient system of the power series development with equation (76):

(162) $(\begin{matrix} {\tilde{a}}_{00} \\ {\tilde{a}}_{01} \\ {\tilde{a}}_{10} \\ {\tilde{a}}_{02} \\ {\tilde{a}}_{11} \\ {\tilde{a}}_{20} \\ {\tilde{a}}_{03} \\ {\tilde{a}}_{12} \\ {\tilde{a}}_{21} \\ .Math. \end{matrix}) = T^{- 1} (N) (\begin{matrix} {\tilde{c}}_{0, 0} \\ {\tilde{c}}_{1, 1} \\ {\tilde{c}}_{1, - 1} \\ {\tilde{c}}_{2, 0} \\ {\tilde{c}}_{2, 2} \\ {\tilde{c}}_{2, - 2} \\ {\tilde{c}}_{3, 1} \\ {\tilde{c}}_{3, - 1} \\ {\tilde{c}}_{3, 3} \\ .Math. \end{matrix}) = T^{- 1} (N) R_{Zernike} (N, α) (\begin{matrix} c_{0, 0} \\ c_{1, 1} \\ c_{1, - 1} \\ c_{2, 0} \\ c_{2, 2} \\ c_{2, - 2} \\ c_{3, 1} \\ c_{3, - 1} \\ c_{3, 3} \\ .Math. \end{matrix}) = T^{- 1} (N) R_{Zernike} (N, α) T (N) (\begin{matrix} a_{00} \\ a_{01} \\ a_{10} \\ a_{02} \\ a_{11} \\ a_{20} \\ a_{03} \\ a_{12} \\ a_{21} \\ .Math. \end{matrix})$

(163) From this, it follows that
R.sub.Pot(N,α)=T.sup.−1(N)R.sub.Zernike(N,α)T(N) (78)
with a block structure of the form

(164) $\begin{matrix} R_{Pot} (15) = (\begin{matrix} 1 & 0 & .Math. & 0 \\ 0 & R_{1} (α) & .Math. \\ R_{2} (α) \\ .Math. & R_{3} (α) & 0 \\ 0 & .Math. & 0 & R_{4} (α) \end{matrix}) & (79) \end{matrix}$
wherein the block matrices are identical with those of equation (70).

(165) FIG. 6 illustrates an exemplary method for individually optimizing a spectacle lens taking higher-order aberrations (HOA) of both the eye and the spectacle lens into consideration. In a step ST12, not only the local aberrations of 2.sup.nd order (S′.sub.xx, S′.sub.xy, S′.sub.yy) but also the aberrations of a higher order (K′.sub.xxx, K′.sub.xxy, K′.sub.xyy etc.) at the vertex sphere are calculated on the basis of wavefront tracing (ST10).

(166) From these, from the local aberrations, the values for sphere, cylinder, and cylinder axis (sph, zyl, A) of the spectacle lens are calculated with the help of Zernike polynomials and/or other suitable metrics, preferably taking the pupil diameter or pupil radius into consideration. Preferably, in a step S14, first of all Zernike coefficients (c.sub.2.sup.0, c.sub.2.sup.2, c.sub.2.sup.−2, . . . ) are determined. Since now also the higher-order local aberrations are known, it is possible to calculate the ideal sph, zyl, A values of the spectacle lens for a finite pupil opening, which preferably correspond to the above-described transformed values. Both the connection between the local aberrations (S′.sub.xx, S′.sub.xy, S′.sub.yy, K′.sub.xxx, K′.sub.xxy, K′.sub.xyy, . . . ) and the Zernike coefficients (c.sub.2.sup.0, c.sub.2.sup.2, c.sub.2.sup.−2, . . . ), as it is particularly referred to in step ST14, and the connection between the Zernike coefficients (c.sub.2.sup.0, c.sub.2.sup.2, c.sub.2.sup.−2, . . . ) and the values for sphere (Sph), cylinder (Zyl bzw. Cyl), and cylinder axis (A or α) are provided as functional connections c.sub.2.sup.0, c.sub.2.sup.2, c.sub.2.sup.−2, . . . )=f(r, ′.sub.xx, S′.sub.xy, S′.sub.yy, K′.sub.xxx, K′.sub.xxy, K′.sub.xyy, . . . ) and Sph, Zyl, A=f(r, c.sub.2.sup.0, c.sub.2.sup.2, c.sub.2.sup.−2, . . . ) in a step ST18, particularly taking the pupil radius r into consideration.

(167) Now, it is preferred that the pupil size r be specified to be variable for every visual point. It is particularly preferred that the pupil size be specified as a function of the object distance, which in turn represents a function of the visual point. This can be based e.g. on the near reflex, so that with near objects the assumed pupil diameter decreases.

(168) Preferably, in the refraction determination (ST20), not only the values for sphere, cylinder, and cylinder axis, particularly for distance and near vision, are determined subjectively, but additionally the higher-order aberrations (c.sub.2.sup.0, c.sub.2.sup.2, c.sub.2.sup.−2, . . . ) are determined with an aberrometer. In a step ST22, the subjective and objective refraction data are combined particularly considering object distance, direction of sight, and pupil diameter. Thus, it is possible to calculate ideal (transformed) prescription values (sph, zyl, A) particularly for different pupil diameters depending on the visual point with suitable metrics. It is particularly preferred that the ideal prescriptions be calculated once and then be deposited as a function of the object distance. Moreover, it is preferred that e.g. with the aberrometer also the individual pupil diameter be determined under photopic (small pupil) and mesopic (large pupil) conditions. Otherwise, standard values from literature have to be used. Subsequently, the spherocylindrical values of the spectacle lens (SL) can be combined with those of the eye (ST24) in a known way (combination SL/eye K: K(Ref,Ast)=SL(Sph,Zyl,Axis)−eye(Sph,Zyl,axis). The target function (ST26), in which particularly the target values S(Ref, Ast) provided in a step ST28 are taken into account, preferably remains unchanged. The differences between the combination values K and the target values S determined in step ST24 are particularly taken into account therein: K(Ref,Ast)−S(Ref,Ast).

LIST OF REFERENCE NUMERALS

(169) 10 main ray 12 eye 14 first refractive surface (front surface of the spectacle lens) 16 second refractive surface (back surface of the spectacle lens) 18 original wavefront 20 propagated wavefront 22 image point 24 neighboring ray ST2, ST4 propagation (and optionally rotation)

Method for calculating and optimizing an eyeglass lens taking into consideration higher-order imaging errors

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G02C7/027

PHYSICS

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G02C2202/22

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G02C7/063

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G02C7/028

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G02C7/02

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International classification

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G02C7/02

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G02C7/06

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