Method, computer program product and device for accelerated wavefront calculation through a complex optical system

Abstract

Computer-implemented method for simulating an optical system by means of wavefront calculation, wherein the optical system is a complex optical system whose effect extends beyond a single refraction, propagation, or reflection, comprising the steps: - computer-implemented setup of at least one wavefront transfer function for the optical system, wherein the wavefront transfer function is designed to assign a corresponding outgoing wavefront to each incident wavefront, taking into account aberrations of an order greater than the order of a defocus; and - Computer-implemented evaluation of at least one wavefront transfer function for at least one wavefront incident on the optical system.

Description

[0001] The invention relates to a method, a computer program product, and a device for simulating an optical system by means of wavefront computation. In particular, the invention relates to a method, a computer program product, and a device for calculating and / or optimizing, as well as manufacturing, a spectacle lens.

[0002] For the production and optimization of spectacle lenses, especially custom lenses, each lens is manufactured to achieve the best possible correction of the refractive error of the wearer's eye for every desired viewing direction or object point. Generally, a lens is considered fully corrective for a given viewing direction when the values ​​for sphere, cylinder, and axis of the wavefront as it passes the vertex sphere match the values ​​for sphere, cylinder, and axis prescribed for the refractive eye. When determining the refraction for a wearer's eye, dioptric values ​​(especially sphere, cylinder, and axis – i.e., spherocylindrical deviations) are measured for a distant (usually infinity) distance and, if necessary (for multifocal lenses), for intermediate distances.For progressive lenses, an addition or a complete near refraction for near distances (e.g., according to DIN 58208) is determined. With modern lenses, object distances other than the standard, which were used in the refraction determination, can also be specified. This establishes the prescription (in particular, sphere, cylinder, axis, and, if applicable, addition or near refraction), which is then sent to a lens manufacturer. Knowledge of a specific or individual anatomy of the respective eye or the actual refractive powers of the refractive error in a given case is not required.

[0003] Complete correction for all directions of vision simultaneously is not normally possible. Therefore, spectacle lenses are manufactured in such a way that they provide good correction of refractive errors and only minimal aberrations, especially in the main areas of use, particularly in the central viewing zones, while allowing for greater aberrations in the peripheral areas.

[0004] To manufacture a spectacle lens in this way, the lens surfaces, or at least one of them, are first calculated to achieve the desired distribution of unavoidable aberrations. This calculation and optimization is typically performed using an iterative variational method by minimizing an objective function. The objective function used is specifically a function F with the following functional relationship to the spherical power S, the magnitude of the cylindrical power Z, and the axis of the cylinder α (also known as the "SZA" combination): F=∑i=1m[gi,SΔ(SΔ,i−SΔ,i,Soll)2+gi,ZΔ(ZΔ,i−ZΔ,i,Soll)2+…]

[0005] In this process, the objective function F at the evaluation points i of the spectacle lens takes into account at least the actual refractive deficits of the spherical power S. Δ,i and the cylindrical action Z Δ,ias well as target values ​​for the refractive deficits of the spherical power S Δ,i,Soll and the cylindrical action Z Δ,i,Soll taken into account.

[0006] Already in DE 103 13 275 A1, it was recognized that it is advantageous to specify the target values ​​not as absolute values ​​of the properties to be optimized, but as their deviations from the regulation, i.e., as the required local mismatch. Thus, the "actual" values ​​of the properties to be optimized in the objective function are not absolute values ​​of these optical properties, but rather the deviations from the regulation. This has the advantage that the target values ​​are independent of the regulation (especially Sph). v ,cylinder v ,Axis v ,P v ,B v ) and the target specifications do not need to be changed for each individual regulation.

[0007] The respective refractive errors at the respective assessment points are preferably weighted using weighting factors g. i,SΔ or g i,ZΔ This is taken into account. The target values ​​for the refractive deficits of the spherical power S are defined here. Δ,i,Soll and / or the cylindrical action Z Δ,i,Soll especially together with the weighting factors g i,SΔ or g i,ZΔ the so-called spectacle lens design. In addition, further residuals, especially further quantities to be optimized, such as coma and / or spherical aberration and / or prism and / or magnification and / or anamorphic distortion, etc., can also be taken into account, which is indicated in particular by the expression “+...” in the above-mentioned formula for the objective function F.

[0008] In some cases, it can significantly improve, especially in the individual fitting of a spectacle lens, if, when optimizing the spectacle lens, not only aberrations up to the second order (sphere, amount of astigmatism and axis position) but also higher order aberrations (e.g. coma, trile leaf error, spherical aberration, etc.) are taken into account.

[0009] It is known from the prior art to determine the shape of a wavefront for optical elements, and in particular for spectacle lenses, which are bounded by at least two refracting interfaces. This can be done, for example, by numerically calculating a sufficient number of neighboring rays, followed by fitting the wavefront using Zernike polynomials. Another approach is based on a local wavefront calculation during refraction (see WO 2008 / 089999 A1). Here, only a single ray (the principal ray) is calculated per viewing point, along with the derivatives of the wavefront's sag with respect to the transverse (perpendicular to the principal ray) coordinates. These derivatives can be calculated up to a certain order, with the second derivatives representing the local curvature properties of the wavefront (such as...).refractive power, astigmatism) describe and the higher derivatives are related to higher-order imaging errors.

[0010] When calculating the path of light through a spectacle lens, the local derivatives of the wavefronts are calculated at a suitable position along the beam path in order to compare them with desired values ​​derived from the refraction of the spectacle wearer. The vertex sphere or, for example, the principal plane of the eye in the corresponding gaze direction can be used as such a position for wavefront evaluation. Alternatively or additionally, the entrance pupil (EP), the exit pupil (AP), and / or, preferably, the plane after refraction at the posterior surface L2 of the lens can be used for wavefront evaluation. It is assumed that a spherical wavefront originates from the object point and propagates to the first lens surface. There, the wavefront is refracted and then propagates to the second lens surface, where it is refracted again.The final propagation then takes place from the second interface to the vertex sphere (or the principal plane of the eye), where the wavefront is compared with predetermined values ​​for correcting the refraction of the eye of the spectacle wearer.

[0011] To perform this comparison based on the refractive data of the respective eye, an established model of the refractive eye is used to analyze the wavefront at the vertex sphere. In this model, a refractive error (refractive deficit) is superimposed on a normally sighted basic eye. This approach has proven particularly useful because it does not require in-depth knowledge of the anatomy or optics of the respective eye (e.g., distribution of refractive powers, axial length, axial ametropia, and / or refractive ametropia). Detailed descriptions of this model, consisting of the spectacle lens and refractive deficit, can be found, for example, in Dr. Roland Enders, "Die Optik des Auges und der Sehhilfen" (Optics of the Eye and Visual Aids), Optische Fachveröffentlichung GmbH, Heidelberg, 1995, pages 25 ff., and in Diepes and Blendowske, "Optik und Technik der Brille" (Optics and Technology of Spectacles), Optische Fachveröffentlichung GmbH, Heidelberg, 2002, pages 47 ff.The correction model described therein by Reiner is used in particular as a proven model.

[0012] Refractive error is defined as the deficiency or excess of refractive power in the optical system of the refractive eye compared to a normal eye of the same length (remaining eye). The refractive power of the refractive error is, in particular, approximately equal to the far point refraction with a negative sign. For complete correction of the refractive error, the spectacle lens and the refractive error together form a telescopic system (afocal system). The remaining eye (the refractive eye without the added refractive error) is considered to have normal vision. A spectacle lens is therefore considered fully corrective for distance vision if its image-side focal point coincides with the far point of the refractive eye and thus also with the object-side focal point of the refractive error.

[0013] In the publication DE 10 2017 007 975 A1 or WO 2018 / 138140 A2, a method and a device are described which allow the calculation or optimization of a spectacle lens to be improved, whereby the spectacle lens is very effectively adapted to the individual requirements of the spectacle wearer using simple measurements of individual optical and ocular anatomical data.

[0014] Furthermore, according to the state of the art, spectacle lens optimization can be achieved by minimizing an objective function that assesses the wavefront aberration within an eye model. The wavefront aberration arises from comparing a reference wavefront with a wavefront determined by wavefront calculation through the refractive components of an eye model. In this process, each wavefront must be alternately refracted and propagated. This approach is based on the publications by G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, D. Uttenweiler: “Derivation of the refractive equations for higher order aberrations of local wavefronts by oblique incidence”, J. Opt. Soc. Am. A 27, 218-237 (2010) and G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, D. Uttenweiler: “Derivation of the propagation equations for higher order aberrations of local wavefronts”, J. Opt. Soc. Am.A 28, 2442-2458 (2011), and is described in particular in the patent specifications DE 10 2012 000 390 A1, US 9,910,294 B2, DE 10 2011 101 923 A1 and WO 2008 / 089999 A1. The above-mentioned publications are expressly referenced herein and their contents are fully incorporated into the present description.

[0015] It is also known that, as an alternative to wavefront calculations, wavefront aberrations can be assessed using a beam of light. This requires calculating the path of each individual light ray of the beam through the lens and the eye model, which takes longer to compute compared to wavefront calculations.

[0016] Even though powerful methods are already available for the individual calculation steps of wavefront refraction and propagation, the sheer number of refractions and propagations results in unacceptably long computation times, especially when new wavefronts have to be repeatedly calculated through the same eye model. Multiple wavefront calculations are necessary, for example, in the case of spectacle lens optimization due to the iterative optimization steps. Furthermore, changing viewing directions and / or object distances can occur during spectacle lens optimization, which can also necessitate multiple wavefront calculations.

[0017] To optimize an optical system via simulation, the passage of light through the system must be physically described and then evaluated according to suitable criteria. Optimization is understood as a targeted modification of the system such that the passing light comes as close as possible to a defined target with regard to the criteria. For example, the passing light can be described by a scalar or a vector electromagnetic field. Furthermore, it is possible to define the surfaces of equal phase within this field as wavefronts and use these as a basis for evaluation.

[0018] If interference effects due to the finite wavelength of light are neglected, then the passage of light can be described using geometric optics instead of wave optics. According to the current state of the art, geometric optics offers methods for calculating ray tracing. A simple condition under which ray tracing can be performed is paraxiality. A particularly simple form of this is Gaussian optics, in which all rays remain within a single meridian. The description then refers only to the optics within that one meridian (in rotationally symmetric systems where each meridian is equivalent, the entire system can then be described vicariously within that single meridian). A more general form suitable for two independent meridians is linear optics.In both Gaussian and linear optics, every system is defined by having an entrance plane perpendicular to the propagation of the principal ray and an exit plane parallel to it. The optical system is characterized by how the coordinates and directions of an exiting ray depend on the corresponding parameters of the entering ray, with this dependence being linear in the paraxial region.

[0019] As an alternative to ray tracing, there is also a conceptually equivalent method using wavefronts (wave tracing), which in geometric optics are defined as the spatial surfaces through which each ray of a beam passes perpendicularly. Geometric optics offers equivalent reformulations of wavefronts. For example, Fourier optics uses planes perpendicular to the propagation of the principal ray instead of spatial surfaces. The optical path difference (OPD) is described as a function of the lateral coordinates of this plane, separating each point on the plane from a predefined reference point.

[0020] Wavefronts, or their equivalent OPD functions, can be described in various ways. For example, these functions can be described by freeform surfaces, such as B-splines. If a portion of the wavefront bounded by a pupil is relevant, wavefronts are described by composing a sufficiently large number of Zernike polynomials (see, e.g., Born and Wolf: "Principles of Optics," Oxford, Pergamon, 1970), weighted with the corresponding Zernike coefficients. A common local description for the vicinity of a principal beam, in turn, consists of Taylor series expansions, i.e., local derivatives of the wavefront with respect to the lateral coordinates, which represent weights with which powers of the coordinates are linearly combined to form the wavefront.

[0021] In particular, it is state of the art to optimize spectacle lenses by calculating wavefronts through the lenses using wave tracing and then comparing them with specific parameters to evaluate an objective function (see, for example, WO 2008 / 089999 A1). This objective function can be designed so that minimizing it leads to an improvement in the spectacle lens.

[0022] In one embodiment of the prior art, only second-order wavefront aberrations (LOA, Lower Order Aberration) are used for the wavefronts. In another embodiment of the prior art, both second-order wavefront aberrations (LOA) and third-order or higher-order wavefront aberrations (HOA, Higher Order Aberration) are used. For calculating wavefronts affected by such HOAs, the prior art includes methods for both refraction (see G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, D. Uttenweiler: “Derivation of the refractive equations for higher order aberrations of local wavefronts by oblique incidence”, J. Opt. Soc. Am. A 27, 218-237 (2010), and WO 2008 / 089999 A1) and propagation (see G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, D.Uttenweiler: “Derivation of the propagation equations for higher order aberrations of local wavefronts”, J. Opt. Soc. Am. A 28, 2442-2458 (2011), and DE 10 2011 101 923 A1). The major technical advantage of these methods is that they are based on analytical formulas and therefore do not rely on the computationally intensive numerical methods of ray tracing.

[0023] Furthermore, the prior art discloses methods according to which wavefronts are calculated only through the spectacle lens itself for the optimization of the spectacle lens (see DE 10 2011 101 923 A1), as well as methods according to which wavefronts are calculated both through the lens to be optimized and through an eye model (see DE 10 2012 000 390 A1, US 9,910,294 B2, DE 10 2017 007 975 B4).

[0024] However, current technology does not offer a method for analytically calculating light in the form of a wavefront, including the HOA, through a complex optical system in general, so that for such a system one has to resort to computationally intensive ray-tracing methods or to repeatedly applying wave-tracing to the individual components, which is also computationally intensive.

[0025] It is therefore an object of the present invention to provide a method for accelerated wavefront computation through a complex optical system. In particular, it is an object of the present invention to provide an efficient method for calculating and / or optimizing a spectacle lens and for manufacturing a spectacle lens. Furthermore, it is an object of the present invention to provide a corresponding computer program and apparatus. These objects are achieved by the subject matter of the dependent claims. Advantageous embodiments are the subject of the sub-claims.

[0026] A first independent aspect for solving the problem concerns a computer-implemented method for simulating an optical system by means of wavefront calculation, wherein the optical system is in particular a complex optical system whose effect goes beyond a single refraction, a single propagation or a single reflection, comprising the steps: - computer-implemented setup of at least one wavefront transfer function for the optical system, wherein the wavefront transfer function is designed to assign a corresponding outgoing wavefront to each incident wavefront, taking into account aberrations of an order greater than the order of a defocus; and - Computer-implemented evaluation of at least one wavefront transfer function for at least one wavefront incident on the optical system.

[0027] For the purposes of this invention, a “complex optical system” is, in particular, an optical system whose effect on any wavefront or, alternatively, on any beam of light cannot be described by refraction or reflection at a single surface or by a single propagation between two different planes. Rather, a complex optical system comprises several components which, when light or a wave or wavefront passes through, result in at least two refractions, and / or at least two propagations, and / or at least one refraction and at least one propagation.

[0028] In the context of the invention, the term "setting up a function" encompasses determining and / or defining the function. Setting up a function is performed automatically or by computer implementation using a processor or computer. In particular, setting up the at least one wavefront transfer function includes assigning coefficients to the at least one wavefront transfer function. Setting up at least one wavefront transfer function can involve setting up a single wavefront transfer function or setting up several, in particular two or three, wavefront transfer functions. For example, a plurality (i.e., two, three, four, five, etc.) of wavefront transfer functions can be set up for a plurality of different configurations of the optical system. Different configurations of an optical system can generally have different parameters that characterize the optical system.describe, encompass. In particular, different configurations of an optical system can also include different positions and / or orientations of components of the optical system relative to each other and / or different positions and / or orientations of the optical system (e.g., relative to another optical system).

[0029] In particular, all established wavefront transfer functions can therefore differ from each other.

[0030] The term "evaluating" the at least one wavefront transfer function for an incident wavefront means, in particular, that the at least one wavefront transfer function is applied to an incident wavefront. In other words, by "evaluating" the at least one wavefront transfer function, a corresponding outgoing wavefront is assigned to a given wavefront incident on the optical system.

[0031] The at least one wavefront transfer function is designed or defined to assign a corresponding outgoing wavefront to each wavefront incident on the optical system, taking into account aberrations with an order greater than the order of a defocus. In particular, the at least one wavefront transfer function is intended to generally describe a change in wavefronts incident on the optical system, taking into account aberrations with an order greater than the order corresponding to a defocus. In particular, the at least one wavefront transfer function is designed and / or defined to transform the at least one incident wavefront into a corresponding outgoing wavefront. In particular, the wavefront transfer function is intended to assign a corresponding outgoing wavefront to each wavefront incident on the optical system.to assign the outgoing wavefront. The wavefront transfer function, at least one of which, also takes into account imaging errors or a component that extends beyond defocus and astigmatism (corresponding to the "order" of a defocus), i.e., depending on the representation, specifically beyond a sphere and a cylinder (including axis position). In a description of imaging errors according to Zernike (i.e., using Zernike polynomials) and / or in a description of imaging errors using a Taylor expansion, this means, in particular, that the wavefront transfer function, at least one of which takes into account imaging errors with an order greater than two, i.e., higher-order aberrations (HOA).In particular, at least one wavefront transfer function thus takes into account not only second-order aberrations (sphere, magnitude of astigmatism and axis position), but also higher-order aberrations, such as spherical aberration, coma, three-leaf error, etc.

[0032] The invention therefore relates in particular to a method for simulating an optical system wherein a wavefront calculation method is used which comprises the following steps: - Setting up a wavefront transfer function for the optical system; - Evaluating the wavefront transfer function for a first wavefront incident on the optical system; - Evaluating the wavefront transfer function for a second wavefront incident on the optical system, which is different from the first wavefront; where the wavefront transfer function - transforms the incoming wavefront into an outgoing wavefront and - generally describes the change in incident wavefronts by the optical system and - taking into account at least one component whose order goes beyond the order of a defocus component and - where the effect of the optical system goes beyond that of a single refraction, a single propagation or a single reflection.

[0033] According to the invention, it has been recognized that a conventional stepwise wavefront calculation unnecessarily repeats the same intermediate steps. In all cases where only the wavefront occurring at the last surface of an optical system (e.g., an eye model) is of interest, the explicit calculation of all wavefronts at the intermediate surfaces is superfluous and can be omitted.

[0034] The present invention therefore breaks with the prior art and proposes instead a method by which an incident wavefront can be transformed into an outgoing wavefront in a single operation. In particular, corresponding transfer functions can be established or defined for any optical system consisting of any number of refracting surfaces and propagations if the parameters of such a complex system are given. The complex optical system, such as a gradient index object (GRIN object), need not even be reducible to a finite number of pure refractions and / or propagations. Instead, a specification of the optical system by the ray transfer function, which uniquely assigns an outgoing ray to each incident ray, is sufficient.The steps for constructing this transfer function may be complex, as they only need to be performed once. Crucially, the computational effort required to evaluate the transfer function must be less than the computational effort required to evaluate the individual steps of propagation and refraction in a complex optical system.

[0035] The current state of the art does not offer a method for analytically calculating light in the form of a wavefront, including the HOA, passing through a complex optical system in general. Therefore, for such a system, one must resort to computationally intensive ray-tracing methods or to time-consuming repeated applications of wave-tracing calculations to the individual components. The present invention, however, solves this problem using an analytical and very computationally efficient method. The technical advantage of the invention lies particularly in the fact that a large number of different wavefronts can be calculated very efficiently through a given complex optical system by exploiting the fact that the optical system itself does not change and the computational step for establishing a wavefront transfer function only needs to be performed once.

[0036] In a preferred embodiment, the invention relates to a method, in particular a computer-implemented method, for optimizing a complete optical system, wherein the optical system represents a second subsystem of the complete optical system and the complete optical system additionally comprises a first subsystem. The first subsystem and / or the second subsystem can be varied, in particular during the optimization process. For the purposes of this invention, "optimization" refers in particular to the calculation and / or optimization of a (to be manufactured) spectacle lens for correcting a refractive error of a spectacle wearer.

[0037] In another preferred embodiment, the first subsystem is a spectacle lens and the second subsystem is a model eye.

[0038] A "model eye" within the meaning of the present invention is, in particular, a data set containing eye model parameters that describe a real eye. Preferably, the model eye also includes provided (especially measured) individual refraction data of a spectacle wearer. The eye model parameters can, for example, be assumed or determined at least partially based on standard or average values. The eye model parameters can also be measured, at least partially.

[0039] In particular, the procedure can include defining an individual eye model, which specifies at least certain requirements regarding the geometric and optical properties of a model eye. For example, an individual eye model can define at least the shape (topography) and / or function of the cornea, especially the anterior corneal surface, and the corneal-lens distance. CL(This distance between the cornea and an anterior surface of the lens in the model eye is also called anterior chamber depth), parameters of the lens in the model eye, which in particular determine at least partially the optical effect of the lens in the model eye, and a lens-retinal distance d LR(This distance between the lens, particularly the posterior surface of the lens, and the retina of the model eye is also referred to as the vitreous length) is determined in a specific manner, in particular such that the model eye exhibits the provided individual refraction data, i.e., that a wavefront emanating from a point on the retina of the model eye corresponds (to a desired accuracy) to the wavefront determined (e.g., measured or otherwise) for the real eye of the spectacle wearer. The parameters of the lens of the model eye (lens parameters) can be, for example, either geometric parameters (shape of the lens surfaces and their spacing) and preferably material parameters (e.g., refractive indices of the individual components of the model eye) that are defined so completely that they at least partially determine the optical effect of the lens.Alternatively or additionally, lens parameters can be defined for the model eye that directly describe the optical effect of the model eye's lens. Regarding the cornea, the shape of the anterior corneal surface is usually measured; however, the effect of the cornea as a whole (without differentiating between the anterior and posterior surfaces) can also be specified. Possibly, the posterior corneal surface and / or corneal thickness can also be specified. Furthermore, if individual intraocular lens data is known or provided, the parameters of the model eye's lens can also be defined based on this intraocular lens data.

[0040] In the simplest case of an eye model, the refraction of the eye is determined by the optical system consisting of the anterior corneal surface, the lens, and the retina. In this simple model, the refraction of light at the anterior corneal surface and the refractive power of the lens (preferably including spherical, astigmatic, and higher-order aberrations), together with their positioning relative to the retina, define the refraction of the model eye. The individual parameters of the model eye are determined based on individual measurements of the wearer's eye, standard values, and / or provided individual refraction data. In particular, some of the parameters (e.g., the topography of the anterior corneal surface, the anterior chamber depth, and / or at least the curvature of a lens surface, etc.) can be provided directly as individual measurements.Other values ​​can also be taken from standard models of a human eye, particularly if they are parameters whose individual measurement is very complex. However, it is not necessary to specify all (geometric) parameters of the model eye from individual measurements or standard models. Instead, one or more (free) parameters can be individually adjusted by calculation, taking the specified parameters into account, so that the resulting model eye exhibits the provided individual refraction data. Depending on the number of parameters contained in the provided individual refraction data, a corresponding number of (free) parameters of the eye model can be individually adjusted (fitted).

[0041] For the calculation or optimization of the spectacle lens, a first surface and a second surface of the spectacle lens can be specified, particularly as starting surfaces with a predetermined (individual) position relative to the model eye. In a preferred embodiment, only one of the two surfaces is optimized. Preferably, this is the back surface of the spectacle lens. Preferably, a corresponding starting surface is specified for both the front and back surfaces of the spectacle lens. In a preferred embodiment, however, only one surface is iteratively modified or optimized during the optimization process. The other surface of the spectacle lens can, for example, be a simple spherical or rotationally symmetric aspherical surface. However, it is also possible to optimize both surfaces.

[0042] Starting from the two given surfaces, the method for calculating or optimizing can include determining the path of a principal ray through at least one viewing point (i) of at least one surface of the spectacle lens to be calculated or optimized into the model eye. The principal ray describes the geometric path of the ray starting from an object point through the two spectacle lens surfaces, the anterior corneal surface, and the lens of the model eye, preferably to the retina of the model eye.

[0043] Furthermore, the calculation or optimization procedure can include evaluating an aberration of a wavefront resulting from a spherical wavefront incident on the first surface of the lens along the principal ray at a reference surface within the model eye, compared to a wavefront converging at a point on the retina of the eye model (reference light). In particular, a spherical wavefront (w0) incident on the first surface (front surface) of the lens along the principal ray can be specified. This spherical wavefront describes the light emanating from an object point (object light). The curvature of the spherical wavefront upon impact with the first surface of the lens corresponds to the inverse of the object distance.Preferably, the method thus comprises defining an object distance model that assigns an object distance to each viewing direction or viewing point of the at least one surface of the spectacle lens to be optimized. This preferably describes the individual usage situation in which the spectacle lens to be manufactured is to be used.

[0044] The wavefront striking the spectacle lens is refracted for the first time, preferably at the lens's front surface. Subsequently, the wavefront propagates along the principal ray within the lens from the front surface to the back surface, where it is refracted a second time. Preferably, the wavefront transmitted through the lens then propagates further along the principal ray to the anterior surface of the cornea, where it is preferably refracted again. Preferably, after further propagation within the eye to the lens, the wavefront is refracted there as well, and finally, preferably, propagates to the retina. Depending on the optical properties of the individual optical elements (lens surfaces, corneal surface, lens), each refraction and each propagation process also leads to a deformation of the wavefront.

[0045] To achieve a precise image of the object point onto a corresponding point on the retina, the wavefront should ideally exit the lens as a converging spherical wavefront whose curvature is exactly the inverse of the distance to the retina. A comparison of the wavefront emanating from the object point with a wavefront (ideally a perfect image) converging at a point on the retina (reference light) thus allows for the evaluation of a misalignment. This comparison, and therefore the evaluation of the object light wavefront in the individual eye model, can be performed at various points along the path of the principal ray, particularly between the second surface of the optimizing lens and the retina. Specifically, the evaluation area can therefore be located at different positions, especially between the second surface of the lens and the retina.Accordingly, the refraction and propagation of the light exiting the object point is preferably calculated for each viewing point in the individual eye model. The evaluation surface can refer either to the actual beam path or to a virtual beam path, such as that used for constructing the exit pupil (EP). In the case of the virtual beam path, the light must be propagated back after refraction through the posterior surface of the lens to a desired plane (preferably to the plane of the EP), whereby the refractive index used must correspond to the medium of the vitreous humor and not to the lens. If the evaluation surface is located behind the lens, or...If the evaluation surface is provided after refraction at the posterior surface of the lens of the model eye, or if the evaluation surface is reached by backpropagation along a virtual beam path (as in the case of AP), then the resulting wavefront of the object light can preferably be easily compared with a spherical wavefront of the reference light. For this purpose, the method preferably comprises specifying a spherical wavefront incident on the first surface of the spectacle lens, determining a wavefront in the at least one eye resulting from the spherical wavefront through the action of at least the first and second surfaces of the spectacle lens, the anterior corneal surface, and the lens of the model eye, and evaluating the aberration of the resulting wavefront compared to a spherical wavefront converging on the retina.If, on the other hand, an evaluation area is to be provided within the lens or between the lens of the model eye and the spectacle lens to be calculated or optimized, a reverse propagation from a point on the retina through the individual components of the model eye to the evaluation area is simply simulated as the reference light in order to compare the object light with the reference light there.

[0046] However, as mentioned at the outset, a complete correction of the eye's refraction simultaneously for all gaze directions, i.e., for all viewing points of the lens surface to be optimized, is generally not possible. Therefore, depending on the gaze direction, a deliberate misadjustment of the lens is preferably introduced. This misadjustment is minimal in the most frequently used areas of the lens (e.g., central viewing points) and somewhat greater in less frequently used areas (e.g., peripheral viewing points), depending on the application. This approach is already known in principle from conventional optimization methods.

[0047] In a further preferred embodiment, the at least one wavefront incident on the optical system is determined on the basis of a predetermined test wavefront passing through the first subsystem.

[0048] In a further preferred embodiment, the method further comprises the step: - Evaluating the overall optical system based on the result of evaluating the at least one wavefront transfer function for the at least one wavefront incident on the optical system, whereby the overall optical system is evaluated by varying the first subsystem until the evaluation meets a predetermined condition.

[0049] In this description, "evaluating" a system refers specifically to evaluating the system using a functional or objective function. In particular, evaluating a system can include minimizing a functional or objective function, for example, using an iterative variational method. In the case of spectacle lens optimization, the objective function could be, for example, the function mentioned at the beginning. F=∑i=1m[gi,SΔ(SΔ,i−SΔ,i,Soll)2+gi,ZΔ(ZΔ,i−ZΔ,i,Soll)2+…] can be used. Since minimizing an objective function and the iterative variation methods used for this purpose are well known to those skilled in the art, they will not be discussed in detail within the scope of this invention. The predetermined condition for the evaluation can, in particular, be a predetermined threshold value of the objective function. If the objective function falls below this threshold value for a specific variation or configuration of the first subsystem, the goal, namely to optimize the overall system, is achieved. Further variation and / or further evaluation is then no longer necessary.

[0050] In particular, the variation of the first subsystem includes a change in at least one refracting surface and / or at least one distance between refracting surfaces of the first subsystem, and / or a tilting and / or displacement of the first subsystem relative to the second subsystem. If the first subsystem is a spectacle lens, a variation of the first subsystem or spectacle lens can, for example, include a change in the shape of at least one spectacle lens surface (front and / or back surface).

[0051] In particular, according to a preferred embodiment, the invention relates to a method for optimizing an overall optical system, wherein the optical system represents a second subsystem of the overall optical system and the overall optical system additionally comprises a first subsystem which can be varied during the optimization process, and wherein the wavefront incident on the optical system is determined by passing a test wavefront through the first subsystem and the overall optical system is evaluated based on the result of the evaluation of the wavefront transfer function for the wavefront incident on the optical system and wherein the first subsystem is varied and the overall optical system is evaluated until the evaluation meets a predetermined condition.

[0052] If the first subsystem is a spectacle lens, then, in order to optimize the spectacle lens, at least one surface of the spectacle lens to be calculated or optimized can be iteratively varied until an aberration of the resulting wavefront corresponds to a predetermined target aberration, i.e., in particular, until it deviates from the wavefront of the reference light (e.g., a spherical wavefront whose center of curvature lies on the retina) by predetermined values ​​of the aberration. The wavefront of the reference light is also referred to here as the reference wavefront. Preferably, the method for this includes minimizing an objective function F, in particular analogous to the objective function already described above.

[0053] In a further preferred embodiment, the evaluation of the overall optical system is performed based on the result of evaluating the at least one wavefront transfer function for a first wavefront incident on the optical system and further on the result of evaluating the at least one wavefront transfer function for a second incident wavefront, wherein the first subsystem is in a first position and orientation relative to the second subsystem when the first wavefront is incident, and wherein the first position differs from the second position and / or the first orientation differs from the second orientation. This approach allows, for example, different viewing points to be considered and evaluated.

[0054] In particular, according to a preferred embodiment, the invention can thus relate to a method for optimizing an overall optical system, wherein the evaluation of the overall optical system includes, in addition to the result of evaluating the wavefront transfer function for the wavefront incident on the optical system, the result of an additional evaluation of the wavefront transfer function for a further incident wavefront. wherein the further wavefront incident on the optical system is determined by the passage of the test wavefront through the first subsystem, wherein the first subsystem is in a second position and orientation to the second subsystem which differs from the position and / or the first orientation from the second position or orientation when evaluating the wavefront transfer function for the incident wavefront.

[0055] In particular, according to a further preferred embodiment, the invention relates to a method for optimizing an overall optical system, wherein the optical system represents a second subsystem of the overall optical system and the overall optical system additionally comprises a first subsystem which can be varied during the optimization process. wherein the first wavefront incident on the optical system is determined by passing a test wavefront through the first subsystem, wherein the first subsystem is in a first configuration, and the overall optical system is evaluated based on the result of the evaluation of the wavefront transfer function for the first wavefront incident on the optical system, and The second wavefront incident on the optical system is determined by the passage of the test wavefront through the first subsystem, wherein the first subsystem is in a second configuration defined based on the evaluation, and the overall optical system is evaluated based on the result of the wavefront transfer function evaluation for the second wavefront incident on the optical system. In particular, the method allows for further variations of the first subsystem and evaluations of the overall optical system until the evaluation meets a predefined condition.

[0056] In particular, the invention may relate to a method for optimizing an overall optical system, wherein the optical system represents a second subsystem of the overall optical system and the overall optical system additionally comprises a first subsystem which can be varied during the optimization process, and wherein the first wavefront incident on the optical system is determined by passing a test wavefront through the first subsystem, wherein the first subsystem is in a first position and a first orientation to the second subsystem, and wherein the second wavefront incident on the optical system is determined by the passage of the test wavefront through the first subsystem, wherein the first subsystem is in a second position and a second orientation relative to the second subsystem, wherein the first position and / or the first orientation differs from the second position or orientation, and wherein The evaluation of the overall optical system includes the result of the evaluation of the wavefront transfer function for the first wavefront incident on the optical system and the result of the evaluation of the wavefront transfer function for the second wavefront incident on the optical system.

[0057] In particular, the evaluation of the overall optical system includes a first evaluation based on the result of the evaluation of at least one wavefront transfer function for a first wavefront incident on the optical system, wherein, upon the incident of the first wavefront on the optical system, the first subsystem is in a first configuration, wherein The evaluation of the overall optical system comprises a second evaluation based on the result of the evaluation of the at least one wavefront transfer function for a second wavefront incident on the optical system, wherein, upon the incident of the second wavefront on the optical system, the first subsystem is in a second configuration, which is determined on the basis of the first evaluation, and wherein preferably Based on the second evaluation, one or more further variations of the configuration of the first subsystem are made, and for each of these further configurations of the first subsystem, the overall optical system is evaluated until the evaluation meets a predefined condition, whereby The different configurations of the first subsystem differ in particular in at least one refracting surface and / or at least a distance between refracting surfaces of the first subsystem, and / or in a position and / or orientation of the first subsystem relative to the second subsystem.

[0058] If the first subsystem is a spectacle lens, a configuration of the first subsystem can be characterized, for example, by the shape of at least one lens surface. A variation of the first subsystem or spectacle lens can, for example, include a change in the shape of at least one lens surface.

[0059] In a further preferred embodiment, the evaluation of the overall optical system is carried out based on the result of evaluating a first wavefront transfer function for a first wavefront incident on the optical system and further on the result of evaluating a second wavefront transfer function for a further, in particular second or third, incident wavefront, wherein the first subsystem is in a first position and orientation relative to the second subsystem when the first wavefront is incident, wherein the first subsystem is in a second position and orientation relative to the second subsystem when the further wavefront is incident, wherein the first position differs from the second position and / or the first orientation differs from the second orientation, and wherein the second wavefront transfer function differs from the first wavefront transfer function.

[0060] In particular, the first wavefront transfer function is set up for a first configuration of the second subsystem and the second wavefront transfer function for a second configuration of the second subsystem that differs from the first configuration of the second subsystem, where the first configuration of the second subsystem describes, for example, a first accommodation of the model eye and the second configuration of the second subsystem describes a second accommodation of the eye or model eye.

[0061] In particular, the invention may, in a preferred embodiment, relate to a method for optimizing a spectacle lens, wherein the evaluation of the overall optical system includes, in addition to the result of the evaluation of the wavefront transfer function for the first wavefront incident on the optical system and, if applicable, in addition to the result of the evaluation of the wavefront transfer function for the second wavefront incident on the optical system, the result of the evaluation of a further wavefront transfer function for a third wavefront incident on the optical system, and wherein the third wavefront incident on the optical system is determined by the passage of the test wavefront through the first subsystem, wherein the first subsystem is in a further position and a further orientation relative to the second subsystem, wherein the first position and / or the first orientation differs from the further position or orientation, and wherein the further wavefront transfer function differs from the wavefront transfer function because the optical system has a different configuration than when evaluating the wavefront transfer function for the first wavefront incident on the optical system, and where the change in the configuration of the optical system can reflect accommodation of the eye.

[0062] This embodiment, which takes into account two configurations of the eye, can advantageously be used in particular for the optimization of progressive lenses.

[0063] In a further preferred embodiment, the first subsystem is a spectacle lens and the second subsystem is a model eye, wherein gaze movements of the model eye that cause a change in the position of the point of penetration of the main ray through the spectacle lens surfaces and / or a change in the angles of incidence on a spectacle lens surface are described as a change in the position and / or orientation of the spectacle lens in the coordinate system of the eye. In this way, different points of view can be taken into account, in particular, by means of an eye movement.

[0064] In another preferred embodiment, the optical system is a GRIN system or comprises at least one GRIN element, where GRIN is the abbreviation for "Gradient Index".

[0065] Preferably, both the incident and the corresponding outgoing wavefronts are represented by coefficients for basis elements of a (common) basis system. The at least one wavefront transfer function assigns the corresponding outgoing wavefront to the incident wavefronts preferably in such a way that, for each basis element represented in the representation of an outgoing wavefront, it determines the coefficient of the outgoing wavefront for that basis element at least as a function of the coefficient of the corresponding incident wavefront for the same basis element.

[0066] Particularly preferably, the basis elements are classified according to at least one order parameter, and the at least one wavefront transfer function assigns the respective corresponding outgoing wavefront to the incident wavefronts in such a way that, for a (especially each) basis element represented in the representation of an outgoing wavefront, it determines the coefficient to this basis element represented in the representation of the outgoing wavefront at least as a function of those coefficients of the corresponding incident wavefront to those basis elements whose value of the order parameter corresponds to the value of the order parameter of the respective basis element represented in the representation of the outgoing wavefront.

[0067] Furthermore, it is particularly preferred if the at least one wavefront transfer function assigns the corresponding outgoing wavefront to the incident wavefronts in such a way that, for each basis element represented in the representation of an outgoing wavefront, it determines the coefficient of that basis element as a function of the coefficients of the corresponding incident wavefront for a plurality (in particular all) of those basis elements whose order parameter value is less than or equal to the order parameter value of the respective basis element represented in the representation of the outgoing wavefront. In particular, coefficients of the corresponding incident wavefront for a plurality of basis elements with different order parameter values ​​can be taken into account.

[0068] In particular, each incident wavefront and its corresponding outgoing wavefront can be represented with respect to a basis system, wherein basis elements of each incident wavefront are classified according to at least one order parameter, and wherein the at least one wavefront transfer function for a given value of a first order parameter is given by transforming, for each wavefront incident on the optical system, at least one of the basis elements whose order parameter is less than or equal to the given value, into a corresponding basis element of the corresponding outgoing wavefront whose order is less than the given value. In particular, for each value of the first order parameter, the basis elements are additionally classified according to at least one second order parameter, the range of which depends on the value of the first order parameter.

[0069] In a preferred embodiment, both the incident and the corresponding outgoing wavefronts are represented by coefficients belonging to basis elements of a basis system, wherein the basis elements are classified according to at least one order parameter, and wherein the at least one wavefront transfer function is given by transforming, for each wavefront incident on the optical system, at least one coefficient assigned to a basis element with an order parameter less than or equal to a predetermined value into a coefficient of the outgoing wavefront belonging to the same basis element. In particular, at least the coefficients of basis elements of the incident wavefront whose order parameter is less than or equal to the predetermined value can be considered for this purpose.In other words, at least the coefficients of the basis elements of the incident wavefront, whose order parameter is less than or equal to the specified value, can be transformed into a coefficient of the outgoing wavefront, whose order parameter is less than or equal to the specified value.

[0070] In particular, at least one wavefront transfer function can be given by transforming, for each wavefront incident on the optical system, at least the coefficients of the basis elements whose order parameter is less than or equal to a given value, into a coefficient of the basis element of the corresponding outgoing wavefront whose order parameter is equal to the given value. The coefficients of basis elements of the incident wavefront whose order parameter is greater than the given value can also be taken into account.

[0071] With a suitable choice of the basis system, mixed terms of different orders can advantageously be neglected, so that a coefficient of the incident wavefront belonging to a basis element of a certain order can be transformed into a coefficient of the outgoing wavefront belonging to the same basis element.

[0072] In a further preferred embodiment, the order parameter is a first order parameter, wherein the basis elements of the basis system are additionally classified according to at least a second order parameter, the range of values ​​of which depends on the value of the first order parameter.

[0073] The at least one wavefront transfer function can be given, for example, for a given order p, by the fact that it includes at least one of the aberrations E2, E3, ..., E pof the first p orders of each wavefront incident on the optical system into one of the aberrations E'2, E'3, ..., E' p the first p orders of the failing wavefront are transferred. Alternatively, aberrations can also be used in three dimensions, in which case the at least one wavefront transfer function for each given order p is given by the fact that it is given for 2 ≤ q x + q y ≤ p the aberrations E qx,qy of the first p orders of each two-dimensional wavefront incident on the optical system into at least one of the aberrations E' nx,ny the first p orders of the failing two-dimensional wavefront are transformed, whereby 2≤nx+ny≤p and qx,qy≥0 and nx,ny≥0.

[0074] In a further preferred embodiment (two-dimensional case), the basis system is thus a decomposition according to aberrations (or the basis system is defined or determined by a decomposition according to aberrations), wherein the order parameter is an order p of the aberrations, wherein the coefficients belonging to the basis elements of the basis system are given by the fact that a coefficient of p-th order of the incident wavefront is an aberration E p of the incident wavefront and that a coefficient of p-th order of the associated outgoing wavefront is an aberration E' p the associated falling wavefront, and where p ≥ 2.

[0075] In another preferred embodiment (three-dimensional case), the basis system is a decomposition according to aberrations (or the basis system is defined or determined by a decomposition according to aberrations), wherein the first order parameter is the sum p of orders px and p y of the aberrations, where the second order parameter is one of the orders p x and p y is, where the coefficients belonging to the basis elements of the basis system are given by the fact that coefficients of the p-th order of the incident wavefront aberrations E px,py of the incident wavefront and that coefficients p-th order of the associated outgoing wavefront aberrations E' px,py of the associated falling wavefront, and where p ≥ 2 and p x ≥ 0 and p y ≥ 0.

[0076] In another preferred embodiment, the at least one wavefront transfer function has the form Ep'=β−r¯10(p−1)∑k1,k2,…,kp−1b¯pk β−Δr¯1(p−1,k*)E2k1E3k2⋯Epkp−1, on, where the indices of the tuple k = (k1, k2,...,k p-1 ) over the range P(k*) ≤ p - 2 and 0 ≤ k1 ≤ 2(p - P(k*) - 2) + δ P(k*),0 running, whereby P(k*)=∑j=1p−2jkj+1, and whereby −r¯10(p−1)=p−δ(p−1),1 and -Δr̅1(p - 1, k*) = (p - 3) + δ (p-1),1 - P(k**) hold, and where β = (-BE2 + A) -1 is given as a function of the at least one incident wavefront and the optical system, and where A, B and the wavefront transfer coefficients b̅ pk are given as a function of the components of the optical system.

[0077] The at least one wavefront transfer function can also be given for any given order p by defining the Taylor derivatives w (2) , w (3) , ..., w (p) the first p orders of each wavefront incident on the optical system into at least one of the Taylor derivatives w' (2) , w' (3) , ..., w' (p)the first p orders of the resulting wavefront. Alternatively, Taylor derivatives can also be used in three dimensions, in which case the at least one wavefront transfer function for each given order p is given by the fact that it is defined for 2 ≤ q x + q y ≤ p the Taylor derivatives w (qx,qy) the first p orders of each two-dimensional wavefront incident on the optical system into at least one of the Taylor derivatives w' (nx,ny) the first p orders of the failing two-dimensional wavefront are transformed, whereby 2≤nx+ny≤p and qx,qy≥0 and nx,ny≥0.

[0078] In a further preferred embodiment (two-dimensional case), the basis system is thus a decomposition in Taylor derivatives (or the basis system is defined or determined by a decomposition in Taylor derivatives), wherein the order parameter is an order p of the Taylor derivatives, wherein the coefficients belonging to the basis elements of the basis system are given by the fact that a coefficient of pth order of the incident wavefront is a Taylor derivative w (p) of the incident wavefront and that the coefficient p-th order of the associated outgoing wavefront is a Taylor derivative w' (p) the corresponding falling wavefront, and where p ≥ 2. The Taylor derivatives can be Taylor derivatives of the wavefront arrow heights or Taylor derivatives of the wavefront OPD (OPD = Optical Path Difference).

[0079] In a preferred embodiment (two-dimensional case), the basis system is thus a decomposition according to Taylor derivatives of the wavefront arrow heights, wherein the order parameter is an order p of the Taylor derivatives of the wavefront arrow heights, and wherein the coefficients belonging to the basis elements of the basis system are given by the fact that a coefficient of pth order of the incident wavefront is a Taylor derivative w (p) the swash level of the incident wavefront and that the coefficient of the pth order of the corresponding outgoing wavefront is a Taylor derivative w' (p) the swash height of the associated falling wavefront, and where p ≥ 2. Here, the notation w means (p) the Taylor derivative of order p of the function w of the incident wavefront, preferably at position 0. Accordingly, w means (p) the Taylor derivative of order p of the function w' of the falling wavefront, preferably at position 0.

[0080] In an alternative preferred embodiment (two-dimensional case), the basis system is a decomposition into Taylor derivatives of a wavefront OPD (optical path difference), wherein the order parameter is an order p of the Taylor derivatives of the wavefront OPD, wherein the coefficients belonging to the basis elements of the basis system are given by the fact that a coefficient of pth order of the incident wavefront is a Taylor derivative OPD (p) of the incident wavefront and that the coefficient p-th order of the associated outgoing wavefront is a Taylor derivative OPD' (p) the associated falling wavefront, and where p ≥ 2. Here, the designation OPD means (p) the Taylor derivative of order p of the OPD function of the incident wavefront, preferably at position 0. Accordingly, OPD' means (p)the Taylor derivative of order p of the OPD' function of the failing wavefront, preferably at position 0.

[0081] In another preferred embodiment (three-dimensional case), the basis system is a decomposition by Taylor derivatives (or the basis system is defined or determined by a decomposition by Taylor derivatives), wherein the first order parameter is the sum p of orders p x and p y of the Taylor derivatives, where the second order parameter is one of the orders p x and p y is, where the coefficients belonging to the basis elements of the basis system are given by the fact that coefficients of the p-th order of the incident wavefront are Taylor derivatives w (px,py) of the incident wavefront and that the coefficients of the p-th order of the associated outgoing wavefront are Taylor derivatives w' (px,py) of the associated falling wavefront, and where p ≥ 2 and px ≥ 0 and p y ≥ 0. Here too, the Taylor derivatives can be Taylor derivatives of the wavefront arrow heights or Taylor derivatives of the wavefront OPD.

[0082] In a preferred embodiment (three-dimensional case), the basic system is thus a decomposition according to Taylor derivatives of the wavefront arrow heights, where the first order parameter is the sum p of orders p x and p y the Taylor derivatives of the wavefront arrow heights, where the second order parameter is one of the orders p x and p y is, where the coefficients belonging to the basis elements of the basis system are given by the fact that coefficients of the p-th order of the incident wavefront are Taylor derivatives w (px,py) the swash level of the incident wavefront and that the coefficients of the p-th order of the associated outgoing wavefront are Taylor derivatives w' (px,py)the arrow height of the associated falling wavefront, and where p ≥ 2 and p x ≥ 0 and p y are ≥ 0. The designation w means... (px,py) the Taylor derivative of order p = p x + p y the function w(x,y) of the incident wavefront, preferably at x = 0, y = 0. Accordingly, w' means (px,py) the Taylor derivative of order p = p x + p y the function w'(x', y') of the failing wavefront, preferably at the position x' = 0, y' = 0. In an alternative preferred embodiment (three-dimensional case), the basis system is a decomposition by Taylor derivatives of a wavefront OPD, where the first order parameter is the sum p of orders p x and p y the Taylor derivatives of the wavefront OPD, where the second order parameter is one of the orders p x and p yis, where the coefficients belonging to the basis elements of the basis system are given by the fact that coefficients of the p-th order of the incident wavefront are Taylor derivatives OPD. (px,py) of the incident wavefront and that the coefficients of the p-th order of the associated outgoing wavefront are Taylor derivatives OPD' (px,py) of the associated falling wavefront, and where p ≥ 2 and p x ≥ 0 and p y are ≥ 0. The term OPD means... (px,py) the Taylor derivative of order p = p x + p y the OPD function of the incident wavefront, preferably at x = 0, y = 0. Accordingly, OPD' means (px,py) the Taylor derivative of order p = p x + p y the OPD' function of the failing wavefront, preferably at the location x' = 0, y' = 0.

[0083] At least one wavefront transfer function can also be given for any given order p by defining its derivatives t. (1) , t (2) , ..., t (i) the first i = p - 1 orders of a directional function t(x) of each wavefront incident on the optical system into at least one of the derivatives t' (1) , t' (2) , ..., t' (i) The first i = p - 1 orders are transformed into a direction function of the emerging wavefront. Alternatively, local directions of the wavefront can also be used in three dimensions, in which case the at least one wavefront transfer function for each given order i = p - 1 is given by the fact that for 2 ≤ j x + j y ≤ i the Taylor derivatives t x (jx,jy) , t y (jx,jy) the first i = p - 1 orders of each two-dimensional wavefront incident on the optical system into at least one of the derivatives t' x(nx,ny) , t' y (nx,ny) the first i = p - 1 orders of the failing two-dimensional wavefront are transformed, where 1 ≤ n x + n y ≤ i and j x ,j y ≥ 0 and n x , n y ≥ 0 applies.

[0084] In a further preferred embodiment (two-dimensional case), the basis system is thus a decomposition in terms of derivatives of directional functions (or the basis system is defined or determined by a decomposition in terms of derivatives of directional functions), wherein the order parameter is an order i of the derivatives of the directional functions, wherein the coefficients belonging to the basis elements of the basis system are given by the fact that a coefficient of the i-th order of the incident wavefront is a derivative t (i) a directional function t(x) of the incident wavefront and that a coefficient of the i-th order of the associated outgoing wavefront is a derivative t(i) a directional function t'(x') of the associated falling wavefront, and where i ≥ 1.

[0085] In another preferred embodiment (three-dimensional case), the basis system is a decomposition by derivatives of direction functions (or the basis system is defined or specified by a decomposition by derivatives of direction functions), wherein the first order parameter is the sum i of orders i x and i y of the derivatives of the directional functions, where the second order parameter is one of the orders i x and i y is, where the coefficients belonging to the basis elements of the basis system are given by the fact that coefficients of the i-th order of the incident wavefront derivatives t x (ix,iy) , t y (ix,iy) of directional functions t x (x,y), t y(x,y) of the incident wavefront and that coefficients i-th order of the associated outgoing wavefront derivatives t' x (ix,iy) , t' y (ix,iy) of directional functions t' x (x',y), t' y (x',y') of the associated falling wavefront, and where i ≥ 1 and i x ≥ 0 and i y ≥ 0 applies.

[0086] Furthermore, Zernike polynomials can be used in two dimensions. In this case, at least one wavefront transfer function for any given order p can be given by specifying the Zernike coefficients Z2, Z3, ..., Z. p the first p orders of each wavefront incident on the optical system into at least one of the Zernike coefficients Z'2, Z'3, ..., Z' pthe first p orders of the emitted wavefront are transferred, where the Zernike coefficients refer specifically to a defined pupil. Alternatively, Zernike polynomials in three dimensions can be used, in which case the at least one wavefront transfer function for each given order p is given by the Zernike coefficients for 2 ≤ q ≤ p. Zqr the first p radial orders of each two-dimensional wavefront incident on the optical system into at least one of the Zernike coefficients Zn'm the first p radial orders of the resulting two-dimensional wavefront are transformed, where 2 ≤ n ≤ p, and where n and q are the radial orders and m and r are the azimuthal orders varying in steps of 2, where -q ≤ r ≤ q and -n ≤ m ≤ n, where the Zernike coefficients refer in particular to a fixed pupil.

[0087] In a further preferred embodiment (two-dimensional case), the basis system is thus a decomposition into Zernike polynomials (or the basis system is defined or determined by a decomposition into Zernike polynomials), wherein the order parameter is a radial order n of the Zernike polynomials, wherein the coefficients belonging to the basis elements of the basis system are given by the fact that a coefficient of nth order of the incident wavefront is a Zernike coefficient Z. n of the incident wavefront and that a coefficient of nth order of the outgoing wavefront is a Zernike coefficient Z' n the associated falling wavefront, where n ≥ 2, and where the Zernike coefficients refer in particular to a specified pupil.

[0088] In a further preferred embodiment (three-dimensional case), the basis system is a decomposition into Zernike polynomials (or the basis system is defined or determined by a decomposition into Zernike polynomials), wherein the first order parameter is a radial order n of the Zernike polynomials, wherein the second order parameter is an azimuthal order m of the Zernike polynomials, and wherein the coefficients belonging to the basis elements of the basis system are given by the fact that coefficients of the nth order of the incident wavefront are Zernike coefficients Z n m of the incident wavefront and that coefficients of the nth order of the associated outgoing wavefront are Zernike coefficients Z' n mof the associated falling wavefront, where n ≥ 2 and -n ≤ m ≤ n, where m is even for even n, where m is odd for odd n, and where the Zernike coefficients refer in particular to a specified pupil.

[0089] Another independent aspect of solving the problem concerns a computer program product comprising machine-readable program code which, when loaded onto a computer, is suitable for executing the method according to the invention described above. In particular, a computer program product is understood to be a program stored on a data carrier. Specifically, the program code is stored on a data carrier. In other words, the computer program product comprises computer-readable instructions which, when loaded into a computer's memory and executed by the computer, cause the computer to perform a method according to the invention described above.The computer program product can, in particular, comprise a computer-readable storage medium containing code stored thereon, wherein the code, when executed by a processor, causes the processor to implement a method according to the invention. In particular, the computer program product can also comprise or be a storage medium with a computer program stored thereon, wherein the computer program is designed, when loaded and executed on a computer, to carry out a method according to the invention. In particular, the invention provides a computer program product, especially in the form of a storage medium or a data stream, containing program code designed, when loaded and executed on a computer, to carry out a method according to the invention, particularly in a preferred embodiment.

[0090] Another independent aspect for solving the problem concerns a device for simulating an optical system by means of wavefront calculation, wherein the optical system is in particular a complex optical system whose effect goes beyond a single refraction, a single propagation or a single reflection, comprising: - a modeling module for providing at least one wavefront transfer function for the optical system, wherein the wavefront transfer function is designed to assign to each incident wavefront to the optical system, taking into account aberrations of an order greater than the order of a defocus, a corresponding outgoing wavefront; - an evaluation module for evaluating at least one wavefront transfer function for at least one wavefront incident on the optical system.

[0091] The modeling module can include an interface and a storage device for providing and storing the wavefront transfer function. Providing the wavefront transfer function specifically means setting up the wavefront transfer function. This is done automatically or by computer implementation. Accordingly, the modeling module can include a computing unit or processor for (automatically) providing or setting up the wavefront transfer function. In particular, the device also includes a data interface for acquiring data from the optical system or the overall system, an evaluation module for evaluating the overall optical system, and / or an optimization module for optimizing the first subsystem.

[0092] In particular, a method for manufacturing a spectacle lens can be provided, comprising the steps: Calculating or optimizing a spectacle lens using a method according to the invention described above; and Providing manufacturing data for the calculated or optimized spectacle lens, and / or manufacturing the calculated or optimized spectacle lens.

[0093] In particular, a device for manufacturing a spectacle lens can be provided, comprising: Calculation or optimization tools designed to calculate or optimize the spectacle lens using a method according to the invention; and Processing tools designed to finish processing the spectacle lens.

[0094] In particular, a spectacle lens produced in this way can be used in a predetermined average or individual position of use of the spectacle lens in front of the eyes of a specific spectacle wearer to correct a refractive error of the spectacle wearer.

[0095] The descriptions of the embodiments of the first aspect given above also apply to the aforementioned further independent aspects and, in particular, to preferred embodiments thereof. In particular, the descriptions given above and below of the embodiments of the other independent aspects also apply to an independent aspect of the present invention and to preferred embodiments thereof.

[0096] The following section describes, by way of example, individual embodiments for solving the problem, illustrated by the figures. Some of the described embodiments exhibit features that are not strictly necessary for carrying out the claimed subject matter, but which provide desirable properties in certain applications. Thus, embodiments that do not possess all the features of the embodiments described below are also considered to be disclosed within the scope of the described technical teaching. Furthermore, to avoid unnecessary repetition, certain features are mentioned only in relation to some of the embodiments described below. It should therefore be noted that the individual embodiments should not only be considered individually, but also in combination.From this overview, the person skilled in the art will recognize that individual embodiments can also be modified by incorporating one or more features from other embodiments. It should be noted that a systematic combination of individual embodiments with one or more features described in relation to other embodiments may be desirable and useful, and should therefore be considered and also be regarded as covered by the description. Brief description of the drawings Fig. Figure 1 shows a schematic sketch of an exemplary optical system; Fig. Figure 2 shows a schematic sketch for the specification of an optical system using a beam transfer function; Fig.Figure 3 shows a schematic sketch of the transformation between wavefronts w(x),w'(x') in the w-representation and the functions t(x),f'(x') in the t-representation according to an embodiment of the present invention; Fig. Figure 4 shows a schematic flowchart according to an embodiment of the present invention based on Taylor series in a meridian; Fig. Figure 5 shows a schematic sketch of the relationship between the OPD τ(x) and the direction of a t(x) ray perpendicular to a wavefront w(x); Fig. Figure 6 shows a schematic sketch of the ray transfer function for propagation; Fig. 7a and Fig. Figure 7b shows schematic sketches of the ray transfer function for refraction; Fig. Figure 8 shows a schematic sketch of a Young diagram for the tuple k = (1,0,2); Fig.Figure 9 shows a schematic sketch of a modified Gullstrand-Emsley eye (mGE eye) to illustrate an embodiment of the present invention. Detailed description of the drawings

[0097] In the context of this invention, the following terms are used as follows (unless otherwise specified, described in one meridian): • Beam

[0098] An infinitesimal beam of light, described as a straight line, half-line, or line segment in space, preferably defined by points of intersection with planes and directional parameters. In the case of a meridian, a ray is described by - x Position parameters of a beam - t Direction parameter of a ray - (x, t) Parameters of an incident ray - (x', t') Parameters of a failing beam

[0099] In the case of two meridians, one ray is described by - x, y position parameters of a ray - t x , t y Directional parameters of a ray - (x,y, t x , t y ) Parameters of an incident ray - (x', y', t' x , t' y ) Parameters of a failing beam • Wavefront

[0100] In the case of a meridian, a wavefront is a curve in the meridian under consideration that is perpendicular to rays. For the sake of simplicity, a curve is also referred to as a surface. - w(x) function to describe a wavefront as a surface - w (p) Derivative of order p of the function w(x), preferably at the point x = 0 - E p = nw (p) Aberration of order p of the wavefront w(x), where n is the refractive index - t(x) Function for describing a wavefront by the dependence of the direction parameter on the position parameter - t(i) Derivative of order i of the function t(x), preferably at the point x = 0 - w(x), w (p) , E p , t(x), t (i) Quantities used to describe the incident wavefront - w'(x'), w' (p) , E' p , t'(x'), t (i) Quantities used to describe the emerging wavefront

[0101] In the case of two meridians, a wavefront is a surface in space that is perpendicular to rays. - w(x, y) function for describing a wavefront as a surface - w (px,py) Derivative of order p = p x + p y the function w(x,y), preferably at the point x = 0, y = 0 - E px,py = nw (px,py) Aberration of order p = p x + p y of the wavefront w(x,y), where n is the refractive index - t x (x,y), t y(x,y) Functions for describing a wavefront by the dependence of the direction parameter on the position parameter - tx(ix,iy),tx(ix,iy) Derivative of order i = i x + i y the functions t x (x,y), t y (x,y), preferably at the position x = 0, y = 0 - w(x, y), w (px,py) , E px,py , t x (x,y) , t y (x,y), tx(ix,iy),ty(ix,iy) Quantities used to describe the incident wavefront - w'(x',y'), w' (px,py) E' px,px , t' x (x',y'), t' y (x',y'), tx,(ix,iy),ty,(ix,iy) Quantities used to describe the emerging wavefront • Beam transfer function

[0102] In the case of a meridian, a function f(x,t) that assigns to an incident ray with parameters (x, t) the parameters (x', t') of an outgoing ray - f prop(x, t) Beam transfer function for pure propagation - f ref (x, t) Beam transfer function for pure refraction

[0103] In the case of two meridians, the function f(x, y, t) x , t y ), which are an incident ray with parameters (x,y,t x ,t y ) the parameters (x', y', t' x , t' y ) of a failing beam - f prop (x,y, t x , t y ) Beam transfer function for pure propagation - f ref (x, y, t x , t y ) Beam transfer function for pure refraction • Wavefront transfer function

[0104] Function determined according to the invention, which for a given beam transfer function f(x,t) or f(x, y, t) x , t y ) assigns an outgoing wavefront to an incoming wavefront • Wavefront transfer coefficients

[0105] Coefficients determined according to the invention, which describe the transfer of a wavefront through the system for a given beam transfer function f(x,t) of an optical system - c̅ ik Coefficient for determining the derivative t' (i) of order i of a failing wavefront, which depends on the derivatives t (1) , ..., t (i) the function of an incident wavefront, where the tuple k = (k1, k2,.., k i ) on the contribution of the product (t (1) ) k1 (t (2) ) k2 ... (t (i) ) ki refers to; - b̅ pk Coefficient for determining the derivative w' (p) or the aberration E' p of order p of a failing wavefront, which depends on the derivatives w (2) , ..., w (p) or from the aberrations E2, ..., E pan incoming wavefront, where the tuple k = (k1,k2,..., k p-1 ) on the contribution of the product (w (2) ) k1 (w (3) ) k2 ... (w (p) ) kp-1 or E2k1E3k2⋯Epkp−1 refers to.

[0106] In a very general form, an optical system is fully specified if there is a rule that uniquely defines an outgoing ray for each incident ray (this rule is called the ray transfer function). All further details of the optical system are then irrelevant for specifying its optical behavior, so the system can also be considered a "black box." If a wavefront accompanies a principal ray to such a system, then, according to the current state of the art, the outgoing wavefront, including its HOA (Higher Area of ​​Analysis), can only be determined by laboriously calculating a sufficiently chosen bundle of neighboring rays using ray tracing and then numerically determining a wavefront from this bundle on the outgoing side, for example, using a fitting procedure.

[0107] A step-by-step analytical calculation from surface to surface using the methods from WO 2008 089 999 A1 and DE 10 2011 101 923 A1 does not solve the problem of the present invention, because not every optical system actually has to consist of individual components that can be described by refracting surfaces or propagation in a homogeneous medium. For example, a GRIN system (gradient index system) has no internal refracting surfaces other than the entrance and exit surfaces and achieves its effect through the inhomogeneity of the material. Such a GRIN system also has well-defined behavior with respect to the imaging of rays, but the wavefront calculation cannot be carried out with a method described in WO 2008 089 999 A1 and DE 10 2011 101 923 A1.

[0108] Even if the complex optical system consists of only a finite number of refracting surfaces and propagations in a homogeneous medium, as schematically depicted in Fig. As shown in Figure 1, the methods from WO 2008 089 999 A1 and DE 10 2011 101 923 A1 do yield a result, but only at the cost of a very high computation time. This is particularly true when a large number of interfaces are present or when an optical system is to be repeatedly calculated with different wavefronts.

[0109] While it might seem obvious to combine existing prior art methods into a single analytical procedure by mutually substituting the analytical formulas, the resulting number of terms would be so high that it would be completely futile to combine them in a way that would result in a computational time advantage. However, this problem is also solved by the present invention, which eliminates the superfluous intermediate steps and establishes a wavefront transfer procedure that is directly based on the beam transfer function from the outset.

[0110] The Fig.Figure 2 schematically shows an optical system for defining the ray transfer function. In linear optics, it is common practice to define a system by an entry plane and an exit plane. More generally, an optical system can also be defined by two non-parallel planes or by two arbitrary surfaces. The essential aspect of the specification, however, is that (initially in the case of a single meridian) there is a coordinate x on the incident surface and a coordinate x' on the reflected surface, which serve to describe the points of intersection of the ray. Furthermore, there must be a unique way to define the direction of an incident ray relative to the entry surface (e.g., by an angle α), as well as to define the direction of the reflected ray relative to the exit surface (e.g., by an angle α'). In the case of two meridians, there must accordingly be two coordinates x and y and two angles α on the incident side.x , α y give, as well as on the falling side coordinates x', y' and two angles α' x , α' y .

[0111] If one then defines a ray vector for a meridian ρ:=(xt):=(xn tan α)ρ':=(x't'):=(x'n' tan α') Then the optical system is uniquely specified if a beam transfer function f: R 2 ↦ R 2 with components f x , f t Given is the transformation p to p': ρ'=f(ρ)⇔(x't')=(fx(x,t)ft(x,t))

[0112] In two meridians, the optical system is uniquely specified if it has a corresponding ray transfer function f: R 4 ↦ R 4 gives.

[0113] It is both in Gaussian optics (f: R 2 ↦ R 2 ) as well as in linear optics (f: R 4 ↦ R 4) State of the art that the effect of a system described by f is described by a system matrix T: ρ'=Tρ,⇔(x't')=(ABCD)(xt) where the 2x2 system matrix (also called transference) is defined by T=Jac(f)=(∂x' / ∂x∂x' / ∂t∂t' / ∂x∂t' / ∂t)=(fx(1,0)fx(0,1)ft(1,0)ft(0,1))=:(ABCD), where J ac denotes the Jacobian matrix, where the entries A, B, C, D are constants that characterize the optical system, and where the notation fm=f(mx,mt)=∂mf=(∂mx / ∂xmx∂mt / ∂tmt)f is used. Furthermore, it is known in the prior art that in a meridian a wavefront with curvature k and vergence S = nk, which enters the system, leads to a wavefront with vergence S' = n'k' upon exiting, wherein the resulting vergence is determined by S'=−C−DSA−BS is given.

[0114] In the case of two meridians, a corresponding relationship can be given; see Qiang L, Shaomin W, Alda J, Bernabeu E: “Transformation of nonsymmetric Gaussian beam into symmetric one by means of tensor ABCD law”, Optic - International Journal for Light and Electron Optics (OPTIK) 85(2): 67-72 (1990). Furthermore, the system can also possess nonlinear components, such as a prismatic effect.

[0115] The description in Eq. (3) for treating rays using a matrix T can be interpreted as a first-order description, while the description in Eq. (5) for treating wavefront vergences can be interpreted as a second-order description. However, there is no prior art description available for treating higher-order wavefront properties (HOA), such as coma or spherical aberration, when only the ray transfer function f is given. This is where the present invention comes in.

[0116] According to the invention, it has been discovered that a wavefront w(x) corresponds to a function t(x) which is obtained from w(x) by a unique and invertible transformation H. The fixed input quantity t, which was originally developed for linear optics, is reinterpreted according to the invention so that it can also be used for the nonlinear description of wavefronts by allowing a dependency t(x). If x is varied, two functions x'(x) and t'(x) are obtained on the output side according to Eq. (2), which implicitly define a relationship t'(x'). This function t'(x') uniquely corresponds to an output wavefront w'(x'), which can be obtained by applying the transformation H inversely to H. -1 receives. This connection is in the Fig. 3 shown schematically.

[0117] Wavefronts can be represented, for example, as symbolic functions or as freeform surfaces. Furthermore, wavefronts can be developed according to base systems, where the order of the wavefronts can function as an order parameter. Order parameters can preferably be used such that contributions decrease in their numerical magnitude with increasing order parameter, so that contributions up to a certain value of the order parameter are taken into account and neglected for even higher values. A similar approach can be used when multiple order parameters are present.

[0118] Wavefronts are preferably represented by Zernike polynomials. In a particularly preferred embodiment of the invention, wavefronts are represented by Taylor series, i.e., they are characterized by their local derivatives at a reference position, preferably at x = 0. In this embodiment, the order parameter is the order p of the derivative w'. (p) (x'), which is to be determined, in two meridians the sum p = p x + p y the orders of the derivative w' (px , py) (x',y').

[0119] In a particular embodiment, optical systems are considered (along a meridian) that have parallel entry and exit planes and whose beam transfer function f(x,t) has the property f(0,0) = 0. Preferably, a beam strikes such a system at x = 0 with direction t = 0, and because f(0,0) = 0, it also exits the system at x' = 0 and t' = 0. According to a further feature of the embodiment, the basic procedure is carried out using a Taylor series for the wavefronts, preferably evaluated at x = 0. The incident and outgoing wavefronts, which must be perpendicular to the beams, then automatically have vanishing first derivatives w. (1) (0) = 0 and w' (1) (0) = 0.

[0120] The Fig.Figure 4 shows a schematic flowchart according to an embodiment of the present invention based on Taylor series in a meridian. In this embodiment, after defining an order p, the ray transfer function f(x, t) must also be determined in the form of all its partial derivatives. fx(1.0)(0.0),fx(0.1)(0.0),ft(1.0)(0.0),ft(0.1)(0.0),fx(2.0)(0.0),… up to order (p - 1). If these are available, then in the next step the procedure for determining the wavefront transfer function, preferably for assigning wavefront transfer coefficients for wavefront calculation, can be carried out directly (see Fig.4) If the derivatives of the beam transfer function are not available, they must be determined beforehand. If the optical system consists of a sequence of refracting surfaces with homogeneous media between them, the derivatives of f(x,t) can be determined from the shape and position of the surfaces and their refractive indices. Preferably, all surfaces are perpendicular to the beam at x = 0; then the derivatives of f(x,t) can be determined from the derivatives of the surfaces and their spacing. However, if no intermediate surfaces are present (e.g., in a GRIN material), the derivatives of f(x,t) must be determined by other means, such as measurement or simulation.

[0121] The procedure can be carried out, for example, by determining the coefficients c̅ ikThe coefficients c̅ are determined for calculation in the t-representation from the derivatives of f(x, t) up to order i = p - 1. This can be done numerically or, preferably, symbolically. In a preferred embodiment, however, the coefficients c̅ are determined from the coefficients. ik using the transformation H, first the coefficients b̅ pk for wavefront calculation in w-representation up to order p directly from the coefficients c̅ ik and determined from the derivatives of f(x,t), whereby this step can again be carried out numerically or preferably symbolically.

[0122] Are the coefficients b̅ pk Once the wavefront calculation has been performed up to order p, the wavefront calculation can be repeated with any number of wavefronts up to order p.

[0123] For wavefront calculation with reversed light direction, the inverse function can alternatively be formed from the beam transfer function f, and then the inventive method for determining coefficients c̅ can be used. ik or coefficients b̅ pk The wavefront transfer function, already determined up to an order p, can be used, or it can be directly inverted for given coefficients to determine the derivatives of the incident wavefront by the derivatives of the outgoing wavefront. Implementation for the transformation H

[0124] Any relationship between a wavefront w(x) and a function t(x) is a transformation. A transformation can be described symbolically or numerically. A preferred embodiment involves expanding the function t(x) into a Taylor series and determining the derivatives t (i) (x) by derivatives w (p)(x) to express, preferably for x = 0.

[0125] The problem of transforming the description of a spatial surface by w(x) into a function t(x) that is referenced to a plane is solved according to the invention as follows. Unlike in the prior art, where the OPD also gives rise to a function τ(x) that has a defined relationship to the wavefront w(x), t(x) denotes a direction and not an OPD. Therefore, a relationship between t(x) and τ(x) must first be established here.

[0126] In the Fig. Figure 5 shows a wavefront together with its reference plane, both of which are pierced by a neighboring ray with direction t. The tangent of the direction angle α, which by definition is given by tan α = t / n, is, on the other hand, equal to the aspect ratio of the wavefront shown in Figure 5. Fig. Figure 5 shows a right-angled triangle. If the length of the hypotenuse is set equal to 1, then the length of the opposite side to α is given by τ. x / n is given (proportional to the derivative of the OPD). This yields the equation tan α=:t(x)n=−τx(x) / n1−(τx(x) / n2)

[0127] Repeated differentiation of Eq. (6) with respect to x and evaluation for x = 0 leads to t(1)=−τ(2)t(2)=−τ(3)t(3)=−τ(4)−3τ(2)3 / n2t(4)=−τ(5)−18τ(2)2τ(3) / n2 ⋮ and substitution of the derivatives τ (2) , τ (3) , τ (4) , ... by the expressions from GI.(B6) in “Appendix B” of the publication by G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, D. Uttenweiler: “Derivation of the refractive equations for higher order aberrations of local wavefronts by oblique incidence”, J. Opt. Soc. Am. A 27, 218-237 (2010), yields for the transformation H: t(1)=−nw(2)t(2)=−nw(3)t(3)=−n(w(4)−6w(2)3)−3(nw(2))3 / n2=−n(w(4)−3w(2) 3)t(4)=−n(w(5)−40w(2)2w(3))−18(nw(2))2(nw(3)) / n2=−n(w(5)−22w(2)2w(3))⋮

[0128] By inverting these equations, one obtains the transformation H.-1 : w(2)=−t(1) / nw(3)=−t(2) / nw(4)=−(t(3)+3t(l)3 / n2) / nw(5)=−(t(4)+22t(l)2t(2) / n2) / n⋮ Assignment of ray transfer functions for elementary propagation and refraction

[0129] The Fig. Figure 6 shows a schematic sketch of the ray transfer function for propagation. A ray propagating from an entry plane to an exit plane at a distance d = τ / n, where τ is the optical direction (OPD) that the light would travel at perpendicular incidence, exits at a different position. The ray transfer function is therefore quite simple because the direction does not change, and thus t' = t. The position component satisfies (x' - x) / (τ / n) = tan α = t / n, so that x' = x + tτ / n 2 The beam transfer function is therefore: (x't')=fprop(x,t)=(x+tτ / n2t),

[0130] The beam transfer function f propRemarkably, the (x, t) for the propagation in Eq. (10) depends only linearly on both x and t. Therefore, the derivatives f prop (nx,nt) (x, t) of f prop (x, t) very simply and vanish for all higher orders n x + n t ≥ 2, as shown in Table 1 below. Table 1: Derivatives of the ray transfer function fprop (x, t) at the point (x, t) = (0,0) Wavefront order Abl.-Ord. n x + n t xOrd.n x tOrd.n t fxprop(nx,nt) ftprop(nx,nt) 0 0 -0 0 0 2 1 1 0 1 0 2 1 0 1 t / n 2 1 All higher orders disappear

[0131] In the special case of a vanishing propagation distance τ = 0, the ray transfer function f must prop (x, t) be the identity with Jacobian matrix Jac f prop = 1. The only entry that distinguishes the derivatives from this trivial case is fxprop(0,1)=τ / n2. refraction

[0132] The Fig.Figure 7 shows a schematic sketch of the ray transfer function for propagation. Refraction is considerably more difficult to treat, and a ray transfer function f ref It is not possible to express (x, t) for closed-form refraction as a function of the properties of the refracting surface. This is because the point of intersection of a neighboring ray through an arbitrary surface can only be determined iteratively. Although there is a way to transform the spatial surface w(x) into a property t(x) for wavefronts using a transformation H, there is no corresponding transformation for the refracting surface itself without incorrectly handling the existing parallax. Therefore, refraction must still occur at a point in space that lies outside the plane of incidence.

[0133] Since no propagation occurs during pure refraction, the exit plane is identical to the entry plane. The problem to be solved according to the invention is therefore: if a neighboring ray (x, t) is given that starts at the entry plane in the direction of a refracting surface, which parameters (x', t') as a function of (x, t) correspond to the refracted ray, with respect to the same plane?

[0134] The preferred application of the method concerns situations in which the ray (x, t) has a unique intersection point with the surface. This intersection point is denoted by (x,z(x)), where the intersection coordinate is given by the unique function 5c(x,t). The direction tangent of the surface normal is denoted by t (see Fig. 7) It is given by the negative of the first derivative of the area, t = -z (1), which is also a function of x and thus also a function t(x, t) = t(z(x, t)). According to Snell's law of refraction, the directions t of the incident ray and t of the surface give rise to a direction t' of the refracted ray, and the backward extension of the refracted ray intersects the plane of entry at the unique position x' (see Fig. 7).

[0135] Snell's law of refraction states that n'sin(arctan(t' / n')−arct¯an t)=n sin(arctan(t / n)−arct¯an t)⇔t'−n't¯1+(t' / n')2=t−nt¯1+(t / n)2 is what, after solving for t', yields t'(x,t)=n'bn−wz¯(1)w+bnz¯(1) where w=n'2−n2b2,b=t+nz¯(1)n2+t21+z¯(1)2 is, and so (1) is evaluated at position x(x, t).

[0136] Once x(x, t) and t'(x, t) are known, it is possible to Fig. 7 can be read directly from the geometric picture that x'(x,t)=x¯(x,t)−t'(x,t)n'z¯(x¯(x,t)) must be. The ray transfer function for refraction is therefore (x't')=fref(x,t)=(x¯(x,t)−t'(x,t)z¯(x¯(x,t)) / n't'(x,t))

[0137] According to the invention, a method for calculating the function x(x, t) is also provided. The direction of the incident ray satisfies the relationship tan α = t / n and therefore tn=x¯(x,t)−xz¯(x¯(x,t))

[0138] Although Eq. (16) cannot be solved closed for x(x, t), one can set up the partial derivatives with respect to (x, t) of Eq. (16) and successively apply these to the derivatives x (1,0) x (0,1) , x (2,0) , x (1,1) , x (0,2) , x (3,0) , etc. solve at position (x, t) = (0,0).

[0139] Another embodiment, which has the advantage of a more compact notation, consists of choosing a suitable approach x Ansatz (x, t) and the introduction of the function x¯(x,t)=x¯approach(x,t)+δx¯(x,t)

[0140] in Eq. (16). The function x Ansatz (x, t) is called the consistent solution of Eq. (16) of order k, and k is called the consistency order of the function x Ansatz (x, t) denotes, if (k + 1) is the lowest order for which one of the derivatives δx (1,0) δx (0,1) , δx (2,0) , δx (1,1) , δx (0,2) , δx (3,0) , etc. does not disappear. The following Table 2 shows the different implementations of the function x. Ansatz (x, t) and their order consistency. Table 2: x Ansatz (x, t) Order consistency x 2 x + t / nz (x) 4 x + t / nz(x + t / nz(x)) 6 x+t / n z¯(x)1−t / n z¯(1)(x) 6

[0141] Analogous to Table 1, the derivatives of the ray transfer function f are shown in Table 3 below. ref (x, t) shown for refraction. In contrast to the case of f prop (x, t) breaks the table for f ref (x, t) ia does not occur at any finite order. Except for fxref(1,0)=1 and ftref(0,1)=1 All entries in Table 3 are proportional to (n - n'). This is to be expected because f ref (x, t) for n' = n must be reduced to the identity using the Jacobian matrix Jac f ref = 1. Table 3: Derivatives of the ray transfer function fref (x, t) at the point (x, t) = (0,0) Order of the wavefront Abl.-Ord. n x + n t xOrd.n x tOrd.nt fxref(nx,nt) ftref(nx,nt) 1 0 0 0 0 0 2 1 1 0 1 -(n' - n)z (2) 2 1 0 1 0 1 3 2 2 0 0 - (n' - n)z (3) 3 2 1 1 0 0 3 2 0 2 0 0 4 3 3 0 (n − n) × 3 / n (2)2 -(n' - n) (z (4) + 3n / n' 2 (n- n')z (2)3 ) 4 3 2 1 (n' - n) / (nn')z (2) -(n' - n) / (nn' 2 (3n) 2 - nn'+ n' 2 z (2)2 4 3 1 2 0 -(n' - n) / (nn' 2 (3n + n')z (2) 4 3 0 3 0 -(n' - n) × 3 / (n 2 n' 2 )(n + n') Actual calculation by the optical system

[0142] The actual goal is to find a function t'(x') for the emitted light that, for a given function t(x), satisfies the equation f(x,t(x))=(fx(x,t(x))ft(x,t(x)))=(x'(x)t'(x'(x))) fulfilled.

[0143] This goal is preferably achieved by introducing intermediate variables. u(x):=x'(x)v(x):=t'(x'(x)) achieved, with which Eq.(18) the form (u(x)v(x))=f(x,t(x)) assumes. A combination of the two equations (19) and (20) yields v(x)=t'(u(x)) which represents the starting point of the procedure. The basic principle consists of repeatedly differentiating Eq. (21) v(1)=u(1)t'(1)v(2)=(u(1))2t'(2)+u(2)t'(1)v(3)=(u(1))3t'(3)+3u(1)u(2)t'(2)+u(3)t'(1)⋮ where, for the sake of readability, the argument '(0)' has been omitted. The first step towards achieving the goal is to successively solve equation (22) for the desired derivatives t'. (1) , t' (2) , t' (3) ,... to solve and thus express in terms of the derivatives of u and v: t'(1)=v(1)u(1)t'(2)=u(1)v(2)−u(2)v(1)(u(1))3t'(3)=u(1)v(3)−u(3)v(1)(u(1))4−3u(2)u(1)v(2)−u(2)v(1)(u(1))5⋮

[0144] The second step is to express the derivatives of u and v in terms of derivatives of the function t(x) (incident light) and derivatives of f(x,t) (properties of the optical system). Differentiating Eq. (20) leads to the following for u (i) on u(1)=fx(0,1)t(1)+fx(1,0)u(2)=fx(1,0)t(2)+fx(2,0)+2fx(1,1)t(1)+fx(0,2)t(1)2u(3)=fx(0,1)t(3)+fx(3,0)+3(fx(2,1 )+fx(1,2)t(1))t(1)+3(fx(1,1)+fx(0,2)t(1))t(2)+fx(0,3)t(1)3⋮v(1)=u(1)(fx→ft)v(2)=u(2)(fx→ft)v(3)=u(3)(fx→ft)⋮ where again the arguments '(0)' and '(0,0)' are omitted. A preliminary result for t' (i) , expressed by t (i) , is then obtained by substituting Eq.(24) into Eq.(23). Solutions in t-representation

[0145] Substituting Eq.(24) into Eq.(23) for t' (i) If one, then with increasing order i, a very large number of similar terms with mixed terms from powers of derivatives t quickly arise. (i) , whose evaluation when numerical values ​​are substituted for t (i)requires a lot of computation time. Therefore, the task of establishing a computationally time-saving method that can be repeatedly evaluated with many different wavefronts is not yet solved by simply substituting Eq. (24) into Eq. (23). Rather, according to the invention, it has been recognized that the symbolic expressions that describe the dependence of the solution t' (i) of the derivatives t (i) describes how, before inserting numerical values, the terms must be sorted and summarized in such a way that only the minimum number of mixed terms consisting of powers of derivatives t remains. (i) must be evaluated numerically.

[0146] The order i = 1 directly results in a fraction. t'(1)=ft(0,1)t(1)+ft(1,0)fx(0,1)t(1)+fx(1,0)=Dt(1)+CBt(1)+A=:β(Dt(1)+C) where, for the sake of brevity, the abbreviation 1β:=u(1)=fx(0,1)t(1)+fx(1,0)=Bt(1)+A can use.

[0147] The next higher order t' (2) Eq. (23) already yields: t'(2)=β3(u(1)v(2)−u(2)v(1))=β3[(Bt(1)+A)(Dt(2)+ft(2,0)+2ft(1,1)t(1)+ft(0,2)t(1)2)−(Dt(1)+C)( Bt(2)+fx(2,0)+2fx(1,1)t(1)+fx(0,2)t(1)2)]=β3[(AD−BC)t(2)+(BD−DB)t(1)t(2)+(Aft(2,0)−Cfx(2,0))+ (Bft(2,0)+2Aft(1,1)−Dfx(2,0)−2Cfx(1,1))t(1)+(Aft(0,2)+2Bft(1,1)−Cfx(0,2)−2Dfx(1,1))t(1)2+(Bft (0,2)−Dfx(0,2))t(1)3]=β3[t(2)+(Aft(2,0)−Cfx(2,0))+(Bft(2,0)+2Aft(1,1)−Dfx(2,0)−2Cfx(1,1))t(1) +(Aft(0,2)+2Bft(1,1)−Cfx(0,2)−2Dfx(1,1))t(1)2+(Bft(0,2)−Dfx(0,2))t(1)3] That is, for example, four contributions to the power t. (1)2 , which can be factored out, and whose prefactors combine to form the prefactor (Aft(0,2)+2Bft(1,1)−Cfx(0,2)−2Dfx(1,1)) summarize.

[0148] According to the invention, the method can be applied regardless of the value of the determinant det T = AD - BC. Preferably, the invention exploits the fact that optical systems are symplectic and satisfy det T = AD - BC = 1.

[0149] Continued substitution and summarizing leads to solutions of the structure t'(1)=β[c¯1,1t(1)+c¯1,0]t'(2)=β3[t(2)+c¯2,0+c¯2,1t(1)+c¯2,2t(1)2+c¯2,3t(1)3]t'(3)=β4[t(3)+β((c¯3,0+c¯3, 1t(1)+c¯3.2t(1)2+c¯3.3t(1)3+c¯3.4t(1)4+c¯3.5t(1)5)+(c¯3.01+c¯3.11t(1)+c¯3.21t(1)2)t(2)+c¯3.02t(2)2)]t'( 4)=β5[t(4)+β2((c¯4.0+c¯4.1t(1)+…+c¯4.7t(1)7)+(c¯4.01+c¯4.11t(1)+…+c¯4.41t(1)4)t(2)+(c¯4.02+c¯4.12t(1)+c ¯4.22t(1)2)t(2)2+c¯4.03t(2)3)+β((c¯4.001+c¯4.101t(1)+c¯4.201t(1)2)t(3)+c¯4.011t(2)t(3))]t'(5)=β6[t(5)+…]

[0150] In general, the solutions are t' (i) given by a summation approach of the form t'(i)=β−r1−0(i)∑k1,k2,…,kic¯ikβ−Δr¯1(i,k*)t(1)k1t(2)k2⋯t(i)ki=β−r1−0(i )[t(i)+∑k1,k2,…,ki−1c¯ikβ−Δr¯1(i,k*)t(1)k1t(2)k2⋯t(i−1)ki−1],i=1,2,3,… with coefficients c ik given, where the bottom line only applies in the symplectic case, and where k = (k1, k2,..., k i ) a tuple k∈N0i from exponents; k* = (k2,..., k i ) is a tuple formed from k by omitting the first element, k** = (k3, .... k j ) is created from k by omitting the first two elements.

[0151] The exponents of β are given by −r1−0(i)=(i+1)−δi1−Δr¯1(i,k*)=(i−2)+δi1−P(k**)

[0152] The coefficients c ik are given in Table 4. For better readability, a shorthand notation can be used. TAC(nx,nt):=Aft(nx,nt)−Cfx(nx,nt)TBD(nx,nt):=Bft(nx,nt)−Dfx(nx,nt) as well as a symmetry transformation X, which is every derivative fx(nx,nt),ft(nx,nt) by X(f x (nx,nt) ) or X(f t (nx,nt) ) replaced and in an exchange of orders n x , n t as well as a sign: X(fx(nx,nt))=(−1)nx+ntfx(nt,nx)

[0153] Eq. (32) directly implies X(A) = -B, X(B) = -A, X(C) = D, X(D) = C, and for the expressions in Eq. (31) X(TAC(nx,nt))=(−1)nx+ntTBD(nt,nx)

[0154] The summation in Eq. (29) is constructed such that the tuple k* over a domain k* ∈ P ∪ (i - 1) runs, which depends only on the order i, where the set P ∪ is defined by P∪(p):={k∈N0p|P(k)≤p}=∪q=1pP(q)where P(p):={k∈N0p|P(k)=p}=∪m=1pP(p,m),where P(p,m):={k∈N0p|P(k)=p∧M(k)=m}

[0155] The numbers used in Eq. (34) are the quantity M(k) and the partition order P(k) and are defined by M(k):=∑v=1pkv=k1+k2+…+kpP(k):=∑v=1pvkv=k1+2k2+…+pkp

[0156] The index k1 runs over the range 0 ≤ k1 ≤ 2(i - P(k*) - 1) + δ P(k*),0 , which depends on the order i and k*.

[0157] As an alternative to representing a tuple by specifying numbers, it can also be represented graphically, preferably using Young diagrams (see Fig. 7) A particularly preferred form of this representation consists of choosing, for a given tuple k, the diagram such that the number of boxes is equal to P(k). If i max If the index (i.e., the order) of the highest non-zero entry of k is given, then the Young diagram contains, as in the Fig. 8 shown, on its left side a rectangle made of i max lines and k imax Columns. All further columns of the diagram are added from the right, next a rectangle made of (i max -1) lines and k imax-1 columns, and the diagram ends on its right side with a rectangle consisting of one row and k1 columns.

[0158] The coefficients c are listed in Table 4 below. ikspecified for general optical systems, as well as for simple propagation over a distance d = t / n, and for propagation through a single surface with surface derivatives z (2) , z (3) ,z (4) , ... between two media with refractive indices n and n'. Table 4: Order Indices Prefactor term coefficient c ik ≡ c i(k1,k*) i k1 k* β -Δii term symbol General case simple propagation, distanceτ / n Refraction at a single surface ( β = 1) 1 0 0 1 1 c 1,0 C 0 - (n'-n)z (2) 1 1 0 1 t (1) c 1,1 X(c 1,1 ) = D 1 1 2 0 0 1 1 c 2,0 TAC(2,0) 0 - (n'-n)z (3) 2 1 0 1 t (1) c 2,1 2TAC(1,1)+TBD(2,0) 0 0 2 2 0 1 (t (1) ) 2 c̅ 2,2 X(c¯2,1)=TAC(0,2)+2TBD(1,1) 0 0 2 3 0 1 (t (1) ) 3 c 2,3 X(c¯2,0)=TBD(0,2) 0 0 2 0 1 1 t (2) c 2,01 1 1 1 3 0 0 β 1 c 3,0 −3TAC(2,0)fx(2,0)+ATAC(3,0) 0 −(n'−n)(z¯(4)+3(n2 / n'2−1)z¯(2)3) 3 1 0 β t (1) c 3,1 −6TAC(2,0)fx(1,1)−3(TBD(2,0)+2TAC(1,1))fx(2,0)+3ATAC(2,1)+BTAC(3,0)+ATBD(3,0) 0 - 3(n'-n)(n'+3n) / n' 2 z (2)3 2 0 β (t (1) ) 2 3 3A(TAC(1,2)+TBD(2,1))+B(3TAC(2,1)+TBD(3,0))−3TAC(2,0)fx(0,2)6(2TAC(1,1)+TBD(2,0))fx(1,1)−3(TAC(0,2)+2TBD(1,1))fx(2,0) 0 - 3(n'-n)(2n'+3n) / (nn' 2 z (2) 3 3 0 β (t (1) ) 3 c 3,3 X(c 3.2 ) 0 - 3(n'-n)(n'+n) / (n 2 n' 2 ) 3 4 0 β (t (1) ) 4 c 3,4 X(c 3,1 ) 0 0 3 5 0 β (t (1) ) 5 c 3,5 X(c 3,0 ) 0 0 3 0 1 β t (2) c 3,01 −3(BTAC(2,0)−ATAC(1,1)+fx(2,0)) 0 0 3 1 1 β t (1) t (2) c 3,11 −3(BTBD(2,0)−ATAC(0,2)+3fx(1,1)) 0 0 3 2 1 β (t (1)2 t (2) c 3,21 X(c 3,01 ) 0 0 3 0 2 β (t (2) ) 2 c 3,02 -3B - 3r / n 2 0 3 0 01 β t (3) c 3,001 1 1 1 4 0 0 β 2 1 c 4,0 15TAC(2,0)(fx(2,0))2 +A(ATAC(4,0)−6TAC(3,0)fx(2,0) −4TAC(2,0)fx(3,0) 0 −(n'−n)(z¯(5)+2(n−n') / n'2×(9n+11n')z¯(2)2z¯(3)) 4 1 0 β 2 t (1) c 4,1 −2(9ATAC(2,1)+5BTAC(3,0)+ATBD(3,0))fx(2,0)+15(2TAC(1.1)+TBD(2,0))(fx(2,0))2−12(ATAC(3,0)−5TAC(2,0)fx(2,0))fx(1,1)+A(4ATAC(3,1)+2BTAC(4,0)+ATBD(4,0)−12TAC(2,0)fx(2,1)−8(TAC(1,1)+TBD(2,0))fx(3,0))fx(1,1) 0 -2(n' - n) (7n' + 18n) / n' 2 With (2) With (3) 4 2 0 β 2 (t (1) ) 2 c 4,2 A2(6TAC(2,2)+4TBD(3,1))+2AB(4TAC(3,1)+TBD(4,0))+60TAC(2,0)(fx(1,1))2−6(ATAC(3,0)−5TAC(2,0)fx(2,0))+B2TAC(4,0)+4(−9ATAC(2,1)−BTAC(3,0)−5ATBD(3,0)+15(2TAC(2,1)+TBD(2,0)fx(2,0)))fx(1, 1)−12ATAC(2,0)fx(1,2)+3((2A(3TAC(1,2)+5TBD(2,1))+2B(TAC(2,1)+TBD(3,0))−5(TAC(0,2)+2TBD(1,1))fx(2,0))fx(2,0)+8(ATAC(1,1)+BTAC(2,0))fx(2,1))−4(ATAC(0,2)+B(4TAC(1,1)+TBD(2,0)))fx(3,0) 0 -2(n' - n) ·(5n' + 9n) / (nn' 2 z (3) 4 3 0 β 2 (t (1) ) 3 c 4,3 A2(4TAC(1,3)+6TBD(2,2))+4AB(3TAC(2,2)+2TBD(3,1))+B2(4TAC(3,1)+TBD(4,0))−ATAC(2,0)ƒx(0,3)+2(−5BTAC(3,0)−A(9TAC(2,1)+TBD(2,0))+30TAC(2,0)ƒx(1,1)+15(2TAC(1,1)+TBD(2,0))ƒx(2,0))ƒx(0,2)+3(20(2TAC(1,1)+TBD(2,0))(ƒx(1,1))2−8(ATAC(1,1)+BTAC(2,0))ƒx(1,2)+(−2(ATAC(0,3)+BTBD(1,2)+5ATBD(1,2)+3BTBD(2,1))+5TBD(0,2)ƒx(2,0))ƒx(2,0)−4(3ATAC(1,2)+BTAC(2,1)+5ATBD(2,1)+BTBD(3,0)−5(TAC(0,2)+2TBD(1,1))ƒx(2,1))−4(ATAC(0,2)+4BTAC(1,1)+BTBD(2,0))ƒx(2,1))−8(ATBD(0,2)+BTx(1,1))ƒx(3,0) 0 0 4 4 0 β 2 (t (1) ) 4 c 4,4 X(c 4,3 ) 0 0 4 5 0 β 2 (t (1) ) 5 c 4,5 X(c 4,2 ) 0 0 4 6 0 β 2 (t (1) ) 6 c 4,6 X(c 4,1 ) 0 0 4 7 0 β 2 (t (1) ) 7 c 4,7 X(c 4,0 ) 0 0 0 1 β 2 4t (2) c 4,01 6A2TAC(1,1)−12ATAC(2,0)ƒx(1,1)+30BTAC(2,0)ƒx(2,0)−18ATAC(1,1)ƒx(2,0)+(ƒx(2,0))2−6ABTAC(3,0)+4Aƒx(3,0) 0 - 6(n'-n)(2n'+3n) / n' 2 z (2)2 4 1 1 β 2 t (1) t (2) c 4,11 −60ATAC(1,1)ƒx(1,1)+12A2TAC(1,2)ƒx(1,1)+30BTBD(2,0)ƒx(2,0)−10B2TAC(3,0)−2ABTBD(3,0)−18ATAC(0,2)ƒx(2,0)−12ATAC(2,0)ƒx(0,2)+24BTAC(1,1)ƒx(2,0)+36BTAC(0,2)ƒx(1,1)+90ƒx(2,0)ƒx(1,1)−18Aƒx(21,1) 0 -6(n' - n) (5n'+ 6n) / (nn' 2 )With (2) 4 2 1 β 2 (t (1) ) 2 t (2) c 4,21 6(A2TAC(0,3)−3B2TAC(2,1)−B2TBD(3,0)−7ATAC(1,1)ƒx(0,2)−8ATAC(0,2)ƒx(1,1)+8BTBD(2,0)ƒx(1,1)+7BTBD(1,1)ƒx(2,0)+20(ƒx(1,1))2−Aƒx(1,2)+AB(ƒx(0,2)ƒt(2,0)−ƒx(2,0)ƒt(0,2))+10ƒx(2,0)ƒx(0,2)−Bƒx(2,1)) 0 -18(n' - n) ·(n'+ n) / (n 2 n' 2 ) 4 3 1 β 2 (t (1) ) 3 t (2) c 4,31 X(c 4,11 ) 0 0 4 4 1 β 2 (t (1) ) 4 t (2) c 4,.41 X(c 4,01 ) 0 0 4 0 2 β 2 (t (2) β 2 c 4,02 3(A2TAC(0,2)−6ABTAC(1,1)+5B2TAC(2,0)−4Aƒx(1,1)+10Bƒx(2,0)) 0 0 4 1 2 β 2 (t (1) )(t (2) ) 2 c 4,12 3(−AB(3TAC(0,2)+2TBD(1,1))+5B2TBD(2,0)−5Aƒx(0,2)+20Bƒx(1,1)) 0 0 4 2 2 β 2 (t (1) ) 2 (t (2) ) 2 c 4,22 6B(−2ATBD(0,2)+2BTBD(1,1)+3ƒx(0,2)) 0 0 4 0 01 β t (3) c̅ 4,001 2(2ATAC(1,1)−2BTAC(2,0)−3ƒx(2,0)) 0 0 4 1 01 β t (1) t (3) c̅ 4,101 4(ATAC(0,2)−BTBD(0,2)−4ƒx(1,1)) 0 0 4 2 01 β (t (1) ) 2 t (3) c̅ 4,201 X(c̅ 4,001 ) 0 0 4 0 3 β 2 (t (2) ) 3 c̅ 4,03 15.B 2 15t 2 / n 4 0 4 0 11 β t (2) t (3) c̅ 4,011 -10B -10t / n 2 0 4 0 001 1 t (4) c̅ 4,0001 1 1 1 5 0 0 β 3 1 c̅ 5,0 ... 0 −(n'−n)×(z¯(6)+(…)z¯(2)5+(…)z¯(2)2z¯(4)+(…)z¯(2)z¯(3)2) ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 5 0 0 1 1 c̅ 5,00001 1 1 1 6 0 0 β 4 1 c̅ 6,0 0 −(n'−n)×(z¯(7)+(…)z¯(2)4z¯(3)+(…)z¯(2)2z¯(5)+(…)z¯(2)z¯(3)z¯(4)+(…)z¯(3)3) Solutions in w-representation

[0159] The solutions for the derivatives of the wavefronts w' (p) obtained by applying the transformation H (see Fig. 3) to find the derivatives w' (2) , w' (3) , w' (4) , ... the failing wavefront as a function of the derivatives w (2) , w (3) , w (4) , ...to describe the incoming wavefront. For this purpose, Eq. (9) can be applied to w' (p) , t' (i) instead of w (p) , t (i) apply and receives w'(2)=−t'(1) / n',w'(3)=−t'(2) / n',m'(4)=−(t'(3)+3t'(1)3 / n'2) / n',….

[0160] After that, all derivatives t' can be calculated. (i) Substitute the solutions of the t-representation from Eq. (28), (29) on the right side of these equations and thus replace them with the derivatives t (k) express this. Finally, one applies the inverse transformation H. -1 by taking all derivatives t (k) through corresponding functions of derivatives w (k) replaced according to Eq.(8).

[0161] The result does not yet solve the problem of optimally short computation time during evaluation, because of the transformations H, H -1 Mixed terms arise, which occur frequently. According to the invention, these can be combined again in the form E'2=β[b¯2,1E2+b¯2,0]E'3=β3[E3+b¯3,0+b¯3,1E2+b¯3,2E22+b¯3,3E23]E'4=β4[E4+β((b¯4,0+b¯4, 1E2+b¯4,2E22+b¯4,3E23+b¯4,4E24+b¯4,5E25)+(b¯4,01+b¯4,11E2+b¯4,21E22)E3+b¯4,02E32)]E'5 =β5[E5+β2((b¯5,0+b¯5,1E2+…+b¯5,7E27)+(b¯5,01+b¯5,11E2+…+b¯5,41E24)E3+(b¯5,02+b¯5,12E2 +b¯5.22E22)E32+b¯5.03E33)+β((b¯5.001+b¯5.101E2+b¯5.201E22)E3+b¯5.011E3E4)]E'6=β6[E6+…] where β=(−BE2+A)−1 is, and where, for easier interpretation using ophthalmic sizes, the notation E' is used. p = n'w' (p) , E p = nw (p) was used (by definition, the lowest non-zero order corresponding to the curvature belongs to order p = 2 in the w-representation, but to i = 1 in the t-representation). A comparison of Eq. (36) with Eq. (28) shows that the solutions E' p for p = i + 1 exhibits exactly the same structure as the solutions t' (i) . In fact, equations (36) can be combined using a summation approach: E'p=β−r¯lo(p−1)∑k1,…,k2,kp−1b¯pkβ−Δr¯1(p−1,k*)E2k1E3k2⋯Epkp−1=β−r¯lo (p−1)[Ep+∑k1,…,k2,kp−1b¯pkβ−Δr¯1(p−1,k*)E2k1E3k2⋯Ep−1kp−2],p=2,3,4,… where the bottom row again only applies in the symplectic case. The coefficients b̅ pk arise from the coefficients c̅ ik as a result of the transformations H, H -1 and can be traced back to these, as shown in Table 5 below. Table 5: Order Indices Prefactor Aberrations Coefficient b̅ pk = b p,(k1,k*) p k1 k* β - Δr1 term symbol General case Simple propagation, distance τ / n Refraction at a single surface (β = 1) 2 0 0 1 1 b̅ 2,0 -C 0 (n' - n)z̅ (2) 2 1 0 1 E2 b̅ 2,1 D 1 1 3 0 0 1 1 b̅ 3,0 −c¯2,0=−TAC(2,0) 0 (n' - n)z̅ (3) 3 1 0 1 E2 b̅ 3,1 +c¯2,1=2TAC(1,1)+TBD(2,0) 0 0 3 2 0 1 E22 b̅ 3,2 −c¯2,2=−X(c¯2,1)=−(TAC(0,2)+2TBD(1,1)) 0 0 3 3 0 1 E23 b̅ 3,3 +c¯2,3=X(c¯2,0)=TBD(0,2) 0 0 3 0 1 1 E3 b̅ 3,01 1 1 1 4 0 0 β 1 b̅ 4,0 -c̅ 3,0 - 3A 2 C 3 / n' 2 0 (n'−n)(z¯(4)−6n(n'−n) / n'2z¯(2)3) 4 1 0 β E2 b̅ 4,1 +c̅ 3,1 + 3AC 2 (2BC + 3AD) / n' 2 0 6(n'- n)(n'- 3n) / n' 2 z̅ (2)2 4 2 0 β E22 b̅ 4,2 -c̅ 3,2 - 3C(B 2 C 2 + 6ABCD + 3A 2 D 2 ) / n' 2 0 6(n'- n)(n'+ 3n) / (nn' 2 )z̅ (2) 4 3 0 β E23 b̅ 4,3 +c̅ 3,3 - 3A / n 2 + 3 D(3B 2 C 2 + 6ABCD + A 2 D 2 ) / n' 2 0 -6(n'- n)(n'+ n) / (n 2 n' 2 ) 4 4 0 β E24 b̅ 4,4 -c̅ 3,4 + 3B / n 2 - 3BD 2 (3BC + 2AD) / n' 2 -3τ / n 4 0 4 5 0 β E25 b̅ 4,5 +c̅ 3,5 + 3B 2 D 3 / n' 2 3τ 2 / n 6 0 4 0 1 β E3 4.01 +c̅ 3,01 0 0 4 1 1 β E2E3 b̅ 4,11 -c̅ 3,11 0 0 4 2 1 β E22E3 b̅ 4,21 +c̅ 3,21 0 0 4 0 2 β E32 b̅ 4,02 -c̅ 3,02 = 3B 3τ / n 2 0 4 0 01 β E4 b̅ 4,001 1 1 1 5 0 0 β 2 1 b̅ 5,0 −c¯4,0−22A2C2TAC(2,0) / n'2 0 (n'−n)(z¯(5)−40n×(n'−n) / n'2z¯(2)2z¯(3)) 5 1 0 β 2 E2 b̅ 5,1 +c¯4,1+22AC(2(AD+BC)TAC(2,0)+AC(2TAC(1,1)+TBD(2,0)) / n'2 0 10(n'−n)(3n'−8n) / n'2z¯(2)z¯(3) 5 2 0 β 2 E22 b̅ 5,2 −c¯4,2−22((A2D2+6ABCD+B2C2)TAC(2,0)+AC×(4(AD+BC)TAC(1,1)+AC(TAC(0,2)+2TBD(1,1))−BCTBD(2,0)−2Dfx(2,0))) / n'2 0 10(n' - n)(n' + 4n) / (nn' 2 )z̅ (3) 5 3 0 β 2 b¯5,3 +c¯4,3+22(2ABD2TAC(2,0)+AC×(2BC(TAC(0,2)+2TBD(1,1))+A(2DTAC(0,2)+CTBD(0,2)))+(A2D2+6ABCD+B2C2)(2TAC(1,1)+TBD(2,0))) / n'2+6A(2BTAC(2,0)−2ATAC(1,1)+3fx(2,0)) / n2 0 0 5 4 0 β 2 E24 b̅ 5,4 -c̅ 4,4 - 0 0 +6(4B2TAC(2,0)−2A2(TAC(0,2)+TBD(1,1))+6Afx(1,1)+Bfx(2,0)) / n2 5 5 0 β 2 E25 b̅ 5,5 +c¯4,5+22((A2D2+6ABCD+B2C2)TBD(0,2)+BD×(BD(6TAC(1,1)+TBD(2,0))+2ADTAC(0,2)+4BCTBD(1,1)+2(Cfx(0,2)−2DBfx(1,1)))) / n'2+6(4A2TBD(0,2)−2B2(TAC(1,1)+TBD(2,0))−Afx(0,2)−6Bfx(1,1)) / n2 0 0 5 6 0 β 2 E26 b̅ 5,6 −c¯4,6−22BD(B(3DTAC(0,2)+2DTBD(1,1)+2CTBD(0,2))−2Dfx(0,2)) / n'2+6B(2BTBD(1,1)−2ATBD(0,2))+3fx(0,2)) / n2 0 0 5 7 0 β 2 E27 b̅ 5,7 +c¯4,7+22B2D2TBD(0,2) / n'2 0 0 5 0 1 β 2 E3 b̅ 5,01 +c̅ 4,11 + 22A 2 C 2 / n' 2 0 10(n'- n)(n'- 4n) / (nn' 2 )z̅ (2)2 5 1 1 β 2 E2E3 b̅ 5,11 -c̅ 4,11 - 44AC(AD + BC) / n' 2 0 10 (n' - n)(3n' + 8n) / (nn' 2 )z̅ (2) 5 2 1 β 2 E22E3 b̅ 5,21 +c̅ 4,21 + 22((A 2 D 2 + 4ABCD + B 2 C 2 ) / n' 2 - A 2 / n 2 ) 0 -40(n'- n)(n'+ n) / (n 2 n' 2 ) 5 3 β 2 E23E3 b̅ 5,31 -c̅ 4,31 + 2B(22D (AD + BC) / n' 2 + 7A / n 2 ) 0 0 5 4 1 β 2 E24E3 b̅ 5,41 +c̅ 4,41 + 2B 2 (11D 2 / n' 2 + 4 / n 2 ) 0 0 5 0 2 β 2 E32 b̅ 5,02 -c̅ 4,02 0 0 5 1 2 β 2 E2E32 b̅ 5,12 +c̅ 4,12 0 0 5 2 2 β 2 E22E32 b̅ 5,22 -c̅ 4,22 0 0 5 0 01 β E4 b̅ 5,001 +c̅ 4,001 0 0 5 1 01 β E2E4 b̅ 5,101 -c̅ 4,101 0 0 5 2 01 β E22E4 b̅5, 201 +c̅ 4,201 0 0 5 0 3 β 2 E33 b̅ 5,03 15 B 2 15π 2 / n 4 0 5 0 11 β E3E4 b̅ 5,011 -10B -10π / n 2 0 5 0 001 1 E5 b̅ 5,0001 1 1 1 6 0 0 β 3 1 b̅ 6,0 ... 0 (n'−n)×(z¯(6)+(…)z¯(2)5+(…)z¯(2)2z¯(4)+(…)z¯(2)z¯(2)3) ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 6 0 0 1 1 b̅ 6,00001 1 1 1 7 0 0 β 4 1 b̅ 7,0 0 ... Example of implementation

[0162] A simple example is the passage of light through a meridian of a Bennett and Rabbetts eye model, i.e., a modified Gullstrand-Emsley eye (mGE eye) adapted for biometric studies (see R. Rabbetts: "Bennett & Rabbetts' Clinical Visual Optics", Butterworth Heinemann Elsevier Health Sciences, 2007, ISBN: 9780750688741). The eyeball contains a cornea with refractive power S C, as well as an eye lens with refractive powers L1 and L2 on the anterior and posterior surfaces, respectively. The radii of curvature of the cornea and the two lens surfaces are r C , r1 and r2. The distance between the cornea and the anterior lens surface is determined by the anterior chamber depth d. CL Given that the lens thickness is determined by d L given, and the vitreous depth by d LR The refractive indices of the chamber fluid are n CL , the lens of the eye L , and that of the vitreous body n LR According to current technology, the following values ​​are commonly used for biometric parameters: dCL=3.6mm,dL=3.7mm,dLR=16.7859082mm,rC=7.8mm,r1=11.0mm,r2=−6.47515m m,nCL=1.336nL=1.422nLR=1.336z¯C(2)=1 / rC=128.2051282m−1,SC=(nCL−1)z¯C (2)=43.076923dptz¯L1(2)=1 / r1=90.9090909m−1,L1=(nL−nCL)z¯L1(2)=7.8181 8181dptz¯L2(2)=1 / r2=−154.436576m−1,L2=(nLR−nL)z¯L2(2)=13.28154560dpt

[0163] In Eq. (39) none of the values ​​with more than three decimal places come from a direct measurement, but from consistency considerations with which the model is exactly emmetropic, i.e. a plane wavefront must be mapped onto a spherical wave that converges exactly on the retina.

[0164] Since both the Gullstrand-Emsley eye and the Bennett-Rabbetts eye are paraxial according to the prior art and therefore only specified up to second-order wavefront aberrations, these models are not directly suitable as embodiments for calculating higher-order aberrations unless a suitable specification for the HOA (Higher Optical Aspect Ratio) is added. For the purposes of the present embodiment, the emmetropic eye property should be retained even for the HOA. Preferably, both lens surfaces, including the HOA, are exact spherical surfaces, whereas the HOA of the cornea must then be selected such that the eye fulfills the emmetropy requirement mentioned above.

[0165] For every spherical surface, the odd derivatives of the function z̅(x) vanish, whereas the even derivatives vanish due to the curvature z̅. (2) are clearly defined in the form of z¯(4)=3(z¯(2))3,z¯(6)=45(z¯(2))5

[0166] Numerically, this means four additional biometric parameters. z¯L1(4)=2.253944×106m−3,z¯L1(6)=2.794145×1011m−5z¯L2(4)=−1.105024×107m−3,z¯L2(6)=−3.953332×1012m−5

[0167] The method according to the invention can now be used to obtain the higher orders. z¯C(4),z¯C(6) to choose the cornea such that the wavefront created after refraction at the back surface of the lens converges exactly onto the retina, i.e., exactly a spherical wave with radius d LR The criterion is that an incident plane wave (w) (k) = 0) on the failing derivatives w'(4)=3(w'(2))3, w'(6)=45(w'(2))5 leads, analogously to Eq. (40). A particularly preferred alternative to derivatives w' (p) Aberrations E' are used to describe wavefronts p = n'w' (p) , so in the case of the spherical shaft E'4 / n'=3(E'2 / n')3,E'6 / n'=45(E'2 / n')5E'4=3nLR−2E'23,E'6=45nLR−4E'25 where n' = n LR . On the other hand, the aberrations E'2, E'4, E'6 are obtained from Eq. (38) and Eq. (36) depending on the optical system, and an incident plane wave means that the aberrations E p = 0 for all orders p. Therefore, for each order p, only the lowest coefficient b̅ carries weight. pk for k = 0 at: E'2=βb¯2.0, E'4=β4βb¯4.0, E'6=β6β3b¯6.0 where b̅ 2,0 = -C from Table 5, and where further b̅ 4,0 a function of the parameter to be determined z¯C(4) is, where b̅ 6,0 a function of the two parameters to be determined z¯C(4),z¯C(6) is. By substituting Eq. (44) into Eq. (43), the requirement is obtained. b¯4.0=3(βnLR)−2b¯2.03b¯6.0=45(βnLR)−4b¯2.05

[0168] Substituting all known numerical values ​​into Table 5 yields the coefficients b¯4.0(z¯C(4))=2.11536×105m−3+0.253296z¯C(6)b¯6.0(z¯C(4),z¯C(6))=2.2 04136×1011m−5−3.36879×104m−2z¯C(4)+3.36306m(z¯C(4))2+0.143949z¯C(6) where β for E2 = 0 according to Eq. (37) is β = A -1 reduced, and the values ​​for A and C are determined using Eq. (49).

[0169] Equations (46) are replaced by z¯C(4)=−2.05536×104m−3,z¯C(6)=−1.51137×1012m−5 solved, so that together with Eq. (41) and Eq. (39) all sixth-order biometric parameters of the mGE eye are determined. The way in which Eq. (47) is derived demonstrates the technical advantages of the invention. Even though the same result can also be obtained by repeated application of the prior art method, the approach via the coefficients b̅ 4,0 , b̅ 6,0 more direct and suitable for determining the dependency on the parameters z¯C(4),z¯C(6) to represent, while the influence of all other refractions and propagations at this moment has already been summarized and pre-evaluated.

[0170] Based on the parameter values, all derivatives of the involved ray transfer functions up to order p = 6 can be combined. Table 6 below shows all derivatives of the ray transfer functions of the refracting components of the mGE eye, Table 7 those for the propagations, and Table 8 those of the entire mGE eye. The derivatives of the ray transfer functions f C,ref , f L1,ref , f L2,ref The refractions can be determined from Table 3 by substituting the values ​​for the beam transfer functions f. e,prop , f dCL,prop , f dL,prop the propagations based on Table 1. Since the mGE eye is symmetrical, these derivatives vanish for odd orders p (i.e., for even values ​​of the sum n). x + n t(of the orders of the derivatives), so that tables 6, 7 and 8 only contain entries for even orders.

[0171] Table 6: Derivatives of the ray transfer functions f C,ref , f L1,ref , f L2,ref the refractions of the components of the eye Order Cornea S C Front surface of lens L1 Rear surface of lens L2 p n x n t ƒxC,ref(nx,nt) ƒtC,ref(nx,nt) ƒxL1,ref(nx,nt) ƒtL1,ref(nx,nt) ƒxL2,ref(nx,nt) ƒtL2,ref(nx,nt) 2 1 0 1 -43.0769 diopters 1 -7.81818 diopters 1 -13.2815 diopters 2 0 1 0 1 0 1 0 1 4 3 0 12401.2 m -2 4.067615*10 5 m - 3 1499.46 m -2 -1.82825*10 5 m -3 -4605.89 m -2 -1.01543*10 6 m -3 4 2 1 32.2432 m -1 -10671.29 m -2 4.11528 m -1 -1440.95 m -2 6.99105 m -1 4809.53 m -2 4 1 2 0 -104.645 m -1 0 -15.7145 m -1 0 -29.3143 m -1 4 0 3 0 -1.31923 0 -0.197152 0 0.197152 6 5 0 -3.93649*10 8 m -4 4.80357*10 11 m -5 1.78842*10 8 m -4 -2.14040*10 10 m -5 -1.72307*10 9 m -4 -3.88671*10 11 m -5 6 4 1 1.24986*10 7 m -3 6.19886*10 8 m -4 1.06288*10 6 m -3 -1.70212*10 8 m -4 5.83689*10 6 m -3 1.82661*10 9 m -4 6 3 2 5.49284*10 4 m -2 -8.19587*10 6 m -3 4694.07 m -2 -1.25104*10 6 m -3 -14721.4 m -2 -8.08185*10 6 m -3 6 2 3 126.595 m -1 -5.87327*10 4 m -2 12.604 m -1 -8231.94 m -2 22.7901 m -1 33476.3 m -2 6 1 4 0 98.8023 m -1 0 -36.5481 m -1 0 -117.301 m -1 6 0 5 0 8.70184 0 0.194345 0 0.194345 Table 7: Derivatives of the ray transfer functions fe,prop, fdCL,prop, fdL,prop of the propagations of the components of the eye Order Vertex distance e Anterior chamber depth d CL Lens thickness d L n x n t ƒxe,prop(nx,nt) ƒte,prop(nx,nt) fxdCL,prop(nx,nt) ftdCL,prop(nx,nt) fxdL,prop(nx,nt) ftdL,prop(nx,nt) 2 1 0 1 0 1 0 1 0 2 0 1 0.013 m 1 2.69461*10 -3 m 1 2.60197*10 -3 m 1 All higher orders disappear

[0172] The derivatives of the component-wise beam transfer functions f C,ref , f L1,ref , f L2,ref , f e,prop f dCL,prop , f dL,prop Within the mGE eye are the starting points for calculating the total ray transfer function f mGE to determine the entire mGE eye. Since f mGE as a chain fmGE(ρ)=fL2,ref(fdL,prop(fL1,ref(fdCL,prop(fC,ref(ρ))))), Since f is defined, all derivatives of f can be mGEup to order p = 6 can be determined using the chain rule from the derivatives of the component-wise ray transfer functions. Table 8: Derivatives of the ray transfer functions fmGE of the modified Gullstrand-Emsley eye Order mod. Gullstrand-Emsley eye mGE p n x n t fxmGE(nx,nt) ftmGE(nx,nt) 2 1 0 0.753858 -60.00001 diopters 2 0 1 5.24176*10 -3 m 0.909314 4 3 0 12221.93 m -2 -9.72751*10 5 m -3 4 2 1 -32.6185 m -1 -10883.5 m -2 4 1 2 -0.595797 -110.553 m -1 4 0 3 -7.82984*10 -3 m -1.56287 6 5 0 7.24284*10 8 m -4 -5.76462*10 10 m -5 6 4 1 1.20722*10 7 m -3 -3.549999*10 8 m -4 6 3 2 10955.5*10 4 m -2 -8.559098*10 6 m -3 6 2 3 -88.57369 m -1 -4.62591 *10 4 m -2 6 1 4 1.407511 1271.8149 m -1 6 0 5 0.056790 m 11.548181

[0173] From the derivatives of the beam transfer function, the prefactor β and the wavefront transfer coefficients b can now be determined. pk of the mGE eye, which are required for the evaluation of the wavefront calculation in Eq.(38),(36). According to Eq. (4): T=(ABCD)=(0.7538580.00524176m−60.00001dpt0.909314)

[0174] Furthermore, the prefactor β according to Eq. (26) is specific for the incident wavefront t (1) = -w (2) = -nE2 = E2, where Eq.(8) and the definition E p = nw (p)This can be exploited, as well as the fact that the refractive index in the space in front of the eye is given by n = 1. The prefactor β can then be determined according to Eq. (37) using Eq. (49) from E2.

[0175] Determining the wavefront transfer coefficients b pk This can finally be done by substituting the numerical values ​​of the derivatives of the beam transfer function from Table 8 into Table 5. The result is shown in Table 9 below. Table 9: Wavefront transfer coefficients bpk for the modified Gullstrand-Emsley eye 1Indices 1 Wavefront order p k* k1 2 3 4 5 6 (0) 0 60,000 diopters 0 2.06320*10 5 m -3 0 3.54733*10 9 m -5 1 0.909314 0 -28692.2 m -2 0 2.16457*10 9 m -4 2 0 597.737 m -1 0 -7.52551*10 6 m -3 3 0 -6.69794 0 -1.03962*10 6 m-2 4 2.26513*10 -2 m 0 23021.1 m -1 5 2.91006*10 - 5 m 2 0 -237.259 6 0 1.70495 m 7 0 -0.009055 m 2 8 2.21606*10 -5 m 3 9 2.38306*10- 9 m 4 (1) 0 1 0 -46287.1 m -2 0 1 0 655.057 m -1 0 2 0 -10.7069 0 3 -0.0343949m 0 4 6.44793*10 -4 m 2 0 5 0 6 0 (2) 0 0.0157253m 0 -2547.464 m -1 1 0 16.42902 2 0 -0.607000 m 3 0.00119945m 2 4 -98305.4 m 4 (0,1) 0 1 0 1216.102 m 5 1 1 0 -17.7499 m 6 2 0 -0.0983983m 7 3 0.00136332m 8 4 -98305.4 m 4 (3) 0 4.12141*10 -4 m 2 0 1 0 2 0 (1,1) 0 0.0524176m 0 1 0 2 0 (0,0,1) 0 1 0 1 0 2 0 (4) 0 1.51224*10 -5 m 3 (2,1) 0 0.00288499m 2 (0,2) 0 0.0524176 m (1,0,1) 0 0.0786265 m (0,0,0,1) 0 1

[0176] The actual goal, namely the repeated calculation of different wavefronts by the mGE eye, can now be achieved on the basis of the values ​​from Eq.(49) and from Table 9.

[0177] The results for E' pThe results for orders p ≤ 6 are shown in Table 10 below. The first column shows the results for a plane incident wavefront, which is mapped onto a spherical wave after construction of the emmetropic mGE eye. The second and third columns refer to an incident wavefront from a distance of 40 cm, where the third column corresponds to an exact spherical wave and the second column to a second-order approximation of this spherical wave. Finally, the fourth column shows an arbitrarily chosen wavefront whose HOA values ​​are only chosen as examples. Table 10: Aberrations E'2, E'3, E'4, E'5, E'6 of the lost wavefront after wavefront calculation of different wavefronts with aberrations E2, E3, E4, E5, E6 by the modified Gullstrand-Emsley eye Aberrations and prefactor plane wavefront Wavefront from a distance of 40 cm arbitrary wavefront with HOA Second-order wavefront (parabola) ball shaft Input E2 0 -2.5 diopters -2.5 diopters -2.5 diopters E3 0 0 0 -1.0*10 3 m -2 E4 0 0 -46.875 m -3 7.0*10 5 m -3 E5 0 0 0 -2.0*10 8 m -4 E6 0 0 -4.39453*10 3 m -5 5.0*10 10 m -5 output β 1.32651 1.26348 1.26348 1.26348 E'2 79.5906 diopters 75.4005 diopters 75.4005 diopters 75.4005 diopters E'3 0 0 0 -1.50974*10 3 m-2 E'4 8.47409*10 5 m -3 1.05563*10 6 m -3 1.05551*10 6 m-3 2.98485*10 6 m-3 E'5 0 0 0 -8.09790*10 8 m -4 E'6 4.51123*10 10 m -5 - 1.88462*10 10 m -5 - 1.88111 *10'°m -5 3.31012*10 11 m -5

[0178] The method according to the invention can preferably be used for optimizing spectacle lenses, wherein, instead of the mGE eye, the actual biometric parameters of the individual eye are used, and the input wavefront for the calculation is preferably a second-order wavefront originating from the spectacle lens to be optimized. The spectacle lens is to be determined such that the output wavefront, with respect to a metric of the spherical wave converging on the retina, is as close as possible.

[0179] In particular, the invention relates to the following points: Item 1: Method for simulating an optical system by means of a wavefront calculation, wherein the optical system is a complex optical system whose effect extends beyond a single refraction, a single propagation or a single reflection, comprising the steps: - Establishing at least one wavefront transfer function for the optical system, wherein the wavefront transfer function is designed to assign a corresponding outgoing wavefront to each incident wavefront, taking into account aberrations of an order greater than the order of a defocus; and - Evaluating the at least one wavefront transfer function for at least one wavefront incident on the optical system. Point 2: Method according to point 1, wherein the method is a method for optimizing an overall optical system, wherein the optical system represents a second subsystem of the overall optical system and the overall optical system additionally comprises a first subsystem, wherein in particular the first subsystem and / or the second subsystem can be varied during the optimization. Point 3: Procedure according to point 2, wherein the first subsystem is a spectacle lens and the second subsystem is a model eye (10). Point 4: Procedure according to point 2 or 3, wherein at least one wavefront incident on the optical system is determined based on a predefined test wavefront passing through the first subsystem. Point 5: Procedure according to one of points 2 to 4, further comprising the step: - Evaluating the overall optical system based on the result of evaluating the at least one wavefront transfer function for the at least one wavefront incident on the optical system, wherein the overall optical system is evaluated by varying the first subsystem until the evaluation satisfies a predetermined condition, wherein The variation of the first subsystem includes, in particular, a change in at least one refracting surface and / or at least one distance between refracting surfaces of the first subsystem, and / or a tilting and / or displacement of the first subsystem relative to the second subsystem. Point 6: Procedure according to point 5, wherein The evaluation of the overall optical system is carried out based on the result of the evaluation of the at least one wavefront transfer function for a first wavefront incident on the optical system and further on the result of the evaluation of the at least one wavefront transfer function for a second incident wavefront, wherein the first subsystem is in a first position and orientation to the second subsystem when the first wavefront is incident, wherein the first subsystem is in a second position and orientation to the second subsystem when the second wavefront is incident, and wherein the first position differs from the second position and / or the first orientation differs from the second orientation. Point 7: Procedure according to one of points 5 to 6, wherein The evaluation of the overall optical system is carried out based on the result of evaluating a first wavefront transfer function for a first wavefront incident on the optical system and further on the result of evaluating a second wavefront transfer function for a further incident wavefront, wherein the first subsystem is in a first position and orientation relative to the second subsystem when the first wavefront occurs, wherein the first subsystem is in a second position and orientation relative to the second subsystem when the further wavefront occurs, wherein the first position differs from the second position and / or the first orientation differs from the second orientation, and wherein the second wavefront transfer function differs from the first wavefront transfer function. Point 8: Procedure according to one of points 2 to 7, wherein the first subsystem is a spectacle lens and the second subsystem is a model eye, and where eye movements of the model eye that cause a change in the position of the point of penetration of the principal ray through the spectacle lens surfaces and / or a change in the angles of incidence on a spectacle lens surface are described as a change in the position and / or orientation of the spectacle lens in the coordinate system of the eye. Point 9: Method according to any of the preceding points, wherein the optical system is a GRIN system or at least includes a GRIN element. Point 10: Procedure according to one of the preceding points, wherein at least one wavefront transfer function has the form E'p=β−r¯10(p−1)∑k1,k2,...,kp−1b¯pk β−Δr¯1(p−1,k*)E2k1E3k2⋯Epkp−1, p=2,3,4,... exhibits and where the indices of the tuple k = (k1,k2,...,k p-1 ) over the range P(k*) ≤ p - 2 and 0 ≤ k1 ≤ 2(p - P(k*) - 2) + δ P(k*),0 running, whereby P(k*)=∑j=1p−2jkj+1, and whereby −r¯10(p−1)=p−δ(p−1),1 and -Δr1(p - 1, k*) = (p - 3) + δ (p-1),1 - P(k**) hold, and where β = (-BE2 + A) -1 is given as a function of the at least one incident wavefront and the optical system, and where A, B and the wavefront transfer coefficients b pk are given as a function of the components of the optical system. Point 11: Method according to one of the preceding points, wherein both the incident and the reflected wavefronts are each represented by coefficients to basis elements of a basis system whose basis elements are classified according to at least one order parameter, and wherein the at least one wavefront transfer function is given by assigning to the wavefronts incident into the optical system the respective corresponding reflected wavefront such that, for a basis element represented in the representation of a reflected wavefront, it determines the coefficient to this basis element represented in the representation of the reflected wavefront as a function of coefficients of the corresponding incident wavefront to a plurality of those basis elements whose value of the order parameter is less than or equal to the value of the order parameter of the respective basis element represented in the representation of the reflected wavefront. Point 12: Procedure according to point 11, wherein the basis system is a decomposition by aberrations, wherein the order parameter is an order p of the aberrations, and wherein the coefficients belonging to the basis elements of the basis system are given by the fact that a coefficient of pth order of the incident wavefront is an aberration E p of the incident wavefront and that a coefficient of p-th order of the associated outgoing wavefront is an aberration E' p the associated falling wavefront, and where p ≥ 2. Point 13: Procedure according to point 11 or 12, wherein the order parameter is a first order parameter, and wherein the basis elements of the basis system are additionally classified according to at least one second order parameter, the range of values ​​of which depends on the value of the first order parameter. Point 14: Procedure according to one of points 11 to 13, wherein the basis system is a decomposition by Taylor derivatives of wavefront arrow heights, wherein the order parameter is an order p of the Taylor derivatives of the wavefront arrow heights, wherein the coefficients belonging to the basis elements of the basis system are given by the fact that a coefficient of pth order of the incident wavefront is a Taylor derivative w (p) the swash level of the incident wavefront and that the coefficient of the pth order of the corresponding outgoing wavefront is a Taylor derivative w' (p) the arrow height of the associated falling wavefront, and where p ≥ 2. Point 15: Procedure according to one of points 11 to 13, wherein the basis system is a decomposition into Taylor derivatives of a wavefront OPD, wherein the order parameter is an order p of the Taylor derivatives of the wavefront OPD, wherein the coefficients belonging to the basis elements of the basis system are given by the fact that a coefficient of pth order of the incident wavefront is a Taylor derivative OPD (p) of the incident wavefront and that the coefficient p-th order of the associated outgoing wavefront is a Taylor derivative OPD' (p) the associated falling wavefront, and where p ≥ 2. Point 16: Procedure according to one of points 11 to 13, wherein the basis system is a decomposition by derivatives of directional functions, wherein the order parameter is an order i of the derivatives of the directional functions, wherein the coefficients belonging to the basis elements of the basis system are given by the fact that a coefficient of the i-th order of the incident wavefront is a derivative t (i) a directional function t(x) of the incident wavefront and that a coefficient of the i-th order of the associated outgoing wavefront is a derivative t' (i) a directional function ť(x') of the associated falling wavefront, and where i ≥ 1. Point 17: Procedure according to one of points 11 to 13, wherein the basis system is a decomposition into Zernike polynomials, wherein the order parameter is a radial order n of the Zernike polynomials, and wherein the coefficients belonging to the basis elements of the basis system are given by the fact that a coefficient of nth order of the incident wavefront is a Zernike coefficient Z n of the incident wavefront and that a coefficient of nth order of the outgoing wavefront is a Zernike coefficient Z' n the associated falling wavefront, where n ≥ 2, and where the Zernike coefficients refer in particular to a specified pupil. Point 18: Procedure according to point 13, wherein the basis system is a decomposition by aberrations, where the first order parameter is the sum p of orders p x and p y of the aberrations, where the second order parameter is one of the orders p x and p yis, where the coefficients belonging to the basis elements of the basis system are given by the fact that coefficients of the p-th order of the incident wavefront aberrations E px,py of the incident wavefront and that coefficients p-th order of the associated outgoing wavefront aberrations E' px,py of the associated falling wavefront, and where p ≥ 2 and p x ≥ 0 and p y ≥ 0. Point 19: Procedure according to point 13, wherein the basis system is a decomposition by Taylor derivatives of wavefront arrow heights, where the first order parameter is the sum p of orders p x and p y the Taylor derivatives of the wavefront arrow heights, where the second order parameter is one of the orders p x and p yis, where the coefficients belonging to the basis elements of the basis system are given by the fact that coefficients of the p-th order of the incident wavefront are Taylor derivatives w (px,py) the swash level of the incident wavefront and that the coefficients of the p-th order of the associated outgoing wavefront are Taylor derivatives w' (px,py) the arrow height of the associated falling wavefront, and where p ≥ 2 and p x ≥ 0 and p y ≥ 0. Point 20: Procedure according to point 13, wherein the basis system is a decomposition by Taylor derivatives of a wavefront OPD, where the first order parameter is the sum p of orders p x and p y the Taylor derivatives of the wavefront OPD, where the second order parameter is one of the orders p x and p yis, where the coefficients belonging to the basis elements of the basis system are given by the fact that coefficients of the p-th order of the incident wavefront are Taylor derivatives OPD. (px,py) of the incident wavefront and that the coefficients of the p-th order of the associated outgoing wavefront are Taylor derivatives OPD' (px,py) of the associated falling wavefront, and where p ≥ 2 and p x ≥ 0 and p y ≥ 0. Point 21: Procedure according to point 13, wherein the basis system is a decomposition by derivatives of directional functions, where the first order parameter is the sum i of orders i x and i y of the derivatives of the directional functions, where the second order parameter is one of the orders i x and i yis, where the coefficients belonging to the basis elements of the basis system are given by the fact that coefficients of the i-th order of the incident wavefront derivatives t x (ix,iy) , t y (ix,iy) of directional functions t x (x,y), t y (x,y) of the incident wavefront and that coefficients i-th order of the associated outgoing wavefront derivatives t' x (ix,iy) , t'y (ix,iy) the directional functions t' x (x',y'), t' y (x',y') of the associated falling wavefront, and where i ≥ 1 and i x ≥ 0 and i y ≥ 0 applies. Point 22: Procedure according to point 13, wherein the basis system is a decomposition into Zernike polynomials, wherein the first order parameter is a radial order n of the Zernike polynomials, wherein the second order parameter is an azimuthal order m of the Zernike polynomials, and wherein the coefficients belonging to the basis elements of the basis system are given by the fact that coefficients of the nth order of the incident wavefront are Zernike coefficients Z n m of the incident wavefront and that coefficients of the nth order of the associated outgoing wavefront are Zernike coefficients Z' n m of the associated falling wavefront, where n ≥ 2 and -n ≤ m ≤ n, where m is even for even n and odd for odd n, and where the Zernike coefficients refer in particular to a specified pupil. Item 23: Computer program product comprising computer-readable instructions which, when loaded into a computer's memory and executed by the computer, cause the computer to perform a procedure according to any one of Items 1 to 22. Item 24: Device for simulating an optical system by means of wavefront calculation, wherein the optical system is a complex optical system whose effect extends beyond a single refraction, propagation or reflection, comprising: - a modeling module for providing at least one wavefront transfer function for the optical system, wherein the wavefront transfer function is designed to assign to each incident wavefront to the optical system, taking into account aberrations of an order greater than the order of a defocus, a corresponding outgoing wavefront; - an evaluation module for evaluating at least one wavefront transfer function for at least one wavefront incident on the optical system. Item 25: Comprehensive method for manufacturing a spectacle lens: - Calculating or optimizing a spectacle lens using a method according to one of points 1 to 22; and - Providing manufacturing data for the calculated or optimized spectacle lens, and / or manufacturing the calculated or optimized spectacle lens. Item 26: Device for manufacturing a spectacle lens comprising: Calculation or optimization tools designed to calculate or optimize the spectacle lens using a method according to any of points 1 to 22; and Processing tools designed to finish processing the spectacle lens. Point 27: Use of a spectacle lens manufactured according to the manufacturing process in accordance with point 25 in a specified average or individual position of use of the spectacle lens in front of the eyes of a specific spectacle wearer to correct a refractive error of the spectacle wearer.

Claims

A computer-implemented method for simulating an optical system by means of wavefront calculation, wherein the optical system is a complex optical system whose effect extends beyond a single refraction, propagation, or reflection, comprising the steps of: - computer-implemented establishment of at least one wavefront transfer function for the optical system, wherein the wavefront transfer function is intended to assign a corresponding outgoing wavefront to wavefronts incident on the optical system, taking into account aberrations of an order greater than the order of a defocus; and - computer-implemented evaluation of the at least one wavefront transfer function for at least one wavefront incident on the optical system. Computer-implemented method according to claim 1, wherein the method is a method for optimizing an overall optical system, wherein the optical system represents a second subsystem of the overall optical system and the overall optical system additionally comprises a first subsystem, wherein in particular the first subsystem and / or the second subsystem can be varied during the optimization. Computer-implemented method according to claim 2, wherein the first subsystem is a spectacle lens and the second subsystem is a model eye (10). Computer-implemented method according to claim 2 or 3, wherein the at least one wavefront incident on the optical system is determined on the basis of a predetermined test wavefront passing through the first subsystem. A computer-implemented method according to one of claims 2 to 4, further comprising the step of: - evaluating the overall optical system based on the result of evaluating the at least one wavefront transfer function for the at least one wavefront incident on the optical system, wherein the overall optical system is evaluated by varying the first subsystem until the evaluation satisfies a predetermined condition, wherein the variation of the first subsystem in particular includes a change in at least one refracting surface and / or at least one distance between refracting surfaces of the first subsystem, and / or a tilting and / or shifting of the first subsystem relative to the second subsystem. Computer-implemented method according to claim 5, wherein the evaluation of the overall optical system is carried out on the basis of the result of the evaluation of the at least one wavefront transfer function for a first wavefront incident on the optical system and further on the basis of the result of the evaluation of the at least one wavefront transfer function for a second incident wavefront, wherein the first subsystem is in a first position and orientation to the second subsystem when the first wavefront is incident, wherein the first subsystem is in a second position and orientation to the second subsystem when the second wavefront is incident, and wherein the first position differs from the second position and / or the first orientation differs from the second orientation. A computer-implemented method according to one of claims 5 to 6, wherein the evaluation of the overall optical system is carried out based on the result of evaluating a first wavefront transfer function for a first wavefront incident on the optical system and further on the result of evaluating a second wavefront transfer function for a further incident wavefront, wherein the first subsystem is in a first position and orientation relative to the second subsystem when the first wavefront is incident, wherein the first subsystem is in a second position and orientation relative to the second subsystem when the further wavefront is incident, wherein the first position differs from the second position and / or the first orientation differs from the second orientation, and wherein the second wavefront transfer function differs from the first wavefront transfer function. A computer-implemented method according to any one of claims 2 to 7, wherein the first subsystem is a spectacle lens and the second subsystem is a model eye, and wherein eye movements of the model eye that cause a change in the position of the point of penetration of the main ray through the spectacle lens surfaces and / or a change in the angles of incidence on a spectacle lens surface are described as a change in the position and / or orientation of the spectacle lens in the coordinate system of the eye. Computer-implemented method according to one of the preceding claims, wherein the optical system is a GRIN system or at least comprises a GRIN element. A computer-implemented method according to one of the preceding claims, wherein the at least one wavefront transfer function has the form E ' p = β − r ¯ 1 0 ( p − 1 ) ∑ k 1 , k 2 ,..., kp − 1 b ¯ pk β − Δ r ¯ 1 ( p − 1, k * ) E 2 k 1 E 3 k 2 ⋯ E pkp − 1 , p = 2,3,4,... exhibits and where the indices of the tuple k = (k 1 ,k 2 ,...,k p-1 ) over the range P(k*) ≤ p - 2 and 0 ≤ k 1 ≤ 2(p - P(k*) - 2) + δ P(k*),0 run, where P ( k * ) = ∑ j = 1 p − 2 jkj + 1 , = and where − r ¯ 1 0 ( p − 1 ) = p − δ ( p − 1 ) ,1 and -Δr 1 (p - 1, k*) = (p - 3) + δ (p-1),1 - P(k**) hold, and where β = (-BE 2 +A) -1 is given as a function of the at least one incident wavefront and the optical system, and where A, B and the wavefront transfer coefficients b pk are given as a function of the components of the optical system. A computer-implemented method according to one of the preceding claims, wherein both the incident and the reflected wavefronts are each represented by coefficients to basis elements of a basis system whose basis elements are classified according to at least one order parameter, and wherein the at least one wavefront transfer function is given by assigning to the wavefronts incident into the optical system the respective corresponding reflected wavefront such that, for a basis element represented in the representation of a reflected wavefront, it determines the coefficient to this basis element represented in the representation of the reflected wavefront as a function of coefficients of the corresponding incident wavefront to a plurality of those basis elements.whose order parameter value is less than or equal to the order parameter value of the respective base element represented in the depiction of the emerging wavefront. Computer-implemented method according to claim 11, wherein the basis system is a decomposition by aberrations, wherein the order parameter is an order p of the aberrations, wherein the coefficients belonging to the basis elements of the basis system are given by the fact that a coefficient of p-th order of the incident wavefront is an aberration Ep of the incident wavefront and that a coefficient of p-th order of the associated falling wavefront is an aberration E'p of the associated falling wavefront, and wherein p ≥ 2. Computer-implemented method according to claim 11 or 12, wherein the order parameter is a first order parameter, and wherein the basic elements of the basic system are additionally classified according to at least one second order parameter, the range of values ​​of which depends on the value of the first order parameter. A computer-implemented method according to any one of claims 11 to 13, wherein the basis system is a decomposition into Taylor derivatives of wavefront arrow heights, wherein the order parameter is an order p of the Taylor derivatives of the wavefront arrow heights, wherein the coefficients associated with the basis elements of the basis system are given in that a coefficient of pth order of the incident wavefront is a Taylor derivative w(p) of the arrow height of the incident wavefront and that the coefficient of pth order of the associated outgoing wavefront is a Taylor derivative w'(p) of the arrow height of the associated outgoing wavefront, and wherein p ≥ 2. A computer-implemented method according to one of claims 11 to 13, wherein the basis system is a decomposition into Taylor derivatives of a wavefront OPD, wherein the order parameter is an order p of the Taylor derivatives of the wavefront OPD, wherein the coefficients belonging to the basis elements of the basis system are given by the fact that a coefficient of pth order of the incident wavefront is a Taylor derivative OPD(p) of the incident wavefront and that the coefficient of pth order of the associated falling wavefront is a Taylor derivative OPD'(p) of the associated falling wavefront, and wherein p ≥ 2. A computer-implemented method according to any one of claims 11 to 13, wherein the basis system is a decomposition into derivatives of directional functions, wherein the order parameter is an order i of the derivatives of the directional functions, wherein the coefficients belonging to the basis elements of the basis system are given in that a coefficient of i-th order of the incident wavefront is a derivative t(i) of a directional function t(x) of the incident wavefront and that a coefficient of i-th order of the associated outgoing wavefront is a derivative t'(i) of a directional function t'(x') of the associated outgoing wavefront, and wherein i ≥ 1. A computer-implemented method according to one of claims 11 to 13, wherein the basis system is a decomposition into Zernike polynomials, wherein the order parameter is a radial order n of the Zernike polynomials, wherein the coefficients belonging to the basis elements of the basis system are given in that a coefficient of nth order of the incident wavefront is a Zernike coefficient of the incident wavefront and that a coefficient of nth order of the outgoing wavefront is a Zernike coefficient of the outgoing wavefront belonging to the outgoing wavefront, wherein n ≥ 2, and wherein the Zernike coefficients refer in particular to a defined pupil. A computer-implemented method according to claim 13, wherein the basis system is a decomposition by aberrations, wherein the first order parameter is the sum p of orders px and py of aberrations, wherein the second order parameter is one of the orders px and py, wherein the coefficients belonging to the basis elements of the basis system are given by the fact that coefficients of p-th order of the incident wavefront are aberrations Epx,py of the incident wavefront and that coefficients of p-th order of the associated falling wavefront are aberrations E'px,py of the associated falling wavefront, and wherein p ≥ 2 and px ≥ 0 and py ≥ 0. A computer-implemented method according to claim 13, wherein the basis system is a decomposition by Taylor derivatives of wavefront arrow heights, wherein the first order parameter is the sum p of orders px and py of Taylor derivatives of the wavefront arrow heights, wherein the second order parameter is one of the orders px and py, wherein the coefficients associated with the basis elements of the basis system are given by the fact that coefficients of the pth order of the incident wavefront are Taylor derivatives w(px,py) of the arrow height of the incident wavefront and that the coefficients of the pth order of the associated outgoing wavefront are Taylor derivatives w'(px,py) of the arrow height of the associated outgoing wavefront, and wherein p ≥ 2 and px ≥ 0 and py ≥ 0. A computer-implemented method according to claim 13, wherein the basis system is a decomposition into Taylor derivatives of a wavefront OPD, wherein the first order parameter is the sum p of orders px and py of Taylor derivatives of the wavefront OPD, wherein the second order parameter is one of the orders px and py, wherein the coefficients associated with the basis elements of the basis system are given by the fact that coefficients of the pth order of the incident wavefront are Taylor derivatives OPD(px,py) of the incident wavefront and that the coefficients of the pth order of the associated falling wavefront are Taylor derivatives OPD'(px,py) of the associated falling wavefront, and wherein p ≥ 2 and px ≥ 0 and py ≥ 0. A computer-implemented method according to claim 13, wherein the basis system is a decomposition into derivatives of directional functions, wherein the first order parameter is the sum i of orders ix and iy of the derivatives of the directional functions, wherein the second order parameter is one of the orders ix and iy, wherein the coefficients belonging to the basis elements of the basis system are given by the fact that coefficients of the i-th order of the incident wavefront are derivatives tx(ix,iy), ty(ix,iy) of directional functions tx(x,y), ty(x,y) of the incident wavefront, and that coefficients of the i-th order of the associated outgoing wavefront are derivatives t'x(ix,iy), t'y(ix,iy) of the directional functions t'x(x',y'), t'y(x',y') of the associated outgoing wavefront, and wherein i ≥ 1 and ix ≥ 0 and iy≥ 0 applies. A computer-implemented method according to claim 13, wherein the basis system is a decomposition into Zernike polynomials, wherein the first order parameter is a radial order n of the Zernike polynomials, wherein the second order parameter is an azimuthal order m of the Zernike polynomials, wherein the coefficients belonging to the basis elements of the basis system are given in that coefficients of order n of the incident wavefront are Zernike coefficients Znm of the incident wavefront and that coefficients of order n of the associated falling wavefront are Zernike coefficients Z'nm of the associated falling wavefront, wherein n ≥ 2 and -n ≤ m ≤ n, where m is even for even n and odd for odd n, and wherein the Zernike coefficients refer in particular to a defined pupil. A computer program product comprising computer-readable instructions which, when loaded into a memory of a computer and executed by the computer, cause the computer to perform a method according to any one of claims 1 to 22. Device for simulating an optical system by means of wavefront calculation, wherein the optical system is a complex optical system whose effect extends beyond a single refraction, propagation, or reflection, comprising: - a modeling module for providing at least one wavefront transfer function for the optical system by computer implementation, wherein the wavefront transfer function is designed to assign a corresponding outgoing wavefront to each wavefront incident on the optical system, taking into account aberrations of an order greater than the order of a defocus; - an evaluation module for evaluating the at least one wavefront transfer function for at least one wavefront incident on the optical system by computer implementation.

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