Spectacle lens design following Hamilton, Maxwell and Keller

Transcription

1 Spectacle lens design following Hamilton, Maxwell and Keller Koby Rubinstein Technion Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 1 / 23

2 Background Spectacle lens design went through a major revolution in the last 3 decades. The main motivation for this was the need to design multifocal (progressive) lenses for the presbyopic population. The advances in understanding spectacle performance, and the introduction of novel optimization methods enabled also the design of high-quality speciality lenses, such as sport lenses. Some of the key ideas were laid down by Hamilton, Maxwell and Keller, as we shall see, although they were motivated by basic science and not by design goals. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 2 / 23

3 Spectacle design Spectacle design is a unique sub-discipline of optical design for three reasons. First, the eye is in general not radially symmetric. Second, the eye scans the visual field, and thus the lens, in many directions and not just in the forward direction. Third, the pupil is relatively small, and therefore only a small wavefront, generated by each object point, needs be considered for any given gaze direction. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 3 / 23

4 Spectacle design Spectacle design is a unique sub-discipline of optical design for three reasons. First, the eye is in general not radially symmetric. Second, the eye scans the visual field, and thus the lens, in many directions and not just in the forward direction. Third, the pupil is relatively small, and therefore only a small wavefront, generated by each object point, needs be considered for any given gaze direction. In fact, for almost all purposes, just the quadratic terms of the entire wavefront suffice to define vision. Indeed, a prescription looks like this: OD: Sph(-3.25), Cyl (0.75), x axis (15) (add (2.5)) Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 3 / 23

5 Spectacle design Spectacle design is a unique sub-discipline of optical design for three reasons. First, the eye is in general not radially symmetric. Second, the eye scans the visual field, and thus the lens, in many directions and not just in the forward direction. Third, the pupil is relatively small, and therefore only a small wavefront, generated by each object point, needs be considered for any given gaze direction. In fact, for almost all purposes, just the quadratic terms of the entire wavefront suffice to define vision. Indeed, a prescription looks like this: OD: Sph(-3.25), Cyl (0.75), x axis (15) (add (2.5)) Mathematically, this is a bad way to express a 2 2 symmetric matrix (the Dioptric Matrix): ( S + C V = 2 cos 2ψ C 2 sin 2ψ ) C 2 sin 2ψ S C 2 cos 2ψ Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 3 / 23

6 Progressive Addition Lenses (multifocal spectacles) Our eye is an autofocus optical element. The eye lens can contract and relax in order to obtain focal distances ranging from a few cm to many meters (accommodation). Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 4 / 23

7 Progressive Addition Lenses (multifocal spectacles) Our eye is an autofocus optical element. The eye lens can contract and relax in order to obtain focal distances ranging from a few cm to many meters (accommodation). However, we gradually lose this flexibility of the lens from the age of about 45. When we look at a near-by object, then, in addition to the contraction of the eye lens, the eyes converge in the nasal direction (convergence). Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 4 / 23

8 Progressive Addition Lenses (multifocal spectacles) Our eye is an autofocus optical element. The eye lens can contract and relax in order to obtain focal distances ranging from a few cm to many meters (accommodation). However, we gradually lose this flexibility of the lens from the age of about 45. When we look at a near-by object, then, in addition to the contraction of the eye lens, the eyes converge in the nasal direction (convergence). The idea behind a Progressive Addition Lens (multifocal lens) is to replace the loss of one dynamical ability (accommodation) with another surviving dynamical ability (convergence). Thus, the goal is to design a lens that will have far vision power in the forward gaze direction, and increasing power as the eye converges in the nasal direction. We recall that the eye keeps moving around. Therefore good vision must be obtained at all gaze directions. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 4 / 23

9 PAL A schematic drawing of a design aimed to achieve good vision for a multitude of object points at different angles and different distances from the observer. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 5 / 23

10 Goals Design method that provides superior focusing of many point objects at many viewing directions and many distances. Design method that provides proper magnification and minimal distortion of extended objects. The design method must be very fast and precise. Additional factors, including production constraints and cosmetics (vanity...) Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 6 / 23

11 Hamilton I shall concentrate on the main building block in our design algorithm - Hamilton s point eikonal function. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 7 / 23

12 Hamilton Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 8 / 23

13 Hamilton Eikonal functions x x (x,h,z) (x,h,z ) z y z y Consider two planes P and P separated by an optical medium (lenses, mirrors,... anything). The Point Eikonal function S(x, y, x, y ) is defined to be the optical distance between two arbitrary points (x, y) P, and (x, y ) P. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 9 / 23

14 Hamilton Eikonal functions x x (x,h,z) (x,h,z ) z y z y Consider two planes P and P separated by an optical medium (lenses, mirrors,... anything). The Point Eikonal function S(x, y, x, y ) is defined to be the optical distance between two arbitrary points (x, y) P, and (x, y ) P. All the geometric information on the optics of the medium is contained in this function. For instance, the direction vector of a ray from (x, y) to (x, y ) is given by n S x,y. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 9 / 23

15 Magnification and Distortion Hamilton also showed the importance of other eikonal (he called them characteristic) functions. For instance, the point-angle eikonal function S M (x, y, ξ, η ) = inf x,y ( S(x, y, x, y ) n x ξ n y η ). Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 10 / 23

16 Magnification and Distortion Hamilton also showed the importance of other eikonal (he called them characteristic) functions. For instance, the point-angle eikonal function S M (x, y, ξ, η ) = inf x,y ( S(x, y, x, y ) n x ξ n y η ). In spite of the importance of the eikonal functions to Hamilton s theory of geometric optics, these functions were not used in practice for optical design. One reason is that S is a function of four variables, and its computation is costly. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 10 / 23

17 Magnification and Distortion Hamilton also showed the importance of other eikonal (he called them characteristic) functions. For instance, the point-angle eikonal function S M (x, y, ξ, η ) = inf x,y ( S(x, y, x, y ) n x ξ n y η ). In spite of the importance of the eikonal functions to Hamilton s theory of geometric optics, these functions were not used in practice for optical design. One reason is that S is a function of four variables, and its computation is costly. Another reason is that the inverse problem is widely open. Namely, given a point eikonal function S(x, y, x, y ), can it be realized by an optical medium? How to realize it? However, we discovered that the problem of quantifying the imaging of extended objects can be done efficiently with these functions. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 10 / 23

18 Local eikonals Since the pupil is small, we do not need the entire eikonal function; just a small portion of the wave emitted by a source object enters it. Instead of the full S, it suffices to compute its second order Taylor expansion S = 1 2 X t SX, about a base ray, where ( ) P Q S = Q t = R Π 11 Π 12 Q 11 Q 12 Π 12 Π 22 Q 21 Q 22 Q 11 Q 21 R 11 R 12 Q 12 p 22 R 12 R 22 Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 11 / 23

19 Local eikonals Since the pupil is small, we do not need the entire eikonal function; just a small portion of the wave emitted by a source object enters it. Instead of the full S, it suffices to compute its second order Taylor expansion S = 1 2 X t SX, about a base ray, where ( ) P Q S = Q t = R Π 11 Π 12 Q 11 Q 12 Π 12 Π 22 Q 21 Q 22 Q 11 Q 21 R 11 R 12 Q 12 p 22 R 12 R 22 P P Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 11 / 23

20 Local eikonals For example, the direction of the rays on both sides of the optical element are determined by nξ nη n ξ n η = Π 11 Π 12 Q 11 Q 12 Π 12 Π 22 Q 21 Q 22 Q 11 Q 21 R 11 R 12 Q 12 p 22 R 12 R 22 x y x y. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 12 / 23

21 Local eikonal functions It turns out that the 4 4 matrix S contains a lot of useful information. For example, the submatrix R is essentially equivalent to the dioptric matrix which is defined by the prescription. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 13 / 23

22 Local eikonal functions It turns out that the 4 4 matrix S contains a lot of useful information. For example, the submatrix R is essentially equivalent to the dioptric matrix which is defined by the prescription. To understand the submatrix Q, let r be the distance between the object and the image. Then we can write ( ξ η ) = rqt n ( x/r y/r ). ) ) Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 13 / 23

23 Local eikonal functions Indeed the submatrix Q has all the information on magnification and distortion induced by the lens for each viewing direction. We express Q in the form Q = UE, where U is a rotation matrix, and E is a symmetric matrix. Then we can define 4 geometric variables embedded in Q. First, let µ 1,2 be the two eigenvalues of E. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 14 / 23

24 Local eikonal functions Indeed the submatrix Q has all the information on magnification and distortion induced by the lens for each viewing direction. We express Q in the form Q = UE, where U is a rotation matrix, and E is a symmetric matrix. Then we can define 4 geometric variables embedded in Q. First, let µ 1,2 be the two eigenvalues of E. Angular magnification: M = r 2n (µ 1 + µ 2 ). Angular Distortion: D = r n (µ 1 µ 2 ). Angular Magnification angle: φ is the angle between the largest magnification axis and the x axis in the plane P. Torsion: φ 1 is the angle defining the rotation matrix U. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 14 / 23

25 Eikonal functions We have a very useful way to characterize magnification and distortion. However, how to compute the point eikonal matrix S? Recall that we need to compute it for many gaze directions, and do so very fast, since it must be done at each iteration of the optimization process. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 15 / 23

26 Eikonal functions We have a very useful way to characterize magnification and distortion. However, how to compute the point eikonal matrix S? Recall that we need to compute it for many gaze directions, and do so very fast, since it must be done at each iteration of the optimization process. Clearly, this cannot be done directly by identifying points in the object and image spaces, and finding their optical distance. It would be way too costly. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 15 / 23

27 Eikonal functions We have a very useful way to characterize magnification and distortion. However, how to compute the point eikonal matrix S? Recall that we need to compute it for many gaze directions, and do so very fast, since it must be done at each iteration of the optimization process. Clearly, this cannot be done directly by identifying points in the object and image spaces, and finding their optical distance. It would be way too costly. Instead, we compute an alternative object - the lens mapping. Using the same notation as in Hamilton s theory, the position and ray direction after passing through an optical medium is defined as a mapping x = x (x, y, ξ, η), y = y (x, y, ξ, η), ξ = ξ (x, y, ξ, η), η = η (x, y, ξ, η). Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 15 / 23

28 Propagation of local ekonals Since we are interested in a neighborhood of the base ray we compute only the Jacobian of the lens mapping: n ξ n η x y ( B A = D C ) nξ nη x y Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 16 / 23

29 Propagation of local ekonals Since we are interested in a neighborhood of the base ray we compute only the Jacobian of the lens mapping: n ξ n η x y ( B A = D C ) Near an object point the lens mapping is initialized as an identity. Solving the eikonal equation, we obtain a propagation equation for it in free space J free : nξ nη x y A A, B B + z/na, C C z n A, D D + z/nc z n B zz nn A. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 16 / 23

30 Propagation of local eikonals The lens mapping due to refraction can be computed exactly, again through a geomtrical solution of the eikonal equation. We use the same notation as in Snell s law, and the lens surface is expressed locally as Z = 1 2 αx 2 + βxy γy 2. Then the Jacobian J refraction is given by the 4 submatrices: A = n cos i ( ) n cos i α β cos i cos i cos i β cos i γ cos i cos i ( ) cos i/ cos i 0 B = 0 1 ( cos i C = ) / cos i ( ) 0 0 D = 0 0 Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 17 / 23

31 Propagation of the point eikonal Since the Jacobians of the lens mapping form a group, we can compute the Jacobian at any point as a product. For instance, after refraction by the two lens surfaces, and propagation up to the eye, the Jacobian is J = J free J refraction J free J refraction J free (Id). Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 18 / 23

32 Propagation of the point eikonal Since the Jacobians of the lens mapping form a group, we can compute the Jacobian at any point as a product. For instance, after refraction by the two lens surfaces, and propagation up to the eye, the Jacobian is J = J free J refraction J free J refraction J free (Id). There is a simple canonical formula that converts J into S. Therefore, once the lens mapping Jacobian is known, we can evaluate the point eikonal. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 18 / 23

33 Hamilton? In retrospect we thought: "This could have been done by Hamilton himself!" Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 19 / 23

34 Hamilton? In retrospect we thought: "This could have been done by Hamilton himself!" However, apparently Hamilton did not see why it would be important... Did somebody else think on doing this? Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 19 / 23

35 Hamilton? In retrospect we thought: "This could have been done by Hamilton himself!" However, apparently Hamilton did not see why it would be important... Did somebody else think on doing this? Yes! Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 19 / 23

36 Maxwell Maxwell, in a little forgotten paper, calculated the point eikonal matrix S for a radially symmetric lens: Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 20 / 23

37 Keller And, Keller (JB and HB!!!) worked out the case of an arbitrary lens, but only for the submatrix R. They formulated their work in a differential geometry framework. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 21 / 23

38 Summary The Hamilton point eikonal function (or rather its 4 4 quadratic form) is a very useful tool in spectacle lens design. The point eikonal matrix (S) was computed by Maxwell in the radial case, by Keller and Keller in part, and by us (and Walther) in the general case Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 22 / 23

39 Summary The Hamilton point eikonal function (or rather its 4 4 quadratic form) is a very useful tool in spectacle lens design. The point eikonal matrix (S) was computed by Maxwell in the radial case, by Keller and Keller in part, and by us (and Walther) in the general case The formula forms a central part in a large design code. The code has been used in the last 15 years by two leading companies to design some of the best lenses in the world. Joint work with Gershon Wolansky, Technion. Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 22 / 23

40 Keller, 1987 Koby Rubinstein (Technion) Spectacle lens design following Hamilton, Maxwell and Keller 23 / 23