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Spectacle lens design following Hamilton, Maxwell
and Keller
Koby Rubinstein
Technion
Koby Rubinstein (Technion)
Spectacle lens design following Hamilton, Maxwell and Keller
1 / 23
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
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
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)
Koby Rubinstein (Technion)
(add (2.5))
Spectacle lens design following Hamilton, Maxwell and Keller
3 / 23
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):
V =
Koby Rubinstein (Technion)
S+
C
2
C
2
C
cos 2ψ
2 sin 2ψ
sin 2ψ
S − C2 cos 2ψ
Spectacle lens design following Hamilton, Maxwell and Keller
3 / 23
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
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
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
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
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
Hamilton - 1824
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
Hamilton - 1824
Koby Rubinstein (Technion)
Spectacle lens design following Hamilton, Maxwell and Keller
8 / 23
Hamilton Eikonal functions
x’
x
(x,h,z)
(x’,h’,z’)
z’
y
z
y’
Consider two planes P and P 0 separated by an optical medium
(lenses, mirrors, ... anything). The Point Eikonal function
S(x, y , x 0 , y 0 ) is defined to be the optical distance between two
arbitrary points (x, y ) ∈ P, and (x 0 , y 0 ) ∈ P 0 .
Koby Rubinstein (Technion)
Spectacle lens design following Hamilton, Maxwell and Keller
9 / 23
Hamilton Eikonal functions
x’
x
(x,h,z)
(x’,h’,z’)
z’
y
z
y’
Consider two planes P and P 0 separated by an optical medium
(lenses, mirrors, ... anything). The Point Eikonal function
S(x, y , x 0 , y 0 ) is defined to be the optical distance between two
arbitrary points (x, y ) ∈ P, and (x 0 , y 0 ) ∈ P 0 .
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 0 , y 0 ) is
given by −n∇Sx,y .
Koby Rubinstein (Technion)
Spectacle lens design following Hamilton, Maxwell and Keller
9 / 23
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 0 , η 0 ) = inf
S(x, y , x 0 , y 0 ) − n0 x 0 ξ 0 − n0 y 0 η 0 .
0 0
x ,y
Koby Rubinstein (Technion)
Spectacle lens design following Hamilton, Maxwell and Keller
10 / 23
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 0 , η 0 ) = inf
S(x, y , x 0 , y 0 ) − n0 x 0 ξ 0 − n0 y 0 η 0 .
0 0
x ,y
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
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 0 , η 0 ) = inf
S(x, y , x 0 , y 0 ) − n0 x 0 ξ 0 − n0 y 0 η 0 .
0 0
x ,y
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 0 , y 0 ), 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
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 = 12 X t SX , about a base ray, where

S=
Koby Rubinstein (Technion)
P
Qt
Q
R
Π11 Π12
 Π12 Π22
=
 Q11 Q21
Q12 p22
Q11
Q21
R11
R12

Q12
Q22 

R12 
R22
Spectacle lens design following Hamilton, Maxwell and Keller
11 / 23
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 = 12 X t SX , about a base ray, where

S=
P
Koby Rubinstein (Technion)
P
Qt
Q
R
Π11 Π12
 Π12 Π22
=
 Q11 Q21
Q12 p22
Q11
Q21
R11
R12

Q12
Q22 

R12 
R22
P’
Spectacle lens design following Hamilton, Maxwell and Keller
11 / 23
Local eikonals
For example, the direction of the rays on both sides of the optical
element are determined by


 

x
Π11 Π12 Q11 Q12
−nξ
 −nη   Π12 Π22 Q21 Q22   y 


 

 n0 ξ 0  =  Q11 Q21 R11 R12   x 0  .
y0
Q12 p22 R12 R22
n0 η 0
Koby Rubinstein (Technion)
Spectacle lens design following Hamilton, Maxwell and Keller
12 / 23
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
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
0 rQ t
ξ
x/r
=
.
η0
y /r
n0
)
)
Koby Rubinstein (Technion)
Spectacle lens design following Hamilton, Maxwell and Keller
13 / 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 µ1,2 be the two eigenvalues of E.
Koby Rubinstein (Technion)
Spectacle lens design following Hamilton, Maxwell and Keller
14 / 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 µ1,2 be the two eigenvalues of E.
Angular magnification: M =
Angular Distortion: D =
r
2n0 (µ1
r
n0 (µ1
+ µ2 ).
− µ2 ).
Angular Magnification angle: φ is the angle between the largest
magnification axis and the x axis in the plane P 0 .
Torsion: φ1 is the angle defining the rotation matrix U.
Koby Rubinstein (Technion)
Spectacle lens design following Hamilton, Maxwell and Keller
14 / 23
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
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
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 0 = x 0 (x, y , ξ, η), y 0 = y 0 (x, y , ξ, η),
ξ 0 = ξ 0 (x, y , ξ, η), η 0 = η 0 (x, y , ξ, η).
Koby Rubinstein (Technion)
Spectacle lens design following Hamilton, Maxwell and Keller
15 / 23
Propagation of local ekonals
Since we are interested in a neighborhood of the base ray we
compute only the Jacobian of the lens mapping:


 0 0 
nξ
nξ

 n0 η 0 
B −A 
 nη 

=
0
 x 
 x 
−D C
0
y
y
Koby Rubinstein (Technion)
Spectacle lens design following Hamilton, Maxwell and Keller
16 / 23
Propagation of local ekonals
Since we are interested in a neighborhood of the base ray we
compute only the Jacobian of the lens mapping:


 0 0 
nξ
nξ

 n0 η 0 
B −A 
 nη 

=
0
 x 
 x 
−D C
0
y
y
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 Jfree :
A → A, B → B + z/nA,
C → C − z 0 n0 A, D → D + z/nC − z 0 n0 B − zz 0 nn0 A.
Koby Rubinstein (Technion)
Spectacle lens design following Hamilton, Maxwell and Keller
16 / 23
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
1 2
αx + βxy + γy 2 .
2
2
Then the Jacobian Jrefraction is given by the 4 submatrices:
n0 cos i 0 − n cos i
α
β cos i
A=
β cos i 0 γ cos i cos i 0
cos i cos i 0
cos i/ cos i 0 0
B=
0
1
0
cos i / cos i 0
C=
0
1
0 0
D=
0 0
Koby Rubinstein (Technion)
Spectacle lens design following Hamilton, Maxwell and Keller
17 / 23
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 = Jfree Jrefraction Jfree Jrefraction Jfree (Id) .
Koby Rubinstein (Technion)
Spectacle lens design following Hamilton, Maxwell and Keller
18 / 23
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 = Jfree Jrefraction Jfree Jrefraction Jfree (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
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
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
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
Maxwell - 1874
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
Keller - 1950
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
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
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
Keller, 1987
Koby Rubinstein (Technion)
Spectacle lens design following Hamilton, Maxwell and Keller
23 / 23