596 J. Opt. Soc. Am. A/Vol. 16, No. 3 /March 1999 Jose B. Almeida

General method for the determinationof matrix coefficients for

high-order optical system modeling

Jose B. Almeida

Universidade do Minho, Escola de Ciencias, P-4709 Braga, Portugal

Received July 14, 1998; accepted October 30, 1998; revised manuscript received November 16, 1998

The nonlinear transformations incurred by the rays in an optical system can be suitably described by matricesto any desired order of approximation. In systems composed of uniform refractive-index elements each indi-vidual ray refraction or translation has an associated matrix, and a succession of transformations correspondsto the product of the respective matrices. A general method is described to find the matrix coefficients fortranslation and surface refraction, irrespective of the surface shape or the order of approximation. The choiceof coordinates is unusual, as the orientation of the ray is characterized by the direction cosines rather than bythe slopes; this is shown to greatly simplify and generalize coefficient calculation. Two examples are shown inorder to demonstrate the power of the method: The first is the determination of seventh-order coefficients forspherical surfaces, and the second is the determination of third-order coefficients for a toroidal surface.© 1999 Optical Society of America[S0740-3232(99)01903-1]

OCIS codes: 220.1010, 220.2740, 080.2730.

1. INTRODUCTIONAn optical system can be effectively modeled in paraxialapproximation by a product of 4 3 4 matrices, each rep-resenting one elementary transformation of the lightrays1; the elementary transformations are either transla-tions of the ray in homogeneous media or the effects ofsurfaces that separate different media. A more accurateapproach implies the consideration of higher-order terms,but the fact that Snell’s law makes use of the sine func-tion rules out the terms of even order; as a consequence,when one wants to improve on the paraxial approxima-tion, one has to consider third-order terms.

Aberrations have already been studied extensively,2,3

but work is still going on to design symbolic models of op-tical systems that computers can use for optimizationpurposes and that humans can look at to gain a better un-derstanding of system performance.

The matrix theory has been extended to deal withhigher-order terms4 through the use of a vector basis thatincorporates two position coordinates and two orientationcoordinates as well as all their third- or higher-order mo-nomials, thus increasing the overall dimension, which be-comes 24 for third-order approximation. It is possible toapply axis symmetry to reduce the matrix dimension byuse of complex coordinates and their higher-order mono-mials; for example, in the third-order the matrices to beconsidered for axisymmetric systems are 8 3 8.4,5

The set of four coordinates normally used to describethe ray consists of the two rectangular coordinates x and yalong with the two ray slopes u 5 dx/dz and v 5 dy/dz.I chose to replace the ray slopes with the direction cosinesrelative to the coordinate axes, s and t, respectively, to al-low for an easier and more elegant formulation of theSnell’s law at a surface but at the expense of renderingthe translation transformation nonlinear.

0740-3232/99/030596-06$15.00 ©

This paper details a general method that can be used toestablish all coefficients needed for any order modeling ofoptical systems built with surfaces of unspecified shape.The power of the method is exemplified by determinationof matrix coefficients for spherical systems in the seventhorder and for a toroidal surface in the third order.

2. CHOICE OF COORDINATESThe optical axis is assumed to lie along the z axis, so theposition of any point is determined by the two coordinatesx and y; when we deal with axis-symmetric systems, thetwo coordinates can be combined in the complex coordi-nate X 5 x 1 iy. The ray orientation is defined by co-sines of the angles with the x and y axes, s and t, respec-tively; again, a complex orientation coordinate can beused, S 5 s 1 it, to simplify the rotational treatment ofaxis-symmetric systems.

Using the same notation of reference,4 x& is the vectorwhose components are the coordinates and their monomi-als up to order n:

x& 5 1xyst

x3

x2yx2sx2txy2

¯

xjyksltm

¯

2 , (1)

1999 Optical Society of America

Jose B. Almeida Vol. 16, No. 3 /March 1999/J. Opt. Soc. Am. A 597

where total order j 1 k 1 l 1 m is n or less and odd,with all exponents greater than zero. By convention, thevector elements are placed in the order of smaller n andlarger four-digit number jklm.

Any transformation of x& into x8& can be representedwith a square matrix S of dimension N equal to the size ofx&, following the equation

x8& 5 Sx&. (2)

The matrix S has the following structure:

S 5 FP434 H43~N24 !

0~N24 !34 E~N24 !3~N24 !G . (3)

The four submatrices that form S have the followingmeaning: P434 contains the paraxial constants of the op-tical system, H43(N24) has 4(N 2 4) high-order coeffi-cients from the series expansion of the transformed coor-dinates, 0(N24)34 is composed of zeros, and finallyE(N24)3(N24) is obtained from the elements of the firstfour lines by a procedure called extension, whereby the el-ements of lines 5 to N from x8& are calculated and allterms of orders higher than n are dropped. More specifi-cally, if we wish to determine the coefficients for the linethat corresponds to the monomial x jyksltm, we take thepolynomial expressions for x, y, s, and t from lines 1–4and raise them to the powers j, k, l, and m, respectively,thus making their product afterward; the result is a poly-nomial of degree n 3 ( j 1 k 1 l 1 m) from which onlythe terms of orders up to n should be considered. Thesubmatrix E(N24)3(N24) can itself be subdivided into com-ponents of different orders and components with all zeroelements.

Although the ray transformation is described by an N3 N matrix, the above considerations show that only the4(N 2 4) elements from the first four lines need be con-sidered, as the extension procedure is standard for alltransformations. In cases in which symmetry exists thenumber of independent coefficients can be greatlyreduced.4,6 It is apparent that the matrix for any giventransformation can be considered to be completely definedwhen the coefficients of the first four lines have beenevaluated, i.e., when the transformed coordinates havebeen expressed in a power series of order n.

The coordinate conventions made above are not thesame as those made in Refs. 4 and 5 and, indeed, by themajority of authors; they may appear to be a poorerchoice, because, as we shall see, they lead to a nonlineartranslation transformation. However, they will simplifythe determination of refraction coefficients. It should benoted, though, that a coordinate change can be seen as atransformation, itself governed by a matrix; it is not dif-ficult to follow the procedure outlined below and thenchange from cosines into slopes, if needed.

3. ELEMENTARY TRANSFORMATIONSA. Translation MatrixThe first elementary transformation that has to be con-sidered is simply the translation of the ray in an homoge-

neous medium; the orientation coordinates do not change,but the position coordinates change according to the equa-tions

x8 5 x 1se

~1 2 s2 2 t2!1/2 ,

y8 5 y 1te

~1 2 s2 2 t2!1/2 , (4)

where e is the distance traveled, measured along the op-tical axis.

The series expansion of the equations is ratherstraightforward; up to the seventh order it is

x8 5 x 1 es 1e2

s3 1e2

st2 13e8

s5 16e8

s3t2 13e8

st4

15e16

s7 115e16

s5t2 115e16

s3t4 15e16

st6,

y8 5 y 1 et 1e2

s2t 1e2

t3 13e8

s4t 16e8

s2t3 13e8

t5

15e16

s6t 115e16

s4t3 115e16

s2t5 15e16

t7. (5)

The above equations give directly the coefficients forlines 1 and 2 of matrix T, which describes the translationtransformation; lines 3 and 4 are made up of zeros, exceptfor the diagonal elements, which are 1; this translates thetwo relations s8 5 s and t8 5 t. The lines 5 to N can beobtained by the extension procedure described above.

If slopes were used instead of direction cosines thetranslation matrix would have a much simpler form, cor-responding to a linear transformation.4

B. Refraction FormulasThe influence of the surface power on the ray coordinatesmust now be considered. It is useful to analyze sepa-rately the changes on the ray orientation and the modifi-cation of the position coordinates. The ray orientationchanges because of the application of Snell’s law when theray encounters the surface of separation between the twomedia; let this surface be defined by the general formula

f ~x, y, z ! 5 0. (6)

Let a ray impinge on the surface on a point of coordi-nates x and y with direction cosines s and t. The normalto the surface at the incidence point is the vector

n 5 grad f ; (7)

the direction of the incident ray is the unit vector

v 5 S st

~1 2 s2 2 t2!1/2D ; (8)

and the refracted ray has a direction that is representedby the vector

v8 5 S s8t8

~1 2 s82 2 t82!1/2D , (9)

where s8 and t8 are the direction cosines after refraction.

598 J. Opt. Soc. Am. A/Vol. 16, No. 3 /March 1999 Jose B. Almeida

One can apply Snell’s law by equating the cross prod-ucts of the ray directions with the normal on both sides ofthe surface multiplied by the respective refractive indices

n v ^ n 5 v8 ^ n, (10)

where n represents the refractive-index ratio of the twomedia.

The algebraic solutions of Eq. 10 can be found for manysurfaces by means of suitable symbolic processing soft-ware such as MATHEMATICA; in certain cases the solutionis easier to find after a suitable change of coordinatesalong the z axis, which will not affect the final result.For instance, for a spherical surface it is convenient toshift the coordinate origin to the center. To find the co-efficients for the matrix R1 , representing the change inthe ray orientation, the explicit expressions for s8 and t8must be expanded in series. This can again be pro-grammed into MATHEMATICA. The examples given belowshow this procedure for two different surfaces.

C. Surface OffsetThe use of only two position coordinates implies thatthese are referenced to a plane normal to the axis. Whena ray is refracted at a surface, the position coordinatesthat apply are those of the incidence point; nevertheless,the translation to the surface is referenced to the vertexplane and thus the position coordinates that come out ofthe translation transformation are those of the intersec-tion with that plane. The difference between the two po-sitions is called the surface offset.

Figure 1 illustrates the problem that must be solved:The reference plane for the position coordinates of the rayin a refraction process is the plane of the surface vertex;the coordinates of the point of intersection of the ray withthis plane are not the same before and after the refrac-tion. The surface matrix must account not only for theorientation changes but also for the position offset intro-duced by the refraction process.

If the origin of the rectangular coordinates is located atthe vertex and (x, y, 0) are the coordinates of the ray

Fig. 1. Ray intersects the surface at a point X1 , which is differ-ent both from the point of intersection of the incident ray withthe plane of the vertex X and from the point of intersection of therefracted ray with the same plane X2 . The surface is respon-sible for three successive transformations: (1) an offset from Xto X1 , (2) the refraction, and (3) the offset from X1 to X2 .

when it crosses the plane of the vertex, the current coor-dinates of a point on the incident ray are given by the twoequations

x8 5 x 1sz8

~1 2 s2 2 t2!1/2 ,

y8 5 y 1tz8

~1 2 s2 2 t2!1/2 . (11)

As the intersection point has to verify both Eq. (6) andEq. (11), it is possible to solve a system of three simulta-neous equations to determine its coordinates. In factthere may be several intersection points, as the ray mayintercept a complex surface at several points; it is not dif-ficult, however, to select the solution of interest by carefulexamination of the surface region that has been hit. Weare usually interested in the solution that has the small-est uz8u value.

As above, a series expansion of the exact expressions ofx8 and y8 must be performed, taking only the terms up tothe nth order; this will yield the coefficients for the ma-trix T1 that describes the transformation from point X topoint X1 , and it is called the forward offset.

The offset from point X1 to point X2 , designated thebackward offset, is the reverse transformation of the for-ward offset, for a ray whose direction coordinates havebeen modified by refraction. We solve Eqs. (6) and (11) interms of x and y and apply the previous procedure to de-termine the coefficients of the transformation matrix T2 .

The matrix that describes the transformations imposedby the surface is the product T2R1T1 , as the ray has tosuffer the three successive transformations: forward off-set, refraction, and backward offset. The surface matrixis called R.

4. STOP EFFECTSThe entrance pupil of the optical system plays an impor-tant role in the overall aberrations; the center of the pupildefines the chief ray, which determines the paraxial im-age point. The ray fan usually extends equally in all di-rections around the chief ray, and the points of intersec-tion of the various rays in the fan with the image planedetermine the ray aberrations. When matrices are usedto model the system, the image appears described interms of the position and orientation coordinates of therays when these are subjected to the first transformation,be it a refraction or a translation; in terms of ray aberra-tions this corresponds to a situation in which the object islocated at infinity and the entrance pupil is placed wherethe first transformation takes place. In fact, the orienta-tion coordinates play the role of object coordinates and theposition coordinates are the actual pupil coordinates, ifthis coincides with the first transformation; the chief raycan be found easily simply by zeroing the position coordi-nate.

Ray aberrations can be evaluated correctly if the imageis described in terms of both object and stop coordinatesor some appropriate substitutes; an object point is thenset by the object coordinates, the paraxial image point isfound when the stop coordinate is zero, and the aberra-

Jose B. Almeida Vol. 16, No. 3 /March 1999/J. Opt. Soc. Am. A 599

tions are described by the differences to this point. Thiscan be done adequately when the stop is located beforethe first surface and so coincides with the entrance pupil.If the object is at infinity the problem is easy to solve, andapplication examples have already been described.6 Ifthe entrance pupil is located at the position zs relative tothe first surface, it means that the rays must perform atranslation of 2zs before hitting it; this can be accommo-dated by means of the right-hand product by a translationmatrix T2zs

, where the index indicates the amount oftranslation.

The case of near objects is more difficult to deal with,because we are faced with a dilemma: If a translationfrom the pupil position to the first surface is applied, theposition coordinates of the incident rays become pupil co-ordinates but the ray origins on the object are lost. Con-versely, we can apply a translation from the object posi-tion, ending up with the reverse problem of knowing theray origins and ignoring their point of passage throughthe pupil. We need the position coordinates at both ob-ject and pupil locations; this suggests that the ray orien-tation should be specified not by its direction cosines butby the coordinates of the point of passage through the pu-pil. In Fig. 2 a ray leaves an object point of coordinates(x, y) and crosses the entrance pupil plane at a point of co-ordinates (xp , yp); then the direction cosines s and t canbe calculated by

s 5xp 2 x

@~xp 2 x !2 1 zp2#1/2

,

t 5yp 2 y

@~ yp 2 y !2 1 zp2#1/2

, (12)

where zp is the position of the pupil relative to the object.The matrix theory developed above was based on direc-

tion cosines rather than pupil coordinates; in conse-quence, a coordinate change is needed before the transla-tion from the object to the first surface is applied to therays. Rather than use Eqs. (12), it is more convenient tolook at them as a coordinate change governed by a matrixTp ; for the determination of its coefficients we apply thestandard procedure of series expansion. The following isthe result for the seventh order:

Fig. 2. Coordinates of the point where the ray crosses the en-trance pupil, together with the object point coordinates, are aconvenient way of defining the ray orientation to incorporate stopeffects.

s 5 21

zpx 1

1

zpxp 1

1

2zp3 x3 2

3

2zp3 x2xp 1

3

2zp3 xxp

2

21

2zp3 xp

3 23

8zp5 x5 1

15

8zp5 x4xp 2

15

4zp5 x3xp

2

115

4zp5 x2xp

3 215

8zp5 xxp

4 13

8zp5 xp

5 15

16zp7 x7

235

16zp7 x6xp 1

105

16zp7 x5xp

2 2175

16zp7 x4xp

3 1175

16zp7 x3xp

4

2105

16zp7 x2xp

5 135

16zp7 xxp

6 25

16zp7 xp

7; (13)

a similar expression exists for t. Equation (13) allows forthe determination of the coefficients for matrix Tp , whichconverts the coordinates (x, y, xp , yp) into (x, y, s, t) be-fore the rays enter the system at the object position. As asymmetrical conversion is not performed when the raysleave the system, there is no place for the use of matrixTp

21, as would be the case in a similarity transformation.A case study would involve the following successive

steps:

1. Determine the matrices for all the translation andrefraction transformations experienced by the ray fromthe object point to the image plane.

2. Determine the matrix Tp , relative to the entrancepupil position.

3. Multiply all the matrices in reverse order, startingwith the translation from the last surface to the imageplane and ending with matrix Tp ; the result is matrix S.

4. Determine a vector base x& with the set of coordi-nates (x, y, xp , yp).

5. Make the product x8& 5 Sx&.

Once x8& has been calculated, its first four elementsare polynomials of the nth degree on the independentvariables (x, y, xp , yp), which model the position and di-rection cosines of the rays on the image plane.

Most optical systems have stops located after the firstsurface, and so the method described above is not ad-equate. One can always find the Gaussian entrance pu-pil of any system and so revert to the previous situation;this will work even if the entrance pupil is found to lie be-yond the first surface, in which case it will give rise to anegative translation. The problem is that the Gaussianentrance pupil is a first-order transformation of the stop,whereas the real entrance pupil is an aberrated image ofthe stop given by the portion of the optical system thatlies before it. Ideally, one should try to describe the im-age in terms of the object coordinates and the point of pas-sage through the stop, which can be rather difficult. Al-ternatively, one can find a reference ray, defined as theray that passes through the center of the stop and is notnecessarily the same as the chief ray, and work with po-sition and orientation coordinates on the image plane,relative to the reference ray. The plot of the relative rayposition versus its relative direction cosines can be usedto describe ray aberrations.7

600 J. Opt. Soc. Am. A/Vol. 16, No. 3 /March 1999 Jose B. Almeida

5. COEFFICIENTS FOR A SPHERICALSURFACEA sphere of radius r with its center at z 5 r is defined bythe equation

x2 1 y2 1 z2 2 2zr 5 0; (14)

this equation will replace Eq. (6) in the algorithm for thedetermination of matrix coefficients. In a previousstudy6 I presented the results for the third order; theseare now extended to the seventh order.

This is a situation in which symmetry must be consid-ered in order to reduce the size of the matrices involved;

Table 1. Seventh-Order Coefficientsfor Spherical Surfaces

Forward BackwardRefraction Offset Offset

Monomial (S8) (X1) (X2)

X (n 2 1)/r 1 1S n 0 0X2X* n (n 2 1)/2r3 0 0X2S* n (n 2 1)/2r2 0 0XX* S n (n 2 1)/2r2 1/2r 21/2rXSS* n (n 2 1)/2r 0 0X* S2 0 0 0S2S* 0 0 0X3X* 2 n (n3 2 1)/8r5 0 0X3X* S* n2(n2 2 1)/4r4 0 0X3S* 2 n2(n2 2 1)/8r3 0 0X2X* 2S n2(n2 2 1)/4r4 1/8r3 21/8r3

X2X* SS* n (2n3 2 3n 1 1)/4r3 0 1/4r2

X2SS* 2 n2(n2 2 1)/4r2 0 0XX* 2S2 n2(n2 2 1)/8r3 0 1/4r2

XX* S2S* n2(n2 2 1)/4r2 1/4r 21/4rXS2S* 2 n (n3 2 1)/8r 0 0X* 2S3 0 0 0X* S3S* 0 0 0S3S* 2 0 0 0X4X* 3 n (n5 2 1)/16r7 0 0X4X* 2S* n2(3n4 2 2n2 2 1)/16r6 0 0X4X* S* 2 3n4(n2 2 1)/16r5 0 0X4S* 3 n4(n2 2 1)/16r4 0 0X3X* 3S n2(3n4 2 2n2 2 1)/16r6 1/16r5 21/16r5

X3X* 2SS* n (9n5 2 10n3 1 1)/16r5 0 3/16r4

X3X* SS* 2 n2(9n4 2 11n2 1 2)/16r4 0 21/8r3

X3SS* 3 3n4(n2 2 1)/16r3 0 0X2X* 3S2 3n4(n2 2 1)/16r5 0 3/16r4

X2X* 2S2S* n2(9n4 2 11n2 1 2)/16r4 1/16r3 27/16r3

X2X* S2S* 2 n (9n5 2 10n3 1 1)/16r3 0 1/4r2

X2S2S* 3 n2(3n4 2 2n2 2 1)/16r2 0 0XX* 3S3 n4(n2 2 1)/16r4 0 21/8r3

XX* 2S3S* 3n4(n2 2 1)/16r3 0 1/4r2

XX* S3S* 2 n2(3n4 2 2n2 2 1)/16r2 3/16r 23/16rXS3S* 3 n (n5 2 1)/16r 0 0X* 3S4 0 0 0X* 2S4S* 0 0 0X* S4S* 2 0 0 0S4S* 3 0 0 0

we use the complex variables X and S defined above,which leads to a complex vector base X& whose elementshave the general form X jX* kSlS* m, with X* and S* rep-resenting the complex conjugates of X and S. Kondo andTakeuchi4 showed that the powers must obey the condi-tion j 2 k 1 l 2 m 5 1; for the seventh order the allow-able combinations are

Table 2. Third-Order Coefficients for a Torous

Forward ForwardRefraction Refraction Offset Offset

Monomial (s8) (t8) (x8) ( y8)

xn 2 1

r10 1 0

y 0n 2 1

r1 1 r20 1

s 0 0 0 0t 0 0 0 0

x3n ~n 2 1!

2r13

0 0 0

x2y 0n ~n 2 1!

2r12~r1 1 r2!

0 0

x2sn ~n 2 1!

r12

01

2r1

0

x2t 0 01

2r10

xy2n ~n 2 1!

2r1~r1 1 r2!20 0 0

xys 0n ~n 2 1!

r1~r1 1 r2!0 0

xytn ~n 2 1!

r1~r1 1 r2!0 0 0

xs2 n ~n 2 1!

2r10 0 0

xst 0 0 0 0

xt2 n ~n 2 1!

2r10 0 0

y3 0n ~n 2 1!

2~r1 1 r2!30 0

y2s 0 0 01

2~r1 1 r2!

y2t 0n ~n 2 1!

~r1 1 r2!20

12~r1 1 r2!

ys2 0n ~n 2 1!

2~r1 1 r2!0 0

yst 0 0 0 0

yt2 0n ~n 2 1!

2~r1 1 r2!0 0

s3 0 0 0 0s2t 0 0 0 0st2 0 0 0 0t3 0 0 0 0

Jose B. Almeida Vol. 16, No. 3 /March 1999/J. Opt. Soc. Am. A 601

1000, 0010, 2100, 2001, 1110, 1011, 0120, 0021,

3200, 3101, 3002, 2210, 2111, 2012, 1220, 1121,

1022, 0230, 0131, 0032, 4300, 4201, 4102, 4003,

3310, 3211, 3112, 3013, 2320, 2221, 2122, 2023,

1330, 1231, 1132, 1033, 0340, 0241, 0142, 0043.

Equation (10) can now be solved, and the resulting ex-pressions for s8 and t8 can be expanded in series up to theseventh order; the results can be combined into one com-plex variable S8. The second column of Table 1 lists thecoefficients for this expansion. Similarly, the forwardoffset coefficients can be found by solving Eqs. (11) andcombining x8 and y8 expansions into one single complexvariable X1 ; the resulting coefficients are listed in column3 of Table 1. The backward offset coefficients are ob-tained in a similar way, although now it is the x and y ex-pansions that must be combined into a single complexvariable X2 .

6. COEFFICIENTS FOR AN ASYMMETRICSURFACEToroidal surfaces are used to correct eye astigmatism;they can be fabricated easily without resorting to sophis-ticated machinery. In this example I aim to show thatthe method is applicable to general surfaces, but the pur-pose here is not to list high-order coefficients for this par-ticular shape. Therefore the study will be restricted tothe third order. Also, as there is no rotational symmetry,complex coordinates will not be used; there is symmetryrelative to two perpendicular planes, and if complex coor-dinates were to be used, the terms should obey the condi-tion j 2 k 1 l 2 m 5 61 as explained in Ref. 5. It maybe clearer not to invoke symmetry at all.

The surface is generated by a circle of radius r1 normalto the xz plane whose center is displaced along a circle ofradius r2 lying on this plane. This surface has curvatureradii of r1 and r1 1 r2 , respectively, on the xz and yzplanes. The surface equation can be written as

~x2 1 y2 1 z2 2 r12 2 r2

2!2 2 4r22~r1

2 2 x2! 5 0. (15)

The solution of Eq. (10) originates the expressions fors8 and t8; these are then expanded in series up to thethird order, and the resulting coefficients are listed in col-umns 2 and 3 of Table 2. The same procedure was ap-plied to the expressions for x8 and y8 that result from Eqs.(11); columns 4 and 5 of the same table list these coeffi-

cients. The backward offset coefficients need not belisted, as in the third order they can be obtained by signreversal from the forward offset coefficients.

7. DISCUSSION AND CONCLUSIONMatrices can be used to model optical systems built withsurfaces of unspecified shape to any desired degree of ap-proximation. The method that was described differsfrom those found in the literature by the choice of direc-tion cosines rather than ray slopes to specify ray orienta-tions. This results in an elegant formulation of Snell’slaw, which can be easily programmed into symbolic com-putation software. In spite of some added complicationin translation matrices, the algorithm is limited only bycomputing power in its ability to determine the matrix co-efficients, irrespective of surface complexity and order ofapproximation. For axis-symmetric systems the matri-ces have a dimension 40 3 40 for the seventh order andcan usually be handled with ordinary personal comput-ers. The influence of stops can also be incorporated intothe matrix description, as was demonstrated. The un-conventional coordinate system that was used can, ifneeded, be converted into the more usual ray slope coor-dinate system through the product with conversion matri-ces of the same dimension.

Two examples were shown for illustration of the meth-od’s capabilities; the first one supplies a list of seventh-order coefficients for systems made with spherical sur-faces and allows the reader immediate use. The secondexample is the determination of third-order coefficientsfor a toroidal surface, showing that complex shapes can beaccommodated.

REFERENCES1. A. Gerrard and J. M. Burch, Introduction to Matrix Meth-

ods in Optics (Dover, New York, 1994).2. M. Born and E. Wolf, Principles of Optics (Pergamon Press,

New York, 1980).3. G. G. Slyusarev, Aberration and Optical Design Theory

(Hilger, Bristol, UK, 1984).4. M. Kondo and Y. Takeuchi, ‘‘Matrix method for nonlinear

transformation and its application to an optical lens sys-tem,’’ J. Opt. Soc. Am. A 13, 71–89 (1996).

5. V. Lakshminarayanam and S. Varadharajan, ‘‘Expressionsfor aberration coefficients using nonlinear transforms,’’ Op-tom. Vis. Sci. 74, 676–686 (1997).

6. J. B. Almeida, ‘‘Use of matrices for third-order modeling ofoptical systems,’’ in International Optical Design Confer-ence, K. P. Thompson and L. R. Gardner, eds., Proc. SPIE3482, 917–925 (1998).

7. D. S. Goodman, in Handbook of Optics, M. Bass, ed.(McGraw-Hill, New York, 1995), Vol. 1, p. 93.