lie algebraic treatment of dioptric power and optical aberrations

Lakshminarayanan et al. Vol. 15, No. 9 /September 1998 /J. Opt. Soc. Am. A 2497

Lie algebraic treatment of dioptric power andoptical aberrations

V. Lakshminarayanan

School of Optometry and Department of Physics and Astronomy, University of Missouri at St. Louis,8001 Natural Bridge Road, St. Louis, Missouri 63121-4499

R. Sridhar and R. Jagannathan

The Institute of Mathematical Sciences, Tharamani, Chennai, India 600113

Received December 12, 1997; revised manuscript received May 11, 1998; accepted May 11, 1998

The dioptric power of an optical system can be expressed as a four-component dioptric power matrix. Wegeneralize and reformulate the standard matrix approach by utilizing the methods of Lie algebra. This gen-eralization helps one deal with nonlinear problems (such as aberrations) and further extends the standardmatrix formulation. Explicit formulas giving the relationship between the incident and the emergent rays arepresented. Examples include the general case of thick and thin lenses. The treatment of a graded-indexmedium is outlined. © 1998 Optical Society of America [S0740-3232(98)03909-X]

OCIS codes: 220.1010, 120.4820.

1. INTRODUCTIONDioptric power of a lens is expressed in the usual repre-sentation as S � CXu, where S is the spherical power, Cis the cylindrical power, and u specifies the cylinder axis.Even though this is perfectly suited for the usual clinicalpurposes, it has been pointed out that ‘‘the future may re-veal wholly new types of refractive behavior for which tra-ditional measure is unsuitable.’’ 1 In particular, this ap-proach is not amenable to mathematical and statisticaltreatments. In an alternative formulation, dioptricpower, usually expressed as a four-component dioptricpower matrix, overcomes these shortcomings. The na-ture and representation of the dioptric power matrix is re-viewed in a recent paper by Harris.1 We refer the readerto this paper and to the extensive references therein.

Because of the reliance on matrix operators, which arelinear, there are certain constraints, specifically, for thetreatment of aberrations (i.e., departures from Gaussianor paraxial optics) and hence a need for a reformulationand extension of the dioptric power matrix method. Amatrix method for doing nonlinear calculations for anyarbitrary conic surface has recently been described byLakshminarayanan and Varadharajan,2,3 based on a pro-posal by Kondo and Takeuchi,4 to calculate aberrationsup to any arbitrary order. This method has been ex-tended to include the case of a general spherocylindricalsurface. However, the drawback of the matrix methods(when extended to the nonlinear regime) is that in gen-eral large matrices are required. For example, if onewere to go up to the fifth order, the size of the matrices isof the order of 125 3 125. The size of the matrices can bereduced by the use of symmetries, but the method is stillcumbersome.

A generalized operator technique would be of great usefor computational purposes, especially for considering any

0740-3232/98/092497-07$15.00 ©

general asymmetrical optical system. With a general op-erator technique it should be possible to evaluate aberra-tion coefficients up to any order by symbolic computationprograms such as MATHEMATICA, MAPLE, or MACSYMA.With this general goal in mind, we have analyzed and ex-tended the formalism of the dioptric power matrix, usingLie algebraic methods.

2. BACKGROUNDA. Lie Algebra in Geometrical OpticsDragt and his co-workers (e.g., Refs. 5–7) have proposednovel methods employing Lie algebraic tools for charac-terizing optical systems. Here the action of each elementof a compound optical system on a ray of light can be rep-resented by an operator. The operator for various ele-ments of the system can be found by use of specific rulesand multiplied together, to yield a resultant operatorcharacterizing the entire system, in a manner similar tothe 2 3 2 system matrix in ordinary Gaussian optics.8

The basic idea of the Lie algebraic technique is thatgeometrical optics can be formulated in terms of a varia-tional principle involving a Lagrangian, and from this onemay find equivalent Hamiltonian equations of motion.9

If the Hamiltonian is known, then, using standard meth-ods of Lie algebra, one can describe geometrical light op-tics in terms of symplectic maps.5–7 These methods pro-vide an operator extension of the linear matrix method tothe general case. A general exposition of Lie algebraicmethods in optics can be found in the papers by Dragt andhis co-workers5,6,10 and by Forest,11 in the book byStavroudis,12 or in the book edited by Mondragon andWolf (Ref. 7).

First, let us briefly recall the basics of the Lie approachto the dynamics of any Hamiltonian system. Let z i de-

1998 Optical Society of America

2498 J. Opt. Soc. Am. A/Vol. 15, No. 9 /September 1998 Lakshminarayanan et al.

note the set of initial 2n phase-space coordinates (gener-alized coordinates and their conjugate momenta) and z f,the final phase-space coordinates. Then, the evolution ofthe system can be described by

z f 5 Mz i (1)

or by

z af 5 Ka 1 (

bRabz b

i 1 (bc

Tabcz bi z c

i 1 ...

~a, b 5 1, 2, ... 2n!,

where M is a transfer map. If small changes are made inthe initial coordinates z i, then there will be correspond-ing changes in the final coordinates. The relationship be-tween these changes can be expressed in terms of theJacobian matrix M:

Ma,b 5]z a

f

]z bi . (2)

Let J be the 2n 3 2n matrix,

J 5 F 0 I

2I 0G , (3)

where I is the identity matrix. It can be proved that theevolution of z is governed by Hamilton’s equations, andthis results in

MTJM 5 J, (4)

where MT is the transpose of M. This is the conditionthat M be a symplectic matrix. Correspondingly, whenM has a Jacobian matrix M that is symplectic at allpoints, the map is symplectic.

In optics, Eq. (4) leads to the well-known lens equation(Ref. 12, p. 245) and is itself called the lens equation, andM is the optical symplectic map. Note that a symplecticmap is a canonical transformation, and vice versa.13 Thesymplectic map can be uniquely defined in terms of a Lietransformation operator and for an optical system con-sisting of many elements can be defined in terms of aproduct of homogeneous polynomials of various degrees inthe variables z i (see Section 3). The fundamental prob-lem of applying Lie algebra to optics becomes one of de-termining which polynomials correspond to various opti-cal elements and which polynomials result fromcascading optical elements, and studying those polynomi-als that correspond to various desired optical properties.

B. Dioptric Power MatrixIn general, the dioptric power can be defined in terms ofthe so-called ray-transfer matrix. Dioptric power definesthe additive contribution of incident position to the emer-gent direction of a ray passing through the system. Ingeneral, if a ray vector r0 is incident upon the entranceplane of an optical system and emerges from the exitplane as a vector r (in some suitable coordinate system),the fundamental Gaussian relationship is

r 5 Sr0 , (5)

where S is a 4 3 4 ray-transfer matrix that contains allthe information about the system and the ray-transfermatrix obeys the symplectic condition14–18 [see Eq. (4)].In general, we write S as

S 5 FA B

C DG , (6)

and the dioptric power (also known as the equivalent di-optric power) for any optical system is given by1,18

F 5 2C. (7)

The dioptric power is a member of a four-dimensional Eu-clidean power space with a three-dimensional symmetri-cal power subspace.

In this paper we reformulate the dioptric power matrixS in terms of Lie operators and extend it to the domain ofnonlinear regimes.

3. FORMALISMConsider an optical system in which a ray of light is inci-dent at the entrance pupil. Below we will use the coor-dinate system and notation used by Harris1 (Fig. 1). LetO be the optical axis of the system and T be the plane ofthe paper, taken perpendicular to O. Let R0 and R beperpendicular projections of the segment of the ray on T.R0 represents the incident ray, and R the outgoing ray.P is the point at which R0 meets T and has position vectory relative to O with respect to a rectangular Cartesiansystem of axes y1 , y2 . More degrees of freedom are nec-essary to completely specify the ray; that is, we need toinclude the direction of propagation. Consider the planey2 5 0 and the perpendicular projection of the ray ontothis plane. Let a1 be the tangent of the angle of projec-tion with the axis O. This is the direction tangent of R inthe direction of y1 . Similarly, by considering the planey1 5 0 and the perpendicular projection of the ray ontothis plane, we can define a2 as the tangent of R in the di-rection of y2 . Therefore we can define (a1 , a2) to be thedirection of R. If n is the index of refraction of the me-dium, then a i 5 nai (i 5 1, 2) defines the so-called re-duced direction of R.

For convenience, let us identify O with the Z direction.Dy1 /Dz and Dy2 /Dz can be seen to be a1 and a2 , respec-tively, so in the continuum limit we can write

Fig. 1. Coordinate system used in the analysis.


a i 5 ndyi

dz. (8)

The elemental path length along the direction of propaga-tion of the ray is simply

ds 5 @~dz !2 1 ~dy1!2 1 ~dy2!2#1/2

5 @1 1 ~ y18 !2 1 ~ y28 !2#1/2dz, (9)

where

yi8 5dyi

dz5

a i

n.

The optical path length is simply

A 5 Ezi

zf

nds 5 Ezi

zf

n@1 1 ~ y18 !2 1 ~ y28 !2#1/2dz. (10)

From Fermat’s principle we know that the path length isan extremum. This implies that, if we take the Lagrang-ian to be

L 5 n@1 1 ~ y18 !2 1 ~ y28 !2#1/2, (11)

then the ray path satisfies the Euler–Lagrangeequations19:

ddz S ]L

]yi8D 2

]L]yi

5 0 ~i 5 1, 2 !. (12)

We can define our canonical coordinates of $ yi , pi (i5 1, 2)% as

pi 5]L

]yi85

na i

~n2 1 a12 1 a2

2!1/2 . (13)

The corresponding Hamiltonian is given by

H 5 2@n2~r ! 2 p12 2 p2

2#1/2, (14)

where n, the refractive-index distribution, is independentof yi8 .

r, the ray vector, is characterized by the four phase-space variables (q1 , q2) and ( p1 , p2) and by its trans-pose rT 5 (q1 , q2 , p1 , p2), where q1 and q2 are identi-fied with y1 and y2 , respectively, used in Eq. (8). Asdescribed in Section 2, the functional relationship be-tween the incident and the emergent ray vectors r i andr f, respectively, is given by the symplectic map [Eq. (1)],where z i and z f are identified with r i and r f, respec-tively. This relationship and its consequences are due tothe fact that Fermat’s principle is equivalent to the state-ment that the incident and emergent rays are related by atrajectory governed by the Hamiltonian. This enables usto introduce the phase space spanned by yi and a i . Wecan also define the Poisson bracket of classicalmechanics20 of two functions f and g defined on the phasespace as

$ f, g% 5 (i

S ]f

]qi

]g

]pi2

]f

]pi

]g

]qiD . (15)

Here, following Dragt et al., we introduce the Lie operatorassociated with f, denoted by :f:, which acts on a generalfunction g as

:f:g 5 $ f, g%. (16)

The powers of the Lie operator :f:n are defined as

:f:2g 5 $ f, $ f, g%%, (17)

with analogous definitions for higher integral powers ofthe Lie operator, with the condition that : f: to the zeropower is the identity operator

:f:0g 5 g,

and we define

exp~ :f: ! 5 (n50

` :f:n

n!,

with the rule

exp~ :f: !g 5 g 1 $ f, g% 112 $ f, $ f, g%%

113! $ f, $ f, $ f, g%%% 1 ... . (18)

With these steps it is possible to write a fundamental re-sult, known as the factorization theorem.19 This theorembasically states that, if M is the symplectic map betweenan initial phase-space variable and the final phase-spacevariable [Eq. (1)], then it can be written uniquely as theproduct

M 5 exp~ :f2 : !exp~ :f3 : !exp~ :f4 : !..., (19)

where fn is a polynomial of degree n in the components ofr. Note that this solution completely satisfies the lensequation. Equation (19) is valid when M maps the originof the phase space into itself. The polynomial f2 repre-sents the paraxial matrix optics; f3 , f4 , etc. represent thedepartures from paraxial optics and reproduce the nonlin-ear effects. The polynomial f4 will represent the third-order aberrations; f6 , the fifth-order aberrations, etc., inthe case of a system with axial rotation and reflectionsymmetries when f3 5 f5 5 ... 5 0. Even if we truncatethe product at a finite number of steps, the product stillrepresents a symplectic map. The reader is referred tothe literature on Lie methods in optics for further details(Refs. 5–7, 21, and 22).

4. EXPRESSIONS FOR ABERRATIONSFROM THE DIOPTRIC POWER MATRIXIn Section 3 we noted that the polynomial f2 in the sym-plectic mapping represents ordinary Gaussian optics.This means that, by retaining f2 only, one can reproduceall the results in the Gaussian approximation by a suit-able choice of f2 . For example, the choice of

f2 5 2l

2

n~a12 1 a2

2!

~n2 1 a12 1 a2

2!5 2

l

2n~ p1

2 1 p22!

(20)

will yield a transit through a medium of uniform refrac-tive index n, through a distance l in the Gaussian ap-proximation. It is easy to show that this is true. Con-sider

qf 5 Mqi 5 exp~21/2n:p2: !qi 5 qi 1 l/npi,

pf 5 Mpi 5 pi.

q


In general, if we assume that the optical system is axi-ally symmetrical about some axis and is also symmetricalwith respect to reflection through some plane containingthe axis of symmetry, then, it can be shown from symme-try considerations that only even indices n of the polyno-mial fn exist. In other words, only polynomials contain-ing powers of the terms p2 (5p1

2 1 p22), p • q

(5p1q1 1 p2q2), q2 (5q12 1 q2

2) will exist, and the oddpower homogeneous polynomials will vanish. In the gen-eral case of a completely asymmetrical system, the otherodd powers will also exist.

Consider third-order aberrations of the system. Be-cause of symmetries, f4 can depend only on the variablesp2, p • q, and q2, and the most general homogeneouspolynomial that can be written is

f4 5 A~ p2!2 1 Bp2~ p • q ! 1 C~ p • q !2

1 Dp2q2 1 E~ p • q !q2 1 F~q2!2. (21)

Comparing with the general Taylor-series representationof the wave front (up to third-order aberrations), we findthat the various terms represent the various Seidel aber-rations. The quantities A –F are arbitrary coefficientswhose values depend on the optical system under consid-eration. The first term in the series represents sphericalaberration; the second term, coma; the third term, astig-matism; the fourth term, curvature of field; and the fifthterm, distortion.23 The sixth term has no effect on imagequality.

We now give the explicit expressions for the various ab-errations in terms of the generalized coordinates (with a5 1, 2):

1. Spherical Aberration

f 5 A~ p2!2,

qaf 5 qa

i 2 4Apai pi2,

paf 5 pa

i .

2. Coma

f 5 Bp2~ p • q !,

qaf 5 qa

i 2 B@2pai ~ pi

• qi! 1 pi2qai #

1B2

2!@4pi2~ pi

• qi!pai 2 pi4qa

i #,

paf 5 pa

i 1 Bpi2pai 1

B2

2!3pi4pa

i .

Terms of order higher than B2 have been ignored.

3. Astigmatism

f 5 C~ p • q !2,

qaf 5 qa

i 2 2C~ pi• qi!qa

i 1 2C2~ pi• qi!2qa

i ,

paf 5 pa

i 1 2C~ pi• qi!pa

i 1 2C2~ pi• qi!2pa

i .

We have neglected terms of the order of C3 and higher.

4. Curvature of Field

f 5 Dp2q2,

af 5 qa

i 2 2Dqi2pai 2 2D2@ pi2qi2qa

i 2 2qi2 ~ pi• qi!pa

i #,

paf 5 pa

i 1 2Dpi2qai 1 2D2pi2@2~ pi

• qi!qai 2 qi2pa

i #.

Here we have neglected terms of order D3.

5. Distortion

f 5 E~ p • q !q2,

qaf 5 qa

i 2 Eqai qi2 1 3

E2

2!qi4qa

i ,

paf 5 pa

i 1 E@ pai qi2 1 2qa

i ~ pi• qi!#

1E2

2!@4qi2~qi

• pi!qai 2 qi4pa

i #.

Terms of order E3 and higher have been neglected.

6. This sixth term has no effect on the quality of theimage. However, it affects the arrival direction a of theray in the image plane.

f 5 F~q2!2,

qaf 5 qa

i ,

paf 5 pa

i 1 4Fqai qi2.

5. GENERAL EXAMPLELet us consider the propagation of a light ray through asystem (Fig. 2) that has constant refractive indices n1 andn2 in regions separated by an interface represented by theequation

zs 5 b2~q12 1 q2

2! 1 b4~q12 1 q2

2!2 1 .... (22)

In this case we can write

n~z ! 5 n1 1 ~n2 2 n1!U@z 2 b2~q12 1 q2

2!

2 b4~q12 1 q2

2!2 2 ...#, (23)

where U is the Heaviside step function. Thus the Hamil-tonian in Eq. (14) is completely known. Since the evolu-tion z i 5 (qi, pi) → z f 5 (qf, pf ) is governed by Hamil-ton’s equations, the map z f 5 Mz i is related to theHamiltonian H as

dMdz

5 M:2H~z i, z !:. (24)

The unique solution to this equation for a given Hamil-tonian H can be written down formally as

M 5 PH expF2:Ezi

z

H~z i, z8!dz8:G J , (25)

where the symbol P denotes z ordering of the exponential.Using this standard procedure (see Refs. 5–7 and 21 for

Fig. 2. Generalized interface between two media. See text fordetails.


details), one can obtain the map M for the given systemwith n(z) as above. After some considerable computa-tion, one finds22 that

M 5 e :f2 :e :f4 :..., (26)

where

f2 5 b2~n2 2 n1!q2,

f4 5 2b23@~n1 2 n2!/n1#$n1@2 2 ~b4 /b2

3!# 2 2n2%~q2!2

1 2b22@~n1 2 n2!/n1#q2~ p • q !

1 b2@~n1 2 n2!/2n1n2#q2p2.

Comparing this f4 with the general expression in Eq. (21),

M 5 e :g4 :eb2~12n !:q2:e2@t/~2n !#:p2:e2@t/~8n3!#:~ p2!2:

3 eg2~12n !:q2:e :f4 :, (28)

where g4 , ( p2)2, and f4 characterize aberrations. Leav-ing out these aberration terms leads to the paraxial(Gaussian) approximation with the corresponding map:

M 5 eb2~12n !:q2:e2~t/2n !:p2:eg2~12n !:q2:. (29)

The effect of this map is

S qW f

qW f D 5 MS qW i

qW i D 5 SS qW i

qW i D , (30)

with S being the ray-transfer matrix [Eq. (6)]:

one can understand the effect of the interface in terms ofaberrations.

Propagation in an axially symmetrical graded-indexmedium can also be treated by this method, since writing

n~z ! 5 n0~z ! 1 n2~z !~q12 1 q2

2! 1 n4~z !~q12 1 q2

2!2

1 ...

determines the Hamiltonian completely, and the corre-sponding map M can be obtained uniquely.

Let us now consider a lens such as that shown in Fig. 3.The lens is bounded by surfaces given by the equations

z 5 z1~q ! 5 g2~q12 1 q2

2! 1 g4~q12 1 q2

2!2 1 ...,

z 5 z2~q ! 5 b2~q12 1 q2

2! 1 b4~q12 1 q2

2!2 1 ....(27)

The refractive index of the material of the lens is assumedto be n, a constant, and the total thickness of the lens ist 5 t1 1 t2 . The map for this lens can be worked out tobe5

Fig. 3. Simple lens of thickness t and of uniform refractive in-dex.

S 5 FA B

C DG 5 F 1 1 2g2~1 2 n !t/n t/n

2~1 2 n !@~b2 1 g2! 1 2~1 2 n !g2tb2 /n# 1 1 2b2~1 2 n !t/nG . (31)

For the case of a spherical lens with surfaces having radii of curvature R1 and R2 , we can identify g2 5 1/2R1 andb2 5 21/2R2 . Then the power matrix S becomes

S 5 F 1 1 ~n 2 1 !t/nR1 t/n

~1 2 n !F S 1R1

21

R2D 1

~n 2 1 !tnR1R2

G 1 2 ~n 2 1 !t/nR2G . (32)

The dioptric power can now be identified as

F 5 2C 5 1/f

5 ~n 2 1 !@~1/R1 2 1/R2! 1 ~n 2 1 !t/nR1R2#,

(33)

which is the well-known result for the power of a thicklens. For a thin lens t 5 0, and we get the usual result,the so-called Lensmaker’s equation.

6. DISCUSSION AND CONCLUSIONSUsing the technique of Lie operator algebra pioneered byDragt and co-workers in the context of geometrical optics,we have generalized the concept of the dioptric power ma-trix useful in ophthalmic and visual optics problems andhave extended the formalism to include aberrations. Theusual methods of Harris and others are not appropriate totreat aberrations, since this is an extension to the non-paraxial regime. This generalization is possible onlywith the identification of the direction tangents (andhence the a’s) with the components of the canonical mo-menta of the optical Hamiltonian. With this crucial step,it is possible to write the Lie algebra operators. It is, ingeneral, not possible to write the explicit relations thatgive the relationship between the incident and the emer-gent rays in the matrix representation, since the relation-ship involves the nonlinear terms. An alternative proce-dure for representing the nonlinear expansion has beengiven elsewhere.2,3 The operator technique is an exten-sion of the matrix methods to the general case. As notedabove, the matrix method of Lakshminarayanan andVaradharajan2 is cumbersome, but with the Lie algebraictechnique it should be possible to evaluate all the aberra-tion terms for any optical system. Actually, the matrix


method of Lakshminarayanan and Varadharajan can berecovered from the Lie algebraic method if the vector r isgeneralized to also include the higher powers of p, q, etc.,and if the matrix relating r i and r f is calculated withtruncation up to the desired order. Note that, with thespecification of the refractive-index distribution n(z), forany optical system, the Hamiltonian is completely deter-mined, and the corresponding symplectic map M describ-ing the effect of the system on any incident ray can becomputed in a unique manner by a systematic procedure.An obvious application of this procedure in visual optics isthe analysis of image formation by the gradient-index eyelens.24 This example, however, is beyond the scope of thepresent paper and will be dealt with elsewhere. Addi-tionally, in the present paper the equations for thick andthin lenses have been derived explicitly from the Lie al-gebraic point of view, and the relationship between theseequations and the well-known ABCD matrix methodologyis shown.

How does the Lie algebraic method compare with other,more common, techniques? There are several computerray-tracing programs that trace a large number of rays byemploying Snell’s law at various interfaces in the systemunder consideration and by searching the lens parameterspace for optimization of the various features of the re-sulting spot diagram. However, as pointed out byDragt,5,6 these search codes tend to run rather blindly.Lie algebraic techniques may provide additional informa-tion regarding the sources of aberrations and the extentto which these may be corrected. It is worthwhile notingthat a computer code for charged-particle beam transportbased on the Lie algebraic techniques, which evaluatesthird-order aberrations, is available.25 We would like toend this comparison of Lie techniques vis-a-vis othermethods by quoting Hawkes,26 who in the context ofcharged-particle optics wrote,

This said, it is in no sense my intention to provoke a po-lemic between the proponents of Lie optical methods andsome (perhaps already fictitious), old guard of eikonal lov-ers: Having myself used characteristic functions for morethan a quarter century, I am delighted to find that famil-iarity with the Lie approach has enabled me to solve atleast one long-standing problem. I am convinced thatthese different approaches must henceforward be re-garded as complementary, some problems being easier totreat by one rather than the other and in practice, bettersoftware often being available one than for the other.Moreover, for complex systems, it is certainly wise to useboth methods in order to check the results. It is thereforehighly desirable that optical designers should be familiarwith both and that the two approaches should be describedfully in future texts on aberrations.

A full-scale comparison between Lie methods and conven-tional methods for general light–optical systems is not yetavailable, to the best of our knowledge.

This paper is an example of the application of Lie alge-braic technique to the study of visual optics in particularand vision science in general. In visual optics, the onlyapplication of group theory that we are aware of is theidentification of the spherocylindrical representation as adihedral group27 D4 . It is possible to further extend thisby the use of Clifford algebras. In the study of visual per-

ception, Lie algebras have been used to provide a meta-language for perceptual processes.28 The method ofgroup extensions has been used to respond to certaincriticisms of a model of mental rotation.29 However, thepowerful tools of group theory, particularly Lie algebraictechniques, which have played a major role in modernphysics, have not been successfully exploited in vision sci-ence. Group theoretical analyses may lead to predictionof new measurable effects or allow newer methods ofanalysis of data and offer greater insight into the clinicalcondition. This should be incentive for further research.

REFERENCES AND NOTES1. W. F. Harris, ‘‘Dioptric power: its nature and its represen-

tation in three and four dimensional space,’’ Optom. VisionSci. 74, 349–366 (1997). This issue of the journal (June1997) is a feature issue on visual optics and contains manyrelated articles.

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

3. V. Lakshminarayanan and S. Varadharajan, ‘‘Calculationof aberration coefficients: a matrix method,’’ in Basic andClinical Applications of Vision Science, V. Lakshminara-yanan, ed. (Kluwer, Dordrecht, The Netherlands, 1997), pp.111–115.

4. M. Kondo and Y. Takeuchi, ‘‘Matrix method for nonlineartransformation and its application to an optical system,’’ J.Opt. Soc. Am. A 13, 71–89 (1996).

5. A. J. Dragt, ‘‘Lie algebraic theory of geometrical optics andoptical aberrations,’’ J. Opt. Soc. Am. 72, 372–379 (1982).

6. A. J. Dragt, ‘‘Elementary and advanced Lie algebraic meth-ods with applications to accelerator design, electron micro-scopes and light optics,’’ Nucl. Instrum. Methods Phys. Res.A 258, 339–354 (1987).

7. A. J. Dragt, E. Forest, and K. B. Wolf, ‘‘Foundations of a Liealgebraic theory of geometrical optics,’’ in Lie Methods inOptics, Vol. 250 of Lecture Notes in Physics, J. SanchezMondragon and K. B. Wolf, eds. (Springer-Verlag, Heidel-berg, 1986), pp. 105–157. This book contains an extensiveoverview of Lie group theory and applications in optics. Seealso the book edited by K. B. Wolf (Ref. 26) for related ar-ticles.

8. See, for example, W. Brouwer, Matrix Methods in OpticalInstrument Design (Benjamin, New York, 1964); A. Gerrardand J. M. Burch, Introduction to Matrix Methods in Optics(Dover, New York, 1994).

9. H. A. Buchdahl, An Introduction to Hamiltonian Optics(Dover, New York, 1993). See also A. K. Ghatak and K.Thyagarajan, Contemporary Optics (Plenum, New York,1978).

10. A. J. Dragt, ‘‘Nonlinear orbit dynamics,’’ in Physics of HighEnergy Particle Accelerators, R. A. Carrigan, ed., AIP Con-ference Proceedings 87 (American Institute of Physics,Woodbury, N.Y., 1982), pp. 147–313.

11. E. Forest, ‘‘Lie algebraic methods for charged particlebeams and light optics,’’ Ph.D. dissertation (University ofMaryland, College Park, Md., 1984).

12. O. Stavroudis, The Optics of Rays, Wavefronts and Caustics(Academic, New York, 1972).

13. For a general treatment of symplectic methods, see V.Guillemin and S. Sternberg, Symplectic Techniques inPhysics (Cambridge U. Press, Cambridge, UK, 1984).

14. G. Nemes, ‘‘Measuring and handling general astigmaticbeams,’’ in Laser Beam Characterization, P. M. Medjias, H.Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (So-ciedad Espanola de Optica, Madrid, 1993), pp. 325–356.

15. E. C. G. Sudarshan, N. Mukunda, and R. Simon, ‘‘Realiza-tion of first order optical systems using thin lenses,’’ Opt.Acta 32, 855–872 (1985).


16. J. A. Arnaud, ‘‘Nonorthogonal optical waveguides and reso-nators,’’ Bell Syst. Tech. J. 49, 2311–2348 (1970).

17. M. J. Bastianns, ‘‘ABCD law for partially coherent Gauss-ian light, propagating through first order optical systems,’’Opt. Quantum Electron. 24, 1011–1019 (1992).

18. W. F. Harris, ‘‘Ray vector fields, prismatic effect and thickastigmatic optical systems,’’ Optom. Vision Sci. 73, 418–423(1996).

19. A. J. Dragt and J. M. Finn, ‘‘Lie series and invariant func-tions for analytic symplectic maps,’’ J. Math Phys. (N.Y.)17, 2215–2227 (1976).

20. H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980).

21. A. J. Dragt and E. Forest, ‘‘Computation of non-linear be-havior of Hamiltonian systems using Lie algebraic meth-ods,’’ J. Math. Phys. (N.Y.) 24, 2734–2744 (1983).

22. G. Rangarajan and M. Sachidanand, ‘‘Spherical aberrationand its correction using Lie algebraic techniques,’’ PramanaJ. Phys. 49, 635–643 (1997).

23. M. Born and E. Wolf, Principles of Optics, 6th ed. (Perga-mon, New York, 1980).

24. D. A. Atchison and G. Smith, ‘‘Continuous gradient indexand shell models of the human lens,’’ Vision Res. 35, 2529–2538 (1995).

25. A computer code, MARYLIE 3.0, a program for charged par-

ticle beam transport based on Lie algebraic methods, hasbeen developed by A. J. Dragt and his colleagues. For in-formation contact A. Dragt, Dynamical Systems and Accel-erator Theory Group, Department of Physics, University ofMaryland, College Park, Maryland 20742-4111.

26. P. W. Hawkes, ‘‘Lie methods in optics: an assessment,’’ inLie Methods in Optics II, K. B. Wolf, ed., Vol. 352 ofSpringer Lecture Notes in Physics (Springer-Verlag,Heidelberg, 1989), pp. 1–17.

27. C. Campbell, ‘‘The refractive group,’’ Optom. Vision Sci. 74,381–387 (1997).

28. See, for example, W. C. Hoffman, ‘‘The Lie algebra of visualperception,’’ J. Math. Psychol. 3, 65–98 (1966). See also P.Dodwell, Visual Pattern Recognition (Holt, Rinehart & Win-ston, New York, 1970). The Lie group approach has beenused in the invariance coding problem by M. Ferraro and T.Caelli, ‘‘Relationship between integral transform invari-ances and Lie group theory,’’ J. Opt. Soc. Am. A 5, 738–742(1988).

29. V. Lakshminarayanan and T. S. Santhanam, ‘‘Representa-tion of rigid stimulus transformations by cortical activitypatterns,’’ in Geometric Representations of Perceptual Phe-nomena, R. D. Luce, M. D’Zmura, D. Hoffman, G. Iverson,and A. K. Romney, eds. (Erlbaum, Mahwah, N.J., 1995), pp.61–69.

lie algebraic treatment of dioptric power and optical aberrations

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