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Appendix A: Geometric approximation and the eikonal equation

The basic concept in the geometric optics approximation in that in the optical range the propagation of waves can be represented (to a very high degree of accuracy) as the transfer of wave energy, which are described by geometric relations. Here we present a transition from the wave equation to an equivalent relation for rays. Consider the scalar wave equation for a wave. This can be written as

(1)

where we have introduced the n=c\v of the medium relative to vacuum. lfn is a constant, for monochromatic waves we assume that

~ =Ij/(r)e -icol where co is the angular frequency. The above solution can be simplified by assuming that if ~ is a wave,

A. A. i(k.r-OJt) 'f' ='f'.e - where k is the wave number ( = 27t = n ~ ) A. c

Vasudevan Lakshminarayanan et al., Lagrangian Optics © Kluwer Academic Publishers 2002 200 Lagrangian Optics

If the direction of k is assumed to be along the z direction, then

ik (nz-ct) = .e • where leo is the wave number in vacuum {k=nko) The plane wave solutions are not physically possible in an inhomogeneous medium because the variation in refractive index in the direction ofpropagation will bend the wave. We assume that the solutions are very nearly plane waves and write:

=eA(r)+ik.(S(x)-CI) (3) where A and S are functions ofpositions. A is the amplitude ofthe wave and S would reduce to nz if n were a constant and can be recognized as being the optical pathlength. Both quantities are real. Taking the above solution, and substituting into the wave equation (I), we can write

Since S and A are real measurable quantities the equation holds only ifthe two terms in the square brackets vanish:

V2A+(VA)2 +k.(n2 _(VS)2)=0 (5) V2 +2VA·V ·S=O we make the assumption that n varies only slowly with distance, and is nearly a constant over distances ofthe order ofwavelength. This implies that A. < / where / is the dimension ofany change in the medium. This is the fundamental geometric optics approximation. 2 The third term k.2(n2 - (VS/) is the most important term since k. = 47t2 IA./ . This leads us to the fundamental equation of called the eikonal equation: Appendix A 201

(6)

The gradient function S is directed along the to the surface S = constant. Therefore, the eikonal describes the constant-phase surfaces of a wave while grad A describes the propagation of light energy along a certain direction at a given point, and leads to the concept of a of light. The ray can formally be defined as a the tangent to which at each point coincides with the vector grad S . The propagation oflight can therefore be considered as the motion oflight energy along the rays. The plane to light rays (i.e., plane S = constant) is nothing but the . The geometric optics (ray) approximation is always valid when 'V2A/A is small in comparison to W..2. That is, the wave amplitude varies significantly only at distances greater that A.. Physically, this implies that this term descnbes the bending of light by material objects- effects. Without going into detail, it can be stated that "classical corresponds to geometric optics limit ofwave motion [1]" in which light rays orthogonal to correspond to particle trajectories orthogonal to surfaces ofconstant S.

Reference:

1. H. Goldstein, , Addison Wesley, Cambridge, MA, (1956) APPENDIX B: FERMAT'S PRINCIPLE FOR A GENERAL MEDIUM OF ARBITRARY ANISOTROPY

Upto this point, we have been dealing with Fermat's principle by assuming an isotropic, non-dispersive medium. Arley (Ref. 1) first showed that Fermat's principle is also applicable to the general case of an anisotropic medium in his studies of the Maxwell equations in the geometrical optical result. A simpler proof was given by Newcomb (Ref. 2) and is derived here.

Consider a simple monochromatic wave, with phase velocity

cP =CO(IjI(X) - t] (1)

where co is a constant and X = (x. y. z). We also define the vector

(2) with components 204 Lagrangian Optics

k 8Iv I 8Iv m 8Iv K =-=-,A.=-=-,f!=-=- ro Bx ro By ro Bz acp acp acp k=-,l=-,m=- (3) Bx By Bz

5~ ro=-- 5t

and 'l'satisfies the equation

F=(K,z)=O (4) where

(5)

This is known as the "reduced eikonal" equation. In the above equation a and b are the principal phase velocities. In a spatially inhomogeneous medium in the z direction they are functions of z. We discard polarization in this analysis. In general ¢ will satisfy a first-order partial differential equation of the form

G(ro,K, X) = 0 (6)

The X dependence takes into account spatial inhomogeneities in the medium, and K(k, I, m) = V¢.The reduced eikonal equation becomes

F(ro,K, X) =0 (7) or F(K, X) =0 since we are dealing with a purely monochromatic wave. Anisotropic Medium 205

It is well-known that the solution of a first order differential equation such as that given above can be reduced to an equivalent set of ordinary differential equations, the characteristic equations:

(8)

where Fk and Fx or the partial derivatives ofFor

X'(s) = vFk (K, X) (9)

K'(s) = -vFx (K,X) (10) where v is an arbitrary multiplier which depends on the choice ofthe parameter s.

The solution curves X =X(s) are the rays. We can also define the ray direction as the direction of signal propagation. Consider the interference between two neighboring solutions of the wave equation corresponding to the same value of co. The two solutions are given by exp(ico 'If) and exp[{ico( ",+ o'lf)] where ",satisfies F(K, X) =0 and 0'"at the same portion satisfies

(11)

The condition for constructive interference occurs at any part 0"'= 0 or alternately in terms ofray direction dX

(12)

This above equation should be satisfied by any admissible oK which satisfies the expression for of. This implies that the vectors dX and FK are parallel. Next, consider the following (use is made ofthe symmetry ofthe tensor VK = VV'If).

VF= Fx +VKFK =0 (13) 206 Lagrangian Optics

K' = X'.VK = vFK.VK = vVKFK = -vFx (14)

This is nothing but a generalized form of the law governing of waves by spatial inhomogeneities in the medium.

Consider the case ofa uniaxial crystal, with optic axis in the z-direction. If, as before, a and b are the principal phase velocities and ¢J the wave phase, the eikonal equation for the E wave is

(15)

where CJ) = aep Ja t.

The characteristic equations are:

Eliminating K, A. and JJ and putting F = 0, we get

(x' / ~2)' =(y' /~2)' =0 (16)

(17)

2 2]1 where j.l = [(,X'2 + y'2)/a + z,2/b 12 . These are the differential equations which describe the rays.

If we make a suitable choice ofthe parameter s such that

b2(x'2+ y'2) + a2z'2 = b2/a2

~j.la2=1 Anisotropic Medium 207

The ray equations reduce to

x' = y' = 0 (18)

(19)

The ray direction is in the direction ofthe Poynting vector.

The wavefront is defined as a moving contour surface of constant phase ¢ . The moving wavefront successively coincides with each of the stationary surfaces of constant lfJ{x) . This means

dljl=dt dljl dt (20)

For the change in '¥ in the time dt following the wavefront. This can also be written as:

dt =dx.VIjI =K.X 'ds (21)

Let Ao and A 1 be any two fixed points on a particular ray R, to which we arbitrarily give the parameter values S = 0 and S = 1, and P denote the intersection of the moving wavefront with R the ray.

The time required for passage of P from Aoand A1 is given by

I T = jK.X'(S) ds (22) o

It should be noted that R is the path of stationary T. R is determined as the path of minimum T between the given end points. This is nothing but Fermat's principle. Note that the "time ofpassage" in the Fermat prinbciple is specifically that ofa point 208 LagrangianOptics

on the ray carryinga constantvalueofthe phase ,p. As a result of , it will in generaldiffer from the measuredtraveltime ofa wavesignal fromAoandA. (Ref. 3).

By solving the system of equations

(23)

We can representk and v as functions of X and X', as

K.X' = L(X,X') (24)

Therefore,the integrandof T is expressibleas a functionof X and X'.

Fermat's principlerequiresus to computethe integralnot only along the ray, but also along various neighboring paths which are not themselves rays. How can this be done? The value of K to be used at a particular point is that which is appropriate to the ray having the same tangent vector, at the point in question as the given path. If we do a total differentiation of F = O.

dF = 0 = Fk .dk + Fx.dx (25)

dL = Lx.dx = L, .dx' = d(k.x')=K.dx' + X' .dK = k.dx' + vF...dK = K .dx' - vFx'dx (26)

The secondorder equationfor x(s) to whichK'(s) = -vFx(K, X) reduces in general:

(27)

This is nothingbut the Euler-Lagrangeconditionfor stationaryT' . Anisotropic Medium 209

In the example ofthe uniaxial crystal given above,

(28)

A differential geometric analysis of light transmission in an anisotropic medium.

For the anisotropic medium, Shen et al [4] have generalized Fermat's principle and the optical metric and have developed a Riemannian geometrical description ofthe space geometry. For an anisotropic medium, the dielectric constant becomes a synunetrical tensor in a three-dimensional space. Shen et al have established the relationship between the metric coefficients ofa three dimensional Riemannian manifold and the dielectric tensor. There is a physical connection between the three dimensional geometry corresponding to geometrical optics and four-dimensional space-time geometry corresponding to metric optics. The geodesic equations in such a space determines the light track and geodesic deviation equations can describe focussing and defocusing ofan optical beam and optical transmission. (see references 5- 7). Consider a static, nonconducting and non magnetic anisotropic medium. It is known that the dielectric tensor E is synunetric and has only 6 instead ofnine components. The dielectric tensor reduces to a diagonal form

(29)

ifthe coordinate system ofprincipal dielectric axes is used. On the other hand for an inhomogeneous anisotropic medium, Eij =Eij (x,x) and the six components Eij ofthe dielectric tensor will vary with x and x.In this case both values ofthe principal dielectric constants and the directions ofprincipal axis will vary. Shen et al made the basic assumption that the anisotropy is globally aligned and Eij _ Eij ( x). This results in a diagonal form ofthe dielectric tensor 210 Lagrangian Optics

Fermat's principle

/5 Jnds=/5 Jndl=O (30)

can be written in this case as

(31)

The element ds characterizes the geometric structure ofa three dimensional manifold associated with the anisotropic medium and

(32)

where gij(x) is the metric tensor ofthe manifold

2 .• 2 n · (i = j) g'J=n./5.. = I I IJ { 0 (i ¢ j)

i, j = 1,2,3 (33)

The corresponding inverse metric tensor has the form Anisotropic Medium 211

{n · -2 i=j ~ i,j = 1,2,3 g ij = j;cj

2 n. (34)

2 n 2

n/

Since the metric has been defined, we can derive the geodesic equation ofthe light track in the static anisotropic medium using standard methods ofdifferential geometry:

(35)

where rj" is the affine connection coefficient and is given by

(36)

substituting for gijetc. we get the connection coefficient as

i=j i .. j=k (37) j;cj .. k,,1d

where n .. = an i I)ax) •

Using the fact (x'x' x') = (x Yz) and substituting into the equation for the geodesics, after much algebraic manipulation we get 212 Lagrangian Optics

(38)

or in form

(39)

where e NZ = Ojk) are unit vectors in the directions ofthe three coordinate axis.

(:ris nothing but the transpose ofthe row matrix : = [(%)(d%s) (%)] This equation is simply the generalized eikonal equation that governs light track in a static anisotropic medium. Ifn;=n, the equation reduces to the familiar eikonal equation (Appendix A).

Ifwe calculate the curvature tensor R ~kl and the curvature R, we can determine the geometrical structure ofthe manifold and the behavior oflight rays in the anisotropic medium. The curvature tensor is defined to be

(40)

where r lk,l is the first order ordinary partial derivative ofthe affine connection rlk with respect to the co-ordinate x' . Differentiating the expression for rlk gives Anisotropic Medium 213

-I -2 nj nj,kJ -nj ni,kni,l (i = j) (41) - 2 - 2 · = 2n·I u J.n J,I..n·l-n· I, I(n J,·lm+n J.n J,u .,,) (h'j=k) 1 o (i;Oj;Ob"i)

substituting these expressions into the equation for the curvature tensor, we get:

in~ 2 ~ I.JJ.. +nJ.n.• )

(42)

and the remaining components are zero. The curvature scalar R defined by

(43) 214 Lagrangian Optics

which is simply

1 -2 - 2 ~~ 2 2) -2 ~ 2 2 2 2] -2 2 2} R =-n4 I)· n . n.I,ll , +n ),11··· -2n I· n 1,1r.nr ) ,1. +(n I,)..) +nk n I,'kn),'k (44)

where the summation is done over the dummy indices ij,k = 1,2,3 and i;tj;tk;ti The curvature tensor and curvature scalar are completely determined by the squared refractive index ofthe anisotropic medium. These quantities completely determine the geometric structure ofthe manifold and hence the behavior of light rays in the medium. The null geodesic equation governs the light track and the geodesic deviation equation will describe the focusing and defocusing oflight rays. The curvature tensor and curvature can also describe birefractive phenomena in the anisotropic medium. The reader is referred to the literature for details .

References

J. N. Arley, K.J. Danslee, V. Denskeb, Selskab, Mat-Fys, 22: I, 1945, M.K1ineand I.W. Kay, Electromagnetic theory and Geometrical optics, Witey-intcrscience, New York, 1965, sec also Yu A. Kravtsov and Yu. I. Ovlov, Geometrical optics ofinhomogeneous media, Sprenger­ verlag, Berlin, (1990) . 2. W.A. Newcomb, Generalized Fermat principle, Am. J. Phys., 51: 338, (1983) 3. F.A. Jenkins and H.E. White, Fundamentals ofoptics , 4" edition, McGraw-Hill, NY (1976), Sec. 12.7. 4. W. Shen, J.Zhang, S.Wang, S.Zhu, Fermat 's principle. the general eikonal equation and space geometry in a SIalic anisotropic medium, J. Opt. Soc. Am. A, 14:2850, (1997) J. W.Shen, S.Zhu, X. Deng, The light tracks in the opticalfibers with 2rypes ofparabolic refractive indices . Chin. J.wers B, 5,516, (1996) 6. H.Guo and X.Deng, Different ial geometric methods in the study ofoptical transmission (scalar theory} I. Stalic transmission case , 1. Opt. Soc. Am. A, 12, 600, (1995); 7. H.Guo and X.Deng, Differentialgeometric methods in the study ofoptical transmiss iontscalar theory) II. time dependent transmission case, J. Opt. Soc. Am. A, 12,607, (1995) APPENDIX C: RAY PROPAGATION AND SYMPLECTIC TRANSFORMATIONS

As usual in Geometric optic consider two planes, the input and output planes, and let the z axis be the optic axis. The optical z axis is the direction oflight propagation. Light enters the optical system from the left at the input plane and exits at the right output plane. Following the notation ofchapter 1, let lqxi ,qYi )be the point of intersection ofray with input plane and corresponding optical direction cosines (PXi , P Yi) be

(1)

where n, is given by the local value ofthe refractive index, nj s nlqx i , q Yi ,z)and

(a Xi ,aYi ) are the angles the ray makes with the q.-z and qy-z planes. Similar notation is used for the out-going ray at the output plane. The fundamental problem in geometrical optics is to determine the input-output relations ofthe optical system.

q, =q.(qj,pd (2) P. =P.(qj,Pi)

These input-output relations corresponds to a trajectory in from the point

PI s (qxi ,qYi ' PXi' PYi ) at Zj to P. == lqx., qy. 'Px. ,PY• ) at Zr and these trajectories are governed by the Hamiltonian and the transformation is a canonical/symplectic transformation and is a straight forward consequence ofthe structure ofthe equations ofmotion.

In chapter 8, we have introduced the fact that this symplectic transformation is a mapping given by the matrix M. Consider the 4 x 4 antisymmetric matrix J (equation 8.8) 216 Lagrangian Optics

(3)

Each entry in the ] is a 2x2matrix. I=(~ ~) (4) o=(~ ~)

The matrix ] has the properties

(5) det J = 1 where ]T denotes the transpose. Let the components ofthe phase space vector be indexed as 1,2,3,4, such that

(6)

The equations ofmotion

--=--dq, cUI dqy = cUI dz dpx dz dpy (7) dpy = cUI dz dqy

can be written succinctly as

du sn -=] '/ ­ j,1 = 1,2,3,4 (8) dz J oUI

Now, ifwe transform the vector U tov

(9) Symplectic Transformations 217

v obeys the equation

(10)

where Mijdenotes the entries ofthe Jacobian matrix associated with the transformation above

M··=-Iov · (11) U 0 Uj

Ifwe require that the transformation leave the Hamiltonian invariant, it implies the matrix relation

(12)

This is the condition that M be a symplectic matrix. It is an easy matter to verify that the 2m x 2m non-singular matrices obeying the above relation forms a group under matrix multiplication. The associativity, identity and unique inverse are readily shown while the closure property also follows from the relationship (i.e., if'M, and M2obey MJMT = I) so will (M1M2)J(M1M2)r = I. In mathematical terms we say that M belongs to the symplectic group Sp(4,R). It is also very easy to show that Mol also obeys the above equation. Also,

(13)

other properties include

detM=±1 (14)

The sign ofthe determinant can be fixed by noting that in a phase space transformation elements ofvolume transform through the determinant ofthe

Jacob ian matrix relative to the transformation according to dVj = det MdV•. Since by Liouville's theorem ofphase space volume invariance, dV. =dV i which puts the condition that det M = ±I. Also ifthe transformation between the vectors V and u is linear the matrix elements if M are just the coefficients ofthe linear combination 218 Lagrangian Optics

4 Vi =LMijuj (15) j-I

Because ofthe symplecticity relation (12) or (13) the 16 entries ofthe 4x4 matrix M are not all independent. Only 10 can be arbitrarily chosen, the others being fixed. This comes from the general property of2mx2m symplectic matrices for which only m(2m+l) elements can be arbitrarily assigned. The rest are fixed by the equation (12) or (13). INDEX

A in homogeneousmedia separated Aberration(s), 107 by surfaces, 156-157 calculation of, 11 0 paraxial equations for, 152 of graded-index media, 134-139 in single refracting surface, Seidel, 118, 169 153-156 in systems with finite discontinuities in in thin , 157-159 refractive index, 139-144 Aberration,spherical, 25, 118, of thin lens, 146-152 121-122,138 Aberration,algebraic treatment of, 161-182 correction of, 175-180 applicationsof, 181-182 definition of, conventional, 178 for correction of , lateral, 121, 149-150 175-180 Lie algebra for, 169 for extension to gradient-index media, longitudinal, 121-122, 150 180-181 Aberration,third-order, 110 formalism and mathematicalbackground algebraic treatment of, 168-170 in, 162-166 in continuously varying for imaging plus telescopic system, inhomogeneousmedium, 167-168 134-139 for lens with planar surface and exit faces, expressions for, 108-118 170-175 Aberrationcoefficients, 118 for symmetric vs. asymmetric systems, for combination of , 152 168 physical significance of, 118-120 for third-order aberrations, 168-170 Aberrationcoefficients A-E. 118 for transit through medium incidenton physical significance of, 118-120 spherical surface, with separating in thin lens, 148-150 medium, 167 Aberrationcoefficients Hij' 133-134 for transit through medium ofunifonn Aberrationcompensation, 159 refractive index, 166-167 Adaptive control processes, Aberration,chromatic, 152-159 mathematicaltheory of, 185 definition of, 152 Adaptive optics, 159 220 Lagrangian Optics

Algebraic treatment ofaberrations. See paraxial equations for, 152 Aberration, algebraic treatment of in single refracting surface, Anisotropic medium 153-156 dielectric constant in, 209 in thin lens, 157-159 static, light track in, 212 Circle ofIeast confusion, 120, Anisotropic medium, light transmission in 121-122,127,128 differential geometric analysis of, 209-214 Coefficients, aberration, 118 generalized eikonal equation for, 212 for combination ofIenses, 152 Astigmatism, 25,118,124-127 physical significance of, 118-120 lack of, 128 Coefficients A-E, 118 Lie algebra for, 169 physical significance of, 118-120 Atmospheric ray trajectory, 29-30 in thin lens, 148-150 Atmospheric refraction, Fermat's principle Coefficients Hi[' 133-134 in, 23-31 Coma, 118, 122-124 Lie algebra for, 169 from set ofparallel rays to axis, B 150-151 Baker-Campbell-Haausdorfftheorem, 168 Compensation, aberration, 159 Barrel distortion, 131 Continuously varying inhomogeneous Bent fiber, ray paths in, 87-90 medium, third-order aberration Bent parabolic-index fiber, skew rays in, in, 134-139 88-90 Control process, 186 Bent parabolic-index slab waveguide Control theoretical problem, 185 for meridional rays at different launch Core cladding interface angles, 88 for meridional rays, 71 for meridional rays at different radii of for skew rays, 72 curvatures, 87 Correction, ofspherical aberration, refracting rays in, 85-86 175-180 Bent slab waveguide, ray paths in, 84-87 Crystal, uniaxial, 206, 209 Bent waveguides, ray paths in. See Ray Curvature offield, 118, 128-130 paths, in bent waveguides Lie algebra for, 170 Biconvex lens, thin, 151-152 Curvature scalar R, 213-214 Busch, Hans, 8 Curvature tensor, 212, 214 Cylindrical coordinates, optical Lagrangian in, 77-79 C Cylindrically symmetric Canonical transformations, 7 refractive-index profiles Caratheodory's theorem, 189-190 optical Lagrangian and ray Cartesian coordinates, ray equation from, equation for, 58-62 34-36 ray paths in, 79-80 Cause, final, 2 in separable form, 81 Caustics, inner and outer, 72 Characteristic equations, 205 Characteristic function, 7 D , 152-159 d variation, 3 definition of, 152 de Maupertuis, Pierre Louis Moreau, in homogeneous media separated by 3-4 surfaces, 156--157 Defocusing, combined effect of, lateral, 158 119-120 longitudinal, 158 Delay function Tth), 25-26 Index 221

Descartes, Rene, 2, 4 Euler, Leonard, 4 Dielectric constant, in anisotropic medium, Euler Lagrange equations, 5 209 Exit pupil plane Dielectric tensor, diagonal form of, 209-210 as pinhole on axis, 132 Dielectric tensor e, 209 rays striking, 119-120 Differential geometric analysis , oflight Extremum, 3 transmiss ion in anisotropic medium, 209-214 Dispersion F on image formation, 152 Factorization theorem, 166 measurement of, 156 Fermat, Pierre de, 2-3 Distortion, 118, 130-132 Fermat's principle, 1,2,7,15-31 barrel, 131 in atmospheric refract ion and Lie algebra for, 170 , 23-31 pincushion, 131 derivation ofLagrange's equations Dynamic al1ocation process, 186 from, 47-49 Dynamic programming, 185-189 Gaussian thin lens and control process in, 186 formulas from, 23-26 definition of, 185-186 law ofreflection from, 17-18 discussion on, 196-197 for nonplanar surface, 20-22 dynamic al1ocation process in, 186 and optical Lagrangian in intuitive principle in, 187 cylindrical coordinates, 77-78 policy in, 187 statement of, 16 optimal, 187 Fermat's principle, for general principle ofoptimality in, 187 medium ofarbitrary anisotropy, recurrence relation in, 187 203-214 Dynamic programming, in optics, 189-197 characteristic equations in, 205 for light propagation in inhomogeneous constructive interference in, 205 medium, 189-191 differential geometric analysis of, minimum pathway in, 191 209-214 for optimum trajectory ofray oflight in interference between neighboring medium, 191-195 solutions ofwave equation in, for waveguide parameters, 195-196 205 optical path length, 195-196 law on refraction ofwaves by ray half-period, 195 spatial inhomogeneities in, ray path length, 195 205-206 ray transit time, 196 ray direction in, 205 ray in direction ofPoynting vector in, 207 E "reduced" eikonal equation in, 204 Economy, I simple monochromatic wave in, Eikonal,7 203-204 Eikonal equation, 6 solution curves as rays in, 205 geometric optics approximation and, time ofpassage in, 207-208 199-201 in uniaxial crystal, 206, 209 for light track in static anisotropic wavefront in, 207 medium , 212 Fermat's principle ofleast time, 1,3 "reduced," 204 Field curvature, 118, 128-130 Ellipse , as projection ofrayon x-y plane, 66 Lie algebra for, 170 Equations ofmotion, Lagrange's, 33-34 Field ray, 112-113 222 Lagrangian Optics

Final cause , 2 simple results for refracting Fish eye, Maxwell, 57-58 surfaces/lenses from, 96-98 Focal length Hamiltonian, in paraxial lens optics, primary, 104, 105 93-106 ofthin lens, 158 application of, 95-98 ultrashort, 134 rays in rotationally symmetric Focal point, primary, 104 optical system , 93-98 Foucault, Leon, 3 for single refracting surface, 98-101 for thick lens, 103-106 G for thin lens, 10I-I03 Gaussian optics, 96 , 8-11 Gaussian thin lens formula, from Fermat's vs. Lagrangian optics, 11 -13 principle, 23-26 Hamilton's characteristic function, Gaussian thin mirror formula, from Fermat's 7-8 principle, 23-26 Hamilton's characteristic function General medium ofarbitrary anisotropy, methods, 162-163 Fermat's principle for, 203-214. See Hamilton's first principle function, 6 also Fermat's principle, for general Hamilton's principle of least , 6, medium ofarbitrary anisotropy 33 Generalized momenta, I0 Heaviside step function, 180 Generalized operator technique, 161 Helical rays, 42-44. See also Skew Geometric analysis, oflight transmission in rays anisotropic medium, 209-214 exact ray paths in parabolic index Geometric optics approximation, 199 fiber of, 67-68 eikonal equation and, 199-20I third-order aberration in, 138-139 Geometrical optics, 6-7 tunneling, in parabolic index fiber, fundamental equation of, 200-201 75-76 fundamental problem in, 162 Hero, 1-2 Geometrical theory ofthird-order Hero ofAlexandria, 16 aberrations. See Third-order Hero's problem, 16 aberrations, geometrical theory of Historical review, 1-6 Gladstone-Dale law, 30 surface, plane, 145-146 Graded-index media, aberrations of, 134-139 I Graded-index multimode media, ray paths Ideal image, 107, 138 in. See Ray paths, in bent waveguides Image formation, dispersion on, 152 Gradient index media, algebraic treatment of Image point, 107 aberrations in, 180-181 Image relays, 134 Greeks, ancient, 1-2 Index fiber, parabolic. See Parabolic index fiber Inhomogeneous media, 139 H continuously varying, third-order Half-period, ray, dynamic programming and, aberration in, 134-139 195 light propagation in, dynamic Hamilton, William Rowan, 5-6 programming for, 189-191 on geometrical optics, 7 Inner caustic , 72 Hamiltonian, 93 Input-ouput relations, ofoptical optical, 93-95 system, 215 for rotationally symmetric system, 108 Intuitive principle , 187-189 Index 223

J Lens Equation , 164 Jacobi, Carl Gustav Jacob, 4-5 Lensmaker's equation, 26 Jacobi theory ofmechanics, 7 Lie algebra. See also Aberration, Jacobi's principle, 4-5, 7 algebraic treatment of in Hamiltonian system dynamics, 163-166 K vs. other techniques, 181-182 Kepler, Johann, 2 Lie transformation association with :f:, 165-166 Light transmission L in anisotropic medium, differential Lagrange, Joseph Louis, 5 geometric analysis of, 209-214 Lagrange multipliers, 5 in anisotropic medium, eikonal Lagrange 's equations, 33-34. See also equation for, 212 Optical Lagrangian in inhomogeneous media, dynamic derivation of, from Fermat's principle, programming for, 189-191 47--49 optimization theory for, 185 for path ofrays in space, 35-36 optimum trajectory of, 191-195 Lagrangian, 5 Light wave, scalar wave equation for, in trajectory ofa particle, 33 199 Lagrangian, optical, 35 Linear refractive index profile, 45 in cylindrical coordinates, 77-79 Liouville's equation, 13 for media with radial symmetry, 55-58 Longitudinal chromatic aberration, and ray equation for cylindrically 158 symmetric profiles, 58-62 Longitudinal spherical aberration, Lagrangian optics, 8-11 150-151 vs. Hamiltonian optics, 11-13 Lateral chromatic aberration, 158 Lateral spherical aberration, 149-1 50 M Law ofreflection M,164 from Fermat's principle, 17-18 Mach, Ernst, 2 with nonplanar surface, 20-22 Mapping M. 164, 215 Hero's derivation of, 16 Marginal , 121 Laws ofmotion, Newton's, I MARYLIE, 181 Laws ofrefraction, from Fermat's principle , Matrix methods, in non-linear regime , 18-20 161 with nonplanar surface, 20-22 Maxwell fish eye, 57-58 Lens(es) Maxwell 's equations, ray equation coaxial system of, 93-94 from, 36 combination of, aberration coefficients Mechanics, optics and, 6-8 and,152 Meridional rays, 40--42 principal planes of, 104-106 all orders ofaberrations in, 138 Lens, thick, 103 in bent parabolic-index slab Hamiltonian for, 103-106 waveguide Lens, thin, 101 for different launch angles, 88 aberration of, 146-152 for different radii ofcurvatures, biconvex, 151-152 87 chromatic aberration for, 157-159 bound,71 focal length of, 158 exact ray paths in parabolic index Hamiltonian for, 101-103 fiber of, 66-67 224 LagrangianOptics

in opticalfiber,69-71 ray paths in, 39-40 third-orderaberrationsin, 137-138 exact, 62-

R in bent slab waveguide, 84-87 Radial symmetry, media with, ray paths in, in cylindrically symmetric 55-58 (z-independent) Radio , through refractive-index profiles, , 45 79-80 Ray(s) in graded-index multimode media complete specification of, 109 (See Ray paths, in bent optimum trajectory of, in media, 191-195 waveguides) in rotationally symmetric optical systems, in media with radial symmetry, 93-106 (See also Paraxial lens 55-58 optics , Hamiltonian in) Ray paths , in bent waveguides, 77-90 Ray(s), bent fiber, 87-90 classification of, 68-76 bent slab, 84-87 meridional, 69-71 cylindrically symmetric skew, 71-76 (z-independent) in space , Lagrange's equations ofmotion refractive-index profiles, for path of, 35-36 79-80 Ray , 49-53 optical Lagrangian in cylindrical Ray direction, 205 coordinates, 77-79 Ray equation, 10, 12-13 ray equations for profiles of form 2(r) 2(r) from Cartesian coordinates, 34-35 n = nI + nl(z), 81-84 for cylindrically symmetric profiles, ray invariants, 81-84 optical Lagrangian and, 58-62 Ray paths, in parabolic index fiber, derivation ofz-component of, 48-49 39-40 from Maxwell's equations, 36 exact, 62-68 Ray equation, optical Lagrangian and, 33-53 helical rays , 42-44 derivation ofLagrange's equations from exact, 67-68 Fermat's principle and , 47-49 meridional rays, 40-42 for helical ray, 42-44 exact, 66-67 inside core ofparabolic index fiber, 39-40 refracting rays, 75 linear refractive index profile and , 45 Ray propagation, symplectic for media with n2 independent ofz transformations and , 215-217 coordinate, 38 Ray trajectory, atmospheric, 29-30 ray acoustics and, 49-53 Ray transit time, dynamic with ray invariant b [[I.c. Gk beta with programming and, 196 tilde over it)) for waveguide, 37-38 Recurrence relation, 187, 197 for ray launched in x-z plane (meridional , law of ray), 40-42 from Fermat's principle, 17-18 for ray launched in y-z plane, 41-42 with nonplanar surface, 20-22 "sech" profile in, 46-47 Hero's derivation of, 16 Ray half-period, dynamic programming and, Refracting rays 195 in bent parabolic-index slab Ray invariant b [[I.c. Gk beta with tilde over waveguide, 85-86 it)), for waveguide, 37-38 in parabolic index fiber, 75 Ray optics, history of,I Refracting surface Ray optics-particle mechanic analogy, 7-8 single, Hamiltonian for, 98-10 I Ray path length, dynamic programming and , spherical, 98-99 195 Refraction, 77 Ray paths Refraction, atmospheric, Fermat's in bent fiber, 87-90 principle in, 23-31 226 Lagrangian Optics

Refraction, laws of Skewness parameter, 61 from Fermat's principle, 18-20 Snell, WiIlebrod, 2 with nonplanar surface, 20-22 Snell's law ofrefraction, 2-3, 19-20, Refractive index, 152 22 atmospheric variation in, 27 in planar atmosphere, 28 for rotationally symmetric system, 96 for road surface mirages, 31 systems with finite discontinuities in, Sound, speed of, 49-53 aberrations in, 139-144 Sound channel, underwater, 52 Refractive index profiles Sound propagation, in ocean, 49 linear, 45 Sound ray propagation, 49-53 "sech," 46-47 Space geometry, Riemannian Refractive index profiles, cylindrically geometrical description of, 209 symmetric Spherical aberration, 25, 118, optical Lagrangian and ray equation for, 121-122,138 58-62 correction of, 175-180 ray paths in, 79-80 definition of, conventional, 178 in separable form, 81 lateral, 121, 149-150 Refractive index variation Lie algebra for, 169 coefficients Hif in terms of, 133-134 longitudinal, 121-122, 150-151 ofsystem of two homogenous media , Spherical mirror, focal length of, 26 139-142 Spherical refracting surface, 98-99 Riemannian geometrical descript ion, of Symmetry, radial, ray paths in media space geometry, 209 with, 55-58 Road surface mirages, Fermat's principle in, Symmetry, rotational, 93 30-31 on even-order terms in expansions, Roemer, Olaf, 2 109-110 Rotational symmetry, 93 Symplectic maps, 164 on even-order terms in expansions, from initial to final conditions, 181 109-110 in Lie approach, 163-166 Rotationally symmetric system, 110 light optics in terms of, 161-162 Hamiltonian for, 108 Symplectic transformations, 7, 215 rays in, 93-98 ray propagation and, 215-217

S T Sagittal field curvature, 129 Tangential field curvature, 129 Sagittal focus, 126 Tangential focus, 126 Scalar wave equation, for light wave, 199 Thick lens, 103 Schroedinger equation, 8 Hamiltonian for, 103-106 "Sech" profile, 46-47 Thin biconvex lens, 151-152 Second paraxial ray, 112-113 Thin lens, 101 Seidel aberrations, 169 aberration of, 146-152 five, 118 chromatic aberration for, 157-159 Single refracting surface, Hamiltonian for, focal length of, 158 98-101 Hamiltonian for, 101 -103 Skew rays, 138, 139 Thin lens formula, Gaussian, from in bent parabolic-index fiber, 88-90 Fermat's principle, 23-26 helical, 39, 42-44 Thin mirror formula, Gaussian, from in optical fiber, 71-76 Fermat 's principle, 23-26 tunneling, in parabolic index fiber, 75-76 Third-order aberrations, 110 Index 227

algebraic treatment of, 168-170 ray, atmospheric, 29-30 in continuously varying inhomogeneous Transit time, ray, dynamic medium, 134-139 programming and, 196 expressions for, 108-118 Tunneling, 77 in helical rays, 138-139 Tunneling helical ray, in parabolic in meridional rays, 137-138 index fiber, 75-76 Third-order aberrations, geometrical theory Tunneling rays, in bent of,107-159 parabolic-index slab waveguide, aberrations in systems with finite 85-86 discontinuities in refractive index, Tunneling skew ray, in parabolic 139-144 index fiber, 75-76 aberration ofthin lens, 146--152 Turning point plane glass surface , 145-146 ofmeridional rays, 71 aberrations ofgraded-index media, ofskew rays, 72 134-139 astigmatism, 124-127 chromatic aberration, 152-159 U in homogeneous media separated by Ultrashort focal lengths, 134 surfaces , 156--157 Underwater sound channel, 52 in single refracting surface, 153-156 Uniaxial crystal, 206, 209 in thin lens, 157-159 Unit magnification planes, 104 coefficients Hij in terms ofrefractive index variation , 133-134 coma , 122-124 V curvature offield, 128-130 Variational principles, I, 2 distortion, 130-132 Velocity profile, for sound, 49-51 expressions for third-order aberration, 108-118 physical significance ofcoeffic ients A-E, W 118-120 Wave optics, 6 spherical aberration, 121-122 Wavefront, 207 Three dimensional manifold Waveguide geometric structure of, 210 periodically segmented, 196--197 inverse metric tensor of, 210-211 ray invariant b for [[I.c. Gk beta metric coefficients of, and dielectric with tilde over it]], 37-38 tensor, 209 Waveguide parameters, dynamic metric tensor of, 210 programming and, 195-196 Time ofpassage optical path length, 195-196 in Fermat's principle , 207- 208 ray half-period, 195 integral in, 208 ray path length, 195 Torre, Amalia , 12 ray transit time, 196 Trajectory atmospheric ray, 29-30 optimum, ofray oflight in medium , Z 191-195 z-independent refractive-index particle, 33 profiles, ray paths in, 79-80