2850 J. Opt. Soc. Am. A/Vol. 14, No. 10 /October 1997 Shen et al.

Fermat’s principle, the general eikonal equation,and space geometry

in a static anisotropic medium

Wenda Shen and Jufang Zhang

Department of Physics, Shanghai University, Shanghai 201800, China

Shitao Wang

Department of Physics, Shanghai Institute of Education, Shanghai 200031, China

Shitong Zhu

Shanghai Institute of Optics and Fine Mechanics, Academia Sinica, P.O. Box 800-211, Shanghai 201800, China

Received September 30, 1996; revised manuscript received March 18, 1997; accepted March 27, 1997

Fermat’s principle and the optical metric are generalized to the case of an anisotropic medium. The metrictensor of a three-dimensional Riemannian manifold is related to the dielectric tensor of the medium. Thegeneral eikonal equation in a static anisotropic medium is derived. The expressions for the curvature tensorand the curvature scalar that characterize the geometrical structure of a three-dimensional manifold aregiven. For an isotropic medium the derived expressions for the curvature tensor and curvature scalar reduceto the previous results. © 1997 Optical Society of America [S0740-3232(97)03209-2]

Key words: Anisotropic medium, Fermat’s principle, Riemannian manifold, optical metric, general eikonalequation, geodesic equation.

PACS numbers: 42., 02.40.K

1. INTRODUCTIONRecently we successfully studied with the help of an opti-cal metric many physical processes and phenomena in theinteraction of light with matter, for example, the gener-ally relativistic ponderomotive force that plays an impor-tant role in intensive-field physics,1 the electron energygain,2 and frequency matching3 in a beat frequency laseraccelerator and the effect of medium background on en-ergy levels of a hydrogen atom.4 In later papers we gavethe Riemannian geometry of a strong-laser plasma5 andthe motion of photons6 and the classical7,8 and thequantum9,10 behavior of free electrons in such a geometry.Now we are looking for the deep significance of the opticalmetric and its further application. In Ref. 11 we re-vealed the connection between the three-dimensionalspace geometry that corresponds to geometrical opticsand the four-dimensional space–time geometry that cor-responds to metric optics. In Ref. 12 the light tracks inoptical fibers with two types of parabolic refractive indexwere examined. It was shown that the geodesic equationdetermines the light track and that the geodesic deviationequation can describe the focusing and defocusing of anoptical beam. A similar idea was used by Guo and Dengto analyze the optical transmission.13,14

However, the research detailed above was confined toan isotropic medium, and the corresponding metric coeffi-cients of the space–time manifold were related to the sca-lar dielectric constant (or the squared refractive index).For an anisotropic medium the dielectric constant (or the

0740-3232/97/1002850-05$10.00 ©

squared refractive index) becomes a symmetrical tensorin a three-dimensional space. The various physical pro-cesses and optical phenomena in an anisotropic mediumare important aspects of physics. Thus it is necessary toconsider to the case of an anisotropic medium and estab-lish the relation between the metric coefficients of a three-dimensional Riemannian manifold and the dielectric ten-sor (or the squared refractive-index tensor) of themedium. In this paper we concentrate on a description ofthe static processes in an anisotropic medium. The treat-ment of the time-dependent processes in an anisotropicmedium will be discussed in a future paper.

2. FERMAT’S PRINCIPLE IN A STATICANISOTROPIC MEDIUM ANDMETRIC COEFFICIENTS OF THREE-DIMENSIONAL SPACEIn a static isotropic medium Fermat’s principle is formu-lated in the form

dE ds 5 d E ndl 5 0, (1)

where n is the scalar refractive index and dl 5 (dx2

1 dy2 1 dz2)1/2. The geometrical structure of a three-dimensional manifold associated with the isotropic me-dium is characterized by the invariant element ds, givenby

ds2 5 n2dl2 5 gij~x!dxidxj. (2)

1997 Optical Society of America

Shen et al. Vol. 14, No. 10 /October 1997 /J. Opt. Soc. Am. A 2851

Here x 5 (xyz) 5 (x1x2x3) and gij(x) is the metric tensorof the three-dimensional manifold, which has the form

gij 5 n2d ij 5 H n2 ~i 5 j !

0 ~i Þ j ! ~i, j 5 1, 2, 3!, (3)

with

d ij 5 H 1 ~i 5 j !

0 ~i Þ j !.

Any dummy indices, such as i in Eq. (2), are understoodto be summed over the values 1, 2, 3 unless otherwisenoted.

Now we consider a nonconducting and nonmagneticstatic anisotropic medium. The material equation be-tween the electric displacement D and the electric vectorE is assumed to have a form that can account for aniso-tropic behavior:

D 5 E • E,

with

E 5 F «11 «12 «13

«21 «22 «23

«31 «32 «33

G .

If we assume that the expression for the electric energydensity @We 5 (1/8p)E • D# retains its validity, we candeduce that the dielectric tensor E must be symmetric,and it has only six instead of nine independentcomponents.15 It is proved in Ref. 15 that when « ij (i, j5 1, 2, 3) are the constants of the medium, the symmet-ric tensor can further reduce to the diagonal tensor be-cause the energy We is positive for any value of the fieldvector. If the coordinate system of principal dielectricaxes is adopted, the diagonal dielectric tensor E has theform

E 5 F «1 0

«2

0 «3

G 5 F n12 0

n22

0 n32G , (4)

where «1 , «2 , and «3 are the principal dielectric constants(or principal permittivities) and n1 , n2 , and n3 are theprincipal refractive indices.

For an inhomogeneous anisotropic medium, « ij5 « ij(x, x), and the six components « ij of the dielectrictensor will vary with x and x. As a result, both the val-ues of the principal dielectric constants «1 , «2 , «3 and thedirections of the principal axes will vary. We restrictourselves to the case in which the anisotropy is globallyaligned and « ij 5 « ij(x). Thus there exists a coordinatesystem of principal dielectric axes fixed in the mediumsuch that the dielectric tensor has the form of Eq. (4).Then Fermat’s principle can be written as

dE ds 5 d E ~n12dx2 1 n2

2dy2 1 n32dz2!1/2 5 0. (5)

The invariant element ds characterizes the geometricalstructure of a three-dimensional manifold associated withthe static anisotropic medium, and ds2 is given by

ds2 5 n12dx2 1 n2

2dy2 1 n32dz2 5 gij~x!dxidxj. (6)

Here gij(x) is the metric tensor of the three-dimensionalmanifold, and it has the form

gij 5 ni2d ij 5 H ni

2 ~i 5 j !

0 ~i Þ j ! ~i, j 5 1, 2, 3!, (7a)

i.e.,

gij 5 F n12 0

n22

0 n32G . (7b)

The determinant of the metric tensor gij is

g 5 det~gij! 5 n12n2

2n32, (8)

and the corresponding inverse metric tensor has the form

gij 5 H ni22 ~i 5 j !

0 ~i Þ j ! ~i, j 5 1, 2, 3!, (9a)

i.e.,

gij 5 F n122 0

n222

0 n322G . (9b)

It is easy to see that Eqs. (7) reduce to Eq. (3) as the me-dium is optically isotropic, namely, n1 5 n2 5 n3 5 n.Therefore Eqs. (4)–(7) are the more general formulism.

We note that, in Hamiltonian foundation of geometricalanisotropic optics,16 Rivera, Chumakov, and Wolf (RChW)start from Fermat’s principle, written as

dEA

B

ds 5 dEA

b

n~x, x!dl 5 0, (10)

where x 5 dx/dl and l is an arc length along the ray.Our treatment of the refractive index can be regarded asthe particular case of RChW:

n~x, x! 5 ~n12x1

2 1 n22x2

2 1 n32x3

2!1/2.

Our case, although it is particular, is physically impor-tant because it is mathematically based on metric consid-eration of space. The dielectric tensor « ij(x) in our modelis understood as the Riemannian metric tensor gij(x).

By the way, Eq. (10) is inadequate for describing theisotropic limit, as was shown in Ref. 16. RChW actuallystart from the following form of Fermat’s principle:

dEzA

zB

dzL@q~z !, z; v~z !# 5 0, (11)

with the Lagrangian function

L~q, z; v ! 5 ~1 1 v2!1/2n~q, z; v !. (12)

To compare RChW’s results with ours we need only toanalyze the Lagrangian function of Eq. (5).

Let x i 5 dxi /dz; then we have

2852 J. Opt. Soc. Am. A/Vol. 14, No. 10 /October 1997 Shen et al.

L 5 F1 1 S n1

n3D 2

x12 1 S n2

n3D 2

x22G1/2

n3~x!

5 ~1 1 x12 1 x2

2!1/2n3~x1 , x2 , x3!

3 H 1 1@~n1 /n3!2 2 1# x1

2 1 @~n2 /n3!2 2 1# x22

1 1 x12 1 x2

2 J 1/2

.

(13)

When n1 5 n2 5 n0 , Eq. (13) can be written as

L 5 ~1 1 v2!1/2H 1 1@~n0 /n3!2 2 1#v2

1 1 v2 J 1/2

n3~q, z !

5 ~1 1 v2!1/2n~q, z; v2! 5 L~q, z; v2!, (14)

with

n~q, z; v2! 5 n3~q, z !H 1 1@~n0 /n3!2 2 1#v2

1 1 v2 J 1/2

5 n3~q, z !F1 1~n0

2 2 n32!

n32 sin2 uG 1/2

,

(15)

where u is the angle between the ray direction q and aprincipal axis of the medium in Ref. 16.

Obviously, Eq. (14) is only a particular case of Eq. (13),and Eq. (14) is also a particular case of Eq. (12) that con-tains only the even powers of v.

From the expression for the refractive index, Eq. (15),we can see that it physically reflects the kind of anisot-ropy expected from the quadrupole and multipole natureif we classify media by multipolarity. The other choicesof refractive indices n(x, x) can be found in Refs. 17 and18. We do not analyze them in this paper because suchan analysis is quite mathematical.

3. GEODESIC EQUATION AND GENERALEIKONAL EQUATIONFrom the mathematical statement of Fermat’s principle,

dE ds 5 dE ~gijdxidxj!1/2 5 0,

we can derive the geodesic equation obeyed by the lighttrack in a static anisotropic medium:

d2xi

ds2 1 G jki

dxi

ds

dxk

ds5 0, (16)

where G jki is the affine connection of the three-

dimensional manifold associated with the anisotropic me-dium and is given by

G jki 5

1

2gilS ]glj

]xk 1]glk

]xj 2]gjk

]xl D . (17)

Substituting Eqs. (7) and (9) into Eq. (17), we obtain

G jki 5 Gkj

i 5 H ni21ni,k ~i 5 j !

2ni22njnj,i ~i Þ j 5 k !

0 ~i Þ j Þ k Þ i !

, (18)

where ni, j is the ordinary partial derivative of ni with re-spect to the coordinate x j, i.e., ni, j 5 ]ni /]x j.

Substituting Eq. (18) into Eq. (16) and noting that(x1x2x3) 5 (xyz), we obtain

dds S n1

2 dxds D 5

12 Fn1, x

2 S dxds D 2

1 n2, x2 S dy

ds D 2

1 n3, x2 S dz

ds D 2G ,dds S n2

2 dyds D 5

12 Fn1, y

2 S dxds D 2

1 n2, y2 S dy

ds D 2

1 n3, y2 S dz

ds D 2G ,dds S n3

2 dzds D 5

12 Fn1, z

2 S dxds D 2

1 n2, z2 S dy

ds D 2

1 n3, z2 S dz

ds D 2G ,(19a)

i.e.,

dds H ~ijk!F n1

2 0

n22

0 n32G F dx/ds

dy/dsdz/ds

G J5

12 S dx

dsdyds

dzds D¹F n1

2 0

n22

0 n32G F dx/ds

dy/dsdz/ds

G . (19b)

Here (ijk) are the unit vectors in the directions of threecoordinate axes, and

¹ [ i]

]x1 j

]

]y1 k

]

]z.

Further, let e 5 (ijk) and

N 2 5 F n12 0

n22

0 n32G 5 E ;

then Eqs. (19) can be written in matrix form:

dds FeN 2S dx

ds D TG 512

dxds

¹N 2S dxds D T

, (20)

where (dx/ds)T is the transposed matrix of the row ma-trix dx/ds 5 @(dx/ds)(dy/ds)(dz/ds)#. Equation (20) isjust the general eikonal equation that governs the lighttrack in a static anisotropic medium. This eikonal equa-tion, which is equivalent to the geodesic equation, can beused to discuss the optical transmission in an anisotropicmedium.

For the isotropic medium, ni 5 n (i 5 1, 2, 3), ds5 ndl, and thus Eqs. (19) or Eq. (20) reduces to the well-known eikonal equation19

ddl S n

dxdl D 5 ¹n. (21)

4. CURVATURE TENSOR AND GEODESICDEVIATION EQUATIONThe curvature tensor Rjkl

i and the curvature scalar R arethe important characteristic quantities of a three-dimensional Riemannian manifold. These quantities de-

Shen et al. Vol. 14, No. 10 /October 1997 /J. Opt. Soc. Am. A 2853

termine the geometrical structure of the manifold and thebehavior of light rays in an anisotropic medium.

The curvature tensor Rjkli is defined as20

Rjkli 5 G jl,k

i 2 G jk,li 1 G jl

mGmki 2 G jk

mGmli , (22)

where G jk,li is the first-order ordinary partial derivative of

the affine connection G jki with respect to the coordinate x8;

i.e., G jk,li 5 ]G jk

i /]x8. A differentiation of Eq. (18) withrespect to x8 yields

G jk,li 5 Gkj,l

i

5 5ni

21ni,kl 2 ni22ni,kni,l ~i 5 j !

2ni22njnj,ini,l

2 ni22~nj,lnj,i 1 njnj,il! ~i Þ j 5 k !

0 ~i Þ j Þ k Þ i !

,

(23)

where ni, jk is the second-order ordinary partial derivativeof the principal refractive index ni with respect to the co-ordinates xj and xk; i.e., ni, jk 5 ]2ni /]xj]xk. InsertingEqs. (18) and (23) into Eq. (22) yields

termined by the dielectric tensor (or the squaredrefractive-index tensor) of an anisotropic medium.

These quantities play an important part in the geo-metrical description that uses the optical metric. It isthrough them that the geometrical structure of a three-dimensional manifold is characterized and the behavior oflight rays in an anisotropic medium is determined. Sincethe optical metric is mathematically equivalent to thegravitational metric, some conclusions drawn from theanalysis for a gravitation field can be applied to investi-gate the optical phenomena. For example, the null geo-desic equation governs the light track, and the geodesicdeviation equation given in Ref. 21,

d2Ni

ds2 1 Rjkli

dxj

dsNk

dxl

ds5 0, (27)

can describe the focusing and defocusing of light rays aslong as Rjkl

i here takes the expression in Eq. (24). N5 (N1N2N3) in Eq. (27) represents the three-dimensional geodesic separation vector. Since the dielec-tric constant (or the squared refractive index) in an aniso-

Rjkli 5 2Rjlk

i 5 512 n22$~njj

2 1 nii2 ! 1 n22@

12 ~np

2!2 2 ~ni2!2 2 ~nj

2!2#% ~i Þ j, i 5 l, j 5 k !

12 n22~njk

2 232 n22nj

2nk2! ~i Þ j Þ k Þ i, i 5 l !

212 n22~nik

2 232 n22ni

2nk2! ~i Þ j Þ k Þ i, j 5 l !

0 otherwise

, (28)

R 5 2n24@~¹2n2! 234 n22~¹n2!2#, (29)

Rjkli 5 2Rjlk

i 5 514 ni

22$2~ni, jj2 1 nj,ii

2 ! 1 np22ni,p

2 nj,p2 2 ni

22@~ni, j2 !2 1 ni,i

2 nj,i2 # 2 nj

22@~nj,i2 !2 2 nj,j

2 ni, j2 #

~i Þ j, i 5 l, j 5 k !

14 ni

22@2ni, jk2 2 ~ni

22ni, j2 ni,k

2 1 nj22ni, j

2 nj,k2 1 nk

22ni,k2 nk, j

2 !# ~i Þ j Þ k Þ i, i 5 l !

214 ni

22@2nj,ik2 2 ~ni

22nj,i2 ni,k

2 1 nj22nj,i

2 nj,k2 1 nk

22nj,k2 nk,i

2 !# ~i Þ j Þ k Þ i, j 5 l !

, (24)

where the remaining components are zero. In the ex-pressions of Rjkl

i [Eq. (24)] the summation is performedfor the dummy index p 5 1, 2, 3 and i Þ p Þ j.

The curvature scalar R is defined as

R 5 Rii 5 gikRik 5 gikRikl

l . (25)

Inserting Eqs. (9) and (24) into Eq. (25) yields

R 514 ni

22nj22$2~ni, jj

2 1 nj,ii2 ! 2 2ni

22@ni,i2 nj,i

2 1 ~ni, j2 !2#

1 nk22ni,k

2 nj,k2 %, (26)

where the summation for the dummy indices i, j, k5 1, 2, 3 and i Þ j Þ k Þ i.

As can be seen from Eqs. (24) and (26), the curvaturetensor Rjkl

i and the curvature scalar R are completely de-

tropic medium is now a tensor, it can be predicted thatthe curvature tensor and the curvature scalar will alsodescribe the birefractive phenomena in the anisotropicmedium.

Because our task in this paper is to generalize the pre-vious treatment of isotropic media to the case of aniso-tropic media with diagonal metric matrices of distinct el-ements, the key is metric coefficients. Provided that themetric is determined, the computation will be similar tothat in the case of isotropic media. Therefore we restrictourselves to studying the general characteristics and hopeto discuss some specific examples of optical problems inour model elsewhere.

For an isotropic medium ni2 5 n2 (i 5 1, 2, 3), the cur-

vature tensor Rjkli and the curvature scalar R [Eqs. (24)

and (26)] reduce to

2854 J. Opt. Soc. Am. A/Vol. 14, No. 10 /October 1997 Shen et al.

where the summation is performed for p like that in Eq.(24) and the Laplace operator ¹2 is

¹2 5]2

]x]x1

]2

]y]y1

]2

]z]z

5]2

]x1]x1 1]2

]x2]x2 1]2

]x3]x3 .

Equations (28) and (29) are in agreement with the resultsin Ref. 13.

5. CONCLUSIONSWe have generalized the Riemannian geometrical de-scription based on Fermat’s principle and on using the op-tical metric to the case of an anisotropic medium. Themetric tensor of a three-dimensional manifold was relatedto the dielectric tensor (or to the squared refractive-indextensor) of the medium. The expression for the affine con-nection was derived, and the general eikonal equationwas obtained from the geodesic equation. For an isotro-pic medium the general eikonal equation reduces to thewell-known eikonal equation. To characterize the geo-metrical structure of a three-dimensional manifold anddetermine the behavior of light rays in an anisotropic me-dium, we discussed the curvature tensor and the curva-ture scalar and derived their expressions. The curvaturetensor and the curvature scalar can describe the opticaltransmission, focusing, defocusing, and birefraction in theanisotropic medium. For an isotropic medium the ex-pressions derived for the curvature tensor and curvaturescalar reduce to the results in Ref. 13.

Further, if any nonlinearity effects are incorporatedinto the classical relativistic treatment, say, if« ij(x, uE(x)u2) is considered, we can study the behavior ofa light beam in a nonlinear anisotropic medium, such asself-focusing and the Kerr effect. We hope to addressthese problems in future studies.

ACKNOWLEDGMENTSThis research is supported by the National Natural Sci-ence Foundation of China and the National Hi-Tech (863–416) Foundation of China.

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