1668 J. Opt. Soc. Am. A/Vol. 19, No. 8 /August 2002 Avendano-Alejo et al.

Huygens’s principle and rays in uniaxialanisotropic media.

I. Crystal axis normal to refracting surface

Maximino Avendano-Alejo and Orestes N. Stavroudis

Centro de Investigaciones en Optica, Loma del Bosque, 115, 37150 Leon, Guanajuato, Mexico

Ana Rosa Boyain y Goitia

Facultad de Ingenierıa Mecanica, Electrica y Electronica, Prolongacion Tampico s/n, 36730 Salamanca,Guanajuato, Mexico

Received December 10, 2001; revised manuscript received March 12, 2002; accepted March 15, 2002

Huygens’s principle is used to derive equations for tracing the extraordinary ray in a uniaxial crystal when thecrystal axis is normal to the refracting surface. Snell’s law is used to trace ordinary rays that lead to an arrayof ordinary spherical wavelets centered on the refracting surface. Each spherical wavelet is then replaced byan extraordinary wavelet in the shape of a rotationally symmetric ellipsoid whose major axis is in the directionof the crystal axis. The envelope of these is the extraordinary wave front; the extraordinary ray is a vectorfrom the center of the wavelet to its point of contact with the extraordinary wave front. The inverse problemis also solved, yielding expressions for the ordinary ray in terms of the extraordinary ray, making it possible totrace the extraordinary ray out of the system by using a fictive ordinary ray. We found the geometrical in-variant for the uniaxial crystals. © 2002 Optical Society of America

OCIS codes: 080.0080, 260.1180, 260.1440.

1. INTRODUCTIONThis is a continuation of a much earlier effort1 to obtainray-tracing formulas for uniaxial anisotropic media inwhich Huygens’s principle was used to set up ordinaryand extraordinary wavelets. The idea was to transformone into the other; this would then induce a transform ofan ordinary ray into an extraordinary ray and vice versa,yielding a formula for the direction cosine vector of the ex-traordinary ray in terms of that of the ordinary ray. Itwas not realized at the time, but a later investigationshowed that the scheme did not work. The transformalso transformed the normal to the refracting surface, sothat it was no longer normal.

In this paper we return to the original statement ofHuygens’s principle; that a wave front is the envelope ofan aggregate of wavelets centered on a previous wavefront in the wave-front train. The idea here is to useSnell’s law to find the ordinary ray vector So , then findthe pattern of Huygens’s wavelets that are centered onthe refracting surface and whose envelope is the ordinarywave front. Then the ordinary wavelets are replaced bythe extraordinary wavelets in the form of rotationallysymmetric ellipsoids that are in contact with the ordinarywavelet at the minor axis. The envelope of this family isthe extraordinary wave front. The extraordinary rayvector Se is then parallel to the vector that extends fromthe center of the wavelet to its point of contact with theextraordinary wave front.

Previous investigations can be summarized as follows:Simon2 arrived at formulas for the extraordinary ray thatagree with ours for the case when the optical axis is nor-

0740-3232/2002/081668-06$15.00 ©

mal to the refracting surface. Her results are based di-rectly on the Maxwell equations and the properties ofuniaxial and biaxial crystals. In a subsequent paper3 sheapplied these results to ray tracing in a Wollaston prism.Cojocaru4 used Fresnel-type calculations, taking into ac-count energy conservation and the continuity of the elec-tric and magnetic field vectors at the refracting surface.Liang5 also used conservation and continuity to obtainformulas for the extraordinary ray. His numerical ex-ample agrees reasonably well with the calculations pre-sented in this paper. The calculations of Shao and Yi6

were supported by their experimental results. Beyerleand McDermid7 also used the properties of the uniaxialcrystal and Fresnel calculations and went further instudying total internal reflection. Their numerical ex-ample agrees very well with ours.

The work presented here differs profoundly from thosecited. First, we use Huygens’s principle to solve theproblem of obtaining the direction cosine vector of the ex-traordinary ray as a function of that of the ordinary ray.Figure 1 below is an attempt to show this. We also invertthese equations to obtain the ordinary ray as a function ofthe extraordinary ray. The scheme is to refract the ex-traordinary ray when it leaves the crystal by creating afictive ordinary ray that obeys Snell’s law.

2. PRELIMINARIESWe begin by assuming that a very narrow bundle of par-allel rays is incident on the surface of a uniaxial crystaland that the bundle subtends so small an area on the

2002 Optical Society of America

Avendano-Alejo et al. Vol. 19, No. 8 /August 2002 /J. Opt. Soc. Am. A 1669

crystal surface that it may be approximated by a plane.We choose the (x, y) coordinate axes to lie on this plane,so that the z axis is in the direction of its normal. We as-sume first that the crystal axis is parallel to the surfacenormal. In what follows we will take the ordinary refrac-tive index to be no and the extraordinary index as ne .

The ordinary ray vector So 5 (jo , ho , zo) is deter-mined by Snell’s law. For example, for a plane refractingsurface, from the ray-tracing equations for refraction,8 wehave

jo 5 mj, ho 5 mh, zo 5 @m2z2 1 ~1 2 m2!#1/2,(1)

where S 5 (j, h, z) is the direction cosine vector of theincident ray, n is the refractive index of the incident me-dium, and m 5 n/no .

If H 5 (h, k, 0) is a point on the refracting surface,then an ordinary ray passing through that point is givenby

Po 5 H 1 ~l/no!So (2)

or, in scalar notation, by

xo 5 h 1 jol/no , yo 5 k 1 hol/no ,

zo 5 zol/no , (3)

where the parameter l is the geometric distance along theray from H to Po . From the third of Eqs. (3), we get

l 5 nozo /zo . (4)

Any ordinary wave front will be normal to So . Fromall possible wave fronts we choose that unique one thatpasses through the coordinate origin. It follows that itsequation is

Po • So 5 xojo 1 yoho 1 zozo 5 0, (5)

so that

zo 5 2~xojo 1 yoho!/zo . (6)

By multiplying the first expression in Eqs. (3) by jo ,the second by ho , and the third by zo and adding, we get

~xo 2 h !jo 1 ~ yo 2 k !ho 1 zozo 5 l/no , (7)

which, because of Eq. (5), becomes

l 5 2no~hjo 1 kho! 5 nozo /zo 5 2no~xojo 1 yoho!/zo2.

(8)Next, from Eq. (2) we get

~Po 2 H!2 5 l2/no2, (9)

the equation of a sphere centered at H with radius l/no .We replace l by the expression in Eq. (8) to get the equa-tion for the ordinary wavelet:

F [ ~xo 2 h !2 1 ~ yo 2 k !2 1 zo2 2 ~hjo 1 kho!2 5 0

[ ~Po 2 H!2 2 ~H • So!2 5 0. (10)

In what follows we make extensive use of a matrixnotation9 in which Eq. (10) becomes

F 5 S xo 2 hyo 2 k D TS xo 2 h

yo 2 k D 1 zo2 2 F S h

k D TS jo

hoD G2

5 0.

(11)

Here the superscript T signals the transpose of a matrix.We next calculate the envelope of this two-parameter

family of wavelets, which will be the equation of thatwave front that passes through the origin. The steps tobe taken10 are first to calculate the partial derivatives ofF with respect to h and k to obtain a pair of simultaneousequations in those parameters. Then we solve these forh and k, which are then substituted back into F [Eq. (11)],resulting in the equation of the envelope. In what fol-lows, X, Y, and Z are unit vectors in the x, y, and z direc-tions.

The partial derivatives of F in Eq. (10) are

]F]h

[ 22~xo 2 h ! 2 2jo~hjo 1 kho! 5 0,

]F]k

[ 22~ yo 2 k ! 2 2ho~hjo 1 kho! 5 0,

(12)

which reduce to

h~1 2 jo2! 2 kjoho 5 xo ,

2hjoho 1 k~1 2 ho2! 5 yo , (13)

a pair of simultaneous equations in h and k. In matrixnotation these become

F1 2 jo2 2joho

2joho 1 2 ho2G S h

k D 5 ~I 2 Em!S hk D 5 S xo

yoD , (14)

where I is the identity matrix and

Em 5 F jo2 joho

joho ho2 G . (15)

The determinant of coefficients of Eq. (14) is

~1 2 jo2!~1 2 ho

2! 2 jo2ho

2 5 1 2 jo2 2 ho

2 5 zo2, (16)

so that its solution is

S hk D 5

1

zo2 ~I 2 En!S xo

yoD , (17)

where En is the adjoint of Em , given by

En 5 F ho2 2joho

2joho jo2 G . (18)

These we substitute back into F [Eq. (11)] to get the equa-tion of the envelope and therefore of the ordinary wavefront. The necessary calculations are, first,

S xo 2 hyo 2 k D 5 2

1

zo2 EmS xo

yoD , (19)

which follows directly from Eq. (17). The product of thiswith its transpose is

S xo 2 hyo 2 k D TS xo 2 h

yo 2 k D 51 2 zo

2

zo4 ~joxo 1 hoyo!2. (20)

This we will need for substitution into Eq. (11). From Eq.(17) we also calculate

1670 J. Opt. Soc. Am. A/Vol. 19, No. 8 /August 2002 Avendano-Alejo et al.

S hk D TS jo

hoD 5 F 1

zo2 S xo

yoD T

~I 2 En!G S jo

hoD 5

1

zo2 ~joxo 1 hoyo!,

(21)

in which (jo , ho) annihilates En . Equations (20) and(21) are substituted into Eq. (11), which becomes

~xojo 1 yoho!2~1 2 zo2! 1 zo

2zo4 5 ~xojo 1 yoho!2,

which reduces to

~xojo 1 yoho!2 2 zo2zo

2 5 0. (22)

This factors into

~xojo 1 yoho 1 zozo!~xojo 1 yoho 2 zozo! 5 0. (23)

The first factor, as promised, is the equation of that ordi-nary wave front that passes through the coordinate origin[Eq. (5)]; the second is also a wave front through the ori-gin but progressing in a retrograde direction and is ig-nored.

3. EXTRAORDINARY CALCULATIONSThe extraordinary wavelet is a spheroid, an ellipsoid ofrevolution with the axis of revolution, the crystal axis, ly-ing on one of the axes of the generating ellipse. The axisof rotation of the ellipsoidal wavelet is in contact with thespherical ordinary wavelet.11 For the moment we takethe vector Z to be the optical axis as well as the normal tothe refracting surface, so that the equation for the spher-oid wavelet is

~xe 2 h !2 1 ~ ye 2 k !2

~l/ne!2 1

ze2

~l/no!2 5 1,

or

ne2@~xe 2 h !2 1 ~ ye 2 k !2# 1 no

2ze2 5 l2. (24)

By substituting for l from Eq. (8), we get

G [ ne2@~xe 2 h !2 1 ~ ye 2 k !2# 1 no

2ze2

2 no2~hjo 1 kho!2 5 0. (25)

In vector form this is

G [ ne2$Z 3 @~Pe 2 H! 3 Z#%2 1 no

2@~Pe 2 H! • Z#2

2 no2~So • H!2 5 0. (26)

Note here that Z • H 5 0, so that the second term of thisequation is independent of H. In matrix form this is

G [ ne2S xe 2 h

ye 2 k D TS xe 2 hye 2 k D 1 no

2ze2

2 no2F S h

k D TS jo

hoD G2

5 0. (27)

As in the case of the ordinary wavelet, we take the par-tial derivatives of G with respect to h and k and solve theresulting simultaneous pair. We use Eq. (25) in thesecalculations and obtain

]G]h

[ 22ne2~xe 2 h ! 2 2no

2jo~hjo 1 kho! 5 0,

]G]k

[ 22ne2~ ye 2 k ! 2 2no

2ho~hjo 1 kho! 5 0,

(28)

which, when rearranged and cast in matrix form, become

Fne2 2 no

2jo2 2no

2joho

2no2joho ne

2 2 no2ho

2G S hk D

5 ~ne2I 2 no

2Em!S hk D 5 ne

2S xe

yeD , (29)

where Em is defined in Eq. (15). The determinant of co-efficients is

~ne2 2 no

2jo2!~ne

2 2 no2ho

2! 2 no4jo

2ho2 5 ne

2do2, (30)

where

do2 5 ne

2 2 no2~1 2 zo

2!. (31)

The solution of Eq. (29) is therefore

S hk D 5

1

do2 ~ne

2I 2 no2En!S xe

yeD , (32)

where En is defined in Eq. (18).In what follows we will rely heavily on the properties of

the two singular symmetric matrices Em and En . First,there are two equations that are easy to verify:

EnEm 5 EmEn 5 0, S jo

hoD T

En 5 EnS jo

hoD 5 0.

(33)

Two other more or less obvious relations are

S jo

hoD T

Em 5 ~1 2 zo2!S jo

hoD T

, (34)

Em2 5 ~1 2 zo

2!Em . (35)

In addition, note that

~1 2 zo2!I 2 En 5 Em . (36)

As above, we substitute h and k from Eq. (32) into theexpression for G in Eq. (27). Directly from Eq. (32) we get

S xe 2 hye 2 k D 5 2

no2

do2 EmS xe

yeD . (37)

The product of this with its transpose is

S xe 2 hye 2 k D TS xe 2 h

ye 2 k D 5ho

4~1 2 zo2!

do4 ~xejo 1 yeho!2.

(38)

We also need

S hk D TS jo

hoD 5 F 1

do2 S xe

yeD T

~ne2I 2 no

2En!G S jo

hoD

5ne

2

do2 ~xejo 1 yeho!. (39)

By substituting Eqs. (38) and (39) into Eq. (27), we getultimately the equation of the envelope and therefore theequation of the extraordinary wave front:

Avendano-Alejo et al. Vol. 19, No. 8 /August 2002 /J. Opt. Soc. Am. A 1671

do2ze

2 2 ne2~xejo 1 yeho!2 5 0, (40)

which factors into

@zedo 1 ne~xejo 1 yeho!#@zedo 2 ne~xejo 1 yeho!# 5 0,(41)

each factor being the equation of a plane passing throughthe coordinate origin and tangent to each of the extraor-dinary wavelets. As in the case of the ordinary wavefront, the first of these is the equation of the extraordi-nary wave front:

zedo 1 ne~xejo 1 yeho! 5 0. (42)

The second factor is a wave front progressing in a ret-rograde direction, which we ignore, as in the case of theordinary wave front.

4. EXTRAORDINARY RAYSuppose that the direction cosine vector of the extraordi-nary ray is Se 5 (je , he , ze). Then a point on this raywill be given by

Pe 5 H 1 tSe , (43)

where Pe 5 (xe , ye , ze). As in the case of the ordinaryray, we eliminate t to obtain

S xe

yeD 5 S h

k D 1ze

zeS je

heD . (44)

This ray must pass through the point where the ex-traordinary wave front and the extraordinary wavelettouch. It follows that if Pe is the point of contact, then itmust satisfy both the equation of the wavelet [Eq. (27)]and that of the wave front [Eq. (42)]. From this we findSe .

Between Eqs. (17) and (32), we eliminate h and k as fol-lows:

S hk D 5

1

zo2 ~I 2 En!S xo

yoD 5

1

do2 ~ne

2I 2 no2En!S xe

yeD . (45)

It is a fairly simple matter to calculate

~ne2I 2 no

2En!21 51

ne2do

2 ~ne2I 2 ho

2Em!. (46)

With this we solve Eq. (45) for xe and ye to get

S xe

yeD 5

do2

zo2 ~ne

2I 2 no2En!21~I 2 En!S xo

yoD

51

ne2zo

2 @ne2~I 2 En! 2 no

2Em#S xo

yoD . (47)

Here we have used Eqs. (33) and (46).The coordinates xe and ye must satisfy the equation for

the extraordinary wave front [Eq. (42)]. First, note thatfrom Eq. (47) xejo 1 yeho can be written as

S xe

yeD TS jo

hoD 5

1

ne2zo

2 H S xo

yoD T

@ne2~I 2 En! 2 no

2Em#J S jo

hoD

51

ne2zo

2 S xo

yoD T

@ne2 2 no

2~1 2 zo2!#S jo

hoD

5do

2

ne2zo

2 ~xojo 1 yoho! 5 2do

2

ne2zo

zo , (48)

where we have used Eqs. (6), (33), and (34). When wesubstitute this into Eq. (42), we get

ze

zo5

do

nezo. (49)

Our next task is to combine Eqs. (32) and (44) by elimi-nating h and k. This results in

S xe

yeD 5

1

do2 ~ne

2I 2 no2En!S xe

yeD 1

ze

zeS je

heD ,

which is rearranged into

S je

heD 5 2

no2ze

do2ze

EmS xe

yeD . (50)

To this we apply Eq. (47) to get

S je

heD 5 2

no2ze

ne2zo

2do2ze

Em@ne2~I 2 En! 2 no

2Em#S xo

yoD

5 2no

2ze

ne2zo

2zeEmS xo

yoD ,

where we have used Eqs. (33) and (35). Next, note that

EmS xo

yoD 5 ~xojo 1 yoho!S jo

hoD 5 2zozoS jo

hoD ,

where we have used Eq. (6), so that

S je

heD 5

no2zezo

ne2zoze

S jo

hoD . (51)

By squaring both sides of this, we obtain

1 2 ze2 5 S no

2zezo

ne2zoze

D 2

~1 2 zo2!. (52)

Here we calculate the square root of both sides and rear-range the factors to get

ne2zeA1 2 ze

2

ze5

no2zoA1 2 zo

2

zo. (53)

Now refer to Fig. 1, in which uo is the angle between theordinary ray and the surface normal and ue is the anglebetween the extraordinary ray and that same normal.Recall that in this section the surface normal and thecrystal axis coincide. Then Eq. (53) becomes

ne2ze tan ue 5 no

2zo tan uo , (54)

an invariance relation between the two rays and their as-sociated parameters. This is illustrated in Fig. 2.

1672 J. Opt. Soc. Am. A/Vol. 19, No. 8 /August 2002 Avendano-Alejo et al.

Fig. 2. Invariance relation. S is the refracting surface, and N is its normal vector. u i is the angle of incidence, and uo is the ordinaryray angle of refraction. ue is the angle between the normal to the refracting surface and the extraordinary ray. ro and re are theordinary and extraordinary rays, and wfo and wfe are the corresponding wave fronts. zo and ze represent distances along the two raysto intersection with their wave fronts. The invariance relation is ne

2ze tan ue 5 no2zo tan uo .

Fig. 1. Ordinary and extraordinary wavelets. wo and we are the ordinary and extraordinary wavelets, wfo and wfe are the corre-sponding wave fronts, and ro and re are the corresponding rays.

Now we refer back to Eq. (51), to which we apply Eq.(49) in order to eliminate zo and ze and get

S je

heD 5

no2ze

nedoS jo

hoD . (55)

Finally, we use the fact that both Se and So are unitvectors. By taking the sum of the squares of both sides ofEq. (55), we find that

1 2 ze2 5

no4ze

2

ne2do

2 ~1 2 zo2!, (56)

which leads us to

ze 5nedo

de, (57)

where

de2 5 ne

4 2 no2~ne

2 2 no2!~1 2 zo

2!. (58)

Avendano-Alejo et al. Vol. 19, No. 8 /August 2002 /J. Opt. Soc. Am. A 1673

This, together with Eq. (55), gives us the direction co-sines of the extraordinary ray in terms of those of the or-dinary ray:

S je

he

ze

D 51

deF no

2 0 0

0 ne2 0

0 0 ne

G S jo

ho

do

D5

1

deF no

2 0 0

0 no2 0

0 0 nedo /zo

G S jo

ho

zo

D . (59)

These results cover the situation where a ray is inci-dent on the surface of a uniaxial anisotropic mediumwhen its optical axis is normal to the refracting surface.The next few paragraphs show how the extraordinary raymay be refracted out of the crystal. Let us assume forthe moment that the exit surface is parallel to the inci-dent surface, so that the crystal axis is perpendicular toboth. Our scheme is to invert Eq. (59) to obtain the or-dinary ray vector in terms of the extraordinary ray vector.Then at the point where the extraordinary ray intersectsthe exit surface we can calculate a fictive ordinary raythat can be refracted by using Snell’s law.

To invert Eq. (59), we first need to find zo in terms ofze . To do this, we use Eqs. (31) and (56) to get

ne2do

2~1 2 ze2! 5 ne

2@ne2 2 no

2~1 2 zo2!#~1 2 ze

2!

5 no4ze

2~1 2 zo2!, (60)

from which comes

1 2 zo2 5

ne4~1 2 ze

2!

no2@ne

2 2 ~ne2 2 no

2!ze2#

. (61)

This in turn leads us to

zo2 5

2ne2~ne

2 2 no2! 1 @ne

4 2 no2~ne

2 2 no2!#ze

2

no2@ne

2 2 ~ne2 2 no

2!ze2#

. (62)

Finally, we use Eq. (57) to calculate first do and thende :

do2 5

ne2no

2ze2

ne2 2 ~ne

2 2 no2!ze

2 , (63)

de2 5

ne4no

2

ne2 2 ~ne

2 2 no2!ze

2 . (64)

Now we can invert Eq. (59):

S jo

ho

zo

D 5 deF 1/no2 0 0

0 1/no2 0

0 0 zo /nedo

G S je

he

ze

D , (65)

where zo in the matrix is given by Eq. (62) and do and deare given by Eqs. (63) and (64).

5. CONCLUSIONSEquations for the extraordinary ray in a uniaxial aniso-tropic medium have been derived in which the crystalaxis is normal to the refracting surface. The result is ex-pressions for the extraordinary ray vector in terms of therefractive indices no and ne , the vector in the direction ofthe crystal axis, and the direction vector of the ordinaryray determined by Snell’s law. The inverse problem isalso solved, yielding formulas for the ordinary ray interms of the extraordinary ray [Eq. (65)]. With this theextraordinary ray can be traced out of the crystal by cal-culating a fictive ordinary ray at the point of incidence ofthe extraordinary ray and the exiting surface. The re-fracted ray can then be calculated by using Snell’s law.

ACKNOWLEDGMENTSIt is with great pleasure that we acknowledge the supportof Consejo Nacional de Ciencia y Tecnologıa under con-tract 28446E and of Sistema Nacional de Investigadores,through experiment 13549.

Correspondence should be addressed to O. Stavroudisat ostavro@foton.cio.mx.

REFERENCES1. O. N. Stavroudis, ‘‘Ray tracing formulas for uniaxial crys-

tals,’’ J. Opt. Soc. Am. 52, 187–191 (1962).2. M. C. Simon, ‘‘Ray tracing formulas for monoaxial optical

components,’’ Appl. Opt. 22, 354–360 (1983).3. M. C. Simon, ‘‘Image formation through monoaxial plane-

parallel plates,’’ Appl. Opt. 27, 4176–4182 (1988).4. E. Cojocaru, ‘‘Direction cosines and vectorial relations for

extraordinary-wave propagation in uniaxial media,’’ Appl.Opt. 36, 302–306 (1997).

5. Quan-Ting Liang, ‘‘Simple ray tracing formulas for uniaxialoptical crystal,’’ Appl. Opt. 29, 1008–1010 (1990).

6. Zhongxing Shao and Chen Yi, ‘‘Behavior of extraordinaryrays in uniaxial crystals,’’ Appl. Opt. 33, 1209–1212 (1994).

7. G. Beyerle and I. S. McDermid, ‘‘Ray-tracing formulas forrefraction and internal reflection in uniaxial crystals,’’ Appl.Opt. 37, 7947–7953 (1998).

8. O. N. Stavroudis, ‘‘Basic ray optics,’’ in Handbook of OpticalEngineering, D. Malacara and B. J. Thompson, eds. (MarcelDekker, New York, 2001), pp. 15–22.

9. P. Lancaster and M. Tismenetsky, The Theory of Matrices,2nd ed. (Academic, Orlando, Fla., 1985), Chap. 2.

10. E. Goursat, A Course in Mathematical Analysis, translatedby E. R. Hedrick (Ginn, Mass., 1904), pp. 459–461.

11. D. H. Goldstein, ‘‘Anisotropic materials,’’ in Handbook ofOptical Engineering, D. Malacara and B. J. Thompson, eds.(Marcel Dekker, New York, 2001), pp. 847–878.