K. V. Gottschalk and M. C. Simon Vol. 18, No. 3 /March 2001/J. Opt. Soc. Am. A 673

Electric polarization and its relationship toBrewster’s angle for

interfaces with uniaxial crystals

Karin V. Gottschalk and Marıa C. Simon*

Departamento de Fısica, Laboratorio de Optica, Facultad de Ciencias Exactas y Naturales, Universidadde Buenos Aires, Ciudad Universitaria-Pabellon I, 1428 Buenos Aires, Argentina

Received February 16, 2000; revised manuscript received August 15, 2000; accepted September 28, 2000

The electric polarization of an interface between an isotropic medium and a uniaxial transparent crystal isanalyzed. The case in which the optical axis lies on the incidence plane is considered. When the incidenceangle is Brewster’s, angle is shown that the effective electric polarization of the interface has the direction ofthe reflected ray. © 2001 Optical Society of America

OCIS codes: 260.1180, 260.1440, 260.5430.

1. INTRODUCTIONThe optical properties of dielectric interfaces between anisotropic medium and a birefringent crystal have beenstudied and analyzed by many authors in recent years.1–3

In a previous paper4 we showed that the electric polar-ization vector, which is not frequently used in optics,might be of interest in describing the Brewster’s-anglephenomenon. In that paper we saw that the electric po-larization vector of the refracted ray coincides with thereflected-ray direction when the incidence angle is Brew-ster’s angle. We also showed5 that when both refractedrays exist, for this incidence angle the polarization-vectorsum has the direction of the reflected ray. In this casethe polarization of the incident ray is such that the result-ing electric-polarization vector in the interface lies on theincidence plane.

In the present paper we analyze the electric polariza-tion for an interface between an isotropic medium withrefractive index ni and a uniaxial transparent crystalwith principal refractive indices ne and no and opticalaxis on the incidence plane.

Owing to the fact that the first medium is not vacuum,we must take into account the electric polarization of bothmedia, and for this reason we define the effective electricpolarization in a way that will be seen in Section 2.

In Section 3 we see that when the incidence angle isBrewster’s angle, the effective electric polarization of theinterface has the direction of the reflected ray.

2. PLANE-WAVE STRUCTUREMaxwell’s equations and the constitutive relationships ofthe medium describe completely the light propagationthrough a medium with well-known electromagneticproperties, since the equations and the relationships yieldthe space–time variations of the electric field E, the elec-tric displacement D, the magnetic induction B, and themagnetic field H. It is well known that if the medium isa birefringent crystal, for any propagation direction there

0740-3232/2001/030673-05$15.00 ©

are two plane waves with two different linear polariza-tions and two different phase velocities. If the crystal isuniaxial, then one of these velocities is equal to the ordi-nary phase velocity uo . The direction of the correspond-ing ordinary wave front coincides with the energy flux di-rection, i.e., the light ray, as it occurs in isotropic media.Furthermore, the electric displacement lies on the planethat is perpendicular to the optical axis, and its directioncoincides with that of the electric field.

The second phase velocity depends on the angle be-tween the normal to the wave front and the optical-axisdirection:

u92 5 ue2 1 ~uo

2 2 ue2!~N9 • z3!2, (1)

where ue is the principal extraordinary phase velocity, N9is a unit vector normal to the wave front, and z3 is unitvector in the optical-axis direction. For this wave, whichis called the extraordinary wave, the ray’s direction rep-resented by the unit vector R9 generally does not coincidewith N9 (Fig. 1). The vectorial formula for the relation-ship between R9 and N9, deduced by Simon and Echarri,6

is

R9 51

fn@ue

2N9 1 ~u02 2 ue

2!~N9 • z3!z3#, (2)

with

fn2 5 @1 2 ~N9 • z3!2#ue

4 1 ~N9 • z3!2ue4. (3)

We now consider an interface between an isotropic me-dium and a uniaxial crystal. The x, y, z coordinate sys-tem characterizes the discontinuity surface in such a waythat the optical axis z3 is (Fig. 2)

z3 5 2x sin u 1 z cos u. (4)

Given the structure of the waves that could propagate inthe crystal, it is clear that if the optical axis lies on theincidence plane, there is a separation in the usual modes(parallel and perpendicular to the incidence plane). Nev-

2001 Optical Society of America

674 J. Opt. Soc. Am. A/Vol. 18, No. 3 /March 2001 K. V. Gottschalk and M. C. Simon

ertheless we study only the case in which the incident rayS is polarized with the electric-field vector parallel to theincidence plane.

To the usual description of the wave propagating in thecrystal, we add the electric polarization vector used inelectromagnetism,7 which for the extraordinary wave canbe written as

P9 5 D9 2 «vE9, (5)

where «v is the dielectric constant of vacuum. The vec-tors D9 and E9 are generally not parallel to each other.Therefore vector P9 coincides neither with D9 nor withE9. In a preceding paper4 we considered an interface be-

Fig. 1. Vectorial diagram for the extraordinary wave. v3 is avector in the optical axis direction: v3 5 (uo

2 2 ue2)(N9

• z3) z3 .

Fig. 2. Reflection and refraction in an isotropic medium–uniaxial crystal interface. The x axis is the inward normal andthe (x, z) plane is the incidence plane. The optical axis z3 is atangle u to z axis. The angle a is the incidence angle and b9 isthe refraction angle. In all figures ˇ corresponds to ˆ in textand → corresponds to boldface.

tween vacuum and an anisotropic medium and saw thatwhen the incidence angle is Brewster’s angle, the direc-tion of P9 coincides with that of the reflected ray. Whenthe first medium has a dielectric constant « i , the latter isno longer valid. This can be explained by taking into ac-count the fact that the electric polarization vector seenfrom this medium is the difference between P9 and thevector Pi proper of isotropic medium corresponding to thesame electric field E9, so we define the effective electricpolarization as

Pe 5 P9 2 Pi , (6)

where

Pi 5 « iE9 2 «vE9, (7)

and from Eqs. (5) and (7), we get

Pe 5 D9 2 « iE9. (8)

It is well known that for isotropic media the electric po-larization, the electric field, and the electric displacementall have the same direction, and from Eqs. (5) and (8) it isclear that P9 and Pe also have the same direction.Therefore when the incidence angle is Brewster’s anglethe angle between the refracted ray and the effective elec-tric polarization is equal to p/2 for any ratio of the refrac-tive indices of the two media. On the other hand, for in-terfaces between isotropic and anisotropic media, P9 andPe do not have the same direction. This is due to the factthat the angle between R9 and N9, and therefore theangle between E9 and D9, depends on the angle betweenN9 and the optical axis [see Eq. (2)]. Since Pe (and P9) isgiven in Eqs. (5) and (8) the angle between N9 and Pe

(and P9) depends on the direction of N9, which changes ifthe refractive index of isotropic medium varies. How-ever, according to Eq. (5) and (8), the projections of P9 andPe on the ray direction are equal to the projection of D9 inthis direction, and

~P9 • R9! 5 ~Pe • R9! 5 ~D9 • R9!. (9)

The scalar product between Eq. (8) and N9 yields

~Pe • N9! 5 2« i~E9 • N9!; (10)

that is, the projection of Pe on the direction of N9 dependson the isotropic medium, and it is opposite to the projec-tion of E9 in the same direction. In Fig. 1 we representthe vectorial diagram that characterizes the structure ofthe extraordinary wave. This wave has a well-definedpolarization determined by the optical-axis direction.The components of the vectors D9 and E9 along the x, y, zaxes satisfy the following relationships:

~D9 • y ! 5 0, (11a)

~D9 • x ! 5 2~N9 • x !

~N9 • z !~D9 • x !, (11b)

K. V. Gottschalk and M. C. Simon Vol. 18, No. 3 /March 2001/J. Opt. Soc. Am. A 675

~E9 • x ! 5 mvFq1 1 q2

~N9 • x !

~N9 • z !G ~D9 • x !,

(12a)

~E9 • y ! 5 0, (12b)

~E9 • x ! 5 2mvFq2 1 q3

~N9 • x !

~N9 • z !G ~D9 • x !,

(12c)

where mv is the magnetic permeability of vacuum and

q1 5 ue2 sin2 u 1 u0

2 cos2 u,

q2 5 ~ue2 2 u0

2!sin u cos u, (13)

q3 5 ue2 cos2 u 1 u0

2 sin2 u.

When we replace the components of D9 and E9 given inEqs. (11) and (12) in the definition of Pe [Eq. (8)], we ob-tain the components of this vector:

~Pe • x ! 5 H 1 2 mv« iFq1 1 q2

~N9 • x !

~N9 • z !G J ~D9 • x !,

(14a)

~Pe • z ! 5 H 2~N9 • x !

~N9 • z !1 mv« i

3 Fq2 1 q3

~N9 • x !

~N9 • z !G J ~D9 • x !.

(14b)

The angle between Pe and the direction normal to the in-terface (x-axis) is termed Ge and is defined by

tan Ge 5 ~Pe • z !/~Pe • x !. (15)

Replacing in Eq. (15) the components of Pe given in Eqs.(14a) and (14b), we obtain an expression for Ge as a func-tion of the refraction angle b 9:

tan Ge 5mv« iq2 tan b9 1 mv« iq3 2 1

~1 2 mv« iq1! tan b9 2 mv« iq2, (16)

where we have replaced (N9 • z)/(N9 • x) with tan b 9(see Fig. 2).

Up to this point, we have analyzed the structure of theplane wave that propagates in the crystal, taking into ac-count the effective electric polarization. In the next sec-tion, this definition will be justified, showing the signifi-cant coincidences that take place when the incidenceangle is Brewster’s angle.

3. BREWSTER’S ANGLEFor the incident wave, polarized parallel to the incidenceplane, the electric displacement and the electric field ob-tained from Maxwell’s equations are

~D • y ! 5 0, (17a)

~D • x ! 5 2~S • x !

~S • z !~D • x !, (17b)

~E • x ! 5 mnui2~D • x !, (18a)

~E • y ! 5 0, (18b)

~E • x ! 5 2mnui2

~S • x !

~S • z !~D • x !, (18c)

with ui being the phase velocity of the isotropic medium.The reflected wave propagates in the isotropic medium

as the incident wave, thus the expressions for the compo-nents of the fields are

~D* • y ! 5 0, (19a)

~D* • x ! 5~S • x !

~S • z !~D* • x !, (19b)

~E* • x ! 5 mn ui2~D* • x !, (20a)

~E* • y ! 5 0, (20b)

~E* • x ! 5 mn ui2

~S • x !

~S • z !~D* • x !. (20c)

With Eqs. (11), (12), and (17)–(20) we solve the boundaryconditions and obtain the following solutions:

~D* • x ! 5~ui

2 2 q2 tan a! tan b9 2 q3 tan a

~ui2 1 q2 tan a! tan b9 1 q3 tan a

~D • x !,

(21)

~D* • x ! 52ui

2 tan b9

~ui2 1 q2 tan a! tan b9 1 q3 tan a

~D • x !,

(22)

where we have replaced (S • z)/(S • x) 5 tan a and(N9 • z)/(N9 • x) 5 tan b 9.

In this case, in which a separation into modes exists,Brewster’s angle, or polarization angle ap is obtained byequalizing to zero the reflection coefficient for the parallelmode

R i 5 ~D* • x !/~D • x !. (23)

In this way we obtain from Eq. (21) the following relation-ship between ap and the corresponding refraction anglebp9 :

tan bp9 5q3 tan ap

ui2 2 q2 tan ap

. (24)

To find an explicit expression for Brewster’s angle, we willuse Snell’s law, which gives us the relationship betweenincidence and refraction angles:

sin b9

u95

sin a

ui. (25)

Substituting this relationship into Eq. (1) for the phasevelocity u9 and using the coordinate rotation [Eq. (4)], weobtain a quadratic equation for tan b 9:

@ui2~1 1 tan2 a! 2 q1 tan2 a# tan2 b9

2 2q2 tan2 a tan b9 2 q3 tan2 a 5 0. (26)

676 J. Opt. Soc. Am. A/Vol. 18, No. 3 /March 2001 K. V. Gottschalk and M. C. Simon

Placing Eq. (24) into Eq. (26) we have the expression forBrewster’s angle as a function of the phase velocities andthe optical-axis direction:

tan2 ap 5ui

2~ui2 2 q3!

ui2q3 2 ue

2uo2 . (27)

[This formula for ap can be also obtained as a particularcase from the formulas written for biaxial crystals (seeRef. 4). Likewise, Lekner8 obtained an explicit formulafor Brewster’s angle in this case].

Fig. 3. Direction of the effective electric polarization relative tothe reflected ray: (a) a . 0; (b) a , 0.

Finally, if we substitute Eqs. (24) and (27) into Eq. (16)for Ge , we find

tan Ge 5 2tan ap , (28)

which shows that vector Pe effectively has the directionthat would correspond to the reflected ray.

To represent the variation of the direction of Pe withthe incidence angle, we define the angle s as the angle be-tween the reflected ray S* and Pe , and in Figs. (3a) and(3b) we show s for a positive and a negative incidenceangle, respectively. Figure 4 represents the angle s as afunction of the incidence angle a, and s passes throughzero at Brewster’s angle.

4. CONCLUSIONWe have analyzed the electric polarization for the ex-traordinary ray when a light ray is refracted on an inter-face between an isotropic medium and a uniaxial crystal.To obtain a separation into the usual modes, we have con-sidered the optical axis contained in the incidence plane.Thus the incident wave that is polarized parallel to theincidence plane causes only one refracted ray, the ex-traordinary one.

We demonstrated in a previous paper4 that when thefirst medium is vacuum and the incidence angle is Brew-ster’s angle, the electric polarization P9 has the directionof the reflected ray. But this is not so when the first me-dium is not vacuum, and we must take into account theelectric polarization of both media. To do this, we definethe effective electric polarization Pe of the interface as thedifference between the electric polarization P9 and theelectric polarization vector corresponding to the isotropicmedium for the same electric field E9. This definitionhas been justified by calculating Brewster’s angle andverifying that for this angle, the effective electric polar-ization has the reflected-ray direction.

Fig. 4. s angle as a function of a, for an isotropic medium–calcite interface (ne 5 1.4865, no 5 1.6584, u 5 20°, ni5 1.1).

K. V. Gottschalk and M. C. Simon Vol. 18, No. 3 /March 2001/J. Opt. Soc. Am. A 677

In particular, for the interface considered in this paper,if the refractive index of the isotropic medium is equal tono , Pe is parallel to the optical axis. And if the refrac-tive index of the isotropic medium takes the value ne , theangle between Pe and the optical axis is (p/2 2 2u). Inboth cases the direction of Pe does not depend on the in-cidence angle.

Furthermore, we can make the following argument foran interface between isotropic media: When the refrac-tive index of the first medium tends to the value of the re-fractive index of the second medium, the electric polariza-tion does not vary but the effective electric polarizationtends to zero. That is, when the interface disappears, Peis identically zero and Pe can be considered to be the elec-tric polarization of the interface.

ACKNOWLEDGMENTSThis work was carried out with the support of ConsejoNacional de Investigaciones Cientıficas y Tecnicas(CONICET, Argentina) and a grant from the Universityof Buenos Aires.

*Carrera del Investigador CONICET.

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angle,’’ Opt. Acta 33(6), 755–770 (1986).4. M. C. Simon and K. V. Gottschalk, ‘‘About the Brewster

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5. M. C. Simon and K. V. Gottschalk, ‘‘Brewster angle in di-electric birefringent media: an explanation by means of di-polar model,’’ Opt. Commun. 126, 113–122 (1996).

6. M. C. Simon and R. M. Echarri, ‘‘Ray-tracing formulas formonoaxial opical components: vectorial formulation,’’Appl. Opt. 25(12), 1935–1939 (1986).

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