Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. A 27

Dispersive effects on the Brewster angle in moving isotropicdielectrics

Jos6 Mario Saca

Departamento de Ffsica, Nicleo de Sucre, Universidad de Oriente, Cuman6, Venezuela

Received January 6, 1984; accepted August 30, 1984

A general method to study dispersive effects on the Brewster angle in moving isotropic semi-infinite dielectrics isdiscussed. In particular, the Brewster angle and the index of refraction of NaCl are calculated as functions of itsvelocity. A law of dispersion without absorption, which is used in ionic crystals and contains the Lyddane-Sachs-

Teller relation, is assumed. Two cases are considered: direction of motion parallel to the dielectric-vacuum inter-face and direction of motion perpendicular to the same interface.

INTRODUCTION

The Brewster angle in uniformly moving isotropic and non-dispersive dielectrics has been studied extensively in the lit-erature.14 When dispersion is taken into account, the dis-persion relation and the Doppler shift in the frequency arenow introduced into the problem. With respect to dispersivemedia, the propagation, reflection, and transmission of elec-tromagnetic waves in moving plasmas have been investigatedthoroughly.57

In this paper, we discuss general effects on the Brewsterangle that are due to the Doppler shift when a semi-infiniteisotropic and dispersive dielectric moves uniformly in a vac-uum. As a particular application, the Brewster angle and theindex of refraction of NaCl are calculated as functions of thedielectric NaCl velocity. This is done by assuming an ideallaw of dispersion (without absorption) containing the Lydd-ane-Sachs-Teller relation used in ionic crystals. Two sit-uations are considered: (1) direction of motion parallel to thedielectric-vacuum interface and (2) direction of motion per-pendicular to the same interface. In case (1), assuming theplane of incidence parallel to the velocity of the medium, wehave for the Doppler shift8

The numerical values"1 for E(O) = n2 (0) = 5.9, E(M) = n2(_)= 2.25, WT = 3.1 X 1013 sec'1, and WL = 5.0 X 1013 sec'1 arethose for NaCl.9 E is the dielectric constant of the diatomiccrystal. Precise definitions of the frequencies WL and WT canbe found in Refs. 9 and 10. WL is the frequency of longitudinaloptical phonons when the wave number k' - 0. The dis-persion relation given by Eq. (4) shows a resonance at w' = WT;

it also shows a forbidden gap: COT < c)' < WL (electromagneticwaves will be totally reflected in this frequency interval be-cause the index of refraction becomes a purely imaginarynumber). In the domain 12 0 < C' < w we use Eqs. (1) and (3)

to obtain

(6)-1 < 1 < -2n(n 2 + 1)1/2/(2n2 + 1)

and 0 1 3 _< 1. In the domain c' > c we have

-2n(n 2 + 1)1/2 /(2n 2 + 1) _ 1 _< 0.

In order to calculate oaB () and n(3), we substitute w' fromEq. (1) into Eq. (4), obtaining

sin aB = 13-1 jI(1)

[ 2 - n2(0) 1/2

- (coY)-' n2 n2 (0)

CT 2(L2J

(8)

(7)

where 13 = v/c and -y = (1 - 32)-1/2. The Brewster angle isgiven byl

sin aB = [1 + n(1 - 2)(1 + n2 )1/2]/[n 2(1 - 12) + 1],(2)

or, after some algebra,

sin aB = [n + 0(n2 + 1)1/2 ]/[nO + (n2 + 1)1/2], (3)

where A' is the frequency in a frame of reference fixed to themedium, w is the frequency in a frame of reference fixed to thelaboratory, and n is the index of refraction. The incidentwave and the Brewster angle aB' are shown in Fig. 1. We usethe dispersion law9 "0

n(co') = n(0)[1 - (O'/wL) 2 I1/2 [1 - (W1//c,.T)2 -1/2, (4)

which contains the Lyddane-Sachs-Teller relation9 "10

(CL 2/cvT 2 ) = [E(0)/e(co)]. (5)

Equating this with the expression for sin aB given by Eq. (3)and solving for 1 = 1(n), we obtain, when X = UT (we madethis choice in order to simplify the algebra)

s = n2 + 12)1+ F2 4-1 i - [F2(n2 + 1) -1]1/2}, (9)h n2 (e+rF2)+F 2 n

where

F = [n2 - n2 ("')]1/2 /[n2- n2(0)]1/2. (10)

In reference to the double signs in Eq. (9), the positive signcorresponds to 1 positive, and the negative to : negative.Figure 2 shows the dependence of the index of refraction andthe Brewster angle on 13 for X = UT. We can see that for thedomain 0 < co' < COT, for each positive value of 1 there is acorresponding value of n, but for negative values of 1 thereis only an extremely short interval of values for : that havecorresponding values of n: -1 _ <13-0.9993. This unusual

0740-3232/85/010027-03$02.00 © 1985 Optical Society of America

Jose Mario Saca

co' = -yco(l - 0 sin a),

28 J. Opt. Soc. Am. A/Vol. 2, No. 1/January 1985 Jos6 Mario Saca

xS

II

VII

z

15

10

(E.,j.)

zI5

Fig. 1. The incident wave and Brewster's angle in the frame S'.

to

IS

150

1I40

,I30

/ Z

N-. -; 20

0 II; WI 4 &,), ~~10

I.u -0.U -0.6 -0.4 -0.2

a022

A 0.2 0.4 0.6 0.8

0 I (a s(

-90

Fig. 2. The Brewster angle and index of refraction versus 13. Di-rection of motion parallel to the interface and to the plane of inci-dence.

situation can be caused only by the Doppler shift, as we cansee from Eqs. (1), (3), and (6). However, for w'> cT, that is,for -2n(n 2 + 1)1/2(2n2 + 1)-1 < 1 < 0, it is only the gap of thedispersion function given by Eq. (4) that causes the gaps inaB and n. The limiting function for 13, defined as L(n) =-2n(n 2 + 1)1/ 2 (2n2 + 1)-' and shown as a reference curve inFig. 2, does not depend on the form of n(').

In case (2), for motion perpendicular to the interface, wehave for the shifted frequency8

Fig. 3. The Brewster angle and index of refraction versus 13. Di-rection of motion perpendicular to the interface. Plane of incidence

0 has an arbitrary orientation.

3 shows the dependence of n and aB on 1 for the n(w') given5 by Eq. (4). The function 13(n) is obtained after eliminating

aB from Eqs. (4), (11), and (12) and when X = UT is givenby

(n. + 1)12n 2(O) ) _J (VL2)

XE+ I 2(o) _ 11) l 1(

where

E = n2n2+ 2) _ 1 + WT 2(n2 1)+ .I.n 2 (0) I + L II.

(14)

CONCLUSIONS

The Brewster angle in moving isotropic dispersive dielectricshas been discussed. As a particular example, the Brewsterangle and the index of refraction of NaCl were calculated asfunctions of the dielectric NaCl velocity for the cases of di-rection motion of the dielectric either parallel or perpendicularto a plane interface formed by a semi-infinite dielectric andvacuum.

ACKNOWLEDGMENTS

I wish to thank the Decanato of Nficleo de Sucre, Universidadde Oriente, for all the help received. I am also in debt to thereviewers for their valuable suggestions.

(' = Y((1 + A cos a) (11)

and the Brewster angle4

Cos aeB = [1 - 13(n2 + 1)1 /21/[(n2 + 1)1/2 - 13 (12)

for -1 < 1 < (n2 + I)-1/2. In this situation we have -1 < 13, OforO<'SoandO4:S (n2+1)-1/2forw'>w. Thisisquite normal, in contrast to the situation of case (1). Figure

REFERENCES

1. V. P. Pyati, J. Appl. Phys. 38, 652 (1967).2. C. Yeh, J. Appl. Phys. 38, 5194 (1967).3. C. W. Chuang and H. C. Ko, J. Appl. Phys. 45,1154 (1974).4. M. Saca, Am. J. Phys. 48, 237 (1980).5. S. N. Stolyarov, Sov. Phys. Tech. Phys. 8,418 (1963).6. C. Yeh, J. Appl. Phys. 37, 3079 (1966).7. C. Yeh, J. Appl. Phys. 38, 2871 (1967).

SI XI

101

IS

10

5

1.0Lo- | 41 -

I A A ^ A ^ A . A A

(13)

Jos6 Mario Saca Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. A 29

8. J. D. Jackson, Classical Electrodynamics (Wiley, New York,1962), Sec. 11.4, pp. 360-364. In Eqs. (1) and (11), a is the angleof incidence of the wave. In our situation, a = aB and a' = aB'-The geometry of the problem is shown in Fig. 1.

9. C. Kittel, Introduction to Solid State Physics, 4th ed. (Wiley,New York, 1971), pp. 181-190.

10. F. C. Brown, The Physics of Solids (Benjamin, New York, 1967),Sec. 8.5.

11. According to Ref. 9, the data were taken at room temperature.12. For a given a, the limiting values 3 - +-1 correspond to a' - 0.

This can be shown by using Eqs. (1) and (3) for the case of parallel

motion [case (1)].