On influence of dichroism on Faraday rotation in biaxial crystals

Volume 41A, number 1

PHYSICS LETTERS

28 August 1972

ON INFLUENCE OF DICHROISM ON FARADAY ROTATION IN BIAXIAL CRYSTALS L. JASTRZI~BSKI Department of Magnetism, Institute of Physics, Polish Academy of Sciences, Warsaw, Poland Received 7 July 1972 The influence of dichroism on Faraday rotation in biaxial crystals is discussed. The matrix, transforming incident wave into wave travelling through the crystal, is given. The formulae applied for interpretation of results of most commonly used experimental techniques are given.

The Faraday rotation in magnetically biaxial crystals has been recently investigated [1, 4—71. Experimental results are usually difficult to interprete due to mixing of pure Faraday rotation and birefringence. The examples of determining pure Faraday rotation as well as birefringence from experimental results are to be found in [1, 21. Unfortunately, dichroism has not been taken into account in these papers, and so the influence of dichroism on pure Faraday rotation can not be estimated from previous experimental data [4—7] This would require some other measurements to be performed. In case of SmFeO3 (magnetization along the c axis) influence of dichroism -

is so strong, that the experimental points can be easily distinguished from the ones obtained when dichroism was neglected [4]. The aim of the present paper is to introduce dichroism into calculations of the Faraday rotation; It is assumed. that the crystal has orthorombic symmetry with the a, b, c, axis parallel correspondingly to the x, y, z, axes and magnetization along the c axis. The conduction tensor is introduced into Maxwell equations because the absorption can not be neglected. The tensors ~ [1,3] are of the form: ~,

~,

ox,io,0 0,12,0

12

,

~ 0, O,~

,

~

iO,Oy,O 0, 0, o~

Solutions of Maxwell’s equations were sought as in [11in form of plane waves propagating along the z direction. There appear two different values for the wave vector k. With the help of these quantities the matrix transforming the incoming wave into the one propagating within the crystals is obtained. In the x, y representation this matrix has the form: 3 z~ exp(—1 [(A2~~B2+l)2+4A2B2]’~~’2 x

—(2A chazsin(~/2)+ 2Bcos(~/2)shaz)+

cos(~/2)[(A2--B2)e°~ +e~]+ 2ABsin(4/2)e~~z + + i(2ABcos(4,/2)e~Z +sin(Ø/2) [e°~—(A2 --B2 )e~ I)

—

\

i(2Bchazsin(~/2)—2A cos(e~/2)sh~z)

x 2A chcrz sin(~/2)+2B cos(~/2)shaz+

cos(Ø/2) [(A2--B2)e°~ +e~Z]+ 2ABe’~sin(~/2) + + i(14Be~cos(~/2) + sin(~/2) [(A2--B2 )e~ e~])

+ i(2Bchazsin(~/2)—14 cos(Ø/2)shcrz)

—

/ (I)

A

=

—2y(e~ ey_a)_2~(41rIw)oE(47rIw)(ox_oy)+bl B = —2y[(4ir/w)(o~--Oy)+bI+2(41r/W)O(x--y--a) --

(e.~eya)2

+ [(47r/w)(o~—o~)+b]2

‘

(~x~y—a)2 +

E(41r/w)(o~—o~)+b] 2

77

Volume 41A, number I

a

PHYSICS LETTERS

Re{(e~—E~)2 (4~)2 (0

h

=

o)2+4~24(4~)o2

Im ~(e~,—c~)2 ~(~)(o~o,)2+4y2

(4~)2

2(y-v)

28 August 1972

~

~~i[

(oy~ox)+4y~oI}J/2

—i[2(e~,--x)~ (o 5,~o~)+4y~O~~½

j3=

~v~ii7~) ~(O~+0~),

C0

=

ct=~w~/~7~) b, Ø=~wv

~(Ev+y),

a •z;

2,

w is the angular of incoming techniques electromagnetic wave. Assuming:y~c~--e~., o~o~--o~, one obtainsA B2,AB~ I. In thefrequency case of experimental applied in the papers [4,51 the formulae which allow to interprete the experimental results are as follows:

1~ I -

for the polarization

(~)—y~= (A sinø

~--

B

cos~) . 2e-2~+

2B (2)

~ 1+~L for the polarization (“Y-~~— = (A sinØ + B cos~) 2e+220+ 2B. It is evident that the influence of dichroism is very significant. If there is no absorption, then B = ct, and matrix (I) is equal to the one obtained in paper [1] Also the formulae (2) become (1÷-I )/I = ±14 sinø, which is the .

same as in the papers [4, 51 The detailed calculations and discussion of formulae (1) and (2) for different materials will be given in a following paper. The author would like to express his gratitude to professor Wadas, professor WardzyiIski and dr Szymczak for guidance and valuable discussion and to mgr Zuberek for interesting remarks.

Reft’rences 11 Wi. Tabor and F.S.

Citen, J. AppI. Phys. 40 (1969) 2760. [21 G.N. Ramachandran and S. Ramaseshan, J. Opt. Soc. Amer. 42 (1952) 49. [3] [.1). Landau and E.M. Lifshitz, Electrodynamics of continuous media (Pergamon Press, Inc. New York, 1960). [41 M.V. Chetkin, Yu. I. Shcherbakov and A. Ya. Chervonkis, Izv. Akad. Nauk SSSR Ser. Liz. 34 (1970) 1041. [51 M.V. Chetkin and Yu. 1. Shcherbakov, Fiz. Tverd. Tela (SSSR) 11(1969)1620. 161 W.J. Tabor, A.W. Anderson and L.G. von Uitert, J. App!. Phys. 41(1970) 3018. [71 A.J. Kurtzig, J. Appl. Phys. 42(1971) 3494.

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