Abstract

The Abelès treatment of refraction in stratified isotropic media is extended to stratified anisotropic media. The present treatment is restricted to linear refraction problems with incident light assumed to be plane waves of infinite extent. A mathematical restriction excludes application to singular cases. Subject to these restrictions, the treatment is quite generally applicable. It is applied to the case of an isotropic semiconductor in an external magnetic field; results agreeing with published experimental data and with an alternative method of calculation for normal incidence are obtained.

© 1970 Optical Society of America

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References

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  1. F. Abelès, Ann. Phys. (Paris) 5, 596, 706 (1950).
  2. See also M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon Press, Inc., New York, 1965), pp. 51 ff.
  3. See also R. Jacobsson, in Progress in Optics, V, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1965), pp. 247 ff.
  4. This may be seen directly by applying the D operator to the determinant of the matrix L(z) in Eq. (4) and then evaluating DL(z) using Eqs. (2).
  5. Note that we follow the coordinate notation used in O. S. Heavens, Optical Properties of Thin Films (Dover Publications, Inc., New York, 1955), Fig. 4.1. This means that the reflected light is described in a left-hand coordinate system if the direction of propagation is taken as the third axis.
  6. This quartic equation corresponds to the Booker equation, familiar in discussions of radio waves in the ionosphere. See, e.g., K. G. Budden, Radio Waves in the Ionosphere (Cambridge University Press, New York, 1961).
  7. See, e.g., E. D. Palik and G. B. Wright, in Free-Carrier Magneto-optical Effects in Semiconductors and Semimetals, III, R. K. Willardson and A. C. Beer, Eds. (Academic Press Inc., New York, 1967), Ch. 10.
  8. E. D. Palik, J. R. Stevenson, and J. Webster, J. Appl. Phys. 37, 1982 (1966).
    [Crossref]
  9. Although the true energy-band structure for the conduction band of PbS is a set of [111] ellipsoids, the simple Drude model with isotropic mass may be used if all frequencies are much larger than the cyclotron frequency. See Ref. 8, Sec. IV.
  10. For the magnetic field in the z direction, the dielectric function for a simple Drude model with isotropic free-carrier mass m*, takes the formɛ=∊∞- 4πiσ/ω,where ∊∞ is the high frequency dielectric constant andσ=(Ne2/m*) [-i(ω-iν)(ω-iν)2-ωc2-ωc(ω-iν)2-ωc20ωc(ω-iν)2-ωc2-i(ω-iν)(ω-iν)2-ωc2000-iω-iν].Here B3 is the applied magnetic field, ω is the angular frequency of the light, ν is the scattering frequency, and ωc=eB3/m*c. For electrons, e=−|e|, where |e| is the magnitude of electronic charge.
  11. We define the ellipsometric variables in the following way. The angle ψ is obtained from tanψ=|Ep|/|Es|. The angle Δ is the difference of phase between the “parallel” and “senkrecht” fields. See, e.g., Ellipsometry in the Measurement of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, and J. Kruger, Eds., Natl. Bur. Std. Misc. Publ. 256 (U. S. Govt. Printing Office, Washington, D. C., 1964).

1966 (1)

E. D. Palik, J. R. Stevenson, and J. Webster, J. Appl. Phys. 37, 1982 (1966).
[Crossref]

1950 (1)

F. Abelès, Ann. Phys. (Paris) 5, 596, 706 (1950).

Abelès, F.

F. Abelès, Ann. Phys. (Paris) 5, 596, 706 (1950).

Budden, K. G.

This quartic equation corresponds to the Booker equation, familiar in discussions of radio waves in the ionosphere. See, e.g., K. G. Budden, Radio Waves in the Ionosphere (Cambridge University Press, New York, 1961).

Heavens, O. S.

Note that we follow the coordinate notation used in O. S. Heavens, Optical Properties of Thin Films (Dover Publications, Inc., New York, 1955), Fig. 4.1. This means that the reflected light is described in a left-hand coordinate system if the direction of propagation is taken as the third axis.

Jacobsson, R.

See also R. Jacobsson, in Progress in Optics, V, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1965), pp. 247 ff.

Palik, E. D.

E. D. Palik, J. R. Stevenson, and J. Webster, J. Appl. Phys. 37, 1982 (1966).
[Crossref]

See, e.g., E. D. Palik and G. B. Wright, in Free-Carrier Magneto-optical Effects in Semiconductors and Semimetals, III, R. K. Willardson and A. C. Beer, Eds. (Academic Press Inc., New York, 1967), Ch. 10.

Stevenson, J. R.

E. D. Palik, J. R. Stevenson, and J. Webster, J. Appl. Phys. 37, 1982 (1966).
[Crossref]

Webster, J.

E. D. Palik, J. R. Stevenson, and J. Webster, J. Appl. Phys. 37, 1982 (1966).
[Crossref]

Wright, G. B.

See, e.g., E. D. Palik and G. B. Wright, in Free-Carrier Magneto-optical Effects in Semiconductors and Semimetals, III, R. K. Willardson and A. C. Beer, Eds. (Academic Press Inc., New York, 1967), Ch. 10.

Ann. Phys. (Paris) (1)

F. Abelès, Ann. Phys. (Paris) 5, 596, 706 (1950).

J. Appl. Phys. (1)

E. D. Palik, J. R. Stevenson, and J. Webster, J. Appl. Phys. 37, 1982 (1966).
[Crossref]

Other (9)

Although the true energy-band structure for the conduction band of PbS is a set of [111] ellipsoids, the simple Drude model with isotropic mass may be used if all frequencies are much larger than the cyclotron frequency. See Ref. 8, Sec. IV.

For the magnetic field in the z direction, the dielectric function for a simple Drude model with isotropic free-carrier mass m*, takes the formɛ=∊∞- 4πiσ/ω,where ∊∞ is the high frequency dielectric constant andσ=(Ne2/m*) [-i(ω-iν)(ω-iν)2-ωc2-ωc(ω-iν)2-ωc20ωc(ω-iν)2-ωc2-i(ω-iν)(ω-iν)2-ωc2000-iω-iν].Here B3 is the applied magnetic field, ω is the angular frequency of the light, ν is the scattering frequency, and ωc=eB3/m*c. For electrons, e=−|e|, where |e| is the magnitude of electronic charge.

We define the ellipsometric variables in the following way. The angle ψ is obtained from tanψ=|Ep|/|Es|. The angle Δ is the difference of phase between the “parallel” and “senkrecht” fields. See, e.g., Ellipsometry in the Measurement of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, and J. Kruger, Eds., Natl. Bur. Std. Misc. Publ. 256 (U. S. Govt. Printing Office, Washington, D. C., 1964).

See also M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon Press, Inc., New York, 1965), pp. 51 ff.

See also R. Jacobsson, in Progress in Optics, V, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1965), pp. 247 ff.

This may be seen directly by applying the D operator to the determinant of the matrix L(z) in Eq. (4) and then evaluating DL(z) using Eqs. (2).

Note that we follow the coordinate notation used in O. S. Heavens, Optical Properties of Thin Films (Dover Publications, Inc., New York, 1955), Fig. 4.1. This means that the reflected light is described in a left-hand coordinate system if the direction of propagation is taken as the third axis.

This quartic equation corresponds to the Booker equation, familiar in discussions of radio waves in the ionosphere. See, e.g., K. G. Budden, Radio Waves in the Ionosphere (Cambridge University Press, New York, 1961).

See, e.g., E. D. Palik and G. B. Wright, in Free-Carrier Magneto-optical Effects in Semiconductors and Semimetals, III, R. K. Willardson and A. C. Beer, Eds. (Academic Press Inc., New York, 1967), Ch. 10.

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Figures (6)

Fig. 1
Fig. 1

Computed values of ψR, where tanψR=|E0p|/|E0s| for normal incidence (solid line) and for 70° angle of incidence (broken line). The incident radiation, represented by the straight line at 45°, is plane polarized with azimuth 45°. The parameters for the unbacked, plane semiconductor sample used are N=2.0×1018 cm−3, ν=90 cm−1, m*=0.16 m0, =17.1, d=4.5 μ. The external magnetic field, B=1.32×105 Oe, is normal to the surface.

Fig. 2
Fig. 2

Computed values of ψT, where tanψT=|Ep+|/|Es+| for light transmitted through the sample, for normal incidence (solid line) and for 70° angle of incidence (broken line), under the same conditions as in Fig. 1.

Fig. 3
Fig. 3

Computed phase difference ΔR, in degrees, between the parallel and the “senkrecht” components of the reflected light, for normal incidence (solid line) and for 70° angle of incidence (broken line), under the same conditions as in Fig. 1. The phase difference for the incident radiation is represented by the straight line at 0°.

Fig. 4
Fig. 4

Computed phase difference ΔT, in degrees, between the parallel and the “senkrecht” components of the transmitted light, for normal incidence (solid line) and for 70° angle of incidence (broken line), under the same conditions as in Fig. 1.

Fig. 5
Fig. 5

Computed reflectance and transmittance, for normal incidence (solid line) and for 70° angle of incidence (broken line), under the same conditions as in Fig. 1.

Fig. 6
Fig. 6

Computed azimuth, χT, of the transmitted light for normal incidence, under the same conditions as in Fig. 1. The azimuth of the incident radiation is represented by the straight line at 45°. The Faraday rotation, χT − 45°, is in good agreement with the results of Palik, Stevenson, and Webster.8

Equations (22)

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E = U ( z ) exp [ i ( ω t - ω S x / c ) ] H = B ( z ) exp [ i ( ω t - ω S x / c ) ] ,
D u 1 = c 11 u 1 + c 12 u 2 + c 14 u 4 D u 2 = u 3 D u 3 = c 31 u 1 + c 32 u 2 + c 34 u 4 D u 4 = c 41 u 1 + c 42 u 2 + c 44 u 4 ,
u 1 = U 1 , u 2 = U 2 , u 3 = B 1 , u 4 = B 2 , D ( c / i ω ) d / d z ,
c 11 = S 31 / 33 , c 12 = S 32 / 33 , c 14 = ( S 2 / 33 ) - 1 , c 31 = 21 - 23 31 / 33 , c 32 = 22 - S 2 - 23 32 / 33 , c 34 = - S 23 / 33 , c 41 = ( 13 31 / 33 ) - 11 , c 42 = ( 13 32 / 33 ) - 12 , c 44 = S 13 / 33 .
L ( z ) [ u 1 a ( z ) u 1 b ( z ) u 1 c ( z ) u 1 d ( z ) u 2 a ( z ) u 2 b ( z ) u 2 c ( z ) u 2 d ( z ) u 3 a ( z ) u 3 b ( z ) u 3 c ( z ) u 3 d ( z ) u 4 a ( z ) u 4 b ( z ) u 4 c ( z ) u 4 d ( z ) ] .
u 1 a ( 0 ) = u 2 b ( 0 ) = u 3 c ( 0 ) = u 4 d ( 0 ) = 1. All other             u j α ( 0 ) = 0.
[ u 1 u 2 u 3 u 4 ] = L ( z ) [ u 1 0 u 2 0 u 3 0 u 4 0 ]             or             U ( z ) = L ( z ) U 0 .
{ E 0 p + exp [ i ( ω t - ω S x / c ) ] }
{ E 0 s + exp [ i ( ω t - ω S x / c ) ] }
a p - E 0 p - + a s - E 0 s - + a p + E 0 p + + a s + E 0 s + = 0 b p - E 0 p - + b s - E 0 s - + b p + E 0 p + + b s + E 0 s + = 0 ,
a p ± = ( l 11 cos θ 0 ± l 14 n 0 ) n 2 - ( l 41 cos θ 0 ± l 44 n 0 ) cos θ 2 a s ± = ( l 12 l 13 n 0 cos θ 0 ) n 2 - ( l 42 l 43 n 0 cos θ 0 ) cos θ 2 b p ± = ( l 21 cos θ 0 ± l 24 n 0 ) n 2 cos θ 2 + ( l 31 cos θ 0 ± l 34 n 0 ) b s ± = ( l 22 l 23 n 0 cos θ 0 ) n 2 cos θ 2 + ( l 32 l 33 n 0 cos θ 0 ) .
[ E 0 p - E 0 s - ] = [ r p p r p s r s p r s s ] [ E 0 p + E 0 s + ] ,
[ r p p r p s r s p r s s ] = 1 a s - b p - - a p - b s - × [ b s - a p + - a s - b p + b s - a s + - a s - b s + - b p - a p + + a p - b p + - b p - a s + + a p - b s + ] .
[ E p + E s + ] = [ t p p t p s t s p t s s ] [ E 0 p + E 0 s + ] ,
t p p = [ l 11 ( cos θ 0 ) ( 1 + r p p ) + l 12 r s p + l 13 n 0 ( cos θ 0 ) r s p + l 14 n 0 ( 1 - r p p ) ] / cos θ 2 t p s = [ l 11 ( cos θ 0 ) r p s + l 12 ( 1 + r s s ) + l 13 n 0 ( cos θ 0 ) ( r s s - 1 ) - l 14 n 0 r p s ] / cos θ 2 t s p = l 21 ( 1 + r p p ) cos θ 0 + l 22 r s p + l 23 n 0 ( cos θ 0 ) r s p + l 24 n 0 ( 1 - r p p ) t s s = l 21 ( cos θ 0 ) r p s + l 22 ( 1 + r s s ) + l 23 n 0 ( cos θ 0 ) ( r s s - 1 ) - l 24 n 0 r p s .
u j ( z ) exp ( - i ω q z / c ) ,
( q + c 11 ) u 1 + c 12 u 2 + c 14 u 4 = 0 q u 2 + u 3 = 0 c 31 u 1 + c 32 u 2 + q u 3 + c 34 u 4 = 0 c 41 u 1 + c 42 u 2 + ( q + c 44 ) u 4 = 0.
u 2 u 1 = [ c 14 c 41 - ( q + c 44 ) ( q + c 11 ) ] [ c 12 ( q + c 44 ) - c 14 c 42 ] r 21 u 3 u 1 = - q r 21 r 31 u 4 u 1 = - c 41 q + c 44 + c 42 q + c 44 r 21 r 41 .
q 4 + q 3 ( c 11 + c 44 ) + q 2 ( c 11 c 44 - c 32 - c 14 c 41 ) + q ( c 34 c 42 + c 12 c 31 - c 32 c 44 - c 11 c 32 ) + c 11 c 34 c 42 + c 12 c 31 c 44 + c 14 c 32 c 41 - c 11 c 32 c 44 - c 12 c 34 c 41 - c 14 c 31 c 42 = 0.
tan 2 χ T = tan 2 ψ T cos Δ T
ɛ=-4πiσ/ω,
σ=(Ne2/m*)[-i(ω-iν)(ω-iν)2-ωc2-ωc(ω-iν)2-ωc20ωc(ω-iν)2-ωc2-i(ω-iν)(ω-iν)2-ωc2000-iω-iν].