Abstract

A general relation is derived between the incident, reflected, and transmitted waves for a multilayer structure containing one or more layers characterized by a magnetooptic dielectric tensor. The method consists of formulating the characteristic matrix for each layer and evaluating the matrix products numerically. From the resulting matrix, desired quantities such as reflectivity, transmission, rotation, and ellipticity are obtained.

© 1984 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).
  2. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965).
  3. H. Walter, “Optik Dunner Schichten,” in Handbuch der Physik, Vol. 25/1 (Springer, Berlin, 1961).
  4. R. Jacobsson, “Light Reflection from Films of Continuously Varying Refractive Index,” Prog. Opt. 5, 249 (1966).
  5. G. N. Ramachandran, S. Ramangeshan, “Crystal Optics,” in Handbuch der physik, Vol. 25/1 (Springer, Berlin, 1966).
  6. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960).
  7. J. H. Judy, “Magnetic Optics Theory,” in Proceedings, Conference on Advances in Magnetic Recording, N.Y. Acad. of Science189, 239 (1972).
  8. M. Mansuripur, G. A. N. Connell, “Signal and Noise in Magneto-Optical Readout,” J. Appl. Phys. 53, 4485 (1982).
    [CrossRef]
  9. D. W. Berreman, “4 × 4-Matrix Formulation for Anisotropic Media,” J. Opt. Soc. Am. 62, 502 (1972).
    [CrossRef]
  10. G. J. Sprokel, “The Reflectivity of a Liquid Crystal Cell in a Surface Plasma on Experiment,” Mol. Cryst. Liq. Cryst. 68, 39 (1981).
    [CrossRef]
  11. D. O. Smith, “Magneto-Optical Scattering from Multilayer Magnetic and Dielectric Films,” Opt. Acta 12, 18 (1965).
  12. M. Freiser, “A Survey of Magneto-Optic Effects,” IEEE Trans. Magn. MAG-4, 152 (1968).
    [CrossRef]
  13. M. Mansuripur, G. A. N. Connell, J. W. Goodman, “Signal to Noise in Magneto-Optic Storage,” Proc. Soc. Photo-Opt. Instrum. Eng. 329, 215 (1982).
  14. D. Cheng, D. Treves, T. Chen, “Static Tests of TbFe Films for Magneto-Optical Recordings,” Proc. Soc. Photo-Opt. Instrum. Eng. 329, 223 (1982).

1982 (3)

M. Mansuripur, G. A. N. Connell, “Signal and Noise in Magneto-Optical Readout,” J. Appl. Phys. 53, 4485 (1982).
[CrossRef]

M. Mansuripur, G. A. N. Connell, J. W. Goodman, “Signal to Noise in Magneto-Optic Storage,” Proc. Soc. Photo-Opt. Instrum. Eng. 329, 215 (1982).

D. Cheng, D. Treves, T. Chen, “Static Tests of TbFe Films for Magneto-Optical Recordings,” Proc. Soc. Photo-Opt. Instrum. Eng. 329, 223 (1982).

1981 (1)

G. J. Sprokel, “The Reflectivity of a Liquid Crystal Cell in a Surface Plasma on Experiment,” Mol. Cryst. Liq. Cryst. 68, 39 (1981).
[CrossRef]

1972 (1)

1968 (1)

M. Freiser, “A Survey of Magneto-Optic Effects,” IEEE Trans. Magn. MAG-4, 152 (1968).
[CrossRef]

1966 (1)

R. Jacobsson, “Light Reflection from Films of Continuously Varying Refractive Index,” Prog. Opt. 5, 249 (1966).

1965 (1)

D. O. Smith, “Magneto-Optical Scattering from Multilayer Magnetic and Dielectric Films,” Opt. Acta 12, 18 (1965).

Berreman, D. W.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Chen, T.

D. Cheng, D. Treves, T. Chen, “Static Tests of TbFe Films for Magneto-Optical Recordings,” Proc. Soc. Photo-Opt. Instrum. Eng. 329, 223 (1982).

Cheng, D.

D. Cheng, D. Treves, T. Chen, “Static Tests of TbFe Films for Magneto-Optical Recordings,” Proc. Soc. Photo-Opt. Instrum. Eng. 329, 223 (1982).

Connell, G. A. N.

M. Mansuripur, G. A. N. Connell, J. W. Goodman, “Signal to Noise in Magneto-Optic Storage,” Proc. Soc. Photo-Opt. Instrum. Eng. 329, 215 (1982).

M. Mansuripur, G. A. N. Connell, “Signal and Noise in Magneto-Optical Readout,” J. Appl. Phys. 53, 4485 (1982).
[CrossRef]

Freiser, M.

M. Freiser, “A Survey of Magneto-Optic Effects,” IEEE Trans. Magn. MAG-4, 152 (1968).
[CrossRef]

Goodman, J. W.

M. Mansuripur, G. A. N. Connell, J. W. Goodman, “Signal to Noise in Magneto-Optic Storage,” Proc. Soc. Photo-Opt. Instrum. Eng. 329, 215 (1982).

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965).

Jacobsson, R.

R. Jacobsson, “Light Reflection from Films of Continuously Varying Refractive Index,” Prog. Opt. 5, 249 (1966).

Judy, J. H.

J. H. Judy, “Magnetic Optics Theory,” in Proceedings, Conference on Advances in Magnetic Recording, N.Y. Acad. of Science189, 239 (1972).

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960).

Mansuripur, M.

M. Mansuripur, G. A. N. Connell, “Signal and Noise in Magneto-Optical Readout,” J. Appl. Phys. 53, 4485 (1982).
[CrossRef]

M. Mansuripur, G. A. N. Connell, J. W. Goodman, “Signal to Noise in Magneto-Optic Storage,” Proc. Soc. Photo-Opt. Instrum. Eng. 329, 215 (1982).

Ramachandran, G. N.

G. N. Ramachandran, S. Ramangeshan, “Crystal Optics,” in Handbuch der physik, Vol. 25/1 (Springer, Berlin, 1966).

Ramangeshan, S.

G. N. Ramachandran, S. Ramangeshan, “Crystal Optics,” in Handbuch der physik, Vol. 25/1 (Springer, Berlin, 1966).

Smith, D. O.

D. O. Smith, “Magneto-Optical Scattering from Multilayer Magnetic and Dielectric Films,” Opt. Acta 12, 18 (1965).

Sprokel, G. J.

G. J. Sprokel, “The Reflectivity of a Liquid Crystal Cell in a Surface Plasma on Experiment,” Mol. Cryst. Liq. Cryst. 68, 39 (1981).
[CrossRef]

Treves, D.

D. Cheng, D. Treves, T. Chen, “Static Tests of TbFe Films for Magneto-Optical Recordings,” Proc. Soc. Photo-Opt. Instrum. Eng. 329, 223 (1982).

Walter, H.

H. Walter, “Optik Dunner Schichten,” in Handbuch der Physik, Vol. 25/1 (Springer, Berlin, 1961).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

IEEE Trans. Magn. (1)

M. Freiser, “A Survey of Magneto-Optic Effects,” IEEE Trans. Magn. MAG-4, 152 (1968).
[CrossRef]

J. Appl. Phys. (1)

M. Mansuripur, G. A. N. Connell, “Signal and Noise in Magneto-Optical Readout,” J. Appl. Phys. 53, 4485 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

Mol. Cryst. Liq. Cryst. (1)

G. J. Sprokel, “The Reflectivity of a Liquid Crystal Cell in a Surface Plasma on Experiment,” Mol. Cryst. Liq. Cryst. 68, 39 (1981).
[CrossRef]

Opt. Acta (1)

D. O. Smith, “Magneto-Optical Scattering from Multilayer Magnetic and Dielectric Films,” Opt. Acta 12, 18 (1965).

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

M. Mansuripur, G. A. N. Connell, J. W. Goodman, “Signal to Noise in Magneto-Optic Storage,” Proc. Soc. Photo-Opt. Instrum. Eng. 329, 215 (1982).

D. Cheng, D. Treves, T. Chen, “Static Tests of TbFe Films for Magneto-Optical Recordings,” Proc. Soc. Photo-Opt. Instrum. Eng. 329, 223 (1982).

Prog. Opt. (1)

R. Jacobsson, “Light Reflection from Films of Continuously Varying Refractive Index,” Prog. Opt. 5, 249 (1966).

Other (6)

G. N. Ramachandran, S. Ramangeshan, “Crystal Optics,” in Handbuch der physik, Vol. 25/1 (Springer, Berlin, 1966).

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960).

J. H. Judy, “Magnetic Optics Theory,” in Proceedings, Conference on Advances in Magnetic Recording, N.Y. Acad. of Science189, 239 (1972).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965).

H. Walter, “Optik Dunner Schichten,” in Handbuch der Physik, Vol. 25/1 (Springer, Berlin, 1961).

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Figures (6)

Fig. 1
Fig. 1

Four-layer film structure containing one magnetooptic layer. Subscript index denotes the layer, superscript + is the incident direction, and superscript − is the direction of the reflected beam.

Fig. 2
Fig. 2

Rotation and ellipticity as a function of g (g = x + iy). Step size for x and y is 0.01. The contour levels for rotation are 0.2–7° in 0.2° steps; for the ellipticity, −0.03 to +0.04 in 0.002 steps. Magnetooptic film thickness is 200 Å.

Fig. 3
Fig. 3

Reflectivity vs magnetooptic layer thickness for the structure in Fig. 1 calculated in 5-Å steps. Normal incidence. Parameters, thickness of antireflection and dielectric layers (Dar,Ddi); curve 1, Dar 1/4λ Ddi:1/4λ curve 2, Dar 1/2λ Ddi 1/4λ; curve 3, Dar 1/4λ Ddi:1/2λ; curve 4, Dar 1/2λ Ddi 1/2λ.

Fig. 4
Fig. 4

Rotation vs magnetooptic layer thickness for the structure in Fig. 1 calculated in 5-Å steps as Fig. 3. Parameters, thickness of antireflection and dielectric layers (Dar,Ddi); curve 1, Dar 1/4λ Ddi:1/4λ curve 2, Dar 1/2λ Ddi 1/4λ; curve 3, Dar 1/4λ Ddi:1/2λ; curve 4, Dar 1/2λ Ddi 1/2λ. Curves 1 and 3 and 2 and 4 coincide for thick m.o. films.

Fig. 5
Fig. 5

Ellipticity vs magnetooptic layer thickness for the structure in Fig. 1 calculated in 5-Å steps as Fig. 3. Parameters, thickness of antireflection and dielectric layers (Dar,Ddi); curve 1, Dar 1/4λ Ddi:1/4λ curve 2, Dar 1/2λ Ddi 1/4λ; curve 3, Dar 1/4λ Ddi:1/2λ; curve 4, Dar 1/2λ Ddi 1/2λ. Curves 1 and 3 and 2 and 4 coincide for thick m.p. films.

Fig. 6
Fig. 6

Reflectivity and rotation vs angle of incidence for p and s polarization. The magnetooptic layer is 1000 Å, essentially a thick FeTb film with a 1/4λ or a 1/2λ antireflection layer.

Equations (56)

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( ω c ) 2 D = ( k · k ) E - k ( k · E ) ;
D = [ ɛ ] E .
ɛ · [ 1 - i g 0 i g 1 0 0 0 1 ] .
[ k 2 - k x 2 - ( ω c ) 2 ɛ - k x k y + i g ( ω c ) 2 ɛ - k x k z - k x k z - i g ( ω c ) 2 ɛ k 2 - k y 2 ( ω c ) 2 ɛ - k y k z - k x k z - k y k z k 2 - k z 2 - ( ω c ) 2 ɛ ] × ( E x E y E z ) = 0 ,
[ k 2 - ( ω c ) 2 ɛ ] 2 + [ k 2 - k z 2 - ( ω c ) 2 ɛ ] g 2 ( ω c ) 2 ɛ = 0
k z 1 + = κ 0 1 + g ω c ɛ κ 0 k z 2 + = κ 0 1 + g ω c ɛ κ 0 k z 1 - = - κ 0 1 - g ω c ɛ κ 0 k z 2 - = - κ 0 1 + g ω c ɛ κ 0 ,
κ 0 2 = ( ω c ) 2 ɛ - ( k x 2 + k y 2 ) .
k z 1 - = - k z 2 + k z 2 - = - k z 1 + .
H x 1 = E y 1 · k z 1 · - ω c ɛ κ 0 2 , H x 2 = E y 2 · k z 2 · - ω c ɛ κ 0 2 , H y 1 = E x 1 · c ω k z 1 , H y 2 = E x 2 · c ω k z 2 .
E x 1 = E y 1 · - i ω c ɛ κ 0 , E x 2 = E y 2 · i ω c ɛ κ 0 .
E 0 y + + E 0 y - = E 1 y + + E 1 y - , H 0 x + + H 0 x - = H 1 x + + H 1 x - , E 0 x + + E 0 x - = E 1 x + + E 1 x - , H 0 y + + H 0 y - = H 1 y + + H 1 y - .
( E 0 y + E 0 y - E 0 x + E 0 x - ) = [ MA 1 ] × ( E 1 y + E 1 y - E 1 x + E 1 x - ) .
[ u 0 + u 1 2 u 0 u 0 - u 1 2 u 0 0 0 u 0 - u 1 2 u 0 u 0 + u 1 2 u 0 0 0 0 0 k 0 z + k 1 z 2 k 0 z k 0 z - k 1 z 2 k 0 z 0 0 k 0 z - k 1 z 2 k 0 z k 0 z + k 1 z 2 k 0 z ] .
E 1 y + exp - i k 1 z + z 1 + E 1 y - exp i k 1 z + z 1 = E 2 y 1 + exp - i k 2 z 1 + z 1 + E 2 y 1 - exp i k 2 z 2 + z 1 + E 2 y 2 + exp - i k 2 z 2 + z 1 + E 2 y 2 - exp i k 2 z 1 + z 1
( E 1 y + E 1 y - E 1 x + E 1 x - ) = [ MA 21 ] × ( E 2 y 1 + exp - i k 2 z 1 + z 1 E 2 y 1 - exp + i k 2 z 2 + z 1 E 2 y 2 + exp - i k 2 z 2 + z 1 E 2 y 2 - exp + i k 2 z 1 + z 1 ) ,
[ MA 21 ] : [ exp i k 1 z z 1 2 u 1 ( u 1 + u 21 u 1 - u 22 u 1 + u 22 u 1 - u 21 ) exp - i k 1 z z 1 2 u 1 ( u 1 - u 21 u 1 + u 22 u 1 - u 22 u 1 + u 21 ) i ω c ɛ exp i k 1 z z 1 2 k 1 z κ 0 ( - k 1 z - k 2 z 1 + - k 1 z + k 2 z 2 + k 1 z + k 2 z 2 + k 1 z - k 2 z 2 + ) i ω c ɛ exp - i k 1 z z 1 2 k 1 z κ 0 ( - k 1 z + k 2 z 1 + - k 1 z - k 2 z 2 + k 1 z - k 2 z 2 + k 1 z + k 2 z 2 + ) ] u 21 = - ω c ɛ k 2 z 1 κ 0 2 , u 22 = - ω c ɛ k 2 z 2 κ 0 2 .
( E 0 y + E 0 y - E 0 x + E 0 x - ) = [ MA 1 ] × [ MA 21 ] × ( E 2 y 1 + exp - i k 2 z 1 + z 1 E 2 y 1 - exp + i k 2 z 2 + z 1 E 2 y 2 + exp - i k 2 z 2 + z 1 E 2 y 2 - exp + i k 2 z 1 + z 1 ) .
E 2 y 1 + exp - i k 2 z 1 + z 2 + E 2 y 1 - exp - i k 2 z 2 + z 2 + E 2 y 2 + exp - i k 2 z 2 + z 2 + E 2 y 2 - exp + i k 2 z 1 + z 2 = E 3 y + exp - i k 3 z z 2 + E 3 y - exp i k 3 z z 2
k 3 z = ( ω c ) 2 ɛ 3 - k 0 y 2 .
[ MA 3 ] × ( E 2 y 1 + exp - i k 2 z 1 + z 2 E 2 y 1 - exp + i k 2 z 2 + z 2 E 2 y 2 + exp - i k 2 z 2 + z 2 E 2 y 2 - exp + i k 2 z 1 + z 2 ) = ( E 3 y + exp - i k 3 z z 2 E 3 y - exp + i k 3 z z 2 E 3 x + exp - i k 3 z z 2 E 3 x - exp + i k 3 z z 2 ) .
( E 0 y + E 0 y - E 0 x + E 0 x - ) = [ MA 1 ] × [ MA 21 ] × [ MA 31 ] × ( E 3 y + exp - i k 3 z z 2 E 3 y - exp + i k 3 z z 2 E 3 x + exp - i k 3 z z 2 E 3 x - exp + i k 3 z z 2 ) .
E 3 y + exp - i k 3 z z 3 + E 3 y - exp + i k 3 z z 3 = E 4 y + exp - i k 4 y z 3 + E 4 y - exp + i k 4 z z 3
( E 3 y + exp - i k 3 z z 3 E 3 y - exp + i k 3 z z 3 E 3 x + exp - i k 3 z z 3 E 3 x - exp + i k 3 z z 3 ) = [ MA 4 ] × ( E 4 y + exp - i k 4 z z 3 E 4 y - exp + i k 4 z z 3 E 4 x + exp - i k 4 z z 3 E 4 x - exp + i k 4 z z 3 ) .
( E 0 y + E 0 y - E 0 x + E 0 x - ) = [ MA 1 ] × [ MA 21 ] × [ MA 31 ] × [ MA 41 ] × ( E 4 y + exp - i k 4 z z 3 E 4 y - exp + i k 4 z z 3 E 4 x + exp - i k 4 z z 3 E 4 x - exp + i k 4 z z 3 ) .
M = [ a i j ] i , j = 1 , 2 , 3 , 4.
( E 0 y + E 0 x + ) = [ a 11 a 13 a 31 a 33 ] × ( E 4 y + E 4 x + )
( E 4 y + E 4 x + ) = 1 D [ a 33 - a 13 - a 31 a 11 ] × ( E 0 y + E 0 x + ) ,
( E 0 y - E 0 x - ) = [ a 21 a 23 a 41 a 43 ] × ( E 4 y + E 4 x + ) , ( E 0 y - E 0 x - ) = 1 D · [ a 21 a 23 a 41 a 43 ] · [ a 33 - a 13 - a 31 a 11 ] × ( E 0 y + E 0 x + ) .
( E 0 y - E 0 x - ) = [ b 11 b 12 b 21 b 22 ] × ( E 0 y + 0 )
E 0 y - = b 11 E 0 y + E 0 x - = b 21 E 0 y +             p polarized .
E 0 y - = b 12 E 0 x + E 0 x - = b 22 E 0 x +             s polarized
E 0 y - = E y exp i ( ω t + θ ) , E 0 x - = E x exp i ( ω t + θ ) , E y = b 11 cos θ θ = arctan b 11 b 11 , E x = b 21 cos ϕ ϕ = arctan b 21 b 21
tan α = E x E y ,
tan 2 ψ = tan 2 α cos ( ϕ - θ ) ,
sin 2 χ = - sin 2 α sin ( ϕ - θ ) ,
R = E y 2 + E x 2 .
E 0 x - E 0 y - = i g ɛ 2 1 - ɛ 2 + Θ ( g 2 ) ,
E 0 y - E 0 y + = 1 - ɛ 2 1 + ɛ 2 + Θ ( g 2 ) ,
R = E 0 x - 2 + E 0 y - 2 E 0 y + 2 = E 0 y - 2 E 0 y + 2 [ 1 + Θ ( g 2 ) ] ,
E y = E y exp i ( ω t + θ ) , E x = E x exp i ( ω t + ϕ )
E y = E y exp i ω t , E x = E x exp i ( ω t + π 2 ) = i E x expi ω t
( E y E x ) = [ cos ψ sin ψ - sin ψ cos ψ ] × ( E y E x ) .
E y = E y cos ψ cos θ + E x sin ψ cos ϕ ( a ) , 0 = E y cos ψ sin θ + E x sin ψ sin ϕ ( b ) ,
E x = E x cos ψ sin ϕ - E y sin ψ sin θ ( c ) , 0 = E x cos ψ cos ϕ - E y sin ψ cos θ ( d ) ,
tan ψ = tan α cos ϕ cos θ = - sin θ tan α sin ϕ ,
tan 2 α sin ϕ cos ϕ = - sin θ cos θ .
tan 2 ψ = 2 tan ψ 1 - tan 2 ψ .
tan 2 ψ = 2 tan α sin ϕ cos ϕ sin ( θ + ϕ ) .
2 tan α = tan 2 α ( 1 - tan 2 α )
tan 2 ψ = tan 2 α · cos ( ϕ - θ ) .
E x E y = tan α sin ϕ - tan ψ sin θ cos θ + tan α tan ψ cos ϕ .
E x E y = tan α sin ϕ cos θ .
E x E y = tan χ ,
sin 2 χ = 2 tan χ 1 + tan 2 χ = 2 tan α sin ϕ cos ϕ cos ( θ + ϕ ) .
sin 2 χ = - sin 2 α sin ( ϕ - θ ) .
tan α = E x E y , tan χ = E x E y , tan2 ψ = tan2 α cos ( ϕ - θ ) , sin 2 χ = - sin 2 α sin ( ϕ - θ ) .

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