Abstract

Multilayer optical structures are considered in which one or more of the layers is magnetotropic. Generalized scattering matrices at the interfaces are derived for the transverse Kerr magnetooptic effect. An algorithm is presented for the calculation of the optical and magnetooptical properties of the structure.

© 1990 Optical Society of America

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References

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  1. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), Sec. 4.8, p. 69ff.
  2. S. Visnovsky, “Magneto-optical Ellipsometry,” Czech. J. Phys. B 36, 625–650 (1986).
    [CrossRef]
  3. K. Balasubramanian, A. S. Marathay, H. A. MacLeod, “Modeling Magneto-optical Thin Film Media for Optical Data Storage,” Thin Solid Films 164, 391–403 (1988).
    [CrossRef]
  4. The author will furnish the source code and documentation for this program to qualified individuals.
  5. G. A. Prinz et al., “Magneto-optic Materials for Biasing Ring Laser Gyros,” Naval Research Laboratory Memorandum Reports 1 (3870), 2 (4198), 3 (4364) (1978–1980).
  6. W. M. Macek, “Ring Laser Magnetic Bias Mirror Compensated for Nonreciprocal Loss,” U.S. Patent3,851,973 (3Dec.1974). See also F. Aronowitz, “The Laser Gyro,” in Laser Applications, M. Ross, Ed. (Academic, New York, 1971), pp. 113–200.
  7. R. P. Hunt, “Magneto-Optic Scattering from Thin Solid Films,” J. Appl. Phys. 38, 1652–1671 (1967). See also R. E. McClure, “An Equivalent Circuit for the Transverse Magneto-Optic Effect in Thin Magnetic Films,” IEEE Trans. Mag. MAG-16, 1185–1187 (1980).
    [CrossRef]
  8. H. T. Minden, “Ellipsometric Measurement of the Kerr Magnetooptic Effect,” Appl. Opt. 18, 813–817 (1979).
    [CrossRef] [PubMed]

1988 (1)

K. Balasubramanian, A. S. Marathay, H. A. MacLeod, “Modeling Magneto-optical Thin Film Media for Optical Data Storage,” Thin Solid Films 164, 391–403 (1988).
[CrossRef]

1986 (1)

S. Visnovsky, “Magneto-optical Ellipsometry,” Czech. J. Phys. B 36, 625–650 (1986).
[CrossRef]

1979 (1)

1967 (1)

R. P. Hunt, “Magneto-Optic Scattering from Thin Solid Films,” J. Appl. Phys. 38, 1652–1671 (1967). See also R. E. McClure, “An Equivalent Circuit for the Transverse Magneto-Optic Effect in Thin Magnetic Films,” IEEE Trans. Mag. MAG-16, 1185–1187 (1980).
[CrossRef]

Balasubramanian, K.

K. Balasubramanian, A. S. Marathay, H. A. MacLeod, “Modeling Magneto-optical Thin Film Media for Optical Data Storage,” Thin Solid Films 164, 391–403 (1988).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), Sec. 4.8, p. 69ff.

Hunt, R. P.

R. P. Hunt, “Magneto-Optic Scattering from Thin Solid Films,” J. Appl. Phys. 38, 1652–1671 (1967). See also R. E. McClure, “An Equivalent Circuit for the Transverse Magneto-Optic Effect in Thin Magnetic Films,” IEEE Trans. Mag. MAG-16, 1185–1187 (1980).
[CrossRef]

Macek, W. M.

W. M. Macek, “Ring Laser Magnetic Bias Mirror Compensated for Nonreciprocal Loss,” U.S. Patent3,851,973 (3Dec.1974). See also F. Aronowitz, “The Laser Gyro,” in Laser Applications, M. Ross, Ed. (Academic, New York, 1971), pp. 113–200.

MacLeod, H. A.

K. Balasubramanian, A. S. Marathay, H. A. MacLeod, “Modeling Magneto-optical Thin Film Media for Optical Data Storage,” Thin Solid Films 164, 391–403 (1988).
[CrossRef]

Marathay, A. S.

K. Balasubramanian, A. S. Marathay, H. A. MacLeod, “Modeling Magneto-optical Thin Film Media for Optical Data Storage,” Thin Solid Films 164, 391–403 (1988).
[CrossRef]

Minden, H. T.

Prinz, G. A.

G. A. Prinz et al., “Magneto-optic Materials for Biasing Ring Laser Gyros,” Naval Research Laboratory Memorandum Reports 1 (3870), 2 (4198), 3 (4364) (1978–1980).

Visnovsky, S.

S. Visnovsky, “Magneto-optical Ellipsometry,” Czech. J. Phys. B 36, 625–650 (1986).
[CrossRef]

Appl. Opt. (1)

Czech. J. Phys. B (1)

S. Visnovsky, “Magneto-optical Ellipsometry,” Czech. J. Phys. B 36, 625–650 (1986).
[CrossRef]

J. Appl. Phys. (1)

R. P. Hunt, “Magneto-Optic Scattering from Thin Solid Films,” J. Appl. Phys. 38, 1652–1671 (1967). See also R. E. McClure, “An Equivalent Circuit for the Transverse Magneto-Optic Effect in Thin Magnetic Films,” IEEE Trans. Mag. MAG-16, 1185–1187 (1980).
[CrossRef]

Thin Solid Films (1)

K. Balasubramanian, A. S. Marathay, H. A. MacLeod, “Modeling Magneto-optical Thin Film Media for Optical Data Storage,” Thin Solid Films 164, 391–403 (1988).
[CrossRef]

Other (4)

The author will furnish the source code and documentation for this program to qualified individuals.

G. A. Prinz et al., “Magneto-optic Materials for Biasing Ring Laser Gyros,” Naval Research Laboratory Memorandum Reports 1 (3870), 2 (4198), 3 (4364) (1978–1980).

W. M. Macek, “Ring Laser Magnetic Bias Mirror Compensated for Nonreciprocal Loss,” U.S. Patent3,851,973 (3Dec.1974). See also F. Aronowitz, “The Laser Gyro,” in Laser Applications, M. Ross, Ed. (Academic, New York, 1971), pp. 113–200.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), Sec. 4.8, p. 69ff.

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of a layered optical structure. In the algorithm the last layer is numbered l, the semi-infinite substrate is l + 1, while the semi-infinite incident medium is 1.

Fig. 2
Fig. 2

Transverse Kerr magnetooptic effect. The direction of magnetization B is perpendicular to the direction of propagation k. The electromagnetic field is decomposed into a perpendicular field Es which is parallel to B and a parallel field Ep which is perpendicular to B. Only the latter field is affected by the magnetization; the effect is to add an electric field component iQEp which points along the direction of propagation.

Fig. 3
Fig. 3

Diagram of the optical electric fields for p-polarization. The dc magnetization is perpendicular to the plane of the paper. The magnetooptic field components iQE are added for the transverse Kerr MO effect: m = exp(2πinm cosθmm). The same diagram holds for the magnetic H fields in the case of s-polarization, except that there are no additional fields due to the magnetooptic effect.

Equations (73)

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θ = ( 1 / 2 ) k Q ( rad / m ) .
δ r ¯ p / r ¯ p
E ¯ m = E m + E m - .
E ¯ m = S ˜ m * E ¯ m + 1 ,
E ¯ m - 1 = S ˜ m - 1 * S ˜ m * E ¯ m + 1 ,
E ¯ 1 = C ˜ * E ¯ l + 1 ;
C ˜ = m = 1 l S ˜ m = c 11 c 12 c 21 c 22 .
E ¯ l + 1 = E l + 1 + 0 .
E 1 + = c 11 E l + 1 + ,
E 1 - = c 21 E l + 1 - ,
r ¯ = E 1 - E 1 + = c 21 c 11 ,
t ¯ = E l + 1 + E 1 + = 1 c 11 .
tan Ψ exp ( i Δ ) = r ¯ p r ¯ s = c p , 21 c s , 21 c s , 11 c p , 11 .
δ r ¯ p = c p , 21 + c p , 11 + - c p , 21 - c p , 11 - ,
r ¯ p = 1 2 ( c p , 21 + c p , 11 + + c p , 21 - c p , 11 - ) .
E ± = E m ± exp [ - i ( k ¯ m ± · r ¯ + ϕ m ) ] .
k ¯ m ± · r ¯ = k m ( ± z cos θ m + x sin θ m ) .
ϕ m = k m d m cos θ m .
ϕ ± ( z ) = k m ( z - d m ) cos θ m .
ϕ z 0 ± = ± k m d m cos θ m .
E ± ( d m ) = E m ± ,
E ± ( 0 ) = m ± 1 E m ± ,
m = exp ( i δ m ) , δ m = k m d m cos θ m .
E m ± and m + 1 ± 1 E m + 1 ± .
E m = E m + 1 ,
H m = H m + 1 ,
μ 0 H ¯ t = - × E ¯ = i ω μ 0 H ¯ .
E m ( z m = d m ) = E m + + E m - ,
E m + 1 ( z m + 1 = 0 ) = m + 1 E m + 1 + + m + 1 - 1 E m + 1 - .
E m + + E m - = m + 1 E m + 1 + + m + 1 - 1 E m + 1 - .
E y , m ± ( r ¯ ) = E m ± exp ( i k m [ ( z - d m ) cos θ m + x sin θ m ] ) .
H x , m ± = 1 i μ 0 ω E y , m ± ( r ¯ ) z = n ¯ m cos θ m Z 0 E y , m ± ( r ¯ ) ,
H x , m ± = n ¯ m cos θ m Z 0 E y , m ± .
H x , m + 1 ± = m + 1 ± 1 n ¯ m + 1 cos θ m + 1 Z 0 E y , m + 1 ± .
H m = n ¯ m cos θ m ( - E m + + E m - ) ,
H m + 1 = n ¯ m + 1 cos θ m + 1 ( m + 1 E m + 1 + + m + 1 - 1 E m - + 1 ) ,
N s , m ( - E m + + E m - ) = N s , m + 1 ( m + 1 E m + 1 + + m + 1 - 1 E m + 1 - ) ,
N s = n ¯ cos θ
E m + = 1 2 N s , m [ ( N s , m + N s , m + 1 ) m + 1 E m + 1 + + ( N s , m - N s , m + 1 ) m + 1 - 1 E m + 1 - ] ,
E m - = 1 2 N s , m [ ( N s , m - N s , m + 1 ) m + 1 E m + 1 + + ( N s , m + N s , m + 1 ) m + 1 - 1 E m + 1 - ] .
S s , m , 11 = ( N s , m + N s , m + 1 ) m + 1 ,
S s , m , 12 = ( N s , m - N s , m + 1 ) m + 1 - 1 ,
S s , m , 21 = ( N s , m - N s , m + 1 ) m + 1 ,
S s , m , 22 = ( N s , m + N s , m + 1 ) m + 1 - 1 ,
T s , m = 2 N s , m .
C ˜ s = m = 1 1 ( 1 T s , m ) S ˜ s , m .
Q ¯ ( - M ) = - Q ¯ ( M ) .
E x + = - E + ( cos θ - i Q ¯ sin θ ) exp [ - i k ( z cos θ + x sin θ ) ] ,
E y + = 0 ,
E z + = E + ( sin θ + i Q ¯ cos θ ) exp [ - i k ( z cos θ + x sin θ ) ] .
E x - = E - ( cos θ + i Q ¯ sin θ ) exp [ - i k ( - z cos θ + x sin θ ) ] ,
E y - = 0 ,
E z - = E + ( sin θ - i Q ¯ cos θ ) exp [ - i k ( - z cos θ + x sin θ ) ] .
( × E ) y = E x z - E z x .
H y ± = n ¯ Z 0 E ± = H ± .
E = E x + + E x - = - E + ( cos θ - i Q sin θ ) + E - ( cos θ + i Q sin θ ) ,
H = H y + + H y - = ( E + + E - ) n ¯ .
[ - E m + ( cos θ m - i Q ¯ m sin θ m ) + E m - ( cos θ m + i Q ¯ m sin θ m ) ] = [ - m + 1 E m + 1 + ( cos θ m + 1 - i Q ¯ m + 1 sin θ m + 1 ) + m + 1 - 1 E m + 1 - ( cos θ m + 1 + i Q ¯ m + 1 sin θ m + 1 ) ] .
γ m ± = 1 ± i Q ¯ m tan θ m ,
cos θ m ( - E m + γ m - + E m - γ m + ) = cos θ m + 1 ( - m + 1 E m + 1 + γ m + 1 - + m + 1 - 1 ) E m + 1 - γ m + 1 + ) .
n ¯ m ( E m + + E m - ) = n ¯ m + 1 ( m + 1 E m + 1 + + m + 1 - 1 E m + 1 - ) .
E m + = cos θ m + 1 2 n ¯ m [ ( N p , m + 1 γ m + + N p , m γ m + 1 - ) m + 1 E m + 1 + + ( N p , m + 1 γ m + - N p , m γ m + 1 + ) m + 1 - 1 E m + 1 - ] ,
E m - = cos θ m + 1 2 n ¯ m [ ( N p , m + 1 γ m - - N p , m γ m + 1 - ) m + 1 E m + 1 + + ( N p , m + 1 γ m - + N p , m γ m + 1 + ) m + 1 - 1 E m + 1 - ] ,
N p , m = n ¯ m cos θ m .
S p , m , 11 = ( N p , m + 1 γ m + + N p , m γ m + 1 - ) m + 1 ,
S p , m , 12 = ( N p , m + 1 γ m + - N p , m γ m + 1 + ) m + 1 - 1 ,
S p , m , 21 = ( N p , m + 1 γ m - - N p , m γ m + 1 - ) m + 1 ,
S p , m , 22 = ( N p , m + 1 γ m - + N p , m γ m + 1 + ) m + 1 - 1 ,
T p , m = 2 N p , m + 1 .
n ¯ m + 1 n ¯ m 1 T p , m .
C ˜ p = m = 1 1 [ ( n ¯ m + 1 n ¯ m ) ( 1 T p , m ) S ˜ p , m ] .
( n ¯ l + 1 n ¯ 1 ) m = 1 1 ( 1 T p , m ) ,
C ˜ p = ( n ¯ l + 1 n ¯ 1 ) m = 1 1 [ ( 1 T p , m ) S ˜ p , m ] .

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