Abstract

After the Mueller matrix of a medium exhibiting a complex Faraday effect is derived, a procedure is developed that permits computation of optical transmission, Faraday rotation, and magnetic circular dichroism (MCD) in planar layered structures when all internal reflections are taken into account. The structures are represented by Mueller matrices and have, therefore, a depolarizing effect on an incident wave of other than circular polarization. The formulas for some common experimental arrangements including ellipticity and azimuth modulation techniques are given and are then applied to three special cases, i.e., a single absorbing plate, a single absorbing layer on a nonabsorbing substrate, and two identical absorbing layers on either side of a nonabsorbing substrate. For magnetic garnet films on nonabsorbing garnet substrates in the region of low absorption, the formulas predict an increase of MCD because of multiple reflections of the order of a few percent. On thin sections of magnetic semiconductors the effect can approach 15%, provided that the absorption is low.

© 1981 Optical Society of America

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References

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  1. Š. Višňovský, V. Prosser, and R. Krishnan, “Effect of multiple internal reflections on Faraday rotation in multilayer structures,” J. Appl. Phys. 49, 403–408 (1978).
    [Crossref]
  2. B. Briat, M. Billardon, and J. Badoz, “Spectroscopic applications of magneto-optics to inorganic materials,” Physica 89B, 27–37 (1977).
  3. C. C. Robinson, “Electromagnetic theory of the Kerr and the Faraday effects for oblique incidence,” J. Opt. Soc. Am. 54, 1220–1224 (1964).
    [Crossref]
  4. R. P. Hunt, “Magneto-optic scattering from thin solid films,” J. Appl. Phys. 38, 1652–1671 (1967).
    [Crossref]
  5. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [Crossref]
  6. J. W. Nielsen, “Bubble domain memory materials,” IEEE Trans. Magn. MAG-12, 327–345 (1976).
    [Crossref]
  7. N. S. Chang and Y. Matsuo, “Magnetostatic surface wave propagation on a periodic YIG film layer,” Appl. Phys. Lett. 35, 352–354 (1979).
    [Crossref]
  8. P. K. Tien, D. P. Schinke, and S. L. Blank, “Magneto-optics and motion of the magnetization in a film-waveguide optical switch,” J. Appl. Phys. 45, 3059–3068 (1974).
    [Crossref]
  9. B. S. Hewitt, R. D. Pierce, S. L. Blank, and S. Knight, “Technique for controlling the properties of magnetic garnet films,” IEEE Trans. Magn. MAG-9, 366–372 (1973).
    [Crossref]
  10. M. Kamin and et al., “Multilayer self-structured bubble memories,” J. Appl. Phys. 50, 2292–2294 (1979).
    [Crossref]
  11. S. H. Wemple and J. A. Seman, “Optical transmission through multilayered structures,” Appl. Opt. 12, 2947–2949 (1973).
    [Crossref] [PubMed]
  12. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chaps. 1 and 2.
  13. J. Badoz and et al., “Sensitive devices to determine the state and degree of polarization of light beam using a birefringence modulator,” J. Opt. (Paris) 8, 373–384 (1977).
    [Crossref]
  14. M. Billardon, “Méthode polarimétrique de précision dans le visible et l’ultra-violet,” Ann. Phys. (Paris) 7, 233–267 (1962).
  15. F. J. Kahn, “Ultraviolet magneto-optics of opaque magnetic media: ferric oxide compounds,” Ph.D. thesis, Harvard University, Cambridge, Mass., 1968 (unpublished).
  16. J. Warner, “The refractive indices of some garnet crystals at 1.15 μ m,” Mat. Res. Bull. 9, 507–510 (1974).
    [Crossref]
  17. P. Hlídek, “Optical properties of CdCr2Se4,” Thesis, Charles University, Prague, Czechoslovakia, 1978 (unpublished).

1979 (2)

N. S. Chang and Y. Matsuo, “Magnetostatic surface wave propagation on a periodic YIG film layer,” Appl. Phys. Lett. 35, 352–354 (1979).
[Crossref]

M. Kamin and et al., “Multilayer self-structured bubble memories,” J. Appl. Phys. 50, 2292–2294 (1979).
[Crossref]

1978 (1)

Š. Višňovský, V. Prosser, and R. Krishnan, “Effect of multiple internal reflections on Faraday rotation in multilayer structures,” J. Appl. Phys. 49, 403–408 (1978).
[Crossref]

1977 (2)

B. Briat, M. Billardon, and J. Badoz, “Spectroscopic applications of magneto-optics to inorganic materials,” Physica 89B, 27–37 (1977).

J. Badoz and et al., “Sensitive devices to determine the state and degree of polarization of light beam using a birefringence modulator,” J. Opt. (Paris) 8, 373–384 (1977).
[Crossref]

1976 (1)

J. W. Nielsen, “Bubble domain memory materials,” IEEE Trans. Magn. MAG-12, 327–345 (1976).
[Crossref]

1974 (2)

J. Warner, “The refractive indices of some garnet crystals at 1.15 μ m,” Mat. Res. Bull. 9, 507–510 (1974).
[Crossref]

P. K. Tien, D. P. Schinke, and S. L. Blank, “Magneto-optics and motion of the magnetization in a film-waveguide optical switch,” J. Appl. Phys. 45, 3059–3068 (1974).
[Crossref]

1973 (2)

B. S. Hewitt, R. D. Pierce, S. L. Blank, and S. Knight, “Technique for controlling the properties of magnetic garnet films,” IEEE Trans. Magn. MAG-9, 366–372 (1973).
[Crossref]

S. H. Wemple and J. A. Seman, “Optical transmission through multilayered structures,” Appl. Opt. 12, 2947–2949 (1973).
[Crossref] [PubMed]

1972 (1)

1967 (1)

R. P. Hunt, “Magneto-optic scattering from thin solid films,” J. Appl. Phys. 38, 1652–1671 (1967).
[Crossref]

1964 (1)

1962 (1)

M. Billardon, “Méthode polarimétrique de précision dans le visible et l’ultra-violet,” Ann. Phys. (Paris) 7, 233–267 (1962).

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chaps. 1 and 2.

Badoz, J.

J. Badoz and et al., “Sensitive devices to determine the state and degree of polarization of light beam using a birefringence modulator,” J. Opt. (Paris) 8, 373–384 (1977).
[Crossref]

B. Briat, M. Billardon, and J. Badoz, “Spectroscopic applications of magneto-optics to inorganic materials,” Physica 89B, 27–37 (1977).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chaps. 1 and 2.

Berreman, D. W.

Billardon, M.

B. Briat, M. Billardon, and J. Badoz, “Spectroscopic applications of magneto-optics to inorganic materials,” Physica 89B, 27–37 (1977).

M. Billardon, “Méthode polarimétrique de précision dans le visible et l’ultra-violet,” Ann. Phys. (Paris) 7, 233–267 (1962).

Blank, S. L.

P. K. Tien, D. P. Schinke, and S. L. Blank, “Magneto-optics and motion of the magnetization in a film-waveguide optical switch,” J. Appl. Phys. 45, 3059–3068 (1974).
[Crossref]

B. S. Hewitt, R. D. Pierce, S. L. Blank, and S. Knight, “Technique for controlling the properties of magnetic garnet films,” IEEE Trans. Magn. MAG-9, 366–372 (1973).
[Crossref]

Briat, B.

B. Briat, M. Billardon, and J. Badoz, “Spectroscopic applications of magneto-optics to inorganic materials,” Physica 89B, 27–37 (1977).

Chang, N. S.

N. S. Chang and Y. Matsuo, “Magnetostatic surface wave propagation on a periodic YIG film layer,” Appl. Phys. Lett. 35, 352–354 (1979).
[Crossref]

Hewitt, B. S.

B. S. Hewitt, R. D. Pierce, S. L. Blank, and S. Knight, “Technique for controlling the properties of magnetic garnet films,” IEEE Trans. Magn. MAG-9, 366–372 (1973).
[Crossref]

Hlídek, P.

P. Hlídek, “Optical properties of CdCr2Se4,” Thesis, Charles University, Prague, Czechoslovakia, 1978 (unpublished).

Hunt, R. P.

R. P. Hunt, “Magneto-optic scattering from thin solid films,” J. Appl. Phys. 38, 1652–1671 (1967).
[Crossref]

Kahn, F. J.

F. J. Kahn, “Ultraviolet magneto-optics of opaque magnetic media: ferric oxide compounds,” Ph.D. thesis, Harvard University, Cambridge, Mass., 1968 (unpublished).

Kamin, M.

M. Kamin and et al., “Multilayer self-structured bubble memories,” J. Appl. Phys. 50, 2292–2294 (1979).
[Crossref]

Knight, S.

B. S. Hewitt, R. D. Pierce, S. L. Blank, and S. Knight, “Technique for controlling the properties of magnetic garnet films,” IEEE Trans. Magn. MAG-9, 366–372 (1973).
[Crossref]

Krishnan, R.

Š. Višňovský, V. Prosser, and R. Krishnan, “Effect of multiple internal reflections on Faraday rotation in multilayer structures,” J. Appl. Phys. 49, 403–408 (1978).
[Crossref]

Matsuo, Y.

N. S. Chang and Y. Matsuo, “Magnetostatic surface wave propagation on a periodic YIG film layer,” Appl. Phys. Lett. 35, 352–354 (1979).
[Crossref]

Nielsen, J. W.

J. W. Nielsen, “Bubble domain memory materials,” IEEE Trans. Magn. MAG-12, 327–345 (1976).
[Crossref]

Pierce, R. D.

B. S. Hewitt, R. D. Pierce, S. L. Blank, and S. Knight, “Technique for controlling the properties of magnetic garnet films,” IEEE Trans. Magn. MAG-9, 366–372 (1973).
[Crossref]

Prosser, V.

Š. Višňovský, V. Prosser, and R. Krishnan, “Effect of multiple internal reflections on Faraday rotation in multilayer structures,” J. Appl. Phys. 49, 403–408 (1978).
[Crossref]

Robinson, C. C.

Schinke, D. P.

P. K. Tien, D. P. Schinke, and S. L. Blank, “Magneto-optics and motion of the magnetization in a film-waveguide optical switch,” J. Appl. Phys. 45, 3059–3068 (1974).
[Crossref]

Seman, J. A.

Tien, P. K.

P. K. Tien, D. P. Schinke, and S. L. Blank, “Magneto-optics and motion of the magnetization in a film-waveguide optical switch,” J. Appl. Phys. 45, 3059–3068 (1974).
[Crossref]

Višnovský, Š.

Š. Višňovský, V. Prosser, and R. Krishnan, “Effect of multiple internal reflections on Faraday rotation in multilayer structures,” J. Appl. Phys. 49, 403–408 (1978).
[Crossref]

Warner, J.

J. Warner, “The refractive indices of some garnet crystals at 1.15 μ m,” Mat. Res. Bull. 9, 507–510 (1974).
[Crossref]

Wemple, S. H.

Ann. Phys. (Paris) (1)

M. Billardon, “Méthode polarimétrique de précision dans le visible et l’ultra-violet,” Ann. Phys. (Paris) 7, 233–267 (1962).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

N. S. Chang and Y. Matsuo, “Magnetostatic surface wave propagation on a periodic YIG film layer,” Appl. Phys. Lett. 35, 352–354 (1979).
[Crossref]

IEEE Trans. Magn. (2)

J. W. Nielsen, “Bubble domain memory materials,” IEEE Trans. Magn. MAG-12, 327–345 (1976).
[Crossref]

B. S. Hewitt, R. D. Pierce, S. L. Blank, and S. Knight, “Technique for controlling the properties of magnetic garnet films,” IEEE Trans. Magn. MAG-9, 366–372 (1973).
[Crossref]

J. Appl. Phys. (4)

M. Kamin and et al., “Multilayer self-structured bubble memories,” J. Appl. Phys. 50, 2292–2294 (1979).
[Crossref]

P. K. Tien, D. P. Schinke, and S. L. Blank, “Magneto-optics and motion of the magnetization in a film-waveguide optical switch,” J. Appl. Phys. 45, 3059–3068 (1974).
[Crossref]

Š. Višňovský, V. Prosser, and R. Krishnan, “Effect of multiple internal reflections on Faraday rotation in multilayer structures,” J. Appl. Phys. 49, 403–408 (1978).
[Crossref]

R. P. Hunt, “Magneto-optic scattering from thin solid films,” J. Appl. Phys. 38, 1652–1671 (1967).
[Crossref]

J. Opt. (Paris) (1)

J. Badoz and et al., “Sensitive devices to determine the state and degree of polarization of light beam using a birefringence modulator,” J. Opt. (Paris) 8, 373–384 (1977).
[Crossref]

J. Opt. Soc. Am. (2)

Mat. Res. Bull. (1)

J. Warner, “The refractive indices of some garnet crystals at 1.15 μ m,” Mat. Res. Bull. 9, 507–510 (1974).
[Crossref]

Physica (1)

B. Briat, M. Billardon, and J. Badoz, “Spectroscopic applications of magneto-optics to inorganic materials,” Physica 89B, 27–37 (1977).

Other (3)

P. Hlídek, “Optical properties of CdCr2Se4,” Thesis, Charles University, Prague, Czechoslovakia, 1978 (unpublished).

F. J. Kahn, “Ultraviolet magneto-optics of opaque magnetic media: ferric oxide compounds,” Ph.D. thesis, Harvard University, Cambridge, Mass., 1968 (unpublished).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chaps. 1 and 2.

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Equations (56)

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( I k - 1 - k I k - 1 + k ) = ( 1 - R k ) - 1 [ ( 1 - 2 R k ) ( D k ) R k ( D k ) - 1 - R k ( D k ) ( D k ) - 1 ] ( I - k + 1 I k + k + 1 ) = ( M k ) ( I k - k + 1 I k + k + 1 ) ,
( S ) = k = 1 n + 1 ( M k ) ,
[ E l ( d k ) E r ( d k ) ] = [ exp ( i δ k l ) 0 0 exp ( i δ k r ) ] [ E l ( 0 ) E r ( 0 ) ] ,
( T k ) = exp [ i ( δ k l + δ k r ) / 2 ] × [ cos ( δ k l - δ k r ) / 2 - sin ( δ k l - δ k r ) / 2 sin ( δ k l - δ k r ) / 2 cos ( δ k l - δ k r ) / 2 ] .
( D k ) = A k ( F k ) ,
A k = exp [ - ( α k l + α k r ) d k / 2 ]
( F k ) = { cos 2 ψ k 0 0 sinh 2 ψ k 0 cos 2 θ k - sin 2 θ k 0 0 sin 2 θ k cos 2 θ k 0 sinh 2 ψ k 0 0 cosh 2 ψ k
( D k ) ( D l ) = [ D ( θ k , ψ k ) ] [ D ( θ l , ψ l ) ] = [ D ( θ k + θ l , ψ k + ψ l ) ] .
θ k = Re [ ( δ k l - δ k r ) / 2 ] = - π ( n k l - n k r ) d k / λ 0 .
( Δ α k ) d k = ( α k l - α k r ) d k = Im 2 ( δ k l - δ k r ) = 4 π ( k k l - k k r ) d k = 4 ψ k .
tan k = tanh [ Im ( δ k l - δ k r ) / 2 ] = tanh [ π ( k k l - k k r ) d k / λ 0 ] = tanh ψ k = tanh [ ( Δ α k ) d k / 4 ] .
cosh 2 ψ k = cos - 1 2 k = cosh [ ( Δ α k ) d k / 2 ] ,
sinh 2 ψ k = tan 2 k = sinh [ ( Δ α k ) d k / 2 ] .
I n + 1 + n + 1 = [ ( S ) 22 ] - 1 I 0 + 1 ,             I n + 1 - n + 1 = 0 ,
( S ) 22 = s = 1 2 n a s ( G s ) ,
( G s ) = k = 1 n ( F k ) μ k s
( S ) 22 = s = 1 2 n - 1 [ a s + ( G s ) + + a s - ( G s ) - ] ,
( S ) 22 = { a 11 0 0 a 14 0 a 22 - a 32 0 0 a 32 a 22 0 a 14 0 0 a 11
( a 11 a 22 ) = s - 1 2 n - 1 ( a s + + a s - ) ( cosh 2 Ψ s cos 2 Θ s ) ,
( a 14 a 32 ) = s = 1 2 n - 1 ( a s + - a s - ) ( sinh 2 Ψ s sin 2 Θ s ) .
[ ( S ) 22 ] - 1 = { b 11 0 0 b 14 0 b 22 b 23 0 0 - b 23 b 22 0 b 14 0 0 b 11 ,
P = [ ( a 11 2 - a 14 2 ) 2 ( a 22 2 + a 32 2 ) - 1 cos 2 2 + ( a 11 sin 2 - a 14 ) 2 ] 1 / 2 × ( a 11 - a 14 sin 2 ) - 1 1
( a 11 a 22 ) = [ ( 1 - R 1 ) ( 1 - R 2 ) ( 1 - R 3 ) ( 1 - R 4 ) A 1 A 2 A 3 ] - 1 × { [ 1 - R 1 ( 1 - 2 R 2 ) ( 1 - 2 R 3 ) R 4 A 1 2 A 2 2 A 3 2 ] × [ cosh 2 ( ψ 1 + ψ 2 + ψ 3 ) cos 2 ( θ 1 + θ 2 + θ 3 ) ] - R 2 [ R 1 A 1 2 + ( 1 - 2 R 3 ) R 4 A 2 2 A 3 2 ] [ cosh 2 ( - ψ 1 + ψ 2 + ψ 3 ) cos 2 ( - θ 1 + θ 2 + θ 3 ) ] - R 2 R 3 ( A 2 2 - R 1 R 4 A 1 2 A 3 2 ) [ cosh 2 ( ψ 1 - ψ 2 + ψ 3 ) cos 2 ( θ 1 - θ 2 + θ 3 ) ] - R 3 [ R 4 A 3 2 + R 1 ( 1 - 2 R 2 ) A 1 2 A 2 2 ] × [ cosh 2 ( ψ 1 + ψ 2 - ψ 3 ) cos 2 ( θ 1 + θ 2 - θ 3 ) ] } .
( a 14 a 32 ) = - [ ( 1 - R 1 ) ( 1 - R 2 ) ( 1 - R 3 ) ( 1 - R 4 ) A 1 A 2 A 3 ] - 1 × { [ 1 + R 1 ( 1 - 2 R 2 ) ( 1 - 2 R 3 ) R 4 A 1 2 A 2 2 A 3 2 ] × [ sinh 2 ( ψ 1 + ψ 2 + ψ 3 ) sin 2 ( θ 1 + θ 2 + θ 3 ) ] - R 2 [ R 1 A 1 2 - ( 1 - 2 R 3 ) R 4 A 2 2 A 3 2 ] [ sinh 2 ( - ψ 1 + ψ 2 + ψ 3 ) sin 2 ( - θ 1 + θ 2 + θ 3 ) ] - R 2 R 3 ( A 2 2 + R 1 R 4 A 1 2 A 3 2 ) [ sinh 2 ( ψ 1 - ψ 2 + ψ 3 ) sin 2 ( θ 1 - θ 2 + θ 3 ) ] - R 3 [ R 4 A 3 2 - R 1 ( 1 - 2 R 2 ) A 1 2 A 2 2 ] × [ sinh 2 ( ψ 1 + ψ 2 - ψ 3 ) sin 2 ( θ 1 + θ 2 - θ 3 ) ] } .
I ± = ( I 0 / 2 ) ( a 11 a 14 ) ( a 11 2 - a 14 2 ) - 1 ,
( I - - I + ) ( I - + I + ) - 1 = - a 14 / a 11 ,
( I - - I + ) ( I - + I + ) - 1 = tanh 2 ψ t ,
I = ( I 0 / 2 ) ( a 11 2 - a 14 2 ) - 1 [ a 11 - a 14 sin ( τ M sin ω t ) ] .
I ( ω ) / I ( 0 ) = - 2 J 1 ( τ M ) a 14 a 11 - 1 sin ω t ,
I ( ω ) / I ( 0 ) = 2 J 1 ( τ M ) tanh 2 ψ t sin ω t .
I ± = I 0 { a 11 ( a 11 2 - a 14 2 ) - 1 + [ a 22 ( a 22 2 + a 32 2 ) - 1 cos 2 β ± a 32 ( a 22 2 + a 32 2 ) - 1 sin 2 β ] cos τ ± a 14 ( a 11 2 - a 14 2 ) - 1 sin τ } / 2 ,
( I - - I + ) ( I - + I + ) - 1 = - [ a 32 ( a 22 2 + a 32 2 ) - 1 sin 2 β cos τ + a 14 ( a 11 2 - a 14 2 ) - 1 sin τ ] [ a 11 ( a 11 2 - a 14 2 ) - 1 + a 22 ( a 22 2 + a 32 2 ) - 1 cos 2 β cos τ ] - 1 ,
( I - - I + ) ( I - + I + ) - 1 = ( sin 2 θ t sin 2 β cos τ + tan 2 t sin τ ) × ( cos - 1 2 t + cos 2 θ t cos 2 β cos τ ) - 1 ,
I = I 0 { a 11 ( a 11 2 - a 14 2 ) - 1 + [ J 0 ( τ M ) + 2 J 2 ( τ M ) cos 2 ω t + ] × [ a 22 ( a 22 2 + a 32 2 ) - 1 cos 2 β + a 32 ( a 22 2 + a 32 2 ) - 1 sin 2 β ] + [ 2 J 1 ( τ M ) sin ω t + ] a 14 ( a 11 2 - a 14 2 ) - 1 } / 2.
I ( ω ) / I ( 0 ) = [ 2 a 14 ( a 11 2 - a 14 2 ) - 1 J 1 ( τ M ) sin ω t ] × { a 11 ( a 11 2 - a 14 2 ) - 1 + [ a 22 ( a 22 2 + a 32 2 ) - 1 cos 2 β + a 32 ( a 22 2 + a 32 2 ) - 1 sin 2 β ] J 0 ( τ M ) } - 1 .
I ( ω ) / I ( 0 ) = - 2 J 1 ( τ M ) sin ω t tan 2 t [ cos - 1 2 t + J 0 ( τ M ) ( cos 2 β cos 2 θ t - sin 2 β sin 2 θ t ) ] - 1 .
I ( ω ) / I ( 0 ) = - 2 J 1 ( τ M ) sin ω t sin 2 t .
I ( ω ) / I ( 0 ) = 2 J 1 ( τ M ) sin ω t ( a 14 / a 11 ) .
I ( 2 ω ) / I ( 0 ) = 2 J 2 ( τ M ) cos 2 ω t [ a 22 ( a 22 2 + a 32 2 ) - 1 cos 2 β + a 32 ( a 22 2 + a 32 2 ) - 1 sin 2 β ] { a 11 ( a 11 2 - a 14 2 ) - 1 + [ a 22 ( a 22 2 + a 32 2 ) - 1 cos 2 β + a 32 ( a 22 2 + a 32 2 ) - 1 × sin 2 β ] J 0 ( τ M ) } - 1 ,
I ( 2 ω ) / I ( 0 ) = 2 J 2 ( τ M ) cos 2 ω t cos 2 ( θ t + β ) × [ cos - 1 2 t + cos 2 ( θ t + β ) J 0 ( τ M ) ] - 1
I ( 2 ω ) / I ( 0 ) 2 J 2 ( τ M ) cos 2 ω t [ a 32 ( a 22 2 + a 32 2 ) - 1 ] × a 11 - 1 ( a 11 2 - a 14 2 ) .
I ( 2 ω ) / I ( 0 ) = 2 J 2 ( τ M ) cos 2 ω t a 11 - 1 ( a 22 2 + a 32 2 ) - 1 ( a 11 2 - a 14 2 ) × ( a 22 cos 2 β + a 32 sin 2 β ) .
I = I 0 { a 11 ( a 11 2 - a 14 2 ) - 1 + [ a 22 ( a 22 2 + a 32 2 ) - 1 sin 2 β + a 32 ( a 22 2 + a 32 2 ) - 1 cos τ cos 2 β + a 14 ( a 11 2 - a 14 2 ) - 1 sin τ cos 2 β ] × [ 2 J 1 ( γ M ) sin ω t + ] + [ a 14 ( a 11 2 - a 14 2 ) - 1 sin τ sin 2 β + a 32 ( a 22 2 + a 32 2 ) - 1 cos τ sin 2 β - a 22 ( a 22 2 + a 32 2 ) - 1 cos 2 β ] × [ J 0 ( γ M ) + 2 J 2 ( γ M ) cos 2 ω t + ] } / 2.
tan 2 β = - [ a 32 a 22 - 1 cos τ + a 14 a 22 - 1 ( a 22 2 + a 32 2 ) ( a 11 2 - a 14 2 ) - 1 sin τ ] .
tan 2 β = tan 2 θ t cos τ + tan 2 t cos - 1 2 θ t sin τ .
tan 2 β = - ( a 32 / a 22 ) ,
a 14 / a 11 - tanh 2 ψ t = - sin 2 t , a 32 / a 22 - tan 2 θ t , a 11 ( a 11 2 - a 14 2 ) - 1 cosh 2 ψ t = cos - 1 2 t , a 14 ( a 11 2 - a 14 2 ) - 1 - sinh 2 ψ t = - tan 2 t , a 22 ( a 22 2 + a 32 2 ) - 1 cos 2 θ t , a 32 ( a 22 2 + a 32 2 ) - 1 - sin 2 θ t .
R 1 = R 2 = R , R 3 = R 4 = 0 , A 1 = A , A 2 = A 3 = 1 , θ 1 = θ , ψ 1 = ψ , θ 2 = θ 3 = ψ 2 = ψ 3 = 0
( a 14 / a 11 ) = ( 1 + R 2 A 2 ) ( 1 - R 2 A 2 ) - 1 tanh 2 ψ .
I ( ω ) / I ( 0 ) = - 2 J 1 ( τ M ) ( 1 + R 2 A 2 ) × ( 1 - R 2 A 2 ) - 1 tan 2 ψ sin ω t .
( I - - I + ) ( I - + I + ) - 1 = - a 32 a 11 - 1 ( a 11 2 - a 14 2 ) ( a 22 2 + a 32 2 ) - 1 = ( 1 + R 2 A 2 ) ( 1 + R 4 A 4 - 2 R 2 A 2 cosh 4 ψ ) sin 2 θ ( 1 - R 2 A 2 ) ( 1 + R 4 A 4 - 2 R 2 A 2 cos 4 θ ) cosh 2 ψ .
R 4 = 0 , A 2 = A 3 = 1 , θ 3 = ψ 2 = ψ 3 = 0.
( a 14 / a 11 ) = - [ 1 + R 1 ( 1 - 2 R 2 ) R 3 A 1 2 + R 2 ( R 1 A 1 2 - R 3 ) ] × [ 1 - R 1 ( 1 - 2 R 2 ) R 3 A 1 - R 2 ( R 1 A 1 + R 3 ) ] - 1 × tanh 2 ψ 1 .
R 1 = R 4 , R 2 = R 3 , A 1 = A 3 , A 2 = 1 , θ 1 = θ 3 , ψ 1 = ψ 3 , ψ 2 = 0.
( a 14 / a 11 ) = - [ 1 + R 1 2 ( 1 - 4 R 2 + 3 R 2 2 ) A 1 4 - R 2 2 ] sinh 4 ψ 1 × { [ 1 - R 1 2 ( 1 - 4 R 2 + 3 R 2 2 ) A 1 4 - R 2 2 ] cosh 4 ψ 1 - 4 R 1 R 2 ( 1 - R 2 ) A 1 2 } - 1 .
( a 14 / a 11 ) - 4 ψ 1 [ 1 + R 1 2 ( 1 - 4 R 2 + 3 R 2 2 ) A 1 4 - R 2 2 ] × [ 1 - R 1 2 ( 1 - 4 R 2 + 3 R 2 2 ) A 1 4 - R 2 2 - 4 R 1 R 2 ( 1 - R 2 ) A 1 2 ] - 1 .