Abstract

The performance of a magneto-optic readout system can be degraded by a phase shift between two basic polarization components of a readout beam. One source of this phase shift is undesired birefringence in the substrate of an optical disk. The effect of disk birefringence on the signal-to-noise ratio of a differential magneto-optic readout is evaluated. It is shown that this effect may be represented by two parameters J2 and J3 that depend only on the birefringence and the numerical aperture of a beam illuminating the disk. Previous papers on this subject either presented involved, open-form expressions for the differential magneto-optic readout or assumed that the beam illuminating the disk is collimated. Here simple closed-form formulas are derived by using parameters J2 and J3. The effects of system conditions that can interact with the disk birefringence, such as an imbalance between the differential data detectors, are also considered.

© 1992 Optical Society of America

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References

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  1. D. Treves, D. S. Bloomberg, “Effect of birefringence on optical memory systems,” in Optical Mass Data Storage II, R. P. Freese, A. A. Jamberdino, M. de Haan, eds. Proc. Soc. Photo-Opt. Instrum. Eng.695, 262–269 (1986).
  2. A. B. Marchant, “Retardation effects in magneto-optic readout,” in Optical Mass Data Storage II, R. P. Freese, A. A. Jamberdino, M. de Haan, eds. Proc. Soc. Photo-Opt. Instrum. Eng.695, 270–276 (1986).
  3. W. A. Challener, T. A. Rinehart, “Jones matrix analysis of magneto-optical media and read-back systems,” Appl. Opt. 26, 3974–3980 (1987).
    [CrossRef] [PubMed]
  4. T. Toda, K. Shigematsu, M. Ojima, M. Yoshihiro, “Analysis of signal-to-noise ratio in magnetooptical disk using a polarization simulator,” Electron. Commun. Jpn. Part 2 72(1), 49–57 (1989).
  5. A. Yoshizawa, N. Matsubayashi, “Analyses of optical anisotropy of PC substrate for M-O disks and its effect on CNR,” in Optical Mass Data Storage II, R. P. Freese, A. A. Jamberdino, M. de Haan, eds. Proc. Soc. Photo-Opt. Instrum. Eng.695, 91–98 (1986).
  6. W. Siebourg, H. Schmid, F. M. Rateike, S. Anders, U. Grigo, “Birefringence—an important property of plastic substrates for magneto-optical storage disks,” Polym. Eng. Sci. 30, 1133–1139 (1990).
    [CrossRef]
  7. B. I. Finkelstein, W. C. Williams, “Noise sources in magneto-optical recording,” in Optical Data Storage, Vol. 10 of OSA 1987 Technical Digest Series (Optical Society of America, Washington, D.C., 1987), pp. 14–17.
  8. P. Spano, S. Piazzolla, M. Tamburrini, “Theory of noise in semiconductor lasers in the presence of optical feedback,” IEEE J. Quantum Electron. QE-20, 350–357 (1984).
    [CrossRef]
  9. K. E. Stubkjaer, M. B. Small, “Noise properties of semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-20, 472–478 (1984).
    [CrossRef]

1990 (1)

W. Siebourg, H. Schmid, F. M. Rateike, S. Anders, U. Grigo, “Birefringence—an important property of plastic substrates for magneto-optical storage disks,” Polym. Eng. Sci. 30, 1133–1139 (1990).
[CrossRef]

1989 (1)

T. Toda, K. Shigematsu, M. Ojima, M. Yoshihiro, “Analysis of signal-to-noise ratio in magnetooptical disk using a polarization simulator,” Electron. Commun. Jpn. Part 2 72(1), 49–57 (1989).

1987 (1)

1984 (2)

P. Spano, S. Piazzolla, M. Tamburrini, “Theory of noise in semiconductor lasers in the presence of optical feedback,” IEEE J. Quantum Electron. QE-20, 350–357 (1984).
[CrossRef]

K. E. Stubkjaer, M. B. Small, “Noise properties of semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-20, 472–478 (1984).
[CrossRef]

Anders, S.

W. Siebourg, H. Schmid, F. M. Rateike, S. Anders, U. Grigo, “Birefringence—an important property of plastic substrates for magneto-optical storage disks,” Polym. Eng. Sci. 30, 1133–1139 (1990).
[CrossRef]

Bloomberg, D. S.

D. Treves, D. S. Bloomberg, “Effect of birefringence on optical memory systems,” in Optical Mass Data Storage II, R. P. Freese, A. A. Jamberdino, M. de Haan, eds. Proc. Soc. Photo-Opt. Instrum. Eng.695, 262–269 (1986).

Challener, W. A.

Finkelstein, B. I.

B. I. Finkelstein, W. C. Williams, “Noise sources in magneto-optical recording,” in Optical Data Storage, Vol. 10 of OSA 1987 Technical Digest Series (Optical Society of America, Washington, D.C., 1987), pp. 14–17.

Grigo, U.

W. Siebourg, H. Schmid, F. M. Rateike, S. Anders, U. Grigo, “Birefringence—an important property of plastic substrates for magneto-optical storage disks,” Polym. Eng. Sci. 30, 1133–1139 (1990).
[CrossRef]

Marchant, A. B.

A. B. Marchant, “Retardation effects in magneto-optic readout,” in Optical Mass Data Storage II, R. P. Freese, A. A. Jamberdino, M. de Haan, eds. Proc. Soc. Photo-Opt. Instrum. Eng.695, 270–276 (1986).

Matsubayashi, N.

A. Yoshizawa, N. Matsubayashi, “Analyses of optical anisotropy of PC substrate for M-O disks and its effect on CNR,” in Optical Mass Data Storage II, R. P. Freese, A. A. Jamberdino, M. de Haan, eds. Proc. Soc. Photo-Opt. Instrum. Eng.695, 91–98 (1986).

Ojima, M.

T. Toda, K. Shigematsu, M. Ojima, M. Yoshihiro, “Analysis of signal-to-noise ratio in magnetooptical disk using a polarization simulator,” Electron. Commun. Jpn. Part 2 72(1), 49–57 (1989).

Piazzolla, S.

P. Spano, S. Piazzolla, M. Tamburrini, “Theory of noise in semiconductor lasers in the presence of optical feedback,” IEEE J. Quantum Electron. QE-20, 350–357 (1984).
[CrossRef]

Rateike, F. M.

W. Siebourg, H. Schmid, F. M. Rateike, S. Anders, U. Grigo, “Birefringence—an important property of plastic substrates for magneto-optical storage disks,” Polym. Eng. Sci. 30, 1133–1139 (1990).
[CrossRef]

Rinehart, T. A.

Schmid, H.

W. Siebourg, H. Schmid, F. M. Rateike, S. Anders, U. Grigo, “Birefringence—an important property of plastic substrates for magneto-optical storage disks,” Polym. Eng. Sci. 30, 1133–1139 (1990).
[CrossRef]

Shigematsu, K.

T. Toda, K. Shigematsu, M. Ojima, M. Yoshihiro, “Analysis of signal-to-noise ratio in magnetooptical disk using a polarization simulator,” Electron. Commun. Jpn. Part 2 72(1), 49–57 (1989).

Siebourg, W.

W. Siebourg, H. Schmid, F. M. Rateike, S. Anders, U. Grigo, “Birefringence—an important property of plastic substrates for magneto-optical storage disks,” Polym. Eng. Sci. 30, 1133–1139 (1990).
[CrossRef]

Small, M. B.

K. E. Stubkjaer, M. B. Small, “Noise properties of semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-20, 472–478 (1984).
[CrossRef]

Spano, P.

P. Spano, S. Piazzolla, M. Tamburrini, “Theory of noise in semiconductor lasers in the presence of optical feedback,” IEEE J. Quantum Electron. QE-20, 350–357 (1984).
[CrossRef]

Stubkjaer, K. E.

K. E. Stubkjaer, M. B. Small, “Noise properties of semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-20, 472–478 (1984).
[CrossRef]

Tamburrini, M.

P. Spano, S. Piazzolla, M. Tamburrini, “Theory of noise in semiconductor lasers in the presence of optical feedback,” IEEE J. Quantum Electron. QE-20, 350–357 (1984).
[CrossRef]

Toda, T.

T. Toda, K. Shigematsu, M. Ojima, M. Yoshihiro, “Analysis of signal-to-noise ratio in magnetooptical disk using a polarization simulator,” Electron. Commun. Jpn. Part 2 72(1), 49–57 (1989).

Treves, D.

D. Treves, D. S. Bloomberg, “Effect of birefringence on optical memory systems,” in Optical Mass Data Storage II, R. P. Freese, A. A. Jamberdino, M. de Haan, eds. Proc. Soc. Photo-Opt. Instrum. Eng.695, 262–269 (1986).

Williams, W. C.

B. I. Finkelstein, W. C. Williams, “Noise sources in magneto-optical recording,” in Optical Data Storage, Vol. 10 of OSA 1987 Technical Digest Series (Optical Society of America, Washington, D.C., 1987), pp. 14–17.

Yoshihiro, M.

T. Toda, K. Shigematsu, M. Ojima, M. Yoshihiro, “Analysis of signal-to-noise ratio in magnetooptical disk using a polarization simulator,” Electron. Commun. Jpn. Part 2 72(1), 49–57 (1989).

Yoshizawa, A.

A. Yoshizawa, N. Matsubayashi, “Analyses of optical anisotropy of PC substrate for M-O disks and its effect on CNR,” in Optical Mass Data Storage II, R. P. Freese, A. A. Jamberdino, M. de Haan, eds. Proc. Soc. Photo-Opt. Instrum. Eng.695, 91–98 (1986).

Appl. Opt. (1)

Electron. Commun. Jpn. Part 2 (1)

T. Toda, K. Shigematsu, M. Ojima, M. Yoshihiro, “Analysis of signal-to-noise ratio in magnetooptical disk using a polarization simulator,” Electron. Commun. Jpn. Part 2 72(1), 49–57 (1989).

IEEE J. Quantum Electron. (2)

P. Spano, S. Piazzolla, M. Tamburrini, “Theory of noise in semiconductor lasers in the presence of optical feedback,” IEEE J. Quantum Electron. QE-20, 350–357 (1984).
[CrossRef]

K. E. Stubkjaer, M. B. Small, “Noise properties of semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-20, 472–478 (1984).
[CrossRef]

Polym. Eng. Sci. (1)

W. Siebourg, H. Schmid, F. M. Rateike, S. Anders, U. Grigo, “Birefringence—an important property of plastic substrates for magneto-optical storage disks,” Polym. Eng. Sci. 30, 1133–1139 (1990).
[CrossRef]

Other (4)

B. I. Finkelstein, W. C. Williams, “Noise sources in magneto-optical recording,” in Optical Data Storage, Vol. 10 of OSA 1987 Technical Digest Series (Optical Society of America, Washington, D.C., 1987), pp. 14–17.

A. Yoshizawa, N. Matsubayashi, “Analyses of optical anisotropy of PC substrate for M-O disks and its effect on CNR,” in Optical Mass Data Storage II, R. P. Freese, A. A. Jamberdino, M. de Haan, eds. Proc. Soc. Photo-Opt. Instrum. Eng.695, 91–98 (1986).

D. Treves, D. S. Bloomberg, “Effect of birefringence on optical memory systems,” in Optical Mass Data Storage II, R. P. Freese, A. A. Jamberdino, M. de Haan, eds. Proc. Soc. Photo-Opt. Instrum. Eng.695, 262–269 (1986).

A. B. Marchant, “Retardation effects in magneto-optic readout,” in Optical Mass Data Storage II, R. P. Freese, A. A. Jamberdino, M. de Haan, eds. Proc. Soc. Photo-Opt. Instrum. Eng.695, 270–276 (1986).

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Figures (6)

Fig. 1
Fig. 1

Schematic of the differential MO readout system.

Fig. 2
Fig. 2

Dependence of the differential MO signal normalized amplitude on the disk vertical birefringence. The vertical birefringence is defined by the difference n z n o of the principal indices of refraction. The variable parameter is the NA of the objective lens.

Fig. 3
Fig. 3

Dependence of the shot noise (normalized by the electronic noise) on the disk vertical birefringence. The vertical birefringence is defined by the difference n z n o of the principal indices of refraction. The variable parameter is the transmittance T X of the x polarization (the polarization that is parallel to the laser diode junction) between the disk and the PBS. The transmittance T Y is assumed to be 1.0. The NA of the objective lens is 0.55.

Fig. 4
Fig. 4

Dependence of the differential MO signal (normalized by the common noise) on the angular misalignment of the PBS. The PBS angle is 45° for perfect alignment. The variable parameters are the disk vertical birefringence n z n o and the transmittance T X , of the x polarization (the polarization that is parallel to the laser diode junction) between the disk and the PBS. The transmittance T Y is assumed to be 1.0. The NA of the objective lens is 0.55.

Fig. 5
Fig. 5

Dependence of the differential MO signal (which is normalized by the common noise) on the PBS transmittance T p for the p polarization. The PBS reflectance R s for the s polarization is assumed to be 1.0. The variable parameters are the disk vertical birefringence n z n o and the transmittance T X of the x polarization (the polarization that is parallel to the laser diode junction) between the disk and the PBS. The transmittance T Y is assumed to be 1.0. The NA of the objective lens is 0.55.

Fig. 6
Fig. 6

The differential MO signal-to-total-noise ratio as a function of the disk birefringence n z n o for a typical MO readout system with a NA of 0.55. The variable parameters are the angular positioning Ω of the PBS and the PBS transmittance T p for the p polarization. The PBS reflectance R s is 1.0 and the transmittances T X and T Y between the disk and the PBS are 0.1 and 1.0, respectively.

Equations (62)

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δ ( ) = ( k h / n z ) [ n z ( n o 2 - sin 2 ) 1 / 2 - n o ( n z 2 - sin 2 ) 1 / 2 ] ,
D ( ) = r ( ) { exp [ i δ ( ) ] Θ ( ) exp [ i τ ( ) ] - Θ ( ) exp [ i τ ( ) ] exp [ - i δ ( ) ] } ,
D ( ) = L ( ) K ( ) L ( ) ,
K ( ) = r ( ) { 1 Θ ( ) exp [ i τ ( ) ] - Θ ( ) exp [ i τ ( ) ] 1 }
L ( ) = { exp [ i δ ( ) / 2 ] 0 0 exp [ - i δ ( ) / 2 ] }
R ( ϕ ) = [ cos ( ϕ ) - sin ( ϕ ) sin ( ϕ ) cos ( ϕ ) ] .
D ( , ϕ ) = R ( ϕ ) D ( ) R T ( ϕ ) .
D ( , ϕ ) = r ( ) [ exp ( i δ ) cos 2 ( ϕ ) + exp ( - i δ ) sin 2 ( ϕ ) i sin ( δ ) sin ( 2 ϕ ) + Θ exp ( i τ ) i sin ( δ ) sin ( 2 ϕ ) - Θ exp ( i τ ) exp ( i δ ) sin 2 ( ϕ ) + exp ( - i δ ) cos 2 ( ϕ ) ] ,
[ x ( , ϕ ) ,             y ( , ϕ ) ] = [ a ( , ϕ ) ,             0 ] D ( , ϕ ) ,
[ x ( , ϕ ) ,             y ( , ϕ ) ] = [ a ( , ϕ ) ,             0 ] D ( , ϕ ) M ,
M = [ t X exp ( i σ / 2 ) 0 0 t Y exp ( - i σ / 2 ) ] ,
C 1 = [ cos ( Ω ) - sin ( Ω ) sin ( Ω ) cos ( Ω ) ] [ t p 0 0 q 1 t p ] = t p [ cos ( Ω ) - q 1 sin ( Ω ) sin ( Ω ) q 1 cos ( Ω ) ] ,
C 2 = [ cos ( Ω ) - sin ( Ω ) sin ( Ω ) cos ( Ω ) ] [ q 2 r s 0 0 r s ] = r s [ q 2 cos ( Ω ) - sin ( Ω ) q 2 sin ( Ω ) cos ( Ω ) ] .
[ a 1 ( , ϕ ) ,             b 1 ( , ϕ ) ] = [ a ( , ϕ ) ,             0 ] D ( , ϕ ) M C 1 ,
[ a 1 ( , ϕ ) ,             b 1 ( , ϕ ) ] = [ a ( , ϕ ) ,             0 ] D ( , ϕ ) M C 1 ,
a 1 ( , ϕ ) = a ( , ϕ ) r ( ) t p exp ( - i σ / 2 ) × [ t X cos ( Ω ) ( exp { i [ δ ( ) + σ ] } cos 2 ( ϕ ) + exp { - i [ δ ( ) - σ ] } sin 2 ( ϕ ) ) + t Y sin ( Ω ) × { Θ ( ) exp [ i τ ( ) ] + i sin [ δ ( ) ] sin ( 2 ϕ ) } ]
a 2 ( , ϕ ) = a ( , ϕ ) r ( ) r s exp ( - i σ / 2 ) × [ - t X sin ( Ω ) ( exp { i [ δ ( ) + σ ] } cos 2 ( ϕ ) + exp { - i [ δ ( ) - σ ] } sin 2 ( ϕ ) ) + t Y cos ( Ω ) × { Θ ( ) exp [ i τ ( ) ] + i sin ( δ ) sin ( 2 ϕ ) } ] ,
b 1 ( , ϕ ) = a ( , ϕ ) r ( ) t p q 1 exp ( - i σ / 2 ) × [ - t X sin ( Ω ) ( exp { i [ δ ( ) + σ ] } cos 2 ( ϕ ) + exp { - i [ δ ( ) - σ ] } sin 2 ( ϕ ) ) + t Y cos ( Ω ) × { Θ ( ) exp [ i τ ( ) ] + i sin ( δ ) sin ( 2 ϕ ) } ] ,
b 2 ( , ϕ ) = a ( , ϕ ) r ( ) r s q 2 exp ( - i σ / 2 ) × [ t X cos ( Ω ) ( exp { i [ δ ( ) + σ ] } cos 2 ( ϕ ) + exp { - i [ δ ( ) - σ ] } sin 2 ( ϕ ) ) + t Y sin ( Ω ) × { Θ ( ) exp [ i τ ( ) ] + i sin ( δ ) sin ( 2 ϕ ) } ] .
I 1 ( , ϕ ) = a 1 ( , ϕ ) a 1 * ( , ϕ ) + b 1 ( , ϕ ) b 1 * ( , ϕ ) ,
I 2 ( , ϕ ) = a 2 ( , ϕ ) a 2 * ( , ϕ ) + b 2 ( , ϕ ) b 2 * ( , ϕ ) .
I 1 ( , ϕ ) = I ( , ϕ ) R ( ) T p × { A ( , ϕ ) T X [ cos 2 ( Ω ) + Q 1 sin 2 ( Ω ) ] + B ( , ϕ ) T Y [ sin 2 ( Ω ) + Q 1 cos 2 ( Ω ) ] - C ( , ϕ ) ( T X T Y ) 1 / 2 sin ( 2 Ω ) ( 1 - Q 1 ) } ,
I 2 ( , ϕ ) = I ( , ϕ ) R ( ) R s × { A ( , ϕ ) T X [ sin 2 ( Ω ) + Q 2 cos 2 ( Ω ) ] + B ( , ϕ ) T Y [ cos 2 ( Ω ) + Q 2 sin 2 ( Ω ) ] + C ( , ϕ ) ( T X T Y ) 1 / 2 sin ( 2 Ω ) ( 1 - Q 2 ) } ,
A ( , ϕ ) = sin 4 ( ϕ ) + 2 cos [ 2 δ ( ) ] sin 2 ( ϕ ) cos 2 ( ϕ ) + cos 4 ( ϕ ) , B ( , ϕ ) = sin 2 [ δ ( ) ] sin 2 ( 2 ϕ ) + 2 Θ ( ) × sin [ δ ( ) ] sin [ τ ( ) ] sin ( 2 ϕ ) + Θ 2 ( ) , C ( , ϕ ) = - sin [ δ ( ) ] sin ( 2 ϕ ) { cos [ δ ( ) ] sin ( σ ) + sin [ δ ( ) ] cos ( σ ) cos ( 2 ϕ ) } - Θ ( ) { cos [ δ ( ) ] cos [ τ ( ) - σ ] + sin [ δ ( ) ] sin [ τ ( ) - σ ] cos ( 2 ϕ ) } .
v ( , ϕ ) = g 2 I 2 ( , ϕ ) - g 1 I 1 ( , ϕ ) ,
v ( , ϕ ) = I ( , ϕ ) R ( ) [ A ( , ϕ ) α + B ( , ϕ ) β + C ( , ϕ ) Γ ] ,
α = T X { g 2 R s [ sin 2 ( Ω ) + Q 2 cos 2 ( Ω ) ] - g 1 T p [ cos 2 ( Ω ) + Q 1 sin 2 ( Ω ) ] } ,
β = T Y { g 2 R s [ cos 2 ( Ω ) + Q 2 sin 2 ( Ω ) ] - g 1 T p [ sin 2 ( Ω ) + Q 1 cos 2 ( Ω ) ] } ,
Γ = ( T X T Y ) 1 / 2 sin ( 2 Ω ) [ g 2 R s ( 1 - Q 2 ) + g 1 T p ( 1 - Q 1 ) ] .
V = ( over circular pupil ) v ( , ϕ ) sin ( ) d [ sin ( ) ] d ϕ = 1 / 2 0 NA 0 2 π v ( , ϕ ) sin ( 2 ) d d ϕ ,
v ( t , , ϕ ) = I ( , ϕ ) R ( ) [ A ( , ϕ ) α + B ( t , , ϕ ) β + C ( t , , ϕ ) Γ ] .
V ( t ) = P R { α ( 1 - J 2 / 2 ) + β [ Θ 2 ( t ) + J 2 / 2 ] - Γ Θ ( t ) J 3 cos ( τ - σ ) } ,
J 2 = 1 / ( NA ) 2 0 NA sin 2 [ δ ( ) ] sin ( 2 ) d ,
J 3 = 1 / ( NA ) 2 0 NA cos [ δ ( ) ] sin ( 2 ) d .
V S ( t ) = P R { α S ( 1 - J 2 / 2 ) + β S [ Θ 2 ( t ) + J 2 / 2 ] - Γ S Θ ( t ) J 3 cos ( τ - σ ) } ,
α S = T X { g 2 R s [ sin 2 ( Ω ) + Q 2 cos 2 ( Ω ) ] + g 1 T p [ cos 2 ( Ω ) + Q 1 sin 2 ( Ω ) ] } / 2 ,
β S = T Y { g 2 R s [ cos 2 ( Ω ) + Q 2 sin 2 ( Ω ) ] + g 1 T p [ sin 2 ( Ω ) + Q 1 cos 2 ( Ω ) ] } / 2.
a = P R Γ Θ J 3 cos ( τ - σ ) ,
b = P R [ ( α + β Θ 2 ) + J 2 ( β - α ) / 2 ] .
a S = P R ( Γ / 2 ) Θ J 3 cos ( τ - σ ) ,
b S = P R [ ( α S + β S Θ 2 ) + J 2 ( β S - α S ) / 2 ] .
R R μ G ,             Θ Θ μ S ,
N L = RIN L ( b 2 + a 2 / 2 ) BW ,
N D = RIN D ( b 2 + a 2 / 2 ) BW .
N SH = 4 q ( b S 2 + a S 2 / 2 ) 1 / 2 BW .
N E = 2 ( R NEP ) 2 BW ,
1 / SNR = 4 { ( RIN / 4 ) [ 1 + 2 ( b / a ) 2 ] + ( q / a ) [ ½ + 4 ( b S / a ) 2 ] 1 / 2 + ( R NEP / a ) 2 } BW ,
δ o = k n o h cos ( o ) ,
δ e = k n e h cos ( e ) .
δ = k h [ n o cos ( o ) - n e cos ( e ) ] .
s x 2 ( 1 / n 2 - 1 / n y 2 ) ( 1 / n 2 - 1 / n z 2 ) + s y 2 ( 1 / n 2 - 1 / n x 2 ) ( 1 / n 2 - 1 / n z 2 ) + s z 2 ( 1 / n 2 - 1 / n x 2 ) ( 1 / n 2 - 1 / n y 2 ) = 0 ,
n 2 = n o 2 ,
s x 2 ( 1 / n 2 - 1 / n z 2 ) + s y 2 ( 1 / n 2 - 1 / n z 2 ) + s z 2 ( 1 / n 2 - 1 / n o 2 ) = 0.
s x 2 + s y 2 = sin 2 e = 1 - cos 2 e ,             s z 2 = 1 - sin 2 e = cos 2 e .
sin 2 e = ( n z 2 / n e 2 ) ( n e 2 - n o 2 ) / ( n z 2 - n o 2 ) ,
cos 2 e = ( n o 2 / n e 2 ) ( n z 2 - n e 2 ) / ( n z 2 - n o 2 ) .
sin ( ) = n e sin ( e ) ,
sin 2 = n z 2 ( n e 2 - n o 2 ) / ( n z 2 - n o 2 ) ;
n e cos ( e ) = ( n o / n z ) ( n z 2 - sin 2 ) 1 / 2 .
sin ( ) = n o sin ( o ) ,
n o cos ( o ) = ( n o 2 - sin 2 ) 1 / 2 .
δ ( ) = ( k h / n z ) [ n z ( n o 2 - sin 2 ) 1 / 2 - n o ( n z 2 - sin 2 ) 1 / 2 ] .

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