Abstract

Crosstalk in optical waveguide switches/modulators caused by having two polarizations simultaneously present is studied in this work. This situation is expected to arise when fiber optical transmission lines are coupled to integrated optical circuits. Modulator/switch performance is found to be strongly affected by having two propagating polarizations. Conditions on device design are found that minimize the polarization sensitivity of switches/modulators, making them suitable for use with fiber optic transmission lines.

© 1976 Optical Society of America

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References

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  1. See, for example, I. P. Kaminow, IEEE Trans. Microwave Theory Tech. MTT-23, 57 (1975).
  2. F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron. QE-8, 222 (1972).
  3. J. H. Hammer, D. J. Channin, M. T. Duffy, Appl. Phys. Lett. 23, 176 (1973).
  4. P. K. Tien, Appl. Phys. Lett. 25, 563 (1974).
  5. I. P. Kaminow, L. W. Stulz, IEEE J. Quantum Electron. QE-11, 633 (1975); C. S. Tsai, P. Saunier, Appl. Phys. Lett. 27, 248 (1975).
  6. J. C. Campbell, F. A. Blum, D. W. Shaw, K. L. Lawley, Appl. Phys. Lett. 27, 202 (1975).
  7. M. Papuchon, Y. Combemale, X. Mathieu, D. Ostrowski, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).
  8. H. Kogelnik, R. V. Schmidt, in Digest OSA Integrated Optics Topical Meeting, Salt Lake City (Optical Society of America, Washington, D.C., 1976).
  9. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).
  10. M. G. Wilson, G. A. Teh, IEEE Trans. Microwave Theory Tech. MTT-23, 85 (1975); R. B. Smith, Electron. Lett. 9, 453 (1973).
  11. W. K. Burns, A. F. Milton, A. B. Lee, E. J. West, Appl. Opt. 15, 000 (June1976).
  12. R. Steinberg, T. G. Giallorenzi (to be published).
  13. D. F. Nelson, J. McKenna, J. Appl. Phys. 38, 4057 (1967).
  14. D. P. Gia Russo, J. H. Harris, J. Opt. Soc. Am. 63, 138 (1973); W. K. Burns, J. Warner, J. Opt. Soc. Am. 64, 441 (1974).
  15. A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
  16. J. A. Arnaud, Bell Syst. Tech. J. 53, 217 (1974).
  17. H. Kogelnik, in Integrated Optics, T. Tamir, Ed. (Springer, Berlin, 1975).
  18. W. E. Martin, Appl. Phys. Lett. 26, 562 (1975).
  19. J. N. Polky, J. Harris, Appl. Phys. Lett. 21, 307 (1972).
  20. R. W. Damon, W. T. Maloney, D. H. McMahon, in Physical Acoustics, W. P. Mason, R. N. Thurston, Eds. (Academic, New York, 1970), Vol. 7.

1976 (1)

W. K. Burns, A. F. Milton, A. B. Lee, E. J. West, Appl. Opt. 15, 000 (June1976).

1975 (6)

M. G. Wilson, G. A. Teh, IEEE Trans. Microwave Theory Tech. MTT-23, 85 (1975); R. B. Smith, Electron. Lett. 9, 453 (1973).

W. E. Martin, Appl. Phys. Lett. 26, 562 (1975).

See, for example, I. P. Kaminow, IEEE Trans. Microwave Theory Tech. MTT-23, 57 (1975).

I. P. Kaminow, L. W. Stulz, IEEE J. Quantum Electron. QE-11, 633 (1975); C. S. Tsai, P. Saunier, Appl. Phys. Lett. 27, 248 (1975).

J. C. Campbell, F. A. Blum, D. W. Shaw, K. L. Lawley, Appl. Phys. Lett. 27, 202 (1975).

M. Papuchon, Y. Combemale, X. Mathieu, D. Ostrowski, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).

1974 (2)

P. K. Tien, Appl. Phys. Lett. 25, 563 (1974).

J. A. Arnaud, Bell Syst. Tech. J. 53, 217 (1974).

1973 (3)

D. P. Gia Russo, J. H. Harris, J. Opt. Soc. Am. 63, 138 (1973); W. K. Burns, J. Warner, J. Opt. Soc. Am. 64, 441 (1974).

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).

J. H. Hammer, D. J. Channin, M. T. Duffy, Appl. Phys. Lett. 23, 176 (1973).

1972 (2)

F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron. QE-8, 222 (1972).

J. N. Polky, J. Harris, Appl. Phys. Lett. 21, 307 (1972).

1969 (1)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

1967 (1)

D. F. Nelson, J. McKenna, J. Appl. Phys. 38, 4057 (1967).

Arnaud, J. A.

J. A. Arnaud, Bell Syst. Tech. J. 53, 217 (1974).

Blum, F. A.

J. C. Campbell, F. A. Blum, D. W. Shaw, K. L. Lawley, Appl. Phys. Lett. 27, 202 (1975).

Borrelli, N. F.

F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron. QE-8, 222 (1972).

Burns, W. K.

W. K. Burns, A. F. Milton, A. B. Lee, E. J. West, Appl. Opt. 15, 000 (June1976).

Campbell, J. C.

J. C. Campbell, F. A. Blum, D. W. Shaw, K. L. Lawley, Appl. Phys. Lett. 27, 202 (1975).

Channin, D. J.

J. H. Hammer, D. J. Channin, M. T. Duffy, Appl. Phys. Lett. 23, 176 (1973).

Combemale, Y.

M. Papuchon, Y. Combemale, X. Mathieu, D. Ostrowski, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).

Damon, R. W.

R. W. Damon, W. T. Maloney, D. H. McMahon, in Physical Acoustics, W. P. Mason, R. N. Thurston, Eds. (Academic, New York, 1970), Vol. 7.

Duffy, M. T.

J. H. Hammer, D. J. Channin, M. T. Duffy, Appl. Phys. Lett. 23, 176 (1973).

Gia Russo, D. P.

Giallorenzi, T. G.

R. Steinberg, T. G. Giallorenzi (to be published).

Hammer, J. H.

J. H. Hammer, D. J. Channin, M. T. Duffy, Appl. Phys. Lett. 23, 176 (1973).

Harris, J.

J. N. Polky, J. Harris, Appl. Phys. Lett. 21, 307 (1972).

Harris, J. H.

Kaminow, I. P.

I. P. Kaminow, L. W. Stulz, IEEE J. Quantum Electron. QE-11, 633 (1975); C. S. Tsai, P. Saunier, Appl. Phys. Lett. 27, 248 (1975).

See, for example, I. P. Kaminow, IEEE Trans. Microwave Theory Tech. MTT-23, 57 (1975).

Kapron, F. P.

F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron. QE-8, 222 (1972).

Keck, D. B.

F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron. QE-8, 222 (1972).

Kogelnik, H.

H. Kogelnik, R. V. Schmidt, in Digest OSA Integrated Optics Topical Meeting, Salt Lake City (Optical Society of America, Washington, D.C., 1976).

H. Kogelnik, in Integrated Optics, T. Tamir, Ed. (Springer, Berlin, 1975).

Lawley, K. L.

J. C. Campbell, F. A. Blum, D. W. Shaw, K. L. Lawley, Appl. Phys. Lett. 27, 202 (1975).

Lee, A. B.

W. K. Burns, A. F. Milton, A. B. Lee, E. J. West, Appl. Opt. 15, 000 (June1976).

Maloney, W. T.

R. W. Damon, W. T. Maloney, D. H. McMahon, in Physical Acoustics, W. P. Mason, R. N. Thurston, Eds. (Academic, New York, 1970), Vol. 7.

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

Martin, W. E.

W. E. Martin, Appl. Phys. Lett. 26, 562 (1975).

Mathieu, X.

M. Papuchon, Y. Combemale, X. Mathieu, D. Ostrowski, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).

McKenna, J.

D. F. Nelson, J. McKenna, J. Appl. Phys. 38, 4057 (1967).

McMahon, D. H.

R. W. Damon, W. T. Maloney, D. H. McMahon, in Physical Acoustics, W. P. Mason, R. N. Thurston, Eds. (Academic, New York, 1970), Vol. 7.

Milton, A. F.

W. K. Burns, A. F. Milton, A. B. Lee, E. J. West, Appl. Opt. 15, 000 (June1976).

Nelson, D. F.

D. F. Nelson, J. McKenna, J. Appl. Phys. 38, 4057 (1967).

Ostrowski, D.

M. Papuchon, Y. Combemale, X. Mathieu, D. Ostrowski, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).

Papuchon, M.

M. Papuchon, Y. Combemale, X. Mathieu, D. Ostrowski, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).

Polky, J. N.

J. N. Polky, J. Harris, Appl. Phys. Lett. 21, 307 (1972).

Reiber, L.

M. Papuchon, Y. Combemale, X. Mathieu, D. Ostrowski, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).

Roy, A. M.

M. Papuchon, Y. Combemale, X. Mathieu, D. Ostrowski, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).

Schmidt, R. V.

H. Kogelnik, R. V. Schmidt, in Digest OSA Integrated Optics Topical Meeting, Salt Lake City (Optical Society of America, Washington, D.C., 1976).

Sejourne, B.

M. Papuchon, Y. Combemale, X. Mathieu, D. Ostrowski, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).

Shaw, D. W.

J. C. Campbell, F. A. Blum, D. W. Shaw, K. L. Lawley, Appl. Phys. Lett. 27, 202 (1975).

Steinberg, R.

R. Steinberg, T. G. Giallorenzi (to be published).

Stulz, L. W.

I. P. Kaminow, L. W. Stulz, IEEE J. Quantum Electron. QE-11, 633 (1975); C. S. Tsai, P. Saunier, Appl. Phys. Lett. 27, 248 (1975).

Teh, G. A.

M. G. Wilson, G. A. Teh, IEEE Trans. Microwave Theory Tech. MTT-23, 85 (1975); R. B. Smith, Electron. Lett. 9, 453 (1973).

Tien, P. K.

P. K. Tien, Appl. Phys. Lett. 25, 563 (1974).

Werner, M.

M. Papuchon, Y. Combemale, X. Mathieu, D. Ostrowski, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).

West, E. J.

W. K. Burns, A. F. Milton, A. B. Lee, E. J. West, Appl. Opt. 15, 000 (June1976).

Wilson, M. G.

M. G. Wilson, G. A. Teh, IEEE Trans. Microwave Theory Tech. MTT-23, 85 (1975); R. B. Smith, Electron. Lett. 9, 453 (1973).

Yariv, A.

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).

Appl. Opt. (1)

W. K. Burns, A. F. Milton, A. B. Lee, E. J. West, Appl. Opt. 15, 000 (June1976).

Appl. Phys. Lett. (6)

W. E. Martin, Appl. Phys. Lett. 26, 562 (1975).

J. N. Polky, J. Harris, Appl. Phys. Lett. 21, 307 (1972).

J. H. Hammer, D. J. Channin, M. T. Duffy, Appl. Phys. Lett. 23, 176 (1973).

P. K. Tien, Appl. Phys. Lett. 25, 563 (1974).

J. C. Campbell, F. A. Blum, D. W. Shaw, K. L. Lawley, Appl. Phys. Lett. 27, 202 (1975).

M. Papuchon, Y. Combemale, X. Mathieu, D. Ostrowski, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).

Bell Syst. Tech. J. (2)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

J. A. Arnaud, Bell Syst. Tech. J. 53, 217 (1974).

IEEE J. Quantum Electron. (3)

I. P. Kaminow, L. W. Stulz, IEEE J. Quantum Electron. QE-11, 633 (1975); C. S. Tsai, P. Saunier, Appl. Phys. Lett. 27, 248 (1975).

F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron. QE-8, 222 (1972).

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).

IEEE Trans. Microwave Theory Tech. (2)

See, for example, I. P. Kaminow, IEEE Trans. Microwave Theory Tech. MTT-23, 57 (1975).

M. G. Wilson, G. A. Teh, IEEE Trans. Microwave Theory Tech. MTT-23, 85 (1975); R. B. Smith, Electron. Lett. 9, 453 (1973).

J. Appl. Phys. (1)

D. F. Nelson, J. McKenna, J. Appl. Phys. 38, 4057 (1967).

J. Opt. Soc. Am. (1)

Other (4)

R. Steinberg, T. G. Giallorenzi (to be published).

H. Kogelnik, in Integrated Optics, T. Tamir, Ed. (Springer, Berlin, 1975).

R. W. Damon, W. T. Maloney, D. H. McMahon, in Physical Acoustics, W. P. Mason, R. N. Thurston, Eds. (Academic, New York, 1970), Vol. 7.

H. Kogelnik, R. V. Schmidt, in Digest OSA Integrated Optics Topical Meeting, Salt Lake City (Optical Society of America, Washington, D.C., 1976).

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Figures (14)

Fig. 1
Fig. 1

Commonly used modulator configurations. In type I modulators the coupling and phase shift regions coincide. In type II devices, the channel coupling regions (labeled 3 dB in the figure) are separated from the phase shift region. At the bottom of the figure are three alternative schemes for realizing the 3-dB couplers needed to construct the type II device.

Fig. 2
Fig. 2

Type I interference modulators—off-state crosstalk. The coefficient κTE (κTM) characterizes the coupling strength between the lowest order x-polarized (y-polarized) modes of the two parallel channel waveguides. The crystal angle measures the angular separation between the waveguide axis and the crystallographic c axis (in the xz plane). The index tensors characterizing the medium properties are that of LiNbO3 and out-diffused LiNbO3 for the substrate and guiding regions, respectively. The guide dimensions and separation were taken as a = b = c = 1 μm: (a) ratio of coupling coefficients and interchannel crosstalk for L = LEx [cf. comment above Eq. (6)]; (b) ratio of coupling coefficients and interchannel crosstalk for L = LEy.

Fig. 3
Fig. 3

Type I interference modulators—off-state crosstalk. This figure shows the effect of channel aspect ratio on the off-state crosstalk. The waveguide height and guide-to-guide spacing are held constant, b = c = 1 μm (cf. Fig. 2 insert). The guidewidth 2a is varied from 2 μm up to 20 μm; the corresponding values of aspect ratio are defined as 2a/b. The interaction length L is taken as LEx [cf. comment above Eq. (6)].

Fig. 4
Fig. 4

Voltage-induced changes in the effective indices of the lowest-order Ex and Ey modes for (a) X-cut LiNbO3 and (b) Y-cut LiNbO3; for X-cut, ΔTE = ΔTM at around 14°; for Y-cut, ΔTE = 3ΔTM at around 82°, and ΔTE = ΔTM at around 45°. These points are found to be of special interest. The applied electric field was taken as 105 V/cm. It was found that the induced index changes were linearly proportional to the applied voltage to a high degree of accuracy.

Fig. 5
Fig. 5

Type I interference modulators—on-state crosstalk. When L = LEx the length of the device has been chosen to achieve zero crosstalk for the Ex mode. (L = LEy is analogous.) The resulting crosstalk, plotted in (a) and (b) for X-cut and in (c) and (d) for Y-cut, is due to incomplete switching. It should be noted that LEx is a function of the crystal angle. The applied field was taken as 105 V/cm in the x-direction and the channel dimensions and channel-to-channel separation as a = b = c = 1 μm

Fig. 6
Fig. 6

Type II modulator with modal interference 3-dB couplers-on-state crosstalk: (a) X-cut; (b) Y-cut. Comparing (a) with the off-state crosstalk curves in Figs. 2(a) and 2(b), we see that 14° is a zero-crosstalk orientation for both switching states. a = b = c = 1 μm (cf. Fig. 2 insert).

Fig. 7
Fig. 7

Type II modulator with tapered velocity 3-dB couplers—off-state crosstalk: (a) L = LEx; (b) L = LEy. Crosstalk approaches zero near 14°. It should be noted that a = b corresponds to an aspect ratio of 2.

Fig. 8
Fig. 8

Type II modulator with tapered velocity 3-dB couplers—on-state crosstalk: (a) L = LEy, X-cut; (b) L = LEx, X-cut. As in Fig. 7, crosstalk approaches zero near 14°, making this a potentially promising candidate for achieving low crosstalk in both the on-state (voltage applied) and off-state (no applied voltage) simultaneously.

Fig. 9
Fig. 9

Type II modulator with tapered velocity 3-dB couplers—on-state crosstalk: (a) L = LEx, Y-cut; (b) L = LEy, Y-cut. Although on-state crosstalk minima occur at 45° and 82° in (b) (i.e., for L = LEy), it is seen from Fig. 7(b) that the off-state crosstalk is substantial at these angles. Thus, Y-cut yields poor devices.

Fig. 10
Fig. 10

Type II modulator with branching waveguide 3-dB couplers—off-state crosstalk. The off-state crosstalk is effectively zero for all angles. This plot was calculated from Eq. (18) in the text, using ΔTM. The same result holds true, i.e., off-state crosstalk is still negligible if the device is first optimized for TE-like propagation, ΔTE.

Fig. 11
Fig. 11

Type II modulator with branching waveguide 3-dB couplers—on-state crosstalk: (a) Y-cut; (b) X-cut. On-state minima occur at 14° for X-cut for both ϕTM = (π/2) and ϕTE = (π/2); minima occur at 45° and 82° for Y-cut with ϕTM = (π/2). Because of the insignificant off-state crosstalk (cf. Fig. 10) all the above minima may be useful. However, the Y-cut minima at 82° looks most promising, as it appears to be least sensitive to the crystal angle.

Fig. 12
Fig. 12

Bragg deflector modulators: (a) applied field distribution of the interdigital electrodes, physical configuration of the planar Bragg modulator, diagram for deriving phasematch conditions; (b) angular spread between the TE and TM modes, which are scattered into different angles from the voltage-induced phase grating. This plot was obtained from Eq. (29) for an assumed film thickness of 2 μm, interelectrode spacing of 15 μm, and interaction length L = 3 mm.

Fig. 13
Fig. 13

Bragg modulator—on-state crosstalk. (a) Optimization for TM waves; Y-cut is not plotted because it has high crosstalk for all angles; X-cut has an on-state crosstalk minima at 14°. (b) Optimization for TE waves; although both X-cut and Z-cut have angles with low crosstalk, the minima appear to be fairly sensitive to the crystal angle.

Fig. 14
Fig. 14

Bragg Modulator—off-state crosstalk. The crosstalk is very small for all small-angle orientations of X-cut and Y-cut.

Tables (1)

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Table I GaAs Channel Waveguide: Electrooptically Induced Changes in the Modal Propagation Indices

Equations (33)

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π / 2 L = - d y ( E a y H b z + E a z H b y - E b y H a z - E b z H a y ) ,
L E x = π 2 ( x 1 z 1 x s 2 ) [ β a ( k x a ) 2 ] × ( γ x a 2 ) [ 1 + ( x s x 1 ) 2 ( k x γ x ) 2 ] exp ( γ x c ) ,
L E y = π 2 [ β a ( k x a ) 2 ] γ x a 2 [ 1 + ( k x γ x ) 2 ] exp ( γ x c ) .
a = a + K x γ x k x k x K x + γ x Γ x K x 2 + Γ x 2 ,
P a ( z ) = P b ( 0 ) 4 κ 2 4 κ 2 + Δ 2 sin 2 [ ( κ 2 + Δ 2 4 ) 1 / 2 z ] ,
Δ P a TM ( L ) = P a TM ( L ) P b TM ( 0 ) = 4 κ TM 2 4 κ TM 2 + Δ TM 2 sin 2 ( κ TM 2 + Δ TM 2 4 ) 1 / 2 L = ρ 2 ρ 2 + 3 ( Δ TM / Δ TE ) E . O . 2 sin 2 { π 2 [ ρ 2 + 3 ( Δ TM ) 2 / ( Δ TE ) E . O , ] 1 / 2 } ,
( κ TE 2 + Δ TE 2 4 ) E . O . L 2 = π 2 = π 2 4 + ( Δ T E 2 ) E . O . 2 L 2 E x ,
Δ TE = ( π 3 ) / ( L E x ) .
P a ( z ) P b ( 0 ) = κ 2 κ 2 + ( Δ 2 / 4 ) sin 2 { 2 [ κ 2 + ( Δ 2 / 4 ) ] 1 / 2 z } cos 2 ϕ + κ 2 Δ 2 4 κ 2 + Δ 2 sin 4 { [ κ 2 + ( Δ 2 / 4 ) ] 1 / 2 z } sin 2 ϕ ,
Δ P a TM ( L ) P b TM ( 0 ) = sin 2 ( ρ π 2 ) cos 2 ( π 2 Δ TM Δ TE )
Δ P b TM ( L ) P b TM ( 0 ) = cos 2 ( ρ π 2 )
ϕ = [ ( 2 π L ) / λ ] E n 3 r ,
[ P a ( z ) ] / [ P b ( 0 ) ] = 1 - exp ( - 2 π ν )
P a ( z ) / P b ( 0 ) = [ 2 exp ( - 2 π ν ) - 1 ] 2 cos 2 ϕ + sin 2 ϕ ,
P B ( z ) / P b ( 0 ) = 4 exp ( - 2 π ν ) [ 1 - exp ( - 2 π ν ) ] cos 2 ϕ .
Δ P a TM P b ( 0 ) = 1 - [ 2 exp ( - 2 π ν TM ) - 1 ] 2 cos ( Δ TM Δ TE π 2 ) + sin 2 ( Δ TM Δ TE π 2 ) .
( Δ P b TM ) / [ P b ( 0 ) ] = [ 2 exp ( - 2 π ν TM ) - 1 ] 2 .
P = 1 2 exp ( - π y ) P incident ,
P 1 = { [ exp ( - π y ) - 1 ] 2 cos 2 ϕ + sin 2 ϕ } P 2 ( 0 ) , P 2 = exp ( - π y ) [ 2 - exp ( - π y ) ] cos 2 ϕ P 2 ( 0 ) ,
( Δ P ) / [ P 2 ( 0 ) ] = [ exp ( - π y ) - 1 ] 2 .
Δ P P 2 ( 0 ) = exp ( - π y ) [ 2 - exp ( - π y ] cos 2 ( Δ TM Δ TE π 2 ) .
Φ = ( 8 V 0 / π 2 ) cos ( K x ) exp ( - K y ) ,
E = ( 8 V 0 K / π 2 ) exp ( - K y ) [ sin ( K x ) x ^ + cos ( K y ) y ^ ] .
Δ P / P 0 = 1 - sin 2 ( Δ TM Δ TE π 2 ) ,
sin 2 [ ( Δ β L ) / 2 ] ( Δ β L / 2 ) 2 [ 1 - 1 12 ( Δ β L ) 2 ] ,
Δ β = [ - K / ( cos θ B ) ] ( sin θ B + K / 2 β TM ) ,
Δ β = - K cos θ B ( - K 2 β TE + K 2 β TM ) = - K 2 2 cos θ B ( β TE - β TM β TM β TE ) .
Δ P / P 0 = K 4 L 2 48 cos 2 θ B ( β TE - β TM β TM β TE ) .
sin θ i + sin θ s = K / β TM .
sin θ s = sin θ i ( 2 β TE - β TM β TM ) .
Δ 0 = 2 ( Δ β / β TM ) tan θ i .
δ θ 1 / 2 = λ TE / D ,     δ ϕ 1 / 2 = Λ / W ,
θ i - 1 ( Δ β ) / ( π β TM cos θ i ) ; Q i = Λ 2 / 2 λ TE L .

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