Abstract

A mode power measure is applied to characterize nonlinear thin film optical waveguides in an approach analogous to that of Chelkowski and Chrostowski. Together with the normalized film thickness and the asymmetry coefficient, it allows us to get a concise overview of the waveguiding properties at a given power. For self-focusing film, we discuss the design conditions under which the degree of asymmetry significantly affects the waveguiding properties.

© 1990 Optical Society of America

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  1. V. E. Wood, E. D. Evans, R. P. Kenan, “Soluble Saturable Refractive-Index Nonlinearity Model,” Opt. Commun. 69, 156–160 (1988).
    [CrossRef]
  2. K. Hayata, M. Koshiba, “Full Vectorial Analysis of Nonlinear-Optical Waveguides,” J. Opt. Soc. Am. B 5, 2494–2501 (1988).
    [CrossRef]
  3. G. I. Stegeman, C. Seaton, “Nonlinear Waves Guided by Thin Films,” Appl. Phys. Lett. 44, 830–832 (1984).
    [CrossRef]
  4. A. Boardman, P. Egan, “Optically Nonlinear Waves in Thin Films,” IEEE J. Quantum Electron., QE-22, 319–324 (1986).
    [CrossRef]
  5. P. M. Lambkin, K. A. Shore, “Asymmetric Semiconductor Waveguide with Defocusing Nonlinearity,” IEEE J. Quantum Electron., QE-24, 2046–2051 (1988).
    [CrossRef]
  6. G. I. Stegeman, J. Ariyasu, C. T. Seaton, T. P. Shen, J. V. Moloney, “Nonlinear Thin-Film Guided Waves in Non-Kerr Media,” Appl. Phys. Lett. 47, 1254–1256 (1985).
    [CrossRef]
  7. U. Langbein, F. Lederer, H.-E. Ponath, “Generalized Dispersion Relations for Nonlinear Slab-Guided Waves,” Opt. Commun. 53, 417–420 (1985).
    [CrossRef]
  8. U. Langbein, F. Lederer, T. Peschel, H. E. Ponath, “Nonlinear Guided Waves in Saturable Nonlinear Media,” Opt. Lett. 10, 571–573 (1985).
    [CrossRef] [PubMed]
  9. G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, Tsae-Pyng Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear Slab-Guided Waves in Non-Kerr Media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
    [CrossRef]
  10. S. Chelkowski, J. Chrostowski, “Scaling Rules for Slab Waveguides with Nonlinear Substrate,” Appl. Opt. 26, 3681–3686 (1987).
    [CrossRef] [PubMed]
  11. H. Kogelnik, V. Ramaswamy, “Scaling Rules for Thin-Film Optical Waveguides,” Appl. Opt. 13, 1857–1862 (1974).
    [CrossRef] [PubMed]
  12. E. D. Rainville, Special Functions, (Chelsea Publishing, New York, 1971), Chap. 21.
  13. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, (Appl. Math Series NBS, Washington, DC, 1972), p. 573.
  14. G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, “Third Order Nonlinear Integrated Optics,” IEEE/OSA J. Lightwave Technol. 6, 953–970 (1988).
    [CrossRef]
  15. G. I. Stegeman, “Guided Wave Approaches to Optical Bistability,” IEEE J. Quantum Electron. QE-18, 1610–1618 (1982).
    [CrossRef]
  16. T. Peschel, P. Dannberg, U. Langbein, F. Lederer, “Investigation of Optical Tunneling Through Nonlinear Films,” J. Opt. Soc. Am. B 5, 29–36 (1988).
    [CrossRef]

1988

P. M. Lambkin, K. A. Shore, “Asymmetric Semiconductor Waveguide with Defocusing Nonlinearity,” IEEE J. Quantum Electron., QE-24, 2046–2051 (1988).
[CrossRef]

V. E. Wood, E. D. Evans, R. P. Kenan, “Soluble Saturable Refractive-Index Nonlinearity Model,” Opt. Commun. 69, 156–160 (1988).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, “Third Order Nonlinear Integrated Optics,” IEEE/OSA J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

T. Peschel, P. Dannberg, U. Langbein, F. Lederer, “Investigation of Optical Tunneling Through Nonlinear Films,” J. Opt. Soc. Am. B 5, 29–36 (1988).
[CrossRef]

K. Hayata, M. Koshiba, “Full Vectorial Analysis of Nonlinear-Optical Waveguides,” J. Opt. Soc. Am. B 5, 2494–2501 (1988).
[CrossRef]

1987

1986

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, Tsae-Pyng Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear Slab-Guided Waves in Non-Kerr Media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

A. Boardman, P. Egan, “Optically Nonlinear Waves in Thin Films,” IEEE J. Quantum Electron., QE-22, 319–324 (1986).
[CrossRef]

1985

G. I. Stegeman, J. Ariyasu, C. T. Seaton, T. P. Shen, J. V. Moloney, “Nonlinear Thin-Film Guided Waves in Non-Kerr Media,” Appl. Phys. Lett. 47, 1254–1256 (1985).
[CrossRef]

U. Langbein, F. Lederer, H.-E. Ponath, “Generalized Dispersion Relations for Nonlinear Slab-Guided Waves,” Opt. Commun. 53, 417–420 (1985).
[CrossRef]

U. Langbein, F. Lederer, T. Peschel, H. E. Ponath, “Nonlinear Guided Waves in Saturable Nonlinear Media,” Opt. Lett. 10, 571–573 (1985).
[CrossRef] [PubMed]

1984

G. I. Stegeman, C. Seaton, “Nonlinear Waves Guided by Thin Films,” Appl. Phys. Lett. 44, 830–832 (1984).
[CrossRef]

1982

G. I. Stegeman, “Guided Wave Approaches to Optical Bistability,” IEEE J. Quantum Electron. QE-18, 1610–1618 (1982).
[CrossRef]

1974

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, (Appl. Math Series NBS, Washington, DC, 1972), p. 573.

Ariyasu, J.

G. I. Stegeman, J. Ariyasu, C. T. Seaton, T. P. Shen, J. V. Moloney, “Nonlinear Thin-Film Guided Waves in Non-Kerr Media,” Appl. Phys. Lett. 47, 1254–1256 (1985).
[CrossRef]

Boardman, A.

A. Boardman, P. Egan, “Optically Nonlinear Waves in Thin Films,” IEEE J. Quantum Electron., QE-22, 319–324 (1986).
[CrossRef]

Chelkowski, S.

Chrostowski, J.

Dannberg, P.

Egan, P.

A. Boardman, P. Egan, “Optically Nonlinear Waves in Thin Films,” IEEE J. Quantum Electron., QE-22, 319–324 (1986).
[CrossRef]

Evans, E. D.

V. E. Wood, E. D. Evans, R. P. Kenan, “Soluble Saturable Refractive-Index Nonlinearity Model,” Opt. Commun. 69, 156–160 (1988).
[CrossRef]

Finlayson, N.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, “Third Order Nonlinear Integrated Optics,” IEEE/OSA J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Hayata, K.

Kenan, R. P.

V. E. Wood, E. D. Evans, R. P. Kenan, “Soluble Saturable Refractive-Index Nonlinearity Model,” Opt. Commun. 69, 156–160 (1988).
[CrossRef]

Kogelnik, H.

Koshiba, M.

Lambkin, P. M.

P. M. Lambkin, K. A. Shore, “Asymmetric Semiconductor Waveguide with Defocusing Nonlinearity,” IEEE J. Quantum Electron., QE-24, 2046–2051 (1988).
[CrossRef]

Langbein, U.

Lederer, F.

Maradudin, A. A.

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, Tsae-Pyng Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear Slab-Guided Waves in Non-Kerr Media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

Moloney, J. V.

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, Tsae-Pyng Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear Slab-Guided Waves in Non-Kerr Media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

G. I. Stegeman, J. Ariyasu, C. T. Seaton, T. P. Shen, J. V. Moloney, “Nonlinear Thin-Film Guided Waves in Non-Kerr Media,” Appl. Phys. Lett. 47, 1254–1256 (1985).
[CrossRef]

Peschel, T.

Ponath, H. E.

Ponath, H.-E.

U. Langbein, F. Lederer, H.-E. Ponath, “Generalized Dispersion Relations for Nonlinear Slab-Guided Waves,” Opt. Commun. 53, 417–420 (1985).
[CrossRef]

Rainville, E. D.

E. D. Rainville, Special Functions, (Chelsea Publishing, New York, 1971), Chap. 21.

Ramaswamy, V.

Seaton, C.

G. I. Stegeman, C. Seaton, “Nonlinear Waves Guided by Thin Films,” Appl. Phys. Lett. 44, 830–832 (1984).
[CrossRef]

Seaton, C. T.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, “Third Order Nonlinear Integrated Optics,” IEEE/OSA J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, Tsae-Pyng Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear Slab-Guided Waves in Non-Kerr Media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

G. I. Stegeman, J. Ariyasu, C. T. Seaton, T. P. Shen, J. V. Moloney, “Nonlinear Thin-Film Guided Waves in Non-Kerr Media,” Appl. Phys. Lett. 47, 1254–1256 (1985).
[CrossRef]

Shen, T. P.

G. I. Stegeman, J. Ariyasu, C. T. Seaton, T. P. Shen, J. V. Moloney, “Nonlinear Thin-Film Guided Waves in Non-Kerr Media,” Appl. Phys. Lett. 47, 1254–1256 (1985).
[CrossRef]

Shen, Tsae-Pyng

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, Tsae-Pyng Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear Slab-Guided Waves in Non-Kerr Media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

Shore, K. A.

P. M. Lambkin, K. A. Shore, “Asymmetric Semiconductor Waveguide with Defocusing Nonlinearity,” IEEE J. Quantum Electron., QE-24, 2046–2051 (1988).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, “Third Order Nonlinear Integrated Optics,” IEEE/OSA J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, Tsae-Pyng Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear Slab-Guided Waves in Non-Kerr Media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

G. I. Stegeman, J. Ariyasu, C. T. Seaton, T. P. Shen, J. V. Moloney, “Nonlinear Thin-Film Guided Waves in Non-Kerr Media,” Appl. Phys. Lett. 47, 1254–1256 (1985).
[CrossRef]

G. I. Stegeman, C. Seaton, “Nonlinear Waves Guided by Thin Films,” Appl. Phys. Lett. 44, 830–832 (1984).
[CrossRef]

G. I. Stegeman, “Guided Wave Approaches to Optical Bistability,” IEEE J. Quantum Electron. QE-18, 1610–1618 (1982).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, (Appl. Math Series NBS, Washington, DC, 1972), p. 573.

Wallis, R. F.

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, Tsae-Pyng Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear Slab-Guided Waves in Non-Kerr Media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

Wood, V. E.

V. E. Wood, E. D. Evans, R. P. Kenan, “Soluble Saturable Refractive-Index Nonlinearity Model,” Opt. Commun. 69, 156–160 (1988).
[CrossRef]

Wright, E. M.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, “Third Order Nonlinear Integrated Optics,” IEEE/OSA J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, Tsae-Pyng Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear Slab-Guided Waves in Non-Kerr Media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

Zanoni, R.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, “Third Order Nonlinear Integrated Optics,” IEEE/OSA J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

G. I. Stegeman, C. Seaton, “Nonlinear Waves Guided by Thin Films,” Appl. Phys. Lett. 44, 830–832 (1984).
[CrossRef]

G. I. Stegeman, J. Ariyasu, C. T. Seaton, T. P. Shen, J. V. Moloney, “Nonlinear Thin-Film Guided Waves in Non-Kerr Media,” Appl. Phys. Lett. 47, 1254–1256 (1985).
[CrossRef]

IEEE J. Quantum Electron.

A. Boardman, P. Egan, “Optically Nonlinear Waves in Thin Films,” IEEE J. Quantum Electron., QE-22, 319–324 (1986).
[CrossRef]

P. M. Lambkin, K. A. Shore, “Asymmetric Semiconductor Waveguide with Defocusing Nonlinearity,” IEEE J. Quantum Electron., QE-24, 2046–2051 (1988).
[CrossRef]

G. I. Stegeman, “Guided Wave Approaches to Optical Bistability,” IEEE J. Quantum Electron. QE-18, 1610–1618 (1982).
[CrossRef]

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, Tsae-Pyng Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear Slab-Guided Waves in Non-Kerr Media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

IEEE/OSA J. Lightwave Technol.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, “Third Order Nonlinear Integrated Optics,” IEEE/OSA J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

U. Langbein, F. Lederer, H.-E. Ponath, “Generalized Dispersion Relations for Nonlinear Slab-Guided Waves,” Opt. Commun. 53, 417–420 (1985).
[CrossRef]

V. E. Wood, E. D. Evans, R. P. Kenan, “Soluble Saturable Refractive-Index Nonlinearity Model,” Opt. Commun. 69, 156–160 (1988).
[CrossRef]

Opt. Lett.

Other

E. D. Rainville, Special Functions, (Chelsea Publishing, New York, 1971), Chap. 21.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, (Appl. Math Series NBS, Washington, DC, 1972), p. 573.

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Figures (15)

Fig. 1
Fig. 1

Nonlinear waveguide bounded by linear dielectrics.

Fig. 2
Fig. 2

Universal dispersion curves for a symmetrical (a = 0) nonlinear thin film waveguide in the case where nf > ns. The curves are labeled with values of bI. The dotted line corresponds to the linear thin film waveguide where bI = 0.

Fig. 3
Fig. 3

Electric field configuration at V = 7.0 for a symmetrical nonlinear waveguide and a symmetrical linear waveguide. (1) bI = 0.01 and b = 1.265; (2) bI = 0.01 and b = 1.117; (3) bI = 0 and b = 0.87.

Fig. 4
Fig. 4

Universal dispersion curves for normalized guided index b < 1 in the case where nf > ns(a = 0). The curves are labeled with values of bI < 0.10. The dotted line corresponds to the linear thin film waveguide where bI = 0.

Fig. 5
Fig. 5

Universal dispersion curves for an asymmetrical nonlinear thin film waveguide in the case where nf > ns. The asymmetry coefficient a = 10. The curves are labeled with values of bI. The dotted line corresponds to the linear thin film waveguide where bI = 0.

Fig. 6
Fig. 6

Universal dispersion curves for normalized guided index b < 1 in the case where nf > ns(a = 10). The curves are labeled with values of bI < 0.10. The dotted line corresponds to the linear thin film waveguide where bI = 0. In the left upper corner, we show the variation with bI of the cutoff normalized frequency of the TE0 mode.

Fig. 7
Fig. 7

Universal dispersion curves for a symmetrical (a = 0) nonlinear thin film waveguide in the case where nf < ns. The curves are labeled with values of bI. The solid lines refer to TE0 modes and the dotted lines to TE1 modes.

Fig. 8
Fig. 8

Universal dispersion curves for an asymmetrical (a = 10) nonlinear thin film waveguide in the case where nf < ns. The curves are labeled with values of bI. The solid lines refer to TE0 modes and the dotted lines to TE1 modes.

Fig. 9
Fig. 9

Relationship between parameter bI and the normalized power flow P/P0 in the TE0 mode when nf > ns for a series of values of V: ○ V = 0.8, ▲ V = 1.0, ● V = 1.5, ■ V = 2.0. The asymmetry coefficient is 0.

Fig. 10
Fig. 10

Relationship between parameter bI and the normalized power flow P/P0 in the TE0 mode when nf > ns and V ≤ 1. The asymmetry coefficient is 10.

Fig. 11
Fig. 11

Relationship between parameter bI and the normalized power flow P/P0 in the TE0 mode when nf > ns and V > 1. The asymmetry coefficient is 10.

Fig. 12
Fig. 12

Relationship between parameter bI and the normalized power flow P/P0 in the TE0 mode when nf < ns and V < 1. The asymmetry coefficient is 0.

Fig. 13
Fig. 13

Relationship between parameter bI and the normalized power flow P/P0 in the TE1 mode when nf < ns, and V > 3. The asymmetry coefficient is 0.

Fig. 14
Fig. 14

Relationship between parameter bI and the normalized power flow P/P0 in the TE0 mode when nf < ns and V > 1. The asymmetry coefficient is 10.

Fig. 15
Fig. 15

Relationship between parameter bI and the normalized power flow P/P0 in the TE1 mode when nf < ns. and V > 3. The asymmetry coefficient is 10.

Equations (32)

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V = 2 π d λ ( n f 2 - n s 2 ) 1 / 2 ,
a = ( n s 2 - n c 2 ) ( n f 2 - n s 2 ) .
b = ( N 2 - n s 2 ) ( n f 2 - n s 2 ) ,
b I = α E 2 ( 0 ) 2 ( n f 2 - n s 2 ) .
E 1 = E 0 exp ( k 1 z ) ,
E 3 = E b exp [ k 3 ( d - z ) ] ,
k 1 2 = k 2 ( N 2 - n s 2 ) ,
k 3 2 = k 2 ( N 2 - n c 2 ) .
C 2 = k 2 E 0 2 [ ( n f 2 - n s 2 ) + ½ α E 0 2 ] ,
C 2 = k 2 E b 2 [ ( n f 2 - n c 2 ) + ½ α E b 2 ] ,
E 2 = p c n [ q ( z + z 0 ) m ] .
q = k ( n f 2 - n s 2 ) 1 / 2 [ ( b - 1 ) 2 + 4 b I ( 1 + b I ) ] 1 / 4 ,
p = ( E 0 2 2 b I ) 1 / 2 { [ ( b - 1 ) 2 + 4 b I ( 1 + b I ) ] 1 / 2 + ( b - 1 ) } 1 / 2 ,
m = 1 2 { 1 + ( b - 1 ) [ ( b - 1 ) 2 + 4 b I ( 1 + b I ) ] 1 / 2 } .
c n ( q d \\ m ) = { ( E 0 p ) ( E b p ) ( 1 - [ b ( b + a ) ( b - 1 ) 2 + 4 b I ( 1 + b I ) ] 1 / 2 ) } { 1 - m [ 1 - ( E 0 p ) 2 ] [ 1 - ( E b p ) 2 ] } .
q d = V [ ( b - 1 ) 2 + 4 b I ( 1 + b I ) ] 1 / 4 .
( E b p ) = { [ ( 1 + a ) 2 + 4 b I ( 1 + b I ) ] 1 / 2 - ( 1 + a ) [ ( b - 1 ) 2 + 4 b I ( 1 + b I ) ] 1 / 2 + ( b - 1 ) } 1 / 2 .
V = 2 π d λ ( n s 2 - n f 2 ) 1 / 2 ,
a = ( n s 2 - n c 2 ) ( n s 2 - n f 2 ) ,
b = ( N 2 - n s 2 ) ( n s 2 - n f 2 ) ,
b 1 = α E 2 ( 0 ) 2 ( n s 2 - n f 2 ) .
q = k ( n s 2 - n f 2 ) 1 / 2 [ ( b + 1 ) 2 + 4 b I ( b I - 1 ) ] 1 / 4 ,
p = ( E 0 2 2 b I ) 1 / 2 { [ ( b + 1 ) 2 + 4 b I ( b I - 1 ) ] 1 / 2 + ( b + 1 ) } 1 / 2 ,
m = 1 2 { 1 + ( b + 1 ) [ ( b + 1 ) 2 + 4 b I ( b I - 1 ) ] 1 / 2 } .
( E b p ) = ( ( 1 - a ) { 1 ± [ 1 + 4 b I ( b I - 1 ) ( 1 - a ) 2 ] 1 / 2 } [ b + 1 ] 2 + 4 b I ( b I - 1 ) ] 1 / 2 + ( b + 1 ) ) 1 / 2 .
q = k ( n s 2 - n f 2 ) 1 / 2 [ ( b + 1 ) 2 + 4 b I ( b I - 1 ) ] 1 / 4 m 1 / 2 ,
m = ( 1 2 { 1 + ( b + 1 ) m [ ( b + 1 ) 2 + 4 b I ( b I - 1 ) ] 1 / 2 } ) - 1 .
P = 1 2 N c 0 - E 2 ( z ) d z .
P / P 0 = b I b 1 / 2 V + [ ( 1 + a ) 2 + 4 b I ( 1 + b I ) ] 1 / 2 - ( 1 + a ) 2 V ( b + a ) 1 / 2 + 2 b I V b 1 / 2 f 1 ( b , b I ) 0 q d [ c n ( u ) + f 1 ( b , b I ) s n ( u ) d n ( u ) d n 2 ( u ) + f 2 ( b , b I ) s n 2 ( u ) ] 2 d u ,
f 1 ( b , b I ) = b 1 / 2 [ ( b - 1 ) 2 + 4 b I ( 1 + b I ) ] 1 / 4 , f 2 ( b , b I ) = b I [ ( b - 1 ) 2 + 4 b I ( 1 + b I ) ] 1 / 2 , P 0 = N c 0 α ( n f 2 - n s 2 ) d .
P / P 0 = b 1 b 1 / 2 V + ( 1 - a ) ± [ ( 1 - a ) 2 + 4 b I ( b I - 1 ) ] 1 / 2 2 V ( b + a ) 1 / 2 + 2 b I V b 1 / 2 f 1 ( b , b I ) 0 q d [ c n ( u ) + f 1 ( b , b I ) s n ( u ) d n ( u ) d n 2 ( u ) + f 2 ( b , b I ) s n 2 ( u ) ] 2 d u ,
f 1 ( b , b I ) = b 1 / 2 [ ( b + 1 ) 2 + 4 b I ( b I - 1 ) ] 1 / 4 , f 2 ( b , b I ) = b I [ ( b + 1 ) 2 + 4 b I ( b I - 1 ) ] 1 / 2 , P 0 = N c 0 α ( n s 2 - n f 2 ) d .

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