Abstract

We examine the behavior of modulational instability (MI) in several classes of high-index glass fibers that are being developed to obtain very high nonlinearities and soften the conditions of generation of highly efficient light sources, namely, telecommunication fibers, air-silica microstructured fibers, tapered fibers, and nonsilica glass fibers. We perform a comparative assessment of their respective performances in MI processes on the basis of three major performance criteria: the level of the input pump power, the fiber length, and the magnitude of the frequency drifts. Indeed, we show that the effectiveness of MI processes in such fibers is not merely influenced by the strength of the nonlinearity, but is also strongly determined by the linear attenuation of waves in the fiber material. In those high-index glass fibers, this attenuation acts as a strong perturbation, causing a frequency drift of the MI sidebands. However, we show that this frequency drift can be totally suppressed by means of a technique based on the concept of a photon reservoir, which feeds in situ the process of MI by continually supplying it the amount of photons absorbed by the fiber.

© 2011 Optical Society of America

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2011 (1)

2010 (4)

2009 (2)

2007 (1)

2006 (2)

2005 (1)

J. H. Chou and R. Wu, “A generalization of the frobenius method for ordinary differential equations with regular singular points,” J. Math. Stat. 1, 3–7 (2005).
[CrossRef]

2004 (1)

2003 (2)

2002 (2)

2001 (1)

2000 (2)

1995 (3)

1993 (1)

1991 (2)

J. L. Coutaz and M. Kull, “Saturation of nonlinear index of refraction in semiconductor-doped glass,” J. Opt. Soc. Am. B 8, 95–98 (1991).
[CrossRef]

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrodinger equation,” Phys. Rev. A 43, 6162–6165(1991).
[CrossRef] [PubMed]

1990 (1)

1989 (2)

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, “Nonlinear optical susceptibilities of high-index glasses,” Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

M. Nakazawa, K. Suzuki, and H. A. Haus, “The modulational instability laser. I. Experiment,” IEEE J. Quantum Electron. 25, 2036–2044 (1989).
[CrossRef]

1988 (2)

L. H. Acioli, A. S. L. Gomes, and J. R. R. Leite, “Measurement of high-order optical nonlinear susceptibilities in semiconductor-doped glasses,” Appl. Phys. Lett. 53, 1788–1790 (1988).
[CrossRef]

C. N. Ironside, T. J. Cullen, B. S. Bhumbra, J. Bell, W. C. Banyai, N. Finlayson, C. T. Seaton, and G. I. Stegeman, “Nonlinear-optical effects in ion-exchanged semiconductor-doped glass waveguides,” J. Opt. Soc. Am. B 5, 492–495 (1988).
[CrossRef]

1987 (2)

P. Roussignol, D. Ricard, J. Lukasik, and C. Flytzanis, “New results on optical phase conjugation in semiconductor-doped glasses,” J. Opt. Soc. Am. B 4, 5–13 (1987).
[CrossRef]

M. J. Potasek and G. P. Agrawal, “Self-amplitude-modulation of optical pulses in nonlinear dispersive fibers,” Phys. Rev. A 36, 3862–3867 (1987).
[CrossRef] [PubMed]

1986 (1)

K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 thz repetition rate by induced modulational instability,” Appl. Phys. Lett. 236–238 (1986).
[CrossRef]

1985 (1)

1984 (1)

1983 (1)

1980 (1)

A. Hasegawa and W. F. Brinkman, “Tunable coherent ir and fir sources utilizing modulational instability,” IEEE J. Quantum Electron. 16, 694–697 (1980).
[CrossRef]

Acioli, L. H.

L. H. Acioli, A. S. L. Gomes, and J. R. R. Leite, “Measurement of high-order optical nonlinear susceptibilities in semiconductor-doped glasses,” Appl. Phys. Lett. 53, 1788–1790 (1988).
[CrossRef]

Aggarwal, I. D.

Agrawal, G. P.

M. J. Potasek and G. P. Agrawal, “Self-amplitude-modulation of optical pulses in nonlinear dispersive fibers,” Phys. Rev. A 36, 3862–3867 (1987).
[CrossRef] [PubMed]

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2008).

Aitken, B. G.

Ambomo, S.

Andres, P.

Banyai, W. C.

Beckwitt, K.

Bell, J.

Bhumbra, B. S.

Birks, T. A.

Borrelli, N. F.

I. Kang, T. D. Krauss, F. W. Wise, B. G. Aitken, and N. F. Borrelli, “Femtosecond measurement of enhanced optical nonlinearities of sulphide glasses and heavy-metal-doped oxide glasses,” J. Opt. Soc. Am. B 12, 2053–2059 (1995).
[CrossRef]

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, “Nonlinear optical susceptibilities of high-index glasses,” Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

Brilland, L.

Brinkman, W. F.

A. Hasegawa and W. F. Brinkman, “Tunable coherent ir and fir sources utilizing modulational instability,” IEEE J. Quantum Electron. 16, 694–697 (1980).
[CrossRef]

Cambrell, G. K.

Cavalcanti, S. B.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrodinger equation,” Phys. Rev. A 43, 6162–6165(1991).
[CrossRef] [PubMed]

Chartier, T.

Chaudhari, C.

Chen, Y. F.

Chernikov, S. V.

Chestnut, D. A.

Chou, J. H.

J. H. Chou and R. Wu, “A generalization of the frobenius method for ordinary differential equations with regular singular points,” J. Math. Stat. 1, 3–7 (2005).
[CrossRef]

Coen, S.

Coutaz, J. L.

Cressoni, J. C.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrodinger equation,” Phys. Rev. A 43, 6162–6165(1991).
[CrossRef] [PubMed]

Cristiani, I.

Cullen, T. J.

da Cruz, H. R.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrodinger equation,” Phys. Rev. A 43, 6162–6165(1991).
[CrossRef] [PubMed]

Dianov, E. M.

Dumbaugh, W. H.

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, “Nonlinear optical susceptibilities of high-index glasses,” Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

El-Amraoui, M.

Emplit, P.

Fatome, J.

Fedoruk, M. P.

Ferrando, A.

Finlayson, N.

Flytzanis, C.

Fontana, F.

Fortier, C.

Franco, P.

Gadret, G.

George, A.

Gomes, A. S. L.

L. H. Acioli, A. S. L. Gomes, and J. R. R. Leite, “Measurement of high-order optical nonlinear susceptibilities in semiconductor-doped glasses,” Appl. Phys. Lett. 53, 1788–1790 (1988).
[CrossRef]

Gong, Y.

Gouveia-Neto, A. S.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrodinger equation,” Phys. Rev. A 43, 6162–6165(1991).
[CrossRef] [PubMed]

Guo, X.

Haelterman, M.

Hall, D. W.

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, “Nonlinear optical susceptibilities of high-index glasses,” Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

Hasegawa, A.

K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 thz repetition rate by induced modulational instability,” Appl. Phys. Lett. 236–238 (1986).
[CrossRef]

A. Hasegawa, “Generation of a train of soliton pulses by induced modulational instability in optical fibers,” Opt. Lett. 9, 288–290 (1984).
[CrossRef] [PubMed]

A. Hasegawa and W. F. Brinkman, “Tunable coherent ir and fir sources utilizing modulational instability,” IEEE J. Quantum Electron. 16, 694–697 (1980).
[CrossRef]

Haus, H. A.

M. Nakazawa, K. Suzuki, and H. A. Haus, “The modulational instability laser. I. Experiment,” IEEE J. Quantum Electron. 25, 2036–2044 (1989).
[CrossRef]

Ironside, C. N.

Jain, R. K.

Jewell, J. L.

K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 thz repetition rate by induced modulational instability,” Appl. Phys. Lett. 236–238 (1986).
[CrossRef]

Jules, J. C.

Kang, I.

Karlson, M.

Kibler, B.

Kito, C.

Krauss, T. D.

Kull, M.

Labruyere, A.

Langebein, U.

Lederer, F.

Leite, J. R. R.

L. H. Acioli, A. S. L. Gomes, and J. R. R. Leite, “Measurement of high-order optical nonlinear susceptibilities in semiconductor-doped glasses,” Appl. Phys. Lett. 53, 1788–1790 (1988).
[CrossRef]

Leon-Saval, S. G.

Liao, M.

Lind, R. C.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, 1983).

Lu, C.

Lukasik, J.

Mamyshev, P. V.

Matos, C. J. S.

Matsumoto, M.

Messaddeq, Y.

Midrio, M.

Miret, J. J.

Misumi, T.

Monteville, A.

Nakazawa, M.

M. Nakazawa, K. Suzuki, and H. A. Haus, “The modulational instability laser. I. Experiment,” IEEE J. Quantum Electron. 25, 2036–2044 (1989).
[CrossRef]

Nakkeeran, K.

Newhouse, M. A.

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, “Nonlinear optical susceptibilities of high-index glasses,” Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

Ngabireng, C.

Ngabireng, C. M.

Nguyen, T. N.

Ohishi, Y.

Peschel, T.

Polacchini, C. F.

Ponath, H. E.

Porsezian, K.

Potasek, M. J.

M. J. Potasek and G. P. Agrawal, “Self-amplitude-modulation of optical pulses in nonlinear dispersive fibers,” Phys. Rev. A 36, 3862–3867 (1987).
[CrossRef] [PubMed]

Prokhorov, A. M.

Qin, G.

Ravi Kanth Kumar, V. V.

Renversez, G.

Ricard, D.

Romagnoli, M.

Roussignol, P.

Rubenchik, A. M.

Russel, P.

Russell, P. St. J.

Sanghera, J. S.

Seaton, C. T.

Shum, P.

Silvestre, E.

Skripatchev, I.

Smektala, F.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, 1983).

Stegeman, G. I.

Suzuki, K.

M. Nakazawa, K. Suzuki, and H. A. Haus, “The modulational instability laser. I. Experiment,” IEEE J. Quantum Electron. 25, 2036–2044 (1989).
[CrossRef]

Suzuki, T.

Sylvestre, T.

Szpulak, M.

Tai, K.

K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 thz repetition rate by induced modulational instability,” Appl. Phys. Lett. 236–238 (1986).
[CrossRef]

Tang, D.

Taylor, J. R.

Tchofo Dinda, P.

Tomita, A.

K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 thz repetition rate by induced modulational instability,” Appl. Phys. Lett. 236–238 (1986).
[CrossRef]

Traynor, N.

Troles, J.

Turitsyn, S. K.

Wadsworth, W. J.

Wang, X. H.

Weidman, D. L.

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, “Nonlinear optical susceptibilities of high-index glasses,” Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

Wise, F. K.

Wise, F. W.

Wu, R.

J. H. Chou and R. Wu, “A generalization of the frobenius method for ordinary differential equations with regular singular points,” J. Math. Stat. 1, 3–7 (2005).
[CrossRef]

Yan, X.

Zambo Abou’ou, M. N.

Appl. Phys. Lett. (3)

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, “Nonlinear optical susceptibilities of high-index glasses,” Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

L. H. Acioli, A. S. L. Gomes, and J. R. R. Leite, “Measurement of high-order optical nonlinear susceptibilities in semiconductor-doped glasses,” Appl. Phys. Lett. 53, 1788–1790 (1988).
[CrossRef]

K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 thz repetition rate by induced modulational instability,” Appl. Phys. Lett. 236–238 (1986).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. Nakazawa, K. Suzuki, and H. A. Haus, “The modulational instability laser. I. Experiment,” IEEE J. Quantum Electron. 25, 2036–2044 (1989).
[CrossRef]

A. Hasegawa and W. F. Brinkman, “Tunable coherent ir and fir sources utilizing modulational instability,” IEEE J. Quantum Electron. 16, 694–697 (1980).
[CrossRef]

J. Lightwave Technol. (1)

J. Math. Stat. (1)

J. H. Chou and R. Wu, “A generalization of the frobenius method for ordinary differential equations with regular singular points,” J. Math. Stat. 1, 3–7 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (8)

M. Karlson, “Modulational instability in lossy optical fiber,” J. Opt. Soc. Am. B 12, 2071–2077 (1995).
[CrossRef]

P. Roussignol, D. Ricard, J. Lukasik, and C. Flytzanis, “New results on optical phase conjugation in semiconductor-doped glasses,” J. Opt. Soc. Am. B 4, 5–13 (1987).
[CrossRef]

C. N. Ironside, T. J. Cullen, B. S. Bhumbra, J. Bell, W. C. Banyai, N. Finlayson, C. T. Seaton, and G. I. Stegeman, “Nonlinear-optical effects in ion-exchanged semiconductor-doped glass waveguides,” J. Opt. Soc. Am. B 5, 492–495 (1988).
[CrossRef]

J. L. Coutaz and M. Kull, “Saturation of nonlinear index of refraction in semiconductor-doped glass,” J. Opt. Soc. Am. B 8, 95–98 (1991).
[CrossRef]

X. H. Wang and G. K. Cambrell, “Simulation of strong nonlinear effects in optical waveguides,” J. Opt. Soc. Am. B 10, 2048–2055 (1993).
[CrossRef]

P. Tchofo Dinda, and K. Porsezian, “Impact of fourth-order dispersion in the modulational instability spectra of wave propagation in glass fibers with saturable nonlinearity,” J. Opt. Soc. Am. B 27, 1143–1152 (2010).
[CrossRef]

I. Kang, T. D. Krauss, F. W. Wise, B. G. Aitken, and N. F. Borrelli, “Femtosecond measurement of enhanced optical nonlinearities of sulphide glasses and heavy-metal-doped oxide glasses,” J. Opt. Soc. Am. B 12, 2053–2059 (1995).
[CrossRef]

Y. F. Chen, K. Beckwitt, F. K. Wise, B. G. Aitken, J. S. Sanghera, and I. D. Aggarwal, “Measurement of fifth- and seventh-order nonlinearities of glasses,” J. Opt. Soc. Am. B 23, 347–352 (2006).
[CrossRef]

Opt. Express (7)

S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express 12, 2864–2869 (2004).
[CrossRef] [PubMed]

V. V. Ravi Kanth Kumar, A. George, and P. Russel, “Tellurite photonic crystal fiber,” Opt. Express 11, 2641–2645(2003).
[CrossRef]

L. Brilland, F. Smektala, G. Renversez, T. Chartier, J. Troles, T. N. Nguyen, N. Traynor, and A. Monteville, “Fabrication of complex structures of holey fibers in chalcogenide glass,” Opt. Express 14, 1280–1285 (2006).
[CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Accumulated MI gain and OMF Ω opt versus distance z.

Fig. 2
Fig. 2

Schematic representation of the MI map in the fiber system for β 2 < 0 and β 4 > 0 , in which the photon reservoir is created by raising the pump power.

Fig. 3
Fig. 3

Plots illustrating the MI processes in the operating conditions of nonsuppression of the frequency drifts ( P 0 = 35 mW ), partial suppression of the frequency drifts ( P 0 = 55 mW ), and total suppression of the frequency drifts ( P 0 = 130 mW ) for the chalcogenide fiber.

Fig. 4
Fig. 4

Schematic representation of the MI map in the fiber system for β 2 < 0 and β 4 > 0 , in which the photon reservoir is created by raising the FOD coefficient β 4 .

Fig. 5
Fig. 5

Plots illustrating the MI process in the operating conditions of nonsuppression of the frequency drifts for a chalcogenide type of fiber with parameters β 2 = 2.6 × 10 3 ps 2 / m , β 4 = 5 × 10 6 ps 4 / m , and α = 1 dB / m . P 0 = 55 mW .

Fig. 6
Fig. 6

Plots illustrating the MI process in the operating conditions of partial suppression of the frequency drifts for a chalcogenide type of fiber with parameters β 2 = 2.6 × 10 3 ps 2 / m , β 4 = 9 × 10 6 ps 4 / m , and α = 1 dB / m . P 0 = 55 mW .

Fig. 7
Fig. 7

Plots illustrating the MI process in the operating conditions of total suppression of the frequency drifts for a chalcogenide type of fiber with parameters β 2 = 2.6 × 10 3 ps 2 / m , β 4 = 15 × 10 6 ps 4 / m , and α = 1 dB / m . P 0 = 55 mW .

Fig. 8
Fig. 8

Plots of the gain spectra for a chalcogenide type of fiber with dispersion parameters β 2 = 2.6 × 10 3 ps 2 / m , β 4 = 9 × 10 6 ps 4 / m , and different values of the absorption parameter α.

Tables (1)

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Table 1 Fiber Parameters and Performance Indicators in the MI Process

Equations (39)

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A z = i β 2 2 A t t + β 3 6 A t t t + i β 4 24 A t t t t + i γ ¯ | A | 2 A 1 + Γ | A | 2 α A 2 ,
q z = i β 2 2 q t t + β 3 6 q t t t + i β 4 24 q t t t t + i γ ¯ exp ( α z ) | q | 2 q 1 + Γ | q | 2 .
d ρ / d z = γ i ρ 3 / [ 1 + Γ ρ 2 exp ( α z ) ] ,
d Φ / d z = γ r ρ 2 / [ 1 + Γ ρ 2 exp ( α z ) ] .
q ( z , t ) = [ ρ + ε ( z , t ) ] exp ( i Φ ( z ) ) ,
ε z = i Ω 2 β 2 2 ε + i Ω 3 β 3 6 ε + i Ω 4 β 4 24 ε + i γ r P 0 ( ε + ε * ) exp ( α z ) .
z [ u s u a * ] = i M [ u s u a * ] ,
M = [ m 11 m 12 m 21 m 22 ] [ D s ( Ω ) + i γ i P Q + ( γ r + i γ i ) P Q ( γ r + i γ i ) P Q ( γ r + i γ i ) P Q D a ( Ω ) + i γ i P Q + ( γ r + i γ i ) P Q ] ,
G ( z , Ω ) = ± 2 γ i 2 P 2 Q 2 ( β 2 2 Ω 2 + β 4 24 Ω 4 + γ r P Q ) 2 + ( γ r P Q ) 2 + 2 γ i P ( Q 1 + Q 1 / 2 ) .
G = 2 ( γ r P 1 Q 1 ) 2 ( β 2 2 Ω 2 + β 4 24 Ω 4 + γ r P 1 Q 1 ) 2 ,
G = | β 2 | Ω 2 | Q 2 | Ω c 2 exp ( α z ) / ( Q 1 Q 2 Ω 2 ) 1 ,
G ( Ω , z ) | β 2 | Ω 2 | Q 2 | Ω c 2 exp ( α z ) / ( Q 2 Ω 2 ) 1 .
G ˜ ( Ω , z ) 0 L g ( Ω , z ) d z .
G ˜ = α L + κ [ W ( Ω , 0 ) tan 1 ( W ( Ω , 0 ) ) ] , for     L > z c ,
G ˜ = α L + κ [ η 1 + tan 1 ( η 1 / η 2 ) ] , for     L < z c ,
κ = α 1 | β 2 | Ω 2 | Q 2 | ,
W ( Ω , x ) = [ ζ exp ( α x ) 1 ] 1 / 2 ,
ζ = Ω c 2 / [ Ω 2 ( 1 Y 2 Ω 2 ) ] ,
η 1 = W ( Ω , 0 ) W ( Ω , L ) ,
η 2 = 1 W ( Ω , 0 ) W ( Ω , L ) .
Ω 1 , 2 = Ω 0 [ 1 ± 1 P 0 / P 0 c L ] 1 / 2 , for     L < z c ,
Ω 1 , 2 = Ω 0 [ 1 ± 1 P 0 / P 0 c ] 1 / 2 , for     L z c ,
d 2 u s d z 2 [ i ( m 11 + m 22 ) + 1 m 12 d m 12 d z ] d u s d z [ i ( d m 11 d z m 11 m 12 d m 12 d z ) + ( m 11 m 22 m 12 m 21 ) ] u s = 0 .
d 2 u s d z 2 + ( α + i β 3 Ω 3 3 ) d u s d z + [ 2 γ r P 0 exp ( α z ) ( β 2 2 Ω 2 + β 4 24 Ω 4 ) + D a D s i α D s ] u s = 0 ,
x = η exp ( α z / 2 ) ,
x 2 d 2 u s d x 2 ϖ x d u s d x ( x 2 μ 2 ) u s = 0 ,
u s = C 01 k = 1 x 2 k + r 1 2 k k ! m = 1 k 1 ( ϑ + 2 m ) + C 02 k = 1 x 2 k + r 2 2 k k ! m = 1 k 1 ( ϑ + 2 m ) ,
r 1 = 1 + ϖ 2 + ϑ , r 2 = 1 + ϖ 2 ϑ ,
u s ( z = 0 ) = u a * ( z = 0 ) = u 0 d u s d z | z = 0 = i ( m 11 + m 12 ) u 0 ,
{ η r 1 C 01 S 1 + η r 2 C 02 S 2 = u 0 , η r 1 C 01 K 1 + η r 2 C 02 K 2 = δ u 0 ,
K 1 = [ r 1 + d 2 ( 1 ) ( r 1 + 2 ) η 2 + d 4 ( 1 ) ( r 1 + 4 ) η 4 + ] , K 2 = [ r 2 + d 2 ( 2 ) ( r 2 + 2 ) η 2 + d 4 ( 2 ) ( r 2 + 4 ) η 4 + ] , S 1 = [ 1 + d 2 ( 1 ) η 2 + d 4 ( 1 ) η 4 + ] , S 2 = [ 1 + d 2 ( 2 ) η 2 + d 4 ( 2 ) η 4 + ] , δ = 2 i α ( m 11 + m 12 ) = 2 i α D 10 , d 2 k ( j ) = 1 2 k k ! m = 1 k 1 ( ϑ + 2 m ) .
C 01 = ( δ S 2 K 2 ) u 0 η r 1 ( K 1 S 2 K 2 S 1 ) , C 02 = ( K 1 δ S 1 ) u 0 η r 2 ( K 1 S 2 K 2 S 1 ) .
g ( Ω , L ) = | u s ( L ) u 0 | 2 .
g A ( Ω , L ) = g ( Ω , L ) α .
P c 1 ( 1 2 Γ ξ Δ ) / ( 2 ξ Γ 2 ) ,
P c 2 ( 1 2 Γ ξ + Δ ) / ( 2 ξ Γ 2 ) ,
P 0 = P c 1 ( β 2 , β 4 ) × exp ( α L ) .
R = P 0 [ 1 exp ( α L ) ] .
P 0 P c 1 ( β 40 ) = R .

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