Abstract

The structure of a Q-modulated distributed feedback laser is designed and simulated. A large reflectivity modulation of the rear reflector is achieved by using an anti-resonant cavity formed by two deep trenches with the one between the modulator and phase section filled by a high index dielectric material. The travelling wave model is presented to analyze the high speed performance of the laser. Due to the effect of the wave propagation in the structure, the modulation extinction ratio decreases with increasing cavity length. It is shown that 40Gb/s RZ signal modulation can be achieved with an extinction ratio of 7dB and 10dB, respectively, for a cavity length of 500μm and 300μm.

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References

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  1. R. M. Spencer, “High speed Direct Modulation of Semiconductor Laser,” Int. J. High Speed Electron. Syst. 8(3), 377–416 (1997).
    [CrossRef]
  2. K. Takagi, S. Shirai, Y. Tasuoka, C. Watatani, T. Ota, T. Takiguchi, T. Aoyagi, T. Nishimura, and N. Tomita, “120°C 10 Gb/s Uncooled Direct Modulated 1.3 AlGaInAs MQW DFB Laser Diodes,” IEEE Photon. Technol. Lett. 16, 2415–2417 (2004).
    [CrossRef]
  3. C. W. Chow, C. S. Wong, and H. K. Tsang, “Reduction of Amplitude Transients and BER of Direct Modulation Laser Using Birefringent Fiber Loop,” IEEE Photon. Technol. Lett. 17(3), 693–695 (2005).
    [CrossRef]
  4. M. Minakata, “Recent Progress of 40 GHz High-speed LiNbO3 Optical Modulator,” Active and Passive Optical Components for WDM Communication,” Proc. SPIE 4532, 16–27 (2001).
    [CrossRef]
  5. P. Tang, A. L. Meier, D. J. Towner, and B. W. Wessels, “High-speed Travelling-wave BaTiO3 Thin-film Electro-optic Modulators,” Electron. Lett. 41(23), 1296–1297 (2005).
    [CrossRef]
  6. Y. Kim, H. Lee, J. Lee, J. Han, T. W. Oh, and J. Jeong, “Chirp Characteristics of 10-Gb/s Electroabsorption Modulator Integrated DFB Lasers,” IEEE J. Quantum Electron. 36(8), 900–908 (2000).
    [CrossRef]
  7. H. Takeuchi, “Ultra-fast Electroabsorption Modulator Integrated DFB Lasers,” Proc. of IEEE International Conference On Indium Phosphide and Related Materials, 428–431 (2001).
  8. M. Suzuki, Y. Noda, H. Tanaka, S. Akiba, Y. Kushiro, and H. Isshiki, “Monolithic Integration of InGaAsP/InP Distributed Feedback Laser and Electroabsorption Modulator by Vapor Phase Epitaxy,” J. Lightwave Technol. 5(9), 1277–1285 (1987).
    [CrossRef]
  9. U. Westergren, M. Chaciński, and L. Thylén, “Compact and efficient modulators for 100 Gb/s ETDM for telecom and interconnect applications,” Appl. Phys., A Mater. Sci. Process. 95(4), 1039–1044 (2009).
    [CrossRef]
  10. J.-J. He, “Proposal for Q-Modulated Semiconductor Laser,” IEEE Photon. Technol. Lett. 19(5), 285–287 (2007).
    [CrossRef]
  11. D. Liu, L. Wang, and J.-J. He, “Rate Equation Analysis of High Speed Q-Modulated Semiconductor Laser,” J. Lightwave Technol. 28, 3128–3135 (2010).
  12. See, for example, L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).
  13. X. Li, Optoelectronic Devices - Design, Modeling and Simulation (Cambridge University Press, 2009).
  14. K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. 15(7), 566–570 (1979).
    [CrossRef]
  15. T. Yu, L. Wang, and J.-J. He, “Bloch Wave Formalism of Photon Lifetime in Distributed Feedback Lasers,” J. Opt. Soc. Am. B 26(9), 1780–1788 (2009).
    [CrossRef]

2010

2009

T. Yu, L. Wang, and J.-J. He, “Bloch Wave Formalism of Photon Lifetime in Distributed Feedback Lasers,” J. Opt. Soc. Am. B 26(9), 1780–1788 (2009).
[CrossRef]

U. Westergren, M. Chaciński, and L. Thylén, “Compact and efficient modulators for 100 Gb/s ETDM for telecom and interconnect applications,” Appl. Phys., A Mater. Sci. Process. 95(4), 1039–1044 (2009).
[CrossRef]

2007

J.-J. He, “Proposal for Q-Modulated Semiconductor Laser,” IEEE Photon. Technol. Lett. 19(5), 285–287 (2007).
[CrossRef]

2005

P. Tang, A. L. Meier, D. J. Towner, and B. W. Wessels, “High-speed Travelling-wave BaTiO3 Thin-film Electro-optic Modulators,” Electron. Lett. 41(23), 1296–1297 (2005).
[CrossRef]

C. W. Chow, C. S. Wong, and H. K. Tsang, “Reduction of Amplitude Transients and BER of Direct Modulation Laser Using Birefringent Fiber Loop,” IEEE Photon. Technol. Lett. 17(3), 693–695 (2005).
[CrossRef]

2004

K. Takagi, S. Shirai, Y. Tasuoka, C. Watatani, T. Ota, T. Takiguchi, T. Aoyagi, T. Nishimura, and N. Tomita, “120°C 10 Gb/s Uncooled Direct Modulated 1.3 AlGaInAs MQW DFB Laser Diodes,” IEEE Photon. Technol. Lett. 16, 2415–2417 (2004).
[CrossRef]

2001

M. Minakata, “Recent Progress of 40 GHz High-speed LiNbO3 Optical Modulator,” Active and Passive Optical Components for WDM Communication,” Proc. SPIE 4532, 16–27 (2001).
[CrossRef]

2000

Y. Kim, H. Lee, J. Lee, J. Han, T. W. Oh, and J. Jeong, “Chirp Characteristics of 10-Gb/s Electroabsorption Modulator Integrated DFB Lasers,” IEEE J. Quantum Electron. 36(8), 900–908 (2000).
[CrossRef]

1997

R. M. Spencer, “High speed Direct Modulation of Semiconductor Laser,” Int. J. High Speed Electron. Syst. 8(3), 377–416 (1997).
[CrossRef]

1987

M. Suzuki, Y. Noda, H. Tanaka, S. Akiba, Y. Kushiro, and H. Isshiki, “Monolithic Integration of InGaAsP/InP Distributed Feedback Laser and Electroabsorption Modulator by Vapor Phase Epitaxy,” J. Lightwave Technol. 5(9), 1277–1285 (1987).
[CrossRef]

1979

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. 15(7), 566–570 (1979).
[CrossRef]

Akiba, S.

M. Suzuki, Y. Noda, H. Tanaka, S. Akiba, Y. Kushiro, and H. Isshiki, “Monolithic Integration of InGaAsP/InP Distributed Feedback Laser and Electroabsorption Modulator by Vapor Phase Epitaxy,” J. Lightwave Technol. 5(9), 1277–1285 (1987).
[CrossRef]

Aoyagi, T.

K. Takagi, S. Shirai, Y. Tasuoka, C. Watatani, T. Ota, T. Takiguchi, T. Aoyagi, T. Nishimura, and N. Tomita, “120°C 10 Gb/s Uncooled Direct Modulated 1.3 AlGaInAs MQW DFB Laser Diodes,” IEEE Photon. Technol. Lett. 16, 2415–2417 (2004).
[CrossRef]

Chacinski, M.

U. Westergren, M. Chaciński, and L. Thylén, “Compact and efficient modulators for 100 Gb/s ETDM for telecom and interconnect applications,” Appl. Phys., A Mater. Sci. Process. 95(4), 1039–1044 (2009).
[CrossRef]

Chow, C. W.

C. W. Chow, C. S. Wong, and H. K. Tsang, “Reduction of Amplitude Transients and BER of Direct Modulation Laser Using Birefringent Fiber Loop,” IEEE Photon. Technol. Lett. 17(3), 693–695 (2005).
[CrossRef]

Han, J.

Y. Kim, H. Lee, J. Lee, J. Han, T. W. Oh, and J. Jeong, “Chirp Characteristics of 10-Gb/s Electroabsorption Modulator Integrated DFB Lasers,” IEEE J. Quantum Electron. 36(8), 900–908 (2000).
[CrossRef]

He, J.-J.

Isshiki, H.

M. Suzuki, Y. Noda, H. Tanaka, S. Akiba, Y. Kushiro, and H. Isshiki, “Monolithic Integration of InGaAsP/InP Distributed Feedback Laser and Electroabsorption Modulator by Vapor Phase Epitaxy,” J. Lightwave Technol. 5(9), 1277–1285 (1987).
[CrossRef]

Jeong, J.

Y. Kim, H. Lee, J. Lee, J. Han, T. W. Oh, and J. Jeong, “Chirp Characteristics of 10-Gb/s Electroabsorption Modulator Integrated DFB Lasers,” IEEE J. Quantum Electron. 36(8), 900–908 (2000).
[CrossRef]

Kim, Y.

Y. Kim, H. Lee, J. Lee, J. Han, T. W. Oh, and J. Jeong, “Chirp Characteristics of 10-Gb/s Electroabsorption Modulator Integrated DFB Lasers,” IEEE J. Quantum Electron. 36(8), 900–908 (2000).
[CrossRef]

Kushiro, Y.

M. Suzuki, Y. Noda, H. Tanaka, S. Akiba, Y. Kushiro, and H. Isshiki, “Monolithic Integration of InGaAsP/InP Distributed Feedback Laser and Electroabsorption Modulator by Vapor Phase Epitaxy,” J. Lightwave Technol. 5(9), 1277–1285 (1987).
[CrossRef]

Lee, H.

Y. Kim, H. Lee, J. Lee, J. Han, T. W. Oh, and J. Jeong, “Chirp Characteristics of 10-Gb/s Electroabsorption Modulator Integrated DFB Lasers,” IEEE J. Quantum Electron. 36(8), 900–908 (2000).
[CrossRef]

Lee, J.

Y. Kim, H. Lee, J. Lee, J. Han, T. W. Oh, and J. Jeong, “Chirp Characteristics of 10-Gb/s Electroabsorption Modulator Integrated DFB Lasers,” IEEE J. Quantum Electron. 36(8), 900–908 (2000).
[CrossRef]

Liu, D.

Meier, A. L.

P. Tang, A. L. Meier, D. J. Towner, and B. W. Wessels, “High-speed Travelling-wave BaTiO3 Thin-film Electro-optic Modulators,” Electron. Lett. 41(23), 1296–1297 (2005).
[CrossRef]

Minakata, M.

M. Minakata, “Recent Progress of 40 GHz High-speed LiNbO3 Optical Modulator,” Active and Passive Optical Components for WDM Communication,” Proc. SPIE 4532, 16–27 (2001).
[CrossRef]

Nishimura, T.

K. Takagi, S. Shirai, Y. Tasuoka, C. Watatani, T. Ota, T. Takiguchi, T. Aoyagi, T. Nishimura, and N. Tomita, “120°C 10 Gb/s Uncooled Direct Modulated 1.3 AlGaInAs MQW DFB Laser Diodes,” IEEE Photon. Technol. Lett. 16, 2415–2417 (2004).
[CrossRef]

Noda, Y.

M. Suzuki, Y. Noda, H. Tanaka, S. Akiba, Y. Kushiro, and H. Isshiki, “Monolithic Integration of InGaAsP/InP Distributed Feedback Laser and Electroabsorption Modulator by Vapor Phase Epitaxy,” J. Lightwave Technol. 5(9), 1277–1285 (1987).
[CrossRef]

Oh, T. W.

Y. Kim, H. Lee, J. Lee, J. Han, T. W. Oh, and J. Jeong, “Chirp Characteristics of 10-Gb/s Electroabsorption Modulator Integrated DFB Lasers,” IEEE J. Quantum Electron. 36(8), 900–908 (2000).
[CrossRef]

Ota, T.

K. Takagi, S. Shirai, Y. Tasuoka, C. Watatani, T. Ota, T. Takiguchi, T. Aoyagi, T. Nishimura, and N. Tomita, “120°C 10 Gb/s Uncooled Direct Modulated 1.3 AlGaInAs MQW DFB Laser Diodes,” IEEE Photon. Technol. Lett. 16, 2415–2417 (2004).
[CrossRef]

Petermann, K.

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. 15(7), 566–570 (1979).
[CrossRef]

Shirai, S.

K. Takagi, S. Shirai, Y. Tasuoka, C. Watatani, T. Ota, T. Takiguchi, T. Aoyagi, T. Nishimura, and N. Tomita, “120°C 10 Gb/s Uncooled Direct Modulated 1.3 AlGaInAs MQW DFB Laser Diodes,” IEEE Photon. Technol. Lett. 16, 2415–2417 (2004).
[CrossRef]

Spencer, R. M.

R. M. Spencer, “High speed Direct Modulation of Semiconductor Laser,” Int. J. High Speed Electron. Syst. 8(3), 377–416 (1997).
[CrossRef]

Suzuki, M.

M. Suzuki, Y. Noda, H. Tanaka, S. Akiba, Y. Kushiro, and H. Isshiki, “Monolithic Integration of InGaAsP/InP Distributed Feedback Laser and Electroabsorption Modulator by Vapor Phase Epitaxy,” J. Lightwave Technol. 5(9), 1277–1285 (1987).
[CrossRef]

Takagi, K.

K. Takagi, S. Shirai, Y. Tasuoka, C. Watatani, T. Ota, T. Takiguchi, T. Aoyagi, T. Nishimura, and N. Tomita, “120°C 10 Gb/s Uncooled Direct Modulated 1.3 AlGaInAs MQW DFB Laser Diodes,” IEEE Photon. Technol. Lett. 16, 2415–2417 (2004).
[CrossRef]

Takiguchi, T.

K. Takagi, S. Shirai, Y. Tasuoka, C. Watatani, T. Ota, T. Takiguchi, T. Aoyagi, T. Nishimura, and N. Tomita, “120°C 10 Gb/s Uncooled Direct Modulated 1.3 AlGaInAs MQW DFB Laser Diodes,” IEEE Photon. Technol. Lett. 16, 2415–2417 (2004).
[CrossRef]

Tanaka, H.

M. Suzuki, Y. Noda, H. Tanaka, S. Akiba, Y. Kushiro, and H. Isshiki, “Monolithic Integration of InGaAsP/InP Distributed Feedback Laser and Electroabsorption Modulator by Vapor Phase Epitaxy,” J. Lightwave Technol. 5(9), 1277–1285 (1987).
[CrossRef]

Tang, P.

P. Tang, A. L. Meier, D. J. Towner, and B. W. Wessels, “High-speed Travelling-wave BaTiO3 Thin-film Electro-optic Modulators,” Electron. Lett. 41(23), 1296–1297 (2005).
[CrossRef]

Tasuoka, Y.

K. Takagi, S. Shirai, Y. Tasuoka, C. Watatani, T. Ota, T. Takiguchi, T. Aoyagi, T. Nishimura, and N. Tomita, “120°C 10 Gb/s Uncooled Direct Modulated 1.3 AlGaInAs MQW DFB Laser Diodes,” IEEE Photon. Technol. Lett. 16, 2415–2417 (2004).
[CrossRef]

Thylén, L.

U. Westergren, M. Chaciński, and L. Thylén, “Compact and efficient modulators for 100 Gb/s ETDM for telecom and interconnect applications,” Appl. Phys., A Mater. Sci. Process. 95(4), 1039–1044 (2009).
[CrossRef]

Tomita, N.

K. Takagi, S. Shirai, Y. Tasuoka, C. Watatani, T. Ota, T. Takiguchi, T. Aoyagi, T. Nishimura, and N. Tomita, “120°C 10 Gb/s Uncooled Direct Modulated 1.3 AlGaInAs MQW DFB Laser Diodes,” IEEE Photon. Technol. Lett. 16, 2415–2417 (2004).
[CrossRef]

Towner, D. J.

P. Tang, A. L. Meier, D. J. Towner, and B. W. Wessels, “High-speed Travelling-wave BaTiO3 Thin-film Electro-optic Modulators,” Electron. Lett. 41(23), 1296–1297 (2005).
[CrossRef]

Tsang, H. K.

C. W. Chow, C. S. Wong, and H. K. Tsang, “Reduction of Amplitude Transients and BER of Direct Modulation Laser Using Birefringent Fiber Loop,” IEEE Photon. Technol. Lett. 17(3), 693–695 (2005).
[CrossRef]

Wang, L.

Watatani, C.

K. Takagi, S. Shirai, Y. Tasuoka, C. Watatani, T. Ota, T. Takiguchi, T. Aoyagi, T. Nishimura, and N. Tomita, “120°C 10 Gb/s Uncooled Direct Modulated 1.3 AlGaInAs MQW DFB Laser Diodes,” IEEE Photon. Technol. Lett. 16, 2415–2417 (2004).
[CrossRef]

Wessels, B. W.

P. Tang, A. L. Meier, D. J. Towner, and B. W. Wessels, “High-speed Travelling-wave BaTiO3 Thin-film Electro-optic Modulators,” Electron. Lett. 41(23), 1296–1297 (2005).
[CrossRef]

Westergren, U.

U. Westergren, M. Chaciński, and L. Thylén, “Compact and efficient modulators for 100 Gb/s ETDM for telecom and interconnect applications,” Appl. Phys., A Mater. Sci. Process. 95(4), 1039–1044 (2009).
[CrossRef]

Wong, C. S.

C. W. Chow, C. S. Wong, and H. K. Tsang, “Reduction of Amplitude Transients and BER of Direct Modulation Laser Using Birefringent Fiber Loop,” IEEE Photon. Technol. Lett. 17(3), 693–695 (2005).
[CrossRef]

Yu, T.

Appl. Phys., A Mater. Sci. Process.

U. Westergren, M. Chaciński, and L. Thylén, “Compact and efficient modulators for 100 Gb/s ETDM for telecom and interconnect applications,” Appl. Phys., A Mater. Sci. Process. 95(4), 1039–1044 (2009).
[CrossRef]

Electron. Lett.

P. Tang, A. L. Meier, D. J. Towner, and B. W. Wessels, “High-speed Travelling-wave BaTiO3 Thin-film Electro-optic Modulators,” Electron. Lett. 41(23), 1296–1297 (2005).
[CrossRef]

IEEE J. Quantum Electron.

Y. Kim, H. Lee, J. Lee, J. Han, T. W. Oh, and J. Jeong, “Chirp Characteristics of 10-Gb/s Electroabsorption Modulator Integrated DFB Lasers,” IEEE J. Quantum Electron. 36(8), 900–908 (2000).
[CrossRef]

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. 15(7), 566–570 (1979).
[CrossRef]

IEEE Photon. Technol. Lett.

K. Takagi, S. Shirai, Y. Tasuoka, C. Watatani, T. Ota, T. Takiguchi, T. Aoyagi, T. Nishimura, and N. Tomita, “120°C 10 Gb/s Uncooled Direct Modulated 1.3 AlGaInAs MQW DFB Laser Diodes,” IEEE Photon. Technol. Lett. 16, 2415–2417 (2004).
[CrossRef]

C. W. Chow, C. S. Wong, and H. K. Tsang, “Reduction of Amplitude Transients and BER of Direct Modulation Laser Using Birefringent Fiber Loop,” IEEE Photon. Technol. Lett. 17(3), 693–695 (2005).
[CrossRef]

J.-J. He, “Proposal for Q-Modulated Semiconductor Laser,” IEEE Photon. Technol. Lett. 19(5), 285–287 (2007).
[CrossRef]

Int. J. High Speed Electron. Syst.

R. M. Spencer, “High speed Direct Modulation of Semiconductor Laser,” Int. J. High Speed Electron. Syst. 8(3), 377–416 (1997).
[CrossRef]

J. Lightwave Technol.

M. Suzuki, Y. Noda, H. Tanaka, S. Akiba, Y. Kushiro, and H. Isshiki, “Monolithic Integration of InGaAsP/InP Distributed Feedback Laser and Electroabsorption Modulator by Vapor Phase Epitaxy,” J. Lightwave Technol. 5(9), 1277–1285 (1987).
[CrossRef]

D. Liu, L. Wang, and J.-J. He, “Rate Equation Analysis of High Speed Q-Modulated Semiconductor Laser,” J. Lightwave Technol. 28, 3128–3135 (2010).

J. Opt. Soc. Am. B

Proc. SPIE

M. Minakata, “Recent Progress of 40 GHz High-speed LiNbO3 Optical Modulator,” Active and Passive Optical Components for WDM Communication,” Proc. SPIE 4532, 16–27 (2001).
[CrossRef]

Other

H. Takeuchi, “Ultra-fast Electroabsorption Modulator Integrated DFB Lasers,” Proc. of IEEE International Conference On Indium Phosphide and Related Materials, 428–431 (2001).

See, for example, L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

X. Li, Optoelectronic Devices - Design, Modeling and Simulation (Cambridge University Press, 2009).

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Figures (8)

Fig. 1
Fig. 1

Schematic of the Q-modulated laser

Fig. 2
Fig. 2

Magnitude (a) and phase (b) of the effective reflectivity spectrum of the rear reflector when the absorption coefficient of the modulator section is 0 or 2000 cm−1.

Fig. 3
Fig. 3

Effective reflectivity of the rear reflector as a function of the absorption coefficient of the modulator waveguide, with the refractive index of the gap filler for Trench 1 as a parameter.

Fig. 4
Fig. 4

Threshold gain (a) and center wavelength (b) versus the phase.

Fig. 5
Fig. 5

Transmission gain spectra of the ON and OFF states

Fig. 6
Fig. 6

Output waveform (left) and frequency chirp and carrier density (right) of the QML under 40Gb/s RZ signal modulation when the injected current into the gain section is 60mA and the reflectivity of the rear reflector changes from 0.3 to 0.85 (a); and when the injected current is 80mA and the reflectivity of the rear reflector varies from 0.3 to 0.5 (b); from 0.3 to 0.85 (c); and from 0.5 to 0.85 (d).

Fig. 7
Fig. 7

Output waveform under 40Gb/s RZ signal modulation when the cavity length is (a) 1000 pitches; (b) 2000 pitches; (c) 3000 pitches; and (d) 4000 pitches.

Fig. 8
Fig. 8

Extinction ratio versus cavity length. The solid line is obtained by fitting the calculated data (circles) with a third-order polynomial.

Tables (2)

Tables Icon

Table 1 Parameters used in the travelling wave model

Tables Icon

Table 2 Parameters used for the four cases of Fig. 7

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

E( z,t )=[F( z,t ) e i β 0 z +R( z,t ) e i β 0 z ] e i w 0 t
1 v g F( z,t ) t + F( z,t ) z =( 1 2 g 1 2 α 0 i δ b i ω 0 c Δn )F( z,t )+iκR( z,t )+ s f
1 v g R( z,t ) t R( z,t ) z =( 1 2 g 1 2 α 0 i δ b i ω 0 c Δn )R( z,t )+iκF( z,t )+ s r
g( z,t )= Γ α g [ N( z,t ) N t ] 1+εP(z,t)
δ b = ω 0 c n eff π Λ
Δn= λ 0 4π Γα α g [ N( z,t )- N t ]
dN(z,t) dt = J ed AN(z,t)BN (z,t) 2 CN (z,t) 3 α g [ N( z,t ) N t ] 1+εP(z,t) v g P(z,t)
s f,r ( z,t ) s f,r * ( z ' ,t) =βK R sp δ(t t ' )δ(z z ' )/ v g
s f,r ( z,t ) s f,r ( z ' ,t) =0
( 1 v g t ± z )[ F( z,t ) R( z,t ) ]=[ A 11 ( z,t ) A 12 ( z,t ) A 21 ( z,t ) A 22 ( z,t ) ][ F( z,t ) R( z,t ) ]+[ S f ( z,t ) S r ( z,t ) ]
t=kΔt,k=0,1,2,.... z=nΔz,n=1,2,3,...
F( z,t ) t = 1 2 ( F n+1,k+1 F n+1,k Δt + F n,k+1 F n,k Δt )
F( z,t ) z = 1 2 ( F n+1,k+1 F n,k+1 Δz + F n+1,k F n,k Δz )
R( z,t ) t = 1 2 ( R n+1,k+1 R n+1,k Δt + R n,k+1 R n,k Δt )
R( z,t ) z = 1 2 ( R n+1,k+1 R n,k+1 Δz + R n+1,k R n,k Δz )
[ A 11 ( z,t ) A 12 ( z,t ) A 21 ( z,t ) A 22 ( z,t ) ][ F( z,t ) R( z,t ) ]+[ C f ( z,t ) C r ( z,t ) ]= 1 4 [ A n+1,k+1 11 A n+1,k+1 12 A n+1,k+1 21 A n+1,k+1 22 ][ F n+1,k+1 R n+1,k+1 ]+ 1 4 [ A n+1,k 11 A n+1,k 12 A n+1,k 21 A n+1,k 22 ][ F n+1,k R n+1,k ]+ 1 4 [ A n,k+1 11 A n,k+1 12 A n,k+1 21 A n,k+1 22 ][ F n,k+1 R n,k+1 ]+ 1 4 [ A n,k 11 A n,k 12 A n,k 21 A n,k 22 ][ F n,k R n,k ]+ 1 4 [ C n+1,k+1 f C n+1,k+1 r ]+ 1 4 [ C n+1,k f C n+1,k r ]+ 1 4 [ C n,k+1 f C n,k+1 r ]+ 1 4 [ C n,k f C n,k r ]
R( 0,t )=F( 0,t ) r eff
F( NΔz,t )=R( NΔz,t )r'
P out F ( t )= dw Γ v g hc λ ( | F( L,t ) | 2 | R( L,t ) | 2 )
Δf= 1 2π Φ F ( L,t ) t
4πn λ L m + Φ 1 + Φ 2 =(2m+1)π

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