Abstract

We propose multiple all-optical logic operations in complementary metal oxide semiconductor compatible silicon-on-insulator waveguides based on three nonlinear phenomena, stimulated Raman scattering, free carrier absorption, and cross phase modulation. The performance of three optical logic operations is simulated by use of the finite-difference time-domain method. We achieved an extinction ratio of approximately 13dB between two logic levels.

© 2009 Optical Society of America

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  1. R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express 11, 1731-1739 (2003).
    [CrossRef] [PubMed]
  2. A.Liu, M. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express 12, 4261-4268 (2004).
    [CrossRef] [PubMed]
  3. T. K. Liang and H. K. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 10, 1149-1153 (2004).
    [CrossRef]
  4. H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low loss silicon waveguide,” Appl. Phys. Lett. 85, 2196-2198 (2004).
    [CrossRef]
  5. M. Khorasaninejad, H. Kaatuzian, and S. S. Saini, “Improved efficiency of silicon Raman lasers with tapered waveguides,” in 21st Annual Lasers and Electro Optics Society Meeting (IEEE, 2008), 842-843.
    [CrossRef]
  6. A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, “Optical amplification and lasing by stimulated Raman scattering in silicon waveguides,” IEEE J. Lightwave Technol. 24, 1440-1455 (2006).
    [CrossRef]
  7. R. Jones, A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Lossless optical modulation in a silicon waveguide using stimulated Raman scattering,” Opt. Express 13, 1716-1723 (2005).
    [CrossRef] [PubMed]
  8. Q. Xu, V. Almeida, and M. Lipson, “Time-resolved study of Raman gain in highly confined silicon-on-insulator waveguides,” Opt. Express 12, 4437-4442 (2004).
    [CrossRef] [PubMed]
  9. Ö. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, “All optical switching and continuum generation in silicon waveguides,” Opt. Express 12, 4094-4102 (2004).
    [CrossRef] [PubMed]
  10. V. E. Perlin and H. G. Winful, “Nonlinear pulse switching using cross-phase modulation and fiber Bragg gratings,” IEEE Photon. Technol. Lett. 13, 960-962 (2001).
    [CrossRef]
  11. Q. Xu and M. Lipson, “All-optical logic based on silicon micro-ring resonators,” Opt. Express 15, 924-929 (2007).
    [CrossRef] [PubMed]
  12. T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171-174 (2006).
    [CrossRef]
  13. V. N. N. Passaro and F. De Leonardis, “All-optical AND gate based on Raman effect in silicon-on-insulator waveguide,” Opt. Quantum Electron. 38, 877-888 (2007).
    [CrossRef]
  14. D. C. Wheeler and D. C. Hall, “Optical interference logic in silicon-on-insulator waveguides,” Proc. SPIE 6130, 61300G (2006).
    [CrossRef]
  15. G. T. Reed and A. P. Knights, An Introduction to Silicon Photonics (Wiley, 2004).
    [CrossRef]
  16. N. Q. Ngo and L. N. Binh, “Novel realization of monotonic Butterworth-type lowpass, highpass, and bandpass optical filters using phase-modulated fiber-optic interferometers and ring resonators,” IEEE J. Lightwave Technol. 12, 827-841 (1994).
    [CrossRef]
  17. S. Diez, R. Ludwig, and H. G. Weber, “Gain-transparent SOA-switch for high-bitrate OTDM add/dropmultiplexing,” IEEE Photon. Technol. Lett. 11, 60-62 (1999).
    [CrossRef]

2007

V. N. N. Passaro and F. De Leonardis, “All-optical AND gate based on Raman effect in silicon-on-insulator waveguide,” Opt. Quantum Electron. 38, 877-888 (2007).
[CrossRef]

Q. Xu and M. Lipson, “All-optical logic based on silicon micro-ring resonators,” Opt. Express 15, 924-929 (2007).
[CrossRef] [PubMed]

2006

D. C. Wheeler and D. C. Hall, “Optical interference logic in silicon-on-insulator waveguides,” Proc. SPIE 6130, 61300G (2006).
[CrossRef]

A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, “Optical amplification and lasing by stimulated Raman scattering in silicon waveguides,” IEEE J. Lightwave Technol. 24, 1440-1455 (2006).
[CrossRef]

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171-174 (2006).
[CrossRef]

2005

2004

2003

2001

V. E. Perlin and H. G. Winful, “Nonlinear pulse switching using cross-phase modulation and fiber Bragg gratings,” IEEE Photon. Technol. Lett. 13, 960-962 (2001).
[CrossRef]

1999

S. Diez, R. Ludwig, and H. G. Weber, “Gain-transparent SOA-switch for high-bitrate OTDM add/dropmultiplexing,” IEEE Photon. Technol. Lett. 11, 60-62 (1999).
[CrossRef]

1994

N. Q. Ngo and L. N. Binh, “Novel realization of monotonic Butterworth-type lowpass, highpass, and bandpass optical filters using phase-modulated fiber-optic interferometers and ring resonators,” IEEE J. Lightwave Technol. 12, 827-841 (1994).
[CrossRef]

Liu, A.

A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, “Optical amplification and lasing by stimulated Raman scattering in silicon waveguides,” IEEE J. Lightwave Technol. 24, 1440-1455 (2006).
[CrossRef]

Abedin, K. S.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171-174 (2006).
[CrossRef]

Almeida, V.

Baets, R.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171-174 (2006).
[CrossRef]

Binh, L. N.

N. Q. Ngo and L. N. Binh, “Novel realization of monotonic Butterworth-type lowpass, highpass, and bandpass optical filters using phase-modulated fiber-optic interferometers and ring resonators,” IEEE J. Lightwave Technol. 12, 827-841 (1994).
[CrossRef]

Bogaerts, W.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171-174 (2006).
[CrossRef]

Boyraz, Ö.

Claps, R.

Cohen, O.

A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, “Optical amplification and lasing by stimulated Raman scattering in silicon waveguides,” IEEE J. Lightwave Technol. 24, 1440-1455 (2006).
[CrossRef]

R. Jones, A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Lossless optical modulation in a silicon waveguide using stimulated Raman scattering,” Opt. Express 13, 1716-1723 (2005).
[CrossRef] [PubMed]

A.Liu, M. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express 12, 4261-4268 (2004).
[CrossRef] [PubMed]

H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low loss silicon waveguide,” Appl. Phys. Lett. 85, 2196-2198 (2004).
[CrossRef]

De Leonardis, F.

V. N. N. Passaro and F. De Leonardis, “All-optical AND gate based on Raman effect in silicon-on-insulator waveguide,” Opt. Quantum Electron. 38, 877-888 (2007).
[CrossRef]

Diez, S.

S. Diez, R. Ludwig, and H. G. Weber, “Gain-transparent SOA-switch for high-bitrate OTDM add/dropmultiplexing,” IEEE Photon. Technol. Lett. 11, 60-62 (1999).
[CrossRef]

Dimitropoulos, D.

Dumon, P.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171-174 (2006).
[CrossRef]

Hak, D.

A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, “Optical amplification and lasing by stimulated Raman scattering in silicon waveguides,” IEEE J. Lightwave Technol. 24, 1440-1455 (2006).
[CrossRef]

R. Jones, A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Lossless optical modulation in a silicon waveguide using stimulated Raman scattering,” Opt. Express 13, 1716-1723 (2005).
[CrossRef] [PubMed]

A.Liu, M. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express 12, 4261-4268 (2004).
[CrossRef] [PubMed]

H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low loss silicon waveguide,” Appl. Phys. Lett. 85, 2196-2198 (2004).
[CrossRef]

Hall, D. C.

D. C. Wheeler and D. C. Hall, “Optical interference logic in silicon-on-insulator waveguides,” Proc. SPIE 6130, 61300G (2006).
[CrossRef]

Han, Y.

Jalali, B.

Jones, R.

A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, “Optical amplification and lasing by stimulated Raman scattering in silicon waveguides,” IEEE J. Lightwave Technol. 24, 1440-1455 (2006).
[CrossRef]

R. Jones, A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Lossless optical modulation in a silicon waveguide using stimulated Raman scattering,” Opt. Express 13, 1716-1723 (2005).
[CrossRef] [PubMed]

Kaatuzian, H.

M. Khorasaninejad, H. Kaatuzian, and S. S. Saini, “Improved efficiency of silicon Raman lasers with tapered waveguides,” in 21st Annual Lasers and Electro Optics Society Meeting (IEEE, 2008), 842-843.
[CrossRef]

Khorasaninejad, M.

M. Khorasaninejad, H. Kaatuzian, and S. S. Saini, “Improved efficiency of silicon Raman lasers with tapered waveguides,” in 21st Annual Lasers and Electro Optics Society Meeting (IEEE, 2008), 842-843.
[CrossRef]

Knights, A. P.

G. T. Reed and A. P. Knights, An Introduction to Silicon Photonics (Wiley, 2004).
[CrossRef]

Koonath, P.

Liang, T. K.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171-174 (2006).
[CrossRef]

T. K. Liang and H. K. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 10, 1149-1153 (2004).
[CrossRef]

Lipson, M.

Liu, A.

Ludwig, R.

S. Diez, R. Ludwig, and H. G. Weber, “Gain-transparent SOA-switch for high-bitrate OTDM add/dropmultiplexing,” IEEE Photon. Technol. Lett. 11, 60-62 (1999).
[CrossRef]

Miyazaki, T.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171-174 (2006).
[CrossRef]

Ngo, N. Q.

N. Q. Ngo and L. N. Binh, “Novel realization of monotonic Butterworth-type lowpass, highpass, and bandpass optical filters using phase-modulated fiber-optic interferometers and ring resonators,” IEEE J. Lightwave Technol. 12, 827-841 (1994).
[CrossRef]

Nicolaescu, R.

H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low loss silicon waveguide,” Appl. Phys. Lett. 85, 2196-2198 (2004).
[CrossRef]

Nunes, L. R.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171-174 (2006).
[CrossRef]

Paniccia, M.

A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, “Optical amplification and lasing by stimulated Raman scattering in silicon waveguides,” IEEE J. Lightwave Technol. 24, 1440-1455 (2006).
[CrossRef]

R. Jones, A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Lossless optical modulation in a silicon waveguide using stimulated Raman scattering,” Opt. Express 13, 1716-1723 (2005).
[CrossRef] [PubMed]

A.Liu, M. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express 12, 4261-4268 (2004).
[CrossRef] [PubMed]

H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low loss silicon waveguide,” Appl. Phys. Lett. 85, 2196-2198 (2004).
[CrossRef]

Passaro, V. N. N.

V. N. N. Passaro and F. De Leonardis, “All-optical AND gate based on Raman effect in silicon-on-insulator waveguide,” Opt. Quantum Electron. 38, 877-888 (2007).
[CrossRef]

Perlin, V. E.

V. E. Perlin and H. G. Winful, “Nonlinear pulse switching using cross-phase modulation and fiber Bragg gratings,” IEEE Photon. Technol. Lett. 13, 960-962 (2001).
[CrossRef]

Raghunathan, V.

Reed, G. T.

G. T. Reed and A. P. Knights, An Introduction to Silicon Photonics (Wiley, 2004).
[CrossRef]

Rong, H.

A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, “Optical amplification and lasing by stimulated Raman scattering in silicon waveguides,” IEEE J. Lightwave Technol. 24, 1440-1455 (2006).
[CrossRef]

R. Jones, A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Lossless optical modulation in a silicon waveguide using stimulated Raman scattering,” Opt. Express 13, 1716-1723 (2005).
[CrossRef] [PubMed]

H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low loss silicon waveguide,” Appl. Phys. Lett. 85, 2196-2198 (2004).
[CrossRef]

Rong, M.

Saini, S. S.

M. Khorasaninejad, H. Kaatuzian, and S. S. Saini, “Improved efficiency of silicon Raman lasers with tapered waveguides,” in 21st Annual Lasers and Electro Optics Society Meeting (IEEE, 2008), 842-843.
[CrossRef]

Thourhout, D. V.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171-174 (2006).
[CrossRef]

Tsang, H. K.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171-174 (2006).
[CrossRef]

T. K. Liang and H. K. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 10, 1149-1153 (2004).
[CrossRef]

Tsuchiya, M.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171-174 (2006).
[CrossRef]

Weber, H. G.

S. Diez, R. Ludwig, and H. G. Weber, “Gain-transparent SOA-switch for high-bitrate OTDM add/dropmultiplexing,” IEEE Photon. Technol. Lett. 11, 60-62 (1999).
[CrossRef]

Wheeler, D. C.

D. C. Wheeler and D. C. Hall, “Optical interference logic in silicon-on-insulator waveguides,” Proc. SPIE 6130, 61300G (2006).
[CrossRef]

Winful, H. G.

V. E. Perlin and H. G. Winful, “Nonlinear pulse switching using cross-phase modulation and fiber Bragg gratings,” IEEE Photon. Technol. Lett. 13, 960-962 (2001).
[CrossRef]

Xu, Q.

Appl. Phys. Lett.

H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low loss silicon waveguide,” Appl. Phys. Lett. 85, 2196-2198 (2004).
[CrossRef]

IEEE J. Lightwave Technol.

A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, “Optical amplification and lasing by stimulated Raman scattering in silicon waveguides,” IEEE J. Lightwave Technol. 24, 1440-1455 (2006).
[CrossRef]

N. Q. Ngo and L. N. Binh, “Novel realization of monotonic Butterworth-type lowpass, highpass, and bandpass optical filters using phase-modulated fiber-optic interferometers and ring resonators,” IEEE J. Lightwave Technol. 12, 827-841 (1994).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

T. K. Liang and H. K. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 10, 1149-1153 (2004).
[CrossRef]

IEEE Photon. Technol. Lett.

S. Diez, R. Ludwig, and H. G. Weber, “Gain-transparent SOA-switch for high-bitrate OTDM add/dropmultiplexing,” IEEE Photon. Technol. Lett. 11, 60-62 (1999).
[CrossRef]

V. E. Perlin and H. G. Winful, “Nonlinear pulse switching using cross-phase modulation and fiber Bragg gratings,” IEEE Photon. Technol. Lett. 13, 960-962 (2001).
[CrossRef]

Opt. Commun.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. V. Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. 265, 171-174 (2006).
[CrossRef]

Opt. Express

Opt. Quantum Electron.

V. N. N. Passaro and F. De Leonardis, “All-optical AND gate based on Raman effect in silicon-on-insulator waveguide,” Opt. Quantum Electron. 38, 877-888 (2007).
[CrossRef]

Proc. SPIE

D. C. Wheeler and D. C. Hall, “Optical interference logic in silicon-on-insulator waveguides,” Proc. SPIE 6130, 61300G (2006).
[CrossRef]

Other

G. T. Reed and A. P. Knights, An Introduction to Silicon Photonics (Wiley, 2004).
[CrossRef]

M. Khorasaninejad, H. Kaatuzian, and S. S. Saini, “Improved efficiency of silicon Raman lasers with tapered waveguides,” in 21st Annual Lasers and Electro Optics Society Meeting (IEEE, 2008), 842-843.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Net Raman gain versus pump intensity when the waveguide is pumped by a 17 ns long Gaussian pulse at wavelength of 1545 nm and a cw probe, Stokes signal at a wavelength of 1680 nm . Solid curve, the modeling result; symbols, experimental results adopted from [2].

Fig. 2
Fig. 2

Modeled profile of a Gaussian pulse with a peak pump intensity of 50 MW / cm 2 and a FWHM of 17 ns along a 4.8 cm SOI waveguide at three different positions.

Fig. 3
Fig. 3

Free carrier density profiles generated by TPA for different pulses with a carrier lifetime of 1 ns . Solid line, free carrier generated due to a 15 ns long Gaussian pulse with a peak power of I = 50 MW / cm 2 ; dashed line, free carrier generated due to a 150 ps long Gaussian pulse with a peak power of I = 50 MW / cm 2 ; dashed–dot line, free carrier generated due to a 150 ps long Gaussian pulse with a peak power of I = 160 MW / cm 2 .

Fig. 4
Fig. 4

Schematic diagram of a SOI waveguide used as an all- optical logic gate media. A and B are the Raman and the control pumps, respectively. Also, C is the output power at the Stokes wavelength.

Fig. 5
Fig. 5

Output power of the logic for different logic values of A and B. A, 100 ps long Gaussian pulse with a peak intensity of 50 MW / cm 2 ; B, 150 ps long Gaussian pulse with a peak power of 10 W .

Fig. 6
Fig. 6

Time transient wavelength shift due to pump pulse B with a FWHM of 50 ps and a peak power of 10 W .

Fig. 7
Fig. 7

Normalized output power for different logic values of A and B after a fifth-order bandpass Butterworth filter with a FWHM of 0.25 nm . A, 5 ps long Gaussian pulse with a peak power of 3 W ; B, 50 ps long Gaussian pulse with a peak power of 10 W .

Equations (12)

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

d I p ( t , z ) d z = α I p ( t , z ) β I p 2 ( t , z ) ( 2 β + g ) I p ( t , z ) I s ( t , z ) σ N ( t , z ) I p ( t , z ) ,
d I s ( t , z ) d z = α I s ( t , z ) β I s 2 ( t , z ) ( 2 β g ) I s ( t , z ) I p ( t , z ) σ N ( t , z ) I s ( t , z ) ,
d N ( t , z ) d t = β 2 h υ p I p 2 ( t , z ) + β 2 h υ s I s 2 ( t , z ) N ( t , z ) τ ,
G = 10 log ( I out I in ) ,
d I p ( t , z ) d z = α I p ( t , z ) β I p 2 ( t , z ) ( 2 β + g ) I p ( t , z ) I s ( t , z ) 2 β I p ( t , z ) ( I p * ( t , z ) + I s * ( t , z ) ) σ N ( t , z ) I p ( t , z ) ,
d I s ( t , z ) d z = α I s ( t , z ) β I s * 2 ( t , z ) ( 2 β g ) I s ( t , z ) I p ( t , z ) 2 β I s ( t , z ) ( I p * ( t , z ) + I s * ( t , z ) ) σ N ( t , z ) I s ( t , z ) ,
d I p * ( t , z ) d z = α I p * ( t , z ) β I p * 2 ( t , z ) ( 2 β + g ) I p * ( t , z ) I s * ( t , z ) 2 β I p * ( t , z ) ( I p ( t , z ) + I s ( t , z ) ) σ N ( t , z ) I p * ( t , z ) ,
d I s * ( t , z ) d z = α I s * ( t , z ) β I s * 2 ( t , z ) ( 2 β g ) I s * ( t , z ) I p * ( t , z ) 2 β I s * ( t , z ) ( I p ( t , z ) + I s ( t , z ) ) σ N ( t , z ) I s * ( t , z ) ,
d N ( t , z ) d t = β 2 h υ p I p 2 ( t , z ) + β 2 h υ s I s 2 ( t , z ) + β 2 h υ p * I p * 2 ( t , z ) + β 2 h υ s * I s * ( t , z ) N ( t , z ) τ ,
Δ n ( t ) = e 2 λ 2 8 π 2 c 2 ε 0 n ( Δ N e ( t ) m c e * + Δ N h ( t ) m c h * ) ,
Δ ϕ ( t ) = 2 π L λ Δ n ( t ) ,
Δ ω ( t ) = d d t Δ ϕ ( t ) .

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