Abstract

Optical spectra of signals at the output of semiconductor optical amplifiers (SOA) provide useful insight into amplifier nonlinearities. In this work, we determine the parameters of an analytical SOA model with a pump-probe experiment by evaluating the measured spectra of the pump and probe pulses at the SOA output. The analytical lumped SOA model considers carrier depletion, carrier recovery, spectral hole burning, two-photon absorption, and we include an additional effect termed ‘two-photon induced free-carrier absorption’, that is responsible for creating an identifiable blue-shifted component in the spectra. We are able to relate the underlying physical nonlinear effects to the spectral peculiarities of the output pump and probe spectra, and give guidelines for the exploitation of these nonlinear effects for optical signal processing. In addition, with a much-simplified SOA model and by replacing the pump pulse with modulated data we show that the output spectrum is altered in a manner consistent with phase patterning effects.

© 2017 Optical Society of America

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References

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2016 (2)

2015 (1)

2013 (3)

2012 (1)

P. Lunnemann, S. Ek, K. Yvind, R. Piron, and J. Mørk, “Nonlinear carrier dynamics in a quantum dash optical amplifier,” New J. Phys. 14(1), 013042 (2012).
[Crossref]

2011 (1)

S. Krishnamurthy, Z. G. Yu, L. P. Gonzalez, and S. Guha, “Temperature and wavelength-dependent two-photon and free-carrier absorption in GaAs, InP, GaInAs, and InAsP,” J. Appl. Phys. 109(3), 033102 (2011).
[Crossref]

2010 (1)

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear Silicon Photonics: Analytical Tools,” IEEE J. Sel. Top. Quantum Electron. 16(26), 200–215 (2010).
[Crossref]

2009 (1)

2008 (2)

2007 (3)

2006 (1)

R. Giller, R. J. Manning, and D. Cotter, “Gain and phase recovery of optically excited semiconductor optical amplifiers,” IEEE Photonics Technol. Lett. 18(9), 1061–1063 (2006).
[Crossref]

2004 (3)

2003 (2)

2002 (2)

2000 (1)

G. Klimeck, R. C. Bowen, T. B. Boykin, and T. A. Cwik, “sp3s* tight-binding parameters for transport simulations in compound semiconductors,” Superlattices Microstruct. 27(5–6), 519–524 (2000).
[Crossref]

1998 (2)

J. Leuthold, P. A. Besse, J. Eckner, E. Gamper, M. Dulk, and H. Melchior, “All-optical space switches with gain and principally ideal extinction ratios,” IEEE J. Quantum Electron. 34(4), 622–633 (1998).
[Crossref]

J. M. Tang and K. A. Shore, “Strong picosecond optical pulse propagation in semiconductor optical amplifiers at transparency,” IEEE J. Quantum Electron. 34(7), 1263–1269 (1998).
[Crossref]

1997 (2)

A. Mecozzi and J. Mørk, “Saturation effects in nondegenerate four-wave mixing between short optical pulses in semiconductor laser amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3(5), 1190–1207 (1997).
[Crossref]

A. Mecozzi and J. Mørk, “Saturation induced by picosecond pulses in semiconductor optical amplifiers,” J. Opt. Soc. Am. B 14(4), 761–770 (1997).
[Crossref]

1996 (2)

1994 (2)

M. G. Kane, I. Glesk, J. P. Sokoloff, and P. R. Prucnal, “Asymmetric optical loop mirror: analysis of an all-optical switch,” Appl. Opt. 33(29), 6833–6842 (1994).
[Crossref] [PubMed]

J. Mørk and A. Mecozzi, “Response function for gain and refractive index dynamics in active semiconductor waveguides,” Appl. Phys. Lett. 65(14), 1736–1738 (1994).
[Crossref]

1992 (1)

1991 (3)

C. T. Hultgren and E. P. Ippen, “Ultrafast refractive index dynamics in AlGaAs diode laser amplifiers,” Appl. Phys. Lett. 59(6), 635–637 (1991).
[Crossref]

M. Willatzen, A. Uskov, J. Mørk, H. Olesen, B. Tromborg, and A.-P. Jauho, “Nonlinear gain suppression in semiconductor lasers due to carrier heating,” IEEE Photonics Technol. Lett. 3(7), 606–609 (1991).
[Crossref]

R. S. Grant and W. Sibbett, “Observations of ultrafast nonlinear refraction in an InGaAsP optical amplifier,” Appl. Phys. Lett. 58(11), 1119–1121 (1991).
[Crossref]

1989 (1)

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25(11), 2297–2306 (1989).
[Crossref]

1982 (1)

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
[Crossref]

1981 (1)

C. H. Henry, R. A. Logan, and K. A. Bertness, “Spectral dependence of the change in refractive index due to carrier injection in GaAs lasers,” J. Appl. Phys. 52(7), 4457–4461 (1981).
[Crossref]

1966 (1)

E. M. Conwell and M. O. Vassell, “High-field distribution function in GaAs,” IEEE Trans. Electron Dev. 13(1), 22–26 (1966).
[Crossref]

Agrawal, G. P.

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear Silicon Photonics: Analytical Tools,” IEEE J. Sel. Top. Quantum Electron. 16(26), 200–215 (2010).
[Crossref]

M. Premaratne, D. Nesic, and G. P. Agrawal, “Pulse amplification and gain recovery in semiconductor optical amplifiers,” J. Lightwave Technol. 26(12), 1653–1660 (2008).
[Crossref]

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25(11), 2297–2306 (1989).
[Crossref]

Anthur, A. P.

Barry, L. P.

Ben Ezra, S.

J. Wang, A. Marculescu, J. Li, P. Vorreau, S. Tzadok, S. Ben Ezra, S. Tsadka, W. Freude, and J. Leuthold, “Pattern effect removal technique for semiconductor-optical-amplifier-based wavelength conversion,” IEEE Photonics Technol. Lett. 19(24), 1955–1957 (2007).
[Crossref]

Bennion, I.

Bertness, K. A.

C. H. Henry, R. A. Logan, and K. A. Bertness, “Spectral dependence of the change in refractive index due to carrier injection in GaAs lasers,” J. Appl. Phys. 52(7), 4457–4461 (1981).
[Crossref]

Besse, P. A.

J. Leuthold, P. A. Besse, J. Eckner, E. Gamper, M. Dulk, and H. Melchior, “All-optical space switches with gain and principally ideal extinction ratios,” IEEE J. Quantum Electron. 34(4), 622–633 (1998).
[Crossref]

Bimberg, D.

Boggess, T.

B. Olson, M. Gehlsen, and T. Boggess, “Nondegenerate two-photon absorption in GaSb,” Opt. Commun. 304, 54–57 (2013).
[Crossref]

Bonk, R.

Bowen, R. C.

G. Klimeck, R. C. Bowen, T. B. Boykin, and T. A. Cwik, “sp3s* tight-binding parameters for transport simulations in compound semiconductors,” Superlattices Microstruct. 27(5–6), 519–524 (2000).
[Crossref]

Boykin, T. B.

G. Klimeck, R. C. Bowen, T. B. Boykin, and T. A. Cwik, “sp3s* tight-binding parameters for transport simulations in compound semiconductors,” Superlattices Microstruct. 27(5–6), 519–524 (2000).
[Crossref]

Cabot, S.

Carroll, J. O.

Conwell, E. M.

E. M. Conwell and M. O. Vassell, “High-field distribution function in GaAs,” IEEE Trans. Electron Dev. 13(1), 22–26 (1966).
[Crossref]

Cotter, D.

R. Giller, R. J. Manning, and D. Cotter, “Gain and phase recovery of optically excited semiconductor optical amplifiers,” IEEE Photonics Technol. Lett. 18(9), 1061–1063 (2006).
[Crossref]

Cwik, T. A.

G. Klimeck, R. C. Bowen, T. B. Boykin, and T. A. Cwik, “sp3s* tight-binding parameters for transport simulations in compound semiconductors,” Superlattices Microstruct. 27(5–6), 519–524 (2000).
[Crossref]

Dagens, B.

M. L. Nielsen, B. Lavigne, and B. Dagens, “Polarity-preserving SOA-based wavelength conversion at 40 Gbit/s using bandpass filtering,” Electron. Lett. 39(18), 1334–1335 (2003).
[Crossref]

de Waardt, H.

Dorren, H. J.

Dorren, H. J. S.

Dorrer, C.

Duill, S. P. Ó.

Dulk, M.

J. Leuthold, P. A. Besse, J. Eckner, E. Gamper, M. Dulk, and H. Melchior, “All-optical space switches with gain and principally ideal extinction ratios,” IEEE J. Quantum Electron. 34(4), 622–633 (1998).
[Crossref]

Eckner, J.

J. Leuthold, P. A. Besse, J. Eckner, E. Gamper, M. Dulk, and H. Melchior, “All-optical space switches with gain and principally ideal extinction ratios,” IEEE J. Quantum Electron. 34(4), 622–633 (1998).
[Crossref]

Ek, S.

P. Lunnemann, S. Ek, K. Yvind, R. Piron, and J. Mørk, “Nonlinear carrier dynamics in a quantum dash optical amplifier,” New J. Phys. 14(1), 013042 (2012).
[Crossref]

Filion, B.

Freude, W.

Gamper, E.

J. Leuthold, P. A. Besse, J. Eckner, E. Gamper, M. Dulk, and H. Melchior, “All-optical space switches with gain and principally ideal extinction ratios,” IEEE J. Quantum Electron. 34(4), 622–633 (1998).
[Crossref]

Gehlsen, M.

B. Olson, M. Gehlsen, and T. Boggess, “Nondegenerate two-photon absorption in GaSb,” Opt. Commun. 304, 54–57 (2013).
[Crossref]

Giles, C. R.

Giller, R.

R. Giller, R. J. Manning, and D. Cotter, “Gain and phase recovery of optically excited semiconductor optical amplifiers,” IEEE Photonics Technol. Lett. 18(9), 1061–1063 (2006).
[Crossref]

Glesk, I.

Gonzalez, L. P.

S. Krishnamurthy, Z. G. Yu, L. P. Gonzalez, and S. Guha, “Temperature and wavelength-dependent two-photon and free-carrier absorption in GaAs, InP, GaInAs, and InAsP,” J. Appl. Phys. 109(3), 033102 (2011).
[Crossref]

Grant, R. S.

R. S. Grant and W. Sibbett, “Observations of ultrafast nonlinear refraction in an InGaAsP optical amplifier,” Appl. Phys. Lett. 58(11), 1119–1121 (1991).
[Crossref]

Guha, S.

S. Krishnamurthy, Z. G. Yu, L. P. Gonzalez, and S. Guha, “Temperature and wavelength-dependent two-photon and free-carrier absorption in GaAs, InP, GaInAs, and InAsP,” J. Appl. Phys. 109(3), 033102 (2011).
[Crossref]

Henry, C. H.

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
[Crossref]

C. H. Henry, R. A. Logan, and K. A. Bertness, “Spectral dependence of the change in refractive index due to carrier injection in GaAs lasers,” J. Appl. Phys. 52(7), 4457–4461 (1981).
[Crossref]

Herrera, J.

Hultgren, C. T.

C. T. Hultgren and E. P. Ippen, “Ultrafast refractive index dynamics in AlGaAs diode laser amplifiers,” Appl. Phys. Lett. 59(6), 635–637 (1991).
[Crossref]

Hutchings, D.

Ippen, E. P.

C. T. Hultgren and E. P. Ippen, “Ultrafast refractive index dynamics in AlGaAs diode laser amplifiers,” Appl. Phys. Lett. 59(6), 635–637 (1991).
[Crossref]

Jaques, J.

Jaques, J. J.

Jauho, A.-P.

M. Willatzen, A. Uskov, J. Mørk, H. Olesen, B. Tromborg, and A.-P. Jauho, “Nonlinear gain suppression in semiconductor lasers due to carrier heating,” IEEE Photonics Technol. Lett. 3(7), 606–609 (1991).
[Crossref]

Jiachuan, L.

Kane, M. G.

Kang, I.

Khoe, G. D.

Klimeck, G.

G. Klimeck, R. C. Bowen, T. B. Boykin, and T. A. Cwik, “sp3s* tight-binding parameters for transport simulations in compound semiconductors,” Superlattices Microstruct. 27(5–6), 519–524 (2000).
[Crossref]

Koonen, A. M. J.

Koos, C.

Krishnamurthy, S.

S. Krishnamurthy, Z. G. Yu, L. P. Gonzalez, and S. Guha, “Temperature and wavelength-dependent two-photon and free-carrier absorption in GaAs, InP, GaInAs, and InAsP,” J. Appl. Phys. 109(3), 033102 (2011).
[Crossref]

Laemmlin, M.

LaRochelle, S.

Lavigne, B.

M. L. Nielsen, B. Lavigne, and B. Dagens, “Polarity-preserving SOA-based wavelength conversion at 40 Gbit/s using bandpass filtering,” Electron. Lett. 39(18), 1334–1335 (2003).
[Crossref]

Leuthold, J.

T. Vallaitis, C. Koos, R. Bonk, W. Freude, M. Laemmlin, C. Meuer, D. Bimberg, and J. Leuthold, “Slow and fast dynamics of gain and phase in a quantum dot semiconductor optical amplifier,” Opt. Express 16(1), 170–178 (2008).
[Crossref] [PubMed]

J. Wang, A. Maitra, C. G. Poulton, W. Freude, and J. Leuthold, “Temporal dynamics of the alpha factor in semiconductor optical amplifiers,” J. Lightwave Technol. 25(3), 891–900 (2007).
[Crossref]

J. Wang, A. Marculescu, J. Li, P. Vorreau, S. Tzadok, S. Ben Ezra, S. Tsadka, W. Freude, and J. Leuthold, “Pattern effect removal technique for semiconductor-optical-amplifier-based wavelength conversion,” IEEE Photonics Technol. Lett. 19(24), 1955–1957 (2007).
[Crossref]

X. Wei and J. Leuthold, “Relation between vestigial-sideband filtering and pi/2 progressive phase shift,” Opt. Lett. 29(14), 1599–1601 (2004).
[Crossref] [PubMed]

J. Leuthold, D. M. Marom, S. Cabot, J. J. Jaques, R. Ryf, and C. R. Giles, “All-optical wavelength conversion using a pulse reformatting optical filter,” J. Lightwave Technol. 22(1), 186–192 (2004).
[Crossref]

J. Leuthold, R. Ryf, D. N. Maywar, S. Cabot, J. Jaques, and S. S. Patel, “Nonblocking all-optical cross connect based on regenerative all-optical wavelength converter in a transparent demonstration over 42 nodes and 16800 km,” J. Lightwave Technol. 21(11), 2863–2870 (2003).
[Crossref]

J. Leuthold, P. A. Besse, J. Eckner, E. Gamper, M. Dulk, and H. Melchior, “All-optical space switches with gain and principally ideal extinction ratios,” IEEE J. Quantum Electron. 34(4), 622–633 (1998).
[Crossref]

Li, J.

J. Wang, A. Marculescu, J. Li, P. Vorreau, S. Tzadok, S. Ben Ezra, S. Tsadka, W. Freude, and J. Leuthold, “Pattern effect removal technique for semiconductor-optical-amplifier-based wavelength conversion,” IEEE Photonics Technol. Lett. 19(24), 1955–1957 (2007).
[Crossref]

Li, Z.

Liu, Y.

Logan, R. A.

C. H. Henry, R. A. Logan, and K. A. Bertness, “Spectral dependence of the change in refractive index due to carrier injection in GaAs lasers,” J. Appl. Phys. 52(7), 4457–4461 (1981).
[Crossref]

Lunnemann, P.

P. Lunnemann, S. Ek, K. Yvind, R. Piron, and J. Mørk, “Nonlinear carrier dynamics in a quantum dash optical amplifier,” New J. Phys. 14(1), 013042 (2012).
[Crossref]

Maitra, A.

Manning, R. J.

R. Giller, R. J. Manning, and D. Cotter, “Gain and phase recovery of optically excited semiconductor optical amplifiers,” IEEE Photonics Technol. Lett. 18(9), 1061–1063 (2006).
[Crossref]

Marculescu, A.

J. Wang, A. Marculescu, J. Li, P. Vorreau, S. Tzadok, S. Ben Ezra, S. Tsadka, W. Freude, and J. Leuthold, “Pattern effect removal technique for semiconductor-optical-amplifier-based wavelength conversion,” IEEE Photonics Technol. Lett. 19(24), 1955–1957 (2007).
[Crossref]

Marom, D. M.

Martin, E.

Maywar, D. N.

Mecozzi, A.

A. Mecozzi and J. Mørk, “Saturation induced by picosecond pulses in semiconductor optical amplifiers,” J. Opt. Soc. Am. B 14(4), 761–770 (1997).
[Crossref]

A. Mecozzi and J. Mørk, “Saturation effects in nondegenerate four-wave mixing between short optical pulses in semiconductor laser amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3(5), 1190–1207 (1997).
[Crossref]

J. Mørk and A. Mecozzi, “Theory of the ultrafast optical response of active semiconductor waveguides,” J. Opt. Soc. Am. B 13(8), 1803–1816 (1996).
[Crossref]

A. Mecozzi and J. Mørk, “Theory of heterodyne pump–probe experiments with femtosecond pulses,” J. Opt. Soc. Am. B 13(11), 2437–2452 (1996).
[Crossref]

J. Mørk and A. Mecozzi, “Response function for gain and refractive index dynamics in active semiconductor waveguides,” Appl. Phys. Lett. 65(14), 1736–1738 (1994).
[Crossref]

Melchior, H.

J. Leuthold, P. A. Besse, J. Eckner, E. Gamper, M. Dulk, and H. Melchior, “All-optical space switches with gain and principally ideal extinction ratios,” IEEE J. Quantum Electron. 34(4), 622–633 (1998).
[Crossref]

Meuer, C.

Mørk, J.

P. Lunnemann, S. Ek, K. Yvind, R. Piron, and J. Mørk, “Nonlinear carrier dynamics in a quantum dash optical amplifier,” New J. Phys. 14(1), 013042 (2012).
[Crossref]

M. L. Nielsen and J. Mørk, “Increasing the modulation bandwidth of semiconductor-optical-amplifier-based switches by using optical filtering,” J. Opt. Soc. Am. B 21(9), 1606–1619 (2004).
[Crossref]

A. Mecozzi and J. Mørk, “Saturation induced by picosecond pulses in semiconductor optical amplifiers,” J. Opt. Soc. Am. B 14(4), 761–770 (1997).
[Crossref]

A. Mecozzi and J. Mørk, “Saturation effects in nondegenerate four-wave mixing between short optical pulses in semiconductor laser amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3(5), 1190–1207 (1997).
[Crossref]

J. Mørk and A. Mecozzi, “Theory of the ultrafast optical response of active semiconductor waveguides,” J. Opt. Soc. Am. B 13(8), 1803–1816 (1996).
[Crossref]

A. Mecozzi and J. Mørk, “Theory of heterodyne pump–probe experiments with femtosecond pulses,” J. Opt. Soc. Am. B 13(11), 2437–2452 (1996).
[Crossref]

J. Mørk and A. Mecozzi, “Response function for gain and refractive index dynamics in active semiconductor waveguides,” Appl. Phys. Lett. 65(14), 1736–1738 (1994).
[Crossref]

M. Willatzen, A. Uskov, J. Mørk, H. Olesen, B. Tromborg, and A.-P. Jauho, “Nonlinear gain suppression in semiconductor lasers due to carrier heating,” IEEE Photonics Technol. Lett. 3(7), 606–609 (1991).
[Crossref]

Naimi, S. T.

Nakamura, S.

Nesic, D.

Ng, W. C.

Nguyen, A. T.

Nielsen, M. L.

M. L. Nielsen and J. Mørk, “Increasing the modulation bandwidth of semiconductor-optical-amplifier-based switches by using optical filtering,” J. Opt. Soc. Am. B 21(9), 1606–1619 (2004).
[Crossref]

M. L. Nielsen, B. Lavigne, and B. Dagens, “Polarity-preserving SOA-based wavelength conversion at 40 Gbit/s using bandpass filtering,” Electron. Lett. 39(18), 1334–1335 (2003).
[Crossref]

O’Duill, S.

Olesen, H.

M. Willatzen, A. Uskov, J. Mørk, H. Olesen, B. Tromborg, and A.-P. Jauho, “Nonlinear gain suppression in semiconductor lasers due to carrier heating,” IEEE Photonics Technol. Lett. 3(7), 606–609 (1991).
[Crossref]

Olson, B.

B. Olson, M. Gehlsen, and T. Boggess, “Nondegenerate two-photon absorption in GaSb,” Opt. Commun. 304, 54–57 (2013).
[Crossref]

Olsson, N. A.

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25(11), 2297–2306 (1989).
[Crossref]

Patel, S. S.

Piron, R.

P. Lunnemann, S. Ek, K. Yvind, R. Piron, and J. Mørk, “Nonlinear carrier dynamics in a quantum dash optical amplifier,” New J. Phys. 14(1), 013042 (2012).
[Crossref]

Poulton, C. G.

Premaratne, M.

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear Silicon Photonics: Analytical Tools,” IEEE J. Sel. Top. Quantum Electron. 16(26), 200–215 (2010).
[Crossref]

M. Premaratne, D. Nesic, and G. P. Agrawal, “Pulse amplification and gain recovery in semiconductor optical amplifiers,” J. Lightwave Technol. 26(12), 1653–1660 (2008).
[Crossref]

Prucnal, P. R.

Raz, O.

Rukhlenko, I. D.

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear Silicon Photonics: Analytical Tools,” IEEE J. Sel. Top. Quantum Electron. 16(26), 200–215 (2010).
[Crossref]

Rusch, L. A.

Ryf, R.

Shi, K.

Shore, K. A.

J. M. Tang and K. A. Shore, “Strong picosecond optical pulse propagation in semiconductor optical amplifiers at transparency,” IEEE J. Quantum Electron. 34(7), 1263–1269 (1998).
[Crossref]

Shu, X.

Sibbett, W.

R. S. Grant and W. Sibbett, “Observations of ultrafast nonlinear refraction in an InGaAsP optical amplifier,” Appl. Phys. Lett. 58(11), 1119–1121 (1991).
[Crossref]

Sokoloff, J. P.

Tajima, K.

Tang, J. M.

J. M. Tang and K. A. Shore, “Strong picosecond optical pulse propagation in semiconductor optical amplifiers at transparency,” IEEE J. Quantum Electron. 34(7), 1263–1269 (1998).
[Crossref]

Tangdiongga, E.

Tromborg, B.

M. Willatzen, A. Uskov, J. Mørk, H. Olesen, B. Tromborg, and A.-P. Jauho, “Nonlinear gain suppression in semiconductor lasers due to carrier heating,” IEEE Photonics Technol. Lett. 3(7), 606–609 (1991).
[Crossref]

Tsadka, S.

J. Wang, A. Marculescu, J. Li, P. Vorreau, S. Tzadok, S. Ben Ezra, S. Tsadka, W. Freude, and J. Leuthold, “Pattern effect removal technique for semiconductor-optical-amplifier-based wavelength conversion,” IEEE Photonics Technol. Lett. 19(24), 1955–1957 (2007).
[Crossref]

Tzadok, S.

J. Wang, A. Marculescu, J. Li, P. Vorreau, S. Tzadok, S. Ben Ezra, S. Tsadka, W. Freude, and J. Leuthold, “Pattern effect removal technique for semiconductor-optical-amplifier-based wavelength conversion,” IEEE Photonics Technol. Lett. 19(24), 1955–1957 (2007).
[Crossref]

Ueno, Y.

Uskov, A.

M. Willatzen, A. Uskov, J. Mørk, H. Olesen, B. Tromborg, and A.-P. Jauho, “Nonlinear gain suppression in semiconductor lasers due to carrier heating,” IEEE Photonics Technol. Lett. 3(7), 606–609 (1991).
[Crossref]

Vallaitis, T.

Van Stryland, E.

Vassell, M. O.

E. M. Conwell and M. O. Vassell, “High-field distribution function in GaAs,” IEEE Trans. Electron Dev. 13(1), 22–26 (1966).
[Crossref]

Venkitesh, D.

Vorreau, P.

J. Wang, A. Marculescu, J. Li, P. Vorreau, S. Tzadok, S. Ben Ezra, S. Tsadka, W. Freude, and J. Leuthold, “Pattern effect removal technique for semiconductor-optical-amplifier-based wavelength conversion,” IEEE Photonics Technol. Lett. 19(24), 1955–1957 (2007).
[Crossref]

Walsh, A. J.

Wang, J.

J. Wang, A. Maitra, C. G. Poulton, W. Freude, and J. Leuthold, “Temporal dynamics of the alpha factor in semiconductor optical amplifiers,” J. Lightwave Technol. 25(3), 891–900 (2007).
[Crossref]

J. Wang, A. Marculescu, J. Li, P. Vorreau, S. Tzadok, S. Ben Ezra, S. Tsadka, W. Freude, and J. Leuthold, “Pattern effect removal technique for semiconductor-optical-amplifier-based wavelength conversion,” IEEE Photonics Technol. Lett. 19(24), 1955–1957 (2007).
[Crossref]

Watts, R. T.

Wei, X.

Willatzen, M.

M. Willatzen, A. Uskov, J. Mørk, H. Olesen, B. Tromborg, and A.-P. Jauho, “Nonlinear gain suppression in semiconductor lasers due to carrier heating,” IEEE Photonics Technol. Lett. 3(7), 606–609 (1991).
[Crossref]

Yu, Z. G.

S. Krishnamurthy, Z. G. Yu, L. P. Gonzalez, and S. Guha, “Temperature and wavelength-dependent two-photon and free-carrier absorption in GaAs, InP, GaInAs, and InAsP,” J. Appl. Phys. 109(3), 033102 (2011).
[Crossref]

Yvind, K.

P. Lunnemann, S. Ek, K. Yvind, R. Piron, and J. Mørk, “Nonlinear carrier dynamics in a quantum dash optical amplifier,” New J. Phys. 14(1), 013042 (2012).
[Crossref]

Zhang, X.

Zhou, R.

Appl. Opt. (1)

Appl. Phys. Lett. (3)

C. T. Hultgren and E. P. Ippen, “Ultrafast refractive index dynamics in AlGaAs diode laser amplifiers,” Appl. Phys. Lett. 59(6), 635–637 (1991).
[Crossref]

J. Mørk and A. Mecozzi, “Response function for gain and refractive index dynamics in active semiconductor waveguides,” Appl. Phys. Lett. 65(14), 1736–1738 (1994).
[Crossref]

R. S. Grant and W. Sibbett, “Observations of ultrafast nonlinear refraction in an InGaAsP optical amplifier,” Appl. Phys. Lett. 58(11), 1119–1121 (1991).
[Crossref]

Electron. Lett. (1)

M. L. Nielsen, B. Lavigne, and B. Dagens, “Polarity-preserving SOA-based wavelength conversion at 40 Gbit/s using bandpass filtering,” Electron. Lett. 39(18), 1334–1335 (2003).
[Crossref]

IEEE J. Quantum Electron. (4)

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25(11), 2297–2306 (1989).
[Crossref]

J. M. Tang and K. A. Shore, “Strong picosecond optical pulse propagation in semiconductor optical amplifiers at transparency,” IEEE J. Quantum Electron. 34(7), 1263–1269 (1998).
[Crossref]

J. Leuthold, P. A. Besse, J. Eckner, E. Gamper, M. Dulk, and H. Melchior, “All-optical space switches with gain and principally ideal extinction ratios,” IEEE J. Quantum Electron. 34(4), 622–633 (1998).
[Crossref]

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

A. Mecozzi and J. Mørk, “Saturation effects in nondegenerate four-wave mixing between short optical pulses in semiconductor laser amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3(5), 1190–1207 (1997).
[Crossref]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear Silicon Photonics: Analytical Tools,” IEEE J. Sel. Top. Quantum Electron. 16(26), 200–215 (2010).
[Crossref]

IEEE Photonics Technol. Lett. (3)

R. Giller, R. J. Manning, and D. Cotter, “Gain and phase recovery of optically excited semiconductor optical amplifiers,” IEEE Photonics Technol. Lett. 18(9), 1061–1063 (2006).
[Crossref]

M. Willatzen, A. Uskov, J. Mørk, H. Olesen, B. Tromborg, and A.-P. Jauho, “Nonlinear gain suppression in semiconductor lasers due to carrier heating,” IEEE Photonics Technol. Lett. 3(7), 606–609 (1991).
[Crossref]

J. Wang, A. Marculescu, J. Li, P. Vorreau, S. Tzadok, S. Ben Ezra, S. Tsadka, W. Freude, and J. Leuthold, “Pattern effect removal technique for semiconductor-optical-amplifier-based wavelength conversion,” IEEE Photonics Technol. Lett. 19(24), 1955–1957 (2007).
[Crossref]

IEEE Trans. Electron Dev. (1)

E. M. Conwell and M. O. Vassell, “High-field distribution function in GaAs,” IEEE Trans. Electron Dev. 13(1), 22–26 (1966).
[Crossref]

J. Appl. Phys. (2)

S. Krishnamurthy, Z. G. Yu, L. P. Gonzalez, and S. Guha, “Temperature and wavelength-dependent two-photon and free-carrier absorption in GaAs, InP, GaInAs, and InAsP,” J. Appl. Phys. 109(3), 033102 (2011).
[Crossref]

C. H. Henry, R. A. Logan, and K. A. Bertness, “Spectral dependence of the change in refractive index due to carrier injection in GaAs lasers,” J. Appl. Phys. 52(7), 4457–4461 (1981).
[Crossref]

J. Lightwave Technol. (7)

M. Premaratne, D. Nesic, and G. P. Agrawal, “Pulse amplification and gain recovery in semiconductor optical amplifiers,” J. Lightwave Technol. 26(12), 1653–1660 (2008).
[Crossref]

J. Wang, A. Maitra, C. G. Poulton, W. Freude, and J. Leuthold, “Temporal dynamics of the alpha factor in semiconductor optical amplifiers,” J. Lightwave Technol. 25(3), 891–900 (2007).
[Crossref]

S. T. Naimi, S. P. Ó. Duill, and L. P. Barry, “All optical wavelength conversion of Nyquist-WDM superchannels using FWM in SOAs,” J. Lightwave Technol. 33(19), 3959–3967 (2015).
[Crossref]

B. Filion, L. Jiachuan, A. T. Nguyen, X. Zhang, S. LaRochelle, and L. A. Rusch, “Semiconductor optical amplifier-based wavelength conversion of Nyquist-16QAM for flex-grid optical networks,” J. Lightwave Technol. 34(11), 2724–2729 (2016).
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Y. Liu, E. Tangdiongga, Z. Li, H. de Waardt, A. M. J. Koonen, G. D. Khoe, X. Shu, I. Bennion, and H. J. S. Dorren, “Error-free 320-Gb/s all-optical wavelength conversion using a single semiconductor optical amplifier,” J. Lightwave Technol. 25(1), 103–108 (2007).
[Crossref]

J. Leuthold, D. M. Marom, S. Cabot, J. J. Jaques, R. Ryf, and C. R. Giles, “All-optical wavelength conversion using a pulse reformatting optical filter,” J. Lightwave Technol. 22(1), 186–192 (2004).
[Crossref]

J. Leuthold, R. Ryf, D. N. Maywar, S. Cabot, J. Jaques, and S. S. Patel, “Nonblocking all-optical cross connect based on regenerative all-optical wavelength converter in a transparent demonstration over 42 nodes and 16800 km,” J. Lightwave Technol. 21(11), 2863–2870 (2003).
[Crossref]

J. Opt. Soc. Am. B (6)

New J. Phys. (1)

P. Lunnemann, S. Ek, K. Yvind, R. Piron, and J. Mørk, “Nonlinear carrier dynamics in a quantum dash optical amplifier,” New J. Phys. 14(1), 013042 (2012).
[Crossref]

Opt. Commun. (1)

B. Olson, M. Gehlsen, and T. Boggess, “Nondegenerate two-photon absorption in GaSb,” Opt. Commun. 304, 54–57 (2013).
[Crossref]

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G. Contestabile, “Wavelength conversion of PAM signals by XGM in SOAs,” in Asia Communications and Photonics Conference, OSA Technical Digest (Optical Society of America, 2016), paper AF4H.4.
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J. Wang, A. Marculescu, J. Li, Z. Zhang, W. Freude, and J. Leuthold, “All-optical vestigial-sideband signal generation and pattern effect mitigation with an SOA based red-shift optical filter wavelength converter,” in Proc. 33rd European Conference on Optical Communication, (ECOC 2008), paper We.2.C.6.
[Crossref]

A. Marculescu, S. Sygletos, J. Li, D. Karki, D. Hillerkuß, S. Ben-Ezra, S. Tsadka, W. Freude, and J. Leuthold, “RZ to CSRZ Format and Wavelength Conversion with Regenerative Properties,” in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, OFC 2009), paper OThS1.
[Crossref]

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Figures (10)

Fig. 1
Fig. 1 Schematic input and output pump power, probe output power, and refractive index change as a function of retarded time τ = t z / v g (t: real time, z: propagation coordinate, vg: group velocity). The dominant process for each time segment is marked. (a) Input pump pulse power (red line, z = 0 ) and output pump pulse power (blue line, z = L ). (b) Probe output light power at z = L . (c) Refractive index change Δn×(τ) according to Eq. (16), averaged over the SOA length. (CD: Carrier depletion; CH: Carrier heating; TPA: Two-photon absorption; SHB: Spectral hole burning; FCATPA: Free-carrier absorption induced by TPA; CC: Carrier cooling; CR: Carrier recovery)
Fig. 2
Fig. 2 Calculated time dependence of SOA gain G = exp(h×). The origin τ = 0 of the retarded time coincides with the maximum of the pump pulse Ps(0,τ), see Fig. 1. With our model we calculate ln (G) = h× and the phase shift Δ φ × = k 0 L Δ n × . (a) Gain and phase shift as a function of retarded time. The inset shows a blow-up. (b) Gain and (c) phase contributions of the various effects. (BF: Band filling; TPA: Two photon absorption; FCATPA: TPA-induced free-carrier absorption delayed by the intervalley scattering time τ 0 = 1 ps ; SHB: Spectral hole burning; CH: Carrier heating).
Fig. 3
Fig. 3 Calculated spectra (log scale) using the parameters from Table 1 and comparison with measured ones. (a) Measured (light gray lines) and calculated (solid black lines) spectra after the SOA of the pump pulse and (b) measured (light gray lines) and calculated (solid black lines) spectra after the SOA of the probe light. The decomposition of the spectra is shown in (c) for the pump pulse and in (d) for the CW pulse light. (BF: Band filling; TPA: Two photon absorption; FCATPA: Free-carrier absorption induced by TPA; SHB: Spectral hole burning; CH: Carrier heating).
Fig. 4
Fig. 4 Optical spectrum for Pin = 0, 3, 7 dBm; Pcw = 2 dBm; I = 400 mA of the (a) pump light and (b) probe light behind the SOA. The frequency origin corresponds to the input pump peak and to the probe light frequency, respectively.
Fig. 5
Fig. 5 Probe light phase shift and optical spectrum. (a) Phase shift dynamics and (b) corresponding induced spectral components in a measured probe light spectrum
Fig. 6
Fig. 6 Scheme of an optical filter based all-optical wavelength conversion system.
Fig. 7
Fig. 7 Measured spectra of a 40 Gb/s wavelength converted signal. Optical spectrum (a) directly after the SOA and (b) after the PROF.
Fig. 8
Fig. 8 Schematic description of a signal evolution. Amplitude (green line) and phase (red line) evolution of (a) an RZ signal ( PPS = 0 ) and (b) a CSRZ signal ( PPS = π ) after a filter supported wavelength conversion. A mark bit decreases the phase by ϕ 1 and a space bit increases the phase by ϕ 2 .
Fig. 9
Fig. 9 Spectral traces of a phase-patterning effect. Calculated spectra for standard (a) RZ and (b) CSRZ signals and for phase patterning affect (c) RZ and (d) CSRZ signals after wavelength conversion, respectively wavelength and format conversion.
Fig. 10
Fig. 10 Schematic of energy band diagram with TPA, intervalley scattering, FCA induced by TPA and FCA by holes. Two photons promote an electron from the light or heavy hole (LH/HH) valence band to the Γ - valley of the lowest conduction band (CB1) at an energy above the X-valley minimum. The photoexcited electron can either remain in the Γ- valley by simultaneous absorption of one photon and one short wave vector phonon, or it can be first scattered by a long wave vector optical phonon into the X-valley and then be lifted to the next higher conduction band (CB2) by absorbing a photon. In the process FCA by holes, an electron from spin orbit (SO) valence band absorbs a photon (without phonon assistance) and occupies a free-hole in the LH/HH valence bands. The one- and two-photon absorptions are shown by red vertical arrows, the phonon absorption by a green arrow and the intervalley scattering by a green dashed arrow.

Tables (2)

Tables Icon

Table 1 List of parameters used to calculate SOA gain and phase shift dynamics. * indicates typical value for the intraband process relaxation time, though not resolved in this work.

Tables Icon

Table 2 Total signal phase change for a bit sequence of a 110100 sequence with a filter set to generate a RZ Signal and a filter offset to generate a progressive phase shifted PPS.

Equations (104)

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

α = 2 k 0 n / N g / N 2 k 0 Δ n Δ g , k 0 = ω 0 c ,
E γ S , × ( z , t ) = P γ S , × ( z , t ) exp [ j φ γ S , × ( z , t ) ] exp [ j ( ω S , × t k 0 n 0 z ) ] .
E γ S , × in ( t ) = P γ S , × in ( t ) exp [ j φ γ S , × in ( t ) ] exp ( j ω S , × t )
E γ S , × out ( t ) = P γ S , × out ( t ) exp [ j φ γ S , × out ( t ) ] exp [ j ( ω S , × t k 0 n 0 L ) ] ,
E γ S , × ( z , t ) = P γ S , × ( z , t ) exp [ j φ γ S , × ( z , t ) ] .
Δ n _ γ S , × ( z , t ) = n _ γ S , × ( z , t ) n 0 = Δ n γ S , × ( z , t ) + j 2 k 0 g γ S , × ( z , t ) , g γ S , × ( z , t ) = g 0 + Δ g γ S , × ( z , t ) .
z E γ S , × ( z , t ) + 1 v g t E γ S , × ( z , t ) = j k 0 Δ n _ γ S , × ( z , t ) E γ S , × ( z , t ) .
P S , × ( z , τ ) = P γ S , × ( z , t ) , g S , × ( z , τ ) = g γ S , × ( z , t ) , φ S , × ( z , τ ) = φ γ S , × ( z , t ) , Δ n S , × ( z , τ ) = Δ n γ S , × ( z , t ) .
P S ( 0 , τ ) = P γ S in ( τ ) = P in ( τ ) , P × ( 0 , τ ) = P γ × in ( τ ) = P × = const , φ S , × ( 0 , τ ) = φ γ S , × in ( τ ) = 0.
P S , × ( L , τ ) = P γ S , × out ( τ + L / v g ) = P S , × out ( τ ) , φ S , × ( L , τ ) = φ γ S , × out ( τ + L / v g ) = φ S , × out ( τ ) .
z P S , × ( z , τ ) = g S , × ( z , τ ) P S , × ( z , τ )
z φ S , × ( z , τ ) = k 0 Δ n S , × ( z , τ ) ,
P S , × ( L , τ ) = P S , × ( 0 , τ ) exp [ h S , × ( τ ) ] , h S , × ( τ ) = 0 L g S , × ( z , τ ) d z .
P S , × ( L , τ ) P S , × ( 0 , τ ) = exp [ h S , × ( τ ) ] .
Δ h S , × ( τ ) = h S , × ( τ ) h 0 = 0 L [ g S , × ( z , τ ) g 0 ] d z .
Δ φ S , × ( τ ) = φ S , × ( L , τ ) φ S , × ( 0 , τ ) = k 0 L Δ n S , × ( τ ) , Δ n S , × ( τ ) = 1 L 0 L Δ n S , × ( z , τ ) d z .
φ S , × out ( τ ) = k 0 0 L Δ n S , × ( z , τ ) d z .
g ( z , τ ) = g 0 + x Δ g x ( z , τ ) , h ( τ ) = h 0 + Δ h ( τ ) , Δ h ( τ ) = x Δ h x ( τ ) , Δ h x ( τ ) = 0 L Δ g x ( z , τ ) d z .
α x = 2 k 0 Δ n x ( z , τ ) Δ g x ( z , τ ) , x= { BF, SHB, CH, TPA } .
Δ n ( z , τ ) = x Δ n x ( z , τ ) + Δ n FCA ( z , τ ) , Δ n x ( z , τ ) = α x 2 k 0 Δ g x ( z , τ ) .
Δ φ x ( τ ) = k 0 0 L Δ n x ( z , τ ) d z = 1 2 α x 0 L Δ g x ( z , τ ) d z , Δ φ FCA ( τ ) = k 0 0 L Δ n FCA ( z , τ ) d z .
φ out ( τ ) = 1 2 x α x Δ h x ( τ ) + Δ φ FCA ( τ ) .
Δ g BF ( z , τ ) = Γ a [ N ( z , τ ) N st ] ,
τ Δ g BF ( z , τ ) = [ g ( z , τ ) 1 2 Δ g TPA ( z , τ ) ] P S ( z , τ ) P sat τ s CD Δ g BF ( z , τ ) τ s CR .
g ( z , τ ) g 0 + Δ g CD ( z , τ ) .
z P CD ( z , τ ) = [ g 0 + Δ g CD ( z , τ ) ] P CD ( z , τ ) .
τ Δ g CD ( z , τ ) = [ g 0 + Δ g CD ( z , τ ) ] P CD ( z , τ ) P sat τ s .
d d τ Δ h CD ( τ ) = exp [ h 0 + Δ h CD ( τ ) ] 1 P sat τ s P in ( τ ) .
Δ h CD ( τ ) = ln { exp ( h 0 ) [ exp ( h 0 ) 1 ] exp [ τ P in ( τ ) d τ / ( P sat τ s ) ] } .
τ R P in ( τ ) d τ = P sat τ s l n [ exp ( h 0 ) 1 exp ( h 0 ) 10 q / 10 ] .
τ Δ g CR ( z , τ ) = Δ g CR ( z , τ ) τ s .
Δ h CR ( τ R ) = Δ h CD ( τ R ) .
Δ h BF ( τ ) = { Δ h CD ( τ ) τ τ R Δ h CD ( τ R ) exp ( τ τ R τ s ) τ > τ R .
Δ g SHB,CH ( z , τ ) = ε SHB,CH [ g ( z , τ ) Δ g TPA ( z , τ ) ] P S ( z , τ ) ,
Δ g SHB,CH ( z , τ ) = ε SHB,CH [ g 0 + Δ g BF ( z , τ ) ] P S ( z , τ ) 1 + ε P S ( z , τ ) .
Δ g SHB,CH ( z , τ ) = ε SHB,CH ε z { ln [ 1 + ε P CD ( z , τ ) ] } .
P CD ( 0 , τ ) = P in ( τ ) and P CD ( L , τ ) = P in ( τ ) exp [ h 0 + Δ h CD ( τ ) ]
Δ h SHB,CH ( τ ) = ε SHB,CH ε ln { 1 + ε P in ( τ ) exp [ h 0 + Δ h CD ( τ ) ] 1 + ε P in ( τ ) } .
Δ g TPA ( z , τ ) = β 2 ( ω S ) A TPA P S ( z , τ ) (degenerate case) .
Δ g TPA, S ( z , τ ) = β 2 ( ω S ) A TPA [ P S ( z , τ ) + β S ( ω S , e S , ω × , e × ) β 2 ( ω S ) P × ( z , τ ) ] .
Δ g TPA, × ( z , τ ) = β 2 ( ω × ) A TPA [ β × ( ω S , e S , ω × , e × ) β 2 ( ω × ) P S ( z , τ ) + P × ( z , τ ) ] .
r TPA = β S , × ( ω 0 , e S , ω 0 , e × ) β 2 ( ω 0 ) ,
Δ g TPA, S ( z , τ ) = β 2 A TPA [ P S ( z , τ ) + r TPA P × ( z , τ ) ]
Δ g TPA, × ( z , τ ) = β 2 A TPA [ r TPA P S ( z , τ ) + P × ( z , τ ) ] .
Δ g TPA, S ( z , τ ) = Δ g TPA ( z , τ ) , Δ g TPA, × ( z , τ ) = r TPA Δ g TPA ( z , τ ) .
Δ h TPA, S ( τ ) = Δ h TPA ( τ ) , Δ h TPA, × ( τ ) = r TPA Δ h TPA ( τ ) .
Δ φ TPA, S ( τ ) = 1 2 α TPA Δ h TPA ( τ ) , Δ φ TPA, × ( τ ) = 1 2 α TPA r TPA Δ h TPA ( τ ) .
ε TPA = β 2 L A TPA h 0
C 1 ( τ ) = | Δ h CD ( τ ) | exp [ | Δ h CD ( τ ) | ] 1 ,
Δ h TPA ( τ ) = ε TPA [ exp ( h 0 ) 1 ] C 1 ( τ ) P in ( τ ) .
Δ n FCA ( z , τ ) = e 2 2 ε 0 n 0 ω 0 2 m FCA N FCA ( z , τ ) ,
N FC A ( z , τ ) = K N e x ( z , τ τ 0 ) , 0 < K < 1.
τ N ex ( z , τ ) = 1 2 A [ Δ g TPA, S ( z , τ ) P S ( z , τ ) ω S + Δ g TPA, × ( z , τ ) P × ( z , τ ) ω × ] N ex ( z , τ ) τ ex .
N ex ( z , τ ) = τ ex 2 A [ Δ g TPA, S ( z , τ ) P S ( z , τ ) ω S + Δ g TPA, × ( z , τ ) P × ( z , τ ) ω × ] .
N ex ( z , τ ) = τ e x Δ g TPA ( z , τ ) P S ( z , τ ) 2 ω 0 A .
Δ n FCA ( z , τ ) = c ω 0 g 0 η FCA ε TPA P S 2 ( z , τ τ 0 ) , η FCA = K e 2 τ e x 4 ε 0 c m FCA ω 0 2 n 0 A
Δ φ FCA ( τ ) = η FCA ε TPA [ exp ( h 0 ) 1 ] 2 C 2 ( τ τ 0 ) P in 2 ( τ τ 0 ) .
C 2 ( τ ) = C 1 ( τ ) [ exp ( h 0 ) 1 ] 1 { exp [ h 0 + Δ h CD ( τ ) ] 1 } exp [ | Δ h CD ( τ ) | ] 1 .
Δ h S ( τ ) = Δ h ( τ ) = Δ h BF ( τ ) + Δ h SHB ( τ ) + Δ h CH ( τ ) + Δ h TPA ( τ ) .
G ( τ ) = exp [ h × ( τ ) ] = exp [ h 0 + Δ h × ( τ ) ]
Δ h × ( τ ) = Δ h BF ( τ ) + Δ h SHB ( τ ) + Δ h CH ( τ ) + r TPA Δ h TPA ( τ ) .
φ S out ( τ ) = φ out ( τ ) = 1 2 [ α BF Δ h BF ( τ ) + α ´CH Δ h CH ( τ ) + α ´TPA Δ h TPA ( τ ) ] + Δ φ FCA ( τ ) .
φ × out ( τ ) = 1 2 [ α BF Δ h BF ( τ ) + α ´CH Δ h CH ( τ ) + r TPA α ´TPA Δ h TPA ( τ ) ] + Δ φ FCA ( τ ) .
Θ S , × ( f ) = | 1 2 P γ S , × out ( t ) exp [ j φ γ S , × out ( t ) ] exp [ j ( k 0 n 0 L + 2 π f t ) ] d t | 2 .
P out ( τ ) = exp [ h 0 + Δ h S ( τ ) ] P in ( τ ) a n d P × out ( τ ) = exp [ h 0 + Δ h × ( τ ) ] P × ,
Θ S , × ( f ) = | 1 2 P S , × out ( τ ) exp [ j φ S , × out ( τ ) ] exp ( j 2 π f τ ) d τ | 2 .
X R = P S out ( τ R ) / P S out,peak = 4 % .
F S , × ( f ) = | Θ S , × measured ( f ) Θ S , × calculated ( f ) | Θ S , × measured ( f ) ,
| Δ g FCA ( z , τ ) | < < | Δ g BF ( z , τ ) | .
Δ g FCA ( z , τ ) = σ FCA N FCA ( z , τ ) .
N FCA ( z , τ ) | N ( z , τ ) N st | < < Γ a σ FCA .
N FCA ( z , τ ) | N ( z , τ ) N st | < < 0.58.
Δ n FCA ( z , τ ) = α FCA 2 k 0 Δ g FCA ( z , τ ) ; Δ n BF ( z , τ ) = α BF 2 k 0 Δ g BF ( z , τ ) .
| Δ n FCA ( z , τ ) | < < | α FCA | α BF | Δ n BF ( z , τ ) | .
Δ φ FCA ( τ ) < < | α FCA | α BF | Δ φ BF ( τ ) | .
α FCA < < Δ φ FCA ( τ dip ) | Δ φ BF ( τ dip ) | α BF .
z P CD ( z , τ ) = g CD ( z , τ ) P CD ( z , τ )
P CD ( 0 , τ ) = P in ( τ )
P CD ( z , τ ) = P in ( τ ) exp [ 0 z g CD ( z , τ ) d z ] .
τ g CD ( z , τ ) = g CD ( z , τ ) P CD ( z , τ ) P sat τ s ,
g CD ( z , ) = g 0 .
h CD ( z , τ ) = 0 z g CD ( z , τ ) d z
τ h CD ( z , τ ) = P in ( τ ) { exp [ h CD ( z , τ ) ] 1 } P sat τ s ,
h CD ( z , ) = g 0 z .
1 exp [ h CD ( z , τ ) ] 1 τ h CD ( z , τ ) = τ ln { 1 exp [ h CD ( z , τ ) ] } .
ln { 1 exp [ h CD ( z , τ ) ] 1 exp ( g 0 z ) } = 1 P sat τ s τ P in ( τ ) d τ .
P ( τ ) = exp [ 1 P sat τ s τ P in ( τ ) d τ ] ,
h CD ( z , τ ) = ln { 1 [ 1 exp ( g 0 z ) ] P ( τ ) } .
P CD ( z , τ ) = P in ( τ ) 1 [ 1 exp ( g 0 z ) ] P ( τ ) .
C n ( τ ) = g 0 [ exp ( h 0 ) 1 ] n 0 L [ P CD ( z , τ ) P in ( τ ) ] n d z ,
C n ( τ ) = g 0 ( G 0 1 ) n 0 L d z { 1 [ 1 exp ( g 0 z ) ] P ( τ ) } n .
u = 1 [ 1 exp ( g 0 z ) ] P .
u | z = L = 1 [ 1 exp ( h 0 ) ] P = G CD 1 ,
g 0 d z = d u u ( 1 P )
C n ( τ ) = ( G 0 1 ) n G CD 1 ( τ ) 1 d u u n [ u 1 + P ( τ ) ] .
d u u ( u b ) = 1 b ( 1 u b 1 u ) d u = 1 b ln ( 1 b u 1 )
b ( G 0 1 ) = G 0 G CD 1 ,
C 1 ( τ ) = [ G 0 G CD ( τ ) 1 ] 1 ln [ G 0 G CD ( τ ) ] .
d u u n ( u b ) = u b + b u n + 1 ( u b ) d u = 1 n u n + b d u u n + 1 ( u b ) .
C n + 1 ( τ ) = [ G 0 G CD ( τ ) 1 ] 1 [ C n ( τ ) G CD n ( τ ) 1 n ( G 0 1 ) n ] .
C 2 ( τ ) = [ G 0 G CD ( τ ) 1 ] 1 [ C 1 ( τ ) G CD ( τ ) 1 G 0 1 ] .
C n ( τ ) > 0.
d d τ C n ( τ ) = n g 0 ( G 0 1 ) n P in ( τ ) P sat τ s P ( τ ) 0 L [ 1 exp ( g 0 z ) ] d z { 1 [ 1 exp ( g 0 z ) P ( τ ) ] } n + 1 .
C n ( ) = G 0 n 1 n ( G 0 1 ) n .

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