Abstract

We discuss an all-optical DPSK wavelength conversion scheme comprising a delay-interferometer demodulation stage followed by a Mach-Zehnder interferometer, the arms of which are formed by nonlinear waveguides. If operated properly, the configuration shows regenerative behaviour. This is true for nonlinear waveguides with a dominant cross-gain nonlinearity (e. g., for an electro-absorption amplitude modulator) as well as for the case of a dominant cross-phase nonlinearity (e. g., for Kerr effect based phase modulator). In addition, we show that nonlinear materials exhibiting cross-gain modulation properties can provide a binary phase response so far only known from the transfer functions of digital electronics.

© 2009 OSA

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2009

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “'All-optical high-speed signal processing with silicon–organic hybrid slot waveguides',” Nat. Photonics 3(4), 216–219 (2009).
[CrossRef]

K. Croussore and G. Li, “Phase-regenerative wavelength conversion for BPSK and DPSK signals,” IEEE Photon. Technol. Lett. 21(2), 70–72 (2009).
[CrossRef]

2008

K. Croussore and G. Li, “Phase and amplitude regeneration of differential phase-shift keyed signals using phase-sensitive amplification,” IEEE J. Sel. Top. Quantum Electron. 14(3), 648–658 (2008).
[CrossRef]

M. Matsumoto and H. Sakaguchi, “DPSK signal regeneration using a fiber-based amplitude regenerator,” Opt. Express 16(15), 11169–11175 (2008).
[CrossRef] [PubMed]

2007

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]

K. Croussore and G. Li, “Amplitude regeneration of RZ-DPSK signals based on four-wave mixing in fibre,” Electron. Lett. 43(3), 177–178 (2007).
[CrossRef]

2006

2005

M. Matsumoto, “Regeneration of RZ-DPSK signals by fiber-based all-optical regenerators,” IEEE Photon. Technol. Lett. 17(5), 1055–1057 (2005).
[CrossRef]

T. Akiyama, M. Ekawa, M. Sugawara, K. Kawaguchi, H. Sudo, A. Kuramata, H. Ebe, and Y. Arakawa, “An ultrawide-band semiconductor optical amplifier having an extremely high penalty-free output power of 23 dBm achieved with quantum dots,” IEEE Photon. Technol. Lett. 17(8), 1614–1616 (2005).
[CrossRef]

2004

S. Schneider, P. Borri, W. Langbein, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Linewidth enhancement factor in InGaAs quantum-dot amplifiers,” IEEE J. Quantum Electron. 40(10), 1423–1429 (2004).
[CrossRef]

2003

A. Bilenca, R. Alizon, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett. 15(4), 563–565 (2003).
[CrossRef]

2002

2001

J. Leuthold, B. Mikkelsen, G. Raybon, C. H. Joyner, J. L. Pleumeekers, B. I. Miller, K. Dreyer, and R. Behringer, “All-optical wavelength conversion between 10 and 100 Gb/s with SOA delayed-interference configuration,” Opt. Quantum Electron. 33(7/10), 939–952 (2001).
[CrossRef]

1992

M. Suzuki, H. Tanaka, and Y. Matsushima, “InGaAsP electroabsorption modulator for high-bit-rate EDFA system,” IEEE Photon. Technol. Lett. 4(6), 586–588 (1992).
[CrossRef]

1982

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

Adolfsson, G.

Akiyama, T.

T. Akiyama, M. Ekawa, M. Sugawara, K. Kawaguchi, H. Sudo, A. Kuramata, H. Ebe, and Y. Arakawa, “An ultrawide-band semiconductor optical amplifier having an extremely high penalty-free output power of 23 dBm achieved with quantum dots,” IEEE Photon. Technol. Lett. 17(8), 1614–1616 (2005).
[CrossRef]

Alizon, R.

A. Bilenca, R. Alizon, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett. 15(4), 563–565 (2003).
[CrossRef]

Arakawa, Y.

T. Akiyama, M. Ekawa, M. Sugawara, K. Kawaguchi, H. Sudo, A. Kuramata, H. Ebe, and Y. Arakawa, “An ultrawide-band semiconductor optical amplifier having an extremely high penalty-free output power of 23 dBm achieved with quantum dots,” IEEE Photon. Technol. Lett. 17(8), 1614–1616 (2005).
[CrossRef]

Baets, R.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “'All-optical high-speed signal processing with silicon–organic hybrid slot waveguides',” Nat. Photonics 3(4), 216–219 (2009).
[CrossRef]

Behringer, R.

J. Leuthold, B. Mikkelsen, G. Raybon, C. H. Joyner, J. L. Pleumeekers, B. I. Miller, K. Dreyer, and R. Behringer, “All-optical wavelength conversion between 10 and 100 Gb/s with SOA delayed-interference configuration,” Opt. Quantum Electron. 33(7/10), 939–952 (2001).
[CrossRef]

Biaggio, I.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “'All-optical high-speed signal processing with silicon–organic hybrid slot waveguides',” Nat. Photonics 3(4), 216–219 (2009).
[CrossRef]

Bilenca, A.

A. Bilenca, R. Alizon, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett. 15(4), 563–565 (2003).
[CrossRef]

Bimberg, D.

S. Schneider, P. Borri, W. Langbein, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Linewidth enhancement factor in InGaAs quantum-dot amplifiers,” IEEE J. Quantum Electron. 40(10), 1423–1429 (2004).
[CrossRef]

Boettger, G.

P. Vorreau, A. Marculescu, J. Wang, G. Boettger, B. Sartorius, C. Bornholdt, J. Slovak, M. Schlak, Ch. Schmidt, S. Tsadka, W. Freude, and J. Leuthold, “Cascadability and regenerative properties of SOA all-optical DPSK wavelength converters,” IEEE Photon. Technol. Lett. 18(18), 1970–1972 (2006).
[CrossRef]

Bogaerts, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “'All-optical high-speed signal processing with silicon–organic hybrid slot waveguides',” Nat. Photonics 3(4), 216–219 (2009).
[CrossRef]

Bornholdt, C.

P. Vorreau, A. Marculescu, J. Wang, G. Boettger, B. Sartorius, C. Bornholdt, J. Slovak, M. Schlak, Ch. Schmidt, S. Tsadka, W. Freude, and J. Leuthold, “Cascadability and regenerative properties of SOA all-optical DPSK wavelength converters,” IEEE Photon. Technol. Lett. 18(18), 1970–1972 (2006).
[CrossRef]

Borri, P.

S. Schneider, P. Borri, W. Langbein, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Linewidth enhancement factor in InGaAs quantum-dot amplifiers,” IEEE J. Quantum Electron. 40(10), 1423–1429 (2004).
[CrossRef]

Chen, J. J.

Croussore, K.

K. Croussore and G. Li, “Phase-regenerative wavelength conversion for BPSK and DPSK signals,” IEEE Photon. Technol. Lett. 21(2), 70–72 (2009).
[CrossRef]

K. Croussore and G. Li, “Phase and amplitude regeneration of differential phase-shift keyed signals using phase-sensitive amplification,” IEEE J. Sel. Top. Quantum Electron. 14(3), 648–658 (2008).
[CrossRef]

K. Croussore and G. Li, “Amplitude regeneration of RZ-DPSK signals based on four-wave mixing in fibre,” Electron. Lett. 43(3), 177–178 (2007).
[CrossRef]

Devgan, P.

Diederich, F.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “'All-optical high-speed signal processing with silicon–organic hybrid slot waveguides',” Nat. Photonics 3(4), 216–219 (2009).
[CrossRef]

Dreyer, K.

J. Leuthold, B. Mikkelsen, G. Raybon, C. H. Joyner, J. L. Pleumeekers, B. I. Miller, K. Dreyer, and R. Behringer, “All-optical wavelength conversion between 10 and 100 Gb/s with SOA delayed-interference configuration,” Opt. Quantum Electron. 33(7/10), 939–952 (2001).
[CrossRef]

Dumon, P.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “'All-optical high-speed signal processing with silicon–organic hybrid slot waveguides',” Nat. Photonics 3(4), 216–219 (2009).
[CrossRef]

Ebe, H.

T. Akiyama, M. Ekawa, M. Sugawara, K. Kawaguchi, H. Sudo, A. Kuramata, H. Ebe, and Y. Arakawa, “An ultrawide-band semiconductor optical amplifier having an extremely high penalty-free output power of 23 dBm achieved with quantum dots,” IEEE Photon. Technol. Lett. 17(8), 1614–1616 (2005).
[CrossRef]

Eggleton, B. J.

Eisenstein, G.

A. Bilenca, R. Alizon, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett. 15(4), 563–565 (2003).
[CrossRef]

Ekawa, M.

T. Akiyama, M. Ekawa, M. Sugawara, K. Kawaguchi, H. Sudo, A. Kuramata, H. Ebe, and Y. Arakawa, “An ultrawide-band semiconductor optical amplifier having an extremely high penalty-free output power of 23 dBm achieved with quantum dots,” IEEE Photon. Technol. Lett. 17(8), 1614–1616 (2005).
[CrossRef]

Esembeson, B.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “'All-optical high-speed signal processing with silicon–organic hybrid slot waveguides',” Nat. Photonics 3(4), 216–219 (2009).
[CrossRef]

Forchel, A.

A. Bilenca, R. Alizon, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett. 15(4), 563–565 (2003).
[CrossRef]

Freude, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “'All-optical high-speed signal processing with silicon–organic hybrid slot waveguides',” Nat. Photonics 3(4), 216–219 (2009).
[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]

P. Vorreau, A. Marculescu, J. Wang, G. Boettger, B. Sartorius, C. Bornholdt, J. Slovak, M. Schlak, Ch. Schmidt, S. Tsadka, W. Freude, and J. Leuthold, “Cascadability and regenerative properties of SOA all-optical DPSK wavelength converters,” IEEE Photon. Technol. Lett. 18(18), 1970–1972 (2006).
[CrossRef]

Fu, L.

Gold, D.

A. Bilenca, R. Alizon, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett. 15(4), 563–565 (2003).
[CrossRef]

Grigoryan, V. S.

Henry, C. H.

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

Johannisson, P.

Joyner, C. H.

J. Leuthold, B. Mikkelsen, G. Raybon, C. H. Joyner, J. L. Pleumeekers, B. I. Miller, K. Dreyer, and R. Behringer, “All-optical wavelength conversion between 10 and 100 Gb/s with SOA delayed-interference configuration,” Opt. Quantum Electron. 33(7/10), 939–952 (2001).
[CrossRef]

Karlsson, M.

Kawaguchi, K.

T. Akiyama, M. Ekawa, M. Sugawara, K. Kawaguchi, H. Sudo, A. Kuramata, H. Ebe, and Y. Arakawa, “An ultrawide-band semiconductor optical amplifier having an extremely high penalty-free output power of 23 dBm achieved with quantum dots,” IEEE Photon. Technol. Lett. 17(8), 1614–1616 (2005).
[CrossRef]

Koos, C.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “'All-optical high-speed signal processing with silicon–organic hybrid slot waveguides',” Nat. Photonics 3(4), 216–219 (2009).
[CrossRef]

Kumar, P.

Kuramata, A.

T. Akiyama, M. Ekawa, M. Sugawara, K. Kawaguchi, H. Sudo, A. Kuramata, H. Ebe, and Y. Arakawa, “An ultrawide-band semiconductor optical amplifier having an extremely high penalty-free output power of 23 dBm achieved with quantum dots,” IEEE Photon. Technol. Lett. 17(8), 1614–1616 (2005).
[CrossRef]

Langbein, W.

S. Schneider, P. Borri, W. Langbein, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Linewidth enhancement factor in InGaAs quantum-dot amplifiers,” IEEE J. Quantum Electron. 40(10), 1423–1429 (2004).
[CrossRef]

Lasri, J.

Leuthold, J.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “'All-optical high-speed signal processing with silicon–organic hybrid slot waveguides',” Nat. Photonics 3(4), 216–219 (2009).
[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]

P. Vorreau, A. Marculescu, J. Wang, G. Boettger, B. Sartorius, C. Bornholdt, J. Slovak, M. Schlak, Ch. Schmidt, S. Tsadka, W. Freude, and J. Leuthold, “Cascadability and regenerative properties of SOA all-optical DPSK wavelength converters,” IEEE Photon. Technol. Lett. 18(18), 1970–1972 (2006).
[CrossRef]

J. Leuthold, B. Mikkelsen, G. Raybon, C. H. Joyner, J. L. Pleumeekers, B. I. Miller, K. Dreyer, and R. Behringer, “All-optical wavelength conversion between 10 and 100 Gb/s with SOA delayed-interference configuration,” Opt. Quantum Electron. 33(7/10), 939–952 (2001).
[CrossRef]

Li, G.

K. Croussore and G. Li, “Phase-regenerative wavelength conversion for BPSK and DPSK signals,” IEEE Photon. Technol. Lett. 21(2), 70–72 (2009).
[CrossRef]

K. Croussore and G. Li, “Phase and amplitude regeneration of differential phase-shift keyed signals using phase-sensitive amplification,” IEEE J. Sel. Top. Quantum Electron. 14(3), 648–658 (2008).
[CrossRef]

K. Croussore and G. Li, “Amplitude regeneration of RZ-DPSK signals based on four-wave mixing in fibre,” Electron. Lett. 43(3), 177–178 (2007).
[CrossRef]

Littler, I. C.

Maitra, A.

Marculescu, A.

P. Vorreau, A. Marculescu, J. Wang, G. Boettger, B. Sartorius, C. Bornholdt, J. Slovak, M. Schlak, Ch. Schmidt, S. Tsadka, W. Freude, and J. Leuthold, “Cascadability and regenerative properties of SOA all-optical DPSK wavelength converters,” IEEE Photon. Technol. Lett. 18(18), 1970–1972 (2006).
[CrossRef]

Matsumoto, M.

M. Matsumoto and H. Sakaguchi, “DPSK signal regeneration using a fiber-based amplitude regenerator,” Opt. Express 16(15), 11169–11175 (2008).
[CrossRef] [PubMed]

M. Matsumoto, “Regeneration of RZ-DPSK signals by fiber-based all-optical regenerators,” IEEE Photon. Technol. Lett. 17(5), 1055–1057 (2005).
[CrossRef]

Matsushima, Y.

M. Suzuki, H. Tanaka, and Y. Matsushima, “InGaAsP electroabsorption modulator for high-bit-rate EDFA system,” IEEE Photon. Technol. Lett. 4(6), 586–588 (1992).
[CrossRef]

Michinobu, T.

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A. Bilenca, R. Alizon, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett. 15(4), 563–565 (2003).
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A. Bilenca, R. Alizon, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett. 15(4), 563–565 (2003).
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Figures (5)

Fig. 1
Fig. 1

DPSK wavelength converter schematic with nonlinear elements (NLE) in the arms of a Mach-Zehnder interferometer (MZI). Symbols () represents 2x2 couplers of the interferometers. An DPSK signal with electric field E in at a wavelength λ in is demodulated by a delay interferometer (DI, time delay difference Δt equals bit period T b) resulting in an OOK signal E up and an inverse OOK signal E low,. An electric field E cnv at the new “converted” wavelength λ cnv passes both NLE resulting in the fields E cnv,up and jE cnv,low in the upper and lower MZI arms. At the difference output port ∆, a filter selects the “converted” wavelength λ cnv resulting in a signal E Δ. A schematic representation of E Δ is sketched along with E cnv,up and E cnv,low for the cases (a) α Η ≠ 0 (mostly XPM, and XGM) and (b) α Η = 0 (XGM only). The optical output signal at port ∆ has been converted to PSK format, and the balanced receiver Rx requires differential encoding for recovering the original data.

Fig. 2
Fig. 2

Power and phase responses of DPSK wavelength converters as a function of input DPSK signal power (left column, (I)) and phase difference ∆Φ (right column, (II)) between consecutive DPSK bits at the input. The power of the input DPSK signal is normalized to a reference input power P in,ref, which is explained in the paragraph before Eq. (26), while the power response is normalized to a value γ2exp(h cnv)/4. The phase difference ∆Φin is specified in Eq. (22). The MZI in the DPSK wavelength converter comprises (A) ideal amplitude modulators, (B) modulators showing amplitude and phase modulation (here an ordinary bulk SOA with αH = 8 and 0 < P eq < ∞), and (C) ideal phase modulators. The labels “1” and “−1” denote the respective ideal DPSK states. The corresponding optimum operating points are marked with short arrows pointing to filled circles (●), whose input powers are used to generate the plots in the right column (II).

Fig. 3
Fig. 3

Constellation diagrams. (a) Input DPSK signal (b) Output wavelength-converted PSK signals for various αH -factors, corresponding to cases (A), (B) and (C) in Fig. 2. Areas in (a) are mapped onto the corresponding constellation areas in (b) having same shading. The amplitude is normalized to the amplitude value at the operating point.

Fig. 4
Fig. 4

Noise suppression of the cascaded XGM-based (αH = 0) DPSK wavelength converter. (a) Schematic 2-stage XGM-based (αH = 0) wavelength converters. Note that the wavelengths of the two converting signals λ cnv,1 and λ cnv,2 are not necessary the same. (b) and (c) show simulation results of the standard deviations of the output amplitude noise σ a out,1(2) after the first and second wavelength converter stages, while the bottom axes are the standard deviation of the input amplitude noise σ a in in front of the first stage. The standard deviations of the input phase noise σph in before the first stage are indicated in the legends. Dash-dotted lines are the borders of the regeneration regions.

Fig. 5
Fig. 5

Noise suppression property of an ideal XPM-based (αH –> ∞) DPSK wavelength converter. (a) Standard deviation of the output amplitude noise σ a out versus standard deviation of input amplitude noise σ a in when the standard deviation of the input phase noise σph in is 0 or 0.25 radians. (b) Standard deviation of output phase noise σph out versus standard deviation of input phase noise σph in when varying the standard deviation of the input amplitude noise σ a in in an interval [0, 0.25]. Dash-dotted lines are the borders of the regeneration regions.

Equations (47)

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n ¯ cnv = n r , cnv j n i , cnv .
n ¯ ( t ) = n r ( t ) j n i ( t ) ,         n r ( t ) = n r , cnv + Δ n r ( t ) ,       and       n i ( t ) = n i , cnv + Δ n i ( t ) ,
α H = Δ n r ( t ) / Δ n i ( t ) .
α H , SOA = n r / N n i / N Δ n r ( t ) Δ n i ( t ) .
T NLE ( t ) = | T NLE ( t ) | exp j φ ( t ) = exp ( j   k 0 0 L n ¯ ( t , z )   d z ) π ,
exp ( h ( t ) 2 ) : = exp [ k 0 0 L n i ( t , z ) d z ] , φ ( t ) : = k 0 0 L n r ( t , z ) d z .
Δ h ( t ) = h ( t ) h cnv = 2 k 0 0 L Δ n i ( t , z ) d z ,           Δ φ ( t ) = φ ( t ) φ cnv = k 0 0 L Δ n r ( t , z ) d z .
T NLE ( t ) = exp ( h cnv 2 ) exp ( Δ h ( t ) 2 ) exp ( j Δ φ ( t ) ) .
Δ φ ( t ) Δ h ( t ) = k 0 0 L Δ n r ( t , z ) d z 2 k 0 0 L Δ n i ( t , z ) d z = α H 2 .
T NLE ( t ) = exp ( h cnv 2 ) exp ( Δ h ( t ) 2 ( 1 + j α H ) ) .
h cnv = h 0 1 + P cnv / P sat , h ( t ) = h 0 1 + ( P cnv + P s ( t ) ) / P sat .
Δ h ( t ) = h ( t ) h cnv = h cnv P s ( t ) / P sat 1 + ( P cnv + P s ( t ) ) / P sat .
Δ φ   =   a H , SOA Δ h / 2 .
h cnv = ( α 0 + α 2 P cnv A eff ) L ,     h ( t ) = ( α 0 + α 2 P cnv + 2 P s ( t ) A eff ) L ,     h 0   =   α 0 L .
P eff = A eff / ( α 2 L ) ,
h cnv = h 0 P cnv / P eff , h ( t ) = h 0 ( P cnv + 2 P s ) / P eff .
Δ h ( t ) = h ( t ) h cnv = 2 P s ( t ) / P eff .
Δ φ ( t ) = k 0 Δ n r ( t ) L = 2 k 0 n 2 L P s ( t ) / A eff = a H , Kerr Δ h / 2   for   α H , Kerr = 2 k 0 n 2 / α 2 .
P eq = { P sat ,   in an SOA-type medium, see Eq .   (11)   , P eff ,   in a Kerr-type medium, see Eq .   (15)   .
α H = { α H , SOA ,   in an SOA-type medium, see Eq .   (4)   , α H , Kerr ,   in a Kerr-type medium, see Eq .   (18)   .
P in ( t ) = A eff c 0 ε 0 n r , 0 | E in ( t ) | 2 / 2   .
Δ Φ in ( t ) = Φ in ( t ) Φ in ( t Δ t ) .
( E up ( t ) E low ( t ) ) = ( 1 / 2 j / 2 j / 2 1 / 2 ) ( δ ( τ - Δ t )exp(j   2 π f in τ ) 0 0 δ ( τ ) ) ( 1 / 2 j / 2 j / 2 1 / 2 ) ( E in ( t τ ) 0 )   d τ .
E up = 1 2 [ A in ( t T b ) exp [ j Φ in ( t T b ) ] A in ( t ) exp [ j Φ in ( t ) ] ] , E low = j 2 [ A in ( t T b ) exp [ j Φ in ( t T b ) ] + A in ( t ) exp [ j Φ in ( t ) ] ] .
T ( t ) = E Δ E cnv = 1 2 γ [ exp ( h cnv 2 ) exp ( Δ h up ( t ) 2 ( 1 + j α H ) ) exp ( h cnv 2 ) exp ( Δ h low ( t ) 2 ( 1 + j α H ) ) ] = γ exp ( h cnv / 2 ) sinh [ 1 + j α H 4 ( Δ h up ( t ) Δ h low ( t ) ) ] exp [ 1 + j α H 4 ( Δ h up ( t ) + Δ h low ( t ) ) ] .
T ( t ) = γ exp ( h cnv 2 ) sinh ( Δ h up Δ h low 4 ) exp ( Δ h up + Δ h low 4 )   .
T ( t ) = γ exp ( h cnv 2 ) sin [ α H 4 ( Δ h up Δ h low ) ] exp [ j α H 4 ( Δ h up + Δ h low ) ]   .
E in = E in,0 + δ E in = A 0 in exp ( j Φ 0 in ) + δ A exp ( j δ Φ ) .
E in ( t n T b ) = E n in ,   with   E n in = ( A 0 in + δ A n in ) exp ( j Φ n in )   and   δ a n   in = δ A n in / A 0 in ,
Δ Φ n in = Φ n + 1 in Φ n in = Δ Φ n , s in + δ Φ n in   .
| E up | 2 = ( A 0 in ) 2 cos 2 ( δ Φ n in 2 ) ( 1 + 2 δ a n + 1 / 2   in ) + ( δ A n in ) 2 + ( δ A n + 1 in ) 2 4 + 1 2 ( δ A n in ) ( δ A n + 1 in ) cos δ Φ n in ,
| E low | 2 = ( A 0 in ) 2 sin 2 ( δ Φ n in 2 ) ( 1 + 2 δ a n + 1 / 2   in ) + ( δ A n in ) 2 + ( δ A n + 1 in ) 2 4 1 2 ( δ A n in ) ( δ A n + 1 in ) cos δ Φ n in .
δ a n + 1 / 2   in = ( δ A n + 1 in + δ A n in ) / ( 2 A 0 in ) .
δ P up P s , 0 = 2 cos 2 ( δ Φ n in / 2 ) δ a n + 1 / 2   in sin 2 ( δ Φ n in / 2 ) , δ P low P s , 0 = sin 2 ( δ Φ n in / 2 ) ( 1 + 2 δ a n + 1 / 2   in ) .
E n out = A n out exp ( j Φ n out ) ,
δ A n out A 0 out = A 0 out A n out A 0 out = | T 0 | | T n | | T 0 | .
Δ h low = 0 ; Δ h up = h cnv P s , 0 / P sat 1 + ( P cnv + P s , 0 ) / P sat .
T 0 = 1 2 γ exp ( h cnv 2 ) [ exp ( Δ h up 2 ) 1 ] .
δ h low = h cnv δ P low / P sat 1 + P cnv / P sat = h cnv δ P low P sat + P cnv ; δ h up = h cnv δ P up / P sat 1 + ( P cnv + P s , 0 ) / P sat = h cnv δ P up P sat + P cnv + P s , 0 .
T n = 1 2 γ exp ( h cnv 2 ) { exp ( Δ h up + δ h up 2 ) exp ( δ h low 2 ) } = 1 2 γ exp ( h cnv 2 ) { exp ( Δ h up 2 ) 1 + [ exp ( Δ h up 2 ) ( exp ( δ h up 2 ) 1 ) ( exp ( δ h low 2 ) 1 ) ] } .
δ A n out A 0 out = ( exp ( δ h up / 2 ) 1 ) exp ( Δ h up / 2 ) ( exp ( δ h low / 2 ) 1 ) exp ( Δ h up / 2 ) 1 ; δ Φ n out = 0 .
T 0 = γ exp ( h cnv 2 ) sin ( α H 4 Δ h up ) exp ( j α H 4 Δ h up ) ,     with     Δ h up = 2 α 2 L P s , 0 / A eff .
Δ φ up = α H 2 Δ h up = π ,   for   P s , 0 = π α H A eff L ,
T 0 = γ exp ( h cnv / 2 ) .
δ h low = Δ h up sin 2 ( δ Φ n in 2 ) ( 1 + 2 δ a n + 1 / 2   in ) ,     δ h up = Δ h up [ 2 cos 2 ( δ Φ n in 2 ) δ a n + 1 / 2   in sin 2 ( δ Φ n in 2 ) ] .
T n = j   γ exp ( h cnv / 2 ) sin [ π 2 cos ( δ Φ n in ) ( 1 + 2 δ a n + 1 / 2   in ) ] exp [ j π δ a n + 1 / 2   in ] .
δ A n out A 0 out = 1 sin [ π 2 cos ( δ Φ n in ) ( 1 + 2 δ a n + 1 / 2   in ) ] ; δ Φ n out = π δ a n + 1 / 2   in .

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