Abstract

Optical parametric amplifiers rely on second-order susceptibility (three-wave mixing) or third-order susceptibility (four-wave mixing) in a nonlinear process where the energy of incoming photons is not changed (elastic scattering). In the latter case, two pump photons are converted to a signal and to an idler photon. Under certain conditions, related to the phase evolution of the waves involved, this conversion can be very efficient, resulting in large amplification of an input signal. As the nonlinear process can be very fast, all-optical applications aside from pure amplification are also possible. If the amplifier is implemented in an optical input-phase-sensitive manner, it is possible to amplify a signal wave without excess noise, i.e., with a noise figure of 0 dB. In this paper, we will provide the fundamental concepts and theory of such amplifiers, with a focus on their implementation in highly nonlinear optical fibers relying on four-wave mixing. We will discuss the distinctions between phase-insensitive and phase-sensitive operation and include several experimental results to illustrate their capability. Different applications of parametric amplifiers are also discussed, including their use in optical communication links.

© 2020 Optical Society of America

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2019 (3)

2018 (5)

2017 (8)

V. Ribeiro, M. Karlsson, and P. Andrekson, “Parametric amplification with a dual-core fiber,” Opt. Express 25, 6234–6243 (2017).
[Crossref]

V. Gordienko, M. F. C. Stephens, A. E. El-Taher, and N. J. Doran, “Ultra-flat wideband single-pump Raman-enhanced parametric amplification,” Opt. Express 25, 4810–4818 (2017).
[Crossref]

K. Bottrill, G. Hesketh, L. Jones, F. Parmigiani, D. Richardson, and P. Petropoulos, “Full quadrature regeneration of QPSK signals using sequential phase sensitive amplification and parametric saturation,” Opt. Express 25, 696–705 (2017).
[Crossref]

L. Li, P. G. Patki, Y. B. Kwon, V. Stelmakh, B. D. Campbell, M. Annamalai, T. I. Lakoba, and M. Vasilyev, “All-optical regenerator of multi-channel signals,” Nat. Commun. 8, 1–11 (2017).
[Crossref]

E. Astra, S. L. Olsson, H. Eliasson, and P. A. Andrekson, “Dispersion management for nonlinearity mitigation in two-span 28 GBaud QPSK phase-sensitive amplifier links,” Opt. Express 25, 13163–13173 (2017).
[Crossref]

M. F. C. Stephens, V. Gordienko, and N. J. Doran, “20 dB net-gain polarization-insensitive fiber optical parametric amplifier with >2 THz bandwidth,” Opt. Express 25, 10597–10609 (2017).
[Crossref]

E. Nazemosadat, A. Lorences-Riesgo, M. Karlsson, and P. A. Andrekson, “Design of highly nonlinear few-mode fiber for C-band optical parametric amplification,” J. Lightwave Technol. 35, 2810–2817 (2017).
[Crossref]

K. Ooi, D. Ng, T. Wang, A. Chee, S. Ng, Q. Wang, L. Ang, A. Agarwal, L. Kimerling, and D. Tan, “Pushing the limits of CMOS optical parametric amplifiers with USRN:Si7N3 above the two-photon absorption edge,” Nat. Commun. 8, 13878 (2017).
[Crossref]

2016 (6)

A. Lorences-Riesgo, P. A. Andrekson, and M. Karlsson, “Polarization-independent phase-sensitive amplification,” J. Lightwave Technol. 34, 3171–3180 (2016).
[Crossref]

A. D. Ellis, M. Tan, M. A. Iqbal, M. A. Z. Al-Khateeb, V. Gordienko, G. S. Mondaca, S. Fabbri, M. F. C. Stephens, M. E. McCarthy, A. Perentos, I. D. Phillips, D. Lavery, G. Liga, R. Maher, P. Harper, N. Doran, S. K. Turitsyn, and S. Sygletos, and P. Bayvel, “4 Tb/s transmission reach enhancement using 10 × 400 Gb/s super-channels and polarization insensitive dual band optical phase conjugation,” J. Lightwave Technol. 34, 1717–1723 (2016).
[Crossref]

A. D. Ellis, M. Tan, M. A. Iqbal, M. A. Z. Al-Khateeb, V. Gordienko, G. S. Mondaca, S. Fabbri, M. F. C. Stephens, M. E. McCarthy, A. Perentos, I. D. Phillips, D. Lavery, G. Liga, R. Maher, P. Harper, N. Doran, S. K. Turitsyn, and S. Sygletos, and P. Bayvel, “4 Tb/s transmission reach enhancement using 10 × 400 Gb/s super-channels and polarization insensitive dual band optical phase conjugation,” J. Lightwave Technol. 34, 1717–1723 (2016).
[Crossref]

F. Parmigiani, K. Bottrill, R. Slavik, D. Richardson, and P. Petropoulos, “Multi-channel phase regenerator based on polarization-assisted phase-sensitive amplification,” IEEE Photon. Technol. Lett. 28, 845–848 (2016).
[Crossref]

S. M. M. Friis, I. Begleris, Y. Jung, K. Rottwitt, P. Petropoulos, D. Richardson, P. Horak, and F. Parmigiani, “Inter-modal four-wave mixing study in a two-mode fiber,” Opt. Express 24, 30338–30349 (2016).
[Crossref]

H. Eliasson, S. L. I. Olsson, M. Karlsson, and P. A. Andrekson, “Mitigation of nonlinear distortion in hybrid Raman/phase-sensitive amplifier links,” Opt. Express 24, 888–900 (2016).
[Crossref]

R. Malik, A. Kumpera, M. Karlsson, and P. A. Andrekson, “Demonstration of ultra wideband phase-sensitive fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 28, 175–177 (2016).
[Crossref]

2015 (9)

C. J. McKinstrie, “Schmidt decompositions of parametric processes III: Simultaneous amplification and conversion,” Opt. Express 23, 16949–16966 (2015).
[Crossref]

A. Kumpera, R. Malik, A. Lorences-Riesgo, and P. A. Andrekson, “Parametric coherent receiver,” Opt. Express 23, 12952–12964 (2015).
[Crossref]

M. F. C. Stephens, I. D. Phillips, P. Rosa, P. Harper, and N. J. Doran, “Improved WDM performance of a fibre optical parametric amplifier using Raman-assisted pumping,” Opt. Express 23, 902–911 (2015).
[Crossref]

M. Karlsson, “Transmission systems with low noise phase-sensitive parametric amplifiers,” J. Lightwave Technol. 34, 1411–1423 (2015).
[Crossref]

M. E. Marhic, P. A. Andrekson, P. Petropoulos, S. Radic, C. Peucheret, and M. Jazayerifar, “Fiber optical parametric amplifiers in optical communication systems,” Laser Photon. Rev. 9, 50–74 (2015).
[Crossref]

H. Eliasson, P. Johannisson, M. Karlsson, and P. A. Andrekson, “Mitigation of nonlinearities using conjugate data repetition,” Opt. Express 23, 2392–2402 (2015).
[Crossref]

T. Umeki, T. Kazama, O. Tadanaga, K. Enbutsu, M. Asobe, Y. Miyamoto, and H. Takenouchi, “PDM signal amplification using PPLN-based polarization-independent phase-sensitive amplifier,” J. Lightwave Technol. 33, 1326–1332 (2015).
[Crossref]

H. Hu, R. Jopson, A. Gnauck, M. Dinu, S. Chandrasekhar, C. Xie, and S. Randel, “Parametric amplification, wavelength conversion, and phase conjugation of a 2.048-Tbit/s WDM PDM 16-QAM signal,” J. Lightwave Technol. 33, 1286–1291 (2015).
[Crossref]

M. Guasoni, “Generalized modulational instability in multimode fibers: wideband multimode parametric amplification,” Phys. Rev. A 92, 033849 (2015).
[Crossref]

2014 (4)

2013 (9)

X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 7, 560–568 (2013).
[Crossref]

C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: Basic theory and simple examples,” Opt. Express 21, 1374–1394 (2013).
[Crossref]

C. J. McKinstrie, J. R. Ott, and M. Karlsson, “Schmidt decompositions of parametric processes II: Vector four-wave mixing,” Opt. Express 21, 11009–11020 (2013).
[Crossref]

C. Lundström, R. Malik, L. Grüner-Nielsen, B. Corcoran, S. L. I. Olsson, M. Karlsson, and P. A. Andrekson, “Fiber optic parametric amplifier with 10-dB net gain without pump dithering,” IEEE Photon. Technol. Lett. 25, 234–237 (2013).
[Crossref]

Z. Tong and S. Radic, “Low-noise optical amplification and signal processing in parametric devices,” Adv. Opt. Photon. 5, 318–384 (2013).
[Crossref]

T. Umeki, M. Asobe, and H. Takenouchi, “In-line phase sensitive amplifier based on PPLN waveguides,” Opt. Express 21, 12077–12084 (2013).
[Crossref]

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nature 7, 354–362 (2013).
[Crossref]

Y. Tian, Y.-K. Huang, S. Zhang, P. R. Prucnal, and T. Wang, “Demonstration of digital phase-sensitive boosting to extend signal reach for long-haul WDM systems using optical phase-conjugated copy,” Opt. Express 21, 5099–5106 (2013).
[Crossref]

R.-J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett. 25, 539–542 (2013).
[Crossref]

2012 (3)

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron. 18, 1016–1032 (2012).
[Crossref]

R. Slavík, A. Borgis, F. Parmigiani, J. Kakande, M. Westlund, M. Sköld, L. Grüner-Nielsen, R. Phelan, D. Syvridis, P. Petropoulos, and D. J. Richardson, “Coherent all-optical phase and amplitude regenerator of binary phase-encoded signals,” IEEE J. Sel. Top. Quantum Electron. 18, 859–869 (2012).
[Crossref]

Z. Tong, A. O. Wiberg, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Broadband parametric multicasting via four-mode phase-sensitive interaction,” Opt. Express 20, 19363–19373 (2012).
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2011 (6)

Z. Tong, C. Lundström, M. Karlsson, M. Vasilyev, and P. A. Andrekson, “Noise performance of a frequency nondegenerate phase-sensitive amplifier with unequalized inputs,” Opt. Lett. 36, 722–724 (2011).
[Crossref]

C. Lundström, Z. Tong, M. Karlsson, and P. A. Andrekson, “Phase-to-phase and phase-to-amplitude transfer characteristics of a nondegenerate-idler phase-sensitive amplifier,” Opt. Lett. 36, 4356–4358 (2011).
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Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
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J. Kakande, C. Lundström, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Multilevel quantization of optical phase in a novel coherent parametric mixer architecture,” Nat. Photonics 5, 748–752 (2011).
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M. E. Marhic, “Polarization independence and phase-sensitive parametric amplification,” J. Opt. Soc. Am. B 28, 2685–2689 (2011).
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T. Richter, R. Elschner, A. Gandhe, K. Petermann, and C. Schubert, “Parametric amplification and wavelength conversion of single- and dual-polarization DQPSK signals,” IEEE J. Sel. Top. Quantum Electron. 18, 988–995 (2011).
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2010 (7)

S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B. Divliansky, N. Alic, S. Mookherjea, and S. Radic, “Mid-infrared wavelength conversion in silicon waveguides using ultracompact telecom-band-derived pump source,” Nat. Photonics 4, 561 (2010).
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X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. Green, “Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides,” Nat. Photonics 4, 557–560 (2010).
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R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
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J. Kakande, C. Lundström, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Detailed characterization of a fiber-optic parametric amplifier in phase-sensitive and phase-insensitive operation,” Opt. Express 18, 4130–4137 (2010).
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J. B. Coles, B.-P. Kuo, N. Alic, S. Moro, C.-S. Bres, J. C. Boggio, P. Andrekson, M. Karlsson, and S. Radic, “Bandwidth-efficient phase modulation techniques for stimulated brillouin scattering suppression in fiber optic parametric amplifiers,” Opt. Express 18, 18138–18150 (2010).
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Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. A. Andrekson, “Noise performance of optical fiber transmission links that use non-degenerate cascaded phase-sensitive amplifiers,” Opt. Express 18, 15426–15439 (2010).
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Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18, 2884–2893 (2010).
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2009 (3)

J. M. C. Boggio, S. Moro, E. Myslivets, J. R. Windmiller, N. Alic, and S. Radic, “155-nm continuous-wave two-pump parametric amplification,” IEEE Photonics Technol. Lett. 21, 612–614 (2009).
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M. Jamshidifar, A. Vedadi, and M. E. Marhic, “Reduction of four-wave-mixing crosstalk in a short fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 21, 1244–1246 (2009).
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2008 (2)

P. A. Andrekson, H. Sunnerud, S. Oda, T. Nishitani, and J. Yang, “Ultrafast, atto-joule switch using fiber-optic parametric amplifier operated in saturation,” Opt. Express 16, 10956–10961 (2008).
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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, 648–658 (2008).
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2007 (5)

R. W. Fung, H. K. Cheung, and K. K. Wong, “Widely tunable wavelength exchange in anomalous-dispersion regime,” IEEE Photon. Technol. Lett. 19, 1846–1848 (2007).
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S. Oda, H. Sunnerud, and P. A. Andrekson, “High efficiency and high output power fiber-optic parametric amplifier,” Opt. Lett. 32, 1776–1778 (2007).
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P. Kylemark, M. Karlsson, T. Torounidis, and P. A. Andrekson, “Noise statistics in fiber optical parametric amplifiers,” J. Lightwave Technol. 25, 612–620 (2007).
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T. Torounidis and P. A. Andrekson, “Broadband single-pumped fiber-optic parametric amplifiers,” IEEE Photonics Technol. Lett. 19, 650–652 (2007).
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P. A. Andrekson and M. Westlund, “Nonlinear optical fiber based high resolution all-optical waveform sampling,” Laser Photon. Rev. 1, 231–248 (2007).
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2006 (5)

2005 (9)

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005).
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R. Tang, J. Lasri, P. S. Devgan, V. Grigoryan, P. Kumar, and M. Vasilyev, “Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input,” Opt. Express 13, 10483–10493 (2005).
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J. M. C. Boggio, S. Moro, E. Myslivets, J. R. Windmiller, N. Alic, and S. Radic, “Experimental and numerical investigation of the SBS-threshold increase in an optical fiber by applying strain distributions,” J. Lightwave Technol. 23, 3808–3814 (2005).
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A. Durécu-Legrand, C. Simonneau, D. Bayart, A. Mussot, T. Sylvestre, E. Lantz, and H. Maillotte, “Impact of pump OSNR on noise figure for fiber-optical parametric amplifiers,” IEEE Photon. Technol. Lett. 17, 1178–1180 (2005).
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P. Kylemark, P. O. Hedekvist, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Correction to ‘noise characteristics of fiber optical parametric amplifiers’,” J. Lightwave Technol. 23, 2192 (2005).
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M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express 13, 7563–7571 (2005).
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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, 1614–1616 (2005).
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N. S. Bergano, “Wavelength division multiplexing in long-haul transoceanic transmission systems,” J. Lightwave Technol. 23, 4125–4139 (2005).
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2004 (10)

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2003 (3)

S. Radic and C. J. McKinstrie, “Two-pump fiber parametric amplifiers,” Opt. Fiber Technol. 9, 7–23 (2003).
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2002 (4)

K. Uesaka, K. K.-Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
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2001 (4)

J. Hansryd and P. A. Andrekson, “Broad-band continuous-wave-pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency,” IEEE Photonics Technol. Lett. 13, 194–196 (2001).
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D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” Pramana J. Phys. 56, 281–285 (2001).
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J. Hansryd, F. Dross, M. Westlund, P. A. Andrekson, and S. N. Knudsen, “Increase of the SBS threshold in a short highly nonlinear fiber by applying a temperature distribution,” J. Lightwave Technol. 19, 1691–1697 (2001).
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J. Kim, O. Boyraz, J. H. Lim, and M. N. Islam, “Gain enhancement in cascaded fiber parametric amplifier with quasi-phase matching: theory and experiment,” J. Lightwave Technol. 19, 247–251 (2001).
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2000 (1)

W. Imajuku, A. Takada, and Y. Yamabayashi, “Inline coherent optical amplifier with noise figure lower than 3 dB quantum limit,” Electron. Lett. 36, 63–64 (2000).
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1999 (4)

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W. Imajuku and A. Takada, “Gain characteristics of coherent optical amplifiers using a Mach-Zehnder interferometer with Kerr media,” IEEE J. Quantum Electron. 35, 1657–1665 (1999).
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W. Imajuku and A. Takada, “In-line optical phase-sensitive amplifier with pump light source controlled by optical phase-lock loop,” J. Lightwave Technol. 17, 637–646 (1999).
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1998 (3)

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1997 (1)

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1996 (5)

A. Takada and W. Imajuku, “Amplitude noise suppression using a high gain phase sensitive amplifier as a limiting amplifier,” Electron. Lett. 32, 677–679 (1996).
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Z. Ahmed, H. Liu, D. Novak, Y. Ogawa, M. Pelusi, and D. Kim, “Locking characteristics of a passively mode-locked monolithic DBR laser stabilized by optical injection,” IEEE Photon. Technol. Lett. 8, 37–39 (1996).
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P. O. Hedekvist, M. Karlsson, and P. Andrekson, “Polarization dependence and efficiency in a fiber four-wave mixing phase conjugator with orthogonal pump waves,” IEEE Photon. Technol. Lett. 8, 776–778 (1996).
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1994 (1)

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1993 (3)

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1991 (4)

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1990 (1)

1989 (3)

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1987 (2)

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1986 (4)

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1985 (4)

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1984 (1)

1982 (2)

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1980 (3)

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1960 (1)

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Other (22)

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge University, 2008).

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R. Tang, P. Devgan, J. Lasri, V. Grigoryan, and P. Kumar, “Experimental investigation of a frequency-nondegenerate phase-sensitive optical parametric amplifier,” in Optical Fiber Communication Conference (OFC) (2005).

M. Jamshidifar, A. Vedadi, and M. E. Marhic, “Continuous-wave one-pump fiber optical parametric amplifier with 270 nm gain bandwidth,” in European Conference and Exhibition on Optical Communication (ECOC) (2009).

E. Nazemosadat, A. Lorences-Riesgo, M. Karlsson, and P. A. Andrekson, “Highly nonlinear few-mode fiber for optical parametric amplification,” in 42nd European Conference on Optical Communication (ECOC) (VDE, 2016).

S. L. I. Olsson, C. Lundström, M. Karlsson, and P. A. Andrekson, “Long-haul (3465 km) transmission of a 10 GBd QPSK signal with low noise phase-sensitive in-line amplification,” in European Conference and Exhibition on Optical Communication (ECOC) (2014).

S. L. Olsson, M. Karlsson, and P. A. Andrekson, “Long-haul optical transmission of 16-QAM signal with in-line phase-sensitive amplifiers,” in European Conference on Optical Communication (ECOC) (IEEE, 2017).

R. Kakarla, J. Schröder, and P. A. Andrekson, “Record-sensitivity Gb/s receiver for free-space applications based on phase-sensitive amplification,” in Conference on Lasers and Electro-Optics (CLEO) (IEEE, 2019).

N. Zhao, B. Huang, R. Amezcua-Correa, X. Li, and G. Li, “Few-mode fiber optical parametric amplifier,” in Optical Fiber Communication Conference (Optical Society of America, 2013), paper OTu2D–5.

S. Takasaka and R. Sugizaki, “Polarization insensitive fiber optical parametric amplifier using a SBS suppressed diversity loop,” in Optical Fiber Communications Conference and Exhibition (OFC) (2016), pp. 1–3.

A. Lorences-Riesgo, T. Eriksson, M. Mazur, P. Andrekson, and M. Karlsson, “Quadrature decomposition of a 20 Gbaud 16-QAM signal into 2x4-PAM signals,” in 42nd European Conference on Optical Communication ECOC) (VDE, 2016).

J. M. C. Boggio, C. Lundström, J. Yang, H. Sunnerud, and P. A. Andrekson, “Double-pumped FOPA with 40 dB flat gain over 81 nm bandwidth,” in European Conference and Exhibition on Optical Communication (ECOC) (2008).

S. Takasaka, Y. Mimura, M. Takahashi, R. Sugizaki, and H. Ogoshi, “Flat and broad amplification by quasi-phase-matched fiber optical parametric amplifier,” in Optical Fiber Communication Conference and Exposition (OFC/NFOEC) (IEEE, 2012).

M. Stephens, A. Redyuk, S. Sygletos, I. Phillips, P. Harper, K. Blow, and N. Doran, “The impact of pump phase-modulation and filtering on WDM signals in a fibre optical parametric amplifier,” in Optical Fiber Communication Conference (Optical Society of America, 2015), paper W2A.43.

H. Sunnerud, S. Oda, J. Yang, T. Nishitani, and P. A. Andrekson, “Optical add-drop multiplexer based on fiber optic parametric amplification,” in 33rd European Conference and Exhibition of Optical Communication (VDE, 2007).

Z. Tong, C. Lundström, E. Tipsuwannakul, M. Karlsson, and P. A. Andrekson, “Phase-sensitive amplified DWDM DQPSK signals using free-running lasers with 6-dB link SNR improvement over EDFA-based systems,” in European Conference and Exhibition on Optical Communication (ECOC) (2010).

T. Richter, B. Corcoran, S. L. Olsson, C. Lundström, M. Karlsson, C. Schubert, and P. A. Andrekson, “Experimental characterization of a phase-sensitive four-mode fiber-optic parametric amplifier,” in European Conference and Exhibition on Optical Communication (ECOC) (Optical Society of America, 2012), paper Th.1.F.1.

G. Kalogerakis, M. E. Marhic, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier with polarization diversity technique,” in Optical Fiber Communication Conference (Optical Society of America, 2006), paper OWT4.

M. Westlund, M. Sköld, and P. A. Andrekson, “All-optical phase-sensitive waveform sampling at 40 GSymbol/s,” in Optical Fiber Communication Conference (Optical Society of America, 2008), paper PDP12.

M. Sköld, M. Westlund, H. Sunnerud, and P. A. Andrekson, “100 GSample/s optical real-time sampling system with Nyquist-limited bandwidth,” in 33rd European Conference and Exhibition of Ogfrptical Communication-Post-Deadline Papers (VDE, 2007).

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Figures (29)

Figure 1.
Figure 1. Typical attenuation spectrum for a state-of-the-art silica fiber. The different amplification bands have been indicated.
Figure 2.
Figure 2. Fiber optic communication link with in-line optical amplifiers.
Figure 3.
Figure 3. Oscillations of a parametrically pumped swing in (a) amplified and (b) deamplified mode. The purple lines show an oscillating but piecewise constant pendulum length $ L(t) $, which is the same in (a) and (b). The blue lines show the corresponding solution $ \phi (t)/\phi (0) $ to Eq. (5) with different initial conditions so that in (a), $ L $ is longer during the downward motion and shorter in the upward motion, and for (b), the opposite holds. The yellow dashed lines show the parametric amplification/damping $ {({L_1}/{L_2})^{ \pm 3t/(2\pi )}} $ in the two cases. The red lines show for reference the unamplified solutions for the constant length $ L(t) = 1 $, which is simply $ \cos (t) $ in (a) and $ \sin (t) $ in (b).
Figure 4.
Figure 4. Different parametric amplifier configurations with signals/idlers in red and pumps in blue. (a) Degenerate one-mode amplifier. (b) A two-mode amplifier with degenerate pump. (c) A four-mode amplifier with one signal and three idler waves. The following three processes jointly form the four-mode process. (d) The modulational instability process, equivalent to the two-mode shown in (b). (e) The Bragg scattering process. (f) The phase conjugation process.
Figure 5.
Figure 5. Possible implementation of a fiber-optic parametric amplifier.
Figure 6.
Figure 6. Parametric gain spectrum at various pump powers with a single pump at 1550 nm, which is 2 nm above the wavelength of zero-dispersion in the HNLF, which is 500 m long with $ \gamma = 16\,{(\text{W}\;\text{km})^{ - 1}} $ and dispersion slope $ S = 0.02\,\,\text{ps}/({\text{nm}^2}\;\text{km)} $.
Figure 7.
Figure 7. Parametric gain spectrum at various pump powers with two pumps operating at 1466.5 nm and 1643.6 nm, respectively. The power in each pump is equal, and the power mentioned reflects the total power of the two pumps. The value of the dispersion curvature ($ {\beta _4} $) was $ 2.4 \cdot {{10}^{ - 5}}{\text{ps}^4}/\text{km} $. The average pump wavelength was 0.3 nm above $ {\lambda _0} $. Other parameters are same as in Fig. 6.
Figure 8.
Figure 8. Parametric gain spectra with a 500 m HLNF having $ \gamma = 16\,{(\text{W}\;\text{km})^{ - 1}} $ (red curve) and a 50 m hypothetical HNLF having $ \gamma = 160\,{(\text{W}\;\text{km})^{ - 1}} $ (blue curve). The total pump power was 800 mW. Red curve is from Fig. 7. For the 50 m case, the pump wavelengths were optimized as 1406.5 nm and 1726.2 nm, and their average wavelength was 0.95 nm above $ {\lambda _0} $. Other parameters are the same as in Fig. 7.
Figure 9.
Figure 9. Measured (symbols) and calculated FOPA gain spectrum in a dual-pump configuration. The pump separation was 103 nm with their average wavelength being near the zero-dispersion of the HNLF used. The total pump power was 2.1 W, and the length and nonlinearity coefficient of the HNLF were approximately 350 m and $ 14\,\,{(\text{W}\;\text{km})^{ - 1}} $, respectively. Reprinted with permission from [124]. Copyright 2008 Optical Society of America.
Figure 10.
Figure 10. (a) Simulated phase matching along an HLNF with and without periodically inserted pump phase correctors. (b) Corresponding signal gain along the HNLF. © 2012 IEEE. Reprinted, with permission, from S. Takasaka et al., Optical Fiber Communication Conference and Exposition (OFC/NFOEC), 2012 [126].
Figure 11.
Figure 11. Top, FOPA with increased SBS threshold using four strained HNLF sections with optical isolators in between, and no phase modulation of the pump. Bottom, measured FOPA gain spectrum. © 2012 IEEE. Reprinted, with permission, from C. Lundström et al., IEEE Photon. Technol. Lett. 25, 234–237 (2013) [85].
Figure 12.
Figure 12. Saturation characteristics of a FOPA. Power evolution versus signal input power (left). Output spectrum with input power being $ - {20}\;\text{dBm}$ (middle). Output spectrum with input power being 4.6 dBm (right). The input pump power was 1 W, and the fiber length was 750 m. Reprinted with permission from [132]. Copyright 2007 Optical Society of America.
Figure 13.
Figure 13. Copier-based approach for generation of a pump, signal, and idler waves needed for operating a PSA. With sufficient loss between the copier and the PSA, the nonideal NF of the copier will not impact the overall link NF.
Figure 14.
Figure 14. Schematic of a copier-PSA optical fiber transmission link.
Figure 15.
Figure 15. Injection locking principle (left) and measured phase-noise variation versus slave laser input power (right). Reprinted with permission from [144]. Copyright 2018 Optical Society of America.
Figure 16.
Figure 16. (a) Maximum transmission reach (in terms of number of 81 km spans) for both PSA and PIA implementations versus the amount of dispersion compensation before the transmission span. (b) Maximum reach in the case when the spans are amplified using distributed Raman amplification. Reprinted with permission from [145]. Copyright 2018 Optical Society of America.
Figure 17.
Figure 17. Schematics of two possibilities to implement polarization-independent operation with parametric amplifiers. Left, diversity approach. Right, vector approach. Reprinted with permission from [151]. Copyright 2016 Optical Society of America.
Figure 18.
Figure 18. Comparison of relative capacity of a PSA versus a PIA amplified system. The solid black line is the capacity for a PIA system.
Figure 19.
Figure 19. Left, geometry and spatial modes of a fiber dispersion-optimized for parametric amplification. Right, resulting simulated gain spectrum using a single pump with total power of 12 W (optimized among the modes) and fiber length of 150 m. The arrow indicates the pump wavelength. Reprinted with permission from [168,169]. Copyright 2016 and 2017 Optical Society of America.
Figure 20.
Figure 20. Measured output spectrum of a four-mode PSA without and with the idlers at the input. The observed signal gain with the idlers is 10.3 dB higher than without them, which is due to the coherent superposition of four waves. Reprinted with permission from [110]. Copyright 2012 Optical Society of America.
Figure 21.
Figure 21. Effect of conjugate twin wave transmission. (a) The initial signal (blue) and its conjugate (red). (b) Both signal after propagation subject to SPM and XPM. (c) As in (b) but with the red signal conjugated. (d) The resulting signal after coherent superposition. The simulation is based on 5000 points, with an SNR of 20 dB and a nonlinear phase shift of 0.8 rad.
Figure 22.
Figure 22. Setup of long-haul transmission of data emulated with circulating loop using PSA as in-line amplifiers. Reprinted with permission from [176]. Copyright 2017 Optical Society of America.
Figure 23.
Figure 23. 10 Gbd, 16-QAM signal constellation diagrams for PIA and PSA in-line amplifier operation in the loop experiment in Fig. 22. The number of round trips is denoted N. Reprinted with permission from [176]. Copyright 2017 Optical Society of America.
Figure 24.
Figure 24. Schematic of a PPLN-based PSA implementation (top) and principle (bottom). Reprinted with permission from [178]. Copyright 2013 Optical Society of America.
Figure 25.
Figure 25. Top, principle of optical phase regeneration in a two-mode PSA. Bottom, calculated gain and output signal phase at increasing pump powers.
Figure 26.
Figure 26. Experimental results of two-mode PSA-based phase regeneration. Top, output phase versus input phase. Bottom, corresponding constellation diagrams. The idler power is increasing from left to right, where it is equal to the signal power. Reprinted with permission from [84]. Copyright 2011 Optical Society of America.
Figure 27.
Figure 27. Schematic of a “black-box” all-optical phase regenerator based on a one-mode PSA. Reprinted with permission from Macmillan Publishers Ltd: R. Slavík et al., Nat. Photonics 4, 690–695 (2010) [79]. Copyright 2010.
Figure 28.
Figure 28. Experimental results of phase regeneration in a one-mode PSA. Reprinted with permission from Macmillan Publishers Ltd.: Slavík et al., Nat. Photonics 4, 690–695 (2010) [79]. Copyright 2010.
Figure 29.
Figure 29. Top, all-optical sampling based pulsed-pump PIA operation. Bottom, examples shown are a pulsed data sequence at 640 Gb/s (left) and a QPSK signal captured by using a coherent received to detect the idler (right). Reprinted with permission from [187,188]. Copyright 2007 Wiley-VCH Verlag GmbH & Co. and 2008 Optical Society of America.

Equations (58)

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N F lumped 1 + 2 m G ( 1 1 G ) 2 m G ,
NF distributed 1 + 2 m ln ( G ) ,
p = m v L = m L 2 ϕ ( t ) .
p ( t ) = m g L sin ( ϕ ) m g L ϕ ,
d d t ( ϕ p ) = ( 0 1 m L 2 ( t ) m g L ( t ) 0 ) ( ϕ p ) = M ( L ) ( ϕ p ) ,
( ϕ ( t ) p ( t ) ) = T ( t , L ) ( ϕ ( 0 ) p ( 0 ) ) ,
T per = ( ( L 1 L 2 ) 3 0 0 ( L 2 L 1 ) 3 ) .
2 E 1 c 2 E t 2 = 1 c 2 P t 2 .
Δ β = 2 β ( ω ) β ( 2 ω ) .
i u z + β ( ω 0 i t ) u + γ | u | 2 u = 0 ,
i u z β 2 2 u t 2 + γ | u | 2 u = 0 ,
d u p d z = i u p ( γ ( 2 P | u p | 2 ) + β p ) + i γ 2 u p u s u i ,
d u s d z = i u s ( γ ( 2 P | u s | 2 ) + β s ) + i γ u p 2 u i ,
d u i d z = i u i ( γ ( 2 P | u i | 2 ) + β i ) + i γ u p 2 u s ,
P = | u p | 2 + | u s | 2 + | u i | 2
Δ β = 2 β p β s β i
d d z ( e s e i ) = i ( κ γ e p 0 2 γ e p 0 2 κ ) ( e s e i ) = K ( e s e i ) .
( e s ( z ) e i ( z ) ) = exp ( K z ) ( e s ( 0 ) e i ( 0 ) ) .
( e s ( z ) e i ( z ) ) = ( μ ( z ) ν ( z ) ν ( z ) μ ( z ) ) ( e s ( 0 ) e i ( 0 ) ) ,
μ ( z ) = cosh ( g z ) + i κ sinh ( g z ) g
ν ( z ) = i γ e p 0 2 g sinh ( g z ) .
2 γ P p Δ β = 0 ,
G PSA = | e s ( 0 ) μ ( z ) + e i ( 0 ) ν ( z ) | 2 | e s ( 0 ) | 2 .
G PSA ( ϕ ) = | μ | 2 + | ν | 2 + 2 | μ | | ν | cos ( ϕ ) ,
G PSA max / min = ( | μ | ± | ν | ) 2 ,
2 ω p = ω s + ω i 2 k p = k s + k i ,
Φ s Φ i = P s h ν s P i h ν i = const ,
ω p 1 + ω p 2 = ω s + ω i ,
g 2 = 4 γ 2 P 1 P 2 ( Δ β γ ( P 1 + P 2 ) 2 ) ,
e s ( z ) = μ e s ( 0 ) + ν e s ( 0 ) ,
( e s ( z ) e i 1 ( z ) e i 2 ( z ) e i 3 ( z ) ) = ( μ ν δ κ . . . . . . . . . . . . . . . . . . ) ( e s ( 0 ) e i 1 ( 0 ) e i 2 ( 0 ) e i 3 ( 0 ) ) = G ( e s ( 0 ) e i 1 ( 0 ) e i 2 ( 0 ) e i 3 ( 0 ) ) ,
S sp = n sp ( G 1 ) h ν ,
σ s sp 2 = 4 R 2 P s S sp Δ f ,
N F = SNR in SNR out ,
NF PIA = ( R P in ) 2 σ s , in 2 σ s , out 2 + σ s sp 2 ( R G P in ) 2 = 2 n sp G 1 G + 1 G ,
n sp = NF G 1 2 ( G 1 ) ,
S sp NF G h ν 2 ,
NF PIA = 2 1 G .
SNR in , s = ( R P s ) 2 2 e R P s Δ f = R P s 2 e Δ f = P s 2 h ν Δ f .
e s ( z ) = μ e s + ν e i + μ n s + ν n i ,
i s = R | μ e s + ν e i + μ n s + ν n i | 2 = R ( P s , out + 2 R e [ ( μ e s + ν e i ) ( μ n s + ν n i ) ] ) ,
G s = P s , out P s = | μ e s + ν e i | 2 P s = | μ e s | e s | + ν e i | e s | | 2 ,
η = P i , out P s = | ν e s | 2 P s = | ν | 2 .
σ s s p 2 = 4 R 2 ( R e [ ( μ e s + ν e i ) ( μ n s + ν n i ) ] ) 2 = 2 R 2 P s , out | μ n s + ν n i | 2 = = 2 R 2 P s , out ( | μ | 2 + | ν | 2 ) ( h ν 2 ) ( 2 Δ f ) = 2 R 2 P s , out ( | μ | 2 + | ν | 2 ) h ν Δ f ,
SNR out = P s , out 2 ( | μ | 2 + | ν | 2 ) h ν Δ f ,
NF = P s ( | μ | 2 + | ν | 2 ) | μ e s + ν e i | 2 = | μ | 2 + | ν | 2 G PSA = 2 G PIA 1 G PSA ,
NF PIA , A = 1 + 2 m ( 1 1 G ) ,
NF PIA , B = 1 + 2 m G ( 1 1 G ) .
NF PSA , A = 1 + m ( 1 1 G ) ,
NF PSA , B = 1 + m G ( 1 1 G ) .
NF C PSA , A = 5 2 + m 2 ,
NF C PSA , B = 3 G 2 + m G 2 .
G ( P s ) = G 0 1 + 2 P s G 0 P p ,
C = B log 2 ( 1 + SNR ) ,
C PSA C PIA = 1 2 log 2 ( 1 + 4 SNR ) log 2 ( 1 + SNR ) ,
y 1 = ( a + n ) exp ( i p | a + n | 2 ) ,
y 2 = ( a + n ) exp ( i p | a + n | 2 ) ,
z = y 1 + y 2 2 = ( a + n ) cos ( p | a + n | 2 ) ,

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