Phase noise in fiber-optic parametric amplifiers and converters and its impact on sensing and communication systems

doi:10.1364/OE.18.021449

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Phase noise in fiber-optic parametric amplifiers and converters and its impact on sensing and communication systems

Slaven Moro, Ana Peric, Nikola Alic, Bryan Stossel, and Stojan Radic »View Author Affiliations

Optics Express, Vol. 18, Issue 20, pp. 21449-21460 (2010)
http://dx.doi.org/10.1364/OE.18.021449

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▸ Article Outline
– Abstract
1. Introduction
2. Theory
3. Experimental Results and Discussion
4. Conclusion
– References and links

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Abstract

We present a theoretical analysis describing the spectral dependence of phase noise in one-pump fiber parametric amplifiers and converters. The analytical theory is experimentally validated and found to have high predictive accuracy. The implications related to phase-coded sensing and communications systems are discussed.

1. Introduction

Fiber-optic parametric amplifiers (FOPAs) are routinely used to amplify and process optical signals in unconventional spectral windows. In addition to the spectral coverage outside the standard rare-earth-doped bands, their operating bandwidths can exceed multiples of conventional telecommunication bands [1

S. Radic, “Parametric amplification and processing in optical fibers,” Laser Photon. Rev. 2(6), 498–513 (2008). [CrossRef]

]. In a simplest FOPA configuration, employing a single pump wave and a typically weaker signal wave co-propagating in a section of dispersion-engineered highly-nonlinear fiber (HNLF), four-photon interaction takes place. Two pump photons are annihilated and a signal and an idler (phase-conjugated replica of the signal) photon are created via third-order nonlinearity. The ultrafast nature of Kerr nonlinearity in silica (characterized by a femtosecond response) is capable of transferring the pump phase and amplitude fluctuations to the signal and the idler wave in a near instantaneous manner.

The pump amplitude fluctuations are typically a combination of pump laser relative intensity noise (RIN), and amplifier noise added during pump wave amplification, with the latter being dominant in most practical situations [2

P. Kylemark, M. Karlsson, T. Torounidis, and P. A. Andrekson, “Noise Statistics in Fiber Optical Parametric Amplifiers,” J. Lightwave Technol. 25(2), 612–620 (2007). [CrossRef]

]. The spectral dependence of the signal and the idler amplitude noise in a one-pump FOPAs has been analyzed recently and shown to include predictable contributions from processes such as the amplified quantum noise (AQN), pump transferred noise (PTN), and Raman phonon seeded excess noise [3

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(3), 2884–2893 (2010). [CrossRef] [PubMed]

]. It was found that AQN dominates the noise performance at input signal levels below −30dBm, whereas PTN tends to limit the signal and idler amplitude fidelity at higher input powers. The PTN originates from the FOPA’s inherent gain sensitivity to instantaneous pump power, causing the amplitudes of the signals positioned at the edges of the gain region to be affected more adversely than those centered closer to the pump. It must be noted that PTN is not an additive but rather a multiplicative noise process, making the standard noise figure definition dependent on the input signal power. Consequently, the standard technique used to calculate the equivalent noise figure in links containing a cascade of amplifiers [4

P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers (Academic Press, 1999).

] is not applicable in FOPA-amplified systems.

The phase fluctuations of the pump wave, on the other hand, are dominated by the necessity for pump phase modulation in order to suppress the stimulated Brillouin scattering (SBS) in nonlinear fiber. In this work, we refer to this phase noise as phase modulation noise (PMN) of the signal and idler wave. The pump phase modulation results in severe spectral broadening of the idler wave that cannot be avoided in single-pumped FOPAs but can be mitigated in the dual-pumped case [5

R. M. Jopson, U.S. Patent 5 386 394 (1994).

–7

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective Suppression of Idler Spectral Broadening in Two-Pump Parametric Architectures,” IEEE Photon. Technol. Lett. 15(5), 673–675 (2003). [CrossRef]

]. Pulsed-pump FOPAs and wavelength converters do not require pump phase modulation, provided that the pulse spectrum is significantly wider than the Brillouin bandwidth (approximately 20MHz in HNLFs). The impact of pump phase modulation on the amplitude fluctuations for on-off-keyed (OOK) signals [8

A. Durécu-Legrand, A. Mussot, C. Simonneau, D. Bayart, T. Sylvestre, E. Lantz, and H. Maillotte, “Impact of pump phase modulation on system performances of fiber optical parametric amplifiers,” Electron. Lett. 41(6), 350–352 (2005). [CrossRef]

P. Kylemark, J. Ren, Y. Myslivets, N. Alic, S. Radic, P. A. Andrekson, and M. Karlsson, “Impact of Pump Phase-Modulation on the Bit-Error Rate in Fiber-Optical Parametric-Amplifier-Based Systems,” IEEE Photon. Technol. Lett. 19(1), 79–81 (2007). [CrossRef]

] and idler phase fluctuations for phase-shift-keyed (PSK) signals [10

R. Elschner, C.-A. Bunge, B. Huttl, A. G. i Coca, C. L. Schmidt, R. Ludwig, C. Schubert, and K. Petermann, “Impact of Pump-Phase Modulation on FWM-Based Wavelength Conversion of D(Q)PSK Signals,” IEEE J. Sel. Top. Quantum Electron. 14(3), 666–673 (2008). [CrossRef]

] have been studied previously. However, the spectral dependence of signal phase noise due to the pump phase modulation has not been studied before, to the best of our knowledge, and is addressed in the reported work. In contrast to signal impairments, idler distortion determines the performance of spectrally mapped systems for either communication or sensing [11

R. Jiang, C.-S. Bres, N. Alic, E. Myslivets, and S. Radic, “Translation of Gbps Phase-Modulated Optical Signal From Near-Infrared to Visible Band,” J. Lightwave Technol. 26(1), 131–137 (2008). [CrossRef]

]. We also investigate pump phase modulation by radio-frequency (RF) noise as a means for spectrally-efficient SBS suppression, and quantify its effects on signal phase noise.

In addition to PMN, the signal and idler phases are modulated by small pump amplitude fluctuations induced by cross-phase modulation (CPM) and four-wave-mixing (FWM) in highly-nonlinear fibers. First discovered by Gordon and Mollenauer [12

J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990). [CrossRef] [PubMed]

], the so-called nonlinear phase noise (NPN) is found to have a major impact in long-haul coherent communication systems [13

K.-P. Ho, Phase-Modulated Optical Communication Systems (Springer, 2005), Chap. 5.

], high power (>100W) amplification [14

S. J. McNaught, J. E. Rothenberg, P. A. Thielen, M. G. Wickham, M. E. Weber, and G. D. Goodno, “Coherent Combining of a 1.26-kW Fiber Amplifier,” Advanced in Solid-State Photonics, paper AMA2 (2010).

], narrow linewidth frequency comb synthesis [15

Y. Kim, S. Kim, Y.-J. Kim, H. Hussein, and S.-W. Kim, “Er-doped fiber frequency comb with mHz relative linewidth,” Opt. Express 17(14), 11972–11977 (2009). [CrossRef] [PubMed]

], and high-quality supercontinuum generation [16

N. Nishizawa and J. Takayanagi, “Octave spanning high-quality supercontinuum generation in all-fiber system,” J. Opt. Soc. Am. B 24(8), 1786–1792 (2007). [CrossRef]

,17

N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs,” J. Opt. Soc. Am. B 24(8), 1756–1770 (2007). [CrossRef]

]. The effect of NPN has been experimentally observed in saturated parametric amplifiers, where noise-loaded PSK signal was partially regenerated using a saturated one-pump FOPA [18

M. Sköld, J. Yang, H. Sunnerud, M. Karlsson, S. Oda, and P. A. Andrekson, “Constellation diagram analysis of DPSK signal regeneration in a saturated parametric amplifier,” Opt. Express 16(9), 5974–5982 (2008). [CrossRef] [PubMed]

,19

M. Skold, M. Karlsson, S. Oda, H. Sunnerud, and P. A. Andrekson, “Constellation diagram measurements of induced phase noise in a regenerating parametric amplifier,” Optical Fiber Communications Conference, paper OML4 (2008).

]. Constellation diagram analysis showed that while the amplitude of a noisy signal was regenerated by the parametric amplitude limiter, the FOPA added excess phase noise to the signal. Not surprisingly, in [19

] the amount of phase noise increased with reduced pump optical signal-to-noise ratio (OSNR) and was also numerically shown to increase with increased pump power [20

M. Matsumoto, “Phase noise generation in an amplitude limiter using saturation of a fiber-optic parametric amplifier,” Opt. Lett. 33(15), 1638–1640 (2008). [CrossRef] [PubMed]

]. Finally, an expression for the variance of NPN in dual-pumped parametric amplifiers has been derived in [21

R. Elschner, and L. Petermann, “Impact of Pump-Induced Nonlinear Phase Noise on Parametric Amplification and Wavelength Conversion of Phase-Modulated Signals,” European Conference in Optical Communications, paper 3.3.4 (2009).

], where the signal SNR penalty due to added NPN was numerically calculated for select PSK formats. The formats possessing smaller separation among symbol phase levels were shown to suffer a larger penalty due to NPN. In this report, we extend the analysis of phase impairment by studying the spectral dependence of NPN in one-pump FOPA both analytically and experimentally. In addition, we discuss the impact of pump optical noise filtering, necessary in all practical systems employing amplified pumps, on the statistics of the nonlinear phase.

In addition to pump-induced sources of phase noise (such as NPN and PMN), the FOPA is also subject to inherent amplified quantum noise (AQN). The AQN arises from parametric amplification of vacuum fluctuations at the signal wavelength as well as parametric conversion of AQN from idler to signal frequency. Unlike the NPN and PMN, the variance of AQN-induced phase noise is dependent on input signal power level. We derive a simple expression for the variance of signal phase noise due to AQN in a FOPA and confirm its validity using a rigorous numerical solver.

2. Theory

The pump propagation in HNLF induces self-phase modulation that can be divided into deterministic and stochastic parts: the average pump power gives rise to the mean nonlinear phase shift, while the pump power fluctuations result in nonlinear phase noise. The HNLF provides practical means for precisely controlled phase matching among all propagating waves, allowing for the NPN to transfer to the signal and the idler via highly efficient CPM and FWM. In order to suppress SBS, the pump wave has to be phase-modulated before entering the HNLF. As a result, the linear phase mismatch between the three interacting waves is varied in time and the signal and idler phases receive temporally varying contribution while propagating along the HNLF. Lastly, the signal and idler phases are also subject to fluctuations due to amplified quantum noise generated during the parametric amplification process, even in the case of a spectrally narrow pump with no intensity fluctuations.

2.1 Statistics of Nonlinear Phase Noise

We begin the phase impairment analysis by considering the FOPA model illustrated in Fig. 1 . A pump wave with optical power P_p and carrier frequency ν_p is amplified in a high-power Erbium-doped fiber amplifier (EDFA), thereby accumulating white Gaussian optical noise. The original pump RIN and laser phase noise are considered to be negligible. The optical amplifier noise, n(t) = n_r(t) + jn_i(t), is a complex white Gaussian random process [22

E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley-Interscience, 2002).

]. The in-phase and quadrature components of the noise have zero mean and variance of N₀Δν/2, where N₀ is the noise power spectral density in one polarization and Δν is the optical bandwidth of interest. The optical signal-to-noise ratio of the pump wave (measured within 0.1nm optical bandwidth) is given by OSNR_0.1nm = P_p/(2N₀ Δν_0.1nm), where Δν_0.1nm is the frequency bandwidth corresponding to 0.1nm at the wavelength of c/ν_p.

Fig. 1 FOPA model schematic used in the derivation of NPN statistics; Acronyms: OBPF - optical band-pass filter; the rest of the symbols are defined within the text.

The optical band-pass filter is introduced to remove excess amplified spontaneous emission (ASE) and the complex pump field after the filter stage can be expressed as

ρ' (t) = \sqrt{P_{p}} + n (t) \otimes h_{i n} (t) = \sqrt{P_{p}} + n' (t),

(1)

where h_in(t) is the optical filter impulse response, and n’(t) is the complex field of the filtered optical noise. The noisy complex pump field then enters HNLF characterized by fiber length L, nonlinear coefficient γ, and negligible intra-channel dispersion. While the assumption that HNLF has no intra-channel dispersion allows derivation of closed-form expressions, it is also justified in most practical cases as the pump positioning in the proximity of the zero-dispersion HNLF frequency is used to maximize the FOPA gain bandwidth [23

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

]. Next, after the propagation through HNLF, neglecting the depletion and transmission loss, the pump complex field becomes:

ρ'' (t) = (\sqrt{P_{p}} + n' (t)) e^{j γ L {| \sqrt{P_{p}} + n' (t) |}^{2}} = (\sqrt{P_{p}} + n' (t)) e^{j ϕ_{N L} (t)} .

(2)

The nonlinear phase shift can further be rewritten as

ϕ_{N L} (t) = γ P_{p} L + 2 γ \sqrt{P_{p}} L n'_{r} (t) + γ L | n' (t) |^{2} .

(3)

The second term in Eq. (3), the pump-noise beat term, dominates the noise performance since practical FOPAs require high pump OSNR. Neglecting the last (noise-noise beat) term, it is clear that the nonlinear phase noise remains a Gaussian-distributed process with zero mean. If we define N₀’ to be the total single-polarization filtered noise power and note that the post-filtering variances of the n_r’(t) and n_i’(t) equal N₀’/2, then the variance of NPN is 2γ²L²P_pN₀’.

Furthermore, it can be shown that the total nonlinear phase, ϕ_NL(t), is non-central χ²-distributed with probability distribution function:

p_{Φ_{N L}} (ϕ_{N L}) = \frac{1}{γ L N_{0}^{'}} e^{- \frac{ϕ_{N L} + γ P_{p} L}{γ L N_{0}^{'}}} I_{0} (\sqrt{\frac{4 P_{p}}{γ L N_{0}^{'}^{2}}} \sqrt{ϕ_{N L}}) .

(4)

The mean nonlinear phase shift is γL(P_p + N₀’), while the variance is γ²L² (2P_pN₀’ + N₀’²). Eq. (4) is an approximation since a closed form analytical solution for the probability density function (PDF) of nonlinear phase exists only for rectangular and Lorentzian optical filter transfer functions [24

N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical preamplifier,” Opt. Express 13(12), 4568–4579 (2005). [CrossRef] [PubMed]

At this juncture, it is important to point out an important difference between nonlinear phase induced in coherent communication systems and nonlinear phase in FOPAs. Firstly, optical filter bandwidths in communication systems typically exceed the signal bandwidths in order to avoid waveform distortions by the filter transfer function. Secondly, optical filters are not commonly used following the amplification process but only prior to final reception of the signal. Finally, any filtering is constrained by the fact that the NPN is distributed uniformly across the spectrum of the modulated signal. In contrast, optical pumps in FOPAs can be very narrowly filtered with bandwidths that must only be large enough to prevent the filter transfer function from converting any pump phase modulation into pump amplitude modulation. Consequently, it is reasonable to insist on spectrally efficient pump phase dithering for any SBS suppression. Following this motivation, we show in Sec. 2.3 that pump phase modulation driven by RF noise source occupying only 1.2GHz of optical bandwidth is sufficient to efficiently suppress SBS and enable more than 30dB of parametric gain. Therefore, the NPN (transferred from the pump to the signal via cross-phase modulation) bandwidth can be smaller than that of the amplified signal – a situation never encountered in coherent communication systems. In practical terms, narrow (sub-10GHz) and high-power-handling optical filters necessary for excess noise filtering of high-power FOPA pumps are not commercially available yet, dictating pump filtering with bandwidths typically exceeding 80GHz. As a consequence, the measurement of NPN becomes directly dependent upon the frequency response of the receiving photodiode and the subsequent electronics. The lower the bandwidth of the receiver, the higher the apparent phase fidelity will be, and vice versa.

2.2 Signal/Idler Phase Noise

As the nonlinear phase noise is added to the pump wave during propagation in HNLF, it is also simultaneously transferred to the signal and idler. The signal and idler phase can be derived analytically by generalizing the analysis reported in [25

R. H. Stolen and J. E. Bjorkholm, “Parametric Amplification and Frequency Conversion in Optical Fibers,” IEEE J. Quantum Electron. 18(7), 1062–1072 (1982). [CrossRef]

], where several assumptions are made: (a) HNLF is lossless, (b) polarization effects are ignored (i.e., the propagating waves are perfectly aligned in polarization at all times), (c) no pump depletion by either the signal or the amplified quantum noise takes place, (d) the nonlinear coefficient γ is frequency-independent, and (e) Raman scattering is neglected. By incorporating these assumptions, the total phase mismatch is time-dependent and given by

κ (t) = Δ β (t) + 2 γ {| \sqrt{P_{p}} + n' (t) |}^{2},

(5)

where the first term represents the linear phase mismatch, and the second term is the nonlinear phase shift due to the noisy pump power. The linear phase mismatch, Δβ(t), owes its time dependence to the phase modulation of the pump wave, as is often the case in continuous-wave FOPAs. The signal and idler amplitudes are affected by NPN and PMN via modulation of the exponential gain constant:

g (t) = \sqrt{{(γ {| \sqrt{P_{p}} + n' (t) |}^{2})}^{2} - {(\frac{κ (t)}{2})}^{2}} .

(6)

The signal and idler phases at the output of HNLF can now be written as:

ϕ_{s} (t, L) = ϕ_{s} (t, 0) + \tan^{- 1} (\frac{κ (t) \sinh [g (t) L]}{2 g (t) \cosh [g (t) L]}) + γ {| \sqrt{P_{p}} + n' (t) |}^{2} L - \frac{Δ β (t) L}{2} + Δ ϕ_{s, A Q N},

(7)

ϕ_{i} (t, L) = [2 ϕ_{p} (t, 0) - ϕ_{s} (t, 0)] + \frac{π}{2} + γ {| \sqrt{P_{p}} + n' (t) |}^{2} L - \frac{Δ β (t) L}{2} + Δ ϕ_{i, A Q N} .

(8)

ϕ_s(t,0) and ϕ_p(t,0) represent the time-dependent phases of the signal and pump lasers before entering HNLF. Since narrow linewidth (<1MHz) lasers are practically available, the initial phase noises tend to be small in comparison to the third and fourth term, which represent NPN and PMN, respectively. The last term represents the AQN contribution to the signal and idler phase noise.

Figure 2(a) shows simulated SNR of nonlinear phase [third term in Eqs. (7) and (8)] as a function of pump OSNR and optical Gaussian filter 3-dB bandwidth. Consistent with Ref [18

], the phase SNR is defined as: SNR_phase = 1/Var{ϕ_NL(t)}. The simulation parameters used are specified in Table 1 ; they correspond to the constructed FOPA used in the experimental section of this work (Sec. 3). As expected from analysis in Sec. 2.1, the phase is corrupted by increased pump optical noise filter’s bandwidth and reduced pump OSNR. The second term in Eq. (7) shows an additional noise source that is present in the signal phase and absent in the idler phase. The inverse tangent term is responsible for wavelength dependence of signal NPN contribution and will be termed chromatic NPN (CNPN) in subsequent discussion. Figure 2(b) depicts the analytically predicted and simulated spectral dependence of signal and idler phase and amplitude fidelity for pump OSNR of 40dB and a fixed optical Gaussian 3-dB bandwidth of 40GHz. The amplitude SNR in this calculation is defined as (Mean{A_s,i(t)})²/Var{ A_s,i(t)}, where A_s,i(t) is the time-varying signal/idler amplitude. All simulations were performed using a commercially available full Generalized Nonlinear Schrodinger Equation (GNLSE) solver (VPItransmissionMaker^TM ). The linear phase mismatch, Δβ = -λ_p ²/(2πc)S(λ_p-λ₀)(2πc/λ_s-2πc/λ_p)² [26

G. P. Agrawal, Nonlinear Fiber Optics (Elsevier, 2007), Chap. 10.

], is considered to be constant (i.e., no pump phase modulation), so that the CNPN term is only a function of pump amplitude noise, n’(t). The signal phase exhibits strong variations at the edge of the parametric gain, significantly impairing the amplifier performance in this spectral region. In continuous-wave-pumped FOPAs, the idler phase is dominated by pump phase modulation and the first term in Eq. (8) dominates the noise properties of the idler phase. However, when pump phase modulation can be avoided, as in the case of pulsed-pump FOPAs, the signal phase still exhibits wavelength dependence, whereas the idler phase possesses purely achromatic properties. The increased stability of the idler’s phase is accompanied by a significant increase in amplitude fluctuations, as can be seen in Fig. 2(b). The noise induced by pump amplitude fluctuations, therefore, is distributed differently (in the two quadratures) for the amplified (signal) and converted (idler) wave. Interestingly, the only spectrally independent quadrature noise component is the phase noise of the idler wave.

Fig. 2 (a) Nonlinear phase SNR vs. pump OSNR and optical Gaussian noise filter 3-dB bandwidth; (b) Signal and idler phase and amplitude SNR spectra due to NPN.

Table 1 FOPA simulation/experimental parameters

Symbol	Description	Value
L	HNLF length	175.0 m
γ	Nonlinear coefficient	15.0 W⁻¹km⁻¹
λ₀	Zero-dispersion wavelength	1562.15 nm
S	Dispersion slope	0.02 ps/nm²-km
λ_p	Pump wavelength	1568.0 nm
P_p	Pump power	1.67 W

2.3 Phase Modulation Noise

Pump phase modulation for SBS suppression in FOPAs has traditionally been implemented by means of one or more RF tones. Unfortunately, this method of phase modulation is bandwidth inefficient, since the frequency space between the original RF tones and their multiples is not utilized. In addition, effective SBS suppression dictates spectral equalization of the tones and the modulation 3-dB bandwidth can easily exceed 10GHz [27

J. Hansryd and P. A. Andrekson, “Broad-Band Continuous-Wave-Pumped Fiber Optical Parametric Amplifier with 49-dB Gain and Wavelength-Conversion Efficiency,” IEEE Photon. Technol. Lett. 13(3), 194–196 (2001). [CrossRef]

]. Recognizing this limitation, we phase-modulate the pump by means of a filtered RF noise source, which provides for significantly narrower pump modulation bandwidths without sacrificing the SBS threshold increase [28

J. B. Coles, B. P.-P. Kuo, N. Alic, S. Moro, C.-S. Bres, J. M. C. Boggio, P. A. Andrekson, M. Karlsson, and S. Radic, “Bandwidth-efficient phase modulation techniques for stimulated Brillouin scattering suppression in fiber optic parametric amplifiers,” Opt. Express 18(17), 18138–18150 (2010). [CrossRef] [PubMed]

]. The use of RF noise source as a phase modulator driver poses specific practical requirement. For a modulator characterized by a specific voltage necessary to induce π-retardation (V_π ), the variance of the driving time-varying voltage, σ² _V(t) , must equal V_π ² in order to optimally suppress the optical carrier. Then, the required RF noise power spectral density is given by S_n,rf = (V_π ²/R_L)/Δf_n,rf , where R_L is the load impedance, and Δf_n,rf is the electrical bandwidth of RF noise. The pump instantaneous frequency, ν_p(t) = c/λ_p + (1/2π)dϕ_p(t)/dt, is a function of the instantaneous pump phase, ϕ_p(t) = πV(t)/V_π , defining the linear phase mismatch as

Δ β (t) = \frac{- 2 π c}{ν_{p}^{2} (t)} S (\frac{c}{ν_{p} (t)} - λ_{o}) {(c / λ_{s} - ν_{p} (t))}^{2} .

(9)

The higher order dispersion coefficients (β₄ , β₆ , etc.) have purposely been omitted since their contributions to the linear phase mismatch are negligible in the bandwidth of interest (100nm). A distant spectral conversion [29

J. M. Chavez Boggio, S. Moro, B. P.-P. Kuo, N. Alic, B. Stossel, and S. Radic, “Tunable Parametric All-Fiber Short-Wavelength IR Transmitter,” J. Lightwave Technol. 28(4), 443–447 (2010). [CrossRef]

] or wide-band parametric amplification and frequency generation would require the inclusion of higher-order dispersive terms. Figure 3(a) shows a contour plot of signal phase SNR as a function of signal wavelength and the electrical noise 3-dB Gaussian bandwidth in the absence of NPN (i.e., n’(t) = 0) and using parameters in Table 1. Figure 3(b) depicts the signal phase SNR spectrum for a 600MHz electrical noise bandwidth. As expected from Eq. (9), the phase SNR reduces as the pump-signal wavelength separation is increased. At the edge of the gain spectrum, the interaction between the Δβ(t)L/2 term and the CNPN term in Eq. (7) results in sharp spectral features. These spectral features are smoothed when some amount of pump amplitude noise (and hence NPN) is present, as is always the case in practical FOPA devices. By comparing Fig. 2(b) and Fig. 3(b), we come to expect the PMN to make at least an order of magnitude smaller contribution to the total phase noise than NPN. The two noise variances become comparable when the pump OSNR exceeds 55dB and pump noise is narrowly (i.e. sub-20GHz) filtered in order to reduce NPN. It is interesting to note that PMN increases with increased HNLF length [see Eqs. (7) and (8)]. Hence, we expect FOPAs employing the combination of longer fiber lengths and reduced pump powers to have a larger contribution of PMN to the total phase noise, thus posing another challenge in devising cost-effective FOPA devices.

Fig. 3 Spectral dependence of signal phase SNR when only PMN is present (a) for a range of RF noise bandwidths, (b) for 600MHz RF noise bandwidth.

2.4 Phase noise due to Amplified Quantum Noise

The exact statistics of phase noise in phase-insensitive inverted-population optical amplifiers (often referred to as linear optical amplifier), where equal amount of Gaussian-distributed noise is added by the amplifier to both quadrature components of the signal, are well known [30

J. W. Goodman, Statistical Optics (Wiley, 1985), Chap. 2.

]. In case of high (>10dB) signal-to-noise ratio at the output of the amplifier, the phase variations are dominated by the imaginary part of complex white Gaussian noise and the phase variance is inversely proportional to the SNR [30

J. W. Goodman, Statistical Optics (Wiley, 1985), Chap. 2.

]. The SNR is given by GP_s/P_n, where G is the amplifier gain, P_s is the input signal power, and P_n is the total noise power in one quadrature and one polarization. For EDFAs, P_n = ½hν_sΔνn_sp(G-1), where h is the Planck’s constant, ν_s is the signal frequency, Δν is the optical bandwidth, and n_sp is the spontaneous emission factor [4

P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers (Academic Press, 1999).

]. Thus, in the limit of high gain (i.e. G>>1), the phase SNR due to ASE becomes 2P_s/(hν_sΔνn_sp).

The parametric amplifier is seeded by zero-point ‘vacuum’ fluctuations with power spectral density of hν/2 in one polarization, as illustrated in Fig. 4 . The vacuum fluctuations at the signal wavelength are amplified by parametric gain G_s = 1 + (γP_p/g)²sinh²(gL) [26

G. P. Agrawal, Nonlinear Fiber Optics (Elsevier, 2007), Chap. 10.

]. In addition, the signal is coupled to the AQN associated with the idler, which is amplified by gain of G_s-1. Adding the two noise contributions, the total power of AQN in one polarization and both quadratures is given by

Fig. 4 Schematic of parametric amplification of quantum noise in a one-pump FOPA. Acronyms: P_s – signal power, ν_s – signal frequency, ν_p – pump frequency, P_p – pump power, OBPF – optical band-pass filter, ν_i – idler frequency, G_s – signal gain, Δν – optical filter bandwidth, AQN – amplified quantum noise.

P_{A Q N} = \frac{h ν}{2} (G_{s} + G_{s} - 1) Δ ν = \frac{h ν}{2} (2 G_{s} - 1) Δ ν .

(10)

The photon number and field-quadrature fluctuations of the phase-insensitive parametric amplifier have been shown to be equivalent to those of the zero-noise-input inverted-population amplifier (e.g. EDFA), provided that complete population inversion is achieved in the case of the latter [31

K. Shimoda, H. Takahasi, and C. H. Townes, “Fluctuations in Amplification of Quanta with Application to Maser Amplifiers,” J. Phys. Soc. Jpn. 12(6), 686–700 (1957). [CrossRef]

–34

A. V. Kozlovskii, “Photodetection of a weak light signal in various quantum states by using an optical amplifier,” Quantum Electron. 36(3), 280–286 (2006). [CrossRef]

]. In other words, the phase-insensitive parametric amplifier adds a circular (in the complex plane) noise cloud to the amplified signal, analogous to its inverted-population counterpart. Consequently, owing to the statistical similarities of the two noise processes (ASE and AQN), it can straightforwardly be concluded that the variance of signal phase noise due to AQN is identical to that of a perfectly inverted EDFA, with signal phase SNR given by

S N R_{p h a s e} = \frac{1}{σ^{2}_{Δ ϕ_{A Q N}}} = \frac{4 G_{s} P_{s}}{h ν (2 G_{s} - 1) Δ ν} .

(11)

The validity of simple theory leading to Eq. (11) was confirmed via simulation employing a full GNLSE solver for FOPA with parameters listed in Table 1. An excellent agreement was found and the results are depicted in Fig. 5 . The analytical approach works even when FOPA is operating near the transparency regime [i.e. at left edge of the gain region in Fig. 5(a)], where the phase fidelity of the input signal(s) is not impaired by the parametric amplifier.

Fig. 5 Analytical/numerical signal phase SNR (when only AQN is present): (a) spectrum for two different input signal powers; (b) dependence on input signal power at the signal wavelength of 1545.0nm.

3. Experimental Results and Discussion

An experimental setup for phase noise characterization in a one-pump FOPA was constructed as shown in Fig. 6 . HNLF and pump parameters are given in Table 1.

Fig. 6 Experimental setup for measurement of phase noise of a FOPA; Inset: electrical waveforms observed after intra-dyne detection of the two output ports of the 90° hybrid. Acronyms: PM – phase modulator, LPF – low-pass (RF) filter, VOA – variable optical attenuator, EDFA – Erbium-doped fiber amplifier, CWDM – coarse wavelength-division multiplexer, HNLF – highly nonlinear fiber, OBPF – optical band-pass filter, LO – local oscillator, ADC – analog-to-digital converter.

The pump, signal, and local oscillator (LO) were standard external cavity lasers and all had 100kHz linewidths. The pump phase was dithered using a 3GHz-bandwidth RF noise source constructed for spectrally efficient SBS suppression and filtered with a 600MHz low-pass electrical filter. The filtered noise bandwidth provided approximately 14dB of SBS threshold increase, which was more than sufficient to fully suppress SBS in the constructed amplifier. A variable optical attenuator, preceding a cascade of low-power and high power EDFAs, was used to vary the OSNR of the pump wave. Pump optical noise was filtered with a combination of a coarse-wavelength-division-mutiplexer (CWDM) filter and a 2nm 3-dB bandwidth band-pass filter with approximately Lorentzian optical power transfer function. The CWDM possessed high extinction, in excess of 45dB, required to guarantee that the filtered amplifier noise at the signal frequency had lower power spectral density than the inherently present quantum noise. The authors in Ref [3

]. refer to the poorly-suppressed (and subsequently amplified by FOPA) amplifier noise as pump residual noise; the experimental setup constructed in this work aimed to control such contribution. Input signal power into the FOPA was kept constant at −20dBm. Following amplification in HNLF, the signal wave was filtered out and combined with an amplified local oscillator (LO) wave in a 90° optical hybrid. Two of the hybrid’s outputs (S + LO and S + jLO) were detected using matched 20GHz linear p-i-n photodiodes with responsivity of 0.95A/W. Incident on the photodiodes, the signal power was kept constant at −20dBm and the LO power at 10dBm. The two beat currents were measured on a 50Gs/s real-time oscilloscope (Tektronix DPO71604B) characterized by 16dB-bandwidth 4th-order Butterworth frequency response.

Optical spectra after the FOPA for two different signal wavelengths are shown in Fig. 7(a) and measured gain spectra are shown in Fig. 7(b). While the gain for the two signal wavelengths in Fig. 7(a) is nearly the same, the OSNR of the farther signal is approximately 6dB lower than the OSNR of the signal closer to the pump. We therefore expect the farther signal to have higher phase noise. It is also interesting to note that pump’s OSNR has reduced by about 15dB following propagation in HNLF, which is equal to the parametric gain in the vicinity of the pump. Indeed, the reduced OSNR is a consequence of added nonlinear phase noise. However, the pump amplitude SNR is unchanged since we know that the pump only acquires a nonlinear phase shift [25

R. H. Stolen and J. E. Bjorkholm, “Parametric Amplification and Frequency Conversion in Optical Fibers,” IEEE J. Quantum Electron. 18(7), 1062–1072 (1982). [CrossRef]

Fig. 7 (a) Optical spectra for two different signal wavelengths, (b) Measured FOPA gain spectrum.

High-resolution optical spectra of pump/signal/idler waves were taken after the FOPA and are depicted in Fig. 8(a) . As expected, the pump and idler frequencies are significantly broadened. The idler spectral broadening is approximately twice as large as that of the pump because two pump photons are involved in the four-photon interaction (2f_p→f_s + f_i ). Looking closely at the high-resolution optical spectrum of the signal wave at different pump OSNRs [Fig. 8(b)], we clearly see the spectral contribution of the narrowband PMN and broadband NPN. Since the measured phase noise will be integrated over 16GHz of electrical bandwidth, we expect the NPN-induced phase noise to dominate the signal phase SNR. The two sharp peaks located 170MHz away from the signal center frequency are the laser cavity sidemodes; they are suppressed by approximately 50dB with respect to the carrier.

Fig. 8 (a) Optical spectra of pump/signal/idler after HNLF; (b) Signal spectrum before and after FOPA. The three disjoint spectral bands centered around the three characteristic wavelengths (signal - 1545.0nm, pump - 1568.0nm, and idler - 1591.7nm) have been shifted for easier comparison.

The waveforms captured by the real-time oscilloscope were processed off-line (as outlined in Ref [18

].), and the wavelength-dependent signal phase SNR and amplitude SNR are plotted in Fig. 9 . The pump OSNR of 40dB was used in order to allow the NPN to dominate the signal phase noise and measure the predicted spectral dependence. As expected from the analysis in Sec. 2.2, the measured phase SNR is lower at the edges of the parametric gain and increases as the signal wavelength approaches that of the pump. An excellent agreement is found among the analytically predicted, the numerically simulated and the experimentally measured signal phase SNR spectra. The signal amplitude SNR is wavelength-dependent owing to wavelength-dependent gain sensitivity to pump power [3

]. The gain sensitivity increases with increased pump-signal separation and the signal suffers larger amplitude noise as evident in Fig. 8.

Fig. 9 Measured, simulated, and analytically predicted signal phase and amplitude SNR spectra for input pump OSNR of 40dB.

4. Conclusion

We have analytically predicted and experimentally verified the spectral dependence of amplified signal and converted idler phase noise in a one-pump fiber-optic parametric amplifier. We demonstrated that, when the pump phase is modulated in a bandwidth efficient manner, the nonlinear phase noise dominates over other parametric amplifier’s noise contributions. This finding has a basic implication as it dictates use of specific means for pump dithering that are similar to RF noise driven modulation used in this work.

Furthermore, it was shown that the phase fidelity exhibits a strong spectral dependence and is significantly degraded at the edges of the gain region. More importantly, the derived analytical theory implies that the idler phase noise exhibits no spectral dependence in the absence of pump phase modulation. However, the idler’s superior phase fidelity is accompanied by an increase in field amplitude fluctuations and casts qualitative new light on the construction of spectrally mapped systems. The main conclusions of this work thus bear significant implications on the design and construction of parametric amplifiers/converters, where frequency and amplitude stability of the newly generated wave(s) is of utmost importance.

Acknowledgements

The authors would like to thank C. J. McKinstrie, M. Karlsson, and A. Radosevic for fruitful discussions and helpful comments. The authors would like to thank Sumitomo Electric Ltd. for providing the nonlinear fiber and Marki Microwave, Inc. for providing some of the necessary RF components. This material is based on research sponsored in part by Lockheed Martin Corporation, Air Force Research Laboratory (AFRL) and the Defense Advanced Research Agency (DARPA) under agreement number FA8650-08-1-7819 Parametric Optical Processes and Systems.

References and links

1.	S. Radic, “Parametric amplification and processing in optical fibers,” Laser Photon. Rev. 2(6), 498–513 (2008). [CrossRef]
2.	P. Kylemark, M. Karlsson, T. Torounidis, and P. A. Andrekson, “Noise Statistics in Fiber Optical Parametric Amplifiers,” J. Lightwave Technol. 25(2), 612–620 (2007). [CrossRef]
3.	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(3), 2884–2893 (2010). [CrossRef] [PubMed]
4.	P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers (Academic Press, 1999).
5.	R. M. Jopson, U.S. Patent 5 386 394 (1994).
6.	M. Ho, M. E. Marhic, K. Y. K. Wong, and L. Kazovsky, “Narrow linewidth idler generation in fiber four-wave mixing and parametric amplification by dithering two pumps in opposition of phase,” J. Lightwave Technol. 20(3), 469–476 (2002). [CrossRef]
7.	S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective Suppression of Idler Spectral Broadening in Two-Pump Parametric Architectures,” IEEE Photon. Technol. Lett. 15(5), 673–675 (2003). [CrossRef]
8.	A. Durécu-Legrand, A. Mussot, C. Simonneau, D. Bayart, T. Sylvestre, E. Lantz, and H. Maillotte, “Impact of pump phase modulation on system performances of fiber optical parametric amplifiers,” Electron. Lett. 41(6), 350–352 (2005). [CrossRef]
9.	P. Kylemark, J. Ren, Y. Myslivets, N. Alic, S. Radic, P. A. Andrekson, and M. Karlsson, “Impact of Pump Phase-Modulation on the Bit-Error Rate in Fiber-Optical Parametric-Amplifier-Based Systems,” IEEE Photon. Technol. Lett. 19(1), 79–81 (2007). [CrossRef]
10.	R. Elschner, C.-A. Bunge, B. Huttl, A. G. i Coca, C. L. Schmidt, R. Ludwig, C. Schubert, and K. Petermann, “Impact of Pump-Phase Modulation on FWM-Based Wavelength Conversion of D(Q)PSK Signals,” IEEE J. Sel. Top. Quantum Electron. 14(3), 666–673 (2008). [CrossRef]
11.	R. Jiang, C.-S. Bres, N. Alic, E. Myslivets, and S. Radic, “Translation of Gbps Phase-Modulated Optical Signal From Near-Infrared to Visible Band,” J. Lightwave Technol. 26(1), 131–137 (2008). [CrossRef]
12.	J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990). [CrossRef] [PubMed]
13.	K.-P. Ho, Phase-Modulated Optical Communication Systems (Springer, 2005), Chap. 5.
14.	S. J. McNaught, J. E. Rothenberg, P. A. Thielen, M. G. Wickham, M. E. Weber, and G. D. Goodno, “Coherent Combining of a 1.26-kW Fiber Amplifier,” Advanced in Solid-State Photonics, paper AMA2 (2010).
15.	Y. Kim, S. Kim, Y.-J. Kim, H. Hussein, and S.-W. Kim, “Er-doped fiber frequency comb with mHz relative linewidth,” Opt. Express 17(14), 11972–11977 (2009). [CrossRef] [PubMed]
16.	N. Nishizawa and J. Takayanagi, “Octave spanning high-quality supercontinuum generation in all-fiber system,” J. Opt. Soc. Am. B 24(8), 1786–1792 (2007). [CrossRef]
17.	N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs,” J. Opt. Soc. Am. B 24(8), 1756–1770 (2007). [CrossRef]
18.	M. Sköld, J. Yang, H. Sunnerud, M. Karlsson, S. Oda, and P. A. Andrekson, “Constellation diagram analysis of DPSK signal regeneration in a saturated parametric amplifier,” Opt. Express 16(9), 5974–5982 (2008). [CrossRef] [PubMed]
19.	M. Skold, M. Karlsson, S. Oda, H. Sunnerud, and P. A. Andrekson, “Constellation diagram measurements of induced phase noise in a regenerating parametric amplifier,” Optical Fiber Communications Conference, paper OML4 (2008).
20.	M. Matsumoto, “Phase noise generation in an amplitude limiter using saturation of a fiber-optic parametric amplifier,” Opt. Lett. 33(15), 1638–1640 (2008). [CrossRef] [PubMed]
21.	R. Elschner, and L. Petermann, “Impact of Pump-Induced Nonlinear Phase Noise on Parametric Amplification and Wavelength Conversion of Phase-Modulated Signals,” European Conference in Optical Communications, paper 3.3.4 (2009).
22.	E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley-Interscience, 2002).
23.	M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators, and Related Devices (Cambridge University Press, 2008).
24.	N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical preamplifier,” Opt. Express 13(12), 4568–4579 (2005). [CrossRef] [PubMed]
25.	R. H. Stolen and J. E. Bjorkholm, “Parametric Amplification and Frequency Conversion in Optical Fibers,” IEEE J. Quantum Electron. 18(7), 1062–1072 (1982). [CrossRef]
26.	G. P. Agrawal, Nonlinear Fiber Optics (Elsevier, 2007), Chap. 10.
27.	J. Hansryd and P. A. Andrekson, “Broad-Band Continuous-Wave-Pumped Fiber Optical Parametric Amplifier with 49-dB Gain and Wavelength-Conversion Efficiency,” IEEE Photon. Technol. Lett. 13(3), 194–196 (2001). [CrossRef]
28.	J. B. Coles, B. P.-P. Kuo, N. Alic, S. Moro, C.-S. Bres, J. M. C. Boggio, P. A. Andrekson, M. Karlsson, and S. Radic, “Bandwidth-efficient phase modulation techniques for stimulated Brillouin scattering suppression in fiber optic parametric amplifiers,” Opt. Express 18(17), 18138–18150 (2010). [CrossRef] [PubMed]
29.	J. M. Chavez Boggio, S. Moro, B. P.-P. Kuo, N. Alic, B. Stossel, and S. Radic, “Tunable Parametric All-Fiber Short-Wavelength IR Transmitter,” J. Lightwave Technol. 28(4), 443–447 (2010). [CrossRef]
30.	J. W. Goodman, Statistical Optics (Wiley, 1985), Chap. 2.
31.	K. Shimoda, H. Takahasi, and C. H. Townes, “Fluctuations in Amplification of Quanta with Application to Maser Amplifiers,” J. Phys. Soc. Jpn. 12(6), 686–700 (1957). [CrossRef]
32.	J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129(1), 481–485 (1963). [CrossRef]
33.	C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13(13), 4986–5012 (2005). [CrossRef] [PubMed]
34.	A. V. Kozlovskii, “Photodetection of a weak light signal in various quantum states by using an optical amplifier,” Quantum Electron. 36(3), 280–286 (2006). [CrossRef]

OCIS Codes
(060.5060) Fiber optics and optical communications : Phase modulation
(190.4970) Nonlinear optics : Parametric oscillators and amplifiers
(270.2500) Quantum optics : Fluctuations, relaxations, and noise

ToC Category:
Nonlinear Optics

History
Original Manuscript: June 28, 2010
Revised Manuscript: September 5, 2010
Manuscript Accepted: September 17, 2010
Published: September 24, 2010

Citation
Slaven Moro, Ana Peric, Nikola Alic, Bryan Stossel, and Stojan Radic, "Phase noise in fiber-optic parametric amplifiers and converters and its impact on sensing and communication systems," Opt. Express 18, 21449-21460 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-21449

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References

S. Radic, “Parametric amplification and processing in optical fibers,” Laser Photon. Rev. 2(6), 498–513 (2008). [CrossRef]
P. Kylemark, M. Karlsson, T. Torounidis, and P. A. Andrekson, “Noise Statistics in Fiber Optical Parametric Amplifiers,” J. Lightwave Technol. 25(2), 612–620 (2007). [CrossRef]
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(3), 2884–2893 (2010). [CrossRef] [PubMed]
P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers (Academic Press, 1999).
R. M. Jopson, U.S. Patent 5 386 394 (1994).
M. Ho, M. E. Marhic, K. Y. K. Wong, and L. Kazovsky, “Narrow linewidth idler generation in fiber four-wave mixing and parametric amplification by dithering two pumps in opposition of phase,” J. Lightwave Technol. 20(3), 469–476 (2002). [CrossRef]
S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective Suppression of Idler Spectral Broadening in Two-Pump Parametric Architectures,” IEEE Photon. Technol. Lett. 15(5), 673–675 (2003). [CrossRef]
A. Durécu-Legrand, A. Mussot, C. Simonneau, D. Bayart, T. Sylvestre, E. Lantz, and H. Maillotte, “Impact of pump phase modulation on system performances of fiber optical parametric amplifiers,” Electron. Lett. 41(6), 350–352 (2005). [CrossRef]
P. Kylemark, J. Ren, Y. Myslivets, N. Alic, S. Radic, P. A. Andrekson, and M. Karlsson, “Impact of Pump Phase-Modulation on the Bit-Error Rate in Fiber-Optical Parametric-Amplifier-Based Systems,” IEEE Photon. Technol. Lett. 19(1), 79–81 (2007). [CrossRef]
R. Elschner, C.-A. Bunge, B. Huttl, A. G. i Coca, C. L. Schmidt, R. Ludwig, C. Schubert, and K. Petermann, “Impact of Pump-Phase Modulation on FWM-Based Wavelength Conversion of D(Q)PSK Signals,” IEEE J. Sel. Top. Quantum Electron. 14(3), 666–673 (2008). [CrossRef]
R. Jiang, C.-S. Bres, N. Alic, E. Myslivets, and S. Radic, “Translation of Gbps Phase-Modulated Optical Signal From Near-Infrared to Visible Band,” J. Lightwave Technol. 26(1), 131–137 (2008). [CrossRef]
J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990). [CrossRef] [PubMed]
K.-P. Ho, Phase-Modulated Optical Communication Systems (Springer, 2005), Chap. 5.
S. J. McNaught, J. E. Rothenberg, P. A. Thielen, M. G. Wickham, M. E. Weber, and G. D. Goodno, “Coherent Combining of a 1.26-kW Fiber Amplifier,” Advanced in Solid-State Photonics, paper AMA2 (2010).
Y. Kim, S. Kim, Y.-J. Kim, H. Hussein, and S.-W. Kim, “Er-doped fiber frequency comb with mHz relative linewidth,” Opt. Express 17(14), 11972–11977 (2009). [CrossRef] [PubMed]
N. Nishizawa and J. Takayanagi, “Octave spanning high-quality supercontinuum generation in all-fiber system,” J. Opt. Soc. Am. B 24(8), 1786–1792 (2007). [CrossRef]
N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs,” J. Opt. Soc. Am. B 24(8), 1756–1770 (2007). [CrossRef]
M. Sköld, J. Yang, H. Sunnerud, M. Karlsson, S. Oda, and P. A. Andrekson, “Constellation diagram analysis of DPSK signal regeneration in a saturated parametric amplifier,” Opt. Express 16(9), 5974–5982 (2008). [CrossRef] [PubMed]
M. Skold, M. Karlsson, S. Oda, H. Sunnerud, and P. A. Andrekson, “Constellation diagram measurements of induced phase noise in a regenerating parametric amplifier,” Optical Fiber Communications Conference, paper OML4 (2008).
M. Matsumoto, “Phase noise generation in an amplitude limiter using saturation of a fiber-optic parametric amplifier,” Opt. Lett. 33(15), 1638–1640 (2008). [CrossRef] [PubMed]
R. Elschner, and L. Petermann, “Impact of Pump-Induced Nonlinear Phase Noise on Parametric Amplification and Wavelength Conversion of Phase-Modulated Signals,” European Conference in Optical Communications, paper 3.3.4 (2009).
E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley-Interscience, 2002).
M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators, and Related Devices (Cambridge University Press, 2008).
N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical preamplifier,” Opt. Express 13(12), 4568–4579 (2005). [CrossRef] [PubMed]
R. H. Stolen and J. E. Bjorkholm, “Parametric Amplification and Frequency Conversion in Optical Fibers,” IEEE J. Quantum Electron. 18(7), 1062–1072 (1982). [CrossRef]
G. P. Agrawal, Nonlinear Fiber Optics (Elsevier, 2007), Chap. 10.
J. Hansryd and P. A. Andrekson, “Broad-Band Continuous-Wave-Pumped Fiber Optical Parametric Amplifier with 49-dB Gain and Wavelength-Conversion Efficiency,” IEEE Photon. Technol. Lett. 13(3), 194–196 (2001). [CrossRef]
J. B. Coles, B. P.-P. Kuo, N. Alic, S. Moro, C.-S. Bres, J. M. C. Boggio, P. A. Andrekson, M. Karlsson, and S. Radic, “Bandwidth-efficient phase modulation techniques for stimulated Brillouin scattering suppression in fiber optic parametric amplifiers,” Opt. Express 18(17), 18138–18150 (2010). [CrossRef] [PubMed]
J. M. Chavez Boggio, S. Moro, B. P.-P. Kuo, N. Alic, B. Stossel, and S. Radic, “Tunable Parametric All-Fiber Short-Wavelength IR Transmitter,” J. Lightwave Technol. 28(4), 443–447 (2010). [CrossRef]
J. W. Goodman, Statistical Optics (Wiley, 1985), Chap. 2.
K. Shimoda, H. Takahasi, and C. H. Townes, “Fluctuations in Amplification of Quanta with Application to Maser Amplifiers,” J. Phys. Soc. Jpn. 12(6), 686–700 (1957). [CrossRef]
J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129(1), 481–485 (1963). [CrossRef]
C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13(13), 4986–5012 (2005). [CrossRef] [PubMed]
A. V. Kozlovskii, “Photodetection of a weak light signal in various quantum states by using an optical amplifier,” Quantum Electron. 36(3), 280–286 (2006). [CrossRef]

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