We propose a phase-sensitive amplifier scheme that balances fiber loss and parametric gain everywhere in a fiber span. We show that, for long links, such a distributed phase-sensitive amplifier has a 3-dB lower noise figure than an ideal distributed phase-insensitive amplifier (e.g. Raman), even if simple direct detection is employed. This sets the ultimate limit for the optimum noise-nonlinearity trade-off in transmission systems.
©2005 Optical Society of America
Phase-sensitive amplifiers (PSAs) have unique properties that allow them to break the 3-dB quantum limit of the optical amplifier noise figure (NF) , as well as provide the phase regeneration leading to suppression of phase, frequency and timing jitters in optical transmission lines [2–4]. The first experimental demonstrations of sub-3-dB NF PSAs were done in χ(2)-based devices [5, 6]. The free-space bulk-crystal PSA also enabled noiseless amplification of images . PSA’s use in optical communication context, however, requires fiber-based approaches. One of such designs, based on a nonlinear optical loop mirror (NOLM), was introduced in , and successfully utilized in  for soliton regeneration in a long-term storage buffer and in  for regeneration of differential phase-shift-keying signals (a similar nonlinear interferometer has also been employed in vacuum squeezing, e.g. [11, 12]). The solutions to synchronization of the pump and incoming signal phases were developed via pump injection locking  and optical phase-locked loop .
Achieving the sub-3-dB NF in fiber-based PSAs, however, is challenged by the Guided Acoustic-Wave Brillouin Scattering (GAWBS)  that introduces uncorrelated phase fluctuations in the counter-propagating NOLM arms, resulting in amplitude noise at the PSA output. The nearly noiseless PSA performance has nevertheless been experimentally demonstrated in [16, 17], using pulses with orthogonal polarizations to cancel common GAWBS noise, followed by , where the GAWBS was avoided by conducting the NF measurements well above this noise’s cut-off of ~2 GHz. Despite these successes, the complexity of GAWBS mitigation clearly limits the potential use of the NOLM-based PSAs. Even more importantly, the NOLM-based PSAs are inherently single-channel devices not compatible with modern wavelength-division-multiplexed (WDM) transmission systems.
The above two drawbacks can be overcome in the new-generation fiber PSAs that are based on non-degenerate four-wave mixing (FWM) in fiber. By exciting a combination of the signal and idler waves at the input, such an optical parametric amplifier can be put into a phase-sensitive regime. Moreover, two independent modes involving signal-idler-wave combinations can be phase-sensitively amplified at the same time, resulting in the same bandwidth utilization as that of the frequency-degenerate PSA. Since all involved waves propagate in the same direction, the GAWBS perturbs all of them equally, with no impact on the PSA gain and noise. In addition, by using the wide-bandwidth parametric amplifiers (e.g. those involving two pumps ), one can simultaneously amplify many WDM channels, provided that their phases are synchronized up to a multiple of π. In-depth analysis of NF of these devices indicates their potential for beating the 3-dB NF quantum limit [20–23]. Recently, a FWM-based phase-sensitive inline amplifier has been experimentally demonstrated [24, 25] using pump modulation sidebands as the signal and idler .
In this paper, we propose a novel distributed PSA that combines the low-noise properties of FWM-based PSAs with the optimal trade-off between optical nonlinearity and spontaneous emission noise provided by balancing the local gain and loss coefficients at every point within the transmission span (distributed gain). Such optimal trade-off, quantified by the minimal product of overall span NF and nonlinear phase shift, has recently been demonstrated for phase-insensitive (Raman) amplifiers (PIAs) using effective-area management in hybrid dispersion-managed fiber spans [27–29], and using the combination of bi-directional and second-order Raman pumping in a single-mode-fiber span . In this paper, we show that the distributed PSA improves the performance by 3 dB over an ideally distributed PIA for a sufficiently long fiber span, which sets the ultimate fundamental performance limit of the amplified transmission link (see Fig. 1).
The proposed distributed PSA scheme uses distributed phase-sensitive parametric amplification of a two-sideband signal with spectrum symmetric around the pump. Uniform distribution of parametric gain in the span is achieved using Raman amplification of the pump, whereas launching both parametric and Raman pumps so that their polarization is orthogonal to that of the two-sideband signal minimizes Raman contribution to the signal gain and noise. Although impact of polarization-mode-dispersion-caused depolarization on the net NF represents an interesting topic, it is beyond the scope of this paper. The acceptance of PSAs as inline amplifiers has so far been hindered by the need for phase-locking between the signal and pump. Our novel distributed configuration greatly simplifies this process, because the pump beam coming from the previous span can be used as a reference for locking the phase of the pump beam of the next span. This concept of a distributed PSA was first mentioned  and then discussed  in our conference presentations. The present paper offers the detailed analysis of this device.
The paper is structured as follows. Section 2 outlines the theoretical principles of the distributed PSA, starting with review of lossless (lumped) FWM-based PSA in Sec. 2.1, and extending this treatment to the lossy medium in Sec. 2.2. Section 3 describes the scheme for realizing a transmission link based on the distributed PSA. Section 4 summarizes the paper.
2. Theory of Distributed Noiseless Amplification
In this section, we will describe the theory of distributed phase-sensitive amplification realized by non-degenerate parametric amplification with energy conservation law of ω1+ω2=ωs+ωI , where ωS,I and ω1,2 are signal/idler and pump frequencies, respectively. We will start with the case of a lumped PSA (lossless nonlinear medium), and then extend our treatment to the distributed case in lossy fiber.
2.1 Non-degenerate parametric amplification in lossless fiber (“lumped” amplifier)
Let us start with a set of parametric amplification equations in fiber with one or two pumps under the undepleted pump approximation:
Here the signal/idler fields AS,I and pump fields A 1,2 are normalized so that the squares of their absolute values yield their respective powers, PS,I and P 1,2. Note that Eqs. (1) apply not only to classical, but also to the quantized fields, in which case AS,I and become annihilation and creation operators, respectively, with commutators [AS ]=[AI] = 1, and the pump quantum noise is neglected. Nonlinear constant γ=n 2ωP/(cA eff) here is related to the nonlinear refractive index n 2 averaged over all polarization states, ∆β=2βP - βS - βI is the wavevector mismatch, and the fiber loss is neglected. Eqs. (1) describe both one- (εP=1/2) and two-pump (εP=1) parametric amplifiers in either polarization maintaining (ε=9/8) or conventional (ε=1) fiber, where both signal and idler are either co-polarized with the pumps (a=b=1), or orthogonal to them (a=b=1/3 for polarization-maintaining and a=1/2, b=1/4 for conventional fiber) . In the one-pump case, we assume that A 1=A 2 and P 1=P 2=P 0/2, where P 0 is the total one-pump power. Note that Eq. (1) applies only to co-polarized pumps (for the case of two cross-polarized pumps, see, e.g. ). The solution of Eqs. (1) has the form of Bogoliubov transformation
where |μ|2-|v|2=1, κ = ∆β - 2γε (P 1+P 2) (2a - εP - 1/2) is the net phase mismatch parameter, θ1 and θ2 are the input pump phases [i.e. e iθ1,2], and is the parametric gain coefficient. Coupled propagation of the signal and idler can be factorized into phase-sensitive amplification A ±(z) = μA ±(0)±v (0) of two independent modes
which can be easily seen by looking at their field quadratures
For quadrature X +, the maximum phase-sensitive gain G PSA=[X +(z)/X +(0)]2=(|μ|+|v|)2 occurs when pump phases add up to θ1+θ2 = -π/2-tan-1 [(κ/2g) tanh gz].
Let us concentrate on the phase-matched case κ=0. If only the signal beam is present at the input, it undergoes traditional phase-insensitive amplification (PIA) with power gain G PIA=|μ|2=cosh2 gz, as shown by Eqs. (2). At the same time, both of its quadrature noise variances go up by 2G PIA-1 because of the noise contributed by the idler mode. The resulting noise figure NFPIA=(2G PIA-1)/G PIA=2-1/G PIA corresponds to the “3-dB” quantum limit of an ideal PIA at high gain (i.e. same as that for highly inverted erbium-doped amplifier). The NF is the same for both homodyned and direct-detected signals, because, at reasonable signal powers, the noise of the latter can be treated as self-homodyning of the amplitude quadrature noise (signal-spontaneous beat noise).
Optimum phase-sensitive amplification takes place when both signal and idler inputs are equally excited. At pump phases θ1+θ2 = -π/2, the maximum parametric power gain
is realized for quadratures X +, Y -, whereas minimum gain
is realized for X -, Y +. In the PSA case, both the mean quadrature value and the noise are amplified by the same factor, yielding the noiseless amplification (NFPSA=1) of X + (if input signal and idler amplitude quadratures are equal) and Y - (if their phase quadratures are equal).
When frequency separation between the signal and idler modes is small, the measurement of quadratures (4), revealing the noiseless amplification, can be done by means of coherent detection with a local oscillator at frequency ωLO= (ωs+ωI)/2, followed by measurements of the photocurrent at frequency ∆ω/2π=(ωs-ωI)/4π. In realistic amplifiers, however, this frequency measures in THz, which far exceeds the electric bandwidth of any photodetector. Nevertheless, the square-law characteristics of a wideband detector, necessary to produce a proper superposition of the signal and idler fields in the photocurrent, can be emulated by nonlinear-optical means, e.g. through cross-phase modulation in a χ(3)-medium such as semiconductor optical amplifier (SOA) or highly-nonlinear fiber (see Fig. 2). Indeed, the nonlinear cross-phase shift imposed on a cw wave by the local oscillator mixed with the signal and idler beams is proportional to the instantaneous intensity ΦNL∝|A LO e iφ+Ase i∆ωt+AI e -i∆ωt|2, and the output electric field of the cw wave is modulated at the beat frequency ∆ω. The magnitudes of the optical spectral components at ω0±∆ω represent the superposition As e -iφ + AI e iφ (or its conjugate) of the signal and idler fields and at φ=0 are equal to X + ± iY -.
If only one of X +, Y - quadratures is encoded with information, the ∆ω-shifted component can be selected by a filter and measured by direct detection. We will call this nonlinear-optics-assisted measurement “quasi-coherent” photodetection. It is worth mentioning that, to maximize the spectral efficiency, both X + and Y - can be independently encoded, in which case they have to be selectively measured by optical homodyne detection. For simplicity of discussion, we will concentrate on the case of only one quadrature encoded with information.
Another approach to quasi-coherent detection is to employ a four-wave-mixing process where one pump photon at frequency ω3 and one idler photon at ωI are annihilated, creating one pump photon at ω4 and one signal photon at ωS (i.e. ω3+ωI=ω4+ωS) . This process resembles beam splitting/combining, either splitting the signal power between signal and idler modes (to prepare the PSA input), or coherently recombining the signal and idler fields into the signal mode (to achieve quasi-coherent detection). We will refer to this process as “nonlinear beam splitter” or BS.
Coherent or quasi-coherent photodetection shows the noiseless properties of PSA for any value of gain. For high values of parametric gain, however, it is possible to observe the benefits of the noiseless amplification by a far simpler direct-detection measurement of either signal or idler mode alone. Indeed, while the quadrature noise of the signal mode As increases by 2G PIA-1 in both PIA and PSA cases, the mean signal gain in PSA case is given by Eq. (5) when the idler mode AI is excited with equal amplitude and appropriate phase. Therefore, the NF for signal-beam amplification is
for G PIA≫1. The seemingly paradoxical 3-dB improvement of the signal-to-noise ratio (SNR) compared to that of the input signal is explained by the observation that, due to strong nonlinear coupling of signal and idler at high gains, the SNR approaches that of the input signal of twice the power (i.e. with sum of input signal and idler powers), which is exactly the same as that for modes (3). Such SNR based on the total input power of the two beams should be considered the effective input SNR, and the effective PSA NF should be given by Eq. (7) increased by 3 dB. Thus, at high PSA gains, the SNR of the signal mode alone does not degrade from the effective input SNR, and noiseless amplification is realized (Fig. 3 left).
2.2 Non-degenerate parametric amplification in lossy fiber (distributed amplifier)
The results discussed in the previous section pertained to the lossless-fiber amplifier, which is a reasonable assumption only for very short fibers (“lumped” parametric amplifiers). Since our goal is to construct an ideal distributed phase-sensitive amplifier that provides optimum trade-off between the spontaneous emission noise and fiber nonlinearity, we need to introduce the fiber loss α by subtracting (α/2) AS,I and (α/2) A 1,2 from the right-hand-side of the top and bottom Eqs. (1), respectively. If we assume that the losses in the pump beams are compensated (e.g. by Raman amplification), so that the pump power is constant along the fiber, then the only change to the solution given by Eqs. (2) will be multiplication of the parameters μ and v by e -αz/2 in the case of classical fields. To calculate the NF, however, we need to consider the evolution of quantized signal and idler fields, in which case we also need to account for the quantum noise introduced by the fiber loss via adding a term √α.cS,I (z) on the right-hand side of the top of Eqs. (1). Here cS and cI are two independent distributed vacuum noise operators with commutators [cS (z),(z′)] = [cI (z),(z′)] = δ(z-z′) describing the quantum fluctuations introduced by random deletion of photons from the signal and idler beams, respectively. Then the signal and idler field operators at the output are given by
where μ and v are still given by Eqs. (2). The above normalization of distributed operators cS,I (z) has been chosen so that, in the absence of parametric gain (μ=1, v=0), the second line of Eq. (8) yields the usual passive-loss noise VS,I , where VS,I are (lumped) vacuum operators with commutators [VS] =[VI] = 1. Although Eq. (8) is valid for any pump power and phase mismatch, the best amplifier performance is achieved in the phase-matched case (κ=0) under the condition that the fiber loss is compensated by the parametric gain at every point in the fiber. This requires the launch values of pump powers to be P 0=α/(2γεb) in one-pump and = α/(4γεb) in two-pump cases, in addition to the phase matching conditions. Assuming this situation and θ1+θ2 = -π/2, we obtain signal power gain G PSA=1 and noise power gain = 1 + αz for the quadratures X +, Y -, and signal power gain G PSA = e -αz and noise power gain = (1 + e -2αz)/2 for the quadratures X -, Y + . Thus, for quadratures X + and Y - the fiber span noise figure is NF=1+αz. This NF represents the fundamental limit of the optical amplifier performance, i.e. it produces the maximum possible SNR for a given level of nonlinear impairments. Same noise-figure expression can be obtained by cascading infinitesimal pieces of optical fiber with attenuation e -α∆z≈1-α∆z and noiseless lumped amplifiers with gain Gj =e α∆z≈1+α∆z, yielding NF= G 1+(G 2-1)+(G 3-1)+… = 1+αz.
Similarly, one obtains that the signal mode alone experiences mean power gain G PSA=1 when the idler mode at the input is also excited with equal magnitude and phase. On the other hand, the variances of both signal quadratures increase by = (3 + e -2αz +2αz)/4, which yields the NF = (3 + e -2αz + 2αz)/4 ≈ αz / 2 for large z. As in the case with lumped amplifier, the factor 1/2 means that the effective input SNR should to be based on the total of signal and idler powers, and that SNR is degraded by a factor ≈αz after long propagation. Thus, after some distance, the NFs for quasi-coherent (corresponding to the fundamental limit for ideal distributed PSA) and direct detection schemes coincide, and the benefit of noiseless amplification can be seen by simple direct detection of either signal or idler beam (see Fig. 3 right). Since in the system with ideal distributed amplification the signal power is constant everywhere, the length of fiber spans only matters for injecting the pump power and optimizing its evolution in the fiber and does not directly affect the NF. Hence, distance z in the NF formulas runs from zero to the length of entire transmission link (e.g. several thousand kilometers) and easily satisfies the condition of being sufficiently large.
We have shown that, for long system reach, the distributed PSA can beat the NF=1+2αz limit  of the ideal distributed PIA by 3 dB. It is important to note that the ideal distributed Raman amplifiers have NF=1+2n spαz, which is, at room temperature, at least 0.5 dB worse than that of the ideal distributed PIA and becomes significantly worse for wide-band systems, owing to the spontaneous emission factor n sp=[1-exp(-h∆v/kT)]-1 reflecting the presence of thermally-excited phonons in the fiber. Thus, the advantage of the distributed PSA over ideal Raman amplifier is at least 3.5 dB. In addition, since the parametric gain is unidirectional, the PSA, unlike the distributed Raman amplifier, does not generate a lot of double Rayleigh backscattering noise. Hence, the Q-factor advantage of distributed PSA over ideal distributed Raman amplifier can actually be several dB higher than the 3.5-dB noise-figure improvement.
3. Distributed-PSA-based transmission link
Figure 4 shows the proposed implementation of a distributed-PSA-based transmission link. Transmitter part consists of a conventional transmitter followed by a PIA or nonlinear beam splitter BS to produce a two-sideband spectrum (signal + idler). At the input of a PSA span, the pump (or pumps in two-pump configuration) is separated from signal/idler and used to phase- and polarization-lock the local pump by means of either optical phase-locked loop (OPLL)  or injection locking of the pump laser . While the previous phase-locking experiments [13, 14] were done on lumped, interferometer-based PSAs, we will deal with a distributed PSA, which permits locking the phase to the strong preceding-span pump rather than to a weak signal. This novel approach is very promising, as it can yield relatively small phase errors. The co-polarized signal and idler are adjusted in phase for maximum gain by means of a fiber stretcher, and are made orthogonal to the parametric pump polarization using a polarization controller. After combining with the pump(s), they enter the fiber span serving as a distributed PSA. By using only the Raman pump(s) co-polarized with PSA pumps, a uniform parametric gain distribution along the fiber is ensured without providing any significant cross-polarized Raman gain to the signal and idler (see inset diagrams in Fig. 4, showing relative polarizations of the signal and idler with respect to the parametric and Raman pumps). Note that the presence of cross-phase modulation from Raman pumps of total power PR modifies the net phase mismatch: κ = ∆β - 2γε [(P 1+P 2) (2a - εP - 1/2) - 2PR (1 -a)]. In addition, the parameters μ and v in Eqs. (2) need to be multiplied by an extra factor e 2iγεPRa z Multiple spans with distributed PSAs can be cascaded, and the end-link receiver may employ coherent, cross-phase-modulation- or BS-based quasi-coherent, or direct detection.
The proposed scheme can potentially transmit multiple channels within the phase-matched bandwidth of the PSA driven by same one or two pumps. Minor phase (or polarization) mismatches among the WDM signal/idler pairs can be corrected by a tunable dispersion (or PMD) compensator. Since the phase-locking is only needed for two pumps (or for one in the single-pump case), its complexity does not increase with the number of WDM channels, which makes this scheme economically feasible. Moreover, the WDM channels can be totally independent, with no need for clock synchronization among them. Apart from low-noise amplification of the amplitude, the PSA by definition performs complete regeneration of binary phase information, such as that in differential or binary phase-shift keying (e.g. see experiment ). Since inter- and intra-channel nonlinearities tend to distort the signal phase / frequency leading to timing jitter and ghost pulse formation [36–38], the continuous regeneration of the phase by the PSA nullifies these effects. This makes a good case for the PSA as an enabler of multichannel all-optical 3R regeneration, particularly, in combination with multichannel amplitude regenerators . It should also be possible to lock the pump phase even in the case when the preceding-span pump’s spectrum has been intentionally broadened by phase modulation to suppress Brillouin scattering . While on the first sight this seems impossible because of very fast phase changes (100 ps for 10 Gb/s modulation), the anti-Brillouin phase modulation is usually carried out by a deterministic signal, such as a pseudo-random bit sequence (PRBS), which allows one to demodulate the pump signal and then lock the local phase to it. Synchronization of the local PRBS pattern can be performed by relatively slow electronics.
One potential challenge in practical realization of the distributed PSA is maintaining phase-matching condition over long spans of fiber. Since the characteristic gain length scale is determined by 1/α, short-distance variations of zero-dispersion wavelength are averaged out, and only variations over tens of kilometers matter. This makes it possible to build the spans from suitable fiber segments with known dispersion, as well as use more advanced concepts such as quasi-phase-matching, etc. Recent PIA experiments in 75-km fiber span  illustrate the feasibility of overcoming the phase-matching issues.
At last, we would like to emphasize the point briefly mentioned in Section 2.1 that the quadratures X + and Y - can carry independent data simultaneously. This means that, even though the PSA with non-degenerate signal and idler uses twice as much bandwidth as a degenerate PSA (e.g. interferometer-based), it also carries twice as much information. Both PSAs have the 3-dB advantage over the PIAs, but neither PSA is capable of reaching the Shannon’s capacity limit because of their phase-sensitive nature. Shannon’s limit requires encoding of both signal quadratures, whereas the PSA amplifies one of them and attenuates the other. This, however, does not make it disadvantaged in practical capacity, because optical transmission systems almost always encode the data onto only one quadrature anyway (either phase- or amplitude-shift keying). This is because the transmission impairments for the two encoding formats are different, making the corresponding system requirements largely incompatible.
We have proposed a scheme for a distributed phase-sensitive amplifier that is capable of reaching the fundamental limit of noise-figure / nonlinearity trade-off of the amplified transmission systems. We have developed a theoretical treatment of this device, and discussed the practical implementation issues. We believe such an amplifier will be able to significantly enhance the performance of communication systems owing to its unique noise-figure and continuous-phase-regeneration properties.
References and links
1. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D. 26, 1817 (1982). [CrossRef]
2. Y. Mu and C. M. Savage, “Parametric amplifiers in phase-noise-limited optical communications,” J. Opt. Soc. Am. B 9, 65 (1992). [CrossRef]
4. J. N. Kutz, W. L. Kath, R.-D. Li, and P. Kumar, “Long-distance propagation in nonlinear optical fibers by using periodically spaced parametric amplifiers,” Opt. Lett. 18, 802 (1993). [CrossRef] [PubMed]
5. J. A. Levenson, I. Abram, T. Rivera, and P. Grainger, “Reduction of quantum-noise in optical parametric amplification,” J. Opt. Soc. Am. B 10, 2233 (1993). [CrossRef]
7. S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless Optical Amplification of Images,” Phys. Rev. Lett. 83, 1938 (1999). [CrossRef]
8. M. E. Marhic, C. H. Hsia, and J.-M. Jeong, Electron. Lett. 27, 210 (1991). [CrossRef]
9. G. D. Bartolini, D. K. Serkland, P. Kumar, and W. L. Kath, “All-optical storage of a picosecond-pulse packet using parametric amplification,” IEEE Photonics Technol. Lett. 9, 1020 (1997). [CrossRef]
10. K. Croussore, I. Kim, Y. Han, C. Kim, G. Li, and S. Radic, “Demonstration of phase-regeneration of DPSK signals based on phase-sensitive amplification,” Opt. Express 13, 3945 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-3945 . [CrossRef] [PubMed]
11. M. Shirasaki and H. A. Haus, “Squeezing of pulses in a nonlinear interferometer,” J. Opt. Soc. Am. B 7, 30 (1990). [CrossRef]
13. A. Takada and W. Imajuku, “In-line optical phase-sensitive amplifier employing pump laser injection-locked to input signal light,” Electron. Lett. 34, 274 (1998). [CrossRef]
14. 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 (1999). [CrossRef]
15. R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244 (1985). [CrossRef]
16. D. Levandovsky, M. Vasilyev, and P. Kumar, “Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier,” Opt. Lett. 24, 984 (1999). [CrossRef]
17. D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” PRAMANA-Journal of Physics 56, 281 (2001). [CrossRef]
18. W. Imajuku, A. Takada, and Y. Yamabayashi, “Low-noise amplification under the 3 dB noise figure in high-gain phase-sensitive fibre amplifier,” Electron. Lett. 35, 1954 (1999). [CrossRef]
19. S. Radic and C. J. McKinstrie, “Two pump fiber parametric amplifiers,” Opt. Fiber Technol. 9, 7 (2003). [CrossRef]
21. P. L. Voss, K. G. Koprulu, and P. Kumar “Raman-noise induced quantum limits for χ(3) nondegenerate phase-sensitive amplification and quadrature squeezing,” submitted to J. Opt. Soc. Am. B.
22. C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12, 5037 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5037 . [CrossRef] [PubMed]
23. C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-13-4986 . [CrossRef] [PubMed]
24. R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-Line Frequency-Nondegenerate Phase-Sensitive Fiber-Optical Parametric Amplifier,” IEEE Photonics Technol. Lett. 17, 1845 (2005). [CrossRef]
25. R. Tang, P. Devgan, V. S. Grigoryan, and P. Kumar, “In-line frequency-non-degenerate phase-sensitive fiber parametric amplifier for fiber-optic communications,” to appear in Electron. Lett.
27. M. Vasilyev, B. Szalabofka, S. Tsuda, J. M. Grochocinski, and A. F. Evans, “Reduction of Raman MPI and noise figure in dispersion-managed fibre,” Electron. Lett. 38, 271 (2002). [CrossRef]
28. M. Vasilyev, “Raman-assisted transmission: toward ideal distributed amplification,” Optical Fiber Communication Conference 2003, Technical Digest (OSA, Washington, D.C.2003), Vol. 1, pp. 303–305, paper WB1.
29. A. F. Evans, A. Kobyakov, and M. Vasilyev, “Distributed Raman transmission: applications and fiber issues,” in Raman Amplifiers in Telecommunications 2: Sub-Systems and Systems, ed. by M. N. Islam, Springer, New York, 2004, pp. 383–412. [CrossRef]
30. J.-C. Bouteiller, K. Brar, and C. Headley, “Quasi-constant signal power transmission,” European Conference on Optical Communication 2002, paper S3.04.
31. M. Vasilyev, “Squeezing and fiber-optic communication,” 9th International Conference on Squeezed States and Uncertainty Relations 2005, May 2005, Besançon, France, paper I 79.
32. Here, conventional fiber is assumed to be averaging the nonlinear interaction over the signal states of polarization while preserving the relative orientation of the pump and signal. Coefficients a, b, ε, and εP are straightforwardly derived from Eq. (4.2.10) of , as done, for example, in .
33. R. W. Boyd, Nonlinear Optics (Academic Press, San Diego, 2003), Chap. 4.2.
34. M. Vasilyev, lecture notes for Nonlinear Optics course (University of Texas at Arlington, 2004).
35. C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express 12, 4973 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4973 . [CrossRef] [PubMed]
36. L. F. Mollenauer, J. P. Gordon, and P. V. Mamyshev, “Solitons in high bit-rate, long-distance transmission,” in Optical Fiber Telecommunications IIIA, I. P. Kaminow and T. L. Koch, eds. (Academic Press, San Diego, 1997), pp. 373–460.
37. I. Shake, H. Takara, K. Mori, S. Kawanishi, and Y. Yamabayashi, “Influence of inter-bit four-wave mixing in optical TDM transmission,” Electron. Lett. 34, 1600 (1998). [CrossRef]
38. A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photonics Technol. Lett. 12, 392 (2000). [CrossRef]
40. G. P. Agrawal, Nonlinear fiber optics (Academic Press, San Diego, 1995).
41. G. Kalogerakis, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Transmission of optical communication signals by distributed parametric amplification,” in Conference on Lasers and Electro-Optics 2005 (Optical Society of America, Washington, DC, 2005), paper CTuT2. [CrossRef]