## Abstract

We theoretically and experimentally analyze the dominant impairment mechanisms affecting the fidelity of optical phase in parametric amplifiers and converters in media characterized by third-order (Kerr) optical nonlinearity. The critical role of narrow-band pump filtering in parametric mixers is quantified with respect to frequency stability of amplified and converted waves. The analysis is generally applicable to all four-photon devices used to generate new frequencies or translate spectral bands.

© 2010 OSA

## 1. Introduction

Four-photon-mixing (FPM) in χ^{(3)} nonlinear media, such as silica optical fibers, can be used to amplify light as well as to create new light frequencies spanning bandwidths of several hundreds of THz [1–3]. In a simplest fiber-optic parametric amplifier/converter (FOPA/C) implementation, a single powerful pump wave and a weak signal wave co-propagate along dispersion-engineered highly-nonlinear fiber (HNLF). Two pump photons are annihilated and a signal and an idler (phase-conjugated replica of the signal) photon are created. The FPM process is subject to inherent quantum-mechanical fluctuations (vacuum noise), which are coupled among the propagating waves via Kerr nonlinearity [4]. In the high gain limit, the resulting photon-number and field-quadrature fluctuations (in case of phase-insensitive parametric amplifiers) have been shown to be equivalent to those of χ^{(2)}-based parametric processes (e.g. sum- and difference-frequency generation) as well as inverted-population (linear) optical amplifiers such as Erbium-doped fiber amplifiers (EDFAs) [4,5].

In addition to quantum noise, parametric processes in χ^{(3)} nonlinear media are subject to pump amplitude and phase fluctuations [6,7]. The Kerr nonlinearity in fused silica is characterized by sub-10fs response time for electronically-dominated nonlinearity subject to a non-resonant light field [8], allowing pump amplitude fluctuations to be nearly instantaneously converted to pump phase fluctuations. This amplitude-to-phase noise conversion was recognized early in transmission systems and dubbed nonlinear phase noise (NPN) by Gordon and Mollenauer [9]. The NPN is a major limitation in narrow linewidth frequency comb synthesis [10], high-fidelity supercontinuum generation [11,12], very high power (>100W) amplification [13], and long-haul coherent communication systems [14].

Owing to nearly perfect phase matching among the propagating waves in HNLF, the NPN is transferred from the pump to the signal and the idler via highly efficient processes of cross-phase modulation (CPM) and four-wave-mixing (FWM). The influence of the NPN has been experimentally observed in saturated parametric amplifiers, where noise-loaded phase-shift-keyed (PSK) signal was partially regenerated using a saturated one-pump FOPA [15,16]. The parametric amplitude limiter added NPN to the amplitude-regenerated signal. The amount of NPN increased with reduced pump optical signal-to-noise ratio (OSNR) and was also numerically shown to increase with increased pump power [17]. Finally, an expression for the variance of nonlinear phase noise in two-pump parametric amplifiers has been derived, accompanied by numerical calculations of SNR penalty for various PSK formats [18]. Not surprisingly, the formats whose closest symbols are the least distant in the phase plane were shown to suffer the largest penalty due to NPN.

All of the aforementioned NPN studies have assumed the impairment source to be the white Gaussian optical noise. In the case of coherent communication systems, the amplitude noise is accumulated during signal wave amplification and converted to phase noise via either self-phase or cross-phase modulation [19]. In either case, it is well understood that only the noise present within the signal bandwidth is relevant for the correct prediction of the NPN statistics. In FOPA, however, a possibility of very narrow optical filtering of the amplified pump wave exists and does not have an equivalent in conventional communications systems in which the filter bandwidth is limited by the channel rate. Thus, the filtered noise bandwidth can be smaller than or larger than the amplified signal bandwidth. In this report, we demonstrate for the first time that the variance of NPN remains unchanged, whereas the variance of the nonlinear chirp (NC) increases with increased optical noise bandwidth (accompanied by decreased noise power spectral density such that the total noise power is constant). The results bear significant practical ramification on the construction and performance of parametric amplifiers and converters.

## 2. Statistics of Nonlinear Phase Noise and Nonlinear Chirp

We begin the analysis by considering the simplest χ^{(3)} parametric amplification architecture shown in Fig. 1
. A pump wave with optical power *P _{p}* and carrier frequency

*ν*is amplified in an optical amplifier (e.g. EDFA), thereby accumulating white Gaussian optical noise. The pump RIN and laser phase noise are considered to be negligible. The optical amplifier noise,

_{p}*n(t) = n*, is a complex white Gaussian random process [20]. The in-phase and quadrature components of the noise have zero mean and variance of

_{r}(t) + jn_{i}(t)*N*, where

_{0}Δν/2*N*is the noise power spectral density in one polarization and

_{0}*Δν*is the optical bandwidth of interest. The optical signal-to-noise ratio of the pump wave (measured in 0.1nm optical bandwidth) is given by

*OSNR*, where

_{0.1nm}= P_{p}/(2N_{0}Δν_{0.1nm})*Δν*is the frequency bandwidth corresponding to 0.1nm at the wavelength of

_{0.1nm}*c/ν*.

_{p}The complex pump field is subsequently band-pass filtered before entering the nonlinear waveguide (e.g. HNLF) characterized by fiber length *L*, and nonlinear coefficient *γ*. We consider intra-channel dispersion of the HNLF at the pump frequency to be negligible in order to allow a closed form derivation. This assumption is justified in most practical cases, where the pump is placed very close to the zero-dispersion wavelength of the HNLF in order to maximize the gain (and/or conversion efficiency) bandwidth. After propagation through HNLF, and neglecting HNLF loss and pump depletion, the pump acquires nonlinear phase noise

*n’(t) = n(t)*⊗

*h*is the complex field of the filtered optical noise and

_{in}(t)*h*is the optical filter impulse response. The signal and idler acquire the same nonlinear phase shift (

_{in}(t)*γP*) as the pump [21], and they are thus subject to the same NPN. The last term in Eq. (1), the

_{p}L*noise-noise beat*term, can be neglected since practical parametric amplifiers/converters require high pump OSNRs in order to minimize the

*pump-transferred noise*[6]. Accordingly, the variance of NPN can be expressed as

*S*is the power spectral density of

_{n}(ω)*n’(t)*.

On the other hand, the noise-induced nonlinear frequency chirp is given by

*n’(t)*is a second-order wide-sense-stationary (WSS) random process [22]. Then, it can be shown that [23]

*E{…}*is the statistical expectation operator and

*R(τ) = E{n’(t)n’(t + τ)}*is the autocorrelation of the optically filtered noise

*n’(t)*. Consequently, the variance of NC is

It is important to reflect on the implications of Eqs. (2) and (5). As would be expected, the variances of both NPN and NC depend on the nonlinear parameters (*γ*, *P _{p}*, and

*L*). From Eq. (2), it is evident that the variance of NPN, a commonly considered quantity in NPN investigation, depends solely on the

*total noise power*(i.e. the noise power integrated over the optical filtered bandwidth). In sharp contrast, in the expression for the variance of NC [Eq. (5)], the noise power spectral density is weighted by the

*ω*term. The angular frequency weighting is a consequence of the temporal change of the statistical properties, mediated by propagation in Kerr media. As a result,

^{2}*the noise spectral width plays a crucial role in the NC statistics*, as illustrated in Fig. 2(a) . The impact of low-power high-frequency noise components of

*S*is exacerbated via multiplication by

_{n,1}(ω)*ω*weighting factor. To illustrate this feature, Figs. 2(b) and 2(c) show the contour plots of standard deviation of NPN and NC, respectively, for a fixed nonlinear phase shift of

^{2}*γP*. The most important feature of plots 2(b) and 2(c) is that the steeper NC contour slope suggests that even when the total noise power is kept constant, the spectrally broader pump optical noise (with appropriately reduced noise power spectral density) will induce a larger spectral broadening than it’s spectrally narrower counterpart. As a direct consequence of Eq. (5), the narrow pump filtering is critical in construction of high-signal-integrity parametric amplifiers and converters.

_{p}L = 5## 3. Experimental Results and Discussion

An experimental setup was constructed in order to characterize the noise-induced NC and validate the analytical findings, as shown in Fig. 3
. The pump wave, centered at 1589.0nm, was amplitude modulated to produce 1ns pulses with 30dB duty cycle. The pump OSNR was varied by varying the input power into the optical amplifier cascade. The optical noise bandwidth was controlled via a flat-top variable-bandwidth OBPF with power transfer functions shown in the left inset of Fig. 3. The amplified pump wave and the surrounding filtered noise were passed through 180m-long HNLF with nonlinear coefficient of 13W^{−1}km^{−1} and a global zero-dispersion wavelength (ZDW) of 1589.0nm (which is exactly equal to the pump wavelength). The right inset in Fig. 3 shows the optical spectrum after HNLF. The broadband amplified quantum noise (AQN) is attributed to parametric gain’s high sensitivity to ZDW fluctuations, especially when the pump is placed at the global ZDW [24]. The pump power entering HNLF was 5.2W, resulting in a total nonlinear phase shift of 12.168 radians. In order to quantify the noise-induced NC, 300m-long standard single-mode fiber (SMF) was inserted to convert pump phase fluctuations into amplitude fluctuations (PM-to-AM) via dispersion (19.239ps/nm-km at 1589.0nm). The VOA preceding the fiber was used to attenuate the pump wave to peak power of 10mW in order to avoid nonlinear effects in SMF. The amplitude fidelity of the pump wave was characterized using an optical sampling oscilloscope with an electrical bandwidth of 500GHz. The oscilloscope bandwidth was much larger than the widest optical noise bandwidth (144GHz), ensuring that no smoothing of the noisy optical waveform by the receiver took place.

Figure 4(a) shows the measured electrical SNR as a function of the optical filter bandwidth. The OSNR required to keep the total optical noise power constant is shown on the right vertical axis. The total noise power at OSNR of 50dB and optical filter bandwidth of 144GHz was used as a reference and kept constant as the filter bandwidth and OSNR were varied for the remaining four data points. An excellent agreement between the semi-analytical model outlined in the Appendix and the measured SNR is recognized in Fig. 4(a). We note that the introduced semi-analytical model enables complete inference of the statistical properties of NPN and NC, shown in Fig. 4(b). As stated previously, the results unambiguously demonstrate that the standard deviation of NPN is unchanged as the filter bandwidth and pump OSNR are varied. In sharp contrast, the standard deviation of NC approximately doubles when the filter bandwidth is quadrupled (e.g. from 36GHz to 144GHz). Thus, the findings are in perfect accord with the analysis in Sec. 2.

## 4. Conclusion

The statistics of nonlinear phase noise and nonlinear frequency chirp arising from amplitude noise to phase noise conversion in a χ^{(3)} nonlinear medium are experimentally and analytically studied and quantified. The study reveals the relative importance of noise power spectral density and noise optical bandwidth. It is found that narrow optical noise filtering, rather than low noise power spectral density, plays the dominant role in successful management of noise-induced nonlinear frequency chirping. The result represents an important step towards the understanding of the impairments associated with spectral broadening of the amplified and the newly-generated waves in fiber parametric mixers. Specifically, the quantified NPN and NC impairments lead to new FOPA(C) construction rules not implemented in the past.

Most importantly, the results of this study state that the statistics of noise-induced nonlinear chirp, rather than those of the nonlinear phase noise, correctly describe the phase degradation of the interacting mixer waves. This conclusion is quite general and not necessarily limited to fiber devices, applying to all processes plagued by the nonlinear phase noise.

## Appendix: Semi-analytical model for PM-to-AM in optical fiber

The semi-analytical model for conversion of phase/frequency fluctuations acquired by pump wave propagation in HNLF to amplitude fluctuations is developed according to the schematic shown in Fig 5 .

Following band-pass optical filtering, the complex pump field is

After propagation in HNLF, neglecting loss and depletion, the pump field acquires a nonlinear phase shift (and therefore a nonlinear frequency chirp):

The optical fiber dispersion is simply treated as a phase shift in the Fourier domain:

where ℑ{…} and ℑ^{−1}{…} represent the Fourier transform and inverse Fourier transform, respectively, *L _{SMF}* is the SMF length, and the dispersion operator is defined as [25]:

The *β _{2}* and

*β*coefficients are related to the dispersion and the dispersion slope of the optical fiber and

_{3}*ω*is the center frequency of the optical pump wave. Following optical detection, the electrical voltage (or current) can be expressed as

_{ref}where *h _{elec}(t)* is the impulse response of the optical sampling oscilloscope. Finally, the electrical SNR measured on the oscilloscope is

with 〈〉 and σ^{2} symbolizing the mean and the variance of the acquired electrical waveform, respectively. Thus, the measurement of electrical SNR allows us to infer the amount of acquired noise-induced nonlinear chirp.

## Acknowledgements

The authors would like to thank A. Radosevic, B. P.-P. Kuo, M. Karlsson, and C. J. McKinstrie for fruitful discussions. This material is based on research sponsored in part by Air Force Research Laboratory (AFRL) and the Defense Advanced Research Projects Agency (DARPA) under agreement number FA8650-08-1-7819 Parametric Optical Processes and Systems. Part of this work was supported in part by Lockheed Martin Corporation. The authors would like to thank Sumitomo Electric Ltd. for providing the nonlinear fiber.

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