## Abstract

Nondegenerate four-wave mixing in fibers enables the tunable and low-noise frequency conversion of optical signals. This paper shows that four-wave mixing driven by pulsed pumps can also regenerate and reshape optical signal pulses arbitrarily.

© 2012 OSA

## 1. Introduction

Parametric devices based on four-wave mixing (FWM) in nonlinear fibers can amplify, frequency convert (FC), phase conjugate, regenerate, sample and switch optical signals in communication systems [1,2]. This letter focuses on FC by the nondegenerate FWM process called Bragg scattering (BS). In this process, which is illustrated in Fig. 1, a sideband (signal) photon and a pump photon are destroyed, and different sideband (idler) and pump photons are created (*π _{p}* +

*π*→

_{s}*π*+

_{q}*π*, where

_{r}*π*represents a photon with carrier frequency

_{j}*ω*.) BS is a versatile process. It can provide tunable [3,4] and low-noise [5,6] FC for classical signals or single photons [7,8]. It can also FC waves with similar (nearby) frequencies [9], or waves with dissimilar (distant) frequencies [10–12]. Not only does BS transfer power (photon flux) from the input signal to the output idler, it also transfers the quantum state [5, 7]. For example, if the input photon is entangled with another quantum degree of freedom, so also will be the output photon. These features make BS useful for conventional communications and quantum information science. In this letter, it will be shown that BS also has the ability to regenerate (clean-up) noisy signals and reshape (reformat) signals arbitrarily.

_{j}## 2. Analysis

In BS, the conservation of energy and momentum are manifested by the frequency- and wavenumber-matching conditions

*ω*denotes a carrier frequency and

_{j}*k*=

_{j}*k*(

*ω*) denotes the associated carrier wavenumber. BS is driven by pump-power-induced nonlinear coupling and suppressed by fiber-dispersion-induced wavenumber mismatch, so it is important to determine the conditions under which BS is wavenumber matched.

_{j}For a wide range of frequencies, the dispersion function *k*(*ω*) can be approximated by the Taylor expansion

*ω*is a reference frequency, the difference frequency

_{a}*ω*is measured relative to the reference frequency and the dispersion coefficient

*β*=

_{n}*d*. It is convenient to let

^{n}k/dω^{n}*ω*be the average frequency of the waves, in which case

_{a}*ω*= −

_{p}*ω*and

_{s}*ω*= −

_{q}*ω*. Define the wavenumber mismatch

_{r}*δ*=

*k*+

_{p}*k*– (

_{s}*k*+

_{q}*k*). Then it follows from Eq. (2) and the preceding definition that

_{r}*β*

_{2}= 0. This condition corresponds to wave frequencies that are perfectly symmetric about the zero-dispersion frequency

*ω*

_{0}. The group-slowness function

*dk*(

*ω*)/

*dω*≈

*β*

_{1}+

*β*

_{3}

*ω*

^{2}/2 depends quadratically on frequency. Hence, the outer pump co-propagates with the signal (

*β*

_{1p}=

*β*

_{1s}), whereas the inner pump co-propagates with the idler (

*β*

_{1q}=

*β*

_{1r}). For high frequencies, the effects of fourth-order dispersion cannot be neglected. However, the waves still co-propagate approximately (

*β*

_{1p}≈

*β*

_{1s}and

*β*

_{1q}≈

*β*

_{1r}) for a wide range of pump frequencies. For example, in a recent experiment [8], |

*β*

_{1r}−

*β*

_{1s}| was 1.36 ps/m, whereas |

*β*

_{1p}−

*β*

_{1s}| and |

*β*

_{1q}−

*β*

_{1r}| were only 0.027 ps/m. These slowness relations enable the reshaping functions described below.

Suppose that the pump and sideband frequencies are chosen so that the matching conditions are satisfied. Then the sideband evolution is governed by the coupled-mode equations

*A*is a slowly-varying wave (mode) amplitude,

_{j}*∂*=

_{z}*∂/∂z*is a distance derivative,

*∂*=

_{t}*∂/∂t*is a time derivative,

*β*is an abbreviation for the group slowness

_{j}*β*

_{1j}and

*γ*is the Kerr nonlinearity coefficient. The effects of intra-pulse dispersion were neglected, because they are weak for a wide range of relevant system parameters. Equations (4) and (5) apply to scalar FWM, which involves waves with the same polarization. Similar equations apply to vector FWM, which involves waves with different polarizations [13, 14].

The effects of nonlinear phase modulation (NPM) are omitted from the following analysis, but will be incorporated numerically. The pumps are not affected by the sidebands, so they convect through the fiber with constant shape. In the low-conversion-efficiency regime, the input signal seeds the growth of a weak idler, whose presence does not affect the signal significantly, so the signal also convects through the fiber with constant shape. Hence, the pump and signal amplitudes can be written in the form

where*a*is a complex constant and

_{j}*f*is a complex shape-function. It is convenient to impose the normalization condition ${\int}_{-\infty}^{\infty}{\left|{f}_{j}(t)\right|}^{2}dt=1$, in which case the amplitude

_{j}*a*is the square root of the pulse energy. The idler equation can be written in the form

_{j}Define the retarded time *τ* = *t –* *β _{r}z*. Then the idler equation becomes

*β*=

_{rs}*β*–

_{r}*β*is the differential slowness (walk-off). By integrating Eq. (8), in which

_{s}*τ*is a parameter, one finds that the output idler

*β*(so the waves collide entirely within the fiber), the integral in Eq. (9) does not depend on

_{rs}l*τ*and the shape-function of the output idler is the conjugate of the shape-function of pump

*q*(which co-propagates with the idler). The interaction between the signal and pump

*p*(which also co-propagate) is strongest if their shape-functions are conjugates of each other, in which case where the strength parameter $\overline{\gamma}=2\gamma {a}_{p}{a}_{q}^{*}/{\beta}_{rs}$ is proportional to the nonlinearity coefficient and the product of the pump amplitudes, and is inversely proportional to the walk-off. (If

*β*differs slightly from

_{p}*β*, the strength parameter is reduced slightly.) Although the amplitude of the output idler is proportional to the amplitude of the input signal, the shape-function of the output idler is specified by the shape-function of the co-propagating pump (not the signal). This pump shape-function could be the same as, or different from, the signal shape-function. Hence, the output idler can be reshaped arbitrarily relative to the input signal. In particular, the amplitude fluctuations associated with a noisy signal can be removed. Similar results apply to the generation of an output signal by pumps and an input idler.

_{s}## 3. Examples

Three examples of idler generation and signal reshaping are now considered. Time is measured in units of the (common) pump width *σ* and distance is measured in units of *σ/β _{r}*. For these conventions, the group slowness is measured in units of

*β*and the interaction length

_{r}*β*is the ratio of the idler transit time to the pump duration. The pumps and signal have Gaussian or super-Gaussian (nearly rectangular) shape-functions. Most of the following results were obtained by integrating Eq. (9) numerically, for cases in which

_{r}l/σ*β*= −1 [because in a frame moving with the average slowness, the (apparent) idler slowness is

_{s}/β_{r}*β*– (

_{r}*β*+

_{r}*β*)/2 =

_{s}*β*/2 and the signal slowness is

_{rs}*β*– (

_{s}*β*+

_{r}*β*)/2 = −

_{s}*β*/2] and

_{rs}*γ̄*= 0.325 (which corresponds to an energy-conversion efficiency of 10%). The idler amplitude is measured in units of

*ia*.

_{s}In the first example, all three input pulses (both pumps and the signal) are Gaussians, and are timed to collide (overlap completely) in the middle of the fiber. The idler generated by these inputs is illustrated in Fig. 2. As the interaction length increases, so also does the idler amplitude and time delay. Increasing the length beyond a value of about 2 delays the idler, but does not change significantly its peak amplitude or shape. Equation (10) omits the effects of signal depletion, which limits the idler growth by reducing the driving strength, and NPM, which limits the idler growth by detuning the FWM process [15]. Numerical solutions of Eqs. (4) and (5) show that neither effect changes the idler magnitude significantly (for low conversion efficiencies). However, NPM does impose a moderate chirp on the output idler (not shown).

In the second example, the signal and its co-propagating pump are Gaussians, whereas the other pump is an 8th-order super-Gaussian. The idler generated by these inputs is illustrated in Fig. 3. As the interaction length increases, the idler amplitude increases and the idler shape becomes more rectangular. Increasing the length beyond a value of about 3 delays the idler, but does not change its shape significantly. Once again, the predictions of perturbation theory are accurate. The output idler is phase-shifted, but unchirped, because the pumps are flat-topped.

In the previous examples, the input signals were perfect noiseless Gaussians. However, real signals are usually degraded by noise, as illustrated in Fig. 4 (top row). It is natural to ask what happens to such signals when they are frequency converted. In the third example, the signals are noisy Gaussians and the pumps are noiseless Gaussians or super-Gaussians. The idlers generated by these inputs are also illustrated in Fig. 4 (middle and bottom rows). In each case, the output idler amplitude at time *t* depends on the input signal amplitude at time *t*′, where *t* – *β _{r}l* ≤

*t*′ ≤

*t*–

*β*. As the interaction length increases, so also does the number of input signal values on which the output idler value depends. Since these input values are statistically independent, the variance of their sum increases (at most) linearly with distance, so the deviation of their sum increases as the square root of distance. Hence, the idler fluctuations increase less rapidly than the mean amplitude, so the idler profile is smoothed. For short distances the back of the (slow) idler is smoothed more than the front, because it has sampled more (fast) signal values, whereas for long distances the whole pulse is smoothed. It should be emphasized that pulsed FC regenerates the pulse shape, but does not regenerate (constrain) the pulse peak-amplitude. This process is complementary to gain-saturated amplification, which removes peak-amplitude variations from a sequence of pulses, but distorts the pulse shapes [16, 17].

_{s}l## 4. Discussion

To regenerate or reshape trains of signals, the fiber (nonlinear medium) should be long enough that each pump–sideband collision is complete, but short enough that the signal (idler) from one collision does not intersect the idler (signal) from the next collision. Suppose that input pump *p* and the input signal are delayed by about 2*σ* relative to the center of the bit slot (and so extend from about 0 to 4*σ*), whereas pump *q* and the virtual idler are advanced by 2*σ* (and so extend from −4*σ* to 0). After a complete (symmetric) collision, the output signal is advanced by 2*σ*, whereas the idler is delayed by 2*σ*. Completeness requires that *β _{rs}l* = 8

*σ*, whereas noninterference between pumps and sidebands in neighboring bit slots requires that max(8

*σ,β*) ≤

_{rs}l*τ*, where

*τ*is the bit duration. As an example, consider a system operating at 10 Gb/s, for which the bit duration is 100 ps. If the pulses have full-widths-at-half-maximum of 20 ps (

*σ*= 12 ps), the bit duration equals 8.3 pulse widths, and if

*β*= 1.4 ps/m, the collision length is 68 m. Speciality fibers with customizable dispersion and lengths of 1–300 m are available, so a variety of useful experiments can be designed.

_{rs}In summary, BS has the potential to regenerate and reshape trains of noisy input signals. Perturbation theory makes accurate predictions for conversion efficiencies up to 10%, which is high enough to be useful. The effects of signal depletion and nonlinear phase modulation on the idler evolution will be studied in detail elsewhere.

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