## Abstract

We propose a systematic approach to evaluating and optimising the wavelength conversion bandwidth and gain ripple of four-wave mixing based fiber optical wavelength converters. Truly tunable wavelength conversion in these devices requires a highly tunable pump. For a given fiber dispersion slope, we find an optimum dispersion curvature that maximises the wavelength conversion bandwidth.

© 2003 Optical Society of America

## 1. Introduction

Future optical networks are likely to operate at data rates in excess of 100 Gb/s. Wavelength conversion will be essential for optical routing [1] and will provide a means to prevent blocking [2] in these networks. Traditionally this involves an intermediate electronic step, however present progress suggests that electronics technology will not be feasible for such speeds.

All-optical signal processing is a promising means to overcome the speed limitations imposed by electronics. The large optical nonlinearities in semiconductor optical amplifiers make them attractive for use in all-optical wavelength converters. However, semiconductor devices have a significant noise figure and require complex configurations [3] to achieve the sub-picosecond response times necessary for operation above 100 Gb/s. Fiber-based nonlinear wavelength converters can operate at much higher bit-rates, because their operation is based on non-resonant glass nonlinearities that have response times well below 1 ps [4].

A major advantage of parametric wavelength converters (PWCs) over other all-optical, fiber-based technologies is that for the ideal case of a truly continuous-wave pump, their operation is independent of the data modulation format and bit rate. This independence arises because the parametric conversion process transfers both the amplitude and phase information of the original signal to the converted wave [5]. In contrast, other all-optical techniques rely on intensity modulation schemes to transfer information to copropagating waves. This makes parametric wavelength conversion more broadly applicable and allows for easier upgrades of transmission equipment than if other fiber-based techniques were used. Additionally, fiber-based PWCs can be polarisation independent [6] and provide broadband conversion [7, 8]. Fiber wavelength converters can also be constructed with low noise-figures [9].

A WDM router must be able to convert the data carried by one wavelength within a band, to any other wavelength within that same band. Conversion from a fixed signal wavelength λ_{s}, to any converted wavelength λ_{c}, within a wavelength range Δλ, can be represented by a vertical line of length Δλ on Fig. 1(a), which is a plot of λ_{c} against λ_{s}. A horizontal line on this diagram represents conversion to a fixed λ_{c} from any λ_{s}. Therefore conversion from any λ_{s} to any λ_{c} within that range is contained within a solid square with side length Δλ. Because here we are only concerned with conversion within the same band, this square must lie symmetrically about the line λ_{c}=λ_{s}, as shown in Fig. 1(c). Furthermore, the conversion efficiency from λ_{s} to λ_{c} should ideally be constant, that is independent of both λ_{s} and λ_{c}.

In a fiber parametric amplifier, a signal and strong pump are launched simultaneously. Power is transferred from the pump to the signal wavelength, resulting in signal gain [10]. A new converted wave is also generated at λ_{c}=1/(2/λ_{p}-1/λ_{s}), where λ_{p} is the pump wavelength. Therefore a tunable λ_{c} is obtained by moving λ_{p}. Different design considerations apply to fiber parametric amplifiers depending on whether their final application is as wavelength converters (possibly with gain) or as broadband amplifiers. PWCs require a tunable λ_{c}, and so need a tunable λ_{p}, whereas parametric amplifiers designed solely for amplification [11] are optimised for gain bandwidth and flatness at a single pump wavelength only.

Work using novel fiber designs [7, 8] has demonstrated PWCs with wide tunability. Contour plots of wavelength converter gain surfaces in (λ_{s},λ_{c}) space have appeared previously [7], but have not, to our knowledge, been used to design devices. Our work maps the response of a fiber PWC on to (λ_{s},λ_{c}) space and systematically investigates methods to maximise the size of the square by tailoring device parameters.

Two spectra for a typical [11] fiber parametric amplifier are shown in Fig. 2(a), with parameters as shown in Table 1. Figure 2(b) shows the ideal profiles that would be required to generate Fig. 1(c). Comparing the shapes in Figs. 2(a) and 2(b), it is evident that typical OPAs used for wavelength conversion (Fig. 2(a)) do not have perfectly flat profiles. The gain ripple in Fig. 2(a) must be quantified so that it can be minimised by design. If the dotted line in Fig. 2(a) is the minimum acceptable gain, *G*
_{thresh}, and the local maxima are denoted *G*
_{max}, then the gain ripple is given by *R*=*G*
_{max}/*G*
_{thresh}.

The intersections of the gain profiles with the minimum acceptable gain can be plotted as a function of (λ_{s},λ_{p}). This plot would show the boundaries of the regions for which the converter gain is above *G*
_{thresh}. The plot is then made compatible with Fig. 1(c) by converting the λ_{p} axis to λ_{c} using λ_{c}=1/(2/λ_{p}-1/λ_{s}), as has been done for Fig. 3. Therefore Fig. 3 shows that a PWC is far from an ideal device because it does not map to a square that is similar to Fig. 1(c). However, by fitting the largest possible square within the solid lines on this figure, one defines a region in which the PWC operates close to an ideal device. This square is a contour enclosing the combinations of λ_{s} and λ_{c} for which the power of λ_{c} is above *G*
_{thresh}. The bandwidth of the PWC is then given by the side length of this inscribed square.

## 2. Theory

Four-wave mixing in fiber parametric amplifiers, which makes use of the cubic nonlinearity in glass, requires a strong pump of frequency ω_{p} and power *P*_{p}
. The four-wave mixing process then generates two photons of frequencies ω_{s} and ω_{c} that are symmetric around ω_{p}, from the annihilation of two photons at ω_{p}. Thus

that is, energy is conserved. This process requires phase matching, and its efficiency depends on the wavenumber mismatch, Δβ, defined by

In a wavelength converter based on four-wave mixing, the process is seeded with a signal of frequency ω_{s} (wavelength λ_{s}), so the frequency of the converted wave ω_{c} (wavelength λ_{c}) is then determined by pump frequency from Eq. (1). Thus ω_{c} can be chosen from a range Δω by varying the pump frequency ω_{p} over a range Δω/2. The gain, *G*, of this process is given by

where γ is the nonlinear parameter of the fiber [10], *L* is the fiber length, and the parametric gain, *g*, is given by

Note that *g* is maximum when Δβ=2γ*P*_{p}
, and the corresponding value fo *G*. and the The maximum value of *g* occurs when Δβ=2γ*P*_{p}
and the corresponding *G* is the maximum gain for a parametric amplifier. We choose this maximum gain to be the *G*
_{max} for PWCs, so therefore

We also note that for Δβ=0, which is true when λ_{c}=λ_{s}=λ_{p}, the gain equals

Over the operating frequency range of these devices, *G*
_{max} and *G*
_{0} can be considered constant because Eqs. (5) and (6) both depend on the fiber only through γ, which is slowly varying with ω. In fact, as is customary in this field of research, γ is assumed constant over the limited wavelength ranges that we are considering (see Table 1). This is consistent with Fig. 2(a), which shows that, for our parameters, the gain in the central minimum (at Δβ=0) and the maximum gain do not depend on λ_{p}.

Here we consider a fiber in which the quadratic dispersion (β_{2}) vanishes at frequency ω_{0} and with cubic (β_{3}) and quartic (β_{4}) dispersions defined at ω_{0}. Ignoring higher-order dispersion, Δβ can be rewritten as

Contours like that in Fig. 3 can be found by inverting Eqs. (3) and (4) to find Δβ, which in general leads to two solutions indicated Δβ (*G*
_{thresh})^{±}. An analytic solution to these equations is possible with careful selection of *G*
_{thresh}. For *G*
_{0}<*G*
_{thresh}<*G*
_{max}, the gain *G*
_{0} at λ_{c}=λ_{s} is, by assumption, below the minimum acceptable gain and so the requirements described by Fig. 1(c) cannot be met. In contrast, for *G*
_{thresh}≤*G*
_{0}, the requirements of Fig. 1(c) can always be satisfied. We choose *G*
_{thresh}=*G*
_{0} because this leads to the smallest ripple,

For *G*
_{thresh}=*G*
_{0}, Eq. (3) gives an analytic solution for g and Eq. (7) is then used to give analytic expressions for the contours. When β_{4}≠0, the contours are:

$${\omega}_{s}={\omega}_{p}\pm \sqrt{6\frac{-\Phi \pm \sqrt{{\Phi}^{2}-\left(\frac{16}{3}\right){\beta}_{4}\gamma {P}_{p}}}{{\beta}_{4}}},$$

where **Φ**=(β_{4}/2)(ω_{p}-ω_{0})^{2}+β_{3}(ω_{p}-ω_{0}). The ± sign inside the square-root in Eq. (9) is independent of the one outside it. Applying Eq. (1) then gives the equivalent expression for ω_{c}. Note that if β_{3}=0 or β_{4}=0, then the last of Eqs. (9) simplifies considerably. We have also found similar equations for the case β_{4}=0.

## 3. Results

Two adjacent sides of the largest square, that fits within the contour described by Eqs. (9), are tangents to the turning points of Eqs. (9) closest to λ_{s}=λ_{0}. These lines intersect with each other and with the original curve. The other two sides can then be found geometrically. We find that when β_{3}=0,

For non-zero β_{3} and β_{4}, however, the critical points and intersections used to find the bandwidth have to be found numerically from Eq. (9).

For a given bandwidth, the value of *G*
_{0}, *G*
_{max} or *R* can be freely chosen. The remaining device parameters are then completely defined by Eqs. (8) and the appropriate solutions to Eq. (10). Equation (10) shows that doubling the bandwidth requires an eight (β_{4}=0) or sixteenfold (β_{3}=0) increase in the γ*P*_{p}
to dispersion-parameter ratio, making it difficult to increase the bandwidth.We have found a novel way to contribute to a further increase in this bandwidth.

The animation in Fig. 4(a) shows the effect of changing β_{4} on the *G*=*G*
_{0} contour, and therefore the bandwidth, of a PWC. ${\mathrm{\beta}}_{4}^{\mathit{\text{opt}}}$ is the value of β_{4} that maximises the conversion bandwidth for a fiber with a β_{3} given in Table 1 and the corresponding maximum bandwidth square is shown in red. The blue square in Fig. 4(a) is the bandwidth for β_{4}=0. The black bandwidth contours fall inside this blue square when β_{4}<0. Therefore only values of β_{4}>0 contribute to an increase in the conversion bandwidth.

Numerical investigations of Eq. (9) reveal a ${\mathrm{\beta}}_{4}^{\mathit{\text{opt}}}$ for each β_{3}. Figure 4(b) shows these ${\mathrm{\beta}}_{4}^{\mathit{\text{opt}}}$ and their corresponding maximum bandwidths. For typical β_{3}, the values of ${\mathrm{\beta}}_{4}^{\mathit{\text{opt}}}$ are close to two orders of magnitude larger than those reported in highly nonlinear fiber [8]. The bandwidth with β_{4}=${\mathrm{\beta}}_{4}^{\mathit{\text{opt}}}$ was consistently found to be 1.4 times greater than when β_{4}=0, in the wavelength range investigated. The red dashed line on Fig. 4 shows the effect on conversion bandwidth when β_{4} is perturbed by 10% from ${\mathrm{\beta}}_{4}^{\mathit{\text{opt}}}$. Such variations from ${\mathrm{\beta}}_{4}^{\mathit{\text{opt}}}$ cause a mean reduction in bandwidth of approximately 8%. The PWCs discussed here operate in the anomalous dispersion regime, bounded on one side by the zero dispersion wavelength (see, e.g., Fig. 4(a)). Therefore to get the most use out of these devices, the fiber zero dispersion wavelength should be located at the normal corner of the desired operating range.

## 4. Discussion and conclusions

This work has investigated ways to increase wavelength conversion bandwidth, while minimising the gain ripple in fiber PWCs. The bandwidths in Eq. (10) increase with increasing γ*P*_{p}
and decrease with increasing dispersion. However, the bandwidth only grows sub-linearly with these parameters. We have also found this to be true in the general case β_{3}≠0 and β_{4}≠0. However, optimisation of fiber dispersion parameters can provide a significant further increase in gain bandwidth. Within the square defining the region of operation, and for the particular choice of *G*
_{thresh}=*G*
_{0}, Eq. (8) shows that the gain ripple of these devices depends on γ*P*_{p}*L*. This allows independent selection of ripple and bandwidth because for a given piece of fiber, the bandwidth is selected by choosing *P*_{p}
and the ripple is fixed by the device length. For other choices of *G*
_{thresh}, this may not be the case.

## Acknowledgements

Justin Blows and Ross McKerracher acknowledge the support of the Australian Photonics Co-operative Research Centre.

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