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.

 

Fig. 1. Schematic of the operation of an ideal wavelength converter. a) Conversion from a fixed λ_s to an arbitrary λ_c, within a range Δλ. b) Conversion from a fixed λ_c to an arbitrary λ_s, within a range Δλ. c) Boundary for the region representing conversion from an arbitrary λ_s to an arbitrary λ_c, both within the same range Δλ.

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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.

 

Fig. 2. a) Theoretical gain spectra for a typical parametric amplifier at two different pump wavelengths. b) Ideal wavelength conversion spectra for two different pump wavelengths. Pump wavelengths are 1553 nm (black) and 1558 nm (red). G _thresh (dotted) was chosen to be the gain at λ_s=λ_p. Other parameters are in Table 1.

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Table 1. Table of parameters used in calculations [11]. λ₀ is the dispersion zero wavelength of the fiber, γ is the fiber nonlinearity, L the fiber length, P_p the pump power, and β₃ and β₄ are the third and fourth order dispersion parameters at λ₀.

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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

 

Fig. 3. Black contours show the intersection of gain curves for a PWC with G _thresh=G ₀. Fiber parameters are shown in Table 1. A square of maximum area (green) has been fitted and indicates the useful device wavelength coverage for a fixed band.

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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=2γP_p , and the corresponding value fo G. and the The maximum value of g occurs when Δβ=2γP_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 (β₂) vanishes at frequency ω₀ and with cubic (β₃) and quartic (β₄) dispersions defined at ω₀. 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 β₄≠0, the contours are:

where Φ=(β₄/2)(ω_p-ω₀)²+β₃(ω_p-ω₀). 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 β₃=0 or β₄=0, then the last of Eqs. (9) simplifies considerably. We have also found similar equations for the case β₄=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=λ₀. These lines intersect with each other and with the original curve. The other two sides can then be found geometrically. We find that when β₃=0,

For non-zero β₃ and β₄, 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 (β₄=0) or sixteenfold (β₃=0) increase in the γP_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 β₄ on the G=G ₀ contour, and therefore the bandwidth, of a PWC. ${\mathrm{\beta }}_4^{\mathit{\text{opt}}}$ is the value of β₄ that maximises the conversion bandwidth for a fiber with a β₃ 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 β₄=0. The black bandwidth contours fall inside this blue square when β₄<0. Therefore only values of β₄>0 contribute to an increase in the conversion bandwidth.

Numerical investigations of Eq. (9) reveal a ${\mathrm{\beta }}_4^{\mathit{\text{opt}}}$ for each β₃. Figure 4(b) shows these ${\mathrm{\beta }}_4^{\mathit{\text{opt}}}$ and their corresponding maximum bandwidths. For typical β₃, 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 β₄=${\mathrm{\beta }}_4^{\mathit{\text{opt}}}$ was consistently found to be 1.4 times greater than when β₄=0, in the wavelength range investigated. The red dashed line on Fig. 4 shows the effect on conversion bandwidth when β₄ 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.

 

Fig. 4. a) Animation showing how G=G ₀ contours (black) change with decreasing β₄ and corresponding device bandwidth (green). The red square is the maximum bandwidth, for β₄=${\mathrm{\beta }}_4^{\mathit{\text{opt}}}$≈4.11×10^-54s⁴m^-1. The blue square represents the bandwidth when β₄=0. b) Plot of ${\mathrm{\beta }}_4^{\mathit{\text{opt}}}$ (black) and corresponding bandwidth maxima (red) for fibers with different β₃ values. The dashed red line shows the worst effect of a 10% perturbation from ${\mathrm{\beta }}_4^{\mathit{\text{opt}}}$. γ, P_p and L are as in Table 1. [Media 1]

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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 γP_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 β₃≠0 and β₄≠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 γP_pL. 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.