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We theoretically study the problem of Raman gain maximization in uniform silicon photonic-crystal waveguides supporting slow optical modes. For the first time, an exact solution to this problem is obtained within the framework of the undepleted-pump approximation. Specifically, we derive analytical expressions for the maximum signal gain, optimal input pump power, and optimal length of a silicon Raman amplifier and demonstrate that the ultimate gain is achieved when the pump beam propagates at its maximum speed. If the signal’s group velocity can be reduced by a factor of 10 compared to its value in a bulk silicon, it may result in ultrahigh gains exceeding 100 dB. We also optimize the device parameters of a silicon Raman amplifier in the regime of strong pump depletion and come up with general design guidelines that can be used in practice.

The rapid progress in the field of silicon photonics realized during recent years draws the reality of all-optical signal processing on a nanoscale closer than ever [^{2}) in the wavelength region near 1550 nm [

Several techniques have been successfully employed to mitigate the nonlinear losses and increase the Raman gain in silicon-on-insulator (SOI) waveguides. The most straightforward method lies in the reduction of free-carrier lifetime. It is typically implemented through a reverse-bias p-i-n diode, which uses silicon waveguide as an intrinsic region and removes carriers from the region near its mode center through the applied static electric field [

Signal gain can also be improved by successively optimizing both the length of the SOI waveguide and the input pump intensity [

Another way of enhancing the amplified signal lies in engineering (tapering) the lateral cross-section of a SRA [

A somewhat more challenging, but quite efficient, gain-improvement technique relies on the slow-light enhancement of nonlinear optical phenomena in silicon photonic crystal (SPhC) waveguides [

Consider amplification of a continuous wave (CW) signal at the frequency

Equations (

Hence, the problem of gain maximization in a slow-light SRA, for a given intensity of the input signal, consists of searching for the optimal values of the waveguide length

The signal gain can be readily optimized analytically in the undepleted-pump regime, which usually holds for relatively short SRAs with ^{2} and for low input signal powers. We now consider this scenario in detail, for it is quite instructive and helps us to illuminate the peculiarities of the optimization problem in the general case.

In the approximation of an undepleted pump, valid as long as

Equation (

To find the values of the three optimal parameters, we use the implicit solution of (

Using (

We maximize

The combination of (

When the slow-down factor of the pump is fixed, (

With the initial and final intensities given by (

It is important to note that the optimal values of

Hence, the optimal values of three parameters that maximize the signal gain for this optimization scheme are given by

If we keep the input pump power constant, then (

Once the optimal quantities

On substitution of (

Consider finally the situation in which the length of an SRA is preset. As we have seen above, in this case (

Since the values of

In Figure ^{2}, and ^{2}. It is seen that the signal gains of >20 dB can be realized with pump powers below 1 GW/cm^{2} if

(a) Ultimate signal gain (dashed curves) and optimal amplifier length (solid curves) plotted as functions of

Equations (

As we saw in the previous section, in the undepleted-pump regime, the signal mode should be slowed down as much as possible, while the pump mode should travel with the highest speed. This is not the case when the intensity of input signal,

Figure ^{2} and ^{2}; the depletion [

Contour plots of signal gain (in dB) for an 8 mm long SRA in the plane formed by the factors ^{2}, ^{2}, ^{2} at point A ^{2} at point B

The gain corresponding to the optimal slow-down factors can be increased by optimizing the input pump power. Red curves in Figures ^{2},

Optimal peak signal gain (solid curves) and slow-down factors,

A further enhancement of gain is generally possible by adjusting the length of the SPhC waveguide. Figure ^{2}, pump power of 10 W is required at the input of the optimized SRA. Since it is very difficult to get such high input power levels in a CW regime, the actual maximum gain will be lower than 31.7 dB and will be set by the available pump power.

Comparison of different curves in Figure

As a concluding remark, it is worth noting that our results are valid not only for SPhC waveguides, but also for any other types of slow-light silicon waveguides, regardless of the relative group delay that occurs upon light propagation through them.

We have theoretically analyzed the problem of gain maximization in silicon Raman amplifiers operating with slow-light pump and signal modes. Using the undepleted-pump approximation, we derived expressions for the maximum signal gain, optimal input pump power, and optimal length of the amplifier with a fixed cross-section. We also showed that the signal gain is maximized when the pump beam moves at its maximum speed. The signal gain grows indefinitely and can exceed 100 dB, provided the signal beam travels more than 10 times slower than its speed in a bulk silicon. In the regime of substantial pump depletion, the parameters of an amplifier were optimized numerically and a relatively large CW gain of 31.7 dB was predicted for an 8 mm long waveguide pumped at an intensity of 3.5 GW/cm^{2}. This result suggests that optimization of slow-light silicon photonic crystal waveguides is a promising approach to the realization of highly efficient Raman amplifiers built using the SOI platform.

This work was supported by the Australian Research Council, through its Discovery Grant scheme under grants nos. DP0877232 and DP110100713. The work of G. P. Agrawal is also supported by the US National Science Foundation award ECCS-0801772.