Photonic integrated circuits are a topic of great interest to researchers interested in producing compact high-performance optical systems. Because of its compatibility with existing electronic systems, silicon-on-insulator (SOI) is one of the most studied types of photonic integrated circuit, and a number of designs for SOI optical components exist in the literature.
Foster et al. (2006)  investigated the properties of an optical amplifier consisting of a silicon waveguide within a silicon dioxide substrate. The input end of the amplifier is fed both the signal and a powerful “pump” wave, on a longer wavelength than the signal. These waves interact inside the amplifier through the degenerate four-wave mixing effect  – an interaction between four photons at three frequencies. Through careful phase matching, the interaction can be set up so that the amplitude of the pump wave is diminished, and the signal is amplified. In addition, another wave – the “idler wave” – forms out of band....
Figure 1 shows a model of the amplifier created in CST MICROWAVE STUDIO® (CST MWS). The degenerate four-wave mixing effect occurs because silicon is a third-order non-linear material, with a χ(3) of around 2 × 10-18 m2/V2 in the infrared spectrum. The χ(3) of the waveguide in the model has been scaled up by 100, and the length of the amplifier reduced by the same amount. This reduces the computational requirements of the simulation without significantly affecting its accuracy.
Defining electric and magnetic symmetry planes reduces the computational complexity of the problem further. To model the combined pump and signal waves, a custom excitation is defined in CST MWS – the sum of two Gaussians in the frequency domain, one corresponding to each input wave. The pump wave has a wavelength of 1.550 µm (193.4 THz) and the signal wave has a wavelength of 1.525 µm (196.6 THz). This signal is introduced by a waveguide port, which ensures that the correct modes are excited (Figure 2). The amplifier is then simulated using the time-domain solver in CST MWS.
As Figure 3 shows, the simulation revealed both the amplification effect and the production of the idler wave. The gain of the amplifier was 5.2 dB. Note that the pump wave is very much larger than the signal – the graph is normalized so that the input pump wave has an amplitude of 1.
The theoretical frequency of the idler wave  is given by:
This works out at 190.2 THz, corresponding to a wavelength of 1.576 µm. This behavior was seen exactly in the simulated model; the idler wave’s calculated wavelength is 1.576 µm.
To illustrate the amplification process, E-field and power monitors were defined corresponding to the frequencies of the signal wave, pump wave and idler wave, as shown in Figures 4, 5 and 6.
 (2006) Mark A. Foster, Amy C. Turner, Jay E. Sharping, Bradley S. Schmidt, Michal Lipson & Alexander L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip”, Nature, vol. 44, p. 960-963
 (2001) Govind Agraval, “Four-wave mixing”, Nonlinear Fiber Optics, pp. 368-423