In this paper, we simulate the propagation of chirped pulses in silicon nanowires by solving the nonlinear Schrodinger equation (NLSE) using the split-step Fourier (SSF) method. The simulations are performed both for the pulse shape (time domain) and for the pulse spectrum (frequency domain), and various linear and nonlinear effects changing the shape and the spectrum of the pulse are analyzed. Owing to the high nonlinear coefficient and a very small effective-mode area, the required length for observing nonlinear effects in nanowires is much shorter than that of conventional optical fibers. The impacts of loss, nonlinear effects, second- and third-order dispersion coefficients and the chirp parameter on pulse propagation along the nanowire are investigated. The results show that the sign and the value of the chirp parameter have important role in pulse propagation so that in the anomalous dispersion regime, the compression occurs for the up- chirped pulses, whereas the broadening takes place for the down-chirped pulses. The opposite situation happens for up- and down-chirped pulses propagating in the normal dispersion regime.
Low cost, availability as well as linear and nonlinear optical properties of silicon has caused silicon to be an ideal material for the production of nanoscale integrated photonic devices [
There are two types of chirped pulses: up-chirp in which the frequency increases with time and down-chirp where the frequency decreases with time. In some sources, chirp is synonymous with swept signal and is used in SONAR, RADAR, communications and high pulse energy amplifiers [
In this paper, the evolution of chirped pulses along the silicon nanowires is simulated and impacts of second- and third-order dispersion coefficients and also loss on pulse evolution are investigated. Moreover, a comparison is made between different cases of chirped and non-chirped input pulses.
In this section, theoretical framework as well as the numerical method for investigation of fem to second chirped pulse propagation in silicon nanowires are presented. The governing equation is the nonlinear Schrödinger equation (NLSE) which is defined as [
where A represents the envelope of the pulse, α is the loss of nanowire and γ denotes The nonlinear parameter. Also, β2 and β3 are the second- and the third-order dispersion coefficients of the nanowire calculated at the pulse wavelength, respectively. Different terms on the right-hand side of Equation (1) are respectively responsible for the attenuation, group-velocity dispersion (GVD), third-order dispersion (TOD) and self-phase modulation (SPM). Basically, the linear effects (including the loss, GVD and TOD) are responsible for changes in pulse shape while nonlinear effects (such as SPM) are responsible for changes of pulse spectrum. As it will be seen, the interplay of linear and nonlinear effects leads to both the temporal and spectral changes of the pulse.
The well-known method for numerically solving the NLSE is the split-step Fourier method (SSFM). We implement a code in MATLAB based on SSFM for different cases of chirped and non-chirped and also the presence or absence of loss and second- and third-order dispersion coefficients. We input a Gaussian pulse with a center wavelength of 1527 nm, a pulse width of 204 fs and a peak power of 40 mW into a single mode nanowire of Si with a cross-section of 445 × 220 nm2 and a length of 4.7 mm [
where A0 is maximum amplitude and C is the chirp parameter. T0 shows half pulse width at 1/e of peak intensity. At first, we start with a simple case where only nonlinear effects (SPM) with second-order dispersion (GVD) are considered in NLSE. We consider γ = 2.2 × 104 (W∙m)−1 and β2 = −3.89 ps2/m, and as is seen from
By adding the loss of 3.5 dB/cm, we see that the power level reduces by a factor of ~2 (
and nonlinear phase shift. In
Now, if we make our model more realistic and consider the input chirp parameter, it can be seen from
Finally, if we complete the model and include the third-order dispersion coefficient β3 = −0.435 ps3/m in NLSE, it is seen that asymmetric variations in pulse shape and pulse spectrum occur which are totally different in comparison with those in
To see the important roles of the initial chirp(C) and the TOD (β3) more explicitly, we plot the output pulse shape and the output pulse spectrum for two different cases: with and without chirp and TOD. This is shown in
GVD and SPM are taken into account. As it is apparent, for both cases, by increasing the initial chirp value, the pulse shape and the spectrum are dramatically broadened. As expected, the symmetry in the pulse shape and the pulse spectrum is preserved in
Silicon nanowires have great potential for exhibiting the nonlinear effects and they are good candidates for applications in very small-scale photonic components. We have shown that when an ultra-short pulse propagates along the Si nanowire, GVD causes symmetric temporal broadening, while TOD changes the pulse shape asymmetrically. The loss decreases the effective length and lowers the power level without a significant change in overall behavior of the pulse shape and spectrum. In particular, the input chirp parameter has a significant role on the evolution of the pulse shape and spectrum along the nanowire. For the up-chirped pulses in the anomalous dispersion regime, where β2C is negative, pulse compression occurs and vice versa. In addition, by increasing the value of the input chirp, changes in both the pulse shape and the pulse spectrum are dramatically increased. The results of this research can be used for pulse shaping and signal processing in silicon nanowires.
The authors would like to thank Dr. Michael Frosz for his assistance with numerical code of NLSE.
Hassan Pakarzadeh,Zeinab Delirian,Mostafa Taghizadeh, (2016) Simulation of Chirped Pulse Propagation in Silicon Nanowires: Shape and Spectrum Analysis. Optics and Photonics Journal,06,53-61. doi: 10.4236/opj.2016.68B010