In this paper, we prove the functionality of PC sections in the particularly case of femtosecond SR structures. We demonstrate that properly designed 2D PCs incorporated in the waveguide of an SR structure can serve both as in-plane reflectors replacing the chip facets and outcouplers of SR pulses in the vertical direction. We simulate the propagation of femtosecond pulses through the PC section of a Super-PCSEL by solving full Maxwell equations using a Finite-Difference Time-Domain (FDTD) method. This technique is a rigorous and powerful tool for modeling optical devices of different types, including PCSELs (Yokoyama and Noda, 2005). It allows for the solution of Maxwell's equations directly without any additional physical approximations. The usual approach followed in FDTD is the one involving staggered grids, for both time and space variables. We deal with Maxwell equations in 2D which are solved using a standard Yee algorithm (Yee, 1966). The system of Maxwell equations to be solved with the fields *E* and *H* is thus (Felbacq, 2016)

$$\frac{\partial {D}_{z}}{\partial t}=c\left(\frac{\partial {H}_{y}}{\partial x}-\frac{\partial {H}_{x}}{\partial y}\right) \frac{\partial {H}_{x}}{\partial t}=-c\frac{\partial {E}_{z}}{\partial y}$$

$$\frac{\partial {H}_{z}}{\partial t}=c\left(\frac{\partial {E}_{y}}{\partial x}-\frac{\partial {E}_{x}}{\partial y}\right) \frac{\partial {H}_{y}}{\partial t}=c\frac{\partial {E}_{z}}{\partial x}$$

1

$$\frac{\partial {D}_{x}}{\partial t}=-c\frac{\partial {H}_{z}}{\partial y} \frac{\partial {D}_{y}}{\partial t}=c\frac{\partial {H}_{z}}{\partial x}$$

We assume here that the PC section is passive (no pumping) and consists of a set of circular air cylinders, with a relative permittivity in the frequency domain given by

$$\epsilon \left(\omega \right)={\epsilon }_{r}+i\frac{\sigma }{{\epsilon }_{0}\omega }$$

2

.

In the time domain, the corresponding relation between *D* and *E* is

$$D\left(t\right)= {\epsilon }_{r}E\left(t\right)+\frac{\sigma }{{\epsilon }_{0}}\underset{0}{\overset{t}{\int }}E({t}^{{\prime }})d{t}^{{\prime }}$$

3

We use 2D perfectly matched layers boundary conditions for the solution of the Eqs. (1)-(3). For simplicity, we consider in this paper the case of the square-lattice PC having a circular unit cell structure with a lattice period of *a*. Figure 3 (a) illustrates the configuration of the PC sections. The width of the PC is supposed to be equal to that of the optical waveguide of the SR device, i.e. *w* = 5–10 µm. It is well known that for effective diffraction in the direction normal to the PC plane the period *a* of the PC should be equal to the emission wavelength in the medium λ (Meier et al, 1999 and Imada et al, 1999). We consider here the most studied so far GaAs/AlGaAs SR heterostructures with the emission wavelength of 870–890 nm. As a result, the number of PC periods in the lateral direction is about *w/a* ~ 20–25. The size of the PC in the direction of the pulse propagation can be varied in a broad range.

We simulate the propagation of a Gaussian pulse with duration in a range of 80-1000 fs through the PC section and calculate the reflection and transmission coefficients as well as the distortion of the initial pulse. The chosen pulsewidths correspond to typical durations of SR pulses which were experimentally observed (Vasil’ev, 2009). Figure 3(b) shows a snap shot of a typical distribution of *E**2**(t)* in the PC section at a certain time. Figure 4 presents the results of the calculations of the propagation of a 120 fs pulse through a PC with 9 by 20 cells.

The distortion of the pulses at this pulsewidth is very small. Indeed, the durations of all tree pulses are equal within 3% accuracy. The bandwidths of the reflected and transmitted pulses are approximately equal to that of the initial pulse. This is the case for all pulsewidths within the range under study.

The reflection/transmission of the generated femtosecond SR pulses by the PC section can be adjusted in a broad range by the variation of *a/λ* ratio. For illustration, Fig. 5 shows the transmission of 80 fs pulses through the PC versus *a/λ*.

The minimum of the transmission coefficient at *a/λ* ≈ 1 corresponds to the maximum of diffraction of the emission in the vertical direction (Sakai et al, 2010).