All-optical switching in a photonic crystal with a defect containing an $\mathsf{N}$-type four-level atomic system

We study the transmission spectra of a one-dimensional photonic crystal with a defect containing a four-level atomic medium that exhibits a greatly enhanced third-order susceptibility while having a vanishing linear susceptibility dependent on the electromagnetically induced transparency. Two ways of controlling the transmission of a photonic crystal are discussed: via absorption, i.e., nonlinear (two-photon) absorption of the probe field enhanced by constructive quantum interference, and via dispersion, which comes down to shifting the resonance frequency of the defect mode for the probe field by varying the refractive index based on the giant Kerr nonlinearity (cross-phase modulation). We demonstrate that such systems enable nonlinear all-optical switching at ultralow intensities of the coupling and switching laser fields.

Figure 1

Energy-level diagram of an $\mathsf{N}$-type four-level atom interacting with a weak probe field (${\omega }_1)$ and two strong fields: the coupling (${\omega }_2)$ and the switching (${\omega }_3)$ fields. ${\delta }_{1,2,3}$ denote the frequency detuning from one-photon resonances for the probe, coupling, and switching fields, respectively.

Figure 2

The dispersive behavior of the imaginary (a),(b) and real (c),(d) parts of the nonlinear susceptibility $\chi \left({\omega }_1\right)$ as the probe frequency detuning varies. ${\Omega }_2={\gamma }_{10}$, ${\delta }_2=0$, ${\gamma }_{10}={\gamma }_{30}=2\pi \times 10$ MHz, ${\gamma }_{20}=0.01{\gamma }_{10}$. (a),(c) ${\delta }_3=0$. (b),(d) ${\Omega }_3={\gamma }_{10}$.

Figure 3

(a) PC defect mode without an EIT medium. The arrows indicate the positions of the interacting wavelengths. (b) Spatial distribution of the intensity $I/I_0$ of the coupling (solid curve) and switching (dashed curve) fields in the defect layer as functions of the normalized (to the layer width $d_{\mathrm{D}})$ coordinate $z$; $z=0$ and $z=1$ correspond to the defect boundaries; $I_0$ is the input intensity.

Figure 4

(a) $\mathrm{Im}n$ dependence on the probe field detuning and the normalized coordinate $z$ in the defect layer for $g_3=0.1{\gamma }_{10}$. (b) $\mathrm{Im}n$ distribution inside the defect layer under resonance conditions ${\delta }_1=0$ for various $g_3$ values. $g_2=0.05{\gamma }_{10}$, ${\gamma }_{10}={\gamma }_{30}=2\pi \times 10$ MHz, ${\gamma }_{20}=0.01{\gamma }_{10}$, ${\chi }_0=8.4\times {10}^{-3}$.

Figure 5

PC transition spectra for various Rabi frequencies of the switching field $g_3$. The rest of the parameters are as in Fig. 4. The inset shows transmission spectra for ${\Omega }_{2,3}=0$ (solid curve) and for $g_3=0$ under EIT (dashed curve).

Figure 6

Transmission coefficient of the probe field versus the Rabi frequency of the switching field: $g_2=0.05{\gamma }_{10}$, ${\gamma }_{10}={\gamma }_{30}=2\pi \times 10$ MHz.

Figure 7

(a) PC transmission spectrum as a function of the probe field frequency detuning and Rabi frequency of the switching field for a fixed detuning ${\delta }_3=30{\gamma }_{10}$; $g_2=0.05{\gamma }_{10}$, ${\gamma }_{10}={\gamma }_{30}=2\pi \times 10$ MHz; (b) Fragments of the transmission spectrum for $g_3=0$ (dotted curve), $g_3=0.42{\gamma }_{10}$ (dash-dotted curve). The solid curve shows the transmission of PC with uniform spatial distribution of the fields. The dashed curve refers to a hypothetical case when spatial non-uniformity of the fields is ignored and ${\gamma }_{30}=0$.

Figure 8

Time dependence of the probe filed pulse. The dotted curve is the input pulse with the pulse length $T_p=10\mu$s. The solid curve is the transmitted pulse under EIT with no switching field. The dash-dot curve refers to the output pulse in the presence of a resonant switching field. The dashed curve refers to the transmitted pulse under off resonant switching field. $g_2=0.05{\gamma }_{10}$.