Goos-Hänchen shift in a standing-wave-coupled electromagnetically-induced-transparency medium

The Goos-Hänchen shift of the system composed by two cavity walls and an intracavity atomic sample is presented. The atomic sample is treated as a four-level double-$\Lambda$ system, driven by the two counterpropagating coupling fields. The probe field experiences the discontinuous refractive index variation and is reflected. Moreover, under the phase-matching condition, the four-wave mixing effect based on electromagnetically induced transparency can cause effective reflection. The Goos-Hänchen shifts appear in both situations and are carefully investigated in this article. We refer to the first one with the incident and reflected light having identical wavelength as the linear Goos-Hänchen shift, and the second one with the reflection wavelength determined by the phase-matching condition as the nonlinear Goos-Hänchen shift. The differences between the two kinds of shifts, such as the incident angle range, conditions for the shift peaks, and controllability, are discussed.

Figure 1

(a) Schematic of the system. The three parts of the system are labeled with arabic numerals 1, 2, and 3. The Greek letters $\alpha$ and $\beta$ label the environments or substrates of the system, ${\theta }_0$ and $s$ stand for the incident angle and the GH shift respectively. (b) The four-level double $\Lambda$ system. ${\Omega }_1$ and ${\Omega }_2$ are the counter-propagating coupling fields. ${\Omega }_i$ stands for the incident field, ${\Omega }_r$ for the reflected field.

Figure 2

GH shift of the (a) TE and (b) TM polarized light in the four-level double-$\Lambda$ system plotted as a function of the incident angle ${\theta }_0.{\delta }_1={\delta }_2=-20$ MHz, and ${\Omega }_1={\Omega }_2=40\text{MHz}$. Other parameters are ${\delta }_i=0,n_{\alpha }=n_{\beta }=1.0,n_1=2.22,n_2=2.22,d_1=d_3=2.3\mu \mathrm{m},d_2=57.5\mu \mathrm{m},{\wp }_{41}=1.336\times {10}^{-29}\text{C}\text{m},{\mathcal{N}}_0=1\times {10}^{14}{\mathrm{cm}}^{-3},{\gamma }_m=2\pi \times 1\mathrm{kHz},{\gamma }_{41}={\gamma }_{31}=2\pi \times 6\mathrm{MHz}$, and ${\omega }_i=2\pi \times 384\mathrm{THz}$.

Figure 3

The resonance condition of the three layers. $\Delta {\lambda }_1$ (or $\Delta {\lambda }_3$, red thicker solid line) and $\Delta {\lambda }_2$ (grey thinner solid line) are plotted as a function of the incident angle ${\theta }_0$. ${\lambda }_i$ is the wavelength of the incident field. All the parameters are identical with those in Fig. 2.

Figure 4

The schematic diagram of the system when the four-wave mixing effect is considered. The length of the second layer is assumed to be long enough for the incident light to be totally reflected.

Figure 5

The nonlinear Goos-Hänchen shifts of the (a) TE and (b) TM polarized light when the four-wave mixing effect based on EIT is considered. ${\lambda }_r$ is the wavelength of the reflected field. The numerical results are calculated under different coupling detunings: ${\delta }_1=-20$ MHz, ${\delta }_2=20$ MHz (solid black line), and ${\delta }_1={\delta }_2=-20$ MHz (red dashed line). The other parameters are ${\delta }_i=-20$ MHz, ${\Omega }_i={\Omega }_2=40$ MHz, $n_{\alpha }=n_{\beta }=1.0,n_1=2.22,d_1=2.3 \mu \mathrm{m},{\wp }_{41}=1.336\times {10}^{-29}$ C m, ${\mathcal{N}}_0=1\times {10}^{14}{\mathrm{cm}}^{-3},{\gamma }_m=2\pi \times 1\mathrm{kHz},{\gamma }_{41}={\gamma }_{31}=2\pi \times 6\mathrm{MHz}$, and ${\omega }_i=2\pi \times 384$ THz.

Figure 6

The Goos-Hänchen shifts of the TE polarized light plotted with the incident angle varying from 0 to $\pi /2$. The nonlinear effect of the four-wave mixing based on EIT is considered. The linear shift is calculated solely from the reflection of the first layer; it is not the linear shift of the three-layer system. The coupling detunings are ${\delta }_1=-20$ MHz and ${\delta }_2=20$ MHz, and the incident detuning is ${\delta }_i=-20$ MHz. The other parameters are identical with those in Fig. 5.