Figure 1

Sketch of a typical optical experiment, showing the pulse incident from vacuum on the boundary with a medium: (a) before interaction ($t=-T\to -\infty$); (b) after interaction ($t=T\to \infty$). Throughout all simulations, we set the linear refractive index $n$ of the nonlinear medium equal to unity in order to avoid any linear Fresnel reflection. A similar goal can be achieved for $n$ different from unity, provided the nonlinear medium is surrounded by a linear medium with the same refractive index (index matching).Reuse & Permissions

Figure 2

Pulse shapes (a) before and (b–d) after interaction with the quadratic medium in the configuration shown in Fig. 1. The input shape is given by $E_{\sin }(z,t=-T)=A\left(z\right)\sin (k_{0}z+{\phi }_0)$ with $A\left(z\right)=\text{sech}[(z-z_0)/z_p]$. Here, ${\omega }_0=k_0=1$ ($k_0$ being the central wave number), $z_p=1.25$, $z_0=300$, $L=50$, $4\pi {\chi }^{\left(2\right)}\max \left(E_{\mathrm{in}}\right)=0.01$, and $2T=1000$. (b) Reflected shape; (c) transmitted shape; and (d) reflected shape after low bandpass frequency filter. Throughout all simulations, linear refractive index of the medium $n=1$. The frequency filter filters away frequencies above $1.5{\omega }_0$. $E_{\mathrm{in}}\left(z\right)$ is the profile of the incident field. The distance (time) is measured in physical units of $k_0$ (${\omega }_0$). The field is normalized in such a way that $\max \left(E\right)={10}^{-2}/{\chi }^{\left(2\right)}$. For the example with GaAs, this means that one unit is equal to 7000 V/m.Reuse & Permissions

Figure 3

Evolution of the total magnetic area $A_H$ (dotted green), the magnetic area ${\left(A_H\right)}_{\mathrm{ref}}$ of the reflected portion of the pulse (dashed red), and the magnetic area ${\left(A_H\right)}_{\mathrm{tr}}$ of the transmitted portion of the pulse (solid black), as functions of time. Three stages are shown. Stage I (from $t=0$ till $t\approx 300$): propagation in vacuum prior to the interaction with the nonlinear medium. Stage II (from $t\approx 300$ till $t\approx 400$, the total interaction time with the nonlinear medium of the forward- and backward-traveling waves being equal to $2L/c$): propagation through the nonlinear medium. Stage III (from $t\approx 400$ further on): propagation of the forward- and backward-traveling waves in vacuum after the interaction has finished. Parameters are as described in the legend of Fig. 2.Reuse & Permissions

Figure 4

Same as described in the legend of Fig. 2 but for the cubic medium, $P={\chi }^{\left(3\right)}{E}^3$, with $4\pi {\chi }^{\left(3\right)}\max \left(\right|E_{\mathrm{in}}{|}^2)=0.01$.Reuse & Permissions

Figure 5

The spectral power of the static component in the quadratic medium with conductivity as a function of the conductivity normalized to the central frequency of the excitation field ${\omega }_0$: $4\pi \sigma \left(0\right)/{\omega }_0$. Red circles (black squares) indicate the value of the spectral power of the static component in the forward- (backward-)traveling wave. Other parameters are: $4\pi {\chi }^{\left(2\right)}{E}_{in}=0.01$, (a) $\nu =0.01{\omega }_0$; (b) $\nu =100{\omega }_0$.Reuse & Permissions

Figure 6

Spectral power of the static component in the quadratic medium with embedded two-level systems as a function of the resonance strength $\alpha$. Red circles (black squares) indicate the value of the spectral power of the static component in the forward- (backward-)traveling wave. The inset shows the spectral power of the second harmonic of the forward-traveling wave for the same parameters as in the main figure. Parameters are: the resonance line is centered at ${\omega }_{\mathrm{res}}=2.5{\omega }_0$, relaxation time of polarization ${\gamma }_{\mathrm{ab}}={\gamma }_a/2=0.005{\omega }_0$, where ${\gamma }_a$ is the relaxation time of the upper level, the lower level is the ground level. Other parameters are as in Fig. 2, except that for figure (b) the duration of the pulse is four times longer than in (a).Reuse & Permissions