Hysteresis in the complex nonlinear refractive index of a homogeneous and isotropic medium

Abstract

We calculate the permittivity, $ε(ω)$, for a medium with a quadratic electro-optic effect, modeling it as a Duffing oscillator. The nonlinear refractive index $n(ω, E(ω))$ and the nonlinear absorption coefficient $α(ω, E(ω))$ exhibit hysteresis when the light intensity is varied [here $E(ω)$ is the electric field strength at angular frequency $ω$], and when the light frequency is varied. $n(ω, E(ω))$ can be negative when the resonances in the permittivity and permeability are close to one another.

Figures (5)

• Figure 1: The real and imaginary parts of the amplitudes ${\mathcal{A}}_{1, a}$, red-solid and red-dashed curves respectively, and ${\mathcal{A}}_{1, b}$, blue-solid and blue-dashed curves respectively, for $\gamma = 0.01 \, \omega_0$ and $\beta = 0.0001 \omega_0^6 / \Gamma^{2}$. The black solid and dashed curves show the real and imaginary parts of ${\mathcal{A}}_{1,0}$ for the linear model (i.e., $\beta = 0$).
• Figure 2: The real and imaginary parts of the amplitudes ${\mathcal{A}}_{1, a}$ and ${\mathcal{A}}_{1, b}$, for $\gamma = 0.05 \, \omega_0$, and $\omega = 1.4 \, \omega_0$.
• Figure 3: The real and imaginary parts of the permittivities $\epsilon_a(\omega)$ and $\epsilon_b(\omega)$ for $\gamma = 0.01 \, \omega_0$, $\Gamma = 0.01 \, \Gamma_0$ and $\omega_p = 1.2 \, \omega_0$.
• Figure 4: The real (solid) and imaginary (dashed) parts of the refractive indices $n_a(\omega)$ (red) and $n_b(\omega)$ (blue) are plotted versus frequency using the parameters $\gamma = 0.01 \, \omega_0$, $\Gamma = 0.01 \, \Gamma_0$, $\omega_p = 1.2 \, \omega_0$, $\gamma_m = 0.005 \, \omega_0$, $\omega_{p,m} = 0.5 \, \omega_0$ and $\omega_{0,m} = 1.1 \, \omega_0$. The solid and dashed black curves show the real and imaginary parts of the refractive index $n_L(\omega)$ for the Lorentz model. The inset is a zoom of the real and imaginary parts of $n_b(\omega)$ for large $\omega$.
• Figure 5: The real and imaginary parts of the refractive indices $n_a(\omega)$ and $n_b(\omega)$ versus field strength for $\omega = 1.2 \, \omega_0$, $\gamma = 0.01 \, \omega_0$, $\omega_p = 1.2 \, \omega_0$, $\gamma_m = 0.005 \, \omega_0$, $\omega_{p,m} = 0.5 \, \omega_0$ and $\omega_{0,m} = 1.1 \, \omega_0$.