General model of optical frequency conversion in homogeneous media: Application to second-harmonic generation in an $\epsilon$-near-zero waveguide

Traditional optical frequency conversion model is well improved in this work. In terms of the dyadic Green's function method, a set of coupled-amplitude equations is reduced under a proposed transition layer assumption, accompanying the simultaneous integral equations. The model, as a generalization of the current frequency conversion theory, is aimed at any one-dimensional thin film or bulk nonlinear structure, allowing for arbitrary optical anisotropy and absorption without pumping and propagating limitations. The assumption reasonably simplifies the strict nonlinear boundary conditions and enables the equations to yield exact radiative field solutions. A field-enhanced phase-matching configuration is designed for second harmonic generation in a lossy $\epsilon$-near-zero material. The high contrast of refractive indices between a substrate (silicon) and the material traps the harmonic wave inside and constructs a natural mirror reflection waveguide. A simulation in the lowest guided mode predicts an efficiency enhancement proportional to the relative wave impedance to the fifth power under a resonant condition.

Figure 1

Schematic of the eigenmodes of the electromagnetic waves generated in a FC process, driven by a planar sheet nonlinear source at $z^{\prime }$ with a fixed wave vector ${\mathbf{k}}_{\mathrm{P}}$, where the forward and backward wave vectors of the four possible modes are illustrated (a) with seriate spherical wavefronts radiated at the sampled secondary source points. The geometries of the transmission modes in the plane of incidence (b) and of the modes (c) in three-dimensional (3D) coordinates are shown, where ${\widehat{\mathbf{a}}}_2$ for ${\widehat{\mathbf{a}}}_{\mathrm{F}2}$ or ${\widehat{\mathbf{a}}}_{\mathrm{B}2}$, $\mathbf{S}$ for a Poynting vector.

Figure 2

Transition layer of an infinitely small thickness $\delta h$ in which the nonlinear polarization grows quickly to the normal from a linear medium to a nonlinear medium. One similar transition layer is on the other side, where the nonlinear polarization goes quickly down to zero.

Figure 3

Clockwise-directed curve $\mathcal{L}$ over the film section $S$, a rectangle, which is divided into four parts as ${\mathcal{L}}_1$ and ${\mathcal{L}}_3$ with a length $2w$ along the $x$ direction, ${\mathcal{L}}_2$ and ${\mathcal{L}}_4$ with a length $l$ along the $z$ direction, respectively.

Figure 4

Schematic of the PM geometry of SHG, shown in the upper diagram, where one harmonic wave vector of two transmission modes, denoted by ${\mathbf{k}}_{\mathrm{F}}$, is phase matched by the wave vectors ${\mathbf{k}}_{\mathrm{F}1}$ and ${\mathbf{k}}_{\mathrm{B}1}$ from two pumping waves, respectively. The lower diagram demonstrates the three wave vectors of nonlinear polarizations, formed by the two pumping waves.

Figure 5

Planar mirror waveguide with a size $w$ for the SHG of the ENZ output, where the ENZ slab with a thickness $l\ll w$ is sandwiched by the substrate with high refractive index, and the harmonic wave grows in the PM direction but at the same time endures slight energy loss in absorption and transmittance.

Figure 6

Contour plots of the gain $g$ scaled by ${10}^{-5}$ for TE (a) and TM (b) modes with ${\varepsilon }^{\prime }={10}^{-2}$ over $\theta \text{−}{\varepsilon }^{″}$ space, and similar plots for TE (c) and TM (d) modes with $W=100$, ${\varepsilon }^{″}={10}^{-4}$ over $\theta \text{−}W$ space, followed with the plots of the logarithmic single-pass gain ${log}_{10}{g}_{\mathrm{s}}$ in the current model (e) and in the extended old model (f). The considered region is limited by $\theta <{\theta }_{\max }$ and $j_{\max }>1$, where the ineffective region is uniformly colored in.