Nonreciprocal Epsilon-Near-Zero-Dielectric Bilayers: Enhancement of Nonreciprocity from a Nonlinear Transparent Conducting Oxide Thin Film at Epsilon-Near-Zero Frequency

We envision the use of an indium tin oxide (ITO) thin film as part of a bilayered silicon-photonics subwavelength device to boost nonlinearity-assisted all-passive and ultrafast nonreciprocal behavior. Asymmetric $p$-polarized oblique excitation of a mode near the epsilon-near-zero frequency, with a highly confined and enhanced normal electric field component and large absorption, allows one to harness the strong subpicosecond nonlinear response of ITO for the generation of notable nonreciprocal performance in an ultracompact two-port element. Although the results are limited by loss, we find the optimal operational point and the maximum attainable nonreciprocal transmittance ratio of the metasurface versus light intensity—including an apparent upper bound slightly over 2—and we perform exhaustive numerical simulations considering nonlinear processes of both anharmonic and thermal nature that validate our predictions, including steady-state and pulsed-laser excitation.

Figure 1

Toy model of nondispersive nonreciprocal bilayer device. (a) Schematic plot of the asymmetric nonlinear bilayer two-port device. (b1) $s$-parameters and $\mathbf{x}$-averaged electric field magnitude inside the ENZ slab as a function of the thickness $D_{\mathrm{Si}}$ of the silicon layer in the linear problem. (b2) Transmission coefficients versus $D_{\mathrm{Si}}$ in the nonlinear nonreciprocal problem [${\chi }^{(3)}=3.33\times {10}^{-21}{\mathrm{(V/m)}}^{-2}$, $E_0={10}^9\mathrm{V/m}$]. (c1),(c2) Magnitude of electric field phasor versus $x$ in the monochromatic nonlinear problem when $D_{\mathrm{Si}}$ minimizes (c1) and maximizes (c2) $|s_{21}|$. (d1),(d2) Electric field at the output plane versus time in the harmonic generation nonlinear problem.

Figure 2

Nonlinearity-induced nonreciprocal response versus $(\lambda ,\theta )$. (a1)–(d1) Linear absorptance, reflectance, normalized $|E_x(x=D_{\mathrm{Si}})|$, and nonlinear-to-linear transmittance ratio, respectively, for excitation from port 1. (a2)–(d2) Similarly for port 2. (e),(f) Linear (reciprocal) transmittance and squared NTR, respectively. (g) Dependence of the nonlinear nonreciprocal response versus incident electric field for the optimal $(\lambda ,\theta )$ point taken from (f), the optimal $\lambda$ being precisely the ENZ wavelength.

Figure 3

Nonreciprocal response for excitation with an ultrafast pulse. (a) Schematic drawing of the setup of the numerical experiment for excitation from port 1, and superimposed incident magnetic field $H_z$ when the maximum of the intensity crosses the middle point of the bilayer ($t=0$). (b) Nonlinearity-induced nonreciprocal response versus incident electric field intensity [in fact, its phasor value halved, for comparison with Fig. 2], assuming both Kerr-like (fast) and thermal (slow) nonlinearities: the blue and red (black and green) curves represent nonlinear-to-linear [nonlinear backward (12)-to-forward (21)] transmittance ratios, respectively.

Figure 4

Nonlinear response of the nonreciprocal device for maximum NTR: $E_0=195\mathrm{MV/m}$. (a1) $\mathbf{x}$-$\mathbf{y}$ maps of (normalized) $E_x$ and $E_y$ at $t=0$ for excitation from both ports, with Kerr nonlinearity only. (a2) 1D sections of $E_x$ in the middle of the ITO film (at fixed $\mathbf{x}$) and $t=0$ vs $\mathbf{y}$, but considering all linear, Kerr, and thermal scenarios. (b) Normalized flux of energy density across the output plane of the bilayer (${\mathrm{\Phi }}_x$ vs $\mathbf{y}$) corresponding to the scenarios in (a2). (c1) Time profile of (normalized) transmitted $H_z$ at $\mathbf{y}=0$ (i.e., very close to the beam axis). (c2) Spectral content of the Kerr signals in (c1), with third and fifth harmonics showing up. (d) Time-varying nonlinear (thermal) plasma frequency at the $\mathbf{x}$-$\mathbf{y}$-center of the ITO film (excitation from opposite inputs included); the dashed lines represent the complement of a normalized measure of the time-average intensity, defined as $1-\{⟨\Vert \tilde{\mathbf{E}}(t){\Vert }^2{⟩}_T/\max [⟨\Vert \tilde{\mathbf{E}}(t){\Vert }^2{⟩}_T]\}\{1-min[{\omega }_{p,h}(t)/{\omega }_p]\}$, at the same point, while the inset shows the second-harmonic ripples (see Appendix pp3).

Figure 5

Optimal values of Si and ITO film thicknesses from analysis of the linear problem for excitation from the ITO-layer side (port 2). (a1)–(c1) Transmittance, absorptance, and normalized (squared) longitudinal electric field magnitude $|E_x(x=D_{\mathrm{Si}})|$, respectively, at the ITO side of the ITO-$\mathrm{Si}$ interface, versus $\mathrm{Si}$ film thickness $D_{\mathrm{Si}}$ and angle of incidence $\theta$, for fixed $D_{\mathrm{ITO}}=100$ nm. (a2)–(c2) map the same quantities versus $(D_{\mathrm{ITO}},\theta )$ for $D_{\mathrm{Si}}=80$ nm.

Figure 6

Nonlinearity-induced nonreciprocal response versus $(\lambda ,\theta )$ if we assume that the collision rate $\mathrm{\Gamma }$ in ITO is ten times smaller. The figure parts are the same as in Fig. 2.

Figure 7

Nonlinear transmittance ratio versus incident electric field magnitude for $\mathrm{\Gamma }=0.0468{\omega }_p$ (black and green lines) and $\mathrm{\Gamma }=0.00468{\omega }_p$ (red and blue lines), each at the corresponding optimal point in the $(\lambda ,\theta )$ plane. Both fast and slow ITO nonlinearities are considered. When $\mathrm{\Gamma }=0.00468{\omega }_p$, the effective thermal susceptibility is reduced according to ${\chi }_{\mathrm{eff}}^{(3)}\propto \mathrm{\Gamma }$ [Eq. (C3)], and so we scale the fast susceptibility ${\chi }^{(3)}$ too (solid blue line). If, on the contrary, we keep the initial value of ${\chi }^{(3)}=6.67\times {10}^{-18}{\mathrm{(V/m)}}^{-2}$, we have the same function compressed by a factor of $\sqrt{10}$ along the $E_0$ axis (dashed blue line). These calculations are performed for steady-state plane-wave excitation in the time domain.

Figure 8

Comparison of the effective nonlinear permittivity $\mathrm{\Delta }ϵ$ versus depth into the ITO film at the optimal $(\lambda ,\theta )$ point of maximum NTR in Fig. 2, considering both Kerr (a) and thermal (b) nonlinearities: similar values of $\mathrm{\Delta }ϵ$ are obtained for both types of nonlinear interactions. Incidence from port 1 (blue) and port 2 (red) is included in both parts.

Figure 9

Nonlinear response of the nonreciprocal device in the saturation regime (low NTR): $E_0=692\mathrm{MV/m}$. The figure parts are the same as in Fig. 4.