Tunable Radiation Patterns on Temperature-Dependent Materials

Abstract

The utilization of optical antennas for active control of far-field radiation at the subwavelength scale is crucial in various scientific and technological applications. We propose a thermally tunable disk design of indium tin oxide (ITO) and aluminum gallium nitride (${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{As}$), enabling a switch between absorption and scattering. Furthermore, the control of far-field radiation pattern can be easily realized by combining ITO and ${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{As}$ to enhance or suppress emission. Our results demonstrate that hybrid structures can be dynamically tuned with temperature variations. In the proposed design, a frequency is achieved at the wavelength of 1240 nm. The thermal tunability of hybrid structures introduces new multifunctional possibilities for light manipulation, thereby enhancing the potential applications of new devices in the near-infrared range.

1. Introduction

With the rapid advancement of miniaturized devices, antenna regulation of antennas has garnered significant attention. Dynamic modulation of the far field to achieve versatile control is highly desirable. Specifically, negative refraction, tunable metasurfaces, optical switches, tunable cavities, and coherent perfect absorbers have been realized using indium tin oxide (ITO) as epsilon-near—zero (ENZ) materials and aluminum gallium arsenide (AlGaAs) [1,2,3,4]. However, there has been no report on temperature-controlled far-field radiation of antennas to date. Significantly, beyond purely optical considerations, the design of temperature-dependent nanoantennas must also account for their electromagnetic response. This can be addressed by solving dynamic regulation problems, such as solutions for exchanging energy and anisotropies. The far-field radiation patterns are determined by the refractive index, which is temperature-dependent. By increasing the temperature of an antenna, multipole decomposition suggests that the temporal and spatial coherence of the far-field radiation can be engineered through careful material selection or dimensional adjustments. The engineering of far-field radiation is of great interest for applications in lighting, thermoregulation, energy harvesting [5,6], tagging, and radiation therapy [7]. Electric regulation has good compatibility and high efficiency, but its structure is complex; light regulation is the fastest, but requires a light source, resulting in low efficiency.

Recent studies have demonstrated that ENZ materials and AlGaAs exhibit an exceptionally large temperature-dependent refractive index (see Figure 1a) [8,9]. Consequently, ENZ and AlGaAs materials offer a new platform for optically tuning the material response within a subpicosecond timescale [2]. One way to increase the temperature modulation speed is to forgo heating the material to obtain hot electrons, which has been demonstrated using ultrafast pump pulses with modulation speeds in the 100 femtoseconds in graphene [10], and several picoseconds in metals such as gold and tungsten [11]. ENZ materials provide a new platform to optically tune the response of the material within a subpicosecond timescale [2].

III-V semiconductors have the advantage of a short electron lifetime compared to silicon, making them suitable for switching applications. ${\mathrm{Al}}_{\mathrm{x}}{\mathrm{Ga}}_{1-\mathrm{x}}\mathrm{As}$ features a broad transparency window ranging from the near to mid-infrared, from 0.9 to 17 μm [12]. Additionally, ${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{As}$ exhibits a high relative permitivity. The scattering response of the ITO disk is weaker than that of the ${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{As}$ disk with the same dimensions due to higher absorption losses of ITO. The Kerr effect, caused by hot electron nonlinearity, plays an important role in changing the induced multipole moments, and thus drastically modulates its scattering, absorption, and extinction as well as its far-field radiation patterns.

In this paper, we utilize ITO and ${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{As}$ to engineer the far-field control of radiation patterns. First, we present the basic formalism describing the refractive indices of ITO and ${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{As}$, and study the scattering and far field properties to realize dynamically reconfigurable far-field radiation pattern based on single nanophotonic structures. We then emphasize the hybridization of ITO and ${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{As}$ to exploit their properties for energy efficiency and sustainability, thereby engineering far-field radiation effectively. The relevance of this study is primarily theoretical, as the relationship between far-field radiation distribution and the transition between absorption and scattering remains largely unexplored using the calibrated temperature-index models. We demonstrate that this aspect, although not explicitly accounted for, may be spatially transferable when applied to a sufficiently large temperature range. Methods to achieve temperature control of far-field radiation have been implemented by heating, adjusting light intensity, or altering voltage as the refractive index changes with electron temperature. Notably, the far-field variation not only changes by half but also shifts the scattering direction from the original x-axis to the y-axis, resulting in a more significant control range. Clearly, nanoscale electromagnetics is one of the technologies with the potential to provide such functionality, including reception, emission, and more generally, the control of light at the nanoscale.

2. Materials and Methods

According to the optical properties of ITO as a function of electron temperature, the real and imaginary components of the refractive index at ${\lambda }_{ENZ}=1240$ nm are shown by the blue lines in Figure 1. The change in the real part of the refractive index with temperature is approximately 0.22, and the linear refractive index is 0.45. Here, the epsilon-near-zero regime of ITO is reached at 1240 nm. The calculated refractive index of ${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.82}$ at ${\lambda }_{ENZ}$ is depicted in Figure 1 by the red line. The change in the real part of the refractive index with intensity is approximately 0.22, while the linear refractive index is 3.32.

The temperature-dependent refractive index of ITO is expressed as

$$ϵ(\omega ,T_e)={ϵ}_{\infty }-\frac{N{e}^2}{{ϵ}_0[m^{*}\left(T_e\right){\omega }^2+ie\omega /\mu \left(T_e\right)]},$$

(1)

where ${\epsilon }_{\infty }$ is the high-frequency permitivity, N is the carrier density, e is the electron charge, and $m^{*}\left(T_e\right)$ and $\mu \left(T_E\right)$ are the electron effective mass and mobility, which vary with electron temperature. The plasma frequency ${\omega }_p=N{e}^2/\left[{\epsilon }_0{m}^{*}\left(T_e\right)\right]$ with the $\gamma =e/\left[m^{*}\left(T_e\right)\mu \left(T_e\right)\right]$. The parameters for the commercial ITO are $N=1.5\times {10}^{21}{\mathrm{cm}}^{-3}$, $m^{*}=0.3964m_e$, ${\epsilon }_{\infty }=3.404$, $E_F=0.8793\mathrm{eV}$, and $C=0.4191{\mathrm{eV}}^{-1}$. The mobility $\mu \left(T\right)=18.3+2.13\times {10}^{-5}{T}^{1.53}$. The real part of the dielectric constant $\epsilon$ of ITO is zero at 1240 nm that is the ENZ wavelength ${\lambda }_{ENZ}$ [13,14]. According to the optical properties of ITO as a function of electron temperature on the model, the real and imaginary components of the refractive index are plotted in Figure 1 in blue at ${\lambda }_{ENZ}$ [9,15]. The change in the real part of the refractive index with intensity is about 0.22, the linear refractive index is 0.45.

The Sellmeier equation of ${\mathrm{Al}}_{\mathrm{x}}{\mathrm{Ga}}_{1-\mathrm{x}}\mathrm{As}$ with temperature dependence can be written as

$$\begin{array}{c}\hfill n(x,\lambda ,T)=\left[10.906-2.92x+\frac{0.97501}{{\lambda }^2+C}\right.\\ \hfill {\left.-0.002467(1.41x+1){\lambda }^2\right]}^{1/2}+\left[(T-{26}^{\circ }\mathrm{C})\right.\\ \hfill \left.\times (2.04-0.3x)\times {10}^{-4}{/}^{\circ }\mathrm{C}\right]\end{array}$$

(2)

The first bracket term is the refractive index at room temperature. The second bracket represents the temperature-dependent Sellmeier coefficient. The temperature T, wavelength $\lambda$ enable us to calculate the index for ${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{As}$, where $C={(0.52886-0.735x)}^2$. The Sellmeier can be used at 1240 $\mathrm{nm}$ and 2000 $\mathrm{K}$ [8]. The calculated index of ${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{As}$ are plotted in Figure 1 in red line at ${\lambda }_{ENZ}$ [8]. The change in the real part of the refractive index with intensity is about 0.22, and the linear refractive index is 3.32.

3. Results

3.1. Tuning the Scattering and Far Field for Temperature-Dependent Materials

The proposed structure, consisting of three-dimensional (3D) ITO disk with height $H=150$ nm and diameter $D=600$ nm, is schematically shown in the inset of Figure 2a. The nonlinear material made of ITO is illuminated by a plane wave with electric field ${\mathbf{E}}_{\mathrm{inc}}\left(\mathbf{r},\omega \right)=(E_0/2)e^{i\left(\mathbf{k}·\mathbf{r}-\omega t\right)}{\mathbf{e}}_x+\mathrm{c}.\mathrm{c}.$, where $\left|\mathbf{k}\right|=2\pi /\lambda$ is the wavenumber, $\omega$ is the angular frequency, $E_0$ is the amplitude of incident field, and c.c. means complex conjugate. $I_0=\frac{1}{2}c{\epsilon }_0{\left|E_0\right|}^2$ is the free-space intensity of the incident plane wave. $\mathbf{E}\left(\mathbf{r},\omega \right)$ is the electric field inside the ITO. The scattering properties of the ITO nanodisk, including its electric and magnetic dipole contributions, are depicted in Figure 2a. The nanodisk is discretized using a 10 nm hexagonal mesh following a convergence test. As shown in Figure 2a, the scattering cross-section exhibits minimal variation with temperature. The primary contribution to the multipole scattering arises from the electric dipoles (ED), while the contributions from magnetic dipole (MD), electric quadrupole (EQ), and magnetic quadrupole (MQ) are relatively minor. Consequently, the far-field radiation pattern, influenced by the interference between the multipoles, remains largely unchanged. Due to the imaginary component of ITO, the antenna also exhibits an absorption cross-section. Figure 2b illustrates that both the absorption and scattering cross-sections vary slightly with temperature, leading to a corresponding change in the extinction cross-section. It is evident from Figure 2b that the absorption cross-section is dominant, approximately twice that of the scattering cross-section.

The optical response of the radiation pattern can be explained by the multipole expansion of the induced displacement current ${\mathbf{J}}_{NL}(r,\omega )=-i\omega \left[{ϵ}_{NL}(\mathbf{r},\omega )-{ϵ}_0\right]\mathbf{E}(\mathbf{r},\omega )$ [16,17]. The excited multipole can be obtained by the induced nonlinear displacement current ${\mathbf{J}}_{NL}$. The exact multipole expansion introduced by [16,17]

$$\begin{array}{cc}\hfill p_{\alpha }& =-\frac{1}{i\omega }\left\{\int d^3\mathbf{r}{J}_{\alpha ,\mathrm{NL}}{j}_0\left(kr\right)\right.\hfill \\ \hfill & \left.+\frac{k^2}{2}\int d^3\mathbf{r}\left[3\left(\mathbf{r}·{\mathbf{J}}_{\mathrm{NL}}\right)r_{\alpha }-r^2{J}_{\alpha ,\mathrm{NL}}\right]\frac{j_2\left(kr\right)}{{\left(kr\right)}^2}\right\},\hfill \\ \hfill m_{\alpha }& =\frac{3}{2}\int d^3\mathbf{r}{\left(\mathbf{r}\times {\mathbf{J}}_{\mathrm{NL}}\right)}_{\alpha }\frac{j_1\left(kr\right)}{kr},\hfill \\ \hfill Q_{\alpha \beta }^{\mathrm{xz}}& =-\frac{3}{i\omega }\left\{\int d^3\mathbf{r}\left[3\left(r_{\beta }{J}_{\alpha ,\mathrm{NL}}+r_{\alpha }{J}_{\beta ,\mathrm{NL}}\right)-2\left(\mathbf{r}·{\mathbf{J}}_{\mathrm{NL}}\right){\delta }_{\alpha \beta }\right]\right.\hfill \\ \hfill & \frac{j_1\left(kr\right)}{kr}+2k^2\int d^3\mathbf{r}[5r_{\alpha }{r}_{\beta }(\mathbf{r}·{\mathbf{J}}_{\mathrm{NL}})-(r_{\alpha }{J}_{\beta ,\mathrm{NL}}+r_{\beta }{J}_{\alpha ,\mathrm{NL}})r^2\hfill \\ \hfill & -r^2(\mathbf{r}·{\mathbf{J}}_{\mathrm{NL}}){\delta }_{\alpha \beta }]\frac{j_3\left(kr\right)}{{\left(kr\right)}^3}\},\hfill \\ \hfill Q_{\alpha \beta }^{yz}& =15\int d^3\mathbf{r}\left\{r_{\alpha }{\left(\mathbf{r}\times {\mathbf{J}}_{\mathrm{NL}}\right)}_{\beta }+r_{\beta }{\left(\mathbf{r}\times {\mathbf{J}}_{\mathrm{NL}}\right)}_{\alpha }\right\}\frac{j_2\left(kr\right)}{{\left(kr\right)}^2},\hfill \end{array}$$

(3)

where $\alpha ,\beta \in x,y,z$, $p_{\alpha }$, $m_{\alpha }$, $Q_{\alpha \beta }^e$, and $Q_{\alpha \beta }^m$ are the electric dipole (ED), magnetic dipole (MD), electric quadrupole (EQ), and magnetic quadrupole (MQ) multipole moments, respectively. $j_n\left(kr\right)$ is the nth spherical Bessel function. The total scattering cross-section can be calculated by [16,17]

$$\begin{array}{ccc}\hfill C_{\mathrm{sca}}& =& \frac{k^4}{6\pi {\epsilon }_0^2{\left|E_0\right|}^2}\left[\sum _{\alpha }\left({\left|p_{\alpha }\right|}^2+{\left|\frac{m_{\alpha }}{c}\right|}^2\right)\right.\hfill \\ & & \left.+\sum _{\alpha ,\beta }\left({\left|k{Q}_{\alpha \beta }^e\right|}^2+{\left|\frac{k{Q}_{\alpha \beta }^m}{c}\right|}^2\right)\right],\hfill \end{array}$$

(4)

which is the contribution of each multipole moment (marked ‘total’ in Figure 2a and Figure 3b). The interference between multipoles can satisfy the generalized Kerker condition. The far field corresponding to the radiation pattern ($\propto {\left|\mathbf{E}\right|}^2$) in space is

$$\begin{array}{ccc}\hfill \mathbf{E}& \approx & \frac{k^2}{4\pi \epsilon }{p}_x\frac{e^{ikr}}{r}\left(-sin\phi \widehat{\phi }+cos\theta cos\phi \widehat{\theta }\right)\hfill \\ & -& \frac{k^2}{4\pi \epsilon }\frac{\sqrt{{\epsilon }_r}{m}_y}{c}\frac{e^{ikr}}{r}\left(sin\phi cos\theta \widehat{\phi }-cos\phi \widehat{\theta }\right)\hfill \\ & -& \frac{k^2}{4\pi \epsilon }\frac{ik{Q}_e}{6}\frac{e^{ikr}}{r}\left(-sin\phi cos\theta \widehat{\phi }+cos2\theta cos\phi \widehat{\theta }\right)\hfill \\ & -& \frac{k^2}{4\pi \epsilon }\frac{ik{Q}_m}{6c}\frac{e^{ikr}}{r}\left(-sin\phi cos2\theta \widehat{\phi }+cos\theta cos\phi \widehat{\theta }\right),\hfill \end{array}$$

(5)

where r, $\theta$, and $\phi$ are the spherical coordinates, k is the wavenumber, and c is the speed of light.

The temperature-induced change in the refractive index results in minor variations in the scattering. These variations are due to the contributions of multipoles, which remain relatively constant and exhibit an almost unchanged far-field radiation pattern, as illustrated in Figure 2(c1–c5). In the study of the optical properties of a single ITO nanostructure, both scattering and absorption are influenced by temperature. However, these changes are minimal, primarily because the refractive index of ITO changes by less than 1, resulting in a very small difference between the refractive index of ITO and the background refractive index (set to 1). As the height of ITO increases at D = 200 nm and D = 1000 nm, the scattering, absorption, and extinction cross-section can be found in Figure A2a,b, respectively. The corresponding multipole decomposition are shown in Figure A2c,d (see Appendix A.2).

Similarly, we use an ${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.72}\mathrm{As}$ antenna of the same size to study its scattering and far-field characteristics. ${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.72}\mathrm{As}$ allows for even greater control over the scattering properties of antennas. As a lossless dielectric, there is no imaginary part of ${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.72}\mathrm{As}$ at 1240 nm. We focus on the scattering cross-section. Figure 3a presents the scattering cross-section as a function of temperature for different height (H) with diameter $D=600$ nm. At a specific diameter and temperature, the scattering cross-section is not proportional to height variations, exhibiting irregular fluctuations. Furthermore, for a fixed size, the pattern of the scattering cross-section changing with temperature is not consistent. In Figure A3, it can be seen that with the increase of H, the trend of the scattering cross-section is different. We have drawn a scattering cross-section of $H=50$ to 600 nm, and we can see from Figure A3 that when $H=150$ nm, the contribution of the multipole changes more than that when $H=50$ and 100 nm, and the drastic change in the multipole can cause abundant far-field radiation phenomena, so we take $D=600$ nm and $H=150$ nm as an example as shown in Figure 3b. In Figure 3b, the scattering cross-section initially decreases and then increases. More diverse changes in different dimensions are shown in Figure A3. For example, when $H=600$ nm, the far-field radiation pattern can be seen in Figure A4(a1–a5) (see Appendix A.3). In Figure 3b, when the temperature is below 866 K, the electric dipole contribution decreases while the magnetic dipole contribution gradually increases, leading to a reduction in far-field radiation in the $\pm y$ direction and an increase in the $+z$ direction. Approaching 866 K, the scattering is the smallest due to the smallest contribution of the electric dipole. Correspondingly, the contribution of the magnetic quadrupole is the largest. When the temperature exceeds 866 K, the electric dipole contribution steadily increases, resulting in forward scattering in the far-field radiation pattern. The significant changes in electric dipoles before and after 866 K profoundly affect the far-field radiation patterns. In Figure 3(c1–c5), the radiation pattern of ${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{As}$ changes with the temperature. Moreover, the radiation patterns change more than that of ITO due to its refractive index changing by 0.22 with temperature. Here, the refractive index of ${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{As}$ is around 3.5, which is significantly different from the background refractive index.

3.2. Tuning the Scattering and Far Field of Hybrid Structures for Opposite Illumination

To realize the switching between tunable far-field radiation, we integrated both materials and investigated their optical characteristics when illuminated by an x-polarized plane wave propagating in two opposite directions, i.e., k = $\pm k_0{\mathbf{e}}_z$. A hybrid nonlinear antenna composed of ITO and lossless ${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{As}$ is depicted in Figure 4a. Owing to the broken inversion symmetry, the hybrid antenna demonstrates a bianisotropic response.

In Figure 4b, it is observed that when light illuminates a material from two opposing directions, the extinction cross-section remains identical. Due to the loss characteristics of ITO, associated with optical bistability, both scattering and absorption cross-sections are contingent upon the direction of incident light. Initially, commencing from room temperature, the scattering cross-section of the hybrid structure exhibits a preference for illumination from below compared to illumination from above. The absorption and scattering can be switched. For bottom illumination, the scattering cross section dominate when $T<677$ K and $T>1244$ K, the absorption cross section dominate when 677 K $<T<$ 1244 K. For top illumination, the scattering cross section dominate when $T<866$ K and $T>1811$ K, the absorption cross section dominate when 866 K $<T<1811$ K. However, nearing approximately 866 K, the scattering cross-sections converge, and with further temperature elevation, a reversal occurs, where the scattering cross-section for illumination from above surpasses that for illumination from below. Figure 4c and Figure 4d depict the multipole decomposition for the bottom and top illuminations, respectively. Under bottom illumination, electric dipoles consistently dominate with variations in temperature. Under top illumination, electric dipoles mainly dominate at $T<866$ K and $T>1777$ K, while both electric and magnetic quadrupoles contribute more at 866 K $<T<1777$ K. The electric field distributions for opposite illuminations are shown in Figure A5 (see Appendix A.4).

The induced electric, magnetic dipoles and quadrupoles interfere constructively (destructively) in the backward (forward) direction. Under bottom illumination (see Figure 5(a1–a5)), the radiation pattern initially consists of two lobes distributed symmetrically about the z-axis without backward illumination at room temperature. As temperature rises, backscattering gradually emerges on the radiation map. Further temperature increase induces bidirectional changes in the radiation pattern, with backward (forward) radiation decreasing (increasing) correspondingly. Ultimately, significant tunability of the induced multipole moments in the hybrid antenna by temperature enables control over radiation patterns from a bidirectional to the unidirectional pattern. Specifically, under top illumination (see Figure 5(b1–b5)), the initially omnidirectional radiation at room temperature transforms into a three-lobe pattern distributed symmetrically about the y-axis and the upper side of the z-axis. With continued temperature elevation, the radiation pattern evolves from three lobes to four, predominantly radiating from the upper side of the z-axis. Subsequently, with further temperature increase, radiation in the y-direction of the four-lobe pattern diminishes, resulting in bidirectional radiation with forward scattering. Consequently, the nonreciprocal characteristics of hybrid structures arise from the distinct electromagnetic couplings generated.

4. Conclusions

In summary, this study investigated the influence of temperature variations on far-field radiation patterns through numerical simulations employing a spatially explicit temperature-index model. A hybrid structure comprising two materials, each exhibiting temperature-dependent refractive indices, was analyzed. The refractive index change from room temperature to 2000 K was determined to be 0.22. The temperature adjustments induce diverse far-field radiation distributions and have a bianisotropic response. Furthermore, the operational wavelength range can be broadened significantly, and the law of refractive index changes with temperature has been given. Moreover, the methodology employed to determine the temperature-dependent refractive indices of ITO and ${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{As}$ at 1240 nm can be extended to other wavelengths, facilitating multi-parameter control of far-field radiation and enhancing regulation replication, thereby expanding the tunable wavelength range.