Ultrafast dynamic switching of optical response based on nonlinear hyperbolic metamaterial platform

Ze Tao Xie; Yanhua Sha; Jiaye Wu; Jiaye Wu; H. Y. Fu; Qian Li

doi:10.1364/OE.457875

1. Introduction

The replacement of electrons with photons is the basis and main promise of optical and quantum information computing. To achieve effective on-chip manipulation of light, electro-optical modulators represented by hybrid optoelectronic approaches are widely adopted in the industry. However, electro-optical devices are energy-hungry and have high latency due to the required electronics [1]. Another potential way to overcome the fundamental limits of electro-optical devices and tailorable control of optical states is all-optical switching [2,3]. All-optical switching can be defined as a device that has a function of ON/OFF conversion for signal light under pump irradiance, with key performance parameters being the modulation depth and switching time. The control of light is based on the pump-intensity-dependent refractive index variations in nonlinear optical materials, namely, the optical Kerr effect [4].

Recently, epsilon-near-zero (ENZ) effects have emerged as a topic of interest due to its unique and exclusive optical properties [5–7]. This has led to many novel phenomena and advanced applications such as electromagnetic energy tunneling [8], electric fields enhancement [9], electro-optical modulators [10], perfect absorbers [11], and light-emitting devices [12], etc. More importantly, extremely large optical nonlinearity has been observed and reported in the ENZ regime of transparent conducting oxides [13,14] and multilayers [15,16]. This property has given rise to numerous nonlinear applications of ENZ materials including efficient frequency conversion [17,18], complex pulse dynamics [19], high-harmonic and terahertz wave generations [20–22], making them promising candidates in the ultrafast all-optical switching systems [23–26].

Artificial metamaterials designed to have a hyperbolic dispersion with a tensor ε = [ε_xx, ε_yy, ε_zz] are known as hyperbolic metamaterial (HMM) [27,28]. With the existence of high-k modes supported by the unique dispersion, HMMs have been used extensively to enhance the large density of photon states and control spontaneous emission [29–31]. In addition, for an HMM, the topologies of the dispersion curve can transit between ellipsoidal topology and hyperboloid topology by different signs of effective dielectric tensor permittivity [32]. This means that HMMs provide an ideal platform for optical topological transition. Recently, strong nonlinearity in multilayer HMMs have become an effective method to achieve dynamic switching of the optical response and topological transition. By considering the large Kerr nonlinearity of metal, it is demonstrated numerically that the nonlinear effect is sufficiently strong to modify the topology of an Ag/TiO₂ multilayer HMM, and results in a significant change in spontaneous emission rate [33]. Also, the optical biostability is realized that depends on the topological transition by the nonlinearity of HMM [34], and the enhanced nonlinearity is experimentally demonstrated in ENZ metal/dielectric multilayer [15], which is further shown to be strongly related to the angle of incidence pump [16]. However, most of the demonstrated all-optical modulation methods rely on high-loss metals with a wavelength-independent third-order susceptibility, and the temporal nonlinear response of HMM induced by free-electron distributions have rarely been studied.

In this work, by taking advantage of the enhanced nonlinearity of ENZ indium tin oxide (ITO), we constructed an all-optical ultrafast tunable multilayer HMM platform with different topologies of the dispersion curve. Unlike previous investigations, the nonlinearity of the proposed HMM is calculated by the electron generation and distributions of ENZ ITO under pump irradiance. The Drude dispersion of ENZ ITO induced by pump fluence results in the ultrafast modulation in the effective permittivity of the HMM, and the HMM exhibits larger changes of the index at its ENZ wavelength than that of ITO. We show that the topology of the HMM can be simply transitioned on the femtosecond time-scale by the energy of pump beam. For the bulk HMM, it is found that a significant peak in the extinction ratio (ER) spectrum arising from Ferrell-Berreman mode at the topological transition point. The position of Ferrell-Berreman mode strongly depends on the pump fluence, which is confirmed by the pump-induced loss function and dispersion relation of the HMM. The pump-induced topological transition of the HMM gives rise to an all-optical modulation approach for spontaneous emission which is characterized by Purcell factor. Moreover, to the best of our knowledge, for the first time we demonstrate the all-optical tunable coupling strength in the light-matter coupling systems based on HMM. By depositing cylindrical antennas on the proposed HMM platform, two obvious peaks are present in the ER spectrum, which attributes to two hybrid resonances arisen by the coupling between the HMM and the antennas. Based on the nonlinearity of the HMM platform, a tuning of two hybrid resonances is observed under various pump fluence and signifies the effective control of the coupling strength. The proposed coupling system can realize a modulation depth more than 15 dB within a 213-fs switching time. The results of this work reveal the potential of ENZ materials-based HMM in photonic topological transitions, spontaneous emission, quantum phenomena, and light-matter coupled systems, making HMM a widely versatile planar platform for all-optical switching.

This paper is structured as follows. In Section 2, the structural design and the topology modeled by effective medium theory (EMT) are introduced. In Section 3, the calculation and analyses on the nonlinearity of the HMM are presented. The physical mechanisms, switching performance, and modulation in Purcell enhancement for the HMM platform are demonstrated in Section 4. The analyses and discussions on the control of coupling strength based on the proposed HMM platform are presented in Section 5. Section 6 summarizes and concludes the paper.

2. Structural design

In this study, the designed HMM based on five pairs of alternating layers of ITO and SiO₂ thin films is schematically shown in Fig. 1(a). We select ITO for its large optical nonlinearity at ENZ wavelength, and SiO₂ for its transparency in the considered spectral region. HMM structure can be fabricated by magnetron sputtering, and the ENZ properties in ITO can be tuned by changing the sputtering power, etc. in the process of magnetron sputtering [35,36]. The permittivity of the proposed multilayer structure is given by a tensor ε = [ε_xx, ε_yy, ε_zz], where in-plane parallel components are defined as ε_xx= ε_yy= ε_||, and out-plane perpendicular component is defined as ε_zz= ε_⊥. Due to the fact that the proposed multilayer structure is composed of subwavelength thickness films, the permittivity can be modeled by the zeroth-order Maxwell-Garnett effective medium theory (EMT):

(1)$$\begin{aligned}{l} {\varepsilon _{||}} &= f{\varepsilon _{\textrm{ITO}}} + \textrm{(}1 - f\textrm{)}{\varepsilon _{\textrm{SiO2}}},\\ {\varepsilon _ \bot } &= \frac{{{\varepsilon _{\textrm{ITO}}}{\varepsilon _{\textrm{SiO2}}}}}{{(1 - f){\varepsilon _{\textrm{ITO}}} + f{\varepsilon _{\textrm{SiO2}}}}},\\f\textrm{ } &= \textrm{ }\frac{{{d_{\textrm{ITO}}}}}{{{d_{\textrm{ITO}}} + {d_{\textrm{Si}{\textrm{O}_2}}}}}, \end{aligned}$$

where ε_SiO2 is the permittivity of silica, ε_ITO is the permittivity of ITO with ENZ wavelength at ∼ 1240 nm [13]; f is the fill fraction, with d_ITO and d_SiO2 representing the layer thicknesses of ITO and silica, respectively.

Fig. 1. (a) Sketch of the proposed HMM consisting of five pairs of ITO (12 nm) and silica (8 nm). (b) Effective medium theory (EMT) of ITO/SiO₂ multilayer with ENZ wavelength at 1454 nm. (c) Optical phase diagram for the HMM. The ENZ wavelength can be situated anywhere from the telecommunication wavelength to the mid-infrared range by varying f.

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According to the dispersion relation ${{(k_x^2 + k_y^2)} / {{\varepsilon _ \bot }}} + {{k_z^2} / {{\varepsilon _{||}}}} = {{{\omega ^2}} / {{c^2}}}$, effective medium exhibits three topologies of the dispersion curves [32]: 1) Type-I HMM (ε_|| > 0, ε_⊥ < 0) with twofold hyperbolic topology; 2) Type-II HMM (ε_|| < 0, ε_⊥ > 0) with onefold hyperbolic topology; 3) isotropic dielectric (ε_|| > 0, ε_⊥ > 0) with ellipsoidal topology. For ε_|| < 0, ε_⊥ < 0, the aforementioned dispersion relation is not satisfied since no real k vector exits, the medium is an effective metal. Note that the topological transitions between different dispersion curves are determined by the fill fraction f and the wavelength.

Figure 1(b) shows numerically calculated the effective permittivity of the HMM with d_ITO =12 nm and d_SiO2 = 8 nm, where ε_|| has a Drude-like curve and ε_⊥ is Lorentzian. The medium behaves as an isotropic dielectric up to 1250 nm (TTP₁), acts as a Type-I HMM between TTP₁ to 1454 nm (ENZ wavelength) and a Type-II HMM to the red of 1654 nm (TTP₂), where TTP₁ and TTP₂ represent the topological transition points from an isotropic dielectric to Type-I HMM and an effective metal to Type-II HMM, respectively. In Fig. 1(c), we show that the optical phase diagram for the HMM with the fill fraction f and wavelength. One can observe that the topological transitions of different types of HMM depends on the geometry of structures, providing limited tunability.

3. Nonlinear properties of HMM

TCO materials such as ITO have been proven to significantly enhance optical nonlinearity at ENZ wavelength, and here we take the advantage of the ultrafast intensity-dependent response of ITO to achieve the nonlinear properties of the proposed HMM. The overall dynamics of the hot electrons can be described by two temperature model (TTM) and its non-parabolic conduction band structure. To achieve an all-optical tunable HMM, we first calculate the electronic dispersion relation in ITO. The complex relative permittivity of ITO can be described by the Drude model:

(2)$$\varepsilon (\omega ) = {\varepsilon _\infty } - \frac{{\omega _p^2}}{{{\omega ^2} + i\gamma \omega }},$$

where ε_∞ is the permittivity at the infinite frequency, ω is the angular frequency of incident light, ω_p is the plasma frequency, γ is the damping coefficient. Here we use experimental values as parameters for Drude model of ITO, which exhibits ENZ wavelength at ∼ 1240 nm [13].

The ultrafast nonlinearity of ITO near ENZ wavelength can be ascribable to modifications in the energy distribution in its non-parabolic conduction band induced by pump, which can be written as the following Kane’s model [37]:

(3)$$\frac{{{\hbar ^2}{k^2}}}{{2m}} = E + C{E^2},$$

where C = 0.4191 eV^-1 describes the non-parabolicity of the conduction band, m = 0.263m₀ is the effective mass of electron (m₀ is the mass of an electron) [38], E is the electron energy, k is the electron wave vector and ħ is the reduced Plank’s constant. After considering the non-parabolicity of the conduction band, the plasma frequency ω_p in the Drude model and the electron density n_e of ITO are derived as [23]:

(4)$${\omega _p}{(\mu ,{T_e})^2} = \frac{{{e^2}}}{{3m{\pi ^2}}}\int_0^\infty {dE{{(\frac{{2m}}{{{\hbar ^2}}}(E + C{E^2}))}^{\frac{3}{2}}}{{(1 + 2CE)}^{ - 1}}( - \frac{{\partial {f_0}(E,{T_{e{\kern 1pt} }})}}{{\partial E}}),}$$

(5)$${n_e}(\mu ,{T_e}) = \frac{1}{{{\pi ^2}}}\int_0^\infty {dE\frac{m}{{{\hbar ^2}}}} (1 + 2CE){(\frac{{2m}}{{{\hbar ^2}}}(E + C{E^2}))^{\frac{1}{2}}}{f_0}(E,{T_e}),$$

where T_e is the free-electron temperature, µ(T_e) is the temperature-dependent electron chemical potential, and f₀(E, T_e) is the Fermi-Dirac distribution. Since the conservation of the electron density n_e under intraband pumping, we can solve Eqs. (6) and (7) with experimental values from Ref. [13]. The calculated temperature-dependent plasma frequency is plotted in Fig. 2(a), and the plasma frequency of ITO decreases with the increment of T_e.

Fig. 2. (a) The calculated plasma frequency ω_p of ITO versus the free-electron temperature curve. (b) Calculated temporal response of electron and lattice temperature of ITO under different pump fluence F. (c) Fluence-dependent effective permittivity of the HMM. The inset shows the change of index Δn of ITO and the HMM for F = 14.17 mJ/cm², while dashed lines represent the respective ENZ points of ITO and the HMM. (d) The nonlinear effective refractive index n₂ and the effective nonlinear absorption coefficient β₂ of HMM.

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Next, we calculate the temporal behavior of the optical properties of the HMM under different pump fluence. The dynamics of the ultrafast thermal process of electrons of ITO can be quantitatively analyzed by the two temperature model (TTM) [13,39]:

(6)$${C_e}({T_e})\frac{{\partial {T_e}(t)}}{{\partial t}} ={-} {g_{ep}}({T_e}(t) - {T_l}(t)) + \frac{{N(t)}}{{{\tau _{ee}}(t)}},$$

(7)$${C_e}({T_e})\frac{{\partial {T_e}(t)}}{{\partial t}} ={-} {g_{ep}}({T_e}(t) - {T_l}(t)) + \frac{{N(t)}}{{{\tau _{ee}}(t)}},$$

(8)$$\frac{{\partial N(t)}}{{\partial t}} ={-} \frac{{N(t)}}{{{\tau _{ee}}(t)}} - \frac{{N(t)}}{{{\tau _{ep}}(t)}} + P(t),$$

where T_l is the lattice temperature, τ_ee (τ_ep) is the electron-electron (electron-phonon) scattering time, g_ep is the electron-phonon coupling coefficient, N represents the non-thermal energy density stored in the excited electrons, C_e is the free-electron heat capacity, and C_l = 2.58×10⁶ Jm^-3K^-1 is the lattice heat capacity which is higher compared to the electron [40]. P(t) is the power of the pump pulse, which is expressed as a Gaussian distribution as follows [41]:

(9)$$P(t) = {I_p}\alpha \exp ( - 2\frac{{{t^2}}}{{{\tau _p}^2}}),$$

where I_p = F/τ_p is the pump peak power density, F is the pump fluence which is independent of time t, τ_p is the pump pulse duration assumed as 100 fs, and α is the attenuation coefficient of ITO [41]. The electron-phonon coupling coefficient g_ep and the free-electron heat capacity C_e can be estimated as [13]:

(10)$${g_{ep}}\textrm{ = }0.562{n_e}\frac{{k_B^2\Theta _D^2{\nu _F}}}{{{L_f}{T_l}{E_F}}},$$

(11)$${C_e} = \frac{{3{\pi ^2}{n_e}{k_B}{T_e}}}{{\sqrt {36T_F^2 + 4{\pi ^4}T_e^2} }},$$

where Θ_D is the Debye temperature, v_F is the Fermi velocity, T_F is the Fermi temperature, E_F is the Fermi level and L_f is the electron mean free path. The parameters used in the modeling in this paper are obtained from Ref. [42]. Figure 2(b) presents the calculated results of TTM under different pump fluence. The occurrence of negative time is attributed to the pulse being centered at zero delay time. Due to the lower electron heat capacity compared to the lattice, T_e rises fast and reaches the maximum within a few hundred femtoseconds, while T_l rises after the electron-electron scattering, and eventually to thermal equilibrium. For the case of F = 14.17 mJ/cm², T_e possesses an ∼ 271-fs rise time and ∼ 583-fs relaxation time, meaning ITO has ultrafast dynamic temperature change between electron and lattice.

By substituting the aforementioned results into the Drude model and the EMT, we can obtain the time-dependent and pump-fluence-dependent permittivity of the HMM. Figure 2(c) depicts the tuning of the effective permittivity of the HMM under different pump fluence, the pump beam is assumed to be normally incident on the HMM in the calculation. The increment of pump fluence leads to a reduction in Im(ε_||) and an increment in Re(ε_||), and the Lorentz-resonance pole in Im(ε_⊥) is weakened for larger pump fluence. By reacting on the pump fluence, the effective permittivity includes ENZ wavelengths can be tuned within the whole telecommunication range. Moreover, the inset shows that the change of index Δn of the HMM is higher than ITO at the respective ENZ wavelengths, evidently, the HMM has larger nonlinearity which is in consensus with the prior reports [15,16]. Such results demonstrate the possibility to finely tune the optical properties of the HMM in telecommunication wavelength to the mid-infrared range by simply controlling the pump fluence. The nonlinear effective refractive index n₂ can be calculated by n₂ = Δn/I_p (Δn = n_pump - n₀ with n_pump being the real part of the HMM refractive index under pump fluence and n₀ being the static real part of the HMM refractive index), and the effective nonlinear absorption coefficient β₂ can be calculated by β₂= 4πΔk/(λI_p) (Δk = k_pump - k₀ with k_pump being the imaginary part of the HMM refractive index under pump fluence and k₀ being the the static imaginary part of the HMM refractive index) [43]. Figure 2(d) illustrates the effective nonlinear coefficient of HMM in the case the pump beam with F = 14.17 mJ/cm² is normally incident on the structure. One can observe that the maximum of n₂ occurs near the ENZ wavelength of HMM [14], the nonlinear effective refractive index n₂ is 5.87×10¹⁶ m²/W and the effective nonlinear absorption coefficient β₂ is -1.27×10⁻⁹ m/W at ENZ wavelength 1454 nm.

We further show that the hyperbolic dispersion phase diagram for the HMM with the pump fluence F and wavelength in Fig. 3(a). We note that the ENZ wavelength, TTP₁ and TTP₂ all cause a redshift in an increasing pump fluence, indicating the topological transition of the HMM can be controlled by the pump energy. And the transient topological transition of the HMM for F = 14.17 mJ/cm² in Fig. 3(b) reveals that all modulation regions of topologies exhibit similar temporal responses within the order of hundred femtoseconds. Figure 3(c) shows the ultrafast temporal response of the HMM for a pump fluence F = 14.17 mJ/cm², based on the modification of the effective permittivity ε_|| at ENZ wavelength of 1454 nm. One can see that the HMM has a 92-fs rise time and a 216-fs recovery time, implying that one modulation conversion can be completely done in an ultrafast time period of 393 fs. Obviously, such nonlinear response is evidence of the ultrafast performance of the tunable HMM, which can be used to investigate the optical switching characteristics and expected to lead to the development of topology transitions based on this platform.

Fig. 3. (a) Topological transition of the HMM for varying pump fluence F. (b) Transient topological transition of the HMM for F = 14.17 mJ/cm². (c) The ultrafast change in real part of the effective permittivity ε_|| at ENZ wavelength. The dashed curve denotes the pulse intensity profile (in arbitrary units).

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4. Switching effect of optical response and Purcell effect in HMM

To explore the dynamic switching properties of the proposed HMM, firstly the static optical response is investigated. The temporal dynamic behavior of HMM is calculated by TTM, and the commercial software Lumerical FDTD Solutions is used for the simulations of HMM under static state and pump fluence [41,44–46]. In order to visually demonstrate the work performance of the proposed HMM, angle-resolved extinction ratio (ER, which is defined as −10Log₁₀T, with T being the transmission) spectra of the HMM with p-polarization are shown in Fig. 4(a). The transmittance spectra of HMM can be measured with a spectrophotometer. An ER peak occurs near TTP₁ at oblique incident angle, and the maximum of the peak becomes larger while its spectral location is nearly unchanged when the incident angle increases. The pinning of the resonance here is not observed at s-polarization, which can be attributed to the excitation of the well-known Ferrell-Berreman mode [47], which is a bulk polarization mode related to the vertical oscillation of volume electrons when excited from free space. Figure 4(b) further displays the calculated dispersion of the HMM as a function of the wave vector k_|| and the corresponding eigen-frequencies. We observe clear hyperbolic modes arising from mutual electromagnetic coupling of surface plasmon polariton modes across multiple ITO-SiO₂ interfaces in the HMM, and a nearly flat dispersion for a large range of k_|| supported by the proposed HMM. The nearly flat dispersion at the left-hand side of the light is located at ∼ 242 THz, which is consistent with the ER peak at ∼ 1250 nm (≈ 239 THz) shown in Fig. 4(a), exhibiting the evidence for the existence of the Ferrell-Berreman mode. The inset of Fig. 4(a) illustrates that the HMM has the maximum absorption (which is defined as 1-R-T, R: reflection) under an incident angle of 65° at the position of Ferrell-Berreman mode. Therefore, we use the maximum absorption angle 65° as the incident angle to study the switching properties of the HMM. The upper inset of Fig. 4(c) reveals the position of Ferrell-Berreman mode is independent of the geometric parameter of the HMM, the increase in the fill fraction f only results in the enhancement and broadening of ER. The origin of the increment of ER can be appreciated by the loss function, which is defined as Im(-1/ε_⊥) [48]. As shown in the lower inset of Fig. 4(c), the loss function shows a peak close to the maximum of ER, and the peak increases and broadens with increasing fill fraction f. These results predict precisely the trends of increasing and broadening the ER peak in the upper inset, confirming that the ER peak is associated with out-plane perpendicular permittivity ε_⊥. Therefore, it can be expected that the ER of the HMM can be control by pump fluence due to the nonlinearity of the HMM. Figure 4(d) illustrates the transmission of the proposed HMM with different numbers of pairs of ITO and silica. One can observe that increasing the number of pairs results in decreasing the transmission over the spectrum. However, the position of the transmission dip is pinned at ∼ 1237 nm, meaning that the position of Ferrell-Berreman mode is independent of the number of pairs of ITO and silica.

Fig. 4. Static optical response of the proposed HMM. (a) ER spectra of the HMM as a function of incident angle. The black dashed line indicates the position of Ferrell-Berreman mode, while the inset shows that the maximum absorption occurs under incident angle of 65°.(b) Color plot of the dispersion relation of the designed HMM. (c) ER spectra for fill fraction f = 0.9, f = 0.6 and f = 0.1 (upper inset). Color map of the loss function as a function of wavelength and f (lower inset). (d) The transmission of HMM with different numbers of pairs of ITO and silica.

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Next, we simulate the switching properties of the HMM by varying pump fluence. The ultrafast nonlinear response can be performed with an optical pump-probe setup. The output beam generated from a laser system is split into two portions, one portion is used as the pump pulse, and the other is fed into an optical parametric amplifier to produce a tunable probe pulse. A photodetector can be used to acquire the transient transmission of the probe pulse, and the time delay between the pump and the probe pulses can be measured by a translation stage and retroreflector [46]. In the upper inset of Fig. 5(a), one can observe that increasing the pump fluence weakens and broadens the peak of ER, accompanied by redshifts. This results from the pump fluence-dependent loss function, which has a reduced loss and a similar trend with the peak of ER, as shown in the lower inset of Fig. 5(a). And the curve of the black circles illustrates that the Ferrell-Berreman mode is red-shifted as the pump fluence increases, and is basically consistent with the position of the ER peak. Figure 5(b) depicts the modulation depth of the HMM for various pump fluence, a maximal modulation depth of -5.19 dB occurs at ∼ 1237 nm when F = 14.17 mJ/cm². Therefore, we proceed with the transient change of transmission for F = 14.17 mJ/cm². The method of measuring transient change of transmission can be calculated by ΔT/T₀, here ΔT = T_on -T₀ with T_on being the transmission under pump fluence and T₀ being the static transmission [14]. The temporal dynamic behavior of the proposed HMM under pump fluence F = 14.17 mJ/cm² is shown in Fig. 5(c). The designed HMM demonstrates a femtosecond transient response, composed of a strong bleach up to 231% at the maximal modulation depth wavelength ∼ 1237 nm, and a weak induced absorption with peak of -48%, which is consistent with the transient response of TTP₁ (white dotted line). The transient nonlinear response of HMM at 1237 nm has a rise time of 62 fs and a recovery time of 160 fs, with a response time of 401 fs. Such results imply that the strong bleach is dependent on static Ferrell-Berreman mode, while the induced absorption is dependent on the dynamic Ferrell-Berreman mode under varying pump fluence. Furthermore, the kinetic traces of the strong bleach under various pump fluence shown in Fig. 5(d) indicate that absorbing more energy causes the increment of the maximum modulation depth of the HMM, while extends its rise time and recovery time. For a pump fluence of 14.17 mJ/cm², the HMM exhibits an ultrafast rise time of 60.38 fs and recovery time of 172.91 fs with a switching response time of 427.35 fs, to achieve a modulation depth of -5.19 dB corresponding to ΔT/T₀= 231%. Note that the asymmetrical gaussian shape of the kinetic trace arises from the small specific heat capacity of the electron and the large specific heat capacity of the lattice. In reality, the losses of the structure mainly arise from the geometry of HMM, namely, induced by the thickness of HMM, additional scattering loss owing to the deviation from an idealized structure, or from the intrinsic loss of ITO, which have influence of the modulation performance and energy consumption of HMM [35], and should be subtly adjusted in actual fabrication, implementation, and applications.

Fig. 5. Switching properties of the proposed HMM. (a) ER spectra of the HMM for various pump fluence F (upper inset). Loss function spectra as a function of F (lower inset), the curve with black circles represent the position of the Ferrell–Berreman mode. (b) Modulation depth under different F. (c) ΔT/T₀ transient spectral map under F = 14.17 mJ/cm². (d) Fluence-dependent modulation depth kinetics at the wavelength of the strong bleach (∼ 1237 nm). All incident light with p-polarization have an incident angle of 65°, and the pump beam is normally incident on the HMM.

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On the other hand, HMM with high local density of optical states has been extensively studied for the engineering of enhancing spontaneous emission, owing to the fact that its hyperbolic dispersion can support optical modes with large wave vectors. Such enhancement can be evaluated with the Purcell factor (PF) which describes the amplification of a photon emitter’s spontaneous emission rate, and is generally defined as the ratio of the radiative rate with a particular electromagnetic structure to that with the homogeneous environment [27,49]. Typically, the change of topology in HMM leads to a significant change in the PF. Recent works focus on modulating the PF of bulk HMM by modifying the fill fraction or thickness [28,30], utilizing phase-change materials, graphene [31], or calculating the switching Purcell effect with Kerr susceptibility (χ⁽³⁾) independent of wavelength [33]. However, the demonstrated PF modulation methods either strongly rely on nanophotonic structures, operating at nanosecond/picosecond switching speed (phase-change materials and graphene), or offer estimation due to wavelength-independent χ⁽³⁾, with fewer researches in the telecommunication to the mid-infrared range.

Here, we further investigate the ultrafast dynamic Purcell effect by simulating the PF from a dipole polarized perpendicularly to the designed HMM and located at the height h = 20 nm with increasing intensity of pump fluence. As illustrated in Fig. 6(a), the proposed HMM can induce significant change in the PF near TTP₁, ENZ wavelength and TTP₂ due to the optical topological transition. As the wavelength increases, an uptick in the PF can be observed at first, and it begins to rise sharply around TTP₁, reaches its maximum near the ENZ wavelength, and then starts to dip until it rises slowly again after TTP₂. The increase in pump fluence leads to a red-shift and enhancement of the PF peak, with a broadening of the PF spectrum. The observed PF peak is likely due to the resonance hybridization associated to the proposed interfaces [50]. The evolution map in Fig. 6(b) further shows that, the trend of TTP₁, ENZ wavelength and TTP₂ are similar to the trend of the PF, with the PF undergoing a dramatic increment near a topological transition TTP₁ and manifesting a peak close to ENZ wavelength. Figure 6(c) depicts the transient change of the Purcell factor ΔPF/PF₀ for F = 14.17 mJ/cm² (ΔPF = PF_on -PF₀, PF_on and PF₀ represent the Purcell factor with the pump turned on and off, respectively). Obviously, the temporal ΔPF/PF₀ is strongly dependent on the topological transition of the HMM. The ΔPF/PF₀ transient spectral map shows remarkable negative and positive regions at the shorter wavelength side and longer wavelength side of the TTP₁ (dotted line in Fig. 6(c)), respectively. The negative regions centered at 1350 nm with a maximal reduced ΔPF/PF₀ of -98%. As for the positive regions, ΔPF/PF₀ is contoured similarly to the temporal ENZ (dashed line in Fig. 6(c)), with a largest enhanced value up to 400% occurring at 1989 nm. The femtosecond kinetic traces of ΔPF/PF₀ at 1989 nm shown in Fig. 6(d) demonstrate the HMM has a rise time of 65 fs and a recovery time of 97 fs with a total response time of 222 fs, achieving 400% of the maximum change in the PF at F = 14.17 mJ/cm². Such results indicate the topological transitions based on the proposed nonlinear HMM platform can help to effectively modulate the Purcell factor, opening the possibility of ultrafast switching the Purcell effect.

Fig. 6. (a) The PF of the designed HMM as a function of wavelength under increasing pump fluence. A dipole is situated at the height h = 20 nm above the HMM (the inset). (b) The evolution of the PF spectra as a function of pump fluence. (c) ΔPF/PF₀ transient spectral map under F = 14.17 mJ/cm². (d) Fluence-dependent ΔPF/PF₀ kinetics at the wavelength of 1989nm.

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5. Dynamic all-optical control of the coupling strength via HMM

Light-matter coupling systems have been a great interest for both fundamental optical phenomena and vast applications in plasmonic devices and quantum information. Currently, there are only a few reports on controlling the coupling strength in an HMM system, and they focus on modifying geometrical parameters such as changing the scale of the plasmonic nanostructures (e.g., Ref. [51]) or the fill fraction of the bulk HMM (e.g., Ref [52].). In this section, to achieve the dynamic control of the coupling strength, we design a nanoscale metasurface-coupled system by depositing commonly known gold cylindrical antennas on the proposed nonlinear HMM platform. The aforementioned natural nonlinearity of the HMM platform allows for tunability of coupling under pump fluence and avoids changes in the geometry of the structure. The schematic of the HMM-based metasurface is shown in the inset of Fig. 7(a), the cylindrical antenna array plays three crucial roles: 1) It efficiently localizes and concentrates the incident electric field inside the HMM at normal incidence, which means that the coupling system can eliminate the restriction of the incident angle compared to the bulk HMM; 2) The cylindrical antennas array as plasmonic resonators excite a plasma resonance at ∼ 1550 nm allowed for the coupling between the antennas array and the bulk HMM; 3) The cylindrical antenna provides a polarization-independent response.

Fig. 7. (a) ER spectra of the HMM-based metasurface for various pump fluence F. The black solid line indicates the resonance of static state. The inset is the 3D schematic of the metasurface. The gold cylindrical antenna thickness is 30 nm and its radius is 320 nm, the period in the x- and y-directions are 900 nm. (b) The evolution of ER spectra of the coupling system as a function of F. (c) Electric field (|E|) distributions calculated with EMT for peak 1 and peak 2, corresponding to the shorter-wavelength hybrid resonance and the longer-wavelength resonance, respectively. (d) Modulation depth of the coupling system under different F. (e) ΔT/T₀ transient spectral map of the coupling system under F = 14.17 mJ/cm². (f) Fluence-dependent modulation depth kinetics at 1432 nm (the position of the short-wavelength hybrid resonances under F = 14.17 mJ/cm²).

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Figure 7(a) shows the all-optical tuning of ER spectra of the HMM-based metasurface. The structure can be fabricated on top of the HMM structure via standard electron beam lithography (EBL) [46]. A spectral splitting with two peaks can be clearly observed in the ER spectrum of the coupling system. For the static state F = 0 (black solid line), the short-wavelength peak has a narrow linewidth and lower value while the longer-wavelength peak has a broad linewidth and higher value. By increasing the pump fluence, a red shift is observed for both two peaks, with the shorter-wavelength peak becoming larger and narrower compared to the longer-wavelength peak. Figure 7(b) further presents the evolution map of the proposed coupling system over a large varying range of pump fluence. We find that the spectral positions and the values of both two peaks depend on the pump fluence. The energy is exchanged reversibly between the two peaks of the spectral splitting when tuning pump fluence, behaving similarly to Rabi oscillation. To understand these peaks, we simulate the electric field (|E|) distributions at F = 7.06 mJ/cm², the results are shown in Fig. 7(c). For the shorter-wavelength peak (peak 1), the field is significantly enhanced and located mainly on the antenna and in the HMM, meaning that the resonance of the antenna can localize the incident electric field into the HMM and leads to the antenna-HMM coupling. As for the longer-wavelength peak (peak 2), the electric field is confined near the antenna and on the lower surface of the HMM, with weaker field enhancement compared to peak 1, showing the main characteristics of localized surface plasmon resonance mode. Thus, the electric fields of two peaks show the coupling between the HMM and the antenna array leads to the two hybrid resonances and manifests as the splitting of ER spectra. We also study the switching performance of the proposed structure. Figure 7(d) shows the modulation depth of the coupling system for various pump fluence. The modulation depth spectra exhibit a spectrally oscillating line-shape, which arises from a pump-induced red-shift of ER spectra in Fig. 7(a).

Four modulation-depth peaks observed in the spectra correspond to the shorter-wavelength resonance for static state and pump irradiance, as well as the longer-wavelength resonance for static state and pump irradiance. The second peak has the largest modulation depth among the four peaks, for example, the structure produces a maximal modulation depth of ∼ 15 dB for F = 14.17 mJ/cm². We calculate the ΔT/T₀ (here, T₀ is the transmission of the structure with F = 0) transient spectral map for F = 14.17 mJ/cm². The results plotted in Fig. 7(e) shows that, similar to the dynamic response in the bulk HMM (c.f. Figure 5(c)), the hybrid resonances of static state give rise to two transient bleaches with a maximum up to 514%, while the resonances for pump irradiance produce two induced absorptions. The induced absorptions with Gaussian shape are attributed to the red-shift of the hybrid resonances under pump fluence. Figure 7(f) shows the femtosecond temporal response of modulation depth for various pump fluence at 1432 nm. The acquired ∼ 213 fs switching time in combination with the ∼ 60 fs rise time and ∼ 100 fs recovery time is evidence of the ultrafast dynamic controlling of coupling strength via the proposed coupling system.

6. Conclusions

We have developed an ultrafast all-optical switch based on nonlinear HMM platform in the telecommunication range. By utilizing the ITO with ENZ-induced nonlinearity, we have achieved dynamic control of the optical properties of the HMM platform under pump irradiance. The calculated results of the non-parabolic conduction band model and TTM indicated that the proposed HMM platform has the largest Kerr nonlinearity at its ENZ wavelength, which is higher than that of the bulk ENZ ITO. The realized modulation in the effective permittivity of the HMM at ENZ wavelength exhibited a 92-fs rise time and a 216-fs recovery time, along with a switching time of 393 fs and a modulation speed of at least 2.5 THz. Also, we showed the femtosecond transient topological transition of the HMM, and that increasing the pump fluence leads to a red-shift in the topological transition points.

For the bulk HMM, the static ER spectrum yields a significant peak arising from Ferrell-Berreman mode at the topological transition point, and its intensity depend strongly on the incident angle. The dynamic ER spectrum demonstrated the increment of pump fluence resulting in the red-shift of Ferrell-Berreman mode, which is confirmed by the loss function. Moreover, the pump-induced topological transition of the HMM lead to a significant modification in Purcell factor of an emitter near the HMM. Simulations show the change of Purcell factor excess of 400% can be happen at a switching time of 222 fs corresponding to 4.5 THz.

We also demonstrated the all-optical control of the coupling strength via depositing nanoscale antennas on the nonlinear HMM platform. It is found that the coupling between the HMM and the designed cylindrical antennas gives rise to two hybrid resonances and manifests as splitting two peaks in ER spectrum. The nonlinearity of the HMM platform allows tuning the optical response of the coupling system. Subsequently, the spectral shift and the energy exchange between two hybrid resonances can be observed under pump fluence, which behaves as a Rabi-like oscillation and signifies controlling the coupling strength. The proposed structure can realize a modulation depth of ∼ 15 dB with a 213-fs switching time or a 4.7-THz modulation speed. The working wavelength of the HMM platform demonstrated in this work can be easily extended into the mid-infrared range by using high mobility TCOs, and the performance can be improved after optimization of the fill fraction and the antennas. The results of this work open up a new avenue to explore the nonlinearity of ENZ multilayer structure and control the topological transition of hyperbolic metamaterials, and can be beneficial in broadening the applications of HMM in quantum phenomena, integrated logical photonic circuits as well as optical communication networks.

Funding

Basic and Applied Basic Research Foundation of Guangdong Province (No. 2021A1515012176); Shenzhen Fundamental Research Program (No. GXWD20201231165807007-20200827130534001); Youth Science and Technology Innovation Talent of Guangdong Province (2019TQ05X227).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Ultrafast dynamic switching of optical response based on nonlinear hyperbolic metamaterial platform

Abstract

Corrections

1. Introduction

2. Structural design

3. Nonlinear properties of HMM

4. Switching effect of optical response and Purcell effect in HMM

5. Dynamic all-optical control of the coupling strength via HMM

6. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (11)

Optics Express