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Abstract
Epsilon-near-zero (ENZ) materials have recently been suggested as excellent candidates for constructing all-optical and electro-optical switches in the infrared. The performance of previously reported ENZ material-based optical switches, however, has been greatly hampered by the low quality– (Q-) factor of the ENZ cavity, resulting in a large required optical pump fluence or applied voltage, a large insertion loss, or a small modulation depth. Here, we propose a solution by integrating the ENZ material into a Bragg microcavity, such that the Q-factor of the coupled cavity can be dramatically enhanced. Using high-mobility Dysprosium-doped cadmium oxide (CdO) as the prototype ENZ material, we numerically show an infrared all-optical switch with its reflectance modulated from near-zero to 94% under a pump fluence of only 7 μJ cm−2, about a 59-time-reduction compared with a state-of-the-art Berreman-type cavity. Moreover, the high-Q coupled cavity can also be adopted to realize a reflective electro-optical switch. Its reflectance can be switched from near-zero to 89%, with a bias electric field well below the breakdown field of conventional gate dielectrics. The switching operation can further be extended to the transmission mode with a slightly modified cavity geometry, with its absolute transmittance modulated by 40%.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Tunable free space optical devices in the infrared are highly desirable, for a wide range of applications including free-space optical communication [1], hyperspectral sensing [2], and controlled thermal emission [3]. Recently, transparent conducting oxides (TCOs), such as indium tin oxide (ITO), have been found promising as the active material for constructing such infrared optical switches [4–14].
TCOs typically have high electron densities ranging from 1019 to 1021 cm−3, resulting in an epsilon-near-zero (ENZ) response, in the near- or mid-infrared spectral range [15]. Light-matter interactions, particularly the local electric field, can be dramatically boosted in TCOs near their ENZ wavelength [16–19]. In addition, the optical properties of TCOs can be actively tuned by optical pumping [4–8] or electric biasing [9–14]. With inter- or intra-band optical pumping, one can modulate the electron density [20] or the effective electron mass [5,7,8] of TCOs, thus tune their ENZ wavelength, leading to a large switching contrast with a response time down to hundreds of fs. However, the large required pump fluence of all-optical switches poses a major challenge for many practical applications. On the other hand, one can use the field effect to modify the electron density in TCOs with an applied voltage [21], and tune their ENZ wavelength within their depletion or accumulation depth typically on the order of 1 nm. Even with such a small tunable volume, large phase tuning has been demonstrated in TCO-based metasurfaces [11,14]. However, a large phase tuning based on TCOs is often accompanied by a large insertion loss. Furthermore, it remains difficult to achieve a large amplitude tuning, either in reflection or transmission geometries. Here, one of the primary impediments is the relatively low quality- (Q-) factors of these ENZ structures, which results from the large intrinsic material absorption, as well as the un-optimized radiative loss of the optical cavity.
In this work, we aim to address the limitations of existing ENZ-material-based optical switches by integrating the ENZ material into a Bragg microcavity, which we refer as “coupled cavity” from here on. The coupled cavity features a dual-band perfect absorption near the ENZ wavelength of the TCO material of choice, Dy-doped cadmium oxide (CdO), with a sharp resonance. We show numerically an all-optical switch and an electro-optical switch, both based on the coupled cavity, that modulate the reflectance of p-polarized incident light from near-zero to near-unity with a low optical pump fluence or a small applied voltage in the mid-infrared. We further present the design of a transmissive electro-optical switch, which also exhibits large switching contrast.
2. Results and discussion
2.1. Static optical response
The proposed coupled cavity is composed of an ENZ thin film sandwiched in a Bragg microcavity, as illustrated in Fig. 1(a). The Bragg microcavity is formed from a front and a back distributed Bragg reflector (DBR), both composed of Si and SiO2, and a defect cavity layer formed from MgO and Dy-doped CdO. The Si and SiO2 multilayer can be grown by low-pressure chemical vapor deposition (LPCVD) [22] or plasma-enhanced chemical vapor deposition (PECVD) [23], and CdO can be grown by high power impulse magnetron sputtering [24]. We use the particle swarm optimization (PSO) algorithm to identify the optimal cavity parameters to achieve the best optical switching contrast (see Methods). The 10-nm-thick ENZ material, Dy-doped CdO (with carrier density Ne = 6.45 × 1019 cm−3), serves as the active material. We choose CdO because of its high electron mobility and low optical loss comparing with other TCOs such as ITO [25,26]. The absorption spectrum of the structure as a function of the incident angle θ is calculated by the transfer matrix method, and is illustrated in Fig. 1(b). The coupling between the CdO layer and the Bragg microcavity leads to a splitting of the cavity resonance near the ENZ wavelength. As a result, narrow-band perfect absorption occurs at two wavelengths. The dual band perfect absorption could be particularly useful for all-optical switching, since it could enable one to separate the pump and probe beam both spectrally and spatially. The Q-factor of the perfect absorption resonance is 274 at the probe wavelength. Figure 1(c) shows the squared magnitude of the electric field in the coupled cavity at the perfect absorption resonance with θ = 45°. A large enhancement factor of 180 is achieved in the CdO layer. Figure 1(d) plots the permittivity of the CdO layer, with its imaginary part to be 0.25 at the ENZ wavelength. In comparison, Fig. 1(e) presents a schematic diagram of a “Berreman cavity” [27,28], which is composed of a 240-nm-thick Dy-doped CdO layer on top of a gold substrate. The primary motivation of this work is to achieve high contrast optical switching. Such that when designing the optical cavity, we set a criterion that the cavity needs to show “perfect” (>99%) absorption in the static case, in order to achieve a good ON/OFF ratio of the switch. We also fix the incident angle at θ = 45° to be fair in comparison. As shown in Fig. 1(f), the Berreman cavity requires the CdO layer to be at least 240-nm-thick in order to achieve >99% absorption. The corresponding absorption spectrum and field distribution are illustrated in Fig. 1(g) and Fig. 1(h), respectively. The Q-factor and the field enhancement factor at the perfect absorption wavelength are 12 and 8, respectively, both much smaller than the corresponding parameters of the coupled cavity.
Fig. 1 Static optical response. (a) Schematic diagram of the high-Q perfect absorber based on the coupled cavity. The dimensions are: tSi = 291.6 nm, tSiO2 = 714.3 nm, tCdO = 10 nm, tf = tb = 586.2 nm. (b) Absorption spectrum of the coupled cavity calculated by the transfer matrix method. The black dashed line indicates the ENZ wavelength of 3.84 μm. The wavelengths and incident angles of the pump and probe beams are labelled by the white arrows. (c) |E|2 distribution (normalized to the incident electric field intensity |E0|2) as a function of the wavelength and position. The inset is the zoom-in view of the white dashed rectangle. (d) The permittivity of CdO used as the ENZ material. (e) Schematic diagram of the perfect absorber based on the Berreman cavity. The thickness of the CdO layer is 240 nm. (f) The maximum absorption of the Berreman cavity as a function of the thickness of CdO. (g) Absorption spectrum of the Berreman cavity calculated by the transfer matrix method. The black dashed line indicates the ENZ wavelength of CdO. The wavelengths and incident angles of the pump and probe beams are labelled by the black arrows. (h) |E|2 distribution (normalized to the incident electric field intensity |E0|2) as a function of the wavelength and position.
We adopt the coupled mode theory [29] to analyze the formation of high-Q perfect absorption of the coupled cavity. For an optical cavity, the stored energy dissipates through either the non-radiative or radiative loss. The non-radiative loss is a result of the intrinsic material absorption, and the radiative loss is due to the incomplete “mirror” reflection. The absorption of an optical cavity can be expressed as
where ω is the angular frequency of the incident light, ω0 is the resonant angular frequency of the cavity, γR and γNR represent the radiative and non-radiative loss rate of the cavity, respectively. Perfect absorption occurs under the critical coupling condition, where γR and γNR are equal, and when the cavity is driven under the resonant condition (ω = ω0). Figure 2(a) shows the loss rate of the coupled cavity as a function of the wavelength, while Fig. 2(b) illustrates corresponding absorption spectrum calculated by Eq. (1). Perfect absorptions occur at the twowavelengths where γR and γNR are equal. The loss rate and the absorption spectrum of the Berreman cavity are shown in Figs. 2(c) and 2(d), in which perfect absorption occurs only at one wavelength. The Q-factor of the perfect absorption resonance in the Berreman cavity is much smaller than the coupled cavity, as a result of the much higher overall loss rate. The large γR of the Berreman cavity results from the low reflectance at the CdO-air interface. The dual-band perfect absorption in the coupled cavity can be explained by the single peak of γNR near the ENZ wavelength. The γNR in the coupled cavity is induced by the absorption of the CdO layer, while the CdO layer itself can be regarded as a cavity, leading to a peak in γNR of the coupled cavity near its ENZ wavelength. Meanwhile, the introduction of the Bragg cavity reduces the radiative loss rate to be comparable with its non-radiative loss rate. As a result, the critical coupling condition can be fulfilled at two wavelengths, which leads to the formation of the dual band perfect absorption.Fig. 2 Theoretical interpretation of the perfect absorption. (a) The radiative and non-radiative loss rate of the coupled cavity. The blue circles mark the pump and probe beam wavelengths. (b) Absorption spectrum of the coupled cavity calculated by the coupled mode theory. (c) The radiative and non-radiative loss rate of the Berreman cavity. The blue circle marks the wavelength of the probe beam. (d) Absorption spectrum of the Berreman cavity calculated by the coupled mode theory.
2.2. All-optical switching
The high-Q coupled cavity can be used for low-power all-optical switching upon intra-band photo-excitation. The physical mechanism is schematically shown in Fig. 3(a), and has been extensively studied in pervious literature [5–8] (also see Methods). In short, the photo-induced heating of the electrons in CdO can lead to a red-shift of its plasma frequency ωp and its ENZwavelength. By assuming a pump beam with wavelength λ = 3.960 μm and θ = 15.9°, we plot in Fig. 3(b) the reflectance spectra of a probe beam with θ = 45°, as a function of the pump fluence. The absolute reflectance of the probe beam at λ = 3.619 μm is modulated from near-zero to 94% with a pump fluence of only 7 μJ cm−2. The near-unity reflectance in the “ON” state implies a negligible insertion loss of the switch. We compare the change in the absolute reflectance ΔR as a function of the pump fluence for the coupled cavity and the Berreman cavity, respectively, as shown in Fig. 3(c). For the Berreman cavity, to achieve a ΔR of the probe beam of 94%, the required pump fluence dramatically increases to 414 μJ cm−2. The lower required pump fluence of the coupled cavity can be primarily attributed to the much thinner CdO layer in order to achieve perfect absorption, as well as the much enhanced cavity Q-factor.
Fig. 3 All-optical switching. (a) Schematic illustration of the electron dynamics in CdO upon intra-band photoexcitation. The black dashed line represents the parabolic approximation. (b) Reflectance spectra of the coupled cavity at a 45° incident angle with different pump fluences. In the coupled cavity, the probe wavelength is 3.619 μm and the pump wave is 3.960 μm. (c) ΔR as a function of the pump fluence for the coupled cavity and the Berreman cavity, respectively. The blue dashed lines indicate ΔR of 94%. The inset is a zoom-in view of ΔR at low pump fluences. In the Berreman cavity, the probe wavelength is 3.802 μm and the pump wavelength is 3.840 μm.
2.3. Electro-optical switching
In order to construct a solid-state electro-optical switch, as shown in Fig. 4(a), we slightly adjust the coupled cavity to facilitate carrier depletion and accumulation in the CdO layer in a metal-oxide-semiconductor (MOS) configuration. Another 10-nm-thick CdO layer with a relatively lower carrier concentration (Ne = 2 × 1019 cm−3) is inserted as an electrode and a 5-nm-thick HfO2 film is added as a gate dielectric. Because the ENZ wavelength of the CdO electrode layer (termed “CdO1”) is far red-shifted from the ENZ wavelength of the active CdO layer (termed “CdO2”), it is essentially transparent at the operation wavelength of the electro-optical switch. We also adjust the thickness of two MgO layers accordingly, to make sure that the modified cavity can still achieve perfect absorption with θ = 45°.
Fig. 4 Electro-optical switching. (a) Schematic diagram of the electro-optical switch based on the coupled cavity. The thickness of the MgO layer is 567.2 nm; the thickness of the two CdO layers are both 10 nm; and the thickness of the HfO2 layer is 5 nm. (b) The purple line illustrates the dependence of λENZ on Ne. The red and black dashed line mark the doping levels of CdO1 and CdO2 layers. The inset shows a schematic diagram of the field effect in the CdO1/ HfO2/CdO2 layered structure. (c) Electron density distribution near the CdO1/HfO2 interface with different applied voltages. (d) Electron density distribution near the CdO2/HfO2 interface with different applied voltages. (e) Reflectance spectra of the reflection-mode electro-optical switch with different applied voltages. (f) The dependence of ΔR on the applied voltage for the coupled cavity, the Berreman cavity with 10-nm-thick CdO, and the Berreman cavity with 240-nm-thick CdO, respectively. (g) Transmittance spectra of the transmission-mode electro-optical switch as a function of the applied voltage.
With a positive applied voltage (as defined in the inset of Fig. 4(b)), electrons are depleted in the active CdO layer, and accumulated in the electrode layer. As shown in Fig. 4(b), the change in Ne of CdO leads to a change of its ENZ wavelength. Figures 4(c) and 4(d) illustrate the simulated electron density distribution in the electrode layer and the active CdO layer as a function of the applied voltage (see Methods). The reflectance spectra of the structure with θ = 45° as a function of the bias voltage are illustrated in Fig. 4(e). With a bias voltage of 3 V, the absolute reflectance at λ = 3.611 μm can be modulated from near zero to 89%. At such a voltage, the corresponding electric field in the HfO2 layer is 3.2 MV cm−1, which is well below the reported breakdown field of HfO2 [30,31]that varies from 4 to 8.5 MV cm−1.
For comparison, we also design similar MOS-type electro-optical switches in two sets of Berreman cavities, with the thickness of the CdO layer set to 10 nm and 240 nm, respectively.
As is evident from Fig. 4(f), the modulation performance of the coupled cavity is far superior. This can be understood as follows: For a Berreman cavity with 10-nm-thick CdO, one could not achieve perfect absorption, since the γR and γNR of the cavity are unbalanced at all incident angles. For a Berreman cavity with 240-nm-thick CdO, the modulation depth is limited by the depletion depth of CdO, which is much smaller than the overall CdO layer thickness.
The operation of the electro-optical switch based on the coupled cavity can be further extended to the transmission mode. The periods of both the front and the back DBRs are optimized to be 4 to allow the efficient modulation of transmittance with θ = 45°. As illustrated in Fig. 4(g), with an applied voltage of 3 V, the transmittance of the device at λ = 3.621 μm can be modulated from 4% to 44%. For a symmetric Bragg microcavity, it allows up to unity transmittance under resonant condition due to the constructive interference of multiple pathways of the incident light. However, in the coupled cavity, the CdO layer introduces appreciable loss near its ENZ wavelength, leading to a low transmittance. With an applied voltage, the CdO layer becomes more transparent, resulting in the increased transmittance.
3. Conclusion
In conclusion, we show that by coupling an ENZ material, Dy-doped CdO, to a Bragg microcavity, the Q-factor of the coupled cavity can be greatly boosted, leading to an improved performance for ENZ-material-based all-optical and electro-optical switches. The proposed planar thin-film stack architecture is relatively straightforward for the experimental implementation. Such a high contrast optical switch can readily find applications in free-space optical communication, narrowband photo detection, and ultrafast control of thermal emission. Similar concept can also be extended to ENZ-material-based switches in guided wave optics [32,33], where the ENZ material can be coupled to high Q micro-ring resonators [34].
4. Methods
4.1. Transfer matrix method
We develop a MATLAB code based on the transfer matrix method [35] to calculate the angle-dependent reflectance and absorption spectra of multilayer stacks. The permittivity of CdO and gold can both be described by the Drude formula:
where ε∞ is the high-frequency dielectric constant, ωp is the plasma frequency and Г = e / (m* μ) is the Drude damping rate. Here e is the elementary charge, m* is the ensemble-averaged effective electron mass, and μ is the electron mobility. We use ε∞ = 5.5, μ = 500 cm2 V−1 s−1 for CdO [25,36], and ε∞ = 1, ωp = 1.372 × 1016 rad s−1, and Γ = 0.405 × 1014 rad s−1 for the gold substrate [37]. The ωp and m* of CdO are fully determined by Eq. (4), Eq. (6) and Eq. (7) as will be detailed in section 4.5. The refractive indices of the dielectric materials used in the calculation are 1.706 for MgO, 3.43 for Si, 1.4 for SiO2, and 2.03 for HfO2.4.2. Coupled mode theory
We use the coupled mode theory [38] to analyze the absorption mechanism of the coupled cavity and the Berreman cavity. The back DBR of the coupled cavity and the Au substrate are regarded as perfect mirrors. Therefore, the radiative loss δR in both cavities can be estimated by δR = −0.25 ln (Rf), where Rf represents the reflectance of the front DBR for the coupled cavity, and Rf represents the reflectance of CdO/air interface for the Berreman cavity.
For the coupled cavity, we calculate the non-radiative loss δNR as δNR = −0.5ln(1-ACdO), where ACdO is the absorption of an individual CdO layer. The single-pass optical path in the MgO/CdO/MgO stack can be expressed as l = nMgO (tf + tb) cosθMgO + (φf + φb + 2φCdO) λ /(4π), where nMgO is the refractive index of MgO, θMgO is the angle of refraction in MgO, φf is the phase shift during the reflection from front DBR, φb is the phase shift during the reflection from back DBR, and φCdO is the phase shift induced by the CdO layer.
The δNR of the Berreman cavity is caused by the absorption of the Au substrate and the CdO layer. We neglect the absorption in Au and estimate the non-radiative loss δNR as δNR = Im(nCdOtCdOcosθCdO), where θCdO is the complex angle of refraction in CdO. The single-pass optical path in CdO can be expressed as l = Re(nCdOtCdOcosθCdO) + (φf + φb) λ /(4π), where φf is the phase shift upon reflection from the CdO/air interface, φb is the phase shift upon reflection from the CdO/Au interface.
The resonance condition is expressed as N λ = l, where N can be any arbitrary integer. The loss rate γ of both the coupled and the Berreman cavities can relate to δ as γ = δ*c/l, where c is the speed of light in vacuum.
4.3. FDTD simulations
The electric field distributions in Figs. 1(c) and 1(h) are calculated using a commercial FDTD simulator by Lumerical Solutions, Inc. We used the Broadband Fixed Angle Source Technique (BFAST) sources, and set BFAST boundary conditions in the x- and y-direction, and perfectly match layer (PML) in the z-direction for all simulations. This technique allows us to attain broadband simulation results at a fixed incident angle. The material properties used in FDTD simulations are consistent with those used in the transfer matrix method.
4.4. Device simulations
The carrier distribution as a function of the applied voltage (Figs. 4(c) and 4(d)) is calculated using a commercial DEVICE simulator by Lumerical Solutions, Inc. The interface between metal electrode and Dy-doped CdO layer is set as Ohmic contact. CdO is set to have a band gap Egap of 2.2 eV, electron affinity of 4.51 eV, and electron mobility of 500 cm2 V−1 s−1. The DC permittivity of CdO and HfO2 are set to be 21.9 and 25, respectively [36,39,40].
4.5 Numerical modelling of all-optical switching in CdO
The dispersion of conduction band electrons in CdO follows the relation [41]:
where is the reduced Planck’s constant, E is electron energy with respect to the conduction band minimum (CBM), k is the electron wave vector, m is the effective mass of electron at the CBM, and 1/Eg is a coefficient that describes the band non-parabolicity (Note Eg does not represent the band gap of CdO.). Here, we use m = 0.12 me and Eg = 2.92 eV for CdO [42]. Electrons in CdO follows the Fermi-Dirac distribution f0 when thermalized. The electron temperature Te, chemical potential μc, electron energy density U, and Ne of CdO are related from the linearized collisionless Boltzmann equation [7]: Once one of these physical parameters is known, we can derive a definite relation among other parameters. Since the electron density Ne is conserved upon intra-band optical pumping, we can obtain the dependence of Te, μc, ωp on the electron energy density U and the pump fluence.Due to the non-parabolicity of the conduction band, the ensemble-averaged effective electron mass m* become k-dependent, and can be given by,
4.6. Particle swarm optimization algorithm
To optimize the performance of the all-optical switch and the transmission-mode electro-optical switch, we use the PSO algorithm to pick the proper parameters for the DBR and CdO. PSO is a population-based stochastic algorithm introduced by Kennedy and Eberhart [43], which is based on social–psychological principles. We develop the PSO algorithm which is suitable for our optimization task based on MATLAB codes released by M. Kelly. The population per iteration is set as 20 and the maximum iteration is set to 40. The optimization is set to converge when the variance is smaller than 10−30. To reduce the required pump fluence for the all-optical switch, we set the variables including ωp of CdO at the static and the photo-excited state, and the periods of the DBR. The target in the optimization is to minimize Δωp for |ΔR| > 90%. To enhance the modulation depth of the transmission-mode electro-optical switch, we only choose the periods of the DBR as the variable, and others parameters remain identical to the corresponding parameters of the reflection-mode electro-optical switch. The optimization target is to maximize the change in the absolute transmittance with an applied voltage of 3 V.
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