Abstract
We show that phase-change materials can be used to switch photonic nanostructures between cloaking and superscattering regimes at mid-infrared wavelengths. More specifically, we investigate the scattering properties of subwavelength three-layer cylindrical structures in which the material in the outer shell is the phase-change material Ge2Sb2Te5 (GST). We first show that, when GST is switched between its amorphous and crystalline phases, properly designed electrically small structures can switch between resonant scattering and cloaking invisibility regimes. The contrast ratio between the scattering cross sections of the cloaking invisibility and resonant scattering regimes reaches almost unity. We then also show that larger, moderately small cylindrical structures can be designed to switch between superscattering and cloaking invisibility regimes, when GST is switched between its crystalline and amorphous phases. The contrast ratio between the scattering cross sections of cloaking invisibility and superscattering regimes can be as high as ∼ 93%. Our results could be potentially important for developing a new generation of compact reconfigurable optical devices.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, investigating the interaction of light with subwavelength structures has attracted a lot of attention, since it could potentially lead to a new generation of photonic devices [1–6]. In particular, the capability to control the scattering of light and achieve invisibility cloaking of subwavelength structures is important for applications in biomedicine, photovoltaics, sensing, optical detection, and near-field imaging [7–13]. In the past few years, the use of plasmonic and dielectric multilayer coatings to drastically reduce the total scattering cross-section of deep subwavelength objects, and thus achieve invisibility cloaking based on scattering cancellation, has been explored [14–18]. In addition to cloaking, it has been demonstrated that subwavelength multilayer core-shell structures can lead to enhanced resonant scattering, so that the scattering cross sections of the original structures are greatly enhanced [19–22]. This phenomenon is commonly referred to as superscattering. Switching between the cloaking and enhanced scattering states could be essential for building compact optoelectronic devices, for reducing the size of optical systems, and for developing reconfigurable optical components [23–25]. Such switching between the enhanced scattering and invisibility cloaking regimes has been demonstrated using nonlinear materials [26] and quantum emitters [27]. An alternative way to achieve this switching could be through the use of materials with tunable optical properties, such as phase-change materials.
Ge2Sb2Te5 (GST) is a phase-change material with an amorphous and a crystalline phase [28]. The covalently bonded amorphous phase of GST corresponds to a disordered material with short-range atomic order. In contrast, the resonantly bonded crystalline phase can be regarded as a semiconductor with orderly aligned atoms. Thus, the optical properties of amorphous GST (aGST) and crystalline GST (cGST) are significantly different. These two phases can be switched reversibly and rapidly by applying external electrical pulses, laser pulses or thermal annealing. Picosecond-order crystallization times have been reported for GST by femtosecond laser pulses [29,30]. Amorphization of GeSbTe has been achieved on subpicosecond timescales with femtosecond laser pulse excitation [31]. In addition to being inexpensive and easy to use in device fabrication processes, GST retains its phase for years after removal of the external excitations. GST has been widely used for non-volatile, rewritable optical data storage devices and for electronic memories [32, 33]. Recently, it has been shown that GST could provide a versatile platform for the realization of optically reconfigurable active photonic devices due to its switchable dielectric properties [34–40].
In this paper, we investigate the scattering properties of a three-layer cylindrical structure with GST at the mid-infrared wavelength of 4 μm. We first consider an electrically small structure. We show that, when GST is switched between its amorphous and crystalline phases, the structure switches between resonant scattering and cloaking invisibility regimes. The contrast ratio between the scattering cross sections of the cloaking invisibility and resonant scattering regimes reaches almost unity. We then consider the case of a larger, moderately small cylindrical structure. In this scenario, we demonstrate that, when GST is switched between its crystalline and amorphous phases, the structure switches between superscattering and cloaking invisibility regimes. The contrast ratio between the scattering cross sections of cloaking invisibility and superscattering regimes can be as high as ∼ 93%. Although here we focus on two-dimensional infinitely long cylindrical structures, the proposed approach is rather general and can be applied to other optical structures.
The remainder of the paper is organized as follows. In Section 2, we employ the full-wave Mie-Lorenz mode-expansion method to describe the scattering properties of the proposed three-layer cylindrical structure. Using this theory, we analyze the switching between the cloaking invisibility and enhanced scattering regimes achieved for electrically small and larger, moderately small cylinders in Subsections 3.1 and 3.2, respectively. Finally, our conclusions are summarized in Section 4.
2. Theory
Our proposed three-layer cylindrical structure is normally illuminated by a TM plane wave propagating in the x direction with the magnetic field polarized along the cylinder (z) axis, as illustrated in Fig. 1. The material in the outer shell is the phase-change material GST. Based on the Mie-Lorenz mode-expansion method [41–43], the expansions for the incident fields are given by
with expansion coefficients where H0 is the strength of the incident magnetic field, k0 is the wave number in free space, and ∊0 is the dielectric permittivity of free space. Mn and Nn are vector cylindrical harmonics of the n-th order [41]. The superscript (1) indicates that for the vector cylindrical harmonics the radial dependence of the fields is given by Bessel functions of the first kind Jn. The expansions of the fields in the core layer (Fig. 1) are where k1 is the wave number in the core layer, and ∊1 is the dielectric permittivity of the material in the core. The expansions of the fields in the inner shell with dielectric permittivity ∊2 (Fig. 1) can be expressed as where k2 is the wave number in the inner shell region. The superscript (2) indicates that for the vector cylindrical harmonics the radial dependence of the fields is given by Bessel functions of the second kind Yn. Similarly, the expansions of the fields in the outer shell region can be written as where k3 is the wave number in the outer shell, and ∊3 is the dielectric constant of GST. The scattered fields outside the three-layer cylindrical structure are given by The superscript (3) in the above equations indicates that for the vector cylindrical harmonics the radial dependence of the fields is given by Hankel functions of the first kind, Hn. By applying the boundary conditions at ρ = ρj, j = 1, 2, 3, we obtain the scattering coefficients an and bn Note that the scattering coefficients an vanish when the plane wave is normally incident on the cylindrical structure [41]. In Eq. (13), and are given by and where , j = 0, 1, 2, 3. All materials are non-magnetic, so that μj = μ0, j = 1, 2, 3. The total scattering cross section (SCS), defined as the ratio of the total scattered power to the intensity of the incident plane wave [12,15,41], is given by where λ0 is the free-space wavelength, and σN is the normalized scattering cross section (NSCS) [44]. Based on Eq. (13), leads to scattering suppression of the nth order multipole. Cloaking invisibility is achieved when all scattering coefficients bn simultaneously approach zero. On the contrary, resonant scattering of the nth order multipole occurs when , which provides an opportunity to dramatically enhance this scattering order. Based on electromagnetic duality, the scattering coefficients bn for a normally incident TE excitation are zero, while , where and can be readily obtained by replacing ∊ with μ in Eqs. (14) and (15), respectively [15,41].3. Results
In this section, we use the phase-change material GST in the three-layer cylindrical structure of Fig. 1, to switch this structure between cloaking and enhanced scattering regimes at the mid-infrared wavelength of λ0 = 4 μm. Mid-infrared radiation in the 3–5 μm wavelength range can propagate through several materials without significant intensity attenuation. Because of this property, there is a wide range of potential military and civil applications in the mid-infrared wavelength regime [45]. For the material in the core of the three-layer cylindrical structure (Fig. 1) we choose zinc oxide (ZnO). Zinc oxide microwires and nanowires have great potential to be used in many commercial applications due to their low cost and simple fabrication process [46–48]. We use the commercial software COMSOL, which is based on the finite-element method, to numerically calculate the SCS of the proposed structures.
3.1. Electrically small cylindrical structures
We first consider electrically small cylindrical structures (k0ρ3 ≪ 1). In this case, by using the asymptotic forms of the Bessel functions, the expressions in Eqs. (14) and (15) for n ≠ 0 can be reduced to the following [49]
and In addition, if the material in the core is dielectric and the structure is electrically small, the scattering cross section for TM excitation is dominated by the terms involving b±1, which are associated with dipolar scattering, and all other terms with n ≠ ±1 are negligible [15,50]. Note that |bn| = |b−n| [41]. Thus, in this case, by setting in Eq. (17), we obtain the following condition to achieve invisibility cloaking where and . Similarly, by setting in Eq. (18), we obtain the following condition to achieve enhanced scattering To realize switching of the three-layer cylindrical structure of Fig. 1 between cloaking and enhanced scattering regimes, requires when GST is in its crystalline phase with dielectric constant ∊3c, and also when GST is in its amorphous phase with dielectric constant ∊3a. In other words, when GST is in its crystalline phase with dielectric constant ∊3c, Eq. (19) should be satisfied, while, when GST is in its amorphous phase with dielectric constant ∊3a, Eq. (20) should be satisfied. Since the left hand sides of Eqs. (19) and (20) are equal, these two conditions can be simultaneously satisfied if ϕu(∊3 = ∊3c)=ϕv(∊3 = ∊3a). To achieve this, we optimize γ1 and the dielectric constant of the inner shell ∊2, to satisfy the following relation at the wavelength of λ0 = 4μmWe use experimental data for the frequency-dependent dielectric constants of aGST and cGST [28]. The refractive indices of aGST and cGST are ñaGST =4.05 and ñcGST =5.9 + 0.16j, respectively, at λ0 = 4μm [28,29]. Thus, at λ0 = 4μm cGST is lossy, while aGST is lossless. The dielectric constant of zinc oxide, which is the material used in the core (Fig. 1), is ∊1 = 8.15 at λ0 = 4μm.
We first neglect the loss of cGST at λ0 = 4μm. In other words, we assume that ∊3a = 4.052 = 16.4 and ∊3c = 5.92 = 34.81. In Fig. 2(a), we show |ϕu(∊3 = ∊3c) − ϕv(∊3 = ∊3a)| as a function of and the dielectric constant of the inner shell ∊2. In Fig. 2(a) we observe three regions in the γ1−∊2 space (marked as I, II, and III) at which Eq. (21) is satisfied. However, since , only corresponds to physical solutions. Figure 2(b) shows as a function of and the dielectric constant of the inner shell ∊2. We observe that region II [Fig. 2(a)] does not correspond to physical solutions, since is negative. On the other hand, regions I and III correspond to physical solutions. We consider a point in region III [indicated by an open circle in Figs. 2(a) and 2(b)] with , ∊2 = −1.25, and . Fig. 3(a) shows the NSCS σN corresponding to this point as a function of the dielectric constant of the outer shell ∊3 calculated with COMSOL. The radius of the ZnO core cylinder ρ1 is set equal to 23 nm, so that the three-layer structure is electrically small (k0ρ3 = k0ρ1γ1γ2 < 0.1) [50]. We observe a peak (σN ≃ 2|b1|2 = 1.9766) and a dip (σN ≃ 2|b1|2 = 1.01 × 10−12) in the scattering cross section for ∊3 ≃ 16.4 (aGST) and ∊3 ≃ 34.81 (cGST), respectively. Thus, the simulation results confirm that for the optimized electrically small cylindrical structure of Fig. 1 switching between cloaking and enhanced scattering regimes can be realized at the mid-infrared wavelength of 4 μm by switching the phase-change material GST between its crystalline and amorphous phases. In Fig. 3(a), we also observe a resonance with asymmetric Fano-like shape at ∊3 = 0. This is due to the strong interference between the cloaking and resonant scattering states within the same structure [51,52]. Based on Eqs. (19) and (20), the cloaking and resonant scattering conditions coincide with each other when ∊3 = 0, and thus a degenerate cloaking-resonant state is formed. Figure 3(b) shows the NSCS σN as a function of for the optimized structure when the phase-change material GST is in its crystalline phase. Although the material loss in cGST affects the total scattering excursion, cloaking invisibility is still achieved (σN = 4.2 × 10−9) when (red circles). The contrast ratio between the cloaking and resonant scattering states, defined as , is almost unity.
The switching between cloaking and enhanced scattering regimes can be observed in the field distributions inside and outside the cylindrical structure for , , ρ1 = 23 nm, and ∊2 = −1.25 (Fig. 4). When the plane wave is normally incident onto the optimized cylindrical structure with GST in its crystalline phase, the plane wave field distribution is not distorted at all by the presence of the structure [Figs. 4(a) and 4(b)]. On the other hand, when the plane wave is incident onto the optimized cylindrical structure with GST in its amorphous phase, the plane wave field distribution is severely distorted by the presence of the structure [Fig. 4(c)], and the incident wave excites a dipole mode supported by the structure [Fig. 4(d)]. In Fig. 2(a), we observe that in both regions I and III, which correspond to physical solutions, the dielectric permittivity of the inner shell ∊2 (Fig. 1) is negative. In general, an electrically small cylindrical structure exhibits a dipolar scattered field due to the electric dipole moment excited by the incident wave. The polarization vector in a shell with negative dielectric constant being antiparallel to those in other layers with positive dielectric constants may lead to the overall cancellation of dipole moment [14]. The dielectric permittivity of the inner shell has therefore to be negative to realize cloaking invisibility for our proposed optimized structure (Fig. 1) with GST in its crystalline phase. A material with dielectric constant ∊ being in the range −10 < ∊ < −0.5 at infrared and visible frequencies can be formed by embedding silver implants in a dielectric host with positive dielectric constant [53]. In addition, there exist natural low loss plasmonic materials with the required negative dielectric permittivity at mid-infrared wavelengths, such as highly doped InP [54]. Its permittivity and plasmonic properties can be tuned through the carrier concentration [54]. In addition, we note that, even in the presence of loss in the inner shell, the contrast ratio is still close to unity [26]. As an example, we found that, for ∊2 = −1.25 + 0.1j, the scattering cross section in the enhanced scattering regime is ∼ 0.1, while the scattering cross section in the cloaking regime is ∼ 8.0 × 10−7.
3.2. Moderately small cylindrical structures
When the physical size of the cylindrical structure increases, other Mie scattering order contributions to the overall scattering cross section also need to be taken into account [55]. Thus, Eq. (21) can no longer be used to determine the optimized geometrical and material parameters of the structure in Fig. 1 for achieving switching between cloaking invisibility and enhanced scattering regimes, when GST is switched from its crystalline to its amorphous phase. In general, cloaking a moderately small structure through multiple scattering orders occurs when the numerators of the corresponding scattering coefficients [Eq. (13)] become zero at the same frequency. In addition, superscattering, which corresponds to the scattering cross section of a moderately small structure being significantly enhanced, occurs when overlap of at least two different scattering resonance modes is achieved at the same frequency [19,20,44]. In this subsection, the radius of the core layer is set equal to 480 nm which is approximately 10 times larger than the radius of the optimized electrically small three-layer cylindrical structure in Subsection 3.1. In addition, the material of the inner shell is chosen to be TiO2 with dielectric constant which can be approximated by the following Sellmeier dispersion equation [56,57]
where the wavelength λ0 is in units of micrometers. Thus, the dielectric constant of TiO2 at the operating wavelength of λ0 = 4 μm is ∼ 5.2. To realize switching between cloaking and superscattering regimes for the moderately small cylindrical structure of Fig. 1, we now use Eqs. (14) and (15) (Section 2). More specifically, we optimize and (Fig. 1) to make the NSCS of the structure [Eq. (16)] as large as possible when GST is in its csystalline phase, and as small as possible when GST is in its amorphous phase at the wavelength of λ0 = 4 μm.In Fig. 5, we show the NSCS σN of the proposed moderately small core-shell cylindrical structure for different scattering orders as a function of and , when GST is in its crystalline phase [Figs. 5(a)–5(d)], and its amorphous phase [Figs. 5(e)–5(h)] at the wavelength of λ0 = 4 μm. For the range of parameters shown, the amplitudes of the higher order coefficients |bn| are negligible for |n| > 3. When the phase-change material GST is in its crystalline phase, the resonant scattering regime is broad in the γ1−γ2 space for n = 0, but narrow for |n| =1, 2, and 3 [Figs. 5(a)–5(d)]. Thus, it is very difficult to achieve overlapping of the dipole (|n| = 1), quadrupole (|n| = 2), and sextupole (|n| = 3) scattering modes. However, superscattering can be realized when there is overlap between the monopole (n = 0) and a higher order multipolar mode (dipole, quadrupole, or sextupole). Thus, superscattering occurs when a higher order multipolar scattering mode is on resonance. When the phase-change material GST is in its amorphous phase, the near-zero scattering regime is narrow in the γ1−γ2 space for n = 0, but quite broad for |n| =1, 2, and 3 [Figs. 5(e)–5(h)]. For the range of parameters shown, the cloaking condition for the proposed moderately small structure can therefore be approximately reduced to |b0|2 = 0. In other words, cloaking occurs when the amplitude of the monopole scattering coefficient |b0| is zero. Switching between superscattering and cloaking regimes can be achieved when superscattering for GST in its crystalline phase and cloaking for GST in its amorphous phase occur at the same point in the γ1−γ2 space. Figure 5 reveals that the optimized moderately small three-layer core-shell structure is in the superscattering regime for GST in its crystalline phase, and in the cloaking regime for GST in its amorphous phase at the wavelength of λ0 = 4μm, when γ1 = 1.99 and γ2 = 1.26 (white circle in Fig. 5). The radius of such a three-layer cylindrical structure is ρ3 = ρ1γ1γ2 = 1023.5 nm, which is .
To further illustrate the superscattering property of the optimized structure, we show the NSCS σN for different scattering orders as a function of with when GST is in its crystalline phase [Fig. 6(a)]. We observe that, when γ1 = 1.99, the monopole, quadrupole and sextupole modes overlap very well. In the lossless cGST case, the overall σN, which is calculated with full-wave finite-element simulations, can reach ∼ 3.29 [Fig. 6(a)]. In the presence of loss, the overall σN is reduced to ∼ 1.85 [Fig. 6(a)]. Similarly, to further illustrate the cloaking property of the optimized structure, we show the NSCS σN for different scattering orders when GST is in its amorphous phase [Fig. 6(b)]. We observe that the overall σN can be drastically suppressed for γ1 = 1.99, where the monopole mode contribution becomes zero, and the higher order multipolar mode contributions are very small. The corresponding overall σN can be as small as ∼ 0.067 [Fig. 6(b)]. The contrast ratio τ between the supercattering and cloaking states is .
The switching between cloaking and superscattering regimes at the wavelength of λ0 = 4 μm can be observed in the field distributions inside and outside the moderately small core-shell cylindrical structure of Fig. 1 for , , and ρ1 = 480 nm (Fig. 7). Figure 7(a) shows the magnetic field profile for a bare ZnO core with radius of ρ1 = 480 nm. In this case, the overall σN is ∼ 0.77. The scattering is dominated by the monopole mode [Fig. 7(b)]. In Figs. 7(c) and 7(d), we show the magnetic field distributions of the optimized three-layer cylindrical structure when GST is in its crystalline phase. When a plane wave is normally incident from the left, the core-shell cylindrical structure induces strong backscattering, and leaves a significant shadow in front of it, where the field strength is reduced [Fig. 7(c)]. The enhanced NSCS (∼ 1.85) is 2.4 times larger than that of the bare ZnO core (∼ 0.77). We observe that, when γ1 = 1.99, the monopole, quadrupole, and sextupole modes overlap very well. The field distribution inside the cylindrical structure is a superposition of the monopole, quadrupole, and sextupole mode fields [Fig. 7(d)], which is consistent with the results associated with superscattering in Fig. 6(a). The monopole modal fields reside mostly in the lossless ZnO core. The strong overlap of the resonant sextupole modal fields with the lossy cGST outer shell leads to a greatly reduced scattering cross section compared to the lossless cGST case [Fig. 6(a)]. Interestingly, even though the material loss in cGST highly affects the sextupole mode, it does not affect much the dipole and quadrupole modes [Fig. 6(a)]. This is due to the weak overlap of the dipole and quadrupole modes with the lossy cGST shell. In addition, the resonant sextupole mode has a higher quality factor than that of the dipole and quadrupole modes [Fig. 6(a)], which results in light being trapped in the structure for a longer duration. This in turn leads to higher power penalty [44,58]. Similarly, in Figs. 7(e) and 7(f) we show the magnetic field distribution of the optimized three-layer cylindrical structure when GST is in its amorphous phase. We observe that in this case there is hardly any scattering. The suppressed NSCS σN (∼ 0.067) is 11.5 times smaller than that of the bare ZnO core (∼ 0.77). Two different resonant modes can be observed in the core and shell regions [Fig. 7(f)]. These two excited resonant modes are out-of-phase and compensate each other in the far-field, resulting in scattering cancellation [16].
4. Conclusions
In this paper, we designed subwavelength three-layer core-shell cylindrical structures with the phase-change material GST for switching between the cloaking invisibility and enhanced scattering regimes at the mid-infrared wavelength of λ0 =4 μm. We used the Mie-Lorenz mode-expansion method and optimized the geometric and material parameters of the structures to achieve the switching. For an electrically small three-layer structure, we optimized the dielectric permittivity of the material in the inner shell and the layer dimensions, to switch between cloaking and resonant scattering by switching the phase-change material GST from its crystalline to its amorphous phase. We found that for the optimized structure the contrast ratio between the cloaking and resonant scattering states is almost unity. For larger, moderately small three-layer structures, we optimized the layer dimensions to minimize the scattering cross section when GST is in its amorphous phase, and maximize the scattering cross section when GST is in its crystalline phase. We found that cloaking occurs when the amplitude of the monopole scattering coefficient is zero, while superscattering occurs when a higher order multipolar scattering mode is on resonance. For the optimized structure, the contrast ratio between the cloaking and superscattering states can be as high as ∼ 93%. We also confirmed these findings with full-wave finite-element simulations.
As final remarks, we first investigated a two-layer cylindrical structure consisting of a core layer and a GST shell layer. We found that such a two-layer structure cannot achieve the same functionality as the three-layer structure. In other words, the inner shell between the core and the GST outer shell is necessary to achieve switching between cloaking and enhanced scattering regimes. This is due to the fact that for electrically small two-layer cylindrical structures, a shell with negative or near-zero permittivity is necessary to cloak a dielectric core based on scattering cancellation [14,15]. Thus, an electrically small two-layer structure consisting of a GST shell and a dielectric core cannot be used to realize switching between the cloaking and resonant scattering regimes. We also note that layered core-shell cylindrical nanostructures can be fabricated by chemical vapor deposition and sputter coating [59]. Light scattered from an individual cylindrical nanostructure can be detected using dark-field microscopy [60]. In addition, switching between cloaking and superscattering regimes using phase-change materials could also be generalized to three-dimensional spherical nanostructures [20]. Our results could be potentially important for developing a new generation of dynamically reconfigurable subwavelength optical devices.
Funding
National Natural Science Foundation of China (NSFC) (61605252); Hunan Provincial Natural Science Foundation of China (2017JJ3375); National Key Research and Development Program of China (2016YFC0102401); National Science Foundation (NSF) (1254934).
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