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Abstract
The wavelength of defect mode in all-dielectric photonic crystals (PCs) with a dielectric defect are blue-shifted as incident angle increases for both transverse electric and transverse magnetic (TM) polarized waves. The blue-shifted property of defect mode limits the design of some optical devices including omnidirectional optical filters and wide-angle polarization selectors. Here we introduce a hyperbolic metamaterial (HMM) layer as a defect into dielectric one-dimensional photonic crystals (1DPCs) to obtain an omnidirectional defect mode for TM polarized waves at near-infrared regimes. Since only one HMM layer is introduced, omnidirectional defect mode with transmittance as high as 71% can be realized. Because of the unusual angle-dependence of propagating phase in the HMM defect, the total phase for satisfying the resonance condition of defect mode can be unchanged in a wide-angle range at a fixed wavelength, which leads to the omnidirectional defect mode. Moreover, the manipulation of propagating phase can be generalized to the case of circularly polarized waves, and we obtain an omnidirectional defect mode for left-handed circularly polarized waves in 1DPCs with a chiral hyperbolic metamaterial defect. Nevertheless, the defect mode for right-handed circularly polarized waves is still blue-shifted. Such spin-selective omnidirectional defect mode can be utilized to greatly enhance circular dichroism in a wide-angle range up to 64.1°. Our structure facilitates the design of omnidirectional optical filters with a high transmittance and circular polarization selectors working in a wide-angle range.
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1. Introduction
Photonic crystals (PCs), artificial microstructures with periodic modulation in the electromagnetic parameters, such as permittivity, permeability or chiral parameter, have attracted researchers’ great interests, since they have significant applications in spontaneous emission control [1], localization of photons [2], and optical circuit devices [3]. The most typical characteristic of PCs is photonic bandgap (PBG), where light is totally reflected by the PCs [4,5]. If a defect is introduced into the PC, a kind of resonant mode called defect mode will occur inside PBGs, and as a result, the electromagnetic waves will be highly localized inside PCs [4,6]. Over the past ten years, the defect mode has been widely utilized in lasers [7], filters [8], fibers [9], nonlinear devices [10,11]. It is known that the defect mode is blue-shifted in all-dielectric one-dimensional photonic crystals (1DPCs) with a dielectric defect layer because of the blue-shifted property of the PBG required by the Bragg condition, i.e., the transmission peak shifts toward shorter wavelength as the incident angle increases. The blue-shifted (angle-dependent) defect modes will bring about some problems. For example, a defect mode that can filter electromagnetic waves in all incident angles at a fixed wavelength is very useful in practice [12], particularly in the case for a source of a Gaussian beam with a certain angular distribution. Therefore, an omnidirectional defect mode that is independent of incident angles is highly desirable in some applications. One feasible way of obtaining the omnidirectional defect mode is using angle-insensitive PBGs. Based on this idea, researchers introduce defects into zero-averaged-index PBGs [13–15] or zero-effective-phase PBGs [16–18], and obtain the omnidirectional defect modes in these angle-insensitive PBGs. However, the formation of these PBGs requires negative-index metamaterials or single-negative metamaterials, which pose a challenge in experiments, especially in near-infrared and visible wavelengths.
Recently, hyperbolic metamaterials (HMMs), as one kind of metamaterials with special iso-frequency curves, can manipulate the light-matter interaction [19–23] and have many potential applications, including long-range dipole–dipole interactions [24], sensor [25], spontaneous emission control [26,27], super-resolution imaging [28,29], wavefront control [30,31], and absorbers [32,33]. HMMs can be mimicked by a one-dimensional metal-dielectric (or plasmonic material-dielectric) stack with subwavelength unit cells at near-infrared or visible region [19]. To date, researchers have theoretically and experimentally realized the angle-insensitive PBGs in 1DPCs containing HMMs based on phase variation compensation effect between HMMs and dielectric [34,35]. This kind of PBGs can be utilized to design omnidirectional absorbers [33] and reflectors [35]. By introducing a dielectric defect into the 1DPCs in which one component of the unit cell is HMM, a nearly omnidirectional filter can be realized [34,36]. Nevertheless, this kind of structure contains multilayers of HMM and the loss from the metallic component of HMM would be large, which strongly decreases the transmittance of the omnidirectional filter.
In this paper, we introduce a HMM layer as a defect into all-dielectric 1DPCs and obtain an omnidirectional defect mode with a high transmittance. We firstly obtain the omnidirectional defect mode for transverse magnetic (TM) polarized waves in 1DPCs with a HMM defect. Since only one HMM layer is introduced, an omnidirectional filter with transmittance as high as 71% is realized at the wavelength of transmission peak at normal incidence (still 54% transmittance at oblique incidence up to 60°). The underlying physics of the omnidirectional defect mode is that the Fabry–Pérot resonance condition can be satisfied in a wide-angle range at a fixed wavelength due to the unusual angle-dependence of propagating phase in the HMM defect. Moreover, the omnidirectional defect mode can be extended to circularly polarized waves and we obtain an omnidirectional defect mode for left-handed circularly polarized (LCP) waves in 1DPCs with a chiral hyperbolic metamaterial (CHMM) defect. The CHMM has a hyperbolic iso-frequency curve for LCP waves and has an elliptical iso-frequency curve for right-handed circularly polarized (RCP) waves [37]. As a result, the CHMM defect is similar to the conventional defect for RCP waves and the corresponding defect mode for RCP waves is still blue-shifted. Such the spin-selective omnidirectional defect mode can be utilized to greatly enhance circular dichroism (CD) with a working angle range up to 64.1°, owing to the fact that the defect mode hardly shifts for LCP waves, while strongly shifts toward shorter wavelength for RCP waves. Our structure facilitates the design of omnidirectional optical filters with a high transmittance and high-efficiency circular polarization selectors working in a wide-angle range, which would be helpful for all-angle phase matching in coherent nonlinear optical process [38], angle-insensitive Rabi splitting [39], and polarization-selective optical components.
2. Omnidirectional defect modes in 1DPCs with a hyperbolic metamaterial defect
The proposed structure is shown in Fig. 1(a), which consists of all-dielectric PCs denoted by (AB)5 and a (chiral) hyperbolic metamaterial defect denoted by C. The whole structure is denoted as (AB)5C(BA)5. Here, A and B represent the layer of titanium dioxide (TiO2) and gallium arsenide (GaAs) with refractive indices of ${n_\textrm{A}}$ and ${n_\textrm{B}}$ and thickness of ${d_\textrm{A}}$ and ${d_\textrm{B}}$, respectively. We set ${n_\textrm{A}} = 2.39$ [40] and ${n_\textrm{B}} = 3.32$ [41] and choose the thickness of each layer as ${d_\textrm{A}} = 137\textrm{ nm}$ and ${d_\textrm{B}} = 112\textrm{ nm}$, respectively. Thickness of C layer is denoted by ${d_\textrm{C}}$. We firstly consider the metamaterial defect is a HMM defect. The iso-frequency curves of dielectrics (A and B) are closed circles, as seen by the blue and cyan solid lines in Fig. 1(b), respectively. However, the iso-frequency curves of HMM for TM polarized waves is a hyperbola, as shown by the red solid line in Fig. 1(b). This hyperbolic iso-frequency curve would provide a way to manipulate the propagating phase inside the defect, as we will see below.
Fig. 1. (a) Schematic of 1DPCs with a metamaterial defect. (b) Iso-frequency curves of dielectrics (A and B) and HMM for TM polarized waves. The cyan, blue and red solid lines represent the iso-frequency curves of A, B and C, respectively.
The HMM defect layer in Fig. 1 can be mimicked by a plasmonic material-dielectric multilayers with subwavelength unit cells, as shown in Fig. 2(a). The effective HMM is denoted by (EF)4, where E represents the GaAs and F represents the indium tin oxide (ITO) plasmonic material in the infrared region, whose relative permittivity can be described by the Drude model as [42]
Fig. 2. (a) Schematic of the HMM composed of subwavelength plasmonic material and dielectric layers. (b) Effective permittivity tensor of the HMM as a function of wavelength.
Based on the transfer matrix method (TMM) [43,44], we calculate the transmittance spectra as a function of wavelengths and incident angles for transverse electric (TE) and TM polarized waves, respectively, as shown in Fig. 3(a). Notice that in TE polarized waves case, the iso-frequency curve of the defect layer is a circle. It can be seen from Fig. 3(a) that the defect mode is almost angle-insensitive for TM polarized waves while it strongly shifts toward shorter wavelengths for TE polarized waves as the incident angles increase from 0° to nearly 90°. Figure 3(b) gives the transmittance spectra at four incident angles of 0°, 30°, 45°, and 60° for TM polarized waves. Remarkably, the peak values of transmittance (at the fixed wavelength of $\mathrm{\lambda =\ }\textrm{1314 nm}$) at 0°, 30°, 45°, and 60° reach 0.71, 0.69, 0.64 and 0.54, respectively, which can be utilized to design a high-efficiency omnidirectional filter. It should be pointed out that we can further tune the period number of PCs on both sides to change the quality factor of the transmission peak, which is very useful in practical applications. Besides, our structure would realize different working wavelength of omnidirectional filter by changing the layer thickness of materials. Moreover, we may introduce materials whose permittivity changes with temperature or external field to realize active tuning of working wavelength for the defect mode.
Fig. 3. (a) Transmittance spectra versus all incident angles for TM and TE polarized waves, respectively. (b) Transmittance spectra at 0°, 30°, 45°, and 60° for TM polarized waves, respectively.
Now we discuss the physical origin of this omnidirectional defect mode. Generally, the formation of the omnidirectional defect mode can be explained by the Fabry–Pérot resonance condition. A defect mode of 1DPCs will appear when the Fabry–Pérot resonance condition is satisfied. The corresponding condition is written as [36,45,46]
where ${\varphi _{left}}$ and ${\varphi _{right}}$ represent the reflection phases of the left and right side of 1DPCs, respectively. $2{\varphi _C} = 2{k_{Cz}}{d_C}$ represents the round-trip propagating phase within the defect layer. For the TM and TE polarized waves, the $2{\varphi _C}$ can be expanded asFig. 4. The phases of the ${\varphi _{left}} + {\varphi _{right}}$, $2{\varphi _C}$, and ${\varphi _{total}}$ as a function of incident angles for (a) TM and (b) TE polarized waves at $\mathrm{\lambda =\ }\textrm{1314 nm}$. The blue, red, and green solid lines represent ${\varphi _{left}} + {\varphi _{right}}$, $2{\varphi _C}$, and ${\varphi _{total}}$, respectively.
Here we compare our work with those in Refs. 36 and 46 from the design, mechanism and device performance. The performance of omnidirectional filter has two aspects. One is the peak value of the transmittance and the other is the quality factor of the transmittance. In general, our structure and the two structures in Refs. 36 and 46 can be considered as Fabry–Pérot cavity with two mirrors. However, the three structures have different properties. The mirrors in Ref. 46 are red-shifted metals and the cavity spacer is blue-shifted dielectric. The mirrors in Ref. 36 are red-shifted photonic band gaps and the cavity spacer (defect) is blue-shifted dielectric. In contrast, the mirrors in our structure are blue-shifted photonic band gaps and the cavity spacer (defect) is red-shifted HMM. The metal mirrors in Ref. 46 can ensure high transmittance, but cannot ensure high quality factor, owing to the limited reflectivity of the metals. Compared to metals, the reflectivity of the photonic band gap can be very high, which can boost the quality factor. Nevertheless, in Ref. 36 the red-shifted photonic band gaps are realized by 1DPC with many lossy HMM layers, which decrease the transmittance. In our structure, the blue-shifted photonic band gaps are realized by all-dielectric lossless 1DPCs. Only defect is one HMM layer, which would ensure high transmittance. Therefore, the mechanism of omnidirectional filter for our structure is different from those in Refs. 36 and 46 and particularly our omnidirectional filter can ensure high transmittance and high quality factor simultaneously. Notice that our structure can only realize omnidirectional defect mode for TM polarized waves. If HMM defect is replaced by negative refractive index material, it would realize omnidirectional transmittance for both TM and TE polarized waves.
In addition, based on TMM we calculate the electric field amplitude distribution $|E |$ of the structure (AB)5(EF)4(BA)5 at $\mathrm{\lambda =\ }\textrm{1314 nm}$ for four incident angles 0°, 30°, 45°, and 60°, respectively, as shown in Figs. 5(a)–5(d). In the calculations, the maximum electric field is normalized, namely ${\rm M}ax\textrm{(}|E |\textrm{)} = \textrm{1}$. The dashed green wireframes represent the HMM defect and the gray shaded regions represent ITO inside the HMM. One can see that the electric field patterns hardly change with the increase of the incident angle. Very interestingly, the peak values of the electric fields are in the dielectric layers rather than in the lossy plasmonic (ITO) layers. This is also the reason that we could obtain a very high transmittance for the omnidirectional defect mode. In practice, this omnidirectional filter with a high transmittance would be fabricated under the current magnetron sputtering [47] or electron-beam vacuum deposition technique [48].
Fig. 5. The electric field distribution in the structure (AB)5(EF)4(BA)5 at four incident angles (a) 0°, (b) 30°, (c) 45°, and (d) 60° corresponding to the defect mode in Fig. 3(b). The dashed green wireframes represent the HMM defect and the gray shaded regions represent ITO inside the HMM.
3. Omnidirectional defect modes in 1DPCs with a chiral hyperbolic metamaterial defect
In section 2, we introduce a HMM defect into 1DPCs and realize the omnidirectional defect mode for TM polarized waves. In fact, the phase tuning mechanism of defect can also be extended to manipulate circularly polarized waves in 1DPCs with a chiral defect. Here the chiral defect can be chosen as CHMM. Because of special iso-frequency curves, CHMM can realize unusual physical phenomena, such as topological photonic phases [49], Weyl points [50], surface waves [30], negative refraction [37], and unusual optical tunneling [51]. In practice, CHMM can be mimicked by a dielectric slab with metallic inclusions [37,49–51] or metallic units [52]. For example, researchers have realized unusual optical tunneling experimentally using metamaterials composed of an array of split ring resonators in conjunction with an array of metallic wires [51], which provides a method to realize a CHMM. It should be pointed out that if the metallic units are electric resonant units, the CHMM will be nonmagnetic and ${\mu _{Cx}} = {\mu _{Cz}} = 1$. The effective permittivity components and chiral parameter $\kappa$ of CHMM could be described as [53]
In Eq. (5), we choose ${\varepsilon _a} = 2$, ${F_e} = {F_\kappa } = 0.4$, ${\omega _e} = {\omega _\kappa } = 1.45\mathrm{\ \times 1}{\textrm{0}^{15}}\textrm{ Hz}$, and ${\gamma _e} = {\gamma _\mathrm{\kappa }} = 3\mathrm{\ \times 1}{\textrm{0}^{13}}\textrm{ Hz}$. Figures 6(a) and 6(b) give the real and imaginary parts of ${\varepsilon _{Cz}}$ and $\kappa$ as a function of wavelength, respectively.
Fig. 6. The effective real and imaginary parts of (a) ${\varepsilon _{Cz}}$ and (b) $\kappa$ as a function of wavelength, respectively.
The dispersion equations of the CHMMs and dielectrics can be given by [32,35]
The ${\varepsilon _i}$ represents the permittivity of A and B. The “$\mathrm{+}$” and “$\mathrm{-}$” represent RCP and LCP waves, respectively. Usually, the iso-frequency curves of dielectrics are circles for RCP and LCP waves. However, according to Eqs. (6) and (7), for CHMMs we can obtain elliptical iso-frequency curve for RCP waves as ${\mathrm{\rho }_ + } > 0$ while hyperbolic iso-frequency curve for LCP waves as ${\mathrm{\rho }_ - } < 0$. Here, we only consider the real parts of the parameters for the iso-frequency curves. In the wavelength region of $\textrm{1351}\mathrm{\ \sim 1800\ }\textrm{nm}$, ${\mathrm{\rho }_ - } < 0$ for LCP waves. Therefore, in this region we can obtain an open hyperbola for LCP waves, which is similar to the red solid line in Fig. 1(b).
Next, we replace the HMM defect in section 2 with the CHMM defect and choose ${d_\textrm{C}} = 385\textrm{ nm}$ to obtain omnidirectional defect mode for LCP waves at $\mathrm{\lambda =\ }\textrm{1383 nm}$. All the other parameters are unchanged. Based on the anisotropic transfer matrix method [54], we calculate the transmittance spectra versus all incident angles for LCP and RCP waves, respectively, as shown in Fig. 7(a). It is seen that the defect mode at $\textrm{1383 }\textrm{nm}$ is almost angle-insensitive for LCP waves. In contrast, the defect mode strongly shifts toward shorter wavelength as the incident angle increases for RCP waves. Such omnidirectional defect mode provides us another way to manipulate circularly polarized waves. Figure 7(b) shows the transmittance at $\textrm{1383 }\textrm{nm}$ for both LCP and RCP waves. It is clearly seen that the transmittance decreases slowly with the increase of the incident angle for LCP waves, while for RCP waves, the transmittance decreases sharply as the incident angle increases and has a low value in a wide-angle range. Figure 7(c) gives the corresponding CD value $[{\textrm{CD = }{{({{T_{LCP}} - {T_{RCP}}} )} / {({{T_{LCP}} + {T_{RCP}}} )}}} ]$. As the incident angle increases from 0° to 90°, the CD value increases rapidly and maintains a large value over a wide-angle range. Simultaneously considering the CD value and the transmittance, we define the angle region where $\textrm{CD} > 0.3$ and ${T_{LCP}} > 0.1$ as the efficient CD region. The efficient $\textrm{CD}$ region ranges from 16.2° to 80.3°, as shown by the red shadow region in Fig. 7(b). The working angle range of $\textrm{CD}$ reaches 64.1° and a maximum CD of 0.894 is obtained at 57.2°. As a result, our structure facilitates the design of omnidirectional optical filters for circularly polarized waves and circular polarization selectors working in a wide-angle range.
Fig. 7. (a) Transmittance spectra versus all incident angles for both LCP and RCP waves. (b) Transmittance at the wavelength of the defect mode ($\mathrm{\lambda =\ 1383}\textrm{ nm}$) as a function of the incident angle for both LCP and RCP waves. (c) Corresponding $\textrm{CD}$ versus incident angle.
4. Conclusions
In summary, we demonstrate omnidirectional defect modes in 1DPCs with a (chiral) hyperbolic metamaterial defect. Because of the unusual angle-dependence of propagating phase in the HMM defect, the total phase for satisfying the resonance condition of defect mode can be invariant in a wide-angle range at a fixed wavelength. The phase manipulation mechanism can be generalized from linearly polarized waves to circularly polarized waves, and we obtain an omnidirectional defect mode for LCP waves. Particularly, since only one HMM layer is introduced, omnidirectional defect mode with a transmittance as high as 71% can be realized. Moreover, our structure can greatly enhance circular dichroism in a wide-angle range up to 64.1°. Our work provides a route to design high-efficiency optical filters working for a Gaussian beam with all-angle components and high-efficiency circular polarization selectors working in a wide-angle range.
Funding
National Natural Science Foundation of China (Nos. 11774261 and 61621001); National Key Research and Development Program of China (Grant No. 2021YFA1400602).
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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