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Abstract
We theoretically and experimentally investigate the angle-dependent omnidirectional photonic bandgap (PBG) in one-dimensional photonic crystals (PCs) comprising hyperbolic metamaterials (HMMs) for TM polarization, which is different from blue-shifted PBG in conventional all-dielectric photonic crystals. The frequency range of PBG increases when the incident angles increase, owing to the red-shift and blue-shift of the long-wavelength and short-wavelength band edges, respectively. The red-shifted band edge originates from the phase-variation compensation mechanism between the HMMs and dielectric material. The experimental values are in good agreement with the simulation results. These nanostructures are ideal for fabricating photonic devices such as omnidirectional reflectors.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Photonic crystals (PCs) [1,2] have gained considerable attention since their development due to their capability to manipulate optical propagation. The photonic bandgap (PBG) is a crucial property of PCs, and the optical propagation is prohibited in the waveband of the PBG. Due to these properties, PCs are used to fabricate optical devices, such as reflectors [3–5], lasers [6,7], fibers [8,9], and tunable optoelectronic devices [10–12]. Omnidirectional reflectors play an essential role in optical systems [3,4,13]. However, the PBG of PCs comprising dielectric films depends on the incident angles. As the incident angle increases, the propagation phase in the dielectric materials decreases. Therefore, the PBG shifts toward shorter wavelengths since the iso-frequency curve of dielectric materials is circular or elliptical. Hence, the applications of PCs with blue-shifted PBGs in fabricating omnidirectional reflectors are limited.
Many studies were conducted to investigate omnidirectional broadband PBGs [14–17]. Metamaterials [18] have extraordinary electromagnetic properties and are used to fabricate PCs to with different PBGs [19–21]. The zero-averaged index gap is omnidirectional in PCs composed of alternative dielectric materials and negative-index metamaterials, independent of the incident angle [16,22]. The zero effective phase gap was angle-independent in PCs comprising alternative permittivity-negative and permeability-negative metamaterials [23,24]. In recent years, hyperbolic metamaterials (HMMs) have attracted significant attention [25–32]. The iso-frequency curve of HMMs is hyperbolic; therefore, their propagation phase increases with an increase in the incident angle, contrary to that in conventional dielectric materials. Based on the phase-variation compensation effect between the HMM and conventional dielectric material, the PCs composed of alternative HMM and dielectric medium have angle-independent and red-shifting PBGs [33,34]. If the propagation phase variation in a unit cell containing an HMM and a dielectric film is opposite for the two band edges of the PBG, the PCs can exhibit a new type of angle-dependent omnidirectional broadband PBG, where the long-wavelength and short-wavelength band edges are red-shifted and blue-shifted, respectively. Compared with two- and three-dimensional nanostructures, one-dimensional PCs have simpler structures which are relatively easier to be fabricated, so we investigate omnidirectional PBG in one-dimensional PCs containing HMMs in this paper.
This paper is organized as follows. Section 2 presents the theoretical investigation of the angle-dependent omnidirectional PBG in PCs composed of an alternative HMM and the dielectric film. Section 3 discusses the fabrication of PC samples and the experimental investigation of omnidirectional PBG relative to the measured transmission spectra. The conclusions are presented in last section.
2. Theoretical investigation of omnidirectional PBG in PCs comprising HMMs
A conventional one-dimensional PC, (AB)N, where N is the periodic number, was considered. The dielectric materials SiO2 and HfO2 were denoted as A and B with refractive indices of nA = 1.43 and nB = 2.10 [35] and thicknesses of dA and dB, respectively. We set nAdA = nBdB = λ0/4, where λ0 = 310.0 nm is the center wavelength of the first PBG. The transmission spectra of truncated PC (AB)10 based on the transfer matrix method [36] with respect to incident angles are presented in Fig. 1, where only transverse magnetic (TM) waves are considered and the magnetic field (H) direction is kept to parallel with the surface of the PC. The band edge wavelengths of the PBG are blue-shifted when the incident angles are increased, which is similar to that of the omnidirectional reflection band in conventional PC; [3] this phenomenon originates from multiple Bragg conditions. Moreover, the mechanism can be explained by the iso-frequency curves of the media in the PC. Media A and B are isotropic dielectrics with dielectric constants εA and εB, respectively. The iso-frequency curves are shown in Fig. 2(a). As the incident angle of light increases, the horizontal component, kx, of the incident wave vector increases, and the components kAz and kBz of the wave vector decrease simultaneously. Therefore, the PBG of PCs composed of isotropic dielectrics is blue-shifted due to an increase in the incident angle, as shown in Fig. 1. Otherwise, the waveband of the PBG narrows with the increase of the incident angle for TM waves. Further, we consider a PC [AB]N comprising HMMs, where medium A is an HMM and B is a conventional dielectric material. The iso-frequency curves of layers A and B are shown in Fig. 2(b). It was observed that kBz in layer B decreased while kAz in layer A increased as kx increased. Therefore, there is a phase variation compensation effect in the unit cell composed of the HMM and dielectric layers, which changes the shifting behavior of the band edge in the PC as the incident angle increases [33,34].
Fig. 2. Iso-frequency curves of two types of media in (a) all-dielectric PC and (b) the PC containing layered HMM and dielectrics.
To examine an entire PBG, the long-wavelength band edge is engineered to be red-shifted, but the short-wavelength band edge is blue-shifted with the increase of the incident angle. The shifting of the band edges can be explained by the Bragg condition for the first PBG, which is given by
where Φ denotes the propagating phase in a unit cell of the PC, kAz and kBz denote the z component of the wave vectors in layers A and B, and λBragg represents the Bragg wavelength. For isotropic dielectric layers A and B, as shown in Fig. 2(a), kA(B)z decreases with the increase of kA(B)x, which leads to a decrease in the band edge wavelengths to satisfy the Bragg condition; thus, the PBG in the PC composed of isotropic dielectrics is blue-shifted, as illustrated in Fig. 1. However, for the type-I electric HMM layer A, as shown in Fig. 2(b), kAz increases as kA(B)x increases. Therefore, the phase variations ($\Delta \Phi \textrm{ = }\Delta {k_{Az}}{d_A} + \Delta {k_{Bz}}{d_B}$) in the unit cell of the PC can be positive or negative with the increase of the incident angle for the TM wave. To examine the angle-dependent omnidirectional PBG, the propagating phase Φ at the long-wavelength band edge must increase with an increase in the incident angle. The following condition must be satisfied to ensure that the band-edge wavelength undergoes a red-shift.In this study, the type-I HMM layer A is a subwavelength metal-dielectric multilayer denoted as (DM)S, as shown in Fig. 3(a), where D and M denote the dielectric material and metal, respectively, and S is the periodic number. The structure of the PC (AB)N containing HMMs is expressed as [(DM)SB]N. The dielectric layers D and B constitute HfO2. The metal layer M is composed of Ag. The Lorentz–Drude model is used to describe the permittivity of Ag, given by
The effective permittivity tensors of HMM (DM)S were obtained from the effective medium theory [25,27,37], as shown in Fig. 3(b). The filling ratio of M layer was 0.58, where the filling ratio is the thickness ratio of M layer and subwavelength unit (DM). The shadow region corresponds to the wavelength range of the HMM, where the conditions εx > 0 and εz < 0 are satisfied within 308–393 nm. If the filling ratio is changed, the wavelength range will be altered. These results indicate that the subwavelength metal-dielectric multilayer is perceived as the HMM layer only in a specific wavelength region. Based on these observations, the PBG in the PC comprising HMMs is considered to be angle-dependent and complete, where the long-wavelength (low-frequency) band edge is red-shifted within the wavelength range of the HMM, and the short-wavelength (high-frequency) band edge is hypsochromic.
Fig. 3. (a) Schematic of a PC containing HMMs. The HMM is denoted by a subwavelength HfO2/Ag multilayer. (b) Effective permittivity tensors of the HMM (DM)S as a function of the wavelength where S is the periodic number.
Based on the phase variation compensation condition [32], appropriate structural parameters can be selected, where the long-wavelength band edge can red-shift with an increase in the incident angle. In the experiment, λBragg = 310 nm was chosen, and the thicknesses of the HMM layer dHMM = 38 nm and dielectric layer dB = 40 nm were obtained according to the phase variation compensation condition [32]. For S = 1, dD = 16 nm, and dM = 22 nm. To systematically analyze the complete PBG, the dispersion relation of the PC composed of HMM and dielectric layers was simulated for TM waves, as illustrated in Fig. 4; the metal loss in the HMM is neglected. An angle-dependent complete PBG appears in the frequency range of 880–1070 THz (wavelength range of 280–340 nm), between the dashed blue lines in Fig. 3(b). As shown in Fig. 4, the high-frequency (short-wavelength) band edge is blue-shifted with the increase in the wave vector component kx, which indicates an increase in the incident angle for TM polarization. However, the low-frequency (long-wavelength) band edge moves toward a lower frequency, contrary to the hypsochromic band edges in the conventional all-dielectric PCs [3], as shown in Fig. 1.
Fig. 4. Dispersion relation of PC [(DM)SB]N for TM polarization. The white and black regions correspond to the PBGs and passbands, respectively. The dashed blue lines indicate the frequency range of omnidirectional PBG.
3. Experimental demonstration of omnidirectional PBG in PCs containing HMMs
According to the formulated parameters, the PC [(DM)B]4 containing the HMMs was fabricated on a SiO2 substrate (denoted by S) using ion-assisted electron-beam evaporation technique under high vacuum conditions. The scanning electron microscopy (SEM) image of the PC sample by a nova NanoSEM 450 is shown in Fig. 5. Employing air (SiO2) as the incident (exit) medium, the simulated transmission spectra of the truncated PC containing the HMMs at different incident angles for TM waves, as presented in Fig. 6(a) were obtained. An angle-sensitive omnidirectional PBG is identified within the range of 265–350 nm in the spectra, and the Q-factor is 3.62. The long-wavelength band edge is red-shifted with an increase in the incident angle but the short-wavelength band edge is blue-shifted. The transmittances of the samples at eight incident angles ranging from 0° to 70° for TM waves are shown in Fig. 6(b). The transmittance and reflectance were measured using a Perkin Elmer LAMBDA 1050 ultraviolet-visible-near-infrared spectrophotometer, where the minimal incident angle is 7.5°for the reflection measurement. Figures 7(a) and 7(b) illustrate theoretical and measured reflection spectra of the truncated PC containing the HMMs at different incident angles for TM waves, respectively. The long-wavelength band edge was seen to be red-shifted. The angle-sensitive omnidirectional PBG was observed in the wavelength region 245–338 nm, and the Q-factor is 3.13. Therefore, the waveband of such omnidirectional PBG in PCs containing HMMs broadens with the increase of the incident angle for TM waves, which is opposite from that of the PBG shown in Fig. 1. In the experimental transmission and reflection spectra presented in Figs. 6(b) and 7(b), the abnormal transmission dips and reflection peaks originate from the effect of the plasma wavelength (approximately 325 nm) of silver, which leads to strong resonant optical absorptance, low transmittance and reflectance. If the negative-permittivity media with low loss are selected to replace the metal and construct perfect HMMs, the PCs containing HMMs can realize an angle-sensitive omnidirectional PBG with perfect reflection broadband.
The measured band edge wavelengths for the incident angles are plotted and depicted by scattered circles in Fig. 8. As the incident angle of TM wave increases, the measured long-wavelength band edge red-shifts from 338 to 342 nm, while the short-wavelength band edge shifts from 245 to 239 nm. The Q-factor changes from 3.13 to 2.82. The background color corresponds with the simulated transmission spectra, and the dashed lines denote the variation trends of the two band edges. The experimental results agree well with the simulated values. The negligible deviations between the simulated and experimental values originate from the errors in monitoring the layer thicknesses during the deposition process and discrepancies in refractive indices between the simulated and experimental studies. Hence, an ideal omnidirectional PBG is developed.
Fig. 8. Comparison of simulated results (background color) and measured band-edges (scattered circles) of the PC containing HMMs for TM waves. All parameters are the same as those in Fig. 6.
4. Conclusion
This paper presents the simulated and experimental demonstrations of the angle-dependent omnidirectional PBG in PCs composed of dielectric materials and HMMs engineered by the subwavelength metal-dielectric multilayers. These PBGs have red-shifted long-wavelengths and hypsochromic short-wavelength band edges. The red-shift of the long-wavelength band edge is due to the phase-variation compensation effect between the HMM and dielectric material. This property can be utilized for the fabrication of novel photonic devices.
Funding
Natural Science Foundation of Shandong Province (ZR2019MA055, ZR2020QA071); National Natural Science Foundation of China (12047536); State Key Laboratory of Surface Physics and Department of Physics (KF2019_05); Key Laboratory of Micro- and Nano-Photonic Structures (Ministry of Education).
Acknowledgments
This work is supported by Physical-Chemical Materials Analytical & Testing Center of Shandong University at Weihai.
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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