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Abstract
In this work, we investigate possibility of engineering photonic density of states (PDOS) in photonic hypercrystals (PHCs). In the course of our analysis, we have demonstrated that it is possible to obtain photonic bandgap for selected polarization of light as well as to achieve significant broadband PDOS enhancement. We have also presented for the first time that anomalous dispersion, that arises from effective resonance of hyperbolic medium constituting the PHC structure, may lead to negative PDOS, which is photonic equivalent of mobility gap, observed in electronic crystals. Furthermore, we have demonstrated that application of PHC structure, instead of standalone hyperbolic medium, allows to obtain more versatile electromagnetic response, such as broadband perfect absorption of adjustable spectral range of operation.
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1. Introduction
Over the last decade, considerable attention has been focused on photonic hypercrystals (PHCs) – artificial optical media that combine the unique material dispersion of hyperbolic metamaterials (HMMs) [1–3] with the band formation from photonic crystals (PCs) [4–6]. The simplest one-dimensional form of hypercrystal structure can be realized with a periodic arrangement of HMM layer and dielectric or (metallic) material layer. Since, in a constituent HMM, the components of the permittivity tensor have opposite signs in two orthogonal directions, and so the high-k propagating waves are supported, the unit cell of the hypercrystal can have subwavelength dimension and still form a photonic band structure [7,8]. This feature results in a number of unique properties, like an unprecedented degree of control of light propagation, deep subwavelength localization of light, possibility to substantially reduce detrimental effect of the material loss in plasmonic devices and systems, etc. [9,10]. Many new interesting applications are enabled by PHCs, including sub-diffraction imaging [11–13], efficient polarizers [14–16], perfect absorbers [17–19], and optical cloaking [20]. Strong field localization in PHC structure that is of great interest in sensing and sub-diffraction imaging applications, can also lead to considerable enhancement of nonlinear optical effects, e.g., [21,22]. It has been demonstrated that the level of nonlinearities achievable in PHCs paves a promising foundation for future optical limiting and optical computing applications [21,22]. The studies on the propagation of electromagnetic surface waves in hypercrystals also revealed unusual properties such as the possibility of electrostatic wave excitation [23] or the existence of an unconventional plasmon-polariton gap that has been exploited to optical bistability used for optical switching [12]. Furthermore, it has been shown that surface wave propagation can be achieved and controlled by graphene-based hypercrystal [24]. The dynamical control of the surface waves has favorable application in the design of various active devices such as optical modulators and switches. It has also been demonstrated that photonic bandgap effects in multiscale hyperbolic metamaterials can be used to control volume plasmon polaritons [8]. Due to that, such structures can be used for far-field subwavelength imaging. Further investigations of propagation plasmon resonances in microstructures has lead to new type of photonic hypercrystals, multilayer trench grating structures, that exhibit relatively low loss propagation, while maintaining strong energy localization [25,26].
Photonic hypercrystals, due to their ability to support electromagnetic states with large momentum (high-$k$ modes), emerged as a very promising platform for second-harmonic (SHG) and third-harmonic generation (THG). More recently, for the first time, the lasing phenomenon in a distributed feedback laser based on PHC was investigated [27]. It has been demonstrated that changing the dispersion type or inducing nonlocality of HMM medium constituting the PHC structure may lead to a number of interesting effects, such as tunable single-mode generation or ultra-low generation threshold [27,28]. Moreover, angle dependence of the photonic bandgaps in a novel kind of 1D PHC entirely composed of HMMs called all-HMM PHC was theoretically realized [29]. As the incident angle increases, the photonic bandgaps in all-HMM PHC exhibit the redshifted PBGs under transverse magnetic polarization while exhibiting the blueshifted photonic bandgaps under transverse electric polarization. Photonic hypercrystals have been proven to modify the emission and absorption properties of quantum emitters [30,31]. It has been experimentally demonstrated that 2D semiconductor monolayers placed in the vicinity of a PHC reveal broadband enhancement of spontaneous emission due to large density of states in the PHC over wide spectral range [30]. It is worth to underline that the photonic density of states (PDOS), like its’ electronic counterpart, is one of the key physical quantities governing a variety of phenomena. However, despite a number of studies, the possibility of controlling PDOS in PHC structure has not been yet investigated.
In our work we present a first comprehensive investigation on controlling the photonic density of states in photonic hypercrystals by changing the effective dispersion of HMM structure constituting the PHC’s basic cell. In line with the approach developed by J. M. Bendickson and J. P. Dowling [32], the PDOS was expressed in terms of the complex transmission coefficient, which may be determined with the help of the transfer matrix method (TMM). Along with the study of the PDOS, we have analyzed the resulting transmission and absorption properties of the PHC structure, that have been also calculated by means of TMM approach. In the course of our analysis, we have demonstrated that controlling the dispersion type of hyperbolic medium via change of fill factor may lead to a number of interesting effects, such as possibility to obtain photonic bandgap for selected polarization of light as well as to achieve significant broadband PDOS enhancement bringing about broadband spontaneous emission enhancement. Moreover, the presence of anomalous dispersion enables to obtain propagation gap that may be utilized to achieve broadband perfect absorption with controllable spectral range of operation.
2. Theory
To investigate the photonic density of states in a photonic hypercrystal, we use the approach suitable for finite, one-dimensional, photonic bandgap structures, see Ref. [32]. In line with the method demonstrated by J. M. Bendickson and J. P. Dowling, we assume that the incident electromagnetic wave has unit amplitude and zero phase. Then, the change of amplitude and phase for wave travelling through the structure is described with a complex transmission coefficient $t(\omega )$, that contains phase information and may be further used to extract the dispersion relation $k = k(\omega )$ and hence the density of modes for a given frequency
Assuming the complex form of transmission coefficient $t(\omega )= x(\omega )+ iy(\omega )= \sqrt T \textrm{exp}({i\varphi } )$, where x and y denote the real and imaginary parts of t, while $\varphi $ is the total phase accumulated by the wave propagating through the structure, we know that
Since the phase can be also written as $\varphi = \textrm{}kd$, where k is the effective wavevector and d is the length of a structure, we get:
Then, by differentiating both sides of Eq. (3) with respect to $\varphi $, we find the relation:
The Eq. (5) allows to express the photonic density of states in terms of the complex transmission coefficient of an arbitrary structure. In the case of photonic hypercrystal, that is consisted of anisotropic media, a complex amplitude transmission coefficient may be determined with the use of transfer matrix method, which is suitable for describing propagation in anisotropic media. For details, see the Supplemental document on transfer matrix method.
3. Results and discussion
3.1 Dispersion properties of hyperbolic metamaterial
Here, we investigate a planar photonic hypercrystal consisting of $N = 10$ basic cells that are composed of isotropic dielectric material and anisotropic hyperbolic metamaterial (see Fig. 1).
For the purpose of our analysis, as one of the materials constituting considered PHC structure, we consider a stoichiometric variant of silicon nitride (Si3N4) that has been proven to reveal dielectric response over NIR spectral range and can be described with well-established Sellmaier formula [28] (see Fig. 1). The other layer in the unit cell of PHC structure is hyperbolic metamaterial composed of periodically arranged layers of isotropic dielectric, i.e., SiO2, and plasmonic material, i.e., ZnO with 6.25% Al doping (AZO), which are described with Sellmaier formula [33] and Drude-Lorentz model [34], respectively. As long as the characteristic dimensions are subwavelength, the complete HMM structure can be described as a homogenous anisotropic medium with uniaxial diagonal permittivity tensor ${\overline{\overline \varepsilon } _{\textrm{HMM}}} = \left[ {{\varepsilon _{xx}},{\varepsilon _{yy}},\; {\varepsilon _{zz}}\left] = \right[{\varepsilon _ \bot },{\varepsilon _{||}},\; {\varepsilon _{||}}} \right]$, composed of components that can be calculated with the help of effective medium theory [1,35]:
Fig. 2. Dispersion map of the considered HMM structure plotted vs. wavelength and fill factor values. Gray dashed lines indicate values of the fill factor selected for analysis.
To fully explore possibility of engineering effective dispersion of hyperbolic metamaterial, we consider four different cases of HMM structures with various fill factor values, i.e., $\rho = $ 0.4, 0.6, 0.75, and 0.95 for constant ${t_{\textrm{AZO}}}$ = 2 nm, constituting, together with Si3N4 layers, the complete photonic hypercrystal. It is worth noting that each of the proposed cases corresponds to different dispersion properties within considered spectral range, which has been indicated in Fig. 2 with gray dashed lines. Finally, the thicknesses of layers constituting the PHC have been scaled to quarter-wave size and calculated as follows:
To unveil effects that arise in photonic hypercrystal rather than in hyperbolic metamaterial, we have conducted the analysis in the following manner: for each case, we have calculated the components of effective permittivity tensor to analyze dispersion properties of HMM structure and then compare power transmission (transmittance), absorption and PDOS spectra for a standalone hyperbolic metamaterial of length ${d_{HMM}} = N \cdot {t_{\textrm{HMM}}}$ (solid blue lines) and the complete PHC structure (dashed orange lines) of total length ${d_{PHC}} = N \cdot ({{t_{\textrm{Si}3\textrm{N}4}} + {t_{\textrm{HMM}}}} )$. In this manner, the analysis has been performed for TE- and TM-polarized waves impinging at angle normal to the surface. Moreover, for the purpose of our analysis, we use normalized photonic density of states:
where ${c_0}$ is speed of light in the vacuum.3.2 Case 1
Firstly, we have investigated a PHC structure composed of metamaterial layers with ρ = 0.95. In this case, due to high content of dielectric material, the HMM structure reveals elliptic dispersion over the whole considered spectral range for both TE and TM polarizations of light (see Fig. 3(a)). However, for TM-polarized waves, the metamaterial reveals anomalous dispersion, i.e., a real part of permittivity decreases with decreasing wavelength, within spectral range of $1.07\; {\mathrm{\mu} \mathrm{m}} < \lambda < 1.16\; {\mathrm{\mu} \mathrm{m}}$ that is directly related to effective resonance of the HMM medium and may be also connected to resonance peak of imaginary part of perpendicular permittivity component $Im({{\varepsilon_ \bot }} )$ (see Fig. 3(b)). This effect is observed only for TM polarization since, due to the arrangement of electric field vector components, propagation properties for TE-polarized waves are solely determined by $Re({{\varepsilon_{||}}} )$, which is positive within the whole considered spectral range.
Fig. 3. Real (a) and imaginary (b) parts of components of effective permittivity tensor plotted vs. wavelength for a hyperbolic metamaterial with ρ ≈ 0.95.
Starting from the optical properties for TE-polarized waves, it can be observed that transmission of the standalone HMM structure reveals Fabry-Perot etalon-like behavior with low absorption (see the interference pattern of the blue curves in Figs. 4(a) and 4(c)) that originates from effective elliptical dispersion of the HMM. On the other hand, the PHC structure reveals a transmission bandgap connected with high reflection around the central wavelength ${\lambda _\textrm{c}} = 1.5\; {\mathrm{\mu} \mathrm{m}}$, which is typical for conventional photonic crystals [32] (see Figs. 4(a),(e)). Due to the elliptical dispersion, the photonic density of states of the PHC reveals bandgap with local peaks at its edges, which is also typical for a conventional photonic crystal (see Fig. 4(g)). Moreover, it can be observed that overall level of photonic density is significantly higher than in the case of the HMM structure. It is worth underlining that such substantial increase of the PDOS is particularly useful in the view of spontaneous emission, which transition rate, according to Fermi’s golden rule, is directly proportional to the density of states. Thus, by appropriate arrangement of the layers in PHC structure, emission of a dipole placed in the vicinity of the structure may be significantly enhanced.
Fig. 4. Transmittance (a),(b), absorption (c),(d), reflectance (e),(f), and PDOS (g),(h) plotted vs. wavelength for TE- (a),(c),(e) and TM-polarized (b),(d),(f) waves travelling in PHC structure with HMM layer with ρ ≈ 0.95.
The PHC structure reveals a similar transmission bandgap for TM-polarized waves that cannot be observed in a standalone hyperbolic metamaterial (see Fig. 4(b)). The central wavelength is shifted with respect to its counterpart for TE-polarized light due to the difference between effective permittivity components (see Fig. 3). Moreover, the PHC structure reveals an additional transmission bandgap, which is a direct consequence of anomalous dispersion of the HMM structure. The “additional” bandgap is connected with anomalous negative photonic density of states (see Fig. 4(h)). This anomalous propagation gap may be perceived as an optical equivalent of a “mobility gap”, i.e., spectral range of zero conductivity, that can be observed in electronic crystals with anomalous dispersion, which leads to absolute absorption [36]. It can be observed that the PHC structure for TM-waves reveals almost negligible reflectance and very high level of absorption ($A \approx 1$) for both bandgaps, which is typical for media revealing anomalous dispersion [37] (see Fig. 4(d),(f)). It is also worth noting that, overall level of PDOS is again significantly higher with respect to the standalone hyperbolic metamaterial (see Fig. 4 ($\lambda > 1.5\; {\mathrm{\mu} \mathrm{m}}$h)).
3.3 Case 2
The second considered PHC structure is based on HMM layers characterized with $\rho = 0.75$. In this case, the elliptic dispersion is dominant within the considered spectral range (see Fig. 5(a)). However, for TE-polarized waves of wavelengths longer than , medium reveals ENZ dispersion and then, for $\lambda > 2.15\; {\mathrm{\mu} \mathrm{m}}$, ${\varepsilon _{||}}$ becomes negative, which results in Type II hyperbolic dispersion. It can also be observed that for TM polarization, the medium reveals much stronger resonance that leads to not only anomalous behavior (within $1.2\; {\mathrm{\mu} \mathrm{m}} < \lambda < 1.3\; {\mathrm{\mu} \mathrm{m}}$) and very high absorption (peak of imaginary part of in Fig. 5(b)), but also to Type I hyperbolic dispersion within $1.1\; {\mathrm{\mu} \mathrm{m}} < \lambda < 1.22\; {\mathrm{\mu} \mathrm{m}}$ spectral range.
Fig. 5. Real (a) and imaginary (b) parts of components of effective permittivity tensor plotted vs. wavelength for a hyperbolic metamaterial with ρ = 0.75.
Due to higher optical losses, introduced by higher relative content of plasmonic material in the hyperbolic medium, the overall level of transmittance is significantly lower than in the previously considered case (compare Figs. 4(a)-(d) and 6(a)-(d)). For TE polarization, transmission bandgaps are observed around $\lambda = 1.5\; {\mathrm{\mu} \mathrm{m}}$ and $\lambda = 0.825\; {\mathrm{\mu} \mathrm{m}}$ with high level of reflectance and significant PDOS enhancement at their edges (see Figs. 6(e) and 6(g)). It is worth to underline that bandgaps are a consequence of Bragg resonance in the structure and cannot be observed in a standalone HMM structure. Additionally, there is a noticeable broadband enhancement of PDOS around $\lambda = 2\; {\mathrm{\mu} \mathrm{m}}$, which may be connected with Type II hyperbolic dispersion of the hyperbolic medium (see Figs. 5 and 6(c)).
Fig. 6. Transmittance (a),(b), absorption (c),(d), reflectance (e),(f) and PDOS (g),(h) plotted vs. wavelength for TE- (a),(c),(e) and TM-polarized (b),(d),(f) waves travelling in PHC structure with HMM layer with ρ ≈ 0.75.
On the other hand, in the case of TM polarization, the overall properties are mostly determined by the dispersion of hyperbolic medium. Due to discrepancy between ${\varepsilon _ \bot }$ and ${\varepsilon _{||}}$ permittivity tensor components, there is no observable photonic bandgap for TM polarization (see Fig. 6(b)). However, the propagation bandgap arising from anomalous dispersion of hyperbolic medium offering substantial PDOS enhancement at its edges is still present (see Figs. 6(b) and 6(h)). In this case, arranging the HMM structure into a PHC allows to obtain broadband PDOS enhancement as well as a higher absorption level that leads to perfect absorption and zero reflection within $1\; {\mathrm{\mu} \mathrm{m}} < \lambda < 1.45\; {\mathrm{\mu} \mathrm{m}}$ spectral range (see Figs. 6(b),(d),(f)). This effect may be explained by the fact that a photonic hypercrystal, due to the existence of Bragg modes, reveals higher coupling between its eigenmodes and freespace modes than an HMM structure [7,30].
3.4 Case 3
Now let’s consider a photonic hypercrystal based on hyperbolic metamaterial with fill factor ρ = 0.6. The effective dispersion of HMM structure for TE polarized waves can be divided into two regions, i.e., positive (${\varepsilon _{||}} > 0$ for $\lambda < 1.7\; {\mathrm{\mu} \mathrm{m}}$) and negative (${\varepsilon _{||}} < 0$ for $\lambda > 1.7\; {\mathrm{\mu} \mathrm{m}}$) permittivity. In the case of TM polarization, the effective resonance of ${\varepsilon _ \bot }$ permittivity tensor component is stronger, wider and shifted towards longer wavelengths with respect to previously considered cases (see Fig. 7(a),(b)). Thus, it can be expected that effects connected with anomalous dispersion and high absorption will reveal more broadband character.
Fig. 7. Real (a) and imaginary (b) parts of components of effective permittivity tensor plotted vs. wavelength for a hyperbolic metamaterial with ρ ≈ 0.6.
In terms of transmittance of the PHC structure for TE-polarized waves, there is only one observable photonic bandgap located around $\lambda = 1\; {\mathrm{\mu} \mathrm{m}}$, while wavelengths longer than $\lambda > 1.5\; {\mathrm{\mu} \mathrm{m}}$ are strongly absorbed and/or reflected (see Figs. 8(a),(c),(e)). However, by analyzing the PDOS spectrum (see Fig. 8(g)), it is possible to observe another photonic bandgap around $\lambda = 1.6\; {\mathrm{\mu} \mathrm{m}}$. Moreover, due to Type II hyperbolic dispersion, the edge at the side of longer wavelengths reveals a substantial enhancement in comparison to the other part of the spectrum. Thus, despite low transmittance, the considered PHC structure still offers significant enhancement of spontaneous emission for dipoles located in its vicinity.
Fig. 8. Transmittance (a),(b), absorption (c),(d), reflectance (e),(f) and PDOS (g),(h) plotted vs. wavelength for TE- (a),(c),(e) and TM-polarized (b),(d),(f) waves travelling in PHC structure with HMM layer with ρ ≈ 0.6.
Again, in the case of TM polarization, the transmittance and PDOS spectra are dominantly influenced by the dispersion of the hyperbolic medium constituting the structure (see Figs. 8(b) and 8(h)). Moreover, due to the shift of the effective resonance, and thus, anomalous dispersion regime, the structure reveals perfect absorption that is achievable in broader spectral range than in the previous case, i.e., within $0.85\; {\mathrm{\mu} \mathrm{m}} < \lambda < 1.75\; {\mathrm{\mu} \mathrm{m}}$. It is worth to reiterate that, due to high reflection, the effect of perfect absorption is not feasible with a standalone HMM structure (see Figs. 8(d),(f)).
3.5 Case 4
Finally, we consider a PHC structure with a basic cell composed of dielectric and metamaterial characterized with fill factor ρ = 0.4. The effective dispersion properties are similar to the previous case (compare Figs. 7(a) and 9(a)). However, the effective resonance of permittivity tensor components is redshifted, while the regime of negative permittivity is blueshifted, which leads to occurrence of metallic dispersion, i.e., ${\varepsilon _ \bot } < 0$ and ${\varepsilon _{||}} < 0$, within $1.4\; {\mathrm{\mu} \mathrm{m}} < \lambda < 1.65\; {\mathrm{\mu} \mathrm{m}}$.
Fig. 9. Real (a) and imaginary (b) parts of components of effective permittivity tensor plotted vs. wavelength for a hyperbolic metamaterial with ρ ≈ 0.4.
Due to high plasmonic material content and, as a consequence, stronger effective resonance (see Fig. 9(b)), the transmission of TE-polarized waves within the considered spectral range is almost nonexistent, i.e., 0.02%, for both standalone HMM and PHC structures (see Fig. 10(a)). Additionally, both structures also reveal similar absorption spectra (see Fig. 10(c)), which are determined by effective dispersion properties of the HMM medium. Moreover, reflectance spectra, with the exception of small small side lobes caused by interference effects in PHC structure, are almost identical (see Fig. 10(e)). Despite many similarities, PHC structures, due to existence of Bragg states, reveal an additional peak of PDOS around , which grants possibility of multiband operation and may substantially enhance spontaneous emission of an emitter operating within this spectral range (see Fig. 10(g)).
Fig. 10. Transmittance (a),(b), absorption (c),(d), reflectance (e),(f) and PDOS (g),(h) plotted vs. wavelength for TE- (a),(c),(e) and TM-polarized (b),(d),(f) waves travelling in PHC structure with HMM layer with ρ ≈ 0.4.
Similarly to the case of TE polarization, both HMM$\lambda = 1.825\; {\mathrm{\mu} \mathrm{m}}$ and PHC structures reveal very low, almost negligible, transmission for TM-polarized waves within the whole considered spectral range (see Fig. 10(b)). Moreover, PDOS in both HMM and PHC are almost identical and reveal remarkable enhancement at the edges of the anomalous dispersion regime (see Fig. 10(h)). However, due to high coupling efficiency between Bragg states and free-space modes, the PHC structure, in contrast to the properties of a standalone hyperbolic metamaterial, reveals perfect absorption ($A \approx 1$) over the whole considered spectral range, i.e., $0.75\; {\mathrm{\mu} \mathrm{m}} < \lambda < 2.25\; {\mathrm{\mu} \mathrm{m}}$ (see Fig. 10(d)), while HMM structure reveals stronger reflection properties over the considered spectral range (see Fig. 10(f)).
4. Conclusions
In this work, we have investigated photonic hypercrystals based on hyperbolic metamaterials of various geometries (defined by fill factor) and dispersion properties. Within our analysis, we have focused on the possibility of controlling photonic density of states, and as a consequence transmittance and absorption, by changing effective dispersion of HMM structure constituting the PHC’s basic cell. In the course of our analysis, we have demonstrated that arranging hyperbolic metamaterials into a photonic hypercrystal opens up new possibilities in shaping optical properties, presented by example of transmittance, absorption, and PDOS. In particular, it is possible to obtain photonic bandgap for selected polarization of light as well as to achieve significant broadband PDOS enhancement, which may lead to versatile, broadband and multiband enhancement of spontaneous emission located in the vicinity of the structure. What is more, we have investigated effects arising from anomalous dispersion that originates from effective resonance that takes place in HMM structure and, until now, have been discussed only in terms of directional properties of PHC structure [14]. In particular, we have demonstrated, for the first time, that anomalous dispersion may be utilized to obtain anomalous propagation bandgap connected with negative density of states and perfect absorption, that may be perceived as a photonic equivalent of “mobility gap”, previously reported only in electronic crystals [36]. Furthermore, we have demonstrated that, by appropriate choice of geometry of the HMM structure constituting the basic cell, it is possible to achieve perfect absorption over almost arbitrary broad spectral range. To sum up, we have reported a number of new, interesting optical effects that may be obtained by arranging a standalone hyperbolic metamaterial into a photonic hypercrystal structure. We believe that the presented versatility of PHCs paves new foundations for further theoretical and experimental research towards practical applications of this class of photonic structures.
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.
Supplemental document
See Supplement 1 for supporting content.
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