Zero-index and Hyperbolic Metacavities: Fundamentals and Applications

As a basic building block, optical resonant cavities (ORCs) are widely used in light manipulation; they can confine electromagnetic waves and improve the interaction between light and matter, which also plays an important role in cavity quantum electrodynamics, nonlinear optics and quantum optics. Especially in recent years, the rise of metamaterials, artificial materials composed of subwavelength unit cells, greatly enriches the design and function of ORCs. Here, we review zero-index and hyperbolic metamaterials for constructing the novel ORCs. Firstly, this paper introduces the classification and implementation of zero-index and hyperbolic metamaterials. Secondly, the distinctive properties of zero-index and hyperbolic cavities are summarized, including the geometry-invariance, homogeneous/inhomogeneous field distribution, and the topological protection (anomalous scaling law, size independence, continuum of high-order modes, and dispersionless modes) for the zero-index (hyperbolic) metacavities. Finally, the paper introduces some typical applications of zero-index and hyperbolic metacavities, and prospects the research of metacavities.


I. INTRODUCTION
Optical devices fundamentally count on the interactions between light and the matter [1]. As one of the most fundamental optical elements, optical resonant cavities (ORCs) design is crucial for effectively enhancing the light-matter interaction [2,3]. The traditional Fabry-Pérot (FP) ORC is composed of two optical mirrors separated by a certain distance, in which light will circulate multiple times, as shown in Fig. 1(a) [4]. The basic physical principle of ORCs is the interference of light, which leads to the formation of standing wave in the cavity. Due to the important role of high-quality factor (Q) and ultra-small ORCs in optics, such as low threshold laser [5,6], high resolution sensor [7][8][9] and so on, it has attracted people's wide attention. In general, in order to realize the ORCs with good electromagnetic (EM) wave confinement, a cavity with high-Q is needed. However, the high-Q ORCs usually need to increase the number of round trips, such as the optical whispering gallery mode (WGM) cavity in Fig. 1(b) [10,11]. This leads to the increase of cavity volume (V), which is not conducive to the miniaturization of the ORCs. On the other hand, although the surface plasmon polaritons (SPPs) can be used to design miniaturized ORCs, the Q of 3 cavities is significantly reduced due to the large inherent loss of metal [12]. Therefore, it is difficult to achieve high-Q cavity mode in small ORCs and it is very meaningful to evaluate the Q/V relation in practical applications. Especially, Q and V correspond to the lifetime and energy density of cavity mode, respectively. With the emergence of photonic crystals (PCs), researchers have proposed PC microcavity [13,14]. The research of PC cavities is mainly focused on one-dimensional (1D) and two-dimensional (2D) PCs, as shown in Figs. 1(c) and 1(d), respectively. Compared with the traditional ORCs in Fig. 1(a), PC microcavity based on photonic band gap has the advantages of high-Q and small-V (the geometric dimension is the same order of magnitude as the wavelength of light). Usually, PC cavities are realized by breaking the symmetry of structure, such as introducing point defects in PCs. As a result, light can be trapped in the position of defect, and cannot propagate to the surrounding, thus forming a PC cavity. So far, PC cavities have been used in low-threshold lasers, high-sensitivity sensors, filters and other optical devices [15]. With the development of science and technology, the traditional FP and even the PC cavities become more and more difficult to meet people's requirements for multi-function optical devices. For example, the field intensity distribution is inhomogeneous because the cavity mode exists in the form of standing wave. For the cavity quantum electrodynamics (CQED), in order to achieve strong coupling between the single atoms and the electromagnetic mode of an ORC, it is necessary to put the atoms exactly in the strongest position of the field. However, due to the small size of quantum dots, it is very difficult to accurately place the quantum dots at the antinode of the cavity mode [16][17][18]. In addition, with the development of on-chip miniaturized photonic devices, the realization of subwavelength optical mode localization becomes a very important scientific problem.
For traditional FP and PC cavities, destructive and constructive interferences appear in different regions due to the determined optical path difference of EM waves, namely local interference. As mentioned above, due to the limitation of interference mechanism, it is difficult to achieve strong coupling between quantum dots and a traditional ORC. However, the EM field in ZIM is uniform, which can be used to achieve the global coherent effect, that is, nonlocal interference phenomenon.
Therefore, ZIMs provide a good platform to study the CQED without precise positioning [76,77].
On the other hand, because of the limitations of the standing-wave formation conditions for FP-type resonance, the miniaturization of ORCs formed using traditional materials is difficult. However, due to the special dispersion curve of HMM, it can also be used to observe the anomalous scaling law and realize size-independent ORCs. Therefore, HMMs provide a good solution for the technologies that require the miniaturization of subwavelength ORCs without reducing the value of Q [93,94].
Zero-index and hyperbolic metacavities greatly deepen our understanding of ORCs, reveal many physical phenomena rarely observed in nature, and offer applications for controlling light, including switching, unidirectional transmission, lasers, sensors, filters, and so on.

II. ZERO-INDEX METACAVITES
As mentioned in the introductory part, ε and/or μ are near zero in ZIMs, which can realize various EM responses and provide new situations to revisit some fundamental phenomena in wave propagations [113]. When both ε and μ are near zero, the ZIM is also called an ε-μ-near-zero (EMNZ) media. Besides, if only ε or only μ is near zero, the ZIM is also called an ε-near-zero (ENZ) or a μnear-zero (MNZ) media. The corresponding EM parameter space of ENZ, MNZ, and EMNZ media are marked respectively by red, blue, and purple in Fig. 2(a). Because the refractive index of ZIMs tends to zero, the wavelength of EM waves tends to infinity. Thus, the phases of EM waves in ZIMs 6 are the same everywhere [114] and the wavelength is greatly stretched [115,116]. As a result, the light-matter interaction in ZIMs can be greatly affected [117]. Especially, because the iso-frequency contour (IFC) of ZIM is approximately a point, it can only support normal incidence EM waves and the oblique incidence EM wave will be reflected, thus it can be used to design high directivity emission [118][119][120]. Besides, diversified transmission characteristics of EM wave have also been verified in ZIMs [121][122][123][124]. For example, EM waves can pass through a narrow ZIM waveguide of arbitrary shape almost without blocking, which can be used to realizing the EM tunneling [125,126].
So far, most of the researches are focused on isotropic ZIMs. But another kind of special ZIM, anisotropic zero-index metamateiral (AZIM), has attracted more and more attention because of its special EM properties. In AZIM, because only one tensor element in the ε or μ tensor is near zero.
In this case, the IFC of AZIM will have many forms, which can be either flat elliptic or hyperbolic [137], as shown in in Fig. 2(d). Therefore, compared with isotropic ZIMs, AZIMs have more abundant EM response characteristics. For example, Luo et al. proposed a near perfect bending waveguides based on AZIMs [138,139]. Interestingly, it has been found that sub-wavelength flux manipulation can be achieved in AZIMs by using the scattered evanescent waves [140], which was confirmed by experiments based on the circuit-based AZIM [141]. In general, ZIM show abundant EM response characteristics when the EM wave interacts with it because of its special EM parameters, which provides a very good platform for realizing more novel physical phenomena and important applications.

Two-dimensional photonic crystal ZIMs
The simplest way to achieve ZIMs is to use the natural dispersion characteristics of materials.
For metal materials, the permittivity can be described by a simple Drude model  [142,143]. Another kind of simple ENZ media can be realized by metal waveguides, as shown in Fig. 2(b) [144][145][146]. When the working frequency is near the cut-off frequency of the guide mode, the effective wavelength of EM waves in the waveguide tends to infinity, thus the waveguide structure can equivalent to the ENZ media.
It should be noted that the impedance of single zero-index media (that is, ENZ or MNZ media) is seriously mismatched with the air, which is unfavorable to the high efficient transmission of EM waves. In 2011, Huang et al. proposed using 2D dielectric PCs to realize matched EMNZ media [147]. By adjusting the permittivity and geometric parameters of the dielectric cylinder in the 2D PC, they found that the accidental degeneracy of electric monopole and electric dipole modes can be realized, and the Dirac-like point can be formed at the center of the Brillouin zone. At this accidental degeneracy case, the 2D PC can be equivalent to an isotropic EMNZ media [147][148][149][150][151][152], which has also been successfully constructed in experiment [153] and implemented on a photonic chip [154]. As shown in Fig. 2(c), each unit consists of a silicon pillar surrounded by polymer. Due to the parallel gold film cladding, the propagating waves are completely consistent in the periodic plane, which is equivalent to the case of using silicon pillar with infinite height [154]. Interestingly, the ZIM realized by the conical dispersion can also be extended to a noncrystalline system [155].

Metal/dielectric multilayered ZIMs
Within an EMT under the condition of long-wave approximation, the periodic arrangement of artificial structures with subwavelength unit-cells can be regarded as an effective homogeneous medium, characterized by macroscopic EM parameters  and  . By designing suitable artificial structures, such as the the AMNZ media can be conveniently engineered based on the split-ring resonaotrs (SRRs) in microwave regime [156,157]. In the visible band, metal/dielectric multilayers have been widely used to create AENZ media [158], as shown in in Fig. 2 [160,161]. In addition, the AENZ media have also been proposed based on the 2D PCs beyond the longwavelength limitation [162]. This optimized EMT will facilitate the design of new metamaterials and show that the AENZ media can indeed be fabricated from a periodic 2D PC structure.

Circuit-based ZIMs
By using transmission lines (TLs), the circuit-based system can be used to construct abundant metamaterials with flexible EM parameters [163,164]. In the circuit-based system, the relationship between the electric and magnetic fields can be easily mapped using the relationship between voltage and current. As a result, the electromagnetic response is equivalent to the circuit parameters. The structure factor of the TL is defined as 0 / eff gZ = , where 0 Z and eff  denote the characteristic impedance and effective wave impedance, respectively. For the general 2D TL model with lumped capacitors and resistors in Fig. 2(f), the impedance and admittance of the circuit are represented by Z and Y, respectively. By mapping the circuit equation (telegraph equation) to Maxwell's equations, the relationship between circuit and electromagnetic parameters can be described by [163,164] where 0  and 0  are the vacuum permittivity and permeability, respectively. The effective permittivity and permeability of the circuit system can be tuned using the lumped elements in the circuit. The above inset of Fig. 2(f) shows an effective circuit model for a circuit-based AENZ media.
In this circuit model, the admittance is and capacitors are loaded in the x direction to realize anisotropic impedance where 0 C and 0 L denote the capacitance and inductance per unit length, respectively. d denotes the size of unit cell. Therefore, the effective electromagnetic parameters of the system are [163,164]  1 .
x z L Cg g According to Eq. (4), we can know that the real part and imaginary part of the z  can be flexibly adjusted by tuning the value of the lumped capacitors C and resistors R. Specially, when , the real part of z  is near to zero and the circuit-based AMNZ media is constructed [165,166]. Inspired by metamaterials, researchers have recently found that resonant structures and nanoparticles can be used to design effective circuit elements in THz [167] and visible light regimes [168], thus the circuit-based ZIMs introduced here may be extended to the high frequency regimes. In addition, due to the similarity of waves, ZIMs and their novel physical properties have been successfully applied to acoustic [169][170][171][172], electronic [173] and thermal [174,175] [78]. The schematics of four ZIM cavities with different shape, size or topology are shown in Fig. 3(a). Interestingly, although the geometries of these cavities are obviously different, they have the eigenmode at the same frequency. For a 2D zero-index resonant cavity in Fig. 3(b), a dielectric particle with cross-sectional area i A , perimeter Under this condition, the eigenfrequencies of the ZIM cavity are determined by the characteristic equation [78]:  Fig. 3(d). However, the quality 11 factor Q of the different ZIM cavities will changes because it depends on the special field intensity distributions in the cavities [78]. Especially, the ZIM cavities not only pave the way to design deformable resonant devices, but also provide a new platform to study the quantum optics [81,176].

Homogeneous fields
Strong fields are helpful for enhancing the interaction between light and the matter. The band gap of PCs [(AB)5C(BA)5)] can be used to confine the EM waves and enhance the strength of the field in the defect region at the center of the PCs, which is shown in Fig. 4 nm), respectively. However, the field profile is always inhomogeneous, which is mainly determined by the characteristics of standing wave field, as shown in Fig. 4(b). This leads to the limitation of some applications, such as the enhancement of nonlinear effects [74]. Based on the Maxwell equations, the fields inside the matched EMNZ media should be homogeneous.
Especially, when the transverse-electric (TE) polarized waves impacting to the MNZ media ) to guaranting the magnetic field is a finite value: Similarly, the magnetic field is homogeneous for the ENZ media under transverse-magnetic (TM) 12 polarized waves. This poses a question: can ZIM be used as special cavity to achieve uniform field enhancement?
In 2011, Jiang et al., propose theoretically and demonstrate experimentally the enhancement of homogeneous fields in a zero-index cavity [74]. The schematic of a 1D PC microcavity with an effective ZIM defect is shown in Fig. 4(c), where the ZIM defect is marked by the purple region.
The corresponding electric field distribution of the ZIM cavity mode is shown in Fig. 4 Fig. 4(d). This enhanced uniform field boosts the average |E| 2 greatly, thus zeroindex cavity provides a good research platform for the field enhancement without increasing the thickness of the reflecting walls. Especially, considering a nonlinear ZIM defect in Fig. 4(c), the localized fields in the nonlinear defect not only can effectively enhance the nonlinear effect, but also limit the damage of inhomogeneous field to nonlinear materials [74].

Inhomogeneous fields
For TM (TE) polarized waves, the electric (magnetic) field in the MNZ (ENZ) meida is a constant. Figure 5(a) shows a 2D MNZ cylindrical cavity surrounded by air [79]. When a line source with TE polarized waves is placed inside this MNZ, it can be cleary seen that the excited electric field in the MNZ media is homogeneous and the isotropic radiation appears in the air, as shown in air core can be used to realizing the inhomogeneous field in ZIMs [79]. This counterintuitive inhomogeneous field will appear when the higher-order modes in the zero-index cavity are excited.
The schematic of a 2D MNZ shell cavity surrounded by air is shown in Fig. 5 where r and  denote the relative distance and angle to the center position, respectively [79].
Moreover, different from the isotropic radiation in Fig. 5(c), the controlling radiation pattern is realized based on the high-order cavity modes in ZIM, in which with numbers of outgoing direction is determined by the angular momentum m. The proposed ZIM cavity may be used to control (enhance or suppress) the radiation of EM waves, to control radiation pattern and to achieve isotropic or directive radiation [79].

EZI cavity realized by two types of single-negative (SNG) media
In Maxwell's theory, the EM parameters of materials are characterized by permittivity ε and permeability μ. The appearance of metamaterials enriches the EM response of materials. People can design artificial materials with arbitrary combination of permittivity and permeability. In addition 16 to left-handed media with ε < 0 and μ < 0, there are also single-negative (SNG) media with negative ε or μ alone. Especially, ε < 0, μ > 0 and ε > 0, μ < 0 correspond to ENG media and MNG media, respectively. ENG and MNG meida are painted respectively pink and blue for see in Fig. 6(a). ENG and MNG media are opaque because their refractive index is imaginary. As a result, the supported EM mode in ENG and MNG media is evanescent wave. But it is interesting that when the structure is composed of matched ENG and MNG media, there is a resonant tunneling mode, thus the combination of the two opaque materials into a new material becomes transparent [177], as shown in Fig. 6(b). The tunneling phenomenon in the heterostructure composed of ENG and MNG media can be realized under the impedance matching and phase matching conditions [177]: where ZENG (kENG) and ZMNG (kMNG) are wave impedances (wave vectors) of ENG and MNG media, respectively. di (i = 1, 2) denote the thickness of the ENG or MNG layer. Considering (7) and (8) can be can be reduced to Under EMT, this heterostructure can be equivalent to a matched EMNZ media with 177,178]. The effective thickness of the cavity is zero because there is no phase accumulation in the cavity. The EM wave can tunnel through the pair defect satisfying Eqs. (7) and (8) without any phase delay since the pair defect is reduced to nihility [177]. Therefore, the matched ENG/MNG structure can be seen a specialy zero-index cavity, where ENG and MNG media act the mirrors. The effective zero-index (EZI) cavity have also been widely studied in the 1D PCs [179][180][181][182][183].
The simplest structure is to insert a pair of ENG/MNG heterostructures directly into 1D PC [179], as shown in Fig. 7(a). The structure denotes as (AB)8CD(AB)7A, where A and B are the normal dielectric materials. C and D denote the ENG and MNG media, respectively. Especially, ENG and MNG media satisfy the matching conditions of Eqs. (7) and (8). In this case, we can find that the inserted ENG/MNG defect will not change the transmission spectrum, which is shown in Fig. 7(b).
The underlying physical mechanism is come from the maintenance of wave interference. Because the ENG/MNG defect is equivalent to a transparent layer with zero effective refractive index, it has on has no effect on the interference of propagating waves in A and B layers [179]. Interestingly, although the the transmission spectrum remains invariant, the field distribution indeed changes noticeably because of the decaying wave in the pair defect. Considering the high gap-edge frequency fH in Fig. 7 It can be clearly seen that there is a band gap exist in the individual left PC: (CD)m and right PC: Fig. 7(f). However, when the zero average conditions satisfied for the heterostructure: (CD)m(C'D')n, the tunneling mode with unit transmittance appears inside the forbidden gap. The corresponding electric fields 2 () Ez distributions of the tunneling mode is shown in Fig. 7(g). In this case, the electric field is mainly localized at the interface of the two PCs. Especially, this unusual tunneling mode realized by the EZI cavity is independent of incident angles and polarizations and have zero phase delay, which can be utilized to design zero-phaseshift omnidirectional filters [180].

Enhanced magneto-optic effect and nonlinear effect
In general, the large magneto-optical (MO) activity and opotical nonlinearity of materials are highly desirable in many applications, such as the isolators, optical switches, etc. In this section, we review that the EZI cavity can be used to significant enhance the MO effect and nonlinear effect.
Compared with the single-layer MO medium, the optical isolator realized by MO PC has the advantages of high transmission, strong Faraday rotation effect and small volume. Transparent yttrium iron garnet (YIG) is one of the most studied MO media. Although the MO activity of MO metal is much larger than that of YIG, the MO metal is opaque, thus the MO activity of MO metal is nearly inaccessible. Similarly, the the third-order nonlinear susceptibility of noble metals is large, but it is difficult to be used. Interestingly, the opaque metal with MO activity or opotical nonlinearity can become transparent when the heterostructurte (i.e., EZI cavity) is formed by matching suitable materials. In addition, the tunneling mode in EZI cavity can realize the strong localization of the field, thus enhancing the interaction between light and matter.Considering a heterostructure consisting of an all-dielectric PC: (AB)6 and a MO metal layer (M), the EZI cavity with tunneling mode is constructed [75]. The corresponding schematic of heterostructure (AB)6MP is shown in Fig.   8(a). B is the SiO2 layer with 2.1 =  . P is the protection film SiO2. A and M are the MO media, which correspond respectively to Bi : YIG and Co6Ag94 media. Under an applied magnetic field is in the z direction, the permittivity tensor of layer A and M is: where  is the off-diagonal element responsible for the strength of MO activity of the medium.
The transmission (solid line) and reflection (dashed line) spectrum of the heterostructure (AB)6MP is shown in Fig. 8 There is a dip of zero reflection at =  631 nm, that is the the tunnelling mode satisfied the matched consitions [75]: As mentioned above, the impedance matching and phase matching conditions are equivalent to the zero average conditions. Therefore, the heterostructure composed of PC and metal layer also belongs to the EZI cavity. Especially, the corresponiding spectrum of the Faraday rotation angles In addition to the MO effect, the EZI cavity can also enhance the nonlinear effect significantly.
For a heterostructure composed of an all-dielectric PC and a metallic film with nonlinearity, the tunneling mode can produce large nonlinear effect because of the strong field localization. The schematic of a 1D heterostructure (AB)7M composed of an all-dielectric PC: (AB)6 and a metal: Ag layer is shown in Fig. 9 where 22 1  is the linaer permittivity and 9 2.4 10 − =  denotes the nonlinear 23 susceptibilities of silver. Figure 9(b) shows the transmission spectrum of the heterostructure (AB)7M without considering the nonlinear susceptibilities of silver (i.e., 0 =  ) . It can be clearly seen a tunneling mode with frequency 0 = f 525 THz appears, which is marked by the dashed line. The corresponding electric and magnetic fields of the tunneling mode are shown in Fig. 9(c). Especially, the EM fields are localized at the interface between the SiO2/TiO2 PC and the silver layer.
Considering the nonlinear susceptibility of silver, the electric field in silver and the frequency of the tunneling mode varies with the electric field intensity switching-down thresholds for bistability, respectively. Moreover, the variance of intensities of thresholds with the frequencies of the incident wave is shown in Fig. 9(e). It can be clearly seen that 520.6 THz corresponds to critical frequency of the incident wave and there is no bistability above 520.6 THz. For comparison, figures 9(f)-9(j) show the results of a tranditional metallo-dielectric PC: (SiO2Ag)7. Especially, the total thickness of the Ag layer in Fig. 9(f) is the same as that of the Ag layer in the heterostructure in Fig. 9(a). Similar to Figs. 9(b), figures 9(g) give the T spectrum of the tranditional metallo-dielectric PC. The frequency of the band-edge mode in Fig. 9(g) is designed equal to the frequency of the tunneling mode in Fig. 9(b). And the corresponding 2 E distribution of the band-edge mode is shown in Fig. 9(h). It is seen that the nodes of the electric field locate at each thin silver layer, thus the tranditional metallo-dielectric PC can enhance transmittance, but the electric field in silver is still weak, which limits the enhancement of the nonlinear effect [184]. The nonlinear response and the threshold strength of 1D tranditional metallo-dielectric PC is shown in Fig. 4(i) and 4(j), respectively. Compared with Fig. 9(e) and 9(j), the critical intensity of the threshold in the EZI cavity is reduced by nearly 2 orders of magnitude than the tranditional metallodielectric PC, which may be used in in many applications, such as bistable switching, light-emitting diodes, etc [184].

EZI cavity with topological characteristics
Recently, the topological photonics has attracted people's great attention due to their great advantages in fundamental topological research and practical applications [185][186][187]. Different from the tight-binding model, 1D PCs with multiple scattering mechanism are also the important topological structures [188][189][190][191][192]. In which, the topological properties can be easily affected by the geometrical settings. Especially, the EM response of materials depends on the permittivity and 25 permeability. When one of them is negative, the material corresponds to the SNG media and they can be seen the light mirrors. By mapping the 1D Maxwell equation to 1D Dirac equation, the topological order of material can be determined by the effective mass m associated with the effective permittivity and permeability [190]: are the effective mass and effective potenstial, respectively. The topological properties of bands or the band gaps can be directly distinguished by the effective EM parameters [193,194]. For ENG media, ε is negative, μ is positive and the effective mass m is negative. However, for MNG media, ε is positive, μ is negative and the effective mass m is positive. Based on this method, the topological edge states in the heterostructure composed of two PCs with different topological orders have been proposed theoretically in visible light band and experimentally verified in microwave band [188][189][190]. So the topological order of electric and magnetic mirrors are different. The circuit system based on the TLs provides a good platform to study the topological structure and the related properties. Figure 10 (a) shows the photograph of a paired structure, which composed of an circuit-based ENG media and a PC [188]. Once the right PC can be effective to MNG media, the EZI cavity with tunneling mode can be realized. In addition, considering two different PCs, the left PC produces the effect of an ENG media and the right that of a MNG media in Fig. 10(b). In Fig. 10(c), the transmission spectra of the right PC, the left PC, and the paired structure are shown by the blue dotted lines, the red dashed lines, and the solid black lines, respectively. For the paired structure, a tunneling mode is at 2.91 GHz in simulation and 3.07 GHz in the experimental measurement, as indicated by the green dotted lines in Fig. 10(c) [188].
Especially, by tailoring the permittivity and permeability of metamaterials, band inversion of the Dirac equation was demonstrated theoretically and experimentally [190]. It has been found that the band inversion accompanies a change of chirality of electromagnetic wave in metamaterials. Three samples: left PC, right PC and the paired structure are constructed based on the circuit-based system.
The density of states (DOS) of the paired structure is simulated and measure in the left and right panels of Fig. 10(e), respectively. The topological edge state (i.e., the tunneling mode in the EZI cavity) is identified by the additional narrow peak appears at =  11.05 GHz within the gap region 26 in Fig. 10(e). Figure 10(f) shows the full-wave simulation of field spatial distribution of the topological edge state. It is clearly seen that the topological edge state is strongly localized at the interface of two PCs with different topological orders. In addition, the measured voltage distribution is meet well with the simulated field distribution, as shown in Fig. 10(g). The EZI cavities with topological characteristics not only provide a proof-of-principle example that EM wave in the metamaterials can be used to simulate the topological order in condensed matter systems, but also are helpful for research into surface modes in PCs and related applications [188][189][190]. The photonic topological edge states based on the 1D PCs have also been demonstrated at high frequency regimes. For the a 1D PC with symmetric unit cell in Fig. 11(a), the topological property of the bands can be determined by the Zak phase [191]. The unit cell is marked by the dashed yellow lines. The light is incident from the left side of the structure. Figure 11 thickness of the unit cell. The band structure of the 1D PC can be obtained by: where q represents the Bloch wavevector. Based on the above parameters, the corresponding band structures of PC1 and PC2 are shown in the middle and right panels of Fig. 11(b). Especially, the Zak phase (0 or π) of bands is marked near the bands. The topological property of the bandgap is painted different colors for see. It can be clearly seen that the 7th band gap of two PCs is topological distinguished. As a result, a topological edge state will occur in the heterostructure PC1-PC2 at the 7th band gap, which can be observed at the calculated transmission spectrum in the left panel of the Fig. 11(b).
Recently, the short-wavelength optical science is undergoing great development. In fact, the refractive indices of all materials are close to 1 in X-ray band, so a single layer cannot be used as a photonic insulator. The X-ray insulator is realized by using the band-gap of multilayer structures.
However, in which the size of unit cell is close to the atomic scale. So the structural fluctuations are unavoidable. A question naturally arises: can the topological properties be applied in the X-ray band to enable new devices? Recently, the topological edge state based on the 1D PCs have also been extened to the the X-ray band [192]. To obtain a gap with strong reflectance, the X-ray is at a grazing incident angle. Considering a PC with symmetric unit cell, the characteristic matrix can be respectively. Based on these effective parameters, the topological order of the band gap can be determined base on the effective mass m. In Fig. 11(e), the reflection phases of PC1 and PC2 belong to the ranges of (−180°, 0) and (0, 180°), which further confirms that the bandgaps of PC1 and PC2 are ENG and MNG gaps, respectively. For the edge state in the EZI cavity, the field maximum is at the interface of the two kinds of PCs. Moreover, from the interface to the left or right end of the structure, the envelope of the field exponentially decays, as shown in Fig. 11(f). The transmission electron microscope (TEM) image of the fabricated PC1-PC2 sample is shown in Fig. 12(a) [192]. The interface region is measured in high-resolution images, as shown in Fig. 12(b). The bright layers are carbon and the dark layers are tungsten. For the individual PC1 and PC2, they will exhibit a Bragg gap. However, once the topologically distinct PC1 and PC2 are combined together, there is a reflection dip in the X-ray bandgap. Which is the topological edge state. Remarkably, this topologically protected edge state is immune to the thickness disorder as long as the zero-average-effective-mass ( = m 0) condition is satisfied [190,192]: where 1 () mx and 2 () mx are the effective masses of the unit cell of PC1 and PC2, respectively.
Length 1 L and 2 L denote the total thicknesses of PC1 and PC2, respectively. The interface position of the two PCs is defined as 0. By adding certain kinds of disorders into PC2, the robustness of X-ray edge state in the hetero-structure is demonstrated in Figs. 12(c)-12(f). It can be seen that when the topological protection condition is still satisfied, the edge state is almost the same as that without disorder. On the contrary, if this condition is not satisfied, the edge state will be greatly affected. Therefore, based on two kinds of 1D PCs with different topological properties, the topological edge state in the X-ray band is demonstrated. Imporantly, this edge state is demonstrated to be robust against thickness disorders as long as the zero-average-effective-mass condition is satisfied. The related results extend the concept of topology to the X-ray band and may provide insightful guidance to the design of novel X-ray devices with topological protections [192]. In addition, the EZI cavity with topological properties also has been extended to the acoustic system [195]. In general, the combination of topology and ZIM will produce more interesting physical properties [196,197], which is worthy of further study in the future. zero-index cavity based on the strong nonlinearity [83]. The switchable ENZ cavity is constructed by two ENZ mirrors, which are implemented in the silicon waveguide, as shown in Fig. 13(a).
According to the EMT, the effective anisotropic ENZ layer is realized by using gold nanowire array [83,198]: where Au  and Si  are the permittivity of Si and Au, respectively. The concentration of gold nanowire is defined by the interdistance p and nanorod diameter 2a as Especially when the cavity mode frequency is tuned to that of the effective ENZ media, the high and low transmission of the structure is obviously affected by the ENZ condition [83]. For the hightransmission state at ENZ condition, the cavity mode is built up between two layers of nanorods.
However, for the low-transmission state deviation from ENZ condition, the mode is mainly reflected.
In addition, as mentioned in the Sec. C, the nonlinear response of the materials can be improved significantly by ZIMs. Therefore, the optical modulator based ons the zero-index cavity provides a double advantage of high mode transmission and strong nonlinearity enhancement in the fewnanorod-based design. In the infrared regime, the permittivity of Au can be written as [83]: zero-index cavity is high, which is marked by the blue line in Fig. 13(b). However, when 3000 = e TK (ON state), the system is deviation from ENZ condition, thus the transmission is low.
Importantly, the all-optical modulation overcomes the barrier that the modulation speed of tranditional modulator is slow, thus enables integrated nanoscale switches and modulators in Si waveguides [83]. The optical switching realized by the large nonlinear response of zero-index cavity can be well used in the applications of optical control [200][201][202][203][204][205][206].
In addition, zero-index cavity with PEC boundary has been used to realize the geometryinvariant resonant cavities in Fig. 3. Especially, this property can be extended to the open ENZ cavity, in which the switching between radiating and non-radiating modes enables a dynamic control of the emission. The schematic of the ENZ cavity for radiation switching is shown in Fig. 13(c), in which a vacuum spherical bubble, containing a quantum emitter (QE), is attached to a membrane.
When the QE is placed in the center 0 = x , the cavity mode corresponds to the non radiative mode, that is, the field will be confined in the cavity, as shown in Fig. 13(d). However, when the QE deviates from the center 3.5 = xm  , the cavity mode corresponds to the radiation mode, and the field radiates strongly to the outside of the cavity, which is shown in Fig. 13

Nonreciprocal transmission
In Sec. C.2, the enhanced MO effect have been presented in the 1D heterostructure, which is associated with the tunneling mode of EZI cavity. Recently, the nonreciprocal properties of MO ZIMs have attracted people's great attention [210,211]. In addition, unpaired Dirac point based on the 2D PCs have been demonstrated can be used to realize the MO ZIM [212]. The nonreciprocal transmission in nonlinear PT-symmetric ZIM has been proposed [213]. Here, we will introduce the realization of enhanced nonreciprocal transmission using MO zero-index cavity [82]. Figure 14(a) shows the schematic of a 1-D magnetophotonic crystal with a magnetized ENZ defect. Under an 34 applied magnetic field is in the y direction, the permittivity tensor of ENZ layer is: the MO effect of ENZ cavity mode is stronger than that of the normal cavity mode. The field enhancement as a function of the layer thickness and the incident angle is shown in Fig. 14 (b).
When the ENZ layer is embed into a 1-D PC, combining the field enhancement effect of the ENZ layer with the confinement effect of the PC barrier, the MO effect will be further enhanced. Based on the transfer matrix method, the transmission spectrum of the 1D PC with MO zero-index cavity with a high transmission, as is illustrated in Fig. 14(f). Therefore, the zero-index cavity can significant enhancement of MO effect and relize the obvious nonreciprocal transmission.

Collective coupling
The interaction between the cavity field and atoms plays an important role in the CQED.
Generally, in order to ensure the strong coupling between the quantum emitter and photons, it is necessary to place the QE at the maximum of the cavity field, which is very challenging in the manufacturing process. Therefore, due to the limited choice of quantum dots and the position uncertainty caused by the inhomogeneity of cavity field, it is difficult to improve the coupling strength between quantum dots and cavity field. In 2012, Jiang et. al., proposed theoretically and verified experimentally that the zero-index cavity with uniformed fields can be used to overcome 36 this limitation [76]. The position-independent normal-mode splitting in cavities filled with ZIMs is demonstrated in the circuit-based system, in which the oscillator is constructed by the metallic SRR. The position-independent mode splitting can be extended to study the collective coupling of randomly dispersed oscillators with zero-index cavity [77]. When nm. The corresponding transmission spectrum is shown in Fig. 16(b). The cavity mode ( 433 = f THz) with exist in the band gap is marked by the red dashed line. In Fig. 16(c), the field of the cavity mode is mainly localized at the cavity region and it decays exponentially with the position away from the cavity. The schematic of a zero-index cavity embedded in 1D PC is shown in Fig. 16(d). It is found that the frequency of cavity mode ( 433 = f THz) does not change after inserting the ZIM layer into the structure in Fig. 16(a). The corresponding field distribution of the cavity mode in the zero-index cavity is shown in Fig. 16(e). It is seen that the enhanced electric fields are uniform in the zeroindex cavity and conforms to Maxwell's equation and boundary conditions. When considering placing multiple oscillators [marked by the white crosses in Fig. 16(e)] into a zero-index cavity, the collective coupling will be verified, as shown in Fig. 16(f). The resonant susceptibility of the oscillator is described by [77,214]: ) and four ( 4 = N ) oscillators, it is seen that the splitting interval of frequency  enlarges when the number of oscillators increases, which is shown in Fig. 16(f). Especially, the mode splitting is proportional to NG and the splitting interval of frequency  for N oscillators with 0 = GG is equal to that for one oscillator with 0 = G NG . Therefore, one can tune the collective coupling between the oscillator can cavity field by varying the number of oscillators in the zero-index cavity. Moreover, the collective coupling has also been studied in a practical system that the ZIM cavity is realized by a 2D dielectric PC at Diraclike point [147]. The schematic of a 1D PC cavity embedded with 2D PC-based ZIM is shown in  In Sec. II, we systematically introduce the realization and physical properties of the zero-index cavity. Moreover, the EZI cavity with zero thickness is presented based on the tunneling mode in the heterostructure. The zero-index cavity not only enhance the interaction between light and matter to increase the nonlinear and MO response, but also realize the position independent coupling and collective coupling effect. Especially, the actively controlled ZIMs, such as electronic control [215], magnetic control [216] and optical pumping [217], greatly enrich the design flexibility of zero-index cavities. In addition, the low-loss ZIMs based on bound states in the continuum (BIC) have been proposed [218][219][220][221][222][223], whose physical mechanism is mainly to suppress the out of plane radiation at the Dirac-like point [221][222][223]. Therefore, the ZIMs and their designed zero-index cavities are gradually attracting people's attention. In addition to the important applications described in this section, they may also be used in many aspects, including absorber [224][225][226], shielding [157,227], splitter [228], sensor [229,230], isolator [231], antenna [232,233], and so on. 40

III. HYPERBOLIC METACAVITES
The interaction between light and matter depends on media dispersion in momentum space, which can be characterized using IFCs. The topological transition of dispersion from a closed sphere in isotropic media to an open hyperboloid in anisotropic media has been achieved by various means [234][235][236][237][238]. Especially, HMMs with hyperbolic dispersion exhibit many intriguing features and attracted people's great attention. One of the most important characteristics of HMM is that it can control the near field effectively. Because of the open IFCs, HMMs support propagating EM waves with large wave vectors [239][240][241]. The capabilities of HMMs have been demonstrated in numerous applications that utilize their exotic high-k modes, such as enhanced spontaneous emission [242][243][244][245][246][247][248], long-range interactions [249][250][251][252][253], superresolution imaging [254][255][256][257], optical pulling forces [258,259], and high-sensitivity sensors [260][261][262][263][264][265][266][267][268]. For electric/magnetic HMM, the principal components of its electric/magnetic tensor have opposite signs. Based on the Maxwell's equations, the dispersion relation of an electric anisotropic material ( can be written as [35][36][37][38][39][40]: where x k , y k , and z k denote the x , y , and z components of the wavevector, 0 k is the wave-vector in free space. //  and  ⊥ denote the permittivity components perpendicular and parallel to the xy palne. The first and second terms describe the EM response of the TE and TM  Fig. 18(c) for example, both //  and  ⊥ are positive in the first quadrant, and the IFC of the media is a closed ellipse or circle. Type II HMM and type I HMM media correspond to the second and fourth quafrants, respectively. In addition, the third quadrant ( 0  ⊥  , // 0   ) represents the ENG media, in which the EM wave will exist in the form of evanescent wave in the media [269,270]. The similar EM parameter distribution of anisotropic media for 0   is shown in Fig. 18(d).

Metal/dielectric multilayer hyperbolic media
According to Eq. (1), HMMs with // 0  ⊥  can be constructed based on the metal/dielectric multilayers. Recently, the rolled-up HMM also have been proposed base on the metal/dielectric multilayers [271]. The corresponding hyperbolic cavity have been widely studied [93,94,102]. Figure 19(a) shows an optical hyperbolic cavity array which is composed of silver/germanium multilayers [93]. Especially, in addition to electric HMM, the magnetic HMM can the associated topological transition can be realized considering the non-local effect of the metal/dielectric multilayer fishnet structure [272,273]. Figure 19(b) shows the schematic and the sample of the multilayer fishnet structure, which can can be used to relize the negative-index media [274] and HMMs [273]. The 3D optical metamaterials offer the effective avenue to explore a large variety of optical phenomena associated with ZIM, negative-index media, and HMMs. 42

Metal nanowire hyperbolic media
Wire metamaterials represent a large class of artificial EM structure, which corresponds the lattices of aligned metal rods embedded in the dielectric host [275]. An important group of subwavelength wire metamaterials possess extreme optical anisotropy based on the EMT: (1 ) where the filling ratio of the metal wires is equal to the radio of cross-sectional areas of the metal wires and the host dielectric shows one sample of the electric HMM in the visible range, which is constructed by an assembly of Au nanorods electrochemically grown into a substrate-supported, thin-film porous aluminium oxide template [260]. The guided mode in HMM with strong field localization is quite similar to the surface plasmon mode of a solid metal film, thus the nanowire hyperbolic media will provide very high sensitivity to refractive index changes [260]. In addition, similar to the metal/dielectric multilayers, metal nanowire structure not only can be used to design electric HMM, but also can realize the magnetic HMM associated with magnetic topological transition of IFC [276].

Circuit-based hyperbolic media
In the 2D TL system, circuit-based metamaterials can realize various electromagnetic parameters under EMT [163,164]. Recently, HMMs with flexbile EM parameters can be well constructed using circuit-based metamaterials in the microwave regime [277][278][279]. For example, the interesting epsilon-near-pole HMM can be easily constructed base on the circuit-based metmaterials [253]. Figure 19(d) shows a circuit-based HMM sample, in which the real part of anisotropic permeability of the system can be tuned by changing the lumped elements [277]. In addition to the real part, the imaginary part of the EM parameters can be directly controlled by adding the lumped resistors [166,280]. The actively controlled HMM has been proposed by using variable capacitance diodes under appling external voltage [137]. Importantly, the hyperbolic topological transition and the novel linear-crossing metamaterials (LCMMs) have been proposed and demonstrated in the circuit-based system [281]. So far, circuit-based HMMs have attracted much attention in terms of various applications: emission pattern control [137], long-range atom-atom interaction [253], collimation [280], super-resolution imaging [281], spin-Hall effect [282], etc.

Anomalous scaling law
Because of the special HMM dispersion, the propagation direction of a wave in a HMM is different from that in a normal anisotropic material with a closed IFC. For the traditional optical cavity made of dielectric, whose IFC is a closed circle and the supported wavevector is limited. In addition, the isotropic property of dielectric making the tranditional optical cavity is directional independent, as shown in Figs. 20(a) and 20(b). The dashed lines denote the IFC at a frequency slightly above the frequency of the solid lines, which indicating the gradient direction of the IFCs of the dielectric [40]. As we all know that the wavevector increase with the frequency increase because of their linear relationship / = k n c  . Specially, when the wavevector in the x direction is fixed (marked by a blue dotted line along the z direction), the wavevector at a higher frequency is 44 larger than one at a lower frequency 21  zz kk , as shown in Fig. 20(a). This positive correlation between wavevector and frequency is marked by the sign '+' for see. Similarly, when the wavevector in the z direction is fixed (which is marked by a blue dotted line along the x direction), the wavevector at a higher frequency is also larger than one at a lower frequency in Fig. 20(b). However, the wavevector property of the HMM is significant different along x and z directions, which is presented in Figs. 20(c) and 20(d). When the wavevector in the x (z) direction is fixed, the wavevector at a higher frequency is larger (smaller) than one at a lower frequency 21  zz kk ( 21  xx kk ). The abnormal wavevector properties of the HMM is marked '-' in Fig. 20(d). Therefore, it can be expected that the abnormal cavity property appears for the HMM in x direction. For tranditional optical cavities composed of dielectric, the frequency increases with the increase of mode order due to the standing wave condition. However, this scaling law of the cavity mode will be modified in the HMMs. Especially, the anisotropic scaling law have been demonstrated in the hyperbolic cavity based on metal/dielectric multilayers [93]. The schematic of the hyperbolic 45 cavity consists of alternating thin layers of silver and germanium is shown in Fig. 21(a). The hyperbolic cavity modes were excited with a TM polarized plane wave propagating along the zdirection. Especially, the first five lowest order modes along z direction are studied, which are labeled (1, 1, 1), (1, 1, 2), (1, 1, 3), (1, 1, 4), and (1, 1, 5), respectively. The corresponding simulated electric field z E distributions of these modes are shown in Fig. 21; here, it can clearly be seen that the higher-order mode is found at a lower resonant frequency in the hyperbolic cavity. The anomalous scaling law of hyperbolic cavity also have been clearly demonstrated based on the circuit-based hyperbolic cavity in the microwave regime [109]. The schematic of the near-field 46 detection system is shown in Fig. 22(a). A subminiature version A (SMA) connector that functions as the source for the system is placed at the center of the sample as a vertical monopole to excite the circuit-based prototype. A small homemade rod antenna is employed to measure the out-of-plane electric field at a fixed height of 1 mm from the planar microstrip. Figure 22

Size independent cavity mode
Hyperbolic cavity exhibits the abnormal scale property along the direction that perpendicular to the optial axis of the HMM. As introduced in Fig. 20 that the abnormal (normal) cavity property appears for the HMM in x(z) direction. Therefore, a question naturally arises: can the size independent cavity mode be realized in hyperbolic cavity by changing the different length in different directions? The dependence of the hyperbolic cavity mode on the structure size in the metal/dielectric multilayer structure (same as the structure in Fig. 21) is studied in Fig. 23. The simulated IFC of the effective HMM for TM polarized wave is obtained from the Fourier spectrum in Fig. 23 Fig. 23(b). It is seen that five cavities with different size support identical optical modes with the same resonant frequency (150 THz) and the same mode order (1, 1, 1) [93]. This property has also been experimentally demonstrated from the transmission spectra in Fig. 23(c). It can be clearly seen that the hyperolic cavity with different size resonate at the same resonant frequency The dependence of the hyperbolic cavity mode on the structure size is also demonsrated based on the circuit-based hyperbolic cavity for TE polarized wave [109]. The size of the hyperbolic cavity is identified by the number of unit cells in the x and z directions. The schematic of the circuit-based hyperbolic cavity with Nx = 7 and Ny = 5 is shown in Fig. 24(a). In fact, the realization of size independent cavity mode in hyperbolic cavity is come from the anisotropic scaling law. Especially, the effect of changing the length of hyperbolic cavity in x and z directions on the cavity mode is presented in Figs. 24(b) and 24(c), respectively. The cavity mode C11 in different cavities are marked by red circles for visualization. It can be clearly seen that the frequency of mode C11 blueshifts 49 (redshifts) with decreasing length in the x(z) direction. Because the dependence of the frequency of cavity mode on the structure's size is opposite to that in the x and z directions, the hyperbolic cavities with different size resonant at same frequency can be realized. Figure 24(d) shows the measured y E spectrum for two circuit-based hyperbolic cavities: Nx = 7, Ny = 5 and Nx = 4, Ny = 4. While the size of the cavity with Nx = 7, Ny = 5 is significantly larger than that of the cavity with Nx = 4, Ny = 4, the frequency of cavity mode C11 is the same for both cavities. Therefore, the size independent cavity mode is experimentally demonstrated in the circuit-based hyperbolic cavity. Especially, the circuit-based hyperbolic cavities not only extend previous research work on hyperbolic cavities to magnetic HMMs, but they also have a planar structure that is easier to integrate and has a smaller loss. The hyperbolic cavities may enable their use in some microwave-related applications, such as in high-sensitivity sensors and miniaturized narrowband filters [109].

Continuum of high-order modes
For conventional cavity mode, any mode order appears at one single frequency. It is because that when the incident angle of the light is fixed, the wavevector in the propagating direction is given. However, once the special wavevector in the propagating direction corresponds to more than one frequency, the mode continuum allowing multiple manifestations of the same order can be realized [103]. The specialy dispersion of HMM not only enables the novel resonant modes with anomalous scaling law [92,93,109] but also may be used to form a mode continuum [103]. The schematic of special hyperbolic cavity for mode continuum is composed of a core layer of HMM, consisting of metal/dielectric multilayers, and two cladding mirrors, as shown in Fig. 25(a). In this case, the the wave-vector component z k solely defines the resonance condition, and thus determines the mode spectrum. The z k of light with incident angle  in the HMM core can be written as [103]: Interestingly, z k is nonmonotonic with angular frequency  at a fixed  in the HMM. As a result, the mode continuum can be realized in the hyperbolic cavity. Figure 25(b) shows the dispersion relation of the microcavity for TM polarization. There are some interesting properties can be found. (1) There are two curves correspond to the cavity mode of the second order. Especially, the slope of the lower frequency second-order mode is negative, which corresponds to a anomalous cavity mode with negative group velocity. Therefore, this configuration gives rise to modes of identical orders appearing at different frequencies.
(2) A novel unconventional zeroth-order mode based on the phase compensation appearing at 61 1.08 10 − = x km . (3) Between the anomalous second-and zeroth-order modes, a cluster of high-order modes exist, forming a mode continuum [103]. The continuum of high-order modes is enlarged at the inset of Fig. 25(b). It is seen that the  Fig. 25(b). Therefore, the continuum of high-order modes is fulled verified and the related results may be applied to a series of practical microcavity applications, such as the switching and fiters [103].

Lasers
One the most important applications of the optical cavity is the lasers. However, it is difficult to miniaturize the conventional cavity, mainly because the FP resonance condition determines that the size of the cavity is comparable to the wavelength. Although the localized surface plasmon resonance of metal can be used for the small volume localization, the intrinsic loss of metal has a great influence on the Q value of the cavity. The high-k modes of HMM provide a new way to surpass these limitations and realize the subwavelength optical cavity with low-loss, which make the HMM cavities attractive candidates for ultra-small low-threshold lasers [99,283,284]. The low threshold spaser based on deep-subwavelength spherical hyperbolic cavities is shown in Fig. 26. Figure 26(a) shows the schematic of the hyperbolic cavity composed of alternating metal and dielectric layers. Take a high-order cavity mode (l=4) for example, the simulated field distributions of effective medium and multilayer structure are shown in Fig. 26(b) and 26(c), respectively. It is seen that the metal/dielectric multilayers can be well used in the hyperbolic cavity structure [94].
For a spherical hyperbolic cavity with 7 alternating layers of silver and dielectric, the extinction efficiency (Qext) spectrum is shown in Fig. 26 In addition to the gain efficiency, the radiation feedback is also an important factor in the design of miniaturized lasing devices. Recently, Shen et al., experimentally demonstrated that hyperbolic cavity can be used to construct deep-ultraviolet (DUV) plasmonic nanolaser without a long structure, 54 which paves the way for the design of subwavelength nanolaser devices [100]. The DUV nanolaser is realized by placing a hyperbolic cavity on a multiple quantum-well (MQW), and the corresponding schematic is shown in Fig. 27(a). Importantly, the designed hyperbolic cavity merges plasmon resonant modes within the cube and provides a unique resonant radiation feedback to the MQW [100]. Figure 27 times, respectively. Therefore, the hyperbolic cavity can be used to design low threshold lasers.
Especially, even when the HMM-MQW sample showed only spontaneous emission, the hyperbolic cavity-MQW sample will show the lasing in the same pump power range [100]. Moreover, figure   27(d) shows the PL spectrum versus pump power of hyperbolic cavity-MQW sample and the PL peak intensity versus pump power is shown in the inset. It is seen that a sharp lasing peak at 289 nm emerges from a broad spontaneous emission spectrum at a pump power around 106 kW cm -2 , which corresponses the lasing threshold of the hyperbolic nanolaser. Therefore, based on the high efficiency radiation feedback of hyperbolic cavity mode, the plasmonic nanolaser using a hyperbolic metacavity on an MQW sample provides a new way to enhance the light-matter interaction. The results in Fig. 27
Especially, the dispersionless bandgap of 1D hypercrystal has been proposed based on the condition of phase variation compensation [286]. One important application of the dispersionless gap is to design the dispersionless cavity mode. Figure 28(a) shows a hypercrystal with a defect layer in the center of the structure. The hypercrystal consists of alternating thin layers of HMM and dielectric, and the effective HMM is realized by the metal/dielectric multilayers, which is shown in the inset of Fig. 28(a). For the hypercrystal [(CD)4B]3B[(CD)4B]3, in which B and C denote the dielectric and D is the metal, the calculated transmittance as a function of the incident angle and frequency is shown in Fig. 28(b). It is seen that the cavity mode inside the dispersionless gap remains nearly invariant with incident angles, which indicates the dispersionless cavity mode is realized in the 1D hypercrystal. Especially, the transmittance spectra of the structure at three representative angles (0 deg, 30 deg and 60 deg) is shown in Fig. 28(c). It is seen that the positions of cavity modes at different incident angles are nearly unchanged, which is in accordance with Fig. 28(b). In addition, the Q factor of cavity modes nearly unchanged at three incident angles. The dispersionless cavity modes of hyperbolic cavity will possess significant applications for all-angle filters. Moreover, considering the semiconductor HMM (tuned dynamically through optical pumping on a picosecond scale) or the graphene-based HMM (tuned dynamically through an external voltage), the working frequency of all-angle filter can be flexibly tuned [293][294][295][296][297][298][299][300]. The dispersionless cavity mode is also very useful in nonlinear wave mixing and phase-matched [301,302]. For the traditional cavity mode, phase matching plays an important role in the coherent nonlinear optical process, which can not meet the requirements of all the incident angles [301]. By contrast, the dispersionless cavity mode may provide all-angle phase matching in coherent nonlinear optical process.

Wide-angle biosensors
The edge states in the 1D heterstructure composed of the metal layer and PCs have the properties of localization and polarization-dependence, which can be used to design ultrasensitive optical sensors based on the singularity of the ellipsometric phase (i.e., reflection phase difference between two orthogonal polarizations  = −

TM TE
 ) [303,304]. The underlying physical mechanism is that the frequency of edge modes of TE and TM polarized waves is not overlapped, thus the ellipsometric phase will changed significantly and it can be used to increase the sensitivity of the sensors. However, the bandgap of tranditional 1D all-dielectric PCs for both TE and TM polarized wavers is blueshift along with the increase of the incident angle, thus the edge states of two polarized waves will blueshit. For the case of small angle incidence, the reflection phase difference of two polarized waves is not obvious, which can not achieve high sensitivity. Therefore, 58 for the traditional 1D PC, the incidence angle range of high sensitivity sensor is small. In the integrated optical system, the incident light often has a certain beam width and angular divergence, which will lead to the further reduction of sensing sensitivity. The special dispersion of hyperbolic cavity mode can be used to realized the red shift band gap of TM polarization and the blue shift band gap of TE polarization in 1D PC with HMMs [265]. The schematic of the heterostructure composed of a metal layer and a 1D hypercrystal (in which the HMM layer is realized by the metal/dielectric multilayers) is shown in Fig. 29(a). The reflectance of hypercrystal as a function of incident angle and wavelength is shown in Fig. 29(b). It is seen that the hyperbolic cavity mode in the hypercrystal can realize the red shift in TM polarization and blue shift in TE polarization, thus greatly broadening the working angle range of high sensitivity biosensor. Considering a thick liquid cell filled with bio-solution, whose refractive index is nBio, the performance of the biosensor based on the hyperbolic cavity at a small incident angle nm. In addition, this biosensor based on the hyperbolic cavity has near-linear response in a relative wide range of refractive index, as shown in Fig. 29(d).
Besides, the sensitivity of the proposed biosensor also has been presented in Fig. 29(d), indicating that the proposed biosensor can work with a high refractive index resolution even at a small incident angle. Therefore, hyperbolic cavity realized by the hypercrystal enable the new designs of biosensors with wide-angle and ultrasensitive properties [265].

IV. CONCLUSIONS AND OUTLOOK
The optical cavity with metamaterial (i.e. metacavity) is of great significance in fundamental and applied physics. On the one hand, the optical cavity can confine the EM wave in a small scale, which can effectively improve the photon DOS thus enhancing the interaction between light and the matter. On the other hand, metamaterials can control the transmission, radiation and coupling of EM wave arbitrarily, so it can realize many novel physical phenomena which are difficult to realize in traditional optical cavities, and can be used to construct more functional optical devices.
In this review, two kinds of special metacavities, zero-index metacavity and hyperbolic metacavity, are systematically introduced. (1) For the zero-index cavity, it can achieve novel geometry (shape, topology, size) independent cavity mode, uniformed field enhancement, and directional radiation. Especially for the matched EZI heterostructure structure, it provides a good 60 solution for the construction of subwavelength optical cavity, and has been proved to significantly enhance the interaction between light and matter, such as nonlinear enhancement, magneto-optical effect enhancement and so on. In addition, it is interesting that the heterostructure composed of two kinds of topological distinguished 1D PCs with symmetrical configuration can realize the topologically protected edge state, which has been confirmed from microwave to ultra-X-ray band.
This is of great significance in the realization of optical devices sensitive to structural errors.
Moreover, the applications of zero-index metacavity in switching, nonreciprocal transmission and collective coupling are introduced in detail.
(2) For the hyperbolic metacavity, it has been observed that it has anisotropic resonance characteristics, and can observe the anomalous scaling law, scale independent cavity modes and continuous cavity modes. In particular, the special dispersion of hyperbolic metacavity can be used to realize miniaturization of low-loss laser. And the special mode distribution of hyperbolic cavity mode can be used to enhance the radiation feedback and reduce the lasing threshold. In addition, for the hypercrystal constructed by HMMs, by adjusting the structure parameters, the dispersionless cavity mode can be realized and used for all-angle filters.
Similarly, based on the dispersion control, hypercrystal can also be used to design highly sensitive biosensors. In general, the zero-index and hyperbolic metacavities provide new and efficient means for EM wave control, and provide new ways for the design of novel optical devices, which are expected to be applied in the future of photonic integration.
As a special kind of anisotropic material, it has the characteristics of both ZIM and HMM. Therefore, the performance of this new metamaterial for optical cavity design needs to be explored in the future.