Photonic Dirac Nodal Line Semimetals Realized by Hyper-crystal

Recently, the gapless Dirac/Weyl nodal semimetals with linear dispersion and topologically protected modes degeneracy are rapidly growing frontiers of topological physics. Especially, type-I, type-II, and critical type-III nodal semimetals are discovered according to the tilt angles of the Dirac/Weyl cones. Here, by introducing hyperbolic metamaterials into one-dimensional photonic crystals, we design the"hyper-crystal"and study the photonic four-fold degenerate Dirac nodal line semimetals (DNLSs) with two types of perpendicularly polarized waves. Moreover, the flexibly controlled photonic DNLSs using the phase compensation effect of hyperbolic dispersion are studied. Our results not only demonstrate a new platform to realize the various photonic DNLSs, where the optical polarization plays the role of electron spin in electronic DNLSs, but also may pave a novel way to explore the abundant Dirac/Weyl physics in the simple classical wave systems.

controlled by the compressive strain [55,57,59], lattice constant [56,58,61] and temperature [60]. All above mentioned modulations for the TPT resort to the change of coupling coefficient between neighbor unit cells, while the unit cells themselves remain unchanged, thus leading to the displacement of Weyl nodes in the process of TPT. This uncontrollable displacement will affect the observation of ideal TPT in photonic fourfold Dirac semimetals.
Photonic crystals (PCs) provide a powerful platform for manipulating the lightmatter interactions [62][63][64][65][66]. In 2021, Hu et al., uncover that double-bowl state in photonic Dirac nodal line semimetal in the one-dimensional (1D) PCs [19]. This pioneering work provides a new mechanism to realize a photonic type-II Dirac nodal line semimetal (DNLS). However, for the conventional 1D PCs composed of two types of dielectrics, the photonic bands will blueshift with the increase of the incident angle, thus the type-I, type-III DNLSs and even the TPT remain elusive in the 1D PCs. The recently emergent hyperbolic metamaterials (HMMs) have attracted people's great attention for their extraordinary optical properties, such as enhanced photonic density of states [73], abnormal coupling effect [74][75][76], unidirectional propagation [77][78][79][80], and so on. As one kind of anisotropic artificial media, HMMs can compensate the propagating phases to an unprecedented extent that usual dielectrics cannot attain [64], which provide the possibility to overcome the general angle-dependent limitation of bands and further explore the various DNLSs and TPT in 1D PCs [67][68][69][70][71][72].
This work is organized as follows: Sec. II covers the design of hyper-crystal composed of electric-magnetic (EM) HMM and dielectric, which satisfies the compensation condition 1 = 2 . Especially, the photonic type-I DNLS is realized based on two type-I Weyl nodal line semimetals (WNLSs) of perpendicularly polarized waves. In addition, the TPT and type-III DNLS are demonstrated by tunning the thickness ratio of EM HMMs and dielectric layers; in Sec. III, hyper-crystal constructed by simple electric HMM and dielectric ( 1 ≠ 2 ) is carried out to study a hybrid photonic DNLS with a type-I WNLS and a type-II WNLS. Finally, Sec. IV summarizes the conclusions of this work.

II. TPT and various photonic DNLSs realized by the hyper-crystal
DNLSs can be described by four-band Hamiltonian, which is expanded in Dirac gamma matrices. Considering the form of tensor product of two Pauli matrices and ( = , , ) acting on two isospin degrees of freedom, the effective Hamiltonian of photonic DNLSs can be written as [19] = 0 ⊗ [( − 1 ) + ( − 2 ) ], where and represent the band index and pseudospin index, respectively. 0 is the identity matrix. For the nodal line semimetals of co-dimension 2, and are the two compound variables of momentum, while 1 and 2 are constants. The effective Hamiltonian in Eq. (1) can be decoupled into two identical blocks, which provide two-fold Weyl nodal lines with different pseudospins α and β, respectively.
The corresponding eigenvalues are given by In order to understand the band structure, figure 1 shows the space of energymomentum in a projection space − − . We can see that the four-fold degenerate nodal line is a 2D where denotes the rotation angle around the axis of . The corresponding energy eigenvalues turn into According to Eq. (4), type-I and type-II DNLSs are realized for ∈ (nπ − /4, nπ + /4) and ∈ (nπ + /4, nπ + 3 /4), which are shown in Fig. 2(a) and Fig. 2  .
Similarly, for the TM waves, the ( 2 + 1 ) ℎ band and the ( 2 + 1 + 1) ℎ band intersect at the frequency The ideal degeneration condition 1 = 2 corresponds to the photonic DNLS with 1 = 2 in the effective Hamiltonian in Eq. (3). According to Eq. (5), this ideal degeneration condition is equal to ̃( 0 ) =̃( 0 ) because ̃=̃ is always satisfied for the isotropic material. Therefore, the EM HMMs will be considered as layer A to realize the various photonic DNLSs. Especially, the EM HMM is mimicked by subwavelength -negative (ENG) media/ -negative (MNG) media/dielectric stacks as (CDE)S in Fig. 3(a). = 50 and = 8 denote the period numbers of the PC and HMM, respectively. Both the permittivity of ENG media and the permeability of MNG media are described by Drude model [66]. The electromagnetic parameters and the thickness of different layers are shown in Table 1.
Although this paper is a theoretical research work, two kinds of single-negative media have been widely studied by periodic arrays composed of meta-atoms [63,81].
Therefore, the results studied in this work are feasible in experimental observation in the future. Based on the effective-medium theory (EMT) in the long-wavelength limitation, the effective electromagnetic parameters of layer A as (CDE)S are given by [64]  Combining Eq. (5) with Eq. (8), we can deduce that the ideal degeneration condition 1 = 2 of photonic DNLS is also equivalent to In order to intuitively determine the frequency of DNLS, ∥ / ⊥ − ∥ / ⊥ is shown by the green dashed line in Fig. 3  The phase variation compensation effect of HMM provides an effective manner to study the various photonic DNLSs [64]. Take the type-I DNLS for example, it can be obtained by Using the transfer-matrix method, we calculate the transmittance spectra of the Nevertheless, this simple 1D hyper-crystal can also be used to study other novel physical properties. In the next section, we will systematically introduce the interesting hybrid photonic DNLS in the simple 1D hyper-crystal composed of electric HMM and dielectric.

III. Hybrid photonic DNLS realized by the hyper-crystal
In above section, the various photonic DNLSs have been demonstrated by the 1D hyper-crystal satisfying the ideal degeneration condition, which indicates the identical rotation angle for TE and TM waves. In fact, the rotation angle of linear dispersion for different pseudospins can also be tuned differently as 1 ≠ 2 In this case, the effective Hermitian model in Eq. (3) can be expressed as . The eigenvalues of Eq.
According to Eq. (12), the hybrid DNLS composed of two WNLSs with different semimetal phases can be obtained, as shown in Fig. 6(a).
where ∞ = 3.9 is the high-frequency permittivity. In addition, ℏ = 2.48 and ℏ = 0.016 . and denote the plasma frequency and damping frequency, respectively. Potassium titanyl phosphate (KTiOPO4, KTP) [83] and Zinc silicon arsenide (ZnSiAs2) [84,85] are selected for dielectric layers B and C, respectively. The refractive indexes of layer B and layer C are close to 1.47 and 3.3, and there is slight dispersion at the working frequency (see Appendix C for details). Because all of the materials are non-magnetic, the permeability of each layer in Fig. 6  For the 1D hyper-crystal which does not satisfy the ideal degenerate condition, that is 1 ≠ 2 , 1 / 1 = 1 / 2 should be taken into account for realizing the hybrid DNLS. As a result, the parameter groups meeting the degeneration become Recently, the non-Hermitian property associated with Dirac point by considering the loss of the system has also attracted people's great attention [86]. Here, the real parts of // and ⊥ are about ten times larger than their imaginary parts, so the loss can be nearly ignored. As a result, multiple concentric Weyl nodal rings can be realized for the TM wave in the 1D hyper-crystal.
At last, we distinguish above introduced two hyper-crystals from the band structures in Fig. 9. The wave vector in the xoy plane is = √ 2 + 2 = 0.76 0 . The band structure of the 1D hyper-crystal [(CD)15B]10 is shown in Fig. 9(a). The results of TE and TM waves are marked by blue and red lines, respectively. We can see that two WNDLs are not coincident on the same . The degenerate points for TE and TM correspond to = 0 and = /Λ, respectively. However, two WNDLs are totally overlapped on the same = 0 for the 1D hyper-crystal [(CD)30B2]10, as shown in Fig.   9(b). As a result, the novel photonic hybrid DNLS is demonstrated for the structure [(CD)30B2]10 in Fig. 9(b).

IV. DISCUSSION
In summary, we reveal the various photonic DNLSs and the associated TPT in 1D hyper-crystal, in which the HMM is composed of two types of single-negative media.
Moreover, considering the widely used electric HMM composed of metal/dielectric stacks, we observe the novel hybrid DNLS in the 1D hyper-crystal. Our results not only demonstrate various DNLSs in optical regime, but also provide a powerful platform for study on relevant Dirac and Weyl physics. Additionally, the photonic DNLS realized by two types of perpendicularly polarized waves may be used to the polarization independent angle filters. Especially, for the TE waves, the ( 1 + 1 ) ℎ band and the ( 1 + 1 + 1) ℎ band degenerate at
More attention is paid to the phases supporting both TE and TM polarized waves.   Considering the 1D hyper-crystal with electric HMM under the condition 1 ≠ 2 , we also study the effectiveness of the EMT for the photonic DNLS. Based on the EMT, the transmission and reflectance spectra of 1D hyper-crystal [AB]10 for TE and TM waves are shown in Fig. 13. Compared the lossless case in Fig. 13(a) and lossy case in Fig. 13(b) for TE wave, we can clearly see that the loss of the system will blur the transmission spectra. However, the photonic WNLS also can be clearly observed in the reflectance spectra, as shown in Fig. 13