Abstract
The Dirac equation for relativistic electron waves is the parent model for Weyl and Majorana fermions as well as topological insulators. Simulation of Dirac physics in threedimensional photonic crystals, though fundamentally important for topological phenomena at optical frequencies, encounters the challenge of synthesis of both Kramers double degeneracy and parity inversion. Here we show how typeII Dirac points—exotic Dirac relativistic waves yet to be discovered—are robustly realized through the nonsymmorphic screw symmetry. The emergent typeII Dirac points carry nontrivial topology and are the mother states of typeII Weyl points. The proposed alldielectric architecture enables robust cavity states at photoniccrystal—air interfaces and anomalous refraction, with very low energy dissipation.
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Introduction
Dirac’s famous equation for relativistic electron waves^{1} is the foundation for both the quantum field theory and the later topological insulators and semimetals.^{2,3,4} There has been a trend in the simulation of relativistic waves and topological states in classical dynamics such as electromagnetic,^{5, 6} acoustic^{7,8,9} and mechanical waves,^{10, 11} mostly in 2D systems. Many novel phenomena in electromagnetism are discovered along this paradigm, such as photonic Zitterbewugung,^{12} zeroindex dielectric metamaterials,^{13} deformation induced pseudomagnetic field for photons,^{14} as well as photonic topological insulators with^{15,16,17,18,19} and without^{5, 20,21,22,23,24} timereversal \(\left( {\cal T} \right)\) symmetry. Recently, such simulation develops from 2D to 3D,^{25,26,27,28,29,30,31,32} exposing to larger wavevector and configuration space that may lead to richer physical phenomena, particularly using \({\cal T}\)invariant materials which are more feasible for highfrequency (e.g., infrared or visible) applications.
Due to its bosonic nature, i.e., \({{\cal T}^{\rm{2}}}\) = 1, the fourfold degenerate photonic Dirac points (DPs) can be created only when Kramers double degeneracy (“spin”) and parityinversion (“orbit”) are synthesized. These two elements are also at the heart of Z _{2} topology in \({\cal P}{\cal T}\)symmetric (\({\cal P}\) is inversion) systems, as revealed in the seminal work of Fu and Kane.^{33} Although there have been a few fine designs^{29, 31, 32} showing the connection between typeI DPs and the Z _{2} topology, typeII DPs [in analog of typeII Weyl Points (WPs),^{34, 35} see Fig. 1] have never been explored in photonics or in other classical/bosonic waves. In this work, we demonstrate the creation and destruction of typeII DPs in PhCs. Besides, we unveil screw symmetry, a fundamental type of nonsymmorphic symmetry, as an effective tool for the creation of DPs.
The distinction between symmorphic (e.g., pointgroup) and nonsymmorphic spatial symmetries in crystals lies in whether the spatial origin can be preserved. Nonsymmorphic symmetries cannot preserve the spatial origin but translate it by a fraction of the crystal period. The screw symmetry, a rotation accompanied with a fraction of lattice translation, is an elementary nonsymmorphic symmetry. So far, the role of screw symmetry on the realization of topological states in classical/bosonic waves has not yet been explored. It is known that screw symmetries lead to double degeneracy for all Bloch states on certain planes in the Brillouin zone (BZ).^{36, 37} Thus the screw symmetries can create a large wavevector space for the simulation of DPs and Z _{2} topology in classical dynamics. The screw symmetries become particularly powerful when there are two orthogonal screw axes, since the product of the two screw rotations is essentially the parity required by the DPs. In this way both the “Kramers” double degeneracy (“spin”) and parityinversion (“orbit”) can be simultaneously synthesized through screw symmetry.
Based on these symmetry considerations we propose an alldielectric tetragonal PhC with screw symmetries for the creation of both typeII and typeI DPs. Our symmetryguided approach is robust: DPs emerge for a variety of geometry and materials. We demonstrate the nontrivial topology of the DPs by studying the edge states. These nonchiral edge states, differing from the chiral edge states of Weyl points (WPs), are below the lightline and form resilient cavity states on PhCair interfaces. Moreover, we show that both typeII and typeI WPs can be derived from these DPs when symmetry is reduced. Anomalous refraction with one or two pairs of opposite refraction angles is predicted for typeII DPs/WPs. To the best of our knowledge, this is the discovery of typeII DPs in photonics and a proposal of typeII WPs in alldielectric PhCs. Our findings may enable unprecedented control of light at optical frequencies using dissipationless materials.
Results
Alldielectric photoniccrystal architecture
We study an alldielectric PhC with tetragonal symmetry of space group P4_{2}/mcm (see Fig. 2) to illustrate the symmetryguided approach. In each unit cell, there are two dielectric blocks (painted as yellow and green in Fig. 2) of the same shape and permittivity ε _{ b }, embedded in a polymer matrix of permittivity ε _{ m } = 1.9. We shall first set ε _{ b } = 16 and the geometry parameters l = 0.5, w = 0.2, and h = 0.5 (lattice constant a ≡ 1). We show later that DPs emerge for other material/geometric parameters as well. These PhCs can in principle be fabricated using layerbylayer methods with the current technology^{38, 39} for infrared frequencies. We use the MIT PHOTONIC BANDS (http://abinitio.mit.edu/wiki/index.php/MITPhotonicBands) to calculate the bulk and surface photonic bands. The tetragonal symmetries crucial to our study are the twofold screw symmetries \({S_x}: = \left( {x,y,z} \right) \to \left( {\frac{1}{2} + x,\frac{1}{2}  y,\frac{1}{2}  z} \right)\) and \({S_y}: = \left( {x,y,z} \right) \to \left( {\frac{1}{2}  x,\frac{1}{2} + y,\frac{1}{2}  z} \right)\) (illustrated in Fig. 2b), and the 180° rotation around the z axis, C _{2}: = (x, y) → (−x, −y). The remaining symmetries are listed and analyzed in the Supplementary Materials.
Photonic Kramers double degeneracy
Antiunitary operators: \({\Theta _i} \equiv {S_i} * {\cal T}\) (i = x, y) are created to elucidate the power of the screw symmetry. The effect of the timereversal operation \({\cal T}\) on a photonic Bloch wavefunction \({\Psi _{n\vec k}}\left( {\vec r} \right) = {\left( {{{\vec e}_{n\vec k}},{{\vec h}_{n\vec k}}} \right)^T}\) is mostly complex conjugation, \({\cal T}{\left( {{{\vec e}_{n\vec k}},{{\vec h}_{n\vec k}}} \right)^T} = {\left( {\vec e_{n\vec k}^ * ,  \vec h_{n\vec k}^ * } \right)^T}\). Since \(\Theta _x^2 = S_x^2 = {T_{100}}\) where T _{100} is a spatial translation by the vector (1, 0, 0), acting Θ_{ x } twice on a photonic Bloch state gives \(\Theta _x^2{\Psi _{n\vec k}}\left( {\vec r} \right) = {e^{i{k_x}}}{\Psi _{n\vec k}}\left( {\vec r} \right)\) (see details in Methods). Θ_{ x } transforms (k _{ x }, k _{ y }, k _{ z }) into (−k _{ x }, k _{ y }, k _{ z }) and is hence invariant on the k _{ x } = π plane, where we find
The above equation, as an analog of the Kramers theorem for fermions, guarantees that all photonic states on the k _{ x } = π plane are doubly degenerate (see Fig. 2c). Similarly, all Bloch states are doubly degenerate on the k _{ y } = π plane due to \(\Theta _y^2 =  1\).
Dirac points
For the creation of DPs, the next important step is to realize parityinversion. Here the parity is defined through C _{2}, which is invariant on the MA line, k _{ x } = k _{ y } = π. The product of the two orthogonal screw rotations yields, S _{ y } S _{ x } = T _{010} C _{2} and S _{ x } S _{ y } = T _{100} C _{2}. On the MA line one hence has
Elegantly, the above algebra reveal that the two degenerate states in any doublet have the same eigenvalue of the C _{2} rotation. Such eigenvalues c _{2} = ±1 precisely represent the parities of the photonic states in the xy plane.
It has been shown in ref. 29 that a DP with synthetic Kramers double degeneracy and parityinversion has nontrivial topological properties. In fact, such DPs are monopoles of the SU(2) Berryphase gauge fields.^{29} The topological charge of a DP is defined by the integral of the SU(2) gauge fields over a tiny sphere containing the DP. It was proved in ref. 40 that in systems with \({\cal P}{\cal T}\) symmetry, the calculation of the topological charge of a DP can be simplified as
where \(c_2^ \) is the parity of the lower branch of the Dirac cone, and \(k_0^ + = {k_0} + {0^ + }\) \(\left( {k_0^  = {k_0}  {0^ + }} \right)\) is the wavevector slightly larger (smaller) than that of the DP on the z direction, k _{0}. Since the total topological charge of photonic bands in the BZ is strictly zero, DPs emerge in pairs with opposite N _{ DP } at opposite wavevectors. Figure 2d shows that there are four DPs in the first six bands, due to the crossing between the p and ddoublets.
Our symmetryguided paradigm provides a robust and effective approach toward topological DPs: Fig. 2e shows that the emergence of DPs is quite robust to the shape and permittivity of the dielectric blocks (more examples are given in the Supplementary Materials), since any crossing between bands of different parities on the MA line leads to DPs.
The spinorbit physics of the Dirac points can be understood via a symmetrybased \(\vec k\) · \(\vec P\) theory (see Supplementary Materials for details). The Hamiltonian can be constructed using the basis of the two doublets, p _{1}, p _{2}, d _{1} and d _{2} [Fig. 2d]. The combination of these states, \(\left {{p_ \pm }} \right\rangle = \frac{1}{{\sqrt 2 }}\left( {\left {{p_1}} \right\rangle \pm i\left {{p_2}} \right\rangle } \right)\) and \(\left {{d_ \pm }} \right\rangle = \frac{1}{{\sqrt 2 }}\left( {\left {{d_1}} \right\rangle \pm i\left {{d_2}} \right\rangle } \right)\), carry finite total angular momenta (TAM) that are opposite for the + and − states (see Supplementary Materials). Emulating fermionic spin and orbit with the TAM and parity, respectively, we find the following photonic Hamiltonian for a DP,
where ω _{0} is the frequency of the DP, v is the characteristic group velocity. \(\hat 1\) is the 2 × 2 identity matrix, \(\widehat {\overrightarrow \sigma }\) is the Pauli matrix vector. The dimensionless \(\vec k\) · \(\vec P\) parameter η here plays an role to distinguish the typeI \(\left( {\left \eta \right < 1} \right)\) and typeII \(\left( {\left \eta \right >1} \right)\) DPs. g _{0} = iαq _{ y }, g _{ x } = −iαq _{ x }, g _{ y } = iβq _{ x }, g _{ z } = −βq _{ y } with \(\vec q \equiv \vec k  \left( {\pi ,\pi ,{k_0}} \right)\), where α and β are the (real) \(\vec k\) · \(\vec P\) coefficients, and \({\cal O}\left( {{q^2}} \right)\) denotes the higherorder quadratic warping terms. Here the spinorbit coupling is emulated by the klinear interaction between the p and d bands due to quasiconservation of the TAM.^{29} The 3D Dirac wave can be regarded as a series of q _{ z }dependent 2D Dirac waves of which the Dirac mass, m _{ D } ≡ vq _{ z }, can be positive, negative, or zero.^{29, 41}
Derived typeII and typeI Weyl Points
A DP can be regarded as composed of a pair of WPs of opposite Chern numbers. Thus when the space symmetry is reduced WPs can emerge from DPs.^{29} To realize the WPs, we deform the unitcell structure in such a way (as displayed in Fig. 3a) that the two screw symmetries S _{ x } and S _{ y }, the three mirror symmetries M _{1}: = (x, y) → (y, x), M _{2}: = (x, y) → (−y, −x), and M _{ z }: = z → −z, as well as the inversion symmetry \({\cal P}\) are broken. However, the C _{2} symmetry is preserved. The removal of the two screw symmetries lifts the double degeneracy on the MA line. However, accidental degeneracy between bands of opposite parity is protected by the C _{2} symmetry. The chiral structure of the PhC results in p _{±} and d _{±}like states in the photonic bands. The crossings between the p and d bands results in WPs of Chern number ±1 (see Supplementary Materials for a \(\vec k\) · \(\vec P\) analysis). We identify six WPs in Fig. 3b (there are more WPs at higher frequency, explaining the nonzero total Chern number). Figure 3b also shows that there are four typeII WPs and two typeI WPs. The 3D dispersions of both typeI and typeII WPs on the lowest dband are shown in Fig. 3c. Our PhC architecture thus allows realization of typeII WPs using dissipationless alldielectric materials.
Robust surface states
According to the bulkedge correspondence principle,^{2,3,4} the (100) surface states of the tetragonal PhC can reveal the Z _{2} topology of the DPs. We then calculate the surface and projected bulk photonic spectrum using a supercell stacking along the x direction [see Methods]. Figure 4a shows a gapless surface band traversing the projected photonic band gap. This surface band is between the upper and lower branches of the typeI DP, but above both branches of the typeII DP. Thus the gapless surface band is induced by the typeI topological DPs. Nonetheless, both typeI and typeII DPs have the same Z _{2} topology (see Fig. 2d). The topological surface states carry finite TAM as indicated in Fig. 4b by the winding profile of the Poynting vectors. The sign of the photonic TAM is changed when the wavevector is reversed (see Fig. 4b). This property is similar to the “spinwavevector locking” on the edges of topological insulators.^{2, 3} We find that the two symmetries, S _{ y } and \({\cal T}\), guarantee that the spectrum in the surface BZ is symmetric under the transformation (k _{ y }, k _{ z }) → (±k _{ y }, ±k _{ z }) (see Methods). It was recently discovered that the surface states of the topological DPs form a doublehelicoid surface states with such spectral symmetry. The nonchiral surface bands of our PhC, are distinctive from the chiral surface states due to WPs.^{25, 27, 28} Moreover, the topological surface states here are below the lightline and hence form cavity states on the PhCair interfaces with no need for additional cladding.
The robustness of the topological surface states can be revealed via their frequency stability against surface modifications. Figure 4c shows that the frequency of the topological surface state is quite robust and insensitive to variations of the thickness of a dielectric slab placed on top of the PhC surface. The change of frequency is within 2.5%, although the field profile has been substantially modified (see Fig. 4d). In contrast, the frequency of a conventional PhC cavity state with woodpilePhC cladding is much more sensitive to the thickness of the slab^{42} (see inset of Fig. 4c and details in Methods), despite the fact that the woodpile PhC has a large complete photonic band gap of δω/ω = 21% while our PhC has no complete photonic band gap. The topological surface states thus form resilient, subwavelength quasi2D photonic systems. The nontrivial topology/Berryphases and the gapless spectrum distinct them from normal PhC surface states.^{43, 44}
Spectral and optical properties
Both typeI and typeII DPs appear in Fig. 2. A more careful study is presented in Fig. 5. From Eq. (4), the spectrum of the DPs in the k _{ x }k _{ z } plane (Fig. 5a) is
where τ = ± stands for the upper and lower branches of the DP, respectively, the dimensionless parameters \(\gamma = \sqrt {{\alpha ^2} + {\beta ^2}}\) and η measure the deformation of the Dirac cone. Particularly, \(\left \eta \right\) > 1 for typeII DPs, whereas \(\left \eta \right\) < 1 for typeI DPs. The isofrequency contour near a typeII DP is a hyperbolic curve (Fig. 5b). In contrast, the isofrequency contours near a typeI DP is of elliptical shapes. For a typeII DP, when ω = ω _{0}, the two branches touch each other and the isofrequency contour becomes a pair of crossing lines (Fig. 5c), between which the angle is \({\theta _{DP}} = 2\,{\rm{arctan}}\left( {\sqrt {\frac{{{\eta ^2}  1}}{{{\gamma ^2}}}} } \right)\). This quantity sets the bounds on the refraction angles near a typeII DP as \(\pm \frac{1}{2}\left( {\pi  {\theta _{DP}}} \right)\).
The dispersion of the typeII DP in the k _{ x }k _{ y } plane is distinctive from the existing typeI DPs^{29, 31, 32} (see Fig. 5d). This spectrum can be understood via the \(\vec k\) · \(\vec P\) Hamiltonian (4) which yields \({\omega _{\tau ,i}}\left( {\vec q} \right) = {\omega _0} + v\eta {q_z} + v\tau \sqrt {q_z^2 + {\gamma ^2}{{\left {{q_i}} \right}^2}} + {\cal O}\left( {{q^2}} \right)\) for i = 1, 2, with τ = ± and q _{1} = q _{ x } + q _{ y } and q _{2} = q _{ y } − q _{ x }. This spectrum is nondegenerate for finite q _{ x } and q _{ y }. The twofold degeneracy is restored only when q _{ x } = 0 or q _{ y } = 0, in accordance with the screw symmetries. The “Vshaped” dispersion in Fig. 5d gives ellipticalshaped isofrequency contours or nonclosing contours in the k _{ x }k _{ y } plane (see Figs. 4f and 5e), depending on the quadratic warping terms.
From the unique spectral properties of the typeII DPs, using frequency and wavevector matching, we derive the anomalous refraction of light: there are two concurrent refraction beams of opposite angles (see schematic in Fig. 5g). An analytic proof is detailed in the Methods section, which is confirmed by the model calculation in Fig. 5h, i for various frequencies, incident angles, and parameters. Interestingly, we find that there is no refraction for η > 1, whereas for η < −1 there are two refraction beams of opposite refraction angles. Since the two DPs at opposite wavevectors have opposite η, the above property can be exploited for selective excitation of typeII DPs. Away from the k _{ x } = π and k _{ y } = π planes, the photonic spectrum is nondegenerate, leading to two pairs of beams with opposite refraction angles, as shown in Fig. 5i by varying the angle of incidence ϕ _{ i } = Arg(q _{ x } + iq _{ y }). Zero refraction angle is realized when ϕ _{ i } is close to \(\frac{\pi }{4},\frac{{3\pi }}{4},\frac{{5\pi }}{4}\), or \(\frac{{7\pi }}{4}\), due to vanishing group velocity in the k _{ x }k _{ y } plane.
The above unconventional optical properties also holds for typeII WPs. Since WPs are twofold degenerate, there can only be one pair of refraction beams. Concurrent positive and negative refraction was found and confirmed by timedomain simulation in a 2D photonic system before.^{45} Here we find, from frequencywavevector conservation, that concurrent negative and positive refraction can also be realized in 3D alldielectric PhCs through typeII DPs/WPs. A timedomain simulation is demanded to further investigate the anomalous refraction, which, however, is beyond the scope of this work.
Discussion
The band topology induced by crystalline symmetries are in the context of topological crystalline states.^{46, 47} Weak disorders that preserve the crystalline symmetry on average should preserve the DPs and their topological surface states.^{29, 47} The topological surface states here can exist on the PhCair interface without further cladding, even though such interface does not preserve the screw symmetries. The robustness of the surface photonic bands show superiority over conventional PhC cavity states. This suggests that topology can be a possible tool to suppress inhomogeneous broadening which is a main obstacle for scalable optical and quantum devices. Our alldielectric topological PhC architecture may inspire future discovery of other 3D topological photonic states in alldielectric photonics, and stimulate future synergy between subwavelength photonic topological materials and optoelectronics on PhC surfaces.
Note added: When this paper was under review for the final round, a proposal of typeII DPs in electronic materials with robust Fermi arcs, has appeared.^{48}
Methods
Symmetry transformation of photonic states
A photonic state \({\Psi _{n\vec k}}\left( {\vec r} \right)\) transforms under the \({\Theta _x} = {S_x} * {\cal T}\) operation as follows,
where \({\hat M_y}\) and \({\hat M_z}\) are the mirror transformation for the electric and magnetic fields:
and the operator \({\hat t_h}\) reverses the sign of the magnetic field. Acting Θ_{ x } twice yields,
Refraction
The photonic dispersion in the medium with refraction index n _{ i } is given by \(\omega = c\left {\vec k} \right{\rm{/}}{n_i}\). We consider a light beam injected from a medium with a refraction index n _{ i } > 1.65 into the PhC to enable frequency and wavevector matching with the Dirac cones. Around the DP at \({\vec k_0}\) = (π, π, k _{0}), the dispersion in the medium can be expressed as \(\omega = c\left {\vec k} \right{\rm{/}}{n_i} = c\left {{{\vec K}_0} + \vec q} \right{\rm{/}}{n_i}\) where \(\vec q = \vec k  {\vec K_0}\). Since the perpendicular wavevector k _{ z } is not conserved during refraction, we can always set
for fixed q _{}, while adjusting k _{ z } to keep a constant frequency. The angle ϕ _{ i } is varied from 0 to 2π. The refraction in the xz plane is determined by matching the frequency and the parallel wavevector, yielding
The perpendicular wavevector q _{ z } in the PhC is determined by the above equation, which has two solutions for η < −1
The refraction angle is determined through the group velocities in the PhC as, \({\theta _r} \equiv  {\rm{arctan}}\left( {\frac{{{v_x}}}{{{v_z}}}} \right)\). Using the dispersion in Eq. (5), we find that \({v_z} = v\left( {\eta + \frac{{\tau {q_z}}}{{\sqrt {{\gamma ^2}q_x^2 + q:z^2} }}} \right)\), \({v_x} = \frac{{\tau v\gamma {q_x}}}{{\sqrt {{\gamma ^2}q_x^2 + q:z^2} }}\). Inserting Eq. (12) into the definition of the refraction angle, we obtain
Refraction for generic \(\vec q\) (i.e., away from the xz or yz plane) is given in details in the Supplementary Materials.
Calculation of surface states
The surface states are obtained by supercell calculations. The supercell is periodic in the yz plane but finite in the x direction. There are seven layers of unit cell along this direction as sandwiched by air layers of length 3a on the left and right, separately. The simple cladding medium (air) used here is nontopological for all polarizations and useful to study topological surface states below the lightline. The supercell structure is set to preserve the S _{ y } symmetry. Since S _{ y } transforms (k _{ y }, k _{ z }) to (k _{ y }, −k _{ z }) in the surface BZ, the surface spectrum is symmetric with respect to k _{ z } = 0 and k _{ z } = π. In addition, the \({\cal T}\) symmetry guarantees that the surface spectrum is invariant under the transformation (k _{ y }, k _{ z }) to (−k _{ y }, −k _{ z }). Therefore the surface photonic dispersion is also symmetric with respect to k _{ y } = 0 and k _{ y } = π. As detailed in ref. 49, although there are other topological degeneracies in our PhC (such as nodal lines), they do not affect the surface states on the (100) and (010) surfaces.
In the calculation of the reference slabdefect states, we have set the permittivity of the slabdefect layer as ε = 8 (the same as that of the dielectric slab on top of the topological PhC). The logs of the woodpile PhCs above and below the slabdefect layer are of width 0.25a, height 0.3a and permittivity of 12 (silicon).
Data availability
All relevant data is available from the corresponding author.
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Acknowledgements
We thank Sajeev John, Zhengyou Liu, Ling Lu, Chen Fang, Huanyang Chen, Yun Lai, Chunying Qiu, and Jie Luo for many inspiring discussions. H.X.W and J.H.J acknowledge supports from the National Science Foundation of China (Grant no. 11675116) and the Soochow university. Z.H.H is supported by National Science Foundation of China (Grant no. 11574226). Y.C and H.Y.K are supported by NSERC of Canada and Center for Quantum Materials at the University of Toronto.
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J.H.J conceived the idea and wrote the manuscript. J.H.J and Z.H.H designed the photonic architecture. H.X.W, Y.C, H.Y.K and J.H.J did the theoretical analysis and calculations. J.H.J guided the research.
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Wang, HX., Chen, Y., Hang, Z.H. et al. TypeII Dirac photons. npj Quant Mater 2, 54 (2017). https://doi.org/10.1038/s415350170058z
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DOI: https://doi.org/10.1038/s415350170058z
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