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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 2 — Jan. 27, 2014
  • pp: 1906–1917
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A semi-Dirac point and an electromagnetic topological transition in a dielectric photonic crystal

Ying Wu  »View Author Affiliations


Optics Express, Vol. 22, Issue 2, pp. 1906-1917 (2014)
http://dx.doi.org/10.1364/OE.22.001906


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Abstract

Accidental degeneracy in a photonic crystal consisting of a square array of elliptical dielectric cylinders leads to both a semi-Dirac point at the center of the Brillouin zone and an electromagnetic topological transition (ETT). A perturbation method is deduced to affirm the peculiar linear-parabolic dispersion near the semi-Dirac point. An effective medium theory is developed to explain the simultaneous semi-Dirac point and ETT and to show that the photonic crystal is either a zero-refractive-index material or an epsilon-near-zero material at the semi-Dirac point. Drastic changes in the wave manipulation properties at the semi-Dirac point, resulting from ETT, are described.

© 2014 Optical Society of America

1. Introduction

Dirac cones in electron systems give rise to many intriguing transport properties, such as Klein tunneling, Zitterbewegung, and anti-localization, because the Ek relation is linear at the corner of the Brillouin zone [1

1. P. R. Wallace, “The band theory of Graphite,” Phys. Rev. 71(9), 622–634 (1947). [CrossRef]

3

3. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009). [CrossRef]

]. Similar dispersion relations, i.e., Dirac or Dirac-like cones, have also been found in classical wave systems [2

2. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007). [CrossRef] [PubMed]

22

22. D. Torrent, D. Mayou, and J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013). [CrossRef]

], which lead to remarkable wave transport behaviors and interesting applications in classical waves. Representative examples are classical analogs of edge states in Quantum-Hall-Effect systems [5

5. S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008). [CrossRef]

], extremal transmission [6

6. R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007). [CrossRef]

,7

7. S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010). [CrossRef] [PubMed]

], classical analogs of Zitterbewegung [8

8. X. Zhang, “Observing Zitterbewegung for Photons near the Dirac Point of a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 100(11), 113903 (2008). [CrossRef] [PubMed]

,9

9. X. Zhang and Z. Liu, “Extremal Transmission and Beating Effect of Acoustic Waves in Two-Dimensional Sonic Crystals,” Phys. Rev. Lett. 101(26), 264303 (2008). [CrossRef] [PubMed]

], and cloaking of an object [10

10. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011). [CrossRef] [PubMed]

].

Very recently, a semi-Dirac cone, a type of unique and unprecedented electronic band dispersion, was discovered [23

23. V. Pardo and W. E. Pickett, “Half-Metallic Semi-Dirac-Point Generated by Quantum Confinement in TiO2/VO2 Nanostructures,” Phys. Rev. Lett. 102(16), 166803 (2009). [CrossRef] [PubMed]

] and studied [24

24. S. Banerjee, R. R. P. Singh, V. Pardo, and W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. 103(1), 016402 (2009). [CrossRef] [PubMed]

]. It was found that, near a point in the Fermi surface in the two-dimensional (2D) Brillouin zone, the dispersion relation is linear along the symmetry line ((1,1) direction) but quadratic in the perpendicular direction [23

23. V. Pardo and W. E. Pickett, “Half-Metallic Semi-Dirac-Point Generated by Quantum Confinement in TiO2/VO2 Nanostructures,” Phys. Rev. Lett. 102(16), 166803 (2009). [CrossRef] [PubMed]

26

26. M. O. Goerbig, “Electronic properties of graphene in a strong magnetic field,” Rev. Mod. Phys. 83(4), 1193–1243 (2011). [CrossRef]

]. This unique feature results in the interesting hybridized property of the associated quasiparticles that are massless along one direction, like those in graphene, but effective-mass-like along the other direction. More interestingly, it is reported that this semi-Dirac point is associated with the topological phase transition between a semi-metallic phase and a band insulator [26

26. M. O. Goerbig, “Electronic properties of graphene in a strong magnetic field,” Rev. Mod. Phys. 83(4), 1193–1243 (2011). [CrossRef]

]. Although semi-Dirac cones in electronic systems have attracted extensive attention, to the best of my knowledge, there has been no report on their classical analogs. If the special peculiarity of a semi-Dirac cone could be transcribed into a classical system, it is possible to envisage various intriguing consequences that could be attributed to this unique dispersion relation. An obvious one is the super anisotropic wave transport behavior near the semi-Dirac point.

The electromagnetic topological transition (ETT), proposed earlier in the context of optical waves, is the electromagnetic equivalent of the “Liftshiz transition”, in which the iso-frequency surface is transformed from a closed to an open geometry as the frequency changes [27

27. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012). [CrossRef] [PubMed]

]. This transition leads to drastic changes in the nature of the electromagnetic radiation, such as an enhanced spontaneous emission rate. Recently, a strongly anisotropic metal-dielectric metamaterial was designed to demonstrate the occurrence of the optical topological transition [27

27. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012). [CrossRef] [PubMed]

]. However, metallic components bring non-negligible losses that might affect the performance of the metamaterial, and, moreover, the size of the metamaterial’s building block is much smaller than (around one twentieth of) the operating wavelength, which means that fabrication of such a material is challenging.

The semi-Dirac point and ETT seem to be two unrelated subjects. In fact, they become connected under certain circumstances. In a 2D dispersive homogeneous anisotropic medium with permittivity ε and uniaxial permeabilityμ=diag(μx,μy), the iso-frequency surface for a transverse-electric (TE) polarized wave, i.e. E=(0,0,Ez), propagating in such a medium follows the relation

kx2μy+ky2μx=ω2ε.
(1)

Thus, altering the sign of one component in the permeability tensor, e.g. μy, would result in a transition in the topology of the iso-frequency surface. Furthermore, if the permittivity ε and μy are simultaneously zero at a particular frequency ω0, it can be shown that the dispersion relation at the center of the Brillouin zone exhibits linear-parabolic behavior near ω0, the peculiar yet defining property of a semi-Dirac cone. ETT is therefore intertwined with the semi-Dirac point under the condition that ε=μy=0 and μx0. Interestingly, this condition indicates that the material is hybridized from a zero-index material (ZIM) with zero permittivity and permeability along the x-direction and an epsilon-near-zero material (ENZM) along the y-direction. Both ZIM [10

10. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011). [CrossRef] [PubMed]

, 28

28. J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010). [CrossRef]

30

30. Y. Wu and J. Li, “Total reflection and cloaking by zero index metamaterials loaded with rectangular dielectric defects,” Appl. Phys. Lett. 102(18), 183105 (2013). [CrossRef]

] and ENZM [31

31. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006). [CrossRef] [PubMed]

34

34. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008). [CrossRef] [PubMed]

] exhibit rich physics that gives rise to unconventional wave manipulation properties [35

35. N. Engheta, “Materials Science. Pursuing Near-Zero Response,” Science 340(6130), 286–287 (2013). [CrossRef] [PubMed]

], such as super-coupling [31

31. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006). [CrossRef] [PubMed]

] and cloaking of an object [10

10. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011). [CrossRef] [PubMed]

]. A material hybridized from ZIM and ENZM is likely to spawn a large variety of possibilities in wave control. Thus, achieving a semi-Dirac point, an ETT, and hybridized properties in just one simple anisotropic material is fundamentally interesting in both theory and application.

The remainder of this paper is organized as follows: the design of the PhC and its band structure, which exhibits a semi-Dirac point and ETT, are presented in Section 2. In Section 3, the perturbation method is developed and exploited to study the dispersion relations and the linear-parabolic behavior of the dispersion relation is affirmed. In Section 4, a boundary effective medium theory is deduced and ETT and the hybridized property of the PhC are subsequently discussed in the context of this theory. Two illustrative examples, beam splitting and beam shaping, are also presented in Section 4 to show the drastic change in the wave manipulation behavior induced by ETT. Conclusions are drawn in Section 5.

2. The photonic crystal system and its dispersion relation

The PhC considered in this study is a square array of elliptical cylinders with a dielectric constant of εs=12.5 embedded in air (ε0=1). The inset of Fig. 1(a)
Fig. 1 (a) The band structure of the 2D PhC composed of a square array of elliptical dielectric cylinders. The inset shows the unit cell of the PhC. A doubly-degenerate state in the center of the Brillouin zone is found near the dimensionless frequency, 0.540, marked as “A”. In the vicinity of this point, the dispersion relation is linear along the ΓXdirection and quadratic along theΓY direction, which is shown more clearly in Fig. 3(d). Near point “A”, there is another state in the center of the Brillouin zone, marked as “B”. The states at points “A” and “B” are used in the perturbation theory. The branches highlighted by black and blue dots are used to compute the effective medium parameters, which are shown in Fig. 4(a). (b) and (c) Enlarged views of the band structure for smaller and larger elliptical cylinders. The doubly-degenerate state shown in (a) splits into two single states, marked as A1 and A2, where A1 corresponds to a dipolar state and A2 corresponds to a monopolar state.
illustrates the unit cell of this PhC. The semi-minor axis of each elliptical cylinder is ra=0.188a, where a is the lattice constant, and the semi-major axis is 1.3 times that of the semi-minor axis, i.e. rb=1.3ra. The electromagnetic wave is transverse electric (TE) polarized and its electric field, E=(0,0,Ez), is always perpendicular to the plane of periodicity.

Figure 1(a) shows the band structure of this PhC calculated by using COMSOL Multiphysics, a commercial package based on finite element simulations. There exists a doubly-degenerate point, marked as “A”, in the Brillouin zone center at the dimensionless frequency, ω˜=ωa/2πc=0.540, where ω is the angular frequency and c is the wave speed in air. This degenerate point is created by accidental degeneracy [10

10. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011). [CrossRef] [PubMed]

, 17

17. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20(4), 3898–3917 (2012). [CrossRef] [PubMed]

20

20. Y. Li, Y. Wu, X. Chen, and J. Mei, “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013). [CrossRef] [PubMed]

] of a monopolar state (see Fig. 3(b)) and a dipolar state (see Fig. 3(c)) at the Γ point, when the frequencies of these two states are deliberately tuned to be identical by adjusting the size or the material of the inclusion. Figures 1(b) and 1(c) show the band structure near the Γ point for smaller and larger dielectric cylinders, whose eccentricity is kept at 1.3 but whose semi-minor axes become 0.180aand 0.195a, respectively. Apparent in both figures are the separated modes at the Γ point marked as “A1” and “A2”, indicating a dipolar and a monopolar mode, respectively. The reversed relative positions of points A1 and A2 for smaller and larger cylinders imply that there must be a case where A1 and A2 coincide, which occurs when ra=0.188a as discussed earlier. This accidental degeneracy produces a very interesting dispersion relation in the vicinity of this degenerate point. Roughly seen in Fig. 1(a) are two linear bands, along the ΓX direction, touching at Point A, and a quadratic band, along the ΓY direction, tangent to a flat band also at Point A. Below the flat band there is a directional gap in the ΓY direction.

Figures 2(a)
Fig. 2 (a) and (b) The three-dimensional band structure of the PhC. The upper surface is a semi-Dirac cone. Near its bottom, it is linear in Δk along all directions except for the ΓY direction, which is quadratic. It touches the lower surface at the Brillouin zone center near the dimensionless frequency, 0.54. The lower surface is flat in one direction and bends down along the other directions. (c) and (d) The iso-frequency surfaces of the lower and higher branches, where hyperbolic and elliptical surfaces are found, respectively.
and 2(b) present simulated three-dimensional dispersion surfaces, from different view angles, in the frequency regime from ω˜=0.45 to 0.70. Two branches are obvious in these figures. The upper branch resembles the semi-Dirac cone discovered in electron systems, and the lower one is shaped like a roof, which is flat in one direction and bends down in the other directions. These two branches touch at a point, which is Point A mentioned earlier. The iso-frequency surface contours for the lower and upper branches are plotted in Figs. 2(c) and 2(d), where respective open hyperbolic and closed elliptical shapes are manifest. The change in the topology of the iso-frequency surface clearly indicates the occurrence of ETT [27

27. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012). [CrossRef] [PubMed]

] at Point A.

3. A perturbation method and the confirmation of the semi-Dirac dispersion

Straightforward linear algebra reveals that the origin of the linear-parabolic dispersion relation resides in the strength of the mode-coupling integral between ψ2 and ψ3, i.e. unitcellψ2*(r)ψ3(r)dr. The linear term disappears as the integral vanishes. Along the ΓY direction, this integration is zero because there is no coupling between ψ2 and ψ3, as is also suggested by the symmetry of these two states. Even though accidental degeneracy is achieved, no linear dispersion is therefore found [20

20. Y. Li, Y. Wu, X. Chen, and J. Mei, “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013). [CrossRef] [PubMed]

]. In fact, if the eigenstates are examined, it is not difficult to find that the underlying physics lies in ψ3, the dipolar mode of the doubly-degenerate state. As shown in Fig. 3(c), the magnetic field of this dipolar mode is polarized vertically, implying that it is a longitudinal mode along the ΓY direction but a transverse mode along the ΓX direction. In electromagnetic waves, the longitudinal branch is localized and almost does not couple to the incident wave and other branches [10

10. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011). [CrossRef] [PubMed]

, 37

37. Y. Wu, J. Li, Z.-Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006). [CrossRef]

]. Thus, a flat band associated with the longitudinal dipolar mode is found in the ΓY direction. However, since ψ3 is transverse to the ΓXdirection, it easily couples to the incident wave and other branches. The coupling between ψ2 and ψ3 is therefore strong in the ΓX direction and a linear dispersion relation is found.

4. Anisotropic effective medium theory and the electromagnetic topological transition

The effective medium results are indeed consistent with the ETT observed earlier. As explained earlier, the iso-frequency surface of an anisotropic material follows the relation described by Eq. (1). For the effective medium of the PhC considered here, in the frequency regime between 0.487 and 0.540, μxeff>0, μyeff<0 and εeff<0. Such a combination of signs in the effective medium parameters not only leads to hyperbolic iso-frequency surfaces, which are close to those in Fig. 2(c), but it also gives rise to a band gap along the ΓY direction and a negative band along the ΓX direction. This is because ky is purely imaginary when kx=0 while nxeff=εeffμyeff<0 when ky=0. Similar analysis can be applied to the higher branch when the frequency is above the semi-Dirac point, where all the material parameters are positive. Elliptical iso-frequency surfaces are therefore expected. The simultaneous zero μyeffand εeffachieved by accidental degeneracy lead to a linear dispersion relation along the ΓX direction in the vicinity of the semi-Dirac point, whereas a single zero in εeffand a positive μxeff make the dispersion relation quadratic along the ΓY direction. All the behaviors predicted by the effective medium parameters are in line with the properties of the simulated band structures.

The ETT results in drastic changes in the wave manipulation properties. In Fig. 5
Fig. 5 A point source is placed inside the center of a square sample of 16-by-16 rods. (a) and (c) show the electric field patterns when the source frequency is below (0.520) and slightly above (0.544) the semi-Dirac point, respectively. Beam splitting and directional beam shaping are observed. (b) The radial flux as a function of the angle for the case simulated in (a). (d) The same as (c) but the sample is replaced by its effective medium. A similar pattern to that shown in (c) is found. Dark red and dark blue indicate the maximum positive and negative values, respectively.
, I demonstrate the radiation properties of a square sample with 16-by-16 rods that are illuminated by a point source located at the center of the sample at two different frequencies. Figures 5(a) and 5(b) show, respectively, the electric field and the flux distributions when the point source radiates at the dimensionless frequency 0.520, which is below the semi-Dirac point. Due to the hyperbolic shape of the dispersion relation at that frequency, the out-going wave splits into four beams, which are consistent with the iso-frequency surface. However, when the frequency is slightly above the semi-Dirac point, i.e. ω˜=0.544, the outgoing beams go mainly along the horizontal direction and the wave front is almost parallel to the vertical surfaces as shown in Fig. 5(c). The field pattern from the same source but inside a homogenous anisotropic medium is plotted in Fig. 5(d), where the effective medium parameters (εeff=0.018, μyeff=0.042, and μxeff=0.495) are obtained from the effective medium theory. Almost the same field pattern is observed in Figs. 5(c) and 5(d), demonstrating the validity of the effective medium theory.

The effective medium theory also reveals the important hybridized feature of the semi-Dirac point studied here. Since μyeffand εeffare simultaneously zero while μxeff is positive, the PhC is a ZIM along the ΓX direction and an ENZM along the ΓY direction. This super anisotropic characteristic is different from previously studied anisotropic zero materials, in which one component of the physical parameters is zero [39

39. H. F. Ma, J. H. Shi, B. G. Cai, and T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14(12), 123010 (2012). [CrossRef]

41

41. Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial Power Combination for Omnidirectional Radiation via Anisotropic Metamaterials,” Phys. Rev. Lett. 108(21), 213903 (2012). [CrossRef] [PubMed]

]. Figures 4(b)4(d) show the electric field pattern of a TE-polarized plane wave with the frequency of the semi-Dirac point impinging on a PhC slab inside a waveguide. The waveguide has boundary walls that are perfect magnetic conductors. Clearly, when the incident wave is propagating in the ΓX direction, the PhC exhibits the typical transmission property of a ZIM [10

10. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011). [CrossRef] [PubMed]

], i.e. total transmission is supported without any phase change [35

35. N. Engheta, “Materials Science. Pursuing Near-Zero Response,” Science 340(6130), 286–287 (2013). [CrossRef] [PubMed]

] inside the material as shown in Figs. 4(c) and 4(d) for the real and imaginary parts of the electric field, respectively. However, when the PhC is illuminated by the same incident wave along the ΓY direction, the transmitted field is very weak, which is consistent with the transmission property of an ENZM. This anisotropic transport feature provides evidence that the PhC has hybridized properties of a ZIM and an ENZM.

5. Conclusions

Appendix

In this appendix, I give a detailed derivation of the perturbation method that is used in Section 2. In two dimensions, the electric field of a TE polarized wave satisfies the following wave equation:
×(1μ(r)×Ezz^)=ω2c02ε(r)Ezz^,
(5)
where ε(r)and μ(r)are the permittivity and permeability, respectively, and c0 is the speed of light in air or vacuum. For periodic systems, the solution of Eq. (5) can be expressed as Bloch wave functions, Ψnk(r)=unk(r)eikr, where unk(r) is a periodic function and k is the Bloch wave vector. The relation between the eigenfrequency (ωnk) and the Bloch wave vector gives the nth branch of the dispersion relations.

In the perturbation theory, the unperturbed Bloch states at k0 are obtained from finite element simulations, which means that Ψnk0(r)=unk0(r)eik0r is known. The Bloch states at k near k0 can be expanded as:
Ψnk(r)=jAnj(k)ei(kk0)rΨjk0(r),
(6)
where the unknown periodic function unk(r) is expressed as linear combinations of unk0(r). By substituting Eq. (6) into Eq. (5) and invoking the orthonormal properties of the Bloch wave functions, i.e. (2π)2ΩunitcellΨlk*(r)ε(r)Ψjk(r)dr=δlj with Ω and δlj denoting the volume of a unit cell and the Kronecker delta, respectively,
j[ωj02ωnk2c02δljPlj(k)]Anj(k)=0
(7)
is obtained.

Here, Plj(k) represents the mode-coupling integrals between two states Ψlk0(r) and Ψjk0(r), and is expressed as
Plj(k)=(kk0)plj(kk0)2qlj,
(8)
whereplj=i(2π)2ΩunitcellΨlk0*(r)[2Ψjk0(r)μ(r)+(1μ(r))Ψjk0(r)]dr and qlj=(2π)2ΩunitcellΨlk0*(r)1μ(r)Ψjk0(r)dr. Since the PhC studied in this work does not involve different magnetic permeability, i.e. μ(r) is 1 everywhere, the expressions for plj and qlj can be greatly simplified.

Acknowledgments

The author is grateful to Prof. Z.Q. Zhang, Prof. C. T. Chan, Prof. J. Li, Prof. J. Mei and Dr. X. Q. Huang for fruitful discussions. Special thanks go to Prof. P. Sheng, Prof. Y. Lai, and Prof. Z. H. Hang for their comments. I also would like to acknowledge insightful comments from anonymous reviewers, which greatly helped to improve the quality of this paper. I thank V. Unkefer for editorial work on this manuscript. This work was supported by KAUST Baseline Research Funds.

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J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012). [CrossRef]

20.

Y. Li, Y. Wu, X. Chen, and J. Mei, “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013). [CrossRef] [PubMed]

21.

D. Torrent and J. Sánchez-Dehesa, “Acoustic Analogue of Graphene: Observation of Dirac Cones in Acoustic Surface Waves,” Phys. Rev. Lett. 108(17), 174301 (2012). [CrossRef] [PubMed]

22.

D. Torrent, D. Mayou, and J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013). [CrossRef]

23.

V. Pardo and W. E. Pickett, “Half-Metallic Semi-Dirac-Point Generated by Quantum Confinement in TiO2/VO2 Nanostructures,” Phys. Rev. Lett. 102(16), 166803 (2009). [CrossRef] [PubMed]

24.

S. Banerjee, R. R. P. Singh, V. Pardo, and W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. 103(1), 016402 (2009). [CrossRef] [PubMed]

25.

G. Montambaux, F. Piéchon, J. N. Fuchs, and M. O. Goerbig, “Merging of Dirac points in a two-dimensional crystal,” Phys. Rev. B 80(15), 153412 (2009). [CrossRef]

26.

M. O. Goerbig, “Electronic properties of graphene in a strong magnetic field,” Rev. Mod. Phys. 83(4), 1193–1243 (2011). [CrossRef]

27.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012). [CrossRef] [PubMed]

28.

J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010). [CrossRef]

29.

V. C. Nguyen, L. Chen, and K. Halterman, “Total Transmission and Total Reflection by Zero Index Metamaterials with Defects,” Phys. Rev. Lett. 105(23), 233908 (2010). [CrossRef] [PubMed]

30.

Y. Wu and J. Li, “Total reflection and cloaking by zero index metamaterials loaded with rectangular dielectric defects,” Appl. Phys. Lett. 102(18), 183105 (2013). [CrossRef]

31.

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006). [CrossRef] [PubMed]

32.

A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013). [CrossRef]

33.

A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007). [CrossRef]

34.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008). [CrossRef] [PubMed]

35.

N. Engheta, “Materials Science. Pursuing Near-Zero Response,” Science 340(6130), 286–287 (2013). [CrossRef] [PubMed]

36.

B. A. Foreman, “Theory of the effective Hamiltonian for degenerate bands in an electric field,” J. Phys. Condens. Matter 12(34), R435–R461 (2000). [CrossRef]

37.

Y. Wu, J. Li, Z.-Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006). [CrossRef]

38.

Y. Lai, Y. Wu, P. Sheng, and Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. 10(8), 620–624 (2011). [CrossRef] [PubMed]

39.

H. F. Ma, J. H. Shi, B. G. Cai, and T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14(12), 123010 (2012). [CrossRef]

40.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012). [CrossRef]

41.

Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial Power Combination for Omnidirectional Radiation via Anisotropic Metamaterials,” Phys. Rev. Lett. 108(21), 213903 (2012). [CrossRef] [PubMed]

OCIS Codes
(260.2030) Physical optics : Dispersion
(160.3918) Materials : Metamaterials
(160.5298) Materials : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: November 26, 2013
Revised Manuscript: January 2, 2014
Manuscript Accepted: January 10, 2014
Published: January 21, 2014

Citation
Ying Wu, "A semi-Dirac point and an electromagnetic topological transition in a dielectric photonic crystal," Opt. Express 22, 1906-1917 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-2-1906


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References

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  19. J. Mei, Y. Wu, C. T. Chan, Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012). [CrossRef]
  20. Y. Li, Y. Wu, X. Chen, J. Mei, “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013). [CrossRef] [PubMed]
  21. D. Torrent, J. Sánchez-Dehesa, “Acoustic Analogue of Graphene: Observation of Dirac Cones in Acoustic Surface Waves,” Phys. Rev. Lett. 108(17), 174301 (2012). [CrossRef] [PubMed]
  22. D. Torrent, D. Mayou, J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013). [CrossRef]
  23. V. Pardo, W. E. Pickett, “Half-Metallic Semi-Dirac-Point Generated by Quantum Confinement in TiO2/VO2 Nanostructures,” Phys. Rev. Lett. 102(16), 166803 (2009). [CrossRef] [PubMed]
  24. S. Banerjee, R. R. P. Singh, V. Pardo, W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. 103(1), 016402 (2009). [CrossRef] [PubMed]
  25. G. Montambaux, F. Piéchon, J. N. Fuchs, M. O. Goerbig, “Merging of Dirac points in a two-dimensional crystal,” Phys. Rev. B 80(15), 153412 (2009). [CrossRef]
  26. M. O. Goerbig, “Electronic properties of graphene in a strong magnetic field,” Rev. Mod. Phys. 83(4), 1193–1243 (2011). [CrossRef]
  27. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012). [CrossRef] [PubMed]
  28. J. Hao, W. Yan, M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010). [CrossRef]
  29. V. C. Nguyen, L. Chen, K. Halterman, “Total Transmission and Total Reflection by Zero Index Metamaterials with Defects,” Phys. Rev. Lett. 105(23), 233908 (2010). [CrossRef] [PubMed]
  30. Y. Wu, J. Li, “Total reflection and cloaking by zero index metamaterials loaded with rectangular dielectric defects,” Appl. Phys. Lett. 102(18), 183105 (2013). [CrossRef]
  31. M. Silveirinha, N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006). [CrossRef] [PubMed]
  32. A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013). [CrossRef]
  33. A. Alu, M. G. Silveirinha, A. Salandrino, N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007). [CrossRef]
  34. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008). [CrossRef] [PubMed]
  35. N. Engheta, “Materials Science. Pursuing Near-Zero Response,” Science 340(6130), 286–287 (2013). [CrossRef] [PubMed]
  36. B. A. Foreman, “Theory of the effective Hamiltonian for degenerate bands in an electric field,” J. Phys. Condens. Matter 12(34), R435–R461 (2000). [CrossRef]
  37. Y. Wu, J. Li, Z.-Q. Zhang, C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006). [CrossRef]
  38. Y. Lai, Y. Wu, P. Sheng, Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. 10(8), 620–624 (2011). [CrossRef] [PubMed]
  39. H. F. Ma, J. H. Shi, B. G. Cai, T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14(12), 123010 (2012). [CrossRef]
  40. J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012). [CrossRef]
  41. Q. Cheng, W. X. Jiang, T. J. Cui, “Spatial Power Combination for Omnidirectional Radiation via Anisotropic Metamaterials,” Phys. Rev. Lett. 108(21), 213903 (2012). [CrossRef] [PubMed]

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