Abstract

Valley-dependent propagation of light in an artificial photonic hexagonal lattice, akin to electrons in graphene, is investigated in microwave regime. Both numerical and experimental results show that the valley degeneracy in the photonic graphene is broken when the frequency is away from the Dirac point. The peculiar anisotropic wave transport property due to distinct valleys is analyzed using the equifrequency contours. More interestingly, the valley-dependent self-collimation and beam splitting phenomena are experimentally demonstrated with the armchair and zigzag interfaces, respectively. Our results confirm that there are two inequivalent Dirac points that lead to two distinct valleys in photonic graphene, which could be used to control the flow of light and might be used to carry information in valley polarized beam splitter, collimator or guiding device.

© 2014 Optical Society of America

1. Introduction

Graphene, the monolayer carbon honeycomb lattice, has attracted a great interest as a promising candidate material for the post-silicon nanoelectronics [13]. Its band structure has two degenerate and inequivalent valleys around Dirac points (K and K'), at the corners of the Brillouin zone. This valley degree of freedom in graphene might be viewed as pseudo-spin and used to construct graphene-based valleytronic devices, referred to as valleytronics [414]. The unconventional edge states in zigzag graphene nanoribbons were predicted but still not observed experimentally so far, since such perfect edges are unstable due to interaction with the substrate and adatoms. In contrary to nature graphene, artificial photonic graphene-like lattices allow tunable strengths of the parameters that are not accessible in nature graphene due to the coarse or impure feature of graphene edges, the strength of the inter-valley coupling.

As predicted by J. L. Garcia-Pomar et al., the Dirac points can be shifted by means of a gate voltage, such that the Fermi level lies in the valence or conduct band, and the trigonal warping (TW) distortion in energy band lifts the degeneracy of two valleys K and K' [6].The band structure show a strong asymmetry between K and K' valleys, which suppresses the weak localization effect. In this situation, electron transport becomes valley dependent, which can be used to design valley beam splitter, collimator or guiding device [6, 7]. However, the electronic properties of practical finite-size graphene are strongly influenced by coarse or impure at edges, which hampers the experimental implementation of the valley-dependent electronic transportation in graphene.

Dirac points can emerge in the band structures of various two-dimensional materials. However, it appears naturally in honeycomb lattices due to the protection of the hexagonal symmetry. Therefore, one can expect that the Dirac points in graphene also exists in its electromagnetic counterpart, the so-called photonic graphene (PG) [1532]. Recent years, PG has been of great interest mainly due to its convenient for experimental investigate Dirac physics. A variety of novel wave transport properties, such as pseudo-diffusion [1518], edge states [1923], zitterbewegung [24, 25], Klein tunneling [26], dynamic localization [2729], and weak anti-localization [30, 31], have been predicted or observed. In addition, Dirac-like point at Γ point (Brillouin zone center) in two dimensional square lattice photonic crystals due to triple degeneracy of modes [32, 33], has aroused increasing attention to the peculiar Dirac cone dispersion. Different from the Dirac-like point at Γ point, there should be two inequivalent Dirac points (K and K') around the corners of the Brillouin zone in PG as predicted in the band structure of graphene. However, the valley degeneracy is preserved at the Dirac point frequency. As such, the wave transport behaviors at K and K' points are not distinguished without using any external device. Therefore, it’s indispensable to experimentally separate the beams belong to different valleys while the valley degeneracy is broken at the frequency away from Dirac points. PG offers a promising platform to design, realize and study the edges, strain and disorder effects on transport property of photons in artificial photonic honeycomb lattices. However, to the best of our knowledge, the experimental demonstration of valley-dependent beams has not been reported so far.

In this paper, we design and carry out experiments to investigate the behavior of valley-dependent beams in PG in microwave regime. In section 2 of this paper, by analyzing the equifrequency contours (EFCs) [3436], we show that the inequivalent valleys (K and K') suffer distinct TW distortion at a frequency away from Dirac point, namely, the valley degeneracy is broken. As such, the component of incident beam associated to different valleys presents different nature. In section 3, we carry out simulations and experiments to verify that, when the beam is incident on the armchair interface, the component belonging to one valley is full self-collimated, and the component belonging to the other valley is split into three beams; when the beam is incident on the zigzag interface, the incident beam splits into two collimated beams, which belong to K and K', respectively. An indirect way to monitor valley-dependent beam propagation in PG structure is also provided, which gives a further proofs of our analysis results. Finally, we conclude in section 4.

2. Analysis of valley-dependent beam

Band structures of the investigated PG calculated by plane wave expansion method for the TE modes (electric fields are parallel to the rods) are shown in Fig. 1(a). The PG is made up of cylindrical alumina rods arranged in a honeycomb lattice, which are immersed in air background. The positions of Dirac points in band structure of the PG can be tuned through changing the parameters of materials and crystal lattice structure. Here, the radius and dielectric constant of the rods are 3 mm and 8.35, respectively. And the distance between two adjacent rods is a = 8.57 mm. In this band structure the band gap become vanishingly small at about 13.65 GHz (normalized frequency of a/λ≈0.39, λ is the wavelength in vacuum). It’s the frequency of Dirac points, which has been experimentally demonstrated in a similar honeycomb lattice PG [31]. As the frequency detuning from Dirac point increases gradually, the EFCs evolve from six small circles to six approximated regular triangles, and the valley degeneracy is gradually broken. It will eventually form a nontrivial TW distortion in band structure that is different for the two adjacent valleys. As is shown in the inset in Fig. 1(a), the EFCs for the particular frequency of 12.52 GHz [(a/λ≈0.36) labeled with the dashed line in Fig. 1(a)] is comprised with six approximated regular triangles which encircle K and K', and point to Γ (the first Brillouin zone center). Obviously, this TW distortion has a distinction between K or K' valleys. It should be mentioned that the Dirac points concerned here is the intersection of the fourth and fifth bands rather than the first and second bands; therefore, the shape of EFCs originated from the TW distortion is slightly different from the results in [6].

 figure: Fig. 1

Fig. 1 (a) Band structures for TE modes in PG with honeycomb lattace. Inset: Brillouin zone and the EFCs at normalized frequency of 0.36 (labeled with the horizontal dashed line). (b) Scheme of valley-dependent wave transport behavior: the blue triangles are the EFCs for the PG at normalized frequency of 0.36. These triangles encircles K or K' points. The blue semicircles are parts of the EFCs for free space at same frequency. The thin gray lines represent the conservation of the parallel wave vector, and the green and the red arrows in the triangles represent the directions of energy flow coming from the K and K' valley, respectively. For the zigzag interface, the refractions in the two valleys have different orientations. For the armchair interface, the beam associated to one of the valleys is fully self-collimated and the beam belonging to the other valley splits into three beams. The energy flows are also indicated with the arrows with different thickness and color in PG structure.

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The distinction of the TW distortion for K and K' points can be explained more clearly in Fig. 1(b). The blue triangles are the EFCs encircle K and K' points in the PG and the blue semicircles are parts of the EFCs for free space at normalized frequency of 0.36. The thin lines and arrows in the triangles represent the conservation of the parallel wave vector. The energy flows in PG structure from different valleys are indicated by the thick arrows with different color [see Fig. 1(b)]. For the zigzag interface, the refractions in the two valleys have different orientations. And each of the splitting beams originates from different valley’s contribution. For the armchair interface, the beam associated to one of the valleys is fully self-collimated and the beam belonging to the other valley splits into three beams. Therefore, the center one of the three beams, as respectively pointed out in [6] and [7], has a much higher intensity than the two side beams, because the center beam has contributions from two valleys, while the side beam has only one-third the contribution from one valley. Here in order to clearly observe beam collimation and splitting phenomena, we suppose wave source from free space has directional and omnidirectional wave vector for armchair and zigzag interface, respectively. So the EFC for free space is separately indicated by solid and partly dotted lines as shown in Fig. 1(b). In a word, the distinctive TW distortion for K and K' valleys will lead to self-collimated and splitting beams in PG with armchair and zigzag incoming interfaces, respectively.

3. Simulations and experiments

In order to confirm the valley-dependent beam transportation, which have been analyzed above, we carry out the numerical simulations (CST Microwave Studio) as well as microwave experiments. Figure 2(a) presents the photographs of the PG sample, which is arranged in a waveguide chamber. Absorbing materials are used to control the width of incident beam, and eliminate the reflection from waveguide walls. In addition, the cylindrical metal defects inserted in PG act as scatterers to monitor valley-dependent beam propagation indirectly. Inset in Fig. 2(a) is the enlarged view of a part of sample with a metal defect (four dielectric rods wrapped by silver paper). Photograph in Fig. 2(b) presents the waveguide chamber, as well as a microwave probe. The PG samples are arranged in close to the edge of the waveguide chamber. Thus the distribution of electric field in the outgoing region can be measured by the microwave probe connected to a network analyzer (Agilent N522A). In our experiments, the probe moves step by step in X-Y plane with fixed lengths of 2 mm.

 figure: Fig. 2

Fig. 2 (a) Photograph of one PG sample in waveguide chamber. The width of incident beam is controlled by absorbing material. The cylindrical metal defects act as scatterers to monitor valley-dependent beam indirectly. Inset is the enlarged view of part of the honeycomb lattice structure where metal defects are inserted. (b) Waveguide chamber and field probe used in experiments.

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In Fig. 3(a), we plot the simulated field distribution when the incident beam impinges normally on the armchair interface of a PG sample without metal defects. Here the PG sample is made up of 616 (28 rows × 22 columns) cylindrical alumina rods. One can clearly find in the PG a collimated beam along Y-direction (ΓK) with almost no diffraction. It is mainly from K' valley. In addition, there are two side beams with very low intensity. They are from the other valley. Measured electric field distribution in the outgoing region [346 mm × 20 mm in the XY plane, the same region encircled by the dashed lines in Fig. 3(a)] is given on the top of Fig. 3(b). The measured and simulated profiles of electric field distributions along the waveguide edge are given on the bottom of Fig. 3(b). One can find our experimental results coincide to the simulations reasonably well.

 figure: Fig. 3

Fig. 3 Beam self-collimation in PG without metal defects. (a)Simulated electric field distribution of PG with armchair interface. Outgoing region is encircled by the red dashed lines; (b) Top: Measured electric field distribution in the outgoing region; Bottom: Comparation of simulated and experimental normalized profiles of electric field distribution along waveguide edge.

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In Fig. 4(a), we plot the simulated field distribution when the incident beam impinges normally on the zigzag interface of a PG sample without metal defects. It shows that the beam is clearly split into two collimated beams, which propagates in ΓΚ and ΓΚ' directions with the intersection angle of 60 degrees. According to the theoretical analysis in Sec. II, the split two beams are origin from Κ and Κ' valleys, respectively. Measured electric field distribution in the outgoing region [384 mm × 20 mm in the XY plane, the same region encircled by the dashed lines in Fig. 4(a)] is given on the top of Fig. 4(b). In addition, the measured and simulated profiles of electric field distributions along the waveguide edge are given on the bottom of Fig. 4(b). We can obtain a relatively consistent field distribution between simulations and experiments. Here we have removed one dielectric rod at the center of the incident interface, as indicated by an arrow in the inset in Fig. 4(b). This defect at interface make the incident beam be scattered partly to ΓΚ and ΓΚ' directions. Otherwise, the beams would be reflected totally due to the wave-vector mismatch.

 figure: Fig. 4

Fig. 4 Beam splitting in PG without metal defects. (a) Simulated electric field distribution of PG with zigzag interface; (b) Top: Measured electric field distribution in the outgoing region, which is encircled by the red dashed lines in Fig. 4(a); Bottom: Comparation of simulated and experimental normalized profiles of electric field distribution along waveguide edge. Inset is the PG model with zigzag interface, in which the middle one of the first row of alumina rods is removed, as indicated by the black arrow).

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So far, we have experimentally investigated the valley dependent self-collimation and beam splitting in PG, by probing the field destitutions in the outgoing region. It is difficult to probing the field in PG sample in our experiments. However, with some embedded metal defects as strong scatterers, we can indirectly detect the propagation of beams in PG. As shown in Fig. 2 (a), four metal rods with radius of about 3.6 mm are inserted in adjacent honeycomb air voids. They constitute metal defects with strong scattering ability. In detail, metal defects on the path of beam would lead to a strong distortion of the measured field distributions in the outgoing region. On the contrary, metal defects away from the beam path would not substantially affect the measurements. In the following, we would present our indirect studies to confirm the valley dependent self-collimation and beam splitting, especially the beam paths in PG.

When the metal defects are embedded into positions accurately on the central axis in y-direction, such as the case shown in Fig. 5(a), the measurements of outgoing fields of self-collimation beam are strongly changed comparing to the situation without the defects [see Fig. 3]. The huge projection of the field profile in the centre disappeared completely. When the defects are far away from the central axis, the measurements of outgoing field seem unchanged with a huge projection in the centre, as shown in Fig. 5(b). Figure 5(c) present the measurements when the defects are a little offset from the centre. One can find the projection remain, but with a little asymmetry. Our measurements consist with the simulations, and confirm the path of valley dependent self-collimation beam in PG clearly. Moreover, calculated field distributions in Fig. 5 reveal that the scattered waves from the defects remain self-collimation beams which propagate along the ΓΚ or the ΓΚ' directions.

 figure: Fig. 5

Fig. 5 Measured profiles of electric field distribution along waveguide edge (top) and simulated electric field distributions in PGs with armchair interface, with metal defects embedded in three different positions (bottom). The waveguide edge is indicated by the red dashed line. (a) Metal defects are embedded into positions accurately on the path of beam; (b) Metal defects are embedded into positions away from the beam path; (c) Metal defects are embedded into positions a little offset from the beam path.

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With the same way, namely measuring the outgoing field of PG with the metal defects embedded in different positions, we can also confirm the paths of valley dependent beam splitting in PG with zigzag interface. If the measurements are significantly different from the previous measurements of PG without the defects, the defects should be on the beam paths. For example, the results in Figs. 6(a-c) are distinguished from the one in Fig. 4(b), which means these defects are on the path. On the contrary, the results in Figs. 6(d) and 6(e), are similar as in Fig. 4(b). Corresponding defects should be away from the beam path. The measurements agree well with the simulations. Still, the scattered waves from the defects remain self-collimation beams along the ΓΚ or the ΓΚ' directions. In addition, we measure the outgoing field of a PG sample where most of the alumina rods between the splitting beam paths are removed, as shown in Fig. 6(f). The results are still similar as in Fig. 4(b), which means the valley dependent beam splitting is robust with respect to the lattices between the splitting beam paths.

 figure: Fig. 6

Fig. 6 Measured profiles of electric field distribution along waveguide edge (top) and simulated electric field distributions in PG with zigzag interface. The waveguide edge is indicated by the red dashed line. Six different defects are applied, respectively. (a) Metal defects are inserted in the right path of the splitting beams in close to the top edge of PG; (b) Metal defects are inserted in the right path of the splitting beams in close to the middle of PG; (c) Metal defects are inserted in the left path of the splitting beams in close to the bottom edge of PG; (d) Metal defects are inserted in a position outside of the paths of splitting beams at the left part of PG; (e) Metal defects are inserted in a position between the paths of the splitting beams; (f) Most of the alumina rods between the splitting beams are removed.

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4. Conclusions

In conclusion, we investigate photonic analogy of valley-dependent transportation of electrons in graphene due to TW distortion in band structure. The TW distortions are distinct between two valleys. Therefore, the valley degeneracy is broken at a frequency away from Dirac points. For the armchair interface, the beam associated to one of the valleys is fully self-collimated and the beam belonging to the other valley splits into three beams. For the zigzag interface, the incident beam splits into two beams, which are related to K and K' valley, respectively. Our experimental observations of the valley-dependent self-collimation as well as the beam splitting phenomena confirm that the valley degeneracy is broken. The valley-dependent beams could be used to control the flow of light, and might be used to carry information in valley polarized beam splitter, collimator or guiding device.

Acknowledgements

This work is supported by the National Basic Research Program (973) of China (No. 2011CB922001), the National Natural Science Foundation of China (Nos. 11234010, 11274207, 11204217), the Innovation Program of Shanghai Municipal Education Commission (No. 14ZZ040), and the Fundamental Research Funds for the Central Universities (No. 2013KJ046).

References and links

1. S. Stankovich, D. A. Dikin, G. H. Dommett, K. M. Kohlhaas, E. J. Zimney, E. A. Stach, R. D. Piner, S. T. Nguyen, and R. S. Ruoff, “Graphene-based composite materials,” Nature 442(7100), 282–286 (2006). [CrossRef]   [PubMed]  

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

3. A. H. Castro Neto, 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]  

4. A. Rycerz, J. Tworzydlo, and C. W. J. Beenakker, “Valley filter and valley valve in graphene,” Nat. Phys. 3(3), 172–175 (2007). [CrossRef]  

5. D. Xiao, W. Yao, and Q. Niu, “Valley-contrasting physics in graphene: magnetic moment and topological transport,” Phys. Rev. Lett. 99(23), 236809 (2007). [CrossRef]   [PubMed]  

6. J. L. Garcia-Pomar, A. Cortijo, and M. Nieto-Vesperinas, “Fully valley-polarized electron beams in graphene,” Phys. Rev. Lett. 100(23), 236801 (2008). [CrossRef]   [PubMed]  

7. Z. Wang and F. Liu, “Manipulation of electron beam propagation by hetero-dimensional graphene junctions,” ACS Nano 4(4), 2459–2465 (2010). [CrossRef]   [PubMed]  

8. F. Zhai, Y. Ma, and K. Chang, “Valley beam splitter based on strained graphene,” New J. Phys. 13(8), 083029 (2011). [CrossRef]  

9. Z. Wu, F. Zhai, F. M. Peeters, H. Q. Xu, and K. Chang, “Valley-dependent brewster angles and goos-hänchen effect in strained graphene,” Phys. Rev. Lett. 106(17), 176802 (2011). [CrossRef]   [PubMed]  

10. D. Gunlycke and C. T. White, “Graphene valley filter using a line defect,” Phys. Rev. Lett. 106(13), 136806 (2011). [CrossRef]   [PubMed]  

11. F. Zhai and K. Chang, “Valley filtering in graphene with a Dirac gap,” Phys. Rev. B 85(15), 155415 (2012). [CrossRef]  

12. K. Behnia, “Condensed-matter physics: polarized light boosts valleytronics,” Nat. Nanotechnol. 7(8), 488–489 (2012). [CrossRef]   [PubMed]  

13. H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotechnol. 7(8), 490–493 (2012). [CrossRef]   [PubMed]  

14. K. F. Mak, K. He, J. Shan, and T. F. Heinz, “Control of valley polarization in monolayer MoS2 by optical helicity,” Nat. Nanotechnol. 7(8), 494–498 (2012). [CrossRef]   [PubMed]  

15. X. Zhang, “Demonstration of a new transport regime of photon in two-dimensional photonic crystal,” Phys. Lett. A 372(19), 3512–3516 (2008). [CrossRef]  

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

17. M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405(14), 2990–2995 (2010). [CrossRef]  

18. 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]  

19. S. Raghu and F. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).

20. S. Bittner, B. Dietz, M. Miski-Oglu, and A. Richter, “Extremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiard,” Phys. Rev. B 85(6), 064301 (2012). [CrossRef]  

21. Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, J. M. Zeuner, S. Nolte, Y. Lumer, N. Malkova, J. Xu, A. Szameit, Z. Chen, and M. Segev, “Observation of unconventional edge states in ‘photonic graphene’,” Nat. Mater. 13(1), 57–62 (2013). [CrossRef]   [PubMed]  

22. M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, D. Song, Z. Chen, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic graphene,” Phys. Rev. Lett. 111(10), 103901 (2013). [CrossRef]   [PubMed]  

23. J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014). [CrossRef]   [PubMed]  

24. 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]  

25. Q. Liang, Y. Yan, and J. Dong, “Zitterbewegung in the honeycomb photonic lattice,” Opt. Lett. 36(13), 2513–2515 (2011). [CrossRef]   [PubMed]  

26. O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010). [CrossRef]   [PubMed]  

27. A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Polychromatic dynamic localization in curved photonic lattices,” Nat. Phys. 5(4), 271–275 (2009). [CrossRef]  

28. A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104(22), 223903 (2010). [CrossRef]   [PubMed]  

29. A. Crespi, G. Corrielli, G. D. Valle, R. Osellame, and S. Longhi, “Dynamic band collapse in photonic graphene,” New J. Phys. 15(1), 013012 (2013). [CrossRef]  

30. R. A. Sepkhanov, A. Ossipov, and C. W. J. Beenakker, “Extinction of coherent backscattering by a disordered photonic crystal with a Dirac spectrum,” Europhys. Lett. 85(1), 14005 (2009). [CrossRef]  

31. X. Wang, H. T. Jiang, C. Yan, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Anomalous transmission of disordered photonic graphenes at the Dirac point,” Europhys. Lett. 103(1), 17003 (2013). [CrossRef]  

32. 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]  

33. 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]  

34. Y. Luo, W. Zhang, Y. Huang, J. Zhao, and J. Peng, “Wide-angle beam splitting by use of positive-negative refraction in photonic crystals,” Opt. Lett. 29(24), 2920–2922 (2004). [CrossRef]   [PubMed]  

35. P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92(12), 127401 (2004). [CrossRef]   [PubMed]  

36. X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83(16), 3251–3253 (2003). [CrossRef]  

References

  • View by:

  1. S. Stankovich, D. A. Dikin, G. H. Dommett, K. M. Kohlhaas, E. J. Zimney, E. A. Stach, R. D. Piner, S. T. Nguyen, and R. S. Ruoff, “Graphene-based composite materials,” Nature 442(7100), 282–286 (2006).
    [Crossref] [PubMed]
  2. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
    [Crossref] [PubMed]
  3. A. H. Castro Neto, 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]
  4. A. Rycerz, J. Tworzydlo, and C. W. J. Beenakker, “Valley filter and valley valve in graphene,” Nat. Phys. 3(3), 172–175 (2007).
    [Crossref]
  5. D. Xiao, W. Yao, and Q. Niu, “Valley-contrasting physics in graphene: magnetic moment and topological transport,” Phys. Rev. Lett. 99(23), 236809 (2007).
    [Crossref] [PubMed]
  6. J. L. Garcia-Pomar, A. Cortijo, and M. Nieto-Vesperinas, “Fully valley-polarized electron beams in graphene,” Phys. Rev. Lett. 100(23), 236801 (2008).
    [Crossref] [PubMed]
  7. Z. Wang and F. Liu, “Manipulation of electron beam propagation by hetero-dimensional graphene junctions,” ACS Nano 4(4), 2459–2465 (2010).
    [Crossref] [PubMed]
  8. F. Zhai, Y. Ma, and K. Chang, “Valley beam splitter based on strained graphene,” New J. Phys. 13(8), 083029 (2011).
    [Crossref]
  9. Z. Wu, F. Zhai, F. M. Peeters, H. Q. Xu, and K. Chang, “Valley-dependent brewster angles and goos-hänchen effect in strained graphene,” Phys. Rev. Lett. 106(17), 176802 (2011).
    [Crossref] [PubMed]
  10. D. Gunlycke and C. T. White, “Graphene valley filter using a line defect,” Phys. Rev. Lett. 106(13), 136806 (2011).
    [Crossref] [PubMed]
  11. F. Zhai and K. Chang, “Valley filtering in graphene with a Dirac gap,” Phys. Rev. B 85(15), 155415 (2012).
    [Crossref]
  12. K. Behnia, “Condensed-matter physics: polarized light boosts valleytronics,” Nat. Nanotechnol. 7(8), 488–489 (2012).
    [Crossref] [PubMed]
  13. H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotechnol. 7(8), 490–493 (2012).
    [Crossref] [PubMed]
  14. K. F. Mak, K. He, J. Shan, and T. F. Heinz, “Control of valley polarization in monolayer MoS2 by optical helicity,” Nat. Nanotechnol. 7(8), 494–498 (2012).
    [Crossref] [PubMed]
  15. X. Zhang, “Demonstration of a new transport regime of photon in two-dimensional photonic crystal,” Phys. Lett. A 372(19), 3512–3516 (2008).
    [Crossref]
  16. R. Sepkhanov, Y. Bazaliy, and C. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
    [Crossref]
  17. M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405(14), 2990–2995 (2010).
    [Crossref]
  18. 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]
  19. S. Raghu and F. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
  20. S. Bittner, B. Dietz, M. Miski-Oglu, and A. Richter, “Extremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiard,” Phys. Rev. B 85(6), 064301 (2012).
    [Crossref]
  21. Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, J. M. Zeuner, S. Nolte, Y. Lumer, N. Malkova, J. Xu, A. Szameit, Z. Chen, and M. Segev, “Observation of unconventional edge states in ‘photonic graphene’,” Nat. Mater. 13(1), 57–62 (2013).
    [Crossref] [PubMed]
  22. M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, D. Song, Z. Chen, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic graphene,” Phys. Rev. Lett. 111(10), 103901 (2013).
    [Crossref] [PubMed]
  23. J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014).
    [Crossref] [PubMed]
  24. 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]
  25. Q. Liang, Y. Yan, and J. Dong, “Zitterbewegung in the honeycomb photonic lattice,” Opt. Lett. 36(13), 2513–2515 (2011).
    [Crossref] [PubMed]
  26. O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
    [Crossref] [PubMed]
  27. A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Polychromatic dynamic localization in curved photonic lattices,” Nat. Phys. 5(4), 271–275 (2009).
    [Crossref]
  28. A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104(22), 223903 (2010).
    [Crossref] [PubMed]
  29. A. Crespi, G. Corrielli, G. D. Valle, R. Osellame, and S. Longhi, “Dynamic band collapse in photonic graphene,” New J. Phys. 15(1), 013012 (2013).
    [Crossref]
  30. R. A. Sepkhanov, A. Ossipov, and C. W. J. Beenakker, “Extinction of coherent backscattering by a disordered photonic crystal with a Dirac spectrum,” Europhys. Lett. 85(1), 14005 (2009).
    [Crossref]
  31. X. Wang, H. T. Jiang, C. Yan, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Anomalous transmission of disordered photonic graphenes at the Dirac point,” Europhys. Lett. 103(1), 17003 (2013).
    [Crossref]
  32. 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]
  33. 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]
  34. Y. Luo, W. Zhang, Y. Huang, J. Zhao, and J. Peng, “Wide-angle beam splitting by use of positive-negative refraction in photonic crystals,” Opt. Lett. 29(24), 2920–2922 (2004).
    [Crossref] [PubMed]
  35. P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92(12), 127401 (2004).
    [Crossref] [PubMed]
  36. X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83(16), 3251–3253 (2003).
    [Crossref]

2014 (1)

2013 (4)

A. Crespi, G. Corrielli, G. D. Valle, R. Osellame, and S. Longhi, “Dynamic band collapse in photonic graphene,” New J. Phys. 15(1), 013012 (2013).
[Crossref]

X. Wang, H. T. Jiang, C. Yan, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Anomalous transmission of disordered photonic graphenes at the Dirac point,” Europhys. Lett. 103(1), 17003 (2013).
[Crossref]

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, J. M. Zeuner, S. Nolte, Y. Lumer, N. Malkova, J. Xu, A. Szameit, Z. Chen, and M. Segev, “Observation of unconventional edge states in ‘photonic graphene’,” Nat. Mater. 13(1), 57–62 (2013).
[Crossref] [PubMed]

M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, D. Song, Z. Chen, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic graphene,” Phys. Rev. Lett. 111(10), 103901 (2013).
[Crossref] [PubMed]

2012 (6)

F. Zhai and K. Chang, “Valley filtering in graphene with a Dirac gap,” Phys. Rev. B 85(15), 155415 (2012).
[Crossref]

K. Behnia, “Condensed-matter physics: polarized light boosts valleytronics,” Nat. Nanotechnol. 7(8), 488–489 (2012).
[Crossref] [PubMed]

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotechnol. 7(8), 490–493 (2012).
[Crossref] [PubMed]

K. F. Mak, K. He, J. Shan, and T. F. Heinz, “Control of valley polarization in monolayer MoS2 by optical helicity,” Nat. Nanotechnol. 7(8), 494–498 (2012).
[Crossref] [PubMed]

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]

S. Bittner, B. Dietz, M. Miski-Oglu, and A. Richter, “Extremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiard,” Phys. Rev. B 85(6), 064301 (2012).
[Crossref]

2011 (5)

Q. Liang, Y. Yan, and J. Dong, “Zitterbewegung in the honeycomb photonic lattice,” Opt. Lett. 36(13), 2513–2515 (2011).
[Crossref] [PubMed]

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]

F. Zhai, Y. Ma, and K. Chang, “Valley beam splitter based on strained graphene,” New J. Phys. 13(8), 083029 (2011).
[Crossref]

Z. Wu, F. Zhai, F. M. Peeters, H. Q. Xu, and K. Chang, “Valley-dependent brewster angles and goos-hänchen effect in strained graphene,” Phys. Rev. Lett. 106(17), 176802 (2011).
[Crossref] [PubMed]

D. Gunlycke and C. T. White, “Graphene valley filter using a line defect,” Phys. Rev. Lett. 106(13), 136806 (2011).
[Crossref] [PubMed]

2010 (5)

Z. Wang and F. Liu, “Manipulation of electron beam propagation by hetero-dimensional graphene junctions,” ACS Nano 4(4), 2459–2465 (2010).
[Crossref] [PubMed]

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405(14), 2990–2995 (2010).
[Crossref]

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]

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
[Crossref] [PubMed]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104(22), 223903 (2010).
[Crossref] [PubMed]

2009 (3)

R. A. Sepkhanov, A. Ossipov, and C. W. J. Beenakker, “Extinction of coherent backscattering by a disordered photonic crystal with a Dirac spectrum,” Europhys. Lett. 85(1), 14005 (2009).
[Crossref]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Polychromatic dynamic localization in curved photonic lattices,” Nat. Phys. 5(4), 271–275 (2009).
[Crossref]

A. H. Castro Neto, 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]

2008 (4)

J. L. Garcia-Pomar, A. Cortijo, and M. Nieto-Vesperinas, “Fully valley-polarized electron beams in graphene,” Phys. Rev. Lett. 100(23), 236801 (2008).
[Crossref] [PubMed]

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

X. Zhang, “Demonstration of a new transport regime of photon in two-dimensional photonic crystal,” Phys. Lett. A 372(19), 3512–3516 (2008).
[Crossref]

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]

2007 (4)

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

A. Rycerz, J. Tworzydlo, and C. W. J. Beenakker, “Valley filter and valley valve in graphene,” Nat. Phys. 3(3), 172–175 (2007).
[Crossref]

D. Xiao, W. Yao, and Q. Niu, “Valley-contrasting physics in graphene: magnetic moment and topological transport,” Phys. Rev. Lett. 99(23), 236809 (2007).
[Crossref] [PubMed]

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

2006 (1)

S. Stankovich, D. A. Dikin, G. H. Dommett, K. M. Kohlhaas, E. J. Zimney, E. A. Stach, R. D. Piner, S. T. Nguyen, and R. S. Ruoff, “Graphene-based composite materials,” Nature 442(7100), 282–286 (2006).
[Crossref] [PubMed]

2004 (2)

Y. Luo, W. Zhang, Y. Huang, J. Zhao, and J. Peng, “Wide-angle beam splitting by use of positive-negative refraction in photonic crystals,” Opt. Lett. 29(24), 2920–2922 (2004).
[Crossref] [PubMed]

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92(12), 127401 (2004).
[Crossref] [PubMed]

2003 (1)

X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83(16), 3251–3253 (2003).
[Crossref]

Bahat-Treidel, O.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
[Crossref] [PubMed]

Bazaliy, Y.

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

Beenakker, C.

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

Beenakker, C. W. J.

R. A. Sepkhanov, A. Ossipov, and C. W. J. Beenakker, “Extinction of coherent backscattering by a disordered photonic crystal with a Dirac spectrum,” Europhys. Lett. 85(1), 14005 (2009).
[Crossref]

A. Rycerz, J. Tworzydlo, and C. W. J. Beenakker, “Valley filter and valley valve in graphene,” Nat. Phys. 3(3), 172–175 (2007).
[Crossref]

Behnia, K.

K. Behnia, “Condensed-matter physics: polarized light boosts valleytronics,” Nat. Nanotechnol. 7(8), 488–489 (2012).
[Crossref] [PubMed]

Bittner, S.

S. Bittner, B. Dietz, M. Miski-Oglu, and A. Richter, “Extremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiard,” Phys. Rev. B 85(6), 064301 (2012).
[Crossref]

Castro Neto, A. H.

A. H. Castro Neto, 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]

Chan, C. T.

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]

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]

Chang, K.

F. Zhai and K. Chang, “Valley filtering in graphene with a Dirac gap,” Phys. Rev. B 85(15), 155415 (2012).
[Crossref]

F. Zhai, Y. Ma, and K. Chang, “Valley beam splitter based on strained graphene,” New J. Phys. 13(8), 083029 (2011).
[Crossref]

Z. Wu, F. Zhai, F. M. Peeters, H. Q. Xu, and K. Chang, “Valley-dependent brewster angles and goos-hänchen effect in strained graphene,” Phys. Rev. Lett. 106(17), 176802 (2011).
[Crossref] [PubMed]

Chen, H.

X. Wang, H. T. Jiang, C. Yan, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Anomalous transmission of disordered photonic graphenes at the Dirac point,” Europhys. Lett. 103(1), 17003 (2013).
[Crossref]

Chen, Z.

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, J. M. Zeuner, S. Nolte, Y. Lumer, N. Malkova, J. Xu, A. Szameit, Z. Chen, and M. Segev, “Observation of unconventional edge states in ‘photonic graphene’,” Nat. Mater. 13(1), 57–62 (2013).
[Crossref] [PubMed]

M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, D. Song, Z. Chen, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic graphene,” Phys. Rev. Lett. 111(10), 103901 (2013).
[Crossref] [PubMed]

Corrielli, G.

A. Crespi, G. Corrielli, G. D. Valle, R. Osellame, and S. Longhi, “Dynamic band collapse in photonic graphene,” New J. Phys. 15(1), 013012 (2013).
[Crossref]

Cortijo, A.

J. L. Garcia-Pomar, A. Cortijo, and M. Nieto-Vesperinas, “Fully valley-polarized electron beams in graphene,” Phys. Rev. Lett. 100(23), 236801 (2008).
[Crossref] [PubMed]

Crespi, A.

A. Crespi, G. Corrielli, G. D. Valle, R. Osellame, and S. Longhi, “Dynamic band collapse in photonic graphene,” New J. Phys. 15(1), 013012 (2013).
[Crossref]

Cui, X.

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotechnol. 7(8), 490–493 (2012).
[Crossref] [PubMed]

Dai, J.

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotechnol. 7(8), 490–493 (2012).
[Crossref] [PubMed]

de Dood, M. J. A.

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]

Derov, J. S.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92(12), 127401 (2004).
[Crossref] [PubMed]

Diem, M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405(14), 2990–2995 (2010).
[Crossref]

Dietz, B.

S. Bittner, B. Dietz, M. Miski-Oglu, and A. Richter, “Extremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiard,” Phys. Rev. B 85(6), 064301 (2012).
[Crossref]

Dikin, D. A.

S. Stankovich, D. A. Dikin, G. H. Dommett, K. M. Kohlhaas, E. J. Zimney, E. A. Stach, R. D. Piner, S. T. Nguyen, and R. S. Ruoff, “Graphene-based composite materials,” Nature 442(7100), 282–286 (2006).
[Crossref] [PubMed]

Dommett, G. H.

S. Stankovich, D. A. Dikin, G. H. Dommett, K. M. Kohlhaas, E. J. Zimney, E. A. Stach, R. D. Piner, S. T. Nguyen, and R. S. Ruoff, “Graphene-based composite materials,” Nature 442(7100), 282–286 (2006).
[Crossref] [PubMed]

Dong, J.

Dreisow, F.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104(22), 223903 (2010).
[Crossref] [PubMed]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Polychromatic dynamic localization in curved photonic lattices,” Nat. Phys. 5(4), 271–275 (2009).
[Crossref]

Fan, S.

X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83(16), 3251–3253 (2003).
[Crossref]

Garanovich, I. L.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104(22), 223903 (2010).
[Crossref] [PubMed]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Polychromatic dynamic localization in curved photonic lattices,” Nat. Phys. 5(4), 271–275 (2009).
[Crossref]

Garcia-Pomar, J. L.

J. L. Garcia-Pomar, A. Cortijo, and M. Nieto-Vesperinas, “Fully valley-polarized electron beams in graphene,” Phys. Rev. Lett. 100(23), 236801 (2008).
[Crossref] [PubMed]

Geim, A. K.

A. H. Castro Neto, 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]

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

Grobman, M.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
[Crossref] [PubMed]

Gunlycke, D.

D. Gunlycke and C. T. White, “Graphene valley filter using a line defect,” Phys. Rev. Lett. 106(13), 136806 (2011).
[Crossref] [PubMed]

Haldane, F.

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

Hang, Z. H.

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]

He, K.

K. F. Mak, K. He, J. Shan, and T. F. Heinz, “Control of valley polarization in monolayer MoS2 by optical helicity,” Nat. Nanotechnol. 7(8), 494–498 (2012).
[Crossref] [PubMed]

Heinrich, M.

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, J. M. Zeuner, S. Nolte, Y. Lumer, N. Malkova, J. Xu, A. Szameit, Z. Chen, and M. Segev, “Observation of unconventional edge states in ‘photonic graphene’,” Nat. Mater. 13(1), 57–62 (2013).
[Crossref] [PubMed]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104(22), 223903 (2010).
[Crossref] [PubMed]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Polychromatic dynamic localization in curved photonic lattices,” Nat. Phys. 5(4), 271–275 (2009).
[Crossref]

Heinz, T. F.

K. F. Mak, K. He, J. Shan, and T. F. Heinz, “Control of valley polarization in monolayer MoS2 by optical helicity,” Nat. Nanotechnol. 7(8), 494–498 (2012).
[Crossref] [PubMed]

Huang, X.

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]

Huang, Y.

Jiang, H. T.

X. Wang, H. T. Jiang, C. Yan, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Anomalous transmission of disordered photonic graphenes at the Dirac point,” Europhys. Lett. 103(1), 17003 (2013).
[Crossref]

Kivshar, Y. S.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104(22), 223903 (2010).
[Crossref] [PubMed]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Polychromatic dynamic localization in curved photonic lattices,” Nat. Phys. 5(4), 271–275 (2009).
[Crossref]

Kohlhaas, K. M.

S. Stankovich, D. A. Dikin, G. H. Dommett, K. M. Kohlhaas, E. J. Zimney, E. A. Stach, R. D. Piner, S. T. Nguyen, and R. S. Ruoff, “Graphene-based composite materials,” Nature 442(7100), 282–286 (2006).
[Crossref] [PubMed]

Koschny, T.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405(14), 2990–2995 (2010).
[Crossref]

Lai, Y.

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]

Li, Y. H.

X. Wang, H. T. Jiang, C. Yan, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Anomalous transmission of disordered photonic graphenes at the Dirac point,” Europhys. Lett. 103(1), 17003 (2013).
[Crossref]

Liang, Q.

Liu, F.

Z. Wang and F. Liu, “Manipulation of electron beam propagation by hetero-dimensional graphene junctions,” ACS Nano 4(4), 2459–2465 (2010).
[Crossref] [PubMed]

Longhi, S.

A. Crespi, G. Corrielli, G. D. Valle, R. Osellame, and S. Longhi, “Dynamic band collapse in photonic graphene,” New J. Phys. 15(1), 013012 (2013).
[Crossref]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104(22), 223903 (2010).
[Crossref] [PubMed]

Lu, W. T.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92(12), 127401 (2004).
[Crossref] [PubMed]

Lumer, Y.

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, J. M. Zeuner, S. Nolte, Y. Lumer, N. Malkova, J. Xu, A. Szameit, Z. Chen, and M. Segev, “Observation of unconventional edge states in ‘photonic graphene’,” Nat. Mater. 13(1), 57–62 (2013).
[Crossref] [PubMed]

Luo, Y.

Ma, Y.

F. Zhai, Y. Ma, and K. Chang, “Valley beam splitter based on strained graphene,” New J. Phys. 13(8), 083029 (2011).
[Crossref]

Mak, K. F.

K. F. Mak, K. He, J. Shan, and T. F. Heinz, “Control of valley polarization in monolayer MoS2 by optical helicity,” Nat. Nanotechnol. 7(8), 494–498 (2012).
[Crossref] [PubMed]

Malkova, N.

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, J. M. Zeuner, S. Nolte, Y. Lumer, N. Malkova, J. Xu, A. Szameit, Z. Chen, and M. Segev, “Observation of unconventional edge states in ‘photonic graphene’,” Nat. Mater. 13(1), 57–62 (2013).
[Crossref] [PubMed]

Mei, J.

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]

Miski-Oglu, M.

S. Bittner, B. Dietz, M. Miski-Oglu, and A. Richter, “Extremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiard,” Phys. Rev. B 85(6), 064301 (2012).
[Crossref]

Nguyen, S. T.

S. Stankovich, D. A. Dikin, G. H. Dommett, K. M. Kohlhaas, E. J. Zimney, E. A. Stach, R. D. Piner, S. T. Nguyen, and R. S. Ruoff, “Graphene-based composite materials,” Nature 442(7100), 282–286 (2006).
[Crossref] [PubMed]

Nieto-Vesperinas, M.

J. L. Garcia-Pomar, A. Cortijo, and M. Nieto-Vesperinas, “Fully valley-polarized electron beams in graphene,” Phys. Rev. Lett. 100(23), 236801 (2008).
[Crossref] [PubMed]

Niu, Q.

D. Xiao, W. Yao, and Q. Niu, “Valley-contrasting physics in graphene: magnetic moment and topological transport,” Phys. Rev. Lett. 99(23), 236809 (2007).
[Crossref] [PubMed]

Nolte, S.

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014).
[Crossref] [PubMed]

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, J. M. Zeuner, S. Nolte, Y. Lumer, N. Malkova, J. Xu, A. Szameit, Z. Chen, and M. Segev, “Observation of unconventional edge states in ‘photonic graphene’,” Nat. Mater. 13(1), 57–62 (2013).
[Crossref] [PubMed]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104(22), 223903 (2010).
[Crossref] [PubMed]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Polychromatic dynamic localization in curved photonic lattices,” Nat. Phys. 5(4), 271–275 (2009).
[Crossref]

Novoselov, K. S.

A. H. Castro Neto, 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]

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

Osellame, R.

A. Crespi, G. Corrielli, G. D. Valle, R. Osellame, and S. Longhi, “Dynamic band collapse in photonic graphene,” New J. Phys. 15(1), 013012 (2013).
[Crossref]

Ossipov, A.

R. A. Sepkhanov, A. Ossipov, and C. W. J. Beenakker, “Extinction of coherent backscattering by a disordered photonic crystal with a Dirac spectrum,” Europhys. Lett. 85(1), 14005 (2009).
[Crossref]

Parimi, P. V.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92(12), 127401 (2004).
[Crossref] [PubMed]

Peeters, F. M.

Z. Wu, F. Zhai, F. M. Peeters, H. Q. Xu, and K. Chang, “Valley-dependent brewster angles and goos-hänchen effect in strained graphene,” Phys. Rev. Lett. 106(17), 176802 (2011).
[Crossref] [PubMed]

Peleg, O.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
[Crossref] [PubMed]

Peng, J.

Pereg-Barnea, T.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
[Crossref] [PubMed]

Peres, N. M. R.

A. H. Castro Neto, 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]

Pertsch, T.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104(22), 223903 (2010).
[Crossref] [PubMed]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Polychromatic dynamic localization in curved photonic lattices,” Nat. Phys. 5(4), 271–275 (2009).
[Crossref]

Piner, R. D.

S. Stankovich, D. A. Dikin, G. H. Dommett, K. M. Kohlhaas, E. J. Zimney, E. A. Stach, R. D. Piner, S. T. Nguyen, and R. S. Ruoff, “Graphene-based composite materials,” Nature 442(7100), 282–286 (2006).
[Crossref] [PubMed]

Plotnik, Y.

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, J. M. Zeuner, S. Nolte, Y. Lumer, N. Malkova, J. Xu, A. Szameit, Z. Chen, and M. Segev, “Observation of unconventional edge states in ‘photonic graphene’,” Nat. Mater. 13(1), 57–62 (2013).
[Crossref] [PubMed]

M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, D. Song, Z. Chen, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic graphene,” Phys. Rev. Lett. 111(10), 103901 (2013).
[Crossref] [PubMed]

Raghu, S.

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

Rechtsman, M. C.

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014).
[Crossref] [PubMed]

M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, D. Song, Z. Chen, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic graphene,” Phys. Rev. Lett. 111(10), 103901 (2013).
[Crossref] [PubMed]

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, J. M. Zeuner, S. Nolte, Y. Lumer, N. Malkova, J. Xu, A. Szameit, Z. Chen, and M. Segev, “Observation of unconventional edge states in ‘photonic graphene’,” Nat. Mater. 13(1), 57–62 (2013).
[Crossref] [PubMed]

Richter, A.

S. Bittner, B. Dietz, M. Miski-Oglu, and A. Richter, “Extremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiard,” Phys. Rev. B 85(6), 064301 (2012).
[Crossref]

Ruoff, R. S.

S. Stankovich, D. A. Dikin, G. H. Dommett, K. M. Kohlhaas, E. J. Zimney, E. A. Stach, R. D. Piner, S. T. Nguyen, and R. S. Ruoff, “Graphene-based composite materials,” Nature 442(7100), 282–286 (2006).
[Crossref] [PubMed]

Rycerz, A.

A. Rycerz, J. Tworzydlo, and C. W. J. Beenakker, “Valley filter and valley valve in graphene,” Nat. Phys. 3(3), 172–175 (2007).
[Crossref]

Segev, M.

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, J. M. Zeuner, S. Nolte, Y. Lumer, N. Malkova, J. Xu, A. Szameit, Z. Chen, and M. Segev, “Observation of unconventional edge states in ‘photonic graphene’,” Nat. Mater. 13(1), 57–62 (2013).
[Crossref] [PubMed]

M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, D. Song, Z. Chen, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic graphene,” Phys. Rev. Lett. 111(10), 103901 (2013).
[Crossref] [PubMed]

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
[Crossref] [PubMed]

Sepkhanov, R.

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

Sepkhanov, R. A.

R. A. Sepkhanov, A. Ossipov, and C. W. J. Beenakker, “Extinction of coherent backscattering by a disordered photonic crystal with a Dirac spectrum,” Europhys. Lett. 85(1), 14005 (2009).
[Crossref]

Shan, J.

K. F. Mak, K. He, J. Shan, and T. F. Heinz, “Control of valley polarization in monolayer MoS2 by optical helicity,” Nat. Nanotechnol. 7(8), 494–498 (2012).
[Crossref] [PubMed]

Shapira, N.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
[Crossref] [PubMed]

Shi, Y. L.

X. Wang, H. T. Jiang, C. Yan, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Anomalous transmission of disordered photonic graphenes at the Dirac point,” Europhys. Lett. 103(1), 17003 (2013).
[Crossref]

Sokoloff, J.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92(12), 127401 (2004).
[Crossref] [PubMed]

Song, D.

M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, D. Song, Z. Chen, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic graphene,” Phys. Rev. Lett. 111(10), 103901 (2013).
[Crossref] [PubMed]

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, J. M. Zeuner, S. Nolte, Y. Lumer, N. Malkova, J. Xu, A. Szameit, Z. Chen, and M. Segev, “Observation of unconventional edge states in ‘photonic graphene’,” Nat. Mater. 13(1), 57–62 (2013).
[Crossref] [PubMed]

Soukoulis, C. M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405(14), 2990–2995 (2010).
[Crossref]

Sridhar, S.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92(12), 127401 (2004).
[Crossref] [PubMed]

Stach, E. A.

S. Stankovich, D. A. Dikin, G. H. Dommett, K. M. Kohlhaas, E. J. Zimney, E. A. Stach, R. D. Piner, S. T. Nguyen, and R. S. Ruoff, “Graphene-based composite materials,” Nature 442(7100), 282–286 (2006).
[Crossref] [PubMed]

Stankovich, S.

S. Stankovich, D. A. Dikin, G. H. Dommett, K. M. Kohlhaas, E. J. Zimney, E. A. Stach, R. D. Piner, S. T. Nguyen, and R. S. Ruoff, “Graphene-based composite materials,” Nature 442(7100), 282–286 (2006).
[Crossref] [PubMed]

Sukhorukov, A. A.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104(22), 223903 (2010).
[Crossref] [PubMed]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Polychromatic dynamic localization in curved photonic lattices,” Nat. Phys. 5(4), 271–275 (2009).
[Crossref]

Sun, Y.

X. Wang, H. T. Jiang, C. Yan, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Anomalous transmission of disordered photonic graphenes at the Dirac point,” Europhys. Lett. 103(1), 17003 (2013).
[Crossref]

Szameit, A.

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014).
[Crossref] [PubMed]

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, J. M. Zeuner, S. Nolte, Y. Lumer, N. Malkova, J. Xu, A. Szameit, Z. Chen, and M. Segev, “Observation of unconventional edge states in ‘photonic graphene’,” Nat. Mater. 13(1), 57–62 (2013).
[Crossref] [PubMed]

M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, D. Song, Z. Chen, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic graphene,” Phys. Rev. Lett. 111(10), 103901 (2013).
[Crossref] [PubMed]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104(22), 223903 (2010).
[Crossref] [PubMed]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Polychromatic dynamic localization in curved photonic lattices,” Nat. Phys. 5(4), 271–275 (2009).
[Crossref]

Tünnermann, A.

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104(22), 223903 (2010).
[Crossref] [PubMed]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Polychromatic dynamic localization in curved photonic lattices,” Nat. Phys. 5(4), 271–275 (2009).
[Crossref]

Tworzydlo, J.

A. Rycerz, J. Tworzydlo, and C. W. J. Beenakker, “Valley filter and valley valve in graphene,” Nat. Phys. 3(3), 172–175 (2007).
[Crossref]

Valle, G. D.

A. Crespi, G. Corrielli, G. D. Valle, R. Osellame, and S. Longhi, “Dynamic band collapse in photonic graphene,” New J. Phys. 15(1), 013012 (2013).
[Crossref]

Vodo, P.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92(12), 127401 (2004).
[Crossref] [PubMed]

Wang, X.

X. Wang, H. T. Jiang, C. Yan, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Anomalous transmission of disordered photonic graphenes at the Dirac point,” Europhys. Lett. 103(1), 17003 (2013).
[Crossref]

Wang, Z.

Z. Wang and F. Liu, “Manipulation of electron beam propagation by hetero-dimensional graphene junctions,” ACS Nano 4(4), 2459–2465 (2010).
[Crossref] [PubMed]

White, C. T.

D. Gunlycke and C. T. White, “Graphene valley filter using a line defect,” Phys. Rev. Lett. 106(13), 136806 (2011).
[Crossref] [PubMed]

Wu, Y.

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]

Wu, Z.

Z. Wu, F. Zhai, F. M. Peeters, H. Q. Xu, and K. Chang, “Valley-dependent brewster angles and goos-hänchen effect in strained graphene,” Phys. Rev. Lett. 106(17), 176802 (2011).
[Crossref] [PubMed]

Xiao, D.

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotechnol. 7(8), 490–493 (2012).
[Crossref] [PubMed]

D. Xiao, W. Yao, and Q. Niu, “Valley-contrasting physics in graphene: magnetic moment and topological transport,” Phys. Rev. Lett. 99(23), 236809 (2007).
[Crossref] [PubMed]

Xu, H. Q.

Z. Wu, F. Zhai, F. M. Peeters, H. Q. Xu, and K. Chang, “Valley-dependent brewster angles and goos-hänchen effect in strained graphene,” Phys. Rev. Lett. 106(17), 176802 (2011).
[Crossref] [PubMed]

Xu, J.

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, J. M. Zeuner, S. Nolte, Y. Lumer, N. Malkova, J. Xu, A. Szameit, Z. Chen, and M. Segev, “Observation of unconventional edge states in ‘photonic graphene’,” Nat. Mater. 13(1), 57–62 (2013).
[Crossref] [PubMed]

Yan, C.

X. Wang, H. T. Jiang, C. Yan, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Anomalous transmission of disordered photonic graphenes at the Dirac point,” Europhys. Lett. 103(1), 17003 (2013).
[Crossref]

Yan, Y.

Yao, W.

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotechnol. 7(8), 490–493 (2012).
[Crossref] [PubMed]

D. Xiao, W. Yao, and Q. Niu, “Valley-contrasting physics in graphene: magnetic moment and topological transport,” Phys. Rev. Lett. 99(23), 236809 (2007).
[Crossref] [PubMed]

Yu, X.

X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83(16), 3251–3253 (2003).
[Crossref]

Zandbergen, S. R.

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]

Zeng, H.

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotechnol. 7(8), 490–493 (2012).
[Crossref] [PubMed]

Zeuner, J. M.

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014).
[Crossref] [PubMed]

M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, D. Song, Z. Chen, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic graphene,” Phys. Rev. Lett. 111(10), 103901 (2013).
[Crossref] [PubMed]

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, J. M. Zeuner, S. Nolte, Y. Lumer, N. Malkova, J. Xu, A. Szameit, Z. Chen, and M. Segev, “Observation of unconventional edge states in ‘photonic graphene’,” Nat. Mater. 13(1), 57–62 (2013).
[Crossref] [PubMed]

Zhai, F.

F. Zhai and K. Chang, “Valley filtering in graphene with a Dirac gap,” Phys. Rev. B 85(15), 155415 (2012).
[Crossref]

Z. Wu, F. Zhai, F. M. Peeters, H. Q. Xu, and K. Chang, “Valley-dependent brewster angles and goos-hänchen effect in strained graphene,” Phys. Rev. Lett. 106(17), 176802 (2011).
[Crossref] [PubMed]

F. Zhai, Y. Ma, and K. Chang, “Valley beam splitter based on strained graphene,” New J. Phys. 13(8), 083029 (2011).
[Crossref]

Zhang, W.

Zhang, X.

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]

X. Zhang, “Demonstration of a new transport regime of photon in two-dimensional photonic crystal,” Phys. Lett. A 372(19), 3512–3516 (2008).
[Crossref]

Zhang, Z.-Q.

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]

Zhao, J.

Zheng, H.

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]

Zimney, E. J.

S. Stankovich, D. A. Dikin, G. H. Dommett, K. M. Kohlhaas, E. J. Zimney, E. A. Stach, R. D. Piner, S. T. Nguyen, and R. S. Ruoff, “Graphene-based composite materials,” Nature 442(7100), 282–286 (2006).
[Crossref] [PubMed]

ACS Nano (1)

Z. Wang and F. Liu, “Manipulation of electron beam propagation by hetero-dimensional graphene junctions,” ACS Nano 4(4), 2459–2465 (2010).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83(16), 3251–3253 (2003).
[Crossref]

Europhys. Lett. (2)

R. A. Sepkhanov, A. Ossipov, and C. W. J. Beenakker, “Extinction of coherent backscattering by a disordered photonic crystal with a Dirac spectrum,” Europhys. Lett. 85(1), 14005 (2009).
[Crossref]

X. Wang, H. T. Jiang, C. Yan, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Anomalous transmission of disordered photonic graphenes at the Dirac point,” Europhys. Lett. 103(1), 17003 (2013).
[Crossref]

Nat. Mater. (3)

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]

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, J. M. Zeuner, S. Nolte, Y. Lumer, N. Malkova, J. Xu, A. Szameit, Z. Chen, and M. Segev, “Observation of unconventional edge states in ‘photonic graphene’,” Nat. Mater. 13(1), 57–62 (2013).
[Crossref] [PubMed]

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

Nat. Nanotechnol. (3)

K. Behnia, “Condensed-matter physics: polarized light boosts valleytronics,” Nat. Nanotechnol. 7(8), 488–489 (2012).
[Crossref] [PubMed]

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotechnol. 7(8), 490–493 (2012).
[Crossref] [PubMed]

K. F. Mak, K. He, J. Shan, and T. F. Heinz, “Control of valley polarization in monolayer MoS2 by optical helicity,” Nat. Nanotechnol. 7(8), 494–498 (2012).
[Crossref] [PubMed]

Nat. Phys. (2)

A. Rycerz, J. Tworzydlo, and C. W. J. Beenakker, “Valley filter and valley valve in graphene,” Nat. Phys. 3(3), 172–175 (2007).
[Crossref]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Polychromatic dynamic localization in curved photonic lattices,” Nat. Phys. 5(4), 271–275 (2009).
[Crossref]

Nature (1)

S. Stankovich, D. A. Dikin, G. H. Dommett, K. M. Kohlhaas, E. J. Zimney, E. A. Stach, R. D. Piner, S. T. Nguyen, and R. S. Ruoff, “Graphene-based composite materials,” Nature 442(7100), 282–286 (2006).
[Crossref] [PubMed]

New J. Phys. (2)

F. Zhai, Y. Ma, and K. Chang, “Valley beam splitter based on strained graphene,” New J. Phys. 13(8), 083029 (2011).
[Crossref]

A. Crespi, G. Corrielli, G. D. Valle, R. Osellame, and S. Longhi, “Dynamic band collapse in photonic graphene,” New J. Phys. 15(1), 013012 (2013).
[Crossref]

Opt. Lett. (3)

Phys. Lett. A (1)

X. Zhang, “Demonstration of a new transport regime of photon in two-dimensional photonic crystal,” Phys. Lett. A 372(19), 3512–3516 (2008).
[Crossref]

Phys. Rev. A (2)

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

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

Phys. Rev. B (3)

S. Bittner, B. Dietz, M. Miski-Oglu, and A. Richter, “Extremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiard,” Phys. Rev. B 85(6), 064301 (2012).
[Crossref]

F. Zhai and K. Chang, “Valley filtering in graphene with a Dirac gap,” Phys. Rev. B 85(15), 155415 (2012).
[Crossref]

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]

Phys. Rev. Lett. (10)

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]

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92(12), 127401 (2004).
[Crossref] [PubMed]

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]

M. C. Rechtsman, Y. Plotnik, J. M. Zeuner, D. Song, Z. Chen, A. Szameit, and M. Segev, “Topological creation and destruction of edge states in photonic graphene,” Phys. Rev. Lett. 111(10), 103901 (2013).
[Crossref] [PubMed]

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, “Klein tunneling in deformed honeycomb lattices,” Phys. Rev. Lett. 104(6), 063901 (2010).
[Crossref] [PubMed]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, S. Longhi, and Y. S. Kivshar, “Observation of two-dimensional dynamic localization of light,” Phys. Rev. Lett. 104(22), 223903 (2010).
[Crossref] [PubMed]

Z. Wu, F. Zhai, F. M. Peeters, H. Q. Xu, and K. Chang, “Valley-dependent brewster angles and goos-hänchen effect in strained graphene,” Phys. Rev. Lett. 106(17), 176802 (2011).
[Crossref] [PubMed]

D. Gunlycke and C. T. White, “Graphene valley filter using a line defect,” Phys. Rev. Lett. 106(13), 136806 (2011).
[Crossref] [PubMed]

D. Xiao, W. Yao, and Q. Niu, “Valley-contrasting physics in graphene: magnetic moment and topological transport,” Phys. Rev. Lett. 99(23), 236809 (2007).
[Crossref] [PubMed]

J. L. Garcia-Pomar, A. Cortijo, and M. Nieto-Vesperinas, “Fully valley-polarized electron beams in graphene,” Phys. Rev. Lett. 100(23), 236801 (2008).
[Crossref] [PubMed]

Physica B (1)

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405(14), 2990–2995 (2010).
[Crossref]

Rev. Mod. Phys. (1)

A. H. Castro Neto, 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]

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Figures (6)

Fig. 1
Fig. 1 (a) Band structures for TE modes in PG with honeycomb lattace. Inset: Brillouin zone and the EFCs at normalized frequency of 0.36 (labeled with the horizontal dashed line). (b) Scheme of valley-dependent wave transport behavior: the blue triangles are the EFCs for the PG at normalized frequency of 0.36. These triangles encircles K or K' points. The blue semicircles are parts of the EFCs for free space at same frequency. The thin gray lines represent the conservation of the parallel wave vector, and the green and the red arrows in the triangles represent the directions of energy flow coming from the K and K' valley, respectively. For the zigzag interface, the refractions in the two valleys have different orientations. For the armchair interface, the beam associated to one of the valleys is fully self-collimated and the beam belonging to the other valley splits into three beams. The energy flows are also indicated with the arrows with different thickness and color in PG structure.
Fig. 2
Fig. 2 (a) Photograph of one PG sample in waveguide chamber. The width of incident beam is controlled by absorbing material. The cylindrical metal defects act as scatterers to monitor valley-dependent beam indirectly. Inset is the enlarged view of part of the honeycomb lattice structure where metal defects are inserted. (b) Waveguide chamber and field probe used in experiments.
Fig. 3
Fig. 3 Beam self-collimation in PG without metal defects. (a)Simulated electric field distribution of PG with armchair interface. Outgoing region is encircled by the red dashed lines; (b) Top: Measured electric field distribution in the outgoing region; Bottom: Comparation of simulated and experimental normalized profiles of electric field distribution along waveguide edge.
Fig. 4
Fig. 4 Beam splitting in PG without metal defects. (a) Simulated electric field distribution of PG with zigzag interface; (b) Top: Measured electric field distribution in the outgoing region, which is encircled by the red dashed lines in Fig. 4(a); Bottom: Comparation of simulated and experimental normalized profiles of electric field distribution along waveguide edge. Inset is the PG model with zigzag interface, in which the middle one of the first row of alumina rods is removed, as indicated by the black arrow).
Fig. 5
Fig. 5 Measured profiles of electric field distribution along waveguide edge (top) and simulated electric field distributions in PGs with armchair interface, with metal defects embedded in three different positions (bottom). The waveguide edge is indicated by the red dashed line. (a) Metal defects are embedded into positions accurately on the path of beam; (b) Metal defects are embedded into positions away from the beam path; (c) Metal defects are embedded into positions a little offset from the beam path.
Fig. 6
Fig. 6 Measured profiles of electric field distribution along waveguide edge (top) and simulated electric field distributions in PG with zigzag interface. The waveguide edge is indicated by the red dashed line. Six different defects are applied, respectively. (a) Metal defects are inserted in the right path of the splitting beams in close to the top edge of PG; (b) Metal defects are inserted in the right path of the splitting beams in close to the middle of PG; (c) Metal defects are inserted in the left path of the splitting beams in close to the bottom edge of PG; (d) Metal defects are inserted in a position outside of the paths of splitting beams at the left part of PG; (e) Metal defects are inserted in a position between the paths of the splitting beams; (f) Most of the alumina rods between the splitting beams are removed.

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