Observation of valley-dependent beams in photonic graphene Fusheng Deng,1,2 Yong Sun,1,* Xiao Wang,1 Rui Xue,1 Yuan Li,1 Haitao Jiang,1 Yunlong Shi,1,2,4 Kai Chang,3 and Hong Chen1 1
MOE Key Laboratory of Advanced Micro-Structured Materials, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China 2 Institute of Solid State Physics, Shanxi Datong University- Datong, China 3 SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, 100083, Beijing, China 4 [email protected] * [email protected]
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 OCIS codes: (230.5298) Photonic crystals; (230.1360) Beam splitters; (230.1150) All-optical devices.
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#220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23605
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1. Introduction Graphene, the monolayer carbon honeycomb lattice, has attracted a great interest as a promising candidate material for the post-silicon nanoelectronics [1–3]. 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 [4–14]. 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
#220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23606
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) [15–32]. 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 [15–18], edge states [19–23], zitterbewegung [24, 25], Klein tunneling [26], dynamic localization [27–29], 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 valleydependent beams in PG in microwave regime. In section 2 of this paper, by analyzing the equifrequency contours (EFCs) [34–36], 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 #220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23607
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].
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 selfcollimated 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.
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.
#220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23608
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.
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. 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.
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 #220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23609
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.
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).
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. 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 ydirection, such as the case shown in Fig. 5(a), the measurements of outgoing fields of selfcollimation 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
#220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23610
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.
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.
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.
#220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23611
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.
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
#220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23612
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).
#220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23613
MOE Key Laboratory of Advanced Micro-Structured Materials, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China 2 Institute of Solid State Physics, Shanxi Datong University- Datong, China 3 SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, 100083, Beijing, China 4 [email protected] * [email protected]
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 OCIS codes: (230.5298) Photonic crystals; (230.1360) Beam splitters; (230.1150) All-optical devices.
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#220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23605
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1. Introduction Graphene, the monolayer carbon honeycomb lattice, has attracted a great interest as a promising candidate material for the post-silicon nanoelectronics [1–3]. 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 [4–14]. 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
#220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23606
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) [15–32]. 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 [15–18], edge states [19–23], zitterbewegung [24, 25], Klein tunneling [26], dynamic localization [27–29], 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 valleydependent beams in PG in microwave regime. In section 2 of this paper, by analyzing the equifrequency contours (EFCs) [34–36], 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 #220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23607
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].
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 selfcollimated 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.
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.
#220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23608
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.
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. 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.
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 #220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23609
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.
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).
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. 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 ydirection, such as the case shown in Fig. 5(a), the measurements of outgoing fields of selfcollimation 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
#220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23610
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.
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.
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.
#220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23611
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.
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
#220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23612
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).
#220251 - $15.00 USD Received 1 Aug 2014; revised 12 Sep 2014; accepted 14 Sep 2014; published 18 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023605 | OPTICS EXPRESS 23613
