ARTICLES PUBLISHED ONLINE: 22 FEBRUARY 2016 | DOI: 10.1038/NMAT4573
Robust reconfigurable electromagnetic pathways within a photonic topological insulator Xiaojun Cheng1,2†, Camille Jouvaud1†, Xiang Ni1,2, S. Hossein Mousavi3, Azriel Z. Genack1,2* and Alexander B. Khanikaev1,2* The discovery of topological photonic states has revolutionized our understanding of electromagnetic propagation and scattering. Endowed with topological robustness, photonic edge modes are not reflected from structural imperfections and disordered regions. Here we demonstrate robust propagation along reconfigurable pathways defined by synthetic gauge fields within a topological photonic metacrystal. The flow of microwave radiation in helical edge modes following arbitrary contours of the synthetic gauge field between bianisotropic metacrystal domains is unimpeded. This is demonstrated in measurements of the spectrum of transmission and time delay along the topological domain walls. These results provide a framework for freely steering electromagnetic radiation within photonic structures.
R
ecent discoveries of novel states of condensed matter characterized by topological order in quantum spin Hall effect (QSHE) systems, topological insulators (TIs), and topological crystalline insulators (TCIs; refs 1–7) have stimulated the search for analogous states in structures supporting classical waves8–13 . This search has been especially fruitful in electromagnetics14 , where numerous theoretical predictions and experimental demonstrations have been made from microwave to optical frequencies15–28 . Whereas early approaches to topological order for photons relied on an external magnetic field to break time reversal (TR) symmetry15–19 , the absence of materials with a strong magnetic response at optical frequencies and the inconvenience associated with the use of strong magnets at microwave frequencies led to increased interest in systems in which TR symmetry is either preserved21–28 or broken by different methods, such as by dynamic modulation of the system parameters20,21,29–31 . Indeed, engineered synthetic gauge fields21–27 , or temporal20,21,29–32 or spatial modulation26 emulating the effect of a magnetic field have proved to be viable alternatives to an actual magnetic field and have yielded desirable topological properties. Although several successful experimental implementations of topological states of light with TR symmetry have been reported recently24,26,28,33 , one important property of true external fields available in condensed matter systems has evaded emulation in photonics—the control of topological states. The possibility of such control through reconfigurable synthetic gauge fields would usher in a broad range of practical applications from spin/helicity filtering to tunable electronic34–36 and analogous photonic devices. Based on this idea, we demonstrate experimentally that electromagnetic radiation can be steered and delivered to any point within a reconfigurable topological metacrystal37–39 without back-reflection. This is achieved by exploiting adjustable gauge fields defining arbitrarily shaped pathways for topological edge states. Synthetic optical media, such as photonic crystals and metamaterials, offer a versatile toolkit for manipulating the properties of electromagnetic waves40,41 . Recently, optical bianisotropic metamaterials have been predicted to support a topologically nontrivial photonic state analogous to the QSHE state of condensed matter25 .
In these systems, bianisotropy, also referred to as magneto-electric coupling42 , serves as a synthetic gauge field leading to a topological transition across the edge. Here, we realize a reconfigurable topological photonic lattice implemented in the parallel plate waveguide43 shown in Fig. 1a,b. The reconfigurable topological structure studied here is formed between two parallel copper plates through which holes are drilled to support a periodic triangular array of copper rods with ring collars. The rods can be moved up or down so that lattices can be formed with the collar touching either the upper or lower copper plate, and structures with arbitrarily shaped boundaries between the two topologically distinct domains can be created.
Reconfigurable topological metacrystal design
The photonic modes of the metacrystal with σz inversion symmetry, which can be created by moving the collars to the midpoint between the plates, can be classified according to their electric field as either symmetric (TE-like) or antisymmetric (TM-like)12,25,43 . Provided that the dispersion of these two families of modes is matched spectrally over an extended frequency band, these photonic modes can be used to emulate the two components of the electronic spin. In fact, by engineering such a degeneracy, one restores electromagnetic duality responsible for the degeneracy between the two polarizations of electromagnetic waves that is typically broken by a material’s response, and the corresponding modes are referred to as polarization or spin-degenerate12,25,43 . Another important requirement for engineering a topological phase is the presence of Dirac-like degenerate points. This is achieved in our system by the lattice symmetry: in the triangular lattice considered here, the lowest Dirac degeneracies are formed at K and K0 points for both TE- and TM-like dipolar modes near a frequency of 21 GHz (Fig. 1c, blue dotted lines). When these requirements are met, reducing the σz symmetry by sliding rods up or down produces an effect analogous to spin–orbit coupling and leads via a topological phase transition to a QSHE-like state12,25,43 . This transition is accompanied by the opening of a complete topological bandgap (Fig. 1c, red dotted curves) and the formation of an ‘insulating state.’ In practice, this reduction of σz inversion symmetry causes
1 Department of Physics, Queens College of the City University of New York, Queens, New York 11367, USA. 2 The Graduate Center of the City University of New York, New York, New York 10016, USA. 3 Microelectronics Research Center, Cockrell School of Engineering, University of Texas at Austin, Austin, Texas 78758, USA. † These authors contributed equally to this work. *e-mail: [email protected]; [email protected]
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2016 Macmillan Publishers Limited. All rights reserved
1
NATURE MATERIALS DOI: 10.1038/NMAT4573
ARTICLES a
m0 m=0 Trivial Non-trivial d0
H Zc = 0
dc
zc
d0
Zc > 0
d Ω (m2)
0
2 0 ky (m − −2 1)
0.0
−0.9
14 K M Wave vector, k
2 −2 0 −1 ) (m K − kx
0
18 Spin down
0 −18
2 0 ky (m − −2 1)
−9
−18 2 0 1 −2 − (m ) k x−K
Top bands Berry curvature
Κ Μ Κ′ Γ
Γ
Spin up
9
22 Zc = 0 Zc ≠ 0
a0
Bottom bands Berry curvature 18
Ω (m2)
Frequency (GHz)
26
dc
Ω (m2)
c
18
H
hc
Γ
−1.8
2 0
Spin up 2 0 ky (m − −2 1)
2 −2 0 −1 (m ) k x−K
−2
Ω (m2)
Zc < 0
hc
1.8 0.9 Spin down 0.0
2 0 ky (m − −2 1)
−2
0 2 1 − (m ) k x−K
Figure 1 | Reconfigurable metacrystal and its bulk band structure. a, Geometric arrangement of metallic rods and collars at different positions inside the parallel plate waveguide: from left to right, the front collars are shifted from the down to up position, leading to topological transitions as indicated by the mass term m. b, Left panel: side view of the metacrystal. Right panel: top view of the metacrystal with the top copper plate and some rods removed to reveal the periodic arrangement of the rods and collars. The field inside the structure is probed via the additional deep subwavelength holes. c, Bulk photonic band structures of the metacrystal for two different configurations: blue dotted curve and red dotted curve correspond to symmetric (no bianisotropy) and asymmetric (with bianisotropy) structures, respectively. d, Berry curvatures for the four bands of interest near the K point calculated from the bulk eigenstates found using the full-wave approach. Calculations of Berry curvature at the K0 point yield identical results and are not shown.
pairwise coupling between the four original Dirac bands—the lower antisymmetric (symmetric) mode and the upper symmetric (antisymmetric) mode, and the formation of new pseudo-spin-up (↑) and pseudo-spin-down (↓) eigenstates. The new eigenstates have their orbital and pseudo-spin degrees of freedom interlocked, with circularly polarized states with TE-like (TM-like) components delayed (advanced) with respect to each other by a phase shift of π/2 (refs 25,43). A first-principles calculation of the topological invariants of the system, the spin-Chern numbers, found from the full-wave numerical eigenstates, confirms that the corresponding bands acquire nonzero spin-Chern numbers when the collars are all moved from the midpoint either up or down, thus substantiating the topological transition. More importantly, as the collars are moved from the lower to the upper plate, the spin-Chern numbers for the ↑ ↑ four dipolar bands reverse from Cu/l = ±1 to Cu/l = ∓1 and from ↓ ↓ Cu/l = ∓1 to Cu/l = ±1, where the subscript indicates the lower (l) or upper (u) band and the superscript denotes the spin state. The Berry curvature in the proximity of the former degeneracy at the K point is shown in Fig. 1d and clearly reveals the four peaks which contribute to the spin-Chern numbers of the individual bands. By means of perturbation theory for electromagnetic waves, the metacrystal can be described by an effective Hamiltonian analogous to the Kane–Mele model4 for graphene with spin– orbital coupling12,25,43 . Indeed, in the basis of circularly polarized hybrid modes, the perturbed Hamiltonian assumes the form b ↑/↓ = vD τˆ0 sˆ0 σb k · δkk + mτˆ3 sˆ3 σˆ 3 , where τˆi and sˆi are the inter-valley H and pseudo-spin Pauli matrices, vD is the group velocity near the Dirac point, and m is the mass term describing the degree of σz inversion-symmetry breaking. First-principles calculations of the spin-Chern numbers also agree with the results of the ↑ effective Hamiltonian description, which yield Cu/l = ±(m/|m|) ↓ and Cu/l = ∓(m/|m|). The ability to independently slide each of the collars between the metal plates makes it possible to controllably introduce a 2
local bianisotropic response which plays the role of the synthetic gauge field within a unit cell and is responsible for the topological phase12,25 . Although the band structures are identical in systems in which all the collars are either up or down, so that they touch one or the other copper plate (Fig. 1c, red dotted curves), these two configurations are topologically distinct, as revealed by opposite sign of their spin-Chern numbers. These distinct topological phases are isolated by the topologically trivial gapless regime of the inversion symmetric structure (Fig. 1c, blue dotted curves).
Observation of topologically protected edge states The key feature of the topologically nontrivial states in condensed matter is their ability to carry charge or spin currents along interfaces while being excluded from the bulk. Topological photonic systems have similar properties with respect to photonic transport, which can be one-way15–21 and spin-polarized22–28 . Although the magnetic or synthetic gauge fields open a bandgap and prevent photons from propagating in the bulk, photonic transport can take place along the boundaries in the form of edge states. The edge states emerge between topologically distinct structures in which the Chern number (or the spin-Chern number) changes across the interface. This can take place either at the boundary between topologically nontrivial and trivial systems, or at an interface separating two topological domains with opposite or different topological indices. Although in this work we focus on the second class of edge states, shown in Fig. 2a, the first class of states, shown in Fig. 2b, is in fact more common in condensed matter systems. It naturally occurs at external boundaries and in photonic systems with TR symmetry broken by the magneto-optical response18,19 , where the domain walls would require a reversal of the magnetization within the structure, which is very challenging to realize in practice. For a topological system with engineered pseudo-spin degrees of freedom, both classes of topological interfaces have been engineered, proving the versatility of this approach24,28 . An example of the first class of interface of the metacrystal with a suitably designed topologically
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2016 Macmillan Publishers Limited. All rights reserved
NATURE MATERIALS DOI: 10.1038/NMAT4573 a
Domain wall
b
m0
23
23
22
22 Frequency (GHz)
Frequency (GHz)
m>0
ARTICLES
21
m=0
21
20
20
19 −π/ π a0
Interface
K′
0 Wavenumber, k
K
19 −π/ π a0
π a0 π/
K′
0 Wavenumber, k
K
π a0 π/
Figure 2 | Topological interfaces in the metacrystal and their edge band diagrams. a, Photonic band structure of a 30 × 1 supercell of the reconfigurable metacrystal with the bianisotropic domain wall along a line bisecting the structure across which the collars are switched from the up to the down position, as shown in the top panel (the parallel plates are not shown). b, Photonic band structure of a 30 × 1 supercell of the non-reconfigurable topological interface in its centre formed between the topological metacrystal and a topologically trivial photonic crystal waveguide made of gapped triangular copper prisms, as shown in the top panel (the parallel plates are not shown). The red and blue dotted lines in a and b correspond to the spin-up (↑) and spin-down (↓) edge modes, respectively. The middle panels in a and b show the confinement of the edge modes to the topological interfaces. The grey dotted lines show the bulk spectrum. The green shaded regions indicate the complete topological bandgaps for the two cases.
trivial structure (see Supplementary Information for details) with the trivial gap opened (the spin-Chern numbers remain zero) by reducing the in-plane inversion symmetry and supporting a pair of edge states is shown in Fig. 2b. Aiming for a reconfigurable system, however, we are more interested here in the second class of topological interface—the domain wall between topologically distinct domains of inverted structures shown in the top panel of Fig. 2a. According to the bulk– boundary correspondence principle44 , such domain walls host twice as many topological modes propagating along the boundary line, which is corroborated by the numerical simulations shown in the two lower panels of Fig. 2a. In contrast to the boundary with the trivial system which cannot be readily reconfigured, domain walls can be defined solely by the gauge field32 responsible for the topological order. In the case of the metacrystal, such domain walls can be created in a controllable manner by displacing the collars in one portion of the structure towards the opposing plate to produce adjoining regions with distinct topological domains across the interface, characterized by spin-Chern numbers of opposite sign and opposite effective mass m. By design, the domain walls can be made in any desired shape for the two-dimensional reconfigurable metacrystal described here. The main property of topological edge states that distinguishes them from their topologically trivial counterparts, such as Tamm states or modes of optical waveguides, is their one-way, chiral or helical character, endowing them with robustness against defects and disorder. In the case of the QSHE emulated here, the edge modes represent helical states with spin (polarization configuration) locked to the propagation direction, as indicated in Fig. 2 (refs 25,43). This property endows the edge states with topological protection against scattering as long as the spin of the edge mode is preserved
by the scattering potentials of the defect or disorder of the crystal. Combined with the reconfigurability of our system, this suggests that it is possible to route topological edge modes confined to the domain wall along arbitrarily shaped paths without backscattering. To test the robustness of photonic transport along reconfigurable pathways enabled by the design of the metacrystal, propagation was measured in samples with a variety of bends and shapes of the topological domain wall, as shown in Fig. 3a–c. The measurements (refer to Methods for details) were performed by means of a vector network analyser. Edge modes are excited by a small dipole antenna at the domain wall on one end of the structure, as indicated by red radiating dots in the left panels of Fig. 3a–c. Transmission was measured by another dipole antenna placed at the opposite end of the domain wall, indicated in blue in Fig. 3a–c. In addition, we performed a discrete near-field scan of the electric field within the structure by inserting the tip of the probe antenna a short distance into deep subwavelength holes at regular intervals along the structure. As shown in Fig. 1b, two additional subwavelength holes were drilled per unit cell so that the triangular symmetry of the lattice is preserved. First-principles numerical simulations confirmed that the overall electromagnetic response of the structure is not compromised by the subwavelength holes. Both the amplitude and phase of the electric field along the domain wall were measured to determine the field distribution and dwell time of the edge modes. Transmission spectra for some of the experimental configurations tested are shown in Fig. 3a–c (left top panels) together with the results of numerical simulations (left bottom panels), which show that the edge modes propagate freely without attenuation due to scattering along curved pathways. In the right-hand panels of Fig. 3a–c the transmission spectra measured for the three configurations (blue solid curves) are compared against the spectrum of
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2016 Macmillan Publishers Limited. All rights reserved
3
NATURE MATERIALS DOI: 10.1038/NMAT4573
ARTICLES Transmission log10(I)
a
−2 −3 −4 −5
d
−6
−1
19
Transmission log10(I)
21
22
−2 −3
−3 −4 −5
Intensity I
−4 −5
y
−6
−6
−7
−7 19
c
20
−2 ln (I/Imax)
Transmission log10(I)
b
0
−7
20
21
22
−8 x
−9
−2
−10
−3 −4
−5
0 Depth, y (cm)
5
10
−5 −6 −7 19
20 21 22 Frequency (GHz)
Figure 3 | Experimental verification of reconfigurable guiding and exponential confinement of the topological edge modes. a–c, Three straight segments of the domain wall separated by two bends of 60◦ , 90◦ and 120◦ , respectively. The left panels show the domain walls separating two crystal structures differing by inversion about a plane midway between the copper plates (top) and the simulations for the corresponding structures (bottom). The right panels show transmission spectra for radiation propagating along the domain wall (blue lines) and through the bulk of the structure (black dashed lines). Green shaded regions indicate the topological bandgap, defined as I < 10−5 for transmission in the bulk. d, Semi-log plot of the variation of the intensity averaged over frequencies near the centre of the topological bandgap (20.75 ± 0.25 GHz) versus distance in the direction perpendicular to the domain wall (as indicated in the inset with white and black dots indicating up and down positions of the collars).
transmission through the bulk of the metacrystal (black dashed curves) with all the collars shifted up. The latter spectrum reveals very low transmission (I < 10−5 ) in the frequency range from 20.1 to 21.7 GHz corresponding to the bandgap, which is surrounded by pass bands for the bulk modes. In contrast to the case of a gapped band structure, the experimental spectra for different domain wall configurations exhibit high transmission, mediated by the edge modes, over the frequency range of the bandgap. The high transmission observed for all domain wall contours, including an irregularly shaped wall with multiple bends, shows that transport is robust. In contrast, for the case of topologically trivial guided modes, one would expect the wave to be reflected each time there is a sharp bend in the path. In addition, the presence of multiple bends would lead to the formation of standing waves in intermediate segments. This would give rise to distinct transmission peaks at Fabry–Perot resonances determined by the length of the segment between the bends43 , which are not observed (Fig. 3a–c). Interestingly, we find that the transmission in the bandgap exceeds the transmission in the pass band. This is a consequence of the confinement of the edge modes to the domain wall, as opposed to the spread of the bulk modes, which fall off roughly as the inverse of the distance. To confirm that the enhanced transmission is by the edge modes, we measured the variation of the electric field in the direction perpendicular to the domain wall. This is seen in Fig. 3d to decay exponentially away from the boundary line.
Spin-locked wave-division in topological junctions To further demonstrate that the edge states are topologically protected and helical and, hence, mimic the edge states of QSHE, 4
we performed further studies to confirm that the spin is locked to the propagation direction. We considered a configuration of the synthetic gauge field distribution with two domain walls crossing in the centre of the structure, as shown in Fig. 4a, so that the four segments of the domain walls are formed with different gradients of the mass term m and the spin-Chern numbers across the walls, as illustrated in Fig. 4b. These gradients define whether a particular spin has positive or negative group velocity along the domain wall. For the case of the spin-down state ψ − , illustrated in Fig. 4b, the left and right walls support only outward propagation (away from the domain walls junction), whereas the top and bottom walls support only inward propagation (towards the junction). For the spin-up state ψ + the situation reverses. Therefore, when the system is excited from the bottom Port 1, only the spin-down ψ − mode is allowed to couple into the system. More importantly, when the spin-down mode ψ − reaches the junction, it cannot excite spin-up edge state ψ + propagating away from the junction along the upper domain wall because of the mismatch in the spin configuration. Thus, transmission through Port 3 seems to be completely suppressed. On the other hand, transmission through the left and right Ports 2 and 4 is allowed because the outward propagation of the spin-down state is permitted along these walls. These conclusions are consistent with the results of numerical simulations shown in the lower panel of Fig. 4a, as well as with measurements of strong transmission at Ports 2 and 4 and its suppression at Port 3, shown in Fig. 3c. It is important to note that such behaviour would not normally take place in a non-topological structure, where the conventional wave-division would be observed with transmission through Port 3 comparable to that through Port 2 because of the similar mutual orientation of the corresponding output waveguides. In contrast,
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2016 Macmillan Publishers Limited. All rights reserved
NATURE MATERIALS DOI: 10.1038/NMAT4573
ARTICLES
a
b
P 3 ((out) Port (out)
Port 3 Port 4
Port 1
Port 2
C↓ = 1
ψ+ ψ−
ψ−
C↓ = 1
C↓ = −11
ψ− Out
Port 4 (out)
Port 2 (out)
C↓ = −1
Port 1 (in)
c Out
Port 2 Port 3 Port 4
Transmission log10(I)
−2
In Out
−3 −4 −5 −6 −7 19.0
19.5
20.0
20.5 21.0 21.5 Frequency (GHz)
22.0
22.5
Figure 4 | Experimental demonstration of spin-locked wave-division of an edge mode at a four-port topological junction. a, Top: four-port configuration formed by the four topological domains with collars shifted up (red colour) and down (blue colour). Bottom: The results of numerical modelling of the spin-locked wave-division. b, Spin locking ensures that the spin-down edge mode (ψ − ) excited at Port 1 can escape only along the two (out of three) domain walls for which the direction of propagation for the edge mode agrees with the spin, leading to coupling only to Ports 2 and 4 with the compete suppression of transmission through Port 3. c, Experimental transmission spectra at output Ports 2, 3 and 4 due to edge modes propagating along the respective domain walls indicated by black dashed lines in a. As before, the green shaded regions indicate the topological bandgap. 5.0
−3
4.5
−4
4.0
−5
3.5
−6 −7 19
b Transmission log10 (I)
c
−2
Dwell time d ϕ / /dω (ns)
Transmission log10(I)
a
20 21 22 Frequency (GHz)
−2 −3
3.0 2.5 2.0 1.5
Straight path 60° bends 120° bends Irregular path Disordered
1.0
−4
0.5
−5
0 0
−6
5
−7 19
10
15 20 25 Path length, s (cm)
νg = 0.93 × 108 m s−1 νg = 0.93 × 108 m s−1 νg = 1.12 × 108 m s−1 νg = 0.95 × 108 m s−1 νg = 0.99 × 108 m s−1 30
35
40
20 21 22 Frequency (GHz)
Figure 5 | Experimental demonstration of ballistic transport of the topological edge modes through randomly shaped domain walls and disordered regions. a, Multiply curved irregular domain wall with random bends. b, The domain wall interrupted by a region with collars randomly distributed between up and down positions within the bounded region (indicated by the dashed line in the left upper inset). All the panels and notations have the same meaning as in Fig. 3a–c. c, The dependence of the dwell time of edge modes on the distance travelled along the domain wall for different path configurations averaged over the bandgap region and the linear fits (straight solid lines). The legend specifies the configuration of the path and the corresponding group velocity found from the slope of the linear fits. The inset shows the configuration of the disordered domain (the same as in b).
the topological character of the splitting at the four-port junction leads to the observed broadband spin-locked wave-division in our case (Fig. 4).
Topological protection against disorder Topological edge modes are robust thanks to their helicity33,45 , the locking of the pseudo-spin to the propagation direction. This
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2016 Macmillan Publishers Limited. All rights reserved
5
NATURE MATERIALS DOI: 10.1038/NMAT4573
ARTICLES
Device demonstration based on reconfigurable gauge field Finally, to demonstrate the proof-of-concept device operation based on the reconfigurability of the synthetic gauge field and robust edge states, we have implemented a topological switch, in which the domain wall shape is modified by a computer-controlled motorized stage, as illustrated by Fig. 6. A subset of the copper posts (positions highlighted by the green colour in Fig. 6a) on one of the sides of the structure was attached and glued to the triangularly shaped plastic sheet, which was moved by a computer-controlled motorized arm, thereby changing the position of the collars inside the metacrystal. As a result, the topological states of the domain 6
a
Port 2 Arm Port 1
b
Port 3
Port 2
log10I
0 −1 −2 0
c
5
10
15 Time (s)
20
25
5
10
15 Time (s)
20
25
Port 3 0 log10I
robustness is ensured as long as the pseudo-spin degree of freedom is preserved by the defects. This is equivalent to the robustness of the edge states in condensed matter topological insulators to all kinds of defects except magnetic impurities giving rise to spinflip processes2,5 . A similar robustness is expected for disordered systems where such ‘nonmagnetic’ defects are randomly distributed throughout the structure. Here we consider two cases of disorder formed by such defects: a randomly shaped domain wall and a disordered region with collars randomly pushed up or down along the path of the edge state. In the first case, the edge mode confined to the one-dimensional domain wall experiences an effective scattering potential at every bend of the wall. It is well known that all states are strongly localized in a one-dimensional random system. Being effectively one-dimensional states, one would normally expect the edge states to experience Anderson localization. Nonetheless, this behaviour seems to be completely upended by the topological nature of the edge states. Because these states do not experience backreflection while propagating along a domain wall, regardless of its shape, standing waves and localized modes cannot be formed along the path46 , as seen in Fig. 5a. Numerical calculations and experimental results are summarized in Fig. 5b for the second case considered: a system with the domain wall interrupted by a region of two-dimensional disorder (within the dashed box in the top panel Fig. 5b and shown in the inset to Fig. 5c), where the collars were randomly shifted up and down. These results confirm unambiguously that, although the edge states may spread into the disordered region, they are immune to backscattering. Thus, for both one- and two-dimensional disorder, topological robustness of the edge states is seen in enhanced transmission over the entire bandgap. To further verify the robustness for these two cases, we examined the dwell time of the edge states as they travel through the system47–50 . The dwell time at a distance s from the source, which equals the derivative of the phase with respect to the angular frequency dϕ(s, ω)/dω, is a powerful tool for characterizing the nature of transport in disordered systems. In the case of ballistic transport, the dwell time scales linearly with propagation distance, s. To this end, we measured spectra of the phase ϕ(s, ω) accumulated by the electric field from the source at the edge to a point a distance s along the domain wall for the paths shown in Fig. 5a,b. In addition, we measured the dwell time in the absence of the disorder—the straight domain wall—and the cases shown in Fig. 3. We found in all cases that the dwell time scales linearly with s with nearly the same slope. This confirms that transport is ballistic in the disordered regions. The results of measurements of the phase for different paths are given in Fig. 5c. The average over the topological bandgap of the inverse of the slopes of the linear fits is given in the legend in Fig. 5c. This average is essentially equal to the group velocity vg of the edge modes of 1.005 × 108 m s−1 calculated from the dispersion of the edge modes found in numerical calculations shown in Fig. 2a. This confirms the ballistic character of transport of topological edge modes along the domain wall33,45 , demonstrating the absence of backscattering in the presence of either one- or two-dimensional spin-preserving disorders.
−1 −2 0
Figure 6 | Demonstration of reconfigurable topological switch and its time-resolved dynamics. a, Configuration of the reconfigurable topological switch, where the green region indicates the domain with the copper posts attached to the computer-controlled motorized arm moving the collars between the down and up positions. b,c, The switch operation defined by one of two possible positions of collars within the reconfigurable topological domain in a. The insets illustrate the change in the power flow due to the change in the synthetic gauge field configuration. The graphs show the time-resolved switching of transmission via the edge states through Ports 2 and 3 for the down and up positions of the reconfigurable domain (indicated by green in a), respectively.
could be automatically switched between two distinct topological configurations. Depending on the configuration selected, up or down, respectively, in the reconfigured area, the electromagnetic radiation was carried by the topological edge states either from Port 1 to Port 2 or from Port 1 to Port 3, as illustrated in the insets to Fig. 6b,c. Measurements of topological switching of transmission are shown in Fig. 6b,c and confirm the possibility of the steering radiation by changing the configuration of the synthetic gauge field.
Summary and outlook In conclusion, we have realized experimentally the concept of robust propagation of electromagnetic edge states along reconfigurable topological domain walls in a topological metacrystal with controllable contours of the synthetic gauge field. This opens up possibilities for routing and manipulating electromagnetic radiation. Such a platform provides a versatile and robust topological approach towards controlling and delivering waves along any desired path to a particular point without back-reflection. In addition, this reconfigurable platform enables the study of a variety of fascinating physical phenomena such as controllable spin filtering and switching. In particular, the pseudo-spin selective wave divider and topological switch demonstrated here prove the versatility of the proposed reconfigurable platform for implementing unique device functionalities. The exploration of the breakdown of robustness in the face of diverse types of disorder and the possibility of Anderson localization in topological systems
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2016 Macmillan Publishers Limited. All rights reserved
NATURE MATERIALS DOI: 10.1038/NMAT4573 are also of great interest. Extending the concept of reconfigurable synthetic gauge fields and domain walls from microwave to optical frequencies32 represents a significant challenge, but would be a milestone in the expansion of the design possibilities of modern photonic devices.
Methods Methods and any associated references are available in the online version of the paper. Received 3 June 2015; accepted 19 January 2016; published online 22 February 2016
References 1. Bernevig, B. A., Hughes, T. L. & Zhang, S. C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006). 2. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010). 3. Kane, C. L. & Mele, E. J. Z2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005). 4. Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005). 5. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011). 6. Fu, L. Topological crystalline insulators. Phys. Rev. Lett. 106, 106802 (2011). 7. Dziawa, P. et al. Topological crystalline insulator states in Pb1−x Snx Se. Nature Mater. 11, 1023–1027 (2012). 8. Prodan, E. & Prodan, C. Topological phonon modes and their role in dynamic instability of microtubules. Phys. Rev. Lett. 103, 248101 (2009). 9. Kane, C. L. & Lubensky, T. C. Topological boundary modes in isostatic lattices. Nature Phys. 10, 39–45 (2014). 10. Yang, Z. et al. Topological acoustics. Phys. Rev. Lett. 114, 114301 (2015). 11. Khanikaev, A. B., Fleury, R., Mousavi, S. H. & Alù, A. Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice. Nature Commun. 6, 8260 (2015). 12. Mousavi, S. H., Khanikaev, A. B. & Wang, Z. Topologically protected elastic waves in phononic metamaterials. Nature Commun. 6, 8682 (2015). 13. Bliokh, K. Y., Smirnova, D. & Nori, F. Quantum spin Hall effect of light. Science 348, 1448–1451 (2015). 14. Lu, L., Joannopoulos, J. D. & Soljaćić, M. Topological photonics. Nature Photon. 8, 821–829 (2014). 15. Haldane, F. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008). 16. Raghu, S. & Haldane, F. D. M. Analogs of quantum-Hall-effect edge states in photonic crystals. Phys. Rev. A 78, 033834 (2008). 17. Wang, Z., Chong, Y., Joannopoulos, J. & Soljačić, M. Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Phys. Rev. Lett. 100, 13905 (2008). 18. Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljačić, M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772–775 (2009). 19. Poo, Y., Wu, R., Lin, Z., Yang, Y. & Chan, C. T. Experimental realization of self-guiding unidirectional electromagnetic edge states. Phys. Rev. Lett. 106, 093903 (2011). 20. Fang, K., Yu, Z. & Fan, S. Microscopic theory of photonic one-way edge mode. Phys. Rev. B 84, 075477 (2011). 21. Fang, K., Yu, Z. & Fan, S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nature Photon. 6, 782–787 (2012). 22. Umucalılar, R. O. & Carusotto, I. Artificial gauge field for photons in coupled cavity arrays. Phys. Rev. A 84, 043804 (2011). 23. Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nature Phys. 7, 907–912 (2011). 24. Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor, J. M. Imaging topological edge states in silicon photonics. Nature Photon. 7, 1001–1005 (2013). 25. Khanikaev, A. B. et al. Photonic topological insulators. Nature Mater. 12, 233–239 (2013). 26. Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013). 27. Lu, L., Fu, L., Joannopoulos, J. D. & Soljačić, M. Weyl points and line nodes in gyroid photonic crystals. Nature Photon. 7, 294–299 (2013).
ARTICLES 28. Chen, W.-J. et al. Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide. Nature Commun. 5, 6782 (2014). 29. Sounas, D. L., Caloz, C. & Alù, A. Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials. Nature Commun. 4, 2407 (2014). 30. Tzuang, L. D., Fang, K., Nussenzveig, P., Fan, S. & Lipson, M. Non-reciprocal phase shift induced by an effective magnetic flux for light. Nature Photon. 8, 701–705 (2014). 31. Estep, N. A., Sounas, D. L., Soric, J. & Alù, A. Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops. Nature Phys. 10, 923–927 (2014). 32. Fang, K. & Fan, S. Controlling the flow of light using the inhomogeneous effective gauge field that emerges from dynamic modulation. Phys. Rev. Lett. 111, 203901 (2013). 33. Mittal, S. et al. Topologically robust transport of photons in a synthetic gauge field. Phys. Rev. Lett. 113, 087403 (2014). 34. Liu, J. et al. Spin-filtered edge states with an electrically tunable gap in a two-dimensional topological crystalline insulator. Nature Mater. 13, 178–183 (2014). 35. Aivazian, G. et al. Magnetic control of valley pseudospin in monolayer WSe2 . Nature Phys. 11, 148–152 (2015). 36. Srivastava, A. et al. Valley Zeeman effect in elementary optical excitations of monolayer WSe2 . Nature Phys. 11, 141–147 (2015). 37. Kwon, D.-H., Wang, X., Bayraktar, Z., Weiner, B. & Werner, D. H. Near-infrared metamaterial films with reconfigurable transmissive/reflective properties. Opt. Lett. 33, 545–547 (2008). 38. Ou, J. Y., Plum, E., Jiang, L. & Zheludev, N. I. Reconfigurable photonic metamaterials. Nano Lett. 11, 2142–2144 (2011). 39. Ou, J.-Y., Plum, E., Zhang, J. & Zheludev, N. I. An electromechanically reconfigurable plasmonic metamaterial operating in the near-infrared. Nature Nanotech. 8, 252–255 (2013). 40. Markos, O. & Soukoulis, C. M. Wave Propagation: from Electrons to Photonic Crystals and Left-Handed Materials (Princeton Univ. Press, 2008). 41. Cai, W. & Shalaev, V. Optical Metamaterials: Fundamentals and Applications (Springer, 2009). 42. Serdyukov, A. N., Semchenko, I. V., Tretyakov, S. A. & Sihvola, A. Electromagnetics of Bi-Anisotropic Materials: Theory and Applications (Gordon and Breach Science, 2001). 43. Ma, T., Khanikaev, A. B., Mousavi, S. H. & Shvets, G. Guiding electromagnetic waves around sharp corners: topologically protected photonic transport in metawaveguides. Phys. Rev. Lett. 114, 127401 (2015). 44. Halperin, B. I. Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev. B 25, 2185–2190 (1982). 45. Khanikaev, A. B. & Genack, A. Viewpoint: light avoids Anderson localization. Physics 7, 87 (2014). 46. Schwartz, T., Bartal, G., Fishman, S. & Segev, M. Transport and Anderson localization in disordered two-dimensional photonic lattices. Nature 446, 52–55 (2007). 47. Avishai, Y. & Band, Y. B. One-dimensional density of states and the phase of the transmission amplitude. Phys. Rev. B 32, 2674–2676 (1985). 48. Iannacone, G. General relation between density of states and dwell times in mesoscopic systems. Phys. Rev. B 51, 4727–4729 (1995). 49. Genack, A. Z., Sebbah, P., Stoytchev, M. & van Tiggelen, B. A. Statistics of wave dynamics in random media. Phys. Rev. Lett. 82, 715 (1999). 50. Davy, M., Shi, Z., Wang, J., Cheng, X. & Genack, A. Z. Transmission eigenchannels and the densities of states of random media. Phys. Rev. Lett. 114, 033901 (2015).
Acknowledgements The implementation of the metacrystal structure benefited greatly from discussions and machining of H. Rose. X. Ma helped carry out the measurements and H. Zheng assisted in the fabrication of the structure. This research was supported by the National Science Foundation (CMMI-1537294 and DMR-1207446).
Author contributions All authors contributed extensively to the work presented in this paper.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to A.Z.G. or A.B.K.
Competing financial interests The authors declare no competing financial interests.
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2016 Macmillan Publishers Limited. All rights reserved
7
NATURE MATERIALS DOI: 10.1038/NMAT4573
ARTICLES Methods Structure design and measurements. The experimental metacrystal is an array composed of 35 × 40 unit cells built from low-loss copper with a lattice constant of a0 = 1.0890 cm in the parallel copper plates, separated by a distance H = 1.0890 cm. The rods of diameter d0 = 0.3175 cm have a collar with diameter dc = 0.6215 cm and height hc = 0.3580 cm centred in the rods. Measurements of field spectra in the range 19–23 GHz were made using an Agilent vector network analyser (N5242A). Radiation is launched and detected using linear dipole antennas which extend 3 mm beyond the outer conductor. Surface measurements are performed through a hexagonal lattice of 1.1-mm-diameter holes in one of the two copper plates. The probe antenna is introduced into these holes to measure the local amplitude and phase of the electric field inside the metacrystal. Measurements were made along the boundary lines between metacrystals, but also along a line perpendicular to the domain wall extending into the bulk medium. The phase was not significantly affected by the depth into the bulk material in the region of the bandgap in which the wave is evanescent.
Numerical results. Numerical simulations were obtained using the finite-element-method software COMSOL Multiphysics and the Radio Frequency module. For bulk (edge) band structure calculations, the periodic boundary conditions were imposed along the edges of the unit cell (supercell) to form an infinite triangular lattice. Large-scale simulations were performed with perfectly matched layer (PML) boundary conditions around the structure. Edge modes are excited by a dipole source placed at the domain wall. R The Chern number was found from the expression C = (1/2π) BZ Ω (k)d2 k by integrating the Berry curvature Ω (k) = ∂kx Ay − ∂ky Ax , where Ak = −i hE |∂k | Ei is the Berry connection, and the normalized eigenstates |Ei were found using the full-wave numerical finite-element-solver COMSOL Multiphysics (RF Module) for each of the bands. The effect of loss in copper was considered by imposing impedance boundary conditions over the metal surface with the resistivity ρ = 1.68 × 10−8 m and was found to have no effect on the topological edge states.
NATURE MATERIALS | www.nature.com/naturematerials
© 2016 Macmillan Publishers Limited. All rights reserved
Robust reconfigurable electromagnetic pathways within a photonic topological insulator Xiaojun Cheng1,2†, Camille Jouvaud1†, Xiang Ni1,2, S. Hossein Mousavi3, Azriel Z. Genack1,2* and Alexander B. Khanikaev1,2* The discovery of topological photonic states has revolutionized our understanding of electromagnetic propagation and scattering. Endowed with topological robustness, photonic edge modes are not reflected from structural imperfections and disordered regions. Here we demonstrate robust propagation along reconfigurable pathways defined by synthetic gauge fields within a topological photonic metacrystal. The flow of microwave radiation in helical edge modes following arbitrary contours of the synthetic gauge field between bianisotropic metacrystal domains is unimpeded. This is demonstrated in measurements of the spectrum of transmission and time delay along the topological domain walls. These results provide a framework for freely steering electromagnetic radiation within photonic structures.
R
ecent discoveries of novel states of condensed matter characterized by topological order in quantum spin Hall effect (QSHE) systems, topological insulators (TIs), and topological crystalline insulators (TCIs; refs 1–7) have stimulated the search for analogous states in structures supporting classical waves8–13 . This search has been especially fruitful in electromagnetics14 , where numerous theoretical predictions and experimental demonstrations have been made from microwave to optical frequencies15–28 . Whereas early approaches to topological order for photons relied on an external magnetic field to break time reversal (TR) symmetry15–19 , the absence of materials with a strong magnetic response at optical frequencies and the inconvenience associated with the use of strong magnets at microwave frequencies led to increased interest in systems in which TR symmetry is either preserved21–28 or broken by different methods, such as by dynamic modulation of the system parameters20,21,29–31 . Indeed, engineered synthetic gauge fields21–27 , or temporal20,21,29–32 or spatial modulation26 emulating the effect of a magnetic field have proved to be viable alternatives to an actual magnetic field and have yielded desirable topological properties. Although several successful experimental implementations of topological states of light with TR symmetry have been reported recently24,26,28,33 , one important property of true external fields available in condensed matter systems has evaded emulation in photonics—the control of topological states. The possibility of such control through reconfigurable synthetic gauge fields would usher in a broad range of practical applications from spin/helicity filtering to tunable electronic34–36 and analogous photonic devices. Based on this idea, we demonstrate experimentally that electromagnetic radiation can be steered and delivered to any point within a reconfigurable topological metacrystal37–39 without back-reflection. This is achieved by exploiting adjustable gauge fields defining arbitrarily shaped pathways for topological edge states. Synthetic optical media, such as photonic crystals and metamaterials, offer a versatile toolkit for manipulating the properties of electromagnetic waves40,41 . Recently, optical bianisotropic metamaterials have been predicted to support a topologically nontrivial photonic state analogous to the QSHE state of condensed matter25 .
In these systems, bianisotropy, also referred to as magneto-electric coupling42 , serves as a synthetic gauge field leading to a topological transition across the edge. Here, we realize a reconfigurable topological photonic lattice implemented in the parallel plate waveguide43 shown in Fig. 1a,b. The reconfigurable topological structure studied here is formed between two parallel copper plates through which holes are drilled to support a periodic triangular array of copper rods with ring collars. The rods can be moved up or down so that lattices can be formed with the collar touching either the upper or lower copper plate, and structures with arbitrarily shaped boundaries between the two topologically distinct domains can be created.
Reconfigurable topological metacrystal design
The photonic modes of the metacrystal with σz inversion symmetry, which can be created by moving the collars to the midpoint between the plates, can be classified according to their electric field as either symmetric (TE-like) or antisymmetric (TM-like)12,25,43 . Provided that the dispersion of these two families of modes is matched spectrally over an extended frequency band, these photonic modes can be used to emulate the two components of the electronic spin. In fact, by engineering such a degeneracy, one restores electromagnetic duality responsible for the degeneracy between the two polarizations of electromagnetic waves that is typically broken by a material’s response, and the corresponding modes are referred to as polarization or spin-degenerate12,25,43 . Another important requirement for engineering a topological phase is the presence of Dirac-like degenerate points. This is achieved in our system by the lattice symmetry: in the triangular lattice considered here, the lowest Dirac degeneracies are formed at K and K0 points for both TE- and TM-like dipolar modes near a frequency of 21 GHz (Fig. 1c, blue dotted lines). When these requirements are met, reducing the σz symmetry by sliding rods up or down produces an effect analogous to spin–orbit coupling and leads via a topological phase transition to a QSHE-like state12,25,43 . This transition is accompanied by the opening of a complete topological bandgap (Fig. 1c, red dotted curves) and the formation of an ‘insulating state.’ In practice, this reduction of σz inversion symmetry causes
1 Department of Physics, Queens College of the City University of New York, Queens, New York 11367, USA. 2 The Graduate Center of the City University of New York, New York, New York 10016, USA. 3 Microelectronics Research Center, Cockrell School of Engineering, University of Texas at Austin, Austin, Texas 78758, USA. † These authors contributed equally to this work. *e-mail: [email protected]; [email protected]
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2016 Macmillan Publishers Limited. All rights reserved
1
NATURE MATERIALS DOI: 10.1038/NMAT4573
ARTICLES a
m0 m=0 Trivial Non-trivial d0
H Zc = 0
dc
zc
d0
Zc > 0
d Ω (m2)
0
2 0 ky (m − −2 1)
0.0
−0.9
14 K M Wave vector, k
2 −2 0 −1 ) (m K − kx
0
18 Spin down
0 −18
2 0 ky (m − −2 1)
−9
−18 2 0 1 −2 − (m ) k x−K
Top bands Berry curvature
Κ Μ Κ′ Γ
Γ
Spin up
9
22 Zc = 0 Zc ≠ 0
a0
Bottom bands Berry curvature 18
Ω (m2)
Frequency (GHz)
26
dc
Ω (m2)
c
18
H
hc
Γ
−1.8
2 0
Spin up 2 0 ky (m − −2 1)
2 −2 0 −1 (m ) k x−K
−2
Ω (m2)
Zc < 0
hc
1.8 0.9 Spin down 0.0
2 0 ky (m − −2 1)
−2
0 2 1 − (m ) k x−K
Figure 1 | Reconfigurable metacrystal and its bulk band structure. a, Geometric arrangement of metallic rods and collars at different positions inside the parallel plate waveguide: from left to right, the front collars are shifted from the down to up position, leading to topological transitions as indicated by the mass term m. b, Left panel: side view of the metacrystal. Right panel: top view of the metacrystal with the top copper plate and some rods removed to reveal the periodic arrangement of the rods and collars. The field inside the structure is probed via the additional deep subwavelength holes. c, Bulk photonic band structures of the metacrystal for two different configurations: blue dotted curve and red dotted curve correspond to symmetric (no bianisotropy) and asymmetric (with bianisotropy) structures, respectively. d, Berry curvatures for the four bands of interest near the K point calculated from the bulk eigenstates found using the full-wave approach. Calculations of Berry curvature at the K0 point yield identical results and are not shown.
pairwise coupling between the four original Dirac bands—the lower antisymmetric (symmetric) mode and the upper symmetric (antisymmetric) mode, and the formation of new pseudo-spin-up (↑) and pseudo-spin-down (↓) eigenstates. The new eigenstates have their orbital and pseudo-spin degrees of freedom interlocked, with circularly polarized states with TE-like (TM-like) components delayed (advanced) with respect to each other by a phase shift of π/2 (refs 25,43). A first-principles calculation of the topological invariants of the system, the spin-Chern numbers, found from the full-wave numerical eigenstates, confirms that the corresponding bands acquire nonzero spin-Chern numbers when the collars are all moved from the midpoint either up or down, thus substantiating the topological transition. More importantly, as the collars are moved from the lower to the upper plate, the spin-Chern numbers for the ↑ ↑ four dipolar bands reverse from Cu/l = ±1 to Cu/l = ∓1 and from ↓ ↓ Cu/l = ∓1 to Cu/l = ±1, where the subscript indicates the lower (l) or upper (u) band and the superscript denotes the spin state. The Berry curvature in the proximity of the former degeneracy at the K point is shown in Fig. 1d and clearly reveals the four peaks which contribute to the spin-Chern numbers of the individual bands. By means of perturbation theory for electromagnetic waves, the metacrystal can be described by an effective Hamiltonian analogous to the Kane–Mele model4 for graphene with spin– orbital coupling12,25,43 . Indeed, in the basis of circularly polarized hybrid modes, the perturbed Hamiltonian assumes the form b ↑/↓ = vD τˆ0 sˆ0 σb k · δkk + mτˆ3 sˆ3 σˆ 3 , where τˆi and sˆi are the inter-valley H and pseudo-spin Pauli matrices, vD is the group velocity near the Dirac point, and m is the mass term describing the degree of σz inversion-symmetry breaking. First-principles calculations of the spin-Chern numbers also agree with the results of the ↑ effective Hamiltonian description, which yield Cu/l = ±(m/|m|) ↓ and Cu/l = ∓(m/|m|). The ability to independently slide each of the collars between the metal plates makes it possible to controllably introduce a 2
local bianisotropic response which plays the role of the synthetic gauge field within a unit cell and is responsible for the topological phase12,25 . Although the band structures are identical in systems in which all the collars are either up or down, so that they touch one or the other copper plate (Fig. 1c, red dotted curves), these two configurations are topologically distinct, as revealed by opposite sign of their spin-Chern numbers. These distinct topological phases are isolated by the topologically trivial gapless regime of the inversion symmetric structure (Fig. 1c, blue dotted curves).
Observation of topologically protected edge states The key feature of the topologically nontrivial states in condensed matter is their ability to carry charge or spin currents along interfaces while being excluded from the bulk. Topological photonic systems have similar properties with respect to photonic transport, which can be one-way15–21 and spin-polarized22–28 . Although the magnetic or synthetic gauge fields open a bandgap and prevent photons from propagating in the bulk, photonic transport can take place along the boundaries in the form of edge states. The edge states emerge between topologically distinct structures in which the Chern number (or the spin-Chern number) changes across the interface. This can take place either at the boundary between topologically nontrivial and trivial systems, or at an interface separating two topological domains with opposite or different topological indices. Although in this work we focus on the second class of edge states, shown in Fig. 2a, the first class of states, shown in Fig. 2b, is in fact more common in condensed matter systems. It naturally occurs at external boundaries and in photonic systems with TR symmetry broken by the magneto-optical response18,19 , where the domain walls would require a reversal of the magnetization within the structure, which is very challenging to realize in practice. For a topological system with engineered pseudo-spin degrees of freedom, both classes of topological interfaces have been engineered, proving the versatility of this approach24,28 . An example of the first class of interface of the metacrystal with a suitably designed topologically
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2016 Macmillan Publishers Limited. All rights reserved
NATURE MATERIALS DOI: 10.1038/NMAT4573 a
Domain wall
b
m0
23
23
22
22 Frequency (GHz)
Frequency (GHz)
m>0
ARTICLES
21
m=0
21
20
20
19 −π/ π a0
Interface
K′
0 Wavenumber, k
K
19 −π/ π a0
π a0 π/
K′
0 Wavenumber, k
K
π a0 π/
Figure 2 | Topological interfaces in the metacrystal and their edge band diagrams. a, Photonic band structure of a 30 × 1 supercell of the reconfigurable metacrystal with the bianisotropic domain wall along a line bisecting the structure across which the collars are switched from the up to the down position, as shown in the top panel (the parallel plates are not shown). b, Photonic band structure of a 30 × 1 supercell of the non-reconfigurable topological interface in its centre formed between the topological metacrystal and a topologically trivial photonic crystal waveguide made of gapped triangular copper prisms, as shown in the top panel (the parallel plates are not shown). The red and blue dotted lines in a and b correspond to the spin-up (↑) and spin-down (↓) edge modes, respectively. The middle panels in a and b show the confinement of the edge modes to the topological interfaces. The grey dotted lines show the bulk spectrum. The green shaded regions indicate the complete topological bandgaps for the two cases.
trivial structure (see Supplementary Information for details) with the trivial gap opened (the spin-Chern numbers remain zero) by reducing the in-plane inversion symmetry and supporting a pair of edge states is shown in Fig. 2b. Aiming for a reconfigurable system, however, we are more interested here in the second class of topological interface—the domain wall between topologically distinct domains of inverted structures shown in the top panel of Fig. 2a. According to the bulk– boundary correspondence principle44 , such domain walls host twice as many topological modes propagating along the boundary line, which is corroborated by the numerical simulations shown in the two lower panels of Fig. 2a. In contrast to the boundary with the trivial system which cannot be readily reconfigured, domain walls can be defined solely by the gauge field32 responsible for the topological order. In the case of the metacrystal, such domain walls can be created in a controllable manner by displacing the collars in one portion of the structure towards the opposing plate to produce adjoining regions with distinct topological domains across the interface, characterized by spin-Chern numbers of opposite sign and opposite effective mass m. By design, the domain walls can be made in any desired shape for the two-dimensional reconfigurable metacrystal described here. The main property of topological edge states that distinguishes them from their topologically trivial counterparts, such as Tamm states or modes of optical waveguides, is their one-way, chiral or helical character, endowing them with robustness against defects and disorder. In the case of the QSHE emulated here, the edge modes represent helical states with spin (polarization configuration) locked to the propagation direction, as indicated in Fig. 2 (refs 25,43). This property endows the edge states with topological protection against scattering as long as the spin of the edge mode is preserved
by the scattering potentials of the defect or disorder of the crystal. Combined with the reconfigurability of our system, this suggests that it is possible to route topological edge modes confined to the domain wall along arbitrarily shaped paths without backscattering. To test the robustness of photonic transport along reconfigurable pathways enabled by the design of the metacrystal, propagation was measured in samples with a variety of bends and shapes of the topological domain wall, as shown in Fig. 3a–c. The measurements (refer to Methods for details) were performed by means of a vector network analyser. Edge modes are excited by a small dipole antenna at the domain wall on one end of the structure, as indicated by red radiating dots in the left panels of Fig. 3a–c. Transmission was measured by another dipole antenna placed at the opposite end of the domain wall, indicated in blue in Fig. 3a–c. In addition, we performed a discrete near-field scan of the electric field within the structure by inserting the tip of the probe antenna a short distance into deep subwavelength holes at regular intervals along the structure. As shown in Fig. 1b, two additional subwavelength holes were drilled per unit cell so that the triangular symmetry of the lattice is preserved. First-principles numerical simulations confirmed that the overall electromagnetic response of the structure is not compromised by the subwavelength holes. Both the amplitude and phase of the electric field along the domain wall were measured to determine the field distribution and dwell time of the edge modes. Transmission spectra for some of the experimental configurations tested are shown in Fig. 3a–c (left top panels) together with the results of numerical simulations (left bottom panels), which show that the edge modes propagate freely without attenuation due to scattering along curved pathways. In the right-hand panels of Fig. 3a–c the transmission spectra measured for the three configurations (blue solid curves) are compared against the spectrum of
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2016 Macmillan Publishers Limited. All rights reserved
3
NATURE MATERIALS DOI: 10.1038/NMAT4573
ARTICLES Transmission log10(I)
a
−2 −3 −4 −5
d
−6
−1
19
Transmission log10(I)
21
22
−2 −3
−3 −4 −5
Intensity I
−4 −5
y
−6
−6
−7
−7 19
c
20
−2 ln (I/Imax)
Transmission log10(I)
b
0
−7
20
21
22
−8 x
−9
−2
−10
−3 −4
−5
0 Depth, y (cm)
5
10
−5 −6 −7 19
20 21 22 Frequency (GHz)
Figure 3 | Experimental verification of reconfigurable guiding and exponential confinement of the topological edge modes. a–c, Three straight segments of the domain wall separated by two bends of 60◦ , 90◦ and 120◦ , respectively. The left panels show the domain walls separating two crystal structures differing by inversion about a plane midway between the copper plates (top) and the simulations for the corresponding structures (bottom). The right panels show transmission spectra for radiation propagating along the domain wall (blue lines) and through the bulk of the structure (black dashed lines). Green shaded regions indicate the topological bandgap, defined as I < 10−5 for transmission in the bulk. d, Semi-log plot of the variation of the intensity averaged over frequencies near the centre of the topological bandgap (20.75 ± 0.25 GHz) versus distance in the direction perpendicular to the domain wall (as indicated in the inset with white and black dots indicating up and down positions of the collars).
transmission through the bulk of the metacrystal (black dashed curves) with all the collars shifted up. The latter spectrum reveals very low transmission (I < 10−5 ) in the frequency range from 20.1 to 21.7 GHz corresponding to the bandgap, which is surrounded by pass bands for the bulk modes. In contrast to the case of a gapped band structure, the experimental spectra for different domain wall configurations exhibit high transmission, mediated by the edge modes, over the frequency range of the bandgap. The high transmission observed for all domain wall contours, including an irregularly shaped wall with multiple bends, shows that transport is robust. In contrast, for the case of topologically trivial guided modes, one would expect the wave to be reflected each time there is a sharp bend in the path. In addition, the presence of multiple bends would lead to the formation of standing waves in intermediate segments. This would give rise to distinct transmission peaks at Fabry–Perot resonances determined by the length of the segment between the bends43 , which are not observed (Fig. 3a–c). Interestingly, we find that the transmission in the bandgap exceeds the transmission in the pass band. This is a consequence of the confinement of the edge modes to the domain wall, as opposed to the spread of the bulk modes, which fall off roughly as the inverse of the distance. To confirm that the enhanced transmission is by the edge modes, we measured the variation of the electric field in the direction perpendicular to the domain wall. This is seen in Fig. 3d to decay exponentially away from the boundary line.
Spin-locked wave-division in topological junctions To further demonstrate that the edge states are topologically protected and helical and, hence, mimic the edge states of QSHE, 4
we performed further studies to confirm that the spin is locked to the propagation direction. We considered a configuration of the synthetic gauge field distribution with two domain walls crossing in the centre of the structure, as shown in Fig. 4a, so that the four segments of the domain walls are formed with different gradients of the mass term m and the spin-Chern numbers across the walls, as illustrated in Fig. 4b. These gradients define whether a particular spin has positive or negative group velocity along the domain wall. For the case of the spin-down state ψ − , illustrated in Fig. 4b, the left and right walls support only outward propagation (away from the domain walls junction), whereas the top and bottom walls support only inward propagation (towards the junction). For the spin-up state ψ + the situation reverses. Therefore, when the system is excited from the bottom Port 1, only the spin-down ψ − mode is allowed to couple into the system. More importantly, when the spin-down mode ψ − reaches the junction, it cannot excite spin-up edge state ψ + propagating away from the junction along the upper domain wall because of the mismatch in the spin configuration. Thus, transmission through Port 3 seems to be completely suppressed. On the other hand, transmission through the left and right Ports 2 and 4 is allowed because the outward propagation of the spin-down state is permitted along these walls. These conclusions are consistent with the results of numerical simulations shown in the lower panel of Fig. 4a, as well as with measurements of strong transmission at Ports 2 and 4 and its suppression at Port 3, shown in Fig. 3c. It is important to note that such behaviour would not normally take place in a non-topological structure, where the conventional wave-division would be observed with transmission through Port 3 comparable to that through Port 2 because of the similar mutual orientation of the corresponding output waveguides. In contrast,
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2016 Macmillan Publishers Limited. All rights reserved
NATURE MATERIALS DOI: 10.1038/NMAT4573
ARTICLES
a
b
P 3 ((out) Port (out)
Port 3 Port 4
Port 1
Port 2
C↓ = 1
ψ+ ψ−
ψ−
C↓ = 1
C↓ = −11
ψ− Out
Port 4 (out)
Port 2 (out)
C↓ = −1
Port 1 (in)
c Out
Port 2 Port 3 Port 4
Transmission log10(I)
−2
In Out
−3 −4 −5 −6 −7 19.0
19.5
20.0
20.5 21.0 21.5 Frequency (GHz)
22.0
22.5
Figure 4 | Experimental demonstration of spin-locked wave-division of an edge mode at a four-port topological junction. a, Top: four-port configuration formed by the four topological domains with collars shifted up (red colour) and down (blue colour). Bottom: The results of numerical modelling of the spin-locked wave-division. b, Spin locking ensures that the spin-down edge mode (ψ − ) excited at Port 1 can escape only along the two (out of three) domain walls for which the direction of propagation for the edge mode agrees with the spin, leading to coupling only to Ports 2 and 4 with the compete suppression of transmission through Port 3. c, Experimental transmission spectra at output Ports 2, 3 and 4 due to edge modes propagating along the respective domain walls indicated by black dashed lines in a. As before, the green shaded regions indicate the topological bandgap. 5.0
−3
4.5
−4
4.0
−5
3.5
−6 −7 19
b Transmission log10 (I)
c
−2
Dwell time d ϕ / /dω (ns)
Transmission log10(I)
a
20 21 22 Frequency (GHz)
−2 −3
3.0 2.5 2.0 1.5
Straight path 60° bends 120° bends Irregular path Disordered
1.0
−4
0.5
−5
0 0
−6
5
−7 19
10
15 20 25 Path length, s (cm)
νg = 0.93 × 108 m s−1 νg = 0.93 × 108 m s−1 νg = 1.12 × 108 m s−1 νg = 0.95 × 108 m s−1 νg = 0.99 × 108 m s−1 30
35
40
20 21 22 Frequency (GHz)
Figure 5 | Experimental demonstration of ballistic transport of the topological edge modes through randomly shaped domain walls and disordered regions. a, Multiply curved irregular domain wall with random bends. b, The domain wall interrupted by a region with collars randomly distributed between up and down positions within the bounded region (indicated by the dashed line in the left upper inset). All the panels and notations have the same meaning as in Fig. 3a–c. c, The dependence of the dwell time of edge modes on the distance travelled along the domain wall for different path configurations averaged over the bandgap region and the linear fits (straight solid lines). The legend specifies the configuration of the path and the corresponding group velocity found from the slope of the linear fits. The inset shows the configuration of the disordered domain (the same as in b).
the topological character of the splitting at the four-port junction leads to the observed broadband spin-locked wave-division in our case (Fig. 4).
Topological protection against disorder Topological edge modes are robust thanks to their helicity33,45 , the locking of the pseudo-spin to the propagation direction. This
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2016 Macmillan Publishers Limited. All rights reserved
5
NATURE MATERIALS DOI: 10.1038/NMAT4573
ARTICLES
Device demonstration based on reconfigurable gauge field Finally, to demonstrate the proof-of-concept device operation based on the reconfigurability of the synthetic gauge field and robust edge states, we have implemented a topological switch, in which the domain wall shape is modified by a computer-controlled motorized stage, as illustrated by Fig. 6. A subset of the copper posts (positions highlighted by the green colour in Fig. 6a) on one of the sides of the structure was attached and glued to the triangularly shaped plastic sheet, which was moved by a computer-controlled motorized arm, thereby changing the position of the collars inside the metacrystal. As a result, the topological states of the domain 6
a
Port 2 Arm Port 1
b
Port 3
Port 2
log10I
0 −1 −2 0
c
5
10
15 Time (s)
20
25
5
10
15 Time (s)
20
25
Port 3 0 log10I
robustness is ensured as long as the pseudo-spin degree of freedom is preserved by the defects. This is equivalent to the robustness of the edge states in condensed matter topological insulators to all kinds of defects except magnetic impurities giving rise to spinflip processes2,5 . A similar robustness is expected for disordered systems where such ‘nonmagnetic’ defects are randomly distributed throughout the structure. Here we consider two cases of disorder formed by such defects: a randomly shaped domain wall and a disordered region with collars randomly pushed up or down along the path of the edge state. In the first case, the edge mode confined to the one-dimensional domain wall experiences an effective scattering potential at every bend of the wall. It is well known that all states are strongly localized in a one-dimensional random system. Being effectively one-dimensional states, one would normally expect the edge states to experience Anderson localization. Nonetheless, this behaviour seems to be completely upended by the topological nature of the edge states. Because these states do not experience backreflection while propagating along a domain wall, regardless of its shape, standing waves and localized modes cannot be formed along the path46 , as seen in Fig. 5a. Numerical calculations and experimental results are summarized in Fig. 5b for the second case considered: a system with the domain wall interrupted by a region of two-dimensional disorder (within the dashed box in the top panel Fig. 5b and shown in the inset to Fig. 5c), where the collars were randomly shifted up and down. These results confirm unambiguously that, although the edge states may spread into the disordered region, they are immune to backscattering. Thus, for both one- and two-dimensional disorder, topological robustness of the edge states is seen in enhanced transmission over the entire bandgap. To further verify the robustness for these two cases, we examined the dwell time of the edge states as they travel through the system47–50 . The dwell time at a distance s from the source, which equals the derivative of the phase with respect to the angular frequency dϕ(s, ω)/dω, is a powerful tool for characterizing the nature of transport in disordered systems. In the case of ballistic transport, the dwell time scales linearly with propagation distance, s. To this end, we measured spectra of the phase ϕ(s, ω) accumulated by the electric field from the source at the edge to a point a distance s along the domain wall for the paths shown in Fig. 5a,b. In addition, we measured the dwell time in the absence of the disorder—the straight domain wall—and the cases shown in Fig. 3. We found in all cases that the dwell time scales linearly with s with nearly the same slope. This confirms that transport is ballistic in the disordered regions. The results of measurements of the phase for different paths are given in Fig. 5c. The average over the topological bandgap of the inverse of the slopes of the linear fits is given in the legend in Fig. 5c. This average is essentially equal to the group velocity vg of the edge modes of 1.005 × 108 m s−1 calculated from the dispersion of the edge modes found in numerical calculations shown in Fig. 2a. This confirms the ballistic character of transport of topological edge modes along the domain wall33,45 , demonstrating the absence of backscattering in the presence of either one- or two-dimensional spin-preserving disorders.
−1 −2 0
Figure 6 | Demonstration of reconfigurable topological switch and its time-resolved dynamics. a, Configuration of the reconfigurable topological switch, where the green region indicates the domain with the copper posts attached to the computer-controlled motorized arm moving the collars between the down and up positions. b,c, The switch operation defined by one of two possible positions of collars within the reconfigurable topological domain in a. The insets illustrate the change in the power flow due to the change in the synthetic gauge field configuration. The graphs show the time-resolved switching of transmission via the edge states through Ports 2 and 3 for the down and up positions of the reconfigurable domain (indicated by green in a), respectively.
could be automatically switched between two distinct topological configurations. Depending on the configuration selected, up or down, respectively, in the reconfigured area, the electromagnetic radiation was carried by the topological edge states either from Port 1 to Port 2 or from Port 1 to Port 3, as illustrated in the insets to Fig. 6b,c. Measurements of topological switching of transmission are shown in Fig. 6b,c and confirm the possibility of the steering radiation by changing the configuration of the synthetic gauge field.
Summary and outlook In conclusion, we have realized experimentally the concept of robust propagation of electromagnetic edge states along reconfigurable topological domain walls in a topological metacrystal with controllable contours of the synthetic gauge field. This opens up possibilities for routing and manipulating electromagnetic radiation. Such a platform provides a versatile and robust topological approach towards controlling and delivering waves along any desired path to a particular point without back-reflection. In addition, this reconfigurable platform enables the study of a variety of fascinating physical phenomena such as controllable spin filtering and switching. In particular, the pseudo-spin selective wave divider and topological switch demonstrated here prove the versatility of the proposed reconfigurable platform for implementing unique device functionalities. The exploration of the breakdown of robustness in the face of diverse types of disorder and the possibility of Anderson localization in topological systems
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2016 Macmillan Publishers Limited. All rights reserved
NATURE MATERIALS DOI: 10.1038/NMAT4573 are also of great interest. Extending the concept of reconfigurable synthetic gauge fields and domain walls from microwave to optical frequencies32 represents a significant challenge, but would be a milestone in the expansion of the design possibilities of modern photonic devices.
Methods Methods and any associated references are available in the online version of the paper. Received 3 June 2015; accepted 19 January 2016; published online 22 February 2016
References 1. Bernevig, B. A., Hughes, T. L. & Zhang, S. C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006). 2. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010). 3. Kane, C. L. & Mele, E. J. Z2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005). 4. Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005). 5. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011). 6. Fu, L. Topological crystalline insulators. Phys. Rev. Lett. 106, 106802 (2011). 7. Dziawa, P. et al. Topological crystalline insulator states in Pb1−x Snx Se. Nature Mater. 11, 1023–1027 (2012). 8. Prodan, E. & Prodan, C. Topological phonon modes and their role in dynamic instability of microtubules. Phys. Rev. Lett. 103, 248101 (2009). 9. Kane, C. L. & Lubensky, T. C. Topological boundary modes in isostatic lattices. Nature Phys. 10, 39–45 (2014). 10. Yang, Z. et al. Topological acoustics. Phys. Rev. Lett. 114, 114301 (2015). 11. Khanikaev, A. B., Fleury, R., Mousavi, S. H. & Alù, A. Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice. Nature Commun. 6, 8260 (2015). 12. Mousavi, S. H., Khanikaev, A. B. & Wang, Z. Topologically protected elastic waves in phononic metamaterials. Nature Commun. 6, 8682 (2015). 13. Bliokh, K. Y., Smirnova, D. & Nori, F. Quantum spin Hall effect of light. Science 348, 1448–1451 (2015). 14. Lu, L., Joannopoulos, J. D. & Soljaćić, M. Topological photonics. Nature Photon. 8, 821–829 (2014). 15. Haldane, F. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008). 16. Raghu, S. & Haldane, F. D. M. Analogs of quantum-Hall-effect edge states in photonic crystals. Phys. Rev. A 78, 033834 (2008). 17. Wang, Z., Chong, Y., Joannopoulos, J. & Soljačić, M. Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Phys. Rev. Lett. 100, 13905 (2008). 18. Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljačić, M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772–775 (2009). 19. Poo, Y., Wu, R., Lin, Z., Yang, Y. & Chan, C. T. Experimental realization of self-guiding unidirectional electromagnetic edge states. Phys. Rev. Lett. 106, 093903 (2011). 20. Fang, K., Yu, Z. & Fan, S. Microscopic theory of photonic one-way edge mode. Phys. Rev. B 84, 075477 (2011). 21. Fang, K., Yu, Z. & Fan, S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nature Photon. 6, 782–787 (2012). 22. Umucalılar, R. O. & Carusotto, I. Artificial gauge field for photons in coupled cavity arrays. Phys. Rev. A 84, 043804 (2011). 23. Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nature Phys. 7, 907–912 (2011). 24. Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor, J. M. Imaging topological edge states in silicon photonics. Nature Photon. 7, 1001–1005 (2013). 25. Khanikaev, A. B. et al. Photonic topological insulators. Nature Mater. 12, 233–239 (2013). 26. Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013). 27. Lu, L., Fu, L., Joannopoulos, J. D. & Soljačić, M. Weyl points and line nodes in gyroid photonic crystals. Nature Photon. 7, 294–299 (2013).
ARTICLES 28. Chen, W.-J. et al. Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide. Nature Commun. 5, 6782 (2014). 29. Sounas, D. L., Caloz, C. & Alù, A. Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials. Nature Commun. 4, 2407 (2014). 30. Tzuang, L. D., Fang, K., Nussenzveig, P., Fan, S. & Lipson, M. Non-reciprocal phase shift induced by an effective magnetic flux for light. Nature Photon. 8, 701–705 (2014). 31. Estep, N. A., Sounas, D. L., Soric, J. & Alù, A. Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops. Nature Phys. 10, 923–927 (2014). 32. Fang, K. & Fan, S. Controlling the flow of light using the inhomogeneous effective gauge field that emerges from dynamic modulation. Phys. Rev. Lett. 111, 203901 (2013). 33. Mittal, S. et al. Topologically robust transport of photons in a synthetic gauge field. Phys. Rev. Lett. 113, 087403 (2014). 34. Liu, J. et al. Spin-filtered edge states with an electrically tunable gap in a two-dimensional topological crystalline insulator. Nature Mater. 13, 178–183 (2014). 35. Aivazian, G. et al. Magnetic control of valley pseudospin in monolayer WSe2 . Nature Phys. 11, 148–152 (2015). 36. Srivastava, A. et al. Valley Zeeman effect in elementary optical excitations of monolayer WSe2 . Nature Phys. 11, 141–147 (2015). 37. Kwon, D.-H., Wang, X., Bayraktar, Z., Weiner, B. & Werner, D. H. Near-infrared metamaterial films with reconfigurable transmissive/reflective properties. Opt. Lett. 33, 545–547 (2008). 38. Ou, J. Y., Plum, E., Jiang, L. & Zheludev, N. I. Reconfigurable photonic metamaterials. Nano Lett. 11, 2142–2144 (2011). 39. Ou, J.-Y., Plum, E., Zhang, J. & Zheludev, N. I. An electromechanically reconfigurable plasmonic metamaterial operating in the near-infrared. Nature Nanotech. 8, 252–255 (2013). 40. Markos, O. & Soukoulis, C. M. Wave Propagation: from Electrons to Photonic Crystals and Left-Handed Materials (Princeton Univ. Press, 2008). 41. Cai, W. & Shalaev, V. Optical Metamaterials: Fundamentals and Applications (Springer, 2009). 42. Serdyukov, A. N., Semchenko, I. V., Tretyakov, S. A. & Sihvola, A. Electromagnetics of Bi-Anisotropic Materials: Theory and Applications (Gordon and Breach Science, 2001). 43. Ma, T., Khanikaev, A. B., Mousavi, S. H. & Shvets, G. Guiding electromagnetic waves around sharp corners: topologically protected photonic transport in metawaveguides. Phys. Rev. Lett. 114, 127401 (2015). 44. Halperin, B. I. Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev. B 25, 2185–2190 (1982). 45. Khanikaev, A. B. & Genack, A. Viewpoint: light avoids Anderson localization. Physics 7, 87 (2014). 46. Schwartz, T., Bartal, G., Fishman, S. & Segev, M. Transport and Anderson localization in disordered two-dimensional photonic lattices. Nature 446, 52–55 (2007). 47. Avishai, Y. & Band, Y. B. One-dimensional density of states and the phase of the transmission amplitude. Phys. Rev. B 32, 2674–2676 (1985). 48. Iannacone, G. General relation between density of states and dwell times in mesoscopic systems. Phys. Rev. B 51, 4727–4729 (1995). 49. Genack, A. Z., Sebbah, P., Stoytchev, M. & van Tiggelen, B. A. Statistics of wave dynamics in random media. Phys. Rev. Lett. 82, 715 (1999). 50. Davy, M., Shi, Z., Wang, J., Cheng, X. & Genack, A. Z. Transmission eigenchannels and the densities of states of random media. Phys. Rev. Lett. 114, 033901 (2015).
Acknowledgements The implementation of the metacrystal structure benefited greatly from discussions and machining of H. Rose. X. Ma helped carry out the measurements and H. Zheng assisted in the fabrication of the structure. This research was supported by the National Science Foundation (CMMI-1537294 and DMR-1207446).
Author contributions All authors contributed extensively to the work presented in this paper.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to A.Z.G. or A.B.K.
Competing financial interests The authors declare no competing financial interests.
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2016 Macmillan Publishers Limited. All rights reserved
7
NATURE MATERIALS DOI: 10.1038/NMAT4573
ARTICLES Methods Structure design and measurements. The experimental metacrystal is an array composed of 35 × 40 unit cells built from low-loss copper with a lattice constant of a0 = 1.0890 cm in the parallel copper plates, separated by a distance H = 1.0890 cm. The rods of diameter d0 = 0.3175 cm have a collar with diameter dc = 0.6215 cm and height hc = 0.3580 cm centred in the rods. Measurements of field spectra in the range 19–23 GHz were made using an Agilent vector network analyser (N5242A). Radiation is launched and detected using linear dipole antennas which extend 3 mm beyond the outer conductor. Surface measurements are performed through a hexagonal lattice of 1.1-mm-diameter holes in one of the two copper plates. The probe antenna is introduced into these holes to measure the local amplitude and phase of the electric field inside the metacrystal. Measurements were made along the boundary lines between metacrystals, but also along a line perpendicular to the domain wall extending into the bulk medium. The phase was not significantly affected by the depth into the bulk material in the region of the bandgap in which the wave is evanescent.
Numerical results. Numerical simulations were obtained using the finite-element-method software COMSOL Multiphysics and the Radio Frequency module. For bulk (edge) band structure calculations, the periodic boundary conditions were imposed along the edges of the unit cell (supercell) to form an infinite triangular lattice. Large-scale simulations were performed with perfectly matched layer (PML) boundary conditions around the structure. Edge modes are excited by a dipole source placed at the domain wall. R The Chern number was found from the expression C = (1/2π) BZ Ω (k)d2 k by integrating the Berry curvature Ω (k) = ∂kx Ay − ∂ky Ax , where Ak = −i hE |∂k | Ei is the Berry connection, and the normalized eigenstates |Ei were found using the full-wave numerical finite-element-solver COMSOL Multiphysics (RF Module) for each of the bands. The effect of loss in copper was considered by imposing impedance boundary conditions over the metal surface with the resistivity ρ = 1.68 × 10−8 m and was found to have no effect on the topological edge states.
NATURE MATERIALS | www.nature.com/naturematerials
© 2016 Macmillan Publishers Limited. All rights reserved
