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The Harper–Hofstadter Hamiltonian and conical diffraction in photonic lattices with grating assisted tunneling

Dubček, Tena, Lelas, Karlo, Jukić, Dario, Pezer, Robert, Soljačić, Marin, Buljan, Hrvoje·2015·New Journal of Physics
DOI 10.1088/1367-2630/17/12/125002
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FIG. 1 (color online) Illustration of the grating assisted tunneling scheme. (a) Sketch of the spatial dependence of the index of refraction δn L ( x, z ) = δn L + δn T + δn G , which creates a synthetic magnetic field in a photonic lattice. A 1D photonic lattice [ δn L = δn L 0 cos 2 ( πx/a )] is superimposed with a linear gradient index in the x direction ( δn T = -ηx , orange dotted line), and an additional small grating potential [ δn G = δn G 0 cos 2 (( q x x -κz ) / 2)], at a small angle ( θ is on the order of 1 ◦ ) with respect to the z axis. In the xz plane we plot the projection of δn L (blue bold lines), and the grating (green dashed lines). (b) A discrete lattice with complex tunneling matrix elements between sites, K x e iφ m , and spatially dependent phases φ m , can model the system in (a). (c) A particular choice of θ (see text), yields a lattice with φ m = πm , and a dispersion with a 1D Dirac cone at k x = 0. (d) The amplitude of the tunneling K x as a function of the strength of the grating δn G 0 (green squares). The tunneling amplitude J x versus δn G 0 for a system without the tilt. (e,f,g) Numerical simulation of the evolution of a wavepacket that initially excites modes close to k x = 0 in the continuous 1D photonic lattice. (e) Diffraction in a periodic photonic lattice δn L . (f) Propagation in the tilted system, δn L + δn T , shows that the tunneling (diffraction) is suppressed. (g) The tunneling is restored by an additional grating potential with q x = π/a . (h) Propagation of the wavepacket in the corresponding discrete model illustrated in (c).
FIG. 1 (color online) Illustration of the grating assisted tunneling scheme. (a) Sketch of the spatial dependence of the index of refraction δn L ( x, z ) = δn L + δn T + δn G , which creates a synthetic magnetic field in a photonic lattice. A 1D photonic lattice [ δn L = δn L 0 cos 2 ( πx/a )] is superimposed with a linear gradient index in the x direction ( δn T = -ηx , orange dotted line), and an additional small grating potential [ δn G = δn G 0 cos 2 (( q x x -κz ) / 2)], at a small angle ( θ is on the order of 1 ◦ ) with respect to the z axis. In the xz plane we plot the projection of δn L (blue bold lines), and the grating (green dashed lines). (b) A discrete lattice with complex tunneling matrix elements between sites, K x e iφ m , and spatially dependent phases φ m , can model the system in (a). (c) A particular choice of θ (see text), yields a lattice with φ m = πm , and a dispersion with a 1D Dirac cone at k x = 0. (d) The amplitude of the tunneling K x as a function of the strength of the grating δn G 0 (green squares). The tunneling amplitude J x versus δn G 0 for a system without the tilt. (e,f,g) Numerical simulation of the evolution of a wavepacket that initially excites modes close to k x = 0 in the continuous 1D photonic lattice. (e) Diffraction in a periodic photonic lattice δn L . (f) Propagation in the tilted system, δn L + δn T , shows that the tunneling (diffraction) is suppressed. (g) The tunneling is restored by an additional grating potential with q x = π/a . (h) Propagation of the wavepacket in the corresponding discrete model illustrated in (c).CC-BY · From the PDF
FIG. 2 (color online) Grating assisted tunneling and conical diffraction in a square photonic lattice. (a) Sketch of the 2D lattice with grating assisted tunneling along the x direction. The resulting nontrivial hopping phases π and 0 are denoted with dashed and solid lines, respectively. A wavepacket that makes one loop around the plaquette accumulates the phase π . (b) The lattice possesses two 2D Dirac cones at ( k x , k y ) = (0 , ± π/ 2 a ) in the dispersion β ( k x , k y ), where β is the propagation constant. (c) Intensity of a beam, which initially excites modes in the vicinity of the Dirac points, after propagation for z = 162 mm. The intensity has two concentric rings corresponding to the conical diffraction pattern. (d) Simulation of the diffraction pattern in the discrete model corresponding to the lattice in (a), which also exhibits conical diffraction (see text for details).
FIG. 2 (color online) Grating assisted tunneling and conical diffraction in a square photonic lattice. (a) Sketch of the 2D lattice with grating assisted tunneling along the x direction. The resulting nontrivial hopping phases π and 0 are denoted with dashed and solid lines, respectively. A wavepacket that makes one loop around the plaquette accumulates the phase π . (b) The lattice possesses two 2D Dirac cones at ( k x , k y ) = (0 , ± π/ 2 a ) in the dispersion β ( k x , k y ), where β is the propagation constant. (c) Intensity of a beam, which initially excites modes in the vicinity of the Dirac points, after propagation for z = 162 mm. The intensity has two concentric rings corresponding to the conical diffraction pattern. (d) Simulation of the diffraction pattern in the discrete model corresponding to the lattice in (a), which also exhibits conical diffraction (see text for details).CC-BY · From the PDF

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Citations

Cites

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Metadata

Authors
Dubček, T., Lelas, K., Jukić, D., Pezer, R., Soljačić, M., & Buljan, H.
Venue
New Journal of Physics
Publisher
IOP Publishing
Published
2015-12-07
Volume
17
Issue
12
Pages
125002
ISSN
1367-2630
DOI
10.1088/1367-2630/17/12/125002
Cited by
17
References
12
Licence
Funders
  • U.S. Army Research Laboratory (W911NF-13-D-0001)
  • Unity through Knowledge Fund (5/13)
  • U.S. DOE (DE-SC0001299 / DE-FG02-09ER46577)

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