Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal

For realistic material parameters, the edge modes occupy a 10% band gap.

“For realistic material parameters, the edge modes occupy a 10% band gap.”

• Also stated in: § Abstract · plain

  “With a 1600 Gauss bias, we obtain a wide MO band gap of 10% relative size and very low dispersion, with the edge mode confined to a few lattice constants.”

The TM band diagram for a square lattice of YIG rods in air contains a quadratic M-point degeneracy between the second and third bands.

“For example, Fig. 2 shows the TM band diagram for a square lattice of YIG rods in air, containing a quadratic M-point degeneracy between the second and third bands.”

Applying an external dc magnetic field lifts the quadratic M-point degeneracy between the second and third bands.

“This degeneracy is lifted upon applying an external dc magnetic field, which introduces magnetic anisotropy in the high-index rods and breaks T (as well as inversion and mirror symmetries.)”

With realistic material parameters, the one-way modes have negligible material loss.

“With realistic material parameters, the one-way modes are laterally confined to a few lattice constants, and occupy a broad (10%) band gap with negligible material loss.”

With realistic material parameters, the one-way modes are laterally confined to a few lattice constants.

“With realistic material parameters, the one-way modes are laterally confined to a few lattice constants, and occupy a broad (10%) band gap with negligible material loss.”

• Also stated in: § Abstract · plain

  “With a 1600 Gauss bias, we obtain a wide MO band gap of 10% relative size and very low dispersion, with the edge mode confined to a few lattice constants.”

Numerical simulations of a one-way waveguide incorporating this crystal show 100% transmission across strong defects.

“Numerical simulations of a oneway waveguide incorporating this crystal show 100% transmission across strong defects.”

• Also stated in: § Abstract · plain

  “In the one-way waveguide, a steady-state source operating at the midgap frequency 0:555 2c=a excites a one-way mode that circumvents the PEC defect, with 100% power transmission throughout the MO band gap.”

The magneto-optical photonic crystal exhibits strong (order unity) time-reversal breaking due to the use of gyromagnetic ferrimagnets.

“The MO photonic crystal exhibits strong (order unity) T breaking due to the use of gyromagnetic ferrimagnets.”

In the one-way waveguide, a steady-state source at midgap frequency excites a one-way mode that circumvents a PEC defect with 100% power transmission throughout the MO band gap.

“In the one-way waveguide, a steady-state source operating at the midgap frequency 0:555 2c=a excites a one-way mode that circumvents the PEC defect, with 100% power transmission throughout the MO band gap.”

With a gyromagnetic linewidth of 0.3 Oe and a dielectric loss tangent of 0.0002, the complex propagation constant is 0.359 + 0.0001i 2π/a in the case of Fig. 3.

“With a gyromagnetic linewidth of 0.3 Oe and a dieletric loss tangent of 0.0002, typical in commericially available monocrystalline YIG [17], the complex propagation constant is 0:359 0:0001i 2=a in the case of Fig. 3.”

• cites [17] 17. E. P. Wohlfarth, in Ferromagnetic Materials (North- Holland, Amsterdam, 1986), Vol. 2, p. 293.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

It is easy to couple to the one-way mode using waves incident on the system boundary.

“Although embedded sources were used in these simulations, we have verified in a separate set of simulations that it is also easy to couple to the one-way mode using waves incident on the system boundary.”

The above results are not substantially altered by material losses and frequency-dependent permeability in a real microwave ferrite.

“The above results are not substantially altered by the material losses and frequency-dependent permeability in a real microwave ferrite.”

Correcting for the frequency dependency of the gyromagnetic permeability reduces the MO band gap to around 6%.

“When correcting this approximation with a complete gyrotropic line shape [11], we find the MO band gap reduces to around 6%.”

• cites [11] 11. D. M. Pozar, Microwave Engineering (Wiley, New York, 1998).Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The imaginary part of the propagation constant corresponds to a decay length of 1300 lattice constants, far exceeding practical structural dimensions.

“The imaginary part corresponds to a decay length of 1300 lattice constants, far exceeding practical structural dimensions.”

The edge modes are immune to scattering from extremely large defects.

“Numerical simulations show that these edge modes are immune to scattering from extremely large defects.”

• Also stated in: § Abstract · boosted

  “Using both frequency and time-domain numerical simulations, we demonstrate that these edge modes are immune to backscattering.”

A Chern-number change in one band is accompanied by an equal and opposite change, −ΔC, in the Chern number of the neighboring band.

“Because of property (ii), a Chern-number change in one band is accompanied by an equal and opposite change, ÿC, in the Chern number of the neighboring band.”

One-way electromagnetic edge modes analogous to quantum Hall edge states can appear in more general settings than 2D photonic crystals with Dirac point-derived band gaps.

“We point out that electromagnetic one-way edge modes analogous to quantum Hall edge states, originally predicted by Raghu and Haldane in 2D photonic crystals possessing Dirac point-derived band gaps, can appear in more general settings.”

• Also stated in: § Abstract · plain

  “In this Letter, we show that one-way modes can be generalized to photonic crystals with gyrotropic constituents without the restriction of having Dirac points in the band structure or the use of gyroelectric materials.”

• cites [6] 6. S. Raghu and F. D. M. Haldane, arXiv:cond-mat/0503588; arXiv:cond-mat/0602501.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: Hatsugai's relation between edge states and Chern numbers, derived using a lattice quantum Hall model, applies to the photonic system.

The Chern number of every band is zero if the Hamiltonian is time-reversal symmetric.

“The key properties of the Chern number are that (i) it is always an integer, (ii) the sum of the Chern numbers over all bands is zero, and (iii) the Chern number of every band is zero if the Hamiltonian is T symmetric [13].”

• cites [13] 13. B. Simon, Phys. Rev. Lett. 51, 2167 (1983).Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The abrupt change in Chern number occurs at the critical point where the band becomes degenerate with a neighboring band, at discrete degeneracy points in k space.

“In fact, this abrupt change occurs at the critical point where the band becomes degenerate with a neighboring band, at discrete ''degeneracy points'' in k space.”

The requirement for Dirac points where bands touch in a linear fashion is not strictly necessary for constructing photonic crystals supporting one-way modes.

“We begin by searching for a T-symmetric band structure with a pair of photonic bands degenerate at discrete k points; the requirement for Dirac points where the bands touch in a linear fashion, which was discussed by Raghu and Haldane [6], is not strictly necessary.”

• cites [6] 6. S. Raghu and F. D. M. Haldane, arXiv:cond-mat/0503588; arXiv:cond-mat/0602501.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

A one-way waveguide using a 2D magneto-optical square-lattice yttrium-iron-garnet photonic crystal operating at microwave frequencies is experimentally feasible.

“Using gyromagnetic materials, we design an experimentally feasible one-way waveguide using a 2D MO square-lattice yttrium-iron-garnet (YIG) photonic crystal (which lacks Dirac points) operating at microwave frequencies.”

As one adiabatically tunes the Hamiltonian, the Chern number of a given band changes abruptly when it changes.

“Property (i) implies that as one adiabatically tunes the Hamiltonian (e.g., varying the permeability tensor by tuning the dc magnetic field bias), the Chern number of a given band changes, if and when it does so, abruptly.”

The 2D waveguide structure can be mapped into an equivalent 3D structure that supports identical field distributions and transport properties.

“The 2D waveguide structure can be mapped into an equivalent 3D structure that supports identical field distributions and transport properties [10].”

• cites [10] 10. M. L. Povinelli et al., Phys. Rev. B 64, 075313 (2001).Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

TM modes in a gyromagnetic photonic crystal can be formally mapped to electronic wave functions in a periodic electromagnetic field.

“We show that the TM modes in a gyromagnetic photonic crystal can be formally mapped to electronic wave functions in a periodic electromagnetic field, so that the only requirement for the existence of one-way edge modes is that the Chern number for all bands below a gap is nonzero.”

• Also stated in: § Abstract · plain

  “To do this, we derive an analytical mapping between the electromagnetic modes in the MO photonic crystal and the wave functions of a nonrelativistic electron in a QH system.”

The topological nature of the edge modes ensures that the interface structure is unimportant, as long as it is sufficiently narrow that higher order modes are avoided.

“The topological nature of the edge modes [6] ensures that the interface structure is unimportant, as long as it is sufficiently narrow that higher order modes are avoided.”

• cites [6] 6. S. Raghu and F. D. M. Haldane, arXiv:cond-mat/0503588; arXiv:cond-mat/0602501.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

With a 1600 Gauss applied field, the permeability tensor elements in YIG at 4.28 GHz are μ=12.4 μ0 and κ=14 μ0.

“With a 1600 Gauss applied field, the tensor elements in YIG at 4.28 GHz are 12:4 0 and 14 0 [11].”

• cites [11] 11. D. M. Pozar, Microwave Engineering (Wiley, New York, 1998).Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

In a square-lattice yttrium-iron-garnet crystal operating at microwave frequencies, which lacks Dirac points, time-reversal breaking is strong enough that the one-way edge mode effect should be easily observable.

“In a square-lattice yttrium-iron-garnet crystal operating at microwave frequencies, which lacks Dirac points, time-reversal breaking is strong enough that the effect should be easily observable.”

With suitable boundary conditions, the mapping between photon states in gyromagnetic crystals and electron wave functions in quantum Hall systems holds for both edge modes and bulk modes.

“With suitable boundary conditions, this mapping holds for both edge modes and bulk modes.”

The only requirement for the existence of one-way edge modes is that the Chern number for all bands below a gap is nonzero.

“We show that the TM modes in a gyromagnetic photonic crystal can be formally mapped to electronic wave functions in a periodic electromagnetic field, so that the only requirement for the existence of one-way edge modes is that the Chern number for all bands below a gap is nonzero.”

• Also stated in: § Abstract · plain

  “This result, together with property (iii), implies a simple and general strategy for constructing photonic crystals supporting one-way modes.”

• cites [9] 9. Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993).Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: Hatsugai's relation between edge states and Chern numbers, derived using a lattice quantum Hall model, applies to the photonic system.

For each value of ω, the existence of unpaired edge modes, as well as their spatial characteristics, can be mapped to a similar problem in a quantum Hall system.

“Thus, for each value of !, the existence of unpaired edge modes, as well as their spatial characteristics, can be mapped to a similar problem in a QH system.”

One-way electromagnetic modes analogous to quantum Hall edge states exist in certain 2D magneto-optical photonic crystals with Dirac point-derived band gaps.

“Raghu and Haldane [6] have recently extended this analogy in a remarkable new direction by predicting the existence of one-way electromagnetic modes similar to the chiral edge states found in the integer quantum Hall (QH) effect [7].”

• Also stated in: § Abstract · plain

  “These modes are confined at the edge of certain 2D MO photonic crystals and possess group velocities pointing in only one direction, determined by the direction of an applied dc magnetic field (the time-reversed versions are not eigenstates of the system).”

• cites [6] 6. S. Raghu and F. D. M. Haldane, arXiv:cond-mat/0503588; arXiv:cond-mat/0602501.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [7] 7. The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin (Springer-Verlag, Berlin, 1987).Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The edge mode immunity to backscattering was corroborated by both frequency-domain and time-domain numerical simulations.

“Using both frequency and time-domain numerical simulations, we demonstrate that these edge modes are immune to backscattering.”

• Also stated in: § Abstract · boosted

  “We corroborated this result with a time-domain calculation in which a temporal Gaussian pulse with a spectral bandwidth of 50% of the band gap and carrier frequency 0:555 2c=a is launched into the waveguide.”