498
19 Dielectric Structures
Fig. 19.15 Transmission of N = 11 quarter-wave stack (n 1 = 1, n 2 = 2) with center n 2 -region of widths 1.8 λ 0 /4
(dashed line) and 2.2 λ 0 /4 (solid line)
Fig. 19.16 Transmission of N = 11 quarter-wave stack (n 1 = 1, n 2 = 2) with center n 2 -regions of widths 3 λ 0 /4 (solid
line), 3.5 λ 0 /4 (dash-dotted line) and 4λ 0 /4 (dashed line)
down magnetic fields. An experimental system similar to this is based on yttrium iron garnet (YIG) as
gyrotropic material for breaking time reversal symmetry and exhibits topological edge modes [1615].
A non-magnetic version which is much more desirable in terms of fabrication and choice of materials
is based on the idea of the Haldane model put forward in [1616]. A next-nearest neighbor coupling t
in the honeycomb lattice is considered and produces topological states if the phase φ
of the (complex)
hopping parameter t
is not zero or π . Further theoretical considerations and modeling of such nonmagnetic, fully dielectric topological resonators were reported in [1617, 1618]. Such resonators have
been realized and investigated in [1619]. The system consists of a square lattice of ring resonators,
which are coupled to each other through link rings. These intermediary links are spatially shifted with
respect to the ring resonators, to introduce an asymmetric set of hopping phases controlling whether the
structure results in topologically trivial φ
= 0 or non-trivial φ
= π/2 situation. Lasing in edge modes
and light transport at the circumference of a 10 × 10 field of resonators have been demonstrated [1619].
A similar concept, also playing with the different hopping parameters to create topologically trivial
and non-trivial two-dimensional photonic band structures has been reported in [1620] (Fig. 19.18).
The dipole (odd parity) and quadrupole (even parity) character of the states changes (does not change)
within a band for the topologically non-trivial (trivial) parts. The optically pumped laser emission
stems from the topologically non-trivial 2D bulk area [1620]. This is attributed to the fact that the
band-inversion around the point leads to a topology-induced mode selection with lower threshold,
19 Dielectric Structures
Fig. 19.15 Transmission of N = 11 quarter-wave stack (n 1 = 1, n 2 = 2) with center n 2 -region of widths 1.8 λ 0 /4
(dashed line) and 2.2 λ 0 /4 (solid line)
Fig. 19.16 Transmission of N = 11 quarter-wave stack (n 1 = 1, n 2 = 2) with center n 2 -regions of widths 3 λ 0 /4 (solid
line), 3.5 λ 0 /4 (dash-dotted line) and 4λ 0 /4 (dashed line)
down magnetic fields. An experimental system similar to this is based on yttrium iron garnet (YIG) as
gyrotropic material for breaking time reversal symmetry and exhibits topological edge modes [1615].
A non-magnetic version which is much more desirable in terms of fabrication and choice of materials
is based on the idea of the Haldane model put forward in [1616]. A next-nearest neighbor coupling t
in the honeycomb lattice is considered and produces topological states if the phase φ
of the (complex)
hopping parameter t
is not zero or π . Further theoretical considerations and modeling of such nonmagnetic, fully dielectric topological resonators were reported in [1617, 1618]. Such resonators have
been realized and investigated in [1619]. The system consists of a square lattice of ring resonators,
which are coupled to each other through link rings. These intermediary links are spatially shifted with
respect to the ring resonators, to introduce an asymmetric set of hopping phases controlling whether the
structure results in topologically trivial φ
= 0 or non-trivial φ
= π/2 situation. Lasing in edge modes
and light transport at the circumference of a 10 × 10 field of resonators have been demonstrated [1619].
A similar concept, also playing with the different hopping parameters to create topologically trivial
and non-trivial two-dimensional photonic band structures has been reported in [1620] (Fig. 19.18).
The dipole (odd parity) and quadrupole (even parity) character of the states changes (does not change)
within a band for the topologically non-trivial (trivial) parts. The optically pumped laser emission
stems from the topologically non-trivial 2D bulk area [1620]. This is attributed to the fact that the
band-inversion around the point leads to a topology-induced mode selection with lower threshold,