500
19 Dielectric Structures
Fig. 19.18 a Schematic different hopping parameters for hexagonal dielectric structures with different aspect ratios of
medium and air. The boundary between the two parts is highlighted in red. b Visualization of the odd (even) parity dipole
(quadrupole) mode. c SEM image of the structure with the boundary between topologically trivial and non-trivial parts
highlighted in red. A magnified view of the area indicated with the dashed white rectangle is depicted in panel d. A
topologically trivial (non-trivial) hexagon is highlighted in green (blue). Adapted from [1620]
This situation is called the strong coupling regime since it leads to anticrossing behavior of the cavity
mode and electronic resonance. In the weak coupling regime for small absorption, the resonances cross
(within their linewidth). For resonance, the emission intensity of the oscillator into the cavity mode is
enhanced and its lifetime is reduced (Purcell effect, cf. Sect. 19.2.2).
The transmission T of a Fabry–Perot cavity with two (equal and lossless) mirrors of transmission
T m = 1 − R m is given by
T (ω) =
T
2
m exp (−2 L α(ω))
|1 − R m exp (i 2 n ∗ L ω/c)|
2
,
(19.38)
with the complex index of refraction n
∗
= n r + iκ =
√ and α = 2ωκ/c (cf. (9.9)). For an empty
cavity, i.e. a (small) background absorption α B and a background index of refraction n r = n B , the
resonances occur when the phase shift 2n B Lω/c is an integer multiple of 2π , i.e. for
ω m = m
π c
n B L
,
(19.39)
with m ≥ 1 being a natural number. In the vicinity of the resonance, i.e. for ω = ω m + δω, we can
expand exp(2n B Lω/c) ≈ 1 + i2n B Lδω/c and obtain from (19.38) a Lorentzian for the transmission
T (ω) ≈
T
2
m exp (−2Lα(ω))
|1 − R m (1 + i2n B L δω/c)|
2
=
(T m /R m )
2 exp (2Lα(ω))
(δω) 2 + γ 2
c
.
(19.40)
The frequency width (HWHM) γ c of the empty-cavity resonance is given by
γ c =
1 − R
R
c
2 n B L
,
(19.41)
19 Dielectric Structures
Fig. 19.18 a Schematic different hopping parameters for hexagonal dielectric structures with different aspect ratios of
medium and air. The boundary between the two parts is highlighted in red. b Visualization of the odd (even) parity dipole
(quadrupole) mode. c SEM image of the structure with the boundary between topologically trivial and non-trivial parts
highlighted in red. A magnified view of the area indicated with the dashed white rectangle is depicted in panel d. A
topologically trivial (non-trivial) hexagon is highlighted in green (blue). Adapted from [1620]
This situation is called the strong coupling regime since it leads to anticrossing behavior of the cavity
mode and electronic resonance. In the weak coupling regime for small absorption, the resonances cross
(within their linewidth). For resonance, the emission intensity of the oscillator into the cavity mode is
enhanced and its lifetime is reduced (Purcell effect, cf. Sect. 19.2.2).
The transmission T of a Fabry–Perot cavity with two (equal and lossless) mirrors of transmission
T m = 1 − R m is given by
T (ω) =
T
2
m exp (−2 L α(ω))
|1 − R m exp (i 2 n ∗ L ω/c)|
2
,
(19.38)
with the complex index of refraction n
∗
= n r + iκ =
√ and α = 2ωκ/c (cf. (9.9)). For an empty
cavity, i.e. a (small) background absorption α B and a background index of refraction n r = n B , the
resonances occur when the phase shift 2n B Lω/c is an integer multiple of 2π , i.e. for
ω m = m
π c
n B L
,
(19.39)
with m ≥ 1 being a natural number. In the vicinity of the resonance, i.e. for ω = ω m + δω, we can
expand exp(2n B Lω/c) ≈ 1 + i2n B Lδω/c and obtain from (19.38) a Lorentzian for the transmission
T (ω) ≈
T
2
m exp (−2Lα(ω))
|1 − R m (1 + i2n B L δω/c)|
2
=
(T m /R m )
2 exp (2Lα(ω))
(δω) 2 + γ 2
c
.
(19.40)
The frequency width (HWHM) γ c of the empty-cavity resonance is given by
γ c =
1 − R
R
c
2 n B L
,
(19.41)