506
19 Dielectric Structures
QD-X
WGM
0
1.322
1.321
1.320
1.319
0
6
0
5
0
3
0
4
0
2
0
1
(b)
7
6
5
4
3
2
1
0
-1.5 -1.0
5
.
0
-
0
.
0
5
.
0
0
.
1
(a)
Fig. 19.26 a Temperature dependence of the energy positions of the whispering gallery mode (WGM) of a 5-µm diameter
(Al,Ga)As/GaAs microdisc (Q = 6500) and the single-exciton resonance of a single InAs quantum dot contained within
the disc. b Intensity of WGM mode as a function of the detuning E WGM − E QD−X from the QD single exciton resonance.
The excitation density was 15 Wcm −2 for all data. Adapted from [1651]
19.2.2 Purcell Effect
According to Fermi’s golden rule (19.1), the probability of an optical transition depends on the density
of available optical modes (final states). If the density of modes is enhanced compared to its vacuum
value (19.2) at a resonance of an optical cavity, the lifetime of the electronic state decreases by the
Purcell factor [1649],
F P =
3
4π 2 Q
(λ/n)
3
V
,
(19.55)
where n is the refractive index of the medium, Q is the quality factor of the cavity resonance and V is the
effective mode volume.
3 Experiments on the emission of quantum dots (that generally provide small
absorption and thus allow for the weak coupling regime) in etched micropillars containing a microcavity
(Fig. 19.25a) have shown that indeed the luminescence decay is faster for cavities with large Purcell
factor (Fig. 19.25b) [1650]. The resonance of cavity mode and emitter leads to an enhanced emission
intensity as shown in Fig. 19.26 for the exciton emission of a single quantum dot in a microdisc [1651].
19.2.3 Deformed Resonators
The whispering gallery modes in circular (or spherical) cavities are long-lived and emission goes into all
angles. Light escape is based only on the exponentially slow process of evanescent leakage (neglecting
disorder effects such as surface roughness). In order to overcome the isotropic light emission, the
resonator needs to be deformed. This can be accomplished with an ellipsoidal shape, i.e.
r (φ) = R [1 + cos φ] ,
(19.56)
where 1+2 is the aspect ratio of the ellipse. The increased radiation leads to a decrease of the Q-factor
as shown in Fig. 19.23b. Also, a new decay process, refractive escape, becomes possible. A ray that is
initially in a whispering gallery trajectory diffuses in phase space until finally an angle smaller than the
critical angle for total reflection (9.11) is reached. The ray dynamics becomes partially chaotic [1652].
3 V is given by the spatial integral of the vacuum field intensity for the cavity mode, divided by its maximum value.
19 Dielectric Structures
QD-X
WGM
0
1.322
1.321
1.320
1.319
0
6
0
5
0
3
0
4
0
2
0
1
(b)
7
6
5
4
3
2
1
0
-1.5 -1.0
5
.
0
-
0
.
0
5
.
0
0
.
1
(a)
Fig. 19.26 a Temperature dependence of the energy positions of the whispering gallery mode (WGM) of a 5-µm diameter
(Al,Ga)As/GaAs microdisc (Q = 6500) and the single-exciton resonance of a single InAs quantum dot contained within
the disc. b Intensity of WGM mode as a function of the detuning E WGM − E QD−X from the QD single exciton resonance.
The excitation density was 15 Wcm −2 for all data. Adapted from [1651]
19.2.2 Purcell Effect
According to Fermi’s golden rule (19.1), the probability of an optical transition depends on the density
of available optical modes (final states). If the density of modes is enhanced compared to its vacuum
value (19.2) at a resonance of an optical cavity, the lifetime of the electronic state decreases by the
Purcell factor [1649],
F P =
3
4π 2 Q
(λ/n)
3
V
,
(19.55)
where n is the refractive index of the medium, Q is the quality factor of the cavity resonance and V is the
effective mode volume.
3 Experiments on the emission of quantum dots (that generally provide small
absorption and thus allow for the weak coupling regime) in etched micropillars containing a microcavity
(Fig. 19.25a) have shown that indeed the luminescence decay is faster for cavities with large Purcell
factor (Fig. 19.25b) [1650]. The resonance of cavity mode and emitter leads to an enhanced emission
intensity as shown in Fig. 19.26 for the exciton emission of a single quantum dot in a microdisc [1651].
19.2.3 Deformed Resonators
The whispering gallery modes in circular (or spherical) cavities are long-lived and emission goes into all
angles. Light escape is based only on the exponentially slow process of evanescent leakage (neglecting
disorder effects such as surface roughness). In order to overcome the isotropic light emission, the
resonator needs to be deformed. This can be accomplished with an ellipsoidal shape, i.e.
r (φ) = R [1 + cos φ] ,
(19.56)
where 1+2 is the aspect ratio of the ellipse. The increased radiation leads to a decrease of the Q-factor
as shown in Fig. 19.23b. Also, a new decay process, refractive escape, becomes possible. A ray that is
initially in a whispering gallery trajectory diffuses in phase space until finally an angle smaller than the
critical angle for total reflection (9.11) is reached. The ray dynamics becomes partially chaotic [1652].
3 V is given by the spatial integral of the vacuum field intensity for the cavity mode, divided by its maximum value.