19.1 Photonic Band Gap Materials
501
where R
= R m exp(−2Lα). Thus, the decay rate (photon loss from the cavity) is proportional to
T m + α B L if both terms are small. The quality factor of the cavity resonance m is given by
Q =
ω m
2 γ c
≈
m π
1 − R
.
(19.42)
Now, the electronic resonance is put into the cavity leading to a change in the dielectric function to (cf.
(D.11))
= n
2
B
1 +
f
1 − (ω 2 + iωω)/ω
2
0
,
(19.43)
where the index of refraction due to the electronic resonance is given by n(ω) =
√ and (D.13a,b).
For resonance of the cavity mode and the electronic oscillator, i.e. ω m = ω 0 , the solution for the cavity
resonance condition 2n r ωL/c = m2π is obtained, using (19.39), from
n r (ω) = m
π c
ω L
= n B
ω m
ω
.
(19.44)
A graphical solution (Fig. 19.19a) yields three intersections of the left and right hands of (19.44). The
very high absorption at the central solution (ω = ω 0 ) results in very low transmission. The other two
solutions
2 yield the frequencies of the coupled normal mode peaks. For f 1, we use (D.13a) in
(19.44) and find for the splitting ± 0 /2 of the two modes
2
0 = f ω
2
0 −
2
.
(19.45)
This frequency is called the Rabi frequency. If the dielectric function of the oscillator is put into (19.38),
the splitting is found to be
2
0 = f ω
2
0 − (( − γ c )
2
.
(19.46)
A splitting will only be observable if 0 γ c . If the two resonances ω c and ω 0 are detuned by
= ω c − ω 0 , the splitting of the transmission peaks shows the typical anticrossing behavior of two
coupled oscillators
2
=
2
0 +
2
.
(19.47)
In the experiment, typically the electronic resonance remains fixed at ω 0 and the cavity resonance is
detuned by variation of the cavity length across the wafer (Fig. 19.19b).
A detailed theory of cavity polaritons is given in [1623]. The nonlinear optics of normal mode
coupling in semiconductor microcavities is reviewed in [1624].
The in-plane dispersion of the cavity polaritons depends on the coupling strength. First, the photon
dispersion is given by
E ph (k) = ω = c k = c
k
2
+ k
2
z
1/2 ,
(19.48)
where k is the in-plane k-vector and k z is given by the resonance condition, k z = ω m /c with (19.39),
k z = m
π
n B L
.
(19.49)
Thus the dispersion relation is no longer linear as for freely propagating light.
2 These solutions only occur for sufficient oscillator strength f > ((/ω 0 ) 2 , i.e. in the strong coupling regime where
2
0 > 0. The absorption coefficient at ω 0 must be larger than n ∞ /c.
501
where R
= R m exp(−2Lα). Thus, the decay rate (photon loss from the cavity) is proportional to
T m + α B L if both terms are small. The quality factor of the cavity resonance m is given by
Q =
ω m
2 γ c
≈
m π
1 − R
.
(19.42)
Now, the electronic resonance is put into the cavity leading to a change in the dielectric function to (cf.
(D.11))
= n
2
B
1 +
f
1 − (ω 2 + iωω)/ω
2
0
,
(19.43)
where the index of refraction due to the electronic resonance is given by n(ω) =
√ and (D.13a,b).
For resonance of the cavity mode and the electronic oscillator, i.e. ω m = ω 0 , the solution for the cavity
resonance condition 2n r ωL/c = m2π is obtained, using (19.39), from
n r (ω) = m
π c
ω L
= n B
ω m
ω
.
(19.44)
A graphical solution (Fig. 19.19a) yields three intersections of the left and right hands of (19.44). The
very high absorption at the central solution (ω = ω 0 ) results in very low transmission. The other two
solutions
2 yield the frequencies of the coupled normal mode peaks. For f 1, we use (D.13a) in
(19.44) and find for the splitting ± 0 /2 of the two modes
2
0 = f ω
2
0 −
2
.
(19.45)
This frequency is called the Rabi frequency. If the dielectric function of the oscillator is put into (19.38),
the splitting is found to be
2
0 = f ω
2
0 − (( − γ c )
2
.
(19.46)
A splitting will only be observable if 0 γ c . If the two resonances ω c and ω 0 are detuned by
= ω c − ω 0 , the splitting of the transmission peaks shows the typical anticrossing behavior of two
coupled oscillators
2
=
2
0 +
2
.
(19.47)
In the experiment, typically the electronic resonance remains fixed at ω 0 and the cavity resonance is
detuned by variation of the cavity length across the wafer (Fig. 19.19b).
A detailed theory of cavity polaritons is given in [1623]. The nonlinear optics of normal mode
coupling in semiconductor microcavities is reviewed in [1624].
The in-plane dispersion of the cavity polaritons depends on the coupling strength. First, the photon
dispersion is given by
E ph (k) = ω = c k = c
k
2
+ k
2
z
1/2 ,
(19.48)
where k is the in-plane k-vector and k z is given by the resonance condition, k z = ω m /c with (19.39),
k z = m
π
n B L
.
(19.49)
Thus the dispersion relation is no longer linear as for freely propagating light.
2 These solutions only occur for sufficient oscillator strength f > ((/ω 0 ) 2 , i.e. in the strong coupling regime where
2
0 > 0. The absorption coefficient at ω 0 must be larger than n ∞ /c.