502
19 Dielectric Structures
(a)
(b)
20
15
10
5
0
-5
-10
-15
5
1
-
0
1
-
-5 0 5
0
1
5
1
0
2
GaAs/
AlGaAs
0
c
Fig. 19.19 a Graphical representation of (19.44) with the two solutions marked with circles for n ∞ = 1 (dashed line),
f = 10 −3 , /ω 0 = 10 −2 and ω 0 = ω m . b Reflectance peak positions (experimental data (circles) at T = 5 K) versus
cavity detuning ω c − ω 0 for a cavity with two GaAs/(Al,Ga)As Bragg mirrors (24/33 pairs for the front/bottom mirror)
and five embedded quantum wells whose resonances are closely matched. Solid lines are a theoretical fit according to
(19.47) with 0 = 4.3 meV. The dashed lines show the electronic resonance ω 0 and the cavity resonance ω c . Part (b)
based on data from [1622]
For small k this leads to an (in-plane) effective photon ‘rest mass’, applying (6.38),
1
m
∗
ph
=
1
2
∂
2 E ph
∂k 2 .
(19.50)
We find
m
∗
ph =
k z
c
=
ω(k = 0)
c 2
.
(19.51)
Now we assume the electronic oscillator to be in resonance with the photon dispersion at k = 0, i.e.
E el = ω(k = 0). The electronic resonance shall have vanishing dispersion for simplicity since the
exciton mass is much larger than (19.51). The eigenwert equation of the coupled system, resembling
(6.61), is
E − E ph
V
V
E − E el
= 0 ,
(19.52)
with two solutions, called the upper and lower cavity polariton branch, visualized in Fig. 19.20. Their
splitting at k = 0 is 2V . Thus the coupling parameter V = 0 /2 corresponds [1623] to the Rabi
frequency (19.45). Experimental values for the splitting of 3–15meV in (In,Al,Ga)As based [1622,
1625–1628], 17–44meV in (Cd,Zn)(Te,Se) based [1629], 6–60meV in (Al,In,Ga)N based [1630–
1634] and 78 meV in ZnO based [1635] microcavities were found. It is possible to condensate cavity
polaritons in the minimum of the dispersion around k = 0 (Bose-Einstein condensation). In [1636]
stimulated scattering and gain from cavity polaritons have been reported. Further details on cavity
polaritons can be found in [1637, 1638].
19.1.10 Hyperbolic Optical Metamaterials
A special class of uniaxial materials, termed hyperbolic metamaterials (HMM), has a (relative) dielectric function of the form (cf. Table 9.2)
19 Dielectric Structures
(a)
(b)
20
15
10
5
0
-5
-10
-15
5
1
-
0
1
-
-5 0 5
0
1
5
1
0
2
GaAs/
AlGaAs
0
c
Fig. 19.19 a Graphical representation of (19.44) with the two solutions marked with circles for n ∞ = 1 (dashed line),
f = 10 −3 , /ω 0 = 10 −2 and ω 0 = ω m . b Reflectance peak positions (experimental data (circles) at T = 5 K) versus
cavity detuning ω c − ω 0 for a cavity with two GaAs/(Al,Ga)As Bragg mirrors (24/33 pairs for the front/bottom mirror)
and five embedded quantum wells whose resonances are closely matched. Solid lines are a theoretical fit according to
(19.47) with 0 = 4.3 meV. The dashed lines show the electronic resonance ω 0 and the cavity resonance ω c . Part (b)
based on data from [1622]
For small k this leads to an (in-plane) effective photon ‘rest mass’, applying (6.38),
1
m
∗
ph
=
1
2
∂
2 E ph
∂k 2 .
(19.50)
We find
m
∗
ph =
k z
c
=
ω(k = 0)
c 2
.
(19.51)
Now we assume the electronic oscillator to be in resonance with the photon dispersion at k = 0, i.e.
E el = ω(k = 0). The electronic resonance shall have vanishing dispersion for simplicity since the
exciton mass is much larger than (19.51). The eigenwert equation of the coupled system, resembling
(6.61), is
E − E ph
V
V
E − E el
= 0 ,
(19.52)
with two solutions, called the upper and lower cavity polariton branch, visualized in Fig. 19.20. Their
splitting at k = 0 is 2V . Thus the coupling parameter V = 0 /2 corresponds [1623] to the Rabi
frequency (19.45). Experimental values for the splitting of 3–15meV in (In,Al,Ga)As based [1622,
1625–1628], 17–44meV in (Cd,Zn)(Te,Se) based [1629], 6–60meV in (Al,In,Ga)N based [1630–
1634] and 78 meV in ZnO based [1635] microcavities were found. It is possible to condensate cavity
polaritons in the minimum of the dispersion around k = 0 (Bose-Einstein condensation). In [1636]
stimulated scattering and gain from cavity polaritons have been reported. Further details on cavity
polaritons can be found in [1637, 1638].
19.1.10 Hyperbolic Optical Metamaterials
A special class of uniaxial materials, termed hyperbolic metamaterials (HMM), has a (relative) dielectric function of the form (cf. Table 9.2)