286
9 Optical Properties
Fig. 9.32 Experimental
two-photon absorption
spectrum of GaAs
(T = 4 K) (dots) plotted as
a function of the difference
of the double-photon
energy 2 from the GaAs
band edge E g . The solid
line is a theoretical
calculation, the dashed
lines represent slopes with
exponent 1/2 and 3/2,
respectively. Adapted
from [901]
10
-2
10
-3
10
-1
1
10
20
10
21
10
19
10
22
T
GaAs
g
1/2
g
g
3/2
(compared to the mK range for atoms). Recent experiments with spatially indirect excitons in coupled
quantum wells lead towards BEC [895, 896]. A sufficiently long lifetime ensures cooling of the
excitons close to the lattice temperature. Another potential candidate for BEC are long-living excitons
(ms-range) in Cu 2 O [897]. The condensation of polaritons (cf. Sect. 9.7.8) in microcavities to welldefined regions of k-space has been discussed in [898] and compared to bosonic condensation in
bulk.
9.7.14 Two-Photon Absorption
So far, only absorption processes that involve one photon have been considered. The attenuation of the
intensity I of a light beam (of frequency ω 0 ) along the z direction can be written as
dI
dz
= −α I − β I
2
,
(9.65)
where α is due to the (linear) absorption coefficient (and possibly scattering) and β is the two-photon
absorption coefficient. A two-photon process can occur in two steps, e.g. via a midgap level, which is
not considered any further here. Here, we consider two-photon absorption (TPA) via the population of a
state at 2 0 higher energy than the initial state with a nonlinear optical process. The TPA coefficient is
related to the nonlinear third-order electric dipole susceptibility tensor [899] χ i jkl . Within the two-band
approximation theory predicts [900]
β ∝
2 0 − E g
3/2 .
(9.66)
The exponent 3/2 is indeed found experimentally, as shown in Fig. 9.32 for GaAs. The strength of
absorption depends on the relative orientation of the light polarization with respect to the main crystallographic directions, e.g. TPA for polarization along 110 is about 20% larger than for the 100
orientation.
9 Optical Properties
Fig. 9.32 Experimental
two-photon absorption
spectrum of GaAs
(T = 4 K) (dots) plotted as
a function of the difference
of the double-photon
energy 2 from the GaAs
band edge E g . The solid
line is a theoretical
calculation, the dashed
lines represent slopes with
exponent 1/2 and 3/2,
respectively. Adapted
from [901]
10
-2
10
-3
10
-1
1
10
20
10
21
10
19
10
22
T
GaAs
g
1/2
g
g
3/2
(compared to the mK range for atoms). Recent experiments with spatially indirect excitons in coupled
quantum wells lead towards BEC [895, 896]. A sufficiently long lifetime ensures cooling of the
excitons close to the lattice temperature. Another potential candidate for BEC are long-living excitons
(ms-range) in Cu 2 O [897]. The condensation of polaritons (cf. Sect. 9.7.8) in microcavities to welldefined regions of k-space has been discussed in [898] and compared to bosonic condensation in
bulk.
9.7.14 Two-Photon Absorption
So far, only absorption processes that involve one photon have been considered. The attenuation of the
intensity I of a light beam (of frequency ω 0 ) along the z direction can be written as
dI
dz
= −α I − β I
2
,
(9.65)
where α is due to the (linear) absorption coefficient (and possibly scattering) and β is the two-photon
absorption coefficient. A two-photon process can occur in two steps, e.g. via a midgap level, which is
not considered any further here. Here, we consider two-photon absorption (TPA) via the population of a
state at 2 0 higher energy than the initial state with a nonlinear optical process. The TPA coefficient is
related to the nonlinear third-order electric dipole susceptibility tensor [899] χ i jkl . Within the two-band
approximation theory predicts [900]
β ∝
2 0 − E g
3/2 .
(9.66)
The exponent 3/2 is indeed found experimentally, as shown in Fig. 9.32 for GaAs. The strength of
absorption depends on the relative orientation of the light polarization with respect to the main crystallographic directions, e.g. TPA for polarization along 110 is about 20% larger than for the 100
orientation.