Broadband Terahertz Spectroscopy
131
ˆ
n =
iωl
c
−
n 2 − n 1
n 2 (n 2 + n 1 )
+ M R
−1 E out (ω)
E out (ω)
.
(10)
Here, l is the path length of the sample, n 1 is the refractive index of the window, n 2 is
the refractive index of the sample, MR is multiple reflection term and E(ω), E(ω)
are pump-induced, and pump-off Fourier transforms of transmitted THz electric
fields, respectively.
Generally, the thin-film approximation is used when the thickness of the film is
so thin that a THz pulse passing through the film acquires a phase shift much smaller
than the typical wavelength, i.e., n f (ω/c)d f 1 where n f is the refractive index
and d f is the thickness of the thin film. Also, the sample is optically much denser
compared to the substrate (n f n s > 1). However, one must be careful in deciding
if the thin-film approximation is applicable or not for a specific experiment. The
complex transmittance is given by equation [70]:
T
∗
(ω) = |T (ω)|e
iϕ(ω)
=
1 + n s
1 + n s + Z 0 σ ∗ (ω)d f
(11)
where Z 0 = 376.7 is the impedance of free space. From the above equation,
complex conductivity (σ*) can be analytically solved:
σ
∗
(ω) =
1 + n s
Z 0 d
cosϕ(ω)
|T (ω)|
− 1 − i
sinϕ(ω)
|T (ω)|
(12)
3.4 Effective Medium Theory
Often THz-TDS and OPTP experiments measure macroscopic permittivity/conductivity of composite systems, particles embedded in host materials, as
shown in Fig. 10. In such a scenario, one needs to use an appropriate effective
medium theory (EMT) approach to determine the intrinsic dielectric response of the
particles of interest. We often encounter this issue especially for studying nanocrystals (NCs) in colloidal dispersion [58]. The scattering effects can be neglected if the
particle size is significantly smaller than the wavelength of THz probe light (1 THz
= 300 μm) [71].
There are several EMTs in the literature [71, 72]. The most commonly used ones
are the Maxwell–Garnett (MG) and the Bruggemann approximation [73, 74]. In MG
theory, the experimentally observed effective dielectric function (ε) is related to the
dielectric functions of the host (ε h ) and the inclusion (ε P ) materials as:
ε − ε h
ε + K ε h
= f
ε P − ε h
ε P + K ε h
(13)
131
ˆ
n =
iωl
c
−
n 2 − n 1
n 2 (n 2 + n 1 )
+ M R
−1 E out (ω)
E out (ω)
.
(10)
Here, l is the path length of the sample, n 1 is the refractive index of the window, n 2 is
the refractive index of the sample, MR is multiple reflection term and E(ω), E(ω)
are pump-induced, and pump-off Fourier transforms of transmitted THz electric
fields, respectively.
Generally, the thin-film approximation is used when the thickness of the film is
so thin that a THz pulse passing through the film acquires a phase shift much smaller
than the typical wavelength, i.e., n f (ω/c)d f 1 where n f is the refractive index
and d f is the thickness of the thin film. Also, the sample is optically much denser
compared to the substrate (n f n s > 1). However, one must be careful in deciding
if the thin-film approximation is applicable or not for a specific experiment. The
complex transmittance is given by equation [70]:
T
∗
(ω) = |T (ω)|e
iϕ(ω)
=
1 + n s
1 + n s + Z 0 σ ∗ (ω)d f
(11)
where Z 0 = 376.7 is the impedance of free space. From the above equation,
complex conductivity (σ*) can be analytically solved:
σ
∗
(ω) =
1 + n s
Z 0 d
cosϕ(ω)
|T (ω)|
− 1 − i
sinϕ(ω)
|T (ω)|
(12)
3.4 Effective Medium Theory
Often THz-TDS and OPTP experiments measure macroscopic permittivity/conductivity of composite systems, particles embedded in host materials, as
shown in Fig. 10. In such a scenario, one needs to use an appropriate effective
medium theory (EMT) approach to determine the intrinsic dielectric response of the
particles of interest. We often encounter this issue especially for studying nanocrystals (NCs) in colloidal dispersion [58]. The scattering effects can be neglected if the
particle size is significantly smaller than the wavelength of THz probe light (1 THz
= 300 μm) [71].
There are several EMTs in the literature [71, 72]. The most commonly used ones
are the Maxwell–Garnett (MG) and the Bruggemann approximation [73, 74]. In MG
theory, the experimentally observed effective dielectric function (ε) is related to the
dielectric functions of the host (ε h ) and the inclusion (ε P ) materials as:
ε − ε h
ε + K ε h
= f
ε P − ε h
ε P + K ε h
(13)