36
2 Terahertz Time-Domain Spectroscopy
2.4.1 Complex Refractive Index
Whilst Eq. 2.10 may always be solved numerically to obtain information about ˜
n s
from the experimentally observed
T (ω) [23–25], a general analytical expression
cannot be found for ˜
n s due to its presence in both ˜
t i j and FP i jk . However, under
certain experimental conditions, the expression for
T (ω) may be simplified and an
analytical expression formed.
The first condition is that in the absence of internal reflections, p = 0 and
FP isi (ω) = 1; this may be achieved experimentally by restricting the width of the
time-domain to only include data up to just before the first internal reflection, as
shown by the shaded area in Fig. 2.5a. The second condition is that we make the
assumption that the frequency-dependence of the refractive index is weak, hence
˜
t i j ( ˜
n) = ˜
t i j ( ˜
n(ω = 0)). Now we may form analytical expressions for the real part of
the complex refractive index, n, and the absorption coefficient, α = 2ωκ/c, as
n(ω) = 1 +
c
ωd
φ(ω),
(2.11)
α(ω) = −
2
d
ln
|
T (ω)|
˜
t is ˜
t si
.
(2.12)
2.4.1.1 Differential Measurements
In some cases, changes in a material property of interest can be small compared
to the background signal, and as such the interesting processes we want to study
may be obscured. Here it is useful to perform a differential measurement, whereby
the change in material property as a function of some control variable is studied.
For example, for a magnetic field-dependent change in a material, the transmission
function becomes
T (ω) = E(B)/E(B = 0). In this situation, both the reference and
sample spectra have propagated through the material under study, and as such are
given by
E r (ω) = ˜
t is ˜
t si E i exp
i
ωd
c
˜
n r
FP isi (ω),
(2.13)
E s (ω) = ˜
t is ˜
t si E i exp
i
ωd
c
˜
n s
FP isi (ω).
(2.14)
An analytical solution for n s in terms of
T (ω) can still be found, provided we make
the assumption that the change in the refractive index is small, ˜
n rs ˜
n r . Now we
may write the complex refractive index of the sample as
˜
n s = ˜
n r + ˜
n rs = (n + iκ) + (δn + iδκ).
(2.15)
2 Terahertz Time-Domain Spectroscopy
2.4.1 Complex Refractive Index
Whilst Eq. 2.10 may always be solved numerically to obtain information about ˜
n s
from the experimentally observed
T (ω) [23–25], a general analytical expression
cannot be found for ˜
n s due to its presence in both ˜
t i j and FP i jk . However, under
certain experimental conditions, the expression for
T (ω) may be simplified and an
analytical expression formed.
The first condition is that in the absence of internal reflections, p = 0 and
FP isi (ω) = 1; this may be achieved experimentally by restricting the width of the
time-domain to only include data up to just before the first internal reflection, as
shown by the shaded area in Fig. 2.5a. The second condition is that we make the
assumption that the frequency-dependence of the refractive index is weak, hence
˜
t i j ( ˜
n) = ˜
t i j ( ˜
n(ω = 0)). Now we may form analytical expressions for the real part of
the complex refractive index, n, and the absorption coefficient, α = 2ωκ/c, as
n(ω) = 1 +
c
ωd
φ(ω),
(2.11)
α(ω) = −
2
d
ln
|
T (ω)|
˜
t is ˜
t si
.
(2.12)
2.4.1.1 Differential Measurements
In some cases, changes in a material property of interest can be small compared
to the background signal, and as such the interesting processes we want to study
may be obscured. Here it is useful to perform a differential measurement, whereby
the change in material property as a function of some control variable is studied.
For example, for a magnetic field-dependent change in a material, the transmission
function becomes
T (ω) = E(B)/E(B = 0). In this situation, both the reference and
sample spectra have propagated through the material under study, and as such are
given by
E r (ω) = ˜
t is ˜
t si E i exp
i
ωd
c
˜
n r
FP isi (ω),
(2.13)
E s (ω) = ˜
t is ˜
t si E i exp
i
ωd
c
˜
n s
FP isi (ω).
(2.14)
An analytical solution for n s in terms of
T (ω) can still be found, provided we make
the assumption that the change in the refractive index is small, ˜
n rs ˜
n r . Now we
may write the complex refractive index of the sample as
˜
n s = ˜
n r + ˜
n rs = (n + iκ) + (δn + iδκ).
(2.15)