2.4 Extracting Sample Properties and Polarisation Information …
35
2.4 Extracting Sample Properties and Polarisation
Information from Time-Domain Data
This section will describe the methods used in this thesis to extract the important
sample properties and information about the polarisation state from the measured
time-domain waveforms. In particular, this section will describe how the complex
transmission function is extracted from the time-domain data, before using it to
extract the complex refractive index in Sect. 2.4.1. The method of extracting the
polarisation state of the THz pulses, in terms of χ and ψ, will be described in
Sect. 2.4.2.
The frequency spectra of the THz pulses detected by EOS are obtained by Fourier
transforming the time-domain data. We can determine the effect the sample has on
the THz pulse by comparing a reference spectrum obtained without the sample in
the spectrometer, E r , to the spectrum obtained with the sample in the spectrometer,
E s . The reference spectrum can be expressed as
E r (ω) = E i exp
i
ωd
c
˜
n i
,
(2.7)
where E i is the electric field at the sample position, and the exponential term describes
the relative phase aquired while propagating a distance d, equal to the thickness of
the sample, through either N 2 -purged air or vacuum with a complex refractive index
˜
n i . After propagation through the sample the spectrum can be expressed as
E s (ω) = ˜
t is ˜
t si E i exp
i
ωd
c
˜
n s
FP isi (ω),
(2.8)
where ˜
n s is the complex refractive index of the sample. This expression must take
into account the losses in the electric field at the entrance and exit surfaces of the
sample, described by the Fresnel transmission coefficients ˜
t i j = 2 ˜
n i /( ˜
n i + ˜
n j ) for
plane waves at normal incidence to a surface. There is an additional contribution from
any internal reflections in the sample, described by the Fabry-Perót term FP isi (ω).
This can generally be expressed for light propagating from medium i to j to k by
FP i jk (ω) =
P
p=0
˜
r jk ˜
r ji exp
i
2ωd
c
˜
n j
p
,
(2.9)
where ˜
r i j = ( ˜
n i − ˜
n j )/( ˜
n i + ˜
n j ) are the Fresnel reflection coefficients at the interfaces. If we divide the sample spectrum by the reference spectrum we obtain the
complex transmission function,
T (ω) = E s (ω)/E r (ω). This allows us to express
T (ω) in terms of ˜
n s as
T (ω) = |
T |e
iφ(ω)
= ˜
t is ˜
t si exp
i
ωd
c
( ˜
n s − ˜
n i )
FP isi (ω).
(2.10)
35
2.4 Extracting Sample Properties and Polarisation
Information from Time-Domain Data
This section will describe the methods used in this thesis to extract the important
sample properties and information about the polarisation state from the measured
time-domain waveforms. In particular, this section will describe how the complex
transmission function is extracted from the time-domain data, before using it to
extract the complex refractive index in Sect. 2.4.1. The method of extracting the
polarisation state of the THz pulses, in terms of χ and ψ, will be described in
Sect. 2.4.2.
The frequency spectra of the THz pulses detected by EOS are obtained by Fourier
transforming the time-domain data. We can determine the effect the sample has on
the THz pulse by comparing a reference spectrum obtained without the sample in
the spectrometer, E r , to the spectrum obtained with the sample in the spectrometer,
E s . The reference spectrum can be expressed as
E r (ω) = E i exp
i
ωd
c
˜
n i
,
(2.7)
where E i is the electric field at the sample position, and the exponential term describes
the relative phase aquired while propagating a distance d, equal to the thickness of
the sample, through either N 2 -purged air or vacuum with a complex refractive index
˜
n i . After propagation through the sample the spectrum can be expressed as
E s (ω) = ˜
t is ˜
t si E i exp
i
ωd
c
˜
n s
FP isi (ω),
(2.8)
where ˜
n s is the complex refractive index of the sample. This expression must take
into account the losses in the electric field at the entrance and exit surfaces of the
sample, described by the Fresnel transmission coefficients ˜
t i j = 2 ˜
n i /( ˜
n i + ˜
n j ) for
plane waves at normal incidence to a surface. There is an additional contribution from
any internal reflections in the sample, described by the Fabry-Perót term FP isi (ω).
This can generally be expressed for light propagating from medium i to j to k by
FP i jk (ω) =
P
p=0
˜
r jk ˜
r ji exp
i
2ωd
c
˜
n j
p
,
(2.9)
where ˜
r i j = ( ˜
n i − ˜
n j )/( ˜
n i + ˜
n j ) are the Fresnel reflection coefficients at the interfaces. If we divide the sample spectrum by the reference spectrum we obtain the
complex transmission function,
T (ω) = E s (ω)/E r (ω). This allows us to express
T (ω) in terms of ˜
n s as
T (ω) = |
T |e
iφ(ω)
= ˜
t is ˜
t si exp
i
ωd
c
( ˜
n s − ˜
n i )
FP isi (ω).
(2.10)