Fig. 2.13] it becomes birefringent with the index of refraction varying with the
driving frequency (ω m ). The incoming polarized IR light propagates with different
speed along the two perpendicular axes (x and y, Fig. 2.13), which arises from the
variation of the refractive index of the optical element. The variation of the refractive
index is wavelength dependent and the maximum variation depends on the V m .
When stress is applied to the PEM along the x axis, due to differences in the speed of
light propagation along the x and y axes, a phase shift of the incoming radiation of a
given wavelength is observed.
When the plane of polarization of the incoming light makes an angle of 45
to the
modulator optical axis (x axis Fig. 2.13), then for radiation of a given wavelength λ 0 ,
and at the time of maximum stress or maximum stretching of the optical element (e.
g. ZnSe), half-wave retardation occurs. Under this condition a switching between
parallel and perpendicular linearly polarized light (δ ¼ 90
) takes place. During one
cycle of the voltage applied to the piezoelectric transducer the half-wave retardation
occurs twice: when the optical element is stressed (compressed) and when it is
stretched (expanded). At this moment of the cycle the output plane of the polarization is orthogonally rotated compared to the plane of polarization of the incoming
radiation. When the optical element is at rest (between compression and expansion)
no retardation of the linearly polarized light occurs. This external modulation of the
incoming IR radiation will allow distinguishing between two signals which are
acquired during a single measurement. The intensity of the radiation passing through
the PEM to the detector [I D (δ)] is given in Eq. (2.22) [2].
I D δ
ð Þ ¼
I s þ I p
2
þ
I s À I p
2
cos δ
ð2:22Þ
The phase shift of the incoming radiation passing through PEM is given in
Eq. (2.23).
δ ¼ δ 0 cos ω m t
ð
Þ
ð
Þ
ð 2:23Þ
It depends on the frequency of the stress modulation of the PEM (ω m ) at time t.
The phase shift is a function of V m and λ. The maximum phase shift (δ 0 ) is equal to,
δ 0 ¼
GV m
λ
ð2:24Þ
where G is a factor which is independent on both, the λ and V m . The phase shift
varies between 0 and π. The maximum phase shift is when δ 0 ¼ π and GV m ¼ πλ 0 for
a certain wavelength λ 0 at which the photoelastic modulator is exposed. In other
words δ is equal to.
28
2 Polarization Modulation Infrared Reflection Absorption Spectroscopy: From. . .
driving frequency (ω m ). The incoming polarized IR light propagates with different
speed along the two perpendicular axes (x and y, Fig. 2.13), which arises from the
variation of the refractive index of the optical element. The variation of the refractive
index is wavelength dependent and the maximum variation depends on the V m .
When stress is applied to the PEM along the x axis, due to differences in the speed of
light propagation along the x and y axes, a phase shift of the incoming radiation of a
given wavelength is observed.
When the plane of polarization of the incoming light makes an angle of 45
to the
modulator optical axis (x axis Fig. 2.13), then for radiation of a given wavelength λ 0 ,
and at the time of maximum stress or maximum stretching of the optical element (e.
g. ZnSe), half-wave retardation occurs. Under this condition a switching between
parallel and perpendicular linearly polarized light (δ ¼ 90
) takes place. During one
cycle of the voltage applied to the piezoelectric transducer the half-wave retardation
occurs twice: when the optical element is stressed (compressed) and when it is
stretched (expanded). At this moment of the cycle the output plane of the polarization is orthogonally rotated compared to the plane of polarization of the incoming
radiation. When the optical element is at rest (between compression and expansion)
no retardation of the linearly polarized light occurs. This external modulation of the
incoming IR radiation will allow distinguishing between two signals which are
acquired during a single measurement. The intensity of the radiation passing through
the PEM to the detector [I D (δ)] is given in Eq. (2.22) [2].
I D δ
ð Þ ¼
I s þ I p
2
þ
I s À I p
2
cos δ
ð2:22Þ
The phase shift of the incoming radiation passing through PEM is given in
Eq. (2.23).
δ ¼ δ 0 cos ω m t
ð
Þ
ð
Þ
ð 2:23Þ
It depends on the frequency of the stress modulation of the PEM (ω m ) at time t.
The phase shift is a function of V m and λ. The maximum phase shift (δ 0 ) is equal to,
δ 0 ¼
GV m
λ
ð2:24Þ
where G is a factor which is independent on both, the λ and V m . The phase shift
varies between 0 and π. The maximum phase shift is when δ 0 ¼ π and GV m ¼ πλ 0 for
a certain wavelength λ 0 at which the photoelastic modulator is exposed. In other
words δ is equal to.
28
2 Polarization Modulation Infrared Reflection Absorption Spectroscopy: From. . .