Chemistry Reference
In-Depth Information
Fig. 11 Time dependence of the light intensity reflected by a sample with anisotropic reflectance
R
ʱ
and
R
ʲ
depending upon the light polarization states
ʱ
and
ʲ
(on the horizontal axis the time t is
reported). In this case,
R
ʱ
>R
ʲ
(
ΔI ¼R
ʱ
R
ʲ
). The polarization of light is switched by the PEM at
the frequency
ˉ
0
between polarization
ʱ
and
ʲ
. The resulting frequency for
ΔI
is 2
ˉ
0
(see text)
explains also that
R
/
R
signal must be analyzed by a lock-in amplifier tuned at the
second harmonic of the modulated signal to extract the physically meaningful
Δ
I
/
I
component (Fig.
11
). A more detailed and rigorous mathematical derivation of the
same result is possible [
18
,
30
]. In this case, the light polarization vector is
represented by a 2-component vector, modified by the progressive application of
2
Δ
2 matrices (Jones matrices) representing the different optical elements along the
optical path [
7
]. The final electric field (entering the detector) is obtained by
successive application of all the 2
2 matrices associated to the optical elements,
in the order with which they appear along the light path. The square of the final
electric field represents the final intensity, with the resulting time dependence.
A significant product of the calculation shows that if the RAS signal is expressed
r
/
r
, where
r
is the complex Fresnel coefficient for reflection (
r
2
as
Δ
¼
R), in the
expression for
r
/
r
, two terms appear, modulated at different frequencies: it is a
complex quantity, whose real part Re(
Δ
Δ
r
/
r
), modulated at 2
ˉ
0
, coincides with
Δ
R
/
R
RAS
(apart from a factor of 2:
Δ
R/R
RAS
¼
2 Re(
Δ
r/r
)) and specifies the light
intensity modulation, while the imaginary part Im(
ˉ
0
, related
to the phase term. Both terms can be measured by opportunely tuning the lock-in at
the correct modulation frequency.
Another important outcome of the complete calculation is that the possible error
sources due to the optical misalignment of the elements (polarizers, sample, PEM,
etc.) enter as a first-order correction in the
Δ
r
/
r
) is modulated at
ˉ
0
contribution, but only as a second
order term in the intensity modulation at 2
ˉ
0
. Then, the real part of the complex
RAS signal, that is,
R
/
R
RAS
, is less sensitive to the eventual lack of accuracy in
setting the apparatus than the imaginary part (as can be easily checked in experi-
ments) [
18
,
30
].
This conclusion has an important practical significance. We have already
reported that RAS was independently developed in the same years in the Bell
Labs (by D. Aspnes) and at the Ioffe Institute (by S. Safarov). We also outlined
how this has meant that two different experimental apparatuses have been projected
and built: the main dissimilarity is represented by the use of just one polarizer in the
Δ
