2.1. Method of Measuring Minimum Angular Deviation of a Prism
Figure 1a shows a typical dispersing prism with an apex angle
and refractive index
n1 (at a specified wavelength of the incident light). A parallel light beam is incident from the left with an incident angle
at the first interface of the prism. It leaves the second interface with an angle
, which is deviated from its original incident direction with an angular deviation
. The symbols
and
indicate the refraction angle and the incident angle for the first and second refraction of the two interfaces, respectively. From the geometry of the prism triangle, we have
Using some algebra and manipulation, we find that the angular deviation
can be written as a function of
and
only. When
reaches its minimum
, the ray traverses the prism symmetrically, that is, parallel to its base (
). Under that situation, we have
where
m in the subscript indicates the minimum value for minimum
, as shown in
Figure 1b.
Applying Snell’s law to the first refraction and assuming the medium around the prism is air or vacuum with a refractive index
= 1.0, the prism’s refractive index
is
Thus, by varying
to find
and knowing
, the refractive index of the prism can be determined.
Figure 1b shows the angular deviation versus the incident for the case of
= 60
0 and
n = 1.5; the minimum
is 37.2
0 at the incident angle
= 48.6
0.
2.2. Method Measuring the Maximum Phase Change in TIR
Now, we consider another scheme proposed by us to measure the refractive index.
Figure 2 illustrates a situation in which a light beam is incident into a transparent semi-circle cylinder (or a semi-sphere) with refractive index
n1, and that of the surrounding medium is
with
. It is well known that when the incident angle is larger than the critical angle
, the total internal reflection (TIR) occurs. As TIR exists, the light beam reflects into the same material without refracting into the medium. Here, the semi-circle shape is chosen for convenience of discussion. The TIR is on the top interface without considering the refraction at the boundaries of the material (i.e., when the light wave enters and leaves the material). It is further assumed that some anti-reflection coating is applied to the boundaries (except the top surface); thus, all of the incident power aimed at the center of the semi-circle with angle
is reflected and leaves the material at the same angle without taking the Fresnel loss at the boundaries into account. We also know that under TIR, the reflected light wave will gain different phase changes for different polarizations as follows [
12]:
where
and
are the phase changes for p and s polarizations, respectively, and
is the refractive index ratio of
and
, which is less than 1.
Figure 2b shows
(blue line) and
(red line) versus
for the case of
and
(
n = 2/3) with the critical angle
. The phase changes start from zero at
and increase monotonically to 180
0 when
reaches 90
0, and
is larger than
in the whole interval. Because the phase differences exist for the two polarizations, there is a polarization change if the incident wave polarization is not parallel with one of the polarizations. For example, as shown in
Figure 3a, if an incident wave is linearly polarized 45
0 with respect to the
x axis, its polarization state will be an elliptical polarization after the TIR. The ellipticity depends on the incident angle and refractive index ratio
. This feature is used as a quarter wave plate that transforms a linear polarization into a circular polarization, called a Fresnel rhomb [
12]. Its advantage over the traditional birefringent wave plate is that it is much less wavelength-sensitive. The phase difference is defined as
, and it can be written as
Figure 3b shows
for the case of
n = 2/3 (
and
). Note that at the two incident angles, 50.23
0 or 53.26
0,
reaches 45
0; thus, after two TAR reflections, the total phase change accumulated is 90
0 (i.e., quarter wave), which is the Fresnel rhomb principle. It is also interesting to find that there is a maximum for the
; in this case, it is
for
, where the (
M) in the subscript indicates the maximum. By differentiating the right side of Equation (5) with respect to
and setting it to zero to find
, we can have
Equation (6) is of value because it gives analytical results for
and
. It is interesting to compare
Figure 3b with
Figure 1b, where both have extreme incident angle values for giving minimum deviation angle or maximum phase change. It is natural to expect this property can be used to measure a unknown refractive index as in the prism case. Since the phase change leads to polarization state change, we need to modify the configuration properly to detect it. Considering
Figure 4, without plotting the hemi-circle boundary, there are two polarizers added outside the material, and the Jones matrices [
16] are employed to analyze the polarization change. As shown in the figure, the light wave is incident along the z direction. There are two polarizers, P
1 and P
2, which make the angle
and
respect their local
x coordinate, respectively. The Jones matrices indicated at the bottom of the figure for
P1, TIR, and
P2 are
,
, and
, respectively. Since we need to polarize the light beam at
to gain
, we can assume, without losing generality, that the Jones vector of the incident light field is
, a unit vector also polarized at
and the T in the superscript is a transport operation. Using the cascade matrices manipulation, the output field is
Finally, we can obtain the output intensity as follows:
It is noted that when
of P
2 is at the angle of 45
0 or 135
0, the output intensities are denoted as
or
, respectively.
Taking
measurement as an example, if we keep increasing the incident angle
from
and detect the intensity
, it reaches its maximum when
, as can be seen from
Figure 3b and Equation (9). This behavior is shown in
Figure 5.
Consequently, by varying the incident angle, the maximum of
can be reached to determine
, and the right half of Equation (6) can be used to calculate
n as
Since
is the refractive index ratio, we need to know the
of the surrounding medium. If we take
for air, the material’s refractive index is
. This is compared with the scheme that determines the refractive index by finding the critical angle, which is not easy to exactly determine. This method works well by finding
, which should be easier to determine more precisely because this angle is usually not close to the critical angle. It is noted that a previous work [
17] calculated the refractive index by measuring the maximum phase change. They used a polarization camera to detect it. In our scheme, the refractive index is calculated from the incident angle that gives maximum phase change. Their method utilized a more complicated setup, but it can reach high accuracy. Our measurement configuration is simpler, and the precision depends on the resolution of the rotation stage and the intensity detector.
2.3. Method for Measuring Ellipticity of TIR (without Scanning the Incident Angle)
In the last section, we successfully find the refractive index by looking for the incident angle that gives maximum phase change in TIR. This is similar to the method in
Section 2.1, which looks for the incident angle that gives the minimum deviation angle. In this section we would like to introduce another possibility in which no incident angle scanning is necessary. This method may be considered an advanced or improved version. This scheme can be seen by examining Equation (5), giving that phase change
only depends on the incident angle
and the index ratio
. Thus, it is possible to find
if
can be found for a given
without using a phase meter or an interferometer. Let us show how it proceeds.
As shown in
Figure 4 and Equation (9), this configuration can be used to measure
and
by simply rotating the
of P2 to 45 and 135 degrees, respectively. Note that by taking the square root of the ratio
/
and using Equation (5), we have
where
is the ellipticity of the polarization ellipse shown in
Figure 3a. The ellipticity is defined as the ratio of the short axis to the long axis of the polarization ellipse of the field, which is the ratio of field amplitude taken at 135
0 and 45
0, respectively. After finding
at any selected incident angle
, the refractive index ratio, from Equation (11), is obtained as
Thus, the material refractive index can be determined. The advantage of this method is that we do not need to vary the incident angle to find n. However, it is suggested to select the incident angle carefully to have a larger value (i.e., bigger ), in order to increase the signal intensity or S/N ratio.
2.4. Dispersion Measurement with Polychromatic Light
In the last section, we present a scheme for obtaining the refractive index by finding the ellipticity of the polarization. Here, we would like to show how to use this feature to obtain the dispersion of the material by employing a polychromatic source [
18]. It is known that material dispersion is a property such that the refractive index depends on the wavelength of the incident light wave, i.e., the refractive index
is actually a function of wavelength
, written as
from now on. Considering that we can prepare an incident flat-top spectrum
in the dispersion interval of interest from
to
, as shown in
Figure 6. As illustrated in
Figure 7, it is incident into a material with
and with
; thus, the index ratio is
. Initially, this spectrum is polarized at 45
0 as in the last section; however, because of the material dispersion, each wavelength component experiences a different amount of the polarization change
. Consequently, the output spectrum
has different ellipticity of polarization ellipse for different wavelengths, as shown in the right of the figure. Similarly, as conducted in the last section, this time, a spectrometer is utilized to detect
by setting the polarization angle of P2 to 45 and 135 degrees, and they are denoted as
and
, respectively. The ellipticity dependence of
, as seen in Equation (11), is the square root of the ratio of
, that is
Consequently, as in Equation (12), the dispersion of the material
is