# Birefringence measurement using the optical heterodyne interferometry

1. Basic principle
Consider passing dual-frequency orthogonal linearly polarized light (STZL oscillation light) through a birefringent sample so that its principal axis coincides with the polarization plane. The speed at which each polarization component passes through the sample is different between the “advance axis” and “retardation axis” orientations of the sample. Therefore, after passing through the sample, there will be a “phase difference” (see birefringence). Therefore, by detecting the phase difference using optical heterodyne interferometry, the amount of birefringence of the sample can be quantitatively measured with high accuracy.
The principle of the birefringence measurement is shown in the figure. A linear polarizer is used to interfere with two lights. In such a configuration, birefringence measurement can be performed with the measurement accuracy of an electric phase meter. Since the measurement accuracy of an electric phase meter is 0.1 degree (or better), the birefringence can be measured with a high accuracy of about 1/4000 of the wavelength of light.
If the refractive indices of the sample are nx and ny, and the thickness at which the light is transmitted is d, the phase lag after transmission $$\phi _x, \phi _y$$ can be expressed as follows
\begin{eqnarray}\phi _x = \frac{2n_x d}{\lambda} \tag{4} \\
\phi _y = \frac{2n_y d}{\lambda} \tag{5} \end{eqnarray}
When the STZL oscillating light is transmitted through the sample, the light intensity signal I obtained by the photodetector is expressed as follows
\begin{eqnarray}I &=& \langle|{E_x + E_y}|^2\rangle\\ &=& \frac{a_x^2+a_y^2}{2}+2a_x a_y cos\{2\pi (f_x – f_y)t + (\phi _x – \phi _y)\}\\ &=& \frac{a_x^2+a_y^2}{2}+2a_x a_y cos\{2\pi f_b t + \Delta\}\\ &=& \frac{a_x^2+a_y^2}{2}+2a_x a_y cos\{2\pi f_b t + \frac{2\pi (n_x – n_y)d}{\lambda}\}\\ &=& \frac{a_x^2+a_y^2}{2}+2a_x a_y cos\{2\pi (f_b t + \frac{\delta n d}{\lambda})\} \tag{6} \end{eqnarray}
where $$\delta$$ is the phase difference of the two component light and $$\delta n$$ is the refractive index difference = the amount of birefringence. From equation (6), we can see that the phase difference between the two lights is converted into the phase difference of the beat signal.
From the above, it can be shown that the amount of birefringence can be measured by measuring the phase of the optical beat signal with an electric phase meter.

2. Simultaneous measurement of principal axis orientation
In the above method, however, (1) the orientation of the birefringence principal axis of the sample must be determined in advance, and (2) the orientation of the principal axis must be precisely aligned with the STZL oscillation polarization plane. Therefore, the phase difference is detected while rotating the polarization plane of the STZL oscillation around the optical axis, and the amount of birefringence and its principal axis orientation are obtained simultaneously.
To rotate the polarization plane around the optical axis, we use the property of the one-half wavelength plate that the outgoing polarization plane is rotated by twice the angle between the incident polarization plane and the main axis of the plate.
An optical system for simultaneously measuring the birefringence and its principal axis orientation using the optical heterodyne method is shown in the figure. In this optical heterodyne interferometer, the two polarization components of the STZL oscillation light pass through exactly the same optical path from the light source to the photodetector. Therefore, the two polarization components are affected by disturbances such as vibrations and air turbulence to exactly the same extent. As a result, any noise due to these disturbances will be cancelled out, and the optical beat signal will be completely unaffected.