# Optical heterodyne interferometry

1. Heterodyne Interferometry
When two waves with slightly different frequencies are superimposed, a “buzz” or “beat” equal to the difference in frequency can be observed. Extracting the necessary information from this “buzz” is called the heterodyne method. Since light is also a wave, it naturally generates “light buzzes” or “light beats. Extracting information from the “optical beat” is called the optical heterodyne interferometry. Interference” refers to the superposition of two lights.　Buzzing can often be experienced when tuning a musical instrument. Tuning is often done with a tuning fork. Let’s think about tuning a guitar with a tuning fork.
At first, when the strings are loose and you pluck the strings with a tuning fork, you will hear the sound of the strings and the sound of the tuning fork separately. As you tighten the strings more and more, you will hear the whole sound as one, but if you listen carefully, you will notice that the sound repeats itself in a very rapid cycle (state 1). As you tighten the strings further, the cycle of the notes becomes longer and longer (state 2), and eventually the intensity of the notes disappears completely (state 3).
This intensity of sound is called “growl”. When the frequencies of two sounds (waves) are different, the difference in frequency is felt. When the difference in frequency between the tuning fork and the string is large (state 1), the period of the hum is fast (the frequency of the hum is large), and when the difference in frequency between the two sounds is small (state 2), the period of the hum is slow (the frequency of the hum is small).
Thus, when two waves are superimposed, a “growl” is created, and the frequency of the growl is equal to the difference between the frequencies of the two waves. In this case, the frequency of the tuning fork is constant, so if we think of it as a reference signal, we can distinguish (=detect) the frequency of the string by the frequency of the buzz, or the period of the intensity of the sound.
The same phenomenon occurs in the case of light as in the case of sound. In other words, when two lights with slightly different frequencies are superimposed, they produce a light buzz equal to the difference frequency. This buzz is called a “light beat,” and the frequency of the light beat is sometimes referred to simply as the “beat frequency. A light beat is detected as a periodic change in light intensity (a change in lightness or darkness).
When some information is given to one of the two lights, the corresponding information appears in the optical beat. The information here refers to some kind of signal in the amplitude, phase, or frequency of the light. In other words, when there is some information in one light, it is possible to superimpose another “reference” light on that light (this is called a reference light) and extract the information from the optical beat signal. This type of signal detection method is called “optical heterodyne interferometry.
The optical heterodyne interferometry has the following features.
• High-precision and high-sensitivity measurements can be made by using a lock-in amplifier or similar device for signal detection.
• When the signal information is only phase information, it is not affected by the intensity fluctuation of the signal light due to disturbance.
• fluctuation of the signal light due to disturbance.
• It is not affected by signal components with different frequencies (noise in general).
• By increasing the intensity of the reference light, it is possible to detect weak signals. etc.

Next, let’s explain the principle of optical heterodyne interferometry using an equation. If we denote the electric field components of the reference light and signal light as Er and Es, respectively, they can be expressed as follows
\begin{eqnarray}E_r=a_r cos(2\pi f_r t + \phi _r) \tag{1} \\
E_s=a_s cos(2\pi f_s t + \phi _s) \tag{2}\end{eqnarray}
where $$a_r, a_s$$ are the amplitudes of the reference light and signal light, respectively. Similarly, $$f_r, f_s, \phi _r, \phi _s$$ represent the frequency and phase, respectively. Superimposing these two lights, the detected light intensity I (since the light intensity is equal to the square of the electric field component) becomes
\begin{eqnarray} I &=& \langle|{E_s + E_r}|^2\rangle\\ &=& \frac{a_s^2+a_r^2}{2}+2a_s a_r cos\{2\pi (f_s – f_r)t + (\phi _s – \phi _r)\}\\ &=& \frac{a_s^2+a_r^2}{2}+2a_s a_r cos\{2\pi f_b t + \Delta\} \tag{3} \end{eqnarray}
where $$\langle\rangle$$ is the time average. $$f_b=f_s – f_r$$ is the “optical beat frequency” and $$\Delta=\phi _s – \phi _r$$ is the “phase difference” between the two optical components.
In the photocurrent component detected by the photodetector, the first and second terms in equation (3) are DC components, and the third term is an AC component that varies sinusoidally with frequency $$f_b$$. The third term is the AC component, which varies sinusoidally with frequency $$f_b$$. This AC signal is called the “optical beat signal.
In optical heterodyne interferometry, the amplitude ($$2a_s a_r$$), frequency ($$f_b$$), or phase(($$\Delta$$) of the optical beat signal is electrically measured, and the information contained in the amplitude ($$a_s$$), frequency ($$f_s$$), or phase ($$\phi_s$$) of the optical signal is extracted.

1. Zeeman Laser
By the way, is there actually such a thing as “two lights with slightly different frequencies”? Such a light can be realized by making a small modification to the laser, or by making a modification to the light to change its frequency afterwards.
The Zeeman laser is a typical example of the former. For the latter, there is a device called a frequency shifter. Both are capable of oscillating dual-frequency light. In practice, it is advisable to distinguish between the advantages and disadvantages.
At Uniopt, we use Zeeman lasers as the light source for our interferometers. This is because (1). The two components do not shift each other due to temperature and other factors, and the optical frequency of the dual-frequency oscillation light is stabilized. In this laser, a static magnetic field is applied to the He-Ne laser to create anisotropy inside the cavity, and two polarization components with different frequencies are oscillated simultaneously. Zeeman lasers are divided into “transverse Zeeman lasers” and “axial Zeeman lasers” according to the direction of the magnetic field applied to the optical axis of the laser.
When a transverse magnetic field is applied to a He-Ne laser, two linearly polarized components with different frequencies and orthogonal to each other are generated. The strength of the static magnetic field applied to the laser tube is determined by the length of the laser tube (=cavity length). Normally, the difference in frequency between the two polarization components of a transverse Zeeman laser is tens to hundreds of kHz, so the “optical beat signal” can be observed by photoelectric detection.
When the laser tube is thermally stabilized, the optical frequency stability is $$10^{-9} – 10^{-10}$$. The stabilized optical frequency is called a “Stabilized Transverse Zeeman Laser” (STZL), where STZL is the first letter of the respective word for Stabilized Transverse Zeeman Laser.