Theory - Frequency-stabilisation

https://sci-hub.wf/10.1364/AO.47.005163

Frequency Modulation Spectroscopy

Frequency Modulation spectroscopy is a technique where the frequency of the laser light is modulated at a relatively high frequency (~MHz) and the detectedlight after absorption is evaluated at the modulation frequency or a harmonic of this frequency. Since technical noise primarily resides at low frequencies, often it depends on the frequency as $1/f$, FM spectroscopy allows efficient noise suppression as the detection is shifted to a high frequency where the noise is significantly lower. This detection can be done either with an analog/digital lock-in amplifier (time domain) or with Fourier analysis (frequency domain)

Lock-in Amplifier

A lock in amplifier is a signal processing instrument that is employed when its favorable to select certain frequency components of a signal. This is clearly the case in FM spectroscopy where the laser light is modulated at a specific frequency. The basis principle of a lock-in amplifier is shown below

Basic schematic of a lock-in amplifier

We can assume a sinusoidal signal at the detector which after amplification becomes $S_\text{sig} (t)=A\sin(2\pi ft+\theta)$. A signal generator produces a reference signal that is sent to the EOM/laser driver to create the frequency modulation. This reference signal is also sent to a mixer where it is multiplied with $S_\text{sig} (t)$. Assuming a sinusoidal reference signal $S_\text{ref} (t) = \sin (2\pi f_\text{ref}t+\phi)$, then the output product of the mixer becomes:

$$S'(t)=A\sin (2\pi f t + \theta)\sin(2\pi f_\text{ref} t + \phi )$$

$$S'(t) = \frac{A}{2} \cos(2\pi(f-f_\text{ref})t+(\theta-\phi))-\frac{A}{2}\cos(2\pi(f+f_\text{ref})t+(\theta+\phi))$$

$S'(t)$ thus contains beats at the sum and difference frequencies after using trigonometric identiities. Since we are modulating the laser frequency with the reference signal, we are only interested in the optical signal generated at the reference frequency. With $f=f_\text{ref}$

$$S'(t)=\frac{A}{2}\cos(\theta-\phi)-\frac{A}{2}\cos(2\pi(2f_\text{ref})t+(\theta+\phi))$$

This signal contains two components, one at zero frequency, and one at high frequency. We can filter out the zero frequency term by using a low-pass filter, resulting in the output signal of

$$S_\text{out}(t) = \frac{A}{2} \cos (\theta-\phi)$$

We can maximiize $S_\text{out}$ by varying $\phi=\theta$.

In practice...

The wavelength modulated laser beam is sent through an absorbing medium and detected with a photodiode connected to a lock-in amplifier. The photodiode measures the intensity of the transmitted light and the lock-in amplifier will output a non-zero signal as soon as there is any signal oscillating at $f_\text{ref}$.

When the laser wavelength is at a point where the absorption profile is steepest (maximum gradient) $f_1$, then it would generate an amplitude modulate with large amplitude and maximum output signal.

We can leverage the output signal to lock our laser to the absorption profile.

Sources

https://pages.physics.wisc.edu/~tgwalker/032.microwave.pdf