Laser Locking For Ultra-High Precision Light.... Pound-Drever-Hall Laser Locking Method....
Used For High Precision Laser Light To Ensure Accurate Measurements, PDH Technique Is Fundamental!
Image Credit:- newscenter.lbl.gov
Introduction
This artcile focuses specifally on a technique used with lasers called locking, it allows scientist to use very fine tuned light.
I'm sure everyone is familiar with a laser, and have probably used one. You will notice that lasers have a single colour, this is because it is a single frequency with a very small range. They are extremely useful in many measuring devices from spectrometers to gravitational wave detectors. Its is they were fine range of frequencies that make them ideal. However in many cases for high precision the laser light must be as close to a single frequency as possible. In order to do this a filtering cavity is first used that only allows a very small frequency band of light through.
The cavity only allows light to enter a very small band also, so to ensure that the light is always at the right frequency to enter and thus leave a locking method must be used. Once this is acheived, the light that leaves the cavity can be used for other purposes like making very precise spectroscopic measurements of atoms or molecules. The gravitational wave detector also works with the same principle described in this article.
The Pound-Drever-Hall locking method is at the heart of Laser experiments.
Overview
The transmissive and reflective behaviour of the cavity strongly depends on the frequency of the laser 
Locking a laser refers to the action of ensuring the laser always matches a resonant mode of the cavity. The Pound-Drever-Hall (P.D.H) method is the favoured technique of locking; established in 1983. Ron Drever envisaged this method that was based on techniques for microwaves used by R.V.Pound. The technique was further developed and implemented by Jan Hall. It's possible with the P.D.H locking method to adjust either the laser frequency or the cavity length by using a piezo actuator. In either case the the resonant mode is locked onto , and maximum transmission occurs. Using this method and carefully isolating any fluctuations it's possible to achieve sub-hertz stability.
The Method
A laser operates with an unstable frequency within a certain bandwidth, the frequency must be actively controlled in order to match the cavity mode. It's imperative to continually measure the difference between the laser frequency and the cavities resonant mode, this difference is 
To date there are no electronics available to directly measure the difference in frequency 
The derivative of the reflection co-efficient
Image Credit:- Ref[1]
Implementation
A laser beam is sent through an electro-optic modulator (EOM) controlled by an oscillator, which modulates the phase and introduces side-bands to the carrier frequency of the laser. The light impinges on the input mirror of the cavity, and the reflected beam is recorded by a photo-detector. The signal from the photo-detector and the oscillator output is then mixed and passed through a servo amplifier (which is a PID controller), which acts to adjust the laser frequency and lock the laser to the cavities resonant mode.
Image Credit:- Ref[1]
Extraction of the Error Signal
The extraction of the error signal is not immediately self evident, but through a mathematical representation it's easier to understand. For the purpose of an introduction, we can consider a Fabry-Perot cavity (seen in Figure 2) to build the formalism.
The magnitude of an incident electric field at the cavity is
where a constant frequency 
where
where 
Image Credit:- Ref[1]
If we pass the laser through an EOM we induce some phase modulation, if the EOM modulates the phase with a frequency of 
which is valid for small modulation depths 
When the modulation depth is small 
The phase modulated laser impinges on the cavity's mirror and some reflection occurs, each component of the frequency will be transformed through reflection, and the electric field becomes
There is still the problem of directly measuring the electric field, however using a photodiode detector we can measure the intensity or power that relates to the electric field as 
The reflected power contains frequency components of the oscillation from modulating the frequency that produces side-bands, the DC power of the carrier, and higher order components from the interaction of side-bands. In order to obtain the phase, oscillating terms must be separated. For fast modulation frequencies the sine term will dominate over the cosine term. It is proportional to the modulation by the EOM, and a mixer can be used to extract it. The mixer multiplies the sine term from the photo-detector with the sinusoidal signal produced by the oscillator, this produces a 
From Figure 4 it can be seen that the error signal is antisymmetric around resonance, there is a large frequency range where the error sign(+ or -) relates to which side of the resonance the laser is on. Also a steep slope near resonance is beneficial for the server loop to produce the locking between the laser and the cavity.
Image Credit:- Ref[1]
Personal PDH Signal
Using the technique described in this article i was able to generate this signal, use it to lock my cavity and detect the light allowed to transmit through the cavity via resonance. In the figure below you will see the PDH signal in blue and the resoance peak of the cavity mode in Red (don't worry about the x-axis in time). I can now use this light to make very small oscillator vibrate and study the motion or even cool them, but thats for another time.
Image Credit:- Self-Produced in Python
I hope you enjoyed the article and found it interesting. Feel free to UPVOTE, FOLLOW and RESTEEM.
*No copyright infringement has occurred. The article was written by me. Pictures are either personal or free to use in the public domain. Figures in this article were extracted from Ref.[1]
References:-
[1] Eric Black:- Notes on the Pound-Drever-Hall technique, 1998
