## Laser Frequency Noise

The roots of modern photonics dates back to the invention of the laser in 1960 by Theodor Maiman. A laser produces a tightly focused and single color beam of light. Its single coloredness has unlocked the quantum world and pushed our society towards the 21st century. Below, we will review some basic terms often found in laser spec-sheets.

Some of the relevant specs in the physics, photonics and engineering -verse are frequency noise and linewidth. When referring to frequency noise or phase noise, the following terms are usually used by colloquially by engineers and physicists: Hz2/Hz, Hz/√Hz, dBc/Hz, rad2/Hz. Linewidth and frequency are always expressed in Hertz. And depending on the scale: Hz, kHz, MHz, GHz. The simplest form of linewidth is measured from the Full Width Half Maximum (FWHM). However, it can also be measured in terms of the nature of the line shape. This is typically the Lorenzian linewidth and, on rare occasions, the Gaussian linewidth. The latter two are obtained by fitting the line shape by the respective functions.

### Laser, color, frequency noise, phase noise, linewidth? What's the deal?

Single color light essentially means that the laser is emitting a very single frequency. The spectral purity of the laser determines the laser's frequency noise. Ideally, if the laser were to emit at a pure single color, it would like something like this. A pure single frequency sine-wave.

In reality, a pure single frequency electromagnetic wave does not exist. The laser will be subject to frequency noise. This results in a fuzziness of the sine-wave when observed over a certain period of time. This fuzziness is a result of sine wave changing frequency or phase and different rates. In layman's terms, the fuzz is generated by laser frequency noise.

The plots above have, so far, been depicted in the time or temporal domain. From the temporal domain, one can move into the frequency domain, where one finds interesting effects. The pure tone depicted above without frequency noise is represented by an infinite thin line (delta function). This is the black curve below, where 0 frequency is the central frequency of the laser.

Now, when frequency noise is present, some interesting features appear. The infinitely thin line shape of the laser has broadened. The nature line shape broadening depends on the type or quality of frequency noise present. Simply-put, it can either be Lorentzian, Gaussian or a convolution of both, Voight.

The frequency noise spectral density plot provides a more complete picture. (See plot below) This is often expressed in Hz2/Hz, Hz/√Hz. It is typically the preferred plot for engineers and physicists. In most cases, physicists will care about the white noise level. This is where the frequency noise transitions into 'flat' white noise level. This white noise level ultimately determines the linewidth of the laser. Physicists have the luxury of locking or stabilization their laser to an atomic transition or an excellent optical tuning fork (optical resonator) to kill all low frequency noise. From the plot below, that is everything below the beta line.

Gianni Di Domenico, Stéphane Schilt, and Pierre Thomann, "Simple approach to the relation between laser frequency noise and laser line shape," Appl. Opt. 49, 4801-4807 (2010). https://doi.org/10.1364/AO.49.004801

For engineers solving real-world problems, the free running linewidth is much more important, as they do not have the luxury or space to lock their laser to an atomic line. To add to the complexity, depending on the timescales of their application, they might need a free running laser that has excellent frequency noise at certain offset frequencies. This means building a free running laser that pushes the beta line to lower frequencies, aka, very long term stable.

From frequency noise and with proper care for units, it is also straight-forward to convert to phase noise following ref 1 and 2 below. Phase noise is typically expressed in either dBc/Hz, rad2/Hz. Phase noise is commonly used to characterize rf and microwave oscillators or oscillators in general. Combining two lasers different only at frequencies that fall in the rf and microwave domain also conveniently transfers all the frequency or phase noise information into that part of the electromagnetic spectrum.

The combined beams can be measured on a photodetector with the appropriate bandwidth. The photodetector signal will contain the rf or microwave signal corresponding to the difference frequencies of the two lasers. This is colloquially called 'Beating' or 'Beat Frequencies' by engineers and physicists. The frequency noise can easily be measured by a frequency counter or a high resolution oscilloscope. It can also be measured conveniently with a modern spectrum analyzers. These often already have a phase measure measurement functionality, including some of the best cross correlation algorithms to reduce measurement noise floor. There are also commercial solutions to measure the frequency noise and phase noise. At Tulon Photonics, our core IP, can also be used to create a laser frequency/phase noise measurement black box without any difficulty. However, that will be a topic for another time.

Lastly, frequency noise and phase noise plots always start from 1 Hz offset frequency and can go as high as 100MHz offset frequency. Below 1 Hz, into the sub-Hz realm, means that the laser is becoming ultra-long term stable. The laser is probably stabilized to an atomic transition or an optical resonator (optical tuning fork). In these cases, the Allan deviation provides a better picture. The allan deviation can be thought of as the inverse of the frequency noise. For a beautiful explaination, see refs 1 and 2 below. :)

Some useful references for frequency noise, phase noise, converting from one to the other and everything else are the beautiful books and papers below.

A Tutorial on Phase Noise by Enrico Rubiola and Claudio E. Calosso https://indico.fnal.gov/event/21836/contributions/64988/attachments/40841/49444/Enrico_LLRF-2019-Tutorial.pdf

Frequency Standards: Basics and Applications by Fritz Riehle

Gianni Di Domenico, Stéphane Schilt, and Pierre Thomann, "Simple approach to the relation between laser frequency noise and laser line shape," Appl. Opt. 49, 4801-4807 (2010). https://doi.org/10.1364/AO.49.004801

## Conversion from Frequency Noise to Phase Noise and Vice Versa

Ref 1 covers the conversion between frequency noise and phase noise very well. The reader is advised to go through it for a better understanding. However, for quick reference, we will quckly go through the highlights. Engineers coming from rf and microwave background often prefer phase noise in dBc/Hz. It is very important to distinguish between double sided, ℒ and single sided Sϕ phase noise spectrum which is related simply by ℒ(f) = Sϕ(f)/2 or ℒ(f) |dB = Sϕ(f) |dB - 3 dB if it is expressed in dB. For completeness Sϕ → rad2/Hz, it follows 10Log10(Sϕ) → dBrad2/Hz. Physicists, on the other hand, prefer frequency noise spectral density either in Hz2/Hz or Hz/√Hz. It should be straightforward to convert from phase noise following Sv=f2Sϕ .

## Laser Frequency Stabilization

In the section above, it has been briefly mentioned in passing that the best way to reduce laser frequency noise is by stabilizing the laser to an atomic transition or an optical resonator or optical tuning fork.

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https://arxiv.org/abs/2203.04550

https://arxiv.org/abs/2212.01266

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