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LASER, as an optical oscillator of a single tone, emits an electrical field that oscillates at the THz frequency level with an amplitude related to its optical power. As we describe in another article, a laser does not exhibit a “pure” emission. The laser emission frequency is not an ideal sine wave in the time domain and not a delta Dirac function in the frequency domain, as well with a constant amplitude. The laser emission frequency fluctuates over time, with different amplitudes at different time scales, as well as its emission power.

In this article, the goal is to give a solid but non exhaustive overview of the metrics and tools that are useful to evaluate the laser emission, such as frequency noise, phase noise, Allan deviation, linewidth, intensity noise, … etc.

TIME DOMAIN METRICS

In this section, we will present three important quantities that can be analyzed in the time domain. These quantities are directly what defines the laser emission $\mathbf{E}(t)$if we restrict it to a simple sine function:

In these equations, we can see three important quantities:

$E(t)$ = Amplitude (Volt, V)

$f(t)$ = Frequency (Hertz, Hz)

$\phi(t)$ = Phase (Radian, rad)

As we explained earlier, a “pure” emission, so a “pure” sine function, is not realistic. Hence, the three above quantities will fluctuate over time around a mean value. We then can write the following:

IMPORTANT: Note that here the frequency fluctuations $\Delta f(t)$ are much slower compared to the laser emission frequency $f_0$, and the amplitude fluctuations $\Delta E(t)$ are much slower compared to the laser emission frequency $f_0$.

Power/Intensity Fluctuations and Mean Value

Regarding a laser radiation, that is an oscillating electro-magnetic field, a common metric to talk about is Power or even Intensity. The Power, that can be for example measured on a photodetector (composed of a photodiode and a current-to-voltage circuit), is proportional to the squared absolute value of the electric field Amplitude denoted $E(t)=E_0+\Delta E(t)$, meaning that Power: $P(t) = P_0 + \Delta P(t)$, where $P_0$ is the mean Power and $\Delta P(t)$ is the Power fluctuations, expressed in Watts (W). Intensity is the optical Power per unit area, so expressed in W/m².

Knowing the laser power and/or intensity is crucial for many applications especially when non-linearity effects are studied or exploited.

For metrology and/or for applications that need as perfect as possible a laser signal, the Power fluctuations is an essential quantity. This Power fluctuations can be analyzed in the time domain to have a simple and first view of it, so a graph with watts over seconds. Then, these power fluctuations could be also reduced by active stabilization, but this is another topic.

Frequency/Phase fluctuations and Frequency mean value

Frequency and Phase are here in the same section as actually both gives “similar” information as the frequency is the time derivative of the phase.

Knowing the laser emission frequency and fluctuations is crucial for many application especially when the laser coherence is concerned and atoms / molecule are involved.

For Metrology and/or for applications that need as perfect as possible laser signal, the frequency / phase fluctuations are essential quantities. This frequency / phase fluctuations can be analyzed in the time domain to have a simple and first view of it, so a graph with Hz or rad over seconds. Then, these frequency / phase fluctuations could be also reduced by active stabilization, but this is another topic.

FREQUENCY DOMAIN METRICS

In this section, we will present important quantities that can be analyzed in the frequency domain. These quantities are directly what defines the laser emission E(t) if we restrict it to a simple sine function, as seen in the section above :

DE(t) = Amplitude fluctuations
Df(t) = Frequency fluctuations
Dphi(t) = Phase fluctuations

The time domain view is interesting to see time-dependent events, but it is most of the time difficult to get all the details of a signal from its time domain view. This is why it is common to plot the quantity in the frequency domain, in order to see what are the frequencies that compose the signal and with which contribution / weight.
Related to a laser emission, the first quantity that is commonly analyzed into frequency domain (or even wavelength domain) is the Power emission through the Optical Spectrum measured typically with an Optical Spectrum Analyzer. However, if we consider a single-frequency laser, this frequency view gives mainly a validation of it, gives the mean power, the central frequency (f0) / wavelength but not really more, as we are limited by the resolution. Since this, other methods are needed to get the three quantity of interest here which are :

• Frequency Noise
• Phase Noise
• Relative Intensity Noise

Phase Noise Power Spectral Density

As defined earlier in this article, knowing the phase fluctuations of a laser is important for different applications. This quantity we wrote it Dphi(t) as a time dependent function. However, by using the following formula that represent the Fourier Transform of the autocorrelation function of the phase fluctuations, we arrive to a frequency dependent quantity called Phase Noise Power Spectral Density (PN-PSD).

Phase Noise Power Spectral Density is a widely used quantity in the Radio-Frequency community to describe phase noise of a MHz / GHz electronic oscillator. It has various units like rad²/Hz, rad/sqrt(Hz) and dBc/Hz when we talk about Spectral Purity : L(f). The Power Spectral Density representation is very important for “energy signal conservation”. This implies that the integration of this signal in the frequency domain remains always the same for any spectral resolution. The PSD view is similar to a classical FFT view with 1Hz resolution bandwidth. For an easy reading and to see all the details, it is common that PN-PSD, FN-PSD and RIN-PSD are in log-log scale.

Frequency Noise Power Spectral Density

Frequency Noise Power Spectral Density is a frequency dependent quantity that shows similar information as PN-PSD. There is a simple and quick relation between the two quantities that is:

Then, there is direct and simple noise relation between frequency and phase as shown later in this article. From FN-PSD and PN-PSD, we can see a lot of things to analyzer the laser performance, as presented in another article. On the plot below, we can see for example “electronic” noise with thin a high peaks on the middle of the spectrum, we can see also “thermal” noise with the increase of the noise when the Fourier frequency is decreasing, and also more “quantum” noise with a white frequency noise floor at high Fourier frequency that is representative to the “Schawlow-Townes” linewidth. FN-PSD is generally expressed in Hz²/Hz or Hz/sqrt(Hz).

Relative Intensity Noise Power Spectral Density

The last quantity we wanted to talk about is the Relative Intensity Noise Power Spectral Density, commonly called RIN. This is a frequency view of the laser power fluctuations normalized by the laser mean power. It is calculated, as the PN-PSD and FN-PSD, by the Fourier transform of the Autocorrelation function of the power fluctuations normalized first by its mean value :

RIN = F [RP(tau)]

Below is a graph of a typical laser RIN, expressed in dB/Hz, or even sometimes written dBc/Hz, 1/Hz and in dBm/Hz, W/Hz when not normalized to the mean power.

TIME/FREQUENCY-INTEGRATION BASED METRICS

On the previous sections, we have presented the basic/main quantities that define an oscillator emission, such as Power, Phase, Frequency. From these quantities, that can be analyzed / observed in the time or frequency domain, some mathematical operation are possible to even get more understanding of the oscillator emission. These operations, at least the ones we will look at, always involve an integration function, over a given time-scale. They are :

• Linewidth & Coherence
• Frequency / Phase / Power standard deviation
• Integrated Frequency / Phase Noise & Timming Jitter
• Frequency Stability

We will not talk about the mean value, for example for the Power or the frequency / wavelength as we already presented them above.

Linewidth & Coherence

Linewidth: such a simple and widely use quantity useful to compare quickly oscillators, but very often misunderstood and misused as a sales pitch that makes no physical sense!

First of all, the definition of the linewidth is very simple and is the full (or half in some cases) width at half maximum: FWHM. And this is true for any shapes (Gaussian, Lorentzian, Voight…). For a single frequency laser oscillator, by looking at the power spectrum is it easy to extract the linewidth (that can also be called -3dB linewidth instead of FWHM). It means, and this is important, that FWHM is an arbitrary definition and as no real mathematical sense.
In that example, if the linewidth is related to the laser emission, it is in Hz as a unit. Even if getting the linewidth from the power spectrum is easy, getting the laser power spectrum itself is challenging !

What is important also to take into account is that the linewidth is an integrated quantity. The linewidth will be different depending on the integration / observation time ! This is why giving just a number of a FWHM as no sense, and the integration time should be always given.
So, the easiest way to get the laser linewidth is to make the beating with another one that is known as much thinner and that emits a frequency close to it in order to be able to detect it with a fast photo-detector. Then, a simple analysis on a Electrical Spectrum Analyzer (ESA) gives the results.
As having another laser at the same wavelength is not always possible, one other possibility is to compute the laser linewidth from its Phase / Frequency Noise PSD. The main challenge here, after being able to measure the laser frequency noise using for example an Optical Frequency Discriminator, is that there is no universal direct mathematical linked between FN-PSD and FWHM as, again, FWHM is an “arbitrary” definition.
There is one famous case where there is a direct mathematical link between FN-PSD and FWHM and it is for a laser with pure white frequency noise, so a laser that has a pure Lorentzian shape as Power Spectrum, and it is given by this formula:
FWHM = pi * Snu
where Snu is the white frequency noise floor in Hz²/Hz. This is commonly called as Lorentzian linewidth, or Schawlow Townes linewidth or even “instantaneous” linewidth as a real laser usually exhibits white frequency noise for Fourier frequencies above 1MHz.
At Silentsys, we call it “commercial” linewidth as it as no physical sense & meaning and this can be confusing and misleading for customers who compare these values for different lasers without having the benefit of hindsight regarding their application’s integration time.
This definition could be fine for lasers that exhibit very high white frequency noise floor, so a Lorentzian linewidth of let’s say 1-10MHz, because this noise contribution is probably the dominant one, especially behind “technical noise” that appear mainly at Fourier frequencies below 1MHz.
IMPORTANT : we advice you to never compare laser based on this “instantaneous” linewidth, especially if the value is >> 1MHz !
To even give you more details about why we have this vision at SILENTSYS, that is from our Time & Frequency background, is that it has no sense to give a number that is not observable because of the Fourier limitation !
For example, a “instantaneous” linewidth of 1kHz integrated on 1us cannot be measured, as if you look at the power spectrum over 1us you will inherently have a resolution limitation of >1MHz ! As you can see from the graph below.
It is then correct to talk about FN-PSD, not FWHM !

As there is no direct mathematical link between FN-PSD and linewidth, as for each and real frequency noise spectrum the power spectrum shape is not a pure Lorentzian function, it is still possible to make estimation, especially based on the called beta-separation line method, developed by Giani XXX + mettre ref.

This method simply divided the FN-PSD datas into two groups, with values below and above a line, to integrates the parts of the spectrum that contributes to the FWHM. This principle is shown on the next plot where we compute the estimated FWHM of a SLIM LINER laser over different time-scale.

Actually, as demonstrated on the next section, we are working at SILENTSYS on a direct mathematical link to better understand how the linewidth comes from the PN/FN-PSD.

Finally for this section, it is also good to keep in mind that the use of a laser during for example 1ms doesn’t mean that all the Frequency Noise below 1kHz will not impact the measurement. The observation time doesn’t act as a pure Heaviside filter, but mainly as a second order high pass filter with the following formula:

where f is the Fourier frequency and Tobs is the observation time.

So, even is the laser is observed over 1ms, all the FN-PSD contributes to its FWHM with less impact for Fourier frequencies below the observation time; which is very important to keep in mind.

Coherence: Instead of linewidth, people usually use “Coherence” as a tool to characterize and compare lasers. It can be Coherence time or Coherence length when related to the propagation speed of the wave. Basically, the Coherence time for an oscillator is the duration for which the correlation between the electrical field and itself at these two different times is equal to exp(-1).

So, in reality, contrary to what is commonly assumed, even is we interfere a laser with itself at two time instants that are a way larger than the coherence time, we will still see interference fringes. However, this is possible by looking at the signal very fastly, so during a very short time scale. If we now look the interferences during a long time (so to integrate the signal) we will not see anymore fringes but just a constant value.

In conclusion to this, the coherence time of a laser is directly related to the Phase / Frequency noise of the laser as we will see later in this article.

Frequency / Phase / Power standard deviation

In our case, looking back to the time-series of the oscillator Phase and Frequency fluctuations, one interesting statistical parameter here is the standard deviation. The standard deviation is a common statistical parameter that we will not develop here. However, one important relation between the Phase / Frequency standard deviation and the Phase / Frequency noise power spectral density is that we can compute this quantity by integrating the spectrum and taking the root mean square as show in the next formula. It is true because of few statistical consideration on the Phase / Frequency fluctuation distribution. This is also what we call Integrated Phase / Frequency noise. It is possible to do the same also for the Power and for the Timing Jitter.

Integrated Frequency / Phase Noise & Timing Jitter

As shown in the previous section, from the Frequency / Phase noise spectrum, we can make an integration to reach the so called Integrated Frequency Noise (in Hz) and the Integrated Phase Noise (in rad). These quantities are often used in the Time and Frequency metrology community. They also are used to compare different oscillators / laser. However, it is also very important to give the Integration Range when giving a value. This quantities are also related to laser linewidth and coherence. Instead of giving only one value, it is common to show the integration over the Fourier frequencies in order to give more details on the noise contributions.

Instead of Integrated Phase noise, another quantity is commonly used and is the Timing Jitter, in second. Simply, it is another view compared to the phase noise by normalizing by the carrier frequency of the signal in order to talk about time and not phase.

Frequency Stability

Now, when the call is to study the frequency “drift” or “long-term” frequency fluctuations of an oscillator, other tools have been developed by the Time and Frequency metrology community. Indeed, the frequency noise PSD introduced in the previous section is a powerful tool to characterize the frequency fluctuations of an oscillator over short timescales (typically shorter than 1 s, corresponding to Fourier frequencies above 1 Hz). The frequency fluctuations over longer timescales (Fourier frequencies typically below 1 Hz) are generally characterized differently, from the time series of the oscillator frequency ν recorded with a frequency counter. This time series is generally normalized by the averaged oscillator frequency ν_0: y=ν/ν_0 to reach relative fluctuations in order to compare easily oscillators (clock) at different carrier frequency. The frequency stability of the oscillator is characterized by the so-called Allan variance (AVAR), which describes how well the oscillator reproduces the same frequency over a given time τ :

where N is the number of frequency samples averaged during the integration time τ. The Allan variance σ_y^2 (τ), or the Allan deviation σ_y (τ), are unitless and represent the relative frequency stability of the oscillator. Other definitions exist in order to better analyze the frequency fluctuations, that are for example HVAR, PVAR, MVAR, TVAR…