β-separation line

The β-separation line is the straight line

$$S_{\beta }(f)=\frac{8\ln 2}{{\pi }^2}f\approx 0.56f$$

drawn on a plot of a laser's frequency-noise spectral density $S_{\delta \nu }(f)$, one-sided and in Hz²/Hz, against the Fourier frequency $f$. Di Domenico, Schilt and Thomann showed that it divides the spectrum into two kinds of noise. Where the noise lies above the line, the frequency excursions are large compared with their Fourier frequency, and they widen the central part of the line; where it lies below, they are small compared with it and only add wings, leaving the width at half maximum unchanged. The full width at half maximum of the linewidth then follows from the area $A$ under the part of the spectrum above the line:

$$\begin{array}{c}{\displaystyle \mathrm{\Delta }\nu \approx \sqrt{8\ln 2A},}\\ {\displaystyle A={\int }_{1\mathrm{/}{T}_0}^{\mathrm{\infty }}H\text{\hspace{0.17em}⁣}[S_{\delta \nu }(f)-S_{\beta }(f)]S_{\delta \nu }(f)df,}\end{array}$$

where $H$ is the unit step function and $T_0$ the observation time, whose inverse is the lowest frequency included.

Why the line rises with frequency

A frequency modulation at a Fourier frequency $f$ with a peak deviation $\delta$ has a modulation index $\delta \mathrm{/}f$. With a large index the optical frequency sweeps slowly across a range of about $\pm \delta$, which broadens the line; with a small index it only puts weak sidebands at $\pm f$. For noise, the mean-square deviation contributed by a band of width about $f$ is about $S_{\delta \nu }(f)f$, so the index squared scales as $S_{\delta \nu }(f)\mathrm{/}f$, and comparing it with a constant gives a line proportional to $f$. The constant $8\ln 2\mathrm{/}{\pi }^2$ makes the rule exact for white frequency noise: a flat level $S_0$ crosses the line at $f={\pi }^2{S}_0\mathrm{/}(8\ln 2)$, the area above it is $S_0$ times that frequency, and the formula returns $\mathrm{\Delta }\nu =\pi S_0$, the width of the Lorentzian line that white noise produces, the intrinsic linewidth of the Schawlow–Townes entry.

Worked example

The example is a laser with the 5.9 kHz intrinsic linewidth of the example in the external cavity laser entry, a white floor $S_0=1.9\times {10}^3$ Hz²/Hz, plus flicker frequency noise $S_{\delta \nu }={10}^9\mathrm{/}f$ Hz²/Hz, which is ${10}^6$ Hz²/Hz at 1 kHz. The flicker noise crosses the line at 42 kHz and lies below it at every higher frequency. The line shape computed in full, from the phase autocorrelation $\exp [-2\int S_{\delta \nu }(f){\sin }^2(\pi f\tau )\mathrm{/}{f}^{2}df]$ with the same lowest frequency $1\mathrm{/}{T}_0$, gives the widths in the last column:

The estimate is 6 to 8% below the full calculation. The width is 25 to 45 times the intrinsic linewidth, and it grows only by a factor of 1.7 over three decades of observation time, since for flicker noise the area above the line grows with the logarithm of $T_0$. For observation times below about 0.1 ms the low-frequency noise no longer washes out the carrier: the total phase fluctuation above $1\mathrm{/}{T}_0$ falls to a few radians, the spectrum becomes a narrow carrier on a broad pedestal, and no single width describes it.

Using it

On the bench, $S_{\delta \nu }(f)$ is measured with a frequency discriminator, such as an unbalanced interferometer or the slope of a cavity resonance, or from a heterodyne beat with a narrower reference (How to measure laser linewidth). The line is drawn over the spectrum, the noise above it is integrated from $1\mathrm{/}{T}_0$ upward, and the result is quoted with the observation time. The intrinsic linewidth $\pi S_0$ is read separately from the white floor, where it appears at high Fourier frequencies. The two numbers describe different time scales, and which one matters depends on the delays and integration times of the application.

Common questions

What is the β-separation line?

The line $S(f)=8\ln (2)f\mathrm{/}{\pi }^2$ on a frequency-noise spectrum, introduced by Di Domenico, Schilt and Thomann. Frequency noise above it determines the laser's linewidth; noise below it only shapes the wings of the line.

How is the linewidth calculated from a frequency-noise spectrum?

Approximately, as $\sqrt{8\ln 2A}$, where $A$ is the integral of the frequency noise over the Fourier frequencies at which it lies above the β-separation line, starting from the inverse of the observation time. Exactly, from the Fourier transform of the field autocorrelation computed from the whole spectrum.

Why does the measured linewidth depend on the observation time?

Because a longer observation includes lower Fourier frequencies, where flicker and drift noise rise above the line and add to the area. In the example above the width grows from 146 kHz at 1 ms to 244 kHz at 1 s.

How accurate is the approximation?

It is exact for white frequency noise. For the white-plus-flicker example above it falls 6 to 8% below the full calculation, and it does not apply at observation times so short that the carrier survives as a separate peak.

References: G. Di Domenico, S. Schilt, and P. Thomann, "Simple approach to the relation between laser frequency noise and laser line shape," Applied Optics 49, 4801 (2010). D. S. Elliott, R. Roy, and S. J. Smith, "Extracavity laser band-shape and bandwidth modification," Physical Review A 26, 12 (1982).