Simple approach to the relation between laser frequency noise and laser line shape

Abstract

Frequency fluctuations of lasers cause a broadening of their line shapes. Although the relation between the frequency noise spectrum and the laser line shape has been studied extensively, no simple expression exists to evaluate the laser linewidth for frequency noise spectra that does not follow a power law. We present a simple approach to this relation with an approximate formula for evaluation of the laser linewidth that can be applied to arbitrary noise spectral densities.

Frequency discriminators for the characterization of narrow-spectrum heterodyne beat signals: Application to the measurement of a sub-hertz carrier-envelope-offset beat in an optical frequency comb

“…This relation has been discussed in detail in the work of Di Domenico et al 9 and we only summarize here the main conclusions of this study: the frequency noise spectrum S δν ( f ) can be divided into two surfaces (the slow and fast modulation areas) by the β-separation line defined as S δν ( f ) = (8Ln(2)/π 2 ) · f . It was shown in Ref.…”

“…It was shown in Ref. 9 that only the slow modulation area of surface A, for which S δν ( f ) > (8Ln(2)/π 2 ) · f , contributes to the linewidth of the signal, FWHM = √ 8Ln(2)A, while the fast modulation area (S δν ( f ) < (8Ln(2)/π 2 ) · f ) only contributes to the wings of the lineshape without affecting the linewidth.…”

“…In this condition, the fraction of the spectral power which is out of the discriminator operating range can be shown to be 0.04% for a Gaussian lineshape (valid for (8Ln(2)/π 2 ) · f BW < S max , see Ref. 9) and 21% for a Lorentzian lineshape (valid for (8Ln(2)/π 2 ) · f BW > S max ). Assuming a white frequency noise in a bandwidth f BW , S δν ( f < f BW ) = S max , the surface of the slow modulation area is simply…”

“…This results from the fact that the additional frequency noise occurring in a bandwidth increment does not contribute to the signal linewidth, as it is entirely below the β-separation line. 9 The upper limit S max of the frequency discriminators has been determined for a white frequency noise; however, one has to keep in mind that a higher frequency noise level acting in a narrower bandwidth would still be measurable if its surface A was smaller than…”

“…Such knowledge is in particular important to determine the feedback bandwidth that is needed to narrow the laser linewidth down to the required level. 9 In this paper, we present a frequency discriminator that we have developed to measure the frequency noise of a radiofrequency (RF) beat signal, as well as different commercially available discriminators, that we have also characterized for comparison. A detailed characterization of these discriminators is presented in Sec.…”