Frequency fluctuations cause the optical field of a real laser to deviate from a pure sine wave. Thus the laser is not purely monochromatic, its power spectrum showing a broadened lineshape rather than a Dirac delta function. The lineshape and linewidth depend on the spectral properties of the frequency fluctuations. This Demonstration calculates the lineshape of a laser for two distinct types of frequency noise.
In the first case (low-pass filter), the frequency noise level is a constant below a cutoff frequency but zero above this threshold. When , the lineshape is Gaussian and the linewidth increases as . When , the lineshape becomes Lorentzian and the linewidth is independent of .
In the second case (high-pass filter) the frequency noise level is a constant above a cutoff frequency but zero below this threshold. When , the lineshape is Lorentzian and the linewidth is given by . When increases and approaches , a sharp peak appears at the center of the lineshape, and the other spectral components are repelled from the center, forming sidebands beyond . When , the sidebands are strongly suppressed and the lineshape is reduced to the central peak whose linewidth is limited by the sampling rate of the discrete Fourier transform.
These observations lead us to separate the frequency noise spectrum into two regions where the effect of noise on the lineshape is radically different: 1) the slow modulation area (left side of the red line), where the noise contributes to the laser linewidth, with a Gaussian shape, and 2) the fast modulation area (right side of the red line), where the noise contributes only to the wings of the line (sidebands) and not to the linewidth, thus transforming the lineshape from Gaussian to Lorentzian.
The calculation of the laser lineshape from the frequency noise spectral density is described in D. S. Elliott, R. Roy, and S. J. Smith, "Extracavity Laser Band-Shape and Bandwidth Modification," Phys. Rev. A, 26, 1982 pp. 12–18. First, one calculates the autocorrelation function of the optical field:
Then the lineshape is given by the Fourier transform of the autocorrelation function:
In the case of low-pass filtered white noise, that is, for a frequency noise spectral density given by
the first integral is easy to evaluate analytically and one obtains the following expression for the autocorrelation function:
where is the sine integral.
In the case of high-pass filtered white noise, the frequency noise spectral density is given by
and one obtains the following autocorrelation function:
However, it is not possible to obtain an analytical expression for the Fourier transform of these autocorrelation functions. Therefore, in both cases, the laser lineshape is evaluated numerically using the discrete Fourier transform.
The red line, separating the frequency noise spectrum into two regions in which the effect of noise on the lineshape is radically different, can be determined by studying the asymptotic behavior of the two limiting cases and . It is given by the following equation: