Lineshape (Lorentzian, Gaussian and Voigt)

The lineshape $g(\nu)$ of a transition describes how its emission, absorption or gain is distributed in frequency around the line center $\nu_0$. It is normalized so that $\int g(\nu)\,d\nu = 1$, and it is characterized by its full width at half maximum (FWHM) $\Delta\nu$. Two shapes cover most cases. Broadening that acts identically on every emitter, such as the finite lifetime of the upper level or collisions, is homogeneous and produces a Lorentzian; a transition with a 10 ns lifetime has a natural Lorentzian width of 15.9 MHz. Broadening that gives different emitters different center frequencies, such as the Doppler shifts of moving atoms, is inhomogeneous and usually produces a Gaussian; the neon line of the helium-neon laser at 632.8 nm is about 1.5 GHz wide at a discharge temperature near 400 K. When both are present the line is a Voigt profile.

Lorentzian: homogeneous broadening

An excited state that decays with lifetime $\tau$ radiates a damped oscillation, whose spectrum is the Lorentzian

$$g_L(\nu) = \frac{\Delta\nu/2\pi}{(\nu-\nu_0)^2 + (\Delta\nu/2)^2},$$

with peak value $2/(\pi\Delta\nu) = 0.637/\Delta\nu$. For decay by spontaneous emission, the natural width is

$$\Delta\nu = \frac{1}{2\pi\tau},$$

so $\tau$ = 10 ns, typical of a strong visible atomic transition, gives 15.9 MHz. If both levels decay, their rates add. Collisions that interrupt the phase of the oscillation also add Lorentzian width in proportion to pressure. For small molecules in air, collisional half widths are typically 0.05–0.1 cm⁻¹ per atmosphere; 0.07 cm⁻¹ is 2.1 GHz, a FWHM of 4.2 GHz at 1 atm, so gas lines measured by TDLAS at atmospheric pressure are mostly Lorentzian. Phonon interactions broaden crystal lines homogeneously: Nd:YAG has a gain linewidth of about 0.5 nm at 1064 nm, 132 GHz.

Gaussian: inhomogeneous and Doppler broadening

An atom moving with velocity component $v$ along the line of sight is shifted by $\nu_0 v/c$. A Maxwell–Boltzmann velocity distribution therefore produces a Gaussian

$$g_G(\nu) = \frac{2}{\Delta\nu_D}\sqrt{\frac{\ln 2}{\pi}}$$ $$\times\,\exp\!\left[-\frac{4\ln 2\,(\nu-\nu_0)^2}{\Delta\nu_D^2}\right],$$

with peak value $0.939/\Delta\nu_D$ and Doppler FWHM

$$\Delta\nu_D = \frac{\nu_0}{c}\sqrt{\frac{8 k_B T \ln 2}{m}}.$$

For neon, $m$ = 20.18 u, at $\nu_0$ = 473.8 THz (632.8 nm), this gives 1.31 GHz at 300 K and 1.51 GHz at 400 K, the value quoted for a helium-neon discharge. Site-to-site variation of the crystal field around rare-earth ions in glass is also inhomogeneous; Nd-doped glass has a largely inhomogeneous fluorescence width of some 20–30 nm.

Voigt profile

Emitters with Lorentzian lines spread over a Gaussian distribution of center frequencies give the convolution of the two, the Voigt profile. Its FWHM follows from the approximation of Olivero and Longbothum, accurate to about 0.02%,

$$\Delta\nu_V \approx 0.5346\,\Delta\nu_L + \sqrt{0.2166\,\Delta\nu_L^2 + \Delta\nu_G^2}.$$

A 1.5 GHz Gaussian with a 100 MHz Lorentzian gives 1.554 GHz, matching a numerical convolution. Methane at 1653.7 nm and 296 K has a Doppler width of 0.56 GHz; combined with a 4.2 GHz collisional width, the Voigt width is 4.28 GHz.

Wings and areas

The two shapes differ most away from the center. At two half widths from line center a Lorentzian is still at 20% of its peak and a Gaussian of the same FWHM at 6.25%; at three half widths the values are 10% and 0.20%, and at ten half widths 1.0% and effectively zero. Absorption far from a line is therefore governed by the Lorentzian component even when the core of the line is Gaussian. The area equals peak × FWHM × 1.571 for a Lorentzian and × 1.064 for a Gaussian, which matters when converting a peak into a line strength.

Laser lineshapes

A free-running single-frequency laser limited only by white frequency noise has a Lorentzian line whose width is the Schawlow–Townes linewidth. Slow frequency wander from 1/f noise and drift adds a roughly Gaussian component that grows with observation time, so a measured laser linewidth is often Voigt-like. In a delayed self-heterodyne measurement the beat spectrum is the self-convolution of the line: twice the optical width for a Lorentzian and $\sqrt{2}$ times for a Gaussian. The shape also sets the relation between linewidth and coherence time: $\tau_c = 0.318/\Delta\nu$ for a Lorentzian and $0.664/\Delta\nu$ for a Gaussian (temporal coherence).

Lineshape and gain

The gain spectrum of a laser medium follows its lineshape, which therefore fixes the gain bandwidth. A homogeneous line saturates uniformly, so one lasing mode can suppress the others; an inhomogeneous line saturates locally through spectral hole burning and supports several modes at once.

Common questions

What is the difference between a Lorentzian and a Gaussian lineshape?

A Lorentzian comes from homogeneous broadening and has wings that fall as $1/(\nu-\nu_0)^2$; a Gaussian comes from inhomogeneous broadening, usually Doppler, and its wings fall off exponentially. For the same FWHM the Gaussian has the higher peak.

When should a Voigt profile be used?

When the Lorentzian and Gaussian widths are within an order of magnitude of each other, as for gas lines at intermediate pressure or lasers with both white and 1/f frequency noise. If one width exceeds the other tenfold, the Voigt FWHM is within 6% of the larger one, though the wings still follow the Lorentzian.

References: A. E. Siegman, Lasers (University Science Books, 1986); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); W. Demtröder, Laser Spectroscopy 1: Basic Principles, 5th ed. (Springer, 2014); J. J. Olivero and R. L. Longbothum, "Empirical fits to the Voigt line width: a brief review," Journal of Quantitative Spectroscopy and Radiative Transfer 17, 233 (1977).