Homogeneous and inhomogeneous broadening

A spectral line is homogeneously broadened when every emitter in the medium has the same resonance frequency and the same line. It is inhomogeneously broadened when the emitters have different center frequencies and the observed line is the envelope of many narrower lines. Homogeneous mechanisms are the finite lifetime of the levels, collisions in a gas and phonon scattering in a solid; they produce a Lorentzian lineshape, with widths from the 15.9 MHz natural width of a transition with a 10 ns lifetime to the 0.45–0.5 nm (120–132 GHz) phonon-broadened gain bandwidth of Nd:YAG at 1064 nm. Inhomogeneous mechanisms are the Doppler shifts of moving atoms, site-to-site variation of the local field around ions in glass and the size distribution of quantum dots; they usually produce a Gaussian, such as the 1.5 GHz Doppler width of the 632.8 nm neon line in a helium-neon laser at a discharge temperature near 400 K.

Homogeneous width and the coherence time

The homogeneous width is set by how long a single emitter keeps a definite phase, the coherence time $T_2$. For a Lorentzian line with FWHM in hertz,

$$\Delta\nu_h = \frac{1}{\pi T_2}.$$

When only spontaneous decay of the upper level acts, $T_2 = 2T_1$ and this reduces to the natural width $1/(2\pi\tau)$: 15.9 MHz for $\tau$ = 10 ns. Collisions and phonons interrupt the phase without ending the excitation and shorten $T_2$. In a semiconductor, intraband carrier scattering gives $T_2$ of about 0.1 ps, so $\Delta\nu_h$ = 3.2 THz, comparable to the gain spectrum itself; semiconductor gain therefore behaves as nearly homogeneous. The spread of center frequencies has its time-domain counterpart in the dephasing time T2*, which is shorter than $T_2$ whenever inhomogeneous broadening is present.

Inhomogeneous width

Doppler broadening is the standard case: for neon at 632.8 nm the FWHM is 1.31 GHz at 300 K and 1.51 GHz at 400 K, growing as $\sqrt{T/m}$. In solids the inhomogeneous width comes from structural disorder: Nd ions in glass occupy sites with different crystal fields, which gives a largely inhomogeneous fluorescence width of some 20–30 nm near 1054 nm (5.4–8.1 THz), against the phonon-limited line of Nd in the ordered YAG crystal. Self-assembled quantum dots differ in size and composition, and the resulting distribution of transition energies spreads the gain of a quantum dot laser over tens of nanometers.

Saturation and laser behavior

A strong field at any frequency inside a homogeneous line reduces the gain of every emitter, so the whole gain curve falls together. With $s = I/I_\text{sat}$ at line center, the gain falls as

$$g = \frac{g_0}{1+s}$$

in a homogeneous medium and as

$$g = \frac{g_0}{\sqrt{1+s}}$$

for a field inside a strongly inhomogeneous line, as treated under gain saturation. At $s$ = 3 the first gives a quarter of the small-signal gain and the second half of it. The slower fall arises because the field depletes only the emitters within roughly a homogeneous width of its frequency, burning a spectral hole and leaving the rest of the line untouched.

Laser mode behavior follows. In an inhomogeneous medium each longitudinal mode draws on its own group of emitters, so several modes oscillate without competing; a helium-neon laser with several modes under its Doppler width and Nd:glass lasers run this way. In a homogeneous medium the mode with the highest net gain clamps the gain for all, which in principle favors a single mode; standing-wave lasers such as the Nd:YAG laser still run on several modes because of spatial hole burning, which a unidirectional ring cavity removes.

Measurement

A single spectrum cannot separate the two contributions. The homogeneous width is found by methods that select a subset of emitters: burning a weak spectral hole and measuring its width (about $2\Delta\nu_h$), saturated-absorption spectroscopy in gases, photon echoes, which measure $T_2$ directly, or spectroscopy of single emitters, such as one quantum dot, whose line is far narrower than the ensemble's. Temperature dependence is also diagnostic: phonon broadening narrows on cooling while static disorder does not.

Pitfalls

The classification depends on temperature and time scale. The erbium line in silica, used in the erbium-doped fiber amplifier, is mostly homogeneous at room temperature, and a strong channel burns a hole only a few tenths of a decibel deep; at 77 K the same fiber is strongly inhomogeneous. Emitters whose frequencies change faster than the measurement, through spectral diffusion or fast carrier scattering, average their differences and act homogeneously.

Common questions

What is the difference between homogeneous and inhomogeneous broadening?

In homogeneous broadening every emitter has the same line, of width set by its coherence time; in inhomogeneous broadening the emitters have different center frequencies. The first gives a Lorentzian that saturates uniformly, the second usually a Gaussian that saturates locally.

Is Doppler broadening homogeneous or inhomogeneous?

Inhomogeneous. Each atom has a definite velocity and so a definite Doppler shift, and a narrowband laser interacts only with the atoms in the matching velocity group.

Why does inhomogeneous broadening allow multimode lasing?

Each mode saturates only the emitters near its own frequency, so modes separated by more than about a homogeneous width draw on different emitters and do not suppress one another.

References: A. E. Siegman, Lasers (University Science Books, 1986); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); O. Svelto, Principles of Lasers, 5th ed. (Springer, 2010); W. Demtröder, Laser Spectroscopy 1: Basic Principles, 5th ed. (Springer, 2014).