Spectral hole burning

Spectral hole burning is what saturation looks like when a transition is inhomogeneously broadened. If every emitter in a medium had the same resonance frequency, saturating the transition with a strong field at any frequency inside the line would reduce the gain or absorption across the whole line together. If instead the line is a sum of many narrower homogeneous lines at different center frequencies, because the emitters sit in different local environments, move with different velocities, or occupy different lattice sites, then a strong field at frequency $\nu_s$ saturates only the emitters whose homogeneous line overlaps $\nu_s$, and the spectrum acquires a dip at that frequency with the rest of the line untouched. The dip is the hole. The T2* entry describes the same distinction in the time domain: the hole exists because the inhomogeneous width exceeds the homogeneous one.

The hole has a width and a depth set by the homogeneous line and the saturation. A saturating field of intensity $s = I/I_\mathrm{sat}$ burns a hole in the population whose full width is $\Delta\nu_h \sqrt{1 + s}$, with $\Delta\nu_h$ the homogeneous linewidth; a weak probe scanned across it sees a dip of width $\Delta\nu_h (1 + \sqrt{1 + s})$, which is twice the homogeneous width at low saturation, because the probe's own homogeneous response convolves with the hole. The depth scales as $s/(1 + s)$ relative to the unsaturated value. Measuring the hole width is therefore one of the standard ways to measure a homogeneous linewidth hidden inside an inhomogeneous envelope, alongside photon echoes; the sub-kilohertz widths of rare-earth ions in crystals, such as the europium line quoted in the T2* entry, have been measured both ways.

The distinction from spatial hole burning is more than initials. Spatial hole burning depletes gain where the standing-wave field is strong and leaves it where the field has nodes, at fixed positions along the cavity; spectral hole burning depletes gain at fixed frequencies regardless of position. Both let a second mode find unsaturated gain and both therefore undermine single-mode operation, but they do it in different coordinates and they respond to different cures. A ring cavity with a traveling wave removes spatial hole burning and does nothing about spectral hole burning; a homogeneously broadened gain medium removes spectral hole burning and does nothing about spatial.

The consequences run through laser physics. In an inhomogeneously broadened gain medium each cavity mode burns its own hole and lases on its own sub-group of emitters, so the laser runs multimode by default: a Doppler-broadened gas or a glass host does this, while a homogeneously broadened crystal or semiconductor tends to single-mode operation once one mode has saturated the shared gain. Nd:glass, with an inhomogeneous width of some 25 nm, and Nd:YAG, with a homogeneous width near 0.5 nm, are the textbook pair. In a standing-wave gas laser the two counter-propagating waves burn two holes in the velocity distribution, at velocities $\pm v$ for a mode detuned from line center, and the holes merge into one at line center where both waves address the same zero-velocity atoms; the resulting dip in output power at the center of the tuning curve is the Lamb dip, roughly one homogeneous width wide, which for a helium-neon laser means of order 100 MHz inside a 1.5 GHz Doppler profile and was the first sub-Doppler frequency reference. In an erbium-doped fiber amplifier the erbium line at room temperature is mostly but not entirely homogeneous, and a strong channel burns a hole a few nanometers wide and a few tenths of a decibel deep around itself, so the gain seen by neighboring channels depends on which other channels are present; at 77 K the same fiber is strongly inhomogeneous and the holes are deep. In a semiconductor the intraband scattering that refills a hole runs in 50 to 100 fs, so a hole exists only transiently, but that transient depletion at the lasing energy is one of the contributions to the nonlinear gain compression that damps relaxation oscillations and limits modulation bandwidth.

A hole that outlives the field that burned it is persistent spectral hole burning, and it turns the effect from a nuisance into a device. In rare-earth-doped crystals at liquid-helium temperatures, ions pumped out of the resonant sub-group can be shelved in a hyperfine level for seconds to days, leaving a hole hundreds of hertz wide at a fixed absolute frequency in a material with no moving parts. A laser locked to such a hole reached a fractional frequency instability of $6 \times 10^{-16}$ (Thorpe et al., 2011), competitive with the best optical cavities and without their thermal-noise floor; the same holes, burned in patterns, are the atomic frequency combs used as quantum memories for light. Both are applications of the fact that the homogeneous width is the resource and the inhomogeneous width is the address space.

References: W. E. Lamb, "Theory of an optical maser," Phys. Rev. 134, A1429 (1964), for the Lamb dip; A. E. Siegman, Lasers (University Science Books, 1986), ch. 30, on hole burning and saturation in inhomogeneous media; M. Bolshtyansky, "Spectral hole burning in erbium-doped fiber amplifiers," J. Lightwave Technol. 21, 1032 (2003); M. J. Thorpe, L. Rippe, T. M. Fortier, M. S. Kirchner and T. Rosenband, "Frequency stabilization to 6 × 10⁻¹⁶ via spectral-hole burning," Nature Photonics 5, 688 (2011). The Doppler width quoted is for neon at 400 K on the 633 nm line.