Spectral hole burning

Spectral hole burning is the frequency-selective saturation of an inhomogeneously broadened transition. When a line is made up of many narrower homogeneous lines at different center frequencies, a strong narrowband field at frequency ${\nu }_s$ saturates only the emitters whose homogeneous lines overlap ${\nu }_s$, and the gain or absorption spectrum acquires a dip at that frequency, the hole, while the rest of the line is left as it was. Probed with a weak field, the hole is $\mathrm{\Delta }{\nu }_h(1+\sqrt{1+s})$ wide, where $\mathrm{\Delta }{\nu }_h$ is the homogeneous linewidth and $s=I\mathrm{/}{I}_{\text{sat}}$ the saturation parameter of the burning field, so it is twice the homogeneous width at low intensity. Holes range from a few nanometers wide and a few tenths of a decibel deep in an erbium-doped fiber amplifier at room temperature down to a few hundred hertz in rare-earth-doped crystals at liquid-helium temperature, and in a semiconductor laser they refill within about 0.1 ps.

Homogeneous and inhomogeneous broadening

If every emitter in a medium had the same resonance frequency, a strong field anywhere inside the line would saturate all of them together and lower the gain or absorption across the whole line, the homogeneous case treated under gain saturation. In many media the emitters differ: atoms in a gas move with different velocities and are Doppler shifted by different amounts, ions in a glass sit in different local environments, and dopants in a crystal occupy slightly different sites. The observed line is then the envelope of many homogeneous lines, each with a width set by the coherence time $T_2$ of a single emitter, spread over an inhomogeneous width set by the distribution of center frequencies. A hole can be burned only where the inhomogeneous width exceeds the homogeneous one; the T2* entry describes the same distinction in the time domain.

Hole width and depth

A saturating field of intensity $I$, with $s=I\mathrm{/}{I}_{\text{sat}}$, removes population from a band of full width

$$\mathrm{\Delta }{\nu }_{\text{hole}}=\mathrm{\Delta }{\nu }_h\sqrt{1+s},$$

the homogeneous width power-broadened by the field, and at the center of that band the population falls by the fraction $s\mathrm{/}(1+s)$ of its unsaturated value. A weak probe scanned across the hole sees it convolved with the probe's own homogeneous response, a dip of width $\mathrm{\Delta }{\nu }_h(1+\sqrt{1+s})$ that tends to $2\mathrm{\Delta }{\nu }_h$ at low saturation; at $s=1$ the population hole is 1.41 homogeneous widths wide and the probed dip 2.41. Holes in the velocity distribution of a gas laser were first analyzed by Bennett and are often called Bennett holes.

Measuring the width of a weakly burned hole is one of the standard ways to find a homogeneous linewidth hidden inside an inhomogeneous envelope, alongside photon echoes, which measure $T_2$ directly. Echoes give a homogeneous linewidth of 122 Hz for europium ions in yttrium orthosilicate at liquid-helium temperature (Equall et al. 1994), so the narrowest hole a weak probe can find in that crystal is about 240 Hz wide.

Spectral vs spatial hole burning

Spectral hole burning differs from spatial hole burning in the coordinate it acts on. Spatial hole burning depletes gain where a 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 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 hole burning.

Multimode lasing and the Lamb dip

In an inhomogeneously broadened gain medium each cavity mode burns its own hole and draws on its own group of emitters, so several modes oscillate at once without competing for the same gain; Doppler-broadened gas lasers and lasers with glass hosts behave this way. A homogeneously broadened medium, such as Nd:YAG at room temperature or a semiconductor, would settle on one mode once that mode had saturated the shared gain, if spatial hole burning did not intervene (longitudinal modes). Nd:glass, whose line is broadened to tens of nanometers by the variation of sites in the glass, and Nd:YAG, whose 0.45 nm (120 GHz) line at 1064 nm is broadened homogeneously by lattice phonons, are the textbook pair.

In a standing-wave gas laser the two counter-propagating traveling waves of a mode interact with different atoms. A mode detuned from line center is resonant, through the Doppler shift, with atoms moving at $+v$ along the axis for one wave and at $-v$ for the other, so it burns two holes in the velocity distribution, placed symmetrically about zero velocity. As the mode is tuned to line center the two holes merge into one at zero velocity, both waves then draw on the same atoms, and the output power dips. This Lamb dip, explained by Lamb's theory of the gas laser and observed in a helium-neon laser by McFarlane, Bennett and Lamb in 1963, is roughly one homogeneous width wide, a small fraction of the 1.5 GHz Doppler width of the 632.8 nm neon line at a discharge temperature of 400 K. Lasers have been frequency-stabilized to its center, and saturated-absorption spectroscopy applies the same principle to an absorbing gas.

Spectral hole burning in erbium-doped fiber amplifiers

The erbium line in a silica erbium-doped fiber amplifier is mostly but not entirely homogeneous at room temperature. A strong channel burns a hole a few nanometers wide and a few tenths of a decibel deep around its own wavelength (Bolshtyansky 2003), so the gain seen by each channel depends on which other channels are present, and a gain spectrum flattened for one channel loading shows ripple under another. Cooled to 77 K, an erbium-doped fiber is strongly inhomogeneous and hole burning is pronounced (Desurvire, Zyskind and Simpson 1990).

The same amplifiers show polarization hole burning, a related saturation that selects by polarization. A polarized signal saturates most strongly the ions whose transition dipoles lie along its field, leaving slightly more gain for the orthogonal polarization, in which amplified spontaneous emission then grows (Mazurczyk and Zyskind 1994). The difference is small in one amplifier but accumulates over the many amplifiers of a long-haul link, and such links scramble the signal polarization faster than the erbium gain can respond.

Spectral hole burning in semiconductor lasers

In a semiconductor the gain at a given photon energy comes from electron-hole pairs at particular energies in the bands. Stimulated emission at the lasing energy removes those carriers faster than scattering among the carriers and with the lattice refills them, which leaves a dip in the carrier distribution at the lasing energy. The refilling takes about 0.1 ps, the intraband relaxation time, so the hole is shallow and lasts only while the field is present. The same scattering sets the homogeneous linewidth: a dephasing time of 0.1 ps corresponds to $1\mathrm{/}(\pi T_2)$ = 3.2 THz, which is 13 meV, or 26 nm at 1550 nm, comparable to the width of the gain spectrum itself, so a semiconductor gain medium behaves as nearly homogeneous.

Pump-probe experiments with femtosecond pulses on InGaAsP amplifiers show the gain recovering in stages: a fast component from spectral hole burning, a slightly slower one as the carriers heated by the pump cool, and the much slower recovery of the carrier density itself (Hall et al. 1990). The fast processes let four-wave mixing in a semiconductor optical amplifier extend to detunings far beyond the bandwidth that the carrier lifetime would allow.

Gain compression and the epsilon factor

In the rate equations of a semiconductor laser the transient hole, together with carrier heating, appears as a gain that falls with photon density at a fixed carrier density:

$$g(N,S)=\frac{g_0(N)}{1+\epsilon S},$$

where $N$ is the carrier density, $S$ the photon density and $\epsilon$ the gain compression factor, of the order of ${10}^{-17}$ cm³ in quantum-well lasers. With $\epsilon =2\times {10}^{-17}$ cm³ and a photon density of $5\times {10}^{14}$ cm⁻³, the value the gain compression factor entry estimates for a 300 µm edge-emitting laser at 10 mW, $\epsilon S$ = 0.01, a 1% compression of the gain. That small term supplies most of the damping of the relaxation oscillations and so sets the ceiling on the intrinsic modulation bandwidth. It also makes the carrier density rise slightly with power above threshold, which shifts the lasing frequency through the linewidth enhancement factor, the adiabatic part of the frequency chirp. Spectral hole burning is one of several contributions to $\epsilon$; values fitted to measured data also absorb carrier heating, the capture of carriers into the wells and spatial hole burning.

Persistent spectral hole burning

A hole that outlives the field that burned it is called persistent. In rare-earth-doped crystals at liquid-helium temperatures, ions optically pumped out of the resonant group can be shelved in another, long-lived hyperfine level of the ground state, leaving a hole at a fixed absolute frequency in a solid with no moving parts. A laser locked to a pattern of such holes in europium-doped yttrium orthosilicate reached a fractional frequency instability of $6\times {10}^{-16}$ at averaging times of 2 to 8 s, and measurements of the holes' sensitivity to environmental perturbations suggest that they can be more stable than the Fabry-Perot cavities of ultrastable lasers (Thorpe et al. 2011). Holes burned in a periodic pattern form the atomic frequency combs used as multimode quantum memories for light (Afzelius et al. 2009). In dye molecules embedded in polymers and glasses at low temperature, photochemical and photophysical changes make holes persistent as well, and they were studied for optical data storage in the frequency domain (Moerner 1988).

Common questions

What is spectral hole burning?

The selective saturation of an inhomogeneously broadened gain or absorption line by a narrowband field. Only the emitters resonant with the field are saturated, so the spectrum develops a dip at the field's frequency, about twice the homogeneous linewidth wide when probed weakly, while the rest of the line is unchanged.

What is the difference between spectral and spatial hole burning?

Spectral hole burning depletes the gain at particular frequencies and needs an inhomogeneously broadened medium; spatial hole burning depletes it at particular positions in a standing-wave field. A traveling-wave ring cavity removes spatial hole burning, and a homogeneously broadened medium removes spectral hole burning.

What is the Lamb dip?

A dip in the output power of a single-mode gas laser tuned to the center of its Doppler-broadened gain line. There the two counter-propagating waves of the mode saturate the same zero-velocity atoms instead of two separate velocity groups, so less gain is available. The dip is about one homogeneous linewidth wide.

Does spectral hole burning occur in EDFAs?

Yes, weakly at room temperature. A strong channel burns a hole a few nanometers wide and a few tenths of a decibel deep, so the gain of each channel depends on which others are present; at 77 K the effect is much stronger.

How does spectral hole burning affect semiconductor lasers?

The hole refills within about 0.1 ps, so it never persists, but while the laser runs it lowers the gain slightly in proportion to the photon density. Together with carrier heating it makes up the gain compression factor $\epsilon$, which damps the relaxation oscillations and limits the modulation bandwidth.

References: W. E. Lamb, "Theory of an optical maser," Phys. Rev. 134, A1429 (1964), for the Lamb dip; W. R. Bennett Jr., "Hole burning effects in a He-Ne optical maser," Phys. Rev. 126, 580 (1962); R. A. McFarlane, W. R. Bennett Jr. and W. E. Lamb Jr., "Single mode tuning dip in the power output of an He-Ne optical maser," Appl. Phys. Lett. 2, 189 (1963); A. E. Siegman, Lasers (University Science Books, 1986), ch. 30, on hole burning and saturation in inhomogeneous media; W. Koechner, Solid-State Laser Engineering, 6th ed. (Springer, 2006), ch. 2, for the Nd:YAG and Nd:glass linewidths; R. W. Equall, Y. Sun, R. L. Cone and R. M. Macfarlane, "Ultraslow optical dephasing in Eu³⁺:Y₂SiO₅," Phys. Rev. Lett. 72, 2179 (1994); E. Desurvire, J. L. Zyskind and J. R. Simpson, "Spectral gain hole-burning at 1.53 µm in erbium-doped fiber amplifiers," IEEE Photonics Technology Letters 2, 246 (1990); M. Bolshtyansky, "Spectral hole burning in erbium-doped fiber amplifiers," J. Lightwave Technol. 21, 1032 (2003); V. J. Mazurczyk and J. L. Zyskind, "Polarization dependent gain in erbium doped-fiber amplifiers," IEEE Photonics Technology Letters 6, 616 (1994), for polarization hole burning; L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012), chs. 4 and 5, for the intraband relaxation time and the gain compression factor; K. L. Hall, J. Mark, E. P. Ippen and G. Eisenstein, "Femtosecond gain dynamics in InGaAsP optical amplifiers," Appl. Phys. Lett. 56, 1740 (1990); 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); M. Afzelius, C. Simon, H. de Riedmatten and N. Gisin, "Multimode quantum memory based on atomic frequency combs," Phys. Rev. A 79, 052329 (2009); W. E. Moerner (ed.), Persistent Spectral Hole-Burning: Science and Applications (Springer, 1988). The Doppler width quoted is computed for neon at 400 K on the 633 nm line.