Spatial Hole Burning and Mode Competition

A single laser mode in a linear cavity is a standing wave, and a standing wave has a texture: bright antinodes every half wavelength separated by dark nodes where the field vanishes. Stimulated emission — the process this whole chapter is built on — depletes the population inversion locally according to the saturation parameter, so the mode consumes more gain near its antinodes and leaves substantially more unsaturated inversion near its nodes. The inversion becomes spatially corrugated, with a pattern of strongly saturated and weakly saturated gain that follows the standing-wave intensity and therefore has spatial period $\lambda/2$ . This is spatial hole burning, and it overturns one of the cleanest predictions in laser physics. That prediction comes from the rate equations for a homogeneously broadened medium: because every atom responds to every frequency, the strongest mode should saturate the gain for all of them and win a winner-take-all contest. Spatial hole burning breaks the contest by making saturation local. The pockets of fresh inversion left at the first mode's nodes are exactly where a second longitudinal mode — shifted a quarter wavelength so its antinodes land there — finds unclaimed gain, reaches threshold, and coexists, so a laser that "should" be single-frequency runs multimode instead. The effect is strongest where the atoms cannot move to average the grating away — solid-state crystals like Nd:YAG and ruby, semiconductor diodes — and it is the reason single-frequency lasers are built as travelling-wave rings or twisted-mode cavities.

Key takeaways

A standing-wave mode has intensity antinodes and nodes; it saturates the inversion strongly at antinodes and barely at nodes, burning a spatial hole of period λ/2 into the gain.
The unsaturated inversion at the nodes is available to a second longitudinal mode whose antinodes fall there, undermining the single-mode selection that uniform saturation would otherwise enforce.
It is strongest when excited atoms or carriers cannot diffuse appreciably during the population-relaxation time, as can occur in solid-state and semiconductor lasers. It is weakened either by eliminating the standing wave — for example with a ring or traveling-wave cavity — or by diffusion when the diffusion length becomes comparable to $\lambda/2$ .
Mode competition for the shared inversion is governed by coupled rate equations with self- and cross-saturation coefficients; in a standing-wave cavity, spatial hole burning can reduce the effective cross-saturation between longitudinal modes relative to self-saturation.
Key idea: a standing-wave field saturates the gain periodically in space with period λ/2, leaving unsaturated inversion at its nodes that other longitudinal modes — whose antinodes fall there — can extract, so this spatial gain grating turns an ideally single-mode homogeneous laser into a multimode one unless the standing wave is eliminated.

The core insight

A standing-wave field saturates the gain periodically in space with period λ/2, leaving unsaturated inversion at its nodes that other longitudinal modes — whose antinodes fall there — can extract; this spatial gain grating converts an ideally single-mode homogeneous laser into a multimode one unless the standing wave is removed.

Where this lives today

Spatial hole burning is the reason single-frequency lasers look the way they do. The workhorse example is the non-planar ring oscillator, or NPRO: a monolithic Nd:YAG resonator whose non-planar geometry, together with magneto-optic Faraday rotation, provides unidirectional operation. With predominantly travelling-wave operation and no standing-wave intensity nodes, the inversion is much more uniformly saturated and spatial hole burning is strongly suppressed, enabling stable single-frequency operation with very narrow linewidth. NPROs seed the injection chains of the LIGO gravitational-wave interferometers, where a hop between longitudinal modes would inject phase noise into a measurement sensitive to a fraction of a proton's width — spatial-hole-burning control is, literally, a precondition for detecting merging black holes. The same physics runs the other way in semiconductor and fibre lasers. In a distributed-feedback (DFB) diode laser the carrier density plays the role of the inversion, and the standing wave burns a carrier grating that reshapes the modal gain and — through the linewidth-enhancement factor — the index, degrading side-mode suppression unless a quarter-wave phase shift is engineered in. In high-power fibre lasers it limits how cleanly a single frequency can be extracted. Wherever a laser must hold one mode against a gain medium that cannot diffuse, the engineer is fighting this hole.

The physical picture

Think of the gain medium as a lawn and the standing-wave mode as a mower whose blades only touch the ground at the antinodes. It cuts stripes spaced λ/2 apart and leaves tall grass at the nodes between them. Bring a second mower whose blades are offset by a quarter wavelength and it finds fresh grass exactly where the first left it standing, so the two run side by side without fighting over the same turf. That offset — antinodes of one mode landing on nodes of the other — is the whole mechanism by which two modes coexist in a medium that "should" support only one. The contrast with uniform saturation is worth stating sharply. Homogeneous gain clamping is global taxation: every photon lowers the gain seen by every mode equally, so the richest mode drains the common account and bankrupts its rivals. Spatial hole burning makes the taxation local: a mode depletes gain only where its own intensity is high, so two modes in different spatial pockets pay into different accounts and both survive — the statement $\beta_{mn} < \beta_{mm}$ in the rate equations.

Figure

A standing-wave mode in blue burns a periodic hole in the inversion in green: at the antinodes where intensity peaks the inversion drops to about N₀/4, while at the nodes it stays near N₀. For Nd:YAG at λ₀ = 1064 nm and n = 1.82 the grating period is λ/2 = 292 nm

The corrugated inversion is a real, physical grating: the modulation of the population is also a modulation of gain and, through the same atomic response, of refractive index, with period λ/2 — the pitch that back-scatters the intracavity field. Whether it matters turns on one question: can the excited atoms hold still? If they diffuse a distance comparable to λ/2 within their lifetime they smear the corrugation flat and restore the single-mode picture; if they cannot move — as in a doped crystal — the grating is frozen and spatial hole burning is at full strength.

Figure

A second longitudinal mode drawn with amber antinodes is shifted a quarter wavelength so its bright spots land on the first mode's nodes, precisely where the green inversion is left unsaturated. In the weak-saturation limit that leftover gain gives the second mode about 5 percent excess gain over the clamped first mode at I₀/I_s = 0.2

The core equations

The starting point is the field itself. A single mode reflecting between two mirrors is a standing wave, and its intensity — what the atoms see and respond to — is the square of that field, modulated at twice its spatial frequency.

E(z,t)=E_0\,\sin(kz)\,\cos(\omega t),\qquad I(z)=I_0\,\sin^2(kz) \qquad (1)

where: $k=2\pi/\lambda$ — wavenumber inside the medium; $I_0$ — peak (antinode) intensity; $z$ — position along the cavity axis; $\lambda$ — wavelength in the medium. Because $\sin^2(kz)$ repeats every $\pi/k$ , the intensity period is λ/2, not a full wavelength: the atoms see a comb of bright fringes, not a uniform flood. The medium responds by saturating: the more intense the local field, the more the inversion is driven down. Steady-state saturation converts the sinusoidal intensity into a modulated inversion — the burned hole, obtained from the rate equation in Derivation A.

N(z)=\frac{N_0}{1+I(z)/I_s}=\frac{N_0}{1+(I_0/I_s)\,\sin^2(kz)} \qquad (2)

where: $N_0$ — small-signal (unsaturated) inversion density; $N(z)$ — local saturated inversion, smallest at the antinodes and near $N_0$ at the nodes; $I_s=\hbar\omega/(\sigma c\tau)$ — saturation intensity. Physical reading: a corrugated reservoir whose troughs sit under the antinodes. A mode does not sample the inversion uniformly: because stimulated emission is proportional to intensity, it feels the inversion weighted by its own intensity profile — so the modal gain is an overlap integral.

g_{\mathrm{mod}}=\frac{\displaystyle\int_0^L \sigma\,N(z)\,I(z)\,\mathrm{d}z}{\displaystyle\int_0^L I(z)\,\mathrm{d}z} \qquad (3)

where: $\sigma$ — stimulated-emission cross section; $L$ — length of the gain medium; $g_{\mathrm{mod}}$ — net gain the mode experiences, weighted by where it is bright. Physical reading: a mode is rewarded for having its antinodes where the inversion is high — why a mode offset from the saturating one can win extra gain. When several modes oscillate together they share one inversion, and their competition is captured by coupled rate equations with self- and cross-saturation coefficients — with the cross terms weaker than the self terms because different modes bleach different pockets.

\frac{\mathrm{d}S_m}{\mathrm{d}t}=\Big(g_m-\frac{1}{\tau_c}\Big)S_m,\qquad g_m=\frac{g_m^{0}}{1+\beta_{mm}S_m+\sum_{n\ne m}\beta_{mn}S_n} \qquad (4)

where: $S_m$ — photon number in mode $m$ ; $g_m^{0}$ — small-signal gain of mode $m$ ; $\tau_c$ — cavity photon lifetime, which sets the loss and hence threshold $1/\tau_c$ ; $\beta_{mm}$ — self-saturation coefficient; $\beta_{mn}$ — cross-saturation coefficient, set by the spatial overlap $\int u_m^2 u_n^2\,\mathrm{d}z$ of the two profiles. Physical reading: spatial hole burning makes $\beta_{mn} < \beta_{mm}$ , so stable two-mode operation follows whenever $\beta_{mm}\beta_{nn} > \beta_{mn}\beta_{nm}$ ; uniform saturation gives $\beta_{mn}=\beta_{mm}$ and one mode wins alone. Finally, whether the grating survives is a race between the upper-level lifetime and how far an excitation can wander — the diffusion length against λ/2.

L_{\mathrm{diff}}=\sqrt{D\,\tau} \qquad (5)

where: $D$ — diffusion coefficient of the excited atoms or carriers; $\tau$ — upper-level lifetime; $L_{\mathrm{diff}}$ — distance an excitation moves before it decays. Physical reading: if $L_{\mathrm{diff}}\gtrsim\lambda/2$ the excitation samples both node and antinode within its lifetime and the grating washes out; if $L_{\mathrm{diff}}\ll\lambda/2$ it is frozen and hole burning bites hard.

Derivation ASpatially saturated inversion from the rate equation

The goal is to obtain the corrugated inversion of equation (2) directly from the population rate equation, so the burned hole is derived rather than asserted and the saturation intensity acquires an explicit microscopic form in terms of the cross section and lifetime. We balance pumping against spontaneous and stimulated loss at steady state, keeping the stimulated rate position-dependent, then read off the constants.

Step 1. Write the steady-state balance for the inversion density at a point $z$ : pumping fills the upper level at rate $R$ , spontaneous processes empty it at $N/\tau$ , and stimulated emission empties it at a rate proportional to the local intensity, $(\sigma c/\hbar\omega)\,I(z)\,N$ . Setting $\mathrm{d}N/\mathrm{d}t = R - N/\tau - (\sigma c/\hbar\omega)\,I(z)\,N = 0$ fixes the local inversion. Why: at steady state the reservoir neither fills nor empties, so the three rates must cancel point by point — and only the stimulated term carries the standing-wave structure.

Step 2. Solve for $N = R\tau/[1 + \sigma c\tau I(z)/\hbar\omega]$ and name the two constants inside: the small-signal inversion $N_0 = R\tau$ that the pump alone sustains, and the saturation intensity $I_s = \hbar\omega/(\sigma c\tau)$ at which stimulated and spontaneous rates match. Why: naming $N_0$ and $I_s$ turns a cluttered fraction into the universal saturation form $N_0/(1+I/I_s)$ that every gain medium obeys.

Step 3. Insert the standing wave $I(z)=I_0\sin^2(kz)$ to obtain $N(z) = N_0/[1+(I_0/I_s)\sin^2(kz)]$ : the inversion falls to $N_0/(1+I_0/I_s)$ under each antinode and rises back to $N_0$ at each node. Why: this is the burned hole — the same saturation law that gives uniform bleaching under a plane wave gives a spatial grating under a standing wave, because the driving intensity is itself periodic.

Go deeper · Derivation B · 1 minWhy a homogeneous laser still runs multimode

Derivation BWhy a homogeneous laser still runs multimode

The goal is to show that even a perfectly homogeneously broadened medium can support more than one longitudinal mode when spatial hole burning reduces cross-saturation. We let mode 1 saturate the inversion, compute the gain mode 2 sees from the overlap of that corrugated inversion with its own intensity profile, and determine whether this modal gain can remain above the cavity-loss threshold because mode 2 samples regions where mode 1 left more inversion unsaturated.

Step 1. Let mode 1 reach threshold first and saturate the medium, so the inversion is $N(z)=N_0/[1+(I_1/I_s)\sin^2(k_1 z)]$ and its own gain is clamped at the threshold $g_{\mathrm{th}}$ set by the cavity loss — mode 1 can grow no further. Why: gain clamping is the standard single-mode result and is correct for mode 1 itself; the question is whether it forecloses everything else.

Step 2. Compute the gain seen by a second mode with $k_2 = k_1+\Delta k$ , chosen so its antinodes fall on mode 1's nodes: the inversion averaged over its profile, $g_2^{\mathrm{eff}}=\sigma\int_0^L N(z)\sin^2(k_2 z)\,\mathrm{d}z\big/\int_0^L\sin^2(k_2 z)\,\mathrm{d}z$ . Why: a mode feels gain weighted by where it is bright, per equation (3), so mode 2's weighting emphasizes exactly the regions mode 1 ignored.

Step 3. Where $\sin^2(k_1 z)\to 0$ — at mode 1's nodes — the inversion returns to its unsaturated value $N_0$ . Because $k_2\neq k_1$ , mode 2 has a different spatial intensity pattern and can sample some of these less-saturated regions. Its average gain can therefore exceed the gain obtained by a mode with the same spatial pattern as mode 1. Why: the key effect is reduced spatial overlap between the two intensity patterns — mode 2 can exploit gain that mode 1 depletes less strongly.

Step 4. Check the limit $k_2=k_1$ (perfect spatial overlap): the two modal gains coincide, $g_2^{\mathrm{eff}}=g_1^{\mathrm{eff}}$ , and the spatial-hole-burning advantage disappears. In this limit, self- and cross-saturation become equal, so the usual homogeneous mode competition is recovered. Why: this confirms that the multimode advantage comes from the spatially varying inversion produced by the standing wave, not from inhomogeneous broadening of the medium.

Go deeper · Derivation C · 1 minThe population grating in the weak-saturation limit

Derivation CThe population grating in the weak-saturation limit

The goal is to expose the spatial grating hidden inside the saturated inversion by expanding it for weak drive, revealing a component that oscillates at twice the field wavenumber. This $2k$ modulation back-scatters the intracavity field and, through the atomic response, imprints a matching index grating — the origin of the coupling between counter-propagating waves and of frequency pulling in semiconductor lasers.

Step 1. Take the weak-saturation limit $I_0\ll I_s$ so that $x=(I_0/I_s)\sin^2(kz)\ll 1$ , and expand the denominator with $1/(1+x)\approx 1-x$ to get $N(z)\approx N_0[1-(I_0/I_s)\sin^2(kz)]$ . Why: below saturation the hole is shallow, and the leading correction already carries the full spatial structure without the clutter of the exact fraction.

Step 2. Apply $\sin^2(kz)=\tfrac12[1-\cos(2kz)]$ to get $N(z)\approx N_0[1-\tfrac{I_0}{2I_s}+\tfrac{I_0}{2I_s}\cos(2kz)]$ — a uniform DC bleaching plus a term oscillating at wavevector $2k$ . Why: separating the flat background from the modulation makes the grating explicit, and $2k$ corresponds to a period π/k = λ/2.

Step 3. Read off the physics: the $\cos(2kz)$ term is a population grating of period λ/2, and since gain and index both track the population, it is at once a gain and index grating with amplitude $\propto I_0/I_s$ . Why: a periodic index of pitch λ/2 is a Bragg grating for the intracavity field, coupling the two travelling-wave components and seeding gain for neighbouring modes — the microscopic engine of mode competition.

Limiting cases

Case	Result	Meaning
Immobile atoms (solid-state, semiconductor)	$L_{\mathrm{diff}}\ll\lambda/2$	Grating frozen; spatial hole burning strong and multimode oscillation favoured. Nd:YAG, ruby, diode lasers.
Fast diffusion (motional averaging)	$L_{\mathrm{diff}}\gg\lambda/2$	Excitations sample nodes and antinodes within a lifetime; grating washes out, restoring uniform saturation and single-mode tendency.
Travelling-wave (ring) cavity	$I(z)=\text{const}$	No standing wave, no nodes; saturation is uniform and homogeneous competition gives single-frequency output. The design cure.
Twisted-mode linear cavity	$\lvert E(z)\rvert^2=\text{const}$	Orthogonal circular polarizations sum to a flat total intensity; the medium sees no grating even inside a linear resonator.
Maximal anti-overlap	$\beta_{mn}\to\text{min}$	Two modes with nodes on each other's antinodes couple most weakly and coexist most easily.

Common trapHomogeneous broadening does not guarantee single-mode oscillation

The tidy conclusion that a homogeneously broadened laser must run single-mode rests on a hidden assumption: that gain saturation is spatially uniform. In a standing-wave cavity it is not — the saturation follows the $\sin^2(kz)$ intensity, so the inversion becomes a grating and the usual homogeneous-mode competition is weakened; under suitable conditions, a homogeneously broadened laser can therefore support several longitudinal modes. A second common confusion is to conflate spatial hole burning with spectral hole burning: the former is a position-space effect from standing waves that occurs even in a homogeneous medium, while the latter is a frequency-space effect requiring an inhomogeneously broadened line. Close analogues, physically distinct.

Worked examples

1 · The grating pitch in Nd:YAG, and is it frozen? (confidence builder)

A Nd:YAG laser oscillates at vacuum wavelength λ₀ = 1064 nm in a crystal of index n = 1.82, upper-level lifetime τ ≈ 230 μs. Find the period of the burned inversion grating, and decide whether diffusion can wash it out.

Step 1. The grating follows the intensity $\sin^2(kz)$ with $k=2\pi n/\lambda_0$ , so its period is half the wavelength in the medium, not in vacuum: $\lambda = \lambda_0/n = 1064/1.82 = 584.6$ nm, giving a pitch $\Lambda = \lambda/2 = 292.3$ nm — about three thousand fringes per millimetre.

Step 2. Can the excited Nd³⁺ ions bridge a fringe within their lifetime? They are locked into fixed lattice sites, so their diffusion coefficient is effectively zero: $L_{\mathrm{diff}}=\sqrt{D\tau}\approx 0 \ll 292$ nm.

Step 3. Because $L_{\rm diff}\ll\Lambda$ , the grating is effectively frozen and spatial hole burning can remain strong. This favors multimode operation by reducing cross-saturation between longitudinal modes, although the actual number of oscillating modes also depends on the gain profile, cavity losses, and pump level. The grating period is $\Lambda=292$ nm, with $L_{\rm diff}\ll\Lambda$ .

2 · How much excess gain does the second mode get?

In the weak-saturation limit, estimate the fractional gain advantage a second longitudinal mode enjoys over the clamped first mode at a peak-to-saturation ratio I₀/I_s = 0.2, assuming the two modes are spatially incommensurate.

Step 1. Use the shallow-hole inversion $N(z)=N_0[1-a+a\cos(2k_1 z)]$ with $a=I_0/(2I_s)$ . Mode 1's modal gain is obtained by averaging the inversion over its own intensity profile, $\sin^2(k_1z)=\frac12[1-\cos(2k_1z)]$ . Because the $\cos(2k_1z)$ inversion grating is correlated with this weighting, the average gives $g_1/g_0=1-\frac32a$ .

Step 2. Mode 2 is weighted by $\sin^2(k_2 z)$ with $k_2\ne k_1$ , so its overlap with the $\cos(2k_1 z)$ grating averages to zero and only the DC bleaching survives: $g_2/g_0 = 1 - a$ . It is penalized by the uniform part of the saturation, not by the grating mode 1 built for itself.

Step 3. The advantage is $g_2/g_0 - g_1/g_0 = (1-a)-(1-\tfrac32 a) = \tfrac12 a = I_0/(4I_s)$ . With $I_0/I_s = 0.2$ this is $0.05$ , and it scales linearly with drive. The second mode sees about 5 percent more gain than the clamped first mode — enough to clear a threshold set by a few-percent round-trip loss, so it oscillates.

3 · Coexistence test in the coupled-mode picture

Two longitudinal modes have equal self-saturation $β_{mm} = β_{nn} = 1.0$ . Compare cross terms $β_{mn} = β_{nm} = 0.6$ (spatial hole burning) against $β_{mn} = β_{nm} = 1.0$ (uniform saturation). Which supports stable two-mode operation?

Step 1. Stable coexistence requires weak coupling, $\beta_{mm}\beta_{nn} > \beta_{mn}\beta_{nm}$ — the self terms must beat the product of the cross terms.

Step 2. Hole-burning case: $\beta_{mm}\beta_{nn}=1.0$ versus $\beta_{mn}\beta_{nm}=0.36$ . Since $1.0 > 0.36$ the inequality holds with room to spare and the modes coexist stably; the reduced cross terms are the signature of modes bleaching different spatial pockets.

Step 3. Uniform case: $\beta_{mn}\beta_{nm}=1.0$ , equal to $\beta_{mm}\beta_{nn}$ . The coupling is exactly critical, so the two-mode solution is marginal and any asymmetry hands the laser to whichever mode leads. Hole burning ( $\beta_{mn} = 0.6$ ) permits stable two-mode operation; uniform saturation ( $\beta_{mn}$ = 1.0) forces single-mode.

Physics of the portrait

Spatial hole burning sits at the junction of the standing-wave resonator, the saturable gain medium, and the semiclassical theory that tied fields and populations together — the last welded into a working theory of mode competition by Willis Lamb.

Origins · 1 minWillis E. Lamb Jr. and how this was discovered

Willis E. Lamb Jr.

1913–2008 · semiclassical laser theory and mode competition

Willis Eugene Lamb Jr. is best remembered for the Lamb shift, the tiny splitting of hydrogen levels he measured in 1947 that launched modern quantum electrodynamics and won him the 1955 Nobel Prize. But his second act shaped laser physics as deeply as his first shaped QED. In his 1964 paper "Theory of an Optical Maser" he built the first thorough semiclassical theory of the laser: atoms treated quantum-mechanically as a saturable medium, the field treated classically as a set of cavity modes, the two coupled self-consistently. Out of that framework fell the machinery this article uses — saturated gain, mode pulling of frequencies, and crucially the self-saturation and cross-saturation coefficients that decide whether two modes compete or coexist. His theory also predicted the Lamb dip, the narrow drop in output power when a Doppler-broadened laser is tuned to line centre and a standing-wave mode burns two merging velocity holes — a spectral cousin of the spatial effect here. The spatial hole itself was pinned down experimentally by Tang, Statz, and deMars in 1963; Lamb's formalism is what let the field turn such observations into quantitative predictions of how many modes a laser runs and how they share the gain.

History

1960

— Maiman demonstrates the first working laser, in ruby

1963

— Tang, Statz, and deMars trace the multimode output of solid-state lasers to spatial hole burning

1964

— Lamb publishes the semiclassical theory of the optical maser, formalizing self- and cross-saturation

1965

— Evtuhov and Siegman introduce the twisted-mode cavity to give axially uniform energy density

1985

— Kane and Byer build the monolithic non-planar ring oscillator for single-frequency Nd:YAG

Going further

Texts. Siegman, Lasers (1986), Ch. 12–13, 25 (spatial hole burning, mode competition) · Svelto, Principles of Lasers, 5th ed., Ch. 7, 12 (mode selection and hole burning) · Saleh and Teich, Fundamentals of Photonics, 3rd ed., Ch. 15–16 (gain saturation) · Demtröder, Laser Spectroscopy, Vol. 1, Ch. 5 (spatial and spectral hole burning) · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 7 · Foot, Atomic Physics, Ch. 7. The core results — the standing-wave intensity, the saturated inversion $N_0/(1+I/I_s)$ with $I_s=\hbar\omega/(\sigma c\tau)$ , the modal-gain overlap integral, the weak-saturation $\cos(2kz)$ grating, the second-mode excess gain $I_0/(4I_s)$ , and the coexistence criterion $\beta_{mm}\beta_{nn} > \beta_{mn}\beta_{nm}$ — are treated in the texts listed above. Historical attributions follow the primary literature (Tang–Statz–deMars 1963; Lamb 1964; Evtuhov–Siegman 1965; Kane–Byer 1985); the Nd:YAG figures λ₀ = 1064 nm, n = 1.82, τ ≈ 230 μs are standard tabulated values.

Builds on: Population Inversion and Optical Gain · Stimulated Emission and the Gain Coefficient · Optical Resonators and Cavity Modes · Longitudinal Modes and Free Spectral Range · Laser Rate Equations. Leads to: Cavity Q Factor and Photon Lifetime · Four-Level Laser Systems · Gaussian Beams.

Ask the tutor: Ask me to walk through why a second longitudinal mode finds gain at the first mode's nodes, or to show how a ring cavity erases the λ/2 grating.

Why is the grating period λ/2 and not a full wavelength λ?

Because the atoms respond to intensity, and intensity is the square of the field. The field $\sin(kz)$ repeats every full wavelength λ, but its square $\sin^2(kz)$ repeats twice as often — every λ/2 — since it does not care whether the field is at a positive or negative crest, only that it is large. Both crests bleach the gain equally, so the inversion troughs sit under both, giving a pitch of λ/2. This is also why the grating scatters light at wavevector $2k$ , the second harmonic of the field.

How is spatial hole burning different from spectral hole burning?

They are analogues in different spaces. Spatial hole burning is a position-space effect: a standing wave saturates the gain only at its antinodes, leaving unsaturated pockets at the nodes, and it happens in any standing-wave cavity even with a homogeneous gain line. Spectral hole burning is a frequency-space effect: in an inhomogeneously broadened medium a monochromatic field saturates only the sub-group of atoms resonant with it, burning a hole in the gain spectrum. The first needs a standing wave; the second needs inhomogeneous broadening. A Doppler-broadened gas laser can show both — the Lamb dip is a spectral hole burnt by the two velocity classes a standing-wave mode addresses.

If the medium is homogeneously broadened, shouldn't the laser be single-mode?

Only if the gain saturates uniformly in space, and in a linear cavity it does not. The single-mode argument says the strongest mode clamps the gain for everyone, but that clamping is uniform only for a travelling wave. A standing-wave mode clamps the gain at its antinodes while leaving the nodes almost fully inverted, and a second mode whose antinodes fall there finds real, above-threshold gain. So homogeneous broadening plus a standing wave gives multimode operation; single-mode output is recovered only when you also remove the standing wave — with a ring or twisted-mode cavity — so that saturation really is uniform.

Why does atomic or carrier diffusion suppress the effect?

Because diffusion carries excitation from the fully inverted nodes into the depleted antinodes, smearing the grating flat. The comparison that matters is the diffusion length $L_{\mathrm{diff}}=\sqrt{D\tau}$ against the pitch λ/2. If an excited atom wanders much farther than λ/2 before it decays, it samples many nodes and antinodes and sees only the average inversion, so the corrugation vanishes and single-mode behaviour returns. If it barely moves — as for dopant ions frozen in a lattice — the grating stands sharp. This is why the effect dominates in solid-state and semiconductor lasers and is milder in gases.

How exactly does a ring laser cure spatial hole burning?

A ring cavity lets the light circulate around a loop. Force oscillation in one direction only — usually with an optical diode built from a Faraday rotator — and the intracavity field is a pure travelling wave whose intensity is constant along the axis. With no nodes there is nothing to burn a spatial hole into: the inversion saturates evenly and the strongest mode clamps the gain for all the rest. The monolithic non-planar ring oscillator is the compact solid-state realization of this idea.

What is the twisted-mode technique, and how does it help in a linear cavity?

The twisted-mode scheme keeps a standing-wave linear resonator but denies the medium the intensity grating. Quarter-wave plates on each side of the gain element make the two counter-propagating waves circularly polarized with opposite handedness. Each is still a standing wave, but their polarizations are orthogonal, so they do not interfere in intensity — the total intensity is uniform along the axis. The atoms see a flat energy density with no nodes, the inversion saturates uniformly, and spatial hole burning is suppressed just as in a ring, without a unidirectional loop.