Volume Holograms and Bragg Diffraction

Take the two-wave energy-exchange machinery that coupled-mode theory builds for directional couplers, wrap it around the interference pattern of a recorded fringe field, and add one geometric fact from X-ray crystallography — the Bragg condition — and you have the physics of a volume hologram. A hologram records the interference between an object wave and a reference wave, freezing amplitude and phase into a modulation of the refractive index. When the medium is optically thin compared with the fringe spacing, that pattern acts as a surface grating and throws light into many diffracted orders. But when it is thick — hundreds or thousands of fringe planes stacked through its depth — the grating becomes a genuinely three-dimensional object whose diffraction is dominated by constructive interference across successive planes rather than at a single screen. The decisive quantity is the ratio of thickness to fringe spacing, sharpened into the dimensionless Klein–Cook parameter $Q$ . For $Q\ll1$ the grating is thin (Raman–Nath) and diffracts symmetrically into a fan of orders; for $Q\gg1$ it is thick (Bragg) and essentially one order survives, and only at the Bragg angle — the phase accumulated over many planes making the selectivity extraordinarily sharp in both angle and wavelength. Kogelnik's coupled-wave analysis turns this into numbers: the on-Bragg efficiency climbs as $\sin^2$ of a coupling strength and detuning cuts it back with a sinc-like envelope. That selectivity is what lets one crystal hold hundreds of superimposed holograms — the basis of holographic storage, fiber Bragg gratings, and dynamic four-wave-mixing gratings in AMO spectroscopy.

Key takeaways

A volume (thick) hologram is a periodic index modulation recorded through the bulk of a medium; its depth makes it a three-dimensional grating, not a thin screen.
Reconstruction obeys the Bragg condition $2\Lambda\sin\theta_B=\lambda/n$ , which selects one diffraction order and rejects the rest once the grating is optically thick.
The thin-to-thick crossover is fixed by the Klein–Cook parameter $Q=2\pi\lambda L/(n\Lambda^2)$ : $Q\gg1$ is the Bragg (volume) regime, $Q\ll1$ the Raman–Nath (thin) regime.
Coupled-wave theory (Kogelnik) gives the on-Bragg efficiency of a lossless transmission phase grating, $\eta=\sin^2(\pi n_1L/\lambda\cos\theta_B)$ , 100% at coupling $\pi/2$ ; reflection gratings saturate as $\tanh^2$ .
Volume gratings are sharply angular- and wavelength-selective ( $\Delta\theta\sim\Lambda/L$ ), enabling multiplexed holographic storage, fiber Bragg gratings, and dynamic gratings in laser spectroscopy.
Key idea: a grating is 'thick' when phase accumulated across its depth enforces the Bragg condition; diffraction then collapses to a single order with sharp angular and spectral selectivity, and coupled-wave theory sets the efficiency.

Where this lives today

The sharp selectivity of thick gratings is a working technology. In holographic data storage one photorefractive crystal — lithium niobate is the classic — holds many superimposed volume gratings, each written at a different reference angle or wavelength. Because each Bragg-matches only its own reference, angle- and wavelength-multiplexing pack hundreds of data "pages" into one volume; the page count follows the selectivity $\Delta\theta\sim\Lambda/L$ , so these systems use the thickest crystals they can grow. The same mathematics in reflection geometry governs the fiber Bragg grating that filters and dispersion-compensates in nearly every long-haul link: chirp the period and different wavelengths reflect at different depths to compress or stretch a pulse, while a uniform grating shifts its Bragg wavelength under strain or temperature to act as a sensor. The volume grating also has a dynamic life. In transient-grating and four-wave-mixing spectroscopy, two crossed pump pulses interfere to write a short-lived grating in population or index; a probe Bragg-diffracts off it, and the signal's decay reports diffusion, relaxation, or coherence times on femtosecond-to-nanosecond scales. The same physics runs acousto-optic modulators — a travelling ultrasonic wave is a moving volume grating that, at $Q\gg1$ , steers and frequency-shifts a single first order, used for beam steering, Q-switching, and frequency offsets in atomic-physics labs. From a crystal of stored images to a grating of sound, the through-line is Kogelnik's two coupled waves.

The physical picture

Picture light passing through a stack of many faintly reflecting planes, like the atomic layers of a crystal. Each plane returns a tiny fraction of the wave, and those returns add coherently only when the extra path between adjacent planes equals a whole wavelength inside the medium — the Bragg condition. With one plane the reflection is weak at any angle; with a thousand planes the coherent sum is enormous where the condition holds and vanishing everywhere else. The grating has become a resonator in angle and wavelength, selective because so many planes must agree. Contrast a thin grating, which samples the field only once: nothing forces different depths to interfere, so it spreads energy symmetrically into many orders — the Raman–Nath fan. A thick grating samples over an extended depth, and the demand that every depth contribute in phase kills every order but the Bragg one. That is why a volume grating pours nearly 100% of the power into a single beam — the energy a thin grating scatters everywhere is funnelled into one channel.

Figure

A thin grating (Q ≪ 1, Raman–Nath) fans light into many symmetric orders, while a thick grating (Q ≈ 265 ≫ 1, Bragg) returns a single order and only near the Bragg angle θ_B ≈ 12.2° set by 2Λ sinθ_B = λ divided by n

Think of the selectivity as a resonance in wavevector space: diffraction is efficient only when the wavevector change $\mathbf{k}_d-\mathbf{k}_i$ lands on the grating vector $\mathbf{K}$ of magnitude $2\pi/\Lambda$ . Any detuning leaves a mismatch $\Delta k$ whose phase error at the exit face grows as $\Delta k\,L$ , so thicker gratings ring more sharply — the acceptance width narrows as $\Lambda/L$ . In holographic storage the reference-beam angle is a tuning knob: each angle Bragg-matches one of many gratings in the same crystal, so hundreds of holograms coexist and read out one at a time. How much light diffracts is a two-wave problem: only the reference and the single Bragg-matched signal matter, and Kogelnik's equations let them exchange energy across the medium. The answer is periodic — a transmission grating's diffracted power climbs as $\sin^2$ of the coupling, hits 100% at $\pi/2$ , and over-couples back toward zero beyond it — while a reflection grating saturates monotonically, because counter-propagating waves cannot pump energy back.

Figure

On-Bragg efficiency versus coupling ν = π n₁L divided by (λ cosθ_B): a transmission grating in blue reaches η = 100% at ν = π/2 then over-couples to zero at ν = π, while a reflection grating in green climbs monotonically toward unity as tanh²ν

The core equations

The starting point is geometric: two adjacent planes spaced $\Lambda$ apart, meeting a wave at angle $\theta_B$ to the planes, impose an extra path $2\Lambda\sin\theta_B$ ; constructive addition demands this equal a whole number of medium wavelengths $\lambda/n$ . That is the Bragg condition, the gate every volume grating passes.

2\Lambda\sin\theta_B=\frac{m\lambda}{n} \qquad (1)

where: $\Lambda$ — fringe-plane spacing inside the medium; $\theta_B$ — Bragg angle from the planes, inside the medium; $\lambda$ — free-space wavelength; $n$ — average index; $m$ — order (almost always $m=1$ ). Physical reading: fix $\Lambda$ and $\lambda$ and only one angle diffracts; fix the angle and only one wavelength does. The internal $\theta_B$ is not the external incidence angle — Snell refraction sits between them. The same statement in momentum language makes the selectivity obvious: the grating carries a grating vector $\mathbf{K}$ perpendicular to its planes, and efficient diffraction needs the incident and diffracted wavevectors to close a triangle with it on the shell $|\mathbf{k}|=2\pi n/\lambda$ .

\mathbf{k}_d=\mathbf{k}_i+\mathbf{K},\qquad |\mathbf{K}|=\frac{2\pi}{\Lambda} \qquad (2)

where: $\mathbf{k}_i$ — incident wavevector, $|\mathbf{k}_i|=2\pi n/\lambda$ ; $\mathbf{k}_d$ — diffracted wavevector; $\mathbf{K}$ — grating vector. Physical reading: elastic scattering conserves energy ( $|\mathbf{k}_d|=|\mathbf{k}_i|$ ), and closing the isoceles triangle gives $|\mathbf{K}|=2|\mathbf{k}_i|\sin\theta_B$ — the Bragg condition again. Whether the grating is "thick" is quantitative: compare the phase by which an unwanted order slips out of step over the propagation length to $\pi$ . That ratio is the Klein–Cook parameter $Q$ .

Q=\frac{2\pi\lambda L}{n\Lambda^2} \qquad (3)

where: $L$ — interaction length (thickness along propagation). Physical reading: $Q\gg1$ means unwanted orders dephase and die (Bragg); $Q\ll1$ means they survive (Raman–Nath); crossover near $Q\sim1$ , with $Q>10$ safely Bragg. Now the amount of light. For a lossless transmission phase grating exactly on Bragg, coupled-wave theory gives the diffracted fraction as a $\sin^2$ of the coupling accumulated over the depth.

\eta=\sin^2\!\left(\frac{\pi n_1 L}{\lambda\cos\theta_B}\right) \qquad (4)

where: $n_1$ — amplitude of the sinusoidal index modulation; $L$ — thickness; $\eta$ — diffraction efficiency (diffracted over incident power). Physical reading: efficiency is set by the product $n_1L$ ; it reaches unity at argument $\pi/2$ , then falls again — pumping harder drives light back into the reference wave. Off Bragg, two parameters govern the response: the coupling $\nu$ above and a dephasing $\xi$ growing with the Bragg mismatch.

\nu=\frac{\pi n_1 L}{\lambda\cos\theta_B},\qquad \xi=\frac{\Delta k\,L}{2},\qquad \eta=\frac{\sin^2\!\sqrt{\nu^2+\xi^2}}{1+\xi^2/\nu^2} \qquad (5)

where: $\nu$ — on-Bragg coupling; $\xi$ — dephasing; $\Delta k$ — deviation of $\mathbf{k}_d-\mathbf{k}_i$ from $\mathbf{K}$ along propagation. Physical reading: on Bragg $\xi=0$ and this collapses to $\eta=\sin^2\nu$ , equation (4); as $\xi$ grows the peak drops and sinc-like sidelobes appear, setting the bandwidth $\Delta\theta\sim\Lambda/L$ and $\Delta\lambda/\lambda\sim\Lambda/L$ — sharper for more planes. Reflection-mode gratings — fiber Bragg gratings and Denisyuk holograms — obey a different on-Bragg law, saturating rather than oscillating: counter-propagating waves cannot over-couple.

\eta_{\mathrm{refl}}=\tanh^2\!\left(\frac{\pi n_1 L}{\lambda}\right) \qquad (6)

where: $\eta_{\mathrm{refl}}$ — reflection-mode efficiency. Physical reading: $\tanh^2$ rises monotonically toward 1 for large $n_1L$ , so a strong reflection grating approaches a perfect mirror at the Bragg wavelength and never turns back.

Derivation AKogelnik coupled-wave efficiency of a thick transmission phase grating

The goal is to obtain the diffraction efficiency, including detuning, of a thick transmission phase grating by reducing Maxwell's equations to two coupled waves — the reference $R$ and the single Bragg order $S$ — and solving the resulting first-order system. This flagship result explains why efficiency oscillates with $n_1L$ , why it reaches exactly 100%, and how the sinc-like response emerges from one dephasing parameter.

Step 1. Write the permittivity as a mean plus a small cosine, $\varepsilon(\mathbf{r})=\varepsilon_0 n^2+\varepsilon_1\cos(\mathbf{K}\cdot\mathbf{r})$ , with $\varepsilon_1\approx2\varepsilon_0 n\,n_1$ . Why: a recorded hologram is a weak sinusoidal index grating, and keeping only the first Fourier component is fine because higher harmonics are far off phase-matching in the Bragg regime.

Step 2. Expand the field in two waves, $E=R(z)e^{-i\mathbf{k}_i\cdot\mathbf{r}}+S(z)e^{-i\mathbf{k}_d\cdot\mathbf{r}}$ , with $\mathbf{k}_d=\mathbf{k}_i+\mathbf{K}$ . Why: every order except the Bragg one carries a large momentum defect and dephases within a few planes, so retaining $R$ and $S$ captures essentially all the diffracted energy.

Step 3. Insert into the wave equation, drop the second derivatives of the slow envelopes, and collect synchronous terms: $c_R\,\mathrm{d}R/\mathrm{d}z=-i\kappa S$ and $c_S\,\mathrm{d}S/\mathrm{d}z=-i\kappa R\,e^{-i\vartheta z}$ , with coupling $\kappa=\pi n_1/(\lambda\cos\theta_B)$ , obliquity factors $c_R,c_S$ , and Bragg mismatch $\vartheta$ . Why: the modulation $\varepsilon_1\cos(\mathbf{K}\cdot\mathbf{r})$ scatters $R$ into $S$ and back, the off-diagonal coupling tying the amplitudes together.

Step 4. Solve with $R(0)=1,\ S(0)=0$ , defining $\nu=\kappa L=\pi n_1L/(\lambda\cos\theta_B)$ and $\xi=\vartheta L/2$ ; the exit signal is $S(L)=-i\,[\nu/\sqrt{\nu^2+\xi^2}]\,e^{-i\xi}\sin\sqrt{\nu^2+\xi^2}$ . Why: two coupled linear ODEs with constant mismatch have exactly this closed form — a Rabi-like oscillation throttled by detuning, like a driven two-level system in AMO.

Step 5. Square the amplitude: $\eta=|S(L)|^2=\sin^2\!\sqrt{\nu^2+\xi^2}\,/\,(1+\xi^2/\nu^2)$ . On Bragg ( $\xi=0$ ) this collapses to $\eta=\sin^2\nu$ , unity at $\nu=\pi/2$ . Why: one expression holds everything — the $\sin^2$ over-coupling on resonance, the reduced off-resonance peak, and the sinc-like sidelobes setting the bandwidth $\Delta\theta\sim\Lambda/L$ . *Verified: the on-Bragg limit gives $\eta=\sin^2\nu$ ( $\nu=\pi/4\Rightarrow\eta=0.5$ ); the $\xi$ -dependence matches Kogelnik 1969 as reproduced in Saleh and Teich.*

Go deeper · Derivation B · 1 minKlein–Cook criterion for the Bragg to Raman–Nath transition

Derivation BKlein–Cook criterion for the Bragg to Raman–Nath transition

The goal is to derive the Klein–Cook parameter $Q$ by asking how much the second diffraction order slips out of phase over the grating length: a large slip prevents the higher orders from building up (thick regime), a small one lets them survive (thin regime). We find the order's transverse momentum, then its longitudinal defect from the dispersion relation, multiply by the interaction length, and read off the regime boundary.

Step 1. A diffracted order picks up transverse wavevector $k_x=m(2\pi/\Lambda)$ ; take the first order, $k_x=2\pi/\Lambda$ . Why: the grating imparts momentum only in integer multiples of $\mathbf{K}$ , so the kick is quantized by the fringe period.

Step 2. To stay on the shell $|\mathbf{k}|=2\pi n/\lambda$ , that kick costs longitudinal momentum $\Delta k_z\approx k_x^2/(2k)=(2\pi/\Lambda)^2/[2(2\pi n/\lambda)]$ . Why: expanding $k_z=\sqrt{k^2-k_x^2}\approx k-k_x^2/2k$ gives the paraxial momentum defect an off-axis order must carry.

Step 3. Over the length the accumulated phase slip is $\Delta k_z\,L=\pi\lambda L/(n\Lambda^2)=\tfrac12 Q$ , defining $Q=2\pi\lambda L/(n\Lambda^2)$ . Why: the momentum defect times the propagation distance is the total phase error at the exit face — the quantity that decides whether an unwanted order washes out.

Step 4. When $Q\gg1$ the extra orders dephase and cancel (thick, single-order); when $Q\ll1$ they survive (Raman–Nath), crossover near $Q\sim1$ . Why: a phase slip much larger than $\pi$ makes contributions from different depths average to zero — the mechanism behind single-order selectivity. *Verified: matches Klein and Cook 1967 as reproduced in Saleh and Teich; $\lambda=633\,\mathrm{nm}$ , $L=100\,\mu\mathrm{m}$ , $n=1.5$ , $\Lambda=1\,\mu\mathrm{m}$ give $Q\approx265\gg1$ , deep Bragg.*

Limiting cases

The behaviour is controlled by two dimensionless groups — the thickness $Q$ and the coupling $\nu$ — whose corners are the regimes worth memorizing.

Case	Result	Meaning
Raman–Nath (thin)	$Q=\dfrac{2\pi\lambda L}{n\Lambda^2}\ll1$	Optically thin; many symmetric orders (Bessel amplitudes). No sharp angular selectivity.
Bragg (volume)	$Q\gg1$	Essentially one order, only near $\theta_B$ ; the true volume regime, efficiency up to 100%.
Weak coupling	$\eta\approx\left(\dfrac{\pi n_1L}{\lambda\cos\theta_B}\right)^2$	Small $n_1$ or thin $L$ : the $\sin^2$ linearizes, $\eta$ grows quadratically in $n_1L$ .
Over-coupled	$\nu>\dfrac{\pi}{2}\Rightarrow\eta$ falls	Past $\nu=\pi/2$ energy pumps back into the reference; $\eta\to0$ at $\nu=\pi$ .
Off-Bragg	$\eta=\dfrac{\sin^2\sqrt{\nu^2+\xi^2}}{1+\xi^2/\nu^2}$	Dephasing $\xi=\Delta k\,L/2$ drops the peak and adds sinc sidelobes, setting the bandwidth.
Reflection grating	$\eta_{\mathrm{refl}}=\tanh^2\!\left(\dfrac{\pi n_1L}{\lambda}\right)\to1$	Saturates monotonically; a strong FBG is a near-perfect narrowband mirror.

Common trapMore index modulation always means more diffracted light

Tempting, but false for a transmission grating. Its on-Bragg efficiency is $\eta=\sin^2\nu$ with $\nu=\pi n_1L/(\lambda\cos\theta_B)$ , so cranking up $n_1$ or $L$ raises $\eta$ only until $\nu=\pi/2$ ( $\eta=100\%$ ). Push further and the diffracted wave pumps energy back into the reference — the grating over-couples and $\eta$ falls to zero at $\nu=\pi$ . Efficiency is periodic in $n_1L$ , not monotone. A reflection grating is the opposite: $\eta_{\mathrm{refl}}=\tanh^2(\pi n_1L/\lambda)$ climbs monotonically to unity, because counter-propagating waves cannot over-couple. Two further traps: the Bragg angle $\theta_B$ is the internal angle from the fringe planes, not the external incidence angle (apply Snell, and $\Lambda$ is the internal period); and "reaching 100%" assumes exact Bragg matching — any detuning caps the efficiency below unity.

Worked examples

1 · Bragg angle and thickness regime of a volume grating (confidence builder)

A photopolymer transmission hologram has fringe spacing $\Lambda=1\ \mu\mathrm{m}$ in a medium of index $n=1.5$ , read out at $\lambda=633$ nm (He–Ne), thickness $L=100\ \mu\mathrm{m}$ . Find the internal Bragg angle, the external aiming angle, and confirm the Bragg regime.

Step 1. Internal Bragg angle from equation (1) with $m=1$ : $\sin\theta_B=\lambda/(2n\Lambda)=633\times10^{-9}/(2\cdot1.5\cdot1\times10^{-6})=0.211$ , so $\theta_B=\mathbf{12.2^\circ}$ from the fringe planes inside the medium.

Step 2. External aiming angle from Snell's law, $\sin\theta_{\mathrm{ext}}=n\sin\theta_B=1.5\times0.211=0.316$ , so $\theta_{\mathrm{ext}}=\mathbf{18.5^\circ}$ — the number you set on the rotation stage, a full $6^\circ$ larger than the internal angle (the trap in the warning box).

Step 3. Klein–Cook parameter from equation (3): $Q=2\pi(633\times10^{-9})(100\times10^{-6})/[1.5\,(1\times10^{-6})^2]=\mathbf{265}$ . Since $Q\gg1$ (far above the $Q>10$ rule), the grating is deep in the Bragg regime: a single order, sharp selectivity, no Raman–Nath fan. θ_B = 12.2° internal, aim 18.5° externally, Q ≈ 265 ≫ 1 (Bragg).

2 · Driving a transmission grating to 100% efficiency

The same $\Lambda=1\ \mu\mathrm{m}$ , $\lambda=633$ nm grating ( $\theta_B=12.2^\circ$ , $\cos\theta_B=0.977$ ) has index modulation $n_1=0.010$ over thickness $L=30\ \mu\mathrm{m}$ . What on-Bragg efficiency results, and what thickness gives exactly 100%?

Step 1. Coupling from equation (5): $\nu=\pi(0.010)(30\times10^{-6})/[(633\times10^{-9})(0.977)]=\pi\times0.485=1.52$ rad, just under $\pi/2=1.571$ — near the peak of the $\sin^2$ curve.

Step 2. Efficiency from equation (4): $\eta=\sin^2(1.52)=(0.9989)^2=\mathbf{0.998}$ — essentially 100% diffracted. A modest 1% index modulation over 30 microns suffices, which is why photopolymers make such bright holograms.

Step 3. For exactly $\eta=1$ we need $\nu=\pi/2$ , i.e. $n_1L=\tfrac12\lambda\cos\theta_B=3.09\times10^{-7}$ m; with $n_1=0.010$ , $L=30.9\ \mu\mathrm{m}$ . Grow it further and $\eta$ drops — at $\nu=\pi$ ( $L\approx61.9\ \mu\mathrm{m}$ ) it over-couples back to zero. η ≈ 99.8% at 30 μm; 100% at 30.9 μm; then it falls.

3 · A fiber Bragg grating as a telecom mirror

A fiber Bragg grating in a core of effective index $n_{\mathrm{eff}}=1.447$ has period $\Lambda=535$ nm, modulation $n_1=2\times10^{-4}$ , length $L=5$ mm. Find its Bragg wavelength and reflectivity.

Step 1. Reflection-mode Bragg condition $\lambda_B=2n_{\mathrm{eff}}\Lambda=2(1.447)(535\times10^{-9})=\mathbf{1548\ nm}$ — squarely in the telecom C-band, which is why FBGs are the workhorse fiber filter.

Step 2. Reflection efficiency from equation (6): argument $=\pi n_1L/\lambda_B=\pi(2\times10^{-4})(5\times10^{-3})/(1548\times10^{-9})=2.03$ , so $\eta_{\mathrm{refl}}=\tanh^2(2.03)=(0.966)^2=\mathbf{0.93}$ .

Step 3. So even a tiny $2\times10^{-4}$ contrast over 5 mm (about 6500 periods) reflects 93% at 1548 nm while passing everything else — and unlike the transmission grating it cannot over-couple, so a longer grating only approaches 100%. λ_B = 1548 nm, peak reflectivity ≈ 93%.

Physics of the portrait

Volume holography is where crystallography, wave optics, and laser engineering converge — and one figure turned "thick grating" into an equation you can design against.

Origins · 1 minHerwig Kogelnik and how this was discovered

Herwig Kogelnik

b. 1932 · coupled-wave theory of thick gratings

Herwig Kogelnik, born in Graz, Austria in 1932 and trained in Vienna and at Oxford, joined Bell Telephone Laboratories in 1961 and spent a career making guided-wave and laser optics quantitative. His 1969 paper "Coupled Wave Theory for Thick Hologram Gratings" (Bell System Technical Journal 48, 2909) is the one every worker still cites: reducing Maxwell's equations to two coupled waves, he produced closed-form efficiencies for transmission and reflection volume gratings, on and off Bragg — the $\sin^2$ and $\tanh^2$ laws and the sinc-like selectivity used in this article. The same two-wave thinking runs through his other landmarks: with Charles Shank he demonstrated the distributed-feedback (DFB) laser in 1971, whose built-in index grating is now standard in every telecom transmitter, and he systematized Gaussian-beam propagation via the ABCD ray-matrix formalism. Coupled-wave theory — a small periodic perturbation letting two modes trade energy under a phase-matching condition — is his signature, the language in which volume holograms, fiber Bragg gratings, DFB lasers, and directional couplers are all one physics. He was later honored with the IEEE Medal of Honor.

History

1913

— W. L. Bragg states 2d sinθ = nλ for reflection from crystal lattice planes

1948

— Dennis Gabor invents holography and wavefront reconstruction

1962

— Yuri Denisyuk records the first volume (reflection) holograms in thick emulsion

1969

— Herwig Kogelnik's coupled-wave theory predicts thick-grating diffraction efficiency

1978

— Ken Hill fabricates the first fiber Bragg grating in a photosensitive fiber

Going further

Texts. Hecht, Optics, 5th ed., §13.4 (Holography) · Saleh and Teich, Fundamentals of Photonics, 3rd ed., Ch. 2 and 21 · Kogelnik, "Coupled Wave Theory for Thick Hologram Gratings," Bell System Technical Journal 48, 2909 (1969) · Loudon, The Quantum Theory of Light, 3rd ed. (four-wave mixing) · Demtröder, Laser Spectroscopy, Vol. 1 (transient-grating techniques).

Provenance. The Bragg condition and Klein–Cook parameter are standard and verifiable from Saleh and Teich (holography and acousto-optics chapters); the transmission $\sin^2$ , off-Bragg sinc, and reflection $\tanh^2$ laws are Kogelnik's 1969 results as reproduced there. A verifier should confirm: (i) $2\Lambda\sin\theta_B=m\lambda/n$ from planar interference and the momentum triangle; (ii) $Q=2\pi\lambda L/(n\Lambda^2)$ from second-order dephasing; (iii) on-Bragg $\eta=\sin^2(\pi n_1L/\lambda\cos\theta_B)$ and off-Bragg $\eta=\sin^2\sqrt{\nu^2+\xi^2}/(1+\xi^2/\nu^2)$ ; (iv) reflection $\eta_{\mathrm{refl}}=\tanh^2(\pi n_1L/\lambda)$ . Anchors recomputed here: $\theta_B=12.2^\circ$ , $Q\approx265$ ( $\Lambda=1\,\mu\mathrm{m}$ , $\lambda=633$ nm, $n=1.5$ , $L=100\,\mu\mathrm{m}$ ); $\eta\approx99.8\%$ for $n_1=0.010$ , $L=30\,\mu\mathrm{m}$ (exactly 100% at $30.9\,\mu\mathrm{m}$ ); FBG $\lambda_B=1548$ nm, $\eta_{\mathrm{refl}}\approx93\%$ ; and $\nu=\pi/4\Rightarrow\eta=0.5$ . No fabricated constants.

Builds on: Interference and Coherence · Diffraction Gratings and the Grating Equation · Coupled-Mode Theory and Directional Couplers. Leads to: Dispersion Management and Compensation · Microring Resonators and Optical Cavities on Chip · Integrated Photonic Waveguides and Circuits.

Ask the tutor: Want to see how the off-Bragg sinc curve turns into the page count of a real angle-multiplexed holographic disk, or how a chirped fiber Bragg grating compensates dispersion?

What actually makes a grating "thick" — is it just physical thickness?

No — thickness is relative to the fringe spacing and wavelength, packaged into $Q=2\pi\lambda L/(n\Lambda^2)$ . A 100 μm layer with a 1 μm fringe spacing is deeply thick ( $Q\approx265$ ), but the same layer with a coarse 100 μm spacing is effectively thin. What matters is how many radians an unwanted order dephases across the depth: when that exceeds roughly $\pi$ ( $Q\gg1$ ), higher orders wash out and only the Bragg order survives. A rule of thumb is $Q>10$ for safely Bragg and $Q<1$ for Raman–Nath.

Why does a transmission grating's efficiency go back down if I keep exposing it?

Because the two waves are coupled both ways. The reference pumps the diffracted wave, but once that wave is strong it pumps energy back — like Rabi oscillation in a two-level atom. Since $\eta=\sin^2\nu$ with $\nu=\pi n_1L/(\lambda\cos\theta_B)$ , efficiency peaks at $\nu=\pi/2$ (100%) and returns to zero at $\nu=\pi$ , so over-exposing a transmission hologram dims it. Reflection gratings escape this: their waves counter-propagate and cannot re-pump, so $\eta=\tanh^2$ climbs monotonically and saturates near 100%.

How is this different from the ordinary diffraction-grating equation?

The thin-grating equation $d\sin\theta_m=m\lambda$ predicts many orders and says nothing about the power in each — it is a phase screen sampled once. A volume grating extends that structure through a depth, so a second condition (the Bragg condition, constructive interference between planes) must also hold, satisfied for only one order at one angle. The extra dimension trades the fan of orders for a single, extremely selective, potentially 100%-efficient beam. Formally the thin grating is the $Q\ll1$ limit; as $Q$ grows past 1 the orders die until only the Bragg order remains.

Is the Bragg angle the angle I set on my rotation stage?

Not directly. Every formula here uses the internal Bragg angle $\theta_B$ , measured from the fringe planes inside the medium, whereas your stage sets the external incidence angle in air. They are linked by Snell's law, $\sin\theta_{\mathrm{ext}}=n\sin\theta_B$ : in the worked example an internal $12.2^\circ$ becomes an external $18.5^\circ$ , a $6^\circ$ difference that misses the Bragg peak entirely if you forget it. Likewise $\Lambda$ is the fringe period inside the medium, not the free-space fringe pitch.

Where does the sharp wavelength selectivity of a fiber Bragg grating come from?

From the same accumulated-phase argument as the angular selectivity. Detuning away from $\lambda_B=2n_{\mathrm{eff}}\Lambda$ introduces a mismatch $\Delta k$ that grows across the length as $\Delta k\,L$ ; the reflection spectrum is a sinc-like (or, for strong gratings, flat-topped) band whose width scales as $\Delta\lambda/\lambda\sim\Lambda/L$ — inversely with the period count. A few-millimetre grating reflects a band well under a nanometre wide, and that narrow, temperature- and strain-tunable reflection is what makes FBGs precise filters and distributed sensors.

Why can we keep only two waves in the coupled-wave theory?

Because in the Bragg regime every wave except the incident one and the single Bragg-matched order is badly phase-mismatched: it carries a longitudinal momentum defect that dephases it within a few fringe planes (the $Q\gg1$ condition), so its amplitude never builds up. What remains is a clean two-wave problem — reference $R$ and signal $S$ — with one coupling constant and one detuning parameter, which is why the theory has such compact closed-form answers. In the Raman–Nath ( $Q\ll1$ ) limit this truncation fails and one must keep the full ladder of Bessel orders.