Bragg condition (Bragg's law)

The Bragg condition, or Bragg's law, gives the angles at which a wave reflects strongly from a set of evenly spaced parallel planes:

$$2d\sin \theta =n\lambda ,$$

where $d$ is the spacing of the planes, $\theta$ the glancing angle between the incoming wave and the planes, $\lambda$ the wavelength and $n$ a whole number, the order. At these angles the reflections from successive planes differ in path by whole wavelengths and add in phase; at other angles they cancel. William Lawrence Bragg derived the condition in 1912 to explain the X-ray patterns of crystals, and he and William Henry Bragg shared the 1915 Nobel Prize in Physics for determining crystal structures with it. The same condition sets the wavelength reflected by the periodic structures of photonics, such as fiber Bragg gratings, DFB lasers and multilayer mirrors, where at normal incidence it becomes ${\lambda }_B=2n_{\text{eff}}\mathrm{\Lambda }$ for a grating of period $\mathrm{\Lambda }$.

Where the condition comes from

A wave reflected from the second plane travels farther than one reflected from the first: $d\sin \theta$ more on the way in and $d\sin \theta$ more on the way out, $2d\sin \theta$ in all. When that extra path is a whole number of wavelengths, the partial reflections from all the planes arrive in phase, and a weak reflection from each plane builds up to a strong one. Two conventions are worth keeping straight. The angle is measured from the planes, not from their normal, and the reflected beam is deviated by $2\theta$ from the incident one, which is why X-ray diffractometers record intensity against $2\theta$. And since $\sin \theta$ cannot exceed 1, only planes with $d\ge n\lambda \mathrm{/}2$ can reflect at all.

X-ray diffraction

Atomic planes in crystals are typically 0.1 to 0.5 nm apart, comparable to X-ray wavelengths and far below visible ones, which is what makes X-ray diffraction by crystals possible. With copper Kα radiation, $\lambda$ = 0.15406 nm:

Measuring the angles of many reflections gives the plane spacings, and from them the crystal lattice; the intensities of the reflections give the positions of the atoms within it. A powder of randomly oriented crystallites produces every allowed reflection at once, as rings, which is the basis of powder X-ray diffraction for identifying materials. Electrons and neutrons of suitable energy have de Broglie wavelengths of the same order and obey the same law. Visible light needs spacings of a few hundred nanometers, found in the ordered silica spheres that give precious opal its colors and in artificial photonic crystals.

Bragg gratings in photonics

In a waveguide or fiber whose refractive index varies periodically along its length, the light travels along the grating axis, at $\theta$ = 90° to the index planes, and its wavelength inside the guide is $\lambda \mathrm{/}{n}_{\text{eff}}$, with $n_{\text{eff}}$ the effective index of the mode. Bragg's law then gives the Bragg wavelength, reflected back along the guide:

$${\lambda }_B=\frac{2n_{\text{eff}}\mathrm{\Lambda }}{m},$$

for order $m$, usually 1. Light near ${\lambda }_B$ couples from the forward mode into the backward one; other wavelengths pass. Typical first-order periods for 1550 nm:

• Fiber Bragg grating: 535.6 nm in a fiber with $n_{\text{eff}}$ = 1.447.
• InP DFB laser: about 242 nm, for $n_{\text{eff}}$ near 3.2.
• Silicon strip waveguide grating: about 318 nm, for the $n_{\text{eff}}$ near 2.44 of a 500 × 220 nm strip.

A uniform grating with coupling coefficient $\kappa$ and length $L$ reflects $R={\tanh }^2(\kappa L)$ at the Bragg wavelength: 58% at $\kappa L$ = 1, 93% at 2 and 99% at 3. A longer grating narrows the reflection band and a stronger index modulation widens it, and apodization of the modulation suppresses its side lobes. The Bragg Wavelength Calculator converts between period and Bragg wavelength.

Stacks of thin films follow the same rule. A multilayer mirror of alternating quarter-wave layers, each of optical thickness $\lambda \mathrm{/}4$, has an optical period of $\lambda \mathrm{/}2$ and so reflects in first order at $\lambda$; this is the distributed Bragg reflector that forms the mirrors of a VCSEL; DBR lasers use the waveguide-grating form above.

Other forms of the condition

The general statement is phase matching: the grating's wavevector, $2\pi \mathrm{/}\mathrm{\Lambda }$, must make up the difference between the incoming and outgoing wavevectors. A grating coupler uses it to send light out of a chip at an angle, $\sin \theta =n_{\text{eff}}-m\lambda \mathrm{/}\mathrm{\Lambda }$ in air. An acousto-optic modulator diffracts light from a sound wave of wavelength $\mathrm{\Lambda }$ at the Bragg angle ${\theta }_B\approx \lambda \mathrm{/}(2\mathrm{\Lambda })$: 0.85° for 1550 nm light on the 52.5 µm acoustic wavelength of TeO₂ driven at 80 MHz.

Temperature and strain

The Bragg wavelength moves with anything that changes the index or the period. Because $n_{\text{eff}}$ also depends on wavelength, the shift is set by the group index $n_g$:

$$\frac{d{\lambda }_B}{dT}=\frac{{\lambda }_B}{n_g}(\frac{\mathrm{\partial }{n}_{\text{eff}}}{\mathrm{\partial }T}+\frac{n_{\text{eff}}}{\mathrm{\Lambda }}\frac{d\mathrm{\Lambda }}{dT})\mathrm{.}$$

In SMF-28 fiber $n_g$ = 1.468 against $n_{\text{eff}}$ = 1.447, but in a 500 × 220 nm silicon strip it is 4.18 against 2.446, so leaving out the dispersion there would overstate the shift by a factor of 1.7.

The waveguide rows follow the resonance drifts of silicon and silicon nitride rings, computed and measured, which depend on the same $d{n}_{\text{eff}}\mathrm{/}dT$ and group index; the difference between them comes mostly from the materials' thermo-optic coefficients. In silica fiber the small thermo-optic term dominates, and stretching the fiber by a strain $\epsilon$ also shifts ${\lambda }_B$, by the fraction $(1-p_e)\epsilon$ with $p_e\approx 0.22$: 1.2 pm per microstrain at 1550 nm, the basis of the FBG sensor.

Common questions

What is Bragg's law used for?

To find the spacing of atomic planes from measured X-ray, electron or neutron diffraction angles, which is how crystal structures are determined, and in photonics to choose the grating period that reflects a given wavelength in a fiber Bragg grating, a DFB or DBR laser, or a multilayer mirror.

Are the Bragg condition and Bragg's law the same thing?

Yes. "Bragg's law" usually refers to the crystallography form, $2d\sin \theta =n\lambda$, and "Bragg condition" to the waveguide form, ${\lambda }_B=2n_{\text{eff}}\mathrm{\Lambda }$; the second is the first at $\theta$ = 90°, with the wavelength taken inside the material.

Why is the angle measured from the planes rather than the normal?

By convention from crystallography, where the planes are the reference. With the angle ${\theta }_i$ measured from the normal, the same condition reads $2d\cos {\theta }_i=n\lambda$.

References: W. H. Bragg and W. L. Bragg, "The reflection of X-rays by crystals," Proceedings of the Royal Society A 88, 428 (1913). T. Erdogan, "Fiber grating spectra," Journal of Lightwave Technology 15, 1277 (1997). A. D. Kersey et al., "Fiber grating sensors," Journal of Lightwave Technology 15, 1442 (1997). J. V. Sanders, "Colour of precious opal," Nature 204, 1151 (1964).