# Diffraction

Diffraction refers to various phenomena which occur when a wave encounters an obstacle.

In classical physics, the diffraction phenomenon is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings.

Similar effects occur when a light wave travels through a medium with a varying refractive index.

Single-slit diffraction

The path difference is given by

d sinθ / 2

so that the minimum intensity occurs at an angle θmin given by

d sinθmin = λ

where

d is the width of the slit,

θmin is the angle of incidence at which the minimum intensity occurs

λ is the wavelength of the light.

A similar argument can be used to show that if we imagine the slit to be divided into four, six, eight parts, etc., minima are obtained at angles θn given by

d sinθn = n λ

where

n is an integer other than zero.

Diffraction grating

It splits and diffracts light into several beams travelling in different directions.

The directions of these beams depend on the spacing of the grating and the wavelength of the light so that the grating acts as the dispersive element.

The form of the light diffracted by a grating depends on the structure of the elements and the number of elements present, but all gratings have intensity maxima at angles θm which are given by the grating equation

D( sinθm + sinθi ) = m λ

where

θi is the angle at which the light is incident

d is the separation of grating elements

m is an integer which can be positive or negative.

Diffraction-limited imaging

The ability of an imaging system to resolve detail is ultimately limited by diffraction. This is because a plane wave incident on a circular lens or mirror is diffracted as described above. The light is not focused to a point but forms an Airy disk having a central spot in the focal plane with radius to first null of

d = 1.22 λ N

where

λ is the wavelength of the light

N is the f-number (focal length divided by diameter) of the imaging optics.

In object space, the corresponding angular resolution is

sinθ = 1.22 λ / D

where D is the diameter of the entrance pupil of the imaging lens (e.g., of a telescope’s main mirror).

Particle diffraction

Quantum theory tells us that every particle exhibits wave properties. In particular, massive particles can interfere and therefore diffract.

The wavelength associated with a particle is the de Broglie wavelength

λ = h / p

where h is Planck’s constant and p is the momentum of the particle (mass × velocity for slow-moving particles).

Bragg diffraction

Diffraction from a three dimensional periodic structure such as atoms in a crystal is called Bragg diffraction. It is similar to what occurs when waves are scattered from a diffraction grating. Bragg diffraction is a consequence of interference between waves reflecting from different crystal planes. The condition of constructive interference is given by Bragg’s law:

m λ = 2 d sinθ

where

λ is the wavelength,

d is the distance between crystal planes,

θ is the angle of the diffracted wave.

m is an integer known as the order of the diffracted beam.