Diffraction Grating - Theory of Operation

Theory of Operation

The relationship between the grating spacing and the angles of the incident and diffracted beams of light is known as the grating equation.

According to the Huygens–Fresnel principle, each point on the wavefront of a propagating wave can be considered to act as a point source, and the wavefront at any subsequent point can be found by adding together the contributions from each of these individual point sources.

Gratings may be of the 'reflective' or 'transmissive' type, analogous to a mirror or lens respectively. A grating has a 'zero-order mode' (where m=0), in which there is no diffraction and a ray of light behaves according to the laws of reflection and refraction the same as with a mirror or lens respectively.

An idealised grating is considered here which is made up of a set of slits of spacing d, that must be wider than the wavelength of interest to cause diffraction. When a plane wave of wavelength λ with normal incidence, (perpendicular to the grating) each slit in the grating acts as a quasi point-source from which light propagates in all directions (although this is typically limited to a hemisphere). After light interacts with the grating, the diffracted light is composed of the sum of interfering wave components emanating from each slit in the grating. At any given point in space through which diffracted light may pass, the path length to each slit in the grating will vary. Since the path length varies, generally, so will the phases of the waves at that point from each of the slits, and thus will add or subtract from one another to create peaks and valleys, through the phenomenon of additive and destructive interference. When the path difference between the light from adjacent slits is equal to half the wavelength, λ/2, the waves will all be out of phase, and thus will cancel each other to create points of minimum intensity. Similarly, when the path difference is λ, the phases will add together and maxima will occur. The maxima occur at angles θ_m, which satisfy the relationship dsinθ_m/λ=|m| where θ_m is the angle between the diffracted ray and the grating's normal vector, and d is the distance from the center of one slit to the center of the adjacent slit, and m is an integer representing the propagation-mode of interest.

Thus, when light is normally incident on the grating, the diffracted light will have maxima at angles θ_m given by:

It is straightforward to show that if a plane wave is incident at any arbitrary angle θ_i, the grating equation becomes:

When solved for the diffracted angle maxima, the equation is:

The light that corresponds to direct transmission (or specular reflection in the case of a reflection grating) is called the zero order, and is denoted m = 0. The other maxima occur at angles which are represented by non-zero integers m. Note that m can be positive or negative, resulting in diffracted orders on both sides of the zero order beam.

This derivation of the grating equation is based on an idealised grating. However, the relationship between the angles of the diffracted beams, the grating spacing and the wavelength of the light apply to any regular structure of the same spacing, because the phase relationship between light scattered from adjacent elements of the grating remains the same. The detailed distribution of the diffracted light depends on the detailed structure of the grating elements as well as on the number of elements in the grating, but it will always give maxima in the directions given by the grating equation.

Gratings can be made in which various properties of the incident light are modulated in a regular pattern; these include

• transparency (transmission amplitude gratings)
• reflectance (reflection amplitude gratings)
• refractive index (phase gratings)
• direction of optical axis (optical axis gratings)

The grating equation applies in all these cases.