# The Grating Equation

When monochromatic light is incident on a grating surface, it is diffracted into discrete directions. We can picture each grating groove as being a very small, slit-shaped source of diffracted light. The light diffracted by each groove combines to form a set of diffracted wavefronts. The usefulness of a grating depends on the fact that there exists a unique set of discrete angles along which, for a given spacing d between grooves, the diffracted light from each facet is in phase with the light diffracted from any other facet, leading to constructive interference.

### Diffraction Grating Geometry

Diffraction by a grating can be visualized from the geometry in Figure 2-1, which shows a light ray of wavelength λ incident at an angle α and diffracted by a grating (of groove spacing d, also called the pitch) along at set of angles {βm}. These angles are measured from the grating normal, which is shown as the dashed line perpendicular to the grating surface at its center. The sign convention for these angles depends on whether the light is diffracted on the same side or the opposite side of the grating as the incident light. In diagram (a), which shows a reflection grating, the angles α > 0 and β1 > 0 (since they are measured counter-clockwise from the grating normal) while the angles β0 < 0 and β–1 < 0 (since they are measured clockwise from the grating normal). Diagram (b) shows the case for a transmission grating. In both cases, the sign conventions are such that, for the m = 0 order, β0 = –α.

Figure 2-1. Diffraction by a plane grating. A beam of monochromatic light of wavelength λ is incident on a grating and diffracted along several discrete paths. The triangular grooves come out of the page; the rays lie in the plane of the page. The sign convention for the angles α and β is shown by the + and – signs on either side of the grating normal. (a) A reflection grating: the incident and diffracted rays lie on the same side of the grating. (b) A transmission grating: the diffracted rays lie on the opposite side of the grating from the incident ray.

By convention, angles of incidence and diffraction are measured from the grating normal to the beam. This is shown by arrows in the diagrams. In both diagrams, the sign convention for angles is shown by the plus and minus symbols located on either side of the grating normal. For either reflection or transmission gratings, the algebraic signs of two angles differ if they are measured from opposite sides of the grating normal. Other sign conventions exist, so care must be taken in calculations to ensure that results are self-consistent.

Another illustration of grating diffraction, using wavefronts (surfaces of constant phase), is shown in Figure 2-2. The geometrical path dif-ference between light from adjacent grooves is seen to be d sinα + d sinβ. [Since β < 0, the term d sinβ is negative.] The principle of constructive interference dictates that only when this difference equals the wavelength λ of the light, or some integral multiple thereof, will the light from adjacent grooves be in phase (leading to constructive interference). At all other angles, the Huygens wavelets originating from the groove facets will interfere destructively.

Figure 2-2. Geometry of diffraction, showing planar wavefronts. Two parallel rays, labeled 1 and 2, are incident on the grating one groove spacing d apart and are in phase with each other at wavefront A. Upon diffraction, the principle of constructive interference implies that these rays are in phase at diffracted wavefront B if the difference in their path lengths, d sinα + d sinβ, is an integral number of wavelengths; this in turn leads to the grating equation. [Huygens wavelets not shown.]

These relationships are expressed by the grating equation

= d (sinα + sinβ)   (2-1)

which governs the angular locations of the principal intensity maxima when light of wavelength λ is diffracted from a grating of groove spacing d. Here m is the diffraction order (or spectral order),which is an integer. For a particular wavelength λ, all values of m for which |mλ/d| < 2 correspond to propagating (rather than evanescent) diffraction orders. The special case m = 0 leads to the law of reflection β = –α.

It is sometimes convenient to write the grating equation as

Gmλ= sinα + sinβ   (2-2)

where G = 1/d is the groove frequency or groove density, more commonly called "grooves per millimeter".

Eq. (2-1) and its equivalent Eq. (2-2) are the common forms of the grating equation, but their validity is restricted to cases in which the incident and diffracted rays lie in a plane which is perpendicular to the grooves (at the center of the grating). Most grating systems fall within this category, which is called classical (or in-plane) diffraction. If the incident light beam is not perpendicular to the grooves, though, the grating equation must be modified:

Gmλ= cosε (sinα + sinβ)   (2-3)

Here ε is the angle between the incident light path and the plane perpendicular to the grooves at the grating center (the plane of the page in Figure 2-2). If the incident light lies in this plane, ε = 0 and Eq. (2-3) reduces to the more familiar Eq. (2-2). In geometries for which ε ≠ 0, the diffracted spectra lie on a cone rather than in a plane, so such cases are termed conical diffraction.

For a grating of groove spacing d, there is a purely mathematical relationship between the wavelength and the angles of incidence and diffraction. In a given spectral order m, the different wavelengths of polychromatic wavefronts incident at angle α are separated in angle:

β(λ) = sin-1(mλ/d - sinα)   (2-4)

When m = 0, the grating acts as a mirror, and the wavelengths are not separated (β = –α for all λ); this is called specular reflection or simply the zero order.

A special but common case is that in which the light is diffracted back toward the direction from which it came (i.e., α = β ); this is called the Littrow configuration, for which the grating equation becomes

= 2d sinα, in Littrow   (2-5)

In many applications a constant-deviation monochromator mount is used, in which the wavelength  is changed by rotating the grating about the axis coincident with its central ruling, with the directions of incident and diffracted light remaining unchanged. The deviation angle 2K between the incidence and diffraction directions (also called the angular deviation) is

2= αβ = constant   (2-6)

while the scan angle Φ, which varies with λ and is measured from the grating normal to the bisector of the beams, is

2Φ = α + β   (2-7)

Note that Φ changes with λ (as do α and β). In this case, the grating equation can be expressed in terms of Φ and the half deviation angle K as

mλ = 2d cosK sinΦ   (2-8)

Here K is called the half deviation angle because the angle between the incident and diffracted beams is 2K. This version of the grating equation is useful for monochromator mounts (see Chapter 7). Eq. (2-8) shows that the wavelength diffracted by a grating in a monochromator mount is directly proportional to the sine of the scan angle Φ through which the grating rotates, which is the basis for monochromator drives in which a sine bar rotates the grating to scan wavelengths (see Figure 2-3).

Figure 2-3. A sine bar mechanism for wavelength scanning. As the screw is extended linearly by the distance x shown, the grating rotates through an angle Φ in such a way that sinΦ is proportional to x.

For the constant-deviation monochromator mount, the incidence and diffraction angles can be expressed simply in terms of the scan angle Φ and the half-deviation angle K via

α(λ) = Φ(λ) + K   (2-9)

and

β(λ) = Φ(λ) – K   (2-10)

where we show explicitly that α, β and Φ depend on the wavelength .

For footnotes and additional insights into diffraction grating topics like this one, download our free MKS Diffraction Gratings Handbook (8th Edition)