A diffraction grating is a device that makes use of the diffraction pattern created by multiple evenly spaced slits. It comprises of an opaque screen with several equidistant slits. When light passes through each slit on a diffraction grating, it undergoes diffraction and gets spread out into multiple beams that interfere with each other and overlap, producing a distinctive pattern of bright and dark bands called a diffraction pattern. Figure 1 shows a setup that uses a diffraction grating.

Figure 1: Setup with a diffraction grating

In practice, a transparent sheet is used to create a diffraction grating by ruling parallel grooves onto it using a diamond point operated with the help of an electronically controlled mechanism. These grooves function as opaque areas. High-quality gratings can feature as many as 10,000 slits per centimeter and the size of the grating can measure up to 50 x 75 cm². Achieving such precision is essential in grating technology. Also, commercially available gratings are plastic replicas of the original ruled surface. Light diffraction using a diffraction grating is shown in figure 2.

Figure 2: Light diffraction using diffraction grating

When a plane light wave with a wavelength λ is incident normally on a diffraction grating, the positions of the principal maxima formed on the other side can be determined by the equation:

This relation, known as the grating equation, is used to calculate the wavelength λ by observing the angle θ_n of the n^th order maxima. The grating element d is a fixed geometric constant. If the light source contains multiple wavelengths, the angle θ_n varies with λ. However, the zero-order maximum occurs at θ_n = 0 for all wavelengths. Higher-order maxima for different wavelengths are separated nearly by the grating. Thus, by measuring the diffraction angles for various colors, we can determine their corresponding wavelengths.

Free Spectral Range

In the case of a polychromatic light source, there is a possibility that at a particular position θ, the maxima of two or more different wavelengths overlap, leading to:

To determine the spectral range (i.e., the range of wavelengths) without overlap between the maxima of orders n and (n+1), consider the corresponding shortest and longest wavelengths, λ₁ and λ₂, respectively. The maximum angle at which the n^th order maxima is formed is due to the longest wavelength, λ₂.

And, the minimum angle at which (n+1)^th order maxima is formed due to shortest wavelength λ₁:

If θ_n+1 is greater than or equal to θ_n, there will be no overlap between the n^th and (n+1)^th order maxima. The equality condition represents the limiting case where:

or

The difference between the values of λ₂ and λ₁ is referred to as the free spectral range of a grating, which varies with the order n and is smaller at higher orders.

Dispersion by a Grating

Dispersion refers to the angular separation of different wavelengths. The angular dispersion of a grating is defined by differentiating the grating equation, dsinθ_n = nλ, as:

Combining grating equation with the above equation gives,

As per the definition, the angular dispersion of a grating is actually independent of the grating element d. However, it increases rapidly with θ_n, resulting in larger angular separation between different wavelengths at higher diffraction angles.

When θ_n is small, i.e., when cos θ_n = 1, it can be shown that (dθ_n/dλ) = n/d is constant. This type of spectrum is called a normal spectrum, where the angular separation dθ between two wavelengths is proportional to the difference dλ between them.

Resolving Power of Grating

The resolving power of a grating is a measure of its ability to separate closely spaced wavelengths and produce distinct peaks of maxima. Mathematically, it can be defined as:

Where ∆λ is the minimum wavelength separation that can be resolved. Therefore, a smaller value of ∆λ implies a greater resolving power for the grating.

The limit of resolution is determined by Rayleigh's criterion, which states that two wavelengths λ and (λ+∆λ) can be considered just resolved if the maximum of (λ+∆λ) falls on the first minimum of the wavelength λ. This criterion is specific to a particular order n of the spectrum.

The position of the n^th order maximum of (λ+∆λ) can be expressed as θ_n, which satisfies the equation:

Therefore, the two wavelengths can be resolved if

or

Thus, the number of slits N ruled on the grating has a linear relationship with the resolving power of the grating. Also, the resolving power is greater for spectra of higher order.

Applications of Diffraction Gratings

Diffraction gratings are commonly used in spectroscopy to separate light into its component wavelengths. This can be used to study the composition of materials, such as in astronomy where the light from stars is analyzed to determine their chemical makeup.

They are used in laser technology to generate laser beams with very specific wavelengths. It is useful in telecommunications, where fiber optic cables use lasers to transmit data.

These gratings can be used to produce holographic images by splitting a laser beam and using one part to illuminate an object, while the other part is used as a reference beam. The interference pattern between the two beams creates a holographic image of the object.

Diffraction gratings are also used in telescopes to study the light from celestial objects. Astronomers can determine the composition and temperature of stars and galaxies by analyzing the spectrum of light.

They can be used in optical communications to separate different wavelengths of light, allowing for multiple channels of information to be transmitted simultaneously over a single fiber optic cable.