Question

Show that the dispersion of a grating is ${\text{D}} = (\tan \theta )/\lambda$.

Answer 1

Dispersion is a phenomena in optics where the phase velocity of a wave is affected by its frequency. Dispersive media are those that have this trait in common.For specificity, the phrase chromatic dispersion is also employed. A rainbow is arguably the most well-known illustration of dispersion.

Complete answer:
The amount of variation in diffraction angle per unit change in wavelength is known as angular dispersion. It's a measurement of the angular spacing between neighbouring wavelengths' beams. Differentiating the grating equation while keeping the angle constant yields a formula for angular dispersion.

Diffraction gratings are used to divide or scatter light into different wavelengths. The relationship between a change in diffraction angle and a slight change in wavelength is known as angular dispersion. Many diffracted orders will be generated by gratings with a low groove frequency. A grating can be used as a beam splitter for monochromatic light, such as from a laser. The maxima (lines) at angles theta come from diffraction by N (multiple) slits,
$d\sin (\theta ) = m\lambda$
where d is the slit-slit spacing.
$D = \dfrac{{d\theta }}{{d\lambda }}$

Differentiate both sides of (1) with respect to theta, and you'll get the dispersion of a grating.
$d\cos (\theta )\dfrac{{d\theta }}{{d\lambda }} = m$
$\Rightarrow \dfrac{{d\theta }}{{d\lambda }} = \dfrac{m}{{d\cos (\theta )}}$
Substituting from $d\sin (\theta ) = m\lambda$ we get,
$\dfrac{{d\theta }}{{d\lambda }} = \dfrac{{d\sin (\theta )}}{{d\cos (\theta )\lambda }}$
$\Rightarrow \dfrac{{d\theta }}{{d\lambda }} = \dfrac{{\tan (\theta )}}{\lambda }$
Hence from the definition of the dispersion we have ${\text{D}} = (\tan \theta )/\lambda$
$\therefore D = \dfrac{{\tan (\theta )}}{\lambda }$
Hence proved.

Note: Diffraction gratings are commonly employed in spectroscopic equipment to convert white light into monochromatic light. This is accomplished by taking use of the grating's capacity to distribute light of various wavelengths into various angles. The well-known grating equation describes the relationship between the incidence and diffraction angles, as well as the wavelength. Many key spectroscopic parameters, including dispersion, resolution, and free spectral range, may be obtained from the grating equation using very simple algebraic operations.