Class 11 > Unit # 13: Physical Optics > Diffraction of Light & Grating

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Diffraction Grating: Definition, Explanation & Derivation

Video Lecture: Diffraction Grating

Watch the complete step-by-step video lecture explaining the structure of a diffraction grating and the derivation of the grating equation:

Definition of Diffraction Grating

“A diffraction grating is an optical device consisting of a very large number of parallel, equidistant slits of equal width.”

Practically, a diffraction grating is constructed on a glass plate by ruling a large number of parallel, fine opaque lines using a diamond point. The transparent space between any two consecutive opaque lines acts as a slit.

Grating Element: The distance between the centers of two adjacent slits (equal to the sum of the width of a transparent slit $b$ and an opaque line $a$) is called the grating element $d$:

$d = a + b$

Alternatively, if a glass plate of length $l$ contains $N$ total lines, the grating element is given by $d = \frac{l}{N}$.

Fig: Schematic representation of a diffraction grating with grating element $d = a + b$.

Explanation and Working Principle

When a parallel beam of monochromatic light is normally incident upon the diffraction grating, secondary wavelets originate from each transparent slit. A convex lens placed in the path focuses these diffracted waves onto a screen.

Waves traveling in a specific direction $\theta$ interfere with one another. If the path difference between waves originating from any two adjacent slits is equal to an integral multiple of the wavelength $\lambda$, they reinforce each other constructively to form a bright principal maximum.

Derivation of the Grating Equation

Consider two corresponding points on adjacent slits separated by the grating element $d = a + b$. The path difference between light rays diffracted at an angle $\theta$ from two adjacent slits is given by:

$\text{Path Difference} = (a + b) \sin\theta = d \sin\theta$

For constructive interference (principal maxima), the path difference must be an integral multiple of the wavelength $\lambda$:

$d \sin\theta = m\lambda \quad \text{or} \quad (a + b) \sin\theta = m\lambda$

where $m$ is the order of the spectrum ($m = 0, \pm 1, \pm 2, \pm 3, \dots$).

• Zero-Order Maximum ($m = 0$): At $\theta = 0^\circ$, the path difference is zero, forming the central bright maximum.
• First-Order Maximum ($m = 1$): Occurs when $d \sin\theta = 1\lambda$.
• Second-Order Maximum ($m = 2$): Occurs when $d \sin\theta = 2\lambda$.
• $m^{\text{th}}$ Order Maximum: Occurs at angle $\theta_m$ given by $d \sin\theta_m = m\lambda$.

A diffraction grating provides very sharp and distinct spectral lines because a large number of slits interfere constructively, making it an extremely powerful tool for precise wavelength measurements.

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