Wave Optics Class 12 Physics Chapter Notes

Key Terms in Wave Optics

Wavefront

The locus of all points in a medium that are vibrating in the same phase. The speed with which the wavefront moves outwards from the source is the speed of the wave.

Huygens' Principle

A principle stating that every point on a wavefront is a source of secondary spherical wavelets, and the new wavefront at any later time is the forward envelope of these wavelets.

Coherent Sources

Two sources of light are said to be coherent if they emit light waves of the same frequency, same wavelength, and have a constant phase difference between them.

Interference

The modification in the distribution of light intensity in the region of superposition of two or more waves. It can be constructive (increased intensity) or destructive (decreased intensity).

Diffraction

The phenomenon of bending of light waves around the sharp corners of an obstacle or aperture and their consequent spreading into the regions of the geometrical shadow.

Fringe Width (β)

The separation between the centers of two consecutive bright (maxima) or dark (minima) fringes in an interference pattern. Formula: β = λD/d.

Polarization

The phenomenon of restricting the vibrations of a transverse wave (like light) to a single direction in a plane perpendicular to the direction of wave propagation.

Brewster's Law

States that for a given medium, the refractive index (μ) is equal to the tangent of the polarizing angle (i_p). At this angle, the reflected light is completely plane-polarized. Formula: μ = tan(i_p).

Huygens' Principle Explained

Huygens' principle is a cornerstone of wave optics, providing a geometrical method to visualize and predict wave propagation. The principle can be broken down into two main parts:

1. Every point on a primary wavefront acts as a source of new disturbance. These new disturbances are called secondary wavelets, which travel outwards in all directions with the speed of the wave in that medium.
2. The new wavefront at any later instant is the forward envelope (the common tangent) of all these secondary wavelets at that instant.

This principle beautifully explains phenomena that ray optics cannot. For example, using this principle, one can derive the laws of reflection and refraction for a plane wave incident on a surface. For reflection, the angle of incidence is proven equal to the angle of reflection (i = r). For refraction, Snell's law (μ₁sin(i) = μ₂sin(r)) is derived by considering the different speeds of the secondary wavelets in the two media. Huygens' construction also naturally leads to the understanding of diffraction, as the secondary wavelets can spread into the geometrical shadow region when encountering an obstacle, explaining the bending of light.

Formula Sheet for Wave Optics

• Phase & Path Difference: Δφ = (2π/λ) × Δx, where Δφ is phase difference and Δx is path difference.
• Constructive Interference (YDSE): Path difference Δx = nλ. Position of bright fringe y_n = nλD/d.
• Destructive Interference (YDSE): Path difference Δx = (n + 1/2)λ. Position of dark fringe y_n = (n + 1/2)λD/d.
• Fringe Width (YDSE): β = λD/d. It is independent of the order 'n'.
• Angular Fringe Width (YDSE): θ = β/D = λ/d.
• Diffraction Minima (Single Slit): a sinθ = nλ, where 'a' is the slit width.
• Diffraction Secondary Maxima (Single Slit): a sinθ = (n + 1/2)λ.
• Width of Central Maximum (Diffraction): Width = 2λD/a. It is twice the width of other secondary maxima.
• Brewster's Law: μ = tan(i_p), where i_p is the polarizing angle.
• Malus' Law: I = I₀ cos²θ, where I₀ is the intensity of polarized light incident on the analyser, and θ is the angle between the pass axes of the polariser and analyser.

Interference vs. Diffraction

Worked Example: YDSE

• {"header":"Calculating Fringe Width","bodyMarkdown":"In a Young's double-slit experiment, the slits are separated by 0.28 mm and the screen is placed 1.4 m away. The distance between the central bright fringe and the fourth bright fringe is measured to be 1.2 cm. Determine the wavelength of light used.\n\nSolution:\n\n Given: d = 0.28 mm = 0.28 × 10⁻³ m, D = 1.4 m.\n Distance of 4th bright fringe from center, y₄ = 1.2 cm = 1.2 × 10⁻² m.\n We know the position of the n-th bright fringe is y_n = nλD/d.\n For the 4th bright fringe (n=4): y₄ = 4λD/d.\n Rearranging for wavelength (λ): λ = (y₄ d) / (4 D)\n λ = (1.2 × 10⁻² m 0.28 × 10⁻³ m) / (4 1.4 m)\n λ = (0.336 × 10⁻⁵) / 5.6\n λ = 0.06 × 10⁻⁵ m = 6 × 10⁻⁷ m = 600 nm.\n\nAnswer: The wavelength of the light is 600 nm."}

Board Exam Traps

Formula Confusion: A common mistake is swapping formulas between interference and diffraction. Remember, for YDSE, d is the distance between the two slits. For single-slit diffraction, a is the width of the single slit. Also, the condition for minima in diffraction (a sinθ = nλ) looks similar to the condition for maxima in interference (d sinθ = nλ). Create a formula chart to keep them separate.

Intensity Graphs: Be prepared to draw and label the intensity distribution graphs for both YDSE and single-slit diffraction. For interference, show equally spaced peaks of equal height. For diffraction, draw a large central peak with much smaller, rapidly decaying secondary peaks on either side.

Practice Questions with Solutions

• What are coherent sources and why are they necessary for observing a stable interference pattern? Coherent sources are sources that emit light of the same frequency with a constant phase difference. They are necessary so that the positions of maxima and minima do not change with time, resulting in a stable, observable pattern.
• On what factors does the fringe width in YDSE depend? Fringe width (β = λD/d) depends directly on the wavelength of light (λ) and the distance to the screen (D), and inversely on the distance between the slits (d).
• State Brewster's Law. Brewster's Law states that the refractive index (μ) of a transparent medium is equal to the tangent of the polarizing angle (i_p). At this angle of incidence, the reflected light is completely plane-polarized. (μ = tan(i_p)).
• Why is the central fringe in a single-slit diffraction pattern so much brighter and wider than the others? The central maximum's width corresponds to the region where wavelets from the entire slit width arrive roughly in phase, leading to strong constructive interference. For secondary maxima, the slit is conceptually divided into sections, and cancellation between wavelets from different sections reduces the overall intensity.

Frequently Asked Questions

Frequently Asked Questions