CBSE Class 12 Physics Revision Notes: Wave Optics

Key Definitions in Wave Optics

Wavefront

A surface connecting all points of a medium oscillating in the same phase at a given instant.

Coherent Sources

Two sources of light are coherent if they emit continuous light waves of the same frequency, same wavelength, and a constant phase difference.

Constructive Interference

When two waves superimpose such that the crest of one falls on the crest of another, resulting in maximum intensity (bright fringe).

Destructive Interference

When two waves superimpose such that the crest of one falls on the trough of another, resulting in minimum intensity (dark fringe).

Diffraction

The bending of light waves around obstacles or through small apertures, spreading into regions where geometric optics predicts a shadow.

Polarisation of Light

The phenomenon of restricting the vibrations of light waves to a single plane perpendicular to the direction of wave propagation.

Unpolarised Light

Light in which electric field vibrations occur in all possible planes perpendicular to the direction of propagation.

Huygens' Principle and Wavefronts

Huygens' principle is a geometrical construction that helps us understand the propagation of light waves. It states that:

1. Every point on a primary wavefront acts as a source of secondary wavelets, which spread out in all directions with the speed of light in that medium.
2. The new wavefront at any later instant is the tangential envelope of all these secondary wavelets.

This principle successfully explains the laws of reflection and refraction. It assumes that light travels in the form of waves and that these waves are transverse in nature. Different types of wavefronts include:

• Spherical wavefront: Originates from a point source of light.
• Cylindrical wavefront: Originates from a linear source (e.g., a slit).
• Plane wavefront: Produced by a distant point source or a distant linear source (when a small portion of spherical or cylindrical wavefront is considered far away from the source).

Interference of Light: Young's Double Slit Experiment (YDSE)

Interference is the phenomenon of redistribution of light energy resulting from the superposition of two or more coherent waves. For sustained interference, the sources must be coherent, monochromatic, and have a constant phase difference.

In Young's Double Slit Experiment (YDSE), light from a single source illuminates two narrow slits, S1 and S2, which act as coherent sources. The waves interfere on a screen placed at a distance D from the slits, forming a pattern of alternating bright and dark fringes.

Conditions for Constructive Interference (Bright Fringes):

• Path Difference (Δx): $nλ$
• Phase Difference (Δφ): $2nπ$

Conditions for Destructive Interference (Dark Fringes):

• Path Difference (Δx): $(n + \frac{1}{2})λ$
• Phase Difference (Δφ): $(2n + 1)π$

Where 'n' is an integer (0, 1, 2, ...), and 'λ' is the wavelength of light.

Fringe Width (β): The distance between two consecutive bright or dark fringes.
$β = \frac{λD}{d}$

• $λ$: wavelength of light
• $D$: distance between slits and screen
• $d$: distance between the two slits

Position of Bright Fringes (y_n):
$y_n = \frac{nλD}{d}$

Position of Dark Fringes (y'_n):
$y'_n = (n + \frac{1}{2})\frac{λD}{d}$

Diffraction of Light: Single Slit

Diffraction is the bending of light waves around the corners of an obstacle or aperture into the region of geometrical shadow. Fraunhofer diffraction occurs when the source and screen are effectively at infinite distances from the obstacle (often achieved using lenses). Fresnel diffraction occurs when the source or screen, or both, are at finite distances.

Single Slit Diffraction: When monochromatic light passes through a single narrow slit of width 'a', it produces a diffraction pattern on a screen. This pattern consists of a central bright maximum (much wider and more intense than subsequent maxima) flanked by alternating dark and bright minima/maxima.

Conditions for Minima (Dark Fringes):
$a \sin θ = nλ$
Where 'n' is an integer (1, 2, 3, ...), $a$ is slit width, $θ$ is the angle of diffraction.

Conditions for Secondary Maxima (Bright Fringes):
$a \sin θ = (n + \frac{1}{2})λ$
Where 'n' is an integer (1, 2, 3, ...).

Width of Central Maximum: $2y_1 = \frac{2λD}{a}$

• $λ$: wavelength of light
• $D$: distance between slit and screen
• $a$: width of the single slit

Intensity Distribution: The intensity of secondary maxima decreases rapidly as 'n' increases, unlike in interference where all bright fringes have nearly equal intensity (for ideal coherent sources).

Polarisation of Light

Polarisation is the phenomenon that demonstrates the transverse nature of light waves. Unpolarised light has electric field vibrations in all possible planes perpendicular to the direction of propagation. A polariser is a device that converts unpolarised light into plane-polarised light by allowing vibrations only in a specific plane.

Brewster's Law: When unpolarised light is incident on an interface separating two transparent media, the reflected light is completely plane-polarised when the reflected and refracted rays are perpendicular to each other. The angle of incidence at which this occurs is called Brewster's angle (i_p).

$μ = \tan i_p$
Where $μ$ is the refractive index of the second medium with respect to the first.

Malus's Law: When plane-polarised light is passed through an analyser (another polariser), the intensity of the transmitted light varies as the square of the cosine of the angle between the transmission axes of the polariser and the analyser.

$I = I_0 \cos^2 θ$
Where $I_0$ is the intensity of the plane-polarised light incident on the analyser, and $I$ is the intensity of the transmitted light. If unpolarised light of intensity $I_u$ is incident on the first polariser, the intensity of light transmitted through it is $I_0 = I_u/2$.

Interference vs. Diffraction

Worked Example: Young's Double Slit Experiment

• {"title":"Fringe Width Calculation","problem":"In a YDSE arrangement, 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 1.2 cm. Determine the wavelength of light used.","solution":"Given: $d = 0.28 \\times 10^{-3} m$, $D = 1.4 m$. Distance to 4th bright fringe, $y_4 = 1.2 \\times 10^{-2} m$.\nFor 4th bright fringe, $y_4 = \\frac{4λD}{d}$.\nRearranging for $λ$: $λ = \\frac{y_4 d}{4D}$\n$λ = \\frac{(1.2 \\times 10^{-2} m) \\times (0.28 \\times 10^{-3} m)}{4 \\times (1.4 m)}$\n$λ = \\frac{0.00336 \\times 10^{-3}}{5.6} = 6 \\times 10^{-7} m = 600 nm$\nAnswer: The wavelength of light used is 600 nm."}

Key Points to Remember for Exams

• Huygens' principle is a geometric construction, not an explanation of the physical origin of light.
• For sustained interference, sources must be coherent (constant phase difference, same frequency).
• Interference fringes are of equal width and nearly equal intensity (in YDSE).
• Diffraction causes the central maximum to be brightest and widest; secondary maxima rapidly decrease in intensity.
• Polarisation is the definitive proof of the transverse nature of light waves.
• Brewster's angle applies to complete polarisation of reflected light when reflected and refracted rays are perpendicular ($μ = \tan i_p$).
• Malus's Law ($I = I_0 \cos^2 θ$) describes the intensity of polarised light passing through an analyser.
• Effect of placing a thin transparent sheet in YDSE: causes fringe shift but no change in fringe width.
• When YDSE is immersed in water, wavelength decreases ($λ' = λ/μ$), so fringe width also decreases ($β' = β/μ$).
• Diffraction is observed when the aperture size is comparable to the wavelength of light.

Exam Tip: Differentiating Interference and Diffraction

A common mistake in exams is confusing interference and diffraction. While both involve superposition, remember the key differences: Interference is typically due to two separate wavefronts from coherent sources, leading to evenly spaced, equally intense fringes. Diffraction involves superposition of secondary wavelets from different parts of the same wavefront passing through a single aperture, resulting in a broad, bright central maximum with rapidly decreasing intensity for subsequent maxima. Pay attention to the conditions for bright/dark fringes and the intensity distribution in both phenomena. Clearly stating the formula for fringe width ($β = \frac{λD}{d}$) for interference and central maximum width ($2β_0 = \frac{2λD}{a}$) for diffraction can earn you full marks.

Practice Questions with Solutions

• Q: What is the condition for destructive interference in terms of path difference? A: The path difference for destructive interference is an odd multiple of half the wavelength, i.e., $(n + \frac{1}{2})λ$.
• Q: How does the fringe width in YDSE change if the entire setup is immersed in water? A: The fringe width decreases because the wavelength of light decreases in water ($λ' = λ/μ$), and fringe width $β$ is directly proportional to $λ$.
• Q: State Brewster's Law. A: Brewster's Law states that when unpolarised light is incident at a specific angle (Brewster's angle) on a transparent surface, the reflected light is completely plane-polarised, and the reflected and refracted rays are perpendicular. Mathematically, $μ = \tan i_p$.
• Q: Why is the central maximum in single-slit diffraction twice as wide as the secondary maxima? A: The central maximum spans from the first minimum on one side ($n=1$) to the first minimum on the other side ($n=-1$), effectively covering twice the angular width compared to the region between a secondary maximum and its adjacent minimum.

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