Introduction

Wave optics explains phenomena that geometric (ray) optics cannot: interference fringes, diffraction around obstacles, and the polarization of light. For NEET, wave optics contributes 2–3 questions per year and tests Young's double slit experiment (YDSE) calculations, single slit diffraction patterns, Brewster's law, and Malus's law. Mastering the fringe width formula $\beta = \lambda D/d$ and its variations is the single highest-yield task in this chapter.

Wavefronts and Huygens' Principle

A wavefront is the locus of all points vibrating in the same phase. A point source produces spherical wavefronts; a linear source produces cylindrical wavefronts; at very large distances, wavefronts become planar. Huygens' principle states that every point on a wavefront acts as a new secondary source of spherical wavelets; the next wavefront is the common tangential envelope of these wavelets. This principle is used to derive the laws of reflection and refraction: when a plane wavefront strikes a refracting surface, the change in wave speed causes the wavefront to tilt, directly producing Snell's law $\frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1}$.

Young's Double Slit Experiment (YDSE)

Two coherent slits $S_1$ and $S_2$, separated by distance $d$, illuminate a screen at distance $D$ ($D \gg d$). The path difference at a point $P$ displaced $y$ from the centre is:

$\Delta = \frac{dy}{D} \quad \text{(small angle approximation)}$

Bright fringes (constructive interference) require $\Delta = n\lambda$, giving fringe positions $y_n = \frac{n\lambda D}{d}$ ($n = 0, \pm1, \pm2, \ldots$).

Dark fringes (destructive interference) require $\Delta = (2n-1)\frac{\lambda}{2}$, giving $y_n = \frac{(2n-1)\lambda D}{2d}$ ($n = 1, 2, 3, \ldots$).

Fringe width (spacing between consecutive bright or dark fringes):

$\beta = \frac{\lambda D}{d}$

Dimensional check: $[\beta] = \frac{[L][L]}{[L]} = [L]$, SI unit: metre. Increasing $\lambda$ or $D$ widens fringes; increasing $d$ narrows them. When the apparatus is immersed in a medium of refractive index $n$, the wavelength becomes $\lambda' = \lambda/n$, so $\beta' = \beta/n$.

Intensity Distribution in YDSE

For two slits of equal intensity $I_0$:

$I = 4I_0\cos^2\!\left(\frac{\phi}{2}\right), \quad \text{where } \phi = \frac{2\pi}{\lambda}\Delta$

$I_{\max} = 4I_0$ at $\phi = 0, 2\pi, \ldots$ (constructive); $I_{\min} = 0$ at $\phi = \pi, 3\pi, \ldots$ (destructive). For unequal slit intensities $I_1$ and $I_2$: $I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$. Coherent sources must have the same frequency and a constant (not necessarily zero) phase difference. Two independent bulbs cannot produce sustained interference because their phases fluctuate randomly.

Single Slit Diffraction

A single slit of width $a$ produces a central maximum of angular half-width $\sin\theta = \lambda/a$, so its linear width on a screen at distance $D$ is $2\lambda D/a$. Secondary minima occur at:

$a\sin\theta = n\lambda \quad (n = 1, 2, 3, \ldots)$

Secondary maxima occur at $a\sin\theta = (2n+1)\lambda/2$ and have width $\lambda D/a$ — exactly half the central maximum width. The central maximum is the brightest; the first secondary maximum carries only about 4.5% of the central intensity.

Polarization

Polarization establishes that light is a transverse wave — the electric field vector oscillates perpendicular to the direction of propagation, and it can be restricted to a single plane.

Brewster's law: At the polarizing (Brewster) angle $\theta_p$, reflected light is completely plane-polarized. The condition is $\tan\theta_p = n$, where $n$ is the refractive index of the denser medium. At this angle, the reflected and refracted rays are perpendicular: $\theta_p + \theta_r = 90°$.

Malus's law: Plane-polarized light of intensity $I_0$ incident on an analyser whose transmission axis is at angle $\theta$ to the polarization direction emerges with intensity:

$I = I_0\cos^2\theta$

At $\theta = 90°$ (crossed polaroids), $I = 0$. Inserting a third polaroid at $45°$ between two crossed polaroids allows light through: passing through all three gives $I = I_0/2 \times \cos^2 45° \times \cos^2 45° = I_0/8$.

Key NEET Takeaways

The fringe width formula $\beta = \lambda D/d$ and the effect of changing the medium are tested almost every year. The three-polaroid problem ($I_0/8$ result) and Brewster angle calculations are frequent numericals. Understand the qualitative difference between YDSE (two-source interference with uniform fringes) and single slit diffraction (one aperture with a central maximum twice as wide as secondary maxima). Coherence is a conceptual favourite — two independent sources cannot produce sustained fringes regardless of their proximity.