12.2 Wave Theory of Light, Interference & Young's Double-Slit Experiment

Wave Theory of Light, Interference & Young's Double-Slit Experiment

In the Intermediate Physics Part I syllabus and the UET ECAT entry test, physical optics represents one of the most conceptually demanding and frequently examined domains. While geometric optics treats light as rectilinear rays, physical optics treats light as an electromagnetic wave phenomenon. ECAT problems require rapid calculation of Young's fringe spacing, thin-film optical path shifts, phase changes upon reflection, and Newton's rings geometry without numerical calculators.

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1. Wavefronts and Huygens' Principle

Definition and Types of Wavefronts

A wavefront is defined as the continuous locus of all adjacent points in a wave field that oscillate with identical phase. A light ray is an oriented line drawn perpendicular to the wavefront representing the local direction of energy propagation.

Wavefront geometries depend on source topology and propagation distance:

1. Spherical Wavefront: Emanates from an idealized isotropic point source in a homogeneous, isotropic medium. As wave energy propagates radially outward, the surfaces of constant phase are concentric spherical shells.
2. Cylindrical Wavefront: Emanates from an extended linear slit source. The surfaces of constant phase form coaxial cylinders.
3. Plane Wavefront: When a spherical or cylindrical wavefront travels a very large distance from its source, any small segment of the expanding curved surface approximates a flat plane. A parallel beam of light (such as sunlight reaching Earth) corresponds to plane wavefronts.

Huygens' Principle

In 1678, the Dutch physicist Christiaan Huygens formulated a geometrical construction to predict the future position of a wavefront:

1. Every point on a primary wavefront acts as a secondary source of spherical wavelets, which spread out in all forward directions with a speed equal to the propagation speed of the wave in that medium.
2. The new position of the wavefront at a later time $t + \Delta t$ is the forward surface of tangency (envelope) that touches all these secondary wavelets.

Applying Huygens' principle explains the laws of reflection ($\theta_i = \theta_r$) and the laws of refraction (Snell's law: $n_1\sin\theta_1 = n_2\sin\theta_2$). When a plane wavefront enters an optically denser medium ($n_2 > n_1$), the wavelets travel slower ($v = \frac{c}{n}$), bending the advancing envelope toward the normal.

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2. Interference of Light & Coherence Conditions

Interference is the physical phenomenon whereby two or more overlapping light waves superpose to produce regions of enhanced brightness (constructive interference) and reduced or zero brightness (destructive interference).

Conditions for Observable, Sustained Interference

To produce a stationary, high-contrast interference pattern on a screen, the interfering beams must satisfy four strict physical criteria:

1. Coherence: The two light sources must maintain a constant phase difference over time. Two independent incandescent lamps, candles, or unlinked LEDs can never produce interference because emission occurs via spontaneous de-excitation of independent atoms lasting roughly $10^{-8}\text{ s}$, creating random phase fluctuations.
2. Monochromaticity: The light waves must possess a single, identical wavelength $\lambda$ (and frequency $f$). If polychromatic white light is used, different wavelengths form overlapping fringes of different widths, washing out contrast except for a few colored bands.
3. Equal or Comparable Amplitudes: If amplitudes $A_1 \approx A_2$, the resultant minimum intensity $I_{\min} \propto (A_1 - A_2)^2 \approx 0$, yielding total darkness and maximum fringe contrast.
4. Small Slit Separation & Large Screen Distance: The slit separation $d$ must be small (fractions of a millimeter) and screen distance $L$ large (a meter or more) to ensure wide, distinguishable fringes.

Path Difference and Phase Difference

If two coherent waves of wavelength $\lambda$ travel distances $x_1$ and $x_2$ to reach a point $P$, their path difference is:

$\Delta x = |x_2 - x_1|$

The corresponding phase difference $\Delta \phi$ is related by:

$\Delta \phi = \frac{2\pi}{\lambda}\Delta x$

• Constructive Interference (Bright Fringes / Maxima): Occurs when the two waves arrive in identical phase (crest meets crest, trough meets trough). This requires the path difference to be an integer multiple of the wavelength:

$\Delta x = m\lambda \quad (m = 0, \pm 1, \pm 2, \pm 3, \dots)$

$\Delta \phi = 2m\pi$

• Destructive Interference (Dark Fringes / Minima): Occurs when the two waves arrive in opposite phase (crest meets trough). This requires the path difference to be an odd half-integer multiple of the wavelength:

$\Delta x = \left(m + \frac{1}{2}\right)\lambda \quad (m = 0, \pm 1, \pm 2, \dots)$

$\Delta \phi = (2m + 1)\pi$

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3. Young's Double-Slit Experiment (YDSE)

In 1801, Thomas Young performed the historic double-slit experiment that conclusively established the wave nature of light. By passing sunlight through a single pinhole and then dividing it between two parallel pinholes $S_1$ and $S_2$, Young created two coherent sources from a single primary source.

Analytical Derivation of Fringe Positions

Consider two narrow parallel slits $S_1$ and $S_2$ separated by distance $d$. A screen is placed at a distance $L$ parallel to the plane of the slits ($L \gg d$). Let $O$ be the central point on the screen equidistant from $S_1$ and $S_2$, and let $P$ be an arbitrary observation point at distance $y$ from $O$.

The optical path difference between the waves arriving at $P$ from $S_2$ and $S_1$ is:

$\Delta x = S_2 P - S_1 P$

Dropping a perpendicular from $S_1$ onto $S_2 P$, for $L \gg d$, the angle $\theta$ between the normal to the slits and the line to $P$ gives:

$\Delta x = d\sin\theta$

From screen geometry, $\tan\theta = \frac{y}{L}$. Because $L \gg y$, the angle $\theta$ is small, permitting the small-angle approximation $\sin\theta \approx \tan\theta \approx \theta$ (in radians):

$\Delta x \approx d\left(\frac{y}{L}\right)$

1. Positions of Bright Fringes (Maxima)

Setting $\Delta x = m\lambda$:

$d\left(\frac{y_m}{L}\right) = m\lambda \implies y_m = m\frac{\lambda L}{d} \quad (m = 0, \pm 1, \pm 2, \dots)$

• Central maximum ($m = 0$): $y_0 = 0$ (at point $O$).
• First bright fringe ($m = 1$): $y_1 = \frac{\lambda L}{d}$.
• Second bright fringe ($m = 2$): $y_2 = \frac{2\lambda L}{d}$.

2. Positions of Dark Fringes (Minima)

Setting $\Delta x = (m + \frac{1}{2})\lambda$:

$d\left(\frac{y_m'}{L}\right) = \left(m + \frac{1}{2}\right)\lambda \implies y_m' = \left(m + \frac{1}{2}\right)\frac{\lambda L}{d} \quad (m = 0, \pm 1, \pm 2, \dots)$

• First dark fringe ($m = 0$): $y_0' = \frac{1}{2}\frac{\lambda L}{d}$.
• Second dark fringe ($m = 1$): $y_1' = \frac{3}{2}\frac{\lambda L}{d}$.

3. Fringe Spacing (Fringe Width $\Delta y$)

The distance between any two consecutive bright fringes or two consecutive dark fringes is defined as the fringe width $\Delta y$:

$\Delta y = y_{m+1} - y_m = (m+1)\frac{\lambda L}{d} - m\frac{\lambda L}{d} = \frac{\lambda L}{d}$

Similarly, for dark fringes:

$\Delta y = y_{m+1}' - y_m' = \left(m + \frac{3}{2}\right)\frac{\lambda L}{d} - \left(m + \frac{1}{2}\right)\frac{\lambda L}{d} = \frac{\lambda L}{d}$

Properties of Young's Fringes

1. Equidistant Spacing: All bright and dark fringes are equally spaced across the center of the pattern.
2. Proportionality to Wavelength: Fringe width is directly proportional to wavelength ($\Delta y \propto \lambda$). Red light ($\lambda \approx 700\text{ nm}$) produces wider fringes than blue light ($\lambda \approx 400\text{ nm}$).
3. Proportionality to Geometry: Fringe width increases with screen distance ($\Delta y \propto L$) and decreases as slit separation increases ($\Delta y \propto \frac{1}{d}$).

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4. Effect of Refractive Index on Interference

When the entire YDSE apparatus is immersed in a liquid of refractive index $n$ ($n > 1$):

1. The frequency $f$ remains constant (governed by the light source).
2. The speed of light decreases to $v = \frac{c}{n}$.
3. The wavelength inside the medium shrinks proportionally:

$\lambda' = \frac{\lambda}{n}$

Consequently, the new fringe spacing $\Delta y'$ inside the medium becomes:

$\Delta y' = \frac{\lambda' L}{d} = \frac{(\lambda / n) L}{d} = \frac{\Delta y}{n}$

ECAT Key Result: Immersing the double-slit apparatus into water ($n = 1.33 = \frac{4}{3}$) compresses the fringe pattern by a factor of $n$: $\Delta y_{\text{water}} = \frac{3}{4}\Delta y_{\text{air}}$.

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5. Stokes' Law: Phase Change on Reflection

In 1849, Sir George Gabriel Stokes established a fundamental wave reflection theorem:

• When a light wave propagating in an optically rarer medium reflects at the boundary of an optically denser medium ($n_2 > n_1$), it undergoes an intrinsic phase reversal of $\pi$ radians ($180^\circ$).
• A phase shift of $\pi$ is equivalent to an extra optical path difference of half a wavelength: $\Delta x_{\text{Stokes}} = \frac{\lambda}{2}$.
• When light reflects from an optically rarer medium ($n_2 < n_1$), no phase change occurs.

This behavior is physically identical to a transverse wave on a string reflecting from a fixed boundary (which inverts phase by $180^\circ$) versus a free boundary (which reflects without phase inversion).

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6. Thin-Film Interference

When light strikes a thin transparent film (such as an oil slick on water or a soap bubble) of physical thickness $t$ and refractive index $\mu$, light reflects from both the top boundary and the bottom boundary. Superposition of these two reflected rays produces interference colors.

Effective Path Difference in a Thin Film

For a ray incident at angle $i$ and refracted at angle $r$, the geometric path difference inside the film is $2\mu t\cos r$. Under near-normal incidence ($\cos r \approx 1$), the mechanical round-trip path is $2\mu t$.

Consider a soap film in air (any thin film whose refractive index is greater than that of the medium on both sides):

1. The top reflection occurs at a denser medium $\implies$ undergoes a $\pi$ phase shift (path addition of $\frac{\lambda}{2}$).
2. The bottom reflection occurs at a rarer medium $\implies$ undergoes zero phase shift.

Because exactly one of the two reflections undergoes a phase inversion, the net optical path difference is:

$\Delta = 2\mu t \cos r + \frac{\lambda}{2}$

Modified Interference Conditions for Reflected Light

• Constructive Interference (Bright Film):

$2\mu t\cos r + \frac{\lambda}{2} = m\lambda \implies 2\mu t\cos r = \left(m - \frac{1}{2}\right)\lambda \quad (m = 1, 2, 3, \dots)$

Relabelling $m - 1$ as $m$ gives the familiar form $2\mu t\cos r = \left(m + \frac{1}{2}\right)\lambda$ with $m = 0, 1, 2, \dots$; both describe the same set of bright films.

• Destructive Interference (Dark Film):

$2\mu t\cos r + \frac{\lambda}{2} = \left(m + \frac{1}{2}\right)\lambda \implies 2\mu t\cos r = m\lambda$

ECAT Key Case (Zero Thickness Film): As a soap bubble becomes extremely thin just before bursting ($t \to 0$), the geometric path $2\mu t \to 0$. The net path difference is solely due to Stokes' reflection: $\Delta = \frac{\lambda}{2}$. Destructive interference occurs for all visible wavelengths, making the top edge of a draining soap film appear completely black in reflected light.

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7. Newton's Rings

When a plano-convex lens with a large radius of curvature $R$ is placed on an optical flat glass plate, a wedge-shaped air film of progressively increasing thickness is formed between them.

When illuminated with monochromatic light at normal incidence, circular concentric alternate bright and dark interference rings—known as Newton's rings—are observed around the point of contact.

Why the Central Spot is Dark in Reflected Light

At the central point of contact between the lens and the glass plate:

• The thickness of the air film is zero ($t = 0$).
• The ray reflecting from the bottom curved surface of the lens undergoes no phase change (glass to air, rarer boundary).
• The ray reflecting from the top surface of the flat plate undergoes a $\pi$ phase change (air to glass, denser boundary).
• The path difference is $\Delta = 0 + \frac{\lambda}{2} = \frac{\lambda}{2}$.

Because the path difference is exactly half a wavelength, destructive interference occurs. Therefore, the central spot is always dark in reflected light.

Quantitative Radii of Newton's Rings

From the geometry of a circle of radius $R$ containing an air film of thickness $t$ at radial distance $r$, by the chord theorem:

$r^2 = t(2R - t) \approx 2R t \implies t = \frac{r^2}{2R}$

Substituting $t$ into the interference conditions for an air film ($\mu = 1$, normal incidence $\cos r = 1$):

1. Dark Rings (Destructive Interference):

$2t = m\lambda \implies 2\left(\frac{r_m^2}{2R}\right) = m\lambda \implies r_m^2 = m\lambda R$

$r_m = \sqrt{m\lambda R} \quad (m = 0, 1, 2, 3, \dots)$

• The radius of the $m$-th dark ring is directly proportional to the square root of the ring order: $r_m \propto \sqrt{m}$.
• Radii ratio of successive dark rings: $r_1 : r_2 : r_3 : r_4 = \sqrt{1} : \sqrt{2} : \sqrt{3} : \sqrt{4} = 1 : 1.414 : 1.732 : 2$.

1. Bright Rings (Constructive Interference):

$2t = \left(m + \frac{1}{2}\right)\lambda \implies r_m' = \sqrt{\left(m + \frac{1}{2}\right)\lambda R} \quad (m = 0, 1, 2, \dots)$

Transmitted Light System

In transmitted light, no phase shift occurs upon transmission. Consequently, the conditions for bright and dark rings are exactly reversed: the central spot in transmitted light is bright, and the transmitted pattern is strictly complementary to the reflected pattern.

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8. Michelson's Interferometer

The Michelson interferometer is a high-precision optical instrument designed by Albert A. Michelson in 1881 to measure tiny displacements and wavelengths of light by splitting a beam into two perpendicular paths.

Working Principle

1. Light from an extended source strikes a half-silvered glass beam splitter oriented at $45^\circ$.
2. Exactly $50\%$ of the light is reflected along Path 1 to movable mirror $M_1$, and $50\%$ is transmitted along Path 2 to fixed mirror $M_2$.
3. A compensating glass plate of identical thickness is placed in Path 2 to ensure both beams traverse identical thicknesses of glass.
4. Both beams reflect normally from their respective mirrors and recombine at the beam splitter, entering an observer's telescope where interference fringes are viewed.

Precision Distance Formula

If mirror $M_1$ is displaced linearly by distance $d$, the optical path length changes by $2d$ (because light travels back and forth). Each time the path changes by one full wavelength $\lambda$, one fringe shifts across the crosshairs of the telescope:

$2d = m\lambda \implies d = m\frac{\lambda}{2}$

This equation allows determination of unknown wavelengths to within nanometer precision, and Michelson used it in 1892–93 to measure the standard metre in wavelengths of cadmium red light (the metre was later defined by a krypton-86 wavelength in 1960 and, since 1983, by the speed of light).

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9. Interference Conditions Reference Table