Superposition of Waves and Interference of Light by Wave Front Splitting and Amplitude Splitting

Definition

Superposition of waves is the principle that when two or more waves overlap at a point, the resultant displacement at that point is the algebraic sum of the individual displacements.

Interference of light is the redistribution of light intensity produced when two coherent light waves superpose, resulting in alternating bright and dark regions.

Wave front splitting is an interference method in which a single wave front from a source is divided into separate parts that travel along different paths and later overlap to form an interference pattern.

Amplitude splitting is an interference method in which a single light beam is divided into two or more beams by partial reflection or refraction, and the beams later recombine to produce interference.

Main Content

1. Superposition Principle and Resultant Wave

• When two waves meet, the displacement at any point is the sum of the displacements due to each wave individually.
• If the waves are in phase, their amplitudes add and the resulting intensity increases; if they are out of phase, they may cancel partially or completely, reducing the intensity.

For two harmonic waves of the same frequency: $$y_1 = a_1 \sin(\omega t), \quad y_2 = a_2 \sin(\omega t + \phi)$$ their resultant wave has amplitude $$A = \sqrt{a_1^2 + a_2^2 + 2a_1a_2\cos\phi}$$ where $\phi$ is the phase difference.

The corresponding intensity is proportional to the square of the amplitude: $$I \propto A^2$$ so the resultant intensity becomes $$I = I_1 + I_2 + 2\sqrt{I_1I_2}\cos\phi$$

• Maximum intensity occurs when $\phi = 0, 2\pi, 4\pi,\dots$, producing constructive interference.
• Minimum intensity occurs when $\phi = \pi, 3\pi, 5\pi,\dots$, producing destructive interference.

This principle is the basis of all interference phenomena in wave optics.

2. Interference of Light and Conditions for Sustained Interference

• Interference of light is observed only when the combining waves are coherent, meaning they have the same frequency and a constant phase difference.
• The light waves must also have the same polarization or at least compatible polarization components to produce stable fringes.

For clear and steady interference patterns, the sources should satisfy:

Coherence

• : constant phase relationship

Monochromaticity

• : nearly the same wavelength

Comparable amplitudes

• : to obtain visible fringe contrast

Small source size

• : to maintain spatial coherence

If the phase difference changes randomly, the bright and dark regions fluctuate and no stable pattern is observed. Ordinary independent light sources, such as two separate bulbs, generally do not produce sustained interference because they are not coherent.

The intensity distribution in interference is often written as: $$I = I_{\max}\cos^2\left(\frac{\phi}{2}\right)$$ for equal intensities, showing periodic variation between bright and dark fringes.

A common example is the Young’s double-slit experiment, where a single source is first divided into two coherent sources, proving the wave nature of light.

3. Methods of Producing Interference: Wave Front Splitting and Amplitude Splitting

Wave front splitting

• uses different parts of the same wave front as coherent sources. The original wave front is divided geometrically, and the parts travel different paths before recombining.

Amplitude splitting

• divides the amplitude of a single beam into separate beams using partial reflection and transmission, after which the beams are recombined to interfere.

Wave front splitting

In this method, the wave front from a source is split by apertures or reflecting surfaces. Since all parts originate from the same wave, the resulting beams are coherent.

Examples include:

• Young’s double-slit experiment
• Fresnel biprism
• Lloyd’s mirror

Characteristics:

• Uses different portions of the same wave front
• Requires narrow slits or appropriate optical arrangement
• Generally produces fringes by path difference between separated beams

Amplitude splitting

In this method, a beam is partially reflected and partially transmitted at an optical surface, creating two or more beams from the same incident beam.

Examples include:

• Thin film interference
• Newton’s rings
• Michelson interferometer
• Fabry–Pérot interferometer

Characteristics:

• Uses a single beam divided in intensity
• Requires partial reflectors or thin films
• Very useful in precision measurements

The major difference is that wave front splitting divides the wave geometrically, whereas amplitude splitting divides the beam by intensity. Both create coherent beams necessary for interference.

Working / Process

1. A light source is arranged so that coherent beams can be produced either by dividing the wave front or by splitting the amplitude of a beam.
2. The two resulting waves travel along different paths and overlap in space, producing a phase difference based on their path difference.
3. At points where the phase difference is an integral multiple of $2\pi$, constructive interference occurs and a bright fringe is formed; where the phase difference is an odd multiple of $\pi$, destructive interference occurs and a dark fringe is formed.

Advantages / Applications

• Interference helps in measuring very small distances, wavelengths, and refractive indices with high precision.
• It is widely used in optical instruments such as interferometers, optical testing devices, and thin film analysis.
• It has important practical applications in understanding coatings, lens testing, surface flatness, and the design of optical devices.

Summary

• Superposition is the addition of wave displacements, and interference is the intensity pattern formed due to superposition of coherent light waves.
• Wave front splitting produces interference by dividing a single wave front into parts, while amplitude splitting produces interference by dividing the beam into different intensity components.
• Both methods are fundamental in wave optics and are used to study the wave nature of light through bright and dark fringe formation.
• Important terms to remember: superposition, coherence, interference, constructive interference, destructive interference, wave front splitting, amplitude splitting, path difference, phase difference, fringe pattern