Interference of Light Waves: Young's Double Slit Experiment, Coherent Sources & Wave Optics Physics HRK Notes

House of Physics October 05, 2025
Physics HRK Notes: Waves and Oscillations - Interference of Light Waves | Complete Guide
Complete Guide to Young's Double Slit Experiment, Coherent Sources, Thin Film Interference, Newton's Rings, Michelson Interferometer, and Fresnel's Biprism
Interference of Light Waves Young's Double Slit Experiment Constructive Interference Destructive Interference Coherent Sources Reading Time: 20 min

📋 Table of Contents

📜 Historical Background

The study of interference of light waves has a rich history that fundamentally changed our understanding of light:

  • Thomas Young (1801): Performed the famous double-slit experiment that demonstrated the wave nature of light
  • Augustin-Jean Fresnel (1815-1820): Developed the mathematical theory of diffraction and interference
  • Albert A. Michelson (1880s): Invented the interferometer and used it for precise measurements
  • Isaac Newton (1666): Observed Newton's rings, though his particle theory couldn't explain them

These developments established the wave theory of light and paved the way for modern optics.

Introduction to Interference of Light Waves

🔬 What is Interference?

Interference is a wave phenomenon that occurs when two waves having the same frequency and moving in the same direction superimpose. This superposition results in either reinforcement or cancellation of the waves, creating a pattern of alternating bright and dark regions.

The phenomenon of interference provides strong evidence for the wave nature of light, as it demonstrates that light waves can interact with each other in ways that particles cannot.

📝 Key Characteristics of Interference

For interference to occur, certain conditions must be met:

  • The waves must have the same frequency
  • The waves must be traveling in the same direction
  • The waves must maintain a constant phase relationship (coherence)
  • The waves must have comparable amplitudes for observable effects

Types of Interference

🌈 Two Types of Interference

Depending on the phase relationship between the interfering waves, interference can be classified into two types:

Constructive Interference

➕ Constructive Interference

If two waves reach a point in phase (crest of one wave falls on the crest of the other wave, and trough of one wave falls on the trough of the other wave), then the two waves reinforce each other and the net wave effect increases.

In constructive interference:

  • The net intensity is greater than the intensity of individual waves
  • The amplitude of the resultant wave is the sum of the individual amplitudes
  • Bright fringes are formed in the interference pattern

Destructive Interference

➖ Destructive Interference

If two waves reach a point out of phase (crest of one wave falls on the trough of the other), then the two waves cancel each other's effect and the net intensity decreases.

In destructive interference:

  • The net intensity is less than the intensity of individual waves
  • The amplitude of the resultant wave is the difference of the individual amplitudes
  • Dark fringes are formed in the interference pattern

Coherent Sources

🔗 What are Coherent Sources?

Two sources are said to be coherent sources if they produce waves of:

  • Same amplitude
  • Same frequency
  • Either no initial phase difference or a constant phase difference

Interference takes place due to the superposition of two exactly similar waves. The two waves interfere only if they have phase coherence, meaning they come from two coherent sources.

💡 Key Insight

In practice, true coherence is difficult to achieve with independent light sources. That's why in interference experiments, we typically derive both interfering waves from a single source using techniques like double slits, biprisms, or beam splitters.

Young's Double Slit Experiment

🔬 Young's Double Slit Experiment

Thomas Young's double slit experiment, performed in 1801, provided the first conclusive evidence for the wave nature of light. It experimentally confirmed Huygens' wave theory of light.

Experimental Setup

⚙️ Young's Double Slit Setup

Monochromatic Source S
↓
Double Slit
S₁ and S₂
↓
Screen with Interference Pattern

Working Principle:

  1. S is a monochromatic source of light (a rectangular slit illuminated by monochromatic light)
  2. This light is divided into two parts by slits S₁ and S₂
  3. Slits S₁ and S₂ act as coherent sources because they are obtained from a single source
  4. d is the distance between slits S₁ and S₂
  5. D is the distance of screen from slits S₁ and S₂, with D ≫ d
  6. Point O is the central point of the screen where waves from S₁ and S₂ cover equal distance, resulting in constructive interference (central maxima)

Path Difference Calculation

🧮 Path Difference Derivation

Step 1: Geometry of the Setup

Consider any point P on the screen at a distance y from central point O. The path difference between the waves arriving at P is:

\[ \text{Path Difference} = S_2P - S_1P \]

Step 2: Approximation for D ≫ d

Since D ≫ d, we can approximate:

\[ \sin \theta \approx \tan \theta \]
\[ \tan \theta = \frac{y}{D} \]

Step 3: Path Difference Formula

From the geometry of the setup:

\[ \text{Path Difference} = d \sin \theta \]
\[ = d \cdot \frac{y}{D} \]

Conditions for Maxima and Minima

🧮 Interference Conditions

Step 1: Constructive Interference (Bright Fringes)

For constructive interference, the path difference must be an integral multiple of wavelength:

\[ \text{Path Difference} = n\lambda \]
\[ d \cdot \frac{y_n}{D} = n\lambda \]
\[ y_n = \frac{n\lambda D}{d} \]

where n = 0, 1, 2, ... and y_n is the position of the nth bright fringe from the center.

Step 2: Destructive Interference (Dark Fringes)

For destructive interference, the path difference must be an odd multiple of half-wavelength:

\[ \text{Path Difference} = (2n - 1)\frac{\lambda}{2} \]
\[ d \cdot \frac{y_n}{D} = (2n - 1)\frac{\lambda}{2} \]
\[ y_n = \frac{(2n - 1)\lambda D}{2d} \]

where n = 1, 2, 3, ... and y_n is the position of the nth dark fringe from the center.

Sample Problems

Sample Problem 1: Fringe Width Calculation

In a double slit experiment, the distance between the slits is 0.5 mm and the screen is 2 m away from the slits. If the wavelength of light used is 5000 Å, calculate the fringe width.

Given Data:
\[ d = 0.5 \, \text{mm} = 0.5 \times 10^{-3} \, \text{m} \]
\[ D = 2 \, \text{m} \]
\[ \lambda = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} = 5 \times 10^{-7} \, \text{m} \]
Fringe width β is given by:
\[ \beta = \frac{\lambda D}{d} \]
\[ = \frac{5 \times 10^{-7} \times 2}{0.5 \times 10^{-3}} \]
\[ = \frac{10 \times 10^{-7}}{0.5 \times 10^{-3}} \]
\[ = 20 \times 10^{-4} \, \text{m} \]
\[ = 2 \, \text{mm} \]
Sample Problem 2: Position of Fringes

In Young's double slit experiment, the distance between the slits is 0.1 mm and the distance of the screen from the slits is 1 m. If the fringe width is 6 mm, find the wavelength of light used.

Given Data:
\[ d = 0.1 \, \text{mm} = 0.1 \times 10^{-3} \, \text{m} \]
\[ D = 1 \, \text{m} \]
\[ \beta = 6 \, \text{mm} = 6 \times 10^{-3} \, \text{m} \]
Fringe width formula:
\[ \beta = \frac{\lambda D}{d} \]
\[ \lambda = \frac{\beta d}{D} \]
\[ = \frac{6 \times 10^{-3} \times 0.1 \times 10^{-3}}{1} \]
\[ = 6 \times 10^{-7} \, \text{m} \]
\[ = 6000 \, \text{Å} \]

Intensity in Young's Double Slit Experiment

💡 Intensity Distribution

The intensity of light at any point on the screen in Young's double slit experiment depends on the phase difference between the two interfering waves. The intensity varies periodically between maximum and minimum values.

Mathematical Derivation

🧮 Intensity Derivation

Step 1: Wave Equations

Let the electric fields of the two waves at point P be:

\[ E_1 = E_0 \sin(\omega t) \]
\[ E_2 = E_0 \sin(\omega t + \phi) \]

where φ is the phase difference between the two waves.

Step 2: Resultant Electric Field

By the principle of superposition:

\[ E = E_1 + E_2 \]
\[ = E_0 \sin(\omega t) + E_0 \sin(\omega t + \phi) \]
\[ = 2E_0 \cos\left(\frac{\phi}{2}\right) \sin\left(\omega t + \frac{\phi}{2}\right) \]

Step 3: Resultant Amplitude

The amplitude of the resultant wave is:

\[ A = 2E_0 \cos\left(\frac{\phi}{2}\right) \]

Step 4: Intensity Calculation

Since intensity is proportional to the square of amplitude:

\[ I \propto A^2 \]
\[ I = 4I_0 \cos^2\left(\frac{\phi}{2}\right) \]

where I₀ is the intensity due to a single source.

Step 5: Phase Difference and Path Difference

The phase difference φ is related to the path difference Δ by:

\[ \phi = \frac{2\pi}{\lambda} \Delta \]

where Δ = d sin θ ≈ d(y/D)

Intensity Distribution

📊 Intensity Pattern

The intensity distribution in Young's double slit experiment follows a cosine-squared pattern:

  • Maximum Intensity: I_max = 4I₀ when cos²(φ/2) = 1, i.e., φ = 2nπ
  • Minimum Intensity: I_min = 0 when cos²(φ/2) = 0, i.e., φ = (2n+1)π
  • Average Intensity: I_avg = 2I₀ (averaged over many fringes)

This pattern of alternating bright and dark fringes is characteristic of interference phenomena.

Phasor Method for Adding Electromagnetic Waves

🔄 Phasor Method

The phasor method is a graphical technique used to add two or more waves with the same frequency but different phases. Each wave is represented as a vector (phasor) whose length represents the amplitude and whose angle represents the phase.

🧮 Phasor Addition

Step 1: Phasor Representation

Consider two waves with amplitudes A₁ and A₂ and phase difference φ. Represent them as vectors:

\[ \vec{A_1} = A_1 \angle 0 \]
\[ \vec{A_2} = A_2 \angle \phi \]

Step 2: Resultant Amplitude

The resultant amplitude is given by the vector sum:

\[ A^2 = A_1^2 + A_2^2 + 2A_1A_2 \cos \phi \]

Step 3: Special Cases

Constructive Interference (φ = 2nπ):

\[ A = A_1 + A_2 \]

Destructive Interference (φ = (2n+1)π):

\[ A = |A_1 - A_2| \]

Interference in Thin Films

🎨 Thin Film Interference

Thin film interference occurs when light waves reflected from the top and bottom surfaces of a thin film interfere with each other. This phenomenon is responsible for the colorful patterns seen in soap bubbles, oil slicks, and anti-reflection coatings.

⚙️ Thin Film Setup

Incident Light
↓
Thin Film
Thickness t
Reflected Waves Interfere

Working Principle:

  1. Light is incident on a thin film of thickness t and refractive index μ
  2. Part of the light is reflected from the top surface
  3. Part of the light is transmitted and reflected from the bottom surface
  4. These two reflected waves interfere with each other
  5. The interference pattern depends on the film thickness, wavelength, and angle of incidence

Conditions for Maxima and Minima

🧮 Thin Film Interference Conditions

Step 1: Path Difference

For near-normal incidence, the path difference between the two reflected rays is:

\[ \Delta = 2\mu t \]

where μ is the refractive index of the film and t is its thickness.

Step 2: Phase Change Consideration

When light reflects from a denser medium, it undergoes a phase change of π (equivalent to a path difference of λ/2).

For a film surrounded by air:

  • Reflection from top surface: Phase change of π
  • Reflection from bottom surface: No phase change (or vice versa depending on relative refractive indices)

Step 3: Conditions for Constructive Interference

For constructive interference (bright fringe):

\[ 2\mu t = (2n - 1)\frac{\lambda}{2} \]

where n = 1, 2, 3, ...

Step 4: Conditions for Destructive Interference

For destructive interference (dark fringe):

\[ 2\mu t = n\lambda \]

where n = 0, 1, 2, ...

Newton's Rings

🌀 Newton's Rings

Newton's rings are a phenomenon in which concentric rings are formed due to interference between light waves reflected from the top and bottom surfaces of the air film between a plano-convex lens and a flat glass plate.

Experimental Setup

⚙️ Newton's Rings Setup

Monochromatic Source
↓
Plano-convex Lens
on Glass Plate
↓
Concentric Rings Pattern

Working Principle:

  1. A plano-convex lens of large radius of curvature is placed on a flat glass plate
  2. An air film of varying thickness is formed between the lens and the plate
  3. Monochromatic light is incident normally on the arrangement
  4. Interference occurs between light reflected from the top and bottom of the air film
  5. Concentric circular fringes (Newton's rings) are observed

Radius of Bright and Dark Rings

🧮 Radius of Newton's Rings

Step 1: Geometry of Air Film

For a lens of radius of curvature R, the thickness t of the air film at a distance r from the point of contact is:

\[ t = \frac{r^2}{2R} \]

This approximation is valid when r ≪ R.

Step 2: Path Difference

The path difference for normally incident light is:

\[ \Delta = 2t + \frac{\lambda}{2} \]

The λ/2 term accounts for the phase change upon reflection from the denser medium.

Step 3: Radius of Dark Rings

For destructive interference (dark rings):

\[ 2t = n\lambda \]
\[ 2 \cdot \frac{r_n^2}{2R} = n\lambda \]
\[ r_n^2 = n\lambda R \]
\[ r_n = \sqrt{n\lambda R} \]

where n = 0, 1, 2, ...

Step 4: Radius of Bright Rings

For constructive interference (bright rings):

\[ 2t = (2n - 1)\frac{\lambda}{2} \]
\[ 2 \cdot \frac{r_n^2}{2R} = (2n - 1)\frac{\lambda}{2} \]
\[ r_n^2 = (2n - 1)\frac{\lambda R}{2} \]
\[ r_n = \sqrt{(2n - 1)\frac{\lambda R}{2}} \]

where n = 1, 2, 3, ...

Michelson's Interferometer

🔧 Michelson's Interferometer

The Michelson interferometer is a precision instrument that uses the interference of light to make extremely accurate measurements of distances, wavelengths, and refractive indices.

Construction and Working

⚙️ Michelson Interferometer Setup

Light Source
↓
Beam Splitter
↓
Two Paths → Interference Pattern

Working Principle:

  1. Light from a monochromatic source is incident on a beam splitter
  2. The beam splitter divides the light into two beams traveling at right angles
  3. Each beam reflects from a mirror and returns to the beam splitter
  4. The recombined beams interfere and form an interference pattern
  5. Moving one mirror changes the path difference, causing the interference pattern to shift

Applications

📏 Measurement of Wavelength

By counting the number of fringe shifts when one mirror is moved a known distance, the wavelength of light can be determined with high precision.

🔍 Measurement of Refractive Index

By placing a transparent medium in one path and measuring the fringe shift, the refractive index of the medium can be calculated.

⚖️ Standard of Length

The Michelson interferometer was used to establish the standard meter in terms of wavelengths of light.

Fresnel's Biprism

🔺 Fresnel's Biprism

Fresnel's biprism is a device used to obtain two coherent sources from a single source by refraction. It consists of two prisms with very small refracting angles placed base to base.

Construction and Working

⚙️ Fresnel's Biprism Setup

Monochromatic Source S
↓
Biprism
↓
Two Virtual Sources S₁ and S₂

Working Principle:

  1. A narrow slit S is illuminated by monochromatic light
  2. The biprism refracts the light, creating two virtual images S₁ and S₂ of the slit
  3. These virtual sources act as coherent sources
  4. The light from S₁ and S₂ overlaps and produces interference fringes
  5. The fringe width and pattern can be used to determine the wavelength of light

Determination of Wavelength

🧮 Wavelength Determination

Step 1: Fringe Width Measurement

The fringe width β is measured on the screen:

\[ \beta = \frac{\lambda D}{d} \]

where D is the distance from the virtual sources to the screen, and d is the distance between the virtual sources.

Step 2: Wavelength Calculation

The wavelength can be calculated as:

\[ \lambda = \frac{\beta d}{D} \]

Step 3: Determination of d

The distance d between the virtual sources can be determined using:

\[ d = 2a(\mu - 1)\alpha \]

where a is the distance from the slit to the biprism, μ is the refractive index of the prism material, and α is the refracting angle of each prism.

Frequently Asked Questions

❓ Why do we need coherent sources for interference?

Coherent sources are necessary for observable interference patterns because:

  • They maintain a constant phase relationship over time
  • This allows the interference pattern to remain stable and observable
  • With incoherent sources, the phase relationship changes randomly, causing the interference pattern to fluctuate too rapidly to be observed

In practice, we typically create coherent sources by splitting light from a single source, ensuring that the two interfering waves have a fixed phase relationship.

❓ What happens to the interference pattern if we use white light instead of monochromatic light?

When white light is used in interference experiments:

  • Different wavelengths produce interference patterns with different fringe widths
  • The central fringe (n=0) is white because all wavelengths have constructive interference at the center
  • Away from the center, the fringes for different colors overlap, creating colored patterns
  • Further away from the center, the patterns for different colors overlap completely, and no distinct fringes are visible

This is why monochromatic light is preferred for clear, distinct interference patterns.

❓ Why is the central fringe in Young's experiment bright?

The central fringe in Young's double slit experiment is bright because:

  • At the central point (O), the path difference between waves from the two slits is zero
  • This means the waves arrive in phase, resulting in constructive interference
  • Since the path difference is zero for all wavelengths, the central fringe is bright even with white light

The central bright fringe is flanked by alternating dark and bright fringes as we move away from the center.

❓ What is the difference between interference and diffraction?

While both interference and diffraction are wave phenomena, they have important differences:

Interference Diffraction
Occurs due to superposition of waves from two or more coherent sources Occurs due to superposition of secondary wavelets from different parts of the same wavefront
All bright fringes have approximately the same intensity The central maximum is much brighter than the secondary maxima
The number of sources is limited and finite Involves an infinite number of sources (points on the wavefront)
Fringes are equally spaced in Young's experiment Fringes are not equally spaced

In practice, most interference patterns also involve some diffraction effects, and the distinction is not always clear-cut.

📚 Master Wave Optics

Understanding interference of light waves is fundamental to wave optics, quantum mechanics, and many areas of modern physics and engineering. Continue your journey into the fascinating world of wave phenomena and their applications.

Read More: Physics HRK Notes of Wave Optics

© House of Physics | HRK Physics: Waves and Oscillations - Interference of Light Waves

Based on Halliday, Resnick, and Krane's "Physics" with additional insights from university physics curriculum

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