Unit 3- Physical Optics | 2nd Semester Bachelor of Optometry

Topic 7: Coherence; Interference; Constructive Interference, Destructive Interference; Fringes; Fringe Width

Introduction

The phenomena of coherence and interference lie at the very heart of wave optics, explaining how waves combine to produce complex patterns of light and dark regions. These effects are fundamental to understanding many optical devices, experiments, and applications such as holography, interferometry, and optical communications.

This article provides a detailed examination of coherence, the principles of interference including constructive and destructive interference, the nature and formation of fringes, and how to calculate fringe width. Each subtopic is treated in detail to build a thorough understanding suitable for advanced studies and exam preparation.

1. Coherence

Definition of Coherence

Coherence 

Coherence describes a fixed and predictable phase relationship between waves at different points in space and time. Two waves are said to be coherent if they maintain a constant phase difference, enabling them to produce stable interference patterns.

Types of Coherence

• Temporal Coherence: Refers to the correlation between the phases of a wave at different times at a single point in space. It is related to the spectral purity or monochromaticity of the source.
• Spatial Coherence: Refers to the correlation between the phases of a wave at different points in space at the same time. It depends on the size and geometry of the source.

Mathematical Description

The degree of coherence can be quantified by the complex degree of coherence, which measures how correlated the electric fields are at two points in space and time. If the phase difference remains constant over the time interval of observation, coherence is said to be high.

Coherence Length and Coherence Time

• Coherence Time (τ): The time duration over which the wave maintains a predictable phase relationship.
• Coherence Length (L): The distance over which the wave remains coherent, given by L = cτ, where c is the speed of light.

Sources of Coherent and Incoherent Light

Ideal coherent sources include lasers and certain interferometer outputs. Natural light sources such as sunlight or incandescent bulbs are typically incoherent due to their broad spectral width and random phase relationships.

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2. Interference

Definition of Interference

Interference patterns of light obtained after passing light through two slits

Interference is the phenomenon where two or more coherent waves superpose to form a resultant wave of greater, lower, or the same amplitude. This superposition leads to regions of reinforcement (constructive interference) and cancellation (destructive interference).

Principle of Superposition

The principle of superposition states that when two or more waves overlap, the resultant displacement at any point is the algebraic sum of the displacements of the individual waves.

Mathematically, if two waves with displacements y₁ and y₂ interfere, the resultant displacement y is:

y = y_1 + y_2

Conditions for Interference

• The waves must be coherent or partially coherent.
• The waves should have the same or nearly the same frequency.
• The waves must overlap in space and time.

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3. Constructive and Destructive Interference

Constructive and Destructive Interference 

Constructive Interference

Constructive interference occurs when the phase difference between the two waves is an integral multiple of 2π (or a path difference equal to an integer multiple of the wavelength). In this case, the waves reinforce each other, resulting in a wave of increased amplitude.

Mathematically, the condition is:

Δφ = 2mπ or Δx = mλ, where m = 0, 1, 2, ...

Here, Δφ is the phase difference, Δx is the path difference, and λ is the wavelength.

Destructive Interference

Destructive interference occurs when the phase difference is an odd multiple of π (or a path difference equal to an odd multiple of half the wavelength). The waves cancel each other out partially or completely, producing a wave of reduced or zero amplitude.

Mathematically:

Δφ = (2m + 1)π or Δx = (m + \frac{1}{2})λ, where m = 0, 1, 2, ...

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4. Fringes

Definition of Fringes

Fringes are alternating bright and dark bands or lines formed on a screen or observation plane as a result of interference between two or more coherent waves. Bright fringes correspond to constructive interference, while dark fringes correspond to destructive interference.

Types of Interference Fringes

• Fresnel Fringes: Produced by interference of light reflected from a surface and a nearby surface (e.g., Newton's rings).
• Young’s Fringes: Produced by interference from two coherent sources, such as the double-slit experiment.
• Haidinger Fringes: Interference patterns formed between two parallel reflecting surfaces.

Formation of Fringes

When two coherent light waves meet on a screen, their phase difference varies with position due to differences in path length, producing an interference pattern of bright and dark fringes.

The pattern depends on the wavelength of light, the geometry of the source setup, and the distance between the source and the screen.

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5. Fringe Width

Definition of Fringe Width

Fringe width (β) is the distance between two successive bright (or dark) fringes in an interference pattern. It quantifies the spatial separation of interference fringes on the observation screen.

Derivation of Fringe Width (Young's Double Slit Experiment)

Consider two slits separated by distance d illuminated by monochromatic light of wavelength λ. The interference pattern is observed on a screen placed at a distance D from the slits.

The path difference between the two waves arriving at a point on the screen at distance y from the central maximum is:

Δx = (d × y) / D

Constructive interference occurs when:

Δx = mλ    where   m = 0, 1, 2, ...

From this, the position of the m^th bright fringe is:

y_m = (m × λ × D) / d

The fringe width β, the distance between two consecutive bright fringes, is:

β = y_m+1 − y_m = (λ × D) / d

Factors Affecting Fringe Width

• Wavelength (λ): Increasing wavelength increases fringe width.
• Distance between screen and slits (D): Increasing distance increases fringe width.
• Distance between slits (d): Increasing slit separation decreases fringe width.

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Summary

Coherence ensures the phase stability needed for interference to occur, which leads to the formation of characteristic patterns of bright and dark fringes on a screen. Constructive interference produces bright fringes while destructive interference produces dark fringes. The spatial distribution of these fringes and their width depend on the wavelength of light, geometry of the setup, and coherence properties of the source.

Understanding these concepts is fundamental in wave optics, allowing us to analyze experiments such as Young’s double slit, design optical instruments, and explore advanced phenomena like holography and interferometry.

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This detailed article with clearly defined subtopics provides a comprehensive understanding of coherence and interference, essential for students preparing for optics and optometry exams.

Topic 8: Double Slits, Multiple Slits, and Gratings

Introduction

The study of light passing through apertures and slits is central to wave optics. This topic explores how light behaves when it encounters double slits, multiple slits, and diffraction gratings, revealing the rich interference patterns that form due to coherent light sources. These phenomena have profound practical implications in spectroscopy, optical engineering, and fundamental physics.

In this detailed article, we will examine the principles of interference arising from double slits, extend the analysis to multiple slits, and finally understand the working and applications of diffraction gratings. Each section is elaborated with theory, mathematical derivations, and practical considerations suitable for advanced optics study.

1. Double Slits

Historical Background

The double slit experiment, first performed by Thomas Young in 1801, was pivotal in establishing the wave nature of light. By shining light through two closely spaced slits, Young observed an interference pattern of bright and dark fringes on a screen, confirming that light behaves as a wave and can interfere constructively and destructively.

Experimental Setup

Double-Slit Experiment 

The classical double slit arrangement consists of:

• A monochromatic coherent light source, often a laser.
• A barrier with two narrow slits separated by a distance d.
• A screen placed at distance D from the slits where the interference pattern is observed.

Formation of Interference Pattern

When coherent light waves pass through the two slits, they act as two coherent point sources, emitting spherical waves. These waves overlap on the screen, producing alternating bright and dark fringes due to constructive and destructive interference.

Effect of Slit Width

In reality, each slit has a finite width a. The diffraction pattern due to each slit modulates the interference pattern. The resulting intensity is the product of the interference pattern from two point sources and the diffraction envelope from single slit diffraction.

This combined pattern is called the interference-diffraction pattern.

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2. Multiple Slits

Conceptual Overview

Multiple Slits Experiment 

Extending from two slits to multiple slits (three or more) results in sharper and more defined interference patterns. Multiple slits act as multiple coherent sources of light, leading to increased contrast and narrower bright fringes.

Intensity Characteristics

• The principal maxima occur when d \sin θ = mλ, where the intensity is maximized.
• Secondary maxima occur at other angles but have much lower intensity.
• The width of the principal maxima decreases as N increases, leading to sharper bright fringes.

3. Diffraction Gratings

Definition and Structure

A diffraction grating consists of a large number of equally spaced parallel slits or grooves on a reflective or transparent surface. It disperses light into its constituent wavelengths by constructive interference of diffracted waves, making it a powerful tool in spectroscopy.

Types of Gratings

• Transmission Gratings: Light passes through the grating slits.
• Reflection Gratings: Light is reflected from grooves etched on a reflective surface.

Working Principle

Each slit in the grating acts as a coherent source emitting secondary wavelets. When light of wavelength λ falls on the grating, the diffracted beams interfere constructively at certain angles satisfying the grating equation

Intensity Distribution

The intensity pattern of a grating is similar to that of multiple slits but with a much larger number of slits, resulting in very sharp and intense principal maxima. 

Resolving Power of a Grating

Resolving power (R) is the ability of a grating to distinguish between two close wavelengths. 

R= λ/dλ= mN

where m is the diffraction order and N is the total number of slits illuminated.

Angular Dispersion

Angular dispersion (D) measures how much the diffraction angle changes with wavelength:

Formula for Angular Dispersion

Angular dispersion (Δ) is given by:

Δ = (θ_v − θ_r) / (λ_v − λ_r)

Where:

• θ_v = Angle of deviation for violet light
• θ_r = Angle of deviation for red light
• λ_v = Wavelength of violet light
• λ_r = Wavelength of red light

A higher angular dispersion means better spectral separation.

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4. Comparison of Double Slit, Multiple Slits, and Gratings

Aspect	Double Slit	Multiple Slits	Diffraction Grating
Number of Slits	2	3 or more	Thousands or more
Pattern Sharpness	Moderate	Sharper	Very Sharp, intense
Applications	Basic interference demonstration	Advanced interference experiments	Spectroscopy, wavelength separation

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5. Applications

Spectroscopy

Diffraction gratings are used to separate light into its component wavelengths for spectral analysis of materials, stars, and chemicals.

Optical Communications

Multiple slit interference principles are used in wavelength division multiplexing (WDM) to handle different wavelengths in fiber optic cables.

Holography

Interference patterns created by multiple beams are foundational in holography to record and reconstruct three-dimensional images.

Measurement of Wavelength

Double slit and grating experiments are classic methods to measure the wavelength of light accurately.

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6. Factors Affecting the Patterns

• Slit Width: Determines diffraction envelope affecting fringe visibility.
• Slit Separation: Influences fringe spacing and order positions.
• Number of Slits: More slits create sharper and more intense maxima.
• Wavelength: Longer wavelengths produce wider fringe spacing.
• Source Coherence: Higher coherence leads to clearer interference fringes.

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7. Summary

Double slits, multiple slits, and diffraction gratings demonstrate the wave nature of light through interference. The number and spacing of slits critically influence the interference pattern's sharpness and intensity. Diffraction gratings, with their large number of slits, provide powerful means to disperse light and analyze its spectral components with high resolution.

Mastering these concepts is crucial for understanding advanced optical phenomena and their practical applications in science and technology.

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This article covers the theoretical foundations, mathematical analysis, and practical applications of double slit, multiple slit interference, and diffraction gratings, fulfilling the need for a detailed and comprehensive understanding for exam preparation.

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