Unit 2: Optical Interference: Principles and Applications

Interference is a phenomenon where two or more waves superimpose to form a resultant wave of greater, lower, or the same amplitude. For sustained interference patterns to be observed, the interfering waves must meet certain conditions:

• Coherent Sources: The sources must emit light waves with a constant phase difference.
• Monochromatic Light: The light must consist of a single wavelength.
• Sources Must Be Close: The sources should be close to each other for observable fringes.
• Same State of Polarization: The waves must have the same polarization.
• Sources Must Originate from a Single Source: This ensures coherence.

Methods of Producing Interference

Interference patterns can be produced by two primary methods:

Division of Wavefront

In this method, light from a single source is split into two coherent waves by passing it through two closely spaced openings or by reflection/refraction arrangements. These waves then travel different paths and interfere. The sources are usually virtual.

Examples:

• Young’s Double Slit Experiment
• Lloyd’s Mirror
• Fresnel’s Biprism

Division of Amplitude

In this method, the amplitude of a single wave is divided into two or more parts by partial reflection and refraction. These divided parts then travel different paths and are later recombined to produce interference.

Examples:

• Thin films
• Newton’s Rings

Young’s Double Slit Experiment

This experiment demonstrates the wave nature of light and the principle of interference by producing an interference pattern from two coherent point sources (slits). Light from a monochromatic source passes through two narrow, closely spaced parallel slits, S₁ and S₂, which act as coherent secondary sources. An interference pattern of alternating bright and dark fringes is observed on a screen placed some distance away.

Path Difference (Δx): For constructive interference (bright fringe), Δx = nλ, where n = 0, 1, 2, ...
For destructive interference (dark fringe), Δx = (n + 1/2)λ, where n = 0, 1, 2, ...
Fringe Width (β): The distance between two consecutive bright or dark fringes.
β = λD / d
Where:

• λ is the wavelength of light
• D is the distance between the slits and the screen
• d is the distance between the two slits

Lloyd’s Mirror

Lloyd’s mirror is an optical setup that produces an interference pattern using a single real source and its virtual image formed by reflection from a plane mirror. The light directly from the source and the light reflected from the mirror interfere.

• The direct and reflected rays appear to originate from two coherent virtual sources.
• A phase change of π (or 180°) occurs upon reflection from a denser medium (the mirror surface).
• Due to this phase change, the central fringe (where path difference is zero) is dark, unlike Young’s experiment where it is bright.
• The fringes are generally localized.

Fresnel’s Biprism

Fresnel’s biprism consists of two thin prisms joined at their bases. When monochromatic light from a narrow slit falls on the biprism, it is refracted to form two coherent virtual images of the slit. These two virtual sources produce an interference pattern on a screen, similar to Young's double-slit experiment.

• The biprism creates two virtual coherent sources (S' and S'').
• The distance 'd' between these two virtual sources can be determined using the geometry of the setup.
• The fringe width can be measured using the formula β = λD / d.

Types of Interference Fringes

Interference fringes can be categorized based on how they are formed and where they are localized.

Fringes of Equal Inclination (Haidinger Fringes)

These fringes are produced when interfering rays have the same angle of inclination with respect to the normal of the film. They are formed when the light from an extended source passes through a film of uniform thickness. They are localized at infinity or in the focal plane of an observing telescope.

• Depend on the angle of incidence (or inclination) of the light rays.
• The thickness of the film is uniform for a given fringe.
• Examples include fringes observed in a parallel-sided thin film with an extended source.

Fringes of Equal Thickness (Fizeau Fringes)

These fringes are produced when the interfering rays correspond to points of equal thickness in the film. They are formed when a film has varying thickness (e.g., a wedge-shaped film or the air film in Newton’s Rings) and are localized within the film itself.

• Depend on the varying thickness of the film.
• The angle of incidence is constant (e.g., normal incidence).
• Examples include fringes observed in a wedge-shaped film and Newton’s Rings.

Interference in Thin Films

When light falls on a thin film, a portion is reflected from the top surface, and another portion is transmitted, then reflected from the bottom surface. These two reflected rays (or transmitted rays) can interfere to produce a pattern.

Optical Path Difference (Δ): For a thin film of refractive index μ and thickness t, for rays incident at an angle r to the normal inside the film:
Δ = 2μt cos r ± λ/2
The ±λ/2 term accounts for the phase change of π (or 180°) that occurs when light reflects from an interface with a denser medium. This phase change is equivalent to an additional path of λ/2. If reflection occurs from a rarer to denser medium, a phase change occurs. If reflection occurs from a denser to rarer medium, no phase change occurs.

Interference in Parallel Thin Films

For a parallel thin film of uniform thickness 't' and refractive index 'μ', light incident at an angle 'i' (refracting at 'r'):

Conditions for Interference in Reflected Light:

• For Bright Fringes (Constructive Interference):
  

  2μt cos r = (2n + 1)λ / 2, where n = 0, 1, 2, ...

• For Dark Fringes (Destructive Interference):
  

  2μt cos r = nλ, where n = 0, 1, 2, ...

(These conditions assume light reflects from a denser medium at the first surface, and from a rarer medium at the second surface, or vice versa, leading to one phase change. If both reflections occur at denser media or both at rarer media, the conditions are reversed.)

Conditions for Interference in Transmitted Light:

The conditions for transmitted light are generally opposite to those for reflected light, as energy conservation dictates that if reflection is maximal, transmission is minimal, and vice versa.

• For Bright Fringes (Constructive Interference):
  

  2μt cos r = nλ, where n = 0, 1, 2, ...

• For Dark Fringes (Destructive Interference):
  

  2μt cos r = (2n + 1)λ / 2, where n = 0, 1, 2, ...

Interference in Wedge-shaped Films

A wedge-shaped film is an air film enclosed between two inclined glass plates. The thickness of the film varies linearly from zero at the line of contact to a maximum at the other end. When monochromatic light is incident normally on such a film, Fizeau fringes (fringes of equal thickness) are observed.

• The fringes are straight lines, parallel to the line of contact of the plates.
• The thickness of the film at a distance 'x' from the line of contact is given by t = xθ, where θ is the angle of the wedge in radians.
• For Dark Fringes: 2μt = nλ (for normal incidence and one phase change)
  Substituting t = xθ, 2μxθ = nλ
• For Bright Fringes: 2μt = (2n + 1)λ / 2 (for normal incidence and one phase change)
  Substituting t = xθ, 2μxθ = (2n + 1)λ / 2
• The fringe width (β) is the distance between two consecutive dark or bright fringes: β = λ / (2μθ)

Newton’s Rings

Newton’s Rings are an interference pattern observed when a plano-convex lens is placed on a flat glass plate. The air film trapped between the curved surface of the lens and the flat plate has a varying thickness, forming Fizeau fringes in the shape of concentric circles.

• The center of the pattern is dark because the thickness of the air film is zero at the point of contact, and a phase change of π occurs upon reflection from the top surface of the glass plate (denser medium).
• The rings become closer as the radius increases.
• They are observed using monochromatic light.

Radii of Newton’s Rings (for reflected light, air film μ=1, and normal incidence)

For Dark Rings:
R_n = √ (nλR)
Where:

• R_n is the radius of the n^th dark ring.
• n = 0, 1, 2, ... (n=0 for the central dark spot)
• λ is the wavelength of light.
• R is the radius of curvature of the plano-convex lens.

For Bright Rings:
R_n = √ ((2n - 1)λR / 2)
Where:

• R_n is the radius of the n^th bright ring.
• n = 1, 2, 3, ... (n=1 for the first bright ring)

Measurement of Wavelength (λ) using Newton’s Rings

The Newton’s Rings experiment can be used to determine the wavelength of monochromatic light.

• Measure the diameters of several dark rings (e.g., D_n and D_m, where m > n).
• The square of the diameter of the n^th dark ring is given by D_n² = 4nλR.
• For the m^th dark ring, D_m² = 4mλR.
• Subtracting these equations: D_m² - D_n² = 4(m - n)λR.

• Therefore, λ = (D_m² - D_n²) / (4(m - n)R)

• By measuring the diameters (D_n and D_m), the difference in the order of rings (m-n), and the radius of curvature of the lens (R), the wavelength λ can be calculated.

Measurement of Refractive Index (μ) using Newton’s Rings

The Newton’s Rings experiment can also be used to determine the refractive index of a transparent liquid.

• First, measure the diameters of dark rings (e.g., D_n) with an air film between the lens and plate.
• Then, introduce the liquid whose refractive index is to be determined between the lens and the plate, ensuring it fills the air film.
• Measure the diameters of the corresponding dark rings (e.g., D'_n) with the liquid film.
• For an air film (μ_air = 1): D_n² = 4nλR.
• For a liquid film (μ_liquid): D'_n² = 4nλR / μ_liquid.
• Dividing the two equations: D_n² / D'_n² = μ_liquid.

• Therefore, μ_liquid = D_n² / D'_n² = (D_m² - D_n²)_air / (D_m² - D_n²)_liquid

• By comparing the squares of the diameters of the same order of dark rings in air and in the liquid, the refractive index of the liquid can be determined.