Fresnel's Biprism experiment is used to produce interference of light and to determine the wavelength of monochromatic light.

A narrow slit S is illuminated by monochromatic light. The light from the slit falls on the Fresnel biprism. Due to refraction through the two halves of the biprism, two virtual images of the slit are formed. These virtual images are represented by S₁ and S₂.

Since S₁ and S₂ originate from the same slit S, they have the same frequency and maintain a constant phase difference. Therefore, they act as coherent sources.

The light waves coming from S₁ and S₂ overlap on the screen EF and produce alternate bright and dark fringes due to interference.

The distance between two consecutive bright fringes or two consecutive dark fringes is called fringe width.

Where,

• β = Fringe width
• λ = Wavelength of light
• D = Distance between virtual sources and screen
• d = Distance between virtual coherent sources S₁ and S₂

The fringe width β is measured using a travelling microscope. The distance D is measured directly. To determine d, the displacement method is used.

In the displacement method, a convex lens is placed between the biprism and the screen. The lens is moved to two positions where sharp images of S₁ and S₂ are formed on the screen.

Let the separations between the images in the two positions be d₁ and d₂ respectively.

After determining d, the wavelength of light is calculated by using:

Thus, by measuring β, d and D, the wavelength of monochromatic light can be determined accurately using Fresnel's Biprism experiment.

When a beam of monochromatic light falls on a thin transparent film, a part of the light is reflected from the upper surface while another part enters the film and is reflected from the lower surface. The two reflected waves travel different optical paths and interfere with each other. This phenomenon is called interference in thin films.

Consider a thin film of thickness t and refractive index μ. Let a monochromatic beam of light be incident on the film at an angle i and refracted at an angle r.

The ray reflected from the upper surface undergoes a phase change of π (equivalent to a path difference of λ/2) because reflection takes place from a denser medium. The ray reflected from the lower surface does not undergo any phase change.

The optical path difference between the two reflected rays is

Condition for constructive interference (Bright Fringe) in reflected light:

Condition for destructive interference (Dark Fringe) in reflected light:

For transmitted light, the conditions are exactly opposite.

Condition for bright fringe in transmitted light:

Condition for dark fringe in transmitted light:

Thus, whenever the reflected system produces a bright fringe, the transmitted system produces a dark fringe, and whenever the reflected system produces a dark fringe, the transmitted system produces a bright fringe.

Hence the reflected and transmitted interference patterns are complementary to each other.

Mathematically,

where I_r and I_t are the intensities of reflected and transmitted light respectively. Therefore, the maxima of one pattern correspond to the minima of the other pattern.

Newton's rings are concentric circular bright and dark fringes produced due to interference of light in the thin air film formed between a plano-convex lens and a plane glass plate.

Monochromatic light from source S is converted into a parallel beam by the convex lens L. A glass plate P inclined at 45° reflects the light downward onto the plano-convex lens C resting on the glass plate G.

The air film between the lens and glass plate has varying thickness. Light reflected from the upper and lower surfaces of this air film interferes and produces circular bright and dark rings known as Newton's rings.

Let,

• R = Radius of curvature of the plano-convex lens
• r_n = Radius of nth ring
• d_n = Diameter of nth ring
• t = Thickness of air film at that point

From the geometry of the figure,

Since the thickness t of the air film is very small compared to R, t² can be neglected.

Since the diameter of the nth ring is d_n,

Substituting this value,

For reflected light, one reflection occurs at a denser medium. Therefore an additional phase difference equivalent to λ/2 is introduced.

For Bright Rings:

Using the value of 2t,

Hence, diameters of bright rings are proportional to the square root of odd natural numbers.

For Dark Rings:

Hence, diameters of dark rings are proportional to the square root of natural numbers.

Therefore, Newton's rings are formed due to interference in a thin air film and appear as alternate bright and dark concentric circular rings.

Given,

For the nth dark ring of Newton's rings,

Substituting the given values,

Therefore, the radius of curvature of the plano-convex lens is

For the thickness of the air film corresponding to the 10th dark ring,

Hence,

The colours observed in thin films such as soap bubbles, oil films on water, and thin transparent sheets are due to interference of light reflected from the upper and lower surfaces of the film.

When white light falls on a thin film, different colours (wavelengths) undergo constructive and destructive interference at different points because the optical path difference depends on the thickness of the film and the wavelength of light.

For a particular thickness, some wavelengths are reinforced while others are cancelled. As a result, different colours are seen at different regions of the film.

Since the thickness of the film varies from point to point, different colours are produced at different positions, giving rise to beautiful coloured patterns.

The following conditions are necessary for observing a sustained interference pattern:

• The two sources must be coherent, i.e., they should maintain a constant phase difference.
• The sources should emit light of the same frequency or wavelength.
• The amplitudes of the two waves should be nearly equal for good contrast between bright and dark fringes.
• The sources should be narrow and close to each other.
• The light should be monochromatic for a clear and stable fringe pattern.
• The interfering waves must have the same state of polarization.

Temporal Coherence: Temporal coherence refers to the ability of a light wave to maintain a constant phase relationship with itself at different instants of time. It determines the duration for which the phase of a wave remains predictable and is related to the monochromaticity of the source.

A source having a long coherence time possesses high temporal coherence.

Spatial Coherence: Spatial coherence refers to the ability of light waves emitted from different points of a wavefront to maintain a constant phase difference. It determines the extent over which interference can be observed across the wavefront.

A small source of light generally possesses high spatial coherence.

Thus, temporal coherence is related to phase correlation in time, whereas spatial coherence is related to phase correlation at different points in space.

A wavefront is an imaginary surface joining all the points of a wave that are in the same phase of vibration at a given instant of time.

Three types of wavefront are:

• Spherical wavefront
• Cylindrical wavefront
• Plane wavefront

The fringe width in Young's double-slit experiment is given by

Hence, fringe width depends on:

• Wavelength (λ) of the light used.
• Distance (D) between the slits and the screen.

Examples of thin film interference are:

• Newton's Rings.
• Colours produced in soap bubbles.

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16. Discuss the phenomena of thin film interference pattern in reflected system. Why does an extremely thin film appear black in reflected system?

Interference in Thin Film (Reflected System)

Consider a thin transparent film of thickness t and refractive index μ. Let a ray PQ of monochromatic light be incident on the upper surface of the film.

A part of the incident light is reflected at the upper surface and forms ray 1. The remaining part enters the film, suffers reflection at the lower surface, and emerges as ray 2. The interference between rays 1 and 2 gives rise to thin film interference.

To obtain the condition for interference, we calculate the path difference between ray 1 and ray 2.

Let the refracted ray travel through QR and RS inside the film.

δ = μ(QR + RS) - QM

From the geometry of the figure,

QR = RT

δ = μ(TL + LS) - QM

Applying Snell's law,

sin i sin r = μ

which gives

QM = μLS

Substituting,

δ = μTL

From triangle QTL,

TL = QT cos r

TL = 2t cos r

Therefore,

δ = 2μt cos r

Since ray 1 is reflected from a denser medium, an additional phase change of π occurs, equivalent to a path difference of λ/2.

Hence the net path difference is

δ' = 2μt cos r + λ 2

Condition for Bright Fringes:

δ' = nλ

2μt cos r + λ 2 = nλ

2μt cos r = (2n - 1) λ 2

t = (2n - 1)λ 4μ cos r

Condition for Dark Fringes:

δ' = (2n + 1) λ 2

2μt cos r = nλ

t = nλ 2μ cos r

Thus, interference in a thin film is produced due to the superposition of light reflected from the upper and lower surfaces of the film.

In an extremely thin film, the thickness of the film is nearly zero (t ≈ 0).

For reflected light, one ray is reflected from the upper surface of the film while another ray is reflected from the lower surface.

The geometrical path difference between the two reflected rays is

δ = 2μt cos r

Since the film is extremely thin,

t = 0

Therefore,

δ = 0

However, the ray reflected from the upper surface undergoes a phase change of π (equivalent to a path difference of λ/2) because reflection occurs from a rarer to a denser medium.

Hence, the effective path difference becomes

δ = λ 2

This corresponds to the condition for destructive interference.

2μt cos r = 0

As a result, the two reflected rays cancel each other and the intensity of reflected light becomes minimum.

Therefore, an extremely thin film appears black in the reflected system.

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2024

(FYUG PHY-DSC201 )

5(a). Explain Huygens' principle of wavefront reconstruction.

5(b). What are the two ways of producing interference? Give one example of each.

Interference can be produced by the following two methods:

1. Division of Wavefront: A single wavefront is divided into two coherent wavefronts which interfere with each other.

Example: Young's Double Slit Experiment and Fresnel's Biprism.

2. Division of Amplitude: A single beam of light is divided into two parts by partial reflection and refraction. The two beams then interfere.

Example: Newton's Rings and Thin Film Interference.

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24. Describe Fresnel's method of biprism for obtaining interference.

same as

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25. How would you determine the wavelength of light with Llyod's mirror experiment?

Lloyd's mirror experiment is a method of producing interference by division of wavefront. It was first demonstrated by Dr. Humphrey Lloyd in 1834.

A monochromatic source S is placed close to a plane mirror MM′. The light from S reaches the screen directly and also after reflection from the mirror. The reflected ray appears to come from a virtual image S₂ of the source S.

Thus, S and S₂ act as two coherent sources and produce interference fringes on the screen AB.

The common region DF on the screen receives light from both S and S₂ and hence interference fringes are formed in this region.

The central fringe is dark because the reflected beam undergoes a phase change of π radians on reflection from the mirror.

Let,

• a = Distance of source S from the mirror
• d = Distance between the coherent sources S and S₂
• D = Distance of the screen from the source
• β = Fringe width

Since S₂ is the virtual image of S,

d = 2a

The fringe width is given by

β = λD d

Substituting d = 2a,

β = λD 2a

Therefore,

λ = 2aβ D

The fringe width β is measured using a travelling microscope. Knowing the values of a and D, the wavelength λ of the monochromatic light can be calculated using the above relation.

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(c) What is Stokes' law for the phase change of reflection? Show that a phase change of π occurs when reflection takes place at the surface of a denser medium.

Stokes' Law: According to Stokes' law, when a light wave is reflected from the surface of a denser medium, the reflected wave undergoes a phase change of π radians (180°). When reflection occurs from a rarer medium, no phase change takes place.

Consider two media of refractive indices μ₁ and μ₂, where

μ₂ > μ₁

A light wave travelling in the rarer medium is incident on the surface of the denser medium.

The amplitude reflection coefficient is given by

r = μ₁ - μ₂ μ₁ + μ₂

Since

μ₂ > μ₁

the numerator is negative. Therefore,

r < 0

A negative value of the reflection coefficient indicates that the reflected wave is reversed in phase with respect to the incident wave.

Thus the phase difference between the incident and reflected waves is

φ = π radians

or

φ = 180°

The corresponding path difference is

Δ = λ 2

Hence, when reflection takes place at the surface of a denser medium, the reflected light undergoes a phase change of π radians (equivalent to a path difference of λ/2).

This result is known as Stokes' law of phase change on reflection.

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(d) Explain the formation of Newton's ring by monochromatic light.

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