Unit 3: Wave Optics and Interference

1. Wave Optics & Wave Fronts

Wave optics (or physical optics) is the study of optical phenomena such as interference, diffraction, and polarization, where the wave-like nature of light is central to the explanation. This differs from geometrical optics, which treats light as a collection of rays traveling in straight lines.

Definition of a Wave Front

A wavefront is defined as the locus of all neighboring particles of a medium vibrating in the exact same phase of vibration at any given instant.

Properties of Wave Fronts

• Perpendicularity of Rays: Light rays representing the direction of propagation of energy are always perpendicular to the wavefront at every point.
• Phase Uniformity: The phase difference between any two points lying on the same wavefront is zero.
• Constant Time of Travel: Light takes the same amount of time to travel from the source to any point on a given wavefront.

Classification of Wavefronts

Depending on the shape of the light source, wavefronts are classified into three types:

2. Huygens' Principle

Huygens' Principle is a geometrical construction method used to determine the position and shape of a new wavefront at any subsequent time, given its initial shape and position.

Postulates of Huygens' Principle

1. Every point on a given wavefront (called the primary wavefront) acts as a source of fresh disturbance. These points emit new spherical wavelets called secondary wavelets.
2. These secondary wavelets travel in all directions with the speed of light in that medium.
3. The forward envelope (the tangential surface touching all these secondary wavelets in the forward direction) at any later instant gives the new position and shape of the wavefront (secondary wavefront) at that instant.

Note on the Backwave: Huygens' principle assumes that secondary wavelets travel only in the forward direction. Mechanically, the amplitude of secondary wavelets in any direction is proportional to (1 + cos(theta)), where theta is the angle with the direction of propagation. For the backward direction, theta = 180 degrees, which makes (1 + cos(180)) = 0. Hence, there is no backward wave.

3. Interference: Division of Amplitude and Wavefront

Interference is the redistribution of light intensity in a medium due to the superposition of two or more coherent light waves.

Coherence

To observe a stable, permanent interference pattern, the two interfering light waves must be coherent. This means they must have the same frequency, wavelength, and a constant or zero phase difference over time. Since independent physical sources cannot maintain a constant phase relationship, coherent sources are produced artificially from a single parent source using two primary methods:

4. Methods of Interference Fringe Production

Fresnel's Biprism

Fresnel's Biprism is an optical instrument that produces interference fringes by dividing the wavefront.

Construction: It consists of two thin prisms joined at their bases. The obtuse angle of the biprism is very large (about 179 degrees), while the acute angles on either side are extremely small (about 30 minutes or 0.5 degrees).

Working: When light from a single monochromatic slit S falls on the biprism, the light passing through the upper half refracts downward, and the light passing through the lower half refracts upward. This produces two virtual coherent sources, S1 and S2, separated by a distance 'd'. The waves originating from these virtual sources overlap on a screen to form interference fringes.

Fringe Width (beta) Formula:
beta = (D * lambda) / d

Where:

• lambda: Wavelength of light
• D: Distance between the slit and the screen
• d: Distance between the virtual coherent sources S1 and S2

Experimental Setup & Measurement of lambda:

1. Measurement of beta: Measured using a micrometer eyepiece by recording the position of several consecutive bright or dark fringes.
2. Measurement of D: Read directly off the graduated scale on the optical bench.
3. Measurement of d (Displacement Method): Since S1 and S2 are virtual, 'd' cannot be measured directly with a scale. A convex lens is placed between the biprism and the eyepiece. For two positions of the lens (conjugate positions), sharp images of the virtual sources are formed. Let the separations of the images in these two positions be d1 and d2. The actual separation 'd' is calculated as:

   d = sqrt(d1 * d2)

Substituting 'd' back into the fringe width equation, the wavelength is given by:

lambda = (beta * sqrt(d1 * d2)) / D

Lloyd's Mirror

Lloyd's Mirror is a simple method to produce interference fringes using a single front-silvered plane mirror.

Working: A narrow monochromatic slit S is placed slightly above the plane of the mirror. Light from S travels directly to a screen (direct beam). Another portion of light strikes the mirror at a grazing angle and is reflected toward the screen. The reflected wave appears to come from a virtual source S' situated behind the mirror. S and S' act as two coherent sources, producing interference fringes in the region of overlap.

The Central Fringe Anomaly:

• In Young's experiment or Fresnel's Biprism, the central fringe (where the path difference is zero) is always bright.
• In Lloyd's Mirror, when the screen is brought into contact with the edge of the mirror, the zero path difference point lies at the contact edge. This central boundary fringe is observed to be completely dark.
• This occurs because the reflected beam undergoes a phase change of pi (corresponding to an extra path difference of lambda / 2) upon reflection from a denser medium (the mirror surface). This phase change is explained by Stokes' Treatment.

5. Phase Change on Reflection: Stokes' Treatment

When light is reflected from the boundary of an optically denser medium, it undergoes a phase change of pi. No such phase change occurs when reflection takes place from a boundary of an optically rarer medium. Stokes proved this using the principle of optical reversibility.

Derivation / Explanation

Consider an incident wave of amplitude 'a' traveling in a rarer medium (refractive index n1) towards a denser medium (refractive index n2). Let:

• r: Reflection coefficient of the boundary for a wave in the rarer medium.
• t: Transmission coefficient of the boundary for a wave in the rarer medium.

The amplitudes of the reflected and transmitted waves are a*r and a*t, respectively.

According to the principle of optical reversibility, if there is no absorption, these two waves can be reversed in direction and should recombine to reconstruct the original incident wave of amplitude 'a' traveling in the opposite direction.

Let's analyze the paths of the reversed waves:

1. The reversed wave a*r reflects at the boundary with amplitude (a*r)*r = a*r² and refracts into the denser medium with amplitude a*r*t.
2. The reversed wave a*t travels from the denser medium back to the boundary. Let r' be the reflection coefficient and t' be the transmission coefficient for light traveling from the denser to the rarer medium. The wave reflects inside the denser medium with amplitude a*t*r' and transmits into the rarer medium with amplitude a*t*t'.

Combining the components in the rarer medium (which must equal the original amplitude 'a'):

a*r² + a*t*t' = a   =>   r² + t*t' = 1

Combining the components in the denser medium (which must cancel out to zero since there was no original wave propagating there):

a*r*t + a*t*r' = 0   =>   r*t + t*r' = 0

Since 't' cannot be zero:

r = -r'

The negative sign mathematically denotes a phase difference of pi (equivalent to a path difference of lambda / 2) between the two reflections. By experiment, this phase change occurs strictly when light reflects at the surface of a denser medium (rarer-to-denser boundary).

6. Interference in Thin Films

A thin film is an optical medium of thickness 't' (on the scale of a few micrometers or nanometers) bounded by two flat, parallel or inclined surfaces. Examples include a soap bubble or an oil layer on water.

Parallel Thin Films

Consider a parallel film of thickness 't' and refractive index 'mu'. A light ray of wavelength 'lambda' is incident at an angle 'i'. It is split into reflected and transmitted components.

Reflected Light System

The path difference between the first two reflected rays is mathematically derived as 2 * mu * t * cos(r), where 'r' is the angle of refraction. However, the first ray reflects at the upper boundary (rarer-to-denser), undergoing a phase shift of pi (path difference of lambda / 2). The second ray reflects at the lower boundary (denser-to-rarer), undergoing no phase change.

The effective optical path difference is:

Delta_eff = 2 * mu * t * cos(r) - lambda / 2

• Condition for Constructive Interference (Bright Fringes):

  2 * mu * t * cos(r) = (2n + 1) * lambda / 2    (where n = 0, 1, 2, ...)

• Condition for Destructive Interference (Dark Fringes):

  2 * mu * t * cos(r) = n * lambda    (where n = 1, 2, 3, ...)

Transmitted Light System

In the transmitted system, neither of the interfering rays undergoes reflection at a rarer-to-denser boundary. Therefore, no phase shift of pi is introduced. The effective optical path difference is:

Delta_eff = 2 * mu * t * cos(r)

• Condition for Constructive Interference (Bright Fringes):

  2 * mu * t * cos(r) = n * lambda    (where n = 0, 1, 2, ...)

• Condition for Destructive Interference (Dark Fringes):

  2 * mu * t * cos(r) = (2n + 1) * lambda / 2    (where n = 0, 1, 2, ...)

Observation: The conditions for bright and dark fringes in the reflected system are exactly opposite to those in the transmitted system. Hence, the two systems are complementary to each other.

Wedge-shaped Thin Films

A wedge-shaped film is formed when two flat surfaces are inclined at a very small angle 'theta'. The thickness of the film increases continuously from the edge of contact.

For reflected light with normal incidence (r = 0) and small angle 'theta', the path difference at any point of thickness 't' is:

Delta_eff = 2 * mu * t - lambda / 2

Conditions for Dark Fringes (Minima):

2 * mu * t = n * lambda

Fringe Width (beta):

The distance between two adjacent dark or bright fringes is given by:

beta = lambda / (2 * mu * theta)

Since 'beta' is constant for a given wedge angle, the fringes are parallel, straight, and equally spaced, running parallel to the line of contact of the wedge.

Comparison of Fringes

7. Newton's Rings

Newton's Rings is an interference pattern produced by the division of amplitude in a variable-thickness wedge-shaped air film formed between a curved lens and a flat plate.

Experimental Setup

A plano-convex lens of large radius of curvature 'R' is placed with its curved surface on a flat glass plate. This encloses a circular wedge of air whose thickness is zero at the point of contact and increases symmetrically outward. When monochromatic light is directed normally onto the lens, alternate dark and bright concentric circular fringes are observed in the reflected light.

Mathematical Derivation

The path difference for reflected light under normal incidence in an air film (mu = 1) of thickness 't' is:

Delta_eff = 2 * t - lambda / 2

From geometry, the film thickness 't' at a distance 'r_n' (radius of the n-th ring) from the point of contact is related to the radius of curvature 'R' of the lens by:

2 * t = r_n² / R    (assuming R is much larger than t)

Substituting this in the path difference equation:

Delta_eff = (r_n² / R) - lambda / 2

1. Dark Rings (Minima)

For destructive interference, the path difference must equal an odd multiple of lambda / 2 (or simply n * lambda after accounting for the reflection phase shift):

r_n² / R = n * lambda
r_n² = n * lambda * R

Since the diameter of the ring is D_n = 2 * r_n, we square both sides to get:

D_n² = 4 * n * lambda * R   =>   D_n = sqrt(4 * n * lambda * R)

Thus, the diameter of dark rings is proportional to the square root of natural numbers (1, 2, 3...).

2. Bright Rings (Maxima)

For constructive interference:

r_n² / R = (2n + 1) * lambda / 2
D_n² = 2 * (2n + 1) * lambda * R   =>   D_n = sqrt(2 * (2n + 1) * lambda * R)

Thus, the diameter of bright rings is proportional to the square root of odd natural numbers (1, 3, 5...).

Why is the Central Spot Dark? At the exact point of contact, the thickness of the air film t = 0. The path difference becomes Delta_eff = -lambda / 2. This satisfies the condition for destructive interference, making the central spot completely dark. If dust is present, the film thickness is not zero, and the central spot may appear bright.

Applications of Newton's Rings

1. Measurement of Wavelength (lambda)

To eliminate errors associated with the point of contact, the diameters of two different rings, say the n-th and (n+p)-th rings, are measured:

D_(n+p)² = 4 * (n + p) * lambda * R
D_n² = 4 * n * lambda * R

Subtracting these two equations gives:

D_(n+p)² - D_n² = 4 * p * lambda * R

Rearranging for wavelength:

lambda = (D_(n+p)² - D_n²) / (4 * p * R)

Where 'p' is an integer, and 'R' is measured using a spherometer.

2. Measurement of Refractive Index (mu) of a Liquid

If a liquid of refractive index 'mu' is placed between the lens and the glass plate, the optical path difference changes, causing the ring diameters to contract:

(D_(n+p)² - D_n²)_liquid = 4 * p * lambda * R / mu

Dividing the air-film equation by the liquid-film equation:

mu = (D_(n+p)² - D_n²)_air / (D_(n+p)² - D_n²)_liquid

This allows direct determination of 'mu' without needing to know 'lambda' or 'R'.

Common Exam Mistakes & Observations

• Phase Corrections: Always remember to subtract or add the extra lambda / 2 path difference when dealing with reflected light at a rarer-to-denser interface.
• Ring Spacing: Fringes of Newton's rings get closer together as the order 'n' increases because the radius is proportional to the square root of 'n'.
• Transmitted Light Setup: If the exam problem states the Newton's rings are observed in transmitted light, the conditions invert: the central spot becomes bright, and there is no Stokes' phase change.