Unit 4: Interferometry and Diffraction

1. Michelson Interferometer

The Michelson Interferometer is an elegant optical device that uses the division of amplitude technique to split a light beam into two paths, reflect them back, and recombine them to produce interference fringes. It has played a historic role in physics and remains a vital tool for high-precision optical measurements.

1.1 Idea of Form of Fringes (No Theory Required)

Depending on the alignment of the mirrors and the nature of the light source, different types of interference fringes are observed:

• Circular Fringes (Fringes of Equal Inclination): These are also known as Haidinger fringes. They form when the two mirrors, M₁ and M₂, are perfectly perpendicular to each other. The virtual air film between them has a uniform thickness. These fringes are localized at infinity and are observed using a telescope focused on infinity.
• Localized Straight or Curved Fringes (Fringes of Equal Thickness): If the mirrors M₁ and M₂ are slightly inclined, a wedge-shaped virtual air film is formed between them. The fringes are straight and parallel to the edge of the wedge. If the path difference is small, they look straight; for larger path differences, they become slightly curved (hyperbolic).

1.2 Determination of Wavelength of Monochromatic Light

The Michelson interferometer can measure the wavelength of a monochromatic light source with high accuracy by tracking the motion of the movable mirror.

Formula: d = N * (λ / 2)

Where:

• d: Distance moved by the movable mirror M₁.
• N: Number of fringes that shift across the field of view.
• λ: Wavelength of the light source.

Step-by-Step Procedure:

1. Set up the interferometer to obtain circular fringes.
2. Focus the cross-wires of the telescope on a specific reference fringe (e.g., the central bright spot).
3. Slowly move the mirror M₁ using the micrometer screw. Count the number of fringes (N) that pass or disappear at the center.
4. Measure the total distance (d) traversed by the mirror.
5. Calculate the wavelength using the rearranged formula: λ = 2d / N.

Exam Note: A common mistake is forgetting the factor of 2. Because light travels to the mirror and back, a physical mirror movement of 'd' changes the total optical path by '2d'.

1.3 Determination of Wavelength Difference

When a light source contains two closely spaced wavelengths, λ₁ and λ₂ (such as the Sodium D-lines where λ₁ ≈ 589.0 nm and λ₂ ≈ 589.6 nm), they produce two separate fringe systems that overlap.

As the mirror M₁ is moved, the two systems alternate between being in-phase (reinforcing each other, leading to maximum fringe contrast/distinctness) and out-of-phase (destructive overlap, leading to minimum contrast/fringe disappearance).

If we measure the distance 'd' between two consecutive positions of maximum distinctness (or minimum visibility):

Formula: Δλ = (λ_avg)² / 2d

Where:

• λ_avg: Mean wavelength (λ₁ + λ₂) / 2
• d: Displacement of mirror M₁ between consecutive fringe disappearances.
• Δλ: Wavelength difference |λ₁ - λ₂|.

1.4 Determination of Refractive Index of a Thin Film

If a thin transparent sheet (such as mica or glass) of thickness 't' and refractive index 'μ' is placed in the path of one of the interfering beams, the optical path length of that beam increases.

The geometric path through the sheet is 't'. Since light travels through the sheet twice (forward and backward), the total path length increases by 2(μ - 1)t.

Formula: 2(μ - 1)t = N * λ

This path change causes a displacement of 'N' fringes across the field. Rearranging this relation allows us to calculate the refractive index:

μ = 1 + (N * λ) / 2t

Or, if μ is known, the thickness 't' of the film can be calculated: t = N * λ / [2(μ - 1)].

1.5 Visibility of Fringes

Fringe visibility (V) measures the contrast of the interference pattern. It depends on the coherence of the light source and the relative intensity of the interfering beams.

Formula: V = (I_max - I_min) / (I_max + I_min)

Where:

• I_max: Intensity of the bright fringe (maximum).
• I_min: Intensity of the dark fringe (minimum).

Physical Interpretation:

• V = 1: Perfect contrast (I_min = 0). The fringes are exceptionally clear.
• V = 0: Complete lack of contrast (I_max = I_min). The fringes disappear, resulting in uniform illumination.

2. Types of Diffraction (Fresnel vs. Fraunhofer)

Diffraction is defined as the bending of light waves around the corners of an obstacle or aperture into the region of geometrical shadow.

Diffraction phenomena are broadly classified into two distinct classes based on the distance of the source and screen from the diffracting element:

3. Kirchhoff's Integral Theorems

Wave propagation was historically explained qualitatively by Huygens' Principle. Kirchhoff put this principle on a rigorous mathematical footing by solving the wave equation using Green's theorem.

3.1 Kirchhoff's Integral Theorem (Statement Only)

Statement: The wave amplitude U(P) at any point P inside a closed volume containing no sources is completely determined by the values of the wave field U and its normal derivative (∂U/∂n) on the bounding closed surface S.

Mathematically expressed as:

U(P) = (1 / 4π) * ∬ [ U * (∂/∂n)(e^ikr / r) - (e^ikr / r) * (∂U/∂n) ] dS

Where 'r' is the distance from point P to the surface element dS, and 'k' is the wave number (2π/λ).

3.2 Fresnel-Kirchhoff Integral Formula (Qualitative Discussion Only)

The Fresnel-Kirchhoff formula is a modification of Kirchhoff's theorem applied specifically to optical diffraction. It provides the foundation for wave optics by calculating the light field at a screen after passing through an aperture.

Key Qualitative Takeaways:

• It explains why light does not travel backwards (a historical limitation of Huygens' theory) by introducing the obliquity factor: F(θ) = [1 + cos(θ)] / 2, where θ is the angle between the normal to the wavefront and the direction of propagation. For backward direction (θ = 180°), F(180°) = 0, mathematically explaining why light does not travel backwards.
• It states that the light disturbance at any point on the screen is the sum of spherical secondary wavelets originating from every point on the aperture, weighted by their amplitude and the obliquity factor.

4. Fraunhofer Diffraction

Fraunhofer diffraction occurs when the incident light is parallel (plane wavefronts) and the diffracted light is focused on a screen placed at an infinite distance (usually achieved using a converging lens).

4.1 Single Slit Diffraction

Consider monochromatic light of wavelength λ incident normally on a long narrow slit of width 'a'.

The path difference between the light rays originating from the two edges of the slit at diffraction angle θ is given by: Δ = a * sin(θ).

Conditions for Intensity Distribution:

• Central Maximum: Occurs at θ = 0. All secondary waves arrive in phase, producing a highly intense central peak.
• Minima (Dark Fringes): Occur when:

  a * sin(θ) = m * λ (where m = ±1, ±2, ±3...)

• Secondary Maxima (Weak Bright Fringes): Occur approximately when:

  a * sin(θ) = (2m + 1) * (λ / 2) (where m = ±1, ±2, ±3...)

4.2 Double Slit Diffraction

Consider two parallel slits, each of width 'a', separated by an opaque space of width 'b'. The distance between corresponding points of the two slits is called the grating element, d = a + b.

The double slit pattern is a superposition of two separate optical phenomena:

1. Diffraction: A single-slit diffraction pattern acting as an "envelope".
2. Interference: Multiple-slit interference fringes inside the diffraction envelope.

Mathematical Conditions:

• Interference Maxima (Principal Maxima):

  (a + b) * sin(θ) = n * λ (where n = 0, ±1, ±2...)

• Interference Minima:

  (a + b) * sin(θ) = (2n + 1) * (λ / 2)

• Diffraction Minima:

  a * sin(θ) = m * λ (where m = ±1, ±2...)

Concept of Missing Orders: If the conditions for an interference maximum and a diffraction minimum are satisfied at the exact same angle θ, that particular order of interference maximum disappears. This is called a missing order:

(a + b) / a = n / m

4.3 Transmission Diffraction Grating

A diffraction grating is an optical component consisting of a large number (N) of parallel, equally spaced slits. If 'a' is the slit width and 'b' is the width of the opaque ruling, then (a + b) is the grating element.

Grating Equation: (a + b) * sin(θ) = n * λ

Where:

• n: Order of the spectrum (n = 1, 2, 3...)
• θ: Angle of diffraction.
• λ: Wavelength of incident light.

If the number of lines per unit length of the grating is 'N_L', then the grating element is (a + b) = 1 / N_L.

5. Resolving Power

The resolving power of an optical instrument represents its ability to form distinctly separate images of two close objects or spectral lines.

Rayleigh's Criterion for Resolution: Two closely spaced spectral lines are just resolved if the central maximum of the diffraction pattern of one wavelength falls exactly on the first minimum of the diffraction pattern of the other wavelength.

5.1 Resolving Power of a Grating

The resolving power (R) of a grating is defined as the ratio of the wavelength λ to the smallest wavelength difference dλ that can be resolved.

Formula: R = λ / dλ = n * N

Where:

• n: Spectral order of observation.
• N: Total number of active lines (rulings) on the illuminated area of the grating.

Exam Note: To increase resolving power, one can either observe higher order spectra (increase n) or use a wider grating containing more lines (increase N).

5.2 Resolving Power of a Telescope

A telescope is used to view distant point objects (such as stars). Due to diffraction through the circular aperture of the telescope's objective lens, the image of a star is not a point but a central bright disk (Airy's disk) surrounded by concentric dark and bright rings.

The angular separation (dθ) between two stars that can just be resolved is called the limit of resolution:

dθ = 1.22 * λ / D

The Resolving Power (RP) is the reciprocal of the limit of resolution:

RP = 1 / dθ = D / (1.22 * λ)

Where:

• D: Diameter of the telescope's objective aperture (lens or mirror).
• λ: Wavelength of light being received.

Physical Significance: To resolve very close stellar objects, telescopes must have very large diameters (D). This is why major astronomical observatories build extremely large aperture telescopes.