1. Introduction to Interference
When light waves from coherent sources meet, they combine according to the principle of superposition. The resultant intensity depends on the phase difference between the waves, which in turn depends on the path difference.
- Light sources must be coherent (constant phase relationship)
- Light sources should be monochromatic (single wavelength)
- The amplitudes of interfering waves should be nearly equal
- Sources should be narrow and close together
Waves meet in phase (crests align with crests)
Path difference = nλ (where n = 0, 1, 2, ...)
Phase difference = 2nπ
Result: Maximum intensity (Bright fringe)
Waves meet out of phase (crests align with troughs)
Path difference = (2n+1)λ/2
Phase difference = (2n+1)π
Result: Minimum intensity (Dark fringe)
2. Interference in Thin Film of Uniform Thickness
When light falls on a thin transparent film (like soap bubble, oil film on water, or anti-reflective coating), it gets partially reflected from the top surface and partially from the bottom surface. These two reflected rays, being derived from the same incident ray, are coherent and produce interference.
- When light reflects from a denser medium (going from lower to higher refractive index), it undergoes a phase change of π (equivalent to path difference of λ/2)
- When light reflects from a rarer medium (higher to lower refractive index), there is no phase change
- This is crucial for determining interference conditions in thin films
Optical Path Difference Calculation
Consider a thin film of refractive index μ and thickness t, with light incident at angle i:
Geometrical path difference: Ray 2 travels an extra distance inside the film
where r is the angle of refraction
Optical path difference: Multiply by refractive index μ
But Ray 1 travels extra in air while Ray 2 is in the film. This horizontal component must be subtracted:
Net optical path difference (geometric):
Additional phase change: Ray 1 reflects from denser medium → extra λ/2 path difference
3. Conditions for Maxima and Minima (Reflected System)
For a thin film surrounded by air (or when μ_film > μ_surrounding), Ray 1 experiences a phase change of π on reflection from the top surface, while Ray 2 does not experience phase change when reflecting from the bottom surface (rarer medium below) or experiences another π change (denser below). The most common case is film in air:
Effective path difference = 2μt cos r - λ/2
(The λ/2 is subtracted because only Ray 1 has phase change)
Condition for Constructive Interference (Bright Fringes/Maxima)
For constructive interference, path difference = nλ (n = 0, 1, 2, ...)
Condition for Destructive Interference (Dark Fringes/Minima)
For destructive interference, path difference = (2n + 1)λ/2
(where we relabel n+1 as n)
When light falls perpendicular to the film surface, cos r = 1:
- Maxima: 2μt = (2n + 1)λ/2
- Minima: 2μt = nλ
For bright film (maxima) in reflected light at normal incidence:
| Condition | Reflected Light | Transmitted Light |
|---|---|---|
| Maxima (Bright) | 2μt cos r = (2n+1)λ/2 | 2μt cos r = nλ |
| Minima (Dark) | 2μt cos r = nλ | 2μt cos r = (2n+1)λ/2 |
When white light falls on a thin film, different wavelengths satisfy the maxima/minima conditions at different thicknesses. At any point with specific thickness t:
- Some wavelengths show constructive interference (appear bright)
- Other wavelengths show destructive interference (get suppressed)
- The remaining wavelengths mix to produce the observed color
This explains the colorful patterns seen in soap bubbles and oil films on water.
4. Wedge-Shaped Film (Qualitative)
Formation of Fringes
When monochromatic light falls normally on a wedge-shaped film, interference occurs between rays reflected from the top and bottom surfaces of the air film. Since the thickness varies along the wedge:
- At positions where 2t = nλ, we get dark fringes
- At positions where 2t = (2n+1)λ/2, we get bright fringes
- Each fringe is a locus of points with the same film thickness
- Fringes are straight and parallel to the contact edge
- Fringes are equally spaced
- Dark fringe at contact edge: At t = 0, path difference = λ/2 (due to phase change), giving destructive interference
- Fringe spacing depends on θ: Smaller wedge angle → wider fringes
- With white light: Colored fringes appear, with edge being dark
Total distance = 10 cm, Number of fringes = 20
Diameter d = L × θ (where L = 10 cm)
5. Newton's Rings
Relation Between Thickness and Radius
For the geometry of a spherical surface touching a flat plate:
Using the property of a chord in a circle:
Since t << R, we can neglect t²:
Conditions for Bright and Dark Rings
For normal incidence on air film, using the thin film interference conditions:
Diameter of Newton's Rings
Squaring both sides:
Applications of Newton's Rings
If the radius of curvature R of the lens is known, the wavelength of monochromatic light can be determined:
For the m-th and n-th dark rings (m > n):
D₅ = 3.36 mm = 0.336 cm, D₁₅ = 5.90 mm = 0.590 cm
R = 100 cm, n = 5, m = 15
When a liquid of refractive index μ is introduced between the lens and plate, the wavelength in the medium becomes λ/μ:
For air film: D_air² = 4nλR
For liquid film: D_liquid² = 4nλR/μ
Taking the ratio:
6. Engineering Applications of Thin Film Interference
- Thin film on lens surfaces
- Thickness = λ/4n for destructive interference
- Reduces reflection, increases transmission
- Used in cameras, eyeglasses, solar cells
- Multilayer thin film structures
- Selective transmission of wavelengths
- Band-pass, high-pass, low-pass filters
- Used in spectrometers, lasers
- Testing flatness of surfaces
- Measuring small thicknesses
- Quality control in optics manufacturing
- Newton's rings for lens testing
For a single-layer AR coating on glass (n_glass = 1.5):
- Optimal coating refractive index: n_c = √(n_air × n_glass) = √1.5 ≈ 1.22
- Optical thickness = λ/4 (quarter-wave plate)
- MgF₂ (n = 1.38) is commonly used
- Reduces reflection from ~4% to <1%
When light travels through a coating of thickness t = λ/(4n), the round-trip path difference is 2nt = λ/2. Combined with the π phase change from the coating-glass interface, the total path difference becomes λ. This means the reflected waves from top and bottom of the coating are exactly out of phase, causing destructive interference and minimizing reflection.