Physics: Coherent Beams and Interference

Coherent Beams

• To obtain a steady interference pattern, the two light waves superimposing must be coherent, meaning their phase difference is constant.
• There are two methods for obtaining coherent beams: division of wavefront and division of amplitude.
• The division of wavefront method divides the incident wavefront into two separate wavefronts, as seen in Young's Experiment.
• The division of amplitude method utilizes partial reflection and refraction to divide the incident wave into two beams, as seen in thin film interference.

Useful Relations

• The phase difference (Φ) between two waves is directly proportional to their path difference: Φ = 2π (path difference).
• A phase change of π (or a path difference of π/2) is introduced when a ray of light is reflected at the surface of a denser medium while traveling through a rarer medium.
• When a light wave travels a distance t in a medium with refractive index μ, the equivalent optical path is given by ut.

Interference in Thin Films

• In thin film interference, a monochromatic light ray is partially reflected and refracted at the surface of a thin transparent film, leading to interference between the reflected rays.
• The path difference between the interfering rays depends on the thickness of the film, the angle of incidence, and the refractive index of the film.
• The total path difference between the reflected rays is given by 2ut cosr ± π/2, where t is the film thickness, u is the refractive index, r is the angle of refraction, and the ±π/2 accounts for the phase change due to reflection from the denser medium.
• For constructive interference (maxima), the path difference must be an even multiple of π/2, leading to the condition: 2ut cosr = (2n - 1) π/2.
• For destructive interference (minima), the path difference must be an odd multiple of π/2, leading to the condition: 2ut cosr = nλ.

Colors in Thin Films

• Thin films like oil on water, soap films, or air films between glass plates exhibit brilliant colors due to interference.
• White light, consisting of a continuous range of wavelengths, interacts with the film, and only those wavelengths satisfying the condition for maxima or minima will be visible with maximum intensity.
• Colors in thin films are a result of the interference of light waves of different wavelengths, where certain wavelengths interfere constructively and others destructively.

Interference in Wedge-shaped Film

• In a wedge-shaped film, the thickness of the film varies linearly, leading to interference patterns that change across the film.
• The path difference between interfering rays is given by Δ = 2ut cos(r + θ) ± π/2, where θ is the angle of the wedge.
• The conditions for constructive and destructive interference in a wedge-shaped film are the same as those for thin films, but with the additional factor of the wedge angle.
• The fringe width (W) in a wedge-shaped film is given by W = λ / 2utθ, where θ is the wedge angle.

Anti-Reflection Coating

• Anti-reflection coatings are thin films applied to optical surfaces to reduce reflection and enhance light transmission.
• These coatings are designed to create destructive interference between the light reflected from the coating and the light reflected from the underlying surface.
• The thickness of the anti-reflection coating is chosen to minimize reflection for a specific wavelength.
• Magnesium Fluoride (MgF2) is commonly used as an anti-reflection coating material due to its refractive index between air and glass.
• The condition for destructive interference in an anti-reflection coating is: 2ut cosr = (2n - 1) π/2, where t is the coating thickness and u is its refractive index.