Interference is a phenomenon where two or more waves superpose to form a resultant wave of greater, lower, or the same amplitude. For sustained and observable interference patterns, the interfering waves must be coherent, meaning they must have the same frequency, constant phase difference, and be nearly monochromatic.
Principles of Interference
Division of Amplitude
In division of amplitude, a single light beam is split into two or more beams of reduced intensity by partial reflection or refraction. These beams then travel different optical paths and are later recombined to produce an interference pattern. This method often results in fringes of equal thickness or equal inclination.
Examples include interference in thin films and Newton's Rings.
Division of Wavefront
In division of wavefront, light from a single point source or narrow slit is divided into two parts by passing it through two closely spaced apertures. These two parts of the wavefront then propagate separately and eventually superpose to create an interference pattern. This method requires a point source or a narrow slit for good coherence.
Examples include Young's Double Slit experiment, Lloyd's Mirror, and Fresnel's Biprism.
Classic Interference Experiments
Young’s Double Slit Experiment (YDSE)
Young's Double Slit experiment is a classic demonstration of the wave nature of light and interference by division of wavefront.
- Setup: Light from a monochromatic point source passes through two narrow, parallel slits (S1 and S2) placed close together. These slits act as two coherent sources, and the light from them interferes on a screen placed some distance away.
- Fringe Formation: A pattern of alternating bright and dark bands (fringes) is observed on the screen. Bright fringes occur where constructive interference takes place, and dark fringes occur where destructive interference takes place.
- Conditions:
- For Constructive Interference (Bright Fringes): The path difference between the waves from S1 and S2 must be an integral multiple of the wavelength (nλ), where n = 0, ±1, ±2, ... .
Path difference = S2P - S1P = nλ - For Destructive Interference (Dark Fringes): The path difference must be an odd integral multiple of half the wavelength [(n + ½)λ], where n = 0, ±1, ±2, ... .
Path difference = S2P - S1P = (n + ½)λ
- For Constructive Interference (Bright Fringes): The path difference between the waves from S1 and S2 must be an integral multiple of the wavelength (nλ), where n = 0, ±1, ±2, ... .
- Fringe Width (β): The distance between two consecutive bright or two consecutive dark fringes.
β = λD / d
Where:
- λ is the wavelength of light.
- D is the distance from the slits to the screen.
- d is the distance between the two slits.
- Central Bright Fringe: For n=0, the path difference is zero, leading to a central bright fringe (maximum intensity).
Lloyd’s Mirror
Lloyd's Mirror is another experiment demonstrating interference by division of wavefront, creating a virtual coherent source.
- Setup: A monochromatic point source (S) is placed close to a highly polished plane mirror. Light directly from S and light reflected from the mirror interfere. The reflected light appears to come from a virtual image source (S') formed behind the mirror. S and S' act as coherent sources.
- Fringe Characteristics: The interference pattern consists of straight, equally spaced fringes. A key characteristic is the central dark fringe at the point where the mirror surface meets the observation screen. This is because light reflected from a denser medium (mirror) undergoes a phase change of π (or path change of λ/2).
Fresnel’s Biprism
Fresnel's biprism uses refraction to create two virtual coherent sources from a single source, enabling interference by division of wavefront.
- Setup: A single narrow slit (S) illuminated by monochromatic light is placed in front of a specially designed prism called a biprism. The biprism consists of two acute-angled prisms joined at their bases.
- Principle: Light from the slit passes through the two halves of the biprism, refracting to appear as if it originates from two virtual coherent sources (S1 and S2) located symmetrically behind the actual slit. These two virtual sources then produce an interference pattern similar to Young's Double Slit experiment.
- Observation: Fringes are observed in the region where light from S1 and S2 overlap.
Types of Interference Fringes
Fringes of Equal Inclination (Haidinger Fringes)
Haidinger Fringes are interference fringes formed by beams that strike a thin film or plate at the same angle of inclination. They are non-localized fringes, observed at infinity (or in the focal plane of a lens) and are typically produced by an extended source.
- These fringes are formed by interference of rays having the same angle of inclination relative to the film, regardless of their point of origin on the extended source.
- They appear as concentric circles, characteristic of interference in thick films or plates, like in Fabry-Perot interferometers.
Fringes of Equal Thickness (Fizeau Fringes)
Fizeau Fringes are interference fringes formed by beams that have traversed paths of equal optical thickness in a thin film. They are localized fringes, observed in the plane of the film itself, and are typically produced by an extended source.
- These fringes are formed when the thickness of the film varies, such as in a wedge-shaped film.
- The fringes follow contours of equal film thickness, providing a map of the thickness variations.
Interference in Thin Films
Interference in thin films occurs when light reflects from the top and bottom surfaces of a thin transparent layer. The two reflected rays interfere, leading to an observable pattern.
- Phase Change on Reflection: When light reflects from an interface with a denser medium (e.g., air to glass), it undergoes a phase change of π (or an additional path difference of λ/2). When reflecting from a less dense medium (e.g., glass to air), there is no phase change.
- Conditions for Interference (Reflected Light): Consider a thin film of thickness 't' and refractive index 'μ', illuminated by light of wavelength 'λ' at an angle 'r' inside the film.
Parallel Films
For a parallel thin film (uniform thickness 't'):
- Optical Path Difference (OPD): The path difference between the ray reflected from the top surface and the ray reflected from the bottom surface is 2μt cos(r).
- Total Phase Change: The ray reflected from the top surface (air-film interface) undergoes a π phase change. The ray reflected from the bottom surface (film-air interface) typically undergoes no phase change. Thus, there is an effective additional path difference of λ/2 due to reflection.
- Conditions for Maxima and Minima:
- For Constructive Interference (Bright):
2μt cos(r) + λ/2 = nλ
2μt cos(r) = (2n - 1)λ/2, where n = 1, 2, 3... - For Destructive Interference (Dark):
2μt cos(r) + λ/2 = (n + ½)λ
2μt cos(r) = nλ, where n = 0, 1, 2, 3...
- For Constructive Interference (Bright):
Wedge-shaped Films
A wedge-shaped film is a thin film where the thickness varies linearly (e.g., two glass plates touching at one end and separated by a thin wire at the other). Interference fringes are observed perpendicular to the edge where the plates meet.
- Optical Path Difference: Similar to parallel films, but 't' varies along the film. The path difference at any point is 2μt cos(r) ± λ/2, where 't' is the thickness at that point.
- Fringe Pattern: Straight, equally spaced fringes are observed, parallel to the line of intersection of the two surfaces of the wedge. These are Fizeau fringes.
- Fringe Width (β): For normal incidence (cos(r) ≈ 1) and small wedge angle θ,
β = λ / (2μθ)
Where θ is the angle of the wedge in radians.
Newton’s Rings
Newton's Rings is an interference phenomenon observed when a plano-convex lens is placed on a plane glass plate, forming an air wedge between them.
- Setup: A plano-convex lens of large radius of curvature is placed on a flat glass plate. Monochromatic light is allowed to fall normally on the lens surface. The air film between the lens and the plate acts as a thin wedge, causing interference.
- Formation: Concentric dark and bright rings are observed, centered at the point of contact. These are fringes of equal thickness (Fizeau fringes). The central fringe is always dark in reflected light.
- Conditions for Interference (Reflected Light): The interference occurs between light rays reflected from the bottom surface of the lens and the top surface of the glass plate. The path difference arises from the varying thickness of the air film. Due to reflection from the denser medium (glass plate), there is a π phase change (or λ/2 path difference) for the ray reflected from the glass plate.
- For Dark Rings: (Constructive interference with effective λ/2 path change)
2t = nλ, where n = 0, 1, 2, 3...
- For Bright Rings: (Destructive interference with effective λ/2 path change)
2t = (n + ½)λ, where n = 0, 1, 2, 3...
- Radii of Rings: If R is the radius of curvature of the plano-convex lens, then for the nth dark ring, the radius rn is given by:
rn2 = nRλ
And for the nth bright ring:
rn2 = (n + ½)Rλ
Measurement of Wavelength (λ) using Newton's Rings
- The radii of the dark rings are measured. For the nth dark ring and (n+p)th dark ring:
- Subtracting these equations:
- Therefore, the wavelength can be calculated as:
- This method provides a precise way to determine the wavelength of monochromatic light.
rn2 = nRλ
r(n+p)2 = (n+p)Rλ
r(n+p)2 - rn2 = pRλ
λ = (r(n+p)2 - rn2) / (pR)<
Measurement of Refractive Index (μ) using Newton's Rings
- To measure the refractive index of a transparent liquid, the experiment is performed in two parts: first with air between the lens and plate, and then by introducing the liquid between them.
- With Air: The radius of the nth dark ring (rn) is given by:
rn2 = nRλ
- With Liquid: When a liquid of refractive index μ is introduced, the wavelength of light in the liquid becomes λ' = λ/μ. The condition for the nth dark ring in the liquid (r'n) becomes:
2t = nλ'
2t = nλ / μ - Using the relation for the radius of the dark ring:
- Therefore, the refractive index of the liquid can be determined by comparing the radii of the rings in air and in liquid:
- More accurately, comparing the difference in squares of radii for (n+p)th and nth dark rings in air and liquid:
r'n2 = nR(λ/μ) = rn2 / μ
μ = rn2 / r'n2
μ = (r(n+p)2 - rn2) / (r'(n+p)2 - r'n2)