Physics: Interference of Light

Here are the study notes in detailed bullet points, focusing on key facts with context:

• Coherent Sources*
• Two sources are said to be coherent if they emit light waves of the same frequency, nearly the same amplitude, and are always in phase with each other.
• There are two types of coherence:
  • Temporal coherence (Coherence in time): a measure of phase relation of a wave reaching at given points at two different times.
  • Spatial coherence (Coherence in space): a measure of phase relation between the waves reaching at two different points in space at the same time.
• Interference*
• Interference occurs when two or more waves superimpose on each other.
• Methods for producing interference patterns:
  • Division of wave front: interference of light takes place between waves from two sources formed due to a single source.
  • Division of amplitude: interference takes place between the waves from the real source and virtual source.
• Interference in Thin Film*
• An optical medium of thickness in the range of 0.5 μm to 10 μm may be considered as a thin film.
• Interference can take place in a thin film by:
  • Reflected light
  • Transmitted light
• Conditions for constructive and destructive interference:
  • Constructive interference: path difference = even multiple of λ/2
  • Destructive interference: path difference = odd multiple of λ/2
• Colors of thin films: only few wavelengths will satisfy the condition of maxima, and corresponding colors will be seen in the pattern.
• Necessity of Broad Source or Extended Source*
• A broad source is necessary to see the whole interference pattern.
• Interference Due to Non-uniform Thin Film (Wedge-shaped Thin Film)*
• Path difference: Δ = 2µt cos (r+θ) - λ/2
• Conditions for maxima and minima:
  • Constructive interference: 2µt cos (r+θ) = (2n + 1) λ/2
  • Destructive interference: 2µt cos (r+θ) = nλ
• Newton's Ring Experiment*
• Newton's rings: a special case of interference in a film of variable thickness, such as that formed between a plane glass plate and a convex lens in contact with it.
• Experimental arrangement: a monochromatic light source, a lens, a glass plate, and a plano-convex lens.
• Path difference: Δ = 2µt - λ/2
• Conditions for maxima and minima:
  • Constructive interference: 2µt = (2n + 1) λ/2
  • Destructive interference: 2µt = nλ
• Diameter of bright and dark rings:
  • Bright rings: Dn = √(2λR) √(2n + 1)
  • Dark rings: Dn = √(4nλR)
• Applications of Newton's Rings*
• Measurement of wavelength of sodium light by Newton's rings.
• Measurement of refractive index of a liquid by Newton's rings.
• Diffraction*
• The phenomenon of bending of light around the corners of an obstacle and their spreading into the geometrical shadow of an object.
• Diffraction pattern: a distribution of light intensity resulting in dark and bright fringes.
• Types of diffraction:
  • Fresnel's diffraction: source of light or screen or both are at a finite distance from the diffracting element.
  • Fraunhofer's diffraction: both the source and the screen are effectively at infinite distance from the diffracting element.
• Fraunhofer Diffraction at a Single Slit*
• Conditions for maxima and minima:
  • Principal maximum (central maxima): α = 0
  • Minima: sin α = 0 => α = ±nπ, where n = 1, 2, 3, ...
  • Secondary maxima: α = ±(2n + 1) π/2, where n = 1, 2, 3, ...
• Intensity distribution:
  • Intensity of central maxima: I = I0
  • Intensity of first secondary maxima: I1 = I0 / 22
  • Intensity of second secondary maxima: I2 = I0 / 62
• Fraunhofer Diffraction at a Double Slit*
• Diffraction pattern: a number of equally spaced interference maxima and minima.
• Resultant amplitude: due to the interference between the two waves of same amplitude and a phase difference φ originating from the middle points of the two slits.### Fraunhofer's Diffraction at a Double Slit
• The resultant intensity at a point on the screen due to a double slit is given by I = 4A^2 cos^2 β / α^2
• The intensity depends on two factors:
  • sin^2 α / α^2, which gives the diffraction pattern due to each single slit
  • cos^2 β, which gives the interference pattern due to two waves of same amplitude (R) originating from the mid-points of their respective slits

Position of Maxima and Minima

• Principal maxima are given by A^2 / α^2, θ = 0
• Position of minima are given by sin α = 0, α ≠ 0, which implies α = ±mπ, where m = 1, 2, 3, ...
• Position of secondary maxima approach to α = ±3π/2, ±5π/2, ±7π/2, ...

Interference Term

• I is maximum when cos^2 β = 1, which implies β = ±nπ, where n = 0, 1, 2, 3, ...
• Intensity is minimum when cos^2 β = 0, which implies β = (2n+1)π/2

Plane Transmission Diffraction Grating (N-Slits Diffraction/Diffraction due to double slits)

• A plane diffraction grating is an arrangement of a large number of close, parallel, straight, transparent, and equidistant slits, each of equal width a, with neighboring slits being separated by an opaque region of width b.
• The spacing (a + b) between adjacent slits is called the diffraction element or grating element.

Theory of Grating

• The diffraction pattern obtained on the screen with a very large number of slits consists of extremely sharp principal interference maximum, while the intensity of secondary maxima becomes negligibly small.
• If we increase the number of slits (N), the intensity of principal maxima increases.
• According to Huygen's principle, all points within each slit become the source of secondary wavelets, which spread out in all directions.
• The resultant amplitude R due to all waves diffracted from each slit in the direction θ is given by R = A sin α, where A is a constant and α = πa sinθ / λ.

Resultant Amplitude and Intensity

• The resultant amplitude R' in the direction θ is given by R' = R sin Nβ / sin β
• The corresponding intensity at P is given by I = R'^2 = A^2 sin^2 α / α^2 sin^2 Nβ / sin^2 β

Principal Maxima, Minima, and Secondary Maxima

• The condition for principal maxima is sin β = 0, i.e., β = ±nπ, where n = 0, 1, 2, 3, ...
• The condition for minima is sin Nβ = 0, but sin β ≠ 0
• The intensity of secondary maxima is proportional to N^2 /[1+(N^2-1) sin^2 β], whereas the intensity of principal maxima is proportional to N^2.

Absent Spectra with a Diffraction Grating

• It may be possible that while the first order spectra is clearly visible, second order may not be visible at all, and the third order may again be visible.
• The condition for the absent spectra is given by (a+ b) /a = n/m, where a is the width of the transparent portion and b is the width of the opaque portion.

Number of Orders of Spectra with a Grating

• The number of spectra that are visible in a given grating can be easily calculated with the help of the equation n = (a+b) sin θ / λ
• The maximum possible value of the angle of diffraction θ is 90°, and therefore sin θ = 1.

Interference

• Coherent sources: two sources emitting light waves of the same frequency, nearly the same amplitude, and always in phase with each other.
  • Temporal coherence (longitudinal coherence): phase relation of a wave at a given point at two different times.
  • Spatial coherence (transverse coherence): phase relation between waves at two different points in space at the same time.
• Methods for producing interference patterns:
  • Division of wavefront: interference between waves from two sources formed from a single source.
  • Division of amplitude: interference between the waves from the real source and virtual source.

Interference in Thin Films

• Thin film: an optical medium of thickness in the range of 0.5 μm to 10 μm.
• Interference in a thin film occurs due to:
  • Reflected light
  • Transmitted light
• Condition for constructive interference: 2 µt cos r = n λ (reflected light) 2 µt cos r = n λ (transmitted light)
• Condition for destructive interference: 2 µt cos r = (2n + 1) λ/2 (reflected light) 2 µt cos r = (2n + 1) λ/2 (transmitted light)
• Colours of thin films: depend on the thickness of the film and the angle of incidence.

Interference Due to Non-Uniform Thin Films (Wedge Shaped Thin Films)

• Condition for constructive interference: 2µcos (r + θ) = (2n + 1) λ/2
• Condition for destructive interference: 2µcos (r + θ) = nλ

Newton's Ring Experiment

• Newton's rings: a special case of interference in a film of variable thickness.
• Experimental arrangement: extended source of light, lens, glass plate, and plano-convex lens.
• Condition for constructive interference: 2 µt = (2n + 1) λ/2
• Condition for destructive interference: 2 µt = nλ

Diffraction

• The phenomenon of bending of light around the corners of an obstacle and their spreading into the geometrical shadow.
• Types of diffraction:
  • Fresnel's diffraction: source of light or screen or both are at a finite distance from the diffracting element.
  • Fraunhofer's diffraction: both the source and the screen are at an infinite distance from the diffracting element.

Fraunhofer's Diffraction at a Single Slit

• Experimental arrangement: source of light, collimating lens, slit, converging lens, and screen.
• Condition for maxima: α = 0 or θ = 0
• Condition for minima: sin α = 0
• Position of minima: α = nπ

Fraunhofer's Diffraction at a Double Slit

• Experimental arrangement: source of light, collimating lens, two parallel slits, converging lens, and screen.
• Condition for maxima: R = 2A sin(πa sinθ/λ)
• Condition for minima: sin(πa sinθ/λ) = 0

Plane Transmission Diffraction Grating (N-Slits Diffraction/Diffraction due to Double Slits)

• A plane diffraction grating consists of a large number of close, parallel, straight, transparent, and equidistant slits.
• Condition for principal maxima: a sin θ = nλ
• Condition for minima: sin(2β) = 0
• Positions of secondary maxima: between two successive principal maxima.

Absent Spectra with a Diffraction Grating

• Condition for absent spectra: a sin θ = mλ (a + b) sin θ = nλ

• Absent spectra occur when the conditions for single-slit pattern and grating spectrum are simultaneously satisfied.### Conditions for Absent Spectra

• The condition for absent spectra in the diffraction pattern is (a+b)/a = n/m.

• When the width of the transparent portion is equal to the width of the opaque portion (a = b), the condition becomes n = 2m, resulting in the absence of 2nd, 4th, 6th, etc. orders of the spectra.

• When b = 2a, the condition becomes n = 3m, resulting in the absence of 3rd, 6th, 9th, etc. orders of the spectra.

Number of Orders of Spectra with a Grating

• The equation to calculate the number of visible spectra in a grating is (a+b) sin θ = n λ.
• The grating element (a+b) is equal to 1/N, where N is the number of lines per inch in the grating.
• The maximum possible value of the angle of diffraction θ is 90°, so sin θ = 1.
• The maximum possible order of spectra (n max) is equal to (a+b)/λ.
• If the grating element (a+b) is between λ and 2λ, then n max is determined by the equation (a+b)/λ.