Wave Optics
Wave optics describes the connection between waves and rays of light. According to the wave theory of light, the light is a form of energy which travels through a medium in the form of transverse wave motion. The speed of light in a medium depends upon the nature of the medium.
Newton’s Corpuscular Theory
- Light consists of very small invisible elastic particles which travel in vacuum with a speed of 3 x 10^{8} m/s.
- The theory could explain reflection and refraction.
Newton's Corpuscular Theory
- The size of corpuscular of different colours of light are different.
- It could not explain interference, diffraction, polarization. photoelectric effect and Compton effect. The theory failed as it could not explain why light travels faster in a rarer medium than in a denser medium.
Wavefront
A wavefront is defined as the continuous locus of all the particles of a medium, which are vibrating in the same phase.
These are three types:
- Spherical wavefront
- Cylindrical wavefront
- Plane wavefront
S = source of light.
AB = wavefront and SP SQ and SR are rays cf light
Huygen’s Wave Theory
- Light travel in a medium in the form of wavefront.
- A wavefront is the locus of all the particles vibrating in same phase.
- All particles on a wavefront behaves as a secondary source of light, which emits secondary wavelets.
Huygen's Principle
- The envelope of secondary wavelets represents the new position of a wavefront.
- When source of light is a point source,the wavefront is spherical.
- Amplitude (A) is inversely proportional to distance (x) i.g., A ∝ 1 / x .
∴ Intensity (I) ∝ (Amplitude)^{2} - When Source of light is linear, the wavefront is cylindrical.
- Amplitude (A) ∝ 1 / √x
∴ Intensity ∝ (Amplitude)^{2} ∝ 1 / x
Huygen’s Principle
- Every point on given wavefront (called primary wavefront) acts as a fresh source of new disturbance called secondary wavelets.
- The secondary wavelets travels in all the directions with the speed of light in the medium.
- A surface touching these secondary wavelets tangentially in the forward direction at any instant gives the new (secondary) wave front of that instant.
Maxwell’s Electromagnetic Wave Theory
- Light waves are electromagnetic waves which do not require a material medium for their propagation.
- Due to transverse nature, light wave undergo polarisation.
- The velocity of electromagnetic wave in vacuum is c = 1 / √μ_{o} ε_{o}.
- The velocity of electromagnetic waves in medium is less than that of light, v < c
v = 1 / √μ_{o} ε_{o} ε_{r} μ_{r} = c / √μ_{o} ε_{r} - The velocity of electromagnetic waves in a medium depend upon the electric and magnetic properties of the medium.
where, μ_{o} = absolute magnetic permeability and
ε_{o} = absolute electrical permittivity of free space.
Electromagnetic Spectrum
- It failed to explain the phenomenon of photoelectric effect, Compton effect and Raman effect.
Max Planck’s Quantum Theory
- Light emits from a source in the form of packets of energy called quanta or photon.
- The energy of a photon is E == hv, where h is Planck’s constant and v is the frequency of light.
- Quantum theory could explain photoelectric effect, Compton effect and Raman effect.
- Quantum theory failed to explain interference, diffraction and polarization of light.
De – Broglie’s Dual Theory
- Light waves have dual nature, wave nature according to Maxwell’s electromagnetic wave theory and particle nature according to Max-Planck’s quantum theory.
- Two natures of light are like the two faces of a coin. In anyone phenomena only its one nature appears.
- Energy of photon = hv = hc / λ
where, h = Planck’s constant 6.6 * 10<sup-34 J / s
de-Broglie wave equation is λ = h / p = h / mv
where h denotes Planck’s constant.
Superposition of Waves
When two similar waves propagate in a medium simultaneously, then at any point the resultant displacement is equal to the vector sum of displacement produced by individual waves.
y = y_{1} + y_{2}
Interference of Light
- When two light waves of similar frequency having a zero or constant phase difference propagate in a medium simultaneously in the same direction, then due to their superposition maximum intensity is obtained at few points and minimum intensity at other few points.
- This phenomena of redistribution of energy due to superposition of waves is called interference of light waves.
- The interference taking place at points of maximum intensity is called constructive interference.
- The interference taking place at points of minimum intensity is destructive interference.
Fringe Width
- The distance between the centers of two consecutive bright or dark fringes is called the fringe width.
- The angular fringe width is given by θ = λ / d.
where λ is the wavelength of light d is the distance between two coherent sources.
Conditions for Constructive and Destructive Interference
➢ For Constructive Interference
- Phase difference, φ = 2nπ
- Path difference, Δx = nλ
where, n = 0, 1, 2, 3,…
➢ For Destructive Interference
- Phase difference, φ = (2n – 1)π
- Path difference, Δx = (2n – 1)π / 2
where, n = 1, 2, 3, …
If two waves of exactly same frequency and of amplitude a and b interfere, then amplitude of resultant wave is given by
R = √a^{2} + b^{2} + 2ab cos φ
where φ is the phase difference between two waves.
R_{max} = (a + b)
R_{min} = (a – b)
Intensity of wave
∴ I = a^{2} + b^{2} + 2ab cos φ
= I_{1} + I_{2} + 2 √I_{1} I_{2} cos φ
where I_{1} and I_{2} are intensities of two waves.
∴ I_{1} / I_{2} = a^{2} / b^{2} = ω_{1} / ω_{2}
Where ω_{1} and ω_{2} are width of slits.
Energy remains conserved during interference.
Interference fringe width
β = Dλ / d
where, D = distance of screen from slits, λ = wavelength of light and d = distance between two slits.
Distance of nth bright fringe from central fringe x_{n} = nDλ / d
Distance of nth dark fringe from central fringe x’_{n} = (2n – 1) Dλ / 2d
Coherent Sources of Light
The sources of light emitting light of the same wavelength, the same frequency having a zero or constant phase difference are called coherent sources of light.
Coherent sources of Light
When a transparent sheet of refractive index μ and of thickness t is introduced in one of the path of interfering waves, then fringe pattern shifts in that direction by a distance Y
Y = D / d (μ – 1) t = β / λ (μ – 1) t
where, β = fringe width.
➢ Fresnel’s Biprism
- It is a combination of two prisms of very small refracting angles placed base to base. It is used to obtain two coherent sources from a single light source.
➢ Llyod’s Mirror
- The shape of interference fringes are usually hyperbolic.
- When screen is held at 900 to the line joining focii of the hyperbola, the fringes are circular.
Llyod's Mirror
- When distance of screen (D) is very large compare to the distance between the slits (d), the Cringes are straight.