Wave optics

 1. Wave Nature of Light: Huygen’s Theory

There are some phenomena like interference, diffraction and polarisation which could not be explained by Newton’s corpuscular theory. These were explained by wave theory first proposed by Huygen.

The assumptions of Huygen’s wave theory are: (i) A source sends waves in all possible directions. The locus of particles of a medium vibrating in the same phase is called a wavefront. For a point source, the wavefront is spherical; while for a line source the wavefront is cylindrical. A distant wavefront is plane. (ii) Each point of a wavefront acts as a source of secondary wavelets. The envelope of all wavelets at a given instant gives the position of a new wavefront.

2. Wavefront

A wavefront is defined as the locus of all the particles which are vibrating in the same phase. The perpendicular line drawn at any point on the wavefront represents the direction of propagation of the wave at that point and is called the ‘ray’.

Types of Wavefronts: The wavefronts can be of different shapes. In general, we experience two types of wavefronts.

(i) Spherical Wavefront: If the waves in a medium are originating from a point source, then they propagate in all directions. If we draw a spherical surface centred at point-source, then all the particles of the medium lying on that spherical surface will be in the same phase, because the disturbance starting from the source will reach all these points simultaneously. Hence in this case, the wavefront will be spherical and the rays will be the radial lines [Fig (a)].

(ii) Cylindrical Wavefront: If the waves in a medium are originating from a line source, then they too propagate in all directions. In this case the locus of particles vibrating in the same phase will be a cylindrical surface. Hence in this case the wavefront will be cylindrical. [Fig. (b)]

(iii) Plane Wavefront: At large distance from the source, the radii of spherical or cylindrical wavefront will be too large and a small part of the wavefront will appear to be plane. At infinite distance from the source, the wavefronts are always plane and the rays are parallel straight lines.

The equation y=asin2π(tT–xλ)

represents the plane wave propagating along positive direction of X-axis.

3. Coherent and Incoherent Sources of Light

The sources of light emitting waves of same frequency having zero or constant initial phase difference are called coherent sources.

The sources of light emitting waves with a random phase difference are called incoherent sources. For interference phenomenon, the sources must be coherent.

Methods of Producing Coherent Sources: Two independent sources can never be coherent sources. There are two broad ways of producing coherent sources for the same source.

(i) By division of wavefront: In this method the wavefront (which is the locus of points of same phase) is divided into two parts. The examples are Young’s double slit and Fresnel’s biprism.

(ii) By division of amplitude: In this method the amplitude of a wave is divided into two parts by successive reflections, e.g., Lloyd’s single mirror method.

4. Interference of Light

Interference is the phenomenon of superposition of two light waves of same frequency and constant phase different travelling in same direction. The positions of maximum intensity are called maxima, while those of minimum intensity are called minima.

Conditions of maxima and minima: If a1 and a2 are amplitudes of interfering waves and φ is the phase difference at a point under consideration, then

Resultant intensity at a point in the region of superposition

I=a21+a22+2a1a2cosφ

=I1+I2+2I1I2−−−−√cosφ

where I1=a21= intensity of one wave

I2=a22= intensity of other wave

Condition of maxima:

Phase difference, φ = 2nπ

or path difference, ∆ = nλ, n being integer

Maximum amplitude, Amax = a1 + a2

Maximum intensity, Imax = A2max = (a1 + a2)2

=a21+a22+2a1a2 =I1+I2+2I1I2−−−−√

Condition of minima: Phase difference, φ = (2n – 1) π

Path difference, ∆=(2n–1)λ2,n=1,2,3,...

Minimum amplitude, Amin = (a1 – a2)

Minimum intensity, Imin=(a1–a2)2=a21+a22–2a1a2

= I1+I2–2I1I2−−−−√

Young’s Double Slit Experiment

Let S1 and S2 be coherent sources at separation d and D be the distance of screen from sources, then path difference between waves reaching at P can be shown as

∆=yndD

For maxima ∆ = nλ

∴ Position of nth maxima yn=nDλd

∴ Position of nth minima yn=(n–12)Dλd

Fringe width: Fringe width is defined as the separation between two consecutive maxima or minima.

Linear fringe width, β=y–n+1–yn=Dλd

Angular fringe width, βθ=βD=λd.

Use of white light: When white light is used to illuminate the slit, we obtain an interference pattern consisting of a central white fringe having few coloured fringes on two sides and uniform illumination.

Remark: If waves are of same intensity,

I1 = I2 = I0 (say) then

I =2I0+2I0cosφ

=2I0(1+cosφ)

=4I0cos2φ2

5. Diffraction of Light

The bending of light from the corner of small obstacles or apertures is called diffraction of light.

Diffraction due to a Single Slit

When a parallel beam of light is incident normally on a single slit, the beam is diffracted from the slit and the diffraction pattern consists of a very intense central maximum, and secondary maxima and minima on either side alternately.

If a is width of slit and θ the angle of diffraction, then the directions of maxima are given by

asinθ=(n+12)λn=1,2,3,...

The position of nth minima are given by

a sin θ = nλ,

where n = ± 1, ± 2, ± 3, ... for various maxima on either side of principal maxima.

Width of Central Maximum

The width of central maximum is the separation between the first minima on either side.

The condition of minima is

a sin θ = ± nλ (n = 1, 2, 3,...).

The angular position of the first minimum (n=1) on either side of central maximum is given by

a sin θ = ± λ

⇒ θ=±sin–1(λa)

∴ Half-width of central maximum, θ=sin–1(λa)

∴ Total width of central maximum, β=2θ=2sin–1(λa)

Linear Width: If D is the distance of the screen from slit and y is the distance of nth minima from the centre of the principal maxima, then

sinθ⋍tanθ⋍θ=yD

Now,

nλ=asinθ⋍aθ

θ=λna=ynD

⇒ yn=nλDa

Linear half-width of central maximum, y=λDa

Total linear width of central maximum, β=2y=2λDa

6. Resolving Power

The resolving power of an optical instrument is its ability to form distinct images of two neighbouring objects. It is measured by the smallest angular separation between the two neighbouring objects whose images are just seen distinctly formed by the optical instrument. This smallest distance is called the limit of resolution.

Smaller the limit of resolution, greater is the resolving power.

The angular limit of resolution of eye is 1′ or (160)°. It means that if two objects are so close that angle subtended by them on eye is less than 1′ or (160)°, they will not be seen as separate.

The best criterion of limit of resolution was given by Lord Rayleigh. He thought that each object forms its diffraction pattern. For just resolution, the central maximum of one falls on the first minimum of the other (Fig. (a)). When the central maxima of two objects are closer, then these objects appear overlapped and are not resolved [Fig. (b)]; but if the separation between them is more than this, they are said to be well resolved.

Telescope: If a is the aperture of telescope and λ the wavelength, then resolving limit of telescope dθ∝λa

For spherical aperture, dθ=1.22λa

Microscope: In the case of a microscope, θ is the well resolved semi-angle of cone of light rays entering the telescope, then limit of resolution =λ2nsinθ

where n sin θ is called numerical aperture.

7. Polarisation

The phenomenon of restriction of vibrations of a wave to just one direction is called polarisation. It takes place only for transverse waves such as heat waves, light waves etc.

Unpolarised Light: The light having vibrations of electric field vector in all possible directions perpendicular to the direction of wave propagation is called the ordinary (or unpolarised) light.

Plane (or Linearly) Polarised Light: The light having vibrations of electric field vector in only one direction perpendicular to the direction of propagation of light is called plane (or linearly) polarised light.

The unpolarised and polarised light is represented as

(a) Unpolarised light

(b) Polarised light

(c) Partially polarised light

Polarisation by Reflection: Brewster’s Law: If unpolarised light falls on a transparent surface of refractive index n at a certain angle iB called polarising angle, then reflected light is plane polarised.

Brewster’s law: The polarising angle (iB) is given by n = tan iB

This is called Brewster’s law.

Under this condition, the reflected and refracted rays are mutually perpendicular, i.e.,

iB + r = 90°

where r is angle of refraction into the plane.

Malus Law: It states that if completely plane polarised light is passed through an analyser, the intensity of light transmitted cos2 θ, where θ is angle between planes of transmission of polariser and analyser i.e.,

I = I0 cos2 θ (Malus Law)

If incident light is unpolarised, then I=I02,

since (cos2 θ)average for all directions = 12.

Polaroid: Polaroid is a device to produce and detect plane polarised light.

Some uses of polaroid are:

(i) Sun glasses filled with polaroid sheets protect our eyes from glare.

(ii) Polaroids reduce head light glare of motor car being driven at night.

(iii) Polaroids are used in three-dimensional pictures i.e., in holography.

Analysis of a given light beam: For this, given light beam is made incident on a polaroid (or Nicol) and the polaroid/Nicol is gradually rotated:

(i) If light beam shows no variation in intensity, then the given beam is unpolarised.

(ii) If light beam shows variation in intensity but the minimum intensity is non-zero, then the given beam is partially polarised.

(iii) If light beam shows variation in intensity and intensity becomes zero twice in a rotation, then the given beam of light is plane polarised.