Revision: 12th Std >> Wave Optics MAH-MHT CET (PCM/PCB) Wave Optics

Wavefront is defined as the locus of all the points in space that reach a particular distance by a propagating wave at the same instant.

A wave front is defined as a surface of constant phase.

The branch of optics that considers light as a wave which can bend around objects, diffract and interfere, etc. is called wave optics.

Coupled time-varying electric and magnetic fields that propagate in space are called electromagnetic waves.

The phenomenon that is based on the fact that light waves are transverse electromagnetic waves is called polarisation.

The branch of optics that is based on rectilinear propagation of light and deals with mirrors, lenses, reflection, refraction, etc. is called ray optics.

When the source of light is linear and all the points equidistant from the source lie on a cylinder, the wavefront which is cylindrical in shape is called a cylindrical wavefront.

When the source of light is a point source, the wavefront which is a sphere with the centre as the source is called a spherical wavefront.

The speed with which the wavefront moves outwards from the source is called the speed of wave.

When a point source or linear source of light is at very large distance and a small portion of the spherical or cylindrical wavefront appears to be plane, such a wavefront is called a plane wavefront.

The secondary light waves emitted in all directions by each point on a wavefront, which travel with the speed of light in the medium, are called wavelets.

The locus of all those particles which are vibrating in the same phase at any instant is called a wavefront; thus a wavefront is a surface of constant phase.

When the displacement of the particle is perpendicular to the direction of propagation of the wave, then it is said to be transverse wave.

The phenomenon of restriction of the vibration of light waves in a particular plane perpendicular to the direction of wave motion is called polarisation of light.

The wave in which the vibration of electric field vectors are confined in one plane and parallel to one unique direction is called linearly polarised wave; it is also referred to as plane polarised wave.

When the plane of vibration of a wave is changed randomly in very short intervals of time, such waves are called unpolarised waves.

A thin film of ultramicroscopic crystals used to produce plane polarised light is called a polaroid.

The phenomenon that occurs when two waves meet while travelling along the same medium is called wave interference.

The phenomenon of redistribution of energy on account of superposition of light waves from two coherent sources is called interference of light.

The points of maximum intensity in the regions of superposition of waves are said to be in constructive interference.

The points of minimum intensity in the regions of superposition of waves are said to be in destructive interference.

The type of diffraction that occurs when the source and the observation screen are far away (effectively at infinite distance) from the diffracting object and fringes are not sharp and well-defined is called Fraunhofer diffraction.

The bending of light near the edge of an obstacle or slit and spreading into the region of geometrical shadow is called diffraction of light.

The type of diffraction that occurs when the source or screen is at a finite distance from the diffracting object and fringes are not sharp and well-defined is called Fresnel diffraction.

The minimum distance of separation between two objects when they can be observed as separate by an optical instrument is called the limit of resolution of that instrument.

The reciprocal of the limit of resolution is called its resolving power.

The reciprocal of the least angular separation between the objects that are just resolved is called the resolving power of the telescope.

The quantity μ sin ⁡θ, where μμ is the refractive index of the medium between the object and the objective, is called the numerical aperture (N.A.) of the objective of the microscope.

The ability of an optical instrument to produce distinctly separate images of two objects very close to each other is called the resolving power of the instrument.

When the separation between the central maxima of two objects is greater than the distance between the central maximum and first minimum of any of the two objects, the images are said to be well resolved.

When the separation between the central maxima of the two objects is just equal to the distance between the central maximum and first minimum of any of the two objects, the images are said to be just resolved.

When the separation between the central maxima of two objects is less than the distance between the central maximum and the first minimum of any of the two objects, the images are said to be 'not resolved' or unresolved.

The condition where the images of two point objects close to each other are regarded as resolved (separated), if the central maximum of one falls on the first minimum of the other, is called Rayleigh's criterion for resolution.

I = I₁ ​+ I₂​ + 2\[\sqrt {I_1​I_2}\] ​​⋅ cos ϕ

When I₁ = I₂ = I₀:

I = \[2I_0(1+\cos\phi)=4I_0\cos^2\left(\frac{\phi}{2}\right)\]

\[\frac{I_{\max}}{I_{\min}}=\left(\frac{a_1+a_2}{a_1-a_2}\right)^2=\left(\frac{\sqrt{I_1}+\sqrt{I_2}}{\sqrt{I_1}-\sqrt{I_2}}\right)^2\]

When two waves of amplitudes a₁ and a_2​ interfere at a point where phase difference is ϕ, the resultant amplitude is:

\[A^2=a_1^2+a_2^2+2a_1a_2\cos\phi\]

R.P. = \[\frac{1}{d\theta}=\frac{D}{1.22\lambda}\]

R.P. = \[\frac {1}{\text {Limit of resolution}}\]

R.P. = \[\frac {1}{d}\] = \[\frac{2\mu\sin\theta}{\lambda}\]

where μ sin⁡ θ is the Numerical Aperture (N.A.) of the objective.

Smallest angular separation dθdθ (Circular aperture):

\[d\theta=\frac{1.22\lambda}{D}\]

where D is the aperture (diameter) of objective of the telescope.

Smallest angular separation (Rectangular aperture):

\[d\theta=\frac{d}{D}\]

where d is slit separation and D is distance.

Limit of resolution (self-luminous objects):

d = \[\frac{1.22\lambda}{2\sin\theta}=\frac{0.61\lambda}{\sin\theta}\]

Limit of resolution (objects illuminated by light of wavelength λ):

d = \[\frac{\lambda}{2\sin\theta}\]

If a liquid of refractive index μμ is between object and objective:

d = \[\frac{\lambda}{2\mu\sin\theta}\]

where θθ is the angle subtended by an object at the objective.

Huygens' Principle states that:

• Each point on a wavefront acts as a secondary source of light emitting secondary light waves called wavelets in all directions.
• These wavelets travel with the speed of light in the medium.
• The new wavefront can be obtained by taking the envelope of these secondary wavelets travelling in the forward direction.
• The wavelets travelling in the backward direction are ineffective.

Additional Points:

• Huygens' principle is a geometrical construction which gives the shape of the wavefront at any time and allows us to determine the shape of the wavefront at a later time.
• It also tells how a wavefront propagates through a medium.
• The energy of the wave travels in a direction perpendicular to the wavefront.

Statement: When plane polarised light is incident on an analyser, the resultant intensity of light transmitted from the analyser varies directly as the square of the cosine of the angle between the plane of transmission axis of the analyser and polariser.

Formula:

Where:

• I₀ = intensity of plane polarised light
• I = intensity of transmitted light from the analyser
• θ = angle between the axis of the polariser and the analyser

Statement: When unpolarised light is incident at polarising angle i_B on an interface separating air from a medium of refractive index μμ, then the reflected light is plane polarised (perpendicular to the plane of incidence), provided:

Additional condition at polarising angle:

i.e., the reflected plane polarised light is at right angles to the refracted light.

Thomas Young first demonstrated the phenomenon of interference of light with the help of a slit, using a monochromatic source and two slits S₁ and S₂​, producing alternating bright fringes (constructive interference) and dark fringes (destructive interference) on a screen.

Statement: According to Lord Rayleigh, the images of two point objects close to each other are regarded as resolved (separated), if the central maximum of one falls on the first minimum of the other.

The reasoning of Rayleigh's criterion is given by considering the intensity distribution in the diffraction pattern produced by two objects.

Three Conditions: