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 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.

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.

At a large distance from the source, a small portion of the spherical wave can be considered as a plane. This is known as a plane wave.

If a point source emits waves uniformly in all directions, the locus of points which have the same amplitude and vibrate in the same phase is a sphere. This is known as a spherical wave.

A wavefront is a surface of constant phase.

The plane of vibration is the plane in which the electric field vector \[\vec{E}\] vibrates or oscillates.

A Polaroid is a thin film of ultramicroscopic crystals used to produce plane-polarised light.

An unpolarised wave is one in which the plane of vibration changes randomly in very short time intervals.

The plane of polarisation is the plane in which vibrations are present — it is perpendicular to the plane of vibration and contains the direction of propagation.

Polarisation is the phenomenon of restricting the vibration of a light wave to a particular plane perpendicular to the direction of propagation of the wave, or confining the electric vector vibrations to one direction perpendicular to the direction of propagation.

A transverse wave is one in which the displacement of particles is perpendicular to the direction of propagation of the wave.

A wave in which the electric field vectors are confined in one plane and are parallel to a unique direction is called a linearly polarised wave or plane polarised wave.

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

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.

Diffraction is the phenomenon of bending of light (or any wave) around the corners or edges of an obstacle or aperture, causing it to spread into the geometrical shadow region and produce alternate dark and bright regions.

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

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.

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.

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.

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.

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.

Statement

When a beam of plane polarised light is incident on an analyser, the intensity of the transmitted light is directly proportional to the square of the cosine of the angle θ between the pass-axis of the analyser and the plane of polarisation of the incident 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:

• Wave optics studies the wave nature of light.
• Newton supported the corpuscular theory of light.
• Huygens proposed the wave theory in 1678.
• Young's 1801 interference experiment supported the wave model.
• Maxwell explained light as an electromagnetic wave.
• Geometrical optics treats light as rays.
• Wave optics includes Huygens' principle, interference, diffraction, and polarisation.