Revision: Class 12 >> Wave Optics NEET (UG) 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.

A wavefront is a surface of constant phase.

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.

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.

A rarer medium is a medium in which light travels faster and whose refractive index is lower compared with the denser medium.

The critical angle is the angle of incidence in the denser medium for which the angle of refraction in the rarer medium becomes 90 degrees.

If i = i_c, then the refracted ray travels along the boundary surface.

Total internal reflection is the complete reflection of light back into the denser medium when light travels from a denser to a rarer medium, and the angle of incidence becomes greater than the critical angle.

When two or more waves travel through the same medium at the same time, the resultant displacement is the sum of the displacements due to the individual waves.

The redistribution of intensity that occurs when two light waves superpose is called interference.

Sources that emit waves of the same frequency and maintain a constant phase difference are called coherent sources.

Sources for which the phase difference changes randomly with time are called incoherent sources.

When two waves meet in the same phase, the resultant amplitude increases and the intensity becomes maximum.

When two waves meet in opposite phase, the resultant amplitude decreases, and the intensity becomes minimum.

A fringe is a bright or dark band formed on the screen due to constructive or destructive interference.

Interference is the phenomenon in which light intensity is modified due to the superposition of two or more light waves.

Fringe width is the distance between two consecutive bright fringes or two consecutive dark fringes.

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.

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

For two waves of equal intensity I₀​, the resultant intensity is:

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

\[\beta=\frac{\lambda D}{d}\]

\[\alpha=\frac{\beta}{D}=\frac{\lambda}{d}\]

\[x_n-x_m=\left[n-\frac{(2m-1)}{2}\right]\beta\]

\[\begin{array} {cc} & x_m=\frac{(2m-1)\lambda D}{2d} \end{array}\]

\[\begin{array} {c}x_n=\frac{n\lambda D}{d}=n\beta \end{array}\]

According to the laws of reflection:

• At the point of incidence, the incident rays, reflected rays, and normal to the reflecting surface all lie in the same plane.
• On opposing sides of the normal are the incident and reflected rays.
• The angle of incidence and the angle of reflection are the same. i.e., ∠i = ∠r.

Explanation:

                Reflection of light

XY: Plane reflecting surface

AB: Plane wavefront

RB₁: Reflecting wavefront

A₁M, B₁N: Normal to the plane

∠AA₁M = ∠BB₁N = ∠i = Angle of incidence

∠TA₁M = ∠QB₁N = ∠r = Angle of reflection

A plane wavefront AB is advancing obliquely towards the plane reflecting surface XY. The AA₁ and BB₁ are incident rays.

When ‘A’ reaches XY at A₁, then the ray at ‘B’ reaches point ‘P’, and it has to cover the distance PB₁ to reach the reflecting surface XY.

Let ‘t’ be the time required to cover the distance PB₁. During this time interval, secondary wavelets are emitted from A₁ and will spread over a hemisphere of radius A₁R, in the same medium. The distance covered by secondary wavelets to reach from A₁ to R in time t is the same as the distance covered by primary waves to reach from P to B₁. Thus, A₁R = PB₁ = ct.

All other rays between AA₁ and BB₁ will reach XY after A₁ and before B₁. Hence, they will also emit secondary wavelets of decreasing radii.

The surface touching all such hemispheres is RB₁ which is the reflected wavefront, bounded by reflected rays A₁R and B₁Q.

Draw A₁M ⊥ XY and B₁N ⊥ XY.

Thus, the angle of incidence is ∠AA₁M = ∠BB₁N = i, and the angle of reflection is ∠MA₁R = ∠NB₁Q = r.

∠RA₁B₁ = 90 − r

∠PB₁A₁ = 90 − i

In ΔA₁RB₁ and ΔA₁PB₁

∠A₁RB₁ = ∠A₁PB₁

A₁R = PB₁    ...(Reflected waves travel an equal distance in the same medium in equal time.)

A₁B₁ = A₁B₁     ....(Common side)

∴ ΔA₁RB₁ ≅ ΔA₁PB₁

∴ ∠RA₁B1 = ∠PB₁A₁

∴ 90 − r = 90 − i

∴ i = r

Also from the figure, it is clear that incident rays, reflected rays, and normal lie in the same plane.

This explains the laws of reflection of light from a plane reflecting surface on the basis of Huygen’s wave theory.

Frequency, wavelength, and speed of light do not change after reflection. If reflection takes place from a denser medium, then the phase changes by π radians.

AB = Incident wavefront

CD = Reflected wavefront

XY = Reflecting surface

If c be the speed of light and t be the time taken by light to go from B to C or A to D or E to G through F, then

t = `(EF)/C + (FG)/C`

= `(AF sin i)/C + (FC sin r)/C`

= `(AC sin r + AF(sin i - sin r))/C`

For rays of light from different parts of the incident wavefront, the values of AF are different. But light from different points of the incident wavefront should take the same time to reach the corresponding points on the reflected wavefront.

So, ‘t’ should not depend upon AF.

This is possible only if sin i – sin r = 0.

i.e., sin i = sin r

⇒ i = r

Hence proved.

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.

Statement:

When unpolarised light is incident at polarising angle i_B on an interface separating air from a medium of refractive index μ, 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.

OR

Statement:

• When the angle of incidence equals the polarising angle (θ_B), the reflected and refracted rays are perpendicular to each other.
• "The refractive index of a medium is equal to the tangent of the polarising angle θ_B."

• 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.