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.

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 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.

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

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

The plane in which vibrations are present is called the plane of polarisation.

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.

or

The phenomenon of confining the vibrations of the electric vector to a particular direction perpendicular to the direction of propagation of light is called Polarization.

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 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.

The plane in which E vibrates/oscillates is known as the plane of vibration.

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

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

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

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

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

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 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 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 (Physical Optics) treats light as a wave, explaining phenomena like interference, diffraction, and polarisation, which Ray Optics cannot explain.

• A wavefront is an imaginary surface where all points of a wave have the same phase (constant phase surface with maximum or minimum value).
• The direction of propagation of a wave is always perpendicular to the wavefront.
• Wavefront of a point source is a sphere; it propagates radially outward.

Types of Wavefronts:

S.No.	Wavefront	Shape of Light Source	Diagram or shape of wavefront	Variation of amplitude (A) with distance	Variation of intensity (I) with distance
1	Spherical	Point source		\[A\propto\frac{1}{r}\]	\[I\propto\frac{1}{r^2}\]
2	Cylindrical	Linear source / Slit		\[A\propto\frac{1}{\sqrt{r}}\]	\[I\propto\frac{1}{r}\]
3	Plane	Extended large source/ Point source at very large distinct		A = constant A ∝ r°	I = constant I ∝ r°

• Every point on a wavefront acts as a secondary source (point source) that emits new spherical wavelets in all directions with the same speed as the original wave.
• The new (forward) wavefront at any later time is the common tangential envelope (tangent surface) to all these secondary wavelets.
• The wavefront in a medium is always perpendicular to the direction of wave propagation.
• Secondary wavelets travel only in the forward direction — backward wavelets are ignored (zero amplitude in backward direction).

Memory: Every point → new source → envelope = new wavefront.

When light travels from optically rarer to denser medium obliquely:

• It bends towards the normal.
• Angle of incidence > angle of refraction.

Speed of light decreases in denser medium; frequency remains unchanged; wavelength decreases.

• Refractive index: \[\mu=\frac{c}{v},\] where c = speed of light in vacuum, v = speed in medium.

• Snell's law: \[\frac{\sin i}{\sin r}=\frac{\mu_2}{\mu_1}=\frac{v_1}{v_2}=\frac{\lambda_1}{\lambda_2}\]

When light travels from optically denser to rarer medium obliquely:

• It bends away from the normal.
• Angle of refraction > angle of incidence.

If angle of incidence exceeds the Critical Angle (C), total internal reflection occurs.

Critical Angle: \[C=\sin^{-1}\left(\frac{\mu_{2}}{\mu_{1}}\right),\mathrm{where~}\mu_{1}>\mu_{2}.\]

At critical angle, refracted ray grazes the surface (angle of refraction = 90°).

Laws of Reflection (proved by Huygens' principle):

• ∠i = ∠r (angle of incidence = angle of reflection)
• Incident ray, reflected ray, and normal all lie in the same plane.

Reflected wavefront is a plane wavefront (mirror image of incident wavefront).

No change in frequency, wavelength, or speed during reflection.

The wavefront after reflection from a plane surface remains planar.

Coherent Sources-

Two sources are coherent if they emit waves of:

• Same frequency
• Constant (fixed) phase difference
• Obtained from a single source

Two coherent sources can be created by:

• Splitting a physical source into two
• Generating two virtual images of the same source

Incoherent Sources-

• Phase difference changes with time → incoherent.
• Two independent monochromatic sources of same frequency are incoherent because atoms cannot emit light waves in same phase simultaneously.

Addition of Waves (Superposition)

Let two waves have amplitudes A₁ and A₂, same frequency, and constant phase difference φ.

• Resultant amplitude: \[A_{res}=\sqrt{A_1^2+A_2^2+2A_1A_2\cos\phi}\]

• Phase angle: \[\theta=\tan^{-1}\left(\frac{A_2\sin\phi}{A_1+A_2\cos\phi}\right)\]

• Since I ∝ A²: \[I=I_1+I_2+2\sqrt{I_1I_2}\cos\phi\]

• Interference = redistribution of energy when two coherent waves superpose.
• Based on energy conservation, total energy remains constant, only redistributed.
• Constructive: I > (I₁ + I₂) → bright fringe
• Destructive: I < (I₁ + I₂) → dark fringe

Conditions for Sustained Interference:

• Sources must be coherent.
• Separation between sources must be small.
• Distance of screen from sources must be large.
• For good contrast: amplitudes of the two waves should be nearly equal.
• Two sources must propagate along same line.

Young's Double Slit Experiment (YDSE):

Setup: Light source → single slit → double slit (S₁ and S₂, separation d) → screen (distance D).

Path difference at point P: \[\delta=S_2P-S_1P=\frac{x_n\cdot d}{D}\]

Bright Fringe (Constructive): Path difference = even multiple of λ/2

δ = nλ, n = 0,1,2,3...

Dark Fringe (Destructive): Path difference = odd multiple of λ/2

• Diffraction = bending and spreading of light waves around obstacles or through narrow openings, producing interference patterns.
• It is due to interference of secondary wavelets from the exposed portion of the wavefront from the slit.
• Key difference from interference: in diffraction, bright fringes have same intensity but bands are of decreasing intensity.

Single Slit Diffraction:

Let a = width of slit, θ = angle of diffraction.

Condition for Minimum (Dark) Intensity:

Condition for Maximum (Secondary Bright) Intensity:

\[a\sin\theta=(2n+1)\frac{\lambda}{2},\quad n=1,2,3...\]

Width of Central Maximum:

For first minima: \[a\cdot\frac{y}{D}=\lambda\Rightarrow y=\frac{\lambda D}{a}\]

\[W=2y=\frac{2\lambda D}{a}\]

Angular width of central maximum:

Linear width of n-th secondary maximum: