Revision: Optics >> Wave Optics Physics (Theory) ISC (Science) ISC Class 12 CISCE

If we draw a surface in a medium such that all the medium particles lying in the surface are in the same phase of oscillation, then the surface is called a 'wavefront'.

The optical path travelled by a light ray is the product of the refractive index of the medium and the actual distance travelled by light in that medium.

The first crystal which polarises the light wave is called ‘polariser'. 

The second crystal which examines the nature of the light emerging from the first crystal, whether it is polarised or not, is called the ‘analyser'.

Unpolarised light is light in which the vibrations of the electric field vector occur in all possible directions in a plane perpendicular to the direction of propagation.

In plane polarised light, the vibrations of the electric vector E occur in a plane perpendicular to the direction of propagation of light, and are confined to a single direction in the plane (do not occur symmetrically in all possible directions).

The plane containing the direction of vibration of the electric vector and the direction of propagation of light is called the 'plane of vibration'.

The plane containing the direction of propagation of light and perpendicular to the plane of vibration is called the ‘plane of polarisation’.

A wave in which the vibrations of the particles of the medium are perpendicular to the direction of propagation.

A wave in which the vibrations of the particles of the medium are parallel to the direction of propagation.

The phenomenon in which the vibrations of the electric field vector of light are restricted to a single direction in a plane perpendicular to the direction of propagation.

Polaroid is a cheap commercial device for producing and detecting plane-polarised light.

The bending of light round the corners of the obstacles, or apertures, is called 'diffraction’.

“The intensity distribution upon the screen is called the ‘diffraction pattern’ of the aperture.”

When two or more waves travel simultaneously in a medium, the resultant displacement at each point of the medium at any instant is equal to the vector sum of the displacements produced by the two waves separately. This principle is called 'principle of superposition'.

The redistribution of light intensity due to the superposition of two light waves is called 'interference of light'.

“If the phase difference between two light waves arriving at a point varies with time in a random way, the wave-sources are said to be incoherent.”

“Two sources are said to be coherent, if they emit light waves having a sharply defined phase difference that remains constant with time.”

“The central white fringe formed when the path difference is zero for all wavelengths is called the zero-order fringe.”

λ_w = \[\frac {λ}{n}\]

\[t=\frac{D}{v}=\frac{D}{c/n}=\frac{nD}{c}\]

OR

d = n D.

e sin θ = ±mλ (m=1,2,3,…)

e sin θ = \[\frac{(2m+1)\lambda}{2}\]

I_{av ​}= \[\frac{I_{\max}+I_{\min}}{2}\]​ = K(a₁²​ + a₂^2​)

Huygens proposed a geometrical construction to explain the propagation of a wavefront in the medium and determined the position of the wavefront after any interval of time. This is known as 'Huygens' principle' and may be stated as follows :

1. Every particle of the medium situated on the wavefront acts as a new wave-source from which fresh waves originate. These waves are called ‘secondary wavelets'.
2. The secondary wavelets travel in the medium in all directions with the speed of the original wave (light) in the medium.
3. The envelope of the secondary wavelets in the forward
   direction at any instant gives the new wavefront at that instant.

Statement

When unpolarised light is incident on the surface of a transparent medium at a particular angle, the reflected light becomes completely plane-polarised.
This angle of incidence is called the polarising angle or Brewster’s angle (i_p).

According to Brewster’s Law, the refractive index n of the medium is related to the polarising angle by:

n = tan ⁡i_p​​

Explanation / Proof

Consider unpolarised light incident on the surface of a transparent medium (e.g., air–glass interface) at the polarising angle i_p​.

Let:

• i_p​ = angle of incidence (polarising angle)
• r = angle of refraction
• n = refractive index of the second medium w.r.t. the first

From Snell’s law:

n = \[\frac {sin ⁡i_p}{sin ⁡r}\]​​

From Brewster’s law:

n = tan⁡ i_p = \[\frac {sin⁡ i_p}{cos⁡ i_p}\]​​

Equating the two expressions for n:

\[\frac{\sin i_p}{\sin r}=\frac{\sin i_p}{\cos i_p}\]

⇒ sin r = cos i_p​

​⇒ sin r = sin(90^∘ − i_p​)

⇒ r = 90^∘ − i_p​

Hence,

i_p + r = 90^∘

Therefore, the reflected ray and refracted ray are mutually perpendicular.

Conclusion

• Brewster’s law establishes a direct relation between refractive index and polarising angle:
  n = tan⁡ i_p​​
• At the polarising angle:
  Reflected light is completely plane-polarised
  Reflected and refracted rays are perpendicular to each other
• This law explains the polarisation of light by reflection and is a strong confirmation of the transverse nature of light waves

Statement

The intensity of plane-polarised light transmitted through an analyser is directly proportional to the square of the cosine of the angle between the transmission axes of the polariser and the analyser.

I = I₀ cos⁡²θ

Explanation / Proof

• Let a beam of completely plane-polarised light of amplitude aaa fall on an analyser.
• Let θ be the angle between the transmission axes of the polariser and analyser.
• The amplitude of light along the analyser’s axis is a cos ⁡θ.
• Since intensity ∝ (amplitude)²,
  I = K(a cos⁡ θ)² = K a² cos⁡² θ
• If I₀ = Ka² is the incident intensity, then:
  I = I₀ cos⁡² θ

Conclusion

Thus, the transmitted intensity depends on the relative orientation of the polariser and analyser and follows the relation

I = I₀ cos⁡² θ​

This relation is known as the Law of Malus.

• According to Huygens’ principle, each point on the incident plane wavefront acts as a source of secondary wavelets, whose forward envelope gives the reflected wavefront.
• The reflected wavefront is obtained by drawing a common tangent to the secondary wavelets, showing that reflection follows wavefront construction.
• Using this construction, the laws of reflection are obtained:
  angle of incidence equals angle of reflection (i = r), and
  incident ray, reflected ray, and normal lie in the same plane.

• Refraction of a plane wavefront can be explained using Huygens’ principle by constructing secondary wavelets in the second medium.
• The refracted wavefront is the forward envelope of secondary wavelets formed in the second medium.
• Rays are normal to wavefronts, so the angles between wavefronts give the angles of incidence and refraction.
• Huygens’ construction leads to Snell’s law, showing that sin⁡i/sin⁡r\sin i / \sin rsini/sinr is constant for two given media.
• Wave theory proves that light travels slower in optically denser media, a result confirmed by Foucault’s experiment.

• In a homogeneous isotropic medium, wavefronts are always perpendicular to the direction of wave propagation.
• Rays are drawn normal to the wavefront and indicate the direction of propagation of the wave.
• A point source produces spherical wavefronts, with rays spreading radially outward.
• A plane wavefront consists of parallel rays, while a linear source produces cylindrical wavefronts.

• At the polarising angle, the reflected light becomes completely plane polarised, while the refracted light is partially polarised.
• Using a pile of parallel plates, repeated refraction and reflection produce almost completely plane-polarised light with vibrations parallel to the plane of incidence.

• Scattering of light occurs when white light passes through very small particles, such as dust or air molecules.
• The scattered light seen perpendicular to the incident beam appears blue.
• Light scattered at right angles is plane-polarised, as shown using an analyser.

• Unpolarised light has electric vectors vibrating randomly in all directions perpendicular to the direction of propagation.
• When unpolarised light passes through an ideal polariser/analyser, the maximum transmitted intensity is 50% of the incident light.
• A Polaroid transmits only those components of light whose electric vectors vibrate parallel to its polarising direction.
• If two Polaroids are parallel, light transmitted by the first passes through the second.
• If two Polaroids are crossed (90°), no light is transmitted, showing complete extinction.

• Polaroids are used to reduce glare from shiny surfaces like wet roads and glass.
• Polarised sunglasses cut off horizontally polarised reflected light and reduce eye strain.
• Polaroids are used in car headlights and windscreens to prevent dazzling from opposite vehicles.
• Crossed Polaroids in cars block headlight glare while allowing safe visibility.
• Polaroids are fitted in microscopes to reduce glare and view minute particles clearly.
• Polaroids in camera lenses help take clear photographs of clouds by reducing scattered light.
• Polaroids are used in trains and aeroplanes to control light intensity through windows.
• Polaroid glasses are used to view three-dimensional (3D) images.
• When a Polaroid is rotated, unpolarised light shows no change in intensity.
• On rotation, plane-polarised light shows maximum and zero intensity, while partially polarised light never becomes zero.

• A single-slit diffraction pattern consists of a bright central band with alternating dark and faint bands on either side.
• The central maximum is the brightest and widest, and most of the incident light is concentrated in it.
• Diffraction becomes more prominent when the slit width is small, especially when it is comparable to the wavelength of light.
• Red light spreads more than blue light in diffraction, showing that diffraction depends on wavelength.
• Narrowing the slit increases the width of the central maximum, while widening the slit reduces diffraction and makes light propagation nearly rectilinear.

• In Young’s double-slit experiment, two narrow slits act as coherent sources, producing an interference pattern of alternate bright and dark fringes on a distant screen.
• Bright fringes are formed by constructive interference when crests meet crests or troughs meet troughs (same phase), resulting in maximum intensity.
• Dark fringes are formed by destructive interference when crests meet troughs (opposite phase), resulting in zero or minimal intensity.
• Fringe width depends on wavelength: red light produces wider fringes than blue light, supporting the wave nature of light.

• Sources must be coherent — they should maintain a constant phase difference for sustained interference.
• The same frequency (monochromatic light) is required; different frequencies cause intensity fluctuations.
• The principle of superposition must apply for interference to occur.
• The separation between sources should be small to obtain sufficiently wide and visible fringes.
• Screen distance should be set to a large value to increase fringe width and visibility.
• Amplitudes of waves should be equal or nearly equal for maximum contrast between fringes.
• Sources (slits) should be narrow to prevent fringe overlap.
• Monochromatic light is essential to avoid mixing and loss of fringe clarity.

• A central bright fringe is formed at point O, where the path difference is zero (S1O = S2O).
• Bright fringes occur when path difference = mλ; dark fringes occur when path difference = (2m − 1)λ/2.
• Positions of bright fringes are given by
  x_m = \[\frac {mDλ}{d}\]​
  and dark fringes lie exactly midway between bright fringes.
• Fringe width (β) is the distance between two successive bright or dark fringes and is the same for all fringes:
  β = \[\frac {Dλ}{d}\]
• Fringe width increases with wavelength; hence, red light produces wider fringes than blue light.

• Interference occurs when waves from two coherent sources superpose, and the resulting intensity depends on the phase difference between them.
• Resultant intensity at a point is given by
  I = I₁ + I₂ + 2\[\sqrt {I_1 I_2}\] cos ⁡ϕ,
  showing its dependence on phase difference ϕ\phiϕ.
• Constructive interference occurs when the waves meet in phase, i.e., ϕ = 2mπ or path difference x = mλ, resulting in maximum intensity.
• Destructive interference occurs when the waves meet in opposite phase, i.e. ϕ = (2m−1)π or path difference x = (2m − 1)\[\frac {λ}{2}\], giving minimum intensity.
• Alternate bright and dark fringes appear on the screen, producing an interference pattern due to the continuous variation in the path difference.