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CISCEISC (Science) ISC Class 12
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Overview: Wave Optics

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Estimated time: 48 minutes
CISCE: Class 12

Definition: Diffraction of Light

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

CBSE: Class 12
Maharashtra State Board: Class 12

Definition: Ray of Light

  • The path along which light travels is called a ray of light.
  • A ray is defined as the path of energy propagation in the limit of wavelength tending to zero.
CISCE: Class 12

Definition: Transverse Wave

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

CISCE: Class 12

Definition: Wavefront

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'.

CISCE: Class 12

Definition: Principle of Superposition

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'.

CISCE: Class 12

Definition: Diffraction Pattern

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

CISCE: Class 12

Definition: Interference of Light

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

CISCE: Class 12

Definition: Optical Path

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.

CBSE: Class 12
Maharashtra State Board: Class 12

Definition: Ray Optics or Geometrical Optics

  • The study of optical phenomena under the assumption that it travels in a straight line as a ray is called ray optics or geometrical optics, as geometry is used in this study.
  • The branch of optics in which one completely neglects the finiteness of the wavelength is called geometrical optics.
CISCE: Class 12

Definition: Longitudinal Wave

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

CISCE: Class 12

Definition: Coherent Sources

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

CISCE: Class 12

Formula: Single Slit Diffraction

e sin θ = ±mλ (m=1,2,3,…)

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Definition: Wave Optics

The branch of optics which uses the wave nature of light to explain the optical phenomena is called wave optics.

CISCE: Class 12

Formula: Optical Path

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

OR

d = n D.

CISCE: Class 12

Definition: Polarisation of Light

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.

CISCE: Class 12

Definition: Polariser

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

CISCE: Class 12

Definition: Incoherent Sources

“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.”

CISCE: Class 12

Formula: Variation of Wavelength in Media

λw = \[\frac {λ}{n}\]

CISCE: Class 12

Formula: Subsidiary Maxima

e sin θ = \[\frac{(2m+1)\lambda}{2}\]

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key Points: Nature of Light

  • Corpuscular theory (Newton): Light consists of particles called corpuscles that travel in straight lines; it explains reflection but fails to account for the correct speed in denser media.
  • Wave theory (Huygens): Light behaves as a wave and accounts for reflection, refraction, interference, diffraction, and polarisation.
  • Wave theory correctly states that the speed of light is lower in denser media, so light bends towards the normal.
  • Geometrical (ray) optics studies light as straight-line rays; wave optics explains light using its wave nature.
  • Dual nature of light: Light exhibits both particle nature (photons) and wave nature under different conditions.
CISCE: Class 12

Principle: Huygens' Wave Theory

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.
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Key Points: Light as a Wave

  • Light is a transverse electromagnetic wave consisting of oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation.
  • Light does not require a medium and travels in a vacuum at the speed of light
    c = 3 × 108 m/sRefractive index n = \[\frac {c}{v}\]
  • Visible light has wavelengths from 400–700 nm; different wavelengths produce different colours and cause dispersion (spectrum formation).
CISCE: Class 12

Definition: Zero-Order Fringe

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

CISCE: Class 12

Key Points: Fraunhofer's Diffraction

  • 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.
CISCE: Class 12

Definition: Analyser

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'.

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Key Points: Light Sources and Wavefronts

  • Primary sources emit their own light (e.g., the Sun, stars, a bulb); secondary sources reflect or scatter light (e.g., the Moon, planets).
  • A wavefront is the locus of all points having the same phase at a given instant of time.
  • The direction of propagation of light is perpendicular to the wavefront (along the rays).
  • A point source produces spherical wavefronts; far from the source, they appear as plane wavefronts.
  • A line source produces cylindrical wavefronts; the wave speed equals the speed at which the wavefront moves.
CISCE: Class 12

Formula: Average Intensity of Interference Pattern

Iav ​= \[\frac{I_{\max}+I_{\min}}{2}\]​ = K(a12​ + a22​)

CISCE: Class 12

Key Points: Wavefront

  • 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.
CISCE: Class 12

Definition: Unpolarised Light

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.

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Law: Huygens' Principle

"Each point on a wavefront acts as a secondary source of light emitting secondary light waves called wavelets in all directions which 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 and is thus, the envelope of the secondary wavelets in forward direction. The wavelets travelling in the backward direction are in effective".

CISCE: Class 12

Key Points: Interference of Light Waves

  • 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.
CISCE: Class 12

Key Points: Plane Wavefront: Reflection and Refraction

Incident Wavefront Medium Nature of Wavefront after Reflection / Refraction
Plane Plane reflecting surface Plane
Plane Plane refracting surface Plane
Plane Prism Plane
Plane Convex lens Spherical (converging)
Plane Concave lens Spherical (diverging)
Plane Concave mirror Spherical (converging)
CISCE: Class 12

Definition: Plane Polarised Light

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).

Maharashtra State Board: Class 12

Law: Laws of Reflection

First Law of Reflection:
i = rThe angle of incidence is equal to the angle of reflection.

Second Law of Reflection:
The incident ray, reflected ray, and the normal at the point of incidence lie in the same plane.

CISCE: Class 12

Definition: Plane of Vibration

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

CISCE: Class 12

Key Points: Reflection of a Plane Wave at a Plane Surface

  • 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.
CISCE: Class 12

Key Points: Constructive and Destructive Interference

  • 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.
CISCE: Class 12

Definition: Plane of Polarisation

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

CISCE: Class 12

Key Points: Refraction of a Plane Wave at a Plane Surface

  • 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.
Maharashtra State Board: Class 12

Key Points: Reflection of Light at a Plane Surface

  • Reflection at a plane surface can be explained using Huygens’ principle and secondary wavelets.
  • The reflected wavefront is formed as the envelope of secondary wavelets produced at the reflecting surface.
  • The distance travelled by incident and reflected waves in the same time is equal (AE = BC = vT).
  • The size of the image formed by a plane mirror is equal to the size of the object.
  • The image formed in a plane mirror shows lateral reversal (right and left are interchanged).
CISCE: Class 12

Key Points: Fringe Formation in Young’s Double-Slit Experiment

  • 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
    xm = \[\frac {mDλ}{d}\]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.
Maharashtra State Board: Class 12

Key Points: Refraction of Light at a Plane Boundary Between Two Media

  • Refraction at a plane boundary can be explained using Huygens’ principle and secondary wavelets.
  • When light enters a denser medium, its speed decreases and the wavefronts become closer.
  • The refracted image is not laterally inverted, but it appears bent (broken) at the boundary for oblique incidence.
  • The wavelength of light changes when it enters a different medium; it decreases in a denser medium.
  • The frequency remains unchanged while passing from one medium to another.
CISCE: Class 12

Key Points: Conditions for Sustained Interference of Light Waves

  • 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.
CISCE: Class 12

Law: Brewster’s Law

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 (ip).

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

n = tan ⁡ip​​

Explanation / Proof

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

Let:

  • ip​ = 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⁡ ip = \[\frac {sin⁡ i_p}{cos⁡ i_p}\]

Equating the two expressions for n:

Hence,

ip + 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⁡ ip​​
  • 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
CISCE: Class 12

Key Points: Polarisation of Light by Refraction

  • 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.
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Definition: Unpolarized Light

Light in which the electric field vectors vibrate in all possible directions perpendicular to the direction of propagation.

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Definition: Plane Polarized Light

Light in which the electric field vectors vibrate only in one particular direction perpendicular to the direction of propagation.

CISCE: Class 12

Key Points: Polarisation by Scattering

  • 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.
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Definition: Polarizer

A material that allows only those light waves to pass whose electric field is along a particular direction (polarizing axis).

CISCE: Class 12

Law: Law of Malus

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.

Maharashtra State Board: Class 12

Definition: Plane of Vibration

The plane containing the electric field vectors of plane polarized light is called the plane of vibration.

CISCE: Class 12

Definition: Polaroid

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

Maharashtra State Board: Class 12

Definition: Plane of Polarization

The plane perpendicular to the plane of the vibration and the electric field vector is called plane of polarization.

CISCE: Class 12

Key Points: Polaroid

  • 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.
CISCE: Class 12

Key Points: Uses of Polaroids

  • 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.
Maharashtra State Board: Class 12

Law: Malus’ Law

It gives the intensity of plane polarized light after passing through a second polarizer, where θ is the angle between the axes of the two polarizers.

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Law: Brewster’s Law

Statement

When unpolarized light is incident on a transparent surface at a particular angle (called Brewster’s angle), the reflected light is completely plane polarised.
At this angle, the reflected and refracted rays are perpendicular to each other.

tan⁡θB = \[\frac {n_2}{n_1}\]

where
θB = Brewster’s angle
n1, n2 = refractive indices of the two media

Proof

At Brewster’s angle,

θB + r = 90∘

From Snell’s law:

n1 sin ⁡θB = n2 sin ⁡r

Since r = 90∘ − θB,

n1 sin⁡θB = n2 cos⁡θB
tan ⁡θB = \[\frac {n_2}{n_1}\]

Conclusion

At Brewster’s angle, the reflected light is completely plane polarized and the reflected and refracted rays are mutually perpendicular.

Maharashtra State Board: Class 12

Definition: Polarization by Scattering

Polarisation by scattering is the phenomenon in which unpolarized light becomes partially or completely plane polarised when it is scattered by small particles such as air molecules or dust particles.

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Definition: Interference

Interference is the phenomenon in which the intensity of light (or any wave) at a point becomes non-uniform due to the superposition of two or more coherent waves, resulting in regions of constructive and destructive interference.

Maharashtra State Board: Class 12

Key Points: Interference

  • Coherent sources emit waves of the same frequency with a constant phase difference.
  • In Young’s double slit experiment, two coherent sources are obtained from a single source.
  • Condition for constructive interference:
    Path difference Δl = nλ
  • Condition for destructive interference:
    Path difference Δl = (n − \[\frac {1}{2}\])λ
  • Position of bright fringe:
    yn = \[\frac {nλD}{d}\]
  • Fringe width:
    W = \[\frac {λD}{d}\](Bright and dark fringes are equally spaced.)
  • For a clear and steady interference pattern:
    Sources must be coherent, monochromatic, of nearly equal amplitude, and slits must be narrow with D ≫ d.
Maharashtra State Board: Class 12

Defintiion: Diffraction of Light

Diffraction of light is the phenomenon in which light spreads into the geometrical shadow region when it passes around the edges of an obstacle or through a narrow aperture whose size is comparable to its wavelength.

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Definition: Fraunhofer Diffraction

Diffraction observed when the source and screen are at large distances from the diffracting element, so that the incident wavefront is plane.

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Definition: Fresnel Diffraction

Diffraction observed when the source or screen (or both) are at finite distances from the obstacle, and the incident wavefront is spherical or cylindrical.

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Formula: Fraunhofer Diffraction at a Single Slit

\[a\sin\theta=\pm\left(n+\frac{1}{2}\right)\lambda\]

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Formula: Width of the Central Bright Fringe

\[W_{c}=2y_{1d}=2W=2\left(\frac{\lambda D}{a}\right)\]

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Definition: Resolving Power

The ability of an optical instrument to distinguish two closely spaced objects as separate and distinct is called its resolving power.

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Definition: Limit of Resolution

The minimum visual angle between two objects that can be just resolved by an instrument is called the limit of resolution.

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Definition: Resolving Power of Telescope

The resolving power of a telescope is then defined as the reciprocal of the least angular separation between the objects that are just resolved.

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Key Points: Resolving Power

  • Resolution depends on diffraction effects caused by the optical instrument's aperture.
  • According to Rayleigh’s criterion, two objects are just resolved when the central maximum of one diffraction pattern coincides with the first minimum of the other.
  • For a single slit (linear objects), the limit of resolution:
    dθ = \[\frac {λ}{a}\]
  • In a microscope, resolving power increases with numerical aperture (N.A. = n sin α) and decreases with wavelength:
    R ∝ \[\frac {N.A.}{λ}\]
  • For self-luminous point objects (microscope):
    a = \[\frac {0.61λ}{N.A.}\]
  • For a telescope, angular resolution is:
    θ = \[\frac {1.22λ}{D}\]where D is the aperture diameter.
  • Resolving power improves when:
    Wavelength is smaller
    Aperture diameter is larger
    The numerical aperture is higher
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