Polarization is the restriction of the vibration of electric field vectors of a transverse wave to a single plane. It provides direct, unambiguous evidence of the transverse nature of light waves.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Basic concepts of transverse nature, unpolarized versus linearly polarized light, Malus's Law, Brewster's Law, and polarization by reflection and simple refraction.
PGT extension
Double refraction in uniaxial crystals, ordinary and extraordinary rays, Nicol prism, production and analysis of circularly and elliptically polarized light, optical activity, and specific rotation.
Prerequisites
Transverse and longitudinal wave kinematics
Maxwell's electromagnetic wave equations
Snell's law and refractive index
Nature of Unpolarized and Polarized Light
Unpolarized light contains rapid, random changes in the orientation of the transverse electric field vector E with equal probability in all directions perpendicular to the direction of propagation.
Linearly polarized light possesses an electric field vector that oscillates along one fixed, specific straight line transverse to the propagation direction.
Natural light sources like the Sun or incandescent lamps emit unpolarized light due to independent atomic emissions with random phases.
Polarization by Reflection and Brewster's Law
When unpolarized light is incident on a transparent dielectric surface at a specific angle called the polarizing angle or Brewster's angle (p), the reflected light is completely plane-polarized.
At Brewster's angle, the reflected ray and the refracted (transmitted) ray are mutually perpendicular to each other.
Brewster's law states that the tangent of the polarizing angle equals the refractive index of the reflecting medium relative to the surrounding medium.
Double Refraction (Birefringence) and Uniaxial Crystals
Anisotropic crystals like calcite and quartz split an unpolarized incident ray into two refracted rays: the Ordinary ray (O-ray) and the Extraordinary ray (E-ray).
The O-ray obeys ordinary Snell's law of refraction and travels with the same speed in all directions within the crystal.
The E-ray does not obey Snell's law, its velocity depends on the direction of propagation inside the crystal, and it has a variable refractive index.
Production and Detection of Polarized Light
Polarizers act as optical filters that transmit only those electric field components parallel to their transmission axis.
Malus's Law governs the intensity of transmitted light when a plane-polarized beam passes through an analyzer.
Quarter-wave plates and half-wave plates introduce phase differences of pi/2 and pi respectively between O-rays and E-rays to manipulate polarization states.
Derivations
Brewster's Law Derivation
Conditions: Light incident from a rarer medium of refractive index n_1 onto a denser dielectric medium of refractive index n_2 at polarizing angle i_p.
Write Snell's law at the interface: n_1 \sin($i_p) = n_2 \sin(r)$, where r is the angle of refraction.
Apply the condition that at Brewster's angle, the reflected ray and refracted ray are perpendicular: i_p + r = 90 degrees, which gives $r = 90 degrees - i_p$.
Substitute r into Snell's law: n_1 \sin($i_p) = n_2 \sin(90^circ - i_p) = n_2 \cos(i_p)$.
Rearrange to find the ratio of sine and cosine: \frac{\sin(i_p)}{\cos($i_p) = \frac{n_2}{n_1}$.
Result: \tan($i_p) = n_{21}$
Malus's Law Derivation
Conditions: Plane-polarized light of initial amplitude E_0 incident on an analyzer whose transmission axis makes an angle theta with the vibration direction of the incident light.
Resolve the incident electric field amplitude E_0 into two rectangular components: E_0 \cos(\theta) parallel to the transmission axis and E_0 \sin(\theta) perpendicular to it.
The analyzer absorbs or blocks the perpendicular component E_0 \sin(\theta), transmitting only the parallel component $E = E_0 \cos(\theta)$.
Recall that the intensity of a light wave is directly proportional to the square of its electric field amplitude: I \propto E^2 and I_0 \propto E_0^2.
Substitute the transmitted amplitude: I \propto (E_0 \cos(\theta))^2, leading directly to $I = I_0 \cos^2(\theta)$.
Result: $I = I_0 \cos^2(\theta)$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Brewster's Law
$$\tan(i_p) = \frac{n_2}{n_1}$$
$i_p = Brewster's polarizing angle$, $n_1 = refractive index of first medium$, $n_2 = refractive index of second medium$ Dimensionless
Valid for non-magnetic dielectric media where relative permeability is approximately unity
Malus's Law
$$I = I_0 \cos^2(\theta)$$
I = transmitted intensity, $I_0 = initial intensity of polarized light$, \theta = angle between polarizer and analyzer axes W/m^2
$I_max = maximum intensity transmitted$ by rotating analyzer, $I_min = minimum intensity transmitted$ Dimensionless
Partially polarized light
Phase Difference in Birefringence
$$\delta = \frac{2\pi}{\lambda} (n_o - n_e) t$$
$\delta = phase difference$, \lambda = wavelength in vacuum, n_o and $n_e = ordinary$ and extraordinary refractive indices, t = thickness of crystal plate Radians
Uniaxial crystal cut parallel to optic axis
Optical Rotation Specific Rotation
$$[\alpha]_t^\lambda = \frac{\theta}{l \cdot c}$$
\theta = angle of rotation, l = length of path in decimeters, $c = concentration in g/cm^3$ degrees \cdot dm^{-1} \cdot (g/cm^3)^{-1}
Optically active solutions at constant temperature and wavelength
Intensity of Unpolarized Light Passing Single Polarizer
$$I = \frac{1}{2} I_0$$
$I_0 = initial unpolarized intensity$, I = transmitted intensity W/m^2
Averaged over all random orientations \co$s^2(\theta) = 1/2$
For negative or positive uniaxial crystals where phase shift is pi/2
Condition for Perpendicular Reflected and Refracted Rays
$$i_p + r = 90^\circ$$
$i_p = angle of incidence (Brewster angle)$, r = angle of refraction Degrees
At maximum polarization by reflection
Graphs and experiments
Malus's Law Verification Graph
Draw: A plot of transmitted intensity I on the y-axis versus \theta (from 0 to 360 degrees) on the x-axis, showing a symmetric wave-like curve peaking at 0, 180, and 360 degrees with zeros at 90 and 270 degrees.
Interpret: Demonstrates the cosine-squared dependence of light intensity passing through crossed or parallel polarizers.
Exam use: Tested via conceptual MCQs asking for intensity values at specific angles like 45 degrees or 60 degrees.
Polarization by Reflection Setup
Draw: Unpolarized ray incident on a glass slab at angle i_p, splitting into a reflected ray at angle i_p and a refracted ray at angle r, with angle between them marked as 90 degrees.
Interpret: Visually proves Brewster's law and the perpendicularity of reflected and refracted beams.
Exam use: Numerical problems calculating refractive index given the angle of incidence or vice versa.
Nicol Prism Construction and Action
Draw: A calcite crystal cut diagonally and cemented with Canada balsam, showing total internal reflection of the O-ray and transmission of the E-ray.
Interpret: Illustrates how a Nicol prism acts as an efficient polarizer by eliminating the ordinary ray.
Exam use: Questions identifying the cement used (Canada balsam) and its refractive index relative to O-ray and E-ray.
Solved numericals
TGT core
Calculate the polarizing angle for glass of refractive index 1.5.
Recall Brewster's law: \tan($i_p) = n$.
Substitute the given value: \tan($i_p) = 1.5$.
Find the inverse tangent: $i_p = \arctan(1.5)$.
Answer: 56.3 degrees
TGT core
Unpolarized light of intensity 32 W/m^2 passes through a polarizer. What is the intensity of the transmitted light?
A polarizer reduces unpolarized light intensity to half its initial value.
Plane-polarized light passes through an analyzer whose axis is rotated by 60 degrees relative to the polarizer. What fraction of the initial polarized light intensity is transmitted?
Calculate intensity ratio: I / $I_0 = (1/2)^2 = 1/4$.
Answer: 0.25 (or 25% of initial intensity)
Assumptions and limitations
Polarizers are assumed to be ideal, absorbing 100% of the orthogonal component and transmitting 100% of the parallel component without internal absorption losses.
Dielectric media are assumed isotropic and non-magnetic for Brewster's law unless specified otherwise.
Common misconceptions
Common error: Sound waves can be polarized just like light waves.
Correct: Sound waves are longitudinal and cannot be polarized.
Polarization requires transverse oscillations occurring perpendicular to the direction of energy propagation.
Common error: Unpolarized light has no electric field oscillations.
Correct: Unpolarized light has rapid, randomly fluctuating electric field oscillations in all transverse directions.
Zero polarization means absence of preferred orientation, not absence of electric fields.
Common error: At Brewster's angle, the reflected and refracted rays have the same intensity.
Correct: At Brewster's angle, the reflected ray is completely polarized, but its intensity is generally much smaller than that of the refracted ray.
Reflectance and transmittance depend on Fresnel's equations, not on equal energy splitting.
Quick revision
Light is a transverse electromagnetic wave, proven conclusively by polarization.
Brewster's law: \tan($i_p) = n_2 / n_1; reflected$ and refracted rays are mutually perpendicular.
Malus's law: Intensity varies as $I = I_0 \cos^2(\theta)$.
Unpolarized light passing through a single ideal polarizer loses exactly half its intensity.
Must remember
Polarization is an exclusive property of transverse waves, confirming that light waves are transverse in nature.
Brewster's Law states that $\tan(i_p) = \frac{n_2}{n_1}$, and at this polarizing angle, the reflected and refracted rays are mutually perpendicular ($i_p + r = 90^\circ$).
When unpolarized light passes through an ideal linear polarizer, its intensity is reduced exactly by half ($I = \frac{1}{2} I_0$).
Malus's Law dictates that transmitted intensity through an analyzer varies as $I = I_0 \cos^2(\theta)$.
Double refraction in uniaxial crystals splits unpolarized light into an Ordinary ray (O-ray, obeying Snell's law) and an Extraordinary ray (E-ray, variable velocity).
A quarter-wave plate introduces a phase difference of $\frac{\pi}{2}$ between the O-ray and E-ray to produce circularly or elliptically polarized light.
Glossary
Unpolarized LightLight with rapid, random changes in the orientation of the transverse electric field vector with equal probability in all perpendicular directions.
Linearly Polarized LightLight possessing an electric field vector that oscillates along one fixed, specific straight line transverse to the propagation direction.
Brewster's Angle ($i_p$)The angle of incidence at which reflected light is completely plane-polarized, equal to the inverse tangent of the relative refractive index.
Ordinary Ray (O-ray)A refracted ray in a birefringent crystal that obeys ordinary Snell's law and travels with a constant speed in all directions.
Extraordinary Ray (E-ray)A refracted ray in a birefringent crystal that does not obey Snell's law and has a velocity and refractive index dependent on propagation direction.
Quarter-Wave PlateAn optical device of specific thickness that introduces a phase difference of $\frac{\pi}{2}$ between O-rays and E-rays.
Specific RotationThe angle of optical rotation produced by a solution of unit length (decimeter) and unit concentration ($g/cm^3$).
Unpolarized vs. Linearly Polarized Light
Feature
Unpolarized Light
Linearly Polarized Light
Electric Field Orientation
Random and changes rapidly in all transverse directions
Confined to a single fixed straight line
Source
Natural thermal sources like Sun and incandescent lamps
Produced via polarizers, reflection at Brewster angle, or scattering
Intensity through rotating analyzer
Remains constant
Varies according to Malus's Law ($I = I_0 \cos^2(\theta)$)
Ordinary Ray (O-ray) vs. Extraordinary Ray (E-ray)
Property
Ordinary Ray (O-ray)
Extraordinary Ray (E-ray)
Snell's Law
Obeys ordinary Snell's law
Does not obey ordinary Snell's law
Refractive Index
Constant ($n_o$)
Variable ($n_e$) depending on direction
Wavefront inside crystal
Spherical wavefront
Ellipsoidal wavefront
Practice Quiz
Choose an option to reveal the explanation.
1. Which of the following waves cannot be polarized?
Explanation: Sound waves in air are longitudinal waves, meaning their oscillations occur along the direction of propagation. Polarization is exclusively a property of transverse waves.
2. Unpolarized light of intensity $I_0$ is incident on a linear polarizer. What is the intensity of the transmitted light?
Explanation: Unpolarized light has electric field components distributed equally in all transverse directions. Averaging $\cos^2(\theta)$ over all random angles yields $\frac{1}{2}$, so the transmitted intensity is $I = \frac{1}{2} I_0$.
3. According to Brewster's law, what is the relation between the polarizing angle $i_p$ and the refractive index $n$ of a dielectric medium (assuming surrounding medium is air with $n_1 = 1$)?
Explanation: Brewster's law states that the tangent of the polarizing angle equals the refractive index of the reflecting medium, i.e., $\tan(i_p) = n$.
4. At Brewster's angle, what is the angle between the reflected ray and the refracted (transmitted) ray?
Explanation: At Brewster's angle, the reflected ray and the refracted ray are mutually perpendicular, making the angle between them exactly $90^\circ$.
5. Malus's Law is expressed mathematically as:
Explanation: Malus's Law states that when completely plane-polarized light is incident on an analyzer, the transmitted intensity varies directly as the square of the cosine of the angle between their transmission axes: $I = I_0 \cos^2(\theta)$.
6. In a birefringent crystal, the Ordinary ray (O-ray) differs from the Extraordinary ray (E-ray) because:
Explanation: The O-ray obeys ordinary Snell's law of refraction and travels with the same speed in all directions inside the crystal, possessing a constant refractive index $n_o$.
7. What phase difference is introduced between the O-ray and E-ray by a quarter-wave plate?
Explanation: A quarter-wave plate is specifically designed to introduce a phase difference of $\frac{\pi}{2}$ (or an optical path difference of $\frac{\lambda}{4}$) between the ordinary and extraordinary rays.
8. If the angle between the polarizer and analyzer transmission axes is $60^\circ$, what fraction of the incident polarized light intensity is transmitted?
Explanation: Using Malus's Law $I = I_0 \cos^2(60^\circ) = I_0 (\frac{1}{2})^2 = \frac{1}{4} I_0$. Thus, a quarter of the intensity is transmitted.
9. Optical rotation specific rotation $[\alpha]_t^\lambda$ depends upon which of the following parameters?
Explanation: Specific rotation is calculated via $[\alpha]_t^\lambda = \frac{\theta}{l \cdot c}$, where $\theta$ is rotation angle, $l$ is path length in decimeters, and $c$ is concentration.
10. Which crystal is commonly cited as a classic example of an uniaxial birefringent crystal exhibiting double refraction?
Explanation: Calcite ($CaCO_3$) and quartz are classic anisotropic uniaxial crystals widely used to demonstrate double refraction.
11. What is the degree of polarization for unpolarized light?
Explanation: Unpolarized light has equal intensity in all transverse directions, making $I_{max} = I_{min}$, leading to a degree of polarization $P = \frac{I_{max} - I_{min}}{I_{max} + I_{min}} = 0$.
12. The refractive index of a glass medium is $1.732$ (which is $\sqrt{3}$). What is its Brewster's polarizing angle?
Explanation: Using $\tan(i_p) = n = \sqrt{3}$, we find $i_p = \arctan(\sqrt{3}) = 60^\circ$.
13. What thickness $t$ is required for a quarter-wave plate for wavelength $\lambda$ where ordinary and extraordinary refractive indices are $n_o$ and $n_e$?
Explanation: Equating the phase shift formula $\delta = \frac{2\pi}{\lambda}(n_o - n_e)t = \frac{\pi}{2}$ gives the thickness $t = \frac{\lambda}{4(n_o - n_e)}$.
14. Which of the following statements regarding unpolarized light is correct?
Explanation: Unpolarized light features rapid, random changes in the orientation of the transverse electric field vector with equal probability in all directions perpendicular to propagation.
15. If a polarizer and analyzer are crossed (their transmission axes are at $90^\circ$ to each other), what is the transmitted intensity of incident plane-polarized light?
Explanation: When polarizer and analyzer are crossed, $\theta = 90^\circ$, so $I = I_0 \cos^2(90^\circ) = 0$, resulting in complete extinction (zero intensity).
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. Light is incident on a transparent medium at Brewster's angle $i_p$. If the refractive index of the medium is $n = 1.5$, the angle of refraction $r$ is:
2. Two polaroids are placed with their transmission axes parallel. By what angle must one polaroid be rotated so that the transmitted intensity becomes one-fourth of the maximum intensity?
Explanation: According to Malus's Law, $I = I_0 \cos^2(\theta)$. We are given $I = \frac{1}{4} I_0$, so $\cos^2(\theta) = \frac{1}{4} \implies \cos(\theta) = \frac{1}{2} \implies \theta = 60^\circ$.
3. Unpolarized light of intensity $16 \text{ W/m}^2$ is passed through a sequence of two ideal polarizers whose transmission axes make an angle of $30^\circ$ with each other. The final transmitted intensity is:
Explanation: After the first polarizer, unpolarized light becomes polarized with intensity $I_1 = \frac{16}{2} = 8 \text{ W/m}^2$. Passing through the second polarizer at $\theta = 30^\circ$: $I_2 = 8 \cos^2(30^\circ) = 8 \times (\frac{\sqrt{3}}{2})^2 = 8 \times \frac{3}{4} = 6 \text{ W/m}^2$.
4. The refractive indices for ordinary and extraordinary rays in a calcite crystal are $n_o = 1.658$ and $n_e = 1.486$ for sodium light of wavelength $\lambda = 589 \text{ nm}$. What is the nature of this crystal?
Explanation: Since $n_e (1.486) < n_o (1.658)$, the velocity of the extraordinary ray is greater than that of the ordinary ray in directions perpendicular to the optic axis, classifying calcite as a negative uniaxial crystal.
5. A beam of unpolarized light falls on a glass plate at Brewster's angle. If the refractive index of glass is $1.5$, what is the angle of incidence?
Explanation: Brewster's angle is defined by $\tan(i_p) = n$, which means the angle of incidence is $i_p = \arctan(1.5)$.
6. If the degree of polarization of a partially polarized beam of light is $0.6$, and the maximum intensity transmitted by a rotating analyzer is $80 \text{ W/m}^2$, what is the minimum intensity transmitted?
Attempt these questions under a ten-minute limit using a separate timer.
1. Which phenomenon provides direct and conclusive evidence of the transverse nature of light waves?
Explanation: Polarization is unique to transverse waves because it requires asymmetry perpendicular to the direction of wave travel, conclusively proving light is transverse.
2. At the polarizing angle, sunlight reflected from a calm lake is:
Explanation: When unpolarized light reflects off a dielectric surface at Brewster's angle, the reflected ray is completely plane-polarized with its electric field vector parallel to the reflecting surface.
3. A half-wave plate introduces what phase difference between the ordinary and extraordinary rays?
Explanation: A half-wave plate introduces a phase difference of $\pi$ (or an optical path difference of $\frac{\lambda}{2}$) between the O-ray and E-ray.
4. In a uniaxial crystal, the ray that travels with a constant velocity in all directions and obeys Snell's law is the:
Explanation: The Ordinary ray (O-ray) obeys ordinary Snell's law of refraction and maintains a constant speed and refractive index in all directions within the crystal.
5. If the angle between the polarizer and analyzer transmission axes is $45^\circ$, what percentage of the incoming plane-polarized intensity is transmitted?
Explanation: Using Malus's Law, $I = I_0 \cos^2(45^\circ) = I_0 (\frac{1}{\sqrt{2}})^2 = 0.5 I_0$, which corresponds to $50\%$ transmission.
6. Which of the following conditions is necessary for Brewster's Law to hold precisely?
Explanation: Brewster's Law $\tan(i_p) = \frac{n_2}{n_1}$ is standardly valid for non-magnetic dielectric media where relative permeability $\mu_r \approx 1$.
7. Specific rotation of an optically active substance is measured in units of:
Explanation: From $[\alpha]_t^\lambda = \frac{\theta}{l \cdot c}$, rotation angle $\theta$ is in degrees, path length $l$ in decimeters, and concentration $c$ in $\text{g/cm}^3$, giving $\text{degrees} \cdot \text{dm}^{-1} \cdot (\text{g/cm}^3)^{-1}$.
8. What happens to the intensity of light passing through a single ideal polarizer when unpolarized light is incident upon it, regardless of the polarizer's orientation?
Explanation: A single polarizer reduces the intensity of unpolarized light by half ($I = \frac{1}{2}I_0$) due to the averaging of transverse electric field components ($\cos^2(\theta) = 1/2$), regardless of orientation.
9. Double refraction in uniaxial crystals splits an unpolarized incident ray into:
Explanation: Anisotropic uniaxial crystals split an unpolarized ray into two orthogonally polarized refracted rays: the Ordinary ray and the Extraordinary ray.
10. If the phase difference $\delta$ between two mutually perpendicular equal-amplitude electric field components is $\frac{\pi}{2}$ and their amplitudes are equal, the resulting polarization state is:
Explanation: Superposition of two perpendicular electric field oscillations of equal amplitude with a phase difference of $\frac{\pi}{2}$ produces circularly polarized light.