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Optics
Production and Detection of Polarized Light
Polarized light refers to transverse electromagnetic waves in which electric field oscillations are restricted to a single plane. The production and detection of polarized light rely on fundamental wave phenomena such as reflection, refraction, double refraction, selective absorption, and scattering.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Basic definitions of polarization, Brewster's law, Malus's law, Polaroid sheets, and polarization by reflection and refraction.
PGT extension
Double refraction in uniaxial crystals, positive and negative crystals, Huygens' explanation of ordinary and extraordinary rays, Nicol prism, quarter-wave and half-wave plates, and analysis of elliptically and circularly polarized light.
Prerequisites
Transverse nature of light waves
Huygens' wave theory of light
Snell's law of refraction and total internal reflection
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, i_p), the reflected light is completely plane-polarized with its electric vector perpendicular to the plane of incidence.
At Brewster's angle, the reflected ray and the refracted ray are mutually perpendicular to each other.
Brewster's law is expressed mathematically as tan($i_p) = n$, where n is the refractive index of the medium.
Selective Absorption and Polaroids
Certain naturally occurring and synthetic crystalline materials possess dichroism, which is the property of selectively absorbing one of the mutually perpendicular components of polarized light while transmitting the other.
Polaroids are manufactured sheets containing microscopic crystals of quinine iodosulfate oriented parallel to one another in a plastic matrix.
When polarized light passes through a polarizer (analyzer), the transmitted intensity varies according to Malus's law: $I = I_0 \cos^{2}(\theta)$.
Double Refraction (Birefringence)
When an unpolarized light ray enters an anisotropic crystal such as calcite (CaCO_3) or quartz, it splits into two refracted rays known as the Ordinary ray (O-ray) and the Extraordinary ray (E-ray).
The O-ray obeys ordinary laws of refraction and travels with the same speed in all directions within the crystal, whereas the E-ray does not obey Snell's law and its speed varies with direction.
In negative crystals like calcite, the velocity of the E-ray is greater than that of the O-ray (v_e > v_o), whereas in positive crystals like quartz, v_o > v_e.
Production and Detection of Elliptically and Circularly Polarized Light
Circularly polarized light is produced when two mutually perpendicular plane-polarized beams of equal amplitude, having a phase difference of odd multiples of π/2, are superimposed.
Elliptically polarized light results from the superposition of two mutually perpendicular plane-polarized beams of unequal amplitudes with a phase difference of (π/2 ± mπ).
Detection involves passing the beam through a quarter-wave plate followed by an analyzer; circular light shows constant intensity while rotated, whereas unpolarized light shows no intensity variation.
Derivations
Brewster's Law Derivation
Conditions: Light incident on a dielectric boundary at polarizing angle i_p
Let unpolarized light be incident at angle i_p such that the reflected ray is completely polarized.
According to Snell's law at the interface, sin(i_p) / sin(r) = n_2 / $n_1 = n$, where r is the angle of refraction.
At the polarizing angle, the reflected ray and refracted ray are perpendicular, so i_p + $r = 90°$, which gives $r = 90° - i_p$.
Substitute r into Snell's law: sin(i_p) / sin(90° - $i_p) = n$.
I_{max} and $I_{min} = maximum$ and minimum intensities observed upon rotating an analyzer dimensionless
Partially polarized light
Refractive Index from Brewster Angle for Glass-Air Interface
$$n = \sin(i_p) / \cos(i_p)$$
$i_p = Brewster angle$ dimensionless
Non-magnetic medium where relative permeability μ_r ≈ 1
Intensity of Unpolarized Light through Single Polaroid
$$I = \frac{1}{2} I_0$$
$I_0 = intensity of incident unpolarized light$, I = transmitted intensity W/m²
Ideal polarizer
Phase Difference in Retardation Plate
$$\delta = \frac{2\pi}{\lambda} (n_e - n_o) d$$
$\delta = phase difference$, d = thickness of plate radians
Normal incidence
Graphs and experiments
Intensity Variation of Polarized Light (Malus's Law Verification)
Draw: A polar plot or Cartesian plot of transmitted intensity I versus angle θ showing a periodic curve proportional to cos²(θ), with maxima at 0°, 180°, 360° and zeroes at 90°, 270°.
Interpret: Demonstrates that light is a transverse wave, as longitudinal waves would show no variation in intensity upon rotation of the analyzer.
Exam use: Numerical problems calculating intensity ratios at specific angles like 30°, 45°, 60°.
Nicol Prism Construction and Action
Draw: A rhombic calcite crystal cut along the diagonal, cemented with Canada balsam, showing total internal reflection of the ordinary ray and transmission of the extraordinary ray.
Interpret: Canada balsam has a refractive index (1.55) intermediate between the O-ray (1.658) and E-ray (1.486) of calcite for sodium light, allowing the O-ray to undergo total internal reflection while the E-ray passes through.
Exam use: Conceptual questions regarding the function of Canada balsam and the elimination of the ordinary ray.
Polarization by Reflection Setup
Draw: A ray of unpolarized light striking a glass plate at angle i_p, splitting into a reflected ray at 90° to the refracted ray, with perpendicular electric vector symbols on the reflected ray.
Interpret: Visually validates Brewster's condition where reflected and refracted rays are perpendicular.
Exam use: Drawing ray diagrams and calculating refractive index given the polarizing angle.
Solved numericals
TGT core
Calculate the polarizing angle for a glass surface of refractive index 1.732. (Given tan(60°) = 1.732)
Recall Brewster's law: tan($i_p) = n$.
Substitute the given value of refractive index: tan($i_p) = 1.732$.
Find the angle whose tangent is 1.732: $i_p = 60°$.
Answer: 60°
TGT core
Unpolarized light of intensity 32 W/m² is incident on a polarizer. The transmitted light is then passed through an analyzer whose transmission axis makes an angle of 45° with the polarizer. Find the intensity of light coming out of the analyzer.
Intensity after passing through the first polarizer (I_0) is half of the incident unpolarized light: $I_0 = 32 / 2 = 16 W/m^{2}$.
Apply Malus's law for the analyzer at angle $\theta = 45°: I = I_0 \cos^{2}(\theta)$.
A quarter-wave plate is designed for sodium light of wavelength 589 nm. Calculate the minimum thickness of the calcite plate if the ordinary and extraordinary refractive indices are $n_o = 1.658$ and $n_e = 1.486 respectively$.
Write the formula for the thickness of a quarter-wave plate: $d = \lambda / [4(n_o - n_e)]$.
Substitute the given values: $\lambda = 589 * 10^-9 m$, $n_o = 1.658$, $n_e = 1.486$.
Calculate the refractive index difference: n_o - $n_e = 1.658 - 1.486 = 0.172$.
Brewster's law assumes non-magnetic dielectric media where relative magnetic permeability μ_r = 1.
Uniaxial crystal formulas assume monochromatic light and normal incidence parallel to specific crystal cuts.
Common misconceptions
Common error: Unpolarized light can be converted into circularly polarized light directly using a single polarizer.
Correct: Circularly polarized light is produced by superimposing two mutually perpendicular plane-polarized beams with a phase difference of π/2.
A single polarizer only produces plane-polarized light from unpolarized light.
Common error: Longitudinal waves can exhibit polarization.
Correct: Only transverse waves can be polarized.
Polarization requires oscillations to occur perpendicular to the direction of propagation, which is impossible in longitudinal waves where oscillations are parallel.
Quick revision
At Brewster's angle, reflected and refracted rays are perpendicular, and tan($i_p) = n$.
Brewster's Law: tan($i_p) = n$
Malus's Law: $I = I_0 \cos^{2}(\theta)$
Must remember
Brewster's Law states that $\tan(i_p) = n$, where $i_p$ is the polarizing angle and $n$ is the refractive index.
At Brewster's angle, the reflected and refracted rays are mutually perpendicular to each other.
Malus's Law dictates that the transmitted intensity through an analyzer is $I = I_0 \cos^2(\theta)$.
Unpolarized light passing through an ideal polarizer has its intensity reduced to half ($I = \frac{1}{2} I_0$).
Anisotropic crystals like calcite and quartz exhibit double refraction, splitting incident light into an ordinary ray (O-ray) and an extraordinary ray (E-ray).
In negative crystals like calcite, the velocity of the extraordinary ray is greater than that of the ordinary ray ($v_e > v_o$).
In positive crystals like quartz, the velocity of the ordinary ray is greater than that of the extraordinary ray ($v_o > v_e$).
Circularly polarized light requires two mutually perpendicular plane-polarized beams of equal amplitude with a phase difference of odd multiples of $\pi/2$.
A quarter-wave plate introduces a phase difference of $\pi/2$ (or path difference of $\lambda/4$) between the O-ray and E-ray.
A half-wave plate introduces a phase difference of $\pi$ (or path difference of $\lambda/2$) between the O-ray and E-ray.
Glossary
Brewster's Angle ($i_p$)The angle of incidence at which reflected light is completely plane-polarized.
Malus's LawLaw stating that the intensity of plane-polarized light transmitted through an analyzer varies as the square of the cosine of the angle between their transmission axes.
Double Refraction (Birefringence)The phenomenon where an unpolarized ray entering an anisotropic crystal splits into two refracted rays: O-ray and E-ray.
Ordinary Ray (O-ray)The refracted ray inside a birefringent crystal that obeys ordinary laws of refraction and travels with the same speed in all directions.
Extraordinary Ray (E-ray)The refracted ray inside a birefringent crystal that does not obey Snell's law and whose speed depends on the direction of propagation.
Quarter-Wave PlateAn optical device made of a birefringent crystal cut parallel to the optic axis that introduces a phase difference of $\pi/2$ between O-ray and E-ray.
Half-Wave PlateAn optical device that introduces a phase difference of $\pi$ between O-ray and E-ray, capable of rotating the plane of polarization.
DichroismThe property of certain materials to selectively absorb one of the mutually perpendicular components of polarized light while transmitting the other.
PolaroidA synthetic sheet containing microscopic quinine iodosulfate crystals oriented in a plastic matrix used to produce polarized light.
Degree of Polarization ($V$)A measure of the proportion of polarized light in a partially polarized beam, defined by $V = \frac{I_{max} - I_{min}}{I_{max} + I_{min}}$.
Ordinary Ray (O-ray) vs Extraordinary Ray (E-ray)
Parameter
Ordinary Ray (O-ray)
Extraordinary Ray (E-ray)
Snell's Law
Obeys ordinary laws of refraction
Does not obey Snell's law
Velocity
Constant in all directions within the crystal
Varies with direction of propagation
Wavefront
Spherical wavefront inside the crystal
Ellipsoidal wavefront inside the crystal
Refractive Index
Constant ($n_o$)
Depends on direction ($n_e$)
Positive vs Negative Birefringent Crystals
Feature
Negative Crystals (e.g., Calcite)
Positive Crystals (e.g., Quartz)
Velocities
$v_e > v_o$
$v_o > v_e$
Refractive Indices
$n_o > n_e$
$n_e > n_o$
Wavefront Relationship
Ellipsoid encloses sphere
Sphere encloses ellipsoid
Practice Quiz
Choose an option to reveal the explanation.
1. Which of the following waves can exhibit the phenomenon of polarization?
Explanation: Polarization is exclusively a property of transverse waves where oscillations occur perpendicular to the direction of propagation. Longitudinal waves oscillate parallel to propagation and cannot be polarized.
2. When unpolarized light is incident on a transparent medium at Brewster's angle $i_p$, the angle between the reflected and refracted rays is:
Explanation: At Brewster's angle, the reflected ray and the refracted ray are mutually perpendicular, making the angle between them exactly $90^\circ$.
3. If the refractive index of a glass medium is $\sqrt{3}$, its Brewster's angle for light incident from air is:
Explanation: According to Brewster's law, $\tan(i_p) = n$. Given $n = \sqrt{3}$, we have $\tan(i_p) = \sqrt{3}$, which implies $i_p = 60^\circ$.
4. Unpolarized light of intensity $I_0$ is incident on an ideal polarizer. The intensity of the transmitted light is:
Explanation: When unpolarized light passes through an ideal polarizer, half of its intensity is absorbed and the transmitted intensity is reduced to $I = \frac{1}{2} I_0$.
5. According to Malus's law, when completely plane-polarized light of intensity $I_0$ passes through an analyzer whose transmission axis makes an angle $\theta$ with the polarizer axis, the transmitted intensity is:
Explanation: Malus's law states that the transmitted intensity is given by $I = I_0 \cos^2(\theta)$.
6. The property of selectively absorbing one of the mutually perpendicular components of polarized light is called:
Explanation: Dichroism is the property possessed by certain crystals and polaroids of selectively absorbing one polarized component while transmitting the orthogonal component.
7. When unpolarized light enters an anisotropic crystal like calcite, it splits into two rays known as:
Explanation: Double refraction causes unpolarized light entering a birefringent crystal to split into an Ordinary ray (O-ray) and an Extraordinary ray (E-ray).
8. In a negative birefringent crystal such as calcite, how do the velocities of the ordinary ray ($v_o$) and extraordinary ray ($v_e$) compare?
Explanation: In negative uniaxial crystals like calcite, the velocity of the extraordinary ray is greater than that of the ordinary ray ($v_e > v_o$). In positive crystals like quartz, $v_o > v_e$.
9. What phase difference is introduced between the O-ray and E-ray by a quarter-wave plate?
Explanation: A quarter-wave plate introduces a phase difference of $\pi/2$ (corresponding to a path difference of $\lambda/4$) between the ordinary and extraordinary rays.
10. Circularly polarized light is produced by superimposing two mutually perpendicular plane-polarized beams of equal amplitude having a phase difference of:
Explanation: Circularly polarized light is formed when two perpendicular plane-polarized waves of equal amplitude are superimposed with a phase difference of odd multiples of $\pi/2$ and appropriate phase alignment.
11. What is the thickness $d$ of a quarter-wave plate for wavelength $\lambda$?
Explanation: The thickness of a quarter-wave plate is given by $d = \frac{\lambda}{4|n_e - n_o|}$, derived from the condition that the path difference is $\lambda/4$.
12. If the angle between the transmission axes of a polarizer and an analyzer is $60^\circ$, what fraction of the incident plane-polarized light intensity is transmitted?
Explanation: Using Malus's law, $I = I_0 \cos^2(60^\circ) = I_0 (1/2)^2 = I_0 / 4$. Thus, $1/4$ of the intensity is transmitted.
13. Elliptically polarized light results from the superposition of two mutually perpendicular plane-polarized beams of:
Explanation: Elliptically polarized light is produced when two orthogonal plane-polarized waves of unequal amplitudes combine with a phase difference of $(\pi/2 \pm m\pi)$.
14. The degree of polarization $V$ for a partially polarized light beam with maximum intensity $I_{max}$ and minimum intensity $I_{min}$ is expressed as:
Explanation: The degree of polarization is defined as $V = \frac{I_{max} - I_{min}}{I_{max} + I_{min}}$.
15. Which of the following crystals is an example of a positive uniaxial crystal?
Explanation: Quartz is a classic example of a positive uniaxial crystal ($v_o > v_e$, $n_e > n_o$), whereas calcite is a negative uniaxial crystal.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The polarizing angle for a medium is $60^\circ$. What is the refractive index of the medium?
2. Two polaroids are placed with their transmission axes parallel to each other. Unpolarized light of intensity $I_0$ is incident on the first polaroid. The intensity of light transmitted through the second polaroid is:
Explanation: The first polaroid converts unpolarized light to plane-polarized light of intensity $I_1 = I_0 / 2$. With parallel axes ($\theta = 0^\circ$), the second polaroid transmits this entire intensity, so $I = I_0 / 2$.
3. A half-wave plate is placed in the path of a beam of plane-polarized light. If the angle between the plane of polarization of the incident light and the optic axis of the plate is $\theta$, the plane of polarization of the emergent light is rotated by an angle of:
Explanation: A half-wave plate rotates the plane of polarization of incident plane-polarized light by an angle of $2\theta$, where $\theta$ is the angle between the incident vibration direction and the optic axis.
4. The refractive indices of calcite for ordinary and extraordinary rays are $n_o = 1.658$ and $n_e = 1.486$ for sodium light of wavelength $5893 \text{ \AA}$. What type of crystal is calcite?
Explanation: Since $n_o (1.658) > n_e (1.486)$, the velocity of the E-ray is greater than that of the O-ray ($v_e > v_o$), which characterizes calcite as a negative uniaxial crystal.
5. At what angle of incidence will the light reflected from glass ($n = 1.5$) be completely polarized?
Explanation: Brewster's angle is given by $i_p = \tan^{-1}(n) = \tan^{-1}(1.5)$.
6. Unpolarized light passes through two polaroids whose axes are inclined at an angle of $30^\circ$ to each other. If the initial intensity is $I_0$, the final transmitted intensity is:
Explanation: First polaroid transmits $I_1 = I_0 / 2$. Applying Malus's law for the second polaroid at $\theta = 30^\circ$: $I_2 = I_1 \cos^2(30^\circ) = (I_0 / 2) \times (\sqrt{3}/2)^2 = (I_0 / 2) \times (3/4) = 3I_0 / 8$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. If unpolarized light is incident on a water surface ($n = 4/3$) at Brewster's angle, what is the angle of refraction?
Explanation: At Brewster's angle $i_p$, the angle of incidence $i_p + r = 90^\circ$, so the angle of refraction $r = 90^\circ - i_p = 90^\circ - \tan^{-1}(n) = 90^\circ - \tan^{-1}(4/3)$.
2. A quarter-wave plate is designed for light of wavelength $\lambda = 589 \text{ nm}$. If $n_o = 1.544$ and $n_e = 1.553$, what is the minimum thickness of the plate?
3. When light travels from an optically rarer medium to a denser medium, the polarizing angle $i_p$ depends on:
Explanation: Brewster's law states $\tan(i_p) = n$, which means the polarizing angle depends solely on the refractive index of the medium (and hence slightly on wavelength due to dispersion).
4. Two crossed polaroids have an unpolarized light of intensity $I_0$ incident on the first. A third polaroid is inserted between them with its axis at $45^\circ$ to both. The final transmitted intensity is:
Explanation: First polaroid: $I_1 = I_0 / 2$. Second polaroid (at $45^\circ$): $I_2 = (I_0 / 2) \cos^2(45^\circ) = (I_0 / 2)(1/2) = I_0 / 4$. Third polaroid (at $45^\circ$ to second, making $90^\circ$ to first): $I_3 = I_2 \cos^2(45^\circ) = (I_0 / 4)(1/2) = I_0 / 8$.
5. Which of the following optical devices can be used to convert elliptically polarized light into circularly polarized light?
Explanation: A quarter-wave plate introduces the appropriate phase retardation to transform polarization states, such as converting elliptically polarized light (with proper orientation) or linear light into circular light.
6. In a calcite crystal, the wavefront of the ordinary ray is:
Explanation: Because the ordinary ray travels with the same speed in all directions within an anisotropic crystal, its wavefront is spherical. The extraordinary ray has an ellipsoidal wavefront.
7. The phenomenon of double refraction was first discovered by:
Explanation: Erasmus Bartholinus discovered double refraction in calcite crystals in 1669.
8. If the maximum and minimum intensities observed when an analyzer is rotated in a beam of light are in the ratio $3:1$, the degree of polarization is:
9. Which wave property is fundamentally responsible for the distinction between polarized and unpolarized light?
Explanation: Polarization relies on restricting the transverse vector vibrations of light waves to a specific plane or locus, which is only possible for transverse waves.
10. What is the primary function of a Nicol prism in optical instruments?
Explanation: A Nicol prism utilizes double refraction and total internal reflection of the O-ray to produce highly pure plane-polarized light and to detect polarization.