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Optics
Plane Polarization of Light
Plane polarization is the restriction of the vibration of the electric field vector of a light wave to a single plane containing the direction of propagation. It provides unequivocal proof of the transverse nature of light waves.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Qualitative understanding of transverse waves, polarization by reflection (Brewster's law), polarization by refraction, and polaroids.
PGT extension
Quantitative treatment of Malus's law, double refraction in uniaxial crystals (ordinary and extraordinary rays), Nicole prism, quarter-wave and half-wave plates, and optical activity.
Prerequisites
Transverse and longitudinal wave kinematics
Maxwell's electromagnetic wave equations (electric and magnetic field vectors)
Snell's law of refraction and total internal reflection
Nature of Unpolarized and Polarized Light
Unpolarized light consists of electromagnetic waves with rapidly and randomly changing planes of electric field vibration perpendicular to the direction of propagation.
Plane-polarized (or linearly polarized) light confines its electric field vector E to a fixed plane throughout its propagation.
The intensity of unpolarized light is equally distributed in all azimuths perpendicular to the wave normal.
Polarization by Reflection and Brewster's Law
When unpolarized light is incident at a specific angle called the polarizing angle (or Brewster's angle, ip) on a transparent medium, the reflected light is completely plane-polarized.
At Brewster's angle, the reflected ray and the refracted ray are mutually perpendicular to each other.
Brewster's law states that the tangent of the polarizing angle is numerically equal to the refractive index of the medium (μ = tan ip).
Double Refraction and Birefringence
Certain 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 laws of refraction and travels with the same speed in all directions within the crystal.
The E-ray does not obey Snell's law, travels with varying speeds depending on the direction, and its refractive index varies with the angle of incidence.
Optical Activity and Specific Rotation
Optically active substances (e.g., quartz, sugar solution, turpentine) rotate the plane of polarization of linearly polarized light passing through them.
The angle of rotation θ is directly proportional to the length of the path (l) and the concentration (c) of the solution.
Specific rotation is defined as the rotation produced by a solution of length 1 decimetre and unit concentration (1 g/cm3) at a specified temperature for a given wavelength.
Derivations
Derivation of Brewster's Law
Conditions: Light incident on a dielectric boundary at polarizing angle ip
Let unpolarized light be incident at Brewster's angle ip such that the reflected ray is completely polarized with electric vibrations perpendicular to the plane of incidence.
At this angle, the angle between the reflected ray and the refracted ray (angle of refraction r) is exactly 90 degrees.
From the geometry of the interface, the angle of incidence ip + 90\deg + $r = 180\deg$, which gives $r = 90\deg - i_{p}$.
Applying Snell's law at the boundary: $\mu = \frac{\sin i_p}{\sin r}$.
Conditions: Plane-polarized light of initial amplitude E0 incident on an analyzer with transmission axis at angle \theta
Let the electric field vector of the incoming plane-polarized light be E0.
When this light passes through an analyzer whose transmission axis makes an angle \theta with the vibration plane of the incident light, the electric vector is resolved into two orthogonal components.
The component parallel to the transmission axis is E0 \cos \theta, and the component perpendicular is E0 \sin \theta.
Only the parallel component is transmitted by the analyzer.
Since intensity is proportional to the square of the amplitude (I \propto E2), the transmitted intensity I is given by $I = I_0 \cos^2 \theta$, where I0 is the initial intensity.
Result: $I = I_0 \cos^2 \theta$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Brewster's Law
$$\mu = \tan i_p$$
\mu = Refractive index of medium, $i_p = Polarizing angle$ Dimensionless
Valid for transparent dielectric media
Malus's Law
$$I = I_0 \cos^2 \theta$$
I = Transmitted intensity, $I_0 = Incident intensity$, \theta = Angle between polarizers W/m2
\lambda = Wavelength in vacuum, n_e, $n_o = Refractive indices$ for E and O rays, t = Thickness of plate Radians
Uniaxial birefringent crystal cut parallel to optic axis
Thickness of Quarter-Wave Plate
$$t = \frac{\lambda}{4(n_e - n_o)}$$
t = Thickness, \lambda = Wavelength, n_e, $n_o = Principal refractive indices$ Metres (m)
Introduces a path difference of \lambda/4 or phase difference of \pi/2
Thickness of Half-Wave Plate
$$t = \frac{\lambda}{2(n_e - n_o)}$$
t = Thickness, \lambda = Wavelength, n_e, $n_o = Principal refractive indices$ Metres (m)
Introduces a path difference of \lambda/2 or phase difference of \pi
Optical Rotation Angle
$$\theta = S \cdot l \cdot c$$
\theta = Angle of rotation, S = Specific rotation, l = Path length, c = Concentration Degrees or radians
Active solution at constant temperature and wavelength
Average Intensity of Unpolarized Light through Polarizer
$$I = \frac{1}{2} I_0$$
I = Transmitted intensity, $I_0 = Unpolarized incident intensity$ W/m2
Averaged over all azimuth angles \theta (since average of \cos^2 \theta is 1/2)
Law of Malus for Amplitude
$$E = E_0 \cos \theta$$
E = Transmitted electric field amplitude, $E_0 = Incident amplitude$ V/m
For linear polarization
Graphs and experiments
Malus's Law Verification Graph
Draw: A plot of transmitted intensity (I) versus \cos^2 \theta (or angle \theta showing a symmetric cosine-squared wave curve between 0 and 360 degrees).
Interpret: The linear straight-line plot of I versus \cos^2 \theta confirms the direct proportionality predicted by Malus's law.
Exam use: Tested via conceptual MCQs regarding maximum and minimum positions when rotating an analyzer.
Polarization by Reflection (Brewster's Angle setup)
Draw: A ray diagram showing unpolarized light incident on a glass slab at ip, with reflected ray completely polarized (perpendicular dots) and refracted ray partially polarized, making a 90-degree angle between them.
Interpret: Visualizes the physical condition where reflected and refracted rays are orthogonal at the polarizing angle.
Exam use: Directly asked in TGT/PGT theoretical questions about the angle between reflected and refracted beams.
Nicol Prism Construction and Action
Draw: A calcite crystal cut diagonally and cemented with Canada balsam, showing total internal reflection of the ordinary ray and transmission of the extraordinary ray.
Interpret: Demonstrates how a single polarizing device isolates a pure extraordinary plane-polarized beam.
Exam use: Question source for identifying crystal cuts and optical cement (Canada balsam refractive index properties).
Solved numericals
TGT core
Calculate the polarizing angle for glass of refractive index 1.5.
Recall Brewster's law: $\mu = \tan i_p$
Substitute the given value: \tan $i_p = 1.5$
Determine the angle: $i_p = \arctan(1.5)$
Answer: i_p \approx 56.3\deg
PGT extension
Unpolarized light of intensity 32 W/m2 is incident on a polarizer. The transmitted light falls on a second polarizer (analyzer) whose transmission axis is oriented at 60 degrees to the first polarizer. Find the final transmitted intensity.
When unpolarized light passes through the first polarizer, its intensity is reduced to half: $I_1 = \frac{1}{2} I_0 = \frac{1}{2} \times 32 = 16 \text{ W/m}^2$.
Apply Malus's law for the second polarizer: $I_2 = I_1 \cos^2 \theta$
A quarter-wave plate is designed for sodium light of wavelength 589 nm. If the principal refractive indices are $n_e = 1.553$ and $n_o = 1.544$, calculate the minimum thickness of the plate.
Use the formula for the thickness of a quarter-wave plate: $t = \frac{\lambda}{4(n_e - n_o)}$
Substitute the given values: $\lambda = 589 \times 10^{-9} \text{ m}$, $n_e = 1.553$, $n_o = 1.544$
Ideal polarizers are assumed to have 100% transmission for parallel electric vector components and 0% for perpendicular components (infinite extinction ratio).
Linear optical media are assumed where refractive indices are independent of light intensity.
Birefringent plates assume uniform thickness and homogeneous crystal structure throughout the active aperture.
Common misconceptions
Common error: Longitudinal waves like sound waves can be polarized.
Correct: Only transverse waves can be polarized.
Polarization requires oscillations to occur perpendicular to the direction of propagation, which is impossible in longitudinal compression waves.
Common error: Unpolarized light has no electric field components.
Correct: Unpolarized light has rapid, randomly fluctuating electric field components in all directions perpendicular to the ray.
Zero time-average value across all azimuths does not mean the instantaneous field vector is zero.
Common error: The angle between the reflected and refracted rays at any incidence angle is 90 degrees.
Correct: The reflected and refracted rays are perpendicular only when incidence is at Brewster's polarizing angle.
Snell's law dictates that r varies with i, and orthogonality is a special geometric consequence unique to $i = i_p$.
Quick revision
Polarization proves the transverse nature of light waves.
Brewster's Law: $\mu = \tan i_p with reflected$ and refracted rays mutually perpendicular.
Malus's Law: $I = I_0 \cos^2 \theta$.
Unpolarized light passing through an ideal polarizer drops to exactly half its initial intensity.
Calcite crystal exhibits double refraction producing O-ray and E-ray.
Quarter-wave plate introduces a phase difference of \pi/2; half-wave plate introduces \pi.
Must remember
Only transverse waves exhibit polarization; longitudinal waves cannot be polarized.
According to Brewster's law, the tangent of the polarizing angle is equal to the refractive index of the medium ($\mu = \tan i_p$).
At Brewster's angle, the reflected and refracted rays are mutually perpendicular.
Unpolarized light passing through an ideal linear polarizer has its intensity reduced by exactly half ($I = \frac{1}{2} I_0$).
Birefringent crystals like calcite split unpolarized light into an Ordinary ray (O-ray) and an Extraordinary ray (E-ray).
A quarter-wave plate introduces a phase difference of $\pi/2$ or a path difference of $\lambda/4$.
A half-wave plate introduces a phase difference of $\pi$ or a path difference of $\lambda/2$.
Optical rotation angle is directly proportional to the path length and concentration of the active solution ($\theta = S \cdot l \cdot c$).
Glossary
Plane-Polarized LightLight in which the electric field vector is restricted to vibrate in a single fixed plane perpendicular to the direction of propagation.
Brewster's Angle ($i_p$)The angle of incidence at which reflected light is completely plane-polarized.
Ordinary Ray (O-ray)A refracted ray in a birefringent crystal that obeys ordinary laws of refraction and travels with 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 refractive index dependent on the angle of incidence.
Specific Rotation ($S$)The rotation of the plane of polarization produced by a solution of unit length (1 dm) and unit concentration (1 g/cm$^3$) at a specified temperature.
Malus's LawLaw stating that the transmitted intensity of polarized light through an analyzer varies as the square of the cosine of the angle between their transmission axes ($I = I_0 \cos^2 \theta$).
Ordinary Ray vs. Extraordinary Ray
Parameter
Ordinary Ray (O-ray)
Extraordinary Ray (E-ray)
Snell's Law
Obeys Snell's law strictly
Does not obey Snell's law
Velocity
Constant in all directions in the crystal
Varies with the direction of propagation
Refractive Index
Constant ($n_o$)
Variable ($n_e$ depending on direction)
Vibration Direction
Perpendicular to the principal plane
Parallel to the principal plane
Quarter-Wave Plate vs. Half-Wave Plate
Parameter
Quarter-Wave Plate (QWP)
Half-Wave Plate (HWP)
Path Difference
$\lambda / 4$
$\lambda / 2$
Phase Difference
$\pi / 2$ radians
$\pi$ radians
Thickness ($t$)
$\frac{\lambda}{4(n_e - n_o)}$
$\frac{\lambda}{2(n_e - n_o)}$
Primary Function
Produces circular/elliptical polarization from linear
Rotates the plane of polarization of linearly polarized light
Practice Quiz
Choose an option to reveal the explanation.
1. Which of the following types of waves cannot exhibit the phenomenon of polarization?
Explanation: Polarization is exclusively a property of transverse waves where oscillations occur perpendicular to the direction of propagation. Sound waves in air are longitudinal compression waves and cannot be polarized.
2. According to Brewster's law, when unpolarized light is incident at the polarizing angle $i_p$ on a transparent medium of refractive index $\mu$, the angle between the reflected and refracted rays is:
Explanation: At Brewster's polarizing 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 $\sqrt{3}$, what is its Brewster's polarizing angle?
Explanation: According to Brewster's law, $\mu = \tan i_p$. Given $\mu = \sqrt{3}$, we have $\tan i_p = \sqrt{3}$, which yields $i_p = 60^\circ$.
4. Unpolarized light of intensity $I_0$ is incident on a linear polaroid. What is the intensity of the transmitted light?
Explanation: Unpolarized light has equal intensity components in all directions. Averaging $\cos^2 \theta$ over all azimuth angles yields $1/2$, so the transmitted intensity is $I = \frac{1}{2} I_0$.
5. Malus's law is represented by the mathematical relation:
Explanation: Malus's law states that when completely plane-polarized light is incident on an analyzer, the transmitted intensity $I$ varies directly as the square of the cosine of the angle $\theta$ between their transmission axes: $I = I_0 \cos^2 \theta$.
6. Which ray in a doubly refracting uniaxial crystal obeys the ordinary laws of refraction (Snell's law)?
Explanation: The Ordinary ray (O-ray) obeys Snell's law of refraction and travels with the exact same speed in all directions within a birefringent crystal.
7. The thickness of a quarter-wave plate for a wavelength $\lambda$ and principal refractive indices $n_e$ and $n_o$ is given by:
Explanation: A quarter-wave plate introduces a path difference of $\lambda/4$, which corresponds to a thickness $t = \frac{\lambda}{4(n_e - n_o)}$.CORRECT.
8. Optical activity in a solution depends on which of the following factors?
Explanation: The angle of optical rotation is directly proportional to both the path length ($l$) and the concentration ($c$) of the active solution: $\theta = S \cdot l \cdot c$.
9. If two crossed polarizers have their transmission axes perpendicular to each other ($\theta = 90^\circ$), the intensity of light transmitted through the second polarizer (analyzer) when illuminated by unpolarized light is:
Explanation: Unpolarized light becomes polarized at $I_0 / 2$ after the first polarizer. When incident on a crossed analyzer at $90^\circ$, Malus's law gives $I = (I_0 / 2) \cos^2(90^\circ) = 0$.
10. Which of the following substances is optically active?
Explanation: Sugar solution is a classic example of an optically active substance that rotates the plane of polarization of linearly polarized light.
11. What is the phase difference introduced by a half-wave plate between the ordinary and extraordinary rays?
Explanation: A half-wave plate introduces a path difference of $\lambda/2$, which corresponds to a phase difference of $\pi$ radians.
12. The specific rotation $S$ of a solution is measured for a path length of:
Explanation: Specific rotation is defined as the rotation produced by a solution of length 1 decimetre ($1 \text{ dm}$) and unit concentration at a specified temperature.
13. Which phenomenon confirms the transverse nature of light waves?
Explanation: Polarization uniquely demonstrates that light vibrations occur transverse to the direction of propagation, as longitudinal waves cannot be polarized.
14. In a uniaxial birefringent crystal, the extraordinary ray (E-ray):
Explanation: Unlike the O-ray, the E-ray does not obey Snell's law and its effective refractive index varies depending on the direction of propagation inside the crystal.
15. If the angle between the transmission axes of a polarizer and an analyzer is $60^\circ$, what fraction of the incident polarized light intensity is transmitted?
Explanation: According to Malus's law, $I = I_0 \cos^2(60^\circ) = I_0 (1/2)^2 = I_0 / 4$. Thus, a fraction of $1/4$ is transmitted.
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. Unpolarized light passes through two polaroids placed with their pass axes at an angle of $45^\circ$ to each other. If the initial intensity is $I_0$, the final transmitted intensity is:
Explanation: After the first polaroid, intensity becomes $I_1 = I_0 / 2$. Applying Malus's law at the second polaroid: $I_2 = I_1 \cos^2(45^\circ) = (I_0 / 2) \times (1/2) = I_0 / 4$.
3. The angle between the reflected and refracted rays when sunlight is reflected from a lake's surface as completely plane-polarized light is:
Explanation: When light is reflected as completely plane-polarized, reflection occurs at Brewster's angle where the reflected and refracted rays are at right angles ($90^\circ$) to each other.
4. A half-wave plate is designed for a wavelength of $600 \text{ nm}$. What is the path difference introduced by this plate for this wavelength?
Explanation: A half-wave plate introduces a path difference of $\lambda/2$. For $\lambda = 600 \text{ nm}$, path difference = $600 / 2 = 300 \text{ nm}$.
5. If the specific rotation of cane sugar is $66^\circ \text{ dm}^{-1} (\text{g/cm}^3)^{-1}$, the rotation produced by $20 \text{ cm}$ of sugar solution of concentration $0.05 \text{ g/cm}^3$ is:
6. Which of the following statements regarding the ordinary and extraordinary rays in a calcite crystal is correct?
Explanation: In uniaxial birefringent crystals, the O-ray travels with the same speed in all directions, whereas the E-ray travels with varying speeds depending on the direction of propagation.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. The refractive index of a medium is $1.5$. The polarizing angle for this medium is given by:
Explanation: From Brewster's law, $\mu = \tan i_p$, which implies $i_p = \tan^{-1}(\mu) = \tan^{-1}(1.5)$.
2. What is the electric field vibration characteristic of plane-polarized light?
Explanation: Plane-polarized light restricts its electric field vector $E$ to a single fixed plane throughout propagation.
3. Unpolarized light of intensity $16 \text{ W/m}^2$ is incident on a polarizer. The transmitted intensity is:
Explanation: When unpolarized light passes through a single polarizer, its intensity is reduced by half: $I = \frac{1}{2} I_0 = \frac{1}{2} \times 16 = 8 \text{ W/m}^2$.
4. The phase difference corresponding to a path difference of $\lambda / 4$ is:
5. Which crystal is commonly used to produce plane-polarized light using double refraction and total internal reflection (Nicol prism)?
Explanation: A Nicol prism is constructed from a calcite crystal cut in a specific manner to isolate the extraordinary ray by total internal reflection of the ordinary ray.
6. If the angle between the pass axes of polaroid P1 and P2 is $30^\circ$, the ratio of transmitted intensity to incident polarized intensity is:
Explanation: Light waves are electromagnetic transverse waves and can be polarized. Sound and pressure waves are longitudinal.
8. The specific rotation of an optically active compound depends upon:
Explanation: Specific rotation is a characteristic property defined for a given wavelength of light and a specified temperature.
9. A quarter-wave plate is placed between crossed polarizers with its optic axis at $45^\circ$ to the polarizer transmission axis. The resulting light is:
Explanation: When linearly polarized light passes through a quarter-wave plate oriented at $45^\circ$ to its optic axis, the orthogonal components have equal amplitudes with a $\pi/2$ phase difference, yielding circularly polarized light.
10. If the refractive indices for ordinary and extraordinary rays in a quartz crystal are $n_o$ and $n_e$ respectively, the condition for a positive uniaxial crystal is:
Explanation: In a positive uniaxial crystal like quartz, the extraordinary refractive index is greater than the ordinary refractive index ($n_e > n_o$).CORRECT.