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Optics
Linear Polarization of Light
Linear polarization is a phenomenon of transverse waves where the electric field vector is restricted to vibrate along a single fixed plane containing the direction of propagation. It provides conclusive evidence of the transverse nature of light waves.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Basic definition of unpolarized and polarized light, Malus's Law, Brewster's Law, polarization by reflection and simple polarizers.
PGT extension
Analytical treatment of electromagnetic waves, Jones calculus representation, double refraction in uniaxial crystals, quarter and half-wave plates, and optical activity.
Prerequisites
Transverse wave kinematics
Maxwell's electromagnetic wave equations
Phase difference and superposition of simple harmonic motions
Nature of Unpolarized and Polarized Light
Unpolarized light consists of rapidly and randomly changing electric field orientations perpendicular to the propagation vector with equal probability.
Linearly polarized light maintains a constant direction of the electric field vector throughout its propagation.
The intensity of unpolarized light passing through an ideal linear polarizer is reduced to exactly half its initial value.
Polarization by Reflection and Brewster's Law
When unpolarized light strikes a dielectric boundary at a specific angle called the polarizing angle or Brewster's angle, 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.
The tangent of the polarizing angle is numerically equal to the relative refractive index of the medium.
Malus's Law
Malus's Law governs the transmission of linearly polarized light through an analyzer.
The transmitted intensity varies directly as the square of the cosine of the angle between the transmission axes of the polarizer and the analyzer.
Maximum transmission occurs at zero degrees, and complete extinction occurs at ninety degrees.
Double Refraction and Birefringence
Anisotropic crystals like calcite split an unpolarized incident ray into two refracted rays: the ordinary (O) ray and the extraordinary (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 ordinary Snell's law and its velocity varies with the direction of propagation inside the crystal.
Derivations
Brewster's Law Derivation
Conditions: Unpolarized light incident on a transparent dielectric boundary at polarizing angle ip
Let ip be the angle of incidence (Brewster's angle) and r be the angle of refraction.
According to Snell's Law, $n = \frac{\sin i_p}{\sin r}$.
At Brewster's angle, the reflected ray and refracted ray are perpendicular, so i_p + $r = 90^{\circ}$, which gives $r = 90^{\circ} - i_p$.
Substitute r into Snell's Law: $n = \frac{\sin i_p}{\sin(90^{\circ} - i_p)} = \frac{\sin i_p}{\cos i_p} = \tan i_p$.
Result: $n = \tan i_p$
Malus's Law Derivation
Conditions: Plane polarized light of amplitude A incident on an analyzer with angle \theta between their transmission axes
Let the electric field vector of the incident polarized light be represented by E = A \sin(\omega t).
The analyzer resolves this vector into two rectangular components: A \cos\theta parallel to the transmission axis and A \sin\theta perpendicular to it.
Only the component parallel to the transmission axis, A \cos\theta, is transmitted by the analyzer.
Since light intensity is proportional to the square of the amplitude (I \propto A^{2}), the transmitted intensity I is given by $I = I_0 \cos^2\theta$, where I_0 \propto A^{2}.
Result: $I = I_0 \cos^2\theta$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Malus's Law
$$I = I_0 \cos^2\theta$$
I = transmitted intensity, $I_0 = initial intensity$, \theta = angle between polarizer and analyzer axes W/m^2
Ideal linear polarizer and analyzer
Brewster's Law
$$n = \tan i_p$$
n = refractive index of the medium, $i_p = polarizing angle$ Dimensionless
$I_0 = initial intensity$, \theta_1 and \thet$a_2 = orientations of successive polarizers$ W/m^2
Sequential transmission
Graphs and experiments
Malus's Law Verification Graph
Draw: A plot of transmitted intensity I versus \cos^2\theta showing a straight line passing through the origin, or I versus \theta showing a symmetric cosine-squared wave curve.
Interpret: Confirms the quadratic dependence of transmitted light intensity on the cosine of the angle between polarizer transmission axes.
Exam use: Numerical and conceptual identification of intensity variations during polarizer rotation.
Polarization by Reflection Setup
Draw: Unpolarized ray incident on a glass slab at angle i_p, showing a reflected ray at angle i_p perpendicular to a refracted ray at angle r.
Interpret: Illustrates Brewster's condition where reflected light is completely plane-polarized with electric vectors perpendicular to the plane of incidence.
Exam use: Answering direct theoretical questions on angle between reflected and refracted rays.
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: Demonstrates how a Nicol prism functions as an effective polarizer and analyzer by eliminating the ordinary ray.
Exam use: Questions regarding optical components used to produce and analyze polarized light.
Solved numericals
TGT core
Unpolarized light of intensity 32 W/m^2 is incident on a polarizer. The transmitted light is then passed through an analyzer whose transmission axis makes an angle of 60 degrees with the polarizer axis. Find the final transmitted intensity.
Intensity after passing through the first polarizer: $I_1 = \frac{1}{2} I_0 = \frac{1}{2} \times 32 = 16 \text{ W/m}^{2}\text{}$.
Apply Malus's Law for the analyzer: $I_2 = I_1 \cos^2\theta$.
Answer: Polarizing angl$e = 58.78^{\circ}$, Angle of refractio$n = 31.22^{\circ}$
PGT extension
A quarter-wave plate is designed for light of wavelength 589 nm. If the ordinary and extraordinary refractive indices for the crystal are 1.544 and 553 respectively (take \Delta n = 0.009), find the minimum thickness of the plate.
Formula for quarter-wave plate thickness: $t = \frac{\lambda}{4(n_e - n_o)}$.
Given $\lambda = 589 \times 10^{-9} \text{ m}$ and |n_e - $n_o| = 0.009$.
Answer: 1.636 \times 10^{-5} \text{ m (or 16.36 } \mu\text{m)}
Assumptions and limitations
Ideal polarizers are assumed to absorb 100 percent of the perpendicular component and transmit 100 percent of the parallel component without internal scattering or absorption losses.
Brewster's law assumes isotropic dielectric media where magnetic permeability is approximately equal to that of free space (mu \approx mu_0).
Common misconceptions
Common error: Sound waves can be polarized just like light waves.
Correct: Sound waves are longitudinal and cannot be polarized.
Polarization is exclusively a property of transverse waves where vibrations occur perpendicular to the direction of propagation.
Common error: The intensity of unpolarized light becomes zero after passing through a single ideal polarizer.
Correct: The intensity becomes exactly half of the initial unpolarized intensity.
An ideal polarizer averages out the random transverse electric field components, transmitting half of the total incident intensity.
Common error: At Brewster's angle, the reflected and refracted rays are parallel.
Correct: At Brewster's angle, the reflected and refracted rays are strictly perpendicular to each other.
The geometric condition derived from Snell's law and boundary conditions proves that i_p + r = 90 degrees.
Quick revision
Unpolarized light intensity drops to half through a single ideal polarizer: $I = I_0 / 2$.
Malus's Law: $I = I_0 \cos^2\theta$.
Brewster's Law: $n = \tan i_p$, with reflected and refracted rays at 90 degrees.
A quarter-wave plate introduces a phase difference of \pi / 2 radians (path difference \lambda / 4).
A half-wave plate introduces a phase difference of \pi radians (path difference \lambda / 2).
Must remember
The intensity of unpolarized light passing through an ideal linear polarizer is reduced to exactly half its initial value: $I = \frac{1}{2} I_0$.
Brewster's Law states that the tangent of the polarizing angle equals the relative refractive index of the medium: $n = \tan i_p$.
At Brewster's angle, the reflected ray and the refracted ray are mutually perpendicular: $i_p + r = 90^\circ$.
Malus's Law dictates that transmitted intensity through an analyzer varies as $I = I_0 \cos^2\theta$.
Uniaxial anisotropic crystals split unpolarized light into two rays: the Ordinary (O) ray and the Extraordinary (E) ray.
A Quarter-Wave Plate introduces a phase difference of $\frac{\pi}{2}$ radians, with thickness $t = \frac{\lambda}{4(n_e - n_o)}$.
A Half-Wave Plate introduces a phase difference of $\pi$ radians, with thickness $t = \frac{\lambda}{2(n_e - n_o)}$.
Sound waves are longitudinal waves and cannot be polarized, whereas light waves are transverse electromagnetic waves.
Glossary
Unpolarized LightLight consisting of rapidly and randomly changing electric field orientations perpendicular to the propagation vector with equal probability.
Linearly Polarized LightLight that maintains a constant direction of the electric field vector throughout its propagation.
Brewster's Angle ($i_p$)The specific angle of incidence at which unpolarized light reflected from a dielectric boundary is completely plane-polarized.
Malus's LawThe physical law governing the transmission of linearly polarized light through an analyzer, given by $I = I_0 \cos^2\theta$.
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 ordinary Snell's law and whose velocity varies with the direction of propagation.
BirefringenceDouble refraction exhibited by anisotropic crystals like calcite, splitting an unpolarized ray into two orthogonally polarized rays.
Ordinary (O) Ray vs. Extraordinary (E) Ray
Parameter
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 ($n_o$)
Direction-dependent ($n_e$)
Velocity
Constant in all directions
Varies with propagation direction inside crystal
Quarter-Wave Plate vs. Half-Wave Plate
Parameter
Quarter-Wave Plate (QWP)
Half-Wave Plate (HWP)
Phase Difference ($\delta$)
$\frac{\pi}{2}$ radians ($90^\circ$)
$\pi$ radians ($180^\circ$)
Thickness Formula ($t$)
$t = \frac{\lambda}{4(n_e - n_o)}$
$t = \frac{\lambda}{2(n_e - n_o)}$
Function
Converts linear polarization to circular/elliptic
Rotates plane of polarization of linearly polarized light
Practice Quiz
Choose an option to reveal the explanation.
1. What is the intensity of unpolarized light after passing through an ideal linear polarizer with initial intensity $I_0$?
Explanation: An ideal polarizer averages out the random transverse electric field components of unpolarized light, transmitting exactly half of the total incident intensity ($I = \frac{1}{2} I_0$).
2. Which of the following waves cannot exhibit polarization?
Explanation: Sound waves are longitudinal waves where particle displacement occurs parallel to the propagation direction, hence they cannot be polarized. Polarization is exclusive to transverse waves.
3. According to Brewster's law, the tangent of the polarizing angle $i_p$ is equal to:
Explanation: Brewster's Law is mathematically expressed as $n = \tan i_p$, where $n$ is the refractive index of the dielectric medium relative to the surrounding medium.
4. At Brewster's angle, what is the geometric relationship between the reflected ray and the refracted (transmitted) ray?
Explanation: At Brewster's angle, the reflected ray and refracted ray are mutually perpendicular ($i_p + r = 90^\circ$). Therefore, the angle between them is $90^\circ$.
5. Malus's Law states that the transmitted intensity $I$ through an analyzer varies with the angle $\theta$ between the transmission axes as:
Explanation: Malus's Law governs analyzer transmission and states that $I = I_0 \cos^2\theta$, where $\theta$ is the angle between the transmission axes of the polarizer and analyzer.
6. Unpolarized light of intensity $I_0$ passes through two ideal polarizers whose transmission axes are parallel ($\theta = 0^\circ$). The final transmitted intensity is:
Explanation: The first polarizer reduces unpolarized intensity to $\frac{1}{2} I_0$ and polarizes it linearly. Since the second polarizer is parallel ($\theta = 0^\circ$), $\cos^2(0^\circ) = 1$, so the final intensity remains $\frac{1}{2} I_0$.
7. What happens to the transmitted intensity of linearly polarized light when the angle between the polarizer and analyzer is set to $90^\circ$?
Explanation: By Malus's Law, when $\theta = 90^\circ$, $\cos(90^\circ) = 0$, leading to complete extinction ($I = 0$). Crossed polarizers block all transmitted light.
8. Anisotropic crystals like calcite split an unpolarized incident ray into two refracted rays known as:
Explanation: Double refraction (birefringence) in anisotropic crystals splits unpolarized light into the Ordinary (O) ray and the Extraordinary (E) ray, which are polarized in mutually perpendicular planes.
9. Which of the following statements is correct regarding the Ordinary (O) ray inside a uniaxial birefringent crystal?
Explanation: The O-ray obeys ordinary laws of refraction (Snell's law) and travels with the same constant speed ($c/n_o$) in all directions within the crystal.
10. What is the phase retardation introduced by a Quarter-Wave Plate (QWP)?
Explanation: A Quarter-Wave Plate introduces a phase difference (retardation) of $\frac{\pi}{2}$ radians ($90^\circ$) between the ordinary and extraordinary wave components.
11. What is the correct formula for the thickness $t$ of a Quarter-Wave Plate?
Explanation: Equating the phase retardation formula $\delta = \frac{2\pi}{\lambda}(n_e - n_o)t = \frac{\pi}{2}$ gives the thickness of a quarter-wave plate as $t = \frac{\lambda}{4(n_e - n_o)}.
12. A Half-Wave Plate introduces a phase shift of:
Explanation: A Half-Wave Plate introduces a phase shift of $\pi$ radians ($180^\circ$) between the O-ray and E-ray components.
13. If the refractive index of a transparent medium is $\sqrt{3}$, what is its Brewster's angle $i_p$?
Explanation: Using Brewster's law $n = \tan i_p$, we have $\tan i_p = \sqrt{3}$, which implies $i_p = \tan^{-1}(\sqrt{3}) = 60^\circ$.
14. When unpolarized light is incident on a transparent glass slab at Brewster's angle, the reflected light is:
Explanation: When unpolarized light strikes a dielectric boundary at Brewster's angle, the reflected ray is completely plane-polarized with its electric vector perpendicular to the plane of incidence.
15. What is the degree of polarization $V$ for unpolarized light?
Explanation: For unpolarized light, $I_{max} = I_{min}$, so the degree of polarization $V = \frac{I_{max} - I_{min}}{I_{max} + I_{min}} = 0$. For completely polarized light, $V = 1$.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. Unpolarized light passes through three successive linear polarizers. The transmission axis of the second polarizer is at $45^\circ$ to the first, and the third is at $90^\circ$ to the first. If initial intensity is $I_0$, the final transmitted intensity is:
Explanation: After the first polarizer, $I_1 = \frac{1}{2} I_0$. After the second polarizer at $\theta_1 = 45^\circ$, $I_2 = I_1 \cos^2(45^\circ) = (\frac{1}{2} I_0)(\frac{1}{2}) = \frac{1}{4} I_0$. After the third polarizer at $\theta_2 = 90^\circ$ relative to the first (so angle with second is $45^\circ$), $I_3 = I_2 \cos^2(45^\circ) = (\frac{1}{4} I_0)(\frac{1}{2}) = \frac{1}{8} I_0$.
2. The refractive index of glass is $1.5$. A ray of light is incident on the glass surface at Brewster's angle. The angle of refraction $r$ is:
Explanation: At Brewster's angle, $i_p + r = 90^\circ$. Since $\tan i_p = n = 1.5$, we have $i_p = \tan^{-1}(1.5)$. Thus, the angle of refraction $r = 90^\circ - i_p = 90^\circ - \tan^{-1}(1.5)$.
3. If the intensity of polarized light transmitted through an analyzer is reduced to one-fourth of its maximum value, the angle between the transmission axes of the polarizer and analyzer is:
Explanation: By Malus's Law, $I = I_0 \cos^2\theta$. Given $I = \frac{1}{4} I_0$, we get $\cos^2\theta = \frac{1}{4}$, which means $\cos\theta = \frac{1}{2}$. Therefore, $\theta = 60^\circ$.
4. A calcite crystal is used to view a dot. Two images (O-ray and E-ray) are observed. As the crystal is rotated about the incident line of sight:
Explanation: The O-ray obeys ordinary refractive laws and forms a stationary image. The E-ray's refractive index varies with crystal orientation, causing its image to rotate in a circle around the stationary O-ray image when the crystal is rotated.
5. What minimum thickness of a quarter-wave plate is required for sodium light of wavelength $\lambda = 589$ nm, given $n_o = 1.544$ and $n_e = 1.553$?
6. Which of the following optical devices can convert unpolarized light directly into circularly polarized light?
Explanation: Unpolarized light passing through a linear polarizer becomes linearly polarized. When this linearly polarized light passes through a quarter-wave plate oriented at $45^\circ$ to its transmission axis, the orthogonal components of equal amplitude acquire a $\frac{\pi}{2}$ phase difference, resulting in circularly polarized light.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. If unpolarized light of intensity $100 \text{ W/m}^2$ is incident on a pair of ideal polarizers with their axes at an angle of $60^\circ$ to each other, what is the transmitted intensity?
Explanation: First polarizer reduces intensity to $\frac{100}{2} = 50 \text{ W/m}^2$. By Malus's Law, second polarizer transmits $I = I_0 \cos^2(60^\circ) = 50 \times (1/2)^2 = 50 \times \frac{1}{4} = 12.5 \text{ W/m}^2$.
2. The Brewster angle for a water-air interface is approximately $53^\circ$. What is the refractive index of water?
Explanation: By Brewster's Law, $n = \tan i_p$. Given $i_p \approx 53^\circ$, the refractive index of water is $\tan(53^\circ)$.
3. In a birefringent crystal, the extraordinary ray (E-ray):
Explanation: Unlike the O-ray, the E-ray's wavefront is spheroidal rather than spherical, meaning its velocity and effective refractive index vary depending on the propagation angle relative to the optic axis.
4. Two polarizers are crossed ($90^\circ$). Unpolarized light of intensity $I_0$ is incident on the first polarizer. What is the final transmitted intensity?
Explanation: The first polarizer transmits $\frac{1}{2} I_0$ and polarizes it. Since the second polarizer is crossed at $90^\circ$ to the first, $\cos^2(90^\circ) = 0$, resulting in zero transmitted light.
5. What is the ratio of transmitted intensities $I_1/I_2$ when a beam of linearly polarized light passes through an analyzer set at angles of $30^\circ$ and $60^\circ$ respectively to the polarizer axis?
6. Which of the following phenomena provides direct experimental evidence that light waves are transverse?
Explanation: Interference and diffraction can be exhibited by both longitudinal and transverse waves. Polarization, however, requires vibrations perpendicular to propagation, directly proving the transverse nature of light.
7. For a positive uniaxial crystal, how do the ordinary and extraordinary refractive indices compare?
Explanation: In positive uniaxial crystals (like quartz), the extraordinary refractive index is greater than the ordinary refractive index ($n_e > n_o$), meaning the O-ray travels faster than the E-ray.
8. A half-wave plate is placed between two crossed polarizers with its optic axis at $45^\circ$ to the transmission axis of the first polarizer. The combination will:
Explanation: A half-wave plate rotates the plane of polarization of linearly polarized light by an angle $2\theta$. If the optic axis is at $45^\circ$ to the incident polarization, the plane of polarization rotates by $90^\circ$, aligning it with the second polarizer and allowing maximum transmission.
9. At what angle of incidence is sunlight reflected from the smooth surface of a calm lake completely polarized, given $n = 1.33$ for water?
Explanation: Complete polarization occurs at Brewster's angle, given by $i_p = \tan^{-1}(n) = \tan^{-1}(1.33) \approx 53.1^\circ$.
10. If the angle between the transmission axes of a polarizer and an analyzer is $\theta$, and the transmitted intensity is one-third of the incident polarized intensity, then $\theta$ is equal to:
Explanation: By Malus's Law, $I = I_0 \cos^2\theta$. Given $I = \frac{1}{3} I_0$, we have $\cos^2\theta = \frac{1}{3}$, which yields $\cos\theta = \frac{1}{\sqrt{3}}$, so $\theta = \cos^{-1}(\frac{1}{\sqrt{3}}).$