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Elliptical Polarization of Light

Elliptical polarization is a state of optical polarization in which the electric field vector traces out an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. It arises when two mutually orthogonal linearly polarized components of unequal amplitudes are superposed with a phase difference that is neither 0, pi, nor an integer multiple of pi.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Understanding the qualitative definitions of unpolarized, linearly, circularly, and elliptically polarized light; action of polarizers and quarter-wave plates; Malus's Law.

PGT extension

Rigorous derivation of the general equation of polarization ellipse from two orthogonal harmonic waves; conditions on amplitudes and phase differences for elliptical polarization; Stokes parameters and Jones calculus basics.

Prerequisites

Superposition principle of orthogonal simple harmonic motions
Transverse nature of light waves
Action of doubly refracting crystals (uniaxial crystals, ordinary and extraordinary rays)

Fundamental Nature of Elliptical Polarization

The electric field vector E has two orthogonal components, E_x and E_y, oscillating in mutually perpendicular planes containing the propagation direction z.
The tip of the electric field vector traces out an ellipse as a function of time at any given point in space.
It represents the most general state of polarization for a monochromatic plane wave, encompassing linear and circular polarization as special cases.

Conditions for Elliptical Polarization

The amplitudes of the two orthogonal components, E_0x and E_0y, must generally be unequal (though unequal amplitudes are not strictly required if the phase difference is not 0, pi/2, etc.).
The phase difference delt$a = \phi_{y} - \phi_{x} between the two components must be different from 0$, pi, 2pi, etc. (which yield linear polarization) and different from +/- pi/2, +/- 3pi/2, etc. with equal amplitudes (which yield circular polarization).
The orientation of the ellipse's axes is inclined at an angle to the coordinate axes depending on the amplitude ratio and phase difference.

Handedness or Chirality

Elliptical polarization can be either right-handed ( dextro-rotatory) or left-handed (laevo-rotatory).
Right-handed elliptical polarization corresponds to the electric vector rotating clockwise when viewed looking into the oncoming light beam.
Left-handed elliptical polarization corresponds to a counter-clockwise rotation of the electric vector.

Production and Detection

Elliptically polarized light is produced when linearly polarized light is incident on a quarter-wave plate (or any arbitrary retarder) with its vibration azimuth oriented at an angle other than 0, pi/2, or multiples of pi/4 relative to the optic axis.
It can be detected using a combination of a quarter-wave plate and an analyzer (polarizer), where rotation of the analyzer yields varying intensity without ever reaching complete extinction.
The minimum and maximum transmitted intensities through a rotating analyzer correspond to the squares of the semi-minor and semi-major axes of the polarization ellipse.

Derivations

Derivation of the General Polarization Ellipse

Conditions: Two orthogonal harmonic waves propagating along the z-axis with arbitrary amplitudes and phase difference delta.

Write the orthogonal electric field components: $E_x = E_{0x} \cos(\omega t - kz)$ and $E_y = E_{0y} \cos(\omega t - kz + \delta)$.
Expand the second equation using trigonometric identity: E_y / $E_{0y} = \cos(\omega t - kz)\cos\delta - \sin(\omega t - kz)\sin\delta$.
Express cos(omega t - kz) as E_x / E_{0x} and sin(omega t - kz) as \sqrt{1 - (E_x / E_{0x})^2}.
Substitute these into the expanded E_y equation, isolate the radical term, square both sides, and rearrange into the standard quadratic form: \frac{E_x^2}{E_{0x}^2} + \frac{E_y^2}{E_{0y}^2} - 2\frac{E_x E_y}{E_{0x} E_{0y}}\cos$\delta = \sin^2\delta$.

Result: The general equation of an ellipse rotated relative to the coordinate axes: \frac{E_x^2}{E_{0x}^2} + \frac{E_y^2}{E_{0y}^2} - 2\frac{E_x E_y}{E_{0x} E_{0y}}\cos$\delta = \sin^2\delta$

Orientation Angle of the Polarization Ellipse

Conditions: Relating the angle psi between the major axis of the ellipse and the x-axis.

Using the general ellipse equation, the orientation angle psi of the major axis with the x-axis is given by the coordinate transformation relation.
Apply the condition for rotation of axes to eliminate the cross-term E_x E_y.
Obtain the relation involving the amplitudes E_{0x}, E_{0y} and the phase difference delta.
Resulting formula: \tan(2\psi) = \frac{2 E_{0x} E_{0y}}{E_{0x}^2 - E_{0y}^2} \cos\delta.

Result: \tan(2\psi) = \frac{2 E_{0x} E_{0y}}{E_{0x}^2 - E_{0y}^2} \cos\delta

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Orthogonal X-Component of Electric Field	$$E_x(z,t) = E_{0x} \cos(\omega t - kz)$$	E_{0x}: Amplitude along x, omega: Angular frequency, k: Propagation constant V/m	Plane wave propagating along z
Orthogonal Y-Component of Electric Field	$$E_y(z,t) = E_{0y} \cos(\omega t - kz + \delta)$$	E_{0y}: Amplitude along y, delta: Phase difference V/m	Plane wave propagating along z
General Equation of Polarization Ellipse	$$\frac{E_x^2}{E_{0x}^2} + \frac{E_y^2}{E_{0y}^2} - 2\frac{E_x E_y}{E_{0x} E_{0y}}\cos\delta = \sin^2\delta$$	E_x, E_y: Instantaneous field components, E_{0x}, E_{0y}: Maximum amplitudes, delta: Phase difference V^2/m^2	Monochromatic wave
Orientation of Major Axis	$$\tan(2\psi) = \frac{2 E_{0x} E_{0y}}{E_{0x}^2 - E_{0y}^2} \cos\delta$$	psi: Angle of major axis with x-axis Radians or degrees	General elliptical state
Ellipticity Parameter	$$\tan\epsilon = \pm \frac{b}{a}$$	a, b: Semi-major and semi-minor axes of the ellipse Dimensionless	-pi/4 <= epsilon <= pi/4
Maximum Intensity	$$I_{max} \propto a^2$$	a: Semi-major axis of the polarization ellipse W/m^2	Measured through a rotating analyzer
Minimum Intensity	$$I_{min} \propto b^2$$	b: Semi-minor axis of the polarization ellipse W/m^2	Measured through a rotating analyzer
Degree of Polarization	$$V = \frac{I_{max} - I_{min}}{I_{max} + I_{min}}$$	I_{max}, I_{min}: Extreme intensities transmitted by analyzer Dimensionless	For fully polarized elliptical light, V = 1

Graphs and experiments

Trace of Electric Field Vector (Polarization Ellipse)

Draw: An ellipse inscribed in a rectangular box of sides 2E_{0x} and 2E_{0y}, with its major axis tilted at angle psi to the horizontal E_x axis.

Interpret: Shows the trajectory traced by the tip of the electric vector in the XY plane over one optical cycle.

Exam use: Used in PGT questions to identify axes intercepts and orientation based on phase difference.

Production of Elliptically Polarized Light via Quarter-Wave Plate

Draw: Unpolarized light passing through a polarizer (producing linear light at angle theta) followed by a quarter-wave plate whose optic axis makes an angle phi with the polarizer transmission axis.

Interpret: Demonstrates how splitting a linear vibration into two orthogonal components with a pi/2 phase shift yields elliptical polarization.

Exam use: Tested in conceptual setup questions regarding wave plates and retarders.

Analyzer Intensity Variation Graph

Draw: A sinusoidal variation of transmitted intensity I versus analyzer rotation angle theta, where the minimum intensity does not drop to zero.

Interpret: Non-zero minima distinguish elliptical polarization from linear polarization (which drops to zero) and circular polarization (which remains constant).

Exam use: Interpreting experimental curves to determine polarization states.

Solved numericals

TGT core

A beam of light has orthogonal electric field components given by $E_x = 3 \cos(\omega t)$ and $E_y = 4 \cos(\omega t + \pi/2)$. Describe the state of polarization.

Identify the amplitudes: $E_{0x} = 3$ and $E_{0y} = 4$.
Identify the phase difference: delt$a = \pi/2$.
Since the phase difference is \pi/2 and amplitudes are unequal (3 != 4), the condition corresponds to elliptical polarization with axes parallel to the coordinate axes.

Answer: Elliptically polarized light with semi-axes 3 and 4 parallel to the coordinate axes.

PGT extension

For an elliptically polarized light wave, the orthogonal amplitudes are $E_{0x} = 2 V/m$ and $E_{0y} = 3 V/m$, and the phase difference is delt$a = \pi/3$. Find the constant term representing sin^2(delta) in the ellipse equation and write the ellipse equation.

Calculate \si$n^2(\pi/3) = (\sqrt{3}/2)^2 = 3/4$.
Substitute $E_{0x} = 2$, $E_{0y} = 3$, and \cos(\pi/3) = 1/2 into the general ellipse equation.
The cross-term coefficient is 2(E_x E_y / (2 \times 3)) \cos(\pi/3) = (E_x E_y / 3) \times (1/2) = E_x E_y / 6.
Substitute into the standard formula: \frac{E_x^2}{4} + \frac{E_y^2}{9} - \frac{E_x E_y}{3} = \frac{3}{4}.

Answer: \frac{E_x^2}{4} + \frac{E_y^2}{9} - \frac{E_x E_y}{3} = \frac{3}{4}

PGT extension

An elliptically polarized light beam passing through a rotating analyzer shows a maximum intensity of 9 units and a minimum intensity of 1 unit. Calculate the ratio of the semi-major axis to the semi-minor axis (a/b) of the polarization ellipse.

Recall that maximum intensity is proportional to the square of the semi-major axis: $I_{max} = k a^2$.
Recall that minimum intensity is proportional to the square of the semi-minor axis: $I_{min} = k b^2$.
Take the ratio: I_{max} / $I_{min} = a^2 / b^2$.
Substitute values: 9 / $1 = a^2 / b^2$, which gives a / $b = \sqrt{9/1} = 3$.

Answer: Ratio of semi-major to semi-minor axis (a/b) = 3:1

Assumptions and limitations

Assumes monochromatic light waves of a single fixed frequency.
Assumes a homogeneous, non-magnetic, and isotropic medium for propagation after polarization.
Limitations apply when dealing with polychromatic light where polarization states vary across frequencies, requiring Stokes parameter integration.

Common misconceptions

Common error: Elliptical polarization requires equal amplitudes for the orthogonal electric field components.

Correct: Elliptical polarization generally occurs with unequal amplitudes; equal amplitudes with a pi/2 phase difference result in circular polarization.

Circular polarization is just a symmetric special case of elliptical polarization where a = b.

Common error: When elliptically polarized light passes through an analyzer, the minimum intensity is always zero.

Correct: The minimum intensity is zero only for linearly polarized light.

For elliptical polarization, the electric vector never vanishes; it only shrinks to the semi-minor axis, yielding a non-zero minimum intensity.

Quick revision

Elliptical polarization is produced by the superposition of two orthogonal linear vibrations with unequal amplitudes and a phase difference other than integral multiples of pi.
Orthogonal X-Component of Electric Field: E_x(z,t) = E_{0x} \cos(\omega t - kz)
Orthogonal Y-Component of Electric Field: E_y(z,t) = E_{0y} \cos(\omega t - kz + \delta)

Must remember

Elliptical polarization represents the most general state of a monochromatic plane electromagnetic wave, where the tip of the electric field vector traces out an ellipse.
The general equation of the polarization ellipse is given by $\frac{E_x^2}{E_{0x}^2} + \frac{E_y^2}{E_{0y}^2} - 2\frac{E_x E_y}{E_{0x} E_{0y}}\cos\delta = \sin^2\delta$, where $\delta$ is the phase difference.
Linear and circular polarizations are special limiting cases of elliptical polarization depending on the amplitudes and phase difference $\delta$.
Right-handed elliptical polarization corresponds to clockwise rotation of the electric vector looking into the beam, while left-handed corresponds to counter-clockwise rotation.
When elliptically polarized light passes through a rotating analyzer, the transmitted intensity varies between a maximum and a non-zero minimum proportional to $a^2$ and $b^2$ respectively.

Glossary

Elliptical PolarizationA state of polarization where the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation.

Phase Difference ($\delta$)The phase angle $\phi_y - \phi_x$ between the two orthogonal electric field components $E_x$ and $E_y$.

Chirality (Handedness)The directional sense of rotation of the electric field vector, classified as right-handed (dextro-rotatory) or left-handed (laevo-rotatory).

Semi-major Axis ($a$) and Semi-minor Axis ($b$)The principal dimensions of the polarization ellipse corresponding directly to the maximum and minimum electric field vector extremities.

Comparison of Polarization States

Polarization State	Amplitude Condition ($E_{0x}, E_{0y}$)	Phase Difference ($\delta$)
Linear Polarization	Any amplitudes ($E_{0x}, E_{0y}$)	$\delta = m\pi$ ($m = 0, \pm 1, \pm 2, \dots$)
Circular Polarization	Equal amplitudes ($E_{0x} = E_{0y}$)	$\delta = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \dots$
Elliptical Polarization	Generally unequal (or equal with non-standard phase)	$\delta \neq m\pi$ and $\delta \neq \pm (2m+1)\frac{\pi}{2}$ (for equal amplitudes)

Practice Quiz

Choose an option to reveal the explanation.

1. What is the most general state of polarization for a monochromatic plane electromagnetic wave?

Explanation: Elliptical polarization is the most general state of polarization, with linear and circular polarization acting as special cases.

2. The general equation of the polarization ellipse for orthogonal components $E_x$ and $E_y$ involves which trigonometric function of the phase difference $\delta$?

Explanation: The general equation involves both $\cos\delta$ in the cross-term and $\sin^2\delta$ on the right-hand side.

3. For right-handed elliptical polarization, how does the electric vector appear to rotate when looking directly into the oncoming light beam?

Explanation: Right-handed (dextro-rotatory) polarization corresponds to a clockwise rotation of the electric vector when viewed looking into the beam.

4. When elliptically polarized light is passed through a rotating analyzer, what does the minimum transmitted intensity correspond to?

Explanation: The minimum intensity transmitted through a rotating analyzer corresponds to the square of the semi-minor axis ($b^2$) of the polarization ellipse.

5. Which condition on the phase difference $\delta$ and amplitudes yields circular polarization instead of general elliptical polarization?

Explanation: Circular polarization requires equal amplitudes ($E_{0x} = E_{0y}$) and a phase difference of odd multiples of $\frac{\pi}{2}$.

6. What is the orientation angle $\psi$ of the major axis of the polarization ellipse relative to the x-axis given by?

Explanation: The orientation of the major axis of the polarization ellipse is governed by $\tan(2\psi) = \frac{2 E_{0x} E_{0y}}{E_{0x}^2 - E_{0y}^2} \cos\delta$.

7. How can elliptically polarized light be experimentally detected?

Explanation: Elliptically polarized light can be detected using a quarter-wave plate combined with an analyzer, showing varying intensity without ever reaching complete extinction.

8. What does the ellipticity parameter $\epsilon$ relate to in terms of the axes of the polarization ellipse ($a$ and $b$)?

Explanation: The ellipticity parameter is defined by $\tan\epsilon = \pm \frac{b}{a}$, where $a$ and $b$ are the semi-major and semi-minor axes.

9. If linearly polarized light is incident on a quarter-wave plate with its vibration azimuth oriented at an arbitrary angle (not $0, \pi/2$, or multiples of $\pi/4$), what state of polarization emerges?

Explanation: An arbitrary orientation of the incident linear polarization relative to the optic axis of a quarter-wave plate produces elliptically polarized light.

10. For a fully polarized elliptical light beam, what is the degree of polarization $V$ defined by extreme intensities ($I_{max}$ and $I_{min}$)?

Explanation: The degree of polarization is given by $V = \frac{I_{max} - I_{min}}{I_{max} + I_{min}}$, which equals 1 for fully polarized light.

11. What is the instantaneous orthogonal component $E_x(z,t)$ for a plane wave propagating along the z-axis?

Explanation: The x-component is represented as $E_x(z,t) = E_{0x} \cos(\omega t - kz)$.

12. Which misconception about elliptical polarization is incorrect?

Explanation: Requiring equal amplitudes is a misconception; elliptical polarization generally occurs with unequal amplitudes.

13. What happens to the polarization state if the phase difference $\delta$ between two orthogonal components of equal amplitude is changed from $\pi/2$ to $0$?

Explanation: A phase difference of $\pi/2$ with equal amplitudes yields circular polarization, while $\delta = 0$ yields linear polarization.

14. What is the unit of the general equation of the polarization ellipse in terms of electric field components?

Explanation: The equation terms are ratios of squared electric fields, making the unit $V^2/m^2$ or dimensionless depending on how normalized, but dimensionally $E^2/E_0^2$ is dimensionless.

15. In the context of UP PGT Physics, what does Stokes parameters analysis help describe?

Explanation: Stokes parameters provide a complete description of any polarization state, including unpolarized and partially polarized light.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. A monochromatic plane wave has orthogonal components $E_x = E_{0x} \cos(\omega t - kz)$ and $E_y = E_{0y} \cos(\omega t - kz + \delta)$. Under what general conditions does this represent elliptical polarization?

Explanation: Elliptical polarization generally requires unequal amplitudes and a phase difference that is neither a multiple of $\pi$ nor an odd multiple of $\pi/2$ with equal amplitudes.

2. When elliptically polarized light is analyzed using a rotating polarizer (analyzer), the ratio of the maximum transmitted intensity to the minimum transmitted intensity is proportional to:

Explanation: Since $I_{max} \propto a^2$ and $I_{min} \propto b^2$, the ratio of intensities is proportional to $\frac{a^2}{b^2}$.

3. What is the phase difference $\delta$ condition required to produce linear polarization from two orthogonal harmonic components?

Explanation: Linear polarization is obtained when the phase difference $\delta$ is an integer multiple of $\pi$ ($0, \pi, 2\pi$, etc.).

4. If the semi-major and semi-minor axes of a polarization ellipse are $a$ and $b$, what does the ellipticity angle $\epsilon$ measure when $\epsilon = 0$?

Explanation: When $\epsilon = 0$, $\tan\epsilon = 0$, meaning $b = 0$, which collapses the ellipse into a straight line representing linear polarization.

5. Which optical device is commonly used for both the production and detection of elliptically polarized light when combined with a polarizer?

Explanation: A quarter-wave plate introduces a $\pi/2$ phase retardation, making it ideal for converting linear polarization to elliptical or circular polarization and vice versa.

6. In left-handed elliptical polarization, looking against the direction of propagation, the electric vector rotates in which direction?

Explanation: Left-handed (laevo-rotatory) elliptical polarization corresponds to a counter-clockwise rotation of the electric vector when viewed looking into the oncoming beam.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. The general equation of the polarization ellipse reduces to which standard equation when $\delta = \pm \pi/2$ and $E_{0x} = E_{0y} = E_0$?

Explanation: Substituting $\delta = \pm \pi/2$ and equal amplitudes into the general ellipse equation yields the circle equation $E_x^2 + E_y^2 = E_0^2$.

2. Which of the following statements about elliptically polarized light passing through an analyzer is correct?

Explanation: For elliptical polarization, the electric vector never vanishes; thus, the analyzer transmits varying intensity with a non-zero minimum.

3. What is the angle $\psi$ of the major axis when $E_{0x} = E_{0y}$ and $\delta = \pm \pi/2$?

Explanation: For circular polarization ($a = b$), the ellipse degenerates into a circle, making the major axis orientation undefined.

4. Calculate the maximum intensity $I_{max}$ relative to the semi-major axis $a$ of a polarization ellipse.

Explanation: Intensity is proportional to the square of the electric field amplitude, so $I_{max} \propto a^2$.

5. If the phase difference $\delta$ between two orthogonal components is $3\pi/2$ and amplitudes are unequal, what type of polarization is produced?

Explanation: Unequal amplitudes with $\delta = 3\pi/2$ generate elliptical polarization.

6. What is the range of the ellipticity parameter $\epsilon$ for an ellipse?

Explanation: The ellipticity parameter is conventionally bounded within $-\pi/4 \le \epsilon \le \pi/4$.

7. In UP TGT syllabus context, recognizing polarization types qualitatively relies on understanding:

Explanation: Polarization types are fundamentally understood through the superposition of two orthogonal harmonic oscillations.

8. What happens to the polarization ellipse if the phase difference $\delta$ changes sign (from $+\delta$ to $-\delta$)?

Explanation: Changing the sign of the phase difference reverses the rotation direction of the electric vector, flipping the handedness.

9. What is the primary reason why unpolarized light cannot be converted into elliptically polarized light by a single quarter-wave plate?

Explanation: Unpolarized light has rapidly fluctuating, random phase differences, preventing a stable elliptical state from forming.

10. If $I_{max} = 9$ units and $I_{min} = 1$ unit for an elliptically polarized beam passing through an analyzer, what is the degree of polarization $V$?

Explanation: Using $V = \frac{I_{max} - I_{min}}{I_{max} + I_{min}} = \frac{9 - 1}{9 + 1} = \frac{8}{10} = 0.8$.