Questions: Polarization States: Linear, Circular, and Elliptical

Question 1

Two perpendicular electric field components of equal amplitude are combined with a phase difference of 45°. What polarization state results?

• A. Circular polarization — equal amplitudes are the only requirement for circular polarization
• B. Linear polarization — the phase difference shifts the oscillation direction but keeps it fixed
• C. Elliptical polarization — both equal amplitudes AND a 90° phase difference are required for circular; 45° produces an ellipse
• D. Unpolarized light — phase differences between components randomize the polarization

Answer: (C)

Circular polarization requires two simultaneous conditions: equal amplitudes AND exactly 90° phase difference. If either condition fails, the E-field tip traces an ellipse instead of a circle. With equal amplitudes and a 45° phase difference, the path is an ellipse — not a circle. The most common misconception is remembering 'equal amplitudes' while forgetting the equally critical 90° phase requirement.

Question 2

Linearly polarized light of intensity I₀ passes through a polarizer whose transmission axis is at 60° to the polarization direction. What intensity emerges?

• A. I₀/2 — because only the cosine component of the electric field is transmitted
• B. I₀ cos(60°) = I₀/2
• C. I₀ cos²(60°) = I₀/4
• D. 0 — because 60° is too close to the crossed-polarizer condition

Answer: (C)

Malus's law states I = I₀cos²θ. At θ = 60°, cos(60°) = 1/2, so cos²(60°) = 1/4, giving I = I₀/4. The cos²θ factor arises because intensity is proportional to amplitude squared: the polarizer transmits the E-field component along its axis (amplitude reduced by cosθ), and since intensity ∝ E², the result is cos²θ. The answer I₀/2 is the classic error from forgetting to square the cosine.

Question 3

Linear polarization and circular polarization are both limiting cases of elliptical polarization.

• T. True
• F. False

Answer: True

Elliptical polarization is the general case for the superposition of two perpendicular components with any amplitudes and any phase difference. When the phase difference is 0° or 180° and amplitudes are arbitrary, the ellipse degenerates to a line — linear polarization. When amplitudes are equal and phase difference is exactly 90°, the ellipse becomes a perfect circle — circular polarization. All three are described by the same framework; linear and circular are special limiting cases.

Question 4

Circularly polarized light can be produced from a single linearly polarized beam by introducing a 90° phase shift to that beam.

• T. True
• F. False

Answer: False

Circular polarization requires two perpendicular components. A single linearly polarized beam has one field component; adding a phase shift to a single component just shifts its phase — you still have one component oscillating in one direction, which remains linear polarization. To produce circular polarization, the beam must first be decomposed into two perpendicular components (by a quarter-wave plate oriented at 45°), and those two components must be given a 90° phase difference with equal amplitudes.

Question 5

What two conditions must both be satisfied for light to be circularly polarized rather than elliptically polarized, and what happens when only one condition is met?

Model answer: Circular polarization requires: (1) the two perpendicular E-field components must have equal amplitudes, and (2) they must have exactly 90° phase difference. If only condition (1) is met — equal amplitudes but phase difference not 90° — the E-field tip traces an ellipse with the phase offset distorting the circle. If only condition (2) is met — 90° phase difference but unequal amplitudes — the tip again traces an ellipse, elongated along the axis of the larger component. Both conditions together give uniform rotation at constant radius: a circle.

The intuition: circular polarization means the E-field vector rotates at constant angular velocity with constant magnitude — like a vector of fixed length spinning steadily. Unequal amplitudes make the magnitude vary as it rotates (ellipse). A non-90° phase difference makes the rotation uneven in angle (also an ellipse). Both constraints together — equal amplitudes, 90° phase — enforce uniform rotation.