---
title: "Production of plane, circular, elliptical light"
subject: "Engineering Physics"
chapter: 4
key: "semester_1/engineering_physics:polarized-light-production-relations"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/polarized-light-production-relations
site: BCT Engineering
---

# Production of plane, circular, elliptical light

Engineering Physics · Chapter 4: Physical Optics

- Asked in: 2073 Chaitra, 5 marks; 2072 Kartik, 5 marks; 2071 Chaitra, 5 marks (asked 3 times)
- Video: [Plane, Circular, Elliptical Polarisation | Plane Circular Elliptical Polarisation | BSc 3rd Sem |...](https://www.youtube.com/watch?v=M4uGIKoUdl4) — Physics Jockey
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/polarized-light-production-relations
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-4

## Question

What is the principle of double refraction? Establish the relations for the production of linearly, elliptically and circularly polarized light (what is polarization?).

## Short answer

Polarization restricts vibrations to a fixed pattern. Double refraction splits light into mutually perpendicular plane-polarized ordinary and extraordinary rays. With phase difference δ = (2π/λ)(μ_o−μ_e)t, δ=0,π gives linear; δ=π/2 and a≠b gives elliptical; δ=π/2 and a=b gives circular.

## Answer

**Polarization** is the property of transverse waves by which the vibrations are restricted to a particular direction or pattern. In **plane (linear)** polarized light the electric vector vibrates in one plane; in **circular** and **elliptical** light its tip traces a circle or ellipse.

#### Principle of double refraction
When unpolarized light enters a doubly refracting (anisotropic) crystal like calcite or quartz, it splits into two rays:
- **Ordinary (o) ray**: obeys Snell's law, has speed v_o in all directions, vibrates perpendicular to the principal section.
- **Extraordinary (e) ray**: does not obey Snell's law, speed varies with direction, vibrates in the principal section.

Both are plane polarized with mutually perpendicular vibrations. Along the optic axis both travel with the same speed. (Huygens: o gives spherical, e gives ellipsoidal wavefront.)

#### Production of polarized light
```text
 plane-polarized   optic axis (OA)
 light, amplitude A  |
 -------->[ crystal plate, thickness t ]--> E and O rays
              angle θ between vibration and OA
```
1. Plane polarized light of amplitude *A* falls normally on a plate cut parallel to the optic axis, vibrating at angle θ to it.
2. It splits into e-component (along axis) **a = A cos θ** and o-component (perpendicular) **b = A sin θ**; they travel with different speeds.
3. On emerging, path difference = (μ_o − μ_e) t, so phase difference **δ = (2π/λ)(μ_o − μ_e) t**.
4. Take x = a cos ωt, y = b cos(ωt + δ). Eliminating t:

```text
x²/a² + y²/b² − (2xy/ab) cos δ = sin²δ
```

5. Special cases:

| Condition | Result |
|---|---|
| δ = 0 or π (path diff 0, λ/2) | y = ±(b/a)x : **linear** |
| δ = π/2 (quarter-wave plate), a ≠ b | x²/a² + y²/b² = 1 : **elliptical** |
| δ = π/2 and a = b (θ = 45°) | x² + y² = a² : **circular** |
| other δ | tilted ellipse (elliptical) |

Thus a quarter-wave plate with θ = 45° gives circular light, with the thickness t = λ/[4(μ_o − μ_e)].

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which statement about the ordinary and extraordinary rays is correct?
   A. Both obey Snell's law
   B. The o-ray obeys Snell's law, the e-ray does not, and they are polarized perpendicular to each other
   C. Both are unpolarized
   D. They have the same speed in all directions
   Answer: B. Only the o-ray obeys Snell's law; the two rays are plane polarized with mutually perpendicular vibrations. (Key point: Double refraction)

2. Light of amplitude A vibrating at angle θ to the optic axis splits into e and o components of
   A. A sin θ and A cos θ
   B. A cos θ and A sin θ
   C. A and A
   D. A/2 and A/2
   Answer: B. The e-component along the axis is A cos θ and the o-component perpendicular is A sin θ. (Key point: Components in crystal plate)

3. The phase difference between the emerging rays is
   A. (2π/λ)(μ_o + μ_e)t
   B. (λ/2π)(μ_o - μ_e)t
   C. (2π/λ)(μ_o - μ_e)t
   D. (μ_o - μ_e)/t
   Answer: C. Path difference is (μ_o - μ_e)t, and phase = (2π/λ) × path difference. (Key point: Phase difference)

4. Circularly polarized light is produced when
   A. δ = π/2 and a = b (θ = 45°)
   B. δ = 0
   C. δ = π
   D. δ = π/2 and a ≠ b
   Answer: A. A quarter-wave plate with equal components gives x² + y² = a². (Key point: Circular light)

## Related in Engineering Physics

- [Distinguishing plane, circular, elliptical light](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adistinguish-polarized-nicol-wave-plate) — Engineering Physics, chapter 4
- [Double refraction, linear and circular light](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adouble-refraction-produce-polarized-light) — Engineering Physics, chapter 4
- [Index difference for circular polarization](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aqwp-refractive-index-difference) — Engineering Physics, chapter 4

## Related in other subjects

- [Two types of polarization with relations](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Atwo-polarization-types) — Electrical Engineering Material, chapter 3
- [Circular well 4 m, light and heavy well](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Acircular-well-4m-light-heavy) — Foundation Engineering, chapter 9