---
title: "Reflected and transmitted interference complementary"
subject: "Engineering Physics"
chapter: 4
key: "semester_1/engineering_physics:thin-film-reflected-transmitted-complementary"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/thin-film-reflected-transmitted-complementary
site: BCT Engineering
---

# Reflected and transmitted interference complementary

Engineering Physics · Chapter 4: Physical Optics

- Asked in: 2072 Chaitra, 5 marks; 2068 Chaitra, 5 marks (asked 2 times)
- Video: [Interference in thin films](https://www.youtube.com/watch?v=mctLGekHLyQ) — Physics Jessy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/thin-film-reflected-transmitted-complementary
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-4

## Question

Define interference. Prove that interference in thin film of reflected and transmitted light are complementary to each other.

## Short answer

Interference is the redistribution of light energy when coherent waves superpose. In a thin film, reflected light has an extra λ/2 phase shift giving bright condition 2μt cos r = (2n+1)λ/2; transmitted light has no extra shift, bright when 2μt cos r = nλ. Thus patterns are complementary: reflected bright corresponds to transmitted dark, conserving energy.

## Answer

**Interference** is the redistribution of light energy in space when two or more coherent waves superpose: the intensity is maximum at some points (constructive) and minimum at others (destructive).

#### Proof for a thin film
Consider a film of thickness *t*, refractive index μ, with refraction angle *r*.

**Reflected light.** The ray reflected from the top surface suffers a phase change of π (λ/2). Path difference between the two reflected rays:

```text
Δ_R = 2μt cos r  ±  λ/2
```

- Bright: 2μt cos r = (2n+1)λ/2
- Dark:   2μt cos r = nλ

**Transmitted light.** The first transmitted ray goes straight through; the second is reflected twice inside the film (at the bottom and then the top surface, both from the film side, film in air). Both are reflections at a rarer medium, so there is no phase change. Path difference:

```text
Δ_T = 2μt cos r
```

- Bright: 2μt cos r = nλ
- Dark:   2μt cos r = (2n+1)λ/2

Compare the conditions:

| Condition | Reflected | Transmitted |
|---|---|---|
| 2μt cos r = nλ | dark | bright |
| 2μt cos r = (2n+1)λ/2 | bright | dark |

Thus the reflected and transmitted patterns are **complementary**: where the reflected system has a maximum, the transmitted has a minimum, and vice versa.

**Energy conservation:** I_reflected + I_transmitted = I_incident (no absorption). So when the reflected intensity is maximum, the transmitted intensity must be minimum. The transmitted fringes, however, have poor contrast because the two transmitted beams have very unequal amplitudes.

## Check yourself

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

1. Interference of light is best described as:
   A. Bending of light around obstacles
   B. Redistribution of light energy when coherent waves superpose
   C. Splitting of light into colours
   D. Absorption of light by a medium
   Answer: B. Coherent waves superpose, giving maxima at some points and minima at others, with energy redistributed. (Key point: Definition of interference)

2. For reflected light from a film, when is the film dark?
   A. 2μt cos r = (2n+1)λ/2
   B. 2μt cos r = nλ
   C. 2μt cos r = λ/2 only
   D. 2μt cos r = 0 never
   Answer: B. The λ/2 phase change in reflection makes 2μt cos r = nλ a dark condition. (Key point: Reflected condition)

3. Why does transmitted light have no extra λ/2 term?
   A. Both internal reflections are at a rarer medium, so no phase change
   B. Transmission always adds 2π
   C. The film absorbs the phase change
   D. Only one ray is transmitted
   Answer: A. The second ray reflects twice from film-to-air boundaries, with no phase change, so Δ_T = 2μt cos r. (Key point: Transmitted path difference)

4. When 2μt cos r = nλ, the reflected and transmitted light are respectively:
   A. bright and bright
   B. dark and bright
   C. bright and dark
   D. dark and dark
   Answer: B. That condition is dark in reflection and bright in transmission, so the patterns are complementary, consistent with energy conservation. (Key point: Complementary patterns)

## Related in Engineering Physics

- [Thin film interference by reflected light](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Athin-film-reflected-interference-theory) — Engineering Physics, chapter 4, asked 2×