---
title: "Condition for achromatism of two lenses"
subject: "Engineering Physics"
chapter: 5
key: "semester_1/engineering_physics:achromatism-condition-dispersive-powers"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/achromatism-condition-dispersive-powers
site: BCT Engineering
---

# Condition for achromatism of two lenses

Engineering Physics · Chapter 5: Geometrical Optics

- Asked in: 2068 Chaitra, 5 marks
- Video: [Condition For Achromatism Of Two Thin lenses- In Contact @studymaterialsfromsai](https://www.youtube.com/watch?v=JcwvOSlTTIc) — Sai
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/achromatism-condition-dispersive-powers
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-5

## Question

Prove that the condition for achromatism for the combination of two lenses of focal length f1 and f2 having dispersive power ω1 and ω2 placed at a separate distance x is (ω1/f1) + (ω2/f2) = (x/f1f2)(ω1 + ω2).

## Short answer

For two thin lenses separated by distance x, achromatism requires the equivalent focal length to be independent of wavelength. Differentiating 1/F = 1/f1 + 1/f2 - x/(f1 f2) and using δ(1/f) = ω/f yields the condition ω1/f1 + ω2/f2 = x(ω1 + ω2)/(f1 f2).

## Answer

**Statement:** two thin lenses of focal lengths f₁, f₂ and dispersive powers ω₁, ω₂, separated by a distance x, form an achromatic combination if ω₁/f₁ + ω₂/f₂ = x(ω₁ + ω₂)/(f₁f₂).

#### Proof
1. The equivalent focal length F of the pair is given by
 1/F = 1/f₁ + 1/f₂ − x/(f₁f₂). (1)
2. For achromatism, F must be the same for all colours (red and violet), so δ(1/F) = 0 when the refractive indices change from μ_R to μ_V. x is fixed.
3. For each lens, 1/f = (μ − 1)(1/R₁ − 1/R₂) = (μ − 1) a, where *a* depends only on the radii. So the change in 1/f for a change δμ is
 δ(1/f) = a δμ = (1/f) δμ/(μ − 1) = ω/f,
 with ω = δμ/(μ − 1) the dispersive power. Hence
 δ(1/f₁) = ω₁/f₁ and δ(1/f₂) = ω₂/f₂.
4. Differentiate (1), treating δ(1/f₁), δ(1/f₂) as the variations:
 δ(1/F) = δ(1/f₁) + δ(1/f₂) − x [ (1/f₁) δ(1/f₂) + (1/f₂) δ(1/f₁) ]
5. Substitute the results of step 3:
 δ(1/F) = ω₁/f₁ + ω₂/f₂ − x [ ω₂/(f₁f₂) + ω₁/(f₁f₂) ]
 = ω₁/f₁ + ω₂/f₂ − x (ω₁ + ω₂)/(f₁f₂)
6. Setting δ(1/F) = 0:

 **ω₁/f₁ + ω₂/f₂ = x (ω₁ + ω₂) / (f₁ f₂)**

#### Special cases
- x = 0 (lenses in contact): ω₁/f₁ + ω₂/f₂ = 0, the achromatic doublet.
- Same material (ω₁ = ω₂): x = (f₁ + f₂)/2.

## Check yourself

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

1. What is required for achromatism?
   A. μ constant
   B. f1 = f2
   C. F must be the same for red and violet, δ(1/F) = 0
   D. d = 0
   Answer: C. The equivalent focal length must not change with colour. (Key point: Basic idea)

2. What is δ(1/f) for a lens?
   A. ω
   B. ωf
   C. 1/(ωf)
   D. ω/f
   Answer: D. δ(1/f) = (1/f) δμ/(μ−1) = ω/f. (Key point: Variation of power)

3. Same material (ω1 = ω2) gives:
   A. x = 2f1f2
   B. x = f1 − f2
   C. x = 0
   D. x = (f1 + f2)/2
   Answer: D. Setting ω1 = ω2 in the condition. (Key point: Special case)

## Related in Engineering Physics

- [Separation of two achromatic lenses](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Atwo-lenses-equivalent-focal-fifty) — Engineering Physics, chapter 5
- [Focal lengths of achromatic lens pair](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aachromatic-lenses-equivalent-thirty-values) — Engineering Physics, chapter 5
- [Equivalent focal length and principal points](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Atwo-lenses-ten-four-principal) — Engineering Physics, chapter 5
- [Cardinal points and equivalent focal length](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acardinal-points-equivalent-focal-length) — Engineering Physics, chapter 5
- [Image by two lenses 30 cm and 40 cm](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Atwo-lenses-thirty-forty-image) — Engineering Physics, chapter 5, asked 3×
- [Principal points and image, two lenses](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Atwo-lenses-principal-points-image) — Engineering Physics, chapter 5

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