Chapter Overview

Dispersion is the separation of polychromatic light into its component wavelengths because a refracting medium generally has different refractive indices for different wavelengths. A prism therefore deviates violet light more strongly than red light.

This chapter covers pure spectrum and dispersive power, chromatic and spherical aberration, and achromatism with its applications.

18.1 Pure Spectrum and Dispersive Power

Dispersion of White Light

White light contains a continuous range of visible wavelengths. When it passes through a prism, different wavelengths undergo different refractions and emerge in different directions, forming a spectrum.

For ordinary glass in the visible region:

• Violet has a larger refractive index and is deviated more.
• Red has a smaller refractive index and is deviated less.

Pure Spectrum

A pure spectrum is a spectrum in which the images corresponding to different wavelengths do not overlap appreciably, so each spectral region is clearly separated from neighboring wavelengths.

A narrow entrance slit, suitable collimation, a dispersing element and proper focusing help produce a clean spectrum.

Angular Dispersion

If δ_v and δ_r are deviations for violet and red light, the angular dispersion between them is:

For a small-angle prism, δ ≈ (μ−1)A, hence:

Dispersive Power

Dispersive power measures the angular spread of colors relative to the mean deviation. Using yellow or a mean refractive index μ as reference:

Equivalently for a small prism:

18.2 Chromatic Aberration

A lens has different refractive indices and therefore different focal lengths for different wavelengths. Violet light is refracted more strongly and normally focuses nearer the lens than red light.

This failure of a lens to bring all colors to the same focus is called chromatic aberration.

Longitudinal Chromatic Aberration

The axial separation between the focal points for red and violet rays is called longitudinal chromatic aberration:

Reduction

• Use an achromatic doublet.
• Use monochromatic light when appropriate.
• Reduce lens aperture in some imaging situations, though this does not remove the underlying wavelength dependence.

Spherical Aberration

Spherical aberration arises because rays passing through different zones of a spherical lens do not generally meet at one point even for monochromatic light. Marginal rays and paraxial rays have different focal positions.

Reduction of Spherical Aberration

• Use a small aperture so mainly paraxial rays pass.
• Use properly designed lens combinations rather than a single spherical surface.
• Use aspheric surfaces in advanced optical systems.

18.3 Achromatism and Its Applications

Achromatism

Achromatism is the reduction or correction of chromatic aberration by combining lenses made from materials with different dispersive properties.

A common achromatic doublet combines a positive crown-glass lens with a negative flint-glass lens so that the color separation produced by one is largely cancelled by the other while a useful net focusing power remains.

Condition for Achromatism of Two Thin Lenses in Contact

For two lens elements whose powers are P₁ and P₂ and dispersive powers are ω₁ and ω₂, a standard achromatism condition is:

Since P = 1/f:

For an ordinary achromatic doublet, one lens is converging and the other diverging, so their focal lengths have opposite signs under the usual sign convention.

Applications

• Camera lenses.
• Telescopes and binoculars.
• Microscopes.
• Projection systems.
• Other optical instruments where color fringes must be minimized.

Dispersive Power and Material Choice

Different transparent materials show different wavelength dependence of refractive index. This is why combining crown and flint glasses can correct color spreading: the materials can be chosen to give useful refracting power with compensating dispersion.

Solved Numerical Examples

Example 1 — Angular Dispersion

Question: A prism deviates violet by 5.2° and red by 4.6°. Find angular dispersion.

Answer: 0.6°.

Example 2 — Small-Prism Color Separation

Question: For a small prism A = 5°, μv = 1.54 and μr = 1.52. Estimate angular dispersion.

Answer: approximately 0.10°.

Example 3 — Dispersive Power

Question: If μv = 1.54, μr = 1.52 and mean μ = 1.53, find ω.

Answer: approximately 0.038.

Example 4 — Chromatic Focal Separation

Question: A lens has fr = 50.5 cm and fv = 49.5 cm. Find longitudinal chromatic aberration.

Answer: 1.0 cm.

Example 5 — Achromatic Condition

Question: Two thin lenses in contact have ω₁ = 0.02 and ω₂ = 0.04. If f₁ = +20 cm, find f₂ for achromatism.

Answer: f₂ = −40 cm.

Important Exam Questions

One-Minute Revision

• Dispersion occurs because refractive index depends on wavelength.
• Violet is usually deviated more than red by glass.
• A pure spectrum minimizes overlap of neighboring wavelength images.
• Angular dispersion = δv − δr.
• For a small prism, angular dispersion ≈ (μv−μr)A.
• Dispersive power compares color spread with mean deviation.
• ω = (μv−μr)/(μ−1).
• Chromatic aberration is color-dependent focusing.
• For a simple converging lens, fv is usually smaller than fr.
• Spherical aberration occurs even with monochromatic light.
• Marginal and paraxial rays may have different focal positions.
• Achromatism reduces chromatic aberration using compensating lens materials.
• A crown positive lens and flint negative lens commonly form an achromatic doublet.
• Achromatism condition: ω₁P₁ + ω₂P₂ = 0.

Diagram Practice

1. Draw a prism dispersing white light into R–O–Y–G–B–I–V.
2. Draw the slit–collimator–prism–lens–screen arrangement for a pure spectrum.
3. Draw chromatic aberration and mark Fv and Fr.
4. Draw spherical aberration with marginal and paraxial foci.
5. Draw an achromatic doublet and nearly common red/violet focus.
6. Sketch the qualitative decrease of refractive index with visible wavelength.

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