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Chromatic Aberration of Lenses

Chromatic aberration is an optical defect in lenses caused by the dependence of refractive index on the wavelength of light, leading to dispersion of white light into constituent colors at different focal points. It prevents a single-element lens from forming a sharp, single-point white-light image.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Definition, cause of chromatic aberration, longitudinal and lateral chromatic aberration, focal length variation for different colors, and condition for achromatism of two thin lenses in contact.

PGT extension

Achromatic combination of two thin lenses separated by a finite distance, dispersive power, achromatism for lenses separated by a distance, and evaluation of radii of curvature for minimizing aberration.

Prerequisites

Snell's Law and Refraction through spherical surfaces
Lens Maker's Formula and thin lens optics
Dispersive power of optical materials and Cauchy's dispersion formula

Origin of Chromatic Aberration

The refractive index (μ) of lens material varies inversely with wavelength (λ) according to Cauchy's relation: $\mu = A + B/\lambda^{2}$.
Violet light (λ is short) experiences a higher refractive index than red light (λ is long), resulting in μ_v > μ_r.
From the Lens Maker's Formula, focal length f is given by 1/$f = (\mu - 1)(1/R_{1} - 1/R_{2})$, causing violet light to focus closer to the lens than red light.

Longitudinal and Lateral Chromatic Aberration

Longitudinal Chromatic Aberration (LCA) is the axial separation between the focal points of extreme colors (violet and red): LC$A = f_{r} - f_{v}$.
Lateral Chromatic Aberration (TCA) is the difference in image sizes for different wavelengths formed in a plane perpendicular to the optical axis.
For an axial object, LCA manifests as a blurred colored patch along the optic axis, whereas TCA causes colored fringes around off-axis image points.

Dispersive Power of Lens Material

Dispersive power (ω) is defined as the ratio of the difference in focal lengths (or refractive indices) for extreme colors to the focal length (or refractive index) for the mean color.
$\omega = (f_{r} - f_{v})/f_{y} = (\mu_{v} - \mu_{r})/(\mu_{y} - 1)$, where subscripts v, r, and y denote violet, red, and yellow (mean) rays.
Dispersive power is a dimensionless quantity characteristic of the optical medium, typically glass.

Achromatic Combination of Lenses

Achromatism aims to bring rays of extreme wavelengths (e.g., red and violet) to a common focus, eliminating chromatic aberration.
Two thin lenses of dispersive powers ω₁ and ω₂ and focal lengths f₁ and f₂ achieve achromatism when ω₁/f₁ + ω₂/f₂ = 0.
To achieve a real, positive focal length for the combination, one lens must be convex (crown glass) and the other concave (flint glass), requiring ω₁ ≠ ω₂.

Derivations

Condition for Achromatism of Two Thin Lenses in Contact

Conditions: Two thin lenses placed coaxially in contact, illuminated by white light.

Write the equivalent focal length F of two thin lenses in contact: 1/$F = 1/f_{1} + 1/f_{2}$.
Differentiate the equation with respect to wavelength to find the change in focal power: d(1/F) = d(1/f₁) + d(1/f₂).
Using the definition of dispersive power $\omega = -df/f$, substitute df/f² = -ω/f into the differentiated expression.
Set d(1/F) = 0 for chromatic aberration to vanish: ω₁/f₁ + ω₂/f₂ = 0.

Result: \frac{\omega_1}{f_1} + \frac{\omega_2}{$f_2 = 0$

Condition for Achromatism of Two Thin Lenses Separated by a Distance

Conditions: Two thin lenses of focal lengths f₁ and f₂ separated by a finite distance d.

Write the equivalent focal length F for two lenses separated by distance d: 1/$F = 1/f_{1} + 1/f_{2} - d/(f_{1}f_{2})$.
Differentiate implicitly with respect to wavelength, noting dispersive powers ω₁ = -df₁/f₁ and ω₂ = -df₂/f₂.
Apply the condition for achromatism by equating the derivative of the reciprocal focal length with respect to wavelength to zero.
Simplify the resulting algebraic expression to obtain the separation d in terms of focal lengths and dispersive powers.

Result: $d = \frac{f_1 + f_2}{2}$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Cauchy Dispersion Formula	$$\mu(\lambda) = A + \frac{B}{\lambda^2}$$	A, B are material constants, λ is wavelength Dimensionless (μ), m² (B)	Valid in spectral regions away from absorption bands
Lens Maker's Formula	$$\frac{1}{f} = ( \mu - 1 ) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$$	f is focal length, μ is refractive index, R₁, R₂ are radii of curvature m (f, R₁, R₂), dimensionless (μ)	Thin lens approximation in air
Dispersive Power	$$\omega = \frac{\mu_v - \mu_r}{\mu_y - 1} = \frac{f_r - f_v}{f_y}$$	μ_v, μ_r, μ_y are refractive indices for violet, red, yellow rays Dimensionless	Standard Fraunhofer lines (C, D, F)
Longitudinal Chromatic Aberration (LCA)	$$LCA = f_r - f_v = \omega f_y$$	f_r, f_v are focal lengths for red and violet light, ω is dispersive power m	Single thin lens
Equivalent Focal Length of Two Lenses in Contact	$$\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2}$$	F is equivalent focal length, f₁, f₂ are individual focal lengths m	Thin lenses in direct contact
Achromatic Condition (Contact Lenses)	$$\frac{\omega_1}{f_1} + \frac{\omega_2}{f_2} = 0$$	ω₁, ω₂ are dispersive powers, f₁, f₂ are focal lengths Dimensionless / m	Minimum chromatic aberration for white light
Equivalent Focal Length for Separated Lenses	$$\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$$	d is separation distance between the two lenses m	Coaxial thin lenses separated by distance d
Achromatic Condition (Separated Lenses)	$$d = \frac{f_1 f_2 (\omega_1 + \omega_2)}{\omega_1 f_2 + \omega_2 f_1} \text{ (general)} \implies d = \frac{f_1 + f_2}{2} \text{ (same material)}$$	d is separation, f₁, f₂ focal lengths, ω₁, ω₂ dispersive powers m	Coaxial system of two lenses of the same material

Graphs and experiments

Focal Length vs Wavelength Plot

Draw: A curve showing focal length (f) on the y-axis against wavelength (λ) on the x-axis, exhibiting an upward monotonic increase from violet to red.

Interpret: Demonstrates that shorter wavelengths focus closer to the optical center than longer wavelengths due to dispersion.

Exam use: Interpreting focal shift and identifying positions of chromatic aberration extremes.

Achromatic Doublet Ray Diagram

Draw: White light incident on a crown-flint glass doublet, showing parallel white rays refracting through both lenses and converging to a single white focal point.

Interpret: Illustrates how combining lenses with compensating dispersive powers corrects longitudinal chromatic aberration.

Exam use: Tracing refracted paths of red and violet components in compound lens systems.

Chromatic Aberration Measurement Bench

Draw: Optical bench setup with a white light source, a pinhole object, coloured filters (red and blue), test lens, and a movable screen.

Interpret: Measures the axial distance between sharp red and blue images to quantify longitudinal chromatic aberration experimentally.

Exam use: Experimental determination of focal differences for distinct spectral lines.

Solved numericals

TGT core

A thin convex lens made of glass has a focal length of 0.30 m for yellow light. If the dispersive power of the glass is 0.02, calculate the longitudinal chromatic aberration between red and violet light.

Identify given values: mean focal length f_y = 0.30 m, dispersive power $\omega = 0.02$.
Use the formula for longitudinal chromatic aberration: LC$A = \omega \times f_{y}$.
Substitute the values: LC$A = 0.02 \times 0.30 = 0.006 m$.

Answer: 0.006 m (or 6 mm)

PGT extension

An achromatic doublet is formed by two lenses in contact. The focal length of the combination for yellow light is 0.40 m. If the dispersive powers of the two glass materials are 0.03 and 0.05, find the individual focal lengths of the two lenses.

Given: Equivalent focal length F = 0.40 m, ω₁ = 0.03, ω₂ = 0.05.
Condition for achromatism in contact: ω₁/f₁ + ω₂/f₂ = 0, which gives f₂ = -(ω₂/ω₁)f₁ = -(0.05/0.03)f₁ = -(5/3)f₁.
Use the combination formula for contact lenses: 1/$F = 1/f_{1} + 1/f_{2} = 1/f_{1} - (3/5)/f_{1} = (2/5)/f_{1}$.
Solve for f₁: f₁ = (2/5) × $F = (2/5) \times 0.40 = 0.16 m$.
Calculate f₂: f₂ = -(5/3) × $0.16 = -0.267 m$.

Answer: f₁ = +0.16 m, f₂ = -0.267 m

PGT extension

Two thin lenses of the same material are separated by a distance d to form an achromatic system. If the focal length of each lens is 0.20 m, calculate the separation d.

For two lenses of the same material (ω₁ = ω₂) separated by distance d, the condition for achromatism is $d = (f_{1} + f_{2})/2$.
Given f₁ = f₂ = 0.20 m.
Substitute the values: $d = (0.20 + 0.20)/2 = 0.20 m$.

Answer: 0.20 m

Assumptions and limitations

The thin lens approximation is assumed, neglecting lens thickness in optical center calculations.
Paraxial ray approximation is used; chromatic aberration formulas do not account for off-axis monochromatic aberrations like coma or astigmatism unless coupled.
Cauchy's dispersion formula is an empirical approximation valid only across visible wavelengths.

Common misconceptions

Common error: Chromatic aberration can be completely eliminated for all wavelengths using a two-lens achromatic doublet.

Correct: An achromatic doublet brings only two extreme wavelengths (usually red and F-blue/violet) to a common focus.

A secondary spectrum remains because the ratio of dispersive powers is not constant across the entire spectrum.

Common error: Concave lenses do not exhibit chromatic aberration.

Correct: All refractive optical elements made of dispersive media exhibit chromatic aberration, regardless of whether they are converging or diverging.

Dispersion is an inherent property of refraction through any material where μ varies with λ.

Quick revision

Chromatic aberration arises due to the wavelength dependence of refractive index (μ ∝ 1/λ²).
Cauchy Dispersion Formula: \mu(\lambda) = A + \frac{B}{\lambda^2}
Lens Maker's Formula: \frac{1}{f} = ( \mu - 1 ) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)

Must remember

Refractive index ($\mu$) of lens material varies inversely with wavelength ($\lambda$) according to Cauchy's relation: $\mu = A + \frac{B}{\lambda^2}$.
Violet light ($\lambda$ is short) has a higher refractive index than red light ($\lambda$ is long), so $\mu_v > \mu_r$.
Longitudinal Chromatic Aberration (LCA) for a single thin lens is given by $LCA = f_r - f_v = \omega f_y$.
Dispersive power ($\omega$) is given by $\omega = \frac{\mu_v - \mu_r}{\mu_y - 1} = \frac{f_r - f_v}{f_y}$.
The condition for achromatism of two thin lenses in contact is $\frac{\omega_1}{f_1} + \frac{\omega_2}{f_2} = 0$.
For two lenses of the same material separated by a distance $d$, the condition for achromatism is $d = \frac{f_1 + f_2}{2}$.

Glossary

Chromatic AberrationFailure of a lens to focus all colors to the same point due to wavelength-dependent refractive index.

Dispersive Power ($omega$)Ratio of the difference in focal lengths for extreme colors to the focal length for the mean color.

Longitudinal Chromatic Aberration (LCA)Axial separation between the focal points of extreme colors (violet and red).

Lateral Chromatic Aberration (TCA)Difference in image sizes for different wavelengths formed perpendicular to the optical axis.

Achromatic DoubletCombination of two lenses designed to bring at least two extreme wavelengths to a common focus.

Longitudinal vs Lateral Chromatic Aberration

Feature	Longitudinal Chromatic Aberration (LCA)	Lateral Chromatic Aberration (TCA)
Orientation	Along the optical axis	Perpendicular to the optical axis (transverse)
Manifestation	Axial spread of focal points ($f_r - f_v$)	Difference in transverse image sizes for different colors
Effect on Image	Blurred colored patch along the axis	Colored fringes around off-axis image points

Practice Quiz

Choose an option to reveal the explanation.

1. According to Cauchy's dispersion formula, how does the refractive index $\mu$ vary with wavelength $\lambda$?

Explanation: Cauchy's relation is given by $\mu = A + \frac{B}{\lambda^2}$, showing that $\mu$ varies inversely with the square of the wavelength.

2. Which of the following colors focuses closest to a converging lens?

Explanation: Violet light has the shortest wavelength and highest refractive index ($\mu_v > \mu_r$), causing it to experience greater bending and focus closer to the lens than red light.

3. What is the formula for Longitudinal Chromatic Aberration (LCA) of a single thin lens?

Explanation: LCA is defined as the axial separation between the focal points of extreme colors, given by $LCA = f_r - f_v$.

4. Dispersive power $\omega$ of a lens material is defined as:

Explanation: Dispersive power is expressed as $\omega = \frac{\mu_v - \mu_r}{\mu_y - 1} = \frac{f_r - f_v}{f_y}$.

5. What is the condition for achromatism of two thin lenses in contact?

Explanation: For two thin lenses in contact to form an achromatic combination, the condition is $\frac{\omega_1}{f_1} + \frac{\omega_2}{f_2} = 0$.

6. To achieve a real, positive focal length for an achromatic doublet in contact, what combination of lenses must be used?

Explanation: Achieving achromatism with a real focal length requires $\omega_1 \neq \omega_2$ and opposite sign focal lengths, meaning one must be a convex lens (crown glass) and the other a concave lens (flint glass).

7. Lateral Chromatic Aberration (TCA) refers to:

Explanation: Lateral or transverse chromatic aberration (TCA) is the difference in image sizes formed for different wavelengths in a plane perpendicular to the optical axis.

8. What is the unit of dispersive power $\omega$?

Explanation: Dispersive power is a ratio of refractive index differences or focal length differences, making it a dimensionless quantity.

9. For two thin lenses of the same material separated by a distance $d$, what is the condition for achromatism?

Explanation: When two lenses of the same material ($\omega_1 = \omega_2$) are separated by a distance $d$, the condition for achromatism is $d = \frac{f_1 + f_2}{2}$.

10. Can concave lenses exhibit chromatic aberration?

Explanation: Dispersion is an inherent property of refraction through any material where $\mu$ varies with $\lambda$, so concave lenses also exhibit chromatic aberration.

11. What is the general condition for achromatism of two thin lenses separated by a distance $d$?

Explanation: The general condition for achromatism of two separated lenses is $d = \frac{f_1 f_2 (\omega_1 + \omega_2)}{\omega_1 f_2 + \omega_2 f_1}$.

12. Why does a standard achromatic doublet fail to eliminate chromatic aberration completely across the entire visible spectrum?

Explanation: A secondary spectrum remains because the ratio of dispersive powers $\frac{\omega_1}{\omega_2}$ is not constant across the entire spectrum, allowing only two extreme wavelengths to be perfectly focused together.

13. If the focal length of a convex lens for yellow light is $0.2\text{ m}$ and its dispersive power is $0.04$, what is the Longitudinal Chromatic Aberration?

Explanation: Using $LCA = \omega f_y$, we get $LCA = 0.04 \times 0.2\text{ m} = 0.008\text{ m}$.

14. In the lens maker's formula $\frac{1}{f} = (\mu - 1)(\frac{1}{R_1} - \frac{1}{R_2})$, what does $f$ represent?

Explanation: In optics, $f$ in the lens maker's formula denotes the focal length of the lens.

15. Which Fraunhofer lines are typically used to define the extreme and mean colors for optical glass calculations?

Explanation: Standard optical calculations use Fraunhofer C line (red, $\lambda \approx 6563\text{ AA}$), D line (yellow, $\lambda \approx 5893\text{ AA}$), and F line (blue/violet, $\lambda \approx 4861\text{ AA}$).

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. Two thin lenses of focal lengths $f_1$ and $f_2$ are placed in contact. The equivalent focal length $F$ is given by:

Explanation: For thin lenses in direct contact, the optical powers add up, leading to $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2}$.

2. An achromatic combination of two lenses in contact has dispersive powers $\omega_1$ and $\omega_2$ and focal lengths $f_1$ and $f_2$. The correct relation is:

Explanation: The standard condition for achromatism of two thin lenses in contact is $\frac{\omega_1}{f_1} + \frac{\omega_2}{f_2} = 0$.

3. The refractive index of glass is highest for which of the following colors of white light?

Explanation: According to Cauchy's relation $\mu = A + \frac{B}{\lambda^2}$, shorter wavelengths correspond to higher refractive indices; thus violet has the highest refractive index.

4. If the dispersive power of crown glass is $0.02$ and that of flint glass is $0.04$, and a crown glass lens of focal length $+10\text{ cm}$ is used, what focal length of flint glass lens is required for an achromatic doublet in contact?

Explanation: Using $\frac{\omega_1}{f_1} + \frac{\omega_2}{f_2} = 0$, we get $\frac{0.02}{10} + \frac{0.04}{f_2} = 0 implies f_2 = -20\text{ cm}$.

5. Longitudinal Chromatic Aberration is measured by:

Explanation: Longitudinal Chromatic Aberration is the axial focal separation between red and violet rays, given by $LCA = f_r - f_v$.

6. Two lenses of the same material are separated by a distance $d$ to form an achromatic system. If their focal lengths are $f_1$ and $f_2$, then $d$ equals:

Explanation: For lenses of the same material ($\omega_1 = \omega_2$), the condition for achromatism when separated by distance $d$ is $d = \frac{f_1 + f_2}{2}$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. What is the primary physical origin of chromatic aberration in lenses?

Explanation: Chromatic aberration arises because the refractive index of the lens material depends on the wavelength of light.

2. If Cauchy's constants are $A$ and $B$, the refractive index for a very long wavelength ($\lambda \to \infty$) approaches:

Explanation: As $\lambda \to \infty$, the term $\frac{B}{\lambda^2}$ approaches zero, leaving $\mu = A$.

3. A thin lens has a focal length of $50\text{ cm}$ for yellow light. If its dispersive power is $0.03$, what is the focal length difference between red and violet light?

Explanation: $f_r - f_v = \omega f_y = 0.03 \times 50\text{ cm} = 1.5\text{ cm}$.

4. Which lens combination satisfies the condition for achromatism and produces a converging equivalent focal length?

Explanation: To make a converging (positive focal length) achromatic doublet, the convex lens (crown, lower dispersive power) must be stronger than the concave lens (flint, higher dispersive power).

5. In an optical system of two separated lenses, the equivalent focal length $F$ for thin coaxial lenses separated by distance $d$ is:

Explanation: The standard formula for the equivalent focal length of two separated thin lenses is $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$.

6. What happens to the Longitudinal Chromatic Aberration of a lens if its focal length for mean color is doubled, keeping dispersive power constant?

Explanation: Since $LCA = \omega f_y$, LCA is directly proportional to $f_y$; doubling $f_y$ doubles the LCA.

7. Which of the following statements regarding chromatic aberration is incorrect?

Explanation: It cannot be completely eliminated for all wavelengths using only a two-lens achromatic doublet due to the secondary spectrum.

8. If $\mu_v = 1.65$, $\mu_r = 1.60$, and $\mu_y = 1.62$, what is the dispersive power of the material?

Explanation: $\omega = \frac{\mu_v - \mu_r}{\mu_y - 1} = \frac{1.65 - 1.60}{1.62 - 1} = \frac{0.05}{0.62} \approx 0.0806$.

9. For an axial point object, how does chromatic aberration manifest in a single lens?

Explanation: Axial object LCA manifests as a blurred colored patch along the optic axis because different colors focus at different points.

10. Two lenses with focal lengths $f_1 = +20\text{ cm}$ and $f_2 = -10\text{ cm}$ are in contact. What is the equivalent focal length of the combination?

Explanation: Using $\frac{1}{F} = \frac{1}{20} + \frac{1}{-10} = \frac{1 - 2}{20} = -\frac{1}{20}$, so $F = -20\text{ cm}$.