Chromatic Aberration: The Defect and Its Correction in Lenses
Understand the cause of chromatic aberration and how achromatic doublet lenses eliminate this optical defect. Complete guide for JEE Physics.
What is Chromatic Aberration?
Chromatic aberration is an optical phenomenon where a lens fails to focus all colors to the same convergence point, resulting in colored fringes around images. This occurs because:
- Different wavelengths of light bend at different angles when passing through a lens
- Refractive index varies with wavelength (dispersion)
- Violet light bends more than red light in glass
- This causes color separation at the focal point
Understanding Chromatic Aberration
White light → Lens → Color separation at different focal points
🔬 Scientific Explanation:
Dispersion Principle: The refractive index (μ) of glass decreases with increasing wavelength:
$μ_v > μ_r$ where $μ_v$ = refractive index for violet, $μ_r$ = refractive index for red
Lens Maker's Formula: $f = \frac{1}{(μ-1)(\frac{1}{R_1} - \frac{1}{R_2})}$
Since $μ_v > μ_r$, focal length for violet light $f_v$ is shorter than for red light $f_r$:
$f_v < f_r$
🎯 Key Points:
Longitudinal Chromatic Aberration: Different colors focus at different distances along the optical axis
Lateral Chromatic Aberration: Different colors focus at different positions in the focal plane
Axial Chromatic Aberration: Variation of focal length with wavelength
Mathematical Analysis of Chromatic Aberration
📐 Lens Formula and Dispersion:
For a thin lens: $\frac{1}{f} = (μ-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$
Differentiating with respect to wavelength:
$\frac{-1}{f^2}\frac{df}{dλ} = \frac{dμ}{dλ}\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$
Chromatic aberration: $df = -f^2 \frac{dμ}{dλ}\left(\frac{1}{R_1} - \frac{1}{R_2}\right) dλ$
Dispersive power: $ω = \frac{μ_v - μ_r}{μ_y - 1}$ where $μ_y$ = refractive index for yellow light
🎯 JEE Application Example:
Problem: A convex lens has focal length 20 cm for red light and 18 cm for violet light. Calculate the dispersive power of the lens material.
Solution:
Mean focal length $f_y = \sqrt{f_r f_v} = \sqrt{20 × 18} = 18.97$ cm
Mean refractive index $μ_y = 1 + \frac{R_1 R_2}{f_y(R_2 - R_1)}$ (using lens maker's formula)
Dispersive power $ω = \frac{f_r - f_v}{f_y} = \frac{20 - 18}{18.97} = 0.105$
Achromatic Doublet: The Solution
Convex (crown glass) + Concave (flint glass) = Achromatic combination
🔧 Working Principle:
Two lenses of different materials:
• Convex lens made of crown glass (lower dispersion)
• Concave lens made of flint glass (higher dispersion)
Condition for achromatism:
$\frac{ω_1}{f_1} + \frac{ω_2}{f_2} = 0$
Where $ω_1$, $ω_2$ are dispersive powers and $f_1$, $f_2$ are focal lengths
Combined focal length: $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2}$
✅ Why This Works:
The convex lens converges light but separates colors
The concave lens diverges light but separates colors in opposite direction
Net effect: Colors recombine at the same focal point while maintaining overall convergence
Result: Sharp, color-free image!
🚀 Problem-Solving Strategies
For Chromatic Aberration:
- Remember: violet focuses closer than red
- Dispersive power ω = (μ_v - μ_r)/(μ_y - 1)
- Longitudinal CA = f_r - f_v
- CA increases with lens power
For Achromatic Doublet:
- Condition: ω₁/f₁ + ω₂/f₂ = 0
- Combined power: P = P₁ + P₂
- Crown glass: lower ω, convex shape
- Flint glass: higher ω, concave shape
Advanced Applications Available
Includes apochromatic lenses, photographic lens design, and JEE Advanced level problems
📝 Quick Self-Test
Try these JEE-level problems to test your understanding:
1. Why does violet light focus closer to a lens than red light?
2. Calculate the focal length of an achromatic doublet with given dispersive powers
3. Explain why a single lens cannot eliminate chromatic aberration completely
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