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2.5.9Optics

Aberrations — chromatic, spherical (concepts)

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🇮🇳 Yeh note Hinglish mein padho →

An aberration is any failure of a real lens or mirror to form a perfect point image of a point object. The ideal "thin-lens" formulas assume paraxial rays (close to the axis) and single wavelength. Aberrations are what happens when those assumptions break.

1. Chromatic Aberration

WHY it happens. Glass is dispersive: nn is larger for blue (short λ\lambda) than for red (long λ\lambda). Since f∝1n−1f\propto \dfrac{1}{n-1}, a bigger nn means a shorter focal length.

Figure — Aberrations — chromatic, spherical (concepts)

Fixing it: the achromatic doublet

2. Spherical Aberration

WHY it happens. A spherical surface is easy to grind but is not the ideal shape. The small-angle approximation sin⁡θ≈θ\sin\theta\approx\theta used to derive 1f\frac1f is only the first term of sin⁡θ=θ−θ36+θ5120−…\sin\theta=\theta-\frac{\theta^3}{6}+\frac{\theta^5}{120}-\dots The θ3\theta^3 term (third-order, "Seidel") is the leading correction. Marginal rays have larger θ\theta, so the neglected −θ3/6-\theta^3/6 makes them bend more than the paraxial formula predicts → they cross the axis early.

Reducing spherical aberration

Recall Three practical cures (try before peeking)
  1. Stop down the aperture (kill marginal rays).
  2. Bend the lens (choose R1,R2R_1,R_2 for fixed ff) — the "Coddington" best-form lens spreads the bending across both surfaces.
  3. Use aspheric / parabolic surfaces, or a doublet that balances positive and negative SA.

Chromatic vs Spherical — the contrast

Recall Feynman: explain to a 12-year-old

A magnifying glass is supposed to gather all the sunlight into one tiny dot. But it cheats in two ways. First, white light is secretly made of colours, and the glass bends blue more than red, so the colours land at slightly different spots — that smears the dot into a tiny rainbow. That's chromatic aberration. Second, the curved edges of the glass are "too curved," so light coming through the rim turns a bit too hard and meets the others too soon, before the light from the middle. That's spherical aberration. To fix the rainbow you glue two kinds of glass together so one undoes the other; to fix the rim problem you either cover the edges or shape the glass like a special curve (a parabola).

Flashcards

Why does blue light focus closer than red in a simple lens?
nn is larger for blue, and f∝1/(n−1)f\propto 1/(n-1), so larger nn → shorter ff.
State the fractional focal shift with wavelength.
dff=−dnn−1\dfrac{df}{f}=-\dfrac{dn}{n-1}.
Define dispersive power ω\omega.
ω=nF−nCnD−1\omega=\dfrac{n_F-n_C}{n_D-1} (blue−red over yellow−1).
What is the achromatic doublet condition?
ω1P1+ω2P2=0\omega_1 P_1+\omega_2 P_2=0, so one lens converges and the other diverges.
Why does an achromat need opposite-sign powers?
Because ω>0\omega>0 always, so ω1P1+ω2P2=0\omega_1P_1+\omega_2P_2=0 forces P1,P2P_1,P_2 to differ in sign.
What approximation's failure causes spherical aberration?
Dropping the −θ3/6-\theta^3/6 term in sin⁡θ≈θ\sin\theta\approx\theta (third-order term).
Do marginal rays focus nearer or farther than paraxial rays?
Nearer (closer to the lens).
How does SA blur scale with aperture radius hh?
Roughly as h3h^3 — halving the aperture cuts blur to 1/8.
What is the circle of least confusion?
The smallest blur spot, located between the marginal and paraxial foci — best screen position.
Is spherical aberration colour-dependent?
No, it is monochromatic — purely a function of ray height/aperture.
Why are telescope mirrors parabolic?
A paraboloid focuses all on-axis parallel rays to one point with zero spherical aberration.
Does stopping down cure chromatic aberration?
No — it cures SA, but colour focal-length differences remain; use a doublet.

Connections

  • Lensmaker's Equation — the source of both "lies" (nn and paraxial)
  • Dispersion and Refractive Index — n(λ)n(\lambda) driving chromatic aberration
  • Achromatic Doublet — practical colour correction
  • Paraxial Approximation — why small angles matter
  • Seidel Aberrations — coma, astigmatism, field curvature, distortion (the other four)
  • Parabolic Mirrors and Telescopes — zero-SA reflectors
  • Fraunhofer Lines — C, D, F reference wavelengths

Concept Map

assumes single n

assumes paraxial rays

causes

blue n larger than red

quantified by

leads to

gives

corrected by

crown plus flint

from

Lensmaker equation

Chromatic aberration

Spherical aberration

Dispersion n depends on wavelength

Blue focuses nearer

df/f = -dn/(n-1)

Dispersive power omega

Longitudinal spread f_C - f_F

Achromatic doublet

dP = 0 across colours

Wide-angle rays bend too much

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, ek perfect lens ka kaam hai ki point object ka point image bane. Lekin asli lens do galtiyan karta hai. Pehli galti: glass har colour ko alag-alag bend karta hai. Blue light ka refractive index nn zyada hota hai, aur kyunki f∝1/(n−1)f \propto 1/(n-1), isliye blue light lens ke paas focus hoti hai aur red door. Isi colour-wise focus difference ko chromatic aberration kehte hain. Iska fix? Ek crown (converging) aur ek flint (diverging) lens ko jodo — flint ka opposite dispersion crown ki galti ko cancel kar deta hai. Condition: ω1P1+ω2P2=0\omega_1 P_1 + \omega_2 P_2 = 0.

Doosri galti: hum derivation mein sin⁡θ≈θ\sin\theta \approx \theta maan lete hain, par asli mein sin⁡θ=θ−θ3/6+…\sin\theta = \theta - \theta^3/6 + \dots. Jo rays lens ke kinaron (marginal zone) se aati hain unka θ\theta bada hota hai, to woh zyada bend hoti hain aur axis ko jaldi (lens ke paas) cross karti hain. Center waali rays door focus karti hain. Single sharp point banta hi nahi — sirf ek circle of least confusion. Yeh hai spherical aberration, aur yeh ek hi colour mein bhi hoti hai.

Yaad rakhne ki baat: chromatic = colour ka chakkar, spherical = aperture height ka chakkar. Aperture chhota karne se (stop down) spherical theek hoti hai kyunki marginal rays hat jaati hain (blur ∼h3\sim h^3, aadha karo to 1/8 blur). Par chromatic theek nahi hoti stop-down se — uske liye doublet chahiye. Aur telescope mirror parabola shape ka isliye banate hain kyunki parabola axial parallel rays ko ekdum ek point pe laata hai, zero spherical aberration ke saath.

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