2.5.9
Ek aberration tab hota hai jab koi real lens ya mirror ek point object ka perfect point image banane mein fail ho jaata hai. Ideal "thin-lens" formulas ye maante hain ki rays paraxial hain (axis ke close) aur single wavelength hai. Aberrations tab hote hain jab ye assumptions toot jaati hain.
Definition Chromatic aberrationFocal length ka wavelength ke saath vary karna , kyunki glass ka refractive index n ( λ ) alag-alag distances par focus karte hain.
YE KYUN HOTA HAI. Glass dispersive hota hai: n blue (chhoti λ red (badi λ f ∝ n − 1 1 n chhota focal length.
Worked example Crown glass, focal length 100 mmDiya hua n C = 1.514 n D = 1.517 n F = 1.522
Step 1 — dispersive power: ω = 1.517 − 1 1.522 − 1.514 = 0.517 0.008 ≈ 0.0155 Ye step kyun? ω
Step 2 — spread: f C − f F = ω f = 0.0155 × 100 ≈ 1.55 Kyun? Ye seedha f f C − f F = ω paas hai.
Intuition Ek jhooth ko khud se cancel karoEk converging crown lens ko ek diverging flint lens ke saath pair karo (flint ki dispersion zyada hoti hai). Flint ka opposite-sign colour spread, crown ke colour spread ko cancel kar deta hai. Net power bachi rehti hai; colour error nahi.
Definition Spherical aberrationEk spherical surface ka ek defect jahan outer (marginal) zones se guzarne wali rays lens ke zyada paas focus karti hain, central (paraxial) zone se guzarne wali rays se. Koi single sharp focus nahi hota, sirf ek "circle of least confusion" hota hai.
YE KYUN HOTA HAI. Ek spherical surface grind karna aasaan hai lekin ye ideal shape nahi hai. Small-angle approximation sin θ ≈ θ f 1 sin θ = θ − 6 θ 3 + 120 θ 5 − … θ 3 θ − θ 3 /6 zyada modnewala banana deta hai → woh axis ko jaldi cross kar lete hain.
Intuition Longitudinal vs lateral
Longitudinal SA : axis ke saath marginal focus aur paraxial focus ke beech ki distance.Lateral SA : paraxial focal plane par blur radius.Sabse chota blur dono ke beech mein hota hai — circle of least confusion , screen rakhne ki best jagah.
Worked example Telescope mirrors parabolic kyun hote hainEk paraboloid saare axial parallel rays ko ek point par focus karta hai — exactly isliye kyunki ye woh shape hai jiske liye optical path length equal hoti hai, on-axis light ke liye koi θ 3 nahi .
Ye kyun kaam karta hai: parabola ek focus tak equal optical path ka locus hai, isliye iske on-axis ke liye zero spherical aberration hai infinity par ek point ke liye. Ek spherical mirror sirf apne centre ke paas iska approximation karta hai.
Recall Teen practical cures (dekhne se pehle try karo)
Aperture stop down karo (marginal rays khatam karo).Lens ko bend karo (fixed f R 1 , R 2 Aspheric / parabolic surfaces use karo, ya ek doublet jo positive aur negative SA balance kare.
Intuition Ek key differenceChromatic colour (wavelength) par depend karta hai aur paraxial ray ke liye bhi hota hai.
Spherical monochromatic hai — ye ek single colour ke saath hota hai, purely ray ki height se.
Ye independent hain aur alag tricks se correct hote hain (colour ke liye doublet, SA ke liye shape/aperture).
Common mistake Classic confusions ko steel-man karna
"Dono dispersion se aate hain." Sahi lagta hai kyunki dono images blur karte hain. Fix: sirf chromatic mein n ( λ ) "Chhota aperture chromatic aberration theek karta hai." Tempting lagta hai — stopping down SA fix karta hai, toh colour kyun nahi? Fix: chromatic ek focal-length shift hai, marginal-ray effect nahi; white light ki thin axial pencil phir bhi colours mein split hoti hai. Tumhe doublet chahiye, iris nahi."Marginal rays door focus karti hain." Aperture bada hai, toh surely zyada door? Fix: − θ 3 /6 ki taraf hai, toh marginal rays jaldi cross karti hain — focus zyada paas hota hai."Parabolic mirrors perfect hain." Ye sirf on-axis perfect hain; off-axis points abhi bhi coma se suffer karte hain, jo ek alag aberration hai.
"C olour C hanges C urvature-of-focus; S phere S hortens for the S ides."
Chromatic = wavelength; Spherical = aperture height. Blue brief hai, red door tak pahuncha. Edges jaldi focus karti hain.
Recall Feynman: 12 saal ke bacche ko explain karoEk magnifying glass ko saari sunlight ko ek tiny dot mein gather karna chahiye. Lekin ye do tarikon se cheat karta hai. Pehla, white light secretly colours se bani hai, aur glass blue ko red se zyada modta hai, toh colours thodi alag jagahon par land karte hain — ye dot ko ek tiny rainbow mein smear kar deta hai. Ye hai chromatic aberration. Doosra, glass ke curved edges "zyada curve" hain, toh rim se aane wali light thodi zyada mur jaati hai aur doosron se jaldi milti hai, middle ki light se pehle. Ye hai spherical aberration. Rainbow theek karne ke liye do tarah ke glass glue karte hain taaki ek doosre ko undo kare; rim problem theek karne ke liye ya toh edges cover karo ya glass ko ek special curve (parabola) ki tarah shape karo.
Ek simple lens mein blue light red se paas kyun focus karta hai? n f ∝ 1/ ( n − 1 ) n f
Wavelength ke saath fractional focal shift batao. f df = − n − 1 d n
Dispersive power ω ω = n D − 1 n F − n C
Achromatic doublet condition kya hai? ω 1 P 1 + ω 2 P 2 = 0
Achromat mein opposite-sign powers kyun chahiye? Kyunki ω > 0 ω 1 P 1 + ω 2 P 2 = 0 P 1 , P 2
Spherical aberration kis approximation ki failure se hota hai? sin θ ≈ θ − θ 3 /6
Marginal rays paraxial rays ke comparison mein nearer focus karti hain ya farther? Nearer (lens ke zyada paas).
SA blur, aperture radius h Roughly h 3
Circle of least confusion kya hai? Sabse chota blur spot, marginal aur paraxial foci ke beech mein — best screen position.
Kya spherical aberration colour-dependent hai? Nahi, ye monochromatic hai — purely ray height/aperture ka function hai.
Telescope mirrors parabolic kyun hote hain? Ek paraboloid saare on-axis parallel rays ko zero spherical aberration ke saath ek point par focus karta hai.
Kya stopping down chromatic aberration theek karta hai? Nahi — ye SA theek karta hai, lekin colour focal-length differences rehte hain; doublet use karo.
Lensmaker's Equation — dono "jhooth" ka source (n Dispersion and Refractive Index — n ( λ ) Achromatic Doublet — practical colour correctionParaxial Approximation — small angles kyun matter karte hainSeidel Aberrations — coma, astigmatism, field curvature, distortion (baaki chaar)Parabolic Mirrors and Telescopes — zero-SA reflectorsFraunhofer Lines — C, D, F reference wavelengths
Dispersion n depends on wavelength
Longitudinal spread f_C - f_F
Wide-angle rays bend too much
