Foundations — Aberrations — chromatic, spherical (concepts)

2.5.9 · D1 · Physics › Optics › Aberrations — chromatic, spherical (concepts)

Yeh page ek toolbox hai. Kisi lens ko fail hote dekhne se pehle, aapko exactly pata hona chahiye ki $\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$ mein har letter ka kya matlab hai, drawing mein wo kaisa dikhta hai, aur yeh topic uske bina kyun nahi chal sakta. Hum inhe order mein build karte hain — har ek sirf upar waale par lean karta hai.

0. "Point image" kya hota hai? (jo cheez galat ho jaati hai)

Figure dekho. Left side par, ek perfect lens hai: baayi taraf ke point se nikalti har ray is tarah bend hoti hai ki wo sab ek jagah daayi taraf cross karti hain. Right side par, ek real lens hai: rays ek doosre ko miss karti hain — wo thodi alag jagahon par cross karti hain, isliye dot ki jagah ek blurry smear milta hai.

1. Ray aur uska angle $\theta$

$\theta$ ki zaroorat kyun hai? Kyunki bending arrival ke angle par depend karti hai. Jo ray glass ko almost flat graze kare (chhota $\theta$) wo barely bend hoti hai; jo ray steeply aakar strike kare (bada $\theta$) wo zyada bend hoti hai. Figure mein, blue near-axis ray ek tiny $\theta$ banati hai; pink rim ray ek bada $\theta$ banati hai. Yahi ek difference spherical aberration ka poora seed hai — isliye $\theta$ pehle apni jagah earn karta hai.

2. Ray height $h$ aur "paraxial"

$h$ aur $\theta$ ek hi fact ke do views hain: ray jitni zyada oopar enter kare (bada $h$), use focus tak pahunchne ke liye utna hi zyada bend karna hoga (bada $\theta$). $R$ radius ke lens ke liye, roughly $\theta$ $h$ ke saath badhta hai.

3. $\sin \theta$ kyun, aur hum use kyun kaatein: small-angle approximation

Light bending Snell's law ko obey karti hai, jisme $\sin \theta$ hota hai — angle ka sine. Sine ek tool hai jo ek angle ko right triangle ke side lengths ke ratio mein convert karta hai (opposite over hypotenuse). Hume iske zaroorat hai kyunki refraction physically us ratio par depend karti hai, raw angle par nahi.

Lekin $\sin \theta$ equations solve karne ke liye awkward hai. Isliye physicists iska Taylor series use karte hain — ek exact tarika hai ek curvy function ko simple powers ki endless sum ke roop mein likhne ka:

$\sin \theta =\theta -\frac{{\theta }^3}{6}+\frac{{\theta }^5}{120}-\dots$

Figure mein true curve $\sin \theta$ (white) ko straight line $\theta$ (blue) ke against plot kiya gaya hai. $\theta =0$ ke paas dono ek doosre ke upar hain — approximation superb hai. Lekin jaise $\theta$ badhta hai, white curve line ke neeche droops ho jaati hai: yeh gap exactly wo $-{\theta }^3/6$ term hai jise humne hata diya.

4. Refractive index $n$

Figure mein ek prism white light ko split kar raha hai. Blue ray sabse zyada bend hoti hai (sabse bada $n$), red sabse kam (sabse chhota $n$). Yahi ek picture chromatic aberration ki janam jagah hai: agar ek lens blue ko red se zyada bend karta hai, to blue ko pehle focus aana hoga.

5. Radii $R_1$, $R_2$ aur shape factor $K$

Inhe kyun alag karein? Kyunki chromatic aberration sirf $n$ change karta hai jabki $K$ fixed rehta hai — yeh separation baad mein differentiate karne mein madad karta hai.

6. Focal length $f$ aur lensmaker's equation

$n-1$ kyun aur $n$ kyun nahi? Kyunki bending surrounding air ($n\approx 1$) ke relative hai. Apne hi index ke medium mein dubo hua lens bilkul bhi light nahi modega — "$-1$" contrast measure karta hai.

7. Power $P$ — wahi cheez, ulti

Yahi additivity exactly hai jis wajah se Achromatic Doublet kaam karta hai — aap do lenses glue karte ho aur unki powers add karte ho, colour errors ko cancel hone dete hue jabki total power survive karti hai.

8. Dispersive power $\omega$ aur Fraunhofer lines

Chromatic aberration fix karne ke liye hume ek number chahiye jis se pata chale "glass kitna colours spread karta hai." Woh number $\omega$ hai.

Prerequisite map

Ise top-down padho: left branch (ray, height, sine cut) spherical aberration ko feed karta hai; middle/right branch ($n$, colour, $\omega$) aur lensmaker's equation chromatic aberration aur uski cure ko feed karta hai. Dono parent topic mein pour hote hain.

Equipment checklist

Daayein side ko cover karo aur har ek ka jawab do. Agar koi stumble kare, to upar woh section dobara padho.

Jab upar ki har cheez second nature ho jaaye, tab aap parent note mein dono jhooton ko toot te dekhne ke liye ready hain: Aberrations — Chromatic & Spherical.