Question bank — Resolving power — Rayleigh criterion

True — ${\theta }_R=1.22\lambda /D$, toh $D$ badhane se resolvable angle chhhota ho jaata hai. Zyada wide aperture kam diffract karta hai, isliye har Airy disc tighter hoti hai aur do sources chhhoti separations par bhi distinguishable rehte hain.

False — yeh do alag gains hain. Light-gathering area ke saath scale karta hai ($\propto D^2$, brightness), jabki resolution $1/D$ ke saath scale karta hai (detail ki fineness); bada telescope dono mein jeetta hai lekin alag-alag physical reasons se.

False — yeh ek agreed convention hai jo mark karta hai jahan combined intensity dip (~26%) detectable banti hai. Nature exactly $1.22\lambda /D$ par ek blob se do blobs mein "switch" nahi karta; criterion sirf ek fair, computable line kheenchta hai.

False — 1.22 sirf circular aperture ke liye hai (Bessel function $J_1$ ka pehla zero). Ek slit plain ${\theta }_R=\lambda /a$ use karta hai; yeh factor ek disc pe integrate karne se aata hai, strip pe nahi.

True — $d_{\min }\propto \lambda$, toh shorter-wavelength (blue) light se smaller resolvable distance milti hai. Isliye UV aur electron microscopes (tiny effective $\lambda$) bahut fine detail resolve kar paate hain.

False — diffraction limit ke baar sirf blur ko enlarge karte ho ("empty magnification"). Detail ${\theta }_R$ ya $d_{\min }$ se cap hoti hai, jo diffraction se set hoti hai, image ko kitna stretch kiya isse nahi.

False (point miss ho raha hai) — transparency zaroori hai lekin asli gain yeh hai ki oil ka zyada refractive index $n$ N.A. $=n\sin \beta$ badhata hai, collecting cone wide hoti hai aur $d_{\min }=0.61\lambda /\text{N.A.}$ chhhota ho jaata hai.

False — Rayleigh ek human-eye convention hai; low-noise detector plus curve-fitting peaks ko thoda aur close se separate kar sakta hai. Criterion "comfortably resolvable" mark karta hai, koi absolute wall nahi.

True — ${\theta }_R=1.22\lambda /D$ sirf ratio $\lambda /D$ par depend karta hai. Dono ko double karne se $(2\lambda )/(2D)=\lambda /D$ milta hai, toh effects exactly cancel ho jaate hain.

Formula radians return karta hai, degrees nahi. Sirf radians mein hi small-angle relation $s=\theta L$ retina ya page par sahi physical spacing deta hai.

Direction galat hai — chhhota $D$ matlab zyada diffraction spreading (${\theta }_R\propto 1/D$), toh constricted pupil thoda worse resolve karta hai. Chhhote pupil ka faida depth of field hai, resolution nahi.

Circular geometry pehle zero ko 1.22 factor se bahar dhakelta hai, giving ${\theta }_R=1.22\lambda /D$. 1.22 drop karna slit result ko disc ke liye use karna hai — yeh ek common galti hai.

Sample ko door karne se cone half-angle $\beta$ chhhota ho jaata hai, N.A. $=n\sin \beta$ kum hoti hai aur $d_{\min }$ badhta hai. Aap chahte ho ki sample paas ho taaki objective wide cone subtend kare.

Ulta hai — green ka $\lambda$ chhhota hota hai, toh ${\theta }_R$ chhhota hota hai aur wahi spacing ab comfortably resolvable hai. Shorter wavelength hamesha help karta hai.

Ulta hai — resolving power $1/{\theta }_R$ hai. Chhhota ${\theta }_R$ (fine angle) matlab high resolving power; bada ${\theta }_R$ matlab instrument sirf coarse, widely-spaced sources separate kar sakta hai.

Finite hole wavefront ke edges kaat deta hai, aur Huygens' principle ke mutabiq har truncated wavefront spread karta hai — wave ko kaat'na hi diffraction hai, toh koi bhi finite lens point ko true point par focus nahi kar sakta.

Ek slit 1-D strip par diffraction integrate karta hai (first null $\lambda /a$ par); ek circle 2-D disc par integrate karta hai, jisse Bessel-function pattern milta hai jiska pehla zero $J_1$ $1.22\lambda /D$ par baith'ta hai — 2-D geometry energy ko bahar redistribute karta hai.

Kyunki first-minimum position pehle se exactly known hai, jo criterion ko compute karna clean banata hai, aur us spacing par ~26% central dip roughly match karta hai jo ek typical aankhein just detect kar sakti hai.

$\lambda \approx 500$ nm aur best oil N.A. $\approx 1.4$ ke saath, $d_{\min }=0.61\lambda /\text{N.A.}\approx 220$ nm. Aap ordinary lenses se isse beat nahi kar sakte kyunki yeh diffraction se set hota hai, lens quality se nahi.

Sirf light-gathering ke liye nahi — bada $D$ directly ${\theta }_R=1.22\lambda /D$ chhhota karta hai, unhe stars (aur planet-star pairs) separate karne deta hai jo chhhote aperture mein ek blob mein merge ho jaate hain.

Arc-length relation $s=\theta L$ sirf radians mein valid hai (yeh circle ke liye $s=r\theta$ se aati hai). Degrees ek spurious factor of $\pi /180$ introduce kar denge.

${\theta }_R=1.22\lambda /D\to 0$ — infinitely wide aperture ek point ko true point par focus karega zero diffraction blur ke saath, yeh ideal (unreachable) limit hai.

${\theta }_R\to 0$, toh arbitrarily fine detail resolvable ho jaati hai — isliye electron microscopes, picometre-scale de Broglie wavelengths ke saath, light microscopes se kahin behtar perform karte hain.

Do peaks ek shallow central dip ke saath jo peaks se lagbhag 26% neeche hai — barely itna deep ki "do hain" signal mile, toh yeh precisely resolved aur merged ki borderline par baithta hai.

$d_{\min }=0.61\lambda /1\approx 305$ nm — oil immersion ke ~220 nm se bura, yeh dikhata hai ki $\lambda$ change kiye bina finer push karne ka ek hi tarika hai ki $n$ ko 1 se upar raise karo.

Nahi — woh do alag sources rehte hain; sirf unke diffraction discs itne heavily overlap karte hain ki central dip gayab ho jaati hai aur detector ek single peak record karta hai. Resolution instrument ki limit hai, objects ki nahi.