Resolving power — Rayleigh criterion

2.5.16 · Physics › Optics

Resolution ko diffraction kyun limit karta hai?

Toh problem yeh hai: do sources, do overlapping Airy discs. Hum unhe kab alag kar sakte hain?

The Rayleigh Criterion

HOW: limiting angle ko scratch se derive karna

Step 1 — Ek single source ki first dark ring kahan hai?

Width $a$ ke single slit ke liye, first minimum tab hota hai jab slit ke across path difference ek wavelength ke barabar ho: $a\sin \theta =\lambda \Rightarrow {\theta }_{\min }\approx \frac{\lambda }{a}$

Yeh step kyun? Slit ki top half ko bottom half se pair karte hue, top se har ray apne partner ko cancel kar deti hai jo $a/2$ neeche hai jab unka path difference $\lambda /2$ ho; poori slit pe sum karne par $a\sin \theta =\lambda$ pe total destructive interference milti hai.

Step 2 — Circular aperture correction.

Ek real lens/aankh/telescope ka circular aperture hota hai (diameter $D$), slit nahi. Ek strip ki jagah disc pe diffraction integrate karne se 1.22 ka ek numerical factor aata hai (Bessel function $J_1$ ka first zero): ${\theta }_{\min }=1.22\frac{\lambda }{D}$

Yeh step kyun? Slit $\lambda /a$ deta hai; circular geometry diffracted energy ko redistribute karta hai aur first dark ring ko thoda bahar push karta hai, $1.22$ se multiply karke. (Slit aperture ke liye, 1.22 hata do aur width $a$ use karo.)

Step 3 — Rayleigh apply karo.

"Peak on first minimum" ka matlab hai do sources ke beech minimum angular separation jo hum resolve kar sakte hain woh exactly ${\theta }_{\min }$ ke barabar hai:

Do important special cases

(a) Telescope / aankh (objects door hain): sources angle $\theta$ se separated hain. Resolvable hai agar $\theta \ge {\theta }_R=1.22\lambda /D$.

(b) Microscope (objects paas hain, smallest distance $d$ se define hote hain): Yahan hum slide pe do points ke beech smallest separation $d$ ki parwah karte hain. Similarly objective ke collecting cone ke saath derive karne par milta hai:

$D$ ki jagah N.A. kyun? Paas ke object ke liye relevant quantity woh cone half-angle hai jo lens collect karta hai, sirf diameter nahi; immersion oil ($n>1$) effective cone ko wide karta hai, $d_{\min }$ ko shrink karta hai — isliye oil-immersion microscopy finer detail resolve karta hai.

Worked Examples

Forecast-then-Verify

Common Mistakes (Steel-manned)

Flashcards

Connections

• Single-slit diffraction — $\lambda /a$ angle ka source jo yahan reuse hota hai.
• Diffraction grating — resolving power — alag formula $R=\lambda /\Delta \lambda =nN$.
• Airy disc and Bessel functions — 1.22 factor ki origin.
• Numerical aperture — microscope resolution drive karta hai.
• Huygens principle — kyun finite apertures diffract karte hain.
• Small angle approximation — ${\theta }_R$ ko physical spacing $s=\theta L$ mein convert karna.

Concept Map