Foundations — Resolving power — Rayleigh criterion

2.5.16 · D1 · Physics › Optics › Resolving power — Rayleigh criterion

Is page pe assume kiya gaya hai ki aapne parent note ke koi bhi symbols pehle nahi dekhe. Hum har ek ko build karenge — simple words, ek picture, aur kyun is topic ko uski zaroorat hai — ek aisi order mein jahan har idea apne pehle wale idea pe lean karta hai.

1. Angle radians mein measure karna — $\theta$

Lekin hum ise yahan degrees mein measure nahi karte. Hum ise radians mein measure karte hain.

Radians kyun aur degrees kyun nahi? Kyunki ek clean fact hai jo aap seedha picture se padh sakte hain: $\text{arc length }s=R\theta (\theta \text{ in radians})$ Arc length sirf radius times angle hai — koi messy conversion factor nahi. Degrees har equation mein ek ugly $\pi /180$ daal dete.

2. Object tak ki distance — $L$, aur angle ko length mein convert karna

Aage badhne se pehle hamen ek aur everyday symbol chahiye.

Topic ko iski zaroorat kyun hai: akela angle aapko millimetres mein real gap nahi batata. Lekin agar aap opening angle $\theta$ aur cheezein kitni door hain ($L$) dono jaante hain, toh radian ki picture directly gap deti hai. Aperture ko circle ka centre maano aur door wali plane ko radius $L$ pe rim maano; tab arc-length rule $s=R\theta$ ban jaata hai $s=\theta L(\theta \text{ in radians}).$ Toh $s$ do points ke beech ka real separation hai (maano, millimetres mein) jo distance $L$ pe tiny angle $\theta$ subtend karta hai. Yeh single line hai jisse har "do dots kitna alag ho sakte hain?" ka jawab milta hai.

3. Wavelength — $\lambda$

Topic ko iski zaroorat kyun hai: jab ek wave kisi hole se squeeze hoti hai toh kitna phailti hai yeh depend karta hai ki wavelength hole size se kaise compare karti hai. Ek long-wavelength wave (bada $\lambda$) bahut zyada fan out hoti hai; ek short-wavelength wave kam fan out hoti hai. Visible light ka $\lambda$ lagbhag $400\text{ nm}$ (violet) aur $700\text{ nm}$ (red) ke beech hota hai, jahan $1\text{ nm}=10^{-9}\text{ m}$ (metre ka ek arabwan hissa). $\lambda =550\text{ nm}$ ke paas green light "average" visible light ka common stand-in hai.

Dekho Huygens principle kyun ek wave phailti hai, aur Single-slit diffraction poore spreading pattern ke liye.

4. Aperture aur uska size — $D$ (ya slit width $a$)

Do shapes matter karte hain:

• Ek circular aperture — diameter $D$ (circle ke seedha across poori width). Aankhhen, lenses, telescopes.
• Ek slit — width $a$ ka ek lamba patla gap. Textbook diffraction mein use hota hai.

Topic ko iski zaroorat kyun hai: diffraction spreading $\propto 1/D$ hai. Ek wide hole light ko kam phailata hai aur ek sharper spot deta hai; ek narrow hole zyada phailata hai aur blur karta hai. Puri resolving-power ki kahani $\lambda$ (phailana chahti hai) aur $D$ (tight rakhna chahta hai) ke beech ki tug-of-war hai.

5. Diffraction aur Airy disc

Ek circular hole ke liye, phiali hui light uniform smear nahi banati — yeh faint rings se ghira ek bright central disc banati hai. Woh central bright disc Airy disc hai, aur uski pehli dark ring centre se ek specific angle pe hoti hai.

Is disc ki detailed shape — aur dark rings kahan girti hain — ek special function se aati hai; dekho Airy disc and Bessel functions.

6. Factor 1.22 aur pehla minimum

$1.22$ kahan se aata hai? Slit ke liye pehla dark fringe angle $\lambda /a$ pe hota hai. Circle ke liye, geometry light ko redistribute karti hai aur pehli dark ring ko thoda aur bahar dhakela jaata hai — exact amount Bessel function $J_1$ ke pehle zero se set hoti hai, jo $1.22$ number deta hai.

7. Rayleigh criterion aur symbol ${\theta }_R$

Upar di gayi sab cheezein us ek rule ke liye set up karti hain jis par parent note bana hua hai.

${\theta }_R$ ko Section 2 ke saath combine karo: distance $L$ pe do dots just separable hain jab unka gap $s={\theta }_{R}L=1.22\lambda L/D$ ho.

8. Numerical aperture — N.A. $=n\sin \beta$

Microscopes ke liye object paas hota hai, isliye relevant size diameter nahi balki cone of light hai jo objective collect karta hai.

Topic ko iski zaroorat kyun hai: paas ke object ke liye smallest resolvable distance hai $d_{\min }=\frac{0.61\lambda }{\text{N.A.}}.$

9. Small-angle shortcut

Topic ko iski zaroorat kyun hai: stars unimaginably door hain, isliye angles microscopic hote hain ($10^{-4}$ rad ya kam). Yahi allow karta hai ki hum $s=\theta L$ (Section 2) seedha use kar sakein trigonometric functions ghisaate phirne ki jagah. Dekho Small angle approximation.

Yeh foundations topic ko kaise feed karte hain

Chain choti hai aur har link ek symbol hai jo ab aapka apna hai:

1. $\lambda$ (Section 3) aur $D$ (Section 4) milke diffraction (Section 5) cause karte hain.
2. Diffraction Airy disc banata hai, jiski pehli dark ring ${\theta }_{\min }=1.22\lambda /D$ pe hoti hai — 1.22 Bessel function se aata hai (Section 6).
3. Rayleigh criterion ek disc ke centre ko dusre ke first minimum ke saath line up karta hai, limiting angle ${\theta }_R={\theta }_{\min }$ aur resolving power $1/{\theta }_R$ deta hai (Section 7).
4. Radians ($s=R\theta$, Section 1), distance $L$ (Section 2) aur small-angle shortcut (Section 9) us angle ko real gap $s={\theta }_{R}L$ mein convert karte hain.
5. Paas ke objects ke liye, numerical aperture $n\sin \beta$ (Section 8) $D$ ki jagah leta hai aur Rayleigh ka $1.22$ ban jaata hai $0.61$, microscope limit $d_{\min }$ deta hai.

Jab ready ho parent pe wapas jao: Resolving power — Rayleigh criterion. Yeh bhi dekho Diffraction grating — resolving power jahan yahi idea spectral lines pe apply hota hai.

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