Diffraction — single slit intensity pattern derivation: Foundations

2.5.14 · D1 · Physics › Optics › Diffraction — single slit intensity pattern derivation

Yeh page parent derivation ke har symbol aur idea ko build karta hai, ek smart 12-saal ke bachche se shuru karke jo pehle kabhi kuch nahi dekha. Neeche kuch bhi assume nahi kiya gaya — har block agla block earn karta hai.

0 — "Wave" aur "wavelength" hote kya hain

Figure dekho: chhote rulers ek pura $λ$ mark karte hain — ek crest (hump ki top) se agले crest tak. Visible light ke liye $λ$ bahut chhoti hoti hai, lagbhag $400$ – $700$ nanometres ( $1$ nm $= 1 0^{- 9}$ m, ek billionth metre).

1 — Phase: ek wave apne cycle mein kitni door hai

Do waves jo "in step" hain (same phase) add hokar ek badi wave banate hain. Do waves jo exactly aadha cycle alag hain ( $π$ out of phase) — ek upar ja rahi hai jab doosri utni hi neeche jaati hai — woh cancel ho jaati hain. Yeh upar-add / opposite-cancel rule pattern ki poori physics hai.

2 — Path difference: phases alag kyun ho jaate hain

Extra distance aur extra phase ke beech ka link poori derivation ka key sentence hai:

3 — Angles aur slit: $θ$ , $a$ , aur $sin θ$

Figure mein, do parallel rays slit ke top aur bottom edges se usi door point ki taraf $θ$ angle par jaati hain. Top ray se bottom ray par ek perpendicular (dashed line) daalo. Jo chhota right triangle banta hai usmein:

Hypotenuse = slit width $a$ ,
angle $θ$ ke opposite side = wo extra path jo bottom ray travel karta hai.

4 — Bahut saari waves add karna: phasors

Jab har strip ka apna phase hota hai, toh hum unki saari waves add karte hain. Wiggly curves ko haath se add karna painful hai, isliye physicists ek picture use karte hain jise phasor kehte hain.

Jab saare phases equal hain, arrows ek hi direction mein point karte hain → woh ek lambe seedhe arrow mein stack ho jaate hain → maximum brightness.
Jab phases evenly fan out hokar poori tarah wapas shuru par curl kar jaate hain, arrows ek closed loop banaate hain → start aur end coincide karte hain → total arrow ki length zero → darkness.

Yeh parent note ke Step 2 ka geometric heart hai. Poori machinery ke liye Phasor Addition of Waves dekho.

5 — Do starring variables: $β$ aur $I$

Recall

$β = 0$ physically kya matlab hai? Zero phase spread — har strip perfectly in step (yeh $θ = 0$ par hota hai). Phasors seedhe stack hote hain, $\frac{s i n β}{β} \to 1$ , aur $I = I_{0}$ : central maximum. ::: $β = 0$ ka matlab hai saare wavelets in phase hain → sabse bright central peak.

6 — Woh tool jo answer mein appear karta hai: $\frac{s i n β}{β}$

Yeh woh geometry hai jo famous ratio produce karta hai, step by step dikhaya gaya hai na ki sirf assert kiya gaya. Tiny equal phasors, har ek pichle se thoda zyada turn hua, ek arc of a circle mein bend ho jaate hain. Us circle ki radius $R$ rakho.

Ab inhe light se connect karo:

Arc length kabhi nahi badlti jab $θ$ vary karta hai — yeh saare chhote phasor lengths ka total hai jo end-to-end rakhkhe hain, jo exactly all-in-phase amplitude hai. Toh $arc = A_{0}$ , deta hai $2 R β = A_{0}$ .
Resultant amplitude $A$ us arc par straight chord hai: $A = chord = 2 R sin β$ .

Unknown radius $R$ eliminate karne ke liye ek ko doosre se divide karo: $\frac{A}{A _{0}} = \frac{2 R s i n β}{2 R β} = \frac{s i n β}{β} ⟹ A = A_{0} \frac{s i n β}{β} .$

$β = 0$ par yeh ratio $\frac{0}{0}$ hai, jo undefined lagta hai — lekin jab $β$ shrink hota hai, $sin β \approx β$ , toh ratio $1$ approach karta hai. Isliye centre finite aur bright hai, koi hole nahi.

Prerequisite map

Parent derivation aur uske neighbours in sabhi par build karte hain: Diffraction — single slit intensity pattern derivation dekho, aur "many sources" ideas ki physics Huygens Principle, Young's Double Slit Experiment, aur baad mein Diffraction Grating mein.

Equipment checklist

Right side cover karo aur check karo ki derivation padhne se pehle har ek ka jawab de sako:

Ek wave ka ek pura upar-neeche jaana kitni distance cover karta hai... ::: ek wavelength $λ$ .
Phase ka ek pura cycle equal hota hai... ::: $2 π$ radians (ya $36 0^{\circ}$ ).
Extra path $Δ$ extra phase mein convert hota hai is factor ke zariye... ::: $\frac{2 π}{λ}$ .
Do waves jo exactly $π$ out of phase hain, jab add hoti hain toh kya hota hai? ::: woh ek doosre ko completely cancel kar deti hain.
Top-edge ray ki compare mein bottom-edge ray ki extra distance hai... ::: $a sin θ$ .
Hum $sin θ$ use karte hain (na $cos$ ya $tan$ ) kyunki hum slit width ko project karte hain... ::: us direction par jis direction mein rays travel karti hain.
Fringe order $m$ count karta hai... ::: kaunsa dark fringe (iske values $\pm 1, \pm 2, \dots$ hain, kabhi $0$ nahi).
Jo dark fringes appear ho sakti hain unki number is condition se limited hai... ::: $∣ m ∣ \leq a / λ$ (kyunki $∣ sin θ ∣ \leq 1$ ).
Ek phasor ek arrow hai jiska length matlab hai ... aur jiska direction matlab hai ... ::: length = amplitude, direction = phase.
Phasors ka closed loop mein curl karna matlab hai total field... ::: zero → ek dark fringe.
$A_{0}$ define hota hai woh amplitude ke roop mein jab... ::: har strip in phase ho ( $θ = 0$ ); yeh phasor curve ki arc length ke equal hoti hai.
Radius $R$ ke circular arc ke liye jo angle $2 β$ subtend karta hai: arc length = ... aur chord = ... ::: arc $= 2 R β$ , chord $= 2 R sin β$ .
$β = \frac{π a sin θ}{λ}$ physically represent karta hai... ::: slit mein total phase spread ka aadha.
Intensity amplitude $A$ se relate hoti hai... ::: $I \propto A^{2}$ (square karo).
Jab $β \to 0$ , ratio $\frac{sin β}{β} \to$ ... ::: $1$ (bright central maximum deta hai).