2.5.15
Ek diffraction grating bas bahut saari parallel slits hoti hain jo ek doosre ke bahut paas hoti hain. Har slit se light har direction mein jaati hai, lekin zyaadatar directions mein alag-alag slits ki waves cancel ho jaati hain. Sirf kuch special directions mein hi saari slits ek saath agree karti hain (constructive interference) — aur wahan light bahut teez, bright principal maxima mein stack ho jaati hai.
YE KYUN MATTER KARTA HAI: Kyunki maxima razor-sharp hote hain aur unka angle wavelength λ spread kar deta hai. Ye spectrometers ka dil hai jo taaron aur molecules ke chemical fingerprints padhne ke liye use hote hain.
Definition Diffraction gratingEk flat surface jisme bahut bada number N grating spacing d n d = n 1 ⇒ d = 500 1 mm = 2000 nm
Intuition HUM KYA COMPARE KAR RAHE HAINMaano light grating par normally (perpendicular) pad rahi hai. Do adjacent slits se angle θ path difference par depend karta hai.
HUM PATH DIFFERENCE KAISE NIKALTE HAIN.
Upar waali slit se neeche waali slit ki ray par ek perpendicular daalo. Jo triangle banta hai usmein slit separation d θ
Δ = d sin θ
CONSTRUCTIVE INTERFERENCE KAISE HOTI HAI.
Adjacent slits reinforce hoti hain jab unka path difference wavelengths ka ek whole number ho — kyunki tab unke crests line up ho jaate hain:
d sin θ = mλ , m = 0 , ± 1 , ± 2 , …
Intuition "Adjacent slits in phase" SAARI slits ke liye kyun kaafi haiAgar neighbour-to-neighbour extra path exactly mλ 2 mλ 3 mλ saari N . Isliye maximum itna intense (∝ N 2
Intuition LIMIT KYUN HOTI HAIsin θ sin θ m = mλ / d m
sin θ = 1 θ = 9 0 ∘
m m a x = ⌊ λ d ⌋
Worked example Example 1 — first-order angle nikalna500 lines/mm wale grating par λ = 600 nm θ 1
Step 1: d = 500 mm − 1 1 = 2.0 × 1 0 − 6 m Kyun? Spacing, lines-per-length ka reciprocal hota hai.
Step 2: d sin θ = mλ m = 1 sin θ 1 = 2.0 × 1 0 − 6 1 ⋅ 600 × 1 0 − 9 = 0.30 Ye step kyun? Humne unknown angle ke liye grating equation rearrange ki.
Step 3: θ 1 = sin − 1 ( 0.30 ) = 17. 5 ∘
Worked example Example 2 — kitne orders?Wahi grating (d = 2.0 × 1 0 − 6 m λ = 600 nm
Step 1: λ d = 600 × 1 0 − 9 2.0 × 1 0 − 6 = 3.33 Kyun? Ye batata hai ki sabse bada integer m sin θ
Step 2: m m a x = ⌊ 3.33 ⌋ = 3
Step 3: Orders hain m = − 3 , − 2 , − 1 , 0 , + 1 , + 2 , + 3 7 maxima .
Ye step kyun? Maxima dono taraf symmetrically aate hain, plus central wala.
Worked example Example 3 — data se grating spacing nikalnaEk laser (λ = 633 nm 40. 0 ∘ d
Step 1: d = sin θ mλ = sin 4 0 ∘ 2 ⋅ 633 × 1 0 − 9 Kyun? d
Step 2: sin 4 0 ∘ = 0.643 d = 0.643 1.266 × 1 0 − 6 = 1.97 × 1 0 − 6 m
Step 3: lines/mm = d 1 = 1.97 × 1 0 − 3 mm 1 ≈ 508 lines/mm
Common mistake Maxima ke liyed sin θ = ( m + 2 1 ) λ Kyun sahi lagta hai: Single-slit aur Young's two-slit problems mein, half-integer values kuch special mark karti hain, toh students woh le lete hain. Fix: Ek grating mein bright (principal) maxima ke liye whole-wavelength path differences chahiye: d sin θ = mλ
Common mistake Ye bhool jaana kid Kyun sahi lagta hai: Data "lines per mm" mein diya hota hai, toh tempting lagta hai us number ko directly equation mein plug kar dein. Fix: Pehle hamesha invert karo: d = 1/ ( lines per unit length )
Common mistake Yeh kehna ki infinitely many orders hainKyun sahi lagta hai: m Fix: sin θ ≤ 1 m ⌊ d / λ ⌋
Common mistake Small-angle approximationsin θ ≈ θ Kyun sahi lagta hai: Young's experiment mein ye kaam karta hai jahan angles bahut chote hote hain. Fix: Grating angles often large hote hain (tens of degrees), isliye poora sin θ
Recall Compute karne se pehle predict karoExample 2 se pehle, guess karo : d / λ = 3.33
Recall Feynman: ek 12-saal ke bachche ko explain karoSocho ek fence hai jisme sau jaisa chote gaps hain, saare ek hi distance par. Uspar ek pure colour ki torch chalaao. Har gap se light nikalti hai aur ripples ki tarah phailti hai. Zyaadatar directions mein alag-alag gaps ki ripples jumble ho jaati hain aur cancel ho jaati hain — andhera. Lekin kuch special directions mein, har ripple ka crest perfectly next gap ke crest se line up hota hai, toh light ek super-bright, super-sharp line mein add up ho jaati hai. Special directions colour par depend karti hain, isliye ek rainbow fan out hoti hai. "Saare crests line up hone" ka rule hai: neighbouring gaps ke beech extra distance wavelengths ka exact number hona chahiye.
Mnemonic Equation yaad rakho"d sine = m lambda" → "D ay S ees M any L ights."
Aur orders count karne ke liye: "d / λ
Principal maxima ke liye grating equation kya hai? d sin θ = mλ m
Grating equation mein d d d = 1/ ( lines per unit length )
Grating maxima two-slit fringes se bahut zyaada sharp aur bright kyun hote hain? Kyunki N ∝ N 2 N
Angle θ Δ = d sin θ
Maximum observable order kya hai? m m a x = ⌊ d / λ ⌋ sin θ ≤ 1
m = 0 Ye θ = 0
d / λ = 3.33 m m a x = 3 − 3.. + 3
Maxima condition d sin θ = ( m + 2 1 ) λ Half-wavelength path differences reinforcement nahi, cancellation cause karte hain.
Young's double-slit experiment — same constructive condition, N Single-slit diffraction — woh envelope set karta hai jo grating peak intensities ko modulate karta hai.Interference and path difference — underlying principle.Resolving power and spectrometers — kyun sharper peaks closer wavelengths resolve karti hain.Wavelength and the visible spectrum — grating white light ko kyun disperse karta hai.
whole number of wavelengths
adjacent implies all N in phase
intensity proportional N squared
Diffraction grating N slits
Two adjacent slits at angle theta
Path difference d sin theta
Constructive interference
Grating equation d sin theta = m lambda
Spreads white into spectrum
Spectrometers reading star fingerprints
Sharp bright principal maxima
