Thin film interference — reflected and transmitted

2.5.12 · Physics › Optics

KYA ho raha hai

Do key physical facts jo hum USE karenge:

Optical path length. Light refractive index $n$ wale medium ke andar slow ho jaati hai. Film ke andar ek geometric distance $d$ ka optical distance $n d$ "cost" hoti hai, kyunki phase ke liye wavelengths ki count matter karti hai, aur medium ke andar $λ_{f i l m} = λ / n$ hoti hai.
Reflection phase flip. Jab light rarer se denser medium mein reflect hoti hai (low $n$ → high $n$ ), toh use $π$ ka extra phase milta hai, yani ek extra half wavelength $λ /2$ . Rarer medium se reflect karne par (high $n$ → low $n$ ) koi flip nahi hoti.

Figure — Thin film interference — reflected and transmitted

KAISE derive karte hain condition (scratch se)

Ek film lo jiska index $n$ hai, thickness $t$ hai, light angle $θ_{i}$ pe incident hai aur andar $θ_{r}$ angle pe refract hoti hai.

Step 1 — Geometric extra path. Ray 2 film ke andar neeche aur wapas upar jaati hai. Ray 1 ke upar extra path (wavefront geometry account karne ke baad) yeh hai:

$Δ_{g eo} = 2 t cos θ_{r}$

Yeh step kyun? Andar ki ray har taraf (neeche aur upar) $t / cos θ_{r}$ slant length travel karti hai, jo $2 t / cos θ_{r}$ slant path deta hai. Lekin dono emerging rays ko ek common wavefront ke along compare karna padta hai (rays ke perpendicular). Exit point se woh wavefront drop karne par slant path ka ek hissa kat jaata hai; in-film travel ka jo hissa actually count hota hai woh film normal pe uski projection hai, $2 t cos θ_{r}$ . Geometrically: $(2 t / cos θ_{r}) - (2 t sin θ_{r} \cdot tan θ_{r}) = 2 t cos θ_{r}$ . Cosine (secant nahi) survive karta hai.

Step 2 — Optical path mein convert karo. Film ke andar light slow hoti hai, isliye $n$ se multiply karo:

$Δ_{o pt} = 2 n t cos θ_{r}$

Kyun? Phase wavelengths count karta hai; film ke andar wavelengths $n$ factor se chhoti hoti hain, isliye optical path geometric path ka $n$ guna hoti hai.

Step 3 — Reflection phase term add karo. Ek film (index $n$ ) consider karo jo ek less dense backing pe rakhi hai, film upar ke medium se denser hai (typical: air–soap–air, ya air–oil–water wale cases alag hote hain — har interface check karo!).

Air–film–air ke liye (jaise soap bubble): top reflection air→film rarer→denser hai ⇒ flip $λ /2$ . Bottom reflection film→air denser→rarer hai ⇒ koi flip nahi. Net extra: ek $λ /2$ .

Toh reflected total path difference yeh hai:

$Δ = 2 n t cos θ_{r} \pm \frac{λ}{2}$

$λ /2$ flip kyun saara game hai

Worked Examples

Common Mistakes

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho light do runners hain jo ek soap bubble par split ho jaate hain. Ek front se bounce karta hai; doosra andar ghusta hai, back se bounce karta hai, aur bahar aata hai. Diver ne thoda extra distance dauda. Saath mein, "harder wall" (denser stuff) se bounce karne par runner ko ek chhota sa hiccup milta hai — aadha step rhythm se bahar. Jab dono runners wapas in step aate hain, toh woh color bright shines karta hai; jab woh out of step hote hain, toh woh color gayab ho jaata hai. Isliye bubbles rainbows dikhate hain: alag colors alag bubble thicknesses par in-step hote hain. Aur ek bubble itna얇a ki pop hone wala ho, black ho jaata hai — kyunki hiccup akela runners ko exactly out of step kar deta hai.

Flashcards

Thickness

t

aur internal angle

θ_{r}

wali film mein lower ray ka geometric extra path kya hai?

2 t cos θ_{r}

Geometric path

2 t cos θ_{r}

kyun hoti hai,

2 t / cos θ_{r}

kyun nahi?

Kyunki rays ko ek common wavefront ke along compare kiya jaata hai; sirf in-film travel ki film normal pe projection count hoti hai, jo

2 t cos θ_{r}

deti hai.

Optical path mein

n

kyun use hota hai, jaise

2 n t cos θ_{r}

mein?

Phase wavelengths count karta hai aur film ke andar

λ_{f i l m} = λ / n

hoti hai, isliye optical path =

n \times

geometric path.

Reflection mein

π

(half-wave) phase shift kab add hoti hai?

Jab light rarer (low

n

) se denser (high

n

) medium mein reflect ho rahi ho.

Soap film in air (one net flip) ke liye reflected BRIGHT condition kya hai?

2 n t cos θ_{r} = (m + \frac{1}{2}) λ

Soap film in air ke liye reflected DARK condition kya hai?

2 n t cos θ_{r} = mλ

Bahut얇i soap film reflection mein black kyun hoti hai?

Geometric path →0, sirf

λ /2

flip bachi rehti hai, jo sabhi colors ke liye destructive interference deti hai.

Reflected aur transmitted patterns ka aapas mein kya relation hai?

Woh complementary (negatives) hain — jahan reflection bright hai, transmission dark hai (energy conservation).

Wavelength

λ

ke liye sabse얇i anti-reflection coating ki thickness kya hai?

t = λ / (4 n_{c})

(quarter-wave, kyunki do flips cancel ho jaate hain aur hum destructive reflection chahte hain).

Formula mein konsa angle jaata hai, incident ya refracted?

Film ke andar refracted angle

θ_{r}

Formula mein

λ

film wavelength hai ya vacuum wavelength?

Vacuum/air wavelength; medium already

n

ke zariye account ho chuka hai.

Connections

Interference of light — iske peeche superposition principle hai
Young's double slit — same coherence/path-difference logic, alag geometry
Newton's rings — varying thickness wali얇i air film (wedge → circular fringes)
Refractive index aur Snell's law — $θ_{r}$ aur optical path dete hain
Phase change on reflection — π-flip rule
Anti-reflection coatings — engineering application
Conservation of energy — kyun reflected/transmitted complementary hain