Birefringence — ordinary and extraordinary rays

2.5.18 · Physics › Optics

Key players

Quantity	Meaning
Optic axis	Woh direction jisme koi birefringence nahin; $n_{o} = n_{e}$ along it
$n_{o}$	Ordinary index (constant)
$n_{e}$	Principal extraordinary index (max deviation, jab $E$ ⟂ optic axis ho)
Positive crystal	$n_{e} > n_{o}$ (e.g. quartz)
Negative crystal	$n_{e} < n_{o}$ (e.g. calcite)

Figure — Birefringence — ordinary and extraordinary rays

DERIVATION 1 — o-ray isotropic kyun hai par e-ray nahin

Ek uniaxial crystal lo jiska optic axis $z$ ke along hai. Anisotropy dielectric tensor mein rehti hai: $ε = ε_{0} n_{o}^{2} 00 0 n_{o}^{2} 0 00 n_{e}^{2}$

Yeh matrix kyun? "Uniaxial" ka matlab hai ki $x$ aur $y$ directions equivalent hain (same $n_{o}$ ), aur sirf $z$ (optic axis) special hai (index $n_{e}$ ).

Ek plane wave $E \propto e^{i (k \cdot r - ω t)}$ ko wave equation satisfy karni chahiye, jo ek given propagation direction $\hat{k}$ ke liye index ellipsoid / Fresnel equation deta hai. Ise $\hat{k}$ ke liye optic axis se angle $θ$ par solve karne par do allowed waves milti hain:

Solution A (o-ray): $E$ $x y$ -plane mein hai (⟂ optic axis). Yeh hamesha index dekhta hai $n_{o}^{2} \Rightarrow v_{o} = \frac{c}{n _{o}} (direction-independent).$ Yeh step kyun? Agar $E$ ka koi $z$ -component nahin hai, toh $ε_{z z} = n_{e}^{2}$ entry kabhi uske upar act nahin karti; sirf $n_{o}$ matter karta hai chahe direction koi bhi ho.

Solution B (e-ray): $E$ ka optic axis ke along ek component hai. Fresnel equation ke through kaam karne par e-ray index formula milta hai (aage derive hoga).

DERIVATION 2 — Extraordinary index $n_{e} (θ)$

Hum $n_{e} (θ)$ chahte hain, woh index jo e-wave dekhti hai jiska wavevector optic axis se angle $θ$ banata hai.

Step 1 — Index ellipsoid set up karo. Ellipsoid hai $\frac{x ^{2} + y ^{2}}{n _{o}^{2}} + \frac{z ^{2}}{n _{e}^{2}} = 1.$ Kyun? Yeh encode karta hai ki index $D$ orientation par kaise depend karta hai: $D$ $z$ ke along → index $n_{e}$ ; $x y$ mein → index $n_{o}$ .

Step 2 — $\hat{k}$ ke perpendicular slice karo. Do allowed indices wo hain jo centre se guzarne wale plane ⟂ $\hat{k}$ se kati ellipse ke semi-axes hain. Ek semi-axis hamesha $n_{o}$ hoti hai (o-ray). Doosri (e-ray) ka index $n (θ)$ tilted ellipse ki geometry se nikalta hai.

Step 3 — Geometry. E-ray ka $D$ optic axis se angle $θ$ banata hai equivalently. Uska index ellipse relation satisfy karta hai: $\frac{1}{n _{e} ( θ ) ^{2}} = \frac{cos ^{2} θ}{n _{o}^{2}} + \frac{sin ^{2} θ}{n _{e}^{2}}$

Yeh form kyun? Unit displacement direction ko ek component $cos θ$ $n_{o}$ -direction ke along aur $sin θ$ $n_{e}$ -direction ke along resolve karo, phir unke inverse-square contributions add karo — bilkul waise jaise " $1/ n^{2}$ ke parallel mein" capacitances/springs combine karte hain.

WORKED EXAMPLES

Common mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Imagine karo ki tum bheed mein daud rahe ho. Agar sab log har direction mein equally packed hain, toh tum ek speed se daudte ho (normal glass). Lekin kuch crystals aise hain jaise ek bheed jo left-right mein tightly packed hai aur up-down mein loosely packed hai. Ab tumhari speed depend karti hai ki tum kis taraf face kar rahe ho! Light bhi same hai: inhi special crystals mein woh do speeds se travel karti hai depending on which way it "wiggles." Toh ek light beam do beams ban jaati hai — ek shant "ordinary" wali jo normal rules follow karti hai, aur doosri rebel "extraordinary" wali jo anokhe tarike se bend karti hai. Woh ek direction jahan sab equally packed hain woh "optic axis" hai, aur wahan se jaane wali light ek single beam rehti hai.

Active-recall flashcards

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Birefringent crystal ko kya define karta hai?

Iske paas do distinct refractive indices hain (

n_{o}

aur

n_{e}

), toh ek unpolarized beam ordinary aur extraordinary rays mein split hoti hai.

Optic axis kya hai?

Woh direction jiske along

n_{o} = n_{e}

hai, toh o- aur e-rays same speed se travel karti hain aur split nahin hoti.

Kaun si ray Snell's law follow karti hai?

Ordinary (o) ray; extraordinary ray nahin karti.

O-ray kaise polarized hoti hai?

E

principal plane (optic axis aur ray wala plane) ke perpendicular hota hai.

Angle

θ

par extraordinary index ka formula?

\frac{1}{n _{e} ( θ ) ^{2}} = \frac{cos ^{2} θ}{n _{o}^{2}} + \frac{sin ^{2} θ}{n _{e}^{2}}

n_{e} (θ)

θ = 0

aur

θ = 9 0^{\circ}

par kya equal hota hai?

n_{e} (0) = n_{o}

(optic axis ke along);

n_{e} (9 0^{\circ}) = n_{e}

(principal value, max splitting).

Positive vs negative uniaxial crystal?

Positive:

n_{e} > n_{o}

(e.g. quartz). Negative:

n_{e} < n_{o}

(e.g. calcite).

Thickness

d

ke baad o aur e rays ke beech phase difference?

Δ ϕ = \frac{2 π}{λ} ∣ n_{o} - n_{e} ∣ d

Quarter-wave plate ki thickness?

d = \frac{λ}{4∣ n _{o} - n _{e} ∣}

(sabse patla,

Δ ϕ = π /2

ke liye).

O-ray ka index constant kyun hota hai?

Uske

E

ka optic axis ke along koi component nahin, toh anisotropic

ε_{z z} = n_{e}^{2}

kabhi act nahin karta — sirf

n_{o}

har direction mein matter karta hai.

Birefringence vs dispersion — difference?

Birefringence polarization se split karta hai (same color); dispersion wavelength se split karta hai.

Connections

Polarization of light — birefringence woh mechanism hai jo polarizers aur wave plates ke peeche hai.
Snell's Law — o-ray ise follow karta hai; e-ray ka violation "extraordinary" define karta hai.
Wave plates (quarter and half wave) — $Δ ϕ$ ka direct application.
Dielectric tensor and anisotropic media — do indices ka origin.
Polaroid and Nicol prism — calcite birefringence use karne wale devices.
Optical activity — related par distinct polarization ka rotation.