Question bank — Birefringence — ordinary and extraordinary rays

False — o-ray ki definition hai ki woh Snell's law ko obey karta hai, seedha jaana zaroori nahi; oblique incidence par yeh kisi bhi normal ray ki tarah refract karta hai, aur normal incidence par hi seedha jaata hai.

True — splitting polarization se hoti hai, wavelength se nahi, isliye perfectly monochromatic light bhi ek o-ray aur ek e-ray dono produce karti hai.

False — bare $n_e$ sirf principal value hai jo $\theta =90^{\circ }$ par hai; actual index direction-dependent $n_e(\theta )$ hoti hai jo ellipsoid relation se milti hai.

True — $\theta =0$ par formula $n_e(0)=n_o$ deta hai, isliye dono waves ek hi speed share karti hain aur kabhi alag nahi hoti.

True — o-ray ki $\vec{E}$ principal plane ke perpendicular hoti hai jabki e-ray ki $\vec{E}$ uss mein hoti hai, isliye dono mutually orthogonal hain (dekho Polaroid and Nicol prism — yahi property ek Nicol prism ko ek ray reject karne deti hai).

False — calcite negative crystal hai ($n_e<n_o$); sign iss baat se decide hota hai ki $n_e$, $n_o$ se bada hai ya nahi, splitting kitni badi dikhti hai uss se nahi.

True — calcite jaise negative crystals ke liye $\Delta n=1.486-1.658=-0.172$; sign sirf record karta hai ki kaun si ray faster hai.

True in practice — glass ko dabane ya garam karne se iske "springs" direction-dependent ho jaate hain (stress birefringence), jo ek isotropic material ko temporarily anisotropic bana deta hai (Dielectric tensor and anisotropic media).

True — o-ray ki $\vec{E}$ mein optic axis ($z$) ke along koi component nahi hota, isliye ${\epsilon }_{zz}=n_e^2$ term kabhi uss par act nahi karta, aur isko constant $n_o$ milta rehta hai.

True for a given crystal and wavelength — thickness hi phase difference set karti hai $\Delta \varphi =\frac{2\pi }{\lambda }|\Delta n|d$; $\pi /2$ se quarter-wave aur $\pi$ se half-wave plate milti hai (Wave plates (quarter and half wave)).

False — birefringence do linearly polarized rays deta hai jo alag speeds par hoti hain; Optical activity ek single polarization ka plane rotate karta hai aur ek alag (chiral) mechanism se aata hai.

Error yeh hai ki bare $n_e$ use kiya gaya — e-ray ki actual index $n_e(\theta )$ hoti hai, isliye local splitting $n_e(\theta )-n_o$ hoti hai aur $\theta =0$ par zero ho jaati hai.

Hum kar sakte hain — e-ray abhi bhi wave equation obey karta hai; bas woh direction-dependent $n_e(\theta )$ use karta hai, isliye ek modified (index-varying) refraction analysis iska direction exactly predict kar sakti hai.

Sine aur cosine swap hain — ${\cos }^2\theta$ ko $n_o^2$ ke saath pair hona chahiye taaki $\theta =0$ par $n_o$ mile, jo "optic axis ke along koi splitting nahi" wali boundary condition se match karta hai.

Nahi — $\theta =0$ par dono indices coincide karte hain, dono waves degenerate hain, aur koi bhi shift nahi hota; crystal uss direction ke liye isotropically behave karta hai.

Relation hai $n=\sqrt{{\epsilon }_r}$, $n={\epsilon }_r$ nahi; index relative permittivity ke square root ke saath scale karta hai.

Yeh ek sahi property hai, lekin yeh isliye hai kyunki iska $\vec{D}$ aur $\vec{E}$ parallel rehte hain (isotropic response); e-ray ka $\vec{D}$ aur $\vec{E}$ alag ho jaate hain, isliye iska ray direction apne wavevector se alag ho jaata hai.

O-ray ki $\vec{E}$ hamesha optic axis ke perpendicular rehti hai, isliye woh sirf $n_o$ wale "springs" sample karta hai; e-ray ki $\vec{E}$ direction ke hisaab se axis ki taraf jhukti hai, $n_o$ aur $n_e$ ka mix sample karti hai.

Kyunki ellipsoid ko slice karne ki geometry $1/n^2$ par relation deti hai; displacement direction $\cos \theta$ aur $\sin \theta$ pieces mein resolve hoti hai jinke $1/n^2$ contributions add hote hain, jaise springs "in parallel" combine hoti hain.

Ek single number sirf isotropic response encode karta hai; tensor material ko alag axes par alag tarah respond karne deta hai, jo exactly woh anisotropy hai jisse do indices produce hote hain (Dielectric tensor and anisotropic media).

Wahan e-ray ki $\vec{E}$ fully perpendicular... correction: $\theta =90^{\circ }$ par propagation axis ke perpendicular hoti hai isliye e-ray ki field fully axis ke along hoti hai, jo $n_e(90^{\circ })=n_e$ deta hai — yeh value $n_o$ se sabse door hai.

Sirf do rays ki speed ka fark hi phase gap $\Delta \varphi =\frac{2\pi }{\lambda }|\Delta n|d$ banata hai; index ka common part dono rays ko equally delay karta hai aur cancel ho jaata hai.

Isotropic glass ka ek hi index hota hai, isliye $\Delta n=0$ aur polarizations ke beech koi bhi relative phase kabhi accumulate nahi hoti — retard karne ke liye kuch hai hi nahi.

Kyunki crystal ke andar dono rays alag speeds par aur generally alag directions mein travel karti hain, isliye ek single incident ray do mein refract hoti hai — doubling refracted rays ki hoti hai, incident ray ki nahi.

Ek — light optic axis ke along travel karti hai ($\theta =0$), jahan $n_e(0)=n_o$, isliye koi splitting nahi hoti.

O- aur e-waves alag speeds par travel karti hain lekin ek hi seedhi line mein, isliye spatially overlap karti hain phir bhi phase difference accumulate karti hain — exactly wave-plate geometry.

Yeh har $\theta$ ke liye $n_o$ ke barabar hota hai — material effectively isotropic hai (koi birefringence nahi), jo formula ki degenerate limit hai.

Unpolarized light sabhi polarizations ka equal mix hoti hai, isliye average mein power o- aur e-rays ke beech roughly evenly split hoti hai; ek pehle se lagaya gaya polarizer (Polaroid and Nicol prism) saari power ek mein bhej sakta hai.

Faster — $n_e<n_o$ matlab ek chhota index aur $v=c/n$, isliye e-ray o-ray se aage nikal jaati hai (positive crystals mein ulta hota hai).

Iska $\vec{E}$ progressively principal plane mein tilt karta hai; $\theta =0$ par yeh o-ray se alag nahi hota, aur $\theta =90^{\circ }$ par fully optic axis ke along ho jaata hai, extreme index $n_e$ feel karta hai.