Aberrations — chromatic, spherical (concepts): Question bank

2.5.9 · D5 · Physics › Optics › Aberrations — chromatic, spherical (concepts)

Ab ye do anchor facts pakad lo — har question inhi mein se ek ko test karta hai:

Chromatic = focal length colour par depend karta hai ( $n$ wavelength $λ$ ke saath badalta hai). Ye axis par ek single thin ray ke liye bhi hota hai.
Spherical = focus ray height $h$ par depend karta hai. Ye ek pure colour mein bhi hota hai.

Definition Char symbols jo is page par baar baar aate hain

$θ$ — axis ke saath ray angle: ek light ray aur lens ki seedhi central line (optical axis) ke beech ka angle. Lens ke edge ke paas takarane wali ray (bada $h$ ) ka $θ$ bada hota hai; beech se guzarne wali ray ka $θ$ bahut chota hota hai. Figure 1 dekho: outer (marginal) ray mota angle banati hai, central (paraxial) ray patla.
$R_{1}, R_{2}$ — glass ki do surfaces ke radii. Har lens face kisi sphere ka ek tukda hai; $R_{1}$ front face ke us sphere ka radius hai, $R_{2}$ back face ka. Flatter face → bada radius. Sign convention: woh surface jo incoming light ki taraf bulge kare, uska radius positive maana jaata hai; jo away cave kare, negative — isi wajah se bi-convex lens ka $R_{1} > 0, R_{2} < 0$ hota hai.
$K$ — un do radii se bana shape factor, $K = \frac{1}{R _{1}} - \frac{1}{R _{2}}$ , jo ek baar fix ho jaata hai jab tum dono faces ki curvature choose karo. Ye lens ki geometry ko $\frac{1}{f} = (n - 1) K$ mein carry karta hai; dekho Lensmaker's Equation.
$P$ — optical power, $P = \frac{1}{f}$ (units: dioptres, $1/ m$ ). "Stronger" lens (chota $f$ ) ka $P$ bada hota hai. Contact mein thin lenses ki powers simply add ho jaati hain, isliye doublet condition $P$ mein likhi jaati hai.
Power ka sign convention: $f > 0$ aur $P > 0$ converging lens ke liye, $f < 0$ aur $P < 0$ diverging lens ke liye. Jis taraf light actually jaati hai (far side) us taraf ki distances positive hoti hain. Ye haath mein rakho — har "diverging" statement isi par tiki hai.

Figure — Aberrations — chromatic, spherical (concepts)

Figure 1 — $θ$ axis ke saath ray angle hai; $h$ ray ki uske upar height hai. Marginal (rim) ray ka $h$ aur $θ$ dono bade hain; paraxial (central) ray ke dono chote hain.

Definition Dispersive power

$ω$ aur Fraunhofer letters — pehle True/False se hi zaroorat padegi Glass alag colours ko alag amounts mein modata hai, isliye $n$ actually $n (λ)$ hai. "Colours kitne spread out hain" ko ek number dene ke liye, opticians teen standard reference colours choose karte hain — solar spectrum mein dark lines jinhe Fraunhofer Lines kehte hain — aur har par $n$ padhte hain:

$n_{C}$ — C line par index (red, hydrogen $α$ ).
$n_{D}$ — D line par index (yellow, sodium).
$n_{F}$ — F line par index (blue, hydrogen $β$ ). Tab dispersive power hai $ω = \frac{n _{F} - n _{C}}{n _{D} - 1} (blue - red, over yellow - 1) .$ Ye colour spread ( $n_{F} - n_{C}$ , blue minus red) ko lens ki overall bending strength ( $n_{D} - 1$ ) ke relative measure karta hai. Kyunki $n_{F} > n_{C}$ aur $n_{D} > 1$ , ye hamesha positive hota hai.

Figure 2 — chromatic aberration: white light enter karta hai, blue (bada $n$ ) zyada bend hota hai aur red ke comparison mein lens ke paas focus karta hai.

Figure 3 — spherical aberration: marginal rays axis ko pehle cross karti hain (near focus), paraxial rays baad mein (far focus); unke beech sabse patli waist circle of least confusion hai.

True or false — justify

Chromatic aberration khatam ho jaata hai agar bilkul paraxial rays use karo.

False. Axial pencil of white light bhi split hoti hai, kyunki

n (λ)

abhi bhi colour se different rehta hai — dekho Dispersion and Refractive Index aur figure 2. Paraxial hona spherical aberration ko khatam karta hai, chromatic ko nahi.

Spherical aberration khatam ho jaata hai agar bilkul monochromatic light use karo.

False. Ye ek monochromatic defect hai: ek colour ke saath bhi, marginal rays (bada

θ

, bada

h

) abhi bhi

- θ^{3} /6

term ki wajah se zyada bend hoti hain aur pehle focus karti hain (figure 3). Colour iske liye irrelevant hai.

Converging lens ke liye, blue light red se zyada door focus karti hai.

False. Blue ka

n

bada hota hai, aur

f \propto 1/ (n - 1)

, isliye blue ka

f

chota hota hai — blue lens ke paas focus karta hai.

Spherical lens mein se marginal rays paraxial rays se zyada door focus karti hain kyunki aperture wider hai.

False. Neglected

- θ^{3} /6

term marginal rays ko axis ki taraf zyada modata hai, isliye wo pehle cross karti hain (paas). Width ka matlab reach nahi hota.

Ek achromatic doublet saari wavelengths ke liye focal length identical bana deta hai.

False. Ye sirf do chosen colours (usually

F

blue aur

C

red) ko coincide karne par majboor karta hai; baaki colours ka ek chota residual bana rehta hai jise secondary spectrum kehte hain.

Kuch glasses ke liye dispersive power

ω

negative ho sakta hai.

False.

ω = (n_{F} - n_{C}) / (n_{D} - 1)

mein

n_{F} > n_{C}

aur

n_{D} > 1

hota hai, isliye

ω > 0

hamesha — wahi positivity ki wajah se doublet ko opposite-sign powers chahiye hoti hain.

Parabolic mirror har tarah ke aberration se zero hoti hai.

False. Ye sirf on-axis parallel light ke liye perfect hai; off-axis point sources abhi bhi coma mein blur hote hain. Dekho Parabolic Mirrors and Telescopes.

Aperture chhota karna (stopping down) spherical aur chromatic dono ki sharpness improve karta hai.

False. Ye spherical aberration ko marginal rays hatakar theek karta hai, lekin chromatic ek axis par focal-length shift hai — chota hole iske liye kuch nahi karta — colour fringes abhi bhi aate hain.

Lensmaker's equation

\frac{1}{f} = (n - 1) K

real thin lens ke liye exact hai.

False. Shape factor

K = \frac{1}{R _{1}} - \frac{1}{R _{2}}

ke saath bhi, ismein do approximations chhipi hain:

n

ek single wavelength ke liye (toote → chromatic) aur

sin θ \approx θ

(toote → spherical). Dekho Lensmaker's Equation aur Paraxial Approximation.

Spherical aberration blur aperture radius

h

ke saath linearly scale karta hai.

False. Ye

θ^{3} \sim h^{3}

term se aata hai, isliye transverse blur roughly

h^{3}

ki tarah badhta hai —

h

aadha karo toh blur lagbhag 8 ke factor se ghata jaata hai.

Spot the error

"Chromatic aur spherical aberration dono light ko colours mein disperse karne ki wajah se hote hain."

Sirf chromatic mein dispersion, yaani

n (λ)

, involved hai. Spherical ek single wavelength ke saath hota hai aur purely ray height aur spherical shape ke baare mein hai.

"Hum chromatic aberration ko iris band karke outer rays rokne se theek karte hain."

Galat tool: outer rays spherical ke villain hain. Chromatic on-axis colours ke beech ek focal-length difference hai; ise Achromatic Doublet se theek karte hain, aperture se nahi.

"Kyunki

\frac{df}{f} = - \frac{d n}{n - 1}

, bade

n

wala lens automatically kam chromatic aberration karta hai."

Chromatic spread

d n

par depend karta hai (kitna

n

colours ke across badalta hai,

ω

mein captured),

n

par nahi. High-

n

flint glass aksar zyada disperse karta hai, kam nahi.

"Doublet colour error cancel karta hai kyunki dono lenses ke equal aur opposite focal lengths hote hain."

Equal-and-opposite

f

se net power zero ho jaayegi (

P = 1/ f

, toh powers cancel ho jaayengi — koi lens hi nahi). Asli condition hai

ω_{1} P_{1} + ω_{2} P_{2} = 0

jisme net power survive kare — colour spreads cancel hoti hain, powers nahi.

"Marginal rays zyada bend hoti hain, toh spherical aberration theek karne ke liye humen lens ko edges par aur zyada curved banana chahiye."

Ulta hai. Edges already light ko zyada hard turn karti hain; cure ye hai ki wahan effective curvature flatten karo — ek aspheric/parabolic surface ya best-form ("Coddington") bending, per Seidel Aberrations.

"Circle of least confusion bilkul paraxial focus par hoti hai."

Nahi. Ye marginal focus (paas) aur paraxial focus (door) ke beech hoti hai, jahan rays ka converging cone sabse patla hota hai — wo best screen position hai (figure 3 dekho).

"Dispersive power numerator mein red aur blue use karta hai aur denominator mein blue."

Denominator yellow reference minus one hai,

n_{D} - 1

ω = (n_{F} - n_{C}) / (n_{D} - 1)

D

yellow hai,

F

blue,

C

red — Fraunhofer Lines.

Why questions

Third-order (

θ^{3}

) term kyun leading spherical-aberration correction hai, second-order kyun nahi?

Expansion

sin θ = θ - θ^{3} /6 + \dots

mein koi even-power term nahi hota, isliye paraxial

θ

ke baad pehla surviving correction

- θ^{3} /6

hai — isliye ise third-order (primary Seidel) aberration kehte hain.

Achromat ke ek lens ko kyun converge karna chahiye jabki doosra diverge kare?

Kyunki

ω > 0

dono glasses ke liye, balance

ω_{1} P_{1} + ω_{2} P_{2} = 0

tab hi hold ho sakta hai jab powers

P_{1}

aur

P_{2}

ke opposite signs hon — aur convention se

P > 0

matlab converging,

P < 0

matlab diverging.

Pinhole cameras mein essentially koi spherical aberration kyun nahi hota?

Pinhole sirf ek patli near-axial pencil pass karta hai (chota

h

, chota

θ

), isliye

h^{3}

marginal error negligible hai — wahi reason hai jisse lens ka aperture chhota karne par image sharp hoti hai.

Distant on-axis star ke liye parabolic mirror spherical mirror se kyun behtar hai?

Ek parabola defined hi aise hota hai ki us par sabhi axial parallel rays apne focus tak equal optical path length travel karti hain, isliye sabhi rays ek point par in-phase pahunchti hain bina kisi

θ^{3}

error ke; sphere sirf apne centre ke paas ye mimic karta hai.

Chromatic aberration mirrors ke liye kyun exist karta... karta hi nahi — kyun?

Reflection

θ_{i} = θ_{r}

follow karta hai bina kisi refractive index ke, isliye koi

n (λ)

nahi aur hence koi colour-dependent focus nahi. Mirrors inherently chromatic aberration se free hote hain (ek bada reason ki bade telescopes mirrors use karte hain).

Ek single high-quality lens itni achchi tarah grind karne par bhi chromatic aberration remove kyun nahi ho sakta?

Grinding shape factor

K

(radii

R_{1}, R_{2}

) control karta hai, jo spherical-type errors fix karta hai, lekin colour spread material ke

n (λ)

se aata hai. Dispersion ko polish karke hataya nahi ja sakta — ek doosra, alag tarah se dispersing glass chahiye.

Edge cases

Diverging (concave) lens ka chromatic aberration kya hota hai, aur sign convention use kaise describe karta hai?

Ye abhi bhi colours split karta hai. Diverging lens ke liye

f < 0

(aur

P < 0

); kyunki

f \propto 1/ (n - 1)

, blue ka bada

n

∣ f ∣

ko chota banata hai, isliye blue ka virtual focus lens ke incoming side par paas hota hai. Displacement converging lens ke opposite direction mein hota hai — exactly isliye ek diverging flint element ek converging crown element ko doublet mein cancel kar sakta hai.

Special ray height

h = 0

par (exactly axis par jaane wali ray), kitna spherical aberration hota hai?

Bilkul zero — on-axis ray ka

θ = 0

hai, isliye

θ^{3}

term vanish ho jaata hai. Spherical aberration tab hi badhta hai jab tum height mein off-axis move karo.

Agar kisi glass ka

n

saari wavelengths par truly constant ho (

d n = 0

), toh chromatic aberration ka kya hoga?

Ye vanish ho jaata hai: chromatic-shift relation

\frac{df}{f} = - \frac{d n}{n - 1} = 0

se (upar formula callout dekho), isliye saare colours ek hi focus share karte hain. Koi real glass aisa nahi hota, lekin ye dikhata hai ki chromatic aberration poora ek dispersion effect hai.

Aperture zero ki taraf shrink hone par circle of least confusion ka kya hota hai?

Longitudinal aur lateral dono spans collapse ho jaate hain (blur

\sim h^{3}

), isliye circle of least confusion paraxial point ki taraf shrink hoti jaati hai — image ideal ban jaati hai lekin dim.

Light jo already monochromatic aur strictly paraxial ho, kya unke in dono aberrations mein se koi bacha rehta hai?

Nahi — paraxial hona spherical ko khatam karta hai aur single-colour hona chromatic ko, isliye dono vanish ho jaate hain. Ye exactly wo idealized regime hai jisme thin-lens formula sahi kaam karta hai.

Agar doublet se do colours coincide karaye jaayein, toh kya baaki wavelengths ka residual colour error har jagah zero hai?

Nahi — uncorrected wavelengths ke liye ek chota secondary spectrum bana rehta hai, kyunki

n (λ)

perfectly linear nahi hota; teen colours correct karna (apochromat) ise ghataata hai lekin kabhi poora khatam nahi karta.

Recall Jaane se pehle ek-line self-test

Chromatic ==colour/wavelength ( $n (λ)$ ) par depend karta hai aur paraxially bhi survive karta hai; spherical ray height $h$ == par depend karta hai aur ek colour mein bhi survive karta hai. Alag causes → alag cures.