Physics 2025 Paper I 50 marks Derive

Q3

(a) Consider a laser system consisting of an active medium placed between a pair of mirrors forming a resonator. Obtain an expression for the threshold population inversion required for the oscillations of laser. (20 marks) (b) For a He – Ne laser system, what will be the magnitude of $\Delta\omega_D$ which represents FWHM of the line shape function g($\omega$), if resonant frequency $\omega_0 = 3 \times 10^{15}$ s$^{-1}$ and temperature T = 300 K ? (10 marks) (c) A cube of mass M and side 'a' is rotating with angular velocity $\omega$ around one of its edges, which is, say, along the x-axis. Obtain the expressions for its angular momentum and kinetic energy. (Given that the $I_{XX} = \frac{2}{3} Ma^2$, $I_{YX} = -\frac{1}{4} Ma^2$ and $I_{ZX} = -\frac{1}{4} Ma^2$) (20 marks)

हिंदी में प्रश्न पढ़ें

(a) एक अनुनादक बनाते दर्पणों के एक युग्म के बीच रखे एक सक्रिय माध्यम के एक लेजर निकाय को लीजिए। लेजर के दोलनों के लिए आवश्यक देहली (थ्रेशोल्ड) जनसंख्या व्युत्क्रमण के लिए एक व्यंजक प्राप्त कीजिए। (20 अंक) (b) एक He – Ne लेज़र निकाय के लिए, यदि अनुनादी आवृत्ति $\omega_0 = 3 \times 10^{15}$ s$^{-1}$ और तापक्रम T = 300 K है, तो $\Delta\omega_D$ का परिमाण क्या होगा जो रेखा आकृति फलन g($\omega$) के FWHM को निरूपित करता है? (10 अंक) (c) द्रव्यमान M और भुजा 'a' का एक घन x-अक्ष के अनुदिश अपने एक किनारे के परितः कोणीय वेग $\omega$ से घूर्णन कर रहा है। उसके कोणीय संवेग और उसकी गतिज ऊर्जा के लिए व्यंजकों को प्राप्त कीजिए। (दिया गया है, $I_{XX} = \frac{2}{3} Ma^2$, $I_{YX} = -\frac{1}{4} Ma^2$ और $I_{ZX} = -\frac{1}{4} Ma^2$) (20 अंक)

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Approach

Derive the threshold population inversion for laser oscillations in part (a) by balancing gain and cavity losses, showing how mirror reflectivity and spontaneous emission enter the condition. For part (b), calculate the Doppler-broadened linewidth using the Maxwell-Boltzmann distribution at 300 K. In part (c), construct the inertia tensor for the cube and use it to find angular momentum components and rotational kinetic energy, noting that ω is along x-axis but L has non-zero y,z components due to non-diagonal elements. Allocate roughly 40% time to (a), 20% to (b), and 40% to (c) based on marks distribution.

Key points expected

  • Part (a): Rate equation analysis showing gain coefficient γ(ν) = (N₂-N₁)B₂₁hνg(ν)/c and threshold condition γₜₕ = α + (1/2L)ln(1/R₁R₂)
  • Part (a): Final threshold population inversion expression (N₂-N₁)ₜₕ = 8πν²τₛₚ/c³g(ν₀) × loss terms, or equivalent with cavity lifetime τc
  • Part (b): Doppler broadening formula ΔνD = (2ν₀/c)√(2kTln2/m) or ΔωD = (2ω₀/c)√(2kTln2/m) for He-Ne with m = 20 amu (Neon)
  • Part (b): Numerical substitution yielding ΔωD ≈ 2π × 1.5 GHz or ~9.4 × 10⁹ rad/s (order of magnitude check essential)
  • Part (c): Recognition that ω⃗ = (ω, 0, 0) and use of Lᵢ = Σⱼ Iᵢⱼωⱼ giving Lx = (2/3)Ma²ω, Ly = -(1/4)Ma²ω, Lz = -(1/4)Ma²ω
  • Part (c): Kinetic energy calculation K = ½ω⃗·L⃗ = ½Iₓₓω² = (1/3)Ma²ω², or equivalently ½ΣᵢⱼIᵢⱼωᵢωⱼ
  • Part (c): Physical explanation that angular momentum is NOT parallel to angular velocity due to non-diagonal inertia tensor (principal axes ≠ coordinate axes)

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies Einstein coefficients relation A₂₁/B₂₁ = 8πhν³/c³ for (a); recognizes Doppler vs natural/homogeneous broadening for (b); understands that products of inertia cause L⃗ ∦ ω⃗ for (c)States threshold condition qualitatively but confuses stimulated emission cross-section with gain; calculates linewidth with wrong mass (He instead of Ne) or omits √ln2; treats cube as diagonal inertia tensorConfuses population inversion with pump power; uses wrong broadening mechanism (collisional instead of Doppler); assumes L⃗ = Iω⃗ scalar relation for rigid body
Derivation rigour25%12.5Step-by-step derivation from cavity mode density and loss rate to threshold condition with proper cavity photon lifetime τc = L/[c(αL+T)]; full tensor multiplication for angular momentum components; dimensional consistency throughoutJumps to final formula without showing balance between gain and round-trip loss; skips intermediate algebra in tensor calculation; minor algebraic errors in exponentsStates formulae without derivation; circular reasoning in threshold condition; fundamental errors in tensor index notation or treats Iᵢⱼ as scalar
Diagram / FBD15%7.5Clear resonator diagram showing mirrors, active medium, with round-trip loss annotations for (a); cube diagram with labeled axes, edge rotation, and principal axes indicated for (c); coordinate system for tensor visualizationBasic resonator sketch without labels; cube drawn but axes unclear; no indication of non-coincident L⃗ and ω⃗ directionsNo diagrams despite geometric complexity; or misleading diagrams showing L⃗ parallel to ω⃗ for (c)
Numerical accuracy20%10Correct substitution in (b): m = 20×1.66×10⁻²⁷ kg, T = 300 K, k = 1.38×10⁻²³ J/K; obtains ΔωD ≈ 9.4×10⁹ rad/s or ΔνD ≈ 1.5 GHz with proper significant figures; order-of-magnitude sanity check against typical He-Ne linewidthCorrect formula but arithmetic error (factor of 2 or √2); uses wrong atomic mass; unit confusion between Hz and rad/sOrder of magnitude error >10²; completely wrong formula (e.g., natural linewidth); no numerical work shown for (b)
Physical interpretation20%10Explains why threshold inversion increases with cavity loss and decreases with stimulated emission cross-section; interprets Doppler width as inhomogeneous broadening limiting single-mode operation; explains dynamic imbalance requiring torque for constant ω rotation in (c)States physical meanings without connecting to laser operation; notes Doppler broadening but not its consequences for mode competition; mentions L⃗ ∦ ω⃗ without explaining dynamical implicationsNo physical interpretation; treats all formulae as purely mathematical; fails to recognize that non-diagonal inertia tensor implies non-principal axis rotation requiring external torque

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