Laser Problems PDF

# Laser ## Problem 1 The wavelength of emission is 7000Å and the coefficient of spontaneous emission is 108s-1. Find the coefficient of stimulated emission. ### Solution Coefficient of stimulated emission (B21) = 38πhv3A21 = h38πν3A21 where A21 = coefficient of spontaneous emission = 108s-1 .. B21 = (7000 × 10-10)3 × 108 8 × 3.14 × 6.626 × 10-34 m. kg-1 = (2.06 × 1021) m. kg-1 ## Problem 2 Find the temperature at which the rate of stimulated emission is equal to the rate of spontaneous emission. Assume λ = 6000Å. ### Solution We know R = rate of stimulated emission/rate of spontaneous emission = 1/ehv/kT - 1 Here, R = 1; So 1/ehv/kT -1 = 1 or, ehv/kT = 2 or, hv/kT = ln2 or, hc/λkT = 0.693 or, T = (6.626 × 10-34) × (3 × 108) / (6000 × 10-10) × (1.38 × 10-23) × 0.693 K = 34642.4 K ## Problem 1 When an electron jumps from an energy level of 5.44 × 10-19J to an energy level of 2.42 × 10-19J, find the wavelength and colour of the photon. ### Solution The wavelength of emitted photon λ = hc/E2-E1 = (6.626 × 10-34 × 3 × 108) / (5.44 × 10-19 – 2.42 × 10-19) m = 6.582 × 10-7m = 658nm The colour of the photon is red. ## Problem 3 Find the energy difference between the two energy levels of neon atoms of a He-Ne gas laser. The transition between these levels gives a light of wavelength 632.8 nm. Also calculate the number of photons emitted per second to give a power output of 2 mW. ### Solution The energy difference between the two energy levels of neon atoms, ΔE = hc/λ = (6.63 × 10-34 × 3 × 108) / (632.8 × 10-9) = 3.143 × 10-19 J = 3.143 × 10-19/1.6 × 10-19 eV = 1.96 eV [: 1 eV = 1.6 × 10-19 J] If n is the number of photons emitted per second, to give a power output of 2 mW, or, P = η × ΔΕ η = P/ΔΕ = 2 × 10-3W/3.143 × 10-19 J = 6.363 × 1015 ## Problem 4 In a lasing process the ratio of population of two energy levels out of which upper one corresponds to metastable state is 1 : 1.009 × 1025. Calculate the wavelength of laser beam at 320 K. ### Solution If the number of atoms per unit volume in the lower energy state and metastable state are N1 and N2 respectively then, N1/N2 = 1 / 1.009 × 1025 or, N2 / N1 = 1.009 × 10-25 T= 320K Now, from Boltzmann relation, [equation (7.5)] we get, N2/N1 = e-(E2-E1)/kT or, 1.009 × 10-25 = e-ΔΕ/kT or, ΔΕ/kT = ln (1.009 × 10-25) or, ΔΕ = 1.38 × 10-23 x 320 x 57.555 [ k = Boltzmann’s constant = 1.38 x 10-23J. K-1] = 2.54 × 10-19 J If λ be the wavelength of the laser beam, then, hc / λ = ΔΕ or, λ = hc/ΔΕ = (6.63 × 10-34J) × (3 × 108 m. s-1)/2.54 × 10-19 J = 7.83 x 10-7 m = 783 x 10-9 m = 783 nm ## Problem 5 In a He-Ne laser transition from 3S to 2P level gives a laser emission of wavelength 632.8 nm. If the 2P level has energy equal to 15.2 × 10-19 J, calculate the pumping energy required. Assume no loss. ### Solution We know, pumping energy = energy of 2P level + energy of photon = 15.2 × 10-19 J + hv = 15.2 × 10-19J + hc/λ = 15.2 × 10-19 J + 6.62 × 10-34 J.s×3×108m. s-1/632.8 x 10-9 m = 15.2 × 10-19 J + 3.13 × 10-19 J = 18.33 × 10-19 J = 11.46 eV [: 1 eV = 1.6 × 10-19 J] ## Problem 6 Find the ratio of populations of the two states in thermal equilibrium at room temperature 27oC. The wavelength corresponding to the energy gap between two states is 6000Å. ### Solution If N2 and N1 are the number of atoms available at the upper energy (E2) state and lower energy (E1) state respectively, then we can write, N2/N1 = eΔΕ/kT, where ΔΕ = E2 - E1 or, N2/N1 = ehv/kT = ehc/λkT = 1.89 x 10-35

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