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5. The figure shows the fundamental absorption band HBr in the near infrared.

Draw an energy-level diagram and show the transitions responsible for theobserved spectrum. [7]

Give a simple theoretical model which accounts for the energy levels, includingmolecular rotation and vibration, and from the data in the figure, extract anapproximate value for the inter-nuclear separation, and the force constant. [10]

From the data in the figure, make an estimate of the temperature of the gassample. [8]

Bromine has a relative atomic mass of 80.

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3. The difference in energy between an upper level 2 and lower level 1 in an atom ish0, and the degeneracies of the upper and lower levels are g2 and g1 respectively.The line-shapes for spontaneous emission, absorption, and stimulated emission are

gA(0), gB(0) andgB0(0) respectively, where in each case the line-shapeis normalised such that

Z

0

gA,B,B0( 0) d= 1.

Define the Einstein coefficients A21,B21and B12for radiative transitions betweenthe two levels in the atom. [3]

Show that the Einstein coefficients are related by

g2B21 = g1B12

and

A21= h3B21

2c3 . [5]

Prove that the three line-shape functions are identical. [3]

What is meant by homogeneous and inhomogeneous line broadening? Give anexample of each. [6]

Spectral brightness is defined as the number of photons per second per steradianper unit area per unit frequency range. An X-ray laser emits pulses of light containing1012 photons per pulse, in a pulse length of 1013 s. The beam has a diameter of300-m and an angular divergence of 106 radians. The photon energy is 8 keV with

a bandwidth (/) of 5 103

. What is the spectral brightness of the X-ray laser?What would the temperature of a blackbody need to be to have the same spectralbrightness at 8 keV as the X-ray laser? [8]

[ The energy density per unit frequency range of blackbody radiation is

() = h3

2c31

exp(h/kBT) 1

at frequency and temperature T. The flux of particles with number density n and ofvelocity c through unit area is nc/4. ]

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2. What is the essential difference between a 4-level and a 3-level solid-state lasersystem? Give one example of each type and explain qualitatively why the pump power

necessary to bring the two different types of laser to threshold can differ widely. [5]

Explain what is meant by Q-switching, and describe briefly one means by whichit is achieved in practice. [5]

Derive the rate equations governing the time evolution of the inversion densityN(t) and the photon number n(t) in the cavity of a 4-level laser that is Q-switched attime t= 0. Assume that the laser medium fills the cavity volume and that the upperand lower laser levels have equal degeneracy.

Show that at the time when n(t) achieves its peak value, the instantaneous valueof the inversion density Np equals the inversion density required to bring the laser tosteady state threshold in the same cavity with the Q-switching element in its low-loss

state. [10]

The energy utilisation factor is defined as

=Ni N

f

Ni,

where Ni is the value ofN(t) at t= 0 and Nf is the value ofN

(t) as t . Showthat

=Np

Niln

NiNf

.

State clearly any assumptions you make. [5]

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