Midsummer Examinations 2016

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1 All candidates Midsummer Examinations 2016 Blackboard version with numerical answers DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR Department PHYSICS AND ASTRONOMY Module Code Module Title Exam Duration (in words) COURSE LASERS AND QUANTUM OPTICS Seventy Five Minutes CHECK YOU HAVE THE CORRECT QUESTION PAPER Number of Pages 4 Number of Questions 3 Instructions to Candidates Candidates should answer TWO questions. The marking scheme for each question is indicated by the numbers in square brackets on the right-hand side of the question. Each question carries 20 marks. NO PAPERS MAY BE REMOVED FROM THE EXAMINATION ROOM. FOR THIS EXAM YOU ARE ALLOWED TO USE THE FOLLOWING: Calculators Books/Statutes provided by the University Are students permitted to bring their own Books/Statutes/Notes? Additional Stationery Casio FX83GTPLUS or Casio FX85GTPLUS Physical Constant and Mathematical Formula sheets No None Page 1 of 4

2 All candidates 1. A monochromatic beam of light is right hand circular polarized and is propagating in a vacuum parallel to the z-axis. The wavelength is λ (m) and the electric field vector amplitude is E 0 (V m 1 ). (a) Write down expressions for (i) components of the wavevector, k, of the beam, [1] (ii) the energy (J) and spin (J s) of each photon in the beam, [2] (iii) the electric field components E x (z, t) and E y (z, t) and [2] (iv) the number density, N (m 3 ), of photons in the beam. [4] (b) The beam is incident on a quarter-wave plate of area A (m 2 ). 95% of the beam passes through the plate and 5% is reflected. Write down expressions for (i) the spin of each photon within the beam that exits the plate and [2] (ii) the spin of each photon within the beam that reflects from the plate. [1] (c) If the area, A, of the quarter-wave plate is fully illuminated by the beam, write down expressions for the forces experienced by the plate. [4] (d) Describe how you would make a quarter-wave plate from a birefringent material. [4] 2. (a) What is Total Internal Reflection and why is it crucial to the way in which optical fibres, used in modern communication systems, are made? [4] (b) Define the numerial aperture and normalised thickness of an optical fibre. [4] (c) What characterises a propagation mode in an optical fibre and how is the number of modes that can propagate related to the normalised thickness? [4] (d) An optical fibre is designed to operate at a wavelength of 1.3 µm. If the refractive indices of the core and cladding are 1.53 and 1.50 respectively, what is the maximum Page 2 of 4

3 All candidates core radius that will give single mode operation? [4] (e) What are the advantages and disadvantages of using a single mode fibre, operating at a wavelength of about 1.3 µm, in a long distance communication system? [4] Page 3 of 4

4 All candidates 3. (a) What is a resonant cavity and why it is important to the operation of most lasers? [4] (b) What type of resonant cavity is used to produce a laser beam with (i) high coherence and (ii) high power? [1] [1] (c) What are the laser modes associated with the performance of a resonant cavity? [1] (d) How is the number of laser modes excited determined by the gain profile function of the electronic laser transition and the length of the resonant cavity? [4] (e) Why are the laser modes which are excited so monochromatic? [2] (f) The line centre of the main transition in the He-Ne laser is nm. A coherent He-Ne laser has a cavity of length 0.5 m. What is the mode separation for the laser and what is the wavelength and number of the mode nearest to the line centre? [3] (g) A laser operating at a wavelength of 532 nm produces pulses which have a duration of 3 ns and a bandwidth of nm. Is the coherence of the beam determined by the duration of the pulses? [2] (h) What is the coherence length of the beam and what is the effective Q factor of the laser? [2] END OF PAPER Page 4 of 4

5 PHYSICAL CONSTANTS elementary charge e = C electron rest mass me = kg proton rest mass mp = kg neutron rest mass mn = kg Planck constant h = 2π = J s speed of light in vacuum c = m s 1 Boltzmann constant k = J K 1 Bohr magneton B = J T 1 Stefan-Boltzmann constant = W m 2 K 4 radiation constant a = 4 / c = J m 3 K 4 Avogadro number N A = mol 1 gas constant R = J mol 1 K 1 gravitational constant G = N m 2 kg 2 permittivity of vacuum o = F m 1 permeability of vacuum o = 4π 10 7 H m 1 atomic mass unit 1 u = kg MeV electronvolt 1 ev = J atmospheric pressure 1 atm = N m 2 ASTRONOMICAL CONSTANTS mass of Sun M O = kg radius of Sun R O = km luminosity of Sun L O = J s 1 one parsec pc = m mean Earth-Sun distance 1 a.u. = m mass of Earth M O = kg mean radius of Earth R O = 6370 km

6 1. Series 1.1 Binomial expansion: (1 x) n MATHEMATICAL FORMULAE N r n x r, where n r r 1.2 Geometric series: x n 1 x N 1 n0 1 x for x Exponential function: e x 1 x x2 2! x3 3! x n n! log(1 x) x x2 2 x (1) n x n...(1 x 1) n 1.5 cos x 1 x2 2! x4 4! sin x x x3 3! x5 5! Trigonometric Identities cos 2 x sin 2 x 1, 1 tan 2 x sec 2 x, cot 2 x 1 cosec 2 x sin2x 2sin xcos x sin( A B) sin Acos B cos Asin B tan A tan B tan( A B) 1 tan Atan B cos A cos B 2cos A B cos A B Integration 3.1 Powers : x 3.2 Rational functions : cos2x 2cos 2 x 1 x1 1 ( 1); x1 log x a 2 x Irrational algebraic functions : (a 2 x 2 ) 1/2 sinh 1 x a n(n 1)...(n r 1) r(r 1)...1 cos( A B) cos Acos B sin Asin B A B sin A sin B 2sin cos A B 2 2 a 1 tan 1 x a (a 2 x 2 ) 1/2 (x 2 a 2 ) 1/2 3.4 Trigonometric functions: sinx cos x cos x sin x 1 a 2 x 2 2a log a x a x sin 1 x a cosh 1 x a cosecx log tan x 2 secx log(sec x tan x) 2 cos 2 2 x sin 2 x 0-0,m n 0 cosm cos nd 2,m n 0 0,m n

7 - sinm cosnd 0 for all m,n sinm sin nd,m n 0-0,m n 3.5 Hyperbolic functions: sinhx cosh x coshx sinh x 3.6 Exponential: e x2 1/2 0 2 cosechx log tanh x 2 sechx tan 1 (sinh x) 4. Statistical Distributions Binomial Poisson Gaussian B(m,n, p) p(m,) m N(x,, ) n! m!(n m)! pm (1 p) nm ; np; 2 np(1 p) m! e ; 2 1 (x )2 exp 1/2 (2) Vector Calculus 5.1 Cartesian coordinates (x,y,z): f f x, f y, f z.a A x x A y y A z z A A z y A y z, A x z A z x, A y x A x y 5.2 Element of Area: Sphere = r 2 sindd Cylinder = rdzd 5.3 Element of Volume: Sphere = r 2 sindrdd Cylinder = rdrdzd 6. Miscellaneous Formulae 6.1 Stirling's formula n!~ n n (2n) 1/2 as n, log n!~ nlog n n as n. e

8 NUMERICAL ANSWERS 1. None. 2. radius a 1.66 µm 3. coherence length 0.09 m, effective Q factor

MIDSUMMER EXAMINATIONS 2001

No. of Pages: 7 No. of Questions: 10 MIDSUMMER EXAMINATIONS 2001 Subject PHYSICS, PHYSICS WITH ASTROPHYSICS, PHYSICS WITH SPACE SCIENCE & TECHNOLOGY, PHYSICS WITH MEDICAL PHYSICS Title of Paper MODULE