PHYS 630: Homework Vdue date: Monday, December 8th, 2008 in my mailbox.You are welcome to work together. If you partially use work from other (e.g.something you might have found in a book or a journal paper), you should properlycredit the author by citing the material used.1. Threshold of a Ruby Laser (20 pts)(a) At the line center of λ0= 694.3 nm transition, the absorption coefficientof ruby in thermal equilibrium (i.e. without pumping) at 300 K is α(ν0) =−γ(ν0) ' 0.2 cm−1. If the concentration of Cr3+ions responsible for thetransition is Na= 1.58×1019cm−3, determine the transition cross sectionσ0≡ σ(ν0). (7 pts)(b) A ruby laser makes use of a 10 cm long rod (with refractive index n =1.76) with a 1 cm2cross section and operates at the transition λ0=694.3 nm. Both ends of the rod are polished and coated so that each hasa reflectance of 80%. Assuming that these are the only sources of losses,determine the resonator loss coefficient αrand the corresponding photonlifetime τp. (7 pts)(c) As the laser is pumped γ(ν0) increases from is initial thermal equilibriumvalue −0.2 cm−1to positive values thereby providing gain. Determinethe population difference threshold Ntfor laser oscillation. (6 pts)2. Spectral Broadening of a Saturated Amplifier (20 pts) Consider ahomogeneously broadened amplifier with a Lorentzian lineshape function withwidth ∆ν:g(ν) =∆ν/(2π)(ν − ν0)2+ (∆ν/2)2.Show that for a photon-flux density ϕ, the amplifier gain coefficient γ(ν) as-sumed a Lorentzian lineshape with width∆νs= ∆νr1 +ϕϕs(ν0)where ϕsis the “saturated flux” defined in the Lecture Notes. (20 pts)13. Frequency pulling effect (30 pts): In the Lecture Notes, we show that thesecond condition for laser oscillation (besides the gain coefficient larger thanthe loss coefficient) was2kd + 2ϕ(ν)d = 2πq.If the first term in the right-hand side is much larger than the second termthen the modes of the laser are given by the resonator modes. We also brieflydiscussed the “frequency pulling” effect occurring when the second term isnon-negligible. The purpose of this exercise is to quantify this effect.(a) Assume ϕ(ν) is given by the phase shift associated to a Lorentzian line-shape function and write down an expression for νq≡ qc/(2d) as a func-tion of ν. (10 pts)(b) Make a sketch for νqas a function of ν and explain how one can graphicallydetermine the frequency of laser oscillation from the plot. The lasermodes have frequencies νq0. (15 pts).(c) Assuming the lasing frequency is very close to the cold resonator model,derive a simple and approximative analytical equation for νq0. (5 pts)4. Numerical investigation of pulsing via Mode-Locking (30 pts): Writea simple computer program to plot the intensity I(t) = |A(t)|2(in arbitraryunits) of a wave whose amplitude A(t) is given by the sumA(t) =MXq=1Aqeiq2πct2d,where d is the length of the resonator. For each the following three casesprovide a plot of |I(t)| for a total number of mode M = 11, 21, and 51. Youwill consider the three following cases for the complex amplitude Aq:(a) Equal magnitudes and same phases. (10 pts)(b) Magnitudes that obey a Gaussian spectral profile |Aq| = exp−12(q/5)2 and the same phases. (10 pts)(c) Equal magnitudes but random phases uniformly distributed in [0, 2π].(10