P. Piot, PHYS 630 – Fall 2008• Consider an atom located in an optical field of flux φ. The probabilityof stimulated emission is• If N1 and N2 are respectively the number of atoms in the lower andupper energy level then– The average density of absorbed photons (number of photon perunit time per unit volume) is– The average density of stimulated photons• The net number of photon gained isGain coefficient I! N1Wi! Wi="#($)! N2Wi! N = (N2" N1)WiP. Piot, PHYS 630 – Fall 2008• N is the population density difference– N>0 population inversion (more atoms in excited states)medium can act as an amplifier– N<0 medium act as an absorber– N=0 medium is transparent• To provide N>0 (the interesting case) we need an external pump thatexcites the atom• External pumping is achieved via radiative or non radiative effects:– Optical pump,– Chemical reaction– Electrical processGain coefficient IIP. Piot, PHYS 630 – Fall 2008• Let ϕ(z) and ϕ(z)+dϕ(z) be thephoton flux entering and existingthe cylinder of length dz• Then dϕ(z) must be the photonflux emitted within the thin sliceof cylinder. So• γ(n) represents the net gain in the photon flux per unit length of themedium:Gain coefficient III! d"= NWidz #d"dz= NWi= N$(%)"z0 dOutputlightInputlightφφ+dφz z+dz! d"dz=#($)"! "(#) = N$28%tspg(#)P. Piot, PHYS 630 – Fall 2008• The solution for the photon flux ϕ(z) is• And correspondingly the optical intensity isso γ(ν) also represents the gain in intensity per unit length of medium• For an interaction region of length d the net gain isGain coefficient IV! "="(0)e#($)z! G(") #$(d)$(0)= e%(")d! I = h"#= I(0)e$(")zP. Piot, PHYS 630 – Fall 2008• If g(ν) assumes the form• Then γ(ν) is also of theBandwidth! g(") =#"/2$("%"0)2+ (#"/2)2γ(ν)/γ(ν0)ν0ν! "(#) ="(#0)($#/2)2(#%#0)2+ ($#/2)2P. Piot, PHYS 630 – Fall 2008• Consider the E-field associated to the wave E∝I1/2• At z+dz the field is• Causality implies (see, e.g., Arfken p. 483) thatthe real and imaginary parts are related viaHilbert transform and the phase shift is givenby (see next slide)Phase shift! E(z) = E (0)e12"(#)ze$i%(z )! E(z + dz) = E(0)e12"(#)[z +dz ]e$i%(z +dz )= E(z)e12"(#)dze$i%dz& E(z)[12"(#) $ i%]dz! "=#$#0%#&(#)ν−ν0ν−ν0γ(ν)ψ(ν)P. Piot, PHYS 630 – Fall 2008• Consider an harmonic oscillator described by• Assume external harmonic force and displacement of the form• The transfer function is• Using the above transfer function one canshow thatrelation between real and imaginary part of the transferfunction -- example using the harmonic oscillator! d2dt2+"ddt+#02$ % & ' ( ) f2(t) = f1(t)! f1(t) = e2i"#tf2(t) = H(#)e2i"#t! H(") =1(2#)21"02$"2+ i"%"! "[H(#)] =$#2(#%#0)&[H(#)]P. Piot, PHYS 630 – Fall 2008Pumping wo amplifier radiation I• To achieve population inversion an external source is needed:such asource is called a “pump”• Pump should provide pumping toexcite the needed state (directlyor indirectly)• Pumping dynamics is described by the rate equations: which providethe change of population densities N1 and N2• If unpumped12τspτnrτ20τ21τ2τ1! 1"2=1"21+1"20=1"nr+1"sp+1"20•eventually all atoms go to unexcited state “0”and N1=N2=0P. Piot, PHYS 630 – Fall 2008• When pumping is provided the rate of increase of populationdensities isR1: rate of pumping atoms out of state 1R2: rate of pumping atoms into state 2“rate” are per unit volume per second• Steady-state condition12τ20τ2τ1! dN2dt= R2"N2#2dN1dt= "R1"N1#1+N2#21τ21R1R2! N0= R2"21#"1"21$ % & ' ( ) + R1"1Steady-statepopulation differencePumping wo amplifier radiation IIP. Piot, PHYS 630 – Fall 2008• Large population difference needed (to have a large gain)• This requires– Large R1 and R2– Long τ2– Short τ1 if R1<τ2/τ21R2• Upper level should be pumped strongly and decay slowly• Lower state should be depumped strongly so it quickly disposes of itspopulation.• If τ21<<τ20 so that τ2=tsp and τ1<<tsp we have! N0= R2"21#"1"21$ % & ' ( ) + R1"1Pumping wo amplifier radiation III! N0= R2tsp+ R1"1P. Piot, PHYS 630 – Fall 2008• Rate equations in presence of “amplifierradiation” are• Population density of level 2 is increased by absorption from level 1to 2 and decrease by spontaneous emission from level 2 to 1• Under steady-statePumping w amplifier radiation I12τ20τ2τ1τ21R1R2! dN2dt= R2"N2#2" N2Wi+ N1WidN1dt= "R1"N1#1+N2#21+ N2Wi" N1Wi! Wi! N =N01+"sWiP. Piot, PHYS 630 – Fall 2008• Steady state population iswith• If τsWi<<1 (small signalapproximation) N~N0• As τsWi increases N→0Pumping w amplifier radiation II! N =N01+"sWi! N0= R2"21#"1"21$ % & ' ( ) + R1"1! "s="2+"11#"1"21$ % & ' ( )