P. Piot, PHYS 630 – Fall 2008• How does one build a laser– Laser medium• Will dictate wavelength, pulse duration, achievable power– Pumping: Amplified Stimulated Emission (ASE)• Multimode in time and space– Resonator: laser oscillator• Mode selectionLaser oscillatorFrom R. TrebinoP. Piot, PHYS 630 – Fall 2008Laser amplification I• Amplification occurs within a frequency bandwidth given by thelinewidth of the considered atomic transition• The laser amplifier is a distributed-gain device characteized by thegain coefficient– When the photon flux is small– When photon flux increases! "0(#) = N0$28%tspg(#)Equilibrium populationdensity differenceTransition crosssection σ(ν)Wavelength in mediumSpontaneous lifetime! "(#) ="0(#)1+$/$s(#)Saturation Photonflux densityP. Piot, PHYS 630 – Fall 2008Laser amplification II• Amplification in a medium alsointroduces a phase shift! "=#$#0%#&(#)ν−ν0ν−ν0γ(ν)ψ(ν)P. Piot, PHYS 630 – Fall 2008Feedback & Loss: Optical resonator• Optical feedback is achieved by placing the gain medium in anoptical resonator• Consider a simple resonator with two mirrors separated by d thephase shift per unit length (= the wave vector) is• The resonator also contributes losses in the system the round triploss is characterized in term of a distributed loss coefficient rα andisthis includes mirrors reflection losses and scattering whilepropagation in the resonator! k =2"#c! exp("2#rd)P. Piot, PHYS 630 – Fall 2008Laser threshold I• Two competing mechanisms– Gain in medium– Loss in resonator• The initiation of laser oscillation requires that the small-signal gaincoefficient is larger that the loss coefficient:• Or equivalently thatNt is the threshold population difference Introducing the photonlifetime αr=1/(cτp).! "0(#) >$r! N0>"r#($)% Nt! Nt=1c"p#($)P. Piot, PHYS 630 – Fall 2008Laser Threshold II• Nt is a function of frequencythe threshold is lowest where the lineshape function is he highest!• Remember thatso g(ν) maximum at ν0, and• If we further assume the linewidth is given by the lifetime only then! Nt=8"#2ctsp$p1g(%)! g(") =#"/2$("%"0)2+ (#"/2)2! Nt=2"#2c2"$%tsp&p! Nt=2"#2c$p=2"%r#2Typical values areNt=105-8 cm-3P. Piot, PHYS 630 – Fall 2008Phase condition• The phase shift imparted to the light in the resonator must be amultiple of 2p:• If φ(v)d is small then• If φ(v)d cannot be neglected needto solve the equation! 2kd + 2"(#)d = 2$qPhase shift dueto resonatorMedium-inducedphase shiftinteger! "="q= qc2d# $ % & ' ( Already derived many timesthis is referred to as “cold resonator” conditionνGainΔνν0Cavity modeΔνcνqνLaser mode! "+c2#"$"0%"&(") ="qThis gives rise to a frequency pullingP. Piot, PHYS 630 – Fall 2008Internal photon flux density I• Provided the threshold andphase conditions are fulfilledlasing will start• Gain initially given by “small-signal” gain, but eventuallydecrease as photon flux in-creases• As long as gain larger thanlosses growth of optical fieldwill prevailGainΔνν0Cold-resonator modesΔνcνq-1νqνq+1Laser oscillation modes! "! "! "P. Piot, PHYS 630 – Fall 2008Internal photon flux density IIγ(ν)/γ0(ν)φ/φs(ν)Laser turn-onαr: loss coefficientSteady statetimetimeGain curveGain clampingP. Piot, PHYS 630 – Fall 2008Internal photon flux III• Equating the large signal gain and loss coefficient gives the photonflux density arising from laser action.• Which can be rewritten in term ofN0 and Nt as! "="s(#)$0(#)%r&1' ( ) * + , ,$0(#) >%r0,$0(#) -%r. / 0 1 0 ! "="s(#)N0Nt$1% & ' ( ) * , N0> Nt0, N0+ Nt, - . / . φφsNt2NtN0Pumping rateNtNtN0Pumping rateNP. Piot, PHYS 630 – Fall 2008Output photon flux• Consider the photon to be “out-coupled” from the cavity by havingone mirror with transmittance TO• The output photon flux is then• The corresponding optical intensity is• And the beam power would be! "O= TO"! IO= h"TO#! PO= IOACross-sectionalarea of the laser