P. Piot, PHYS 630 – Fall 2008Difference frequency generationω1ω1ω3ω2 = ω3 − ω1Parametric Down-Conversion(Difference-frequency generation)Optical ParametricOscillation (OPO)ω3ω2"signal""idler"By convention:ωsignal > ωidlerω1ω3ω2Optical ParametricAmplification (OPA)ω1ω1ω3ω2Optical ParametricGeneration (OPG)Difference-frequency generation takes many useful forms.mirror mirrorP. Piot, PHYS 630 – Fall 2008Optical parametric amplificationOptical parametric generation/amplification allows frequencytunability over a large bandwidthP. Piot, PHYS 630 – Fall 2008Optical parametric amplificationω1ω3ω2Optical ParametricAmplification (OPA)Assume ω3 is a “strong wave”The wave equation gives the coupledequationsSo both E(ω) and E(2ω) are solutions of an ODE of the formSo general solution of the formE(2ω)=0 a z=0 and E(ω) arbitrary gives! "zE(#) = AE*(2#)E (3#)"zE(2#) = BE*(#)E (3#)$ % & ! "z2E (#,2#) = DE(#,2#)! M sinh(Dz) + N cosh(Dz)! E (2") = X sinh(Dz)E(") = Y cosh(Dz)# $ %P. Piot, PHYS 630 – Fall 2008Optical parametric amplification & oscillatorsE(2ω)=0 at z=0 and E(ω) arbitrary givesFor large z field amplitude growths as exp(Dz)Such scheme can be used as an amplifier! E (2") = X sinh(Dz)E(") = Y cosh(Dz)# $ % ωiωpωsmirror mirrorThe big advantage here is the possibility to tune ωi and ωs, e.g. via phase matchingP. Piot, PHYS 630 – Fall 2008Example of useP. Piot, PHYS 630 – Fall 2008Third order polarizationE(2ω)=0 at z=0 and E(ω) arbitrary givesFirst let’s consider the simple case of third harmonic generation(THG). TakeThen the third order polarization is! PNL"= 4#(3),$%&"E$E%E&! E = E0cos("t)! E = 4"(3)E03cos3(#t)= 4"(3)E0314cos(3#t) +34cos(#t)$ % & ' ( ) 3rd order generationNonlinear contributionthat affects the fundamentalfrequencyP. Piot, PHYS 630 – Fall 2008Third order polarizationThe ω term in the polarization is of the formTherefore we can define an intensity-dependent refractive index:Which is often written asThis is known as the optical Kerr effect! PNL(") =#(1)E(") + 3#(3)| E(") |2E(")=#effE(")! n2= 1+"eff= 1+"(1)+ 3"(3)2ZI= n02+ 6Z"(3)I! n = n0+ n2INonlinear refractive index!
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