P. Piot, PHYS 630 – Fall 2008Nonlinear optics• To date we considered dielectric media characterized by a linearrelation between polarization and E-fied• Now we consider media characterized by a nonlinear relationbetween E and P! P ="0#E! P = a1E +12a2E2+16a3E3+ ...="0#E +#(2)E2+#(3)E3+ ...( )="0#E + 2dE2+ 4#(3)E3+ ...2nd order 3rd orderP. Piot, PHYS 630 – Fall 2008Nonlinear wave equation I• From Maxwell’s equation and now considering• We obtain a nonlinear wave equationwhere P is usually written as! "2E #1c02$t2E =µ0$t2P! P ="0#E + PNL! PNL= 2dE2+ 4"(3)E3+ ...! D ="0E + PP. Piot, PHYS 630 – Fall 2008Nonlinear wave equation II• Using• The nonlinear wave equation can be rewritten with• Born approximation consist in adopting an iterative solution for theNL wave equation! "2E #1c2$t2E = SS =µ0$t2PNL! c = c0/ nn2= 1+"“source”radiationS ES(E)P. Piot, PHYS 630 – Fall 2008• Assume higher order than second order are negligible• Take• Corresponding polarization isSecond harmonic generation (SHG) I! PNL= 2dE2! E =12E (")ei"t+ c.c.( )! E = 2d14E (")ei"t+ c.c.( )E*(")e#i"t+ c.c.( )= dEE*+ d EEe2i"t+ c.c.( )dcoptical rectification2ω2nd harm. generationEP= +P. Piot, PHYS 630 – Fall 2008• Use to frequency-double lasersSecond harmonic generation (SHG) IIP.A. Franken, et al, Physical Review Letters 7, p. 118 (1961)P. Piot, PHYS 630 – Fall 2008Second harmonic generation (SHG) III• The actually published results…Input beamThe second harmonicNote that the very weak spot due to the second harmonic is missing.It was removed by an overzealous Physical Review Letters editor,who thought it was a speck of dirt.P. Piot, PHYS 630 – Fall 2008• Consider a plane wave propagating along z, and write the paraxial Hemholtzequation• Can be approximated aswhere Δk=k(2ω)-k(ω) so finallyneed Δk=0 to achieve maximumE(2ω) electric field (this was nottaken care in the 1961 experiment)• This is referred to as “phase matching”Phase matching I! "#2E (2$) % 2ik&zE (2$)[ ]e%k(2$)z=µ0&t2dE2($)( )e%k($)z! "zE (2#) = AdE2(#)e$kz! E(2") #sin$kz$kzPoor phase matchingGood phase matchingP. Piot, PHYS 630 – Fall 2008Phase matching IIwhich will only be satisfied when:Unfortunately, dispersion prevents this from ever happening!So we’re creating light at ωsig = 2ω. 02 2 ( )polk k nc!!= =0 0(2 )( ) (2 )sigsig sigk n nc c!!! != =sig polk k=!2!FrequencyRefractive indexAnd the k-vector of the polarization is: The phase-matching condition is:The k-vector of the second-harmonic is:! n(2") = n(")P. Piot, PHYS 630 – Fall 2008Phase matching IIIWe can now satisfy the phase-matching condition.Use the extraordinary polarizationfor ω and the ordinary for 2ω.Birefringent materials have different refractive indices for different polarizations. Ordinary and extraordinary refractive indicescan be different by up to ~0.1 for SHG crystals. !2!FrequencyRefractive index! neonne depends on propagation angle, so we can tune for a given ω.Some crystals have ne < no, so the opposite polarizations work.! no(2") = ne(")P. Piot, PHYS 630 – Fall 2008Phase matching IV0ˆ2 cos2 ( ) cospolpolk k k k zk nc!"" !#= + =$ =r r r2(2 )sigok nc!!=ˆˆcos sink k z k x! != "rˆˆcos sink k z k x! !"= +rzxBut:So the phase-matchingcondition becomes:θθ! n(2") = n(")cos#•Phase matching via non-colinear overlap oftwo ω beamsP. Piot, PHYS 630 – Fall 2008Application of noncollinear phase matching•Noncollinear phasematching can be usedto measure the durationof ultra-short