Chapter 7 Four-Wave Mixing phenomena

Transcription

1 Chapter 7 Four-Wave Mx phenomena We wll dcu n th chapter the general nonlnear optcal procee wth four nteract electromagnetc wave n a NLO medum. Frt note that FWM procee are allowed n all meda (nveron or non nveron ymmetry Mult-reonant procee can dramatcally enhance the magntude of. In general, when ( allowed, >>, we therefore wll conder NLO ( meda wth nveron ymmetry uch that ( effect can be neglected. Ueful applcaton wth procee Tunable I or UV coherent ource n all meda Wave front recontructon wth DFWM (e.g., Phae Conjugate Optc Powerful pectrocopc technque 7. Thrd-Order Nonlnear Optcal Suceptblte ( n a medum wth nveron ymmetry where 0, the lowet nonlnear optcal proce. cont of 8 term Near reonance, a few term of energy denomnator are reonantly enhanced through the Note: vertce on the left: C (, vertce on the left: C (, vertce on the left: C (, 0 vertce on the left: C (, 0 C (, + C (, +C (, +C (,0 8 Permutaton of three vertce! 6! 8 8 In prncple, the reonant part of part through reonant dperon can be eparated from the non reonant

2 + N 7.. Sly eonant Cae For the three nput pump frequence,,,, le reonance of can occur when any of the three frequence or ther algebrac um approach a tranton frequency of the medum, uch a ( reonant wth E g g In th cae, + N a * N ( M g j ( M g kl ( g + ( Γ [ ( a ] jke g a a er n n er g er n n er g g er n n er g g g a where ( M g j [ ] ( ( + ( M [ kl n a j j k e ere n n erk g ] ( ( + n

3 The expreon for the I-th dagram N N [ ( a + ] I term Tr( µ µ * * gn EE E EE E n where H( H ( H( I n g g n n g + (0 ( gg ( + ( + Γ ( g g The expreon for the II-th dagram ( II (I + H( H ( (0 H( n n g gg n ( ( + Γ ( + g g (0 gg ( er g n ( er ( er g n ( er n er g gg n ( g g + Γ g g + a (0 gg { M g g} n er g ( + Γ gg n g g g g a Smarly, (III IV (0 n era g ( + Γ ( + gg { M g} g g g

4 7.. Doubly eonant Cae reonant wth g g molecular vbratonal tranton, plu le-photon reonance (.e., two-photon proce reonantly couple wth the Note the follow photon energy conervaton relaton: + and + ( a j k l Ne rgm rmg rg nr ( ( + Γ ( + Γ m mg gg g g ( b j k l Ne rgm rmg rg nr ( ( + Γ ( + Γ m mg g g g g [ ( ] jkl ( a + b j j k l Ne rgm rmg rgm r mg rg nr + m ( mg ( + mg ( + Γ ( g g + Γg g II. Tme-order Permutaton

5 ( c e k j (0 Ne rg mrmg rgn r gg m ( mg ( + Γ ( g g + Γg g ( ( a+ c Ne r r r r r r l l k j (0 g m mg g m mg gn gg m ( mg ( + mg ( + Γ ( g g + Γg g r k j l Ne (0 gn rnm rmn ( d gg m ( + Γ ( mg + Γ r l j k k j Ne r (0 gn r rnm rmn rnm r mn ( a+ d { } gg + m ( + Γ ( + Γ wmg mg 7.. Trply eonant Cae All three energy denomnator n are near reonant

6 ( + Ne ( + Γ ( Γ ( + Γ ( Γ r r r r + T j k e (0 n g n g gn gn n g gg ( + Γ ( Γ n n ( + Γ ( + / g ( + Ne r j k e n r n g r gn r n (0 gg ( n g Γ n g ( n g + Γ n g + ( n + Γ n ( g Γ g( n g + Γ n g ( n + Γ n ( + / T n The effectve of an enemble of molecule or on hould be a weghted average of over the dtrbuton of the reonant frequence ( f ( H 7- General Theory of Four-Wave Mx For a cubc or otropc medum

7 E P c [ + ε ( ] π ( where / Aum SVA Neglgble pump depleton Smplfy boundary condton d ε π ( P e dz Z k z ε (, z ε ε ε ( e e π ( jkl lj k l ( z where α z Now further aum Output Feld n the ame mode a one of the nput feld,.e. E El Then E l wll experence gan or lo nduced by the nonlnear wave nteracton & ε π k z jkl ε lj ε k ε e dz d π d ε z ε ε e ε ( z ( jkl lj k k z

8 ε ( z ε (0 e g ( z α z π Z where g ( z ( e jklε j ε k ( z epecally when E E & 0 π lj * k ( Im( jkl ε j g z z The reult mlar to the cae of tmulated aman gan. Backward Parametrc Amplfcaton In th cae, two pump beam are ued ntead of one feld a n optcal parametrc amplfcaton baed on ( effect. Let u conder (a perfect phae match: 0 (b neglgble pump depleton: dε z dz π ε * ε ε ε π * * * ε ε ε d dz

9 ε ε ε * ( z 0 ( z l / co( gol / + (0tan( gol / ε ( z l ε ( ltan( g l / + ε (0 / co( g l / * * * o π where g0 eff g l π Onet of Ocllaton! o o 7. Degenerate Four-Wave Mx (DFWM f, then P ( compoed of three term: P ( PA ( +, + PB (, + P ( + +, C where P ( +, ( : E ( E ( E ( * A P ( +, ( : E ( E ( E ( * B P ( +, ( : E ( E ( E ( * C

10 ( at leat ly reonant ar from two-photon zero frequency (-0 reonance ( - + /T or two-photon reonance f + 0. when 0, trply reonant. Th mple very large reonant enhancement n DFWM. We can undertand DFWM a follow: Two of the three nput feld nterfere and form a tatc grat, the thrd nput wave cattered by the grat to yeld the output wave, e.g. grat formed by and catter the wave to yeld ± ( when. + Snce ε / c, the three output wave are not phae-matched n general. But f, then alway phae matched,.e. (, ( : E( E ( E ( * A( E E E + B( E E E + C( E E E * * *

11 B( θ A( π θ mov grat tatc grat By properly arra the polarzaton of the nput feld, only one term can reman. Intal condton: ε ( l 0 ε (0 ε object wave o from oluton of the coupled wave eq. ε * (0 ε (0tan( o / * o g l ε (0 tan( g l / complex conjugate wave of ε (0 8- Phae Conjugaton by DFWM Output wave complex conjugate to the phae of an nput wave Can appear n SFG, DFG, OPA and FWM If the PC output propagate n the backward drecton wth repect to the correpond nput wave, t can be ued to correct the aberraton encountered by the nput wave, uch a Method to acheve PC DFWM by u Photorefractve crytal; tmulated Brlloun catter 7.5 Tunable I & UV Generaton Objectve: u FWM to extend laer output frequency to I or UV. Conder atomc vapor ~ 0 eu / atom Sle reonance ~ 0 eu / atom Can be doubly or trply reonant to ncreae further.

12 E.g. Tunable I output wth by vary Ne ( A r p p r 5 5 r 5 p 5 p r jkl p j k e p where A p wth ( + Γ ( + Γ ( 5 p 5 p 5 5 p 50 cm, Γ 0. cm - 5 p 5 p 5000 cm / N ~ 6 0 eu / atom - 7 when atom cm eu 7 0 N~ 0 / ~ 6 0! Tunable UV output

13 e.g., FWM for tme-reolved nfrared pectrocopy The converon from I to the vble effectvely unform over a relatvely wde pectral rae. The up-converted gnal thu provded a fathful reproducton of the pectral ntenty.