New approach to Fully Nonlinear Adiabatic TWM Theory

Transcription

1 New approach to Fully Nonlnear Adabatc TWM Theory Shunrong Qan m preentng a new elegant formulaton of the theory of fully nonlnear abatc TWM (FNA-TWM) n term of ellptc functon here. Note that the lnear cae of SFG and DFG n the undepleted pump approxmaton decrbed by the FVH repreentaton ha been exploted everal year ago. For the ake of completene, preent the peudo-fvh repreentaton to decrbe OPA. Moreover, m tryng to dplay an overvew of TWM procee and how that both the lnear cae, the lnear abatc SFG(DFG) and the lnear OPA, are only the pecal cae of my theory. Fnally alo pont out that the geometrc mage of the o-called abatc ba act a the geodec lne of the generalzed Bloch phere. Keyword: Frequency converon, Adabatc Three-wave-mxng Proce. ntroduton A one type of the mplet nonlnear optcal proce, three-wave-mxng (TWM) procee ha been tuded ntenely and wdely ued n frequency converon. n the low-varyng ampltude approxmaton and plane-wave approxmaton, the couplng equaton n the contnuou plane wave cae wrte: E z E z E E exp( k z),, * z E E exp( k z) z () where the couplng coeffcent () eff (,,) nc. Note that f we et E= E E, E E, and, SHG can be decrbed by equaton (). Now et z E A exp( k z),,, () yeld: where A (,,) the normalzed ampltude. Subcrbng () nto () A A A A A,, * A AA ()

2 where the gn of effectve nonlnear coeffcent, and, z, k /. For mplcty, et A and arg(a ),,,. then t not hard to obtan the Manley-Rowe relaton: K,, (4) K where K (,,) are Manley-Rowe contant. n the pecal cae when couplng coeffcent and phae-mmatch k are both contant, the analytc oluton ha been preented by J. A. Armtrong et al. n 96 []. However, nce the phae mmatch act a a ghot to nterfere the frequency converon, everal dfferent method were put forward to wept the ghot out, among whch the theory of QPM the mot hopeful and powerful one to provoke reearcher to patal modulated the couplng coeffcent and phae-mmatch k n dfferent way. An nteretng way the abatc method, frtly poted by H. Suchowk n 8 []. Snce then, abatc method become more and more popular n nonlnear optc [][4][5]. Moreover, the attempt to remove the undepleted pump approxmaton from TWM alo hatened the theory of fully nonlnear abatc TWM poted by G. Porat and A. Are n [6]. Here the author want to preent a new elegant formulaton of that theory n term of ellptc functon wthout the ntroducton of Hamltonan formulaton. For completene, to begn, the author decde to brefly preent the SFG(DFG) and OPA n the undepleted pump approxmaton, whch namely, n hort, lnear SFG(DFG) and OPA n th artcle.. The Lnear SFG(DFG) and OPA A. The FVH repreentaton Accordng the theory of H. Suchowk, the Schrodnger equaton of lnear SFG wrte: Hˆ h (5) () where ˆ Hh (Re( ) m( ) k ), [A,A] T eff, Ep, c nn n z n z A E exp( k z ), A E exp( k z ), and,, are

3 three Paul matrce, wrte:,. Clearly, the Schrodnger equaton of lnear DFG hare the ame form wth that of lnear DFG. Note that can be decrbed by Bloch vector { UVW,, } n the FVH repreentaton, where U t not hard to ee that * Re(AA) decrbed by the well-known Bloch Equaton: z, * V m(aa) W A (6) A ndeed fall on the Bloch phere whoe evoluton where {Re( ), m( ), k} the torque vector whoe norm equal to the (7) ntantaneou rotatonal angular velocty of about,.e. k. Table The egenytem of Hamltonan H ˆ h Egenvalue k Egenvector {n( ), co( )} {co( ), n( )} k Aume that the couplng coeffcent real. The egenvector hown n Table ued for the abatc ba n the cae of lnear SFG, where the mxng angle and tan( ), whle k n the cae of the lnear DFG. Clearly, to acheve the full frequency converon from gnal to dle wave, the mxng angle need to change from to durng the whole TWM proce. Accordng to quantum abatc theorem, the mxng angle hould atfy the abatcty condton of lnear SFG(DFG), whch wrte: (8) k

4 ..8 ntenty Fgure Adabatc traectory (, ) of SFG and t mage n Bloch phere B. The Peudo-FVH repreentaton n a mlar way, the Schrodnger equaton of lnear OPA wrte: Hˆ nh z (9) () ~ ~ ~ where ˆ Hnh (Re( ) m( ) k ), [A,A] T eff, Ep, c nn n z n z ~ ~ ~ A E exp( k z ), A E exp( k ( ) ), and,, are three peudo-paul matrce, wrte: ~ ~, ~, Note that can be decrbed by Bloch vector { UVW,, } repreentaton, wrte: Set * Re(AA) n the FVH * U, V m(aa), W A () A K A A. t not hard to ee that, fall on one heet of the bparted hyperbolod when K, whle on the cone when K. We only focu on the former cae, and call that urface peudo-bloch phere. Note that alo ha t Bloch Equaton, wrte: z where {Re( ), m( ), k}, the bnary operator repreent a pecal product of two three-dmenonal vector, named peudo-cro product, whoe defnton : () 4

5 AB A B e, and,,,, k k k a bt mlar to the well-known Lev-Cvta ymbol, k,, whch wrte:, k,,(, k, ) (,,),(,,),(,,), (, k, ) (,, ), (,, ), (,, ), otherwe Such a Bloch Equaton, whch can be derved from quantum Louvlle equaton ung the algebrac properte of three peudo-paul matrce ntroduced above, mple that the evoluton of mlar to Lorenz rotaton n the pecal theory of relatvty. Suppoe that purely magnary wth the potve magnary part and q. Then focu on the cae when k q. Accordng to the theory of S. báñez et al. [7], to completely decrbe the non-hermtan Hamltonan need the borthogonal egen-equaton: Hˆ n En n ˆ * H n En () n where n and n are adont vector, whch atfy the followng orthonormal relaton: m n mn,. Table The egenytem of Hˆ nh and ˆ nh H when k q Egenvalue k q K q Egenvector ˆ nh H { nh( ),coh( )} {coh( ), nh( )} ˆ H nh { nh( ),coh( )} {coh( ), nh( )} Note that the egenvector {coh( ), nh( )} hown n Table q certanty the abatc ba, where tanh( ). Repreent the ntenty of k abatc gnal and dle wave a a functon of :, () 5

6 4 ntenty Fgure Adabatc traectory (, phere ) of DFG and t mage n peudo-bloch Clearly, to acheve the full frequency converon from gnal to dle wave, the mxng angle need to change from to durng the whole TWM proce. Accordng to the non-hermtan quantum abatc theorem, hould atfy the abatcty condton of lnear SFG(DFG), whch wrte [7] : k (4) Set the lnear abatc parameter r l to be rl k. One can ee that a approache, r l wll become much larger than, and that to ay, the abatc traectory would break down. Of coure, a workaround may be uper-abatc method [8][9], but there no need to ntroduce t here. However, n the cae of fully nonlnear abatc OPA, t would become better (ee Secton.B). Htherto, two type of abatc procee have been ntroduced here, one lnear abatc SFG (DFG), and another lnear abatc OPA. n the next ecton, the reader may fnd that the two cae are both the pecal cae of fully nonlnear TWM proce. One may alo ee that the geometrc repreentaton of the o-called abatc ba act a the geodec lne of the generalzed Bloch phere.. Fully Nonlnear Adabatc TWM (FNA-TWM) Theory A. Fully Nonlnear Adabatc SFG(ncludng SHG) and DFG Let dcu abatc SFG frt. Note that abatc SHG can be vewed a a pecal cae of abatc SFG. Aume that K K. Accordng the Manley-Rowe relaton (4), one can 6

7 parameterze A (,,) by u: A K dn( ue ) A K cn( ue ) A K n( ue ) (5) where n( u ), cn( u ), dn( u ) are Jacob ellptc functon harng common parameter whch wrte m K / Kand u a functon of. One hould note that the leat common perod of the abolute value of ellptc functon ntroduced above T K( m), where K( m ) repreent the complete ellptc ntegral of frt knd: K( m) mn. Defne n( u)dn( u) dn( u)cn( u) cn( u)n( u) J( u) m. Subcrbng (5) nto cn( u) n( u) dn( u) (), and then eparatng the real and magnary part, yeld the couplng equaton of u and : u K n( ) K co( ) Ju ( ) (6) where J( u) J( u),. Snce Ju ( ) alo a functon wth a perod of T,.e. Ju ( T) Ju ( ), we only need to focu on the followng regon: R ( u, ) u T /, (named the reduced ( u, ) plane). Now let calculate the tatonary pont: (, ) u, whch atfyng the ytem of equaton: u. One can ealy get the root equaton of (6): of the frt, (7) Havng obtaned the value of, u whch ay u (u ) correpondng to ( ) repectvely, can be obtaned from olvng the econd equaton of (6): co( ) KJu ( ) (8) For the ake of mplcty, et. Though the value of u and u can only 7

8 be calculated numercally, one hould note that accordng to equaton (7) and (8), Fgure (a) n fact how u the abatc traectory dcued below. Through patal modulatng from a mall negatve value to a large potve one ncreangly, we can obtan the potve abatc traectory ( u, ), whch actually ft the proce of SFG(SHG), nce a u whle, and a () u T / whle. Clearly, the parameter u correpond to mxng angle n lnear cae. To acheve the full frequency converon from gnal to dle wave, the mxng angle TWM proce. u need to change from to T / durng the whole (a) The graph of functon J( u ) (b) The abatc traectory ( u, ) Fgure Now try to derve the abatcty condton. Frtly, uppoe that there a mall dplacement of, wrte ( u, ) ( u, ) ( u, ) ( u, ) (9) n the frt order approxmaton of ( u, ) at the pont (, ) u, (6) become: u u K u KJ ( u ) () together wth ntal condton ( u(), ()) (,). Followng G. Porat and A. Are, one can obtan that, ' u u co ( ') ' n u () 8

9 where we obtan: wrte: K. Set the nonlnear abatcty condton to be u, then J( ) u u For the cae when m, the abatcty condton of SFG n term of () u u 4K () Recall that n the undepleted pump approxmaton, the abatcty condton wrte n a mlar form of (6) (ee Secton.A): / z k (4) Note that u u, one can obtan: K (5) Set the nonlnear abatcty parameter to be r nl : rnl K (6) The nonlnear abatcty parameter r nl characterze the magntude of the dabatc couplng of two dfferent abatc paenge,.e. ( u, ) and ( u, ). ntenty r nl 5 5 (a) The evoluton of and n SFG (b) Nonlnear abatcty parameter Fgure 4 Adabatc SFG wth K/ K and ( ) Now that the general cae a m [,] have been hown, let oberve what on 9

10 earth happen n the lmtng cae.e. when m at one of the end pont of the nterval [,]. ()When m, the pump wave never deplete, and the abatc traectory ( u, ) wrte: K co ( ) K u n ( ) K u (7) whch the reult hown n Secton.A. ()When m, a a cae that can alo decrbe SHGthe abatc traectory ( u, ) wrte: where K K K. K u ech ( ) K u (8) tanh ( ) Fgure 5 The abatc traectory ( u, ) of SHG For the cae when u, on one hand, one can ee from Fgure 5(a) that, become teady at, on the other hand, from Fgure 5(b) that ( ) u become very large. Clearly, once u larger than, r nl become much larger than, and the abatcty condton (6) would be volated, o the abatc traectory may break down, whch mlar to the cae of lnear abatc OPA (ee Secton.B). ntenty r nl (a) The evoluton of and n SHG (b) Nonlnear abatcty parameter

11 Fgure 6 Adabatc SHG wth 4 For the cae of DFG, n a mlar way, aume that K K. Accordng the Manley-Rowe relaton (4), we can parameterze A (,,) by u A K dn( ue ) A K cn( ue ) A Kn( ue ) (9) where n( u ), cn( u ), dn( u ) are Jacob ellptc functon whoe parameter namely m K/ K and u a functon of. Snce the dervaton of the abatc traectory of DFG completely mlar to that hown above, the author chooe to omt t here. B. Fully Nonlnear Adabatc OPA Followng the approach howed above, let turn to the theory of abatc OPA. Aume K K, and parameterze A (,,) by u: A K n( ue ) A K cn( ue ) A K dn( ue ) () where n( u ), cn( u ), dn( u ) are Jacob ellptc functon harng common parameter whch wrte m K / K, and u a functon of. Note that n the general cae the leat common perod of ellptc functon ntroduced above T K( m) K( m) and K( m) K( m). Subcrbng () nto (), yeld:, where K( m ) defned n Secton.A, u K co( ) K n( )J( u) () where J( u) J ( u),. Snce J( u ) a functon wth a perod of T, the only need to focu on the regon: R( u, ) u T /, (named the reduced ( u, ) plane). Then let calculate the tatonary pont: (, ) u, whch atfyng the

12 ytem of equaton: u. One can ealy get the root equaton of (6): from the frt /, / () Havng obtaned the value of, u can be obtaned from olvng the frt equaton of (6), ay u (u ) correpondng to ( ), repectvely: K u () n( )J( ) For mplcty, et. Clearly, to acheve the full frequency converon from gnal to dle wave, the mxng angle u need to change from ( ) u to u T/ durng the whole TWM proce. Followng the ame approach, et the nonlnear abatcty condton of OPA n term of u to be: u (4) where K. For the cae when m, the abatcty condton of J ( ) u SFG n term of u wrte: u 4K (5) Recall that the ntal ntenty of pump lght much larger that the gnal, the abatcty condton wrte (ee Secton.B): / z k q, for k q (6) Set nonlnear abatcty parameter to be r nl : rnl K (7)

13 Fgure 7 The abatc traectory ( u, ) of OPA ntenty r nl (a) The evoluton of and n OPA (b) Nonlnear abatcty parameter Fgure 8 Adabatc OPA wth K/ K and 4( 5) Fnally, accordng to G. G. Luther et al [], [A, A, A] T can be geometrcally decrbed by the generalzed Bloch vector { UVW,, }, where * U Re(AA A), V m(aa A), W a A (8) * and a (,,) are any number atfyng the relaton: aa a. Therefore, fall on the generalzed Bloch phere: U V W W ( ) (9) W ak ak, W ak ak, W ak ak. where

14 Fgure 9 The generalzed Bloch phere of SFG, wth W,, () () () Fgure The generalzed Bloch phere of OPA, wth W,, () () () Obvouly, n the undepleted pump approxmaton, for the cae of lnear abatc SFG(DFG), the normalzed ampltude of pump wave A A (). Set a, a, a, generalzed Bloch phere become Bloch phere. And for that of lnear abatc OPA, the normalzed ampltude of pump wave A A (). Set a, a, a and :, and generalzed Bloch phere become peudo-bloch phere. Moreover, one can repreent the generalzed Bloch vector n term of u U co( ), Vn( ), W a (4) where,, and are functon of u whch have been defned above for 4

15 dfferent cae. t eay to ee from (7) and () that the abatc traectore howed above are ut the generatrce of the generalzed Bloch phere, or n other word, the geodec lne between the bottom and top of the urface. V. Summary n concluon, have howed that the two cae,.e. lnear abatc SFG(DFG) and OPA, are clearly the lmtng one of fully nonlnear TWM proce and that the geometrc mage of the o-called abatc ba act a the hortet path (geodec lne) connectng ntal tate and target tate.e. the generatrce of the generalzed Bloch phere. The author glad to receve any ueful comment on th artcle from reader. Reference [] J. A. Armtrong, N. Bloembergen, J. Ducung and P. S. Perhan, nteracton between Lght Wave n a Nonlnear Delectrc, Phy. Rev., 7: 98 (96) [] H.Suchowk, D. Oron, A. Are, and Y. Slberberg, Geometrcal Repreentaton of Sum Frequency Generaton and Adabatc Frequency Converon, Phy. Rev. A,78, 68 (8) [] H.Suchowk, V. Prabhudea, D. Oron, A. Are, Y. Slberberg, Robut abatc um frequency converon, Opt Expre., Jul ;7(5):7-4 (9) [4] H.Suchowk, B.D.Bruner, A. Ganany-Padowcz, et al. Adabatc frequency converon of ultrafat pule, Appl. Phy. B,5: () [5] A. A. Rangelov, N. V. Vtanov, Broadband um-frequency generaton ung cacaded procee va chrped qua-phae-matchng, Phy Rev A,85,4584 () [6] G. Porat and A. Are, Effcent, broadband, and robut frequency converon by fully nonlnear abatc three-wave mxng, J. Opt. Sc. Am. B, 4 5 () [7] S. báñez, S. Martínez-Garaot, X Chen, E. Torrontegu, and J. G. Muga, Shortcut to abatcty for non-hermtan ytem, Phy. Rev. A 84, 45 () [8] M.V. Berry, Trantonle quantum drvng, J. Phy. A: Math. Theor (9pp) (9) [9] S. báñez, S. Martínez-Garaot, X Chen, E. Torrontegu, and J. G. Muga, Shortcut to abatcty for non-hermtan ytem, Phy.Rev. A 84, 45 () [] G. G. Luther, M. S. Alber, J. E. Marden, and J. M. Robbn, Geometrc analy of optcal frequency converon and t control n quadratc nonlnear meda, J. Opt. Soc. Am. B 7, 9-94 () 5