Retarded Interaction of Electromagnetic field and Symmetry Violation of Time Reversal in Non-linear Optics

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1 Rardd Inracon of Elcromagnc fld and Symmry Volaon of Tm Rrsal n Non-lnar Opcs M Xaocun (Insu of Torcal Pyscs n uzou, Cna, E-mal:mxc@6.com Absrac Basd on Documn (, by consdrng rardd nracon of radaon flds, rd ordr ranson probabls of smulad radaons and absorpons of lg ar calculad. T rsd formulas of nonar polarzaons ar prodd. T rsuls sow a a gnral procsss of non-lnar opcs ola m rrsal symmry. T pnomna of non-lnar opcs olang m rrsal symmry jus as sum frquncy, doubl frquncy, dffrn frquncs, doubl sabl sas, slf-focusng and slf-dfocusng, co pnomna, as wll as opcal slf-ransparnc and slf absorpons and so on ar analyzd. PACS numbrs: 4., 4.5.-p, f, k Ky Worlds: Elcromagnc Inracon, Quanum Mcancs, Lasr, Non-lnar Opcs, Symmry Volaon of m rrsal, Irrrsbl Procsss. Tm rrsal of rd ordr procss of doubl poons As prod n Documn (, afr rardd ffc of radaon fld s consdrd, procsss of lg s smulad radaons and absorpons ola m rrsal symmry. In s papr, w connu o dscuss problms of m rrsal n procsss of non-lnar opcs. T problm of doubl poons s dscussd a frs. Accordng o formula ( n documn (, by consdrng roaon wa approxmaon, wn w a ranson probably amplud of scond ordr procss of doubl poon smulad absorpon ( a m ( ( ( ( ll mm ( So oal ranson probably of smulad absorpon of doubl poons n frs and scond procsss n un m s ( π 4 W ll mm 4 R * ( ( ll mm δ ( ( Hr R rprsns ral par of funcon. Sarly, w can oban m rrsal of ranson probably of smulad absorpon of doubl poons n scond procss

2 ( a Tm ( ( ll ( ( By rlaons,, ( * *, *, ( and *, m rrsal of oal ranson probably of smulad absorpon of doubl poons n frs and scond procsss s mm ( ( W T π 4 ll mm * [ ( ll mm ] δ ( 4 R Accordng o formulas (8, (9 and ( n Documn (, w a as wll as, * * ( ( ( ll mm ll mm (4 (5, ll mm ll mm, w a ( ( W. W T So r s no symmry olaon of m rrsal n frs and scond ordr procsss of doubl poons. W sould consdr rd ordr procss. In lg of ( n Documn (, ranson probably amplud of rd ordr procss s a m n Ĥ an d ( ( ( ( ( Ĥ a ( d n n (6 By ag ngral of formula, w can oban probably amplud of rd ordr procsss. T rsul s sown n appndx. or doubl poon absorpon procsss w, by roaon wa approxmaon, probably amplud and ranson probably n un m ar nddually ( a m ( ( ( ll mm ( ll ll (7 4 ( W ( π 4( 4A l ll mm 4 R * ( Al ( ( ll mm δ ( (8 Hr Al ll ll. On or and, by ag k j n (4 n Documn (, w can g m rrsal of (6

3 Tm l Tlj j j j ( ( ( ( jl ( a ( Ĥ a ( d Ĥ a ( a Tlj j jl (9 By consdrng rlaons * * H Tlj H jl, H Tlj H jl, ( * *,, ( * * and as wll as by sam mod o do ngral and ak roaon wa approxmaon,, w g m rrsals of probably amplud and ranson probably n rd ordr procss of doubl poon absorpon nddually 4 ( ( a Tm( ( ( ( ll mm mm mm W ( * * * * 4 ( ll mm ( mm mm ( ( ( π 4( 4A T m ll mm * * * [( Am ( ll mm δ ( 4 R ] ( Hr Am mm mm. Comparng w (8, w ow a dffrnc Al A m lads o m rrsal symmry olaon. Bu f g ordr mulpl momn ffcs ar omd, symmry olaons of m rrsals n rd ordr procsss would no xs.. Sum frquncy procss and s m rrsal T procss of sum frquncy procss n non-lnar opcs s a an lcron ranslas no gr nrgy lr m from low nrgy ll l by absorbng wo poons w frquncs and nddually, n ms ou a poon w frquncy and ranslas from gr nrgy ll m no low nrgy ll l agan. Suppos a ncdn lg s paralll on conanng frquncs, and, and srng of lcrcal fld s E, nracon Haonans bwn lcron and radaon fld ar λ λ λ λ Ĥ ( λ λ Ĥ ( λ λ λ λ λ ( Hr λ qe kλ λ μ R p qe kλ R λ p ( λ μ

4 q E k R λ λ λ μ q E kλ R λ q E λ (4 λ μ μ λ and ak sum or ndx Wn w calcula probably amplud, rsul corrsponds o l λ λ n ( n Documn ( or n formula n appndx of s papr. or procss an lcron absorbs wo poons w frquncs and, ranss from low nrgy ll l no g nrgy ll m, ranson probably corrsponds o l n (8 of doubl poon absorpon procss. Wn lcron ranss back from g nrgy ll m no low nrgy ll l by mng a poon w frquncy, afr rardd ffc and g ordr procsss ar consdrd, accordng o (47 n Documn (, ranson probably s ( π Am W δ ( (5 Trfor, for sum frquncy procss a an lcron ranslas no sa m from sa l by absorbng wo poons w frquncs and, n ranslas back no orgnal sa l from sa m by mng ou a poon w frquncs, oal ranson probably s W ( ( π 4( 4A l W ll mm 4 R * ( A ( ( δ ( l ll mm π Am δ ( (6 Manwl, accordng calculaon n (8 n Documn (, afr rardd ffcs and g ordr procsss ar consdrd, ranson probably a an lcron ranss from sa l no sa m by absorbng a poon w frquncy s ( π A l WT δ ( (7 So for m rrsal of sum frquncy procss a an lcron absorbs a poon w frquncy and ranss from sa l no sa m, n ms wo poons w frquncs and, and ranss back no orgnal sa l, oal ranson probably s W ( ( π 4( 4A m T WT ll mm * * * [( A ( ] δ ( 4 R m ll mm 4

5 π Al δ ( (8 Sarly, bcaus of A A, sum frquncy procss olas m rrsal symmry. l m By sam mod, w can pro a or procsss of non-lnar opcs jus as doubl frquncy, dffrnc frquncy, paramrc amplfcaon, Smulad Raman scarng, Smulad Brlloun scarng and so on ar also asymmrc undr m rrsal. T rason s sam a lg s g ordr smulad radaon and absorpon procsss ar asymmrc undr m rrsal afr rardd ffc of radaon flds ar a no accoun. Wa s dscussd abo s basd on quanum mcancs. Bu n mos praccal problms of non-lnar opcs, w calcula problms basd on classcal quaons of lcromagnc flds. So w nd o dscuss rsd non-lnar polarzaons wn rardd ffc of radaon fld s a no accoun. Accordng o currn ory of nonar opcs, polarzaons ar rgardd uncangd undr m rrsal. Ts dos no concdn w ral obsraons. T rason s a w oy consdr dpolar approxmaon wou consdrng rardd ffcs of radaon fld and g ordr procsss of lg s smulad radaon and absorpon. W now dscuss rson of non-lnar polarzaons afr rardd ffc of radaon flds and g ordr prurbaon procsss ar a no accoun, as wll as a gra numbr of pnomna olang symmry of m rrsal n non-lnar opcs.. Non-lnar polarzaons and s symmry olaon of m rrsal In lg of (54 n Documn (54, w l D λ D rprsn rsd dpolar momn afr rardd ffc s consdrd, D lm λlm Dlm rprsn m rrsal of D. W a λ λ lm and D Dlm n gnral. Trfor, as long as w l D D n currn formula ( n of non-lnar polarzaons, w oban rsd formula afr rardd ffc of radaon χ j...k flds and g ordr prurbaon procsss ar a no accoun. Corrspondngly, w l D ( n and oban m rrsal formula χ. I s obous a non-lnar polarzaons can no lm D lm Tj...k kp uncangd undr m rrsal. or xampl, for non-lnar polarzaons of scond ordr procsss, w a Is m rrsal s χ χ ( (,, jk ( (,, Tjk In gnral, w a D D lm ( D ( ( βn D j D n n k n β ( ( N 4ε nn p n β nβ ( D ( ( n β D j D n n k β n ( ( N 4ε nn p n β nβ (9 ( and χ ( χ (. So n gnral suaons w a ( n χ χ ( n. jk Tjk Trfor, polarzaon formula of lcrcal mdum and s m rrsal ar ( ( ( P ε χ E χ E E E E E ( j j χjk j k j...k Tj... k ( 5

6 P T ( ( ( ( χ E χ E E E E E ε T Tj j χtjk j k ( W a P T P n gnral. So moon quaon of classcal lcrcal fld and s m rrsal ar also asymmrcal n gnral w forms E ε E μ P μ ( μ E E P ε μ T Bcaus ( and ( ar basc quaons of nonar opcs, w can say a gnral nonar opcal procsss ola m rrsal symmry. In fac, oy by analyzng nonar opcs pnomna wou complx calculaons, w can sow rrrsbly of nonar opcal procsss drcly undr m rrsal. Toug rrrsbly concp of procsss s no complly sam w a of asymmry of m rrsal, y ar concdn n ssnc. So l s analyz sam praccal xampls o xposur furr rrrsbly of nonar opcs procsss blow. 4. Irrrsbly of nonar opcs procsss As w ow a procsss of lnar opcs jus as lg s propagaons, rflcon, rfracon, polarzaon and so on n unform mdums ar rrsbl. or xampl, lg s focusng causd by a common conx mrror a pon A sown n g.5. Wn a bam of common lg s projcd no a conx mrror, would b focusd a focus O. If w pu a sam conx mrror a pon B, and O s also focus of conx mrror B, lg sn ou from O pon would bcom a bam of paralll lg agan wn rans ou conx mrror B. T procss a lg mong from O B can b rgardd as m rrsal procss of lg mong from A O. I s obous a procss s rrsbl. T scond xampl s a a bam of w sug can b rsold no a spcrum w dffrn colors by a prsm. Wn s lgs w dffrnc colors ar rflcd back no prsm along sam pas, w sug would b formd agan. T rd xampl s a a bam of lg can bcom wo dffrn polarzaon lgs w dffrn propagaon drcons wn lg s projcd no a doubl rfracon crysal. If s wo polarzaon lgs ar rflcd back no crysal along sam pa agan, orgnal lg s formd. All of s procsss ar rrsbl. Bu n procsss of non-lnar opcs, rrsbly dos no xs. Som xampls ar sown blow. 4. Lg s mulpl frquncy, dffrnc frquncy and paramr amplfcaon A bam of lasr w frquncy s projcd no a propr mdum and propr pas macng cnology s adopd. T lg w mulpl frquncs would b found n pnrang lg bsds orgnal lg w frquncy as sown n g.. If lgs w frquncs and ar rflcd back no sam mdum, as sown n g., y can b complly synszd no orgnal lg w a sngl frquncy. Som lg w frquncy would bcom mulpl frquncy lg agan by mulpl frquncy procss. Som lg w frquncy would bcom lg w frquncy by dffrnc frquncy procss. Manwl, som lg w frquncs and would pnra mdum wou bng cangd as sown n g. So orgnal npu lg can b rcord and rrsbly of procss s bro. T suaons ar sam for sum frquncy, dffrnc frquncy and paramr amplfcaon procsss and so on. 6

In polarzaon and magnzaon procsss of frrolcrcs and frromagncs, lcromagnc flds cangng along pos drcons can b rgardd as m rrsal of flds cangng along nga drcons. Tr xs lcrc and pas ysrss.

7 ( 4. Bsably of opcs As sown n g. and g.4, procsss of opcal bsably ar jus as polarzaon and magnzaon procsss of frrolcrcs and frromagnc. In procsss ysrc loops ar formd bwn ncdn and ougong lcrcal fld srngs. In polarzaon and magnzaon procsss of frrolcrcs and frromagncs, lcromagnc flds cangng along pos drcons can b rgardd as m rrsal of flds cangng along nga drcons. Tr xs lcrc and pas ysrss. T ysrc loops ar sar o a ngn cyclng loops. Afr a cyclng, a dsspaon s producd and rrsbly of procss s olad. g.. Procss of lg s mulpl frquncy g.. Tm Rrsal procss of Lg s mulpl frquncy g. Opcal bsably of nrobnzn g 4. Opcal bsably of mxng yp 4. Slf-focusng and slf-dfocusng procsss of lg ( Mdum s rfrac ndx would b cangd nonarly wn a bam of lasr w unn dsrbuon on s cross scon jus as Gauss dsrbuon s projcd no a propr mdum. T rsul s a mdum sms o bcom a conx or conca mrror so a paralll lg would b focusd or dfocusd. Ts s jus slf-focusng and slf-dfocusng of lg. T saonary slf-focusng procss s sown n g.6. Paralll lg s focusd a pon O. Tn bcoms a n bam of lg projcng ou mdum. Comparng w common focusng procss sown n g.5, f slf-focusng procss s rrs bl, lg focusd a pn O would bcom paralll lg agan wn projcng ou mdum as sown n dod lns n g 6. Bu dos no acually. So slf-focusng procsss ar rrrsbl, and so do for slf-dfocusng procss of lg. 7

8 g 5. ocusng procss of lg g 6. Slf-focusng procss of lg 4.4 Doubl and mul-poon absorpon ( 4 In doubl absorpon procss of poons, an lcron n low-nrgy ll would absorb wo poons a frquncs and and ranss o g- nrgy ll. Bu f lcron a g-nrgy ll ranss back o low-nrgy ll, r gs ou oy a poon a frquncy, or wo poons a frquncs,. In gnral would no g ou wo poons w orgnal frquncs and. So doubl poon absorpon procsss ar rrrsbl. And so do for mul-poon absorpon. ( Poon co pnomna Undr a cran condons of mpraur and magnc fld, a bam of lasr can b rsold no wo lgs w a m dffrnc by usng a m rgulaor of opcs. Tn wo lgs ar md no a propr crysal. Tus r lg sgns can b obsrd wn y pass roug crysal. T las sgn s poon co. Ts s a d of nsan corn pnomna of lg. If s r lgs sgns ar mpord no sam mdum agan, y can rurn no orgn wo lgs. Er r sgns ar obsrd (no now co s c ausd or mor sgns ar obsrd (r ar nw sgns ar causd. In fac, bsds poon co, r ar lcron spn co, frromagnc co and plasma co and so on. All of m ar rrrsbl and ola m rrsal symmry. 4.6 Lg s sponanous radaon procsss As w own a r xss wo ds of dffrn procsss for lg s radaons,.., sponanous radaon and smulad radaon, bu r xss oy on d of absorpon procss,.., smulad radaon wou sponanous absorpon n naur. An lcron can oy ransform from g nrgy ll no low nrgy ll by mng a poon sponanously, bu can no ransform from low nrgy ll 8

9 no g nrgy ll by absorbng a poon sponanously. So procsss of lg s absorpons msls ar obously asymmrcal undr m rrsal. 5. Rardd nracon and non-qulbrum corrcon funcon n sascal pyscs In Documn (, auor ad dscussd rason a symmry olaon of m rrsal n g ordr smulad radaon and absorpon of lg. In w of s problm s sgnfcanc, w dscuss s problm furr n s papr. As w own a n currn sascal pyscs, consra forc s usd. Basd on, Loull quaon s symmrcal undr m rrsal. By consdrng fac a oluon procsss of mos maral sysms ar rrrsbl, n ordr o dscrb rrrsbly of non-qulbrum sysms, w a o nroduc som concps pnomnscally rla o rrrsbly and dsspaon jus as non-qulbrum corrlaon funcons and flucuaon corrlaon funcons and so on. W wll pon ou a non-qulbrum corrlaon funcons and flucuaon corrlaon funcons ar rla o rardd nracon blow. or non-qulbrum procsss, nroducon of non-qulbrum corrcon funcon s ssnal. As long as s drmnd, non-qulbrum oluon of sysm would b drmnd n cran conn. In sascal mcancs, non-qulbrum corrlaon funcon s gnrally dfnd as ( 6 G AB ( x,r, τ, < A( x, B( x r, τ > (4 T formula ndcas a alu of pyscal quany A a spac pon x and momn s rla o alu of pyscal quany B a spac pon x r and momn τ and corrla dgr s rprsn by G AB. As w own accordng o spcal rlay, propagaon spd of nracon s lmd. Suppos a propagaon spd of nracon s lg s spd n acuum, l τ r / c <, nflunc of B on A s rardd. If τ r / c >, nflunc of B on A s ladng. If corrlaon funcons ar no local,.., for arbrary pon x, corrlaon funcons ar sam, w can wr (4 as G AB ( r, τ, < A( B( r, τ >< A( B( r, ± / c > r (5 Undr gnral suaon, (4 and (5 ar asymmrcal undr m rrsal. or qulbrum sas, w do sascal arag of (4 or probably dsrbuon funcon of qulbrum sas, corrlaon funcon can bcom ang nong o do w m and coordna x w G AB τ > (6 ( r, < A( x, B( x r, τ > < A( x, B( x r, ± r / c Oy n s cas, oluon of sysm would b symmry undr m rrsal. or quanum sascal procsss of non-qulbrum, by consdrng non-commua caracr of opraors, corrspondng non-qulbrum corrlaon funcon s dfnd as G AB [ ] ( x,r, τ, Tr ρ Â( x, B ( x r, τ B ( x r, τ Â( x, I s sam a nracon among dffrn pars s also rardd or ladng. L s dscuss a xampl,.., Casmr-Poldr nracon o s concr form of corrlaon funcon. By consdrng flucuaon f unform lcromagnc fld, l R r r, paral quany of (7 9

10 dpolar momn a pon r can b wrn as ( 7 d Hr D R ( R, sp R sp R sp ( d α ( E ( r, α ( D ( R, d α ( D ( R, d α ( E ( r, j j j s a funcon rla o rardd Grn s funcon. T frs m on rg sd s sponanous flucua on dpolar momn, scond s dpolar momn nducd by flucuaon lcrc fld ( r, E, rd s dpolar momn nducd by rardd lcrc fld wc propagas from r o propagang o pon r, wc s causd by dpolar momn a s dduc fld a pon r, and so on. Trfor, oal lcrc fld a pon r s E T j j j (8 r, for s dpolar nducd by rardd lcrc fld ( r, R sp R sp ( r, E ( r, D ( R, d D ( R, d α ( E ( r, j j j j E j d by flucuaon lcrc j sp R sp ( R, d α ( D ( R, d R Dj j jk k (9 T frs m on rg sd of formula s flucuaon lcrc fld a r pon. T scond s rardd fld a s causd by flucuaon dpolar momn a pon r n s propagad o pon r. T rd s rardd lcrc fld a s nducd by dpolar momn wc s causd by flucuaon lcrc fld a pon r n s propagad o pon r, and so on. By usng formulas, w oban flucuaon corrlaon funcon T R < d E r, > α D R, < E r, E r, ( ( ( ( ( ( > α j α R R sp sp ( D ( R, D ( R, < d ( d ( > j jk j k α, R ( α ( D ( R, < E ( r, E ( r > j j R R sp sp ( (, D ( R, < d ( d ( > α D ( j R jk k I s obous a corrla on funcon s rla rardd nracon closly. By mans of corrlaon funcon, Casmr-Poldr ponal can b wrn as j V d π ( R < d E ( r, > T ( So Casmr-Poldr ponal s also rla o rardd nracon. W can also us concp of acuum flucuaon o dduc Casmr-Poldr nracon nrgy mor smply. Suppos a acuum flucuaon lcrc fld of Sysm wc locad a pon r s composd of E ( r, and E ( r,. E ( r, s rardd lcrc fld nducd by sponanous dpolar momn d sp α E( r, a pon r and n r sp s propagad o pon and d s causd by acuum flucuaon lcrc fld E ( r, a pon r. W can l approxm aly

11 αe ( ( r, R r, g R c E ( Hr g s a dmnsoss funcon. Trfor, w a I s obous a E ( r, by lcrc fld ( r, E ( E E ( r, ( r, E ( r, s rla o rardd ffc of lcrc fld. T dpolar momn nducd and nracon nrgy ar nddually [ ] ( r, α E ( r, ( r, d (4 E ( R, α ( [ E ( r, E ( r, ] α ( E (, E ( r, V r (5 Tus, dpolar momn nducd by acuum flucuaon lcrc fld and nracon nrgy of acuum flucuaon lcrc fld ar rla o rardd ffcs. By comparng dscusson abo and calculang rsuls of usng quanum mcancs and g ordr rardd ffc of radaon fld n s papr, w can oban som concluson as sown blow. In currn non-lnar and non-qulbrum sascal mcancs, rrrsbly nroducd roug flucuaon and non-qulbrum corrlaon funcons ar rla closly o rardd nracon. In s way, calculaon can b graly smplfd. Howr, s d of mod s pnomnscal on, no sars from orgnal prncpl, oband rsuls would b sascal and approxma ons. Tr xss som randomcy and uncrany. or xampl, oy dpolar momns ar nold n formulas abo, no mulpl momns o xs. All ffc of mulpl momns s subsud by som paramrs wc ar rmand o b drmnd. Bu accordng o mod n s papr, w calcula problm basd on orgnal prncpl of quanum pyscs. No oy dpolar momn, bu also mulpl momns ar nold. W can calcula any g ordr procsss n prncpl. W can calcula any g ordr procss n prncpl and all paramrs n pnomnscal ory can b drmnd. So asymmry of m rrsal dducd n s papr coms from orgnal dynamc prncpl of pyscs w ssnal sgnfcanc. T asymmry olaon nroducd by non-qulbrum corrlaon and flucuaon corrlaon funcon s causd by sascal mod w assump lmn, no comng from orgnal dynamc prncpl of pyscs. Toug rsuls dducd from wo mods ar qualn, and mod of s papr s complx and dffcul comparng w sascal mod. In fac, concps of flucuaon and dsspaon ar oy suabl o qulbrum or nar qulbrum sysms. T dducd rrrsbly s also lmd. or sysms far away from qulbrum sas, s unsuabl for us usng flucuaon ory o xplan rrrsbly. Bu mod of s papr s gnrally suabl. On or and, basd on yposs of qualn probably and nsmbl, sascal pyscs of qulbrum sa as bn sablsd a prsn. Bu as w own a sascal pyscs of non-qulbrum sa as no bn complly sablsd up o nw. T rason s a w do no ow dynamc orgn of rrrsbly n non-qulbrum procsss. All rrrsbly n non-qulbrum ory s nroducd by pnomnscal mod. Basd on rardd nracon of lcromagnsm, w can fnd dynamc orgn of rrrsbly for

12 non-qulbrum sascal pyscs and rac unfd dscrpon for bo qulbrum and non-qulbrum pyscs. In s papr, w dscuss problms basd on non-rla quanum and rardd lcromagnc nracon. T rardd f nracon ndcas a propagaon spd of nracon s lmd. So afr rardd ffc s consdrd, nracon Haonan s rla on n ssnc. T oy problm s a w a no usd quanzd opraors. By src consdraon, w sould dscuss problm basd on quanum fld ory. Bcaus Haonan of lcromagnc nracon s symmrcal undr m rrsal, n lg of common undrsandng a prsn, procsss of lg s smulad and absorpon would b symmry undr m rrsal. Tn, wr or no rsuls oband n papr a procsss of lg s g ordr smulad and absorpon and non-lnar opcs ola m rrsal symmry conradcs w judgmn of quanum fld ory? T auor as xpland s problm n Documn (. T rason s a n nracon bwn boundng aoms and poons, nrgy lls of boundng aoms ar no connuous. By rsrcon of nrgy consraon law, cran flal or paral ranson procsss ar forbddn or can b acd. Ts rsrcons would caus symmry olaon of m rrsal of or paral ranson procsss wc can b acualzd rally. Ts ralzabl flal or paral procsss wc ola m rrsal symmry gnrally ar jus praccally obsrabl pyscal procsss. or lcromagnc nracon bwn non-boundng aoms and radaon flds, r s no s d of symmry olaon of m rrsal. T calculaons sow a olaon of m rrsal aks plac n scond ordr procss. So wn srng of radaon fld s bg noug, olaon of symmry would b gra. Bu f all ralzabl and forbddn procsss ar a no accoun, oal procsss of lcromagnc nracon ar sll symmrcal undr m rrsal. So rsuls n s papr do no conradc w judgmn of quanum fld ory. On or and, n quanum fld ory, w can dscuss m rrsal n bo coordna spac and momnum spac. T nracon Haonan n coordna spac conans all possbl procsss acually, so symmry of m rrsal s also for sum of all procsss. In momnum spac, owr, w calcula concr flal or paral ranson procsss, n sad of oal procsss acually. I can b prod n lg of quanum fld ory a normalzaon procss of rd ordr rx angls n momnum spac for sngl procss also olas m rrsal symmry, oug sum of all flal procsss s sll symmrc n coordna spac. Howr, olaon of m rrsal n 5 normalzaon procss of rd ordr rx angls s ry small oy w magnud of, w can no fnd m n lg of currn prcson of xprmns w magnud of a mos. W nd mor accura xprmns o rfy xsnc of symmry olaon of m rrsal n g ordr normalzaon procsss of quanum fld ory. Ts problm s nglcd a prsn. T auor wll dscuss s problm n dal lar. T auor grafully acowldgs aluabl dscussons w Profssors Qu Ysn n Pyscal and Opcal Tcnology Collg, ujan Normal Unrsy and Zng Sbao n Pyscal Dparmn, uzou Unrsy.

13 Rfrncs. M Xaocun, Elcromagnc Rardd Inracon and Symmry Volaon of Tm Rrsal n Lg s Hg Ordr Smulad Radaon and Absorpon Procsss, Scnc n Cna, Srs G, Volum 5, Numbr, (8. arx/pyscs/87.. S.L.McCall, H.M.Gbbs and T.N.C.Vaasn, J. Op. Soc. Am. 65, 84 (975. D.A.B.Mllr, D.S.Ca, T.C.Dmn, C.Gossard, W.Wgmann, T.H.Wood and C.A.Burrus, Appl. Pys.l.45, 8 (984. T.Kobayas, N.C.Koar and H.Uck, Pys.R. A 9, 77 (984. A.K.Kar, J.G.H. Maw al., Appl. Pys. L. 4, 4 (98.. M.M.T.Loy and Y.R.Sn, Pys.R.L, 5, (97; Appl.Pys.L. 9, 85 (97. G.K.L.Wong and Y.R.Sn, Pys R. L, 57 ( W.Kasr and C.G.B.Garr, Py.R.L.7, 9 ( N.A.Kurn, I.D.Ablla and S.R.Harman, Pys.R.L,, 567 ( R. Balscu, Sascal Mcancs of Equlbrum and Non-qulbrum Sas, ( Scond Volum, Guangx Normal Unrsy Publsng Hous,, ( Zang L, Gmoln, ronr Problms of Quanum Mcancs, Tsngua Unrsy Publsng Hous, 6, (. Appndx T ranson absorpon probably amplud of rd ordr procss for lg s smulad radaon and a ( ( m n ( ( ( ( ( ( ( ( ( (

14 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( k n, ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 4

15 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( T m rrsal of ranson probably amplud of rd ordr procss of lg s smulad radaon and absorpon a ( ( Tm n ( ( ( ( ( ( 5

16 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( k n, ( ( ( ( ( ( ( ( ( ( ( ( ( ( ' ' ' ( ( ( ( ( ( ( ( ( 6

17 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 7

t=0 t>0: + vr - i dvc Continuation

t=0 t>0: + vr - i dvc Continuation hapr Ga Dlay and rcus onnuaon s rcu Equaon >: S S Ths dffrnal quaon, oghr wh h nal condon, fully spcfs bhaor of crcu afr swch closs Our n challng: larn how o sol such quaons TUE/EE 57 nwrk analys 4/5 NdM