Simulating beam-shape effects in non-collinear second harmonic generation

Transcription

1 Simulatig bam-shap ffcts i o-colliar scod harmoic gratio Simulació d fctos d prfil d haz gració d sgudo armóico o colial Bjamí Aloso (1,*), Javir R. Vázquz d Aldaa (1), Luis Roso (1,) 1. Dpartamto d Física Aplicada, Uivrsidad d Salamaca, P. d la Mrcd s/, 378 Salamaca (Spai).. Ctro d Lásrs Pulsados Ultracortos Ultraitsos, CLPU, 378 Salamaca, (Spai). (*) mail: b.aloso@usal.s Rcibido / Rcivd: 31/1/8. Vrsió rvisada / Rvisd vrsio: 17//9. Acptado / Accptd: 5//9 ABSTRACT: W prst a simplifid modl for th simulatio of Scod Harmoic Gratio (SHG) with two fudamtal bams that propagat i diffrt dirctios: o-colliar SHG. I spit of its simplicity (diffractio is ot icludd ad th slowly varyig vlop approximatio i spac is assumd) it ca b usd i may ralistic situatios. It has b implmtd o Mathmatica ad is frly availabl to th radr. Bam-shap ffcts, covrsio fficicis, iput itsity, phas-matchig ad othr paramtrs ca b studid with our cod i a virtual xprimt. W hav applid th modl to SHG of 164 m radiatio i KDP crystal as a xampl. Ky words: Noliar Optics, Scod Harmoic Gratio, No-Colliar, Simulatio. RSUMN: S prsta u modlo scillo para la simulació d la Gració d Sgudo Armóico (SHG) l caso qu los dos hacs fudamtals s propaga distita dircció: SHG o colial. A psar d su simplicidad (o icluy difracció y s asum aproximació d volvt ltamt variabl l spacio), s pud usar muchas situacios ralistas. S ha implmtado Mathmatica y stá dispoibl para l lctor. u xprimto virtual, ustro código prmit studiar fctos d prfil dl haz, ficicia d covrsió, itsidad d trada, ajust d fass y otros parámtros. Como jmplo, hmos aplicado l modlo a la SHG d radiació d 164 m u cristal KDP. Palabras clav: Óptica No Lial, Gració d Sgudo Armóico, No Colial, Simulació. RFRNCS AND LINKS [1] P. Frak, A. Hill, C. D. Ptrs, G. Wirich, Gratio of optical harmoics, Phys. Rv. Ltt. 7, (1961). [] J. Giordmai, Mixig of light bams i crystals, Phys. Rv. Ltt. 8, 19- (196). [3] J. A. Armstrog, N. Blombrg, J. Ducuig, P. S. Prsha, Itractios btw light wavs i a oliar dilctric, Phys. Rv. 17, (196). [4] C. Médz, J. R. Vázquz d Aldaa, G. A. Torchia, L. Roso, Itgratd-gratig-iducd cotrol of scod-harmoic bams i frqucy-doublig crystals, Opt. Ltt. 3, (5). [5] F. Zrik, J.. Midwitr, Applid Noliar Optics, Joh Wily & Sos, Nw York (1973). [6] J. M. Cabrra, F. Agulló, F. J. Lópz, Óptica lctromagética, Vol. II, Addiso Wsly, Madrid (). [7] R. W. Boyd, Noliar Optics, Acadmic Prss, Bosto (199). [8] Y. Sh, Th Pricipls of Noliar Optics, Wily, Nw York (1984). [9] B.. A. Salh, M. C. Tich, Fudamtals of Photoics, pp. 1-17, Joh Wily & Sos, Nw York (1991). [1] G. Idbtouw, T. Zukowski, Noliar optical ffcts i absorbig fluids: som udrgraduat xprimts, ur. J. Phys. 5, (1984). Opt. Pura Apl. 4 (1) (9) Socidad spañola d Óptica

2 . [11] I. Ruddock, Noliar optical scod harmoic gratio, ur. J. Phys. 15, (1994). [1] V. G. Dmitriv, G. G. Gurzadya, N. D. Nikogosya, Hadbook of Noliar Optical Crystals, Sprigr, Hidlbrg (1991). [13] K. Yag, S. Tripathy, J. Kumar, Gralizd xprssios of ffctiv oliar optical cofficit for ocolliar phas matchig i uiaxial ad cubic mdia, Mol. Cryst. Liq. Crys. B: Noliar Opt. 19, (1998). [14] R. Broso,. J. Brdstir, Schaum's Outli of Diffrtial quatios, pp , McGraw-Hill, USA (3). [15] M. Abramowitz, I. A. Stgu, Hadbook of Mathmatical Fuctios with Formulas, Graphs, ad Mathmatical Tabls, p. 896, Dovr, Nw York (1974). [16] P. Pliszka, P. P. Barj, Noliar trasvrs ffcts i scod-harmoic gratio, J. Opt. Soc. Am. B 1, (1993). 1. Itroductio Noliar procsss hav b broadly studid i diffrt aras of physics. I particular, Scod Harmoic Gratio (SHG) is o of th most basic procsss i oliar optics, arisig wh a lasr bam its ough propagats through crtai optical matrials. Sic th first studis [1- ], SHG has attractd a lot of itrst du to th possibility to grat lasr bams oscillatig at w frqucis othrwis uavailabl. Frqucy doublig of lasr bams is owadays a vry wll stablishd tchiqu that is usd for may practical applicatios. Th simplst xprimtal stup ivolvs oly o icidt fudamtal bam that covrts its frqucy to th scod harmoic (colliar SHG). I practic, oly a fw matrials ar covit for SHG bcaus of th propagatio ffcts: th icidt lasr (at th fudamtal frqucy) ad th scod harmoic bam (with frqucy two tims that of th fudamtal) will propagat, i gral, with diffrt phas vlocitis du to th disprsiv proprtis of th oliar mdium. This mismatch of th phas vlocitis is rsposibl for th dstructiv itrfrc of th scod harmoic wavs gratd at diffrt poits of th propagatio dirctio that fially kills th gratd scod harmoic at a giv propagatio lgth. As it has b widly studid, th coditio o th sam phas acquird durig th propagatio (kow as phas matchig) is cssary to hav fficit frqucy covrsio. Nowadays th cocpt of phas matchig is xprimtally sttld by usig diffrt polarizatios i aisotropic matrials (th ordiary ad th xtraordiary, playig with th crystal axs oritatio rspct to th propagatio to match thir idxs), quasi-phas-matchig [3] or th gomtris whr svral (o-colliar) fudamtal bams ar ivolvd [4]. I th o-colliar SHG th agl btw bams ca b usd to achiv th phas matchig, as will b s blow, which ops a altrativ to th us of aisotropic crystals. Th us of ocolliar gomtris is thus cssary i may practical situatios. For istac, it allows th itrisic spatial sparatio of th fudamtal ad th scod harmoic bcaus thy propagat i diffrt dirctios (th SHG ca b asily slctd with a diaphragm). I mor complx xprimts, it ca play th rol of a spctromtr sic it slct a wavlgth for ach lgth wh workig with thick crystals. Morovr, if studyig othr scod ordr oliar procsss (such as sum or diffrc frqucy gratio), it is commo havig madatory o-colliar cofiguratios. I may situatios of itrst, th thortical study of th procss rquirs umrical simulatio. May sophisticatd modls of SHG from cotiuous bams to ultra-short pulss hav b dvlopd i this cotxt ladig to succssful rsults wh comparig to xprimts. Both colliar ad o-colliar cass hav b solvd icludig all kid of corrctios, for istac diffractio, disprsio of th rfractiv idx or group-vlocity disprsio. Th cocpt of phas matchig has b dply studid ad udrstood. Most of ths sophisticatd modls ar vry computatioally dmadig ad hardly accssibl bcaus of its complxity. O th othr sid, SHG with moochromatic ad colliar pla wavs ca b dscribd i a straightforward way udr crtai approximatios, as it is do i most rfrc books [5-7]. Mor advacd books dscrib mor complx procsss ad situatios [8]. I this ss, Opt. Pura Apl. 4 (1) (9) Socidad spañola d Óptica

3 aalytical xprssios ca b obtaid assumig prfct phas matchig or i th low covrsio limit (udpltd pump). Howvr, to our kowldg, thr is a wid gap btw SHG with pla moochromatic wavs ad mor ralistic cass. Th aim of this papr is to prst a asy way to aalyz SHG for moochromatic wavs cosidrig trasvrs ffcts ad o-colliar gomtris (th spatial profil of th two fudamtal iput bams is arbitrary). As i most of th umrical modls, th slowly varyig vlop approximatio (SVA) i spac is assumd. Although it is fudamtal, th ffct of diffractio is ot icludd i th propagatio bcaus of th charactristics of th modl. Howvr, it ca b thought to b ralistic i may practical situatios whr larg ufocusd (or loosly focusd) bams ar ivolvd, bams that ar owadays availabl with itsity larg ough to produc frqucy covrsio. Thus, w will assum that th bam propagats without divrgc alog th crystal lgth (typically with thicksss of about a fw millimtrs). Th modl is frly availabl (attachd to th publicatio) ad ca b asily modifid by itroducig diffrt trasvrs mods (Lagurr-Gaussia ad Hrmit- Gaussia bams [9]), pak itsitis, lgth ad oliar suscptibility of th crystal ad othr paramtrs. Th algorithms could b also xtdd to pulsd bams, but takig ito accout th tim dpdc is far from th scop of this work. This mthod is much simplr tha othr modls usd i ivstigatio icludig all rlvat ffcts both i tim ad i spac, but o th cotrary, it is simpl, traspart, it ca b fast udrstood ad usd i may situatios, yildig maigful rsults. Morovr, it givs us th chac of chckig diffrt iputs wh tryig to gt a crtai output. Th problm quatios ar proposd icludig th first oliar trm of th polarizatio (quadratic ffcts). Aftr maipulatios w foud thr coupld quatios i first ordr partial drivativs. Thrfor, w adaptd stadard umrical algorithms to itgrat such quatios whr th fild is propagatd i subsqut plas. Du to th rlvac of oliar optics i may applicatios, th iclusio of simpl xprimts i udrgraduat laboratoris is bcomig mor ad mor importat [1,11]. Th modl ad th softwar prstd i our work ar proposd to b a virtual xprimt o oliar optics vry suitabl for udrgraduat lvl. Th studts ar xpctd to lar th mtiod fudamtal cocpts of oliar optics ad to familiariz with simulatio ad umrical algorithms. A udrgraduat lvl of optics is rquird i th studts bfor usig this matrial.. Phas matchig i uiaxial crystals Th most commo procdur for achivig phas matchig i SHG is to mak us of th birfrigc (dpdc of th rfractiv idx o th dirctio of polarizatio) of aisotropic crystals. I particular, uiaxial crystals lik KDP (KH PO 4 ), ar widly usd i this cotxt. I this sctio, w brifly discuss th gral physics of uiaxial crystals applid to achivig phas matchig for SHG. Th particular cas of a KDP crystal usd to grat th scod harmoic of a λ =1.64 µm lasr bam (λ =.53 µm) will b giv as xampl. I o-colliar gomtris, th two fudamtal wavs will b assumd to propagat isid th crystal with agls ±α rspct to th z-axis (without loss of grality), i th pla x-z, thus rsultig i scod harmoic bam propagatig alog th z-axis as it ca b s i Fig. 1 (ot that outsid th crystal both fudamtal bams will propagat with a largr agl rspct to th z-axis i agrmt with Sll s law). Fig. 1. Gomtry ivolvd i th o-colliar SHG. Th diagram of Fig. 1 shows, i fact, th phas matchig coditio for ay paramtric procss lik SHG: th sum of th wav vctors of th icomig (fudamtal) wavs must b qual to th wav vctor of th gratd wav (scod harmoic), that is k =k + +k. From this vctorial coditio w gt k =cosα (k + +k ) ad fially th followig coditio for th rfractiv idics: =(k + +k )cosα. It ca b xprimtally achivd by choosig a suitabl agl α i o- Opt. Pura Apl. 4 (1) (9) Socidad spañola d Óptica

4 colliar SHG or by matchig th fudamtal ad scod harmoic idxs i colliar SHG thaks to th matrial birfrigc. Both cass will b discussd blow. Uiaxial crystals ar charactrizd by havig a particular axis with diffrt rfractiv idx, kow as th optic axis of th crystal. This causs that a pla wav will propagat with diffrt rfractiv idx dpdig o th dirctio of propagatio of th phas (giv by its wav vctor k r ). I ths crystals, light ca propagat with two diffrt bhaviours: ordiary ad xtraordiary. Th ordiary wav, polarizd i th pla orthogoal to th optic axis (isotropy pla), travls with a rfractiv idx o, whras th ffctiv rfractiv idx for th xtraordiary wav (with polarizatio prpdicular to th ordiary wav) dpds o th propagatio dirctio (s, for istac, [6,7]) z ( θ) =. (1) si ( θ) + cos ( θ) whr θ is th agl btw th wav vctor ad th optic axis, ad th idxs z ad o ar th pricipal valus of th rfractiv idxs of th crystal. Notic that w hav calld z=(θ=9º), th rfractiv idx for th xtraordiary wav wh its phas propagats prpdicular to th optic axis of th crystal. O th othr had, wh th agl is θ=º (paralll to th optic axis), th xtraordiary idx is o ad th wav propagats lik th ordiary o. As a rsult, th ordiary ad th xtraordiary wavs hav diffrt polarizatio. I th particular cas of KDP, th pricipal valus of th rfractiv idics for λ =1.64 µm ad λ =.53 µm ca b calculatd from th Sllmir quatios [1]: o, o, = 1.494, = 1.519, z, z, z = 1.463, = () Dpdig o th rol playd by th fudamtal or th scod harmoic, two typs of phas matchig ar dfid. Wh both fudamtal wavs ar ordiary or xtraordiary, th phas matchig procss is amd typ-i. Wh th fudamtal wavs hav th opposit polarizatio (o is ordiary ad th othr xtraordiary), th phas matchig procss is amd typ-ii. W will focus our work o typ-i phas matchig with both fudamtal wavs havig ordiary polarizatio (thus, th gratd scod harmoic will b xtraordiary). Th optic axis of th crystal will b cotaid i th pla dfid by th dirctio of propagatios of th wavs. Du to birfrigc, th wav vctor ad th rgy (Poytig vctor) of th xtraordiary wav (th scod harmoic) will propagat isid th crystal alog slightly diffrt dirctios [6,7,1]. Th agl btw phas ad rgy propagatio is amd walk-off agl ad ca b calculatd from with θ dfid as ta θ θwalk off = θ θ, (3) = ta θ. o,z, Th oliar ffct is producd by a o-zro scod ordr suscptibility χ () i th matrial givig iducd oliar polarizatio. Th oliar propagatio icrass with this suscptibility, which is rsposibl for th SHG. Typically, th ffctiv oliar cofficit dff dfid as dff=χ () is usd istad. Th ffctiv oliar cofficit dff of th frqucy covrsio dpds o th xprimt gomtry ad it is rlatd to th kow paramtrs of th crystal. Th xprssio i a crystal class 4m (poit group) is th followig [13] ( θ ) ( φ + φ ) d ff = d 36 si si +. (4) whr θ is th agl btw th optic axis of th crystal ad Poytig vctor of th scod harmoic wav, ad φ + is th azimuthal agl of th wav + rspct to th crystal axs startig at positiv x- axis. W assum that th crystal has b cut i th bst situatio to produc oliar scod ordr ffcts, that is, wh th azimuthal agls with th crystal axs ar φ + =φ =π/4. Th oliar cofficit is tak from th litratur [1] ad is d36(1.64 µm)=.39 pm/v. Sic w ar workig i th CGS systm of uits, w hav to covrt it to th appropriat uits usig th ratio 1 statvolt=99.8 V, thus th cofficit usd is d36= cm/statvolt. First of all, i th o-colliar SHG, w us th agl btw icidt bams dirctios to achiv th phas matchig coditio (or at last small phas mismatch xprimtally) so w ca obtai th phas matchig agl αpm giv by cosαpm=, (θ)/ o,. Morovr, w cosidr that th optic axis agl of th crystal is θ=9º bcaus th ffctiv oliar cofficit will b th maximum ad th scod harmoic (xtraordiary Opt. Pura Apl. 4 (1) (9) Socidad spañola d Óptica

5 wav) will ot hav walk-off (s Fig. (a)). Th, th dsird paramtrs ar: d α, ff PM ( θ) = z, = = 1.13º. = 1.479; 8 θ = 9º ; cm/statvol t; (5) Fig.. Optic axis (O.A.) oritatio ad bam propagatio dirctio i o-colliar (a) ad colliar (b) gomtry. Th walk-off agl givs th dirctio of Poytig vctor for colliar scod harmoic (du to th particular choic of th O.A. oritatio i th ocolliar stup th walk-off is zro i this cas.) Th idics show i th schm rprst th rfractio idics xpricd by ach wav. Cocrig th colliar situatio, th variabl usd to rach colliar phas matchig is th agl of th optic axis rspct to th propagatio (s Fig. (b)). Sic w ar solvig typ I gratio whr th xtraordiary wav is scod harmoic, th agl ca b calculatd from th phas matchig PM θ =, ad is giv by [7] coditio ( coll ) o, PM ( ) o, o, si coll = z, z, θ θ PM coll = º. (6) Assumig agai that th crystal has b dsigd i th most favourabl situatio, w hav φ + =φ =φ =π/4 ad th agl of th optic axis is θ PM coll, so th paramtrs tak th valus θ d, ff ( θ) = o, = 4.86º ; = = 1.494; 9 cm/statvolt. (7) I this cas, th scod harmoic has a agular walk-off θwalk-off=1.64º, calculatd from q. (3). W hav kpt a prcisio of.1º for th agls i ths calculatios just for covic. Th prcisio rquird i lab xprimts for th agls is difficult to assss providd that it is strogly dpdt o svral paramtrs as th moochromaticity of th fudamtal wav, th divrgc of th bam or th crystal lgth (i stadard goiomtrs th prcisio is usually much smallr, typically º1' =.17º ). If th phas matchig coditio is ot xactly satisfid, th powr of th gratd scod harmoic sigal will dramatically dcras. This ffct is accoutd r r for through r th r phas mismatch, dfid as k = k + + k k, that projctd alog th propagatio dirctio of th scod harmoic givs k=k + cosα+k cosα k. It ca b show [6,7] that, i th low covrsio limit (udpltd pump) i colliar gomtry ad cosidrig pla moochromatic wavs, th SHG itsity dpds with th phas mismatch as k L I I ( ) L sic, (8) π whr L is th crystal lgth (fixd), I () is th fudamtal itsity iput ad th sic-fuctio is dfid as sic(x)=si(πx)/(πx). This fuctio has a maximum wh k= ad oscillats btw zro ad scodary maxima (s Fig. 3). Th first zro aroud th mai maximum appars at k=±π/l. For a giv SHG gomtry with prfct phas matchig, if th wavlgth of th fudamtal bam or th phas matchig agl slightly chags, th k chags accordigly ad th covrsio fficicy dcrass. Sic th first zro positio is proportioal to L 1, th spctral ad agular admittacs (rag of wavlgths ad phas matchig agls for which th SHG fficicy drops to.5 of th maximum valu) ar largr for shortr crystal lgths. O th othr had, th fficicy is also proportioal to L, so th optimum crystal lgth for a giv xprimt must b foud as a commitmt btw fficicy ad admittacs. Fially, otic that th SHG itsity dpds o I ( ), so it icrass oliarly (as xpctd) with th fudamtal itsity. A aalogous xprssio to q. (8) ca b foud for o-colliar SHG ad, thrfor, th abov cosidratios ca b xtdd to this cas. 3. Modl 3.a. Filds ad st out Th proposd problm has b st out cosidrig icidt bams with arbitrary trasvrs mods ad a dfid propagatio dirctio i th crystal, so w hav to solv quatios with th thr spatial variabls. Opt. Pura Apl. 4 (1) (9) Socidad spañola d Óptica

6 whr c is th spd of light. Th dilctric prmittivity =1+χ (1) icluds th first ordr suscptibility χ (1) arisig from th liar polarizatio trm. I our scop, w hav a quatio for ach wav with its corrspodig oliar polarizatio trm PNL. I th ocolliar cas, th oliar polarizatios ar giv by [7]: P P ± () * = χ m = χ () + it,. it (1) Fig. 3. Scod harmoic itsity as a fuctio of th phas mismatch: sic-fuctio. I ordr to simplify th fial xprssios of th quatios, w dfi ± (fudamtal bams corrspodig to agls ±α, s Fig. 1) ad (th scod harmoic) as th complx vlops of th filds (that is, th lctric filds xcpt for th tmporal ad th rapidly oscillatig spatio-tmporal phas) that oly dpd o th positio r={x,y,z}: ± i k ( ± x si α + z cos α ) ( r, t) = ± ( r) i ( r, ) ( r) [ k ] z t t =. [ t ] ; (9) I th particular cas of colliar SHG th agl is α=, codsig th dfiitios i ( r, t) = ( r) xp{ i[ kz t] }; ( r, t) = ( r) xp i[ k z ] { }. t 3.b. quatios obtaiig (1) I this study, w do ot rgard th vctorial charactr of th lctric fild, assumig that ach fild has th corrct polarizatio accordig to th phas-matchig coditios i th crystal (cssary i aisotropic matrials). W will cosidr oly typ-i phas matchig that mas that th two fudamtal bams hav th sam polarizatio, but ca b dirctly xtdd to othr cass. Th systm of uits usd is th CGS. W cosidr Maxwll s quatios icludig th first oliar trm associatd to th scod ordr suscptibility χ (), which is th rsposibl for th SHG. Th rsultig wav quatio is 4π PNL =, (11) c t c t Th scod ordr suscptibility taks diffrt valus dpdig o th mix-frqucy procsss that it ivolvs. From this momt, w dfi th scod ordr suscptibility for th SHG procss as χ () χ () (, ). Obviously, it has b cosidrd as a umbr ot as a tsor (i aisotropic matrials it is a tsor, but a ffctiv scalar valu is always usd). I th colliar itractio w cosidr that th filds + ad dgrat i oly o calld. Sic it is physically cssary that th itsity of this wav is th sum of th fudamtal o-colliar itsitis wh α= (big ach of that half th total itsity), th amplitud of th colliar fudamtal wav will b th sam xcpt for a factor qual to th root squar of two. This is th raso why th cofficit two disappars i th oliar polarizatio oscillatig at doubl frqucy (it is absorbd i th fild amplitud): P P = χ = χ () () * it, it ( ). (13) Th, w itroduc ths cosidratios i th wav quatios for th ad filds, ad w assum th followig mai approximatios. Th first assumptio is th slowly varyig vlop approximatio (SVA) i spac. It cosists of glctig th scod ordr spatial drivativ of th vlops alog th propagatio dirctio. It ca b assumd i most of th situatios i which homogous mdia ar ivolvd ad o itrfac is prst i th propagatio. Th scod o supposs that th diffractio is gligibl i th propagatio lgth. It is rasoabl for short propagatio lgths ad soft bam-shaps. This approach also implis that bams propagat without big strogly focusd. Th rsultig quatios that w hav solvd both i th o-colliar cas ad th colliar o ar: ± ta α + ± = x z (14a) () 4iχ π * = xp{ iz k}. m c k cos α Opt. Pura Apl. 4 (1) (9) Socidad spañola d Óptica

7 z () 16iχ π = c k + iz k +. (14b) whr w hav dfid th phas mismatch k = k cos( α) k. Th full drivatio of th propagatio quatios ca b foud at th hadr of th Mathmatica script. Notic th high symmtry of th quatios ivolvig th vlops. Th thr coupld quatios ca b particularizd to th colliar gratio: z z () 4iχ π = c k () 8iχ π = c k iz k + iz k *, ( ). (15) It is fudamtal to raliz that th sourcs of th oliar procsss (th right-had sid of th quatios) ar modulatd by a phas icludig th phas mismatch k. Wh this quatity is zro, both fudamtal ad gratd wavs propagat with th sam phas vlocity projctio alog th z- axis givig costructiv itrfrc. This is xactly th mtiod coditio o th propagatio vlocity of th wavs (phas matchig coditio), which implis th rlatio ()cos(α)=(). Th oliar polarizatio trms ar th sourc of th filds ad bcom from th thr sstial procsss that ca produc th wavs mixig i all possibl dirctios, bcaus th oliar procss is rvrsibl ad dpdig o th phas matchig coditios it ca b rgy rturig to th fudamtal wavs from that covrtd to scod harmoic (dow-covrsio). W hav dpdc o th thr spatial variabls i th vlops, but it is possibl to rcovr th pla wav cas assumig that th vlops oly dpd o th z-coordiat. This problm has b solvd i classical txtbooks as Zrik t al [5] ad may mor sic th. As it has b said i that simplifid situatio, aalytical xprssios ca b foud udr crtai coditios. 4. Solvig mthod 4.a. Basic cosidratios Th fial quatios form a systm of coupld quatios i partial drivativs of first ordr. Th philosophy of th algorithm is always to calculat th fild distributio i a complt pla z=z+ z usig th valus of th lctric-filds i th prvious pla z=z. Th crystal is sampld i a thr-dimsioal Cartsia grid. At th bgiig, w usd th ulr s mthod [14] of first ordr f(x+ x) f(x)+ x[ f(x)/ x], which ca b asily applid to th fild propagatig alog th z-axis. Howvr, th drivativs of th icidt bams (th fudamtal bams) ar with rspct to a combiatio of th variabls x ad z. Th way to propagat ths filds is by mas of th itgratio of th z-drivativ (i th z-dirctio) ad umrical approximatio of th x-drivativ as f(x)/ x [f(x+ x) f(x)]/ x. Th umrical itgratio of th quatios is giv by: ± () 4iχ π * x z xp{ i k z }[ ] ( ). m (16a) r c k cos α ± (, y, z + z) [ ± ] ( ) m z ta α r x ( r ) (, y, z + z) [ ] ( ) z xp{ + i k z }[ ] ( ). () 16iχ π x r + r c k (16b) Th ffctiv oliar cofficit dff is commoly usd istad of th scod ordr oliar suscptibility dff χ (). Th filds hav b mad adimsioal dividig by th vlop pak of o icidt bam. Morovr, th procdur was improvd i th o-colliar cas by substitutig th first ordr mthod for th scod ordr Modifid ulr s Mthod [14] (also kow as trapzoidal formula [15]). To v mor icras th accuracy without dig dsr samplig (ad logr calculatio tims), w us th 4th ordr Rug-Kutta mthod i th colliar cas [14]. Th bam width paramtr (i th calculatio) is tak to b high ough to compltly iclud th trasvrs distributio of rgy for both fudamtal bams. 4.b. Iitial coditios W choos z as th dirctio i which th scod harmoic bam propagats, big x th dirctio Opt. Pura Apl. 4 (1) (9) Socidad spañola d Óptica

8 cotaiig th othr compot of th o-colliar bams ad y th trasvrs dirctio. Bfor solvig th diffrtial quatios that govr th procss, w hav to dfi th iitial coditios, that is, th valus of th lctric filds at th bgiig of th crystal (z=zmi) for th whol trasvrs pla (iitializ th filds). W cosidr that i th pla z=zmi oly xits fudamtal fild. Although this is ot a cssary coditio that ca b chagd by th usr, ot havig scod harmoic iput is th most commo situatio. Furthrmor, i o-colliar SHG w d to chag th coordiats from th icidt bams itrisic systm to th coordiats dscribd abov ad th impos th iitial coditios at z=zmi. W us Lagurr-Gaussia or Hrmit-Gaussia bams [9] as iput. 4.c. Cotrol of th accuracy I spit of implmtig scod ad fourth ordr umrical mthods, w must b vry carful with th choic of th samplig i ordr to sur th accuracy of th solutio. Th stp siz i z-dirctio must b small ough to avoid divrgc i th ulr s mthod. This is roughly giv by th coditio: l z <<, (17) () χ max whr w hav usd th dfiitio l=c k cosα/4π ad max is th maximum valu of th lctric fild rachd at a crtai poit of th spatial samplig. Thrfor, w kow that th product btw th adimsioal magitud χ () max ad th stp z/l will play th rol of th cssary coditio for th itgratio. Additioally, th accuracy achivd by th calculatio ca b chckd by tstig th lctric rgy of th itractig wavs. Th total rgy is proportioal to = + ik si α x + * + R{ }, + (18) ad must rmai costat from a z-pla to th followig o. Du to th itrfrc trm btw th two fudamtal bams (ot that thy hav th sam polarizatio bcaus typ-i phas matchig is cosidrd) th total rgy oscillats i th x-dirctio but its itgral givs a avrag that is a costat. Th priod of such oscillatios is π/(k siα), so w ca chck if th rsolutio of th samplig is high ough to dscrib it. Th total rgy cosrvatio is vrifid durig th calculatio ad, if th stp is ot small ough, th program sds us a warig mssag to abort valuatio bcaus calculatios will b wrog. Th cosrvatio rror limit is st by dfault to 1% from iitial rgy, but it ca b maually chagd. 5. Applicatio to Scod Harmoic Gratio i KDP 5.a. KDP crystal ad bam paramtrs I ordr to show a applicatio of th modl, w us as icidt bams th output of th Nd:YAG lasr, with a fudamtal wavlgth λ =164 m i th ifrard ad th gr scod harmoic λ =53 m. This covrsio is vry oft usd. Nd:YAG lasrs produc pulss of typically fw aoscods. Th study of SHG with moochromatic wavs is thus justifid du th vry log puls duratio. Th calculatio of th optimum paramtrs of th o-colliar ad colliar SHG is xplaid i dtail i Sctio. Th simulatio cosidrs typ I gratio. I o-colliar covrsio, w assum that th agl of th optic axis rspct to th scod harmoic propagatio is θ=9º. Such a agl mas that th optic axis is prpdicular to th pla cotaiig th thr itractig bams (pla x-z). Th rfractiv idxs ar o, =1.494 ad, (θ)= z, =1.479, ad th o-colliar agl for phas matchig is αpm=1.13º. Th ffctiv oliar cofficit rsults dff= cm/statvolt. O th othr had, i colliar gratio th phas matchig coditio imposs θ=41.3º ad, (θ)= o, = Th ffctiv oliar cofficit is dff= cm/statvolt. W obtai th maximum valu of th complx vlop modul for th icidt bams from th pak itsity of th bams i th air. Th, w must covrt it to CGS uits = ± µ max statvolt ( ) 1 / 4 cm I = max W cm, (19) Opt. Pura Apl. 4 (1) (9) Socidad spañola d Óptica

9 whr w hav cosidrd th diffrc btw th complx amplitud (that w us i th modl) ad th ral fild. 5.b. No-colliar SHG with Gaussia bams Th fudamtal wavs ar th Gaussia bams (mod of th Lagurr-Gaussia bam [9]). I th o-colliar itractio, th poit whr fudamtal bams crosss is tak as th origi of coordiats, ad th problm is solvd i a symmtric rgio cotaiig th suprpositio zo whr th oliar procss taks plac. Thus, th computatio volum is dtrmid from th icidt bams width ad th o-colliar agl α. Th crystal dimsios i ctimtrs ar (.8,.14,.8), rspctivly to (x,y,z)-dirctios, corrspodig to a bam width.141 cm, ad th itgratio stp usd is z=.1 cm (surig th accuracy). Th ctrs ar sparatd a distac.143 cm (th product of th crystal lgth by taα) at th trac of th crystal. Th pak itsity valu is Imax=6 1 7 W/cm, which implis ± max =79. statvolt/cm accordig to q. (19). W ittioally us a o-colliar agl slightly diffrt from αpm, so thr is a small phas mismatch k= cm 1. Th program implmtd o Mathmatica is attachd to th oli publicatio as Mdia 1. I Fig. 4 th total rgy of th thr itractig wavs is show alog th crystal (it is itgratd i th y-dirctio). Th ctr of th two fudamtal bams (propagatig with agls ±α) crosss at z= ad itrfrc frigs ca b s whr fudamtal bams ovrlap, as prdictd i q. (18). Th itractio btw both bams grats a scod harmoic bam that propagats alog th z-axis. As th two fudamtal bams sparat thy carry much lss rgy du to th covrsio to SHG. Th outputs of th program ar th lctric-fild distributios ovr th whol samplig. Itsitis, rgis ad phas-maps ar calculatd from ths filds. W show as xampl th trasvrs itsity ad phas distributio of th scod harmoic for th paramtrs giv abov bam at diffrt sctios of th crystal i Fig. 5 (th tir vido is availabl oli as supplmtary matrial: Mdia ). Th scod harmoic itsity is widr i th y- dirctio. This asymmtric shap rflcts th importac of th suprpositio zo: sic th icidt bams propagat with opposit x- compots th mai spatial suprpositio btw both bams is achivd aroud x=. Morovr, w ca lar that th spatial phas varis fastr i th y- dirctio tha th x-dirctio, ad its diffrc icrass as th bam propagats. Fig. 4. Dsity of total lctromagtic rgy itgratd i y-dirctio. Fig. 5. Itsity (up) ad phas (dow) of th scod harmoic at diffrt sctios of th crystal. Th x- dirctio is th vrtical axis ad th y-dirctio is th horizotal axis. Fig. 6. Trasvrs distributio of th ormalizd total rgy at th bgiig of th crystal (lft), at th ctr (middl) ad at th d (right). Th maximum at th ctrs of th fudamtal bams is causd by dowcovrsio (du to phas mismatch). Th x-dirctio is th vrtical axis ad th y-dirctio is th horizotal axis. Opt. Pura Apl. 4 (1) (9) Socidad spañola d Óptica

10 This is mor clarly rprstd i Fig. 6, whr trasvrs distributios (sctios at diffrt z=costat plas) of th dsity of total lctromagtic rgy ar plottd (whol vido attachd oli: Mdia 3). Th sctio at th bgiig of th crystal (lft) shows th gaussia icidt bams bcaus iitially thr is ot scod harmoic wav. Th, both fudamtal wavs propagat ad cross at th ctr of th crystal (z=), givig itrfrc frigs ar this poit. I z= thr is also th gratd scod harmoic (o oscillatig cotributio). Th rgy output is th rmaidr of th fudamtal bams at th top ad at th bottom, ad th scod harmoic lavig th crystal at th ctr (s also Fig. 4). i Fig. 8. Th maxima ad miima rflct th diffrt rats of th covrsio (rfrrd to th propagatio dirctio z) causd by th iitial trasvrs distributio of rgy. 5.c. Colliar SHG for Hrmit-Gaussia mod 1 Th icidt wav is th mod 1 of th Hrmit- Gaussia bams [9], which has a ull miimum i th li y=. I this cas, th computatio rgio is agai symmtric rspct to th origi of coordiats. Th crystal lgth is.8 cm, ad w hav chos a bam width.8 cm, ad th itgratio stp usd is z=. cm, which surs th accuracy. Th vlops ar ormalizd rspct to th valu max =895 statvolt/cm that would giv pak itsity Imax=1 9 W/cm if it wr th Hrmit-Gaussia mod (gaussia bam). W iduc a cosidrabl phas mismatch k=.176 cm 1 arisig from a agl of th optic axis of th crystal diffrt from θpm. Th Mathmatica program corrspods to Mdia 4. Trasvrs sctios of scod harmoic itsity at diffrt positios of th crystal ar rprstd i Fig. 7 (vido availabl oli: Mdia 5). Th phas mismatch causs th chag i th dirctio of th rgy covrsio, altratig from fudamtal to scod harmoic ad vic vrsa. Sic scod harmoic gratio is a oliar procss, it occurs fastr whr th iitial lctric fild is highr, ladig to a arlir chag from upcovrsio to dow-covrsio at th two maxima of th icidt bam, which producs th rig structur at ach lob of th bam. Ths phoma hav alrady b simulatd with mor complicatd modls [16] icludig diffractio ad obtaiig similar rsults. Th ffct of th bam shap o phas mismatch is clarly show if w rprst th propagatio alog th oliar matrial (sctio x=) as s Fig. 7. Trasvrs sctios of scod harmoic itsity. Th phas mismatch causs th chag i th dirctio of th rgy covrsio altratig from fudamtal to scod harmoic wavs. Th x-dirctio is th vrtical axis ad th y-dirctio is th horizotal axis. Fig. 8. Scod harmoic itsity at a sctio x=. 6. Coclusios W hav prstd a algorithm for o-colliar scod harmoic gratio with moochromatic wavs. Th mai assumptios of th modl (slowly varyig vlop approximatio i spac ad diffractio gligibl) allow a rasoably simpl tratmt of th problm but it rmais usful i may practical applicatios. ffcts of th trasvrs bam shap, itsity, phas matchig, Opt. Pura Apl. 4 (1) (9) Socidad spañola d Óptica

11 crystal lgth ad oliar proprtis, ca b asily udrstood with th cod. W propos th us of such cod as virtual xprimt o oliar optics that could b vry suitabl for udrgraduat lvl. Th studts ar xpctd to lar fudamtal cocpts of oliar optics ad to familiariz with simulatio ad umrical algorithms. Ackowldgmts This work was fudd by th Spaish Miistry of Scic ad ducatio (Projcts No. FIS ad No. CSDC ). Bjamí Aloso is gratful to th Spaish Miistry of Scic MICINN for th fiacial support. Th program is ot vry computatioally dmadig ad ca b ru o typical dsktop computrs without spcific hardwar rquirmts. This is o of th goals of our simplifid modl. Fast calculatios ca b do with duratios i th ordr of a coupl of miuts. Opt. Pura Apl. 4 (1) (9) Socidad spañola d Óptica