Apodization of chirped quasi-phasematching devices

Transcription

1 Phllps et al. Vol. 3, No. 6 / June 23 / J. Opt. Soc. Am. B 55 Apodzaton of chrped quas-phasematchng devces C. R. Phllps,,2, * C. Langrock, D. Chang, Y. W. Ln, L. Gallmann, 2 and M. M. Fejer Edward L. Gnzton Laboratory, Stanford Unversty, Stanford, Calforna 9435, USA 2 Department of Physcs, Insttute of Quantum Electroncs, ETH Zurch, Zurch 893, Swtzerland *Correspondng author: cphllps@phys.ethz.ch Receved February 25, 23; accepted March 28, 23; posted Aprl 7, 23 (Doc. ID 85964); publshed May 5, 23 Chrped quas-phasematchng (QPM) optcal devces offer the potental for ultrawde bandwdths, hgh converson effcences, and hgh amplfcaton factors across the transparency range of QPM meda. In order to properly take advantage of these devces, apodzaton schemes are requred. We study apodzaton n detal for many regmes of nterest, ncludng low-gan dfference frequency generaton (DFG), hgh-gan optcal parametrc amplfcaton (OPA), and hgh-effcency adabatc frequency converson (AFC). Our analyss s also applcable to secondharmonc generaton, sum frequency generaton, and optcal rectfcaton. In each case, a systematc and optmzed approach to gratng constructon s provded, and dfferent apodzaton technques are compared where approprate. We fnd that nonlnear chrp apodzaton, where the polng perod s vared smoothly, monotoncally, and rapdly at the edges of the devce, offers the best performance. We consder the full spatal structure of the QPM gratngs n our smulatons, but utlze the frst order QPM approxmaton to obtan analytcal and semanalytcal results. One applcaton of our results s optcal parametrc chrped pulse amplfcaton; we show that specal care must be taken n ths case to obtan hgh gan factors whle mantanng a flat gan spectrum. 23 Optcal Socety of Amerca OCIS codes: (9.436) Nonlnear optcs, devces; (23.745) Wavelength converson devces; (9.497) Parametrc oscllators and amplfers; (32.78) Ultrafast devces. INTRODUCTION Chrped (aperodc) quas-phasematchng (QPM) gratngs have receved attenton for varous optcal frequency converson schemes, ncludng dfference frequency generaton (DFG), optcal parametrc amplfcaton (OPA), sum frequency generaton (SFG), optcal parametrc oscllators (OPOs), and many other applcatons [ 23]. Ther man role so far has been to broaden the phasematchng bandwdth compared to conventonal perodc QPM gratngs, wthout the need to use short crystals wth reduced converson effcency, tghter focusng, or hgher ntenstes. Ths broadenng can be understood through a smple spatal frequency argument: due to dsperson, there s a mappng between phasematched frequency and gratng k-vector; n chrped QPM gratngs, the gratng k-vector s swept smoothly over the range of nterest, thereby broadenng the spatal Fourer spectrum of the gratng and hence the phasematchng bandwdth. For contnuous wave (cw) nteractons nvolvng the generaton of a weak wave from two undepleted waves, the generaton of the output wave can be descrbed n terms of the spatal Fourer transform of the QPM gratng evaluated at the spatal frequency correspondng to the phase msmatch assocated wth the three-wave nteracton [2]. Ths type of nteracton corresponds, for example, to neglgble pump depleton and low sgnal amplfcaton n OPA, or to secondharmonc generaton (SHG) wth neglgble pump depleton. Snce the nonlnear polarzaton s abruptly turned on at the edges of the nonlnear crystal, ths QPM transfer functon exhbts an nterference effect assocated wth the correspondng hgh spatal frequency components. Such a spectral rpple s hghly undesrable n most applcatons. In order to remove the spectral rpple, apodzaton technques may be employed to brng some measure of the effectve nonlnearty smoothly to zero [,24 27]. Several schemes have been proposed, ncludng deleted domans, duty cycle varaton, wavegude taperng [24], the use of nonlnear chrp profles [], and step-chrp desgns [26]. In the context of apodzaton, we take lnearty to mean that the output wave of nterest s lnear n the nonlnear coeffcent dz. In many cases of practcal nterest, the assumpton of lnearty does not hold. For example, n hgh-gan OPA employng chrped QPM gratngs, apodzaton s partcularly mportant []. Furthermore, relatvely recently t has been shown that saturated nonlnear nteractons n chrped QPM gratngs can exhbt hgh effcences due to an adabatc followng process [7,8]. As a result of ths process, for three-wave mxng (TWM) processes nvolvng nput pump and sgnal waves and a generated dler wave, the rato of pump output and nput ntenstes approaches wth respect to both the nput sgnal and pump ntenstes,.e., approaches % pump depleton. Ths behavor, termed adabatc frequency converson (AFC), occurs for nteractons that are both plane wave and monochromatc, provded that the QPM gratng s suffcently chrped. In ths paper, we study apodzaton for varous dfferent types of operatng regmes of nterest to chrped QPM devces, ncludng low-gan and low-effcency nteractons such as DFG, hgh-gan OPA nteractons, and even AFC. We show how apodzaton profles can be constructed systematcally, and how ther constructon can be connected wth the underlyng structure of the TWM process n an ntutve way. Ths approach enables hgh performance, lmted only by the /3/655-8$5./ 23 Optcal Socety of Amerca

2 552 J. Opt. Soc. Am. B / Vol. 3, No. 6 / June 23 Phllps et al. nherently dscrete QPM gratng structure. We analyze all of the desgns presented n detal numercally, n partcular takng nto account the full nonlnear evoluton of the coupled waves n the actual dscrete QPM structure (as opposed to assumng a smplfed frst order QPM nteracton). Our study wll enable the contnued development of many dfferent chrped QPM devces and technologes. In Secton 2, we establsh the CWEs that wll be used n the remanng sectons. In Secton 3, we solve these equatons for the smple case of DFG n a chrped QPM gratng, hghlghtng the connecton between these solutons and the egenmodes of the relevant CWE. In Secton 4, we determne and compare apodzaton profles for the lnear cases [correspondng to DFG, SFG, SHG, optcal rectfcaton (OR), etc.]. In Secton 5, we consder the case of hgh-gan OPA n chrped QPM devces, and establsh and compare apodzaton technques. In Secton 6, we ntroduce AFC, and show n detal how these nteractons can be analyzed and understood usng the geometrcal analyss of TWM processes developed n [28]. In Secton 7, we develop apodzaton procedures for AFC va ths geometrcal analyss. We gve an example apodzaton profle desgned for the case of a moderate-gan, hgh-pumpdepleton OPA devce. Our results show that for all of the above types of nteractons, and partcularly for OPA and AFC, nonlnear chrp apodzaton offers sgnfcant advantages over other approaches, such as deleted doman apodzaton (DDA). Last, we conclude and dscuss several mportant practcal aspects of our results n Secton COUPLED WAVE EQUATIONS In ths secton, we ntroduce the equatons governng arbtrary TWM processes n QPM devces. We consder plane wave, quas-cw nteractons, for whch each sgnal frequency mxes wth a sngle pump and dler frequency. Even for non-cw nteractons, ths assumpton s very useful for studyng TWM processes, for example, n optcal parametrc chrped pulse amplfcaton (OPCPA) systems. A. Three-Wave Mxng In the quas-cw approxmaton, the evoluton of the electrc feld n the QPM gratng s gven by the coupled wave equatons (CWEs) [29]: da dz ω d n c dza s A p e Δk z ; da s dz ω sd n s c dza A pe Δk z ; da p dz ω pd n p c dza A s e Δk z ; (a) (b) (c) where subscrpts, s, and p denote quanttes assocated wth the dler, sgnal, and pump envelopes, respectvely. The normalzed nonlnear coeffcent dz s defned n terms of the spatally varyng nonlnear coeffcent dz as dz dz d, where d s the nonlnear coeffcent n the unmodulated materal. ω j s the angular optcal frequency of wave j (these satsfy ω p ω ω s ), and n j s refractve ndex of wave j. The materal phase msmatch Δk s gven by Δk k p k s k, where k j ω j n j c s the wavevector for wave j. The electrc feld envelopes A j are assumed here for smplcty to contan only a sngle frequency component, and are defned such that the total electrc feld s gven by Ez; t X A 2 j z expω j t k j z c:c:; (2) j where c.c. denotes complex conjugate. Note that when pulsed nteractons are analyzed wthn the quas-cw lmt, a set of CWEs smlar to Eqs. () should be found, and expressed n the frequency doman, snce the couplng coeffcents ω j d n j c and phase msmatch Δk are frequency dependent. B. Quas-Phasematchng Equatons () apply for arbtrary spatal profles of the nonlnear coeffcent dz, provded that backward waves can be neglected. In a QPM gratng, d d. Because of ths constrant, t s possble to wrte arbtrary QPM gratng profles n a Fourer seres: dz dz d sgncosϕ G z cosπdz (3a) 2Dz X m m 2 snπmdz expmϕ πm G z; (3b) where Dz and ϕ G z are the gratng duty cycle and phase functons, respectvely. Often, Dz.5 by desgn, due to QPM fabrcaton lmtatons [3], mnmzaton of photorefractve effects [3 33], and also n order to maxmze the ampltude of the frst Fourer order (m ) n Eq. (3), but we consder more general cases here. In chrped QPM gratngs, the phase functon can be expressed as Z z ϕ G z ϕ G z K g z dz ; (4) z where K g z s the smooth and contnuous local gratng k-vector (or local spatal frequency), and ϕ G z s a chosen ntal phase. z and z f denote the postons of the nput and output ends of the gratng. Note that we do not assume z, and therefore the exp k j z phase factors n Eq. (2) must be accounted for n determnng the nput envelopes A j z gven a known nput electrc feld Ez ;t. In an nteracton where only one Fourer order of the QPM gratng s close to the phasematchng condton, dz can be approxmated by consderng only the mth terms n Eq. (3b). In partcular, for a frst order QPM nteracton, Eqs. () become da dz κ za s A p e ϕ z ; da s dz κ sza A pe ϕ z ; (5a) (5b)

3 Phllps et al. Vol. 3, No. 6 / June 23 / J. Opt. Soc. Am. B 553 da p dz κ pza A s e ϕ z : (5c) In these equatons, the couplng coeffcents κ j z are gven by κ j z ω jd 2 n j c π snπdz; (6) and the accumulated phase msmatch ϕ z s gven by Z z ϕ z ϕ z Δk z dz ; (7) z where the local phase msmatch s gven by Δk m z k p k s k mk g z; (8) for nteger m (the QPM order), and ϕ z corresponds to the relatve phase between the three envelopes and ther drvng terms at the nput of the devce. Usually, one of the envelopes s ntally zero, and n such cases ϕ z has no effect on the dynamcs. The frst order QPM approxmaton s often very accurate n practcal stuatons, as we wll show later n ths paper. C. Normalzed Coupled Wave Equatons For the purposes of analyss and numercal smulatons, t s often useful to normalze Eqs. (5). The photon flux of wave j s proportonal to n j ω j ja j j 2. Motvated by energy conservaton, and n partcular by the Manley Rowe relatons, we ntroduce dmensonless envelopes a j whose square magntudes are proportonal to these photon fluxes. These envelopes are mplctly defned wth rs ω j X n A j n ja ω n j 2 a j : (9) n n j n In these defntons, A n s the envelope of wave n at the nput to the gratng. Wth these defntons, Eqs. (5) become da dz gzγa s a p e ϕ z ; da s dz gzγa a pe ϕ z ; da p dz gzγa a s e ϕ z ; (a) (b) (c) where gz snπdz, and the couplng coeffcent γ s gven by rs ω ω s ω p X n j γ ja n n s n p ω j j 2 2d j πc : () ϕ z s defned n Eq. (7). The nput condtons for Eqs. () satsfy, n all cases, X ja j z j 2 : (2) j j Furthermore, we could also ntroduce a dmensonless propagaton coordnate ζ γz; however, for clarty we work wth physcal unts nstead. An analogous set of normalzed CWEs could be obtaned, accountng for the full spatal dependence of the QPM gratng [ dz ]. The only dfferences would be substtutng gz π dz 2, and ϕ z Δk z [see Eqs. ()]. 3. LOW-GAIN, LOW-DEPLETION DEVICES To begn our study, we frst consder cases n whch only the generated dler wave n Eqs. () changes substantally; the other two envelopes propagate lnearly,.e., wthout depleton, amplfcaton, or nonlnear phase shfts, and hence are constant. Ths type of confguraton, whle qute smple, s applcable to many dfferent types of devces, and can help gude ntuton for more complcated cases. We wll always assume n ths paper that one of the waves s zero at the nput of the gratng. For defnteness we choose ths to be the dler wave, but our results apply to other cases, wth mnor modfcatons. The generated wave, found by ntegratng Eq. (a), can be expressed as A z f ω d n c A s A p F dzδk ; (3) where the F denotes the Fourer transform, defned as Ff zk R f z exp kzdz. Eq. (3) holds for arbtrary QPM structures, for whch dz wthn the nterval z ;z f and dz elsewhere. An mportant consequence of ths Fourer transform relaton s that n a devce of fnte length, the dler spectrum acqures a rpple, due to nterference assocated wth the abrupt changes n dz at z and z f. The same argument apples f we consder only the frst Fourer order of the gratng, as n Eqs. (5) and (). In Secton 4, we dscuss apodzaton functons to suppress such spectral rpples. In ths secton, we show solutons for the partcular case of a lnearly chrped QPM gratng, and use that soluton to ntroduce a heurstc for constructng apodzaton functons for general chrped QPM gratngs. A. Analytcal Soluton For a lnearly chrped gratng, gven by K g z K g z Δk z z for constant chrp rate Δk. In ths case, the phase ϕ z n Eqs. () s gven by ϕ z ϕ z Δk 2 z z pm 2 z z pm 2 ; (4) where z pm s the phasematched pont, satsfyng K g z pm Δk. Assumng ϕ z and a constant QPM duty cycle [gz g constant for z z z f ], we ntegrate Eq. (a) to obtan a normalzed output dler feld under the frst order QPM approxmaton. The result s s a z 2 a s a p e π 4 e Δk z pm z 2 2 2πγ 2 g 2 Δk p p erf 2 e π 4 Δk zpm z p p erf 2 e π 4 Δk z zpm ; (5)

4 554 J. Opt. Soc. Am. B / Vol. 3, No. 6 / June 23 Phllps et al. where erf s the error functon. Based on Eq. (5), the generated dler (at z z f ) exhbts a sgnfcant rpple n ampltude and phase as a functon of z pm, and hence as a functon of phase msmatch Δk (based on the mappng between Δk and z pm ), and thus on optcal frequency ω (based on the mappng between Δk and ω). A mathematcally smple way to suppress ths rpple s by extendng the QPM gratng length L to nfnty,.e., z and z f. In these lmts, the (asymptotc) output dler feld s gven by a L 8 s < : 2πγ 2 g 2 jδk e π sgnδk 4 e j Δk 2 z pm z 2 9 γg Δk z f e Δk 2 z pm z 2 z f z pm 2 γg = Δk z ; a s a p ; (6) where Δkz f as z f, and smlarly Δkz as z. The second and thrd terms n ths equaton, whose phase oscllates rapdly compared to that of the frst term, eventually vansh, yeldng a magntude that s ndependent of z pm (and hence frequency). In ths case, the nonlnear nteracton s turned on and off smoothly by the gradual transton from large to small phase msmatch; the result s an apodzed nteracton. Further nsght can be ganed by examnng Eq. (a). In the case of a constant gratng duty cycle and a constant, fnte phase msmatch, Eq. (a) supports dler egenmodes: solutons whose magntude s ndependent of z. We defne these egenmodes n the general case [wth varyng gz and Δkz, but stll mantanng the assumpton of constant sgnal and pump envelopes] as a eg z γgz Δk z a s a p e ϕz : (7) If Δkz and gz are constant, ths egenmode s a soluton to Eq. (a). Based on Eq. (6), we also defne the zeroth-order q dler as a 2πγ 2 g 2 jδk ja s a p e π sgnδk 4. Wth these defntons, Eq. (6) can be wrtten n the followng form: a L z f a e ϕ z pm a eg z f a eg z ; (8) wth spectral rpples essentally orgnatng from the nonzero values of the dler egenmode at the nput and output ends of the devce (a eg z and a eg z f, respectvely). Ths form usually apples even for gratngs wth monotonc but spatally varyng chrp rates and can, n these more general cases, be understood va the statonary phase approxmaton [34]. Apodzaton can thus be vewed as reducng the magntude of dler egenmode to zero (or close to zero) suffcently slowly. The connecton between egenmodes of the TWM nteracton and apodzaton s qute general, and even apples for other, more complcated types of nteractons [7], as we dscuss n Secton 7. Such egenmodes are also connected wth the cascaded phase shfts acqured durng phase msmatched χ 2 nteractons [35]. To llustrate Eq. (8) and ts accuracy, we show n Fg. the propagaton of the dler as a functon of z for a partcular lnearly chrped gratng. a Exact dler Estmate k /2 (z-z pm ) Fg.. Evoluton of the dler a n a lnearly chrped QPM gratng, for a DFG nteracton. The sold (blue) lne shows the numercal soluton of Eq. (a), correspondng to (and ndstngushable from) the analytcal soluton gven by Eq. (5). The dashed (black) lne shows the asymptotc soluton correspondng p to Eq. (8). A normalzed propagaton coordnate, gven by ξ jδk jz z pm, s used. The dler n each curve s normalzed to ja j 2 [defned by Eqs. (6) and (8)]. The dashed (black) curve shows a eg z a eg z for ξ <.5, and a eg z a eg z a exp φ z pm for ξ >.5. The nput, phasematchng, and output ponts correspond to ξ 5, ξ, and ξ 5, respectvely. Because of the way the fgure has been normalzed, there are no other free parameters n the calculaton. B. Adabatcty Heurstc In ths subsecton, we wll use Eqs. (7) and (8) to develop a heurstc crteron by whch apodzaton profles can be constructed. These apodzaton regons wll be appended to the ends of a nomnal gratng profle (e.g., lnearly chrped); we llustrate dfferent types of apodzaton n Secton 4. The essental dea s to mpose changes n gz and Δk z such that the rato gz jδk zj at z z and z z f,and for ths change to be slow enough that the dler egenmodes Eq. (7) are stll accurate local solutons to Eq. (a). Inspectng Eq. (7), away from z pm, the egenmodes have a slowly varyng ampltude and rapdly varyng phase. Our heurstc s to mantan ths structure (phase varyng much more rapdly than relatve changes n ampltude), whch leads to the condton γg d γg d Δk dz Δk dz e ϕ : (9) In a QPM devce supportng a wde spectral bandwdth, ths condton must be met for all the spectral components of nterest, and hence for all values of z pm,atallpostonsz wthn both the nput and output apodzaton profles. Evaluatng the dervatves n Eq. (9) and replacng the by ϵ for small ϵ >, we thus fnd, wthn the apodzaton regons, dg max ω g dz dδk Δk dz ϵjδk j ; (2) where maxmzaton wth respect to ω s performed over the spectral range of nterest. Ths equaton can be used to construct dfferental equatons for apodzaton profles n whch K g z, gz, or both, or a related quantty, are vared nonlnearly wth poston. In Secton 4, we consder specfc apodzaton examples based on ths adabatcty equaton.

5 Phllps et al. Vol. 3, No. 6 / June 23 / J. Opt. Soc. Am. B K >< ϵs c z z b; K b; K for K g z <K b; : K g z K nom z for K b; K g z K b; : (25) >: K ϵs c z z b; K b; K for K g z >K b; : 4. QPM APODIZATION TECHNIQUES In ths secton, we use Eq. (2) to construct apodzaton regons for nomnally chrped QPM devces, and compare dfferent apodzaton technques. We assume that there s a nomnal, unapodzed chrp profle K nom z that has already been chosen, and show how to determne apodzaton profles that are appended to the ends of ths nomnal profle n order to suppress the spectral rpples that would otherwse occur. A. Nonlnear Chrp (NLC) Apodzaton Frst, we consder nonlnear chrp apodzaton (NLCA). In ths case, there s a constant duty cycle, dg dz, but a spatally varyng chrp rate dδk dz. Therefore,Eq.(2) canbe wrtten as s c dk g dz ϵ mn ω fδk ω K g z 2 g; (2) where s c sgnδk s the sgn of the gratng chrp rate, and we have replaced the wth (assumng that the nequalty s strct for the extrema of the spectrum). Eq. (2) can be expressed n a more explct form by dstngushng between the nput and output ends of the gratng, and by treatng K g as the ntegraton varable rather than z (whch s possble snce we assume monotonc chrp functons): dk g ϵk s c dz K g 2 f Δkz; ω > ; ϵk K g 2 f Δkz; ω < ; (22) where K mn ω Δk ω and K max ω Δk ω (wth mn and max performed over the frequency range of nterest). Wthn each apodzaton regon, Eq. (22) can be solved analytcally, yeldng the followng mplct equaton: ϵs K g z K K b; K c z z b; ; (23) where z b; and K b; represent boundary condtons. These boundary condtons can be determned from the fact that we append apodzaton regons to a nomnal QPM profle, under the assumpton that K g and dk g dz must be contnuous at the apodzaton boundares. These boundares are therefore the ponts at whch the chrp rate n Eq. (22) equals the nomnal chrp rate. Explctly, for a nomnal profle K nom z and chrp rate dk nom dz K nom;z, z b; and K b; are the solutons to the followng equatons: K nom;z z b; ϵs c K K b; 2 ; K b; K nom z b; : (24a) (24b) Gven the soluton to Eq. (24), we can now specfy the full form of the gratng: Note that the equaton for K g z n the apodzaton regons dverges wth respect to z. Therefore, for a real gratng, ntal and fnal values of K g z must be chosen. In choosng these values, we must bear n mnd that the fnal values of γωgz Δk z; ω determne the fdelty of the apodzaton, but only to the extent that the frst order QPM contrbuton (m ) domnates. The exstence of other QPM orders means that, eventually, ncreases n Δk no longer suppress spectral rpple. In an extreme case of K g z passng through phasematchng for thrd-order QPM, for example, the spectral rpple could actually be made worse. The lmts on K g z are thus determned by a trade-off between apodzaton fdelty, fabrcaton constrants, hgher-order-qpm contrbutons, and potentally other ssues as well. We expect that n most practcal cases, these ssues wll not substantally lmt the performance of the apodzed devce. Note also that, n a practcal devce, t may be useful to reduce the chrp rate at the edges of the gratng so that the range of K g z s not senstve to changes n crystal length that occur durng polshng. We dscuss ths further n Subsecton 8.B. In Subsectons 4.B and 4.C, we dscuss two other apodzaton technques. In Subsecton 4.D, we show example apodzaton profles and correspondng dler spectra (Fg. 2). B. Duty Cycle Apodzaton Another approach to suppressng spectral rpples s by apodzng gz nstead of Δk. In prncple, changes n gz can be mplemented va changes n the QPM duty cycle Dz. Therefore, we refer to ths approach as duty cycle apodzaton (DCA). For ths case, we can agan assume a known, nomnal chrp functon (e.g., a lnear chrp), and substtute ths functon nto Eq. (2). Analogously to Eq. (2), we fnd the followng dfferental equaton for gz, sutable for a fnte and monotonc nomnal chrp rate: s g dg g dz ( dk g s g Δk ω K g z dz " s ϵ 2 Δk #) K g z 4 dk g dz 2 ; (26) mn ω where s g sgndg dz. Ths equaton can be used to determne gz n a smlar way to the NLCA case dscussed n Subsecton 4.A, but we omt the mathematcal detals here. C. Deleted Doman Apodzaton In lthographc polng, t s often challengng n practce to obtan a custom duty cycle functon, due to the dynamcs of the polng process [3], partcularly for MgO:LNbO 3 polng [36]. Instead, the voltage waveform used for polng s usually chosen to yeld as close to a 5% duty cycle as possble. Such gratngs are also advantageous n order to suppress photorefractve

6 556 J. Opt. Soc. Am. B / Vol. 3, No. 6 / June 23 Phllps et al. effects [3], ncludng recently dentfed pyroelectrcally nduced beam dstortons [32,33]. If duty cycle modulaton s not possble, a contnuously varyng frst order QPM coeffcent gz cannot be acheved. An alternatve approach s to use a dscrete approxmaton to the desred contnuous gz profle, as demonstrated n [24]. In ths scheme, QPM domans are deleted (not nverted) n order to reduce the effectve duty cycle of the gratng, averaged over many perods. Therefore, we call ths approach deleted doman apodzaton (DDA). To express ths approach mathematcally, we assume that the gratng has N perods wth center postons z n and length l n n ; ;N, and ether 5% (g n ) or%(g n ) duty cycle. The spatal profle of the gratng s then gven by dz X 2g n Π ln 2z z n ; (27) n where Π l z s the rectangle functon (wth wdth l and center ). The ntegral of d should then be a good approxmaton to the target profle gz: X Z z g n 2Π ln 2z z n dz n z Z z z gz dz : (28) A smple way to obtan each g n from ths approxmate relaton s to ntally assume g n, and then reverse the sgn of g n whenever, for a poston z z n l n 2, the left hand sde of Eq. (28) exceeds the rght hand sde. D. Comparson and Dscusson In ths subsecton, we compare the three apodzaton technques descrbed n Subsectons 4.A 4.C (NLCA, DCA, and DDA, respectvely). We choose an example wth the followng parameters. The gratng center spatal frequency K g 3 2 mm ( 2 μm perod), nomnal length L nom mm, nomnal bandwdth Δk BW 2 mm, and postve chrp rate Δk Δk BW L nom 2 mm 2. Apodzaton profles are found va the precedng dfferental equatons, and we choose ϵ.5 n each case (for ϵ >, the apodzaton fdelty decreases). For the NLCA example, the range of k-vectors, jk g z f K g z j 2 mm, and the apodzed gratng length s L z f z 4 mm. For the DCA and DDA examples, gz f gz.52, and the apodzed gratng length s L z f z 6 mm. The resultng dler spectrum for phase msmatches n the vcnty of frst order QPM s shown n Fg. 2. The NLCA profle s found accordng to Eq. (25); the DCA profle accordng to Eq. (26), and the DDA example s determned from the DCA profle combned wth Eq. (28). The NLCA and DCA cases show the best rpple suppresson, whle the DDA case stll has a substantal rpple n ths example. The NLCA rpple s lmted by the range of QPM perods and presence of hgher order QPM terms. In cases where precse control of the duty cycle s not possble over a wde range of perods, even orders of the gratng wll contrbute as well, so second-order QPM contrbutons represent one possble lmtng factor for the NLCA approach. These ssues are dffcult to quantfy n general, so we have restrcted the example to a range of K g z far from hgher-order QPM. The DCA rpple wll be lmted by the range of duty cycles that can be fabrcated relably. Ths range s typcally qute restrcted, a a.2..9 (a) k / k L nom.5 (b) K g (mm - ) g(z) 4 (c) Kg: Poston.5 NLCA..2.3 DCA NLCA.4 NLCA DCA Unapodzed DCA DDA k / k L nom K g (mm - ) (c) K g : NLCA K g : DCA g: DCA g: NLCA Poston Fg. 2. Apodzaton examples for DFG, calculated numercally va Eq. (a), usng the full d gratng structure. (a) Nonlnear chrp apodzaton (NLCA), (b) duty cycle apodzaton (DCA), (c) deleted doman apodzaton (DDA) wth a doman profle determned from the DCA example n (b). The parameters are gven n the text. and therefore the hgh performance of the DCA example may not be achevable n practce. Furthermore, the length of the apodzaton regon wll usually be longer for DCA compared wth NLCA. Whle DDA s not lmted by duty cycle fabrcaton ssues, the DDA example exhbts poorer performance than DCA, due to the non-neglgble k-space bandwdth, compared to the nomnal gratng k-vector. Consequently, the mplct assumpton of a small change n relatve phase between the dler and ts drvng polarzaton between the remanng undeleted domans does not hold. More specfcally, to acheve a low effectve nonlnearty, there must be a large gap between undeleted domans. Snce all of the spectral components of nterest are phase msmatched n the apodzaton regons for a chrped gratng, ths gap can correspond to a large relatve phase shft between the dler and ts drvng polarzaton. Such large phase shfts lkely prevent the nterference that would, for the case of a smooth duty cycle modulaton, (almost) completely suppress the nput and output egenmodes (and hence the spectral rpples). Ths ssue s partcularly mportant for gratngs wth a broad k-space bandwdth, such as the example shown n Fg. 2. On the other hand, for perodc QPM gratngs, such as those dscussed n [24], the underlyng assumpton of a slowly varyng relatve phase holds very well, and DDA s effectve n such cases..5 g(z)

7 Phllps et al. Vol. 3, No. 6 / June 23 / J. Opt. Soc. Am. B 557 Snce DCA s challengng n terms of fabrcaton and DDA exhbts reduced performance, the NLCA approach wll be favorable n most chrped QPM-gratng cases. These conclusons are especally vald n more complcated nteractons nvolvng hgh gan or pump depleton. An example of the poorer performance of DDA compared wth NLCA n a hgh-gan nteracton s shown n Secton 5. In each of the above apodzaton approaches, the absolute scale of the ampltude rpple remanng n the apodzed devce s determned by fabrcaton and QPM constrants, such as the presence of hgher-order QPM contrbutons, and s ndependent of gratng length. In contrast, the scale of the zerothorder term [a n Eq. (8)] ncreases wth gratng length for a gven bandwdth, snce n a longer devce the chrp rate can be reduced. Therefore, the relatve scale of the rpples s reduced n longer devces. Note, however, that the range of group delays assocated wth any remanng spectral rpples s not reduced, and may ncrease. 5. OPTICAL PARAMETRIC AMPLIFICATION We now turn our attenton to hgh-gan and hgh-effcency devces. In ths secton, we consder chrped QPM OPA. Such devces have several advantageous propertes, ncludng the potental for hgh gan, almost arbtrary gan bandwdth, talorable gan and phase spectra, and hgh converson effcency [,4,5,7]. They have been used n a md-nfrared OPCPA system, enablng broad bandwdth, hgh power, and hghrepetton rate operaton [9,]. A crtcal consderaton n obtanng hgh-fdelty amplfcaton from these devces s apodzaton. Wthout apodzaton, there s a pronounced rpple n gan and phase; ths rpple can be much more severe than n the lnear cases (e.g., DFG), due to the hgh gan nvolved. The presence of such a rpple can be explaned heurstcally by the abrupt turn-on and turnoff of nonlnear couplng between the sgnal and dler felds at the edges of the devce, n analogy to the smpler case of DFG. Here, we buld on the theoretcal work presented n [] to show how optmal OPA apodzaton profles can be constructed. Our approach s smlar n sprt to the one presented n Secton 4 for DFG apodzaton. A. Overvew of Chrped QPM OPA We frst gve a bref theoretcal descrpton of chrped QPM OPA nteractons, under the assumpton of a cw (or quascw) pump wave, whch s undepleted and much stronger than the sgnal and dler waves (ja j ja p j and ja s j ja p j). In ths case, the amplfcaton of each sgnal spectral component (and correspondng dler component) s governed by Eqs. (a) and (b). By defnng new envelopes b j accordng to a j z gz 2 exp ϕ z 2b j z for j and j s, the followng second-order equaton can be obtaned from Eqs. (a) and (b) []: where the potental Qz s gven by d 2 b s dz 2 Qzb s; (29) Q γg 2 d g Δk 2 dz g 2 g 2; 4 g Δk (3) where f df dz for functon f z. Ths potental s poston dependent (va gz and K g z) and frequency dependent (va γω and Δk ω). Equaton (29), whch s n standard form, s amenable to complex Wentzel Kramers Brlloun (WKB) analyss. Ths analyss yelds several mportant results for devce operaton []. In partcular, for smoothly chrped gratngs, the sgnal ntensty gan can be approxmated accordng to Z s zf Δk lng s ω 2 Re gzγω 2 ω K g z 2 dz; z 2 (3) whch, n the case of a constant gratng chrp rate (Δk Δk z) and a 5% duty cycle (g ), yelds G s exp2πγ 2 jδk j for spectral components wthn the gan bandwdth. Equaton (3) shows that gan occurs for each spectral component ω over the spatal regon for whch the sgnal-dler couplng, gzγω, s suffcently large compared wth the phase msmatch, Δkz; ω Δk ω K g z. Outsde ths spatal regon, the sgnal and dler waves are oscllatory. The ponts where the ntegrand n Eq. (3) s zero are called turnng ponts. The gan bandwdth can be determned from Eq. (3) as the range of frequences for whch both turnng ponts le wthn the gratng. For the case of a lnear gratng chrp, ths bandwdth s gven mplctly as the range of frequences ω whose phase msmatch Δk ω les wthn the followng range: ω Δk ω K g jδk Lj 2 2γω; jδk Lj 2γω ; 2 (32) for center gratng k-vector K g. Outsde the gan regon, the waves are oscllatory versus poston, leadng to fluctuatons of the output waves versus frequency, and consequently a rpple n the output spectrum (snce the oscllatons of dfferent spectral components have dfferent phase). These propertes are llustrated n Fg. 3, where we show the propagaton of a sngle spectral component. The fgure shows both an unapodzed and an apodzed case. Apodzed and unapodzed gan spectra are shown later, n Fg. 4, after we have dscussed an apodzaton scheme. B. Apodzaton Constrant for Chrped QPM OPA The oscllatory behavor shown n Fg. 3 can be understood as nterference assocated wth the two complex WKB global asymptotc solutons of Eqs. (29) (see, for example, the appendces of []). The role played by these global solutons n the context of ampltude oscllatons s twofold: frst, the assumed nput condtons to the devce (zero dler, fnte sgnal) mply that the sgnal conssts of a complex lnear superposton of these two global solutons. Second, the global solutons themselves are oscllatory. Gven such oscllatons, frequencydependent changes n the phasematched pont lead to oscllatons n the gan spectrum. These spectral oscllatons can be suppressed by suppressng oscllatons n the WKB solutons, whch s acheved by the condton jνzj for z z and z z f, where ν s defned as

8 558 J. Opt. Soc. Am. B / Vol. 3, No. 6 / June 23 Phllps et al. 5 (a) 35 3 (a) a j 2 a j 2-5 Idler (unapodzed) Sgnal (unapodzed) Idler (apodzed) Sgnal (apodzed) (z z pm ) / L deph 3.5 x 6 (b) (z z pm ) / L deph Fg. 3. Propagaton example for OPA n a lnearly chrped QPM gratng, showng the sgnal and dler as a functon of normalzed poston. The gan factor λ R 2.2. The poston has been normalzed to the dephasng length, L deph 2γ jδk j. The normalzed gratng length s gven by L L deph 8. The phasematchng pont z pm s located at the mddle of the gratng. The dashed lnes show the evoluton of the sgnal and dler n a apodzed gratng wth the same parameters; oscllatons n a z and a s z near z z and z z f are suppressed n ths case. For ths example, we apodze va the chrp rate and duty cycle smultaneously to reveal the dler evoluton under dealzed nput condtons; (b) shows the felds on a lnear scale to better ndcate how the oscllatons are suppressed near the output of the gratng n the apodzed case, but not the unapodzed case; the sgnal and dler magntudes are ndstngushable on ths lnear scale due to the hgh gan nvolved. νz Δk z 2γgz : (33) The jνz j condton ensures that the nput condtons correspond to one of the two global solutons (very small contrbuton from the other soluton) [Eqs. (C4) and (C5) of []]. The jνz f j condton suppresses rpples on ths global soluton [Eq. (C3) of []]. These condtons are also qute closely related to the dler egenmodes of Eq. (7). Indeed, for large phase msmatches, Eqs. (a and b) support two local sgnal-dler egenmodes whose ampltudes would be constant wth respect to poston n the case of a constant and large phase msmatch and constant duty cycle. These egenmodes may be determned by substtutng a a exp R z z ϕ z dz and a s a s exp R z z ϕ s z dz nto Eqs. (a and b), neglectng dervatves of a j z, and solvng for ϕ z, ϕ s z, and ja s z a zj 2. Whle these egenmodes are not solutons to Eqs. (a and b) n the case of a varyng phase msmatch, they provde some nsght nto the more complcated dynamcs supported by the WKB solutons descrbed above. For large phase msmatches, one of these two egenmodes has a small sgnal component, whle the other has a small dler Gan K g (mm - ) k /γ Poston (mm) x 5 (c) 6 (b) 3 3 (K g - K g ) / γ DDA DCA NLCA NLCA: Predcted Range Wavelength (nm) Fg. 4. Chrped-QPM OPA apodzaton examples. (a) Gratng k-vector profle for a NLCA example, (b) normalzed gratng chrp rate correspondng to (a), and (c) sgnal gan spectrum. The dashed lnes n (b) show mnδk γ 2 ϵ, wth mnmzaton performed wth respect to sgnal wavelength (restrcted to the target gan bandwdth that spans the nm range). Ths fgure ndcates that the optmal normalzed chrp rate n the apodzaton regon approxmately satsfes jδk jmnδk 2 ϵ. The dashed lnes n (c) ndcate analytcal estmates of the fluctuatons n the gan due to the fnte value of jγ Δk j at the ends of the gratng, as descrbed n the text. The gan spectra for DCA and correspondng DDA examples are also shown n (c), for comparson. component. The rate of sgnal-phase accumulaton, φ s z, dffers between these two egenmodes, leadng to ampltude oscllatons unless the sgnal and dler felds correspond to only one of the egenmodes. In the lmt where νz, the nput condtons (zero dler) are matched to one of the two egenmodes, thereby suppressng rpples near the start of the gratng [regon z z z pm of the apodzed example n Fg. 3] snce the magntude of ths egenmode vares slowly as ν s decreased, but does not vary over the fast Δk length scale assocated wth the local phase msmatch Δk. Note that the dfferent length-scales n phase-msmatched TWM nteractons have been analyzed usng multple-scale analyss n [37].

9 Phllps et al. Vol. 3, No. 6 / June 23 / J. Opt. Soc. Am. B 559 After the amplfcaton regon (z z pm ), the felds can be thought of as consstng of a lnear combnaton of the two local egenmodes. In the lmt where νz f, the sgnal-component of one of the two egenmodes s brought to zero at the end of the gratng, thereby suppressng sgnalampltude oscllatons [regon z pm z z f of the apodzed example n Fg. 3]. The jνz j and jνz f j condtons can be acheved, n prncple, by extendng the nomnal gratng chrp profle to nfnty. In practce, ths condton can be acheved (wth hgh fdelty) by apodzaton. In performng such apodzaton, jνzj n Eq. (33) should be ncreased slowly enough that the local WKB solutons reman accurate; f νz s changed too rapdly, there wll be couplng between the two asymptotc solutons of Eqs. (), agan leadng to nterference. As such, a suffcent crteron s to mantan the valdty of the WKB approxmaton wthn the apodzaton regons. Ths condton corresponds, for an ndvdual spectral component ω, to the followng nequalty [38]: dq dz ϵ; (34) Q 3 2 for small ϵ. Ths condton should be mantaned for all spectral components of nterest [.e., all wthn the amplfcaton bandwdth assocated wth Eq. (3)]. C. Nonlnear Chrp Apodzaton for OPA In ths subsecton we use Eq. (34) to determne apodzaton profles for undepleted-pump, chrped QPM OPAs. As before, we start from a nomnal gratng desgn, such as a lnear chrp profle. Apodzaton regons are then appended to ths nomnal desgn, wth the gratng k-vector and duty cycle properly matched at the jonng ponts. Wth Eq. (34), these apodzaton regons can be made as short as possble. Ths condton wll help wth meetng any maxmum gratng length constrants. In the context of OPAs, an even greater beneft s that optmzed apodzaton profles wll ensure that parastc effects, such as pump SHG and other unwanted processes, can be avoded or suppressed n the apodzaton regons by havng a chrp rate that s as hgh as possble wthn as short a length as possble. Equaton (34) yelds an mplct dfferental equaton n z whch, as stated, s dffcult to solve. A much smpler equaton s obtaned by wrtng gz and the chrp rate K z dk g dz as functons of K g, and ntegratng versus K g. Ths approach s possble because we assume that K g z s monotonc, and gz s monotonc wthn each ndvdual apodzaton regon. For notatonal convenence, n the followng analyss we suppress the g subscrpt on K g. We manly consder NLCA rather than DCA or DDA for the reasons dscussed n Subsecton 4.D, and therefore assume that gz s constant versus poston. For smplcty, we also assume that γω γ s frequencyndependent, although n practce ths assumpton s not necessary. Wth the above assumptons, the followng equaton can be obtaned from Eqs. (3) and (34): dk z dk 2 mn Δk 4ϵ 2 Δk 2 γ 2 2 K z 3 2 Δk 2 K 2 z mn Δk ff Δk ;K;K z g; (35) where Δkz; ω Δk ω Kz. The mnmzaton n Eq. (35) wth respect to Δk ω s performed over the desred amplfcaton bandwdth of the devce. The functon f has been ntroduced as a shorthand to dentfy the dependence of the nequalty on Δk (and hence on optcal frequency). Close to the turnng ponts satsfyng Δk K 2γ, f Δk ;K;K z s negatve (unless ϵ s chosen to be too large, e.g., ϵ ), whch means that jdq dzj > ϵjq 3 2 j. If we move far enough from these turnng ponts (toward the edges of the nomnal gratng profle) then equalty s obtaned. We wsh to fnd a par of gratng k-vectors at whch to begn the apodzaton regon; these are denoted K apod;s and K apod;e for the startregons and end-regons of the gratng, respectvely. These wll be the k-space ponts at whch nequalty n Eq. (35) s satsfed for the whole spectral range of nterest. The mnmzaton n Eq. (35) wll correspond to ether the mnmum or maxmum value of Δk ω across ths spectral range. Beyond these k-vectors, t s possble to chrp the gratng more rapdly whle stll satsfyng Eq. (34). We therefore assume that the nverse gratng profle, zk, corresponds to a nomnal chrp profle z nom K for K between K apod;s and K apod;e, and s determned va Eq. (35) outsde ths regon. Based on the above dscusson, we frst determne K apod;s and K apod;e by solvng the followng par of equatons: mn Δk ff Δk ;K apod;j ;K z;nom g dk z;nom 2; dk mn Δk jδk K apod;j j 2γ ; (36a) (36b) for j s and j e, where subscrpt nom denotes the nomnal gratng profle. For a lnearly chrped gratng, K z;nom K s constant. Eq. (36a) arses from Eq. (35), whle Eq. (36b) ensures that the K apod;j le outsde the amplfcaton regon for all of the spectral components nvolved. It s convenent to defne the k-space doman of the nomnal gratng: domk nom fk:mnk apod;j K maxk apod;j g: (37) j j We now use K apod;j to fnd the entre gratng profle by frst solvng the followng equatons for Kz and K z K: Kz K nom z;k domk nom ; dk z dk s p K z mn f ;K domk nom ; Δk (38a) (38b) where s K z sgndk z;nom dk, and wth the ntal condtons for Eq. (38b) gven by K z K apod;j K z;nom K apod;j : (39) Equatons (38a) and (38b) yeld K z K over the entre gratng spatal frequency profle. We can thus determne zk by

10 56 J. Opt. Soc. Am. B / Vol. 3, No. 6 / June 23 Phllps et al. Z K f zk K z K dk; (4) K and then nvert ths functon to fnd K g z. An example mplementaton of ths desgn procedure s shown n Fg. 4. For ths example, we select a specfc expermental confguraton. We assume an MgO:LNbO 3 OPA devce at 5 C, desgned to amplfy sgnal components between 45 and 65 nm usng a narrow-bandwdth 64 nm pump. The correspondng range of dler wavelengths s 3 4 nm. For llustraton purposes, we assume a nondspersve couplng coeffcent γ, wth γω γ. We choose a gan coeffcent λ R;p γ 2 jδk j2.2. The range of phase msmatches correspondng to the sgnal bandwdth s Δk BW 6.6 mm. Ths value orgnates from the materal phase msmatches of Δk 29.7 and 23. mm for sgnal wavelengths of 45 and 65 nm, respectvely. Accordng to Eq. (3), a lnear chrp rate Δk and nomnal gratng length L nom satsfyng jδk L nom jδk BW 4γ are requred n order to fully amplfy ths bandwdth. Therefore, gven values of L nom, Δk BW, and λ R;p, the values of γ and jδk j can be determned. Here, we choose L nom mm, whch yelds γ 2.4 mm and Δk 2.62 mm 2. To construct an apodzaton profle, we assume a postve chrp rate (Δk > ), a symmetrc gratng profle, set ϵ, and solve Eq. (38). It s useful to ntroduce a parameter δ to descrbe the range of gratng k-vectors, K g z f K g z 2γ δ; (4) where we choose δ 45 for the present example. Ths value s chosen to yeld large values of jνj defned n Eq. (33) at the ends of the devce, whle stll remanng far from hgher-order QPM. For our example, K g z 39.5 mm and K g z f 3.3 mm. The k-vector profle s shown n Fg. 4(a). In Fg. 4(b), the gratng chrp rate s compared to Δk 2, llustratng that optmal normalzed chrp rate s approxmately proportonal to the square of the mnmum phase msmatch. The output gan spectrum as a functon of wavelength s shown n Fg. 4(c). We show DCA and DDA examples for comparson to the NLCA case. For each of these smulatons, we assume the full gratng structure ( d ) and ntegrate Eqs. () numercally, for each spectral component, wthn each successve QPM doman. The fnte value of δ defned above as well as hgher-order QPM contrbutons result n a rpple n the gan for the NLCA example. Under the assumpton of a frst order QPM nteracton, the rpple s such that j lng s 2πλ R;p j 4δ for frequences wthn the amplfcaton regon []. In a real gratng, there are also small contrbutons from the other orders of the QPM gratng. The dashed black lnes n Fg. 4 are bounds on the gan rpple found by analytcally summng all such contrbutons, showng excellent agreement wth the full numercal NLCA smulatons. The gan spectra for the DCA and DDA examples are also ncluded. The DCA example was specfed heurstcally, wth an ampltude profle determned va hyperbolc tangent functons; t shows comparable performance to NLCA, but requres large modulaton of the QPM duty cycle. The DDA example was derved from the DCA example, and shows a sgnfcant reducton n apodzaton qualty compared wth both DCA and NLCA. For these DCA and DDA examples, the chrp rate s lnear. Note that wth no apodzaton, the rpple s huge, wth gan varyng by a factor of 4 across the passband (from.5 6 to 2 6 ). 6. ADIABATIC FREQUENCY CONVERSION We next consder adabatc frequency converson (AFC). Ths type of nteracton can occur n chrped QPM gratngs when the couplng between the three waves s suffcently strong, and enables hgh pump converson effcency across a very broad range of phase msmatches and ntenstes. In ths secton, we frst show how AFC can be analyzed and understood n general TWM confguratons. In Secton 7, we develop an apodzaton procedure for AFC devces va an approach smlar to the DFG and OPA cases consdered above. In AFC, nstead of reachng a maxmum at a certan nput ntensty, the pump depleton can ncrease monotoncally wth respect to the nput ntensty of ether the pump or sgnal wave, or both [7,8]. Rather than back-convert after the pont of maxmum pump depleton, the felds adabatcally follow a local nonlnear egenmode that evolves wth the gratng perod. If the gratng s suffcently chrped, then at the nput to the devce, the relevant egenmode corresponds to zero dler (.e., to the nput condtons), whle at the output ths egenmode corresponds to zero pump (the desred output condton of full pump depleton). Ths adabatc followng process s possble provded that the couplng rate γ between the felds s strong enough (at a gven chrp rate Δk ), or f the chrp rate s slow enough (at a gven couplng rate). The requred couplng rate also depends on the nput condtons, as we wll dscuss. Before consderng AFC and nonlnear egenmodes n more detal, we dscuss n Subsecton 6.A a reformulaton of Eqs. () based on [28], n whch TWM nteractons are descrbed geometrcally. Ths geometrc analyss provdes many nsghts nto the structure of varous TWM processes, especally AFC, as we wll show. A. Geometrc Analyss of Three-Wave Mxng Interactons In [28], the geometrcal analyss was motvated by the Hamltonan structure of the TWM equatons. Furthermore, [28] consdered general QPM nteractons, correspondng to d. Here, we brefly recaptulate the formulaton gven n more detal n [28], and gve the modfcatons requred for frst order QPM nteractons. Frst, reduced feld varables X, Y, and Z can be defned accordng to X Y a a s a pe ϕ z Z ja p j 2 ; (42) The phase of X Y specfes the relatve phase between the envelopes a j and ther drvng polarzatons n the CWEs [Eqs. ()], and hence the drecton of energy transfer. The remanng varable Z specfes the pump photon flux. These varables can be treated as specfyng a real-valued poston vector W n an abstract 3-space, defned as W X;Y;Z T : (43) Durng propagaton, W evolves accordng to the evolvng complex envelopes a j, but s constraned to le on a surface whose

11 Phllps et al. Vol. 3, No. 6 / June 23 / J. Opt. Soc. Am. B 56 shape s determned by the nput condtons. Ths surface s gven by the mplct relaton φ, where φ X 2 Y 2 ZZ K p Z K sp ; (44) and where the constants K p and K sp are Manley Rowe nvarants, gven by K jp ja j j 2 ja p j 2 ; (45) for j and j s. Wth the envelope defntons and nput condtons consdered here, wth zero nput dler, K sp and K p ja p z j 2. It s convenent to ntroduce a parameter ρ descrbng the rato of nput photon fluxes: ρ ja s z a p z j 2. For cases wth an nput sgnal that s much larger than the pump (ρ, and hence Z K p ), φ s (approxmately) a sphere. For other nput condtons (ρ ), the conserved surface φ s not sphercal, but remans closed and convex [28]. The evoluton of W s gven by dw H φ; (46) dγz where X; Y; Z T, and where the local Hamltonan H, whch s dscussed n more detal n [28], can be expressed as H gx Δk 2γ Z K p K sp : (47) Ths local Hamltonan s poston-dependent n the case of a chrped QPM gratng. Eq. (46) mples that W s constraned to reman on the mplct surface φ, snce the force actng on W s perpendcular to the local surface normal φ. B. Soluton for Unform QPM Gratngs The geometrcal approach of [28], an mportant result of whch s reproduced n Eq. (46), greatly smplfes the nterpretaton of many TWM problems. For example, durng propagaton n a unformly phasematched medum, W s constraned to le on the ntersecton between a plane (H constant) and a convex surface (φ ), and hence on a rng (or a sngle pont). The dstance requred to fully traverse the rng s the perod assocated wth the Jacob-ellptc analytcal solutons of the there-wave mxng problem, derved n [39]. Much of the complcated structure of these analytcal solutons can thus be vsualzed wth ths geometrcal constructon. Furthermore, Eq. (46) reveals the exstence of the local nonlnear egenmodes dscussed above. The exstence of such egenmodes s well known [35], but the geometrc descrpton provdes a partcularly convenent framework for ther study, and the nterpretaton of ther role n spatally nonunform structures, such as chrped gratngs. These egenmodes, whch we denote as W m, satsfy dw dz, and hence correspond to the two ponts for whch H s normal to the surface φ (the ponts where φ s n the same drecton as H). Because φ s closed and convex, there are two and only two such ponts. These egenmodes can also be vewed as generalzatons of the egenmodes we dscussed n Subsectons 3.A and 5.B for the specfc cases of DFG and OPA, respectvely. To llustrate the nonlnear egenmodes and the evoluton of W, we show n Fg. (5) a propagaton example for a unformly phasematched devce. In ths example, the black arrow ponts n the drecton of H; ts locaton has been chosen so that the pont where t touches the conserved surface φ corresponds to a nonlnear egenmode. Snce the medum s unform (constant g and Δk, and other QPM orders are neglected), ths egenmode s fxed, and hence represents a true egenmode of the TWM nteracton. In a chrped devce, the drecton of H, and hence the local nonlnear egenmode W m, would be swept from the top to the bottom of φ as ν Δk 2γg [see Eq. (33)] s swept from ν to ν. C. Soluton for Chrped QPM Gratngs We next consder the AFC solutons supported by chrped QPM gratngs, for whch the reduced feld vector Wz can follow the nonlnear egenmodes W m z. Snce W m X m Y m Z m T are ponts where H φ, they can be found, for any gven value of ν, by solvng the followng set of equatons: Z= a p a j 2 (a) -.2 X (b) Y Sgnal Pump Poston (normalzed) Fg. 5. (a) Soluton to an example TWM problem, vsualzed wth the geometrc descrpton of [28] [and Eq. (46) n partcular]. The parameters for ths example are K p ja p z j 2.9, g (5% duty cycle), and Δk γ. The surface shown s φ, and the curve (blue) represents the trajectory of W. Snce Δk and g are both constant n ths example, the curve les on a plane H constant. W s ntally at the top of the surface (Z ja p j 2 K p.9), and the drecton of W (wth ncreasng z) s shown by the blue arrow on the curve. The drecton of H s also shown (black arrow). The pont where ths arrow touches the surface s a nonlnear egenmode assocated wth the chosen parameters (.e., a pont where H s n the drecton of the surface normal to φ, φ). In ths unchrped example, the feld vector W orbts around the fxed egenmode W m. In (b), the photon fluxes ja j j 2 are shown for comparson. One perod of these fluxes corresponds to a complete traversal of the blue curve n (a); poston s normalzed to γ, defned n Eq. ().