Femtosecond-rate space-to-time conversion

Transcription

1 Marom et al. Vol. 17, No. 10/Otober 000/J. Opt. So. Am. B 1759 Femtoseond-rate spae-to-time onversion an Marom,* mitriy Panasenko, Pang-Chen Sun, and Yeshaiahu Fainman epartment of Eletrial and Computer Engineering, University of California, San iego, 9500 Gilman rive, La Jolla, California Reeived June 1, 000 A real-time spatial temporal proessor based on asaded nonlinearities onverts spae-domain images to time-domain waveforms by the eration of spetrally deomposed ultrashort pulses and spatially Fouriertransformed images arried by quasi-monohromati light waves. We use four-wave mixing, ahieved by asaded seond-order nonlinearities with type II nonollinear phase mathing, for femtoseond-rate proessing. We present a detailed analysis of the nonlinear mixing proess with waves ontaining wide temporal and angular bandwidths. The wide bandwidths give rise to phase-mismath terms in eah proess of the asade. We define a omplex spatial temporal filter to haraterize the effets of the phase-mismath terms, modeling the deviations from the ideal system response. New experimental results that support our findings are presented. 000 Optial Soiety of Ameria [S (00) ] OCIS odes: , , , INTROUCTION Methods for ultrafast optial waveform synthesis (also referred to as pulse shaping) have been investigated for diverse appliations in the natural sienes and engineering. Reviews of various pulse-shaping tehniques an be found in Refs. 1 and. Based on linear-system theory, filtering the temporal-frequeny omponents of an ultrashort optial pulse results in high-resolution waveform synthesis. The filtering an be ahieved by spatially dispersing the frequeny omponents in a spetral proessing devie (SP), a free-spae optial setup onsisting of diffration gratings and lenses, and inserting a spatial mask with the enoded amplitude and phase information. After the spetral deomposition wave 3 (SW) has been filtered by the spatial mask, the frequeny omponents are reomposed in the SP to generate the synthesized temporal waveform. Early pulse-shaping experiments used prefabriated masks to filter the SW, 4 whih later were replaed by ative-filtering devies suh as spatial light modulators 5 and aousto-opti modulators. 6 This approah may not yield an adequate time response for adaptive ontrol of the synthesized waveform needed for some appliations, as it is limited by omputation time, signal propagation delay, and modulator response time. Additionally, most modulators operate with either phase or amplitude modulation, requiring a ompliated asade of two devies for omplete omplex-amplitude filtering. 7 An alternative approah for generating the spetral filter uses in situ optial holographi reording of the erferene of the spatial Fourier transform (FT) of a spatial image and a referene po soure. 8,9 The reorded spatialfrequeny information serves as a spetral filter for the SW of an inident ultrashort referene pulse. The synthesized waveform is orrelated to the spatial image used in the reording, resulting in a spae-to-time onverted image. This approah an be erpreted as a four-wave mixing proess between two waves arrying spatial FT information and two SW s with the temporal FT information, resulting in the exhange of information between the spatial and temporal hannels. The holographi reording medium s harateristis determine the performane of the information exhange between the four erating waves. For example, using holographi film will require a long reording and proessing time. Thik holograms formed in bulk photorefrative rystals yield high diffration effiieny but also require a long reording time owing to slow eletron mobility essential in buildup of a spae-harge field 9 (from tens of miroseonds to several minutes). Multiple-quantum-well (MQW) semiondutor photorefratives an perform holographi reording with a miroseond response time yet result in low diffration effiieny owing to the short eration length. 10,11 For ompatibility with the requirements of ultrafast appliations suh as high-speed optial ommuniation with ultrashort pulses and quantum ontrol of atomi and moleular vibrational states, the synthesized waveform must be updated, or modulated, at high rates with both amplitude and phase information. Suh a femtoseondrate response time an be provided only by parametri proesses that involve bound-eletron nonlinearities. Our spae-to-time onversion sheme, first reported in Ref. 1, exhanges the information from a spatial image to a temporal waveform by a four-wave mixing proess in a () nonlinear rystal. The asaded seond-order nonlinearity (CSN) arrangement we are utilizing onsists of a frequeny-sum generation proess followed by a frequeny-differene generation proess that satisfies the type II nonollinear phase-mathing ondition. 13 The nonlinear wave mixing ours in the Fourier domain of the temporal and spatial hannels (see Fig. 1). The frequeny-sum proess mixes the SW of the input ultrashort pulse and the spatial FT of the spatial-image information illuminated with a quasi-monohromati plane wave. The resultant ermediate wave is determined by the produt of the omplex amplitudes of the temporal and spatial FT fields. The asaded frequeny-differene proess mixes the ermediate wave with the spatial FT of a po soure illuminated by the same quasi /000/ $ Optial Soiety of Ameria

2 1760 J. Opt. So. Am. B/ Vol. 17, No. 10/ Otober 000 Marom et al. Fig. 1. Femtoseond-rate spae-to-time onversion setup based on nonlinear wave mixing with nonollinear type II asaded seondorder nonlinearities in the Fourier-domain plane of the temporal and spatial hannels. A frequeny-sum proess between waves U 1 and U gives rise to the wave U. A frequeny-differene proess between the waves U and U 3 generates the desired output wave U 4. monohromati plane wave. The seond spatial wave ontains no information, in either the spae or the time domain, and its funtion is to downonvert the arrier frequeny of the ermediate wave generated by the first nonlinear proess. The resultant field in the CSN proess is the filtered SW, whih is reomposed to yield the synthesized output temporal waveform. In this paper we desribe and analyze in depth the spae-to-time onversion proess employing CSN, ahieving a femtoseond response time and high onversion effiieny. In ontrast to our initial analysis in the time domain, 1 a temporal-frequeny-domain analysis is presented in Setion, in whih we develop the expression for the synthesized temporal waveform under the assumption of an ideal CSN proess with perfet phase mathing. In Setion 3 we investigate the effets of the phase-mismath terms in the asaded proesses and haraterize them with a omplex spatial temporal transfer funtion. The derivation of the phase-mismath terms dependene on the temporal- and spatial-frequeny bandwidths is deferred to the appendix. Experimental results verifying the findings of our analysis are presented in Setion 4, illustrating the ability to onvert both amplitude and phase information with this tehnique. We onlude with a summary and disussion in Setion 5.. ANALYSIS OF THE SPACE-TO-TIME CONVERTER The nonlinear eration between the SW of the input ultrashort pulse and the quasi-monohromati waves at the Fourier plane of the optial setup is analyzed in this setion under the following assumptions: (i) The nonlinear wave-mixing proess is weak, suh that the amplitude of the generated wave is proportional to the produt of the omplex amplitudes of the mixed waves. (ii) The erating waves are phase mathed for all temporal frequenies ontained in the bandwidth of the short pulse and spatial frequenies of the spatial image. We defer the justifiation and disussion on the validity of this last assumption to Setion 3, in whih a omplex filtering response funtion is rodued to haraterize the effet of phase mismath on the onversion proess. Sine the spae-to-time onversion ours for one-dimensional spatial signals, the y-axis information is omitted. In the following, we first analyze the input temporal- and spatialinformation hannels, followed by an analysis of the CSN wave-mixing proess giving rise to the synthesized temporal waveform. A. Input Optial Channels An ultrashort optial pulse, with a temporal envelope waveform of p(t), is utilized at the input temporal hannel of the SP. The pulse is propagating in free spae toward a diffration grating, at an angle relative to the grating normal (see Fig. 1). The input ultrashort pulse is haraterized in the propagating pulse s oordinate system (x 1, z 1 )as E pulse x 1, z 1 ; t w 1 x 1 p t t 0 z 1 exp j 0 z 1 0 t, (1) where w 1 ( ) defines the transversal field distribution or the spatial mode of the pulse, is the speed of light in vauum, 0 is the enter optial frequeny, and t 0 is an arbitrary time referene. The pulse is propagating in the z 1 diretion at a group veloity and a phase veloity of the speed of light. We assume the spatial mode size of the pulse is suffiiently large to ignore diffration effets in the region of erest near the grating for the representation of Eq. (1) to be valid. Sine we are assuming that the nonlinear wave mixing of the spae-to-time onverter operates in the linear regime, the signal analysis may be performed in the temporal-frequeny domain. We Fourier transform (FT) Eq. (1) to perform a temporal-frequeny deomposition of the short pulse, yielding Ẽ pulse x 1, z 1 ; w 1 x 1 exp j z 1 p 0 exp j 0 t 0, ()

3 Marom et al. Vol. 17, No. 10/Otober 000/J. Opt. So. Am. B 1761 where the tilde oversript denotes a FT, e.g., p () p(t)exp( jt)dt. We perform a rotation of the oordinate system from (x 1, z 1 )to(x, z) for ompatibility with the oordinate system of the SP, yielding Ẽ pulse x, z; w 1 z sin x os exp j z os x sin p 0 exp j 0 t 0. (3) To find the inident field on the input grating of the SP, the field of Eq. (3) is evaluated at z 0. The effet of the grating diffration an be modeled by adding the grating momentum, k g, in the x diretion to the k vetor that haraterizes the propagation diretion of the wave, resulting in Ẽ input x, 0; w 1 x os exp j sin k g x p 0 exp j 0 t 0. (4) The grating momentum, k g, and the inidene angle,, are hosen suh that the enter frequeny 0 will diffrat in the diretion of the optial axis of the system. Setting 0 sin / k g and substituting the grating s k vetor, k g /, where is the grating period, yields sin 0 /, where 0 is the enter wavelength. To further simplify the notation, we define a new spatialaperture funtion suh that w(x) w 1 (x os ). Without loss of generality we also shift the input aperture of the beam from the optial axis of the system by /. This is done to satisfy the nonollinear beam-propagation requirement in the Fourier plane of the SP for phase mathing in the CSN proess. The input temporalhannel field to the optial proessor, u t1, is therefore haraterized by ũ t1 x; w x x exp j 0 p 0 exp j 0 t 0. (5) We may inverse FT Eq. (5) to generate the input signal in the time domain, yielding u t1 x; t w x p t t 0 x expj 0 t. (6) Equation (6) desribes the short pulse sanning aross the fixed aperture at a veloity of / in the x diretion. For onveniene we may wish to set the time-delay parameter suh that the traveling pulse is at the enter of the aperture at t 0. This is ahieved by setting t 0 /. The temporal-hannel field of Eq. (5) is spatially Fourier transformed by a lens of foal length f, yielding the SW U 1 of the short pulse, U 1 x; p 0 exp j 0 t 0 w x x exp j 0 exp j xx f dx, (7) where we defer solving this Fourier egral to the nonlinear wave-mixing analysis of Setion 3. The two spatial hannels, one ontaining the enoding information and the seond a po soure, are modeled next. A mask ontaining spatial-domain information is plaed in the input plane of the proessor, alongside the diffration grating used for the input temporal hannel. The mask is shifted from the optial axis by / to satisfy the requirement of nonollinear beam propagation in the Fourier plane of the SP. The information mask of the spatial hannel is illuminated with a quasimonohromati light soure at enter frequeny 1. The first input spatial hannel of the proessor, u s1, an be expressed as u s1 x, z 0; t m x expj 1 t, (8) where m(x) is the spatial-information mask. The first input spatial-hannel field is spatially Fourier transformed by the lens, yielding the field U, whih we express in the temporal-frequeny domain by taking the temporal FT, generating U x; 1 exp j xx m x 1 fdx, (9) where ( ) is the ira delta funtion. The spatial wave ontains no temporal-frequeny bandwidth owing to our illumination with a quasi-monohromati light soure. The seond spatial hannel onsists of a po soure at the input plane that is illuminated by the same quasimonohromati light soure as the first spatial hannel. The po soure is shifted from the optial axis by /, as the first spatial hannel has been. A polarizationseletive beam splitter (see PBS in Fig. 1) is used for effiient superposition of the two spatial hannels neessary for the wave-mixing proess. The seond input spatialhannel field to the proessor, u s, an be expressed as u s x; t x expj 1 t. (10) The seond input spatial-hannel field is also spatially Fourier transformed by the lens, yielding the field in the Fourier plane U 3, expressed in the temporal-frequeny domain as

4 176 J. Opt. So. Am. B/ Vol. 17, No. 10/ Otober 000 Marom et al. U 3 x; 1 exp j xx x 1 fdx. (11) opropagate. The eration of U 3 and U gives rise to a fourth wave U 4 at the output of the nonlinear rystal, whih is our signal of erest, given by U 4 x; eff U x; U 3 *x; d We next examine the spae time onversion proess by nonlinear wave mixing the three waves defined in Eqs. (7), (9), and (11). B. Four-Wave Mixing by the Casaded Seond-Order Nonlinearity Proess A nonlinear rystal exhibiting a large nonlinear suseptibility () is plaed at the Fourier plane of the SP (see Fig. 1). The two optial waves, U 1 and U, from the input temporal hannel and the first spatial hannel erat within the rystal, giving rise to an ermediate wave U in a frequeny-sum proess. The type II nonollinear phase-mathing ondition is satisfied by (i) adjusting the polarization diretions of the temporal and spatial hannels to oinide with the rystal s ordinary and extraordinary diretions, respetively, and (ii) setting the erating input waves to propagate nonollinearly. The seond ondition is satisfied by the spatial separation of the input hannels, resulting in a propagationdiretion-differene angle of /f between U 1 and U. The omplex amplitude of the generated ermediate wave is proportional to the nonlinear polarization arising from the two fundamental waves, yielding U x; eff U 1 x; U x; d eff U 1 x; 1 U x; 1, (1) where () eff is the effetive nonlinear-suseptibility oeffiient at the propagation diretions of the erating waves, and we assume it to be independent of in the temporal-frequeny bandwidth of erest. The onvolution egral desribing the nonlinear polarization is trivial to solve sine U has no temporal-frequeny bandwidth. The ermediate wave is therefore shifted up in frequeny by 1, suh that U is osillating at a enter frequeny of 0 1 and is polarized in the extraordinary diretion owing to the type II eration. Next, we onsider the eration of the third wave U 3 with the other waves in the nonlinear rystal. Sine the rystal exhibits type II phase mathing, there is no eration between the wave from the temporal hannel U 1 and the wave from the seond spatial hannel U 3, as they are both polarized in the ordinary-axis diretion of the rystal. The ollinear waves from the two spatial hannels produe a seond-harmoni wave, but it is of no erest for our study of the spatial temporal proessor. The signal of erest is generated from the eration of the wave from the seond spatial hannel U 3 and the ermediate wave U generated in the frequeny-sum proess. These two waves are orthogonally polarized and satisfy the nonollinear phase-mathing ondition for a frequeny-differene generation proess. The nonollinear phase-mathing ondition is automatially satisfied beause U and U 3 have the same optial frequeny and eff U x; 1 U 3 *x; 1 eff U 1 x; U x; 1 U 3 *x; 1. (13) The nonlinear polarization has the form of a orrelation egral in the frequeny-differene proess. This egral is again trivial to solve, as the seond spatial hannel is also quasimonohromati. The output wave is equivalent to a four-wave mixing proess ahieved by asaded seond-order nonlinearities. The resultant wave, U 4,is the SW of the synthesized waveform with enter frequeny 0, generated by filtering the input SW U 1 by a spatially modulated wave. It is opropagating with U 1 and polarized in the extraordinary diretion (as U is extraordinary and U 3 is ordinary). Polarization optis may be used to separate the opropagating waves U 1 and U 4 (see Fig. 1). The generated field U 4 is spatially Fourier transformed by a lens of foal length f, yielding the optial field on the output diffration grating, given by u t x; U 4 x; exp xx j dx x f p 0 exp j 0 t 0 x 1 w x 1 x exp 1 j 0 x m x x 3 x 3 exp jx x x f x 1 f x 1 f x 3 1 f dxdx 1 dx dx 3. (14) Integrating over the variables x, x, and x 3 yields u t x; p 0 exp j 0 t 0 x 1 w x 1 m 1 x x 1 x exp 1 j 0 dx 1. (15) Assuming 0 for typial ultrashort pulses, we an also eliminate the wavelength dependene in the enoding mask by replaing with 0. It is possible to ondut the analysis without this assumption, resulting in weakly hirped output signals. 14 We next perform a hange of variable by defining a new egration variable,, suh that x x 1 /. The resultant field on the grating is given by

5 Marom et al. Vol. 17, No. 10/Otober 000/J. Opt. So. Am. B 1763 u t x; p 0 exp j 0 t 0 x exp j 0 w x m 1 0 exp j 0 d. (16) The effet of diffration from the output grating of the SP is rodued o Eq. (16), yielding u out1 x; p 0 exp j 0 t 0 exp j 0 gx k w m x 1 0 exp j 0 d. (17) The momentum of the diffration grating, k g 0 /, anels the angular-dispersion term of the output field, suh that all the frequeny omponents of the short pulse opropagate at an angle relative to the optial axis. To propagate away from the grating plane, we rerodue the longitudinal dependene as appears in Eq. (3) [using the loal z oordinate (see Fig. 1) and negleting a onstant phase fator from the shift from z to z]. We also replae the aperture funtion w( ) by the original spatialmode funtion of the input waveform, w 1 ( ) [see Eq. (4)]. Additionally, we eliminate the / shift in the pupil funtion, whih was rodued to satisfy the phase-mathing ondition in the Fourier plane of the proessor, yielding u out1 x, z, p 0 exp j 0 t 0 exp j x sin z os w 1 x os z sin m os 1 0 exp j 0 d. (18) We perform a rotation of oordinate systems from the optial system s (x, z) to that of the output pulse (x, z ) (see Fig. 1) and take the inverse temporal FT to get the output in spae time oordinates, E out x, z ; t exp j 0 z 0 t w 1 x m os 1 0 p t t 0 z d. (19) Equation (19) desribes the output temporal pulse as an egral that mixes the information from the input temporal-pulse struture, the spatially enoded mask, and the spatial-aperture funtion. This type of assoiation between the spatial and temporal harateristis has been shown for other ases of ultrafast-waveform proessing. 4,15,16 When the beam size of the input shortpulse signal is very narrow, the desired funtionality will not be ahieved. Therefore in most pratial ases of erest we operate the SP in the high-resolution limit. In this ase, the spatial-mode extent (or, more preisely, time-of-flight duration) w 1 ( os /) is muh longer than the duration of the ultrashort pulse p(). Thus the egral an be approximated by simply evaluating the input pupil at t t 0 z /, whih is the enter oordinate of the short pulse p(t), and taking it out of the egral, yielding E out x, z ; t exp j 0 z 0 t w 1 x os t t 0 z y t t 0 z, (0) where y(t) is defined by the onvolution of the input pulse and a time-saled enoding mask, suh that yt m 1 0 pt d m 1 0 t pt. (1) Equation (0) desribes an ultrafast waveform propagating in free spae in the z diretion. The temporal harateristi of the output waveform, y(t), is determined by the onvolution operation of Eq. (1). The spatial mode of the output waveform is now spae time dependent, as is harateristi in filtering experiments of spatially dispersed temporal-frequeny omponents. 17,18 This dependene travels along with the output waveform and represents a linear skew of the spatial mode relative to the time position of the output waveform y(t). Thus there may be variations of the observed waveform at different spatial loations, a phenomenon that may influene po proesses suh as detetion, spatial filtering, and oupling of the output signal o a single-mode fiber. We next analyze the onversion proess in the nonlinear rystal, taking o aount the effet of the phasemismath terms of the CSN. 3. CASCAE SPATIAL TEMPORAL WAVE MIXING The real-time spatial temporal wave mixing is performed by asaded femtoseond-rate parametri erations in a rystal exhibiting a strong () oeffiient. The CSN proess uses a ombination of frequeny uponversions and frequeny downonversions and has to satisfy the onditions of energy and momentum onservation for effetive energy transfer. Owing to the broad temporal bandwidth of ultrashort pulses and the broad angular

6 1764 J. Opt. So. Am. B/ Vol. 17, No. 10/ Otober 000 Marom et al. bandwidth of the spatial-information hannel, it will be pratially impossible to satisfy the phase-mathing requirement neessary to ahieve high onversion effiieny. Therefore nonuniform signal onversion aross the temporal and spatial bandwidths will be observed, as phase mathing annot be supported in every situation. In this setion we fous on the analysis of the nonlinear onversion proess in the rystal. To simplify the analysis, we assume that the generated waves in the three-wave mixing proesses of the asade are operating in the linear-onversion regime (i.e., weak eration leading to nondepleting fundamental waves). The linear eration allows us to deompose the waves to their temporal- and spatial-frequeny domains, to treat eah frequeny omponent individually, and to egrate over the temporal and spatial bandwidths to get the output waveform. By deomposing the waves, we may use the monohromati plane-wave solution of the wavemixing proess. The plane-wave solution will be valid as long as the nonlinear rystal is within the onfoal parameter of the SW. Sine we are erested in highresolution proessing, whih has a short onfoal parameter, the allowable rystal length will be limited, thereby reduing the onversion effiieny. In general, there will be a trade-off between the resolution and the onversion effiieny, whih is not within the sope of this study. We express the spatially and temporally deomposed input fields inside the nonlinear rystal, followed by a solution to the oupled-wave equations under the nondepletingpump approximation. We alulate the phase-mismath terms in the appendix. A. eomposed Temporal and Spatial Fields Inside the Nonlinear Birefringent Crystal The spetrally deomposed field from the input temporal hannel is found by evaluating the Fourier egral of Eq. (7), yielding U 1 x; p 0 w x f exp j f x 0 exp j 0, () where the spatial FT of the input-pupil funtion, w ( ), has mixed spae- and temporal-frequeny variables, haraterizing the spatial dispersion, and we use t 0 /. The spatial width of w also defines the onfoal parameter, whih limits our rystal length. The spatial linear phase term in Eq. () determines the propagation diretion, owing to the spatial shift of the input pupil by /. We denote the angle between the z axis and the k vetor of the propagating wave as 1 / f, using the paraxial approximation. The SW is inident upon the nonlinear rystal in the FT plane of the lens. At the rystal erfae, loated at z 0 (see Fig. 1), the wave is refrated o the rystal in aordane to Snell s law. The SW in the rystal, when negleting the Fresnel refletion at the boundary and aounting for field attenuation in a higher-index material, an be expressed as Û 1 x; p 0 w no x f exp j ˆ n o f x 0 exp j 0, (3) () where n o is the refrative index for ordinary-polarized light at frequeny, and the hat oversript denotes variables inside the rystal (e.g., ˆ /n () o ). Note that the linear phase terms of Eqs. () and (3) are equal in magnitude, owing to the onservation of momentum in the x diretion (i.e., kinemati ondition). The propagation diretion inside the rystal is frequeny dependent, owing to the dispersive nature of the refrative index of the rystal. The mask ontaining the spatial-domain information generates the field U in the FT plane, found by evaluating the Fourier egral of Eq. (9). For the purpose of this analysis we model the spatial-information hannel by (x ), a ira delta funtion shifted from the enter of the information mask by the parameter. The synthesized waveform from an arbitrary mask m(x) is alulated by multiplying the delta-funtion response by m() and egrating over the parameter (yielding the onvolution egral). By use of the delta-funtion model the first input spatial hannel generates an extraordinarypolarized plane wave with a single spatial frequeny, U x, ; 1 exp j 1 f fx. (4) The angle between the z axis and the propagation diretion of this wave is denoted by () (/ )/f. The refrated field in the rystal, Û, preserves the momentum in the x diretion and attenuates the field by the extraordinary refrative index. Evaluating the propagation angle and k vetor is ompliated by the refrativeindex dependene on the propagation diretion and is performed in the appendix. The seond spatial hannel, onsisting of the delta funtion at x /, generates an ordinary-polarized plane wave in the Fourier plane of the proessor. Evaluating Eq. (11) and aounting for the refration o the rystal yields Û 3 x; 1 1 exp n 1 o j ˆ 1 n 1 o x f, (5) ( where n 1 ) o is the ordinary refrative index at frequeny 1. The angle between the z axis and the propagation diretion of the field before refration is denoted by 3 / f. The nonlinear eration of the input fields Û 1, Û, and Û 3 is analyzed next.

7 Marom et al. Vol. 17, No. 10/Otober 000/J. Opt. So. Am. B 1765 B. Solution of Coupled-Wave Equations Charaterizing Casaded Proesses The general solution to the oupled-mode equations governing nonollinear three-wave proesses an be very omplex. 19 When we limit the analysis to weak erations with nondepleting input waves and slowly varying funtions in spae and time (whih holds for the SW), an analyti solution to the proess is possible. Owing to the nondepleting-wave approximation, only two differential equations remain, whih desribe the evolution of the ermediate wave, Û, generated by a frequeny-sum proess, and the resultant output wave, Û 4, generated by a frequeny-differene proess. Geometrial fators are rodued to the oupled equations owing to the nonollinear arrangement (where we ignore the effet of the walk-off angle between the k vetor and the Poynting vetor in the propagation of extraordinary-polarized fields in an anisotropi rystal). We denote the angle between the z axis and the k vetor of the ermediate wave as ˆ and the angle between the z axis and the k vetor of the output field as ˆ 4 (both defined inside the rystal). Both these angles vary as a funtion of and (the dependene is evaluated in the appendix). Negleting absorption in the rystal, the differential equations governing wave mixing of monohromati plane waves are given by 0 equations desribing the phase mismath in the wavemixing proesses are nonlinear. To simplify the phasemismath representation, we perform a small-signal analysis about 0 and 0 and linearize the relationship. The details of the linearization proess and evaluation of the two phase-mismath terms are presented in the appendix. A simple analyti solution to the oupled differential equations an be obtained by arguing that sine the generated wave Û 4 is weaker than the input waves, we an neglet the seond nonlinear polarization term in Eq. (6a) (i.e., Û 1 Û Û 3 Û 4 ). This assumption deouples the differential equations, implying that the rate at whih the ermediate wave is depleted by the seond proess of the asade is muh smaller than the rate at whih it is being generated. Therefore Eq. (6a) of the ermediate field is dependent on the two known input fields and an be solved diretly. This solution is inserted o Eq. (6b) desribing the evolution of the output wave, whih an also be solved diretly now, as the driving waves are ompletely haraterized. The solution yields dû x,, z; dz j 0 1 osˆ d Û 1 x, z; 1 Û x,, z; 1 exp jk 1, z Û 3 x, z; 1 Û 4 x,, z; 1 exp jk, z, (6a) dû 4 x,, z; dz 0 j dû x,, z; 1 osˆ 4 4 Û 3 *x, z; 1 expjk, z, (6b) where d is the seond-order nonlinear optial oeffiient. The right-hand side of the equations have been simplified by substituting the enter-frequeny value for the propagation-diretion angles, ˆ and ˆ 4, the eletri permittivity and magneti permeability, and, and the optial arrier in the term preeding the nonlinear polarization. The phase-mismath terms for the frequenysum and frequeny-differene proesses, k (1) (, ) and k () (, ), respetively, are given by k 1, kˆ 1 kˆ kˆ,, (7a) k, kˆ 3 kˆ, kˆ 4,. (7b) Figure illustrates a graphial representation of the phase-mismath terms of Eqs. (7a) and (7b). The Fig.. Graphial representation of the phase mismath in nonollinear mixing in an anisotropi rystal for (a) the uponversion proess and (b) the downonversion proess.

8 1766 J. Opt. So. Am. B/ Vol. 17, No. 10/ Otober 000 Marom et al. Û 4 x,, z; Û 1 x; Û x, ; 1 Û 3 *x; 1 H NL,, z, (8) where the omplex transfer funtion, H NL (,, z), haraterizes the response of the asaded wave-mixing proess and is given by exp H NL,, z jk1, k, z 1 k 1, k, k 1, expjk, z 1. (9) k 1, k, The omplex transfer funtion of Eq. (9) depends on the temporal-frequeny omponent of the input pulse, on the spatial frequeny arising from the amount of shift of the delta funtion in the enoding mask, and on the length of the nonlinear rystal. Sine the omplex transfer funtion is temporal-frequeny dependent, the output waveform will not be transform limited. However, any predominant quadrati-phase term an be eliminated by longitudinal adjustment of the output grating. Reroduing the enoding-mask information, the SW of the synthesized waveform at the edge of the rystal is defined by Û 4 x, L ; Û 1 x; Û 3 *x; 1 mû x, ; 1 H NL,, L d, (30) where we replaed the eration distane z by the thikness of the rystal, L. To find the output temporal waveform, Eq. (30) needs to be spatially Fourier transformed to the output plane, diffrated from the output grating, and inverse temporally Fourier transformed. Equation (30) annot be solved analytially exept for the most trivial ases, thus requiring a omputation program to alulate the preise spae-to-time onverted waveform. Suh a program ould also ompute the onversion effiieny, whih will be information dependent. For pratial reasons it is desirable for the spae-totime proessor to operate effiiently. The onversion effiieny of the wave-mixing proess with CSN an be alulated with Parseval s theorem, as the total power in the spetral representation is equal to the total power in the time domain. Sine Eq. (30) defines the SW of the synthesized waveform, the power spetrum is equal to the spatial ensity distribution at the output fae of the rystal. The power spetrum, or output ensity, an be evaluated analytially for a simple mask suh as m(x) (x), yielding I 4 x; P 1 P P 3 p 0 w x f 0 H NL, 0,L, (31) where we rodue the optial powers P 1, P, and P 3 of the input temporal hannel, the spatial-information hannel, and the spatial referene hannel, respetively, and H NL, 0,L L k 1 sin k 1, 0 k, 0 L sin k, 0 L sin k 1, 0 k, 0 L sin k, 0 L os k1, 0L. (3) The transfer funtion will perform spetral filtering, as is evident from Eq. (31). The onversion effiieny, defined as the ratio of the powers of the output synthesized waveform to the input temporal hannel, depends linearly on the power level of eah spatial pump hannel. Our findings were experimentally validated and are presented in the following setion. 4. EXPERIMENTAL RESULTS OF SPACE-TO- TIME CONVERSIONS The operation of the spatial temporal proessor of Fig. 1 was verified by synthesizing various ultrafast temporal waveforms. In the experiments we used a ommerial laser system onsisting of a Ti:sapphire ultrashort pulse osillator and a regenerative amplifier. The system generates ultrashort laser pulses of 100-fs duration at a enter wavelength of 800 nm with an energy level of 1 mj per pulse. For the ense quasi-monohromati light soure required by the spatial hannels we used 90% of the emitted output-pulse power and strethed the pulse to a several-pioseond duration by a grating pair. The strethed pulse was split o two beams for implementing the two quasi-monohromati spatial hannels. Sine the CSN proess ours with a femtoseond-sale time response, the slow temporal-frequeny variation in the strethed pulses does not affet the wave-mixing proess, as long as the temporal frequenies of the two spatial hannels are instantaneously equal. The remaining 10% of the short-pulse laser output power was used as the referene ultrashort pulse in the input temporal hannel. The SW U 1 is generated by a 600-lines/mm blazed grating that provides an angular-dispersion parameter of 0.48 and a lens of f 375 mm foal length. The spatial temporal wave mixing by the () media was performed in a -mm-thik type II -barium borate rystal. Several experiments were onduted to illustrate this real-time proessing tehnique, demonstrating its ability to ontrol amplitude and phase in the output temporal waveform by a omplex-amplitude spatial-information hannel. All the synthesized waveforms were observed with a real-time pulse-imaging setup. 1 The pulse-imaging tehnique generates a spatial signal at the output plane that is proportional to the temporal onvolution of the synthesized waveform and a referene ultrashort pulse. The referene-pulse soure used in the pulse imager was the residual input pulse of the spatial temporal proessor, after separating it from the synthesized output wave-

9 Marom et al. Vol. 17, No. 10/Otober 000/J. Opt. So. Am. B 1767 form with a polarizing beam splitter. We viewed the output signal s ensity with a harge-oupled devie and extrated the temporal information from the image. A. Conversion Effiieny and Spetral Response For initial haraterization of the spatial temporal wavemixing proess by CSN, the two monohromati waves were foused by ylindrial lenses to form line soures at the input spatial hannels. The resulting spatial plane waves U and U 3 erat with the SW, U 1, of an input transform-limited short pulse. Sine the plane waves ontain no information, in either the spae or the time domain, their funtion is to uponvert, then downonvert, the arrier frequeny of the SW, generating by the CSN proess the new SW U 4. The downonverted SW U 4 was generated only in the presene of the waves from the two spatial hannels, verifying that the output SW is produed by a CSN proess. The dependene of the synthesized waveform s output power to the optial power in eah of the spatial hannels was haraterized by plaing variable neutral-density filters in the spatial hannels (see Fig. 3). The linear dependene of the output power to eah spatial hannel is verified by the exellent fit to the 10-dB/de line. When both spatial hannels powers are varied simultaneously, the output power variation fits the 0-dB/de line. The linear dependene onfirms our assumption of the nondepleting pump approximation in Setion 3. However, we observed that the dependene is sublinear when the spatial hannels powers were weak. This effet is more pronouned when only one of the spatial hannels is attenuated. We believe this is due to depletion of the attenuated pump wave, as our analysis of Setion 3 did not take o aount strong eration that may lead to suh fundamental-wave depletion. The maximum onversion effiieny of the spae-to-time proessor, measured as the ratio of the input-pulse optial power to the output-pulse optial power, was 16%, illustrating the ad- Fig. 3. Output power measurements as a funtion of spatial pump-beam powers. Varying one spatial hannel while keeping the seond onstant illustrates the linear dependene of the output temporal power on eah spatial hannel. Varying the power of the spatial hannels simultaneously displays a quadrati dependene, as expeted. Fig. 4. (a) Measured power spetrum of the input pulse (solid urve) and the generated output pulse (dashed urve) exhibiting spetral filtering owing to a spetrally dependent phase mismath in the proessor. Also shown is the theoretial output power spetrum (dot-dash urve) alulated by applying the spetral filter to the measured input power spetrum. (b) Pulse images of the input pulse (solid urve) and the generated output pulse (dashed urve), demonstrating the inreased duration of the output pulse owing to spetral filtering. vantage of the CSN approah as opposed to onventional (3) nonlinearity for four-wave mixing. The power spetrum of the output pulse was ompared with that of the input pulse (see Fig. 4a). The observed spetral narrowing is due to an inreasing phase mismath as the frequenies deviate from the enter frequeny. A theoretial urve of the expeted output-pulse power spetrum [based on Eq. (31)] shows a lose orrespondene to the atual observed spetrum. The theoretial urve was generated by evaluating the phase mismath for the two proesses in the asade with published -barium borate parameters 3 for type II nonollinear eration with a 4 ernal angle between the fundamental waves (assuming both proesses are phase mathed at 0 and 0). We adjusted the enter frequeny of the theoretial urve to math the measured data. Owing to the redution of the available bandwidth from spetral filtering, we observed an aompanying inrease in the temporal duration of the output pulse (see Fig. 4b). The measured full width at half-maximum of the deteted image in the time-to-spae onverter inreased from 18 to 88 fs.

10 1768 J. Opt. So. Am. B/ Vol. 17, No. 10/ Otober 000 Marom et al. B. Paket of Ultrafast Pulses Novel uses for bursts of ultrashort pulses, suh as terahertz-rate exitation and optial ommuniation with ultrashort data pakets, have inreased the erest in developing tehniques for generation of suh ultrashortpulse sequenes. With the femtoseond-rate spae-totime onverter, suh a pulse sequene an be generated by a one-dimensional mask ontaining an array of po soures. The mask information an be expressed by mx N/ A n x nx, (33) nn/ where x is the fixed pith between adjaent po soures, and A n is a salar weight term that may be a binary bit (value of 1 or 0), a real number, or a omplex number. By use of this information mask in Eq. (1), the temporal output signal takes the form yt N/ nn/ A n N/ nn/ A n p t 1 0 nx pt t 1 0 nx. (34) In the spae-to-time mapping, the spatial pith x in the mask has been onverted to a temporal separation of t 1 x/ 0. Various masks ontaining narrow slits have been used in the spae-to-time setup to generate ultrafast pulse pakets. Previously, we reported results using a mask with a 0.8-mm pith. 1 A finer mask with a 0.4-mm pith was used to generate a denser ultrafast pulse paket (see Fig. 5). A ylindrial lenslet array was used to fous the spatial beam o the mask for inreased light throughput. The spatial beam mode shape resulted in nonuniform power on eah spatial slit. This power distribution has been transferred to the ultrafast pulse paket. The spae-to-time mapping oeffiient transforms the 0.4-mm spatial pith to a time-domain separation of t 0.64 ps (with 1 0 and 0.48). The Fig THz-rate ultrashort-pulse paket generated by a spatial-information mask onsisting of a sequene of po soures separated by 0.4 mm. measured temporal separation was ps with a resolution of 33 fs, resulting in a 1.56-THz-rate pulse paket. C. Square Pulse Generation of synthesized square pulses by spetral filtering is not a simple task. The mask should ontain amplitude information over a wide dynami range as well as phase information for implementing negative values. Spae-to-time mapping tehniques simplify the task of omplex mask preparation, while the requirement for a wide dynami range in the reording medium remains. In photorefratives, where the reorded spae-harge field is dependent on the modulation funtion, the reording proess exhibits a nonlinear dependene on the reording beams. While this enables nonlinear proessing, 4 for generation of high-fidelity square pulses a weak information beam must be used, giving rise to shallow modulation depths and low diffration effiieny. In our wave-mixing approah with CSN the amplitude of the synthesized wave is dependent on the amplitudes of the three input waves (as long as wave depletion and phase mismath are negligible). This linear relationship gives rise to high-fidelity onversion from spae to time. In the ase of a square pulse the synthesized signal may be expressed as yt ret t T pt, (35) where T 1 X/ 0 is the duration of the ret funtion and X is the width of the slit. An adjustable slit was plaed in the spatial-information hannel to experiment with several synthesized squarepulse durations. The generated waveforms (see Fig. 6) were modified in real time by hanging the slit width, and the square pulses were immediately observed with the pulse imager on a monitor. At wide slit widths, nonuniformity aross the top of the square pulse has been observed. We have verified that the reason is due to nonuniformity in the illumination of the spatial hannel. An image of the spatial-information hannel in the spae-totime proessor mathed well with the observed image of the synthesized waveform in the time-to-spae onverter. There are two limitations to the maximum slit width that an be used: the resolution of the SP [see Eq. (19)], and the optial power from the tight fous of the spatial hannel in the Fourier plane may exeed the damage threshold of the nonlinear rystal.. Spherial Wave Front To demonstrate the ability to enode phase information, a po soure, generated by fousing the spatial beam with a ylindrial lens, was used as the spatial-information hannel. Translating the lens longitudinally away from the input plane formed a spherial wave front with variable quadrati phase on the input plane of the spatialinformation hannel. As the translation of the line soure from the input plane is inreased (in either the positive or the negative diretion), the output SW aquires a larger positive or negative quadrati phase and after reomposing on the output grating, emerges as a

11 Marom et al. Vol. 17, No. 10/Otober 000/J. Opt. So. Am. B 1769 hirped pulse (either a positive or negative hirp). The measured synthesized hirped pulses exhibited, as expeted, broader pulses along with a redution of the peak ensity. 1 The effet of a longitudinal translation of the po soure away from the input plane of the proessor is expressible as a onvolution with a quadrati-phase funtion 5 (free spae propagation), resulting in the field distribution in the spatial-information hannel, exp jk mx j 1 exp j 1 x, (36) where is the displaement of the fous plane of the ylindrial lens from the input plane of the proessor. The phase information desribed in Eq. (36) gives rise to the synthesized waveform exp jk 1 0t 1 yt expj pt, (37) j 1 whih desribes a hirp funtion (quadrati temporalfrequeny variation) imposed on the input pulse. A Fourier-domain representation of the output synthesized signal is easier to erpret in this ase, yielding ỹ exp j p 0. (38) Fig. 7. Linear orrespondene between translation of the po soure away from the input plane of spae-to-time onverter and translation of the image-formation plane in the pulse imager. The solid line orresponds to a theoretial urve of slope 1/. From Eq. (38) it is lear that the amount of hirp is linearly dependent on the displaement. To measure the amount of hirp that is present in the synthesized waveform, we utilize a unique property of the pulse imager. Sine the pulse imager generates the omplex-amplitude information of the temporal onvolution operation in spatial oordinates, the resulting spatial quadrati phase in the image of a hirped pulse an be used to gauge the amount of hirp. By longitudinally translating the output plane, a loation an be found where the wave fouses to its tightest spot size, eliminating the spatial quadrati phase. 1 Therefore as the pulse hirp is inreased (either positively or negatively), the loation of the plane at whih the wave fouses hanges (either farther or loser). The spatial output signal of the pulse imager an be expressed as 1 bx p x p x y p expj 1 x x 0 1 x, (39) whih orresponds to the onvolution of the ultrashort pulses with a quadrati-phase funtion. Using the kernel of free-spae propagation at frequeny 0, exp j( 0 /)(x /Z), where Z is the displaement from the image plane, we seek the value of Z that will anel the quadrati-phase terms in Eq. (39). This value of Z orresponds to the loation where the signal fouses to its tightest spot size. The value of Z is given by Z 1 0. (40) Fig. 6. Superimposed images of square pulses generated by varying the width of a square aperture in the spatial hannel. This saling rule is dependent on the ratio of the frequenies used in the spatial and temporal hannels of the spae-to-time onverter and a fator of owing to the doubled-frequeny output of the pulse imager. We have verified experimentally the relation in Eq. (40) by finding the loation where the output of the pulse imager fouses, for eah translated loation used in the spatial hannel of the spae-to-time onverter (see Fig. 7). In our experiment the saling rule redues to 1/ sine 1 0. Thus we have a orrespondene between the longitudinal translation of the line soure in the input spatial hannel of the spatial temporal proessor and the longitudinal translation of the fous plane of the pulse imager.

12 1770 J. Opt. So. Am. B/ Vol. 17, No. 10/ Otober 000 Marom et al. 5. CONCLUSION AN ISCUSSION We have rodued and experimentally demonstrated femtoseond-rate spae-to-time onversion of amplitude and phase information from a quasi-monohromati spatial-image hannel to an ultrafast temporal waveform. Relative to other spatial temporal proessing tehniques, the CSN four-wave mixing approah provides femtoseond-rate proessing owing to the fast boundeletron nonlinearity and high effiieny on aount of a relatively large () oeffiient. Suh apabilities are required for quantum ontrol experiments or ultrafast data modulation in optial ommuniations, where rapid, highfidelity waveform manipulations are required. To model limitations imposed by phase-mathing onsiderations in the nonlinear wave-mixing proess, we have rodued a omplex transfer funtion to haraterize the system response. The phase mismath in the wave-mixing proess arises from the broad temporal- and spatial-frequeny ontent of the input temporal-pulse and spatial-image information, respetively. The transfer funtion an be used to ompute the preise spae-to-time onverted waveform and assist in optimal seletion of a nonlinear rystal for real-time pulse shaping. The onversion effiieny from the input referene pulse to the output synthesized waveform was found to depend on the power levels of the two spatial hannels. Several experiments were onduted to illustrate the ability to onvert amplitude and phase information from the spae domain to the time domain with high effiieny. The spetrum redution in the synthesized waveform owing to the omplex transfer funtion was observed and was in good agreement with the predited spetrum. All the experiments were onduted on a single-shot basis, highlighting the benefit of proessing with a fast nonlinear effet. The spae-to-time onversion proess that we have demonstrated uses an idential quasi-monohromati light soure in the two spatial hannels. This requirement is typial for holographi proessing, where the two waves need to be mutually oherent during the long reording phase. Sine our proessor is based on a realtime four-wave mixing proess, this requirement an be alleviated. The neessary ondition for a faithful signal onversion by nonlinear wave mixing is that eah of the spatial hannels be implemented with a monohromati light soure, with no temporal bandwidth. With this ondition the nonlinear polarization term an be trivially alulated [as we have done in Eqs. (1) and (13)]. It is therefore feasible to perform wavelength tuning of the synthesized temporal waveform by use of different optial frequenies in the two spatial hannels. The enter frequeny of the synthesized temporal waveform is shifted by the frequeny differene of the two soures used in the spatial hannels. Sine quasi-monohromati light soures are used in reality, we require that the oherene time of the light soures be muh longer than the time window of the SP (whih is a very simple requirement to satisfy). The propagation diretions of the fields from the two spatial hannels may require adjustment, suh that the phase-mathing ondition is satisfied for the two proesses of the asade. The hypothesized wavelengthtuning property of the real-time four-wave mixing proessor is unique to our spae-to-time tehnique and annot be aomplished by holographi reording means. APPENIX A: EVALUATION OF PHASE-MISMATCH TERMS The output signal of the CSN proess was shown in Subsetion 3.B to depend on the phase-mismath terms in the oupled-wave equations haraterizing the generation of the new fields. The phase mismath ours sine the phase-mathing requirement for effiient energy transfer annot be supported for the temporal bandwidth of the SW and the angular bandwidth of the spatialinformation hannel. In this appendix we derive the phase-mismath terms for the two proesses of the asade, using the information-hannel model of a delta funtion shifted by an amount. The phase-mismath terms obtained in this appendix are inserted o Eq. (9), desribing the omplex transfer funtion of the proessor. The first nonlinear eration of the asade, the frequeny-sum proess, generates the ermediate wave, Û. (Note that all properties of the waves inside the rystal are with the hat oversript.) The phase mismath of the first proess of the asade is defined by the vetor equation, Eq. (7a), where the phase mismath must lie in the z diretion owing to the kinemati ondition (onservation of momentum in the x diretion). Sine the erating waves are propagating at very small angles relative to the z axis, we an use the paraxial approximation (note that Fig. shows greatly exaggerated angles), simplifying the phase-mismath expression to the sum of the k-vetor magnitudes, i.e., k (1) (, ) kˆ 1() kˆ () kˆ (, ). We next find these k-vetor magnitudes for evaluation of the first phase-mismath term. The first input wave, Û 1, from the input temporal hannel, is polarized parallel to the rystal s ordinary axis. Therefore the wave-vetor magnitude is independent of its propagation diretion and is given by kˆ 1() k( 0 ) (k/)( 0 ). To find the wave-vetor magnitude of the ermediate wave, Û, and its propagation diretion, we apply the kinemati ondition using the paraxial approximation, yielding kˆ, ˆ, kˆ 1ˆ 1 kˆ ˆ k 1 1 k, (A1) where the seond equality results from invoking the kinemati ondition on the input fields aross the rystal erfae. (Before refration o the rystal, 1 is not frequeny dependent, as all the omponents propagate in the same diretion [see Eq. ()], and the magnitude of k does not vary with the shift of in, as it is a quasimonohromati wave.) To analyze the effet of varying the parameters 0 and, we perform a smallsignal perturbation about 0 and 0 by defining the following: kˆ, kˆ kˆ, ˆ, ˆ ˆ,

13 Marom et al. Vol. 17, No. 10/Otober 000/J. Opt. So. Am. B 1771 k 1 0 0, f f, (A) kˆ ˆ ˆ 1 n 1 e ˆ ˆ 1 n n ˆ ˆ where kˆ and ˆ are the k-vetor magnitude and angle of propagation, respetively, for the ermediate wave evaluated at 0 and 0. Inserting the perturbed expressions o Eq. (A1), negleting seond-order terms of the form kˆ ˆ, and simplifying the expression by removing the onstant terms that satisfy the equality on both sides of the equation yields 1 f f, (A6) where ˆ is the angle between the propagation diretion and the z axis for no spatial shift of the delta funtion. By negleting seond-order terms of the form n ˆ, and expressing the index hange as n (dn e ( 1 ) /d)ˆ, we find kˆ ˆ kˆ ˆ f 1 f, (A3) where we used 1 / f and k 1 /. We an further simplify the expression by setting kˆ (kˆ /) (kˆ /)ˆ, resulting in 1 f v g ˆ kˆ ˆ ˆ kˆ kˆ 1 f, ˆ kˆ (A4) where we used the definition of the group veloity, v g (kˆ /) 1. Using this result, we an express the k vetor of the ermediate wave as kˆ, kˆ v 1 g kˆ 1 f v g kˆ ˆ kˆ ˆ f ˆ, (A7) dn 1 e n ˆ d ˆ where the derivative of the refrative index an again be alulated. 6 The k vetor of the field is given by 1 (n n )/, yielding k 1 n 1 e ˆ 1 dn e 1 d ˆ f. (A8) dn 1 n 1 e e ˆ ˆ d ˆ 0 kˆ kˆ 1 ˆ f. kˆ (A5) The term kˆ / is evaluated by ( 0 1 )/ ( (n 0 1 ) e /), where the rate of hange of the extraordinary refrative index at angle ˆ an be found by differentiating the equation governing the refrative-index dependene on angle of an anisotropi rystal. 6 Next, we find the k-vetor properties of the wave from the spatial-information hannel, Û, whih is also polarized in the diretion of the extraordinary axis. Sine this wave is quasimonohromati, the propagation diretion inside the rystal determines the wave-vetor magnitude. We find the propagation diretion by using the kinemati ondition aross the rystal erfae together with a small-signal perturbation, yielding Fig. 8. Illustration of phase mismath owing to broad temporal bandwidth (spatially dispersed along x) and angular bandwidth. The vetor sum of the two fundamental waves generates a k-vetor front that is ompared with the available k-vetor front of the frequeny-sum wave. Exat phase mathing is possible at eah temporal frequeny with different omponents of the spatial angular bandwidth.