Femtosecond pulse imaging: optical oscilloscope

Transcription

1 Sun et al. Vol. 14, No. 5/May 1997/J. Opt. So. Am. A 1159 Femtoseond pulse imaging: optial osillosope ultrafast P. C. Sun, Y. T. Mazurenko,* and Y. Fainman Department of Eletrial and Computer Engineering, University of California at San Diego, La Jolla, California Reeived July 30, 1996; aepted November 4, 1996; revised manusript reeived Deember 12, 1996 A nonlinear optial proessor that is apable of real-time onversion of a femtoseond pulse sequene into its spatial image is introdued, analyzed, and experimentally haraterized. The method employs nonlinear spetral domain three-wave mixing in a rystal of LiB 3 O 5, where spetral deomposition waves of a shaped femtoseond pulse are mixed with those of a transform-limited pulse to generate a quasi-monohromati seond-harmoni field. By means of this nonlinear proess, the temporal-frequeny ontent of the shaped pulse is diretly enoded onto the spatial-frequeny ontent of the seond-harmoni field, produing a spatial image of the temporal shaped pulse. We show that, unlike the ommonly used autoorrelator, suh time-tospae onversion arries both amplitude and phase information on the shape of the femtoseond pulses Optial Soiety of Ameria [S (97) ] 1. INTRODUCTION Ultrashort-pulse laser tehnology has reently experiened signifiant advanes, produing high-peak-power pulses of optial radiation a few femtoseonds in duration, orresponding to only a few yles of its fundamental frequeny. 1 4 The future progress of this area is inevitable beause of the unique properties of ultrashort laser pulses that are ruial for various siene and engineering appliation areas inluding optial ommuniations, 5,6 medial and biomedial imaging, 7 11 hemistry, and physis A ommon feature of these appliations relies on our ability to ontrol the shape of the ultrashort pulses as well as our ability to detet the shape of the ultrashort pulses that an be seen as an ultrafast osillosope. In this paper we desribe a new tehnique for imaging of femtoseond optial pulses based on spetral domain three-wave mixing in nonlinear optial rystals. This tehnique will be disussed in onjuntion with terabit optial ommuniation network appliations, although our results are general and useful for various other appliations. The bandwidth and the effiieny of fiber-opti ommuniation systems exeed those of eletrial able systems. However, urrently, we are far from realizing the potential performane of optial networks. Eletroni devies and systems onneted to optial networks may reah bit rates of the order of 1 10 Gbit/s. In ontrast, the maximum bit rate of a photoni network may exeed 1 Tbit/s. The 2 3-order-of-magnitude mismath between fiber and eletroni devie apaity an be used to inrease speed, seurity, and reliability in the data transmission. To implement these appliations, it will be neessary to onstrut an all-optial preproessor at the transmitter and a postproessor at the reeiver, whih will perform multiplexing and demultiplexing, respetively. The multiplexer performing the spae-to-time transformation will ombine relatively slow parallel eletroni hannels into an ultrahigh-bandwidth serial fiber-opti hannel (i.e., parallel-to-serial onversion), whereas the demultiplexer will perform the inverse time-to-spae transformation for eletroni detetion (i.e., serial-to-parallel onversion). For effiient bandwidth utilization these proessors need to be operated at rates determined by the bandwidth of the optial pulses. Suh spae time optial proessors have been onstruted and applied for pulse shaping, filtering, 20,21 and spae-to-time multiplexing 22,23 and time-to-spae demultiplexing There exist various tehniques to implement the multiplexer. For example, pulse-shaping devies have been used to modify the temporal shape of a femtoseond optial pulse by temporal spetral filtering ,19 In priniple, suh a multiplexer an be operated with fixed or programmable masks that need to hange at rates of the order of the time aperture of the system (i.e., nanoseond rate). For demultiplexing, both three-wave and fourwave mixing tehniques have been demonstrated. The initial experiments of pulse-shape onversion based on the nonlinear optial three-wave mixing were demonstrated experimentally for nanoseond 26,27 and pioseond 28 pulses, whereas in this paper we introdue, analyze, and experimentally demonstrate an all-optial proessor used for imaging of femtoseond pulses. We also show that this method is apable of providing not only amplitude but also phase information on the femtoseond pulse. In the next setion we will briefly review the spetral domain pulse-shaping devie that will be used to produe the input shaped pulses for our femtoseond pulse imaging system. In Setion 3 we will analyze the femtoseond pulse imaging tehnique based on spetral domain threewave mixing and use a few speifi examples to show that this method provides not only the shape but also the phase information on femtoseond pulses. Setion 4 will introdue and desribe the experimental apparatus that /97/ $ Optial Soiety of Ameria

2 1160 J. Opt. So. Am. A/Vol. 14, No. 5/May 1997 Sun et al. we have onstruted and applied for imaging of femtoseond pulses, whereas in Setion 5 we will report experimental results on imaging femtoseond pulses by using a few speifi examples. In the last setion we will provide a summary. 2. SPECTRAL DOMAIN PULSE-SHAPING DEVICE The spetral domain pulse-shaping tehnique is desribed by a shemati diagram shown in Fig. 1. The devie is based on spetral deomposing a transform-limited laser pulse into the spetral domain, modifying the pulse spetrum by using a one-dimensional (1-D) fixed or programmable spatial mask or hologram, and reombining the modified temporal spetrum bak into the time domain to form a shaped pulse. Consider an input pulse and an output pulse u i t ū i texp j t u s t ū s texp j t, (1a) (1b) where is the enter arrier frequeny of the input pulse, t is the time oordinate, and ū i (t) and ū s (t) are the omplex shape funtions of the input and the output pulse, respetively. The spatial mask introdued in the spetral deomposition plane of Fig. 1 will at as a temporal filter with a transfer funtion M ( ), with its inverse Fourier transform orresponding to the time domain impulse response mt m texp j t M exp jtd. (2) Under the paraxial approximation the output pulse from the pulse-shaping devie of Fig. 1 an be desribed by a onvolution between the input pulse and the time domain impulse response of the pulse-shaping devie, 15,16 given by u s t u i t mt Ū i M exp jtd Ū s exp jtd, (3) where denotes the onvolution operation and we have used the Fourier transform pairs Ū i ū i texp j texpjtdt, (4a) Ū s ū s texp j texpjtdt, (4b) where here and heneforth we neglet onstant terms in our derivations. It should be noted that Eq. (3) was derived under the assumption that the transverse spatial extent of the shaped pulses is very large, and therefore we an neglet the effet of the spatially dependent spetral Fig. 1. Shemati diagram of the pulse-shaping devie with a spetral domain spatial filter. Fig. 2. rystal. Shemati diagram of the femtoseond pulse imaging system based on nonlinear spetral domain three-wave mixing in a LBO

3 Sun et al. Vol. 14, No. 5/May 1997/J. Opt. So. Am. A 1161 Fig. 3. Illustration of (a) spetral deomposition waves (SDW s) and (b) orresponding time-dependent wave vetors of a transformlimited pulse in the Fourier plane of the pulse imaging system. dispersion as desribed in Ref. 29. The output of the pulse-shaping devie (see Fig. 1) desribed by Eq. (3) will be used in our analysis and experimental demonstration of the femtoseond pulse imaging tehnique introdued in the next setion. 3. FEMTOSECOND PULSE IMAGING METHOD The femtoseond pulse imaging tehnique is being investigated for quantitative visualization of a shaped pulse in real time. The method performs time-to-spae onversion by employing spetral domain three-wave mixing in a nonlinear optial rystal. The shemati diagram explaining the priniple of operation of the femtoseond pulse imaging tehnique is shown in Fig. 2. For detailed analysis we will examine the optial field distribution at eah stage in the proessor. We will onsider in our analysis only spatial (x) and temporal (t) oordinates and their orresponding Fourier transform oordinates in the spatial-frequeny (f x ) and temporal-frequeny () domains. Consider a shaped pulse produed by the pulseshaping devie to be inident onto the first refleting grating of the pulse imaging system of Fig. 2 at an inidene angle [see also Fig. 3(a)]. The grating is arranged suh that the first diffration order for the spetral omponent at the enter arrier frequeny will propagate into the diretion of the optial axis of the system. Therefore, when the shaped pulse arrives onto plane 1, a single spetral omponent of the inident field at an arbitrary frequeny, Ū s ( )exp( j t), will result in a diffrated field given by Ū 1s x;, t Ū s exp jt exp j x wx, (5) where we define sin, w(x) is the pupil funtion of the optial field on the refleting grating, is the speed of light, and the seond exponential term aounts for the grating diffration phenomena experiened by the spetral omponent of the inident wave of frequeny. Considering Fourier optis analysis to aount for the field propagation from plane 1, through the lens, onto foal plane 2, we simply apply the spatial Fourier transform, yielding U 2s f x ;, t Ū s exp jt W f x 2 Ū s exp jt W x 2F 2, (6) where f x x/2f, W(f x ) is the spatial Fourier transform of the pupil funtion w(x), and F is the foal length of the Fourier transform lens in Fig. 3(a). Equa-

4 1162 J. Opt. So. Am. A/Vol. 14, No. 5/May 1997 Sun et al. tion (6) shows that the input optial pulse is spatially dispersed in the Fourier transform plane, where eah spetral omponent of the inident pulse oupies a width equal to the width of the funtion, W(f x ). The optial field in plane 2 is found by integrating the spetra of the fields for eah spetral omponent of the inident shaped pulse, yielding u 2s x; t U 2s x;, td Ū s W x 2F exp jtd. 2 (7) In our derivations we have negleted the small hromati dispersion in the lens material. For simpliity, we assume that the funtion Ū s ( ) is varying slowly in an interval orresponding to the width of the funtion, W(f x ). This assumption is valid for most pratial ases beause the variations in Ū s ( ) are determined by the feature sizes in the spetral filter M() of the pulseshaping devie, while, as a result of the Fourier transform relation, the width W(f x ) is inversely proportional to the pupil size w(x). Therefore, for a system with a large illumination window w(x) on a large-aperture grating, Eq. (7) an be approximated by moving the funtion Ū s ( ) out of the integral while evaluating its value at the enter of the funtion Wx/2F ( )/2, yielding u 2s x; t Ū s x x F W x 2F 2 exp jtd Ū s x x F w tf x F Ft exp j (8a) x F, where the enter of the funtion Wx/2F ( )/2] is F xf. (8b) Under the paraxial approximation, F x, relation (8a) an be further simplified to F x t exp j 1 F x t. (9) u 2s x; t Ū s x F w 1 Relation (9) shows that the frequeny of the field in spetral plane 2 hanges linearly along the x axis. This field is alled the spetral deomposition wave (SDW) of the shaped pulse. Figure 3(a) also shows the time evolution of the SDW wave fronts for a transform-limited optial pulse as the inident shaped pulse, i.e., Ū s (x /F) is real. For a transform-limited pulse the wave fronts of the SDW ontinuously hange along the time axis, with the instantaneous spatial frequeny of the SDW defined by k s x t sx,t x t F, (10) where s (x, t) is the instantaneous phase funtion of the SDW. Figure 3(b) shows the distribution of wave vetors of a transform-limited pulse in time. From relation (9) and Fig. 3(a), we observe that the spatial and temporal shapes of the SDW are determined by the spetral funtion of the pulse, Ū s (), and the illumination window funtion w(x), respetively. Next we onsider a time-delayed referene pulse u r (t t) with the same enter arrier frequeny and a relative time delay t with respet to the shaped pulses. The referene pulse and its orresponding Fourier transform are desribed by u r t t ū r t texp j t, Ū r expjt ū r t texp j t (11a) expjtdt. (11b) The referene pulse is inident onto the grating at an angle, as shown in Fig. 4(a). Following derivation proedures similar to those desribed above, we an alulate the spetral plane field distribution of the referene pulse, yielding u 2r x; t Ū r x F w 1 F x t t exp j 1 F x t t. (12) The SDW of the referene pulse is similar to that of the shaped pulse, but with the temporal frequeny hirped in the opposite diretion on aount of the negative inidene angle. Again, if the referene pulse is transform limited, the instantaneous spatial frequeny is given by k r x t rx,t x tt. (13) Fsin As shown in Fig. 4(a), a nonlinear optial rystal is introdued into plane 2 of the system. The rystal optial axis is oriented to ahieve type I phase mathing 30,31 for generation of the seond-harmoni field that results from mixing the SDW s of the shaped and referene pulses. For type I phase mathing, the resultant seond-harmoni field an be viewed as the response of a linear system, whih an be desribed by the onvolution between the system impulse response funtion and the driving term, i.e., the multipliation of the omplex amplitudes of the two fundamental inident fields. 31 The system impulse response is proportional to the group mismath of the two inident fundamental fields and the nonlinear interation

5 Sun et al. Vol. 14, No. 5/May 1997/J. Opt. So. Am. A 1163 length between the fields within the rystal. In our arrangement of the experiment, we use the nonollinear geometry for type I phase mathing that provides a nonlinear interation length of approximately a fration of a millimeter. For a moderate group mismath of pulses in the nonlinear rystal, the system impulse response funtion is a very narrow funtion, and the nonlinear seondharmoni interation takes plae almost instantaneously. Therefore the indued seond-harmoni field an be onsidered to be proportional to the produt of the two SDW s given by relations (9) and (12) at eah transverse oordinate x, yielding u II x, t Ku 2s x; tu 2r x; t Kw1 w1 F x t F x t t Ū x s F Ū r x F exp j 2 x F t 2 exp j2 t Kax; tbxexp j2 t, (14) where we assume that the aperture of the rystal in the x diretion is larger than the transverse size of the SDW s of the two pulses to prevent the utoff of the spatial-frequeny ontent of the generated seondharmoni field, K is a onstant aounting for the onversion effiieny of the seond-harmoni-generation proess, and the funtions a(x, t) and b(x) are defined by ax, t w 1 F x t w1 F x t t, (15a) bx Ū s x F Ū r x F exp j 2 x F t 2. (15b) The resulting seond-harmoni field desribed by Eq. (14) onsists of a long-pulse of temporal duration determined by the funtion a(x, t) defined in Eq. (15a). This funtion sets a time window that is entered at the temporal position of the referene pulse at the input of the pulse imaging system. If the width of the shaped pulse is extended beyond this time window, then the part of the shaped pulse outside this time window annot be seen by the system, beause the nonlinear interation in the nonlinear rystal annot take plae. Equation (14) also shows that the resultant seond-harmoni field is a quasimonohromati wave of a onstant enter arrierfrequeny 2. The most important term resulting from the generation of the seond-harmoni field in the femtoseond pulse imaging system is desribed by the funtion b(x). This term registers spatially the temporal interation between the two pulses. Consider using the paraxial approximation and negleting the small term x/f in the funtion a(x; t), so that Eq. (15a) an be rewritten as Fig. 4. (a) Illustration of nonlinear three-wave interation of the SDW s of the two transform-limited pulses in the nonlinear rystal produing a seond-harmoni field with a onstant temporal frequeny and a linear spatial arrier frequeny depending on the relative delay between the two pulses, (b) k-vetor diagram showing the instantaneous phase-mathing ondition for three-wave mixing.

6 1164 J. Opt. So. Am. A/Vol. 14, No. 5/May 1997 Sun et al. ax, t at w t t t w. (16) Under this assumption the funtion b(x) remains the only spatially dependent term in Eq. (14), and we an finally apply the spatial Fourier transformation that is implemented with a seond lens in Fig. 2, yielding an output image u II x, t Bxatexp j2 t, (17) where the funtion B(x) is the spatial Fourier transform of b(x) given by Bx bxexp j2f x xdx Ū s x F Ū r x F exp j 2 x F t 2 exp j2f x xdx ū s 2 x ū r 2 x x, (18) with f x x/f, x (/2)t, and x t/2. Equation (18) desribes a spatial domain image that is equivalent to the temporal onvolution between the timeinverted shaped pulse and the referene pulse. The spatial image is entered at a oordinate x x depending on the relative time delay between the two pulses. In imaging systems terminology the funtion ū r (2/)(x x)] of Eq. (18) an be seen as a point-spread funtion of the ultrashort-pulse imaging system entered at x x. The width of the referene pulse determines the maximum resolution of the pulse imaging system; the resolution an be optimized by using transform-limited referene pulses that an be obtained from a pulsed laser soure with dispersion ompensation. The onversion proess from time domain to spae domain within the nonlinear rystal possessing (2) nonlinear suseptibility an be best desribed by using the momentum and energy onservation relations that govern the three-wave mixing proess. Beause the two input SDW s have idential but inverted spetra dispersed in the transverse spatial diretion, the energy onservation ondition requires that the third wave generated at the output of the nonlinear rystal be a quasi-monohromati wave with a onstant frequeny 2 [see Eq. (14)]. However, to ahieve an effiient three-wave mixing proess, we also need to satisfy the instantaneous phase-mathing ondition k 2 k s k r, (19) where k s and k r are the wave vetors of the two relatively inverted SDW s of the shaped and the referene pulse, respetively [see Eqs. (10) and (13)], and k 2 is the wave vetor of the generated seond-harmoni field. Figure 4(b) depits a graphial example of the k-vetor phasemathing ondition in the spetral plane when both inident pulses are transform limited and possess a relative time delay t. This figure shows that although the wave vetors of eah SDW of the input pulses are ontinuously hanging (i.e., rotating) in time, the instantaneous sum of the two wave vetors beomes independent of absolute time. At any time instane t, the generated seondharmoni field has a wave vetor that points to a ommon diretion. For our example on three-wave mixing of two transform-limited pulses with time delay t, we an determine the spatial frequeny of the resulting seondharmoni field by adding Eqs. (10) and (13), yielding k 2 x k s x t k r x t t F. (20) The result of the last equation shows that the generated seond-harmoni field has a onstant k-vetor diretion depending only on the time delay between the two transform-limited pulses. In the following subsetions, we onsider two examples on ultrashort-pulse imaging. In these examples we will use the same transform-limited pulses for produing the shaped pulses with the system of Fig. 1 and for the referene pulses with the system of Fig. 2. With this assumption the orresponding funtions are given by ū s t pt m t, ū r t pt, (21a) (21b) where p(t) is the real-value shape funtion of the transform-limited pulse. A. Example 1: Sequene of Transform-limited Pulses For this example the impulse response of the pulseshaping devie orresponds to m (t) n A n (t t n ), providing at the output (see Fig. 1) a sequene of transform-limited pulses ū s t n A n pt t n, (22a) where A n and t n are the amplitude and the time delay for the nth pulse in the sequene. Substituting Eq. (22a) into Eq. (18), we obtain the orresponding image, Bx n A n p 2 x x n * p 2 x x (22b) where x n t n /2 and * denotes the orrelation operation. The image desribed by Eq. (22b) onsists of a 1-D sequene of autoorrelation funtions in spae, where the image orresponding to the nth pulse in the sequene is loated at oordinate x x x n. Note that the spatial image of the pulse sequene that an be displayed by the system is still limited by the time window funtion a(t), whih defines the field of view of the pulse imaging system. The result of Eq. (22b) enables us to perform serial-to-parallel demultiplexing by onverting a 1-D time sequene of pulses into a 1-D spatial image with one-toone orrespondene. One suh a onversion is performed in real time (i.e., at femtoseond rate), the teh-

7 Sun et al. Vol. 14, No. 5/May 1997/J. Opt. So. Am. A 1165 nique will be useful for serial-to-parallel demultiplexing that needs to be developed for the next generation of terabit optial ommuniation networks. B. Example 2: A Chirped Pulse A hirped pulse an be produed from a pulse-shaping devie by using a quadrati phase mask (e.g., a ylindrial lens) as the spetral filter, resulting in an impulse response m (t) F 1 exp j( 2 /2) expj(t 2 /2), where is the group-veloity dispersion oeffiient and is inversely proportional to the foal length of the ylindrial lens, and F 1 is the inverse Fourier transform operator. The pulse resulting at the output of the pulse-shaping devie is ū s t pt exp j t 2 2. (23a) The orresponding pulse image an be found by substitution of Eq. (23a) into Eq. (18), providing an image Bx p 2 p x exp j 2 x x p 2 x * p 2 exp j 22 x 2 22 x2 2 x x 2. (23b) The first equality in Eq. (23b) shows that the resultant image orresponds to ross orrelation between the timeinverted hirped pulse with the transform-limited referene pulse. The resultant image orresponds to the envelope funtion of the hirped pulse, whih ontains no information about the sign and the magnitude of dispersion of the hirp pulse. However, from the seond equality of Eq. (23b), this same image is also equivalent to the onvolution between the orrelation of two transformlimited pulses with a spatial quadrati phase funtion. With the use of basi priniples of Fourier optis, 32 suh a onvolution orresponds to free-spae propagation of the optial field over a distane. z 2 2 k (23) measured from the image plane of the pulse imaging system of Fig. 2. This means that if we translate the observation plane along the longitudinal diretion by a distane z, we will obtain an image idential to that obtained with transform-limited pulses. Note that the diretion and the distane of this translation are diretly related to the sign and the magnitude of the groupveloity dispersion oeffiient of the hirped pulse. Therefore, by measuring these parameters (i.e., the diretion and the distane of the observation plane), we an determine quantitatively the group dispersion properties of a hirped pulse. 4. DESCRIPTION OF THE EXPERIMENTAL APPARATUS The optial setup implementing the femtoseond pulse imaging tehnique is depited shematially in Fig. 2, where we use ultrashort pulses of 200 fs at a enter wavelength of 920 nm that are generated from a mode-loked Ti:sapphire laser (Mira 900, Coherent In., Palo Alto, Calif.). The laser output is split into two beams, one to be used as a referene beam and the other introdued into the pulse-shaping devie shown in Fig. 1 to reate a shaped pulse. The shaped pulse from the output of the pulse-shaping devie and the referene pulse beams are then introdued into the pulse imaging system of Fig. 2. Both beams are spatially expanded and ollimated to produe a large illumination area on the gratings of the spetral deomposition devies, providing a wider time window for time-to-spae onversion. In our experiment the expanded beams of approximately 3 m in diameter are inident onto gratings of 2-in. (5.08-m) aperture sizes, with an inident angle of 33.5, providing a time window of approximately 66 ps in duration. The two metalli blazed gratings of 600 lines/mm are both arranged ollinearly in the vertial diretion (see Fig. 2), but with their blaze angles oriented in opposite diretions from eah side of the grating normal. The opposite orientation of the blaze is neessary to produe the relative spetrum inversion of the SDW s orresponding to the referene and shaped pulses. A vertial spatial separation between the shaped pulse and referene pulse beams is introdued to produe a vertial spatial-frequeny arrier between their orresponding SDW s in the Fourier transform plane. These two spatial-frequeny arrier-shifted SDW s are introdued into a LiB 3 O 5 (LBO) nonlinear rystal (Super Tehnology In., Los Angeles, Calif.) for generation of the seond-harmoni field at 460 nm. The two Fourier lenses in the pulse imaging system have idential foal lengths of 375 mm. The rystal ut and the rystal orientation are designed to satisfy the type I phase-mathing ondition for nonollinear seond-harmoni generation. With suh rystal preparation the generated seond-harmoni field will propagate in a vertial bisetor diretion, whih in our experiments oinides with the diretion of the optial axis of the system. The onversion effiieny of the three-wave mixing is measured to be approximately 0.1%. A horizontal slit and a narrow-band olor filter are plaed behind the rystal to filter out the fundamental frequeny light of the two input beams. The seond lens is used to perform a spatial Fourier transform of the generated seond-harmoni field, produing an image deteted by a CCD amera. Note that sine the SDW s of the ultrashort pulses appear foused vertially in the Fourier plane, the generated seond-harmoni field is onfined in the spatial-frequeny plane in the vertial diretion, produing in the image plane a spatially unonfined and uniform-intensity image in the vertial diretion. However, in the horizontal diretion, the generated image shows the temporal shape of the input shaped pulse. In

8 1166 J. Opt. So. Am. A/Vol. 14, No. 5/May 1997 Sun et al. the following setion we disuss the experiments on imaging the ultrashort pulses by using our femtoseond pulse imaging system. 5. EXPERIMENTAL RESULTS In our experiments with the femtoseond pulse imaging system of Fig. 2, we first alibrate the sale of time-tospae onversion. In this experiment a transformlimited pulse is split into two pulses implementing the shaped pulse and the referene pulse. The relative time delay between the two pulses is ontrolled by a delay line introdued into the path of the referene beam. For this ase the pulse image is a narrow vertial line orresponding to the autoorrelation funtion of a transform-limited pulse, where the enter of the autoorrelation image is determined by the time delay between the two pulses. During the alibration proess we hange and measure the time delay between the two pulses and at the same time observe the spatial movement of the pulse image by using a CCD amera and a TV monitor. The motion of the image follows the adjusted delays instantaneously. The intensity of the image gradually dereased as the time delay between the two pulses inreased and, finally, disappeared when the delay exeeded the time window of the pulse imaging system. Figure 5 shows two pulse images obtained by hanging the optial path-length differene by 6.35 mm (i.e., 1/4 in.), whih orresponds to a temporal shift of the referene pulse by ps. Based on the onversion relation x t/2, we estimate the total movement of the image on the CCD amera plane to be 5.75 mm. The two images are measured to be separated by 442 CCD pixels, thus orresponding to a time-to-spae onversion sale of fs per pixel. Under the assumption of Gaussian-shape pulses, we estimate that the generated output image has a full width at half-maximum (FWHM) value of approximately 227 fs, whih is slightly wider than the initial width of the pulse from the laser soure (200 fs). We expet that the small inrease of the pulse width might our as a result of the spatialfrequeny utoff by the limiting aperture of the LBO rystal used in our experiment. Next we performed imaging experiments with various shaped pulses obtained at the output of the pulse-shaping devie. In priniple, the temporal shape of the pulse obtained from the pulse-shaping devie is proportional to the temporal Fourier transform of the transmittane Fig. 5. Reorded pulse images that are equivalent to the autoorrelation funtion of a transform-limited pulse. The two images are obtained by hanging the relative delay of the referene pulse by 6.35 mm, orresponding to ps. Fig. 6. Images of shaped pulses obtained with the femtoseond pulse imaging system of Fig. 2. The shaped pulses are obtained by using the pulse-shaping devie of Fig. 1 with spetral filter implemented by (a) a binary phase grating and (b) a Ronhi grating (50/50). funtion of the spetral filter. For example, when the spetral filter is implemented by a simple grating, the resultant shaped pulse will onsist of a sequene of equally spaed pulses, eah with a different amplitude, depending on the speifi grating struture (see example 1 of Setion 3). We used spetral filters implemented by two gratings, a binary phase grating and a binary amplitude grating. The binary phase grating of period m possesses a magnitude of phase modulation that produes strong 1, 0, and 1 orders and negligibly weak higher diffration orders (this was verified by performing a onventional spatial Fourier transform at the same wavelength). We employed this phase grating in the pulseshaping devie and used the output shaped pulse as an

9 Sun et al. Vol. 14, No. 5/May 1997/J. Opt. So. Am. A 1167 input to our femtoseond pulse imaging system. The resultant shaped pulse image [see Fig. 6(a)] onsists of three uniform-intensity stripes orresponding to shaped pulses of three pulses separated by 2.06 ps. This value is found to be onsistent with the alulated value of 2 ps (see Appendix A). The seond spetral filter that we used in the pulseshaping devie was a 50/50 binary amplitude grating (i.e., the Ronhi grating) with a 250-m grating period. Suh a grating has the unique property that its Fraunhofer diffration pattern does not have even diffration orders exept for the 0 order. The pulse image [see Fig. 6(b)] learly shows that the even-order pulses in the sequene (exept the 0 order) do not appear. The three entral pulses (i.e., 1, 0, and 1 orders) are lose together, while the other pulses (orresponding to higher orders) are separated by twie the distane. We estimate the distanes between the enter pulses to be 1.75 ps, whih is onsistent with the alulated value of 1.69 ps (see Appendix A). In the third experiment we examine example 2 of Setion 3, where our main result shows that the pulse image of a hirped pulse an be transferred into an image of the orresponding transform-limited pulse by a longitudinal translation of the observation plane. In this experiment we use a 1-D binary phase diffrative optial lens (250-mm foal length for the 1 and 1 diffration orders at the design wavelength of 800 nm) as the spetral filter. The element was designed to produe uniform diffration effiienies for the 1, 0, and 1 diffration orders, representing transmittane funtions of a negative quadrati phase, a onstant linear phase, and a positive quadrati phase, respetively. In order to separate the output pulses orresponding to these three diffration orders, the diffrative lens was translated off axis. This off-axis shift introdues a onstant linear phase differene be- Fig. 7. (a) Shemati diagram of a setup for imaging of negative hirped, transform-limited, and positive hirped pulses produed with a 1-D diffrative lens used as the spetral filter in the pulse-shaping devie. (b) The three images obtained at the three observation planes demonstrate that the hirp pulses in the time domain an our as images of transform-limited pulses in the spae domain by variation of the longitudinal oordinate of the observation plane.

10 1168 J. Opt. So. Am. A/Vol. 14, No. 5/May 1997 Sun et al. tween the adjaent diffration orders, whih in turn introdues a relative time delay between the orresponding pulses at the output of the pulse-shaping devie. The output shaped pulse onsists of a sequene of three timeseparated pulses: negative hirped, transform-limited, and positive hirped pulses orresponding to the 1, 0, and 1 diffration orders, respetively. The image of the shaped pulse is shown in Fig. 7. When the observation plane oinides with the exat image plane [see Fig. 7(a), plane 2], we observe a transform-limited pulse image in the middle between the images of the positive and negative hirped pulses [see Fig. 7(b)]. The FWHM widths for these two hirped pulse images are measured to be approximately idential and equal to 2.39 ps, whih is a few perent less than the theoretially predited value of 2.51 ps (see Appendix A). This ould also be due to the slight spatial-frequeny utoff by the limiting aperture of the LBO rystal. As we move the observation plane toward the lens [see Fig. 7(a), plane 1], the image orresponding to the negative hirped pulse beomes narrower, with a width orresponding to that of a transform-limited pulse, while the two other pulses beome wider [see Fig. 7(b)]. Similarly, as we move the observation plane further from the image plane [see Fig. 7(a), plane 3], the image orresponding to the positive hirped pulse beomes narrow, with a width very lose to that of a transform-limited pulse [see Fig. 7(b)]. The distanes between the three observation planes are measured to be idential and equal to 150 mm, where theoretially we found the value to be 144 mm (see Appendix A). The experimental result demonstrates that this pulse imaging tehnique an determine the sign and the magnitude of the temporal hirp of the shaped pulse by analyzing a sequene of images deteted in a few observation planes. In ontrast, using a onventional autoorrelator tehnique will provide idential time envelope information for both negative and positive hirped pulses. 6. CONCLUSION We introdued, analyzed, and evaluated experimentally a femtoseond pulse imaging tehnique that is useful in various appliations. Our method, based on three-wave mixing in nonlinear LBO rystals, allows us to onvert the omplex amplitude of an ultrashort temporal pulse signal to a orresponding spatial image that resembles the temporal signal in spae. We show that, unlike the ommonly used autoorrelator, suh time-to-spae onversion arries both amplitude and phase information on the pulses. In partiular, we have demonstrated that by examining the longitudinal oordinate of the observation plane at the output of the femtoseond pulse imaging system, we an determine the sign and the magnitude of temporal hirping of pulses. The experimental results are found to be in good agreement with the analyti results. The measured onversion effiieny of the threewave mixing is relatively low (i.e., 0.1%) beause of the spatial hirping of the pulse, whih redues the peak power density neessary for nonlinear onversion. However, in our experiments, the energy of the single pulse is of the order of 1 10 nj with the laser operated at a 77- MHz rate, and the image intensity of the seondharmoni field was found suffiient for CCD detetion with video rate integration time. In priniple, this pulse imaging system should be able to operate with singlepulse detetion, 33 but a higher-sensitivity 1-D CCD array may be required. The femtoseond pulse imaging system is operating at femtoseond rates and therefore an be useful for demultiplexing in terahertz-bandwidth optial fiber networks. APPENDIX A In this appendix we desribe the pulse-shaping devie shown in Fig. 1. To employ the expression desribing the output of the pulse-shaping devie [see Eq. (3)], we need first to find the relation between eah temporal-frequeny omponent at the input and the output of the pulseshaping devie and its spatial oordinates in the spetral plane. The relation between the temporal frequeny and the spatial transverse oordinate in the spetral filter plane, x p, is similar to that of Eq. (8b) and is given by p F p x p p F, (A1) p where p sin p and p and F p are the inident angle of the input pulse beam and the foal length of the Fourier lenses in the pulse-shaping devie, respetively. Using the last equation, we an define x p, (A2) p F p where again the paraxial approximation is applied. Consider a spetral domain spatial filter with transmittane funtion M (x p ). We first perform a oordinate transformation of M(x p ) from the spatial domain to the temporalfrequeny domain by using Eq. (A2), yielding u s t exp j t Ū i M pf p exp jtd ū i t m t p F pexp j t. (A3) The term inside the square brakets of Eq. (A3) will determine the shape of the output pulse. In the following subsetions we evaluate the pulse-shaping devie used to prepare pulses for examples 1 and 2 of Setion Pulse Shaping with a Grating Filter The transmittane funtion of the spetral domain spatial filter implemented by a grating is desribed by M x p G x p x p omb x p x p, (A4) where G(x p /x p ) is the grating s omplex transmittane funtion defined on a single period x p and omb is used to denote an infinite 1-D train of funtions. Consider a transform-limited pulse ū i (t) p(t) at the input of the pulse-shaping devie with a grating filter given by Eq. (A4). The output of the pulse-shaping devie is determined from Eq. (A3), yielding

11 Sun et al. Vol. 14, No. 5/May 1997/J. Opt. So. Am. A 1169 u s t pt g x pt 2 p F p omb x p 2 p F p t exp j t pt g t t p omb t t p exp j t, (A5) where we used the oordinate transformation of Eq. (A2), g(t) is the inverse Fourier transform of the funtion G(), and t p (2 p F p )/(x p ) is defined as the temporal separation between the pulses determined by the different diffration orders of the grating. In our experiments with the pulse-shaping devie (i.e., F p 250 mm, p 0.552, and 920 nm), we used two types of grating, a Ronhi grating with x p 250 m and a binary phase grating with x p m. The resultant shaped output pulses are separated by 1.69 ps and 2 ps for the Ronhi grating and the binary phase grating, respetively. 2. Pulse Shaping with a Cylindrial Lens The transmittane funtion of the spetral domain spatial filter implemented by a ylindrial lens of foal length f is given by M x p exp j f x p 2. (A6) The output shaped pulse is determined from Eq. (A5) by t u s t pt exp 2 j j 2exp t, (A7) where 2 p F 2 p /f. This expression desribes a hirped pulse. For a Gaussian-shape transform-limited pulse p(t) with FWHM of, the width of the hirped pulse is /2. (A8) In our experiment we used a binary phase diffrative ylindrial lens of foal length f mm (250 mm at the design wavelength of 800 nm). We onsider only the 1, 0, and 1 diffration orders generated from the diffration optial element. These three orders will produe a sequene of three pulses at the output of the pulseshaping devie: negative hirp, transform-limited, and positive hirp pulses. Using the experimental parameters of the pulse-shaping devie with an inident transform-limited pulse of 200 fs and the parameters of our diffrative element, we alulate the pulse width of 2.51 ps for the output hirped pulses (both positive and negative hirps). The longitudinal distane between the imaging planes for the transform-limited pulse and the hirped pulses an also be estimated, by using Eq. 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