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pectral Fraunhofer regime: time-to-frequencyonversion by the action of a single time lens onn optical pulse

ose Azana, Naum K. Berger, Boris Levit, and Baruch Fischer

We analyze a new regime in the interaction between an optical pulse and a time lens �spectral Fraunhoferregime�, where the input pulse amplitude is mapped from the time domain into the frequency domain�time-to-frequency conversion�. Here we derive in detail the conditions for achieving time-to-frequencyconversion with a single time lens �i.e., for entering the spectral Fraunhofer regime� as well as theexpressions governing this operation. Our theoretical findings are demonstrated both numerically andexperimentally. A comparative study between the proposed single-time-lens configuration and theconventional dispersion � time-lens configuration for time-to-frequency conversion is also conducted.Time-to-frequency conversion with a single time lens can be used for applications similar to thosepreviously proposed for the conventional time-to-frequency converters, e.g., high-resolution measurementof fast optical temporal waveforms. Moreover, our results also indicate that the spectral Fraunhoferregime provides additional capabilities for controlling and processing optical pulses. © 2004 OpticalSociety of America

OCIS codes: 070.2590, 320.5540, 060.5060, 200.3050, 250.5530.

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. Introduction

pace–time duality is based on the analogy betweenhe equations that describe the paraxial diffraction ofeams in space and the first-order temporal disper-ion of optical pulses in a dielectric.1–8 The dualityan also be extended to consider imaging lenses:he use of quadratic phase modulation on a temporalaveform is analogous to the action of a thin lens on

he transverse profile of a spatial beam.1–7 The timeens can be implemented in practice with an electro-ptic phase modulator driven by a sinusoidal radiorequency �RF� signal,2,4,6 by mixing the originalulse with a chirped pulse in a nonlinear crystal3,5

sum-frequency generation�, or by means of cross-hase modulation of the original pulse with an in-ense pump pulse in a nonlinear fiber.7 Optical

J. Azana �azana@inrs-emt.uquebec.ca� is with the Institut Na-ional de la Recherche Scientifique �INRS�—Energie, Materiaux etelecommunications, 800 de la Gauchetiere Ouest, Suite 6900,ontreal, Quebec H5A 1K6, Canada. N. K. Berger, B. Levit, and. Fischer �fischer@ee.technion.ac.il� are with the Department oflectrical Engineering, Technion—Israel Institute of Technology,aifa 32000, Israel.Received 15 May 2003; revised manuscript received 19 Septem-

er 2003; accepted 14 October 2003.0003-6935�04�020483-08$15.00�0© 2004 Optical Society of America

ignal processing operations based on a time lensnclude real-time Fourier transformation,1,6 temporalmaging,2,3 and time-to-frequency conversion.4–7

Here we conduct a detailed investigation of a newegime in the interaction between optical pulses andime lenses. In particular, we demonstrate thathen a time lens operates on an optical pulse thisulse can enter a regime where the input pulse am-litude is mapped from the time domain into therequency domain �time-to-frequency conversion�.s is schematically shown in Fig. 1, this regime cane interpreted as the frequency-domain dual of theemporal Fraunhofer regime8 �frequency-to-time con-ersion or real-time Fourier transformation by tem-oral dispersion� and, as a result, we refer to it as thepectral Fraunhofer regime.Among other potential applications, time-to-

requency conversion can be used for the measure-ent of the intensity temporal profile of ultrashort

ptical pulses �by simply measuring the spectrum ofhe transformed signal�. In contrast with other ap-roaches, time-to-frequency conversion provides aast, noniterative �single-shot� and unambiguouseasurement of the temporal waveform, being espe-

ially suited for the measurement of long single-ransient events with ultrafast temporal details or,quivalently, signals with a large time–bandwidthroduct. Time-to-frequency conversion was previ-

10 January 2004 � Vol. 43, No. 2 � APPLIED OPTICS 483

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usly demonstrated with a system comprising a timeens preceded by a suitable dispersive device4–7 �sche-

atic shown in Fig. 2�. Our proposal �using a singleime lens operating in the spectral Fraunhofer re-ime� simplifies the design and implementation ofime-to-frequency converters, since it avoids the usef a dispersive device preceding the time lens.In this paper we derive the conditions for achieving

ime-to-frequency conversion with a single time lensi.e., conditions for entering the spectral Fraunhoferegime� as well as the expressions governing thisperation. Our theoretical findings are confirmedy means of numerical simulations, and an experi-ental demonstration of the phenomenon is also pro-

ided. A comparative study between the single-lensonfiguration and the dispersion–time-lens configu-ation for time-to-frequency conversion is also com-leted.

. Theory

. Temporal Fraunhofer Regime in a Dispersive Medium

s is shown in Fig. 1, the spectral Fraunhofer regimen a time lens can be interpreted as the frequency-omain dual of the temporal Fraunhofer regimereal-time Fourier transformation� in a dispersiveedium. We first briefly review the theory of tem-

ig. 1. Dual Fraunhofer regimes: �a� frequency-to-time conver-ion using dispersion, �b� time-to-frequency conversion using aingle time lens.

ig. 2. Schematic of the dispersion–time-lens configuration forime-to-frequency conversion.

84 APPLIED OPTICS � Vol. 43, No. 2 � 10 January 2004

oral Fraunhofer in dispersive media.8 In what fol-ows, the involved signals are assumed to bepectrally centered at the optical frequency �0; weork with the complex temporal envelope of the sig-als, and we ignore the average delay introduced byhe components �dispersion or time lens�. The tem-oral Fraunhofer regime can be observed when anptical pulse is temporally stretched by first-orderispersion. The first-order dispersion coefficient ofhe dispersive medium is defined as �� � ��2������2��0, where ���� is the spectral phase response ofhe medium. Notice that the variable � is the base-and frequency variable, i.e., � � �opt �0, whereopt is the optical frequency variable. If this disper-ion coefficient �� satisfies the condition8

�������t2�8 , (1)

t being the total duration of the unstretched opticalulse, then the output pulse envelope b�t� is propor-ional to the spectrum of the input pulse A��� �seeig. 1�a�, i.e. �b�t�� � �A�� � t�����. Inequality �1� issually referred to as the temporal Fraunhofer con-ition, since it is the time-domain analog of the well-nown Fraunhofer condition in the problem of spatialiffraction.

. Spectral Fraunhofer Regime in a Time Lens

time lens is a phase-only modulator with a phaseodulation function2

m�t� � exp� j��t� � exp� j��t�2�t2, (2)

here �t � ��2��t���t2t�0 will be referred to as thehase-factor of the time lens. Let us now evaluatehe action of the time lens over a given arbitraryptical pulse c�t�. The output pulse d�t� from theime lens in response to the input pulse c�t� is giveny d�t� � c�t�m�t�. In the frequency domain, theroduct can be described as a convolution D��� ���� � M���, where D��� and C��� are the Fourier

ransforms of d�t� and c�t�, respectively, and M��� ishe Fourier transform of the time-lens modulationunction, M��� � exp�j�1�2�t��

2. We derive

D��� � C��� � M��� � ���

C���exp�j�1�2�t���

� ��2d� � M��� ���

C���exp�j�1�2�t��2

� exp� j�1��t���d�, (3)

here �� is the total spectral bandwidth of the inputulse c�t�. If this bandwidth is sufficiently narrowhat it satisfies the condition

��t�����2�8 , (4)

hen the phase term exp�j�1�2�t��2 within the last

ntegral in relation �3� can be neglected, since ��1�� ��2� � �1�2�� ������2�2 �� �we remind the

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eader that the variable � is a base-band frequency�.n this case, relation �3� can be approximated by

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here the last integral has been solved by consider-ng exp� j�1��t��� the kernel of a Fourier transfor-

ation. Relation �5� indicates that under theonditions of inequality �4� the spectrum of the outputulse D��� is, within a phase factor M���, propor-ional to the input temporal waveform c�t� evaluatedt the instant t � ���t. In other words, the spectralnergy distribution of the output optical pulse �D����2s an image of the temporal intensity distribution ofhe input optical pulse �c�t��2, i.e., �D����2 � �c�t ���t��

2 �as represented in Fig. 1�b�. Inequality �4� ishe frequency-domain dual of the temporal Fraun-ofer condition in inequality �1�. We refer to this

nequality as the spectral Fraunhofer condition.ote that inequality �4� conditions the phase-factor

chirp� of the time lens ��t�, depending on the fastestemporal feature of the input optical signal to beeasured, �t � 2 ��� �i.e., depending on the re-

uired temporal resolution �t�. Specifically, accord-ng to inequality �4�, the shorter �faster� the temporaleature to be resolved, the larger the phase factor ofhe time lens must be. In particular, an estimationf the temporal resolution �t provided by the systeman be obtained from inequality �4� by simple math-matical manipulation. We first derive that ��t� ��2 ��t�2�8 � ��2�t2�, which translates into a con-ition for the shorter temporal feature that can beesolved by the system, �t2 �� ��2��t��. Strictly,his condition can be written as

�t � �5 ���t� � 4����t�. (6)

owever, our experimental and simulation resultsave demonstrated that this condition is too restric-ive, much as the similarly derived Fraunhofer-zoneistance in the problem of spatial diffraction is far tooestrictive in practice. A more precise estimation ofhe temporal resolution provided by the system isiven by the following expression:

�t � 1����t�. (7)

. Brief Comparison with the Dispersion–Time-Lensonfiguration

ime-to-frequency conversion of optical pulses haseen previously demonstrated with a system com-rising a dispersive medium followed by a time lens,rranged in the appropriate balance.4–7 A sche-atic of this configuration is shown in Fig. 2. Spe-

ifically, the system must be designed so that theollowing relation is satisfied:

� � � 1, (8)

� t

here �� is the first-order dispersion coefficient ofhe dispersive medium �as defined above�. If condi-ion �8� is satisfied, then it can be shown that thenput–output relation in the system is also given byq. �5�; i.e., again, the spectrum of the pulse at theutput of the system D��� is, within a phase factor���, proportional to the input temporal waveform

�t� evaluated at the instant t � ���t.Obviously, the main advantage of the configuration

roposed herein as compared with the dispersion–ime-lens configuration is that our proposal avoidshe use of an input dispersive device, and, as a result,t represents a much simpler and practical alterna-ive for implementing time-to-frequency conversion.t is also important to compare the two describedonfigurations for time-to-frequency conversion inerms of the temporal resolutions that they can pro-ide. As discussed above, the temporal resolutionorresponding to the single-lens configuration can bestimated from inequality �7�. For the dispersion–ime-lens configuration, the temporal resolution isimited mainly by the equivalent effects of diffractionrising from the finite aperture. In the temporalontext, the aperture corresponds to a finite time win-ow through which the fields pass on their wayhrough the time lens.2 Briefly, if the input pulse isoo short �broadband�, it is stretched in excess by theispersion preceding the time lens and overfills thenite time aperture of the time lens. As mentionedbove, this effect essentially limits the achievableemporal resolution, i.e., the shortest pulse that cane processed with the system. Mathematically, theemporal duration of the pulse after dispersion can bestimated as �tout � ������ � 2 ������t, where ��epresents the total bandwidth of the input pulse tohe system, and �t represents the fastest temporaleature of this input pulse �temporal resolution�. Andequate processing of the pulse is ensured only if itsemporal duration after dispersion �i.e., at the inputf the time lens� is shorter than the time aperture �af the time lens, �tout � �a. This fixes a limit for thechievable temporal resolution according to the fol-owing expression:

�t �2

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ote that in deriving inequality �9� we have takennto account the relation between the dispersion andhe time-lens phase factor given by Eq. �8�. We nowave closed-form expressions for the temporal reso-

ution provided by the single-time-lens system �in-quality �7� and by the dispersion–time-lens systeminequality �9�. Based on these closed-form expres-ions, the corresponding approaches for time-to-requency conversion can be now quantitativelyompared in terms of achievable temporal resolu-ions. For the two proposed configurations, the tem-oral resolution can always be improved �i.e., �t cane shortened� by increasing the magnitude of thehase-factor ��t�. �This tendency can be clearly de-uced from Eqs. �7� and �9�. In fact, in the single-

10 January 2004 � Vol. 43, No. 2 � APPLIED OPTICS 485

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ime-lens configuration the temporal resolutionepends only on this phase-factor ��t�, whereas in theispersion–time-lens system the temporal resolutionlso depends on the temporal aperture �a of the timeens �the longer the time aperture, the faster theemporal feature that can be processed�.

For comparative purposes, let us assume that weant to achieve a given temporal resolution �t. In

he case of the single-time-lens configuration, thehase-factor ��t� of the time lens must be fixed tonsure that ��t� � 1��t2 �see inequality �7� whereas inhe case of the dispersion–time-lens configuration thehase-factor ��t� of the time lens must be fixed so that

�t� � 2 ���a�t� �see inequality �9�. These require-ents indicate that the single-time-lens configura-

ion will require a lower phase factor ��t� �asompared with the dispersion–time-lens configura-ion� in order to achieve a given temporal resolutiont only if this temporal resolution is slow enough thatt satisfies �t � �a�2 . To be able to solve fasteremporal features than �a�2 , the single time lensill require a larger phase factor than theispersion–time-lens configuration. Thus, based ex-lusively on time-resolution criteria, in principle, theingle-time-lens configuration would be the preferredne for processing pulses with temporal featureslower than �a�2 , whereas the dispersion–time-lensonfiguration would be more convenient for process-ng pulses with temporal features faster than �a�2 .his is only partially true, because it is also impor-

ant to note that the resolution limit that we aresing to compare the systems under study is the fun-amental resolution limit of the system; i.e., it isntrinsic to the system configuration itself �ideal sys-em�. There are, however, other important aspectshat can affect the resolution provided by the de-cribed systems, which we did not take into accountn our previous discussion. These other aspects in-lude the finite temporal resolution of the final signalecording system, as well as temporal aberrationsnduced by deviations from the ideal quadratic phaseesponse in the dispersive medium or in the timeens.9 Regarding these temporal aberrations, ithould be mentioned that one of the advantages ofsing the single-time-lens configuration instead ofhe conventional dispersion–time-lens approach forime-to-frequency conversion is that we eliminate onef the sources of aberrations in the system, namely,he dispersive medium9 �as mentioned above, thether source of aberrations is the time lens�. Notehat the aberrations induced by the dispersive me-ium �associated with deviations from its ideal qua-ratic phase spectral response� will become moreignificant for temporally shorter �spectrally broader�ulses. As a result, it is expected that the elimina-ion of the dispersive medium in the temporal systemill have a positive effect on the overall performance

f this system �in terms of temporal resolutions�.his point nonetheless requires further investiga-

ion, but its detailed study is beyond the scope of thisork.

86 APPLIED OPTICS � Vol. 43, No. 2 � 10 January 2004

. Simulation Results

he introduced theory first has been confirmed byeans of numerical simulations. Figure 3 shows re-

ults from simulations in which the time lens is con-gured to operate within the spectral Fraunhoferegime. Specifically, the pulse incident upon theime lens �Fig. 3�a� and the output pulse from theime lens �Fig. 3�b� are shown. The plot at the bot-om of each figure represents the temporal waveformaverage optical intensity�, the plot at the left showshe corresponding spectral energy density of the sig-al, and the contour two-dimensional image showshe joint time-frequency energy distribution of theignal �darker regions in the image correspond toigher intensities�. The joint time–frequency repre-entation provides information on the temporal loca-

ig. 3. Simulation results for pulse propagation through a timeens operating within the spectral Fraunhofer regime. Plots athe bottom show the signal in the temporal domain, plots at the lefthow the signal in the frequency domain, and the larger 2D imageshow the joint time-frequency representation of the signal. �a�nput pulse to the time lens; �b� output pulse from the time lens.

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ion of the signal’s spectral components10 and allows uso get a deeper insight into the structure of the signaltself as well as into the nature of the processes undernalysis. For the time–frequency distributions, weave used a combined wideband–narrowband spectro-ram,11 which, in contrast to other distributions, pro-ides good resolution in both time and frequencyomains simultaneously without introducing signifi-ant interference terms. The input signal �Fig. 3�a�onsists of two consecutive and partially overlappedransform-limited Gaussian pulses of different inten-ity: each individual pulse has a FWHM time widthf �8 ps, and the two pulses are separated by �25 ps.he total bandwidth of this input signal is estimated toe �� � 2 � 100 GHz. We simulated an ideal time-ens process, i.e., an ideal quadratic phase modulationrocess. According to inequality �4�, to ensure an ef-cient time-to-frequency mapping, the phase factor ofhe time lens must satisfy ��t� �� ��2�8 � 15.7 � 103

ad GHz2. Specifically, the phase factor of the timeens used in this first simulation is fixed to �t � 22.5 �04 rad GHz2. Note that this value can be achievedith current technology. In particular, phase factorsp to �t � 5.623 THz2 rad have been reported.3 Re-ults in Fig. 3 are in excellent agreement with thoseheoretically predicted. As expected, the energy spec-rum of the output pulse �Fig. 3�b� is a replica �image�f the input pulse temporal shape. Note that theime-lens process does not affect the temporal pulsehape �in intensity�. However, the spectral compo-ents of the input signal are redistributed in referenceo the time axis according to the instantaneous fre-uency characteristic of the time lens �straight lineith a slope equal to the phase factor �t�. The spec-

ral Fraunhofer condition ensures that this spectraledistribution occurs in such a way that only a singleominant frequency term exists at a given instant ofime. This effect can be clearly observed in the time–requency representation of the output signal in Fig.�b�: The energy of the signal is distributed in theime–frequency plane along a straight line, corre-ponding to the instantaneous frequency characteris-ic of the time lens. This is also consistent with theime–frequency relation between the temporal objectnd the spectral image defined by relation �5� �thiselation is given by the phase factor of the time lens,.e., by the slope of the instantaneous frequency char-cteristic�. It is worth noting that, as in the case ofhe real-time Fourier transformation process, the di-ect correspondence between the time and the fre-uency domains in the output pulse providesdditional capabilities to further control and modifyhe pulse temporal �spectral� energy distribution �seeef. 8 for a more detailed discussion on this point�.For comparison, Fig. 4 shows the output signal

rom a time lens that provides an strength insuffi-ient to ensure time-to-frequency conversion �i.e., theime lens operates out of the spectral Fraunhoferegime�, assuming the same input signal as in therevious example. In particular, the time lens usedor the simulations in Fig. 4 has a phase factor �t �

� 104 rad GHz2, which does not satisfy spectral

raunhofer condition �4�. In this last case, the spec-ral components of the signal are again redistributedccording to the instantaneous frequency character-stic of the time lens, but the process is now insuffi-ient to ensure an effective separation of thesepectral components in the temporal domain �i.e., tonsure an efficient time-to-frequency conversion pro-ess�. The joint time–frequency representationhows that the output signal energy is distributed inwider frequency band at each instant of time so that

he energy spectrum does not replicate the shape ofhe temporal waveform.

For completeness, we finally conducted a set of sim-lations on time-to-frequency conversion by using theonventional dispersion–time-lens configuration.e assume the same input pulse as in the previous

xamples and the same time lens process as in theast example �Fig. 4�. The dispersive medium is de-igned to fulfill condition �8�, i.e. �� � 1��t � 50s2�rad. Figure 5 shows the time–frequency repre-entations corresponding to the signal at the outputf the dispersive medium �Fig. 5�a� and the signal athe output of the time lens �Fig. 5�b�. As expected,he energy spectrum of the output pulse from theystem �Fig. 5�b� is an image of the input pulse tem-oral shape �Fig. 3�a�. The use of a dispersive de-ice in the system introduces a significant distortionn the temporal shape of the output pulse as com-ared with that of the input pulse. In fact, in con-rast to the case of the single-time-lens configuration,here is no direct correspondence between the tem-oral and spectral domains in the output pulse.

. Experimental Results

igure 6 shows a schematic of our experimental ar-angement used to observe the spectral Fraunhoferegime. An actively mode-locked laser diode with an

ig. 4. Simulation results for pulse propagation through a timeens, assuming the same input pulse as in Fig. 3�a� but operatingutside the spectral Fraunhofer regime. The representationhows the output pulse from the time lens, with the same defini-ions as for Fig. 3�a�.

10 January 2004 � Vol. 43, No. 2 � APPLIED OPTICS 487

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xternal resonator based on a uniform fiber Braggrating �FBG-1� was used as the optical pulse source.he source generated optical pulses at a repetition ratef 0.99 GHz centered at a wavelength of 1548 nm.he generated pulses were non-transform-limited

chirped� nearly Gaussian pulses. These pulses wereubsequently compressed to a time width of �17.5 psy use of dispersion-compensating fiber. The com-ressed pulses were conveniently reshaped by meansf a FBG-based optical pulse shaper, consisting of twoonsecutive uniform FBGs �FBG-2 and FBG-3�. Inarticular, the FBGs were specifically designed to gen-rate a nonsymmetric double pulse �see Fig. 7�b� fromhe input Gaussian pulses. The gratings FBG-2 andBG-3 were written in a boron-doped photosensitiveber by cw UV radiation �� � 244 nm� by use of the

ig. 5. Simulation results for pulse propagation through a con-entional time-to-frequency converter �dispersion–time-lens sys-em� configured to implement time-to-frequency conversion �ashown in Fig. 2�, assuming the same input pulse as in Fig. 3�a�.a� Output pulse from the dispersive medium, with the same def-nitions as for Fig. 3�a�. �b� Output pulse from the time lenssystem�, with the same definitions as for Fig. 3�a�.

88 APPLIED OPTICS � Vol. 43, No. 2 � 10 January 2004

hase-mask technology. The FBGs were 0.3 mm longnd were spaced apart by 0.3 mm. The measuredeflectivities of the gratings were 5.3% and 4.8%, re-pectively. To obtain the desired temporal opticalaveform �nonsymmetric double pulse, as shown inig. 7�b�, it was necessary to introduce a phase shiftetween the reflection coefficients corresponding to thewo individual gratings, FBG-2 and FBG-3. For thisurpose the grating structure was conveniently pro-essed with UV radiation after its fabrication �i.e., therimming process�. During the measurement stage,he required fine adjustment was achieved by temper-ture control of the gratings with a thermoelectricodule.A LiNbO3 electro-optic modulator, driven by a sinu-

oidal RF modulation signal, was used as the time-lensechanism.2,4,6 The modulation frequency and mod-lation index �amplitude� were fixed to �m � 2 9.9Hz rad and A � 2.4 rad, respectively. With thesealues, the phase factor of the time lens can be esti-ated2 as ��t� � A�m

2 � 9286.27 GHz2 rad. Notehat the two RF signal generators used to drive theode-locked laser diode and the electro-optic phaseodulator �time lens�, respectively, were synchronized

y means of a signal operating at 10 MHz. We alsoote that the required synchronization between the

ncoming optical pulses and the modulation RF signaln the time lens was ensured by use of a RF phasehifter. Finally, the optical signals were measured inhe temporal domain with a fast photodetector �PD�ollowed by a sampling oscilloscope, both providing aandwidth of �50 GHz. For their measurement inhe spectral domain, we used a conventional opticalpectrum analyzer providing a resolution of �� � 0.015m. Figure 7�a� shows the measured energy spec-rum of the input pulse �before the time lens�. Theotal bandwidth of the input optical pulse is estimatedo be �� � 0.5 nm ��� � 2 62.5 GHz rad�. For thearameters used, the spectral Fraunhofer conditioninequality �4� is satisfied, but in the form ��t� � ��2� . Figure 7�b� shows the measured energy spectrum

ig. 6. Schematic of the experimental setup. Solid �dotted� linesre used for optical �electrical� signals. FBG, fiber Bragg grating;DFA, erbium-doped fiber amplifier; C, fiber coupler; PC, polar-

zation controller; PD, photodetector; Synch, synchronization.

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f the output pulse �after the time lens, solid curve�.or comparison, the measured temporal waveform ofhe input �output� pulse is also shown in Fig. 7�b� �dot-ed curve�. The time and wavelength scales in Fig.�b� are related according to relation �5� by t t0 ��2 c��0

2�t��� �0�, where c is the speed of light inacuum and t0 and �0 are the central instant and theavelength of the optical pulse, respectively. Our ex-erimental results confirmed our theoretical predic-ions. As expected, the energy spectrum of the opticalulse at the output of the time lens is an image of theemporal optical waveform at the input of the timeens, according to the relation given by inequality �4�.n other words, an efficient time-to-frequency conver-ion process is achieved. Note that the sign of theime-to-frequency scale conversion is fixed by the signf the phase factor �t; i.e., it depends on the half-periodhat is chosen for the modulation RF signal in the timeens.

ig. 7. Experimental results: �a� measured spectrum of the in-ut pulse to the time lens; �b� measured spectrum of the outputulse from the time lens �solid curve, top scale� and measuredemporal waveform of the input pulse to �output pulse from� theime lens �dotted curve, bottom scale�.

As previously discussed, the temporal resolutionrovided by our imaging system can be estimatedrom inequality �7�. Specifically, the time lens usedn our experiments provides a phase factor ��t� �286.27 GHz2 rad, which fixes a temporal resolutionf �t � 1����t� � 10 ps. This resolution is of therder of that estimated from our experimental re-ults. Note that the following nonidealities in theystem introduce additional errors: �i� the spectralraunhofer condition �inequality �4� is not strictlyatisfied; i.e., as mentioned above, it is satisfied as ��t�

��2�8 ; and �ii� the temporal duration of the inputptical pulse ��75 ps� exceeds the time aperture �a ofhe time lens2 ��a � 1��m � 16 ps�. As mentionedbove, the temporal resolution provided by the sys-em can be improved by increasing the magnitude ofhe phase factor ��t�. In general, alternative tech-ologies for implementing the time lens, such ashose based on nonlinear processes,3,5,7 permitchieving much larger phase factors than the electro-ptic technology used herein and consequently havehe potential for further improving the achievableemporal resolutions. As an example, a time lensuch as that reported in Ref. 3 ��t � 5.623 THz2 rad�ould be used to implement time-to-frequency conver-ion of optical pulses with a temporal resolution of therder of a few hundreds of femtoseconds ��420 fs, asetermined by inequality �7�.

. Conclusions

n summary, we have analyzed in detail a new regimethe spectral Fraunhofer regime� in the interactionetween optical pulses and time lenses. In this re-ime the temporal waveform of the input pulse isapped into the spectral domain by the action of a

ingle time lens. With this idea, simplified and op-imized time-to-frequency converters for measuringltrafast optical waveforms in the spectral domainan be implemented. More importantly, the con-epts explored herein should prove to be very usefulor other applications in the areas of optical signalrocessing and optical pulse shaping.

This research was supported by the Fonds de re-herche Nature et Technologies of the Quebec Gov-rnment �Canada�, by the Division for Researchunds of the Israel Ministry of Science.

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