Information processing with longitudinal spectral ...emerald.ucsd.edu/Docs/SapersteinAO2008.pdf · Information processing with longitudinal spectral decomposition of ultrafast pulses

Information processing with longitudinal spectraldecomposition of ultrafast pulses

Robert E. Saperstein* and Yeshaiahu FainmanElectrical and Computer Engineering Department, University of California, San Diego, 9500 Gilman Drive, La Jolla,

California 92037-0409, USA

*Corresponding author: [email protected]

Received 23 April 2007; revised 9 August 2007; accepted 10 August 2007;posted 14 August 2007 (Doc. ID 82277); published 27 September 2007

We describe what we believe to be novel methods for waveform synthesis and detection relying onlongitudinal spectral decomposition of subpicosecond optical pulses. Optical processing is performed inboth all-fiber and mixed fiber–free-space systems. Demonstrated applications include ultrafast opticalwaveform synthesis, microwave spectrum analysis, and high-speed electrical arbitrary waveform gen-eration. The techniques have the potential for time–bandwidth products of �104 due to exclusive relianceon time-domain processing. We introduce the principles of operation and subsequently support these withresults from our experimental systems. Both theory and experiments suggest third-order dispersion asthe principle limitation to large time–bandwidth products. Chirped-fiber Bragg gratings offer a route toincreasing the number of resolvable spots for use in high-speed signal processing applications. © 2008Optical Society of America

OCIS codes: 320.1590, 320.5540.

1. Introduction

Fourier methods are widely used in optical informa-tion processing for time-domain waveform synthesisand detection. In such approaches an input pulse isspectrally decomposed and modulated by an appro-priately designed spectral filter. In general, variousultrafast applications may need either generation orcharacterization of waveforms depending on whethertheir frequency response is or is not known a priori.Ultrafast pulses prove useful in driving or probingsuch systems because their broad, deterministic spec-tral amplitude sees many Fourier coefficients in thesystem transfer function. Traditionally, the space do-main is used to perform optical Fourier processing,where pairs of gratings and lenses are used to decom-pose the spectral content of the pulsed, optical wave-forms [1]. Liquid-crystal and acousto-optic (AO)modulators provide means for introduction of some-what arbitrary system transfer functions [2,3]. Morerecently, compact waveguide-based approaches haveused arrayed waveguide gratings (AWGs) to perform

a spectral decomposition analogous to those achievedwith traditional, free-space coupled gratings [4].

Information processing arrangements using thespace domain for transverse spectral decompositionbenefit from the maturity of such techniques, withrespect to basic understanding, component quality,and the excellent phase control that free-space ma-nipulations provide. Component insensitivity to op-tical power permits the shaping and detecting ofhigh-peak-power pulses used to drive nonlinear opti-cal systems. Furthermore, nonlinear optical elementscan be introduced directly into these processors toprovide unique ultrafast waveform synthesis capa-bilities [5–7] with response times in the range offemtoseconds. Drawbacks to free-space approachesinclude scaling in volume to achieve large time–bandwidth products (TBWPs) and limiting the tem-poral extent of waveforms coupled back into asingle-mode fiber (SMF). The origin of this last re-straint is the time–space interrelation inherent in tra-ditional pulse-shaping devices [8]. Spatial–temporalpulse shapers utilizing diffraction gratings and lenseshave a spatial offset between advanced and delayedcomponents in the transverse mode of the output op-tical waveform. A subsequent launch into the output

0003-6935/08/0400A21-11$15.00/0© 2008 Optical Society of America

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fiber windows the spatial mode and thus also win-dows (i.e., truncates) the created pulse shape in time.AWG approaches rely on the complex waveguide el-ements to circumvent large processor volumes andtime–space coupling issues, but they too are practi-cally limited by fabrication requirements. The num-ber of resolvable spots in the output waveform forboth Fourier synthesis and direct space-to-time AWGapproaches [4,9] is equal to the number of waveguidechannels. An alternative approach to Fourier pro-cessing relying on a single transverse spatial modecan better integrate with fiber systems and will scalein length only for the achievement of large TBWPs.

Our current research concentrates on creatingfiber-based and fiber-integrated processing tech-niques for optical waveforms that utilize standardoptical communications equipment. A Fourier syn-thesis approach succeeds in the time domain by usingchromatic dispersion for longitudinal spectral decom-position. Drawing on the identical mathematicaltreatments of diffraction in space and dispersion intime [10], an approximate Fourier transform (FT) ofan incident optical signal is achieved via chromaticdispersion after reaching the so called far-field ap-proximation. With such an amount of dispersion, thetemporal waveform closely resembles the spectrum[11]. We call such a waveform a longitudinal spectraldecomposition wave (SDW). A time-variant element(e.g., a modulator) filters this SDW, and conjugatingthe dispersion-based optical system through propa-gation in a dispersive source matched (i.e., of oppositesign but equal magnitude) to the first dispersive ele-ment allows us to recompose the pulsed waveform.The number of resolvable spots that the processoroffers then relies on a suitable modulation schemeand the experimental ability to disperse a subpico-second pulse using second-order dispersion. Thefiber-based approach offers a route to an in-linepulse-shaping operation spanning long propagationdistances as detailed in Section 2. The potential forlarge TBWPs under the extended time apertures oflongitudinal SDWs is broadly appealing and is high-lighted in the high-speed microwave-photonic pro-cessing applications of Sections 3 and 4. Section 5introduces distortions affecting all longitudinal SDWprocessors and offers an alternative dispersion tech-nology for TBWP maximization. Conclusions are pro-vided in Section 6.

2. Optical Pulse Shaping

A number of science and engineering applications,including coherent quantum control and optical com-munications, motivate research in optical pulse shap-ing [12,13]. Fourier synthesis methods are broadlyimplemented to achieve desired shaped pulsed wave-forms with subpicosecond temporal features. In suchan approach the produced optical signal is composedof a waveform proportional to the input pulse con-volved with the FT of the applied system transferfunction. Transverse SDWs that use free-space com-ponents offer an attractive level of power handlingand phase stability for coherent control applications,

but their time aperture scales with the volume of thesystem and may suffer from space–time coupling.Our all-fiber-based pulse shaper offers an alternativeroute to creating shaped waveforms, in which largeTBWPs are achieved through the manipulation ofwaveforms in dispersion-compensated links. Experi-mentally altering dispersion-induced SDWs to per-form pulse shaping has been described [14] andshown in mixed fiber–free-space systems [15]. Herewe demonstrate an all-fiber realization for pulseshaping operating over dispersion-compensated fiberlinks [16].

A schematic of our approach to pulse shaping usinglongitudinal SDWs is shown in Fig. 1. To understandthe technique, consider a subpicosecond transform-limited pulse at z � 0 with its spectrum U�0, �� prop-agating along the z axis of a fiber. Here � representsa shifted spectrum around the carrier frequency, i.e.,� � � � �c if � is the absolute frequency. The opticalfield at the output of the first dispersive fiber element,assumed to provide anomalous dispersion (�2 � 0,where �2 is the second-order term in the Taylor ex-pansion of the propagation constant), is representedthrough the multiplication of the spectrum of thepulse at z � 0 with a quadratic phase, yielding

U�z, �� � U�0, ��exp��j12 �2z�2�. (1)

The strength of the quadratic phase term dependson the value of �2 and on z, the propagation length. Inthe time domain, pulse propagation [i.e., Eq. (1)] isrepresented by a Fresnel integral, which relaxes tothe so called far-field approximation when the condi-tion |�2z| �� �2�2, where � is the transform-limitedpulse half-width, is satisfied. In this regime the time-domain signal represents the spectral amplitudeclosely. Thus the relationship |�2z| �� �2�2 indicatesthe propagation requirements needed to achieve aSDW in the time domain. With dispersion in SMF of�2 � �20 ps2�km and pulses of �0.1 ps duration, thefar field is achieved within only a few meters of fiber.

Modulating the temporal SDW with ideal signalv�t� � A B cos��0t� using an electro-optic (EO) de-vice and then propagating through a dispersion-compensating element providing matched standarddispersion leads to an output waveform in the timedomain [16]:

Fig. 1. (Color online) General pulse-shaping approach using dis-persive fiber and an electro-optic modulator for signal processing.

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Uout�z, t� � AU�0, t�exp�j�ct� B exp��j12 �2z�0

2� U�0, t � �2z�0�expj��c �0�t

B exp��j12 �2z�0

2�U�0, t �2z�0�

expj��c � �0�t. (2)

We make an assumption that the period of the mod-ulating signal, T0 � 2���0, is less than the timeaperture of the SDW. In Eq. (2) the formation of threepulses is recognized: one advanced and one delayedfrom a central pulse. Ignoring the constant phaseterms, Eq. (2) shows that the temporal shifts of thesatellite pulses are proportional to the strength of �2,z, and �0. Larger chromatic dispersion or higher mod-ulation frequency leads to greater shifts in time. Be-cause the output of a longitudinal SDW FT pulseshaper is the input pulse convolved with the FT of themodulating spectral filter, intuitively, the pulses inEq. (2) correspond to the three frequency componentsof the modulating signal, v(t). Thus, by controlling thespectrum of the electronic signal applied to the mod-ulator, an arbitrary temporal optical waveform canbe created. A linear phase term modulating each sat-ellite pulse is seen to upshift and downshift the tem-poral bandwidth of the delayed and advanced pulses,respectively, by the modulation frequency. With theuse of an EO modulator for signal filtration, theseDoppler shifts may approach tens of GHz. Althoughfrequency shifts from the carrier are present in thesatellite pulses created in traditional pulse-shapingmethods as well, such shifts are smaller becauseslower modulators are utilized. To reduce the fre-quency shifts of the satellite pulses in our technique,a larger dispersion-induced time aperture is needed.Although not critical to all pulse-shaping applica-tions, this dependence of frequency shifts on temporalshifts requires consideration when performing signalprocessing.

The intensity waveform resulting from a simula-tion that follows the model developed by Eq. (2) isshown in Fig. 2(a). The modulating signal is identical

to v(t) used in Eq. (2) where the coefficients are suchthat the dc component is stronger than the micro-wave signal �A � B�. The launched 150 fs pulse wasstretched to and compressed from 1 ns with purelysecond-order dispersion, and the modulating toneis set such that �0 is equal to 2� 7 Grad�s. Satellitepulses should appear at temporal locations, t � �2z�0, which correspond to roughly 2.3 ps as �2z� 52 ps2. Because of the difficulties in measuringoptical waveforms with subpicosecond features, Fig.2(b) shows the modeled results of a practical solutionto detection: an intensity autocorrelation (with back-ground) of the signal in Fig. 2(a).

The experimental approach to realize pulse shap-ing follows the schematic in Fig. 1. A Coherent MIRApumped optical parametric oscillator delivers 150 fstransform-limited pulses at 76 MHz and 1 nJ of en-ergy per pulse. The matched dispersive elements arespools of SMF and a dispersion-compensating fiber(DCF) module. The operating wavelength is tuned to1.55 �m to minimize attenuation and realize sub-stantial dispersion. A grating stretcher precedes cou-pling into the SMF to avoid optical nonlinear effectsassociated with the highest peak intensities. Approx-imately 2.6 km of SMF are employed to disperse thepulse to a nanosecond FWHM. The DCF module pro-vides �44 ps�nm of dispersion compensation. Exper-imentally, launching pulses of the order of 100 fsrequires that all fibers and prechirping devices bematched with high precision. The length of SMFmust be tailored to within a few meters of the ideallength to measure a subpicosecond pulse at the out-put. Spectral filtering via modulation is performed bya 10 GB�s LiNbO3 Mach–Zehnder intensity modula-tor. The output waveform from the DCF is measuredusing a Michelson-interferometer-based autocorrela-tor. To view the interferometric autocorrelation trace(ACT), an amplified silicon photodiode is operated ina nonlinear detection mode via two-photon absorp-tion (TPA) [17]. The TPA detection scheme pro-vides useful phase sensitivity that gives experimentalverification of full pulse recompression after propaga-tion in the DCF. The nonlinear detection also facili-tates detection of the time–frequency-shifted satellitepulses since the cross correlation of these pulses(within the autocorrelation of the entire waveform)will average to the background level with a slow lin-ear detector. The intensity requirements of the TPAprocess, however, require that an erbium-doped fiberamplifier (EDFA) be used in the system.

Detected results from the experimental setup de-scribed above are given in Fig. 3. The two intensityautocorrelations are generated through digital post-processing of measured interferometric ACTs. Inboth cases the radio-frequency (rf) signal generatorsupplied a 7 GHz sinusoid with 15 dBm electricalpower. In the experiments represented in Fig. 3(a)the bias to the modulator is controlled such that astrong dc (unmodulated) component is allowed topass. Relating to Eq. (2), the optical field can be de-scribed with coefficient A greater than coefficient B.

Fig. 2. (a) Modeled intensity of a temporal waveform output fromthe system as described by Eq. (2). Time is designated relative tothe central pulse corresponding to the constant (dc) componentof the modulating signal v(t). The satellite pulses relate to themicrowave sidebands, 7 GHz. Here the microwave signal isweaker than the dc. (b) Modeled intensity autocorrelation trace(with background) of the optical waveform in (a). The small pulsesoccurring at 4.7 ps are cross correlations of the pulses at 2.3 psin (a). The cross correlation of the dc and satellite pulses in (a) formthe pulses at 2.3 ps in (b).

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The temporal shift, t � �2z�0, should correspond to 2.3 ps. Correspondingly, Fig. 3(a) shows close sim-ilarity to the modeling output of Fig. 2(b). Here againthe small pulses at 4.7 ps in the ACT are formed bythe cross correlation of satellite pulses at 2.3 psin the output optical waveform. The strong centralpulse in the output field cross correlating with a sat-ellite pulse leads to the pulses at 2.3 ps in the ACT.In Fig. 3(b) the modulator is null biased to completelyremove the unmodulated optical carrier component.The output field is described by setting A to zero inEq. (2). Confirmation of the near full modulationdepth is given by the complete removal of visiblepulses at 2.3 ps in the detected intensity ACT. Theelevation of the signal background in Fig. 3(b) from 1to just less than 1.2 is evidence of an uncompressedbackground arising from self-phase modulation anddispersion. The background is estimated to containapproximately 75% as much power as the signal.Here the localized loss of the null-biased modulatorled to a compensating increase of power above thenonlinear threshold at another location in the sys-tem. This background is undesirable and can beavoided, as seen in Fig. 3(a), with stricter power bud-geting and avoidance of nonlinear interactions. Ofcentral importance, the trace in Fig. 3(b) points to anunavoidable distortion that creates a departure fromthe 2:1 ratio of the central pulse to side pulses. Twoequally intense pulses should produce a 2:1 ratiowhen autocorrelated. Higher-order dispersion termsin the fiber propagation introduce signal-dependentbroadening leading to a distortion mechanism fur-ther detailed in Section 5.

The figure of merit to describe the pulse shaper isits TBWP. For the fiber-based pulse-shaping method,the maximum TBWP relates to the ratio of the timeaperture of the SDW to the transform-limited pulseduration (bandwidth equivalent). For nanosecond-scale time apertures, this number exceeds 103. How-ever, a maximum modulator speed of �100 GHzlimits the number of resolvable spots in the output byreducing the maximum temporal shifts. TBWP couldthen be defined as the ratio of maximum modulatorspeed to minimum achievable modulation speed (de-fined as the inverse of the time aperture of the SDW).

For nanosecond scale SDWs our TBWP would exceed102. To return to 1000 resolvable spots, �10 ns timeapertures are required. Limiting progress is the chal-lenge of recompressing large-time-aperture SDWswith DCF. The ultimate observed TBWP limitationcomes from the presence of aberrations associatedwith higher-order dispersion. As mentioned above,and clarified further in Section 5, higher-order dis-persion drastically reduces the usable range of tem-poral shifts.

The pulse shaper creates copies of the originalpulse at temporal shifts corresponding to the spectralcomponents of the modulating signal, indicating thatthe processor can function as a high-bandwidth mi-crowave spectrum analyzer [18]. Here the spectralresolution figure of merit is determined by two satel-lite pulses that can be temporally resolved. Thisrequirement is satisfied to first order if their modu-lation frequency difference is greater than the in-verse of the time aperture of the SDW. For microwavespectrum analysis applications as described in [18],one can determine the microwave spectral content ofan unknown signal v(t) by examining the temporalwaveform output from the system while scaling timeto frequency by � � t���2z�.

3. Microwave Spectrum Analysis

Analysis of microwave signals in the gigahertz toterahertz range is of critical importance to a varietyof fields including radar and lidar, imaging, and as-tronomical detection [19,20]. Commonly employedelectronic techniques are limited in speed and rely onchannelization of the signal bandwidth to providehigh-resolution spectrum analysis. Microwave pho-tonics methods for spectrum analysis offer the po-tential to achieve wide instantaneous bandwidth incompact designs [18,21]. Recognizing the utility ofthe pulse-shaping method in Section 2 for microwavespectrum analysis, we modify that technique to dem-onstrate an approach for microwave spectrum anal-ysis that exploits the bandwidth of longitudinalSDWs to process microwave signals without the tech-nical challenge of recompressing the ultrafast opticalsignal. Specifically, a straightforward method for mi-crowave spectrum analysis is realized with ultimatepotential for analyzing signals with bandwidths ap-proaching 100 GHz and with resolution less than 10MHz [22].

A schematic diagram of our approach is shown inFig. 4. The front end is similar to that of the fiber-based pulse shaper. After large initial dispersion, theSDW is separated into two copies, whereby one ismodulated by the microwave signal of interest, andthe other copy is given an adjustable temporal delay.Upon recombination and square-law detection, thedelayed, dispersed pulse copies create an intermedi-ate frequency (IF) in the generated photocurrent. Tofirst order, a unique temporal beat frequency existsbetween the two SDWs in relation to the time delayand strength of dispersion. For microwave signal pro-cessing, the IF serves as a delayed-tuned local oscil-

Fig. 3. Experimental intensity autocorrelation traces of signalswith 7 GHz modulation. The rf power is maintained at 15 dBmwhile bias voltage is varied. (a) ACT in the presence of a strong dccomponent in the modulating signal v(t). Experimental resultshows agreement with the trace obtained from our linear model inFig. 2(b). (b) ACT when the modulator was null biased. The auto-correlation process leads to the formation of three pulses as theinner two satellite pulses drop out.

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lator (LO) sweeping over the modulation spectrum.Using a time-averaged detector as a low-pass filter(LPF), the system isolates that microwave signalcomponent that is homodyned by the variable LO. Atrace over all delays, equivalent to a cross correlationof the two pulses, shows correlation peaks for allspectral components of the microwave signal. Thisarrangement is analogous to a rf superhomodyne re-ceiver.

An analytic description of the microwave spec-trum analyzer proceeds in the time domain. Linearchromatic dispersion introduces a quadratic spectralphase, and therefore in the time domain an ultra-short pulse is convolved with a quadratic phase ofstrength proportional to �2 and z. Equation (3) de-scribes this Fresnel transform on the ultrashort pulseafter it is relaxed by the Fraunhofer approximation inthe temporal far-field:

U�z, t� � ejt2

2�2z���

�

U�0, t��e�jt

�2zt�dt�

� ejt2

2�2z F�U�0, t�� t2��2z

. (3)

The pulse described by Eq. (3) is split into two copies:The amplitude of one copy is modulated by a micro-wave signal, s(t), while a delay, �, is introduced ontothe second one. Recombining the two pulses yields asignal:

Ur�z, t� � ej�t���2

2�2z F�U�z, t�� t��

2��2z

ejt2

2�2z s�t�F�U�z, t�� t2��2z

. (4)

For small delays the pulses overlap nearly com-pletely, and the SDW envelope shape becomes of sec-ondary importance. Detection of the received signalwith a square-law photodiode produces a cross termwith a linear phase proportional to the delay. Timeaveraging the cross term by the detector for time, T,leads to a signal:

1T�

0

T

�F�U�0, t� t2��2z�2s�t�cos�2�� �

2��2z�t ��2

2�2z�dt.

(5)

Thus, as the time integration is lengthened, the de-tection will isolate the component of s(t) oscillating ata microwave frequency, ���2��2z�. By varying thedelays, all the spectral components of s(t) can be de-termined from the correlation process. The additionalconstant phase related to �2 in the cosine term isnegligibly small.

Figure 5 shows experimental results from a proof-of-concept experiment. We implement the system byfirst generating 150 fs pulses at 1.55 �m in a MIRAoptical parametric oscillator laser system. The pulsesare dispersed in a free-space grating stretcher to thepicosecond scale to lower peak power before they arelaunched into 25.2 km of SMF. An optical spectrumanalyzer is used to monitor and confirm no change inthe spectrum of the pulse at the output of the fiber toensure that the propagation is linear. The generatedlongitudinal SDW is over 6 ns in duration. Thepolarization of the SDW is set with a polarizationcontroller and a fiber-coupled linear polarizer. Thiswaveform enters a free-space Mach–Zehnder ar-rangement where it is split; one copy is controllablydelayed while the other copy enters a 1 GHz LiNbO3modulator. The modulator is driven by a rf source,which produces a 760 MHz tone locked in phase tothe 76 MHz repetition rate of the laser (see the fol-lowing discussion). A bias accompanies the rf tone.Detection of the cross correlation is recorded in anInGaAs photodiode. The upper axis of Fig. 5 providesthe correlation delays as recorded, whereas the loweraxis makes use of the conversion of delay to fre-quency, f � ���2��2z�. The corresponding spectrum of

Fig. 5. Correlation trace from experimental proof of concept.The rf signal consists of dc and 760 MHz components. Results arefiltered in postprocessing to remove background fluctuations. Theupper axis gives temporal correlation delays as recorded. Thelower axis shows a scaling of the delay axis to frequency �f �

��2��2z�.

Fig. 4. (Color online) Schematic diagram of the microwave spec-trum analyzer shown. The signal voltage is a single rf tone with abias. The correlation output shows the two sidebands and thecentral dc spike.

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s(t), as read from the lower axis, confirms the dc termand oscillatory terms at 760 MHz. Viewing the up-per axis, the trace has peaks at the central point and 2.4 ps (2��2zfm � 2.4 ps if �2 � 20 ps2�km, z �25.2 km, and fm � 7.6 10�4 THz).

The requirement of time averaging introduces anexperimental trade-off. In order that the LPF opera-tion remains narrowband, long time integration isnecessary. This additional time not only introduceslatency but also implies that more pulses from thepulse train will fall on the detector in a given inte-gration window. The spectrum of the ultrashort pulsesubsequently becomes discretized where the combfunction multiplying the optical spectrum has a fi-nesse related to the integration time. With respect tothe operation of the microwave spectrum analyzer,the LO is discretized because the beat between apulse pair can only constructively add to that of thenext pair in the train at frequencies corresponding tomultiples of the laser repetition rate. Ultimately,spectrum analysis produces a sampled version (Fou-rier series) of the microwave spectrum of s(t). To pro-duce a measurable signal when multiple pulses areaveraged, our demonstrator analyzes a rf tone de-rived from a 10 multiplication of the repetition rateof the laser. However, low-repetition-rate mode-locked laser pulses with large subsequent dispersioncan generate SDWs that fill the entire integrationwindow. If integration is matched to the pulse repe-tition period, the probability of intercept goes to unitywhile LO spectral discretization and the signal phas-ing requirements vanish.

The resolution of the method is locked to the timeaperture created by the SDW and hence the length ofpropagation in the dispersive element. To achieve 10MHz resolution 100 ns SDWs are needed. Althoughthe LO can be tuned to frequencies approaching thebandwidth of the ultrashort pulse, at such delays thepulse pair is incapable of providing full LO resolution.The lack of pulse overlap reduces the effective timeaperture and, by the uncertainty principle, degradesresolution. Current technologies limit EO modulationto the 100 GB�s and slower range. For such micro-wave signals, the amount of delay required to gener-ate the LO is less than 10% of the pulse width, thusensuring large pulse overlap. More significant thanpulse overlap, higher-order dispersion can reduce theTBWP by orders of magnitude as the LO resolutiondegrades quickly at low frequencies. The effect is de-tailed further in Section 5. Note that using a purelylinear dispersive element, the TBWP of such a devicecould exceed 10,000 resolvable spots. To avoid theneed for scanning and fully take advantage of thelarge instantaneous bandwidth short pulses offer,the space domain must be exploited. In such a man-ner, variable delays are introduced in parallel andarrayed detection is employed to achieve correlationin space [23].

4. High-Speed Microwave Signal Generation

Many applications can exploit the high-speed IF gen-erated through the interference of two delayed, dis-

persed ultrafast optical pulses. In contrast to themicrowave spectrum analysis approach pursued inthe former section, a high-speed photodection systemcan convert the IF into an oscillating photoelectriccurrent [24]. Such an approach is shown as a pathtoward linear synthesis of ultrawideband (UWB) mi-crowave signals with arbitrary waveform.

UWB signals are receiving much attention fortheir potential use in both radar systems and high-speed data links [25]. Their large fractional bandwidthprovides multipath immunity, low probability of inter-cept, and high-precision ranging capabilities. Low-frequency spectral components also offer significantground and material penetration. Conventional meansfor generation of UWB signals are limited due to therelatively slow speed of electronics for digital-to-analogconversion. Currently available arbitrary waveformgeneration devices operate up to only a few GHz. Thedesire to create UWB signals with carrier frequen-cies in the tens of GHz and fractional bandwidthsapproaching 100% suggests investigating microwave-photonic approaches. A number of hybrid optical–electronic techniques for microwave- and millimeter-wave arbitrary waveform generation have beenexplored. Pulse waveform compression using disper-sion was used to spectrally shift a 10 GHz tone to 15GHz [26] but was limited in achievable frequencyshift due to rf signal distortion. A similar approachwith adaptive processing and an optical supercon-tinuum source produced rf signals in the 1–12 GHzrange [27]. These methods were further modified foroperation with the Talbot effect to produce highlycompressed periodic waveforms [28]. However, allthree of these methods are limited to generation ofmicrowave sinusoids where signal modulation is re-stricted to frequency variation. Space-to-time opticalmethods offer an alternative approach to microwavearbitrary waveform generation [29–31], but thesemethods are limited to the TBWP of spatial pulseshapers. In these techniques, delayed and weightedoptical pulses serve as samples for a desired electricalwaveform, which is generated through subsequentlow-pass filtering via detection of the pulse train en-velope in a photodiode. A promising approach produc-ing microwave tones through isolation and mixing ofcomb lines of a mode-locked optical source has dem-onstrated generation of a 12.4 GHz microwave carrierwith nanosecond-scale envelope features [32]. Thisapproach suggests a system to modulate the individ-ual comb lines and mix via photodetection for com-plete microwave arbitrary waveform generation.However, the underlying shortcoming is that all themicrowave tones are locked to the repetition rate ofthe original mode-locked source. Thus the achieve-ment of an arbitrary carrier frequency is greatly lim-ited unless a serrodyne technique is employed. Thesystem also loses power contained in all unused comblines.

Our work demonstrates a method for electricalarbitrary waveform generation with independentenvelope and carrier control through the manipu-lation of longitudinal SDWs in the time-domain pro-

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cessor of Fig. 6 [33]. As in Section 3, the basicprocessor unit is an asymmetric Mach–Zehnder in-terferometer. Whereas the microwave spectrumanalysis technique relied on a slow integrating de-tector, the arbitrary waveform generation tech-nique requires a fast photodiode to convert the IFinto an electronic carrier frequency. Thus, to firstorder, at the output of the processor, photodetectionand filtering creates a bipolar electrical signal withcarrier, fm � ���2��2z�. An EO modulator in one armof the Mach–Zehnder interferometer defines the mi-crowave envelope. The time aperture of the longitu-dinal SDW sets a lower limit on the slowest realizablefeature size or carrier frequency. Concurrently, thelarge optical bandwidth ��1 THz� makes fast carriergeneration achievable; however, this number is prac-tically limited by available photodetector speed. En-velope features imprinted on the pulse copy with theEO modulator are limited by the speed of the signalgenerator and modulator. An extension to signal gen-eration with large fractional bandwidth for UWB isachievable if the processor were expanded to includemultiple channels. As an example linear synthesis ispossible with the introduction of a second modulatedand delayed path, as suggested in Fig. 6, creating atthe output a superposition of three modulated micro-wave carriers.

Our experimental approach is identical to that ofSection 3 except for the use of an inline EDFA afterthe fiber spool and detection using a sampling oscil-loscope with a 30 GHz optical detector head. Addi-tionally, a high-pass filter is used to isolate the signalof interest from the dc term associated with the SDWenvelope. Because of the nature of the sampling os-cilloscope, the modulator is again driven by sinusoidsderived from a harmonic of the laser repetition rate.Figure 7(a) shows an oscilloscope trace of the detected

waveform for a delay of � � 14.5 ps. The generatedcarrier is approximately 4.5 GHz. The subcarrier at760 MHz is clearly visible. Large modulation depth isachieved by controlling the bias to the modulator.Figure 7(b) shows a fast Fourier transform (FFT) ofthe detected signal samples.

The technique demonstrates high-speed micro-wave signal generation. The system offers indepen-dent control of the carrier and envelope. The TBWPof such a system is limited by the achievable timeaperture (10–100 ns) and maximum photodetectorspeed ��100 GHz� to values in the range of 103–104.Practical limitations to TBWP stemming from higher-order dispersion are even more restrictive. These willbe discussed in Section 5. Additional channels wouldgreatly improve the fractional bandwidth of generatedsignals, ultimately providing a path toward microwavearbitrary waveform generation.

5. Distortion to Longitudinal Spectral DecompositionWave Systems

The principle distortion source for systems process-ing longitudinal SDWs is the nonuniform mapping ofthe optical spectrum to the temporal position withinthe SDW. The problem arises if the dispersive me-dium that utilizes quadratic dispersion (i.e., �2) toconvert a transform-limited ultrashort pulse into alinearly chirped longitudinal SDW has significanthigher-order dispersion terms in the Taylor expan-sion of the propagation constant (i.e., �3, �4, etc.). Forthe case of standard single-mode and dispersion-compensating fibers employed in our demonstrators,at our specific operating wavelength and bandwidththe most significant of these is �3. The resulting dis-tortion brings significant reduction in TBWP to eachof our waveform synthesizers and analyzers dis-cussed above. To restore the TBWPs of our proces-sors, a dispersion-free source of higher-order termsmust be employed. In this direction we experimen-tally highlight the capabilities of chirped fiber Bragggratings (CFBGs) to provide pure second-order dis-persion, and we address the principle hindrance totheir employment, strong second-order polarizationmode dispersion (PMD).

For our systems, Eq. (1) should be modified to re-flect the presence of �3:

Fig. 6. (Color online) Schematic of our proposed method for elec-trical arbitrary waveform generation. Longitudinal SDWs are de-layed and modulated before mixing in a fast square-law detector.A bandpass filter separates the dc terms to create a bipolar signal.The superposition of channels leads to a linear synthesis of mod-ulated microwave carriers at the output.

Fig. 7. (a) Oscilloscope recorded waveform from single-channelrealization of an electrical arbitrary waveform generation tech-nique. A 3 GHz high-pass filter removes the slow dc signal pro-ducing a bipolar waveform. (b) FFT of recorded waveform showsthe 4.5 GHz carrier and 760 MHz envelope feature.

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U�z, �� � U�0, ��exp��j12 �2z�2�exp��j

16 �3z�3�.

(6)

The new phase term induces a quadratic mappingof the spectral component to the temporal position inthe longitudinal SDW. This term then distorts theotherwise linear mapping provided by the �2 term.Because the effect on the dispersive stretch-and-compress processor is conceptually different from thestretch-only methods, an examination of each typewill elucidate the role of �3 in TBWP reduction.

The optical pulse shaper creates delayed copies ofthe original pulse through frequency shifting the sig-nal at the last precompensation point of a dispersion-compensated link. Although we introduced thetechnique for ultrafast optical waveform synthesis,the general approach is receiving attention as amethod for slow-light optical buffering of longer, op-tical communication pulses [34–37]. This communitylikes to refer to frequency shifters (EO modulators) astime prisms. Recall that the principle of operation,introduced in Section 2, is to introduce a variablegroup delay through upshifting or downshifting thecarrier frequency before the compensating fiber,where �2 gives the desired group velocity adjustmentto the frequency-shifted signal. Similarly, if �3 ispresent, it results in a group velocity dispersion ad-justment. Thus time delayed or advanced pulses willsee residual uncompensated group velocity disper-sion, which will cause pulse broadening. Detailedmathematical treatment of these processes is de-scribed in [16]. The net result is that pulse time shiftsgreater than a certain duration will result in a con-volution smearing of the satellite pulse energy tobackground levels. As a figure of merit we introducethe pulse shift (delay or advance) that results in pulsewidth doubling as

Ts � �3To2��2��3�. (7)

For our fibers where �2 � �20 ps2�km, �3 � 0.1ps3�km, and To � 0.1 ps, then Ts � 3.4 ps. Thus apulse shifted by 34 pulse widths will encounter apulse width doubling. Note that longer initial pulsewidths like those used in optical communications in-cur reduced �3 spreading, but their time delay comesat the expense of larger �2z products and thus in-creased system latency and loss. Experimental evi-dence of the �3 delay-induced spreading is shown inthe four overlaid intensity autocorrelation traces ofFig. 8. These ACTs are similar to the result in Fig.3(b) with modulation frequencies of 3, 4, 5, and6 GHz. The EO modulator is null biased to producetwo pulses in the output field and three pulses in theACT. Incurred nonlinear penalties lead to an uncom-pressed background as is detailed in Section 2 for thedescription of Fig. 3(b). In the ACTs of Fig. 8, largerfrequency or temporal shifts introduce larger tempo-ral spreads in the detected pulse pairs. The departure

from the 2:1 central-pulse-to-satellite-pulse ratio inthe ACTs provides evidence that the advanced pulsein the field was less broadened than the delayedpulse. This effect arose through interplay betweenthe �3 effect and residual uncompensated �2. At theoutput, spreading from uncompensated �2 is thesame for all shifted pulses, but spreading due to �3 ispulse position dependent. As the two effects super-pose, a single pulse shift will see minimum spread-ing. Thus, although �3 is generally an aberration, itcould be useful in creating asymmetric waveforms.

The microwave spectrum analyzer and waveformsynthesizer of Sections 3 and 4 operate by creating asingle-frequency beat tone in the microwave rangethrough the interference of delayed copies of the dis-persed SDW. When �3 is present, a cubic spectralphase produces a higher-order mapping of the spec-trum to time in the SDW. Thus the beat betweendelayed longitudinal SDWs will not form from a con-stant frequency offset at all points under the SDWenvelope. The result is that this IF will be time vari-ant, and the resulting LO for the spectrum analyzeror the carrier for the generated microwave signal willsuffer an increased linewidth. Furthermore, when in-troducing larger delays to increase the LO or carrierfrequency, the linewidth will increase as well, result-ing in rapid resolution decrease and loss of signalstrength. This issue is illustrated schematically inFig. 9.

A potential solution to the problems stemming fromthe use of optical fibers is to employ instead CFBGswith controllable dispersion parameters. Writing tech-niques for long CFBGs are improving, and commer-cially available gratings offer differential group delayswell in excess of 100 ps (shortest path to longest pathdelay relates to grating length) [38,39]. Gratings alsooffer the benefits of low loss, low latency, short inter-action length, and small footprint. A number of works

Fig. 8. (Color online) Overlaid intensity autocorrelations showingthe effect of increased modulation frequency on a pulse-shapedwaveform. The temporal broadening of the 6 GHz up- and down-shifted pulses is apparent when comparing their FWHM to thosepulses originating from 3 GHz modulation.

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have demonstrated passive optical pulse shapingthrough the use of superstructures and spectral filterswritten directly into fiber Bragg gratings [40–43].When using designed CFBGs, the resulting complica-tions of �3 give way to the simplified analysis of Sec-tions 2, 3, and 4. As evidence of their performanceimprovement, Fig. 10 demonstrates the linewidth im-provement to LO–microwave carrier generation of-fered by a CFBG over standard SMF-28. Both aSMF spool and a CFBG were used to disperse a sub-picosecond pulse to �5 ns. The grating offers 340ps�nm of dispersion and was created to have a purelyquadratic spectral phase response, i.e., with �3 � 0.The spool of SMF-28 was 25.2 km long. The time-domain signals created through the interference oftwo delayed pulse copies were recorded using a fastp-i-n diode. No subsequent filtration was per-formed. The FFTs from both SMF (solid curve) andCFBG (dashed curve) are overlaid in Fig. 10(b).Figure 10(a) is a modeling result that is well repli-

cated in Fig. 10(b). Longer delays (higher beat fre-quency) result in greater broadening for SMF butshow little measurable difference from short delays(lower beat frequency) for the CFBG. The improve-ment to microwave carrier linewidth is clearly ob-served.

While CFBGs offer a plan toward processor im-provement with regard to �3 distortion, they are notyet suitable for integration into our demonstrators. Aprinciple hindrance to further usage of CFBGs is thewell-known problem of their birefringence [44]. Thisbirefringence is fundamental and is induced duringthe writing process. Because the birefringent axisrotates along the length of the long grating, it can beconsidered a concatenation of many wave plates. Ad-ditionally, each region of the grating is responsible forreflecting a different spectral component, leading to anet result of spectrally dependent polarization rota-tion. In practice, the effect looks like all-order PMD.Compensating for the polarization rotation requireseither a fast time domain polarization rotator or finestress modifications to the grating. Both approachesprove experimentally challenging. A method for com-plete PMD mitigation through design strategies isdesirable.

The net result of second-order PMD is the poorefficacy of any subsequent polarization-dependentdevices or processes. The most important limitationin our processors is the incompatibility of longitudi-nal SDWs from CFBGs with EO modulators. Thepolarization variation results in both spectrally de-pendent transmittance and spectrally dependentmodulation depth. The severity of the second-orderPMD can be seen in Fig. 11. For visualization, polar-ization states of 30 cw wavelength signals between1535 and 1565 nm are recorded and displayed on thePoincaré sphere. The samples are spaced by 1 nmand have identical initial polarization. Figure 11(a)shows a control where the tunable laser is passedonly through SMF patch cables. Figure 11(b) showsthe polarization state of the light at different wave-lengths after reflection from the CFBG. Clearly, thedistribution of polarization is undesirable. Our cur-rent efforts are focused on obtaining and character-izing CFBGs written in polarization-maintaining(PM) fibers. We hope that the large PM fiber birefrin-

Fig. 11. (Color online) Polarization states displayed on a Poincarésphere for cw signals between 1535 and 1565 nm with 1 nm step.Propagation is through (a) SMF patch cables and an optical circu-lator and (b) SMF patch cables, the circulator, and the 3M CFBG.

Fig. 9. (Color online) Time–frequency plot visualizations of lon-gitudinal SDWs with �3 distortion. The interference of two delayedcopies (upper portion) produces the beat tone under the SDWenvelope (lower portion). For larger delays the detected photocur-rent has a large microwave chirp. This chirp broadens the line-width of the microwave signals used in the techniques of Sections3 and 4.

Fig. 10. (Color online) Overlaid FFTs collected from time-domaininterference of delayed longitudinal SDW copies. Solid curvescorrespond to SMF-produced SDWs and dashed curves are CFBG-originated SDWs. An �40 ps delay results in �13 GHz beat fre-quency between SDW copies, and the �10 ps delay produces an �3GHz beat. (a) Modeling results created with 25.2 km of SMF, �2

� �20 ps2�km, and �3 � .1 ps3�km. CFBG has a �2z value of430 ps2�km. (b) Experimental results agree well with modeling.

1 February 2008 � Vol. 47, No. 4 � APPLIED OPTICS A29

gence will allow all spectral components of the signalto align in a single principle state of polarization.Subsequent processing can make use of the availabil-ity of pure �2 to improve the TBWP of all longitudinalSDW Fourier techniques.

6. Summary

In summary, we have presented optical signal pro-cessing methods based on longitudinal spectral de-composition of ultrashort pulses using chromaticdispersion. Our techniques perform ultrafast opticalpulse shaping, microwave spectrum analysis, and mi-crowave electronic signal generation. With the matu-rity of space-domain techniques for ultrafast opticalsignal processing, the motivation to pursue time-domain approaches stems from the ease of systemscalability to large time apertures and the compati-bility with fiber-optic equipment. Ultimately thecreation of nanosecond to microsecond, longitudinalSDWs implies huge TBWPs for these processors. Ac-cessing the large TBWP requires some technologicaladvancement, and we are currently pursuing CFBGsas a means of achieving ideal, long SDWs.

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