. INTRODUCTIONhe space-time analogy establishes a formal link betweenhe diffraction of one-dimensional paraxial monochro-atic light beams and the temporal dispersion of plane-ave pulses in first-order dispersive media [1–3]. A keyptoelectronic element is the time lens [4], which ideallymparts a quadratic temporal phase modulation on theomplex field of a short light pulse.
In the earlier experiments, time lenses were imple-ented with high-bandwidth electro-optic phase modula-
ors (EOPMs). These modulators work based on the Pock-ls effect, where the refractive index of an electro-opticaterial changes as a result of the application of an elec-
ric field. In a typical traveling-wave configuration, thelectric field is applied in the perpendicular direction ofhe light propagation, and the refractive index is propor-ional to the applied field. Assuming a phase-velocity andmpedance mismatch, the instantaneous optical phase athe output of the waveguide is just proportional to the in-tantaneous electric field. Therefore, a sinusoidal radio-requency (RF) clock signal can provide quadratic phaseodulation locally [4]. These electro-optic time lensesave been used for several ultrafast optical signal pro-essing applications, such as waveform magnification [5]ased on temporal imaging [6]; waveform diagnosis basedn time-to-frequency conversion [7]; elimination of linearistortions [8] based on optical Fourier transformation9]; or arbitrary waveform generation [10] based on multi-avelength pulse compression [11], just to name a few ex-mples. However, time lenses implemented in EOPMs re-uire the input pulse width to be limited to a smallraction of the signal period (around 15%) if higher-order
erms in the cosine function expansion are to be disre-arded [12]. The deviation from the parabolic profile (oremporal aberrations) degrades the performance of thebove mentioned ultrafast processors. Besides, the maxi-um phase achievable in electro-optic time lenses is lim-
ted by the maximum voltage acceptable by the device.Research on alternative time-lens implementations
as done through non-linear optics. These include para-etric processes such as sum- or difference-frequency
eneration in ��2� materials [13]. Recent studies consid-red periodically poled lithium niobate waveguides [14].lternatively, time lenses implemented in highly non-
inear fibers based on cross-phase modulation using para-olic pump pulses have achieved better results than thelectro-optic counterpart configurations in, e.g., opticalourier transformation [15,16]. More recently, investiga-
ions in silicon photonics have led to the first implemen-ation of a time lens in a silicon-on-insulator platform, ex-loiting the four-wave mixing effect in strongly non-linearano-waveguides [17]. The silicon time-lens features si-ultaneously high maximum phase value and large tem-
oral window, providing the key to implement for the firstime or upgrade previously reported ultrafast optical pro-essors [18–21].
Despite these impressive advances, the electro-opticime lens has several characteristics that still make itery attractive for several applications, e.g., flat fre-uency comb generation [22,23]. Features inherent tolectro-optic time lensing, such as high-speed operation�10 GHz�, wavelength preserving operation, and easyeconfiguration, are very difficult to achieve using para-etric devices.
010 Optical Society of America
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Munioz-Camuniez et al. Vol. 27, No. 10 /October 2010 /J. Opt. Soc. Am. B 2111
In this work, we provide some guidelines to improvehe performance of electro-optic time lenses through aearly parabolic phase modulation over a great fraction ofhe period of the driving signal. As in the pioneer paper byan Howe et al. [11], the key for increasing the temporalindow is to consider additional harmonic terms in thexpansion of the parabolic profile. Of course, in practice,nly a limited number of RF tones can be employed. Ourimulations show that for high chirp lenses, at least inulse compression applications, the use of a truncatedourier series results in the appearance of the Gibbs phe-omenon, which seriously deteriorates the system perfor-ance. To avoid the unwanted phase modulation intro-
uced by Gibbs oscillations, we propose here a differentarmonic expansion where the coefficients of the trun-ated series are determined through a non-linear leastquares (NLSS) fit of the parabolic modulation over aser-defined fraction of the period. We restrict our studyo three clock harmonics just to ensure a practical imple-entation. In this way, we show that it is possible to
chieve an aberration-free behavior over a time windowovering the 70% of the driving period while keeping aigh chirping rate. The scheme of our system is shown inig. 1. We propose the use of a single-tone RF signal fol-
owed by frequency multiplication, delaying, weighting,nd mixing in the electrical domain. Finally, as an illus-rative example, we test the method by means of some nu-erical simulations involving a pulse compression experi-ent with a low-V� EOPM.
. THEORYet us first review conventional time-lens operation basedn electro-optic phase modulation. As is usual in practice,e assume that the time lens is implemented through anOPM driven with a time-dependent voltage, V�t�V0 cos�2�ft�. In this equation V0 is the maximum ampli-
ude voltage, and f is the clock frequency. The driving sig-
ig. 1. (Color online) Schematic setup for the implementation ofhe nearly aberration-free time lens. The input periodic opticalignal acquires a fully undistorted quadratic phase modulationdashed line) while preserving the intensity profile (solid line).ince the optical phase acquired by propagating through theOPM is proportional to the electrical signal driving it, tochieve this goal we have to synthesize properly the electricalignal. This is done by delaying, weighting (with variable electri-al amplifiers), and summing a common clock signal.
al only provides a locally quadratic phase around theusp of the cosine function. Therefore, if the optical signals temporally limited to a small fraction of the period T1/ f, the pulse acquires a phase shift that is nearly qua-ratic [6]. Mathematically,
��t� =�V0
2V�
cos�2�t
T � = �� cos�2�t
T � � − 2���2t2
T2 , �1�
hich corresponds to the second-order expansion of theosine function. The constant term in the expansion of thecquired phase ��t� has been disregarded, as it only shiftshe phase mean value. In the above equation �� is the so-alled modulation index, and V� is related to the peak-to-eak drive for achieving a � phase shift. Equation (1) cor-esponds to a lens with a temporal focal length �f±1/ ����2�f�2�. The � sign depends on whether theaximum or the minimum of the cosine function is con-
idered. Concerning temporal aberrations, for a pulsepanning over T /2 we get a maximum phase error at theulse edge of �� /5. These deviations lead to phase errorshat result in an elongated pulse for compression applica-ions.
Our goal is to increase significantly and in a simpleay the time window in which the time lens operatesith a low aberration. In a first stage, inspired by vanowe et al. [11], we consider the harmonic expansion
��t� = ���q=0
n
Aq cos�q2�t
T � , �2�
here n is the order of the expansion. Note that the zero-rder term is again irrelevant as it just introduces a con-tant phase. In [11], the coefficients Aq were calculated inccordance with Fourier analysis. We try to match thearabolic modulation in a time window covering thehole period. However, owing to the truncation of theourier series, Gibbs oscillations with a peak-to-peak am-litude about �� /10 occur. As we shall show in the follow-ng section, this leads to the formation of spurious side-obes in a pulse compression experiment.
To avoid this drawback, we propose reaching the para-olic behavior by calculating the Aq coefficients by fittinghe discretized function −2���2t2 /T2 with a NLSSethod. The key point is to minimize the sum
S = �i=1
N �− 2�2ti2
T2 − �q=0
n
Aq cos�q2�ti
T��2
�3�
nside a user-defined fraction �01� of the entire pe-iod. In the above equation, N is the number of selectedoints resulting from sampling the ideal parabolic profile.his approximation reduces phase errors inside the tem-oral window T. When =1, the coefficients match thosebtained from the Fourier expansion. The accuracy isarger as n increases. Because the NLSS method is per-ormed for a particular choice of , in principle, the coef-cients Aq are -dependent. In Fig. 2, we plot the calcu-
ated dependence with the aperture parameter for theptimized coefficients calculated from the NLSS. Forractical reasons, we have restricted ourselves to the case=3. We found that the above numerical curves can be fit-
ed by the following curves (also shown in the plot):
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2112 J. Opt. Soc. Am. B/Vol. 27, No. 10 /October 2010 Munioz-Camuniez et al.
A1 = 1.5 + 0.542,
A2 = − 0.125 − 0.0176e3.19,
A3 = 0.011 + 0.00062e6.07. �4�
n the above calculations we have fixed N=40. We observehat some of the coefficients achieve negative values.rom a practical perspective, this can be easily achievedy introducing a � phase shift (i.e., a delay of T /2) in theorresponding harmonic.
Realistic EOPMs have an intricate frequency depen-ence of the V� parameter. This can be taken into accountn our synthesis design by writing Aq=�Vq / �2V�q�, where
�q is the voltage necessary to achieve a � phase shiftith the qth-harmonic. Thus, we can still use the previ-us analysis if we modify accordingly the peak-to-peakoltage of the qth-harmonic, V�q, in order to obtain thealculated Aq coefficient. For this aim, one needs to cali-rate the frequency dependence of V� in the range of thearmonics to be used, for which standard procedures exist24,25].
Next, in Fig. 3 we illustrate the dissimilarity betweenhe parabolic curve and three different scenes: the single-one approximation, the above NLSS fit �n=3� with 0.7, and the third-order Fourier harmonic expansion
i.e., A0=−�2 /6, A1=2, A2=−1/2, and A3=2/9). In math-matical terms, the deviation D in Fig. 3(b) is defined as
D�t� = f�t� + 2���2t2
T2 , �5�
here f�t� is the approximation used instead of the idealarabolic profile. Over the selected time window our ap-roach shows a maximum phase error of only �� /70. Thisesult is better than the deviation �� /10 that correspondso the maximum separation in the interval �−T /4 ,+T /4�or the Fourier approximation. The oscillations in theourier expansion shown in Fig. 3(b) are due to the series
runcation. We note that the Aq coefficients calculatedith the NLSS method minimize the oscillations inside
he designed time window when compared to the Fourierxpansion approach.
In our proposal there is a trade-off between the maxi-um time window and the minimum quadratic phase de-
iation. The optimum choice for must be set in terms ofhe temporal extension and shape of the input pulse. Aentative criterion is provided by the deviation Deighted by the pulse shape ��t�. So can be selected byinimizing the mathematical expression
ig. 2. (Color online) Plot of the values of the parameter Aq in2���2t2 /T2 with only the first three terms of the harmonic seri
��� =1
T−T/2
+T/2 ��t���q=0
n
Aq��cos�q2�t
T� + 2�2t2
T�dt.
�6�
rom a practical point of view, we assume a linear rela-ionship between the applied voltage and the accumu-ated phase in the EOPM. Then, a nearly quadratic phasean be obtained by a single modulator driven with the sig-al achieved by frequency multiplication and mixing inhe electrical domain from the fundamental clock signal.ifferent weighting factors are accomplished by properlectrical amplification/attenuation of the different har-onic channels following the prescriptions in Eq. (4).
. SIMULATION RESULTSo illustrate how the phase errors affect to time-lens be-avior, we numerically simulate the compression of a
s of the fraction coefficient to reproduce the parabolic profileose fundamental period is T.
ig. 3. (Color online) (a) Comparison among the lens profilesorresponding to single-cosine approximation (long-dashedurve), Fourier expansion (dashed-dotted curve), NLSS fit withoefficients provided by Eq. (4) (solid curve), and ideal parabolicrofile (dotted curve); (b) relative deviation between the para-olic profile and the above three curves.
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Munioz-Camuniez et al. Vol. 27, No. 10 /October 2010 /J. Opt. Soc. Am. B 2113
uper-Gaussian pulse, with envelope exp�−�t / T�2m�, and=4, from a train at 10 GHz repetition rate. The param-
ter determines the pulse width. We consider three dif-erent pulse durations with full width at half-maximum:8, 67, and 86 ps. They, respectively, cover 40%, 70%, and0% of the whole period �T=100 ps�. The three phaseodulations performed in the EOPM are just the same as
hose already discussed in the previous section (single-osine approximation, Fourier harmonic expansion, andarmonic expansion by the NLSS fit with =0.7). Afterodulation, the pulse is propagated through a single-ode optical fiber up to the temporal focus. In opticalourier transformation, pulse compression occurs afterropagating a distance Z=�f /�2=1/�2���2�f�2 [9], where2 is the group-velocity-dispersion parameter of the fiber.or the sake of clarity, only the first-order dispersion isonsidered in the simulation.
For aberration-compensated lenses, the EOPM isriven with three harmonic signals of frequencies 10, 20,nd 30 GHz. On the other hand, driving voltages are de-ermined through the coefficients Aq. In this way, we en-ure an identical temporal focal length in the three cases.owever, according to Fig. 2, note that the maximum
oltage is different in each case. An alternative compari-on is carried out if an identical maximum driving signals set for the three procedures. In this case, the maximumchievable focal power is different. We simulate compres-ion in two ways: the first one with the same focal poweror the three lenses and, in the second way, with the sameaximum voltage.
. Compression by Lenses with the Same Focal Lengthn this example we choose two different modulation indi-es, ��=� rad and ��=2.5� rad, which are realistic for aow-V� phase modulator. In the single-cosine approxima-ion, the maximum voltage is given by V0=��2/����V�,ith �=1. According to the plot of A1 in Fig. 2, �=2 for the
tandard Fourier series, and �=1.76 for the NLSS fit.hen, for ��=� rad, the different maximum amplitudeoltages are 2V�, 4V�, and 3.52V�. For ��=2.5� rad, theoltages are 2.5 times greater.
Numerical results are shown in Fig. 4. Our proposalrovides compressed pulses close to the aberration-freeituation for input pulses whose widths fall into the tem-oral window 0.7T [Figs. 4(b) and 4(e)]. Compared withhe conventional Fourier series expansion, there are noignificant differences for the lowest �� value. For ��2.5� rad (right column in Fig. 4), the NLSS method al-ays provides better results than the Fourier series. In
he latter case we note the appearance of sidelobes thateteriorate the output pulse. The presence of the aboveidelobes is attributed to the Gibbs phenomenon, an effecthat becomes more severe for higher compression ratios.
. Compression by Lenses with the Same Voltageext we consider the more realistic case in which theOPM has a fixed maximum achievable phase value (forimplicity we consider it to be independent of the drivingrequency). Now the corrected lenses have different focalengths, and thus the propagation distance in the opticalber to the temporal focus is also different. This is indeed
he case reported in [11]. For our example, we choose twoifferent maximally achievable voltages: 2V� and 8Vp.The results are shown in Fig. 5. For shorter pulses the
ompression with the single-cosine approximation is bet-er than in any of the two harmonic expansions [Fig. 5(a)],ue to the higher optical power in the single-tone case.owever, if the temporal length of the pulse approaches0% of the period or goes beyond, both harmonic expan-ions work better than the single-cosine approximationsee Figs. 5(b), 5(c), 5(e), and 5(f)]. Likewise, it is apparenthat in all situations harmonic expansion by the NLSS fitorks better than the Fourier expansion due to theigher focal length in the latter case and the lack of side-
obes. It is interesting to note that, in principle, the Gibbs
ig. 4. (Color online) Left and right columns show results forulse compression results throughout EOPM lensing and fiberropagation for ��=� rad and ��=2.5� rad, respectively. Theurve codes are the same as those in Fig. 3. The initial pulse du-ations are (a) 38, (b) 67, and (c) 86 ps.
ig. 5. (Color online) Same as in Fig. 4 but for V0=2V� (left col-mn) and V =8V (right column).
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2114 J. Opt. Soc. Am. B/Vol. 27, No. 10 /October 2010 Munioz-Camuniez et al.
henomenon does not appear in the improved parabolicrofile reported in [11] due to the relatively low index-odulation value.
. DISCUSSIONne can note that delaying, weighting, and mixing of thearmonics in the electrical domain are a route to achieveser-defined electrical signals. The procedure presented
n this work is aimed to optimizing the temporal aberra-ions of electro-optic time lenses, for which we have fo-used on the algorithms that optimize an electrical para-olic profile with the minimum number of resources. Buthe NLSS method could be used to achieve, if necessary,lternative optical phase profiles.On the other hand, we have concentrated on removing
he temporal aberrations. However, in temporal imagingystems and optical Fourier transformation other kinds ofberrations—namely, spectral aberrations—appear, ow-ng to the negative impact of third-order dispersion in theispersive elements employed [12]. Because the lens ac-ive element is a phase-only modulator, in principle, weannot compensate for the higher-order dispersion termsn temporal imaging or optical Fourier transformers.owever, following the work of Pelusi et al. [26], we re-ind that a phase-only modulator can be inserted in an
ltrafast temporal shaping system [27] to actively controlhe spectral components of a short light pulse. In thisase, the NLSS method could be used to calculate the op-imum choice of harmonic parameters in order to compen-ate simultaneously for the first- and higher-order disper-ions in a dispersive transmission link.
. SUMMARY AND CONCLUSIONSe have proposed a method for correcting temporal aber-
ations in time lenses implemented with an electro-optichase modulator (EOPM). It makes use of a harmonic ex-ansion with coefficients calculated by a non-linear leastquares (NLSS) algorithm fit to the ideal parabolic modu-ation profile. The proposed approach permits one todapt the optimized parabolic region to a user-definedemporal window, which results are very useful when theemporal duration of the pulse is known. By restrictinghe temporal window to a fraction of the period of theulse train, the weighting factors obtained through theon-linear fit ensure lower power consumption with re-pect to the case in which the whole period is considered.e numerically demonstrate nearly aberration-free be-
avior for pulse widths about 70% of the driving periodith only three harmonic components. Our optimization
echnique avoids the unwanted Gibbs phenomenon thatppears at high modulation indices. In view of the rapiddvances in lithium niobate technology and the fabrica-ion of low-V� EOPMs, we believe that the general expres-ions achieved in this work for time-lens optimization cane used for several applications when an accurate para-olic profile at high speed is needed, such as flat fre-uency comb generation or high-speed optical Fourierransformation of ultrashort light pulses.
CKNOWLEDGMENTShis research was funded by the Dirección General de In-estigación Científica y Técnica, Spain and FEDER, un-er the project FIS2007-62217, and the Convenio Univer-itat Jaume I-Fundació Caixa Castelló, under project P1B2007-09. V. Torres-Company acknowledges the Spanishinistry of Science and Innovation and the Spanishoundation for Science and Technology for a postdoctoral
ellowship. J. Ojeda-Castañeda gratefully acknowledgesnancial support from the University of Valencia, Spain,nder the Program “Estancias Temporales para Investi-adores Invitados.”
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