February 15, 2010 / Vol. 35, No. 4 / OPTICS LETTERS 535

Diffractive pulse shaper for arbitrarywaveform generation

Omel Mendoza-Yero,1,2,* Gladys Mínguez-Vega,1,2 Jesús Lancis,1,2 and Vicent Climent1,2

1GROC·UJI, Departament de Física, Universitat Jaume I, 12080 Castelló, Spain2Institut de Noves Tecnologies de la Imatge, Universitat Jaume I, 12080 Castelló, Spain

*Corresponding author: omendoza@uji.es

Received August 19, 2009; revised November 30, 2009; accepted January 11, 2010;posted January 20, 2010 (Doc. ID 115972); published February 10, 2010

We propose an all-diffractive pulse shaper for arbitrary waveform generation in the femtosecond regime.This optical device improves in several aspects the performance of our previous quasi-direct pulse shaperreported in Mínguez-Vega et al. [Opt. Express 16, 16993 (2008)]. In the present implementation, by usinggrayscale masks we can achieve arbitrary temporal waveforms. Additionally, the holographic reconstructionof the above masks by means of phase holograms allows for a high-efficiency shaping process. The behaviorof the pulse shaper is tested by numerical simulations. © 2010 Optical Society of America

OCIS codes: 050.1965, 320.5540.

Waveforms generated by femtosecond lasers may beunsuitable for scientific or technological applicationsunless light pulses are modified in a timely manner.In accordance with practical specifications, complexwaveforms can be synthesized by different pulse-shaping methods. Among them, the pulse shapingbased on the Fourier transform of a given spatial pat-tern onto the dispersed optical spectrum of the pulsehas been widely used by many researchers since the1970s. In the very beginning, Desbois et al. [1] andAgostinelli et al. [2] carried out the spatial filtering ofdispersed optical-frequency components of the pulseto achieve specific waveforms in the picosecond re-gime. The synthesis of arbitrarily shaped femtosec-ond pulses by spatial filtering through a nondisper-sive grating apparatus has been also proved byWeiner et al. [3]. More recently, a quite extensive re-view of femtosecond pulse shaping by using differenttypes of spatial light modulators (SLMs) based onFourier synthesis methods was published [4]. In thiscontext, dynamic and fast operations usually de-manded in photonics, quantum dynamics, orultrafast telecommunications have inspired research-ers to look for better pulse-shaper designs. Hence,SLMs, such as a liquid crystal SLM or an acoustic-optic modulator, are now used to create program-mable pulse shapers. In more sophisticated pro-cesses, optical waveforms are synthesized with thehelp of computer-generated holograms (CGHs) or dy-namic real-time holographic materials.

The Fourier transform method is not always themost convenient way of generating shaped opticalwaveforms. The main reason is the time expendedduring the Fourier transform computation of the spa-tial patterns, which is critical for particular applica-tions. In the telecommunication area, suited opticalpackets can be synthesized by means of a direct(rather than a Fourier transform) space-to-time(DST) pulse shaper [5]. In this case, the shape of theoptical waveform is simply derived from the mappingof some scaled spatial pattern. In recent years, DSTpulse shaping has been successfully implemented byusing different optical elements, including i.e., micro-

electromechanical system micromirror arrays [6] or a

0146-9592/10/040535-3/$15.00 ©

combination of a reflective arrayed waveguide grat-ing and a patterned mask [7]. Recently, we showedthat the strong chromatic behavior associated with akinoform diffractive lens can be exploited to designan optical device for pulse shaping in the femtosec-ond regime [8]. This optical device is basically com-posed of a circular symmetry binary amplitude maskand a kinoform diffractive lens that are facing eachother. The output optical waveform is obtainedwithin a pinhole located at the focal point of the dif-fractive lens. The time profile of the resultant wave-form is given by the convolution of an incident shortpulse with the binary amplitude mask, representedin the squared radial coordinate r2 and adequatelytransformed to the time domain. Because the map-ping is carried out in r2 instead of r, the pulse shap-ing can be thought as a quasi-direct space-to-time(QDST) process. We showed that our QDST pulseshaper can generate bursts of flattop pulses with po-tential application, i.e., as optical packet headers inpacket-switched networks and for the photonicallyassisted generation of microwave and millimeter-wave arbitrary waveforms.

In this Letter we propose a diffractive pulse shaperfor arbitrary waveform generation with high-efficiency operation. By arbitrary waveforms, wemean pulse profiles having any shape in time or,equivalently, in the spectral domain. This pulseshaper improves the performance of our previous de-vice in two main aspects. First, in order to achieve ar-bitrary waveforms, we use grayscale masks insteadof binary amplitude masks. Second, to increase theefficiency of the pulse-shaping process, the grayscalemasks are implemented by phase holograms that arereconstructed just in front of the diffractive lens.Here, it should be also pointed out that the inherentdiffractive nature of our system makes it suitable forcontrolling and manipulating pulses in the extremeultraviolet (XUV) or x-ray spectral region, where re-fractive lenses cannot be used owing to the absorp-tion of materials.

Figure 1 shows a schematic representation of theproposed QDST pulse shaper. In the simplest setup

(right part of Fig. 1), spatial patterns just before the

2010 Optical Society of America

536 OPTICS LETTERS / Vol. 35, No. 4 / February 15, 2010

kinoform lens are given by circular symmetric gray-scale masks. The frequency-dependant focal length ofthe kinoform diffractive lens can be expressed byZ���=Z0� /�0, where �0 is the carrier frequency ofthe pulse. The amplitude of the temporal waveformuout��� at the center of the output pinhole is approxi-mately assessed by the following convolution expres-sion [8]:

uout��� � u0��� � �q��/��exp�− i�0���. �1�

In Eq. (1), u0��� is the amplitude of the incidentpulse and q��� is the transmittance of the mask de-fined as a function of the variable �=r2. The conver-sion constant of the QDST pulse shaper is �=1/ �2cZ0�, where the variable c is the velocity of lightin vacuum. From Eq. (1), when u0��� is much shorterthan the minimum feature size of the scaled mask,the square of the output temporal intensity �uout����2is mainly the function �q�� /���2. This condition is usedto design the grayscale masks. Hence, the transmit-tance of a mask along its radial direction is deter-mined from the user-defined temporal intensity ex-pressed as a function of gray levels. In addition, theoutput temporal window �t�a2 / �2cZ0� can be tunedby changing either the radius of masks, a, or the focallength of the diffractive lens, Z0, for the carrier fre-quency of the pulse, �0.

The physical behavior of the QDST pulse shaper isstudied by means of numerical simulations. Usingthe generalized Huygens–Fresnel integral we deter-mine the field amplitude at the pinhole plane, foreach frequency component of the pulse. Then, theamplitude of temporal waveforms is assessed fromthe Fourier transform of the field amplitude. Let’s as-sume first that we are dealing with the optical setupbasically composed of a grayscale mask, and thekinoform diffractive lens (right part of Fig. 1). Figure2(b) shows an arbitrary temporal waveform synthe-

Fig. 1. (Color online) Diffractive optical device for arbi-trary waveform generation.

sized by means of the above optical setup together

with its corresponding grayscale mask, Fig. 2(a). Thismask originates a sequence of three pulses of differ-ent heights with Gaussian, square, and complex pro-files, respectively. The solid curve holds for thewanted temporal profile that is obtained on-axis atthe pinhole plane. The dotted curve is the average oftemporal waveforms within the pinhole. The simula-tions were carried out for an ultrashort pulse of 30 fsFWHM and a diffractive kinoform lens of focal dis-tant f=105 mm, for the central wavelength of thepulse �800 nm�. In addition, the radius of the pinholeis 50 �m, whereas the maximum extent of the maskis a=8 mm.

The total efficiency of the pulse shaper is estimatedafter multiplying the transmittance of the mask bythe ratio between the energy passed through the pin-hole and the total diffracted field energy at the pin-hole plane. For the parameters in Fig. 2, the total ef-ficiency is about 22.9%. Apart from the masktransmittance, the efficiency of the pulse shaper de-pends on the pinhole size. This factor can be opti-mized after determining the spatial extension of thefocus at the pinhole plane, which is roughly esti-mated from the Fourier transform of the mask trans-mittance. For practical applications, one may acceptcertain compromise between the total efficiency ofthe pulse shaper and the temporal shape of the re-sultant waveform.

In the previous configuration, the efficiency of thepulse shaper is limited by the transmittance of thegrayscale masks. To improve this aspect, the left partof Fig. 1 is considered. Now, the light coming from afemtosecond laser impinges onto a phase hologram.The phase front of the pulse is modified to generatethe required light distribution over the diffractivelens. Here, it should be recalled that phase holo-grams produced with conventional photolithographictechniques can reach efficiencies of the order of 80%[9]. In contrast, the angular dispersion introduced bythe phase hologram and the free-space propagation ofpulse-frequency components may cause degradationin the reconstructed hologram owing to the spatialchirp [10]. On the other hand, phase holograms aretypically calculated for a single wavelength. There-fore reconstructed holograms are expected to formhigh-quality images under femtosecond illuminationonly in the long-pulse-duration regime.

To mitigate the above unwanted effects, a Fresnel

Fig. 2. (a) Grayscale mask. (b) Optical waveform for theon-axis case and for a pinhole of radius 50 �m.

hologram will be designed for near-field reconstruc-

February 15, 2010 / Vol. 35, No. 4 / OPTICS LETTERS 537

tion �z�2a�. These holograms are calculated, for thecarrier frequency of the pulse �3� /4�1015 Hz�, usingan iterative Fourier transform algorithm. Ourmethod is based on the well-known Gerchberg–Saxton algorithm but is carried out in two stages asproposed by Wyrowski [11]. Figure 3(a) shows a digi-tal reconstructed CGH of the mask in Fig. 2(a), forthe distance z=0.15 m between the planes of thephase hologram and the diffractive lens. In this case,the resultant temporal waveforms for the same pin-hole as before appear in Fig. 3(b). To determine theabove curves, both amplitude and phase of the recon-structed CGH were taken into account. After consid-ering 80% efficiency for the hologram, the total effi-ciency of the pulse shaper increases in value to45.5%. So space-to-time conversion is now carried outwith an additional 22.6% of the incident pulse energy.Additionally, the correlation coefficient between tem-poral waveforms in Figs. 2(b) and 3(b), for the on-axiscase, indicates that curves have a similarity of 91%.Small differences in the temporal waveforms are aconsequence of the loss of quality in the digital recon-structed CGH due to the increasing of the spatialchirp with free-space propagation. That is, the CGHis reconstructed not only for the carrier frequency butalso for the remaining frequency components of theultrashort pulse. Therefore the resultant irradiancepattern seems to be blurred, and consequently space-to-time conversion is affected.

We can modify spatial masks to compensate for theeffects of the pinhole size on the temporal shape ofoptical waveforms. For applications requiring a fixedpinhole size, we can modify the spatial masking func-tion to maintain high pulse-shaping efficiency. Toshow that, the simplest pulse-shaper optical setup isused for a femtochemistry application addressed toobtain molecular alignment [12]. In particular, thetemporal shape of the revivals in the nitrogen mol-ecule is controlled by applying asymmetric pulseshapes like those shown in Fig. 4(b). To generate theabove sequence of pulses, we calculate the grayscalemasks given in Fig. 4(a). The mask in Fig. 4(a) topgives the expected temporal profile (solid curve) justin the center of the pinhole, as previous examples inFigs. 2 and 3. The mask in Fig. 4(a) bottom allows usto obtain the same profile (dotted curve), but thistime within a pinhole radius of 25 �m. That is, thebottom mask was designed to yield the same average

Fig. 3. (a) Digital reconstructed phase CGH of the mask inFig. 2(a), at the plane z=0.15 m. (b) Optical waveform forthe on-axis case and for a pinhole of radius 50 �m.

temporal waveform within the pinhole as the topmask does for the on-axis case. In this way the pulseshaper generates the desired temporal waveformwith maximum efficiency for the present application.

Finally, one might realize that if the phase holo-gram is implemented onto a liquid crystal SLM, thetemporal waveforms can be dynamically changed. Wealso indicate that the QDST pulse shaper is useful inmany scientific and technological applications, in-cluding pulse-train generation for telecommunica-tions; multiple-pulse excitation of atoms, molecules,and solids; coherent control of chemical reactions;molecular alignment; or generation of terahertz ra-diation.

This research was funded by the Spanish Ministe-rio de Educación y Ciencia (MEC), Spain, throughConsolider Programme SAUUL CSD2007-00013. Wealso acknowledge partial support from the Conselle-ria de Empresa, Universitat i Ciència, GeneralitatValenciana, under the project FIS2007-62217.

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Fig. 4. (a) Grayscale masks designed to achieve the ex-pected temporal profiles at the center of the pinhole (top),and within a pinhole of radius 25 �m (bottom). (b) Corre-sponding optical waveforms.