2122 J. Opt. Soc. Am. B/Vol. 19, No. 9 /September 2002 Malvezzi et al.

Second-harmonic generation in reflection anddiffraction by a GaAs photonic-crystal waveguide

Andrea Marco Malvezzi

Istituto Nazionale per la Fisica della Materia, and Dipartimento di Elettronica, Universita degli Studi di Pavia,Via Ferrata 1, I-27100 Pavia, Italy

Francesco Cattaneo, Gabriele Vecchi, Matteo Falasconi, Giorgio Guizzetti, and Lucio Claudio Andreani

Istituto Nazionale per la Fisica della Materia, and Dipartimento di Fisica ‘‘A. Volta,’’ Universita degli Studi diPavia, Via Bassi 6, I-27100 Pavia, Italy

Filippo Romanato, Luca Businaro, and Enzo Di Fabrizio

National Nanotechnology Laboratory, TASC–Istituto Nazionale per la Fisica della Materia NanolithographyBeamline at Elettra Synchrotron Light Source, S.S14 Km 163.5, I-34012 Basovizza-Trieste, Italy

Adriana Passaseo and Massimo De Vittorio

National Nanotechnology Laboratory, Istituto Nazionale per la Fisica della Materia, and Dipartimento di Ingegneriadell’Innovazione, Universita degli Studi di Lecce, via per Arnesano, I-73100 Lecce, Italy

Received November 30 2001; revised manuscript received March 11, 2002

Nonlinear reflection and diffraction measurements have been performed on a GaAs/AlGaAs photonic-crystalwaveguide patterned with a square lattice: The basis in the two-dimensional unit cell consists of rings of airin the dielectric matrix. The measured angles of diffracted second-harmonic beams agree with those pre-dicted for nonlinear diffraction conditions. Results for second-harmonic intensities as a function of incidenceangle, polarization, and pump wavelength show that the reflected second-harmonic signal is dominated by thecrystalline symmetry of GaAs, whereas nonlinear diffraction is determined by the photonic-crystal structure.© 2002 Optical Society of America

OCIS codes: 190.4720, 190.4400, 190.2620, 190.4390.

1. INTRODUCTIONSince the first prediction of nonlinear diffraction in spa-tially modulated systems,1 the subject of second-harmonicgeneration (SHG) in diffraction from nonlinear gratingshas received much attention both experimentally andtheoretically.2–16 A second-harmonic (SH) beam gener-ated by the nonlinear material and diffracted by the grat-ing occurs at directions given by nonlinear diffraction con-ditions (if the material is rough, diffracted SHG occursalso in a diffusion cone1). The diffracted SH signal is ob-viously small for a shallow grating and increases withgroove depth. Many papers3–6 deal with one-dimensional metallic or metallodielectric gratings andwith resonance effects related to surface plasmons;others7–12 exploit SH diffraction by a monolayer gratingon top of a centrosymmetric medium for the study of sur-face phenomena (adsorption, diffusion, etc.) without theneed to consider effects of the substrate. Recently, non-linear diffraction during in-plane propagation in two-dimensional (2-D) photonic crystals with spatially vary-ing x (2) was predicted17 and experimentally observed.18

One- and two-dimensional photonic-crystal slabs basedon GaAs (Refs. 19 and 20) are well suited for SHG studiesof reflection and diffraction because the second-order sus-

0740-3224/2002/092122-07$15.00 ©

ceptibility of GaAs is large and photonic lattices at opticalwavelengths can be defined and patterned deeply into thesamples by lithography and etching. The main differ-ence between a photonic crystal and a conventional grat-ing is that in the former the etch depth is large, leading tostrong efficiencies for linear diffraction.21,22 As is wellknown, for well-defined photonic modes with reasonablylow propagation losses, the photonic pattern must be im-pressed in a planar waveguide, and the average dielectricconstant of the claddings must be lower than that of thecore layer after the patterning: For symmetric GaAs/AlGaAs or asymmetric air/GaAs/AlGaAs photonic-crystalslabs, this means that the lower cladding has to be etchedto a certain depth.

In this paper, we report on SHG measurements of re-flection and diffraction from an air/GaAs/AlxGa12xAsphotonic-crystal waveguide patterned deeply with a 2-Dsquare lattice of air rings. The lattice has a low air frac-tion (;12%) and an etch depth of more than 1mm: In-deed, linear reflectance measurements at various angleson this sample23 showed narrow resonances, which pointto the existence of low-loss photonic modes, as were firstshown in triangular lattices of holes in the research re-ported in Ref. 24. We detected the diffracted SH signal

2002 Optical Society of America

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from a waveguide excited with a Ti:sapphire laser andverified the nonlinear diffraction rules by performing ex-periments at various angles of incidence. Also, we mea-sured the reflected and the diffracted intensities as func-tions of the wavelength (in the spectral range allowed bytuning of the Ti:sapphire oscillator) and of the polariza-tion of the incident beam. These measurements, whencompared with reflection SHG on the unpatterned wave-guide area, point to high diffraction intensities and to po-larization properties that are characteristic of thephotonic-crystal waveguide.

This paper is organized as follows: We first give a de-scription of the sample and of the experimental setup andillustrate the nonlinear diffraction rules for the crystalstructure and experimental geometry that we have cho-sen. We then present and discuss the experimental re-sults for nonlinear reflection and diffraction. Section 5contains concluding remarks.

2. EXPERIMENTALThe waveguide investigated in this study was grown bymetal–organic chemical vapor deposition upon a (100)semi-insulating GaAs substrate. The structure consistsof a 200-nm GaAs buffer layer, a 1500-nm Al0.25Ga0.75Aslower cladding, and a 500-nm GaAs core layer (nominallyundoped): Thus the upper layer is GaAs and the air/GaAs/Al0.25Ga0.75As waveguide is asymmetric. We pat-terned the waveguide by x-ray lithography, followed byreactive-ion etching. The mask for x-ray lithography wasgenerated by electron-beam lithography on a 4 mm3 4 mm area, and it consisted of a square lattice ofsquare holes (282 nm on a side) rotated by 45° with re-spect to the axes of the lattice. Proximity x-ray lithogra-phy performed at the LILIT beam line at the ELETTRASynchrotron (Trieste, Italy) was employed to transfer thepatterning to the sample surface. A diffraction effect inx-ray lithography, controlled by the distance between themask and the stepper, resulted in partial filling of the airpillars and in a reduction of the air fraction. Reactive-ion etching with SiCl4 was then performed to pattern thephotonic lattice deeply into the waveguide. The etchdepth was more than 1 mm. Two scanning electron mi-crographs of the sample are shown in Fig. 1. The Bra-vais lattice is a 2-D square lattice with lattice constant ofa 5 500 nm, and the basis within the unit cell consists ofa ring of air in the dielectric matrix. The air fraction ofthe photonic lattice is approximately 12%. Note that theorientation of the mask was chosen in such a way that thesquare axes of the photonic lattices coincide (within asmall uncertainty) with the crystallographic axis [110] ofGaAs: The [100] axis of the microscopic lattice (whichdetermines the symmetry properties of the x (2) tensor)and the [10] axis of the photonic lattice (which producesthe diffractive effects) are rotated by 45° with respect toeach other.

In Fig. 2, we show the linear reflectance of the sampleat various angles of incidence for s-polarized light inci-dent along the G –M orientation.23,24 Complex oscilla-tions that resulted from Fabry–Perot interference in thewaveguide structure are apparent. At 0.6–1 eV, narrowresonant structures with a well-defined angular disper-

sion can be seen. They are due to the excitation of pho-tonic modes of the waveguide that were coupled to the in-coming field by the patterning. The energy positions ofthese modes are indicated on the u 5 5° and the u5 60° curves by vertical bars. Figure 2 shows that thesample behaved as a photonic-crystal slab with well-defined photonic modes: It also shows that the energywindow in which these modes were resonantly coupled toan external field was 0.6–1 eV. For higher energies nostructures associated with photonic modes were observed.Similar results were obtained for other polarizations andsample orientations. The Fabry–Perot oscillations weremodeled to be consistent with the 1-mm etched depth ofthe photonic structures.23

The source of excitation for SHG measurements is aTi:sapphire laser system that provides pulses of an'130-fs duration at an 80-MHz repetition rate centeredat wavelengths of 750–840 nm, with an average power onsample of as much as 150 mW. The ultrashort-pulsed la-ser was used here to provide a sufficiently high intensityfor obtaining a detectable SH signal, while minimizingthe thermal load to the sample. Pulses were focused to a40–50-mm diameter with a 20-cm focal-length lens onto

Fig. 1. Scanning electron micrographs of the sample studied inthe research reported here. (a) Front view, (b) cross sectionshowing vertical air structures. The lattice constant of thesquare Bravais lattice is a 5 500 nm.

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the sample after filtering and intensity and polarizationcontrols. The SH radiation was collected by an optical fi-ber (400-mm core diameter) and delivered in front of acooled photomultiplier after passing through a combina-tion of color filters plus a wideband interference filter thatrejected the radiation at the pump frequency. No polar-ization selection of the SH signal was implemented. Theoptical fiber was of UV–visible type, transmitting morethan 95% of the radiation at 400 nm.

The geometry of the optical layout is sketched in Fig. 3,where the notation for the angles involved in the experi-ment is also shown. It includes the angle of incidence uof the pump radiation, angle f formed by the G –X direc-tion of the photonic crystal to the plane of incidence, polardiffraction angle u8 measured from the normal n to thesample, and the azimuthal angle f 8 of the diffractedbeam (also measured from the plane of incidence). Ofthese four angles all are variable in the experimental

Fig. 2. Linear variable-angle reflectance for s-polarized light (s-pol) incident in the G –M direction. The angle of incidence var-ies from 5° to 60°, in steps of 5°. The curves are shifted verti-cally for clarity. The vertical bars denote the energy positions ofthe photonic resonances for u 5 5° and u 5 60°.

Fig. 3. Experimental geometry for SHG at the sample. The in-cident (i), reflected (r), and diffracted (d) wave directions are rep-resented by thick lines. n is the normal to the sample surface,and the relevant angles are indicated.

setup except f 8, which we kept fixed at 0° or 180°. Thus,only in-plane diffraction beams (forward and backward)were being detected, whereas u8 and f were remotely con-trolled in 0.225°/step increments. Optical alignment ofthe setup requires that a set of translation stages for theazimuthal axis of rotation of the sample (defining f) andits polar axis of rotation (defining u) cross at the laser fo-cal spot on the sample surface. Moreover, the rotation ofthe optical fiber (defining the angle u8) has to share thesame vertical axis of rotation.

The SH signal is detected with a light chopper, whichmodulates the pump radiation up to a few kilohertz. Thephotomultiplier signal is then recorded by an averagingoscilloscope, so the amplitude of the signal with respect tobackground is evaluated.

3. NONLINEAR DIFFRACTIONCONDITIONSFor a given incidence angle and sample orientation a SHbeam is generated in all directions that satisfies the fol-lowing conditions:

ki8 5 2ki 1 G, (1)

where ki (ki8) is the parallel wave vector of the incident(SH) beam and G is a reciprocal lattice vector of the pho-tonic lattice. For G 5 0, we have the usual SHG on re-flection. One can view the diffracted SH beam [G Þ 0 in

Fig. 4. Nonlinear diffraction angles for light incident along theG –X orientation of the photonic crystal, assuming that l5 800 nm and a 5 500 nm: (a) polar angle u8, (b) azimuthalangle f 8. Solid (dotted) curves, diffraction in the plane (out ofthe plane) of incidence. The integer numbers (n1 and n2) thatlabel the diffraction order [Eqs. (2)] are indicated on each curve.

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Eq. (1)] as being due to SHG on reflection followed by dif-fraction at the harmonic frequency or else as being due todiffraction at the pump frequency followed by SHG: Thetwo processes are physically indistinguishable.

We now specify the nonlinear diffraction conditions bymaking reference to the experimental scheme and theangles shown in Fig. 3. The reciprocal lattice vectors canbe expressed as G 5 (2p/a)(n1 ı 1 n2), where ı and are unit vectors along the [1, 0] and the [0, 1] axes of thephotonic lattice and n1 , n2 5 0, 61, 62 ,... . Thus Eq.(1) becomes

sin u8 cos~f 8 1 f ! 5 sin u cos f 1 n1

l

2a,

sin u8 sin~f 8 1 f ! 5 sin u sin f 1 n2

l

2a, (2)

where l is the wavelength of the pump beam. It can beseen that Eqs. (2) follow from the analogous equations forlinear diffraction with the replacement l → l/2. Thewavelength at the cutoff for nonlinear diffraction isdouble that in the analogous linear process: As a conse-quence, nonlinear diffraction can occur even when lineardiffraction is forbidden.

In general, most diffracted beams lie out of the plane ofincidence. Only for an incident beam along the G –X di-rection of the lattice (f 5 0°) or along the G –M direction

Fig. 5. Nonlinear diffraction angles for light incident along theG –M orientation of the photonic crystal, assuming that l5 800 nm and a 5 500 nm: (a) polar angle u8, (b) azimuthalangle f 8. Solid (dotted) curves, diffraction in the plane (out ofthe plane) of incidence. The integer numbers (n1 and n2) thatlabel the diffraction order [Eqs. (2)] are indicated on each curve.

(f 5 45°) are there diffracted beams in the plane of inci-dence. These are the only beams that are detected by ourexperimental setup. To illustrate the physical conditionsimplied by Eq. (1), in Fig. 4, we show the angles u8 and f 8of all SH diffracted beams for light incident along theG –X direction, and, in Fig. 5, we show similar results forthe G –M orientation. We chose the ratio l/a 5 1.6, cor-responding to a lattice constant of a 5 500 nm and apump wavelength of l 5 800 nm. Diffraction in theplane of incidence is indicated by solid curves; diffractionout of the plane is marked by dotted curves. We resolvedthe ambiguity in the definitions of u8 and f 8 by imposingthe condition that 2p/2 , f 8 , p/2: In this way a posi-tive (negative) u8 means that diffraction occurs in the for-ward (backward) half-space. [This choice produces thejumps in Figs. 4(a) and 5(a), which correspond to dif-fracted beams with an angle f 8 passing through 90° thatchanges from the backward to the forward half-space.]The curves are marked by the integers (n1 and n2) of Eqs.(2), which label the diffraction order.

The following conclusions can be drawn from Figs. 4and 5: For light incident along G –X and considering allangles of incidence, there are as many as three SH beamsdiffracted in the plane of incidence, one of which is in theforward direction [the (1, 0) curve] and the other two ofwhich are in the backward direction [the (21, 0) and (22,0) curves]. The (21, 0) nonlinear diffraction occurs for allangles of incidence. The other four diffracted SH beams(dotted curves) are out of the plane of incidence. Forlight incident along G –M, there is only one in-plane dif-fracted beam [corresponding to (21, 21)], which occursfor all incidence angles larger than approximately 8°, andthe six other diffracted beams are out of the plane of in-cidence. In general, it can be seen that nonlinear diffrac-tion occurs preferentially in the backward geometry.

4. RESULTS AND DISCUSSIONWith the apparatus described above, we measured the SHsignal generated by the photonic crystal by fixing thesample orientation and varying the angle u or the wave-length of the excitation radiation. The pump intensity onthe sample was measured before each data-taking sessionby use of a calibrated powermeter and a knife-edge scanof the laser spot at the sample position. We calculatedthe intensity of the resultant SH radiation absolutely byconsidering the geometry of the apparatus, the detectorresponse, and the transmission of the optical fiber and ofthe filters used in the spectral interval considered. As isdiscussed below, the calibration was checked against theSH generated in reflection from bulk crystalline GaAssamples. The SH signals detected both in reflection andin diffraction follow a quadratic law in the input fluence.The nonlinear radiation appears to be spatially coincidentwith the pump in reflection.

The diffraction angles of the SH signal are shown inFig. 6 relative to the angle of incidence u of the pump fora wavelength of 814 nm. The SH diffracted signals weremeasured for both s and p polarizations of the incident ra-diation and for both the G –X [f 5 0°, curve (21, 0) ofFig. 4] and the G –M [f 5 45°, curve (21, 21) of Fig. 5]photonic-crystal orientations. In the G –X orientation a

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further in-plane diffraction mode (22, 0) is predicted at astrongly negative (290° , u8 , 2 35°) diffractionangle, which unfortunately could not be reached with theexperimental setup. The theoretical SH diffractioncurves with l 5 814 nm are also shown for the two cases.Agreement between the experimental results and the the-oretical curves derived from Eqs. (1) and (2) is very good.The uncertainty of the incidence angles is approximately1° and is the result of the alignment procedures both ofthe sample and of the incident laser beam. In the setupthe optical fiber’s angular position is not absolutely cali-brated but is obtained by measurement of the position ofthe SH reflected signal. Thus the uncertainty in theangle of the SH reflected and diffracted beams is of thesame order as the uncertainty in the incident angle.

For both reflected and diffracted SH measurements thenonlinear reflection and diffraction coefficients, defined asRNL 5 IR(2v)/I(v)2 and DNL 5 ID(2v)/I(v)2, respec-tively, were evaluated. In Fig. 7(a), nonlinear reflectioncoefficients RNL measured at several angles of incidence uare shown for various combinations of selected input po-larization and photonic-crystal orientation at a pumpwavelength of l 5 814 nm. These data are in goodagreement with the theoretical calculation and the ex-perimental results of the nonlinear reflection coefficientfor a bulk nonpatterned (001) GaAs surface.25–27 Indeed,the air-filling factor of the photonic crystal amounts toonly 12% in this sample, leading to a nonlinear responsethat is close to that of bulk GaAs. A value of the order of;5 3 10224 m2 W21 for s polarization and the G –X crys-tal orientation (which corresponds to light incident alongthe [110] crystallographic direction of GaAs) was obtainedand compares favorably with that obtained from the mea-sured value of x (2).27 Similarly, consistent values werealso obtained for the p polarization (;1 3 10222 m2 W21

with G –M crystal orientation for both theoretical and ex-perimental results). In fact, the SH reflected signal forincident p polarization is greater than for incident s po-larization; for s polarization, G –M signals (light incidentalong the [100] axis of GaAs) are lower than G –X signals(light incident along the [110] axis of GaAs), whereas for ppolarization the trend is the opposite. The occurrence ofa large difference in the SH amplitudes on reflection with

Fig. 6. Measured and calculated diffraction angles for light in-cident along the G –X and the G –M crystal orientations and forthe two polarizations (s-pol and p-pol) of the pump beam as afunction of the angle of incidence u. The pump wavelength is814 nm, and this value is also used for the theoretical curves.

s and p input polarization is a useful test, in this case, ofthe mutual orientation of the GaAs crystal and the pho-tonic lattice. Figure 7(a) therefore stresses the domi-nance, in nonlinear reflection, of the crystal symmetry ofGaAs over the photonic lattice. The maximum values ofRNL that occur for u > 50° for the p-polarized pump aredue to increasing coupling of the input in proximity to thepseudo-Brewster angle of incidence. The inset in Fig.7(a) shows the angular behavior of RNL for p-polarizedincident light calculated from the formula RNL

} ue2v(u)x (2)ev(u)ev(u)u2, where ev (e2v) is the linear(nonlinear) Fresnel factor25 evaluated at the angle of in-cidence: This formula applies to SH reflection from asemi-infinite medium. The calculated curves are verysimilar to the measured ones, showing that neither thewaveguide structure nor the patterning with low air frac-tion appreciably alters the angular dependence of thenonlinear reflection coefficient.

The nonlinear response determined by the photonic-crystal structure emerges instead from the study of thenonlinear diffraction coefficients. These are shown inFig. 7(b) as a function of incident angle u for l5 814 nm pump wavelength, again for incident p and spolarization and for the two photonic-crystal orientations.First, we can observe that nonlinear diffracted coefficientdata with s polarization are comparable with RNL resultsfor the same polarization, and this points to the impor-tance of the diffraction process for this polarization. In

Fig. 7. Nonlinear (a) reflection and (b) diffraction coefficients asa function of angle of incidence for a pump wavelength of l5 814 nm, for the crystal orientations and input polarizations(p-pol and s-pol) considered. Note the break in the vertical scalein (a). Insets, results of an approximate theoretical model (seetext).

Malvezzi et al. Vol. 19, No. 9 /September 2002 /J. Opt. Soc. Am. B 2127

addition to the angular dependence, it is interesting tocompare the values of the coefficients in thiss-polarization case. The diffraction efficiency h5 DNL/(DNL 1 RNL) is of the order of 30% for bothphotonic-crystal orientations, indicating that a consistentfraction of the SH power is diverted by the photonic-crystal structure into the diffraction mode. (If nonlineardiffraction into all modes, including those out of the planeof incidence, were considered, the value of the diffractionefficiency would be even higher.) A distinctive indicationof a photonic crystal is the u dependence shown in thecase of p polarization, which is characteristic of a macro-scopic structure superimposed upon a GaAs substrate.In contrast to the behavior of nonlinear reflection, thenonlinear diffraction coefficient for p polarization has adecreasing behavior with angle of incidence in the G –Xorientation, whereas a maximum near 50° is measuredfor the G –M photonic orientation. In an attempt to in-terpret the measured diffraction curves, we calculated theangular dependence of the nonlinear diffraction coeffi-cient from the formula DNL } u«2v(u8)x (2)«v(u)«v(u)u2,where now the nonlinear Fresnel factor «2v was calcu-lated at the diffraction angle u8. This expression is exactin the case of shallow nonlinear gratings8–11 for which thespatial variation of the linear susceptibility has a negli-gible effect and SH diffraction originates only from amodulation of x (2): In the present case, both linear andnonlinear susceptibilities are spatially modulated in thewhole waveguide structure, and the theory of nonlineardiffraction is considerably more complex. The DNL

curves that arise from the formulas above are shown inthe inset of Fig. 7(b), again for p polarization and for thetwo crystal orientations, and show striking similarities tothe corresponding experimental curves. In particular,the maximum of DNL for an incidence angle of u ' 50° forp polarization and G –M orientation is reproduced. Thismaximum arises from the combined effects of the pseudo-Brewster angle [expressed by the Fresnel factor «v(u) andalso affecting the nonlinear reflection] and of the nonlin-ear Fresnel factor «2v(u8) at the diffraction angles u8shown in Fig. 6. The remaining differences from themeasured angular dependences of DNL can be attributedto the waveguide structure and to the resultant Fabry–Perot oscillations.

Fig. 8. Nonlinear reflection and diffraction coefficients as afunction of pump wavelength for the G –X sample orientation,angle of incidence u 5 45°, and s polarization (s-pol).

In Fig. 8, we show the measured nonlinear reflectionand diffraction coefficients as a function of the pumpwavelength for s polarization and an incidence angle ofu 5 45° with G –X crystal orientation. These data fur-ther emphasize the fact that nonlinear diffraction by thephotonic-crystal structure has completely different prop-erties from nonlinear reflection. Indeed, a change of afactor of approximately 2 in RNL is observed when thepump wavelength is changed from 826 to 839 nm,whereas an almost constant behavior of DNL is observed.Comparing with the linear reflectance curves of Fig. 2, wesee that SH measurements are made in a frequency re-gion where no photonic mode of the patterned waveguideis observed but where pronounced Fabry–Perot oscilla-tions still remain. Thus any theory of nonlinear reflec-tion and diffraction must take into account the patternedwaveguide structure with full modulation of linear andnonlinear optical susceptibilities in three dimensions.

5. CONCLUSIONSSecond-harmonic diffraction from a GaAs/AlGaAsphotonic-crystal waveguide with a square lattice has beenmeasured at the wavelengths of a Ti:sapphire laser. Theangles of diffracted SH beams agree well with those pre-dicted by nonlinear diffraction conditions. Results for re-flected SH intensities are close to those for a (001) GaAscrystal surface, with the 45° rotation between the squarephotonic lattice and the [100] crystallographic axis ofGaAs taken into account. This is so because of the lowair fraction of the photonic crystal and shows that the re-flected SH signal is essentially determined by the micro-scopic crystalline symmetry of GaAs rather than by thephotonic lattice. The dependence of diffracted SH inten-sities on incidence angle, input polarization, and pumpwavelength, however, is different from that of SH reflec-tion owing to the properties of the photonic lattice. Thediffraction-to-reflection ratio is large and points to the im-portance of nonlinear diffraction in photonic crystals, justas for the analogous linear phenomenon. Theoretical cal-culations of the diffraction intensities are in progress andwill contribute to a detailed understanding of the nonlin-ear process.

ACKNOWLEDGMENTSThis study was supported in part by the Istituto Nazion-ale per la Fisica della Materia (INFM) PAIS 2001 project‘‘2DPHOCRY’’ and by the INFM Femtosecond Lab. Theauthors are grateful to Y. Chen and D. Peyrade for per-forming part of the processing and to M. Galli and M.Patrini for linear reflectance measurements of thesample.

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