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Yan et al. Vol. 26, No. 1 /January 2009 /J. Opt. Soc. Am. A 135

Bragg diffraction of multilayer volumeholographic gratings under ultrashort laser

pulse readout

Aimin Yan,* Liren Liu, Yanan Zhi, De’an Liu, and Jianfeng Sun

Information Laboratory, Shanghai Institute of Optics and Fine Mechanics, The Chinese Academy of Sciences,Shanghai 201800, China

*Corresponding author: yanaimin@mail.siom.ac.cn

Received August 8, 2008; revised November 16, 2008; accepted November 17, 2008;posted November 21, 2008 (Doc. ID 99975); published December 19, 2008

The multilayer coupled wave theory is extended to systematically investigate the diffraction properties ofmultilayer volume holographic gratings (MVHGs) under ultrashort laser pulse readout. Solutions for the dif-fracted and transmitted intensities, diffraction efficiency, and the grating bandwidth are obtained in transmis-sion MVHGs. It is shown that the diffraction characteristics depend not only on the input pulse duration butalso on the number and thickness of grating layers and the gaps between holographic layers. This analysis canbe implemented as a useful tool to aid with the design of multilayer volume grating-based devices employed inoptical communications, pulse shaping, and processing. © 2008 Optical Society of America

OCIS codes: 320.5540, 090.7330, 160.5320, 260.1960.

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. INTRODUCTIONolume holographic gratings (VHGs) are of wide interest

n many applications [1–3] because of their properties ofigh diffraction efficiency, excellent wavelength selectiv-

ty, and angular selectivity. Lately they have receivedide attention as a class of novel diffraction elements inhich multiple layers of VHGs are separated by opticallyomogeneous intermediate layers (as shown in Fig. 1).hanks to the unique properties of VHG, multilayerHGs (MVHGs) have become an ideal candidate for vari-us promising technological applications such as opticalnterconnects [4], wavelength division multiplexes andemultiplexes [5], and optical filters [6,7]. Therefore, anowledge of the diffraction behaviors of such a systemould be very valuable for characterizing and optimizing

uch volume diffractive optical elements, and several au-hors have considered this problem using various ap-roaches [8–12].The most commonly used theory for diffraction by a

HG is the coupled-wave theory of Kogelnik [13]. The re-ultant widely used simple analytical expressions haveeen found to yield extremely good results when the un-erlying assumptions are satisfied. However, these theo-etical analyses cannot directly be applied to MVHGsince the system is multiple layers. Yakimovich [8] intro-uced layer transmission matrices to Kogelnik’s [13]oupled-wave theory and presented diffraction analyses ofultilayer holograms. We denoted it as the multilayer

oupled wave theory. Based on the theory, Spariosu et al.4] developed a type of multi-layer volume diffractive op-ical element, which can simulate segmented slantedrating structures with variable element thickness. Ko-otskii and Nikulin [14] presented a theoretical analysis

f the diffraction of a Gaussian optical beam by a systemf two diffraction gratings and Vre and Hesselink [15]

1084-7529/09/010135-7/$15.00 © 2

nalyzed the properties of photorefractive stratified vol-me holographic optical elements. Although many studiesave been reported in this field, all of them, to my knowl-dge, assume a continuous wave as a light source. Fewtudies have touched on the use of an ultrashort laserulse as a readout beam.The ultrashort optical pulse has many useful applica-

ions in optics because of its abundant spectrum and highandwidth. The research on diffraction of ultrashort opti-al pulses by VHGs has recently attracted considerablettention. Ding et al. [16] have shown both theoreticallynd experimentally that it is possible in the frequency do-ain to achieve a large enough diffraction bandwidth ofHG for a bandwidth of 100 fs pulses mainly by increas-

ng the grating period and decreasing the grating thick-ess. Recently Wang et al. [17] have studied the pulsehaping properties of a VHG read by an ultrashort pulseonsidering the combined effects of the grating param-ters, the dispersion and optical anisotropy of the media,nd the polarization state of the incident light. We [18]lso extended the two-dimensional coupled-wave theoryo study the diffraction of an ultrashort pulsed finiteeam by a VHG. However, the detailed diffraction proper-ies of an ultrashort optical pulse by a system of MVHGsave not been performed. In practical applications, it is

mportant to carry out systematic investigations in thiseld, because a system of MVHGs and ultrashort opticalulses both have a variety of novel applications, and theirombination has enormous potential. No research has soar been reported on the combination of a system of

VHGs and pulsed light. This also provides the motiva-ion behind our work presented here. In this paper, con-idering the dispersion effect of the VHG, we extend theoupled-wave theory of the multilayer structure to studyhe Bragg diffraction properties of ultrashort optical

009 Optical Society of America

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136 J. Opt. Soc. Am. A/Vol. 26, No. 1 /January 2009 Yan et al.

ulses, and present a systematically theoretical analysisn the spectrum distribution of the diffracted intensities,he diffraction bandwidth, and the total diffraction effi-iency of a system of MVHGs.

In Section 2, the modified multilayer coupled-waveheory used to analyze the diffraction properties of a sys-em of MVHGs is derived, and expressions of the spectralntensity distributions of the transmitted and diffractedulsed beams are given. Numerical analysis of the diffrac-ion properties of the system of MVHGs in the transmis-ion geometry for s-polarized fields is given in Section 3. Aummary of the results and the conclusions are presentedn Section 4.

. MODIFIED MULTILAYER COUPLED-AVE ANALYSIS

he configuration analyzed is shown in Fig. 1. The systemf MVHGs is composed of N VHGs separated by N−1 in-ermediate layers. In general, all layers have differenthicknesses Ti and gaps di.

All VHGs are assumed to extend infinitely in the x–ylane and have the same grating material and gratingrientation with the grating period �. And they are all as-umed to be thick: the Q parameter, Q=2�Ti� / �n0�2�,rovides an evaluation of the grating thickness with re-pect to the condition [12] Q�1, where n0 is the averageefractive index of the medium and � is the free-spaceavelength of the reading beam.Considering the dispersion effect of the grating mate-

ial, its refractive index is described by

n��� = n0��� + n1 cos�K · r� n1 � n0���, �1�

here n1 is the amplitude of the refractive index modula-ion, �=2�c /� is the angular frequency of the input pulse,is the speed of light, and K is the grating vector with

rating slant angle � and magnitude K=2� /�. It is as-umed n1�n0���. All gaps have the same refractive indexqual to the corresponding mean values n0���. An ul-rashort pulse with a field amplitude of U0��� in the fre-uency domain and the propagation vector kr=��sin xcos z� is incident on the system of MVHGs, where �2�n��� /� is the propagation constant and is the inci-ent angle.For each frequency component of the input ultrashort

ulse, we can use the multilayer coupled-wave theoryhat Yakimovich [8] developed. In this paper, since an ul-rashort pulse contains a broad optical frequency spec-rum, we extend the existing multilayer coupled-wave

Fig. 1. Model of a system of multilayer volume

heory to analyze polychromatic fields. In the Bragg-iffraction regime, only the transmitted and diffractedeams are present [19]. We also do not take absorptionnto account. Here the total electric field in the layers ofhe holographic grating region may be written as

E�x,z,�� = erRi�z,��exp�− jkr · r� + esSi�z,��exp�− jks · r�,

�2�

here Ri�z ,�� and Si�z ,�� are the complex amplitudes ofhe transmitted and diffracted beams in the ith layer and= �x ,y ,z�. Here er and es are the relevant beam unit po-

arization vectors, whereas ks is the propagation vector ofhe diffracted beam. The phase-matching condition foriffraction process is given by ks=kr−K.Similarly, in the ith intermediate layer the field is

E�x,z,�� = erRi���exp�− jkr0 · r� + esSi���exp�− jks0 · r�,

�3�

here Ri��� and Si��� are constants for a given frequency,r0=ks0=�0�sin x+cos z�, �0=2�n0��� /�.Substituting Eq. (2) into the scalar wave equation

2E+k2E=0 (where k=2�n /�), we can get the transmit-ed and diffracted beams on the right-hand boundary ofhe ith layer Rir and Sir connected with the amplitudes ofhese beams on the left-hand boundary Ril and Sil by theollowing matrix equation:

�Rir

Sir� = �mi11 mi12

mi21 mi22� �Ril

Sil� , �4�

here mi11 = �cos�VTi� + j�

Vsin�VTi��exp�− j�Ti�, �5a�

mi12 = − j�

V�CS

CRsin�VTi�exp�− j�Ti�, �5b�

mi21 = − j�

V�CR

CSsin�VTi�exp�− j�Ti�, �5c�

mi22 = �cos�VTi� − j�

Vsin�VTi��exp�− j�Ti�. �5d�

In Eq. (5), CR=cos ; CS=cos − �K /�� cos �; V=��2+�2;=�n1�er · es� /��CRCS is the coupling constant, where

phic gratings read by an ultrashort laser pulse.

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Yan et al. Vol. 26, No. 1 /January 2009 /J. Opt. Soc. Am. A 137

er · es�=1 for s polarized and for p polarization �er · es�−cos 2�−�� (here the s-polarized wave defines an inci-ent beam whose polarization is perpendicular to the in-ident plane, whereas the p-polarized wave has a polar-zation parallel to the plane); and �= /2cS is off-Braggarameter and =K cos��−�− �K2c /2�n����.At the boundary between the ith intermediate layer

nd the i+1 grating, we can also find the following matrixquation:

�R�i+1�l

S�i+1�l� = �Di� �Rir

Sir� , �6�

here the transmission function

�Di� = �1 0

0 exp− j di

CS� .

For a system of transmission MVHGs, on the input sur-ace �z=0�, there is no diffracted beam and S0�0,��=0 andhe input beam is R0�0,��=U0���. Then, at the outputoundary, the amplitudes of the diffracted beam S�Td ,��nd transmitted beam R�Td ,�� form the following matrixquation:

�R�Td,��

S�Td,��� = �MC� �R0�0,��

S0�0,��� . �7�

here

Td = �i=1

N

Ti + �i=1

N−1

di,

�MC� = MNDN−1MN−1 . . . DiMi . . . D1M1,

re, respectively, the total thickness and the optical trans-ission function of the system; Mi and Di can be obtained

n Eqs. (4) and (6).According to Eq. (7), the intensity distributions of the

ransmitted and diffracted pulsed beams are

IR�Td,�� = R�Td,�� 2, IS�Td,�� = S�Td,�� 2. �8�

e can see that the distributions of the transmitted andiffracted pulsed beams will be greatly influenced by theumber and thickness of grating layers and gaps betweenhem, the grating period, the pulse duration of the inputltrashort pulse, and the dispersion effect of the gratingaterials. Taking the LiNbO3 crystal and Gaussian-

rofile pulse in the time domain as examples, the numeri-al analysis of the distribution will be presented in Sub-ection 3.B.

For each frequency component the diffraction efficiencys defined as

����� = � CS IS�Td,���/�CRI0�0,���, �9�

here I0�0,��= R0�0,�� 2. Their maximum values are forhe center frequency component that satisfies the Braggondition of VHG; they are reduced to the well-knownxpressions of Yakimovich [8] for the MVHGs read by

continuous wave, namely, ����0�= � CS sin2�V��0���i=1

N Ti��� /CR for transmission geometry. But the totaliffraction efficiency � of the system of MVHGs be-

Tol

omes more complicated than it. Here we define �Tol as

�Tol = CS

CR

� IS�Td,��d�

� I0�0,��d�

. �10�

. CALCULATED RESULTS ANDISCUSSIONSetailed results for the system of MVHGs in the un-

lanted transmission geometry are presented in this sec-ion. In all cases the electric fields are s polarized. Heree assume that the system of MVHGs is recorded in ahotorefractive LiNbO3 crystal, which is widely used inractical applications.Here we neglect the dispersion of n1, since n1�n���,

nd

n1 = − n03��0��13Esc/2, �11�

here �13=8.610−12 mV−1 is the electro-optic coefficientnd Esc is the space-charge field produced by the linearhotorefractive effect in the process of transport of photo-nduced charges in the crystal (diffusion, drift, photovol-aic effect, etc. [19]]. We assume that the space-chargeeld Esc is parallel to the grating vector K. We choose theentral wavelength �0=1.06 �m, �=3 �m, and Esc=5.010−6 V m−1. Considering the dispersion effect of the

rystal, the wavelength dependence of its refractive index20] is given by

n2��� = a0 +a1

�2 − a2− a3�2, �12�

here a0=4.9048, a1=0.117680, a2=0.047500, a30.027169, and � is in micrometers.As an example, we assume that the ultrashort pulse

as a Gaussian distribution in the time domain, so thatts temporal amplitude can be generally expressed as

u0�t� = exp�− j�0t�exp− 2 ln 2t2

�2 , �13�

here � is the FWHM (intensity) pulse duration. The Fou-ier transform U0��� of this pulse is

U0��� =� ��2

2 ln 2exp�−

�2

2 ln 2

�� − �0�2

4 � , �14�

nd the corresponding FWHM bandwidth of its powerpectrum U0��� 2 is ��in=�0

2CB / �c��, where CB=0.44121].

. Half-Power Bandwidth of MVHGshe Bragg selectivity is a notable feature of VHG. Its se-

ective bandwidth is a critical parameter for a pulsed lighteadout, because the incident light is no longer the usualonochromatic, but involves a broad spectrum that corre-

ponds to different frequency. Therefore, for a fixed grat-ng and Bragg incident angle, it is impossible for all spec-rum components simultaneously to satisfy the Bragg

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138 J. Opt. Soc. Am. A/Vol. 26, No. 1 /January 2009 Yan et al.

ondition except the central wavelength �0. So the off-ragg parameter exists in Eq. (5). The first-order ap-roximation of the Taylor series of for a slight deviationf the input wavelength ��=�0+��� but without angle de-uning is

= −�vp

�2n��0�vg��, �15�

here

vp =c

n��0�,

vg =c

n��0� − �0�dn���

d��

�=�0

re the phase and group velocities, respectively. If the dis-ersion effect of the grating material is not considered, weave vp=vg, and then this expression in this paper can beeduced to that of Yakimovich’s [8].

By using a method similar to that of [16] for the singleHG, we can get a wavelength range ��G, beyond which

he diffraction efficiency of each frequency componentithin the spectrum of the ultrashort pulse will drop be-

ow half of its maximum value. With the expression [inq. (15)] of the given Bragg mismatch parameter , ��Gan be approximately expressed as

��G =16��n��0�cos

K2

vg

vp, �16�

here � is the average value of � when the diffraction ef-ciency drops to half of its maximum value with smallhanges of �, which can be calculated numerically from���� for MVHGs.From Eq. (16), we easily see that the grating band-

idth ��G is inversely proportional to the square of therating vector K and increases with decreasing incidentngle . The conclusions are consistent with [16] for a con-inuous wave. But there also exist some differences. Con-idering the dispersion effect of the LiNbO3 crystal, ��Gill be smaller because vg�vp, which can be calculated

rom Eqs. (12) and (15). It means that the input ul-rashort pulse is broadening in the temporal domain afteraving been diffracted by the MVHGs with the dispersionffect. In addition, ��G depends on the average value ofhe off-Bragg parameter �. If any one of the parametersi, di, the total thickness Td or the number of layershanges, the values of � and ��G will greatly different.hese differences eventually result in the different spec-rum distributions of the diffracted and transmittedulsed beams. Thus in practical applications, if the geom-try parameters of the system of MVHGs and input con-itions are appropriately chosen, we can obtain an appro-riate Bragg selectivity bandwidth �G to diffract theeeded spectrum components of an ultrashort pulse with

ufficient diffraction efficiency. This phenomenon is espe-ially useful for the applications of pulsing shaping, pro-essing, and filter.

. Diffracted Spectrum and Diffracted Bandwidthere, for the first time to our knowledge, diffraction by a

ystem of MVHGs is discussed with ultrashort pulseeadout and the dispersion effect of the grating materialaken into account. Equation (8) is used to determine thentensity distributions of the diffracted pulse beams. Fig-re 2 shows the spectral distributions of the normalizediffraction intensity of a two-layer system for various val-es of the thicknesses of the grating layers and the gapsetween them, and the grating period, provided that thenput pulse is a Gaussian shape with a pulse duration of=50 fs. The diffraction bandwidth is regarded as theWHM of the diffracted intensity in the frequency do-ain.In Fig. 2(a) the grating-layer thickness is held con-

tant, T1=T2=0.5T0, and the relative distance betweenayers d1 /T0=0,0.3,0.6,0.9, where T0=1 mm is the thick-ess of the whole grating without intermediate layers.he spectral distributions change greatly with increasing1/T0 and the intensities of the sidelobe increase. This in-icates that the diffracted bandwidth decreases and morend more frequency components of the input pulse cannote diffracted effectively by the system of MVHGs due tohe excellent Bragg selectivity of the grating. This is be-ause additional phase mismatching is produced betweennteracting beams with the increase of the intermediate-ayer thickness, which results in the enhanced Bragg se-ectivity of the system. In Fig. 2(b), a gap betweenratings and the total thickness of the grating layersre fixed, d1=0.5T0, T1+T2=T0, and the relativehickness between two gratings T1 /T20.1/0.9,0.2/0.8,0.3/0.7,0.5/0.5. It is shown that when

he intermediate-layer thickness and the summed thick-ess of two gratings remain constant, the smaller theelative thickness or the larger the absolute value of T1T2 becomes, the larger the diffraction bandwidth. For

he MVHGs used in Figs. 2(a) and 2(b), the correspondingiffraction bandwidth can be obtained from Figs. 3(a) and(b) at �=50 fs. Figure 4(c) shows the relation betweenhe diffraction intensity spectrum and the grating periods, T1=T2=0.5T0, and d1=0.1T0 ,�=1, 3, and 5 �m. It is

lear that when the grating period increases fromto 5 �m, the decreasing speed of spectral distributions

ecomes slower and the bandwidth of the diffraction in-reases because of the weaker Bragg selectivity ofVHGs.The normalized intensity distributions of the two-, five-

and eleven-layer systems of MVHGs are illustrated inig. 4. The three systems are composed of identical holo-raphic gratings separated by gaps of equal magnitudeTi=0.5T0 and di=0.1T0). It is shown that as the numberf layers increases, the profiles of the diffraction intensitypectrum also changes dramatically. In addition, the dif-racted intensity of the 11-layer system differs more fromhe Gaussian profile and is even split into two Gaussianobes. This distortion is more visible as the number of lay-rs increases. The effect is due to the stronger coupling ofnergy between the transmitted and diffracted beams

Ftalt=a

Fwith the same parameters as Figs. 2(a) and 2(b).

Ftwm

Yan et al. Vol. 26, No. 1 /January 2009 /J. Opt. Soc. Am. A 139

ig. 2. Normalized spectral distributions of the diffraction in-ensity in the two-layer system read by an ultrashort laser pulset �=50 fs: (a) T1=T2=0.5T0, and the relative distance betweenayers d1 /T0=0,0.3,0.6,0.9, (b) d1=0.5T0, T1+T2=T0, andhe relative thickness between two gratings T1 /T20.1/0.9,0.2/0.8,0.3/0.7,0.5/0.5, (c) T1=T2=0.5T0, d1=0.1T0,

nd the grating period �=1, 3, and 5.

ig. 3. Variations of the diffraction bandwidth as a function of �

ig. 4. Normalized spectral distributions of the diffraction in-ensity in the two-, five-, and eleven-layer systems of MVHGsith identical holographic gratings separated by gaps of equalagnitude (T =0.5T and d =0.1T ) at �=50 fs.

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140 J. Opt. Soc. Am. A/Vol. 26, No. 1 /January 2009 Yan et al.

uring the Bragg diffraction process with the large valuef the total thickness including grating layers and gaps.

For the grating parameters used in Figs. 2, if only theulse duration � changes, for example, � increases from0 to 250 fs, Fig. 3 shows the variations of diffractionandwidth as a function of �. We give the typical valuesvery 20 fs. The corresponding input bandwidths are plot-ed in these figures for comparison. We see that the dif-raction bandwidths and input bandwidths decrease withncreased �, and the diffraction bandwidth is close to thenput bandwidth for all cases. In addition, for a givenalue of �, the diffraction bandwidth for the d1=0 case inig. 4(a) and the T1 /T2=0.1/0.9 case in Fig. 3(b) are the

argest. From Figs. 2–4 we systematically show the dif-racted beam properties including the spectral intensityistribution and the diffraction bandwidth based on theultilayer coupled-wave theory, which could not be de-

cribed by the theory of single VHG given in [13,17].

. Total Diffraction Efficiencyhe definition of the total diffraction efficiency �Tol for aystem of MVHGs read by an ultrashort pulse is given inq. (10). In Figs. 5(a) and 5(b) we show the variations of

ig. 5. Variation of the total diffraction intensity �Tol in the two-ayer system as a function of � with the same parameters as Figs.(a) and 2(b).

Tol with the input pulse duration � for the two-layer sys-em. The grating parameters are the same as those usedn Figs. 2(a) and 2(b).

As is shown in Fig. 5, �Tol increases with increased �nd reaches a constant when � is large enough so that thenput pulse can be considered as a continuous wave. Forxample, at T1=T2=0.5T0 and d1=0.3T0, �Tol is close to.4. The reason is that the diffraction efficiency of pulsedeam is influenced by pulse duration � (or the input band-idth ��in). When the pulse duration increases, the inci-ent pulsed beam involves fewer frequency componentshat deviate from the Bragg condition. The narrower thenput bandwidth ��in is, the more effective is the diffrac-ion, and hence the larger is the diffraction efficiency. Forfixed value of �, the smaller d1 is, the larger is �Tol. In

ddition, it is interesting in Fig. 5(b) that the larger T1T2 is, the larger is �Tol.Taking the MVHGs used in Fig. 4, Fig. 6 shows the to-

al diffraction efficiency �Tol as a function of � for the two-,ve-, and, eleven-layer systems of MVHGs. It is clear thator a given value of �, �Tol of the 11-layer system is not theargest, but is smaller than that of the five-layer systemnd larger than that of the two-layer system. For ex-mple, at �=150 fs, a two-layer system can diffract asuch as 40%, an 11-layer system in excess of 58%, and ave-layer system more than 70%. For the system of 11-

ayer MVHGs, increasing the number of grating layersas actually decreased the diffraction efficiency. This isecause the variation of diffraction efficiency of a singlerequency is the sin �x� function, for example,����0�� CS sin2�V��0���i=1

N Ti��� /CR. The maximum diffractionfficiency occurs when the product of the parameter V���nd the total thickness satisfy the condition, �V��0���i=1

N Ti��=m�� /2� (m is an odd integer) for the centerrequency component that satisfies the Bragg condition ofHG. Therefore, although the total thickness of 11-layerVHGs is larger than that of the five-layer system, the

iffraction efficiency may be smaller because of the prod-ct of �V��0���i=1

N Ti�� deviating too much from m�� /2�.Therefore, in order to get a large diffraction bandwidth

r a shaped ultrashort pulse with suitable total diffrac-ion efficiency in practical applications, the optimal com-ining of the thicknesses of the grating and intermediate

ig. 6. Variation of the total diffraction intensity �Tol as a func-ion of � in the two-, five-, and eleven-layer systems of MVHGsith the same parameters as Fig. 3.

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Yan et al. Vol. 26, No. 1 /January 2009 /J. Opt. Soc. Am. A 141

ayers, the number of modulation layers of the system ofVHGs, and the input pulse conditions should be se-

ected. Such a characterization will be useful for evaluat-ng the performance of optical systems employing

VHGs, determining the optimal parameters at the de-ign phase, and improving existing optical systems.

. CONCLUSIONShe Bragg diffraction properties of a system of MVHGsnder an ultrashort laser pulse readout have been sys-ematically investigated based on the modified multilayeroupled-wave theory. With variations of the input pulseuration, the diffraction bandwidth and the total diffrac-ion efficiency are analyzed, which could not be consid-red by an analysis based on the coupled-wave theory of aontinuous wave.

The normalized intensity distributions of the diffractedeam, the diffraction bandwidth, and the total diffractionfficiency are obtained. All of them are greatly affected byhe the input pulse duration, the number and thickness ofrating layers, the gaps between holographic layers, andhe total thickness of the system of MVHGs. When theulse duration of an input ultrashort pulse is increased,he total diffraction efficiency is close to the diffraction ef-ciency for the same MVHGs illuminated by a continuousave. With the increase of the number of grating layers,

he total diffraction efficiency does not always increase,ut maintains the existing oscillation. So, to obtain theistributions of the diffracted pulsed beam with approxi-ate bandwidth and enough total diffraction efficiency,

he optimal combination of the input conditions and theVHGs parameters should be selected. The analysis and

bservations of this paper will be valuable for the accu-ate analysis of the interaction of ultrashort opticalulses and a variety of periodic structures, thus facilitat-ng the design and the investigation of novel optical de-ices based on MVHGs.

CKNOWLEDGMENThe authors gratefully acknowledge support from the Na-ional Natural Science Foundation of China (NSFC)grant 60708018).

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