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1554 J. Opt. Soc. Am. B/Vol. 22, No. 7 /July 2005 G. Lévêque and R. Mathevet

Blazed atom grating

Gáetan Lévêque and Renaud Mathevet

Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes, Laboratoire Collisions, Agrégats etRéactivité, Unité Mixte de Recherche, Centre National de la Recherche Scientifique 5589, Université Paul

Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France

Received July 26, 2004; accepted November 22, 2004

We present a method to create a blazed atomic diffraction grating by use of a periodical optical potential. Likeits optical counterpart, the blazed atomic diffraction grating distributes intensity into a specific nonzero dif-fraction order. Total internal reflection of a laser beam coupled to the nanostructured surface of a prism resultsin transverse modulation of the intensity responsible for atomic diffraction. For specific illumination param-eters and periodicity of the pattern, the long-range potential interacting with the atoms has an asymmetricsawtooth shape. Analytic and numerical calculations show that population diffracted in the +1 order can beoptimized to approximately 55%, with almost no population into the −1 order. © 2005 Optical Society ofAmerica

OCIS codes: 020.0020, 015.1950.

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. INTRODUCTIONoherent matter-wave optics is a most direct manifesta-

ion of quantum behavior and dates back to the earlyuantum mechanics with, for example, the famouslectron-diffraction experiment of Davisson and Germern 1927.1 Since then, diffraction has been observed for at-ms, molecules, and other particules. Neutron diffractions nowadays routinely used in material science.

Neutral atoms are interesting because they are weaklyoupled to the environment and easy to produce. Further-ore, with now well-established techniques, they can also

e slowed, cooled, and coherently controlled with their in-ernal degrees of freedom. These developments havepened a new field of physics called atom optics. As withight optics, two main domains can be distinguished. Onhe one hand, in geometrical atom optics, experiments areesigned to deflect, guide, and, especially, focus atoms forpplications such as nanolithograhy. On the other hand,oherent atom optics deals with atom interferences andiffraction (see Refs. 2 and 3 for a paper and book). Theatter process has received a lot of attention as it can besed to create beam splitters for atom interferometersith separated arms. It is observed with either transmis-

ion material gratings4 or light standing waves in freepace.5 More elaborate configurations by use of an addi-ional magnetic field,6 bichromatic light,7,8 or doubletanding waves9 have been used or proposed to create alazed grating. One can then control the relative intensi-ies of the diffraction orders. In addition to these trans-ission gratings, reflection gratings have been realizedith evanescent standing waves10,11 or simple time modu-

ation of the evanescent field12 above a prism under totalnternal reflection.

Finally, in the past ten years, a strong tendency towardntegration and miniaturization has developed with theim of achieving degenerate quantum gases. Thanks toighly developed microelectronics planar architectures,

0740-3224/05/071554-7/$15.00 © 2

toms can now be directly manipulated near surfaces,mplementing the so-called atom chips.13 Up to now, elec-rical and magnetical fields have mainly been used. An al-ernate approach is to use the optical near field emergingrom a nanostructured substrate. It combines several ad-antages: The design is flexible, no metal is required, theagnetic field remains a free parameter, atoms never hit

r pass through the surface so clogging is avoided, and aingle beam is required, which greatly simplifies align-ent.In a previous paper14 we studied both theoretically and

umerically atom diffraction when a one-dimensional orwo-dimensional dielectric pattern is engraved on theubstrate. Then we introduced15 a graphical analysis as aesign aid for the nanostructured array and for under-tanding the diffraction pattern. In this paper we applyhis graphical analysis to predict and optimize a blazedtom grating.A blazed grating sends higher intensity into some spe-

ific diffraction channel. In light optics it is obtained withsymmetrical reflecting grooves. The groove angle is cho-en so that the reflection angle matches the diffractionngle for the design order and wavelength. The same ef-ect is obtained in atom optics when the isopotential linesxhibit some kind of left–right asymmetry in a sawtoothashion. Nevertheless, in contrast light optics, we will seehat such potentials cannot be generated simply from ansymmetry in the shape of the grating features: They re-ult from a careful choice of materials, period, and illumi-ation of the nanostructure.This paper is organized as follows. In Section 2 we

resent briefly the geometry of the nanostructure andive a graphical discussion of the optical potential. Ana-ytic atomic diffraction theory from such a potential ishen briefly reviewed in Section 3. We then show how touild an asymmetrical potential from the graphical dis-ussion previously introduced. The analytical theory is af-

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G. Lévêque and R. Mathevet Vol. 22, No. 7 /July 2005 /J. Opt. Soc. Am. B 1555

erward applied in Section 4 and finally compared with aumerical simulation in Section 5. The last section is de-oted to summarizing and concluding remarks.

. GEOMETRY AND OPTICAL POTENTIALn this part we discuss briefly the proposed geometry andhe principle of atomic diffraction. For more details, theeader is referred to Ref. 14. For the sake of simplicity, weonsider a two-level atom of resonant frequency v0, line-idth g, and saturation intensity Is. It interacts with aonochromatic light field of frequency v and intensity

srd. The detuning of the laser with respect to the atomicransition is defined as d=v−v0. If the laser intensity isufficiently weak or the detuning is large enough so thatbsorption can be neglected or both, it can be shown16

hat the interaction between the atom and the light fieldeduces to the dipole potential:

Vsrd ="g2

8d

Isrd

Is. s1d

his potential is proportional to the local-field intensitynd repulsive (attractive) if the laser is blue (red) detunedrom the atomic transition.

We are interested in the field diffracted by a subwave-ength grating of period L in the x direction, depositednto a dielectric prism of refraction index n. The angle ofncidence u of the incoming light is assumed large enougho that light experiences a total internal reflection at therism surface. The azimuthal orientation with respect tohe grating will be denoted w (see Fig. 1). As the boundaryonditions are periodic, the field above the prism can beourier expanded:

Esrd = om

Em expsikm · ldexps− kmzd. s2d

or an incoming wave vector ki, momentum and energyonservation read as

km = nkii + m

2p

Lex,

ig. 1. Grating of parallel stripes of width a and period L. Illu-ination is made at incidence and azimuthal angles su ,wd, and

ight experiences total internal reflection into the prism. Theear field above the prism is evanescent and modulated by theurface corrugation.

km2 = km

2 − ki2, s3d

ith kii=ki

xex+kiyey, l=xex+yey as parallel components

nd z as the direction normal to the prism surface. Thendex m labels the Fourier orders of the field.

If no grating is present, the field simply reduces to anvanescent wave Esrd=E0 expsinki

i · ldexps−k0zd and thushe potential Vsrd=V0 exps−2k0zd, where V0 depends onhe laser intensity and detuning. The detuning is chosenositive so the potential is repulsive, and, if the intensitys strong enough, V0 can exceed the kinetic energy Ek ofn atom impinging in normal incidence onto the prism inhe z direction. Such an atom specularly bounces off therism acting as an atomic mirror. The corresponding clas-ical turning point, denoted zr in the following, reads asr= s1/2k0dlnsV0 /Ekd. Nevertheless, it has been shown11

hat the atom must reflect at least 100–150 nm away fromhe prism to keep van der Waals attraction of the surfaceegligibly small.With a periodically modulated surface, the situation is

ifferent as many Fourier orders of the field are presentnd their range Lm=1/km varies with the order. The re-ult is a composite field structure that evolves with theeight z above the prism. However, as the atom has toounce rather far away from the prism to avoid stronglyttractive van der Waals forces, only the harmonics withhe longer ranges contribute significantly to the diffrac-ion process. To better understand this Fourier-order fil-ering effect with height, we used in Ref. 15 a graphicalnalysis of the dispersion relation Eqs. (3), as illustratedn Fig. 2.

The reciprocal vectors Qm= sm2p /Ldex are representedlong the x axis by Qm=A0Am

W , with Am as points of ab-cissa m2p /L. The projection of the incoming wave vectorn the prism plane (Oxy) is represented by AA0

W=nkii. As

he azimuthal angle w is varied, A describes a circleround A0. The parallel component of the diffracted waveectors km is thus represented by nki

i+Qm=AA0W+A0AmW

AAmW . Let us draw a circle of radius ki around each Amnd denote Hm the point of the circle such that AHmAm istriangle with a right angle in Hm. The dispersion rela-

ion km2 =km

2 −ki2 is simply the Pythagorean theorem in

his triangle, and thus km corresponds to AHm. In thisepresentation it is easy to see which are the dominantomponents at long distances (shortest AHm) and howhey change when the angle u of the incident wave vectori, the orientation w of the plane of incidence, or the grat-

ng period L is varied.

ig. 2. Graphical presentation of the dispersion relation, Eqs.3). The figure is plotted for the following parameters: w=60°, l850 nm, n=1.46, and L=250 nm.

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Three remarks are now in order. First, if the gratingeriod is sufficiently subwavelength, that is, L,l /2pki, the diameter 2ki of the circles is smaller than theirpacing 2p /L, and the different circles do not intersect.econd, light experiences total internal reflection, whicheans that k0 is real, and thus nuki

iu.ki. The point A thusies outside the circle centered on A0. Finally, we will as-ume that A lies outside every circle. If the point A liesithin the circle centered on Am, the related mth diffrac-

ion order would no longer be evanescent. The situationsepicted in Figs. 2 and 3 are thus typical, and we can nowurn to the study of the diffraction process.

. DIFFRACTION FROM A SINGLETANDING-WAVE POTENTIALigure 3 corresponds to the following parameters: L336 nm, n=1.46, u=60°, l=850 nm and w=0°. Both of

he ranges L0,1 of the harmonics 0 and +1 are equal to 176m. The ranges L−1,2 are 37 nm, and higher orders areven shorter. Thus, if the classical turning point is ap-roximately 150 nm above the surface, only the zerothnd first orders remain, and the potential reads as

Vsx,zd < V0 exps− 2k0zdF1 + e1szdcosS2p

LxDG ,

ith

e1szd = e1 expfsk0 − k1dzg, e1 = 2uE0 · E1

*u

E02 .

or this specific case, k0=k1, and the contrast e1 is inde-endent of z. In general, the contrast e1szd depends oneight z, and, if k0,k1, the atomic mirror (associatedith the zeroth order) extends far away from the prism,hereas the grating remains located near the prism.arying the incidence azimuthal angle w, one can adjusthe ranges and thus the contribution of the two orders:he mirror and grating functions can be varied somewhat

ndependently.In Fig. 3, A is exactly between A0 and A1, k0=k1 and

he contrast e1szd is z independent. This arrangement cor-esponds to L=l /2n sin u and is exactly the case of a barerism with a partially standing evanescent wave as pre-iously studied.11

A detailed description of different theories of atomic dif-raction from partially standing evanescent waves can beound in Ref. 17. Among them, the thin-phase grating ap-

ig. 3. When L=l /2n sin u, A is in the middle of the segmentA0A1g, and then k0=k1 for w=0. This situation gives the sameeld structure as a partial standing evanescent wave.

roximation gives a simple physical insight into the phys-cs involved in diffraction, at least at normal incidence. Inhis semiclassical approximation, the interaction betweenhe modulated part of the potential and the atom is as-umed weak enough so that classical trajectories are un-ffected. The atomic phase distortion is then evaluated byne’s computing the classical action divided by Planck’sonstant along the classical unperturbed trajectories.

Applying this approximation to a sinusoidally modu-ated potential results in an atomic wave-function phase

odulated at the same period. It is then well known fromrequency modulation theory that the diffraction patternonsists of peaks whose amplitudes are given by theessel functions of the first kind. Extension of the expres-ion for the population spectrum Pm given in Ref. 17 tohis more general case leads to18

Pm = Jm2fe1szrdP Bsa1dg,

ith

a1 = k1/k0,

P = pa/"k0,

zr =1

2k0lnSV0

EkD ,

Bsa1d = 2a1−1GSa1 + 1

2D2Y Gsa1 + 1d, s4d

here G is the Euler’s gamma function, zr is the classicalurning point, P is the dimensionless incoming atomicomentum pa normalized to "k0, a1 is the grating–mirror

ecoupling parameter, and B is a form factor that ac-ounts for the exponential decays of the mirror and grat-ng. We see from the e1szrd factor that what matters is theorrugation of the potential at the classical turning point.n the following, the indices m or n label atom diffractionrders.

In Fig. 4 are represented the diffraction probabilitiesm for the orders m=−2 to m= +2 as a function of the re-uced incoming atomic momentum P. A few points can beoted. First, the diffraction probabilities in opposite or-ers are always equal to P =P .19 Second, maximum dif-

ig. 4. Analytic atomic diffraction spectrum, Eqs. (4), as a func-ion of atomic incident normalized momentum P for a subwave-ength grating. Only specular reflection sm=0d and the first twohannels sm= ±1, ±2d are shown.

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raction efficiency is of the order of 30% in each of the ±1opulation. Approximately 40% of the population is lost inhe remaining 0, ±2,… orders. Last, the diffraction spec-rum can be tuned to some extent by one’s simply varyinghe angle w of the plane of incidence. This changes k1 andhus a1. To go beyond the thin-phase grating approxima-ion’s numerical calculations of the field and diffractedopulations would be required; at normal incidence, noajor difference has been found.14

After this summary of the geometry, fields, and theoret-cal background, we can now turn to the central part ofhis paper: how to improve diffraction efficiency in a defi-ite order.

. ATOMIC BLAZED GRATING. Grating Geometry and Illuminationo create an atom blazed grating, we can think at first tose sawtooth grooves, analogous to a light grating, as de-icted in Fig. 5. Unfortunately, this approach does notork under usual conditions of illumination. Just above

he structure, the optical potential will mimic somewhathe left–right symmetry breaking of the material bound-ry conditions. The asymmetric modulation is constructedrom Fourier components of higher order s±2, ±3,…d thatre dephased with respect to the first order. Nevertheless,hese higher orders have a short range Lm and will makenegligible contribution at the classical turning point of

he atom trajectory. A more fruitful approach is to choosepecial illumination conditions so that two nonzero Fou-ier components of the field dominate at the classicalurning point. These conditions are then easily foundrom the diagrams introduced before: A should lie some-here in between A1 and A2 (Fig. 2), so orders 1 and 2ave approximately the same range. As to be shown inubsection 4.B, the potential has then the expected saw-ooth shape in Fig. 6. To achieve this condition, however,equires a particular choice of the grating period.

For typical glasses whose indices of refraction rangerom 1.5 sSiO2d to 2.2 sSi3N4d, A0A=nki sin u is onlylightly larger than ki=2p /l. Inspection of Fig. 3 showshat it could be desirable to have A0A,3p /L, approxi-ately midway between A1 and A2. As the choice of re-

raction index is limited, A0A is not readily tunable. Theistance between Am points must then be decreased, andherefore the period L must increase. As a consequence,he diffraction angle will be smaller. On the diagram plot-ed on Fig. 7, the distances between the centers of the

Fig. 5. Schematic view of a dielectric blazed grating.

ircles are then shorter than their radii. They mutuallyntersect, and w has to be far enough from 0 so that A liesutside any circle, thus ensuring all orders are evanes-ent. Moreover, the ranges L1,2 of the orders 1 and 2 arereater than the one of zeroth order, and the contrast ofhe potential increases with height. This implies that ei-her the depth of the grooves or the index contrast muste small so that, at the classical turning point, the actualontrast of the potential is not too great. Otherwise, manyiffraction channels will be opened,17 which is undesir-ble for our purpose of optimizing diffraction in one chan-el. Finally, we will assume for the sake of simplicity thatand w are chosen so that A is closer to A0 than to A3 so

hat the third order can be neglected. Analytical calcula-ions of the diffraction spectrum are then straightfor-ard.

. Potential and Diffraction Spectrumnder the previous assumptions, the potential simply

eads as

Vsx,zd = V0 exps− 2k0zdF1 + e1szdcosS2p

LxD

+ e2szdcosS4p

Lx + f0DG , s5d

ith e1szd=e1 expfsk0−k1dzg and e2szd=e2 expfsk0−k2dzg.he relative phase of the first and second Fourier compo-

ig. 6. Left scale: potential transverse variation at the classicalean turning point Vsx ,zrd. Right scale: departure of the actual

lassical turning point from its mean value, Dz=zsxd−zr. It giveshe section of an equivalent atomic hard mirror. Dotted line, andeal sawtooth profile.

ig. 7. Graphical plot of the dispersion relation, Eqs. (3), for alazed grating parameter set. If w=45°, all harmonics are eva-escent. In contrast, the first, second, and third orders are radia-ive for w=0°.

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ents of the field is f0. This relative phase emerges notrom the shape of the grooves, which are symmetrical inhe following calculations, but from the symmetry break-ng induced by the orientation w of the plane of incidence.his symmetry breaking vanishes for w=p /2 (plane of in-idence parallel to the grooves). The azimuthal angle whus helps to optimize the diffraction pattern by tuning0.The diffraction spectrum is simply obtained by our con-

idering that the total phase modulation Ftot of the out-oing atomic wave C is the sum of the phase modulationsm associated with each potential order. Thus exp

iFtotdC=expsiSmFmdC with, in our case, m restricted to 1nd 2. Then the diffraction spectrum is the convolution ofhe diffraction spectra associated with each individualarmonic m. These are Bessel functions of argument

1,2= e1,2szrdP Bsa1,2d. Then, after some algebra,

Pm = Uon

in expsinf0dJm−2nfe1szrdP Bsa1dg

3Jnfe2szrdP Bsa2dgU2. s6d

The interpretation is clear: Atomic diffraction in the or-er m corresponds to a net momentum exchange ofs2p /Ld. The factor Jnfe2szrdP Bsa2dg is the nth diffractionrder of the second harmonic sm=2d of the potential: Itorresponds to a net momentum exchange of nf2s2p /Ldg.imilarly, Jn8fe1szrdP Bsa1dg is the n8th diffraction order ofhe first harmonic sm=1d of the potential and correspondso a momentum exchange of n8s2p /Ld. The product of thewo gives an overall momentum exchange sn8+2nds2p /Ldnd thus n8=m−2n for an overall exchange of ms2p /LdFig. 8).

Equation (6) states that all the different paths with netomentum exchange ms2p /Ld interfere with a tunable

hase shift f0 that can thus be optimized to maximize P1,or example. From what we learned before with a singlearmonic in the potential (Fig. 4 and Section 3), we canxpect F1,2 to be small unless the population is spreadver many channels. The diffraction spectra associatedith each individual harmonic m are then limited to the 0nd ±1 orders. When n and n8 are scanned through −1, 0,nd 1, the only terms that survive in P1 are

P1 < uJ0sF2dJ1sF1d + i expsif0dJ1sF2dJ−1sF1du2

= uJ1sF1dfJ0sF2d − i expsif0dJ1sF2dgu2. s7d

he quantity in brackets is obviously maximum when0=p /2 and

ig. 8. First-order atomic diffraction path corresponding to thexchange of one quantum of momentum with the second har-onic of the potential (2) and one quantum in the opposite direc-

ion with the first harmonic of the potential (1). First-ordertomic diffraction is the coherent sum of all such paths.

P1opt < uJ1sF1dfJ0sF2d + J1sF2dgu2. s8d

oreover, the contrasts can also be optimized. J1sF1d isaximum for F1<2, and J0sF2d+J1sF2d is maximumhen F2<1. One should thus verify F1 /F2<2e1szrd / e2szrd because a1<a2⇒ Bsa1d< Bsa2d. Atomic dif-

raction in the first order is thus maximized if the con-rast at the classical turning point of the first harmonic ofhe potential is twice that of the second and they areephased by p /2. Under these conditions, the potential athe classical turning point is as shown in Fig. 6. We canhen estimate an upper limit to the diffracted populationsy

Pm ø Hon

uJnfe2szrdP Bsa2dgJm−2nfe1szrdP Bsa1dguJ2, s9d

hich numerically gives P1ø60%. The optimized diffrac-ion spectrum is shown in Fig. 9. It clearly confirms thelaze effect with the first-order population exceeding allther orders including 0 and a diffraction efficiency of5%, nearly the maximum achievable.To summarize, we can say that this analytical approach

ields physical insight into the relevant parameters of therating period, groove depth, and illumination and allowsne to choose appropriate values for blazing. Neverthe-ess, it is easy to see that if Eq. (6) is extended to morehan two harmonics in the potential, many terms then in-erfere with great sensitivity to the actual relative phasesnd amplitudes. Thus, to take into account all field con-ributions and the internal states of a real atom, we musturn to numerical calculations.

. NUMERICAL SIMULATIONSumerical simulations of this experiment require, first,

he calculation of the optical near field and, second, theotion of a multilevel atom in the associated potential.he optical field is obtained with the differential theory ofratings20 that naturally gives the Fourier harmonics ofhe vector electric and magnetic fields above theanostructure.21 Wave-packet propagation is then per-ormed with the time-split operator technique.22 Well-ocalized incident wave packets are used so they contain aarge energy distribution. The outgoing wave packet is

ig. 9. Analytic atomic population, expression. (8), as a functionf atomic-incident-reduced momentum P for an optimized blazedrating (f0=p /2, F1=1.84, F2=0.82). Only specular reflection (=0, solid curve) and m= ±1 channels are shown (m= +1, long-ashed curve; m=−1, short-dashed curve).

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hen projected on the diffraction states for each energy toet the diffraction spectrum. Details of the algorithms cane found elsewhere.14

The specific structure is a grating of period L=964 nm,f rectangular stripes of height h=20 nm engraved in ai3N4 prism whose refraction index is n=2.16. The indexf the stripes is n=2.06. To have a weak index contrastbecause of the increase of the field contrast with height),e fill the empty space between stripes with the prismaterial. The incident propagation vector is set at an in-

idence angle u=60° with a wavelength l=850 nm nearhe resonance transition of the cesium atom. For w=45°,he ranges of the different relevant orders are

m = 0, 1/k0 = 86 nm,

m = + 1, 1/k1 = 139 nm,

m = 2, 1/k2 = 139 nm,

m = 3, 1/k3 = 86 nm,

m = − 1, 1/k−1 = 57 nm.

n contrast what was assumed in the analytical analysis,he third-order range is not shorter than that of the ze-oth. Depending on the third-order contribution to the po-ential at z=0, i.e., the actual shape of the grooves, atomiciffraction patterns might depart significantly from the

ig. 10. Atomic diffraction population as a function of atomic-ncident-reduced momentum P. Top panel, full numerical calcu-ation; central panel, numerical calculation but with the poten-ial spectrum artificially truncated to the zeroth, first, andecond orders; lower panel, analytical calculation, expression (7).nly specular reflection (m=0, solid curve) and m= ±1 channelsre shown (m= +1, long-dashed curve; m=−1, short-dashedurve).

nalytic ones. The width of the stripes is a=300 nm. Itan be verified that, at any height above the structure,he intensity modulation is no longer a pure sine curve;or comparison with the analytical model, we can extracthe following parameters at z=0:

e1 = 1.3 3 10−2, e2 = 5.8 3 10−3, f0 < 68 ° .

The corresponding diffraction spectra as a function ofhe incident normalized momentum of the particle areiven in Fig. 10. The top panel shows the full numericalalculation. The central panel is a numerical calculationut with the potential spectrum artificially truncated tohe zeroth, first, and second orders. The bottom panelhows the analytical calculation expression (7) with theame orders. The blaze effect is then fully confirmed byhe more realistic numerical calculations that show onlyinor changes from the analytical ones. The only change

etween the top and the middle curves is the shape of the1st order, which proves that the third order of the fieldlays a minor role in the main diffracted population.Parameters of the grating were chosen to optimize the

tructure, that is, e1szrd / e2szrd=2.2. Then the efficiency isood, 53%, and is practically only due to the two first har-onics. Even if the phase is slightly different from p /2,

he maximal expected value is almost reached.Finally, we can compare, as in light optics, the diffrac-

ion angle and the reflection angle onto the potential. Theiffraction angle is simply the ratio of the atom de Broglieavelength to the grating period, which is

udiffsPd = ldB/L = Q/k0. s10d

To estimate the reflection angle, we adopt the hard-wallicture.17 To a certain approximation, the optical evanes-ent potential can be replaced by an infinitely high poten-ial (i.e., a perfect mirror) with a spatially modulated po-ition. The mirror surface is the place where the potentialquals the incident kinetic energy Vsx ,zd=Ek. The opticalnalogy is then obvious. As the contrast is small, the lat-er equation can be expanded around the mean classicalurning point zr to give the mirror postion zsxd:

2k0fzsxd − zrg = 2k0Dzsxd

< e1szrdcossQxd + e2szrdcoss2Qx + f0d,

s11d

hich is drawn, together with a sawtooth approximationn the dotted line, in the right scale of Fig. 6. The reflec-ion angle is then twice the inclination angle g of the fac-ts:

urefsPd = 2g =9Î3

8p

Q

k0e1fzrsPdg, s12d

or the optimized parameters. Note that, in contrast to op-ics, the reflection angle depends on the momentumwavelength) as faster atoms penetrate deeper in the po-ential and contrast is height dependent. A blaze effect ishus expected when these two angles are equal udiffsPduref. This equation can then be numerically solved toive P<78 for the dimensionless momentum of maximumlaze, which compares well with diffraction spectra (Fig.

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0). This simple calculation confirms that the same quali-ative reasoning applies to atomic as well as light blazeiffraction.

. CONCLUSIONe have shown that a periodic nanostructure engraved

n the surface of a prism can play the role of an atomiclazed grating. Efficiency as great as 55% is expected. Araphical representation of the dispersion relation and annalytical simplified model have proven to be useful inhe search and optimization of the geometry. The ultimateoal of this approach is to tailor the diffraction spectrumor applications in atom optics: for example, getting twoand, if possible, only two) diffracted beams to function asrobust beam splitter for atom interferometry.

CKNOWLEDGMENTShe authors thank John Weiner for discussion and com-ents on the manuscript. We have benefited from the

omputing facilities provided by the massively parallelenter CALmip of Toulouse. This research was supportedy the French Ministère de l’Éducation Nationale, de’Enseignement Supérieur et de la Recherche, the Fondsational de la Science, ACI Nanoscience, and Europeannion Nanocold contrast IST-2001-32264.

R. Mathevet, the corresponding author, can be reachedy e-mail at [email protected].

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Pritchard, “An interferometer for atoms,” Phys. Rev. Lett.66, 2693–2696 (1988).

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1. See also J. C. Weeber, “Diffraction en champ procheoptique. Analyse des images de microscopie à effet tunnelphotonique,” Ph.D. thesis (Université de Bourgogne, Dijon,France, 1996).

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blazed atom grating

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