Numerical and experimental studyof grating efficiency for synchrotron monochromators

Alessandro Mirone, Eric Delcamp, Mourad Idir, Gilles Cauchon, Francois Polack,Pierre Dhez, and Claudine Bizeuil

The development of third-generation synchrotron sources has stimulated efforts toward high-resolutionmonochromators. A good knowledge of grating efficiency is needed to achieve an optimal compromisebetween resolution and photon flux. Because simple geometric models fail to describe correctly thegratings properties in the UV–to–soft-X-ray range, we have developed a simulation software based ondifferential theory. A simplified R-matrix propagation algorithm assures the numerical stability of thecode for deep gratings. Our numerical results are compared with previous research on deep gratings.Experimental and numerical studies have been performed on some test cases at a synchrotron source.Very good agreement between numerical prediction and measurement has been found. © 1998 OpticalSociety of America

OCIS codes: 050.1950, 360.6720, 300.6190.

1. Introduction

The blossoming of third-generation synchrotronsources in recent years has pushed forward the ex-ploitation of high-resolution synchrotron beamlinessignificantly. The great brilliance of these sourcesand their low emittance ~defined as the product of theangular aperture and the source size! match the ac-ceptance of high-resolution monochromators effi-ciently and permit a considerable reduction in themonochromator slit widths and aperture angleswhile maintaining useful photon fluxes. The scien-tific community has taken great advantage of thesenew sources. In a wide range of disciplines, such aschemistry, solid-state physics, and biology, more andmore accurate measurements are necessary. As aconsequence, user demand in terms of flux and reso-lution is rapidly increasing. Flux and resolution aretwo parameters that are usually difficult to compro-

A. Mirone, E. Delcamp, M. Idir, G. Cauchon, F. Polack, and P.Dhez are with the Laboratoire pour l’Utilisation du RayonnementElectromagnetique, Batiment 209D, Centre Universitaire 91405Orsay, France. A. Mirone is also with the Institute d’OptiqueTheorique et Appliquee, Unite de Recherche Associee 14 du CentreNational de la Recherche Scientifique, BP 147, 91493 Orsay,France. C. Bizeuil is with the Commissariat a l’Energie Atom-ique, BIII, BP 12 91680 Bruyeres le Chatel, France.

Received 6 January 1998; revised manuscript received 9 April1998.

0003-6935y98y255816-07$15.00y0© 1998 Optical Society of America

5816 APPLIED OPTICS y Vol. 37, No. 25 y 1 September 1998

mise in the design process. This compromise de-pends on a number of configuration parameters, allaffecting flux and resolution. For a given gratingand geometry, the best performances are obtained atthe low-energy end of the spectral range when lightstrikes or leaves the grating at nearly grazing inci-dence. However, the grating efficiency cutoff is sosharp that nobody dares to venture far. Severalsimple geometric models are commonly used for eval-uation purposes.1 These models fail to describecorrectly the grating performance in the UV–to–soft-X-ray range at low grazing angles2 especially nearthe cutoff and for gratings with a high number oflines ~.600ymm!. We note, by the way, that in thebeamline design process one tends usually to increasethe number of grating lines to compensate by greaterdispersion for the resolution loss due to the slopeerrors of the optics andyor the instability and inac-curacy of the tuning movements.3 To make opti-mized designs of synchrotron beamlines, one needs amore realistic method for electromagnetic modelinggratings.4 We have developed a numerical codebased on differential theory.5 Our numericalmethod addresses successfully the problem of numer-ical stability for deep gratings6–8 by a simplifiedR-matrix propagation algorithm.8 The software isnumerically stable for grating efficiency simulationsin the whole range of visible to x-ray radiation. Thecode is written in object oriented language and is easyto adapt to any grating shape.

In Section 2 we describe our numerical method.

wt

1

c

~rt

et

c

Fto

The numerical stability of our algorithm is studied inSection 3, where we compare our code with previousresults.8 Finally, in Section 4 we compare our code

ith experimental measurements on test gratingshat we performed at the Super-ACO storage ring.

2. Numerical Method

Our code applies to periodic gratings of any shape inclassical diffraction ~ruled perpendicular to the inci-dence plane!. The grating is described by the dielec-tric function e~x, z! ~see Fig. 1! satisfying theperiodicity condition

e~x, z! 5 e~x 1 d, z!, (1)

where d is the grating period. The grating is com-posed of a substrate at z , 0 described by a constantdielectric function e~x, z! 5 es and is limited by vac-uum, e~x, z! 5 1, at z . h. The differential theory inits simplest form consists of propagating the Fouriercomponents of the electromagnetic field from the top~z 5 h! to the bottom ~z 5 0! of the grating, asexplained in Subsection 2.A. This approach, al-though applying formally to any kind of grating, suf-fers from numerical problems. In the general case,with lossy media and evanescent waves, the presenceof both exponentially growing and damped solutionsruins the numerical precision of the code for suffi-ciently thick gratings. For extending the range ofvalidity of the differential theory, several methodshave been proposed.6–8

We have adopted a simplification of the R-matrixpropagation algorithm.8 In Subsection 2.A we recallthe principles of the differential theory, and in Sub-section 2.B we present our method.

A. Differential Theory

Let us consider a layer described by a scalar dielectricfunction e~x, z! satisfying the periodicity condition,Eq. ~1!. We assume that the layer is bound by vac-uum at z 5 h and by the substrate at z 5 0 ~see Fig.!. Let us consider a monochromatic Bloch solution

Fig. 1. Differential theory of grating studies of the propagation ofthe electromagnetic field in a medium ~the grating! described by aperiodic dielectric function e~x, z! 5 e~x 1 d, z!. The grating isbound by a substrate at z 5 0 and by vacuum at z 5 h.

1

~x 1 d, z! 5 exp~ik\d!c~x, z! to Maxwell equations,corresponding to waves propagating in the xz planeclassical diffraction!. The following statement is aesult of the general properties of the Maxwell equa-ions.

Statement:The solution satisfies a second-order differential

quation and is completely determined for s polariza-ion ~p polarization!, once Ey~x, z! and ]zEy~x, z!

@By~x, z! and Ex~x, z!# are assigned over a plane z 5onst ~E and B being the electric and the magnetic

fields, respectively!.In the differential theory for gratings, each func-

tion g~x, z! is represented by a truncated Fourierseries over x:

g~x, z! 5 (j52n

j5n

gj~z!expFiSk\ 12pjd DxG. (2)

The function g~x, z! is known within the accuracylimits of the truncated expansions if we know at eachz the vector g~z! of its Fourier components:

g~z! 5 1g2n~z!

· · ·g0~z!· · ·

gn~z!2 . (3)

Within this formalism we can express Ey~x, z! and]zEy~x, z! ~for s polarization! as a superposition ofplane waves propagating upward and downward onthe z axis:

Ey~z! 5 @U~z! 1 D~z!#,

]zEy~z! 5 @iKzU~z! 2 iKzD~z!#, (4)

where U and D are the upward and the downwardplane-wave coefficients and Kz is a diagonal matrixwhose elements are defined as the z components ofthe wave-number vectors of the plane waves in vac-uum:

Kz~m, m! 5 Fk02 2 Sk\ 1

2mp

d D2G1y2

, (5)

where k0 is the wave-vector modulus in vacuum.or future reference we also define the diagonal ma-rix of the z components of the wave-number vectorsf the plane waves in the substrate:

Kzs~m, m! 5 Fes k0

2 2 Sk\ 12mp

d D2G1y2

. (6)

In this paper we discuss only s polarization. Theformalism for p polarization is analogous. In fact,there is a symmetry between s- and p-polarizationformalisms.9

We can now express the statement as

Ey~z! 5 MEy

1~z!U~h! 1 MEy

2~z!D~h!,

]zEy~z! 5 M]zEy

1~z!U~h! 1 M]zEy

2~z!D~h!, (7)

September 1998 y Vol. 37, No. 25 y APPLIED OPTICS 5817

ˆ ˆ

bt

c

~f

pEtwdsU

te

o

lgWcatc~

5

where U~h! and D~h! are the upward and the down-ward plane-wave components at the top of the layerand the matrices MEy

6~z! and M]zEy

6~z! describe thepropagation of the field along the z axis. These ma-trices obey the following differential equation, whichis obtained by Fourier transforming the Helmotzequation:

]z MEy

6 5 M]zEy

6,

]z M]zEy

6 5 Kz2MEy

6 2 @e#~z! 2 I#k02MEy

M6, (8)

where I is the identity matrix and e#~z! is the matrixuildup from the Fourier transform components ofhe dielectric function as

e#m,n~z! 51d *

0

d

e~x, z!expF2i~n 2 m!2p

dxGdx. (9)

The solution to Eq. ~8! is determined by the initialonditions obtained from Eqs. ~4! and ~7! at z 5 h:

MEy

6~h! 5 I,

M]zEy

6~h! 5 ~ 6 !iKz, (10)

where I is the identity matrix. Equations ~8! and10! are the bases of differential theory. They can beormally applied to any kind of grating.

The calculation of the grating efficiency consists inropagating the field from z 5 h to z 5 0 by means ofqs. ~8! and then imposing at z 5 0 the condition of

he outgoing waves. Such a condition is imposed byriting the field at z 5 0 as a superposition of theownward plane waves propagating into the sub-trate. By equating the field given by Eq. ~4! forˆ ~0! 5 0 to the one given by Eqs. ~7! at z 5 0, one gets

D~0! 5 MEy

1~0!U~h! 1 MEy

2~0!D~h!,

2iKzsD~0! 5 M]zEy

1~0!U~h! 1 M]zEy

2~0!D~h!, (11)

where D~0! is the vector whose elements are the co-efficients of downward plane waves in the substrate.We can calculate on the basis of this equation thereflectivity matrix R defined by

U~h! 5 RD~h!,

R 5 2@iKzsMEy

1 ~0! 1 M]zEy

1 ~0!# 2 1

3 @M]zEy

2 ~0! 1 iKzsMEy

2 ~0!#. (12)

Equations ~12! can be successfully used only if thelayer is sufficiently thin that the differential theorycan be applied without numerical problems.8 Fordeep lossy media, for example, the M2 and M1 ma-rices in Eqs. ~12! tend to infinity while in the case ofvanescent waves the matrices M6 have small and

large terms. In these cases it becomes impossible tomanipulate them without loss of information.

To extend the range of applicability of the differ-ential methods, one must adopt a stabilization algo-rithm.

We describe our method in Subsection 2.B. It con-sists basically of splitting the grating into a stack of

818 APPLIED OPTICS y Vol. 37, No. 25 y 1 September 1998

thin layers and then propagating the reflectivity ma-trix from the bottom ~substrate! to the top ~vacuum!f the substrate.

B. Reflectivity-Matrix Propagation Algorithm

Let us consider a deep grating as a stack of N thinayers piled on the substrate ~see Fig. 2!. Each sin-le layer satisfies the periodicity condition of Eq. ~1!.e assume that the thickness of each layer is suffi-

iently small for the methods of Subsection 2.A topply without problems. If the whole stack is toohick for the differential theory to apply, we can stillalculate the reflectivity matrix R defined by Eqs.12! by the following recursive method:

• Calculate the reflectivity matrix R0 for the sub-strate only ~only Fresnel equations for plane waves!.

• Consider the next thin layer j ~ j 5 1 for the firststep! and calculate the matrices M]zEy

6~0!j andMEy

6~0!j for this layer by the method described inSubsection 2.A.

• Calculate four matrices, Aj11, IAj

12, IAj21,

and Aj22, by which one can express the upward and

the downward wave coefficient Uj21IDj21 at the bot-tom of the thin layer j as functions of the coefficientsUj, Dj at the top of the layer:

Uj21 5 Aj11Uj 1 Aj

12Dj,

Dj21 5 Aj21Uj 1 Aj

22Dj. (13)

The A matrices are a linear combination of M ma-trices,

Ajs1,s2 5 @MEy

s2~0!j 1 s1~iKz!21M]zEy

s2~0!j#y2, (14)

where s1 and s2 run over 1 and 2.

• Calculate the reflectivity matrix Rj for the grat-ing formed by the substrate and the layers i # j by the

Fig. 2. In the reflectivity–matrix propagation algorithm the grat-ing is split into a stack of thin layers. Each of them is sufficientlythin for the differential theory to apply to without numerical di-vergences. The reflectivity matrix is then propagated from thebottom to the top of the stack by the methods described in Sub-section 2.B.

1

wi1F

following recursive formula that is obtained by sub-

stituting Um 5 RmDm in Eqs. ~13!:

Rj 5 ~Aj11 2 Rj21 Aj

21!21~Rj21 Aj22 2 Aj

12!. (15)

This recursive procedure can be applied to thewhole stack. One ends up with the reflectivity RN ofthe grating. There is no problem of numerical insta-bility because one propagates the reflectivity matrixR, which is bound by energy conservation. Eventhough some elements of R may become large whensurface waves are excited, there is no divergent be-havior of R. We safely treat the divergent behaviorof the evanescent waves by propagating the matricesM]zEy

6~z!j and MEy

6~z!j through sufficiently thin lay-ers.

Our method is a simplified version of that in Ref. 8.The R-matrix propagation method proposed by theseauthors propagates a matrix that describes the grat-ing as a whole. Both reflection and transmissionefficiencies can be obtained from this matrix, whilewith our method only reflectivity is propagated.

3. Test Cases

We compare our code results with known results. InSubsection 3.A we give examples of grating compu-tations in the visible range while in Subsection 3.Bwe deal with gratings in the x-ray range.

A. Test Cases for Deep Gratings

We have chosen two test cases from Ref. 8 and onefrom Ref. 10 for comparison of our code results withknown results from stabilization algorithms. Thefirst example ~Fig. 5 in Ref. 8! is a 2000-grooveymmgold grating in the Littrow mount with a symmetricaltriangular profile. The grating is used in TE polar-ization; the incident wavelength is 0.6 mm. The re-fractive index of gold at this wavelength is n 5 0.2 1i2.897. We show in Fig. 3 the evolution of the 21-and 0-order efficiencies as a function of groove depth.The calculations were made with a Fourier series

Fig. 3. Evolution of the 21- and 0-order efficiencies as a functionof the groove depth for a 2000-grooveymm gold grating in a Littrowmount with a symmetrical triangular profile. The grating is usedin TE polarization; the incident wavelength is 0.6 mm. The re-fractive index of gold at this wavelength is 0.2 1 i2.897. Thiscurve should be compared with Fig. 5 of Ref. 8.

1

truncated at n 5 6 @Eq. ~3!#. Our calculation resultsreproduce the literature data exactly.8

The second example ~Fig. 6 in Ref. 8! deals with asinusoidal silver grating covered with a dielectriccoating. The groove spacing is 1 mm, the groovedepth is 0.3 mm, the wavelength is 0.8 mm, the inci-dence angle is 30°, and the polarization is TE. Therefractive index is n 5 0.09 1 i5.45 for silver and 1.5for the dielectric coating. We show in Fig. 4 the0-order efficiency as a function of the thickness e ofthe dielectric layer. We find the numerous sharpfeatures that the authors of Ref. 8 attribute to theexcitation of guided modes with the addition of someextra features whose visibility depends on the reso-lution of the graphics.

The last example for the visible range deals with acoated sinusoidal silica ~SiO2, n 5 1.46! grating in theLittrow mount and TM polarization with a0.3333-mm period at a 1200-Å sinusoidal corrugationand a coating formed by 15 alternating layers of zincsulfite ~ZnS, n 5 2.37! and cryolite ~Na3AlF6, n 5.35! with the ZnS layer adjacent to the substrate.All the coating layers have the same normalized

thickness r defined by

r 5djnj

l, (16)

where dj is the thickness of the layer j and nj is itsrefractive index. We report in Fig. 5 on the first-order efficiency as a function of the normalized thick-ness for a wavelength, l 5 0.59 mm. The results arein quantitative agreement with those of Ref. 10.

B. Comparison with Known Results in the X-Ray Range

We have computed the diffraction efficiencies for ablazed multilayer to compare with the literature datain the x-ray range. We report in Table 1 the calcu-

Fig. 4. Efficiency of 0 order as a function of thickness e of thecoating for a sinusoidal silver grating with a dielectric coating.The groove spacing is 1 mm, the groove depth is 0.3 mm, the

avelength is 0.8 mm, the incidence angle is 30°, and the polar-zation is TE. The refractive index is 0.09 1 i5.45 for silver and.5 for the dielectric coating. This curve should be compared withig. 6 of Ref. 8.

September 1998 y Vol. 37, No. 25 y APPLIED OPTICS 5819

3n1b1i~ftcw

pwpoio

t

chc

5

lation results for a blazed gold grating with a period,d 5 0.36 mm, and a blaze angle, c 5 0.5°, coated by a0-period tungstenycarbon multilayer. The thick-esses of the carbon and tungsten layers were 22 and1 Å, respectively, and the refractive indices for car-on tungsten and gold were assumed to be 0.999425i0.000654, 0.997774 1 i0.00947 and 0.99768714 1

0.0011879, respectively, at a wavelength of 13.3 ÅRef. 11!. ~Actually these values are quite differentrom Henke data.! The grazing angle is 11.7°, andhe calculations have been performed with 25 Fourieromponents @n 5 12 in Eq. ~3!#. Our results agreeith Table 1 of Ref. 11.

Fig. 5. Efficiency of order 21 as a function of the normalizedthickness e for a coated sinusoidal silver grating with a dielectricoating in the Littrow mount and TM polarization. The gratingas a 0.3333-mm period, 1200-Å sinusoidal corrugation, and aoating formed by 15 alternating zinc sulfite ~ZnS, n 5 2.37! and

cryolite ~Na3AlF6, n 5 1.35! layers with the ZnS layer adjacent tothe substrate. The wavelength is 0.59 mm.

Table 1. Absolute Efficiency of the Blazed Multilayer Grating

m Absolute Efficiency

26 8.250812e206

25 1.216429e205

24 1.907575e205

23 3.314356e205

22 6.969795e205

21 1.329813e201

0 4.268160e204

1 1.901793e204

2 1.027458e204

3 7.003769e205

4 6.318391e205

5 7.793800e205

Table 2. Properties of the Two Gratings from Jobin–Yvon

SampleRc

a

~mm!hb

~Å! Duty CycleGrating

~linesymm!Coating ~Au!

~Å!

U2291 6791 68 0.59 2400 800U2286 4998 106 0.62 1200 300

aCurvature radius.bGroove depth.

820 APPLIED OPTICS y Vol. 37, No. 25 y 1 September 1998

In conclusion to this subsection note that the pro-osed algorithm is stable for TE polarization in thehole range of visible to x-ray radiation. The TMolarization case was formally equivalent to the TEne, and the algorithm was stable in the cases ofnterest in this paper. However, note, as shown byther authors,8 that algorithms based on the differ-

ential theory may fail when the optical index has alarge imaginary part ~much larger than unity! in thegeneral TM case.

The calculations in these cases may show instabil-ities or even diverge. This behavior ~which is neverencountered in the x-ray range! is beyond the scope ofhis paper.

4. Experimental Tests

At the Super-ACO synchrotron source we have per-formed experimental tests on two sample gratingsfrom Jobin–Yvon. Both gratings, called U2291 andU2286, had rectangular profiles. We obtained themby holographic printing and ion etching a silica sub-strate, the former at 2440 linesymm with a duty cycleof 0.59 and the latter at 1200 linesymm with a dutycycle of 0.62. The groove depths were 68 and 106 Å,respectively, and both gratings were coated, the formerwith 800-Å-thick gold and the latter with 300 Å.

The grating parameters are summarized in Table2. The measurements have been performed at pho-ton energies of 800 and 1000 eV for U2991 and1000 eV for U2286. The wavelength selection wasmade by a double-crystal monochromator with twoparallel beryl crystals. The third Bragg order fromthe crystals was filtered out by electronic threshold-ing of the signal of the proportional counter. Weshow in Figs. 6–8 the experimental and numericalefficiencies of the diffracted orders, 21 and 11, as afunction of the grazing angle. Figures 6 and 7 showsample U2991 at 800 and 1000 eV, respectively, andFig. 8 shows sample U2286 at 1000 eV. In the nu-merical calculations we considered the nominal grat-ing parameters given by Jobin–Yvon ~see Table 1!with the exception of the roughness of the gold–vacuum interface, which was the only free parameter

Fig. 6. Experimental and numerical efficiencies of the diffractedorders of 21 and 11 for the U2991 sample at 800 eV.

nezd

Gt

b

in the simulation. We considered two roughness pa-rameters: the roughness st at the top of the rectan-gular profile and the roughness sb at the bottom.For sample U2991 we found good agreement with st5 sb 5 11 Å, and for sample U2286 we found st 5 15Å and sb 5 10 Å. To introduce the roughness in our

umerical model we smoothed the dielectric profile~x, z! by convoluting with a Gaussian function on theaxis. Because our model can treat an arbitrary

ielectric function e~x, z!, we can treat any roughnessmodel.

The experimental curves in Figs. 6 and 8 show aslight violation of the reciprocity relation. In fact,for perfectly rectangular profiles, the orders of 21and 11 should have the same peak height. Thesmall discrepancy that one observes may depend onseveral factors. The small but nonnegligible curva-ture of the grating can be the source of a systematicexperimental error when the grating is aligned in ourexperimental setup. On the other hand, the grat-ings were just intermediate samples from the optimi-zation of the fabrication process at Jobin–Yvon, andsmall imperfections should be taken into account.The agreement between experiment and our numer-ical modeling is nonetheless excellent, even whenperfect rectangular profiles in the simulation are as-sumed.

A check of the predictions of simple geometric mod-els for this kind of grating is interesting. We showin Fig. 9 the 21- and 1-order efficiencies at 800 eV asa function of incidence angle. We compare our codewith the geometric model with and without correctionfor shadowing effects ~Chap. 7 of Ref. 1!. The effi-ciency of grazing angle a for the reflected diffractionorder m is given in the geometric models by

rSa 1 b

2 D 3 U1d (*t1

t2

expS2im2px

d Ddx 1 exp$ih@sin~a!

1 sin~b!#k0% *b1

b2

expS2im2px

d Ddx)U2

, (17)

Fig. 7. Experimental and numerical efficiencies of the diffractedorders of 21 and 11 for the U2991 sample at 1000 eV.

1

where r~u! is the reflectivity for a plane substrate ata grazing angle u and b is the observation angle fororder m given by the grating law. The integrationlimits in the first integral are t1 5 0 and t2 5 d ~1 2

!, where G is the duty cycle. In the second integralhe limits are b1 5 t2 and b2 5 d if shadowing is

neglected, whereas when shadowing is consideredthey are b1 5 t2 1 h cot~a! and b2 5 max@d 2 h cot~b!,

1#. The calculations are made for sample U2991 at800 eV without consideration of roughness. One cansee that the simple geometric models break down inthe most interesting part of the spectrum when graz-ing geometry is reached.2–12

5. Conclusions

The differential theory for the grating has been ap-plied to the modeling of synchrotron monochromatorgratings. We implemented in our code a stabiliza-

Fig. 8. Experimental and numerical efficiencies of the diffractedorders of 21 and 11 for the U2286 sample at 1000 eV.

Fig. 9. Efficiency of 21 and 1 orders at 800 eV versus the grazingangle for the U2991 sample. We compare our code ~solid curve!with geometric modeling with ~dotted–dashed curve! and without~dashed curve! correction for shadowing effects ~Chap. 7 of Ref. 1!.The calculations are made without considering the roughness.One can see that the geometric models break down in the mostinteresting part of the spectrum when grazing geometry isreached.

September 1998 y Vol. 37, No. 25 y APPLIED OPTICS 5821

w

McKinney and C. A. Palmer, eds., Proc. SPIE 3150, 48–57

5

tion algorithm that functions in difficult test cases.Experimental tests have been performed on two sam-ples, and an excellent agreement with our numericalmodel has been found. The height position and theshape of the diffracted orders peaks as a function ofthe incidence angle are well estimated by our code.The rigorous electromagnetic treatment was provednecessary for the design of synchrotron beamlinemonochromators because the behavior of diffracted-order efficiency and especially the efficiency cutoff atlow grazing angles defines the operational geometryof the gratings. Simple geometric models, on theother hand, fail to calculate correctly the efficiency forthe most interesting configurations of high-resolutionmonochromators.

We thank David Weber and the other authors ofthe C11 matrix classes uMatrix under GPL, which

e have used intensively.

References1. A. G. Michette, Optical Systems for Soft X Rays ~Plenum, New

York, 1986!.2. M. Neviere, J. Flamand, and J. M. Lerner, “Optimization of

gratings for soft-x-ray monochromators,” Nucl. Instrum. Meth-ods 195, 183–189 ~1982!.

3. E. Delcamp, F. Polack, and B. Lagarde, “SM-PGM monochro-mators design for large spectral range,” in Gratings and Grat-ing Monochromators for Synchrotron Radiation, W. R.

822 APPLIED OPTICS y Vol. 37, No. 25 y 1 September 1998

~1997!.4. E. DelCamp, B. Lagarde, F. Polack, and J. Susini, “Optimiza-

tion software for X-UV monochromators,” in Optics for HighBrightness Synchrotron Radiation Beamlines, L. E. Bermanand J. Arthur, eds., Proc. SPIE 2856, 120–128 ~1996!.

5. M. Neviere, P. Vincent, and R. Petit, “Sur la theorie du reseauconducteur et ses applications a l’optique,” Nouv. Rev. Opt. 8,65–74 ~1974!.

6. D. M. Pai and K. A. Awada, “Analysis of dielectric gratings ofarbitrary profiles and thickness,” J. Opt. Soc. Am. A 8, 755–762~1991!.

7. M. Neviere and F. Montiel, “Deep gratings: a combination ofthe differential theory and the multiple reflection series,” Opt.Commun. 108, 1–7 ~1994!.

8. F. Montiel and M. Neviere, “Differential theory of gratings:extension to deep gratings of arbitrary profile and permittivitythrough the R-matrix propagation algorithm,” J. Opt. Soc. Am.A 11, 3241–3252 ~1994!.

9. R. Petit, ed., Electromagnetic Theory of Gratings ~Springer-Verlag, Berlin, 1980!.

10. L. Li, “Multilayer-coated diffraction gratings: differentialmethod of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11,2829–2836 ~1994!.

11. M. Neviere, “Multilayer coated gratings for x-rays diffrac-tion: differential theory,” J. Opt. Soc. Am. A 8, 1468–1473~1991!.

12. J. B. West and H. A. Padmore, “Optical engineering,” in Hand-book of Synchrotron Radiation, Vol. 2, Ge. Marr, ed. ~North-Holland, Amsterdam, 1987!.