thin-film interference effects on the efficiency of a normal-incidence grating in the 100–350-Å...

Thin-film interference effects on the efficiency ofa normal-incidence grating in the 100–350-Åwavelength region

John F. Seely, Leonid I. Goray, William R. Hunter, and Jack C. Rife

Thin-film interference effects were observed in the normal-incidence efficiency of a 2400-grooveymmreplica grating. The efficiency was measured in the 100–350-Å wavelength range and had an oscillatorybehavior that resulted from the presence of a thin SiO2 coating. The thicknesses of the SiO2 and theunderlying oxidized aluminum layers were inferred from computer modeling of the zero-order efficiency.The efficiencies in the diffracted orders were calculated with the modified integral approach and ac-counting for the multilayer coating and the groove profile derived from atomic force microscopy. Thecalculated and measured efficiencies were in good agreement. © 1999 Optical Society of America

OCIS codes: 050.1950, 050.1960, 260.3160, 260.7200, 310.6860.

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1. Introduction

Multilayer-coated diffraction gratings that have rel-atively high efficiencies at normal incidence in theextreme ultraviolet region have been developed.Peak normal-incidence efficiencies as high as 16%have been achieved.1 The multilayer coating ismatched to the groove profile so that the efficiency isoptimal in a wavelength band of interest.

As part of the design process for the optimization ofthe multilayer coating and the groove profile, it isuseful to have a computer program that models theefficiency of the multilayer-coated grating. The pro-gram should account for the finite conductivity of thecoating, the irregular shape of the groove profile, themicroroughness of the groove facets, and the two po-larization orientations of the incident radiation.Such rigorous calculations demand significant com-puter resources, and numerical algorithms that rap-idly converge are required if accurate results are to beobtained in a reasonable time.

Neviere2,3 has adapted the differential formalism

J. F. Seely and J. C. Rife are with the Naval Research Labora-tory, Washington, D.C. 20375-5352. L. I. Goray is with Interna-tional Intellectual Group, Incorporated, P.O. Box 335, Penfield,New York 14526. W. R. Hunter is with Sachs-Freeman Associ-ates, Incorporated, 1401 McCormick Drive, Landover, Maryland20785.

Received 16 September 1998; revised manuscript received 16November 1998.

0003-6935y99y071251-08$15.00y0© 1999 Optical Society of America

to calculate the efficiency of a multilayer grating.Calculations were presented for ideal blazed grooveprofiles and only for the polarization orientation withthe electric field vector parallel to the grooves.

In an effort to develop a computer program thatcalculates grating efficiencies on a small personalcomputer, Goray and Chernov4,5 have used a modi-fied integral method. In more recent research,6 thiscomputer program was used to model the efficiency ofa 2400-grooveymm blazed grating. The calculationaccounted for the groove profile as determined fromatomic force microscopy, the optical properties of theSiO2 surface, and the polarization of the incident ra-diation. The calculated efficiencies in the 0, 61, and

2 orders were compared with the normal-incidencefficiencies measured using synchrotron radiation inhe 125–225-Å wavelength range. The calculatednd measured efficiencies of this uncoated gratingere in good overall agreement.In the present research, the measured and calcu-

ated efficiencies of a replica of the 2400-grooveymmrating that was analyzed in Ref. 6 are presented.s a result of the replication process, the grating hadn oxidized aluminum surface. A thin SiO2 over-

coating was applied to the replica grating for thepurpose of reducing the microroughness of the groovefacets. The efficiencies in the 0, 61, and 62 orders,measured using synchrotron radiation, have an in-teresting oscillatory behavior that results from thepresence of the thin SiO2 layer on the highly reflect-ing oxidized aluminum surface. The thicknesses ofthe SiO2 and Al2O3 layers were inferred from the

1 March 1999 y Vol. 38, No. 7 y APPLIED OPTICS 1251

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frequency and amplitude of the zero-order efficiencyin the 100–350-Å wavelength range. Using theselayer thicknesses and the groove profile that was de-rived from atomic force microscopy, we calculated theefficiencies in the 0, 61, and 62 orders using anenhanced version of the computer program that isbased on the modified integral method4,5 and ac-counts for the multilayer coating. The calculatedefficiencies are in good overall agreement with themeasured efficiencies. The oscillatory behavior ofthe efficiency as a function of wavelength, resultingfrom thin-film interference in the SiO2 layer, is mod-eled accurately. This research demonstrates thatthin-film interference effects in the efficiency of agrating with several layers can be modeled accuratelyon a small personal computer in a reasonable amountof time.

2. Replica Grating and Atomic Force Microscopy

The master grating was fabricated by Spectrogon UKLimited ~formerly Tayside Optical Technology!.

he groove pattern was fabricated in fused silica us-ng a holographic technique. The groove patternas ion-beam etched to produce an approximately

riangular, blazed groove profile. The grating has400 groovesymm, a concave radius of curvature of.0 m, and a patterned area of size 45 mm by 35 mm.he master grating was uncoated.The replica of the master grating was produced byyperfine, Inc. As a result of the replication pro-

ess, the replica grating had an aluminum surface.thin SiO2 coating was applied to the aluminum

surface for the purpose of reducing the microrough-ness.

The surface of the replica grating was character-ized using a Topometrix Explorer Scanning ProbeMicroscope, a type of atomic force microscope ~AFM!.

he AFM images typically had 500 3 500 pixels andscan range of 1–20 mm ~pixel size 20–400 Å!. The

ilicon probe had a pyramid shape. The base of theyramid was 3–6 mm in size, the height of the pyra-id was 10–20 mm, and the height-to-base ratio was

pproximately 3. The tip of the pyramid had a ra-ius of curvature less than 200 Å. The AFM scansere performed using the noncontact resonatingode in which the change in the oscillation ampli-

ude of the probe is sensed by the instrument.A surface topology reference sample was used to

ptimize the AFM scanning parameters, to calibratehe height scaling of the instrument, and to evaluatehe performance of the AFM. This was essential forhe accurate characterization of the replica grating.he surface topology reference sample consisted of anrray of approximately square holes fabricated on theilicon dioxide surface of a silicon die by VLSI Stan-ards, Inc. The top surface of the die was coatedith a thin layer of platinum. The hole array had aitch of 3 mm and a hole depth of 180 Å.A typical image of the surface topology reference

ample is shown in Fig. 1. The size of the image is 4m. The scan positions across the edges of the holere indicated in Fig. 1, and the scan data across these

252 APPLIED OPTICS y Vol. 38, No. 7 y 1 March 1999

our edges are presented in Figs. 2~a!–2~d!. The cor-responding line-spread functions ~LSF’s!, the deriva-tives of the scan data, are presented in Figs. 2~e!–2~h!.The widths of the LSF range from 390 to 790 Å.These four LSF’s are indicative of the range of the LSF

Fig. 1. AFM image of the surface topology reference sample.The image size is 4 mm. The scan positions across the edges of thehole are indicated.

Fig. 2. ~a!–~d! AFM scan data across the edges of a hole on thesurface topology reference sample. ~e!–~h! The LSF’s.

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derived from a number of scans of the surface topologyreference sample. Assuming that the sides of theholes on the sample are essentially vertical, as speci-fied by the manufacturer, the LSF is characteristic ofthe probe tip size and the instrument response func-tion. Because the LSF widths ~390–790 Å! are two tofour times the radius ~#200 Å! of the probe tip, we canonclude that the overall instrument response functions several times wider than the probe tip radius.

The power spectral densities ~PSD’s! derived fromwo images of the surface topology reference samplere shown in Fig. 3. The curves labeled ~a! and ~b! inig. 3 are the PSD’s derived from images of size 20nd 4 mm, respectively. The rms microroughness,erived from the area under the PSD curve, is 13 Å inhe frequency range 2–40 mm21.

In general, the measured LSF and PSD depend onthe depth and the lateral pitch of the surface featuresand the instrumental scanning parameters. Thus itis difficult to directly transfer the information gainedfrom the surface topology reference sample to thecharacterization of the replica grating. For exam-ple, the replica grating has rounded grooves with asmaller depth ~85 Å! and a smaller lateral pitch~0.417 mm! compared with the surface topology ref-erence sample ~180 Å and 3 mm, respectively!. How-ever, the surface topology reference sample wasessential for calibrating the height measurements,for adjusting the instrumental scanning parameters,and for characterizing the performance of the pyra-midal scanning probe.

An AFM image of two grooves of the replica gratingis shown in Fig. 4. The scan was performed acrossthe grooves over a range of 1 mm ~20-Å pixels!. Theertical scale in Fig. 4 was expanded to reveal theexture of the grating surface. The PSD derivedrom a 2-mm-size image spanning nearly five grooves

Fig. 3. PSD of the surface topology reference sample derived fromAFM images of sizes ~a! 20 mm and ~b! 4 mm.

s shown in Fig. 5. The peak in the 2–3-mm21 fre-quency range results from the 0.4167-mm groove pe-iod. The rms microroughness is 7 Å in the 4–40-m21 frequency range.By comparison, the microroughness of the master

grating measured by the same type of AFM instru-ment was 3.2 Å,6 and this implies that the replicagrating is significantly rougher than the master grat-ing. This may result from the replication process,which for a concave grating is at least a two-stepprocess. Furthermore, the master grating was fab-ricated on a fused-silica surface by a holographictechnique and was ion-beam polished, whereas thealuminum surface of the replica grating may contrib-ute to its larger microroughness. The replica grat-ing without the SiO2 coating was not characterized byatomic force microscopy.

A typical groove profile derived from the AFM im-

Fig. 4. AFM image of two grooves across the 2400-grooveymmreplica grating. The image size is 1 mm by 1 mm. The horizontaland vertical scales are indicated.

Fig. 5. PSD of the 2400-grooveymm replica grating derived fromn AFM image of size 2 mm.


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age ~1 mm in size! of the replica grating is shown inFig. 6. The groove profile is approximately triangu-lar in shape with rounded corners and troughs andwith facet angles of 3.4° and 6.2°. The averagegroove depths derived from the AFM images are inthe range of 85–95 Å. These values of the facet an-gles and the groove depth are larger than the corre-sponding values for the master grating of 2.5° and5.5° facet angles and 75 Å average groove depth.Thus the grooves of the replica grating are deeperand the facet angles are steeper compared with thoseof the master grating.

3. Measured Efficiencies

We measured the efficiency of the replica gratingusing the Naval Research Laboratory beamline X24Cat the National Synchrotron Light Source at theBrookhaven National Laboratory. The synchrotronradiation was dispersed by a monochromator thathad a resolving power of 600.7,8 Thin filters sup-pressed the radiation from the monochromator in thehigher harmonics. The wavelength scale was estab-lished by the geometry of the monochromator and theabsorption edges of the filters.

The replica grating was mounted in the reflecto-meter so that the dispersed radiation from the mono-chromator was incident on the grating at an angle of15.2° measured from the normal to the surface of thegrating. At fixed wavelengths, the detector wasscanned in angle about the grating. We performedmeasurements at 23 wavelengths in the 125–182-Årange using a silicon filter and at 17 wavelengths inthe 172–225-Å range using an aluminum filter. Theincident radiation was approximately 90% polarizedwith the electric field vector in the plane of incidence~p polarization!. In this orientation, the electricfield vector was perpendicular to the grating grooves.

Fig. 6. Typical groove profile derived from the AFM image of the2400-grooveymm replica grating. The average peak-to-valleygroove depth is 85 Å. The blaze angles, measured from the hor-izontal, of the left and right facets are 3.4° and 6.2°, respectively.


A typical detector scan is shown in Fig. 7 for awavelength of 187.9 Å, where the inside ~m . 0! andoutside ~m , 0! diffraction orders are identified.

he width of the orders was determined primarily byhe detector slit that was 0.5° wide in the dispersionirection.The grating was oriented so that the 3.4° facetsere facing the incident radiation. In this orienta-

ion, the inside orders were closest to satisfying then-blaze condition for these facets, where the radia-ion is specularly reflected from the facets. In Fig. 7,he efficiencies in the inside orders ~11 and 12!, dif-

fracted at angles closest to the incident radiation, aremore intense than the outside orders ~21 and 22!.The scattered light level between the orders is low fordiffraction angles in the off-blaze direction ~.15.2°!.The scattered light level is higher in the on-blazedirection ~,15.2°!.

We determined the measured efficiencies by fittinga background curve and five Gaussian profiles to the0, 61, and 62 orders as shown by the curves in Fig.8. The background and the five Gaussian profileswere fitted to the data points using a least-squarestechnique. The heights of the Gaussian profiles~above the background curve! were determined for

Fig. 7. Measured grating efficiency, as a function of the diffrac-tion angle, for a wavelength of 187.9 Å and an angle of incidence of15.2°. The inside ~m . 0! and outside ~m , 0! diffraction ordersare indicated.

Fig. 8. Fit of Gaussian profiles ~smooth curves! to the measuredgrating efficiencies ~data points!. The wavelength of the incidentradiation was 187.9 Å and the angle of incidence was 15.2°.

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each of the 40 detector scans that were performed atfixed wavelengths in the 125–225-Å wavelengthrange.

Although the angular positions of the diffractedorders changed with the wavelength of the incidentradiation, the angular position of the zero order wasfixed at an angle equal to the angle of incidence~15.2°!. The detector was positioned on the zero or-

er, and we measured the efficiency in the zero ordern the 100–138-Å range using a boron filter, in the25–175-Å range using a silicon filter, and in the70–340-Å range using an aluminum filter. Theero-order efficiency is shown by the data points inig. 9.The zero-order efficiency, as well as the efficiencies

n the diffracted orders that are discussed below, hasn interesting oscillatory behavior. In contrast, thefficiency of the uncoated master grating variedmoothly with wavelength.6 This oscillatory behav-or is attributed to the presence of the thin SiO2 layer

on the oxidized aluminum surface of the replica grat-ing.

To understand the oscillatory behavior, the reflec-tance of a thin SiO2 layer on an oxidized aluminumsurface was calculated as a function of wavelength.The computational model consisted of a SiO2 layerand an Al2O3 layer on opaque aluminum. The re-flectance was calculated for an angle of incidence of15.2° and for p-polarized incident radiation. Thewavelength positions of the extrema in the reflec-tance curve depended on the assumed thicknesses ofthe SiO2 and Al2O3 layers. These two layer thick-nesses were determined by matching the wavelengthpositions of the extrema of the calculated reflectancecurve to the extrema of the measured zero-order ef-ficiency curve of the replica grating.

Shown in Fig. 10 is the calculated reflectance of aSiO2 layer of variable thickness and an Al2O3 layer of30-Å thickness on opaque aluminum. The opticalconstants were derived from the compilations of Henkeet al.9 The three curves in Fig. 10 were calculated forhe wavelengths of three of the maxima in the zero-rder efficiency curve ~127.5, 169.0, and 217.0 Å!.

Fig. 9. ~a! Measured grating efficiency in the zero order at anangle of incidence of 15.2°. ~b! The reflectance of 743 Å of SiO2

and 30 Å of Al2O3 on opaque aluminum calculated at an angle ofncidence of 15.2°.

The three curves in Fig. 10 have locally coincidingmaxima at a SiO2 thickness of 743 Å. The threecurves have locally coinciding maxima for no otherSiO2 thickness. Additional calculations indicatedthat the local maxima in the calculated reflectancecurves matched those in the measured zero-order effi-ciency curve only for a SiO2 thickness of 743 6 5 Å andan Al2O3 thickness of 30 6 5 Å. In addition, the pres-ence of the Al2O3 layer was necessary for the goodagreement between extrema in the calculated reflec-tance curve and the measured zero-order efficiencycurve. The inferred thicknesses of the SiO2 layer andthe Al2O3 layer depend on the accuracy of the assumedoptical constants.9

In Fig. 9, curve ~b! is the calculated reflectance ofhe 743-Å SiO2 layer and the 30-Å Al2O3 layer on

opaque aluminum. The assumed angle of incidenceis 15.2°, the incident radiation is 90% p polarized, andthe microroughness is 7 Å. The extrema in the cal-culated reflectance curve ~b! in Fig. 9 and the mea-ured zero-order efficiency curve ~a! occur at the sameavelengths throughout the 100–300-Å wavelength

ange. The oscillations are thin-film interferenceeatures resulting from the presence of the SiO2 coat-

ing. The oscillations diminish in amplitude at thelonger wavelengths where SiO2 is less transmissive.The wavelength resolution of the measurements andthe spectral resolution of the monochromator werenot sufficient to observe the high-frequency, small-amplitude oscillations at wavelengths longer thanthe aluminum L edge at 170 Å.

Similar calculations were performed assuminghat the coating was SiO rather than SiO2. In this

case, the reflectance was lower than the measuredzero-order efficiency curve at the longer wavelengths.Because the grating efficiency cannot exceed the re-flectance of the coating, we infer that the compositionof the coating is primarily SiO2 rather than SiO.

The measured efficiencies in the 0, 61, and 62rders as functions of the incident wavelength arehown by the curves with data points in Fig. 11.lso shown is the total efficiency that is the sum of

Fig. 10. Calculated reflectance of a layer of SiO2 of variable thick-ess and a 30-Å layer of Al2O3 on opaque aluminum for the three

wavelengths: ~a! 217.0 Å, ~b! 169.0 Å, and ~c! 127.5 Å. The ver-tical dashed line indicates the only SiO2 thickness ~743 Å! at whichhe maxima of the three reflectance curves coincide.


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the measured efficiencies in the five orders. The ef-ficiencies in all the orders have an oscillatory behav-ior as a function of wavelength. The period of theoscillation is the same in all orders and is related tothe thicknesses of the SiO2 and Al2O3 layers.

4. Calculated Efficiencies

The curves without data points in Fig. 11 are thecalculated efficiencies. The efficiencies were calcu-lated by the modified integral method4,5 generalizedfor the case of a multilayer grating. Instead of ex-panding the groove profile in a Fourier series,10 theactual points of the groove profile derived from atomicforce microscopy ~Fig. 6! were used as the collocationpoints. This approach resulted in good convergencein the extreme ultraviolet and x-ray wavelength re-gions and improved accuracy for calculations usingan arbitrary groove profile. In addition, instead ofthe usual representation of the Green functions andtheir derivatives as Hankel functions, expansionsalong the exponential function series were used.This resulted in a higher accuracy of the solution andshorter calculation times.

Fast convergence was achieved for the case ofa multilayer grating with a nonideal, arbitrarygroove profile operating in the extreme ultravioletand x-ray wavelength regions. These results wererealized notwithstanding the fact that the spacingbetween the collocation points was equivalent to less

Fig. 11. Comparison of the measured grating efficiency ~dataoints! and the calculated efficiency ~curves without data points!

for the diffraction orders ~a! 11, ~b! 12, ~c! 21, ~d! 22, ~e! 0, and ~f !sum of all orders.


than one or, in some cases, a few wavelengths of theradiation.

The computer program, written in the C11 lan-guage, implements several mathematical and pro-gramming improvements. For the purpose of usingthe computer memory more economically for the caseof large numbers of collocation points and layers, thematrices are inverted along one row. The calcula-tion of the real and imaginary parts of the complexvariables, relating particularly to the exponentialfunctions, are carried out separately from each other.The complicated mathematical functions are codedwith the Assembler programming language.

For 210 collocation points ~the dimension of thesquare matrix!, the calculation of the multilayer grat-ing efficiency for one wavelength required only 128 son a rather small personal computer ~133-MHz Pen-tium with 64-Mbyte RAM!. Typically only several

egabytes of RAM were actually used.The calculation of the multilayer grating efficiencyas performed for wavelengths in the 125–225-Å

ange and for an angle of incidence of 15.2° ~withrespect to the normal to the grating surface!. Thegrating facets were oriented identically to that duringthe measurements, with the shallower facet ~blazeangle 3.4°! facing the incident radiation. The elec-ric field vector was perpendicular to the grooves.he surface layers were specified to be a SiO2 layer of

thickness 743 Å and an Al2O3 layer of thickness 30 Åon opaque aluminum. The 0, 61, 62, 63, 64, and65 orders were included in the calculation. The ef-ficiencies of the orders higher than u3u were quite low.

As shown by the curves without data points in Fig.11, the calculated efficiencies have the same oscilla-tory behavior as the measured values. The calcu-lated and measured efficiencies in the 61 orders arein excellent agreement, whereas for the 62 orders thecalculated efficiencies tend to be higher than the mea-sured efficiencies. The zero-order calculated effi-ciency is also higher than the measured efficiency forthe shorter wavelengths. Considering that thereare no free parameters in the calculation of the effi-ciencies, the overall agreement between the calcu-lated and measured efficiencies is considered to beremarkably good.

The explanation of the differences between the cal-culated and the measured efficiencies will requireadditional studies that are beyond the scope of thepresent research. These differences may resultfrom an incomplete modeling of the microroughnessof the groove profile and variations in the groovedepth with spatial periods larger than the groovespacing that are not accounted for in the presentcalculation. It is possible that the microroughnessof the SiO2 layer, the oxidized aluminum layer, andthe grating substrate may be different, and this wasnot accounted for in the computational model.

The measured and calculated efficiencies areshown in Figs. 12~a! and 12~b!, respectively. Theseplots reveal the relationship of the efficiencies in thevarious orders as functions of wavelength. For ex-ample, the 61 efficiency is highest at the longer

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wavelengths, whereas the 12 order is highest at theshorter wavelengths. This is consistent with theblaze condition ml 5 2d sin w sin u, where d is thegroove spacing, w is the blaze angle, and u is thegrazing angle on the facet.11 In Fig. 12, the mea-sured and calculated efficiencies in the 21 and 12orders cross at wavelengths of 204 and 210 Å, respec-tively. The measured and calculated efficiencies inthe 11 and 12 orders cross at wavelengths of 162 and182 Å, respectively.

The groove profile used in the calculation ~Fig. 6!as an average peak-to-valley depth of 85 Å. Byerforming calculations using the same groove profileut scaled to different assumed groove depths, it wasound that the wavelengths of the crossings of the 21

and 12 orders and of the 11 and 12 orders weresensitive to the assumed groove depth. For exam-ple, the 11 and 12 crossing occurred at wavelengthsof 152, 182, and 203 Å for assumed groove depths of75, 85, and 95 Å.

In general, the calculated efficiencies in the 11 and21 orders tend to decrease with increasing groovedepth. This is illustrated in Fig. 13 for the 11 order.As the groove depth increases, the calculated efficien-cies in the 12 and 22 orders increase at the longerwavelengths and decrease at the shorter wavelengths.These trends are consistent with the blaze condition inwhich the blaze angle w and the blaze wavelength mlincrease with groove depth. The best overall agree-ment between the calculated and measured efficien-cies was achieved with an average peak-to-valleygroove depth of 85 Å as shown in Fig. 6.

Fig. 12. ~a! Measured and ~b! calculated grating efficiencies in theindicated orders.

5. Conclusions

The oscillatory behavior ~as a function of wavelength!f the measured normal-incidence efficiency of a rep-ica grating was identified as a thin-film interferenceffect resulting from the presence of a thin SiO2 layer

on a highly reflecting oxidized aluminum surface.The SiO2 and Al2O3 layer thicknesses were inferredfrom the wavelengths of the extrema of the measuredzero-order efficiency.

The modified integral method described in Refs. 4and 5 was extended to accurately calculate the effi-ciency of the multilayer-coated grating in the softx-ray region. The calculation used a realistic grooveprofile derived from the AFM images and accountedfor the rounding of the corners of the groove profileand the microroughness of the facets. The thick-nesses of the SiO2 and Al2O3 layers, on the opaquealuminum layer, that were used in the calculationwere those independently derived from the oscilla-tions in the measured zero-order efficiency. Thusthere were no free parameters in the calculation, andthe comparison of the calculated and measured effi-ciencies was a severe test of the validity of the com-putational modeling.

The calculated and measured efficiencies in the 11and 21 orders were in excellent agreement. Thecalculated efficiencies in the 12 and 22 orderstended to be larger than measured.

The wavelengths of the crossings of the efficiencycurves were sensitive functions of the assumed groovedepth, and this constrained the computational modeland further tested its validity. The wavelength cross-ings of the calculated and measured efficiencies werein good agreement for an average peak-to-valleygroove depth of 85 Å that was consistent with the AFMcharacterization of the grating surface.

The efficiency calculations were performed on asmall personal computer. Considering the goodoverall agreement between the calculated and mea-sured efficiencies and the reasonable computationtime on a small computer, the modified integralmethod appears to be a useful tool for the design ofmultilayer gratings.

Fig. 13. Efficiency in the 11 order calculated for assumed groovedepths of 75, 85, and 95 Å. The measured 11 order efficiency isshown by the data points.


ods for x-ray and XUV grating diffraction analysis,” in X-Ray

1

This research was supported by NASA projectW19193. Part of the research was performed at theNational Synchrotron Light Source, which is spon-sored by the U.S. Department of Energy under con-tract DEAC02-76CH00016. The AFM images wererecorded in collaboration with Kaj Stolt of AnalyticalAnswers, Inc. We thank B. Bach of Hyperfine, Inc.for information concerning the replication of the grat-ing.

References and Notes1. J. F. Seely, M. P. Kowalski, R. G. Cruddace, K. F. Heidemann,

U. Heinzmann, U. Kleineberg, K. Osterried, D. Menke, J. C.Rife, and W. R. Hunter, “Multilayer-coated laminar gratingwith 16% normal-incidence efficiency in the 150-Å wavelengthregion,” Appl. Opt. 36, 8206–8213 ~1997!.

2. M. Neviere, “Multilayer-coated gratings for x-ray diffraction:differential theory,” J. Opt. Soc. Am. A 8, 1468–1473 ~1991!.

3. M. Neviere, “Bragg–Fresnel multilayer gratings: electromag-netic theory,” J. Opt. Soc. Am. A 11, 1835–1845 ~1994!.

4. L. I. Goray, “Numerical analysis for relief gratings working inthe soft x-ray and XUV region by the integral equation meth-od,” in X-Ray and UV Detectors, R. B. Hoover and M. W. Tate,eds., Proc. SPIE 2278, 168–172 ~1994!.

5. L. I. Goray and B. C. Chernov, “Comparison of rigorous meth-


and Extreme Ultraviolet Optics, R. B. Hoover and A. B. C.Walker, eds., Proc. SPIE 2515, 240–245 ~1995!.

6. M. P. Kowalski, J. F. Seely, L. I. Goray, W. R. Hunter, and J. C.Rife, “Comparison of the calculated and the measured efficien-cies of a normal-incidence grating in the 125–225-Å wave-length range,” Appl. Opt. 36, 8939–8943 ~1997!.

7. J. C. Rife, H. R. Sadeghi, and W. R. Hunter, “Upgrades andrecent performance of the gratingycrystal monochromator,”Rev. Sci. Instrum. 60, 2064–2067 ~1989!.

8. W. R. Hunter and J. C. Rife, “An ultrahigh vacuum reflecto-meterygoniometer for use with synchrotron radiation,” Nucl.Instrum. Methods A 246, 465–468 ~1986!.

9. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interac-tions: photoabsorption, scattering, transmission, and reflec-tion at E 5 50–30,000 eV, Z 5 1–92,” At. Data Nucl. Data Tables54, 181–342 ~1993!. Updated optical constants were obtainedfrom the internet site cindy.lbl.govyoptical_constants.

10. D. Maystre, “A new general integral theory for dielectric coatedgratings,” J. Opt. Soc. Am. 68, 490–495 ~1978!.

11. J. F. Seely, M. P. Kowalski, W. R. Hunter, J. C. Rife, T. W.Barbee, Jr., G. E. Holland, C. N. Boyer, and C. M. Brown,“On-blaze operation of a MoySi multilayer-coated, concave dif-fraction grating in the 136–142-Å wavelength region and nearnormal incidence,” Appl. Opt. 32, 4890–4897 ~1993!.

thin-film interference effects on the efficiency of a normal-incidence grating in the 100–350-Å...

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