806 J. Opt. Soc. Am. B/Vol. 1, No. 6/December 1984 Tatchyn et al.

Outline of a variational formulation of zone-plate theory

Roman Tatchyn

Stanford Synchrotron Radiation Laboratory, Stanford University, Stanford, California 94305

Paul L. Csonka

Institute of Theoretical Science, Department of Physics, University of Oregon, Eugene, Oregon 97403

Ingolf Lindau

Stanford Synchrotron Radiation Laboratory and Stanford Electronics Laboratories, Stanford University,Stanford, California 94305

Received March 22, 1984; accepted August 22, 1984

Transmission zone plates composed of materials describable by the complex index of refraction [ = (1 - ) +ik] modulate both the amplitude and the phase of the light passing through them. In this paper, a general varia-tional approach to the design of zone-plate profiles that maximize system transfer functions, such as W(m)/IN or(m)/lOUT (where IIN and IOUT are the total input and output powers and Ifm) is the mth-order diffracted power),

is presented in terms of the real and imaginary components of fi. The variational problem, as formulated here, isshown to generate some important classical zone-plate configurations as special solutions and to lead naturally tothe derivation of the most general zone-plate configurations possible. Some specific optimum solutions are derivedfor gold in the soft-x-ray range and are tabulated for convenient reference.

1. INTRODUCTION

In recent years, various suggestions for zone-plate configu-rations have been advanced for use in the VUV and soft-x-rayranges. At least two of these suggestions, the first by Kirz'and the second by Ceglio and Smith,2 dealt explicitly withmaterials whose translucency to x rays depends on the com-plex index of refraction [ = (1 - ) + ik]. Both of thesearticles examined the effects of the specific thickness profilesof the zone-plate material on the throughput efficiency intothe diffracted orders; Kirz also investigated the effects of somenonuniform profiles on improving the diffracted output effi-ciency I(')/IOUT, and Ceglio and Smith attempted to find, onthe basis of intuitive arguments, the specific profile that wouldmaximize the diffracted power into the first order. In thispaper, a more general approach is employed to maximize thediffraction efficiencies of zone plates: The desired diffractionefficiency is expressed as a functional of the zone-platethickness profile, and a variational analysis is used to identifythe optimum profile that maximizes the diffraction efficiency.As is shown, some well-known zone plates appear naturallyas solutions to this variational analysis (for limiting values ofk or ), and the most general efficiency-maximizing profilesare obtained as well. Once the solutions are derived, somespecific numerical examples are tabulated for gold in the100-600-eV range. Throughout the following analysis, k isrestricted to nonnegative values (i.e., k 0) to stress the ap-plication to passive absorbing materials in the soft-x-rayrange, but, in fact, the analysis is easily extended to arbitraryvalues of k, as long as suitable care is taken in interpreting theformulas. As far as the values of are concerned, althoughthey are typically small (i.e., I 61 << 1) for most materials in thesoft-x-ray range, their allowed range in the present analysis

is extended to 61 < , since this introduces no discontinuitiesinto the derived formulas and, more importantly, permits theanalysis to be transposed easily to analogous physical systemswhose active elements may be capable of inducing variationsin the 6 parameter that are much larger than those associatedwith materials in the soft-x-ray range. Nevertheless, pri-marily owing to the onset of strong reflectivity and refractioneffects in materials in which 6 becomes appreciable, the for-mulas developed in this paper are to be applied carefullyoutside the soft-x-ray range.

2. DEFINITIONS

For the analysis presented in this paper, the following defi-nitions will be useful (see Fig. 1):

A (r) = [1 - (r)] + ik(r) is the complex index of refractionas a function of the zone-plate radial dimension r. 6 is thephase-delay constant and k is the attenuation constant.

r is the zone-plate radial dimension, measured in units ofnumbers of wavelengths.

L is the distance from the zone plate center to the point ofobservation. The focal length L is measured in units ofnumbers of wavelengths.

ai is the radius of the ith zone, measured in units of numbersof wavelengths.

di is the open-aperture dimension in the ith zone in unitsof numbers of wavelengths.

r = (L2 + r2)1/2.m is the number of the diffracted order. Positive m's cor-

respond to the real foci, and negative m's correspond to thevirtual foci.

0740-3224/84/060806-06$02.00 © 1984 Optical Society of America

Vol. 1, No. 6/December 1984/J. Opt. Soc. Am. B 807

h(r) is the thickness profile of the zone-plate material as afunction of r, measured in units of numbers of wavelengths.

I(m)/IIN is the throughput efficiency, the ratio of powerdiffracted into the mth order divided by the total inputpower.

I(m)/IOUT is the output efficiency, the ratio of power dif-fracted into the mth order divided by the total outputpower.

A is the set of real numbers.X is the wavelength of the source light.

3. FORMULATION

In setting up the variational problem, we are free to define and

maximize any suitable system transfer function that can beexpressed as a functional of the zone-plate geometry and A (r).

(Note that, although we are restricting ourselves here to

maximizing our transfer functions in terms of h, we could just

as well maximize them with respect to any of the other pa-rameters listed in the choices above.) For the analysis in this

paper, the entire effect of the zone-plate geometry is repre-sented by the parameter h(r), which is the thickness of thematerial presented to the light at radius r: This reduction ofthe geometry is warranted by the fact that interface effectssuch as refraction and reflection may be disregarded for most

materials in the soft-x-ray range,3 and it also means that ouroptimum solutions will be thickness profiles only and not

specific shapes. Choosing the appropriate mathematicalrelation between [h(r), A(r)] and the desired system transferfunction, we are also free to select all the associated parame-

ters as well. The principal choices for the problem at handare as follows:

1(m)/IIN1. System transfer function (m)/OUT

- others

n constant with r2. h(r) Ia unrestricted with r

3. Relation between F

zone-plate geometryand system transfer Ffunction I

4. Refraction {rgrdlInclude

Jh(r) restrictedh(r) lh(r) unrestricted

'resnel-Kirchhoff integralrelation

lull-scale vector-integralrelation

)thers possible

5 Standard zone-plate geometry6. Zone-area geometry (see Fig. 2) others

7. k

8.

= 0

{E3S>

= 00

=O0

= 00

-r r .

h h(ri p

d y a2 .

Fig. 1. Side view of an optimum zone-plate profile with focal lengthL. The incoming wave on the left is a monochromatic unit plane waveoriginating at infinity. The first-order (m = 1) focus is at P. Eachzone i has radius a, and an open-aperture area of decremental radiusdi. The zone-plate geometry is defined by arbitrarily making itsright-hand face flat and by specifying its thickness as a function ofr with the thickness-profile function h(r). All dimensions are in unitsof numbers of wavelengths, and (L2 + al 2)1 /2 - L = 1.

There are clearly an infinite number of different cases thatmay be analyzed, and other choices for other parameters could

have been listed as well. For this paper, however, we are in-terested only in a limited number of cases selected from theabove listings. These are given in Table 1. The table as-sumes (1) the scalar Fresnel-Kirchhoff relation, 4 (2) thestandard zone-plate geometry (Fig. 2), (3) A constant with r,(4) a plane-wave source at infinity, and (5) no refraction in thezone-plate material. The constants have been partitionedas shown to correspond to materials that are transparent,translucent, and opaque, respectively.

In setting up the appropriate form of the Fresnel-Kirchhoffintegral,4 we will determine the phase and the amplitude ofthe electric field at each point on the zone area by tracing eachincoming ray through the zone material and thereby de-scribing its phase and amplitude relative to all the other raysby the factor exp[-27rih(r)(6 - ik)]. This is a valid procedurewhenever the zone radii are at least several wavelengths longand when the total zone-plate radius is much greater than thezone-plate thickness, conditions well fulfilled by most zoneplates in the soft-x-ray range. In addition, we will restrict L

to be sufficiently far away from the zone plate that the usualcosine factor in the integrand of Eq. (1) may be neglected. 4 Itis important to note that all these restrictions in no way limitthe validity and universality of the variational approach setforth in this paper and are invoked primarily to obtainclosed-form solutions in the present analysis.

The choice of zone geometry is depicted in Fig. 2. In thescheme shown, the successive zone radii are given by a 1

2 = 2L+ 1, a2

2= 4L + 4, a3

2 = 6L + 9, a 42

= 8L + 16, etc., where L

is the first-order focal length. For L >> 1, all zone areas areapproximately equal, and the higher real orders (m = 2, 3, 4,... ) are focused approximately at L/2, L/3, L/4, etc. Thischoice of zone dimensions is necessary for the decompositionof the integral in Eq. (1) into a factor depending on the firstzone geometry only and a factor depending only on the num-ber of zones [see Eq. (2)]. In this choice of zone geometry,therefore, each zone contributes equally to the amplitude atP.

The desired relation, subject to all the previously statedchoices and restrictions is, finally,

I(m) f exp(2wiro) dS 12.Ifz ro

9. Source JPlane wave at IX others

(1)

Tatchyn et al.

808 J. Opt. Soc. Am. B/Vol. 1, No. 6/December 1984 Tatchyn et al.

Table 1. Zone-Plate Cases Individually Suitable for Optimizationak =, k >0, ke i>O, k=co

Maximize 6 E l? = 0 5 e 5E

h(r) Restricted(m)/IIN Case 1 Case 2 Case 3 Case 4

1(m)/lOUT Case 5 Case 6 Case 7 Case 8h(r) Unrestricted

(m)/IIN Case 9 Case 10 Case 11 Case 121(m)/lOUT Case 13 Case 14 Case 15 Case 16

a Separate columns for cases such as (k = , = 0), (k = A, C = 0) have not been included because of their obvious physical interpretations and to reduce redun-dancy.

FRONT IEW - SIDE VIEWFig. 2. The standard zone-plate zone geometry, with all dimensionsin units of numbers of wavelengths. The successive radii are a 2 =2L + 1 a2 2 = 4L + 4, a32 = 6L + 9, etc. If L >> 1, then each zone's areais approximately the same.

For m = 1, we can rewrite Eq. (1) as

I(1) AJ al-di 2r exp[2wi [ro - h(r)(6 - ik)]1 rdrr ro

+ Hal exp(2diro)r2 2 (2)a1 -ds ro

where A is a function of the number of zones but not of h(r),and the constraint on L is (L2 + a 2)'/2 - L = 1 (see Fig.1).

In all cases, the power emerging from the plate's first zoneis

IOUT1 = exp[-47rh(r)k]27rrdr

+ w[a,2 - (al - dl)21

and the total power impinging upon the first zone is

IN1 = 7ra,2 .

Note that Eqs. (2) and (3) are both functionals of h (r).In order to proceed with the analysis in the next section, it

will be convenient to rewrite the quantity enclosed by abso-lute-value signs in Eq. (2) as (C2 + S2), where

C fal-di 2wr exp(-27rhk)k~O ro

X cos 27r[(L2 + r2 )1/2 - h6 + B]rdr

+ 2 sin 7r[(L2 + a, 2 )'/ 2 - [L2 + (al - dl)2]1l/2]), (5)

I (al-di 2w exp(-2-7rhk)

O ro

X sin 27r[(L2 + r 2)1/2 - h5 + B]rdri (6)

and

B = -1/2 1(L2 + a, 2)1/2 + [L2 + (a, - d) 2]/2}. (7)

4. ANALYSIS FOR CONSTANT h(r)

The thickness function h(r) may be restricted in an infinitenumber of ways before the variational analysis. Indeed, thesymmetry restriction on h (r)

J l 2r [ ro + B)exp[-2 7rih(r)(- ik)]rdr = 0

may be shown5 to lead to the variational derivation of theconstant-thickness zone plate as well as of the Gabor zoneplate,' but this is not done in this paper (see, however, Ref. 6).Rather, we predetermine h(r) to be constant, this being azone-plate configuration of significant importance. If h(r)is constant, the quantity C2 + S2 becomes

2 I.,rSt zone)al= C2 + S2

= 4 [exp(-4rhk)sin2(wrj[L2 + (al - dl)2 ]/ 2 - L)+ 2 exp(-2rhk)sin(wr[L2 + (al -dl)2]l/2- L)sin wr(L2 + a 2)1/2 - [L12 + (al -dl)2]/2I

X (cos 2r h - 1/2 [L2 + a 2)/ 2]1)+ sin2 7rj(L2 + a 2)1/2 - [L2 + (al - d) 2 ]/ 2

1] (8)

We can now maximize .. t zone) with respect to both d andh by taking the ordinary derivatives of it with respect to theseparameters and setting them to zero. This will yield

[L2 + (al - dl)2 ]'/ 2 - L

= (L2 + a12 )1/2 - [L2 + (a, - d,)2]'/ 2 = 2

and

exp(-2whk) = cos(2wh-0,m, (10)cos y

where

k _

cos a' = (k2 + 2)1/2' sin or = (k2 + 62)1/2 (11)

(9)

Vol. 1, No. 6/December 1984/J. Opt. Soc. Am. B 809

Table 2. Optimum Profiles Corresponding to the Cases Given in Table 1 (for m = 1) I

k 0, ke >0, ke >0, k=,Maximize E? 6 = 0 3> E R 1?

h(r) ConstantVVI()IN Rayleigh Fresnel Generalized Fresnel

phase plate zone plate phase plate zone plate

(Case 1) (Case 2) (Case 3) (Case 4)

I(')/IOUT Rayleighphase plate(Case 5) (Case 6) (Case 7) (Case 8)

h(r) UnrestrictedI(1)/IIN Blazed Fresnel Generalized Fresnel

Rayleigh zone plate blazed zone plate

phase plate (Case 10) zone plate (Case 12)

(Case 9) (Case 11)

I(')/IOUT Blazed GeneralizedRayleigh output-efficiencyphase plate (Case 14) zone plate (Case 16)

(Case 13) (Case 15)

a Some well-known classical zone plates are seen to appear as optimum solutions.

Once the optimum h, ho, is found from Eq. (10), the maxi-mum efficiency I(')/IIN]MAX will be given by

[I( 1 )/IIN]MAX =2 (sin2 27rhob) [1 + (Ž)21

k 0, 3 $ 0, a. (12)

Given Eqs. (8)-(11), we can fill in the cases in the first row

of Table 1. We first see that relation (9) holds for all valuesof k and I. For k = (opaque material), we therefore clearly

end up with the Fresnel zone plate (see Tables 1 and 2). Forother limiting values of k and 3, Eq. (8) should be consultedin conjunction with Eqs. (10)-(12).

For 3 = 0, we see that h should be infinitely thick to provide

complete attenuation in the material-filled areas, resultingonce again in the Fresnel zone plate (Case 2). For k = 0, the

equations enforce h = 1/(2), which yields the well-known

Rayleigh phase plate. For k, 3 e X, (k > 0), we obtain the

generalized phase plate, whose thickness is given by Eq. (10)

for a given (k, 3) and which has also been analyzed by Kirz.1

To solve Cases 5-8 in Table 1, we need to take

d ( a,2(C2 + S2) 0.h) d 42 h) -(al - dl02[l - exp(-47whk)]1

(13)

For k = 0, the optimum solution is again the Rayleigh phase

plate. The specific solutions for Cases 6-8 are not presented

in this paper, but their determination, based on Eq. (13), isstraightforward.

5. ANALYSIS FOR UNRESTRICTED h(r)

To find the optimum unrestricted functions ho(r) we take thevariation of the system transfer functions I(')/IIN and I(')/

IOUT [which are functionals of h(r)] with respect to h(r) andset them equal to zero.6 This will yield two classes of func-

tions that will make the system transfer functions stationarywith respect to ho(r). Some two of these functions will

maximize I(')/IIN and I(')/IOUT-

The two stationary conditions may be expressed as

g[I()] = 0

h(r) h(r)=ho(r)

and

b[IW('OUT] I5h~r H h(r)=ho(r)

(14)

=0

or

(15)F I(1) r rUT rh = r6h W ) 5Ah(r) ] h(r)=ho(r)

where

(16)IOUT| h(r)=ho(r)

In solving the perturbations, we will use the suitably sim-plified implication that

[4h If gR [h(r)]dr

+J(I[h(r)]drfl = - f f(R[h(r)]dr1 OR

+ {f I[h(r)]dr} ) dr =o0]

Given this relation and expressions (2)-(7), (11), and

Ccos V' (C 2 + S2)1/2

(17)

(18)

we straightforwardly derive the stationary condition for ex-tremization of IP1)IIIN to be

-4r2(C 2 + S 2)1/2 (k2 + 2 )/ 2 [exp(-27rhk)/ro]

X cosl27r[(L 2 + r 2)1/2 - h6 + B] + y - v = 0 (19)

and the stationary condition for extremization of I()/IoUT tobe

Tatchyn et al.

810 J. Opt. Soc. Am. B/Vol. 1, No. 6/December 1984

cos{2r[(L2 + r 2)1/2 - h6 + B] + y -

4r cos Yrro exp(-27rhk)

a, 2 (C2 + S2)1 /2

For Eq. (19), the maximizing solutions will occur when theargument of the cosine is set equal to 7r/2 ± M7r, where M isany integer. If we specify the boundary condition

ho(0) = 0, (21)

the optimum profile in the first zone will be given by

ho(r) = 0; [al - di(opt)] r < al,ho(r) = 1/6[(L2 + r 2)l/ 2

- L]; 0 r < [al - dl(opt)],(22)

where d(opt) may be derived from the transcendentalequation

exp[-27r - (F + 1)] = sny (23)

and the relation

F = [L2 + (a1 - d1)2 ]1/2 - (L2 + a1

2)/2. (24)

Once the optimum F, Fopt, is found from Eq. (23), themaximum possible efficiency for the optimum zone plate willbe given by [cf. Eq. (12)]

[IP)/IINIMAX = 2 (sin4 rFopt) [1 + * (25)

At this point, we can examine the solutions for Cases 9-12in light of Eqs. (22)-(25). When k = 0, Eqs. (23) and (24) statethat d(opt) = 0, and we obtain what may be called the blazedRayleigh phase plate (under certain conditions referred to asthe Fresnel lens). For k = and = 0, just as in the case ofconstant h(r), the solutions generate the Fresnel zone plate.For the case most relevant to the soft-x-ray range, where k,

e Al (k > 0), we find that d(opt) varies with both k and 6,and the profile and dimensions are specified by Eqs. (22)-(24).The structure in this case (Case 11) may be called the "gen-eralized blazed zone plate".

Tatchyn et al.

For Eq. (20), a closed-form solution is, in general, impos-sible. Solutions may be generated on the computer by setting

(20) the boundary condition

ho(0) = 0 (26)

and using the slope relation derived from Eq. (20):(1 + r- + 27r tanf27r[(L2 + r2)1/2 -ho + B] + - v

dho ro /dr 27rw tan{27r[(L2 + r 2 )1/ 2

- hob + B] + y - + 2rk

(27)

to construct ho and to find the optimum d, d1(opt).For k = 0, however, we can clearly see that the solution for

Case 13 will also be the blazed Rayleigh phase plate, and, fork = , the solution for Case 16 will be identical with that forCase 8. For = 0, we see from Eq. (20) that h(r) will varyroughly as ln(cosl27r[(L2 + r2)1/2 + B + y - v]). The struc-ture corresponding to this case (Case 15) may be referred toas the "generalized output-efficiency zone plate".

It should be pointed out that the above results constitutea complete solution to the problem addressed by Kirzl in hisown comprehensive monograph. Numerical computationsof ho for this case indicate efficiencies [I('/IouT]MAx of asmuch as 82% for gold in the low end of the soft-x-ray range(-150 eV).

6. NUMERICAL EXAMPLES

In this section, we tabulate optimum parameters and ef-ficiencies for a gold zone plate ( constant with r) designedto maximize I()/IIN (i.e., Cases 3 and 11). Some other usefulnumerical parameters and figures of merit for some of theconstant-thickness zone plates (Cases 1-8) may be found inRef. 1. The numbers for Cases 3 and 11 are given in Table 3and have been derived from recent optical-constant compi-lations.7 Needless to say, the numbers are not to be taken asfinal, as the true values of the optical constants used in thistable may yet prove to be significantly different.3

It is of interest to compare these numbers with those de-rived by Ceglio and Smith.2 In that paper, the authors at-

Table 3. Optimum Parameters and Figures of Merit for Gold Zone Plates with Constant and Unrestricted Profilesover the Range 100-600 eVa

I(1 )IIIN)MAXh(r) h(r)

eV -F0 pt Unrestricted Constantb k 6 k/5100 0.288 0.27 0.173 0.0454 0.111 0.409120 0.2416 0.38 0.22 0.0178 0.071 0.25140 0.238 0.39 0.22 0.0109 0.045 0.24150 0.25 0.36 0.21 0.0101 0.037 0.273160 0.209 0.31 0.19 0.0101 0.03 0.33180 0.309 0.23 0.15 0.0107 0.021 0.51200 0.337 0.19 0.13 0.0115 0.017 0.676240 0.36 0.16 0.12 0.0113 0.013 0.87280 0.3621 0.16 0.12 0.0097 0.011 0.88300 0.372 0.15 0.116 0.009 0.0091 0.99350 0.368 0.15 0.119 0.0073 0.008 0.91500 0.368 0.15 0.118 0.0047 0.005 0.942600 0.358 0.16 0.123 0.0034 0.004 0.85

a Equations (10)-(12) and (23)-(25) and a hand calculator have been used for the computations.b Fpt = -0.5.

Vol. 1, No. 6/December 1984/J. Opt. Soc. Am. B 811

tempted to specify intuitively a profile function that theybelieved would maximize I 1 /IIN and, on that basis, ended upwith less than optimum throughput efficiencies. For exam-ple, one of their conclusions was that it would be of "no value"

to blaze a zone plate when "k/5 > 0.2," whereas we see from

Table 3 that doing so can actually yield efficiency gains of

30-80% over the generalized phase plate (Case 3) over the

entire 100-600-eV range.

7. HIGHER ZONES AND HIGHER ORDERS

The reader will have noticed that for the optimum unre-stricted profile our solutions specified only the first zone [Eq.

(22)] and were presented for m = 1. This was done to con-

serve space and because the extension to higher zones andorders is easily made. By using Eq. (19), the general profilefor the ith zone becomes [cf. Eqs. (22)]

hoi(r) = 0; [ai - di(opt)] < r < ai,

hoi(r) = - [(L2 + r 2 )1/2 - (L2 + ai-12 )1/2];

ai- 1 r < [ai - di(opt)]; ao = 0.

For the constant-profile zone plates, Caseslowing relations hold:

(L2 + ai2)1 /2 - [L2 + (ai - di)2]1/2 = 2;2

[L2 + (as - di)2]1/2 - (L2 + ai-12)1/2 = 1;

For the higher orders, the entire analysis is sti]more general constraint imposed:

(L2+a, 2 )l/2 -L = m; m = 2,3,4

Similar extensions are obtained for all the othand cases.

8. CONCLUSIONS

In the foregoing discussion, we have attempte4the problem of maximizing system transfer funcbe expressed as functionals of zone-plate geo:most general way possible. As a result, nonechoices and restrictions made in the foregoing,,or reduces the generality of our approach. Fostead of a plane-wave source at infinity, we rcluded a point source a finite distance away iiKirchhoff integral [Eq. (1)].4 For that mattervector analog of the Fresnel-Kirchhoff integibeen employed. With either choice, we coulda zone geometry that would have enabled us t

integral diffraction relation [as with Eqs. (1)

(28)

,1-4, the fol-

text] or we could have selected a more arbitrary geometry.Zone-plate geometries on curved surfaces could have been

considered, and more-general optical effects, such as reflectionand refraction, or ii nonconstant6 with r, could have been in-cluded. With each of these refinements, the analysis mightwell become more complex, but it should always be possible

to extract the solutions with a computer.One interesting follow-up investigation that comes to mind

would be to extend the present analysis to include refraction,as this becomes a strong effect in the VUV range for manymetals. 3 Much work remains to be done in finding zone-area

geometries (note that arbitrary transfer functions can also be

maximized with respect to the zone-area geometry Z) andappropriate transfer functions that would lead to optimalmatching of sources such as synchrotrons and undulators(wigglers) to desired image points P. Finally, the maximi-zation of other types of system functions (such as the systemresolution) is also an important area for future investigation.

In summary, it should be clear from these observations, aswell as from the multiplicity of choices listed in Section 4, that

there is ample opportunity for the extension and further ap-plication of the analysis developed in this paper.

ACKNOWLEDGMENT

Portions of this work were performed at the Stanford Syn-chrotron Radiation Laboratory, which is supported by theU.S. Department of Energy.

ao = 0. (29) R. Tatchyn is also an affiliate member, Berkeley Center for

X-ray Optics, LBL, Berkeley, California 94720.[1 valid, with a

REFERENCESL,... (30) 1. J. Kirz, "Phase zone plates for x-rays and the extreme UV," J. Opt.

erpaamters 2.Soc. Am. 64, 301-309 (1974).

rer parameters 2. N. M. Ceglio and H. I. Smith, "An efficient lensing element for

x-rays," presented at the Eighth International Conference onX-Ray Optics and Microanalysis, Boston, Mass., August 18-24,

1977.3. R. Tatchyn, "The introduction and development of a new method

for measuring the optical constants of metals in the soft x-raydi to formulate range," Ph.D. dissertation (Stanford University, Stanford, Calif.,tions that may 1982), Chap. III.metries in the 4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,of the specific New York, 1968), Chap. III.

analyses limits 5. R. Tatchyn, P. L. Csonka, and I. Lindau, "The constant-thicknesszone plate as a variational solution," Opt. Acta (to be pub-

r example, in- lished).night have in- 6. R. Tatchyn, P. L. Csonka, and I. Lindau, "Optimization of planar

a the Fresnel- metallic nonrefracting transmission-grating profiles for mth-order

r, the full-scale intensity maximization in the soft-x-ray range," J. Opt. Soc. Am.

ral could have 72, 1630-1638 (1982).havecouldecave 7. H. J. Hagemann, W. Gudat, and C. Kunz, "Optical constants froml have selected the far infrared to the x-ray region: Mg, Al, Cu, Ag, Au, Bi, C, and

o separate our A12 03 ," Rep. No. 74/7 (Deutsches Elektronen-Synchrotron,

and (2) in the Hamburg, Federal Republic of Germany, 1974).

Tatchyn et al.