Modelling of planar transmission-type diffraction gratings compatible with synchrotron radiation sources

Nuclear Instruments and Methods 172 (1980) 287-291 © North-Holland Publishing Company

MODELLING OF PLANAR TRANSMISSION-TYPE DIFFRACTION GRATINGS COMPATIBLE WITH SYNCHROTRON RADIATION SOURCES * R. T A T C H Y N and I. L I N D A U Stanford Synchrotron Radiation Laboratory, Stanford University, Stanford, CA 94305 USA

This paper describes a comprehensive modelling scheme for idealized rectangular transmission gratings of arbitrary dimensions placed at arbitrary angles to the irradiating beam. The beam parameters are those of an arbitrary synchrotron source radiating past a series of mirrors and slits. The geometrical effects of the mirrors and slits on the beam characteristics are calculated by the methods of phase-space analysis. The particular aspects of grating performance that are determined by the computer program are the output intensity spectrum and the power loading in the grating structure. Though gratings of any material may be analyzed by the program, the material analyzed and utilized to date has been gold. Two items of particular interest that are discussed in this paper are modelled blazing effects and modelled thermal loading of the grating structure under high flux conditions.

was a p p r o x i m a t e d algebraically by:

1. I n t r o d u c t i o n

y = 3 . 1 4 1 4 3 3 8 6 2 ( x - _~)9 + 3 2 . 9 6 3 0 4 1 6 9 ( x - ¼)8

The potential use o f free-standing ultrafine transmission gratings as m o n o c h r o m a t o r elements in soft X-ray s y n c h r o t r o n b e a m lines has led to the modelling w o r k presented in this paper. In particular, the gratings' spectra versus rotation and degrees o f thermal stress under different s y n c h r o t r o n ring conditions have been investigated. A description o f the program analysis and some i m p o r t a n t results follow.

+ 145.985015(x - ¼)7 + 3 5 4 . 8 7 2 8 8 8 5 ( x - I) 6 + 5 1 5 . 1 0 9 3 3 3 ( x - ¼)s + 4 5 4 . 3 9 8 2 2 2 3 ( x - 1)4 + 2 3 5 . 8 9 4 4 5 5 ( x - ~)3 + 6 3 . 6 4 5 8 4 1 2 6 ( x - I) z + 4 . 6 4 8 7 6 1 9 0 4 ( x - ~) + 0 . 5 5 .

(1)

Effects o f o t h e r mirrors and slits in the system are m o d e l l e d b y formulas derived f r o m phase-space analysis [ 1 ] o f the s y n c h r o t r o n light.

2. S y n c h r o t r o n source Figure 1 shows the universal s y n c h r o t r o n radiation curve c o m m o n to all synchrotrons. Since the radiation incident on the grating will have passed a mirror, energies above 3 0 0 0 eV will n o t reach the grating. Due to this fact, the s p e c t r u m (fig. 1) was partitioned into 300 10 eV increments, w i t h grating p e r f o r m a n c e evaluable at each f r e q u e n c y . The total p o w e r dissipated in the sum o f the powers dissipated at each frequency. To facilitate c o m p u t e r w o r k , the curve in fig. 1

1,5

~-%

1.c

H

0.5

O -2

I

-1.5

I

-1

I

-0.5

I

I

0

05.

109 (w/w c )

* This work was supported by the National Science Foundation under Contract No. DMR 77-27489.

Fig. 1. Universal synchrotron radiation curve. "t"= Beam Energy/mo c2 ; p = ring radius; w c = 3@ (c/p). 287

VI. PERFORMANCE OF GRATINGS AND MIRRORS

R. Tatchyn, L Lindau / Planar transmission-type diffraction gratings

288

~.

•

\'

y'

..~

%

,/,~//

\x i !

"\

Fig. 2. Grating dimensions shown through the bals' cross sections.

LIGHT \,

\ \,,

3. The grating The grating structure and geometry are shown in figs. 2, 3 and 4. The ideal rectangular shape is quite valid as long as the aspect ratio (w/(a - d ) ) -1 is large [2] @7). This has been verified by application of this modelling program [3] to experimental results obtained with real gratings [4]. For small aspect ratios, more general, polygonal contours are needed to model grating performance. The field distribution across the top of the grating is (fig. 4), with a, d and w expressed in wavelengths of the source light [e.g., a -= a(A)/X(A)] :

rl(x [a j=-(N-1)/2 (~)

+R

{ 1

(N--1)/2

* -

a j=-(N-l)/2

L

ka\X

~!(

~

}

Fig. 4. Geometrical parameters of tilted grating. Y = X sin 0, E = w sec 0, 0 = grating angle, D = w sin 0, L = w tan 0, B = ( a - d ) cos0 - w s i n 0 , M = ( a - d ) - w t a n 0 , C =wsin0,

A=d-wtanO,

F=dcosO-sinO,

R=wtanO,

c=

Ax csc 0 = AX'(2 CSC20).

( a j = - ( N _ l)/2

+ A ( x ) • 1 (N-1)/2 ~ -

6 -1 x-

w tan 0

VV

(2)

X

La\X

+ (}d-w tan0)

]) R

M *

The spectrum is the absolute value of the Fourier transform of (2) squared; lap (s)12 = (sin ~sNat2 [exp(i@~)L exp [-2~is(a - d \ sin ~rsa / + w tan 0)/21 + exp(iq~R) R exp [2ds(a - d

i

+ W tan 0)/2] + 2 ~ + e x p ( - d a s ) A I2 .

(3)

In (3), c)L and 0R are refraction corrections for the edges. Computing L, R, M, A as Fourier transforms of L, R, M, A with surface effects [5] included in the factors BL, BR, BM gives [for h = n + ik = (1 - 6) + ik]: __

~_

]

-

rEi

:TM

I

1'

LIGHT

Fig. 3. Detailed breakdown of light incident on tilted grating.

= (I cos OBD v2 X

1 - exp{-27rw tan 0 [k csc 0 + i(s +8 csc 0)] } 2rr[k csc 0 + i(s + 6 csc 0)] (4)

R. Tatchyn, L Lindau /Planar transmission-type diffraction gratings

(B BRsin20(e-4ZrkwsecO-- -- 1)) PR = P1N R lW sin 0 + 8Irk

= ( / c o s OBR)1/2

×

1 - exp (-27rw tan 0 [k csc 0 - i(s - ~ csc 0)] } 2rr[k csc 0 - i(s - 5 csc 0)]

(5) = (I cos OBM)l/2exp [--2nw sec 0(5 -- ik)] X (a - d - w tan 0) sinc(a - d - w tan 0) s

(10)

PM=P1N(a -- d - w tan 0) cos O(BR1 BM e -47rwsec 0).

(6)

The optical constants h are currently taken from Hagemann et al., DESY Report No. SR 74/7.

4. Results

Power dissipated in the grating Under various ring conditions is shown in fig. 5. To estimate thermal loading, we need the following constants: (1) emissivity ~0.025 ; (2) thermal conductivity ~3.1 W K -1 cm -1 ; (3) density -~19.3 gm cm-3; (4) coefficient of linear expansion ~13.9 X 10-6

The match between the experimental results and the computer model has been reported elsewhere [4]. Some interesting observations are given here: (1) the dispersion varies approximately as sec 0. To tune over one octave starting at normal incidence therefore requires rotation from 0 - 6 0 ° . (2) At rotated positions, the spectrum is asymmetrically distributed. If 0o+ is the anglebetween the zeroth and either first order, 0 is the grating angle (fig. 4), then the Fourier shift theorem yields the relation 0o_+,

(11)

-

= (I cos 0)1/2(d - w tan 0) sinc(d - w tan 0) s . (7)

0 = sin-1 (-+ aX-+sin 0 )

289

(8)

where a is the grating period (A) and X is the source wavelength (A). (3) For a rectangularly shaped grating that is practically opaque to the source light, excessive rotation will change the effective bar/aperture ratio in such a way that the power spreads into all orders, diminishing first-order power tremendously in the limit. For non-rectangular shapes, this effect may be minimized greatly; but, as long as the grating has a finite thickness, the effect will be always present at large enough angles.

K-1 -(1/0 (al/aT).

Partitioning the grating as shown in fig. 6, we compute thermal distortion (buckling for the grating bars in Cell 1 and blistering of the grid in Cell 2). For the bars in Cell 1, the lateral buckling is A --~ 3.738 X 10-4 x/g-oL 2 ( c m ) ,

and, for the grid in Cell 2, the "blistering" displacement is A'~- 13.5 × 10 =4 x/~(NL/2) 2 ( c m ) ,

4

2-

Given the surface effect correcting factors BL, BL1, BR, BR1, BM, we get the following expressions for power dissipated in the grating bars:

C O

(9)

(13)

where go is the power per cubic centimeter in the bars and g' is the power density in the grid structure. Computations show negligible bar distortion under severe grating conditions, given current values of L, but rather severe "blistering". The indications are that designing gratings with more massive grids (to decrease g') and smaller N's (finer support girds) should make operation under high-flux conditions (but not greatly exceeding total intensities of the order of 1 w c m - 2 ) feasible, as long as active heat

5. Power and thermal loading

PL =P1N(BL1 w sin0 + BL sin 20(e_4~rkwsecO - 1)) 87rk

(12)

i 1

2 3 RIN6 ENE•6¥ (GeV)

4

5

F i g . 5. P o w e r d i s s i p a t e d in g r a t i n g v e r s u s b e a m e n e r g y a n d c u r r e n t , a = 2 d = 1 0 4 A , i l l u m i n a t e d a r e a is 6 m m b y 18 m m , w = 2 0 0 0 A , p o w e r d e n s i t y is ( 9 . 2 6 X 1 0 4 W c m - 3 , g r a t i n g angle = 0° , occlusion factor ~0.4.

VI. PERFORMANCE

OF GRATINGS AND MIRRORS

R. Tatchyn, L Lindau / Planar transmission-type diffraction gratings

290

\\,

/ /

,/E

uJ

C,<

= (WO_~)

~ BLAZE SLOPE

j

I

~V-~

--~

i I

Fig. 7. Canonical blazing scheme.

u , ~ . ( O l RSE GRID CELL 2

[

iA

I L ~'-~ GRATING

c]

I

,

. . . . . . . . . . . . . .

~ [ ] ~

,

~. . . . . . . . . . . . .

I-lFir-]

CELL,' FINE

G R I D "j

Fig. 6. Schematic of grating structure (not to scale and geometrically distorted).

For d = 0 , 0 ~ k ~ 8 , eq. (14)indicates close to 100% efficiency into first order for o~= 1laB. Most elemental metals do not fulfill the above condition very well over any large portion of the soft X - r a y range (due to the inherent nature of h), but present research on multilayer structures may possibly provide materials that do. For various other parameters and energies, fig. 8 shows the ratios / ( - a)blazed /(-1)unblazed ;

sinks are provided to establish correct boundary conditions.

6. Blazing If a canonical blazing scheme is given as in fig. 7, we get an exact expression for the intensity spectrum with surface effects ignored:

I(s). =

/(°)blazed /(O)unblazed ;

/(a)blazed /(1)unblazed " From the figures it is clear that blazing can significantly improve the first order efficiency of an unblazed grating. Once reliable optical constants are compiled for a grating's material over the applicable

....

te_4~o k

200 eV 600 eV 1000 eV

......

- -

Io

i

x (COsh 2rr(a- d) kc~- cos 2~r(a- d ) ( s - o£ ) )

.~>>..~

o. s

+

'x

2 e-2rrw°k(cos nsa) d sinc ds

,',~~.~

rr[s 2 - 2~8s + oP(a 2 + k 2 ) ]

~2

o~6)

i

X (s - ~a) cosh 7r(a - d) k~ - (s - o~a)

o

0'2 0'4 o. ALPHA

X cos rr(a - d)(s - aS) sinh 7r(a - d) ka]} (14)

/

/

i

/

/ /'

o

L //"

~",

1

0.2

X sinh rr(a - d) kol] L sin 2n-Woa [ko~ sin ~(a - d)

+d2sinc2ds](SinrrsNa] 2 " \ sin rrsa /

3 o~

X {cos 27rWo8[(s - czS) sin 7r(a - d)(s - o~6) X cosh rr(a - d) ka - ka cos rr(a - d)(s -

i

°'.2 #4 o~ ALPHA

i

//

012 015. • 0.6 ALPHA

Fig. 8. Blazing effects on the - 1 s t , 0th, and 1st orders of an unblazed gold grating. The curves e x t e n d i n g out to a = 0.4 are for gratings with (a/d) = 4 and those e xt e ndi ng out to a = 0.6 are for gratings with (a/d) = 2. we = 1 5 0 0 A ; a = 10 000 A ; alpha -= blaze slope.

R. Tatchyn, L Lindau / Planar transmission-type diffraction gratings

energy range, curves such as those in fig. 8 may be generated f o r different geometrical (blazing) parameters to determine an optimized grating geometry for some appropriate energy range.

7. Conclusions In order to determine whether or not ultrafine transmission gratings are suitable for use as diffraction elements in synchrotron beam lines, both their mechanical and optical performances must be evaluated. In this paper we have presented a comprehensive computer model of an arbitrary rotatable grating of rectangular (and/or blazed) profile and have used the computer-generated results to evaluate the grating's spectral and thermal loading performances under high flux conditions. In order that a complete estimation of the grating's mechanical performance may be effected, it will also be necessary to model the effects of radiation-induced electrostatic charge on the grating bars under high irradiation. If modelling effects prove inconclusive, the effect of radiation4nduced electrostatic charge on the grating structure must be determined by experiment. These final relevant aspects of grating performance are presently being investigated.

291

In conclusion, if we assume that electrostatic radiation4nduced stresses are negligible below a certain "critical" level of irradiative flux, then we can deduce from the results presented in this paper that, below this "critical" level, proper design and blazing of ultrafine transmission gratings can yield optical dispersing elements of outstanding efficiency and quality at low to moderate intensities of synchrotron ring light.

References [1] P. Pianetta and I. Lindau, J. Electr. Speetr. Rel. Phen. 11 (1977) 13. [2] R.O. Tatchyn, I. Lindau and E. K/fllne, Proc. Synchrotron radiation instrumentation (NBS, Washington, DC, June 1979), these Proceedings, p. 315. [3] R.O. Tatchyn, Stanford Synchrotron Radiation Laboratory Report #78/04, May 1978, pp. VI67. [4] J.C. Delvaille, H.W. Schnopper, E. K//ltne, I. Lindau, R.O. Tatchyn, R.A. Gutcheck and R.Z. Baehrach, Proc. Synchrotron radiation instrumentation (NBS, Washington, DC, June 1979), these Proceedings, p. 281. [5] M. Born and E. Wolf, Principles of optics (Pergamon Press, New York, 1965) ch. XIII.

VI. PERFORMANCE OF GRATINGS AND MIRRORS