Superfine holograhic X-ray gratings may be produced by transferring onto metal an interference pattern generated by two branches of a sufficiently coherent X-ray beam, emitted in the form of synchrotron radiation from high energy electron storage rings. Generation of the coherent beam requires restrictions on the beam size. A calculation is presented which allows an exact evaluation of this restriction. The effect of defocusing optics on the expected resolution is also studied

Csonka, P. L.

English

NASA Technical Reports Server

N86-24518
1985
NASA/ASEE SUMMER FACULTY RESEARCH FELLOWSHIP PROGRAM
MARSHALL SPACE FLIGHT CENTER
THE UNIVERSITY OF ALABAMA
CALCULATIONS ON THE PRODUCTION AND USE OF SUPERFINE
HOLOGRAPHIC X-RAY GRATINGS FOR ASTROPHYSICAL OBSERVATIONS
Prepared by:
Academic Rank:
University and Department:
NASA/MSFC:
Laboratory:
Division:
Branch:
NASA Counterpart:
Date:
Contract No.:
Paul L. Csonka, Ph.D.
Professor
University of Oregon
Institute of Theoretical Science
Space Science Laboratory
Astrophysics
High Energy Physics
Martin C. Weisskopf
August 30, 1985
NGT 01-008-021
The University of Alabama
XI
https://ntrs.nasa.gov/search.jsp?R=19860015047 2020-03-20T15:40:34+00:00Z
CALCULATIONS ON THE PRODUCTION AND USE OP SUPERFINE
HOLOGAPHIC X-RAY GRATINGS FOR ASTROPHYSICAL OBSERVATIONS
by
Paul L. Csonka
Professor of Physics
Institute of Theoretical Science
University of Oregon
Eugene, Oregon 97403
ABSTRACT
Superfine holographic x-ray gratings may be produced by trans-
ferring onto metal an inteference pattern generated by two branches
of a sufficiently coherent x-ray beam, emitted in the form of
synchrotron radiation from high energy electron storage rings. Gene-
ration of the coherent beam requires restrictions on the beam size.
A calculation is presented which allows an exact evaluation of this
restriction. The effect of defocusing optics on the expected resolu-
tion is also studied.
Xl-i
1 . INTRODUCTION/OBJECTIVES
Superfine holographic x-ray gratings are to be produced by
transferring onto metal an interference pattern generated by two
branches of a sufficiently monochromatic and coherent x-ray beanr .
Such an x-ray beam can be obtained by selecting an appropriately
monochromatized and coherent portion of a synchrotron radiaton beam
emitted by circulating electrons in a high energy electron storage
ring, such as SSRL at Stanford.
Selection of the appropriate beam portion requires first of all
adequate monochromatization. Denoting by A the wavelength of the
recording x-ray radiation, and by Ai the maximum pathlength dif-
ference between the two branches of the imprinting beam, one has to
have
Here f1 is a constant between zero and unity, to be specified below.
Second, the angular divergence, A9, needs to be restricted as
where 0 <_ f <^ 1 is a constant. Third, the cross-section of the beam
has to be limited to achieve adequate coherence, which, in turn
requires that the effective size of the radiation source (here circu-
lating electrons) be constrained depending on the size of the area
over which the interference pattern is to be established. That, of
course, will reduce the number of photons which can be used to pro-
duce interference, and increase the time which has to elapse while
the pattern is imprinted on a photosensitive material (such as PMMA) .
It is, therefore, important to accurately evaluate the necessary beam
size, to be able to restrict the source area as required, but no
more. A calculation will be described here which enables one to
evalute the necessary restriction in beam size under quite general
conditions, as illustrated in Fig. 1. .Superfine holographic x-ray
gratings are expected to achieve line densities itO lines/mm. For
an incident wavelength of X = 100 A at normal incidence on the
grating, when the radiation is collected by an optical element with 1
arcsec accuracy, the intrinsic grating resolution is of the order
of AA/X =10 . The question will be investigated to what extent
this resolution can be approached in an actual instrument.
XI-1
2. COHERENCE REQUIREMENTS
The notation is explained in Figs. 1 and 2. For any vector V,
we will write |^ | « v.
Referring to Fig. 2, one observes
L2,, - f-(L 4- P , ) + ?. I2 ; i, k =» 1, 2 (1)ik L <• sk' ij '
L., - {IE + P , I2 - 2(L + P , ] P. + P ,2\2 . (2)ik s)c sk J. i
Introduce the notion
ak = L + P3k, (3)
and write a, and a, for the y and z component of a. . Since theKy kz J t~ -v
coorindate axes are so chosen that the z component of P^ and
vanishes, one can rewrite (2) as
ik
The
alz
Next we find the maximum value of AL.. Clearly AL. will reach
its highest value for any chosen point P/
XI-2
when the point P, is such that for it L21 is maximum, and at the
same time P., is such that I^ -j is minimum.
o
If P j is located in Region 1, then the minimum value of L11
occurs for that P.J, for which P1 = aiv, and then L.... =aiz* ^
that P- for which L-1 is maximum, note that for the choice i = 2,
^ ^ I
Eq. (4) reaches extremum as a function of P~, when
P_= -a, ,2 1y
2
and this extremum is a minimum. Since L is a second order poly-
nomial in ?2/ L?1 can have no other extremum. Its maximum value as
a function P- is therefore reached at the endpoints of the allowed
range of variation for P-, i.e., at P_ =» ±D /2. By direct inspec-
tion, the correct choice is
" °* (7,2 '
so that the maximum of L-,i is
and the maximum over P- and P2 of AL1 , for a given P is
P2J tt, S1y * L2lalyl
Next, consider the case when P - is located in Region 2. We
already know that L^ can have no 'extremun of any type in Region 2,
nor can I^v Therefore, that P1 for which L11 is smallest, once P ^
is fixed, must lie on the boundaries of its allowed region of varia-
tion, i.e., at ±D /2. Similarly, that P,, for which L-^ is maximum,
must also lie there. By directly comparing the two possible choices,
one finds
XI-3
so that the maximum over P^ and P~ of AL for a given P'^ is
]/2max
D" _ 1,
¥ i2iV
(10)
(11)
From Eqs. (9) and (11)
lalyl + 2 °yl ^  Reg* 1
2 a. D in Reg. 2
(12)
where £ is defined by the above.
The maximum of AL11 over all possible P.J, P2 and Pg^ can now be
easily evaluated. Denoting the maximum value of |a1 | by |a | ,
one finds
max
l^y'max T
 D
2 y
(13)
2 a ;in Reg. 2
So far we dealt with the source and interference planes as if
they were one dimensional. The second surface dimension, along the x
axis, can be easily incorporated, by allowing PJ and "a^ to have x
components. Then by similar reasoning
XI-4
max
n=x,y
-{1 ,lz E
n=z,y
f|«u -1D .2l1/25 n J J,In1 2 (14)
•»- 2 a E £lz1 n=x,y
and the definition of £ is obtained from that of £ by changing all
subscripts y to x.
In the special case when L is parallel to z and both the source
plane and the interference plane are perpendicular to L, then Bq.
(13) reduces to the known formula
*ax[V ^Jto^
1
la, |>« ' 2|a, 11
 1z ' ' 1z '
j l l D j * D| J
|D.| l°yl
6
; if D SO
s
; if D >D
s
(14a)
and £ goes over into the known function A .
n n
XI-5
III. OEFOCUSING ELEMENTS
Assume that the colletor has an intrinsic angular inaccuracy of e.
For AXAF e is of the order of 2 x 10 radians. It will also be
assumed that the detector size can be made arbitrarily small, so that
the instrument resolution is not limited by detector size. For AXAF,
that assumption is not automatically valid, but for superfine holo-
graphic x-ray gratings, in many cases, instrument resolution is
limited by the location of the grating, and that is the topic to be
studied here.
The geometry and notation is explained in Fig. 3. The angular
deviation introduced by the grating will be detectable over the
"noise" caused by finite e and I, if
p* 0
tan i|» i — tan e + — tan 9 . (15)
L L
The deviation in the detector plane induced by the grating in m
order near normal incidence for wavelength X will differ from that
for X + AX by
„., dtanii dcosi|» ,._ ±m dtani|> AX __. .. / i c \LAX f ~r;— = AXL —- *• = ~T L f ( V ) ( io)dcosip dX d dcosif* X r
where f is defined as
2 2f (<JO = -(cos ij> - tan \|>J/siniJ) .
Therefore, the image of a single point source in m order, near
normal incidence, formed by two different wavelengths, X and
X + AX, will be distinguishable, provided that
+ L tan 9 (17)
For AXAF, one has F = 103 cm, e = 2 x 10~ radians,
and 9 = 0.1. Then
XI-6
if L and £ are measured in centimeters. Now clearly the best choice
is i = 0, and then
M > 2 x 1(T2 1
X * L (in cm) fdji) ' v '
For superfine gratings with d = 10 on, and X =• 100 A, one has
f (ill)- 1/tfT, and
AX 1.41 x 10 ... -„„ . ,-,».
—r i —r—r-. — ; for X = 100 A . ( 2 0 )X L (in cm)
For gratings with diameter^20 on in AXAF, one has L < 100 cm, which
-4
would give AX/X > 1.4 x 10 for X = 100 A. This resolution is
clearly far from the intrinsic resolution of the grating, which is
about 5 x 10 for this wavelength.
To increase the instrument resolution, one can either increase
the grating diameter, D , or increase L.
The increase in resolution with D is linear, and we will not
concern ourselves further with that variation.
To increase the resolution while keeping D fixed, one may in-
g
troduce a defocusing element at a distance I before the original
focal plane. If that element changes 9 by a factor h~ (h > 1), then
the new focal plane will be farther from the defocusing element than
the original focal plane was, by a factor h. Then the wavelength
resolution of the new instrument will be given by Eq. (17), provided
that in it one substitutes for F, L, SL, and 9 the new effective
values: F', L1, V , and 6*. Assuming F, L > £ > i, one has
F' = (F - i ) + hi =• F -h I (h - 1)g g g
L' = L -i- £ (h - 1)g
£' = h i
8 = h"1 9
and the new resolution is
XI-7
h£ tan (h 9)
When £ = 0, and the defocusing element is located close behind
the grating, I •» L, theng
Clearly, for F/L large, the resolution will improve essentially
linearly with h, up to values h » F/L. Beyond that, the resolution
tapers off, and approaches its asymptotic value for large h:
which is, of course, the intrinsic resolution of the grating.
For AXAF, it appears possible to reflect the beam at least once,
maybe twice (once backward and once forward) along the axis of the
focusing telescope. In that way one could increase the wavelength
resolution for F - L = 900 cm, h = 21, to 66% of its intrinsic value;
i.e., to AX/X = 7.3 x 10 at X =» 100 A. Reflecting the beam in this
manner causes no difficulty in this wavelength range, since the re-
flectivity of multilayer mirrors can approach 66%. However, the use
of multilayer reflection optics will reduce the bandwidth of the
instrument.
An alternative option would be to place the same instrument into
Spacelab. Defocusing could be done with grazing incidence mirrors,
which would alleviate the problem of wavelength reduction. Addi-
tional reflections would not be required if the dimensions of
Spacelab are to be as envisioned today.
XI-8
ACKNOWLEDGEMENTS
I wish to thank Mike Hettrick of BNL and Martin Weisskopf of
MSFC for interesting discussions on this topic.
REFERENCES
1. Csonka, Paul L., "Holographic X-Ray Gratings Generated by
Synchrotron Radiation", J. Appl. Phys., 52, 2692 (1981).
XI-9
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I-12


Calculations on the production and use of superfine holographic X-ray gratings for astrophysical observations

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