Download Phase zone plates for x rays and the extreme uv - X

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA
VOLUME 64, NUMBER 3
MARCH 1974
Phase zone plates for x rays and the extreme uv
Janos Kirz*
Laboratory of Molecular Biophysics,Department of Zoology, Oxford University,Oxford, OXI 3PS, England
(Received 12 May 1973)
Phase-reversal zone plates can be designed even for regions of the electromagnetic spectrum where the
index of refraction is complex, with a real part close to 1.0. These devices are superior to Fresnel zone
plates both in their light collection, and in their signal-to-noise characteristics. Materials with suitable
optical and mechanical properties exist throughout most of the 1-800-A wavelength range for their
construction. Imperfections in fabrication, such as incorrect plate thickness, sloping zone edges, or an error
in the width of alternate zones result in only moderate deterioration in optical performance.
Index Headings: Zone plates; Diffraction; X rays; Ultraviolet; Image formation.
Although Fresnel zone plates have been known for
about 100 years,1 they have not enjoyed much popularity for four reasons.
(i) Their light-collecting
efficiency is poor (only 1/7r2
of the power incident on the zone plate participates in
forming the primary image).
(ii) They suffer from high background (4 of the light
incident is undiffracted and creates a continuous background in the image plane).
(iii) They are highly chromatic (f (c1/X).
(iv) They are difficult to make. (Ring spacing becomes very small for visible light, smaller yet for
shorter wavelengths.)
show how a device of this type can be made for the ex-
treme uv and the x-ray range.
THE RAYLEIGH-WOOD PHASE-REVERSAL
ZONE PLATE
The Rayleigh-Wood
device consists of a series of
concentric ring-shaped zones. Alternate zones are open,
the others advance (or retard) the phase of the incident
radiation by 7r radians. This is done by adding (or
subtracting) X/2 to the optical path of the rays in these
zones, using refractive material of the appropriate
thickness.
Consider plane waves incident on this zone plate. At
the
primary focus, the resultant amplitude from each
*To overcome the first two difficulties, Lord Rayleigh2 zone must add in phase. In the direction of increasing
proposed the phase-reversal zone plate in 1888 and
radius along the zone plate, the path of the rays to the
Wood demonstrated its advantages over Fresnel zone
focus increases. Over a pair of zones, the path increases
plates 10 years later. Even this device is of little practi- by X and the phase 0 by 27r.In the first of these two
cal use in those parts of the electromagnetic spectrum zones, the phase may be considered to increase from 0
where lenses or mirrors are available.
to 7r,with the resultant amplitude having phase 7r/2.
Interest in Fresnel zone plates has been revived by In the next zone, the phase moves from wr to 27r,with
the possibility of their use with x rays by Myers4 and
the resultant at 3wr/2.This would interfere destructively
by Baez.' In the extreme ultraviolet and the x-ray
with the contribution of the previous zone were it not
domain the choice of optical elements is limited, and
for the phase shift =4 17r introduced by the zone-plate
Fresnel zone plates have found increasing use here. In
material.
1961, Baez demonstrated the use of a self-supporting
In the next pair of zones, the phases repeat with 0
Fresnel zone plate with open transparent zones with
larger by 27r; the resultant amplitude for each zone
ultraviolet radiation' (X=2537 A). Recently Pfeifer
therefore adds in phase at the primary focus, as reet al. extended the use of this device down to 565 A.6
quired. Each time the path increases by X/2 (and 0 by
In the use of Fresnel zone plates in x-ray astronomy,7' 8 7r),a new zone starts; this requirement determines the
9
and spectroscopy, the high background, due primarily
zone radii,
to the undiffracted radiation, can be removed by central
n2X2
stops.'" The resulting zone ring however has an even
rn2 = nfX+
(1)
worse light-collecting
efficiency than that of the zone
plate. We propose to improve the light-collecting
efficiency and to suppress the background in these
devices using the Rayleigh-Wood scheme for replacing
opaque zones by phase-shifting ones. We show that this
procedure is useful and possible even though the refractive index of all mabterial is complex, with the real part
very close to 1.0.
4
where rnis the radius of the nth zone, and f is the focal
length. In most devices built to date nX<<f, and we
have the simple expression
rn; (nfX)i.
(2)
In this approximation, all zones have equal areas and
In what follows, we will briefly review the properties contribute equally to the irradiance at the focus.
The zone plate acts as a lens for finite object distances
of the Rayleigh-Wood phase-reversal zone plate and
301
JANOS KIRZ
302
p and image distances q, satisfying the relation
-+-=-,
(3)
Recently, zone plates with very large numbers of
zones have been made by use of the interference pattern
between two spherical wave fronts from the same
laser.8 "" 6" 7 . This technique results in zones with approximately sinusoidal amplitude-transmittance profiles. The optical properties of such devices, known as
Gabor zone plates, have been described by Waldman,"8
Horman and Chau,'9 and Markus.2 0 Their performance
characteristics are also shown in Table I. They are
somewhat inferior to Fresnel zone plates, as only 1/16
of the incident flux appears in the real image. The
perfect Gabor zone plate has no higher-order foci.
If the absorber is replaced by a phase shifter in the
Gabor zone plate (as in the change from the Fresnel
to the Rayleigh-Wood zone plate) the performance, as
shown in Table I, improves considerably.'5 2'
where f =r?2 /X. The expression for the radii Eq. (1), is
accurate only for the object (or image) at infinity. Using
this zone plate for finite object distances introduces
spherical aberrations. For specificapplications, however,
the second term in Eq. (1) can be adjusted to eliminate
these. 1"
2
In general,
qnX
2
XnX(q-f)+4fq\
nX(q-f)+2q
n2X2 nX(q-f)+4fq\
2
2
16 nX(q-f)+2q2
If we expand this expression in powers of nX,we obtain
rn2=
n2X2
3f(q-Of)
of v+-- 1+00-.
n3X3\
(4)
PHASE-REVERSAL ZONE PLATE
WITH ABSORPTION
For q = f the higher-order terms vanish, and we recover
Eq. (1). For q=p=2f the higher-order terms vanish
once again, and we obtain
r, 2 =nfx+
In the far-ultraviolet and x-ray regions of the electromagnetic spectrum, all materials become absorbing, and
the real part of the refractive index approaches the value
of 1.0. It is therefore not possible to build a genuine
Rayleigh-Wood device with transparent phase-shifting
zones. It is possible, however, to approach it by careful
n2X2
16
In general, the second term in Eq. (4) becomes signifi-
choice of the material used. We will derive here the flux
cant if n2 > f/X. The higher-order terms remain negligible for all object distances unless nl>5f/X.
The resolving power of the zone plate is very similar
to that of a lens of the same diameter.'3
In addition to the primary real focus, the device has
distribution from a zone plate made using a material of
refractive index
n=1-6-ik,
has an identical series of virtual foci, allowing the use of
the zone plate as a diverging element as well as a con-
verging one. The distribution of the incident flux
among these foci has been considered by Horman'4 and
by Dammann". In Table I, we contrast the way this
flux is distributed in the conventional Fresnel zone plate
and in the Rayleigh-Wood phase-reversal zone plate.
In the latter, all foci gain by a factor of 4, and both the
background from the zero-order undiffracted beam and
the absorption in (and possible damage to) the plate
have been eliminated.
Zone plate
1st order
k
"1=-.
'5
_
(6)
It is sufficient to consider a pair of zones, over which
the optical path to the image increases by X (and the
phase 0 by 27r), for, apart from obliqueness and other
normally small corrections, all such pairs contribute
equally to the image. As we shall show later, the conventional choice of zone boundaries always gives
I. Distribution of the flux incident on zone plates.
Real imagesa
2nd, 3rd, etc., orders
Undiffracted
1/Xr2
0, 1/(37r)2, 0, 1/(57r),
. .
Rayleigh-Wood
4/,2
0, 4/(37r2), 0, 4/(57r)2 ,
...
(Phase reversal)
Gaboro
1/16
(No higher-order images)
Gabor
0.34
0.10, 0.01, 0.0006,
...
Absorbed
1
0.45
0
0
1
4
8
0.10
0
116
1
Flux in first- and higher-order real images and their sum in units of the incident flux. Virtual images have identical fluxes.
Including virtual images.
I Sinusoidal amplitude-transmittance variation.
d The amplitude of phase modulation for maximum first-order irradiance,
= 1.84 radians (peak to peak), is assumed.
a
b
Totalb
Total
Fresnel
(Opaque zones)
(Phase
shift)dI
(5)
where a is the deviation of the real part from unity, and
k is the imaginary part of the index of refraction.
We will find that the performance of the zone plate
depends on the ratio
higher-orderfoci at f/(2m+ 1), m= 1, 2, .... It also
TABLE
Vol. 64
March 1974
PHASE
ZONE PLATES
FOR X RAYS AND euv
303
If we use the parameter 77defined in Eq. (6), the exponent becomes
27rkt/X=7
and the intensity,
C2
I1l=-(l+e721'7-2e-170 coso).
(7)
72
Contributions to the higher-order images are easily
obtained, leading to the general expression
r C2
|(I
+ 62,7,A-2e717 coso),
Im=i
m = 41, ±-3, 4±5, ..
0,
m=4t2, 4±4,±6, ....
This expression gives the results shown in Table I
for the Rayleigh-Wood zone plate for 7=0 and 0=7r
and the results for the Fresnel zone plate for a-b>oo.
We are interested in choosing the material and the
thickness that will maximize the flux. The choice of
material is straightforward:
Since k is constrained to be
non-negative by the requirement that materials attenuate rather than amplify the wave, we have to choose 7
as small as possible.
Fig. 1. Graphical method for finding amplitude in primary
image from one pair of zones, using wavelets. A, is the amplitude
from the open zone, A, from the-zone with shifter, introducing
phase shift k.
maximum flux in the primary image. Therefore, we
adopt it here.
We leave the first zone of our pair open, and cover
the second with a thickness t of our refractive material.
This material attenuates the flux by a factor e-4rktlx,
the amplitude by e-27rktIX, and retards its phase with
respect to the open zone by 4 = 2irt6/X.
To obtain the contribution of our pair of zones to the
amplitude at the primary image, we find the vector
sum of the wavelets. This is shown graphically in Fig. 1.
We can write the amplitude from the open zone at
this image as
C
r
iC
AXP=-
eidO=-,
27r Jo
Let us consider now the choice of 0 for a given material. To maximize the flux we set
C2
=-(-2
ail
2
-qe+2ne-'7 cos0+2e7hk sin0)=0.
(8)
The solution q5optof this equation, as a function of 77,
is shown in Fig. 2. As X approaches 0, fopt approaches
7r as expected.
For a given choice of r and X, the flux absorbed in
TT
'
I
I
I
'
I
'
I
.
O.9TT-
0.8TT
en
Cd
.-
2
where C = Ii,, is the total flux incident on this pair of
'a
zones.
The amplitude due to the shifted zone is
A,
C
=-e-2rkt/A
27r
0 .6 r
(ri
I ei(0 -O)dO=--e-2rktlX,-ix
Jrr
The flux due to the pair of zones at the image
o .5Tr
0
2
0.4
0.6
0.8
11
C
1I= A p+A. j 2=-(1+e-4rkt/X
0.2
-2e-rkti/X coso).
FIG.
2. The phase shift 0 .pt, which provides maximum
image flux as a function of t7.
1.0
JANOS KIRZ
304
our pair of zones is
C2
Iabs,=_(l-e72n+),
(9)
2
and the flux in the zero-order undiffracted beam is
Io= -
dO+e-
j
refraction
e-idOj
2
2
";+2e'70
(ii) The thickness necessary for obtaining the desired
phase shift 4 must be smaller than or comparable to
the smallest zone spacing.
(iii) They must have good mechanical properties, to
permit fabrication and use of these thin devices.
To find material with suitable 7jvalues, we need to
examine the real and imaginary part of the index of
C
=-(I+e4
(10)
cosO).
Vol. 64
Once again the expressions in Eqs. (9) and (10) reduce
to the entries given in Table I in the appropriate limits.
In Fig. 3, we present the fraction of the incident flux
that reaches the primary image for 'k4opt as a function
of q. We also show the fraction absorbed and that
transmitted undiffracted.
So far, our aim has been to maximize the flux in the
primary image. This solution does not maximize the
signal-to-noise parameter I,/I1, except for q=0. We
shall return to the problem of reducing the background
in the section on distorted profiles.
as a function
of the wavelength
7 to 190 A.26 Many of these data, however, have been
derived from reflection, rather than transmission,
measurements; this may introduce some systematic
errors.27 At shorter wavelengths, and away from absorption edges, the free-electron expression
Nere
3=
-X
27r
2
is a good guide. [Here Ne,=ZeffpN/A
DESIGN
CONSIDERATIONS FOR EXTREME
uv AND X-RAY USE
A. Choice of Materials
The requirements for acceptable materials are
(i) They must have small q values in the wavelength
range of interest.
0.5
0.4
a
for each
candidate.
There is a considerable wealth of information available
on absorption coefficients both at x ray22' 23 and at
ultraviolet2 4 wavelengths. Much less is known about the
real part of the refractive index. Aluminum has been
studied extensively throughout the spectrum,22 and
some data have been obtained on several materials from
(11)
is the (effective)
number of electrons/cm3 , and re =e 2 /mc 2 is the classical
radius of the electron.] More information would clearly
be desirable. It is to be noted, however, that optical
properties of thin films often vary depending on the
process of fabrication and on surface impurities; therefore, some experimentation may be needed in any
specific case.
In Table II, we present data on some materials that
seem suitable in the wavelength range from 1 to 800 A.
Table II is by no means complete. It is meant to illustrate, however, that materials exist with <710.2, which
will provide the necessary phase shift with a thickness
in the 0.07-3.2-jim range, in the wavelength interval
considered.
B. Construction Considerations
0.3
.-
0
so
Fresnel zone plates for use with visible light have
traditionally been made by photography, with a photoemulsion supported on an optically flat glass plate.', 3
For use with ultraviolet and x rays, however, the zone
plate has to be on a substrate that transmits the incident
radiation without serious loss of flux. Ideally there
would be no substrate at all; the zone plate designed
0.2
'i
0.1
0
0
0.2
0.4
0.6
0.8
1.0
11
Fig. 3. Optical performance of zone plate as a function of n with
phase shifter set for maximum image flux (Fig, 2). The fraction
of incident flux in the primary image (solid curve) and in the undiffracted zero-order background (dashed curve) are shown, as
well as the fraction absorbed by the zone-plate material. The
points q= co correspond to conventional Fresnel zone plate with
opaque zones. The Rayleigh-Wood zone plate has -q= 0.
by Baez' is in fact of this nature. The open zones are
literally open, except for the four or five radial struts
that hold the absorbing rings in place. Plates of this
design are commercially available,2 8 and have formed the
basis of some further development over the past
decade.29 Construction of a self-supporting plate with a
different design is described by Bol Raap et al.30 These
designs are suitable for our phase-reversal approach in
PHASE
March 1974
ZONE PLATES
TABLE
Material and
wavelength
range
Au
1.1-2.0
FOR X RAYS AND euv
305
II. Optical properties of materials for zone-plate construction.'
k
X
(A)
6
6
SX10
1.2
1.5
2.0
kX1O
11=-
lb
a
(jUn)
Comments
30
45
81
2.1
5
13
0.07
0.11
0.16
2.0
1.6
1.2
8 from Eq. (I1),
k from Ref. 22
2
3
4
5
41
92
164
256
1.5
6.6
20
43
7.5
10
15
20
25
121
216
480
860
1350
1.4
4.5
23
70
170
2.4
1.6
1.2
0.9
3.1
2.3
1.5
1.1
0.9
a from Eq. (11)0
Be
7-25 A
0.04
0.07
0.12
0.17
0.01
0.02
0.05
0.08
0.12
Al
8-25 A
10
15
20
25
23.6
31.4
310
700
1250
1950
14
64
185
410
0.05
0.09
0.15
0.21
1.6
1.0
0.74
0.58
8 from Eq. (11),
k from Ref. 23
1140
1860
0.10
0.11
1.0
0.8
a and k from
0.05
0.10
0.18
0.71
0.35
0.24
a and k from
0.19
0.090
0.090
0.32
0.12
0.08
8 and k from
Ref. 25
A
Cu
1.5-5 A
LiF
20-40 A
Polystyrene
44-160A
Bed
130-300 A
Al
300-800 A
67
113
150
4700
15 400
27 000
110
200
230
1540
5000
300
500
700
42 000
200 000
450 000
8200
17 0000
40 0000
Uncertainties of 8 are typically
k from Ref. 22
8 from (Eq. 11),
k from Ref. 23
Ref. 26
Ref. 26
415%, of k410-40%.
b Thickness for maximum image flux.
¢ To find 8 from Eq. (11), we used Zeff=Z
d Be is probably suitable in the 130-300-A
for X below the K edge, Zeff=Z-2 above.
range, although no consistent set of optical constants has been found in the literature for
this wavelength interval.
I These values correspond to pure aluminum. For material with oxidized surfaces, they have to be increased by a factor of about
1.6-1.8.
those regions of the spectrum where the material and
the thickness required provide enough rigidity for
self-support.
In many regions of the spectrum, the parameters
given in Table II result in structures with insufficient
rigidity of their own. In these cases, the zone plates
must be prepared by electroplating or evaporating onto
a suitable substrate. This technique has been used with
success by Schmahl and Rudolph.'" In Table III, we
present a list of some materials that may be useful as
TABLE
substrates in
complete lists
To form the
similar to the
the 1-300-A wavelength range. Moreare available in the literature.2 4
zones would require the use of techniques
ones developed for constructing conven-
tional Fresnel zone plates. The necessary pattern may be
created in photoresist by contact printing a zone mask
on it, then developing and etching. The electron-optical
technique developed by the TUbingen group29 is another
way to copy or reduce the scale of a high-quality
master.
III. Transmittance of substrate materials.3
Transmittance
Material
Au
Cu
Be
Al
Polystyrene
Other Plastics
Based on Refs. 22-24.
Wavelength
range (A)
through
1000-4 layer
Comments
<2
<5
<27
>90%
>90%
>90%
< 20
> 90%
170-300
>50%
Transmits >70% for 170<X<400A
>90%
>80%
if protected from oxidation
Other hydrocarbon plastics behave similarly
E.g., formvar, collodion, zapon, mylar
44-90
44-80
Transmits>70% for X<40A
Probably also transmits >70% for 115<X<300 A
if protected from oxidation
JANOS KIRZ
306
Vol. 64
Optical techniques8", that can produce plates with
many zones are certainly applicable, although the
resulting zone profiles will not necessarily be rectangular
in cross section. The effects of such deviations from the
performance
model that we have been considering will be examined
will then
in the next section.
The radius of the central zone is simply determined by
the wavelength, and the desired focal length: From Eq.
(1), r1~ (fX)i. The maximum number of zones, however,
depends on the resolution of the construction process.
If the narrowest ring that can be formed has a width
Ar, then the maximum number of zones in the plate
Within our pair of zones (corresponding to an interval
will be (assuming
hnax<
f/X)
generality to give us some insight into the optical
to be expected in a wide variety of cases.
We shall start by writing down the expressions for the
amplitudes in the more-general case of Fig. 4(b), and
consider
some
interesting
special
cases.
of 27rin 0), we distinguish four regions.
Region I is the open region. (0<0<01), with ,=0.
Region III is the shifting region, with a phase shift
of 00,and amplitude attenuation ergo
(02<0<03).
Regions II and IV are transition regions; for region
II, 01<0<02, and 4= (Oo/Ca)(0-01),whereas for region
IV, 03a<0<27r,and ' = (Oo/a)(27r-0), where a =02-0
-03, is the width of the transition regions.
Remembering that for our materials the phase is
retarded with respect to the vacuum, we can write for
the amplitude at the primary image
=27r
fax
4(Ar)2
For X = 50 A, f = 20 cm, and Ar =1 ,m, zone plates with
250 zones can be formed. This zone plate is 1 mm in
C Jf
over-all diameter. Because of the quadratic dependence
on Ar, a modest improvement
in technique
A 1=-
le-'7
M1
°ei[°-0t°OUd.
(12)
27r J o
results in
significant gains that are especially important for use at
the shorter wavelengths, or with shorter focal lengths.
The image flux is porportional to Hmax, whereas the
angular resolution is about 1.22 Ar/f.
Whatever the process of fabrication, the thickness
and the uniformity of thickness of the zone material
has to be carefully controlled. To provide a quantitative
guide to the tolerances in the construction process, we
2
The integration gives
C
a2
Ai=-e
i(61/2) 1-A
I
2
e
a +40o(n7+j)2
\
a
0,
0
X [sin- - eh0o07+0 sins+ -)
shall turn now to a discussion of the effects of imper-
fections on optical performance.
+
ZONE PLATES WITHI- DISTORTED
pPI10FILES
The zone plates discussed so far had open and shifting
zones with sharply defined boundaries between these at
the radii r., given by Eqs. (1) or (4). As already mentioned, the open and shifting zones are then nearly
equal in area.
In this section, we shall consider zone plates with
more-general profiles. We continue to insist that the
pattern repeat for all zone pairs, but we allow the boundary within the zone pair to be in an arbitrary position
[Fig. 4(a)]. We shall also allow the transition from open
to shifting regions to take place over finite transition
regions [Fig. 4(b)]. These considerations will allow us
to examine the effects of imperfections in zone formation. They will also lead us to the design of zone plates
with the undiffracted zero-order background entirely
eliminated.
The flux distribution from zone plates with moregeneral transmission profiles (but no phase shift) has
been obtained by Waldman," Markus,"0 and by Horman
and Chau.lU The distributioll from zone plaLes witlh more
general phase-shift profiles has been obtained by
Dammann."5 Our choice of a profile made up of linear
segments simplifies the algebra, but retains enough
ako(q+i)
F
al2+¢,2(,q+i)2L
0,
[Cos-_e-o(0+i)cos+Ž
2
)]}. (13)
The undiffracted, zero-order amplitude Ao is given by
Ao=-
c
I
r
e-1(G)e-if(O)dO
2rJo
2a
C
(1-e-fo(0X+i)
02+
27r
0o(07+i)
+2(7r-a)e-0-0)+0 |.
(14)
The flux absorbed is given by
C
2
2
7r
abs=-
27r C
=
C2
l
(--
1(0))d
(1-e-2
(j-e-2q00)
e-2
7r
i'_700
170l-)]
l-
(15)
2,70o
These expressions become quite simple for the case
of rectangular profiles. In that case a =0, and we get,
ZONE PLATES
PHASE
March 1974
FOR X RAYS AND euv
307
(a)
shifted
'
o
.~ ---.
!~
0.4
(c)
0.4 -
*0
0.2
0.2
open
0
open
I
0,
2n
T
0
0.8,
n
1.2,
0
'..
0.8, n
.
0.2 -
0.8n
1.2n
n
. .2n
01 +2r
01
5. Optical performance of zone plate with rectangular
profile as a function of the boundary position 01between open and
shifting areas (see inset). Fraction of incident flux in primary
image (a), in undiffracted background (b), and in zone-plate
absorption (c) are shown for oo=qtkopt. The curves marked n==
refer to Fresnel zone plates with opaque zones, 7=0 refers to
Rayleigh-Wood design.
FIG.
(b)
a
a
0
0
ground in the primary-image plane. Some of these can
be reduced or eliminated with the aid of stops or apertures in specific applications.
Let us allow now the transition from open to shifting
I
I
8,
82 "
03
2Tr
8t+2n
e
FIG. 4. Zone profiles. The thickness is shown in terms of the
phase shift 4. (a) Rectangular profile. Oo=o7r
corresponds to conventional zone plate with equal open and shifting zone areas.
(b) Distorted profile with sloping transition regions II and IV
between open (I) and shifting (III) areas.
for the flux in the primary image,
zones to be continuous. As long as the slopes are steep,
the general features of optical performance change little.
For a/37r<0.1 and q<0. 2 , the curves of Fig. 5 and 6
would have to be shifted by less than 0.01 scale unitsa hardly noticeable amount.
Zone plates with more-gently sloping profiles such as
may result from optical fabrication are inferior in their
C2
0,
performance.
To illustrate the general trend, we present
I,=
2A,!'
+- sin2-(1+e-2'7o-2e-'?o cosqo). (16)
in Fig. 8 the optimal phase shift, and in Fig. 9 the flux
7r2
2
as a function of a for the case of equal open and shifting
Note that the only difference between this expression zones (01+a=7r).
and our previous result Eq. (7) is the factor sin2(01/2).
All of the zone profiles that we have considered so far
As expected, the flux is greatest for the conventional have real and virtual images with identical radiances.
choice of boundaries (01=7r). It is also apparent from
This identity results from the symmetry of the transiEq. (16) that we have already found the optimum value tion regions. If, in Fig. 4(b), we increase region II with
of 00 for any value of 01 by solving Eq. (8).
respect to region IV, the radiance of the real image
We shall first examine the effect of small departures
increases relative to the virtual image, and vice versa.
from the optimized profile. In Fig. 5, we show the flux
in the primary image, in the zero-order background, and
the absorbed
flux as a function
of 01, the boundary
position between open and shifting regions. In Fig. 6
these quantities are presented as a function of the thick-
(We assume here that the real part
of the index of
refraction is less than 1; otherwise the converse is true.)
A special case of this asymmetric
arrangement
with
,q=0, 0o=27r, 01=0, 02=22r is the phase plate,3" phase
ness of the shifting material. As is evident from the
figures, the image flux is relatively insensitive to small
changes of these variables. It may therefore be desirable
I
I
.
I
I (a)
II
I
I.,
0.6-
(C
in many applications to choose values of 01 and qkothat
minimize the background,
at a small cost of image
irradiance. In fact, the zero-order flux can be eliminated
entirely by choosing 'o=7r, and
01
2e-1r
0.4 -
0.2 0
10
o
0 .8
I
n
..
.
M.
1.0
1
.2
0.8
1.0
1.2
(17)
7r
l+e-4t
The image flux and absorption obtained with this choice
of parameters are shown in Fig. 7.
Although, in this design, there is no undiffracted
background, the contribution from higher-order and
virtual images will still constitute an undesirable back-
FIG. 6. Optical performance of zone plate with rectangular
profile as a function of the thickness of the shifting layer in units
of tkopt (see inset). Fraction of incident flux in primary image (a),
in undiffracted background (b), and in zone-plate absorption (c)
are shown for equal open and phase-shifted areas (0,= ir). The
curves marked 77= -o refer to the Fresnel zone plates with opaque
zones, 7= 0 refers to Rayleigh-Wood design.
JANOS KIRZ
308
1.0
*-,
.-
0.8 1
U
0.5
rr
~I .
'
Vol. 64
0.4
0.81T
0.6
0.6T
C
0.3
J
absorbed_..
0
-imaged
0.4
U
.-
0.2 [
0
I
I
0.1
0.2
Fresnel lenss, or Kinoform lens,33 with which all of the
incident flulxgoes into one primary image. Properties of
zone plates ; with various profiles approximating this
ideal have' been discussed by Dammann."5
1ARY AND FURTHER COMMENTS
We have demonstrated that the choice of materials
for zone-phateceliminated
construction is governed by the parameter
,
\
1.4w
,
.
,
.
/must
/.
0
T
0.2T
0.6T
0.4OT
0.8Tr
Tr
Ca
Fig. 7. Characteristics of zone plate with rectangular profile,
designed to eliminate zero-order undiffracted background, as a
function of q. The curves represent the position of the zone
boundary (01), the fraction of the incident flux in the primary
image (imaged) and in zone plate absorption (absorbed). For this
design, ko= 7r.
SUMI
0.1
0
11
k
0.2
0
0
cU
0
-`
FIG. 9. Fraction of the incident flux in the primary image (solid
curves), in the undiffracted background (dashed curves), and in
zone-plate absorption (dotted curves) for various values of 71for
zone plates with sloping profiles. Profile shape and phase thickness
are shown in Fig. 8. (Absorption is 0 for q=O.)
-q. Almost throughout the wavelength range 1-800 A
there are materials for which q<0. 2 ; with such materials
it is possible to approach the optical performance of the
Rayleigh-Wood phase-reversal zone plate.
Compared to the conventional Fresnel zone plate
with opaque zones, the image flux is improved by a
factor between 2 and 4. The undiffracted zero-order
background can be reduced by a factor of at least 4, or
entirely at a small cost of image flux. The
energy absorbed by the zone plate depends strongly on
,q. For -q=0.2, the absorption is reduced by only 30%;
however, for q=0.05, the reduction is 75%.
We have also shown that the optical performance of
our zone plate will not deteriorate substantially if the
zones are formed with somewhat distorted profiles.
To obtain good imaging characteristics, the zone plate
be used with a monochromatic or nearly monochromatic
a 2Tr
source. When used with a wavelength
dif-
ferent from the one for which it was designed, the image
will suffer both from chromatic and spherical aberrations,"' owing to the way in which our expression for
zone radii Eq. (4) depends on f and X. In addition, for
the phase-reversal zone plate, the optimal zone thick-
1.2w
w
ness is also a function of X, and broad-band
1 -0.2
illumination
will therefore inevitably result in some undiffracted
background. Nevertheless, it has been demonstrated" 6
0.8w
0
0.2w
0.4w
0.6M
0.8n
l
Ca
+o,
FIG. 8. PIaase thickness
giving maximum flux in primary
image of zolne plate with sloping transition regions (see inset).
The profile iis rectangular for a = 0, and triangularfor a = 7r. The
fully open, and fully shifted regions are of the same width
(O+aw=7r).
that Fresnel zone plates are useful with sources of
continuous but peaked or filtered spectra. In such
applications, phase-reversal zone plates will lead to
significant improvement.
Although the narrowness of the zone-plate bandwidth
is certainly a disadvantage in some applications, it can
also be exploited to isolate pre-selected spectral lines.9
March 1974
PHASE
ZONE PLATES
7 30
In addition to their use in x-ray telescopes, ' zoneplates have been suggested for use in microscopes and
use. In all of
monochromators for vacuum-ultraviolet
these applications, phase reversal will be useful. Moreover, in linear devices such as transmission gratings for
use in the x-ray region,34 phase reversal may be used to
improve the efficiency over a limited spectral range.
Although we have been emphasizing applications in
the uv and x-ray regions of the electromagnetic spectrum, all the discussions and results concerning zoneplate performance and distortions have a much wider
range of validity. As long as reflections at zone-plate
is the case if the index
surfaces may be neglected-as
of refraction is near 1.0-these
results apply equally
well to the use of zone plates with visible light, micro-
waves, and all other electromagnetic radiation, and to
their possible use with thermal neutrons, sound, and
other wave phenomena as well.
ACKNOWLEDGMENTS
This investigation grew out of a conversation with
Dr. David Sayre. His continued interest and useful
comments are greatfully acknowledged. It is a pleasure
to thank Professor Luis W. Alvarez for his encourage-
ment and advice, and Professor Dorothy C. Hodgkin
for her kind hospitality in Oxford.
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