An Exact Theory of Imaging

1
An Exact Theory of Imaging with a Parabolic
Continuously Refractive X-ray Lens
Victor G. Kohn Russian Research Center
Kurchatov Institute E-mail
kohn_at_kurm.polyn.kiae.su Internet
http//kohnvict.chat.ru
2
1996, Nature, v. 384, p. 49 A. Snigirev, V.
Kohn, I. Snigireva, B. Lengeler "A compound
refractive lens for focusing high-energy X-rays"
30 holes of 600 ?m diameter in Al show the ESRF
optics beamline (BM5) source size projection 150
?m / 20 ? 8 ?m, rd 1.8 m Energy 14 keV Top
electron microscope image Bottom focus line
profile
3
1998, Applied Optics, v.37, p.653 A. Snigirev,
V. Kohn, I. Snigireva, A. Souvorov, B.
Lengeler 200 holes of 500 ?m diameter in Al in
cross geometry (spacing 50 ?m, length 11 cm)
show 2D image of source with FWHM ? 8 ?m x 18 ?m
(V x H) ESRF, BM5, E 30 kev, r d 2.2 m
4
Microcapillary X-ray lens Yu. I. Dudchik, N. N.
Kolchevsky, et al. Nucl. Instr. Meth. A, 1999,
v.421, p.361 Rev. Sci. Instr., 1999, v.70,
p.4161 Nucl. Instr. Meth. A, 2000, v.454,
p.512 Proc. SPIE, 2001, v.4145, p.235
Air bubbles in epoxy glue inside the glass
capillary. Capillary diameter ? 0.8 0.4 mm
Design and Ray tracing

1 glass capillar, 2 epoxy glue or
glycerine, 3 air bubbles, 4 long injector
needle, 5 compressed air
Visible light microscope image of the
microcapillary x-ray lens, The diameter of the
capillary is 0.8 mm Experiment at Spring-8
shows significant aberrations.
5
Microcapillary X-ray lens Y. Kohmura, K. Okada,
T. Ishikawa, et al. Nucl. Instr. Meth. A, 2001,
467-468, p.881
Resent experiment shows a good resolution
Experiment at Spring-8 Lens of 185 air bubbles E
18 keV, F 0.48 m, M 10 Test sample of
Tantalum stripe pattern, thickness 0.5 ?m, line
0.4 ?m, space 0.4 ?m
(arrows) ? transmissivity of ? 94 Source
BL47XU undulator Detector fluorescence screen
and CCD (6 ?m pixel) Exposure time 20 ms
6
Alligator x-ray lens with variable focal
length B. Cederstrom, M. Danielsson, M.
Lundqvist, et al. Nature, 2000, v. 404, p.951
Proc. SPIE, 2001, v.4145, p.294 ( Be ) D. A.
Arms, E. M. Dufresne, N. R. Pereira, et al.
Rev. Sci. Instrum., 2002, v.73, p.1492 ( Li )
Focusing is not good. The deflection of 10 keV
narrow beam (slit) shows 3-d harmonic of 30 keV
?? ? 2N? h 0.5 mm
7
Planar parabolic refractive lenses made from
Silicon V.V. Aristov, M.V. Grigoriev, S.M.
Kuznetsov, et al. Opt. Comm., 2000, v.177,
p.33 Appl. Phys. Lett., 2000, v.77, p.4058 Proc.
SPIE, 2001, v.4145, p.285
PPRL can work with laboratory source. However,
the aperture is small ? 100 ?m.
8
Parabolic compound refractive lenses made from
Aluminium B. Lengeler, C. G. Schroer, A.
Snigirev, et al. Appl. Phys. Lett., 1999, v.74,
p.3924 J. Synchr. Rad., 1999, v.6, p.1153 - - -
many other - - - J. Synchr. Rad., 2002, v.9, p.119
Parameters of element R 0.2 mm, d 0.01
mm p 1 mm, a 1 mm Such lens is a good
imaging tool.
9
Imaging with the Lengelers lens, example 1
from "X-Ray Microscopy" Proc. 6-th Intern.
Conf., AIP Conf. Proc., 2000, v.507, p.340
X-ray micrograph of an insect, E 23.5 keV, M
12 N 62, F 1.65 m, L 1 1.79 m, L 2
21.44 m (slight defocusing)
10
Imaging with the Lengelers lens, example 2
from Nucl. Instr. Meth. A, 2001, v.467- 468,
p.966 FZP of 200 ?m diameter, 169 zones (IESS,
Italy)
E 14.4 keV, M 24, N 50, F 0.718 m, L 1
0.748 m, L 2 18 m
E 23.5 keV, M 12, N 62, F 1.65 m, L
1 1.79 m, L 2 21.44 m
11
Parabolic compound refractive lenses made from
Berillium B. Lengeler, C. G. Schroer, A.
Snigirev, et al. Proc. SPIE, 2002, 4783, p.10-18
(http//www.xray-lens.de) H. R. Beguiristain,
J. T. Cremer, M. A. Piestrup, et al. Opt. Lett.
2002, 27, N.9, p. 778-780
12
Quasi-parabolic compound lenses made of plastic
Stanford SRL V- size from 440 to 27.5
?m Distance is smaller than focal distance
M.A. Piestrup, J.T. Cremer, H.R. Beguiristain, et
al. Rev. Sci. Instrum., 2000, v.71, p.4375 Y.
Ohishi, A.Q.R. Baron, T. Ishikawa, et al. Nucl.
Instr. Meth. A, 2001, v.467-468, p.962 ?. ?.
????????, ?. ?. ??????????, ?. ?. ???, ?. ?.
?????????, et al. ???????? ??? ????????????
????????, ???-6524/14, 2002
reduce beam size for high-presure experiments
gain 12 spot size 100 ?m
V-size 76 ?m
13
Parabolic equation and parabolic medium
single lens ? compound lens ? continuous
lens
Wave field of radiation has slowly varying
transverse part
which is a solution of parabolic equation
14
Analytical formula for a long PCR lens propagator
Empty space (Kirchhoff propagator)
Thin (short) lens (transmission function)
Long lens new propagator ( JETP Letters,
2002, v. 76, p. 701 rus)
v. 76, p. 600 eng
15
Analytical formula for the image propagator

JETP, 2003, v. 124, p. 224 rus
v. 97, p. 204 eng
The integral is not a convolution
16
Focused imaging of micro-objects, new lens formula
Parameters of image propagator can be written in
a standard form
through generalized distances
Focused image condition
Neglecting absorption one has
17
Ray Tracing plano-concave lens (plane wave
focusing)
Parabolic profile of the boundary can not focus
rays for all aperture. Various parts of aperture
focus rays on different distances. Dispersion of
focus points L
18
Ray Tracing bi-concave lens (plane wave
focusing)
19
Ray Tracing bi-concave lens (imaging)
The lens formula is met for a central part of the
lens aperture . However the rays near the edge
of aperture are not focused.
20
Ray Tracing PCR lens (imaging)
The lens with a proper curvature radius R or a
critical length L? has no aberration and
focus point to point for any deviation.
21
Ray Tracing PCR lens (the rays trajectories)
?he ray trajectory in a paraxial approximation is
described by a simple equation
The lens looks like a wave guide. It can focus x
rays many time inside itself.
The solution is simply obtained
22
PCR lens effective aperture and linear size of
focus
The lens restricts a front of plane wave due to
effective aperture and focuses all the
transmitted wave on the distance
The intensity peak has Gaussian shape
The effective aperture is equal to the integral
intensity. The linear size of the focus spot is
FWHM of the intensity peak
23
Computer simulation of images with PCR-lens
The program was elaborated as the Igor-Pro
application. It allows one to calculate the
images of some simple objects within the standard
experimental setup which include distance from
source to object rs distance from object to
lens ro distance from lens to 2-D detector ri
Igor-Pro is an interactive environment for
preparing the input data, calculation and
graphics, made by WaveMetrics company.
As an example of program possibilities,
the images of the small hole of 3 µm diameter in
the Si plate of 3 µm thickness is considered. The
hole is located at different places respective
to the lens centre. The energy E 25 keV, the
phase shift -0.294, absorption coeff. 0.001,
rs 40 m
Si
24
x, y (Fmin, Fmax)
100 Lengelers lenses L 10 cm, F 115.7
cm, FL 117.4 cm ri ro 229.23 cm E 25
kev rs 40 m Resolution 0.4 µm Such lens is
used in experiments
25
x, y (Fmin, Fmax)
400 Lengelers lenses L 40 cm, F 28.925
cm, FL 36.85 cm ri ro 51.024 cm E 25
kev rs 40 m Resolution 0.2 µm Such lens is
not used in experiments
26
x, y (Fmin, Fmax)
400 Lengelers lenses L 40 cm, F 28.925
cm, FL 36.85 cm ri ro 51.024 cm E 25
kev rs 40 m The same lens and the Same
experiment condition as in the previous
slide, but the object is Si sphere of 3
mcm diameter. The contrast is reverse at the
boundary of aperture and smaller. At the centre
a difference is small.
27
The main reason of focused imaging transparent
object is absorption inside the long (thick)
x-ray lens. The object change the ray
trajectories due to refraction. Different
trajectories have different ray path inside the
lens where absorption is not negligible.
28
Conclusion
1. Long parabolic compound refractive lens is a
useful tool for x-ray optical imaging of
micro-objects 2. Such lenses have no aberration
and image point to small spot 3. The theory of
long compound lens is developed completely 4.
Standard lens formula exists in terms of
generalized distances 5. The aperture and focus
spot decrease with increasing lens length,
but a ratio resolution/aperture ß/d 6.
Completely transparent objects can be imaged if
it produces a phase gradient. Even a sign of
phase gradient is visible 7. Large intensity
gain is under question due to many dams (thin
parts between concave boundaries) which absorb
x-ray radiation.