Download Focusing X-Ray Beams to Nanometer Dimensions

Focusing X-Ray Beams to Nanometer Dimensions
C. Bergemann1, H. Keymeulen2, and J.F. van der Veen1,2
für Festkörperphysik, ETH-Zürich, Switzerland
2
Paul Scherrer Institut, Villigen, Switzerland
1Laboratorium
θi
ϑ
r
r
φ
z
φ
z
φ0
y
y
x
Fig. 1: (a) planar wedge, (b) hollow circular capillary. Angle and distances
are not to scale.
The propagation of e.m. radiation through the wedge is described by the Helmholtz
equation
∆u + n2k 2u = 0,
(a)
0
(1)
where u is the z-component of the electric field E = (0, 0, Ez). Assuming u = 0 at the
boundary we find for the mth eigenmode of Eq. (1) in polar coordinates:
um (r,φ) = Hm(1π) / φ0 (kr) sin(mπφ / φ0 ),
(2)
(1)
where the Hankel function H mπ / φ0 of the first kind can be expanded into
H m(1π) / φ0 ( kr ) ~

2
m 2π 2 mπ 2 π 
exp i  kr + 2 −
− .
πkr
2
φ0 kr 2φ0 4 

(b)
Incident mode TE #
1
2
3
4
5
0
8
7
6
0.10
5
4
3
0.05
2
1
0
0.00
0.00
(c)
0.02
Exit angle (degrees)
Because θ c ∝ λ , Wc is independent of λ . To a good approximation: Wc ≈ 30 nm / ρ ,
with ρ the density of the confining material in g/cm3. The lowest mode remains
bound within the wedge down to zero gap, but its evanescent amplitude within the
confining material diverges as W → 0, see Fig. 2a. The beam can be focused down
to a FWHM spot size of 0.64Wc, which for SiO2 equals ~13 nm. Cladding the SiO2
surface with e.g. a layer of Au would reduce Wc to 8 nm.
The focusing of an X-ray beam is equivalent to confining a quantum wavefunction
inside a potential well of maximum height V0 ≡ . In analogy with the uncertainty
principle, ~Wc can be recognized as the minimum focus size.
2
3
4
5
1
2
3
4
5
0.15
8
7
6
5
4
3
0.5
2
1
0
0.0
0.1
0.2
0.3
Incident angle (Degrees)
(d)
Incident mode TE #
6
0
8
1
2
3
4
5
.015
7
6
0.10
5
6
.010
5
4
4
3
0.05
2
3
.005
2
1
1
0
0.01
0.02
0.03
6
8
7
0.00
0.00
6
1.0
0.03
Incident mode TE #
0
In reality, u ≠ 0 in the confining material, since the refractive index n =1 − δ
inside the wedge deviates only slightly from unity. From the phase in Eq. (3), the
local radial wavevector is k (1 − m2π 2 / 2φ02k 2r 2 ).When this becomes comparable to
the wavevector k (1 − δ ) in the medium, the waves start to leak out of the waveguide,
preventing further focusing. This happens at m times the critical gap width Wc = λ / 2θ c ,
with θ c = 2δ the critical angle for total reflection.
1
1.5
0.0
0.01
Incident angle (Degrees)
(3)
Incident mode TE #
6
0.15
Exit mode TE #
(b)
x
Exit mode TE #
(a)
Exit
angle
(degrees)
Exit
mode
TE #
ϑi
Generally, the incident wave excites several modes m in the waveguide, and the mode
distribution is found by matching the amplitude of the incoming wavefield to that of a
linear combination of eigenmodes. Even when the standing wave perfectly fits within
the entrance gap, a phase difference between the cylindrically curved wavefront of
the wedge eigenmodes and the plane incident waves will remain (Fig. 1a). This gives
rise to mode mixing and interference. The magnitude of the phase mismatch can be
expressed by the parameter ξ ≡ kriφ 02 / π . Of particular interest is the case ξ ≈ 1,
where a few excited modes are interfering to form a pattern of strong localised
maxima. Fig. 2c shows for the TE2 input mode (m = 3) how such mode interference
can be exploited to achieve a small spot of ~20 nm width without having to produce a
small exit gap. For ξ > 1, interference between many modes generally results in a
complicated intensity pattern, see Fig. 2d.
For higher input modes m, the influence of the evanescent waves becomes more
significant, since the critical gap width mWc is much larger. In Fig. 2e a virtually pure
TE7 mode is excited at the entrance: essentially all radiation intensity disappears out
of the gap at W ≈ (m − 1)Wc . The wedge therefore acts as a low-pass mode filter.
The field profile across the exit plane of the wedge gives rise to a distinct far-field
pattern and this is how the predicted beam evolution can be verified. The diffracted
intensity distribution I (θi ,θ e ) has been measured as a function of θi and the exit angle
θe [1]. The diffraction includes a post-reflection of the exiting waves from the
extended lower plate [2]. Fig. 3a shows the measured distribution. Diffraction
patterns have been calculated for ξ = 0.30, 0.94 and 3.00, and only the one at ξ = 0.94
(Fig. 3c) resembles the experimental one.
Exit angle (degrees)
At synchrotron radiation sources, focusing devices such as refractive lenses, Fresnel
zone plates, KB-mirrors and tapered capillaries are in use. What is the smallest spot
size that these devices can reach?
Consider compression of the beam in a planar wedge, see Fig. 1a. In our
experimental setup the wedge has an angle φ0 = 7.71×10−5, a length of 4.85 mm, and
gap widths of W1 = 538 nm and W2 = 164 nm at its entrance and exit. A prereflection of the incoming plane wave results in a standing wavefield that can be
made to match the gap width at the entrance.
0
.000
0.000
Incident angle (Degrees)
0.001
0.002
0.003
Incident angle (Degrees)
Fig. 3: Logarithmic intensity plots of the intensity I (θi ,θ e ) diffracted from the exit
of a planar SiO2 wedge, with λ = 0.093 nm: (b) - (d) are calculations for geometries
similar to the respective panels of Fig. 2; (a) shows the measured intensity
distribution, to be compared with (c). The intensity distributions along the vertical
lines labelled as incident mode #2 are the diffracted fields for the TE2 input mode as
shown in Figs 2b-d.
The theory for the planar wedge can readily be extended to the tapered circular capillary
(Fig. 1b). We find for its eigenmodes
ulm ( r,ϑ ,φ ) = hl(1) ( kr) ⋅ l −m J m (lϑ ) ⋅ cos(mφ ),
(1)
the
l
with h
(4)
spherical Hankel function of the first kind and Jm the Bessel function. Here,
l = jms / ϑ0 , with jms the sth zero of the Bessel function of mth order and ϑ0 the half-angle
of the cone. Mode (m, s) starts to leak out at a critical diameter ~jmsλ/ c. For the lowest
mode (0,1), this is 1.53Wc ≈ 46 nm/ ρ and the spot reaches a minimum diameter
Fig. 2: Calculated X-ray intensity distribution inside a planar SiO2 wedge, with λ =
0.093 nm and with the incident standing wave matched to various input modes, for
different values of the phase mismatch parameter ξ. (a) Evolution of input mode TE0
at low mismatch (ξ = 0.3). (b) - (d) Configuration mimicking the experiment, with a
TE2 input mode, and ϕ0 adjusted to give: (b) ξ = 0.3; (c) ξ = 0.96, close to the
experimental geometry; (d) ξ = 3.0. (e) Low-pass filter: the TE7 mode (with low
mismatch, ξ = 0.3) ceases to be transmitted and radiates out of the waveguide as W
falls below 7-8Wc.
(FWHM) of 0.74Wc.
The beam size limit ~Wc, derived here for waveguiding geometries, also applies to other
X-ray focusing devices. For example, the ultimate spot size achievable with a Fresnel
zone plate is determined by the outermost zone width. Making the latter smaller than ~Wc
causes the beam passing through it to become more delocalized. Note also, that the
expression Wc = λ / 2θ c is analogous to the expression λ / 2 NA for the diffraction limited
spatial resolution of an optical lens system, with NA the numerical aperture.
[1] M.J. Zwanenburg et al., Physica B 283, 285 (2000).
[2] M.J. Zwanenburg et al., Phys. Rev. Lett. 82, 1696 (1999).