X-ray waveguiding studies of ordering phenomena in confined fluids - Chapter 5 Focusing of coherent x rays with tapered waveguides

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X-ray waveguiding studies of ordering phenomena in confined fluids

Zwanenburg, M.J.

Publication date

2001

Link to publication

Citation for published version (APA):

Zwanenburg, M. J. (2001). X-ray waveguiding studies of ordering phenomena in confined

fluids.

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Chapterr 5

Focusingg of coherent x rays with

taperedd waveguides

FocusingFocusing of x rays in one dimension is achieved in a planar waveguide with linear taper.taper. The waveguide consists of two plates with a variable tilt angle and air gap betweenbetween them. Compression of the beam and coherent mode coupling inside the waveguidewaveguide results in a line focus of 26 nm height at the exit.

5.11 Introduction

Inn the past decade various optical devices have been developed for the focusing of xx rays. These developments largely took place at synchrotron radiation sources, becausee the brilliance of these sources generally made such investments pay off. Focusingg optics for hard x rays include curved crystals and mirrors [45], Fresnel andd Bragg-Fresnel zone plates [46], refractive lenses [47], capillary optics [48] and planarr waveguides in combination with a resonant beam coupler [39, 49]. These devicess either focus the beam in one or in two dimensions. Spot sizes down to a feww hundred nanometer can be obtained. Applications of focusing optics include tracee element mapping on small samples using x-ray fluorescence, microdiffraction andd phase contrast microscopy.

Thee high brilliance of third-generation synchrotron radiation sources enables extractionn of a transversely coherent beam of sufficiently high intensity that stud-iess of speckle patterns [50] and x-ray photon correlation spectroscopy (XPCS) [51]

experimentss can be performed. Usually, a coherent beam is selected with a pinhole att some distance from the sample. Diffraction at the pinhole opening then results inn a broadened beam at the sample position. For some applications one would like too select a coherent beam and subsequently compress it to nanometer dimensions att the sample position, thus allowing for, e.g., XPCS or phase contrast imaging studiess [52] on a very small object. Here we present a tapered planar waveguide withh which we obtained a line focus of 26 nm height at the waveguide exit. Inter-ferencee between local modes of the waveguide gives rise to an intensity maximum off a width substantially smaller than the gap width at the exit. Hence, the ac-tuall intensity gain is higher than that resulting from compression of an incoherent beamm by the same device.

Sectionn 5.2 discusses the propagation of modes through the tapered waveguide andd section 5.3 describes measurements of the angular distribution of intensity exitingg the waveguide, confirming the coherent focusing properties of the device.

5.22 Mode propagation in tapered planar

wave-guide e

Considerr a planar waveguide with tunable air gap (see chapter 3), accepting a highlyy collimated synchrotron x-ray beam as shown in Fig. 5.1a. The horizontal polarizationn direction of the beam is parallel to the plane of the waveguide. With thee plates set parallel , the principle of operation of the waveguide is as follows. Interferencee between the incident plane waves and the waves reflected from the bottomm plate in front of the waveguide yields a sinusoidal standing wave pattern. Forr specific angles of incidence 0», a node of the standing wave coincides with the surfacee of the upper reflecting plate. For these angles the standing wave matches preciselyy the gap width and the wave becomes a propagating transverse electric (TE)) mode of the waveguide. The incidence angle corresponding to excitation of modee TE™ equals 0* = 9m = (m + l)X/2W. Here, A is the photon wavelength

andd W the distance between the plates. The integer mode number m refers to thee number of nodes in the gap between the plates, not counting the nodes at thee plate surfaces. In the ray analogue of waveguiding, the 'mode angle' 6m is

FocusingFocusing of coherent x rays with tapered waveguides 89 9

Figuree 5.1 Schematic of the waveguide set-up and the scattering geometry. The

waveguidewaveguide is shown in (a) with the plates set parallel, and in (b) with the upper plateplate tilted. For illustration, the field profiles of the TE2 mode at the entrance andand exit planes are sketched. In a parallel plate geometry, the modes propagate

undisturbed.undisturbed. In the tilted geometry, the exiting field profile is much different from thethe profile at the entrance. Angles and distances are not to scale.

waveguidee with parallel plates, a single mode which starts at the entrance passes downn the waveguide undisturbed and emerges at the exit with the same sinusoidal fieldfield distribution (Fig. 5.1a). One identifies the field distribution across the exit gapp by measuring the angular distribution of intensity in the far field, as described inn the previous chapters.

Noww assume that the upper plate is tilted downward in the direction of propa-gationn by an angle 9t, see Fig. 5.1b. The plate distance W decreases linearly with

thee coordinate z along the propagation direction:

W{z)W{z) = W1-9t-z, (5.1)

wheree W\ is the gap width at the entrance. The gap width at the exit is given byby W2 = Wi — 6t L, with L the length of the waveguide. The wavefield of a

specificc TEm mode, excited at the entrance, no longer matches the smaller gap

thee spacing between nodes of the wavefield, the wave field is simply compressed andd retains approximately its sinusoidal form. If, however, the change in gap width iss much larger, scattering also occurs into other local modes. The resulting electric fieldfield amplitude \I>(x, z) = Ey(x,z) is skew along the transverse coordinate x and

inn general is a complicated function of the coordinates (x, z) within the waveguide. Thee field amplitude is most easily found by numerically solving the scalar wave equationn (see chapter 3)

V2*(x,, z) + k$n(x, z)2V(x, z) = 0, (5.2) withh V2 = d2/dx2 + d2/dz2, ko = 2n/X the wave number and with \I>(x, z)

satis-fyingg the boundary condition

tf(0,z)tf(0,z) = 0 f o r 0 < z < o o ,

V(W{z),z)V(W{z),z) = 0 f o r 0 < z < L . (5.3) Thesee boundary conditions can be applied, provided the evanescent field penetrates

thee plate material over a depth negligibly small compared to the gap width, which iss the case for mode angles 9m not too close to the critical angle for total reflection.

Forr the case of zero tilt angle (W^ = Wi), the TEm modes

tt(x,tt(x, z) = sm(ko9mx)e-i0mZ (5.4)

aree solutions of Eq. (5.2). Here, /3m = kocos9m is the propagation constant and

99mm = (m + l)n/koWi. For non-zero tilt angle, the solutions of Eq. (5.2) have

aa more complex form. Rather than searching for analytical solutions, we solved Eq.. (5.2) numerically by use of a finite difference beam propagation method (FD-BPM)) as described in chapter 2. The input wave field at the entrance is a single TEmm mode which matches the local gap width W\:

*(x,0)) = sm(kadmx). (5.5)

Ass an example we show how a TE2 input mode propagates in a waveguide of length LL = 4.85 mm, with a gap width decreasing from W\ = 538 nm at the entrance to WW22 = 164 nm at the exit. This corresponds with a tilt angle 6t = 0.00442°. We

performedd the calculations assuming a wavelength A = 0.093 nm. These values correspondd to our experimental conditions, see section 5.3. A contour plot of the

FocusingFocusing of coherent x rays with tapered waveguides 91 1

00 1 2 3 4 5

zz (mm)

Figuree 5.2 Distribution of the field intensity within the waveguide, numerically

calculatedcalculated for X = 0.093 nm, W\=538 nm, W%= 164 nm and L = 4-85 mm. As inputinput mode, the TE% mode was chosen (0i= 0.0149°). Note that distances along thethe horizontal axis are given in millimeters and along the vertical axis in nanome-ters. nanome-ters.

calculatedd intensity I(x, z) = \^(x, z)\ within the waveguide is shown in Fig. 5.2. Nearr the entrance, the nodes and antinodes of the starting TE2 mode are clearly

seen.. Further downstream, the node spacing is seen to be compressed in proportion withh the decreasing gap width. However, the intensities in the antinodes become unevenlyy distributed across the gap, oscillating between the upper and lower plate withh ever growing amplitude and shortening period. These oscillations are the resultt of interference between waves which are reflected from the upper and lower platess at larger and larger angles. In the calculation, the mode angle at the entrancee is ö2 = 0.0149°. In the ray analogue, the number of bounces from the

platess equals ~ (6m + 8t) L/W2 ~ 9.

Returningg to the contour plot of Fig. 5.2, we observe a strong intensity maxi-mumm at 0.1 mm beyond the exit plane. This maximum, which is 6 times stronger thann the intensity in an antinode of the TE2 input mode, may serve as a focus

forr scattering experiments on a small sample positioned just behind the waveguide exit.. The intensity gain is higher than the gain resulting from beam compression alone,, which is simply the ratio between the gap widths at the entrance and the

exitt (W1/W2 = 3.28). In the example given, the antinode of the compressed wave fieldd which is closest to the bottom plate is located 26 nm above the bottom plate andd has full-width-at-half-maxima of 24 nm along the ;e-axis and 0.15 mm along thee 2-axis. Two less intense neighbouring maxima lie upstream, at a distance of 0.111 mm and 0.22 mm along the z-axis. If required, the position and the spacing off the maxima along the z-axis can be modified by changing the waveguide length orr the tilt angle or by choosing a different input mode. For example, lowering thee input mode while keeping the other parameters constant will result in a larger spacingg along z, at the cost of a larger focus width along the x-direction.

Thee wave field can be compressed further by reducing the width of the exit gap. Forr a given entrance gap width, a limit to the focus size at the exit is eventually set byy the bouncing angle becoming larger than the critical angle for total reflection, inn which case the local modes become radiation modes. Of course, this limit can bee overcome by reducing also the entrance gap width while keeping the tilt angle thee same, but then the beam intensity accepted by the device is correspondingly reduced. .

5.33 M e a s u r e m e n t of waveguiding p r o p e r t i e s

Wee investigate the propagation of modes in the tapered waveguide by measuring, forr a large set of incidence angles 0j, the intensity I(9i,9e) emerging from the

waveguidee as a function of exit angle 9e. We compare the measured intensity

distributionn with the calculated one using the Fourier transform 2 2

WMWM =

PW2 PW2

II ^ (x,L) sin (k9ex)dx

Jo Jo (5.6) )

Here,, R is the distance from the waveguide exit to the detector and ^(x, L) is the numericallyy calculated field profile at the exit, corresponding to the input wave fieldd ^(x, 0) = sm(ko9ix). (Note, that 9i may deviate from an input mode angle 99mm).). The comparison between measured and calculated intensity distributions

formss a stringent test of the coherent waveguiding properties of the device. Thee x-ray beam in our experiments was generated by the undulator of beamline IDD 10A at the European Synchrotron Radiation Facility [24]. Before the beam enteredd the waveguide, it was monochromatised using the (111) reflection of a

FocusingFocusing of coherent x rays with tapered waveguides 93

0.000 0.01 0.02 0.03 0.00 0.01 0.02 0.03 Incidencee angle 6i (degrees)

Figuree 5.3 Logarithmic contour plots of the intensity 1(6 v9e) diffracted from the

exitexit of the tapered waveguide as a function of 9i and 6e. Measured and calculated

intensitiesintensities are shown in panels (a) and (b), respectively. No data were taken for 6i<6i< 0.004° and the lower left corner of (a). Both measurements and calculations werewere performed for A = 0.093 nm, W\= 538 nm, W2= 164 nm and L = 4-85 mm.

siliconn crystal (bandwidth AA/A = 1.4 x 10~4). A wavelength A = 0.093 nm wass selected. Given the small synchrotron source size and the gap width at the entrancee of the waveguide, the beam accepted by the waveguide can be considered too be fully coherent along the x-axis. The longitudinal coherency condition is met ass well. The waveguide plates are flat fused-silica disks, the bottom one having aa much larger diameter than the upper one. The upper plate is mounted on a tripodd of inchworm motors and an additional piezo-driven translator. By moving thee motors together or independently we set the plate distance and the tilt angle. Thee pair of plates form an optical interferometer, so that the distance and the tiltt angle can be monitored during the waveguiding experiments. For a detailed descriptionn of the waveguide set-up and the scattering geometry, see chapter 3.

Thee upper plate was positioned such that the gap width at the entrance was W\W\ = 538 nm and the gap width at the exit W2 = 164 nm. A beam of 0.1 mm

widthh in the horizontal plane was defined by a pair of slits in front of the waveguide set-up.. The beam intensity accepted by the waveguide was typically ~ 4 x 108

photons/s.. TE modes with increasingly higher mode number were excited at the entrancee by changing the angle of incidence 6i in small steps. At each $i value the intensityy ƒ(#;, 0e) diffracted from the waveguide exit was recorded as a function of

thee exit angle 9e with the use of a position-sensitive CCD detector. The measured

intensityy distribution I(9il9e) is shown in Fig. 5.3a in the form of a logarithmic

grey-scalee plot. The plot was made by integrating the CCD image, taken at each incidencee angle 9{, horizontally over the width of the beam.

Thee measured intensity distribution I(9i, 9e) shows a sequence of intensity

max-imaa at a discrete set of angle pairs (0j, 9e). While the maxima for a non-tapered

waveguidee are located exactly along the diagonal 9e = 9i [53], for our tapered

wa-veguidee they are scattered around the line 9e — (W1/W2) &i (this relation being a

consequencee of Liouville's theorem). In addition, each maximum is elongated along thee #e-axis by the same ratio W1/W2. We conclude that the tapered waveguide

indeedd compresses the input modes. However, interference causes the intensity to bee unevenly distributed over the antinodes of the compressed wave field, giving risee to the complex diffraction pattern shown in Fig. 5.3.

Wee have compared the experimentally obtained diffraction pattern data with numericall calculations of I(9i,9e) using Eq. (5.6). The calculated pattern is shown

inn Fig. 5.3b. The positions of the measured and calculated intensity maxima are seenn to match quite well, except for some subsidiary maxima at 9t values larger

thann ~ 0.025°. The latter disagreement is probably caused by small deviations fromm the assumed geometry of an abruptly ending flat upper plate. Especially the compressionn of the higher input modes is sensitive to such deviations.

Wee have not yet performed a systematic search for the experimental conditions yieldingg optimal focusing at the exit plane. The generally good agreement between measuredd and calculated intensity distributions, together with the reproducibility inn the nanopositioning of our waveguide components, will make it possible to search forr optimized wave fields through calculations such as those shown in Fig. 5.2 and too implement them in the experiment.

Inn conclusion, we have demonstrated how a tilted planar waveguiding geome-tryy is used to generate a very small line focus of coherent x rays. Our method of focusingg is based on compression of the wave field and the exploitation of multi-modee interference effects. The measurements and the calculations demonstrate the feasibilityy of shaping the compressed wave field in a controlled way. An obvious

FocusingFocusing of coherent x rays with tapered waveguides 95 drawbackk is the small entrance gap, accepting only a small part of the

synchro-tronn beam. Prefocusing of the synchrotron beam onto the waveguide entrance iss possible, but its effect on the coherent waveguiding properties has yet to be investigated. .