- Email: [email protected]
Focusing of coherent X-rays in a tapered planar waveguide
Physica B 283 (2000) 285}288
Focusing of coherent X-rays in a tapered planar waveguide M.J. Zwanenburg*, J.H.H. Bongaerts, J.F. Peters, D. Riese, J.F. van der Veen Van der Waals}Zeeman Institute, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
Abstract Focusing of X-rays in one dimension is achieved in a planar waveguide with linear taper. The waveguide consists of two plates with a variable tilt angle and an air gap between them. Compression of the beam and coherent mode coupling inside the waveguide result in a line focus of 26 nm height at the exit. 2000 Elsevier Science B.V. All rights reserved. PACS: 61.10.!i; 07.85.Qe; 41.50.#h Keywords: Focusing of X-rays; X-ray waveguide; Coherent X-ray scattering
1. Introduction In the past decade various optical devices have been developed for the focusing of X-rays. These developments largely took place at synchrotron radiation sources, because the brilliance of these sources generally made such investments pay o!. Focusing optics for hard X-rays include curved crystals and mirrors [1], Fresnel and Bragg}Fresnel zone plates [2], refractive lenses [3], capillary optics [4] and planar waveguides in combination with a resonant beam coupler [5,6]. These devices either focus the beam in one or in two dimensions. Spot sizes down to a few hundred nanometer can be obtained. Applications of focusing optics include trace element mapping on small samples using X-ray #uorescence, microdi!raction and phase contrast microcopy. The high brilliance of third-generation synchrotron radiation sources enables extraction of a transversely coherent beam of su$ciently high intensity that studies of speckle patterns [7] and X-ray photon correlation spectroscopy (XPCS) [8] experiments can be performed. Usually, a coherent beam is selected with a pinhole at some distance from the sample. Di!raction at the pinhole opening then results in a broadened beam at the sample position. For some applications one would like to select
* Corresponding author. Fax: 31-20-525-5102. E-mail address: [email protected] (M.J. Zwanenburg)
a coherent beam and subsequently compress it to nanometer dimensions at the sample position, thus allowing for, e.g., XPCS or phase contrast imaging studies [9] on a very small object. Here we present a tapered planar waveguide with which we obtained a line focus of 26 nm height at the waveguide exit. Interference between local modes of the waveguide gives rise to an intensity maximum of a width substantially smaller than the gap width at the exit. Hence, the actual intensity gain is higher than that resulting from compression of an incoherent beam by the same device. Section 2 discusses the propagation of modes through the tapered waveguide and Section 3 describes measurements of the angular distribution of intensity exiting the waveguide, con"rming the coherent focusing properties of the device.
2. Mode propagation in tapered planar waveguide Consider a planar waveguide with tuneable air gap [10], accepting a highly collimated synchrotron X-ray beam as shown in Fig. 1a. The horizontal polarisation direction of the beam is parallel to the plane of the waveguide. With the plates set parallel, the principle of operation of the waveguide is as follows. Interference between the incident plane waves and the waves re#ected from the bottom plate in front of the waveguide yields a sinusoidal standing wave pattern. For speci"c angles of incidence h , a node of the standing wave coincides with
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 0 0 3 - 7
286
M.J. Zwanenburg et al. / Physica B 283 (2000) 285}288
tion of the coordinates (x, z) within the waveguide. The "eld amplitude is most easily found by numerically solving the scalar wave equation [11] RW(x, z) RW(x, z) # #kW(x, z)"0, Rx Rz
(2)
with k"2p/j the wave vector and with W(x, z) satisfying the boundary condition Fig. 1. Schematic representation of the waveguide set-up and the scattering geometry. The waveguide is shown in (a) with the plates set parallel, and in (b) with the upper plate tilted. For illustration, the "eld pro"les of the TE mode at the entrance and exit planes are sketched. In a parallel plate geometry, the modes propagate undisturbed. In the tilted geometry, the exiting "eld pro"le is much di!erent from the one excited at the entrance. Angles and distances are not to scale.
the surface of the upper re#ecting plate. For these angles the standing wave matches precisely the gap width and the wave becomes a propagating transverse electric (TE) mode of the waveguide. The incidence angle corresponding to excitation of mode TE equals h "h ,
(m#1)j/2=. Here, j is the photon wavelength and = the distance between the plates. The integer mode number m refers to the number of nodes in the gap between the plates, not counting the nodes at the plate surfaces. In the ray analogue of waveguiding, the &mode angle' h is the angle at which the rays bounce success ively from the plate surfaces. In a waveguide with parallel plates, a single mode which starts at the entrance passes down the waveguide undisturbed and emerges at the exit with the same sinusoidal "eld distribution (Fig. 1a). One identi"es the "eld distribution across the exit gap by measuring the angular distribution of intensity in the far "eld, as described in Ref. [10]. Now assume that the upper plate is tilted downward in the direction of propagation by an angle h , see Fig. 1b. The plate distance = decreases linearly with the coordinate z along the propagation direction =(z)"= !h z,
W(0, z)"0 for 0)z)R, W(=(z), z)"0 for 0)z)¸.
(3)
These boundary conditions can be applied, provided the evanescent "eld penetrates the plate material over a depth negligibly small compared to the gap width, which is the case for mode angles h not too close to the
critical angle for total re#ection. For the case of zero tilt angle (= "= ), the TE modes
W(x, z)"sin(kh x)e\ @ X (4)
are solutions of Eq. (2). Here, b "k cos h is the propa
gation constant and h "(m#1)p/k= . For non-zero
tilt angle, the solutions of Eq. (2) have a more complex form. Rather than searching for analytical solutions, we solved Eq. (2) numerically by use of a "nite-di!erence beam propagation method (FD-BPM) as described in Ref. [12]. The input wave"eld at the entrance is a single TE mode which matches the local gap width =
W(x,0)"sin(kh x). (5)
As an example we show how a TE input mode propa gates in a waveguide of length ¸"4.85 mm, with a gap width decreasing from = "538 nm at the entrance to = "164 nm at the exit. This corresponds with a tilt angle h "0.004423. We performed the calculations as suming a wavelength j"0.093 nm. These values correspond to our experimental conditions (see Section 3). A contour plot of the calculated intensity I(x, z)" "W(x, z)" within the waveguide is shown in Fig. 2. Near
(1)
where = is the gap width at the entrance. The gap width at the exit is given by = "= !h ¸, with ¸ the length of the waveguide. The wave"eld of a speci"c TE
mode, excited at the entrance, no longer matches the smaller gap width further downstream. If the change in gap width is very small compared to the spacing between nodes of the wave"eld, the wave"eld is simply compressed and retains approximately its sinusoidal form. If, however, the change in gap width is much larger, scattering also occurs into other local modes. The resulting electric "eld amplitude W(x, z) is skew along the transverse coordinate x and in general is a complicated func-
Fig. 2. Distribution of the "eld intensity within the waveguide, numerically calculated for j"0.093 nm, = "538 nm, = " 164 nm and ¸"4.85 mm. As input mode, the TE mode was chosen. Note that distances along the horizontal axis are given in millimeters and along the vertical axis in nanometers.
M.J. Zwanenburg et al. / Physica B 283 (2000) 285}288
the entrance, the nodes and antinodes of the starting TE mode are clearly seen. Further downstream, the node spacing is seen to be compressed in proportion with the decreasing gap width. However, the intensities in the antinodes become unevenly distributed across the gap, oscillating between the upper and lower plate with ever growing amplitude and shortening period. These oscillations are the result of interference between waves which are re#ected from the upper and lower plates at larger and larger angles. In the calculation, the mode angle at the entrance is h "0.01493. In the ray analogue, the number of bounces from the plates equals &(h #h )¸/= K9.
Returning to the contour plot of Fig. 2, we observe a strong intensity maximum 0.2 mm beyond the exit plane. This maximum, which is 6 times stronger than the intensity in an antinode of the TE input mode, may serve as a focus for 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 exit (= /= "3.28). In the example given, the antinode of the compressed wave"eld which is closest to the bottom plate is located 26 nm above the bottom plate and has full-width-at half-maxima of 24 nm along the x-axis and 0.15 mm along the z-axis. Two less intense neigbouring maxima lie upstream, at a distance of 0.11 and 0.22 mm along the z-axis. If required, the position and the spacing of the maxima along the z-axis can be modi"ed by changing the waveguide length or the tilt angle or by chosing a di!erent input mode. For example, lowering the input mode while keeping the other parameters constant will result in a larger spacing along z, at the cost of a larger focus width along the x-direction. The wave"eld can be compressed further by reducing the width of the exit gap. For a given entrance gap width, a limit to the focus size at the exit is eventually set by the bouncing angle becoming larger than the critical angle for total re#ection, in which case the local modes become radiation modes. Of course, this limit can be overcome by reducing also the entrance gap width while keeping the tilt angle the same, but then the beam intensity accepted by the device is correspondingly reduced.
3. Measurement of waveguiding properties We investigate the propagation of modes in the tapered waveguide by measuring, for a large set of incidence angles h , the intensity I(h , h ) emerging from the waveguide as a function of exit angle h . We compare the measured intensity distribution with the calculated one using the Fourier transform
4 5 I(h , h )" W(x, ¸)sin(kh x) dx . =
(6)
287
Here, W(x, ¸) is the numerically calculated "eld pro"le at the exit, corresponding to the input wave"eld W(x,0)" sin(kh x). (Note, that h may deviate from an input mode angle h .) The comparison between measured and cal culated intensity distributions forms a stringent test of the coherent waveguiding properties of the device. The X-ray beam in our experiments was generated by the undulator of beamline ID10A at the European Synchrotron Radiation Facility [13]. Before the beam entered the waveguide, it was monochromatised using the (1 1 1) re#ection of a silicon crystal (bandwidth *j/j" 1.4;10\). A wavelength j"0.093 nm was selected. Given the small synchrotron source size and the gap width at the entrance of the waveguide, the beam accepted by the waveguide can be considered to be fully coherent along the x-axis. The longitudinal coherency condition is met as well. The waveguide plates are #at fused-silica disks, the bottom one having a much larger diameter than the upper one. The upper plate is mounted on a tripod of inchworm motors and an additional piezo-driven translator. By moving the motors together or independently we set the plate distance and the tilt angle. The pair of plates form an optical interferometer, so that the distance and the tilt angle can be monitored during the waveguiding experiments. For a detailed description of the waveguide set-up and the scattering geometry, see Refs. [10,14]. The upper plate was positioned such that the gap width at the entrance was = "538 nm and the gap width at the exit = "164 nm. A beam of 0.1 mm width in the horizontal plane was de"ned by a pair of slits in front of the waveguide set-up. The beam intensity accepted by the waveguide was typically &4;10 photons/s. TE modes with increasingly higher mode number were excited at the entrance by changing the angle of incidence h in small steps. At each h value the intensity I(h , h ) di!racted from the waveguide exit was recorded as a function of the exit angle h with the use of a position sensitive CCD detector. The measured intensity distribution I(h , h ) is shown in Fig. 3a in the form of a linear grey-scale plot. The plot was made by integrating the CCD image, taken at each incidence angle h , horizon tally over the width of the beam. The measured intensity distribution I(h , h ) shows a sequence of intensity maxima at a discrete set of angle pairs (h , h ). While the maxima for a non-tapered waveguide are located exactly along the diagonal h "h [10], for our tapered waveguide they are scattered around the line h "(= /= )h (this relation being a consequence of Liouville's theorem). In addition, each maximum is elongated along the h -axis by the same ratio = /= . We conclude that the tapered waveguide indeed compresses the input modes. However, interferences cause the intensity to be unevenly distributed over the antinodes of the compressed wave"eld, giving rise to the complex di!raction pattern shown in Fig. 3.
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M.J. Zwanenburg et al. / Physica B 283 (2000) 285}288
gap, accepting only a small part of the synchrotron beam. Prefocusing of the synchrotron beam onto the waveguide entrance is possible, but its e!ect on the coherent waveguiding properties has yet to be investigated.
Acknowledgements
Fig. 3. Logarithmic contour plots of the intensity I(h , h ) dif fracted from the exit of the tapered waveguide as a function of h and h . Measured and calculated intensities are shown in panels (a) and (b), respectively. No data were taken for h (0.0043 and the lower left corner of (a). Both measurements and calculations were performed for j"0.093 nm, = " 538 nm, = "164 nm and ¸"4.85 mm.
We have compared the experimentally obtained diffraction pattern data with numerical calculations of I(h , h ) using Eq. (6). The calculated pattern is shown in Fig. 3b. The positions of the measured and calculated intensity maxima are seen to match quite well, except for some subsidiary maxima at h values larger than &0.0253. The latter disagreement is probably caused by small deviations from the assumed geometry of an abruptly ending #at upper plate. Especially the compression of the higher input modes is sensitive to such deviations. We have not yet performed a systematic search for the experimental conditions yielding optimal focusing at the exit plane. The generally good agreement between measured and calculated intensity distributions, together with the reproducibility in the nanopositioning of our waveguide components, will make it possible to search for optimised wave"elds through calculations such as those shown in Fig. 2 and to implement them in the experiment. In conclusion, we have demonstrated how a tilted planar waveguiding geometry is used to generate a very small line focus of coherent X-rays. Our method of focusing is based on compression of the wave"eld and the exploitation of multi-mode interference e!ects. The measurements and the calculations demonstrate the feasibility of shaping the compressed wave"eld in a controlled way. An obvious drawback is the small entrance
We thank the sta! of ESRF, in particular F. Zontone, for their assistance during the measurements. Discussions with G.H. Wegdam are gratefully acknowledged. This work was part of the research programme of the Foundation of Fundamental Research on Matter (FOM) and was made possible by "nancial support from the Netherlands Organisation for Scienti"c Research (NWO).
References [1] A.A. Macdowell, R. Celestre, C.-H. Chang, K. Franck, M.R. Howells, S. Locklin, H.A. Padmore, J.R. Patel, R. Sander, Proceedings SPIE 3152, July 1997, San Diego. [2] A.A. Snigirev, Rev. Sci. Instr. 66 (1995) 2053. [3] A.A. Snigirev, V.G. Kohn, I.I. Snigireva, B. Lengeler, Nature 384 (1996) 49. [4] D.H. Bilderback, S.A. Ho!man, D.J. Thiel, Science 263 (1994) 201. [5] Y.P. Feng, S.K. Sinha, H.W. Deckman, J.B. Hastings, D.P. Siddons, Phys. Rev. Lett. 71 (1993) 537. [6] W. Jark, S. Di Donzo, S. Lagomarsino, A. Cedola, E. di Fabrizio, A. Bram, C. Riekel, J. Appl. Phys. 80 (1996) 4831. [7] M. Sutton, S.G.J. Mochrie, T. Greytak, S.E. Nagler, L.E. Berman, G.A. Held, G.B. Stephenson, Nature 352 (1991) 608. [8] T. Thurn-Albrecht, G. Meier, P. MuK ller-Buschbaum, A. Patkowski, W. Ste!en, G. GruK bel, D.L. Abernathy, O. Diat, M. Winter, M.G. Koch, M.T. Reetz, Phys. Rev. E 59 (1999) 642. [9] S. Lagomarsino, A. Cedola, P. Cloetens, S. Di Fonzo, W. Jark, G. SoullieH , C. Riekel, Appl. Phys. Lett. 71 (1997) 2557. [10] M.J. Zwanenburg, J.F. Peters, J.H.H. Bongaerts, S.A. de Vries, D.L. Abernathy, J.F. van der Veen, Phys. Rev. Lett. 82 (1999) 1696. [11] D. Marcuse, Theory of Dielectric Waveguides, Academic Press, San Diego, 1991. [12] R. Scarmozzino, R.M. Jr Osgood, J. Opt. Soc. Am. B 8 (1991) 724. [13] G. GruK bel, J. Als-Nielsen, A.K. Freund, J. Phys. IV (France) 4 (1994) C9}27. [14] M.J. Zwanenburg, J.F. van der Veen, H.G. Ficke, H. Neerings, Rev. Sci. Instr., accepted.
Focusing of coherent X-rays in a tapered planar waveguide M.J. Zwanenburg*, J.H.H. Bongaerts, J.F. Peters, D. Riese, J.F. van der Veen Van der Waals}Zeeman Institute, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
Abstract Focusing of X-rays in one dimension is achieved in a planar waveguide with linear taper. The waveguide consists of two plates with a variable tilt angle and an air gap between them. Compression of the beam and coherent mode coupling inside the waveguide result in a line focus of 26 nm height at the exit. 2000 Elsevier Science B.V. All rights reserved. PACS: 61.10.!i; 07.85.Qe; 41.50.#h Keywords: Focusing of X-rays; X-ray waveguide; Coherent X-ray scattering
1. Introduction In the past decade various optical devices have been developed for the focusing of X-rays. These developments largely took place at synchrotron radiation sources, because the brilliance of these sources generally made such investments pay o!. Focusing optics for hard X-rays include curved crystals and mirrors [1], Fresnel and Bragg}Fresnel zone plates [2], refractive lenses [3], capillary optics [4] and planar waveguides in combination with a resonant beam coupler [5,6]. These devices either focus the beam in one or in two dimensions. Spot sizes down to a few hundred nanometer can be obtained. Applications of focusing optics include trace element mapping on small samples using X-ray #uorescence, microdi!raction and phase contrast microcopy. The high brilliance of third-generation synchrotron radiation sources enables extraction of a transversely coherent beam of su$ciently high intensity that studies of speckle patterns [7] and X-ray photon correlation spectroscopy (XPCS) [8] experiments can be performed. Usually, a coherent beam is selected with a pinhole at some distance from the sample. Di!raction at the pinhole opening then results in a broadened beam at the sample position. For some applications one would like to select
* Corresponding author. Fax: 31-20-525-5102. E-mail address: [email protected] (M.J. Zwanenburg)
a coherent beam and subsequently compress it to nanometer dimensions at the sample position, thus allowing for, e.g., XPCS or phase contrast imaging studies [9] on a very small object. Here we present a tapered planar waveguide with which we obtained a line focus of 26 nm height at the waveguide exit. Interference between local modes of the waveguide gives rise to an intensity maximum of a width substantially smaller than the gap width at the exit. Hence, the actual intensity gain is higher than that resulting from compression of an incoherent beam by the same device. Section 2 discusses the propagation of modes through the tapered waveguide and Section 3 describes measurements of the angular distribution of intensity exiting the waveguide, con"rming the coherent focusing properties of the device.
2. Mode propagation in tapered planar waveguide Consider a planar waveguide with tuneable air gap [10], accepting a highly collimated synchrotron X-ray beam as shown in Fig. 1a. The horizontal polarisation direction of the beam is parallel to the plane of the waveguide. With the plates set parallel, the principle of operation of the waveguide is as follows. Interference between the incident plane waves and the waves re#ected from the bottom plate in front of the waveguide yields a sinusoidal standing wave pattern. For speci"c angles of incidence h , a node of the standing wave coincides with
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 0 0 3 - 7
286
M.J. Zwanenburg et al. / Physica B 283 (2000) 285}288
tion of the coordinates (x, z) within the waveguide. The "eld amplitude is most easily found by numerically solving the scalar wave equation [11] RW(x, z) RW(x, z) # #kW(x, z)"0, Rx Rz
(2)
with k"2p/j the wave vector and with W(x, z) satisfying the boundary condition Fig. 1. Schematic representation of the waveguide set-up and the scattering geometry. The waveguide is shown in (a) with the plates set parallel, and in (b) with the upper plate tilted. For illustration, the "eld pro"les of the TE mode at the entrance and exit planes are sketched. In a parallel plate geometry, the modes propagate undisturbed. In the tilted geometry, the exiting "eld pro"le is much di!erent from the one excited at the entrance. Angles and distances are not to scale.
the surface of the upper re#ecting plate. For these angles the standing wave matches precisely the gap width and the wave becomes a propagating transverse electric (TE) mode of the waveguide. The incidence angle corresponding to excitation of mode TE equals h "h ,
(m#1)j/2=. Here, j is the photon wavelength and = the distance between the plates. The integer mode number m refers to the number of nodes in the gap between the plates, not counting the nodes at the plate surfaces. In the ray analogue of waveguiding, the &mode angle' h is the angle at which the rays bounce success ively from the plate surfaces. In a waveguide with parallel plates, a single mode which starts at the entrance passes down the waveguide undisturbed and emerges at the exit with the same sinusoidal "eld distribution (Fig. 1a). One identi"es the "eld distribution across the exit gap by measuring the angular distribution of intensity in the far "eld, as described in Ref. [10]. Now assume that the upper plate is tilted downward in the direction of propagation by an angle h , see Fig. 1b. The plate distance = decreases linearly with the coordinate z along the propagation direction =(z)"= !h z,
W(0, z)"0 for 0)z)R, W(=(z), z)"0 for 0)z)¸.
(3)
These boundary conditions can be applied, provided the evanescent "eld penetrates the plate material over a depth negligibly small compared to the gap width, which is the case for mode angles h not too close to the
critical angle for total re#ection. For the case of zero tilt angle (= "= ), the TE modes
W(x, z)"sin(kh x)e\ @ X (4)
are solutions of Eq. (2). Here, b "k cos h is the propa
gation constant and h "(m#1)p/k= . For non-zero
tilt angle, the solutions of Eq. (2) have a more complex form. Rather than searching for analytical solutions, we solved Eq. (2) numerically by use of a "nite-di!erence beam propagation method (FD-BPM) as described in Ref. [12]. The input wave"eld at the entrance is a single TE mode which matches the local gap width =
W(x,0)"sin(kh x). (5)
As an example we show how a TE input mode propa gates in a waveguide of length ¸"4.85 mm, with a gap width decreasing from = "538 nm at the entrance to = "164 nm at the exit. This corresponds with a tilt angle h "0.004423. We performed the calculations as suming a wavelength j"0.093 nm. These values correspond to our experimental conditions (see Section 3). A contour plot of the calculated intensity I(x, z)" "W(x, z)" within the waveguide is shown in Fig. 2. Near
(1)
where = is the gap width at the entrance. The gap width at the exit is given by = "= !h ¸, with ¸ the length of the waveguide. The wave"eld of a speci"c TE
mode, excited at the entrance, no longer matches the smaller gap width further downstream. If the change in gap width is very small compared to the spacing between nodes of the wave"eld, the wave"eld is simply compressed and retains approximately its sinusoidal form. If, however, the change in gap width is much larger, scattering also occurs into other local modes. The resulting electric "eld amplitude W(x, z) is skew along the transverse coordinate x and in general is a complicated func-
Fig. 2. Distribution of the "eld intensity within the waveguide, numerically calculated for j"0.093 nm, = "538 nm, = " 164 nm and ¸"4.85 mm. As input mode, the TE mode was chosen. Note that distances along the horizontal axis are given in millimeters and along the vertical axis in nanometers.
M.J. Zwanenburg et al. / Physica B 283 (2000) 285}288
the entrance, the nodes and antinodes of the starting TE mode are clearly seen. Further downstream, the node spacing is seen to be compressed in proportion with the decreasing gap width. However, the intensities in the antinodes become unevenly distributed across the gap, oscillating between the upper and lower plate with ever growing amplitude and shortening period. These oscillations are the result of interference between waves which are re#ected from the upper and lower plates at larger and larger angles. In the calculation, the mode angle at the entrance is h "0.01493. In the ray analogue, the number of bounces from the plates equals &(h #h )¸/= K9.
Returning to the contour plot of Fig. 2, we observe a strong intensity maximum 0.2 mm beyond the exit plane. This maximum, which is 6 times stronger than the intensity in an antinode of the TE input mode, may serve as a focus for 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 exit (= /= "3.28). In the example given, the antinode of the compressed wave"eld which is closest to the bottom plate is located 26 nm above the bottom plate and has full-width-at half-maxima of 24 nm along the x-axis and 0.15 mm along the z-axis. Two less intense neigbouring maxima lie upstream, at a distance of 0.11 and 0.22 mm along the z-axis. If required, the position and the spacing of the maxima along the z-axis can be modi"ed by changing the waveguide length or the tilt angle or by chosing a di!erent input mode. For example, lowering the input mode while keeping the other parameters constant will result in a larger spacing along z, at the cost of a larger focus width along the x-direction. The wave"eld can be compressed further by reducing the width of the exit gap. For a given entrance gap width, a limit to the focus size at the exit is eventually set by the bouncing angle becoming larger than the critical angle for total re#ection, in which case the local modes become radiation modes. Of course, this limit can be overcome by reducing also the entrance gap width while keeping the tilt angle the same, but then the beam intensity accepted by the device is correspondingly reduced.
3. Measurement of waveguiding properties We investigate the propagation of modes in the tapered waveguide by measuring, for a large set of incidence angles h , the intensity I(h , h ) emerging from the waveguide as a function of exit angle h . We compare the measured intensity distribution with the calculated one using the Fourier transform
4 5 I(h , h )" W(x, ¸)sin(kh x) dx . =
(6)
287
Here, W(x, ¸) is the numerically calculated "eld pro"le at the exit, corresponding to the input wave"eld W(x,0)" sin(kh x). (Note, that h may deviate from an input mode angle h .) The comparison between measured and cal culated intensity distributions forms a stringent test of the coherent waveguiding properties of the device. The X-ray beam in our experiments was generated by the undulator of beamline ID10A at the European Synchrotron Radiation Facility [13]. Before the beam entered the waveguide, it was monochromatised using the (1 1 1) re#ection of a silicon crystal (bandwidth *j/j" 1.4;10\). A wavelength j"0.093 nm was selected. Given the small synchrotron source size and the gap width at the entrance of the waveguide, the beam accepted by the waveguide can be considered to be fully coherent along the x-axis. The longitudinal coherency condition is met as well. The waveguide plates are #at fused-silica disks, the bottom one having a much larger diameter than the upper one. The upper plate is mounted on a tripod of inchworm motors and an additional piezo-driven translator. By moving the motors together or independently we set the plate distance and the tilt angle. The pair of plates form an optical interferometer, so that the distance and the tilt angle can be monitored during the waveguiding experiments. For a detailed description of the waveguide set-up and the scattering geometry, see Refs. [10,14]. The upper plate was positioned such that the gap width at the entrance was = "538 nm and the gap width at the exit = "164 nm. A beam of 0.1 mm width in the horizontal plane was de"ned by a pair of slits in front of the waveguide set-up. The beam intensity accepted by the waveguide was typically &4;10 photons/s. TE modes with increasingly higher mode number were excited at the entrance by changing the angle of incidence h in small steps. At each h value the intensity I(h , h ) di!racted from the waveguide exit was recorded as a function of the exit angle h with the use of a position sensitive CCD detector. The measured intensity distribution I(h , h ) is shown in Fig. 3a in the form of a linear grey-scale plot. The plot was made by integrating the CCD image, taken at each incidence angle h , horizon tally over the width of the beam. The measured intensity distribution I(h , h ) shows a sequence of intensity maxima at a discrete set of angle pairs (h , h ). While the maxima for a non-tapered waveguide are located exactly along the diagonal h "h [10], for our tapered waveguide they are scattered around the line h "(= /= )h (this relation being a consequence of Liouville's theorem). In addition, each maximum is elongated along the h -axis by the same ratio = /= . We conclude that the tapered waveguide indeed compresses the input modes. However, interferences cause the intensity to be unevenly distributed over the antinodes of the compressed wave"eld, giving rise to the complex di!raction pattern shown in Fig. 3.
288
M.J. Zwanenburg et al. / Physica B 283 (2000) 285}288
gap, accepting only a small part of the synchrotron beam. Prefocusing of the synchrotron beam onto the waveguide entrance is possible, but its e!ect on the coherent waveguiding properties has yet to be investigated.
Acknowledgements
Fig. 3. Logarithmic contour plots of the intensity I(h , h ) dif fracted from the exit of the tapered waveguide as a function of h and h . Measured and calculated intensities are shown in panels (a) and (b), respectively. No data were taken for h (0.0043 and the lower left corner of (a). Both measurements and calculations were performed for j"0.093 nm, = " 538 nm, = "164 nm and ¸"4.85 mm.
We have compared the experimentally obtained diffraction pattern data with numerical calculations of I(h , h ) using Eq. (6). The calculated pattern is shown in Fig. 3b. The positions of the measured and calculated intensity maxima are seen to match quite well, except for some subsidiary maxima at h values larger than &0.0253. The latter disagreement is probably caused by small deviations from the assumed geometry of an abruptly ending #at upper plate. Especially the compression of the higher input modes is sensitive to such deviations. We have not yet performed a systematic search for the experimental conditions yielding optimal focusing at the exit plane. The generally good agreement between measured and calculated intensity distributions, together with the reproducibility in the nanopositioning of our waveguide components, will make it possible to search for optimised wave"elds through calculations such as those shown in Fig. 2 and to implement them in the experiment. In conclusion, we have demonstrated how a tilted planar waveguiding geometry is used to generate a very small line focus of coherent X-rays. Our method of focusing is based on compression of the wave"eld and the exploitation of multi-mode interference e!ects. The measurements and the calculations demonstrate the feasibility of shaping the compressed wave"eld in a controlled way. An obvious drawback is the small entrance
We thank the sta! of ESRF, in particular F. Zontone, for their assistance during the measurements. Discussions with G.H. Wegdam are gratefully acknowledged. This work was part of the research programme of the Foundation of Fundamental Research on Matter (FOM) and was made possible by "nancial support from the Netherlands Organisation for Scienti"c Research (NWO).
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