X-Ray Propagation in Tapered Waveguides: Simulation and Optimization
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J a n X-Ray Propagation in Tapered Waveguides: Simulation andOptimization
Sebastian Panknin a , Alexander K. Hartmann b and Tim Salditt a a Institut f¨ur R¨ontgenphysik,Universit¨at G¨ottingen,Friedrich-Hund-Platz 1, 37077 G¨ottingen, Germany b Institut f¨ur Physik, Universit¨at Oldenburg,26111 Oldenburg, Germany
Abstract
We use the parabolic wave equation to study the propagation of x-rays in tapered waveguides by numercial simulationand optimization. The goal of the study is to elucidate how beam concentration can be best achieved in x-rayoptical nanostructures. Such optimized waveguides can e.g. be used to investigate single biomolecules. Here, wecompare tapering geometries, which can be parametrized by linear and third-order (B´ezier-type) functions and canbe fabricated using standard e-beam litography units. These geometries can be described in two and four-dimensionalparameter spaces, respectively. In both geometries, we observe a rugged structure of the optimization problem’s “gainlandscape”. Thus, the optimization of x-ray nanostructures in general will be a highly nontrivial optimization problem.
1. Introduction
X-ray waveguides are promising novel optical devices to generate x-ray quasi-point sources for hard x-rayimaging [1,2]. X-ray waveguides (WG) have been designed and fabricated as planar layered systems (1D-WG)[3,4,5] or as lithographic channels (2D-WG) [6,7]. In both cases (guiding layer or guiding channel) mainlywaveguides with a core of constant cross section have been considered. Beam propagation in such structures,which are translationally invariant along the propagation axis z , is by now well studied and understood[8,9,10,11]. As optical elements, such waveguides serve mainly two purposes: Firstly, the radiation modesare damped out and the cross section of the exit beam is thus reduced significantly with respect to theincident beam. Secondly, in the case of single-mode waveguides [12], the phase and amplitude of the wave is Preprint submitted to Elsevier 22 October 2018 etermined exactly, and therfore presents an ideal wave for coherent imaging. In other words, the waveguideacts as a spatial and coherence filter, but not to collimate or to concentrate the beam. The flux densitynecessary for nanometer-sized beam with high enough intensities for practical applications must then beachieved by focusing elements placed in front of the waveguide, such as Kirkpatrik-Baez (KB) mirrors [13,7],Fresnel zone plates [14] or compound refractive lenses [15,16].Here we address the question to which extent tapering of the waveguide can be used to concentrate thebeam and to filter it at the same time. This would make pre-focusing optics in front of the waveguide obsoletefor some applications. A first experimental realization was demonstrated by tapered air-gap waveguides[17], but was restricted to the multi-modal regime. Bergemann et al. have then studied linearly taperedx-ray waveguides by analytical and numerical calculation [18], concentrating on minimum beam width atthe exit. Here we want to generalize the numerical studies to non-linear tapering profiles. Rather thanthe minimum achievable beam width, which has been addressed before [18], we address the maximum fluxdensity enhancement as function of tapering parameters, and the associated question of an optimum interfacegeometry. As in all problems of reflective optics, e.g. bent mirrors, one would assume that the shape functionis a crucial parameter in the problem. As a first step, we consider both linear and third order polynominalinterface geometries. Note that the parameters studied correspond well to structures which can also befabricated by e-beam lithography. The propagation in the stuctures is studied by numerical solution of theparabolic wave equation (PWE), as used previously for standard waveguides [19,10]. The use of the PWEwill be briefly explained in the next section, followed by the simulation results.
2. The simulation method
The propagation of monochromatic x-rays in materials is described by the Helmholtz equation[ △ + k n ]Ψ = 0 . (1)Ψ denotes an electric field component in this scalar wave equation, k = πλ the wavenumber, and λ thewavelength. The index of refraction is given by n = 1 − δ − i β , where the imaginary part accounts forabsorption. The Laplacian is written as △ = ( ∂ x + ∂ z ) for the two-dimensional case which is of interesthere. The direct solution of this elliptical differential equation is problematic due to numerical instabilityand the fact that the region of interest is large compared to the wavelength (micrometers up to millimeterscompared to ˚Angstr¨om). A numerically much more stable alternative is provided by the parabolic waveequation, which can be used in controlled and excellent approximation for forward propagation problemsin the x-ray regime [20,21,22]. To this end, a function u ( x, z ) is defined by Ψ( x, z ) =: u ( x, z ) exp( − i kz ).Neglecting terms of order O ( ∂ z u ) which is justified if the z -axis is almost parallel to the wavevector, theHelmholtz equation is fullfilled for Ψ, under the condition that u solves the parabolic wave equation2 − k∂ z + ∂ x + k ( n − u = 0 (2)Mathematically, this equation is of the same form as the stationary Schr¨odinger equation for a massiveparticle (without spin) in a potential. After renaming of variables, the potential in the Schr¨odinger equationis equivalent to the the refraction index n profile in the optical case considered here.In order to simplify the expressions and to find a suitable length scale for numerical calculations, weuse the natural units δk for the distance along the propagation direction, and W = πk √ δ for the lateraldimension, respectively. Note that W is the critical width at which a waveguide becomes mono-modal, i.e.the width at which the waveguide supports only a single mode [18,23]. At the same time, a waveguide ofwidth d = W leads to the highest possible wave confinement. The intensity distribution of the wave broadensboth for smaller and larger d . While the second case is trivial, it is important to note that the nearfieldintensity distribution is broadened by evanescent waves in the cladding, if d is reduced below W . Using thesenatural units Z := δkz, X := xW , (3)(2) can be rewritten as ∂ Z u = − i π ∂ X u + ( i − βδ ) u, in material0 , in vacuum . (4)The ratio βδ remains the only material dependent parameter in this equation.The equation is solved using a Crank-Nicolson finite-difference scheme, which was already proved to givereliable results for x-ray waveguides and Fresnel zone plates [22,24]. For the initial values, a plane wavepropagating along z was assumed. The lateral boundary conditions are given by a damped plane wavepropagating in the material far away from the region of interest.For field propagation in vacuum, the Fresnel-Kirchhoff integral for two-dimensional beam propagation [25]was used [26], Ψ( x, z ) = i √ λ Z d x ′ Ψ( x ′ ,
0) e − i kr √ r cos α , (5)where α is the angle between the z -axis and the vector ( x − x ′ , z ). The corresponding distance to the originis r = z + ( x − x ′ ) .
3. Tapered x-ray waveguides: shape optimization
We have considered two generic types of tapering geometry: (i) the linear taper, and (ii) a taperingfunction parameterized by B´ezier curves. For comparison with experimental values, we have chosen ma-terial parameters and photon energy E according to recent experiments and present fabrication methods.Vaccuum channels in Si have been simulated at a photon energy E = 12 keV. Taking air/vaccuum as theguiding material simplifies simulations because only the index of refraction of the cladding differs from unity.3ptically, the structures are still very close to the experiments with polymer channels in Si [7,27], and arecompletely equivalent to the second generation of x-ray waveguide channels fabricated by e-beam lithogra-phy, ion etching and subsequent wafer bonding of a cap wafer. For both tapering geometries, the waveguideend width was held constant at d = 1 , W = 20 nm which is the critical width W = πk √ δ for single-modewaveguides [18,23]. At this value, the resolution limit of waveguides is reached, corresponding to a full widthat half maximum FWHM min = 0 . W for the intensity profile in the channel of width W [18].The goal of the optimization was to quantify the field propagation and confinement, for given open-ing width D and end width d or for given d only. In general, one can expect that optimization of x-raynanostructures might be highly non-trivial, as for many optimization problems occurring in physics [28,29].Nevertheless, since we restricted ourself here to the two above mentioned tapering geometries, we could usea quite simple optimization algorithm, see below. The two tapering geometries were parameterized as follows(see Fig. 1):– In the linear case, a straight interface seperates the channel from the cladding at an opening angle ϑ . Thelength of the waveguide is denoted by L .– In the case of a B´ezier curve interface, the shape is controlled by two vectors P = ( x , y
1) and P =( x , y L and angle ϑ . Fig. 2 shows the resulting peak intensities I normalized to the incident intensity I , i.e. the gain I/I , at the end of the waveguide (exit plane) as a function of L and ϑ . Hence, this plotdisplays the “gain landscape” of the optimization problem. Even for this two-parameter space, a very ruggedgain landscape is visible: With increasing opening width D ≃ L tan( ϑ ), I first increases, in proportion tothe geometric acceptance. The initial increase of I ( ϑ, L = const) crosses over to a surprisingly broad rangeof angles where the radiation transport and collimation raimains high, but is modulated by an oscillationoccuring with a periodicity at which small. Finally, the intensity decreases when ϑ becomes too large, leadingto a leaking of the intensity out of the confining interfaces. This leakage occurs for multireflections of highorder n . The condition to prevent the leakage is simply given by nϑ ≤ ϑ c , where ϑ c is the angle of totalreflection, as expected from ray optics. Nevertheless, at the end of the waveguide a dicrete set of rod-likeintensity spikes are observed in the simulation, travering or ’leaking’ from the interface, see again Fig. 1.These rods or spikes originate from interference effects which would correspond to transmitted beams aftermultiple reflections in the ray-optical description. When ϑ is varied, these rods move and give rise to anoscillatory behavior in the exit plane. This effect is characterized by the oscillation between either sharpand intense profile at the exit, i. e. high I and small with (full width at half intensity FWHM) or a smeared4 ig. 1. Shape functions and field distribution of taperd waveguides: the top row shows the schematics, the bottom row thecalculated intensity distribution for the optimal parameters of each kind. The waveguides consists of vacuum channels em-bedded in Silicon at a photon energy of E = 12 keV. (bottom left) Linear taper with an opening angle ϑ = 0 . ϑ c , trans-porting radiation collected within a width of D = 0 . µ m over a propagation distance of L = 8 mm down to the criticalexit cross section of d = 19 .
88 nm, corresponding to the critical width W of Si . (bottom right) Waveguide tapered with aB´eziercurve interface. The lengths d , D and ℓ as above. The two control vectors are P = ( x = 3 .
45 mm , y = 766 . P = ( x = 7 .
25 mm , y = 176 . profile with low I and large FWHM. In summary, I ( ϑ, L = const) possesses an optimum angle ϑ , while I ( ϑ = const , L ) increases with the length L , if ϑ is chosen accordingly, at least over the simulated range in L . We chose L = 8 mm as an experimentally achievable upper limit, see Fig. 2. For this length, the optimalangle is ϑ ≃ . ϑ c leading to an opening width of D = 0 . µ m and I/I = 51 . y and y at fixed x = L and x = L . As in the linear case the exit was set to the criticalwidth d = W . D and L were kept constant. 5
0 0,04 0,08 0,12 0,16 1 2 3 4 5 6 7 8 0 10 20 30 40 50 60 I / I ϑ / ϑ c L [mm] I / I I / I w [ n m ] ϑ / ϑ c intensityhalf width w Fig. 2. (top) Maximum intensity I ( ϑ, L ) at the exit normalized to the incident intensity I , as a function of opening angle ϑ and waveguide length L for linear taper. (bottom) Maximum intensity I ( ϑ, L = 8mm) (left axis) and beam width FWHM(right axis) at the waveguide exit, as a function of ϑ . The oscillations are phase shifted with respect to each other. I / I y [ µ m] y [ µ m ] -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 2,5-0,5 0 0,5 1 1,5 2 2,5 3 Fig. 3. Maximum intensity I (normalized by I ) of a waveguide with B´ezier-type tapering parameterized by P = (1 / L, y )und P = (2 / L, y ). The function I ( y , y ) exhibits a well pronounced global maximum. The results, see Fig. 3 showed again a rugged gain landscape with a couple of local minima. Nevertheless,the maximum intensity seems to occur close to the linear case, and a few local minima are present inthis particular region. Therefore, a simple approach was feasible: using the amoebae algorithm [31], theoptimum parameters P , P were searched, starting from the solution of the linear case P = ( L, D − A )and P = ( L, D − A ) with A = ( D − d ). Note that using other starting values, the algorithm alsofound different local maxima, proving the complexity of the underlying gain landscape and the usefullnessof our pre-scanning of the parameter space. As expected, the maximum intensity is higher for the moregeneral B´ezier case than for the linear taper, namely 82 . . I/I , and F = 72 . .
27 for the integrated flux, see Tab. 1. However, this difference is surprisingly small. The resultare tabulated in Tab. 1, showing maximum intensity F , integrated flux J = R d xI , width w (FWHM) ofintensity in the exit plane, as well as the angular acceptance on the incidence side, defined as the width(FWHM) of I ( α ), where α is the incidence angle α . Note that if not otherwise stated, the angle α betweenincident beam and waguide axis was kept constant at α = 0. In addition to the two tapering geometries, theplanar waveguide is included as a reference. As expected, the angular acceptance ω is much smaller for thetapered waveguides due to conservation of photon density in phase space (brilliance).7 able 1Simulation results: maximum intensity F , flux J , width (FWHM) of intensity at the exit w , and FWHM for the angulardependency of the intensity (angular acceptance) ω are tabulated for (a) the plane, (b) linear tapered and (c) a waveguide withinterfaces parameterized as B´ezier curves (values obtained after parameter optimization).(a) planar (b) linear (c) B´ezierintensity II , · − . , JI d , · − w ωϑ c
4. Summary and Conclusion
We have studied beam propagation and concentation in tapered x-ray waveguides by simulations of finite-difference equations and subsequent parameter screening and optimization. The local slope of the waveguideinterfaces and the associated tapering geometry is limited by the angle of total reflection, which restrictsthe tapering geometry to very elongated shapes. Over a device length on the order of 10mm, an incomingbeam with a width of 0 . − µ m can be concentrated to the minimum width (FWHM) w = 0 . W ≃
13 nmfor silicon. The associated expected flux enhancement in the range of 40 −
80 for one-dimensional focusing isobserved in the simulation, confirming that most of the radiation is indeed transported to the exit aperture.For two-dimensional focusing, the gain would be squared, and would thus range up to about 5 × , if10mm long structures are available. Note that both lithographic wafer processing and hollow glass fiberswith well controlled shapes are conceivable even on length scales of 100mm. However, let us stay withing amore conservative estimate based on a 10mm long waveguide. As a result, the flux density gain, the size ofthe entrance aperture, and the angular acceptance ω are fully compatible with an optical scheme, where asynchrotron source is imaged by a lense of moderate numerical aperture (mirror system or CRL) onto theentrance aperture of a tapered waveguide. Let us consider realistic values for 3rd generation synchrotronsources: a source on the order of 100 µ m cross section is demagnified by CRL or KB optics of focal length f ≃ −
5m positioned 100m behind the source. In this case, the focal spot and the convergence angleof the prefocussing optics can be matched to the entrance aperture and the angular acceptance ω of thetapered waveguide, respectively. In this case the combined gain of the pre-focussing optics and the taperedwaveguide would lead to an intense and collimated virtual source, which is unlikely to be achieved by asingle optical device. Most importantly, the simulation results show that the performance of the taperedwaveguide does not depend on the shape function in a very sensitive way. Indeed, the beam can be funneledto the exit aperture surprisingly well already with a linear taper. In sharp contrast to single bounce mirror8ptics, the tapered waveguide almost forces the convergence of the beam, and is thus much less sensitiveto figure errors. Nevertheless, we have also observed a rather rugged gain landspace for this optimizationproblem. Since the optimization was only for a few-parameter space, a simple algorithm could be used here.We expect that for the optimization of more complex x-ray nanostructures, like layered materials, moresophisticated algorithms have to be applied.AcknowledgmentWe thank Christian Fuhse for helpful discussions and the foundation work concerning the finite differenceequations simulations in our group. We gratefully acknowledge funding by Deutsche Forschungsgemeinschaft(DFG) through Sonderforschungsbereich 755 Nanoscale Photonic Imaging .References [1] S. D. F. W. Jark, S. Lagomarsino, C. Giannini, L. D. Caro, A. Cedola, M. M¨uller, Non-destructive determination of localstrain with 100-nanometre spatial resolution, Nature 403 (2000) 638–640.[2] C. Fuhse, C. Ollinger, T. Salditt, Waveguide-based off-axis holography with hard x rays, Physical Review Letters 97 (25)(2006) 254801.URL http://link.aps.org/abstract/PRL/v97/e254801 [3] E. Spiller, A. Segm¨uller, Propagation of x rays in waveguides, Appl.Phys.Lett. 24 (1973) 60–61.[4] Y. P. Feng, S. K. Sinha, H. W. Deckman, J. B. Hastings, D. P. Siddons, X-ray flux enhancement in thin-film waveguidesusing resonant beam couplers, Phys. Rev. Lett. 71 (1993) 537–540.[5] W. Jark, A. Cedola, S. D. Fonzo, M. Fiordelisi, S. Lagomarsino, N. V. Kovalenko, V. A. Chernov, High gain beamcompression in new-generation thin-film x-ray waveguides, Appl.Phys.Lett. 78 (2001) 1192–1194.[6] F. Pfeiffer, C. David, M. Burghammer, C. Riekel, T. Salditt, Two-dimensional x-ray waveguides and point sources, Science297 (2002) 230–234.[7] A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, T. Salditt, Two-dimensional hard x-ray beam compression bycombined focusing and waveguide optics, Phys. Rev. Lett. 94 (2005) 074801.[8] M. J. Zwanenburg, J. F. Peters, J. H. H. Bongaerts, S. A. de Vries, D. L. Abernathy, J. F. van der Veen, Coherentpropagation of x rays in a planar waveguide with a tunable air gap, Phys. Rev. Lett. 82 (1999) 1696–1699.[9] L. D. Caro, C. Giannini, S. D. Fonzo, W. Yark, A. Cedola, S. Lagomarsino, Spatial coherence of x-ray planar waveguideexiting radiation, Optics Commun. 217 (2003) 31–45.[10] C. Fuhse, T. Salditt, Finite-difference field calculations for two-dimensionally confined x-ray waveguides, Appl. Opt. 45 (19)(2006) 4603–4608.URL http://ao.osa.org/abstract.cfm?URI=ao-45-19-4603 [11] C. Fuhse, T. Salditt, Propagation of x-rays in ultra-narrow slits, Opt.Comm. 265 (2006) 140–146.
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