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X-ray waveguiding studies of ordering phenomena in confined fluids
Zwanenburg, M.J.
Publication date
2001
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Zwanenburg, M. J. (2001). X-ray waveguiding studies of ordering phenomena in confined
fluids.
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X-rayy waveguiding studies
off ordering phenomena
inn confined fluids
PHENOMENAA IN CONFINED FLUIDS
ACADEMISCHH PROEFSCHRIFT
TERR VERKRIJGING VAN DE GRAAD VAN DOCTOR AANN DE UNIVERSITEIT VAN AMSTERDAM,
OPP GEZAG VAN DE R E C T O R MAGNIFICUS
P R O F .. DR. J.J.M. FRANSE
TENN OVERSTAAN VAN EEN DOOR HET COLLEGE VOOr pTOTIOtieS INGESTELDEE COMMISSIE, IN HET OPENBAAR TE VERDEDIGEN
INN DE AULA DER UNIVERSITEIT
OPP DONDERDAG 1 FEBRUARI 2001, TE 10.00 UUR
DOOR R
Michell Jacobus Zwanenburg
Promotiecommissie e
Promotorr Prof. dr. J.F. van der Veen Overigee leden Prof. dr. F.R. de Boer
Dr.. W.K. Kegel Prof.. dr. A. Lagendijk
Prof.. dr. H.B. van Linden van den Heuvell Prof.. dr. M.A.J. Michels
Prof.. dr. M.K. Smit Dr.. G.H. Wegdam
Faculteitt der Natuurwetenschappen, Wiskunde en Informatica
Vormgevingg omslag: Edwin Veer ISBNN 90-5776060-6
Thee work described in this thesis was carried out at the European Synchrotron Radiationn Facility (France) and at the Van der Waals-Zeeman Institute of the Universityy of Amsterdam. The work is part of the research program of the Foundationn for Fundamental Research on Matter [ Stichting voor Fundamenteel Onderzoekk der Materie (FOM)] and was made possible by financial support from thee Netherlands Organization for Scientific Research [Nederlandse Organisatie voorr Wetenschappelijk Onderzoek (NWO)].
T h i ss t h e s i s is b a s e d o n t h e following p u b l i c a t i o n s
C H A P T E RR 3 M . J . Z W A N E N B U R G , H. N E E R I N G S , H . G . F I C K E , AND J . F . VAN D E RR V E E N ,
AA planar x-ray waveguide with a tunable air gap for the structural investigationinvestigation of confined fluids,
Rev.. Sci. Instr. 7 1 , 1723 (2000).
C H A P T E RR 4 M . J . Z W A N E N B U R G , J . F . P E T E R S , J.H.H. B O N G A E R T S , S.A. D EE V R I E S , D . L . A B E R N A T H Y , A N D J . F . VAN D E R V E E N ,
CoherentCoherent propagation of x rays in a planar waveguide with a tun-ableable air gap,
Phys.. Rev. Lett 8 2 , 1696 (1999).
C H A P T E RR 5 M . J . Z W A N E N B U R G , J.H.H. B O N G A E R T S , J . F . P E T E R S , D . O . R I E S E ,, AND J . F . VAN D E R V E E N ,
FocusingFocusing of coherent x rays in a tapered planar waveguide,
Physicaa B 283, 285 (2000).
C H A P T E RR 6 M . J . Z W A N E N B U R G , J . H . H . B O N G A E R T S , J . F . P E T E R S , D . O . R I E S E ,, AND J . F . VAN D E R V E E N ,
X-rayX-ray waveguiding studies of ordering phenomena in confined fluids, fluids,
J . F .. VAN D E R V E E N , M . J . Z W A N E N B U R G , AND W . J . H U I S M A N ,
LayeringLayering at the solid-liquid interface,
Synchrotronn Radiation News 12, 47 (1999).
S.A.. D E V R I E S , P . G O E D T K I N D T , S.L. B E N N E T T , W . J . H U I S M A N , M . J . Z W A N E N B U R G ,, R. F E I D E N H A N S ' L , D . - M . S M I L G I E S , A. S T I E R L E , J . J . D E Y O R E O ,, W . J . P . VAN E N C K E V O R T , P . B E N N E M A , A N D E . V L I E G ,
X-rayX-ray diffraction studies of KDP crystal surfaces, 3.3. Crystal Growth 2 0 5 , 202 (1999).
S.A.. D E V R I E S , W . J . H U I S M A N , P . G O E D T K I N D T , M . J . Z W A N E N B U R G , S.L. B E N N E T T ,, I.K. R O B I N S O N , A N D E . V L I E G ,
SurfaceSurface atomic structure of the (\/3xy/3)R3(P-Sb reconstructions of Ag(lll)
andand Cu(lll),
Surf.. Sci. 414, 159 (1998).
S.A.. D E V R I E S , W . J . H U I S M A N , P . G O E D T K I N D T , M . J . Z W A N E N B U R G , S.L. B E N N E T T ,, AND E . V L I E G ,
FloatingFloating stacking fault during homoepitaxial growth of Ag(lll),
Phys.. Rev. Lett. 8 1 , 381 (1998).
S.A.. D E V R I E S , P . G O E D T K I N D T , S.L. B E N N E T T , W . J . H U I S M A N , M . J . Z W A N E N B U R G ,, D . - M . S M I L G I E S , A. S T I E R L E , J . J . D E Y O R E O , W . J . P . VAN E N C K E V O R T ,, P . B E N N E M A , A N D E . V L I E G ,
SurfaceSurface atomic structure of KDP crystals in aqueous solution: an explanation ofof the growth shape,
Phys.. Rev. Lett. 8 0 , 2229 (1998).
W . J .. H U I S M A N , J . F . P E T E R S , M . J . Z W A N E N B U R G , S.A. D E V R I E S , T . E . D E R R Y ,, D . L . A B E R N A T H Y , A N D J . F . VAN D E R V E E N ,
LayeringLayering of a liquid in contact with a hard wall,
Naturee 390, 379 (1997).
G . F .. Z H O U , M . J . Z W A N E N B U R G , A N D H. B A K K E R ,
AtomicAtomic disorder and phase-transformation in Ll(2)-Ni%Al by mechanical milling,
11 Introduction 11
1.11 Background 11 1.22 Structure determination 15
1.33 The thesis 18
22 Theory of wave propagation in planar x-ray waveguides 21
2.11 Introduction 21 2.22 Modes in a planar waveguide 23
2.33 Mode coupling 29 2.44 Beam propagation method 32
2.55 Numerical methods 33 2.5.11 Finite difference mode solver (FD-MS) 33
2.5.22 Solving coupled mode equations 36 2.5.33 Finite difference beam propagation method (FD-BPM) . . . 38
2.66 Propagation of waves emerging from the waveguide 41
33 A tunable x-ray waveguide 45
3.11 Introduction 45 3.22 X-ray waveguiding 46
3.2.11 Mode excitation and propagation 46 3.2.22 Requirements on the x-ray source 53 3.2.33 Requirements on the waveguide 55 3.2.44 Detection of far-field angular intensity distribution 57
3.33 Apparatus 57 3.3.11 Design 57 3.3.22 Measurement of plate distance and tilt angle 60
3.44 X-ray waveguiding experiments 64
44 Transmission properties of the waveguide with air gap 69
4.11 Introduction 69 4.22 Mode excitation and propagation 71
4.33 Experimental 73 4.44 Results and discussion 74
4.4.11 Far-field diffraction patterns 74 4.4.22 Transmission measurements 76 4.4.33 Multi-mode interference 79
4.55 Conclusions 85
55 Focusing of coherent x rays with tapered waveguides 87
5.11 Introduction 87 5.22 Mode propagation in tapered planar waveguide 88
5.33 Measurement of waveguiding properties 92
66 Ordering phenomena in confined colloids 97
6.11 Introduction 97 6.22 Principle 98 6.33 Experimental 101 6.44 Results and discussion 103
6.55 Conclusions 108
AA On the asymmetry of the angular intensity distribution 109
Referencess 113 Summaryy 119 Samenvattingg 123 Dankwoordd 127
Introduction n
1.11 Background
Inn this thesis we present a method for the structural analysis of fluids in a con-finedd geometry. An everyday example of a confined fluid is a lubricant between twoo sliding surfaces. It is a well-known fact that the properties of a lubricating filmm strongly depend on its thickness, i.e., the degree of confinement. For exam-ple,, very thin lubricating films are known to induce stick-slip motion in which the slidingg object alternates between stick and slip [1]. During these transitions the lubricantt is believed to transform repeatedly from an ordered, solid-like, state to aa fluid state (see Fig. 1.1) and vice versa. In the solid-like state the adhesion betweenn the sliding surfaces may be large enough to cause damage to the surfaces. Fromm an economical viewpoint, this is highly unwanted. Besides being of techno-logicall relevance, confinement-induced ordering phenomena are interesting from a fundamentall point of view.
Lett us illustrate our point by first considering the arrangement of the fluid's constituentss at the interface with a single solid wall. Close to the wall, the fluid orderss in layers parallel to the wall. As a result, the density oscillates with increas-ingg distance from the wall around the value of the density corresponding to that off the bulk fluid. The period of the oscillations is roughly equal to the diameter off the liquid's constituents. At some distance from the wall, the amplitude of the oscillationss decays to zero. At this point, the fluid no longer 'feels' the presence of thee wall and it attains bulk properties.
12 2 ChapterChapter 1
(a) )
(b) )
Figuree 1.1 During stick-slip motion the lubricant alternates between (a) a solid-likelike state and (b) a liquid or fluidized state.
Inn a confined fluid, an additional solid-fluid interface is present. For large distancess between the two surfaces, the region which is influenced by the presence off a surface does not extend far enough into the fluid that it influences the ordering off the fluid at the opposite wall (see Fig. 1.2a). As the wall separation is reduced, thesee regions start to overlap, causing the molecules to 'feel' the influence of both walls.. In this case, the density variations induced by both walls may 'interfere' as iss illustrated in Figs. 1.2b-d. The precise shape of the resulting density profile is determinedd by the nature of the interaction between the fluid's constituents and betweenn the wall and the fluid. A detailed description of the effects of various interactionss is found in Ref. [2], along with references to other work.
Experimentally,, layering at the single solid-liquid interface has been observed inn various fluids. In the case of colloidal suspensions this was achieved by means of opticall microscopy (see Fig. 1.3). Recently, x-ray scattering studies have revealed layeringg on the molecular scale at the interface between liquid gallium and a di-amondd surface [3]. Studies of confined fluids, however, are more difficult because thee region of interest is less well accessible. In general, ordering phenomena on the molecularr scale have been studied with the "surface force apparatus" (SFA) [2]. Thee SFA enables confinement of a fluid between two atomically smooth surfaces whichh may be separated by distances as small as a few tenths of a nanometer. Onee of the surfaces is attached to a spring enabling measurement of the normal forcess acting on it, and in some designs an additional spring is attached for studies
(a) )
R O Q O o n p o Q O o O O
8öo°o
_ 0
oOoo
0
ö^ ^
8 0 0 0 0
oooo o
JOOOO O
o o o o
o
0
o o
ooo o
ooo o
.ooo o
(b) ) (c) ) (d) )Figuree 1.2 (a) Structure of a confined fluid for large wall separations and the
correspondingcorresponding density profile p{x). (b)-(d) Structure models and corresponding densitydensity profiles for three wall separations. From left to right the distance decreases fromfrom approximately 4 to 3 particle diameters. The figures in the center correspond
toto a wall separation of 3.5 particle diameters. In the plots of the density, the dashed curvescurves show the density profiles for a single solid-liquid interface. The solid curve showsshows the sum of the dashed curves.
14 4 ChapterChapter 1
Figuree 1.3 Differential interference contrast microscopy image of the ordering of colloidalcolloidal spheres at a smooth hard wall of a glass capillary. The spheres are made of PMMA-PHSPMMA-PHS and have a diameter of 1 jim. They are dissolved in a refractive-index andand density-matched solvent. The volume fraction of the spheres is approximately
O.48,O.48, just below the freezing transition at 0.494- (By courtesy of W. K. Kegel) off friction and lubrication. The normal forces may have an oscillatory character ass a function of the distance. For the detection of these oscillations extremely fiatt surfaces are required. As the surface separation decreases, the amplitude of thesee oscillations becomes larger. This indicates an increasing degree of order [2]. Theree is believed to be a strong correlation between these force oscillations and the constructivee and destructive 'interference' between the density profiles discussed previously. .
Mostt of the above results were confirmed by computer simulations. Only for colloidall systems which consist of particles with diameters in the range of optical wavelengths,, direct experimental verification was possible (see above). Recently, thee SFA was adapted so as to allow for studies of the structure of confined fluids byy means of x-ray scattering [4]. In this device the x-ray beam passed through thee confined fluid in the direction normal to the confining surfaces (see Fig. 1.4a). Inn such a scattering geometry, however, the scattered intensity is dominated by aa background originating from scattering by the confining walls. Moreover, the sensitivityy to the structure of the fluid is highest in the plane of the surfaces. In orderr to reduce the background and to determine the structure of the fluid across thee gap, we let the x-ray beam enter the device from the side (see Fig. 1.4b). In thiss way, the x rays may be confined to the gap while they propagate along the directionn parallel to the walls. Effectively, the gap acts as a planar waveguide for xx rays.
Figuree 1.4 (a) Normal-incidence scattering geometry employed in the adapted surfacesurface force apparatus, (b) Our scattering geometry, in which the x-ray beam entersenters the device from the side.
Thee subject of the thesis is the exploitation of the waveguiding properties of thee confining geometry for the determination of the structure of a confined colloid. Forr a colloidal suspension of ~ 100 nm particles, confinement effects can already bee expected at gap sizes of a few hundred nanometers. This makes the system is welll suited for analysis by our waveguiding method.
1.22 Structure determination
XX rays are widely used for imaging opaque objects. On airports the content of our suitcasess is screened while in the hospital doctors use them to check the contents off our body. These applications are based on the fact that x rays penetrate matter deeplyy and that different materials have different absorption cross sections. Due too their small wavelength, x rays are also suitable for the determination of the microscopicc structure of matter.
Thee conventional imaging techniques mentioned above rely on absorption con-trastt and totally neglect the potential of phase contrast. In phase-contrast imaging onee illuminates the object (or part of it) with a (partially) coherent wave field. As thee wave field traverses the object, phase differences may arise between different partss of the wave front, which enhances the contrast in the image. Such phase differencess are due to variations in the refractive index of the object, which for
16 6 ChapterChapter 1
(a)) (b)
Figuree 1.5 Imaging of a locust using a) absorption contrast and b) phase contrast. FromFrom Ref. [5j.
x-rayy wavelengths is given by
nn = l-S + i/3. (1.1) Here,, (3 = a\/4-ir incorporates the absorption of the x rays, with A the wavelength
off the x rays and \i the absorption coefficient. This term affects the amplitude of thee waves. The term 6 = \2rene/2w, with re the classical radius of the electron
andd ne the electron density of the material, incorporates the refraction of waves
andd affects the phase of the x rays. For sufficiently small wavelengths (A < 0.1 nm),, (3 may be as small as 1CT9 whereas 8 is of the order of 1CT6. Thus in the x-rayy regime it is possible to observe phase contrast without having to deal with absorptionn contrast.
Ann advantage of being sensitive to phase contrast is that already the smallest variationss in the refractive index can be detected. For this reason phase-contrast x-rayy imaging has great potential for application in medicine and in biology, enabling onee to differentiate between different kinds of soft tissue (see Fig. 1.5). In confined fluids,, the variations in the refractive index due to layering effects, as discussed above,, may also be very small.
Inn our waveguiding geometry (see Fig. 1.6) the x rays are confined to the gap whereinn the fluid resides. This is achieved if the refractive index of the fluid is higherr than that of the walls, i.e., if the density of the fluid is lower than that of
Figuree 1.6 Schematic of the waveguiding geometry (not to scale). The fluid within
thethe gap between the confining walls has ordered in six layers. The layering gives riserise to pronounced oscillations in the density in the direction perpendicular to the confiningconfining plates. Due to these variations the incoming waves are scattered into differentdifferent directions. As a result of the layering, the transmitted intensity distribu-tion,tion, which is measured in the far field, exhibits pronounced peaks at different exit anglesangles 6e.
thee walls. In this way we do not have to be concerned with background scattering fromfrom the confining walls. Whenever there are variations in the refractive index n(x,z),n(x,z), the incident wave front experiences phase changes and is refracted into otherr directions. The refracted waves interfere with all the other waves present inn between the walls, resulting in a strongly modified field distribution at the exit off the waveguide. For these experiments an x-ray source with a sufficiently high spatiall coherence is required. Such a source is available at, e.g., the European Synchrotronn Radiation Facility in Grenoble (France), where we have performed thee experiments discussed in this thesis.
Forr imaging an object, one generally places a position-sensitive detector behind it.. Depending on the distance R between the object and the detector, three regimes forr detection can be distinguished. In the 'contact' or 'near-contact' regime, in whichh the detector is placed directly behind the object, the observed contrast in the imagee is due to absorption. In the Fresnel diffraction regime for R < W2/\, where WW is the size of the object, phase contrast is present in the image, which depends
18 8 ChapterChapter 1 onn the distance R. In the Fraunhofer diffraction regime, which corresponds to distancess R > W2/X, the observed contrast is independent of R. In the latter regime,, a Fourier transform of the transmission function of the phase object is observed.. In our experiments W is the distance between the confining walls, which iss typically smaller than 600 nm. Given a wavelength A = 0.1 nm and a detector att more than 1 m distance from the exit of the waveguide, all of our experiments weree performed in the Fraunhofer regime (R » 3.6 mm).
Wee deduced the refractive-index profile of a confined colloid from measured Fraunhoferr diffraction patterns, i.e., from far-field intensity distributions as a func-tionn of the exit angle 8e (see Fig. 1.6). The patterns were typically measured for
aa range of incidence angles 0*, which results in a measured intensity distribution I{@t,I{@t, #e)- A direct inversion of I(0j, 6e) into the refractive-index distribution n(x, z)
iss difficult because information on the phase of the field within the waveguide is lostt in the measurement (the well-known 'phase problem' [6]). Here we also note thatt the confined colloid is a thick phase object:
66maxmaxLL « l O x A (1.2)
forr a maximum refractive-index contrast within the fluid of <5max ~ 1 x 10~6 and
ann object thickness equal to the waveguide length L — 5 mm. The large opti-call thickness makes a direct inversion a 'tour de force'. We therefore have taken recoursee to the following model-dependent analysis. Assuming a model for the refractive-indexx distribution n(x, z) we calculate the field at the exit of the wave-guide.. From this, we derive a far-field diffraction pattern which we compare with thee measured diffraction patterns. If there is no match between the measurements andd the calculation we adjust the model and repeat the procedure until matching iss achieved (see chapter 6).
1.33 The thesis
Thee outline of the thesis is as follows. In chapter 2 the theory of propagation of x rayss through planar waveguides is discussed. In addition, a few numerical methods aree presented for the calculation of wave fields within planar waveguides having variouss refractive-index profiles. Chapter 3 describes the experimental setup of thee x-ray waveguide together with the requirements imposed on the x-ray source.
Inn a first essential step, we have studied the coherent propagation of x rays in the absencee of a fluid. This is the subject of chapter 4. In chapter 5 we show that it is possiblee to focus x rays into an extremely fine line by tapering a planar waveguide. Inn chapter 6 we present a determination of the structure of a confined colloidal suspension,, using the waveguiding method discussed in the preceding chapters.
Theoryy of wave propagation in
planarr x-ray waveguides
InIn this chapter we calculate the electromagnetic field amplitudes within a planar
waveguide.waveguide. Emphasis is put on methods applicable to waveguides having guiding layerslayers which are homogeneous, inhomogeneous or tapered. The underlying physical principlesprinciples of these methods and their numerical implementation will be discussed. Far-fieldFar-field diffraction patterns from the exit of the waveguide are determined from the calculatedcalculated field distributions in order to compare them with experimentally obtained diffractiondiffraction patterns.
2.11 Introduction
Thee propagation of electromagnetic (e.m.) waves through waveguides plays an importantt role in everyday life. For example, phone calls and television broadcasts reachh us via networks of guided-wave devices such as optical fibers and integrated-opticss devices. These technologies could be further developed thanks to the effort whichh was put into understanding their operation. As a result, a wide variety of mathematicall methods exists for the calculation of e.m. field amplitudes within these,, sometimes, very complex devices [7].
Inn general, these methods consist of solving Maxwell's equations which describe thee propagation of e.m. waves through a medium. Our aim here, is to apply some off these methods to the propagation of x rays in planar waveguides. We mainly
n,>n2 2
ChapterChapter 2
(b) )
(c)) (d)
Figuree 2.1 Schematics of the waveguiding geometries encountered in experiments:
(a)(a) empty waveguide, (b) waveguide filled with a layered medium (c) inhomoge-nouslynously filled waveguide and (d) waveguide with a tapered guiding layer. The arrows pointingpointing from left to right in (a) indicate the propagation direction of the x rays. focuss on how the amplitude and phase of the e.m. fields are affected by variations off the refractive index, i.e. the density variations, within the guiding layer of thee waveguide. For the waves propagating within the waveguide, these refractive-indexx variations act as scattering centers. In integrated optics such variations, whichh are mainly caused by errors in the production process, are unwanted, since theyy induce power losses and cross-talk. Most of the mathematical methods were thuss applied in order to quantify and reduce these scattering effects. By contrast, wee aim at maximizing these effects so as to obtain the spatial distribution of the refractive-indexx variations.
Forr this purpose, we need to calculate both the amplitude and phase of the e.m.. field. Often used simplifications of the mathematical treatment, which only yieldd the power within the waveguide [8], may not be applied since they average overr the phases. However, the complication of calculating both the amplitude and thee phase is largely compensated by simplifications which arise from working at x-rayy wavelengths. For example, we may neglect backward scattering of waves fromm refractive-index variations.
Wee discuss methods which are applicable to the waveguiding geometries en-x en-x
f f
n;f W W ni i
counteredd in our experiments (see Fig. 2.1). The planar geometries shown in Figs. 2.1aa and 2.1b are invariant under translations along the z-direction (and the y-direction),, whereas those corresponding to Figs. 2.1c and 2.1d are not, i.e. the refractive-indexx profiles depend explicitly on z.
Inn the former case, the e.m. field distribution within the waveguide is elegantly describedd in terms of modes [9]. These are special solutions of Maxwell's equations whichh form a complete and orthogonal set. Hence, any e.m. field distribution withinn these waveguides may be written as a superposition of modes. In the latterr case, strictly speaking, no modes exist. However, the e.m. field may be describedd in terms of modes belonging to the undisturbed waveguide (Fig. 2.1a) whichh are coupled due to the refractive-index variations. Although the coupled modee description is convenient for understanding the effect of these variations on thee propagating waves, it is less suited for the numerical calculation of the e.m. fieldfield amplitude within the waveguide. Instead, we employ the numerical Beam Propagationn Method (BPM), which calculates the total e.m. field directly instead off performing a decomposition into modes.
Inn the following sections, we discuss the underlying physical principles of the methodss in the above order, after which we consider their numerical implementa-tion.. The chapter concludes with a section dedicated to the propagation of waves emergingg from the waveguide into the free space behind the waveguide.
2.22 M o d e s in a p l a n a r waveguide
InIn linear, dielectric, non-magnetic and source-less media the propagation of waves iss described by the Maxwell equations (in SI units) [9],
V x EE = - / i0— ; V-eon2E = 0,
(2.1) ) VV x H = e0n2—; V /z0H = 0,
wheree E is the electric field, H the magnetic field, n the refractive index, e0 the
free-spacee permittivity, /i0 the magnetic permeability and t the time variable. If we
assumee the x rays to be monochromatic, so that the time dependence of the fields iss e™*, where a? = 27rc/A = k0c is the radial frequency, with A the wavelength,
24 4 ChapterChapter 2 vacuum,, then separation of the time and space coordinates leads to the Helmholtz equationss [7],
AEAE + kln2E = - V { V ( l n n2) - E ) (2.2a) AHH + k20n2K = - V ( l n n2) x V x H. (2.2b)
Here,, A is the vectorial Laplace operator.
Lett us first consider the propagation of waves through a planar waveguide with aa homogeneous guiding layer and cladding (see Fig. 2.1a), i.e., Vn2 = 0 and Vn2, = 0,, respectively. Then, the right-hand side of Eqs. (2.2a) and (2.2b) vanishes, i.e. thee vector components of the electric and magnetic fields are decoupled. However, thee step in the refractive index across the interface between the guiding layer andd cladding imposes boundary conditions on the fields. Across the interface, the followingg components of the vector fields have to be continuous: Ey, Ez, Hx, Hy,
andd Hz [9]. The field component Ex has to fulfil the condition n\Ex = n\Ex at
thee interface, where the superscripts (1) and (2) refer to the field in the guiding layerr and the cladding, respectively. These boundary conditions lead to two types off solutions to Eqs. (2.2): transverse-electric (TE) and transverse-magnetic (TM) waves.. As their names indicate, both solutions consist of field components that are transversee to the propagation direction. As depicted in Fig. 2.1a, we choose our coordinatee system such that the waves propagate along the z-direction. Then, for thee TE waves, Ez — 0 holds everywhere and Hz is continuous across the interfaces.
AA TE wave consists of the following non-zero field components: Ey, Hx, Hz. The
TMM waves fulfil Hz = 0 everywhere while Ez is continuous across the interfaces.
Forr TM waves the field components Hy, Ex and Ez are non-zero.
Iff inhomogeneities are present (Vn^ / 0 or Vn2, ^ 0), all the field components aree coupled and the solutions to Eqs. (2.2) cannot be expressed in terms of TE or TMM waves only. However, if An/n -C 1, where An is the variation of the refractive indexx over a distance of one wavelength, the coupling may be neglected [7]. This iss the case at x-ray wavelengths, since An ~ 10- 6 and n ~ 1. Since the x rays in ourr experiments are TE polarized, we focus on the TE waves only.
Forr TE waves and in the absence of coupling, the Helmholtz equations (2.2) reducee to VV22EEyy + k2n2Ey = 0, (2.3a)
"-"- = -£;& <
2
-
3b
>
* . - — ^ .. (2.3c) (J,(J,00UJUJ oxwheree V2 = d2/dx2+d2/dy2+d2/dz2. At this point, we assume that the refractive-indexx profile is independent of the ^-coordinate and that the fields do not vary in thee y-direction. Then, Eq. (2.3a) can be further simplified by separating the variabless x and z, while assuming a travelling-wave character of the TE waves alongg the z-direction:
V{x,z)V{x,z) = <f>{x)e-i(ix, (2.4) wheree ty(xyz) denotes Ey(x,z) for ease of notation. The so-called propagation
constantt /3 of the TE wave [8], is the ^-component of the wave vector k with |k|| =nifc0. Substitution of Eq. (2.4) into Eq. (2.3a) results in the differential
equation: :
gg + (fcoV-/?2)0 = O. (2.5)
Sincee Ey and Hz should be continuous across the interface between the guiding
layerr and cladding, <j>(x) should be continuous and differentiable (see Eq. (2.3c)) att the interface. If we require that the fields vanish at infinity, then, by solving Eq.. (2.5), we obtain the so-called guided modes [8] (Fig. 2.2a). If we do not apply thee latter boundary condition, we obtain the so-called radiation modes [8] (Fig. 2.2b). .
Lett us now consider, as an example, the properties of the modes corresponding too a piecewise constant refractive-index profile as depicted in Fig. 2.2. A guided modee solution of Eq. (2.5) can generally be written as
4>(x)4>(x) =
C e - ^ ll x < 0
Ae~Ae~ikik**** + BeikxX 0 ^ x < W , (2.6) (Ae~(Ae~ikik**ww + Beik*w)e-^x-w) x ^ W
26 6 ChapterChapter 2
n22 n1
x=W W
x=0 0
n(x)) (a)
(b) )
Figuree 2.2 (a) A guided mode and (b) a radiation mode for a waveguide with a
piece-wisepiece-wise constant, stepped, refractive-index profile.
wheree kx = \/n\k$ — 01 and 7 = \J'ft2 - n\k% [8]. Within the guiding layer
thee solution is a superposition of two plane waves counter-propagating in the x-direction,, whereas within the cladding it decays to zero exponentially as x —» , providedd 7 is real and positive. Accordingly, the guided modes have a propagation constantt in the range n2fco < l/3| <
«afco-Inn order to obtain A/B, C/B (where B may be determined from the power off the e.m. field) and (3, we apply the requirement that 0(x) be continuous and differentiablee at x = 0 and x = W (see text below Eq. (2.5)). This leads to the followingg system of non-trivial equations:
AA + B = C,
-ik-ikxxAA + ikxB = jC,
ZfCx-riG G ++ ikxBe ikikxxW W == — 'yAe -ik-ikxxW W -/Be -/Be ikikxxW W
Solvingg this set of equations, we obtain A A B B C_ C_ B B ikikxx - 7 ikikxx + 7 ' (2.7a) ) (2.7b) ) (2.7c) ) (2.8a) ) (2.8b) ) ikikxx + 7
Thee two ratios correspond to the complex amplitude reflection and transmission coefficients,, respectively [10]. By substitution of Eq. (2.8a) into Eq. (2.7c) we
n, ,
Figuree 2.3 Illustration of the ray approach to waveguiding. All rays that travel in thethe same direction belong to the same plane wave.
obtainn the equation
| ^ - e - ,, (2.9) whichh determines the values of /?, given ri\, n2 and k0. This equation has only a
finitefinite number of solutions j3 = /3m, where m = 0,1,2,..., mmax is called the mode
numberr [8]. The corresponding guided-mode profile is denoted as <f>m(x).
Forr |/3| < n2fco, 7 is imaginary. In this case the field profile in Eq. (2.6) would
consistt of a standing wave within the guiding layer and outward propagating waves withinn the claddings. Such a solution, however, is not normalizable. In Ref. [8] thiss issue is addressed in detail. Here, it suffices to note that for these values of (3,(3, the solutions to Eq. (2.5) are called radiation modes and that they form a continuumm instead of a discrete set.
Inn a ray-optics approach the modes are described in terms of rays instead of wavee fields. Rays are the lines which cross the surfaces of constant phase of the wavee field at right angles, see Fig. 2.3. In order to derive the mode properties, wee consider the phase changes that a ray acquires upon propagation through a waveguidee with a piece-wise constant refractive-index profile.
Propagatingg across the guiding layer at an angle 9, a ray picks up a phase changee konjW/ sinö. Upon reflection from the interfaces an additional angle-dependentt phase change <p is picked up as the complex amplitude reflection coef-ficient,ficient, given by Eq. (2.8a), is written as,
28 8 ChapterChapter 2 where e
tan(vV2)) = i/ikx. (2.11)
Here,, \r\ is the intensity reflection coefficient. The discreteness of the guided modess follows from combining the phase changes if we take into account that the rayy from A to B is not reflected whereas the one from C to D is reflected twice [8]. Becausee points on the same phase front must be in phase, the optical path length differencee between the rays AB and CD must be a multiple of 2n, i.e.
A;0n1(s22 — si) — 2(f — 2irm, (2-12)
wheree m is an integer and «i and s2 are the distances AB and CD, respectively.
Althoughh derived from a different point of view, Eq. (2.12) is identical to Eq. (2.9).. Assuming that the waves do not penetrate the cladding, i.e. 7 —> 00, we findd from Eq. (2.11) that if = IT. Inserting s2 — S\ = 2W sin# « 2W6, it is possible
too solve Eq. (2.12) for the angles corresponding to the guided modes, __ (m+l)7T _ (m+l)7T
VmVm
~~ n.koW ~ k0W ' [2A6)
wheree m is the mode number. With j3m — nikocos6m we obtain the solutions to
Eq.. (2.9).
Forr a guided mode, the mode angle 9m is smaller than the critical angle for
totall internal reflection 6C. Hence, the total number of guided modes is
mm a xx = k0Wec/7r - 1. (2.14)
Thee radiation modes correspond to rays propagating at angles 9 > 6C. Since there
iss no total internal reflection from the interfaces, a radiation mode is not confined too the guiding layer.
Althoughh the modes for a waveguide with a piece-wise constant refractive-index profilee may be completely different from those for a waveguide with a varying pro-filee n(x), they have one important property in common. Since they are both solutionss of Eq. (2.5), which, together with the boundary conditions, is a Sturm-Liouvillee type boundary-value problem, they constitute an orthogonal set [11]. Byy combining this property with the fact that the guided modes and the radia-tionn modes form a complete set, we may express any field distribution within the
waveguidee as a superposition of guided and radiation modes [8], i.e.
*(*,, z)=Yi cm<j>m{x)e-i^ + / c^We-^dp, (2.15)
m=00 J°
wheree p labels the continuum of radiation modes. The coefficients \cm\ and \cp\
aree proportional to the power carried by the respective modes. The value of cm,
andd similarly that of cp, is given by the projection of the field profile at the entrance
off the waveguide onto the modes,
em=em=
m * . ^ *
(2'
16)Inn most cases, the contribution of radiation modes may be neglected. Once a radiationn mode is excited, it loses its power due to absorption in the cladding, so thatt it will not reach the end of the waveguide.
2.33 Mode coupling
Densityy variations within a waveguide cause its refractive index to depend on the spatiall coordinates like n — n(x) or n = n(x,z). In the first case, the wave fieldfield within the waveguide can be described in terms of modes as discussed in thee previous section. Nevertheless, it is sometimes more convenient to treat the refractive-indexx variation along re as a perturbation of the piece-wise constant refractive-indexx profile corresponding to the empty waveguide and to describe the wavee field in terms of the empty waveguide modes. In this way it becomes easier too understand the effect of the refractive-index variations on the wave field within thee waveguide. In the second case, strictly speaking no modes exist. However, wee can still write the wave field as a superposition of orthogonal modes. Usually, wee take the modes belonging to the empty waveguide or the modes belonging to,, e.g., n(x,0), as our basis set. The latter is more convenient for studying the influencee of a 2-dependence if n(x, z) = f(x)g(z). In both cases, the modes of thee 'undisturbed' waveguide are coupled to each other due to the refractive-index variations,, i.e., upon propagation they exchange power continuously.
Inn order to determine the coupling strengths between modes for the most generall case, we write the field amplitude within the perturbed waveguide, with
30 0 ChapterChapter 2 refractive-indexx profile n(x,z), as a superposition of the modes 0m belonging to
thee undisturbed waveguide having a refractive-index profile no(x):
00 0
* ( * ,, z) = Y, cm{z)4>m{x)e-i^\ (2.17)
m=0 0
wheree the propagation constant j3m corresponds to the mode </>m. The expansion
coefficientss {(^(z)} now depend on z. We have simplified the notation of Eq. (2.15)) by setting the upper limit of the summation to infinity, so as to indicate the contributionn of radiation modes. The linear combination is only in terms of forward propagatingg modes; at x-ray wavelengths there is negligible backward scattering off waves from refractive-index variations because An/n <C 1. Substitution of Eq. (2.17)) into Eq. (2.3a) results in
m=00 ^ / \ /
(2.18) ) Iff we now make use of the fact that the modes 4>m fulfil Eq. (2.5) (with n = no(x)),
wee obtain
(2.19) ) Multiplyingg both sides in Eq. (2.19) with <j>*k and by making use of the
orthogo-nalityy of the modes, gives d?cd?ckk dck kl °° 2 ~ 2 l ^^ = -yk n KK m=0 where e yoo o '22 dx (2.21) ^^ - 2i0k^ = - f £ ^ " ^ j ~ & [n(x, zf - n0(*)2] ^mdx, (2.20) / oo o \4>k{x)t \4>k{x)t -oo o
iss the power in mode k. Now we neglect the second-order derivative in z, since [12] ]
Thiss reduces Eq. (2.20) to the following set of coupled differential equations:
^ « e u ^ r ^ f z K ^ - ^ , ,
(2.23) ) m=0 0 where e mk mk kk 22 f°° »» = 2i(3°PkJ (l)*k{x)[n(x,z)2-no(x)2]<l)m(x)dx, knkn«« 2ik /_„
0ï(ar)^ *
)2"
n o (*
) 2] ^
^ (
2-
24)
givess the strength of the coupling between mode m and k. The initial condition ccmm{z{z — 0) is provided by projecting the wave field which is incident onto the
waveguidee entrance onto mode m, see Eq. (2.16).
Itt is seen that the cm's oscillate as a function of z. This results in a field
distri-butionn which strongly depends on the spatial coordinates due to the interference off the modes caused by the term exp [i(@k — j3m)z]. In the case that only mode m
andd k are coupled, the period of the oscillations of |cm(2)|2 is given by [12]
LLcc = . * — , (2.25)
wheree L^ = n/({3m — /3k) is the beating length of the modes m and k. As the
strengthh of the refractive-index variations increases, the coupling length Lc
de-creasess and the variations in the amplitude of the field are more rapid.
Inn order to solve the coupled-mode equations we distinguish two limits. The firstt is the weak-coupling limit in which the strength of the refractive-index vari-ationss is such that Lc is much larger than the length L of the waveguide. In this
case,, a perturbation solution to Eq. (2.23) can be found [8]. It is assumed that thee amplitude of the modes do not change much over the length of the waveguide. Byy treating it approximately constant, we may integrate Eq. (2.23) and obtain thee approximate solution to Eq. (2.23):
cckk(L)(L) « cfc(0) + J2 c-(°) / Tmk(z)é^-^zdz. (2.26)
Thee second limit is that of strong coupling. In this case the coupled-mode equa-tionss (2.23) are solved numerically, as will be discussed in section 2.5.2. It is in thiss regime that one obtains maximum sensitivity to the spatial distribution of the refractivee index.
322 Chapter 2
2.44 Beam propagation method
Soo far we have discussed methods that are based on the propagation of modes. Byy contrast, the beam propagation method (BPM) describes the propagation of wavess in terms of total fields. This powerful numerical method was originally developedd for use in underwater acoustics and seismology, before it was adapted too the simulation of wave propagation in optical devices [13]. It solves the wave equation,, Eq. (2.3a), directly, given the refractive-index profile n(x, z) and the fieldd profile incident on the waveguide entrance. In its basic form, the use of the BPMM is restricted to wave propagation problems which involve scalar fields and forwardd propagating waves only. Alternative formulations of the BPM exist, which aree able to handle vector fields [14, 15] and even back-reflections [16].
Inn practice, we apply the BPM whenever the refractive index is ^-dependent andd absorption is non-negligible. The latter is taken into account by adding a imaginaryy part to the refractive index, i.e. n — 1 — S + ia, where a = A^/47r and fifi the wavelength-dependent absorption coefficient [17]. The BPM is also used to checkk the consistency of results obtained with the modal techniques. It is even possiblee to obtain the relative strengths {cm} of the modes and their propagation
constantss {/?m} from the BPM calculations [18].
Wee derive the principle of the BPM from the Helmholtz equation for TE waves:
VV22VV + k%n2{x, z)tf = 0, (2.27)
wheree V2 = d2/dx2 + d2/dz2 and # = Ey(x,z). First, we apply the so-called
slowlyy varying envelope approximation [19] by substituting
V{x,z)V{x,z) = *!j{x,z)e-iez, (2.28) intoo Eq. (2.27), which removes the rapid variations from the field. The constant
(3(3 is the reference propagation constant and denotes any representative value of kk00n(x,n(x, z) [19]. In our calculations we always used $ = ko since n{x, z) « 1. Now,
iff we neglect the d2^/dz2-teim1 we obtain
WW « f d \ *,-2 „.,. (2.29)
dzdz 2k0
-U-U
dxdx-
*»v
- D*
2 2Thee essence of the BPM is to integrate Eq. (2.29) numerically. It calculates ip(x,ip(x, z + Az), given ip(x, z). This may be achieved in different ways [7, 19]. Section 2.5.33 presents a detailed account of the numerical implementation we have used.
Byy neglecting the d2ijj/dz2-term we applied the so-called paraxial or parabolic approximationn [19], which is equivalent to the approximation (2.22). This approx-imationn is accurate if variations in the refractive index are small, i.e. An/n <C 1, andd if the angular spectrum of the fields is narrow, i.e. Akx/ko «C 1 [19], where AkAkxx is the spread in kx. As discussed before, the first condition is easily met
forr x rays. In most cases, the second condition is also easily met. If, however, AkAkxx is large, it is advisable to approximate d2iip/dz2 using a method of successive
approximationn [20].
2.55 Numerical methods
2.5.11 Finite difference mode solver (FD-MS)
Onlyy for very few refractive-index profiles n = n(x) can the modes be solved analytically,, i.e. for stepped refractive-index profiles and parabolic refractive-index profiless [7]. In the other cases they are found by numerically solving:
0+[fc
o2n(x)
2-/?
2]00 = O. (2.30)
Thee ^-coordinate is discretized such that (J)(XQ + iAx) — <f>(xi) = <f>t, where i =
0,...,, N — 1, XQ the lower limit of x and Ax the step size. The derivatives are approximatedd by difference expressions [21], which changes Eq. (2.30) into
^ ' w ' ^ ^ ^ ' ^ ^^
(2
-
31)
Noww we group the <^'s into a column vector (f> = (<^0,0l5... , <t>N_2,0jv_i) and write
Eq.. (2.31) as a matrix eigenvalue equation
MM • <}> =/32<f>, (2.32)
wheree M is a N x N tri-diagonal matrix with the following non-zero elements: 2 2
MMitiiti = -j^^ + klnixi)2, (2.33a)
MM_xx = Miti+1=-~. (2.33b)
Manyy numerical procedures are available for finding the eigenvalues and eigenvec-torss of the matrix M. In this thesis, we applied the powerful QR-algorithm after
34 4 ChapterChapter 2 transformingg M to a Hessenberg form [22]. Once the eigenvectors and eigenvalues, i.e.. the modes and propagation constants, are known, the field distribution within thee waveguide is calculated using Eq. (2.15).
Thee matrix M, with the matrix elements (2.33), includes the assumption 4>_4>_11 = 0 and <j)N — 0. The boundaries of the computational window must
there-foree be sufficiently far away from the guiding layer so that the amplitudes of the guidedd modes have decayed to zero. The obtained radiation modes are those of thee continuous spectrum that fulfil these boundary conditions. Therefore, it is advisablee to use the FD-MS only for calculations which do not require an accurate descriptionn of radiation phenomena.
Thee step size Ax should be much smaller than the oscillation period of the mode: :
Axx < „:, (2.34) wheree ni is the refractive index of the guiding layer and j3m the propagation
con-stantt corresponding to the mode <pm. The convergence of the numerical method
wass checked for a waveguide with a stepped refractive-index profile. The calcu-latedd mode profiles <j>m(x) are shown in Fig. 2.4 for a few step sizes. The lowest
modee 0O is found to converge already for Ax = 24 nm, whereas for cf>6 a step size
off Ax << 24 nm is needed. However, more stringent is a convergence test of the propagationn constants {Pm}, see Fig. 2.5. The relative error (/?m - ^ ) / / 5 ^ in the
propagationn constants, with (3^ the value to which j3m converges, decreases with
stepp size as shown in Fig. 2.5b. It is seen that the difference in relative errors for mm = 0 and m = 13 reduces to only one order of magnitude for Ax < 3 nm. In conclusion,, the higher modes and corresponding propagation constants should be calculatedd with much smaller step size than the lower ones.
Thee finite-difference mode solver has been used to calculate field distributions withinn the waveguide. An example of such a calculation is shown in Fig. 2.6. The x-rayy beam entered the waveguide from the left and the field profile at the entrance $(x,$(x, 0) was chosen to excite both the TE2 and TE3 modes. The strong intensity
variationss along the x and z directions are the result of interference between modes duee to the difference in their propagation constants (see chapter 4 for a detailed discussion).. In Fig. 2.6a only guided modes were taken into account, whereas inn Fig. 2.6b also radiation modes were considered. In both cases we did not use
(a) ) o o Ax=6 6 Ax=24^ ^ A x = 4 8 , , -4000 -200 0 200 400 xx (nm) CO O -4000 -200 0 200 400 xx (nm) E E -4000 -200 0 200 400 xx (nm) 0.0 0 -200.0 0
F i g u r ee 2.4 Mode profiles <f>m(x) calculated for a waveguide with a stepped
refractive-indexrefractive-index profile, with nx= 1, n2= 1 — 2.57 x 10~6, W = 400 nm and
XX = 0.0931 nm. The dependence of the mode profiles on the step size Ax is shownshown for the modes (a) m = 0, (b) m = 4 and (c) m = 6. The vertical lines indicateindicate the boundaries between the guiding layer and the cladding, (d) The prop-agationagation constant /3m as a function of mode number for Ax = 6 nm. The size of
36 6 ChapterChapter 2 E E «a, , 55 10 15 20 25 Axx (nm) 55 10 15 20 25 Ax(nm) )
Figuree 2.5 (a) f3m calculated as a function of the step size Ax for mm = 0,1,2,..., 15 (mmax= 19), using the same parameters as in Fig. 2.4- (b) RelativeRelative error of the propagation constant, (/3m—/3^n)//3^n, as a function of Ax for mm = 0 and m = 13. For {Fm we took the 0m value found for Ax = 1.5 nm.
thee second term on the right-hand-side of Eq. (2.15) but only the first term with mm MM = 19 and mmax = 40, respectively. As expected, the intensity in Fig. 2.6a is
confinedd within the guiding layer. In Fig. 2.6b, also intensity outside the guiding layerr is observed. This is due to the excitation of radiation modes by the incident fieldd profile. In the absence of absorption, the radiation modes exit the waveguide withoutt attenuation.
Comparingg Figs. 2.6a and 2.6b, we only observe small differences between the intensityy distributions within the waveguide. This indicates that the field within thee guiding layer is accurately described by an expansion of the wave field in terms off guided modes only.
2.5.22 Solving coupled mode equations
Wee describe waveguiding in terms of coupled modes in order to understand the couplingg mechanism due to variations in the refractive-index profile. In the follow-ingg we solve the set of coupled differential equations Eq. (2.23). Let us consider a refractive-indexx profile n = n(x). In this case the coupling coefficients rmt in Eq.
800 0 400 0 E E c c -400 0 800 0 400 0 E E x x -400 0
F i g u r ee 2.6 Calculated intensity distribution \fy(x,z)\ within a waveguide having
aa stepped refractive index profile (ri\— 1 and r\%= 1 — 2.57 x 10~6) and a guiding layerlayer of width W = 400 nm. The incident field, with A = 0.0931 nm, was chosen soso as to excite both TE3 and TEA. In (a) only guided modes were taken into account
andand in (b) also radiation modes were considered. The step sizes are Ax = 6 nm andand Az = 0.01 mm and the computational window ranged from —1000 nm to 12001200 nm.
38 8 ChapterChapter 2 Inn order to solve the set of equations Eq. (2.23)
dcdck k dz dz
-T-T
LL= Y,
TmkCm exP W * - 0m)*). (2-35)
m=0 m=0 wee substitute Thiss results in Cm(z)Cm(z) = am(z)eif3™z. (2.36) ^ ^^ = f ) (Tmk ~ 0m6mk) am(z). (2.37)Thiss is a matrix equation, dz dz
m=0 m=0
^ = - « Q - a ( z ) ,, (2.38) withh the z-independent matrix elements defined as
QQmkmk = iTmk + 0m8mk (2.39)
andd the column vector a given by a(z) = (ao(z), ai(z), • )• The solution to Eq. (2.38)) is given by
oo o
a(z)) = exp (-iQz) a(0) = £ e""»* [vj • a(0)] vp, (2.40)
wheree {Ap} and {vp} are the eigenvalues and eigenvectors of the matrix Q,
respec-tively.. Numerically, these are obtained as discussed in section 2.5.1. The electric fieldd amplitude follows from
oo o
*(ar,z)) = 5]am(2r)0ra(a:), (2.41)
m=0 0
wheree {am(z)} are the components of a(z) as calculated from Eq. (2.40), and
{</>m}} the modes for a waveguide with a refractive-index profile n = no(x).
2.5.33 Finite difference beam propagation method (FD-BPM)
Inn order to evaluate the field profile within a waveguide having a refractive-indexx profile n — n(x,z), we have to solve the differential equation (2.29) nu-merically.. This is accomplished by approximating the derivatives with finite-differencee expressions. Therefore, we discretize the field amplitude, ip{x,z) —ip(xuip(xu + iAx, ZQ + jAz) = tpl, with XQ and ZQ the lower limits of x and z, respec-tively,, and i = 0,l,...,N —I and j = 0,l,...,Nz — l. Applying the discretisation,
ass given in Eq. (2.31), to Eq. (2.29), we obtain the following matrix equation
¥ ^
M** <
2 4 2>
wheree ij>j = (^Q, ..., V'JV-I) • The Nx N matrix M' is tri-diagonal with the following non-zeroo matrix elements:
KiKi = T A * - *ï ["O*)
2" 1] , (
2'
43a)
(Ax) ){Ax) {Ax)
Iff we would assume the fields to be zero for x_i and xjv, unwanted reflections of wavess propagating towards the boundaries of the computational window would occur.. In order to avoid this, we employed so-called transparent boundary condi-tionss [23]. These are valid under the assumption that close to the boundary the fieldfield is a plane wave, i.e.
^^ = ^ = exp(ikxAx) = q0 (2.44)
andd similarly for the boundary values for i = N — 1. The boundary conditions are implementedd by replacing MQ0 and M'N_1 N-l with the following expressions:
M
ó.oo = 7 7 ^ 7 5 - * o [ » » ( * o )2- l ] + * > (2.45a) (Aar) )
M'N-I,N-IM'N-I,N-I = ——ï-kZInixN-tf-ïl+qN-i (2.45b)
(Ax) )
Byy integrating Eq. (2.42) with the use of the implicit Crank-Nicholson scheme [19,, 22] we obtain a stable algorithm [22]. It consists of solving
forr ^+ 1. In essence, the solution found for z = Zj is propagated forward to zz = Zj+i by means of
// „ A ^ \ _ 1
(
I+
£*)~~
* ™
40 0 ChapterChapter 2 E E c c 800 800 400 0 -400 0 22 3 zz (mm)
Figuree 2.7 Intensity distribution \ty(x, z)|2, calculated with the FD-BPM,BPM, within a waveguide having a stepped refractive index profile (m= 1, rara22== i - 2-57 x 10~6+il.25 x iO"8 and W = ^#0 nrnj. 77ie step sizes Ax and
AzAz were 6 nm and 0.01 mm, respectively. The computational window ranged fromfrom -1000 nm to 1200 nm and the incident field, with \= 0.0931 nm, was
chosenchosen to excite both the modes TE3 and
wheree I denotes the identity matrix. With Eq. (2.47), the problem of solving a complexx differential equation is reduced to inverting the matrix I + («Az/4feo)M'. Numericallyy this is not a difficult task.
Ass for the FD-MS, we have to check the convergence of the obtained solution. Here,, three parameters influence the convergence: Ax, Az and the size of the computationall window. In practice, we repeat the calculations with reduced step sizess until the solution converges. The size of the computational window must be sufficientlyy large so as to avoid reflections of waves from its boundaries, which may occurr in spite of the use of transparent boundary conditions.
Forr illustration, we consider the same waveguiding conditions as were assumed forr the calculations shown in Fig. 2.6. We added a imaginary part to the refractive indexx of the cladding so as to include absorption. The intensity distribution within thee guiding layer is the same as the distribution obtained with FD-MS, compare
4>.(X) )
II e
¥(x,L) )
Figuree 2.8 (a) The spherical waves emitted by a point within the exit plane of the
waveguidewaveguide are also reflected, causing interference between the reflected and non-reflectedreflected wave fronts, (b) The far-field angular intensity distributions are calculated fromfrom the overlap between the field distribution at the waveguide exit ^(x, L) and
thethe 'modes' <pe(x) allowed within the half-space behind the waveguide exit.
Fig.. 2.7 and Fig. 2.6b. However, because of absorption, waves propagating within thee cladding are attenuated and do not reach the waveguide exit.
Extremelyy small step sizes Ax and Az result in unacceptably long computa-tionn times. Although this problem did not occur for the refractive index profiles consideredd in this thesis, there are ways to reduce the computation time to an acceptablee level. One of them is to approximate d2ip/dz2 (see Section 2.4), which allowss for greater step sizes. Another option is to use a non-uniform discretisation suchh that a small step size Ax or Az is chosen in regions where the refractive index variess rapidly over small distances in the x or z direction, respectively.
2.66 P r o p a g a t i o n of waves emerging from t h e
waveguide e
Experimentally,, we identify the modes exiting the waveguide by measuring the intensityy of the outgoing wave field as a function of the angle 9e (see Fig. 2.8a).
Withh the detector placed in the far field, such angular intensity distributions corre-spondd to Fraunhofer diffraction patterns. Comparing the measured patterns with
42 2 ChapterChapter 2 patternss deduced from calculated field distributions at the exit of the waveguide ty(x,ty(x, L), in combination with a search for the best fit, we are able to deduce the refractivee index profile of the waveguide (see e.g. chapter 6).
Inn order to calculate the Fraunhofer diffraction patterns correctly we have to takee into account the reflections of outgoing waves from the lower surface (see Fig. 2.8a).. Therefore, let us consider, within the exit plane of the waveguide, a ribbon alongg the y-direction at a position x with an infinitesimal width dx. Such a ribbon emitss a cylindrical wave. For angles 6e > 0, the far-field amplitude due to the
cylindricall waves emitted from positions x > 0, is given by [6]
A(9A(9ee e-ikoR f y(x,L)-e-ikoe*xdx, (2.48)
withh R ;§> W2/\ the distance between the waveguide exit and the detector. The far-fieldd amplitude due to the reflected waves is given by
AArr{9{9ee)) = f-£-\ e-ikoR f°° tf (z, L) reiko0*xdx. (2.49)
Here,, r is the complex amplitude reflection coefficient of the reflecting surface, whichh is given by Eqs. (2.10) and (2.11). Due to the evanescent waves, the field amplitudee is not exactly zero within the lower plate. Hence, we also have to include thee cylindrical waves originating from positions x < 0. Their far-field amplitude afterr transmittance through the surface is given by
// \ l / 2 y-0
AAtt(6(6ee)) = I — J e-ikoR ƒ V(x, L) te-ik°e°xdx, (2.50)
wheree t — 1 + r is the complex amplitude transmission coefficient [10]. By adding thee three contributions, we obtain the following expression for the far-field angular intensityy distribution [6]:
We)We) ^ (~)(~) e-ikoR \ r ^{x, L) • (e-iko6^ + reik°e°x) dx
7
VV [J°
2(2.51)
++ / tf(x,L) te-ik°e°xdx\\ .
J—J—oooo J
Forr illustration, we assume that the penetration depth of the x rays into the lower surfacee is zero, i.e. <p = TT. In this case, r = — 1 and t = 0, which reduces Eq.
(a)) (b)
0.000 0.01 0.02 eee (degrees)
0.03 3
Figuree 2.9 (a) Calculated intensity of the wave field (X = 0.0931 nm) in the vicinityvicinity of the waveguide exit (n\= 1, n2= 1 — 2.57 x 10~6 and W = 400 nm). (b)(b) Fraunhofer diffraction pattern calculated using Eg. (2.53).
(2.51)) to
I(0I(0ee)) = 4 (—(— \ , '
;;
\RX \RX ^(x,^(x, L) sm(k0dex)dx (2.52) ) Examiningg this expression, we note that sin(fc0öe:r) gives the ^-dependence of the
'modes'' allowed within the half-space behind the waveguide exit (see Fig. 2.8b). Inn general, the intensity distribution is proportional to the overlap of the field amplitudee at the exit of the waveguide and the modes <f>e(x), i.e.
2 2
m m
^(x,L)4>^(x,L)4>ee(x)dx (x)dx (2.53) )Thee (j)e(x) may be interpreted as the modes of a waveguide with infinite gap width W.W. The angular spacing between the corresponding mode angles A9m = X/2W
(seee Eq. (2.13)) is infmitesimally small, so that the modes form a continuum. Too illustrate the above, we calculated the intensity of the field ^(a:, 2)| near thee exit of an empty waveguide (Fig. 2.9a). We excited a TE2 mode at the entrance off the waveguide, which reaches the exit undisturbed. The angle 0e at which the
intensityy is seen to exit the waveguide, equals the mode angle 02 of the TE2 mode.
Thiss is also clear from the diffracted intensity shown in Fig. 2.9b, which exhibits aa maximum at 9e = 62- As expected, there is no intensity at other mode angles.
44 4 ChapterChapter 2 Thee subsidiary diffraction maxima are due to the finite extent of the wave field acrosss the exit plane.
AA tunable x-ray waveguide
InIn this chapter we present a planar x-ray waveguide with a tunable air gap as the guidingguiding medium. Discrete transverse-electric modes excited in the air gap propagate almostalmost undisturbed. Filling the air gap with a fluid allows for studies of ordering phenomenaphenomena in a confined geometry. Since the guided modes are mainly confined to thethe guiding layer, background scattering from the plates is very low. Starting from thethe propagation characteristics of the modes in the empty waveguide, requirements onon the x-ray source and on the positioning accuracy of the plates are derived. The
constructionconstruction of the waveguide is described and measurements of the far-field an-gulargular distributions of intensity exiting the waveguide are presented which illustrate thethe waveguide's properties.
3.11 Introduction
Wee have designed a device which confines a fluid between two plates in a gap tunablee down to tens of nanometers. The distance is set by a combination of piezo-drivenn actuators and optical interferometry. The structure of the confined fluidfluid is studied using a x-ray beam incident on the device as shown in Fig. 3.1a. Sincee at x-ray wavelengths the refractive index of the fluid is higher than that off the confining plates, the device acts as a waveguide for x-rays and part of the incidentt beam is confined to the fluid-filled gap.
Becausee the field of a guided mode is evanescent within the plate material, the backgroundd of scattered intensity from the sample environment is very small. In addition,, the sample material is positioned in a wave field which is coherent in the
46 6 ChapterChapter 3 confiningg direction and has a known amplitude and phase at each point. Varia-tionss in the sample's density then give rise to mode coupling effects as discussed inn the previous chapter. While in optical transmission technology such intermodal scatteringg phenomena are generally undesired, here they are exploited for the de-tectionn of density variations within the confined sample (see chapter 6). Scattering experimentss can be performed both in and out of the plane of the waveguide, see Figs.. 3.1a and 3.1b.
Too excite a single mode, the incident beam should have a sufficient spatial co-herence.. This is achieved when the source is sufficiently small and far away from thee waveguide. The source should also have a high intensity because the wave-guidee entrance intercepts only a small part of the beam in the vertical direction. Bothh requirements are met at a high-brilliance undulator of a third-generation synchrotronn radiation source.
Forr the device to function as a waveguide and sample container, it should have extremelyy flat and parallel plates. The gap has to be set and monitored with nanometerr precision. This chapter describes how this is achieved. In section 3.2 wee treat the propagation of x rays through a planar waveguide and discuss the requirementss imposed on the x-ray source and the device. In section 3.3, the technicall layout of the device is given. Results of waveguiding experiments are presentedd in section 3.4.
3.22 X-ray waveguiding
3.2.11 Mode excitation and propagation
Wee consider the propagation of transverse-electric (TE) modes [8] through a wa-veguidee as depicted in Fig. 3.1. The bottom plate, having a much larger diameter thann the upper one, acts as both as pre-reflector and a post-reflector. The plates aree horizontal, the plane of incidence of the beam is vertical. The electric field vectorr is perpendicular to the plane of incidence, i.e., in the horizontal plane. The modess have a standing-wave character across the gap and are evanescent in the confiningg plates. The wavelength of the x rays is of the order A ~ 0.1 nra and the reflectionn angles are grazing (typically less than 0.1°). Where applicable, the small anglee approximation is used in trigonometric relations.
(a)) detector
Figuree 3.1 Schematic of the waveguide and the scattering geometry in (a) side
viewview and (b) top view. The fluid (shaded) is confined in a gap of width W. The anglesangles of incidence and exit are 9j and 0e. The reflections from the bottom plate inin front of the waveguide entrance and behind the exit are indicated as well. The detectordetector is rotatable in the vertical and horizontal planes. Angles and distances are notnot to scale.
48 8 ChapterChapter 3
Figuree 3.2 The reflection of a plane wave from (a) a single surface and (b) two
parallelparallel surfaces. Due to interference of the direct and reflected waves, a standing wavewave pattern is formed above the surface, see (a). The presence of a parallel surface atat a node of the standing-wave pattern makes the wave propagate in the direction parallelparallel to the surfaces. The internodal spacing is related to the wavelength and thethe angle of incidence 0i as shown. Angles and distances are not to scale.
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