X-ray waveguiding studies of ordering phenomena in confined fluids - Chapter 4 Transmission properties of the waveguide with air gap

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X-ray waveguiding studies of ordering phenomena in confined fluids

Zwanenburg, M.J.

Publication date

2001

Link to publication

Citation for published version (APA):

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

fluids.

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Transmissionn properties of t h e

waveguidee with air gap

WeWe present a study of mode propagation in planar x-ray waveguide having an air gapgap as the guiding medium. Individual transverse electric modes were found to propagatepropagate through the waveguide essentially undisturbed and with negligible scat-teringtering losses to other modes. If different modes are excited simultaneously at the waveguidewaveguide entrance, then the phase relation between these modes as given by their propagationpropagation constants is found to be preserved over the entire length of the wave-guide. wave-guide.

4.11 Introduction

Almostt twenty years ago, Spiller and Segmüller [37] made and tested the first planarr waveguides for x rays. Their aim was to show that the emerging field of integrated-opticss could be extended from the visible to x-ray wavelengths, so as too overcome the strong absorption in the ultraviolet. The devices they developed consistedd of a boron nitride guiding layer sandwiched between AI2O3 claddings. Althoughh the guiding layer material had a low Z-number, substantial absorption att x-ray wavelengths caused considerable power losses. In order to overcome these losses,, they suggested that the ultimate guiding layer would be a vacuum (or air) gap.. Six years later, such a waveguide for x rays was realized by Fischer and Ulrich [38].. They demonstrated self-imaging of the modes within the waveguide and

detector r x-rays s chromium m

t t

Figuree 4.1 Schematic of the waveguiding geometry. Angles and distances are not

toto scale.

showedd that the coherence of the incident field can be preserved upon propagation. Thiss property plays an important role in the structure determination of confined fluidss (see chapter 6).

Withh the advent of third-generation synchrotron ray sources, interest for x-rayy waveguides increased [39, 40], because of their potential as nanometer-sized sourcess of extremely coherent x-ray beams. Low-loss transport of x-rays was achievedd in hollow glass capillaries [41], which in tapered form are used as fo-cusingg devices. However, because of their large acceptance area and their varying crosss section, such devices do not allow for propagation of single modes.

Inn this chapter, we demonstrate the excitation and propagation of a single mode inn a planar x-ray waveguide (see Fig. 4.1). The waveguide is designed to confine a fluidfluid within the gap between the plates, enabling the determination of the arrange-mentt of the fluid's constituents by coherent scattering. In a first, essential, step towardss achieving this goal, we discuss results obtained from waveguiding exper-imentss in the absence of the fluid. Measurements of far-field diffraction patterns fromm the waveguide exit, and the transmitted intensity are compared with model calculations.. Finally, we discuss measurements of the specular reflected intensity (0jj = 0e), showing that coherence is preserved upon propagation.

4.22 Mode excitation and propagation

Thee incident field is determined by the geometry of our experiment as shown in Fig.. 4.1. Let z and x be the coordinates along the propagation direction and along thee normal to the plates, respectively. A plane e.m. wave, with wave number ko andd with the electric field polarized perpendicular to the plane of incidence, is incidentt onto the device at a grazing angle. In front of the waveguide entrance the wavee is totally reflected from the large bottom plate. The resulting field across thee entrance plane (z = 0) is a standing wave due to interference of the incident wavee and the reflected wave. The standing wave field has a node at the surface of thee bottom plate (x = 0) and is given by Eq. (3.1),

<f>(x)<f>(x) « sin(fco^i^) (4.1)

wheree $i is the angle of incidence, and we have neglected the evanescent wave in thee plate. For a gap of width W the field (j>(x) will also have a node at the position off the upper plate if 0, = 9m, with 6m = (m + l)n/koW and m = 0,1,2,.... This

wavee field, which excites the mth transverse-electric (TE) mode of the waveguide, willl propagate through the waveguide undisturbed. The corresponding expression forr the amplitude of the wave field within the waveguide is given by Eq. (3.3),

*m( x ,, z) = <t>m{x) expH/?mz), (4.2)

wheree (3m = nifeo cos#m « fco(l — 0^/2) is the propagation constant of the mode

(seee section 2.2) with nj = 1 the refractive index of the guiding layer. The mode

4>m(4>m(xx))isis gi v e n °y

4>4>mm{x){x) = sm(k0emx). (4.3)

Inn order to illustrate the effect of neglecting the evanescent waves, we have calculatedd the amplitude profiles of the nine lowest-order modes (see Fig. 4.2) by solvingg Eq. (2.5) for the experimental conditions considered here. The validity off the assumption is seen to hold up to the eight mode (0j = 0$). In order to determinee until which incidence angle 9i our assumption holds, we have calculated thee penetration depth as a function of the mode angle (see Fig. 4.3) with

ChapterChapter 4

n n 2 2 n(x) )

W+2£ £

Figuree 4.2 Real mode profiles corresponding to a waveguide with the following

parameterparameter values: W = 486 nm, A = 0.0751 nm, nt= 1 and n$= 1 - 5 x 10~6.

wheree 6C is the critical angle for total internal reflection [8]. Figure 4.3b shows the

dependencee of the penetration depth on the refractive index contrast between the guidingg layer and the cladding, 6 = fij - n2 = 1 - rc2. Clearly, for small angles

Oi,Oi, large 6 and a gap width of a few hundred nanometer, we have ^ C l l7. In this case,, we may neglect the contribution of the evanescent field to the propagation off the modes.

Forr angles of incidence 9i ^ 6>m, the wave field amplitude 4>(x) at the entrance

planee makes a sharp drop to zero at x = W. Therefore, a coherent superposition off guided modes is needed to match the incident field. The corresponding wave fieldfield is given by the Fourier expansion (3.9):

mmax x V{X,Z)=V{X,Z)= J2Cm(ei)^m(x,z), m=0 m=0 with h Cm{6i) Cm{6i) 2(-l)2(-l) mm 99mmsm(ksm(koo00iiW) W) kaW kaW (4.5) ) (4.6) ) Here,, tym(x,z) is given by Eq. (4.2) and mmax = k0W9c/ir — 1 is the maximum

0.000 0.05 0.10 0.15 0.20 #ii (degrees) E E c c .. 10 22 4 <5(x10"6) )

Figuree 4.3 (a) Penetration depth £ as a function of the angle of incidence (with

00CC== 0.18°) and (b) as a function of the refractive-index contrast between the air

gapgap and the cladding, 8 8 = 1 — n^. The wavelength A = 0.0751 nm.

largerr than mmax, i.e. the radiation modes [8], are not confined to the guiding layer

andd will be absorbed by the plate material. For an incident angle 0, in between twoo consecutive guided-mode angles, the amplitude is distributed predominantly overr the two neighboring modes.

4.33 Experimental

Ourr waveguide was made of two fused-silica plates with an optical flatness of <A/20,, coated with a thin metal film. Chromium layers of 30 nm thickness were sputter-depositedd onto the bottom plate (0 25 mm) and thermally evaporated ontoo the upper plate ( 0 5.2 mm). The surfaces had an r.m.s. roughness of 0.3 nmm and 0.4 nm, respectively. These values were determined from x-ray reflectivity measurementss on each of the plates. The gap width was controlled by piezo-driven motorss and measured by means of optical interferometry as discussed in chapter 3. AA photon energy of 16.5 keV (A = 0.0751 nm) was selected using the (111) reflectionn of a diamond crystal monochromator. The monochromator is followed byy a mirror for suppression of higher harmonics from the undulator. The intensity off the beam of 0.1 mm width passing through a vertical gap of 500 nm was typically

2.44 x 107 photons/s. The transverse coherence length £„ of the beam in the vertical planee along the coordinate x is determined by £v = XD/av = 144 /im, where

DD « 45 m is the distance from the source to the sample and av = 23.5 /im. the

verticall source size (full-width-at-half-maximum) of the beam. The incident field iss fully coherent across the gap since £v is much larger than the typical gap sizes

usedused in the experiment. In the horizontal plane the source size equals 928 /mi, whichh yields a transverse coherence length £h — 3.6 /mi. As £h is much smaller

thann the horizontal beam width of 0.1 mm, the beam has incoherent properties in thiss direction. The longitudinal coherence length equals £t = A2/AA = 1.5 /im,

withh A A/A = 5 x 10- 5 the monochromator bandwidth [24]. £j is to be compared withh the maximum path length difference PLDmax « L{82c — 9Q)/2 between the

highestt and lowest modes after travelling over the length of the waveguide L. For

LL = 5.2 mm we find PLDmax « 26 nm and we conclude that £z ;§> PLDmax. Hence,

thee non-zero bandwidth of the monochromator does not affect the coherent phase relationn between different guided modes.

Givenn an angular spacing A0m = n/koW between modes of typically 0.005°

andd a vertical beam divergence being much smaller than this value, it is possible too excite only one mode at a time. The total number of guided modes mmax is

determinedd by the critical angle for total reflection from the chromium layer which equalss 6C = 0.18° (n2 = 1 - 5.0 x 10- 6). For a gap width of 400 nm we find

4.44 Results and discussion

4.4.11 Far-field diffraction patterns

Thee modes propagating through the waveguide for a given incidence angle 0* are identifiedd by measurement of the far-field angular distribution of intensity exiting thee waveguide. The diffracted intensity was recorded as a function of the exit anglee 6e by a Nal scintillating detector which can be rotated in the vertical plane.

AA slit in front of the detector fixes the vertical opening angle at 0.0005°. The presencee of the reflecting bottom plate behind the exit plane causes interference betweenn the direct and specularly reflected waves emerging from the exit, making itt the time-reversed case of the interference occurring at the front of the waveguide.

0.000 0.01 0.02 0.03 0.04 0.05 Incidencee angle 6t (degrees)

F i g u r ee 4.4 Logarithmic contour plot of the intensity diffracted from the exit of

thethe waveguide as a function of 9i and 9e. The measurements were performed at a

wavelengthwavelength A = 0.0751 nm and for W = 486 nm and L = 5.2 mm.

Wee measured the diffraction patterns for a range of fixed incidence angles 0$ up too a value corresponding to excitation of the 1 1t h T E mode. Fig. 4.4 shows a logarithmicc contour plot of the intensity over a mesh of angle pairs (6i,6e), in

stepss of A9, = 0.001° and A0e = 0.0005°. The peaks along the diagonal at

modee angles 0i = 6m, are the unperturbed guided modes. Their angular spacing

(A6(A6mm w 0.0044°) corresponds to a gap width W of 486 nm, which confirms the

interferometricallyy measured gap width. The off-diagonal peaks in between mode angless are subsidiary diffraction maxima associated with the finite width of the gap.. For 6i in between mode angles 9m and 9m+1, the field amplitude within the

waveguidee is distributed over a complete set of modes, but mainly the neighboring ones.. Figure 4.5 illustrates this for patterns along the vertical lines in Fig. 4.4, whichh were taken at 9t values equal to 97, (97 + 08)/2 and 9&. The corresponding

amplitudee distributions within the entrance plane are shown as well.

propagationn model based on the wave field amplitude as given by Eq. (4.5). The diffractedd intensity in the far-field limit, including the post-reflection at the lower plate,, is given by Eq. (3.13b):

2 2

WW2 2 m=0 0

wheree R is the distance from the waveguide exit to the detector. The calculated positionss and heights of the diffraction minima are in good agreement with the measurementss (dashed curves in Fig. 4.5). However, the observed phase contrast iss smaller than calculated. This probably relates to a partial incoherence of the beamm in the vertical plane, which is caused by the optical elements along the incidentt beam path [42]. We find a better fit to the measured diffraction patterns iff I(6ii6e) as given by Eq. (4.7) is convoluted with a Gaussian distribution in 0»

havingg a full-width-at-half-maximum of 0.0033° (solid curves in Fig. 4.5). It is as yetyet unclear which optical elements are responsible for the reduced phase contrast.

Deviationss between measurements and calculations are found at higher sub-sidiaryy maxima of the diffraction pattern for 0; = (67 + 9%)(2 (Fig. 4.5b). This

partt of the spectrum is sensitive to the change of the field amplitude at x = W, wheree it has to drop sharply to zero. Given the good fits, there is no indication thatt modes are excited by surface imperfections as was previously found in solid waveguidingg structures [43].

4.4.22 Transmission measurements

Thee power transmitted by the waveguide is determined by opening the detector soo as to capture all of the intensity emerging from the exit. This is equivalent to integratingg over vertical fines in the contour plot of Fig. 4.4. In this way, we have measuredd the transmitted power as a function of 8j. The measurements shown in Fig.. 4.6a were performed on a waveguide with SiC>2 plates (see chapter 3), set at gapss of 727 nm and 371 nm. The wavelength was A — 0.0931 nm.

Ass expected, the transmitted power decreases proportionally with decreasing gapp width. For both measurements, the period of the oscillations equals the mode spacingg A6m. The shoulder at 9{ ta 0.0035° in the top curve (W = 727 nm) is

aa suppressed peak. This is a geometrical effect, caused by the finite size of the lowerr plate. For very small angles of incidence, the footprint of the incident beam

ii ' i ' i r

0.022 0.03 0.04 0.05 0.06 0ee (degrees)

F i g u r ee 4.5 Diffraction patterns from the exit of the waveguide for different angles

ofof incidence 9t corresponding with the vertical lines in Fig. 4-4- The values of' 0,

areare (a) 9r, (b) (9T+9s)/2, and (c) 0%. The dashed curves are patterns calculated

withwith the use of Eq. (4-V- The solid curves have been obtained by convoluting I(9I(9vv99ee)) in Eq. (4-7) with a Gaussian intensity distribution in 9i, see text.

becomess larger than the length Le of the lower plate in front of the waveguide

entrance.. This reduces the flux of the incident beam at the waveguide entrance forr angles 9i < W/Le. Inserting the parameter values for our geometry, Le « 11.5

mmm and W — 727 nm, we indeed find 9i < 0.0036°. Note that the same happens whenn the x rays exit the waveguide.

Lett us now compare the transmission curve measured for W — 371 nm, with a calculationn of the transmitted power. The latter is given by

Jo Jo I{0i,OI{0i,Oee)RdB)RdBee.. (4.8)

Itt can be readily verified, by substituting Eq. (3.12) into Eq. (3.13a), that

oo rW

// I(9i,9e)Rd9e= \V(x,L)\2dx. (4.9)

JoJo Jo

Iff we now substitute Eq. (4.5) into the right-hand side of Eq. (4.9), we obtain Eq. (3.10b): :

TITT T"roax

P{0i)P{0i) = y E

- ^ )

- (

°)

m=0 0

Heree it is assumed that the radiation modes, with m > mmax, are absorbed by the

claddingg before they reach the exit of the waveguide. The calculated transmitted powerr is depicted by the thick solid curve shown in Fig. 4.6. The powers P (0i), multipliedd with a constant scaling factor, match well with the data. At angles

9i9i > 0.02°, the transmitted power becomes lower than the calculated one. The

differencee arises from absorption in the cladding, which becomes non-negligible as thee penetration depths increase for larger angles.

Thee oscillations observed in the transmitted power are dominated by the fact thatt the incident field profile is clipped by the top plate at the waveguide entrance forfor 9i i=- 9m: i.e. the field drops sharply to zero at x = W. Besides being clipped,

thee incident field is filtered. The waveguide works as a low-pass spatial filter for the incidentt field since only the guided modes (m < mmax) contribute to the wave field

att the waveguide exit. Even if mmax —> oo, P(9i) still oscillates with a substantial

amplitudee as can be seen from

ww

°° r

limm P{9i) - -K-Ycrntfi)^ sin2(fcoMcb = P^) (4.11a) m=00 JU

WW sm{2koQiW)

0.000 0.01 0.02 0.03 0.04 0.05 0;; (degrees)

Figuree 4.6 Transmitted power as a function of the incidence angle 8ir measured for

twotwo different gap sizes (top to bottom): W = 727 and 371 nm. The line through thethe data for W = 727 nm is to guide the eye. The thick solid and dashed curves areare the powers calculated with the use of Eq. (4-10) and Eq. (4.11b), respectively.

Here,, Pi(&i) is the power of the incident field across the gap, which is shown as the dashedd curve in Fig. 4.6. Comparing the curves for P;(ö;) and P{0i) we conclude thatt the effect of filtering is much smaller than the effect of clipping of the incident field. field.

4.4.33 Multi-mode interference

Thee specular reflectivity of the waveguide, i.e. the diffracted intensity at 9 = 9e =

99i}i} has its maximum value at each mode angle 9m. At other angles the reflectivity

iss smaller because of destructive interference between modes, see the intensity alongg the diagonal of the contour plot in Fig. 4.4. Angle-dependent reflectivity curvess are shown in Fig. 4.7 for three different values of W. For the mode spacing, thee relation A9m = ir/koW is confirmed. Also a longer-period variation of the

reflectivityy is present, which is due to multi-mode interference. We have calculated thee reflectivity using Eq. (4.7), see dashed curves. The function 1(9,9) multiplied

80 0

Figuree 4.7 Diffracted intensities measured along the diagonal 9 = 9e= 9i in Fig.

4-4,4-4, for L = 5.2 mm and gap widths W of (a) 391 nm, (b) 478 nm, and (c) 506 nm.nm. The dashed curves are intensities calculated with the use of Eq. (4-V- The solidsolid curves have been obtained by convoluting Eq. (4-7) with a gaussian intensity distribution,distribution, see text.

byy a constant scaling factor, reproduces the measured reflectivity curves very well, exceptt at the lowest angles where surface irregularities at the entrance may have affectedd the measurements.

Again,, the observed interferences are weaker than calculated and the convo-lutionn as described above provides a better fit (solid curves). Our observation thatt the slow periodic variations in the specular reflectivity are in good agreement withh the multi-mode propagation theory [44], is direct proof that the coherence is preservedd over the entire length of the waveguide. The latter is quite remarkable. Whilee the number of oscillations made by the e.m. field of a single mode over the

0.5L L

F i g u r ee 4.8 Intensity of the field within a waveguide, calculated with the

useuse of Eq. (4-V> w^i ^=0.0751 nm, W = 44% nm> L=10.4 mm and

0.== 2.5öo= 0.0091°.

distancee L is of the order PmL/2-rr ~ 108, the difference (/3m — (3n)L/2n ~ 1 in

thee number of oscillations between (not too distant) modes m and n is found to remainn well-defined.

Thee minima of the reflected intensities shown in Fig. 4.7 exhibit a strong dependencee on the angle 6. However, for certain combinations of A, W and L, theree may be no dependence on 6 at all. In the following it is shown t h a t for these parameters,, the field profile across the incidence plane is directly imaged onto thee exit plane of the waveguide, i.e. ^(x,L) = ^(x, 0). Alternatively, an inverted imagee with respect to the line x = W/2 may be formed, i.e. ^(x, L) = ty(W—x, 0). Inn order to derive the conditions for which this self-imaging occurs, we take the phasee of the TEo mode as a common factor out of the sum in Eq. (4.5) [44], so t h a t t

*(*,L)== £ c

<t>

e

^>-^

'.

m=0 0

Thee phase factor in Eq. (4.12) may be worked out as follows:

(00-PJL (00-PJL feo(l-feo(l-7rA A m m 0|> > ? ? (m.(m. 4

i *

--el --el

82 2

Byy defining the 'self-imaging' length as

AWAW2 2

UU = ^ , (4.14)

wee rewrite the phase factor as

WoWo__0m)L0m)L = ^n^L ( 4 1 5 )

Now,, we make a distinction between the even and odd modes in the summation inn Eq. (4.12). For the even modes, m(m + 2) is also even. This results in

eei{(3i{(300-(3-(3mm)L)L = j for L = pL^ ^1 6j

withh p = 0,1,2,.... For the odd modes, ra(m + 2) is odd and

Ci ( f t - A J L=// ! forL = 2pLs

\\ - 1 iorL = {2p+l)Ls.

Hence,, if L is an even multiple of Ls, then the phase changes of all the modes

alongg L differ by multiples of 2-K. In this case, all the guided modes interfere with thee same relative phases as in z = 0. This results in a direct image of the incident field,field, i.e., ^ ( z , L) = ^(x,0). If on the other hand L is an odd multiple of Ls,

thee even and odd modes will be in antiphase. Because in a symmetric waveguide

<j><j>mm{x){x) — <f>m(W — x) for the even modes and 4>m{x) = —<f>m(W — x) for the odd

modes,, it is readily verified that in that case the resulting field is an inverted image off the incident field, i.e., ty(x, L) = ty(W — x, 0). For illustration, we calculated the intensityy of the field within a waveguide |^(x, z)\ having a length L = Ls = 10.4

mm,, see Fig. 4.8. Indeed, at z = 10.4 mm an inverted image of the incident field iss formed.

Forr values of L half-way in between multiples of Ls, multiple images can be

found.. For example, for L = (p + |)LS with p = 0,1,2,...., the total field is given

by y

* ( * ,, (p + -)Ls) = Y, ^ ™ ei m ( m + 2 ) ( p +*) 7 r. (4.18) m=0 0

Forr even and odd modes, the phase factor assumes two different values:

9Mm+2)&H-i)irr = ƒ X for m = 2fc ( 4 1 9 )

withh k = 0,1,2,.... This results in

*(x,(p*(x,(p + s) = £c2fc</>2fc(*) " H )P + 1$ >2 f c + 1< ^+ 1( W - x ) , (4.20)

kk k

wheree we have used 02&+i(x) = ~4>2k+i(W — x)- It is easy to show that

££ cfcfc* = \ [*(*, 0) + # ( W - x, 0)j (4.21) and d

Hence, ,

EE c2fc+102Jt+1 = i [*(W - x, 0) - * ( * , 0)]. (4.22)

*(x,, (p + -)L.) = l2 J *(s,0) + y ^ - * ( W - x, 0). (4.23)

Thiss equation represents a pair of images of *(x, 0). One is erect, the other is inverted.. In Fig. 4.8 we can distinguish both images at z = L3/2, although they

overlap. .

Now,, we derive an approximate expression for the intensity of the minima in

1{6,6).1{6,6). These occur for angles 0 in between mode angles:

Aninn = I(0m+l/2,6m+l/2) (4.24a)

W W

~~ H\ K i ^ i ^ e " ^ 1 + c2m+1(em+1/2)e-^L\' , (4.24b)

wheree we havee taken into account only the contribution of nearest-neighbor modes. Byy making use of Eq. (4.6), it is readily verified that:

Cm(6>m+i/2)) = - 3 (4.25)

TrmTrm +1

and d

CCm+m+l(0l(0m+1/2m+1/2)) = -j. 4.26)

7TTO++ 1

Iff we assume c^(0TO+1/2) » c^+ 1(0m + 1 / 2) = 4/-7T2 for m ^ 4, which is valid to

withinn 14 %, we may rewrite Eq. (4.24b) as 32 2

0.00 0 0.01 1 0.022 0.03

66 (degrees)

0.04 4 0.05 5

Figuree 4.9 Specular intensity measured for A = 0.0751 nm, L = 5.2 mm and a

gapgap width of W = 44% nm- The solid curve are intensities calculated with the use ofEq.ofEq. (4-7).

withh /o = I(9m,em) = W2/RX. Equation (4.27) is valid to within 11% for mode

numberss m ^ 4. The difference in propagation constants is approximated as follows: :

PPm+m+ll ~Pm = ^0 COS 6m+i - k0 COS Or, .. XTT «« -(2m + 3)-4W4W22' ' 3)TTL L (2mm + 3)7rLl

} }

Substitutionn into Eq. (4.24a) leads to 322 f

IminImin = h—7 < 1 + COS

7TT (_

Fromm this expression it becomes clear that the intensities of the minima are con-stantt as a function of m whenever the parameters L, W and A fulfil the self-imaging conditionss discussed previously. It is readily verified that

64/O/TT44 for L = 2pLs

00 forL = ( 2 p + l ) Ls (4.31)

Thee last case is illustrated in Fig. 4.9. It shows intensities I{0,0) which were measuredd and calculated with the parameter values W — 442 nm, A = 0.0751 nm andd L = 5.2 mm ~ Ls/2. Indeed, the minima are constant as a function of 8. The

measuredd intensities agree very well with the calculated intensities. Once again, wee conclude that the coherent properties of the incident wave field are preserved uponn propagation through the waveguide.

4.55 Conclusions

Thee measured far-field diffraction patterns show that, due to the high spatial co-herencee of the incident wave field, single TE modes are excited at the entrance off the waveguide for angles of incidence 6i ~ 9m. For other incidence angles we

observedd the excitation of multiple modes. These were found to interfere contin-uouslyy upon propagation. The preservation of longitudinal coherence is indicated byy the good agreement between the measured and calculated reflected intensities. Measurementss of the transmitted power as a function of 9{ showed oscillations

withh a period A9m. These are due to the clipping of the incident field profile by

thee top plate. The clipping was found to have a bigger effect on the transmitted powerr than the loss of power due to the excitation of radiation modes.