X-ray waveguiding studies of ordering phenomena in confined fluids

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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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Appendixx A

Onn the asymmetry of t h e angular

intensityy distribution

Wee discuss the asymmetry that the angular intensity distribution I(6i,9e) may

displayy around the diagonal 0, = 9e. Consider, for example, the contour plots

shownn in Figs. 5.3, 6.4 and 6.5. These intensity distributions were all obtained for waveguidingg geometries in which the refractive-index profile of the waveguide is asymmetricc with respect to z = L/2. In the following we derive an expression for

I{9i,I{9i, @e) which clearly shows that this is at the origin of the observed asymmetry.

Lett us consider a waveguide of total length L which is divided into three sec-tionss (see Fig. A.la). We assume that these sections each have a ^-independent refractive-indexx profile, i.e. ni — rii(x) and n2 = rt2(a;). In order to obtain an

expressionn for 7(0j, 0e), we first derive an expression for the field amplitude at the

exitt of the waveguide. According to Eq. (2.15), the field at z = L\ is given by

*0r,LOO = £ c ; M ^ ) e -i / J~L l, (A.1) with h

& & == -11 f°° m fc(s)02>(*)<faf (A.2)

where e

PP

11

^^ = J~J&\x)\

2

dx (A.3)

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110 0 AppendixAppendix A

iss the power in mode m. Here, (j>i{x) = <p{(9i,x) denotes the incident wave field

andd the superscripts (1) refer to section 1. In Eq. (A.2) we assumed, for ease off notation and without loss of generality, that the mode profiles (f)$(x) are real-valuedd functions. The upper limit of the summation in Eq. (A.l) is set to infinity soo as to indicate the contribution of the radiation modes. The field amplitude at thee end of the second section is given by:

00 0

(,, Li +L

2

**) = J2 c g W O * ) * - ^ , (A.4)**

m=0 0 with h 11 r°° C ™ = ^ ) // M^L,)^(x)dx. (A-5)

Substitutingg Eq. (A.l) into Eq. (A.5), we can rewrite Eq. (A.4) as:

ooo oo

,, + L

2

) = £ 52<$

CC )

**'-«F'C2?<l&(x)e-& *, (A.6)**

kk 6 m=0m=0 fc=0 with h // oo o ^\x')^(x')dx'.^\x')^(x')dx'. (A.7) -oo o

Byy repeating the above steps for the third section, we finally obtain the field amplitudee at the exit of the waveguide:

oo o

*(*,i)== £ c<»e-'^Ct2)e-'^C^(z)e-<^. (A.8)

l,m,k=0 l,m,k=0

Thee intensity of the exiting wave field measured in the far-field at an angle 6e

iss given by Eq. (2.52):

2 2

I{9i,9I{9i,9ee)=4 )=4

\ l / 2 , 0 0

— JJ *-***]_ m,x,L)4>e(9e,x)da (A.9) )

Onn the right-hand side of Eq. (A.9), we added 9i and 9e in order to indicate the

implicitimplicit dependence of ^(x,L) and <j>e{x) on these parameters. Making use of

time-reversall symmetry of the incoming and outgoing waves, i.e.,

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OnOn the asymmetry of the angular intensity distribution _{111 1} z=L/2 2 \ \ Öii ƒ"* ; ; n** ri^t i L ii 4 n. . ^x ^x

s\ s\

L3 3 \ \ > > z= = n ii I. L, , L/2 2 %% jn. w w - ^ 0 . . L22 U detector r M(e„e.) ) (a) ) (b) )

Figuree A . l Asymmetric geometries corresponding to the intensity distributions

I{$i,9I{$i,9ee)) given by (a) Eg. (A. 12) and (b) Eg. (A. 13).

wee rewrite Eq. (A.9) as:

I{0I{0uu00ee)) = 4

// - \ l / 2 ,00

\R\)\R\) e'ik0RJ y(ei^L)<f>We,x)dx (A.11) ) Byy inserting Eq. (A.8) into Eq. (A.11) and performing the integration, we obtain

m , » . ) - Ê 41> ( 9j) e - ^ "l' C ^ > e - ^ » ^ c S1> e - ^ "i» c < " ( e « )2. (A.12)

l,m,k=0 l,m,k=0

Byy interchanging 0j and 9e, we obtain

m A )) = RX

l,m,k=0 l,m,k=0

.. (A.13)

Iff the sections are distributed asymmetrically with respect to z = L/2, i.e., L\ ^ L3,, then Eqs. (A.12) and (A.13) are not equivalent (see Fig. A.l). Hence, the

intensityy distribution I{Q^ 9e) is then asymmetric with respect to the diagonal line

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X-ray waveguiding studies of ordering phenomena in confined fluids - Appendix A On the asymmetry of the angular intensity distribution

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