Coherence approach in neutron, x-ray, and neutron spin-echo reflectometry

Coherence approach in neutron, x-ray, and neutron spin-echo reflectometry

Victor O. de Haan,

*

Jeroen Plomp, M. Theo Rekveldt, and Ad A. van Well

Department Radiation, Radionuclides & Reactors, Faculty of Applied Sciences, Delft University of Technology, Mekelweg 15, 2629 JB Delft, The Netherlands

Robert M. Dalgliesh and Sean Langridge

Rutherford Appleton Laboratory, Chilton, Oxfordshire OX11 0QX, United Kingdom

Amarante J. Böttger and Ruud Hendrikx

Department Materials Science and Engineering, Faculty of Mechanical, Maritime and Materials Engineering, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands

共Received 15 December 2009; revised manuscript received 1 March 2010; published 23 March 2010兲

Based on the application of coherence theory to neutron scattering a description is given of the propagation of neutrons or x-rays through a reflectometer. Important coherence effects at the sample position are discussed. Further, an outline is given how to determine the measured count rate in a detector on the basis of this method including neutron-polarization effects. It is shown in what way the Born approximation and distorted-wave Born approximation can be used within this theory. An outline is given of the phase-object approximation, describing the specular and diffuse scattering from a surface with large surface structure, extending over the existing capabilities of the distorted-wave Born approximation. The incorporation of neutron-polarization effects enables the detailed discussion of neutron spin-echo coding techniques applied to reflectometry. DOI:10.1103/PhysRevB.81.094112 PACS number共s兲: 61.05.fj, 42.25.Kb, 68.49.⫺h, 42.25.Fx

I. INTRODUCTION

The coherence theory was advanced for neutron scattering by Gähler et al.1_{They showed that coherence theory can be} used to describe the propagation of neutrons through instru-ments. Rauch et al.2 _{and Pushin et al.}3 _{applied coherence} theory for their measurements with Bonze-Hart interferom-eters. Moreover, in a recent paper4 _{it was shown that by} using the coherence matrix formalism it is possible to de-scribe the propagation of both up- and down-spin states of a neutron wave. The principles of this method and the appli-cation of coherence theory to the propagation of neutrons through a reflectometer are described in the following sec-tion. As the reflectivity of x-rays can be described in a simi-lar way, the description of the propagation of nonposimi-larized neutrons also holds for x-rays. Further, the development of spin-echo angular labeling techniques successfully applied in spin-echo small-angle neutron scattering5,6 _{can be used to} code the off-specular neutron scattering, which in turn yields information about in-plane inhomogeneities. In contrast to the conventional off-specular methods, where experiments are performed in reciprocal space, the spin-echo method probes directly in real space共on length scales from approxi-mately 20 nm– 20 ␮m兲. The main object of this paper is the theoretical description and the physical interpretation of the specular and diffuse scattering in reciprocal space as ob-tained by neutron or x-ray scattering theory and in real space as obtained by the neutron spin-echo technique. The theoret-ical frame work will be described in terms of wave functions. For the x-ray case a completely analog derivation can be made by changing the wave function into an appropriate electric or magnetic field quantity.7

II. COHERENCE THEORY

Coherence theory describes the correlation of a field at different locations in space and at different times. For an

accurate description the reader can consult Ref. 8. The field considered here is the neutron-wave function. Let⌿共rជ, t兲 de-note a neutron-wave function characterizing the field at point r

ជ at time t. For a realistic neutron source it will be a fluctu-ating function of time and may be regarded as a typical member of an ensemble consisting of all possible neutron-generating events. It consists of a large number of Fourier components which are independent of each other so that their superposition gives rise to a fluctuating field which is only describable in statistical terms. For a statistically stationary beam it can be constructed from its constituting monochro-matic waves9

⌿共rជ,t兲 =

冕

␺kជ共rជ兲e−i␻ktd3k, 共1兲

where kជ is the wave vector or propagation vector,␻k=vpk

where vp=បk/2m equals the phase velocity of the wave

function, half of the neutron velocity, and m equals the neu-tron mass. For x-rays the phase velocity in vacuum is equal to the speed of light. We will use the formalism of the co-herence matrix, ⌫ˆ as developed in Ref. 4 to describe the propagation of the polarized neutron beam trough the instru-ment. This will be discussed in Sec. IV. For nonpolarized neutrons the formalism reduces to the propagation of the mutual coherence function, ⌫. This is used in Sec. III. Fur-ther definitions of the parameters are equal to those in Ref.4.

III. REFLECTOMETRY

For reflectometers the typical instrument geometry is shown in Fig.1. A neutron created by a source x = x₀, repre-sented by a mutual coherence function⌫0propagates through

space toward the sample position x = xs. At the sample

posi-tion the incident neutron is represented by the mutual coher-ence function ⌫in and interacts with the sample. After the

interaction the scattered neutron is represented by the mutual coherence function,⌫sc.1

A. Propagation from source to sample

For a completely homogeneous incoherent source with area A0the mutual coherence function at the source position

is given by Eq.共31兲 of Ref.4. Substituting this in Eq.共30兲 of Ref.4yields for the mutual coherence function at the sample position ⌫in共rជ1,rជ2,␶兲 = J0 2vp e−ikvp␶ 4␲

冕

A₀ eik共R2−R1兲 R2R1 d2r₁

⬘

, 共2兲 where for reflectometry in the far-zone cos␪i⬇1. If rជ2= rជ1

+ rជ, rជ=共⌬x,⌬y,⌬z兲T_{, and rⰆr}

1than R2− R1can be

approxi-mated by R2− R1⬇ − ⌬x共1 −␨z 2_{/2 −␨} y 2_{/2兲 − ⌬y␨} y−⌬z␨z, 共3兲

where␨z=共rជ−r1

⬘

ជ1兲·eជz/r1and␨y=共rជ−r1

⬘

ជ1兲·eជy/r1. The

denomi-nator in the integral of Eq. 共2兲 can be reduced to r₁2 without introducing a large error. If the source aperture is rectangular with a height Wy and a width Wz and the middle of the

aperture is situated at y = H and z = 0 the integral can be evaluated as ⌫in共rជ1,rជ1+ rជ,␶兲 ⬇ J₀ 2vp eik共⌬x−vp␶兲 4␲r₁2 e i␲/2共␬_y/␬_x兲2E关␬x共␬y/␬x 2 + H − y1+ Wy/2兲兴 − E关␬x共␬y/␬x 2 + H − y1− Wy/2兲兴 ␬x ⫻ ei␲/2共␬_z/␬_x兲2E关␬x共␬z/␬x2− z1+ Wz/2兲兴 − E关␬x共␬z/␬x2− z1− Wz/2兲兴 ␬x 共4兲 with E共x兲 = C共x兲 + iS共x兲, 共5兲 where C共x兲 and S共x兲 are the cosine and sine Fresnel integrals10 _{and ␬} x 2_{= k}⌬x ␲r12, ␬y= k⌬y ␲r1, and ␬z= k⌬z ␲r1. Note that

limx→⬁E共x兲=共1+i兲/2−i exp共−i␲x2/2兲/␲x + O共x−3兲. This

limit can be used to find the mutual coherence function for 兩⌬x兩Ⰶ⌬y2_{k ,⌬z}2_k ⌫in共rជ1,rជ1+ rជ,␶兲 ⬇ J0 2vp WyWze ik_共⌬x−vp␶兲 4␲r₁2 sin

冉

␲ 2␬yWy

冊

␲ 2␬yWy sin

冉

␲ 2␬zWz

冊

␲ 2␬zWz . 共6兲 Further, note that limx→0E共x兲=x关1+O共x4兲兴−ix3关␲/6

+ O共x4_{兲兴. This limit can be used to find the neutron density at}

the sample surface

⌫in共rជ,rជ,0兲 =

J₀ 2vp

WzWy

4␲r₁2. 共7兲

Examples of the amplitude of the normalized mutual coher-ence function are shown in Fig. 2. The parameters used in the calculations are given in Table I. Figure A shows the function along the ⌬z direction 共⌬x=0兲. There is no strong dependence on z1 nor on x1. Figure B shows the function

along the⌬x direction 共⌬z=0兲 at the sample surface for sev-eral positions on the sample z1= 0 cm, z1= 1 cm, and z1

= 2.5 cm. Note the difference in scale for the ⌬x and ⌬z directions. This is due to the difference in diaphragm width and due to the small glancing angles ␪i. The width of the

mutual coherence function along the ⌬x direction reduces when the distance to the center of the sample increases. This is due to a small rotation z1/r1of the mutual coherence func-FIG. 2. 共A兲 Amplitude of the normalized mutual coherence function,⌫ cut along the ⌬z axis 共⌬x=0兲 for z₁= 0, 1, and 2.5 cm 共all coincide on black line兲. 共B兲 The same function cut along the ⌬x axis 共⌬z=0兲 at the sample surface for z1= 0 共full line兲, z1= 1 cm

共long-dashed line兲, and z1= 2.5 cm共short-dashed line兲. The

instru-ment parameters used are given in TableI. FIG. 1. Notation relating to the propagation of the mutual

tion around the y axis, originating from a slightly different direction of the beam incident on the sample. To elucidate this, a two-dimensional plot of the mutual coherence func-tion along the sample surface as funcfunc-tion of both⌬x and ⌬z at the middle of the sample共z1= 0兲 is shown in Fig.3. Note

again the difference in scales in the ⌬x and ⌬z directions. Interesting fact is that the mutual coherence function has an extended tail in the direction 兩⌬z兩⬇兩⌬x共Wz/4+z1兲/r1兩. This

shows that if the coherence properties of the beam are split into two separate contributions for the⌬x and ⌬z directions, the coherence of the beam might be under or over estimated. Further, the coherence properties of the incident beam strongly depend on the position on the sample in the direc-tion perpendicular to the beam. The coherence properties vary much less as function of the position on the sample in the direction parallel to the beam.

B. Propagation from sample to detector

The count rate in the detector is due to the scattered mu-tual coherence matrix propagated from the sample position to the detector and can be found by an integral of the neutron flux Jជ共rជd兲=2vជp␳共rជd兲 over the detector area, Ad at a position

r

ជd, according to Eq.共40兲 of Ref.4

Id= 2vp

冕

A_d

⌫共rជd,rជd,0兲d2rd, 共8兲

where it was assumed that the detector area is perpendicular to the beam direction. Equation 共30兲 of Ref. 4 is used to calculate the propagation to the detector when Ri=兩rជd− rជi兩 is

taken. For the integration area the sample plane is taken 共y = 0兲 and cos␾i= yd/Ri, where 共xd, yd, zd兲T= rជd. If, again, rជ2

= rជ1+ rជ, rជ=共⌬x,0,⌬z兲T, and rⰆR1 the result is

⌫共rជd,rជd,0兲 =

冕

y=0 cos2␾1 R₁2␭2

冕

y=0 e−iqជd1·rជ⌫ sc共rជ1,rជ1+ rជ,0兲d2rd2r1, 共9兲 where qជ_d1= k共rជd− rជ1兲/R1. The inner integral of this formula

can be interpreted as a Fourier transform of the mutual co-herence function at position rជ1 on the sample plane. The

outer integral is an averaging of that Fourier transform over the whole sample. In the far-zone approximation the angles

␾iare almost␲/2 radians 共see Fig.1兲 and Riin the cosine

factors and the denominator inside the integral can be taken constant: Ri⬇rds and cos␾1= yd/rds. If the detector is

lo-cated at the sample’s horizon共i.e., yd= 0兲 the neutron density

due to the scattered beam becomes 0. The explanation for

TABLE I. Parameters for calculation of examples of properties of a reflectometer.

Symbol Description Quantity Unit

H Height diaphragm above sample surface 4 cm

W_y Width diaphragm in y direction 1 mm

Wz Width diaphragm in z direction 5 cm

r₁ Distance between diaphragm and sample 4 m

Hd Height detector diaphragm above sample surface 4 cm

D_y Width detector diaphragm in y direction 1 mm

Dz Width detector diaphragm in z direction 5 cm

r_ds Distance between sample and detector 1 m

␭ Wavelength used in calculations 0.2 nm

FIG. 3. 共Color online兲 Amplitude of the normalized mutual coherence function at the sample surface for different position on the sample

z1= 0 cm共left兲, z1= 1 cm共mid兲, and z1= 2.5 cm共right兲. The instrument parameters used are given in TableI. Note the different scales for

the axes. For negative z₁the function is mirrored in the⌬z=0 axis with respect to the function for a positive value of z₁. The white arrows indicate the directions of the cuts of Fig.2. The color scale is linear ranging from 1 in the middle of the graphs共red兲 to 0 at the outer limits of the graphs共blue兲.

this is that the sample area as seen by a detector at this position is zero. If the detector area is large and the scattering is in the specular direction so that yd/rdscan be taken equal

to y1/r1, the total count rate in the detector is given by

Id= 2vp

冕

y=0 y₁2 r₁2⌫sc共rជ,r1 ជ,0兲d1 2_r 1. 共10兲

If the scattered mutual coherence function is sufficiently nar-row qជ_d1· rជcan be approximated by

qជd1· rជ⬇ k⌬x − k⌬x yd2+共zd− z1兲2 2共xd− x1兲2 − k⌬zzd− z1 xd− x1 . 共11兲 If further, the second and third term can be neglected 共Ⰶ1兲, the above integral Eq.共9兲 reduces to a Fourier transform of

the x direction of the mutual coherence function. The mum value of the second or third term is given by the maxi-mum of k⌬x or k⌬z and the maximum of either yd/共xd− x1兲

or 共zd− z1兲/共xd− x1兲. The position of the detector should be

approximately in the position of the reflected beam. This limits the maximum values of yd/共xd− x1兲 and 共zd− z1兲/共xd

− x₁兲. The maxima for ⌬x and ⌬z are determined by the spatial resolution of the reflectometer under consideration. If for the maximum of ⌬x the first maximum in Fig. 2共B兲 is taken and for⌬z the first 0 of Eq. 共6兲 is taken, the maximum

of the second and third term is on the order of 1. Hence, if the sample correlations extend over much smaller distances than the resolution of the reflectometer as determined by the entrance slit and the sample size, one can safely interpret Eq. 共9兲 as mentioned. However, for sample correlations

extend-ing toward and over the resolution of the reflectometer the above complete integral equation should be used to calculate the detector count rate.

For the interpretation of reflectivity measurements the transition at the sample surface from the incident mutual coherence function to the scattered one is needed. This tran-sition cannot be described by means of the coherence theory as the wave function itself is scattered by the optical poten-tial of the scattering object. Hence, to make the transition first it must be known how the scattered wave function is related to the incident wave function. Then, the definition of the mutual coherence function can be applied to determine the transition of the incident mutual coherence function into the scattered one.

C. Specular reflection

If the optical potential Vr共rជ兲 of the sample is statistically

stationary and only a function of the direction perpendicular to the sample surface 共here the y direction兲 the three-dimensional time-dependent Schrödinger equation can be re-duced to three one-dimensional time-independent equations. The first two differential equations are linear second-order differential equations in x and z and can easily be solved. The last differential equation depends on the potential Vr共y兲.

In the region y⬎0 above the sample the potential is 0 and this equation also reduces to a linear second-order differen-tial equation with a general solution11

␺k_y共y兲 = e−ikyy+␳eikyy. 共12兲

The first term at the right-hand side corresponds to the inci-dent beam and the second term to the 共specularly兲 reflected beam, where ␳ is the reflectance. A quasimonochromatic beam at the sample surface can be thought of as an ensemble average of plane waves. The scattered mutual coherence function can be calculated by realizing that the propagation formula for the mutual coherence function 关Eq. 共30兲 of Ref.

4兴 consists of two integrals. One integrating the incident

wave function reaching point rជ1 from rជ1

⬘

and the other

inte-grating the incident wave function reaching point rជ2from rជ2

⬘

.

To find the scattered mutual coherence function one has to multiply共before integrating兲 the incident wave functions by the appropriate reflectance. For a completely homogeneous incoherent source the scattered mutual coherence function at the sample surface reduces to

⌫sc共rជ1,rជ2,␶兲 = J0 2vp e−ikvp␶ 4␲ ⫻

冕

A₀ ␳共q1兲ⴱ␳共q2兲 cos␪1cos␪2eik共R2−R1兲 R2R1 d2r₁

⬘

, 共13兲 where qi= k共rជ·ei

⬘

ជy兲/Ri. If the mutual coherence function is

sufficiently narrow qi⬇q1so that␳共q1兲ⴱ␳共q2兲=R共q1兲, the

re-flectivity. If the source aperture is rectangular the integral can be evaluated as ⌫sc共rជ1,rជ1+ rជ,␶兲 = R˜共q,⌬q,␣兲⌫in共rជ1,rជ1+ rជ,␶兲, 共14兲 where q = kH/r1,⌬q=kWy/2r1,␣=

冑

⌬x/␲k, and R ˜ 共q,⌬q,␣兲 =

冕

q−⌬q q+⌬q R共␩兲e−i共␲/2兲␣2_␩2 d␩

冕

q−⌬q q+⌬q e−i共␲/2兲␣2␩2d␩ 共15兲

is a folded reflectivity. Note that for ⌬x=0 this reduces to

R

˜ 共q,⌬q,0兲 =

冕

q−⌬q

q+⌬q

R共␩兲d␩

2⌬q . 共16兲

If the scattering 共or reflection兲 is mainly in the specular di-rection and the detector area is large enough the total count rate in the detector is given by Eq.共10兲

Id= 2vp

冕

A_s q2 k2R˜ 共q,⌬q,0兲⌫in共rជ1,rជ1,0兲d 2_r 1, 共17兲

which using Eq. 共7兲 reduces for an incoherent source to

Id= J0WzWy

冕

A_s q2 k2 R ˜ 共q,⌬q,0兲 4␲r₁2 d 2_r 1 共18兲

Id= J0AsWzWy 4␲r₁2k2

冕

q−⌬q q+⌬q ␩2_{R共␩兲d␩} 2⌬q . 共19兲

Note that the total reflected beam is detected by the detector and it is assumed that the sample is small so that the reso-lution is due to the entrance diaphragm and the distance to the sample only.

D. Scattering

In case that the mutual coherence function at the sample position can be considered homogeneous, it was shown4_that using the coherence theory the count rate at the detector given by Eq. 共8兲 can be expressed as an integral over the

sample volume of a product of three factors

Id= 2vp

冕

Rout共rជ1,rជ1+ rជ兲Gr共rជ兲⌫in共rជ1,rជ1+ rជ,0兲d3r, 共20兲

where Gr共rជ兲 is the sample correlation function determined by

the scattering of the neutron-wave function at the scattering object and Rout共rជ1,rជ1+ rជ兲 =

冕

A_d e−pជ·rជ 兩rជd− rជ1兩2 d2rd 共21兲

with pជ= k共rជd− rជ1兲/兩rជd− rជ1兩. This function behaves in a similar

way as the incoming mutual coherence function at the sample position for a completely incoherent source given by Eq.共4兲. The relevant dimensions in this case are the distance

between sample and detector and the detector pixel position and size. We would like to emphasis that Eq.共20兲 only holds

if the mutual coherence function at the sample position can be taken homogeneous, i.e., independent of rជ1. This

condi-tion can be relaxed when an average over the sample area is taken. In that case the mutual coherence function needs to be homogeneous over the correlation length of the sample, which can be inferred form the extension of the sample cor-relation function, Gr. Then, this equation can be converted

into a convolution of an instrumental resolution function and the sample structure factor

Id共rជd兲 = RFT共Qជ兲 ⴱ Sk共Qជ兲, 共22兲

where Qជ= pជ− kជ= k共rជd− rជs兲/兩rជd− rជs兩−k共rជs− rជ

⬘

兲/兩rជs− rជ

⬘

兩 is the

wave-vector transfer. The instrumental resolution function is determined by instrument details. For a complete incoherent source RFT_共Qជ_{兲 =}

冕冕

A₀

冕

A_d J₀e−Qជ ·rជ 4␲兩rជd− rជs兩2兩rជs− rជ

⬘

兩2 d2_r dd2r

⬘

d3r. 共23兲 The sample structure factor is defined as the Fourier trans-form of the sample correlation function

Sk共Qជ兲 =

冕

e−iQជ ·rជGr共rជ兲d3r. 共24兲

One can use the Lipmann-Schwinger equation9,12 _{to solve} the Schrödinger equation, resulting in the first Born

approxi-mation or distorted-wave Born approxiapproxi-mation.13–15 _Another way to solve the scattering of the neutron-wave function is to use the phase-object approximation.4

The Lippmann-Schwinger equation can be iterated result-ing in an infinite series for⌿sc共rជ兲 which can be rewritten in

the form

⌿sc共rជ兲 = ⌿in共rជ兲 −

2m

ប2

冕

G共+兲共rជ,rជs兲Vr共rជs兲⌿in共rជs兲d3rs,

共25兲 where G共+兲共rជ, rជs兲 is a specialized 共potential-dependent兲

Green’s function describing the scattering into a scattered-wave function and defined by the solution of the following Schrödinger equation

冋

ⵜ2_{+ k}2₋ 2mVr共rជs兲

ប2

册

G共+兲共rជ,rជs兲 =␦共rជ− rជs兲, 共26兲

where the ⵜ2 _{operator represents the derivatives to rជ}

s. This

equation describes the scattered wave as a superposition of waves produced by many scattering events occurring at dif-ferent elements of the sample. This Green’s function can be approximated by taking the Green’s function for the undis-turbed potential at position 共x,z兲 along the sample surface. For a layer with a variable height, H on top of a substrate of a different material the Green’s function is derived in the Appendix. Inserting Eq. 共A17兲 and the incident-wave

func-tion given by

␺in共kជ,rជ兲 =␺0共kជ储兲eikជ·rជ 共27兲

in the scattered-wave function in Eq.共25兲 yields

␺sc共kជ,rជ兲 =␺in共kជ,rជ兲 −␺0共kជ储兲

冕

0 ⬁ _eip_ជ·rជ 2␲ipy ⫻

冕

e−iQជ ·rជs_eipyys⌿ p关ys,H共rជs,储兲兴 mVr共rជs兲 2␲ប2 d 3_r sd2p储, 共28兲 where again Qជ is the wave-vector transfer defined as Qជ= pជ − kជ. The integral over rជscan be split in an integral over rជs,储

and ys, yielding ␺sc共kជ,rជ兲 =␺in共kជ,rជ兲 +␺0共kជ储兲 ⫻

冕

0 ⬁ _eip_ជ·rជ 2␲py

冕

e−iQជ储·rជs,储⌰共k y, py,rជs,储兲d2rs,储d2p储, 共29兲 where ⌰共ky, py,rជs,储兲 =

冕

−⬁ ⬁ eikyys⌿ p关ys,H共rជs,储兲兴 mVr共rជs兲 2␲iប2 dys. 共30兲 The scattered-wave function in Eq. 共29兲 must be integrated

over all possible contributions 关Eq. 共1兲兴 and inserted in the

definition of the mutual coherence function关Eq. 共2兲 of Ref.

4兴 to find the scattered mutual coherence function close to

the incident beam and the reflected/refracted beam and as-suming the scattered mutual coherence function close to the sample surface is homogeneous, the sample structure factor can be found as a two-dimensional Fourier transform of the sample surface-correlation function

Sk共pជ,kជ兲 =

冕

e−iQ

ជ_储·rជ储_G

s共pជ,kជ,rជ储兲d2r储 共31兲

defined by共details can be found in Ref.16兲

Gs共pជ,kជ,rជ储兲 =

冕

⌰ⴱ共ky, py,rជs,储兲⌰共ky, py,rជs,储+ rជ储兲d2rs,储.

共32兲 Note that now the sample structure factor is a two-dimensional Fourier transform of a sample surface-correlation function. Essentially this sample surface-correlation function is the one-dimensional Fourier transform 共y direction兲 of the sample correlation function

Gs共pជ,kជ,rជ储兲 =

冕

e−iQyyGr共rជ储+ yeជy兲dy. 共33兲

The dependence of Routand⌫in on y in Eq.共20兲 is the same

as the factor e−iQyy _{so that the integral over y reduces this}

equation to Id= 2vp

冕

y=0 Gs共pជ,kជ,rជ储兲R共rជ1,rជ1+ rជ储兲d2r, 共34兲 where R共rជ1,rជ1+ rជ储兲 = Rout共rជ1,rជ1+ rជ储兲⌫in共rជ1,rជ1+ rជ储,0兲.

Equations 共31兲 and 共34兲 hold for all Green’s functions that

can be cast in the form of Eq. 共A17兲, not only the one

de-rived here. In case of a substrate with a certain height profile 共the layer is of the same material as the substrate兲, ⌿p关ys, H共rជs,储兲兴 reduces to the phase-object approximation

⌿p共ys,H兲 =␶ a,s_共p y兲eiH共py s −py兲−ipy s_y s y_s⬍ H

e−ipyys_{+ e}ipy共ys−2H兲␳a,s共p

y兲 ysⱖ H 共35兲 so that ⌰共ky, py,rជs,储兲 = py 2␲ py− py s ky− py se iH_{共rជs,储兲共ky}−py兲_. 共36兲

For the First Born approximation one finds

⌿p共ys,H兲 = e−ipyys 共37兲 so that ⌰共ky, py,rជs,储兲 = ␳b ky− py eiH共rជ_s,储兲共ky−py兲_, 共38兲

where ␳b is the scattering length density 关4␲␳b= py

2_−共p

y s_兲2_兴.

For the distorted-wave Born approximation as advocated by Sinha14

⌿p共ys,H兲 =␶a,s共ky兲␶s,a共py兲ei共ky

s −p_ys−ky兲ys 共39兲 so that ⌰共ky, py,rជs,储兲 =␶a,s共ky兲␶s,a共py兲 ␳b ky s − py se iH共rជ_s,储兲共ky s −py s 兲_{. 共40兲}

The count rate at the detector can be found by first determin-ing the sample surface-correlation function by insertdetermin-ing this function in Eq. 共32兲 and inserting it in Eq. 共34兲 or by first

calculating the Fourier transform to determine the sample structure factor in Eq. 共31兲 and fold it with the instrumental

resolution according to Eq.共22兲.

IV. SPIN-ECHO NEUTRON REFLECTOMETRY Rekveldt considered combining the neutron spin-echo technique and reflectometry.17 _{In the same way as in} small-angle neutron scattering5_{it is possible to use the precession} of the neutron spin to code the angle of the neutron path through magnetic-flux-density regions. The influence of the magnetic-flux density is described by means of its influence on the precession along the classical neutron path through the instrument.4

To describe the neutron path in case of neutron reflecto-metry four angles are important: ␣k, the angle between the

path of the incident neutron 共wave vector kជ兲 and the sample surface and the same for the not-scattered or transmitted neu-tron. ␤k, the angle the path of the incident neutron makes

with the xy plane. ␣p the angle between the path of the

off-specularly scattered neutron共wave vector pជ兲 and the sample surface and ␤p the angle between the path of the

off-specularly scattered neutron and the xy plane. These angles are schematically shown in Fig. 4.␪s is the angle between

the path of the off-specularly scattered neutron and the path of the not-scattered neutron. Using these definitions, the wave vector of the incident neutron is represented by

k ជ= k

冢

cos␣kcos␤k − sin␣k cos␣ksin␤k

冣

共41兲 and the wave vector of the scattered neutron by

FIG. 4. 共Color online兲 Definition of angles in spin-echo neutron reflection geometry.␣kis the angle between the path of the incident

neutron 共represented by the green arrow and vector kជ, or the red arrow where the start of the vector has been shifted to the sample surface兲 and the sample surface. ␤_k is the angle the path of the incident neutron makes with the xy plane.␣pis the angle between

the path of the off-specularly scattered neutron 共blue arrow and vector pជ兲 and the sample surface and ␤pis the angle between the

path of the off-specularly scattered neutron and the xy plane.␪sis

the angle between the path of the off-specularly scattered neutron and the path of the incident neutron.

pជ= k

冢

cos␣pcos␤p

sin␣p

cos␣psin␤p

冣

. 共42兲

The polarization precession angle acquired by a neutron trav-eling through parallelogram shaped magnetic-flux-density re-gions 共with length, L and strength, B兲 is given by

␾I共rជ

⬘

,rជ兲 = c␭BL共1 −␪Itan␪0兲 共43兲

and

␾II共rជ

⬘

,rជ兲 = − c␭BL共1 −␪IItan␪0兲, 共44兲

where c = 4.63209⫻1014 _T−1_m−2 _{and ␪}

0 is the angle

be-tween the x axis and the faces of the magnets. The angle that is coded by the precession angle 共␪_I or ␪_II兲 is the angle the neutron path makes with the x axis in the plane of the paral-lelogram共see also Fig. 6 of Ref.4兲. The coded angle before

the sample is ␤k and after the sample is either ␤k or ␤p

depending on which neutron path is considered共the transmit-ted or the scattered one, respectively兲. Here, for a not-scattered or specularly reflected neutron the precession angle acquired in region I is exactly balanced by the one in region II, producing a perfect spin echo. If␤p is different from␤k

by scattering in the z direction this exact balance is canceled and a net precession angle remains. This enables the probing of the sample surface structure in the z direction. The in-plane structure of the sample can also be determined by means of off-specular reflection as was discussed in the pre-vious chapter. Then, the sample surface-structure factor is determined from the measurement of the neutron count rate at the relevant detector positions. As all direct scattering techniques this is done in momentum transfer-space or Q space. Using the spin-echo technique this is converted to real space by an appropriate Fourier transform using the preces-sion angle coding of the momentum transfer. This was first experimentally tested by Felcher et al.18 _{in 2002 and} re-peated in 2003.19

As only elastic scattering is addressed here, the momen-tum transfer is completely determined by the scattering angle

␪s, or the difference between␤kand␤p. The propagation of

the mutual coherence matrix is given by Eq. 共27兲 of Ref.4

where the matrices Dˆ are defined in Eq. 共26兲 of Ref. 4 and describe the magnetic-flux-density interaction with the neutron-wave function traveling from rជ or r₁

⬘

ជ on the neutron₂

⬘

source to rជ1or rជ2on the detector via the sample surface. As

shown by de Haan4_{the device matrix between polarizer and} flipper for the spin-echo instrument here is given by

Eˆ 共rជ,rជ

⬘

兲 = Rˆ共rជ

⬘

,rជ兲†Tˆz关␾II共pជ兲兴Tˆz关␾I共kជ兲兴Rˆ共rជ

⬘

,rជ兲, 共45兲

where Rˆ 共rជ

⬘

, rជ兲 is the device matrix of the rotator before re-gion I and Rˆ 共rជ

⬘

, rជ兲† _{the same for the rotator after region II}

and␾I and␾IIare determined by Eqs.共43兲 and 共44兲 taking ␪I=␤k and ␪II=␤p. If the sample scatters nonmagnetic the

device matrix for region I, the sample and region II is just Tˆz共lជse· Qជ兲, where as before Qជ= pជ− kជis the wave-vector

trans-fer at the sample position, the direction of lជseis the coding

direction 共here eជz兲 and lse= c␭2BL tan␪0/2␲ is called the

spin-echo length. The constant cBL tan␪₀/2␲ is called the spin-echo length constant. The polarizing factor of Eˆ 共rជ, rជ

⬘

兲 is

PE共rជ,rជ

⬘

兲 = 关1 − PR

2_共r

ជ

⬘

,rជ兲兴cos兵lជse· Qជ其 + PR

2_共r

ជ

⬘

,rជ兲, 共46兲 where PR is the polarizing factor of Rˆ 共rជ

⬘

, rជ兲. For a perfect

spin-echo instrument the factor before the cosine must be maximal and the other term minimal hence PR= 0. The

com-plete device matrix is found by including the polarizer, flip-per, and analyzer

Dˆ 共rជ,rជ

⬘

兲 = DˆA共rជ,rជ

⬘

兲Fˆ共rជ,rជ

⬘

兲Eˆ共rជ,rជ

⬘

兲DˆP共rជ,rជ

⬘

兲, 共47兲

where DˆP corresponds to the device matrix of the polarizer,

DˆA that of the analyzer, and Fˆ that of the flipper just before

the analyzer.4_{In the following it is assumed that these device} matrices are constant. Hence Dˆ only depends on pជ− kជ. The detector count rate can be found by Eq.共8兲. The detector area

is assumed to be perpendicular to vជp and assumed large

enough to capture all scattered or reflected neutrons. This yields16 _{for a completely homogeneous, incoherent, and} un-polarized source with area A0

Id=

冕

A_d

冕

A₀ J0⍀d共pជ− kជ兲 4␲rds2rs02 Sk共pជ,kជ兲d2r

⬘

d2rd, 共48兲 where ⍀d共pជ− kជ兲 = 1 2Tr关Dˆ共pជ− kជ兲Dˆ共pជ− kជ兲 †_{兴 共49兲}

and kជ= k共rជs− rជ

⬘

兲/rs, pជ= k共rជd− rជs兲/rds, and rជ

⬘

is the source

po-sition. It was assumed that the sample dimensions were small compared to the distance between sample and source and between detector and sample. The two-shim count rate can be determined by replacing ⍀d in the above equation by

TPTA/4 where TP and TA are the transmissions of polarizer

and analyzer, respectively,

Is= TPTA 4

冕

A_d

冕

A₀ J0 4␲rds2rs2 Sk共pជ,kជ兲d2r

⬘

d2rd. 共50兲

The two-flip count rate is found by replacing ⍀d by

TPTAPPPAPE共pជ− kជ兲/4. The integral over the detector area

can be transformed to an integral over py and pz and the

integral over the source area can be transformed to an inte-gral over kyand kz

If= TPTAPPPA 4 J0 4␲k4 ⫻

冕冕冕冕

Sk共pជ,kជ兲PE共pជ− kជ兲dpzdpydkzdky. 共51兲

This integral can be evaluated by realizing that in the case considered PEonly depends on the difference pz− kz. Further,

the sample structure factor only depends on pជ− kជ and kyand

py. If the detector is wide enough the integral over pz goes

from −⬁ to +⬁ and can be replaced by an integral over Qz

If= TPTAPPPA 4 J₀ 4␲k2

冕

dkz

冕

S ˜ k共pជ,kជ兲PE共Qz兲dQz, 共52兲 where S ˜ k共pជ,kជ兲 = 1 k2

冕冕

Sk共pជ,kជ兲dkydpy 共53兲

and the measured polarization becomes

Pm= PPPA

冕

S˜k共pជ,kជ兲PE共Qz兲dQz

冕

˜Sk共pជ,kជ兲dQz

. 共54兲

The factor PE共Qz兲 is given by Eq. 共46兲. In the ideal spin-echo

instrument PR= 0 and the integral over Qzrepresents a cosine

transform 共or the real part of a Fourier transform兲 of the sample structure factor hence the above equation becomes

Pm= PPPA

G共lse兲

G共0兲, 共55兲

where

G共lse兲 = k2R

冋

冕

˜Sk共pជ,kជ兲eilseQzdQz

册

共56兲

comparable to the result obtained by spin-echo small-angle neutron scattering.4_{G共r兲 is a one-dimensional sample} corre-lation function. For standard neutron-reflection geometry 共see Fig. 1兲 the range of integration of ky is determined by

position H and the width Wyof the entrance diaphragm and

the range of integration of pyby position Hd and the width

Dy of the diaphragm before the detector. In general these

ranges are quite small and the integration over kyand pycan

be interpreted as a resolution effect. Ignoring these resolution effects one finds

Pm PPPA =

冕

e−iQxx_G s共pជ,kជ,lseeជz+ xeជx兲dx

冕

e−iQxx_G s共pជ,kជ,xeជx兲dx , 共57兲

which reduces for a one-dimension grating perfectly aligned with the beam direction to

Pm

PPPA

=Gs共pជ,kជ,lse兲 Gs共pជ,kជ,0兲

. 共58兲

One should realize that the validity of Eq.共56兲 depends on

several conditions, which were mentioned during its deriva-tion. The most important ones are the quality of the spin-echo instrument, which determines PE共Qz兲 and the small size

of the sample compared to the distance between sample and source and between sample and detector. In 共spin-echo兲 small-angle neutrons-scattering instruments this condition is almost always met. For neutron reflectometers large sample sizes are more common. In case were the sample size is too large deviations might occur and one should use Eq. 共48兲

instead. Another important condition is that the sample

cor-relation length that is probed 共lse兲 is smaller than

冑

rds/k.

Otherwise the equations used to calculate the scattered co-herence function should be calculated using its definition and the scattered-wave functions. Finally, the magnetic-flux den-sity should be homogeneous over the probed length, other-wise the way the coherence matrix propagates is not accu-rately described by Eq. 共30兲 of Ref. 4. In that case no analytical solution has been found for the propagation of the mutual coherence matrix or共which is the same兲 polarization of the neutron beam.

V. EXPERIMENTS

Several experiments were performed on two samples. The sample shape is shown in Fig.5. The shape was designed as an asymmetric silicon block profile on a silicon substrate with period, L of 900 nm and height, H of 80共sample A兲 and 40 nm 共sample B兲, respectively. The width, W of the block compared to the total period was designed for both samples 2:9 hence W equals 200 nm. The sample areas were approxi-mately 2 cm2_{. Atomic force microscopy 共AFM兲}

measure-ments were done to check the design parameters at the real samples. The results are shown in Table II. An example is shown as an inset in Fig. 5. The period and width of the sample are close to their design values but the heights of the samples deviate considerably. This is probably due to the etching process used to manufacture the sample. The sample surface-correlation function for these block-shaped samples according to the phase-object approximation determined by inserting Eq.共36兲 in Eq. 共32兲 is given by

FIG. 5. 共Color online兲 Top view and side view of the samples. x is the beam direction, y is perpendicular to the sample surface, and

z is perpendicular to the beam direction and parallel to the sample

plane. They are asymmetric silicon block profiles on a silicon sub-strate with period of L = 900 nm and height, H = 80 nm共sample A兲 and 40 nm 共sample B兲, respectively. The width, W of the block compared to the total period is for both samples 2:9. The in-plane angle between beam direction and ridges is␣, resulting in an effec-tive period L_eff= L/sin␣ and width W_eff= W/sin␣. The sample ar-eas were approximately 2 cm2. The red ellipses show the coherence areas of the spin-up and spin-down wave functions.

Gs共rជ储兲 = py 2 As 4␲2RPO共pជ,kជ兲gs共rជ储兲, 共59兲 where RPO共pជ, kជ兲=兩共py− py s_兲/共k y− py s_兲兩2_{and g} s共rជ储兲 is given by 1 − 2rជ⬜· nជ储 L 关1 − cos共QyH兲兴 0 ⬍ rជ储· nជ⬜+ mL⬍ W 1 −2W L 关1 − cos共QyH兲兴 otherwise

with Qy= py− ky, nជ⬜the unit vector perpendicular to the

di-rection of the grating and m the largest integer smaller than r

ជ储· nជ⬜/L. The corresponding sample structure factor can be

split in a specular and diffuse part

Sk共pជ,kជ兲 = SS共pជ,kជ兲 + SD共pជ,kជ兲 共60兲 given by SS共pជ,kជ兲 = py 2 AsRPO共pជ,kជ兲␦2共Qជ储兲 ⫻

再

1 −2W L

冉

1 − W L

冊

关1 − cos共QyH兲兴

冎

and SD共pជ,kជ兲 = py2AsRPO共pជ,kជ兲␦共Qជ储· nជ储兲2关1 − cos共QyH兲兴 ⫻

兺

m=−⬁,m⫽0 m=⬁ ␦

冉

Qជ储· nជ⬜− 2␲m L

冊

sin2

冉

m␲W L

冊

m2␲2 , where nជ储 is the unit vector parallel to the direction of the

grating. Note that for the specular part RPO共pជ, kជ兲 reduces to

the Fresnel reflectivity of the air-substrate interface.

A. Neutron reflectivity

Neutron-reflectivity measurements were performed on re-flectometer CRISP of ISIS/RAL.20_{Measurements were done} on sample A at an incident angle of 3.5 mrad and three

angles of the ridges compared to the incident beam 共this in-plane angle between the incident-beam angle and the ridges, ␣ was 6.7°, 3.5° and 2.3°, respectively兲. The details of the measurements are shown in Table II. The measure-ments and theoretical calculations according to the phase-object approximation are shown in Fig.6. A further measure-ment was done on this sample at an incident angle of 7.0 mrad and an in-plane angle of 6.7°. The measurement and theoretical calculations are shown in Fig. 7.

The theoretical calculations were done according to Eq. 共22兲 by inserting Eq. 共60兲 and using a simplified Gaussian

instrumental resolution function. In contrast to Eq. 共4兲 the

instrumental resolution function is taken as the product of two orthogonal resolution functions to simplify the calcula-tions. A Gaussian resolution function in Qx was used. The

standard deviation of the distribution was given by the spread in incident angles, ⌬␪ 共given in Table II兲 so that

␴Q_x= 2␲␪⌬␪/␭. Hence, the corresponding mutual coherence

function was represented by a Gaussian distribution with a standard deviation 共or effective coherence length兲 rx

=共␴Q_x兲−1. The coherence length in the direction

perpendicu-lar to the x direction in the sample plane was taken very small and the corresponding instrumental resolution function represented by a␦共Qz兲 function.

Measurements of both the specular and the off-specular intensity show qualitatively the same features as calculations according to the phase-object approximation. As is men-tioned in the Appendix the calculation of the Green’s func-tion is strictly speaking only valid under condifunc-tions that are not always satisfied, especially for the specular part. The specular part SS _{of the scattering can be substituted by the}

specular reflectivity calculated by assuming a layer of the same height as the samples ridges and an average scattering potential so that SS共pជ,kជ兲 = py 2 As

冏

r₀F+ r₁Fe2iklH 1 + r₀Fr₁Fe2iklH

冏

2 ␦2_共Qជ 储兲, 共61兲

where klis the perpendicular component of the wave vector

inside the layer, r₀F is the Fresnel reflection coefficient from

TABLE II. Neutron-reflectometry results, measurements details, and sample parameters共coherence length for␭=0.15 nm兲.

Measurement 87 88 95 96 98

Sample A A A A B

␪, incident angle/mrad 3.53 6.98 3.5 3.59 3.48

⌬␪, resolution incident angle/mrad 0.07 0.26 0.10 0.13 0.10

␣, in-plane angle 6.7° 6.7° 3.9° 2.1° 2.7° H, designed height/nm 80 80 80 80 40 AFM height 100共2兲 100共2兲 100共2兲 100共2兲 61共3兲 Fitted height 98共2兲 98共2兲 99共2兲 99共2兲 56共2兲 L, period/ ␮m 0.90 0.90 0.90 0.90 0.90 W, designed width/ ␮m 0.20 0.20 0.20 0.20 0.20 AFM width 0.30共5兲 0.30共5兲 0.30共5兲 0.30共5兲 0.30共5兲 Fitted width 0.21共5兲 0.23共7兲 0.23共5兲 0.21共5兲 0.23共9兲 Coherence length/ ␮m 100 14 70 50 70

air to layer and r₁F the reflection coefficient from layer to substrate. For the above measurements this gives a more ac-curate description of the specular reflectivity. The fitted pa-rameters are shown in Table II. The fitted heights agree within the errors to the heights determined with AFM surements. The fitted widths are smaller than the AFM mea-surements but within error bars equal to the design values. Probably the AFM width is overestimated due to the size of the AFM tip共⬇10 nm兲. The phase-object approximation

de-viates close to the critical edge. The calculations in this re-gion are very sensitive to the exact resolution function 共or mutual coherence function兲 at the sample position and of the sample surface-correlation function.

B. X-ray reflectivity

X-ray reflectivity measurements were performed with a Panalytical Xpert Pro MRD X-Ray diffractometer at Mate-rial Science and Engineering Department of the Delft Uni-versity of Technology. The used setup is shown in Fig.8. See its caption for component details. Several measurements were done with different in-plane angles. Figure9shows the measurement and theoretical calculations for sample A. The angle between the incident beam and the profile was approxi-mately 10°. The scattered intensity was calculated in a simi-lar way as described for the neutron-reflectivity measure-ments, using the parameters shown in Tables II and III. Figure 10shows a similar measurement and theoretical cal-culation on sample B. The relevant features of the measure-ments are clearly reproduced by the phase-object approxima-tion. The different positive and negative orders are at the same position and their intensities relative to each other are calculated quite accurately. The intensity oscillations along

FIG. 6.共Color online兲 Normalized neutron count rate at detector as measured共left兲 and calculated 共right兲 for sample A and incident angle of 3.5 mrad and in-plane angle 6.7°, 3.5°, and 2.3° as function of scattering angle ␪ and wavelength 共the wavelength scale was renormalized to constant ⌬␭/␭ to increase the statistics 共note the logarithmic wavelength scale兲. Note the intensity is located at Q_x = 2␲n/Lefffor orders n = 0, −1, −2, and −3. The angle-independent

structure between␭=0.6 and 0.8 nm is due to increased background of the共insufficiently blocked兲 second pulse from the ISIS target.

FIG. 7.共Color online兲 Normalized neutron count rate at detector as measured共left兲 and calculated 共right兲 for sample A and incident angle of 7.0 mrad and in-plane angle 6.7° as function of scattering angle ␪ and wavelength ␭. Note the intensity is located at Q_x = 2␲n/Lefffor orders n = 0, −1, and −2. The same comments hold as

in the caption of Fig.6.

FIG. 8. 共Color online兲 X-ray measurement setup. From left to right: the x-ray tube Cu with line focus 共PW3373/00兲 used as the source, slit 1共width 0.5°兲, x-ray mirror 共PW3088/60兲 with an equa-torial divergence⬍0.05°, an automatic beam attenuator 共PW 3087/ 60兲, an asymmetrical four-crystal Ge 220 monochromator 共PW3098/27兲 used to obtain Cu K␣1 radiation 共⌬␭/␭=23 ppm兲

with beam divergence of 0.005°, a crossed slit collimator共PW3143/ 20兲 with axial height of 5 mm and equatorial width of 1 mm, the goniometer with a radius of 360 mm 共PW3050/65兲, slit 2 共0.016°2␪兲, and a proportional gas-filled detector 共PW3011/20兲.

FIG. 9. 共Color online兲 Measured 共left兲 and calculated 共right兲 x-ray intensity at detector for sample A. Note the color scale is logarithmic.

the orders due to the sample height and optical phase match-ing of the reflection at the top and bottom of the profile, have the same position and frequency. Some deviations occur. Es-pecially close to and below the critical edge and in the re-gions between the orders, and the relative intensity of the orders with respect to the specular order. This could be due to several factors such as the sample surface roughness, ape-riodicity of the grating or inaccurate estimate of the incident-beam-resolution function. Further due to the limited size of the diaphragm before the detector it is possible that some x-rays are scattered out of the beam and not detected. In the calculations a self-affine roughness21 _{was added using a} roughness standard deviation of 1.1 nm for sample A and 0.6 nm for sample B with a correlation length of 1000 nm and a jaggedness of 1. Although this roughness clearly shows up as off-specular intensity between the orders it only roughly cor-responds to the measured values indicating that the actual sample roughness deviates from the simple model considered here. This also obstructs further detailed comparison between measurements and calculations.

C. Neutron spin-echo reflectivity

Neutron spin-echo measurements were performed with sample B on reflectometer OffSpec of ISIS/RAL.22–24_{As this} is a time-off-flight reflectometer, the spin-echo length is pro-portional to the square of the wavelength. For the measure-ments here the spin-echo length constant was set at 3.05 ␮m/nm2_{. Again measurements for several in-plane}

angles were done for a fixed incident angle, ␪= 3.4 mrad. The normalized shim count rate for ␣= 0° and ␣= 0.8° are shown in Fig. 11. For ␣= 0.8° the specular reflectivity and also two negative orders and one positive order can be seen.

When␣ is reduced to 0, the orders merge toward the same position as the specular reflectivity. The increased width of the specular reflectivity is due to the divergence in ␣.

The normalized measured polarization is also measured and shown in Fig. 12. For ␣= 0.8° the polarization of the specular part is 1. This is due to the fact that for the truly specular part Qz= 0 and hence ␤p=␤k so that the rotation

built up in the first arm of the spin-echo spectrometer is completely reversed in the second arm. The polarization along the orders oscillates between +1 and −1 as shown for the −1 order in Fig.13. This can be understood by looking at the diffuse part of the surface-scattering function given by Eq. 共60兲. Inserting this function in Eq. 共54兲 yields

Pm

PPPA

= cos

冉

2␲mlsecos␣

L

冊

共62兲

under the condition that Qx= 2␲m sin␣/L, which is fulfilled

for the mth order. Hence, the polarization along order m os-cillates as a cosine function of the spin-echo length with a period of L/m 共as cos␣⬇1兲 and an amplitude of 1. This also explains the large oscillations in the polarization in the re-gion just below or above the specular ridge for the measure-ments at␣= 0. Note that for a one-dimensional height profile the shape and period of these oscillations are completely in-dependent from the precise form of the sample structure fac-tor 共and hence also from the scattering theory used兲. It is

TABLE III. X-ray measurement details 共coherence length for ␭=0.15 nm兲.

Parameter Sample A Sample B

In-plane angle/degrees 10.5 9.7

Angular resolution/mrad 0.077 0.077

Coherence length/ ␮m 91 91

Isotropic roughness/nm 0.6 1.1

FIG. 10. 共Color online兲 Measured 共left兲 and calculated 共right兲 x-ray intensity at detector for sample B. Note the color scale is logarithmic.

FIG. 11. 共Color online兲 Measured normalized shim count rate for sample B and incident angle of 3.4 mrad and in-plane angles 0° 共left兲 and 0.8° 共right兲 共estimated in-plane angle divergence 0.15°兲 as function of scattering angle␪ and wavelength ␭. The arrow shows the position of the cut of Fig.14.

FIG. 12. 共Color online兲 Measured normalized polarization for sample B under the same conditions as Fig.11. Everywhere in the plot where the error bars on the polarization were larger than the polarization, the polarization was changed to 0 for display purposes. The black arrow shows the position of the cut of Fig.14.

only needed that the different orders do not overlap. Hence, these gratings are extremely suitable for calibrating the spin-echo length constant of these kinds of spectrometers.

For the measurement at␣= 0° the specular cut of the nor-malized polarization as function of spin-echo length is shown in Fig. 14. The black continuous line is a model cal-culation using the phase-object approximation for the off-specular and off-specular scattering, determined by inserting Eq. 共59兲 in Eq. 共58兲. The fitted values for the model were H

= 50共1兲 nm and W=0.27共2兲 ␮m. Although the fit is good, the fitted parameters deviate considerably from the corre-sponding AFM values. This shows again that conditions un-der which the phase-object approximation holds are not com-pletely satisfied. The共red兲 dashed line is a model calculation using the phase-object approximation for the off-specular and the reflectivity of the average scattering-length-density profile for the specular scattering as was done for the calcu-lations of the normal off-specular measurements as shown before in Sec.V A. This can be done by inserting Eq.共60兲 in

Eq. 共54兲 and using Eq. 共53兲. Interestingly, now the

phase-object approximation for the specular reflectivity is more ac-curate than the values from the average scattering-length-density profile. The reason for this must be sought in the fact

that the distance between the ridges becomes larger共toward infinity due to ␣→0兲 and the averaging distance for the scattering-length-density profile is smaller than this. One should realize that this averaging distance is not related to the coherence length of the neutron beam but to the extinc-tion length of the neutron in the sample material.25 Only the sample volume within the extinction length contributes con-siderably to the reflected beam. The extinction length is on the order of共␭␳b兲−1, where␳bis the scattering-length density

of the sample material. For silicon and a wavelength of 0.2 nm, the extinction length is some 20 ␮m, hence one should average the scattering-length-density profile over this length scale. As the grating period is 900 nm, the region for the transition angle is at arcsin共0.9/20兲=2.6° for 0.2 nm neu-trons and increasing linearly with wavelength. This explains also why, even when the conditions of the Appendix are not completely satisfied, the phase-object approximation pro-duces correct results for the normalized polarization when the structures in the direction of the beam are longer than this extinction length.

The instrumental resolution has been applied in the same way as discussed for the previous experiments. The fact that one can still measure the sample correlations over distances much larger than the coherence length of the beam is due to the splitting of the wave function in a spin-up and spin-down wave function. The splitting is accomplished by the first spin-echo arm of the spectrometer. The distance on the sample between the coherence areas of the spin-up and the spin-down wave functions is equal to the spin-echo length, lseand is much larger than the coherence length of the

neu-tron beam in that direction. This is shown schematically as the red shaded areas in Fig.5. Both wave functions are still able to interfere at the detector position as the splitting is reversed in the second spin-echo arm of the spectrometer, resulting in overlapping coherence areas.

VI. CONCLUSIONS

It is possible to use the coherence theory to describe the propagation of neutrons through neutron or x-ray reflectome-ters. The coherence theory describes the propagation of the ensemble average of the neutron-wave function. The coher-ence theory as adopted here, only considers neutron-wave functions, having approximately an equal amount of total energy, denoted by (quasi-) monochromatic wave functions. This is due to the dispersion relation of matter waves and hence the interference between nonmonochromatic wave functions can, in general, be ignored. X-rays in vacuum have no dispersion, however the same limitation holds as the x-rays are dispersive in a medium with finite-scattering po-tential. For reflectometry the coherence theory makes clear which approximations are used to obtain the standard results. It has been made clear which conditions must be fulfilled to be able to interpret the detector intensity as a folding of the sample correlation function with the instrumental resolution function. Especially the homogeneous coherence function at the sample position and the limited size of the sample surface-correlation function ensure that these conditions are met.

FIG. 13. Measured 共dots and error bars兲 and calculated 共line兲 normalized polarization along the order −1共indicated in Fig.12as white arrow兲 as function of spin-echo length for sample B for an in-plane angle of 0.8°.

FIG. 14. 共Color online兲 Measured 共dots and error bars兲 and cal-culated 共lines兲 normalized polarization as function of spin-echo length for sample B. Black line: completely according to phase-object approximation, red dashed line: specular reflectivity calcu-lated from average scattering-length-density profile. Grating is per-fectly aligned with incident beam.

Further it was shown that neutron-polarization effects can be described by using the coherence matrix approach. The interpretation of the neutron spin-echo reflectivity signal as a one-dimensional sample correlation function has been de-rived, again with the conditions needed for this interpreta-tion.

The phase-object approximation has been extended for neutron and x-rays reflectivity geometry and compared with measurements of neutron reflectivity, x-ray reflectivity, and neutron spin-echo reflectivity on silicon gratings. It has been found that in the description of neutron spin-echo reflectivity measurements, the extinction length of the neutrons inside the sample material might play an important role.

ACKNOWLEDGMENTS

This research project has been supported in part by the European Commission under the 6th Framework Program through the Key Action: Strengthening the European Re-search Area, ReRe-search Infrastructures 共Contract No. RII3-CT-2003-505925兲 and by the Austrian Science Fund 共Project No. F 1514兲. This work was financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).

APPENDIX: SINGLE-LAYER GREEN’S FUNCTION In the most general case for elastic scattering, propagation of the wave function itself in the sample is governed by the stationary Lippmann-Swinger equation.9 _{The basis of the} Lippmann-Schwinger equation is the time-dependent Schrödinger equation applied to a time-dependent potential. The potential can be factorized in a time-dependent part and a space-dependent part: V共rជ, t兲=Vr共rជ兲Vt共t兲. By varying Vt共t兲

in a specific way 共0 for t→ ⫾⬁ and 1 during the scattering process兲, the channel state can be identified as the solution of the Schrödinger equation before the scattering process and the scattered state as the solution after the scattering process.12 _{The Lippmann-Schwinger equation can also be} derived from the stationary Schrödinger equation assuming appropriate boundary conditions共an incident plane wave and a scattered spherical wave at the sample position兲. The result is9

⌿sc共rជ兲 = ⌿in共rជ兲 −

2m

ប2

冕

G0共rជ− rជs兲Vr共rជs兲⌿sc共rជs兲d3rs,

共A1兲 where ⌿in共rជ兲 represents the channel state or incident wave

function,⌿sc共rជ兲 the scattered state or wave function, Vr共rជs兲

the scattering potential of the sample and G₀共rជ兲 is the free-particle Green’s function26given by

G0共rជ兲 =

eikr

4␲r. 共A2兲

In general the scattered-wave function is unknown. However, the Lippmann-Schwinger equation can be iterated resulting in an infinite series for⌿sc共rជ兲 which can be rewritten in the

form

⌿sc共rជ兲 = ⌿in共rជ兲 −

2m

ប2

冕

G共+兲共rជ,rជs兲Vr共rជs兲⌿in共rជs兲d3rs,

共A3兲 where G共+兲共rជ, rជs兲 is a specialized 共potential dependent兲

Green’s function describing the scattering into a scattered-wave function and defined by the solution of the following Schrödinger equation

冋

ⵜ2_{+ k}2₋ 2mVr共rជs兲

ប2

册

G共+兲共rជ,rជs兲 =␦共rជ− rជs兲, 共A4兲

where the ⵜ2 operator represents the derivatives to rជs. This

equation describes the scattered wave as a superposition of waves produced by many scattering events occurring at dif-ferent elements of the sample. This Green’s function can be approximated by taking the Green’s function for the undis-turbed potential at position 共x,z兲 along the sample surface. For a layer with a variable height, H on top of a substrate of a different material the Green’s function is derived below. To find the solution a precise estimate of the Green’s function and the undisturbed scattered-wave function is needed. Here the Green’s function for a single layer on a substrate is de-rived and implemented to find the scattered wave. The Green’s function for scattered waves can be approximated by taking the Green’s function for scattered waves for the un-disturbed potential13,27,28_{defined by}

冋

ⵜ2_{+ k}2₋2mV共0兲共rជs兲

ប2

册

G共+兲共rជ,rជs兲 =␦共rជ− rជs兲, 共A5兲

where theⵜ2operator represents the derivatives to rជs. It can

be solved by assuming that the Green’s function can be fac-torized G共+兲共rជ,rជs兲 = 1 4␲2

冕

₀ ⬁ eipជ储·共rជ−rជs兲g共p y,y,ys兲d2p储, 共A6兲

where pជ is a wave vector with length k and g共py, y , ys兲 is a

one-dimensional Green’s function perpendicular to the sur-face. Note, that if g共py, y , ys兲 would be equal to eipy共y−ys兲/2ipy

this formula can be converted to Weyls representation of a spherical wave, which in the far zone reduces to

G共+兲共rជ,rជs兲 = G0共rជ− rជs兲 =

eik兩rជ−rជs兩

4␲兩rជ− rជs兩

. 共A7兲

Inserting the Green’s function in Eq. 共A5兲 and inverse

Fou-rier transforming, yields

⳵2_g ⳵y2+ py 2 g =␦共y − ys兲 yⱖ H ⳵2_g ⳵y2+共py l_兲2_{g =␦共y − y} s兲 0 ⱕ y ⬍ H ⳵2_g ⳵y2+共py s_兲2_{g =␦共y − y} s兲 y ⬍ 0, 共A8兲

where y = 0 is the position of the substrate-layer interface, y = H is the position of the layer-air interface, py

l =

冑

py2−共kc l_兲2 and py s =

冑

py 2 −共kc

s_兲2_{. The positive sign is taken as p}