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\begin{document}
\draft
\title{Off-specular M\"{o}ssbauer Reflectometry}
\author{L. De\'{a}k, L. Botty\'{a}n, and D.L. Nagy}
\address{KFKI Research Institute for Particle and Nuclear Physics,\\
P.O.B. 49, H-1525 Budapest, Hungary}
\author{H. Spiering}
\address{Institut f\"{u}r Anorganische und Analytische Chemie, Johannes Gutenberg\\
Universit\"{a}t Mainz, Staudinger Weg 9, D-55099 Mainz, Germany}
\date{Submitted to Physical Review B on February xx, 2003}
\maketitle

\begin{abstract}
Synchrotron M\"{o}ssbauer Reflectometry (SMR), the grazing incidence nuclear
resonant scattering of synchrotron radiation, can be applied to perform
depth-selective phase analysis and to determine the isotopic and magnetic
structure of thin films and multilayer. Off-specular SMR provides
information from the lateral structure of multilayer. A theoretical approach
of off-specular SMR is presented.
\end{abstract}

\pacs{PACS: 42.25-p,68.35.-p,76.80.+y,78.66.-w}

\preprint{HEP/123-qed}

\narrowtext

\section{Introduction}

The specularly reflected radiation from layered systems does not depend on
the lateral structure; it depends only on the lateral averages of material
parameters.\cite{Lax51} For studying lateral inhomogenities, such as
magnetic domains etc., off-specular reflectometry is necessary. Domain
structure of AF-coupled MLs is an issue of both theoretical and
technological importance. Off-specular non-polarized \cite{Langridge00} and
polarized neutron reflectometry,\cite{Felcher93,Lauter00} soft-x-ray
resonant magnetic diffuse scattering \cite{Hase00} and, very recently,
off-specular Synchrotron M\"{o}ssbauer Reflectometry (SMR) have been used to
estimate the AF-domain-size distribution in magnetic MLs.\cite{Nagy02}

The theory of the off-specular neutron reflectometry based on the Distorted
Wave Born Approximation\cite{Sinha88} (DWBA) has been published earlier.\cite
{Toperverg01,Lauter94,Ruhm99} DWBA solves the problem of diffuse
(off-specular) field as perturbation around the specular field, which latter
field can be easily calculated, even for general stratified media, by
suitable matrix methods.\cite
{Toperverg01,Felcher87,Paratt54,Majkrzak89,Deak01} In case of off-specular
SMR the DWBA can also be applied and similarly the solution of the specular
problem is well known in the literature.\cite
{Deak01,Afanasjev65,Hannon69,Hannon85,Irkajev93,Irkajev93b,Deak96,Rohlsberger94}
Because SMR works in the time domain, this approach is not straightforward.
Our aim is the implementation of off-specular reflectometry into the data
evaluation program EFFI (Environment For FItting).\cite{Spiering00}

Starting from the general theory of Lax \cite{Lax51} and from the common
optical formalism of polarized neutron and M\"{o}ssbauer reflectometry,\cite
{Deak01} an expression for diffuse scattering of multiple component
(electromagnetic and/or quantum mechanical particle) waves on laterally
inhomogeneous stratified media is obtained. As Lax has pointed out, from the
point of view of specular reflection, the set of discrete atomic scattering
centers can be described/replaced by homogeneous index of refraction $n$ and
solving the scattering problem is equivalent to solving the wave equation.
This approach is valid only if the direction of the scattered wave is far
from the direction of higher order atomic Bragg reflections. In this paper
we study the grazing angle incidence limit only, for which the index of
refraction approximation is valid.\cite{Hannon85,Deak96,Rohlsberger94}

Our model systems are stratified media, multilayer and thin films, having
lateral inhomogeneity on the mesoscopic scale, being much larger then the
atomic distances. In each homogeneous part around position ${\bf r}$ an
index of refraction $n\left( {\bf r}\right) $ is defined. Since $n$ for both
slow neutrons and X-rays hardly differs from the $2\times 2$ unit matrix $I$%
, the susceptibility defined by $\chi \left( {\bf r}\right) =2\left[ n\left( 
{\bf r}\right) \text{-}I\right] $ is conveniently used (see also Ref. \cite
{Deak01}).

\section{Off-specular scattering}

Using the index of refraction approximation in each homogeneous part of the
system the solution of the inhomogeneous wave equvation

\begin{equation}
\left[ \Delta +k^2I\right] \Psi \left( {\bf r}\right) =-k^2\chi \left( {\bf r%
}\right) \Psi \left( {\bf r}\right)   \label{1}
\end{equation}
yields $\Psi \left( {\bf r}\right) $, representing the two components of the
electromagnetic field or the quantum mechanical spinor state of neutron at
position ${\bf r,}$ with $k$ being the wave number in vacuum. In stratified
medium we compose the susceptibility function 
\begin{equation}
\chi \left( {\bf r}\right) =\sum_{l=1}^S\chi _l\left( {\bf r}_{\parallel
}\right) ,  \label{2}
\end{equation}
as the sum of the susceptibility functions of the individual layers $l$ ($l$ 
$=1$..$S$, the last layer $S$ is the substrate) depending on the lateral
coordinate ${\bf r}_{\parallel }$ only.

If the homogeneous parts of the system are large compared to the wavelength,
we may assume that the exact solution $\Psi \left( {\bf r}\right) $ is close
to the solution $\Psi _{{\rm coh}}\left( {\bf r}\right) $ of the coherent
(specular) field equation\cite{Lax51} 
\begin{equation}
\left[ \Delta +k^2I\right] \Psi _{{\rm coh}}\left( {\bf r}\right) =-k^2\sum 
\overline{\chi }_l\Psi _{{\rm coh}}\left( {\bf r}\right) \text{,}  \label{3}
\end{equation}
which is obtained from Eq. (\ref{1}) by replacing the susceptibilities $\chi
_l\left( {\bf r}_{\parallel }\right) $ by the average susceptibility $%
\overline{\chi }_l=\left\langle \chi _l\left( {\bf r}_{\parallel }\right)
\right\rangle $ of each layer $l$. In order to arrive at a perturbation
equation, the sum $-k^2\sum \overline{\chi }\Psi \left( {\bf r}\right) $ is
added and subtracted at the right hand side of Eq. (\ref{1}) 
\begin{equation}
\left[ \Delta +k^2I\right] \Psi \left( {\bf r}\right) =-k^2\sum \overline{%
\chi }_l\Psi \left( {\bf r}\right) -k^2\sum \left( \chi _l\left( {\bf r}%
_{\parallel }\right) -\overline{\chi }_l\right) \Psi \left( {\bf r}\right) 
\text{.}  \label{4}
\end{equation}
For homogeneous layers $\chi _l\left( {\bf r}_{\parallel }\right) =\overline{%
\chi }_l$ the second sum vanishes on the right, so that Eq. (\ref{4}) is
equivalent to Eq. (\ref{3}), the basic equation of the specular
reflectometry.\cite{Ruhm99,Deak01,Deak96,Spiering00}

We look for the general solutions

\begin{equation}
\Psi \left( {\bf r}\right) =\Psi _{{\rm coh}}\left( {\bf r}\right) +\Psi _{%
{\rm off}}\left( {\bf r}\right)   \label{5}
\end{equation}
of Eq. (\ref{4}) as the sum of the coherent field $\Psi _{{\rm coh}}\left( 
{\bf r}\right) $, which vanishes out of the specular direction, and the
off-specular field $\Psi _{{\rm off}}\left( {\bf r}\right) $. Substituting
Eq. (\ref{4}) into Eq. (\ref{3}) gives
\begin{equation}
\left[ \Delta +k^2I\right] \Psi _{{\rm off}}\left( {\bf r}\right) =-k^2\sum
\left( \chi _l\left( {\bf r}_{\parallel }\right) -\overline{\chi }_l\right)
\Psi _{{\rm coh}}\left( {\bf r}\right) -k^2\sum \chi _l\left( {\bf r}%
_{\parallel }\right) \Psi _{{\rm off}}\left( {\bf r}\right) \text{.}
\label{6}
\end{equation}

By the optical method \cite{Deak01} the coherent field $\Psi _{{\rm coh}%
}\left( {\bf r}\right) $, the solution of Eq. (\ref{3}), is obtained

\begin{equation}
\Psi _{{\rm coh}}\left( {\bf r}\right) =T\left( {\bf r}_{\perp }\right) \Psi
^{{\rm in}}{\rm \exp }\left( i{\bf k}_{\parallel }\cdot {\bf r}_{\parallel
}\right)  \label{7}
\end{equation}
(see Appendix), where ${\bf r}_{\perp }$ and ${\bf r}_{\parallel }$ are the
perpendicular and lateral components of ${\bf r}$, respectively and $\Psi ^{%
{\rm in}}$ the amplitude of the incident plane wave

\begin{equation}
T\left( {\bf r}_{\perp }\right) =L^{\left[ 21\right] }\left( {\bf r}_{\perp
}\right) \left( I-R_{{\rm sp}}\right) +L^{\left[ 22\right] }\left( {\bf r}%
_{\perp }\right) \left( I+R_{{\rm sp}}\right)   \label{8}
\end{equation}
is the transmission factor of the coherent field between the top surface $%
{\bf r}_{\perp }=0$ and an arbitrary depth ${\bf r}_{\perp }$. $R_{{\rm sp}}$
is the $2\times 2$ specular reflectivity matrix of the system \cite{Deak01}, 
$L^{\left[ 21\right] }\left( {\bf r}_{\perp }\right) $ and $L^{\left[
22\right] }\left( {\bf r}_{\perp }\right) $ are the $2\times 2$ submatrices
of the $4\times 4$ characteristic matrix $L$ at depth ${\bf r}_{\perp }$.%
\cite{Deak01,Deak96}

Eq. (\ref{6}) can be solved up to arbitrary accuracy applying higher order
Born approximations. As our first approximation the second term of the right
hand side is neglected, so the solution can be given by using the Green
function technique, 
\begin{equation}
\Psi _{{\rm off}}\left( {\bf r}\right) =\frac{k^2}{4\pi }\sum_l%
%TCIMACRO{\dint }
%BeginExpansion
\displaystyle \int 
%EndExpansion
\frac{\exp \left( ikR\right) }R\left( \chi _l\left( {\bf r}_{\parallel
}^{\prime }\right) -\overline{\chi _l}\right) \Psi _{{\rm coh}}\left( {\bf r}%
^{\prime }\right) {\bf d}^3{\bf r}^{\prime }\text{,}  \label{9}
\end{equation}
where $R=\left| {\bf r}-{\bf r}^{\prime }\right| .$ The approximation
requires $\left\| \Psi _{{\rm coh}}\left( {\bf r}\right) \right\| \gg
\left\| \Psi _{{\rm off}}\left( {\bf r}\right) \right\| $, which condition
is expected to be valid for magnetic domains in multilayer, where large
enough homogeneous domains are observed,\cite{Nagy02,Lauter94} so that the
exact solution $\Psi \left( {\bf r}\right) $ is close to the coherent field $%
\Psi _{{\rm coh}}\left( {\bf r}\right) $ according to the above condition.

Far from the scattering system the Fraunhofer approximation

\begin{equation}
\frac{\exp \left( ikR\right) }R\approx \frac{\exp \left( ikr\right) }r\exp
\left( -i{\bf k^{\prime }\cdot r}^{\prime }\right)   \label{10}
\end{equation}
is applied and the final result reads , 
\begin{equation}
\Psi _{{\rm off}}\left( {\bf r}\right) =\sqrt{\frac \pi 2}\frac{k^2}r\exp
\left( ikr\right) \sum_lS_l\left( {\bf K}_{\parallel }\right) T_l\left( {\bf %
k}_{\perp }^{\prime }\right) \Psi ^{{\rm in}}\text{,}  \label{11}
\end{equation}
where

\begin{equation}
T_l\left( {\bf k}_{\perp }^{\prime }\right) =\frac 1{\sqrt{2\pi }}%
\int\limits_{Z_l}\exp \left( -i{\bf k}_{\perp }^{\prime }{\bf r}_{\perp
}\right) T\left( {\bf r}_{\perp }\right) {\bf dr}_{\perp }  \label{12}
\end{equation}
is the Fourier integral over the one dimensional interval $Z_l$ of layer $l$;

\begin{equation}
S_l\left( {\bf K}_{\parallel }\right) =\frac 1{2\pi }\int \exp \left( -i{\bf %
K}_{\parallel }{\bf r}_{\parallel }\right) \left( \chi _l\left( {\bf r}%
_{\parallel }\right) -\overline{\chi _l}\right) {\bf d}^2{\bf r}_{\parallel }
\label{13}
\end{equation}
is the two dimensional Fourier transform of $\left( \chi _l\left( {\bf r}%
_{\parallel }\right) -\overline{\chi }\right) $ and ${\bf K}_{\parallel }$
is the lateral component of the momentum transfer vector ${\bf K}={\bf k}%
^{\prime }-{\bf k}$.

The off-specularly scattered intensity $I_{{\rm off}}=\left( \Psi _{{\rm off}%
},\Psi _{{\rm off}}\right) $reads 
\begin{equation}
I_{{\rm off}}=\frac{\pi k^4}{2r^2}\sum_{ll^{\prime }}\left( \Psi ^{{\rm in}%
},T_l^{+}\left( {\bf k}_{\perp }^{\prime }\right) S_l^{+}\left( {\bf K}%
_{\parallel }\right) S_{l^{\prime }}\left( {\bf K}_{\parallel }\right)
T_{l^{\prime }}\left( {\bf k}_{\perp }^{\prime }\right) \cdot \Psi ^{{\rm in}%
}\right) \text{,}  \label{14}
\end{equation}
which for arbitrary incident polarization is written as 
\begin{equation}
I_{{\rm off}}=\frac{\pi k^4}{2r^2}\sum_{ll^{\prime }}{\bf Tr}\left[
T_l^{+}\left( {\bf k}_{\perp }^{\prime }\right) S_l^{+}\left( {\bf K}%
_{\parallel }\right) S_{l^{\prime }}\left( {\bf K}_{\parallel }\right)
T_{l^{\prime }}\left( {\bf k}_{\perp }^{\prime }\right) \cdot \rho \right] 
\text{,}  \label{15}
\end{equation}
where $\rho $ is the polarization density matrix of the incident radiation.%
\cite{Blume68} From the convolution theorem follows that the Fourier
transform $C_{ll^{\prime }}\left( {\bf R}_{\parallel }\right) $ of 
\begin{equation}
C_{ll^{\prime }}\left( {\bf K}_{\parallel }\right) =\frac 1{2\pi }%
S_l^{+}\left( {\bf K}_{\parallel }\right) S_{l^{\prime }}\left( {\bf K}%
_{\parallel }\right) \text{,}  \label{16}
\end{equation}
is the cross correlation function of the susceptibilities between layers $l$
and $l^{\prime }$ 
\begin{equation}
C_{ll^{\prime }}\left( {\bf R}_{\parallel }\right) =\frac 1A\int \left( \chi
_l\left( {\bf R}_{\parallel }+{\bf r}_{\parallel }\right) -\overline{\chi }%
_l\right) ^{+}\left( \chi _{l^{\prime }}\left( {\bf r}_{\parallel }\right) -%
\overline{\chi }_{l^{\prime }}\right) {\bf d}^2{\bf r}_{\parallel }
\label{17}
\end{equation}
$A$ is the area of the surface $\left( A\rightarrow \infty \right) $. The
final result then becomes 
\begin{equation}
I_{{\rm off}}=\frac{\pi k^4}{2r^2}\sum_{ll^{\prime }}{\bf Tr}\left[
T_l^{+}\left( {\bf k}_{\perp }^{\prime }\right) C_{ll^{\prime }}\left( {\bf K%
}_{\parallel }\right) T_{l^{\prime }}\left( {\bf k}_{\perp }^{\prime
}\right) \cdot \rho \right] \text{,}  \label{18}
\end{equation}
which expression is very convenient in case of randomly distributed lateral
inhomogenities.

\section{Off-specular SMR}

The result of Eq. (\ref{18}) is valid for neutron and x-ray reflectometry
working in the energy domain. However, in case of SMR we detect the time
response after the synchrotron pulse,\cite{Trammell79} which is 
\begin{equation}
\Psi _{{\rm off}}\left( {\bf r,}t\right) =\frac 1{\hbar \sqrt{2\pi }}%
\int\limits_{-\infty }^\infty \Psi _{{\rm off}}\left( {\bf r,}E\right) \exp
\left( -iEt/\hbar \right) {\bf d}E\text{,}  \label{19}
\end{equation}
the Fourier transform of the off-specular field. As both the
susceptibilities $\left( \chi _l\left( {\bf r}_{\parallel }^{\prime
},E\right) -\overline{\chi _l}\left( E\right) \right) $and the coherent
field $\Psi _{{\rm coh}}\left( {\bf r}^{\prime },E\right) $ strongly depend
on the energy\cite{Trammell79} $E$ and according to the definitions of Eqs. (%
\ref{8}), (\ref{12}) and (\ref{13}), the energy dependence also regards to $%
S_l\left( {\bf K}_{\parallel },E\right) $ and $T_l\left( {\bf k}_{\perp
}^{\prime },E\right) $, Eq. (\ref{18}) can not been applied.

In order to work out Eq. (\ref{19}) we define a geometric distribution
function $\Omega _l^\mu \left( {\bf r}_{\parallel }\right) $ of each
homogeneous region $\mu =1$,..,$M$ in layer $l.$ $\Omega _l^\mu \left( {\bf r%
}_{\parallel }\right) $ describes the homogeneous area with an energy
dependent susceptibility $\chi _l^\mu \left( E\right) $ . Over the region of 
$\chi _l^\mu \left( E\right) $ the distribution function $\Omega _l^\mu
\left( {\bf r}_{\parallel }\right) =1$ otherwise $\Omega _l^\mu \left( {\bf r%
}_{\parallel }\right) =0$ , so that the total inhomogeneous susceptibility
is the sum 
\begin{equation}
\chi \left( {\bf r,}E\right) =\sum_{\mu =1}^M\sum_{l=1}^S\Omega _l^\mu
\left( {\bf r}_{\parallel }\right) \chi _l^\mu \left( E\right) \text{,}
\label{20}
\end{equation}
where the space and energy dependent parts of $\chi \left( {\bf r,}E\right) $
have been separated. The average susceptibility inside layer $l$ reads 
\begin{equation}
\overline{\chi _l}\left( E\right) =\frac 1A\sum_{\mu =1}^MA_l^\mu \chi
_l^\mu \left( E\right) \text{,}  \label{21}
\end{equation}
where $A_l^\mu =\int \Omega _l^\mu \left( {\bf r}_{\parallel }\right) {\bf d}%
^2{\bf r}_{\parallel }$ is the total area of the homogeneous part $\mu $
inside layer $l$. Starting from Eqs. (\ref{13}), (\ref{20}) and (\ref{21}) 
\begin{equation}
S_l\left( {\bf K}_{\parallel },E\right) {\bf =}\sum_{\mu =1}^MG_l^\mu \left( 
{\bf K}_{\parallel }\right) \overline{\chi }_l^\mu \left( E\right) \text{,}
\label{22}
\end{equation}
with 
\begin{equation}
G_l^\mu \left( {\bf K}_{\parallel }\right) =\frac 1{2\pi }\int \exp \left( -i%
{\bf K}_{\parallel }{\bf r}_{\parallel }\right) \left( \Omega _l^\mu \left( 
{\bf r}_{\parallel }\right) -\frac 1AA_l^\mu \right) {\bf d}^2{\bf r}%
_{\parallel }\text{,}  \label{23}
\end{equation}
we finally get the energy dependent off-specular field

\begin{equation}
\Psi _{{\rm off}}\left( {\bf r,}E\right) =\sqrt{\frac \pi 2}\frac{k^2}r\exp
\left( ikr\right) \sum_{l,\mu }G_l^\mu \left( {\bf K}_{\parallel }\right)
\chi _l^\mu \left( E\right) T_l\left( {\bf k}_{\perp }^{\prime },E\right)
\Psi ^{{\rm in}}  \label{24}
\end{equation}
and the off-specular intensity 
\begin{equation}
I_{{\rm off}}\left( E\right) =\frac{k^4}{4r^2}\sum_{ll^{\prime }\mu \mu
^{\prime }}C_{l^{\prime }l}^{\mu ^{\prime }\mu }\left( {\bf K}_{\parallel
}\right) {\bf Tr}\left[ \left( \Gamma _{l^{\prime }}^{\mu ^{\prime }}\right)
^{+}\left( {\bf k}_{\perp }^{\prime },E\right) \Gamma _l^\mu \left( {\bf k}%
_{\perp }^{\prime },E\right) \cdot \rho \right] \text{.}  \label{25}
\end{equation}
The matrices for area $\mu $, $l$%
\begin{equation}
\Gamma _l^\mu \left( {\bf k}_{\perp }^{\prime },E\right) =\chi _l^\mu \left(
E\right) T_l\left( {\bf k}_{\perp }^{\prime },E\right) ,  \label{26}
\end{equation}
are the products of the homogeneous solution $T_l\left( {\bf k}_{\perp
}^{\prime },E\right) $ and the susceptibility $\chi _l^\mu \left( E\right) $
of that area.

\begin{equation}
C_{l^{\prime }l}^{\mu ^{\prime }\mu }\left( {\bf K}_{\parallel }\right) =%
\frac 1{2\pi }G_l^{\mu ^{\prime }}\left( {\bf K}_{\parallel }\right)
^{*}G_l^\mu \left( {\bf K}_{\parallel }\right)   \label{27}
\end{equation}
is the geometrical correlation function between layers $l$ and $l^{\prime }$%
, homogeneous parts $\mu $ and $\mu ^{\prime }$. In Eq. (\ref{25}) the
geometrical correlation function is separated from the energy dependent
functions, and can therefore be applied for time domain and energy domain
measurements, as well. Now the Fourier transformation can be worked out.
With Eq. (\ref{19}) the time dependent intensity becomes 
\begin{equation}
I_{{\rm off}}\left( t\right) =\frac{k^4}{4r^2}\sum_{ll^{\prime }\mu \mu
^{\prime }}C_{l^{\prime }l}^{\mu ^{\prime }\mu }\left( {\bf K}_{\parallel
}\right) {\bf Tr}\left[ \left( \Gamma _{l^{\prime }}^{\mu ^{\prime }}\right)
^{+}\left( {\bf k}_{\perp }^{\prime },t\right) \Gamma _l^\mu \left( {\bf k}%
_{\perp }^{\prime },t\right) \cdot \rho \right] \text{,}  \label{28}
\end{equation}
where

\begin{equation}
\Gamma _l^\mu \left( {\bf k}_{\perp }^{\prime },t\right) =\frac 1{\hbar 
\sqrt{2\pi }}\int\limits_{-\infty }^\infty \Gamma _l^\mu \left( {\bf k}%
_{\perp }^{\prime },E\right) \exp \left( -iEt/\hbar \right) {\bf d}E
\label{29}
\end{equation}

\section{Time integrated off-specular SMR}

One possibility for off-specular reflectometry is the geometry called '$%
\omega $-scan', where we set the value of $2\theta $ (we fix the height of
the detector) and vary the sample orientation $\omega $ (leaving the
constraint of specular reflection $\omega $ $=\theta $). In the special case
of $\omega $- scan the lateral components of the momentum transfer vector $%
{\bf K}_{\parallel }$ and the perpendicular component of the wave vector of
the out coming wave ${\bf k}_{\perp }^{\prime }$can be expressed by the
grazing angle $\theta $ and by $\omega $

\begin{equation}
{\bf K}_{\parallel }=2k\sin \theta \sin \left( \theta -\omega \right)
\label{30}
\end{equation}
\begin{equation}
{\bf k}_{\perp }^{\prime }=k\sin \omega \text{.}  \label{31}
\end{equation}
The time-integrated intensity is $I\left( \omega ,\theta \right)
=\int\limits_{t^1}^{t^2}I_{{\rm off}}\left( t\right) {\bf d}t,$ where $t_1$
and $t_2$ define the time window of the measurements after the synchrotron
bunch. Applying Eqs. (\ref{28}) and (\ref{29}) the final form reads 
\begin{equation}
I\left( \omega ,\theta \right) =\frac{N\tau }{t_{{\rm Bunch}}}\frac{k^4}{%
4\hbar r^2}\sum_{n=-\infty }^\infty S_n\int\limits_{-\infty }^\infty {\bf d}%
E\sum_{ll^{\prime }\mu \mu ^{\prime }}C_{l^{\prime }l}^{\mu ^{\prime }\mu
}\left( {\bf K}_{\parallel }\right) {\bf Tr}\left[ \left( \Gamma _{l^{\prime
}}^{\mu ^{\prime }}\right) ^{+}\left( {\bf k}_{\perp }^{\prime },E+n\hbar
\Omega \right) \Gamma _l^\mu \left( {\bf k}_{\perp }^{\prime },E\right) \rho
\right] \text{,}  \label{32}
\end{equation}
where $N$ is the number of photons in the incident synchrotron bunch, $\tau $
is the duration of the synchrotron bunch ($\tau \approx 50$ ps at ESRF), $t_{%
{\rm Bunch}}$ is the time interval between the synchrotron bunches, $\Omega =%
\frac{2\pi }{t_{{\rm Bunch}}}$ and $S_n$ are the discrete Fourier components
of the periodical time window function $S\left( t\right)
=\sum\limits_{n=-\infty }^\infty S_n\exp \left( in\Omega t\right) $of the
experiment defined by $S\left( t\right) =1$ for $t_1<t<t_2$, otherwise $%
S\left( t\right) =0$ after each synchrotron bunch.

\section{Conclusion}

Off-specular SMR is sensitive to the lateral structure of the hyperfine
fields, so it can be used for studying magnetic domains in multilayer.
Because of the underlying common optical approach the same theory can be
applied for SMR, x-ray reflectometry, spin polarized neutron reflectometry 
\cite{Deak01} and x-ray resonance exchange scattering \cite{Hannon88,Deak03}
for both specular and off-specular reflection. Using perturbation theory
(DWBA), the off-specularly reflected intensity of SMR has been expressed by
the geometrical correlation functions of the lateral structure and by the
specular field profile in the layers. A newer version of the data evaluation
computer program {\bf EFFI} ({\bf E}nvironment {\bf F}or {\bf FI}tting),
based on this calculus, will be freely soon available.\cite{Spiering00,EFFI}

\section{Acknowledgements}

This work was partly supported by the Hungarian Scientific Research Fund
(OTKA) under Contract No. T029409 and by the Hungarian Academy of Sciences
(Contact No. AKP 2000-103 2,3). The financial support by the Deutscher
Akademischer Austauschdienst (DAAD) is gratefully acknowledged.

\appendix 

\section*{General solution of the coherent field equation}

The solution of Eq. (\ref{3}) was given in Refs. \cite
{Deak01,Deak96,Spiering00}, where instead of the second order differential
equation regarding to $\Psi _{{\rm coh}}\left( {\bf r}\right) $, after
defining the derivative field $\Phi \left( {\bf r}_{\perp }\right) =\left(
ik\sin \Theta \right) ^{-1}\Psi ^{\prime }\left( {\bf r}_{\bot }\right) $, a
system of first order differential equations was used.\cite{Deak01} The
solution was given in terms of the $4\times 4$ characteristic matrix $L$ of
the system.\cite{Deak01,Deak96,Spiering00} 
\begin{equation}
%TCIMACRO{
%\binom{\Phi \left( {\bf r}_{\perp }\right) }{\Psi \left( {\bf r}_{\perp }\right) }}
%BeginExpansion
{\Phi \left( {\bf r}_{\perp }\right)  \choose \Psi \left( {\bf r}_{\perp }\right) }
%EndExpansion
=L\left( {\bf r}_{\perp }\right) 
%TCIMACRO{\binom{\Phi \left( 0\right) }{\Psi \left( 0\right) }}
%BeginExpansion
{\Phi \left( 0\right)  \choose \Psi \left( 0\right) }
%EndExpansion
\text{,}  \label{A1}
\end{equation}
which equation has the physical meaning that there is linear connection
between the fields in depth ${\bf r}_{\perp }=0$ and in arbitrary depth $%
{\bf r}_{\perp }$ the characteristic matrix $L$ being the matrix
coefficient. Taking into account the boundary conditions, the field at the
top surfaces of the system $\left( {\bf r}_{\perp }=0\right) $ is 
\begin{equation}
\Psi \left( 0\right) =\Psi ^{{\rm in}}+R_{{\rm sp}}\Psi ^{{\rm in}}\text{,}
\label{A2}
\end{equation}
the sum of the incident $\Psi ^{{\rm in}}$ and the reflected $R_{{\rm sp}%
}\Psi ^{{\rm in}}$ waves so Eq. (\ref{A1}) reads 
\begin{equation}
%TCIMACRO{
%\binom{\Phi \left( {\bf r}_{\perp }\right) }{\Psi \left( {\bf r}_{\perp }\right) }}
%BeginExpansion
{\Phi \left( {\bf r}_{\perp }\right)  \choose \Psi \left( {\bf r}_{\perp }\right) }
%EndExpansion
=L\left( {\bf r}_{\perp }\right) 
%TCIMACRO{
%\binom{\Psi ^{{\rm in}}-R_{{\rm sp}}\Psi ^{{\rm in}}}{\Psi ^{{\rm in}}+R_{{\rm sp}}\Psi ^{{\rm in}}}}
%BeginExpansion
{\Psi ^{{\rm in}}-R_{{\rm sp}}\Psi ^{{\rm in}} \choose \Psi ^{{\rm in}}+R_{{\rm sp}}\Psi ^{{\rm in}}}
%EndExpansion
\text{,}  \label{A3}
\end{equation}
where the concept of impedance tensors was used,\cite{Borzdov76} taking into
account that the fields at ${\bf r}_{\perp }=0$ are in vacuum (see Eqs. (21)
and (22) of Ref. \cite{Deak01}). Expressing the second component from Eq. (%
\ref{A3}), the field at arbitrary depth ${\bf r}_{\perp }$ reads 
\begin{equation}
\Psi \left( {\bf r}_{\perp }\right) =\left\lfloor L^{\left[ 21\right]
}\left( {\bf r}_{\perp }\right) \left( I-R_{{\rm sp}}\right) +L^{\left[
22\right] }\left( {\bf r}_{\perp }\right) \left( I+R_{{\rm sp}}\right)
\right\rfloor \Psi ^{{\rm in}}\text{,}  \label{A4}
\end{equation}
and using the notation $T\left( {\bf r}_{\perp }\right) =L^{\left[ 21\right]
}\left( {\bf r}_{\perp }\right) \left( I-R_{{\rm sp}}\right) +L^{\left[
22\right] }\left( {\bf r}_{\perp }\right) \left( I+R_{{\rm sp}}\right) $the
solution of the three dimensional homogeneous wave equation is 
\begin{equation}
\Psi _{{\rm coh}}\left( {\bf r}\right) =T\left( {\bf r}_{\perp }\right) \Psi
^{{\rm in}}\exp \left( i{\bf k}_{\parallel }\cdot {\bf r}_{\parallel
}\right) \text{.}  \label{A5}
\end{equation}

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\end{document}
