%% 10/1/98 dd Most of the following was copied %% from ~davis/src/scatter/tex/spie_v4.tex.gz %% and slightly modified. %% Double comments put in by dd \clearpage \section{Scattering Model} \label{sec:scat_theory} \begin{quotation} {\it Objective:} Understand and model the ``anomalous scatter'' seen in XRCF tests of the HEG gratings, Section~\ref{sec:scatter}. {\it Publication(s):} Davis {\it et al.}~\cite{davis98} \end{quotation} \subsection{Introduction} %{{{ XRCF tests of the HEG gratings showed anomalous scattering of monochromatic radiation, in particular the gratings showed events with significant deviations from the integer grating orders and the events were concentrated along the dispersion direction, Section~\ref{sec:scatter} and Marshall {\it et al.}~\cite{marshall97}. In this section, we present a general overview of grating scatter as a result of fluctuations in the grating bar geometry. The measured HEG grating scatter is shown to be consistent with what one expects from scatter due to deviations in the grating bar geometry from perfect bars. For the purposes of modeling, a rectangular bar model is adopted and bar parameters are deduced via fitting the model to the scattering data. The correlations deduced from this model lead to a simple physical picture of grating bar fluctuations where a small fraction of the bars, e.g., $1$ in $200$ are leaning. We treat the problem of scattering from a disordered diffraction grating, and from first principles arrive at an equation for the probability distribution of scatter from such a grating. Without making any reference to a specific geometric model for the grating bar shapes, we show that the anomalous scattering observed at XRCF (e.g., see \Figure{fig:nocorr}) shares many of the features that one would expect to be produced as a result of grating disorder. In particular, the strong peaks that appear symmetrically about the various diffraction orders can be explained as a natural consequence of one or more correlations between neighboring bars with a correlation length that is characterized by the width of the peaks. The correlation length is estimated to be roughly three grating periods. \begin{figure} \begin{center} \epsfig{file=1-3nocorr-1.775.ps,height=10cm,angle=270} \caption{\label{fig:nocorr} \small Measured HEG scatter at 1.775~keV. The scattering-per-bin normalized to the first-order intensity is plotted versus diffraction order. The solid curve is the best fit to this data using a simple rectangular grating model with no bar-to-bar correlations. } \end{center} \end{figure} A more detailed description of the correlations based on a rectangular model for the bar shapes is presented in the section \ref{sec:rect}. Although the rectangular model has been shown to be inadequate for modeling the grating efficiency to the one percent calibration goal\cite{markert95}, it performs well at the ten to twenty percent level. Its most redeeming feature is its simplicity, which allows many of the calculations to be performed analytically. Within the context of this model, expressions for the correlations and the scattering probability are derived and then fitted to the experimental data. The resulting fit, while not perfect, does reflect many of the salient features of the data, although the resulting effective rectangular parameters are somewhat larger than the values implied by modeling of the grating efficiencies, Figure~\ref{fig:grat_cross_secs}. %% dd changed from Obs Guide ref to figure ref in last sentence above. In the final section we present a discussion of the correlations as deduced from the modeling. Based upon the form of the correlations, a physical picture of the gratings emerge in which a small fraction (e.g., $1$ in every $200$) of the grating bars are leaning. Finally, the major results of this section are summarized and future modeling improvements are suggested. %}}} %%\subsection{Theoretical Overview} %{{{ \subsection{General Formulation} %{{{ Consider a diffraction grating composed of $N$ lines, or bars, with period $d$, and a photon with wave number $k=2\pi/\lambda$ incident upon the grating. It follows from Huygen's principle that the probability for scattering into an angle between $\theta$ and $\theta+\d{\theta}$ with respect to the normal of the grating is given by \begin{equation} \label{pNmicro} p(k, s)\d{s} = \frac{kd}{2\pi N} \bigg|\sum_{j=0}^{N-1} e^{ijksd} F_j(k, s)\bigg|^2 \; \d{s} , \end{equation} where $s=\sin\theta$ and the {\em structure factor} $F_j(k,s)$ is defined by \begin{equation} \label{Fj} F_j(k, s) = \frac{1}{d} \int_0^d \d{x} e^{iksx + i\phi_j(k,x)} . \end{equation} The shape and composition of an individual grating bar determines the complex phase shift $\phi_j(k,x)$ that, in general, depends upon energy and, possibly, the polarization of the photon. Assuming the validity of scalar diffraction theory, and neglecting reflection and refraction, the phase shift can be written in the form \begin{equation} \phi_j(k,x) = -k (\delta - i\beta) z_j(x), \end{equation} where $\delta$ and $\beta$ are energy dependent functions related to the dielectric constant of the grating bars. The function $z_j(x)$ represents the path length of the photon as it passes though the $j$th grating bar; it is sometimes called, rather loosely, the ``grating bar shape'', and more rigorously, the ``path-length function''. In general, the structure factor $F_j(k, s)$ will vary with the grating bars because the ``bar shape'' will vary from bar to bar. Since an exact computation of the scattering probability would require complete knowledge of all the bar shapes, and since there are millions of bars that contribute to the scattering, we must resort to a statistical means of computing \eq{pNmicro}. This is completely analogous to the situation in statistical physics where one deals with systems that consist of large numbers of degrees of freedom, and whose complete characterization requires a knowledge of the initial conditions for each degree of freedom. These problems are treated statistically where one introduces the notion of an ensemble of macroscopically identical systems and makes an ergodic hypothesis that at any given instant the microscopic state of the system is represented by one of the systems in the ensemble. Hence the assumption of ergodicity allows one to compute the time average of a macroscopic quantity by averaging that quantity over the ensemble. In a similar vein we shall introduce the idea of an ensemble of gratings and make the assertion that the observed scattering does not depend upon the exact ``microscopic'' state of the grating, but that it is an indicator of the ``macroscopic'' state. In other words, we shall assume that the scattering is a macroscopic property of the ensemble and may be computed via \begin{equation} \label{pN} p(k, s)\d{s} = \frac{kd}{2\pi N} \bigg\langle \bigg|\sum_{j=0}^{N-1} e^{ijksd} F_j(k, s)\bigg|^2 \bigg\rangle \; \d{s} , \end{equation} where $\langle\cdot\rangle$ denotes averaging over the ensemble. The specification of the ensemble involves knowing the probability for a particular configuration of grating bars, denoted by the ordered set of bar shapes $\{z_j\}$. Let this probability be represented by the probability density ${\cal P}(z_0,z_1,{\ldots},z_{N-1})$ such that the ensemble average of some function $f(z_0,{\ldots}z_{N-1})$ is given by \footnote{% Since the bar shape $z_j(x)$ is a function, the integrals in this equation are functional integrals and the symbol ${\cal D}z(x)$ is called the {\em Weiner measure}. See the book by Feynman and Hibbs\cite{feynman65} for a simple introduction to functional integration. Alternatively, one can view such an integral as a multiple integral over the parameters defining a shape, e.g., the bar width and height for a rectangular bar. } \begin{equation} \langle f(z_0,{\ldots},z_{N-1})\rangle = \int {\cal D}z_0(x)\cdots{\cal D}z_{N-1}(x) f(z_0,{\ldots}, z_{N-1}) {\cal P}(z_0,{\ldots},z_{N-1}). \end{equation} Furthermore, we shall assume that the ensemble is such that $\ave{f(z_j)}$ is independent of $j$, and that $\ave{f(z_j, z_l)}$ depends only upon the difference $j-l$. For obvious reasons, a probability distribution with these properties is said to be {\em homogeneous} or {\em stationary}\cite{stratonovich63}. With the assumption of a homogeneous probability distribution for bar shapes, denote the ensemble averaged structure factor by \begin{equation} \Fbar(k, s) = \bra F_j(k, s) \ket \label{fbar} \end{equation} and define $f_j(k, s)$ to be the deviation of the $j$th bar from this value via \begin{equation} F_j(k, s) = \Fbar(k, s) + f_j(k, s). \label{fj} \end{equation} Substituting the above definitions into \eq{pN} yields for the probability density \begin{equation} p(k,s) = \frac{kd\sin^2(\half Nksd))}{2\pi N \sin^2(\half ksd)} |\Fbar(k,s)|^2 + \frac{kd}{2\pi N} \sum_{j,l} e^{iksd (j - l)} \ave{f_j f_l^*}. \end{equation} For very large $N$, the first term goes to zero like $1/N$ except when $ksd$ is a multiple of $2\pi$ where it becomes proportional to $N$. In fact, one can show that the factor involving $N$ in first term is a representation of a series of delta functions: \begin{equation} \frac{kd\sin^2(\half Nksd)}{2\pi N \sin^2(\half ksd)} = \sum_n \delta_N(s - \frac{2\pi n}{kd}) . \end{equation} This equivalence allows the probability density to be written \begin{equation} \label{pN2} p(k,s) = |\Fbar(k,s)|^2 \sum_n \delta_N(s-\frac{2\pi n}{kd}) + \frac{kd}{2\pi N} \sum_{j,l} e^{iksd (j - l)} \ave{f_j f_l^*}. \end{equation} Thus, the first term represents the familiar interference pattern that consists of a series of sharp peaks, weighted by $|\Fbar(k,s)|^2$, whose height varies with $N$. The peaks occur at values of $ksd$ that are multiples of $2\pi$, or equivalently when \begin{equation} n \lambda = d \sin\theta, \end{equation} which is the familiar grating equation. The second term is the scattering term and is more complicated. Superficially, it appears to vanish in the limit of a large number of bars; however, owing to the double sum (hence $N^2$ terms), this is not necessarily true. The quantities $\ave{f_j f_l^*}$ represent correlations between the various grating elements. If the shape of each grating bar element is independent of the others, then $\ave{f_j f_l^*}$ vanishes except when $j = l$. More explicitly, if no correlations exist between the fluctuations of the grating bars then \begin{equation} \ave{f_j f_l^*} = \ave{|f_j|^2} \delta_{jl} . \label{nocorr} \end{equation} If there are correlations between the various grating bars, then \eq{nocorr} is not correct. We make the assumption that if any correlations are present, then they have a short range. This is really an assumption about the fabrication method, but one could imagine the presence of long range correlations from e.g., a holographic fabrication process in which a non-monochromatic light source would create a beating effect. Similarly, for a ``ruled'' grating, the mechanical system could introduce periodic long range correlations. Nevertheless, we shall assume that $\ave{f_j f_l^*}$ is zero for $|j - l|$ greater than some number $M$. For example, if $M = 1$, then only nearest neighbor correlations may be present. If $M \ll N$, then one may neglect edge effects in the sum and write the scattering term in \eq{pN2} as \begin{equation} \frac{kd}{2\pi N} \sum_{j=0}^{N-1}\sum_{m=-M}^{M} e^{imksd} \ave{f_j f_{j+m}^*}. \end{equation} Since we have assumed that the ensemble governing the probability distributions is homogeneous, by definition we know that the correlations are translation invariant, i.e., $\ave{f_j f_{j+m}^*}$ does not depend upon $j$. Hence, the scattering term reduces to \begin{equation} \label{complexscat} \frac{kd}{2\pi} \sum_{m=-M}^{M} e^{imksd} c_m(k, s) , \end{equation} where $c_m$ has been defined by \begin{equation} \label{cm-def} c_m (k, s) = \ave{f_j(k, s) f_{j+m}^*(k,s)}. \end{equation} Using the property that $c_{-m}=c_m^*$, which follows from the definition of $c_m$, and separating out real and imaginary parts, it follows that \eq{pN2} may be written \begin{equation} \label{INall} \begin{split} p(k,s) =& |\Fbar(k,s)|^2 \sum_n \delta_N(s-\frac{2\pi n}{kd}) + \frac{kd}{2\pi} c_0(k,s) \\ &+ \frac{kd}{\pi} \sum_{m=1}^{M} \bigg\{ \mbox{Re}[c_m(k,s)] \cos (mksd) - \mbox{Im}[c_m(k,s)] \sin(mksd) \bigg\}. \end{split} \end{equation} This equation shows that the intensity pattern will consist of sharp diffraction peaks on top of a weak but diffuse scattering background. The actual angular dependence of the diffuse scattering will depend upon the precise nature of the correlations and upon the correlation length given by $M$. Nevertheless, this equation permits us to make several observations about the scattering without reference to a detailed model of the grating bar shapes. First of all, note that if the correlations $c_m(k,s)$ are slowly varying functions of $s$, then the scattering will be quasi-periodic with a period of $2\pi/kd$. That is, the scattering will roughly repeat itself between diffraction orders. The scattering data shown in \Figure{1775keVdata} appears to share this feature. For example, strong scattering peaks appear just to the right of the first, second, and third diffraction orders. In addition, there is a peak just to the left of first order and there appears to be some evidence for peaks just to the left of the second and third orders, although the existence of the peak at the left of second order is questionable. On the other hand, the fact that the scattering is not strictly periodic is just an indication that $c_m(k,s)$ depends upon $s$. %}}} \subsection{Displacement Variations} %{{{ One might think that since the scattering is relatively weak near zeroth order with no strong peaks in that region, the apparent quasi-periodicity reflected in the data around first, second, and third orders is coincidental. However, an important prediction of \eq{INall} is that for fluctuations involving simple displacements of the grating bars, there can be no scattering near zeroth order. In fact, this is a well known result from the theory of X-ray diffraction from crystals\cite{warren90}. To prove this in the context of a diffraction grating, consider a model of $N$ grating bar shapes that differ from one another only by a shift, or displacement from their ideal lattice positions. That is, the path-length function for the $j$th bar is assumed to be given by $z_j(x)=z(x+a_j)$ where $a_j$ denotes the shift of the bar from its ideal position in the unit cell. Furthermore, assume that $z(x)$ is non-zero only for values of $x$ such that $|a_j| \le x \le d-|a_j|$ for all $j$. Physically, this means that the center of the grating bar whose shape is described by $z(x)$ lies near middle of the cell. Then, the structure factor $F_j$ for the $j$th unit cell can be written \begin{equation} \begin{split} F_j(k, s) &= \frac{1}{d} \int_0^d \d{x} e^{iksx + i\phi(x + a_j)} \\ &= \frac{1}{d} e^{-iksa_j} \int_{a_j}^{d+a_j} \d{x} e^{iksx + i\phi(x)} \\ &= \frac{1}{d} e^{-iksa_j} \bigg( \int_0^d + \int_d^{d+a_j} - \int_0^{a_j} \bigg) \d{x} e^{iksx + i\phi(x)} . \end{split} \end{equation} With the assumption that $z(x)$, and hence $\phi(x)$, vanishes for $x<|a_j|$ or for $x \ge d-|a_j|$, the latter two integrations can be performed with the result \begin{equation} F_j(k,s) = e^{-iksa_j} F(k,s) + \frac{1}{iksd} (1 - e^{-iksa_j}) (e^{iksd} - 1) . \end{equation} Here, $F(k,s)$ has been defined \begin{equation} F(k,s) = \frac{1}{d}\int_0^d \d{x} e^{iksx + i\phi(x)}. \end{equation} From this, it trivially follows from \eq{fbar}, that \begin{equation} \Fbar(k,s) = \ave{e^{-iksa_j}} F(k,s) + \frac{1}{iksd} (e^{iksd} - 1) (1 - \ave{e^{-iksa_j}}) \end{equation} and we also deduce from \eq{fj} that \begin{equation} f_j(k,s) = \bigg[e^{-iksa_j} - \ave{e^{-iksa_j}}\bigg] \bigg[F(k,s) - \frac{1}{iksd}(e^{iksd} - 1)\bigg] . \end{equation} Thus, from this equation it is immediately clear that a model of grating bar {\em fluctuations consisting of pure displacements produces no scattering in the immediate vicinity of zeroth order}. This result is quite general since no further assumptions were made about about the actual shape of the grating bars. To see the impact of these fluctuations on the diffraction orders, assume that the shift parameters $a_j$ are independently Gaussian distributed with a mean of zero and a variance of $\sigma^2$. For such a distribution, one can show that \begin{equation} \ave{e^{-i2\pi n a_j/d}} = e^{-\half (2\pi n\sigma/d)^2} \end{equation} from which it follows \begin{equation} |\Fbar(k,s_n)|^2 = |F(k,s_n)|^2 e^{-(2\pi n \sigma/d)^2}, \end{equation} where $s_n=2n\pi/kd$ is the value of $s$ at the $n$th diffraction order. This equation explicitly shows that the fluctuations reduce the power in the diffraction orders, as one would expect from considerations of flux conservation. %}}} \subsection{Bar Geometry Variations} %{{{ The geometry of the $j$th grating bar may be specified by a number of geometric quantities. Let the $\mu$th such quantity for the $j$th bar be denoted by $\xi_{\mu}^{(j)}$. For example, a rectangular grating bar may be specified by three quantities: the width ($\xi_1$), height ($\xi_2$), and displacement ($\xi_3$). The structure factor $F_j$ will be a function of these quantities and indicated symbolically by $F_j(k,s,\xi)$. Let $\Delta\xi_{\mu}^{(j)}$ be the deviation of the $\mu$th parameter from its mean, i.e., \begin{equation} \xi_{\mu}^{(j)} = \ave{\xi_{\mu}^{(j)}} + \Delta\xi_{\mu}^{(j)} . \end{equation} Assuming that the fluctuations $\Delta\xi_{\mu}^{(j)}$ are small, we can approximate the structure factor by \begin{equation} F_j(k,s,\xi) = F(k,s,\ave{\xi}) + \sum_{\mu} \frac{\partial F}{\partial\xi_{\mu}} \Delta\xi_{\mu}^{(j)}, \end{equation} where $F(k,s,\ave{\xi})$ is independent of $j$ by virtue of the assumed homogeneity of the ensemble. It then follows from \eq{fj} \begin{equation} f_j(k,s,\xi) = \sum_{\mu} \frac{\partial F}{\partial\xi_{\mu}} \Delta\xi_{\mu}^{(j)} \end{equation} and from \eq{cm-def} \begin{equation} \label{cm-small} \begin{split} c_m(k,s) &= \ave{f_j f_{l+m}^{*}} \\ &= \sum_{\mu\nu} \frac{\partial F}{\partial\xi_{\mu}} \frac{\partial F^{*}}{\partial\xi_{\nu}} \ave{\Delta\xi_{\mu}^{(j)}\Delta\xi_{\nu}^{(j+m)}} . \end{split} \end{equation} Since the ensemble is assumed to be homogeneous, the correlations $\ave{\Delta\xi_{\mu}^{(j)}\Delta\xi_{\nu}^{(j+m)}}$ are independent of $j$, and will be denoted by $G_{\mu\nu}(m)$, i.e., \begin{equation} G_{\mu\nu}(m) = \ave{ \Delta\xi_{\mu}^{(j)} \Delta\xi_{\nu}^{(j+m)} }. \end{equation} Moreover, they are independent of $k$ and $s$, and assuming left-right invariance of the grating, they satisfy the symmetry relations \begin{equation} G_{\mu\nu}(m) = G_{\mu\nu}(-m) = G_{\nu\mu}(m) . \end{equation} From this symmetry it follows that $c_m(k,s)$ is real. Since $G_{\mu\nu}(m)$ is an even function of $m$, it may be represented as Fourier cosine series. In addition, from physical considerations the correlations must fall off with increasing $|m|$. For simplicity, assume that the correlations $G_{\mu\nu}(m)$ may be represented in the form \begin{equation} \label{eq:gmunu} G_{\mu\nu}(m) = \sigma_{\mu\nu} e^{-\alpha_{\mu\nu} |m|} \cos (2\pi m \Omega_{\mu\nu}). \end{equation} In terms of these correlations, the probability density for scattering from \eq{complexscat} may be expressed as \begin{equation} \frac{kd}{2\pi} \sum_{\mu\nu} \sigma_{\mu\nu} \frac{\partial F}{\partial\xi_{\mu}} \frac{\partial F^{*}}{\partial\xi_{\nu}} \bigg[ \sum_{m=-\infty}^{\infty} e^{imksd} e^{-\alpha_{\mu\nu} |m|} \cos (2\pi m\Omega_{\mu\nu}) \bigg] . \end{equation} In deriving this equation, the exponential fall off of the correlations have allowed the sum to extended to all $m$. The sum over $m$ may be performed by writing the cosine in its complex representation and recognizing that the resulting sum is geometric. It is left as an exercise for the reader to show that \begin{equation} \label{sumoverm} \begin{split} \sum_{m=-\infty}^{\infty} e^{imksd} e^{-\alpha |m|} \cos (2\pi m\Omega) & \\ = \frac{1+e^{-\alpha}}{2} \bigg[ & \frac{1-e^{-\alpha}}{(1-e^{-\alpha})^2 +2e^{-\alpha}(1-\cos(ksd+2\pi\Omega))} \\ &+ \frac{1-e^{-\alpha}}{(1-e^{-\alpha})^2 + 2e^{-\alpha}(1-\cos(ksd-2\pi\Omega))} \bigg] \end{split} \end{equation} The right hand side of this equation shows that the correlations will produce peaks whenever $\cos(ksd \pm 2\pi\Omega)$ is unity, i.e., at values of $s$ when $ksd/2\pi = n \pm \Omega$. In other words, {\em the scattering will consist of peaks that are symmetrically located about the diffraction orders}, although the strength of the peaks will depend upon the exact form of $c_m(k,s)$. A quick glance at \Figure{1775keVdata} shows that the peaks immediately to the left and right of first order have this property. In addition, the data has a peak just to the right of third order and it appears to have a small peak symmetrically placed at the left of third order. It is harder to make the argument about second order because if there is a peak to the left of second order then it is suppressed. The location of the peaks and their widths provide valuable information about the parameters $\Omega$ and $\alpha$. In particular the location of the peaks give information about $\Omega$, and their widths are governed by $\alpha$. To see this, suppose that $\alpha$ and $\theta=ksd+2\pi\Omega$ are small. Then the first term term on the right hand side of \eq{sumoverm} may be approximated by \begin{equation} \frac{\alpha}{\alpha^2 + \theta^2}. \end{equation} This function has a maximum at $\theta=0$ and the half-width at half-maximum is $\alpha$. Hence, the width of the scattering peak provides information about $\alpha$, which in turn dictates the range of the correlation responsible for the peak. Armed with this knowledge one can make a rough estimate of $\Omega$ and $\alpha$ for the correlation associated with the peaks closest to the diffraction orders of \Figure{1775keVdata}. We estimate $\Omega$ to be around $0.1$ and $\alpha$ to be roughly $0.05\times 2\pi\approx 0.3$. This implies that the correlations have a length of $1/\alpha$ or about $3$ grating bars. %}}} %}}} \subsection{Correlations in the Rectangular Model} %{{{ \label{sec:rect} In this section, we assume that the grating bar shapes may be represented by a simple rectangle with height $h$ and width $w$. The center of the bar is assumed to offset a distance $a$ from the center of the lattice node (see \Figure{figrect}). \begin{figure} \begin{center} \epsfig{file=bar.eps,height=6cm} \caption{\label{figrect} \small Figure showing the geometric parameters for a rectangular grating bar of width $w$ and height $h$ in a unit cell with period $d$. The displacement of the bar from the center of the unit cell is given by $a$.} \end{center} \end{figure} The rectangular geometry readily permits the integral in \eq{Fj} to computed with the result \begin{equation} F_j(k,s) = \frac{\sin(\half ksd)}{\half ksd} + e^{iksa_j}(e^{-kh_j\beta-ikh_j\delta} - 1) \frac{\sin(\half ksw_j)}{\half ksd} . \end{equation} As in the previous section, we assume that the correlations are small so that \eq{cm-small} is valid. Expressing \eq{cm-small} in terms of $h$, $w$, and $a$, results in \begin{equation} \begin{split} c_m(k,s) =& d^2 \bigg|\deldel{F}{a}\bigg|^2 G_{aa}(m) +h^2 \bigg|\deldel{F}{h}\bigg|^2 G_{hh}(m) +d^2 \bigg|\deldel{F}{w}\bigg|^2 G_{ww}(m) \\ &+2 hd \; \mbox{Re} \bigg[ \deldel{F}{a}\deldel{F^{*}}{h} \bigg] G_{ah}(m) +2 hd \; \mbox{Re} \bigg[ \deldel{F}{w}\deldel{F^{*}}{h} \bigg] G_{wh}(m) \\ &+2 d^2 \; \mbox{Re} \bigg[ \deldel{F}{a}\deldel{F^{*}}{w} \bigg] G_{aw}(m), \end{split} \end{equation} where factors of $d$ and $h$ have been introduced to make the correlation functions $G_{\mu\nu}(m)$ unitless, e.g., \begin{equation} \begin{split} G_{wh}(m) &= \ave{\frac{\Delta w}{d}\frac{\Delta h}{h}} \\ &= \sigma_{wh} e^{-\alpha_{wh}|m|} \cos(2\pi m\Omega_{wh}). \end{split} \end{equation} The partial derivatives of the structure factor $F(k,s)$ are rather straightforward to carry out with the result: \begin{equation} \begin{split} \bigg|\deldel{F}{a}\bigg|^2 &= \frac{4}{d^2} [1 - 2e^{-kh\beta} \cos (kh\delta) + e^{-2kh\beta}] \sin^2(\half ksw) \\ \bigg|\deldel{F}{w}\bigg|^2 &= \frac{1}{d^2} [1 - 2e^{-kh\beta} \cos (kh\delta) + e^{-2kh\beta}] \cos^2(\half ksw) \\ \bigg|\deldel{F}{h}\bigg|^2 &= k^2(\delta^2+\beta^2) e^{-2kh\beta} \bigg[\frac{\sin(\half ksw)}{\half ksd}\bigg]^2 \\ \mbox{Re}\bigg[\deldel{F}{w}\deldel{F^{*}}{h}\bigg] &= \frac{k}{d} e^{-kh\beta} \bigg[ \beta\cos(kh\delta) + \delta \sin(kh\delta) - \beta e^{-kh\beta} \bigg] \frac{\sin(ksw)}{ksd} \\ \mbox{Re}\bigg[\deldel{F}{a}\deldel{F^{*}}{h}\bigg] &= \frac{4k}{d} e^{-kh\beta} \bigg[ -\beta \sin(kh\delta) + \delta\cos(kh\delta) - \delta e^{-kh\beta} \bigg] \frac{\sin^2(\half ksw)}{ksd} \\ \mbox{Re}\bigg[\deldel{F}{a}\deldel{F^{*}}{w}\bigg] &= 0, \end{split} \end{equation} This model contains $17$ independent parameters that include the bar width $w$ and height $h$ as well as the $15$ parameters that define the correlation functions $G_{\mu\nu}(m)$. A $\chi^2$ minimization procedure based upon the Marquardt algorithm\cite{bevington92} was used in deducing the parameters. Since the scattering contains energy-dependent terms, a global fit to the scattering data at $1.384$, $1.775$, and $2.035$ keV was performed. These particular data were used because we feel that they have the the least experimental uncertainties and would provide the best estimates for the parameters. We first considered the case involving no correlations. This reduced the parameter space from $17$ dimensions down to $5$. \Figure{fig:nocorr} shows the best fit of the rectangular model with no correlations to the data $1.775$ keV. From the figure, one can immediately see that the model without correlations fails to reproduce any of the rich structure seen in the data. The lack of any prominent scattering features is a strong indication that the grating scatter must be the result of correlations. \Figure{fig:en0-2} shows the result of a simultaneous fit to the $1.384$, $1.775$, and $2.035$ keV scattering data when correlations were included. The parameters deduced from the fit are presented in \Table{tbl:corrparms}. Fluctuations involving the bar displacement $a$ were not included in this fit, reducing the number of free parameters down to $11$ (including them did not substantially improve the fit so we felt that it was better to restrict the model to $11$ parameters). \begin{figure} \begin{center} \epsfig{file=1-3wh-en0-2.ps,height=20cm,angle=0} \caption{\label{fig:en0-2} \small This figure shows results of a global fit to the $1.384$, $1.775$, and $2.035$ keV scattering data. The model included width-width, height-height, and width-height correlations. } \end{center} \end{figure} \begin{table}[t] \renewcommand{\arraystretch}{1.25} \begin{equation*} \begin{array}{|rll||rcl||rcl|} \hline h &=& 0.64\mbox{$\mu$m} & w/d &=& 0.84 & & & \\ \hline \sigma_{ww} &=& 6.5\times10^{-5}& \alpha_{ww} &=& 0.27 & \Omega_{ww} &=& 0.092 \\ \hline \sigma_{hh} &=& 5.0\times10^{-5} & \alpha_{hh} &=& 0.60 & \Omega_{hh} &=& 0.50 \\ \hline \sigma_{wh} &=& 4.2\times10^{-5} & \alpha_{wh} &=& 0.30 & \Omega_{wh} &=& 0.91 \\ \hline \end{array} \end{equation*} \caption{ \label{tbl:corrparms} \small This table shows the best fit parameter values as deduced from fitting the scattering data at $1.384$, $1.775$, and $2.035$ keV. Displacement variations were not considered. \Figure{fig:en0-2} shows the resulting fit.} \end{table} We can immediately see that including correlations results in a dramatic improvement of the fit. Although not perfect, this model shows all the main features in the scattering data at these energies. This is especially true for our best data set, the data at $1.775$ keV. The main areas of disagreement are near the half order locations, and at the second and third order scattering peaks where it underestimates the scattering. Also the average effective-rectangular bar width resulting from the fit is about $35$ percent larger than the accepted average width: $w/d$ is $0.84$ compared to $w/d\approx 0.60$ based on detailed vertex-model fits\cite{markert95} to the facet data\cite{obsguide}. Nevertheless, we feel that the fit is remarkable considering the simplicity and known shortcomings of the model. \Figure{fig:en3-6} shows the predictions of the model at higher energies ($5-8$ keV) overlaid with scattering data at those energies. The model is not too bad at predicting the scatter between the first and third orders in the high energy data, although it performs poorly when extrapolated past third order. This should not be too surprising since the data used to determine the model's parameters did not extend past third order. \begin{figure} \begin{center} \epsfig{file=1-3wh-en3-6.ps,height=16cm,angle=270} \caption{\label{fig:en3-6} \small These figures show the predictions of the correlation model of Figure{fig:en0-2} evaluated at at $5$, $6$, $7$, and $8$ keV overlaid with the scattering data at these energies. } \end{center} \end{figure} %}}} \subsection{Discussion} %{{{ While the fit adequately reproduces many of the features observed in the data, the real physics lies in the form of the correlation functions deduced from the fit. A plot of the three correlation functions $G_{ww}(m)$, $G_{hh}(m)$, and $G_{wh}(m)$ is shown in \Figure{fig:corr}. The dominate correlation is $G_{ww}(m)$, i.e., the width-width correlation. As can been seen from the figure, the shape of this correlation implies that the grating bar widths of the first and second nearest neighbor bars are positively correlated whereas the widths of the fourth and fifth nearest neighbors are anti-correlated. Physically, this correlation could arise by groups of two or three grating bars leaning in the same general direction. It is this correlation that appears to be responsible for the strong peaks closest to the diffraction orders. \begin{figure} \begin{center} \epsfig{file=1-3wh-corr.ps,height=13cm,angle=270} \caption{\label{fig:corr} \small This figure shows a plot of the correlation functions parameterized in \eq{eq:gmunu} using the parameters from \Table{tbl:corrparms}. } \end{center} \end{figure} The form of the height-height correlations, $G_{hh}(m)$, indicate that heights of nearest neighbors are anti-correlated. This has the effect of doubling the grating period and it is this correlation that contributes to the scattering near the half-orders. The simplest physical picture of this type of correlation is that of the left and right nearest neighbors leaning in the direction of the central bar. One must not come to the conclusion that these physical pictures of the width-width and height-height correlations are mutually exclusive. Indeed there is nothing that prevents the gratings from having both types of disorder because each type may occur at disjoint parts of the grating. Thus the physical picture that emerges from these correlations is that the grating consists of localized groups of leaning bars and that the groups may be classified according to at least two varieties. The first type consists of several bars that lean in the same direction, possibly three or four together. The second type of fluctuation involves a group of three bars where the left and right bar leans on the central one. The final correlation, $G_{wh}(m)$, can be though of as the result of a combination of these two physical pictures. If the explanation of the source of the correlations is correct, then the size of the correlations may be used to deduce the frequency of such fluctuations. For example, consider the width-width correlation, which is roughly $10^{-4}$ and implies that the fluctuation $\Delta w/d$ is $0.01$ or about $1$ percent of the period. This is consistent with $1$ in $200$ grating bars having a fluctuation in width by half a period, while the other $199$ bars do not fluctuate at all. Finally, if the scattering is due to the presence of leaning bars, then the absence of any detectable scattering by the MEG gratings is very easy to understand. The MEG gratings have a bar width of about $0.2$ microns, and a bar height of about $0.36$ microns. Hence, compared to the HEG gratings, which have a height to width ratio of about $5$, the MEG gratings have a height to width ratio that is less than $2$, and would be much less likely to lean. %}}} \subsection{Conclusion and Next Steps} %{{{ Despite the apparent simplicity of both the rectangular model and the assumed form of the correlations, the resulting fits to the data are relatively good. The primary goal of this work was to demonstrate that the observed scatter could be the result of imperfections in the gratings and not to some other more exotic physical explanation. We are quite content that this goal has been achieved and we believe that the rectangular model has served us quite well in this regard. Nevertheless, the fits are far from perfect and there are several discrepancies that should be accounted for by a more accurate and sophisticated model. Over $144$ distinct grating facets contributed to the scatter and it is not inconceivable that a better fit could be achieved by using more than one set of bar parameters. This is perhaps the most trivial extension of the simple rectangular model used here. Also, we assumed that the deviations in the bar parameters from their ideal values were small enough that only first order deviations were important. Although the average fluctuation was found to be very small, any one bar parameter could deviate significantly from its mean, with the implication that higher order terms in the expansion could be significant. Since the actual bars have path-length functions that are more trapezoidally shaped than rectangular, it makes more sense to consider a model involving trapezoidal bars. Such a model would also permit a more direct study of leaning bars--- something that is not possible within the confines of the rectangular model. Finally, there are other components of the gratings that have been totally ignored in this work, and the fluctuations in these quantities could contribute to scatter. A more complete treatment would also include these fluctuations, e.g., fluctuations in the plating base as well as in the thickness of the polyimide film. %}}} %%\section*{ACKNOWLEDGMENTS} %{{{ %%This work benefited from the encouragement of Claude %%Canizares and the other members of the MIT/ASC and HETG teams. We %%also thank the PSU ACIS group for making the ACIS XRCF data sets %%conveniently available. %%The work was supported in part by NASA under the HETG contract %%NAS-38249 and the AXAF Science Center contract to the Smithsonian %%Astrophysical Observatory, NAS8-39073. %}}}