# Very High Resolution Solar X-ray Imaging Using Diffractive Optics

B. R. Dennis1G. K. Skinner2,3M. J. Li4A. Y. Shih1 1 Code 671, NASA Goddard
Space Flight Center email<EMAIL_ADDRESS>
2 Code 661 email<EMAIL_ADDRESS>
3 CRESST and Univ of Maryland, College Park
4 Code 553 email: %**** ̵DiffractiveXrayOptics˙21May2012.tex ̵Line ̵100
<EMAIL_ADDRESS>

###### Abstract

This paper describes the development of X-ray diffractive optics for imaging
solar flares with better than 0.1 arcsec angular resolution. X-ray images with
this resolution of the $\geq 10$ MK plasma in solar active regions and solar
flares would allow the cross-sectional area of magnetic loops to be resolved
and the coronal flare energy release region itself to be probed. The objective
of this work is to obtain X-ray images in the iron-line complex at 6.7 keV
observed during solar flares with an angular resolution as fine as 0.1 arcsec
- over an order of magnitude finer than is now possible. This line emission is
from highly ionized iron atoms, primarily Fe xxv, in the hottest flare plasma
at temperatures in excess of $\approx$10 MK. It provides information on the
flare morphology, the iron abundance, and the distribution of the hot plasma.
Studying how this plasma is heated to such high temperatures in such short
times during solar flares is of critical importance in understanding these
powerful transient events, one of the major objectives of solar physics. We
describe the design, fabrication, and testing of phase zone plate X-ray lenses
with focal lengths of $\approx$100 m at these energies that would be capable
of achieving these objectives. We show how such lenses could be included on a
two-spacecraft formation-flying mission with the lenses on the spacecraft
closest to the Sun and an X-ray imaging array on the second spacecraft in the
focal plane $\approx$100 m away. High resolution X-ray images could be
obtained when the two spacecraft are aligned with the region of interest on
the Sun. Requirements and constraints for the control of the two spacecraft
are discussed together with the overall feasibility of such a formation-flying
mission.

###### keywords:

Solar flares, X-rays, lenses, imaging

##  1 Introduction

Sect:Introduction

High resolution X-ray imaging has been discussed extensively in astrophysics
over many years. One of the most exciting prospects with the 0.1 to 1 micro-
arcsecond resolution that should be possible with diffractive optics is to be
able to probe the space-time at the event horizon of super-massive black
holes, e.g. Skinner (2001, 2009). However, achieving this resolution requires
lenses with focal lengths of hundreds or thousands of km. Their use would
require flying X-ray detectors on one spacecraft and lenses on a second
spacecraft separated from the first by the focal length in the direction of
the source of interest. Formation flying with such large spacecraft
separations and the stabilization and alignment knowledge necessary to
generate such high-resolution images would be a major advance on existing
capabilities.

We show here that a less demanding form of the same technology can contribute
in heliophysics to the frontier area of solar flare research. The high
resolution X-ray imaging of the Sun possible with diffractive optics on a
scale more modest than that considered for astrophysical applications offers
the possibility of obtaining significantly finer angular resolution than is
possible with the conventional diffraction-limited reflective optics used at
longer wavelengths. Even a 1-m diameter mirror is diffraction limited at
optical wavelengths at $\approx 0.1$ arcsec. The Advanced Technology Solar
Telescope (ATST) in Hawaii will operate at the diffraction limit of a 4-m
diameter mirror in the near infra-red wavelength range of 900 - 2500 nm to
achieve 0.03 arcsec resolution at 500 nm and 0.08 arcsec at 1.6 $\mu$m over a
field of view of 2 to 3 arcmin (Keil et al., 2011). Because of the much
shorter wavelengths of X-rays (1 Å = 0.1 nm), diffraction-limited X-ray optics
of only 1 cm in diameter could equal or even improve on this angular
resolution.

High resolution imaging of solar X-rays would be invaluable in both soft
X-rays ($\lesssim$10 keV) from hot plasma and hard X-rays ($\gtrsim$ 10 keV))
from nonthermal distributions of energetic electrons. Soft X-ray observations
with 0.1 arcsec resolution of the $\gtrapprox$10 MK plasma in solar active
regions and solar flares would allow the cross-sectional area of magnetic
loops to be resolved and could allow the coronal flare energy release region
itself to be probed on physically meaningful spatial scales. At higher
energies, high resolution imaging would be possible of hard X-rays produced by
beams of nonthermal electrons as they stream down from the acceleration site
in the corona into the higher density regions of the lower corona and
chromosphere. The structure of the compact footpoint X-ray sources at the 0.1
arcsec level will provide information on the details of the coronal
acceleration process itself. The ability to detect the relatively weak coronal
emission from the electron beams in the presence of the bright chromospheric
footpoint sources (dynamic range requirement of $\gtrapprox$100:1) will also
allow the acceleration site to be studied in unprecedented detail.

Although there is great interest in X-ray imaging over a broad energy range
extending into the nonthermal domain above 10 keV, technical issues limit the
technique using diffractive X-ray optics to a very narrow energy range.
Consequently, for this initial demonstration of high resolution solar X-ray
imaging we have chosen to concentrate on the soft X-ray range, specifically
the group of Fe emission lines between 6.6 and 6.7 keV. The strongest line is
typically the ‘w’ Fe xxv resonance line at 6.699 keV (1.851Å) as seen in
Figure fig:BCS_spectra. By using detectors with relatively modest energy
resolution or by absorbing lower energy photons, it is possible to arrange
that the diffractive optics image of a flare is dominated by emission in this
line complex. In this way we are able to make images almost free from blurring
due to photons at other energies that come to a focus at different distances
due to the chromatic aberration intrinsic to diffractive optics. We propose to
use this line complex to facilitate the first demonstration of high resolution
imaging using diffractive X-ray optics and readily available detectors. It
will be possible to extend this technique to higher energies in the future but
because the solar flare X-ray emission above 10 keV is all bremsstrahlung
continuum with no narrow lines, detectors with high energy resolution will be
required to isolate the energies in the narrow range that come to a focus at
the detector.

---
Figure 1.: High resolution X-ray spectra of a solar flare showing the complex
of lines between 6.5 and 6.75 keV. The lines are from highly ionized iron
atoms in a plasma at a temperature of $\approx$20 MK, with the most prominent
line from Fe xxv at 6.699 keV that is labeled with a ’w’ in the lower plot
using the naming convention of Gabriel (1972). The spectrum was recorded with
the Bent Crystal Spectrometer (BCS) on the Solar Maximum Mission during a
flare in 1980 June 29. fig:BCS˙spectra

###  1.1 Solar Flare X-ray Emission

Sect:Science

X-rays from solar flares in the energy range between $\approx$1 and 10 keV are
emitted from hot plasma with temperatures from $\approx$$10^{6}$ K (1 MK) to
sometimes in excess of 50 MK. Detailed studies of this radiation, both
spectroscopically and by imaging, provide crucial information on the nature of
the heated plasma, and can provide clues to help us understand how such a
large volume of plasma can be heated to such high temperatures so rapidly
during a flare.

The solar spectrum in this energy range is a combination of both line and
continuum emission. The many narrow lines are emitted by transitions of atoms
of the different elements of the plasma in the solar atmosphere in various
stages of ionization; the continuum emission is from both free-free and free-
bound interactions of electrons with atomic nuclei that produce bremsstrahlung
and recombination radiation, respectively. The most detailed catalog of lines
in the 3-10 keV range is given by Phillips (2004, 2008). Both line and
continuum spectra can be synthesized using the CHIANTI database and software
(Dere et al., 1997, 2009) for a range of possible temperatures, plasma
abundances, and excitation and ionization conditions. The line of greatest
interest here is the Fe xxv resonance line at 6.699 keV (1.851 Å). It is the
strongest and most prominent line of the complex of iron lines between 6.5 and
6.7 keV (Figure fig:BCS_spectra). This line was named the ‘w’ line by Gabriel
(1972) and arises from the $1s^{2~{}1}S_{0}-1s2p^{1}P_{1}$ transition of
helium-like iron nuclei, i.e., with just two electrons remaining of the
original 26 of an unionized iron atom. It is emitted from plasma at
temperatures above $\approx$10 MK, and hence provides information on the
hottest plasma generated in a solar flare. During the flare impulsive phase,
the line shows a broadening that is commonly attributed to turbulence in the
plasma resulting from the impact of nonthermal electrons. The broadening may
amount to 0.03 Å (0.01 keV or 10 eV), equivalent to a few hundred km s-1.
Nevertheless, the line is always sufficiently narrow to allow imaging with
better than 0.1 arcsec resolution using diffractive optics.

It is clear from images taken with existing X-ray instruments and those in
different wavelength bands that there are features that are unresolved at the
currently resolvable angular scale of $\approx$1 arcsec. The ubiquitous
magnetic loops observed before, during, and after flares appear to be
significantly narrower than 1 arcsec. The magnetic reconnection process that
is believed to lead to the energy release from the coronal magnetic field that
heats the plasma takes place on scales much smaller than 1 arcsec. The total
energy and density of the thermal plasma can be estimated from the emission
measure and source volume as revealed by the X-ray images but the values so
obtained are subject to a large, order-of-magnitude, uncertainty because of
the unknown “filling factor,” the ratio of the apparent volume of the plasma
to the actual volume. Only by making images with higher angular resolution can
this uncertainty be diminished and ultimately eliminated once the finest
plasma structures are resolved. These unknowns then set the goal of the
current effort to image in soft X-rays with better than 0.1 arcsec resolution,
fully an order of magnitude better than what has been achieved to date.

###  1.2 Current Capabilities and Limitations

It is clear that the finest possible angular resolution is required to fully
understand the heating of the flare plasma. To date, the highest angular
resolution for solar observations in this energy range is $\approx$1 arcsec
(full width at half maximum, FWHM) achieved with the X-ray Telescope (XRT) on
Hinode (Golub et al., 2007; Weber et al., 2007) using a single monolithic
grazing-incidence mirror with two reflections. With grazing incidence optics,
angular resolution is limited by surface figure and, where nested optics are
employed, by alignment considerations. As pointed out by Davila (2011), for
example, conventional high energy optics techniques cannot obtain the required
$<$0.1 arcsec resolution.

The Ramaty High Energy Solar Spectroscopic Imager (RHESSI) has achieved
$\approx$2 arcsec angular resolution using Fourier-transform imaging at
energies extending from $\approx$3 keV up to 100 keV, and with progressively
poorer resolution up to 17 MeV (Lin et al., 2002). Bi-grid rotating modulation
collimators are used with 1.5 m separation between grids and a finest slit
pitch of 34 $\mu$m. Higher resolution would be possible with finer slits
and/or greater grid separation but at the expense of field of view. In
addition, the achievable dynamic range in any one image using this technique
is limited to less than $\approx$50:1 by side lobes in the response function.
This makes it difficult, for example, to image the relatively weak and
extended coronal X-ray sources in the presence of intense compact footpoint
sources.

##  2 Diffractive X-ray Optics

Sect:Diff_Opt The angular resolution possible with reflecting optics is
critically dependent on the accuracy of the polishing of the mirror surfaces
and on their alignment. In contrast, the tolerances necessary to manufacture
diffractive lenses capable of ultra-high angular resolution are much more
relaxed. This is because diffractive focusing depends on modulation of the
phase and/or amplitude of radiation by an optical element working in
transmission at normal incidence. This means that the lenses can be relatively
thin, and alignment precision is not a serious issue.

Davila (2011) has proposed the use of a photon sieve, a form of Fresnel zone
plate, to achieve high angular resolution in the extreme ultraviolet. We
concentrate here on Fresnel lenses that can offer higher efficiency than zone
plate variants and suffer less from diffraction into spurious orders. The
principles of phase Fresnel lenses and their development for X-ray and gamma-
ray astronomy have been discussed in a series of papers (Skinner, 2001;
Krizmanic et al., 2005; Krizmanic et al., 2007; Skinner, 2009).

###  2.1 Lens Design Considerations

sect:design

The focussing ability of diffractive optics requires that radiation that
passes through different parts of the lens must all arrive at the focal point
with identical phase. This can be achieved if the thickness, $t$, of the lens
with a focal length, $f$, is the following function of the radial distance,
$r$, from its center:

$\displaystyle t(r)$ $\displaystyle=$
$\displaystyle\left(r^{2}/(2f\delta)\pmod{\lambda}\right)+t_{0}$ (1)
$\displaystyle=$
$\displaystyle\left(r^{2}/(2f\lambda)\pmod{1}\right)t_{2\pi}+t_{0},$ (2)

where $\lambda$ is the wavelength and the refractive index is $\mu=1-\delta$
(see Krizmanic et al., 2007). The addition of the $t_{0}$ term reflects the
fact that, apart from absorption, the presence of a constant thickness
substrate does not affect the focussing properties, just the throughput. In
Equation eqn:t2pi, we have written $t_{2\pi}$ for the thickness of material,
$\lambda/\delta$, that changes the phase by $2\pi$. Conveniently, in the soft
X-ray band, this quantity is typically a few $\mu$m leading to structures well
matched to fabrication by Micro-Electro-Mechanical Systems (MEMS) engineering
technology.

The ideal profile corresponding to Equation eqn:t2pi, as illustrated in Figure
fig:profilesa, produces an optic known as a phase Fresnel lens (PFL). It can
be replaced by a multi-step approximation (Figure fig:profilesb), the limiting
case being where there are just two thicknesses - $t_{0}$ (which may be zero)
and $t_{0}+t_{2\pi}/2$ \- as illustrated in Figure fig:profilesc. In a further
approximation, it is possible simply to block the radiation for which the
phase is not within $\pm\pi/2$ of the ideal, leading to the zone plate
structure shown in Figure fig:profilesd. For that reason, a two-level
diffractive lens is often referred to as a phase zone plate or PZP. The ideal
efficiencies, defined as the flux in the first order focus as a fraction of
the total incident flux, are listed in Table table:eff for the different
cases.

---
Figure 2.:  (a) The ideal cross-section through a Phase Fresnel Lens (PFL)
with a maximum thickness (t) of $t_{0}$ (which may be zero) $+\hskip
3.61371ptt_{2\pi}$. For X-rays, the refractive index is less than one
($\delta$ is positive) so the configuration shown corresponds to a converging
lens. (b) A four-level approximation to a PFL profile with a thickness of
$t_{0}+0.75\hskip 3.61371ptt_{2\pi}$. (c) A two-level approximation (phase
zone plate, PZP) with a thickness of $t_{0}+0.5\hskip 3.61371ptt_{2\pi}$. (d)
A zone plate, in which alternate zones are opaque to the radiation.
fig:profiles Table 1.: The ideal efficiency of diffraction into the first-
order focus for lenses with profiles such as those illustrated in Figure
fig:profiles. Absorption is assumed to be negligible.

| phase |  |  | phase |
---|---|---|---|---|---
Approximation | Fresnel lens |  $n$-levels | 4-levels | zone plate | zone plate
| (PFL) |  |  | (PZP) | (ZP)
Figure fig:profiles | (a) |  | (b) | (c) | (d)
Efficiency | 1 | $\left[\frac{n}{\pi}\sin(\frac{\pi}{n})\right]^{2}$ | 0.811 | 0.405 | 0.101

table:eff

Irrespective of the profile adopted, at the edge of the lens the structure is
periodic with the minimum period given by

$p_{\mathrm{min}}=2f\lambda/d,\ilabel{eqn:pmin}$ (3)

where $d$ is the lens diameter. The focal length $f$ can be written in terms
of the photon energy, $E$, in the range that is of particular interest here
as:

$f=102\left(\frac{p_{\mathrm{min}}}{1.9\mbox{
$\mu$m}}\right)\left(\frac{E}{6.65\mbox{ keV}}\right)\left(\frac{d}{20\mbox{
mm}}\right)\mbox{ m}\ilabel{eqn:f}.$ (4)

For a diffraction-limited lens, the angular diameter of the Airy disk is :

$\theta_{\mathrm{d}}=2.44\>\lambda/d\>=1.22\>p_{\mathrm{min}}/f.$ (5)

For an ideal lens, the Airy disk contains 84% of the energy in the first order
focus. With typical parameters, its angular size is much smaller than the
resolution targeted here:

$\theta_{\mathrm{d}}=4.7\left(\frac{E}{6.65\mbox{
keV}}\right)^{-1}\left(\frac{d}{20\mbox{ mm}}\right)^{-1}\>{\mbox{milli-
arcsec}}\ilabel{eqn:diff}.$ (6)

For our purposes, a more important consideration than the diffraction limit is
blurring due to chromatic aberration. Equation eqn:f shows that the focal
length is proportional to photon energy so in practice the use of such lenses
is restricted to observations in which a narrow range of energies is dominant
or where the bandwidth is limited in other ways. If the point spread function
(PSF) of the lens is approximated by the best-fit Gaussian and the spectral
line to be imaged is also taken to be Gaussian with a FWHM of $\Delta E$, then
the chromatic aberration contribution to the FWHM angular resolution of the
lens is found to be

$\theta_{\mathrm{c}}=5.2\left(\frac{\Delta
E}{E}\right)\;\left(\frac{d}{20\mbox{ mm}}\right)\;\left(\frac{f}{100\mbox{
m}}\right)^{-1}\>{\mbox{arcsec}}.\ilabel{eqn:chrom}$ (7)

Note that the PSF is strongly cusped and the central peak is much sharper than
this implies. On the other hand, if the criterion adopted is the width that
contains 84% of the energy (the fraction within the central peak of an Airy
disk), then the numerical factor in Equation eqn:chrom is more than doubled.

A typical line width due to thermal broadening of the Fe xxv 6.699 keV line is
about 2 eV, corresponding to a Doppler velocity of $\approx$100 km s-1. There
is often additional broadening due to turbulence, and line shifts due to bulk
motion, both of which can be several times larger than this. Furthermore,
although this spectral line is frequently dominant, it is just one line of
many lines in a complex that can be approximated by a Gaussian with a
characteristic width of about 100 eV. If the detector is energy resolving and
can be used in a photon counting mode, one can attempt to select just those
photons that fall within a narrow energy band. However, for Silicon CCDs and
similar detectors, the attainable FWHM resolution $\Delta E$ at 6.7 keV is
limited to about 150 eV – wider than the line complex. If we take a width of
100 eV and the reference parameters used in Equations eqn:f and eqn:diff, the
limit due to chromatic aberration is 80 milli-arcsec FWHM.

A further limit to the angular resolution potentially arises because of the
spatial resolution of the detector. For a pixel size $\Delta x$ the
corresponding limit is

$\theta_{\mathrm{s}}=\frac{\Delta x}{f}=21\left(\frac{\Delta x}{10{\mbox{
$\mu$m}}}\right)\left(\frac{f}{100\mbox{ m}}\right)^{-1}\>{\mbox{milli-
arcsec}.\ilabel{eqn:detlim}}$ (8)

The work of Young (1972) on aberrations in zone plate imaging can be applied
to Fresnel lens optics. It shows that, for systems of interest for solar
physics or astrophysics, geometric aberrations are very small, and the only
limit on the field of view will be that imposed by the size of a practical
detector.

Thus, none of these limits prevent an angular resolution of the order of 100
milli-arcsec from being obtained in imaging observations in the Fe-line
complex provided that lenses with $p_{\mathrm{min}}$ of 1.5 – 2 $\mu$m can be
made and used in a configuration with a focal length of $\approx$100 m.

Figure 3.: Photograph of a 3-cm diameter silicon phase zone plate fabricated
in the Detector Development Laboratory (DDL) at Goddard Space Flight
Center.fig:PZPphoto

Figure 4.: Scanning electron microscope image of the central area of a
demonstration lens showing the circular slits and the radial support ribs.
fig:Lens˙SEMimage

Figure 5.: Scanning electron microscope image of the outer section of a lens
showing the finest slits and part of a radial support rib. fig:SEMslits

###  2.2 Lenses Fabrication

sect:fab

We have initially concentrated on the fabrication of PZP lenses with the
cross-section shown in Figure fig:profilesc. For a given $t_{2\pi}$, it is
easier to obtain a small $p_{\mathrm{min}}$ with this profile than to
fabricate the tapered PFL profile (Figure fig:profilesa) or a multi-level
approximation to one (Figure fig:profilesb). For a particular focal length, a
smaller $p_{\mathrm{min}}$ allows the lens diameter to be larger (per
Equations (eqn:pmin) and (eqn:f)), and the increased geometric area can
compensate for the lower efficiency. We have fabricated lenses with parameters
given in Table table:params that are similar to the reference values in
Equation (eqn:f) needed to meet the requirements for a flight instrument. A
photograph of such a phase zone plate is shown in Figure fig:PZPphoto with
scanning electron microscope images of the central area in Figure
fig:Lens_SEMimage and the finest slits at the edge in Figure fig:SEMslits.

Table 2.: Comparison of the parameters of the demonstration lenses with a
possible flight design.

Parameter | Demonstration | Possible flight |
---|---|---|---
|  | laboratory lenses | design |
Energy | $E$ | 5.411 | 6.65 | keV
Focal length | $f$ | 110.4 | 100 | m
Diameter | $d$ | 30 | 20 | mm
Finest pitch | $p_{\mathrm{min}}$ | 1.7 | 1.9 |  $\mu$m
Profile height | $t$ | 8.2 | 8.2 |  $\mu$m

table:params

We chose to make the lenses of silicon since MEMS techniques are most advanced
for this material and it is acceptable for lenses working in the X-ray energy
range considered here. At 6.699 keV, the thickness of Si needed to give a
$2\pi$ phase change ($t_{2\pi}$) is 16.8 $\mu$m. This is the peak thickness,
$t$, of the profile for an ideal PFL (Figure fig:profilesa); for a PZP, only
half this thickness is required so $t=8.4$ $\mu$m. In fact we have adopted
$t=8.2$ $\mu$m as by reducing the depth slightly imperfect phase matching is
traded for reduced absorption and a small improvement in overall efficiency is
obtained, while making fabrication marginally easier.

Fabrication of the lenses was conducted in the Detector Development Laboratory
(DDL) at GSFC, a fully equipped semiconductor processing facility center with
Class 10 clean-room capabilities. Phase zone plate lenses (Figure
fig:profilesc) were designed using the DW2000 software tool for mask layout.
Standard photolithography and advanced Deep Reactive Ion Etching (DRIE)
processes were employed with a UV mask aligner (SUSS MicroTec MA-6) and a
high-rate etcher (STS).

Each lens was fabricated from a 4-inch diameter Silicon-On-Insulator (SOI)
wafer. As shown in Figure fig:fab, an SOI wafer consists of three layers - (1)
a thin Si layer on top called the device silicon layer, (2) a thin silicon
dioxide (SiO2) insulating layer, and (3) a thicker Si layer called the handle
silicon layer.

---
Figure 6.:  Cross-sections of SOI wafers for illustration of the lens
fabrication process. Top: SOI wafer showing the top device silicon layer, a
thin insulating layer of silicon oxide, and the bottom thicker layer called
the handle silicon layer; middle: lens etched out from the device silicon
layer; bottom: support structure etched from the silicon substrate or handle
silicon layer. fig:fab

The sequence of operations is illustrated in Figure fig:fab. The lens pattern
is first formed on the front (device) side of the SOI wafer by UV exposure of
a photo-resist layer through a chromium-on-glass mask. This is followed by
development and DRIE etching down to the silicon oxide insulating layer. The
SOI wafer is then attached to a glass wafer with wax to protect the front-side
silicon lens features. The procedure is repeated on the backside to produce
the spider-web support structure, again with DRIE etching as far as the oxide
layer.

This DRIE process allows slits with the required high-aspect ratios of up to
20:1 to be etched in silicon. However, the parameters of the etch process had
to be tuned in order to obtain vertical silicon walls with the required width
of remaining silicon between adjacent slits. An anti-reflection coating was
used to make good optical contact between the SOI wafer and the glass mask,
allowing a precise lens pattern with features at the submicron level to be
obtained.

Techniques exist that allow the fabrication of lenses with multiple levels
(Figure fig:profilesb) or even good approximations to the ideal continuous
profile (Figure fig:profilesa) – see for example di Fabrizio et al. (1999) and
Krizmanic et al. (2005). If lenses could be made with the same diameter and
focal length as our two-level lenses but with, say, a 4-level stepped profile,
then the effective area could, in principle, be more than doubled (see Table
table:eff). Fabricating such structures would require etching features finer
by at least a factor of two, and somewhat deeper. In conjunction with the Army
Research Laboratory (ARL) in Adelphi, MD, we are investigating the feasibility
of using e-beam lithography to obtain four-level profiles. Because of the long
writing times involved, initially the objective is to make a sample 1 mm wide
strip across the radius of a 30 mm diameter lens.

###  2.3  Test Results

section:TestResults

Several lenses have been tested at the NASA-GSFC X-ray Interferometry Testbed.
This facility provides a 600-m evacuated path between source and detector
stations with an intermediate station for the focusing optics. Because of the
relatively low melting point of iron and the need for high surface brightness,
it was not possible to use an iron target X-ray tube, which would have
produced the Fe K$\alpha$ line at 6.4 keV, close to the solar Fe xxv $w$ line
at 6.699 keV. Instead, a chromium target X-ray tube was used, producing the Cr
K$\alpha$ line at 5.411 keV. Consequently, although the lenses were designed
to have the finest pitch $p_{\mathrm{min}}$, thickness $t$, and focal length
$f$ similar to those needed for a 20 mm diameter flight lens working at 6.699
keV, the diameter was actually 30 mm. The focal length was chosen to be 110.4
m at the Cr line energy so that, when the lens was positioned at the ‘optics’
station of the testbed 146 m from a 5-$\mu$m wide tungsten slit placed
directly in front of the source, an enlarged image of the slit was formed in
the plane of the detector, 452 m from the lens (see Figure fig:TestSetup).
This magnifying configuration has the advantage that the 13 $\mu$m pixels of
the Roper CCD camera used as a detector were capable of resolving the image of
the slit. The camera was used in a photon-counting mode, accepting events in a
narrow energy range (5.3$-$5.6 keV) containing the Cr K$\alpha$ line while
excluding the K$\beta$ line at 5.947 keV.

Figure 7.: Test setup at Goddard’s 600-m X-ray Interferometry Testbed. A CCD
X-ray camera with 13 $\mu$m pixels was used to record the magnified image of a
5-$\mu$m slit, whose width at a distance of 146 m corresponds to 7 milli-
arcsec. Not to scale. fig:TestSetup

An image produced with this configuration is shown in Figure fig:image and a
cross-section though it in Figure fig:xsect. The core of the response is 66
$\mu$m FWHM which corresponds to 30 milli-arcsec, or 28 milli-arcsec when
allowance is made for broadening by the detector pixelization and the finite
width (5 $\mu$m) of the slit. There is a significant amount of power in the
wings that are rather broader than the core but even measuring the width at
one tenth maximum, the width of 70 milli-arcsec is better than the target of
100 milli-arcsec.

---
Figure 8.:  An image of a 5 $\mu$m slit obtained with the test setup in Figure
fig:TestSetup. The pixels correspond to an angular size of 6 milli-arcsec.
fig:image
---
Figure 9.: Intensity as a function of angular offset transverse to the slit
for the image in Figure fig:image. fig:xsect

We have estimated the effective area of the lens by comparing the count rate
in the peak with that when the lens was replaced by a plate with a small hole
of known size. To avoid pulse pile-up effects and to allow any variation of
efficiency across the lens to be investigated, some of the measurements were
made with a 5 mm aperture in front of the lens. The position of the aperture
could be controlled in two axes with stepper-motor drives. We find an average
efficiency of 24% with very little radial variation, leading to an effective
area of just over 2 cm2. This can be compared with the theoretical value for
an ideal phase zone plate of 40.5%, which is expected to be reduced to 26.8%
when absorption, obscuration by the radial support ribs, and the fact that the
profile is optimized for an energy different from the test energy, are taken
into account. A similar calculation leads to an ideal efficiency of 33.7% at
$\approx$6.7 keV.

##  3 Space Mission Concept

Sect:MissionConcept

The major problem associated with flying such phase zone plate lenses in space
and obtaining X-ray images of the Sun with sub-arcsecond resolution is the
required focal length. The lenses we have developed and demonstrated have a
focal length $f$ of $\approx$ 100 m. Extensible booms up to 60 m in length
have been flown in space (Farr et al., 2007) and longer ones proposed (Johnson
et al., 2010, e.g.) but for such distances formation flying of two spacecraft
with lenses on one and detectors on the other becomes an attractive
possibility.

###  3.1 Two-Spacecraft Formation Flying

For our purposes, the formation-flying concept requires control of the
relative positions of the two spacecraft and of their orientations. Table
table:requirements summarizes the technical requirements. Note that the
spacecraft station-keeping and attitude control requirements are relatively
lax. This is because images can be built up from a series of ‘snapshots’ and
milli-arcsec imaging can be achieved providing only that milli-arcsec
knowledge of the lens-detector alignment with respect to the Sun at the time
of each snapshot is available. Aspect control of the individual spacecraft is
required only to a fraction of a degree. The transverse station keeping
requirement is dictated only by the need to keep the region of interest imaged
on the detector. The axial requirement is determined by the necessity of
keeping the out-of-focus blurring to a negligible level. Precise measurement
of the actual alignment at all times then allows the image to be reconstructed
taking into account the actual determined attitude without compromising the
angular resolution. If the detector is a CCD or other device that integrates
between readouts and does not allow accurate time-tagging, then there is a
requirement that the alignment remain stable on the timescale of the
integration – perhaps one or a few second(s).

Table 3.:  The principal parameters and station-keeping requirements for a
solar flare X-ray imager using diffractive X-ray optics on a formation-flying
mission. table:requirements Lens - detector requirements
---
Lens - detector separation |  $\approx$100 m |  Larger separations would allow larger lenses, but would reduce the field of view possible with a reasonable detector size
Pixel size for 0.1” resolution |  25 $\mu$m |  X-ray CCD or X-ray pixel detector
Field of View |  3 arcmin |  Assumes 10 cm detector diameter to cover a typical active region - could be larger.
Station keeping requirements
Control - transverse |  2 cm |  Requirement is to keep image on detector - could move detector or lens independently.
Control - axial |  25 cm |  Equivalent to 20 eV energy shift
Stability - transverse |  $<$0.1 arcsec = 25 $\mu$m at 100 m over 1 s |  Alignment should not change between read-outs
Knowledge - transverse |  $<$0.1 arcsec = 25 $\mu$m at 100 m |  Alignment of line joining lens and detector must be known relative to the direction to Sun center to an accuracy better than the desired resolution.
Knowledge - axial |  10 cm |  To determine the energy of the focussed X-rays to $<$10 eV.
Attitude requirement
Control and knowledge |  10 arcmin |  Tilt and tip of lens and detector are not critical

###  3.2 Orbit Considerations

To maintain the configuration of the two spacecraft, a quasi-continuous thrust
will be needed on one, or both, spacecraft to overcome differential
accelerations due to gravity gradients, radiation pressure, and drag. For
spacecraft separations in the range under consideration here of order 100 m,
solar gravity forces are never an important consideration. However, within the
Earth-Moon environment, gravity gradient forces will generally dominate. At a
distance, D, from the center of the Earth, a spacecraft of mass
$M_{\mathrm{sat}}$ will require a thrust, $\Delta F_{\mathrm{g}}$, to keep at
a constant direction and separation $f$ ($f\ll D$) relative to a passive
reference spacecraft. The required thrust is given by –

$\Delta F_{\mathrm{g}}=8\left(\frac{M_{\mathrm{sat}}}{1,000\mbox{
kg}}\right)\left(\frac{D}{10,000\mbox{
km}}\right)^{-3}\left(\frac{f}{100\mbox{ m}}\right)\mbox{
mN.}\ilabel{eqn:diffgrav}$ (9)

The $D^{-3}$ term in Equation (eqn:diffgrav) means that orbits in which the
spacecraft spend most of the time far from Earth are strongly preferred.
Highly eccentric orbits with periods of several days duration are probably a
viable choice, though observations will be interrupted during perigee and the
subsequent time necessary to reconfigure the formation. Krizmanic et al.
(2005) have discussed propulsion solutions for astrophysical missions with
lenses having much longer focal lengths and higher thruster requirements than
implied by Equation (eqn:diffgrav).

Far from the Earth and Moon, the dominant disturbance force is likely to be
differential radiation pressure. At 1 AU, this is only 4.6 to 9.2 $\mu$N per
m2 of difference in effective areas of the two spacecraft normal to the Sun,
depending on the reflection properties of the surfaces. Stationing the
spacecraft pair close to the Sun-Earth L1 Lagrangian point would allow for
almost continuous aligned observations.

We note that ESA’s Proba-3 mission (Vives et al., 2010; Lamy et al., 2010) is
designed to demonstrate many of the capabilities needed for a mission to
perform high angular resolution solar X-ray imaging with diffractive lenses.
Proba-3 will control two spacecraft separated by 25–250 m so that a
coronagraph occulter on the front spacecraft will appear directly in front of
the Sun when viewed by instruments on the other spacecraft. This technology
demonstration mission will be in a 24-h eccentric Earth orbit and so the
required alignment will only be maintained for limited times. The precision of
the specification for the attitude determination falls only a little short of
that needed to take full advantage of the capabilities of the lenses under
discussion here.

A coronagraph could be a natural companion instrument to a diffractive optics
X-ray imager. A lens (or multiple lenses) could be mounted within the
coronagraph occulting disk. Light blocks would ensure that they are opaque to
visible/UV radiation while having negligible X-ray absorption (such light
blocks will, in any case, be desirable for thermal control).

##  4 Simulations of Capabilities

Sect:sims

In order to demonstrate the potential of the X-ray imaging technique
considered here, we have performed Monte Carlo simulations of the operation of
an instrument with a diffractive lens on one spacecraft and a CCD imaging
detector on the other. We used the lens parameters for the possible flight
design given in Table table:params. The assumed lens performance is based on
the lenses that we have developed and tested as described in Sections sect:fab
and section:TestResults. We have modeled a flare of similar intensity and
temperature to the 1980 June 29 event (see Figure fig:BCS_spectra), a class M4
flare, and used the spatial distribution shown in Figure fig:SimImage(top) in
which a loop is assumed to be composed of a large number (30) of fine
intertwined threads, inspired by a model illustrated in Figure 1.18 of
Aschwanden (2005).

For the spectrum of the emission, we have used a CHIANTI simulation of a
plasma with coronal abundances, a temperature of 18.8 MK, and a volume
emission measure of $10^{48.84}~{}\mathrm{cm}^{-3}$ based on the GOES X-ray
data of the flare. This gives approximately the spectral form and
normalization measured with BCS shown in Figure fig:BCS_spectra. In order to
follow changes in the flare loop structures, we would like to be able to
obtain an image in, say, 10 s. We assume that, because of the limitations of a
CCD detector, a 10 s accumulation is composed of 1 s observation periods each
followed by a 1 s readout time, leading to 50% live time.

The simulations demonstrate an important consideration when making
observations of relatively bright, potentially fast-changing objects. The
event rate in all the bright regions of the image precludes operation in the
photon counting mode since there will be multiple photons per pixel in each
1-s frame. However, because the emission is concentrated in a narrow energy
band, the total charge collected provides a good measure of the signal. For
operation in this analog mode, it is important to attenuate the lower energy
photons as much as possible, and for this purpose we assume that a
25-$\mu$m-thick layer of Si and a 10-$\mu$m layer of Fe is placed in the line
of sight, perhaps forming a lens substrate.

Figure fig:SimImage(middle) shows a simulated image derived from the model
image shown in Figure fig:SimImage(top). The image quality is more affected by
photon statistics than by the optics. Figure fig:SimImage(bottom) shows a
corresponding simulation of a 10 times longer observation or a 10 times
stronger flare.

---
Figure 10.: (top) The spatial distribution model used as input to the
simulations shown below. (middle) The image resulting from a 10 s (5 s
effective) observation of the above source distribution. (bottom) The
corresponding image with 10 times the exposure or for a 10 times stronger
flare.fig:SimImage

##  5 Summary and Conclusions

Sect:conclusions

The results presented here show that diffractive X-ray lenses can be made that
are capable of solar imaging in the Fe xxv $w$ emission line and nearby lines.
With such lenses, the capability exists for making X-ray images over narrow
energy ranges with angular resolutions as fine as 0.1 arcsec. This high
angular resolution and the effective area of the demonstration lenses already
fabricated would allow hot plasma generated during solar flares to be studied
on unprecedentedly fine spatial scales and determine the multi-threaded nature
of hot coronal magnetic loops. The detailed structure of the coronal energy
release sites could also be explored in detail during solar flares.

We note that the efficiency of a PZP lens can be improved by using a
multilevel approximation to the ideal Phase Fresnel Lens profile rather than
just the two levels of the lenses we have fabricated. In developments not
reported here, we have demonstrated the capability of using grey-scale
technology to generate a 4-level profile which should double the efficiency of
the lens and give twice the sensitivity. Alternatively, the higher efficiency
could be used to achieve a higher bandwidth with the same sensitivity by
fabricating a smaller lens with a higher focal ratio. The technique proposed
here can be extended to higher energies, though even longer focal distances
would be needed to achieve a useful effective area unless lenses with even
finer slits and higher aspect ratios can be developed. On the other hand, the
lower absorption losses at higher energies may allow the bandwidth to be
increased with achromatic combinations (Skinner, 2010).

Detailed design of a mounting and support structure and demonstration that the
lens can withstand launch and operational environments are planned, as are
further studies of the formation-flying aspects of a possible mission. More
detailed development of the complete instrument concept is underway. These
activities are designed to bring a high resolution X-ray imager concept to the
point where it could be proposed for flight, perhaps as a science demonstrator
on a formation-flying technology mission.

#### Acknowledgements

We thank John Krizmanic and Keith Gendreau for supporting the lens design and
testing effort, Kenneth Phillips for providing the spectra shown in Figure
fig:BCS_spectra and for help with the solar objectives, and Amil Patel, Gang
Hu, and Thitima Suwannasiri for their work fabricating the lenses in Goddard’s
Detector Development Lab. This project was supported with funding from the
Goddard Internal Research and Development (IRAD) program. CHIANTI is a
collaborative project involving George Mason University, the University of
Michigan (USA), and the University of Cambridge (UK).

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