MAXIM Interferometer Tolerances and Tradeoffs

Ann Shipley^a^, Webster Cash^a^, Keith Gendreau^b^, and Dennis
Gallagher^c^

^a^ University of Colorado, ^b^NASA/Goddard Space Flight Center, ^c^Ball
Aerospace

1.  

Maxim (Micro-Arcsecond X-ray Imaging Mission) consists of thirty-two
individual grazing incidence interferometer channels that act, in
combination, like a high-resolution imaging telescope. In this paper, we
will describe an optical design for Maxim and calculate principal
optical tolerances. These tolerances offer advantages that make
anticipated engineering challenges more soluble and affordable within
the limitations of current technology. We also discuss key design
tradeoffs that contribute to a preliminary tolerance budget.

1.  Black hole, diffraction, grazing incidence, interferometer, MAXIM,
    pathlength, phase, SEU, tolerance, x-ray

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1.  INTRODUCTION

MAXIM has the potential to image the x-ray sky at the micro-arcsecond
level and is part of the advanced planning for NASA's Structure and
Evolution of the Universe Space Science theme.^1^ A Pathfinder mission
is also under study that will be capable of 100 micro-arcsecond
resolution.^2,3,4^ Imaging resolutions at these levels can open the door
to understanding fundamental phenomena hidden in an accretion disk and
even at the event horizon of a black hole.^5,6^

As we continue our work in grazing incidence x-ray interferometry,
variations of the optical configuration continue to evolve.

Since we first demonstrated x-ray interference fringes using an optical
design we call the "X-configuration" (see Figure 1) we have carefully
evaluated alignment tolerances, subsystem requirements, and estimated
costs to develop a mission.^7,8,9,10^ In this paper, we will discuss the
most recent changes to the interferometer optical design, evaluate
tolerances, and present advantages the new "periscope configuration"
offers.

2.  Periscope configuration

The periscope interferometer configuration creates two diffraction
limited wavefronts and mixes them at a detector in a similar fashion to
the X-configuration interferometer. Each periscope module contains a
collecting primary mirror and a secondary mirror to steer the beam onto
the detector. Figure 2 shows a plane wavefront from infinity projected
onto a flat primary mirror in each of two periscope modules. The
wavefront is then reflected onto a second flat within each periscope
module and steered so that the two beams converge at the detector plane
to create interference fringes.

![](media/image1.wmf){width="4.861111111111111in"
height="1.9333333333333333in"}

Figure 3 shows one periscope module with its two flat mirrors mounted
onto a platform. The primary mirror is provided with control of three
degrees of freedom for alignment, while the secondary mirror is mounted
in a stationary position on the optical bench. The entire module will
also require alignment capability in roll, pitch, and yaw rotations at
the interferometer level. Rotational alignment mechanisms for the entire
module are not illustrated in Figure 3.

Figure 3: Periscope configuration mirror module

3.  principal optical tolerances

In this section we derive equations to describe the amount of allowable
error in the position of a periscope module. The interferometer's
performance is dependent upon maintaining phase information of x-rays at
the focal plane after they pass through each periscope module. The
distance each x-ray travels from the time it reflects off the primary
mirror and passes through the periscope module onto the detector is
defined as its pathlength. Phase information is controlled in this
analysis by defining a limitation on the pathlength difference between
module channels. We set this pathlength difference limit at one tenth of
an x-ray wavelength.

1.  Module X-direction sensitivity

Lateral motion of one periscope module in the X-direction causes a phase
change at the focal plane. The diagram in Figure 4 represents a plane
wavefront reflecting through two periscope modules at grazing incidence
and projected onto a focal plane. One periscope module is moved a
distance δ out of position in the --X direction. This causes the exit
beam to also shift by a distance δ consequently resulting in an optical
path difference (OPD) of δsinψ. The quantity sinψ is approximately half
the baseline (D) divided by the focal length (F). If we limit the OPD to
λ~xray~/10, the allowable lateral error δ becomes λF/5D.

![](media/image2.wmf){width="2.1338582677165356in"
height="1.2736220472440944in"}

2.  Module Y-direction lateral sensitivity

![](media/image3.wmf){width="0.9807086614173228in"
height="0.4188976377952756in"}

Periscope module position errors in the Y-direction do not cause a phase
change at the focal plane. Theoretically, the module could drift any
distance in the Y-direction as long as the angle to the target is still
within line of sight. For practical purposes we set a limit at one tenth
the mirror width. The mirror width is taken as a function of the mirror
length (m) in order to preserve a square projected aperture. Therefore,
we define the mirror width as msin(g), where g is the graze angle
between the incoming beam and the primary mirror. As a result, the
allowable motion becomes δ=msin(g)/10 or about ±1mm for a 30cm long
mirror.

3.  Module Z-direction lateral sensitivity

![](media/image4.wmf){width="2.001574803149606in"
height="1.7110236220472441in"}Translation of one periscope module in the
Z-direction changes the optical pathlength in that channel and
consequently causes a phase change at the focal plane. If one module is
shifted in the Z-direction its optical pathlength increases by δ where
the incoming beam strikes the primary mirror. At the same time the
pathlength is decreased by δcosψ near the focal plane as shown in Figure
6. Therefore, the optical path difference is the sum of the two
pathlength changes: δ−δcosψ. This equation can be reduced algebraically
by factoring out δ and applying a series expansion to cosψ. We limit the
OPD \< λ~xray~/10 and solve for δ. A simple substitution for ψ yields
our final result: δ\<4λF^2^/5D^2^.

4.  Module X-axis Rotation Sensitivity (yaw error)

![](media/image5.wmf){width="2.3736220472440945in"
height="1.1015748031496062in"}A yaw rotation, φ, of the periscope module
about the center point between its flats does not cause a phase change
at the focal plane. However, this type of rotation error can reduce the
amount of light reflected through the periscope. In this analysis, we
limit the rotation such that the projection of light reflected from the
primary mirror will fill 90% of the secondary mirror thickness. The
mirror edge translates some amount, δ, when the module is rotated about
its X-axis by an angle φ. The distance δ is defined as msinφ, where m is
the length of a mirror. Next, we limit the translation error to one
tenth the thickness of a mirror: msin(g), where g is the graze angle
between the incoming beam and the mirror. A simple substitution for δ
leads us to the conclusion that a yaw error should be limited to no more
than one tenth of the graze angle.

5.  Module Y-rotation Sensitivity (pitch error)

Rotation of a periscope module about its Y-axis induces a pathlength
change within the module, and therefore, a phase change at the focal
plane. Figure 8 shows a periscope module that is rotated about the
midpoint of ray L~2~ (halfway between the two mirrors). The wavefront
travels along incoming ray L~1~ from the source to the primary mirror,
is reflected from the primary as ray L~2~ onto the secondary mirror, and
follows outgoing ray L~3~ reflected from the secondary mirror onto the
detector. Rays L~1~, L~2~, and L~3~ each reflect from the mirrors at
graze angle θ~1~. The distance, h, between the mirrors represents the
closest proximity the mirrors may have to one another without vignetting
the reflected beam. If the distance between the mirrors is held constant
and the mirror set is rotated by an angle φ about the midpoint of L~2~,
the pathlength will change. The new path is represented by heavy dashed
lines, and the new graze angle is defined as θ~2~.

Figure 9 shows how we derive the equation to represent allowable pitch
error for a periscope module rotated as described above. The diagram in
this figure is simplified to represent a general case for the
pathlength. We define this pathlength, P, as the sum of the incoming ray
L~1,~, internal reflection ray L~2~, and outgoing ray L~3.~ A vertical
reference line is perpendicular to incoming ray L~1~ and d~1~ represents
the extra distance L~1~ would have to travel to reach the reference
line. L~1~\* is the total distance from the source to the reference
line. The circumstance for L~3~ is the same as L~1~, and the length of
L~2~ is h/sinθ. The actual lengths of L~1~\* and L~3~\* are not
important and we take them to zero. If we sum L~1~, L~2~, and L~3~ we
quantify the pathlength as (h/sinθ)(1-cos2θ). We can find an optical
path difference by using the information we defined in Figure 8. The
variable h was defined above as a constant during a pitch rotation, but
the graze angle, θ, changes. Therefore, we can find the OPD by applying
θ~1~ and θ~2~ to the pathlength equation and taking the difference
between them. Next, we simplify by applying a series expansion to the
cos2θ terms. Since the graze angle is small we can further simplify by
assigning each Sinθ is approximately equal to its respective θ.
Additionally, the difference θ~2~-θ~1~ is shown in Figure 8 to be equal
to the pitch error angle φ. The OPD is now reduced to the quantity 2hφ.
If we restrict the OPD to be less than λ/10, we can define φ\<λ/20h. A
final substitution puts h in terms of the mirror length and graze angle:
φ\<λ/20msinθ.

![](media/image6.wmf){width="6.416535433070866in" height="3.25in"}

6.  Module Z-rotation Sensitivity (roll error)

A module roll about its Z-axis causes a rotation and shift of the
interference fringes at the focal plane. Figure 10 shows how a rotation,
φ, of one module about the center point between its two flats creates a
displacement of the projected aperture at the focal plane. In the X-Y
plane, the reflected beam follows an arc of radius D/2, and its
displacement, δ, at the focal plane is defined as Dsinφ/2. The resulting
fringe shift, ε, is shown in the magnified circle of Figure 9 and has
length δsinφ. The optical path difference is the sinψ projection of ε
onto the Z-X plane: OPD=δsinφsinψ. If we limit the optical path
difference to one tenth an x-ray wavelength and redefine sinψ in terms
of the system baseline (D) and focal length (F), δ must be less than
λF/5Dsinφ. Finally, we substitute our original definition for δ into
this equation and solve for φ.

![](media/image7.wmf){width="1.0216535433070866in"
height="0.5137795275590551in"}![](media/image8.wmf){width="1.8846456692913385in"
height="0.4389763779527559in"}![](media/image9.wmf){width="1.752755905511811in"
height="0.4582677165354331in"}![](media/image10.wmf){width="2.2874015748031495in"
height="0.4645669291338583in"}![](media/image11.wmf){width="2.04251968503937in"
height="0.23622047244094488in"}![](media/image12.wmf){width="1.2165354330708662in"
height="0.4574803149606299in"}

4.  configuration Tradeoffs

The periscope optical configuration offers several advantages over our
original X-configuration. First, the periscope design offers redundancy
between modules. This is good for two reasons: it is very cost effective
to build identical components, and redundant components in a system
reduce operational risk. Our thirty-two channel interferometer design
includes several periscope modules on each spacecraft. The operational
loss of one module would somewhat degrade the image at the detector, but
would not cause a loss of the mission. Another advantage the periscope
design offers is relaxation of some positional stability tolerances
required for formation flying. In order to achieve our high-resolution
goal, both optical designs require a large baseline in the
interferometer. The required distance between primary mirrors may vary
between 2 meters to 200 meters. Large distances between optics and the
ability to vary resolution capability based on a change in baseline
makes formation flying an attractive alternative in the mission design.
Decidedly, the easiest possible attitude control and stability
requirements are very desirable for this mission. In the X-configuration
design primary mirrors formation fly separately, while the secondary
mirrors are all mounted together on another spacecraft. This
configuration forces each primary mirror spacecraft to maintain
stability required for a single mirror in the system. In this section we
will show that spacecraft stability in both the X and Z translations
would be extremely challenging to maintain. However, the same mirror
stability is reasonable to maintain within a single periscope module.
The module concept also provides flexibility to allow for varied spacing
between an aligned pair of primary and secondary mirrors. In effect, we
can create a delay line by applying a high-resolution actuator to the
X-direction spacing between mirrors. A delay line can be used to correct
for pathlength differences attributed to the location of a periscope
module within the interferometer array. This simple alignment capability
enhances the system by allowing for a better distribution of periscope
modules across the UV plane and ultimately reduces the energy resolution
requirements on the detector.^2^

One tradeoff that we make in choosing the periscope design is a loss of
magnification at the focal plane. The distance between the detector and
optics is driven in both designs by the number of pixels per fringe
width at the focal plane. Good image resolution requires that the width
of each interference fringe be covered by several pixels on the
detector. Although the use of flat mirrors has several advantages in
this application, one disadvantage is that magnification at the focal
plane is not very great. Distances in each design range from hundreds of
meters to hundreds of kilometers and therefore require the detector be
mounted on a separate spacecraft. The X-configuration beams cross and
approach the focal plane in a quasi-parallel nature that causes fringe
amplification. Periscope module beams do not cross before arriving at
the detector, but approach at a smaller angle resulting in lower
magnification. This means the periscope design dictates that the
detector must formation fly a longer distance from the optics. A larger
distance between formation flying spacecraft can make communication and
metrology more complicated. However, we feel this is a soluble problem
and worth the periscope design gain in formation flying position.
Although the periscope design offers relief in positional stability
requirements, it does not appear to offer the same advantage in module
pointing. We discuss periscope rotational stability and phase
sensitivity later in this section.

1.  Mirror Tolerances

In a previous paper we derived the equations shown in Table 1.^4^ Each
equation represents the maximum mirror position error allowable if the
optical path difference is limited to λ~xray~/10. The analytic solution
column quantifies allowable error based on each equation, and the
raytrace column lists results from an analysis of mirror errors that
were performed at Ball Aerospace. The raytrace model consists of
thirty-two periscope modules arranged in a circular array with a 1.4m
diameter baseline. A uniformly random error was applied to each degree
of freedom in one mirror from each periscope module and optical
sensitivities were derived using a Monte Carlo approach. Several runs
were averaged and the tolerance for each error type was varied to derive
a system sensitivity curve for each error source. Allowable error in the
raytrace results is limited at an 80% relative Strehl ratio. This ratio
is defined as the peak intensity of a perturbed system divided by the
peak intensity of a point source with a perfectly aligned no-loss
system. Errors in the linear Y-direction and rotation about the X-axis
are not modeled in the raytrace because they do not induce phase errors
in the system and are only limited by the clear aperture.

2.  Periscope Module Tolerances and Advantages

The principal optical module tolerances derived in section 3.0 of this
paper are assembled in Table 2 next to the mirror tolerances for
comparison. The tolerances for each case have been calculated using
parameters currently defined for the MAXIM pathfinder mission.^2,3,4^
Table 2 clearly illustrates the periscope design tolerance advantages in
several degrees of freedom. The linear stability required for mirrors in
both the X and Z directions is very tight in the nanometer range. This
tolerance is challenging, but within the grasp of current technology if
the mirrors can be mounted on a single optical bench or periscope
module. However, nanometer range stability would be required for each
spacecraft carrying a primary mirror in the X-configuration design. A
periscope module must be held within ±20μm in the X-direction: still a
difficult task, but within reach in the near future. The Z-direction
stability requirement for a periscope module is significantly relaxed
from ±41nm to eight meters! Additionally, the range of allowable roll is
brought from an arc-minute for mirrors up to one quarter of a degree for
a module. Unfortunately, the periscope design does not offer relaxed
tolerances in all degrees of freedom. Yaw is tightened by a factor of
two, but still remains at a reasonable level. Our tightest and most
difficult pointing challenge remains at the arcsecond level in pitch
rotation. Despite all the advantages the periscope design has to offer,
it does not compensate for phase sensitivity associated with pitch
error. Nonetheless, the periscope configuration is a clear improvement
as we progress toward our goal of establishing a successful formation
flying mission. The application of these tolerances at the mission level
is discussed in detail by Gendreau et al.^2^

5.  CONCLUSION

A direct comparison of tolerances from the original X-configuration and
new periscope design lead to the conclusion that the new design is
clearly superior. We provided a thorough review of module tolerances
outlining how the periscope design works and why many of the tolerances
are relaxed. Mirror tolerances from a previous paper have been verified
with an independent raytrace and yielded excellent results. Yet, the
final analysis comes down to phase sensitivity due to pitch error.
Despite all the advantages that the periscope design has to offer, the
pitch error requirement remains at the milli-arcsecond level. We are
currently working to improve the optical design further to relax the
pitch error requirement. Some current ideas include an active delay
control, additional mirrors within a module, and reducing the size and
separation of the mirrors.

**ACKNOWLEDGEMENTS**

The authors wish to thank Mike Leiber and Nick White for contributions
to the development of this work.

This work was supported by NASA National Institute for Advanced Concepts
and NASA grant 153-4121.

**REFERENCES**

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