Cassegrain Shutter Exposure

Patrick Jelinsky

04/21/2004

Introduction

The CCD detectors on SNAP require a shutter to block the starlight while
they are being read out. The current SNAP baseline has a large shutter
between the focal plane and the folding flat. There are many problems
with this shutter. It has large angular momentum and therefore must be
counterbalanced very well. A non counterbalanced shutter will cause the
observatory to move over 1" (about 50 times the pointing requirement).
It crosses from a cold (140K) to warm boundary (20 C), and therefore
causes heat losses to the focal plane. The detectors are passively
cooled by a radiator which can only dissipate a limited amount of power.
Figure 1 shows the fraction of the passive radiator budget that a gap in
the thermal shield causes. A shutter will need a gap between 60 and 70
mm and therefore use up about 1/3 of the radiator budget. The shutter is
warm and close to the detectors which causes a significant background in
the longest HgCdTe filter. The current prediction is that the background
will be 0.25 e-/sec with the shutter closed, about 2/3 of the zodiacal
background. For these an other reasons, other shutter options are being
considered.

![](media/image1.wmf)

Figure 1 Heat loss versus size of gap in the thermal shield.

The most current idea is to move the shutter to the cassegrain focus of
the telescope. The cassegrain focus of the telescope is a focal surface
formed before the folding flat. An inherent disadvantage of a shutter at
this location is that the exposure will be non-uniform. Figure 2
schematically demonstrates why the exposure will be non-uniform. When
the shutter is partially open some parts of the focal plane will receive
100% of the light and others no light.

![](media/image2.png){width="5.9840277777777775in" height="5.3625in"}

Figure 2 Schematic of Cassegrain shutter in the open, partially open,
and closed positions. The blue is the cassegrain mask, the red are the
shutters, and the black are the paths of the light.

The purpose of this memo is to calculate the exposure non-uniformity
versus position and the required angular accuracy of the shutter for a
given exposure knowledge.

# Theory

The cassegrain shutters move through an angle near 90°. A schematic of
the shutter motion for an exposure is shown in Figure 3. The shutter
starts opening at 0 seconds, and moves for the first Tmove seconds, then
it is open for Topen seconds, finally it takes Tmove seconds to close.

![](media/image3.png){width="5.998611111111111in"
height="3.5708333333333333in"}

Figure 3 Typical Cassegrain Shutter motion.

The exposure for any given position on the focal plane is:

$$\text{Exp}(x,y) = \int_{0}^{T_{\text{exp}}}{\text{Tran}(x,y,\theta(t))\text{dt}}$$

where Exp(x,y) is the exposure at location x,y; Tran(x,y,θ) is the
transmission of the shutter at focal plane location x,y, at the angle θ;
θ(t) is the angle at time t, and Texp is the total time from start of
opening to end of closing the shutter. If the opening and closing is
symmetric then from the equation above and Figure 3:

$$\text{Exp}(x,y) = T_{\text{open}} + 2\int_{0}^{T_{\text{move}}}{\text{Tran}(x,y,\theta(t))\text{dt}}$$

Therefore to calculate the exposure for any location, the function
Tran(x,y,θ) must be know and the function θ(t) must be known.
Tran(x,y,θ) will be determined from a ray trace of the telescope in the
next section. And θ(t) will be determined from an analysis of the blade
motion in the section following the ray trace section.

# Ray trace

A non-sequential ray trace was used to determine the transmission of the
shutter for various positions on the focal plane and angles of the
shutter. Figure 4 shows the 25 locations on the focal plane that were
ray traced through the system. Because of symmetry, these should map the
maximum and minimum of the exposure map. For each focal plane position
the shutter angle was moved and the number of rays that hit the focal
plane recorded. Over 100, 000 rays were traced at each location and
shutter angle. The grid of shutter angles was set so that the exposure
could be mapped to batter than 0.1%.

![](media/image4.png){width="3.8555555555555556in"
height="4.043055555555555in"}

Figure 4 Ray trace focal plane locations are shown in blue. Because of
symmetry these locations should map the maximum and minimum exposure
excurtions.

Figure 5 is a plot of the transmission of the shutter versus angle for
the five locations with X near zero. The blue curve is for Y at 132.6mm,
red for Y at 163.4 mm, green for Y at 194.2 mm, black for Y at 225.0 mm,
and cyan for Y at 255.8 mm. An example transmission map is shown in
Figure 6. This is the transmission when the blade is at 60°. It is an
interpolation of the coarser ray trace data so there are some artifacts.
For the actual science data a much finer grid of positions would be
done.

![](media/image5.wmf)

Figure 5 Transmission of the shutter versus angle for the five focal
plane locations near X = 0. The legend is the Y location on the focal
plane in mm.

![](media/image6.png){width="5.990972222222222in"
height="2.8541666666666665in"}

Figure 6 Transmission of the shutter when the blade is at 60°.

# Blade Motion

The last bit of information needed for the exposure calculations is the
position of the shutter blade versus time. The equation of motion for
the shutter blade is:

$$\overset{¨}{\theta} = \frac{\tau(t)}{I}$$

where θ is the angular position of the blade, τ is the torque applied at
time t, and I is the moment of inertia of the blade about the axis of
rotation. Every motor will have a maximum torque that it can apply. The
shutter will move quickest if a positive maximum torque is applied for
half the time and then a negative maximum torque is applied for the rest
of the time. This is a square wave in torque. It may have some
implications with the observatory, since the torque and the slope of the
torque are discontinuous and it has more high frequency components. For
this analysis the torque was modeled as a rounded square wave of torque.
The function of torque is given as:

$$\tau(t) = \begin{matrix}
\tau_{\text{max}}\text{sin}^{2}\left( \frac{2\pi t}{\text{fT}} \right) & 0 < t < \frac{\text{fT}}{4} \\
\tau_{\text{max}} & \frac{\text{fT}}{4} < t < \frac{T(2 - f)}{4} \\
\tau_{\text{max}}\text{cos}\left( \frac{2\pi}{\text{fT}}\left( t - \frac{T(2 - f)}{4} \right) \right) & \frac{T(2 - f)}{4} < t < \frac{T(2 + f)}{4} \\
 - \tau_{\text{max}} & \frac{T(2 + f)}{4} < t < \frac{T(4 - f)}{4} \\
\tau_{\text{max}}\left\lbrack \text{sin}^{2}\left( \frac{2\pi}{\text{fT}}\left( t - \frac{T(4 - f)}{4} \right) \right) - 1 \right\rbrack & \frac{T(4 - f)}{4} < t < T
\end{matrix}$$

where *τ~max~* is the maximum torque the motor can apply, *T* is the
time it takes to move the shutter 90°, f is the fraction of time the
torque is slewing (*f* = 0 is a square wave, and *f* = 1 is nearly a sin
wave). The time *T* can be showed to be:

$$T = \sqrt{\frac{2\pi I}{\tau_{\text{max}}\left( 1 - \frac{f}{2} + \frac{f^{2}}{8}\left( \frac{\text{12}}{\pi^{2}} - 1 \right) \right)}}$$

where *I* is the moment of inertial of shutter. The shutter takes 38%
longer to open with f=1 than a square wave. Figure 7 shows the torque
versus time for various values of *f*.

![](media/image7.png){width="4.252083333333333in"
height="3.188888888888889in"}

Figure 7 Torque versus time for various values of f.

![](media/image8.png){width="5.0in" height="3.75in"}The power spectrum
of the torque will give an idea of the high frequency force components
that may stimulate the observatory to shake. Figure 8 shows the square
root of the power spectrum of the torque for both f=0 and f=1. The
square root was plotted to more easily show the higher frequency
components. The f=1 curve has much less energy at the higher
frequencies.

Figure 8 The power spectrum of the torque for square wave (blue) and f=1
(red).

Exposure Uniformity

The uniformity of the exposure was calculated at the 25 locations on the
focal plane using the torque function given in the previous section.
David Pankow showed that with an appropriate motor the shutter would
move 90° in about 0.077 seconds with a square wave torque function. The
exposure uniformity was calculated for both f=0 (square wave) and an f=1
(smooth torque) to bound the problem. An opening time for a square wave
was assumed to be 0.08 seconds. Figure 9 shows the non-uniformity of the
exposure versus exposure time. The blue curve is for a torque function
with f=0 (square), and the red curve is for f=0. The maximum exposure
deviation is 5.9% for a one second exposure. Figure 10 shows an exposure
deviation map for a 1 second exposure using a torque function with f=1.

![](media/image9.png){width="5.188888888888889in"
height="3.8777777777777778in"}

Figure 9 Exposure Non-Uniformity versus exposure. The blue curve is for
torque with f=0(square), and the red curve is for f=1.

![](media/image10.png){width="5.247222222222222in"
height="2.5104166666666665in"}

Figure 10 Exposure Deviation map for a 1 second exposure (f=1).

Exposure Knowledge

The knowledge of the exposure is limited by the readout of the position
of the shutter. The knowledge (as a percentage) improves with longer
exposure times (as does the uniformity). The shutter readout was modeled
as an encoder with n counts per revolution read out on a periodic
interval. Figure 11 is a plot showing the real location of the shutter
versus time and the encoder value versus time. A linear and cubic spline
was fit to the encoder locations and this was used for the exposure
determination. The knowledge of the exposure was determined by comparing
the exposure determined from the readout locations and that determined
from the actual positions. This was done for a 1 second exposure, which
should be the worse case. Many phases of the encoder steps were modeled
and the worse was used.

![](media/image11.png){width="4.875694444444444in"
height="3.634027777777778in"}

Figure 11Position of shutter. The blue line is the actual position, the
red line is the redout from the encoder, and the green line is a linear
interpolation of the encoder position.

Figure 12 shows the results using a linear interpolation of the encoder
positions. The knowledge of the exposure is better tan 0.1% for an
encoder with a resolution better than 256 counts/revolution and sampled
more than 5000 times a second. And the knowledge can be better than
0.01% for an encoder with more than 512 counts/revolution sampled 20,000
times per second. Figure 13 shows the results using a cubic spline
interpolation. The knowledge of the exposure is better tan 0.1% for an
encoder with a resolution better than 256 counts/revolution and sampled
more than 5000 times a second. And the knowledge can be better than
0.01% for an encoder with more than 256 counts/revolution sampled 20,000
times per second. Most encoders or resolvers will have more than 1024
counts per revolution. It should also be easy to read them up to 10,000
to 20,000 times per second, so the exposure knowledge should be well
below 0.05%.

![](media/image12.wmf)

Figure 12 Exposure knowledge for a 1 second exposure using a linear
interpolation. Each line is for a different sampling of the encoder. The
blue line is sampled every millisecond and the pink line is sampled
every 50 microsecond.

![](media/image13.wmf)

Figure 13 Exposure knowledge for a 1 second exposure using a cubic
spline interpolation. Each line is for a different sampling of the
encoder. The blue line is sampled every millisecond and the pink line is
sampled every 50 microsecond.

# Conclusion

The uniformity of the exposure from a shutter at the cassegrain focus
looks good. It should be verified to make sure it meets the. The
knowledge of the exposure from a shutter at the cassegrain focus also
looks good. But again it needs to be checked against a requirement.

There are several studies that need to be completed in the future. The
requirement for limiting energy at high frequencies needs to be
quantified and then the shutter mechanical model should be checked to
make sure it is satisfactory. A detailed mechanical layout of the
cassegrain mask and shutter needs to be done to make sure that it fits
and can be made. Finally, a finer grid of points in the focal plane will
need to be ray traced to create an acceptable exposure calculation for
the mission. It is not worth doing this until the telescope, baffles and
detector locations are more solid.

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