that we only need to think about photons arriving on the flat surface of the
principal-aperture-plane. Figure 8.1 shows how we can think of the calibrated
light-field-sequence $\mathcal{L}$. All our further interpretation of the
light-field-sequence $\mathcal{L}$ are based on the mindset presented by the
Equation 8.6, and Figure 8.1.

Figure 8.1: The photons in the light-field-sequence $\mathcal{L}$. Black dots
are photons with arrows indicating their direction of motion along their
dashed trajectories. The calibration of the plenoscope abstracts away all the
quirks of the large imaging-reflector, the lenses, the small cameras and the
photo-sensors. What is left are single photons traveling along rays with
respect to the principal-aperture-plane. Note that photon 1 and photon 2
occupy the same ray $k_{1}=k_{2}$, but at different positions because of their
different arrival-times ${t_{\text{pap}}}_{1}\neq{t_{\text{pap}}}_{2}$. Here
$c$ is the speed of light.

####  8.3 Isochronous imaging

We defined the light-field-geometry $G$ with respect to the principal-
aperture-plane. However, if one intends to calculate classic image-sequences
$\mathcal{I}$ from the recorded light-field-sequence $\mathcal{L}$, the
arrival-time-delays ${\overline{t}_{\text{pap}}}$ must not be used for
calibration. One can use no arrival-time-delays and just use the raw recorded
arrival-times $t_{\text{raw}}$ to get a good timing resolution for imaging.
This is because the coarse surface of Portal’s large imaging-reflector follows
a parabola. However, a parabola can only do good isochronous imaging in the
central part of the image, but the plenoscope can do better. The plenoscope
can synthesize the perfect isochronous image in post for the entire field-of-
view. Isochronous imaging means that incoming light-fronts, where the photons
travel together in a plane perpendicular to their direction of motion, will
arrive in the same pixel at the same moment. Figure 8.12 shows the path-
length-delays which are added to the calibration used for the principal-
aperture-plane. The Figures 8.8 and 8.9 show the time-delays
${\overline{t}_{\text{img}}}$ which are added to the raw arrival-times
$t_{\text{raw}}$ in order to obtain perfect isochronous images.

####  8.4 Light-field-geometry of Portal

We visualize the light-field-geometry of Portal when the large imaging-
reflector and the light-field-sensor of Portal are both in their target-
geometry with respect to each other. We propagate $10^{9}$ photons to estimate
Portal’s light-field-geometry.
In Figure 8.2 we find the magnitude of the spread in incident-directions
versus the magnitude of the incident-directions. This is an inverse
representation of the point-spread-function discussed on telescopes. As
expected from the large imaging-reflector, the inner region has a smaller
spread in incident-directions. The finite aperture of the small cameras in the
light-field-sensor alone is expected to induce a spread in the order
111Standard-deviation of a uniform distribution in a limited range:
$0.067^{\circ}/\sqrt{12}$. of $\approx 0.02^{\circ}$. Even in the outer region
of the field-of-view, the spread in incident-directions is below the spacing
of the small cameras.

Figure 8.2: Histogram of the mean magnitude of the incident-directions versus
the spread magnitude of the incident-directions for all photo-sensors. The
darkening in the histogram goes with the square-root of the number of lixels.

The Figures 8.3 to 8.11 show the arrangement of the photo-sensors inside the
$12.1\,$m diameter light-field-sensor of Portal. We draw each photo-sensor as
a small hexagon and color it according to its value. We also show close-ups of
the same sensor-plane in order to see the structures inside the small cameras.
Compare the arrangement of the photo-sensors with Figure 5.11, and 7.3.
Figure 8.3 shows the incident-directions in both $\overline{c_{x}}$, and
$\overline{c_{y}}$ for all photo-sensors. Same as with an image-sensor in a
telescope, the positions of the photo-sensors on the sensor-plane correspond
directly to the incident-directions.

Figure 8.3:  The photo-sensors colored according to their incident-directions
$\overline{c_{x}}$ (top) and $\overline{c_{y}}$ (bottom).

The Figures 8.4 and 8.5 show the support-positions $\overline{x}$, and
$\overline{y}$ for all the photo-sensors. Unlike a telescope, the additional
projection provided by the lenses in the small cameras allow the photo-sensors
inside a small camera to sample the support-positions across the full $71\,$m
aperture of the large imaging-reflector. The close-ups in the Figures 8.4 and
8.5 show clearly the repeating pattern of the small cameras.

Figure 8.4:  The photo-sensors colored according to their support-position
$\overline{x}$. Full sensor-plane on top, close-up at the bottom.

Figure 8.5:  The photo-sensors colored according to their support-position
$\overline{y}$. Full sensor-plane on top, close-up at the bottom.

Figures 8.6 and 8.7 show the collection-efficiency $\overline{\eta}$ of all
the photo-sensors. As can be seen from the close-ups, the central photo-
sensors inside a small camera are less efficient than the outer ones. This is
because the central photo-sensor is partly blocked due to the housing of the
light-field-sensor itself. Depending on the position of the small camera on
the sensor-plane, the shadow of the housing of the light-field-sensor moves
accordingly to the opposite direction. Since the observations of air-showers
approach a single photon regime, the recorded light-field-sequences are always
subject to quantization noise. This makes an application of the collection-
efficiency $\overline{\eta}$ difficult or even impossible for the correction
of photon-intensities. However, we use it extensively to speed up the
simulations of Portal when we inject night-sky-background-photons into the
light-field-sensor, see also Section 18.3.

Figure 8.6:  The photo-sensors colored according to their collection-
efficiency $\overline{\eta}$. Full sensor-plane on top, central close-up at
the bottom.

Figure 8.7:  Same as Figure 8.6. Close-ups on the left and upper regions of
the the sensor-plane.

The Figures 8.8 and 8.9 show the time-delays $\overline{t}_{\text{img}}$ of
the photo-sensors which are needed for a perfect isochronous image. The
overall spread of the delays is only $\approx 1\,$ns and thus already
sufficient for the observation of air-showers even if we do not correct for
it. Just as one would expect from a telescope with a parabolic imaging-
reflector, the spread in the central part of the sensor-plane is lowest.

Figure 8.8:  The photo-sensors colored according to their arrival-time-delay
for the image-plane $\overline{t}_{\text{img}}$. Full sensor-plane on top,
central close-up at the bottom.

Figure 8.9:  Same as Figure 8.8. Close-ups on left and upper regions of the
sensor-plane.

The Figures 8.10 and 8.11 show the time-delays $\overline{t}_{\text{pap}}$ of
the photo-sensors which are needed for the principal-aperture-plane.

Figure 8.10:  The photo-sensors colored according to their arrival-time-delay
for the principal-aperture-plane $\overline{t}_{\text{pap}}$. Full sensor-
plane on top, central close-up at the bottom.

Figure 8.11:  Same as Figure 8.10. Close-ups on left and upper regions of the
sensor-plane.  Figure 8.12: Time-of-flight offsets for isochronous imaging.
Parallel rays carry photons arriving on the principal-aperture-plane $x,\,y$.
For isochronous imaging we add the signed time-of-flight offsets (thick
arrows) between the light-front (dashed-line) and the principal-aperture-plane
to the arrival-times of the photons in pixel $(c_{x}\,,c_{y})$.

####  8.5 Multiple light-field-geometries on Portal

Since in Portal the geometry between the large imaging-reflector and the
light-field-sensor might not be fix during all time, the light-field-geometry
for Portal depends on the actual geometry between the two. This is a trade-off
that we have to do in order to build a large $71\,$m imaging-reflector with a
decoupled light-field-sensor. During the observations with Portal, the cable-
robot-mount will always try to bring the large imaging-reflector and the
light-field-sensor into their desired target geometry
$\hat{T}_{\text{target}}$. However, small deviations from the desired target
geometry $\hat{T}_{\text{target}}$, which the cable-robot-mount can not
correct for fast enough, might occur. The reason for such fast deviations
might be wind-gusts. The actual geometry $\hat{T}_{\text{actual}}$ between the
large imaging-reflector and the light-field-sensor is recorded during the
observations so that dedicated light-field-geometries
$G(\hat{T}_{\text{actual}})$ can be applied accordingly. We describe $\hat{T}$
with the homogeneous transformation from the large imaging-reflector’s frame
into the light-field-sensor’s frame.

####  8.6 Conclusion

The light-field-geometry is a powerful tool to describe and simulate any
optical instrument sensing photons. The light-field-geometry allows us to
mentally decouple the implementation-details of an instrument from its actual
perception. With the light-field-geometry we can abstract away all the quirks
and features of Portal’s large imaging-reflector, and its lenses in the light-
field-sensor. All these quirks and features are taken care of in the light-
field-geometry. The light-field-geometry allows us to focus on what actually
matters: Individual photons traveling along their trajectories in a three-
dimensional space, see Figure 8.1.
Having the light-field-geometry of an instrument is enough to estimate the
response of that instrument for a given light-field-input. And the other way
around, when we have the response and the light-field-geometry of an
instrument, we can reconstruct the light-field which the instrument observed.
Together with the operations formulated in Chapter 9, the light-field-geometry
allows us to overcome aberrations, see Chapter 10, and to compensate
misalignments, see Chapter 11.

###  Chapter 9 Interpreting the light-field-sequence

The light-field-sequence $\mathcal{L}$ of the plenoscope can be approached in
various ways to reconstruct the air-shower in order to learn about the cosmic
particle. Here we demonstrate the methods we have investigated so far. First,
we show three different representations of the light-field-sequence
$\mathcal{L}$. Second, we demonstrate imaging, areal-sampling, refocused
imaging, synthetic apertures, light-fronts. Three-dimensional tomography we
discuss separately in Chapter 15.

####  9.1 Representing the light-field-sequence

There are different ways to represent the light-field-sequence $\mathcal{L}$.
We use three different representations. Each representation has certain
advantages. Our three representations can be converted into each other without
loss.

##### Lixel-time-histogram

First, we represent the light-field-sequence as a two-dimensional histogram

$\displaystyle\mathcal{L}[k,\,t]$ $\displaystyle=$
$\displaystyle\begin{bmatrix}i_{0,\,0}&i_{0,\,1}&i_{0,\,2}&\ldots&i_{0,\,T-1}\\\
i_{1,\,0}&i_{1,\,1}&i_{1,\,2}&\ldots&i_{1,\,T-1}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
i_{K-1,\,0}&i_{K-1,\,1}&i_{K-1,\,2}&\ldots&i_{K-1,\,T-1}\\\ \end{bmatrix}$
(9.1)

which we call lixel-time-histogram. The columns correspond to the $K$ lixels
in the plenoscope, and the rows correspond to the $T$ time-cells. Each entry
$i_{k,\,t}$ is the intensity of photons which arrived in lixel $k$ at time-
cell $t$. On Portal the shape of $\mathcal{L}[k,\,t]$ is
$((K=515,023{})\times(T=100))$. Portal uses $T=100$ time-cells of $500\,$ps
duration each. Usually the population of $\mathcal{L}[k,\,t]$ is very sparse.

##### Pixel-paxel-time-histogram

For the specific design of Portal, the two-dimensional histogram
$\mathcal{L}[k,\,t]$ can also be written as a three-dimensional histogram
$\mathcal{L}[n,\,m,\,t]$, where the lixels $k$ are addressed using the pixels
$n$, and paxels $m$ which they approximatively correspond to, see section 8.1.

$\displaystyle\mathcal{L}[n,\,m\,,t]$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\begin{bmatrix}i_{0,\,0,\,0}&i_{0,\,1,\,0}&\ldots&i_{0,\,M-1,\,0}\\\
i_{1,\,0,\,0}&i_{1,\,1,\,0}&\ldots&i_{1,\,M-1,\,0}\\\
\vdots&\vdots&\ddots&\vdots\\\
i_{N-1,\,0,\,0}&i_{N-1,\,1,\,0}&\ldots&i_{N-1,\,M-1,\,0}\\\ \end{bmatrix},\\\
\\\ \begin{bmatrix}i_{0,\,0,\,1}&i_{0,\,1,\,1}&\ldots&i_{0,\,M-1,\,1}\\\
i_{1,\,0,\,1}&i_{1,\,1,\,1}&\ldots&i_{1,\,M-1,\,1}\\\
\vdots&\vdots&\ddots&\vdots\\\
i_{N-1,\,0,\,1}&i_{N-1,\,1,\,1}&\ldots&i_{N-1,\,M-1,\,1}\\\ \end{bmatrix},\\\
\vdots\\\
\begin{bmatrix}i_{0,\,0,\,T-1}&i_{0,\,1,\,T-1}&\ldots&i_{0,\,M-1,\,T-1}\\\
i_{1,\,0,\,T-1}&i_{1,\,1,\,T-1}&\ldots&i_{1,\,M-1,\,T-1}\\\
\vdots&\vdots&\ddots&\vdots\\\
i_{N-1,\,0,\,T-1}&i_{N-1,\,1,\,T-1}&\ldots&i_{N-1,\,M-1,\,T-1}\\\
\end{bmatrix}\end{bmatrix}$ (9.2)

When we compute the sum along two of the three axis of the three-dimensional
histogram $\mathcal{L}[n,\,m\,,t]$, we can directly obtain an image, an areal-
sample, or an intensity-sequence. The Equations 9.3 to 9.9 illustrate the
various processing options of summing up the photon-intensities along certain
axes of $\mathcal{L}[n,\,m\,,t]$.

$\displaystyle\sum_{n,\,m,\,t}\mathcal{L}[n,\,m,\,t]$
$\displaystyle\rightarrow$ $\displaystyle\mathcal{L},\,\,\text{Intensity,
number of photons}$ (9.3) $\displaystyle\sum_{m,\,t}\mathcal{L}[n,\,m,\,t]$
$\displaystyle\rightarrow$ $\displaystyle\mathcal{L}[n],\,\,\text{Classic
image of telescope}$ (9.4) $\displaystyle\sum_{n,\,t}\mathcal{L}[n,\,m,\,t]$
$\displaystyle\rightarrow$ $\displaystyle\mathcal{L}[m],\,\,\text{Areal
intensity on aperture}$ (9.5)
$\displaystyle\sum_{n,\,m}\mathcal{L}[n,\,m,\,t]$ $\displaystyle\rightarrow$
$\displaystyle\mathcal{L}[t],\,\,\text{Intensity sequence, light-curve}$ (9.6)
$\displaystyle\sum_{t}\mathcal{L}[n,\,m,\,t]$ $\displaystyle\rightarrow$
$\displaystyle\mathcal{L}[n,\,m],\,\,\text{Static light-field}$ (9.7)
$\displaystyle\sum_{m}\mathcal{L}[n,\,m,\,t]$ $\displaystyle\rightarrow$
$\displaystyle\mathcal{L}[n,\,t],\,\,\text{Classic image-sequence, video}$
(9.8) $\displaystyle\sum_{n}\mathcal{L}[n,\,m,\,t]$ $\displaystyle\rightarrow$
$\displaystyle\mathcal{L}[m,\,t],\,\,\text{Areal intensity-sequence on
aperture}$ (9.9)

On Portal the shape of $\mathcal{L}[n,\,m\,,t]$ is
$((N=8,443{})\times(M=61{})\times(T=100))$.

##### Stream of photons

We can also describe the light-field-sequence $\mathcal{L}$ as a stream

$\displaystyle\mathcal{L}[j]$ $\displaystyle=$
$\displaystyle\begin{bmatrix}k_{0}&,&{t_{\text{raw}}}_{0}\\\
k_{1}&,&{t_{\text{raw}}}_{1}\\\ \vdots&&\vdots\\\
k_{J-1}&,&{t_{\text{raw}}}_{J-1}\\\ \end{bmatrix}$ (9.10)

of all the $J$ photons in it. Here each row in $\mathcal{L}$ describes a
single photon. The first column addresses the lixel $k_{j}$ which absorbed the
$j$-th photon, and the second column is the arrival-time
${t_{\text{raw}}}_{j}$ of the $j$-th photon. The stream of photons is
practical to label individual photons as it is done in the classification of
Cherenkov-photons and night-sky-background-photons. Further, the stream of
photons does not assume that lixels are organized hierarchical so that the
light-field $\mathcal{L}[n,m]$ becomes a rectangular histogram in $n$ and $m$.
Personally, we mostly ended up using the stream of photons to represent the
light-field-sequence $\mathcal{L}$ because of its generality and efficiency
due to its natural zero-suppression.

####  9.2 Directional sampling – Imaging

The plenoscope can do imaging exactly like a conventional telescope. In the
special case of Portal, we simply sum up the photon-intensities of all lixels
which belong to the same pixel, see Equations 9.4 and 9.8. In words closer to
the implementation of Portal, we simply sum up the photon-intensities of all
photo-sensors inside of each small camera, see Figure 2.2. Imaging can be
described as a linear-combination

$\displaystyle\mathcal{I}[n]$ $\displaystyle=$ $\displaystyle
U_{\text{imaging}}[n,\,k]\cdot\mathcal{L}[k]$ (9.11)

of the photon-intensities in the light-field. Photon-intensities from several
lixels $k$ in $\mathcal{L}[k]$ are added up in a particular pixel $n$ in the
image $\mathcal{I}[n]$. This linear-combination can be described using the
imaging-matrix

$\displaystyle U_{\text{imaging}}[n,\,k]$ $\displaystyle=$
$\displaystyle\begin{bmatrix}u_{0,\,0}&u_{0,\,1}&\ldots&u_{0,\,K-1}\\\
u_{1,\,0}&u_{1,\,1}&\ldots&u_{1,\,K-1}\\\ \vdots&\vdots&\ddots&\vdots\\\
u_{N-1,\,0}&u_{N-1,\,1}&\ldots&u_{N-1,\,K-1}\\\ \end{bmatrix}.$ (9.12)

Each matrix-element $u_{n^{\prime},\,k^{\prime}}$ describes how much of the
photon-intensity from the lixel $k^{\prime}$ in the light-field
$\mathcal{L}[k]$ is added up into the pixel $n^{\prime}$ in the image
$\mathcal{I}[n]$. In the special case of Portal, we can compute the matrix-
elements using Equation 8.3

$\displaystyle u_{n,\,k}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}1&\text{if}\,\,n=\text{floor}(k/M)\\\
0&\text{else}\\\ \end{array}\right.{}.$ (9.15)

Figure 9.1 visualizes the matrix-elements on Portal to synthesize the central
pixel $n=4,221$ in the image. In general, the matrix-elements in
$U_{\text{imaging}}[n,\,k]$ can be computed using the light-field-geometry
$G$. We can compute the matrix-elements of $U_{\text{imaging}}[n,\,k]$ using
the angular distances between the incident-directions of the pixels $n$ in
$\mathcal{I}[n]$ and a lixel $k$ in $G$.

$\displaystyle u_{n,\,k}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}1&\text{if}\,\,\sqrt{({c_{x}}_{n}-{c_{x}}_{k})^{2}+({c_{y}}_{n}-{c_{y}}_{k})^{2}}\leq
c_{\text{pixel-radius}}\\\ 0&\text{else}\\\ \end{array}\right.{}$ (9.18)

where $c_{\text{pixel-radius}}$ is e.g. the angular radius of a pixel in the
image $\mathcal{I}[n]$. The matrix-elements do not need to be strictly binary
(0 or 1) as shown in the Equations 9.15, and 9.18. The weights can depend on
the distances between the incident-directions, or on the integrated overlap
between pixels to reduce aliasing.

Figure 9.1:  Close-up on the sensor-plane of Portal. Black photo-sensors
(lixels) are summed up to synthesize the central pixel ($n=4,221$) in the
image. Black is a weight of $1$, and white is a weight of $0$.

The Figures 6.4, 6.8, 6.12, 6.16, and 6.20 show images projected from Portal’s
light-field using Equation 9.15.

####  9.3 Areal-sampling

The plenoscope can do areal-sampling of the photon-intensities on ground
similar to the modified solar-concentrators for dense areal-sampling presented
in Section 19.2. An areal-sample is the photon-intensity-histogram along the
support-positions $x$, and $y$ on the principal-aperture-plane. With the
special design of Portal, we just sum the light-field-sequence
$\mathcal{L}[n,\,m,\,t]$ according to the Equations 9.5, or 9.9. In general,
areal-sampling can also be formulated as a linear-combination

$\displaystyle\mathcal{A}[m]$ $\displaystyle=$ $\displaystyle
U_{\text{areal}}[m,\,k]\cdot\mathcal{L}[k]$ (9.19)

similar to Equation 9.11. Here $\mathcal{A}[m]$ is the one-dimensional photon-
intensity-histogram along the $m$ paxels on the principal-aperture-plane, and

$\displaystyle U_{\text{areal}}[m,\,k]$ $\displaystyle=$
$\displaystyle\begin{bmatrix}u_{0,\,0}&u_{0,\,1}&\ldots&u_{0,\,K-1}\\\
u_{1,\,0}&u_{1,\,1}&\ldots&u_{1,\,K-1}\\\ \vdots&\vdots&\ddots&\vdots\\\
u_{M-1,\,0}&u_{M-1,\,1}&\ldots&u_{M-1,\,K-1}\\\ \end{bmatrix}$ (9.20)

is the areal-sampling-matrix. The matrix-element $u_{m^{\prime},\,k^{\prime}}$
expresses how much of the photon-intensity of lixel $k^{\prime}$ from
$\mathcal{L}[k]$, goes into paxel $m^{\prime}$ in $\mathcal{A}[m]$. In the
special case of Portal, the matrix-elements

$\displaystyle u_{m,\,k}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}1&\text{if}\,\,m=k-\text{floor}(k/M)\times
M\\\ 0&\text{else}\\\ \end{array}\right.{}$ (9.23)

are calculated using the Equations 8.3, and 8.4.

Figure 9.2:  Close-up on the sensor-plane of Portal. Black photo-sensors
(lixels) are summed up to synthesize a single paxel ($m=54$) on the principal-
aperture-plane. Black is a weight of $1$, and white is a weight of $0$.

Figure 9.2 shows the matrix-elements for the particular row
$U_{\text{areal}}[m=54,\,k]$ in the special case of Portal. In words close to
Portal’s implementation, we sum up the intensities of each $m$-th photo-
sensors in all small cameras, compare Figure 2.3. In general, the weights

$\displaystyle u_{m,\,k}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}1&\text{if}\,\,\sqrt{({x}_{m}-{x}_{k})^{2}+({y}_{m}-{y}_{k})^{2}}\leq
d_{\text{paxel-radius}}\\\ 0&\text{else}\\\ \end{array}\right.{}$ (9.26)

can be estimated using the distance between the support-position of paxel $m$
in $\mathcal{A}$ and the support-position of lixel $k$ in the light-field-
geometry $G$. The radius of the paxels $d_{\text{paxel-radius}}$ can be used
as a threshold here.

####  9.4 Refocusing images in post

The plenoscope can project its light-field onto images which are refocused to
different object-distances in post, this means after the light-field was
recorded. In Chapter 6 we demonstrate such refocused images on the example of
the Portal Cherenkov-plenoscope, see the Figures 6.7, 6.11, 6.15, 6.19, and
6.23. Here we present a geometric description for refocusing in post which is
motivated by [Ng et al., 2005] and Figure 6.2. First, we discuss the
trajectories of the photons after they passed the principal-aperture-plane and
call those trajectories image-rays. Second, we use the image-rays and their
intersections with virtual sensor-planes to formulate a projection of the
light-field onto images focused to different object-distances.

##### Image-rays

Image-rays describe the trajectories of the photons after those have passed
the principal-aperture-plane, see Figure 9.3. In Figure 6.2, the image-rays
are defined by their support-positions on the principal-aperture-plane $a$,
$a^{\prime}$, and $a^{\prime\prime}$ together with their absorption-positions
on the sensor-plane $i$, $i^{\prime}$, and $i^{\prime\prime}$.

Figure 9.3:  A ray $\vec{r}(\lambda)_{k}$ is reflected on the principal-
aperture-plane at the bottom of the figure. We call the reflected ray
$\vec{\rho}(\lambda)_{k}$ the image-ray. Both rays share the same support-
position $\vec{s}_{k}$.

Every ray $\vec{r}(\lambda)_{k}$ according to Equation 2.1 has a corresponding
image-ray

$\displaystyle\vec{\rho}(\lambda)_{k}$ $\displaystyle=$
$\displaystyle\vec{s}_{k}+\lambda\frac{\vec{\delta}_{k}}{|\vec{\delta}_{k}|_{2}}.$
(9.27)

Both rays $\vec{r}(\lambda)_{k}$, and $\vec{\rho}(\lambda)_{k}$ share the same
support-vector

$\displaystyle\vec{s}_{k}$ $\displaystyle=$
$\displaystyle\left(x_{k},\,\,y_{k},\,\,0\right)^{T}$ (9.28)

such that they both intersect on the principal-aperture-plane for $\lambda=0$

$\displaystyle\vec{r}(0)_{k}$ $\displaystyle=$
$\displaystyle\vec{\rho}(0)_{k}.$ (9.29)

The direction-vector $\vec{\delta}_{k}$ of the image-ray
$\vec{\rho}(\lambda)_{k}$ is calculated using the Thin-lens-equation 6.1 and
Figure 6.2. We first calculate where the image-ray would intersect with the
focal-plane in focal-distance $f$ of the imaging-system

$\displaystyle\vec{b}_{k}$ $\displaystyle=$ $\displaystyle
f\left(-\tan({c_{x}}_{k}),\,\,-\tan({c_{y}}_{k}),\,\,1\right)^{T},$ (9.30)

and second we calculate the direction-vector

$\displaystyle\vec{\delta}_{k}$ $\displaystyle=$
$\displaystyle\vec{b}_{k}-\vec{s}_{k}$ (9.31)

by subtracting the support-position $\vec{s}_{k}$ from the intersection point
$\vec{b}_{k}$ in the sensor-plane.

##### Refocusing on the plenoscope

We express the refocusing as a projection

$\displaystyle\mathcal{I}_{g}[n]$ $\displaystyle=$ $\displaystyle
U_{\text{imaging}}(g)[n,\,k]\cdot\mathcal{L}[k]$ (9.32)

of the light-field onto an image, similar to regular imaging in Equation 9.11.
However, this time the imaging-matrix

$\displaystyle U_{\text{imaging}}(g)[n,\,k]$ $\displaystyle=$
$\displaystyle\begin{bmatrix}u_{0,\,0}(g)&u_{0,\,1}(g)&\ldots&u_{0,\,K-1}(g)\\\
u_{1,\,0}(g)&u_{1,\,1}(g)&\ldots&u_{1,\,K-1}(g)\\\
\vdots&\vdots&\ddots&\vdots\\\
u_{N-1,\,0}(g)&u_{N-1,\,1}(g)&\ldots&u_{N-1,\,K-1}(g)\\\ \end{bmatrix}$ (9.33)

depends on the object-distance $g$ where we want to focus on. The matrix-
elements $u_{n,\,k}(g)$ are calculated using the $k$-th image-ray
$\vec{\rho}(\lambda)_{k}$, and the corresponding $x$, and $y$ positions of the
$n$-th pixel on a virtual sensor-plane in image-distance $b$. First, we use
the Thin-lens-equation and our desired object-distance $g$ to calculate the
image-distance

$\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{1}{\frac{1}{f}-\frac{1}{g}}$ (9.34)

where of the virtual sensor-plane. Second, we calculate how far

$\displaystyle\lambda_{g}$ $\displaystyle=$
$\displaystyle\frac{b}{\vec{z}\cdot\vec{\delta}_{k}}$ (9.35)

we have to travel along the direction-vector $\vec{\delta}_{k}$ of our image-
ray $\vec{\rho}(\lambda)_{k}$ to intersect with our virtual-sensor-plane in
distance $b$. Third, we calculate the intersection-position

$\displaystyle\vec{i}_{k,g}$ $\displaystyle=$
$\displaystyle\vec{\rho}(\lambda_{g})_{k}$ (9.36)

of the image-ray on the virtual-sensor-plane. Fourth, we compute the position

$\displaystyle\vec{p}_{n,g}$ $\displaystyle=$
$\displaystyle\left(x_{n},\,y_{n},\,b\right)^{T}$ (9.37)

of the $n$-th pixel in our virtual-sensor-plane. Here $x_{n}$, and $y_{n}$
correspond to the $x$ and $y$ positions of the $n$-th pixel on the sensor-
plane. Fifth, we compute the distance

$\displaystyle o_{n,k,g}$ $\displaystyle=$
$\displaystyle|\vec{i}_{k,g}-\vec{p}_{n,g}|_{2}$ (9.38)

between the pixel-position $\vec{p}_{n,g}$ and the intersection-position
$\vec{i}_{k,g}$ in the sensor-plane. Finally, we calculate the matrix-elements

$\displaystyle u_{n,\,k}(g)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}1&\text{if}\,\,o_{n,k,g}\leq\text{pixel-
radius}\\\ 0&\text{else}\\\ \end{array}\right.{}$ (9.41)

by comparing the distance $o_{n,k,g}$ with e.g. the pixel-radius to decide if
the $k$-th lixel is participating to the photon-intensity in the $n$-th pixel.
In Figure 9.1 we see a visual representation of the matrix-elements for the
$n$-th pixel in Portal when we do not refocus. Now we can visualize the
matrix-elements for the $n$-th pixel in Portal when we refocus in post to
different object-distances $g$, see Figure 9.4. In Portal, the default sensor-
distance $d$ is equal to the focal-length $f$, such that the default object-
distance where Portal focuses on is in $g=\infty$. 111In Section 12.6, we
discuss possible benefits for the Chrenkov-plenoscope’s trigger when setting
the default object-distance not to infinity, but to the air-shower-maximum.
Find in Figure 9.4, that the refocusing to infinity yields exactly the same
matrix-elements we obtain in the case for imaging without refocusing in Figure
9.1.
Figure 9.6 shows the interplay of neighboring pixels in the resulting
refocused image. We color the photo-sensors according to the seven neighboring
pixels they will be added up into in the refocused image. We find that indeed
all the photo-sensors contribute to the resulting refocused images. In the
Figures 9.4, 9.5, and 9.6 we find that the patterns of photo-sensors is not
perfectly symmetric in all refocusing scenarios. Due to the binary decision
made in Equation 9.41, already small statistical fluctuations in the
estimation of the light-field-geometry $G$ can cause asymmetric patterns in
this first implementation.

Figure 9.4: Continues in Figure 9.5. Part 1 of 2. Black photo-sensors (lixels)
are summed up to synthesize a single pixel focused to different object-
distances. The object-distance focused to is written on the right, where the
rays converge. Here we see the central pixel $n=4,221$ on Portal.

Figure 9.5:  Continuation of Figure 9.4. Part 2 of 2.

Figure 9.6: Same as Figure 9.4, but including the neighboring pixels. Central,
black pixel id is $4,221$, green is $4,124$, blue is $4,222$ is, red is
$4,220$ is, cyan is $4,125$ is, magenta is $4,317$ is, and orange is $4,318$
is.

####  9.5 Synthetic apertures

We interpret the plenoscope’s light-field-sequence as a set of multiple image-
sequences observed with an array of telescopes. In Chapter 6 we demonstrate
this on the example of Portal in the Figures 6.5, 6.9, 6.13, 6.17, and 6.21.
Arrays of Cherenkov-telescopes observe gamma-rays successfully and
efficiently. Thus splitting the light-field-sequence into multiple image-
sequences might be a first step to approach the light-field and the
plenoscope. We formulate the synthetic creation of telescope apertures as a
projection of the light-field

$\displaystyle\mathcal{I}_{m}[n]$ $\displaystyle=$ $\displaystyle
U^{m}[n,\,k]\cdot\mathcal{L}[k].$ (9.42)

On Portal, we can slice the light-field-sequence $\mathcal{L}[n,\,m,\,t]$
along its $M=61{}$ paxels to obtain the $M=61{}$ image-sequences
$\mathcal{I}_{m}[n,\,t]$ observed by $M=61{}$ synthetic telescopes

$\displaystyle u^{m^{\prime}}_{n^{\prime},\,k}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}1&\text{if}\,\,n^{\prime}=n\,\,\text{and}\,\,m^{\prime}=m\\\
0&\text{else}\\\ \end{array}\right.{}.$ (9.45)

Here we used the Equations 8.3, and 8.4 to compute $n$, and $m$ from $k$.
In general we calculate the matrix-elements $u^{m}_{n,\,k}$ based on the
support-positions of the rays defined in the light-field-geometry $G$. First,
we calculate the distance

$\displaystyle o_{k,\,m}$ $\displaystyle=$
$\displaystyle\sqrt{(\overline{x}_{k}-x_{m})^{2}+(\overline{y}_{k}-y_{m})^{2}}$
(9.46)

between the support-position of the $k$-th ray on the principal-aperture-plane
at $\overline{x}_{k}$, $\overline{y}_{k}$ and the center of the $m$-th
synthesized telescope-aperture $x_{m}$, $y_{m}$. Second, we can compute the
matrix-elements

$\displaystyle u^{m}_{n^{\prime},\,k}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}1&\text{if}\,\,n^{\prime}=n\,\,\text{and}\,\,o_{k,\,m}\leq
r_{\text{m}}\\\ 0&\text{else}\\\ \end{array}\right.{}$ (9.49)

by comparing the distance $o_{k,\,m}$ with e.g. the radius $r_{\text{m}}$ of
the $m$-th synthesized telescope-aperture. Again we calculate $n$ from $k$
using Equation 8.3. Here we describe a synthetic aperture with the shape of a
disc with a radius $r_{\text{m}}$, but one might also use other shapes for the
synthetic apertures as e.g. hexagons, squares or triangles. In Chapter 6 we
synthesize seven large telescopes by using the mask in Figure 6.3 and Equation
9.49.

####  9.6 Light-front

The plenoscope can reconstruct the light-front of the Cherenkov-photons in the
three-dimensional space above its aperture, as it is shown in Figure 8.1. This
might be obvious as the light-front corresponds to our very definition of the
light-field-sequence in Chapter 8 and thus does not need further
interpretation, but we list it here explicitly because it opens up so many
possibilities. We use the light-front in Chapter 14 to reconstruct the
incident-direction of very low energetic $1\,$GeV cosmic gamma-rays.

###  Chapter 10 Overcoming aberrations, enlarging the field-of-view

Real imaging-optics such as lenses or imaging-reflectors suffer from
aberrations which reduce the angular resolution for imaging. The plenoscope
can overcome these aberrations when they originate from the aperture of the
imaging-optics being extended [Hanrahan and Ng, 2006]. On a perfect imaging-
optics, the absorption-position of a photon on the sensor-plane only depends
on the incident-direction $c_{x}$, $c_{y}$ of the photon relative to the
optical-axis of the imaging-system. On real imaging-optics however, the
photon’s absorption-position on the sensor-plane also depends on the support-
position $x$, $y$ where the photon passed the aperture-plane. This results in
pixels to not only collect photons with incident-directions associated with
these pixels, but to also collect photons with different incident-directions.
Here we will demonstrate how far plenoptic perception can overcome aberrations
on the example of a spherical imaging-reflector. We will briefly inspire the
drastic implications for the enlarging of the field-of-views on e.g.
Cherenkov-telescopes by about one order-of-magnitude.
Overcoming aberrations will turn out to be already included in the light-
field-geometry $G$ introduced in Chapter 8, and is closely related to the
compensation of misalignments between the imaging-reflector and the light-
field-sensor discussed in Chapter 11.

####  10.1 Using the light-field-geometry

Figure 10.1 shows the four parallel photons D, E, F, and G on a plenoscope. In
Figure 10.1 the imaging-reflector suffers from aberrations, in contrast to the
imaging-reflector shown in Figure 2.2 which is free of aberrations. The
aberrations of the imaging-reflector cause the photons D, and G to enter
different small cameras in the light-field-sensor although the four photons D,
E, F, and G have all the same incident-directions.

Figure 10.1:  Four photons D, E, F, and G with the same incident-directions on
a plenoscope. In contrast to the imaging-reflector shown in Figure 2.2, here
the imaging-reflector suffers from aberrations. To visualize the slight off-
axis incident-direction, we put a fifth photon in light gray close to the
optical-axis of the imaging-reflector. On the imaging-reflector, short lines
at the intersection-positions of the photons indicate the surface-normal of
the imaging-reflector. In Figure 10.3 we show a close-up of the light-field-
sensor.

Figure 10.2:  Close up of the light-field-sensor shown in Figure 10.1.

Figure 10.3:  The light-field-geometry of the plenoscope shown in the Figures
10.1 and 10.3 visualized in the light-field-sampling-graph, compare Figure
19.1 Chapter 19.

On a classic image-sensor, we can not resolve that the photons D, and G have
the same incident-directions as the photons E, and F. The image is blurred.
But with a light-field-sensor shown in Figure 10.1 we can resolve the true
incident-directions of all the photons if we know the light-field-geometry $G$
of our plenoscope. The light-field-geometry $G$, which is introduced in
Chapter 8, tells us for each photo-sensor in the light-field-sensor which
solid-angle of incident-directions $c_{x}$, $c_{y}$, and which area of
support-positions $x$, $y$ are covered by this photo-sensor. In the close-up
Figure 10.3 we see the two-dimensional addressing of the photo-sensors in the
light-field-sensor. Analog to Figure 2.3, we see in Figure 10.3 how the photo-
sensors sample the light-field. In contrast to the sampling of the plenoscope
free of aberrations in Figure 2.3, here we find that the sampling of the
photo-sensors is not a rectangular grid in the space of incident-directions
and support-positions but a skewed grid instead. The exact arrangement of the
photo-sensors in the space of incident-directions and support-positions is
what is stored in the light-field-geometry $G$. So the sampling of the photo-
sensors in Figure 10.3 is a visual representation of the light-field-geometry
$G$ for the plenoscope presented in Figure 10.1 which has an imaging-reflector
with aberrations. In Figure 10.3, we find that the photo-sensors (0, 0), (1,
1), (1, 2), and (2, 3) sample similar incident-directions as they are arranged
on a horizontal line in the space of incident-directions and support-
positions. So when we project the recorded light-field $\mathcal{L}$ onto an
image using the light-field-geometry $G$, we obtain an image where the effects
of the aberrations from the imaging-reflector are reduced. The projection can
be visualized in Figure 10.3 when the intensity of the photo-sensors is
projected onto the axis of incident-directions. For details on projecting the
light-field onto images see the Sections 9.2 and 9.4.

####  10.2 Example – spherical imaging-reflector

Spherical imaging-reflectors are one of the simplest to be fabricated.
Spherical imaging-reflectors are often used to have large field-of-views [Fors
et al., 2013, Cortina et al., 2016], but like other imaging-reflectors they
suffer from aberrations. We simulate the imaging on a spherical imaging-
reflector which has the same focal-length/diameter$=1.5$ ratio as Portal’s
large imaging-reflector. One after the other, we simulate the exposures of a
classic image-sensor and three different light-field-sensors. The image-sensor
and the light-field-sensor have the same pixel-sizes of $0.067^{\circ}$. The
light-field-sensors only differ in the number of paxels they use to segment
the principal-aperture-plane. We have light-field-sensors with 3, 9, and 27
paxels on the diagonal of the aperture. This simulation is in two dimensions,
and assumes that the light-field-sensors are perfect and only limited by their
discrete number of paxels.
Before we reconstruct images from the responses of the different sensors, we
estimate the light-field-geometry of each sensor, even the light-field-
geometry $G_{\text{spherical imaging-reflector + image-sensor}}$ of the
classic image-sensor. This makes the comparison fair. Now both the image of
the classic image-sensor and the images of the light-field-sensors are
corrected for distortions caused by the spherical imaging-reflector.
Distortions are when the absorption-position of a photon on the sensor-plane
is not linear proportional to the incident-direction of the photon.
Distortions can be, and usually are, corrected for when reconstructing images
from image-sensors. The four Figures 10.4, 10.5, 10.6, and 10.7 show the
images reconstructed by both the image-sensor and the light-field-sensors for
various incident-directions. Finally, in Figure 10.8 we summarize our findings
and show how good light-field-sensors can overcome aberrations of spherical
imaging-reflectors.

##### Description for the Figures 10.4, 10.5, 10.6, and 10.7

Each figure consists of six panels. The two panels at the top show the
spherical imaging-reflector, the optical-axis, the sensor-plane, and the
trajectories of the photons. In the left panel at the top, we find an overview
of the setup, and in the right panel at the top we find a close-up on the
sensor-plane. In the left panel at the top, the close-up is indicated with a
black square. Below we have four panels showing the four one-dimensional
images reconstructed by the four different sensors. The upper most one-
dimensional image is from the image-sensor, the lower ones are from the light-
field-sensor with different numbers of paxels. The four panels showing the
one-dimensional images have a vertical, dashed line to indicate the true
incident-direction of the photons.

Figure 10.4:  $0.0^{\circ}$ incident-direction.

Figure 10.5:  $2.3^{\circ}$ incident-direction.

Figure 10.6:  $4.6^{\circ}$ incident-direction.

Figure 10.7:  $7.0^{\circ}$ incident-direction.

##### Discussion

From the reconstructed, one-dimensional images shown in the Figures 10.4,
10.5, 10.6, and 10.7, we find: First, that the spread of the reconstructed
incident-directions of the photons is reduced when we use light-field-sensors
instead of image-sensors. Second, that the spread of the reconstructed
incident-directions of the photons becomes smaller the more paxels are used in
the light-field-sensor. And third, we find that the asymmetry of the
reconstructed incident-directions of the photons is reduced on the light-
field-sensors. The binning-artifacts in the reconstructed images seen in
Figure 10.7 do not significantly effect our findings presented here.
In Figure 10.8, we find that even light-field-sensors with low numbers of
paxels provide a strong reduction of the effect of aberrations on imaging.
From a geometric point-of-view it is only natural, that when the number of
paxels is increased further, aberrations of imaging-optics are overcome
completely. This might become more clear when we describe the light-field-
geometry $G$ with other words: In the case of the plenoscope composed from an
light-field-sensor and an imaging-reflector, the light-field-geometry $G$
describes the surface-normal of the aperture, see the small surface-normals
indicated in Figure 10.1. So when we have more paxels in the light-field-
sensor, we have a more dense sampling of the aperture’s surface-normals, and
so we can correct better for aberrations which originate from these surface-
normals.

Figure 10.8:  The spread of the reconstructed incident-directions of photons
versus the true incident-direction of the photons. The number of paxels here
is the number of paxels on the diagonal of the aperture. For comparison, our
Portal Cherenkov-plenoscope has $9$ paxels across the diagonal of its large
imaging-reflector.

####  10.3 Enlarging the field-of-view

In Figure 10.8 we find that the spread of the reconstructed incident-
directions is much reduced and becomes more constant over the range of true
incident-directions when we use light-field-sensors instead of image-sensors.
Of course more detailed studies are needed, but Figure 10.8 suggests that a
Cherenkov-plenoscope could be build with an enormous $16^{\circ}$ diagonal
field-of-view. A $201\,$deg2 field-of-view would be an improvement of more
than one order-of-magnitude over the field-of-views on todays Cherenkov-
telescopes with about $16\,$deg2, see Figure 5.12. And it would still be an
improvement by a factor of $4$ over the planned Schwarzschild-Couder-
Cherenkov-Telescopes which are dedicated for wide field-of-views [Acharya et
al., 2013] and also reach angular resolutions in and below the $0.1^{\circ}$
regime.
The bottom-line is that Cherenkov-plenoscopes become interesting independent
of their benefits for observations of low energetic gamma-rays, but because
they can enlarge the field-of-view which we currently can not achieve by other
means.

####  10.4 Upgrading the Cherenkov-telescope

We suggest to equip the Cherenkov-telescope111In general. This includes all
existing and future Cherenkov-telescopes. with a light-field-sensor not to
gain plenoptic perception, but to improve image-quality. With imaging-
reflectors with diameters below $\approx 23\,$m, there is not much plenoptic
perception-power for air-showers to be gained anyhow, compare Chapter 6.
However, the size of the field-of-view, the sharpness of the images, and
especially the consistency of the images across the field-of-view would be
improved dramatically. In case one fears the costs for additional read-out-
channels in a light-field-sensors, we propose a way to mutilate the light-
field-sensor in order to reduce the number of read-out-channels to the same
number needed for image-sensors, while still profiting from corrections for
imaging provided by plenoptic perception. We can hardwire the light-field-
geometry $G$ into the light-field-sensor to sum up the photo-sensors which
belong to a reconstructed pixel before these photo-sensors are fed into
individual read-out-channels. Projecting the light-field onto an image is
about summing up the correct photo-sensors as we show in the Sections 9.2, and
9.4. To be clear: The number of individual photo-sensors will increase to the
number of lixels, but the number of read-out-channels for digitization will
stay the lower number of pixels. A Cherenkov-telescope equipped with such a
mutilated light-field-sensor would have the optics of a plenoscope, but the
read-out of a telescope. The resulting Cherenkov-telescope would not record
light-fields, but it could record images with much larger field-of-views, and
much reduced aberrations.

####  10.5 Overcoming aberrations in Portal

The imaging-reflector of Portal is not free of aberrations. We can visualize
this aberrations using the light-field-geometry of Portal. When we describe
the projection of the light-field onto images as a matrix-multiplication, as
we do in Section 9.2, we can visualize the columns of the imaging-matrix
$U_{\text{imaging}}[n,\,k]$ just as we do in Figure 9.1. The aberrations
should show up as asymmetric patterns in the visualizations of the columns of
the imaging-matrix $U_{\text{imaging}}[n,\,k]$. Now in the Figures 9.1, 9.4,
and 9.5, we hardly see any hint for this asymmetry because the columns
visualized in these figures correspond to the central pixel $n=4,221$ in the
field-of-view. For the central pixel, the aberrations induced by the imaging-
reflector of Portal are only minor. However, aberrations can be seen for the
outer regions in the field-of-view. Here we visualize six of these matrix-
columns for six pixels ranging from the center of the field-of-view at
$0.0^{\circ}$ to the outer perimeter of the filed-of-view at $3.0^{\circ}$. We
compute the imaging-matrix $U_{\text{imaging}}[n,\,k]$ according to Equation
9.18. Table 10.1 shows the incident-directions of the six different pixels.
Figure 10.9 shows the photo-sensors participating to the six pixels on the
photo-sensor-plane of the light-field-sensor in Portal. The Figures 10.9, and
10.10 do not show the point-spread-function of the imaging-reflector, but they
visualize which photo-sensors are summed to synthesize a pixel because their
corresponding lixels have near-by incident-directions as the pixel. Based on
the light-field-geometry for Portal, we can overcome a large part of the
aberrations induced by the large imaging-reflector. In Chapter 11,
reconstructed images of a phantom-source observed with the Portal plenoscope,
and a telescope are shown side-by-side. The reduction of aberrations can be
seen in the reconstructed images recorded by the plenoscope, in contrast to
the images reconstructed from the image-sensor of the telescope.

$n$ |  $c_{x}$/deg |  $c_{y}$/deg |  $\sqrt{{c_{x}}^{2}+{c_{y}}^{2}}$/deg |  $x$/m |  $y$/m
---|---|---|---|---|---
4,221 | 0.00 | 0.00 | 0.0 | -0.00 | 0.00
3,541 | 0.42 | 0.42 | 0.6 | -0.79 | -0.81
2,779 | 0.85 | 0.85 | 1.2 | -1.61 | -1.60
2,135 | 1.27 | 1.27 | 1.8 | -2.41 | -2.41
1,448 | 1.70 | 1.70 | 2.4 | -3.22 | -3.22
902 | 2.12 | 2.12 | 3.0 | -4.01 | -4.01
Table 10.1:  The six example pixels with indexes $n$ represent various
incident-directions $c_{x}$, and $c_{y}$. Here $x$ and $y$ are the average
positions of the corresponding photo-sensors in the photo-sensor-plane, see
Figure 10.9.  Figure 10.9:  Black photo-sensors are summed to synthesize the
six pixels with incident-directions $0.0^{\circ}$, $0.6^{\circ}$,
$1.2^{\circ}$, $1.8^{\circ}$, $2.4^{\circ}$, and $3.0^{\circ}$. The black
boxes indicate the range of the close-ups shown in Figure 10.10.

Figure 10.10:  Black photo-sensors are summed to synthesize pixels in the
image because their corresponding lixels have incident-directions close to
these pixels. Figure 10.9 shows the summation of all the six different pixels
at once. Upper, left shows $0.0^{\circ}$, upper, right shows $0.6^{\circ}$,
middle, left shows $1.2^{\circ}$, middle, right shows $1.8^{\circ}$, lower,
left shows $2.4^{\circ}$, and lower, right shows $3.0^{\circ}$.

###  Chapter 11 Compensating misalignments between the light-field-sensor and
the imaging-reflector

On telescopes, the alignment of the image-sensor with respect to the imaging-
optics has a strong effect on image-quality. Building telescope-mounts and
support-structures which keep the optics aligned becomes an expensive
challenge on larger telescopes. However, if a light-field-sensor is used
instead of an image-sensor, and if the actual alignment of the light-field-
sensor with respect to the imaging-optics is known, the quality of the images
reconstructed from the light-field is little to none effected by
misalignments.
Compensating the misalignments of the light-field-sensor is a key-feature of
the Portal Cherenkov-plenoscope. It allows us to mechanically separate the
light-field-sensor from the large imaging-reflector in order to defer the
physical limit of the square-cube-law, and to build larger apertures, see
Chapter 16. We show that the compensation of misalignments is already included
in the light-field-geometry $G$ which we introduce in Chapter 8, and is
closely related to the overcoming of aberrations discussed in Chapter 10.
First, we organize the misalignments into four components and discuss their
effects on imaging. Second, we show how the light-field-geometry $G$ can be
used to compensate such misalignments. And third, we simulate the imaging of
the Portal Cherenkov-plenoscope and a same sized Cherenkov-telescope when both
observe a three-dimensional phantom-source located up in the atmosphere while
the sensor-planes of both the telescope and the plenoscope are misaligned in
various ways.

####  11.1 Organizing components of misalignments

We organize the misalignments between the sensor-plane and the imaging-optics
into four components listed below. We use the frame of the imaging-optics as a
reference, and translate and rotate only the image-sensor.

#####  Translations parallel to the optical-axis

Figure 11.1:  Translations parallel to the optical-axis. The photons
$\overline{ea}$, $\overline{ea^{\prime}}$, and $\overline{ea^{\prime\prime}}$
only converge in a single point on the image-sensor when the sensor-plane-
distance $d$ is equal to the image-distance $b$. But when the image-sensor is
translated away from its target-position, the absorption-positions $i$,
$i^{\prime}$, and $i^{\prime\prime}$ do not converge and the image of $e$ is
blurred.

When we translate the sensor-plane parallel to the optical-axis of the
imaging-optics, see Figure 11.1, we change the sensor-plane-distance $d$
introduced with the thin-lens in Figure 6.2. A parallel translation
corresponds to focusing the image to different object-distances. On Cherenkov-
telescopes we have to choose one optimal object-distance to focus on [Hofmann,
2001, Trichard et al., 2015] before the image of the air-shower is recorded.
On image-senors, we can not correct for parallel translations, not even when
we know the actual parallel translation. Image-sensors do not measure the
trajectories $\overline{ia}$, $\overline{i^{\prime}a^{\prime}}$, and
$\overline{i^{\prime\prime}a^{\prime\prime}}$, but only the absorption-
positions $i$, $i^{\prime}$, and $i^{\prime\prime}$. However, light-field-
sensors on the other hand do measure the trajectories $\overline{ia}$,
$\overline{i^{\prime}a^{\prime}}$, and
$\overline{i^{\prime\prime}a^{\prime\prime}}$ and thus can reconstruct the
intersection-positions of such trajectories on a virtual sensor-plane to
restore the image. We discuss the restoring of images focused to any object-
distance in Section 9.4.
We estimate the acceptable tolerances for parallel translations before the
effect shows up in the reconstructed images for a Cherenkov-telescope of the
same size as the Portal Cherenkov-plenoscope. When we target to focus on an
object-distance $g_{\text{target}}=10.0\,$km, our target-sensor-plane-distance
is $d_{\text{target}}=107.64\,$m according to the Thin-lens-equation 6.1.
According to the Depth-of-field-equation 6.2 the depth-of-field spans from
$g_{-}=9.18\,$km to $g_{+}=10.8\,$km. The corresponding sensor-plane-distances
to focus exactly on the start and end of the depth-of-field are
$d_{-}=107.75\,$m, and $d_{+}=107.56\,$m respectively. We identify the
difference between our targeted sensor-plane-distance $d_{\text{target}}$ and
the sensor-plane-distances which would move our focus beyond our depth-of-
field $d_{\pm}$ as the acceptable tolerance for parallel translations

$\displaystyle\Delta_{\parallel}^{\text{trans}}$ $\displaystyle=$
$\displaystyle d_{\text{target}}-d_{\pm}$ (11.1) $\displaystyle d_{\pm}$
$\displaystyle=$ $\displaystyle\frac{1}{\frac{1}{f}-\frac{1}{g_{\pm}}}.$
(11.2)

The sensor-plane must not be translated further than $+104\,$mm and $-88\,$mm
parallel to the optical-axis before the image differs recognizably. In units
of the focal-length this tolerance is $8.5\times 10^{-4}\,f$.

#####  Translations perpendicular to the optical-axis

Figure 11.2:  Translations perpendicular to the optical-axis. The photons
$\overline{ea}$, $\overline{ea^{\prime}}$, and $\overline{ea^{\prime\prime}}$
will still converge in the absorption-points $i=i^{\prime}=i^{\prime\prime}$.
Although the photons enter different photo-sensors when the image-sensor is
translated perpendicular to the optical-axis, they still can be assigned to
the pixel corresponding to their incident-directions.

When we translate the sensor-plane perpendicular to the optical-axis of the
imaging-optics, see Figure 11.2, the image will simply show a different region
of incident-directions. As long as the actual perpendicular translation is
known, and the field-of-view is large enough so that our region-of-interest
does not drop out of it, then translations perpendicular to the optical-axis
do not effect the image-quality. On real imaging-optics unfortunately,
distortions and aberrations dilute the angular resolution the larger the
incident-direction of the photons becomes. So on real telescopes, we expect
that translations perpendicular to the optical-axis have an effect on the
image-quality which becomes larger, the larger the perpendicular translation
becomes. However, with light-field-sensors on the other hand we can even
compensate the increasing aberrations of the imaging-optics, see Chapter 10,
so that the quality of the images reconstructed from light-field-sensors is
not effected by perpendicular translations.
If we would not correct for perpendicular translations, a Cherenkov-telescope
of the same size of Portal could tolerate perpendicular translations up to
about the extend $p$ of the projection of a pixel onto the sensor-plane

$\displaystyle\Delta_{\perp}^{\text{trans}}$ $\displaystyle=$
$\displaystyle\pm\frac{p}{2},$ (11.3)

which gives $\Delta_{\perp}^{\text{trans}}\approx\pm 63\,$mm, and is related
to $\Delta_{\parallel}^{\text{trans}}$ by a factor of $f/D$. In units of the
focal-length, this tolerance is $5.8\times 10^{-4}\,f$.

#####  Rotations parallel to the optical-axis

Figure 11.3:  Rotations parallel to the optical-axis. All photons coming from
$e$ still converge in one absorption-position $i=i^{\prime}=i^{\prime\prime}$.
Just as for perpendicular translations, shown in Figure 11.2, the photons
might be absorbed by a different photo-sensor now, but they still can be
assigned to the pixel corresponding to their incident-directions.

When we rotate the image-sensor parallel to the optical-axis of the imaging-
optics, see Figure 11.3, we simply end up with an image rotated around its
center. We can fully compensate rotations parallel to the optical-axis in post
when we know the actual parallel rotation.
Again we can estimate the tolerance for a parallel rotation angle

$\displaystyle\Delta_{\parallel}^{\text{rot}}$ $\displaystyle=$
$\displaystyle\frac{\Delta_{\perp}^{\text{trans}}}{f\tan(\theta_{\text{FoV}}/2)}$
(11.4)

when we would not correct for it in post. We use the acceptable tolerance for
perpendicular translations $\Delta_{\perp}^{\text{trans}}$ and divide it by
the radius of the sensor-plane. For the example of the Cherenkov-telescope of
the same size as Portal we get $\Delta_{\parallel}^{\text{rot}}=0.59^{\circ}$.

#####  Rotations perpendicular to the optical-axis

Figure 11.4:  Rotations perpendicular to the optical-axis. The absorption-
positions $i$, $i^{\prime}$, and $i^{\prime\prime}$ are effected in a similar
way as shown in the component for parallel translations, see Figure 11.1.

When we rotate the sensor-plane perpendicular to the optical-axis of the
imaging-optics, one region on the sensor-plane will be closer to the imaging-
optics than the opposite region on the sensor-plane, see Figure 11.4. The two
regions effectively have different sensor-plane-distances, what corresponds to
the two regions being focused to different object-distances. So we expect
similar effects on the image-quality as for translations parallel to the
optical-axis, but more complicated since the image is not evenly out-of-focus,
but differently for different regions. Just as for the parallel translations,
on image-sensors we can not compensate the effects of this perpendicular
rotations in post, not even when we know the actual perpendicular rotation.
Again, this is because an image-sensor does not measure the trajectories
$\overline{ia}$, $\overline{i^{\prime}a^{\prime}}$, and
$\overline{i^{\prime\prime}a^{\prime\prime}}$, but only the absorption-
positions $i$, $i^{\prime}$, and $i^{\prime\prime}$. For the same argument we
pointed out for parallel translations, a light-field-sensor can compensate
perpendicular rotations when the rotation is known.
We can estimate the tolerance for perpendicular rotation angles

$\displaystyle\Delta_{\perp}^{\text{rot}}$ $\displaystyle=$
$\displaystyle\frac{\Delta_{\parallel}^{\text{trans}}}{f\tan(\theta_{\text{FoV}}/2)}$
(11.5)

based on the tolerance for parallel translations
$\Delta_{\parallel}^{\text{trans}}$ and the radius of the sensor-plane. This
gives us a tolerance for perpendicular rotations of $\pm 0.83^{\circ}$ for a
Cherenkov-telescope with the size of Portal before the image differs
recognizably.

##### Summarizing misalignment-components

Based on the Figures 11.1, 11.2, 11.3, and 11.4, we summarize our findings in
Table 11.1. However, Table 11.1 addresses perfect light-field-sensors. As we
will see next, real light-field-sensors are limited in resolution or in the
acceptance-angles for incoming photons.

misalignment- | acceptable tolerances | can be compensated on:
---|---|---
component | without compensation | image-sensor | light-field-sensor
|  $\Delta_{\parallel}^{\text{trans}}=-88\,$mm to $+104\,$mm | No | Yes
|  $\Delta_{\perp}^{\text{trans}}=\pm 63\,$mm | Yes, partly | Yes
| $\Delta_{\parallel}^{\text{rot}}=\pm 0.59^{\circ}$ | Yes | Yes
| $\Delta_{\perp}^{\text{rot}}=\pm 0.83^{\circ}$ | No | Yes
Table 11.1:  Summary of the four misalignment-components. The acceptable
tolerances are estimated for a Cherenkov-telescope with the size of Portal.
The restriction ’partly’ here refers to aberrations caused by real imaging-
optics.

####  11.2 Using the light-field-geometry

Figure 11.5 shows a plenoscope observing the three photons A, B, and C while
its light-field-sensor is misaligned with respect to the imaging-reflector.
Despite the misalignment, the three photons are absorbed by the photo-sensors
inside the light-field-sensor, see Figure 11.7. When we know the actual
alignment $\hat{T}_{\text{actual}}$ between the light-field-sensor and the
imaging-reflector, we can estimate the specific light-field-geometry
$G(\hat{T}_{\text{actual}})$ for the whole misaligned plenoscope, see Figure
11.7. Although the photons A, and C have the same incident-directions, they
enter different small cameras in the light-field-sensor. But using the actual
light-field-geometry $G(\hat{T}_{\text{actual}})$, we can still reconstruct
the actual incident-directions of A, and C. In the Figures 11.7, and 11.7 we
also show the target-alignment $\hat{T}_{\text{target}}$ of the light-field-
sensor in light gray. We find that in Figure 11.7, the black Photons A, B, and
C which are detected by the misaligned light-field-sensor are close to their
light gray counterparts which are detected by the light-field-sensor in its
target-alignment.
From our thoughts summarized in Table 11.1, we already concluded that the
superior knowledge of the light-field-sensor about the trajectories of the
photons will allow it to compensate misalignments. In Figure 11.7 we see how
this superior knowledge shows up as a distortion of the light-field-geometry
$G(\hat{T}_{\text{actual}})$. But from the plenoscope shown in the Figures
11.5, and 11.7 we also see that real light-field-sensors have limited
resolutions. They have a limited number of small cameras, and a limited number
of photo-sensors inside the small cameras. In the next Section 11.3 we take a
look at the actual compensation-power for misalignments on the Portal
Cherenkov-plenoscope which has such limited resolutions as it is meant to be
fabricated in a cost-effective way.

Figure 11.5:  The same three photons A, B, and C from Figure 2.2, but this
time the light-field-sensor is misaligned in $\hat{T}_{\text{actual}}$ with
respect to the imaging-reflector. In bright gray we indicate the target-
alignment $\hat{T}_{\text{target}}$ of the light-field-sensor. Figure 11.7
shows a close-up of the light-field-sensor.

Figure 11.6:  Close-up of Figure 11.5. Figure 11.7 shows the resulting light-
field-geometry $G(\hat{T}_{\text{actual}})$.

Figure 11.7:  A visual representation of the light-field-geometry
$G(\hat{T}_{\text{actual}})$ corresponding to the misalignment
$\hat{T}_{\text{actual}}$ shown in Figures 11.5, and 11.7.

####  11.3 Imaging a phantom-source with Portal

To get a qualitative feeling for the effects of misalignments on imaging, and
to get a qualitative feeling for the compensation-power for misalignments of
Portal’s light-field-sensor, we reconstruct images of a phantom-source. The
phantom-source has a well defined shape in the three-dimensional space, and is
described in the Figures 11.9, and 11.9. The phantom-source extends from an
object-distance of $2.5\,$km up to $20.0\,$km, and all its photons reach the
principal-aperture-plane in a $1\,$ns time-window, similar to Cherenkov-
photons produced in air-showers. There are no night-sky-background-photons,
but only the photons from the phantom-source. In our simulation we set up the
Portal Cherenkov-plenoscope and a Cherenkov-telescope of the same size as
Portal. The plenoscope records a light-field-sequence over a period of
$50\,$ns, and the telescope records an image-sequence also over a period of
$50\,$ns. Both instruments take $2\times 10^{9}$ samples per second. However,
for the static flash of photons from the phantom-source we just look at the
integrated light-field and image respectively. Both instruments have their
sensor-plane-distances set to $d_{\text{target}}=107.65\,$m to focus to an
object-distance of $g_{\text{target}}=10\,$km. The images reconstructed from
the telescope are corrected as much as possible for the given situation, e.g.
they are corrected for perpendicular translations, parallel rotations, and
distortions caused by the imaging-reflector. The images reconstructed from the
light-field from the plenoscope are in addition also corrected for the
remaining components of misalignments such as parallel translations and
perpendicular rotations. One might argue that with a target object-distance of
$g_{\text{target}}=10\,$km, the Cherenkov-telescope can never produce a fully
sharp image of any layer in the phantom-source, since there is no layer
exactly at this object-distance, and thus a side-by-side comparison of the
telescope and the plenoscope is not fair. Yes, it is not fair, but not because
we designed the phantom-source this way, but because the plenoscope is the
superior instrument. The purpose of these instruments is to record air-
showers, and just as air-showers, our phantom-source spans a wide range of
object-distances which simply exceeds the depth-of-field of a Cherenkov-
telescope of the size of Portal, see Chapter 6. Focuses on different object-
distances of the air-shower offer different advantages for the reconstruction
of different properties of the air-shower [Hofmann, 2001], and thus any focus
chosen on a Cherenkov-telescope of the size of Portal ends up to be an
inferior compromise. Actually some of the misalignments will by chance even
improve the imaging on the telescope.

##### What we expect to see in the images

  1. 1.

We expect the images reconstructed from the light-field recorded by the
plenoscope to always look the same regardless of the actual scenario of
misalignment of the light-field-sensor.

  2. 2.

We expect the plenoscope to be able to project its light-field onto images
focused to different object-distances to e.g. find the sharpest response for
the spiral, the sun-symbol, and the smiley in the phantom-source.

  3. 3.

We expect the images from the telescope to be only one single image for each
scenario of misalignment, and not multiple images taken with different
focuses. This might be obvious, but we want to remind ourselves that like the
plenoscope, the telescope has only one single chance to record the air-shower.
The telescope has to choose one focus which it can not optimize in advance for
the individual air-shower.

##### Scenarios of misalignments

We set up four different scenarios of misalignments, see Table 11.2. First, we
look at images of the phantom-source when the light-field-sensor is in its
target-alignment for an object-distance of $10\,$km. Second, we look at images
taken with three different perpendicular rotations of up to $8.31^{\circ}$.
Third, we look at images taken with four different parallel translations from
$-877\,$mm to $+1,035\,$mm. And fourth, we look at images taken when all
components of misalignment come together.

$\includegraphics[angle={90},width=33.04076pt]{figures_D_compensating_misalignments_D_symbol_trans_para.png}{}/$mm |  $\includegraphics[angle={90},width=33.04076pt]{figures_D_compensating_misalignments_D_symbol_trans_perp.png}{}/$mm |  $\includegraphics[angle={90},width=33.04076pt]{figures_D_compensating_misalignments_D_symbol_rot_para.png}{}/$deg |  $\includegraphics[angle={90},width=33.04076pt]{figures_D_compensating_misalignments_D_symbol_rot_perp.png}{}/$deg | Figures
---|---|---|---|---
0 | 0 | 0 | 0 |  11.10, 11.11, 11.12
0 | 0 | 0 | 2.70 | 11.13
0 | 0 | 0 | 5.54 | 11.14
0 | 0 | 0 | 8.31 | 11.15
-877 | 0 | 0 | 0 | 11.16
-240 | 0 | 0 | 0 | 11.17
+398 | 0 | 0 | 0 | 11.18
+1,035 | 0 | 0 | 0 | 11.19
+1,200 | 800 | 17.00 | 8.00 | 11.20
Table 11.2:  Overview of the simulated scenarios of misalignments. The
magnitude of the misalignments is up to one order-of-magnitude above the
acceptable tolerances listed in Table 11.1. We concentrate on the components
of misalignment which can not be compensated by image-sensors.

##### Description for the Figures 11.10 to 11.20

The Figures 11.10 to 11.20 show images of the phantom-source. The top of the
figure shows the type and magnitude of the misalignment, compare with Table
11.2. The figures are split into two columns. The left column named
’Telescope’, shows the single image recorded by the Cherenkov-telescope. The
right column named ’Plenoscope’ shows three images projected from the light-
field recorded by the Portal Cherenkov-plenoscope. The three reconstructed
images are focused such that the spiral, the sun-symbol and the smiley are
sharp in the image. The choice of three images here is arbitrary and could
always be extended to a more detailed series of refocused images like the
series shown in the Figures 11.11, and 11.12. Always keep in mind that the
images $\mathcal{I}[c_{x},c_{y}]$ we look at are only incomplete projections
of the light-field $\mathcal{L}[c_{x},c_{y},x,y]$, and that the only reason
for doing so is to allow our limited visual perception to get an impression.

##### Discussing the reconstructed images

When we look at the images in the right plenoscope-column of the Figures
11.10, to 11.20, we find that they all look remarkably similar despite the
different scenarios of strong misalignments. Even misalignments which exceed
the acceptable tolerances for image-sensors by one order-of-magnitude are
apparently not an issue for the light-field-sensor in Portal. The spiral, the
sun-symbol, and the smiley can in all scenarios be focused on sharply. Neither
the sun-symbol nor the spiral show signs of asymmetries or aberrations, in
contrast to the images of the telescope in the left columns. This is because
the images reconstructed from the light-field are naturally also corrected for
aberrations, see Chapter 10. In fact, the images look so alike that it is hard
to belief they were reconstructed from light-fields recorded in scenarios of
different misalignments. But when we take a close look on the scenario of
strong perpendicular rotations of $8.31^{\circ}$ along the $y$-axis in Figure
11.15, we find that the hexagonal pixels in the images have small gaps in
between them for negative incident-directions in $c_{x}$, and are compressed
for positive incident-directions in $c_{x}$. This irregular grid of the pixels
is caused by the rotation of the sensor-plane. Also when we look at the
composition of several misalignments in Figure 11.20, we see that the
reconstructed images are from a light-field recorded during this specific
misalignment as we see that the sun-symbol is not fully included anymore in
the field-of-view. Also the grid of pixels is rotated in Figure 11.20 to
compensate the parallel rotation. In all reconstructed images from the light-
field we can see that it is possible to nail the focus on one particular part
of the phantom-source. In all the refocused images focused on the smiley we
can clearly see its mouth and eyes. In all the refocused images focused on the
sun-symbol we can clearly see the flares of the sun-symbol and even see the
gap between the flares and the ring. And in all the refocused images focused
on the spiral, the spiral is evenly glowing, the parts of the spiral close to
the center of the image are as sharp as the parts of the spiral in the outer
realm of the image. There are no signs of aberrations.
Not so for the images of the telescope. All images of the telescope show
asymmetries in sharpness across the field-of-view. Already in the target-
alignment shown in Figure 11.10, the mouth and the eyes of the smiley are
washed out and the spiral is almost gone. The flares of the sun-symbol are
hardly recognizable. During the perpendicular rotation along the $y$-axis in
the Figures 11.13, to 11.15, the sensor-plane happens to reach sensor-plane-
distances which favor the imaging of the smiley and the sun-symbol.
The bottom-line here is that although the images from the telescope are not
completely useless during misalignments, the images reconstructed from the
light-field of the plenoscope simply nail the focus in every scenario.
However, the serious argument for the plenoscope is not that its images are
sharper, but the fact that we can reconstruct the object-distances of the
individual parts of the phantom-source. Equation 9.34 states, that for any
sensor-plane-distance of a virtual sensor-plane there is a corresponding
object-distance. The plenoscope can reconstruct the phantom-source in the
third dimension, a dimension which is beyond the perception of the telescope.
We discuss the full potential for three-dimensional reconstructions in Chapter
15 about tomography.

Figure 11.8:  The phantom-source in three dimensions consists of a triangle-
wire, a spiral-wire, and a sun-symbol-wire. The wires emit photons. At the
bottom is the principal-aperture-plane of the imaging-reflector. The symbols
scale with with the object-distance they are located in. This way, they appear
with similar sizes in the images. Note, the axis of the object-distance is
compressed, and not to scale.

Figure 11.9:  The symbols are located in different object-distances above the
aperture:
$g_{\text{triangle}}=2.5\,$km
$g_{\text{spiral}}=4.2\,$km
$g_{\text{sun-symbol}}=7.1\,$km
$g_{\text{smiley}}=11.9\,$km
$g_{\text{cross}}=20.0\,$km

Target-alignment
Telescope Plenoscope image-sensor light-field-sensor

Figure 11.10:  See a detailed series of refocused images from the plenoscope
in the Figures 11.11, and 11.12.

Figure 11.11:  A series of images showing the phantom-source. The images are
projections of the light-field recorded by the Portal Cherenkov-plenoscope.
Each image is focused to a different object-distance $g$, shown in the lower,
left corner of each image. The series continues in Figure 11.12.

Figure 11.12:  Continuation of Figure 11.11.

Rotation perpendicular $y$-axis $2.77^{\circ}$
Telescope Plenoscope image-sensor light-field-sensor

Figure 11.13:  By chance, the rotation helps the imaging of the smiley and the
sun-symbol on the telescope.

Rotation perpendicular $y$-axis $5.54^{\circ}$
Telescope Plenoscope image-sensor light-field-sensor

Figure 11.14:  Again, the rotation helps with the imaging of the smiley and
the sun-symbol on the telescope, see Figure 11.13.

Rotation perpendicular $y$-axis $8.31^{\circ}$
Telescope Plenoscope image-sensor light-field-sensor

Figure 11.15:  Note the tiny white gaps between the pixels on the left half of
the images. The regular grid of pixels is distorted due to the strong
rotation.

Translation parallel $-877\,$mm
Telescope Plenoscope image-sensor light-field-sensor

Figure 11.16:  Sensor-plane-distance is $d=d_{\text{target}}-877\,$mm
$=106.77\,$m, and corresponds to a focus on an object-distance of $42.2\,$km.

Translation parallel $-240\,$mm
Telescope Plenoscope image-sensor light-field-sensor

Figure 11.17:  Sensor-plane-distance is $d=d_{\text{target}}-240\,$mm
$=107.41\,$m, and corresponds to a focus on an object-distance of $12.6\,$km.

Translation parallel $+398\,$mm
Telescope Plenoscope image-sensor light-field-sensor

Figure 11.18:  Sensor-plane-distance is $d=d_{\text{target}}+398\,$mm
$=108.04\,$m, and corresponds to a focus on an object-distance of $7.5\,$km.

Translation parallel $+1,035\,$mm
Telescope Plenoscope image-sensor light-field-sensor

Figure 11.19:  Sensor-plane-distance is $d=d_{\text{target}}+1,035\,$mm
$=108.68\,$m, and corresponds to a focus on an object-distance of $5.3\,$km.

+ + + Composition
Telescope Plenoscope image-sensor light-field-sensor

Figure 11.20:  The misalignment moves and rotates the image in the space of
incident-directions $c_{x}$, and $c_{y}$. For example, the sun-symbol is not
any longer fully included in the image.

###  Chapter 12 Triggering the Cherenkov-plenoscope

The continuous stream of sensor-responses in a Cherenkov-telescope or
plenoscope is too large to be analyzed in detail. Limited processing-resources
force us to reject large parts of the observed sensor-responses, and only
analyze a small fraction of it. The logic which decides what parts to reject,
and what parts to analyze is called the trigger. At an expected trigger-rate
of $\approx 59\times 10^{3}\,$s-1, and an exposure time of 50 ns for each
event, the trigger on Portal rejects 99.705 % of the continuously observed
light-field-sequence, and copies only 0.295 % of it to a more permanent-
storage for further reconstructions of air-showers, compare Figure 2.9. A
smart, yet simple trigger is crucial for the cost-effective detection of low
energetic cosmic gamma-rays. With current technology we can read-out all the
photo-sensors in real-time, but we can not store the resulting light-field-
sequence for further reconstructions. We can only buffer the light-field-
sequence for a short duration of $\approx 100\,$ns up to a few $1\,$us. This
is possible because all the read-out-channels can write to their own, fast but
small memory-buffer in parallel. On Cherenkov-telescopes, there exist digital,
and analog implementations of this memory-buffer. Digital implementations can
use flash-analog-to-digital-converters [Weinstein et al., 2007, Bulian et al.,
1998] followed by a digital memory-buffer. Analog implementations can use e.g.
a sequence of sample-and-hold-circuits111Also called domino-ring-sampler
[Ritt, 2008], or switched-capacitor-array, or analog-ring-sampler. [Funk et
al., 2004, Sitarek et al., 2013], or even a delay-line [Kildea et al., 2007].
All these implementations permanently overwrite the oldest part of the
recorded image-sequence with the latest one coming from the photo-sensors.
Just as in the Cherenkov-telescope, the trigger in the Cherenkov-plenoscope
has to decide whether the current light-field-sequence inside the memory-
buffer contains Cherenkov-photons from an air-shower worthwhile to analyze.
In this Chapter: First, we discuss how established triggers in Cherenkov-
telescopes make their decision based on the density of photons in image-
sequences. Second, we propose to adopt triggers acting on image-sequences for
the Cherenkov-plenoscope, but we propose to implement multiple of such image-
sequences refocused to different object-distances. Third, we motivate the
trigger-threshold chosen for the simulations of Portal, and show the
performance of the simulated trigger.

####  12.1 Triggering Cherenkov-telescopes

Two methods are used to make the trigger in a Cherenkov-telescope more
sensitive to air-showers, and less sensitive to fluctuations of the night-sky-
background-photons. First, triggers look for a high density of photons in
time. The compact $\approx 10\,$ns arrival of Cherenkov-photons on ground
clearly separates these from the steady arrival of night-sky-background-
photons. Second, triggers look for spatial patterns in the image-sequences.
Cherenkov-photons are likely to populate nearby pixels in the image-sequences
because their emission-positions in the air-shower populate a rather compact
volume. Both methods are often used in parallel or in stages to make the final
trigger-decision. However, since the atmospheric Cherenkov-method approaches
the regime of single-photons, there is an additional aspect relevant for the
trigger-decision. The observable fluctuations in the arrival of night-sky-
background-photons can be large because the pixels sample only small solid-
angles, and integrate only over short periods of $\approx 5\,$ns. This
limitation is known as the Law of Large Numbers [Poisson, 1837]. Because of
this, the threshold for the trigger has to be larger then the anticipated
threshold deduced from the average flux of the night-sky-background-photons.
Otherwise the trigger will store too many image-sequences only containing
fluctuations of the night-sky-background. The larger the number of photons
used for the trigger-decision becomes, the smaller becomes the relative
statistical fluctuation of this number. Therefore, some Cherenkov-telescopes
sum up multiple read-out-channels before comparing the result to a threshold
[Delagnes et al., 2006, Vogler et al., 2011, Garcia et al., 2014]. This is
called a sum-trigger. Other Cherenkov-telescopes do compare the density of
photons on individual pixels, but then demand compact regions of triggered
pixels in a second stage [Bulian et al., 1998, Weinstein et al., 2007, Kildea
et al., 2007] to counteract the fluctuation of the night-sky-background-
photons. With these methods, air-showers induced by lower energetic particles
are more likely to be recorded while fluctuations of the night-sky-background-
photons are more likely to be rejected.

Figure 12.1: The differential flux of night-sky-background-photons. The solid
line is used in the simulations of Portal. We assume that the flux of night-
sky-background-photons at the observatory on Roque de los Muchachos on Canary
island La Palma, Spain is representative for most sites where Portal might be
build. Numerical values of solid line are taken from [Gaug et al., 2013],
which is based on [Benn and Ellison, 1998]. For comparison, the dashed line is
taken from [Preuss et al., 2002].

####  12.2 A light-field-trigger for Portal

For this study of the Portal Cherenkov-plenoscope, we adopt aspects of both
trigger-implementations on Cherenkov-telescopes and create a trigger with two
stages. In the first stage, we adopt the summation of the established sum-
trigger to reduce the effect of statistical fluctuations as early as possible.
We sum up $\approx 427$ lixels each into large, and overlapping trigger-
pixels. These trigger-pixels form a trigger-image-sequence. The intensity in
these trigger-pixels is then integrated continuously over a period of 5 ns
where the integral is compared with a predetermined trigger-threshold.
In a second stage, we adopt the search for coincident patterns and thus demand
that at least two neighboring trigger-pixels must exceed their predetermined
threshold in order to trigger the read-out.
For this, the light-field-sequence needs to be projected onto an trigger-
image-sequence in real-time. This projection can be written as a matrix-
multiplication, see Equation 9.32 in Sections 9.2, and 9.4. Each trigger-pixel
in the resulting image-sequence is computed from the sum of corresponding
lixels (read-out-channels) in the light-field-sequence. This means, that we
can hardwire a real-time-projection of the light-field-sequence into the
housing of the light-field-sensor itself using e.g. operational amplifiers to
sum up the analog output-potentials of the corresponding photo-sensors. This
is for the same reasons as discussed in Section 10.4. From the implementation
point-of-view, the sum-trigger for the Cherenkov-plenoscope is the same as the
sum-trigger for the Cherenkov-telescope. Both are summing up certain read-out-
channels into trigger-pixels. In the simplest case, this summation into a
trigger-pixel can look like e.g. the patterns shown in the Figure 12.2.

Figure 12.2:  A possible trigger-pixel for the center of the field-of-view.
Focus is set to infinity. A total of 427 black photo-sensors are summed up to
form a trigger-pixel. Just like Figure 9.1, this is created using Equation
9.41 but with a wider pixel-radius of $1.1\times 0.067^{\circ}$.

A single trigger-pixel thus contains on average $7\times 61{}=427$ read-out-
channels (lixels). As we discuss in Chapter 10, the (trigger-)image-sequence
has much reduced aberrations and distortions. In contrast to Cherenkov-
telescopes, the Portal Cherenkov-plenoscope profits from its plenoptic-
perception here. To take even more advantage of the plenoptic-perception, we
actually implement three layers of summations, creating three (trigger-)image-
sequences from one light-field-sequence in parallel. The individual
(trigger-)image-sequences are focused to object-distances of $7.5\,$km,
$15\,$km, and $22.5\,$km each. Figure 12.3 shows the three different
summations implemented for the central trigger-pixel ($n=4,221$) in the field-
of-view. We create trigger-pixels which include all lixels within a radius of
$110\%\times 0.067^{\circ}$ in incident-directions. Portal’s trigger is able
to differentiate between air-showers which reach their maximum Cherenkov-
photon-production on different altitudes. It partly overcomes the depth-of-
field-limit on the trigger-level.

Figure 12.3:  Signals from black photo-sensors (lixels) will be summed up to
form the central trigger-pixel in the field-of-view of Portal.

Table 12.1 summarizes the basic specifications of Portal’s trigger.

Duration of sliding integration | 5 ns
---|---
Num. of refocused image-sequences | 3
Object-distances of refocused image-sequences | 7.5, 15.0, and 22.5 km
Num. of trigger-pixels per trigger-image-sequence | 8,443
Num. of lixels summed-up into one trigger-pixel | 427
Avg. nsb-photons in time-integrated trigger-pixel | 53.5 photons
Threshold for time-integrated trigger-pixel | 103 photons
Min. num. of neighboring trigger-pixels above threshold | 2
Expected trigger-rate due to air-showers |  $59\times 10^{3}\,$s-1
Expected trigger-rate due to nsb-fluctuations |  $\mathcal{O}(22\,$s${}^{-1})$
Avg. rate of nsb-photons in lixel |  $25\times 10^{6}\,$s-1
Table 12.1:  The basic specifications of Portal’s trigger. Here nsb is short
for night-sky-background. We use the night-sky-background shown in Figure
12.1.

####  12.3 Portal’s trigger-threshold

As the rate of air-showers with energies from $1\,$GeV to $10\,$GeV is very
high, the expected trigger-rates of $\approx 59\times 10^{3}\,$s-1 on Portal
will be higher then the trigger-rates of existing Cherenkov-telescopes, see
Figures 2.7, and 2.9. A trigger-rate of $59\times 10^{3}\,$s-1 on Portal gets
close to the limits of todays implementations on Cherenkov-telescopes
[Delagnes et al., 2006] which are designed to reach $100\times 10^{3}\,$s-1.
As a consequence, we apply a rather high trigger-threshold so that in addition
to the trigger-rate induced by air-showers, the trigger-rate induced by
fluctuations in the night-sky-background is low. We set the trigger-threshold
such that there are no accidental triggers caused by the night-sky-background-
photons in all of the $\approx 45\,$ms of simulated observations in which the
trigger is exposed. From this we conclude that the accidental trigger-rate
will not drastically exceed $22\,$s-1. During the dark night, there are $53.5$
photons in a trigger-pixel integrated over $5\,$ns. In the simulations of
Portal we use the night-sky-background-flux shown in Figure 12.1. The
predetermined threshold for an integrated trigger-pixel is $103$ photons,
which is $\approx 6.7$ standard-deviations above the fluctuations of the
night-sky-background. Figure 12.10 shows Portal’s trigger-rate depending on
the trigger-threshold.

####  12.4 Performance of Portal’s trigger

In the Figures 12.4, 12.6, and 12.8, we show Portal’s trigger-probability
depending on the number of true and detected Cherenkov-photons. Although both
Figures 12.4 and 12.6 show the trigger-probability for electromagnetic air-
showers, the gamma-rays in Figure 12.4 come from a point-source, while the
electrons in Figure 12.6 come from a diffuse source. In Figure 12.4 we find
that for gamma-rays coming from a point-source within Portal’s field-of-view,
the air-showers which have $\approx 100$ of their Cherenkov-photons detected
also have a $50\%$ chance to trigger. We also find that the probability to
trigger vanishes for air-showers with only 20 or less Cherenkov-photons. When
comparing the trigger-probabilities for diffuse electrons in Figures 12.6 with
the trigger-probability for diffuse protons in Figure 12.8, we find that
protons need about twice as much Cherenkov-photons to be detected in order to
trigger Portal. This is probably due to the wider spread of secondary
particles in hadronic air-showers. This wider spread also widens the spread of
the Cherenkov-photons, and this leads to a lower density of Cherenkov-photons
in the trigger-pixels. Because of this, the correlation of the number of true
and detected Cherenkov-photons with the true energy of the cosmic particle is
more clear for gamma-rays in Figure 12.5, and electrons in Figure 12.7, then
for protons in Figure 12.9.

Figure 12.4:  The trigger-probability for gamma-rays from a point-source
versus the number of true and detected Cherenkov-photons.  Figure 12.5:
Number of true and detected Cherenkov-photons versus the true energy of gamma-
rays coming from a point-source. The histogram in the upper panel has the sum
along its columns normalized to one. The lower panel shows the number of
events populating the columns in the upper histogram.  Figure 12.6:  The
trigger-probability for electrons from a diffuse source versus the number of
true and detected Cherenkov-photons.  Figure 12.7:  Number of true and
detected Cherenkov-photons versus the true energy of electrons coming from a
diffuse source. The histogram in the upper panel has the sum along its columns
normalized to one. The lower panel shows the number of events populating the
columns in the upper histogram.  Figure 12.8:  The trigger-probability for
protons from a diffuse source versus the number of true and detected
Cherenkov-photons.  Figure 12.9:  Number of true and detected Cherenkov-
photons versus the true energy of protons coming from a diffuse source. The
histogram in the upper panel has the sum along its columns normalized to one.
The lower panel shows the number of events populating the columns in the upper
histogram.  Figure 12.10:  The trigger-rate versus the trigger-threshold for
the Portal Cherenkov-plenoscope. This is often called rate-scan. Based on our
simulations. The solid black line is the sum of triggers caused by
fluctuations of the night-sky-background-photons according the flux shown in
Figure 12.1, and charged cosmic-rays (protons, electrons, and positrons)
according to the fluxes shown in Figure 2.7. The dotted line is only the
triggers caused by the night-sky-background. For low trigger-thresholds below
$\approx 85$ photon-equivalents, our simulation runs out of statistics which
is why the trigger-rate seems to saturate at $2\times 10^{7}\,$s-1. The
vertical gray line marks the trigger-threshold of $103\,$photo-equivalents
which we use for Portal’s performance-estimate in Section 2.6. The instrument-
response-functions in Figures , 2.4 and 2.5 correspond to a trigger-threshold
of $103\,$photo-equivalents.

####  12.5 Conclusion

Our first trigger for the Portal Cherenkov-plenoscope can be implemented with
established technology, and allows Portal to reach an energy-threshold for
gamma-rays of $1\,$GeV. The sum-trigger for Portal effectively reduces
accidental triggers caused by fluctuations in the night-sky-background.
Portal’s trigger makes its decision based on image-sequences just like the
trigger in a Cherenkov-telescope, but Portal’s trigger profits from reduced
aberrations and distortions provided by the plenoptic-perception. By summing
up the responses of individual read-out-channels (lixels), it is possible to
implement the projection of the light-field-sequence onto image-sequences in
real-time. The light-field-sequence can even be projected onto multiple
refocused image-sequences in real-time. Choosing between multiple layers of
refocused image-sequences might allow to correct for misalignments between the
light-field-sensor and the large imaging-reflector in real-time on the
trigger-level. We choose a high trigger-threshold to eliminate accidental
triggers. The lower trigger-rate due to the high trigger-threshold will ease
the trigger’s implementation and also the further processing of the light-
field-sequences.

####  12.6 On Portal’s target-object-distance

The patterns shown in Figure 12.3 can be improved. In Figure 12.3, the target-
geometry for Portal is such that the sensor-plane of the light-field-sensor is
exactly in a distance of one focal-length to the principal-aperture-plane of
the large imaging-reflector. Using Equation 9.12 without the refocusing used
in Equation 9.33, this corresponds to a target-object-distance on infinity.
However, to optimize the trigger for closer object-distances, the light-field-
sensor’s target-position can be adjusted to the corresponding sensor-plane-
distance in the first place. This way, the summation-patterns shown in Figure
12.3 would be much more dense in the sensor-plane what probably will ease
their implementation. In contrast, focusing to infinity would yield a much
more spread summation-pattern than the one shown in Figure 12.2.

###  Chapter 13 Classifying Cherenkov-photons

About $99.9\%$ of the photons recorded by Portal, are night-sky-background-
photons which do not carry information about the origins of the cosmic
particles. It is important to tell apart Cherenkov-photons from the pool of
night-sky-background-photons. This can only be done based on the structure and
density of the photons in the recorded light-field-sequence. In this first
investigation, we fall back to established methods used on Cherenkov-
telescopes where photons are classified based on their density in image-
sequences. This is in the three-dimensional space of incident-directions
$c_{x}$, $c_{y}$, and arrival-times $t$. So we project Portal’s light-field-
sequence $\mathcal{L}[c_{x},c_{y},x,y,t]$ onto an image-sequence
$\mathcal{I}[c_{x},c_{y},t]$ and then apply the exact same methods shown in
Chapter 26 of Part II in this thesis. The projection of the light-field is
done using Equations 9.32 and 9.41. The only difference with the Portal
Cherenkov-plenoscope is, that we repeat the classification multiple times for
a single light-field-sequence by each time refocusing the image-sequence to a
different object-distance. In the end, we classify all the dense photons found
in all refocused image-sequences to be the Cherenkov-photons using a logical
AND-operation. Probably it will be worth to investigate novel methods in the
future which take the light-field into account more naturally, but for this
first investigation we are pleased with the classification-performance.

###  Chapter 14 Estimating the angular resolution for gamma-rays at 1 GeV

A high angular resolution for cosmic gamma-rays is key for gamma-ray-
astronomy. In this Chapter, we show how we estimate the angular resolution of
Portal which we present in Figure 2.8. In this early estimate of the angular
resolution, we focus on gamma-ray-energies close to the energy-threshold of
Portal at $1\,$GeV, since those events will be most abundant in the
observations. For higher energies, we argue that Portal will at least reach
the angular resolutions reached by arrays of Cherenkov-telescopes, because the
light-field-sequences recorded by Portal can always be interpreted as multiple
image-sequences recorded by an array of Cherenkov-telescopes. Also in this
early estimate, we choose to estimate the incident-direction of cosmic gamma-
rays not using images, as it is usually done on Cherenkov-telescopes, but
using the orientation of the Cherenkov-light-front in the recorded light-
field-sequence, compare Figure 8.1. Our intent here is not to present the best
method, but to present a method which makes most use of the Cherenkov-
plenoscope’s novel possibilities.

####  14.1 Cherenkov-light-front

When the read-out of Portal is triggered, see Chapter 12, Portal records a
light-field-sequence. In this light-field-sequence, we classify Cherenkov-
photons and night-sky-background-photons, see Chapter 13. Next we reconstruct
the three-dimensional movement of the Cherenkov-photons above the principal-
aperture-plane, see Figure 8.1. The position of a photon $j$ on its
trajectory, or more precisely the position of a photon on the ray
$\vec{r_{k}}(\lambda=-c(t-{t_{\text{pap}}}_{j}))$ describing the $k$-th lixel
which contains the $j$-th photon, depends on the global time $t$. The rays
$\vec{r_{k}}(\lambda)$ are defined by the light-field-geometry $G$, see
Chapter 8. We choose $t$ such that the point-cloud of Cherenkov-photons is
centered around the principal-aperture-plane. Figure 14.1 shows the Cherenkov-
light-front created by a gamma-ray and reconstructed by the Portal Cherenkov-
plenoscope.

####  14.2 Limitations due to timing-resolution

We simulate Portal’s timing-resolution (standard-deviation) for the arrival-
time of single photons to be $\Delta_{t}\approx 1\,$ns according to our
findings for the timing-resolution of todays Cherenkov-telescopes, see Chapter
25 in Part II of this thesis. This finite timing-resolution sets a limit to
the angular resolution for the reconstruction of a Cherenkov-light-front’s
surface-normal. The paxels with the furthest distance in the Portal Cherenkov-
plenoscope have a baseline of $L\approx 65\,$m. Within the duration of the
timing-resolution, a photon travels a distance of $\Delta l=c\Delta_{t}\approx
0.3\,$m along its trajectory. Here $c$ is the speed-of-light. Thus the angular
resolution will be limited to about $\arctan(\Delta l/L)=0.26^{\circ}$.
Although the statistics of many Cherenkov-photons in the light-front help to
improve the angular-resolution, we find in Figure 14.2 an angular resolution
of $0.31^{\circ}$ which is already close to the limitations caused by a finite
timing-resolution.

Figure 14.1:  The Cherenkov-light-front reconstructed by the Portal Cherenkov-
plenoscope for a gamma-ray with energy 997 MeV, incident-direction
$c_{x}=1.87^{\circ}$, $c_{y}=-0.28^{\circ}$, and core-position $x=34\,$m, and
$x=-95\,$m. The same event in all three images, only the perspective is
slightly rotated each time. Each blue point is a photon recorded by Portal and
classified to be a Cherenkov-photon from within a pool of $\approx 0.5\times
10^{6}$ night-sky-background-photons.  Figure 14.2:  The residual incident-
directions of cosmic particles inducing electro-magnetic air-showers. The true
energies of the cosmic particles here is restricted from $750\,$MeV to
$1,500\,$MeV. The true incident-directions of the particles are diffusely
distributed inside and beyond the field-of-view of Portal. A total of $68\%$
of the reconstructed events are left of the vertical black line at
$0.31^{\circ}$. This vertical black line marks the $1\sigma$-containment-
radius which is shown in Figure 2.8.

###  Chapter 15 Reconstructing air-showers using tomography

To reconstruct the type, energy, and direction of a cosmic particle from the
light-field-sequence we have multiple options. We might perform a feature-
generation similar to the generation of Hillas-features for images [Hillas,
1985] but in higher dimensions for the light-field. We also might have a look-
up-table111Also called ’template-matching’, or ’log-likelihood-fitting’. [Le
Bohec et al., 1998] where a recorded light-field is compared to a collection
of simulated light-fields for which the cosmic particle’s properties are
known. Both these methods work fine with imaging on Cherenkov-telescopes and
are probably a good starting-point.
However, with the observation-power of the Cherenkov-plenoscope it might be
possible to do particle-physics within the air-showers themselves as
individual jets emerge. For particle-physics, we propose to use tomography to
reconstruct the air-shower itself in three spatial dimensions. From the light-
field-sequence, we already know the three-dimensional trajectories and the
arrival-times of the Cherenkov-photons. But we still do not know the photon’s
production-positions. We try to estimate the Cherenkov-photon’s production-
positions using an assumption. We assume that a volume-cell in the atmosphere
in which many trajectories of Cherenkov-photons come close to each other, has
a high probability to also contain the production-positions of these
Cherenkov-photons. The method to estimate this probability from a light-field
is known as tomography222Also called ’focus-stack-deconvolution’, or ’light-
field-microscopy’, or ’narrow-angle-tomography’. [Levoy et al., 2006].
Here we present our first steps into tomography with the Cherenkov-plenoscope.

####  15.1 Motivation

The reconstruction of three-dimensional densities from either a stack of
refocused images [McNally et al., 1999], or from a light-field itself [Levoy
et al., 2006] is a well established method in microscopy. In microscopy, the
density of light-emitting structures of biological cells is studied.
Biological cells are so small that their structures appear to be almost
transparent when imaged with a microscope. Often the cells are not shine
through with an external light-source, but are made to shine themselves by
fluorescence. In a light-field-mindset, fluorescence-emission in biological-
cells is very similar to Cherenkov-emission in air-showers. In both scenarios
the vast majority of the observed medium is transparent with only small
regions emitting photons. In both scenarios, we are interested in the
production-positions of the photons. In both scenarios, large apertures induce
narrow depth-of-fields. Here large does not mean the absolute size, but the
size relative to the objects to be observed. Microscopy motivates us to
investigate tomography for the Cherenkov-plenoscope. However, there are also
differences between fluorescence-emission and Cherenkov-emission. Most of all,
unlike the isotropic emission of fluorescence-photons, Cherenkov-photons are
only emitted in a narrow cone. Also in Cherenkov-astronomy, we might profit
from the measurement of the photons arrival-times.

####  15.2 Basic Method

Tomography starts with a light-field-sequence $\mathcal{L}[c_{x},c_{y},x,y,t]$
recorded by a plenoscope, or an array of telescopes333In tomography this is
also called a line-integral, as it represents the intensity integrated along
the light-field-cells which can be approximated by rays., and a three-
dimensional structure inside a three-dimensional volume which has emitted the
photons recorded in $\mathcal{L}$. Now to reconstruct the three-dimensional
structure which has emitted the photons, we have to relate the light-field-
geometry $G$ of our instrument with the volume it is observing. We divide the
three-dimensional volume into densely packed volume-cells, and we approximate
the light-field-cells in $G$ using rays, see Equations 2.1, and 8.6. Figure
15.1 shows a simplified light-field-geometry $G$ approximated with $9$ rays
facing a volume divided into $4$ volume cells.

Figure 15.1: Basic tomography with rays and volume-cells. Equation 15.1 shows
the corresponding system-matrix.

We relate the rays with the volume-cells by calculating their overlap, and
represent it in a matrix

$\displaystyle S[v,\,k]$ $\displaystyle=$
$\displaystyle\begin{bmatrix}{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&1.1&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&1.0&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}\\\
{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&1.0&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}\\\
{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&0.6&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&1.0&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}\\\
{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&0.4&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&0.5&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}&1.0&{\color[rgb]{0.7,0.7,0.7}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0.7,0.7}0}\\\
\end{bmatrix},$ (15.1)

which is often called system-matrix in tomography. Here $S[v,\,k]$ has $V=4$
rows for the volume-cells, and $K=9$ columns for the rays. In our
representation $S[v,\,k]$ contains the Cartesian-distance a ray travels
through a given volume-cell, with the edge of the volume-cell having length
$1$. From here on, we can follow different methods to reconstruct the volume-
cell’s intensities $D[v]$ using the system-matrix $S[v,\,k]$, and the recoded
light-field $\mathcal{L}[k]$. Here $D[v]$ is a one-dimensional vector
containing the reconstructed photon-emission-intensity in each of the $V$
volume-cells, and $\mathcal{L}[k]$ is the a one-dimensional vector containing
the measured intensities in each of the $K$ light-field-cells. For the
reconstruction of $D[v]$, we first explored filtered-back-projection, and then
went on to iterative methods such as the (simultaneous)-algebraic-
reconstruction-technique [Andersen and Kak, 1984]. Figure 15.4 shows
reconstructions using our first implementation of filtered-back-projection.
For Cherenkov-astronomy it is important to note that, with the exception of
recent advances in positron-emission-tomography [Conti, 2009], established
tomography in microscopy, biology, and medicine does not take into account the
photon’s time-of-flight. Equation 15.1 does not contain the distance between a
volume-cell and the support-position of a ray on the aperture-plane.
Also, Equation 15.1 is only approximative as the light-field-cells are only
approximated using single rays. In the future this can be improved by
approximating a light-field-cell with multiple rays contained in it.

####  15.3 Portal’s reconstruction-power

Figure 15.2 shows Portal’s three-dimensional reconstruction-power relative to
the atmospheric volume. The atmospheric cube with a volume of $2\times 2\times
20$ km3 above Portal is divided into $200\times 200\times 200$ volume-cells.
Each volume-cell is $10\times 10\times 100$ m3. The figure shows the number of
stereo-baselines in each volume-cell. The number of stereo-baselines in a
volume-cell is

$\displaystyle N_{\text{Baselines}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(N_{\text{Supports}}^{2}-N_{\text{Supports}})$ (15.2)

where $N_{\text{Supports}}$ is the number of unique support-positions that
emit rays into this volume-cell. For Portal the maximum number of stereo-
baselines in a volume-cell is 1,830 when all of Portal’s 61 support-positions
(paxels) emit at least one ray into this volume-cell.

Figure 15.2: Portal’s three-dimensional reconstruction power. The number of
stereo-baselines in the volume-cells above Portal’s aperture-plane. Portal’s
aperture-plane goes along the bottom of the figure at object-distance = 0 km.
Here we see only the few volume-cells in the $x$-object-distance-slice for
$y=0$. Each volume-cell is $10\,$m in $x$, $10\,$m in $y$, and $100\,$m in
object-distance. The darkening here is proportional to the square-root of the
number of stereo-baselines $N_{\text{Baselines}}$, which is proportional to
the number of unique support-positions in a volume-cell $N_{\text{Supports}}$,
see Equation 15.2.

In Figure 15.2 we find that Portal’s reconstruction-power vanishes towards
larger object-distances. At the given binning of the atmospheric-volume,
Portal’s reconstruction-power is optimal up to object-distances of $\approx
10\,$km. When pointing to zenith, this corresponds to an altitude of $15\,$km
a.s.l. when building Portal on $5,000\,$m a.s.l. We also find that there is no
reconstruction-power in the first few $100\,$m above Portal’s aperture as the
bundles of rays originating in each support-position have not yet overlapped.
Naturally we find that there is only reconstruction-power within Portal’s
view-cone. This very asymmetric population of rays inside volume-cells
increases the chance for unresolvable artifacts, and is often discussed as
’narrow-angle-tomography’. From Figure 15.2 we can already see possible
limitations to Portal’s three-dimensional reconstruction-power. Air-showers
which emit Cherenkov-photons in volume-cells with low numbers of stereo-
baselines will be more difficult to be reconstructed. Further, even if an air-
shower emits Cherenkov-photons within a volume-cell, this does not yet mean
that those Cherenkov-photons will reach Portal’s aperture as the emission of
Cherenkov-photons is not isotropic.

####  15.4 Reconstructing air-showers in the refocused-image-space

In Figure 15.2 we find that the volume-cells in the Cartesian-space above the
atmosphere are populated very asymmetrically. Some have many, others have no
rays passing through them. This asymmetry can be overcome when the tomography
is done in the three dimensional space of a refocused stack of images [McNally
et al., 1999]. Instead of the Cartesian-rays defined in Equation 2.1, we use
the image-rays introduced in Section 9.4. Instead of subdividing the
Cartesian-volume above the Cherenkov-plenoscope, we subdivide the three
dimensional space of refocused images in incident-directions $c_{x}$, and
$c_{y}$, as well as image-distances $b$, compare Equations 9.34, and 9.37. The
advantage of this refocused-image-space is, that the density of rays in the
cells is almost the same for all the cells. The plenoscope’s vanishing
reconstruction-power for larger object-distances is taken into account here
naturally. In our first tests with iterative reconstructions, this reduces
many artifacts when estimating the emission-density of Cherenkov-photons
$D[v]$. Again, the overlap of image-rays and the pixels in the refocused-
image-space can be represented in a system-matrix as shown in Equation 15.1,
and is often called three-dimensional-point-spread-function. Figure 15.3
illustrates the basic idea of the refocused-image-space.

Figure 15.3:  A light-source in object-distance $g$ is observed by the
plenoscope with focal-length $f$. The dashed line is the optical-axis and the
horizontal line at the bottom is the principal-aperture-plane. When we refocus
a stack of images from the light-field (right part of figure), the image of
the light-source will be sharpest in the image for image-sensor-distance $b$
which corresponds to a focus on object-distance $g$. Other images for image-
sensor-distances $\neq b$ will show the blurring (Bokeh) induced by the narrow
depth-of-field caused by the large imaging-reflector. In this figure, the
lowest image in the refocused stack is for a image-sensor-distance of $f$,
which corresponds to a focus on $\infty$.

Figure 15.4: Figures taken from [Engels, 2017]. A tomographic reconstruction
of an air-shower induced by a hadronic cosmic-ray. Upper figures show a
$x$-$y$-slice of volume-cells at object-distance $=10.9$ km, and lower figures
show a $x$-$y$-slice of volume-cells at object-distance $=12.4$ km. Compare
the bright structures in both the simulation-truth on the left, and in the
reconstruction on the right. The filtered-back-projection used in this figures
is developed and implemented by the author of this thesis (S.A.M.). In his
master-thesis, Axel Arbet Engels investigates the reconstruction of direct
Cherenkov-photons using tomography on a perfect Cherenkov-plenoscope with an
enormous $150\,$m aperture-diameter. The goal is to push the limits of
chemical composition-measurements using direct Cherenkov-photons emitted by
the cosmic nuclei itself.

####  15.5 Conclusion

We see great potential for tomographic reconstructions of air-showers using
the Cherenkov-plenoscope, or a powerful array of Cherenkov-telescopes such as
the Cherenkov-Telescope-Array (CTA). The big similarity with tomography in
microscopy allows the atmospheric-Cherenkov-method to adopt many well
established techniques. For the $150\,$m diameter Cherenkov-plenoscope
investigated by Axel Arbet Engels, the reconstruction-power for air-showers
using the a simple filtered-back-projection is extraordinary, see Figure 15.4.
Tomography is the generalization of the stereo-technique and will potentially
enable us to observe the inner structure of air-showers.

###  Chapter 16 Pointing the Cherenkov-plenoscope using the cable-robot-mount

We propose the cable-robot-mount to point the Cherenkov-plenoscope to
different gamma-ray-sources in the sky. Although it might be possible to mount
the 71 m diameter Portal on a conventional altitude-azimuth-mount, we decide
to investigate the cable-robot-mount as it potentially offers a more cost-
effective solution. The concept of the cable-robot-mount is proposed by the
author of this thesis (S.A.M.), while the details are worked out in close
collaboration with the department for civil, environmental and geomatic
engineering at ETH-Zurich.

  1. 1.

We motivate a dedicated cable-robot-mount to point the Cherenkov-plenoscope.

  2. 2.

We list technical specifications, identify states of operation, and
investigate environmental properties of possible sites.

  3. 3.

We briefly describe how civil-engineer Spyridon Daglas together with the
author of this thesis (S.A.M.) simulate the cable-robot-mount.

  4. 4.

We briefly summarize the findings of Spyridon Daglas on the performance and
feasibility of cable-robot-mounts ranging from $30\,$m to $100\,$m in
diameter.

####  16.1 Motivation – A mount for the plenoscope

To reach an energy-threshold for gamma-rays of $1\,$GeV, the Portal Cherenkov-
plenoscope needs a large imaging-reflector of $71\,$m in diameter. Established
Cherenkov-telescopes only have mounts for imaging-reflectors up to $28\,$m in
diameter, see Section 19.6. Although the conventional altitude-azimuth-mount
of the Cherenkov-telescope might be an applicable mount for the Portal
Cherenkov-plenoscope, we see the potential for a more cost-effective mount. On
the Cherenkov-plenoscope, the demand for a rigid alignment between the
imaging-reflector and the light-field-sensor is much more relaxed in contrast
to the demand for a rigid alignment between the imaging-reflector and the
image-sensor on a Cherenkov-telescope, see Chapter 11. As long as the actual
alignment is known, the Cherenkov-plenoscope can compensate misalignments
between its imaging-reflector and its light-field-sensor. While on the
Cherenkov-telescope rigid support-structures between the imaging-reflector and
the image-sensor are mandatory, on the Cherenkov-plenoscope this rigid
support-structures can be omitted. Rigid support-structures shadow the
aperture, and add up weight to the moving parts. But the main problem that we
identify with rigid support-structures is, that all the forces needed to hold
the image-sensor are guided through the support-structure of the imaging-
reflector, compare Figures 19.7, 19.10, 19.11, 19.16, 19.17, and 19.19. Since
the forces holding the image-sensor change with the pointing-direction of the
Cherenkov-telescope, also the shape of the imaging-reflector changes. And such
deformations of the imaging-reflector must be minimized for high optical
quality. On small Cherenkov-telescopes such deformations might be too small to
effect optical quality, but on larger telescopes it becomes increasingly
difficult to find cost-efficient solutions in order to keep the deformations
below an acceptable limit, or to actively correct for them [Biland et al.,
2007]. The physical limit of the square-cube-law [Galilei, 1638] makes the
forces needed to hold a structure grow quicker than the areas which have to
tolerate these forces. As a consequence, we can not simply scale up an
existing mount.
Another drawback of the altitude-azimuth-mount is that it can not move fast
near the zenith what conflicts the hunt for transient phenomena in the highly
variable gamma-ray-sky. The inverse kinematics of the altitude-azimuth-mount
has a near-zenith-singularity [Borkowski, 1987] which causes unacceptably
large accelerations and forces when moving close to the zenith during a
repositioning.
Motivated by the Arecibo radio-telescope [Altschuler and Nieves, 2002], the
NIST cable-robot-manipulator [Albus et al., 1993], and the cable-robot-
simulator [Miermeister et al., 2016], see Figures 16.1, and 16.2, we decided
to investigate a different design for the mount of Portal.

Figure 16.1:  Figure taken from [Miermeister et al., 2016]. The cable-robot-
simulator by Max-Planck-Institute for Biological Cybernetics in cooperation
with the Fraunhofer-Institute for Manufacturing, Engineering, and Automation.
This impressive motion simulator can accelerate payloads of up to $500\,$kg
with up to $14\,$ms-2. The design-goal of the cable-robot-simulator is not to
slowly move the large components of a Cherenkov-plenoscope, but to be a motion
simulator for up to two passengers with an ’extraordinary’ [Miermeister et
al., 2016] power to weight ratio for maximum acceleration. See also Figure
16.2. For the Cherenkov-plenoscope, we adopt the icosahedron-shape of the
moving platform to mount the light-field-sensor. Compare Figure 5.8.  Figure
16.2:  Figure taken from [Miermeister et al., 2016]. A prototype of the cable-
robot-mount is implemented in a so called ’crossed over’ configuration and has
eight powerful actuated winches in each corner of a large room. See also
Figure 16.1. Compare this configuration with the cable-robot-mount for the
light-field-sensor shown in Figure 2.1.

We call it the cable-robot-mount. Our cable-robot-mount for Portal is
separated into two independent mounts. One mount is only actuating the large
imaging-reflector, and the second mount is only actuating the light-field-
sensor. The two independent mounts always try to establish the desired target-
alignment between the light-field-sensor and the large imaging-reflector,
while we compensate possible misalignments due to e.g. wind-gusts later in
software, see Chapter 11. The independent actuation of the two components
reduces the forces running through the support-structure of the large imaging-
reflector and thus allows us to build larger imaging-reflectors. The cable-
robot-mount supports the imaging-reflector, and the light-field-sensor with
many cables in parallel which are actuated by winches in order to minimize the
mass of the moving structures. The cable-robot-mount extensively uses
robotics, which is the study of controlling the actuated winches using a
computer that is aware of the parallel, elastic, and inverse kinematics of the
entire mount. The cable-robot-mount is meant to trade large and costly
support-structures, as well as large and expensive mechanical joints in change
for computer-control and parallel actuation. We suspect, that in contrast to
the costs for support-structures and mechanical joints, the costs for
computer-control and parallel actuation scale less rapidly with the size of
the mount. The cable-robot-mount only has one moving support-structure which
is the dish holding the mirror-facets of the imaging-reflector. The remaining
support-structures in the cable-robot-mount are fixed towers where the cables
and winches are attached to. Such large towers are common in e.g. overhead-
power-lines what reduces the costs for their engineering and production.
Figures 16.3, and 16.4 show conceptual drawings of Portal and its cable-robot-
mount.

Figure 16.3: A possible parking position for the Portal Cherenkov-plenoscope
and its cable-robot-mount.

Figure 16.4: Portal and its cable-robot-mount pointing to different zenith-
distances of $0.0^{\circ}$, $22.5^{\circ}$, and $45.0^{\circ}$.

In summary, we expect the cable-robot-mount to reduce the forces running
through the support-structure of the imaging-reflector which shall allow the
construction of larger imaging-reflectors. We expect the cable-robot-mount to
reposition the Cherenkov-plenoscope faster during its hunt for transient
phenomena because its inverse kinematic is free of singularities. And further,
we expect the cable-robot-mount to be very cost-effective because all its
fixed towers and actuated winches are mass produced components which are
commonly used in e.g. overhead-power-lines, wind-turbines, and harbor-cranes.

####  16.2 Technical specifications

To find a cost-effective design and to communicate our needs, we formulate the
technical specifications for the Cherenkov-Plenoscope’s mount. With these
specifications in mind, Figure 16.5 illustrates what components of the cable-
robot-mount have potential to be investigated. Because a light-weight design
for the imaging-reflector is crucial, we investigate the dimensions and masses
of existing mirror-facets in Table 16.1. We want to reach zenith-distances of
up to $45^{\circ}$.

Figure 16.5: This is based on the author’s interpretation of the potential for
civil-engineering after discussions and studies with the experts. The tall
masts are considered to be standard components used in overhead-land-lines.
The cage for the light-field-sensor is also considered to be fairly standard.
The steel-cables are considered higher standard, but still well within range
of commercial products for e.g. harbor-cranes. The concrete-pillars are
considered to be simple from a load-point-of-view, but are worth to
investigate cost-effective, and wind-shielding designs. Clearly, the civil-
engineers consider the large imaging-reflector’s support-structure holding the
mirror-facets to be most worthy to investigate in order to find cost-effective
designs.

##### States of operation

We identify three states of operation for the Portal Cherenkov-plenoscope.

  1. 1.

Parking. During the day, the large imaging-reflector and the light-field-
sensor are both lowered to their own pedestals for parking, see Figure 16.3.
When the large imaging-reflector rests on its pedestal pointing to zenith, the
intense111Solar-constant 1.3 kW m-2 $\times$ aperture-area of imaging-
reflector $3,684\,$m2 $\div$ area of projected image of the Sun $0.68\,$m2 7.4
MW m-2 image of the Sun can not be projected onto objects on ground.

  2. 2.

Fast pointing. The Cherenkov-plenoscope points fast from one source in the sky
to another source. Both the large imaging-reflector and the light-field-sensor
are moved independent of each other. Since there are no near-zenith-
singularities, the cable-robot-mount always rotates the Cherenkov-plenoscope
along one single axis only. For fast pointing, we desire angular velocities of
$90^{\circ}$min-1. In this state, the Cherenkov-plenoscope does not have to
reach its target-geometry, as it will not record air-showers during fast
pointing.

  3. 3.

Slow tracking. Only here the Cherenkov-plenoscope tries to reach its target-
geometry. Only during slow tracking, the Cherenkov-plenoscope records air-
showers. To counteract the rotation of the earth, an angular velocity of
$0.25^{\circ}$min-1 is mandatory. To continuously spread the signal-, and
background-regions across different regions inside the field-of-view [Finnegan
et al., 2011]222E.g. so called orbit-mode implemented in the VERITAS
Cherenkov-telescopes., we desire an angular velocity of $0.75^{\circ}$min-1.
Together, we conclude that low angular velocities during slow tracking can be
approximated using static load-scenarios.

##### Site and environment – Atacama-desert, Chile

We choose to adopt the environmental properties of Atacama-desert in Chile on
$\approx 5,000\,$m a.s.l.. The environmental properties of this site are well
documented by e.g. the European-Southern-Observatory’s ALMA-telescope, and the
Cherenkov-Telescope-Array. Further, the Atacama-desert fulfills our demands
for exceptional dark, and clear skies, and it is high in altitude. Although
the geomagnetic cutoff-rigidity of $\approx 10\,$GV is not the highest,
compare Figure 2.6, it is representative for our estimate of Portal’s
performance. Figure 16.6 shows the landscape of Atacama-desert. And the
Figures 16.7, and 16.8, show typical temperatures and wind-speeds for this
site. Temperature, and changes in temperature effect the choice of materials.
Wind speeds are used to formulate a worst-case load-scenario during
observations. Occasional wind-gusts are used to formulate a survival-load-
scenario. We want observations to be possible up to wind-speeds of $15$ ms-1
(54 kmh-1), and we want the mount to survive wind-speeds of up to $55\,$ms-1
(200 kmh-1).

Figure 16.6: Figure taken from [Radford et al., 2008]. Atacama-desert in Chile near the summit of Cerro Chajnantor on $5,612\,$m a.s.l..  Figure 16.7: Figure taken from [Miyata et al., 2008]. Temperatures over the years 2006, and 2007 at the summit of Cerro Chajnantor on $5,612\,$m a.s.l., Chile. Temperatures for the nearby Pampa la Bola plateau on $\approx 5,000\,$m a.s.l. are reported to be $\approx 5^{\circ}$C higher on average.  Figure 16.8: Figure taken from [Miyata et al., 2008]. Wind-speeds over the years 2006, and 2007 at the summit of Cerro Chajnantor on $5,612\,$m a.s.l., Chile. Wind-speeds for the nearby Pampa la Bola plateau on $\approx 5,000\,$m a.s.l. are reported to be $\approx 5\,$ms-1 lower on average.  type-designation | shape | area | spacing | mass | areal-density | areal-density
---|---|---|---|---|---|---
|  |  |  |  |  | with actuators
|  | m2 | m | kg | kg m-2 | kg m-2
MAGIC 1st gen. | square | 0.25 | 0.50 | 4.0 | 16.0 | 21.0
MAGIC 2nd gen. | square | 0.96 | 0.96 | 13.4 | 14.0 | 19.2
VERITAS | hexagon | 0.32 | 0.61 | 10.4 | 32.4 | 48.0
FACT/HEGRA | hexagon | 0.32 | 0.61 | 5.5 | 17.6 | 33.7
CTA-MST, INAF | hexagon | 1.25 | 1.20 | 25.0 | 20.0 | 24.0
CTA-LST, INAF | hexagon | 1.97 | 1.50 | 45.0 | 22.8 | 25.3
Table 16.1: Mirror-facets on Cherenkov-telescopes. Here the spacing is the
flat-to-flat-, or inner-diameter of the facets. We assume that the weight of a
two-axis-actuator for the orientational fine-alignment is $\approx
5\,$kg/facet, regardless of the area of the facet. The MAGIC, and FACT facets
we measured ourselves. The geometry of VERITAS is taken from [Perkins et al.,
2007], while we assume that the density of the glass is $\approx 2,800\,$kg
m-3. Cherenkov-Telescope-Array (CTA) specifications are taken from [Pareschi
et al., 2013a]. Mass of the CTA-LST facet is taken from [Teshima et al.,
2011]. Here LST is short for Large-Size-Telescope, MST is short for Medium-
Size-Telescope, and INAF is short for Instituto-Nazionale-di-AstroFisica.

####  16.3 Designing a support-structure for the imaging-reflector

As indicated in Figure 16.5, we focus on investigating the imaging-reflector’s
support structure. Spyridon proposes to use a space-truss to support the
mirror-facets. He proposes to use a rectangular space-truss-lattice to ease
the mounting of the hexagonal mirror-facets as it is shown in Figure 16.9. In
this configuration, the corners of the mirror-facets line up with the nodes of
the space-truss. He disfavors a ribbed-dome space-truss, as it makes the
mounting of the mirror-facets more complicated, and deforms in potentially
worse shapes harming the optical quality. After initial simulations, he
identified that stiffening the outer realm of the imaging-reflector could
prevent tensions from running into the inner parts. A tension-ring in the
outer realm will potentially lower the overall weight and deformations. Such a
tension-ring-design can be implemented by e.g. using stiffer bars in between
the outer nodes of the space-truss.
Therefore, our focus is on a space-truss with an outer tension-ring. But for
example another proposal by Prof. Mario Fontana suggests to use large
inflatable air-bags between the bars of the space-truss-lattice to increase
the stiffness of the overall imaging-reflector.

####  16.4 Simulating the cable-robot-mount

We implement a simulation to investigate the deformations of Portal’s large
imaging-reflector under various load-scenarios, as well as the impact of these
deformations on the optical quality. The simulation is implemented in equal
parts by Spyridon Daglas, and the author of this thesis (S.A.M.). We quantify
the optical quality using the standard-deviation of the imaging-reflector’s
point-spread-function for incoming photons running parallel to the optical-
axis.
The imaging-reflector is simulated in a fully parametrized way. This allows us
to investigate the optics when we change parameters such as the diameter, the
focal-length, or the type of curvature. But it also allows us to investigate
the mechanics when we change parameters such as mirror-facet-size, space-
truss-designs, and material-properties.
We also define load-scenarios. We simulate pointing the cable-robot-mount to
various zenith-distances. We simulate constant wind, wind-gusts, and even
earth-quakes.
For each tuple of load-scenario, and imaging-reflector:

  1. 1.

We simulate the deformation of the imaging-reflector under gravity when
pointing to zenith. This is the default-load-scenario.

  2. 2.

We estimate the optimal orientational fine-alignment for each mirror-facet