# Focal ratio degradation for fiber positioner operation in astronomical
spectrographs

Brent Belland${}^{\text{1,5}}$ James Gunn2 Dan Reiley1 Judith Cohen1 Evan
Kirby1 Antonio Cesar de Oliveira3 Ligia Souza de Oliveira3 Mitsuko Roberts1
and Michael Seiffert${}^{\text{1,4}}$ 1Physics Department, Caltech, 1200 E.
California Blvd., Pasadena, CA 91125, USA
2Astronomy Department, Princeton University, 4 Ivy Lane, Princeton, NJ 08544,
USA
3Laboratório Nacional de Astrofisica, MCTI, Rua Estados Unidos, 154, Bairro
das Nações, Itajubá, MG, Brazil
4Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
<EMAIL_ADDRESS>
((to be inserted by publisher); (to be inserted by publisher); (to be inserted
by publisher))

###### Abstract

Focal ratio degradation (FRD), the increase of light’s focal ratio between the
input into an optical fiber and the output, is important to characterize for
astronomical spectrographs due to its effects on throughput and the point
spread function. However, while FRD is a function of many fiber properties
such as stresses, microbending, and surface imperfections, angular
misalignments between the incoming light and the face of the fiber also affect
the light profile and complicate this measurement. A compact experimental
setup and a model separating FRD from angular misalignment was applied to a
fiber subjected to varying stresses or angular misalignments to determine the
magnitude of these effects. The FRD was then determined for a fiber in a fiber
positioner that will be used in the Subaru Prime Focus Spectrograph (PFS). The
analysis we carried out for the PFS positioner suggests that effects of
angular misalignment dominate and no significant FRD increase due to stress
should occur.

###### keywords:

Focal ratio degradation, optical fiber, telescope, stress, misalignment.

5Corresponding author.

; ; ;

##  1 Introduction

Optical fibers serve an important role in modern astronomical instrumentation.
They are a key element in a number of current multi-object spectrographs
(e.g., Smee _et al._ (2013); Lewis _et al._ (2002)), where they relay light
from astronomical objects at the telescope focal surface to a remotely located
spectrograph. Optical fibers will find continued use in essentially all next-
generation massively-multiplexed astronomical spectrographs. Projects under
development include the Subaru Prime Focus Spectrograph (PFS) Sugai _et al._
(2015), DESI (Flaugher & Bebek, 2014), MOONS (Cirasuolo _et al._ , 2011),
4MOST (de Jong _et al._ , 2012), and WEAVE (Dalton _et al._ , 2012), among
others. This paper is motivated by focal ratio degradation (FRD) testing for
the Subaru PFS, utilizing the same fiber and fiber positioning system that
will be used in the PFS.

FRD is the decrease of the focal ratio of a beam traversing a fiber Ramsey
(1988). The light power distribution thus varies due to FRD after transmission
through a fiber, causing issues for subtracting night sky lines in fiber-fed
spectrographs in astronomical implementations Clayton (1989); Bolton &
Schlegel (2010). These issues must be addressed in order to limit the sky
subtraction residuals to less than 0.5% of the sky continuum, as is the goal
for the PFS (Tamura _et al._ , 2016). The FRD of the fiber is a function of
many components, potentially varying as the fiber experiences different
stresses during positioning and telescope operation. FRD in the PFS has a
dynamically changing effect on the point spread function, as will be discussed
below. Furthermore, FRD can cause a decrease in throughput if it causes some
light to scatter to a focal ratio that is smaller than the acceptance focal
ratio of the spectrograph. An example of how FRD can affect a beam is
demonstrated schematically in Figure 1.

Figure 1:  Simulated effect of FRD on a uniform tophat profile. Above, the
radial profile of the image when input and output from the fiber is shown.
Below is a two dimensional image that demonstrates the effect FRD has on the
point spread function has on FRD. Notably, an instrument that could only
accept up to the initial tophat would lose throughput due to the FRD-induced
increased profile size. Even if the instrument can accept faster inputs,
increasing FRD causes increasing illumination of the outer parts of the pupil,
which for the fast spectrograph cameras in all the projects discussed in the
introduction brings increasing aberrations and degradation of the point spread
function.

The causes of FRD include any effect that can scatter the light, including
polishing irregularities at the fiber surface, microbending introducing
irregularities to the core/clad interface, diffraction, bending, twisting, and
squeezing among other causes Clayton (1989); Oliveira _et al._ (2005). It
should therefore be minimized to diminish the throughput loss and quantified
to understand its effect on the light’s point spread function.

However, the key to precise atmospheric background subtraction for PFS is the
accurate modeling of the point spread function, not just knowledge of the FRD.
This in turn requires an accurate model of the intensity as a function of
angle when light exits the fiber and enters the spectrograph. In particular,
any change with the instrument configuration, such as the position in the
focal plane or the position of individual fibers within their patrol region
that affects the PSF will need to be determined. With this in mind, we
conducted a series of experiments to evaluate changes in FRD in a fiber due
specifically to its fiber positioning hardware which we call a Cobra. For
these purposes, we wish to know not only the net effect, but also the
components due to applied fiber stress (that is, the bending and twisting
inside the Cobra due to its motion), angular misalignments (both static and
dynamic). The repeatability and predictability of the FRD is also of foremost
importance.

There are two primary methods to measure FRD: a cone test,also known as a
solid angle test, and a ring test, also known as a collimated beam test Haynes
_et al._ (2011); Yan _et al._ (2018). The cone test involves a uniform beam of
known solid angle that fills a fiber. The output profile is measured and
typically the enclosed energy at varying f-ratios is found Oliveira _et al._
(2005); dos Santos _et al._ (2014); relative enclosed energies are compared to
determine FRD. The cone test is able to probe the whole input cone of the
fiber, but it is sensitive to misalignments. In contrast, the ring test
involves a beam input at a known angle into a fiber, quantifying FRD by the
FWHM of the resulting ring output Allington-Smith _et al._ (2013); Finstad _et
al._ (2016). While ring FRD can be quantified by its angular size, it is
restricted to a small range about the input f number and characterize the
light power distribution over the entire fiber outside of this range.
Comparison of results between these two tests is complicated by the fact that
the measurement outputs are different.

In this paper, a method of determining absolute FRD from the cone test is
proposed. Such a system would be able to probe the whole input range of a
measured fiber, being sensitive to input profile variations in a single test,
while still quantifying the FRD in a way comparable to the ring test. In
particular, the misalignment component of the cone profile can be extracted,
yielding both fiber FRD and angular misalignment at the same time.

This paper is structured to discuss the FRD extraction model considered here
in Section 2 and then hardware setup in Section 3. Afterward, various forms of
stress (bending, squeezing, and twisting), and thus potentially of FRD, are
measured in a Polymicro fiber in Section 4, and angular misalignment in a
fiber is interpreted using the cone test in Section 5 to compare to the
angular component of the model. Finally, each of these components is
considered in the fiber positioning system, the Cobra, which will be used in
the Subaru PFS, to determine the relative magnitude and contributions of
stress vs. angle in Section 6.

##  2 Model

FRD is the effective smearing of light due to imperfections of the fiber
core/clad interface from fabrication and from stresses such as from bending,
as well as from fiber end face preparation. Assuming these defects are
scatterers of a random angle and over the length of the fiber many such
defects are encountered, by the Central Limit Theorem, the angle scattered
should roughly be describable by a Gaussian distribution with standard
deviation $\sigma$. In practice, Haynes _et al._ (2011) found using the ring
test that modal diffusion due to effects such as microbending behaved in a
Gaussian way, aperture diffraction could be approximated as a Gaussian, and
end face surface roughness could be modeled by a Gaussian or Lorentzian
profile with Lorentzian component decreasing with smoother fiber surface. In
this paper, the Gaussian effects are assumed to dominate FRD.

To first order we wish to characterize FRD by the $\sigma$ of a Gaussian that,
when convolved with a tophat input light distribution, yields the observed
output profile. Without any other outside influences, this $\sigma$ would
characterize FRD, but angular misalignments can confound this analysis. Let
the $\sigma$ value due to FRD be $\sigma_{FRD}$ whereas $\sigma_{TOT}$ will be
used to characterize the $\sigma$ that would create an equivalent width of the
entire dropoff of power over angle. We can find $\sigma_{TOT}$ directly from
analyzing a fiber’s output profile, but $\sigma_{FRD}$ can also be extracted
due to the different effects FRD and angular misalignment have on the profile.
In the case of no angular misalignment, $\sigma_{TOT}=\sigma_{FRD}$.

To verify the initial premise of this model, namely that scattering from FRD
can be quantified with a value for $\sigma$, previous work must be considered.
FRD due to scattering of light causing light coupling to higher modes within
the fiber has been analyzed by Gloge (1972), who developed a model to describe
the power distribution $P$ of light from a fiber of length $L$ due to
microbending, as a function of angle from normal incidence $\theta$:

$\frac{\partial P}{\partial
L}=-A\theta^{2}P+\frac{D}{\theta}\frac{\partial}{\partial\theta}\left(\theta\frac{\partial
P}{\partial\theta}\right)$ (1)

with $A$ and $D$ corresponding to absorption and coupling coefficients
respectively.

This power distribution PDE was solved by Gambling _et al._ (1975) for a plane
wave at an angle of incidence $\theta_{i}$:

$P(\theta)\simeq\frac{1}{bL}\exp\left({-\frac{\theta^{2}+\theta_{i}^{2}}{4DL}}\right)I_{0}\left(\frac{\theta\theta_{i}}{2DL}\right)$
(2)

for $b=4\sqrt{AD}$ and $I_{0}$ representing a Bessel function of zeroth order,
for $bL\ll 1$. Effectively, this predicts the ring test output. Notably, this
expression asymptotically approaches Gaussian behavior in the limits
$(\frac{\theta\theta_{i}}{2DL})\ll 1$ and $(\frac{\theta\theta_{i}}{2DL})\gg
1$, both with standard deviation

$\sigma=\sqrt{2DL}.$ (3)

In order to relate the results of Gambling et al. to the cone test, due to the
simple nature of the plane wave, it is possible to integrate to create an
input profile for a cone of light; integrating over an input profile
$G(\theta,\theta_{i})$ yields an output power $F$ of

$F(\theta)=\int_{0}^{2\pi}\int_{0}^{\pi}G(\theta^{\prime},\phi^{\prime})P(\theta,\theta^{\prime})\sin(\theta^{\prime})d\theta^{\prime}d\phi^{\prime}.$
(4)

Assuming a uniform cone-like beam with rotational symmetry about the fiber
axis, the overall output power $F$ from the fiber then becomes

$F(\theta)\propto
2\pi\int_{0}^{\theta_{i}}P(\theta,\theta^{\prime})\sin(\theta^{\prime})d\theta^{\prime}.$
(5)

This integral approximates a normal error function with the same $\sigma$ as
the one corresponding to that of the ring test.

Thus, the FRD-only model aligns well with a $\sigma_{FRD}$ characterizing its
output, just as utilized in this model.

However, in general, there is more than FRD affecting the profile;
$\sigma_{TOT}>\sigma_{FRD}$ due to angular misalignments. If the chief ray is
not normal to the fiber face, a tophat input does not appear to be a tophat at
the fiber. Instead, the optical system is at a nonzero angle relative to the
fiber axis.

This has the effect of convolving each point in the tophat with a ring of size
corresponding to its angular misalignment. That is, the effect of angular
misalignment is not to convolve the profile with a Gaussian as microbends do;
by analyzing the shape of a profile dropoff, the effect of angular
misalignment can be separated from that of microbend FRD. In 2D, such an
angularly misaligned profile from rings is difficult to calculate
analytically; however, in 1D, the convolution kernel of a profile with an
angular misalignment $a$ would behave as

$k(\theta)\propto\frac{1}{\sqrt{1-(\frac{\theta}{a})^{2}}}$ (6)

Note how this convolution kernel is bimodal at $\theta=\pm a$, yielding a
dropoff distinct from a gaussian convolution. Such behavior has a distinct
effect on the point spread function compared to FRD and must be considered
independently.

To first order, the net width of the dropoff, $\sigma_{TOT}$, is then the sum
of the rms angular misalignment kernel width (of $\frac{a}{\sqrt{2}}$) and the
Gaussian kernel width (independent of misalignment, called $\sigma_{FRD}$).
That is, $\sigma_{TOT}^{2}\approx\sigma_{FRD}^{2}+\frac{a^{2}}{2}$. A more
thorough analysis of the convolved functions approximates an analytic FRD
$\sigma_{TOT}$ of

$\sigma_{\text{TOT}}^{2}\approx\sigma_{\text{FRD}}^{2}\left(1+\frac{1+A^{2}}{2+1.5A^{2}}A^{2}\right)$
(7)

where $A=\frac{a}{\sigma_{\text{FRD}}}$. This estimate of $\sigma_{TOT}$
varies by $<1\%$ from the numerical integration result for angular
misalignments $A\leq 6$.

Figure 2 visualizes how angular misalignment may affect the profile of the
dropoff using a model of a uniform input profile, for a given $\sigma_{TOT}$.
Notably, the error function dropoff from a purely Gaussian convolution becomes
bimodal with large angular misalignments, and the effective width of the
profile for a given FRD increases with angular misalignment. Note that holding
$\sigma_{TOT}$ constant for a range of angular misalignments allows for a
comparison of angular misalignments, but for a fixed $\sigma_{FRD}$ angular
misalignments increase the width of the dropoff.

Also from Figure 2, note that from the model of a uniform beam with f number
2.8 (that is, numerical aperture, or NA, of 0.1785), the amount of light
cutoff at f number 2.5 (NA of 0.2) can also be seen and the light cutoff can
be determined. At 8.9 mrad of $\sigma_{TOT}$ and no angular misalignment, only
0.03% of the light is cut off, whereas for comparison, at 20 mrad of
$\sigma_{TOT}$ and no angular misalignment, about 1.9% of the light is
blocked.

Figure 2:  Dropoff in pixel intensity for the angular misalignment model with
uniform input beam of f number 2.8 (numerical aperture of 0.1785). The effect
on the profile for the same net width $\sigma_{TOT}=8.9$ mrad are shown for
varying angular misalignments, which corresponds to a different gaussian
contribution from FRD (large angles are shown to emphasize the effect of
angular misalignment). Also shown is the f number of 2.5 corresponding to the
spectroscope cutoff. The light lost at the cutoff varies at a given
$\sigma_{TOT}$ for different angular misalignments.

Again, this model should be related back to Gloge’s modal diffusion theory;
angular misalignments can be factored in with an appropriate choice of
$G(\theta^{\prime},\phi^{\prime})$ using equation (4). Detailed analysis is
left for future work; however, the feasibility of this paper’s model can be
tested using a simple $G(\theta^{\prime},\phi^{\prime})$. A circle in a plane
tilted at an angle with respect to a viewer appears to be an ellipse. Thus,
consider the profile of an ellipse for $G(\theta^{\prime},\phi^{\prime})$.
Because only $G$ depends on $\phi^{\prime}$ in Equation (4), we can substitute
it with a rotationally averaged over $\phi^{\prime}$ function of
$\bar{G}(\theta^{\prime})$; the 1D profile of a rotationally averaged ellipse
has an approximately bimodal distribution similar to a step function convolved
with the kernel in equation (6). An example of such a convolution is shown in
figure 3. Overall, the simplified model is consistent with FRD results in the
literature and allows for a simple quantification of angular misalignment that
is not unreasonable.

Figure 3:  Simple example of an ellipse profile (left) that to first order
approximates a tilted circle. A figure demonstrating what a 2D profile with
the rotationally averaged profile is shown in the middle panel, with its 1D
profile from the center shown on the right. Notably, the dropoff in the 1D
profile does not appear to behave like an error function; instead, it is
similar to an image convolved with a ring. This suggests how our model could
be related to Gloge’s power distribution model.

##  3 Hardware

In order to measure FRD well, the input illumination into the fiber must be
well-characterized, and the camera must have enough precision to determine
small variations in the output profile. This experiment is designed to
compactly measure FRD variations in a fiber illuminated with uniform intensity
up to a given numerical aperture. Furthermore, the experiment is designed so
that the illumination is uniform over a spot of a $\gtrsim 9.5$mm size,
putting a less strict requirement on fiber and illuminator alignment. But the
primary purpose of this design is to enables FRD measurement of a fiber
throughout its range of motion in a Subaru fiber positioner without constant
realignment; as stated in Section 6, the fiber positioner patrol region size
is 9.5mm and fits within the illumination spot size.

The illuminator system generates a 10mm diameter telecentric image with
uniform pupil illumination. Light from a 450nm LED source is collimated by a
100mm focal length, 50mm diameter planoconvex lens. An aperture stop is formed
by an EDC-15-08160-A engineered diffuser from RPC Photonics, which uniformly
distributes the collimated beam into a $15^{\circ}$ cone. At the aperture
stop, an 18-vane adjustable iris provides a round aperture of adjustable
diameter. Light is focused to the image plane with a 100mm focal length, 50mm
diameter doublet, located near the aperture stop. A 100mm focal length, 15mm
diameter plano-convex singlet is used as a field lens, ensuring the
illumination is across the image plane. All optics were mounted into an off-
the-shelf cage system, and baffling was used to shield the illuminator from
external light. The experimental illuminator and camera are shown in Figure 4
and in a Zemax schematic in Figure 5.

Figure 4:  Picture of the main camera and illuminator setup of the experiment,
with fiber extending off the right side of the image. Components of the
illuminator and camera setup are labeled. Figure 5:  Cutout schematic view
from above of the apparatus used in the experiment, with blue rays outlining
the optical path through the system. The illuminator is at the top of the
figure consisting of the LED through the field lens, while the camera is at
the bottom with the achromat and sensor. The fiber in a Cobra fiber positioner
was also fitted with a microlens to map a f/2.2 beam to an f/2.8 beam. The
illuminator and camera were mounted securely to an optical breadboard and the
camera was covered with a shroud during measurement.

The illumination color was chosen to be approximately monochromatic to avoid
concerns about FRD variations with wavelength, because there is literature
indicating that FRD effects increase with increasing wavelength. In
particular, Gloge’s coupling coefficient $D$ (see Section 2) is proportional
to $\lambda^{2}$, causing the predicted FRD of his model (equation (3)) to be
proportional to wavelength, and there are groups that have experimentally
found such a dependence (e.g., Poppett & Allington-Smith (2007)). As a result,
FRD in the red to near infrared bands measured in the PFS may be larger than
the values calculated here. Wavelength dependence with FRD is not fully
established, however; for instance, see also Murphy _et al._ (2008).

The light from the illuminator is incident upon a Polymicro fiber with model
number FBP127165190 (part number 1068020148) indicating a FBP fiber with a 127
$\mu$m core diameter, 165 $\mu$m clad diameter, and 190 $\mu$m buffer
diameter; the same type of fiber as will be used in the PFS. FBP fiber is a
low hydroxyl (-OH) fiber well-suited for astronomy, designed to accept
broadband illumination from near infrared to ultraviolet; additional
measurements on wavelength-dependent throughput and FRD of this fiber are in
dos Santos _et al._ (2014). The fiber end is terminated and either fitted with
a zirconium ferrule or, when in a Cobra fiber positioner, fitted with a
microlens in the Cobra fiber arm (See section 6). The microlens takes an f/2.2
beam at the edge of the acceptance range in the fiber and increases its f
number to f/2.8. The f/2.8 beam is slow enough to comfortably be accepted by
the f/2.5 acceptance cone of the spectrograph in the final system, minimizing
throughput losses. While the microlens may have its own effect on FRD
affecting the final image, this was not observed in our tests.

The image from the fiber is projected onto a CMOS camera system, with a 15mm
diameter, 25mm focal length achromatic lens to focus the fiber light onto the
camera sensor. Although the camera was mounted onto an optical breadboard, is
was not aligned with the fiber end by cage system. Alignment was completed
visually with a Newport LPV-1 5-axis mount that held the fiber end (terminated
with a zirconium ferrule) based on the output image from the camera. During
operation the camera mount was covered with a black shroud to prevent stray
light from entering the camera.

Image data was analyzed via a Matlab program interfacing with the camera. The
center of the profile was determined after fitting for its center after
subtracting out a background to remove scattered light and detector offset.
Additionally, flats were taken for the camera pointed at a white background.
While analysis was done without the flat correction initially, a later flat
was used to determine the camera response and vignetting. This correction had
a minimal effect (on the order of 0.1 mrad) on the calculation of
$\sigma_{TOT}$. The center of the circle was found by calculating “white” and
“dark” values by fitting Gaussians to histograms of number of instances for
each brightness occurrence, then finding all points with 30%, 50% and 70% of
the “white” minus “dark” values, fitting each contour to the equation for a
circle. Afterward, the azimuthal average of points at the same distance
rounded to the nearest pixel from the center was taken and combined into a 1D
profile, from which the FRD was analyzed. Small shifts on the order of a pixel
could affect FRD measurements by 0.1 mrad, a nontrivial variation, so center
finding was analyzed carefully. To verify that the center was indeed found, in
initial images the center was tested by shifting the center within a 5px x 5px
square, with the effects on the 1D profile (and in particular, $\sigma_{TOT}$)
analyzed.

To characterize the dropoff of the profile due to FRD, we found the width
($\sigma_{TOT}$) of a 2D Gaussian that, when convolved with a uniform tophat
profile, was the same width as the observed dropoff (section 2). An example of
this procedure is demonstrated in Figure 6, which shows the 2D profile and
radial profile, and fit for $\sigma_{TOT}$ depicted in Figure 7. In practice,
the width of the dropoff (from 85% of max brightness to 15% of max brightness)
was calculated and scaled to $\sigma_{TOT}$. Fitting for the 85% and 15%
allows for a fit that is less biased from small deviations from a normal at
large angles due to scattered light, such as due to the small bump at large
radius visible in Figure 6. Furthermore, this fitting method is more
meaningful when angular misalignment is more prominent; each profile for a
variety of angular misalignments is normalized to the width of the dropoff
between the 85% and 15% brightness levels, as described in more detail in
section 5.

Figure 6:  Example data analysis image output from 2D image (left) scaled
according to counts at the camera to radial profile extracted from the fitted
center. The darker spots visible in the image were dust particles and are not
intrinsic to the profile, though radially averaging minimizes the effect this
has on the data.  Figure 7:  A zoomed-in image of the radial profile around
the region of interest, the dropoff. A model normal error function calculated
from the difference between the 85% and 15% brightness levels (dashed lines)
is in good agreement with the observed profile. The width, as indicated in the
figure, can be extracted and is proportional to $\sigma_{TOT}$.

Measurement of the widths of the profile dropoffs are made in pixels and then
converted into equivalent angular size. However, this means that measurement
precision is dictated by pixel size; in practice a pixel corresponded to
approximately 0.1 mrad. Multiple images from the same experimental state had
dropoff width varying with $\pm 1$ pixel, corresponding to a $\pm 0.1$ mrad
uncertainty from width calculation.

An uncertainty in the calculation in this method was caused by the calculation
of the 85% and 15% brightness radial locations. In particular, the uncertainty
for these points was taken to be half the range of radii with the 85% and 15%
values within one standard deviation of the average value (scaled for counting
statistics with number of uncorrelated data points, respectively; see section
1.2.1).

The basic data analysis was verified with a 100 $\mu$m pinhole replacing the
fiber to visualize the input to the fiber. The resulting image is shown in
Figure 8. Notably, the pinhole behaves like a zero length fiber, so it should
have a sharp dropoff, whereas there is a nonzero width dropoff actually
observed. This effect is expected to add in quadrature with the FRD to give a
final width, but the sharpness of the profile was such that this effect was
assumed to be negligible. Also, the irregularities seen in the 2D pinhole
image demonstrate a pattern with characteristic size of about 20 pixels; in
the calculation of the uncertainty of the 85% and 15% brightness levels, the
standard deviation of the mean was calculated by dividing the calculated
standard deviation at a given radius by the square root of the effective
number of uncorrelated pixels at that radius (that is, number of pixels at
that radius divided by 20).

Figure 8:  Measurement with the fiber replaced by a pinhole. This measurement
represents the instrumental limit for the low FRD condition. The correlated
speckles in the 2D profile are likely due to fabrication variations across the
surface of the diffuser. The width of the dropoff in the radial profile is
much lower than the dropoff measured from fibers; the nonzero width may
indicate a systematic - though small - overestimation of FRD.

##  4 Stresses

It is informative to examine potential sources of FRD due to the fiber itself.
Ideally, a perfect fiber would transmit the exact same signal out as was
input, but imperfections in the fiber geometry can induce coupling to higher-
order modes and “spread out” the input light causing FRD. In this subsection,
we investigate three sources of stress on a fiber and their effect on FRD
using an effective f/2.8 beam. Effects from bending, twisting, and squeezing
are presented and related by calculating stress $\sigma$ on the fiber, with
subscripts differentiating stress and FRD. However, only a perfunctory attempt
has been made in this work to model FRD as a function of stress, because
length over which stress is applied, degree of imperfections in the core/clad
interface, fiber cladding Young’s modulus, direction of stress applied, and
many other factors complicate such an analysis.

We first examined stress due to bending. A Polymicro fiber was subject to a
loop of a measured, decreasing radius while measuring FRD, with FRD results in
Figure 9 and example loop inset. As the literature indicates Clayton (1989),
large-scale bending (with respect to the 127$\mu m$ core of the fiber) does
not significantly contribute to the FRD of this system. In particular, no FRD
change was found until a bend radius of about $1.25$ cm, where it increased by
1 mrad.

Figure 9:  FRD as a function of bend curvature, with fiber bent as shown in
the inset. No observable change in FRD occurred until a bend radius of 1.3 cm.

The stress due to a loop of radius $R$ and a fiber with fiberglass core/clad
of core radius $r_{core}$ depends linearly on its Young’s modulus $E$, as
shown in Equation (8). Note that the Young’s modulus for fiberglass used is
large relative to other materials but standard for optical fibers, while fiber
radius (with buffer included) is taken to be small relative to loop radius.

$\sigma_{bend}\approx Er_{core}\frac{1}{R}$ (8)

Hefferon (1987) where $\sigma_{bend}$ corresponds to the induced stress on the
fiber. In particular, in this fiber the stress where FRD notably increased
would be about 500 MPa as a result.

It is worth considering twist as a potential contribution to FRD in comparison
to bending, because twisting is expected to be present in the Cobra fiber
positioner system up to a full $360^{\circ}$ range of motion. It would be
informative to determine the stress on the fiber due to twist to predict if
there will be an observable effect on FRD. With shear modulus $G$ of clad
radius $r_{clad}$, Equation (9) applies for twist of angle $\theta=\pi$

$\sigma_{twist}=Gr_{clad}\frac{\theta}{L}$ (9)

Hefferon (1987) over length $L$ dictated by the Cobra system, notably similar
to the bending stress equation (Equation (8)). In this experiment, the fiber
is held in place with epoxy at the fiber arm (see Figure 13) at the end of the
fiber positioner on one side (see Section 6), but is not otherwise
significantly constrained (allowing $L$ to be on the order of a meter). Thus,
the fiber is able to twist over the order of the length of the fiber, and the
twist stress is proportionally orders of magnitude lower than that considered
in the bend test above. It is expected that twist will not have a significant
effect on FRD in this system. In the PFS, the distance between the fiber arm
and a segmented tube over which twisting occurs is shorter at about 28 cm, so
twist will contribute a larger stress than that from an unconstrained fiber
but still significantly smaller than the maximum bending stress tested in this
paper.

Beyond bending and twisting that may be induced in fibers during operation of
the Subaru telescope, pinching is another possible additional sources of
stress. There are strain relief boxes to alleviate this kind of issue, but it
is still expected that some amount of applied stress will change during system
operation.

To test the effect of pinching on a fiber, a Polymicro fiber was subject to an
increasing weight on an aluminum base over a $L=1.5$ cm length of fiber, up to
a few kilograms. The weight can be converted to an average pressure after
calculating the effect of compression of the fiber itself due to the weight.
In particular, the half-width $b$ of the fiber due to the compression is given
by

$b=\sqrt{\frac{4F\left(\frac{1-\nu_{1}^{2}}{E_{1}}+\frac{1-\nu_{2}^{2}}{E_{2}}\right)}{\pi
L\left(\frac{1}{R_{1}}+\frac{1}{R_{2}}\right)}}$ (10)

Bamberg (2006) where $F$ is the force applied to the fiber, $\nu$ is the
Poisson’s ratio for the two interfacing objects, denoted by the subscripts
(let object 1 be the aluminum base and object 2 be the polyimide buffer of the
fiber), $E$ is Young’s modulus, and $R$ is the radius of each object.

Average pressure $P$ on the buffer thus is proportional to $\sqrt{F}$. A plot
demonstrating the $\sigma_{TOT}$ dependence on $P$ is shown in Figure 10.
There is an approximately linear relationship between pressure on the buffer
and $\sigma_{TOT}$. However, light propagates through the core and depends on
the core/clad interface, so the buffer pressure alone may not characterize the
physical nature of this FRD increase.

Figure 10:  Variation of FRD with applied radial pressure to the fiber buffer.
FRD appears to grow roughly linearly with applied buffer pressure, although
extrapolation to lower pressures may not be accurate. Scatter also increases
with applied weight, indicating how small variations in pressure distribution
can significantly affect FRD measurements as weight increases. Error bars in
this figure indicate the standard deviation of all measurements at a given
pressure.

##  5 Angular misalignment

Angular misalignment is known to contribute to the power distribution fiber
output. Due to this, measurements of FRD often rely on optical systems that
are carefully aligned. This holds especially true for the cone test. While
ideally in practice every fiber would be coaxial with a uniform beam, in
practice, a fiber in a Cobra fiber positioner will move through a range of
angular misalignments during operation due to impossibility of perfect angular
alignment. Furthermore, due to the design of fibers throughout the focal
plane, some fibers necessarily receive a non rotationally symmetric profile,
violating the simplification that permits $\sigma_{FRD}$ to characterize the
point spread function without additional parameters. Thus, angular
misalignments’ effects on FRD must be well understood to correct for
foreground sky signal throughout telescope operation for any fiber
positioning.

A test was performed with a f/2.8 beam using two Polymicro fibers (with
ferrules on both ends) to determine the effect of angular misalignment on the
measurement of FRD. To investigate the effects of angular misalignment, the
FRD from each fiber (taken individually) was minimized with respect to the two
angular degrees of freedom in a Newport LPV-1 mount. The minimum was taken to
correspond to the least angular misalignment. Then, adjustments to this
position were taken in $\approx\frac{1}{8}$ mm steps in the screws holding the
mount’s angular position (corresponding to about 3 milliradian angular
adjustments) with FRD measured at each stop. The angular misalignment was
determined from the geometry of the mount relative to its calculated minimum.
The results from this test as well as fits to equation (7)’s prediction are
shown in Figure 11.

Figure 11:  Variation of FRD with geometric angular misalignment from fitted
minimum for two fibers. The effect is approximately hyperbolic, fitted for
$\sigma_{FRD}$ (see Section 5) and an x-axis offset. The bottom panel shows
the residuals, which are subtracted from the fit. The minimum $\sigma_{TOT}$,
roughly corresponding to the $\sigma_{FRD}$ of each fiber, varies between the
two but is within expected variation. While the fit is good for just
effectively fitting to a minimum, there appears to be a nontrivial residual
for the fiber of larger $\sigma_{FRD}$.

However, angular misalignment can also be determined directly from the
profile. By taking the width between the 85% and 15% brightness positions, a
family of 1D profiles with angular misalignment ratios $A$ from 0 to 6 were
calculated with 0.1 step increments (See section 5 for details about $A$) and
fitted to the profile, with the final value of $A$ selected from the best fit.
Because both $A$ and $\sigma_{TOT}$ are calculated from this model, using
equation (7), $\sigma_{FRD}$ and thus angular misalignment $a$ can also be
found. The geometric angle can be compared to the angle predicted from the
model to determine the model’s efficacy, as shown in Figure 12

Figure 12:  Angular misalignment between the geometric misalignment and the
algorithmically calculated misalignment, with errors calculated from
$\sigma_{TOT}$ and the range of angular misalignment with sum of least squares
within 5% of the minimum fit. Agreement between the angles is low at lower
angular misalignments, possibly due to nonuniform profile effects that
dominate misalignment effects for low misalignment, but is good at large
angles. This result indicates that the misalignment extraction from the model
is reasonable.

Verification of the angular misalignment effect on FRD with angle is
particularly valuable when interpreting the FRD with Cobra fiber position,
because an imperfectly aligned Cobra will traverse a range of angular
misalignments that can noticeably affect FRD measurements.

##  6 Cobra FRD and Discussion

The Cobra Fiber Positioner is a $\theta$-$\phi$ eccentric-axis fiber
positioner that permits a fiber to be located anywhere within a 9.5mm patrol
region. The Cobra units were constructed by New Scale Technologies, and the
positioning of a Cobra was controlled using New Scale Technology’s Pathway
program with pulses of specified number of steps and length. However, mapping
from steps to angular position is not perfectly accurate which does cause an
uncertainty in position that is not calibrated for. To minimize angular error,
the Cobra fiber positioner was mounted to a machined Thorlabs LCP01 piece to
be aligned using the Thorlabs cage system. Figure 13 depicts a Cobra used in
the conducted experiment.

Figure 13:  Schematic cutaway of a Cobra fiber positioner, with various parts
and parameters marked. Figure from the Subaru PFS collaboration’s Cobra Fiber
Positioner Manufacturing Readiness Review. Also see Fisher _et al._ (2014) for
a Cobra figure and more discussion of the Cobra.

The FRD of a fiber fitted in a Cobra fiber positioner with a microlens was
measured throughout the range of the positioner’s motion in both stages with a
f/2.2 beam, which becomes f/2.8 after the microlens. Due to the nature of the
Cobra mount used, angular alignment could not be adjusted to be minimized. The
FRD was measured in steps of 500 steps from 0 to 3500 in stage 1 and back
again to determine hysteresis. The measured FRD is shown in Figure 14.

Figure 14:  Profile $\sigma_{TOT}$ of a fiber with microlens throughout the
range of motion in both stages of the Cobra positioner. There is a significant
variation of $\sigma_{TOT}$ through stage 1 ($\theta$ stage) and stage 2
($\phi$ stage) motion. The sinusoidal behaviour across stage 1 indicates an
angular misalignment effect. Uncertainties of about $\pm 50$ steps in stage 1,
corresponding to $\pm 5^{\circ}$ angular precision, were also present though
not depicted in this figure.

The FRD through the positionings in stage 1 ($\theta$ stage) appears to
increase and then decrease, completing a cycle in the $360^{\circ}$ range of
stage 1. Furthermore, while FRD varies across stage 1 motion, there is no
noticeable systematic increase in FRD after the full $360^{\circ}$ twist is
imposed on the fiber, which would increase twist while minimizing net angular
misalignment change. Thus, twist appears to not contribute noticeably to FRD
as was expected due to its low imposed stress (see Section 4). Thus, FRD
variations across stage 1 appear to be due to angular misalignment.

Across positions in stage 2, there is a more significant increase in
$\sigma_{TOT}$. However, the amplitude of the variations across stage 1 also
increase, indicating a larger angular misalignment. This complicates the
analysis of bend stress vs. angular misalignment as the primary cause of the
FRD increase.

The expected bend stress induced due to the Cobra fiber positioner can be
calculated from Equation (8). By substituting in $R=3.12$ cm, the minimum
inverse curvature of the fiber in a Cobra after approximating its path as a
helix, the resulting stress is lower than the stress in the bending test where
$\sigma_{TOT}$ notably increased.

The angular misalignment can be measured by applying Equation (7) to the range
of FRDs through stage 1 or stage 2 motion and calculating the variation of
angular misalignments required to generate the resulting plot. This angular
misalignment measurement can help give insight about the possible effect on
FRD during Cobra operation. The extracted angular data agrees with the above
predictions, as seen in Figure 15.

Figure 15:  Angular misalignment as calculated from the power distribution
model in this paper (section 2). The sinusoidal behavior across stage 1
suggests that the $\sigma_{TOT}$ variation in stage 1 is indeed due to angular
misalignment rather than stress, and angular misalignment due to the $\phi$
stage is also apparent.

The angular misalignment extracted from the profiles thus allows for
$\sigma_{FRD}$ to be extracted directly from the profiles in this system, as
shown in Figure 16. These values of $\sigma_{FRD}$ are broadly consistent with
a constant $\sigma_{FRD}$ of about 6.5 mrad throughout all fiber positions,
indicating that the stresses during operation did not significantly increase
FRD. From the values of $\sigma_{TOT}$ alone, it would be impossible to
determine the source of profile broadening.

Figure 16:  FRD extracted from the fiber profiles after considering angular
misalignment. The FRD can be well-characterized in a single experimental setup
across varying configurations due to this angular misalignment extraction, and
is consistent with minimal stress on the fiber during Cobra operation.

In conclusion, a model of fiber profile due to FRD and angular misalignment
was presented and used to analyze cone profiles of a fiber under stress or
angular misalignment. While high stresses were found to increase FRD, FRD did
not significantly increase for low stress. Angular misalignment was found to
have a significant effect on the profile width, which was fitted by the model
considered. The FRD component of the model is represented as a Gaussian as
seen in previous literature results (e.g. Haynes _et al._ (2011)), whereas the
angular misalignment component of the profile is simplified into one
parameter. Some stresses that would be present in the Subaru PFS’s Cobra fiber
positioner system are considered and expected to be low during operation. The
angular misalignment from the presented model is compared to a test with
angular misalignment calculated geometrically and found to be consistent for
angular misalignments larger than about 5 milliradians. Thus, the model could
then be applied to the Cobra fiber positioner throughout its range of motion,
indicating minimal FRD increase, if any, that would have otherwise been
obscured by the angular misalignment present in the system. An experimental
setup such as used in this paper combined with the model used should simplify
analysis of FRD in dynamic systems without the need to realign for each
configuration to be tested.

## Acknowledgments

We are grateful to the Kavli Institute for the Physics and Mathematics of the
Universe, the National Astronomical Observatory of Japan, and the Subaru PFS
collaboration for their feedback and recommendations throughout this project.
We would especially like to thank Naoyuki Tamura and Yuki Moritani for
discussions and direction for this project, as well as Leandro dos Santos for
fiber preparation. This material is based upon work supported by the National
Science Foundation under MSIP grant no. 1636426.

This research has made use of NASA’s Astrophysics Data System.

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