# Optimum telescope focal ratios for microlens-to-fiber coupled integral field
spectrographs

Sabyasachi Chattopadhyay  South African Astronomical Observatory, 1
Observatory Rd, Observatory, Cape Town, 7925, South Africa  Matthew A.
Bershady  South African Astronomical Observatory, 1 Observatory Rd,
Observatory, Cape Town, 7925, South Africa  University of Wisconsin,
Department of Astronomy, 475 North Charter Street, Madison, WI 53706, USA
Department of Astronomy, University of Cape Town, Private Bag X3, Rondebosch
7701, South Africa  Marsha J. Wolf  University of Wisconsin, Department of
Astronomy, 475 North Charter Street, Madison, WI 53706, USA  Michael P. Smith
University of Wisconsin, Department of Astronomy, 475 North Charter Street,
Madison, WI 53706, USA

###### Abstract

We describe the optimum telescope focal ratio for a two-element, three
surface, telecentric image-transfer microlens-to-fiber coupled integral field
unit within the constraints imposed by micro-optics fabrication and optical
aberrations. We create a generalized analytical description of the micro-
optics optical parameters from first principals. We find that the optical
performance, including all aberrations, of a design constrained by an analytic
model considering only spherical aberration and diffraction matches within
$\pm$4% of a design optimized by ray-tracing software such as Zemax. The
analytical model does not require any compromise on the available clear
aperture; about 90% mechanical aperture of a hexagonal microlens is available
for light collection. The optimum telescope f-ratio for a 200$\upmu$m core
fiber fed at f/3.5 is between f/7 and f/12. We find the optimum telescope
focal ratio changes as a function of fiber core diameter and fiber input beam
speed. A telescope focal ratio of f/8 would support the largest range of fiber
diameters (100 to 500$\upmu$m) and fiber injection speeds (between f/3 and
f/5). The optimization of telescope and lenslet-coupled fibers is relevant for
the design of high-efficiency dedicated survey telescopes, and for retro-
fitting existing facilities via introducing focal macro-optics to match the
instrument input requirements.

###### keywords:

Integral Field Spectroscopy; Microlens-Fiber IFU; Microlens Optical Design:
Telescope f-ratio.

* Further author information: (Send correspondence to Sabysachi Chattopadhyay)
E-mail<EMAIL_ADDRESS>Telephone: +27 78 298 6850

##  1 Introduction

Modern day spectroscopic surveys employ collecting apertures running from
10m-class telescopes (VIRUS[1, 2], PFS[3], MSE[4]), to modest 4m-class
telescopes (e.g. DESI[5], 4MOST[6]), down to even a few hundred mm (LVM[7]).
At the heart of each of these surveys is fiber-optic coupled spectroscopy.
Fiber optics serve as a convenient bridge between the mobile telescope system
and the more stable spectrograph mounts, as well as a image reformatting
system from the telescope focal plane to spectrograph entrance slit. While
Galactic and extragalactic surveys have traditionally employed single fibers
to multiplex redshift and spectral diagnostic measurements of stars, distant
galaxies and quasars, more recently there has been an additional focus on
mapping resolved sources (nearby galaxies and Galactic nebulae) using
integral-field spectroscopy (IFS). In general, fiber-fed IFS (IFUs; e.g.,
SparsePak[8], PPak[9], VIRUS-P[1], VIRUS-W[2], MaNGA[10], and MEGARA[11]) are
flexible, cost-effective and suitable for optical and near IR wavelengths.
Fibers also lend themselves well to multi-object IFS of distributed and/or
extended sources contained in large (often many deg2) telescope fields of view
(e.g., SAMI[12] on the 4m AAT and MaNGA[13] on the 2.5m Sloan Telescope).
Consequently fiber-fed IFS remains an appealing approach along side other
flavours of IFS (e.g., image-slicing with SPHERE[14], MUSE[15] and KMOS[16] on
VLT, or KCWI[17] on Keck).

Despite fiber optics being a popular and cost-effective mode of coupling
telescopes to spectrographs, fibers are well-known to degrade the input focal-
ratio upon output. This focal ratio degradation (FRD) via stepped index,
multi-mode fibers[18, 19, 20] affects the efficiency of observation because it
is in essence an injection of entropy into the information gathering system.
It is observed [21, 22] that the effect of FRD can be minimized by injecting a
faster beam into the fiber.

A contemporary measurement of this effect is illustrated in Figure 1 using a
differential reimaging system similar to [8, 10, 23]. The focal ratio
degradation was measured for a 2-m length of 400$\upmu$m diameter core step-
index, broad-band fiber (MOLEX/Polymicro FBP 400:440:470) fed with a centered,
reimaged near-field spot of 100$\upmu$m in diameter. The input focal ratio was
varied by changing the aperture of an intermediate pupil-stop. The output
focal ratio was measured at a radius covering 98% of the normalized encircled
energy. The salient feature of this measurement was that it was performed with
a well-aligned inject beam, telecentric to within 0.1$\degree$ in tip/tilt
similar to that of [23]. The upper limit on the angular deviation limits
geometric FRD (gFRD) to no faster than f/286; over the range of injection
speeds this is inconsequential. While the left hand panel of Figure 1 shows
the well-known non-linear ‘saturation’ of the output beam speed, in the right
panel we present the same measurements as a ratio of out vs input solid angle.
This has a simple, linear behavior with input numerical aperture from which
entropy gain due to fiber FRD can be easily estimated. The qualitative
behavior motivates why present-day fiber-coupled spectrographs tend to have
faster collimators (see the references above).

Figure 1: Focal ratio degradation by a 2m long 400$\upmu$m diameter core fiber
for various input focal ratio measured at a normalized encircled energy of
98%. Measurements details are described in the text. The figure depicts the
effect of focal ratio degradation via fiber increases with slower input beam
in the left panel while the right panel shows the linear loss of solid angle
on sky with decreasing input numerical aperture.

It is also the case that for IFS using bare, densely packed circular fibers,
the fill factor is typically only 60% [10] given the inherent geometry plus
the minimum requirements for cladding and mechanical buffer. Some gains can be
had by removing the buffer and lightly fusing the fibers, but is technically
challenging and comes at the cost of more FRD [24]. Consequently, it is often
desirable to couple fibers to the telescope focal plane via lenslet arrays.
One purpose of lenslet coupling is to increase the fill-factor of the fiber
array. In principle, circular lenslets with hexagonal packing or hexagonal
lenslets with circular chromium coating can increase the fill-factor up to
90%. We consider this lenslet geometry in this paper. However, when coupling
lenslets to fibers, a further challenge arises: The clear aperture in a micro-
lens system falls quickly compared to the available mechanical aperture with
faster input beams from telescope [25, 26]. Thus in an effort to reduce FRD
one would struggle to attain an acceptable fill factor on sky [25].

The aim of this study is to resolve the tension between the optimum input
focal ratios for lenslet and fiber optics. This is part of the broader
question i.e. given a survey being planned, either using an existing telescope
or a new telescope, what would be the ideal telescope focal ratio at the input
of the microlens for a set of fiber input speed, fiber diameter and fabricable
microlens properties. To do so we develop a generic analytical description of
an image transfer IFU optical model to optimize the telescope focal ratio
within the constraints posed by the microlens array fabrication. We assume the
spectrograph design accommodates a fast fiber output beam or, alternatively,
uses additional microlenses at the output end of the fiber to modulate the
speed of the spectrograph collimator. We start in Section 2 by describing the
optical parameters of an image transfer IFU system and the parameter space
available for optimization given fabrication limitations and FRD. In Section 3
we describe the analytical model to define the IFU optical parameters.
Following sections discuss two distinct regimes where on-axis aberrations are
either dominated by diffraction or spherical aberration (SA) (Section 4) and
the design choices that minimize light loss from aberrations in general
(Section 5). We confirm the reliability of our model compared to ray-trace
calculations in the non-diffraction-limited regime in Section 5 as well. In
Section 6 we describe the acceptable range of telescope beam speed defined by
the fabrication limits. Section 7 summarizes our results and presents our
conclusions.

The result of our analysis calls for much slower telescope beams than
developed for telescopes optimized for bare-fiber feeds, such as HET and SALT,
which are close to f/4. The optimum telescope f-ratio we find is closer to
f/8, although in detail we will show this depends on the fiber core diameter.
The optimum f-ratio can be achieved on an existing telescope via a macro-focal
reducer or expander [27, 28, 29]. However, for a dedicated survey facility we
suggest optimizing the native telescope beam speed to minimize the total
number of optical elements [30, 31].

##  2 Design choices and constraints

###  2.1 Image transfer microlens system

We describe telecentric reimaging (or image-relay) microlens systems in this
paper, and note differences with pupil-transfer microlens systems below. The
basic optical concept is to re-image the focal plane on to the fiber cores
through a set of two MLAs. Conceptually, the first MLA elements act as field
lenses while the second produces telecentric images. In practice, to optimize
image quality the optics deviate slightly from this idealized prescription
[see for example VIRUS2 design and Section 5 here][32]. A bi-convex MLA (BC)
with appropriate radius of curvature (RoC) and thickness creates a pupil at
its exit surface, and the exit beam is collimated for all field points. A
suitable gap (depending on the RoC of the second MLA) of the BC and plano-
convex MLA (PC) ensures a telecentric image at the flat back surface of the
PC.

One of the free design parameter is the micro-image size at the input of the
fiber. However, in our previous study [26], we found that the micro-image
diameter should be $\sim$97% of the fiber core diameter in order to minimize
étendue gain. Thus the fiber core diameter and micro-image diameter are used
interchangably in our analysis. To maximize grasp (area–solid-angle product,
or A$\Omega$), the ideal micro-image diameter ($\rm d_{m}$) should be equal to
the diameter of the fiber core. However, due to fiber positioning and
alignment issues $\rm d_{m}$ needs to be smaller than fiber core. Based on our
own photo-lithographic work and comparison to commercially available products
we estimate current technology permits fibers to be positioned within $\rm\pm
3~{}\upmu$m RMS at any desired spacing[26].

Figure 2: Arrangement of a reimaging microlens array system that captures a
part of the telescope focal plane ($\rm d_{a}$) and injects this into the
fiber core. Key design parameters are: (a) Field 0: Central field point; (b)
Field 1: Edge field point; (c) $\rm{d_{a}}$: Diameter of the input beam/Bi-
convex microlens (BC) clear aperture diameter; (d) $\rm{r_{b}}$: BC radius of
curvature; (e) $\rm{D_{b}}$: BC thickness; (f) d: Field 0 pupil diameter; (g)
$\rm{r_{p}}$: plano-convex microlens (PC) radius of curvature; (h)
$\rm{D_{p}}$: PC thickness; (i) $\rm{D_{G}}$: Air gap width between PC and BC;
(j) $\rm{d_{m}}$: micro-image diameter/fiber core diameter. Details about
implemented microlens properties are discussed in [33].

Figure 2 illustrates a typical image-relay microlens system where telescope
light is fed at a focal ratio of $\rm f_{\rm tel}$. This beam enters the bi-
convex microlens with $\rm r_{b}$ radius of curvature, $\rm D_{b}$ thickness,
and $\rm d_{a}$ clear aperture. The thickness is defined such that the beam
forms its pupil at the exit surface of the bi-convex microlens. The pupil is
then focused to an image plane via a plano-convex lens with $\rm r_{p}$ radius
of curvature and $\rm D_{p}$ thickness, sitting at a distance of $\rm D_{G}$
from the bi-convex lens. $\rm D_{p}$ and the beam diameter of the central
field defines the injected f-ratio ($\rm f_{\rm fib}$) into the fiber. Since
the pupil is positioned at the exit surface of the bi-convex MLA, one can
define $\rm D_{p}$ as $\rm n_{g}\times r_{p}$/($\rm n_{g}$ \- 1) so that the
plano-convex microlens produce telecentric micro-image at its planar back
plane. Thus $\rm f_{\rm fib}$ and hence $\rm d$ and $\rm D_{p}$ define $\rm
r_{p}$ as well. On the other hand, $\rm D_{G}$ defines the MLA clear aperture
diameter which in turn defines the sky sampling size (via plate scale). We
provide mathematical prescriptions of all these dependencies in the following
section 3 and more details about their derivation can be found in Appendix A.

###  2.2 Other microlens systems

Other microlens relay options include transfer of the image with a pupil onto
the fiber entrance aperture. These so-called pupil-transfer systems have the
advantage of requiring only one MLA. In the simplest case using a single
plano-convex MLA can be bonded directly to the fiber to minimize reflection
losses. Unfortunately, the pupil image onto the fiber is not telecentric,
leading to potentially significant gFRD. Because the fibers azimuthally
scramble the input signal, the non-telecentric feed cannot be corrected at the
fiber output. It is straightforward to correct for this by replacing the
plano-convex MLA with something similar to the biconvex MLA seen in Figure 2.
Consequently pupil-transfer systems will have similar, but simpler
descriptions and constraints compared to the image-relay microlens systems.
For this reason we break out the constraints from both the bi-convex and
plano-convex elements.

###  2.3 The merit of telecentric designs

While the re-imaging MLA system does not require additional air-glass surfaces
to create a telecentric design, it does place constraints on the lenslet
design as we will see below. Further, as noted above a telecentric pupil-
imaging MLA system does require the addition of two air-glass surfaces.
Broadly, then it is worth considering whether imposing telecentric designs are
worth the cost in either design constraints or reflection losses. It is
easiest to understand this in the context of the pupil-imaging MLA system as
follows.

As a general, qualitative statement for the telecentric pupil-imaging MLA
case, given modern multi-layer anti-reflection coatings with broad-band
performance achievable at 0.5-1% reflection loss per air-glass surface over
more than an octave in wavelength, the bar is not set very high to gain by
adding a two air-glass surfaces to eliminate gFRD.

Equation 3 of [34] can be used to compute the throughput loss due to gFRD. The
fraction of the light lost is given by $L\sim(0.4/B)(\delta/u)$ where $u$ is
the angle of the marginal ray with respect to the apex of the telecentric
beam, $\delta$ is the angle of the marginal ray in the non-telecentric beam
with respect to the apex of the telecentric beam at the field edge, and $B$ is
the fractional area of the telescope primary mirror unobscured by the
secondary mirror. The angles in a PC MLA pupil-imaging system are illustrated
in Figure 3. As two representative extrema, $B$ has a value of about $0.95$
for the Gemini 8m telescope [35] and a value of about 0.7 for the Sloan 2.5m
telescope [36].

Figure 3: Ray schematic for the non-telecentric PC MLA pupil-imaging fiber
system (left). The telescope focal plane is formed at the location of the
left-most vertical dashed line. Blue rays trace the telescope pupil formed by
the lenslet and also the fiber-input light cones at the edge of the fiber. The
chief ray of these cones is at an angle $\delta$ with respect to the
telecentric angle. Red rays (dashed) trace the input cone on the fiber face at
the field center, with a marginal ray angle $u$. The geometric FRD loss based
on [34] is shown at right for two telescope central obstructions of 5% (black)
and 30% (red), and two fiber input beam speeds (f/3 and f/5 in air, bottom and
top curves, respectively for each color). The lenslet and fiber index of
refraction is assumed to be matched at $\rm n_{g}=1.46$.

Figure 3 shows that gFRD already leads to 10-30% throughput loss for $f_{\rm
tel}=10$ (depending on fiber injection speed and telescope central
obstruction), and this increases rapidly for faster telescopes. For any
plausible telescope f-ratio, uncorrected gFRD losses never fall below 1%. This
makes it clear that the telecentric design is preferable in all relevant cases
for pupil-imaging. For reference the GMOS IFU on Gemini [37] has a PC MLA
lenslet system fed at f/50 and injecting light into the fibers at f/4. This
system should have $\sim$5% thoughtput loss due to gFRD.

###  2.4 Critical design parameter space for image-relay microlens systems

The primary design parameters that have bearing on manufacturing limitations
include the surface curvature given by $\rm r_{p}$ and $\rm r_{b}$, and the
MLA thickness $\rm D_{b}$ or $\rm D_{p}$. Based on discussion with, e.g.
A$\upmu$S111Advanced Micro-Optic Systems, https://www.amus.de, we specify
these fabrication-based limits for these parameters:

  1. 1.

Maximum manufacturing limitation on the thickness ($\rm D_{b}$ or $\rm D_{p}$)
of a microlens of 10 mm and a minimum of 0.3/0.6 mm for PC and BC MLA
respectively.

  2. 2.

Fabrication limitation on the minimum radius of curvature of a microlens ($\rm
r_{b}$ or $\rm r_{p}$) of 0.02 mm.

The specific values for these parameters depend on the fiber input f-ratio
$\rm f_{\rm fib}$ and micro-image diameter $\rm d_{m}$ as well as $\rm f_{\rm
tel}$. We also bound $\rm f_{\rm fib}$ and $\rm d_{m}$ in our analysis, as
follows, leaving $\rm f_{\rm tel}$ unbounded. We constrain $\rm d_{m}$ by the
range of off-the-shelf multi-mode fiber size, with 50 $\rm\upmu$m as the
smallest, and 600 $\rm\upmu$m as the largest that has seen practical use [38].

The range of $\rm f_{\rm fib}$ depends on the range of acceptable f-ratios for
the spectrograph input (the collimator). Traditional spectrograph designs have
preferred slow collimators to increase spectral resolution by decreasing the
angular size of the slit while keeping the overall beam size as small as
possible. Slower input beams offer the added opportunity of demagnification
via fast camera optics to optimize matching to detector focal-planes. However,
with fiber coupling, FRD requires collimator speeds of f/5 to prevent
significant loss of etendue, as seen in successful fiber-fed spectrograph
designs (Hectoechelle[39]; GIRAFF[40]; upgraded WIYN Bench[41];
SDSS/BOSS[42]). Indeed, in more recent designs (e.g., DESI-f/3.7[5],
M2FS-f/3.4[43], WEAVE-f/3.1[44], MEGARA-f/3[11], PFS-f/2.8[3],
DOTIFS-f/4.5[45]), collimators are substantially faster to further minimize
FRD. Spectrograph collimators are able to accept f/2.5 with standard off the
shelf optics while custom modification may able to grasp faster beams[46]
approaching the fiber numerical aperture (typically f/2.2 for standard, high-
transmission broad-bandpass step-index fiber). We take $\rm 2.5<f_{\rm fib}<5$
to be a reasonable range for this parameterin order to keep FRD-induced
entropy increase minimal to modest (see Figure 1) in standard step-index fiber
optics with numerical apertures of 0.22, most commonly used in astronomical
instrumentation.

In summary, then:

  1. 3.

Fiber diameter $\rm d_{m}$ between 50 to 600 $\rm\upmu$m.

  2. 4.

Fiber input beam speed $\rm f_{\rm fib}$ between f/2.5 and f/5.

##  3 Analytic model of micro-lens optical parameters

In what follows it is useful to define the parameter $\rm\upeta$ as the ratio
of telescope to fiber focal-ratios:

$\upeta\ \equiv\ \rm f_{\rm tel}/\rm f_{\rm fib}.$ (1)

Appendix A then presents derivations of equations for all of the microlens
design parameters (surface curvatures $\rm r_{b},r_{p}$; thickness and spacing
$\rm D_{b},D_{G},D_{p}$) as a function of the telescope input and fiber input
focal-ratios, the size of the micro-image ($\rm d_{m}$) and the glass index
$\rm n_{g}$. As before, we equate the micro-image ($\rm d_{m}$) with the fiber
core diameter.

For a fiber-microlens based IFU, the fiber radius can be chosen based on the
camera and detector size and usually ranges from 50 to 600 $\rm\upmu$m. Thus
for each $\rm d_{m}$ and $\rm f_{\rm fib}$ we should be able to find a $\rm
f_{\rm tel}$ where the microlens radius of curvature is manufacturable while
providing adequate clear aperture as well as acceptable spot size even on the
edge field. At this point we can vary the telescope f-ratio and the “aperture
ratio” of the beam diameter at the PC array ($\rm d$) to the available
aperture diameter ($\rm d_{a}$). All these quantities are shown in the Figure
2.

##  4 Model Considerations: On-Axis Aberrations

The dominant on-axis aberrations in the micro-lens optical system are due to
sphericity and diffraction. Minimizing these as well as higher-order off-axis
aberrations is desirable to maximize the encircled energy injected into a
finite fiber core. While our analytical model can deal with geometric
aberrations (such as SA), the effort to minimize these depends on whether they
dominate. First we show that diffraction, even for micro-lens systems, remains
a minimal effect, which allows us to focus on the impact of minimizing SA on
the lenslet design. These calculations are done at an effective wavelength of
800 nm which is the logarithmic mid-point for good fiber transmission
(400-1600nm), and the worse-case limit for diffraction in visible-wavelength
systems.

To compare diffraction to SA we compare the impact on the radius enclosing 90%
of the encircled energy (EE90). In case of SA, the effect can be estimated by
the transverse component i.e. the radius of circle of least confusion (CoC)
which can be roughly estimated (in radians) as $\rm d^{3}/(16\rm r^{3})$,
where d is the diameter of the entry aperture and r is the radius of
curvature. Using equations 3 and 7 for $\rm d$ and $\rm r_{p}$ from Appendix
A, we can express this as $1/(16{\rm f_{fib}}(\rm n_{g}-1))^{3}$. The increase
in edge field EE90 due to diffraction is simply the first minimum of the airy
disk which is measured as $1.22\uplambda/\rm d$ radians. For a system which
converts an f/11 telescope beam into an f/3 fiber input within a microimage
diameter of 100$\upmu$m, the values of $\rm r_{p}$ and d are 0.365 mm and
0.267 mm respectively. At 800 nm, the EE90 due to SA is more than 6 times
greater than the same for diffraction. Figure 4 shows the range of $\rm
f_{fib}$ and $\rm d_{m}$ (note again: we are assuming $\rm d_{m}$ is the same
as the fiber core diameter) where diffraction or SA dominates for different
telescope beam speed. It is interesting to note that the system tend to be
more diffraction limited with slower $\rm f_{fib}$ for a given $\rm d_{m}$ and
$\rm f_{tel}$. Slower $\rm f_{fib}$ would also (1) introduce additional FRD;
(2) reduce the grasp of the system, (3) reduce the radius of curvature of the
optical elements; and hence are not advisable to use. Thus for practical
purpose we would concentrate on the range of $\rm f_{fib}$, $\rm d_{m}$ and
$\rm f_{tel}$ where SA dominates. Our f-ratio selection range also shows that
beyond $\rm f_{tel}/7$, most of our optimization space comes under systems
that are mostly SA dominated.

Figure 4: Range of fiber input beam speed and micro-image diameter (matched to
the fiber core diameter)

for which diffraction or spherical aberration dominates for different
telescope beam speeds. For practical purposes, the fiber core diameter will be
close to the micro-image diameter. Curves indicate the ratio of the airy disk
first minimum for a wavelength of 800 nm to the circle of least confusion at
values of unity (solid line), 1:3 (dashed; strongly spherical-aberration
limited) and 3:1 (dot-dash; strongly diffraction-limited). The aperture and
radius of curvature are computed using equations 3 and 7 from Appendix A. The
blue shaded region is the design target for fiber input f-ratio.

##  5 Constraining the telescope focal-ratio: minimizing throughput loss due
to aberrations

Given the dominance of SA over diffraction in most applications, we use our
analytic model to minimize the effect of spherical aberration on diminishing
throughput. We then check if this analytic model is accurate when considering
higher-order aberrations, and conclude on the limits our calculations place on
the telescope focal-ratio.

To minimize the effects of light-loss from SA, we assume the thickness of
microlens positions the CoC on the exit surface of the PC microlens (the fiber
input core location). We can estimate this PC thickness as the distance from
the entry into MLA to the point of crossing the optical axis by a marginal ray
given the separation between the chief and the marginal ray and lenslet radius
of curvature is known. This is an analytical description of the longitudinal
SA. We have used this description to derive the relative constraint between PC
MLA thickness and radius of curvature for a given $\rm d_{a}$ and $\rm d_{m}$
(hence $\rm\upeta$). Likewise for a given radius of CoC (depending on how fast
is the PC MLA) we can work out the losses for a fiber with a diameter of $\rm
d_{m}$. If the prescription on the PC MLA thickness and radius become too
severe we can define the trade-off in photon loss outside of $\rm d_{m}$ if
the CoC is not placed on the exit surface of the PC MLA.

Appendix B and C describe the analytical model to define the longitudinal and
transverse SA (LSA and TSA). We found that analytical solutions to be
cumbersome to define the location and radius of CoC and hence we used the
numerical approach. It may be noted that the result of the analytical model is
still important as an initial condition for performing optical simulation to
predict precise SA losses in specific IFU design studies.

We used d/2 as the marginal ray height and normalized with $\rm r_{p}$. Their
ratio only depends on $\rm f_{fib}$ and $\rm n_{g}$. On the other hand, the
normalized paraxial focus location is $\rm n_{g}/(n_{g}-1)$ as described in
Appendix B. Given this, we compute the radius and location of CoC and marginal
focus for a grid of $\rm r_{2}/r_{p}$ (refer to Figure 13 in Appendix C) from
0 to 0.15 with 101 samples while the position along the optical axis had 51
samples between 3 and 3.3 mm. The result is tabulated in Table 1 which
establishes the well known facts (47): (a) the radius of the circle of least
confusion, $\rm r_{coc}$, is $\sim$ 0.25 times the beam radius at marginal ray
height; (b) the distance of CoC from paraxial focus is $\sim$ 0.75 times the
distance of marginal focus from the paraxial focus. The minor variation in TSA
and LSA due to fiber input beam speed is negligible.

$\mathbf{f_{fib}}$ | pf | mf | coc | $\mathbf{\frac{pf-coc}{pf-mf}}$ | $\mathbf{r_{coc}}$ | $\mathbf{r_{pf}}$ | $\mathbf{\frac{r_{coc}}{r_{pf}}}$
---|---|---|---|---|---|---|---
2.5 | 3.19 | 3.04 | 3.075 | 0.76 | 0.0056 | 0.0223 | 0.25
3 | 3.19 | 3.088 | 3.1125 | 0.76 | 0.003 | 0.0124 | 0.24
3.5 | 3.19 | 3.116 | 3.135 | 0.74 | 0.0019 | 0.0076 | 0.25
4 | 3.19 | 3.133 | 3.1475 | 0.75 | 0.0013 | 0.005 | 0.26
4.5 | 3.19 | 3.146 | 3.157 | 0.75 | 0.0009 | 0.0034 | 0.26
5 | 3.19 | 3.154 | 3.1625 | 0.76 | 0.0007 | 0.0025 | 0.28
Table 1: Effect of fiber input beam speed on the location and radius of circle
of least confusion. From left to right, $\rm f_{fib}$ is the fiber injection
speed, pf is the paraxial focal length, mf is the focal length from the
marginal ray, coc is the distance to the circle of least confusion, $\rm
r_{coc}$ is the radius of the circle of least confusion, and $\rm r_{pf}$ is
the beam radius at the paraxial focus. All the quantities are normalized by
the plano-convex radius of curvature. All dimensions are in mm except for
ratios and f-number.

###  5.1 Accuracy of the analytic model

At this point it is important to note that there is also SA as well as off-
axis aberration introduced by the biconvex lens entry and exit surfaces. The
analytical optimization of multi surface SA is non-trivial and hence Zemax is
used for this purpose. We used optical parameters from our analytic model as
initial conditions for the Zemax optimization and looked for how these
parameters change after detailed optimization. This optimization accounts for
higher order aberration such as coma, astigmatism, color etc. We used two
configurations, one each in regimes where SA or diffraction dominate (refer to
Figure 4):

  1. 1.

SA dominated regime: a f/11 telescope beam is relayed to a f/3.7 fiber input
beam for a micro-image diameter of 100$\upmu$m.

  2. 2.

Diffraction-limited regime: a f/7 telescope beam is relayed to a f/4 beam for
a micro-image diameter of 50 $\upmu$m.

We use equations 2 to 16 for $\rm d_{a}$ and $\rm r_{b}$ from Appendix A to
predict the optical parameters of the above mentioned configurations. The
analytical model is built on the assumption that the pupil should be formed at
the exit surface of the BC MLA and the PC MLA is going to form the CoC at its
exit surface. We then use Zemax optimization to further optimize both the
configurations. Table 2 shows the comparison chart of the parameters and
performance before and after optimization. The design parameters change little
in the SA-limited case (as we might expect), but significantly in the
diffraction-limited regime. Nonetheless, we find that the analytical
prediction produces optical performance similar to that of Zemax optimized
design within 4% for both the SA dominated regime and the diffraction
dominated scenario based on EE98 estimates. Since the fiber azimuthally
scrambles the spatial information at its input, RMS spot radius do not provide
the required estimate of performance.

|  | dimensions
---
are in $\upmu$m
| Spherical Aberration
---
dominated
| Diffraction
---
dominated
Configuration |  (a) $f_{tel}$ | 11 | 7
(b) $f_{fib}$ | 3.7 | 4
(c) $d_{m}$ | 100 | 50
|  | Model | Simulation | Model | Simulation
Optical parameter |  (d) $r_{p}$ | 333 | 327 | 68.4 | 114
(e) $D_{p}$ | 1050 | 994 | 218.4 | 305
(f) $D_{G}$ | 730 | 469 | 150 | 103
(g) $r_{b}$ | 999 | 887 | 126 | 141
(h) $D_{b}$ | 3160 | 3119 | 382 | 578
Performance |  (i) $d_{f0}$ | 1.207 | 1.121 | 0.317 | 0.4
(j) $d_{f1}$ | 3.424 | 2.919 | 3.555 | 1.24
(k) EE98 | 52 | 50 | 24.5 | 25
Table 2: Comparison of design parameters and performance of the reimaging
microlens system based on our analytic model (Model) and after optimization
using Zemax (Simulation) for different configurations in spherical aberration
(SA) dominated and diffraction dominated case. The different configuration,
optical and performance parameters are: (a) $\rm f_{tel}$: telescope focal
ratio; (b) $\rm f_{fib}$: fiber input focal ratio; (c) $\rm d_{m}$: micro-
image diameter at the fiber input face; (d) $\rm r_{p}$: plano-convex
microlens radius of curvature; (e) $\rm D_{p}$: plano-convex microlens
thickness; (f) $\rm D_{G}$: air gap thickness between plano-convex and bi-
convex microlens; (g) $\rm r_{b}$: bi-convex microlens radius of curvature;
(h) $\rm D_{b}$: bi-convex microlens thickness; (i) $\rm d_{f0}$: RMS spot
size at the field center of the fiber input surface; (j) $\rm d_{f1}$: RMS
spot size at the field edge of the fiber input surface; (k) EE98: radius of
98% of the normalized encircled energy at the fiber input surface. This
corresponds to the reimaged patch of $d_{a}$ = $d_{m}\times\upeta$. All
dimensions, except for focal ratios, are provided in $\upmu$m. Model and
Simulation refer to analytical model predicted value and Zemax optimization
provided value respectively.

###  5.2 Limiting telescope beam speed in the presence of micro-lens
aberrations

For a single spherical lens system (e.g., just the PC MLA), intuitively one
would place the CoC produced by the lens on the fiber input aperture, which is
what our analytic model assumes. For a compound lens system the situation is
less obvious, however our Zemax optimization indicates (Table 2) that placing
the CoC produced by the PC MLA very close to the fiber entrance aperture
remains optimal when spherical aberration dominates over diffraction on-axis.

We have used Zemax simulation to define the photon loss for a fiber capturing
the CoC at its entry surface for a range of telescope and fiber input f-ratio.
The throughput (provided as percentage of input photons) captured within the
micro-image diameter of 100 $\rm\upmu$m for different $\rm f_{\rm tel}$ and
$\rm f_{\rm fib}$ is plotted in figure 5 including the effects of diffraction
and off-axis aberrations. The Figure shows that $\rm f_{\rm tel}$ of 6, 7, 7,
8 and 8 are optimum for $\rm f_{\rm fib}$ of 3, 3.5, 4, 4.5 and 5
respectively. Although the plot is for 100 $\rm\upmu$m microimage diameter,
the optimum telescope f-ratio of f/8 or slower holds true for other micro-
image diameters ranging from 50 to 600 $\rm\upmu$m. This discussion can be
used as a constraint on $f_{tel}$ coming from aberrations and can be
summarized as the following:

  1. 1.

The faster the telescope beam, the larger the SA, as well as off-axis
aberrations, hence the larger the losses.

  2. 2.

At fast telescope beams, a fast fibre input beam helps to minimize the losses.

Figure 5: Percentage of throughput captured within the micro-image diameter
($\rm d_{m}$, taken to be the same as the fiber diameter) against telescope
focal-ratio for different fiber input beam speed for plano-convex MLA assuming
collimated input beam in the presence of diffraction, spherical aberration and
all higher-order aberrations. Results are based on Zemax modeling. Similar
results are found for micro-image diameters ranging from 50 to 600
$\rm\upmu$m.

##  6 Constraining the telescope focal-ratio: microlens manufacturing
limitations

Aberrations are not the only component that set a bound on the usable
telescope focal ratio, $\rm f_{\rm tel}$. Several optical parameters, such as
the radius of curvature and thickness of PC and BC microlenses also pose
constraints. In this section we describe the limiting conditions originating
from such manufacturing limitations for various micro-image diameters,
$\rm{d_{m}}$, and fiber injection speeds, $\rm{f_{fib}}$. These limits are
computed using our analytical model in Section 3. We use openly available
information provided by A$\upmu$S as a benchmark for fabrication.

###  6.1 Limits on $f_{tel}$ from lenslet radii of curvature

Figures 6 and 7 show the effect of $\rm f_{\rm fib}$, and $\rm d_{m}$ and $\rm
f_{\rm tel}$ on the required radii of curvature (RoC) for bi-convex ($\rm
r_{b}$) and plano-convex ($\rm r_{p}$) MLAs, respectively. Clearly the
fabrication limit on $\rm r_{b}$ does not constrain $\rm f_{tel}$ for any
fiber size and injection speed consider. Similarly, for On the other hand, the
fabrication limit on $\rm r_{p}$ shows that for $\rm 50~{}\upmu$m fiber it is
necessary for $\rm f_{\rm tel}\geq 6$ for $\rm f_{\rm fib}=5$, but it does not
provide any restriction for faster fiber injection speeds $\rm f_{\rm fib}<4$.
For larger fibers ($\geq 100\upmu$m) we can conclude broadly that there is no
limiting $\rm f_{\rm tel}$ originating from fabrication constraints on the MLA
RoC.

Figure 6: Dependence of bi-convex microlens (BC) radius of curvature ($\rm
r_{b}$) on telescope f-ratio for a range of fiber input f-ratios (different
colored lines defined the bottom-right panel key) and fiber diameters, assumed
here to equal to $\rm d_{m}$. From top-left to bottom right panels correspond
to $\rm 50~{}\upmu$m to $\rm 600~{}\upmu$m core fiber. Black dashed lines
denote 0.02 mm, the minimum possible radius of curvature. Figure 7: Dependence
of plano-convex microlens (PC) radius of curvature ($\rm r_{p}$) on telescope
f-ratio, fiber diameter, and fiber input f-ratio as in Figure 6. Curves are
defined as in Figure 6.

###  6.2 Limits on $f_{tel}$ from MLA thickness

Apart from the microlens radius of curvature, the microlens thicknesses ($\rm
d_{p}$ and $\rm d_{b}$) also constrain the telescope f-ratio range. Equations
5 and 10 in Appendix A show both $\rm d_{p}$ and $\rm d_{b}$ increase linearly
with $\rm d_{m}$ and hence the manufacturable thickness could pose constraints
on $\rm f_{\rm tel}$. We have computed the dependence of $\rm f_{\rm tel}$ on
MLA thickness for different fiber injection speeds and micro-image diameters
in Figures 8 and 9.

Figure 8: Dependence of bi-convex microlens required thickness ($\rm D_{b}$)
on telescope f-ratio, fiber diameter $\rm d_{m}$, and fiber input f-ratio
(different colored lines defined the bottom-right panel key). The dash-dot and
the dashed black lines represent the higher and the lower limit of thickness
respectively. Figure 9: Dependence of plano-convex microlens required
thickness $\rm D_{p}$ on telescope f-ratio, fiber diameter $\rm d_{m}$, and
fiber input f-ratio (different colored lines defined the bottom-right panel
key). The dashed black lines represent the thickness lower limit; the upper
limit is at the upper limit of the plotting range.

For fabrication thickness upper limits, a PC MLA can have arbitrarily large
thickness by bonding it to a flat substrate. However, for the BC MLA, the
registration error between curved faces increases with thickness and hence a
manufacturing limit of 10 mm is used by the vendors. It should be noted that
with additional effort of alignment and bonding, two PC MLAs and arbitrarily
thick glass plate can be used to eliminate the thickness upper limit.
Consequently the BC MLA thickness upper limit is taken to be a soft limit. In
general MLA thickness upper limits tend to drive the telescope speed to be
faster. As can be seen in Figures 8 and 9, a 10 mm thickness upper limit, the
bi-convex MLA sets $\rm f_{\rm tel}\leq 7(11)$ for $\rm f_{\rm fib}=2.5(5)$,
respectively, for $\rm 600~{}\upmu$m fibers. For smaller fibers with diameter
$\leq\rm 100~{}\upmu$m the thickness does not limit the $\rm f_{tel}$ within
the range of telescope focal ratios that is being considered. The PC MLA
thickness imposes no limits on $\rm f_{\rm tel}$ as it can be increased
arbitrarily.

For fabrication thickness lower limits, the manufacturer has to ensure that
the array is flat and thus it is difficult to reduce lenslet thickness below
0.3 and 0.6 mm for PC and BC MLA respectively. In general MLA thickness lower
limits tend to drive the telescope speed to be slower. As can be seen in
Figures 8 and 9, the minimum thickness forces $\rm f_{tel}\geq 9(11)$ for 50
$\upmu$m fiber for $\rm f_{fib}=2.5(5)$ but does not limit fibers with $\geq
400\upmu$m cores.

###  6.3 Summary of limitations on $\rm f_{tel}$

Table 3 compiles all the limitations on $\rm f_{\rm tel}$ from different
criteria presented in this and the previous section (4). In the final columns
we take the most conservative option as our final limit. In most cases the
lower limit is set by ‘SA,’ which includes the impact of all aberrations on
EE98. This means that this limit is unaffected by any potential future
improvements in fabrication limits for spherical lenslet figures. Aspherics
may alter this picture, and are worthy of future consideration. In constrast,
the upper limit on $\rm f_{tel}$ is constrained solely by the BC MLA thickness
upper limit. As described above, this upper limit may be surmountable and as
such there is no hard upper limit on the telescope beam speed.

It is evident and intuitive that faster telescope beams require the optical
dimensions of the microlens arrays to be smaller and hence lossier for fiber-
coupling due to increased aberrations. Less intuitive is the fact that for
larger fiber core diameter, the limiting telescope f-ratio gets faster. The
effect is most pronounced going from the diffraction-limited to SA dominated
regimes (in the range $\rm 50~{}\upmu m<d_{m}<100~{}\upmu m$). Indeed, in the
SA dominated regime the limiting telescope f-ratio is constant for a given
fiber input beam speed, and only changes by 33% for a factor of two change in
fiber injection speed.

$\rm d_{m}$ ($\upmu$m) | $\rm f_{fib}$ | Thickness | RoC | Abr | Final
---|---|---|---|---|---
BC | PC | BC | PC
L | U | L | U | L | U | L | U | L | U | L | U
50 | 2.5 | 6 | - | 9 | - | - | - | - | - | 6 | - | 9 | -
3 | 7 | - | 9 | - | - | - | - | - | 6 | - | 9 | -
3.5 | 8 | - | 10 | - | - | - | - | - | 7 | - | 10 | -
4 | 9 | - | 10 | - | - | - | - | - | 7 | - | 10 | -
4.5 | 9 | - | 11 | - | - | - | 6 | - | 8 | - | 11 | -
5 | 10 | - | 11 | - | - | - | 6 | - | 8 | - | 11 | -
100 | 2.5 | - | 14 | 6 | - | - | - | - | - | 6 | - | 6 | 14
3 | 6 | - | 7 | - | - | - | - | - | 6 | - | 7 | -
3.5 | 6 | - | 7 | - | - | - | - | - | 7 | - | 7 | -
4 | 7 | - | 7 | - | - | - | - | - | 7 | - | 7 | -
4.5 | 8 | - | 8 | - | - | - | - | - | 8 | - | 8 | -
5 | 8 | - | 8 | - | - | - | - | - | 8 | - | 8 | -
200 | 2.5 | - | 10 | - | - | - | - | - | - | 6 | - | 6 | 10
3 | - | 11 | - | - | - | - | - | - | 6 | - | 6 | 11
3.5 | - | 12 | - | - | - | - | - | - | 7 | - | 7 | 12
4 | 6 | 13 | 6 | - | - | - | - | - | 7 | - | 7 | 13
4.5 | 7 | 14 | 6 | - | - | - | - | - | 8 | - | 8 | 14
5 | 7 | - | 7 | - | - | - | - | - | 8 | - | 8 | -
300 | 2.5 | - | 8 | - | - | - | - | - | - | 6 | - | 6 | 8
3 | - | 9 | - | - | - | - | - | - | 6 | - | 6 | 9
3.5 | - | 10 | - | - | - | - | - | - | 7 | - | 7 | 10
4 | 6 | 11 | - | - | - | - | - | - | 7 | - | 7 | 11
4.5 | 6 | 12 | 6 | - | - | - | - | - | 8 | - | 8 | 12
5 | 7 | 13 | 6 | - | - | - | - | - | 8 | - | 8 | 13
400 | 2.5 | - | 7 | - | - | - | - | - | - | 6 | - | 6 | 7
3 | - | 8 | - | - | - | - | - | - | 6 | - | 6 | 8
3.5 | - | 9 | - | - | - | - | - | - | 7 | - | 7 | 9
4 | - | 10 | - | - | - | - | - | - | 7 | - | 7 | 10
4.5 | 6 | 11 | 6 | - | - | - | - | - | 8 | - | 8 | 11
5 | 6 | 12 | 6 | - | - | - | - | - | 8 | - | 8 | 12
500 | 2.5 | - | 7 | - | - | - | - | - | - | 6 | - | 6 | 7
3 | - | 8 | - | - | - | - | - | - | 6 | - | 6 | 8
3.5 | - | 8 | - | - | - | - | - | - | 7 | - | 7 | 8
4 | - | 9 | - | - | - | - | - | - | 7 | - | 7 | 9
4.5 | 6 | 10 | 6 | - | - | - | - | - | 8 | - | 8 | 10
5 | 6 | 11 | 6 | - | - | - | - | - | 8 | - | 8 | 11
Table 3: Summary of limits on telescope beam speed for all configurations of
varying micro-image diameter at the fiber input ($\rm d_{m}$) and fiber input
beam speed ($\rm f_{fib}$). L and U denotes lower and upper limits of
telescope beam speed respectively while no limits are represented with dash.
BC and PC represents bi-convex and plano-convex microlens while Abr and RoC
stands for optical aberrations and radius of curvature, respectively. All
constraints are from Section 6 except for optical aberrations (Section 4)

We provide a visual summary of the Table 3 in Figure 10 where the final range
of acceptable telescope focal ratios are plotted against fiber diameter for
three fiber input beam speeds. In terms of conserving grasp, it is important
to note that due to FRD it is preferable to use fiber injection speeds closer
to $\rm f_{fib}=3$ than $\rm f_{fib}=5$. Overall, however, it is important to
keep in mind that the grasp (A $\Omega$) does not depend on the telescope
focal-ratio but rather only on the square of the telescope diameter $\rm
D_{tel}$ (collecting area, A), and the product of $\rm{d_{m}/(D_{tel}\times
f_{fib})}$ (solid angle, $\Omega$). While the solid angle can be re-written as
$\rm{d_{a}/(D_{tel}\times f_{tel})}$, fundamentally the grasp is driven by the
telescope size and the considerations of the spectrograph injection speed and
aperture, hence the fiber size $\rm{d_{m}}$ and the fiber injection speed.

As a common practice in slit spectroscopy, astronomers often need to modulate
the slit width at fixed focal ratio to optimize the resolution-throughput
product of their observations given the requirements of the scientific
program, the observing conditions, and the angular scale of their sources. For
fiber spectroscopy, including lenslet-coupled IFUs, this is equivalent to
changing the fiber core diameter at fixed focal ratio. In this context Figure
10 shows that for fiber sizes smaller than $100~{}\upmu$m there is a limited
range in fiber size that can be accommodated at fixed telescope and fiber
injection speeds. Conversely, for fiber core diameters of 100 $\upmu$m and
larger lenslet coupling can be achieved with a fixed telescope focal ratio of
f/8 for fiber injection at f/3 (or f/9 to f/10 for fiber injection at f/4,
etc.). Similarly, for fiber cores $>100~{}\upmu$m, there is a broader range of
fiber injection speeds at a fixed telescope focal ratio in the range of 8 to
10, with f/8 providing the largest range of fiber injection speeds. On this
basis we suggest f/8 as the optimum telescope f-ratio in terms lenslet-fiber
coupling flexibility for fibers with core sizes $\geq 100~{}\upmu$m. The use
of smaller fibers may apply to small telescopes (below 4m) where the science
requires matching of the fiber core to the seeing disk. While Figure 10
indicates that telescope focal ratios faster than f/8 would allow for faster
injection speeds for larger fibers, the faster telescope focal ratios will
limit the dynamic range in fiber core size over which this is possible.

Figure 10: Summary of table 3 in a visual form. Range of acceptable $\rm
f_{tel}$ is plotted against different fiber sizes for $\rm f_{fib}$ of f/3,
f/4 and f/5.

##  7 Summary

In this paper we presented the trade-offs in optical design of micro-lens
reimaging systems constrained to link telescope focal planes to fiber optics.
Specifically, we have examined the constraints on telescope focal ratio for a
realistic range of fiber core diameters and fiber injection speeds given (i)
manufacturing limitations on micro-lens arrays and (ii) the need to have high-
efficiency coupling. For this trade study we have developed an analytical
model from first principals and two basic conditions: (a) the BC MLA should
produce the pupil at its exit surface; (b) the image produced by the PC MLA
must be telecentric to eliminate geometric FRD. We evaluate the efficacy of
the model by considering the salient performance parameter of fibers acting as
photon buckets: the percentage of energy enclosed with the fiber core by the
compound lenslet system. We adopt this metric rather than RMS spot radius
since fibers scramble the spatial information. In this regard the performance
of the model-predicted optical design matches within 4% of optimized models
using ray-trace software. We have also characterized in detail the different
regimes in which these MLA systems are dominated (on axis) by diffraction
versus SA. While the detailed micro-lens parameters (thickness, spacing,
radius of curvature) are more accurately prescribed by our analytic model in
the SA limited regime, in all cases, the model designs provide better than 98%
throughput apart from Fresnel losses at the glass-air boundaries in all
regimes.

We computed the overall upper and lower limits on telescope focal ratio set by
the fabrication limit of BC and PC MLA as well as light-losses from SA and
off-axis aberrations. Lower limits on $\rm f_{tel}$ are set by considering
light-losses into the fiber, while upper limits are set by the BC MLA
thickness. This upper limit of $\rm f_{tel}$ may be relaxed via additional
effort in replacing a monolithic BC MLA with two back-facing PC MLAs, possibly
separated by a flat glass plate.

We find the telescope focal ratio lower-limit falls precipitously from the
smallest fibers considered ($50~{}\upmu$m) to $100~{}\upmu$m, which
corresponds to the transition between diffraction to SA dominated regimes. For
fibers larger than $100~{}\upmu$m the limiting telescope focal-ratio is given
roughly by $\rm f_{tel,lower}=0.8f_{fib}+4$ and $\rm
f_{tel,upper}=2f_{fib}+3$. For the smallest $50~{}\upmu$m fibers $\rm
f_{tel,lower}=1.5f_{fib}+4$, with no upper limit. The case for the
$100~{}\upmu$m fiber core is intermediate to these two limiting cases.

For a spectrograph fed directly by fibers injected with $\rm 3<f_{fib}<5$, we
find that an f/8 telescope would support fiber diameters from 100 to 500
$\upmu$m within the optimized range. This enables a wide range of spatial and
spectral resolution for survey optimization with the same telescope and
spectrograph hardware. Finally we comment that while it is always preferable
to reduce the number of optical elements when possible, the native telescope
focal ratio can often be reimaged by a suitable focal reducer or expander. On
the basis of our analysis we suggest such additional optics should be
designed, for most applications, to deliver a focal surface feeding micro-lens
arrays at f/8. MLA designs with additional, or aspheric surfaces may alter
these conclusions, but such augmentation incurs additional cost and complexity
in design, fabrication and assembly.

## Acknowledgments

This research was supported by funds from the University of Wisconsin-Madison
Graduate School, the U.S. National Science Foundation grants AST-1517006 and
AST-1814682, and the South African National Research Foundation SARChI
program.

##  Appendix A Derivation of model parameters

Here we calculate the relationships between the critical MLA parameters
(surface curvature, thickness and spacing) as a function of the telescope
input and fiber input focal-ratios and the size of the micro-image, which
again we take to be just slightly under-sized with respect to the fiber core
diameter. The fiber input f-ratio $\rm f_{\rm fib}$ is defined in the air
while $\rm D_{p}$ and $\rm d$ defines the f-ratio in glass. Hence $\rm f_{\rm
fib}$ is usually multiplied with refractive index of glass ($\rm n_{g}$) at
the system wavelength.

###  A.1 Optical diameters

Relevant diameters can be computed by using the conservation of entropy
(grasp) to relate the bi-convex MLA clear aperture $\rm d_{a}$ then can be
formulated as a function of micro-image diameter $\rm d_{m}$:

$\rm d_{a}\ =\ \frac{\rm f_{\rm tel}}{(\frac{\rm f_{\rm fib}}{\rm d_{m}})}\
=\rm d_{m}\ \upeta.$ (2)

From this, the beam diameter for a single field point at the input of the
Plano-convex MLA (PC) is $\rm d\leq d_{a}-d_{m}$. This follows by construction
in Figure 2 due to the constraints that (i) the PC MLA diameter cannot not
exceed that of the BC MLA diameter, and (ii) the telecentricity requirement
stipulates the principle ray for the edge field-point must fall at a height of
$d_{m}$/2 from the optical axis while the distance between the marginal and
principle ray is d/2. Hence d/2 + $d_{m}$/2 has to be equal to $d_{a}$/2,
namely the clear aperture diameter of the PC and BC lenslets. Although $\rm d$
can be less than $\rm d_{a}-d_{m}$, this reduces $\rm r_{p}$. Hence maximizing
$\rm d$ helps minimize SA, so we take $\rm d=d_{a}-d_{m}$. This condition for
$\rm d$ can be rewritten with equation 2 as

$\rm d\ =\rm d_{m}\ (\upeta-1).$ (3)

###  A.2 PC MLA thickness and radius of curvature

From fiber input focal ratio, one can deduce that the plano-convex lens
thickness is simply:

$\rm f_{\rm fib}\ n_{g}=\frac{\rm D_{p}}{d}.$ (4)

Note that since f-ratio is determined in air, refractive index has to be
multiplied to get the f-ratio in the glass. This can be rewritten using
equation 3 as:

$\rm D_{p}\ =\rm d_{m}\rm n_{g}\rm f_{\rm fib}\ (\upeta-1).$ (5)

Using the lens makers equation for plano convex lens with $\rm r_{p}$ is the
radius curvature of the curved surface and the other side is flat, we can
state that,

$\frac{1}{\rm D_{p}/\rm n_{g}}=(\rm
n_{g}-1)(\frac{1}{r_{p}}-\frac{1}{\infty}).$ (6)

Again note the use of $\rm n_{g}$ as a dividing factor as the focal length is
usually defined in air but in our case the focus is the flat back plane of the
MLA. Hence the thickness of PC lenslets should in air focal length multiplied
by the refractive index of the glass. Hence, the plano-convex MLA (PC) radius
of curvature can then be written as:

$\rm r_{p}\ =\rm D_{p}\rm(n_{g}-1)/\rm n_{g}\ =\rm{d_{m}}\rm f_{\rm
fib}\rm(n_{g}-1)(\upeta-1).$ (7)

###  A.3 MLA spacing

Again, using the lens makers equation for plano convex lens we can find that,

$\frac{1}{\rm D_{G}}=(\rm n_{g}-1)(\frac{1}{r_{p}}-\frac{1}{\infty}).$ (8)

Since $\rm D_{G}$ is defined in air we don’t need to include $\rm n_{g}$.
Hence the gap between the bi-convex and plano-convex MLA can be derived as

$\rm D_{G}\ =\rm r_{p}/\rm(n_{g}-1)\ =\rm{d_{m}}\rm f_{\rm fib}\ (\upeta-1).$
(9)

###  A.4 BC MLA thickness and radius of curvature

Similar to plano-convex MLA, the bi-convex MLA (BC) thickness can be derived
as:

$\rm D_{b}\ =\rm n_{g}\rm d\rm f_{\rm tel}\ =\rm d_{m}\rm n_{g}\rm f_{\rm
tel}\ (\upeta-1).$ (10)

The bi-convex MLA (BC) radius of curvature can be derived from the path of
chief ray of the edge field inside the BC MLA as shown in Figure 11, as
follows. From Snell’s law, $\rm\sin{\uptheta}=n_{g}\sin{\upbeta}$, and from
Figure 11 we have $\rm\uptheta=\upalpha+\upbeta$.

Figure 11: Propagation path of edge field chief ray inside the bi-convex
lenslet.

From this, we can easily derive

$\tan{\upbeta}=\frac{\sin{\upalpha}}{n_{g}-\cos{\upalpha}},$ (11)

while from the right angle triangle we can define $\rm\sin{\upalpha}$ as

$\sin{\upalpha}\approx\frac{\rm d_{a}/2}{\sqrt{\rm(d_{a}/2)^{2}+\rm
D_{b}^{2}}},$ (12)

and similarly,

$\sin(\upalpha+\upbeta)=\frac{\rm d_{a}/2}{\rm r_{b}},$ (13)

which we can combine to write:

$\tan{\upbeta}=\frac{\rm d_{a}/2}{\sqrt{(\rm n_{g}\rm r_{b})^{2}-(\rm
d_{a}/2)^{2}}}.$ (14)

From equations 14 and 11 we can easily define the radius of the BC MLA as,

$\rm r_{b}\ =\ \frac{1}{\rm n_{g}}\left[\left(\rm
n_{g}\sqrt{\rm(d_{a}/2)^{2}+\rm{D_{b}}^{2}}-\rm
D_{b}\right)^{2}+\rm(d_{a}/2)^{2}\right]^{1/2}$ (15)

for which equations 2 and 10 can be used to express $\rm r_{b}$ in terms of
$\rm f_{\rm tel}$, $\rm f_{\rm fib}$ (or $\rm f_{\rm tel}$ and $\rm\upeta$),
and $\rm d_{m}$:

$\rm r_{b}\ =\rm d_{m}\ \upeta\left[\left(\sqrt{1/4+(\rm n_{g}\ \rm f_{\rm
fib}\ (\upeta-1))^{2}}-\rm f_{\rm fib}\ (\upeta-1)\right)^{2}+1/(4{\rm
n_{g}^{2}})\right]^{1/2}.$ (16)

##  Appendix B Refraction on the spherical surface: Dependence of microlens
thickness on beam diameter.

Here we define the PC MLA thickness, f (sag plus flat thickness), with radius
of curvature R, such that the ray of an object at infinity at height r crosses
the optical axis at the exit surface. Due to SA, this thickness depends on
height. We have defined the thickness for an object at infinity and in line
with our requirement of treating the PC MLA which is the imaging lens in our
system. This generic description of MLA thickness will help us define the PC
lenslet parameters requried to locate the CoC on the PC MLA exit surface in
the next subsection. Using the derivation from Hecht[48] for an object at a
distance $\rm S_{o}$ in front of the MLA, illustrated in Figure 12,

Figure 12: Derivation of plano-convex microlens focal length.

their equation 5.4, given here as:

$\frac{\rm n_{1}(\rm S_{o}+\rm R)}{\rm l_{o}}=\frac{\rm n_{2}(\rm S_{i}-\rm
R)}{\rm l_{i}}.$ (17)

can be rewritten for an object at infinity ($\rm S_{o}=l_{o}=\infty$, $\rm
S_{i}=f$, $\rm n_{1}$ = 1 and $\rm n_{2}=n_{g}$):

$\rm n_{g}(\rm f-\rm R)=\rm l_{i}.$ (18)

It is straight forward to construct:

$\rm l_{i}=[\rm R^{2}+(\rm f-\rm R)^{2}+2\rm R(\rm f-\rm
R)^{2}cos\upphi]^{1/2},$ (19)

where $\rm\cos\phi=\sqrt{1-(\frac{r}{R})^{2}}$. Solving for f - R we arrive at
an expression for f:

${\rm f}={\rm R}\left[1+\frac{1}{\sqrt{\rm n_{g}^{2}-(\frac{\rm r}{\rm
R})^{2}}-\sqrt{1-(\frac{\rm r}{\rm R})^{2}}}\right]$ (20)

which has the expected paraxial approximation (r/R$\ll$1) of $\rm
f=R(n_{g}/(n_{g}-1)$. The dependence of the microlens thickness f on (r, R,
$\rm n_{g}$) is simply the manifestation of SA on a collimated beam.

##  Appendix C Effect of spherical aberration on microlens focal length:
Longitudinal and Transverse spherical aberration

The longitudinal SA (LSA) is defined by the distance between paraxial focus
and the focus of the marginal ray while the transverse SA (TSA) is the
marginal ray height at the paraxial focus. However, to minimize the effect of
SA the location and radius of circle of least confusion (CoC) is important to
understand. The location would directly define the thickness of the PC lenslet
while the CoC radius would impose condition on the lenslet properties in order
to restrict the micro-image to form within the fiber core to reduce photon
loss at the MLA-fiber junction. Increasing the fiber core diameter arbitrarily
to engulf entire micro-image would lead to increase in optical entropy which
can be equated to loss of observation time.

Now having the analytical description of thickness of PC lenslet in our
arsenal, we will describe the radius and position of circle of least confusion
and what should be the strategy for minimizing the radius. In figure 13 we
define a plano-convex microlens of radius of curvature R fed with a collimated
beam. The height of the ray from the optical axis is r; $\rm\upalpha$ and
$\rm\upbeta$ are the angles of incidence and refraction at the input surface,
respectively; q and $\rm\updelta$ are the distances of the crossing of optical
axis inside the microlens and chief ray entry point to the arbitrary ray entry
point respectively. Suffixes 1 and 2 denote angles and dimensions associated
with marginal and paraxial ray, respectively. The radius of the circle of
least confusion (CoC) is defined as $\rm r_{3}$ while the distance from circle
of least confusion from the marginal ray focus is $\rm\upgamma$.

Figure 13: Derivation of position and radius of circle of least confusion.

It is easy to find that $\rm\updelta_{2}=R-\sqrt{R^{2}-r_{2}^{2}}$. We can
also deduce the following,

$\frac{\rm r_{3}}{\upgamma}=\frac{\rm r_{1}}{\sqrt{\rm R^{2}-\rm r_{1}^{2}}},$
(21)

and,

$\frac{\rm r_{2}}{\rm r_{3}}=\frac{\rm f(\rm r_{2})-\updelta_{2}}{\rm f(\rm
r_{2})-\rm f(\rm r_{1})-\upgamma}.$ (22)

Solving equation 17 and 18 we get,

$\rm r_{3}=\frac{\rm r_{1}\rm r_{2}\left(\rm f(\rm r_{2})-\rm f(\rm
r_{1})\right)}{\rm r_{1}\left[\rm f(\rm r_{2})-\rm R+\sqrt{\rm R^{2}-\rm
r_{2}^{2}}\right]+\rm r_{2}\sqrt{\rm R^{2}-\rm r_{1}^{2}}},$ (23)

and,

$\upgamma=\frac{\rm r_{2}\sqrt{\rm R^{2}-\rm r_{1}^{2}}(\rm f(\rm r_{2})-\rm
f(\rm r_{1}))}{\rm r_{1}\rm f(\rm r_{2})-\rm r_{1}(\rm R-\sqrt{\rm R^{2}-\rm
r_{2}^{2}})+\rm r_{2}(\rm R-\sqrt{\rm R^{2}-\rm r_{1}^{2}})}$ (24)

One can differentiate equation 23 and find the maxima to identify the location
of the CoC. However, this analytical approach is quite cumbersome and hence it
is better to solve the problem numerically.

## References

  * [1] G. J. Hill, P. J. MacQueen, M. P. Smith, et al., “Design, construction, and performance of VIRUS-P: the prototype of a highly replicated integral-field spectrograph for HET,” in Ground-based and Airborne Instrumentation for Astronomy II, I. S. McLean and M. M. Casali, Eds., 7014, 2408 – 2422, International Society for Optics and Photonics, SPIE (2008).
  * [2] M. H. Fabricius, S. Barnes, R. Bender, et al., “VIRUS-W: an integral field unit spectrograph dedicated to the study of spiral galaxy bulges,” in Ground-based and Airborne Instrumentation for Astronomy II, I. S. McLean and M. M. Casali, Eds., 7014, 2445 – 2454, International Society for Optics and Photonics, SPIE (2008).
  * [3] H. Sugai, N. Tamura, H. Karoji, et al., “Prime Focus Spectrograph for the Subaru telescope: massively multiplexed optical and near-infrared fiber spectrograph,” Journal of Astronomical Telescopes, Instruments, and Systems 1(3), 1 – 11 (2015).
  * [4] A. Hill, N. F. III, A. McConnachie, et al., “Maunakea Spectroscopic Explorer (MSE): instrumentation suite,” in Ground-based and Airborne Instrumentation for Astronomy VII, C. J. Evans, L. Simard, and H. Takami, Eds., 10702, 487 – 499, International Society for Optics and Photonics, SPIE (2018).
  * [5] B. Flaugher, “The Dark Energy Survey,” International Journal of Modern Physics A 20, 3121–3123 (2005).
  * [6] R. de Jong, S. Barden, O. Bellido-Tirado, et al., “4most: 4-metre multi-object spectroscopic telescope,” Proceedings of SPIE - The International Society for Optical Engineering (2014).
  * [7] N. P. Konidaris, N. Drory, C. S. Froning, et al., “SDSS-V local volume mapper instrument: overview and status,” in Ground-based and Airborne Instrumentation for Astronomy VIII, C. J. Evans, J. J. Bryant, and K. Motohara, Eds., 11447, 233 – 250, International Society for Optics and Photonics, SPIE (2020).
  * [8] M. A. Bershady, D. R. Andersen, J. Harker, et al., “SparsePak: A formatted fiber field unit for the WIYN telescope bench spectrograph. i. design, construction, and calibration,” Publications of the Astronomical Society of the Pacific 116, 565–590 (2004).
  * [9] A. Kelz, M. Verheijen, M. Roth, et al., “Pmas: The potsdam multi‐aperture spectrophotometer. ii. the wide integral field unit ppak,” Publications of the Astronomical Society of the Pacific 118(839), 129–145 (2006).
  * [10] N. Drory, N. MacDonald, M. A. Bershady, et al., “The MaNGA Integral Field Unit Fiber Feed System for the Sloan 2.5 m Telescope,” The Astronomical Journal 149, 77 (2015).
  * [11] A. G. de Paz, E. Carrasco, J. Gallego, et al., “MEGARA, the new intermediate-resolution optical IFU and MOS for GTC: getting ready for the telescope,” in Ground-based and Airborne Instrumentation for Astronomy VI, C. J. Evans, L. Simard, and H. Takami, Eds., 9908, 420 – 439, International Society for Optics and Photonics, SPIE (2016).
  * [12] S. M. Croom, J. S. Lawrence, J. Bland-Hawthorn, et al., “The Sydney-AAO Multi-object Integral field spectrograph,” MNRAS 421, 872–893 (2012).
  * [13] K. Bundy, M. A. Bershady, D. R. Law, et al., “Overview of the SDSS-IV MaNGA Survey: Mapping nearby Galaxies at Apache Point Observatory,” apj 798, 7 (2015).
  * [14] E. Giro, R. U. Claudi, J. Antichi, et al., “BIGRE: a new double microlens array for the integral field spectrograph of SPHERE,” in Ground-based and Airborne Instrumentation for Astronomy II, I. S. McLean and M. M. Casali, Eds., 7014, 1254 – 1265, International Society for Optics and Photonics, SPIE (2008).
  * [15] R. Bacon, M. Accardo, L. Adjali, et al., “The MUSE second-generation VLT instrument,” in Ground-based and Airborne Instrumentation for Astronomy III, I. S. McLean, S. K. Ramsay, and H. Takami, Eds., 7735, 131 – 139, International Society for Optics and Photonics, SPIE (2010).
  * [16] R. Sharples, R. Bender, R. Bennett, et al., “Kmos: A multi-object deployable-ifu spectrometer for the eso vlt,” New Astronomy Reviews 50(4), 370 – 373 (2006).  Integral Field Spectroscopy: Techniques and Data Production.
  * [17] P. Morrissey, M. Matuszewski, C. Martin, et al., “The Keck Cosmic Web Imager: a capable new integral field spectrograph for the W. M. Keck Observatory,” in Ground-based and Airborne Instrumentation for Astronomy IV, I. S. McLean, S. K. Ramsay, and H. Takami, Eds., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 8446, 844613 (2012).
  * [18] A. D. Eigenbrot, M. A. Bershady, and C. M. Wood, “The impact of surface-polish on the angular and wavelength dependence of fiber focal ratio degradation,” in Ground-based and Airborne Instrumentation for Astronomy IV, SPIE 8446, 84465W (2012).
  * [19] J. Allington-Smith, C. Dunlop, U. Lemke, et al., “End effects in optical fibres,” Monthly Notices of the Royal Astronomical Society 436, 3492–3499 (2013).
  * [20] E. Carrasco and I. R. Parry, “A method for determining the focal ratio degradation of optical fibres for astronomy,” Monthly Notices of the Royal Astronomical Society 271, 1–12 (1994).
  * [21] I. R. Parry, “Optical fibres for integral field spectroscopy,” New Astronomy Reviews 50(4), 301–304 (2006).  Integral Field Spectroscopy: Techniques and Data Production.
  * [22] L. W. Ramsey, “Focal ratio degradation in optical fibers of astronomical interest.,” in Fiber Optics in Astronomy, S. C. Barden, Ed., Astronomical Society of the Pacific Conference Series 3, 26–39 (1988).
  * [23] J. D. Murphy, P. J. MacQueen, G. J. Hill, et al., “Focal ratio degradation and transmission in VIRUS-P optical fibers,” in Advanced Optical and Mechanical Technologies in Telescopes and Instrumentation, E. Atad-Ettedgui and D. Lemke, Eds., 7018, 1020 – 1031, International Society for Optics and Photonics, SPIE (2008).
  * [24] J. J. Bryant, J. Bland-Hawthorn, L. M. R. Fogarty, et al., “Focal ratio degradation in lightly fused hexabundles,” MNRAS 438, 869–877 (2014).
  * [25] S. Chattopadhyay, M. A. Bershady, M. J. Wolf, et al., “Optimization of telescope focal ratios for MLA-fiber coupled integral field units,” in Advances in Optical and Mechanical Technologies for Telescopes and Instrumentation IV, R. Navarro and R. Geyl, Eds., 11451, 1241 – 1250, International Society for Optics and Photonics, SPIE (2020).
  * [26] S. Chattopadhyay, M. A. Bershady, M. J. Wolf, et al., “Fiber positioning in microlens-fiber coupled integral field unit,” Journal of Astronomical Telescopes, Instruments, and Systems 6(2), 1 – 20 (2020).
  * [27] H. Lee, G. J. Hill, B. L. Vattiat, et al., “VIRUS-2 for the Harlan J. Smith telescope of the McDonald Observatory,” in Ground-based and Airborne Instrumentation for Astronomy VII, C. J. Evans, L. Simard, and H. Takami, Eds., 10702, 2609 – 2617, International Society for Optics and Photonics, SPIE (2018).
  * [28] H. Lee, G. J. Hill, N. Drory, et al., “Mix and match as you go: beam switch module that splits wavelengths, scrambles beams, and switches fibers in multiplexed replicated astronomical spectrograph systems such as VIRUS2,” in Advances in Optical and Mechanical Technologies for Telescopes and Instrumentation IV, R. Navarro and R. Geyl, Eds., 11451, 444 – 456, International Society for Optics and Photonics, SPIE (2020).
  * [29] G. J. Hill, H. Lee, B. L. Vattiat, et al., “VIRUS2: a next generation replicated integral field spectrograph with wide field and broad wavelength coverage,” in Ground-based and Airborne Instrumentation for Astronomy VIII, C. J. Evans, J. J. Bryant, and K. Motohara, Eds., 11447, 208 – 221, International Society for Optics and Photonics, SPIE (2020).
  * [30] T. S. Chonis, H. Lee, G. J. Hill, et al., “Design and construction progress of LRS2-B: a new low resolution integral-field spectrograph for the Hobby-Eberly Telescope,” in Ground-based and Airborne Instrumentation for Astronomy IV, I. S. McLean, S. K. Ramsay, and H. Takami, Eds., 8446, 997 – 1018, International Society for Optics and Photonics, SPIE (2012).
  * [31] H. Lee, T. S. Chonis, G. J. Hill, et al., “LRS2: a new low-resolution spectrograph for the Hobby-Eberly Telescope,” in Ground-based and Airborne Instrumentation for Astronomy III, I. S. McLean, S. K. Ramsay, and H. Takami, Eds., 7735, 2674 – 2683, International Society for Optics and Photonics, SPIE (2010).
  * [32] G. J. Hill, H. Lee, B. L. Vattiat, et al., “VIRUS2: a next generation replicated integral field spectrograph with wide field and broad wavelength coverage,” in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 11447, 1144716 (2020).
  * [33] D. Lee, R. Haynes, D. Ren, et al., “Characterization of lenslet arrays for astronomical spectroscopy,” Publications of the Astronomical Society of the Pacific 113, 1406–1419 (2001).
  * [34] C. G. Wynne, “Telecentricity in fibre-fed spectrographs,” MNRAS 260, 307–316 (1993).
  * [35] F. C. Gillett, M. Mountain, R. Kurz, et al., “The Gemini Telescopes Project (Invited Paper),” in Revista Mexicana de Astronomia y Astrofisica Conference Series, E. Falco, J. A. Fernandez, and R. F. Ferrero, Eds., Revista Mexicana de Astronomia y Astrofisica Conference Series 4, 75 (1996).
  * [36] J. E. Gunn, W. A. Siegmund, E. J. Mannery, et al., “The 2.5 m Telescope of the Sloan Digital Sky Survey,” AJ 131, 2332–2359 (2006).
  * [37] J. Allington-Smith, G. Murray, R. Content, et al., “Integral Field Spectroscopy with the Gemini Multiobject Spectrograph. I. Design, Construction, and Testing,” PASP 114, 892–912 (2002).
  * [38] A. Eigenbrot and M. A. Bershady, “Vertical Population Gradients in NGC 891\. I. $\nabla$Pak Instrumentation and Spectral Data,” Astrophysical Journal 853, 114 (2018).
  * [39] A. H. Szentgyorgyi, P. Cheimets, R. Eng, et al., “Hectochelle: a multiobject echelle spectrograph for the converted MMT,” in Optical Astronomical Instrumentation, S. D’Odorico, Ed., 3355, 242 – 252, International Society for Optics and Photonics, SPIE (1998).
  * [40] Flores, H., Puech, M., Hammer, F., et al., “Giraffe multiple integral field units at vlt: A unique tool to recover velocity fields of distant galaxies*,” A&A 420(3), L31–L34 (2004).
  * [41] M. Bershady, S. Barden, P.-A. Blanche, et al., “WIYN bench upgrade: a revitalized spectrograph,” in Ground-based and Airborne Instrumentation for Astronomy II, I. S. McLean and M. M. Casali, Eds., 7014, 211 – 222, International Society for Optics and Photonics, SPIE (2008).
  * [42] S. A. Smee, J. E. Gunn, A. Uomoto, et al., “The Multi-object, Fiber-fed Spectrographs for the Sloan Digital Sky Survey and the Baryon Oscillation Spectroscopic Survey,” Astronomical Journal 146, 32 (2013).
  * [43] M. Mateo, J. I. B. III, J. Crane, et al., “M2FS: the Michigan/Magellan Fiber System,” in Ground-based and Airborne Instrumentation for Astronomy IV, I. S. McLean, S. K. Ramsay, and H. Takami, Eds., 8446, 1686 – 1704, International Society for Optics and Photonics, SPIE (2012).
  * [44] G. Dalton, S. C. Trager, D. C. Abrams, et al., “WEAVE: the next generation wide-field spectroscopy facility for the William Herschel Telescope,” in Ground-based and Airborne Instrumentation for Astronomy IV, I. S. McLean, S. K. Ramsay, and H. Takami, Eds., 8446, 220 – 231, International Society for Optics and Photonics, SPIE (2012).
  * [45] H. Chung, A. N. Ramaprakash, A. Omar, et al., “Dotifs: a new multi-ifu optical spectrograph for the 3.6-m devasthal optical telescope,” Proceedings of SPIE 9147, 91470V (2014).
  * [46] R. Yan, M. A. Bershady, M. P. Smith, et al., “The prototype telescope and spectrograph system for the AMASE project,” in Ground-based and Airborne Instrumentation for Astronomy VIII, C. J. Evans, J. J. Bryant, and K. Motohara, Eds., 11447, 1936 – 1947, International Society for Optics and Photonics, SPIE (2020).
  * [47] D. J. Schroeder, Astronomical optics / Daniel J. Schroeder., Academic Press, San Diego ; (1987).
  * [48] E. Hecht, Optics, Pearson (2012).