Preliminary optical design for the WEAVE two-degree prime focus corrector

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Nov 15, 2013 (3 years and 8 months ago)

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Preliminary optical design for the
WEAVE

two
-
degree prime focus
corrector


Tibor Agócs
*
a
, Don Carlos Abrams
b
,
Diego Cano Infantes
b
, Neil O'Mahony
b
, Kevin Dee
c
,
Jean
-
Baptiste Daban
d
, Carole Gouvret
d
, Sebastien Ottogalli
d

a
NOVA Optical Infrared Instrumentation Group at ASTRON, P.O. Box 2, 7990 AA Dwingeloo, The
Netherlands
;

b
Isaac Newton Group of Telescopes, Apartado

de correos 321, 38700 Santa Cruz de L
a Palma
,
Canary Islands, Spain
;

c
Engineering & Project Solutions Ltd, Daresbury Science Park, Warrington, WA4 4FS, United
Kingdom;

d
Laboratoire Lagrange, UMR7293, Université de Nice Sophia
-
Antipolis, CNRS, Observatoire de la
Côte d’Azur, F
-
06304 Nice, France
.

ABSTRACT

We present the preliminary optical design for
a new
two
-
degree refractive prime focus corrector for the 4.2m William
Herschel Telescope optimised for the wide
-
field multi
-
object
spectrograph
, WEAVE (WHT Enhanced Area Velocity
Explorer). From the two concep
tual

designs described previously

[1]
, the counter
-
rotating atmospheric dispersion
corrector
approach
was selected and further optimized to meet the flat image sur
face requirement. The preliminary
design provides good polychromatic image quality. The PSF
does not exceed

0.6 arcsec (80% encircled energy) over a
wavelength range from 370 to 1000nm

covering

a two degree FOV for zenith angles up to 65 degrees. We descri
be the
corrector’s performance and the trade
-
off between telecentricity and
the requirement for a
flat image surface. We present
the results of the tolerance and thermal analyses, ghost and scattered light calculations and the finite element analysis
that
are necessary to establish the PSF error budget for the corrector.

Keywords:

astronomy, telescope,
preliminary,
optical design, wide
-
field, corrector, ADC

1.

INTRODUCTION

WEAVE (WHT Enhanced Area Velocity Explorer) is the proposed wide
-
field highly
-
multiplexed spectrograph for the
prime focus of the William Herschel Telescope. It will provide unique capabilities in the northern hemisphere for
scientific exploitation of upco
ming European
-
led and international imaging surveys. The science cases for WEAVE are
summarised by Balcells et al.
[2]

and more recently by Dalton et al.

[3]
.

The key requirements for the new corrector for the WHT prime focus are: a field of view of
two

degrees in diameter;
transmission over a broad range of wavelengths (requ
irement 390
-
1000nm, goal 370
-
1000nm); and a polychromatic PSF
smaller than 0.6 arcsec (80% ee)
throughout

the
full
field of view. The throughput of the optics should be as high as
possible;

a reasonable goal would be to achieve throughputs of about 0.6 at
360nm, 0.8 at 400nm

and

greater than
0.9 at
wavelengths above 500nm.

Previously we presented two conceptual designs for the
new
WHT
field corrector
[1]
, each with
a different type of
Atmospheric Dispersion Corrector (ADC). One
included

a counter
-
rotating cemented doublet pair with curved surfaces,
while the other
was
based on a doublet that
could
be laterally shifted
in and
out
of
the beam (Subaru
-
type ADC) in order

to cancel out atmospheric dispersion.


*agocs
@astron.nl;

phone +31 (0)52 159 51 73;
http://www.astron.nl/





Following the conceptual phase, both optical designs were re
-
optimized to provide a flat image surface; this requirement
being driven by the fibre positioner.
At the end of the conceptual phase a decision was made that the counter
-
rotating
ADC would be fur
ther investigated as the baseline design due to the following factors:

i.

Its

better image quality throughout the whole field, especially at large off
-
axis angles.

ii.

Its

better image quality at large zenith angles (50
-
65 degrees).

iii.

Apart from the flat image su
rface requirement, telecentricity
was
also an important goal when considering
fibre injection. Although neither of the two designs
were
truly telecentric, the counter
-
rotating ADC design
had
smaller chief ray angles at the edge of the field

and this was

ad
vantageous for the fibre system (
this is
discussed

further

in section
2
).

We present the preliminary design for the new refractive corrector with the counter
-
rotat
ing ADC. In section 2 issues
related to interfaces and manufacturing are discussed, the trade
-
off between telecentricity and the image surface
curvature is described and finally the image quality, throughput and vignetting of the preliminary design are pre
sented.
In section 3 the tolerance analysis and the PSF error budget are discussed and section 4 explains the ghost and scattered
light analyses. In section 5 the thermal effects on the lenses and mechanical structures are discussed. Section 6 reviews
the
results from the Finite Element Analysis (FEA) and in the last section the conclusions are summarized and activities
of the next phase outlined.

2.

PRELIMINARY OPTICAL
DESIGN

2.1

Interfaces and manufacturability

The preliminary optical design is based on two co
unter
-
rotating prism pairs that have curved surfaces (
Figure
1
). The
pairs are rotated through an elevation
-
dependent angle (0 to 90 degrees) in opposite directions a
bout the telescope
optical
axis. The result of the rotation is that the dispersion vector is increased to compensate for the atmospheric dispersion
over different zenith angles (0 to 65 degrees).



Figure
1

-

The layout of the corrector. On the left a 2D view shows the components of the prime focus corrector,
on the right the
same arrangement is depicted but with

the WHT pri
mary mirror.

The major difference between the conceptual design and the preliminary design is that the image surface of the latter is
flat. The flat image surface appeared as an interface requirement after a study showed that the additional degrees of
fre
edom necessary for the pick and place fibre positioner
,

in the case of a curved image surface
,

would significantly
increase
instrument
complexity and cost. Other optical design
-
related changes included an increa
sed back focal distance
(260mm)

which was nee
ded for the correct functioning of the fibre positioner (
an adaptation of the
AAT’s 2dF design)
.
L
arger lens dimensions

were also incorporated
. After communication with lens manufacturers it became apparent that
X
Y
Z
Lens 1

Lens 2
-
3 & 4
-
5

Lens 6

ADC

Asphere

4.2m/F2.5
WHT primary

Prime focus
corrector





the lenses needed to be oversized by approx
imately 20mm in radius with respect to the clear aperture of the beam. Also
the lens thicknesses needed to be increased to at least 10% of
the
lens diameter in order to reduce risks and
manufacturing
costs.

The desig
n contains one aspheric surface

which i
s located on the convex surface of the last lens. The maximum deviation
from the best fit sphere over the clear aperture of the beam is less than 0.8mm, which is still large and at the limit of wha
t
can

be tested using interferometry. Another possibility f
or testing the aspheric surface is
to use
a 3D measurement based
on contact profilometry which can achieve approximately 300nm RMS accuracy. By distributing errors in the system
and tightening tolerances on certain parameters, the 300nm RMS surface accurac
y could be suitable for the measurement
of the aspherical surface (
this is discussed further

in section
3
).

The wavelength range used during the optimization
was
370
-
1000nm. Initially all wavelengths were represented with the
same weights during optimization but
after
taking into account the scie
nce requirements
,

a trade
-
off was made and the
370
-
390nm range
was

weighted less.
The Gaussian

Quadrature (GQ)

[4]

method was used to calculate the different
weight
s for different wavelengths. The
results are shown in the following figures and demonstrate

that
the
image quality
for

the 390
-
1
000nm range can be improved
by reducing the

image quality over
the
370
-
390nm

range
.



Figure
2

-

Comparison of image quality. The

same weights are used during the optimization over the 370
-
1000nm
range (left) vs.
the
GQ method (lower weights at the edge of the 370
-
1000nm wavelength range).
The
X axis
represents field positions (
zero to one
degree) and the Y axis shows RMS spot radius in μm (maximum value is 20
μm). On the left graph the uppermost three lines correspond to 370nm, 382nm and 402nm.


2.2

Telecentricity vs. flat image surface

During the re
-
optimization of the corrector design, to ac
commodate the flat image surface requirement, it became
apparent that the design would not be telecentric. Telecentricity is desired because the image surface has to satisfy the
coupling requirements of the fibre modules and c
oupling efficiency

can be maxi
mized if the
chief rays
are
normal

to
the
fibre coupling surfaces. In the current design the chief ray angle increases from zero (on
-
axis) to
four

degrees (one
degree off axis) as shown in
Figure
3
.

Optically the fibre system consists of microlenses, right
-
angle prisms and the fibres themselves. The microlenses convert
the F
-
ratio of the corrector into the F
-
ratio of the spectrograph, the prisms fold th
e light path 90 degrees and finally the
fibres guide the light to the Nasmyth platform, where the spectrograph is located. The prisms are based on total internal
reflection (TIR) and
thus
they can be used to correct the non
-
telecentricity of the prime focu
s field corrector. If the
corrector were perfectly telecentric the prism angle would be 45 degrees. Based on different configurations summarized
in
Table
1

the non
-
tel
ecentricity can be compensated in order to decrease F ratio and improve fibre coupling.



0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
2
4
6
8
10
12
14
16
18
20
+Y Field in Degrees
Diffraction Limit
20120523_Weave_PFC_PD_cemented_370-1000.ZMX
Configuration 1 of 12
RMS Spot Radius in µm
RMS Spot Radius vs Field
WHT PF CORR 2 deg FOV design - counter rot ADC
25-5-2012
Poly
0.37
0.382
0.402
0.43
0.466
0.512
0.568
0.637
0.721
0.818
0.92
1
Reference: Centroid
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
2
4
6
8
10
12
14
16
18
20
+Y Field in Degrees
Diffraction Limit
20120523_Weave_PFC_PD_cemented_370-1000_withGQ.ZMX
Configuration 1 of 12
RMS Spot Radius in µm
RMS Spot Radius vs Field
WHT PF CORR 2 deg FOV design - counter rot ADC
25-5-2012
Poly
0.37
0.382
0.402
0.43
0.466
0.512
0.568
0.637
0.721
0.818
0.92
1
Reference: Centroid




Table
1

-

Possible right angle prism configurations.

Number of
separate
fields

TIR surface angles
of the prisms
(degrees)

Field distribution
(ranges
defined by off
-
axis field
angles)

Additional angle to couple
in the fibre on the top of
the F2.7 beam

F ratio necessary
for fibre coupling

1

47

0
-
1

+/
-
2

F2.3

2

46; 48

0
-
0.6; 0.6
-
1

+/
-
1

F2.5

4

45.5; 46.5; 47.5;
48.5

0
-
0.3; 0.3
-
0.6; 0.6
-
0.8; 0.8
-
1

+/
-
0.5

F2.6

If the
image plane of the corrector
is divided
into two areas
, for example one area which includes the
central field up to
0.6 degree
s

and
a second area which includes

field angles
from

0.6 to 1 degree
,

the TIR surface angle
for

the prisms wi
ll
be 46 and 48 degrees respectively. The prism
s

with the 46 degree

angle

will cover the chief ray angle deviation
s

from

zero

to

two

degrees and the 48 degree
prisms
will cover chief ray
angle
deviation
s

from

two

to

four

degrees (
Figure
3
).
Using this method,
the non
-
telecentric light can be
more efficiently
coupled into the fibres. The use of different prisms
with different angles requires correct prism orientation so

that the incidence plane is perpendicular to the TIR surface of
the prism
s
.

In order to minimise light loss
es
, t
he microlenses
will
have to be designed to accept a slightly faster beam
.
Instead of the nominal F2.7 beam a F2.5 beam has to be taken into
consideration

hence the

additional angle


parameter

(±1
o

for

two
fields
) noted

in Table 1
. In
the
case
where
there is only one
set of
prism
s, all

with
the same
TIR angle, the
microlenses have to convert
the light
from
an
F2.3

beam

to
an
F3.2

beam

that
wil
l
be fed into the spectrograph.




Figure
3

-

On the left
,

chief ray angles are shown with respect to
the
surface normal (non
-
telecentricity)
throughout

the field. The prism configuration with two separate fields is also illustrated

(46
o

and 48
o

TIR)
. On the
right the marginal ray angles are plotted with respect to the surface normal.


2.3

Image quality

The image quality satisfies the requirement that 80% of

the encircled energy (ee80) of the polychromatic light is within

a

0.6 arcsec diameter circle over the whole
two

degree FOV (
Table
2
). From the limited number of glass blanks that are
available with the required size, the best candidates for the ADC are the N
-
BK7 and LLF1 glasses.
However

the
dispersion correction is not perfect and the ee80 diameter is slightly increased with increasi
ng off
-
axis field position.
The
e
e80 diameter maps are calculated for various zenith distances

and are shown in Figure 4.


0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Chief ray angle (degrees)

Off axis field angle (degrees)

Chief ray angle (deg)

6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Marginal ray angle (degrees)

Off axis field angle (degrees)

Marginal ray angles (deg)

46
o

TIR prism

48
o

TIR prism





Table
2

-

Polychromatic (370
-
1000nm) ee80 diameter va
lues for zenith angles 0 (left)
and 65 degrees (right) i
n μm
and arcsec. The table contains
the
averag
e values over the radial field. D
ue to the tilted surfaces of the ADC, the
ee80
diameter
varies for the same radius

depending on the angular position within the FOV
.

Off
-
axis field angle (deg.)

Average ee80 d
iameter

Zenith angle 0 deg.

Average ee80 diameter

Zenith angle 65 deg.

μm

arcsec

μm

arcsec

0.00

27.80

0.50

30.18

0.54

0.09

27.92

0.50

30.37

0.54

0.18

28.35

0.51

30.77

0.55

0.27

28.68

0.51

31.08

0.56

0.36

28.53

0.51

31.10

0.56

0.45

27.68

0.50

30.65

0.55

0.55

26.57

0.48

30.08

0.54

0.64

26.55

0.48

30.10

0.54

0.73

27.72

0.50

31.16

0.56

0.82

29.16

0.52

32.56

0.58

0.91

29.47

0.53

32.73

0.59

1.00

31.15

0.56

34.21

0.61


0
o

zenith angle

28
o

zenith angle



57
o

zenith angle

65
o

zenith angle



Figure
4

-

The e
e80 diameter maps are shown for di
fferent zenith angles

throughout the 2
o

FOV
.
The
top
of the
scale represents 0.65 arcsec ee80 diameter.

0.0000
0.0650
0.1300
0.1950
0.2600
0.3250
0.3900
0.4550
0.5200
0.5850
0.6500
20120523_Weave_PFC_PD_cemented_ee80.ZMX
Configuration 1 of 12
ee80 map
WHT PF CORR 2 deg FOV design - counter rot ADC
26-5-2012
Polychromatic EE80 values
wavelength range: 370nm-1000nm
0.0000
0.0650
0.1300
0.1950
0.2600
0.3250
0.3900
0.4550
0.5200
0.5850
0.6500
20120523_Weave_PFC_PD_cemented_ee80.ZMX
Configuration 5 of 12
ee80 map
WHT PF CORR 2 deg FOV design - counter rot ADC
26-5-2012
Polychromatic EE80 values
wavelength range: 370nm-1000nm
0.0000
0.0650
0.1300
0.1950
0.2600
0.3250
0.3900
0.4550
0.5200
0.5850
0.6500
20120523_Weave_PFC_PD_cemented_ee80.ZMX
Configuration 10 of 12
ee80 map
WHT PF CORR 2 deg FOV design - counter rot ADC
26-5-2012
Polychromatic EE80 values
wavelength range: 370nm-1000nm
0.0000
0.0650
0.1300
0.1950
0.2600
0.3250
0.3900
0.4550
0.5200
0.5850
0.6500
20120523_Weave_PFC_PD_cemented_ee80.ZMX
Configuration 12 of 12
ee80 map
WHT PF CORR 2 deg FOV design - counter rot ADC
26-5-2012
Polychromatic EE80 values
wavelength range: 370nm-1000nm





2.4

Throughput

The throughput of the corrector takes into account the internal
transmittance and surface Fresnel or coating losses of the
lenses (vignetting is discussed in section
2.5
). In the wavelength range below 380nm the internal transm
ittance of the
glasses dominates

and

above 380nm the Fresnel losses determine the throughput limit (
Figure
5
). Compared to the
conceptual design the glass materials
are the same, the glass thicknesses on average are increased by 30% and the AR
coating specifications are also changed. The average loss of transmission
, per surface,

ove
r the required wavelength
range
is 2% and this reflects a more realistic scenario rega
rding the coating of glasses with diameters

in the order of
600mm
.

The first lens is an exception in that

the current baseline design
has the lens

uncoated in order to reduce risks
and costs.

2.5

Vignetting

Vignetting was investigated in detail considering a
ll possible contributors. Vignetting due to the fibre positioner (current
baseline outer diameter is 1750mm), the Nasmyth turret and the spiders are inevitable
. V
ignetting
was

calculated and
is
shown together with the corrector’s throughput
in
Figure
5
.




Figure
5

-

Throughput of the corrector without the primary mirror
and with no

vignetting (left). Throughput of
the corrector with the primary mirror and vignetting due to secondary obscuration and spiders is shown (right).


Another source of vignetting that was investigated
was

the placement of a circular
aperture

just before lens
1 (primary
mirror side)
.
A beam diameter of 1100mm

falling onto lens 1, with a 1000mm circular aperture, results in vignetting on
the image surface

which starts at 0.65 degrees off axis and increases to only 6% at the edge of the two degree FOV

(
Figure
6
)
.


Transmission vs. Wavelength
20120523_Weave_PFC_PD_cemented.zmx
Configuration 1 of 12
WHT PF CORR 2 deg FOV design - counter rot ADC
27-5-2012
Field Pos: 0.0000, 0.0000 deg
Field is unpolarized.
0.000
0.37000
0.500
0.68500
1.000
1.00000
Transmission
Wavelength in µm
Transmission vs. Wavelength
20120523_Weave_PFC_PD_cemented_VIGN.ZMX
Configuration 1 of 12
WHT PF CORR 2 deg FOV design - counter rot ADC
27-5-2012
Field Pos: 0.0000, 0.0000 deg
Field is unpolarized.
0.000
0.37000
0.500
0.68500
1.000
1.00000
Transmission
Wavelength in µm





Figure
6

-

Vignetting by a circular aperture with an inner diameter of 1000mm just before lens 1.


Without the circular apertur
e the maximum allowable size
for

lens 1 (~1000mm) drives the design and although
sufficient image quality can be achieved, the non
-
telecentricity is affected by this dimensional limitation.
The following
advantages are seen when this vignetting is introduc
ed into the design
:

i.

Non
-
telecentricity can be reduced by 1.4
-
1.5 degrees.

ii.

Image quality can be improved slightly. In essence,
on

average
a

0.05 arcsec reduction of the ee80 diameter
is observed

throughout the 2
o

FOV
. This is a natural consequence of redu
cing the off
-
axis rays that usually
contribute more to geometrical aberrations.

iii.

The first lens can be shifted towards to the primary mirror by about 200mm, which means that mechanical
stiffness and stability can be improved

in this design
.

At the time of

writing,
it
has not been decided whether the vignetting
introduced
by the
above
mentioned 1000mm
circular aperture
,

just before lens 1
,

will be implemented or not. Further investigation
s are

needed

to fully understand the
impact of such a change on the in
strument design

and
the consequences
on the WEAVE science cases.

3.

TOLERANCE ANALYSIS

3.1

Sensitivity analysis and PSF budget

T
he PSF budget was produced
(
Table
3
)
by taking into account all possible sources of errors
. The tolerances in the table
were continuously iterated (through communication with
the
mechanical designers and
the
optical component
m
anufacturers) until the image quality requirement was satisfied. According to the
most recent

version of the PSF error
budget the optical design satisfies the 0.6 arcsec ee80 diameter requirement and the rest of the errors do not contribute
more than an 0.
86 arcsec increase, so that the overall image quality (RSS of design and errors) is
better than
1 arcsec. A
contingency of 0.3 arcsec (0.83 arcsec calculated RSS), which is between the overall ee80 diameter and the fibre core
size ensures that the correcto
r satisfies the science requirements.








Table
3

-

PSF budget of the corrector

Error source

RSS contribution to
ee80 diameter

(arcsec)

Notes

Design

0.55

Optical design ee80 diameter (polychromatic, average)

at zenith
angle
zero

degrees

Manufacturing I.

0.30

Refractive index, Abbe
-
number, power, irregularity, thickness,
wedge

Manufacturing II.

0.15

Homogeneity, stress birefringence

Alignment of corrector

0.40

Decentres, tilts, axial displacements, ADC alignment

Alignment of
telescope to
corrector

0.40

Decentre, tilt
, axial displacement

Thermal stability

0.30

Expansion of telescope structure, mounts, etc., radius, thickness,
refractive index change due to temperature

Positional stability
(flexures)

0.20

Flexures due to
gravity load, estimation

Positional stability (rest)

0.25

V
ibrations, mounting stiffness, etc.

Primary mirror figuring

0.20


Total

RSS

0.99


Contingency

0.30


Fibre core size

1.30


The PSF budget contains the manufacturing errors of the lenses, the surface regularities and irregularities, the centre
thicknesses and wedges as well as errors related to material properties like homogeneity and stress birefringence. With
respect to the m
aterial properties, a melt fit approach was presumed and thus the sensitivities were calculated after
curvatures, axial thicknesses and ADC surface tilts had been re
-
optimized. The alignment errors consist of the axial
displacement, decentre and tilt of al
l optical components and also the alignment errors of the ADC components during
cementing. Alignment errors at the ADC component level can be reduced during the alignment of the whole corrector
assembly. Contributors to instability errors include vibration

and wind shake, which cannot be corrected for with
compensators. Another source of instability comes from the changing gravity vector. The instrument
will

operat
e

between zenith angles 0 and 65 degrees. FEA shows that the deformation of the lenses due to
gravity is negligible, but
flexure of the prime focus assembly and bending of the whole telescope structure play an important role

in establishing
the PSF budget

and contribute to significant
increases in
spot size. Compensation for mechanical flexures wil
l be
carried
out
during the alignment phase
. T
he prime focus assembly will be aligned such that it performs optimally at a zenith
angle of 32.5 degrees and the performance
will be

reduced, due to flexures, when at zenith or at a maximum of 65
degrees zeni
th angle. There
will also be the
possibility to include different pre
-
loads

where the spiders are connected to
the
top
-
end

ring. This will allow for better balance to be achieved and for the gravity load to be compensated. Another
important contributor to
PSF degradation comes from thermal effects, namely from the glass thickness and curvature
changes of the lenses, thermal expansion of the mechanical structure and temperature inhomogeneity
-
induced refractive
index changes and transient birefringence. All t
hese
factors were taken into account

when compiling the
most recent

PSF
budget.
Nevertheless communication with
the
lens manufacturers is still on
-
going

and thus the data presented in table 3

reflects the current state of the tolerancing process
which will

need to be finalised
.
The worst offenders
in the PSF error
budget
are described in section
3.2
.

The sensitivity analysis was
carried out in Zemax
©
. Sensitivities were calculated with scripts so
that
the repetitive tasks
could be
performed

relatively easily. Using the calculated sensiti
vities
for
the considered
degrees of freedom
the PSF
error budget was generated and the iterations to distribute th
e errors between manufacturing and alignment

etc. could be
performed in an efficient
manner
.







3.2

Worst offenders

3.2.1

Manufacturing

The worst off
enders are the largest contributors to the PSF error budget.
T
aking into
consideration
today’s
manufacturing
capabilities

the power and irregularity requirements
can

be satisfied relatively easily
with the
except
ion of

the irregularity of the aspherical su
rface.
The manufacturing accuracy
of

the aspheric lens
will be

driven by the
measurement accuracy and
will

depend on the applied technique.
F
or example
, if this is measured

using

stitching
interferometry, the surface irregularity could be
better
controlled than
if a less accurate 3D measurement was used.

3.2.2

Alignment

When c
onsidering alignment
,

the most important tolerances are
those related

to
the first doublet of the ADC
,

possibly
as
a result of

the relatively short radius of
the

last surface.
T
he

fibre positioner plate

alignment is
also important
, the tip and
tilt of this surface should be
preferably
better than
0.01 deg
ree
s
. D
ue to the lack of compensation
for
this error the same
tolerance should be taken into account
when considering

stability

(
including
both
positional and thermal stability)
.

The
alignment of the
corrector with regards to the primary mirror is one of t
he most critical tolerances.
T
o
some degree

the
axial displacement

can be compensated
for
by refocusing with the last lens
,
but
the
negative
effect
s

of tip/tilt

and
especially decentr
ing

type misalignments
on polychromatic spot size
are

substantial.

3.2.3

Stability

Stability can be divided into three subgroups: thermal stability, positional stability from flexures and positional
stability
from other factors such as vibrations

and

mounting stiffness etc.
Since there are two surfaces where fibres are positioned
and these surfaces are interchanged during the night, the stability of the field plate is an important
issue
.
S
olving

stabi
lity
issues, not only for the corrector

unit
, but
for
the whole
top
-
end

of the telescope is a challenging task.
T
he biggest
concern is displacement and tilt of the corrector
unit
with regards to the primary mirror and although
some
corrections
can be
appli
ed
,
vibrations
and flexures
of the telescope structure
de
grade image quality.
Currently
this contribution to
the
PSF error budget is estimat
ed

and thus
the flexures will

need to

be modelled in greater detail
. The largest contribution in
the thermal
component
of the PSF budget is
caused by
the inhomogeneity of the glass blanks, however some
compensation can be
applied such as

the focus
term can

be compensated

for

in the polishing phase.

3.3

Single value decomposition (SVD)
of the sensitivity matrix

The S
VD of the sensi
tivity matrix
for

the corrector
is a powerful tool
for analysing

the system’s misalignments

and
instabilities
.
The correlation between
various
degrees of freedom
(
DOF
)

of the system and its aberrations

can be
determined.
Here w
e present SVD on the sensitivity matrix
based on methods demonstrated elsewhere (
[5]
,
[6]

and
[7]
)
.

The sensitivity matrix is a Jacobian matrix that can be defined as












where


is the aberration vector that is form
ed by decomposing the
wavefront error (
WFE
)

into Zernike polynomials

and



is a small change
in

DOF of the system
. SVD is a linear algebra technique to
decompose

the sensitivity matrix into
the following three matrices:







where
U

and
V

are column orthogonal matrices and
S

is a diagonal matrix.

Essentially

these
three
matrices indicate the
type of orthogonal aberrations that can result from misalignments and they
are
list
ed

from
the
easiest to produce to
the
hardest to produce.
The

diago
nal

matrix

S

contains the singular values (



,


, etc.)

which measure the sensitivity of the
sys
tem to a particular aberration. Also









etc.
and




corresponds to the most sensitive aberration mode

which is then
followed by the other modes w
ith decreasing sensitivity.
The

U

matrix
contains the Zernike coefficients for
a given aberration mode and
V

lists the misalignments that produced those aberrations.
For example,
the most sensitive
aberration is given by the first
column vector from
U

that
results

from the misalignments that are listed in the first
column vector of
V

and
has the strength of



.

In the following
,

the r
esults of the SVD are presented for the corrector.
The design was evaluated at
633nm

for
9 field positions
where

the
AD
C configuration corresponded to
zero

degree
zenith angle.

This wavelength was chosen because it is close to the
midpoint of the
wavelength range that the corrector is
designed to operate over and
it
corresponds to the wavelength of the helium
-
neon laser.





Table
4

-

The
most important
orthogonal aberration modes (2
nd

column) and the misalignments that cause them
(3
rd

column)

are shown
.

M
isalignments
are

given
in
units of
DOF adjustments

(
Y axis
)

as a function of

DOF
numbers (
X axis
)
.

Modes

Column of
U
matrix

(aberrations for 9 field
positions)

Column of
V

matrix

(misalignments of the 40 DOF)

Notes

1



The first orthogonal aberration
mode
has

primary astigmatism,
primary coma and
defocus as a
result of decentring and tilting

the corrector components.


D
ue to the system

s symmetry
,
the second mode is similar to
the first and thus is not shown
.

3



The
third

mode
consist
s

mostly

of
primary
coma
.
This

is

originated

from a different
linear combination of the
decentres and til
ts

than
those
seen
in mode 1 and 2
.


The fourth mode
is
similar to
the third
and thus is not shown
.

5



The fifth mode
has

some
de
focus

and primary spherical
aberration
s
.

A
t large off axis
posit
ions
,

astigmatism and coma
start
s

to dominate.
This mode
is

primarily
caused by axial
displ
a
cement after refocus
ing

and some additonal axial
displacement instability.


6



The sixth mode

is

dominated by

defocus from the axial
displacement of the components
due to instab
i
lity.





5
10
15
20
25
30
35
40
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
V configuration singular vector #1
Comp number
Value
5
10
15
20
25
30
35
40
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
V configuration singular vector #3
Comp number
Value
5
10
15
20
25
30
35
40
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
V configuration singular vector #5
Comp number
Value
5
10
15
20
25
30
35
40
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
V configuration singular vector #6
Comp number
Value




In
Table
4
,

the 2
nd

column
contains the wavefront aberrations
derived
from the column vectors of the
U

matrix (
basically
the Zernike coefficients for all field positions)

and t
he 3
rd

column
plots

the misalignments
that caused them
(misalignments
along the

Y axis and DOF number
along the

X axis
)
.
The modelled DOF
were

derived from

decentr
ing

and tilt
ing

the corrector (DOF number 1
-
4) and
decentring

and tilt
ing

all components incl
uding
the image plane (DOF
number 5
-
26
). The last DOF are the axial
separations between components of which
27
-
33

are with alignment
(refocusing) and

34
-
40

are without refocusing.
This indicate
s

that
alignment and stability
were

both modelled in the
sensit
ivity matrix and SVD calculation. M
isalignments
(Y axis)
are given in the units of DOF adjustments,
which

are

summarized in
Table
5
.
The values in the
t
able were calculated during the
sensitivity analysis. Since the only possible
compensator is the axial disp
lacement of lens 6, the decentring

and tilt
ing

type
s

of
DOF cannot be corrected

for

and so
these numbers in the table correspond to
both

alignment a
nd stability. The axial position of the lenses
was

taken into
account twice, once for alignment (applying refocusing) and once for stability (without refocusing).


Table
5

-

Units of DOF adjust
ments used for the SVD analysis are de
rived from the sensitivity analysis.


Components

Alignment/

Stability

DOF type

Corrector

Lens 1

All other components

A & S

Decentre x/y

500
µm

100
µm

50
µm

A & S

Tilt x/y

0.005
o

0.01

o

0.01

o

S

Axial
displacement


(without refocus)

50
µm

10
µm

10
µm

A

Axial
displacement

(with refocus)

1000
µm

500
µm

250
µm


The most important
results of the SVD
-
based analysis
are

summarized as follow
s
:

i.

SVD
confirms the

results
of

the sensitivity analysi
s and shows the same worst offenders.

ii.

The largest aberrations can arise from the
x and y
decentres and tilts
of the whole corrector assembly,

and
from
the other independent components (modes 1
-
2 and 3
-
4). Image quality improvement can be achieved
by further tightening the tolerance
s

on the who
le corrector unit or on the first ADC component.
Another
possibility
would be

to investigate other compensator DOF, e.g. tip
-
tilt of lens 6
which could partially

compensate
for

the errors
induced by

decentr
ing and tilting

of other

elements
.

iii.

Aberrations that are presented in
(
ii
)

are p
aired

and
they have similar magnitudes

in the SVD, which
reflects the symmetry of the system
with respect to

x and y decentres and tilts.

iv.

The linear combination of axial displ
acement type
errors

in

alignment (
fo
r example where
refocusing is
applied) is a significant contributor to image quality degradation (mode 5).
When i
nvestigating the
aberration maps it
was
clear

that refocusing
was

not
effective

for
the one degree off
-
axis field positions
.
A
lthough the imag
e quality
could

be maintained at the centre of the field,
image quality
degradation
was

observed

due to astigmatism and coma

at the field edge
.

v.

A
xial displacement

stability

is also important.
Mechanical analysis
will
be used to verify

that

the

values

used

to represent the system flexure
are realistic, especially
with regard

to

the stabili
ty of the whole
corrector unit.








Figure
7

-

T
he diagonal values of the S matrix are shown. They measure the sensitivity of the system to a
particular orthogonal aberration.
T
he

aberration modes
, which

are in pairs
,

reflect
the symmetry between
x
tilt
/decentre

and
y
tilt
/decentre

misalignments.


4.

GHOS
T AND SCATTERED LIGH
T ANALYSIS

4.1

Ghost light analysis

Ghosts arise from multiple reflections between optical surfaces. In this context the focus and pupil ghosts were
investigated, primarily concentrating on any double bounce reflections between optical
surfaces. The only non
-
optical
surface involved in the analysis was the field plate of the fibre positioner (the image surfac
e of the corrector).
The field
plate in the current WEAVE design is made of

steel

and
,

as such
,

has a finite reflection over a broa
d wavelength range
that

creates ghosts.
During the ghost analysis

it became apparent that
the fi
e
l
d plate
did

not
pose

any serious risk
s to
issues such as
fibre cross talk
.
T
he ghost light intensity levels
were

low.
T
he closest focus
sed

ghost
was

33mm awa
y
from the image plane and
originate
d

from
a
reflection between
the
last and first surfaces of the first ADC
component
.

All
surfaces will have an AR coating with an average reflection
of less than 2% (except lens 1)
.

T
aking

into account the
2.7
F
-
ratio
,

th
e ghost light was at a level of
10
-
4

w
hen compared to the parent ray.

T
he closest pupil ghost
was

69mm away
from the image plane and
or
i
ginate
d

from reflection
s

between the image plane and the last surface of the second ADC
doublet.

With proper calibration
,

the effect of the ghost pupil can be compensated
for
during observations.


4.2

Scattered light analysis

S
cattered light
was
modelled only at a basic level.
M
ounting surfaces between the lenses were modelled and, assuming
various surface scattering characte
ristics, the approximate scattered light levels were calculated. Initial analyses showed
that baffling
would be
required between all lenses, especially between lens 1 and the ADC.
Furthermore it

was
also
demonstrated that
the edges of all lenses

would need

to
be grounded and painted ma
t
t black in order to reduce scattered
light from these surfaces.




5
10
15
20
25
30
35
40
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
SVD singular values for the original sens. matrices
Singular vector number (n)
Singular value (S)
Mode 1 and 2

Mode 3 and 4

Mode 5

Mode 6





5.

THERMAL ANALYSIS

The
operating
temperature range for the
new
prime focus corrector is between
-
5 and 25
o
C (
the mean temperature at the
Roque de Los Muchachos
is 11
o
C). Thermal effects that perturb the performance of the corrector include the following:

i.

The glass thicknesses and curvatures of the lenses
will
change due to the
CTE differences

of the glass
materials over the operating temperature range

ii.

The therm
al expansion of the mechanical structure changes the axial separations between the lenses.
Refocusing can be applied as compensation, but there will always be some residual image quality
degradation that is inevitable.

iii.

The temperature inhomogeneity
-
induced

refractive index change will degrade image quality.

iv.

The temperature inhomogeneity
-
induced transient birefringence will degrade image quality.

6.

MECHANICAL SYSTEM AN
D
FINITE ELEMENT ANALY
SIS

6.1

Mechanical system

The design of the mechanical
system

for

the corrector has
been initiated

(
Figure
9
)
. The main
mechanical structures

that
are pertinent
to the corrector’s design are:
the
top
-
end

ring
,
the
vanes and the stru
cture that contains the lenses

(
Figure
8
)
.
Also important, but not strictly

related

to the design of the corrector are the
instrument
turntable and the mechanical
str
ucture of the fibre positioner. A new
top
-
end
assembly, including the
ring

and
vane
s
,
will be designed to accommodate
the new corrector
on

the WHT.
With regards to

the mechanical structure for the lenses
the following
two de
sign concept
s

were investigated.

1.

Design 1 integrates all the optical components
, one by one,

into

the
mechanical structure

of the
top
-
end
.

2.

Design 2 separates the central section

(a cylindrical barrel supported by the vanes)

from

the prime focus
corrector.

After careful consideration, des
ign 2 was selected for further development because the alignment of the prime focus
corrector can be carried out off the telescope as a single unit. Mounting the corrector into the central section will requir
e
a simpler alignment procedure than if each le
ns was incorporated one at a time.

The disadvantage of d
esign 2
is that it
has a larger
vignetting
footprint i.e.
a
diameter
of

1700mm
compared to 1450mm for
design

1.



Figure
8

-

The mechanical design of the corrector (left)
and the ADC with the drive motors and encoders (right).







Figure
9

-

The
top
-
end

of the William Herschel Telescope with
the WEAVE

subsystems.


6.2

FEA

Finite Element Analysis (FEA) is an essential tool
for investigating

the effects of
gravity loading and
thermal expansion
.

Lens deformation due to gravity

loads

and
stresses due to
mounting

determine stress
birefringence and

will be
determined from this study. More importantly, given the sensitivity on image quality due to flexures
, the whole telescope
structure including the

top
-
end

ring and mounting arrangement
s

for

the corrector
w
ill
be analysed under varying gravity
vectors. Thermal effects due to thermal expansion and differences in
the
CTE of materials in the mechanical structure
will
also
be
analysed
, although to some
degree

compensation will be achieved by refocusing. Static (
a given
temperature)
as well as

dynamic modelling (changing the
rmal environment
) are important

and

require
different
approach
es
. These analyses will
be
carried out

during

a later stage of the project.

7.

CONCLUSIONS AND THE
NEXT PHASE

T
he preliminary
optical
design
for

the new prime focus corrector for the WHT

demonstrates acceptable
polychromatic
image quality
as required by

WEAVE. T
he ee80 diameter does not exceed 0.6 arcsec over a wa
velength range from 370
to 1000
nm covering a two degree FOV for ze
nith angle
s up to 65 degrees. T
he results of the
initial
optical
tolerance

analysis along with the
ghost and scattered light analyses
do not present any major technical issues

although more
scattered light analyses need to be performed
.
In the next phase
of the project detailed mechanical modelling
will be
carried out

in order to verify that
the alignment tolerances
, positioning and thermal stability are achievable.
The o
ptical
system issues
related

to the manufacturing and testing of the aspherical compon
ent and the
optical interface

with the
fibre system

will be carefully analysed before the design is finalised.



New top
-
end ring

New vanes

Prime focus turntable

Cable wrap

Prime focus
corrector
integrated into
central section

Fibre positioner






REFERENCES

[1]

Agócs, T., Balcells, M., Benn, C. R., Abrams, D. C., Cano Infantes, D., "Two
-
degree FOV prime focus
corrector and ADC concepts for
the 4.2m WHT," Proc. SPIE 7735, 225 (2010).

[2]

Balcells, M., Benn, C. R., Carter, D., Dalton, G. B., Trager, S. C., Jarvis, M., Percival, W., Feltzing, S., Walton,
N., Helmi, A., Brown, A., Verheijen, M. A. W., Abrams, D. C., Cano, D., Agócs, T., Peletier, R.

F., Pérez
-
Fournon, I., Evans, C., McLure, R., Sharples, R. M., Trujillo, I., "Design drivers for a wide
-
field multi
-
object
spectrograph for the WHT," Proc. SPIE 7735, 275 (2010).

[3]

Dalton, G., B., Carter, D., Trager, S., C., Bonifacio, P., Aguerri, A., Abra
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, Lewis, I., J.,
Middleton, K., F., Mottram, C., J., "WEAVE: the next generation wide
-
field spectroscopy facility for the
William Herschel Telescope," Proc. SPIE 8446, 23 (2012).

[4]

Forbes, G., W., "Optical system assessment for design: numerical ray tracing
in the Gaussian pupil," J. Opt.
Soc. Am. A, 5, 1943 (1988).

[5]

Chapman, H., N., Sweeney, D., W., "A rigorous method for compensation selection and alignment of
microlithographic optical systems," Proc. SPIE 3331, 102 (1998).

[6]

Hvisc, A., M., Burge, J., H., "Ali
gnment analysis of four
-
mirror spherical aberration correctors," Proc. SPIE
7018, 19 (2008).

[7]

Agócs, T., Venema, L., Korkiakoski, V., Kroes, G., "Optimizing optical systems with active components," Proc.
SPIE 8450, 202 (2012).