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A Novel Wavefront Sensor for Interferometry
Nazim A. Bharmal a , David F. Buscher a , Christopher A. Hani# a , and Justin I. Read b
a Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK
b Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK
ABSTRACT
A new design for a wavefront sensor suitable for low­order low­light correction is shown. The hybrid modal
sensor, the Nine Element (NE) sensor, is compared with a curvature sensor and quadcell under single aperture
applications. The design of the NE sensor allows the use of readily­available array detectors. We discuss the
optimisation of the design to maximise its performance with respect to the number of Zernike polynomials to
detect and optical parameters, using a simulated annealing technique. Numerical simulations show the good
SNR response low­light levels, and indicate a reduction in wavefront variance from 6.41 rad 2 to 2.01 rad 2 . The
sensitivity to tip/tilt errors is demonstrated to be comparable to a quadcell. Successful closed feedback loop
operation results in corrected Strehl ratios of over 0.5. Improvements and future work is discussed.
Keywords: adaptive optics, optical interferometry, wavefront sensing
1. INTRODUCTION
Adaptive Optics has matured technologically in the last decade to become a integral part of new telescopes.
From initial applications on 4m sized apertures, the science and engineering drivers for new developments
are now focused on optimising image quality for apertures sizes of 8m and greater, a necessary step toward
application on the next generation of large telescopes. Optical interferometers achieve their high resolution by
an array of small telescopes. The first prototypes used apertures that were e#ectively immune to atmospheric
phase errors (beyond piston and tip/tilt). However, new arrays under construction will extend aperture sizes
to approximately 1m, where phase errors become an important issue in the design.
Since an interferometer does not su#er the loss of resolution from phase errors that a single aperture telescope
would, the reduction in visibility, and subsequent drop in SNR, is the major problem to overcome. Simply
retrofitting an existing AO system to an array telescope would restrict targets to the brightest due to minimum
flux requirements. This requirements would e#ectively reduce the SNR of observations.
While spatial filters 1 can be a solution to the atmospheric phase errors, one problem is the coupling losses
from phase aberrations. This can reduce throughput significantly, lowering the SNR. Most of the power in
turbulence induced phase fluctuations is in the lowest spatial frequencies. This is where AO for an interferometer
will provide the biggest gains, by correcting those errors which have the largest e#ect on beam propagation, 2
in conjunction with spatial filtering.
Compared to AO systems currently installed on single aperture telescopes, an interferometric environment
will have a reduced number of photons to work with. This is partially due to the consequence of having small
apertures, and we also wish to maximise the throughput into the science instruments. The regime in which we
wish to apply AO can therefore be described as the low­light low­correction regime.
Wavefront sensors (WFSs) are the key to successful low­light low­correction AO, since these are the sub­
systems which are a#ected most by both constraints. Two categories of sensor exist, Modal and Zonal. The
zonal approach is best typified by the Shack­Hartmann 3 sensor which measures the phase in separate spatial
zones. Modal sensors operate by separating the phase into orthogonal spatial frequency modes and measuring
the amplitude of each one, an example being the curvature wavefront sensor. 4
In this paper we use numerical simulations to show how a new modal design, the NE sensor, obtains low­light
low­correction, with the emphasis on stable feedback loop. Performance in comparison with existing wavefront
sensor designs are examined and a discussion of possible further developments are made.
Further information contact: e­mail: Nazim Bharmal nab26@mrao.cam.ac.uk

intra-focal extra-focal
plane
focal plane
Wavefront with positive defocus aberration
plane
intra-focal extra-focal
plane
focal plane
plane
Wavefront with negative defocus aberration
Figure 1. The degeneracy when attempting to measure defocus aberrations at the focal plane. At the focal plane,
the same image is formed with either defocus aberration. The degeneracy is broken when taking measurements before
(intra­focal) and after (extra­focal) the focus on the optical axis.
2. DESIGN AND IMPLEMENTATION
2.1. Theory
Over a circular aperture with no obstructions, we can decompose a wavefront using Zernike circle polynomials, 5
with a useful expressions given by Noll 6 for the typical variance given by each polynomial, see table 1.
Table 1. Residual variance after correction of successive Zernike polynomials, # i in units of (D/r0 ) 5/3 rad 2
# 1 = 1.03 # 2 = 0.58 # 3 = 0.13 # 4 = 0.11 # 5 = 0.09 # 6 = 0.07
Current interferometers use a tip/tilt system, which measures and corrects the first two Zernike polynomials
(Z 2 ,Z 3 ) by using a Quadcell. This type of sensor is a 2x2 array detector which cannot detect further modes.
The next mode, defocus (Z 4 ), is detectable as an aberration at the focal plane but a degeneracy exists in the
sign of the amplitude (figure 1) since an equal but opposite defocus produces the same intensity distribution.
This degeneracy can be broken by making measurements out of focus, similar in concept to using a phase
diversity technique. Changing the defocus is relatively easy to perform with optical components making it a
practical implementation to simulate. The two equal but opposite defocused planes are called the intra­focal
plane and extra­focal plane.
Using the defocus aberration as an example, it can be seen that a zero amplitude aberration gives two equally
defocused images. A positive (in the sense of positive amplitude of the Z 4 term) defocus will produce a smaller,

Z 2
Z 3
Z 4
Z 6
Z 5
Intrafocal Extrafocal
Figure 2. Distortion expected with the first 5 Zernikes after piston for the intra­focal and extra­focal planes. The solid
line shows the outline of an Airy disc for a given aberration and the dotted line shows the same with no aberrations.
Note that for Z2 and Z3 , the aberrated image gives the same signal in both planes.
and thus more intense, image in the intra­focal plane and a larger, and less intense, image in the extra­focal
plane. The converse is true for a negative defocus, a larger, less intense image in the intra­focal plane, and a
smaller, more intense image in the extra­focal plane. By taking a di#erential measurement of the intensity in
the two planes the amplitude of the aberration can therefore be determined.
A similar result applies to the astigmatic aberrations (Z 5 ,Z 6 ). By defining complementary areas on each
plane, we can distinguish between particular aberrations and a pictorial summary of the signals measurable is
shown in figure 2. We define the areas to be square pixels arranged as a 3 â 3 array, with the width of a pixel
being of the order of half an Airy diameter, which corresponds to # 0.5#f , where f is the focal ratio of the
optics concerned. This can allow easy use of an arbitrary sized array detector by simply altering the f ratio.
The pixels can then correspond directly to elements of an array detector such as a CCD. Changing the size of
a pixel with fixed f ratio changes both the sensitivity and linearity of the sensor.
This new configuration, dubbed the Nine Element (NE) sensor (figure 3) now has the ability to detect the
changes in size of the image in horizontal, vertical, and both diagonal axes. The first two Zernike polynomials,
Z 2 and Z 3 , give the same displacement of the image in the intra­focal and extra­focal planes. The di#erential
measurement between corresponding detector pixels is therefore zero for these aberrations. However, the sum
of intensity in both planes will null the other aberrations and maintain the displacement. By measuring the
di#erence between pixels either side of the central pixel, for vertical and horizontal directions, the displacement
can be obtained (which we call #x and #y). The method chosen here actually returns three displacements for
each direction.
0 @
#x 1
#x 2
#x 3
1 A = 0 @
P 3 - P 1
P 6 - P 4
P 9 - P 7
1 A ,
0 @
#y 1
#y 2
#y 3
1 A = 0 @
P 1 - P 7
P 2 - P 8
P 3 - P 9
1 A (1)

f
l l
p
4
7
1
2
3
6
9
8
5
Figure 3. Optical layout of NE sensor, the parameters l and p are as described in the text, the circle represents the
outline of an Airy disc at the focus. The diagram is not shown to scale. The number refer to individual detector elements,
referred as Pn in the text.
where P i refers to the summed intensity of array element i in both planes. From the sensor, a measurement
vector a of length 15 is returned, which contains 9 di#erential and 6 displacement values. Compared to the
Quadcell, we have increased the pixels over which the light is distributed from 4 to 18 (9 in each plane), and
have increased the number of detectable Zernike polynomials to 5.
2.2. Linearity
To recover the Zernikes amplitudes z from the measurements a, we first make the assumption of linearity. For
small amplitudes (< 1 rad), this is valid but we must be careful to assess the range for which this is true. These
assumption imply there exists a matrix M such that z = Ma.
Constructing M is more di#cult than its inverse M -1 , which can be obtained from
a = M -1 z (2)
and letting z i = 0.1# ik . The index k then refers to a list of Zernikes we wish to attempt to measure. We can
construct M -1 row­wise using each Zernike polynomial individually with a suitably small amplitude of 0.1 rad.
The inverse of M may not exist, and this can be determined by the singularness of M -1 . The inversion of
this matrix and the measure of its singularity can be done simultaneously by a Singular Value Decomposition 7
(SVD) algorithm. This procedure decomposes any matrix, A, into three components, U, W, and V. Two
matrices, U and V, are unitary and W is pseudo­diagonal (some elements on the diagonal may be zero).
Inversion is done by standard techniques except setting elements which are zero on the diagonal of W to zero
in W # -1 , equivalent to reducing the range of M # . The resultant matrix give the following relationship,
z # = M # a (3)
where z # # z as a # 0.
The reduced range of M # corresponds to aberrations which are unreachable through the measurement space
and hence were not useful anyway. This singularity information can be used to therefore select those Zernikes
which can be usefully detected i.e. not marginally detected. and those which are best ignored.
The range of linearity for a given Zernike polynomial is obtained by inputting that aberration with varying
input amplitudes and comparing this to detected amplitudes. This range is taken as adhoc; as long as the sensor
does not return amplitude values larger or opposite in sign from those input then they will not contribute to
instabilities in a feedback loop.

2.3. Noise
Using SVD, the algorithm assumes a variance of 1 on each row of the initial matrix and subsequently performs a
least­squares fit equivalent with the decomposed matrices. The decomposition allows an estimate of the variance
on each element of the returned vector z # since the noise adds in quadrature and depends only on the scaling
of vectors by M -1 . By scaling rows of M # -1 appropriately, e#ects of detector noise and photon noise can be
obtained quickly without the need for Monte Carlo­type simulations. Early optimisation steps can be performed
quickly to remove those modes which may su#er from excess noise, implying low sensitivity, as mentioned in
the previous section. The quoted noise values are for a signal of 0.5 photons per defocused image plane i.e. an
input of one photon.
2.4. Analysis & optimisation
Once the matrix M # has been obtained, the NE sensor's performance can be quantified. Noise on measurement
has been described above, as has linearity in the measurement of individual Zernikes. Two further parameters
are critical to successful operation; the cross­talk between detectable modes and the aliasing of higher frequency
modes onto those detected.
The wavefront variance can be better quantified by the Strehl ratio which is a standard measure of wavefront
aberration in an imaging system. As is usual in AO analysis, imaging to obtain the Strehl ratio and the wavefront
sensing are done at two di#erent wavelengths to take advantage of the # 6/5 scaling.
Evolving the phase screen temporally, by using Taylor's hypothesis of a frozen layer of turbulence moving
past an aperture, can be used in conjunction with a closed loop (feedback) system to obtain a realistic opera­
tional model. Iterating allows an averaged corrected and uncorrected Strehl ratio with other parameters to be
determined. The simulation code was written in a modular fashion, allowing the change of wavefront sensor type
without changing other components. This was used to perform the comparison of the results against established
designs.
By fixing the focal length of the system at f/30 with a 1 m diameter aperture, the only two parameters to
optimise upon were sensor pixel size, p, and length of defocus, l. Using a fixed size computational grid with a
fixed aperture on the grid, the sensor pixel size given is in fractions of an Airy disk diameter and the defocal
length in metres.
The optimisation process was carried out using a simulated annealing algorithm 7 which used an adhoc cost
function. To calculate the cost, a generic algorithm was employed which took a set of consecutive Zernikes, from
Z 2 to Z 16 with the same amplitude, and input each as wavefront phase individually. From this a M â N matrix
Q of values was created, each row of which consists of the di#erence between measured and input amplitudes
for the measured Zernikes. The dimension M equals the 14 Zernikes input and dimension N equals the number
of measured Zernikes.
Q ij = |z # j,Z i - z j,Z i | (4)
with z j,Z i
= # ji â a, given an amplitude a for an input of Z i .
Considering just the the measurable Zernikes, we can take the upper N rows to form a N â N matrix Q # .
The trace of Q # will reveal the departure from linearity and the o#­diagonal components will be non­zero for
cross­talk between measurable polynomials.
The remaining M -N â N matrix Q ## is representative of aliasing and any components which are non­zero
are a measure of this e#ect. This places the requirement that M > N which is clearly achieved for M = 14
with this sensor. A sum of components this matrix is a measure of aliasing.
Q =
0 B B B B B B @
0 @
#
Q #
#
1 A
0 @
#
Q ##
#
1 A
1 C C C C C C A

To calculate the linearity cost we use an amplitude of 1 rad per Zernike and return the trace of Q # , and for
aliasing we use an amplitude of 2 rad and return the sum of the elements of Q ## . Cross­talk was ignored since
it was found to be negligible even at high amplitudes (# 1 rad). The total cost is the sum of both costs in
quadrature.
3. RESULTS
3.1. Optimisation
3.1.1. Noise
From noise results the first 5 Zernikes, Z 2 to Z 6 are most favourably detected with a noise of 2.47 rad. Including
coma (Z 7 and Z 8 ) increases the noise to 14.93 rad and modes higher than Z 9 would be undetectable due to the
limited number of pixels. The explanation for the poor sensitivity with coma is that the Z 2 and Z 3 polynomial
terms are present in the Z 8 and Z 7 polynomial terms, respectively, creating significant confusion between them.
Table 2. Noise in rad 2 for the first 5 Zernikes. The input signal is 1 photon. Total noise is 2.47 rad rms
Z 2 Z 3 Z 4 Z 5 Z 6
1.74 1.74 0.20 1.63 0.79
3.1.2. Simulated annealing
The optimisation procedure used was not guaranteed to give the optimal solution. However using the parameters
returned, table 3, to examine linearity and in closed­loop simulations, gave results that were no worse than those
found by manually searching parameter space. They can be taken as acceptably optimal but clearly further
work remains in this area.
Table 3. Parameters used in numerical simulations of the NE sensor. Pixel size p is in Airy disc diameters
p 0.5 computational array size 128 â 128
l 1.5 m detection wavelength 700 nm
f/# 30 imaging wavelength 1 µm
aperture diameter, D 1 m r 0 D/3.0
wind velocity, v 3 ms -1 feedback iteration time, T r 0 /10v
detector noise 10e -
3.1.3. Linearity
The linearity is shown in figure 4 and can be seen to be better for Z 2 and Z 3 than the other polynomials.
However, the turnover occurs for the astigmatic modes, Z 5 and Z 6 at 1.4 rad and 1.1 rad respectively and for Z 4
at 0.9 rad. These may seem initially disappointing, but in a closed feedback loop the amplitudes being detected
are planned to be < 1 rad and therefore these limits are acceptable. The di#culty may come from initial
measurements to close the loop and dealing with large departures from the predicted amplitudes. Although not
investigated, an adaptive algorithm may have to be employed to prevent instabilities with strong turbulence.
3.2. Closed­loop
A wavefront sensor is typically engaged in a positive closed feedback loop, and to simulate this the NE sensor
was input with a temporally evolving phase screen. The screen simulates turbulence with parameters in table
3 corresponding to those for a site and instrument such as COAST. 8 The aperture phase is extracted using
a circular mask, o#set from the origin, on a single square phase screen (length # aperture mask diameter),

Figure 4. Measured amplitudes versus input amplitudes for the first 5 Zernike polynomials, Z2 to Z6 .
constructed via a Fourier transform technique. 9 The mask is defined as the circular aperture (no obstructions)
and the o#set is updated with each iteration to simulate the passing of the phase screen at constant velocity
across the aperture. The correction ``element'' is a simply a sum of Zernikes. For comparison, using identical
turbulence and temporal parameters, a quadcell (QC) and a curvature wavefront sensor (CWFS), with 19 radial
sensor elements, were also simulated.
Table 4. Comparison of the NE sensor with quadcell and curvature wavefront sensor for varying photon numbers. The
noise incorporated both detector and photon noise terms. The photon rates are those input into the sensor, not at the
imaged planes. The uncorrected variance is 6.41 rad 2 . The corrected Strehl ratios are those for residual tip/tilt after
correction and with the residual removed. The CWFS and NE failed to converge with a 10 2 photon flux.
Sensor Number Corrected Strehl Variance remaining
photons/iteration with Z 2/3 without Z 2/3 rad 2
CWFS 10 6 0.532 0.792 1.57
10 3 0.435 0.700 2.36
10 2 ­
QC 10 6 0.474 0.589 1.89
10 3 0.473 0.589 1.90
10 2 0.458 0.584 2.12
NE 10 6 0.545 0.643 2.02
10 3 0.518 0.624 2.01
10 2 ­
The results, table 4, show the expected intermediate performance of the NE sensor. Compared to quadcell,
the Strehl ratios are higher but the variance is lower. The NE sensor failed to close the loop for 10 2 pho­

tons/iteration, as did the CWFS. That intensity of photons is the same as or below the simulated detector
readout noise of 10 e - per sensor pixel for the two sensors so the failure is not surprising.
The CWFS shows better performance at high light levels than the NE sensor since it can detect 19 Zernikes,
but the tip/tilt performance is relatively poor. Considering that these are the aberrations with the highest
amplitudes, we can see the QC, which is just a tip/tilt detector, has a similar improvement to the NE sensor
with the removal of remaining tip/tilt errors. This suggests that the NE sensor is at least as good as the quadcell
in correcting tip/tilt. The drop in performance when operating with 10 3 photons/iteration is greater with the
CWFS than the NE sensor, suggesting the NE sensor is more robust to noise and therefore better suited to
low­light conditions.
3.3. Aliasing and higher­order Zernikes
Although not explicitly considered here, the e#ect of high­order Zernikes present in the atmospheric phase
screen do not alias to an extent that causes errors preventing the closure of the feedback loop. This can be
understood by considering the aberrations caused in the focal plane, close to where we are taking images. The
high­order aberrations are converted to high frequency aberrations which are averaged out over the size of the
detector pixels we use. Therefore by choosing a relatively low­resolution array, we are biased toward those low
frequency aberrations we wish to detect and can ignore otherwise noisy signals.
4. CONCLUSION
The numerical simulations performed with the NE sensor show that this design should be suitable for use in
the low­light level environment of an interferometer to perform low­order correction. The detection of the first
5 Zernike polynomials can be made reliably with a relatively straightforward design, which is implementable
with commonly available array detectors, such as CCD cameras. The design does not have any major impact
on the optical design of a telescope, requiring only an input of the beam received from the sky.
The performance of the sensor is expected to work best in conjunction with a spatial filtering system where
the combination will maximise the useful light throughput and SNR, compared to application of either technique
alone.
The optimisation technique used did not necessarily converge on the optimal solution and prior to work
on a prototype, this problem must be solved or at least the range of error in optimisation more quantitatively
established.
ACKNOWLEDGMENTS
Thanks are due to Justin Read for initial code used in the development of this paper and Fran›cois Rigaut for
making publicly available his CWFS simulations on the Internet.
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