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couples or bad bed-fellows?
aAstrophysics Group, Cavendish Laboratory, JJ Thomson Avenue, CB3
0HE, UK
ABSTRACT
Enhancing the limiting sensitivity of optical/infrared
interferometry is one of the “holy grails” of interferometric
research. While the use of adaptive optics is in principle
attractive, a number of issues suggest that its ability to enhance
the sensitivity of ground-based arrays is less clear. Indeed, the
ultimate sensitivity of an array may be limited by any of the
multiple active and photon-hungry subsystems that comprise its
whole. In this paper we investigate the limiting sensitivity of
interferometer arrays using unit telescopes of moderate size (i.e.
with D ≤ 4 m) equipped with natural guide star adaptive optics
systems. We focus on how to realise the best limiting sensitivity
for observations in the near-infrared. We find that for Vega-type
targets, i.e. those that have similar magnitudes at all
wavelengths, the use of an adaptive optics system can provide
enchancements in limiting sensitivity of up to 1.5 magnitudes.
However, for redder targets this improvement can decrease
dramatically, and very similar sensitivity (mlimiting ≤ 0.5) can be
obtained with arrays using 1.5m-class apertures and tip-tilt
correction alone.
Keywords: Optical Interferometry, Adaptive Optics, Fast Tip-Tilt,
Sensitivity
1. INTRODUCTION
The relatively poor sensitivity limit of ground-based
optical/infrared (IR) interferometers, in comparison to
conventional ground-based telescopes, is often seen as a
significant shortcoming. Not only has it hindered interferometric
methods from becoming part of the routine apparatus of
observational astronomy, but it has also limited the scientific
exploitation of optical/IR interferometry in, for example, studies
of extra-galactic targets and for studies in which surveys of
substantial numbers of targets are necessary.
Since the last SPIE meeting the interferometric literature has been
dominated by scientific results from the VLTI and CHARA arrays.
These two “facility-class”∗ interferometers demonstrate an
interesting feature of contemporary arrays, i.e., that their
sensitivity may not be determined by what at first sight one might
expect. In the case of these two arrays, the 64-fold larger area of
the unit telescopes at the VLTI is not matched by a similarly
enhanced sensitivity: in low resolution mode in the near-IR both
arrays have a publicised limiting sensitivity of roughly mK '
8.1,2
The augmentation of ground-based arrays with adaptive optics (AO)
has often been seen as an obvious stepping stone for an enhanced
capability.3 In comparison to the use of arrays with small
telescopes and tip-tilt correction alone, it is usually assumed
that modern AO systems, which routinely operate using guide stars
as faint as mR ∼ 16,4 should permit the use of larger
interferometric collectors without incurring a penalty from higher
order wavefront perturbations. However, while this would certainly
be true for very bright targets where an “extreme” AO system might
limit residual wavefront errors to an arbitrarily small level, at
the sensitivity limit the benefits of a natural guide star (NGS) AO
system are less easy to infer.
Further author information: (Send correspondence to A.D.R.) A.D.R.:
E-mail: adr34@mrao.cam.ac.uk, Telephone: +44 (0) 1223 337345
C.A.H.: E-mail: cah@mrao.cam.ac.uk, Telephone: +44 (0) 1223 337307
∗Here, we use the term facility-class simply to identify arrays
that are operated for a community from a range of user
institutions and for a significant fraction of the year.
1
In this study we have investigated the faint-source performance of
a ground based near-IR interferometer utilising active tip-tilt and
group-delay fringe tracking sub-systems, with and without NGS AO
correction of the unit telescopes. Our analysis was originally
motivated as part of the conceptual design work for the Magdalena
Ridge Observatory Interferometer (MROI) but our results are
relevant for future arrays that may utilise unit telescopes in the
2 m to 4 m-class size range. We outline the assumed interferometer
implementation in Section 2, and describe the details of our
numerical performance model in Section 3. Our results are presented
and discussed in Section 4, where we focus on the degradation of
the interferometric performance as a function of target brightness
and the limiting sensitivity of the non-AO and AO-augmented arrays.
Finally we summarise our conclusions in Section 5.
2. SYSTEM MODEL
2.1 Functional description
In order to ground our study to existing optical/IR interferometric
arrays, we have focused our analysis on an interferometer securing
science data in one of the H or K near-IR bands. Each unit
telescope is assumed to have a fast tip-tilt system as well as an
optional NGS AO system, both located at the telescope. We have not
investigated the use of off-axis reference stars and so assume that
both of these systems use light from the interferometric target
itself. In order to realise the maximum sensitivity, we have
assumed dichroic separation of the light for each of the active
interferometer sub-systems rather than splitting an individual
bandpass three- ways. A typical allocation is shown in Fig. 1,
where the reddest light (the near-IR K band) has been sent to the
science beam combiner, the near-IR H band is used to drive a
fringe-tracking beam combiner, while the visible R and I bands are
used for tip-tilt sensing and the near-IR J band for the AO.
This particular choice of dichroic separation has been informed by
three rules-of-thumb:
• First, because the interferometer sensitivity will be compromised
if its fringe tracker loses lock, it will generally be helpful to
run the fringe tracker at as long a wavelength as possible. This
will not only reduce the probability that atmospheric fluctuations
lead to temporary visibility dropouts, but more importantly, will
ensure that the target will appear as unresolved as possible on the
tracked baselines;
• Second, in order to limit chromatic effects, e.g. those due to
atmospheric dispersion, it will generally be helpful to limit the
separation between the wavelengths used for sensing the atmospheric
fluctuations and those at which the correction is applied;
• Finally, it will generally be of value to run the tip-tilt system
at a wavelength at which the impact of any residual uncorrected
wavefront perturbations will be minimised.
The last of these three guidelines can be complicated by the
spectral energy distribution (SED) of the target. For any given set
of tip-tilt sensor characteristics and target SED there will be a
trade-off between minimising losses
Figure 1. The assumed division of light for the different active
interferometer subsystems analysed in this study. Here we have
assumed that only the near-IR H and K bands are propagated to the
beam combining laboratory, while the redder visible and near-IR J
band are split off at the unit telescope for the tip-tilt and AO
systems respectively.
2
in tip-tilt performance due to residual wavefront errors and due to
photon flux. As a result, the optimisation of the tip-tilt sensing
wavelength has to be optimised on a target by target basis. For the
test cases presented here, the relevant choice is between utilising
the bandpasses for the tip-tilt and AO subsystems identified in
Fig. 1 or instead feeding the J-band light to the tip-tilt system
and the visible R and I bands to the AO sensor. We present results
for both of these scenarios below.
A key element of our system model is that we assume the use of a
group-delay fringe tracker to stabilise the array from atmospheric
piston fluctuations. The photon flux needed for group delay fringe
tracking is several magnitudes smaller than that needed for
sub-wavelength fringe stabilisation or “phase”-tracking. The
former, which seeks to only maintain the white light fringe
position to within a few fringes of its “ideal” location is
sometimes referred to as “coherencing” and its use has to be
coupled with the incoherent addition of, e.g., bispectral
measurements from successive short exposure interferograms.
However, for interferometers that cannot capitalise on off-axis
fringe stabilisation, i.e. arrays that do not have a dual-feed
capability, the faintest targets visible will be those that are too
faint for high-accuracy phase tracking but still bright enough to
be tracked successfully with a group-delay fringe tracker.
2.2 Implementational details
In order to allow our analysis to be as useful as possible, we
summarise below the main implementational-specific assumptions we
have made in our modelling. In large part we have attempted to
address two shortcomings of previous work. More specifically,
wherever possible, we have aimed to:
1. Utilise reasonable and realistic values for the performance
characteristics of the hardware used for the three main active
systems being modelled, i.e. the tip-tilt, AO and fringe tracking
sub-systems;
2. Incorporate as complete as possible an enumeration of the
throughput and visibility loss budget associated with the fully
interferometric optical train.
The relevance of this strategy is crucial because the absolute
sensitivity levels we are aiming to determine depend sensitively on
these assumptions, and previous assessments of the limiting
sensitivity of interferometric arrays may have been
over-optimistic. Throughout the discussion we assume that the first
component the light meets after exiting the telescope optics is a
diffraction-limited atmospheric dispersion corrector, so that the
subsequent active subsystems are all delivered an instantaneous
image corrected for differential atmospheric refraction. Further
details of the assumed performance characteristics of each of the
three key elements of our model are presented below.
2.2.1 Tip-tilt implementation
For convenience we have assumed a tip-tilt sensor and corrector
architecture that resembles that implemented at the MROI. This uses
a dichroic pickoff at the telescopes feeding a silicon- or
HeCdTe-based photon-limited detector via an achromatic focusing
optic and a pair of fold mirrors. We assume the use of either
electron- multiplying CCD or near-IR avalanche photo-diode arrays,
both of which are relatively mature technologies. The tip-tilt
sensor is assumed to drive an active mirror located such that it
introduces neither pupil shear nor piston fluctuations into the
light travelling to the science and fringe tracking beam combiners.
At the MROI this function is realised with the UT secondary mirror,
but any similarly specified mirror will suffice.
We further assume that the tip-tilt system can operate at any
closed-loop bandwidth up to 50 Hz, the exact value being determined
by optimising the system signal-to-noise at the light level under
consideration. Since we are interested in studying the low-light
level performance of the interferometer, the typical optimum
closed- loop bandwidths can easily be delivered with existing
high-speed hardware and software. A summary of our assumptions that
determine the effective photon rate delivered to the tip-tilt
system is given in Table 1.
3
Table 1. Summary of assumed transmissions and efficiencies for the
different interferometer subsystems and the atmo- sphere. All the
detectors are assumed to have no readout noise and no significant
thermal background. The penultimate entry in the table refers to
the factor by which the fringe visibility of a source with an
intrinsic visibility of unity is reduced due to
instrument-dependent sources, e.g. static optical wavefront errors,
delay line jitter, finite exposure time etc.
Component Attribute Bandpass Value
” ” J band 0.90
” ” H band 0.95
Tip-tilt system Mean throughput from M1 to detector 600-950 nm
0.60
” ” J band 0.60
” ” J band 0.70
AO system Mean throughput from M1 to detector 600-950 nm 0.60
” ” J band 0.60
” ” J band 0.70
Fringe tracker system Mean throughput from M1 to detector H band
0.25
Mean detector QE ” ” 0.70
Intrinsic target visibility ” ” 0.75
2.2.2 AO implementation
For this study we haved assumes that an N × N Shack-Hartmann AO
system is present at each of the unit telescopes, where N is to be
optimised for the light level under consideration. As for the
tip-tilt subsystem, we assume a dichroic pickoff feeding light from
the target to a lenslet wavefront sensor utilising either a
silicon- or HeCdTe-based photon-limited detector. We note that the
choice of a Shack-Hartmann as opposed to a curvature wavefront
sensor is unlikely to impact our results significantly. Independent
analyses (see, e.g. Rigaut et al 19975) confirm that for comparably
specified detectors, the limiting sensitivity of realistic
implementations of these two are identical to within a few tenths
of a magnitude.
In view of its close proximity to and similar number of optical
components as the tip-tilt system, we associate an identical value
for the AO system throughput as for the tip-tilt system.
Furthermore, since these two systems run independently, the
exposure time and close-loop bandwidth of the AO system have been
jointly optimised to give the lowest possible residual wavefront
error as a function of light level. A summary of our assumptions
that determine the effective photon rate delivered to the AO system
can be found in Table 1.
2.2.3 Interferometric implementation
A third aspect of our system model that deserves review is the
assumed implementation of the interferometer itself. We have again
used the architecture of the MROI to guide this, but most of the
assumptions we have made would be typical for most modern array
designs.
From the point of view of sensitivity, the two most important
aspects of any interferometric implementation will be its
throughput and the apparent fringe visibility at the fringe
tracking beam combiner. We have adopted a value in the near-IR H
band of 25% for the total throughput from the unit telescope
primary mirrors to the fringe tracking detector. This is an
aggressive figure, but consistent with free-space beam relay from
the array elements to the beam combiner along an evacuated path and
the use of an efficient optical layout. Similarly, we have adopted
a demanding value of 60% for the rms visibility for an unresolved
target observed in the H band.
4
100
101
102
103
104
White Seyfert 1 T Tauri
Figure 2. Photon rates measured in number per coherence volume (=
r20 × t0) expected above the Earth’s atmosphere for a “white”
Vega-like star, a core dominated AGN, and a typical T Tauri star.
In the main body of the text this unit of photon rate is referred
to as α. Canonical values of 10 cm and 3.14 ms have been used for
r0 and t0 at 500 nm respectively and rectangular bandpasses from
600-950 nm, 1165–1325 nm, 1500–1780 nm, and 2030–2360 nm were
assumed. At each bandpass the intrinsic spectral energy of the
source, as well as the λ6/5 scaling of the spatial and temporal
atmospheric scale sizes, have been considered. All the targets have
the same magnitude in the R and I bands of 12.5.
This figure represents the factor by which instrumental factors are
expected to have reduced the measured fringe contract from its
expected value of unity. This figure does not include the effects
of the atmospheric spatial perturbations uncorrected by the
tip-tilt and AO systems — these are included elsewhere — but
incorporates effects such as the static high order wavefront errors
from the instrument optics, the effects of static and slowly
varying optical misalignments, and fringe decorrelation due to
atmospheric and instrumental piston jitter during finite
integration times.
As mentioned in sub-section 2.1, we have assumed the use of a
group-delay fringe tracker to establish the interferometric
limiting sensitivity. Such a sub-system basically interrogates the
power spectra of sequences of successive short-exposure dispersed
interferograms so as to estimate the displacement of the
white-light fringe from its nominal position (see, e.g. Basden and
Buscher6). The relatively coarse resolution associated with this
approach means that the spectra from many tens of short-exposure
interferograms can typically be incoherently integrated before the
white light fringe position has moved by an appreciable amount.
Here we have assumed individual interferogram exposure times of 1.6
t0, i.e. optimum for photon-limited data, and allowed for an
incoherent integration time of 20× this, and hence an improvement
in SNR of
√ 20 ∼ 4.5. This factor is
appropriate for the MROI where the near-IR H band will be split
into five spectral channels from 1.5µm to 1.8µm. The simulations by
Buscher7 suggest that an overall signal-to-noise ratio of 4 is
enough to ensure reliable group-delay tracking and so we have used
this threshold to estimate the light level at which interferometric
observations cease to be feasible.
2.3 Target details
Since the rationale for interferometric observations is usually to
resolve the target under study, we have chosen not to follow usual
practice where the source is assumed to provide fringes of unit
contrast. Rather, we have taken it as a given that the
interferometer incorporates separate science and fringe-tracking
correlators and that it incorporates baseline-bootstrapping, i.e.
utilises a quasi-redundant array layout so that fringe tracking on
nearest-neighbour telescope pairs can serve to stabilise the
fringes on the longer baselines. We have therefore assumed a more
realistic intrinsic source visibility on the fringe-tracker
baselines of 0.75 so as to indicate that on the longer baselines
fed to the science beam combiner the target will be resolved.
5
In order to facilitate comparisons between observations of
different types of scientific targets, we show in Fig. 2 the
expected number of photons (above the Earth’s atmosphere) arriving
from three different types of targets: (a) a “white” Vega-like star
with equal magnitudes in all the photometric bandpasses; (b) a
typical T-Tauri star with R−J , J −H, and H−K colours of 1.5, 0.8
and 0.5 respectively; and (c) an AGN with R−J , J − H, and H − K
colours of 1.3, 0.9 and 0.8 respectively. The data are presented in
units of the number of photons arriving per coherence volume, i.e.
per r0-sized patch per t0: hereafter we shall refer to this
quantity as α. The values of α have been computed incorporating
both the scaling of the atmospheric spatial and temporal scales
with wavelength and the typical SEDs of the targets. Two features
of the Fig. 2 are immediately apparent. First, the large increases
in α as the observing wavelength gets longer, and second, the
additional increase in α associated with the very red SEDs of some
of the targets. As we shall see later, the latter can be
particularily important in certain situations.
3. PERFORMANCE MODEL
To determine how the performance of our array degrades with target
brightness we have used the instantaneous group-delay
fringe-tracker signal-to-noise as our primary metric. More
specifically, we identify the limiting sensitivity as the target
brightness at which the instantaneous fringe-tracker SNR falls
below 0.88, corresponding a value of 4 after incoherently averaging
over 20× 1.6 t0 interferograms. Throughout we have assumed values
of 10 cm and 3.14 ms for r0 and t0 at 500 nm, corresponding to a
wind speed of 10 ms−1.
The dependency of the instantaneous fringe-tracker SNR on the
tip-tilt and AO systems, as well as other experimental factors is
described in the flow diagram of Fig. 3. This also makes clear how
the performances of two sub-systems themselves depend on aspects of
the facility implementation and site and target qualities. In broad
terms, the target brightness will affect both the quality of the
tip-tilt and and the higher order wavefront correction, which in
turn will reduce the apparent contrast of the interferometric
fringes. Since the fringe-tracking signal-to-noise is a function of
both the apparent fringe contrast and the detected photon rate, as
the target brightness reduces we expect a runaway decline in
interferometric performance below some threshold light level.
A key element of our analysis is that we have decoupled the choice
of the operating points of the tip-tilt and AO systems such that
for each we have optimised parameters such as the exposure time
and/or Shack-Hartmann sub-aperture size etc independently. This is
reasonable given the fact that they use different bandpasses and
independent cameras. Furthermore, at low light levels the
cross-coupling of the two systems† becomes irrelevant because at
these low signal levels the AO correction becomes increasingly
poor.
In the following sub-sections we describe how we have modelled the
low light level performance of these three key sub-systems in the
array, and how we have combined these to estimate the overall
fringe-tracking SNR. Our treatment follows closely the approach
taken by Hardy8 in his system analysis of single telescope AO
systems, and the reader is referred to his comprehensive study for
details of many of the formulae presented below.
3.1 Tip-tilt performance
For tip-tilt systems that use on-axis natural guide stars, it is
usual to consider four major sources of performance degredation.
Each of these is enumerated below and formulae presented that allow
each to be characterised in terms of an effective mean-square
wavefront error measured in radians squared. We have assumed that
these errors are independent, and so to get the total effective
wavefront error contributed by the tip-tilt system we sum these in
quadrature.
3.1.1 Measurement Error
This is the error associated with the inability to measure the
position of the target precisely due to two factors: first, the
finite number of detected photons and, second, the fact that for
certain aperture sizes and seeing conditions the instantaneous
image may be speckled. The one-axis rms tilt error for a target
unresolved by an individual unit telescope is given by
σTTSNR = 3π2
SNR , (1)
†By this we mean that the point spread function seen by the
tip-tilt system is a function of the quality of the AO
correction.
6
Measurement
Figure 3. Flow chart showing the dependencies associated with the
tip-tilt, adaptive optics and fringe-tracker systems, and how the
properties of the target, atmosphere and system-implementation
impact the performance of these three systems.
where D the unit telescope diameter, SNR is the signal-to-noise
ratio of the measurement, and the error is expressed in terms of
radians of phase at the tip-tilt sensing wavelength. For the
photon-limited detectors we have asssumed, the latter will be given
by
√ N , where N is the number of photons collected in a single
exposure.
The factor χ is associated with the ratio of the sampling rate at
which measurements are made and the closed- loop bandwidth of the
tip-tilt system.9 Here, we assume sampling at 10× the closed-loop
bandwidth‡, and so χ has a value of approximately 0.5.
In the regime where the interferometer unit telescope diameter, D,
is less than r0, the factor of D r0
takes a limiting value of unity. On the other hand, if D is greater
than this value and an AO system is operating, then r0 can be
replaced with a larger effective Fried parameter, which we denote
as ρ0. This accommodates the beneficial impacts of additional
high-order wavefront correction on the PSF compactness. The value
of ρ0 was
‡This is a rule of thumb commonly used to relate a first-order
servo’s desired bandwidth with the sampling frequency required to
successfully close a control loop.
7
computed in a similar fashion to that used by Cagigal and Canales10
based on the wavefront errors remaining after the AO correction.
However, we have found that this correction factor has a negligible
impact on the use of Eq. 1 since at photon rates well before the
limiting sensitivity is reached ρ0 has already tended to r0.
3.1.2 Temporal Errors
Two temporal factors provide limits to the quality of the tip-tilt
corrections that can be delivered. The first of these is associated
with the finite correction bandwidth of the tip-tilt servo-system
and is usually referred to as the “bandwidth” error. The second is
related to any pure time delays caused by, e.g. data transfer or
processing times. The latter is generally much smaller than the
former, particularly at low light levels, and so we have only
included the former in our analysis. The bandwidth error can be
expressed as
σTTBW = fT fC
2 , (2)
where fT is the Tyler tilt-tracking frequency, and fC is the closed
loop bandwidth, which as mentioned above we have set to be ten
times lower than the sampling frequency of the FTT system. If we
assume a single-layer atmosphere with a wind speed v, the Tyler
frequency can be written as 0.08v/r0. This leads to an expression
for the one-axis bandwidth error in radians of phase at the
tip-tilt wavelength of
σTTBW =
1 6 fC
3.1.3 Centroid Anisoplanatism Error
At a fundamental level, any tip-tilt system works by measuring some
function of the instantaneous image bright- ness distribution and
using this to estimate the relative contributions of tip and tilt
to the total wavefront error. Higher-order wavefront errors with
the same angular dependence as tip and tilt — usually referred to
as pure coma terms — induce apparent shifts in the image
center-of-mass that mimic the effects of pure tip and tilt. For
sensors that utilise simple centroid measurements, this leads to an
error in the tip and tilt determination that is usually referred to
as “centroid anisoplanatism”. The one-axis value for this error
(again, at the tip tilt wavelength) can be written as
σTTCA = 0.086
(4)
and so is simply dependent on the ratio of the unit telescope
diameter to Fried’s parameter. If an AO system is present and is
able to measure and correct N orders of coma perfectly, then this
error is reduced and lowered by a factor of (N + 1)−
7 6 .
3.1.4 Higher Order Errors
Since tip tilt systems make no attempt to correct for higher-order
components of the atmospheric perturbations, the wavefront quality
delivered to the the fringe tracking subsystem must become
increasingly poorer as the unit telescope size increases. The
magnitude of these residual perturbations is well established11 and
is given by
σTTHO = 0.37
(5)
and which can be computed at any wavelength by scaling r0
accordingly. This term is considerably larger than the centroid
anisoplanatism error, and straightforwardly identifies the well
known rule of thumb that the maximum benefit from low-order
adaptive correction occurs at a telescope diameter of just over
3r0.
If an AO system is running – and working well – then the impact of
these higher order wavefront errors may be largely mitigated. In
this case Eq. 5 can be replaced by an equivalent expression that
captures the strength of the residual perturbations after adaptive
correction. The enumeration of the rms magnitude of these remaining
wavefront errors is the subject of the next sub-sections.
8
3.2 AO performance
For interferometers with on-axis NGS AO sub-systems, there will be
three principal factors contributing to the delivered wavefront
quality, analogous to the four effects considered in sub-section
3.1. As for the contributions from the tip-tilt system, these have
been treated as independent and so to get the total residual AO
wavefront error these have been summed in quadrature. Each of these
three contributions is described below.
3.2.1 Measurement Error
This error is the exact analog of the tip-tilt measurement error
described in sub-section 3.1.1 and is associated with the
imprecision in measuring the centroid of a lenslet sub-image due to
the finite light level and the presence of any speckles. The
expression for the one-axis rms phase error takes the same form as
Eq. 1 except with D, the aperture size, replaced with d, the
Shack-Hartmann sub-aperture size, viz.:
σAOSNR =
3π2
32
d
SNR , (6)
where all the terms have their usual meaning and the error is given
in radians at the AO sensing wavelength.
3.2.2 Temporal errors
The quality of the AO correction will be limited in exactly the
same way as the tip-tilt system by a bandwidth and a pure time
delay term. And again, at low light levels it will be former that
dominates. In this case there is an equivalent expression for the
one-axis phase error as Eq. 2 but with the Tyler tilt frequency
replaced by the Greenwood frequency, fG. For our single layer
atmosphere fG will be given by 0.427 v
r0 leading to the following
expression for the residual one-axis wavefront error contribution
at the AO sensing wavelength:
σAOBW = 0.427 v
3.2.3 Fitting error
The final contribution to the residual AO wavefront error will
arise from the finite size of the sub-apertures comprising the
Shack-Hartmann wavefront sensor and the efficacy of the deformable
mirror in taking up any subsequently commanded shape. For a
deformable mirror where each subaperture is mapped onto a region
with three (piston, tip and tilt) degrees of freedom, a suitable
expression for the residual wavefront error is
σAOFIT = 0.37
, (8)
where d is the sub-aperture size. The insightful reader will note
the similarity of this equation to Noll’s expression for the
residual wavefront error associated with a tip-tilt system alone,
i.e. Eq. 5.
3.3 Interferometric signal-to-noise
The group-delay SNR is computed using the standard expression for
the two-beam power spectrum signal-to-noise in the photon limited
limit:12
FTSNR =
√ M
4
2NV 2 , (9)
where N is the total number of photons detected per interferogram,
V is the apparent visibility of the fringes being tracked, and M
the number of interferograms that are incoherently added§.
§For simplicity we have ignored the terms associated with any
“double frequency” power and the high-light-level variance in the
fringe pattern itself.
9
0
2
4
6
8
10
12
14
WFE Visibility Contribution=1 WFE Visibility Contribution=0.5 WFE
Visibility Contribution=0.25 WFE Visibility Contribution=0.1
Figure 4. Fringe tracker SNR as a function unit telescope size for
four different levels of residual atmospheric wavefront error and
assuming an intrinsic target fringe contrast of 75%, an
instrumental coherence loss factor of 60% and a throughput
consistent with the values in Table 1. The term “Visibility
contribution” can be roughly interpreted as the Strehl ratio
delivered to the beam combiner. The values of SNR have been
computed assuming the incoherent addition of 20 interferograms and
a value for the photon rate per coherence volume, α, in the H band
of 10. This figure excludes the efficiency of transmission through
the atmosphere. Even small reductions in the delivered wavefront
quality lead to a diminishing ability to group delay fringe track.
The typical threshold for success, where SNR≥ 4, is shown as a
horizontal line.
Eq. 9 is straightforward to compute for any given values of M , V ,
target brightness and telescope size and has been plotted for a
range of different values of V in Fig. 4. The most important
feature to note here is the very strong reduction in
fringe-tracking signal-to-noise ratio with apparent visibility, V .
In short, if the residual wavefront errors after tip-tilt and AO
correction are too large — and hence the fringe contrast degraded —
the ability to track on the source fringes will be dangerously
compromised.
While Fig. 4 can provide some useful insights on its own, a
critical step in our modelling has been the computation of the
fringe contrast degradation due to residual atmospheric
perturbations as a function of light level. Formally, the mean
square fringe visibility can be defined as
⟨ V 2 ⟩
⟩ , (10)
where T (r) is the diffraction-limited telescope transfer function,
B (r) is the short exposure partially corrected atmospheric
transfer function, and the angle brackets refer to an average over
multiple realisations of the atmo- sphere. If the residual
atmospheric perturbations are assumed to be homogeneous¶ and in the
near-field one can replace B (r) with an expression involving the
structure function of the residual phase perturbations, viz.:
B (r) = exp
where Dφ (r) is the residual phase structure function.
¶While this is not strictly true for a partially corrected
wavefront (see, e.g. Heidbreder 196713), in the worst case — where
only tip and tilt corrections are assumed to have been made —
simulations have shown7 that the error arising from this assumption
leads to roughly a 8% reduction in the estimated RMS visibility as
compared to the correct value. This error decreases as the level of
wavefront correction improves.
10
We have estimated the residual phase structure function using the
results of Dai (1995)14 who presents analytical residual phase
structure functions for various level of modal wavefront
correction. We have used the magnitude of the overall rms residual
wavefront error (determined from Eqs. 6, 7, and 8) and the values
of D and r0 to select which of Dai’s structure functions to use,
interpolating between them if necessary. In addition, we have
included a further contribution to the residual structure function
based on the level of correction of the tip and tilt modes (as
estimated by the quadrature sum of the terms given by Eqs. 1, 2,
and 4). In this way we have been able to estimate the fringe
visibility coherence loss arising from the increasingly poorer
wavefront correction that fainter targets give rise to.
4. RESULTS & DISCUSSION
It is useful to first examine the relative magnitudes of the
different terms contributing to the residual wavefront errors seen
by the fringe tracker as a function of light level. Fig. 5 shows
these contributions for a tip-tilt system and an AO system using
light in the 600-950 nm bandpass for a 3 m diameter telescope. The
tip-tilt errors almost exclusively originate from the higher-order
wavefront errors beyond tip and tilt, whereas the AO errors are
dominated by similar contributions from the measurement and
bandwidth errors. While the bandwidth error has no direct
dependence on target brightness, optimisation of the overall error
causes it to track the bandwidth error, and both rise
near-quadratically as a function of α.
These two figures highlight a very important point, i.e. that for
telescopes that are considerably larger than 3 m in diameter — in
this case the ratio of D/r0 is approximately 7 at the fringe
tracker wavelength — there is a significant benefit to be had from
using an AO system to reduce the higher order residual wavefront
errors. Moreover, even at photon rates corresponding to values of α
of as low as ∼ 3 this may be beneficial. On the other hand, this
minimum photon rate will increase as D becomes smaller and the
magnitude of the higher order errors diminishes.
The impact of these trade-offs on the limiting sensitivity of our
interferometer as a function of unit telescope size is shown in
Fig.6. The left hand panel shows the behaviour for a hot Vega-like
target, in which its magnitude is the same in all bandpasses, and
can be understood relatively straightforwardly. For values of D
smaller than ∼ 3r0, the high-order wavefront error is small and so
there is no advantage to be gained by using an AO system.
100 101 102 103
Fitting Error
Total Error
Figure 5. Individual wavefront error contributions as seen by the
fringe tracker in the H band arising from the tip-tilt (left) and
and AO (right) systems for a 3 m diameter telescopes with the
overall system throughput determined by the values given in Table
1. The abcissa, α, shows the number of photons arriving at the unit
telescope primary mirror in the 600-950 nm bandpass per coherence
volume assuming values of r0 and t0 appropriate for a wavelength in
the center of the band and unit atmospheric transmission. The
ordinate gives the mean square wavefront error in radians measured
at 1.65µm, i.e. as seen by the fringe tracker.
11
10.5
11.0
11.5
12.0
12.5
13.0
13.5
u d e
TT in RI only TT in J only TT in RI, AO in J TT in J, AO in
RI
2 3 4 5 6 7 8 9 D/r0@FT Wavelength
1.0 1.5 2.0 2.5 3.0 3.5 4.0 Telescope Diameter / m
12.0
12.5
13.0
13.5
14.0
14.5
u d e
TT in RI only TT in J only TT in RI, AO in J TT in J, AO in
RI
2 3 4 5 6 7 8 9 D/r0@FT Wavelength
Figure 6. The magnitude in the 600-950 nm bandpass of the faintest
target that allows an H-band fringe tracker SNR of 4 to be achieved
as a function of UT diameter, D, for (a) a Vega-like target (left)
and (b) a Seyfert 1 AGN. In each panel the four curves show the
behaviour for differing choices of tip-tilt and AO sensor
wavelength. The small steps in the curves describing the behaviour
of the AO-enhanced arrays locate the values of D where the number
of Shack-Hartmann lenslets across the telescope aperture increases
by an integer.
However, as D increases, these uncorrected modes become more
important and so to realise a fixed fringe-tracker SNR a brighter
target is required. For these larger telescope sizes, the use of an
AO system reduces these high order errors usefully, and so the
interferometer can actually access targets that are between 1 and
1.5 magnitudes fainter than an optimised tip-tilt-only array.
At the sensitivity limits shown in the left hand panel of Fig.6,
the typical number of photons being delivered to each
Shack-Hartmann lenslet per AO exposure time is between 3 and 6, the
larger value being associated with an AO sensor operating in the J
band. We can identify this differential in performance with the
lower fitting error (see sub-section 3.2.3) that comes from
performing the wavefront sensing at a shorter wavelength. This more
than compensates for the three-fold increase in the value of α in
the J band as compared to the 600-950 nm bandpass, and highlights
the importance of correctly selecting the bandpasses sent to the
tip-tilt and AO sub-systems.
The behavior of an array studying a redder source, for example an
AGN core, is shown in the right panel of Fig.6. This reveals two
new features. First, the limiting sensitivity of the non-AO
enhanced array is substantially improved, by a factor of more than
two magnitudes. This arises for such implementations because the
dominant contribution to the fringe tracker signal-to-noise is
independent of the number of photons being sent to the tip-tilt
system. For arrays that have no AO correction, the tip-tilt
wavefront residuals are wholly limited by the fixed high-order
error, and so a red target can be proportionally fainter in the
bluer tip-tilt bandpass but still be delivering a sufficient number
of photons to the fringe-tracker to meet the signal-to-noise
target. The increase in sensitivity by a factor of roughly 7 is
exactly what would have been expected from the relative fluxes
shown in Fig. 2.
Interestingly, this behaviour is not shared by AO-augmented arrays
that utilise large unit telescopes. Our results suggest that, while
for small values of D there is a similar two magnitude enhancement
in sensitivity, this is not matched when D/r0 exceeds about three.
Were the sensitivity enhancement to be independent of D, then the
number of photons arriving at each Shack-Hartmann lenslet per AO
exposure time would drop below a few and under these circumstances
the AO measurement error would be expected to render the AO
correction nugatory. This is indeed what we find, i.e. a gradual
saturation of the AO-enhanced limiting sensitivity at a level only
some 20% (0.2 magnitudes) higher than an optimised non-AO-corrected
array.
While we have only presented results for these two types of
targets, our modelling has shown that the optimisation of an array
to realise the best possible sensitivity is a complex task. The
role of the wavelength
12
dependence of the target being observed plays a perhaps
surprisingly important role, and in certain cases there can be a
strong argument not to deploy AO for limiting sensivity reasons
alone.
What is clear though, is that for brighter targets, AO will be of
value, both for observing targets in narrow bandpasses, but also
for allowing more rapid collection of data and, therefore, access
to science opportunities that demand large sets of interferometric
data to be secured.
It is worthwhile to note that the typical magnitude limits we see
here are, for our white target, between 12 and 13.5 — significantly
different from the 16th magnitude we see AO systems used down to on
other, single-aperture, facilities. The reason for this apparent
discrepancy is that interferometry requires a high (e.g. 50-60%)
Strehl ratio, and an adaptive optic system using targets fainter
than 12th magnitude is not delivering a sufficiently corrected
wavefront for our interferometric purposes.
5. CONCLUSIONS
We have conducted an investigation into the limiting sensitivity of
an interferometric array equipped with a fast tip-tilt system and
an optional adaptive optics system, where either can operate with J
band flux or the combined flux of the R and I bands. A
fringe-tracking system operates is assumed to operate using H band
flux in all cases. By using realistic values for the efficiencies
of each element of the system and including the effect of colours
of one class of targets of interest, our results are of practical
use for the future expansion of current interferometric facilities
and the eventual design of next-generation arrays.
The main conclusions we have drawn from this analysis are:
1. Adaptive optics will provide a useful (more than a magnitude)
gain in limiting sensitivity for an interfero- metric array with
aperture sizes larger than 2 m when observing white targets.
2. If the target being observed has a redder SED e.g. an AGN, then
the sensitivity gain is greatly diminished to only 0.2 magnitudes,
even for aperture sizes up to 4 m, when compared to a facility with
smaller aperture sizes optimised for tip-tilt correction
only.
3. It is clear that it is very important to consider the colour of
the targets of interest, as this can lead to performance gains that
might be otherwise be missed. Most notably, we show that for a 4 m
aperture a gain in limiting sensitivity of nearly a magnitude can
be achieved for a white target simply by choosing the correct
wavebands to feed to the tip-tilt and adaptive optics
systems.
4. Realising the limiting sensitivity we find for the array
configurations and observations considered here of 14-14.5 would be
a significant achievement for interferometric arrays — regardless
of whether we choose to do it with or without adaptive
optics.
ACKNOWLEDGMENTS
The authors would like to thank Barney McGrew for his steady stream
of sensible comments.
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