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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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
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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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