Simulating a dual beam combiner at SUSI for narrow-angle astrometry
Yitping Kok, Vicente Maestro, Michael J Ireland, Peter G Tuthill, J Gordon Robertson
aa r X i v : . [ a s t r o - ph . I M ] F e b Experimental Astronomy manuscript No. (will be inserted by the editor)
Simulating a dual beam combiner at SUSI fornarrow-angle astrometry
Yitping Kok · Vicente Maestro · Michael J. Ireland · Peter G. Tuthill · J. Gordon Robertson
Received: date / Accepted: date
Abstract
The Sydney University Stellar Interferometer (SUSI) has two beamcombiners, i.e. the Precision Astronomical Visible Observations (PAVO) andthe Microarcsecond University of Sydney Companion Astrometry (MUSCA).The primary beam combiner, PAVO, can be operated independently and istypically used to measure properties of binary stars of less than 50 milliarc-sec (mas) separation and the angular diameters of single stars. On the otherhand, MUSCA was recently installed and must be used in tandem with the for-mer. It is dedicated for microarcsecond precision narrow-angle astrometry ofclose binary stars. The performance evaluation and development of the datareduction pipeline for the new setup was assisted by an in-house computersimulation tool developed for this and related purposes. This paper describesthe framework of the simulation tool, simulations carried out to evaluate theperformance of each beam combiner and the expected astrometric precision ofthe dual beam combiner setup, both at SUSI and possible future sites.
Keywords computer simulation · optical interferometry · visible wavelength · phase-referencing · astrometry Y. Kok, V. Maestro, P. G. Tuthill, J. G. RobertsonSydney Institute for Astronomy,School of Physics, University of Physics,NSW 2006, Australia.E-mail: [email protected]. J. IrelandDepartment of Physics and Astronomy,Macquarie University,NSW 2109, Australia. Kok et al.
A dual beam combiner setup was recently installed in SUSI. The main role ofthe new setup is to perform high precision narrow-angle astrometry of closebinary stars. The relative position of one star on the celestial sphere withrespect to another in a binary system can be determined by measuring theseparation (in optical delay) of their fringe packets formed by an optical longbaseline interferometer like SUSI. The accuracy of the projected separationof the binary star systems obtained from this method is determined by theuncertainty of the optical delay measurement. A more accurate measurementof the optical delay of a fringe packet can be made by measuring the phasedelay of the fringes instead of the group delay (Lawson et al, 2000).Now, if the measurements are to be carried out from the ground then theposition of the pair of fringe packets must first be stabilized because theirpositions are not static but constantly changing due to atmospheric turbu-lence. This can be achieved using a technique called phase-referencing (PR)(Colavita, 1992; Shao and Colavita, 1992). In this technique, two beam com-biners are required. The phase delay of one fringe packet is measured accuratelyin the presence of atmospheric turbulence with one beam combiner (usuallycalled the fringe tracker) and then fed-forward into another companion beamcombiner to stabilize the position of the same or another fringe packet. Thistechnique was demonstrated with PHASES (Muterspaugh et al, 2010) at PTI(Colavita et al, 1999) in the near infra-red wavelengths which had achievedan astrometric precision of 35 µ as (with separation less than 1 ′′ close binaries)within 70 minutes of observation time (Lane and Muterspaugh, 2004).The dual beam combiner setup in SUSI is specifically designed to do thesame (phase-referencing observations). The main beam combiner at SUSI,namely PAVO is used as the fringe tracker to measure the phase delay ofthe fringes of the primary star in the binary system in real time and thecompanion beam combiner, namely MUSCA, is used to simultaneously recordeither the fringes of the primary or the secondary star. This setup is similar toPHASES where both beam combiners receive the same pair of starlight beamsfrom the siderostats and observe the same field of view ( < ′′ ) of the sky. How-ever in many other ways it is different. Firstly, PAVO and MUSCA operate inthe visible wavelengths. Secondly, each beam combiner operates at a slightlydifferent bandwidth compared to the other. Thirdly, the phase-referencing ofstellar fringes are carried out in post-processing which eliminates the need fora feedback servo loop in MUSCA. Lastly, MUSCA observes only one stellarfringe packet at a time but can switch between a pair of fringe packets of abinary star during observation.With the introduction of MUSCA into SUSI, the existing data reductionpipeline was also upgraded to support the dual beam combiner configuration.The software development, which mainly involved putting in additional fea- a fringe packet is an interference pattern produced by a light source of finite bandwidth(e.g. starlight) whereby the fringe visibility diminishes quickly to zero as the optical pathdifference (OPD) producing the fringes deviates away from zero.imulating a dual beam combiner at SUSI for narrow-angle astrometry 3 PAVOsimMUSCAsim PipelineIPipelineII PR'edfringesV , dPRsim?V, d& etc.VS. ~ ~No Yes simulators(this work) code-under-test Fig. 1: Logical flow diagram of the data reduction pipeline test bench.tures to estimate phase delay of stellar fringes and to carry out a non-realtime phase-referencing operation, was greatly assisted by an in-house com-puter simulation framework. The framework and its usage are described inthis paper. Firstly, Section 2 gives a general overview of the simulation frame-work. Subsequently, Sections 3, 4 and 5 describe models of fringes employedby the simulators to generate the test data sets while Section 6 describes amethod that was used to include the effect of atmospheric turbulence in thesimulation. Lastly Section 7 shows the output of several dual beam combinersimulations and the expected performance of the instruments.
Two simulators, which are computer models of the PAVO and MUSCA beamcombiners, are developed to generate a set of simulated interferograms of eachbeam combiner based on user-specified inputs. The simulators were written inthe Interactive Data Language (IDL) but the design concepts and algorithmsdescribed here can be implemented in any other languages. Both simulatorscan read the same set of user-specified inputs and by doing so allow users tosimulate a dual beam combiner operation in which stellar fringes are recordedby the actual instruments simultaneously in real time. The simulated interfer-ograms can then be used to test the fringe visibility squared ( V ) estimationand the phase-referencing algorithm of the upgraded data reduction pipeline(Kok et al, 2012). By comparing the user-specified input and the simulatoroutput, especially the estimated V since it is the main science observable ofPAVO, the accurateness of the estimation and the performance of the dualbeam combiner setup can be assessed. Fig. 1 illustrates the data reductionpipeline test bench which shows the usage of the two simulators developed inthis work.An overall logical flow of both simulators is illustrated in Fig. 2. Simulationbegins with a generic model of two or more pupils. They are then customized Kok et al. addphasenoisestart processuservariables finish convertintensityto photoncounts Yes Nocustomizepupil outputfiles makeD-frame?
V, d& etc.moreframes?addnoise? AA makeS-/F-/R-frame convertphotonsto ADU add CCDreadnoisemakegenericpupil YesNo NoYes
Fig. 2: Logical flow diagram of the PAVO and MUSCA simulators.according to the optics of individual beam combiner. Additional and optionalphase noise to simulate the effect of atmospheric turbulence can be includedbefore the customization of the pupils. Then user-specified inputs are pro-cessed, e.g. to determine the number of interferograms (referred to as framesin the flow diagram) to be generated or the visibility and phase delay of thefringes to be simulated. After generating the required frame either by coherentor incoherent combination of the pupils, depending on the type of frame (referSection 3.3), e.g. science (S-), foreground (F-), ratio (R-) or dark (D-) frame,the amplitude of the combined pupil is converted into photon counts and sub-sequently into the detector read-out units (ADU). The detector read-out noiseis also included into the simulator output. The simulated data is then savedinto a file of appropriate format. The simulation finishes when all the requiredframes are generated. The details of each stage of the logical flow are discussedin the subsequent sections.
The PAVO beam combiner is a multi-axially aligned Fizeau-type interferom-eter. But unlike a typical Fizeau interferometer, PAVO forms spatially mod-ulated interference fringes in the pupil plane of the interferometer and thenspectrally disperses the fringes with an integral field unit. It also employs spa-tial filtering in its image plane and an array of cylindrical lenslet to utilize thefull multi-r aperture of the siderostats at SUSI. The lenslet array fragmentsthe pupil of the siderostats into several segments so that fringes from differentparts of the pupil can be measured separately. The schematic diagram of thePAVO beam combiner is shown in Fig. 3. It combines starlight beams from any imulating a dual beam combiner at SUSI for narrow-angle astrometry 5 InterferometerFromBeams Achromat AchromatKnife−Edge Lenslet ArrayOptimized FragmentationPupil AchromatStarlight DetectorEMCCDSplicingImage−PlaneMirror SystemImage−Plane Prism Re−imaging and Dispersive OpticsFocusing Optics Spatial FilterMask
Fig. 3: Schematic diagram of the PAVO beam combiner at SUSI.2 of the 11 siderostats (Davis et al, 1999) at one time. Despite that, the PAVOsimulator developed in this work is able to simulate higher order beam com-bination (i.e. 3 and more beams simultaneously) because the optical design ofPAVO at SUSI is an adaptation of a twin instrument at the CHARA array onMount Wilson, California (McAlister et al, 2005), which has the capability ofcombining starlight beams from up to 3 telescopes. The original optical designof PAVO at CHARA and the modified version at SUSI have been discussed indetail by Ireland et al (2008) and Robertson et al (2010) respectively.3.1 Input and output of the simulatorThere are more than 20 input parameters to the PAVO simulator with themore critical ones listed in Table 1. Some input parameters are common tothe MUSCA simulator which will be described in Section 4. The main outputof the simulator is a set of FITS formatted files which contain images ofsimulated interferograms, hereinafter referred to as frames, as well as headerinformation (e.g. timestamp, fringe lock status, etc) as would be recorded bythe PAVO camera. There are four different types of frames recorded by thecamera during actual observations. There are the science, ratio, foreground anddark frames. Examples of each of these frames are shown in Fig. 4. Each frame,except the dark, contains images of the spectrally dispersed (horizontally inthe figure) pupil as sampled by an array of lenslets. The number of lenslets, N LL , is different between PAVO at SUSI and at CHARA. The example shownin Fig. 4 is of the former which has the lenslets arranged in a one-dimensionalarray and there are 4 lenslets per pupil. The left pupil is for science while themiddle and right pupils are used for tip-tilt correction and therefore are ignoredin the simulation. Frames of PAVO at CHARA contains only the science pupil.3.2 Model pupilBefore the pupils are combined to form fringes, they are first spatially filteredin the image plane with square apertures (hereinafter referred to as the PAVO http://fits.gsfc.nasa.gov Kok et al.
Table 1: Input parameters for the PAVO (P) and MUSCA (M) simulators
Name Simulators Description t START
P,M Start of simulation in Julian date t STEP
P,M Exposure time of camera/photodetector N S − FITS
P Number of science type FITS to generate N R − FITS
P Number of ratio type FITS to generate N F − FITS
P Number of foreground type FITS to generate N D − FITS
P Number of dark type FITS to generate N MED
P,M Number and types of optical media ζ P,M Astrometric OPD in m N TEL
P,M Number of telescopes B P,M Details of telescopes (e.g. baselines) to be used m V P,M Magnitude of source in V band V P,M Model complex visibility of source D , d P,M Offset of ζ in m r P,M Fried parameter in m σ − r P,M Wavelength in which r is specified ( µ m) τ P,M Coherence time in milliseconds L P,M Outer scale of atmospheric turbulence in m t STEP
M Time interval between steps N STEP
M Number of steps per scan N SCAN
M Number of scan to simulate L SCAN
M Length of a scan in µ m mask), one for each beam, to remove high spatial frequency noise arising as aconsequence of atmospheric turbulence and optical aberration. A pupil froma telescope is modeled by a square matrix, ˜ P , of size N FFT × N FFT ,˜ P = (cid:26) J N FFT , N FFT ; perfect wavefront J N FFT , N FFT ◦ exp ( i ϕ ) ; corrugated wavefront (1)where J represents a unit matrix consisting of all 1s of size (row × column)indicated by its subscript and the phase component is either zero or ϕ so asto represent a perfect or corrugated wavefront. A corrugated wavefront due toatmospheric turbulence will be elaborated in Section 6. The notation ◦ is theHadamard or the element-by-element multiplication operator of two matrices.The model of a spatially filtered pupil, P ′ , is then, P ′ = FT − n (cid:3) ◦ FT n (cid:13) ◦ ˜ P oo (2)where (cid:13) and (cid:3) are square matrices of the same size which define a circularpupil and the spatial filter respectively. Each element in the matrices is definedas, (cid:13) u,v = (cid:26) p ( u − N FFT / + ( v − N FFT / ) ≤ N DIA /
20 ; otherwise (cid:3) x,y = (cid:26) | x − N FFT / | ≤ N MASK / | y − N FFT / | ≤ N MASK /
20 ; otherwise (3) imulating a dual beam combiner at SUSI for narrow-angle astrometry 7(a)(b) (c)
Fig. 4: (a) An example of real on-sky data of α Gru recorded by the PAVOcamera. It contains images of spectrally dispersed (horizontally) pupils as sam-pled by an array of four lenslets. The science pupil on the left (indicated bythe dotted box) of the frame is where fringes are formed by a pair of over-lapped pupils. The pupils in the middle and left of the frame are used fortip-tilt control. They are ignored and will not be generated during simulation.The images of the science pupil for different frame types are shown in (b) and(c). The images, from left to right and top to bottom, belong to four differenttypes of frames, namely science, foreground, ratio and dark respectively. Theimages in (b) are recorded by the camera while those in (c) are generated bythe simulator. The convention adopted by the mathematics in this paper takesthe vertical axis, which is the fringes direction, as the x -axis and the horizontal axis, which is the spectral channel direction, as the y -axis.where N DIA and N MASK are relative sizes of the diameter of the telescopesand the width of the PAVO mask respectively. N FFT , N DIA and N MASK takeonly integer values. The ratio of the width of the mask to the size of the imagevaries with the wavelength of light and is defined in Eq. (4). In this paper thewavelength of light is always represented by its reciprocal, or wavenumber, σ or Kok et al. σ j , where j is the index of vector σ which represents a range of wavenumbersapplicable to PAVO. The wavelength scaling factor in Eq. (4) is derived basedon the f -ratio of the beam and the distance of the PAVO mask from thepupil plane. The different values between PAVO at SUSI and CHARA meansthe optical setup at the two interferometers are not exactly the same. Thesimulator keeps the denominator of the left-hand side (LHS) of Eq. (4) constantand varies N MASK at different wavelengths of light to satisfy the equation. N MASK N FFT /N DIA ≈ (cid:26) . σ j ; SUSI8 . σ j ; CHARA (4)With N FFT and N DIA typically set to 256 and 114 respectively, the range ofvalues of N MASK is 5–7 for SUSI and 22–30 for CHARA.Before a model pupil is used to form fringes, it is down-sampled to the sizeof an actual image taken by the PAVO camera, which is N X × N LL , where N X is the number of pixels in the x -axis and N LL is the number of lenslets inthe y -axis. The values of each parameter are listed in Table 2. As a result, themodel pupil becomes, P = 1 N X N LL I TN FFT , N X P ′ I N FFT , N LL (5)where I M , N is a M × N matrix, I M , N = J M / N , · · · J M / N , · · · · · · J M / N , (6)and J M / N , is a M/N × M is an integer multiple of N . For example, to down-sample a 6 × P ′ to a 2 × P , the following operation can beapplied, P = 16 T P ′ , P ′ , · · · P ′ , P ′ , P ′ , · · · P ′ , ... ... . . . ... P ′ , P ′ , · · · P ′ , (7)where the first and the last matrices on the right-hand side (RHS) are I , and I , respectively.The values of the physical parameters in Eq. (4), Eq. (5) and other equa-tions in this section are listed in Table 2. imulating a dual beam combiner at SUSI for narrow-angle astrometry 9 Table 2: Specification of PAVO
Parameter Notation PAVO@CHARA PAVO@SUSINumber of spectral channels N σ
19 21Spectral range ( µ m − ) σ N LL
16 4Number of pixels in x-axis N X
128 32FOV to pupil size ratio ˜ R t α = t START + 50 × αt STEP (8)where α is the number of FITS files already generated before the current one.The typical value of t STEP for PAVO at SUSI and CHARA is 5ms and 8msrespectively. The number of files to be generated for each type of frame isdetermined by the user.
The science frames are generated using two or three pupils, depending on thenumber of telescopes in use. Simulation of PAVO at CHARA can use up tothree. In reality the pupils are aligned to overlap each other and combinedto produce spatially modulated fringes across the pupils. The model intensityacross the overlapped pupils at one particular wavelength is given as, F = w j N TEL X θ =1 N TEL X ˜ θ =1 V θ, ˜ θ a θ P ( θ ) ◦ a ˜ θ P (˜ θ ) ◦ exp (cid:16) i Φ ( θ, ˜ θ ) (cid:17) (9)where the notation P represents the complex conjugate of the variable P .The indices θ and ˜ θ denote one of the several pairs of telescopes used in thesimulation ( N TEL ) while a θ denotes the weighted amplitude of P ( θ ) such that, a + a + · · · + a N TEL = 1 (10)This condition has no physical reason but is imposed for the convenience ofscaling the normalize intensity to the right photon rate in Section 5. The term w j states the relative intensity of the summation in Eq. (9) at one wavelengthwhile the vector w which w j is a part of describes the spectrum of the lightsource and the bandpass profile of PAVO, w = (cid:2) w w · · · w N σ (cid:3) ; where wJ N σ , = 1 (11) This term can easily be customized by user according to the need of a simula-tion. However, the results shown in Section 7 were simulated with a smooth-edged top hat function for w . The model of fringes, F , is expressed in thisform in order to allow a model of complex fringe visibility, V θ, ˜ θ , to be appliedto the pairs of pupils. The complex fringe visibility matrix, which is a usersupplied input, is defined as, V = V ( σ j B , ) · · · V ( σ j B ,N TEL ) V ( σ j B , ) 1 · · · V ( σ j B ,N TEL )... ... . . . ... V ( σ j B ,N TEL ) V ( σ j B ,N TEL ) · · · (12)where each off diagonal element represents the fringe visibility of a model lightsource at a given wavelength and baseline. The term B θ, ˜ θ is the magnitude ofa baseline vector B θ, ˜ θ formed by a pair of telescopes θ and ˜ θ . Lastly the matrix Φ represents an additional phase difference between the two pupils and it isused to model the difference in piston and tilt in the wavefront of the pupils. Φ ( θ, ˜ θ ) = 2 πσ j (cid:16) ζ θ, ˜ θ + [ N ] j, ∗ z ( θ, ˜ θ ) + D θ, ˜ θ + N j, S θ, ˜ θ ˜ R xJ , N LL (cid:17) (13)The first three terms in Eq. (13) represent the piston term. The first term ζ θ, ˜ θ is the astrometric OPD due to the position of the target star with respectto the baseline B θ, ˜ θ while the second term [ N ] j, ∗ z ( θ, ˜ θ ) is the optical path ofthe delay line used to compensate the astrometric OPD. ζ = s · B , · · · ˆ s · B ,N TEL − ˆ s · B , · · · ˆ s · B ,N TEL ... ... . . . ... − ˆ s · B ,N TEL − ˆ s · B ,N TEL · · · (14)The optical delay line can comprise of various optical media. The types ofoptical media are specified by the user but practically it is not more than4 different types (e.g. vacuum, air and two types of glass, BK7 or F7). Therefractive indices for each medium at the wavenumbers of PAVO are calculatedusing values and constants obtained from Tango (1990). The notation [ N ] j, ∗ represents the j -th row of the refractive indices matrix, N , where, N = n ( σ ) n ( σ ) · · · n N MED ( σ ) n ( σ ) n ( σ ) · · · n N MED ( σ )... ... . . . ... n ( σ N σ ) n ( σ N σ ) · · · n N MED ( σ N σ ) (15)and n i is the refractive index of one optical medium. In order to set a con-vention, the first medium ( i = 1) is air. Each element in the column vector z represents the optical path length of each medium and to optimally compen-sate a given astrometric OPD the values of z are calculated using the method imulating a dual beam combiner at SUSI for narrow-angle astrometry 11 described by Tango (1990). The third term in Eq. (13) is an offset term toallow users to simulate a non-optimally compensated astrometric OPD. Theuser input matrix D is defined as, D = d , · · · d ,N TEL − d , · · · d ,N TEL ... ... . . . ... − d ,N TEL − d ,N TEL · · · (16)where d , for example is the OPD offset for baseline B , .The last term in Eq. (13) is the OPD caused by the differential wavefronttilt between a pair of pupils at the pupil plane. It is proportional to the sep-aration of the apertures on the PAVO mask and inversely proportional to thedistance of the pupil plane from the mask. S θ, ˜ θ is the ratio of the separation ofthe apertures on the PAVO mask to the width of each aperture and is definedas, S θ, ˜ θ = 2 | θ − ˜ θ | (17)˜ R is the ratio of the field of view (FOV) of the PAVO camera to the diameterof one pupil at σ = 1 µm − and its value is given in Table 2. Lastly, x ∈ R : − / ≤ x i ≤ / N X which represents thepixels across the field of view of the camera along the direction of the tilt.This direction is also the axis where interference fringes are formed across thecamera and is referred to as the x -axis by convention.Now, F is defined at just one wavelength. In reality, the combined pupils arespectrally dispersed by a prism. In order to model this F is evaluated N σ times,each time at a different wavelength within the PAVO spectral bandwidth.Multiple F matrices are then rearranged in the following order to mimic theactual interferogram recorded by the camera using only their real parts (asdenoted by the notation ℜ ), F = N PHOTONS × ℜ nh F (1) ∗ , F (2) ∗ , · · · F ( N σ ) ∗ , F (1) ∗ , · · · F ( N σ ) ∗ , F (1) ∗ ,N LL · · · F ( N σ ) ∗ ,N LL io + ǫ (18)The superscripts in parentheses represent the matrices evaluated at differ-ent wavelengths within the spectral bandwidth. The interferogram, F , whichtakes only the real part of the F , is padded with columns of zeros, . Thisis done according to a PAVO parameter definition file. The file describes thepixels locations of a spectral channel within the camera’s field of view whichwas determined through calibration with up to two lasers. The scaling factor, N PHOTONS , in Eq. (18) converts intensity to energy in terms of number ofphotons. This factor is proportional to the brightness of the target star, num-ber of telescopes used and the exposure time, all given by the user. A noiseterm, ǫ , is added to the interferogram to simulate photon noise, multiplication noise and read noise of the EMCCD camera. It is not purely an additive termas suggested because the expression of the noise term in Eq. (18) is simplis-tic. Physical models are used in the simulation to generate the photon andmultiplication noise components based on the number of photons. Foreground frames are generated in a very similar way to the science frames.Instead of setting D to values within the coherent length of the PAVO spectralbandwidth, it is set to a very large number (e.g. 1m) so that no fringes areformed across the pupils. Furthermore the visibility of the fringes are set tozero. In the actual beam combiner, foreground frames are recorded by givingthe optical delay line a large offset from its last position where fringes werefound. Ratio frames are generated using only one pupil at a time. In the actual beamcombiner, such frames are recorded when one of the many beams is blockedfrom reaching the camera. This type of frame is used by the data reductionpipeline to determine the intensity of pupil from each telescope. With only onepupil, the F matrix becomes, F = a θ P ( θ ) (19) Unlike previous types, dark frames are generated without any pupil. In the ac-tual beam combiner, dark frames are recorded when all the beams are blockedfrom reaching the camera. The interferogram contains only the noise termwhich in this case made up of only the read noise of the camera. This type offrame is used by the data reduction pipeline to subtract the noise floor in theinterferogram. F = ǫ (20) The MUSCA beam combiner is a co-axially aligned pupil-plane Michelsoninterferometer. It combines only two beams at one time, each from one sidero-stat. There are no spatial fringes in the image plane. If the image is diffractionlimited, the Airy disk will be completely dark when the pupils are out of phaseand bright when the pupils are in phase. MUSCA produces interference fringesby varying the difference in piston between the two pupils through time. This imulating a dual beam combiner at SUSI for narrow-angle astrometry 13 A P D A P D SN scanningmirror IR cut-off filter S17MN17M S18MLN4LS4 APD1APD0N18M BS
Fig. 5: Schematic diagram of the MUSCA beam combiner at SUSI. The beam-splitter in the diagram is labeled BS while the APDs are single pixel avalanchephotodiodes used for photon-counting.Table 3: Specification of MUSCA
Parameter Notation Typical valuesSpectral range ( µ m − ) σ µ m) L SCAN
30, 140Number of steps per scan N STEP t STEP is done by changing the optical path length of one pupil in air by moving a mir-ror in discrete steps between two locations back and forth rapidly. The mirror,hereinafter referred to as the scanning mirror (N18M in Fig. 5), makes a scanby moving from one extreme position to another. The physical parametersrelated to the scanning mirror and the operational spectral range of MUSCAare listed in Table 3. The schematic diagram of the MUSCA beam combiner isshown in Fig. 5. Fringes are formed at the beamsplitter (BS) and are recordedby a pair of avalanche photodiodes (APDs), one on each sides. However thedetails of its optical design are discussed in another paper (Kok et al, 2012).4.1 Input and output of the simulatorThe format of the input to the MUSCA simulator is exactly the same as theinput to the PAVO simulator. Some input parameters are common to bothsimulators but there are some parameters which are applicable only to theMUSCA simulator. The parameters are listed in Table 1.Instead of FITS, however, the output of the MUSCA simulator is a plaintext file, which has the same format as one generated by the actual beamcombiner. The file contains a time series of photon counts recorded by the
30 40 50 60 70 80 90−1000−50005001000 up − sc an s OPD ( µ m)30 40 50 60 70 80 90−1000−50005001000 do w n − sc an s OPD ( µ m) (a) −30 −20 −10 0 10 20 30−1000−50005001000 up − sc an s OPD ( µ m)−30 −20 −10 0 10 20 30−1000−50005001000 do w n − sc an s OPD ( µ m) (b) Fig. 6: An example of two sets of time series of photon counts (a) recorded bythe actual MUSCA beam combiner and (b) generated by the simulator plottedas separate scans. Each scan is plotted with an offset for visualization. The x -axis shows the relative OPD within the scan range of the scanning mirror.The top and bottom plots are scans of different direction.APDs in the image plane as the scanning mirror periodically scans through apredetermined scan range. Each photon count has a timestamp with a precisionof 10 microseconds. Fig. 6 shows an example of the photon counts recorded bythe actual instrument as well as a set generated by the simulator.4.2 Model pupilThe model of a pupil in MUSCA is straightforward as there is no spatial filterin MUSCA. Suppose the model pupil is represented again by the matrix P ofsize N FFT × N FFT , then, P = (cid:13) ◦ ˜ P (21)where ˜ P and (cid:13) are defined earlier in Eq. (1) and Eq. (3) respectively.4.3 Simulating the photon counts in a scanThe temporal fringes in MUSCA are generated using a pair of pupils, P (1) and P (2) . In the physical instrument the pupils are aligned and combined coaxiallyat a beamsplitter to produce two sets of output pupils. In the simulator theoutput pupils are modeled as two N σ × N STEP matrices, F ( L ) = 1 + 2 a a ℜ (cid:8) (( w · v ) T J , N STEP ) ◦ Q ◦ exp ( i Φ ) (cid:9) F ( R ) = 1 − a a ℜ (cid:8) (( w · v ) T J , N STEP ) ◦ Q ◦ exp ( i Φ ) (cid:9) (22) imulating a dual beam combiner at SUSI for narrow-angle astrometry 15 The matrices and their superscripts ( L ) and ( R ) represent the intensity of theoutput pupil on the left and right side of the beamsplitter. The variables a , a and w are previously defined in Eq. (10) and Eq. (11). The vector v , which issimilar to the matrix V in Eq. (12), is the model fringe visibility of the sourcefor the baseline defined by telescopes of pupil P (1) and P (2) . v = (cid:2) V ( σ B , ) V ( σ B , ) · · · V ( σ N σ B , ) (cid:3) (23)Each element of Q is, Q j,k = P N FFT u =1 P N FFT v =1 P (1 ,j,k ) u,v P (2 ,j,k ) u,v Area of circular aperture in (cid:13) (24)The real part of Eq. (24),
ℜ { Q j,k } , represents the normalized intensity atwavenumber σ j due to the sum of pupils P (1) and P (2) at k -th step of a scan. Φ is the additional phase difference between two telescope pupils at variouswavelengths, Φ = 2 π (cid:0) σ T ζ + J N , σ NZ + σ T d (cid:1) (25)and it is used to model the difference in piston between the pupils at each stepof a scan.The structure of Eq. (25) is very similar to Eq. (13). The first term inEq. (25), ζ , is the astrometric OPD per scan of the baseline B , . Each element, ζ k , is evaluated at time t α,k , ζ k = | ˆ s · B , | t = t α,k (26)where, t α,k = t START + t STEP ( αN STEP + k ) (27)and α is the number of elapsed scans before the current one and k is theindex of a step within a scan. The typical value of t STEP for MUSCA is 0.3ms.The second term in Eq. (25), NZ , is the optical path of the delay line usedto compensate the astrometric OPD. The matrix, N , previously defined inEq. (15), denotes the refractive indices of each optical medium in the pathwhile the matrix, Z , denotes the path length of each medium at every step ina scan. Z = z ( t α, ) z ( t α, ) · · · z ( t α,N STEP ) z ( t α, ) z ( t α, ) · · · z ( t α,N STEP )... ... . . . ... z N MED ( t α, ) z N MED ( t α, ) · · · z N MED ( t α,N STEP ) + ℓ , N STEP ... , N STEP (28)The values of elements in the first term of the RHS of Eq. (28) are calculatedusing the method described by Tango (1990). The vector ℓ in Eq. (28) describes the relative change in the optical path length of air at every step in a scan dueto the motion of the scanning mirror. ℓ = L SCAN N STEP − × (cid:2) − N STEP / − N STEP / · · · − · · · N STEP / − (cid:3) (29)Elements in the same column of Z have the same timestamp as an elementwith the same index in ζ . The last term in Eq. (25), d , is the user-specifiedoffset at each step to simulate a non-optimally compensated astrometric OPD. d = (cid:2) d d · · · d N STEP (cid:3) (30)After the pupils are combined and have formed fringes, it is assumed thatthe entire image of the pupil falls within the active area of the photodiodesand all photons are detected. The photodetectors in MUSCA have only onepixel and are unable to resolve any spatial variation in intensity at the imageplane. Therefore only an average intensity is recorded, hence the term Q inEq. (22). In addition to that the photodetectors in MUSCA are unable toresolve intensity variation across wavelengths. Therefore the number of photoncounts recorded by the photodetectors is the sum of photons across the entireMUSCA operating bandwidth, F = N PHOTONS × J , N σ F + ǫ (31)Similar to Eq. (18), N PHOTONS is a scaling factor that converts the intensityof the output pupil to the number of photons expected from the source givenits brightness in magnitude scale and ǫ is a noise term included to simulatephoton noise and the dark count noise of the detector. The scaling factor N PHOTONS in Section 3 and Section 4 is estimated based onthe expected throughput of the beam combiner to be simulated, the efficiency(Q.E.) of the APDs and a calibrated magnitude-to-flux scaling factor, F ν , byBessell (1979). The first two factors collectively describe the efficiency of theinstrument, η , which is found to be ∼
3% for PAVO and ∼
1% for MUSCA. Thelower efficiency in MUSCA is possibly due to the aluminium coated mirrorsused in SUSI and a silvered beamsplitter in MUSCA. The values of F ν atdifferent photometric bands are listed in Table 4, which is reproduced fromBessell (1979). From F ν , the number of photons from a m V magnitude starcollected by a telescope with an area of A tel m in ∆t seconds can be estimated.Putting the factors together, N PHOTONS ≃ η × . × × − m V / . × F ν ∆ (1 /σ ) ∆λ A tel ∆t (32) imulating a dual beam combiner at SUSI for narrow-angle astrometry 17 Table 4: Absolute flux calibration of α Lyrae (Bessell, 1979)
Filter λ eff ∆λ Flux density, F ν band ( µ m) ( µ m) ( × − W m − Hz − )U 0.36 0.076 ∗ ∗ ∗ C C ∗ where the ∆ (1 /σ ) is the bandwidth of the beam combiner and ∆λ is thebandwidth referred from Table 4 which has an effective wavelength close tothat of the beam combiner. As an example, in the case of the MUSCA beamcombiner, ∆ (1 /σ ) ≃ . µ m, ∆λ = 0 . µ m, λ eff = 0 . A tel = 7 × − m and ∆t ≃ × A turbulent atmosphere introduces amplitude and phase fluctuation to anotherwise plane wavefront of light from a distant star. Although amplitudefluctuation can be simulated, both PAVO and MUSCA simulators make theassumption that the atmospheric phase fluctuation only gives rise to pertur-bation in the phase of a wavefront. This approximation by discarding theamplitude (scintillation) term, also known as the near-field approximation,holds very well for pupils larger than 2.5cm under typical turbulence condi-tion (Roddier, 1981).The phase fluctuation in the atmosphere is simulated using a large ( N ATM × N ATM ) two-dimensional array of random phasor, φ , which has a power spec-trum given by (Roddier, 1981; Glindemann, 2011), (cid:12)(cid:12)(cid:12) ˆ φ u,v (cid:12)(cid:12)(cid:12) = C (cid:0) L − + ( u + v ) (cid:1) − / (33)The notation ˆ φ denotes the Fourier transform of φ , u and v are the indices ofthe array and L is the outer scale of the turbulent structure of the model at-mosphere, which is set to a very large number in the simulation. The phase fluc-tuation is recovered by taking the inverse Fourier transform of (cid:12)(cid:12)(cid:12) ˆ φ u,v (cid:12)(cid:12)(cid:12) exp( iε )where ε is the phase of the Fourier transform and it is a random variable. Therandomness of the generated phase fluctuation, φ , is controlled by adjustingthe scaling factor, C . It is tweaked so that the structure function of φ , h| φ u,v − φ u + r u ,v + r v | i = 6 . p r u + r v r ! / (34)has its characteristic Fried parameter, r , set to the value specified by the user.Fig. 7(a) shows an image of the random phases generated using this method.The grayscale of the images indicate the value of the phases. Fig. 7: (a) An array of random phases generated with the inverse Fouriertransform method. The value of the phases is indicated by the shades of grayin the images. (b) Small portions of the larger array in (a) sampled at slightlydifferent positions at each time step showing the progress through time. Timeincreases from left to right and top to bottom.On the other hand the phase fluctuation of a wavefront across a telescope,or ϕ in Eq. (1) and Eq. (21), is simulated by extracting a small portion( ∼ × φ array. The typical size of the φ array is2048 × r is specified at a certain wavelength, σ − r , and the phasefluctuation in φ is generated according to r , the value extracted from thelarger array must be scaled by a factor of σ j /σ r , where σ j is the desiredwavenumber for simulation, before applying it to the model pupils in Eq. (1)and Eq. (21). The position of this small sampling window is displaced acrossthe larger array after every time step of the simulation. This simulates theeffect of Taylor’s (1938) hypothesis of a frozen atmosphere drifting across theaperture of the telescope. The rate of displacement of the sampling windowdepends on the wind speed which is estimated from,¯ v = 0 . r τ (35)where τ is the coherence time of the phase fluctuation and the time step of thesimulation, t STEP . Both parameters are specified by the user. The direction ofthe displacement is random but remains constant throughout the simulation.Fig. 7(b) shows several snapshots of phase variation over a small portion of alarger array drifting across the sampling window.A separate array of φ is generated for each telescope. By doing this, it isassumed that the phase fluctuations over individual telescopes are uncorre-lated. As an effect, the low-frequency phase fluctuations do not increase withbaseline in the simulation. In practice this scenario is true if the baseline islonger than the outer scale of turbulence, L , which is in the order of 100m inthe troposphere (Roddier, 1981). Therefore this approach of having a separate imulating a dual beam combiner at SUSI for narrow-angle astrometry 19 array of φ for each telescope does not simulate the piston term of an aber-rated pupil due to atmospheric turbulence realistically. In order to addressthis shortcoming in the PAVO and MUSCA simulator, the input variable, D or d (refer Table 1), can be used to add an extra differential piston betweenpupils from two apertures.Another shortcoming of this method in simulating the phase variationacross a turbulent atmosphere is a structure function that deviates from itstheoretical value at large distances. Fig. 8 shows comparison between a theo-retical structure function and one calculated from the φ . The plots in the figureare ( D φ / . / versus r and normalized to the size of the array, N ATM . Thetheoretical Kolmogorov (KM) model curve is a straight line with unity slope( r = 1). The deviation from the theoretical value is very pronounced espe-cially at large distances. This is not surprising because the power spectrum, | ˆ φ | , is undersampled at low frequencies, where most of the energy resides.A practical solution to this problem, which is the approach implemented inthis work, is to sub-sample the phase fluctuation from a much larger array(McGlamery, 1976; Shaklan, 1989; Lane et al, 1992). With a telescope aper-ture model of ∼
5% of the size of the larger phase array, such approximationproduces phase fluctuation that is at most 20% off the theoretical KM model.This is shown in Fig. 8. Also shown in the figure is a theoretical von Karman(VK) model which has a structure function very similar to that of the gener-ated phase fluctuation. The outer scale of turbulence of the model is ∼
44% of N ATM and at this outer scale of turbulence, given the telescope aperture usedin the simulation model, the variance of the tip-tilt angle of an image formedwith the phase fluctuation is reduced by ∼
40% from its expected value basedon a KM model which has an infinite outer scale (Sasiela and Shelton, 1993).This reduction in tip-tilt fluctuation simulates the active tip-tilt correction atSUSI which stabilizes the image of a star.
The main objective of this simulation framework is to test the data reductionpipeline of the beam combiners. It is also a good tool to investigate softwarebugs in a data reduction pipeline during its development stage. The followingsections discuss some testcases using the simulators.Several testcases were carried out to demonstrate the functionality of thesimulator and at the same time the accuracy of the data reduction pipeline forboth PAVO and MUSCA. Testcase I is carried out to verify the extraction ofvisibility squared, V , of a set of fringes by the PAVO pipeline. Testcase II-IVwere carried out to probe the lower bound phase error of the phase-referencedfringes constructed by the PAVO and MUSCA phase-referencing pipeline. Ontop of that they are also used to verify the PAVO V pipeline. Table 5 showsthe input for each testcase. All input parameters for Testcase II-IV are keptthe same except for the fringe visibility parameter. ATM )0.00.20.40.60.81.0 S t r u c t u r e f un c t i on r a t i o s KMVK (L =0.44)SimulationError ATM )0.000.050.100.150.20 S t r u c t u r e f un c t i on r a t i o s Fig. 8: Plots of ( D φ / . / against r where D φ is either a theoretical, basedon a Kolmogorov (KM) model, Eq. (34), or a von Karman (VK) model (Valley,1979)), and a simulated structure function. Both axes are normalized to thesize of the random phases array. The dashed lines in the plots are the differ-ence between the solid (KM) and dash dot (simulation) lines relative to theformer. If the simulated structure function is exactly the same as the theoret-ical structure function then the dash lines will be horizontal at 0. The rightplot is the zoomed in version of the left.7.1 To verify the PAVO V reduction pipelineTestcase I-IV are used to verify the V reduction pipeline of PAVO at SUSI.In these testcases the PAVO simulator is used to investigate the pipeline’sability to reproduce the square of the visibility of a given model. The inputfringe visibility for Testcase I is a sinusoidal function which models a binarystar system with the primary and secondary stars almost equal in brightness(contrast ratio of ∼ ∼ ′′ using a15m baseline. On the other hand, the input fringe visibilities for Testcase II-IV are constant to model an unresolved single star but have different valuesof instrument visibilities. Since PAVO at SUSI uses only two telescopes atany one time, ζ and D each reduces to a 2 × ζ , and d , , are shown in Fig. 9. Alltestcases have 3 sub-cases (A, B and C) where the photon rate is varied tosimulate an observation of a zeroth, 2nd or 4th magnitude star. The numberof photons in a generated PAVO frame is adjusted to match the number inan actual frame recorded by the PAVO camera at a gain of 5, for a star ofmagnitude m V = 0 . m V = 2 .
0, and 25, for a star of magnitude m V = 4 . V of each simulated data set extracted by the PAVO imulating a dual beam combiner at SUSI for narrow-angle astrometry 21 Table 5: Input parameters for all testcases
Testcases → I II III IVNames ↓ Section Section 7.1 Section 7.1 & 7.2 t START t STEP
PAVO: 5.0msMUSCA: 0.2ms N S − FITS N R − FITS N F − FITS N D − FITS N STEP – 1024 N SCAN – 150 L SCAN – 140 µ m N MED ζ See Fig. 9 N TEL B | B , | ∼ m V Sub-case A: 0.0Sub-case B: 2.0Sub-case C: 4.0 V D , d See Fig. 9 r σ − r µ m τ L Fig. 9: The value of ζ , and d , used in the simulation versus the framenumber. The latter is a pseudo-random variable where the difference in valuebetween each step is d , ( t α ) − d , ( t α − ) ∼ N (0 , . µ m. σ B (10 rad −1 ) V (a) σ B (10 rad −1 ) TF (b) Fig. 10: A comparison between user specified models of V (solid line) and theestimated V produced by the PAVO V reduction pipeline. The top plot of (a)shows the comparison between a wavelength dependent V model (input) andthe (output) values estimated by the pipeline at high photon rate (sub-case A)while the bottom plot of (a) represents a similar comparison at lower photonrate (sub-case B). The ratio of estimated to model V for Testcase I (+ , × ),II ( ⋄ , △ ), III ( ▽ , ⊲ ) and IVB ( ⊳ ) are plotted against spatial frequency in (b).Each pair of symbols in the parentheses except for Testcase IV represent sub-case A and B respectively. Also in (b), the ratios are shown when atmosphericphase noise was excluded in the simulation (no ATM).pipeline are plotted against the models fed into the simulator in Fig. 10. Alsoplotted for comparison, in Fig. 11, are the estimated V of Testcase II-IV.Both figures show that the estimated values are consistent and within 20% ofthe model. The expected reduction in visibility of the fringes, which is morepronounced at higher spatial frequencies (right half of the graphs), is due tothe atmospheric phase noise. The observed trend in Fig. 11(b), is due to awavelength-dependent scaling factor in the order of unity which is in turncaused by the shape of a Fourier domain windowing function used in thedata reduction pipeline. This scaling factor can be calibrated out because itis identical regardless of the visibility function, as seen in Fig. 10(b) whichplots the ratios between the estimated and the model V (or TF) across spa-tial frequency. Data points with TF significantly larger than 1 is an effect ofcalculating a ratio with the denominator that has a very small value.In summary, these testcases show that the V reduction pipeline has somedependence on the seeing condition, which can be calibrated out, but no mea-surable bias as a function of target brightness and the square of the visibilityof the fringes. imulating a dual beam combiner at SUSI for narrow-angle astrometry 23 σ B (10 rad −1 ) V (a) σ B (10 rad −1 ) V (b) Fig. 11: Similar to Fig. 10(a), the top, middle and bottom plots of (a) and (b)show the comparison between a constant V model (input) and the estimated(output) of the pipeline for Testcase II, III and IV respectively. The plots onthe left column are from sub-case A where the photon rate is higher thansub-case B which plots are on the right column. The estimated V values ofsub-case B of Testcase IV are very noise but are still unbiased. The plots in(a) are simulated with the presence of atmospheric phase noise while plots in(b) are simulated without. Frames (x1000) E rr o r ( µ m ) −2−1012 190190 269269 −2−1012 192192 104610460 1 2 3−2−1012 917917 0 1 2 3 49654965 (a) Frames (x1000) E rr o r ( µ m ) −2−1012 3030 9696 −2−1012 4747 1571570 1 2 3−2−1012 170170 0 1 2 3 44794479 (b) Fig. 12: The errors of group delay estimates obtained from the PAVO reductionpipeline for Testcase II-IV (from top to bottom). The left and right columnsare errors for sub-cases A and B respectively. The standard deviation of theerrors, in unit of nanometers, are stated in the plots. The data for the plotsin (a) and (b) are obtained with and without atmospheric phase noise in thesimulations.7.2 To verify the PAVO and MUSCA phase-referencing pipelineIn addition to the goal mentioned in previous section, the main aim of Test-case II-IV is to test a newly written PAVO and MUSCA phase-referencingpipeline and to probe the lower bound phase error of the phase-referencedfringes. Both simulators are fed with the same input for this purpose. Someinputs which are common to both simulators, e.g. ζ and D for PAVO, areresampled at a different rate for MUSCA because the time steps of the twosimulators are different. Inputs which are not common to both inputs, e.g. aredefined separately.The role of the PAVO data reduction pipeline in these testcases is to provideestimates of the group and phase delay of the fringes. The group delay of thefringes is defined by the user-specified input variables, D and d . The errors ofthe group delay estimates are shown in Fig. 12. The standard deviation of theerror of the group delay estimates increases as the visibility of the fringes andthe photon rate decrease. This is expected as the signal becomes weaker thanthe photon noise and read noise of the camera. Since the phase delay of thefringes is estimated from the group delay estimate, residuals of group delayerrors which are larger than one wavelength of the fringes produce unreliablephase delay estimates. Only reliable estimates of phase delay are chosen andapplied to stabilize the position of a fringe packet generated by the MUSCAsimulator (Kok et al, 2012). The percentage of reliable estimates within anobservation is a function atmospheric seeing, brightness of the target star andthe visibility of the stellar fringes. imulating a dual beam combiner at SUSI for narrow-angle astrometry 25 scan length (microns) up − sc an s −15 −10 −5 0 5 10 1510203040506070 scan length (microns) do w n − sc an s −15 −10 −5 0 5 10 1510203040506070 (a) scan length (microns) up − sc an s −15 −10 −5 0 5 10 1510203040506070 scan length (microns) do w n − sc an s −15 −10 −5 0 5 10 1510203040506070 (b) Fig. 13: Simulated MUSCA fringes in a waterfall plot format showing the effectof (a) without and (b) with phase-referencing. The vertical axis is time and thehorizontal axis is the optical delay. The top and bottom plots are of differentscan direction.Waterfall plots in Fig. 13 show the position of the fringe packet throughtime. They clearly show that the fringe packet has a continuously changingposition and a relatively constant position before and after the phase delay es-timates are applied. The phase variation of the phase-referenced fringes acrossmultiple scans, as seen in the plot, can be estimated from the h| C | i metric,which is a measure of the coherence of the fringes (Kok et al, 2012), and isgiven as, s x ≈ − h| C |i N SC N SC − (cid:18) π ˜ σ M (cid:19) (36)where N SC is the number of good scans used in the estimation and ˜ σ M is themean wavenumber of MUSCA fringes, which has a typical value of 1.2 µ m − .The factor N SC N SC − is used to obtain an unbiased estimate of the sample vari-ance. Table 6 shows the value of h| C | i , N SC and s x for all testcases.For a scan to be considered good, there must be a continuously reliablephase delay estimate throughout at least of the time to make the scan. Thenumber of good scans as well as the value of h| C | i decrease in the same trendas the standard deviation of the errors of the group delay estimates increasesbecause residuals of the errors which are larger than one wavelength of thefringes produce unreliable phase delay estimates. Testcase IVB is an exampleof an extreme case as there are no good scans found.The summation of the phase-referenced fringes in all good scans reproducesthe fringe packet without the distortion introduced by the atmospheric phasenoise. The effect of coherent summation of the fringes is shown in Fig. 14. Theposition of the fringe packet within the scan range of the scanning mirror canbe determined to an uncertainty defined by s x / √ N SC . For example, in order to Table 6: Results from PAVO and MUSCA phase-referencing algorithm with ATM without ATMTestcases ↓ m V → V = 0 . N SC
74 74 72 74 74 74 h| C | i s x (nm) 126 188 179 84 143 156III ( V = 0 . N SC
74 58 21 74 74 74 h| C | i s x (nm) 127 220 275 100 188 189IV ( V = 0 . N SC
58 0 0 74 0 0 h| C | i s x (nm) 218 N/A N/A 198 N/A N/A −20 −15 −10 −5 0 5 10 15 20−1−0.500.51x 10 OPD ( µ m) a r b . un i t s −20 −15 −10 −5 0 5 10 15 20−1−0.500.51x 10 OPD ( µ m) a r b . un i t s u−scans (PR off), N=74u−scans (PR on), V=0.16 (a) wavenumbers (1/ µ m) F r i nge po w e r ( a r b . un i t s ) N=74 (PR off) wavenumbers (1/ µ m) F r i nge po w e r ( a r b . un i t s ) N=74 (PR on) <|C| >=0.0139<|C| >=0.6407 (b) Fig. 14: The top and bottom plots of (a) show the result of summation ofmultiple scans of incoherent and coherent fringes respectively. Similarly thetop and bottom plots of (b) show the huge (about an order of magnitude)difference in the power spectrum of the incoherently and coherently summedfringe packets respectively. The number of scans used for the summation andthe h| C | i metric are indicated in (b).determine the fringe packet position to an uncertainty of 5nm, which translatesto an astrometric uncertainty of 10 µ as with 100m baseline, in Testcase IIIB,at least 1900 good scans are required. This number is not impractical as aperiod of good seeing at SUSI can produce over 1000 good scans (Kok et al,2012) in about 10 minutes of observation (a total of 3600 scans) with β Cru,which has a V magnitude of 1.3 and an uncalibrated V range of 0.02–0.20 asmeasured by PAVO.Although the results in Table 6 were simulated for the setup at SUSI, theycan be extrapolated to estimate the performance of a similar setup at differentobservation sites. For example, the scenario in Testcase IIC (4th magnitude imulating a dual beam combiner at SUSI for narrow-angle astrometry 27 star, seeing condition of ∼ ′′ and diameter of light collecting aperture of 14cm)is equivalent to observing a ∼ ∼ ′′ seeing(e.g. at NPOI or in Antarctica) but with twice the aperture size (28cm). Insuch scenario, at least 1000 good scans are required to achieve an astrometricprecision of 10 µ as with a 100m baseline. The phase-referencing performancesin Testcase IIB and IIC are similar because a higher EMCCD gain was usedto compensate the lower flux in the latter. The simulators developed in this work not only have been a very useful tool totest the data reduction pipeline developed for the new setup and observationtechnique at SUSI but have also shown the feasibility of visible wavelengthphase-referencing at sub-wavelength OPD uncertainty. The PAVO–MUSCAsetup at SUSI can determine the position of a fringe packet of a 2-4th mag-nitude star to an accuracy of 5nm by coherently integrating 1000-2000 goodfringe packet scans. The simulated performance of the dual beam combiner canbe extrapolated to estimate performance of a similar setup at future possiblesites (e.g. NPOI and Antarctica). In addition to that, due to the selection ofinput parameters the simulators were designed to accept they could also beused to perform simulation for many other functions (e.g. to explore the op-tion of expanding the capability of PAVO at CHARA from a 3-telescope to a4-telescope beam combiner or to investigate the effect of optical aberration oflenses on the performance of the beam combiners) which is beyond its mainrole described in this paper. The IDL code for the simulators can be obtainedfrom the corresponding author via email.
Acknowledgements
Y.K. would like to acknowledge the support from the University ofSydney International Scholarship (USydIS).
References
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