Broadband Fizeau Interferometers for Astrophysics
aa r X i v : . [ a s t r o - ph . C O ] J un Astronomy&Astrophysicsmanuscript no. aap35˙arxiv c (cid:13)
ESO 2018October 30, 2018
Broadband Fizeau Interferometers for Astrophysics
Siddharth S. Malu and Peter T. Timbie Raman Research Institute, Sadashivanagar, Bangalore 560 080, India. e-mail: [email protected] University of Wisconsin-Madison, Madison WI 53706, USA. e-mail: [email protected]
Received ; accepted
ABSTRACT
Context.
Measurements of the 2.7 K cosmic microwave background (CMB) radiation now provide the most stringent constraintson cosmological models. The power spectra of the temperature anisotropies and the E -mode polarization of the CMB are explainedwell by the inflationary paradigm. The next generation of CMB experiments aim at providing the most direct evidence for inflationthrough the detection of B -modes in the CMB polarization, presumed to have been caused by gravitational waves generated duringthe inflationary epoch around 10 − s. The B -mode polarization signals are very small ( ≤ − K) compared with the temperatureanisotropies ( ∼ − K). Systematic e ff ects in CMB telescopes can cause leakage from temperature anisotropy into polarization.Bolometric interferometry (BI) is a novel approach to measuring this small signal with lower leakage. Aims.
If BI can be made to work over wide bandwidth ( ∼ − Methods.
For subdividing the frequency passsband (‘sub-band splitting’ henceforth), we write an expression for the output from everybaseline at every detector in the focal plane as a sum of visibilities in di ff erent frequency sub-bands. For operating the interferometersimultaneously as an imager, we write the output as two integrals over the sky and the focal plane, with all the phase di ff erencesaccounted for. Results.
The sub-band splitting method described here is general and can be applied to broad-band Fizeau interferometers across theelectromagnetic spectrum. Applications to CMB measurements and to long-baseline optical interferometry are promising.
Key words. cosmology: cosmic background radiation — techniques: interferometric — instrumentation: interferometers
1. Introduction
Interferometry has a long history for astronomical measurementsat radio, millimeter, submillimeter, IR and optical wavelengths.In 1868 Hyppolyte Fizeau (Fizeau 1868) described how the di-ameters of stars could be measured by optical interferometry.He proposed ‘masking’ the aperture of a telescope to create in-terference fringes in the focal plane, similar to a Young’s doubleslit interference experiment. Today aperture masks are used toovercome atmospheric ‘seeing’ e ff ects to reach the di ff ractionlimit of single aperture telescopes (Tuthill et al. 2000). For evengreater angular resolution, beam combination from widely sep-arated apertures is used in long baseline optical interferometers.Michelson used this technique to measure, for the first time, thediameters of stars. He used flat mirrors to reflect the beams fromseparated apertures into a single telescope, which acted as thebeam combiner.This type of beam combination, in which beams are com-bined in the image plane of a telescope, is called ‘Fizeau inter-ferometry,’ or ‘adding interferometry.’ An alternate approach in-volves combining the beams in the pupil plane of a telescope andis called ‘Michelson interferometry’ after the technique used inthe Michelson-Morley experiment (Monnier 2003). Both typesof combiners are used in long-baseline optical interferometers(Traub 2000). Conventional radio interferometers can also bethought of as Michelson interferometers. They typically mix theRF signals from each antenna in an array to lower frequencies(heterodyning) and interfere (multiply) them electronically onepair at a time with either an analog or digital correlator to mea-sure visibilities. This approach is sometimes called ‘multiplying’ interferometry. These techniques are widely used for radio wave-lengths to sub-mm wavelengths. As the number of antennas, N ,increases the number of correlations to be performed grows as N ( N − /
2. Although radio correlators are improving rapidly,they are currently limited to combining signals from about 100antennas and bandwidths of a few GHz (Lawrence et al. 2008).In this paper we describe a Fizeau ‘adding’ interferometerthat overcomes the bandwidth and large- N limitations of hetero-dyne interferometers. The instrument is optimized for precisionmeasurements of the temperature and polarization anisotropyof the CMB and is sometimes called a ‘bolometric interferom-eter’ (Timbie et al. 2006; Tucker et al. 2008; Charlassier et al.2008; Hamilton et al. 2008; Hamilton & Charlassier 2010).Interferometers are less sensitive to some kinds of systematic ef-fects found in imaging instruments (Bunn 2007). The techniquecan be used at any wavelength.In particular, we focus on an approach to broadening thebandwidth of a Fizeau interferometer. Spectral resolution is notrequired for many applications where the source has a contin-uum spectrum (such as the CMB) and signal averaging overbroad bands is required to uncover faint signals. As one increasesthe bandwidth the di ff erent wavelengths within the passband willsmear out the interference fringes and reduce the sensitivity ofthe instrument. Furthermore, this ‘fringe washing’ reduces theresolution of the interferometer in the u-v plane. In this paperwe present a simple and powerful technique, which we call ‘sub-band splitting’ to overcome the limitations caused by large band-widths in Fizeau interferometers. We discuss how this approachcan improve parameter estimation in the specific case of obser- Siddharth S. Malu and Peter T. Timbie: Broadband Fizeau Interferometers for Astrophysics vations of the CMB, where sensitivity and u-v space resolutionare critical for constraining CMB power spectra.We begin by motivating the need for adding interferometryin §
2, followed by the reason sub-band splitting is required. § §
2. Cosmological Context: The CMB and AddingInterferometry
We now have a ‘standard model’ of cosmology in which the in-flationary paradigm describes many aspects of the universe ac-curately through well-quantified parameters. Precision measure-ments of the CMB are the most powerful probes for determin-ing these parameters. In particular, measurements of the angularpower spectrum of the CMB temperature and E -mode polariza-tion anisotropy have yielded a wealth of information about theearly universe (see Komatsu et al. 2009, for example). However,inflation is driven by physics that we do not currently understand(Baumann et al. 2009). The most direct way to probe inflation isthrough the so-called B -modes in CMB polarization. The am-plitude of the B -mode signal is directly related to the energyscale of the particle interactions that occurred during inflation.However, the energy scale of inflation is not known. Recent mea-surements (Gupta et al. 2010; Chiang et al. 2010), are approach-ing the level at which B -modes are expected to appear in modelsin which inflation occurred at the GUT scale. However, inflationmay have involved lower-energy interactions; there is no lowerbound to the amplitude of the B -modes.Current (Takahashi et al. 2010; Hinderks et al. 2009) andplanned CMB instruments (Bock et al. 2009) all use some typeof imaging technique. With focal-plane arrays of hundreds ofbackground-limited detectors they are capable of detecting the B -mode signals predicted in the most optimistic models, at thelevel of ∼ − K. Systematic e ff ects have been extensivelystudied for imaging polarimeters in the context of CMB mea-surements and appear to be controllable at this level as well(Hu et al. 2003; Bock et al. 2006). However, at some level all in-struments can ‘mix’ the relatively large temperature anisotropyand E -mode polarization signals into B -modes.Di ff erent systematic e ff ects are found in interferometers. Itis for this reason that heterodyne interferometers have been usedfor many years to study the CMB temperature and polarizationpower spectra and the Sunyaev-Zel’dovich e ff ect. These instru-ments multiply together the RF signals from all possible pairsof antennas (baselines) in an array to measure a set of visibili-ties, which are related to the sky image through a Fourier trans-form (Rohlfs & Wilson 2004). In fact, the first detection of CMB E -mode polarization was made by a multiplying interferome-ter: DASI (Kovac et al. 2002). DASI had 13 single-mode anten-nas and performed pairwise correlation of signals across the Kaband, from 26 - 36 GHz.Several groups have studied the possibility of building a newgeneration of mm-wave interferometers specifically to search forthe small polarization signals in the CMB (Timbie et al. 2006).Compared to existing mm-wave interferometers, these new in-struments would have to do the following: 1) collect more modesof radiation from the sky by including more antennas ( >
100 );2) operate with broader bandwidth ( ∼ >
10 GHz), and 3) operate over a broader range of center fre-quencies, at least up to 90 GHz, to be able to detect and reject astrophysical foreground sources by their spectral signatures.Because pairwise correlation requires multiplying N ( N − / B -mode polarization. The technique is com-patible with either coherent receivers (amplifiers) or incoherentdetectors (bolometers).In a 2-element adding interferometer the electric field wave-fronts from both antennas are added and then squared in a detec-tor (Rohlfs & Wilson 2004). ( See Fig. 1.) The result is a constantterm proportional to the intensity plus an interference term. Theconstant term is an o ff set that is removed by phase-modulatingone of the signals. Phase-sensitive detection at the modulationfrequency recovers both the in-phase and quadrature-phase inter-ference terms and reduces susceptibility to low-frequency drifts(1 / f noise) in the detector and readout electronics. The addinginterferometer recovers the same visibilities as a multiplying in-terferometer: V ( u ) = Z Z I ( ˆn , ν ) G ( ˆn ) e i π u · ˆn J ( ν ) d ν d ˆn (1)where I ( ˆn , ν ) ∝ | E ( ˆn , ν ) | is the incident intensity and E ( ˆn , ν ) isthe incident electric field as a function of position of the sourceon the sky, ˆn , and frequency, ν . G ( ˆn ) is the primary beam (power)pattern of the antennas (assumed identical and for simplicity as-sumed independent of frequency). u is the vector between thecenters of the antenna apertures, measured in wavelengths, andhas components u and v. The frequency bandpass of the instru-ment is J ( ν ).To combine signals from N > ff erences (and therefore phase di ff erences) forrays traveling from a source to a detector: one path di ff erence oc-curs outside the instrument, and the other inside the instrument.Compare this to a conventional interferometer (also shown in 1-d, though the extension to 2-d is straightforward) as shown inFig.(4), where rays only undergo a phase di ff erence before theyenter the antennas. (Note that long-baseline optical interferome-ters usually include a ‘delay line’ between the apertures and thebeam combiner. This element introduces an equal path length forall rays entering an aperture. In contrast, the internal path di ff er-ences we are concerned with are di ff erent for each pixel in thefocal plane.)Let us explore what this combination of path di ff erencesachieves. We start by noting that the ‘external’ phase di ff erences,which are present in any interferometer, are the reason that thevisibility function is a Fourier transform of the image on the sky.The visibility measured by a single baseline essentially selectsone Fourier mode from the image. In the Fizeau system, we havean additional set of phase di ff erences. Without loss of general-ity, we may assign a negative sign to the phases introduced inside iddharth S. Malu and Peter T. Timbie: Broadband Fizeau Interferometers for Astrophysics 3 the instrument. Now, if we sum over both the phases, we get aFourier transform followed by an inverse Fourier transform - butthis is the image itself! Thus, Fizeau combination enables imag-ing in an interferometer. The image formed in the focal plane ofthe Fizeau combiner is equivalent to the ‘dirty image’ measuredby conventional radio interferometers. This topic is discussedlater in this paper in Appendix A.In addition, we show that, given enough detectors on the fo-cal plane, we can extract some spectral information from the in-terference fringes and determine the visibilities in several sub-bands. The Fizeau system enables extraction of spectral infor-mation via geometry, without additional components like filters.Without spectral information an interferometer with a largebandwidth su ff ers from a large radial width of each pixel in theu-v plane. This is shown in Fig.(3). Let ν be the center frequencyand ∆ ν the bandwidth. Then, a baseline of length B will measurethe CMB power spectrum over a band centered on spherical har-monic number ℓ = π B λ = πν Bc = π √ u + v (2)where the width in ℓ -space is ∆ ℓ = π ∆ ν Bc . (3)As mentioned above, the additional spectral information that isavailable to us can be used to sub-divide the band in the u-vplane. We discuss this aspect in detail in § ff erentbaselines can be distinguished from each other (Charlassier et al.2009; Hyland et al. 2009). In principle this technique could workwith multimode horns or antennas as well.
3. The Fizeau combiner output and its relation tovisibilities
In this section, we study the output of the adding interferometerand its dependence on instrument parameters: number of detec-tors on the focal plane, number of antennas, etc. and relate thisoutput to visibility from an interferometer, in order to describebandwidth splitting in the following section. In what follows, wedenote the output at the detectors as O . A simple adding interfer-ometer with two antennas / apertures (single baseline) is shown inFig.(1). A generalized Fizeau adding interferometer is shown inFig.(5). In an adding interferometer, electric fields are added and thensquared at the detector. If E ( ˆn , ν ) and E ( ˆn , ν ) are electric fields Fig. 1.
Schematic of an adding interferometer with N = A the electric field is E , and at A it is E exp i ϕ , where ϕ = π u · ˆn and | u | = | B | / λ . | B | is the length of the baseline, and ( B · ˆn ) / | B | is the angle of the source with respect to the symmetry axis of the base-line. (For simplicity consider only one wavelength, λ , and ignore timedependent factors.) In a multiplying interferometer the in-phase outputof the correlator is proportional to E cos ϕ . For the adding interferom-eter, the output is proportional to E + E cos( ϕ + ∆ ϕ ( t )). Modulationof ∆ ϕ ( t ) allows the recovery of the interference term, E cos ϕ , which isproportional to the visibility of the baseline. Fig. 2.
Block diagram of an adding interferometer with N >
2. Eachphase shifter is modulated in a sequence that allows recovery of theinterference terms (visibilities) by phase-sensitive detection at the de-tectors. The signals are mixed in the beam combiner and detected. Thebeam combiner can be implemented either using guided waves (e.g. ina Butler combiner) or quasioptically (Fizeau combiner), as above. Thetop triangles represent corrugated conical horn antennas. For the caseof an interferometer using coherent receivers, amplifiers and / or mixerscould be placed before the beam combiner. incident at antennas 1 and 2 respectively, then, denoting the pri-mary beam of the outward-facing antennas by G ( ˆn ), an addinginterferometer will detect G ( ˆn ) | E ( ˆn , ν ) + E ( ˆn , ν ) | (4)from a certain direction ˆn . Now E ( ˆn , ν ) and E ( ˆn , ν ) di ff er onlyby a phase factor ϕ = π B · ˆn / λ , so that we can write the electricfields as E ( ˆn , ν ) and E ( ˆn , ν ) exp( i ϕ ), and the output at a single Siddharth S. Malu and Peter T. Timbie: Broadband Fizeau Interferometers for Astrophysics frequency for the simple arrangement shown in Fig.(1) is G ( ˆn ) | E ( ˆn , ν ) + E ( ˆn , ν ) exp( i ϕ ) | = G ( ˆn ) | E ( ˆn , ν ) | (1 + cos ϕ ) (5) ≡ G ( ˆn ) | E ( ˆn ) , ν | (cid:0) + ℜ [exp( i ϕ )] (cid:1) (6) E ( ˆn , ν ) ∝ I ( ˆn , ν ) where I ( ˆn , ν ), the intensity, is a linear combi-nation of Stokes parameters (see Lin & Wandelt 2006, e.g.). Toisolate the interference term, a time–varying phase modulationis applied, as shown in Fig.(1). Then, the demodulated output isgiven by G ( ˆn ) I ( ˆn , ν ) cos ϕ ≡ G ( ˆn ) I ( ˆn , ν ) ℜ [exp( i ϕ )] (7)where ϕ = π B · ˆn / λ = π u · ˆn . The general arrangement of a Fizeau interferometer is shown inFig.(5). Let B k be the baseline formed by antennas at A p and A l , such that B k = A p − A l ( k ∈ [1 . . . N ( N − /
2] and p , l ∈ [1 . . . N ]), and ˆn is a direction in the sky, as shown in Fig.(5).Then, u k = B k / λ , so that the external path di ff erence is ( A p − A l ) · ˆn = B k · ˆn , as shown in Fig. (5).The di ff erence between this arrangement and that of the sim-ple adding interferometer in Fig.(1) is the set of internal phasedi ff erences introduced inside the instrument due to the fact thatthere is more than one detector, and the geometry of the arrange-ment. Let x j denote the position of the j th detector on the focalplane. Then, the internal path di ff erence between the rays fromantennas p and l , (which form baseline k ), is x jk ( x ) = x j · B k asshown in Fig.(5). Thus, the intensity at the detector at x j con-tributed by baseline k is G ( ˆn ) I ( ˆn , ν ) ℜ [exp i ( ϕ k + πλ x jk )] . (8)where ϕ k = π B k · ˆn / λ = π u k · ˆn .If we denote the output at the j th detector from the k th base-line as O jk , and integrate over all directions in the sky and thebandwidth ∆ ν , we get O jk = Z Z G ( ˆn ) I ( ˆn , ν ) ℜ [exp i ( ϕ k + πλ x jk )] J ( ν ) d ˆn d ν, (9)where J ( ν ) is a bandpass weighting. There is an integration overdetector area as well in eq.(9), which will be implicit until § ff ect of the detector area and make asuitable and practical approximation . Notice that x jk does notdepend on the direction ˆn . For now, we make the (crude) ap-proximation that λ ≡ λ (the central wavelength), so that this‘internal phase factor’ may be taken out of the integral over ν as well. As discussed in § ffi cient way to deal with this issue in § O jk = ℜ " exp( i πλ x jk ) × Z Z G ( ˆn ) I ( ˆn , ν ) exp (cid:2) i ( ϕ k ) (cid:3) J ( ν ) d ν d ˆn (10) = ℜ " exp( i πλ x jk ) × Z Z G ( ˆn ) I ( ˆn , ν ) exp [ i π u k · ˆn ] J ( ν ) d ν d ˆn | {z } . (11)The quantity indicated by the underbrace in eq.(11) is the Visibility (defined in eq.(1)) from the k th baseline, V ( u k ) ≡ V k .We can now denote the ‘internal phase di ff erences’ (2 π/λ ) x jk as φ jk , so that O jk = ℜ h exp( i φ jk ) V k i . (12)In Appendix A, we show that by integrating over the entire focalplane, we can recover the image convolved with a “dirty beam”,as in radio interferometry.In the next section, we consider the net signal from a singlebaseline and describe how it can be split into ‘sub–bands.’ Forthe sake of simplicity, the index k corresponding to baseline k isdropped, and all equations in §
4. Spectral information from an interferometerusing a Fizeau approach
The output measured at the detectors in a Fizeau interferome-ter contains the following phase information integrated over theentire bandwidth:1. phase introduced because of the path di ff erence between anytwo rays that arrive from the same part of the sky on the twooutward-facing antennas that make up a baseline; and2. phase introduced because of the path di ff erence between anytwo rays that arrive from two di ff erent antennas on to thesame point in the focal plane.The phase in point 1 is due to the fact that we are considering aradio interferometer, and so the visibility that we measure must,by definition, include this phase. However, the phase in 2 aboveintroduced by the beam combiner needs to be factored out torecover visibility from each bolometer. If there were a way tocalculate the ‘internal phase’ introduced by the beam combinerover the whole bandwidth, then all we would need to do is todivide the output at each point in the focal plane by this ‘internalphase’, and we would get visibility directly . We can think of the e ff ect of the instrument on the visibilitiesin the following way. Let us divide the entire bandwidth of the A range of values of ν will produce a range of ℓ ’s, or a band in ℓ -space (as discussed in §
2, see eqs.(2,3)). A finite-bandwidth interfer-ometer thus measures what is called a ‘bandpower’ instead of a singlevalue of the power spectrum at one value of ℓ . But the power spec-trum is just the variance of the visibilities for a circle (ring) in the u–vplane (see Malu 2007; Charlassier et al. 2010, Fig. 5.8, Figs. 1&2 re-spectively). And so we get di ff erent bandpowers for the same baselineand orientation but for di ff erent frequencies.iddharth S. Malu and Peter T. Timbie: Broadband Fizeau Interferometers for Astrophysics 5 instrument into m sub-bands and let ν , ν ... ν m be the centre-frequencies of each one. Then for one baseline, one orientation,and one detector position, these will correspond to visibilities V , V ... V m and to phase di ff erences φ j , φ j ... φ jm (where j rep-resents the detector). If we represent the output at the j th detectoras O j then we get (as in the previous section, but dropping theindex k for baseline, since we’re considering just one baseline): O j = m X α = ℜ h V α exp i φ j α i . (13)Given just one detector, it is impossible to extract every V α forevery sub-band, even though we know precisely what the φ j α ’sare. However, if we have m detectors, then we can easily writethe following system of equations for each baseline: O = ℜ [ V exp i φ + V exp i φ + · · · + V m exp i φ m ] O = ℜ [ V exp i φ + V exp i φ + · · · + V m exp i φ m ] . . . O m = ℜ [ V exp i φ m + V exp i φ m + · · · + V m exp i φ mm ] . (14)Now, as discussed in the preceding section and in Fig.(1), wecan apply a unique phase shift to each baseline. If we denotethis phase shift by ∆ ϕ (it is understood that ∆ ϕ = ∆ ϕ ( t )) and theoutput after applying the phase shift as O ′ . . . O ′ m , then we getanother set of m equations: O ′ = ℜ [ V exp i ( φ + ∆ ϕ ) + · · · + V m exp i ( φ m + ∆ ϕ )] O ′ = ℜ [ V exp i ( φ + ∆ ϕ ) + · · · + V m exp i ( φ m + ∆ ϕ )] . . . O ′ m = ℜ [ V exp i ( φ m + ∆ ϕ ) + · · · + V m exp i ( φ mm + ∆ ϕ )] . (15)This is a system of 2 m equations with 2 m unknowns - ℜ [ V ], ℜ [ V ] . . . ℜ [ V m ] . . . ℑ [ V ], ℑ [ V ] . . . ℑ [ V m ], and so we can solvefor the values for each one of these ‘sub-band visibilities’. Thebeam combiner thus achieves far more than just separating thereal and imaginary parts of visibilities. We mentioned in § / estimated. However, calcu-lating this internal phase is not easy, since integration over thebandwidth complicates the calculation, as seen in § § ff erences inthe Fizeau interferometer create fringe patterns on the focalplane. This is exactly the same as saying that the visibilitiesin each sub-band are modulated by a fringe which depends onbaseline length. To extract these visibilities, we need to separatethe fringes. In order to do so, we need to realize that what we ob-serve at every detector is the visibility on the sky times the fringesummed over the area of the detector as well as bandwidth.The fringe pattern is di ff erent for every frequency in thebandwidth. Visibility is also di ff erent for di ff erent frequencies.Since we can compute the fringe corresponding to each fre-quency, it is also straightforward to sum up these fringes overa small ‘sub-band’ over the area of a single detector. The outputat each detector from a baseline is known, and this can be writtenas a sum over the product of visibility for a ‘sub-band’ and thefringe for the respective ‘sub-band’. This system of equationscan be solved for each baseline to yield the sub-band visibilities. We have demonstrated this in § ff ectively, this amounts tonot using the crude approximation in § ff ect of the fringe pattern. Let us account for these e ff ects inthe following way. Let A be the e ff ective collecting area of eachdetector. Let f ( x , ν α ) be the value of the fringe pattern (see dis-cussion at the beginning of this section) at a point on the focalplane x and in a frequency sub-band marked by α . Then, equa-tions (14,15) become O j = m X α = ℜ "Z V α exp i φ j α ( x ) f ( x , ν α ) d x O ′ j = m X α = ℜ "Z V α exp i ( φ j α ( x ) + ∆ ϕ ) f ( x , ν α ) d x (16)where it is understood that integration is done over the area ofthe detector.This leaves us with an issue - that of deconvolving the V ’sfrom the integrals. However, if the area of the detector is smallcompared to the width of fringes, then we can assume that thephase di ff erences remain roughly constant over the collectingarea of one detector, so that we may write O j = A m X α = ℜ h V α exp i φ j α ( x ) F ( x , ν α ) i O ′ j = A m X α = ℜ h V α exp i ( φ j α ( x ) + ∆ ϕ ) F ( x , ν α ) i (17)where F ( x , ν α ) represents an “average” value of the fringe pat-tern.Equations (17) again have 2 m variables and can be solved toget 2 m quantities: the real and imaginary parts of m visibilitiesover the bandwidth. Application to CMB cosmology:
The QUBIC collabora-tion is implementing the technique described in this paper(Charlassier et al. 2010).
5. Conclusions
1. The Fizeau system makes it possible to recover spectral in-formation without the need for filters.2. The Fizeau system acts naturally as an imager.In addition, by introducing phase modulators discussed in(Hyland et al. 2009; Charlassier et al. 2009), we can measurevisibilities for all baselines in a Fizeau system.While it is possible to divide the bandwidth into many di ff er-ent sub-bandwidths, it isn’t possible to do this indefinitely. Thebeam for a single antenna determines the FOV of the instrumentand limits the resolution in the u–v plane, as shown in Fig.(3).It is also possible to operate the interferometer simultane-ously as an imager. The additional modulation mentioned aboveopens up a range of possibilities, including the simultaneousmeasurement of visibilities and images. This is described inAppendix A.In conclusion, the Fizeau system introduced here is poten-tially powerful tool for astrophysics: it could allow the recov-ery of more information than is possible with traditional inter-ferometers or imagers and does not need significantly more re- Siddharth S. Malu and Peter T. Timbie: Broadband Fizeau Interferometers for Astrophysics sources to build. Its application in CMB cosmology is straight-forward and can be demonstrated in future version of QUBIC(Hamilton & Charlassier 2010).
Acknowledgements.
We thank the members of the MBI and QUBIC collabora-tions for many fruitful discussions on bolometric interferometry.
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The u–v plane coverage of one baseline of an interferom-eter for a single pointing in a single baseline orientation angle.Radial spread in a single pixel in the u–v plane due to bandwidthis shown. Resolution in the u–v plane is determined by the pri-mary beam or size of field-of-view. The minimum size of eachsub–band is also determined by this resolution, as shown. iddharth S. Malu and Peter T. Timbie: Broadband Fizeau Interferometers for Astrophysics 7
Fig. 4.
A Fizeau adding interferometer used as a beam combiner for long-baseline optical interferometry (figure reproduced fromTraub (2000)). The baseline B is the separation of the apertures, which can be widely spaced, and determines the angular resolutionof the instrument. The Fizeau beam combiner has a di ff erent and much smaller baseline length (not labeled). The‘external’ phasedi ff erences are marked B · ˆn . Fig. 5.
A simple 1-d Fizeau system, similar to a figure in Traub (2000) . O denotes the origin of the co-ordinate system, and is thecenter of the observation plane. x i ’s denote positions of detectors on the focal plane, and A l ’s denote positions of apertures, where l ∈ [1 . . . N ]. Notice that there are two sets of phase di ff erences, marked ( A p − A l ) · ˆn ≡ B k · ˆn and x jk ( x ) = x j · ( A p − A l ) for externaland internal phase di ff erences respectively. The geometrical variation of x jk ( x ) = x j · ( A p − A l ) allows “sub–band splitting”. Siddharth S. Malu and Peter T. Timbie: Broadband Fizeau Interferometers for Astrophysics
Appendix A: Imaging in a Fizeau system
In eq.(9), if we now integrate over the focal-plane area, omittingthe integral over bandwidth, which is implicit (and rememberingthat ϕ ≡ π B · / λ ): O = Z Z ℜ G ( ˆn ) I ( ˆn , ν ) exp " i2 π B · ˆn − x i j λ d ˆn d x , (A.1)where we have changed the sign on the ‘internal’ phase di ff er-ences. This can be done without loss of generality, since theinternal and external phase di ff erences are independent of eachother.Let us consider just one term in the expression ℜ ( ... ): O = Z Z I ( ˆn , ν ) exp " i2 π B · ˆn λ d ˆn exp (cid:20) − i2 π x i j λ (cid:21) d x (A.2)If we include the e ff ect of the primary beams of the inward–facing antennas ( G ( x )) and adopt I = I ( ˆn , ν ), O = Z G ( x ) Z G ( ˆn ) I exp " i2 π B · ˆn λ d ˆn | {z } exp (cid:20) − i2 π x i j λ (cid:21) d x (A.3)The quantity in underbrace is clearly a fourier transform, and theexpression can be written as O = Z G ( x ) F ( GI ) exp (cid:20) − i2 π x i j λ (cid:21) d x (A.4)If the distance from the inward-facing antennas to the focal plane ≫ the collecting area for each bolometer, O = F − ( G F ( GI )) (A.5)The beam needs to be deconvolved from the above expression inorder to obtain an image from the instrument.Now, eq(A.4) can be split up over the focal plane: O = N X i = Z i G F ( GI ) exp (cid:20) − i2 π x i j λ (cid:21) d x (A.6)where 1 . . . N are labels for bolometers on the focal plane.Each of the bolometer outputs then represents a pixel in im-age space. The total number of pixels depends on the resolutionof the instrument, and not the number of bolometers on the focalplane. Therefore, if the number of bolometers on the focal planeare greater than the number of pixels in the image, we need to“repixelize” the image obtained, so that all pixels are indepen-dent of each other.In general, this is how the beam is convolved with the imageon the sky for the Fizeau beam combiner: O = F − ( G F ( GI )) (A.7) = h(cid:16) F − G (cid:17) ∗ ( GI ) i (A.8)In traditional interferometry, eq.(A.7) would read O = F − ( F ( GI )) (A.9) = F − ( F G ) ∗ ( F I ) (A.10) ≡ GI (A.11)In eq.(A.10), the u–v space beam F G needs to be multi-plied by u–v coverage, which is a “mask”, say M ( u , v ). Then, F − ( F G × M ( u , v )) is called the “dirty beam” in traditional in-terferometry. In eq.(A.7), the factor M ( u , v ) is included. Eq.(A.8) thus tells us that the dirty beam for the image produced by theFizeau combiner is more involved than the traditional interfer-ometer dirty beam, but remains conceptually equivalent.There are two assumptions inherent in the foregoing discus-sion:1. The focal plane is large enough to receive most of the powerfrom the inward-facing antennas2. There are no “blank” areas on the focal plane for which theincident power is not absorbed by a bolometerIt is possible to detect the correlated signal from a pair of an-tennas as well. In order to separate every unique baseline, time–varying phase–shifts can be applied to the signal at the base ofthe skyward–facing antennas. The “correlated signal” from eachpair of antennas is simply the visibility from a baseline, with onecrucial di ff erence: there is an “internal phase” added to every vis-ibility due to the relative positions of antennas on the observationplane and detectors on the focal plane. These phase di ff erencesare geometrical. Since the output from all N antennas is inci-dent on every detector, we can say that the output from a singledetector contains information about N ( N − //