Issue 
A&A
Volume 550, February 2013



Article Number  A59  
Number of page(s)  11  
Section  Astronomical instrumentation  
DOI  https://doi.org/10.1051/00046361/201220429  
Published online  28 January 2013 
Selfcalibration: an efficient method to control systematic effects in bolometric interferometry
^{1}
APC, Astroparticule et Cosmologie, Université Paris Diderot,
Bâtiment Condorcet, 10 rue Alice Domon et Léonie
Duquet,
75205
Paris Cedex 13,
France
email: mabigot@apc.univparis7.fr
^{2}
TuringSolutions, 25 rue Dauphine, 75006
Paris,
France
Received:
21
September
2012
Accepted:
4
December
2012
Context. The QUBIC collaboration is building a bolometric interferometer dedicated to the detection of Bmode polarization fluctuations in the cosmic microwave background.
Aims. We introduce a selfcalibration procedure related to those used in radiointerferometry to control a wide range of instrumental systematic errors in polarizationsensitive instruments.
Methods. This procedure takes advantage of the need for measurements on redundant baselines to match each other exactly in the absence of systematic effects. For a given systematic error model, measuring each baseline independently therefore allows writing a system of nonlinear equations whose unknowns are the systematic error model parameters (gains and couplings of Jones matrices, for instance).
Results. We give the mathematical basis of the selfcalibration. We implement this method numerically in the context of bolometric interferometry. We show that, for large enough arrays of horns, the nonlinear system can be solved numerically using a standard nonlinear leastsquares fitting and that the accuracy achievable on systematic effects is only limited by the time spent on the calibration mode for each baseline apart from the validity of the systematic error model.
Key words: instrumentation: polarimeters / instrumentation: interferometers / cosmic background radiation / inflation / methods: data analysis
© ESO, 2013
1. Introduction
The quest for the Bmode of the polarization of the cosmic background is one of the scientific priorities of observational cosmology today. Observing this mode appears to be the most powerful way to constrain inflation models. However, detecting such a weak signal is a real experimental challenge. In addition to a high statistical sensitivity (a huge number of horns and bolometers required), future experiments will need excellent quality of foreground removal and unprecedented control of instrumental effects.
Currently, most projects are based on the experimental concept of an imager. A promising alternative technology is bolometric interferometry. This is the project of the QUBIC instrument (QUBIC collaboration 2010). A first module is planned for installation at the FrancoItalian Concordia Station in Dome C, Antarctica in 2014. The aim is to combine the advantages of an imager in terms of sensitivity with those of an interferometer in terms of controlling systematic effects. The statistical sensitivity of the QUBIC instrument is comparable to that of an imager with the same number of horns covering the same sky fraction. The full QUBIC instrument (six modules) will comprise three frequencies (97, 150, and 220 GHz) and aims to constrain, at the 90% confidence level, a tensortoscalar ratio of 0.01 with one year of data.
The aim of this article is to introduce a new method, specific to bolometric interferometry, called selfcalibration, and to give an example of application with the QUBIC instrument. This method allows a wide range of instrumental systematic effects to be controlled.
This selfcalibration technique is based on the redundancy of the receiver array (Wieringa 1991). It uses the need for redundant baselines of the interferometer to measure exactly the same quantity in the absence of systematic effects. For a real instrument, these measurements will be different because of systematics. The small differences can be used to calibrate parameters that characterize the instrument completely for each channel and estimate the instrumental errors.
In the case of a bolometric interferometer, the square horn array will provide a large number of redundant baselines. In this way, a bolometric interferometer can be selfcalibrated thanks to a calibration mode during which it will separately measure the n_{h}(n_{h} − 1)/2 baselines or a fraction of the n_{h}(n_{h} − 1)/2 baselines with n_{h} the number of horns observing an external polarized source.
This method is inspired by traditional interferometry (Pearson & Readhead 1984) where signal phases are often lost due to atmospheric turbulence. Standard calibration procedures exist in radiointerferometry and are similar to those used for imaging techniques based on observations of an unresolved source whose flux is assumed to be known. We emphasize that the word selfcalibration refers, by opposition, to a procedure in which no knowledge of the observed source is required (most of the time, the object which is scientifically studied is itself used as the calibration source). Most of these selfcalibration techniques are based on the evaluation of socalled closure quantities phases or amplitudes. A set of unknown phases can, for instance, be iteratively reconstructed by forming quantities where they are nullified (the product of the three visibilities that can be formed with three antennas). The use of redundant baselines for calibration is in contrast rather uncommon in radiointerferometry (see Wieringa 1991; Noordam & de Bruyn 1982). This comes from most radiointerferometers having very few redundant baselines achieve very high angular resolution, it is indeed better to arrange a given number of antennas in order to optimize the uvplane sampling, rather than to maximize redundancy.
A new kind of alldigital radiointerferometer the “omniscope”, dedicated to 21 cm observations, has recently been proposed by Tegmark & Zaldarriaga (2009; 2012); its concept can be summarized by the following five steps:

1)
signals collected by antennas are digitized right afteramplification,

2)
a temporal fast Fourier transform (FFT) is performed in order to split them into frequency subbands,

3)
a spatial FFT is performed on each subband set,

4)
the square modulus of the FFT result is computed,

5)
an inverse spatial FFT is performed to recover the visibilities.
There are conceptual similarities with the bolometric interferometer concepts, since in this instrument, steps two and three are performed in an analogical way by the beam combiner and the bolometers, respectively. Because antennas have to be located on a grid in order for the FFTs to be performed, an omniscope will possess many redundant baselines, and Liu et al. (2010) have shown how this allows selfcalibrating the complex gains of the antennas. In both cases (standard radiointerferometer or omniscope), the aim is to calibrate the complex gains of the antennas and there is actually a mathematical trick to get a linear system of equations from which these gains can be obtained. The selfcalibration procedure explained in this paper is inspired by methods used in the paper of Liu et al. (2010) even if the design of the QUBIC instrument is different.
This paper is organized as follows. Section 2 introduces the selfcalibration method with the Jones matrix formalism and the Mueller matrix formalism for radiointerferometry and the omniscope. We show in Sect. 3 how the procedure can be applied to the QUBIC bolometric interferometer, and finally we describe the possible selfcalibration algorithm and its results.
2. General principle
2.1. Instrumental systematics modelization with Jones matrices
In this section, we use the notation proposed by (O’Dea et al. 2007). With the electric field of an incident radiation at a frequency φ defined as , and choosing two basis vectors and orthogonal to the direction of propagation , all the statistical information is encoded in the coherence matrix C(1)where E_{x}, E_{y} are complex amplitudes of the transverse electric field and I, Q, U, and V are the Stokes parameters.
The propagation of an incident radiation through a receiver can be described by a Jones matrix J such that the electric field after passing through the receiver is (2)where the Jones matrix J is a 2 × 2 complex matrix. It describes how the instrument linearly transforms the twodimensional vector representing the incoming radiation field into the twodimensional vector of the outgoing field .
For an instrument with several components, the Jones matrix is the product of the Jones matrices for each component. For example, the ideal Jones matrix for an instrument in which the incident radiation passing through a rotating halfwave plate before propagating through the horns is (3)where ω is the angular velocity of the halfwave plate, J_{rot} the rotation matrix and J_{hwp} the ideal Jones matrix of the halfwave plate.
After passing through the receiver, ideal orthogonal linear detectors measure the power in two components To model systematic errors within a polarizationsensitive interferometer, the Jones matrix can be described by introducing diagonal terms: the complex gain parameters g_{x} and g_{y} and nondiagonal terms: the complex coupling parameters e_{x} and e_{y} associated to the orthogonal polarizations (4)Systematic errors arising from the halfwave plate and from the square horn array can be modeled by
The electric field propagated through the halfwave plate and the horn i becomes (7)
2.2. Instrumental systematics modelisation with Mueller matrix
The Jones matrix expresses the transformation of the electric field in the x and ydirections and the Mueller matrix describes how the different polarization states transform. The Jones matrix is a 2 × 2 matrix, whereas the Mueller matrix is a 4 × 4 matrix. The 4 × 4 Mueller matrix can thus be written as the direct product of the 2 × 2 Jones matrices.
We calculate the tensor product of the outgoing field given by Eq. (2) (8)In general, the direct product gives the vector defined as (9)With Eq. (8), one can write the transmission of the electric field through an instrument described by its Jones matrix using the vector of this electric field using the matrix direct product^{1}
(10)Accordingly Eq. (1), the polarization state of this electric field can be described by the Stokes vector defined by (11)where I, Q, U, and V are the Stokes parameters.
One can obtain the expression of the Stokes vector from the vector defined in Eq. (9) (12)where Substituting Eq. (12) into Eq. (10), it follows that the outgoing Stokes vector can be written as (13)where M = A(J ⊗ J^{ ∗ })A^{1} is the Mueller matrix that describes how the Stokes parameters transform.
2.3. Polarized measurement equation with Mueller formalism
A polarizationsensitive interferometer measures the complex Stokes visibilities from all baselines defined by the horns i and j in an array of receivers (14)These vectors could reduce to a scalar or a vector with 2, 3, or 4 elements depending on the Stokes parameters the instrument is sensitive to. One can define a = 1,2,3,4 as the number of Stokes parameters the instrument allows to be measured.
The n_{h}(n_{h} − 1)/2 baselines of the interferometer can be classified into sets s_{β} of redundant baselines (same length, same direction) indexed by β. In the absence of systematic errors, the redundant visibilities should have exactly the same values (15)For a real instrument, however, redundant visibilites will not have exactly the same values because of systematic errors and statistical (photon) noise, and one can write the following system of a × n_{h}(n_{h} − 1)/2 complex equations (16)where n_{ij} are statistical noise terms and where M_{ij} are a kind of complex Mueller matrices that reduce to the identity matrix for a perfect instrument: (17)The elements of these matrices are not independent and can be expressed in terms of the diagonal and the nondiagonal terms of the Jones matrix.
The first order Mueller matrix for a polarization sensitive experiment is (18)with (19)(20)(21)(22)
3. Application to the QUBIC bolometric interferometer
3.1. Observables in bolometric interferometry
In this section, we derive the expression for the power received in the focal plane in the case of bolometric interferometry. The bolometric interferometer proposed with the QUBIC instrument (the QUBIC collaboration 2010) is the millimetric equivalent of the first interferometer dedicated to astronomy: the Fizeau interferometer − see Fig. 1 for the design of the QUBIC instrument.
The receptors are two arrays of n_{h} horns: the primary and secondary horns backtoback on a square grid behind the optical window of a cryostat. Filters and switches are placed in front and between the horn array. The switches will be only used during the calibration phase. The polarization of the incoming field is modulated using a halfwave plate located before the primary horns. This location of the halfwave plate avoids a leakage from the Stokes parameter I to the Stokes parameters Q and U if the halfwave plate has no inhomogeneities.
Signals are correlated together using an optical combiner. The interference fringe patterns arising from all pairs of horns, with a given angle, are focused to a single point on the focal plane. Finally, a polarizing grid splits the signal into x and ypolarizations, each being focused on a focal plane equipped with bolometers. These bolometers measure a linear combination of the Stokes parameters modulated by the rotating halfwave plate.
With an interferometer, the correlation between two receivers allows for direct access to the Fourier modes (visibilities) of the Stokes parameters I, Q, and U. In the case of a bolometric interferometer, the observable is the superposition of the fringes formed by the sky electric field passing through a large number of backtoback horns and then focused on the detector plane array. The image on the focal plane of the optical combiner is the synthesized image, because only specific Fourier modes are selected by the receiving horns array. A bolometric interferometer is therefore a synthesized imager whose beam is the synthesized beam formed by the array of receiving horns.
Fig. 1 Design of the QUBIC instrument. 
In the following section, we begin by defining the formalism for the QUBIC instrument in an ideal case, that is to say, without systematics. The electric field passing through a halfwave plate that rotates at angular speed ω and modulates the two orthogonal polarizations η ∈ {x,y} becomes, as a function of the observed direction , (23)For an ideal instrument, the electric field, after passing through the halfwave plate, collected by one horn i located at for the polarization η ∈ {x,y} when all the primary horns are looking at the same radiation field as a function of the observed direction and reaching the bolometer q at the direction viewed from the optical center of the beam combiner is (24)where is the product matrix of the matrices α_{iq} and .
We have introduced the matrices and α_{iq} which characterize completely the instrument. A matrix is defined for each channel of pointings (p) and horns (i) and includes the primary beam , the horn position and the direction of pointing (25)with (26)and (27)A matrix α_{iq} is defined for each channel of horns (i) and bolometers (q) and includes the geometrical phases induced by the beam combiner, the beams of the secondary horns , and the gain g_{q} of the bolometer^{2}(28)with (29)and (30)The detector direction as viewed from the optical center of the optical combiner is given by the unit vector , and λ is the wavelength of the instrument. The integrations are over the surface of each individual bolometer modeled with the tophat like function ^{3}, and on bandwidths of the instrument J(ν) with ν the frequency.
We assume that there is no crosspolarization because the matrices and α_{iq} are written for an ideal case so that there is no dependence on the direction of polarization. The power measured by one polarized bolometer located at in the focal plane of the beam combiner is then (31)where is given in Eq. (24).
Without systematics, Eq. (31) can be written as (32)where the synthesized beam for the detector q is formed by the arrangement of the primary horns array as (33)The synthesized beam depends on the sky direction , so the synthesized image is the convolution of the sky and of the electric field through the synthesized beam.
One can rewrite Eq. (31) to exhibit the modulation of the polarization induced by the halfwave plate as (34)where ϵ^{x} = 1 for the polarization x, ϵ^{y} = −1 for the polarization y, and where are the synthesized images on the focal plane for each Stokes parameter X = {I,Q,U}. The cosine and sine coefficients come from the modulation induced by the rotating halfwave plate.
Systematic effects arising at any level of the detection can be modeled by associating a Jones matrix to each horn i J_{horn,i} and a Jones matrix of the halfwave plate J_{hwp}. They can be introduced as defined in Eq. (7), and Eq. (24) becomes (35)where is the incident electric field, J_{horn,i} the Jones matrix of the horn i, J_{hwp} the Jones matrix of the halfwave plate, and J_{rot} the rotation matrix induced by the rotation of the halfwave plate.
3.2. Selfcalibration procedure
During the selfcalibration mode, which is distinct from the ordinary datataking mode, the instrument scans a polarized source and measures the n_{h}(n_{h} − 1)/2 synthesized images one by one from all baselines or only a fraction of them. In the QUBIC design, this can be achieved using switches located between the backtoback horns. The switches are used as shutters that are operated independently for all channels and are only required during the calibration phase. One can modulate on/off a single pair of horns while leaving all the others open in order to access the synthesized images measured by each pair of horns alone. This procedure requires the knowledge of the individual primary beams of each horn. The maps of the primary beams can be obtained independently through scanning an external unpolarized source.
By repeating this with all baselines, all bolometers, and all directions of pointing, one can construct a system of equations whose unknowns are

1.
the complex coefficients , defined for each horn i, each bolometer q, and for each polarization η which correspond to 4n_{h}n_{q} parameters,

2.
the horn location (2n_{h} parameters),

3.
the direction of pointing (2n_{p} parameters),

4.
the complex horns systematic effects g_{x,i}, g_{y,i}, e_{x,i}, e_{y,i} defined for each horn i and for each polarization η (8n_{h} parameters),

5.
the complex halfwave plate systematic effects h_{x}, h_{y},ξ_{x},ξ_{y} (8 parameters).
For an instrument with n_{h} horns, n_{q} bolometers and for a scan of n_{p} pointings, the number of unknowns is (36)During the selfcalibration procedure, the number of constraints is given by the measurements, i.e. the synthesized images. One has the n_{h}(n_{h} − 1)/2 measured synthesized images for each bolometer n_{q}, each pointing n_{p}, each Stokes parameter, and the two focal planes. The number of constraints is given by (37)The problem becomes easily overdetermined: for an instrument with n_{h} = 9, n_{q} = 4, and n_{p} = 10, the number of constraints is 8640 and the number of unknowns is 262. It can be solved with a least squares algorithm.
The first module of the QUBIC instrument will consist of 400 primary horns and two 1024 element bolometer arrays. The number of constraints could be reduced if the selfcalibration is performed not on the n_{h}(n_{h} − 1)/2 baselines but on a fraction of baselines. This is shown in the following.
Closing all the switches except two would actually dramatically change the thermal load on the cryostat, which could affect the bolometric measurements. Fortunately, there is a trick explained in Appendix A that allows to be indirectly measured while minimally changing the thermal load. One can show that (38)where is the quantity measured by a bolometer q when all the switches are open except the i and j, and , are the powers measured when all the switches are open except respectively i or j. Measuring these three terms therefore allows measuring while keeping the thermal load almost constant. However, this also increases the noise. The noise on each term is therefore where NET is the noise equivalent temperature of the bolometers, and T the temperature of the 100% polarized source.
3.3. Numerical simulation
We have numerically implemented the method to check if the nonlinear system could be solved. We generate the instrument with a set of ideal parameters (horn locations, directions of pointing, primary and secondary beams, detector locations, etc.), and a set of parameters randomly corrupted by systematic errors (horn location errors, pointing errors, assymetries of beams, bolometer location errors, diagonal and nondiagonal terms of the Jones matrices, etc.). The widths of the random deviation of all corrupted parameters around their ideal value are given in Table 1. These values of corruption are independent Gaussian errors added to each parameter and each range of error is fixed according to the tolerance we impose on components.
Range for systematic errors for each parameter.
The primary beams are assumed to be known perfectly so their values are not varied in the simulation; however, the matrix varies as the horn locations and the pointing directions are unknowns of the system. In the simulation, the angular position of the halfwave plate is drawn at random. A random angular position of the halfwave plate is given for each measurement.
To get a solvable system, one must add some normalization constraints for the coefficients and , which do not change the modeling of systematic errors. They mean that the selfcalibration only allows for relative calibration of these parameters. One can add

1.
an absolute calibration of the global gain of the instrument,.

2.
a convention on the phase of the coefficients, . A rotation of global phase φ_{q} applied to the coefficients for one bolometer q does not modify Eqs. (24) and (31) and therefore the observations.

3.
a convention on the primary beams, . Multiplying the coefficients by a term defined as c_{i}e^{φi} and dividing the coefficients by the same term does not modify Eqs. (24) and (31) and therefore the observations.

4.
a convention on the phase of the coefficients, . A rotation of global phase applied to the coefficients for one pointing p does not modify Eqs. (24) and (31) and the observations.

5.
an absolute calibration of the overall gain of the horns, g_{x,0} = 1. It means that the selfcalibration procedure only allows a relative calibration of the gain terms.
We compute the corrupted synthesized images and add Gaussian statistic noise given by (39)where the noise equivalent temperature of the bolometers NET is taken to be 300 μK s^{1/2}, the temperature of the 100% polarized source is T = 100 K, and the time spent on each baseline on the calibration mode is t_{b} = 1 s. The usual convention is to give the NET for unpolarized detectors, but it is convenient in our case to use quantities with polarization. In this case, the NET is given by .
We solve the nonlinear system with a standard nonlinear leastsquares method based on a LevenbergMarquardt algorithm. The ideal coefficients (without systematic errors) are used as starting guess for the different parameters.
Fig. 2 Results of the selfcalibration simulation for the synthesized images , , and for the X focal plane on the right and Y focal plane on the left for an instrument with nine primary horns, nine bolometers, and ten pointings for a time spent on calibration mode for each baseline of t_{b} = 1 s and 100 realizations. These plots represent scatter plots of ideal vs. real synthesized image in red and of recovered vs. real in blue. The red plots show the corruption after adding the systematic effects defined in Table 1. The blue plots show that the corruption is solved after applying the selfcalibration method. 
3.4. Results
To be able to have a large number of realizations, we run the simulation for an array of nine primary horns, nine bolometers, and ten pointings for 100 realizations. Figure 2 shows the result of the selfcalibration simulation. The six plots are scatter plots of ideal vs. real synthesized images in red and of recovered vs. real synthesized images in blue for the X and Y focal planes and each Stokes parameter I, Q, and U. The synthesized images are computed with Eq. (31), the ideal synthesized images are the synthesized images without systematic effects, the real synthesized images are the simulated measurements, and the recovered synthesized images are computed with the output parameters of the selfcalibration simulation. The six plots show the advantage of the selfcalibration method.
This method allows access to the systematic effects of the horns and of the halfwave plate. It also allows calibrating the parameters and that completely characterize the instrument for each channel of pointings, horns, and bolometers.
In running the simulation, one can find that the residual error on each output parameter will depend on the number of horns, bolometers, pointings, baselines per pointings, and on the time spent measuring each baseline. Adding more horns, pointings, bolometers, and baselines increases the mathematical constraints on a given measurement, it allows to form new baselines and adds redundancy on the horn array. Figure 3 shows that the residual diagonal term error of the Jones matrix of the halfwave plate improves as the number of baselines per pointing is higher. This result was obtained for a simulation with nine primary horns, nine bolometers and ten pointings for 40 realizations and for a time spent on each baseline t_{b} = 1 s. The baselines measured for each pointing are chosen randomly. Similar plots are obtained with the other parameters defined in Table 2.
One can put together these variables and define a power law that allows calculating the residual error on each parameter defined in Table 1(40)with c a constant and α, β, γ, Φ the exponent of the number of horns n_{h}, bolometers n_{q}, pointings n_{p}, and the percent of baselines per pointing n_{bs}.
The values for each index are summarized in Table 2 for two different measuring times per baseline t_{b} = 1 and t_{b} = 100 s.
Fig. 3 Results of the selfcalibration simulation for the diagonal terms of the Jones matrix of the halfwave plate and a time spent per baseline t_{b} = 1 s. This plot represents the residual error on these parameters as a function of the percent of baselines per pointing. The red line represents a power law of the shape with c a constant, n_{bs} the number of baselines per pointing and Φ the index of the power law given in Table 2. This law gives the limit of the relative accuracy that can be achieved on the systematic parameters. The green line represents the input error on the systematic effects of the halfwave plate in the simulation given in Table 1. 
One can observe in Table 2 that the error on the different reconstructed parameters is better when the time spent on each baseline is longer. Figure 4 represents the residual halfwave plate gain error as a function of the time spent on each baseline during the calibration mode. It shows that the limitation of the accuracy achieved on the systematic parameters is given by the time spent on calibration mode for each baseline t_{b}.
The first QUBIC module will contain 400 primary horns, or 79800 baselines; therefore, we need to spend about 22h on calibration in order to measure all the baselines during one second. This lapse of time could, however, be much reduced with a small information loss if the selfcalibration procedure was not performed on all baselines. The accuracy on the output parameters also depends on the number of baselines per measurement as illustrated in Fig. 3. It will be important to determine which is the most interesting strategy for the QUBIC instrument.
Using the law given by Eq. (40), one can extrapolate the result given in Table 2 to the residual error for the QUBIC instrument with 400 horns, 2 × 1024 bolometers, and 1000 pointings for two different measuring times per baseline t_{b} = 1 and t_{b} = 100 s. The result is given in Table 3. The values of the standard deviation between the corrupted and reconstructed parameters are obtained by replacing in Eq. (40) the values of exponent given in Table 2 applied to the design of the QUBIC instrument. It shows a very significant improvement on the level of the residual systematics after selfcalibration, even for 1 s.
Results of the selfcalibration simulation for an instrument with 9 horns, 9 bolometers, and 10 pointings.
Fig. 4 Results of the selfcalibration simulation for the diagonal terms of the Jones matrix of the halfwave plate. We show the residual error (in blue) on these parameters as a function of the measuring time spent on each baseline on the calibration mode. The red line represents a power law of the shape with NET the noise equivalent temperature of the bolometers, T the temperature of the polarized source, and t_{b} the measuring time spent on each baseline. This law gives the limit of the relative accuracy that can be achieved on the systematic parameters according to the measuring time spent per baseline. The green line represents the input error on the systematic effects of the halfwave plate in the simulation given in Table 1. 
3.5. Finding limits
Accordingly Eq. (34), in an ideal case, in the absence of systematic effects, the powers measured on the x and y polarized focal planes after demodulation of the halfwave plate are (41)with ϵ^{x} = 1 for the polarization x, ϵ^{y} = −1 for the polarization y, T_{O}, T_{C}, and T_{S} refer to a constant, cosine, and sine terms obtained after the demodulation of the halfwave plate, S_{I}, S_{Q}, and S_{U} refer to the synthesized images for each Stokes parameter I, Q, and U and the synthesized beam. The integrations are performed over the bandwidth of the instrument, over the surface of the bolometer, and over the sky direction.
For a real instrument, in the case where the halfwave plate is located before the horns, leakages from Q to U and from U to Q appear. Using the selfcalibration simulation, one can estimate the leakage from Q into U and from U into Q by calculating the standard deviation of the difference between the ideal and corrupted parameters , , , , and (without selfcalibration) and of the difference between the corrupted and recovered parameters , , , and (with selfcalibration).
In the case without selfcalibration, Eq. (41) becomes (42)In the case with selfcalibration, Eq. (41) becomes (43)With Eqs. (42) and (43), one can observe that instrument errors and systematic effects induce leakage from Q to U and from U to Q but also alter the polarization amplitude. It is assumed that there is no correlation of the errors in the synthesized beam across the bolometers, and the simulation is realized for a point source so that the sky convolution can be ignored.
Results of the selfcalibration simulation for the QUBIC instrument with 400 horns, 2 × 1024 bolometers array, 1000 pointings, and all baseline measurements.
It is interesting to focus on the Bmode power spectrum in order to estimate the EB mixing and to constrain the Emode leakage in the Bmode power spectrum. In general, one can define the vector that defines the errors on Stokes parameters (44)where , and is the vector of measured Stokes parameters and is the vector of the Stokes parameters obtained with the selfcalibration method. For the QUBIC instrument, there is no leakage of I to Q and U because the halfwave plate is in front of the instrument so, Eq. (44) becomes (45)where Δ_{QQ}, Δ_{QU}, Δ_{UQ}, and Δ_{UU} are the errors between the synthesized beam without the selfcalibration method and the one after applying the selfcalibration method. Following Appendix B and Eq. (45), one can write the error terms as Δ_{QQ} = 1 + ϵ, Δ_{QU} = ρ, Δ_{UQ} = −ρ, and Δ_{UU} = 1 + ϵ where the complex term ϵ changes the amplitude of polarization and ρ mixes both the Q and U Stokes parameters. One can define the error matrix on the Q and U Stokes parameters as M_{s} − 11 where the matrix M_{s} is given by (46)To go further, one can estimate the leakage from E to Bmode and give a constraint on the value of the tensortoscalar ratio r. An error in diagonal terms Δ_{QQ} and Δ_{UU} will affect the amplitude of the E and Bmode power spectrum. The nondiagonal terms Δ_{QU} and Δ_{UQ} result in a leakage from the E to Bmode power spectrum (or the B to Emode power spectrum). To have a constraint on the Bmode, as the Emode is far above that of the Bmode in amplitude, one can use the equation derived by (Rosset et al. 2010) obtained with the firstorder approximation (47)where ΔCl^{BB} is the error on the Bmode power spectrum Cl^{BB} and Cl^{EE} is the Emode power spectrum, and we suppose ϵ is small. One can refer to Appendix B for the explicit details of this equation. It shows that the uncertainty on parameter ρ must be lower than 0.5% to have a leakage from E to Bmode lower than 10% of the expected Bmode power spectrum Cl^{BB} for a tensortoscalar ratio of r = 0.01 for l < 100.
From Eq. (47), one can estimate the leakage from the Emode into the Bmode power spectrum given by the term ρ^{2}Cl^{EE}. Figure 5 represents the error on the Bmode power spectrum ΔC_{l} as a function of the multipoles. The leakage from the Emode into the Bmode is therefore significantly reduced by applying the selfcalibration procedure, even with a modest 1s per baseline (corresponding to a full day dedicated to selfcalibration). The leakage can be further reduced by spending more time on selfcalibration.
Fig. 5 ΔC_{l} due to leakage from Emode for different times on measurements per baseline t_{b} = 1 s, 10 s and, 100 s for the QUBIC instrument. The colored solid lines represent the leakage after applying the selfcalibration method for different measuring times per baseline. The dashed line represents the error on the Bmode power spectrum without selfcalibration method. The black lines are the primordial Bmode spectrums for r = 10^{1}, r = 10^{2}, and r = 10^{3}. 
4. Conclusion
Bolometric interferometry differs considerably from standard radio interferometry in the sense that its primary goal is not to reach a good angular resolution but to achieve high statistical sensitivity and good control of systematic effects. In this perspective, redundancy turns out to be the crucial property to fulfill these two objectives, as shown in Charlassier et al. (2009) and in the present article.
In this paper, we have shown that with a polarized calibration source and the use of the successive observation of this source with all pairs of horns of the interferometer (selfcalibration), one can have low and controllable instrumental systematic effects. Redundant baselines should give the same signal if they are free of systematics. By modeling the instrument systematics with a set of parameters (Jones matrices, location of the horns, beams), one can use the measurements of the different baselines to solve a nonlinear system that allows the systematic effects parameters to be determined with an accuracy that, besides the correctness of the modelling, is only limited by the photon noise, hence by the time spent on selfcalibration. The more horns and bolometers in the array, the more efficient the selfcalibration procedure.
The resolution of the system is CPUintensive for large bolometric interferometers and should be implemented on massively parallel computers in the future. Using simulations with various horns and bolometer arrays of moderate sizes, we have obtained a scaling law that allows us to extrapolate the accuracy of selfcalibration to the QUBIC instrument with 400 horns, 2 × 1024 bolometer arrays and 1000 pointing directions towards the calibration source. We find that with a few seconds per baseline (corresponding to a few days spent on selfcalibration), knowledge of the instrumental systematic effects parameters can be improved by at least two orders of magnitude, allowing minimization of the leakage from E into B polarization down to a tolerable level. This can be improved by spending more time on selfcalibration.
The idea of developing bolometric interferometry was motivated by bringing together the imager exquisite sensitivity allowed by bolometer arrays and the ability to handle instrumental systematic effects allowed by interferometers. Bolometric interferometers have been shown to have a sensitivity similar to that of imagers (Hamilton 2008; QUBIC collaboration 2010), while we have shown in the present article that the self calibration allows achieving an excellent handling of systematic effects that has no equivalent with an imager.
Acknowledgments
The authors are grateful to the QUBIC collaboration and Craig Markwardt for MPFIT. They would like to thank Gael Roudier for his help and the referee for an attentive reading. This work was supported by Agence Nationale de la Recherche (ANR), Centre National de la Recherche Scientifique (CNRS), and la région d’Ile de France.
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Appendix A: Measuring the bolometer power for two opened horns is equivalent to measuring the bolometer power when all horns are open except the horn i and j
The power collected by the bolometer q for the opened horns i and j without polarization is (A.1)which can be written as (A.2)The total power measured for all the baselines can be expressed as (A.3)The power measured by a bolometer q for all horns opened except the horn i is (A.4)The power measured by a bolometer q for all horns opened except the horn j is (A.5)The power measured by a bolometer q for all baselines opened, except the baseline formed by the horns i and j, is (A.6)with Finally, one can find (A.7)So: (A.8)
Appendix B: Error on E and Bmode power spectra
One can define the Stokes parameters in spin2 spherical harmonics base (B.1)where Q and U are defined at each direction .
It is convenient to introduce the linear combinations One can define the two scalar fields Using the coefficients and , one can construct the angular power spectrum and as One can express the Q and U Stokes parameters as a function of the coefficients and Using Eqs. (B.4), (B.5), (B.8), and (B.9), one can express the coefficients and as a function of the Stokes parameters Q and U(B.10)where the integration is taken over the whole sky, and (B.11)In the case of a bolometric interferometer, globar errors on synthesized beam will affect the amplitude of polarization and mix the Q and U Stokes parameters.
To model systematic errors, one can introduce a Jones matrix that describes the propagation of radiation through a receiver (B.12)where the gain g_{η} and the leakage e_{η} are complex values.
From this relation, one can construct the Mueller matrix M, which tells us how the Stokes vector transforms (B.13)and is the outgoing Stokes vector and M = A(J ⊗ J^{ ∗ })A^{1} is the resulting error matrix.
We are only interested in the Q and U Stokes parameters. In this case, the error matrix M becomes (B.14)The first order of this matrix M is One can generalize this error matrix for Jones matrix of any component and rewrite it as M_{s} − 11 where (B.15)where the complex term ϵ describes the error of the amplitude of polarization and the complex term ρ mixes the Q and U Stokes parameters.
In this case, Eq. (B.10) becomes (B.16)where the coefficients and include systematic effects.
In terms of power spectra, an error in polarization amplitude will affect the amplitude of the E and Bmode power spectra,
and an error that mixes the Q and U Stokes parameters leads to a leakage from the E to Bmode (and from the B to Emode).
One can easily find that with systematic effects, the coefficients and can be expressed as where ϵ is the error in amplitude and ρ the error of polarization leakage. Using Eqs. (B.6) and (B.7), one can obtain (B.19)(B.20)where we suppose ϵ is small, Cl^{EE,meas} and Cl^{BB,meas} are the E and Bmode power spectra including systematic effects, Cl^{EE} and Cl^{BB} the input power spectra, the complex term ϵ describes the error of amplitude of the Bmode power spectrum, and the complex term ρ results in a leakage from the E to Bmode power spectrum.
To focus on the Bmode, one can define the error on Cl^{BB} power spectrum (B.21)
All Tables
Results of the selfcalibration simulation for an instrument with 9 horns, 9 bolometers, and 10 pointings.
Results of the selfcalibration simulation for the QUBIC instrument with 400 horns, 2 × 1024 bolometers array, 1000 pointings, and all baseline measurements.
All Figures
Fig. 1 Design of the QUBIC instrument. 

In the text 
Fig. 2 Results of the selfcalibration simulation for the synthesized images , , and for the X focal plane on the right and Y focal plane on the left for an instrument with nine primary horns, nine bolometers, and ten pointings for a time spent on calibration mode for each baseline of t_{b} = 1 s and 100 realizations. These plots represent scatter plots of ideal vs. real synthesized image in red and of recovered vs. real in blue. The red plots show the corruption after adding the systematic effects defined in Table 1. The blue plots show that the corruption is solved after applying the selfcalibration method. 

In the text 
Fig. 3 Results of the selfcalibration simulation for the diagonal terms of the Jones matrix of the halfwave plate and a time spent per baseline t_{b} = 1 s. This plot represents the residual error on these parameters as a function of the percent of baselines per pointing. The red line represents a power law of the shape with c a constant, n_{bs} the number of baselines per pointing and Φ the index of the power law given in Table 2. This law gives the limit of the relative accuracy that can be achieved on the systematic parameters. The green line represents the input error on the systematic effects of the halfwave plate in the simulation given in Table 1. 

In the text 
Fig. 4 Results of the selfcalibration simulation for the diagonal terms of the Jones matrix of the halfwave plate. We show the residual error (in blue) on these parameters as a function of the measuring time spent on each baseline on the calibration mode. The red line represents a power law of the shape with NET the noise equivalent temperature of the bolometers, T the temperature of the polarized source, and t_{b} the measuring time spent on each baseline. This law gives the limit of the relative accuracy that can be achieved on the systematic parameters according to the measuring time spent per baseline. The green line represents the input error on the systematic effects of the halfwave plate in the simulation given in Table 1. 

In the text 
Fig. 5 ΔC_{l} due to leakage from Emode for different times on measurements per baseline t_{b} = 1 s, 10 s and, 100 s for the QUBIC instrument. The colored solid lines represent the leakage after applying the selfcalibration method for different measuring times per baseline. The dashed line represents the error on the Bmode power spectrum without selfcalibration method. The black lines are the primordial Bmode spectrums for r = 10^{1}, r = 10^{2}, and r = 10^{3}. 

In the text 
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