Kernel nullers for an arbitrary number of apertures
AAstronomy & Astrophysics manuscript no. main c (cid:13)
ESO 2020August 19, 2020
Kernel nullers for an arbitrary number of apertures
Romain Laugier , Nick Cvetojevic , and Frantz Martinache Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, FranceAugust 19, 2020
ABSTRACT
Context.
The use of interferometric nulling for the direct detection of extrasolar planets is in part limited by the extremesensitivity of the instrumental response to tiny optical path differences between apertures. The recently proposed kernel-nuller architecture attempts to alleviate this effect with an all-in-one combiner design that enables the production ofobservables inherently robust to residual optical path differences ( (cid:28) λ ). Aims.
Until now, a unique kernel nuller design has been proposed ad hoc for a four-beam combiner. We examine theproperties of this original design and generalize them for an arbitrary number of apertures.
Methods.
We introduce a convenient graphical representation of the complex combiner matrices that model the kernelnuller and highlight the symmetry properties that enable the formation of kernel nulls. The analytical description ofthe nulled outputs we provide demonstrates the properties of a kernel nuller.
Results.
Our description helps outline a systematic way to build a kernel nuller for an arbitrary number of apertures.The designs for three- and six-input combiners are presented along with the original four-input concept. The combinergrows in complexity with the square of the number of apertures. While one can mitigate this complexity by multiplexingnullers working independently over a smaller number of sub-apertures, an all-in-one kernel nuller recombining a largenumber of apertures appears as the most efficient way to characterize a high-contrast complex astrophysical scene.
Conclusions.
One can design kernel nullers for an arbitrary number of apertures that produce observable quantitiesrobust to residual perturbations. The designs we recommend are lossless and take full advantage of all the availableinterferometric baselines. They are complete, result in as many kernel nulls as the theoretically expected number ofclosure-phases, and are optimized to require as few outputs as possible.
1. Introduction
The last 25 years have seen the detection of more than 4,000exoplanets (Schneider et al. 2011). Despite the indirect na-ture of most detections, existing observations already pro-vide us with a wealth of information on the properties of ex-oplanetary systems: their mass, size, and orbital elements.Yet direct detection of a planet’s reflected, or radiated light,for its direct spectral analysis for a large sample of targets,remains an exciting prospect that will contribute to furthercharacterize individual planets, in particular the propertiesof their atmospheres (Marois et al. 2008; Zurlo et al. 2016).The use of coronagraphic instruments is now leading tothe detection of young giant planets in wide orbits aroundnearby stars (Macintosh et al. 2015; Chauvin et al. 2017;Mesa et al. 2019). This success is carried by the continuedimprovements of extreme adaptive optics systems (Sauvageet al. 2016; Lozi et al. 2018; Boccaletti et al. 2020). Forsmaller separations approaching the diffraction limit (be-low ∼ λ/D ), small residual wavefront errors still dominatethe error budget and coronagraphic solutions become lessfavorable.Lacour et al. (2019) have demonstrated the advantagesbrought by long baseline interferometry for the character-ization of extrasolar planets. This observing mode takesadvantage of the spatial filtering provided by the resolvingpower of each of the 8 meter telescopes of the VLTI, coupledinto single mode fibers to reach the required contrast. Inter-ferometric nullers (Bracewell 1978; Colavita et al. 2009; Ser-abyn et al. 2019; Hoffmann et al. 2014; Defrère et al. 2015;Norris et al. 2020) offer the possibility to explore smaller an- gular separations through the use of fragmented aperturesand long-baseline interferometry. Some combining solutionshave been found that optimize the rejection of resolved stars(Angel & Woolf 1997; Guyon et al. 2013). The exploitationof these instruments is still limited by their vulnerability tooptical path differences (OPD) errors, and requires sophis-ticated statistical analysis, like those proposed by Hanotet al. (2011), Defrère et al. (2016) and more recently usedby Norris et al. (2020) to disentangle the off-axis astrophys-ical signal from the effects of unwanted OPD.Classical long-baseline and Fizeau interferometry makeextensive use of the production of robust observables, likeclosure phases (Jennison 1958) and their generalized form,kernel phases (Martinache 2010), to sidestep the limitationsbrought by the OPD residuals. This approach has providedreliable performance at very small separations, down to oneresolution element and below.Bringing together the robustness of interferometric ob-servables and the photon-noise suppression of nulling, isan exciting perspective as it opens a novel high-contrast,high-precision regime. The double Bracewell architecture(Angel & Woolf 1997) was remarked to offer such robust-ness (Velusamy et al. 2003) when implemented with theadequate phase shift between the two stages (called "sin-chop"). A different approach was later proposed by Lacouret al. (2014), exploiting the measurement of fringes in theleakage light.Martinache & Ireland (2018) introduced an alternative,more efficient solution for a four-telescope beam-combinerarchitecture that produces six nulled outputs. By analyzingthe response of these outputs to parasitic OPDs (instrumen- Article number, page 1 of 11 a r X i v : . [ a s t r o - ph . I M ] A ug &A proofs: manuscript no. main tally or atmospherically induced phase error), the authorsidentify linear combinations of outputs that are robust tothese aberrations to second order. The solution they pro-pose concerns a four-input nuller that provides three nulledkernel observables.In this paper, we look for the properties ensuring that acombiner will produce kernel nulls. They help outline a gen-eral strategy for the design of kernel nullers for an arbitrarynumber of apertures.
2. Analysis of the existing four-input kernel-nullingarchitecture
Using the nuller architecture laid out by Martinache & Ire-land (2018) for a four-beam interferometer as a startingpoint, and reexamining its properties, we look into ways ofgeneralizing this special case to a wider range of configura-tions, involving different numbers of apertures.The inner structure of a homodyne interferometric com-biner (nulling or not) is conveniently represented by a com-biner matrix M that acts on a vector z of input electricfields and leads to the production of an output electric fieldvector x . x = M · z . (1)Eventually, a detector records the intensity associated tothe square norm of this output electric field.The fact that only the square norm of the field isrecorded has two consequences. The first is that the re-sponse of the combiner is insensitive to the absolute phase ofthe input electric field: one of the sub-apertures can there-fore be arbitrarily picked as a reference, and the phasesof the different electric fields sampled by the other sub-apertures are measured relative to that reference. The sec-ond is that the output intensity is equally insensitive to anyglobal phase shift φ applied to any row m of the matrix M describing the combiner. This is of consequence when iden-tifying distinct combinations (or rows).Assuming that the recombiner is fed by a balanced ar-ray of identical sub-apertures, the complex amplitude of theinput electric field can be described by a vector of phasors.We will further assume that the combiner benefits from afringe tracker that, although not perfect, brings the systemclose to its nominal state. The fringe tracking residuals areassumed to be small and all phasors e − jϕ k are here approx-imated using the following expansion: z k = e − jϕ k ≈ − jϕ k , (2)where j is the imaginary unit.Since only intensities are measured, the overall responseof the system is a quadratic function of the perturbationphase vector. Martinache & Ireland (2018) thus use thisdescription to look at the properties of the second orderderivative of the intensity relative to the phase. One of the n a sub-apertures being used as a phase reference, there are n a − n d = n a ∗ ( n a − / n o × n d ma-trix A called the matrix of second order derivatives, where n o is the number of relevant outputs. Linear combinationsof rows of A that equal 0 cancel out the second order in-tensity deviations caused by small input phase errors. The same linear combination applied to the intensity measuredafter the recombiner will be equally insensitive to small in-put phase errors. We refer to these linear combinations askernel outputs or kernel nulls when applied to a collectionof nulled outputs.The rank of A and the possibility of forming such robustobservables rely entirely on the properties of the matrix M ,and therefore does not depend on the geometry of the in-put array. However, the question of whether a kernel nullcarries astrophysically relevant information also depends onthe configuration of the array. Throughout this work, phaseand amplitude contributions are considered independently,but their coupled contribution is neglected. For now, wewill further examine the properties of the combiner and in-troduce a convenient visual representation of the structureof M . The effect of the matrix M on the complex amplitude ofthe input electric field can be conveniently visualized by aseries of plots of the complex plane. For a given combiner,each input is represented by a colored arrow which, in theabsence of environmental perturbation, is aligned with thereal axis. Each plot illustrates the effect of a row of M onsuch inputs: the resulting electric field is the sum of all col-ored arrows present in the plot. A nuller is characterized byseveral outputs for which the sum of the arrows, associatedto the electric fields, sum up to zero. These complex matrixplots (CMP) will be used throughout this work to describeseveral nuller designs of varying complexity. The architecture of the kernel nuller described in Marti-nache & Ireland (2018) builds from an initial all-in-onefour-beam nuller whose overall effect can be described bythe following matrix: N = 1 √ − − − − − − (3)This matrix is real. Each nulled row of N recombines dis-tinct arrangements of the four input electric fields such thatthe coefficients on the corresponding rows sum up to zero,as represented in Fig. 1, with arrows aligned with the realaxis: two positive (or not phase-shifted), and two negative(or phase-shifted by π ). As discussed in this reference, thisnuller does not allow the formation of kernels: the output in-tensities it produces are a degenerate function of the targetinformation and input phase perturbations. The outputs ofthis nuller can however be fed to a second stage, describedby the following matrix: S = 1 √ e j π e j π e j π e j π e j π e j π , (4) Article number, page 2 of 11omain Laugier et al.: Kernel nullers for an arbitrary number of apertures
Input 0Input 1Input 2Input 3
Fig. 1.
CMP of the matrix N of Eq. (3) representing a four-input nulling combiner. The first row constitutes the brightchannel, with all inputs combined constructively. Note how eachoutput is a contribution of all the inputs, and not just a pair ofthem, which prevents the direct interpretation through the uvplane. which leaves the bright output untouched but further splitsthe nulled ones, and selectively introduces π/ M = S · N . The CMPs of this modified combiner, shown in Fig.2, offer a more easily readable description of its effect, withcomponents of the output electric field no longer simplyaligned with the real axis, but spanning the complex plane.The new complex configuration enables the larger diver-sity that is required to disentangle the otherwise degenerateeffect that environmental perturbations have on the inputelectric fields. The modified nuller indeed features more out-puts than inputs, and a close examination of the CMPs ofFig. 2 shows that all six combinations offer a distinct ar-rangement of the four input fields. The construction of alarger number of distinct nulls is one of the requirementsfor the existence of a non-empty left null space for A de-scribed in Sec. 2.1. In effect, pairs of outputs produce thesame response to environmental effects, while still produc-ing different response to off-axis light.
3. Properties of conjugate pairs of nulls
Identifying the kernel-forming combinations of outputs no-longer requires building the second order derivative matrix A but can be achieved by examination of the CMP repre-sentation of the nuller. Figure 2 lays out, side by side, thetwo outputs leading to one kernel-null observable. We callthese outputs enantiomorph: close examination of any suchpair of outputs reveals that the electric field combinationpatterns are the mirror image of one-another. Input 0Input 1Input 2Input 3
Fig. 2.
CMP of the S · N combination. The first output is thebright channel for which all the inputs add-up constructively.The vectors are staggered for readability. Pairs of nulled rowsrepresented side-by-side are mirror images of each-other (enan-tiomorph). Note that the amplitude of the phasors is reducedcompared to Fig. 1 due to additional splitting. Given that the measured intensity associated to any out-put is insensitive to a global phase shift, one can alwaysapply such a shift so as to align the arrow corresponding tothe phase reference input with the real axis, and point ittowards the positive direction. After such a rotation is ap-plied, enantiomorph outputs simply become complex con-jugate. This makes it possible to write simple equationsthat describe the two key properties of kernel nulls: theirrobustness to small phase perturbation, and the antisym-metric nature of the signal they provide.
Considering m and m , a conjugate pair of null rows of M : m = m ∗ . (5)A corresponding kernel null κ ( z ) writes as the difference ofthe two measured intensities: κ ( z ) = | m z | − | m z | = m z ( m z ) ∗ − m z ( m z ) ∗ . (6) Article number, page 3 of 11 &A proofs: manuscript no. main
Using (5) and (6) gives: κ ( z ) = m z ( m z ) ∗ − m ∗ z ( m ∗ z ) ∗ . (7)In the case of the approximation mentioned in Eq. (2): κ ( z ) = m ( a + j ϕ )( m ( a + j ϕ )) ∗ − m ∗ ( a + j ϕ )( m ∗ ( a + j ϕ )) ∗ , (8)where a is a vector of ones. Developing this expression,since m a = 0 and m ∗ a = 0, the only terms left are theones containing only the imaginary perturbation term j ϕ : κ ( z ) = m j ϕ ( m j ϕ ) ∗ − m ∗ j ϕ ( m ∗ j ϕ ) ∗ . (9)Distributing the conjugate operator gives: κ ( z ) = − m j ϕ m ∗ j ϕ + m ∗ j ϕ m j ϕ , (10)and therefore κ ( z ) = 0 due to the commutativity. Thisshows that the subtraction of intensity of complex conju-gate pairs of nulled outputs always produces a kernel nullthat is robust to arbitrary imaginary phasors, to which thesmall input phase aberrations are approximated.This property also applies to arbitrary purely real inputelectric fields that would correspond to pure photometricerror generated by fluctuations of the coupling efficiencies.Considering a purely real input vector a : κ ( z ) = m a ( m a ) ∗ − m ∗ a ( m ∗ a ) ∗ . (11)Distributing the conjugate operator gives: κ ( z ) = m am ∗ a − m ∗ am a , (12)and therefore κ ( z ) = 0 due to the commutativity.At any instant, the subtraction of the signals recordedby conjugate (or more generally enantiomorph) outputsforms a kernel null. Conjugate pairs of nulls allow the for-mation of kernel nulls. This property generalizes to enan-tiomorph pairs of nulls through the rotation by a singlecommon phasor. A complementary approach for the iden-tification of robust combinations of outputs is the use ofthe singular value decomposition (SVD) of the second or-der derivative matrix A , as mentioned by Martinache &Ireland (2018), which ensures the identification of all therobust combinations of outputs.This behavior can be illustrated by adding differentphased contributions to the inputs, and plotting the result-ing electric field on top of the original perfectly cophasedCMP (kept in dashed lines). The first panel of Fig. 3 usesthis representation of the combined light to illustrate howsmall input phase aberrations affect the amplitude (andtherefore the intensity) of the combiner’s outputs. In par-ticular it shows how, for small phase errors, conjugate pairsof nulls suffer the same leakage light intensity. The second and third panels of Fig. 3 show how input lightcoming from a significantly off-axis source (input phases φ ≥ z and z are twoinput electric field vectors coming from sources located atsymmetric positions in the field of view, then: z = z ∗ . (13)Considering again a conjugate pair of null rows m and m ,and by substitution of (13) into (6) we get: κ ( z ) = m z m ∗ z − m z m ∗ z . (14)After substitution of (5), this becomes: κ ( z ) = m ∗ z m z − m ∗ z m z . (15)This leads to the conclusion that the response is antisym-metric: κ ( z ) = − κ ( z ) . (16)Conversely, one may also extract from this pair of nullsthe complementary observable: τ ( z ) = m z ( m z ) ∗ + m z ( m z ) ∗ (17)whose response is symmetric. The observables κ and τ therefore carry complementary information on the targetfield, much like the amplitude and phase of complex visi-bility in classical interferometry.Although τ does not have the same robustness to aber-rations, there may be ways to use it with the processingmethods employed by Hanot et al. (2011) and Norris et al.(2020) so as to provide additional information on the targetin different science cases. κ is best suited for the study ofhigh-contrast non-symmetrical features such as planetarycompanions, while τ may be used to study brighter sym-metrical features such as debris disks or stellar envelopes.Combining both types of observables could enable imagereconstruction.The τ observables carry some information about theinput phase errors. One can use their values over the courseof a scan or modulation of the OPDs to locate the setpointof the kernel-nuller, for which they will reach a minimum.
4. Construction of new nullers
The properties used in Sects. 3.2 and 3.3 to demonstratethe robustness of kernel nullers to small phase perturba-tions may be used as constraints to guide the design of anarbitrary kernel nuller matrix. For the output of any row l to provide an on-axis null, the matrix coefficients mustsatisfy: n a − X k =0 M k,l = 0 . (18) Article number, page 4 of 11omain Laugier et al.: Kernel nullers for an arbitrary number of apertures
Small error Partially resolved Null peak
Input 0Input 1Input 2Input 3Output
Input 0Input 1Input 2Input 3Output
Output 0Input 0Input 1Input 2Input 3
Fig. 3.
CMP for a four-input combiner, representing in dashed lines the coefficients of the combiner matrix M , functionallyequivalent to M , and in solid arrows the contributions of the complex amplitude of an example input electric field to the outputelectric field represented in black. Most dashed lines are hidden under the arrows. A black circle of radius equal to the modulus ofthis output is plotted for visual cue, its area being proportional to the corresponding intensity. Enantiomorph pairs that generatekernel combinations by subtraction are represented side-by-side. Like Martinache & Ireland (2018), we use the example of theVLTI UT configuration observing at zenith at a wavelength of 3 . µm . On the left-hand panel, a source located at 0.2 mas from theoptical axis and for which the corresponding input phase shifts are within the small phase approximation. As a result, the outputintensities within each pair are fully correlated and result in no kernel signal. On the central panel, the source is located 1.1 masoff-axis which generates larger phase shifts. As a result, the null intensities from the enantiomorph pairs begin to decorrelate andgenerate kernel-null signal. On the right-hand panel, the source is located 4.3 mas from the optical axis, in the position where thefirst nulled output peaks. At this position, the second output gets to zero. Output intensities are unchanged when the coefficients ofa row are all multiplied by a common phasor. We thereforeapply one such phasor so as to get Arg( M ,l ) = 0. We alsoset output M k, ) = 0.Simple solutions to Eq. (18) for a balanced array can befound by picking arrangements of uniformly spaced phasevalues in the [0 , π ] interval as can be seen of Figs. 2, 4 and6. The phase of each coefficient is therefore a multiple ofΦ = 2 π/n a . On the CMPs seen thus far, this would resultin the rotation of all of the arrows on the nulled outputsuntil the one associated with input n a − n max = ( n a − . (19)The phase term φ k,l writes: φ k,l = c k,l Φ , (20) where c k,l a is the k -th term of the l -th possible combinationon the circle. In general, a complex coefficient of M willtherefore write: M k,l = a l · e jφ k,l , (21)where a l is a real coefficient, normalizing the matrix, sothat M represents a lossless beam-combiner for which eachcolumn vector is of unit norm. As mentioned in appendixA, this condition on the norm is necessary (but not suffi-cient) to ensure that the matrix represents a lossless beamcombiner, and one solution for it is to have: ( a l = √ n a for the bright output a l = √ n a q n a − n null for the nulled outputs (22)where n null is the number of nulled outputs. Normalizationis not mandatory to study the qualitative properties of thecombiner, but it is necessary to study their throughput ina quantitative manner and their practical implementation.The matrix M obtained with Eq. (21), represents a com-biner for which pairs of complex conjugate nulls can be sub-tracted to build the kernel nulls that are the focus of thiswork. Article number, page 5 of 11 &A proofs: manuscript no. main
Inputs Distinct nulls Indep. nulls Kernel nulls n a n max n indep. n kn Table 1.
Growth of kernel-nuller combiners with the number ofapertures.
As shown by Eq. (19) and Table 1, the number of nulledoutputs that would result from a strict application of theseblueprint rules rapidly grows as the factorial of the numberof inputs. However, for numbers of apertures larger thanfour, although all the nulls produced with the presentedscheme are distinct, some of them do not carry new infor-mation on the target, as their response function to off-axissignal is a linear combination of the response function ofother nulls.Here, we analyze this property empirically by examiningthe response maps (analogous to Fig. 5 and 7 of Martinache& Ireland (2018)) and assembling them as vectors of a setof nulled outputs and a set of kernel outputs. The ranks ofthese sets provide the number of independent observablesproduced by the combiner. Although we were not able tolink this property to particular traits of the combinations,the largest number of independent kernel nulls obtainableby a given non-redundant array of apertures was always thesame as the number of independent closure-phases, whichis in agreement with the expectations set by Martinache &Ireland (2018). For any non-redundant array of apertures,this number is: n kn ( n a ) = (cid:18) n a (cid:19) − ( n a −
1) = ( n a − n a − . (23)The underlying relationship between the baselines and ournew observables is non-trivial but will be assumed to holdfor any non-redundant array. For redundant arrays, thisnumber decreases. We will call complete a nuller that pro-vides the aforementioned maximum number n kn of inde-pendent observables. The number of independent nulls inthe full set is n indep. = 2 × n kn . These results obtainedempirically for up to seven inputs are shown in Table 1,along with their expected progression for larger numbers ofinputs.As seen in Eq. (22), an increase in the number of nulledrows decreases the normalization coefficients a i , as in prac-tice fewer splittings of the input light are necessary to ob-tain the fewer combinations. Our goal may therefore be toconstruct complete combiners using the minimum numberof nulled combinations from the full matrix M , with theintent of increasing its throughput. Expecting this numberto be twice the number of kernel nulls (if we consider onlypairwise kernel nulls), this produces a very large number n crops ( n a ) of possible combinations: n crops ( n a ) = (cid:18) ( n a − n a − n a − (cid:19) (24)Only for the cases of three and four inputs is the solutionunique ( n crops ( n a ) = 1), and all null rows must be kept. For more inputs, this number grows rapidly. Although a largefraction of them are complete, fewer satisfy the conditionsdetailed in appendix A for conservation of energy.The following characteristics are shared by the three-and four-input combiners, as well as all the lossless realiza-tions of the cropped five-input combiner: – All nulls appear in conjugate (or enantiomorph) pairs,which implies that robust observables can be con-structed by subtraction. – Each phasor appears in each column the same numberof times (except for the one of phase zero which serves asthe reference). Equation (23) implies that for a combinerproducing n o = 2 n kn nulls, each phasor is used ( n a −
5. Examples of combiners
The simplest practical example of this architecture appearsfor the combination of three inputs. Here, the algorithm re-sults in the formation of two enantiomorph nulled outputs.Those two outputs will, by subtraction, produce one robustobservable. The resulting combiner matrix writes: M = 1 √ e jπ e jπ e jπ e jπ . (25)The combinations offered by this matrix are illustrated inFig. 4.As an example, we built a response map of the robustobservable produced by this combiner fed by three of theVLTI (von der Lühe 1997) unit telescopes (UTs) observingat zenith. Figure 5 shows the values of the kernel-null ob-servable represented as a two-dimensional function of therelative position of a source normalized by the flux of oneaperture. While simpler than the one provided in Fig. 7 ofMartinache & Ireland (2018) for the four-input combiner,this pattern retains the same antisymmetric property. Article number, page 6 of 11omain Laugier et al.: Kernel nullers for an arbitrary number of apertures
Input 0Input 1Input 2
Fig. 4.
CMP for a three-input kernel nuller of Eq. (25). Thefirst row corresponds to the bright channel with the overlappingphasors staggered for readability. The two nulled outputs arecomplex conjugates of one-another and will form a kernel null.
20 40 60 80 100East position (m)20304050607080 N o r t h p o s i t i o n ( m ) UT2 UT3 UT4Input 0Input 1Input 2
20 10 0 10 202015105051015200.0 0.2 0.4 0.6 0.8 1.0On-sky RA position (mas)0.00.20.40.60.81.0 O n - s k y D E C p o s i t i o n ( m a s ) K e r n e l - nu ll v a l u e ( s i n g l e a p e r t u r e f l u x ) Fig. 5.
Top: the three telescope configuration picked for theexample and corresponding to the position of three of the VLTIUTs. Bottom: the value of the kernel null as a function of therelative position of a source of unit contrast at 3 . µm , normal-ized to the throughput of one aperture. The map is relevant fora target observed at Zenith and would evolve with the projectedaperture map. Note the antisymmertric nature of the response,as demonstrated in Sect. 3.3 Assuming the practical implementation of the combineritself can be manufactured either with bulk or integratedoptics, this configuration would allow the production of ro-bust high-contrast observables with the least amount of in-frastructure. Drawing a parallel between this type of com-binations and closure triangles used for closure phases istempting but misleading. Here, as the combination must bedone optically rather than in post-processing, kernel nullingdoes not scale in the same way. The advantages and draw-backs of using these simple combiners as building blocks isbriefly discussed in Sect. 6.1.
We also outline a solution for a kernel-nulling recombinerfor six telescopes that could, for example, be used at thefocus of the CHARA array. The initial algorithm produces acombiner matrix M with 121 rows with redundancy in theoff-axis response. It is cropped to M using the guidelinesoffered in Sect. 4.2 to reduce it to the minimum of 21 rowswhile making sure the number of independent kernel nulls n kn is preserved. Furthermore, by enforcing the propertiesoutlined in Appendix A, we make sure that M remainsthe matrix of a lossless beam combiner.The matrix describing this 6-input combiner writes: M = 1 √ √ e jπ e jπ − e jπ e jπ e jπ e jπ − e jπ e jπ − e jπ e jπ e jπ e jπ − e jπ e jπ e jπ e jπ e jπ e jπ e jπ e jπ − e jπ e jπ e jπ e jπ − − e jπ e jπ e jπ e jπ − e jπ e jπ e jπ e jπ e jπ e jπ e jπ e jπ − e jπ e jπ e jπ e jπ − e jπ e jπ e jπ − e jπ e jπ e jπ e jπ − e jπ e jπ e jπ − e jπ e jπ e jπ e jπ − e jπ e jπ e jπ e jπ e jπ − e jπ e jπ e jπ e jπ − e jπ e jπ − e jπ e jπ e jπ e jπ − e jπ e jπ e jπ e jπ − e jπ e jπ e jπ e jπ − e jπ e jπ e jπ . (26)This combiner offers a total of 20 independent nulls, and10 independent kernel nulls. The corresponding CMP isshowed in Fig. 6 and highlights how each aperture con-tributes to all of the outputs.To illustrate the astrophysical information gathered bythe larger number of kernel nulls, we construct responsemaps of the kernel-null observables. These plots, shown inFig. 7 display the response of each of the observables forthe combiner being fed by the CHARA array observing atarget at zenith in the 3 . µm wavelength. The patterns re-flects the richness of the uv coverage provided by an array Article number, page 7 of 11 &A proofs: manuscript no. main
Input 0Input 1Input 2Input 3Input 4Input 5
Fig. 6.
Representation of the 20 nulled outputs of a six inputbeam combiner proposed in Eq. (26). The conjugate pairs thatform the 10 kernel nulls are represented side-by-side. like CHARA, and the fact that each output uses informa-tion collected by every telescope. As a consequence, eachmap covers the field of view differently, and brings newconstraint on the properties of the astrophysical scene ob-served.
6. Discussion
No active long-baseline optical interferometer currently pro-vides more than six sub-apertures. However, the maskingof monolithic apertures to produce interferometric arraysis an established practice (Tuthill et al. 2010; Jovanovicet al. 2012) that may be used in conjunction to nullinginterferometry (Norris et al. 2020). Therefore, the use ofeven larger combiners may prove to be a viable alternativesto small inner working angle coronagraphs (Guyon et al.2006). Their robustness to small aberrations might provideunprecedented contrast performance in the 1 − λ/D regimein the near infrared. Instead of building an all-in-one combiner, which may bedifficult to construct for a large number of apertures, analternative approach would be to multiplex several inde-pendent nullers, that each recombines a smaller number ofapertures. For example, instead of a six-input nuller pro-ducing 20 nulled outputs, one conservative option would be N o r t h p o s i t i o n ( m ) O n - s k y D E C p o s i t i o n ( m a s ) K e r n e l - nu ll v a l u e ( s i n g l e a p e r t u r e f l u x ) Fig. 7.
Top: the six telescope configuration for the CHARA ar-ray used as example. Bottom: the value of all 10 kernel nulls asa function of the relative position of a source at the wavelength3 . µm observed at zenith. The transmission is normalized bythe flux of a single aperture. Again, each map remains antisym-metric.Article number, page 8 of 11omain Laugier et al.: Kernel nullers for an arbitrary number of apertures to use two three-input kernel nullers, identical to the onepresented in Sect. 5.1, side by side producing four nulledoutputs.While the latter of these two options results in a reas-suring higher throughput per output, it can only producedistinct robust observables, where the M combiners of-fers 10. Moreover, this multiplexed option also results intwo bright channels, where some of the off-axis light is alsolost, further reducing the overall efficiency of the combiner.In between these two extreme scenarios, intermediate solu-tions can be imagined to alleviate some of their risks anddeficiencies, with a modular design multiplexing the nullersmuch like a number of ABCD combiners are multiplexedinside the beam combiner of VLTI/GRAVITY.It was already argued by Guyon et al. (2013) that effi-cient nulling solutions concentrate the most starlight intothe smallest number of outputs, which favors the all-in-onecombiner over the multiplexed versions. If manufacturabil-ity or operational constraints were to prevent the deploy-ment of an all-in-one combiner at the focus of a specific ob-servatory, one way to alleviate this inefficiency could be torecombine the light from the multiple bright outputs, intoan additional nulling stage so as to extract additional usefulobservables. This type of architecture, in part inspired bythe hierarchical fringe tracker idea of Petrov et al. (2014),might prove a necessary compromise to the implementationof a kernel nuller at the focus of a very long baseline observ-ing facility such as the envisioned Planet Formation Imager(Monnier et al. 2016), for which a distributed hierarchicalrecombination mode seems particularly apt. In addition to trade-off considerations between the totalnumber of observables and the throughput efficiency of theavailable options, one must also consider whether the num-ber of inputs has an impact on the phase-noise rejectionperformance of a kernel nuller.To evaluate this risk, we trace the evolution of the noiseaffecting the outputs and their kernels as a function of theamount of phase noise affecting the inputs. We do this forthe 3T, 4T and the 6T designs described in the previoussections. For simplicity, this study assumes that the phasenoise affecting all inputs is Gaussian, non-correlated, andcharacterized by a single rms value equally affecting all in-puts. Following a Monte Carlo approach, random realiza-tions of input piston errors are drawn, propagated throughthe different combiner matrices, and the standard devia-tion of the output intensities are evaluated. Figure 8 thusshows the evolution of the standard deviation of the out-put intensities of the nulls and of the kernel nulls of thedifferent architecture normalized by the peak null intensity I peak from their response map. This therefore constitutesa noise-to-signal ratio of sorts. The simple Bracewell nulleris also added to this study for comparison, as modelled bythe combiner matrix: M B = (cid:20) − j j (cid:21) . (27)This plot shows that the larger kernel-nulling combinersprovide a rejection of the phase noise that is very similarto the smaller ones, if not slightly better. The improvementon the raw observables may be credited to a manifesta-tion of the central limit theorem affecting the distribution input (radians)10 / I p e a k Bracewell nullRaw null 3TKernel null 3TRaw null 4TKernel null 4TRaw null 6TKernel null 6T
Fig. 8.
Propagation of phase noise from the input to the nulledintensities for different kernel-nulling architectures. Values arenormalized by the peak transmission of an off-axis signal. Theraw nulls (dashed lines) are compared with their correspondingkernel-null observable (solid lines), showing the suppression of2nd order phase noise (by a few orders of magnitude) for smallinput phase error. This effect decreases as the input phase errorsdepart from the small phase approximation. The behavior of theBracewell nuller is shown in dashed blue for reference. of the sum of a larger number of complex intensities. Fur-ther interpretation of this plot must be undertaken withcaution. Indeed, while the distribution of kernel nulls un-der such conditions is close to Gaussian (Martinache & Ire-land 2018), the distribution of null intensities is not (Hanotet al. 2011) and is therefore poorly described by its stan-dard deviation. While a full performance comparison of thedifferent designs lies outside the scope of the present paperand would include the coupled effects of phase and ampli-tude fluctuations (Lay 2004), these elements already indi-cate that kernel nullers recombining a large number of sub-apertures are intrinsically at least as robust to phase noiseas their smaller, simpler counterparts. This is an encourag-ing prospect for single telescope applications of the kernelnuller for which a potentially large number of sub-aperturescan be used.
7. Conclusions
In this work, we offer a new description of the kernel-nullerdesign introduced by Martinache & Ireland (2018). This isdone by introducing a new graphical representation of thecomplex matrix that models the nuller and the transfor-mations it operates on the input electric field. Combinedwith an analytical description of the outputs used to forma kernel, these representations explain the origin of a ker-nel nuller’s main properties: their intrinsic robustness tosmall input piston and amplitude fluctuations, and theirsensitivity to asymmetric features of the observed scene.We incidentally show that the same outputs can also besummed so as to fall back on the original outputs of an all-in-one N nuller stage, while not robust to perturbations,can nevertheless provide further astrophysical information.Our graphical and analytical representations help de-vise a systematic way to build a kernel nuller as a combinerfeaturing pairs of channels that are enantiomorph in thecomplex plane. It is this feature that makes two channelsequally sensitive to perturbations although they respond Article number, page 9 of 11 &A proofs: manuscript no. main differently to the presence of an off-axis structure. This ap-proach allow us to design kernel nullers for an arbitrarynumber of apertures, which we here apply to three- andsix-aperture arrays.We discuss the possibility of simplifying kernel nullersthat grow in complexity when they recombine a larger num-ber of input beams, for instance, using distinct nullers op-erating in parallel over a subset of input beams. For a giventotal number of inputs, a global architecture, giving accessto a larger number of high-contrast observables is more effi-cient and offers the means to explore and characterize com-plex astrophysical scenes. For the same number of inputs,we can also note that the total number of outputs for a ker-nel nuller (exactly twice the number of theoretical closure-phases) is in fact less than that of non-nulling combinersdesigned to measure the complex visibility of all baselines.Integrated optical circuits in particular already enable theimplementation of such complicated designs in small andstable packages, and are a very promising avenue for theconstruction of these larger kernel-nulling combiners.While no existing long baseline optical interferometricfacility currently offers the simultaneous combination ofmore than six apertures, a kernel nuller sampling the pupilof a single telescope could prove to be a valuable comple-ment to a coronagraph, producing high contrast observa-tions near one resolution element that would be insensitiveto the small but ever present adaptive optics residuals. Theevaluation of performance in practical implementations in-cluding the contribution of coupled phase an amplitude con-tributions and the consideration of relevant science caseswill be the topic of future theoretical and experimentalwork.
Acknowledgements.
We thank Alban Ceau and Coline Lopez for theirsuggestions to improve the manuscript. KERNEL has received fund-ing from the European Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation program (grant agree-ment CoG - 683029).
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Appendix A: Lossless combiners
As emphasized by Loudon (2000) in section 3.2, the conser-vation of energy in a single beamsplitter cube imply thatits matrix is unitary. This property can be generalized tolarger combiners. The required condition is that the sum ofintensities of the inputs be the same as the sum of intensi-ties at the outputs. As the output intensities are gatheredin the vector x , this sum also writes: n outputs X i =0 | x i | = x H x , (A.1)with H designating the Hermitian adjoint (conjugate trans-pose) operator. Based on Eq. (1) this sum writes: x H x = ( Mz ) H Mz , (A.2)which develops as follows: x H x = z H M H Mz . (A.3)As a consequence we obtain: ∀ z ∈ C n a n outputs X i =0 | x i | = n a X i =0 | z i | ⇐⇒ M H M = I . (A.4)As a consequence, the following propositions are equiva-lent: – M is the matrix of a lossless beam combiner. – M is semi-unitary on the left. – M H is the left inverse of M . – The columns of M are orthonormal. – All the singular values of M are equal to one.are equal to one.