Three dimensional photonic combiner for optical astrointerferometry Stefano Minardia,b , Thomas Pertscha , Ralph Neuh¨auserb a Institute
of Applied Physics, Friedrich Schiller University Jena, Max-Wien-Platz 1, 07743 Jena - Germany; b Astrophysical Institute and University Observatory, Friedrich Schiller University Jena, Schillerg¨asschen 2, 07745 Jena, Germany ABSTRACT
We put forward an innovative scheme allowing multiple beam combination and closure-phase retrieval by means of a three dimensional array of coupled optical waveguides. Light propagation in an array of evanescently coupled waveguides is similar to conventional diffraction, however it is bound to a system with finite degrees of freedom. We have demonstrated that the latter feature allows relating uniquely the intensity pattern at the end of a 3x3 waveguide array to the amplitude and relative phases of three monochromatic fields coupled to suitable input waveguides. The method is scalable to arbitrary large arrays of telescopes and baselines. Keywords: Integrated optics, coupled waveguides arrays, optical astrointerferometry
1. INTRODUCTION High resolution imaging of astronomical objects with astronomical interferometers requires a dense sampling of the coherence function of their light. This is accomplished by measuring the complex visibilities of interference fringes over a large number of different baselines. The current trend in optical interferometry is to perform multiple beam combination providing simultaneous measurements of complex visibilities over all possible baselines subtended by an array of several telescopes. This strategy is particularly suited for the observation of fast events such as nova/supernova explosions,1 photospheres of stars2 or transits of exoplanets.3 In this context, integrated optics is an established technological solution to perform multiple beam combination in astronomical interferometers. The high stability of integrated optical interferometers against vibrations and thermal variations is the key of their high performance in terms of visibility and phase measurement.4 Recently, an integrated chip allowing the measurement of the phase over 6 baselines among 4 telescopes has been realized and tested.5 The complexity of the integrated optical components grows rapidly as more telescopes and baselines are added, requiring a rather intricate design if planar geometries are considered. Three dimensional geometries, such as those enabled by laser-writing techniques,6, 7 could greatly simplify the design of beam combiners in multiple telescopes facilities, e.g. by allowing fiber cross-overs. Here we present an innovative and general method exploiting a 3D fiber geometry to perform interferometric multiple beam combination.8 The proposed scheme is based on the properties of periodic two-dimensional arrays of evanescently coupled waveguides. To exemplify the concept, we discuss the paradigmatic case of beam combination with 3 telescopes. The general discussion for the combination of more beams will be deferred to forthcoming publications. In this paper, we present the general theoretical background (Section 2), and then discuss a possible implementation of the concept in an astronomical interferometer (Section 3). Before conclusions are drawn, in Section 4 we report on the result of numerical simulations which prove the potential of the discrete beam combiner to reach performances comparable to existing interferometric beam combiners. Further information: (Send correspondence to S.M.) S.M.: E-mail:
[email protected], Telephone: +49 (0) 3641 947 848
2. THEORY 2.1 Discrete optics In an array of closely spaced single mode optical fibers, the overlapping of the evanescent fields of the modes pertaining to neighboring waveguides is responsible for a power leakage from waveguide to waveguide. The modes of these arrays are said to be mutually coupled. A system on N waveguides is characterized by the set of complex amplitudes An≤N pertaining to the fundamental mode of each waveguide n, hence the name discrete optics used to describe arrays of waveguides. The evolution of the fields {An } (hereafter described as the ’state’ of the array) along the longitudinal direction z of the fibers is described by a set of coupled wave equations: N
i
∂An (z) X = Cn,k Ak (z) ∂z
(1)
k=1
where {Cn,k } is a matrix with 0 diagonal coefficients. In the case of a linear array of waveguides where only the coupling between neighboring waveguides is significant, {Cn,k } takes the following form: Cn,k =
π δ|n−k|−1 2Lc (λ0 )
(2)
where δ is the usual Kr¨ oneker symbol, and Lc (λ0 ) is the wavelength dependent coupling length, which defines the propagation scale over which the power leakage between neighboring waveguides occurs. Integration of Eqn. (1) leads to the construction of an Hermitian matrix {an,k } describing the transformation of the input state Ak into the output state Un : N X Un = an,k Ak (3) k=1
For a coupling matrix like in Eqn. (2) and an infinite array, the coefficients can be calculated exactly:9 {an,k } = in−k Jn+k (2πz/Lc )
(4)
In the case of finite two-dimensional arrays, the use of virtual sources allows the analytical evaluation of the transfer matrix.10 Given the field amplitudes in each waveguide at the end of the array, Eqn. (3) shows that the input field amplitudes can be retrieved inverting the matrix {an,k }. In the following section, we show that under appropriate conditions, a measurement of the power carried by each waveguide can be related trough a linear operator to the mutual first order coherence functions of the fields coupled to 3 selected input waveguides of a 3x3 array. As will be clear later on, this fact enables the application of arrays of fibers for beam combination.
2.2 Complex visibilities from a discrete set of intensity measurements Consider 3 fields Ak injected in 3 separate sites of a N = 9 sites waveguide array. By squaring equation 3 and taking the time average we can evaluate the intensity of the modes in each waveguide at the end of the array: 3 2 X 2 In =< |Un | >=< an,f (k) Ak >, (5) k=1
where < · > indicates the time averaging and f (k) is a function mapping 1...3 onto the excited sites of the 9 waveguides array. The development of the sum on the right-hand-side leads to a sum of 9 terms: In = |an,f (1) |2 < |A1 |2 > +|an,f (2) |2 < |A2 |2 > +|an,f (3) |2 < |A3 |2 > +an,f (1) a∗n,f (2) < A1 A∗2 > +an,f (2) a∗n,f (1) < A2 A∗1 > +an,f (1) a∗n,f (3) < A1 A∗3 > +an,f (3) a∗n,f (1) < A3 A∗1 >
(6)
+an,f (2) a∗n,f (3) < A2 A∗3 > +an,f (3) a∗n,f (2) < A3 A∗2 > If we take into account the definition of complex coherence function (complex visibility): Γk,l =< Aj A∗k >= γj,k |Aj | |Ak |
(7)
it is clear from Eqn. (7) that the intensity of the field in each waveguide is a linear combination of the mutual coherence functions Γk,l of the input fields. Equation (7) can be cast in the form of a real-valued matrix multiplication N X In = αn,k Jk , (8) k=1
provided that we substitute the complex coherence terms in Eqn. (7) with combinations the quadrature components Jk of the complex visibilities: J1 =< |A1 |2 >, J4 + iJ7 =< J5 + iJ8 =< J7 + iJ9 =<
A1 A∗2 A1 A∗3 A2 A∗3
J2 =< |A2 |2 >,
>= Γ12
J3 =< |A3 |2 >
J4 − iJ7 =< A2 A∗1 >= Γ∗12
>= Γ13
J5 − iJ8 =<
>= Γ23
J7 − iJ9 =<
A3 A∗1 A3 A∗2
>= >=
(9)
Γ∗13 Γ∗23
With this change of variables, the elements of the array {αn,k } are: αn,1 αn,4 αn,7
|an,f (1) |2 αn,2 = |an,f (2) |2 αn,3 = |an,f (3) |2 = 2Re an,f (1) a∗n,f (2) αn,5 = 2Re an,f (2) a∗n,f (3) αn,6 = 2Re an,f (1) a∗n,f (3) = −2Im an,f (1) a∗n,f (2) αn,8 = −2Im an,f (1) a∗n,f (3) αn,9 = −2Im an,f (2) a∗n,f (3) =
(10)
The non-singularity of the {αn,k } is the minimal condition to be fulfilled in order to derive the complex visibilities from intensity measurements at the end of the waveguide array. Moreover, because any measurement implies a finite precision, an additional requirement is that the matrix is well conditioned, i.e. the solution Jk of Eqn. (8) should not be too sensitive to the errors in the data In . A good quality parameter to gauge the sensitivity to noise of the solution of linear systems of equations is the condition number11 κ defined as: κ=
σmax (α) σmin (α)
(11)
Where σmax and σmin represent the maximum and minimum modulus of the eigenvalues of the matrix {α}. As discussed below, these conditions can be fulfilled in a two dimensional array of fibers with a specific length by selecting appropriate injection points for the fields An .
3. APPLICATION TO ASTROINTERFEROMETRY A possible application of the result discussed in the previous section is the multiple beam combination in an interferometric array of 3 telescopes. Following the scheme of Figure 1, the image of the stars are sampled by single mode fibers connected to specific sites of a two-dimensional array of N = 9 waveguides. Coupling to the fibers can be enhanced by adaptive optics and can be as high as ≈ 80%.12 To insure coherent mixing of the input fields inside the waveguide array, the optical path difference between the various telescopes should be adjusted by means of delay lines (not shown) placed in the relay between the telescope pupil and the focal plane (where the coupling into the fibre takes place). The readout of output intensity (power) pattern of the waveguide array can be performed in several ways. For nearly monochromatic input, the output of the array can be imaged onto a CCD camera or linked to 9 single-pixel detectors by means of optical fibers. The quadrature components of the coherence function and the in-coupled power by each telescope can be thus retrieved for all possible baselines of the 3 telescopes array. For spectrally broadband input, the chromatic dispersion of the waveguide coupling coefficients may be significant, therefore a spectrally resolved readout of the matrix is required. To this end, the output sites of the two dimensional array can be linked to a linear array of fibers placed at the entrance slit of an imaging monochromator coupled to a CCD array. This configuration allows for spectrally resolved interferometry as for
Figure 1. Schematic view of an astrointerferometer featuring a discrete 3-beam combiner. The whole-field output pattern of array of waveguides can be monitored with a CCD camera, or an array of 9 single-pixel detectors. The 3x3 waveguide array can be demultiplexed into a linear array of waveguides and fed to an imaging spectrometer to obtain complex visibilities resolved in wavelength.
the AMBER and MIDI instruments at VLTI, thus potentially providing precious astrophysical information e.g. about the distribution of moving masses in astronomical targets.13 The array of waveguides could be fabricated by means of laser-writing techniques.6, 7 This technology allows the fabrication of complex three-dimensional refractive structures in transparent media by controlled damage of the material operated by tightly focused ultrashort laser-pulses. The technique can be used on a wide variety of materials, therefore potentially allowing the fabrication on materials where traditional microfabrication techniques such as photolithography cannot be used. The array size is determined by the coupling strength between waveguides. This parameter can be tailored by controlling i) the mode size supported by the waveguides and/or ii) the separation between the waveguides. The first task can be accomplished by tayloring the refractive index modulation depth of the waveguide by controlling the power of the writing laser. Typical arrays of single mode fibers feature waveguide separations of a few 10 µm and coupling lengths of a few centimeters. A possible implementation of the discrete beam combiner could occupy a volume of 1x1x100 = 100 mm3 , which makes the proposed device especially interesting for future space-borne interferometers.14
4. NUMERICAL RESULTS We have tested the potential of our scheme to retrieve complex visibilities for the paradigmatic case of an array of three telescopes. This is in fact the minimal configuration allowing closure phase measurements.15 The optimal arrangement of the three sites on the 9 waveguides array was tested by evaluating the conditional number of the matrix {αn,k } for each of the 84 possible configurations and for sample lengths ranging from 0 to 4 · Lc = 100 mm. The search was simplified by removing from the list of configurations those which are equivalent due to symmetries. The results for a few selected configurations are shown in Figure 2. The majority of the configurations lead to very large conditional numbers (> 1012 ) indicating that the matrices αn,k are nearly singular. A group of configurations shows however low condition numbers (κ ≈ 10) for limited ranges of sample lengths. To asses the performance of the discrete beam combiner, we simulated an observation of the triple star depicted in Figure 3(a). The two off axis components of the system have a separation of 7 mas and 10 mas from the central star and a flux (normalized to the central star) of 1.3 and 1.75 magnitudes, respectively. A gaussian profile has been assumed for all the stars with a 0.6 mas FWHM diameter. The object was observed at a wavelength of λ0 = 1.6µm by means of the telescope array depicted in figure 3(b) during 6 hours. For simplicity, it has been assumed that the star is located along the direction of the rotation
Figure 2. Condition number of the matrix {αn,k } as a function of the waveguide array length and calculated for selected input configurations (inset). The chosen coupling length is Lc = 25 mm.
Figure 3. Simulation of the observation of a triple-star target. (a) The simulated triple star system. Star diameters: 0.6 mas. Secondary component: flux relative to primary = 1.3 mag; separation = 7 mas. Tertiary component: flux relative to primary = 1.75 mag; separation = 10 mas. (b) Arrangement of the 3 telescope array used to observe the simulated system. (c) Amplitude of the coherence function in the u, v plane. The traces of the baselines during a 6 hours observation are overlaid to the plot. The observation wavelength was chosen to be λ0 = 1.6µm.
axis of the Earth, and the telescope array is located at the pole where the star is observable. The traces of the sampled points in the (u, v) plane during the observation are superimposed to Figure 3(c), which represents the calculated amplitudes of the visibilities of the triple star system. For the chosen baselines, the amplitude of the visibilities and the closure phase (sum of the phases of the three coherence functions15 ) during the observation time are shown in Figure 4(a) and (b), along with the estimated values and measurement errors obtained from the realistic modeling of the discrete beam combiner. In the model, we calculate for the selected observation points the expected intensities at the output of the discrete beam combiner {In }. We then model the error in the intensity measurement by adding to each {In } a pseudorandom number representing the photon shot noise of the detected intensity. We assume that the detector has a minimum shot noise of 0.5%, which is typical for 16 bit detectors working near the saturation level, which we set equal to the maximum value of the {In }. For each of the 100 noise realizations we calculate the retrieved values of {Jn }. These sets are then used to calculate the average value and the standard deviation of the amplitude of
Figure 4. Retrieved (points) and true (lines) visibilities (a) and closure phases (b) for the simulated star system of Figure 3(a) observed with the hypothetical observatory depicted in Fig. 3(b).
the visibilities and the closure phases. The simulations point out that the proposed beam combiner could reach a precision of the normalized visibility measurement of ±0.025, while the the closure phase could be determined with a error below ±0.5 degree. While these values are in line with the current performance of existing interferometers, we point out that the performance of the proposed wave coherence meter can be improved mainly by the use of detectors with larger dynamical range. Moreover, the optimization of the geometry of the fiber array could lead to a reduction of the condition number of matrix {αn,k }, and a consequent reduction of the error propagation effect.
5. CONCLUSIONS In this work, we have shown that the mutual coherence functions of 3 fields can be evaluated by measuring the power delivered by individual waveguides forming a coupled 3x3 array. A simple linear system of equations relates the power pattern at the end of the array to the quadrature components of the complex coherence function of the fields. The method can be applied to determine the complex visibility of an astronomical target observed by an interferometric array of 3 telescopes. The scheme naturally provides the complex visibilities over all possible baselines. A realistic simulation of the observation of a triple star system reveals that the discrete beam combiner could afford a performance comparable with existing beam combiners. In the simulation, the main limitation to the precision of the retrieved visibilities arises from the limited dynamic of the detectors. Further improvement can derive by a reduction of the condition number of the system of equations linking the measurements to the quadrature components of the coherence function. We remark that the devised scheme is not limited by the number of chosen telescope and therefore can be virtually scaled up to arbitrary of telescopes and baselines.
REFERENCES 1. O. Chesneau, D. P. K. Banerjee, F. Millour, N. Nardetto, S. Sacuto, A. Spang, M. Wittkowski, N. Ashok, R. K. Das, C. Hummel, S. Kraus, E. Lagadec, S. Morel, M. Petr-Gotzens, F. Rantakyro, and M. Schoeller, “Vlti monitoring of the dust formation event of the nova v1280 scorpii,” A&A 487, pp. 223–235, 2008. 2. J. D. Monnier, M. Zhao, E. Pedretti, N. Thureau, M. Ireland, P. Muirhead, J.-P. Berger, R. Millan-Gabet, G. Van Belle, T. ten Brummelaar, H. McAlister, S. Ridgway, N. Turner, L. Sturmann, J. Sturmann, and D. Berger, “Imaging the Surface of Altair,” Science 317(5836), pp. 342–345, 2007. 3. G. T. van Belle, “Closure phase signatures of planet transit events,” Pub. Astr. Soc. Pac. 120, pp. 617–624, 2008. 4. J.-B. L. Bouquin, P. Labeye, F. Malbet, L. Jocou, F. Zabihian, K. Rousselet-Perraut, J.-P. Berger, A. Delboulbe, P. Kern, A. Glindemann, and M. Schoeller, “Integrated optics for astronomical interferometry . vi. coupling the light of the vlti in k band,” A&A 450, pp. 1259–1264, 2006.
5. M. Benisty, J.-P. Berger, L. Jocoul, P. Labeye, F. Malbet, K. Perraut, and P. Kern, “An integrated optics beam combiner for the second generation vlti instruments,” A&A 498, pp. 601–613, 2009. 6. T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tuennermann, “Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,” Opt. Lett. 29, pp. 468–470, 2004. 7. R. R. Thomson, A. K. Kar, and J. Allington-Smith, “Ultrafast laser inscription: an enabling technology for astrophotonics,” Opt. Exp. 17, pp. 1963–1969, 2009. 8. S. Minardi and T. Pertsch, “Astrointerferometry with discrete optics,” arXiv:1005.5000v1 [physics.optics] . 9. A. L. Jones, “Coupling of optical fibers and scattering in fibers,” J. Opt. Soc. Am. 55, pp. 261–271, 1965. 10. A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tuennermann, U. Peschel, and F. Lederer, “Light evolution in arbitrary two-dimensional waveguide arrays,” Phys. Rev. A 75, p. 053814, 2007. 11. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. D. Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK User’s Guide, SIAM, Philadelphia, 1999. http://www.netlib.org/lapack/lug/lapack lug.html. 12. S. Shaklan and F. Roddier, “Coupling starlight into single-mode fiber optics,” Appl. Opt. 27, pp. 2334–2338, 1988. 13. G. Weigelt, T. Driebe, K.-H. Hofmann, S. Kraus, R. Petrov, and D. Schertl, “Amber/vlti observations of η carinae with high spatial resolution and spectral resolutions of λ/δλ = 1500 and 12000,” New Astr. Rev. 51, pp. 724–729, 2007. 14. C. V. M. Fridlund, “The darwin mission,” Advances in Space Research 34, pp. 613–617, 2004. 15. D. H. Rogstad, “A technique for measuring the visibility phase with an optical interferometer in the presence of atmospheric seeing,” Appl. Opt. 7, p. 585, 1968.
