- Email: [email protected]
Imaging laboratory tests on a fiber linked telescope array
1 February 1999
Optics Communications 160 Ž1999. 27–32
Imaging laboratory tests on a fiber linked telescope array L. Delage a , F. Reynaud a , E. Thiebaut ´ b
b
a IRCOM (CNRS UMR 6615), Equipe optique, 123, rue A. Thomas, F-87060 Limoges Cedex, France Centre de Recherche Astronomique de Lyon (CNRS UMR 5574), 9, aÕenue Charles Andre, ´ F-69561 Saint Genis LaÕal Cedex, France
Received 3 August 1998; revised 10 November 1998; accepted 12 November 1998
Abstract We report laboratory experimental results obtained with end to end image reconstruction breadboard including a 2D object, a highly birefringent fiber linked telescope array, an interferometric mixing assembly and an image restoration algorithm. The full control of birefringent effects allows the acquisition of the object complex Fourier spectrum Žphase and modulus.. The results consist of an accurate image reconstruction compared with the original source. The image restoration algorithm seeks for the object brightness distribution that satisfies two criteria: goodness of fit with the data and regularization. Future improvements of both the instrumental setup and image restoration algorithm are discussed. q 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Interferometers; Image processing techniques; Interferometric techniques
1. Introduction Following the idea proposed by Froehly w1x, few teams have investigated the optical fiber use for the implementation of very large interferometers in high angular astronomical resolution w2–4x. In the frame of an all fiber device, Shaklan w5x has shown preliminary results demonstrating the possibility of using standard fibres for imaging restoration. Unfortunately, the polarisation status control with this type of fiber is very difficult to achieve and a bad correction of this polarisation status can induce loss of interferometer calibration. In this paper, we propose an alternative solution consisting in using a highly birefringent ŽHB. fiber in a laboratory stellar interferometer devoted to demonstrate the HB fiber capabilities in the frame of stellar high resolution image reconstruction. The main features of the coherent propagation of light in HB optical fibers have been demonstrated earlier w6,7x. Highly contrasted fringes have been obtained in a three fiber interferometer including 25 m arms. In this experiment the source was randomly polarised point like radiating a broadband spectrum Ž650–850 nm. in order to calibrate the interferometer w8,9x. Dispersion and birefrin-
gent differential effects were cancelled and a servo system was able to control the optical path of each interferometric arm with a typical 2 nm accuracy. The following step, reported in this paper, consists in implementation of a breadboard allowing to test the object Fourier spectrum acquisition in a laboratory stellar interferometer and to confront the image restoration algorithm with real experimental data. This end to end experiment has been designed in order to be robust, leading to a full control of the experimental parameters. This experiment is designed with a quasimonochromatic source in order to avoid wavelength dependence of spatial frequencies sampled by the telescope arrays. Care has been taken to implement an object with an accurate control of the intensity distribution over large pattern variety not restricted to binary intensity distribution. Using separated lasers to feed each pixel, the spatial incoherence of the source is accurately fulfilled. The telescope array uses a set of pupils with periodic spacing that is not optimized to reduce the number of acquisitions but yields a simple determination of the whole phase and modulus of the object Fourier spectrum. This
0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 6 3 3 - 6
28
L. Delage et al.r Optics Communications 160 (1999) 27–32
procedure is derived from the method initially proposed by Rogstad w10x and developed by Greenaway w11x. It allows to eliminate phase distortion due to turbulence by using the phase closure technique. Moreover, the determination of the fringe phase in an interferometer makes it necessary to build an experimental set up with interferometric arms characterized by a unique optical path. The requirement makes the use of HB fiber mandatory w12x to avoid crosstalk between the two polarization modes and to control the polarization status along interferometric arms whatever the environmental conditions and the bandwidth spectrum source used. Our image restoration algorithm is derived from those developed for bispectral and radio-astronomical data. We further regularize the inversion problem by making use of a smoothing constraint and we extend the definition of the likelihood term to account for two different kind of data: Fourier modulus data and Fourier phase or closure data.
2. Experimental setup The experimental set up is shown in Fig. 1 and consists of a three telescope array with a fiber linkage. The waveguides are optical maintaining polarization singlemode fibers HB600 from York Corporation.
In order to simplify the experimental work, linearly polarized and monochromatic elementary sources have been used. For a real astronomical observation, the stellar interferometer will have to include additional functions such as dispersion and birefringence compensation and an internal metrology w6,7x. In the frame of this laboratory demonstration, those functions are not necessary. Moreover, since this laboratory experimental work is devoted to image reconstruction, the use of a wide spectral bandwidth source is not necessary and becomes a drawback because the spatial frequency is wavelength dependent. Consequently, a wide spectral bandwidth would necessitate a spectroscopic mode analysis. This would not present a significant novelty but would introduce strong experimental constraints and require significant extra cost. Conversely the polarization control is mandatory and achieved by using HB fibers taking care of aligning the wave-guide principal axes w13x. The source is a two-dimensional laboratory object. Each pixel of the object consists of a monomode fiber fed by a He–Ne laser radiating at wavelength l s 633 nm. The fibers are glued side by side in order to define a pixel matrix. Using independent He–Ne laser emitting uncorrelated optical fields, the object is spatially incoherent. The object intensity distribution is tuned by controlling the launching power efficiency for the different fibers. The
Fig. 1. Optical setup
L. Delage et al.r Optics Communications 160 (1999) 27–32
29
Fig. 2. Fringes produced in the image plane Žleft. and the corresponding Fourier spectrum Žright..
outline of the object is typically a square of 400 mm side. It illuminates a 5 m focal length collimator with a 40 mm clear aperture. Moreover, a direct imaging allows the monitoring of the object intensity. The resolution of the object recorded corresponds to the full u–Õ coverage sampled by the synthesis array. The recorded object intensity stability is in the range of 1% for the brightest source and 3% for the faintest source because of the laser photometric stability and detector noises. Three gradient index lenses act as three telescopes Žhenceforth denoted by T1, T2 and T3. of a real instrument sampling the far field of the object. These 1.8 mm diameter apertures are aligned along the analysis direction. Apertures T1 and T2 remain steady for all the configurations of the input pupil. They are separated by a distance called reference basis Ž b 2 y b1 s D b s 3 mm.. The phase and modulus of the object intensity spectrum at the spatial frequency analyzed by the reference basis are respectively close to 0 and 1. Assuming this condition, the reference basis does not resolve the object. The third aperture is moved along the analysis direction by a step equal to the reference basis Ž b 3 y b 1 s nD b with n integer.. The output pupil is reconfigured by using the optical fiber flexibility. The fiber outputs are glued side by side to give rise to a two-dimensional output pupil array with triangular configuration. This avoids to observe fringe patterns exhibiting a redundant spatial frequency and yields good sampling of the interference pattern. The output fringe patterns are recorded on a CCD camera with a thermocooler ŽFig. 2.. The 8 bits analog to digital conversion of the image intensity leads to 1% intensity error. The Fourier transform of the recorded fringe pattern yields three complex visibility data ŽFig. 2.. A full u–Õ plane coverage is obtained by rotating the object. The data used
in this paper were obtained by using six object rotations by 308 steps and nine baseline configurations per object orientation.
3. Data provided by the instrument After calibration of the visibility attenuation terms 1 and for a given array configuration and object orientation, the fringe pattern ŽFig. 2. resulting from the interferences of pupil i and pupil j yields a complex measurement r i, j expŽiCi, j . Žwhere i s 'y 1 . related to the object Fourier spectrum:
r i , j s O˜e Ci , j s ue
ž ž
b j y bi
l b j y bi
l
/ /
, q wj y wi ,
Ž1. Ž2.
where bi and bj are the corresponding telescope positions Žon a plane perpendicular to the line of sight., l is the wavelength, w i Ž w j . is the phase distortion at pupil i Ž j . and < O˜ Ž u.< and u Ž u. are the modulus and the phase of the object spectrum at frequency u Žsubscripts e and r stand for experimental and restored respectively.. Phase distortion induced by static Žor slowly variable. optical defaults can be calibrated. For a ground based telescope array, phase distortion however includes a random contribution due to turbulence. Since this random contribution cannot be estimated, phase distortion terms must therefore be cancelled in order to recover relevant 1
This attenuation is mainly due to the bad field overlapping in the interferometric recombination.
L. Delage et al.r Optics Communications 160 (1999) 27–32
30
phase information. This is achieved by using phase closure:
4. Regularized image restoration
b Ž u,z . s C 1,2 q C 2,3 y C 1,3
Given the interferometric data D, image reconstruction consists in finding the best values X which parameterize the object brigthness distribution Or Ž x .. For instance: X k s Or Ž x k . where the x k are discrete spatial coordinates. A common procedure to solve such an inverse problem is to minimize the discrepancy between the model and the data. The so-called likelihood of the data given the model provides a convenient measure of this goodness of fit:
s ue Ž u . q ue Ž z . y ue Ž u q z . ,
Ž3.
with u s Ž b 2 y b 1 .rl and z s Ž b 3 y b 2 .rl. Finally one array configuration yields three modulus measurements r Ž u., r Ž z . and r Ž u q z . and one phase closure b Ž u,z . which only depend on the Fourier spectrum of the object brigthness distribution. In the case of a single large telescope, bispectrum is a speckle interferometry technique which aims at getting rid of phase distortions. Bispectrum phase data are the analog of the data provided by phase closure techniques for an array of small apertures. For that reason, methods for recovering the object Fourier phase from phase closure data are almost identical to those developed for the bispectrum w14–17x In the particular experiment considered here, a given configuration provides measurements at frequencies: u s D u, z s Ž n y 1. D u and u q z s nD u with D u ' D brl. In this mono-dimensional case, the phase closure is:
b 1,ny1 ' b Ž u, Ž n y 1 . u . s u 1 q uny1 y un ,
Ž4.
L Ž X . A ylog Ž Pr Ž D < X . . , where PrŽ D < X . is the probability of the data given the model. In practice, minimizing LŽ X . solely provides a solution which may not be unique Žthe problem is ill-posed. and that, at least, displays artifacts due to noise or rounding errors Žthe problem is ill-conditioned.. Besides, all solutions X should be considered as compatible with the data as soon as LŽ X . , Lexpected where Lexpected is the expected value of LŽ X .: Lexpected s L Ž X . Pr Ž D < X . d D ) min X Ž L Ž X . . .
H
Ž6.
where u i ' ueŽ i D u.. Using recurrence, the phase of the object spectrum can be recovered at sampled frequencies up to the phase offset u 1 and modulo-2p uncertainty by the recurrence:
In fact, having LŽ X . - Lexpected results in overfitting the data Žfitting the noise.. One has to account for additional information in order to select a single solution among all those that are compatible with the data. This is obtained by minimizing a penalizing function w19x:
unq1 s u 1 q un y b 1 , n q 2 k np .
Q Ž X . s LŽ X . q m R Ž X . ,
Ž5.
As noted by other authors Žsee e.g. Ref. w14x., using recurrence to recover the phase has the drawback of propagating the noise from the low to the high frequencies. A better processing of the data would be to operate a global image inversion by fitting the phase closures directly w18x as well as the modulus data. Once again, we have preferred simplicity over performance; it however should be noted that our image restoration algorithm can be modified to account for phase closure data rather than phase data. Besides, owing to the high signal-to-noise ratio in our experiment Žstandard deviation of our phase data was sb , 0.06 rad., we do not expect severe effects resulting from noise propagation in recurrence Ž5.. A further simplification occurs since, in our particular setup, the object is unresolved by the small base, i.e. u 1 , 0. Anyway the u 1 value would only induce a lateral shift in the reconstructed image. Finally, phase recovery necessitates to find the k i Ža process called phase unwrapping .; this problem has been addressed by several authors w15–17x. In our experiment, most of the modulus measures are redundant: r 1 s r Ž u. is measured n times, r i s r Ž i = u. Žfor i s 2, . . . , n. are measured twice and the last one rnq1 only once. The redundant modulus data are averaged to improve their signal-to-noise ratio. The signal-to-noise ratio of our averaged modulus measures was SNR r , 9.
Ž7.
where RŽ X . is a regularization term which accounts for the a priori information. The Lagrange multiplier m is justified by the fact that one wants to minimize QŽ X . subject to the constraint that LŽ X . s Lexpected to avoid overfitting of the data. If the measurement noise follows a normal law, the likelihood term is given by the so-called chi-square of the data: LŽ X . ' x 2. For our experimental data, the chi-square can be expanded in two terms: x 2 s xr2 q xu2 where xr2 measures the agreement of the restored and experimental object spectrum moduli over all measured frequencies:
xr2 s Ý u
< O˜r Ž u . < y < O˜e Ž u . < var Ž < O˜e Ž u . < .
2
.
Ž8. 2
Similarly we can write: xu2 s Ý u w ur Ž u . y ue Ž u . x r varw ueŽ u.x but, following Haniff w16x, it is possible to avoid phase unwrapping by rewritting xu2 as:
xu2 s
Ý u
mod 2p w ur Ž u . y ueX Ž u . x var w ueX Ž u . x
2
,
Ž9.
where the function mod 2p Ž . returns its argument modulo 2p in the range Žyp ,q p x and where the phase data ueX Ž u. are now obtained from Eq. Ž5. with all k i set to 0.
L. Delage et al.r Optics Communications 160 (1999) 27–32
31
Fig. 3. True object Žleft. and restored one Žright..
Using the phasors, one can obtain an equivalent expression: X
xu2 s
Ý
var Ž ueX Ž u . .
u
,
Ž 10 .
which further avoids possible problems due to the discontinuity of mod 2p Ž . at "p w17,20x. Actually the discontinuity problem should only matter when the standard deviation of the phase data becomes of the order of one radian. Indeed, we noticed no difference in our results when we used definition Ž9. or Ž10.. The purpose of minimizing the term RŽ X . is to account for sufficient a priori information to make the minimization of O Ž X . well-conditioned and to avoid overfitting of the data. For instance, we must avoid unphysical solutions Že.g. the object brigthness distribution must be positive. and we prefer a solution with limited bandwidth Žsince there is a cutoff frequency in the measurements.. Besides, we expect that the Fourier spectrum of the solution smoothly interpolates missing data between discrete measured frequencies Žthis is equivalent to enforcing a support constraint for the solution: the field-of-view width is typically V ; 1rDu where D u s D brl is the typical distance between measured frequencies.. All these constraints can be enforced thanks to a suitable regularizing function RŽ X .. In the domain of radio astronomy, maximum entropy methods have proved to be efficient in restoring images from discrete and sparse Fourier data Žsee e.g. Ref. w21x for a review.. In maximum entropy methods, the regularizing function is the negentropy w22x: R Ž Or . s Ý Or Ž x . log w Or Ž x . rP Ž x . x y 1 ,
Ž 11.
x
where the so-called prior P Ž x . is the brightness distribution that is the solution in the absence of data. For instance, one commonly uses P Ž x . s c where c is the
expected Žuniform. background level. This criterion prevents negative values and tends to reduce the numberrsize of structures that significantly emerge from the background. In order to further enforce the smoothness of the solution, Horn w23x proposed to use a floating prior that is the seeked object brightness convolved by some smooth function: P Ž x . s Ž S) Or . Ž x . . Since we wanted to prevent spurious high frequencies in the result, we used that latter definition with P given by the discrete convolution of X by the smoothing matrix: Ss
1 2 16 1 1
2 4 2
1 2 . 1
Finally, in order to minimize the penalizing function QŽ X . subject to the constraint LŽ X . s Lexpected , we wrote an algorithm which is similar to that of Skilling and Bryan w24x but which further accounts for the dependence of the prior P with the sought solution X. The restored image along with the original object is shown in Fig. 3 and Table 1 shows that the restored relative positions and photometry compare favorably with Table 1 Relative position and photometry of the 3 faintest stars with respect to the brightest one in the original Žtop row. and restored images Žbottom row. 2nd star
3rd star
4th star
x y Dm
y88.2"0.1 29.6"0.1 0.30"0.01
y55.7"0.1 38.4"0.1 0.64"0.01
y34.9"0.1 y1.3"0.1 0.84"0.01
x y Dm
y88.7"0.8 29.7"0.3 0.4"0.1
y57.1"0.8 39.4"0.7 0.7"0.1
y35.4"0.9 y0.1"0.1 0.9"0.2
L. Delage et al.r Optics Communications 160 (1999) 27–32
32
that of the original distribution. We have noticed that augmenting m so as to over-regularize the solution leads to a bias that mainly affects the photometry of the solution. w3x
5. Perspectives and conclusion This preliminary experiment has demonstrated the possibility to reconstruct images accurately using a fiber linked telescope array. This technique is fully compatible to a real stellar interferometer as long as different functions for coherent propagation of light are implemented w7x. The experiment has been achieved with 0.7 nW collected in the focal plane of each telescope. Such a power is equivalent to a 0 magnitude star observed by three 1.6 m telescopes over a 100 nm spectral bandwidth around 550 nm w25x. The operating wavelength can be chosen over the silica fiber transparency window Ž400–1800 nm. in order to take advantage of the high technological development in the fiber telecommunication domain. Of course this experimental configuration is preliminary and many improvements are currently under study. In order to simplify the detection assembly and to make the interferometer easy to be servo controlled, the experimental set up is being modified in order to display fringes versus time using a temporal linear optical path modulation. This technique is derived from that of Delaire and Reynaud w26x. We also want to minimize the number of telescope configurations following, e.g., Mugnier et al. w27x. Besides, we plan to modify our image restoration algorithm in order to directly fit the closure phase data. This could be simply achieved by rewriting xu2 as:
xu2
w4x
w5x w6x w7x
w8x
w9x
w10x w11x w12x
w13x
w14x w15x
sÝ
mod 2p w ur Ž u . q ur Ž z . y ur Ž u q z . y be Ž u,z . x
u ,z
var w be Ž u,z . x
2
,
and would improve the performance of the algorithm when dealing with noiser data.
w16x w17x w18x w19x w20x
Acknowledgements This study has been financially supported under Ultimatech grant No. 96N89r0062.
References w1x C. Froehly, Coherence and interferometry through optical fiber, in: Ulrich, Kjar ¨ ŽEds.., Proc. ESO Conf. High Angular Resolution, 1981, p. 285. w2x P. Connes, F. Reynaud, Fiber tests on a radiotelescope, in: F.
w21x w22x
w23x w24x w25x w26x w27x
Merkle ŽEd.., Proc. NOAO–ESO Conference on High Resolution Imaging by Interferometry, European Southern Observatory, Garching bei Munchen, Germany, 1988, pp. 1117– 1129. V. Coude´ Du Foresto, Interferometrie astronomique in´ ´ frarouge par optique guidee ´ monomode, Ph.D. thesis, Universite´ Paris VII, 1994. S.R. Restaino, J. Baker, R.A. Carreras, J.L. Fender, G.C. Loos, R.J. McBroom, D. Nahrstedt, D.M. Payne, Meas. Sci. Technol. 8 Ž1997. 1105. S. Shaklan, Opt. Eng. 29 Ž1990. 684. J. Alleman, F. Reynaud, P. Connes, Appl. Opt. 34 Ž1995. 2284. F. Reynaud, J. Alleman, H. Lagorceix, Interferometric fiber arms for stellar interferometry, in: International Symposium on Space Optics, SPIE Proc., vol. 2209, Garmish–Partenkirchen, FRG, 1994, pp. 431–435. F. Reynaud, H. Lagorceix, Stabilization and control of a fiber array for the coherent transport of beams in a stellar interferometer, in: P. Kern, F. Malbet ŽEds.., Conf. Astrofib’96, Integrated Optics for Astronomical Interferometry, 1996, pp. 249–257. H. Lagorceix, Application des fibres optiques unimodales a` l’interferometrie stellaire, Ph.D. thesis, Universite´ de Limo´ ´ ges, 1995. D.H. Rogstad, Appl. Optics 7 Ž1968. 585. A.H. Greenaway, Optics Comm. 42 Ž1982. 157. L. Delage, F. Reynaud, Analysis of polarisation requirement in a fiber linked stellar interferometer, in: P. Kern, F. Malbet ŽEds.., Conf. Astrofib’96, Integrated Optics for Astronomical Interferometry, 1996, pp. 37–43. L. Delage, Mise en oeuvre et applications d’interferometres ` stellaires a` trois voies utilisant des fibres optiques unimodales en silice, Ph.D. thesis, Universite´ de Limoges, 1998. J.D. Freeman, J.C. Christou, F. Roddier, D.W. McCarthy, M.L. Cobb, J. Opt. Soc. Am. A 5 Ž1988. 406. J.C. Marron, P.P. Sanchez, R.C. Sullivan, J. Opt. Soc. Am. A 7 Ž1990. 14. C. Haniff, J. Opt. Soc. Am. A 8 Ž1991. 134. C.L. Matson, J. Opt. Soc. Am. A 8 Ž1991. 1905. A. Glindemann, J.C. Dainty, J. Opt. Soc. Am. A 10 Ž1993. 1056. D.M. Titterington, Astron. Astrophys. 144 Ž1985. 381. E. Thiebaut, Imagerie astrophysique a` la limite de diffraction ´ des grands telescopes. Application a` l’observation des objets ´ froids, Ph.D. thesis, Universite´ de Paris 7, 1994. R. Narayan, R. Nityananda, ARA&A 24 Ž1986. 127. J. Skilling, Maximum Entropy and Bayesian Methods, chapter Classic maximum entropy, Kluwer Academic, 1989, pp. 45–52. K. Horne, MNRAS 213 Ž1985. 129. J. Skilling, R. Bryan, MNRAS 211 Ž1984. 111. C. Allen, Astrophysical Quantities, The Athlone Press, University of London, 3rd ed, 1976. E. Delaire, F. Reynaud, Electron. Lett. 29 Ž1993. 1718. L. Mugnier, G. Rousset, F. Cassaing, J. Opt. Soc. Am. A 13 Ž1996. 2367.
Optics Communications 160 Ž1999. 27–32
Imaging laboratory tests on a fiber linked telescope array L. Delage a , F. Reynaud a , E. Thiebaut ´ b
b
a IRCOM (CNRS UMR 6615), Equipe optique, 123, rue A. Thomas, F-87060 Limoges Cedex, France Centre de Recherche Astronomique de Lyon (CNRS UMR 5574), 9, aÕenue Charles Andre, ´ F-69561 Saint Genis LaÕal Cedex, France
Received 3 August 1998; revised 10 November 1998; accepted 12 November 1998
Abstract We report laboratory experimental results obtained with end to end image reconstruction breadboard including a 2D object, a highly birefringent fiber linked telescope array, an interferometric mixing assembly and an image restoration algorithm. The full control of birefringent effects allows the acquisition of the object complex Fourier spectrum Žphase and modulus.. The results consist of an accurate image reconstruction compared with the original source. The image restoration algorithm seeks for the object brightness distribution that satisfies two criteria: goodness of fit with the data and regularization. Future improvements of both the instrumental setup and image restoration algorithm are discussed. q 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Interferometers; Image processing techniques; Interferometric techniques
1. Introduction Following the idea proposed by Froehly w1x, few teams have investigated the optical fiber use for the implementation of very large interferometers in high angular astronomical resolution w2–4x. In the frame of an all fiber device, Shaklan w5x has shown preliminary results demonstrating the possibility of using standard fibres for imaging restoration. Unfortunately, the polarisation status control with this type of fiber is very difficult to achieve and a bad correction of this polarisation status can induce loss of interferometer calibration. In this paper, we propose an alternative solution consisting in using a highly birefringent ŽHB. fiber in a laboratory stellar interferometer devoted to demonstrate the HB fiber capabilities in the frame of stellar high resolution image reconstruction. The main features of the coherent propagation of light in HB optical fibers have been demonstrated earlier w6,7x. Highly contrasted fringes have been obtained in a three fiber interferometer including 25 m arms. In this experiment the source was randomly polarised point like radiating a broadband spectrum Ž650–850 nm. in order to calibrate the interferometer w8,9x. Dispersion and birefrin-
gent differential effects were cancelled and a servo system was able to control the optical path of each interferometric arm with a typical 2 nm accuracy. The following step, reported in this paper, consists in implementation of a breadboard allowing to test the object Fourier spectrum acquisition in a laboratory stellar interferometer and to confront the image restoration algorithm with real experimental data. This end to end experiment has been designed in order to be robust, leading to a full control of the experimental parameters. This experiment is designed with a quasimonochromatic source in order to avoid wavelength dependence of spatial frequencies sampled by the telescope arrays. Care has been taken to implement an object with an accurate control of the intensity distribution over large pattern variety not restricted to binary intensity distribution. Using separated lasers to feed each pixel, the spatial incoherence of the source is accurately fulfilled. The telescope array uses a set of pupils with periodic spacing that is not optimized to reduce the number of acquisitions but yields a simple determination of the whole phase and modulus of the object Fourier spectrum. This
0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 6 3 3 - 6
28
L. Delage et al.r Optics Communications 160 (1999) 27–32
procedure is derived from the method initially proposed by Rogstad w10x and developed by Greenaway w11x. It allows to eliminate phase distortion due to turbulence by using the phase closure technique. Moreover, the determination of the fringe phase in an interferometer makes it necessary to build an experimental set up with interferometric arms characterized by a unique optical path. The requirement makes the use of HB fiber mandatory w12x to avoid crosstalk between the two polarization modes and to control the polarization status along interferometric arms whatever the environmental conditions and the bandwidth spectrum source used. Our image restoration algorithm is derived from those developed for bispectral and radio-astronomical data. We further regularize the inversion problem by making use of a smoothing constraint and we extend the definition of the likelihood term to account for two different kind of data: Fourier modulus data and Fourier phase or closure data.
2. Experimental setup The experimental set up is shown in Fig. 1 and consists of a three telescope array with a fiber linkage. The waveguides are optical maintaining polarization singlemode fibers HB600 from York Corporation.
In order to simplify the experimental work, linearly polarized and monochromatic elementary sources have been used. For a real astronomical observation, the stellar interferometer will have to include additional functions such as dispersion and birefringence compensation and an internal metrology w6,7x. In the frame of this laboratory demonstration, those functions are not necessary. Moreover, since this laboratory experimental work is devoted to image reconstruction, the use of a wide spectral bandwidth source is not necessary and becomes a drawback because the spatial frequency is wavelength dependent. Consequently, a wide spectral bandwidth would necessitate a spectroscopic mode analysis. This would not present a significant novelty but would introduce strong experimental constraints and require significant extra cost. Conversely the polarization control is mandatory and achieved by using HB fibers taking care of aligning the wave-guide principal axes w13x. The source is a two-dimensional laboratory object. Each pixel of the object consists of a monomode fiber fed by a He–Ne laser radiating at wavelength l s 633 nm. The fibers are glued side by side in order to define a pixel matrix. Using independent He–Ne laser emitting uncorrelated optical fields, the object is spatially incoherent. The object intensity distribution is tuned by controlling the launching power efficiency for the different fibers. The
Fig. 1. Optical setup
L. Delage et al.r Optics Communications 160 (1999) 27–32
29
Fig. 2. Fringes produced in the image plane Žleft. and the corresponding Fourier spectrum Žright..
outline of the object is typically a square of 400 mm side. It illuminates a 5 m focal length collimator with a 40 mm clear aperture. Moreover, a direct imaging allows the monitoring of the object intensity. The resolution of the object recorded corresponds to the full u–Õ coverage sampled by the synthesis array. The recorded object intensity stability is in the range of 1% for the brightest source and 3% for the faintest source because of the laser photometric stability and detector noises. Three gradient index lenses act as three telescopes Žhenceforth denoted by T1, T2 and T3. of a real instrument sampling the far field of the object. These 1.8 mm diameter apertures are aligned along the analysis direction. Apertures T1 and T2 remain steady for all the configurations of the input pupil. They are separated by a distance called reference basis Ž b 2 y b1 s D b s 3 mm.. The phase and modulus of the object intensity spectrum at the spatial frequency analyzed by the reference basis are respectively close to 0 and 1. Assuming this condition, the reference basis does not resolve the object. The third aperture is moved along the analysis direction by a step equal to the reference basis Ž b 3 y b 1 s nD b with n integer.. The output pupil is reconfigured by using the optical fiber flexibility. The fiber outputs are glued side by side to give rise to a two-dimensional output pupil array with triangular configuration. This avoids to observe fringe patterns exhibiting a redundant spatial frequency and yields good sampling of the interference pattern. The output fringe patterns are recorded on a CCD camera with a thermocooler ŽFig. 2.. The 8 bits analog to digital conversion of the image intensity leads to 1% intensity error. The Fourier transform of the recorded fringe pattern yields three complex visibility data ŽFig. 2.. A full u–Õ plane coverage is obtained by rotating the object. The data used
in this paper were obtained by using six object rotations by 308 steps and nine baseline configurations per object orientation.
3. Data provided by the instrument After calibration of the visibility attenuation terms 1 and for a given array configuration and object orientation, the fringe pattern ŽFig. 2. resulting from the interferences of pupil i and pupil j yields a complex measurement r i, j expŽiCi, j . Žwhere i s 'y 1 . related to the object Fourier spectrum:
r i , j s O˜e Ci , j s ue
ž ž
b j y bi
l b j y bi
l
/ /
, q wj y wi ,
Ž1. Ž2.
where bi and bj are the corresponding telescope positions Žon a plane perpendicular to the line of sight., l is the wavelength, w i Ž w j . is the phase distortion at pupil i Ž j . and < O˜ Ž u.< and u Ž u. are the modulus and the phase of the object spectrum at frequency u Žsubscripts e and r stand for experimental and restored respectively.. Phase distortion induced by static Žor slowly variable. optical defaults can be calibrated. For a ground based telescope array, phase distortion however includes a random contribution due to turbulence. Since this random contribution cannot be estimated, phase distortion terms must therefore be cancelled in order to recover relevant 1
This attenuation is mainly due to the bad field overlapping in the interferometric recombination.
L. Delage et al.r Optics Communications 160 (1999) 27–32
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phase information. This is achieved by using phase closure:
4. Regularized image restoration
b Ž u,z . s C 1,2 q C 2,3 y C 1,3
Given the interferometric data D, image reconstruction consists in finding the best values X which parameterize the object brigthness distribution Or Ž x .. For instance: X k s Or Ž x k . where the x k are discrete spatial coordinates. A common procedure to solve such an inverse problem is to minimize the discrepancy between the model and the data. The so-called likelihood of the data given the model provides a convenient measure of this goodness of fit:
s ue Ž u . q ue Ž z . y ue Ž u q z . ,
Ž3.
with u s Ž b 2 y b 1 .rl and z s Ž b 3 y b 2 .rl. Finally one array configuration yields three modulus measurements r Ž u., r Ž z . and r Ž u q z . and one phase closure b Ž u,z . which only depend on the Fourier spectrum of the object brigthness distribution. In the case of a single large telescope, bispectrum is a speckle interferometry technique which aims at getting rid of phase distortions. Bispectrum phase data are the analog of the data provided by phase closure techniques for an array of small apertures. For that reason, methods for recovering the object Fourier phase from phase closure data are almost identical to those developed for the bispectrum w14–17x In the particular experiment considered here, a given configuration provides measurements at frequencies: u s D u, z s Ž n y 1. D u and u q z s nD u with D u ' D brl. In this mono-dimensional case, the phase closure is:
b 1,ny1 ' b Ž u, Ž n y 1 . u . s u 1 q uny1 y un ,
Ž4.
L Ž X . A ylog Ž Pr Ž D < X . . , where PrŽ D < X . is the probability of the data given the model. In practice, minimizing LŽ X . solely provides a solution which may not be unique Žthe problem is ill-posed. and that, at least, displays artifacts due to noise or rounding errors Žthe problem is ill-conditioned.. Besides, all solutions X should be considered as compatible with the data as soon as LŽ X . , Lexpected where Lexpected is the expected value of LŽ X .: Lexpected s L Ž X . Pr Ž D < X . d D ) min X Ž L Ž X . . .
H
Ž6.
where u i ' ueŽ i D u.. Using recurrence, the phase of the object spectrum can be recovered at sampled frequencies up to the phase offset u 1 and modulo-2p uncertainty by the recurrence:
In fact, having LŽ X . - Lexpected results in overfitting the data Žfitting the noise.. One has to account for additional information in order to select a single solution among all those that are compatible with the data. This is obtained by minimizing a penalizing function w19x:
unq1 s u 1 q un y b 1 , n q 2 k np .
Q Ž X . s LŽ X . q m R Ž X . ,
Ž5.
As noted by other authors Žsee e.g. Ref. w14x., using recurrence to recover the phase has the drawback of propagating the noise from the low to the high frequencies. A better processing of the data would be to operate a global image inversion by fitting the phase closures directly w18x as well as the modulus data. Once again, we have preferred simplicity over performance; it however should be noted that our image restoration algorithm can be modified to account for phase closure data rather than phase data. Besides, owing to the high signal-to-noise ratio in our experiment Žstandard deviation of our phase data was sb , 0.06 rad., we do not expect severe effects resulting from noise propagation in recurrence Ž5.. A further simplification occurs since, in our particular setup, the object is unresolved by the small base, i.e. u 1 , 0. Anyway the u 1 value would only induce a lateral shift in the reconstructed image. Finally, phase recovery necessitates to find the k i Ža process called phase unwrapping .; this problem has been addressed by several authors w15–17x. In our experiment, most of the modulus measures are redundant: r 1 s r Ž u. is measured n times, r i s r Ž i = u. Žfor i s 2, . . . , n. are measured twice and the last one rnq1 only once. The redundant modulus data are averaged to improve their signal-to-noise ratio. The signal-to-noise ratio of our averaged modulus measures was SNR r , 9.
Ž7.
where RŽ X . is a regularization term which accounts for the a priori information. The Lagrange multiplier m is justified by the fact that one wants to minimize QŽ X . subject to the constraint that LŽ X . s Lexpected to avoid overfitting of the data. If the measurement noise follows a normal law, the likelihood term is given by the so-called chi-square of the data: LŽ X . ' x 2. For our experimental data, the chi-square can be expanded in two terms: x 2 s xr2 q xu2 where xr2 measures the agreement of the restored and experimental object spectrum moduli over all measured frequencies:
xr2 s Ý u
< O˜r Ž u . < y < O˜e Ž u . < var Ž < O˜e Ž u . < .
2
.
Ž8. 2
Similarly we can write: xu2 s Ý u w ur Ž u . y ue Ž u . x r varw ueŽ u.x but, following Haniff w16x, it is possible to avoid phase unwrapping by rewritting xu2 as:
xu2 s
Ý u
mod 2p w ur Ž u . y ueX Ž u . x var w ueX Ž u . x
2
,
Ž9.
where the function mod 2p Ž . returns its argument modulo 2p in the range Žyp ,q p x and where the phase data ueX Ž u. are now obtained from Eq. Ž5. with all k i set to 0.
L. Delage et al.r Optics Communications 160 (1999) 27–32
31
Fig. 3. True object Žleft. and restored one Žright..
Using the phasors, one can obtain an equivalent expression: X
xu2 s
Ý
var Ž ueX Ž u . .
u
,
Ž 10 .
which further avoids possible problems due to the discontinuity of mod 2p Ž . at "p w17,20x. Actually the discontinuity problem should only matter when the standard deviation of the phase data becomes of the order of one radian. Indeed, we noticed no difference in our results when we used definition Ž9. or Ž10.. The purpose of minimizing the term RŽ X . is to account for sufficient a priori information to make the minimization of O Ž X . well-conditioned and to avoid overfitting of the data. For instance, we must avoid unphysical solutions Že.g. the object brigthness distribution must be positive. and we prefer a solution with limited bandwidth Žsince there is a cutoff frequency in the measurements.. Besides, we expect that the Fourier spectrum of the solution smoothly interpolates missing data between discrete measured frequencies Žthis is equivalent to enforcing a support constraint for the solution: the field-of-view width is typically V ; 1rDu where D u s D brl is the typical distance between measured frequencies.. All these constraints can be enforced thanks to a suitable regularizing function RŽ X .. In the domain of radio astronomy, maximum entropy methods have proved to be efficient in restoring images from discrete and sparse Fourier data Žsee e.g. Ref. w21x for a review.. In maximum entropy methods, the regularizing function is the negentropy w22x: R Ž Or . s Ý Or Ž x . log w Or Ž x . rP Ž x . x y 1 ,
Ž 11.
x
where the so-called prior P Ž x . is the brightness distribution that is the solution in the absence of data. For instance, one commonly uses P Ž x . s c where c is the
expected Žuniform. background level. This criterion prevents negative values and tends to reduce the numberrsize of structures that significantly emerge from the background. In order to further enforce the smoothness of the solution, Horn w23x proposed to use a floating prior that is the seeked object brightness convolved by some smooth function: P Ž x . s Ž S) Or . Ž x . . Since we wanted to prevent spurious high frequencies in the result, we used that latter definition with P given by the discrete convolution of X by the smoothing matrix: Ss
1 2 16 1 1
2 4 2
1 2 . 1
Finally, in order to minimize the penalizing function QŽ X . subject to the constraint LŽ X . s Lexpected , we wrote an algorithm which is similar to that of Skilling and Bryan w24x but which further accounts for the dependence of the prior P with the sought solution X. The restored image along with the original object is shown in Fig. 3 and Table 1 shows that the restored relative positions and photometry compare favorably with Table 1 Relative position and photometry of the 3 faintest stars with respect to the brightest one in the original Žtop row. and restored images Žbottom row. 2nd star
3rd star
4th star
x y Dm
y88.2"0.1 29.6"0.1 0.30"0.01
y55.7"0.1 38.4"0.1 0.64"0.01
y34.9"0.1 y1.3"0.1 0.84"0.01
x y Dm
y88.7"0.8 29.7"0.3 0.4"0.1
y57.1"0.8 39.4"0.7 0.7"0.1
y35.4"0.9 y0.1"0.1 0.9"0.2
L. Delage et al.r Optics Communications 160 (1999) 27–32
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that of the original distribution. We have noticed that augmenting m so as to over-regularize the solution leads to a bias that mainly affects the photometry of the solution. w3x
5. Perspectives and conclusion This preliminary experiment has demonstrated the possibility to reconstruct images accurately using a fiber linked telescope array. This technique is fully compatible to a real stellar interferometer as long as different functions for coherent propagation of light are implemented w7x. The experiment has been achieved with 0.7 nW collected in the focal plane of each telescope. Such a power is equivalent to a 0 magnitude star observed by three 1.6 m telescopes over a 100 nm spectral bandwidth around 550 nm w25x. The operating wavelength can be chosen over the silica fiber transparency window Ž400–1800 nm. in order to take advantage of the high technological development in the fiber telecommunication domain. Of course this experimental configuration is preliminary and many improvements are currently under study. In order to simplify the detection assembly and to make the interferometer easy to be servo controlled, the experimental set up is being modified in order to display fringes versus time using a temporal linear optical path modulation. This technique is derived from that of Delaire and Reynaud w26x. We also want to minimize the number of telescope configurations following, e.g., Mugnier et al. w27x. Besides, we plan to modify our image restoration algorithm in order to directly fit the closure phase data. This could be simply achieved by rewriting xu2 as:
xu2
w4x
w5x w6x w7x
w8x
w9x
w10x w11x w12x
w13x
w14x w15x
sÝ
mod 2p w ur Ž u . q ur Ž z . y ur Ž u q z . y be Ž u,z . x
u ,z
var w be Ž u,z . x
2
,
and would improve the performance of the algorithm when dealing with noiser data.
w16x w17x w18x w19x w20x
Acknowledgements This study has been financially supported under Ultimatech grant No. 96N89r0062.
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