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SPIE Proceedings [SPIE Optical Science and Technology, SPIE's 48th Annual Meeting  San Diego, CA (Sunday..
SPIE Proceedings [SPIE Optical Science and Technology, SPIE's 48th Annual Meeting  San Diego, CA (Sunday 3 August 2003)] Astronomical Adaptive Optics Systems and Applications  Simulations of a longbaseline interferometer with adaptive optics
Ting, Chueh, Giles, Michael K., Voelz, David, Tyson, Robert K., LloydHart, MichaelJak bardzo podobała Ci się ta książka?
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Tom:
5169
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2003
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DOI:
10.1117/12.503492
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Simulations of a LongBaseline Interferometer with Adaptive Optics Chueh Ting*, Michael K. Giles, David Voelz The Klipsch School of Electrical and Computer Engineering New Mexico State University Las Cruces, NM, USA 880038001 ABSTRACT Simulation results of a longbaseline optical interferometer with adaptive optics are presented in this paper. Longbaseline optical interferometers have become useful tools for obtaining detailed stellar information and highresolution images in the astronomy community. Several interferometric systems have been implemented successfully without adaptive optics; however, adaptive optical systems may be needed for a new generation of longbaseline interferometers with large telescopes such as those being developed for the Magdalena Ridge Observatory (MRO). A longbaseline optical interferometer in the turbulent atmosphere is modeled first, then an optical interferometer with an adaptive optics system (AOS) is modeled and the resulting fringe patterns for different input turbulence scales are interpreted. Finally, the performance of a long baseline optical interferometer with and without an AOS is carefully evaluated and recommendations are made for the implementation of adaptive optics in the 1.5meter MRO telescopes. Keywords: stellar interferometry, adaptive optics, atmospheric turbulence. 1. INTRODUCTION During the past decade, longbaseline stellar interferometers have become powerful tools for making accurate measurements of stellar astrophysical parameters in the astronomy community.1,2 Image degradation by the atmosphere is an important issue for a single large telescope or a stellar interferometer with large telescopes. For a single large telescope, adaptive optics has become a promising technology to reduce image degradation due to atmospheric turbulence.3,4,5 For a stellar interferometer, low order adaptive optic correction (e.g., tilt) has been investigated and implemented,6,7 but the advantages of adaptive optics for correction of high order aberration in a stellar; interferometer with large telescopes have not been fully established. Due to the high cost of implementing an adaptive optics system, we use a computer simulation to perform preliminary tradeoff analyses and to evaluate whether or not the performance of a stellar interferometer improves when an adaptive optics system is used for highorder correction. The main goal of this paper is to establish a simulation methodology to evaluate the system performance of a stellar interferometer with and without adaptive optics. P P P P P P The method used to simulate a stellar interferometer with and without an adaptive optical system is based on an analysis of the optical system in the presence of atmospheric turbulence. We first present the geometry and nomenclature used to analyze a simple stellar interferometer comprising two separated telescopes in a turbulent environment. We perform a coherent analysis of the fringe visibility obtained by superimposing the beams from the two telescopes for different scales of turbulence and show that the use of an AOS to correct the degradation of the modulation transfer function (MTF) caused by atmospheric turbulence may indeed improve the overall system performance of a stellar interferometer with large telescopes. System performance is evaluated with and without adaptive optics for different strengths of atmospheric turbulence using two system performance metrics, the system coherence loss or visibility degradation factor, K, and the system Strehl ratio, S. The results obtained using these two metrics are compared, and the similarities of the two metrics are discussed. The methodology and related theories used in this study are introduced in Section 2, the simulation results are presented in Section 3, and the conclusions are discussed in Section 4. *chting@nmsu.edu; phone 1 505 6461911 ext 5; fax 1 505 6461435; lens.nmsu.edu U 250 U Astronomical Adaptive Optics Systems and Applications, edited by Robert K. Tyson, Michael LloydHart, Proceedings of SPIE Vol. 5169 (SPIE, Bellingham, WA, 2003) 0277786X/03/$15 · doi: 10.1117/12.503492 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 05/26/2015 Terms of Use: http://spiedl.org/terms 2. METHODOLOGY The methodology is based on the analysis of an optical system in the presence of atmospheric turbulence. The optical characteristics of the stellar interferometer are presented, the atmospheric MTF is discussed, and then the visibility loss function is derived and compared with the system Strehl ratio. Finally, the visibility loss and the Strehl rato are used as benchmarks to evaluate system performance. The performance with adaptive optics is estimated by selectively removing the degrading effects of tilt and higherorder aberrations from the atmospheric MTF. 2.1 Characteristics of the Stellar Interferometer We analyze a stellar interferometer that consists of two wellcorrected, diffraction limited optical telescopes, each having clear aperture diameter D and focal length f. The baseline distance between the two telescopes is d0 , therefore, the spatial frequency corresponding to the baseline distance is [ 0 = d 0 / Og. [, Using the spatial frequency variable, we define T ([ ) as the wellknown MTF for a single circular telescope: T ([ ) where 2º ª °S «cos 1 §¨ Of[ ·¸ Of[ 1 §¨ Of[ ·¸ » ®2 « © D ¹ D © D ¹ »¼ ° ¬ 0 ¯ [ D Of , (1) D [! Of O is the wavelength. The atmospheric MTF, A([) has been derived by Fried.8 P § § O f[ · ¸¸ A([ ) exp¨ 3.44¨¨ ¨ © r0 ¹ © 5 3 § ¨1 D §¨ Of[ ·¸ ¨ © D ¹ © 1 3 · ·¸ , ¸ ¸¸ ¹¹ (2) where r0 is Fried’s parameter or coherence length. Values for the parameter D are defined in Fried’s paper8 as zero for P P long exposure and nonzero for short exposure. Long exposure corresponds to the uncorrected atmospheric MTF and short exposure to the MTF with tiptilt correction. 2.2 Performance Metrics – Visibility Loss Function and Strehl Ratio Since a stellar interferometer measures the degree of coherence (i.e., the visibility of interference fringes) of the light propagating from an observed source, the coherence loss (i.e., the visibility loss) function is a good measure of its performance. Similarly, the Strehl ratio is a commonly used performance metric for an imaging telescope. Previous studies have shown a direct relationship between Strehl ratio and coherence loss;6,7 however, high order aberration analysis has not been reported explicitly in these studies. In this paper we develop the relationship between the Strehl ratio and the coherence loss, and we investigate the effect of higher order aberrations on each performance metric. P P In the following analysis, we use a twodimensional notation in which the single variable [ actually represents two orthogonal variables. Thus the integrals are calculated in both dimensions. In all cases, the pupil (aperture), wave, and phase functions are analyzed in spatial frequency space as mentioned above in Section 2.1. The complex waves, u1 and u2, passing through the two apertures of the stellar interferometer are combined to produce interference fringes in the image plane (or the pupil plane) of the interferometer. The complex fringe visibility function corresponding to the interference of these two waves can be represented as9 B B B B P V12 P V 1 ³ U 1U 2*\ 1\ 2*W ([ )d 2[ , (3) where W([) is the clear aperture function of each coherent aperture and of the superimposed apertures, V is the clear jI * jI aperture area, U1 and U2 are the magnitudes of the two complex waves, and \ 1 e 1 and \ 2 e 2 represent the phase functions of the two complex waves (zero mean complex Gaussian random variables). Assuming the product B B B B Proc. of SPIE Vol. 5169 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 05/26/2015 Terms of Use: http://spiedl.org/terms 251 *12 U 1U 2* is constant across the superimposed apertures, i.e., the source is unresolved by a single aperture, the complex visibility loss can be defined as9 P V12 *12 K & & V 1 ³ W (r )\ 1 ([ )\ 2* ([ )d 2[ . (4) If the atmospheric turbulence obeys the Kolmogorov model and is stationary and the baseline distance between apertures is sufficient to assume uncorrelated turbulenceinduced phase functions in the two apertures, the mean square value of the complex visibility loss is  K 2 & V 2 & ³ ³ W (r )W (r [ )\ 1 & & & & & & (r )\ 2* (r )\ 1* (r [ )\ 2 (r [ )d 2 rd 2[ V 1 ³ A 2 ([ )T ([ ) d 2[ . & & & ¢\ 1 (r )\ 1* (r [ )² In equation (5), r is a dummy variable of integration, A([ ) (5) & & & ¢\ 2* (r )\ 2 (r [ )² is the atmospheric MTF, and T([) is the MTF for a single telescope which is the normalized aperture autocorrelation function. Therefore W ([ ) W ([ ) VT ([ ) where denotes autocorrelation. Here we have used the following relation for the complex Gaussian process: & & & & & & & & & & & & ¢\ 1 (r )\ 2* (r )\ 1* (r [ )\ 2 (r [ )² = ¢\ 1 (r )\ 1* (r [ )² ¢\ 2* (r )\ 2 (r [ )² & & * & & & * & + ¢\ 1 ( r )\ 2 ( r )² ¢\ 1 (r [ )\ 2 ( r [ )² . (6) The Strehl ratio for a stellar interferometer, in terms of atmospheric MTF and the single telescope MTF, can be represented as:6 P P ³ A([ ) T ([ ) d ³ T ([ ) d [ S 2 [ . 2 (7) 2.3 Stellar Interferometry with Adaptive Phase Correction The method used to simulate the strength of the wavefront aberration produced by the atmosphere follows Fried10 and Wang11. If we assume the residual phase distribution is independent of the compensated phase distribution, the atmospheric MTF with adaptive phase compensation may be written as11: P P P P P m([ ) APC § § O f[ · ¸ M exp¨ 3.44¨¨ ¨ r0 ¸¹ © © 5 3 P · ¸ dT udu exp§¨ Q(u,T ) ·¸ , ¸ ³³ © 2 ¹ ¹ (8) where 2 Q(u, T ) 32 u 6.88 § Of[ · 2 2 ¨ ¸ 0.10243 0.25428u 0.12712u cos 2(T I ) . 5 S r0 3 © D ¹ > @ (9) The physical interpretation for each term in Q(u,T) is explained as follows: the first term is tiptilt, the second term combines tiptilt, defocus and astigmatism, and the last term is defocus. 252 Proc. of SPIE Vol. 5169 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 05/26/2015 Terms of Use: http://spiedl.org/terms 3. Simulation Results To understand how adaptive optics can improve the performance of a stellar interferometer system, we simulate the system first in terms of the visibility loss function and then in terms of the Strehl ratio variation for different strengths of turbulence. The variation of visibility loss with turbulence strength is shown in Figure 3.1. The simulation results show that tiptilt correction improves the visibility loss function for large turbulence strengths. Even more important for the stellar interferometer, a significant additional improvement is obtained when both tiptilt and two higher order wave front terms are corrected. The Strehl ratio variation as a function of turbulence strength is shown in Figure 3.2. Simulation results show that the Strehl ratio of the system improves with correction as the strength of turbulence increases. That is, image quality is improved by using the adaptive optical system. The results of these simulations indicate that an adaptive optical system used in a stellar interferometer can dramatically improve the image quality when the strength of turbulence (D/r0) is large. 4. Conclusions and Discussion A methodology to simulate a stellar interferometer with and without adaptive optics in the presence of atmospheric turbulence has been established and demonstrated. According to the simulation results, the atmospheric MTF may be modified with adaptive optics to improve the system performance of a stellar interferometer. The Strehl ratio and visibility loss function both improve dramatically with adaptive optics for large strength of turbulence. Since the Strehl ratio and visibility loss function are closely related, it is possible to predict the performance of a twotelescope interferometer system in terms of visibility loss by measuring the Strehl ratio in one of the telescope systems. 5. Acknowledgments The authors are grateful to Dr. William J. Tango for helpful discussions and suggestions. 6. References 1. M. Shao, M. M. Colavita, “LongBaseline optical and infrared stellar interferometry,” Rev. Astron. Astrophys., 30, 457, 1992. 2. A. Quirrenbach, “Optical Interferometer,” Annu. Rev. Astron. Astrophys., 39, 353, 2001. 3. D.M. Alloin, J. M. Mariotti, DiffractionLimited Imaging with Very Large Telescope, NATO ASI series, 1988. 4. R.K. Tyson, Principles of Adaptive Optics, 2nd edition, Academic Press, 1998. 5. F. Roddier, Adaptive Optics in Astronomy, Cambridge University Press, 1999 6. W. J. Tango, R. Q. Twiss, “Michelson Stellar Interferometry,” Progress in Optics XVII, 241, 1980. 7. T.A. ten Brummelaar, W.G. Bagnuolo, Jr., and S. T. Ridgway, “Strehl ratio and visibility in longbaseline stellar interferometer,” Opt. Lt., 20, 521, 1995. 8. D.L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am., 56, 1372, 1966. 9. W.J. Tango, Private communication. 10. D.L. Fried, “Statistics of geometric representation of wavefront distortion,” J. Opt. Soc. Am., 56, 1372, 1966. 11. J.Y. Wang, “Optical resolution through a turbulent medium with adaptive phase components,” J. Opt. Soc. Am., 67, 383, 1977. Proc. of SPIE Vol. 5169 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 05/26/2015 Terms of Use: http://spiedl.org/terms 253 Figure 3.1 Coherence loss (visibility loss) for the interferometer for different turbulence strengths with and without an adaptive optical system Figure 3.2 Strehl ratio for the interferometer for different turbulence strengths with and without an adaptive optical system 254 Proc. of SPIE Vol. 5169 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 05/26/2015 Terms of Use: http://spiedl.org/terms