Adaptive optics performance model for optical interferometry

Transcription

1 Adptive optics performnce model for opticl interferometry D. Mozurkewich, 1 S. R. Restino, 2 J. T. Armstrong, 2, * nd G. C. Gilbreth 2 1 Sebrook Engineering, 9310 Dubrry Avenue, Sebrook, Mrylnd 20706, USA 2 U.S. Nvl Reserch Lbortory, 4555 Overlook Avenue SW, Wshington, DC 20375, USA *Corresponding uthor: tom.rmstrong@nrl.nvy.mil Received 12 October 2006; revised 28 Februry 2007; ccepted 2 Mrch 2007; posted 16 Mrch 2007 (Doc. ID 76022); published 20 June 2007 The opticl interferometry community hs discussed the possibility of using dptive optics (AO) on pertures much lrger thn the tmospheric coherence length in order to increse the sensitivity of n interferometer, lthough few quntittive models hve been investigted. The im of this pper is to develop n nlytic model of n AO-equipped interferometer nd to use it to quntify, in reltive terms, the gins tht my be chieved over n interferometer equipped only with tip tilt correction. Functionl forms re derived for wvefront errors s function of sptil nd temporl coherence scles nd flux nd pplied to the AO nd tip tilt cses. In both cses, the AO nd fringe detection systems operte in the sme spectrl region, with the shring rtio nd subperture size s djustble prmeters, nd with the interferometer bems ssumed to be sptilly filtered fter wvefront correction. It is concluded tht the use of AO improves the performnce of the interferometer in three wys. First, t the optiml perture size for tip tilt system, the AO system is s much s 50% more sensitive. Second, the sensitivity of the AO system continues to improve with incresing perture size. And third, the signlto-noise rtio of low-visibility fringes in the bright-str limit is significntly improved over the tip tilt cse Opticl Society of Americ OCIS codes: , , , Introduction Extending the imging cpbilities of opticl interferometry, either to finter objects or to the low-fringe visibilities tht chrcterize detiled structure, requires lrger pertures. Mking use of lrger pertures depends in turn on dptive optics (AO) of higher order thn the tip tilt correction used on current interferometers. In this pper, we model the ppliction of AO to opticl interferometry (OI) in order to obtin quntittive estimte of the improvement in sensitivity tht cn be chieved in n AO-equipped interferometer over tip tilt-equipped interferometer. We hve lso strted n experimentl progrm [1] to evlute the combintion of AO nd OI t the Nvy Prototype Opticl Interferometer (NPOI) /07/ $15.00/ Opticl Society of Americ The key results of our modeling re, first, tht combining AO nd OI produces significnt gin in sensitivity over tip tilt correction, even t perture sizes D of 3 to 4 times the tmospheric coherence length r 0 where the performnce of tip tilt system peks [2] nd, second, tht the sensitivity of AO-corrected OI continues to improve s the perture size grows, rther thn declining s in the cse of tip tilt correction. The first of these results grees with the conclusion of Bldwin nd Hniff [3] tht combining AO nd OI should produce fctor of 2 increse in sensitivity for D 3r 0, but the second result demonstrtes tht the benefits of pplying AO to OI cn be significntly greter. A third result, which follows from the first two, is tht low-visibility fringes from bright, resolved strs re lso significntly improved by the use of AO. Opticl interferometry is limited by the sptil nd temporl dimensions of tmospheric turbulence, chrcterized by r 0, the coherence length defined by Fried [4] nd t 0, the tmospheric coherence time (we 10 July 2007 Vol. 46, No. 20 APPLIED OPTICS 4413

2 Report Documenttion Pge Form Approved OMB No Public reporting burden for the collection of informtion is estimted to verge 1 hour per response, including the time for reviewing instructions, serching existing dt sources, gthering nd mintining the dt needed, nd completing nd reviewing the collection of informtion. Send comments regrding this burden estimte or ny other spect of this collection of informtion, including suggestions for reducing this burden, to Wshington Hedqurters Services, Directorte for Informtion Opertions nd Reports, 1215 Jefferson Dvis Highwy, Suite 1204, Arlington VA Respondents should be wre tht notwithstnding ny other provision of lw, no person shll be subject to penlty for filing to comply with collection of informtion if it does not disply currently vlid OMB control number. 1. REPORT DATE REPORT TYPE 3. DATES COVERED to TITLE AND SUBTITLE Adptive optics performnce model for opticl interferometry 5. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Nvl Reserch Lbortory,4555 Overlook Avenue, SW,Wshington,DC, PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR S ACRONYM(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public relese; distribution unlimited 13. SUPPLEMENTARY NOTES 14. ABSTRACT 11. SPONSOR/MONITOR S REPORT NUMBER(S) 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT. REPORT unclssified b. ABSTRACT unclssified c. THIS PAGE unclssified Sme s Report (SAR) 18. NUMBER OF PAGES NAME OF RESPONSIBLE PERSON Stndrd Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18

3 dopt the definition dvocted by Buscher [5]). Once the collecting perture or the integrtion time exceeds few times r 0 or t 0, the interference fringe signl-to-noise rtio S begins to decrese. AO is subject to the sme limittion; we need to correct the wvefront over lengths tht scle with r 0 nd over times tht scle with t 0. Becuse the number of photons needed for trcking fringes is comprble to the number needed to correct the wvefront, the sensitivity of the two techniques, i.e., the minimum number of photons N per coherence volume t 0 r 2 0 needed to rech given signl-to-noise rtio for wvefront or fringe sensing, should be pproximtely equl. This is the essence of Bldwin nd Hniff s clim tht the benefits of combining AO nd OI re limited. However, this rgument misses key point: The number of photons needed to trck fringes is gthered from the whole perture, not just n pproximtely r 0 -sized ptch. Thus incresing the overll perture size decreses the demnd for fringe-trcking photons from ech subperture, mking more photons vilble for AO for ech subperture. In more qulittive vein, the occurrnce of mximum in fringe sensitivity t modest multiple of r 0 clerly indictes tht those photons re not being used in n optimum wy. Incresing the perture increses the number of photons vilble for tip tilt sensing but decreses wvefront qulity, so some of the photons llocted for tip tilt should be used for sensing higher-order berrtions insted. Our AO OI model, lthough bsed in prt on numericl simultions of tip tilt errors, expresses the dependence of sensitivity on such fctors s flux, AO servo bndwidth, nd AO subperture size in nlytic terms. Although not s ccurte s numericl pproch with more error terms, n nlytic model fvors intuitive understnding. This pproch lso hs the dvntge of clerly showing wht ssumptions we mde nd where pproximtions were used. To keep the model resonbly uncoupled from ssumptions bout the source nd instrument, we ssume tht the interferometer nd the AO operte in the sme spectrl region, with frction f of the flux sent to the AO system nd the rest pssed to the fringe-detection system. Although hving the AO nd OI work in different spectrl regions will ffect the bsolute sensitivity, it should hve no effect on the point of the pper, which is the qulittive behvior of sensitivity s function of perture size. We brek the modeling into severl pieces. In Section 2, we strt by quntifying the dependence on subperture size, integrtion time, nd servo bndwidth of the wvefront vrince for n AO system bsed on Shck Hrtmn wvefront sensor. We then discuss, in Section 3, optimizing the wvefront by vrying subperture size nd integrtion time. In Section 4, we consider the ppliction of AO to interferometry. Under the ssumption tht the AO nd fringe detection systems shre the sme wvelength rnge, we determine how the optiml division of light between the two systems vries with N. We then clculte the resulting overll sensitivity, including the effect of sptilly filtering both interferometry bems fter wvefront correction. For comprison, in Section 5 we exmine the cse of n interferometer with only tip tilt correction. We finish with discussion in Section 6, including systems in which f or the number of subpertures cross the telescope dimeter cnnot be vried, nd summry in Section Adptive Optics: Modeling Wvefront Vrince We strt by modeling the propgtion of wvefront through n AO system on single perture of dimeter D, operting with subpertures of dimeter A nd servo time constnt t s. We generte wvefront using stndrd Kolmogorov representtion of the tmosphere, in which the vrince of the phse difference (in rdins) between two points seprted by B is given by [4] 2 B 6.88 B r (1) Our conceptul AO system uses Shck Hrtmn sensor for ech subperture to detect locl tilt nd to drive tip tilt controller for tht subperture. We ssume tht continuity of the deformble mirror will control differentil piston effects between subpertures. After the AO correction, the wvefront is not perfectly flt. The remining vritions re ttributble to three contributions: residul tilt error within ech subperture, distortion t higher sptil frequencies within ech subperture fter tilt nd piston terms re removed, nd time evolution of the wvefront between mesuring the errors nd pplying the corrections. We will ssume tht these errors re independent nd dd their vrinces. Our decomposition of the errors is similr to the pproch tken by Angel [6], lthough tht work is in the high-strehlrtio regime. Our model is not perfect. First, it neglects correltions between the three sources of error. Second, it lcks ny tretment of scintilltion. We rgue tht scintilltion is unimportnt here: Ares in the wvefront tht contribute few photons to the AO lso contribute few photons to the interferometry. Including the effects of scintilltion is importnt for modeling high-strehl-rtio system but not for the low-strehlrtio system we describe here. We next describe ech of the terms in sequence. A. Residul Tilt Error The sensor for ech subperture is qud cell. Determining how the vrince of the position mesurement depends on flux entiled generting simulted wvefronts nd clculting the resulting imges through subperture of size A r 0. For ech of 1000 simultions, we clculted the vrition dq dx of the error signl Q due to n ngulr error x. This procedure ws repeted for rnge of vlues of A r 0. These simultions give residul tilt vrince Q 2, in rdi APPLIED OPTICS Vol. 46, No July 2007

4 ns, tht is well fit over the rnge 0 A r 0 5by 2 Q A 2 Q 1 Q 2 A r 0 2, (2) n with Q nd Q , where n is the number of photons in the integrtion. To convert Q into phse vrince, consider circulr perture of dimeter A centered t the origin of the x, y plne. A tilt of rd long the x xis produces dely error cross the perture equl to x. The men dely error is zero. The dely vrince is given by 2 d A x 2 dxdy A 2 4, (3) where the re of the integrl is bounded by the unit circle. Replcing with Q nd noting tht the phse vrince 2 tt d, we find tht 2 tt 2 2 A Q, (4) 2 Q 2 1 Q 2 A r 0 2. (5) n The number of detected photons should be normlized to stndrd integrtion time nd perture. We will use the coherence time nd length, t 0 nd r 0, for these stndrds. Using N for the number of photons per coherence volume t 0 r 0 2 gives n N t t 0 A 2 r 0. (6) For our purposes, t t s, the servo time constnt, giving 2 tt 4N 2 t 1 s t 0 Q 1 A 2 r 0 Q 2. (7) B. Higher-Order Distortion Becuse the AO tkes out the low-sptil-frequency vritions, the vrince of the higher-order distortions for the entire perture is the sme s the vrince for single subperture. Noll [7] clcultes these prtilly corrected wvefront vrinces. For the cse in which piston, tip, nd tilt re completely removed, the relevnt vrince is 3 from Noll s Tble IV: with ho A 5 3 r 0, (8) C. Wvefront Evolution The AO system genertes wvefront correction for wvefront errors corresponding to the middle of the integrtion. By the time this correction is pplied, it is out of dte by pproximtely t s, the servo time constnt, nd the wvefront hs evolved. However, it is only the evolution of the tip tilt of the subperture tht mtters; piston hs no effect on the AO, nd the higher-order vritions re not corrected. All three chnge the wvefront, but only tip tilt chnges lter the vrince. For the ske of hving closed-form expression, we replce the tip tilt power spectrum for the subperture with the power spectrum for the phse difference between two points seprted by the subperture dimeter A. This form hs the correct totl power but somewht overestimtes the highfrequency contribution, conservtive ssumption; the exct form flls off s 11 3 t high frequencies [8]: where 2 p 0 t for (9) p 0 t for, (10) dr 0 At 0, (11) the frequency corresponding to sptil scles comprble to A. The constnt d is derived from Buscher et l. [5] while p 0 is 8 2 times the vlue in Buscher et l., which converts it to units of rdins 2 Hz for one-sided, rther thn two-sided, power spectrum. The effect of first-order servo with time constnt t s on the tip tilt power cn be pproximted by multiplying the power spectrum by t s 2 for frequencies less thn 1 t s. For t s 1, the power tht is not corrected by the ngle trcking servo is the integrl of this filtered power spectrum nd ccounts for the time evolution term, given by 2 E t 5 3 s t 0 A r t s t 0 2, (12) with 18p 0 5 nd 18p 0 d Finlly, we combine the three contributions to the wvefront vrince. To simplify the nottion, we define A r 0, t s t 0, 2 Q 1 4, nd q 2 2 Q 2 4. The vlues work out to be 3.43, q , (s bove), 0.081, nd Then 2 N 2 q 2 N (13) 10 July 2007 Vol. 46, No. 20 APPLIED OPTICS 4415

5 3. Minimizing Wvefront Vrince In this section we solve the problem of djusting nd to produce minimum vrince wvefront t given N, the number of photons per coherence volume. We strt with the prtil derivtives of Eq. (13): 2 N 2 q 2 2 N N , (14) 4 3. (15) Setting these two prtils equl to zero produces two equtions in three unknowns, i.e., N nd the optiml vlues of nd : N q 2 2 N N , (16) 1 3. (17) These cn be solved for (N) nd (N). We first define, where nd re t their optiml vlues. We then solve for N ; the lgebric detils cn be found in Appendix A. We find tht N 3 6 q (18) (19) Although we do not hve n exct inversion of this eqution, N N (20) gives N to better thn 1% for N 250. Figure 1 shows the fit. The optiml subperture size nd the integrtion time cn be found by inverting Eq. (16) to yield N q 5 3 1, (21) N (22) From this expression, we determine the normlized servo time constnt,, from nd the definition of. The optimum subperture nd integrtion time s function of N re shown in Fig. 2. The finl step is to clculte the minimum phse vrince cross subperture from Eq. (13) using the optiml vlues of nd. The result is plotted in Fig. 3. It is cler tht AO cn be useful with only few Fig. 1. Vrition of s function of N, the number of photons delivered to the dptive optics system. The points re the evlution of Eq. (18). The curve is the pproximtion of Eq. (20) APPLIED OPTICS Vol. 46, No July 2007

6 Fig. 2. Optimum vlues for A r 0 nd t s t 0 s function of the number of photons per coherence volume N delivered to the dptive optics system. detected photons per coherence volume. We show the dependence of the wvefront vrince on deprtures from the optimum vlues for nd in Fig. 4. To simplify the computtions in the next section, we pproximte the vrince with bn c, (23) N 0.450, (24) where the prmeters 0, b, nd c re defined by the first line. This form fits the exct expression to better thn 2% for N Interferometry Model In this section, we pply our wvefront vrince results to our model interferometer in order to derive the best vlue of f, the frction of incoming light tht is sent to the AO system, nd to clculte the functionl form of the sensitivity of the interferometer. Our model interferometer hs two pertures, ech of dimeter D, ech collecting N photons per coherence volume, nd ech with perfect sptil filtering. In everything tht follows, we will ssume tht the integrtion time for the interferometer, t I, is equl to the tmospheric coherence time t 0. Chnging this ssumption does nothing but multiply the right-hnd side of Eq. (25) by fctor tht my depend on the spectrum of the str. This fctor hs no effect on the functionl form of performnce versus str brightness nd so does not chnge the min point of this pper. We will lso ssume tht the fringe-detection system is photon-noise limited, nd tht it is in the photon-rich regime N I V 2 10, where N I is the number of photons delivered to the bem combiner). Note tht the photon-rich-regime ssumption is consistent with the results of the previous section, in which we conclude tht AO cn be beneficil with only smll number of photons per coherence volume, becuse the fringe-detection system receives light from severl AO subpertures. The result of the photon-noise-limit nd photonrich-regime ssumptions is tht the fringe mplitude signl-to-noise rtio S 1 V N I, where the fctor 1 is the normliztion for four-bin fringe smpling. Thus we hve e 2 S V2 N 1 f t I t 0 D2 4r 0 2, (25) in which the leding 2 is from the two pertures, nd the fctor e 2 reflects the effect of sptil filtering, which converts wvefront irregulrities in the two interferometer bems into flux vritions. The phse vrince in ech bem is 2, so in the bsence of sptil filtering the phse vrince between bems would be 2 2, which would reduce V 2 in the combined bem, nd thus S 2, by fctor of e 2 2. Sptil filtering insted reduces the flux in ech bem, nd thus the vlue of S 2, by fctor of only e 2. The performnce of the AO system is identicl to tht derived in the previous section except tht the number of photons per coherence volume N is replced with fn. Eqution (23) becomes b fn c, (26) 10 July 2007 Vol. 46, No. 20 APPLIED OPTICS 4417

7 Fig. 3. Phse vrince 2 cross subperture s function of the number of photons per coherence volume N delivered to the dptive optics sensor for the cse in which nd re optimized. so d 2 df bc N c f c 1. (27) Mximizing S 2 with respect to f (nd ssuming the str is unresolved so V 1) yields ds 2 df 2 N t I t 0 D 2 r 0 e 2 N 2 1 f t I t 0 D r 0 d 2 2 2e df, (28) 0, (29) which leds to N c bc 1 f c 1. (30) f The optiml frction f is plotted versus N in Fig. 5. Once gin, n exct inversion is lcking, but f log N log 2 N log 3 N (31) is ccurte to better thn 1% over the relevnt rnge of N. We re finlly in position to evlute the sensitivity of n AO-equipped OI s function of perture size. Strt by returning to Eq. (25) with t I t 0 nd note tht 2 depends on fn [Eq. (26)], nd f depends only on N [Eq. (31)]. Thus S D depends only on N.We ssume minimum vlue of S 5 for fringe trcking, clculte S D, then clculte the vlue of D needed to chieve our ssumed vlue of S. The results of this procedure re plotted in Fig. 6, but before discussing these results, we will clculte the sensitivity of OI in some comprison cses the most importnt being tht of tip tilt correction only using the sme ssumptions s in the AO cse. 5. Opticl Interferometry with Tip Tilt Only A system with only tip tilt correction is the most importnt comprison cse since most current interferometers operte in this mode. In this cse, the fringe trcking signl-to-noise rtio is still governed by Eq. (25), but now D r 0 is replced by. Agin tking t I t 0, we hve S 2 N 2 1 f 2 e 2. (32) Since there is no optimiztion for subperture size in this cse, we hve to exmine the behvior of 2 by returning to Eq. (13) nd replcing N with fn. Mximizing S 2 with respect to f nd now gives nd S N 2 1 f 2 e, 0, (33) 4418 APPLIED OPTICS Vol. 46, No July 2007

8 Fig. 4. Phse vrince cross subperture normlized to the vrince t the optiml vlues of nd. () Normlized vrince is plotted s function of, the subperture size in units of r 0, normlized to its optiml vlue, opt. The curves, from upper to lower, re clculted for N 100, 10, nd 1. (b) Normlized vrince is plotted s function of, the AO servo time constnt in units of t 0, normlized to its optiml vlue, opt. The curves, from upper to lower, re clculted for N 1, 10, nd 100. where S 2 f N 2 2 e 2 N 2 1 f 2 e 2 2 f, 0, (34) 2 fn 2 q 2 2 fn , (35) f f 2 N 2 q 2 f 2 N. (36) Eqution (33) reduces to 2 0 nd cn be rerrnged to give fn q , (37) while combining Eqs. (34), (33), nd (35) gives 1 f (38) We solved this set of equtions itertively for N(), the number of photons per coherence volume needed to ttin the trget vlue of S for fixed vlue of. First, for the given vlue of, we guess vlue of nd use Eqs. (37) nd (38) to clculte f nd N. We use these vlues in Eq. (32) to find S. We then djust the vlue of nd repet the process until we obtin the desired vlue of S. As with the AO cse, the results re plotted in Fig Discussion Figure 6 shows the photons per coherence volume needed to obtin signl-to-noise rtio S 5 with n opticl interferometer for both the dptive optics nd tip tilt cses. Two importnt points should be mde. The first point is tht the sensitivity for the tip tilt cse hs pek; for lrger pertures, sensitivity ctully decreses. Our clcultions give D r for the perture size t pek sensitivity. This result is not new; the literture cites vlues in the rnge of 3 to 5r 0 for the optiml perture size [2,9]. The second point to be mde is tht the sensitivity of n AO-endowed system does not pek t some vlue of D r 0 but continues to improve s the perture size increses. This behvior my seem surprising t first, but it cn be explined by considering tht lrger perture cn devote lrger frction of its photons to wvefront correction while continuing to gther the totl number of photons needed for fringe detection. The fct tht sensitivity continues to 10 July 2007 Vol. 46, No. 20 APPLIED OPTICS 4419

9 Fig. 5. The frction of light to send to the dptive optics system tht optimizes the interferometric signl-to-noise rtio s function of the number of photons per coherence volume presented to the whole system (full curve). For comprison, we plot the sme quntity for the tip tilt cse with D r 0 3.9, the vlue for mximum sensivity (dshed curve). increse with perture size is importnt for the design of future interferometers. Note lso tht the AO-equipped system is up to 50% more sensitive thn the tip tilt system t the optiml tip tilt perture size. Consider tip tilt system with the optiml perture dimeter, so tht the sensitivity does not vry with smll chnge in perture, or equivlently with smll chnge in the number of detected photons. This lck of dependence of the sensitivity on photon rte implies Fig. 6. Photons per coherence volume needed to obtin fixed signl-to-noise rtio with n opticl interferometer, s function of perture size, for n dptive-optics equipped interferometer nd for n interferometer with tip tilt correction only APPLIED OPTICS Vol. 46, No July 2007

10 Fig. 7. Signl-to-noise rtio t V 1 s function of perture size in the bright-str limit. tht we re not using those photons in n optiml wy. In fct, looking t the contributions to the wvefront vrince bers tht conclusion out. The tip tilt error, the only contribution to the vrince tht depends on the number of photons, is only pproximtely one third s lrge s the contribution from higher-order berrtions t low flux N 4 nd signl-to-noise rtio S 1, nd is smller still t higher flux levels. We should expect tht using more photons to estimte higher-order berrtions, reducing ho 2 t the expense of incresing tt 2, should improve system performnce. The dvntges of n AO-equipped interferometer do not depend criticlly on the bility to djust f or to mtch conditions. For instnce, if f remins fixed t 0.2, roughly the optiml vlue for fint sources, the sensitivity is within few percent of the optiml AO result. If the AO system hs s few s four subpertures cross the telescope dimeter, the sensitivity t the fint limit is within 10% of the fully djustble system. Our discussion hs centered so fr on the question of the fintest source tht cn be observed with n interferometer in the fully djustble AO nd tip tilt cses. However, for strs tht re bright enough for either system, it is still dvntgeous to use AO. The benefit comes in the improved signl-to-noise rtio, which leds directly to more ccurte visibility mesurements. Highly ccurte visibilities re importnt for precise ngulr dimeter mesurements nd re criticl for limb-drkening nd surfce structure observtions. To mke the comprison for this limit, we exmine the performnce of the two systems t fixed vlue of N. As before, the strting point is Eq. (25): S 2 NV2 2 1 f D 2 r 0 e 2. (39) For the tip tilt cse, the choice of N determines 2. We follow the sme procedure s in the fint-str limit, using Eqs. (37) nd (38) to find n eqution for N,. We then invert this eqution to find for fixed vlues of N nd, nd use Eq. (39) to determine S. For the AO cse, we determine f s before from Eq. (31). Agin using the fct tht f nd 2 cn be expressed s functions of N lone, we clculte S directly from Eq. (39). Figure 7 shows the signlto-noise rtio of the two systems t fringe visibility V 1. To evlute the sensitivity of the two systems to low visibilities, we inverted Eq. (39) to find V(S, N). The overll picture is very similr to the fint-str limit: The optiml perture size for the tip tilt cse is 4r 0, the AO-equipped system outperforms the tip tilt system, nd S continues to increse with D r 0 for the AO cse. 7. Summry This pper presents n nlytic model for ssessing the performnce of n opticl interferometer fitted with dptive optics (AO), nd uses the model to compre the performnce of n AO-equipped interferometer with one using only tip tilt correction. The AO-equipped interferometer performs better for ll perture sizes lrger thn pproximtely 0.5r 0, where r 0 is the coherence length of the tmosphere. The best performnce of the tip tilt corrected system is t perture sizes ner 4r 0, but even for those p- 10 July 2007 Vol. 46, No. 20 APPLIED OPTICS 4421

11 erture sizes, the AO-equipped system is superior by fctor of pproximtely 2. In ddition, the AOequipped system continues to gin in sensitivity with incresing perture size, symptoticlly pproching mximum in performnce. At perture sizes of 10r 0, the AO-equipped system is 10 times s sensitive s the tip tilt system s sensitivity t 4r 0, where it performs best. For bright strs, those within rech of both systems, the AO-equipped system hs significnt dvntges in fringe signl-to-noise rtio nd in the smllest detectble visibility. These results re preliminry; clerly the model cn be improved by including prmeteriztion bsed on both detiled numericl nlysis nd experimentl vlues. Nevertheless, it is cler tht dding AO to n OI cn significntly improve its limiting mgnitude, s well s its bility to mesure visibilities with high precision. Appendix A. Derivtion of Optimiztion of nd To find the integrtion time nd subperture size tht minimize the wvefront vrince 2, we strt with Eqs. (16) nd (17): N 2 q 2 N , N , (A1) (A2) which cn be solved for (N) nd (N). First eliminte from Eq. (A1): q 2 N (A3) Set nd eliminte from Eqs. (A2) nd (A3): Now eliminte : N , (A4) q N (A5) N 3 14 q N nd solve for N: N 9 56 q , (A6) (A7) N 3 6 q (A8) We thnk the nonymous reviewers for comments tht helped clrify this pper. This reserch ws supported by the the Office of Nvl Reserch. References 1. S. R. Restino, J. Andrews, J. T. Armstrong, T. Mrtinez, C. Wilcox, nd D. Pyne, Nvy prototype opticl interferometer upgrde with light-weight telescopes nd dptive optics: A sttus updte, Proceedings of 2006 AMOS Technicl Conference (2006), pp D. F. Buscher, Getting the most out of C.O.A.S.T., Ph.D. disserttion (Cmbridge University, 1988). 3. J. E. Bldwin nd C. A. Hniff, The ppliction of interferometry to opticl stronomicl imging, Philos. Trns. R. Soc. London Ser. A 360, (2002). 4. D. L. Fried, Opticl resolution through rndomly inhomogeneous medium for very long nd very short exposures, J. Opt. Soc. Am. 56, (1966). 5. D. F. Buscher, J. T. Armstrong, C. A. Hummel, A. Quirrenbch, D. Mozurkewich, K. J. Johnston, C. S. Denison, M. M. Colvit, nd M. Sho, Interferometric seeing mesurements on Mount Wilson: Power spectr nd outer scles, Appl. Opt. 34, (1995). 6. J. R. P. Angel, Ground-bsed imging of extrsolr plnets using dptive optics, Nture 368, (1994). 7. R. J. Noll, Zernike polynomils nd tmospheric turbulence, J. Opt. Soc. Am. 66, (1976). 8. C. B. Hogge nd R. R. Butts, Frequency spectr for the geometric representtion of wvefront distortions due to tmospheric turbulence, IEEE Trns. Antenns Propg. 24, (1976). 9. F. Roddier, The effects of tmospheric turbulence in opticl stronomy, Prog. Opt. XIX, (1981) APPLIED OPTICS Vol. 46, No July 2007