Field-of-view limitations of phased telescope arrays

Field-of-view limitationsof phased telescope arrays

James E. Harvey and Christ Ftaclas

The optical performance of imaging phased telescope arrays is degraded by various design, manufactur-ing, and operational errors. Perhaps the most basic and fundamental of these error sources are theresidual aberrations of the optical design chosen for the individual telescopes. We show that third-orderfield curvature and distortion, which are rather benign aberrations in a conventional telescope, result inrelative phase and tilt errors between the individual telescopes making up the array. The field-dependent image degradation caused by these relative phase and tilt errors is then predicted for differentsubaperture configurations and telescope design parameters. For phased arrays made up of simpletwo-mirror telescopes, distortion limits the field of view to less than 5 arcmin for small subapertures 1D ,

0.5 m2, and field curvature limits the field of view to less than 1 arcmin for subaperture diameters greaterthan 2 m. Quantitative parametric results yielding tolerances for residual field curvature as the phasedarray is scaled up in size are presented graphically. If a 0.5-deg field of view is desired for telescopediameters greater than 2 m, complex telescope configurations are necessary to satisfy the rather tighttolerances on both field curvature and distortion.Key words: Phased telescope arrays, synthetic aperture imaging, aberrations.

1. Introduction

Previous studies have indicated that pupil-mappingerrors and off-axis aberrations of the individual tele-scopes of a phased telescope array can rapidly de-grade image quality for increasing field angles, thusseverely limiting the useful field of view 1FOV2 of thephased telescope array.1–6 Correcting the pupil map-ping errors poses an engineering challenge, but theresidual design aberrations of the individual tele-scopes are more fundamental in nature.The results of preliminary calculations performed

in Ref. 5 indicate that for visible light and 2.0-m-diamsubapertures, either of the above error sources maylimit the useful FOV to a few tens of arcseconds forsome subaperture configurations of interest. Thosepreliminary indications of severe FOV limitationsprovided the impetus for the more detailed analysispresented in this paper.

J. E. Harvey is with The Center for Research and Education inOptics and Lasers, The University of Central Florida, 12424Research Parkway, Orlando, Florida 32826. C. Ftaclas is withHughes Danbury Optical Systems, Inc., 100 Wooster Heights Road,M@S 807, Danbury, Connecticut 06810.Received 16 March 1995; revised manuscript received 2 May

1995.0003-6935@95@255787-12$06.00@0.

r 1995 Optical Society of America.

System issues that need some initial definition tobound the problem are: image-quality criterion, FOV,aperture configuration, individual telescope configura-tion, and operating wavelength.The proper criterion for evaluating optical perfor-

mance depends on a number of factors: the nature ofthe source or object to be imaged, the nature of thedetector or sensor to be used, and the goal of theparticular application. For simplicity we use theStrehl ratio as an image-quality criterion in thisstudy. Because we are interested primarily in thedegradation of image quality with field angle forphased telescope arrays, the particular image-qualitycriterion is of less concern than its variation with fieldangle. We recognize fully that for imaging applica-tions where fine detail in extended objects is ofconcern, some characteristic of the modulation trans-fer function 1MTF2 is a more appropriate performancecriterion than the Strehl ratio.Current interest in phased telescope arrays falls

into three main categories that are characterized bytheir FOV requirements: 112 Laser beam transmis-sion applications are essentially zero-field systems1unless target acquisition or tracking is also per-formedwith the transmitting telescope2.7–12 122Astro-nomical observatories require a finite but modestFOV.13 132 Surveillance applications require a sub-stantial FOV.14–18 We investigate image quality para-metrically as a function of FOV for various subaper-

1 September 1995 @ Vol. 34, No. 25 @ APPLIED OPTICS 5787

ture configurations and telescope configurations.The nominal FOV requirement or goal has been takenas 15 arcmin. This corresponds to a full field of 0.5deg.The choice of a subaperture configuration can dras-

tically affect the diffraction-limited imaging character-istics of a phased telescope array.19 The images maycontain artifacts and spurious images that are unde-sirable for certain applications involving point sources.Also, if direct-imaging phased telescope arrays are tobe used for applications where it is necessary to studyfine detail in extended objects, the subaperture con-figurationmust not be so dilute as to produce substan-tial areas of zero modulation within the cutoff fre-quency of the MTF plane.20Applications involvingmonochromatic imagingwith

active illumination as well as applications of broad-band 1white-light2 imaging with phased telescope ar-rays are of interest. Wavelength has been treated asa study parameter; however, because some param-etersmust be held constant to keep the volume of dataat a manageable level, a nominal value of 0.5 µm hasbeen chosen for calculating quantitative results inthis paper. Calculating polychromatic image qualityis a straightforward extension of the calculationsperformed in this paper.Because quantitative image-quality predictions de-

pend on a specific subaperture configuration and thisis not easy to vary parametrically, we need someinitial definition of the number, size, shape, andposition of the subapertures making up the array.An appropriate number of individual telescopes mak-ing up an array, for the purposes of this study, isassumed to be between 4 and 12.Figure 1 illustrates 112 an arrangement of four

subapertures similar to the configuration of the Na-tional New Technology Telescope 1NNTT2 studied atthe National Optical Astronomy Observatories andthe Multipurpose Multiple Telescope Testbed 1MMTT2at the Air Force Phillips Laboratory, 122 six subaper-tures similar to the Multiple Mirror Telescope 1MMT2operated by the Smithsonian Astrophysical Observa-tory on Mt. Hopkins south of Tucson, Ariz., 132 atwo-dimensional nonredundant array of six subaper-tures referred to as a Golay-6 configuration,21 142 aCircle-N configuration where N is any desired num-ber 1most likely 7, 9, or 112, 152 another two-di-

Fig. 1. Subaperture configurations considered in this study.

5788 APPLIED OPTICS @ Vol. 34, No. 25 @ 1 September 1995

mensional nonredundant array of nine subaperturesreferred to as a Golay-9 configuration, and 162 aneight-element Mill’s Cross consisting of two orthogo-nal one-dimensional nonredundant arrays of foursubapertures each. Another class of arrays not illus-trated here is the Cornwell arrays. It achieves themaximum uniformity in the MTF plane when thesubapertures are constrained to lie on a circle.22 TheEuropean Very Large Telescope23 and the Keck Ar-ray24 are large astronomical telescope arrays cur-rently under construction.The dilution ratio 1the ratio of the separation of

adjacent subapertures to their diameter2 of the arrayis also a study parameter. Note that compact two-dimensional arrays are desirable for continuous cover-age in the MTF plane as illustrated in Fig. 2; fr is theminimum spatial frequency at which the MTF falls tozero. A dilution ratio greater than 2 assures theexistence of areas of zero modulation in the MTF.In this study we limit our attention to phased

arrays of identical axially symmetric telescopes thatare phased and combined to produce a direct image inthe focal plane with a resolution characteristic of thediameter spanned by the array. This configurationis illustrated schematically in Fig. 3. The locationand size of the nth telescope making up the array arespecified by rn andDn, respectively; rn8 andDn8 are thecorresponding location and size of the various subap-ertures in the entrance pupil of the beam-combiningtelescope. MT is thus the transversemagnification ofthe individual telescopes. The magnification of thebeam-combining optics Mc must be precisely equal toMT if significant image degradation from pupil-mapping errors is to be avoided.5 The modular as-pect of phased arrays of independent telescopes shouldprovide substantial advantages in the construction ofa large telescope in space because each individualtelescope can be fabricated, assembled, and alignedon the ground. The complete module can then belaunched into orbit and fitted into the array, thusminimizing the necessary amount of on-orbit assem-bly.A systems engineering approach is employed to

determine quantitatively the FOV limitations of sucharrays as they are scaled up in size for imagingapplications in space. The development and applica-tion of a detailed error-budget tree are discussed inSection 2.

Fig. 2. Compact two-dimensional arrays providing continuouscoverage in MTF plane. This is desirable if fine detail is to beobserved in extended objects.

Fig. 3. Configuration for a phased array of afocal telescopes complete witih phase-adjusting mirrors and a beam-combining telescope.

Conventional astronomical telescopes are correctedfor spherical aberration and coma.25 For wide-angleapplications, astigmatism must also be corrected.The field curvature can be characterized as a field-dependent defocus that varies as the square of thefield angle 1which merely shifts the image axiallywithout loss of definition2. Its effects can thereforebe eliminated if a curved focal surface is utilized.Distortion is similarly a rather benign aberration thatcan be characterized as a field-dependent magnifica-tion error that varies as the third power of the fieldangle 1i.e., it merely displaces the image of a pointsource laterally without loss of definition2.However, these aberrations lose their benign na-

ture when conventional telescopes are phased up toform an imaging array. They now introduce relativephase and pointing errors 1between the elementsmaking up the array2 that do produce a field-dependent image degradation of point sources, thuslimiting the FOV of phased telescope arrays. Aquantitative analysis of the image degradation exhib-ited by phased telescope arrays caused by field curva-ture and distortion of the individual telescopes ispresented in Section 3. The results of extensiveparametric performance predictions are then summa-rized, and conclusions are reached concerning thefundamental FOV limitations of phased telescopearrays.

2. Development of an Error-Budget Tree

The detailed design of a large space-based phasedtelescope array depends critically on the ability tomake realistic performance predictions in the pres-ence of error sources and to determine the tolerancesnecessary to meet given system performance require-ments. The primary requirement for the purposesof this study is to obtain near-diffraction-limited

behavior 1Strehl ratio . 0.82 over FOV’s as high as 0.5deg.A comprehensive list of potential error sources that

might degrade the optical performance of imagingphased telescope arrays was tabulated first. Theseerror sources were then categorized and put into fourprimary groups that follow the chronological develop-ment of a phased-telescope-array program from de-sign through fabrication, assembly, and alignment tolaunch and operation in the space environment.The next step in the development of an error-budget

tree for imaging phased telescope arrays was toconvert the top-level requirement expressed in termsof Strehl ratio to a requirement in terms of the moreconventional rms wave-front error. When the follow-ing expression for the Strehl ratio is used,26

Strehl ratio 5 exp3212psw@l224, 112

the top-level off-axis optical-performance require-ment corresponds to an rms wave-front error of lessthan 0.075 waves at the operational wavelength.A preliminary tops-down error-budget allocation

was assigned to each error-source category. Theseallocations were then further broken down among theerror sources in each category. This is merely thestarting point for a bottoms-up error-budgeting pro-cess involving reallocation by application of engineer-ing experience and detailed analysis of the individualerror sources in an attempt to equalize the difficulty ofachieving the performance required by each allocation.This is the optimum condition that reduces risk anddevelopment costs for the program. The resultingerror-budget tree is illustrated in Fig. 4.State-of-the-art experience with the optical fabrica-

tion of large-precision optics was applied to thereallocation of the optical-fabrication errors. As an


Fig. 4. Off-axis error-budget tree resulting from a detailed bottoms-up reallocation of individual error sources.

example, the Hubble Space Telescope 1HST2 errorbudget allowed an rms wave-front error of 0.04 waves1l 5 0.6328 µm2 for optical-fabrication errors for theprimary and secondary mirror combined.27 Thus wechose this as a realistic state-of-the-art allocation foroptical-fabrication errors for a phased telescope array.The relative fraction of this allocation given to low-spatial-frequency figure errors and mid-spatial-frequency surface errors was based on engineeringexperience. Themicroroughness would undoubtedlybe given a specification based on separate scatteringrequirements.The intratelescope alignment errors and assembly

deformations were similarly taken from engineeringexperience on such programs as the HST. Theallocation of sw , 0.0250 waves for phasing errors isbased on the successful laboratory demonstration ofthe PHASAR Telescope of the Air Force WeaponsLaboratory being phased to approximately l@15 andthe assumption that this performance can be im-proved on in the future.16 The error-budget alloca-tion of sw , 0.0150 waves for relative pointing andfocus errors was also chosen to be consistent withpreviously referenced studies and experiments.7,8,15,16The final two error sources in the assembly andalignment category are the pupil-mapping errors andrelative magnification errors previously discussed asbeing particularly important in wide FOV imagingapplications utilizing phased telescope arrays. Theseare not fundamental problems as are the residual


optical design aberrations; however, they pose severeengineering challenges in the positioning and opera-tion of these complex optical systems. Root-mean-square wave-front error allocations of sw , 0.0150waves were given to these two error sources. Whenthese error-budget allocations are root sum squared,they result in a total allocation of sw , 0.0404 forassembly and alignment errors as indicated in Fig. 4.Finally, the allowable contributions from error

sources in the category of environmental errors wereadjusted from their preliminary values. The alloca-tion of sw , 0.0200 for thermal deformations is againconsistent with error-budget allocations on the HST.In addition to the usual allocation for image jitter anddrift, for phased telescope arrays we must also put alimit on pupil jitter and drift 1dynamic pupil-mappingerrors2. An rms wave-front error of sw , 0.0150 wasthus estimated for these two error sources. Aprelimi-nary allocation of sw , 0.0100 for gravity releaseerrors from built-in stresses caused by assembly andalignment of the individual telescopes in a 1-g gravita-tional field is provided 1an estimate to be verified byfurther analysis2. These allocations root sum squareto a value of sw , 0.0309 for all environmental errors.The allowable allocation remaining for residual

optical-design errors 1because of off-axis aberrationsof the individual telescopes2 is given by an rmswave-front error of sw , 0.0375. The effects ofthese aberrations are studied parametrically in the

following sections to determine whether wide fieldscan be achieved 1as constrained by this error budgetallocation2 as the phased telescope arrays are scaledup in size.Clearly these error-budget allocations will be ad-

justed in the future as technological developments aremade or additional knowledge is acquired concerningthese very complex optical systems.

3. Effects of Telescope Aberrations on Phased-ArrayPerformance

The absence of a perturbing atmosphere in space-based applications permits the possibility of achiev-ing high resolution over a substantial FOV withphased telescope arrays if the error sources discussedin Section 2 are adequately controlled. Many ofthese error sources have been discussed in the litera-ture as shown by our fairly extensive references.Ironically, the most frequently ignored potential errorsource is also perhaps the most basic and fundamen-tal, i.e., the inherent aberrations of the optical designchosen for the individual telescopes. It is the FOVlimitations of these residual optical-design errors1field curvature and distortion2 that are emphasizedbelow.

A. Diffraction-Limited Performance

We should understand the diffraction-limited perfor-mance of an optical system before discussing thedegradation of that diffraction-limited performanceby aberrations. As stated above, the Strehl ratio hasbeen chosen as the image-quality criterion for thisstudy. The Strehl ratio is defined as the peak irradi-ance of an aberrated point-spread function 1PSF2divided by the peak irradiance of the diffraction-limited PSF for a given optical system. The diffrac-tion-limited PSF of a synthetic aperture array madeup of circular subapertures of diameterD is given by

I1x,y2 5 1pD2@4l f 2232 J11prD@l f 2@1prD@l f 24

3 0on51

N

exp3i2p1xxn 1 yyn2@l f 402 . 122

Note that this is merely an interference term multi-plied by an envelope function given by the irradiancedistribution that would be produced by a singlecircular subaperture. The choice of a subapertureconfiguration can drastically affect the diffraction-limited imaging characteristics of phased telescopearrays.19 Harvey et al. have calculated and com-pared the PSF profiles and fractional encircled energycurves for a variety of both dilute and close-packedcircular subaperture configurations.5,28

B. Residual Design Errors 1Aberrations2

The wave-front aberration function for a rotationallysymmetric optical system can be written as

W 5 W020a2 1 W111ba cos c 1 W040a4 1 W131ba3 cos c

1 W222b2a2 cos2 c 1 W220b

2a2 1 W311b3a cos c

1 higher-order terms, 132

where b is the normalized field parameter, a is thenormalized pupil radius, and c is the azimuthal pupilparameter.29 The first two terms in this aberrationexpansion are the first-order aberrations commonlyreferred to as defocus and tilt 1or a lateral magnifica-tion error2. The next five terms are the third-order1Seidel2 aberrations that have traditionally been calledspherical aberration, coma, astigmatism, field curva-ture, and distortion.We start our parametric study with an array of

two-mirror afocal telescopes of the Mersenne design1confocal paraboloids2 as shown in Fig. 3. If theindividual telescopes comprising the array are prop-erly fabricated, assembled, and pointed, there are nodefocus or tilt errors. Furthermore this particulartelescope design is also inherently free of third-orderspherical aberration, coma, and astigmatism. How-ever, any two-mirror telescope of nonunit magnifica-tion will exhibit field curvature. Hence only fieldcurvature and distortion exist through third order forMersenne telescopes.The magnitude of the field-curvature coefficient is

determined by the primary mirror diameter 1Dp2, its fnumber 1Fp2, the telescope magnification 1m2, and thefield angle 1u02. For Mersenne telescopes30

field curvature 5 W20 5 3Dp@116Fp 24311 2 m2@m4u02.

142

Similarly, the magnitude of the distortion coefficientis given by 30

distortion 5 W11 5 1Dp@82313m 1 121m 2 12@m24u03.

152

Because the field curvature varies as u02 and distor-

tion varies as u03, it is clear that field curvature

dominates distortion for small field angles and distor-tion dominates field curvature at sufficiently largefield angles. Figure 5 shows quantitatively that thiscrossover point occurs at a field angle of ,1 deg for a2-m-diam telescope with an [email protected] primary mirrorand a magnification of 0.1. Furthermore for fieldangles of less than 5 arcmin, the field curvaturecoefficient is at least an order of magnitude larger

Fig. 5. Field curvature dominating distortion at small field angles.


than distortion. Thus we first concern ourselveswith field curvature in the preliminary analysis ofphased arrays of two-mirror telescopes.In conventional telescopes, field curvature is a

rather benign aberration because it displaces only theimage of a point source axially by an amount propor-tional to the square of the field angle rather thansmearing it as does spherical aberration, coma, orastigmatism. Therefore, when a curved focal surfaceis used, no image degradation is produced.However, in a phased telescope array, the field

curvature of the individual telescopes results in awave-front error over the nth subaperture given by

Wn1x, y2 5 W20r2 5 W201x2 1 y 22. 162

When referenced to the array coordinate system,

Wn1x8, y 82 5 W2031x8 2 xn22 1 1 y 8 2 yn224. 172

where xn and yn are the coordinates of the nthsubaperture as shown in Fig. 6. This equation canbe rewritten as

Wn 5 W20rn2 2 2W201x8xn 1 y 8yn2 1 W20r82. 182

The first term is a relative phase 1piston2 error be-tween the various subapertures, and the second termis a relative wave-front tilt 1or pointing error2 betweenthe subapertures. These two terms can rapidly de-grade image quality for increasing field angles, thusseverely limiting the useful FOV of the phased tele-scope array. The third term is a conventional fieldcurvature for the array that can be partially compen-sated by utilizing a curved focal surface for thebeam-combining telescope.Note that the first two terms of Eq. 182 depend on the

position coordinates of the subapertures; hence theachievable FOV may vary substantially with thesubaperture configuration. Also, the first term canbe canceled by repositioning the phase-adjusting mir-rors; however, this can be done for only one fieldposition. Hence this does not increase the FOV, butit does provide a convenient mechanism for agilepointing of the phased telescope array over a limitedangular range without slewing the array 1or even theindividual telescopes2.

C. Computational Techniques

Three different techniques of calculating the imagedegradation of phased telescope arrays caused by fieldcurvature of the individual telescopes were employed

Fig. 6. Relationship between subaperture and array coordinates.


in this study: 112 An analytical expression for theStrehl ratio was derived for some situations. 122 Ananalytical expression for the rms wave-front error swwas derived for the case in which field curvature is theonly aberration present. 132 A computer code wasdeveloped to calculate the rmswavefront error numeri-cally in the presence of arbitrary aberrations.

1. Analytical Solution for the Strehl RatioThis first method of calculating the Strehl ratio of aphased telescope array was used in Ref. 5 in anattempt to describe the effects of field curvature of theindividual telescopes. Overly pessimistic resultswere obtained because the last term of Eq. 182 wascompletely discarded, so we can argue that one couldcompensate for this term by invoking a curved focalsurface for the beam-combining telescope.31 Thisresulted in the following analytical solution for theStrehl ratio:

Strehl ratio 51

N2 0on51

N

somb12W20rnDp 2

3 exp12i2pW20rn22 02 , 192

where

somb1r@D2 5 2J11pr@D2@1pr@D2, 1102

as defined by Gaskill,32 and W20 5 W20@l, rn 5 rn@l,D5 D@l, etc.However, completely discarding the array field cur-

vature term was equivalent to choosing a nonopti-mum curved focal surface for the beam-combiningtelescope. This effect is discussed in detail in Subsec-tion 3.C.2.

2. Analytical Solution for Wave-Front VarianceRecognizing the nonoptimum results predicted by themethod discussed above we proceed to derive ananalytical expression for the wave-front variance of aphased telescope array resulting from field curvatureof the individual telescopes. If we sum the contribu-tions of the wave-front error described in Eq. 182 overthe subapertures making up the array and add anarbitrary amount of array field curvature 1W2082 corre-sponding to a curved focal surface in the beam-combining telescope, we obtain the following expres-sion for the wave-front error for the entire array:

W 5 on51

N

Wnr2Cn 2 on51

N

W208r82Cn, 1112

where

Cn 5 cyl31r8 2 rn2@D4 1122

is a shifted cylinder 1top-hat2 function of diameter Das defined by Gaskill.32When we substitute Eq. 182 into Eq. 1112 and

rearrange them, we obtain

W 5 on51

N

W20rn2Cn 2 on51

N

2W201xnx8 1 yn y82Cn

1 on51

N

1W20 2 W2082r82Cn. 1132

Factoring outW20 and setting

q 5 W208@W20, 1142

we obtain

W 5 W203on51

N

rn2Cn 2 2 on51

N

1xnx8 1 yn y82Cn

1 11 2 q2 on51

N

r8Cn4 . 1152

We now proceed to calculate the wave-front variance:

sw2 5 W2 2 W2 5

1

NA e2`

` e W 2dx8dy8

2 1 1NA e2`

` e Wdx8dy822

, 1162

whereN is the number of subapertures and

A 5 area of subaperture 5 pD2@4 5 pr02. 1172

Substituting Eq. 1152 into Eq. 1162 and performingsome tedious algebra, we finally obtain the followingexpression for the wave-front variance in terms of thesubaperture configuration parameters:

sw2 5 1W20r02223q2rms 1 q2sr

2 1 11 2 q22@124, 1182

where

rms 5 1rn@r022 1192

is the normalized mean square radius of the subaper-ture array and

sr2 5 1rn@r024 2 1rn@r022

2

5 1rn@r024 2 rms2 1202

is the variance in the square of the subaperture radialposition about that mean square radius as illustratedin Fig. 7a. Because sw

2 consists of three positivedefinite terms, it follows that, for any value of q fi 0,sw

2 is a minimum when sr2 is zero, i.e., when all the

subapertures are centered on a circle as illustrated inFig. 7b.Recall that parameter q is the ratio of the field

curvature equivalent to the curved focal surface andthe field curvature of the individual telescopes.Taking the derivative of sr

2 with respect to parameterq and setting the resulting quantity equal to zero, we

can find an extremum for sr2:

≠sw2@≠q 5 1W20r022232qrms 1 2qsr

2 2 211 2 q2@124 5 0.

1212

Because the second derivative is positive,

≠2sw2@≠q2 5 21W20r02221rms 1 sr

2 1 1@122 . 0, 1222

that extremum is in fact a minimum. Equation 1212can now be solved for the optimum value of theparameter q:

qopt 5 11@122@1rms 1 sr2 1 1@122. 1232

Note that qopt is always less than unity. Substitutingthis value of q back into Eq. 1182 provides the mini-mum achievable wave-front variance:

swmin2 5 1W20r02223qopt2rms 1 qopt2sr

2 1 11 2 qopt22@124,

1242

where qopt corresponds to the optimum focal surface.This maximum performance of each subaperture

configuration thus occurs at the respective values ofqopt as determined by Eq. 1232. These values aretabulated below for various subaperture configura-tions.However, when q 5 0 1the condition that corre-

sponds to a flat focal surface2 Eq. 1182 reduces to

sw2 5 1W20r0222@12, 1252

and the wave-front variance of a phased telescopearray depends only on the field curvature of theindividual subapertures and their size and does notdepend on the subaperture configuration. Note thatthis is the well-known variance of a defocused wavefront over a single circular pupil, i.e., the wave-frontvariance of an aperture made up of N identicalcircular subapertures, each with the same amount offield curvature, is just the wave-front variance of oneof the subapertures.For a monolithic circular aperture 1conventional

telescope2, sr2 5 0 and rms 5 0; hence Eq. 1182 reduces

to

sw2 5 1W20r022211 2 q22@12 1262

Fig. 7. a, Deviation of subaperture radial position from rmsradius indicated. b, Circle-N subaperture configuration wheresr

2 5 0.


when q 5 1, which corresponds to an optimally curvedfocal surface for a monolith:

sw2 5 0. 1272

Hence there is no image degradation on this optimallycurved focal surface because of the field curvature forconventional monolithic imaging systems.The image quality as a function of parameter q is

illustrated graphically in Fig. 8 for a variety ofsubaperture configurations including the monolithiccircular aperture. The Strehl ratio is calculatedfrom the wave-front variance by involving Eq. 112.Note that, although the monolith improves monotoni-cally as q goes from zero 1the flat focal plane2 to unity1the optimally curved focal surface for the monolith2,the phased telescope arrays experience a slight im-provement for small values of q followed by a declinein image quality. This phenomenon is illustratedmore clearly in the expanded view for small values ofq provided in Fig. 9.It can be seen from both Table 1 and Fig. 9 that the

optimum focal surface for phased telescope arraysdiffers only slightly from the flat focal plane, and onlyinfinitesimal improvement is realized by going to acurved focal surface.

Fig. 9. Strehl ratio versus q illustrating that the maximumperformance occurs at q , 0.05 for all the phased-array configura-tions that were studied.

Fig. 8. Strehl ratio versus q for various subaperture configura-tions.


The analytical solution for wave-front varianceexpressed by Eq. 1182 thus provides a great deal ofinsight into the way in which the subaperture configu-ration affects the optical performance of phased ar-rays of telescopes exhibiting field curvature. Further-more this equation can very easily be used to performother parametric studies and sensitivity analyses.When we set q 5 1, which corresponds to the analyti-cal solution for the Strehl ratio expressed by Eq. 192,we obtained excellent agreement between these twocomputational techniques. This serves as a check onboth computational techniques.

3. Numerical Solution for Wave-Front VarianceThe third computational technique is the brute forcenumerical calculation of wave-front variance. Thisdoes not provide as much insight as the previousanalytical solution; however, it is not limited to aparticular aberration such as field curvature. Forexample, whatever the aberrationmight be, the wave-front variance can be calculated directly by numeri-cally evaluating Eq. 1162, which is repeated here forconvenience:

sw2 5 W 2 2 W2 5

1

NA e2`

` e W 2dx8dy8

2 1 1NA e2`

` e Wdx8dy822

. 1282

Acomputer code was written and implemented with anumerical sampling density sufficient to achieve theclosed-form solution to ,0.02%. The results wereshown to be in excellent agreement with the results ofthe other two techniques.All three of these computational techniques are

complementary and have been used to obtain resultsin various parametric studies and sensitivity analy-ses.

D. Inherent Field Curvature of Two-Mirror Telescopes

The following parametric performance predictionspertain to phased arrays of two-mirror telescopes thatsuffer from the field curvature inherent in suchdesigns. The magnitude of the field-curvature coeffi-cient for a Mersenne design is given by the primarymirror diameter, its f number [email protected], the telescope

Table 1. Optimum Focal Surface for Various SubapertureConfigurations

Configuration qopt

Monolith 1.0000Circle-3 0.0416NNTT 0.0281MMT 0.0143Circle-9 0.0067Golay-6 0.0039Golay-9 0.0005Mill’s cross 0.0002

magnification, and the field angle as previously ex-pressed in Eq. 142We have shown above that there is virtually noth-

ing to be gained in terms of performance by going to acurved focal surface for phased telescope arrays;hence the following parametric curves are all for flatfocal planes. We have also already shown that for aflat focal plane 1q 5 02 the wave-front variance andtherefore the Strehl ratio will be the same for allsubaperture configurations when the subaperturesize is held fixed.Figure 10 graphically illustrates image degrada-

tion expressed in terms of the Strehl ratio as afunction of the field angle in arcminutes and itssensitivity to the subaperture diameter. The wave-length is held constant at l 5 0.5 µm, the primarymirror F@no. is set at 2.5, the telescope magnificationis assumed to be 0.1, and the subaperture arraydilution ratio is 1.2.Recall that our error-budget allocation for off-axis

optical-design errors corresponds to a Strehl ratio of0.95. Note from Fig. 10 that subaperture diametersof less than 0.25 m meet the error-budget allocationfor a semifield angle of 30 arcmin and subaperturediameters of less than 0.5 m meet the error-budgetallocation for a semifield angle of 10 arcmin. For1.0-m-diam subapertures, the semifield angle satisfy-ing the error-budget drops to ,3.5 arcmin. If thediameter of the telescopes making up the array ismade even larger, the FOV within which the arraysatisfies the established error-budget allocation con-tinues to drop rapidly. For example, an array madeup of 4-m-diam telescopes performs satisfactorily overfield angles of less than 0.5 arcmin, and 8-m tele-scopes yield a semifield angle of only ,10 arcsec.Our nominal goal of a 15-arcmin semifield angle canbe satisfied with a subaperture diameter of ,0.40 m.The sensitivity of the phased-array performance to

other telescope design parameters is also of interest.It should be evident that for a flat focal plane 1a 5 02,the wave-front variance and therefore the Strehl ratiois the same for all dilution ratios when the subaper-ture size is held fixed. This fact was verified bycalculations for various dilution ratios that wereobserved to give identical results. Figures 11 and 12

Fig. 10. Sensitivity of Strehl ratio to subaperture size indicatingsubarc minute FOV’s for telescope diameters greater than 2 m.

demonstrate the sensitivity of the FOV to telescopemagnification and the primary mirror F@no., respec-tively. Note that phased telescope arrays are rela-tively insensitive to modest changes in the F@no. ofthe primary mirror; however, changes in the telescopemagnification can make a significant change in theuseful FOV of the array.A different perspective on the comparison between

the subaperture configurations is obtained if we holdthe total collecting area of the array constant ratherthan the diameter of the subapertures. Dependingon the application, this may be the more meaningfulcomparison. Figure 13 dramatically illustrates thesuperiority in FOV of using a larger number ofsmaller telescopes for equal area arrays.

E. Residual Field Curvature of Corrected Systems

The previous parametric curves have indicated that,although a 15-arcmin FOV may be achieved with a40-cm-diameter laboratory workingmodel of a phasedarray of two-mirror telescopes; the inherent fieldcurvature of such designs does not allow them to bescaled up for large space applications. We now pro-ceed to determine parametrically how well the fieldcurvature must be corrected if a 0.5-deg FOV is to beachieved with a phased array made up of 1 to4-m-diam individual telescopes.

Fig. 11. Sensitivity of the Strehl ratio to telescope magnification.

Fig. 12. Sensitivity of the Strehl ratio to the primary mirrorF@no.


Figure 14 illustrates the predicted Strehl ratio at afield angle of 15 arcmin as a function of the fractionalresidual field curvature after correction for a varietyof individual telescope diametersDn. For example, aphased array made up of 2-m-diam telescopes whosefield curvature has been corrected to 0.0075 of itsinherent value would just meet the error-budgetallocation of a 0.95 Strehl ratio at a field angle of 15arcmin. Similarly, a phased array made up of 4-m-diameter telescopes will meet the requirement if thefield curvature is corrected to slightly better than0.001 of its inherent uncorrected value.

F. Image Degradation from Distortion

In Subsection 3.E we showed how well the fieldcurvature must be corrected to not be a dominanterror source preventing wide-field imaging withphased telescope arrays. We must now look at thenext term in the aberration expansion shown in Eq.132. Third-order distortion in the individual tele-scopes will have the form

Wn1x, y2 5 W11r cos c 5 W11ry. 1292

However, from Fig. 6 we see that when we go to thearray coordinate system y 5 y8 2 yn and

Wn1x8, y82 5 W111 y8 2 yn2, 1302

Fig. 13. Sensitivity of Strehl ratio to the subaperture configura-tion 1equal area arrays2.

Fig. 14. Improved performance of field-flattened telescope de-signs.


for a phased telescope array, we have

W1x8, y82 5 on51

N

W11y8 cyl31r8 2 rn2@D4

conventionaldistortion

2 on51

N

W11yn cyl31r8 2 rn2@D4. 1312

The first term is just conventional distortion that willnot degrade the image of a point source; however, thesecond term is again field-dependent relative phaseerrors between the various subapertures that willseverely degrade image quality.Taking W11 from Eq. 152, calculating the wave-front

variance by the numerical technique discussed above,then substituting it into Eq. 112 to determine theStrehl ratio permit us to predict the Strehl ratioversus field angle for different design parameters.Figure 15 illustrates the predicted Strehl ratio versusfield angle for different subaperture diameters.For the design parameters indicated [email protected]; m 5

0.1; dilution ratio, 1.2; l 5 0.5 ,mm2, the third-orderdistortion inherent in a two-mirror telescope wouldlimit the semifield angle to ,7 arcmin for 0.25-m-diam telescopes to less than 5 arcmin for 1.0-m-diamtelescopes and to ,2 arcmin for 8.0-m telescopes.It is therefore clear that the distortion must also becorrected if a 0.5-deg FOV is to be achieved with thephased arrays of large telescopes.

G. FOV Limitations because of Field Curvature andDistortion

Both field curvature and distortion inherent in simpletwo-mirror telescopes have been individually shownto limit severely the achievable FOV of imagingphased telescope arrays. Figure 16 illustrates thecombined effect of the these two residual aberrationson the performance of a four-element array of Mer-senne telescopes for different subaperture diameters.When Fig. 16 is compared with Figs. 10 and 15, it isevident that the field curvature dominates distortionin large subapertures 1FOV , 1.0 arcmin2 and distor-

Fig. 15. FOV limitation resulting from uncorrected distortion.

tion dominates the field curvature for small subaper-tures 1FOV , 10 arcmin2.Miao and Shannon33 have discussed a design ap-

proach and investigated several design configurationsfor wide FOV phased telescope arrays. They de-scribed a complex, folded, five-mirror array systemthat eliminates all aberrations except for a small andsomewhat controllable residual distortion.Stuhlinger34 has also developed several possible

design configurations for wide-angle phased telescopearrays. We stated that correction of the field curva-ture is relatively straightforward; however, control ofsubaperture distortion was the driving factor in thesubtelescope design and is crucial to the opticalphasing of the arrays. He also stated that, typically,the necessary design configurations result in ninereflections per ray path including folding mirrors anda beam-combining telescope. The only subaperturetelescope design presented in his study that achievedthe necessary field curvature and distortion correc-tion with fewer than four mirrors was a totallyobscured system

4. Summary and Conclusions

We have considered the problems of obtaining a wideFOV 10.5 deg2 with large, space-based, direct-imagingphased telescope arrays. After defining some of thesystem’s critical issues, we reviewed and summarizedprevious relevant research in the literature. Re-sidual optical-design errors 1off-axis aberrations2 ofthe individual telescopes making up the array stoodout as a fundamental limitation largely unexplored todate.We then made an extensive list of potential error

sources and categorized them in the form of anerror-budget tree including errors in optical design,optical fabrication, assembly and alignment, and envi-ronment. After choosing a top-level image-qualityrequirement as a goal, first we performed a prelimi-nary tops-down error-budget allocation; then, basedon engineering experience, detailed analysis, or datafrom the literature, we performed a bottoms-up error-

Fig. 16. FOV limitation resulting from uncorrected field curva-ture and distortion. Note that the field curvature dominates thedistortion for large subapertures and distortion dominates the fieldcurvature for the small subapertures.

budget reallocation in an attempt to achieve anequitable distribution of difficulty in satisfying thevarious allocations. This exercise provided a realis-tic allocation for residual off-axis optical-design errors1sw , 0.03752 in the presence of state-of-the-art opticalfabrication and alignment errors.The effects of telescope aberrations on phased-

array performance was then discussed in detail, andit was shown that the somewhat benign 1for conven-tional optical systems2 aberrations of field curvatureand distortion result in field-dependent relative phase1piston2 and pointing 1tilt2 errors that rapidly degradethe image quality as the FOV is increased.Three different computational techniques were de-

veloped for computing the image degradation of phasedtelescope arrays from the aberrations of the indi-vidual telescopes: 112 An analytical expression forthe Strehl ratio was derived for field curvature. 122An analytical expression was derived for the rmswave-front error sw from field curvature. 132 A com-puter code was developed to calculate numerically therms wave-front error in the presence of arbitraryaberrations. Technique 122 was found to provide agreat deal of insight concerning the optimally curvedfocal surface and the variation 1or lack thereof 2 inperformance for different subaperture configurations.A flat focal surface is almost optimum, and for thatflat focal surface there is no difference in the imagedegradation from field curvature for different configu-rations. All three computational techniques weretested against one another and found to yield identi-cal results for given test cases involving only fieldcurvature. These comparisons validate both compu-tational techniques. Technique 132 was then used forthe calculations of image degradation from distortion.Parametric studies and sensitivity analyses were

then performed for a variety of subaperture configu-rations and telescope design parameters in an at-tempt to determine how the off-axis performance of aphased telescope array varies as the telescopes arescaled up in size. It was quickly learned that theinherent field curvature of two-mirror telescopesprevents the attainment of a 0.5-deg FOV for indi-vidual telescope diameters greater than ,40 cm.A phased array made up of 4-m telescopes wouldsatisfy our image-quality requirement only over afield angle of ,0.5 arcmin. Furthermore, even if thefield curvature were completely corrected, distortionwould limit the FOV to ,7 arcmin for telescopes assmall as 25 cm in diameter.Thus specific telescope designs that are corrected

for both field curvature and distortion must be devel-oped if large space-based phased telescope arrays withlarge FOV’s are to be a reality.

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Field-of-view limitations of phased telescope arrays