Abstract

This work addresses the physical basis of the measurement process for object-based phasing of an array of telescopes. In this regard an enhanced least-squares estimator that is capable of differentiating among three families of array aberrations in an object-based phasing system is developed. In a system of this nature the system to be phased illuminates the object of interest and the return radiation is detected. Telescope aberrations, atmospheric aberrations, and speckle-induced aberrations are all reported by the estimator to facilitate correction of telescope and atmospheric aberrations. This is accomplished by proper handling of the unobservable modes and recognizing that the five global aberrations—telescope array piston, atmospheric array piston and tilt, and speckle array piston and tilt—cannot be measured accurately so they need to be projected out of the estimated piston commands. Except for these relatively benign array aberrations, the disturbances for all three families of array aberrations are estimated exactly. An interesting feature of the speckle array aberrations is that a synthetic aperture is developed that is almost twice as large as the array of telescopes under consideration.

© 2012 Optical Society of America

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References

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  1. J. D. Barchers, Nutrionics, Inc., 4665 Nautilus Ct. S., Suite 500, Boulder, Colorado, 80301 (private communication, 2001).
  2. T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-synchronous and self-referenced coherent beam combination for large optical arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 480–486 (2007).
    [CrossRef]
  3. M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
    [CrossRef]
  4. This is a “fast” way to obtain the correct result. Here the derivative is with respect to pT assuming that pT is independent of p. In actuality, the derivative of the scalar nTn is taken with respect to the elements of p resulting in a set of multiple equations with multiple unknowns. When this process is completed Eq. (29) is obtained.
  5. It is well known that the quantity H0 is equivalent to the pseudoinverse of G. This is the solution with minimum norm. Although the result expressed in Eq. (38) could be determined immediately, the process illustrated here provides significant insight into what will be required in the next section to obtain an appropriate estimator. For this application a naive application of Eq. (38) will result in a deficient estimator.
  6. T. Rhoadarmer, “Phased Array Test and Evaluation (PhATE) Project,” Contract FA9451-06-C-0374, 2009 Annual Summary, Science Applications International Corporation (2009).

2009

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[CrossRef]

2007

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-synchronous and self-referenced coherent beam combination for large optical arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 480–486 (2007).
[CrossRef]

Aschenbach, K.

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[CrossRef]

Baker, J. T.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-synchronous and self-referenced coherent beam combination for large optical arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 480–486 (2007).
[CrossRef]

Barchers, J. D.

J. D. Barchers, Nutrionics, Inc., 4665 Nautilus Ct. S., Suite 500, Boulder, Colorado, 80301 (private communication, 2001).

Benham, V.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-synchronous and self-referenced coherent beam combination for large optical arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 480–486 (2007).
[CrossRef]

Beresnev, L. A.

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[CrossRef]

Carhart, G. W.

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[CrossRef]

Liu, L.

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[CrossRef]

Lu, C. A.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-synchronous and self-referenced coherent beam combination for large optical arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 480–486 (2007).
[CrossRef]

Pilkington, D.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-synchronous and self-referenced coherent beam combination for large optical arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 480–486 (2007).
[CrossRef]

Rhoadarmer, T.

T. Rhoadarmer, “Phased Array Test and Evaluation (PhATE) Project,” Contract FA9451-06-C-0374, 2009 Annual Summary, Science Applications International Corporation (2009).

Sanchez, A. D.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-synchronous and self-referenced coherent beam combination for large optical arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 480–486 (2007).
[CrossRef]

Shay, T. M.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-synchronous and self-referenced coherent beam combination for large optical arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 480–486 (2007).
[CrossRef]

Vorontsov, M. A.

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[CrossRef]

Weyrauch, T.

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-synchronous and self-referenced coherent beam combination for large optical arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 480–486 (2007).
[CrossRef]

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[CrossRef]

Other

This is a “fast” way to obtain the correct result. Here the derivative is with respect to pT assuming that pT is independent of p. In actuality, the derivative of the scalar nTn is taken with respect to the elements of p resulting in a set of multiple equations with multiple unknowns. When this process is completed Eq. (29) is obtained.

It is well known that the quantity H0 is equivalent to the pseudoinverse of G. This is the solution with minimum norm. Although the result expressed in Eq. (38) could be determined immediately, the process illustrated here provides significant insight into what will be required in the next section to obtain an appropriate estimator. For this application a naive application of Eq. (38) will result in a deficient estimator.

T. Rhoadarmer, “Phased Array Test and Evaluation (PhATE) Project,” Contract FA9451-06-C-0374, 2009 Annual Summary, Science Applications International Corporation (2009).

J. D. Barchers, Nutrionics, Inc., 4665 Nautilus Ct. S., Suite 500, Boulder, Colorado, 80301 (private communication, 2001).

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Figures (15)

Fig. 1.
Fig. 1.

Speckle measurements create a synthetic aperture that is almost twice as large as the array. The diagrams in this figure illustrate the nature of the speckle return from the object.

Fig. 2.
Fig. 2.

The influence matrix, G, for a five-element linear array. The influence matrix illustrated here is partitioned to more clearly illustrate the relationship between the 25 measurements identified by the transmitter and receiver combinations and the 19 disturbances to be estimated (five telescope aberrations, five atmospheric aberrations and nine speckle aberrations). As an example, the twenty-first row of this matrix corresponds to transmission from aperture E and receiving in aperture A. The measurement consists of the sum of the telescope piston, E, the atmospheric piston e+a and the speckle piston S5.

Fig. 3.
Fig. 3.

The eigenvalues of GTG. Since three eigenvectors of GTG have eigenvalues equal to zero, three unobservable modes are present and GTG is not invertible.

Fig. 4.
Fig. 4.

The unobservable modes associated with a five-element linear array. These modes are the eigenvectors of GTG with zero eigenvalue. Three unobservable modes are present and are illustrated by a line type.

Fig. 5.
Fig. 5.

The observable modes associated with a five-element linear array. These mode are the eigenvectors of GTG with nonzero eigenvalues. The modes are ordered from small eigenvalue to largest. These three figures illustrate the 16 observable modes available in the five element linear array. Each line type corresponds to a different eigenvector.

Fig. 6.
Fig. 6.

The matrix product HG illustrated in Eq. (52). The figure illustrates that the matrix, HG, is block diagonal as the upper-right and lower-left elements are equal to zero.

Fig. 7.
Fig. 7.

The figure illustrates the nondiagonal nature of the normalized noise covariance matrix, HHT. The diagonal term illustrates the error associated with the estimate of a single piston.

Fig. 8.
Fig. 8.

The figure illustrates the diagonal elements of HHT. This is the normalized noise-induced error when each piston is estimated. The figure illustrates that the speckle estimator exhibits the most error.

Fig. 9.
Fig. 9.

(a), (c), (e), and (g) illustrate the unobservable modes for array sizes ranging from two to five elements, respectively. (b), (d), (f), and (h) illustrate the eigenvalues of GTG for array sizes ranging from two to five elements, respectively. In all cases three unobservable modes are present. In addition, the three unobservable modes consist of the same global aberrations of interest, which include telescope array piston, atmospheric array piston and array tilt, and speckle array piston and array tilt.

Fig. 10.
Fig. 10.

Matrix product HG for a two through five element linear array. (a) through (d) correspond to linear arrays consisting of two through five elements, respectively. In all cases HG is block diagonal but the information content increases as the array size increases. (a) illustrates that the turbulence aberrations for the two-element array are not measured since all the elements in the “missing” center block are equal to zero. The upper-left block illustrates that telescope array tilt can be measured but telescope array piston is removed. The lower-right block corresponds to speckle. Since both array piston and array tilt are removed here only V-mode (a mode proportional to 1, 2, 1) can be measured because only three degrees of freedom are available (i.e., three elements are in the synthetic aperture). (b) illustrates that the situation is improved somewhat for the three-element linear array. Although many speckle modes are available, only atmospheric V-mode can be measured. (c) and (d) pertain to the four- and five-element linear arrays, respectively, and illustrate that a significant amount of information can be measured.

Fig. 11.
Fig. 11.

Noise gain for linear array consisting of two through five elements. To obtain these results, Eqs. (44) and (45) were used. The noise gain is reduced as the number of telescopes is increased. It is roughly equal to 1N, where N is the number of telescopes.

Fig. 12.
Fig. 12.

Eigenvalues of GTG for a 19 aperture hexagonal array. Inset: enlargement of the left side of the figure illustrating that four unobservable modes are present.

Fig. 13.
Fig. 13.

Four unobservable modes associated with a 19-telescope hexagonal array. Each column corresponds to one of the four unobservable modes. Top row illustrates the speckle component, the middle row illustrates the atmospheric component, and the bottom row corresponds to the telescope component. The much larger synthetic aperture associated with the speckle component is clearly visible. As in the one-dimensional case, these unobservable modes can be expressed as superpositions of the appropriate global modes. The bottom row illustrates that the telescope component of the null space is telescope array piston. In addition to array piston the atmospheric and speckle components of the null space require both x- and y-tilt because this is now a two-dimensional problem. A careful analysis of the ±60° tilts present in the first two columns illustrate that they are linear superpositions of x- and y-tilts. As a consequence, the null space is spanned by the seven global modes: telescope array piston, atmospheric array piston and array x-, y-tilt, and speckle array piston and array x-, y-tilt.

Fig. 14.
Fig. 14.

The matrix product HG for a (a) seven and (b) 19 element hexagonal array. As in the one-dimensional case, these matrices are block diagonal and illustrate that the three families of aberrations—telescope, atmosphere, and speckle—can be determined independently.

Fig. 15.
Fig. 15.

The noise gain for a number of hexagonal arrays consisting of seven to 127 telescopes. The noise gain is proportional to 1N, where N is the number of telescopes.

Equations (54)

Equations on this page are rendered with MathJax. Learn more.

E2=A1drP(r)[ϕ(r)ϕ¯]2,
A=drP(r),
P(r)=i=1NW[(rri)/R],
W(r)={1,for |r|1;0,otherwise.
ϕ(r)=n=1Nϕn(r),
ϕn(r)=p=0NzapnZpn(r),
Zpn(r)=W[(rrn)/R]Zp[(rrn)/R].
drP(r)Zpn(r)Zpn(r)=ANδppδnn,
ϕ¯=A1drP(r)ϕ(r).
ϕ¯=A1n=1Np=0NzapndyP(r)Zpn(r).
Z0n(r)=W[(rrn)/R]Z0(r/R),
Z0(r/R)=1,
ϕ¯=A1n=1Np=0NzapndrP(r)Zpn(r)Z0n(r).
ϕ¯=1Nn=1Na0n.
E2=A1drP(r)ϕ2(r)ϕ¯2.
E2=A1n=1Np=0Nzn=1Np=0NzapnapndrP(r)Zpn(r)Zpn(r)1N2n=1Nn=1Na0na0n.
E2=1Nn=1Np=0Nz(apn)21N2n=1Nn=1Na0na0n.
E2=1Nn=1N(a0na¯0)2+1Nn=1Np=1Nz(apn)2,
a¯0=1Nn=1Na0n.
1Nn=1Np=1Nz(apn)2=0,
ϕn=a0na¯0.
E2=1Nn=1Nϕn2.
ϕn=ϕTn+ϕAn
E2=1Nn=1N(ϕTn2+2ϕTnϕAn+ϕAn2).
m=Gp+n.
n=mGp.
nTn=(mTpTGT)(mGp),
nTn=mTmmTGppTGTm+pTGTGp.
dnTndpT=GTm+GTGp.
GTGp=GTm.
GTGen=λnen,
GTG=UDUT.
λn=0for1nN0,
enTp=0for1nN0,
enenTp=0for1nN0.
(GTG+n=1N0enenT)p=GTm.
p^=H0m,
H0=(GTG+n=1N0enenT)1GT.
p^=Hm,
H=(IWWT)(GTG+n=1N0enenT)1GT.
p^=Hn.
p^p^T=HnnTHT,
nnT=σn2I,
p^p^T=σn2HHT.
E2/σn2=1Nn=1N(ϕTn2/σn2+2ϕTnϕAn/σn2+ϕAn2/σn2).
m=Gp,
n=0.
W=[0.4472135954999580.0000000000000000.0000000000000000.0000000000000000.0000000000000000.4472135954999580.0000000000000000.0000000000000000.0000000000000000.0000000000000000.4472135954999580.0000000000000000.0000000000000000.0000000000000000.0000000000000000.4472135954999580.0000000000000000.0000000000000000.0000000000000000.0000000000000000.4472135954999580.0000000000000000.0000000000000000.0000000000000000.0000000000000000.0000000000000000.4472135954999580.6324555320336760.0000000000000000.0000000000000000.0000000000000000.4472135954999580.3162277660168380.0000000000000000.0000000000000000.0000000000000000.4472135954999580.0000000000000000.0000000000000000.0000000000000000.0000000000000000.4472135954999580.3162277660168380.0000000000000000.0000000000000000.0000000000000000.4472135954999580.6324555320336760.0000000000000000.0000000000000000.0000000000000000.0000000000000000.0000000000000000.3333333333333330.5163977794943220.0000000000000000.0000000000000000.0000000000000000.3333333333333330.3872983346207420.0000000000000000.0000000000000000.0000000000000000.3333333333333330.2581988897471610.0000000000000000.0000000000000000.0000000000000000.3333333333333330.1290994448735810.0000000000000000.0000000000000000.0000000000000000.3333333333333330.0000000000000000.0000000000000000.0000000000000000.0000000000000000.3333333333333330.1290994448735810.0000000000000000.0000000000000000.0000000000000000.3333333333333330.2581988897471610.0000000000000000.0000000000000000.0000000000000000.3333333333333330.3872983346207420.0000000000000000.0000000000000000.0000000000000000.3333333333333330.516397779494322],
WTW=I,
(IWWT)en=0forn=1,2,3.
UTW=[0.0006426234291170.1286958468865130.3514827394276290.3444650246693160.8609533649933120.6625442015053610.5121778798575650.0949250184233470.4854211435212980.2325178589614910.6172801374471330.0333346423225610.1015197892970580.7387219776086560.2486716825726520.0474410294268670.0948820588537330.0000000000000010.0353604555736780.0000000000000020.0000000000000000.0000000000000000.1737502696333880.0000000000000010.0709332505454650.0141252978477000.0282505956954010.0000000000000020.0105283753966300.0000000000000010.0000000000000010.0000000000000010.1668333871069820.0000000000000010.0681094450787210.0034085133613930.0068170267227870.0000000000000190.0025405558594310.0000000000000080.0000000000000000.0000000000000010.4045199174779450.0000000000000000.1651445647689540.0211716366459810.0423432732919610.0000000000000020.0157804062451120.0000000000000010.0000000000000000.0000000000000010.0176806720720000.0000000000000000.0072181041476460.0262631327111970.0525262654223940.0000000000000000.0195753833481110.0000000000000010.0000000000000000.0000000000000000.0539633162995610.0000000000000010.0220304316270560.0292786557386290.0585573114772580.0000000000000000.0218230215071290.0000000000000000.0032157424939070.0064314849878140.0000000000000000.0023968729381700.0000000000000000.0000000000000000.0000000000000000.0724549229379820.0000000000000000.0295795984251220.0198965172287970.0397930344575930.0000000000000000.0148299883463620.0000000000000000.0000000000000000.0000000000000000.7918393755846680.0000000000000000.3232670714044130.4184934459543410.8369868919086810.0000000000000000.3119265977640130.000000000000000].
p^=HGp.
E2/σn2=0.22.
E2/σn=0.0033687+2.1529N,

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