Abstract

We extend the redundant spacings calibration method for finding piston coefficients affecting the elements of a dilute aperture array so that tilt phase coefficients can also be calculated and corrected without the need for assumptions about the object. The tilt coefficient retrieval method is successfully demonstrated in simulation, and the specifics of correction by image sharpness are discussed, showing that in dilute aperture systems this method does not necessarily produce a unique image.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. H. Greenaway, “Terrestrial optical aperture synthesis technique (TOAST),” Opt. Commun. 58, 149–154 (1986).
    [CrossRef]
  2. J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
    [CrossRef]
  3. A. M. Johnson, R. J. Eastwood, and A. H. Greenaway, “Optical aperture synthesis,” in Proceedings of 3rd EMRS DTC Technical Conference, http://www.emrsdtc.com/conferences/2006/downloads/pdf/conference_papers/B016.pdf .
  4. R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).
  5. P. M. Blanchard, A. H. Greenaway, R. N. Anderton, and R. Appleby, “Phase calibration of arrays at millimeter and optical wavelengths,” J. Opt. Soc. Am. A 13, 1593–1600 (1996).
    [CrossRef]
  6. R. J. Eastwood, A. M. Johnson, and A. H. Greenaway, “Calculation and correction of piston phase aberration in synthesis imaging,” J. Opt. Soc. Am. A 26, 195–205 (2009).
    [CrossRef]
  7. M. Borne and E. Wolf, Principles of Optics (Pergamon, 1959).
  8. C. M. Harding, R. A. Johnston, and R. G. Lane, “Fast simulation of Kolmogorov phase screen,” Appl. Opt. 38, 2161–2170 (1999).
    [CrossRef]
  9. R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. 64, 1200–1210 (1974).
    [CrossRef]

2009

1999

1998

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

1996

1986

A. H. Greenaway, “Terrestrial optical aperture synthesis technique (TOAST),” Opt. Commun. 58, 149–154 (1986).
[CrossRef]

1974

Anderton, R. N.

Appleby, R.

Armstrong, J. T.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Benson, J. A.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Blanchard, P. M.

Borne, M.

M. Borne and E. Wolf, Principles of Optics (Pergamon, 1959).

Bowers, P. F.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Buffington, A.

Buscher, D. F.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Clark, J. H.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Eastwood, R. J.

R. J. Eastwood, A. M. Johnson, and A. H. Greenaway, “Calculation and correction of piston phase aberration in synthesis imaging,” J. Opt. Soc. Am. A 26, 195–205 (2009).
[CrossRef]

A. M. Johnson, R. J. Eastwood, and A. H. Greenaway, “Optical aperture synthesis,” in Proceedings of 3rd EMRS DTC Technical Conference, http://www.emrsdtc.com/conferences/2006/downloads/pdf/conference_papers/B016.pdf .

Elias, N. M.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Greenaway, A. H.

R. J. Eastwood, A. M. Johnson, and A. H. Greenaway, “Calculation and correction of piston phase aberration in synthesis imaging,” J. Opt. Soc. Am. A 26, 195–205 (2009).
[CrossRef]

P. M. Blanchard, A. H. Greenaway, R. N. Anderton, and R. Appleby, “Phase calibration of arrays at millimeter and optical wavelengths,” J. Opt. Soc. Am. A 13, 1593–1600 (1996).
[CrossRef]

A. H. Greenaway, “Terrestrial optical aperture synthesis technique (TOAST),” Opt. Commun. 58, 149–154 (1986).
[CrossRef]

A. M. Johnson, R. J. Eastwood, and A. H. Greenaway, “Optical aperture synthesis,” in Proceedings of 3rd EMRS DTC Technical Conference, http://www.emrsdtc.com/conferences/2006/downloads/pdf/conference_papers/B016.pdf .

Ha, L.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Harding, C. M.

Hummel, C. A.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Hutter, D. J.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Johnson, A. M.

R. J. Eastwood, A. M. Johnson, and A. H. Greenaway, “Calculation and correction of piston phase aberration in synthesis imaging,” J. Opt. Soc. Am. A 26, 195–205 (2009).
[CrossRef]

A. M. Johnson, R. J. Eastwood, and A. H. Greenaway, “Optical aperture synthesis,” in Proceedings of 3rd EMRS DTC Technical Conference, http://www.emrsdtc.com/conferences/2006/downloads/pdf/conference_papers/B016.pdf .

Johnston, K. J.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Johnston, R. A.

Lane, R. G.

Ling, L.-C.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Mozurkewich, D.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Muller, R. A.

Rickard, L. J.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Simon, R. S.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).

White, N. M.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

Wolf, E.

M. Borne and E. Wolf, Principles of Optics (Pergamon, 1959).

Appl. Opt.

Astrophys. J.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy prototype optical interferometer,” Astrophys. J. 496, 550–571 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

A. H. Greenaway, “Terrestrial optical aperture synthesis technique (TOAST),” Opt. Commun. 58, 149–154 (1986).
[CrossRef]

Other

M. Borne and E. Wolf, Principles of Optics (Pergamon, 1959).

A. M. Johnson, R. J. Eastwood, and A. H. Greenaway, “Optical aperture synthesis,” in Proceedings of 3rd EMRS DTC Technical Conference, http://www.emrsdtc.com/conferences/2006/downloads/pdf/conference_papers/B016.pdf .

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1.
Fig. 1.

Schematic showing how the diffraction pattern of a pair of subapertures is constructed from the summation of differential components located at the end points of a vector spacing. The shading of aperture k indicates a linear phase function across it, while aperture j has zero phase (to simplify the concept). Only spacings running parallel to that of the two subapertures are shown in the figure. However, contributions from all spacings with the same magnitude and direction (but different locations relative to the subapertures) are involved, provided both their end points are simultaneously within one or the other subaperture.

Fig. 2.
Fig. 2.

Diagrammatic illustration of positioning of a single spatial frequency component in the image plane resulting from the following: (a) a pair of subapertures with tilt phase difference, and (b) a pair of subapertures with piston phase difference. Note that this diagram does not show resultant superposition, and so no enveloping function is present. The intention is only to illustrate that the phase difference between the ends of each frequency component vector (due here to the phase function in the right-hand subaperture as indicated by gray level) results in a corresponding fringe shift when the phase changes, and not when it does not.

Fig. 3.
Fig. 3.

Diagrammatic illustration of how the correlation function of subapertures j and k is formed as they are displaced from one another.

Fig. 4.
Fig. 4.

Dilute aperture used for simulations. This array has 12 subapertures comprising 18 total redundant spacings with 9 independent and none present more than twice, satisfying the requirements of the RSC theory. The scale is chosen to sample the wavefront received at a 50 cm diameter telescope—an application that has been mentioned by potential sponsors; the cross indicates the origin.

Fig. 5.
Fig. 5.

(a) Dilute aperture of Fig. 4 is shown pixelated for use in the simulations, with resolution as required to make the computational burden tractable. (b) Four subapertures from (a) that form a parallelogram redundant set; the heterogeneity in size and shape that leads to inaccuracies in the coefficient calculations can clearly be seen. (c) Higher resolution representation of the subapertures used to calculate interferograms in the simulations using the array theorem.

Fig. 6.
Fig. 6.

Example of the phase component of an auto-correlation, calculated by taking the Fourier transform of an interferogram from one of the simulations, generated using the configuration of Fig. 4. The circular phase patches can be seen against the random background, with discontinuous transitions between black and white indicating the presence of a modulo 2π wrapping.

Fig. 7.
Fig. 7.

(a) Linear loci between two of the measurement points in the cross-correlation phase functions of a pair of redundant spacings and the third, which is common to both paths. Along these lines, samples are taken of the phase and combined to form the phase function free of object information, according to the relative orientation of the spacing vectors—subtracted (as in this case) if parallel, added if antiparallel. (The figure shows the subtraction over all points in the phase patches for illustration only.) (b) One-dimensional unwrapping of the combined redundant phase. The differences between the values at the unwrapped end points and the common point are then used in the redundant conditions phase vector.

Fig. 8.
Fig. 8.

(a) Example of the phase functions applied to the subapertures; (b) for comparison, residual tilts after correction on the same scale.

Fig. 9.
Fig. 9.

Results of the simulations as shown in (a) a frequency distribution of residual subaperture phase rms as a percentage of the uncorrected rms (mean 1.13%), and (b) a scatter plot indicating the relationship between original and residual rms phase values, together with an axis scaled to show the Strehl ratio of the residual aberration.

Fig. 10.
Fig. 10.

Complex addition of cross-correlation components of redundant spacings, Rjk and Rlm, dependent only on piston phase differences φkφj and φmφl. Both redundant components are shown maximized, hence RjkRlm, because piston phases have no effect on them—it is this that leads to piston components having no influence on the correlation magnitude of nonredundant spacing. Clearly, |Rjk+Rlm| is maximized if and only if the arguments of Rjk and Rlm are identical, i.e., (φkφj)(φmφl)=0.

Fig. 11.
Fig. 11.

(a) Unaberrated PSF of the 12 subaperture array of Fig. 4, and (b) PSF of the same aperture array when the piston part of its phase function defines an arbitrary plane across the array and the individual subaperture tilts have the same (as each other) tilt orientation.

Fig. 12.
Fig. 12.

First row: (a) a simple circular source, (b) the unaberrated image of (a) produced by the 12 subaperture array of Fig. 4, and (c) the image resulting from the PSF of Fig. 11(b). Second row: (d) an object that is imaged by the same 12 subaperture array producing a large point spread function to illustrate the image aliasing, unaberrated (e), and with the same PSF as in (c) to give (f).

Equations (20)

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

RL(α)=L*(r)L(r+α)dr,
RL(α)=jAkAmj(rsj)exp[iϕj(rsj)]mk(rsk+α)exp[iϕk(rsk+α)]dr
RL(αx,αy)=jAkAexp[iϕj(xxj,yyj)]exp[iϕk(xxk+αx,yyk+αy)]dxdy=jAkAexp[iΔL(x,y|αx,αy)]dxdy,
RL(αx,αy)=j,kARjk(αxxjk,αyyjk)
Δjk(x,y|αx,αy)=ϕk(x+αx,y+αy)ϕj(x,y),
Δjk(x,y|αx,αy)=Δ¯jk(αx,αy)+Δ˜jk(x,y|αx,αy).
Rjk(αx,αy)={exp[iΔ˜jk(x,y|αx,αy)]dxdy}exp[iΔ¯jk(αx,αy)]=Mjk(αx,αy)exp[iΔ¯jk(αx,αy)],
Mjk(αx,αy)=exp(iΔ˜jk(x,y|αx,αy))dxdy.
Mjk(αx,αy)=cos(Δ˜jk(x,y|αx,αy))dxdy.
Δ¯jk(αx,αy)=(ak+aj)2αx(bk+bj)2αy+ckcj.
θjk(αx,αy)(ak+aj)2αx(bk+bj)2αy+ckcj=μjk(αx,αy),
(ak+aj)2αx(bk+bj)2αy+ckcj+ψjk(αx,αy)=μjk(αx,αy)+ψjk(αx,αy)θjk(αx,αy).
(ak+ajamal)2αx(bk+bjbmbl)2αy+ckcjcm+cl=μjk(αx,αy)μlm(αx,αy).
(akaj)2αx(bkbj)2αy+ckcj+ψjk(αx,αy)=μjk(αx,αy).
(akaj)2αx(bkbj)2αy+ckcj+ψjk(αx,αy)=(akaj)2αx(bkbj)2αy+ckcj
12π|I(ξ)|2dξ=|RL(α)|2|U(α)|2dα,
(a1+a2)2a12=0(a1+a3)2a13=0(an1+an)2an1n=0(b1+b2)2b12=0(b1+b3)2b13=0(bn1+bn)2bn1n=0.
F[g(ξ)f(ξγ)]=F[g(ξ)]*F[f(ξγ)]=F[g(ξ)]*exp(iγα)F[f(ξ)],
cj=xjγx+yjγy,
cjk=ckcj=(xkxj)γx+(ykyj)γy,

Metrics