Abstract

The principle of redundant spacings calibration has previously been described for the purpose of calibrating piston phase aberration affecting elements of a dilute aperture array using a system of linear equations in terms of the aperture phases as well as object phase information. Here we develop matrices for the correction of piston phase aberration by use of image sharpness and also by phase retrieval. These are both presented in wavefront sensor formulation in order to draw analogy between the approaches. We then discuss solution ambiguity affecting both methods and describe array design criteria to prevent such ambiguity. The problem of increased image aliasing under image sharpness correction is also highlighted.

© 2008 Optical Society of America

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References

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  1. R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).
  2. A. H. Greenaway, “Terrestrial Optical Aperture Synthesis Technique (TOAST),” Opt. Commun. 58, 149-154 (1986).
    [CrossRef]
  3. J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias II, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark III, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550-571 (1998).
    [CrossRef]
  4. M. J. Creech-Eakman, E. J. Bakker, D. F. Buscher, T. A. Coleman, C. A. Haniff, C. A. Jurgenson, D. A. Klinglesmith III, C. B. Parameswariah, V. D. Romero, A. V. Shtromberg, and J. S. Young, “Magdalena Ridge Observatory Interferometer: status update,” Proc. SPIE 6268, 62681V (2006).
    [CrossRef]
  5. A. M. Johnson, R. J. Eastwood, and A. H. Greenaway, “Optical aperture synthesis,” in presented at the 3rd Electro Magnetic Remote Sensing Defence Technology Centre (EMRS DTC) Technical Conference, Edinburgh, Scotland, July 13-14, 2006. Available at http://www.emrsdtc.com/conferences/2006/confeṟmaterial.htm (last accessed December 15, 2008).
  6. A. H. Greenaway, “Self-calibrating dilute-aperture optics, in digital image synthesis and inverse optics,” Proc. SPIE 1351, 738-748 (1990).
    [CrossRef]
  7. M. Born and E. Wolf, Principles of Optics (Pergamon, 1995).
  8. R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. 67, 1200-1210 (1974).
    [CrossRef]
  9. P. M. Blanchard, A. H. Greenaway, R. N. Anderton, R. Appleby, “Phase calibration of arrays at optical and millimeter wavelengths,” J. Opt. Soc. Am. A 13, 1593-1600 (1996).
    [CrossRef]
  10. J. H. Hubbard and B. B. Hubbard, Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (Prentice-Hall, 2002).
  11. R. Kress, Numerical Analysis (Springer, 1998).
    [CrossRef]

2006 (1)

M. J. Creech-Eakman, E. J. Bakker, D. F. Buscher, T. A. Coleman, C. A. Haniff, C. A. Jurgenson, D. A. Klinglesmith III, C. B. Parameswariah, V. D. Romero, A. V. Shtromberg, and J. S. Young, “Magdalena Ridge Observatory Interferometer: status update,” Proc. SPIE 6268, 62681V (2006).
[CrossRef]

1998 (1)

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias II, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark III, 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 (1)

1990 (1)

A. H. Greenaway, “Self-calibrating dilute-aperture optics, in digital image synthesis and inverse optics,” Proc. SPIE 1351, 738-748 (1990).
[CrossRef]

1986 (1)

A. H. Greenaway, “Terrestrial Optical Aperture Synthesis Technique (TOAST),” Opt. Commun. 58, 149-154 (1986).
[CrossRef]

1974 (1)

R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. 67, 1200-1210 (1974).
[CrossRef]

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 II, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark III, L. Ha, L.-C. Ling, N. M. White, and R. S. Simon, “The Navy Prototype Optical Interferometer,” Astrophys. J. 496, 550-571 (1998).
[CrossRef]

Bakker, E. J.

M. J. Creech-Eakman, E. J. Bakker, D. F. Buscher, T. A. Coleman, C. A. Haniff, C. A. Jurgenson, D. A. Klinglesmith III, C. B. Parameswariah, V. D. Romero, A. V. Shtromberg, and J. S. Young, “Magdalena Ridge Observatory Interferometer: status update,” Proc. SPIE 6268, 62681V (2006).
[CrossRef]

Benson, J. A.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias II, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark III, 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.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1995).

Bowers, P. F.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias II, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark III, 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.

R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. 67, 1200-1210 (1974).
[CrossRef]

Buscher, D. F.

M. J. Creech-Eakman, E. J. Bakker, D. F. Buscher, T. A. Coleman, C. A. Haniff, C. A. Jurgenson, D. A. Klinglesmith III, C. B. Parameswariah, V. D. Romero, A. V. Shtromberg, and J. S. Young, “Magdalena Ridge Observatory Interferometer: status update,” Proc. SPIE 6268, 62681V (2006).
[CrossRef]

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

Coleman, T. A.

M. J. Creech-Eakman, E. J. Bakker, D. F. Buscher, T. A. Coleman, C. A. Haniff, C. A. Jurgenson, D. A. Klinglesmith III, C. B. Parameswariah, V. D. Romero, A. V. Shtromberg, and J. S. Young, “Magdalena Ridge Observatory Interferometer: status update,” Proc. SPIE 6268, 62681V (2006).
[CrossRef]

Creech-Eakman, M. J.

M. J. Creech-Eakman, E. J. Bakker, D. F. Buscher, T. A. Coleman, C. A. Haniff, C. A. Jurgenson, D. A. Klinglesmith III, C. B. Parameswariah, V. D. Romero, A. V. Shtromberg, and J. S. Young, “Magdalena Ridge Observatory Interferometer: status update,” Proc. SPIE 6268, 62681V (2006).
[CrossRef]

Eastwood, R. J.

A. M. Johnson, R. J. Eastwood, and A. H. Greenaway, “Optical aperture synthesis,” in presented at the 3rd Electro Magnetic Remote Sensing Defence Technology Centre (EMRS DTC) Technical Conference, Edinburgh, Scotland, July 13-14, 2006. Available at http://www.emrsdtc.com/conferences/2006/confeṟmaterial.htm (last accessed December 15, 2008).

Elias, N. M.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias II, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark III, 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.

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

A. H. Greenaway, “Self-calibrating dilute-aperture optics, in digital image synthesis and inverse optics,” Proc. SPIE 1351, 738-748 (1990).
[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 presented at the 3rd Electro Magnetic Remote Sensing Defence Technology Centre (EMRS DTC) Technical Conference, Edinburgh, Scotland, July 13-14, 2006. Available at http://www.emrsdtc.com/conferences/2006/confeṟmaterial.htm (last accessed December 15, 2008).

Ha, L.

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

Haniff, C. A.

M. J. Creech-Eakman, E. J. Bakker, D. F. Buscher, T. A. Coleman, C. A. Haniff, C. A. Jurgenson, D. A. Klinglesmith III, C. B. Parameswariah, V. D. Romero, A. V. Shtromberg, and J. S. Young, “Magdalena Ridge Observatory Interferometer: status update,” Proc. SPIE 6268, 62681V (2006).
[CrossRef]

Hubbard, B. B.

J. H. Hubbard and B. B. Hubbard, Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (Prentice-Hall, 2002).

Hubbard, J. H.

J. H. Hubbard and B. B. Hubbard, Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (Prentice-Hall, 2002).

Hummel, C. A.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias II, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark III, 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 II, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark III, 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.

A. M. Johnson, R. J. Eastwood, and A. H. Greenaway, “Optical aperture synthesis,” in presented at the 3rd Electro Magnetic Remote Sensing Defence Technology Centre (EMRS DTC) Technical Conference, Edinburgh, Scotland, July 13-14, 2006. Available at http://www.emrsdtc.com/conferences/2006/confeṟmaterial.htm (last accessed December 15, 2008).

Johnston, K. J.

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

Jurgenson, C. A.

M. J. Creech-Eakman, E. J. Bakker, D. F. Buscher, T. A. Coleman, C. A. Haniff, C. A. Jurgenson, D. A. Klinglesmith III, C. B. Parameswariah, V. D. Romero, A. V. Shtromberg, and J. S. Young, “Magdalena Ridge Observatory Interferometer: status update,” Proc. SPIE 6268, 62681V (2006).
[CrossRef]

Klinglesmith, D. A.

M. J. Creech-Eakman, E. J. Bakker, D. F. Buscher, T. A. Coleman, C. A. Haniff, C. A. Jurgenson, D. A. Klinglesmith III, C. B. Parameswariah, V. D. Romero, A. V. Shtromberg, and J. S. Young, “Magdalena Ridge Observatory Interferometer: status update,” Proc. SPIE 6268, 62681V (2006).
[CrossRef]

Kress, R.

R. Kress, Numerical Analysis (Springer, 1998).
[CrossRef]

Ling, L.-C.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias II, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark III, 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 II, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark III, 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.

R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. 67, 1200-1210 (1974).
[CrossRef]

Parameswariah, C. B.

M. J. Creech-Eakman, E. J. Bakker, D. F. Buscher, T. A. Coleman, C. A. Haniff, C. A. Jurgenson, D. A. Klinglesmith III, C. B. Parameswariah, V. D. Romero, A. V. Shtromberg, and J. S. Young, “Magdalena Ridge Observatory Interferometer: status update,” Proc. SPIE 6268, 62681V (2006).
[CrossRef]

Rickard, L. J.

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

Romero, V. D.

M. J. Creech-Eakman, E. J. Bakker, D. F. Buscher, T. A. Coleman, C. A. Haniff, C. A. Jurgenson, D. A. Klinglesmith III, C. B. Parameswariah, V. D. Romero, A. V. Shtromberg, and J. S. Young, “Magdalena Ridge Observatory Interferometer: status update,” Proc. SPIE 6268, 62681V (2006).
[CrossRef]

Shtromberg, A. V.

M. J. Creech-Eakman, E. J. Bakker, D. F. Buscher, T. A. Coleman, C. A. Haniff, C. A. Jurgenson, D. A. Klinglesmith III, C. B. Parameswariah, V. D. Romero, A. V. Shtromberg, and J. S. Young, “Magdalena Ridge Observatory Interferometer: status update,” Proc. SPIE 6268, 62681V (2006).
[CrossRef]

Simon, R. S.

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias II, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark III, 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 II, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark III, 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. Born and E. Wolf, Principles of Optics (Pergamon, 1995).

Young, J. S.

M. J. Creech-Eakman, E. J. Bakker, D. F. Buscher, T. A. Coleman, C. A. Haniff, C. A. Jurgenson, D. A. Klinglesmith III, C. B. Parameswariah, V. D. Romero, A. V. Shtromberg, and J. S. Young, “Magdalena Ridge Observatory Interferometer: status update,” Proc. SPIE 6268, 62681V (2006).
[CrossRef]

Astrophys. J. (1)

J. T. Armstrong, D. Mozurkewich, L. J. Rickard, D. J. Hutter, J. A. Benson, P. F. Bowers, N. M. Elias II, C. A. Hummel, K. J. Johnston, D. F. Buscher, J. H. Clark III, 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. (1)

R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. 67, 1200-1210 (1974).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

A. H. Greenaway, “Terrestrial Optical Aperture Synthesis Technique (TOAST),” Opt. Commun. 58, 149-154 (1986).
[CrossRef]

Proc. SPIE (2)

M. J. Creech-Eakman, E. J. Bakker, D. F. Buscher, T. A. Coleman, C. A. Haniff, C. A. Jurgenson, D. A. Klinglesmith III, C. B. Parameswariah, V. D. Romero, A. V. Shtromberg, and J. S. Young, “Magdalena Ridge Observatory Interferometer: status update,” Proc. SPIE 6268, 62681V (2006).
[CrossRef]

A. H. Greenaway, “Self-calibrating dilute-aperture optics, in digital image synthesis and inverse optics,” Proc. SPIE 1351, 738-748 (1990).
[CrossRef]

Other (5)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1995).

J. H. Hubbard and B. B. Hubbard, Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (Prentice-Hall, 2002).

R. Kress, Numerical Analysis (Springer, 1998).
[CrossRef]

A. M. Johnson, R. J. Eastwood, and A. H. Greenaway, “Optical aperture synthesis,” in presented at the 3rd Electro Magnetic Remote Sensing Defence Technology Centre (EMRS DTC) Technical Conference, Edinburgh, Scotland, July 13-14, 2006. Available at http://www.emrsdtc.com/conferences/2006/confeṟmaterial.htm (last accessed December 15, 2008).

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

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Figures (9)

Fig. 1
Fig. 1

Showing the overlapping region C ( α ) of two aperture functions relatively shifted by α . The area of this region is M j k ( α ) in Eq. (8).

Fig. 2
Fig. 2

Perspective diagram of a tilt plane bisecting the centers of R L piston phase patches—the lighter side of the patch is before/above the tilt plane (partially transparent in the illustration), while the darker half is behind/below it. The angles β and γ indicate the two dimensions of tilt, with respect to the horizontal and vertical reference axes.

Fig. 3
Fig. 3

Showing in (a) a 12-element dilute aperture (center marked with a cross), (b) its unaberrated PSF, and (c) the PSF subject to a piston-defined tilt plane.

Fig. 4
Fig. 4

Showing on the first row: (a) a simple circular source, (b) the unaberrated image of (a) produced by the 12-aperture array of Fig. 3a, and (c) the image resulting from the PSF of Fig. 3c. The second row shows an object which is (d) imaged by the same 12-aperture array producing a large PSF to illustrate the image aliasing, unaberrated (e), and with the same PSF as in (c) to give (f). The third row shows the object repeated in (d) for convenience, (g) imaged unaberrated using again the same array with a higher resolution PSF, and (h) with the same piston-defined tilt plane PSF.

Fig. 5
Fig. 5

Showing two nine-aperture redundant spacing arrays with similar but subtly different configurations. The redundancies in (a) are made solely from three-aperture linear arrangements as in (c), whereas in (b) there are many parallelogram redundancies, collapsed as shown in (d) such that sides of the parallelogram formed by black dots all become parallel as one set of dots is shifted to lie co-linear with the others (becoming gray).

Fig. 6
Fig. 6

Progression of linear redundancy information by Gaussian elimination, resulting in factors of 2 in the triangular form of block D. The figure illustrates a redundancy condition (in block C) that can be used in combination with measured phase data (rows through blocks A and B) to produce an upper-triangular matrix. Addition and subtraction of the rows from A, B set the correspoding elements in C to zero, but lead to a matrix element with value 2 in block D due to the elements identified by the lozenge in block B.

Fig. 7
Fig. 7

Experimental setup of the laboratory demonstration of visibility on linearly redundant baselines. A collimated laser source is projected onto a liquid crystal SLM which is programmed to modulate the phase of a disk corresponding to the selected inner aperture of the RSC mask (indicated). This light then passes through the aperture array and the resulting interferogram is imaged on a CCD camera.

Fig. 8
Fig. 8

Laboratory demonstration of visibility on linearly redundant baselines. The array configuration is that of Fig. 5a, and one of the three innermost apertures is the one modulated. This is central (and hence involved twice in the redundancy equations) in a single pair of linearly redundant spacings, so the visibility changes at twice the rate of the two other linearly redundant baselines, where it is situated at the end. Visibilities from nonredundant baselines remain constant (within experimental error) and clustered around the center of the plot, while those due to redundant baselines whose phase is not being modulated are constant and clustered toward the top. The displacement seen between the single cycling visibilities is due to additional aberrations in the optical setup, and the spread of visibilities is because of nonuniformity in the light source.

Fig. 9
Fig. 9

Combination of complex cross correlation components for redundant spacings s j k = s l m and s j l = s k m with s j k s j l . Where the illumination is the same on all apertures (a), the autocorrelation magnitude peaks will vary according to Eq. (15) between the maximum when the phasors are parallel and zero when they are antiparallel. If the illumination is different (but constant across any particular aperture) (b), the relative maximum peak height will be reduced as compared with (a), and the minimum will be greater.

Equations (30)

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

I ( ξ ) = R L ( α ) U ( α ) exp ( i ξ α ) d α = F [ R L U ] ,
R L ( α ) = L * ( r ) L ( r α ) d r .
A ( r ) = j A a ( r r j ) ; { a ( s ) = 1 s ρ = 0 s > ρ } ,
L ( r ) = L ( r ) j A a ( r r j ) = j A m j ( r r j ) exp [ i φ ̃ j ( r r j ) ] ,
R L ( α ) = L * ( r ) L ( r α ) d r = j A k A m j ( r r j ) exp [ i φ ̃ j ( r r j ) ] m k ( r r k α ) exp [ i φ ̃ k ( r r k α ) ] d r ,
R L ( α ) = j A k A exp [ i ( φ ̃ k φ ̃ j ) ] m j ( r r j ) m k ( r r k α ) d r .
R L ( α ) = j A k A R j k ( α s j k ) ,
R j k ( α ) = exp [ i ( φ ̃ k φ ̃ j ) ] m j ( r ) m k ( r α ) d r = M j k ( α ) exp [ i ( φ ̃ k φ ̃ j ) ] ,
φ j = φ ̃ j φ ̂ j .
φ k φ j = ψ j k .
φ ̂ k + φ ̂ j = ψ j k φ ̃ k + φ ̃ j .
[ 1 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 ] [ φ ̂ 1 φ ̂ 2 φ ̂ N ]
= [ ψ 1 , 2 φ ̃ 2 + φ ̃ 1 ψ 1 , 3 φ ̃ 3 + φ ̃ 1 ψ N 1 , N φ ̃ N + φ ̃ N 1 0 ] .
φ ̂ k φ ̂ j φ ̂ m + φ ̂ l = φ ̃ k φ ̃ j φ ̃ m + φ ̃ l .
R L ( α ) = M j k ( α s j k ) exp [ i ( φ k φ j ) ] = M j k ( α s j k ) exp [ i ( φ k φ j ) ] + M l m ( α s l m ) exp [ i ( φ m φ l ) ] + = M j k ( α s j k ) { exp [ i ( φ k φ j ) ] + exp [ i ( φ m φ l ) ] + } ,
R L ( α ) = 2 M j k ( α s j k ) cos ( φ k φ j 2 φ m φ l 2 ) .
R L ( α ) = M j k ( α s j k ) ,
1 2 π I ( ξ ) 2 d ξ = R L ( α ) 2 U ( α ) 2 d α ,
[ R L ( α ) U ( α ) ] = [ R j k ( α s j k ) U ( α ) ] = φ ̃ k φ ̃ j + θ j k ,
φ ̃ k φ ̃ j + θ j k = μ j k .
φ ̃ k φ ̃ j = μ j k θ j k
φ ̃ k φ ̃ j φ ̃ m + φ ̃ l = μ j k μ l m ,
φ ̃ k φ ̃ j φ ̃ m + φ ̃ l = μ j k μ l m ,
φ ̃ k φ ̃ j + θ j k = μ j k .
x j = y j m = j + 1 n a j m x m a j j ,
μ 12 = θ 1 2 + φ 1 φ 2 ± 2 n 12 π ,
μ 13 = θ 1 3 + φ 1 φ 3 ± 2 n 13 π ,
μ 23 = θ 2 3 + φ 2 φ 3 ± 2 n 23 π ,
μ 34 = θ 3 4 + φ 3 φ 4 ± 2 n 34 π ,
M 2 ± n π = φ 2 ,

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