Abstract

A digital piston phase correction method is presented for partially coherent synthesis with active or passive illumination. An anamorphic pupil relay causes a linear shift of a two subaperture array between the entrance and exit pupils. This shift separates the subapertures’ cross- and auto-correlations while retaining their common spatial frequency information. Digital analysis of these common frequencies finds the separation distance and piston phase error of the cross-correlations and enables lossless correction of phase error. Corrected images are diffraction limited. Partial coherence affects the contrast of each spatial frequency, causing optical path difference tolerances to relax as system bandwidth decreases.

© 2017 Optical Society of America

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References

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  1. N. J. Miller, M. P. Dierking, and B. O. Duncan, “Optical sparse aperture imaging,” Appl. Opt. 46, 5933–5943 (2007).
    [Crossref]
  2. D. Rabb, D. Jameson, A. Stokes, and J. Stafford, “Distributed aperture synthesis,” Opt. Express 18, 10334–10342 (2010).
    [Crossref]
  3. A. H. Greenaway, “Diffraction-limited pictures from single turbulence-degraded images in astronomy,” Opt. Commun. 42, 157–161 (1982).
    [Crossref]
  4. S. Lacour, E. Thiebaut, and G. Perrin, “High dynamic range imaging with a single-mode pupil remapping system: a self-calibration algorithm for redundant interferometric arrays,” Mon. Not. R. Astron. Soc. 374, 832–846 (2007).
    [Crossref]
  5. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).
  6. W. A. Traub, “Combining beams from separated telescopes,” Appl. Opt. 25, 528–532 (1986).
    [Crossref]
  7. J. Cheng, The Principles of Astronomical Telescope Design (Springer, 2009).
  8. J. W. Goodman, Statistical Optics (Wiley-Interscience, 1986).
  9. A. M. Tai, “Passive synthetic aperture imaging using an achromatic grating interferometer,” Appl. Opt. 25, 3174–3190 (1986).
    [Crossref]
  10. “Preliminary: sensors unlimited ga1280jsx,” Princeton University, UTC Aerospace Systems Sensors Unlimited, Inc., Doc. No: (2014).
  11. J. R. Fienup and A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,” J. Opt. Soc. Am. A 7, 450–458 (1990).
    [Crossref]
  12. S. Krug, “Digital phase correction of a partially coherent sparse aperture system,” Masters thesis (University of Dayton, 2015).

2010 (1)

2007 (2)

S. Lacour, E. Thiebaut, and G. Perrin, “High dynamic range imaging with a single-mode pupil remapping system: a self-calibration algorithm for redundant interferometric arrays,” Mon. Not. R. Astron. Soc. 374, 832–846 (2007).
[Crossref]

N. J. Miller, M. P. Dierking, and B. O. Duncan, “Optical sparse aperture imaging,” Appl. Opt. 46, 5933–5943 (2007).
[Crossref]

1990 (1)

1986 (2)

W. A. Traub, “Combining beams from separated telescopes,” Appl. Opt. 25, 528–532 (1986).
[Crossref]

A. M. Tai, “Passive synthetic aperture imaging using an achromatic grating interferometer,” Appl. Opt. 25, 3174–3190 (1986).
[Crossref]

1982 (1)

A. H. Greenaway, “Diffraction-limited pictures from single turbulence-degraded images in astronomy,” Opt. Commun. 42, 157–161 (1982).
[Crossref]

Cheng, J.

J. Cheng, The Principles of Astronomical Telescope Design (Springer, 2009).

Dierking, M. P.

Duncan, B. O.

Fienup, J. R.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley-Interscience, 1986).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

Greenaway, A. H.

A. H. Greenaway, “Diffraction-limited pictures from single turbulence-degraded images in astronomy,” Opt. Commun. 42, 157–161 (1982).
[Crossref]

Jameson, D.

Kowalczyk, A. M.

Krug, S.

S. Krug, “Digital phase correction of a partially coherent sparse aperture system,” Masters thesis (University of Dayton, 2015).

Lacour, S.

S. Lacour, E. Thiebaut, and G. Perrin, “High dynamic range imaging with a single-mode pupil remapping system: a self-calibration algorithm for redundant interferometric arrays,” Mon. Not. R. Astron. Soc. 374, 832–846 (2007).
[Crossref]

Miller, N. J.

Perrin, G.

S. Lacour, E. Thiebaut, and G. Perrin, “High dynamic range imaging with a single-mode pupil remapping system: a self-calibration algorithm for redundant interferometric arrays,” Mon. Not. R. Astron. Soc. 374, 832–846 (2007).
[Crossref]

Rabb, D.

Stafford, J.

Stokes, A.

Tai, A. M.

A. M. Tai, “Passive synthetic aperture imaging using an achromatic grating interferometer,” Appl. Opt. 25, 3174–3190 (1986).
[Crossref]

Thiebaut, E.

S. Lacour, E. Thiebaut, and G. Perrin, “High dynamic range imaging with a single-mode pupil remapping system: a self-calibration algorithm for redundant interferometric arrays,” Mon. Not. R. Astron. Soc. 374, 832–846 (2007).
[Crossref]

Traub, W. A.

Appl. Opt. (3)

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

Mon. Not. R. Astron. Soc. (1)

S. Lacour, E. Thiebaut, and G. Perrin, “High dynamic range imaging with a single-mode pupil remapping system: a self-calibration algorithm for redundant interferometric arrays,” Mon. Not. R. Astron. Soc. 374, 832–846 (2007).
[Crossref]

Opt. Commun. (1)

A. H. Greenaway, “Diffraction-limited pictures from single turbulence-degraded images in astronomy,” Opt. Commun. 42, 157–161 (1982).
[Crossref]

Opt. Express (1)

Other (5)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

“Preliminary: sensors unlimited ga1280jsx,” Princeton University, UTC Aerospace Systems Sensors Unlimited, Inc., Doc. No: (2014).

J. Cheng, The Principles of Astronomical Telescope Design (Springer, 2009).

J. W. Goodman, Statistical Optics (Wiley-Interscience, 1986).

S. Krug, “Digital phase correction of a partially coherent sparse aperture system,” Masters thesis (University of Dayton, 2015).

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Figures (14)

Fig. 1.
Fig. 1.

Assuming a point source target, (a) entrance and (b) separated exit pupils with their spatial frequency responses.

Fig. 2.
Fig. 2.

Diagram of the beam expander and the diffuser.

Fig. 3.
Fig. 3.

Diagram of the anamorphic pupil relay.

Fig. 4.
Fig. 4.

Image of the laboratory setup for anamorphic pupil relay. The target is located 2.6    m before the front set of lenses.

Fig. 5.
Fig. 5.

(a) Image and (b) FT of a star target.

Fig. 6.
Fig. 6.

(a) Part of the 3    lp / mm Ronchi ruling image and (b) the FT.

Fig. 7.
Fig. 7.

Separated auto- and cross-correlations of a 3    lp / mm Ronchi ruling.

Fig. 8.
Fig. 8.

Element multiplication of a shift-corrected cross-correlation and the complex conjugate of the auto-correlation (left). The amplitude and phase of the complex product (right).

Fig. 9.
Fig. 9.

Piston corrections and recombination of the frequency response of a star target.

Fig. 10.
Fig. 10.

Illustration of the physical OPD correction method.

Fig. 11.
Fig. 11.

Piston varying over wavelength.

Fig. 12.
Fig. 12.

High-frequency center regions of the shift-corrected images with (a) no piston correction and (b) piston correction.

Fig. 13.
Fig. 13.

Piston (a) blurred and (b) corrected.

Fig. 14.
Fig. 14.

Experimental (thin) and analytical (bold) FDC envelopes for bandwidths of 0 nm (solid), 27 nm (dashed), and 50 nm (dotted). Letters (a)–(f) correspond to frequencies of 1 6    lp / mm , respectively.

Tables (2)

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Table 1. Measurements for the Pupil Relay System with Focal Lengths Measured for a Beam at 1560 nm

Tables Icon

Table 2. Experimental Registration Valuesa

Equations (12)

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f sub = D sub λ z ,
f total = D array λ z .
U array ( ξ , η ) = U 1 ( ξ , η + a / 2 ) + U 2 ( ξ , η a / 2 ) exp ( j Δ φ ) ,
G ( ζ , η ) = U 1 ( ζ , η ) U 1 * ( ζ , η ) + U 2 ( ζ , η ) U 2 * ( ζ , η ) + U 1 ( ζ , η ) U 2 * ( ζ , η ) δ ( ζ , η + a ) exp ( j Δ φ ) + U 2 ( ζ , η ) U 1 * ( ζ , η ) δ ( ζ , η a ) exp ( j Δ φ ) .
Δ φ = ( f x f y cross 21 auto * ) ,
G ( ξ , η ) = U 1 ( ξ , η ) U 1 * ( ξ , η ) + U 2 ( ξ , η ) U 2 * ( ξ , η ) + U 1 ( ξ , η ) U 2 * ( ξ , η ) + U 1 * ( ξ , η ) U 2 ( ξ , η ) .
Δ φ 1 Δ φ 2 = 2 π OPD ( 1 λ 1 1 λ 2 ) ,
OPD λ ¯ 2 2 π Δ φ 1 Δ φ 2 Δ λ .
V fringe = sinc ( Δ v τ ) ,
V = sinc ( OPD l ) .
auto + ( cross 12 + cross 12 ) rect ( f y d i λ ¯ 2 a Δ λ ) V ,
FDC = I auto + V sinc ( ya Δ λ d i λ ¯ 2 ) ( I 12 + I 21 ) .

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