Abstract

The standard two-point resolution criteria for a single-aperture optical system are not appropriate for multiaperture systems. In this paper we propose a new two-point resolution criterion based on the idea of thresholding the irradiances of the resulting far-field diffraction patterns of multiaperture optical systems. Theoretical data of irradiance versus point separation for various multiaperture optical systems are presented. The two-point resolution for these configurations was analyzed. A new two-point resolution criterion in which thresholds are used is demonstrated for coherent and incoherent illumination. Three different multiaperture incoherent systems are compared by using this threshold criterion.

© 1988 Optical Society of America

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References

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  1. J. S. Fender, “Synthetic apertures: an overview,” in Synthetic Aperture Systems, J. S. Fender, ed., Proc. Soc. Photo-Opt. Instrum. Eng.440, 2–7 (1974).
    [CrossRef]
  2. G. M. Sanger, T. E. Hoffman, M. A. Reed, “Some design aspects of a multiple-mirror telescope,” in Instrumentation in Astronomy I, L. Larmore, R. W. Pondexter, eds., Proc. Soc. Photo-Opt. Instrum. Eng.28, 161–170 (1972).
    [CrossRef]
  3. R. V. Shack, J. D. Ramcourt, H. Morrow, “Effects of dilution on a six-element synthetic aperture,” Appl. Opt. 10, 257–259 (1971).
    [CrossRef] [PubMed]
  4. A. B. Meinel, M. P. Meinel, N. J. Woolf, “Multiple aperture telescope diffraction images,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9, pp. 149–201.
    [CrossRef]
  5. V. N. Sintsov, A. F. Zapryagaev, “Aperture synthesis in optics,” Sov. Phys. Usp. 17, 931–941 (1975).
    [CrossRef]
  6. Lord Rayleigh, Collected Papers (Cambridge U. Press, Cambridge, 1902), p. 384.
  7. G. Sparrow, “On spectroscopic resolving power,” Astrophys. J. 44, 76–86 (1961).
    [CrossRef]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  9. J. P. Mills, B. J. Thompson, “Effect of aberrations and apodization on the performance of coherent optical systems. II. Imaging,” J. Opt. Soc. Am. A 3, 704–716 (1986).
    [CrossRef]
  10. B. J. Thompson, G. B. Parrent, Contemporary Optics— Physical Optics Notebook (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1971).

1986 (1)

1975 (1)

V. N. Sintsov, A. F. Zapryagaev, “Aperture synthesis in optics,” Sov. Phys. Usp. 17, 931–941 (1975).
[CrossRef]

1971 (1)

1961 (1)

G. Sparrow, “On spectroscopic resolving power,” Astrophys. J. 44, 76–86 (1961).
[CrossRef]

Fender, J. S.

J. S. Fender, “Synthetic apertures: an overview,” in Synthetic Aperture Systems, J. S. Fender, ed., Proc. Soc. Photo-Opt. Instrum. Eng.440, 2–7 (1974).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hoffman, T. E.

G. M. Sanger, T. E. Hoffman, M. A. Reed, “Some design aspects of a multiple-mirror telescope,” in Instrumentation in Astronomy I, L. Larmore, R. W. Pondexter, eds., Proc. Soc. Photo-Opt. Instrum. Eng.28, 161–170 (1972).
[CrossRef]

Meinel, A. B.

A. B. Meinel, M. P. Meinel, N. J. Woolf, “Multiple aperture telescope diffraction images,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9, pp. 149–201.
[CrossRef]

Meinel, M. P.

A. B. Meinel, M. P. Meinel, N. J. Woolf, “Multiple aperture telescope diffraction images,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9, pp. 149–201.
[CrossRef]

Mills, J. P.

Morrow, H.

Parrent, G. B.

B. J. Thompson, G. B. Parrent, Contemporary Optics— Physical Optics Notebook (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1971).

Ramcourt, J. D.

Rayleigh, Lord

Lord Rayleigh, Collected Papers (Cambridge U. Press, Cambridge, 1902), p. 384.

Reed, M. A.

G. M. Sanger, T. E. Hoffman, M. A. Reed, “Some design aspects of a multiple-mirror telescope,” in Instrumentation in Astronomy I, L. Larmore, R. W. Pondexter, eds., Proc. Soc. Photo-Opt. Instrum. Eng.28, 161–170 (1972).
[CrossRef]

Sanger, G. M.

G. M. Sanger, T. E. Hoffman, M. A. Reed, “Some design aspects of a multiple-mirror telescope,” in Instrumentation in Astronomy I, L. Larmore, R. W. Pondexter, eds., Proc. Soc. Photo-Opt. Instrum. Eng.28, 161–170 (1972).
[CrossRef]

Shack, R. V.

Sintsov, V. N.

V. N. Sintsov, A. F. Zapryagaev, “Aperture synthesis in optics,” Sov. Phys. Usp. 17, 931–941 (1975).
[CrossRef]

Sparrow, G.

G. Sparrow, “On spectroscopic resolving power,” Astrophys. J. 44, 76–86 (1961).
[CrossRef]

Thompson, B. J.

J. P. Mills, B. J. Thompson, “Effect of aberrations and apodization on the performance of coherent optical systems. II. Imaging,” J. Opt. Soc. Am. A 3, 704–716 (1986).
[CrossRef]

B. J. Thompson, G. B. Parrent, Contemporary Optics— Physical Optics Notebook (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1971).

Woolf, N. J.

A. B. Meinel, M. P. Meinel, N. J. Woolf, “Multiple aperture telescope diffraction images,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9, pp. 149–201.
[CrossRef]

Zapryagaev, A. F.

V. N. Sintsov, A. F. Zapryagaev, “Aperture synthesis in optics,” Sov. Phys. Usp. 17, 931–941 (1975).
[CrossRef]

Appl. Opt. (1)

Astrophys. J. (1)

G. Sparrow, “On spectroscopic resolving power,” Astrophys. J. 44, 76–86 (1961).
[CrossRef]

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

Sov. Phys. Usp. (1)

V. N. Sintsov, A. F. Zapryagaev, “Aperture synthesis in optics,” Sov. Phys. Usp. 17, 931–941 (1975).
[CrossRef]

Other (6)

Lord Rayleigh, Collected Papers (Cambridge U. Press, Cambridge, 1902), p. 384.

A. B. Meinel, M. P. Meinel, N. J. Woolf, “Multiple aperture telescope diffraction images,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9, pp. 149–201.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. S. Fender, “Synthetic apertures: an overview,” in Synthetic Aperture Systems, J. S. Fender, ed., Proc. Soc. Photo-Opt. Instrum. Eng.440, 2–7 (1974).
[CrossRef]

G. M. Sanger, T. E. Hoffman, M. A. Reed, “Some design aspects of a multiple-mirror telescope,” in Instrumentation in Astronomy I, L. Larmore, R. W. Pondexter, eds., Proc. Soc. Photo-Opt. Instrum. Eng.28, 161–170 (1972).
[CrossRef]

B. J. Thompson, G. B. Parrent, Contemporary Optics— Physical Optics Notebook (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1971).

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Figures (11)

Fig. 1
Fig. 1

Diagrams of (a) a single large optic aperture and (b) three-aperture, (c) four-aperture, and (d) six-aperture systems.

Fig. 2
Fig. 2

Impulse response of (a) a single large optic aperture and (b) a six-aperture system of equivalent diameter. The aperture configuration of each system is depicted in the upper right-hand corner of each plot.

Fig. 3
Fig. 3

Example of a multiaperture system with an aperture–origin separation of pn and subaperture radius of a.

Fig. 4
Fig. 4

Far-field irradiance patterns for a six-aperture system with aperture-origin separations of (a) 2.00a, (b) 3.00a, and (c) 4.00a.

Fig. 5
Fig. 5

Irradiance patterns for incoherent two-point illumination of (a) a single large aperture and of six-aperture systems with aperture–origin separations of (b) 2.00a and (c) 4.00a. Each irradiance pattern in each series corresponds to point separations of 0.00, 0.16 (single-aperture Sparrow limit), 0.19 (single-aperture Rayleigh limit), and 0.50, from left to right.

Fig. 6
Fig. 6

Sample irradiance-versus-point-separation plot for a six-aperture system with an aperture–origin separation of 2.00a, illuminated incoherently. Symbols: ——, central value;— — —, side-lobe maxima; • • • • •, coordinates of point A described in the text.

Fig. 7
Fig. 7

Diffraction patterns of two coherent point sources limited by threshold values of (a) 0.1, (b) 0.3, (c) 0.5, and (d) 0.9.

Fig. 8
Fig. 8

Diffraction patterns of a six-aperture system with an aperture–origin separation of 3.00a, illuminated by two coherent point sources and limited by a threshold value of 0.5 at point separations of (a) 1.50, (b) 0.70, (c) 0.65, and (d) 0.30.

Fig. 9
Fig. 9

Illustration of the threshold criterion for a coherently illuminated six-aperture system with an aperture–origin separation of 3.00a and threshold values of (a) 0.1, (b) 0.3, (c) 0.7, and (d) 0.9. Symbols: ——, central value;— — —, sidelobe maxima; • • • • •, threshold value.

Fig. 10
Fig. 10

Sample threshold plot for a coherently illuminated six-aperture system (threshold values are shown next to corresponding curves) compared with the two-point resolution performance (using the Sparrow criterion) of a single large aperture (• • • •) of equivalent diameter.

Fig. 11
Fig. 11

Threshold plots for comparing the incoherent two-point resolution performances of the three-, four-, and six-aperture systems (curves are indicated by numerals) at threshold values of (a) 0.3, (b) 0.5, (c) 0.7, and (d) 0.9 and the performance (using the Sparrow criterion) of a single large aperture (• • • •) of equivalent diameter.

Equations (10)

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U = 2 π a x 1 / λ f ,
V = 2 π a y i / λ f .
P ( x , y ) = circ ( r / a ) * 1 N δ ( x p n cos θ n , y p n sin θ n ) ,
h ( U , V ) = F { P ( x , y ) } = F { circ ( r / a ) } F { 1 N δ ( x p n cos θ n , y p n sin θ n ) } = { 2 π α 2 J 1 [ ( U 2 + V 2 ) 1 / 2 ] ( U 2 + V 2 ) 1 / 2 } × { 1 N exp [ i ( U p n cos θ n + V p n sin θ n ) / a ] } ,
2 π a 2 J 1 [ ( U 2 + V 2 ) 1 / 2 ] ( U 2 + V 2 ) 1 / 2 ,
u 0 ( x 0 , y 0 ) = δ ( x 0 b , y 0 ) + δ ( x 0 + b , y 0 ) .
u i ( U , V ) = 2 π a M λ f [ δ ( U b M 2 π a / λ f , V ) + δ ( U + b M 2 π a / λ f , V ) ] * h ( U , V )
= 2 π a M λ f [ h ( U b M 2 π a / λ f , V ) + h ( U + b M 2 π a / λ f , V ) ] = u i + b + u i b ,
I i ( U , V ) = | u i b + u i + b | 2 = | u i ( U , V ) | 2 ,
I i ( U , V ) = | u i b | 2 + | u i + b | 2 = I i b + I i + b ,

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