Abstract

When a large aperture is synthesized with an array of smaller subapertures for high-resolution imaging applications, it is important not only to arrange the subapertures to achieve minimal spatial frequency redundancy but also to choose the size of the subapertures (i.e., the dilution ratio) necessary to achieve the best possible image quality. Spurious or ghost images often occur even for nonredundant dilute subaperture arrays. We show that array configurations producing a uniform modulation transfer function will not exhibit these undesirable ghost images. A prescription that is unique and original (to the best of our knowledge) is then presented for constructing both one-dimensional and two-dimensional configurations of dilute subaperture arrays that results in a uniform spatial frequency response with an arbitrarily high spatial resolution for reciprocal path-imaging applications.

© 1995 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]

1994 (1)

J. E. Harvey, A. Kotha, “Sparse array configurations yielding uniform MTF's in reciprocal path imaging configurations,” Opt. Commun. 106, 178–182 (1994).
[CrossRef]

1992 (2)

1991 (2)

D. A. de Wolf, “Backscatter enhancement: random continuum and particles,” J. Opt. Soc. Am. A 8, 465–471 (1991).
[CrossRef]

C. J. Solomon, J. C. Dainty, R. G. Lane, “Double passage imaging through a random phase screen using a nonredun-dant aperture,” J. Mod. Opt. 10, 1993–2008 (1991).
[CrossRef]

1990 (4)

T. Mavroidis, J. C. Dainty, “Imaging after double passage through a random screen,” Opt. Lett. 15, 857–859 (1990).
[CrossRef] [PubMed]

G. Welch, R. L. Phillips, “Simulation of enhanced backscatter by a phase screen,” J. Opt. Soc. Am. A 7, 578–584 (1990).
[CrossRef]

M. Nieto-Vesperinas, “Enhanced backscattering,” Opt. Photon. News 1, 50–52 (1990).
[CrossRef]

R. Barakat, “Dilute aperture diffraction imagery and object reconstruction,” Opt. Eng. 29, 131–139 (1990).
[CrossRef]

1989 (2)

1988 (2)

E. Jakeman, “Enhanced backscattering through a deep random phase screen,” J. Opt. Soc. Am. A 5, 1638–1648 (1988).
[CrossRef]

J. E. Harvey, R. A. Rockwell, “Performance characteristics of phased array and thinned aperture optical telescopes,” Opt. Eng. 27, 762–768 (1988).

1986 (2)

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

G. W. Swenson, “Radio-astronomy precedent for optical interferometer imaging,” J. Opt. Soc. Am. 3, 1311–1319 (1986).
[CrossRef]

1985 (1)

J. E. Harvey, M. J. MacFarlane, J. L. Forgham, “Design and performance of ranging telescopes: monolithic versus synthetic aperture,” Opt. Eng. 24, 183–188 (1985).

1984 (2)

1980 (1)

1979 (1)

1978 (1)

1971 (3)

1970 (5)

1967 (1)

J. S. Preston, Retro-reflexion by diffusing surfaces,” Nature 213, 1007–1008 (1967).
[CrossRef]

1963 (1)

1961 (1)

E. Ingelstam, “Nomenclature for Fourier transforms of spread functions,” J. Opt. Soc. Am. 51, 1441 (1961).

1956 (1)

Barakat, R.

R. Barakat, “Dilute aperture diffraction imagery and object reconstruction,” Opt. Eng. 29, 131–139 (1990).
[CrossRef]

Beckers, J. M.

J. M. Beckers, “Field of view considerations for telescope arrays,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 255 (1986).
[CrossRef]

Bogaturov, A. N.

Boyce, B. M.

B. M. Boyce, “The role of imaging processing in synthetic aperture systems and aberrated axially symmetric systems,” in A Symposium on Sampled Images (Perkin-Elmer, Norwalk, Conn., 1971), pp. 313–327.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), Chap. 4, p. 51.

Bunner, A. N.

J. E. Harvey, A. B. Wissinger, A. N. Bunner, “A parametric study of various synthetic aperture telescope configurations for coherent imaging applications,” in Infrared, Adaptive, and Synthetic Aperture Optical Systems, J. S. Fender, R. B. Johnson, W. L. Wolfe, eds., Proc. Soc. Photo-Opt. Instrum. Eng.643, 194–207 (1985).

Dainty, J. C.

C. J. Solomon, J. C. Dainty, “Use of polarisation in double passage imaging through a random screen,” Optics. Commun. 87, 207–211 (1992).
[CrossRef]

A. N. Bogaturov, A. S. Gurvich, V. A. Myakinin, J. C. Dainty, C. J. Solomon, N. J. Wooder, “Use of polarization in interferometry after double passage through turbulence,” Opt. Lett. 17, 757–759 (1992).
[CrossRef] [PubMed]

C. J. Solomon, J. C. Dainty, R. G. Lane, “Double passage imaging through a random phase screen using a nonredun-dant aperture,” J. Mod. Opt. 10, 1993–2008 (1991).
[CrossRef]

T. Mavroidis, J. C. Dainty, “Imaging after double passage through a random screen,” Opt. Lett. 15, 857–859 (1990).
[CrossRef] [PubMed]

Davis, W. F.

W. A. Traub, W. F. Davis, “Coherent optical system of modular imaging collectors (COSMIC) telescope array: astronomical goals and preliminary image reconstruction results,” in Advanced Technology Optical Telescopes, L. D. Barr, G. Burbidge, eds., Proc. Soc. Photo-Opt. Instrum. Eng.332164–175 (1982).
[CrossRef]

de Wolf, D. A.

Dummer, R. S.

Forgham, J. L.

J. E. Harvey, M. J. MacFarlane, J. L. Forgham, “Design and performance of ranging telescopes: monolithic versus synthetic aperture,” Opt. Eng. 24, 183–188 (1985).

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 7, p. 208.

Golay, M. J. E.

Goodman, J. W.

F. D. Russell, J. W. Goodman, “Nonredundant arrays and postdetection processing for aberration compensation in incoherent imaging,” J. Opt. Soc. Am. 61, 182–191 (1971).
[CrossRef]

J. W. Goodman, “Synthetic aperture optics,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1970), Vol. 8, pp. 3–48.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, pp. 13–21.

Gori, F.

F. Gori, G. Guattari, “Imaging systems using linear arrays of nonequally spaced elements,” Phys. Lett. A 32, 38–39 (1970).
[CrossRef]

F. Gori, G. Guattari, “Optical analog of a nonredundant array,” Phys. Lett. A 32, 446–447 (1970).
[CrossRef]

Gu, Z. H.

Guattari, G.

F. Gori, G. Guattari, “Optical analog of a nonredundant array,” Phys. Lett. A 32, 446–447 (1970).
[CrossRef]

F. Gori, G. Guattari, “Imaging systems using linear arrays of nonequally spaced elements,” Phys. Lett. A 32, 38–39 (1970).
[CrossRef]

Gurvich, A. S.

Gush, H. P.

Harvey, J. E.

J. E. Harvey, A. Kotha, “Sparse array configurations yielding uniform MTF's in reciprocal path imaging configurations,” Opt. Commun. 106, 178–182 (1994).
[CrossRef]

J. E. Harvey, R. A. Rockwell, “Performance characteristics of phased array and thinned aperture optical telescopes,” Opt. Eng. 27, 762–768 (1988).

J. E. Harvey, M. J. MacFarlane, J. L. Forgham, “Design and performance of ranging telescopes: monolithic versus synthetic aperture,” Opt. Eng. 24, 183–188 (1985).

J. E. Harvey, A. B. Wissinger, A. N. Bunner, “A parametric study of various synthetic aperture telescope configurations for coherent imaging applications,” in Infrared, Adaptive, and Synthetic Aperture Optical Systems, J. S. Fender, R. B. Johnson, W. L. Wolfe, eds., Proc. Soc. Photo-Opt. Instrum. Eng.643, 194–207 (1985).

Hoffman, T. E.

G. M. Sanger, T. E. Hoffman, M. A. Reed, “Some design aspects of a multiple mirror telescope,” in Instrumentation in Astronomy, L. Larmore, R. W. Poindexter, eds.,Proc. Soc. Photo-Opt. Instrum. Eng.28, 161–171 (1972).
[CrossRef]

Ingelstam, E.

E. Ingelstam, “Nomenclature for Fourier transforms of spread functions,” J. Opt. Soc. Am. 51, 1441 (1961).

Ishimaru, A.

Jakeman, E.

Kohl, R. H.

Kotha, A.

J. E. Harvey, A. Kotha, “Sparse array configurations yielding uniform MTF's in reciprocal path imaging configurations,” Opt. Commun. 106, 178–182 (1994).
[CrossRef]

Krim, M. H.

M. H. Krim, “Applications of replicated glass mirrors to large segmented optical systems,” in Large Optics Technology, G. M. Sanger, ed., Proc. Soc. Photo-Opt. Instrum. Eng.571, 60–75 (1986).

Kuga, Y.

Lane, R. G.

C. J. Solomon, J. C. Dainty, R. G. Lane, “Double passage imaging through a random phase screen using a nonredun-dant aperture,” J. Mod. Opt. 10, 1993–2008 (1991).
[CrossRef]

Love, T. J.

Low, F. J.

A. B. Meinel, R. R. Shannon, F. L. Whipple, F. J. Low, “A large multiple-mirror telescope project,” in Instrumentation in Astronomy, L. Larmore, R. W. Poindexter, eds., Proc. Soc. Photo-Opt. Instrum. Eng.28, 155–160 (1972).
[CrossRef]

MacFarlane, M. J.

J. E. Harvey, M. J. MacFarlane, J. L. Forgham, “Design and performance of ranging telescopes: monolithic versus synthetic aperture,” Opt. Eng. 24, 183–188 (1985).

Maradudin, A. A.

Mavroidis, T.

McGurn, A. R.

Meinel, A. B.

A. B. Meinel, “Aperture synthesis using independent telescopes,” Appl. Opt. 9, 2501–2504 (1970).
[CrossRef] [PubMed]

A. B. Meinel, R. R. Shannon, F. L. Whipple, F. J. Low, “A large multiple-mirror telescope project,” in Instrumentation in Astronomy, L. Larmore, R. W. Poindexter, eds., Proc. Soc. Photo-Opt. Instrum. Eng.28, 155–160 (1972).
[CrossRef]

Montgomery, W. W.

Myakinin, V. A.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, “Enhanced backscattering,” Opt. Photon. News 1, 50–52 (1990).
[CrossRef]

Phillips, R. L.

Preston, J. S.

J. S. Preston, Retro-reflexion by diffusing surfaces,” Nature 213, 1007–1008 (1967).
[CrossRef]

Reed, M. A.

G. M. Sanger, T. E. Hoffman, M. A. Reed, “Some design aspects of a multiple mirror telescope,” in Instrumentation in Astronomy, L. Larmore, R. W. Poindexter, eds.,Proc. Soc. Photo-Opt. Instrum. Eng.28, 161–171 (1972).
[CrossRef]

Rockwell, R. A.

J. E. Harvey, R. A. Rockwell, “Performance characteristics of phased array and thinned aperture optical telescopes,” Opt. Eng. 27, 762–768 (1988).

Russell, F. D.

Sanger, G. M.

G. M. Sanger, T. E. Hoffman, M. A. Reed, “Some design aspects of a multiple mirror telescope,” in Instrumentation in Astronomy, L. Larmore, R. W. Poindexter, eds.,Proc. Soc. Photo-Opt. Instrum. Eng.28, 161–171 (1972).
[CrossRef]

Shack, R. V.

Shannon, R. R.

A. B. Meinel, R. R. Shannon, F. L. Whipple, F. J. Low, “A large multiple-mirror telescope project,” in Instrumentation in Astronomy, L. Larmore, R. W. Poindexter, eds., Proc. Soc. Photo-Opt. Instrum. Eng.28, 155–160 (1972).
[CrossRef]

Smith, F. D.

Solomon, C. J.

C. J. Solomon, J. C. Dainty, “Use of polarisation in double passage imaging through a random screen,” Optics. Commun. 87, 207–211 (1992).
[CrossRef]

A. N. Bogaturov, A. S. Gurvich, V. A. Myakinin, J. C. Dainty, C. J. Solomon, N. J. Wooder, “Use of polarization in interferometry after double passage through turbulence,” Opt. Lett. 17, 757–759 (1992).
[CrossRef] [PubMed]

C. J. Solomon, J. C. Dainty, R. G. Lane, “Double passage imaging through a random phase screen using a nonredun-dant aperture,” J. Mod. Opt. 10, 1993–2008 (1991).
[CrossRef]

C. J. Solomon, Blackett Laboratory, Imperial College, London SW7 2BZ, UK (personal communication), 1994.

Som, S. C.

Stockham, L. W.

Swenson, G. W.

G. W. Swenson, “Radio-astronomy precedent for optical interferometer imaging,” J. Opt. Soc. Am. 3, 1311–1319 (1986).
[CrossRef]

Swindell, W.

Tapster, P. R.

Traub, W. A.

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

W. A. Traub, W. F. Davis, “Coherent optical system of modular imaging collectors (COSMIC) telescope array: astronomical goals and preliminary image reconstruction results,” in Advanced Technology Optical Telescopes, L. D. Barr, G. Burbidge, eds., Proc. Soc. Photo-Opt. Instrum. Eng.332164–175 (1982).
[CrossRef]

Trowbridge, T. S.

Tsang, L.

Weeks, A. R.

Welch, G.

Whipple, F. L.

A. B. Meinel, R. R. Shannon, F. L. Whipple, F. J. Low, “A large multiple-mirror telescope project,” in Instrumentation in Astronomy, L. Larmore, R. W. Poindexter, eds., Proc. Soc. Photo-Opt. Instrum. Eng.28, 155–160 (1972).
[CrossRef]

Wissinger, A. B.

J. E. Harvey, A. B. Wissinger, A. N. Bunner, “A parametric study of various synthetic aperture telescope configurations for coherent imaging applications,” in Infrared, Adaptive, and Synthetic Aperture Optical Systems, J. S. Fender, R. B. Johnson, W. L. Wolfe, eds., Proc. Soc. Photo-Opt. Instrum. Eng.643, 194–207 (1985).

Wooder, N. J.

Appl. Opt. (5)

J. Mod. Opt. (1)

C. J. Solomon, J. C. Dainty, R. G. Lane, “Double passage imaging through a random phase screen using a nonredun-dant aperture,” J. Mod. Opt. 10, 1993–2008 (1991).
[CrossRef]

J. Opt. Soc. Am. (9)

J. Opt. Soc. Am. A (6)

Nature (1)

J. S. Preston, Retro-reflexion by diffusing surfaces,” Nature 213, 1007–1008 (1967).
[CrossRef]

Opt. Commun. (1)

J. E. Harvey, A. Kotha, “Sparse array configurations yielding uniform MTF's in reciprocal path imaging configurations,” Opt. Commun. 106, 178–182 (1994).
[CrossRef]

Opt. Eng. (3)

J. E. Harvey, R. A. Rockwell, “Performance characteristics of phased array and thinned aperture optical telescopes,” Opt. Eng. 27, 762–768 (1988).

R. Barakat, “Dilute aperture diffraction imagery and object reconstruction,” Opt. Eng. 29, 131–139 (1990).
[CrossRef]

J. E. Harvey, M. J. MacFarlane, J. L. Forgham, “Design and performance of ranging telescopes: monolithic versus synthetic aperture,” Opt. Eng. 24, 183–188 (1985).

Opt. Lett. (3)

Opt. Photon. News (1)

M. Nieto-Vesperinas, “Enhanced backscattering,” Opt. Photon. News 1, 50–52 (1990).
[CrossRef]

Optics. Commun. (1)

C. J. Solomon, J. C. Dainty, “Use of polarisation in double passage imaging through a random screen,” Optics. Commun. 87, 207–211 (1992).
[CrossRef]

Phys. Lett. A (2)

F. Gori, G. Guattari, “Imaging systems using linear arrays of nonequally spaced elements,” Phys. Lett. A 32, 38–39 (1970).
[CrossRef]

F. Gori, G. Guattari, “Optical analog of a nonredundant array,” Phys. Lett. A 32, 446–447 (1970).
[CrossRef]

Other (17)

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), Chap. 4, p. 51.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, pp. 13–21.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 7, p. 208.

Synthetic Aperture Optics (National Academy of Sciences–National Research Council, Washington, D.C., 1967).

M. H. Krim, “Applications of replicated glass mirrors to large segmented optical systems,” in Large Optics Technology, G. M. Sanger, ed., Proc. Soc. Photo-Opt. Instrum. Eng.571, 60–75 (1986).

W. A. Traub, W. F. Davis, “Coherent optical system of modular imaging collectors (COSMIC) telescope array: astronomical goals and preliminary image reconstruction results,” in Advanced Technology Optical Telescopes, L. D. Barr, G. Burbidge, eds., Proc. Soc. Photo-Opt. Instrum. Eng.332164–175 (1982).
[CrossRef]

J. E. Harvey, A. B. Wissinger, A. N. Bunner, “A parametric study of various synthetic aperture telescope configurations for coherent imaging applications,” in Infrared, Adaptive, and Synthetic Aperture Optical Systems, J. S. Fender, R. B. Johnson, W. L. Wolfe, eds., Proc. Soc. Photo-Opt. Instrum. Eng.643, 194–207 (1985).

J. S. Fender, ed., Synthetic Aperture Systems, Proc. Soc. Photo-Opt. Instrum. Eng.440, 1–172 (1984).

R. B. Johnson, W. L. Wolfe, J. S. Fender, eds., Infrared, Adaptive, and Synthetic Aperture Optical Systems, Proc. Soc. Photo-Opt. Instrum. Eng.643, 121–243 (1986).

J. M. Beckers, “Field of view considerations for telescope arrays,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 255 (1986).
[CrossRef]

J. S. Fender, ed., Multiple-Aperture Optical Systems, special issue of Opt. Eng.27, 705–800 (1988).

M. W. Stockton, ed., “Symposium on synthetic aperture optics,” Tech. Rep. 58 (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1970).

J. W. Goodman, “Synthetic aperture optics,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1970), Vol. 8, pp. 3–48.
[CrossRef]

B. M. Boyce, “The role of imaging processing in synthetic aperture systems and aberrated axially symmetric systems,” in A Symposium on Sampled Images (Perkin-Elmer, Norwalk, Conn., 1971), pp. 313–327.

A. B. Meinel, R. R. Shannon, F. L. Whipple, F. J. Low, “A large multiple-mirror telescope project,” in Instrumentation in Astronomy, L. Larmore, R. W. Poindexter, eds., Proc. Soc. Photo-Opt. Instrum. Eng.28, 155–160 (1972).
[CrossRef]

G. M. Sanger, T. E. Hoffman, M. A. Reed, “Some design aspects of a multiple mirror telescope,” in Instrumentation in Astronomy, L. Larmore, R. W. Poindexter, eds.,Proc. Soc. Photo-Opt. Instrum. Eng.28, 161–171 (1972).
[CrossRef]

C. J. Solomon, Blackett Laboratory, Imperial College, London SW7 2BZ, UK (personal communication), 1994.

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Figures (11)

Fig. 1
Fig. 1

Relationship between the complex pupil function, OTF, and the PSF.

Fig. 2
Fig. 2

One-dimensional, four-element, nonredundant array of subapertures spanning a total distance of L, with dilution ratios of a, 1.00; b, 0.50; c, 0.25; d, 0.125.

Fig. 3
Fig. 3

Graphic illustration of the pupil function and the corresponding diffraction-limited PSF for dilution ratios of 1.00, 0.50, 0.25, and 0.125.

Fig. 4
Fig. 4

Graphic illustration of the pupil function and the corresponding diffraction-limited MTF for dilution ratios of 1.00, 0.50, 0.25, and 0.125.

Fig. 5
Fig. 5

Illustration of the object, the PSF, and the image produced by apertures of width a, max; b, 2 max; c, 4 max; d, 7 max = L.

Fig. 6
Fig. 6

Graphic illustration of the pupil function and the corresponding diffraction images of a two-bar target for dilution ratios of 1.00, 0.50, 0.25, and 0.125.

Fig. 7
Fig. 7

Conjugate-wave formation by reciprocal path scattering through a random phase screen or turbulent medium.

Fig. 8
Fig. 8

Schematic of RPI through a random screen.

Fig. 9
Fig. 9

One-dimensional nonredundant polarizing array of square subapertures and the corresponding MTF for RPI applications.

Fig. 10
Fig. 10

a, Nine-element one-dimensional, nonredundant, polarizing imaging array of square subapertures; b, the corresponding uniform MTF for RPI applications.

Fig. 11
Fig. 11

a, Two-dimensional polarizing array of square subapertures; (b) its associated MTF exhibiting a uniform response over an extended range in spatial frequency space.

Equations (28)

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OTF ( ξ , η ) = P ( x , y ) P ( x x , y y ) d x d y | P ( x , y ) | 2 d x d y , x = λ f ξ , y = λ f η ,
PSF ( x , y ) = | A ( ξ , η ) | ξ = x / λ f , η = y / λ f 2 , Where A ( ξ , η ) = { P ( x , y ) } .
dilution ratio β = / max .
fill factor = area of subapertures area of full aperture spanned by subapertures .
= β max = β L / 7 ,
S = ( L ) / 6 .
P ( x , y ) = rect ( x ) * [ δ ( x + 3 S ) + δ ( x + 2 S ) + δ ( x S ) + δ ( x 3 S ) ] ,
P ( x , y ) = rect ( x ) * [ δ ( x + 3 S ) + δ ( x 3 S ) ] + rect ( x + S / 2 ) * [ δ ( x + 3 S / 2 ) + δ ( x 3 S / 2 ) ] .
A ( ξ ) = ( 1 i λ f ) { sinc ( ξ ) 2 cos ( 2 π 3 S ξ ) + sinc ( ξ ) exp [ i 2 π ( S / 2 ) ξ ] 2 cos [ 2 π ( 3 / 2 ) S ξ ] } ,
A ( ξ ) = ( 2 i λ f ) sinc ( ξ ) { cos ( 2 π 3 S ξ ) + exp [ i 2 π ( S / 2 ) ξ ] cos [ 2 π ( 3 / 2 ) S ξ ] } .
PSF ( x ) = [ 4 2 λ 2 f 2 sinc 2 ( x / λ f ) ] { cos 2 ( 2 π 3 Sx / λ f ) + cos 2 [ 2 π ( 3 / 2 ) Sx / λ f ] + 2 cos ( 2 π 3 Sx / λ f ) cos [ 2 π ( 3 / 2 ) S x / λ f ] × cos [ 2 π ( S / 2 ) x / λ f ] } .
MTF ( ξ ) = tri ( ξ / λ f ) * { δ ( ξ ) + 1 4 n = 1 6 [ δ ( ξ n S λ f ) + δ ( ξ + n S λ f ) ] } .
Object ( x ) = rect ( x 0.5 0.5 ) + rect ( x + 0.5 0.5 ) .
( n 2 ) 2 for even n ,
n + 1 2 n 1 2 for odd n ,
n ( n 1 ) 2 for any n .
( n 2 ) 2 λ f for even n ,
( n + 1 2 n 1 2 ) λ f for odd n .
ξ c = ( L ) / λ f ,
n = 4 ( L ) / .
rectangle function rect ( x ) = { 0 for | x | > 2 , 1 2 for | x | = 2 , 1 for | x | < 2 ; triangle function tri ( x ) = { 1 | x | for | x | < , 0 for | x | ; sinc function sinc ( x ) = sin ( π x / ) π x / ; Dirac delta function δ ( x a ) .
f ( x ) * g ( x ) = f ( α ) g ( x α ) d α ,
f ( x ) g ( x ) = f ( α ) g ( α x ) d α .
F ( ξ , η ) = f ( x , y ) exp [ i 2 π ( x ξ + y η ) ] d x d y , f ( x , y ) = F ( ξ , η ) exp [ i 2 π ( x ξ + y η ) ] d ξ d η .
f ( x , y ) F ( ξ , η ) , or F ( ξ , η ) = { f ( x , y ) } , rect ( x , y ) sinc ( ξ , η ) , tri ( x , y ) sinc 2 ( ξ , η ) , 2 cos ( 2 π ax ) δ ( ξ a ) + δ ( ξ + a ) , δ ( x ) 1 .
given f ( x ) F ( ξ ) , then f ( x / a ) | a | F ( a ξ ) ;
given f ( x ) F ( ξ ) , then f ( x x 0 ) exp ( i 2 π x 0 ξ ) F ( ξ ) ;
given f ( x ) F ( ξ ) , g ( x ) G ( ξ ) , then f ( x ) * g ( x ) = F ( ξ ) G ( ξ ) .

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