Abstract

The use of a Fresnel zone plate as a coded aperture for imaging incoherent radiation such as gamma rays has been previously reported. The coded image is in many respects similar to a hologram and can be decoded or reconstructed with a coherent optical system. In this paper, the general theory of coded-aperture imaging is presented, first for an arbitrary code and then for an on-axis zone plate, an off-axis zone plate, and a one-dimensional zone plate (or linear chirp). With the on-axis plate, a matched imaging condition is suggested as a guide to optimizing image contrast. With the off-axis zone plate and the linear chirp, it is necessary to use a half-tone screen to spatially heterodyne the object spectrum into the passband of the aperture. In all three cases, expressions for the resolution, depth of field, field of view, and relative efficiency are derived. A simplified noise analysis is presented, and some practical system constraints are discussed.

© 1973 Optical Society of America

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References

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  1. H. O. Anger, in Instrumentation in Nuclear Medicine, J. Hine, Ed. (Academic, New York, 1967), Vol. 1, p 485.
  2. M. A. Bender, M. Blau, Nucleonics 21, 52 (1963).
  3. J. R. Mallard, R. J. Wilks, in Medical Radioisotope Scintigraphy, (IAEA, Vienna, 1969), Vol. 1, p. 3.
  4. F. D. Thomas, W. H. Thomas, G. F. Knoll, Ref. 3, p. 43.
  5. N. O. Young, Sky Telesc. 25, 8 (1963). L. Mertz, N. O. Young in Proc. Internat. Conf. on Optical Instruments (Chapman and Hall, London, 1961), p. 305.
  6. H. H. Barrett, J. Nucl. Med. 13, 382 (1972).
    [PubMed]
  7. R. W. Wood, Physical Optics (Dover, New York, 1957), p. 37.
  8. G. L. Rogers, Nature 166, 237 (1950).
    [CrossRef] [PubMed]
  9. H. H. Barrett, D. T. Wilson, G. D. DeMeester, Opt. Commun. 5, 398 (1972).
    [CrossRef]
  10. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957, p. 360.
  11. A. R. Shulman, Optical Data Processing (Wiley, New York, 1970), p. 387.
  12. D. J. Stigliani, R. Mittra, R. G. Semonin, J. Opt. Soc. Am. 57, 610 (1967).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 441.
  14. J. R. Klauder, A. C. Price, S. Darlington, W. J. Aibersheim, Bell Syst. Tech. J. 39, 745 (1960).
  15. E. Leith, J. Upatniecks, J. Opt. Soc. Am. 52, 1123 (1962).
    [CrossRef]
  16. H. H. Barrett, Proc. IEEE 60, 723 (1972).
    [CrossRef]
  17. R. H. Tancreil, M. B. Schulz, H. H. Barrett, L. Davis, M. G. Holland, Proc. IEEE 57, 1211 (1969).
    [CrossRef]
  18. W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), p. 112.
  19. J. W. Goodman in Modern Optics, Jerome Fox, Ed., Microwave Research Inst. Sym. Series (Polytechnic Press, Brooklyn, 1967), Vol. 17, p. 573.
  20. M. Young, J. Opt. Soc. Am. 62, 972 (1972).
    [CrossRef]
  21. L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 88.
  22. H. Eininghammer Thesis, Tubingen U. (1968) NTIS Document N68 34135.
  23. W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nucl. Med. 13, 612 (1972).
    [PubMed]
  24. V. L. Hirschy, J. P. Aldridge, Rev. Sci. Instrum. 42, 381 (1971).
    [CrossRef]
  25. K. S. Pennington, P. M. Will, Opt. Commun. 2, 167 (1970).
    [CrossRef]
  26. W. E. Koek, Proc. IEEE 58, 1773 (1970).
    [CrossRef]
  27. R. H. Dicke, Astrophys. J. 153, L101 (1968).
    [CrossRef]
  28. G. W. Stroke, G. S. Hayat, R. B. Hoover, J. H. Underwood, Opt. Commun. 1, 138 (1969).
    [CrossRef]
  29. K. Biedermann, Optica Acta 17, 631 (1970).
    [CrossRef]
  30. K. S. Pennington, P. M. Will, Proc. IEEE 60, 669 (1972).
    [CrossRef]
  31. H. H. Barrett, K. Garewal, D. T. Wilson, Radiology 104, 429 (1972).
    [PubMed]
  32. M. J. E. Golay, J. Opt. Soc. Am. 39, 437 (1949) and J. Opt. Soc. Am. 41, 468 (1951); IRE Trans. Inf. Theory IT-7, 82 (1961).
    [CrossRef] [PubMed]
  33. C. E. Cook, M. Bernfeld, Radar Signals (Academic, New York, 1967), p. 245.

1972 (7)

H. H. Barrett, J. Nucl. Med. 13, 382 (1972).
[PubMed]

H. H. Barrett, D. T. Wilson, G. D. DeMeester, Opt. Commun. 5, 398 (1972).
[CrossRef]

H. H. Barrett, Proc. IEEE 60, 723 (1972).
[CrossRef]

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nucl. Med. 13, 612 (1972).
[PubMed]

K. S. Pennington, P. M. Will, Proc. IEEE 60, 669 (1972).
[CrossRef]

H. H. Barrett, K. Garewal, D. T. Wilson, Radiology 104, 429 (1972).
[PubMed]

M. Young, J. Opt. Soc. Am. 62, 972 (1972).
[CrossRef]

1971 (1)

V. L. Hirschy, J. P. Aldridge, Rev. Sci. Instrum. 42, 381 (1971).
[CrossRef]

1970 (3)

K. S. Pennington, P. M. Will, Opt. Commun. 2, 167 (1970).
[CrossRef]

W. E. Koek, Proc. IEEE 58, 1773 (1970).
[CrossRef]

K. Biedermann, Optica Acta 17, 631 (1970).
[CrossRef]

1969 (2)

G. W. Stroke, G. S. Hayat, R. B. Hoover, J. H. Underwood, Opt. Commun. 1, 138 (1969).
[CrossRef]

R. H. Tancreil, M. B. Schulz, H. H. Barrett, L. Davis, M. G. Holland, Proc. IEEE 57, 1211 (1969).
[CrossRef]

1968 (1)

R. H. Dicke, Astrophys. J. 153, L101 (1968).
[CrossRef]

1967 (1)

1963 (2)

M. A. Bender, M. Blau, Nucleonics 21, 52 (1963).

N. O. Young, Sky Telesc. 25, 8 (1963). L. Mertz, N. O. Young in Proc. Internat. Conf. on Optical Instruments (Chapman and Hall, London, 1961), p. 305.

1962 (1)

1960 (1)

J. R. Klauder, A. C. Price, S. Darlington, W. J. Aibersheim, Bell Syst. Tech. J. 39, 745 (1960).

1950 (1)

G. L. Rogers, Nature 166, 237 (1950).
[CrossRef] [PubMed]

1949 (1)

Aibersheim, W. J.

J. R. Klauder, A. C. Price, S. Darlington, W. J. Aibersheim, Bell Syst. Tech. J. 39, 745 (1960).

Aldridge, J. P.

V. L. Hirschy, J. P. Aldridge, Rev. Sci. Instrum. 42, 381 (1971).
[CrossRef]

Anger, H. O.

H. O. Anger, in Instrumentation in Nuclear Medicine, J. Hine, Ed. (Academic, New York, 1967), Vol. 1, p 485.

Barrett, H. H.

H. H. Barrett, D. T. Wilson, G. D. DeMeester, Opt. Commun. 5, 398 (1972).
[CrossRef]

H. H. Barrett, Proc. IEEE 60, 723 (1972).
[CrossRef]

H. H. Barrett, K. Garewal, D. T. Wilson, Radiology 104, 429 (1972).
[PubMed]

H. H. Barrett, J. Nucl. Med. 13, 382 (1972).
[PubMed]

R. H. Tancreil, M. B. Schulz, H. H. Barrett, L. Davis, M. G. Holland, Proc. IEEE 57, 1211 (1969).
[CrossRef]

Beierwaltes, W. H.

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nucl. Med. 13, 612 (1972).
[PubMed]

Bender, M. A.

M. A. Bender, M. Blau, Nucleonics 21, 52 (1963).

Bernfeld, M.

C. E. Cook, M. Bernfeld, Radar Signals (Academic, New York, 1967), p. 245.

Biedermann, K.

K. Biedermann, Optica Acta 17, 631 (1970).
[CrossRef]

Blau, M.

M. A. Bender, M. Blau, Nucleonics 21, 52 (1963).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 441.

Cook, C. E.

C. E. Cook, M. Bernfeld, Radar Signals (Academic, New York, 1967), p. 245.

Darlington, S.

J. R. Klauder, A. C. Price, S. Darlington, W. J. Aibersheim, Bell Syst. Tech. J. 39, 745 (1960).

Davenport, W. B.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), p. 112.

Davis, L.

R. H. Tancreil, M. B. Schulz, H. H. Barrett, L. Davis, M. G. Holland, Proc. IEEE 57, 1211 (1969).
[CrossRef]

DeMeester, G. D.

H. H. Barrett, D. T. Wilson, G. D. DeMeester, Opt. Commun. 5, 398 (1972).
[CrossRef]

Dicke, R. H.

R. H. Dicke, Astrophys. J. 153, L101 (1968).
[CrossRef]

Eininghammer, H.

H. Eininghammer Thesis, Tubingen U. (1968) NTIS Document N68 34135.

Garewal, K.

H. H. Barrett, K. Garewal, D. T. Wilson, Radiology 104, 429 (1972).
[PubMed]

Golay, M. J. E.

Goodman, J. W.

J. W. Goodman in Modern Optics, Jerome Fox, Ed., Microwave Research Inst. Sym. Series (Polytechnic Press, Brooklyn, 1967), Vol. 17, p. 573.

Han, K. S.

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nucl. Med. 13, 612 (1972).
[PubMed]

Hayat, G. S.

G. W. Stroke, G. S. Hayat, R. B. Hoover, J. H. Underwood, Opt. Commun. 1, 138 (1969).
[CrossRef]

Hirschy, V. L.

V. L. Hirschy, J. P. Aldridge, Rev. Sci. Instrum. 42, 381 (1971).
[CrossRef]

Holland, M. G.

R. H. Tancreil, M. B. Schulz, H. H. Barrett, L. Davis, M. G. Holland, Proc. IEEE 57, 1211 (1969).
[CrossRef]

Hoover, R. B.

G. W. Stroke, G. S. Hayat, R. B. Hoover, J. H. Underwood, Opt. Commun. 1, 138 (1969).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957, p. 360.

Jones, L. W.

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nucl. Med. 13, 612 (1972).
[PubMed]

Klauder, J. R.

J. R. Klauder, A. C. Price, S. Darlington, W. J. Aibersheim, Bell Syst. Tech. J. 39, 745 (1960).

Knoll, G. F.

F. D. Thomas, W. H. Thomas, G. F. Knoll, Ref. 3, p. 43.

Koek, W. E.

W. E. Koek, Proc. IEEE 58, 1773 (1970).
[CrossRef]

Leith, E.

Mallard, J. R.

J. R. Mallard, R. J. Wilks, in Medical Radioisotope Scintigraphy, (IAEA, Vienna, 1969), Vol. 1, p. 3.

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 88.

Mittra, R.

Pennington, K. S.

K. S. Pennington, P. M. Will, Proc. IEEE 60, 669 (1972).
[CrossRef]

K. S. Pennington, P. M. Will, Opt. Commun. 2, 167 (1970).
[CrossRef]

Price, A. C.

J. R. Klauder, A. C. Price, S. Darlington, W. J. Aibersheim, Bell Syst. Tech. J. 39, 745 (1960).

Rogers, G. L.

G. L. Rogers, Nature 166, 237 (1950).
[CrossRef] [PubMed]

Rogers, W. L.

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nucl. Med. 13, 612 (1972).
[PubMed]

Root, W. L.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), p. 112.

Schulz, M. B.

R. H. Tancreil, M. B. Schulz, H. H. Barrett, L. Davis, M. G. Holland, Proc. IEEE 57, 1211 (1969).
[CrossRef]

Semonin, R. G.

Shulman, A. R.

A. R. Shulman, Optical Data Processing (Wiley, New York, 1970), p. 387.

Stigliani, D. J.

Stroke, G. W.

G. W. Stroke, G. S. Hayat, R. B. Hoover, J. H. Underwood, Opt. Commun. 1, 138 (1969).
[CrossRef]

Tancreil, R. H.

R. H. Tancreil, M. B. Schulz, H. H. Barrett, L. Davis, M. G. Holland, Proc. IEEE 57, 1211 (1969).
[CrossRef]

Thomas, F. D.

F. D. Thomas, W. H. Thomas, G. F. Knoll, Ref. 3, p. 43.

Thomas, W. H.

F. D. Thomas, W. H. Thomas, G. F. Knoll, Ref. 3, p. 43.

Underwood, J. H.

G. W. Stroke, G. S. Hayat, R. B. Hoover, J. H. Underwood, Opt. Commun. 1, 138 (1969).
[CrossRef]

Upatniecks, J.

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957, p. 360.

Wilks, R. J.

J. R. Mallard, R. J. Wilks, in Medical Radioisotope Scintigraphy, (IAEA, Vienna, 1969), Vol. 1, p. 3.

Will, P. M.

K. S. Pennington, P. M. Will, Proc. IEEE 60, 669 (1972).
[CrossRef]

K. S. Pennington, P. M. Will, Opt. Commun. 2, 167 (1970).
[CrossRef]

Wilson, D. T.

H. H. Barrett, K. Garewal, D. T. Wilson, Radiology 104, 429 (1972).
[PubMed]

H. H. Barrett, D. T. Wilson, G. D. DeMeester, Opt. Commun. 5, 398 (1972).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 441.

Wood, R. W.

R. W. Wood, Physical Optics (Dover, New York, 1957), p. 37.

Young, M.

Young, N. O.

N. O. Young, Sky Telesc. 25, 8 (1963). L. Mertz, N. O. Young in Proc. Internat. Conf. on Optical Instruments (Chapman and Hall, London, 1961), p. 305.

Astrophys. J. (1)

R. H. Dicke, Astrophys. J. 153, L101 (1968).
[CrossRef]

Bell Syst. Tech. J. (1)

J. R. Klauder, A. C. Price, S. Darlington, W. J. Aibersheim, Bell Syst. Tech. J. 39, 745 (1960).

J. Nucl. Med. (2)

H. H. Barrett, J. Nucl. Med. 13, 382 (1972).
[PubMed]

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, J. Nucl. Med. 13, 612 (1972).
[PubMed]

J. Opt. Soc. Am. (4)

Nature (1)

G. L. Rogers, Nature 166, 237 (1950).
[CrossRef] [PubMed]

Nucleonics (1)

M. A. Bender, M. Blau, Nucleonics 21, 52 (1963).

Opt. Commun. (3)

H. H. Barrett, D. T. Wilson, G. D. DeMeester, Opt. Commun. 5, 398 (1972).
[CrossRef]

G. W. Stroke, G. S. Hayat, R. B. Hoover, J. H. Underwood, Opt. Commun. 1, 138 (1969).
[CrossRef]

K. S. Pennington, P. M. Will, Opt. Commun. 2, 167 (1970).
[CrossRef]

Optica Acta (1)

K. Biedermann, Optica Acta 17, 631 (1970).
[CrossRef]

Proc. IEEE (4)

K. S. Pennington, P. M. Will, Proc. IEEE 60, 669 (1972).
[CrossRef]

W. E. Koek, Proc. IEEE 58, 1773 (1970).
[CrossRef]

H. H. Barrett, Proc. IEEE 60, 723 (1972).
[CrossRef]

R. H. Tancreil, M. B. Schulz, H. H. Barrett, L. Davis, M. G. Holland, Proc. IEEE 57, 1211 (1969).
[CrossRef]

Radiology (1)

H. H. Barrett, K. Garewal, D. T. Wilson, Radiology 104, 429 (1972).
[PubMed]

Rev. Sci. Instrum. (1)

V. L. Hirschy, J. P. Aldridge, Rev. Sci. Instrum. 42, 381 (1971).
[CrossRef]

Sky Telesc. (1)

N. O. Young, Sky Telesc. 25, 8 (1963). L. Mertz, N. O. Young in Proc. Internat. Conf. on Optical Instruments (Chapman and Hall, London, 1961), p. 305.

Other (12)

H. O. Anger, in Instrumentation in Nuclear Medicine, J. Hine, Ed. (Academic, New York, 1967), Vol. 1, p 485.

J. R. Mallard, R. J. Wilks, in Medical Radioisotope Scintigraphy, (IAEA, Vienna, 1969), Vol. 1, p. 3.

F. D. Thomas, W. H. Thomas, G. F. Knoll, Ref. 3, p. 43.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957, p. 360.

A. R. Shulman, Optical Data Processing (Wiley, New York, 1970), p. 387.

L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 88.

H. Eininghammer Thesis, Tubingen U. (1968) NTIS Document N68 34135.

R. W. Wood, Physical Optics (Dover, New York, 1957), p. 37.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), p. 112.

J. W. Goodman in Modern Optics, Jerome Fox, Ed., Microwave Research Inst. Sym. Series (Polytechnic Press, Brooklyn, 1967), Vol. 17, p. 573.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 441.

C. E. Cook, M. Bernfeld, Radar Signals (Academic, New York, 1967), p. 245.

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Figures (10)

Fig. 1
Fig. 1

Use of a Fresnel zone plate with a scintillation camera.

Fig. 2
Fig. 2

Fresnel zone plate.

Fig. 3
Fig. 3

Off-axis zone plate.

Fig. 4
Fig. 4

Geometry used in the general equations for coded-aperture imaging.

Fig. 5
Fig. 5

Illustration of the focusing properties of a zone plate.

Fig. 6
Fig. 6

Focusing properties of a zone plate in combination with a lens.

Fig. 7
Fig. 7

Plot of the magnitude of the Fourier transform of one term in the zone plate transparency function.

Fig. 8
Fig. 8

Focusing properties of an off-axis zone plate.

Fig. 9
Fig. 9

Focusing properties of an off-axis zone plate in combination with a lens.

Fig. 10
Fig. 10

Light distribution in the Fourier transform plane of an off-axis zone plate and lens. A very low contrast film was used in order to show many orders.

Equations (132)

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r = r + ( s 2 / s 1 ) ( r - r ) ,
h ( r ) = τ 4 π ( s 1 + s 2 - z ) - 2 g ( r ) f ( r , z ) d 2 r d z .
f ( r , z ) = f ( r ) δ ( z ) ,
h ( r ) = C f ( r ) g ( a r + b r ) d 2 r ,
a s 1 / ( s 1 + s 2 ) ,
b s 2 / ( s 1 + s 2 ) ,
C ( τ / 4 π ) ( s 1 + s 2 ) - 2 .
F ( k ) 1 2 π - f ( r ) exp [ i ( k · r ) ] d 2 r ,
G ( k ) 1 2 π - g ( r ) exp [ i ( k · r ) ] d 2 r ,
H ( k ) 1 2 π - h ( r ) exp [ i ( k · r ) ] d 2 r ,
g ( r ) = 1 2 π - G ( k ) exp [ - i ( k · r ) ] d 2 k
( 2 π C / a 2 ) F ( k ) G ( - k / b ) = H ( - a k / b ) .
f ( r ) = a 2 4 π 2 C - H ( - a k / b ) G ( - k / b ) exp [ - i ( k · r ) ] d 2 k .
r n = r 1 n             n = 1 , 2 , .
g ( r ) = { 1 if sin ( π r 2 / r 1 2 ) 0 , 0 if sin ( π r 2 / r 1 2 ) < 0 ,
g ( r ) = 1 2 - 2 π sin ( π r 2 / r 1 2 ) - 2 3 π sin ( 3 π r 2 / r 1 2 ) ,
g ( r ) = 1 2 - 1 π i p = - ( odd values only ) 1 p exp ( i π p r 2 / r 1 2 ) .
f p = r 1 2 / λ p ,
exp ( i π r 2 / λ f ) .
A ( r ) = [ 2 J 1 ( π D r / f λ ) ] / ( π D r / f λ ) ,
d = β λ f / D ,
δ z = β λ ( f / D ) 2 ,
1 / f = ( 1 / f l ) + ( 1 + f p ) .
A ( r ) G ( 2 π r / λ f l ) .
G ( k ) = 1 2 δ ( k ) + p ( odd ) G p ( k ) ,
G p ( k ) = ( i / 2 π 2 p ) d 2 r exp [ i ( k · r + π p r 2 / r 1 2 ) ] .
g ( r ) = 1 / 2 + p g p exp [ i ϕ p ( r ) ] ,
g p = - ( π i p ) - 1 , ϕ p = π p r 2 / r 1 2 ,
k p ( r ) d ϕ p / d r = 2 π p r / r 1 2 .
G p ( k ) const · exp ( - i k 2 r 1 2 / 4 π p ) ,             0 k k p ( r N ) .
ν ( r ) k ( r ) / 2 π = r / r 1 2 ,
ν c = k c / 2 π = r c / r 1 2 .
r 1 = r 1 m / a
D = D m / a = 2 r N m / a .
d = β f λ / D = β r 1 2 m / 2 a r N .
m ( s 2 / s 1 ) δ = β r 1 2 m / 2 a r N
δ = β [ ( s 1 + s 2 ) / s 2 ] ( r 1 2 / 2 r N ) .
Δ r N r N - r N - 1 r 1 / 2 N = r N / 2 N ,
δ = β [ ( s 1 + s 2 ) / s 2 ] Δ r N .
η z p η p h = transparent area of zone plate area of equivalent pinhole = 1 2 ( 2 r N ) 2 ( β Δ r N ) 2 = 8 N 2 β 2 = 1 2 D 4 β 2 r 1 4 .
Δ s 1 / s 1 = ( β / 8 N ) [ ( s 1 + s 2 ) / s 2 ] .
D shadow = 2 r N [ ( s 1 + s 2 ) / s 1 ] = D x tal ,
Δ r N [ ( s 1 + s 2 ) / s 1 ] d cam ,
D shadow = 4 N d cam .
FOV = D x tal ( s 1 / s 2 ) .
( 2 π C / a 2 ) F ( - k b / a ) G ( k / a ) = H ( k ) .
ν obj = ( s 1 / s 2 ) ν obj = ( a / b ) ν obj .
ν z p = [ s 1 / ( s 1 + s 2 ) ] ν z p = a ν z p .
ν ^ z p = r N / r 1 2 = D / 2 r 1 2 .
ν ^ z p = ν ^ obj ,
r N r 1 2 · s 1 s 1 + s 2 = ν ^ obj s 1 s 2 .
l min 1 / ( 2 ν ^ obj ) .
Δ r N [ ( s 1 + s 2 ) / s 2 ] optimum = l min .
l min ( s 1 / s 2 ) optimum d cam ,
δ = β l min .
Δ ν z p = D / r 1 2 .
ν obj min = [ ( r c - 1 2 D ) / r 1 2 ] · [ s 2 / ( s 1 + s 2 ) ] .
f ( r ) f ( r ) t ( r )
F ( k ) F ( k ) T ( k ) ,
t ( r + n λ ½ x ) = t ( r ) ,             n = 0 , ± 1 , ± 2 , ,
T ( k ) = m = - T m δ ( k - m k ½ ) ,
k ½ = k ½ = 2 π ν ½ = 2 π / λ ½ .
F ( k ) T ( k ) = m = - T m F ( k - m k ½ ) .
ν ½ = ν c
ν ½ ( s 1 / s 2 ) = ν c [ s 1 / ( s 1 + s 2 ) ] .
δ = β r 1 2 D · s 1 + s 2 s 2 = β r c D ν c · s 1 + s 2 s 2 ,
δ = β r c / D ν ½ .
Δ s 1 s 1 = β 8 ( D 2 / 4 r 1 2 ) s 1 + s 2 s 2 = β r c 2 D 2 ν c · s 1 + s 2 s 2 = β r c / 2 D 2 ν ½ .
η z p / η p h = D 4 / ( 2 β 2 r 1 4 ) .
[ r 1 2 / ( D + 2 r c ) ] · [ ( s 1 + s 2 ) / s 1 ] d cam .
δ off - axis δ on - axis = ( r 1 2 ) off - axis ( r 1 2 ) on - axis = 1 + 2 r c D .
ν ^ obj = 1 2 Δ ν z p ,
Δ ν z p = ( D / r 1 2 ) [ s 1 / ( s 1 + s 2 ) ] optimum .
ν ^ obj = ( D / 2 r 1 2 ) [ s 2 / ( s 1 + s 2 ) ] optimum .
ν ^ obj = ( D / 2 r c ) ν ½
ν ½ = r c / D l min .
D / r c = ( Δ ν z p ) / ν c = ( Δ ν z p ) / ν c .
D / r c 1.
g ( r ) = g x ( x ) g y ( y ) ,
g y ( y ) = { 1 if y h / 2 , 0 if y > h / 2 ,
g x ( x ) = { 1 if sin π x 2 / r 1 2 0 and L 1 x L 2 , 0 otherwise .
L L 2 - L 1 ,
Δ ν = ν ( L 2 ) - ν ( L 1 ) = L / r 1 2 .
ν ½ s 1 s 2 = L 1 + 1 2 L r 1 2 · s 1 s 1 + s 2 .
δ x = β Δ ν · s 1 + s 2 s 2 = β L ξ · s 1 + s 2 s 2 ,
ξ = L Δ ν .
η z p / η p h = ξ / 2 ,
Δ s 1 / s 1 ( 2 / ξ ) · [ ( s 1 + s 2 ) / s 2 ] .
I ( t ) = I 0 ( 1 2 - 2 π sin α t 2 ) , t 1 t t 2 ,
< i 2 > = e < I > Δ ν ,
i = I - < I > .
ξ = ( t 2 - t 1 ) Δ ν Δ T Δ ν .
( S / N ) a = ( 2 ξ ½ I 0 / π ) / ( < i 2 > ½ )
( S / N ) obs = [ ( S / N ) a 2 ] / { [ 1 + 2 ( S + N ) a 2 ] 1 / 2 } .
( S N ) obs 1 2 ( S N ) a = 2 ξ ½ I 0 π < I 2 > ½ .
n 0 = I 0 Δ T / 2 e .
( S / N ) obs S / N = [ ( 2 2 ) / π ] ( n 0 ) ½ ,
S / N ( n 0 ) ½ .
S N = 2 2 π ( n 0 M ) ½ = 2 2 π M ( n t ) ½ ,
N d obj / δ             for d obj ( s 2 / s 1 ) < < L ( s 1 + s 2 ) / s 1
M ξ             for d obj ( s 2 / s 1 ) > L ( s 1 + s 2 ) / s 1 .
( S / N ) z p ( S / N ) p h = ( 2 2 / π ) ( n 0 / M ) 1 / 2 ( n 0 η p h / η z p ) 1 / 2 = 2 2 π ( ξ 2 M ) 1 / 2
ξ = 2 η z p / η p h .
S / N δ 2 n t 1 / 2 .
n t δ - 4 .
M z p ( k ) = G ( - k / b ) M cam ( - a k / b ) ,
θ max = tan - 1 [ ( D x tal / 2 ) / ( s 1 + s 2 ) ] .
t x tal tan θ max < < d cam ,
t x tal < < [ 2 d cam ( s 1 + s 2 ) ] / D x tal .
t z p tan θ max < < Δ r N = ( 2 ν ^ z p ) - 1 .
ν c - v ½ < < Δ ν z p
ν z ( s 1 s 2 ) - ν ½ ( s 1 s 1 + s 2 ) < < ν z D r c · s 1 s 2 ,
δ ( s 1 / s 2 ) < < ( D / r c ) [ 1 + ( s 1 / s 2 ) 0 ] ,
δ ϕ < < D / 2 r c .
( x - x ) / = ( x - x ) / ( + s 1 ) = ( x - x ) / ( + s 1 + s 2 ) .
h ( r ) = C f ( r ) t ( r ) g ( r ) d 2 r ,
C ( τ / 4 π ) ( s 1 + s 2 + ) - 2 .
r = a 1 r + b 1 r
r = a 2 r + b 2 r ,
a 1 = ( + s 1 ) / ( + s 1 + s 2 ) ,
a 2 = / ( + s 1 + s 2 ) ,
b 1 = s 2 / ( + s 1 + s 2 ) ,
b 2 = ( s 1 + s 2 ) / ( + s 1 + s 2 ) .
h ( r ) = C f ( r ) t ( a 2 r + b 2 r ) g ( a 1 r + b 1 r ) d 2 r .
H ( k ) = C a 1 2 m T m F [ - b 1 a 1 k - m ( b 2 - b 1 a 2 a 1 ) k ½ ] × G ( k a 1 - m a 2 a 1 k ½ ) ,
lim 0 H ( k ) = ( C / a 2 ) m T m F ( - b k / a - m k ½ ) G ( k / a ) .
G ( k a 1 - m a 2 a 1 k ½ ) const . exp [ - i r 1 2 4 π ( k a 1 - a 2 a 1 k ½ ) 2 ] .
exp { [ ( - i r 1 2 a 2 ) / ( 2 π a 1 2 ) ] k · k ½ } ,
Δ x = - r 1 2 a 2 k ½ / 2 π a 1 2 .
ν c · [ ( + s 1 ) / ( + s 1 + s 2 ) ]
ν ½ · [ ( / ( + s 1 + s 2 ) ] .
k = - m k ½ ( s 1 / s 2 ) .

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