Abstract

Interferometric synthetic aperture arrays (SAA) can be refocused at any range via digital processing of the raw data, called the visibility function. Such systems are sensitive, however, to ranging errors in digital refocusing. They, therefore, exhibit a limited depth of focus, because these errors are equivalent to introducing defocus in the system. We derive an analytical expression for the Strehl ratio of synthetic aperture arrays, which accounts for both the antenna voltage pattern and the focal shift factor encountered in imaging systems with a low Fresnel number. Assessment of the depth of focus of short-range imaging arrays is allowed and is illustrated for common array designs.

© 2010 Optical Society of America

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References

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  1. A. R. Thompson, J. M. Moran, and G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (Wiley, 2001).
    [CrossRef]
  2. F. Torres, A. Camps, J. Bará, and I. Corbella, Radio Sci. 32, 629 (1997).
    [CrossRef]
  3. A. R. Harvey and R. Appleby, Aeronaut J. 107, 87 (2003).
  4. B. J. Davis, T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, in Proceedings of the IEEE International Conference on Image Processing 2007 (IEEE, 2007), Vol. IV, pp. 145–148.
  5. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  6. J. Ojeda-Castañeda, M. Martínez-Corral, P. Andrés, and A. Pons, Appl. Opt. 33, 7611 (1994).
    [CrossRef] [PubMed]
  7. Y. Li and E. Wolf, Opt. Commun. 39, 211 (1981).
    [CrossRef]
  8. Y. Li and E. Wolf, J. Opt. Soc. Am. A 1, 801 (1984).
    [CrossRef]
  9. Y. L. Chow, IEEE Trans. Antenn. Propag. 20, 30 (1972).
    [CrossRef]
  10. E. Keto, Astrophys. J. 475, 843 (1997).
    [CrossRef]
  11. B. M. Lucotte, B. Grafulla-González, and A. R. Harvey, Radio Sci. 44, RS1006 (2009).
    [CrossRef]

2009 (1)

B. M. Lucotte, B. Grafulla-González, and A. R. Harvey, Radio Sci. 44, RS1006 (2009).
[CrossRef]

2003 (1)

A. R. Harvey and R. Appleby, Aeronaut J. 107, 87 (2003).

1997 (2)

F. Torres, A. Camps, J. Bará, and I. Corbella, Radio Sci. 32, 629 (1997).
[CrossRef]

E. Keto, Astrophys. J. 475, 843 (1997).
[CrossRef]

1994 (1)

1984 (1)

1981 (1)

Y. Li and E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

1972 (1)

Y. L. Chow, IEEE Trans. Antenn. Propag. 20, 30 (1972).
[CrossRef]

Andrés, P.

Appleby, R.

A. R. Harvey and R. Appleby, Aeronaut J. 107, 87 (2003).

Bará, J.

F. Torres, A. Camps, J. Bará, and I. Corbella, Radio Sci. 32, 629 (1997).
[CrossRef]

Boppart, S. A.

B. J. Davis, T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, in Proceedings of the IEEE International Conference on Image Processing 2007 (IEEE, 2007), Vol. IV, pp. 145–148.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Camps, A.

F. Torres, A. Camps, J. Bará, and I. Corbella, Radio Sci. 32, 629 (1997).
[CrossRef]

Carney, P. S.

B. J. Davis, T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, in Proceedings of the IEEE International Conference on Image Processing 2007 (IEEE, 2007), Vol. IV, pp. 145–148.

Chow, Y. L.

Y. L. Chow, IEEE Trans. Antenn. Propag. 20, 30 (1972).
[CrossRef]

Corbella, I.

F. Torres, A. Camps, J. Bará, and I. Corbella, Radio Sci. 32, 629 (1997).
[CrossRef]

Davis, B. J.

B. J. Davis, T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, in Proceedings of the IEEE International Conference on Image Processing 2007 (IEEE, 2007), Vol. IV, pp. 145–148.

Grafulla-González, B.

B. M. Lucotte, B. Grafulla-González, and A. R. Harvey, Radio Sci. 44, RS1006 (2009).
[CrossRef]

Harvey, A. R.

B. M. Lucotte, B. Grafulla-González, and A. R. Harvey, Radio Sci. 44, RS1006 (2009).
[CrossRef]

A. R. Harvey and R. Appleby, Aeronaut J. 107, 87 (2003).

Keto, E.

E. Keto, Astrophys. J. 475, 843 (1997).
[CrossRef]

Li, Y.

Lucotte, B. M.

B. M. Lucotte, B. Grafulla-González, and A. R. Harvey, Radio Sci. 44, RS1006 (2009).
[CrossRef]

Marks, D. L.

B. J. Davis, T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, in Proceedings of the IEEE International Conference on Image Processing 2007 (IEEE, 2007), Vol. IV, pp. 145–148.

Martínez-Corral, M.

Moran, J. M.

A. R. Thompson, J. M. Moran, and G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (Wiley, 2001).
[CrossRef]

Ojeda-Castañeda, J.

Pons, A.

Ralston, T. S.

B. J. Davis, T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, in Proceedings of the IEEE International Conference on Image Processing 2007 (IEEE, 2007), Vol. IV, pp. 145–148.

Swenson, G. W.

A. R. Thompson, J. M. Moran, and G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (Wiley, 2001).
[CrossRef]

Thompson, A. R.

A. R. Thompson, J. M. Moran, and G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (Wiley, 2001).
[CrossRef]

Torres, F.

F. Torres, A. Camps, J. Bará, and I. Corbella, Radio Sci. 32, 629 (1997).
[CrossRef]

Wolf, E.

Y. Li and E. Wolf, J. Opt. Soc. Am. A 1, 801 (1984).
[CrossRef]

Y. Li and E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Aeronaut J. (1)

A. R. Harvey and R. Appleby, Aeronaut J. 107, 87 (2003).

Appl. Opt. (1)

Astrophys. J. (1)

E. Keto, Astrophys. J. 475, 843 (1997).
[CrossRef]

IEEE Trans. Antenn. Propag. (1)

Y. L. Chow, IEEE Trans. Antenn. Propag. 20, 30 (1972).
[CrossRef]

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

Opt. Commun. (1)

Y. Li and E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

Radio Sci. (2)

F. Torres, A. Camps, J. Bará, and I. Corbella, Radio Sci. 32, 629 (1997).
[CrossRef]

B. M. Lucotte, B. Grafulla-González, and A. R. Harvey, Radio Sci. 44, RS1006 (2009).
[CrossRef]

Other (3)

A. R. Thompson, J. M. Moran, and G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (Wiley, 2001).
[CrossRef]

B. J. Davis, T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, in Proceedings of the IEEE International Conference on Image Processing 2007 (IEEE, 2007), Vol. IV, pp. 145–148.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

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Figures (2)

Fig. 1
Fig. 1

S ( W 20 ) / K ( W 20 ) for the power-law Y array (Y), Reuleux triangle array (R), circular array (C), and real circular aperture system (Real). ρ a 0.0091 and N F = 20 .

Fig. 2
Fig. 2

S ( W 20 ) for the same systems as in Fig. 1 (with the same legend). (a) N F = 20 and (b) N F = 5 .

Equations (18)

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W 20 = a 2 z 2 λ f ( f + z ) ,
P 0 ( ρ ) = p ( ρ ) exp [ j 2 π W 20 ρ 2 ] ,
p ( ρ ) = { 1 | ρ | 1 , 0 | ρ | > 1 , .
P ( ρ , W 20 ) = e j 2 π W 20 ρ 2 k = 1 k = N δ ( ρ ρ k ) * * p ( ρ ρ a ) ,
P ^ ( r , W 20 ) = j e j k z a 2 λ f ( f + z ) 1 + 1 1 + 1 P ( ρ , W 20 ) × exp [ j 2 π a λ f ( ξ x + η y ) ] d ξ d η ,
P ^ ( r , W 20 ) = N F 2 W 20 f k = 1 k = N p ( ρ k ρ a ) * * exp [ j 2 π ( W 20 ρ k 2 + u ξ k + v η k ) ] ,
= N F 2 W 20 f k = 1 k = N F k ( r , W 20 ) × exp [ j 2 π ( W 20 ρ k 2 + u ξ k + v η k ) ] ,
F k ( r , W 20 ) = V k ( r , W 20 ) p ( ρ ρ a ) exp ( j 2 π W 20 ρ 2 ) × exp [ j 2 π ( ξ α + η β ) ] d ξ d η ,
F k ( r , W 20 ) = 2 π V k ( r , W 20 ) 0 ρ a exp ( j 2 π W 20 ρ 2 ) × J 0 ( 2 π ζ ρ ) ρ d ρ ,
V k ( r , W 20 ) 2 J 1 [ 2 π b γ ( N F 2 W 20 ) / a 2 ] 2 π b γ ( N F 2 W 20 ) / a 2 ,
F k ( r , W 20 ) = A k exp ( j θ k ) .
h ( 0 , W 20 ) = 2 ( N F 2 W 20 ) 2 f 2 m = 1 m = N n = m + 1 n = N A m A n × cos [ 2 π W 20 ( ρ m 2 ρ n 2 ) + θ m θ n ] .
S ( W 20 ) = h ( 0 , W 20 ) h ( 0 , 0 ) = | P ^ ( 0 , W 20 ) | 2 | P ^ ( 0 , 0 ) | 2 .
S ( W 20 ) = K ( W 20 ) ( π ρ a 2 ) 2 B m = 1 m = N n = m + 1 n = N A m A n × cos [ 2 π W 20 ( ρ m 2 ρ n 2 ) + θ m θ n ] ,
K ( W 20 ) = ( N F 2 W 20 ) 2 N F 2 ,
S real ( W 20 ) = K ( W 20 ) sinc 2 ( W 20 ) ,
S 0 ( W 20 ) = 2 K ( W 20 ) N ( N 1 ) m = 1 m = N n = m + 1 n = N cos [ 2 π W 20 ( ρ m 2 ρ n 2 ) ] .
S cir ( W 20 ) = 2 K ( W 20 ) N ( N 1 ) V 0 ( 0 , 0 ) 1 ( π ρ a 2 ) 2 m = 1 m = N n = m + 1 n = N A m A n × cos ( θ m θ n ) .

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