Abstract

The problem of determining the number of degrees of freedom (NDF) of the field radiated by an electric current supported over a bounded rectilinear domain and observed over multiple bounded domains parallel to the source is addressed. The analysis is achieved by means of the singular value decomposition of the radiation operator so that the NDF is identified as the number of “significant” singular values. The aim is to analyze whether the multidomain observation allows to increase the available NDF. By analytical arguments, we show that collecting data over multiple domains shapes the singular value behavior but it still presents a steep decay in correspondence to an index dictated by the observation domain that subtends the largest observation angular sector.

© 2007 Optical Society of America

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References

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2007

2004

2001

2000

1998

D. A. B. Miller, Opt. Lett. 23, 1645 (1998).
[CrossRef]

F. Soldovieri and R. Pierri, Inverse Probl. 14, 321 (1998).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1998).

1996

T. Isernia, G. Leone, and R. Pierri, IEEE Trans. Antennas Propag. 44, 701 (1996).
[CrossRef]

1993

D. Blackbell and C. J. Oliver, J. Phys. D 26, 1364 (1993).
[CrossRef]

E. Scalas and G. A. Viano, J. Opt. Soc. Am. A 10, 991 (1993).
[CrossRef]

1985

1961

D. Slepian and H. O. Pollak, Bell Syst. Tech. J. 40, 43 (1961).

Barakat, R.

Blackbell, D.

D. Blackbell and C. J. Oliver, J. Phys. D 26, 1364 (1993).
[CrossRef]

Burvall, A.

Friberg, A. T.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1998).

Isernia, T.

T. Isernia, G. Leone, and R. Pierri, IEEE Trans. Antennas Propag. 44, 701 (1996).
[CrossRef]

Leone, G.

R. Solimene, G. Leone, and R. Pierri, J. Opt. Soc. Am. A 21, 1402 (2004).
[CrossRef]

T. Isernia, G. Leone, and R. Pierri, IEEE Trans. Antennas Propag. 44, 701 (1996).
[CrossRef]

Liseno, A.

Martinsson, P.

Miller, D. A. B.

Newsam, G.

Oliver, C. J.

D. Blackbell and C. J. Oliver, J. Phys. D 26, 1364 (1993).
[CrossRef]

Pierri, R.

R. Solimene, G. Leone, and R. Pierri, J. Opt. Soc. Am. A 21, 1402 (2004).
[CrossRef]

R. Pierri, A. Liseno, F. Soldovieri, and R. Solimene, J. Opt. Soc. Am. A 18, 352 (2001).
[CrossRef]

F. Soldovieri and R. Pierri, Inverse Probl. 14, 321 (1998).
[CrossRef]

T. Isernia, G. Leone, and R. Pierri, IEEE Trans. Antennas Propag. 44, 701 (1996).
[CrossRef]

Piestun, R.

Pollak, H. O.

D. Slepian and H. O. Pollak, Bell Syst. Tech. J. 40, 43 (1961).

Scalas, E.

Slepian, D.

D. Slepian and H. O. Pollak, Bell Syst. Tech. J. 40, 43 (1961).

Soldovieri, F.

Solimene, R.

Viano, G. A.

Appl. Opt.

Bell Syst. Tech. J.

D. Slepian and H. O. Pollak, Bell Syst. Tech. J. 40, 43 (1961).

IEEE Trans. Antennas Propag.

T. Isernia, G. Leone, and R. Pierri, IEEE Trans. Antennas Propag. 44, 701 (1996).
[CrossRef]

Inverse Probl.

F. Soldovieri and R. Pierri, Inverse Probl. 14, 321 (1998).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. D

D. Blackbell and C. J. Oliver, J. Phys. D 26, 1364 (1993).
[CrossRef]

Opt. Lett.

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1998).

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Figures (2)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Eigenvalue behavior of the operator in Eq. (4) (solid curve) and of the operator in Eq. (5) (dotted curve) normalized to λ. X s = X 1 = X 2 = 20 λ and z 1 = 110 λ . z 2 = 180 λ (top panel); z 2 = 118 λ (bottom panel).

Equations (9)

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E i ( x i ) = 1 z i X S X S exp [ j β ( x i x ) 2 ( 2 z i ) ] J ( x ) d x ,
A : J L 2 ( S ) E = ( E 1 , E 2 ) L 2 ( O 1 × O 2 ) ,
γ n u n = A A u n ,
γ n u n ( x ) = λ X S X S sin [ β X 1 z 1 ( x x ) ] π ( x x ) exp [ j β 2 z 1 ( x 2 x 2 ) ] u n ( x ) d x + λ X S X S sin [ β X 2 z 2 ( x x ) ] π ( x x ) exp [ j β 2 z 2 ( x 2 x 2 ) ] u n ( x ) d x ,
γ n u ̃ n ( x ) = λ X S X S sin [ β X 1 z 1 ( x x ) ] π ( x x ) u ̃ n ( x ) d x + λ X S X S sin [ β X 2 z 2 ( x x ) ] π ( x x ) u ̃ n ( x ) d x ,
A A 2 λ P S B Ω 1 P S + λ P S B Ω 2 P S + λ P S B Ω 3 P S ,
u ̃ k n ( x ) = ϕ n ( x , c k ) exp ( j α k x ) ,
γ k n = η k γ n P ( c k ) λ ,
P S B Ω k P S u ̃ h n 0 ,

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