Abstract

The problem of diagnosing a grid of small (in terms of the probing wavelength) dielectric scatterers is considered. The aim is to detect and locate possible defects occurring within a known grid when one (or more) scatterer is removed/missing (fault). The study is developed for the canonical case of a TM scalar two-dimensional geometry with the scatterers consisting of dielectric cylinders of small circular cross section. The scattering by a fault is modeled by relaying only to a priori information about the complete grid which leads to a numerically effective inversion procedures as the bulk of the numerical effort is to be done only once. Inversion is achieved by a truncated singular value decomposition scheme and results are provided in terms of closed form expressions for the probability of detection and of false alarm. This allows us to foreseen the achievable performance and to highlight the role of scattering configuration parameters. Numerical examples are also enclosed to corroborate theoretical outcomes. The case of two or more faults is considered as well. For such a case it is numerically shown that detection method still works well even though multiple scattering (occurring between faults) is neglected.

© 2015 Optical Society of America

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References

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    [Crossref]
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2013 (1)

R. Solimene and G. Leone, “MUSIC algorithms for grid diasgnostics,” IEEE Geosc. Rem. Sens. Lett. 10(2), 226–230 (2013).
[Crossref]

2008 (1)

2005 (1)

R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antenn. Propag. 53 (9), 3019–3029 (2005).
[Crossref]

2004 (1)

2003 (1)

2002 (2)

S. Enoch, B. Gralak, and G. Tayeb, “Enhanced emission with angular confinement from photonic crystals,” Appl. Phys. Lett. 81, 1588–1590 (2002).
[Crossref]

P. Lalanne and A. Talneau, “Modal conversion with artificial materials for photonic-crystals waveguide,” Opt. Express 10, 354–359 (2002).
[Crossref] [PubMed]

1998 (2)

1997 (1)

1993 (1)

1976 (1)

C. Elachi, “Waves in active and passive periodic structures: a review,” IEEE Trans. Microwave Theory Tech. 24(12), 1666–1698 (1976).

Balanis, C.

C. Balanis, Advanced Engineering Electromagnetics (Wiley and Sons, 1989).

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics Publishing, 1998).
[Crossref]

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics Publishing, 1998).
[Crossref]

Brancaccio, A.

Brown, E.R.

Caloz, C.

C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley-IEEE Press, 2005).

Elachi, C.

C. Elachi, “Waves in active and passive periodic structures: a review,” IEEE Trans. Microwave Theory Tech. 24(12), 1666–1698 (1976).

Enoch, S.

S. Enoch, B. Gralak, and G. Tayeb, “Enhanced emission with angular confinement from photonic crystals,” Appl. Phys. Lett. 81, 1588–1590 (2002).
[Crossref]

Gralak, B.

S. Enoch, B. Gralak, and G. Tayeb, “Enhanced emission with angular confinement from photonic crystals,” Appl. Phys. Lett. 81, 1588–1590 (2002).
[Crossref]

Groby, J.P.

Harrington, R.F.

R.F. Harrington, Time Harmonic Electromagnetic Fields (Wiley and Sons, 2001).
[Crossref]

Itoh, T.

C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley-IEEE Press, 2005).

Joanopulos, J.D.

J.D. Joanopulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton U. Press, 1995).

Lalanne, P.

Leone, G.

Lesselier, D.

Leviatan, Y.

Liseno, A.

R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antenn. Propag. 53 (9), 3019–3029 (2005).
[Crossref]

Ludwig, A.

Maystre, D.

Meade, R.D.

J.D. Joanopulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton U. Press, 1995).

Parker, C.D.

Pierri, R.

R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antenn. Propag. 53 (9), 3019–3029 (2005).
[Crossref]

A. Brancaccio, G. Leone, and R. Pierri, “Information content of Born scattered fields: results in the circular cylindrical case,” J. Opt. Soc. Am. A 15(7), 1909–1917 (1998).
[Crossref]

Romano, J.

R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antenn. Propag. 53 (9), 3019–3029 (2005).
[Crossref]

Solimene, R.

R. Solimene and G. Leone, “MUSIC algorithms for grid diasgnostics,” IEEE Geosc. Rem. Sens. Lett. 10(2), 226–230 (2013).
[Crossref]

R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antenn. Propag. 53 (9), 3019–3029 (2005).
[Crossref]

A. Brancaccio, G. Leone, and R. Solimene, “Fault detection in metallic grid scattering,” J. Opt. Soc. Am. A 28(12), 2588–2599 (1998).
[Crossref]

Talneau, A.

Tayeb, G.

S. Enoch, B. Gralak, and G. Tayeb, “Enhanced emission with angular confinement from photonic crystals,” Appl. Phys. Lett. 81, 1588–1590 (2002).
[Crossref]

G. Tayeb and D. Maystre, “Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities,” J. Opt. Soc. Am. A 14(12), 3323–3332 (1997).
[Crossref]

Winn, J.N.

J.D. Joanopulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton U. Press, 1995).

Yablonovitch, E.

Yasumoto, K.

K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals (Optical Engineering, CRC Press, 2005).
[Crossref]

Appl. Phys. Lett. (1)

S. Enoch, B. Gralak, and G. Tayeb, “Enhanced emission with angular confinement from photonic crystals,” Appl. Phys. Lett. 81, 1588–1590 (2002).
[Crossref]

IEEE Geosc. Rem. Sens. Lett. (1)

R. Solimene and G. Leone, “MUSIC algorithms for grid diasgnostics,” IEEE Geosc. Rem. Sens. Lett. 10(2), 226–230 (2013).
[Crossref]

IEEE Trans. Antenn. Propag. (1)

R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antenn. Propag. 53 (9), 3019–3029 (2005).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

C. Elachi, “Waves in active and passive periodic structures: a review,” IEEE Trans. Microwave Theory Tech. 24(12), 1666–1698 (1976).

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

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

Opt. Express (1)

Other (6)

J.D. Joanopulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton U. Press, 1995).

K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals (Optical Engineering, CRC Press, 2005).
[Crossref]

C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley-IEEE Press, 2005).

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics Publishing, 1998).
[Crossref]

R.F. Harrington, Time Harmonic Electromagnetic Fields (Wiley and Sons, 2001).
[Crossref]

C. Balanis, Advanced Engineering Electromagnetics (Wiley and Sons, 1989).

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Figures (10)

Fig. 1
Fig. 1 Scatterers location and numbering.
Fig. 2
Fig. 2 Grid of N = 121 scatterers. The observation points are taken along the dot circle. Stars denote the source positions. Cylinders are numbered form the upper side from left to right. Red and green points indicate position m = 30 and m = 61 respectively.
Fig. 3
Fig. 3 Normalized singular values for the configuration of Fig. 2.
Fig. 4
Fig. 4 Spatial content for NT = 60 (left) and NT = 87 (right).
Fig. 5
Fig. 5 Side lobes content for NT = 60: central fault at m = 61 (left); peripheral fault m = 6 (right). Note that the gray scale is enhanced with respect to Fig. (4) due to the lower value of side lobes content.
Fig. 6
Fig. 6 PD(61) (solid lines) and [PFA(n,61)]max(n) (dashed lines) versus Ath for different values of the truncation index: SNR = 10dB (top); SNR = 5dB (bottom).
Fig. 7
Fig. 7 Results for a fault whose actual position is denoted by the red circle. White circles denote detected faults for SNR = 10, NT = 60 and different Ath.
Fig. 8
Fig. 8 Results for a fault whose actual position is denoted by the red circle. White circles denote detected faults for SNR = 10, NT = 112 and different Ath.
Fig. 9
Fig. 9 Results for two faults whose actual position is denoted by the red circles. White circles denote detected faults for SNR = 10, NT = 60 and Ath = 0.3.
Fig. 10
Fig. 10 Results for three (left) and five (right) faults whose actual position is denoted by the red circles. White circles denote detected faults for SNR = 10, NT = 60 and Ath = 0.3.

Equations (42)

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E s ( r , r s ) = E N 1 ( r , r s ) E N ( r , r s ) ,
E s ( r ) = j β ζ Ω m J ( r ) G N ( r , r ) d r ,
E s ( r ) = β 2 ( 1 ε r ) E N 1 ( r m , r s ) Ω m G N ( r , r ) d r ,
G N ( r , r ) = j 4 [ H 0 ( 2 ) ( β | r r | ) + n = 1 N a n ( r ) H 0 ( 2 ) ( β | r r n | ) ] ,
G N ( r , r ) = j 4 [ b m ( r ) J 0 ( β ε r | r r m | ) ] ,
[ A ¯ ¯ 0 B 0 I ¯ ¯ A ¯ ¯ 1 B 1 I ¯ ¯ ] [ a ¯ b ¯ ] = [ y ¯ 0 y ¯ 1 ]
y n i = J i ( β a ) H i ( 2 ) ( β a ) H 0 ( 2 ) ( β | r r n | ) n = 1 , , N
A n m i = { 1 if n = m J i ( β a ) H i ( 2 ) ( β a ) H 0 ( 2 ) ( β d n m ) if n m
B i = [ ( ε r 1 ) i + 1 ] J i ( β ε r a ) H i ( 2 ) ( β a ) i = 0 , 1.
Ω G N ( r , r ) d r = j 4 b m ( r ) 0 2 π 0 a J 0 ( β ε r ρ ) ρ d ρ d θ = j π a 2 β ε r b m ( r ) J 1 ( β ε r a )
E N 1 ( r m , r s ) = β ζ I 4 [ H 0 ( 2 ) ( β | r m r s | ) + n = 1 , n m N a n ( m ) ( r s ) H 0 ( 2 ) ( β d n m ) ] ,
E s ( r , r s ) = C b m ( r ) [ H 0 ( 2 ) ( β | r m r s | ) + n = 1 , n m N a n ( r s ) H 0 ( 2 ) ( β d n m ) + n = 1 , n m N Δ a n ( m ) ( r s ) H 0 ( 2 ) ( β d n m ) ]
E s ( r , r s ) = C b m ( r ) [ D b m ( r s ) + n = 1 , n m N Δ a n ( m ) ( r s ) H 0 ( 2 ) ( β d n m ) ]
D = H 0 ( 2 ) ( β a ) J 0 ( β a ) [ J 0 ( β ε r a ) + F ]
F = j π β a 2 ( J 0 ( β ε r a ) J 1 ( β a ) ε r J 0 ( β a ) J 1 ( β ε r a ) )
A ¯ ¯ a ¯ = y ¯
A ¯ ¯ = ( A ¯ ¯ 0 B 0 B 1 A ¯ ¯ 1 ) / ( 1 B 0 B 1 )
y ¯ = ( y ¯ 0 B 0 B 1 y ¯ 1 ) / ( 1 B 0 B 1 )
Δ A i j ( m ) = { 0 if i m and j m A i j if i = m or j = m 0 if i = j = m .
a ¯ ( m ) = [ A ¯ ¯ ( m ) ] 1 y ¯
Δ a ¯ ( m ) = ( A ¯ ¯ Δ A ¯ ¯ ( m ) ) 1 Δ A ¯ ¯ ( m ) a ¯
Δ A ¯ ¯ ( m ) a ¯ ( r s ) = A ¯ ¯ ( m ) a m ( r s ) + X ¯ ( m )
n m Δ a n ( m ) ( r s ) H 0 ( 2 ) ( β d n m ) = T ¯ ( m ) T Δ a ¯ ( m ) ( r s )
n m Δ a n ( m ) ( r s ) H 0 ( 2 ) ( β d n m ) = T ¯ ( m ) T ( A ¯ ¯ Δ A ¯ ¯ ( m ) ) 1 A ¯ ( m ) a m ( r s )
a m = F b m
E s ( r , r s ) = K b m ( r ) b m ( r s ) Γ m
Γ m = 1 + F D [ T ¯ ( m ) T ( A ¯ ¯ Δ A ¯ ¯ ( m ) ) 1 A ¯ ( m ) ]
E s ( r o , r s ) = K n = 1 N b n ( r o ) b n ( r s ) γ ( n )
γ ( n ) = { Γ n if n = m 0 if n m
E ¯ s = L ¯ ¯ γ ¯
E s ( r o , r s ) = n = 1 N β 2 ( 1 ε r ) E N 1 ( r n , r s ) Ω n G N ( r o , r ) d r δ n m
E ¯ s = L ¯ ¯ ˜ δ ¯
γ ¯ ^ = min γ ¯ S E ¯ s L ¯ ¯ γ ¯
γ ¯ ^ = min γ ¯ C N 2 E ¯ s L ¯ ¯ γ ¯ 2
γ ¯ = i = 1 N T c i v ¯ i
E ¯ s = L ¯ ¯ γ ¯ + η ¯
γ ¯ ˜ = i = 1 N T c i v ¯ i + i = 1 N T u ¯ i H η ¯ σ i v ¯ i
S C ( n ) = Γ n i = 1 N T | v i ( n ) | 2
S L ( m , n ) = Γ m i = 1 N T v i H ( m ) v i ( n ) n m
p | γ ˜ | ( α ) = α var η ( n ) exp ( α 2 + | m e a n γ ¯ ˜ ( n ) | 2 2 var η ( n ) ) I 0 ( α | m e a n γ ¯ ˜ ( n ) | 2 var η ( n ) )
P F A ( m , n ) = Q ( | S L ( m , n ) | v a r η ( n ) , A t h v a r η ( n ) )
P D ( n ) = Q ( | S C ( n ) | v a r η ( n ) , A t h v a r η ( n ) )

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