Abstract

This paper applies the multiple signal classification (MUSIC) imaging method to determine the locations of a collection of small anisotropic spherical scatterers in the framework of total internal reflection tomography. Multiple scattering between scatterers is considered, and the inverse scattering problem is nonlinear, which, however, is solved by the proposed fast analytical approach where no associated forward problem is iteratively evaluated. The paper also discusses the role of propagating and evanescent waves, the polarization of incidence waves, separation of scatterers from the surface of the substrate, and the level of noise on the resolution of imaging.

© 2008 Optical Society of America

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References

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  1. E. Wolf and J. T. Foley, "Do evanescent waves contribute to the far field?" Opt. Lett. 23, 16-18 (1998).
    [CrossRef]
  2. P. S. Carney and J. C. Schotland, "Near-field tomography," in Inside Out: Inverse Problems and Applications, G.Uhlman, ed. (Cambridge U. Press, 2003), pp. 133-168.
  3. D. G. Fischer, "The information content of weakly scattered fields: implications for near-field imaging of three-dimensional structures," J. Mod. Opt. 47, 1359-1374 (2000).
    [CrossRef]
  4. J. Sun, P. S. Carney, and J. C. Schotland, "Near-field scanning optical tomography: a nondestructive method for three-dimensional nanoscale imaging," IEEE J. Sel. Top. Quantum Electron. 12, 1072-1082 (2006).
    [CrossRef]
  5. P. S. Carney and J. C. Schotland, "Theory of total-internal-reflection tomography," J. Opt. Soc. Am. A 20, 542-547 (2003).
    [CrossRef]
  6. P. S. Carney and J. C. Schotland, "Three-dimensional total internal reflection microscopy," Opt. Lett. 26, 1072-1074 (2001).
    [CrossRef]
  7. F. Simonetti, "Multiple scattering: the key to unravel the subwavelength world from the far-field pattern of a scattered wave," Phys. Rev. E 73, 036619 (2006).
    [CrossRef]
  8. K. Belkebir, P. C. Chaumet, and A. Sentenac, "Superresolution in total internal reflection tomography," J. Opt. Soc. Am. A 22, 1889-1897 (2005).
    [CrossRef]
  9. G. Y. Panasyuk, V. A. Markel, P. S. Carney, and J. C. Schotland, "Nonlinear inverse scattering and three-dimensional near-field optical imaging," Appl. Phys. Lett. 89, 221116 (2006).
    [CrossRef]
  10. G. Gao and C. Torres-Verdín, "High-order generalized extended Born approximation for electromagnetic scattering," IEEE Trans. Antennas Propag. 54, 1243-1256 (2006).
    [CrossRef]
  11. M. Cheney, "The linear sampling method and the MUSIC algorithm," Inverse Probl. 17, 591-595 (2001).
    [CrossRef]
  12. A. Kirsch, "The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media," Inverse Probl. 18, 1025-1040 (2002).
    [CrossRef]
  13. A. J. Devaney, E. A. Marengo, and F. K. Gruber, "Time-reversal-based imaging and inverse scattering of multiply scattering point targets," J. Acoust. Soc. Am. 118, 3129-3138 (2005).
    [CrossRef]
  14. H. Ammari, E. Iakovleva, D. Lesselier, and G. Perruson, "MUSIC-type electromagnetic imaging of a collection of small three-dimensional bounded inclusions," SIAM J. Sci. Comput. (USA) 29, 674-709 (2007).
    [CrossRef]
  15. L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
    [CrossRef]
  16. M. Lax, "Multiple scattering of waves," Rev. Mod. Phys. 23, 287-310 (1951).
    [CrossRef]
  17. L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley-Interscience, 2000).
    [CrossRef]
  18. Y. Zhong and X. Chen, "MUSIC imaging and electromagnetic inverse scattering of multiply scattering small anisotropic spheres," IEEE Trans. Antennas Propag. 55, 3542-3549 (2007).
    [CrossRef]
  19. C. F. Bohren and D. R. Huffman, Absorption and Scatttering of Light by Small Particles (Wiley, 1998).
    [CrossRef]
  20. J. D. Jackson, Classical Electrodynamics, 3rd ed., (Wiley, 1998).
  21. J. A. Kong, Electromagnetic Wave Theory (EMW, 2000).
  22. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, 1986).
  23. E. A. Marengo and F. K. Gruber, "Noniterative analytical formula for inverse scattering of multiply scattering point targets," J. Acoust. Soc. Am. 120, 3782-3788 (2006).
    [CrossRef]
  24. X. Chen and Y. Zhong, "A robust noniterative method for obtaining scattering strengths of multiply scattering point targets," J. Acoust. Soc. Am. 122, 1325-1327 (2007).
    [CrossRef] [PubMed]

2007 (3)

H. Ammari, E. Iakovleva, D. Lesselier, and G. Perruson, "MUSIC-type electromagnetic imaging of a collection of small three-dimensional bounded inclusions," SIAM J. Sci. Comput. (USA) 29, 674-709 (2007).
[CrossRef]

Y. Zhong and X. Chen, "MUSIC imaging and electromagnetic inverse scattering of multiply scattering small anisotropic spheres," IEEE Trans. Antennas Propag. 55, 3542-3549 (2007).
[CrossRef]

X. Chen and Y. Zhong, "A robust noniterative method for obtaining scattering strengths of multiply scattering point targets," J. Acoust. Soc. Am. 122, 1325-1327 (2007).
[CrossRef] [PubMed]

2006 (5)

E. A. Marengo and F. K. Gruber, "Noniterative analytical formula for inverse scattering of multiply scattering point targets," J. Acoust. Soc. Am. 120, 3782-3788 (2006).
[CrossRef]

J. Sun, P. S. Carney, and J. C. Schotland, "Near-field scanning optical tomography: a nondestructive method for three-dimensional nanoscale imaging," IEEE J. Sel. Top. Quantum Electron. 12, 1072-1082 (2006).
[CrossRef]

F. Simonetti, "Multiple scattering: the key to unravel the subwavelength world from the far-field pattern of a scattered wave," Phys. Rev. E 73, 036619 (2006).
[CrossRef]

G. Y. Panasyuk, V. A. Markel, P. S. Carney, and J. C. Schotland, "Nonlinear inverse scattering and three-dimensional near-field optical imaging," Appl. Phys. Lett. 89, 221116 (2006).
[CrossRef]

G. Gao and C. Torres-Verdín, "High-order generalized extended Born approximation for electromagnetic scattering," IEEE Trans. Antennas Propag. 54, 1243-1256 (2006).
[CrossRef]

2005 (2)

K. Belkebir, P. C. Chaumet, and A. Sentenac, "Superresolution in total internal reflection tomography," J. Opt. Soc. Am. A 22, 1889-1897 (2005).
[CrossRef]

A. J. Devaney, E. A. Marengo, and F. K. Gruber, "Time-reversal-based imaging and inverse scattering of multiply scattering point targets," J. Acoust. Soc. Am. 118, 3129-3138 (2005).
[CrossRef]

2003 (1)

2002 (1)

A. Kirsch, "The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media," Inverse Probl. 18, 1025-1040 (2002).
[CrossRef]

2001 (2)

P. S. Carney and J. C. Schotland, "Three-dimensional total internal reflection microscopy," Opt. Lett. 26, 1072-1074 (2001).
[CrossRef]

M. Cheney, "The linear sampling method and the MUSIC algorithm," Inverse Probl. 17, 591-595 (2001).
[CrossRef]

2000 (1)

D. G. Fischer, "The information content of weakly scattered fields: implications for near-field imaging of three-dimensional structures," J. Mod. Opt. 47, 1359-1374 (2000).
[CrossRef]

1998 (1)

1951 (1)

M. Lax, "Multiple scattering of waves," Rev. Mod. Phys. 23, 287-310 (1951).
[CrossRef]

Appl. Phys. Lett. (1)

G. Y. Panasyuk, V. A. Markel, P. S. Carney, and J. C. Schotland, "Nonlinear inverse scattering and three-dimensional near-field optical imaging," Appl. Phys. Lett. 89, 221116 (2006).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Sun, P. S. Carney, and J. C. Schotland, "Near-field scanning optical tomography: a nondestructive method for three-dimensional nanoscale imaging," IEEE J. Sel. Top. Quantum Electron. 12, 1072-1082 (2006).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

G. Gao and C. Torres-Verdín, "High-order generalized extended Born approximation for electromagnetic scattering," IEEE Trans. Antennas Propag. 54, 1243-1256 (2006).
[CrossRef]

Y. Zhong and X. Chen, "MUSIC imaging and electromagnetic inverse scattering of multiply scattering small anisotropic spheres," IEEE Trans. Antennas Propag. 55, 3542-3549 (2007).
[CrossRef]

Inverse Probl. (2)

M. Cheney, "The linear sampling method and the MUSIC algorithm," Inverse Probl. 17, 591-595 (2001).
[CrossRef]

A. Kirsch, "The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media," Inverse Probl. 18, 1025-1040 (2002).
[CrossRef]

J. Acoust. Soc. Am. (3)

A. J. Devaney, E. A. Marengo, and F. K. Gruber, "Time-reversal-based imaging and inverse scattering of multiply scattering point targets," J. Acoust. Soc. Am. 118, 3129-3138 (2005).
[CrossRef]

E. A. Marengo and F. K. Gruber, "Noniterative analytical formula for inverse scattering of multiply scattering point targets," J. Acoust. Soc. Am. 120, 3782-3788 (2006).
[CrossRef]

X. Chen and Y. Zhong, "A robust noniterative method for obtaining scattering strengths of multiply scattering point targets," J. Acoust. Soc. Am. 122, 1325-1327 (2007).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

D. G. Fischer, "The information content of weakly scattered fields: implications for near-field imaging of three-dimensional structures," J. Mod. Opt. 47, 1359-1374 (2000).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Lett. (2)

Phys. Rev. E (1)

F. Simonetti, "Multiple scattering: the key to unravel the subwavelength world from the far-field pattern of a scattered wave," Phys. Rev. E 73, 036619 (2006).
[CrossRef]

Rev. Mod. Phys. (1)

M. Lax, "Multiple scattering of waves," Rev. Mod. Phys. 23, 287-310 (1951).
[CrossRef]

SIAM J. Sci. Comput. (USA) (1)

H. Ammari, E. Iakovleva, D. Lesselier, and G. Perruson, "MUSIC-type electromagnetic imaging of a collection of small three-dimensional bounded inclusions," SIAM J. Sci. Comput. (USA) 29, 674-709 (2007).
[CrossRef]

Other (7)

L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

C. F. Bohren and D. R. Huffman, Absorption and Scatttering of Light by Small Particles (Wiley, 1998).
[CrossRef]

J. D. Jackson, Classical Electrodynamics, 3rd ed., (Wiley, 1998).

J. A. Kong, Electromagnetic Wave Theory (EMW, 2000).

A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, 1986).

P. S. Carney and J. C. Schotland, "Near-field tomography," in Inside Out: Inverse Problems and Applications, G.Uhlman, ed. (Cambridge U. Press, 2003), pp. 133-168.

L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley-Interscience, 2000).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

MUSIC pseudospectrum in the z = 0.1 λ plane for TM incidences. (a) The test dipole is x ̂ oriented. (b) The test dipole is y ̂ oriented. (c) The test dipole is z ̂ oriented. (d) The test dipole is 1 3 ( x ̂ + y ̂ + z ̂ ) oriented.

Fig. 2
Fig. 2

MUSIC pseudospectrum in the z = 0.1 λ plane for TM incidences, with the test dipole y ̂ oriented. The y coordinate of the first sphere is slightly shifted to 0.001 λ .

Fig. 3
Fig. 3

MUSIC pseudospectrum in the z = 0.1 λ plane for TE incidences. (a) The test dipole is x ̂ oriented. (b) The test dipole is y ̂ oriented. (c) The test dipole is z ̂ oriented. (d) The test dipole is 1 3 ( x ̂ + y ̂ + z ̂ ) oriented.

Fig. 4
Fig. 4

MUSIC pseudospectrum in the z = 0.1 λ plane for a mixture of TE and TM incidences. The test dipole is x ̂ oriented. The pseudospectra generated from test dipoles oriented in y ̂ , z ̂ , and 1 3 ( x ̂ + y ̂ + z ̂ ) , almost identical to the shown pseudospectrum generated from the x ̂ -oriented dipole, are not repeated here.

Fig. 5
Fig. 5

MUSIC pseudospectrum in the y = x plane (upper figures) and y = x plane (lower figures) for four types of incidences. 30 dB white Gaussian noise is added. The spheres are located at ± 0.1414 λ in the transverse ( t ) direction. (a) Incidence type A. (b) Incidence type B. (c) Incidence type C. (d) Incidence type D.

Fig. 6
Fig. 6

MUSIC pseudospectrum in the y = x plane (upper figures) and y = x plane (lower figures) for four types of incidences. 50 dB white Gaussian noise is added. The spheres are located at ± 0.1414 λ in the transverse ( t ) direction. (a) Incidence type A. (b) Incidence type B. (c) Incidence type C. (d) Incidence type D.

Fig. 7
Fig. 7

MUSIC pseudospectrum in the y = x plane (upper figures) and y = x plane (lower figures) for four types of incidences. 30 dB white Gaussian noise is added. The spheres are located at ± 0.1414 λ in the transverse ( t ) direction. The substrate in the half-space z < 0 is replaced by free space. (a) Incidence type A. (b) Incidence type B. (c) Incidence type C. (d) Incidence type D.

Equations (20)

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ϵ ̿ m = Ξ ̿ m 1 diag { ϵ m ( 1 ) , ϵ m ( 2 ) , ϵ m ( 3 ) } Ξ ̿ m ,
E t in ( r k ) = E 0 in ( r k ) + m k i ω μ 0 G ̿ t ( r k , r m ) ξ ̿ m E t in ( r m ) ,
ξ ̿ m = i 4 π k a m 3 ϵ 0 μ 0 Ξ ̿ m 1 diag [ ϵ m ( 1 ) ϵ 0 ϵ m ( 1 ) + 2 ϵ 0 , ϵ m ( 2 ) ϵ 0 ϵ m ( 2 ) + 2 ϵ 0 , ϵ m ( 3 ) ϵ 0 ϵ m ( 3 ) + 2 ϵ 0 ] Ξ ̿ m ,
G ̿ t ( r , r ) = G ̿ 0 ( r , r ) + G ̿ r ( r , r )
ψ ¯ t in = ψ ¯ 0 in + Φ ̿ Λ ̿ ψ ¯ t in ,
ψ ¯ t in = [ E t in ( r 1 ) T , E t in ( r 2 ) T , , E t in ( r M ) T ] T ,
ψ ¯ 0 in = [ E 0 in ( r 1 ) T , E 0 in ( r 2 ) T , , E 0 in ( r M ) T ] T ,
Λ ̿ = diag [ ξ ̿ 1 , ξ ̿ 2 , , ξ ̿ M ] ,
Φ ̿ ( m , m ) = { i ω μ 0 G ̿ t ( r m , r m ) m m , 0 m = m .
E i sub ( r ) = α y ̂ exp ( i k s k ̂ i sub r ) + ( 1 α ) y ̂ × k ̂ i sub exp ( i k s k ̂ i sub r ) ,
E 0 in ( r ) = [ T TE α y ̂ + T TM ( 1 α ) y ̂ × k ̂ i ] exp ( i k 0 k ̂ i r ) ,
ψ ¯ t in ( i ) = ( I ̿ 3 M Φ ̿ Λ ̿ ) 1 ψ ¯ 0 in ( i ) ,
E ( j , i ) = G ̿ ( j ) Λ ̿ ψ ¯ t in ( i ) ,
G ̿ F ( r , r ) = exp ( i k 0 r ) 4 π r [ exp ( i k 0 r r ̂ ) ( I ̿ 3 r ̂ r ̂ ) + exp ( i k 0 r ̃ r ̂ ) ( R E p E p ̃ E + R M p M p ̃ M ) ] ,
K ̿ = R ̿ Λ ̿ ( I ̿ 3 M Φ ̿ Λ ̿ ) 1 T ̿ ,
Φ ( r ) = log 10 ( σ p = 0 u ¯ p * f ̂ ( r ) 2 p 0 ) 1 2 ,
J ̿ = R ̿ K ̿ ,
J ̿ = Λ ̿ ( T ̿ + Φ ̿ J ̿ ) .
Λ ̿ i = J ̿ i [ ( T ̿ + Φ ̿ J ̿ ) ] i , i = 1 , 2 , , M ,
E = ( i = 1 M Λ ̿ i Λ ̿ i 2 ) 1 2 ( i = 1 M Λ ̿ i 2 ) 1 2 × 100 % ,

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