Abstract

A nonlinear signal processing method is applied to the design of strongly scattering objects to realize a defined angular response. Investigated as the complement of inverse scattering problems, k-space design methods are combined with cepstral filtering to obtain a permittivity distribution that scatters with the desired response. Starting with the rigorously computed angular spectrum of the scattering amplitude of an object of simple geometric shape, the corresponding k-space is modified to provide the desired scattering behavior. In order to account for strong scattering, cepstral filtering is applied to map the associated distribution of secondary sources to a unique permittivity distribution. The inversion process results in a structure that exhibits the desired properties and which can be interpreted as a perturbation of the initial structure. Simulation results are presented which illustrate the usefulness of this method. In particular, objects are modified to enhance forward scattering and suppress scattering in all other direction. Results are verified using a rigorous finite-difference frequency-domain scheme to simulate scattering. The method is demonstrated as a novel means for designing invisible objects that act as electromagnetic cloaks.

© 2007 Optical Society of America

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References

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    [CrossRef] [PubMed]
  7. A. Alù and N. Engheta, "Optical nanotransmission lines: synthesis of planar left-handed metamaterials in the infrared and visible regimes," J. Opt. Soc. Am. B 23, 571 - 583 (2006).
    [CrossRef]
  8. R. C. Rumpf and E. G. Johnson, "Fully three-dimensional modeling of the fabrication and behavior of photonic crystals formed by holographic lithography," J. Opt. Soc. Am. A 21, 1703 - 1713 (2004).
    [CrossRef]
  9. R. Rumpf and E. G. Johnson, "Comprehensive modeling of near-field nano-patterning," Opt. Express 13, 7198 - 7208 (2005).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  11. L. Fatone, M. C. Recchioni, and F. Zirilli, "A method to solve an acoustic inverse scattering problem involving smart obstacles," Waves Random Complex Media 16, 433-455 (2006).
    [CrossRef]
  12. U.  Leonhardt, "Optical Conformal Mapping," Science 312, 1777 - 1780 (2006).
    [CrossRef] [PubMed]
  13. U.  Leonhardt, "Notes on Conformal Invisibility Devices," New. J. Phys.  8, 118 (2006).
    [CrossRef]
  14. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science Express 314, 977-980 (2006).
  15. D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794 (2006).
    [CrossRef] [PubMed]
  16. S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
    [CrossRef]
  17. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
    [CrossRef] [PubMed]
  18. E.  Wolf and T.  Habashy, "Invisible bodies and uniqueness of the inverse scattering problem," J. Mod. Opt. 40, 785 - 792 (1993).
    [CrossRef]
  19. G. Gbur, "Nonradiating sources and other ‘invisible’ objects," E. Wolf, ed., in Progress in Optics (Elsevier, Amsterdam, 2003), Vol. 45, pp. 273 - 315.
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    [CrossRef]
  21. U. Shahid, M. Testorf and M. A. Fiddy, "Minimum-phase-based inverse scattering algorithm applied to Institute Fresnel data," Inverse Probl. 21, S153 - 164 (2005).
    [CrossRef]
  22. M. A. Fiddy and M. Testorf, "Inverse scattering method applied to the synthesis of strongly scattering structures," Opt. Express 14, 2037 - 2046, (2006).
    [CrossRef] [PubMed]
  23. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. 14, pp. 302-307, 1966.
    [CrossRef]
  24. A. Taflove and S. C. Hagness, Computational Electrodynamics the Finite-Difference Time-Domain Method, 3rd ed, (Artech House, 2005).
    [PubMed]
  25. R. C. Rumpf, "Design and optimization of nano-optical elements by coupling fabrication to optical behavior," PhD dissertation, University of Central Florida, pp. 60-81, 2006.

2006 (10)

L. Fatone, M. C. Recchioni, and F. Zirilli, "A method to solve an acoustic inverse scattering problem involving smart obstacles," Waves Random Complex Media 16, 433-455 (2006).
[CrossRef]

U.  Leonhardt, "Optical Conformal Mapping," Science 312, 1777 - 1780 (2006).
[CrossRef] [PubMed]

U.  Leonhardt, "Notes on Conformal Invisibility Devices," New. J. Phys.  8, 118 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science Express 314, 977-980 (2006).

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
[CrossRef] [PubMed]

M. A. Fiddy and M. Testorf, "Inverse scattering method applied to the synthesis of strongly scattering structures," Opt. Express 14, 2037 - 2046, (2006).
[CrossRef] [PubMed]

A. Alù and N. Engheta, "Optical nanotransmission lines: synthesis of planar left-handed metamaterials in the infrared and visible regimes," J. Opt. Soc. Am. B 23, 571 - 583 (2006).
[CrossRef]

A. Mehta, R. C. Rumpf, Z. Roth, and E. G. Johnson, "Nanofabrication of a space-variant optical transmission filter," Opt. Lett. 31, 2903 - 2905 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794 (2006).
[CrossRef] [PubMed]

2005 (2)

R. Rumpf and E. G. Johnson, "Comprehensive modeling of near-field nano-patterning," Opt. Express 13, 7198 - 7208 (2005).
[CrossRef] [PubMed]

U. Shahid, M. Testorf and M. A. Fiddy, "Minimum-phase-based inverse scattering algorithm applied to Institute Fresnel data," Inverse Probl. 21, S153 - 164 (2005).
[CrossRef]

2004 (2)

M. Testorf and U. Gibson, "Design of thin-film-coated diffractive optical elements with frequency variant transmission functions," Proc. SPIE 5515, 158 - 169 (2004).
[CrossRef]

R. C. Rumpf and E. G. Johnson, "Fully three-dimensional modeling of the fabrication and behavior of photonic crystals formed by holographic lithography," J. Opt. Soc. Am. A 21, 1703 - 1713 (2004).
[CrossRef]

1999 (1)

1998 (1)

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679 - 1682 (1998).
[CrossRef] [PubMed]

1995 (1)

1993 (1)

E.  Wolf and T.  Habashy, "Invisible bodies and uniqueness of the inverse scattering problem," J. Mod. Opt. 40, 785 - 792 (1993).
[CrossRef]

1986 (1)

M. A. Fiddy, "Inversion of Optical Scattered Field Data." J. Phys. D 19, 301-317 (1986).
[CrossRef]

1978 (2)

1966 (1)

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. 14, pp. 302-307, 1966.
[CrossRef]

Alù, A.

Case, S. K.

Chambers, D. M.

Chen, C.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679 - 1682 (1998).
[CrossRef] [PubMed]

Cummer, S. A.

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science Express 314, 977-980 (2006).

Dallas, W. J.

Engheta, N.

Fan, S.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679 - 1682 (1998).
[CrossRef] [PubMed]

Fatone, L.

L. Fatone, M. C. Recchioni, and F. Zirilli, "A method to solve an acoustic inverse scattering problem involving smart obstacles," Waves Random Complex Media 16, 433-455 (2006).
[CrossRef]

Fiddy, M. A.

M. A. Fiddy and M. Testorf, "Inverse scattering method applied to the synthesis of strongly scattering structures," Opt. Express 14, 2037 - 2046, (2006).
[CrossRef] [PubMed]

U. Shahid, M. Testorf and M. A. Fiddy, "Minimum-phase-based inverse scattering algorithm applied to Institute Fresnel data," Inverse Probl. 21, S153 - 164 (2005).
[CrossRef]

M. A. Fiddy, "Inversion of Optical Scattered Field Data." J. Phys. D 19, 301-317 (1986).
[CrossRef]

Fink, Y.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679 - 1682 (1998).
[CrossRef] [PubMed]

Friesem, A. A.

Gibson, U.

M. Testorf and U. Gibson, "Design of thin-film-coated diffractive optical elements with frequency variant transmission functions," Proc. SPIE 5515, 158 - 169 (2004).
[CrossRef]

Habashy, T.

E.  Wolf and T.  Habashy, "Invisible bodies and uniqueness of the inverse scattering problem," J. Mod. Opt. 40, 785 - 792 (1993).
[CrossRef]

Joannopoulos, J. D.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679 - 1682 (1998).
[CrossRef] [PubMed]

Johnson, E. G.

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science Express 314, 977-980 (2006).

Leonhardt, U.

U.  Leonhardt, "Optical Conformal Mapping," Science 312, 1777 - 1780 (2006).
[CrossRef] [PubMed]

U.  Leonhardt, "Notes on Conformal Invisibility Devices," New. J. Phys.  8, 118 (2006).
[CrossRef]

Mait, J. N.

Mehta, A.

Michel, J.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679 - 1682 (1998).
[CrossRef] [PubMed]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science Express 314, 977-980 (2006).

Nordin, G. P.

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science Express 314, 977-980 (2006).

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794 (2006).
[CrossRef] [PubMed]

Peri, D.

Popa, B.-I.

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Recchioni, M. C.

L. Fatone, M. C. Recchioni, and F. Zirilli, "A method to solve an acoustic inverse scattering problem involving smart obstacles," Waves Random Complex Media 16, 433-455 (2006).
[CrossRef]

Roth, Z.

Rumpf, R.

Rumpf, R. C.

Schurig, D.

D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science Express 314, 977-980 (2006).

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
[CrossRef] [PubMed]

Shahid, U.

U. Shahid, M. Testorf and M. A. Fiddy, "Minimum-phase-based inverse scattering algorithm applied to Institute Fresnel data," Inverse Probl. 21, S153 - 164 (2005).
[CrossRef]

Smith, D. R.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science Express 314, 977-980 (2006).

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794 (2006).
[CrossRef] [PubMed]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science Express 314, 977-980 (2006).

Testorf, M.

M. A. Fiddy and M. Testorf, "Inverse scattering method applied to the synthesis of strongly scattering structures," Opt. Express 14, 2037 - 2046, (2006).
[CrossRef] [PubMed]

U. Shahid, M. Testorf and M. A. Fiddy, "Minimum-phase-based inverse scattering algorithm applied to Institute Fresnel data," Inverse Probl. 21, S153 - 164 (2005).
[CrossRef]

M. Testorf and U. Gibson, "Design of thin-film-coated diffractive optical elements with frequency variant transmission functions," Proc. SPIE 5515, 158 - 169 (2004).
[CrossRef]

Thomas, E. L.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679 - 1682 (1998).
[CrossRef] [PubMed]

Winn, J. N.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679 - 1682 (1998).
[CrossRef] [PubMed]

Wolf, E.

E.  Wolf and T.  Habashy, "Invisible bodies and uniqueness of the inverse scattering problem," J. Mod. Opt. 40, 785 - 792 (1993).
[CrossRef]

Yee, K. S.

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. 14, pp. 302-307, 1966.
[CrossRef]

Zirilli, F.

L. Fatone, M. C. Recchioni, and F. Zirilli, "A method to solve an acoustic inverse scattering problem involving smart obstacles," Waves Random Complex Media 16, 433-455 (2006).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. 14, pp. 302-307, 1966.
[CrossRef]

Inverse Probl. (1)

U. Shahid, M. Testorf and M. A. Fiddy, "Minimum-phase-based inverse scattering algorithm applied to Institute Fresnel data," Inverse Probl. 21, S153 - 164 (2005).
[CrossRef]

J. Mod. Opt. (1)

E.  Wolf and T.  Habashy, "Invisible bodies and uniqueness of the inverse scattering problem," J. Mod. Opt. 40, 785 - 792 (1993).
[CrossRef]

J. Opt. Soc. Am. A (3)

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

J. Phys. D (1)

M. A. Fiddy, "Inversion of Optical Scattered Field Data." J. Phys. D 19, 301-317 (1986).
[CrossRef]

New. J. Phys. (1)

U.  Leonhardt, "Notes on Conformal Invisibility Devices," New. J. Phys.  8, 118 (2006).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. E (1)

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Proc. SPIE (1)

M. Testorf and U. Gibson, "Design of thin-film-coated diffractive optical elements with frequency variant transmission functions," Proc. SPIE 5515, 158 - 169 (2004).
[CrossRef]

Science (3)

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679 - 1682 (1998).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
[CrossRef] [PubMed]

U.  Leonhardt, "Optical Conformal Mapping," Science 312, 1777 - 1780 (2006).
[CrossRef] [PubMed]

Science Express (1)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science Express 314, 977-980 (2006).

Waves Random Complex Media (1)

L. Fatone, M. C. Recchioni, and F. Zirilli, "A method to solve an acoustic inverse scattering problem involving smart obstacles," Waves Random Complex Media 16, 433-455 (2006).
[CrossRef]

Other (3)

G. Gbur, "Nonradiating sources and other ‘invisible’ objects," E. Wolf, ed., in Progress in Optics (Elsevier, Amsterdam, 2003), Vol. 45, pp. 273 - 315.

A. Taflove and S. C. Hagness, Computational Electrodynamics the Finite-Difference Time-Domain Method, 3rd ed, (Artech House, 2005).
[PubMed]

R. C. Rumpf, "Design and optimization of nano-optical elements by coupling fabrication to optical behavior," PhD dissertation, University of Central Florida, pp. 60-81, 2006.

Supplementary Material (2)

» Media 1: AVI (2403 KB)     
» Media 2: AVI (2219 KB)     

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Figures (8)

Fig. 1.
Fig. 1.

k-space interpretation of scattering in the first order Born approximation: (a) geometry for plane wave scattering of a permittivity distribution; (b) k-space representation of the incident wave and one scattered plane wave component. The scattering amplitude is proportional to the object spectrum at the Ewald circle.

Fig. 2.
Fig. 2.

(a) Rigorous solution of plane wave scattering off a homogeneous cylinder, (b) k-space constructed from the far field for a set of different incident field directions.

Fig. 3.
Fig. 3.

Magnitude of the inverse Fourier transform of the k-space distribution shown in Fig. 2(b).

Fig. 4.
Fig. 4.

(a) Reconstruction of object permittivity contrast based on (a) cepstral filtering, and (b) spectral filtering.

Fig. 5.
Fig. 5.

Reconstruction of object permittivity contrast after k-space engineering based on (a) cepstral filtering, and (b) spectral filtering.

Fig. 6.
Fig. 6.

(a) Refractive index distribution of scattering object. Here the object is a perfect cylinder with n=2.0. (b) (Movie 2403kb) Total-field computed by FDFD simulation. [Media 1] (c) Scattered-field computed by FDFD simulation. (d) Pattern of energy scattered from object. Lobes correspond to preferred directions of scattering.

Fig. 7.
Fig. 7.

(a) Refractive index distribution of filtered scattering object. (b) (Movie 2219kb) Total-field computed by FDFD simulation. [Media 2] (c) Scattered-field computed by FDFD simulation. (d) Pattern of energy scattered from object. Lobes correspond to preferred directions of scattering.

Fig. 8.
Fig. 8.

Angular spread of the far field for plane wave. The finite simulation window for the near field in (a) results in a finite angular spread of the far field pattern predicted by the numerical diffraction model.

Equations (3)

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Ψ s r k r ̂ 0 = exp ( ik r ) 8 πk r D V ( r ' ) Ψ ( r , k r ̂ 0 ) exp ( ik r r ̂ 0 ) d r 2
V B r k r ̂ 0 V ( r ) Ψ r k r ̂ 0 Ψ 0 r k r ̂ 0
log ( V Ψ ) = log V + log Ψ + i [ arg ( V ) + arg ( Ψ ) ] ,

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