Abstract

An iterative numerical approach to the approximate computation of the bistatic scattering width for weakly nonlinear dielectric infinite cylinders is proposed. The cylinders are assumed to be arbitrarily shaped and illuminated by a transverse-magnetic wave. The bistatic scattering width is calculated with an iterative numerical technique that can use both the classic first-order Born approximation and the distorted-wave Born approximation to provide the starting internal field distribution. Several numerical results are presented concerning Kerr-like nonlinearities. Circular and square cylinders are considered, as well as shells and scatterers with irregular cross sections. The effects of the nonlinearities, of the incident-field amplitude, and of the ratios between linear dimensions and wavelengths are analyzed, and the convergence of the iterative approach is evaluated in some significant cases.

© 1995 Optical Society of America

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  1. A. F. Peterson, “Analysis of heterogeneous EM scatterers: research progress of the past decade,” IEEE Proc. 79, 1431–1453 (1991).
    [Crossref]
  2. L. J. F. Broer, “Wave propagation in nonlinear media,” Z. amgew. Math. Phys. 16, 18–26 (1965).
    [Crossref]
  3. K. M. Leung, “Scattering of transverse-electric electromagnetic waves with a finite nonlinear film,” J. Opt. Soc. Am. B 5, 571–574 (1988).
    [Crossref]
  4. N. Bloembergen, Nonlinear Optics (W. A. Benjamin, Reading, Mass., 1965).
  5. S. Caorsi, M. Pastorino, “Integral equation formulation of electromagnetic scattering by nonlinear dielectric objects,” Electromagnetics 11, 357–375 (1991).
    [Crossref]
  6. D. Mihalache, D. Mazilu, M. Bertolotti, C. Sibilla, “Exact solution for nonlinear thin-film guided waves in higher-order nonlinear media,” J. Opt. Soc. Am. B 5, 565–570 (1988).
    [Crossref]
  7. S. Caorsi, A. Massa, M. Pastorino, “Bistatic scattering width evaluation for nonlinear isotropic infinite circular cylinders,” Microwave Opt. Technol. Lett. 7, 639–642 (1994).
    [Crossref]
  8. M. A. Hasan, P. L. E. Uslenghi, “Electromagnetic scattering from nonlinear anysotropic cylinders. Part I: Fundamental frequency,” IEEE Trans. Antennas Propag. AP-38, 323–333 (1990).
  9. J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross-section shape,” IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
    [Crossref]
  10. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  11. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).
  12. S. Caorsi, A. Massa, M. Pastorino, “A numerical approach to full-vector electromagnetic scattering by three-dimensional nonlinear bounded dielectrics,” IEEE Trans. Microwave Theory Tech. MTT-43, 428–436 (1995).
    [Crossref]
  13. R. J. Wombell, R. D. Murch, “The reconstruction of dielectric objects from scattered field data using the distortedwave Born approximation,” J. Electromagn. Waves Applic. 7, 687–702 (1993).
    [Crossref]
  14. W. C. Chew, Y. M. Wang, “Reconstruction of twodimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
    [Crossref]
  15. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  16. M. J. Hagmann, R. L. Levin, “Convergence of local and average values in three-dimensional moment-method solutions,” IEEE Trans. Microwave Theory Tech. MTT-33, 649–654 (1985).
    [Crossref]
  17. M. Slaney, A. C. Kak, L. E. Larsen, “Limitation of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–873 (1984).
    [Crossref]
  18. M. J. Hagmann, O. P. Gandhi, C. H. Durney, “Upper bound on cell size for moment-method solutions,” IEEE Trans. Microwave Theory Tech. MTT-25, 831–832 (1977).
    [Crossref]

1995 (1)

S. Caorsi, A. Massa, M. Pastorino, “A numerical approach to full-vector electromagnetic scattering by three-dimensional nonlinear bounded dielectrics,” IEEE Trans. Microwave Theory Tech. MTT-43, 428–436 (1995).
[Crossref]

1994 (1)

S. Caorsi, A. Massa, M. Pastorino, “Bistatic scattering width evaluation for nonlinear isotropic infinite circular cylinders,” Microwave Opt. Technol. Lett. 7, 639–642 (1994).
[Crossref]

1993 (1)

R. J. Wombell, R. D. Murch, “The reconstruction of dielectric objects from scattered field data using the distortedwave Born approximation,” J. Electromagn. Waves Applic. 7, 687–702 (1993).
[Crossref]

1991 (2)

A. F. Peterson, “Analysis of heterogeneous EM scatterers: research progress of the past decade,” IEEE Proc. 79, 1431–1453 (1991).
[Crossref]

S. Caorsi, M. Pastorino, “Integral equation formulation of electromagnetic scattering by nonlinear dielectric objects,” Electromagnetics 11, 357–375 (1991).
[Crossref]

1990 (2)

W. C. Chew, Y. M. Wang, “Reconstruction of twodimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[Crossref]

M. A. Hasan, P. L. E. Uslenghi, “Electromagnetic scattering from nonlinear anysotropic cylinders. Part I: Fundamental frequency,” IEEE Trans. Antennas Propag. AP-38, 323–333 (1990).

1988 (2)

1985 (1)

M. J. Hagmann, R. L. Levin, “Convergence of local and average values in three-dimensional moment-method solutions,” IEEE Trans. Microwave Theory Tech. MTT-33, 649–654 (1985).
[Crossref]

1984 (1)

M. Slaney, A. C. Kak, L. E. Larsen, “Limitation of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–873 (1984).
[Crossref]

1977 (1)

M. J. Hagmann, O. P. Gandhi, C. H. Durney, “Upper bound on cell size for moment-method solutions,” IEEE Trans. Microwave Theory Tech. MTT-25, 831–832 (1977).
[Crossref]

1965 (2)

L. J. F. Broer, “Wave propagation in nonlinear media,” Z. amgew. Math. Phys. 16, 18–26 (1965).
[Crossref]

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross-section shape,” IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
[Crossref]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).

Bertolotti, M.

Bloembergen, N.

N. Bloembergen, Nonlinear Optics (W. A. Benjamin, Reading, Mass., 1965).

Broer, L. J. F.

L. J. F. Broer, “Wave propagation in nonlinear media,” Z. amgew. Math. Phys. 16, 18–26 (1965).
[Crossref]

Caorsi, S.

S. Caorsi, A. Massa, M. Pastorino, “A numerical approach to full-vector electromagnetic scattering by three-dimensional nonlinear bounded dielectrics,” IEEE Trans. Microwave Theory Tech. MTT-43, 428–436 (1995).
[Crossref]

S. Caorsi, A. Massa, M. Pastorino, “Bistatic scattering width evaluation for nonlinear isotropic infinite circular cylinders,” Microwave Opt. Technol. Lett. 7, 639–642 (1994).
[Crossref]

S. Caorsi, M. Pastorino, “Integral equation formulation of electromagnetic scattering by nonlinear dielectric objects,” Electromagnetics 11, 357–375 (1991).
[Crossref]

Chew, W. C.

W. C. Chew, Y. M. Wang, “Reconstruction of twodimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[Crossref]

Durney, C. H.

M. J. Hagmann, O. P. Gandhi, C. H. Durney, “Upper bound on cell size for moment-method solutions,” IEEE Trans. Microwave Theory Tech. MTT-25, 831–832 (1977).
[Crossref]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Gandhi, O. P.

M. J. Hagmann, O. P. Gandhi, C. H. Durney, “Upper bound on cell size for moment-method solutions,” IEEE Trans. Microwave Theory Tech. MTT-25, 831–832 (1977).
[Crossref]

Hagmann, M. J.

M. J. Hagmann, R. L. Levin, “Convergence of local and average values in three-dimensional moment-method solutions,” IEEE Trans. Microwave Theory Tech. MTT-33, 649–654 (1985).
[Crossref]

M. J. Hagmann, O. P. Gandhi, C. H. Durney, “Upper bound on cell size for moment-method solutions,” IEEE Trans. Microwave Theory Tech. MTT-25, 831–832 (1977).
[Crossref]

Hasan, M. A.

M. A. Hasan, P. L. E. Uslenghi, “Electromagnetic scattering from nonlinear anysotropic cylinders. Part I: Fundamental frequency,” IEEE Trans. Antennas Propag. AP-38, 323–333 (1990).

Kak, A. C.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitation of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–873 (1984).
[Crossref]

Larsen, L. E.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitation of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–873 (1984).
[Crossref]

Leung, K. M.

Levin, R. L.

M. J. Hagmann, R. L. Levin, “Convergence of local and average values in three-dimensional moment-method solutions,” IEEE Trans. Microwave Theory Tech. MTT-33, 649–654 (1985).
[Crossref]

Massa, A.

S. Caorsi, A. Massa, M. Pastorino, “A numerical approach to full-vector electromagnetic scattering by three-dimensional nonlinear bounded dielectrics,” IEEE Trans. Microwave Theory Tech. MTT-43, 428–436 (1995).
[Crossref]

S. Caorsi, A. Massa, M. Pastorino, “Bistatic scattering width evaluation for nonlinear isotropic infinite circular cylinders,” Microwave Opt. Technol. Lett. 7, 639–642 (1994).
[Crossref]

Mazilu, D.

Mihalache, D.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Murch, R. D.

R. J. Wombell, R. D. Murch, “The reconstruction of dielectric objects from scattered field data using the distortedwave Born approximation,” J. Electromagn. Waves Applic. 7, 687–702 (1993).
[Crossref]

Pastorino, M.

S. Caorsi, A. Massa, M. Pastorino, “A numerical approach to full-vector electromagnetic scattering by three-dimensional nonlinear bounded dielectrics,” IEEE Trans. Microwave Theory Tech. MTT-43, 428–436 (1995).
[Crossref]

S. Caorsi, A. Massa, M. Pastorino, “Bistatic scattering width evaluation for nonlinear isotropic infinite circular cylinders,” Microwave Opt. Technol. Lett. 7, 639–642 (1994).
[Crossref]

S. Caorsi, M. Pastorino, “Integral equation formulation of electromagnetic scattering by nonlinear dielectric objects,” Electromagnetics 11, 357–375 (1991).
[Crossref]

Peterson, A. F.

A. F. Peterson, “Analysis of heterogeneous EM scatterers: research progress of the past decade,” IEEE Proc. 79, 1431–1453 (1991).
[Crossref]

Richmond, J. H.

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross-section shape,” IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
[Crossref]

Sibilla, C.

Slaney, M.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitation of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–873 (1984).
[Crossref]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Uslenghi, P. L. E.

M. A. Hasan, P. L. E. Uslenghi, “Electromagnetic scattering from nonlinear anysotropic cylinders. Part I: Fundamental frequency,” IEEE Trans. Antennas Propag. AP-38, 323–333 (1990).

Wang, Y. M.

W. C. Chew, Y. M. Wang, “Reconstruction of twodimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[Crossref]

Wombell, R. J.

R. J. Wombell, R. D. Murch, “The reconstruction of dielectric objects from scattered field data using the distortedwave Born approximation,” J. Electromagn. Waves Applic. 7, 687–702 (1993).
[Crossref]

Electromagnetics (1)

S. Caorsi, M. Pastorino, “Integral equation formulation of electromagnetic scattering by nonlinear dielectric objects,” Electromagnetics 11, 357–375 (1991).
[Crossref]

IEEE Proc. (1)

A. F. Peterson, “Analysis of heterogeneous EM scatterers: research progress of the past decade,” IEEE Proc. 79, 1431–1453 (1991).
[Crossref]

IEEE Trans. Antennas Propag. (2)

M. A. Hasan, P. L. E. Uslenghi, “Electromagnetic scattering from nonlinear anysotropic cylinders. Part I: Fundamental frequency,” IEEE Trans. Antennas Propag. AP-38, 323–333 (1990).

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross-section shape,” IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
[Crossref]

IEEE Trans. Med. Imaging (1)

W. C. Chew, Y. M. Wang, “Reconstruction of twodimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[Crossref]

IEEE Trans. Microwave Theory Tech. (4)

M. J. Hagmann, R. L. Levin, “Convergence of local and average values in three-dimensional moment-method solutions,” IEEE Trans. Microwave Theory Tech. MTT-33, 649–654 (1985).
[Crossref]

M. Slaney, A. C. Kak, L. E. Larsen, “Limitation of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–873 (1984).
[Crossref]

M. J. Hagmann, O. P. Gandhi, C. H. Durney, “Upper bound on cell size for moment-method solutions,” IEEE Trans. Microwave Theory Tech. MTT-25, 831–832 (1977).
[Crossref]

S. Caorsi, A. Massa, M. Pastorino, “A numerical approach to full-vector electromagnetic scattering by three-dimensional nonlinear bounded dielectrics,” IEEE Trans. Microwave Theory Tech. MTT-43, 428–436 (1995).
[Crossref]

J. Electromagn. Waves Applic. (1)

R. J. Wombell, R. D. Murch, “The reconstruction of dielectric objects from scattered field data using the distortedwave Born approximation,” J. Electromagn. Waves Applic. 7, 687–702 (1993).
[Crossref]

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

Microwave Opt. Technol. Lett. (1)

S. Caorsi, A. Massa, M. Pastorino, “Bistatic scattering width evaluation for nonlinear isotropic infinite circular cylinders,” Microwave Opt. Technol. Lett. 7, 639–642 (1994).
[Crossref]

Z. amgew. Math. Phys. (1)

L. J. F. Broer, “Wave propagation in nonlinear media,” Z. amgew. Math. Phys. 16, 18–26 (1965).
[Crossref]

Other (4)

N. Bloembergen, Nonlinear Optics (W. A. Benjamin, Reading, Mass., 1965).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

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Figures (6)

Fig. 1
Fig. 1

BSW of a nonlinear circular cylinder: k1a = 0.6π; Ψ0 = 1.0 (V/m), ϕ0 = 0, L = 1.1, M = 225. DWBA: (a) β = 0.1, (b) β = 0.2, (c) β = 0.3, (d) β = 0.5, (e) β = 0.8. Born approximation: (f) β = 0.1, (g) β = 0.2. (h) Effects of the nonlinear index β and analytical and numerical computations for the linear case (β = 0).

Fig. 2
Fig. 2

Effects of the nonlinear parameter β on the BSW of a nonlinear cylindrical shell: k1a1 = 0.6π, k1a2 = 0.49π, Ψ0 = 1.0 (V/m), ϕ0 = 0, L = 1.15, M = 40.

Fig. 3
Fig. 3

Effects of the propagation direction of the incident uniform plane wave on the BSW of a nonlinear cylindrical half-shell: k1a1 = 0.6π, k1a2 = 0.49π, θ0 = 1.05π, Ψ0 = 1.0 (V/m), ϕ0 = 0, L = 1.5, M = 21, β = 0.2. k is the propagation vector.

Fig. 4
Fig. 4

BSW of a nonlinear square cylinder: k1l = 0.96π; Ψ0 = 1.0 (V/m), ϕ0 = 0, L = 1.1, M = 144. (a) k = 1,(b) k = 2, (c) k = 10 (k is the iteration number), (d) residual error Ξ(k) versus number of iterations. Numerical computation for the linear case (β = 0).

Fig. 5
Fig. 5

Effects of the ratio between the wavelength (at the fundamental frequency) and the dimensions of the cylinder cross section. BSW of a square cylinder: Ψ0 = 1.0 (V/m), ϕ0 = 0, L = 1.15, β = 0.1, M = 144. (a) k1l = 0.8π, (b) k1l = 1.6π, (c) k1l = 2.4π, (d) k1l = 3.2π.

Fig. 6
Fig. 6

Effects of the amplitude Ψ0 of the incident uniform plane wave. BSW of an infinite cylinder of irregular cross section: ϕ0 = 0, k1s = 0.53π, ∊L = 1.2, β = 0.08, M = 80. (a) Ψ0 = 0.5 (V/m), (b) Ψ0 = 0.75 (V/m), (c) Ψ0 = 1.0 (V/m), (d) Ψ0 = 1.5 (V/m).

Tables (1)

Tables Icon

Table 1 Residual Errors Ξ(k) [Eq. (14)] Concerning the Simulations of the Iterative Process for the BSW Computation, Assuming Different Values of the Ratio between the Wavelength and the Linear Dimensions of the Cross Section of a Nonlinear Square Cylinder (Fig. 6)

Equations (18)

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P ( r , t ) = χ 1 · E ( r , t ) + χ 2 : E ( r , t ) E ( r , t ) + χ 3 : E ( r , t ) E ( r , t ) + . . . ,
NL ( r , t ) = 0 [ L ( r ) + β | E ( r , t ) | 2 ] ,
NL ( r , t ) = 0 ( L ( r ) + G { E ( r , t ) } ) ,
σ 2 D ( ϕ ) = lim ρ [ 2 π ρ | E s ( ρ , ϕ ) | 2 | E i ( ρ , ϕ ) | 2 ] ,
NL ( x , y ) = 0 ( L ( x , y ) + g { Ψ t ( x , y , t ) } ) ,
( t 2 + k 2 m ) Ψ z s ( m ) ( x , y ) = k 2 m [ L ( x , y ) 1 ] Ψ z t ( m ) ( x , y ) k 2 m Π z ( m ) ( x , y ) ,
Π z ( m ) ( x , y ) = p = q = τ m p q U p ( x , y ) Ψ z t ( q ) ( x , y ) ,
E i ( x , y , t ) = Ψ i ( x , y ) exp ( j ω 0 t ) = Ψ 0 exp [ j ( k 1 y + ω 0 t + ϕ 0 ] ,
σ 2 D ( ϕ ) = lim ρ [ 2 π ρ | Ψ s ( 1 ) ( ρ , ϕ ) | 2 Ψ 0 2 ] .
σ 2 D ( ϕ ) σ 2 D NL L ( ϕ ) = lim ρ [ 2 π ρ | Ψ s ( DWBA ) ( ρ , ϕ ) | Ψ 0 2 2 ] = lim ρ [ 2 π ρ | j ( k 1 2 / 4 ) S { [ L ( x , y ) 1 ] Ψ z t ( 1 ) ( x , y ) + z ( 1 ) NL L ( x , y ) } H 0 ( 2 ) ( k 1 d ) d x d y | Ψ 0 2 2 ] ,
σ 2 D ( k + 1 ) NL ( ϕ ) = lim ρ [ 2 π ρ | Ψ ( k + 1 ) s ( DWBA ) ( ρ , ϕ ) | 2 Ψ 0 2 ] = lim ρ [ 2 π ρ | j ( k 1 2 / 4 ) S { [ L ( x , y ) 1 ] Ψ z ( k ) t ( 1 ) ( x , y ) + Π z ( k ) ( 1 ) NL L ( x , y ) } H 0 ( 2 ) ( k 1 d ) d x d y | Ψ 0 2 2 ] ,
[ G ] M × M Ψ _ M × 1 L = Ψ _ M × 1 i ,
σ 2 D ( k + 1 ) NL ( ϕ ) = ( π k 1 b / | Ψ 0 | 2 ) | p = 1 p { [ L ( m ) 1 ] × Ψ z ( k ) t ( 1 ) ( x m , y m ) + Π z ( k ) ( 1 ) N L ( x m , y m ) } × J 1 ( k 1 b ) exp { j k 1 ( x m cos ϕ + y m sin ϕ ) | 2 ,
Ξ ( k + 1 ) = 1 M 2 m = 1 M Ψ z ( k + 1 ) t ( 1 ) ( x m , y m ) Ψ i ( x m , y m ) + j ( k 1 2 / 4 ) S [ ( L ( x , y ) 1 ) Ψ z ( k + 1 ) t ( 1 ) ( x , y ) + Π z ( k + 1 ) ( 1 ) N L ( x , y ) ] H 0 ( 2 ) ( k 1 d m ) d x d y ,
g m n = j ( k 1 2 / 4 ) S n [ ( L ( x , y ) 1 ) H 0 2 ( k 1 d m ) d x d y , m = 1 , . . . , M , n = 1 , . . . , M ,
g m n = ( j / 2 ) [ L ( m ) 1 ] [ π k 1 b H 1 ( 2 ) ( k 1 d m n ) 2 j ] m = n ,
g m n = ( j / 2 ) [ L ( m ) 1 ] π k 1 b J 1 ( k 1 b ) H 0 ( 2 ) ( k 1 d m n ) m n .
k p l = 2 ,

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