Abstract

The problem of scattering from a perfectly conducting cylindrical reflector is examined with the method of the modified theory of physical optics. In this technique the physical optics currents are modified by using a variable unit vector on the scatterer’s surface. These current components are obtained for the reflector, which is fed by an offset electric line source. The scattering integral is expressed by using these currents and evaluated asymptotically with the stationary phase method. The results are compared numerically by using physical optics theory, geometrical optics diffraction theory, and the exact solution of the Helmholtz equation. It is found that the modified theory of physical optics scattering field equations agrees with the geometrical optics diffraction theory and the exact solution of the Helmholtz equation.

© 2007 Optical Society of America

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References

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  1. M. Martinez-Burdalo, A. Martin, and R. Villar, "Uniform PO and PTD solution for calculating plane wave backscattering from a finite cylindrical shell of arbitrary cross section," IEEE Trans. Antennas Propag. 41, 1336-1339 (1993).
    [Crossref]
  2. G. Hyde, "Studies of the focal region of a spherical reflector: stationary phase evaluation," IEEE Trans. Antennas Propag. AP-16, 646-656 (1968).
    [Crossref]
  3. W. L. Stutzman and G. A. Thiele, Antenna Theory and Design (Wiley, 1988).
  4. G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (IEE, Peregrinus, 1976).
  5. P. Ya. Ufimtsev, "Method of edge waves in the physical theory of diffraction," Air Force System Command, Foreign Tech. Div. Document ID FTD-HC-23-259-71 (1971).
  6. S. W. Lee, "Comparison of uniform asymptotic theory and Ufimtsev's theory of electromagnetic edge diffraction," IEEE Trans. Antennas Propag. AP-25, 162-170 (1977).
  7. Y. Z. Umul, E. Yengel, and A. Aydin, "Comparison of physical optics integral and exact solution for cylinder problem," in Proceedings of ELECO 2003, Third International Conference on Electrical and Electronics Engineering, December 3-7, 2003, Bursa, Turkey, pp. 245-248, Y. Z. Umul, E. Yengel, and A. Aydin, "Comparison of physical optics integral and exact solution for cylinder problem," in Proceeding of ELECO 2003 Third International Conference on Electrical and Electronics Engineering, Bursa, Turkey,31-34.
  8. Y. Z. Umul, "Modified theory of physical optics," Opt. Express 12, 4959-4972 (2004).
    [Crossref] [PubMed]
  9. Y. Z. Umul, "Modified theory of physical optics approach to wedge diffraction problems," Opt. Express 13, 216-224 (2005).
    [Crossref] [PubMed]
  10. U. Yalçin, "Asymptotic evaluation of edge diffracted electromagnetic fields from a concave conducting surface by physical optics approach," presented at the Union Radio-Scientifique Internationale-Turkey 2002 First National Conference, Istanbul, Turkey, September 18-20, 2002 (in national language).
  11. M. Idemen and G. Uzgoren, "Some diffraction coefficients related to diffractions at the edges of curved reflectors," in Proceedings of the 12th European Microwave Conference, September 13-17, 1982, pp. 225-299.
  12. M. Idemen, "Diffraction of an obliquely incident high-frequency wave by a cylindrically curved sheet," IEEE Trans. Antennas Propag. AP-34, 181-187 (1986).
    [Crossref]

2005 (1)

2004 (1)

1993 (1)

M. Martinez-Burdalo, A. Martin, and R. Villar, "Uniform PO and PTD solution for calculating plane wave backscattering from a finite cylindrical shell of arbitrary cross section," IEEE Trans. Antennas Propag. 41, 1336-1339 (1993).
[Crossref]

1986 (1)

M. Idemen, "Diffraction of an obliquely incident high-frequency wave by a cylindrically curved sheet," IEEE Trans. Antennas Propag. AP-34, 181-187 (1986).
[Crossref]

1977 (1)

S. W. Lee, "Comparison of uniform asymptotic theory and Ufimtsev's theory of electromagnetic edge diffraction," IEEE Trans. Antennas Propag. AP-25, 162-170 (1977).

1968 (1)

G. Hyde, "Studies of the focal region of a spherical reflector: stationary phase evaluation," IEEE Trans. Antennas Propag. AP-16, 646-656 (1968).
[Crossref]

Aydin, A.

Y. Z. Umul, E. Yengel, and A. Aydin, "Comparison of physical optics integral and exact solution for cylinder problem," in Proceedings of ELECO 2003, Third International Conference on Electrical and Electronics Engineering, December 3-7, 2003, Bursa, Turkey, pp. 245-248, Y. Z. Umul, E. Yengel, and A. Aydin, "Comparison of physical optics integral and exact solution for cylinder problem," in Proceeding of ELECO 2003 Third International Conference on Electrical and Electronics Engineering, Bursa, Turkey,31-34.

Y. Z. Umul, E. Yengel, and A. Aydin, "Comparison of physical optics integral and exact solution for cylinder problem," in Proceedings of ELECO 2003, Third International Conference on Electrical and Electronics Engineering, December 3-7, 2003, Bursa, Turkey, pp. 245-248, Y. Z. Umul, E. Yengel, and A. Aydin, "Comparison of physical optics integral and exact solution for cylinder problem," in Proceeding of ELECO 2003 Third International Conference on Electrical and Electronics Engineering, Bursa, Turkey,31-34.

Hyde, G.

G. Hyde, "Studies of the focal region of a spherical reflector: stationary phase evaluation," IEEE Trans. Antennas Propag. AP-16, 646-656 (1968).
[Crossref]

Idemen, M.

M. Idemen, "Diffraction of an obliquely incident high-frequency wave by a cylindrically curved sheet," IEEE Trans. Antennas Propag. AP-34, 181-187 (1986).
[Crossref]

M. Idemen and G. Uzgoren, "Some diffraction coefficients related to diffractions at the edges of curved reflectors," in Proceedings of the 12th European Microwave Conference, September 13-17, 1982, pp. 225-299.

James, G. L.

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (IEE, Peregrinus, 1976).

Lee, S. W.

S. W. Lee, "Comparison of uniform asymptotic theory and Ufimtsev's theory of electromagnetic edge diffraction," IEEE Trans. Antennas Propag. AP-25, 162-170 (1977).

Martin, A.

M. Martinez-Burdalo, A. Martin, and R. Villar, "Uniform PO and PTD solution for calculating plane wave backscattering from a finite cylindrical shell of arbitrary cross section," IEEE Trans. Antennas Propag. 41, 1336-1339 (1993).
[Crossref]

Martinez-Burdalo, M.

M. Martinez-Burdalo, A. Martin, and R. Villar, "Uniform PO and PTD solution for calculating plane wave backscattering from a finite cylindrical shell of arbitrary cross section," IEEE Trans. Antennas Propag. 41, 1336-1339 (1993).
[Crossref]

Stutzman, W. L.

W. L. Stutzman and G. A. Thiele, Antenna Theory and Design (Wiley, 1988).

Thiele, G. A.

W. L. Stutzman and G. A. Thiele, Antenna Theory and Design (Wiley, 1988).

Ufimtsev, P. Ya.

P. Ya. Ufimtsev, "Method of edge waves in the physical theory of diffraction," Air Force System Command, Foreign Tech. Div. Document ID FTD-HC-23-259-71 (1971).

Umul, Y. Z.

Y. Z. Umul, "Modified theory of physical optics approach to wedge diffraction problems," Opt. Express 13, 216-224 (2005).
[Crossref] [PubMed]

Y. Z. Umul, "Modified theory of physical optics," Opt. Express 12, 4959-4972 (2004).
[Crossref] [PubMed]

Y. Z. Umul, E. Yengel, and A. Aydin, "Comparison of physical optics integral and exact solution for cylinder problem," in Proceedings of ELECO 2003, Third International Conference on Electrical and Electronics Engineering, December 3-7, 2003, Bursa, Turkey, pp. 245-248, Y. Z. Umul, E. Yengel, and A. Aydin, "Comparison of physical optics integral and exact solution for cylinder problem," in Proceeding of ELECO 2003 Third International Conference on Electrical and Electronics Engineering, Bursa, Turkey,31-34.

Y. Z. Umul, E. Yengel, and A. Aydin, "Comparison of physical optics integral and exact solution for cylinder problem," in Proceedings of ELECO 2003, Third International Conference on Electrical and Electronics Engineering, December 3-7, 2003, Bursa, Turkey, pp. 245-248, Y. Z. Umul, E. Yengel, and A. Aydin, "Comparison of physical optics integral and exact solution for cylinder problem," in Proceeding of ELECO 2003 Third International Conference on Electrical and Electronics Engineering, Bursa, Turkey,31-34.

Uzgoren, G.

M. Idemen and G. Uzgoren, "Some diffraction coefficients related to diffractions at the edges of curved reflectors," in Proceedings of the 12th European Microwave Conference, September 13-17, 1982, pp. 225-299.

Villar, R.

M. Martinez-Burdalo, A. Martin, and R. Villar, "Uniform PO and PTD solution for calculating plane wave backscattering from a finite cylindrical shell of arbitrary cross section," IEEE Trans. Antennas Propag. 41, 1336-1339 (1993).
[Crossref]

Yalçin, U.

U. Yalçin, "Asymptotic evaluation of edge diffracted electromagnetic fields from a concave conducting surface by physical optics approach," presented at the Union Radio-Scientifique Internationale-Turkey 2002 First National Conference, Istanbul, Turkey, September 18-20, 2002 (in national language).

Yengel, E.

Y. Z. Umul, E. Yengel, and A. Aydin, "Comparison of physical optics integral and exact solution for cylinder problem," in Proceedings of ELECO 2003, Third International Conference on Electrical and Electronics Engineering, December 3-7, 2003, Bursa, Turkey, pp. 245-248, Y. Z. Umul, E. Yengel, and A. Aydin, "Comparison of physical optics integral and exact solution for cylinder problem," in Proceeding of ELECO 2003 Third International Conference on Electrical and Electronics Engineering, Bursa, Turkey,31-34.

Y. Z. Umul, E. Yengel, and A. Aydin, "Comparison of physical optics integral and exact solution for cylinder problem," in Proceedings of ELECO 2003, Third International Conference on Electrical and Electronics Engineering, December 3-7, 2003, Bursa, Turkey, pp. 245-248, Y. Z. Umul, E. Yengel, and A. Aydin, "Comparison of physical optics integral and exact solution for cylinder problem," in Proceeding of ELECO 2003 Third International Conference on Electrical and Electronics Engineering, Bursa, Turkey,31-34.

IEEE Trans. Antennas Propag. (4)

M. Martinez-Burdalo, A. Martin, and R. Villar, "Uniform PO and PTD solution for calculating plane wave backscattering from a finite cylindrical shell of arbitrary cross section," IEEE Trans. Antennas Propag. 41, 1336-1339 (1993).
[Crossref]

G. Hyde, "Studies of the focal region of a spherical reflector: stationary phase evaluation," IEEE Trans. Antennas Propag. AP-16, 646-656 (1968).
[Crossref]

S. W. Lee, "Comparison of uniform asymptotic theory and Ufimtsev's theory of electromagnetic edge diffraction," IEEE Trans. Antennas Propag. AP-25, 162-170 (1977).

M. Idemen, "Diffraction of an obliquely incident high-frequency wave by a cylindrically curved sheet," IEEE Trans. Antennas Propag. AP-34, 181-187 (1986).
[Crossref]

Opt. Express (2)

Other (6)

U. Yalçin, "Asymptotic evaluation of edge diffracted electromagnetic fields from a concave conducting surface by physical optics approach," presented at the Union Radio-Scientifique Internationale-Turkey 2002 First National Conference, Istanbul, Turkey, September 18-20, 2002 (in national language).

M. Idemen and G. Uzgoren, "Some diffraction coefficients related to diffractions at the edges of curved reflectors," in Proceedings of the 12th European Microwave Conference, September 13-17, 1982, pp. 225-299.

W. L. Stutzman and G. A. Thiele, Antenna Theory and Design (Wiley, 1988).

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (IEE, Peregrinus, 1976).

P. Ya. Ufimtsev, "Method of edge waves in the physical theory of diffraction," Air Force System Command, Foreign Tech. Div. Document ID FTD-HC-23-259-71 (1971).

Y. Z. Umul, E. Yengel, and A. Aydin, "Comparison of physical optics integral and exact solution for cylinder problem," in Proceedings of ELECO 2003, Third International Conference on Electrical and Electronics Engineering, December 3-7, 2003, Bursa, Turkey, pp. 245-248, Y. Z. Umul, E. Yengel, and A. Aydin, "Comparison of physical optics integral and exact solution for cylinder problem," in Proceeding of ELECO 2003 Third International Conference on Electrical and Electronics Engineering, Bursa, Turkey,31-34.

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Figures (4)

Fig. 1
Fig. 1

Scattering geometry from the PEC and aperture surfaces.

Fig. 2
Fig. 2

x y -plane cross section of the source and PEC cylinder surface.

Fig. 3
Fig. 3

x y -plane cross section of the image source and the PEC cylinder and S 2 aperture surfaces.

Fig. 4
Fig. 4

RCS for the MTPO, PO, and exact solution for ϕ 0 = π 4 .

Equations (67)

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J e s = ( n 1 × H t ) S 1 ,
J e s = ( n 2 × H i ) S 2 ,
J m s = ( n 2 × E i ) S 2 ,
n 1 = sin ( u α ) t cos ( u α ) n ,
n 2 = cos ( v + α ) t + sin ( v + α ) n ,
E s = E i s + E r s ,
E i s = j ω μ 0 4 π S 2 n 2 × H i S 2 e j k R 2 R 2 d S + 1 4 π S 2 × ( n 2 × E i ) S 2 e j k R 2 R 2 d S ,
E r s = j ω μ 0 4 π S 1 ( n 1 × H t ) S 1 e j k R 1 R 1 d S
ρ = a ,
E i = ω μ 0 I 4 H 0 ( 2 ) ( k ρ 1 ) e z ,
H i = H i ( sin α e ρ + cos α e ϕ ) ,
ρ 1 = [ ρ 2 + ρ 0 2 2 ρ ρ 0 cos ( ϕ ϕ s ) ] 1 2
J M T P O = e z 2 H i cos u = e z 2 H i cos ( α + β 2 )
J PO = e z 2 H i cos α .
d S = a d ϕ d z .
E r s = e z j k 2 Z 0 I a 8 π ϕ = ϕ 0 ϕ 0 z = cos ( α + β 2 ) H 0 ( 2 ) ( k ρ 1 ) e j k R 1 R 1 d ϕ d z
E r s = e z k 2 Z 0 I a 8 ϕ = ϕ 0 ϕ 0 cos ( α + β 2 ) H 0 ( 2 ) ( k ρ 1 ) H 0 ( 2 ) ( k ρ 2 ) d ϕ
J e s M T P O = e z H i cos v ,
J m s M T P O = E i [ e ρ sin ( v + α ) e ϕ cos ( v + α ) ] ,
E i s = e z k 2 Z 0 I a 8 ϕ = ϕ 0 ϕ 0 sin ( α β 2 ) H 0 ( 2 ) ( k ρ 1 ) H 0 ( 2 ) ( k ρ 2 ) d ϕ .
E s = E i s + E r s ,
E s = e z k 2 Z 0 I a 8 [ ϕ = ϕ 0 ϕ 0 cos ( α + β 2 ) H 0 ( 2 ) ( k ρ 1 ) H 0 ( 2 ) ( k ρ 2 ) d ϕ + ϕ = ϕ 0 ϕ 0 sin ( α β 2 ) H 0 ( 2 ) ( k ρ 1 ) H 0 ( 2 ) ( k ρ 2 ) d ϕ ]
H 0 ( 2 ) ( k υ ) 2 π e [ j k υ + j ( π 4 ) ] k υ ,
E s = e z j k Z 0 I a 4 π [ ϕ = ϕ 0 ϕ 0 cos ( α + β 2 ) e j k ( ρ 1 + ρ 2 ) ρ 1 ρ 2 d ϕ + ϕ = ϕ 0 ϕ 0 sin ( α β 2 ) e j k ( ρ 1 + ρ 2 ) ρ 1 ρ 2 d ϕ ]
ψ ( ϕ ) = ρ 1 + ρ 2 = ρ 0 cos σ + a cos α + ρ cos γ + a cos β ,
ψ ( ϕ ) = ρ 0 sin σ d σ d ϕ a sin α d α d ϕ ρ sin γ d γ d ϕ a sin β d β d ϕ ,
σ = π ϕ s ± ϕ α ,
γ = π ± ϕ ϕ ± β ,
d σ d ϕ = ± 1 d α d ϕ , d γ d ϕ = 1 ± d β d ϕ .
ρ 0 sin α = a sin σ , ρ sin β = a sin γ .
ψ ( ϕ ) = a sin α ± a sin β = 0 ,
α s = β s ,
ψ ( ϕ ) = a cos α d α d ϕ a cos β d β d ϕ .
d α d ϕ = 1 + a ρ 1 cos α ,
d β d ϕ = 1 a ρ 2 cos β ,
ψ ( ϕ ) = a cos α ( ρ 1 + a cos α ρ 1 ) a cos β ( ρ 2 a cos β ρ 2 ) .
I 0 f ( ϕ s ) e j k ψ ( ϕ s ) 2 π j k ψ ( ϕ s )
ρ 1 s = l 0 = ( ρ 0 cos σ + a cos α s ) ,
ρ 2 s = l 1 = ( ρ cos γ + a cos α s ) ,
f ( ϕ s ) = j k Z 0 I a cos α s 4 π l 0 l 1 ,
ψ ( ϕ s ) = l 0 + l 1 ,
ψ ( ϕ s ) = a cos α s [ a cos α s ( l 0 + l 1 ) 2 l 0 l 1 l 0 l 1 ] ,
E r = e z E i s D ( α s ) e j k l 1
E i s = k Z 0 I 8 π e [ j k l 0 + j ( π 4 ) ] k l 0
D ( α s ) = l 0 α cos α s a cos α s ( l 0 + l 1 ) 2 l 0 l 1
α = α e , β = β e , ϕ e = ϕ 0 ,
l 0 = ( ρ 0 cos σ + a cos α e ) ,
l 1 = ( ρ cos γ + a cos β e ) .
E d e z 1 j k f ( ϕ 0 ) ψ ( ϕ 0 ) e j k ψ ( ϕ 0 ) ,
f ( ϕ 0 ) = j k Z 0 I a 4 π cos ( α e + β e 2 ) l 0 l 1 ,
ψ ( ϕ 0 ) = l 0 + l 1 ,
ψ ( ϕ ) = a sin α e a sin β e
E r d = e z Z 0 I 4 π cos ( α e + β e 2 ) sin α e sin β e e j k ( l 0 + l 1 ) l 0 l 1
f ( ϕ 0 ) = j k Z 0 I a 4 π sin ( α e β e 2 ) l 0 l 1 ,
ψ ( ϕ 0 ) = l 0 + l 1 ,
ψ ( ϕ ) = a sin α e + a sin β e
E id = e z Z 0 I 4 π sin ( α e β e 2 ) sin α e sin β e e j k ( I 0 + I 1 ) l 0 l 1
E d = E r d + E i d = e z Z 0 I 4 π cos ( α e + β e 2 ) sin ( α e β e 2 ) sin α e sin β e e j k ( I 0 + I 1 ) l 0 l 1 ,
D e d = e j ( π 4 ) 2 π cos ( α e + β e 2 ) sin ( α e β e 2 ) sin α e sin β e .
E d = e z E i ϕ 0 D ed e j k l 1 k l 1 ,
E i ϕ 0 = k Z 0 I 8 π e j k l 0 + j ( π 4 ) k l 0 ,
E r = e z E i s R ρ 1 ρ 1 + l 1 e j k l 1 ,
1 ρ 1 = 1 l 0 2 a cos α
E r = e z E i s l 0 a cos α s a cos α s ( l 0 + l 1 ) 2 l 0 l 1 e j k l 1
T e = e j ( π 4 ) 2 π 1 sin ϕ 0 1 + sin ϕ sin ϕ sin ϕ 0 ,
α e = ϕ 0 , β e = ϕ
D PO = e j ( π 4 ) 2 π cos α e sin α e sin β e

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