Abstract

The backscatter cross sections for randomly oriented metallic flakes are derived using the full wave approach. The metallic flakes are characterized by their surface height spectral density function. Both specular point and Bragg scattering at optical frequencies are accounted for in a self-consistent manner. It is shown that the average normalized backscatter cross sections (per unit volume) for the randomly oriented metallic flakes are larger than that of metallic spheres.

© 1983 Optical Society of America

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References

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  1. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (McMillan, New York, 1963).
  2. S. O. Rice, Commun. Pure Appl. Math 4, 351 (1951).
    [CrossRef]
  3. D. E. Barrick, “Rough Surfaces” in Radar Cross Section Handbook (Plenum, New York, 1970), Chap. 9.
  4. A. Ishimaru, Wave Propagation and Scattering in Random Media in Multiple Scattering, Turbulence, Rough Surfaces and Remote Sensing, Vol. 2 (Academic, New York, 1978).
  5. G. S. Brown, IEEE Trans. Antennas Propag. AP-26, 472 (1978).
    [CrossRef]
  6. E. Bahar, D. E. Barrick, M. A. Fitzwater, IEEE Trans. Antennas Propag. AP-31, 698 (1983).
    [CrossRef]
  7. E. Bahar, Radio Sci. 16, 331 (1981a).
    [CrossRef]
  8. E. Bahar, Radio Sci. 16, 1327 (1981b).
    [CrossRef]
  9. B. G. Smith, IEEE Trans. Antennas Propag. AP-15, 668 (1967).
    [CrossRef]
  10. M. I. Sancer, IEEE Trans. Antennas Propag. AP-17, 577 (1969).
    [CrossRef]
  11. T. Hagfors, J. Geophys. Res. 71, 279 (1966).
  12. G. L. Tyler, Radio Sci. 11, 83 (1976).
    [CrossRef]
  13. H. Ehrenreich, IEEE Spectrum 2, 162 (1965).
    [CrossRef]
  14. J. W. Wright, IEEE Trans. Antennas Propag. AP-14, 749 (1966).
    [CrossRef]
  15. J. W. Wright, IEEE Trans. Antennas Propag. AP-16, 217 (1968).
    [CrossRef]
  16. G. R. Valenzuela, Radio Sci. 3, 1051 (1968).

1983

E. Bahar, D. E. Barrick, M. A. Fitzwater, IEEE Trans. Antennas Propag. AP-31, 698 (1983).
[CrossRef]

1981

E. Bahar, Radio Sci. 16, 331 (1981a).
[CrossRef]

E. Bahar, Radio Sci. 16, 1327 (1981b).
[CrossRef]

1978

G. S. Brown, IEEE Trans. Antennas Propag. AP-26, 472 (1978).
[CrossRef]

1976

G. L. Tyler, Radio Sci. 11, 83 (1976).
[CrossRef]

1969

M. I. Sancer, IEEE Trans. Antennas Propag. AP-17, 577 (1969).
[CrossRef]

1968

J. W. Wright, IEEE Trans. Antennas Propag. AP-16, 217 (1968).
[CrossRef]

G. R. Valenzuela, Radio Sci. 3, 1051 (1968).

1967

B. G. Smith, IEEE Trans. Antennas Propag. AP-15, 668 (1967).
[CrossRef]

1966

T. Hagfors, J. Geophys. Res. 71, 279 (1966).

J. W. Wright, IEEE Trans. Antennas Propag. AP-14, 749 (1966).
[CrossRef]

1965

H. Ehrenreich, IEEE Spectrum 2, 162 (1965).
[CrossRef]

1951

S. O. Rice, Commun. Pure Appl. Math 4, 351 (1951).
[CrossRef]

Bahar, E.

E. Bahar, D. E. Barrick, M. A. Fitzwater, IEEE Trans. Antennas Propag. AP-31, 698 (1983).
[CrossRef]

E. Bahar, Radio Sci. 16, 1327 (1981b).
[CrossRef]

E. Bahar, Radio Sci. 16, 331 (1981a).
[CrossRef]

Barrick, D. E.

E. Bahar, D. E. Barrick, M. A. Fitzwater, IEEE Trans. Antennas Propag. AP-31, 698 (1983).
[CrossRef]

D. E. Barrick, “Rough Surfaces” in Radar Cross Section Handbook (Plenum, New York, 1970), Chap. 9.

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (McMillan, New York, 1963).

Brown, G. S.

G. S. Brown, IEEE Trans. Antennas Propag. AP-26, 472 (1978).
[CrossRef]

Ehrenreich, H.

H. Ehrenreich, IEEE Spectrum 2, 162 (1965).
[CrossRef]

Fitzwater, M. A.

E. Bahar, D. E. Barrick, M. A. Fitzwater, IEEE Trans. Antennas Propag. AP-31, 698 (1983).
[CrossRef]

Hagfors, T.

T. Hagfors, J. Geophys. Res. 71, 279 (1966).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media in Multiple Scattering, Turbulence, Rough Surfaces and Remote Sensing, Vol. 2 (Academic, New York, 1978).

Rice, S. O.

S. O. Rice, Commun. Pure Appl. Math 4, 351 (1951).
[CrossRef]

Sancer, M. I.

M. I. Sancer, IEEE Trans. Antennas Propag. AP-17, 577 (1969).
[CrossRef]

Smith, B. G.

B. G. Smith, IEEE Trans. Antennas Propag. AP-15, 668 (1967).
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (McMillan, New York, 1963).

Tyler, G. L.

G. L. Tyler, Radio Sci. 11, 83 (1976).
[CrossRef]

Valenzuela, G. R.

G. R. Valenzuela, Radio Sci. 3, 1051 (1968).

Wright, J. W.

J. W. Wright, IEEE Trans. Antennas Propag. AP-16, 217 (1968).
[CrossRef]

J. W. Wright, IEEE Trans. Antennas Propag. AP-14, 749 (1966).
[CrossRef]

Commun. Pure Appl. Math

S. O. Rice, Commun. Pure Appl. Math 4, 351 (1951).
[CrossRef]

IEEE Spectrum

H. Ehrenreich, IEEE Spectrum 2, 162 (1965).
[CrossRef]

IEEE Trans. Antennas Propag.

J. W. Wright, IEEE Trans. Antennas Propag. AP-14, 749 (1966).
[CrossRef]

J. W. Wright, IEEE Trans. Antennas Propag. AP-16, 217 (1968).
[CrossRef]

G. S. Brown, IEEE Trans. Antennas Propag. AP-26, 472 (1978).
[CrossRef]

E. Bahar, D. E. Barrick, M. A. Fitzwater, IEEE Trans. Antennas Propag. AP-31, 698 (1983).
[CrossRef]

B. G. Smith, IEEE Trans. Antennas Propag. AP-15, 668 (1967).
[CrossRef]

M. I. Sancer, IEEE Trans. Antennas Propag. AP-17, 577 (1969).
[CrossRef]

J. Geophys. Res.

T. Hagfors, J. Geophys. Res. 71, 279 (1966).

Radio Sci.

G. L. Tyler, Radio Sci. 11, 83 (1976).
[CrossRef]

E. Bahar, Radio Sci. 16, 331 (1981a).
[CrossRef]

E. Bahar, Radio Sci. 16, 1327 (1981b).
[CrossRef]

G. R. Valenzuela, Radio Sci. 3, 1051 (1968).

Other

D. E. Barrick, “Rough Surfaces” in Radar Cross Section Handbook (Plenum, New York, 1970), Chap. 9.

A. Ishimaru, Wave Propagation and Scattering in Random Media in Multiple Scattering, Turbulence, Rough Surfaces and Remote Sensing, Vol. 2 (Academic, New York, 1978).

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (McMillan, New York, 1963).

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Figures (8)

Fig. 1
Fig. 1

Plane of incidence, scattering plane, and reference (x,z) plane.

Fig. 2
Fig. 2

Local plane of incidence and scatter and local coordinate system with unit vectors n ̅ 1, n ̅ 2, n ̅ 3.

Fig. 3
Fig. 3

Randomly oriented flake.

Fig. 4
Fig. 4

σHHθ=〈σVVθ for the composite surface h as a function of θF.

Fig. 5
Fig. 5

σHHθ=〈σVVθ for the large-scale filtered surface hl as a function of θF.

Fig. 6
Fig. 6

σHHθ = 〈σVVθ for the small-scale surface hs as a function of θF.

Fig. 7
Fig. 7

σVHθ = 〈σHVθ for the composite surface h as a function of θF.

Fig. 8
Fig. 8

σVHθ = 〈σHVθ for the small-scale surface hs as a function of θF.

Equations (53)

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r ̅ s = r ̅ l ( x , h l , z ) + n ̅ h ̅ s .
W ( υ x , υ z ) = 1 π 2 h h exp ( i υ x x d + i υ z z d ) d x d d z d ,
r ̅ d = ( x x ) a ̅ x + ( z z ) a ̅ z = x d a ̅ x + z d a ̅ z .
k F < k = ( υ x 2 + υ z 2 ) 1 / 2 < k d ,
k d < k = ( υ x 2 + υ z 2 ) 1 / 2 < k c ,
σ P Q = σ P Q l + σ P Q s ,
σ P Q l = | χ s ( v ̅ n ̅ s ) | 2 σ P Q
χ s ( v ̅ n ̅ s ) = χ s ( υ ) = exp i υ h ̅ s ,
v ̅ = k ¯ f k ¯ i = k 0 ( n ̅ f n ̅ i ) υ = | v ̅ | .
n ̅ = f / | f | = ( h x a ̅ x + a ̅ y h z a ̅ z ) / ( h x 2 + 1 + h z 2 ) 1 / 2 sin γ cos δ a ̅ x + cos γ a ̅ y + sin γ sin δ a ̅ z ,
f = y h ( x , z ) , h x = h / x , h z = h / z ,
n ̅ s = v ̅ / υ .
σ P Q = 4 π υ y 2 k 0 2 [ | D P Q n ̅ a ̅ y | 2 P 2 ( n ̅ f , n ̅ i n ̅ ) p ( n ̅ ) ] n ̅ s ,
σ P Q s = m = 1 σ P Q s m ,
σ P Q s m = 4 π k 0 2 | D P Q | 2 P 2 ( n ̅ f , n ̅ i | n ̅ ) n ̅ a ̅ y × exp ( υ y ̅ 2 h ̅ s 2 ) [ υ y ̅ 2 ] 2 m W m ( υ x ̅ , υ z ̅ ) m ! p ( h x , h z ) d h x d h z .
v ̅ = υ x ̅ n ̅ 1 + υ y ̅ n ̅ 2 + υ z ̅ n ̅ 3 ,
n ̅ 1 = ( n ̅ × a ̅ z ) / | n ̅ × a ̅ z | , n ̅ 2 = n ̅ , n ̅ 3 = n ̅ 1 × n ̅ .
W m ( υ x ̅ , υ z ̅ ) 2 2 m = 1 ( 2 π ) 2 h ̅ s h ̅ s m exp ( i υ x ̅ x ̅ d + i υ z ̅ z ̅ d ) d x ̅ d d z ̅ d = 1 2 2 m W m 1 ( υ x ̅ , υ z ̅ ) W 1 ( υ x ̅ υ x ̅ , υ z ̅ υ z ̅ ) d υ x ̅ d υ z ̅ = 1 2 2 m W m 1 ( υ x ̅ , υ z ̅ ) W 1 ( υ x ̅ , υ z ̅ ) .
n ̅ f = n ̅ i = a ̅ y .
a ̅ y = sin θ F cos ϕ F a ̅ x + cos θ F a ̅ y + sin θ F sin ϕ F a ̅ z .
a ̅ z = ( n ̅ i × a ̅ y ) / | n ̅ i × a ̅ y | and a ̅ x = a ̅ y × a ̅ z .
n ̅ i = ( n ̅ i a ̅ x ) a ̅ x + ( n ̅ i a ̅ y ) a ̅ y = sin θ 0 i a ̅ x cos θ 0 i a ̅ y ,
cos θ 0 i = n ̅ i a ̅ y = cos θ F .
cos ψ F i = a ̅ z ( n ̅ i × a ̅ y ) | n ̅ i × a ̅ y | = [ a ̅ z a ̅ y a ̅ y ] | n ̅ i × a ̅ y | = a ̅ x a ̅ y | n ̅ i × a ̅ y | = cos ϕ F ,
sin ψ F i = a ̅ z × ( n ̅ i × a ̅ y ) n ̅ i | n ̅ i × a ̅ y | = sin ϕ F .
ψ F f = ψ F i .
T F i = [ cos ψ F i sin ψ F i sin ψ F i cos ψ F i ] .
T F f = [ cos ψ F f sin ψ F f sin ψ F f cos ψ F f ] .
T F f = T F i .
D = C 0 i n T f F T i ,
C 0 i n = n ̅ i n ̅ = cos γ cos θ F sin γ sin θ F cos δ .
D F = T F f DT F i = C 0 i n T F f T f FT i T F i C 0 i n T T f FT T i ,
T T i T i T F i = [ cos ( ψ i + ϕ F ) sin ( ψ i + ϕ F ) sin ( ψ i + ϕ F ) cos ( ψ i + ϕ F ) ] [ C T S T S T C T ] .
cos ψ i = [ cos γ sin θ F + sin γ cos θ F cos δ ] / S 0 i n ,
sin ψ i = sin γ sin δ / S 0 i n ,
S 0 i n = [ 1 ( C 0 i n ) 2 ] 1 / 2 .
D F = C 0 i n [ C T 2 F V V S T 2 F H H C T S T ( F V V + F H H ) C T S T ( F V V + F H H ) C T 2 F H H S T 2 F V V ] .
σ P Q ¯ = 0 π / 2 0 2 π σ P Q p ( θ F , ϕ F ) d ϕ F d θ F 0 π / 2 σ P Q o p ( θ F ) d θ F ,
p ( θ F , ϕ F ) = sin θ F 2 π p ( θ F ) 2 π .
| D F P Q ¯ | 2 1 2 π 0 2 π | D F P Q | 2 d ϕ F = ( C 0 i n ) 2 2 π 0 2 π C T 2 S T 2 | F V V + F H H | 2 d ϕ F = ( C 0 i n ) 2 8 | F V V + F H H | 2 ,
| D F P P ¯ | 2 1 2 π 0 2 π | D F P P | 2 d ϕ F = ( C 0 i n ) 2 2 π 0 2 π [ C T 4 | F V V | 2 + S T 4 | F H H | 2 2 C T 2 S T 2 Re ( F V V F H H * ) ] d ϕ F = ( C 0 i n ) 2 [ 3 | F V V | 2 + 3 | F H H | 2 2 Re ( F V V F H H * ) ] = ( C 0 i n ) 2 ( 4 | F V V | 2 + 4 | F H H | 2 | F V V + F H H | 2 ) ,
W ( υ x ̅ , υ z ̅ ) = { 2 π B / k 4 , k F k k c 0 , k > k c and k < k F ,
B = 0.016 k 2 = υ x ̅ 2 + υ z ̅ 2 ( cm ) 2 k F = 2 π / L F and L F = 0.002 cm k c = 0.45 × 10 6 cm 1 } .
λ 0 = 0.555 × 10 4 cm ( k 0 = 2 π λ 0 = 1.132 × 10 5 cm 1 ) .
ε r = 40 i 12 ,
h ̅ s 2 0 2 π k d k c W ( k ) 4 kdkd ϕ B 2 ( 1 k d 2 1 k c 2 ) = 0.195 × 10 10 cm 2 = 1 4 k 0 2
k d = [ 2 B k 0 2 k c 2 / ( k c 2 + 2 B k 0 2 ) ] 1 / 2 = 0.202 × 10 5 ( cm ) 1 .
| χ s ( υ y ̅ ) | 2 = exp ( υ y ̅ 2 h ̅ s 2 , and χ s ( υ ) = exp ( β ) .
h l 2 = 0 2 π k f k d W ( k ) 4 kdkd ϕ = B 2 ( 1 k F 2 1 k d 2 ) = 0.791 × 10 9 cm 2 .
σ l t 2 = h l t 2 = 0 2 π k F k d W ( k ) 4 k 3 dkd ϕ = B ln ( k d k F ) = 0.298 × 10 1 .
p ( n ̅ ) = 1 π σ l t 2 exp ( h x 2 + h z 2 σ l t 2 ) .
σ P Q θ = 1 2 π 0 2 π σ P Q d ϕ F .
σ P Q s m 4 π k 0 2 [ | D P Q | 2 n ̅ a ̅ y × exp ( υ y ̅ 2 < h ̅ s 2 ) ( υ y ̅ 2 ) 2 m W m ( υ x ̅ , υ z ̅ ) m ! ] n ̅ = a ̅ y ,

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