Abstract

This paper presents a new formulation of the 3D Kirchhoff approximation that allows calculation of the scattering of vector waves from 2D rough surfaces containing structures with infinite slopes. This type of surface has applications, for example, in remote sensing and in testing or imaging of printed circuits. Some preliminary calculations for rectangular-shaped grooves in a plane are presented for the 2D surface method and are compared with the equivalent 1D surface calculations for the Kirchhoff and integral equation methods. Good agreement is found between the methods.

© 2008 Optical Society of America

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References

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  1. T. A. Germer, “Effect of line and trench profile variation on specular and diffuse reflectance from a periodic structure,” J. Opt. Soc. Am. A 24, 696-701 (2007).
    [CrossRef]
  2. M. Rovira-Laparra, N. C. Bruce, O. G. Rodriguez-Herrera, J. G. Bañuelos-Muñeton, E. V. Basiuk, and J. M. Saniger-Blesa, “Characterisation of chemically etched indium phosphide surfaces with light scattering,” Waves Random Complex Media 17, 221-231 (2007).
    [CrossRef]
  3. R. A. Depine and D. C. Skigin, “Scattering from metallic surfaces having a finite number of rectangular grooves,” J. Opt. Soc. Am. A 11, 2844-2850 (1994).
    [CrossRef]
  4. Y.-L. Kok, “General solution to the multiple-metallic-grooves scattering problem: The fast polarization case,” Appl. Opt. 32, 2573-2581 (1993).
    [CrossRef] [PubMed]
  5. M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigourous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068-1076 (1995).
    [CrossRef]
  6. A. Mendoza-Suárez and E. R. Méndez, “Light scattering by a reentrant fractal surface,” Appl. Opt. 36, 3521-3531 (1997).
    [CrossRef] [PubMed]
  7. D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. 16, 449-454 (1968).
    [CrossRef]
  8. K. Tang, R. A. Dimenna, and R. O. Buckius, “Regions of validity of the geometric optics approximation for angular scattering from very rough surfaces,” Int. J. Heat Mass Transfer 40, 49-59 (1997).
    [CrossRef]
  9. N. C. Bruce, “Calculations of the Mueller matrix for scattering from two-dimensional surfaces,” Waves Random Media 8, 15-28 (1998).
    [CrossRef]
  10. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces [Pergamon, 1963 (reprinted by Artech House, 1987)].
  11. A. Ishimaru and J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: A theory based on the modified Kirchhoff approach,” Waves Random Media 1, 21-34 (1991).
    [CrossRef]
  12. N. C. Bruce and J. C. Dainty, “Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 1471-1481 (1991).
    [CrossRef]
  13. N. C. Bruce, “Scattering from infinitely sloped surfaces by use of the Kirchhoff approximation,” Appl. Opt. 42, 2398-2406 (2003).
    [CrossRef] [PubMed]
  14. N. C. Bruce, “Control of the backscattered intensity in random rectangular-groove surfaces with variations in the groove depth,” Appl. Opt. 44, 784-791 (2005).
    [CrossRef] [PubMed]
  15. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (IOP, 1991).

2007 (2)

M. Rovira-Laparra, N. C. Bruce, O. G. Rodriguez-Herrera, J. G. Bañuelos-Muñeton, E. V. Basiuk, and J. M. Saniger-Blesa, “Characterisation of chemically etched indium phosphide surfaces with light scattering,” Waves Random Complex Media 17, 221-231 (2007).
[CrossRef]

T. A. Germer, “Effect of line and trench profile variation on specular and diffuse reflectance from a periodic structure,” J. Opt. Soc. Am. A 24, 696-701 (2007).
[CrossRef]

2005 (1)

2003 (1)

1998 (1)

N. C. Bruce, “Calculations of the Mueller matrix for scattering from two-dimensional surfaces,” Waves Random Media 8, 15-28 (1998).
[CrossRef]

1997 (2)

K. Tang, R. A. Dimenna, and R. O. Buckius, “Regions of validity of the geometric optics approximation for angular scattering from very rough surfaces,” Int. J. Heat Mass Transfer 40, 49-59 (1997).
[CrossRef]

A. Mendoza-Suárez and E. R. Méndez, “Light scattering by a reentrant fractal surface,” Appl. Opt. 36, 3521-3531 (1997).
[CrossRef] [PubMed]

1995 (1)

1994 (1)

1993 (1)

1991 (3)

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (IOP, 1991).

A. Ishimaru and J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: A theory based on the modified Kirchhoff approach,” Waves Random Media 1, 21-34 (1991).
[CrossRef]

N. C. Bruce and J. C. Dainty, “Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 1471-1481 (1991).
[CrossRef]

1968 (1)

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. 16, 449-454 (1968).
[CrossRef]

1963 (1)

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces [Pergamon, 1963 (reprinted by Artech House, 1987)].

Bañuelos-Muñeton, J. G.

M. Rovira-Laparra, N. C. Bruce, O. G. Rodriguez-Herrera, J. G. Bañuelos-Muñeton, E. V. Basiuk, and J. M. Saniger-Blesa, “Characterisation of chemically etched indium phosphide surfaces with light scattering,” Waves Random Complex Media 17, 221-231 (2007).
[CrossRef]

Barrick, D. E.

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. 16, 449-454 (1968).
[CrossRef]

Basiuk, E. V.

M. Rovira-Laparra, N. C. Bruce, O. G. Rodriguez-Herrera, J. G. Bañuelos-Muñeton, E. V. Basiuk, and J. M. Saniger-Blesa, “Characterisation of chemically etched indium phosphide surfaces with light scattering,” Waves Random Complex Media 17, 221-231 (2007).
[CrossRef]

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces [Pergamon, 1963 (reprinted by Artech House, 1987)].

Bruce, N. C.

M. Rovira-Laparra, N. C. Bruce, O. G. Rodriguez-Herrera, J. G. Bañuelos-Muñeton, E. V. Basiuk, and J. M. Saniger-Blesa, “Characterisation of chemically etched indium phosphide surfaces with light scattering,” Waves Random Complex Media 17, 221-231 (2007).
[CrossRef]

N. C. Bruce, “Control of the backscattered intensity in random rectangular-groove surfaces with variations in the groove depth,” Appl. Opt. 44, 784-791 (2005).
[CrossRef] [PubMed]

N. C. Bruce, “Scattering from infinitely sloped surfaces by use of the Kirchhoff approximation,” Appl. Opt. 42, 2398-2406 (2003).
[CrossRef] [PubMed]

N. C. Bruce, “Calculations of the Mueller matrix for scattering from two-dimensional surfaces,” Waves Random Media 8, 15-28 (1998).
[CrossRef]

N. C. Bruce and J. C. Dainty, “Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 1471-1481 (1991).
[CrossRef]

Buckius, R. O.

K. Tang, R. A. Dimenna, and R. O. Buckius, “Regions of validity of the geometric optics approximation for angular scattering from very rough surfaces,” Int. J. Heat Mass Transfer 40, 49-59 (1997).
[CrossRef]

Chen, J. S.

A. Ishimaru and J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: A theory based on the modified Kirchhoff approach,” Waves Random Media 1, 21-34 (1991).
[CrossRef]

Dainty, J. C.

N. C. Bruce and J. C. Dainty, “Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 1471-1481 (1991).
[CrossRef]

Depine, R. A.

Dimenna, R. A.

K. Tang, R. A. Dimenna, and R. O. Buckius, “Regions of validity of the geometric optics approximation for angular scattering from very rough surfaces,” Int. J. Heat Mass Transfer 40, 49-59 (1997).
[CrossRef]

Germer, T. A.

Grann, E. B.

Ishimaru, A.

A. Ishimaru and J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: A theory based on the modified Kirchhoff approach,” Waves Random Media 1, 21-34 (1991).
[CrossRef]

Kok, Y.-L.

Méndez, E. R.

Mendoza-Suárez, A.

Moharam, M. G.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (IOP, 1991).

Pommet, D. A.

Rodriguez-Herrera, O. G.

M. Rovira-Laparra, N. C. Bruce, O. G. Rodriguez-Herrera, J. G. Bañuelos-Muñeton, E. V. Basiuk, and J. M. Saniger-Blesa, “Characterisation of chemically etched indium phosphide surfaces with light scattering,” Waves Random Complex Media 17, 221-231 (2007).
[CrossRef]

Rovira-Laparra, M.

M. Rovira-Laparra, N. C. Bruce, O. G. Rodriguez-Herrera, J. G. Bañuelos-Muñeton, E. V. Basiuk, and J. M. Saniger-Blesa, “Characterisation of chemically etched indium phosphide surfaces with light scattering,” Waves Random Complex Media 17, 221-231 (2007).
[CrossRef]

Saniger-Blesa, J. M.

M. Rovira-Laparra, N. C. Bruce, O. G. Rodriguez-Herrera, J. G. Bañuelos-Muñeton, E. V. Basiuk, and J. M. Saniger-Blesa, “Characterisation of chemically etched indium phosphide surfaces with light scattering,” Waves Random Complex Media 17, 221-231 (2007).
[CrossRef]

Skigin, D. C.

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces [Pergamon, 1963 (reprinted by Artech House, 1987)].

Tang, K.

K. Tang, R. A. Dimenna, and R. O. Buckius, “Regions of validity of the geometric optics approximation for angular scattering from very rough surfaces,” Int. J. Heat Mass Transfer 40, 49-59 (1997).
[CrossRef]

Appl. Opt. (4)

IEEE Trans. Antennas Propag. (1)

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. 16, 449-454 (1968).
[CrossRef]

Int. J. Heat Mass Transfer (1)

K. Tang, R. A. Dimenna, and R. O. Buckius, “Regions of validity of the geometric optics approximation for angular scattering from very rough surfaces,” Int. J. Heat Mass Transfer 40, 49-59 (1997).
[CrossRef]

J. Mod. Opt. (1)

N. C. Bruce and J. C. Dainty, “Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 1471-1481 (1991).
[CrossRef]

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

Waves Random Complex Media (1)

M. Rovira-Laparra, N. C. Bruce, O. G. Rodriguez-Herrera, J. G. Bañuelos-Muñeton, E. V. Basiuk, and J. M. Saniger-Blesa, “Characterisation of chemically etched indium phosphide surfaces with light scattering,” Waves Random Complex Media 17, 221-231 (2007).
[CrossRef]

Waves Random Media (2)

N. C. Bruce, “Calculations of the Mueller matrix for scattering from two-dimensional surfaces,” Waves Random Media 8, 15-28 (1998).
[CrossRef]

A. Ishimaru and J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: A theory based on the modified Kirchhoff approach,” Waves Random Media 1, 21-34 (1991).
[CrossRef]

Other (2)

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (IOP, 1991).

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces [Pergamon, 1963 (reprinted by Artech House, 1987)].

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Figures (9)

Fig. 1
Fig. 1

Geometry of a single surface segment for a 1D surface.

Fig. 2
Fig. 2

Geometry of a segment of a 2D surface.

Fig. 3
Fig. 3

Geometry used for testing the vector-wave scattering method. d is the groove depth, a is the groove width, L is the size of the square test surface, and θ inc is the incidence angle. The H and V polarizations are defined relative to the plane of the surface without the groove.

Fig. 4
Fig. 4

3D log scale intensity maps of the light scattered from a single groove in a plane. Top left, HH scattering; top right, VV; bottom left, HV; bottom right, VH. Surface parameters: L = 10 λ , a = 4 λ , and d = 0.52 λ . The incidence angle is θ inc = 20 ° .

Fig. 5
Fig. 5

As Fig. 4 for d = 1.75 λ .

Fig. 6
Fig. 6

Cross sections through the graphs for polarization HH in Fig. 4 (left-hand graph) and Fig. 5 (right-hand graph) (indicated by the crosses) and the VV polarization curves for the same cases (open triangles) compared with the 2D Kirchhoff results for the same cases (points) and the integral equation results (solid curves). The cross section corresponds to the value of the scatter angle in the x - z plane having a value of 0 ° . The x axis in this figure is the value of the scatter angle in the y - z plane of Figs. 4, 5.

Fig. 7
Fig. 7

3D log scale intensity maps of the light scattered from four grooves in a perfectly conducting plane. Top left, HH scattering; top right, VV; bottom left, HV; bottom right, VH. Surface parameters: Average groove width 2 λ with a rectangular probability distribution of width 1 λ ; average groove depth 2 λ with a rectangular probability distribution of width 1 λ ; and average groove separation (end of one groove to the start of the next) 2 λ with a rectangular probability distribution of width 1 λ . The incidence angle is θ inc = 20 ° .

Fig. 8
Fig. 8

As Fig. 7 but for six grooves in a perfectly conducting plane.

Fig. 9
Fig. 9

Cross sections through the graphs for polarization HH in Fig. 7 (top) and Fig. 8 (bottom) (indicated by the crosses) and the VV polarization curves for the same cases (open triangles) compared with the 2D Kirchhoff results for the same cases (solid curves). The cross section corresponds to the value of the scatter angle in the x - z plane having a value of 0 ° . The x axis in this figure is the value of the scatter angle in the y - z plane of Figs. 7, 8.

Equations (16)

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n = sin β x + cos β y = d y d S x + d x d S y .
Φ sc = 1 4 π Φ inc ( x s , y s ) { ( 1 + R ) [ h ( x ) sin θ sc cos θ sc ] H 1 ( 1 ) ( k r ) i ( 1 R ) [ h ( x ) sin θ inc cos θ inc ] H 0 ( 1 ) ( k r ) } d x ,
Φ sc = 1 4 π Φ inc ( x s , y s ) [ ( 1 + R ) sin θ sc H 1 ( 1 ) ( k r ) i ( 1 R ) sin θ inc H 0 ( 1 ) ( k r ) ] d y 1 4 π Φ inc ( x s , y s ) [ ( 1 + R ) cos θ sc H 1 ( 1 ) ( k r ) + i ( 1 R ) cos θ inc H 0 ( 1 ) ( k r ) ] d x ,
E sc ( r ) = i k exp ( i k r ) 4 π r k sc × S [ n × E sur ( r ) η k sc × n × H sur ( r ) ] exp ( i k sc r ) d S ,
n × E = E i n c { ( 1 R V ) ( n k i n c ) ( k i n c × e ) + ( 1 R V ) ( n k i n c ) ( e t ) ( t × k i n c ) + ( 1 + R H ) ( e t ) ( n × t ) } ,
η n × H = E i n c [ ( 1 + R V ) ( n e ) k i n c ( 1 R H ) ( n k i n c ) e + ( R V + R H ) ( e t ) ( k i n c n ) t ] ,
x = ( cos β , 0 , sin β ) ,
y = ( 0 , cos α , sin α ) ,
n = x × y x × y .
n = 1 sin θ ( sin β cos α , cos β sin α , cos β cos α ) .
sin β = d h x d S x , cos β = d x d S x ;
sin α = d h y d S y , cos α = d y d S y ;
n = 1 sin θ ( d y d S y d h x d S x , d x d S x d h y d S y , d x d S x d y d S y ) ,
n = 1 d S x d S y sin θ ( d y d h x , d x d h y , d x d y ) .
1 d S x d S y sin θ = 1 dS x × dS y = 1 d S ,
n d S = ( d y d h x , d x d h y , d x d y ) .

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