Abstract

From elementary theoretical considerations, a simple formula is obtained for representing the surface current density generated by an incident wave illuminating a two-dimensional perfectly conducting rough surface. This formula can be considered a correction of the Kirchhoff approximation in that it takes into account the radius of curvature of the surface. It is shown from comparisons with rigorously obtained results on circular cylinders, diffraction gratings, and nonperiodic rough surfaces that the precision of the new method is much better than that given by the Kirchhoff approximation.

© 1989 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 (Pergamon, New York, 1963).
  2. P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North-Holland, Amsterdam, 1967), pp. 53–97.
    [CrossRef]
  3. D. Maystre, O. Mata Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
    [CrossRef]
  4. T. V. Vorburger, E. C. Teague, “Optical techniques for online measurement of surface topography,” Precis. Eng. 3, 61–83 (1981).
    [CrossRef]
  5. D. Maystre, “Electromagnetic scattering from perfectly conducting rough surfaces in the resonance region,”IEEE Trans. Antennas Propag. AP-31, 885–895 (1983).
    [CrossRef]
  6. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984), pp. 1–67.
    [CrossRef]
  7. J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1964), p. 376.
  8. L. Schwartz, Methodes Mathematiques pour les Sciences Physiques (Hermann, Paris, 1965), p. 76.

1983 (2)

D. Maystre, O. Mata Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
[CrossRef]

D. Maystre, “Electromagnetic scattering from perfectly conducting rough surfaces in the resonance region,”IEEE Trans. Antennas Propag. AP-31, 885–895 (1983).
[CrossRef]

1981 (1)

T. V. Vorburger, E. C. Teague, “Optical techniques for online measurement of surface topography,” Precis. Eng. 3, 61–83 (1981).
[CrossRef]

Beckmann, P.

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

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North-Holland, Amsterdam, 1967), pp. 53–97.
[CrossRef]

Mata Mendez, O.

D. Maystre, O. Mata Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
[CrossRef]

Maystre, D.

D. Maystre, “Electromagnetic scattering from perfectly conducting rough surfaces in the resonance region,”IEEE Trans. Antennas Propag. AP-31, 885–895 (1983).
[CrossRef]

D. Maystre, O. Mata Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
[CrossRef]

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984), pp. 1–67.
[CrossRef]

Roger, A.

D. Maystre, O. Mata Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
[CrossRef]

Schwartz, L.

L. Schwartz, Methodes Mathematiques pour les Sciences Physiques (Hermann, Paris, 1965), p. 76.

Spizzichino, A.

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

Teague, E. C.

T. V. Vorburger, E. C. Teague, “Optical techniques for online measurement of surface topography,” Precis. Eng. 3, 61–83 (1981).
[CrossRef]

Van Bladel, J.

J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1964), p. 376.

Vorburger, T. V.

T. V. Vorburger, E. C. Teague, “Optical techniques for online measurement of surface topography,” Precis. Eng. 3, 61–83 (1981).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

D. Maystre, “Electromagnetic scattering from perfectly conducting rough surfaces in the resonance region,”IEEE Trans. Antennas Propag. AP-31, 885–895 (1983).
[CrossRef]

Opt. Acta (1)

D. Maystre, O. Mata Mendez, A. Roger, “A new electromagnetic theory for scattering from shallow rough surfaces,” Opt. Acta 30, 1707–1723 (1983).
[CrossRef]

Precis. Eng. (1)

T. V. Vorburger, E. C. Teague, “Optical techniques for online measurement of surface topography,” Precis. Eng. 3, 61–83 (1981).
[CrossRef]

Other (5)

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

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North-Holland, Amsterdam, 1967), pp. 53–97.
[CrossRef]

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984), pp. 1–67.
[CrossRef]

J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1964), p. 376.

L. Schwartz, Methodes Mathematiques pour les Sciences Physiques (Hermann, Paris, 1965), p. 76.

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Figures (11)

Fig. 1
Fig. 1

Presentation of the problem for an arbitrary surface.

Fig. 2
Fig. 2

The particular case of circular cylinders.

Fig. 3
Fig. 3

Comparison of the rigorously obtained values of the corrections to the normal derivative of the field with those obtaimed from the BR theory [actually, with their approximations as given by Eqs. (39′) and (50′)]. The abscissa represents the λ/R ratio. The angle of incidence θ0 is equal to 0° (point O in Fig. 2). The left-hand and right-hand panels represent results for TE and TM polarizations, respectively. The modulus m of the corrections is shown in the top panels, and the phase a is shown in degrees in the bottom panels. The solid curves represent the rigorously obtained values, and the dashed curves represent the approximate values.

Fig. 4
Fig. 4

The same as Fig. 3 but for the point P of Fig. 2 (θ0 = 30°).

Fig. 5
Fig. 5

The same as Fig. 3 but for the point Q of Fig. 2 (θ0 = 60°).

Fig. 6
Fig. 6

Profile of the nonperiodic rough surface for H = 0.4 μm.

Fig. 7
Fig. 7

Comparison of the surface current densities deduced from the B, BR, and rigorous theories. The solid curves represent the modulus D of the difference between the rigorously obtained surface current and that deduced from the BR theory, and the dotted curves represent the difference between rigorously obtained results and those obtained from the B theory. The left-hand panels correspond to λ = 1 μm, and the right-hand panels correspond to λ = 4 μm. The values of the depth H of the surface are 0.1 μm (top), 0.4 μm (middle), and 1 μm (bottom).

Fig. 8
Fig. 8

The same as Fig. 7 but for diffraction patterns: the solid curves represent the rigorously obtained results, the dotted curves represent the results deduced from the B theory, and the dashed curves represent the results obtained from the BR theory.

Fig. 9
Fig. 9

(Left-hand panel) Difference D between the rigorously obtained surface current density and that obtained from (dotted curves) the B theory or (solid curves) the BR theory for λ = 1 μm and H = −0.4 μm. (Right-hand panel) Corresponding diffraction pattern, with the same conventions as in Fig. 8.

Fig. 10
Fig. 10

The same as Fig. 9 but for H = +0.4 μm and an incidence of 45°

Fig. 11
Fig. 11

The same as Fig. 9 but for H = 0.4 μm and TM polarization.

Tables (6)

Tables Icon

Table 1 Comparison of the Efficiencies, in the First Order, of a Perfectly Conducting Sinusoidal Grating Illuminated at Normal Incidence with λ = 0.3 μma

Tables Icon

Table 2 The Same as Table 1 but for λ = 0.6 μm

Tables Icon

Table 3 The Same as Table 1 but for λ = 1 μm

Tables Icon

Table 4 The Same as Table 1 but for TM Polarization

Tables Icon

Table 5 The Same as Table 2 but for TM Polarization

Tables Icon

Table 6 The Same as Table 3 but for TM Polarization

Equations (58)

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f ( 0 ) = d f / d x ( 0 ) = 0.
E i ( x , y ) = - k + k p ( α ) exp ( i α x - i β y ) d α ,
β = ( k 2 - α 2 ) 1 / 2
- k + k ( α - α 0 ) p ( α ) d α = 0.
u ( x ) = - E i [ x , f ( x ) ] exp ( - i α 0 x )
= - - k + k p ( α ) exp [ i ( α - α 0 ) x - i β f ( x ) ] d α ,
u ( 0 ) = P 0 ,
u ( 0 ) = d u / d x ( 0 ) = i P 1 = 0 ,
u ( 0 ) = d 2 u / d x 2 ( 0 ) = - P 2 - f ( 0 ) I ,
P n = - - k + k ( α - α 0 ) n p ( α ) d α ,
I = d E i / d n ( 0 ) = - i - k + k β ( α ) p ( α ) d α ;
β ( α ) = β 0 + β 0 ( α - α 0 ) + β 0 2 ( α - α 0 ) 2 + ,
I = i ( β 0 P 0 + β 0 P 1 + β 0 2 P 2 + ) .
E d ( x , y ) = - + q ( α ) exp ( i α x + i β y ) d α .
D = d E d / d n ( 0 ) = - + i β ( α ) q ( α ) d α .
Q n = - + ( α - α 0 ) n q ( α ) d α
D = i ( β 0 Q 0 + β 0 Q 1 + β 0 2 Q 2 + ) .
E d [ x , f ( x ) ] = - E i [ x , f ( x ) ] .
- + q ( α ) exp [ i ( α - α 0 ) x + i β f ( x ) ] = u ( x ) .
Q 0 = u ( 0 ) = P 0 .
D = i β 0 Q 0 = i β 0 u ( 0 ) ,
D = - i β 0 E i ( 0 , 0 ) .
I = i β 0 P 0 = - i β 0 E i ( 0 , 0 )
d E / d n ( 0 ) = 2 I ,
Q 0 = u ( 0 ) ,
i Q 1 = u ( 0 ) = 0 ,
- Q 2 + f ( 0 ) D = u ( 0 ) .
D = i ( β 0 Q 0 + β 0 Q 1 + β 0 2 Q 2 ) .
D = i { β 0 u ( 0 ) + β 0 2 [ f ( 0 ) D - u ( 0 ) ] } ,
D = [ i β 0 u ( 0 ) - i β 0 2 u ( 0 ) ] / [ 1 - i β 0 2 f ( 0 ) ] .
D = { i β 0 P 0 + i β 0 2 [ P 2 + f ( 0 ) I ] } / [ 1 - i β 0 2 f ( 0 ) ] .
E i ( x , y ) = E i ( 0 , 0 ) exp ( i α 0 x - i β 0 y ) ,
α 0 = k sin ( θ 0 ) ,
β 0 = k cos ( θ 0 ) .
p ( α ) = E i ( 0 , 0 ) δ ( α - α 0 ) ,
P 0 = E i ( 0 , 0 ) ,
P i = 0 ,             1 i ,
I = i β 0 E i ( 0 , 0 ) .
D = I [ 1 + i β 0 2 f ( 0 ) ] / [ 1 - i β 0 2 f ( 0 ) ] ,
d E / d n ( 0 ) = 2 I / [ 1 - i β 0 2 f ( 0 ) ] .
β 0 = - k 2 / β 0 3 = - 1 / [ k cos 3 ( θ 0 ) ] ,
d E / d n ( 0 ) = 2 I C E ,
C E = 1 / { 1 + i / [ 2 k R cos 3 ( θ 0 ) ] } .
a = - tan - 1 { 1 / [ 2 k R cos 3 ( θ 0 ) ] } .
- + [ β - α f ( x ) ] q ( α ) exp [ i ( α - α 0 ) x + i β f ( x ) ] d α = v ( x ) .
v ( x ) = - k + k [ β + α f ( x ) ] p ( α ) exp [ i ( α - α 0 ) x - i β f ( x ) ] d α
- + β 0 q ( α ) d α = β 0 Q 0 = v ( 0 ) .
H ( 0 , 0 ) = 2 H i ( 0 , 0 ) ,
β 0 Q 0 + β 0 Q 1 + β 0 2 Q 2 = v ( 0 ) ,
- α 0 f ( 0 ) Q 0 + [ i β 0 - f ( 0 ) ] Q 1 + i β 0 Q 2 = v ( 0 ) ,
[ α 0 f ( 0 ) - i β 0 2 f ( 0 ) ] Q 0 + [ 4 i α 0 f ( 0 ) + f ( 0 ) ] Q 1 + [ β 0 + 3 i f ( 0 ) ] Q 2 = - v ( 0 ) .
H ( 0 , 0 ) = H i ( 0 , 0 ) + Q 0 , H ( 0 , 0 ) = 2 H i C H ,
C H = 4 k 2 f ( 0 ) + 3 i β 0 f 2 ( 0 ) - i α 0 f ( 0 ) - i β 0 3 7 2 k 2 f ( 0 ) + i k 2 β 0 ( 9 2 - 2 k 2 β 0 2 ) f 2 ( 0 ) - 3 2 i k 2 α 0 β 0 2 f ( 0 ) - i β 0 3 .
C H = 1 + i / ( 2 k R cos 3 θ 0 ) .
C ˜ E = 1 - i / ( 2 k R cos 3 θ 0 ) ,
C ˜ H = 1 / [ 1 - i / ( 2 k R cos 3 θ 0 ) ] ,
f ( x ) = 0             if x [ - 2 , + 2 ]
f ( x ) = H exp [ 3 - 3 1 - ( x 2 / 4 ) ]             otherwise .

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