Abstract

When light is incident on a mirror, it induces a current density on its surface. This surface current density emits radiation, which is the observed reflected field. We consider a monochromatic incident field with an arbitrary spatial dependence, and we derive an integral equation for the Fourier-transformed surface current density. This equation contains the incident electric field at the surface as an inhomogeneous term. The incident field, emitted by a source current density in front of the mirror, is then represented by an angular spectrum, and this leads to a solution of the integral equation. From this result we derive a relation between the surface current density and the current density of the source. It is shown with examples that this approach provides a simple method for obtaining the surface current density. It is also shown that with the solution of the integral equation, an image source can be constructed for any current source, and as illustration we construct the images of electric and magnetic dipoles and the mirror image of an electric quadrupole. By applying the general solution for the surface current density, we derive an expression for the reflected field as an integral over the source current distribution, and this may serve as an alternative to the method of images.

© 2008 Optical Society of America

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References

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  1. J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere Publishing, 1987).
  2. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  3. A. Sommerfeld, “Mathematische Theorie der Diffraction,” Math. Ann. 47, 317-374 (1896).
    [CrossRef]
  4. P. C. Clemmow, “A method for the exact solution of a class of two-dimensional diffraction problems,” Proc. R. Soc. London, Ser. A 205, 286-308 (1950).
  5. G. D. Maliuzhinets, “Excitation, reflection and emission of surface waves from a wedge with given face impedances,” Sov. Phys. Dokl. 3, 752-755 (1958).
  6. T. B. A. Senior and J. L. Volakis, “Scattering by an imperfect right-angled wedge,” IEEE Trans. Antennas Propag. 34, 681-689 (1986).
    [CrossRef]
  7. B. Budaev, Diffraction by Wedges (Longman Scientific, 1995).
  8. R. F. Harrington, Field Computation by Moment Methods (IEEE Press, 1993).
    [CrossRef]
  9. A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE Press, 1998).
  10. H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, and H. Blok, “Creation and annihilation of phase singularities near a sub-wavelength slit,” Opt. Express 11, 371-380 (2003).
    [CrossRef] [PubMed]
  11. H. F. Schouten, T. D. Visser, D. Lenstra, and H. Blok, “Light transmission through a subwavelength slit: Waveguiding and optical vortices,” Phys. Rev. E 67, 036608-1-4 (2003).
    [CrossRef]
  12. A. W. Maue, “Zur Formulierung eines allgemeinen Beugungsproblems durch eine Integralgleichung,” Z. Phys. 126, 601-618 (1949).
    [CrossRef]
  13. S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409-418 (1982).
    [CrossRef]
  14. P. K. Murthy, K. C. Hill, and G. A. Thiele, “A hybrid-iterative method for scattering problems,” IEEE Trans. Antennas Propag. 34, 1173-1180 (1986).
    [CrossRef]
  15. K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media, Part I: Theory,” IEEE Trans. Antennas Propag. 38, 335-344 (1990).
    [CrossRef]
  16. K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media, Part II: Implementation and results for contiguous half-spaces,” IEEE Trans. Antennas Propag. 38, 345-352 (1990).
    [CrossRef]
  17. J.-M. Jin and J. L. Volakis, “A finite element-boundary integral formulation for scattering by three-dimensional cavity-backed apertures,” IEEE Trans. Antennas Propag. 39, 97-104 (1991).
    [CrossRef]
  18. D. D. Reuster and G. A. Thiele, “A field iterative method for computing the scattered electric fields at the apertures of large perfectly conducting cavities,” IEEE Trans. Antennas Propag. 43, 286-290 (1995).
    [CrossRef]
  19. W. D. Wood, Jr., and A. W. Wood, “Development and numerical solution of integral equations for electromagnetic scattering from a trough in a ground plane,” IEEE Trans. Antennas Propag. 47, 1318-1322 (1999).
    [CrossRef]
  20. D. Torrungrueng, H.-T. Chou, and J. T. Johnson, “A novel acceleration algorithm for the computation of scattering from two-dimensional large-scale perfectly conducting random rough surfaces with the forward-backward method,” IEEE Trans. Geosci. Remote Sens. 38, 1656-1668 (2000).
    [CrossRef]
  21. H. F. Arnoldus and J. T. Foley, “The dipole vortex,” Opt. Commun. 231, 115-128 (2004).
    [CrossRef]
  22. H. F. Arnoldus, “Surface currents on a perfect conductor, induced by a magnetic dipole,” Surf. Sci. 601, 450-459 (2007).
    [CrossRef]
  23. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 122.
  24. J. van Bladel, Singular Electromagnetic Fields and Sources (Clarendon Press, 1991).

2007

H. F. Arnoldus, “Surface currents on a perfect conductor, induced by a magnetic dipole,” Surf. Sci. 601, 450-459 (2007).
[CrossRef]

2004

H. F. Arnoldus and J. T. Foley, “The dipole vortex,” Opt. Commun. 231, 115-128 (2004).
[CrossRef]

2003

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, and H. Blok, “Creation and annihilation of phase singularities near a sub-wavelength slit,” Opt. Express 11, 371-380 (2003).
[CrossRef] [PubMed]

H. F. Schouten, T. D. Visser, D. Lenstra, and H. Blok, “Light transmission through a subwavelength slit: Waveguiding and optical vortices,” Phys. Rev. E 67, 036608-1-4 (2003).
[CrossRef]

2000

D. Torrungrueng, H.-T. Chou, and J. T. Johnson, “A novel acceleration algorithm for the computation of scattering from two-dimensional large-scale perfectly conducting random rough surfaces with the forward-backward method,” IEEE Trans. Geosci. Remote Sens. 38, 1656-1668 (2000).
[CrossRef]

1999

W. D. Wood, Jr., and A. W. Wood, “Development and numerical solution of integral equations for electromagnetic scattering from a trough in a ground plane,” IEEE Trans. Antennas Propag. 47, 1318-1322 (1999).
[CrossRef]

1995

D. D. Reuster and G. A. Thiele, “A field iterative method for computing the scattered electric fields at the apertures of large perfectly conducting cavities,” IEEE Trans. Antennas Propag. 43, 286-290 (1995).
[CrossRef]

1991

J.-M. Jin and J. L. Volakis, “A finite element-boundary integral formulation for scattering by three-dimensional cavity-backed apertures,” IEEE Trans. Antennas Propag. 39, 97-104 (1991).
[CrossRef]

1990

K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media, Part I: Theory,” IEEE Trans. Antennas Propag. 38, 335-344 (1990).
[CrossRef]

K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media, Part II: Implementation and results for contiguous half-spaces,” IEEE Trans. Antennas Propag. 38, 345-352 (1990).
[CrossRef]

1986

P. K. Murthy, K. C. Hill, and G. A. Thiele, “A hybrid-iterative method for scattering problems,” IEEE Trans. Antennas Propag. 34, 1173-1180 (1986).
[CrossRef]

T. B. A. Senior and J. L. Volakis, “Scattering by an imperfect right-angled wedge,” IEEE Trans. Antennas Propag. 34, 681-689 (1986).
[CrossRef]

1982

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409-418 (1982).
[CrossRef]

1958

G. D. Maliuzhinets, “Excitation, reflection and emission of surface waves from a wedge with given face impedances,” Sov. Phys. Dokl. 3, 752-755 (1958).

1950

P. C. Clemmow, “A method for the exact solution of a class of two-dimensional diffraction problems,” Proc. R. Soc. London, Ser. A 205, 286-308 (1950).

1949

A. W. Maue, “Zur Formulierung eines allgemeinen Beugungsproblems durch eine Integralgleichung,” Z. Phys. 126, 601-618 (1949).
[CrossRef]

1896

A. Sommerfeld, “Mathematische Theorie der Diffraction,” Math. Ann. 47, 317-374 (1896).
[CrossRef]

IEEE Trans. Antennas Propag.

T. B. A. Senior and J. L. Volakis, “Scattering by an imperfect right-angled wedge,” IEEE Trans. Antennas Propag. 34, 681-689 (1986).
[CrossRef]

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409-418 (1982).
[CrossRef]

P. K. Murthy, K. C. Hill, and G. A. Thiele, “A hybrid-iterative method for scattering problems,” IEEE Trans. Antennas Propag. 34, 1173-1180 (1986).
[CrossRef]

K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media, Part I: Theory,” IEEE Trans. Antennas Propag. 38, 335-344 (1990).
[CrossRef]

K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media, Part II: Implementation and results for contiguous half-spaces,” IEEE Trans. Antennas Propag. 38, 345-352 (1990).
[CrossRef]

J.-M. Jin and J. L. Volakis, “A finite element-boundary integral formulation for scattering by three-dimensional cavity-backed apertures,” IEEE Trans. Antennas Propag. 39, 97-104 (1991).
[CrossRef]

D. D. Reuster and G. A. Thiele, “A field iterative method for computing the scattered electric fields at the apertures of large perfectly conducting cavities,” IEEE Trans. Antennas Propag. 43, 286-290 (1995).
[CrossRef]

W. D. Wood, Jr., and A. W. Wood, “Development and numerical solution of integral equations for electromagnetic scattering from a trough in a ground plane,” IEEE Trans. Antennas Propag. 47, 1318-1322 (1999).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

D. Torrungrueng, H.-T. Chou, and J. T. Johnson, “A novel acceleration algorithm for the computation of scattering from two-dimensional large-scale perfectly conducting random rough surfaces with the forward-backward method,” IEEE Trans. Geosci. Remote Sens. 38, 1656-1668 (2000).
[CrossRef]

Math. Ann.

A. Sommerfeld, “Mathematische Theorie der Diffraction,” Math. Ann. 47, 317-374 (1896).
[CrossRef]

Opt. Commun.

H. F. Arnoldus and J. T. Foley, “The dipole vortex,” Opt. Commun. 231, 115-128 (2004).
[CrossRef]

Opt. Express

Phys. Rev. E

H. F. Schouten, T. D. Visser, D. Lenstra, and H. Blok, “Light transmission through a subwavelength slit: Waveguiding and optical vortices,” Phys. Rev. E 67, 036608-1-4 (2003).
[CrossRef]

Proc. R. Soc. London, Ser. A

P. C. Clemmow, “A method for the exact solution of a class of two-dimensional diffraction problems,” Proc. R. Soc. London, Ser. A 205, 286-308 (1950).

Sov. Phys. Dokl.

G. D. Maliuzhinets, “Excitation, reflection and emission of surface waves from a wedge with given face impedances,” Sov. Phys. Dokl. 3, 752-755 (1958).

Surf. Sci.

H. F. Arnoldus, “Surface currents on a perfect conductor, induced by a magnetic dipole,” Surf. Sci. 601, 450-459 (2007).
[CrossRef]

Z. Phys.

A. W. Maue, “Zur Formulierung eines allgemeinen Beugungsproblems durch eine Integralgleichung,” Z. Phys. 126, 601-618 (1949).
[CrossRef]

Other

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 122.

J. van Bladel, Singular Electromagnetic Fields and Sources (Clarendon Press, 1991).

J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere Publishing, 1987).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

B. Budaev, Diffraction by Wedges (Longman Scientific, 1995).

R. F. Harrington, Field Computation by Moment Methods (IEEE Press, 1993).
[CrossRef]

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE Press, 1998).

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Figures (3)

Fig. 1
Fig. 1

Electromagnetic field is incident on the surface of a perfect mirror. The surface of the mirror is the x y plane, and the z axis is directed toward the incident field. The incident field induces a current density i ( r , t ) on the surface, and this vector field determines a field line pattern on the surface.

Fig. 2
Fig. 2

Current density j ( r ) is the source of the incident field, and is confined in the z direction as shown.

Fig. 3
Fig. 3

Field lines of the current density i ( r , t ) in the mirror for the case where the source of the incident radiation is an electric quadrupole with a quadrupole tensor given by Eq. (43), and for a fixed time t. The field lines are in the radial direction, either inward or outward, and they change direction across the singular circles, indicated by thin lines.

Equations (70)

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E ( r , t ) = Re [ E ( r ) e i ω t ] ,
B ( r ) sc = μ o 4 π × d S i ( r ) g ( r r ) ,
g ( r r ) = e i k o r r r r ,
E ( r ) sc = i c 2 ω × B ( r ) sc .
g ( r r ) = i 2 π d 2 k 1 β e i k ( r r ) + i β z z ,
β = { k o 2 k 2 , k < k o i k 2 k o 2 , k > k o } .
K ± = k ± β e z ,
B ( r ) sc = μ o 8 π 2 d 2 k 1 β e i K ± r K ± × I ( k ) ,
I ( k ) = d S i ( r ) e i k r ,
E ( r ) sc = 1 8 π 2 ϵ o ω d 2 k 1 β e i K ± r K ± × [ K ± × I ( k ) ] ,
i ( r ) = 1 4 π 2 d 2 k I ( k ) e i k r ,
E ( r ) sc , = E ( r ) inc , .
1 8 π 2 ϵ o ω d 2 k 1 β e i k r { k o 2 I ( k ) k [ k I ( k ) ] } = E ( r ) inc , .
μ o 8 π 2 d 2 k 1 β e i k r k × I ( k ) = B ( r ) inc , .
B ( r ) s = μ o 4 π × d V j ( r ) g ( r r ) ,
g ( r r ) = i 2 π d 2 k 1 β e i K ± ( r r ) .
B ( r ) s = μ o 8 π 2 d 2 k 1 β e i K ± r K ± × J ± ( k ) .
J ± ( k ) = d V j ( r ) e i K ± r
E ( r ) s = 1 8 π 2 ϵ o ω d 2 k 1 β e i K ± r K ± × [ K ± × J ± ( k ) ] .
E ( r ) inc , = 1 8 π 2 ϵ o ω d 2 k 1 β e i k r { k o 2 J ( k ) k [ K J ( k ) ] }
d 2 k 1 β e i k r [ k o 2 ( I + J , ) k ( k I + K J ) ] = 0 ,
k ( k I ) k o 2 I = k o 2 J , k ( K J ) ,
k 2 I = k ( k I ) k × ( k × I ) ,
I ( k ) = J ( k ) 1 β [ e z J ( k ) ] k
I ( k ) = J ( k ) 1 β [ e z J ( k ) ] K ,
I ( k ) = 1 β e z × [ K × J ( k ) ] .
i ( r ) = 1 4 π 2 e z × d 2 k 1 β e i k r [ K × J ( k ) ] ,
d 2 k 1 β K e i K ( r r ) = 2 π g ( r r ) , z > z ,
i ( r ) = 1 2 π e z × d V j ( r ) × [ g ( r r ) ] .
f ( r ) = 1 r 2 ( i k o 1 r ) e i k o r ,
i ( r ) = 1 2 π e z × d V f ( r r ) ( r r ) × j ( r ) .
j ( r ) = i ω d δ ( r r o ) ,
i ( r ) = i ω 2 π f ( r r o ) e z × [ ( r r o ) × d ] .
j ( r ) = p × δ ( r r o ) .
f ( r r o ) = ( r r o ) h ( r r o ) ,
h ( r ) = 1 r 3 ( k o 2 + 3 i k o r 3 r 2 ) e i k o r .
i ( r ) = 1 π f ( r r o ) e z × p + 1 2 π h ( r r o ) e z { ( r r o ) × [ ( r r o ) × p ] } ,
i ( r ) = 1 2 π k ( r r o ) e z × p + 1 2 π h ( r r o ) [ ( r r o ) p ] e z × ( r r o ) ,
k ( r ) = 1 r ( k o 2 + i k o r 1 r 2 ) e i k o r ,
j ( r ) = i ω 6 Q δ ( r r o ) ,
i ( r ) = i ω 12 π h ( r r o ) e z × { ( r r o ) × [ ( r r o ) Q ] } .
α , β Q α β e α × e β = 0 , α , β = x , y , z ,
Q Q o 6 [ 1 0 0 0 1 0 0 0 2 ] , Q o > 0 ,
i ( r ) = Q o i ω H 4 π 6 r h ( r r o ) ,
i ( r , t ) = Q o c k o 6 H 4 π 6 1 q 1 3 [ 3 q 1 cos ( q 1 ω t ) + ( 1 3 q 1 2 ) sin ( q 1 ω t ) ] r ,
q 1 = k o r r o
tan ( q 1 ω t ) = 3 q 1 3 q 1 2 ,
B ( r ) sc = μ o 8 π 2 d 2 k 1 β e i K r K × J ( k ) , z < 0 .
B ( r ) r = μ o 8 π 2 d 2 k 1 β 2 e i K + r K + × { e z × [ K × J ( k ) ] } ,
B ( r ) r = μ o 8 π 2 d 2 k 1 β e i K + r K + × J + ( k ) im .
J + ( k ) im = J ( k ) J ( k ) .
1 β K + × { e z × [ K × J ( k ) ] } = K + × J + ( k ) im ,
J ± ( k ) = i ω d e i K ± r o ,
B ( r ) s = i ω μ o 8 π 2 d 2 k 1 β e i K ± ( r r o ) K ± × d .
J + ( k ) im = i ω d im e i K r o ,
d im = d d ,
r o im = r o , r o , ,
B ( r ) r = i ω μ o 8 π 2 d 2 k 1 β e i K + ( r r o im ) K + × d im ,
J ± ( k ) = i K ± × p e i K ± r o ,
J + ( k ) im = i K + × p im e i K + r o im ,
p im = p p
J ± ( k ) = ω 6 K ± Q e i K ± r o .
J + ( k ) im = ω 6 K + Q im e i K + r o im ,
Q = α , β Q α β e α e β , α , β = x , y , z ,
Q im [ Q x x Q x y Q x z Q y x Q y y Q y z Q z x Q z y Q z z ]
B ( r ) r = μ o 8 π 2 × [ e z × d V A ( r , r ) × j ( r ) ] .
A ( r , r ) = d S g ( r r ) ( r r ) f ( r r ) .
A ( r , r ) = i d 2 k 1 β 2 K e i K + r e i K r , z , z > 0 ,
A ( r , r ) = i d 2 k 1 β 2 K e i K + ( r r im ) , z , z > 0 .
B ( r ) r = i ω μ o 8 π 2 × { e z × [ A ( r , r o ) × d ] } ,

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