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P. Lalanne and H. Liu
that this difference becomes smaller and smaller as the metal conductivity increases
(or λ increases) [14].
This property has an important consequence if one neglects the small residual
difference. When a subwavelength indentation is illuminated by a TM polarized
light, the scattered field can be seen as the total field radiated by two line sources,
J x and J z , and the relative amplitudes of the line sources are arbitrary: they for
instance depend on the incident illumination (its angle of incidence for instance
if it is a plane wave), on the actual geometry of the indentation, on the dielectric
and metal permittivities… A priori two independent radiation problems should be
considered, but since the quasi-cylindrical waves associated to the two line source
polarizations are identical in shape, any arbitrary sub-indentation illuminated by any
incident electromagnetic field will launch a unique field (the quasi-cylindrical wave)
on a metallic surface, in addition to the SPP. Also note that the mixing ratio between
the SPP and the quasi-CW radiated by the two line sources are approximately the
same [14], and we finally conclude that this mixing ratio is fixed for the field scattered
by any sub-indentation.
10.4.2 SPP and Quasi-CW Launched by a Dipole Point-Source on
a Metal Surface (3D case)
The field radiated by point-source or sub-λ antennas in the vicinity of metallic surfaces has been of long-standing interest in classical electromagnetism. In particular
for long-distance radio propagation and for remote sensing, the problem was analyzed in detail by Sommerfeld [29, 30], Norton [21–23] and others for a half-space
conductor with a finite conductivity (the sea surface for instance). The conclusions
were that the radiated field can be calculated as an integral along a contour in the complex plane and is composed of two contributions, the Zenneck mode (corresponding
to the pole, the analog of the SPP at visible frequencies) and a “direct” wave (corresponding to branch integrals, the analog of the quasi-CW). In an intermediate region
and near the surface, the field is well approximated by that of the cylindrical Zenneck
mode; but then, as the distance increases further, the long-distance propagation is
mainly due to the direct wave that is often referred to as the Norton wave. The latter,
whose amplitude asymptotically decays as 1/r 2 , overcomes the Zenneck mode at
large distances because of the additional exponential damping factor exp(ik SP r ) of
the Zenneck mode. These issues are discussed in great detail in the review article by
R.E. Collin [6] or in the book by Baños [2].
Hereafter we simply show an example for the sake of illustration. For a vertical dipole, perpendicular to the interface, the in-plane component of the radiated
field is radially polarized and isotropic. The situation is more interesting for an inplane dipole (let us say parallel to the x-axis). Both the SPP and quasi-CW fields
on the surface are anisotropic. Figure 10.6 shows the radial electric fields radiated
by such a dipole. The SPP field is proportional to r −1/2 exp(ik SP r ) and its electric
P. Lalanne and H. Liu
that this difference becomes smaller and smaller as the metal conductivity increases
(or λ increases) [14].
This property has an important consequence if one neglects the small residual
difference. When a subwavelength indentation is illuminated by a TM polarized
light, the scattered field can be seen as the total field radiated by two line sources,
J x and J z , and the relative amplitudes of the line sources are arbitrary: they for
instance depend on the incident illumination (its angle of incidence for instance
if it is a plane wave), on the actual geometry of the indentation, on the dielectric
and metal permittivities… A priori two independent radiation problems should be
considered, but since the quasi-cylindrical waves associated to the two line source
polarizations are identical in shape, any arbitrary sub-indentation illuminated by any
incident electromagnetic field will launch a unique field (the quasi-cylindrical wave)
on a metallic surface, in addition to the SPP. Also note that the mixing ratio between
the SPP and the quasi-CW radiated by the two line sources are approximately the
same [14], and we finally conclude that this mixing ratio is fixed for the field scattered
by any sub-indentation.
10.4.2 SPP and Quasi-CW Launched by a Dipole Point-Source on
a Metal Surface (3D case)
The field radiated by point-source or sub-λ antennas in the vicinity of metallic surfaces has been of long-standing interest in classical electromagnetism. In particular
for long-distance radio propagation and for remote sensing, the problem was analyzed in detail by Sommerfeld [29, 30], Norton [21–23] and others for a half-space
conductor with a finite conductivity (the sea surface for instance). The conclusions
were that the radiated field can be calculated as an integral along a contour in the complex plane and is composed of two contributions, the Zenneck mode (corresponding
to the pole, the analog of the SPP at visible frequencies) and a “direct” wave (corresponding to branch integrals, the analog of the quasi-CW). In an intermediate region
and near the surface, the field is well approximated by that of the cylindrical Zenneck
mode; but then, as the distance increases further, the long-distance propagation is
mainly due to the direct wave that is often referred to as the Norton wave. The latter,
whose amplitude asymptotically decays as 1/r 2 , overcomes the Zenneck mode at
large distances because of the additional exponential damping factor exp(ik SP r ) of
the Zenneck mode. These issues are discussed in great detail in the review article by
R.E. Collin [6] or in the book by Baños [2].
Hereafter we simply show an example for the sake of illustration. For a vertical dipole, perpendicular to the interface, the in-plane component of the radiated
field is radially polarized and isotropic. The situation is more interesting for an inplane dipole (let us say parallel to the x-axis). Both the SPP and quasi-CW fields
on the surface are anisotropic. Figure 10.6 shows the radial electric fields radiated
by such a dipole. The SPP field is proportional to r −1/2 exp(ik SP r ) and its electric