Abstract

The results of computer simulations based on the Finite Difference Time Domain method with local space and time grid refinement, are presented for an elliptical aperture in a thin metal film illuminated by a normally incident, monochromatic plane wave. Both cases of incident polarization parallel and perpendicular to the long axis of the ellipse are considered. An intuitive description of the behavior of the electromagnetic fields is developed in each case, and simulation results that exhibit patterns similar to those expected from this qualitative analysis are presented. The simulations reveal, in quantitative detail, the amplitude and phase behavior of the E- and B-fields in and around the aperture.

© 2004 Optical Society of America

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References

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  9. Q. Cao and P. Lalanne, �??Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,�?? Phys. Rev. Lett. 88, 57403 (2002).
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J. Opt. Soc. Am. A (2)

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. (1)

H. A. Bethe, �??Theory of diffraction by small holes,�?? Phys. Rev. 66, 163 (1944).
[CrossRef]

Phys. Rev. Lett. (2)

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, �??Transmission resonances on metallic gratings with very narrow slits,�?? Phys. Rev. Lett. 83, 2845�??48 (1999).
[CrossRef]

Q. Cao and P. Lalanne, �??Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,�?? Phys. Rev. Lett. 88, 57403 (2002).
[CrossRef]

Other (4)

J. D. Jackson, Classical Electrodynamics (2nd edition, Wiley, New York, 1975).

A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House 2000).

Jin Au Kong, Electromagnetic Wave Theory (EMW Publishing, Cambridge, Massachusetts, 2000).

M. Born and E. Wolf, Principles of Optics (7th edition, Cambridge University Press, UK, 1999).

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Figures (24)

Fig. 1.
Fig. 1.

Grid refinement near elliptical aperture in a thin metal film. The base grid has cell size Δ=16nm, refined to Δ/4=4nm around the aperture.

Fig. 2.
Fig. 2.

(a) E-field lines of a static electric dipole p emerge from the positive pole and disappear into the negative pole. (b) Oscillating electric dipole emanates E-lines that reverse direction on spherical shells separated by λ/2. The curl of the E-field creates B-field lines that surround the dipole in closed circular loops. (c) Static magnetic dipole m is a closed loop of electrical current whose B-field pattern is similar to the E-field of an electric dipole. (d) An oscillating magnetic dipole behaves similarly to an electric dipole, albeit with the roles of E and B reversed.

Fig. 3.
Fig. 3.

Normally incident plane-wave on a perfect conductor (yellow slab) induces a surface current Is , which radiates two equal-amplitude plane waves in ±Z-directions. In the lower half-space the induced beam cancels out the incident beam. In the upper half-space, the incident and reflected beams interfere, creating standing-wave fringes of both E- and B-fields.

Fig. 4.
Fig. 4.

A small elliptical aperture in the system of Fig. 3, with its major axis parallel to the surface current Is , distorts the current distribution by diverting its path to avoid the hole. The surface currents in the vicinity of the aperture deposit opposite charges around the sharp corners of the ellipse, causing the E-lines to break up at these corners.

Fig. 5.
Fig. 5.

(a) The B-field above the aperture of Fig. 4, without breaking up, thins down and sags into the hole. (b) The E-field, whose strong fringe is not immediately above the aperture but a distance of Δz=λ/4 away, is squeezed toward the center of the hole, while, at the same time, leaking some of its energy into the aperture. The E-lines can originate or terminate on the charges deposited by the surface current Is on the sharp corners of the ellipse. Note that the charge polarity is such that the E-lines above have the same direction as those inside and below the aperture.

Fig. 6.
Fig. 6.

(a) Surface current distribution obtained when the (uniform) surface current of Fig. 3 is subtracted from that of Fig. 4. Charges appear in regions where the current’s divergence is non-zero. (b) The net effect of the aperture on the uniform surface current of Fig. 3 is the addition of an electric dipole p and two loops of current that circulate in opposite directions; each current loop is a magnetic dipole ±m. (c) Combined radiation pattern of the electric and magnetic dipoles in the vicinity of the aperture.

Fig. 7.
Fig. 7.

Computed plots of Ex ,Ey ,Ez in an XY-plane located a short distance (Δz=20 nm) above the surface of the conductor in the system of Fig. 4. Top row: amplitude, bottom row: phase. The silver film is 124 nm-thick, the aperture is 800 nm-long and 100 nm-wide, and the radiation wavelength is λ=1µm.

Fig. 8.
Fig. 8.

Computed amplitude and phase plots of Ey ,Ez in the central YZ-plane in the system of Fig. 4. The silver film’s cross-section is indicated with dashed lines.

Fig. 9.
Fig. 9.

Computed plots of Hx ,Hy ,Hz in the XY-plane 20 nm above the conductor surface in the system of Fig. 4. Top row: amplitude, bottom row: phase.

Fig. 10.
Fig. 10.

Amplitude and phase plots of Hx ,Hz in the central XZ-plane in the system of Fig. 4.

Fig. 11.
Fig. 11.

Computed plots of Ex ,Ey ,Ez in the XY-plane 20 nm below the bottom facet of the conductor in the system of Fig. 4. Top row: amplitude, bottom row: phase.

Fig. 12.
Fig. 12.

Computed plots of Hx ,Hy ,Hz in the XY-plane 20 nm below the bottom facet of the conductor in the system of Fig. 4. Top row: amplitude, bottom row: phase.

Fig. 13.
Fig. 13.

Profiles of the magnitude |S| of the Poynting vector in various cross-sections of the system of Fig. 4. The superimposed arrows show the projection of S in the corresponding plane. (a) Central XZ-plane. (b) Central YZ-plane. (c, d) XY-planes located 20 nm above and below the aperture.

Fig. 14.
Fig. 14.

When the incident E-field is parallel to the minor axis of an elliptical aperture, the surface currents Is deposit charges at and around the long side-walls of the aperture. These oscillating charges radiate as an electric dipole flanked by a pair of magnetic dipoles, creating circulating magnetic fields around the ellipse’s minor axis that push the incident B-lines upward and sideways.

Fig. 15.
Fig. 15.

(a) Surface currents and charges produced by the aperture of Fig. 14. (b) The current loops in (a) are equivalent to a pair of magnetic dipoles, ±m, while the charges constitute an electric dipole p. (c) In the XY-plane above the aperture, the E-field is dominated by the electric dipole p. (d) In the XY-plane immediately above the aperture, the B-field profile is shaped by competition between the electric dipole p and the magnetic dipoles ±m.

Fig. 16.
Fig. 16.

With reference to Fig. 15(b), the B-field of the electric dipole p combines with that of the magnetic dipoles ±m to produce closed loops in and around the aperture. The solid B-lines bulge above and below the metal film, while the dashed B-lines hug the conductor’s top and bottom surfaces.

Fig. 17.
Fig. 17.

Cross-sections of the system of Fig. 14 in YZ- and XZ-planes. (a) The charges accumulating on the aperture’s side-walls produce an E-field opposite in direction to the incident field. (b) The dipoles p, +m, and -m of Fig. 15(b) push the B-fringe above the aperture upward and sideways.

Fig. 18.
Fig. 18.

Computed plots of Ex ,Ey ,Ez in the XY-plane 20 nm above the conductor’s surface in the system of Fig. 14. Top row: amplitude, bottom row: phase.

Fig. 19.
Fig. 19.

(Left) amplitude and phase of Ey in the central XZ-plane; (right) amplitude and phase of Ey ,Ez in the central YZ-plane in the system of Fig. 14. The fringes in the two panels are differently colored because the color scale for Ey in the YZ-plane has been greatly expanded by two (barely visible) hot spots on the sidewalls near the bottom of the hole.

Fig. 20.
Fig. 20.

Computed plots of Hx ,Hy ,Hz in the XY-plane 20 nm above the surface of the conductor in the system of Fig. 14. Top row: amplitude, bottom row: phase.

Fig. 21.
Fig. 21.

(Left) amplitude and phase of Hx ,Hz in the central XZ-plane; (right) amplitude and phase of Hx in the central YZ-plane in the system of Fig. 14.

Fig. 22.
Fig. 22.

Computed plots of Ex ,Ey ,Ez in the XY-plane 20 nm below the bottom facet of the conductor in the system of Fig. 14. Top row: amplitude, bottom row: phase.

Fig. 23.
Fig. 23.

Computed plots of Hx ,Hy ,Hz in the XY-plane 20 nm below the bottom facet of the conductor in the system of Fig. 14. Top row: amplitude, bottom row: phase.

Fig. 24.
Fig. 24.

Profiles of the magnitude |S| of the Poynting vector in various cross-sections of the system of Fig. 14. The superimposed arrows show the projection of S in the corresponding plane. (a) Central XZ-plane. (b) Central YZ plane. (c, d) XY-planes located 20 nm above and below the aperture.

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