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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
1. Introduction
The apertures of classical optics simply block those parts of an incident wavefront that fall outside the aperture, allowing everything else to go through intact. Moreover, multiple apertures act upon an incident beam independently of each other, polarization effects are usually negligible (i.e., scalar diffraction), and it is not necessary to keep track of both the electric- and the magnetic-field components of the beam [1]. All of the above assumptions break down when apertures shrink to dimensions comparable to or smaller than a wavelength [2, 3, 4]. For example, transmission through two small adjacent apertures cannot be treated by assuming that only one aperture is open at a time, then adding the fields transmitted by the individual apertures. (This is because the electric charge and current distributions in the vicinity of one aperture are influenced by the radiation pattern of the other aperture.) Polarization effects are extremely important for small apertures as exemplified by the case of a normally incident beam going through an elliptical aperture in a thin metal film: whereas in the case of polarization (i.e., E-field) parallel to the long axis of the ellipse there is negligible transmission, when the incident polarization is rotated 90° to point along the ellipse’s minor axis, the aperture transmits a substantial fraction of the incident light. Finally, to analyze the interaction of light with small apertures, it is generally necessary to keep track of both E and B components of the electro-magnetic wave, as the modification of one of these fields produces non-trivial changes in the other field’s distribution [5].
This paper presents the results of computer simulations based on the Finite Difference Time Domain (FDTD) [6] method for an elliptical aperture in a thin metal film illuminated by a normally incident, monochromatic plane wave. In Section 2 we describe the space and time local grid refinement FDTD method used in the computations. Some properties of Maxwell’s equations and dipole fields are discussed in Section 3. The cases of incident plane-wave polarization parallel (Section 4) and perpendicular (Section 5) to the long axis of the ellipse are considered. The summary of results and conclusions are presented in Section 6.
2. FDTD method with local grid refinement
In this work, the time dependent Maxwell equations are solved numerically using an FDTD non-conformal grid refinement method in Cartesian space coordinates. At the root of the mesh refinement tree is a coarse grid covering the entire computational domain. Each successive level consists of a collection of rectangular regions with cell sizes refined by a factor of two and nested within their parent grids. Inside of each rectangular region the standard second-order accurate FDTD algorithm is applied, while at the interfaces between the coarse and fine grids an interpolation in space and time must be used to coordinate solutions at different levels. Figure 1 shows a three level grid structure used to refine a region around a small elliptical aperture, with four-fold reduction of the cell size Δ/4 on the most refined grid. Maximum improvement in the computational cost is possible when using both space and time grid refinement, when each grid refinement level advances solution with its own time step, computed from the stability criteria for the FDTD method with given spatial cell size. The grid refinement method, with mesh structure shown in Fig. 1, requires about ten times less processor time, compared to the uniform grid with cell size Δ/4 and the same spatial extent as the base grid in 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.
3. Maxwell’s equations, dipole fields and metallic mirrors
We begin developing a qualitative description of the behavior of the electromagnetic fields in the vicinity of the aperture by describing in the next three subsections some properties of Maxwell’s equations, radiating dipole fields and light reflection from a highly conducting flat mirror, as an aid for the discussions in the following sections.
3.1. Maxwell’s equations
In developing an intuitive understanding of the electromagnetic field distribution around an aperture, we rely heavily on Maxwell’s divergence equations, ∇·D=ρ and ∇·B=0, where D=ε0εE, B=µ 0 H, and ρ is the electric charge density [1, 5]. (ε0 and µ 0 are the permittivity and permeability of free-space, while ε is the relative permittivity of the local environment.) The divergence-free nature of the magnetic field simply means that the B-field lines cannot be interrupted; they can go around in loops or they can form unbroken infinite lines, but they cannot originate, nor can they terminate, at specific points in space. A similar argument applies to D-field lines, except in locations where electric charges exist. When charges are present, lines of D originate on positive charges and terminate on negative charges; everywhere else the D-lines can twist and turn in space, but they cannot start or stop.
The other two of Maxwell’s equations, ∇×H=J+∂ D/∂t and ∇×E=-∂ B/∂t, are necessary not only for generating the E and B fields from electrical currents (J is the local current density), but also to sustain these fields in source-free regions of space [5] When highly conducting media (e.g., metallic bodies) are present in a system, surface currents Is develop that support the magnetic field H immediately above the conducting surfaces. Aside from these electrical currents that act as sources of the H-field, time variations of the E-field are needed at each point of space to maintain the local B-field. In a similar vein, aside from electric charges that act as sources and sinks for the D-field, time variations of B are necessary to maintain the local E-field. The lines of the current density J remain divergence-free, except in those locations where they deposit electrical charges, that is, ∇·J=-∂ρ/∂t [5, 7].
Inside an electrical conductor J=σ E, where σ is the conductivity of the material. Good conductors (e.g., metals) have large conductivities, which means that the E-field must all but vanish from the interior of such bodies. When the fields are oscillatory, any magnetic fields inside a good conductor will produce, by virtue of the Faraday law, ∇×E=-∂ B/∂t, a local electric field. Since E-fields are not allowed inside a conductor, time-varying magnetic fields, being intimately associated with the electric fields, must also be absent. The interior of good conductors thus remains free of charges, currents, and time-varying electromagnetic fields. Charges and currents, however, can and do develop on the conductor’s surface, where they give rise to E and B fields in the vicinity of the surface outside the conductor.
The fifth equation of classical electrodynamics, the Lorentz law of force F=q(E+V×B), expresses the force F experienced by a particle of charge q and velocity V [1]. This equation is occasionally useful in developing a qualitative picture of the current distribution in the vicinity of small apertures. For example, within the skin depth of a conductor, the directions of E and B would indicate the sense in which local surface currents are affected by the Lorentz force acting on the charge carriers. Typically, the E-field is the dominant factor in this regard, as evidenced by the constitutive relation J=σ E. Any transverse deflections of the current by the B-field are generally neglected, unless the Hall conductivity of the medium is explicitly included in the constitutive relations.
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.
3.2. Radiation by an oscillating dipole
With reference to Fig. 2(a), a static electric dipole p creates, in its surrounding environment, electric-field lines that emerge from the positive pole and disappear into the negative pole. A periodically oscillating electric dipole emanates E-field lines that reverse direction at half-period intervals. The constant speed of light in all directions in space then dictates that these E-field reversals occur on spherical shells separated by a half-wavelength (λ/2) from their adjacent shells. The zero-divergence requirement imposed on the E-lines by the first Maxwell equation thus requires the existence of the closed lines of field depicted in Fig. 2(b). The curl of the E-field gives rise to B-field lines that encircle the dipole in closed loops, sustaining the E-field oscillations while simultaneously being generated by them. In the space between adjacent spherical shells separated by λ/2, the E-lines are not parallel to these shell surfaces, but bend inward or outward as shown to maintain the divergence-free condition of the E-field [7].
A static magnetic dipole m, shown in Fig. 2(c), is a closed loop of electrical current whose B-field pattern is similar to the E-field of an electric dipole. Figure 2(d) shows an oscillating magnetic dipole, which behaves in much the same way as an electric dipole does, albeit with a role reversal for E and B [7]. These examples indicate that by direct appeal to Maxwell’s equations, especially the divergence laws, it is possible to obtain an intuitive picture of the electromagnetic field distribution. In the discussions that follow, we will use the dipole radiation patterns sketched in Figs. 2(b) and 2(d) to elucidate the nature of transmission through subwavelength apertures in a thin metal film.
3.3. Plane wave reflection from a (highly conducting) flat mirror
Figure 3 shows the case of a normally incident plane wave on a perfect conductor (yellow slab at the bottom). The incident beam induces a surface current Is in the conductor, which creates equal-amplitude plane waves propagating in the ±Z-directions [5, 7]. In the half-space below the conductor, the induced and incident plane-waves cancel each other out. In the half-space above the conductor, interference between the incident and reflected beams creates standing-wave fringes of the electric-field E and the magnetic field B. The B-field is strongest at the surface of the conductor, reversing sign at intervals of Δz=λ/2, where its adjacent peaks are located. The peaks of the E-field, also located at λ/2 intervals, are staggered relative to the B-field peaks, thus coinciding with planes of vanishing magnetic field.
At the upper surface of the conductor, where the E-field is zero, the B-field is sustained by the surface current Is. (Although Is is shown anti-parallel to the standing-wave’s E-field at Δz=λ/4, in reality Is is 90° behind this E-field, reaching maximum when the E-field directly above the surface is going through zero on its way to the peak.) In the half-space above the conductor, in the absence of any electrical charges and currents, the E-field is sustained by the time-variations of the B-field, and vice-versa.
In an imperfect conductor, where conductivity is large but finite, the E- and B-fields penetrate slightly beneath the surface, producing a Lorenz force on the moving charges that comprise the surface current. While the E-field provides the current’s driving force, the magnetic component of the Lorentz force attempts to drive the surface current further down into the conductor (radiation pressure). In general, the surface current Is need not be in-phase with the penetrating E-field, since, at optical frequencies, the electrical conductivity σ is a complex number.
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. 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.
4. Elliptical aperture illuminated with plane-wave polarized along the long axis
The presence of a small (subwavelength-sized) elliptical aperture in the system of Fig. 3 distorts the surface current Is in the vicinity of the aperture by diverting the current’s path to avoid the hole, as shown in Fig. 4. The B-lines within the fringe immediately above the mirror surface reorient in such a way as to remain perpendicular to the lines of Is, thus bending toward the center of the aperture. The B-lines directly above the aperture, lacking support from an underlying surface current, drop into the hole on the left side and re-emerge on the right side. (The B-lines, of course, cannot break up because ∇·B=0 everywhere; they can only bend locally and change direction, but must remain continuous at all times.)
The lines of surface current Is that begin and end on the ellipse’s sharp corners deposit electric charges around these corners, which charges then act as sources and sinks for the E-lines in their neighborhood. Elsewhere, lack of any significant amount of charge means that the E-lines cannot break up, but rather they must twist and turn continuously as they adjust to the new environment created by the presence of the hole. The E-field in and around the aperture must be distributed in a way that would support the B-field (through the curl equations), but, because a parallel E-field cannot exist on conducting surfaces, it must also stay away from the interior walls of the hole. Figure 4 shows a possible way for the E-lines just above the aperture to dodge the side-walls and concentrate near the center, as they drop into the hole from above. The bundle of E-lines in the middle of the hole (parallel to the ellipse’s long-axis) then acts as a source of circulating magnetic fields that wrap around the long axis (∇×H=∂ D/∂t), thus supporting the B-field above, below, and inside the aperture.
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. (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.
Figure 5(a) shows that, in the central XZ cross-section of the aperture, the B-lines above the aperture, without breaking up, thin down and sag toward and into the hole. Magnetic energy thus leaves the mid-section of the strong B-fringe above the hole and leaks into the hole and beyond. The behavior of the E-field in the central YZ-plane is depicted in Fig. 5(b). Here the strong fringe, which is not immediately above the aperture but a distance of Δz=λ/4 away, is squeezed laterally toward the hole’s center, while, at the same time, leaking some of its energy into the aperture. Some of the E-lines originate or terminate on the charges deposited by the surface current Is on the sharp corners of the ellipse. (The dashed lines in Fig. 5(b) represent the bending of the E-field out of the YZ-plane toward charges that reside on the side-walls near these sharp corners.) Note that the charge polarity is such that the E-lines above have the same direction as those inside and below the aperture. It is important to recognize that the surface current Is lags 90° behind the E-field of the first fringe. Thus, when the E-field directly above the aperture reaches its maximum along the negative Y-axis, Is, which has been traveling in the positive Y-direction until that moment, has stopped and is beginning to reverse direction. This explains why the charges reach their maximum strength when the E-field immediately above the aperture is at a peak, and also clarifies the reasoning behind the polarity chosen for the charges in Fig. 5(b).
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. 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.
Aside from the incident beam, which is fixed at the outset, all other radiation in the system of Fig. 4 is generated by the surface currents Is (and the charges deposited by Is around the sharp corners of the aperture). The same is true of the system of Fig. 3, with its uniform current confined to the upper surface of the conductor. Any differences between the radiation fields in the systems of Fig. 3 and Fig. 4 must therefore arise from the difference between the two surface current distributions. Subtracting the (uniform) surface current of Fig. 3 from that of Fig. 4 yields the distribution sketched in Fig. 6(a). Far from the aperture, of course, the perturbation caused by the aperture is small and the two surface currents must cancel out. In the vicinity of the aperture we find two loops of current circulating in opposite directions, as well as positive and negative charges in those regions where the divergence of the local current is non-zero. As shown in Fig. 6(b), these circulating currents are equivalent to a pair of oppositely oriented magnetic dipoles +m and -m (i.e., a magnetic quadrupole, assuming their separation is much less than a wavelength); the charges localized on the aperture’s sharp corners give rise to an oscillating electric dipole p. Thus, adding the dipoles p and ±m to the system of Fig. 3 should transform it over to the system of Fig. 4.
Figure 6(c) shows that, in the vicinity of the aperture, the combined radiation pattern of the electric dipole and the magnetic quadrupole consists of a circulating B-field around the major axis of the ellipse, and an E-field distribution that tends to stay away from the long side-walls of the aperture. These fields, when added to the E- and B-fringes of Fig. 3, produce the field profiles of Figs. 4 and 5. The circulating magnetic field around the ellipse’s major axis in Fig. 6(c) is responsible for the bending of the B-lines toward and into the hole, as sketched in Figs. 4 and 5(a). Similarly, superposition of the E-field pattern of Fig. 6(c) with the uniform E-fringe that exists above an apertureless mirror gives rise to the E-field pattern of Fig. 4 in the XY-plane immediately above the aperture.
In practice, the metallic film has a finite thickness, and the combined radiation by the dipole p and quadrupole ±m of Fig. 6(b) must vanish within the body of the film. To this end, the magnetic dipoles may have to tilt sideways, one to the right and the other to the left, so that everywhere inside the metal film their E- and B-fields will be cancelled by the corresponding fields of the electric dipole. Physically, the sideways tilt of the ±m dipoles is a consequence of the induced surface currents on the interior side-walls of the aperture, which currents also help to support the B-field adjacent to these side-walls; see Fig. 5(a).
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.
All in all, the primary source of radiation through the aperture of Fig. 4 seems to be the ±m quadrupole depicted in Fig. 6(b); the induced dipole p in this system is relatively weak and plays a secondary role, namely, canceling the quadrupole’s radiation inside the metal film. (The weakness of the dipole p is borne out by the results of our numerical simulations. It is also evident that, in the limit of large aspect ratio, when the ellipse approaches an infinitely long slit, the dipole p must vanish.) In general, quadrupolar sources are weak radiators, thus accounting for the weakness of transmission through an elliptical aperture illuminated by a plane wave whose polarization direction coincides with the major axis of the ellipse.
Figure 7 shows computed plots of Ex,Ey,Ez in the XY-plane located 20 nm above the surface of the conductor in the system of Fig. 4. The simulated conductor is a 124 nm-thick film of silver (n+ik=0.226+i6.99 at λ=1.0µm) having an 800 nm-long, 100 nm-wide elliptical aperture. The magnitude of each field component is plotted in the top row of Fig. 7, and the corresponding phase profile appears below it. For our purposes, the main utility of the phase distribution is to indicate the relative orientation of the various field components. For instance, if the phase of Ey at a given location happens to be ϕo, then if the phase of Ex at that location turns out to be equal (or nearly equal) to ϕo, we will know that Exx̂+Eyŷ oscillates back and forth between the first and third quadrants of the XY-plane. However, if the phase of Ex hovers around ϕo±180°, then Exx̂+Eyŷ oscillates between the second and fourth quadrants.
The E-field distribution of Fig. 7 is consistent with the qualitative behavior sketched in Figs. 4, 5(b), and 6(c). The Ex component bends the central field lines toward the middle of the aperture, and pushes the peripheral lines further way, thus ensuring that the long side-walls repel the parallel E-field. The Ey component is strengthened near the center of the aperture because the field lines are pushed upward and squeezed laterally toward the center. Finally, the Ez component confirms the presence of charges of opposite sign at and around the sharp corners of the aperture. These pictures are consistent with the presence of a weak electric dipole and a magnetic quadrupole in and around the elliptical aperture.
Computed amplitude and phase plots of Ey,Ez in the central YZ-plane of the aperture are shown in Fig. 8. The bands of Ey above the aperture are the standing-wave fringes created by the interference between the incident and reflected beams. The weak nature of transmission through the aperture is evident in the very small perturbation of the fringes, as they sag ever so slightly to fill the top of the aperture. The profile of Ez, once again, confirms the accumulation of electric charges around the sharp corners of the hole. Moreover, it shows that the charges on the top facet of the metal film, while much stronger than those on the bottom facet, have the same sign as the charges on the bottom; in other words, the top and bottom charges are both positive at one end of the ellipse, and both negative at the opposite end.
Figure 9 shows plots of Hx,Hy,Hz in the XY-plane 20 nm above the surface of the conductor, while amplitude and phase plots of Hx and Hz in the central XZ-plane appear in Fig. 10. As expected from the preceding discussion of Figs. 4 and 5, the magnetic fringe nearest the surface is seen to leak into the aperture by bending the H-lines near the corners of the ellipse toward the center and down into the hole.
Computed plots of Ex,Ey,Ez in the XY-plane 20 nm below the conductor are shown in Fig. 11, and the corresponding H-field distributions appear in Fig. 12. While the profiles of these fields confirm the behavior expected from our earlier qualitative analysis, their small magnitudes testify to the weak nature of radiation by the ±m quadrupole (and the accompanying p dipole) induced by the incident beam in the vicinity of the aperture of Fig. 4.
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.
Figure 13 shows distributions of the magnitude |S| of the Poynting vector in various cross-sections of the system of Fig. 4. The superimposed arrows on each plot show the projection of S in the corresponding plane. For instance, in the XZ cross-section depicted in Fig. 13(a), the arrows represent Sxx̂+Szẑ, whereas in the YZ cross-section of Fig. 13(b) the arrows correspond to the projection Syŷ+Szẑ of the Poynting vector on the YZ-plane. The plots in Figs. 13(c) and (d) show the distributions of |S| in the XY-planes immediately above and below the aperture. In the absence of an aperture, S is essentially zero everywhere, as the reflected beam cancels out the incident beam’s energy flux. When the aperture is present, however, the fields are redistributed in such a way as to draw the incident optical energy toward the aperture. In the present case, the energy flows in from the periphery, fails to find a way through the aperture, bounces back and returns toward the source in the region directly above the aperture. In the process, several vortices are formed, where the incoming energy makes a sharp turnaround and heads back toward the source.
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.
Figure 13(d) shows that the Poynting vector S=1/2 Real (E×H*) at the bottom of the hole has a magnitude |S|~2.5×10-6 W/m 2, consistent with the transmitted E-field of ~0.06 V/m and H-field of ~2.3×10-4 A/m, considering the large phase difference of Δϕ~70° between the E- and H-fields near the bottom of the aperture; see Figs. 11 and 12. Since the incident planewave is assumed to have Ey=1.0 V/m, Hx=Ey/Zo=2.65×10-3 A/m (free-space impedance Zo~377Ω), which correspond to an incident energy density ~1.32×10-3 W/m 2, the power transmission efficiency η at the center of the elliptical aperture of Fig. 4 is seen to be just under 0.2%. For apertures or slits, η is defined as the ratio of |S| at the aperture’s center just below the conductor to the incident plane-wave’s optical power density, |S inc|~1.32×10-3 W/m 2. We will see in the next section that when the incident polarization is rotated 90° (to point along the minor axis of the ellipse), the transmission efficiency through the aperture increases to η~93%, a nearly 500-fold improvement.
5. Elliptical aperture illuminated with plane-wave polarized along the short axis
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, as shown in Fig. 14. 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. In the area surrounding the hole, the E-field produced by these dipoles bends the Is lines toward the mid-section of the aperture as shown in Fig. 14, and as required for self-consistency.
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.
Aside from the incident beam, all the radiation in the system of Fig. 14 is produced by the surface currents Is and the charges created by these currents. Subtracting the (uniform) surface current in the system of Fig. 3 from that in Fig. 14 thus yields the current distribution of Fig. 15(a), which is responsible for the difference between the radiation patterns in the two systems. When added to the uniform current of Fig. 3, the currents of Fig. 15(a) produce the Is pattern shown in Fig. 14. The current loops of Fig. 15(a) are equivalent to a pair of oppositely oriented magnetic dipoles, +m and -m, while the charges deposited on the long sides of the aperture constitute an electric dipole p; see Fig. 15(b). Figure 15(c) shows that, in the XY-plane immediately above the aperture, the E-field is dominated by the electric dipole p. The contribution of the magnetic dipoles is to enhance the E-field at the center of the aperture, while weakening it in the outer regions.
Figure 15(d) shows that in the XY-plane directly above the aperture, the B-field profile is shaped by competition between the electric dipole p and the magnetic dipoles ±m. The electric dipole dominates near the center but, further away, the magnetic dipoles dictate the B-field’s behavior. The dotted B-lines near the sharp corners of the ellipse in Fig. 15(d) show the field leaving the XY-plane to enter/exit the hole vertically (i.e., in the Z-direction). Although not shown in this figure, B-lines that enter the hole from above, close the loop by circling beneath the metal film and returning through the hole to reconnect with the B-lines above the film; see Fig. 16.
The surface charges and currents of Fig. 15(a) create magnetic fields in the free-space regions inside the hole as well as those above and below the metal surface. The B-field of the electric dipole p combines with that of the magnetic dipoles ±m to produce closed loops in the vicinity of the aperture, as shown in Fig. 16. The solid B-lines in this figure bulge above and below the metal surface, while the dashed lines hug the conductor’s top and bottom surfaces. (The B-field cannot penetrate into the conductor, but, as it emerges from the hole, it bends above and below the surface in such a way as to bring the field lines close to the metallic surface.) In all cases, the lines of B must form closed loops to guarantee the divergence-free nature of the field. Since neither E- nor B-fields can exist within the conductor, the fields radiated by the electric dipole p must cancel out those of the magnetic dipoles ±m everywhere inside the metallic medium. The radiation emanating from these dipoles, however, permeates the interior of the hole as well as the free-space regions on both sides of the conductor. To get in and out of the hole, the B-lines of Fig. 16 appear to descend through one of the current loops that constitutes a magnetic dipole in Fig. 15(b), then return through the other loop. Note the change of direction of the magnetic field at the upper surface of the elliptical aperture: the direction of B just above the hole is opposite to that beneath the hole’s upper surface. This 180° phase shift, dictated by the presence of the (uniform) Is on the top surface of the elliptical aperture in Fig. 15(a), will disappear when the fringes of Fig. 3 are added to the fields produced by p, m and -m to yield the total field in and around the aperture. The induced electric charges on the surfaces surrounding the aperture produce an oscillating E-field in the short gap between the long side-walls as well as in the regions immediately above and below the aperture. The time rate of change of this field, ∂ D/∂ t, which is equivalent to an electric current density J across the gap, creates circulating magnetic fields around the short axis of the ellipse [5]. These B-fields by themselves, however, are not sufficient to explain the field profile depicted in Fig. 16, and must be augmented by the fields produced by the circulating currents around the ellipse’s sharp corners (i.e., the±m dipoles) to yield a complete picture. Moreover, inside the metallic medium, the E- and B-fields of the p dipole cannot vanish without the compensating contributions of the ±m dipoles.
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. (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. 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. 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. 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. (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.
Figure 17 shows cross-sections of the system of Fig. 14 in YZ- and XZ-planes. Since Is lags 90° behind the incident E-field immediately above the aperture, the accumulating charges on and around the side-walls produce electric fields opposite in direction to the incident E-field. The E-lines may now start on positive charges and end on negative charges (∇·D=ρ), as shown in Fig. 17(a). This change of direction of the E-field causes a 180° phase shift in Ey from above to below the aperture. The E-fringe just above the aperture thus becomes weaker, sharing some of its energy with the E-field inside and below the aperture.
The XZ cross-section of the system of Fig. 14 depicted in Fig. 17(b) shows how the oscillating electric dipole p and magnetic dipoles ±m push the B-fringe above the aperture upward and sideways to make room for circulating B-fields that surround the short axis of the elliptical aperture. The resulting redistribution of the magnetic energy of the B-fringe above the hole thus makes it possible for some of the energy stored in this fringe to leak into the hole as well as the space below the hole. (The B-field distribution inside the aperture and in the region below the metal film is the same as that in Fig. 16, since the added fringes contribute only to the half-space above the conductor.) The divergence-free nature of the B-lines requires their continuity, which is evident in Fig. 17(b), in contrast to the E-lines of Fig. 17(a), which break up whenever they meet electrical charges.
Figure 18 shows computed plots of Ex,Ey,Ez in the XY-plane 20 nm above the surface of the conductor in the system of Fig. 14 (top row: magnitude, bottom row: phase). The strong z-component of E indicates the presence of significant amounts of electric charge on the conducting surfaces in the vicinity of the hole; the sign-reversal of Ez from one side of the hole to the other shows that the charges on the two sides have opposite signs. Figure 19, left panel, shows the amplitude and phase of Ey in the central XZ-plane, while the right panel shows Ey,Ez in the central YZ-plane. Inside and below the aperture Ey is seen to be strong, and to have reversed direction relative to the E-field immediately above the aperture; its energy appears to have been extracted from the E-fringe directly above the hole. The distribution of Ez shows, once again, the presence of electric charges on the top and bottom surfaces of the conductor; these charges have the same sign on the top and bottom surfaces on either side of the hole, but their sign is reversed in going from the left-side to the right-side.
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. (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.
Computed plots of Hx,Hy,Hz in the XY-plane 20 nm above the conductor’s surface appear in Fig. 20. Figure 21, left panel, depicts the amplitude and phase of Hx,Hz in the central XZ cross-section, while the right panel shows the distribution of Hx in the central YZ-plane. The magnetic field’s behavior in these pictures is in accord with the qualitative behavior sketched in Fig. 17(b). Note, in particular, that the profile of Hz in Fig. 21 resembles the z-component of the circulating B-field in Fig. 17(b). Note also the draining of magnetic energy out of the B-fringe above the hole, and its redistribution not only in the form of magnetic fields inside and below the aperture, but also in the enhanced values of Hx directly above the conductor’s surface.
Plots of Ex,Ey,Ez in the XY-plane 20 nm below the bottom surface of the conductor are shown in Fig. 22, and the corresponding magnetic-field plots appear in Fig. 23. These pictures are in full agreement with the qualitative diagrams of Figs. 15–17.
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. 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.
Figure 24 shows distributions of the magnitude |S| of the Poynting vector in various cross-sections of the system of Fig. 14. The superimposed arrows on each plot show the projection of S in the corresponding plane. For instance, in the XZ cross-section depicted in (a) the arrows represent Sxx̂+Szẑ, whereas in the YZ cross-section of (b) the arrows correspond to the projection of the Poynting vector on the YZ-plane, namely, Syŷ+Szẑ. The plots in Figs. 24(c) and (d) show the distributions of |S| in the XY-planes 20nm above and below the aperture. In the absence of an aperture, S is essentially zero everywhere, as the reflected beam cancels out the incident beam’s energy flux. When the aperture is present, however, the fields are redistributed in such a way as to draw the incident optical energy toward the aperture. The energy flows in from the region directly above as well as from the periphery of the hole in every direction. In addition to the straight-ahead energy, some of the peripheral energy also goes through the hole, thus enhancing the overall transmission. Further away from the aperture, especially in the YZ-plane (which contains the ellipse’s short axis), the peripheral incoming energy turns away from the aperture and returns to the source.
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.
The magnitude of the Poynting vector in the center at the bottom of the hole is |S|~1.23×10-3 W/m 2, which is consistent with the transmitted E- and B-fields of ~1.6 V/m and ~2.14×10-3 A/m, with a phase difference Δϕ=ϕE-ϕB~45° (see Figs. 22 and 23). The transmission efficiency of the optical power density at the center of this aperture is thus η~93%, which is nearly 500 times greater than that obtained when the incident polarization was parallel to the ellipse’s long axis. (η is the ratio of |S| at the aperture’s center just below the conductor to the incident plane-wave’s optical power density, |S inc|~1.32×10-3 W/m 2.) Several factors appear to have contributed to this strong performance (compared to the case of parallel polarization), among them, more electrical charges and stronger surface currents (especially on the bottom facet of the conductor), and a greater separation between the ±m magnetic dipoles, which tend to cancel each other out when they are close together.
6. Concluding remarks
We have analyzed the transmission of light through small elliptical apertures in a thin silver film at λ=1.0µm. Both cases of incident polarization parallel and perpendicular to the major axis of the ellipse were considered. The transmission efficiency η was found to be low for parallel polarization and high for perpendicular polarization.
The hallmark of the low-transmission case was a weak excitation of electric and magnetic dipoles on the upper surface of the metal film, which produced even weaker excitations on the lower surface. Although not described here, we have observed similar behavior for a circular aperture (diameter =100nm, silver film thickness =124nm, η=0.06% at the center of the aperture 20nm below the conductor), and also for an infinitely long, 100nm-wide slit (η=0.14% at the center of the slit 36nm below the bottom facet; incident polarization parallel to the slit). For the elliptical hole under low-transmission conditions, η drops rapidly with an increasing film thickness h, from 0.2% at h=124nm, to 0.008% at h=186nm, and to below 0.001% at h=248nm. It appears that the elliptical hole, when considered as a waveguide,[8, 9, 11] does not support any guided mode whose E-field is predominantly aligned with the ellipse’s long axis.
The high-transmission ellipse revealed the excitation of fairly strong electric and magnetic dipoles on the upper surface of the metal film, which induced even stronger dipoles on the film’s lower facet. In this case η remains high for thicker films as well (η=93% for h=124nm, 86% for h=186nm, and 136% for h=248nm), indicating propagation through the hole (along the Z-axis) of a guided mode whose E-field is largely parallel to the ellipse’s short axis. We also found that an infinitely long, 100nm-wide slit exhibits strong transmission for an incident polarization aligned with the narrow dimension of the slit (η~69% at the center of the slit, 36nm below a 124nm-thick silver film).
It thus appears that achieving a large η requires an aperture that can excite strong oscillator(s) on the upper surface of the film, which would then induce strong oscillations on the lower facet, thereby creating the conditions for the passage of a substantial amount of electro-magnetic energy through the subwavelength opening in the metal film. The ability of a hole (or slit) to support a guided mode that can be excited by the incident polarization appears to be critical for achieving large transmission, especially for thicker films. We mention in passing that the relative strength of the induced charges on the top and bottom surfaces of the silver film was seen to vary with the film thickness h. For thick films, we suspect the coupling strength of the guided mode both into and out of the aperture (i.e., at the entrance and exit facets, located on opposite sides of the film) to determine the relative strength of the induced charges and currents at the opposite rims. Recent reports of various aperture designs that have significant throughputs (compared with simple circular or square-shaped apertures)[4, 10, 12] indicate that the aforementioned principles, far from being specific to elliptical holes in thin metal films, have a broad range of application.
Acknowledgments
The authors are grateful to Dennis Howe, Pavel Polynkin, Ewan M. Wright, and Pierre Meystre of the Optical Sciences Center, and to M. Brio of the Department of Mathematics for many helpful discussions. Thanks are also due to Professor Motoichi Ohtsu and Dr. K. Kobayashi of the Tokyo Institute of Technology for their valuable comments on our manuscript. This work has been supported by AFOSR contract F49620-03-1-0194.
References and links
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