Transmission of polarized light through sub-wavelength slit apertures is studied based on the electromagnetic field distributions obtained in computer simulations. The results show the existence of a cutoff for E ‖ and a strong transmission (with no cutoff) for E ⊥; here ‖ and ⊥ refer to the direction of the incident E-field relative to the long axis of the slit. These observations are explained by the standard waveguide theory involving inhomogeneous plane waves that bounce back and forth between the interior walls of the slit aperture. We examine the roles played by the slit-width, by the film thickness, and by the real and imaginary parts of the host material’s dielectric constant in determining the transmission efficiency. We also show that the slit’s sharp edges can be rounded to eliminate highly-localized electric dipoles without significantly affecting the slit’s throughput. Finally, interference among the surface charges and currents induced in the vicinity of two adjacent slits is shown to result in enhanced transmission through both slits when the slits are separated by about one half of one wavelength.
©2004 Optical Society of America
Since the experiments of Ebbesen et al showed extraordinary transmission through 2D arrays of holes in metallic films , the goal of theoretical research in this area has been to achieve an understanding of the transmission mechanism in 1D and 2D configurations [2–5]. In the meantime, experiments in the microwave regime have also demonstrated high transmission efficiency for a sub-wavelength slit aperture in a metallic absorber . This implies that the transmission mechanisms for gratings and single holes may be closely related, and that the periodicity of the structure may not be a requirement for efficient transmission, as originally stated by Ebbesen et al . The single-aperture transmission mechanism, however, has not received much attention despite its potential applications in photolithography, scanning microscopy, optical data storage, and light-emitting diode (LED) design.
Bravo-Abad et al have suggested a guiding mode mechanism for single apertures . Zakharian et al used Finite Difference Time Domain (FDTD) simulations to investigate subwavelength elliptical apertures in thin metallic films . They found low transmission when the incident E-field was parallel, and high transmission when the E-field was perpendicular to the long axis of the ellipse. These studies hinted at the existence of a cutoff for E ‖ illumination, while E ⊥ was found to excite a guided mode within the aperture. In Sections 3 and 4 we demonstrate the validity of this conjecture for slit apertures under E ‖ and E ⊥ illumination. Although there appears to be a consensus in the literature that the high transmission observed for periodic (i.e., grating-like) structures is caused by the excitation of surface plasmons in the metallic film [1,3], the details remain sketchy. We will see in Sections 4, 5 and 6 that, for a single slit, significant transmission occurs when the conduction electrons are localized at and around the edges of the aperture; the E- and H-fields then couple to surface charges and currents that collectively dictate the transmission properties of the slit.
The perfect conductor assumption, which is often invoked in theoretical treatments of the gratings, ignores the absorption of light by the metallic medium whose permittivity ε has finite real and imaginary parts, ε=ε′+iε″. In Section 7 we show that lack of significant absorption plays an important role in the visible and near-infrared range of wavelengths, where |ε″| has smaller values than in the infra-red and microwave regime. A detailed picture of absorption in the slit walls and vicinity emerges from the simulation results presented in Section 8. In Section 9 we invoke the standard waveguide theory to explain the existence of a cutoff for E ‖ illumination, and the high transmission efficiency of E ⊥ illumination. By varying the degree of sharpness of the slit’s edges in Section 10, we examine the role of edge sharpness in producing E-field hot spots, and verify that the edge profile does not substantially affect the efficiency η of transmission. To bridge the gap between a single slit and a periodic array of such slits (i.e., gratings), several two-slit configurations are simulated in Section 11, where, for given film thickness, η is found to depend on the separation between the two slits.
2. The simulation setup
A Gaussian beam of light, having full-width at half-maximum intensity FWHM=1.5µm at the waist (located at z=1200nm) is seen in Fig. 1 to propagate along the negative z-direction. The vacuum wavelength of the light, λo=1.0 µm, is in the near-infrared range (frequency ν=c/λo=3.0×1014 Hz). A silver film of thickness t centered at z=0 has a slit aperture of width W. The subwavelength aperture (i.e., W < λo) transmits a small fraction of the incident energy, while most of the energy is reflected back toward the source. Since illumination is uniform in the x-direction and the setup is independent of x, Maxwell’s equations are decoupled into E ‖ and E ⊥ modes, as indicated in Fig. 1.
The E ‖ mode consists of Ex, Hy and Hz field components (Ex is along the length of the slit). The E ⊥ mode consists of Hx, Ey and Ez , with Ey being along the width of the slit. We investigate the slit’s transmission mechanism by analyzing the computed field distributions in and around the aperture. The polarization state (E ‖ or E ⊥), film thickness t, slit-width W, dielectric constant ε, and the slit edge sharpness are used as variable parameters to study the corresponding variations in the transmission efficiency η of the slit. Using the z-component Sz of the Poynting vector S=½Real (E×H*), the efficiency η is defined as the ratio of the integrated Sz over the output aperture to that over the input aperture. Occasionally, we use the value of Sz in the middle of the output aperture (i.e., at y=0, z=-½ t) divided by the incident beam’s Sz at the center of the input aperture (i.e., at y=0, z=+½ t) as a measure of the slit’s transmission efficiency. It turns out, in the case of E ⊥ illumination, that the transmitted light is fairly uniform across the aperture and drops sharply to zero outside the slit-width. The integrated Sz is thus very nearly the same as Sz at the mid-point of the aperture (to within 1%), and the two definitions of efficiency yield essentially the same results.
In our FDTD simulations the permittivity ε of the metallic film is obtained from Debye’s polarization model, namely,
For silver at λ o=1.0 µm, τ=8.3692 fs, ε ∞=1.0, Δε=-12428.5, σ=1.31488×107 (ωm)-1. This material’s permittivity has a large negative real part, ε′, and a small imaginary part, ε″. To understand the relation between the efficiency η and the material’s permittivity ε, we will introduce in Section 7 different (artificial) materials with permittivities that differ from that of silver in either the real part, or the imaginary part, or both.
3. Cutoff for sub-wavelength slits under E‖ illumination
Figure 2 shows computed field distributions for a subwavelength slit (W=0.4λo) under E ‖ illumination. The weak field below the aperture indicates that almost no light passes through. (In contrast, when W>½λo, there is not much attenuation and the wave propagates along the slit waveguide, as will be seen in Section 5.)
The total electric field =+ at the top air-to-silver interface is essentially zero (see the |Ex | plot), while Hy in this region is at a maximum. The reason is that very little E-field is needed to set the conduction electrons in motion, and thus to establish the surface current Jx , at the top facet of the conductor. The surface current is equal to the magnetic field component Hy just above the conductor (Ampere’s law). Maxwell’s equations require Ex to be continuous at the interface, and negligible inside the metal. We thus find ≈0 at the interface. In contrast, continuity at the interface is not required for Hy because of the presence of the surface current Jx .
The and carry the electromagnetic energy in the -z direction, while the metallic film sends a strong reflected beam back in the +z direction. For a metal film with no apertures, these counter-propagating beams form a standing wave (period=½λo) above the upper surface; see the |Ex | and |Hy | plots above and away from the aperture. When a slit is cut through the film, there will be no local currents in the air-gap; then the incident Ex , no longer balanced by the reflected E-field, falls into the slit. The penetrating Ex creates a surface current Jx along the slit walls, which, in turn produces an Hz on the walls’ exterior surfaces. Since the currents on both slit-walls are in-phase, the phase of Hz on the two walls must differ by π, as seen in the Hz phase plot of Fig.2. Around the slit corners on the incidence-side of the aperture, the current Jx exists on the top metal surface (i.e., at z=½t) and also on the slit walls (y=±½W); the profiles of these currents may be inferred from the |Hy | and |Hz | plots in Fig.2. The H-fields at and near the surfaces are always in-phase with Jx , while Ex is ~90° ahead of the current Jx ; see the phase plots of Fig.2. The weak E-fields near the metallic surfaces indicate that very little E-field is needed to produce the currents that sustain the local H-fields.
These and similar simulations for E ‖ illumination revealed the existence of a cutoff at W≈½λ o for guided waves through the slit. When W<½λ o, very little light could get through the slit, and the incident optical energy, aside from a small fraction that was absorbed within the metal’s skin depth, was reflected back toward the source.
4. Transmission of E⊥ light through subwavelength slits
The simulations described in this and the following sections demonstrate the absence of a cutoff for E ⊥ illumination, even for an aperture as narrow as W=λ o/150. Transmission is seen to be the result of strong E- and H-fields that propagate along the slit walls, being supported by the appropriate distribution of surface charges and currents on these walls.
Figure 3 shows computed plots of Ey, Ez, Hx for a 100nm-wide slit in a 700 nm-thick silver film. As before, very little Ey is needed on the top surface to sustain the surface current Jy , which supports the magnetic field Hx immediately above the surface. The reflected Ey and Hx interfere with the corresponding incident fields to produce standing waves above the top surface. Jy stops abruptly at the edges of the slit, giving rise to accumulated charges at the sharp corners, where these (oscillating) charges on opposite edges of the slit behave as an electric dipole. The surface charges — being proportional to the perpendicular component of the local E-field at the metal surface — may play a role in enhancing transmission through the slit. (Surface charge profiles can be examined through the Ez -field on horizontal surfaces and the Ey -field on vertical surfaces. Similarly, the surface currents Jy and Jz can be inferred from the distribution of the Hx -field near the metal surfaces.) Needless to say, the charge density ρ and the current density J are related through the continuity equation ∇·J+∂ρ/∂t=0 .
Inside the slit, the surface charges and currents carry the traveling beam along the negative z-direction. When this beam reaches the bottom of the aperture, it creates a second electric dipole, which generates an upward-traveling wave inside the slit. The counter-propagating beams then form a standing wave along the z-axis, which redistributes the charges and currents in accordance with the interference pattern of the E- and H-fields.
In the time-harmonic steady state, the relative motion of the charges and currents can be inferred from the phase plots of Fig.3. The charges are accumulated in three places on each side of the slit: in the middle of the wall (at z=0 in the |Ey | plot) and at the upper and lower corners of the slit (at z=±½ t in the |Ey | and |Ez | plots). The currents are especially strong along the walls at z=±¼ t, and on the top surface of the film; see the |Hx | plot. Since the E-field starts on positive charges and ends on negative charges, the charges on the left and right sides of the slit have opposite signs; see the Ey, Ez phase plots. The phase of Ez (or Hx ) below z=0 being 180° shifted relative to that above z=0 indicates that the conduction electrons move back and forth between the center of the wall (z=0) and the top and bottom corners (z=±½t). At the upper left corner of the aperture, for instance, electrons are deposited by the current on the left slit-wall as well as that on the top film surface. The 90° phase delay between the currents and the corresponding charges is in accord with the charge continuity equation. The phase of the charges on the slit corners (see the Ez phase plot) differs by ~90° from the phase of the current along the slit walls (see the Hx phase plot), while it differs by less than 90° from the current on the top/bottom surfaces. Thus, the dipoles derive their charge more from the slit walls’ current than from the current on the top/bottom surfaces. Also, the current on the slit walls is apparently stronger than that on the top surface; see the |Hx | plot.
The case of t=700nm is not the only one with large transmission. At t=300nm, for which the field profiles are shown in Fig.4, strong electric dipoles once again appear at the sharp edges of the slit (see plots of |Ey | and |Ez |), and large transmission is subsequently observed. The difference with the case of t=700nm is that, at z=0, the charges disappear while the wall surface current in that region reaches a maximum; see the |Hx | plot. Also, between the slit’s top and bottom, Ey shows a phase-shift of ~180°. It is evident that a strong current along the slit walls feeds the top and bottom dipoles. As before, the currents on the top surface of the film are weaker than those on the slit walls, but they contribute to the localized charges near the slit edges as well.
The case of t=500nm is shown in Fig.5, where only one strong dipole is observed (at the bottom of the slit). The charges that produce the dipole at the top of the aperture have diminished (see plots of |Ey | and |Ez |), and the transmission efficiency is substantially reduced compared to the preceding cases of t=700nm and 300nm.. The current on the top surface of the film, which brings charges to the upper corners and spreads them over the slit walls down to z=0, is the dominant current in the upper half of the slit. In the lower half, the approximately 90° phase difference between the wall current and the concentrated charges at the bottom corners identifies the wall current as the source of the charges. It appears that the destructive interference between the two counter-propagating beams within the slit is responsible for the reduced transmission efficiency in this case.
5. Dependence of E‖ and E⊥ transmission on film thickness
In Fig.6(a) we have plotted the component Sz of the Poynting vector at the output aperture for films of differing thickness t; the-slit width is fixed at W=100nm and illumination is E ⊥. From the figure the transmission efficiency η is seen to be ~200% when t=300nm and 700nm, but only ~70% when t=100nm, 500nm, and 900nm.
Figure 6(b) shows plots of Sz at the output aperture under E ‖ illumination for several values of the film thickness, when W=400nm<½λo (i.e., below cutoff). The graphs indicate that, although there is some transmission for very thin films, Sz decays rapidly with an increasing film thickness until transmission essentially drops to zero beyond t ~ λo. Increasing the slit-width above W=½λo, however, removes all obstacles to transmission. As shown in Fig.6(c), when W=600nm, transmission efficiency η in the middle of the slit is ~200%, and the dependence of Sz on film thickness is rather insignificant.
Figure 6(d) is a summary of the results for both E ‖ and E ⊥ illumination; here the integrated Sz immediately after the apertures studied in (a)–(c) is displayed as function of film thickness t. For E ⊥, the transmitted optical energy is seen to vary periodically with thickness (period ~400 nm); the curve’s envelope drops gradually because of the absorption in the slit walls. For E ‖, the throughput of a narrow aperture (W=400 nm) drops exponentially with film thickness, but remains fairly constant for an aperture above the cutoff (W=600nm).
6. Enhanced transmission efficiency for narrow slits
Integrated transmission efficiency versus the silver film thickness t was computed under E ⊥ illumination for different slit widths ranging from W=100nm (see Fig. 6(d)) to W=50nm, 20nm, 10nm, and 6nm; see Fig. 7. In each case, maximum transmission is obtained by fine-tuning t, which is a specially sensitive adjustment for narrower slits. The efficiency is seen to increase with a decreasing W, although not fast enough to make the total amount of light that passes through the slit an increasing function of the slit-width. These findings are in agreement with the theoretical results of Bravo-Abad et al , who reported enhanced transmission efficiencies in the microwave regime for subwavelength slits in a perfectly conducting host material. A high-quality-factor resonator appears to have been set up within the slit, and a traveling wave trapped between the entrance and exit apertures of the waveguide. For narrow slits, the impedance mismatch between the slit waveguide and the exterior (free-space) regions is so great that the waveguide becomes a high-Q cavity, thus enhancing the efficiency of transmission by promoting strong oscillations within the slit.
7. Effect of material parameters
One conclusion from the preceding E ⊥ simulations is that the surface currents create a pair of strong electric dipoles (at the top and bottom edges of the slit), which dipoles then radiate into the slit and beyond. It is entirely possible, however, that the strong dipoles near the top and the bottom of the slit are related to the bulk modes (i.e., modes excited in the bulk of the host material). The bulk modes are confined at the top and bottom surfaces of the silver film within the material’s skin depth. Under such circumstances, the transmission efficiency of the slit is only indirectly affected by the localized dipoles near the slit’s edges, namely, to the extent that the bulk modes and the guided (slit) modes collectively determine the coupling efficiency of the incident beam to each and every mode. Either way, the propagating mode within the slit is sustained by the charges and currents of the conduction electrons in and around the slit walls.
Dissipation of the surface currents within the slit walls — resulting in optical-to-thermal energy conversion — depends on the imaginary part ε″ of the host material’s permittivity. The effects of both the real part ε′ and the imaginary part ε″ of ε may be studied by comparing materials of differing permittivities. (In FDTD simulations there is no need to restrict the material to a perfect conductor, a shortcoming of many theoretical investigations.) In this section we study the contributions of ε′ and ε″ to transmission efficiency by varying their values separately as well as jointly. Table 1 lists the parameters of silver and three related (artificial) materials used in our next set of simulations. Both the real and imaginary parts of ε contain the effects of bound as well as conduction electrons, since these are indistinguishable in the macroscopic Maxwell equations . (It may be argued that the artificial materials in Table 1 cannot be subjected to direct experimental tests and, as such, it would be better to stick with silver and vary the wavelength from the intraband region to the interband absorption region — while at the same time scaling the slit-width — in order to study the effects of interest in an experimentally verifiable way. This approach, however, will not be able to clarify the separate roles played by ε′ and ε″, which is our primary goal in the present study.)
At normal incidence, the reflectivity R of a plane surface having dielectric constant ε is given by
Silver has a large, negative ε′ together with a fairly small ε″, which makes it a good reflector with very little absorption; its reflectivity for a plane wave at normal incidence is R=98.2%, with the remaining 1.8% of the incident energy being absorbed within the skin depth. Material_I has the same ε′ as silver but a much larger ε″ ; its reflectivity at normal incidence is R=81.8%. Material_II has a much weaker ε′ but the same ε″ as silver, and a poor reflectivity of R=41.6%. The permittivity of Material_III is three times that of silver (both real and imaginary parts), and its reflectance is R=99%.
Figure 8 shows the simulation results for the four materials listed in Table 1 under E ⊥ illumination; in all cases the slit-width W is 100nm. The film thickness t is 700nm for Silver, Material_I, and Material_II, while it is 750nm for Material_III; this choice of t allows for maximum throughput in all cases. For Material_I, the induced dipoles are relatively weak and transmission efficiency is substantially below that of silver. This is caused by the greater absorption of Material_I compared to silver, which could be caused, for instance, by the material’s larger electrical conductivity. For Material_II, almost no light gets through to the other side, even though Ey is quite strong at the entrance side (the induced dipole is visible on the upper edges of the aperture). The effect of large absorption is clearly seen not only in the reduced transmission through this slit, but also in the deeper penetration of the light into the film’s various exposed surfaces. For Material_III, which has a slightly better reflectivity than silver, the slit’s throughput is perceptibly improved. This is reminiscent of most metals in the microwave regime , where an extremely large electrical conductivity renders the material essentially a perfect reflector.
8. Absorption along the walls and near the edges of the slit
In all cases that exhibit high throughput under E ⊥ illumination, strong electric dipoles are seen to have been created at the sharp edges of the slit aperture. However, transmission is not always large when strong dipoles are present, as is the case, for instance, for Materials I and II in Fig. 8. For these materials, loss from absorption along the slit walls is plotted in Fig. 9 and compared with the corresponding loss for silver. Because the behavior of the Poynting vector S near the corners is complicated, we plot only Sy — the component of S perpendicular to the slit wall — for one of the walls in the range z=[-270 : 270 nm].
The fact that Sy in Fig. 9 is positive indicates that optical energy flows from the air-gap into the slit wall. Absorption peaks and valleys in Fig. 9 occur at the same locations where the H-field of the slit has maxima and minima (these also coincide with the minima and maxima of the E-field). Large absorption thus occurs where the surface current Jz is intense, whereas low absorption is the hallmark of a weak local Jz . Interference between counter-propagating waves (originated at slit’s top and bottom) is responsible for the oscillation of the absorption curves of Silver and Material_I in Fig. 9. For Material_II, the absorption curve does not exhibit oscillatory behavior, because the reflected mode at the bottom of the slit is fairly weak.
The energy flow mechanism into the aperture can be explained by the profile of the Poynting vector S(y, z) in and around the slit. Figure 10 shows logarithmic plots of |S (y, z)| for Silver, Material_I, and Material_II. (Logarithmic plots mitigate the enormous variations of S between the air and the material regions.) In all cases, the optical energy in the incidence space is seen to flow from the nearby regions of the slit toward and into the slit. In Silver, absorption is weak and the energy flow inside the walls, pumped by the light that passes through the slit, seems to be opposite in direction to the flow within the slit itself. The energy flux into Material_I is perpendicular to the air-metal interfaces, and the power entering the host medium is seen to be rapidly absorbed. For Material_II the index contrast between the air and the medium is small, resulting in a fairly strong flux of optical energy into the slit walls. Within the slit, the mode’s energy declines rapidly along the propagation path; the lost energy being fed to the walls and converted to heat.
9. Slit as a waveguide
For E ‖ illumination, the boundary conditions require that Ex =Hy ≈0 on the slit walls at y=±½W. Apparently, it is not possible for these fields to build up to any substantial levels in the middle of the aperture when the slit width W is small. In contrast, for E ⊥ illumination, the main field components are Ey and Hx , both of which can assume large values on the slit walls because the boundary conditions relate these fields to the charge and current densities on the walls’ metallic surfaces. The E- and H-fields of E ⊥ illumination thus propagate down the slit aperture, with mild attenuation caused by absorption in the walls . For E ⊥ illumination, therefore, as long as ε′≪0 and |ε″| is not too large, a propagating slit mode exists and strong transmission occurs, as was seen to be the case for Silver and Material_III. The coupling efficiency η is dependent on the wavelength λo, film thickness t, slit-width W, and the permittivity ε of the slit’s host medium.
Maxwell’s equations can be set up and solved for the modes of a waveguide comprising a pair of flat, metallic mirrors separated by an air-gap . For E ‖ illumination, the field components of a plane-wave, whether in the air-gap or in the metallic region, are given by
In general, σy =σ′y+iσ″y and σz =σ′z+iσ″ z are complex numbers, thus encompassing the range of homogeneous and inhomogeneous plane-waves (including evanescent waves). For a beam propagating in the positive z-direction, σ″z must be positive. In the air-gap region of the slit, Maxwell’s equations require +=1, yielding . Inside the metallic mirrors, σ z has the same value as in the air-gap, but σ y must satisfy the relation +=ε, namely, , where the σy on the right-hand-side is . For an even mode of the air-gap the field profile in the interval -½W≤y≤½W is the superposition of two plane-waves of the type given by Eq.(3), that is,
with similar expressions for Hy and Hz . Matching the boundary conditions at y=½W yields:
Given the values of λo, ε, and W, one can solve Eq. (5) numerically to obtain acceptable value(s) of σy for the guided mode(s). In the E ‖ cases studied in Section 5, for instance, when W=0.5λo we find σy =0.917-0.00245i. This leads to σ z =0.3989+0.0056i, whose small imaginary part implies a weakly attenuated guided mode. However, for W=0.1λo we find σy =3.402-0.0366i, which leads to σz =0.0383+3.252i and strong attenuation. Similarly, the results for W=0.4λo (where σy =1.123-0.0036i, σz =0.0079+0.51i) and for W=0.6λo (where σy =0.775-0.0017i, σz =0.632+0.0021i) agree with the findings of Section 5.
A similar analysis for E ⊥ illumination shows that the field profile for a plane-wave in the waveguide’s metallic walls is given by
Similar expressions describe plane-waves in the air-gap, provided that we set ε=1.0 and , whereas in the metal . Once again we write the even mode of the air-gap as a superposition of two plane-waves, as in Eq. (4), and match the boundary conditions at y=½W to obtain the following guiding condition:
Here, as in Eq. (5), σy stands for . In contrast to the E ‖ case discussed earlier, we now find that inserting W=0.1λo into Eq. (7) yields σy =0.0117-0.685i. This corresponds to σz =1.2121+0.0066i, whose small imaginary part implies a weakly attenuated mode. The real part of σ z represents an oscillation period λo/1.2121=825nm for the guided mode, consistent with the simulation results. Similarly, for W=0.5λo under E ⊥ illumination, we find σy =0.0056-0.315i, which yields σz =1.049+0.0017i, for another weakly attenuated mode.
As for the various materials discussed in Section 6 in conjunction with a W=0.1λo slit under E ⊥ illumination, we find for Material_I that σz =1.1333+0.0733i. Since the imaginary part of this σz is more than an order-of-magnitude greater than that for silver, it should come as no surprise that the slit in Material_I exhibits a greater attenuation. For Material_II we find σz =1.5512+0.4764i, in line with its substantial rate of decay. Finally, σz =1.1253+0.0039i obtained for Material_III is consistent with its observed strong transmission.
10. Sharpness of the edge
Up to this point, in all our simulation results with E ⊥ illumination, the E-field plots as well as plots of the Poynting vector component Sz at the output facet show two spikes (often referred to as “hot spots”) near the sharp edges of the aperture. We reduced the degree of sharpness at these corners and studied the effect on the field distribution as well as on the transmission efficiency of the slit. Figure 11 shows computed plots of Hx for a 100nm-wide slit in a 700nm-thick silver film for four different radii of curvature (r=0, 30nm, 40nm, 60nm). It must be emphasized that, since in the non-uniform mesh used in our FDTD simulations the pixel-size in the region of the slit was 2.0 nm, the rounded slit edges were well resolved.
Overlapped on each frame in Fig. 11 is a plot of Sz at the exit facet. It is readily observed that, with the increasing roundness of the corners, the sharp spikes in Sz disappear. (We also confirmed the disappearance of the E-field’s hot spots.) The spikes are thus mere artifacts associated with the slit’s sharp edges. The Sz profiles in Fig.11 are seen to become broader, while their peak values (at the center of the aperture) drop with the increasing roundness of the corners. The total transmitted optical power through the aperture, however, is not much affected until the corner radius reaches ~60nm, at which point the slit’s throughput is down by ~20%. For the range of corner radii studied, we found the effect on the phase profile of the fields (not shown) to be insignificant.
11. Pair of adjacent slits
Since the extraordinary transmission of subwavelength hole arrays was reported by Ebbesen et al , the relation between single-aperture transmission and that of a periodic array of such apertures (i.e., grating-like structures) has been a topic of debate. In this section we confine our attention to the interaction between a pair of slits in a fairly thick metal film, and present preliminary simulation results that clarify the nature of cooperation between adjacent apertures; a cooperation that results in enhanced transmission. More detailed studies of a periodic array of such slits will be the subject of a forthcoming paper.
Figure 12 shows the distribution of Ez in the case of E ⊥ illumination for two identical 100nm-wide slits in a 700nm-thick silver film. From left to right, the center-to-center spacing of the slits is d=200nm, 500nm, and 900nm. The white curves beneath each slit show Sz (y) at the exit facet of the apertures. When d=200nm, the separation is so small that the positive and negative charges around the edges of the two slits tend to cancel each other out, thus weakening the throughput of both slits. As the distance between the slits widens to ~500nm, the interaction between the surface charges and currents near the edges of the slits intensifies and the transmission reaches its maximum, which is ~50% greater than that for each slit alone. Beyond d=500nm, the interaction weakens and, by the time d reaches ~900nm, the two slits have begun to act more or less independently.
Figure 13 shows the distributions of Ey, Ez, Hx (top row: magnitude, bottom row: phase) for a pair of 100nm-wide slits in a 700nm-thick silver film under E ⊥ illumination. The center-to-center spacing of the slits is d=600 nm. In addition to the asymmetric distribution of charge near the slits’ edges (visible in the |Ez | plot), note, in the |Hx | plot, the presence of a strong surface current in between the two slits on the film’s bottom facet. Inside each slit the fields have left-right symmetry, which indicates the absence of excited odd modes.
The aforementioned interaction between adjacent dipoles on the same surface of a film containing two or more slits raises a similar question concerning the interaction between the two dipoles on the opposite ends of a single slit. The latter case, however, is not easy to investigate because adjusting the distance between these dipoles requires changing the film thickness, which, as observed in Section 5, tends to disturb the interference pattern between the upward and downward propagating modes of the slit waveguide.
12. Summary and concluding remarks
Our FDTD simulations confirmed the polarization dependence of transmission through slit apertures in metallic films. For E ‖ illumination (i.e., incident E-field aligned with the slit’s long axis), the transmission efficiency η decays rapidly with film thickness t if the slit-width W<½λo. On the other hand, η does not vary much with the film thickness if W>½λo. For E ⊥ illumination, where the incident E-field is perpendicular to the slit direction, there is no cutoff for any value of W and, for silver at λo=1.0 µm and W=100 nm, for example, η can be as high as 200% for certain values of the film thickness t. Even larger values of η can be achieved by reducing W and optimizing t. For W=6.0 nm and t=195 nm, for instance, we found η≈1700%. Enhanced efficiency, however, is not synonymous with increased throughput, as the slit-width W shrinks faster than the efficiency η can rise.
Under E ⊥ illumination, transmission efficiency η varies with the film thickness t. The efficiency attains a maximum when t is such that the slit can hold an integer number of the standing-wave fringes within its depth. Under these circumstances, large amounts of electric charge accumulate near the top and bottom edges of the slit, while traveling waves constructively interfere within the slit waveguide.
Absorption within the slit walls was found to play a role in determining the slit’s efficiency by reducing the effective transmission, especially for thicker films . Simulations based on different material parameters indicate that, in the visible/near-infrared range of frequencies, high transmission should be easier to achieve with some materials than with others: Assuming that everything else is equal, the host material having a larger electrical conductivity exhibits greater losses. However, if conductivity becomes extremely large, as it does for good metals in the microwave regime, then the material approaches a perfect reflector, rendering transmission losses through the slit waveguide negligible.
The sharpness of the aperture’s edges were found to be of little consequence in determining the optical throughput of subwavelength slits. The accumulated charges near the sharp edges correspond to an excited mode of the slit waveguide that decays rapidly and thus fails to reach the opposite end of the slit. These rapidly decaying modes, whose presence is indeed essential for matching the boundary conditions at the entrance and exit facets of the aperture, tend to disappear when the sharp edges are rounded. Their absence, however, does not seem to substantially affect the efficiency of transmission through the subwavelength slit.
Double-slit simulations showed the existence of electromagnetic cross-talk between the two slits. In the particular case studied, transmission through both adjacent slits was enhanced when the separation between them reached ~½λo [14,15]. Judging by the appearance of the electromagnetic field profiles within the slits, this enhancement is not caused by the excitation of additional (e.g., odd) modes in the slits. Rather, we believe the efficiency improvement is due to an increase in the effective coupling coefficient between the incident beam and the guided mode of each slit.
The authors are grateful to Alexandre Brolo and Reuven Gordon of the University of Victoria, British Columbia, for their insightful comments on the manuscript. This work has been supported by the AFOSR contract F49620-03-0194 and by the NSF contract DMS 0335101.
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