A directional coupling effect of surface plasmon waves (SPW) in subwavelength metallic slits is studied. The p-polarized wave generates two SPW modes in the subwavelength slit. As the angle of incidence is changed, coupling arises between both SPW modes. The coupling length increases exponentially with the width of the slit but is independent of the angle of incidence. At a coupling length and an incident angle of ~30°, light is emitted from one side of the slit. The single side emission light has a width smaller than 50nm and double peak intensity than at normal incidence. The SPW coupling effect reveals a simple way for producing a nanometer light source of high transmission intensity.
©2005 Optical Society of America
Strong relationships exist among biological structures, fluorescent information and the physiological function of bio-molecules. J.O. Tegenfeldt et al.  used a micro-nano fluidic device to cause fluorescent beads to flow over a near-field light source, a subwavelength slit, and thus break the optical diffraction limit in a fast detection scheme. This near-field bio-scanner, which operates in a manner that is the inverse of near-field scanning optical microscopy (NSOM) with a fixed near-field source, has the potential to map genomes directly on DNA molecules of chromosomal length in a robust manner and at high speed without complex scanning system. Although a near-field slit can provide a subwavelength resolution of the order of 200nm, it does not suffice for mapping information of the genome or other biological structures. Reducing the slit width reduces the resolution to 50nm by confining the light wave to a 30nm-wide slit.  Such a narrowing also greatly reduces the optical throughput and raises difficulties in detection. Most of the energy is transformed into heat that is dissipated to the ambient environment. The extraordinary transmission phenomenon recently discovered by Ebbesen et al.  increases the throughput by many orders of magnitude when surface plasmon resonances occur in periodic metallic structures. A near-field light source cannot be achieved with ultra-high optical resolution and high optical intensity when a nanometer-sized slit is used without periodic surroundings.
P-polarized and s-polarized light waves have very different characteristics when they propagate through a subwavelength slit. The electric field component of the p-polarized light wave that is normal to the metal-dielectric interface excites a surface plasmon wave (SPW) that propagates along this interface , but s-polarized light cannot yield an SPW because the boundary condition is different and it has no electric field component that is normal to the interface. SPWs cause p-polarized light to have much higher transmission efficiency than s-polarized light in nano metallic slits . Unfortunately, the presence of two peaks at the silt opening is associated with a detrimental effect. The SPW increases the FWHM (full-width half-maximum) of the p-polarized light. The FWHM can be greatly reduced if the SPW can be emitted from only one side of the slit.
Most researchers have studied the normal incidence of light waves through nano metallic slits. – Symmetrically incident light causes equivalent SPWs to arise at slit edges. However, the symmetric condition does not apply when light is obliquely incident. This change results in an interesting phenomenon and supports the possibility to solve the problem associated with a near-field light source. This work explores the propagation behavior of light through a subwavelength slit at various angles of incidence. Finite-difference time-domain (FDTD)  studies demonstrate a directional coupling between two SPW modes. The directional coupling effect occurs when two identical waveguides are very close. There are two modes, fundamental symmetrical mode and first-order asymmetrical mode, exist in this dual waveguides system. Due to different velocities of both modes, optical energy is transferred from one waveguide to the other as shown in Fig. 1(a). In a metallic slit, the metal-air interfaces are waveguides for SPWs. Because of the subwavelength separation, both SPW waveguides are coupled together. Normal incidence only excites symmetrical mode. Off-angle incidence can excite both symmetrical and asymmetrical modes and cause SPW coupling as shown in Fig. 1(b). The SPW coupler takes a zigzag path. We found at a coupling length and an incident angle of ~30°, most of the optical power in the slit can be emitted from single edge of the slit. The single edge emission forms a nanometer light source without the loss of too much transmission intensity through a subwavelength slit.
2. Theoretical simulation
The simulation model is a 5μm-thick aluminum film with w-wide single slit, infinite extent along the y-axis on the glass substrate. Therefore, 2D FDTD computation is feasible, as shown in Fig. 2. The refractive index of the glass substrate is 1.45. Air is above the metallic slit. We used a commercial code (FullWAVE, RSoft Photonics) to perform the simulations. The Drude dispersion model is applied to simulate the metallic structure.
Where ωp is the plasmon frequency, and vc is the collision frequency. Both values are determined by the complex refractive index (0.8884+6.466i for Al film) at the reference wavelength (λ=532nm). The simulation domain is 0.8μm wide and 6μm high. The grid size is 2nm for Δx and 2nm for Δz. The time step is 1nm. The perfect matching layer (PML) conditions are set for all boundaries. It is noted that a guiding mode is usually launched for a conventional waveguide coupler. In our simulations, the guiding SPW mode is not used for the input field. Instead, a continuous plane wave with different angle of incidence and polarization is launched from the glass substrate. The reason is that the slit is much smaller than the focused spot of a laser, whose amplitude and phase are uniform in the focal plane. Therefore, the plane wave incidence is reasonably applied for real cases. Besides, the surplus modes in a conventional waveguide coupler would propagate a long distance and interfere with guiding modes. However, in the subwavelength metallic slit, those surplus modes are under cut-off conditions and can not propagate in the slit. Only SPW waves can guide along the slit walls. Hence, the plane wave is used as the input field.
Figure 3 presents the simulated results for various slit widths. The slit widths were 100nm (<λ/2), 300nm (<λ) and 600nm (>λ). The incident angle was 30°. The light was p-polarized, and so stimulated collective electron oscillations and produced SPW that propagated along both walls. Light is symmetrically distributed in the 100nm-wide slit, but exhibited a zigzag pattern in the 300nm-wide slit. Figure 3(d) shows the movie of the propagation of light in the 300nm-wide slit at various stop time. The light follows a zigzag path. When the propagation length is ~0.3 μm, optical energy is transferred from one side to the other side. If the thickness of the slit equals this length, then light can be emitted from just one side of the slit. Oblique incidence causes only single-side emission rather than the two peaks at the slit opening formed when the light is normally incidence. Such single emissive light can be used as an ideal nanometer light source. Notably, when the slit width exceeds the wavelength, not only SPW modes but also other guiding modes are present in the slit as shown in the 600nm-wide slit. The optical transmission efficiency is high, but the spatial resolution is poor.
The concept of coupling between two optical waveguides  is exploited to illustrate above zigzag propagations of light through a single slit. Figure 4(a) depicts a directional waveguide coupler, in which the metal-air interfaces behave as two optical waveguides for the SPW. When two waveguides are sufficiently close to each other, their evanescent waves overlap, causing considerable energy transfer. Based on normal-mode theory for a waveguide coupler, the slit is regarded as a waveguide system that has only two guiding modes -symmetric (SPW(0)) and asymmetric (SPW(1)). The electric field in the slit is a linear combination of both waveguide modes, and the optical intensity is described as,
Where Δβ is the difference between the propagation constants of both SPW modes and α is the absorption constant. For normally incident light, only the SPW(0) mode can be excited, yielding a symmetric distribution in the slit. An obliquely incident light waves can excite both SPW(0) and SPW(1) modes. Both modes propagate through the slit with various velocities, so their interference yields nodes in the slit and causes energy to flow from one side to the other side. However, when the slit width is less than half of the wavelength, only the fundamental symmetric mode is retained. Changing the incident angle does not excite the SPW(1) mode. Therefore, only a symmetric field is present in the 100nm slit, as shown in Fig. 3(a). For a larger slit, both SPW(0) and SPW(1) modes are excited. Their interference yields a zigzag path in the slit, as shown in Fig. 3(b). The length that causes the energy to be transferred from one side to the other side is defined as the coupling length (Lc), and equals π/Δβ. When w=300nm, Lc is ~0.3μm and Δβ is ~10.47μm-1. If the slit is widened further, not only the SPW modes but also the propagation wave in the slit are involved in coupling, yielding a complicated power distribution as shown in Fig. 3(c). Therefore, the width that yields only two SPW modes in the slit is estimated to be between λ/2 and ~λ. In this range, the slit can be modeled as a directional SPW coupler. For a conventional waveguide coupler, the coupling length is exponentially increased with the waveguide separation. Figure 4(b) plots the calculated periods (twice coupling length) for slit widths from 260nm to 440nm. The coupling length data can be well fitted by Lc = A exp(B×w), where w is the width, A=76.889nm and B=0.007nm-1.
Another important characteristic of the waveguide coupler is that the coupling length is independent of the method of excitation, indicating that the coupling length is independent of the incident angle. Figure 5 displays the optical power distribution in the slit for oblique angles of incidence of 10°, 20° and 40°. The width of the slit was 300nm and the incident wavelength was 532nm. The zigzag path is indeed independent of the incident angle, confirming the coupling behavior. If the zigzag path is caused by to-and-fro reflections of light in slit walls, then the path must vary with the incident angle. Figure 5(d) shows the optical power distribution for s-polarized wave with an incident angle of 30°. Light is confined in the slit and is symmetrical at the output. No node pattern exists in the optical field. It is known that s-polarized wave can not excite SPW. Hence we conclude that coupling of SPW modes is the mechanism for the zigzag propagations. Notably, the coupling length does not vary with the incident angle, but the magnitudes of SPW(0) and SPW(1) are strongly related to the angle of incidence. Ensuring that the magnitudes of both SPW modes are similar is important for a completely transfer of light from one waveguide to the other. When the slit thickness equals to the coupling length, a suitable incident angle can make light emitting only from single edge of the slit opening. To find the suitable angle, we plot the relative optical intensities at various angles of incidence. Figure 5(e) shows the result. The relative optical intensity is defined as the peak optical density on the right side over the peak density on the left side at a coupling length. It is found that incident angle of between 30° and 40° can achieve a relative magnitude of over fifty. At these angles, most of the light is confined to one side of the slit.
Above calculations show that slit width determines the coupling length and the angle of incidence decides optical power distribution between slit walls. Because the zigzag path in the slit is due to the coupling effect of SPWs, it is expected that kinds of metals and surrounding medium also affect the coupling behavior of SPWs. Figures 6(a), 6(b) and 6(c) show the optical power distributions in three different metallic slits: Ag, Al and Au slits. The slit width is 300nm and the surrounding medium is air. The optical wavelength is 532nm with 30° incident angle. It can be seen that coupling length is dependent on metals. The alumina slit has a little shorter coupling length than that of a silver slit. The gold has the longest coupling length. However, in gold slit, light decays greatly after propagating for ~0.7μm. The coupling length is also dependent on the surroundings. For example, Fig. 6(d) shows the calculation result for alumina slit in water surrounding. Compared with Fig. 6(b), the coupling length is increased from 0.3μm to ~0.6μm.
The coupling behaviors for different metals and surroundings can be elucidated by the coupled-mode theory.  The coupling coefficient is related to the overlap of evanescent waves between two neighboring waveguides. A large overlap of evanescent waves causes a strong waveguide coupling and results in a short coupling length. In a metallic slit, the metal/air interfaces are regarded as two identical waveguides for SPWs. The propagation constant of SPW is calculated by , where εm is the dielectric constant of metal, β is the real part of propagation constant and α is the absorption coefficient. The depth of the evanescent wave is calculated as . A smaller β has a larger dsp, which results in a larger overlap of evanescent waves and a shorter coupling length. Notably, the propagation length of SPW is short due to large absorption of metal. The length is estimated by Lsp =1/2α. In order to have a low-loss single-side emission from a subwavelength slit, the propagation length should be much longer than a coupling length. Table 1 shows dielectric constants, the calculated propagation constants and propagation lengths of SPWs at 532nm wavelength for different metals and surroundings. The metals are Ag, Al and Au. The surrounding mediums are air and water. Comparing the propagation constants, it is found that Al slit has longest evanescent wave. Hence its coupling length is the shortest among these metals. The result is consistent with the FDTD simulations. When the slit is put in water, the β is substantially increased. For example, β of SPW is ~12 at the Al/air interface. It is increased to ~16 at the Al/water interface. The large β confirms that the coupling length is greatly increased in the water environments as indicated in Fig. 6(d). For the propagation length, SPWs in Al and Ag slits have lengths much longer than a coupling length. However, SPW in a Au slit has a propagation length of ~0.7μm, close to a coupling length. The gold slit has a large absorption at 532nm wavelength. It can not be used as a coupler as demonstrated in Fig. 6(c).
We have demonstrated that SPWs in a 300nm-wide Al or Ag slit has a coupling length as short as ~300nm. The absorption in the slit can be neglected due to the long propagation lengths of SPWs. Hence, either Al or Ag is a suitable metal for studying the SPW coupling effect in a subwavelength slit. If the slit thickness is equal to a coupling length, it is expected that light is emitted only from one side of slit opening. To demonstrate the advantages of the single side emission, we calculate the optical intensity distribution near the slit opening. Figure 7 shows optical intensity distribution at 10nm away from the slit surface. The metal is Al. The slit width is 300nm and its thickness is equal to a coupling length (i.e. 300nm for air, 600nm for water). The dashed purple line at top image is the result for normal incidence. It shows two peaks at the slit edges. These two peaks come from the symmetric SPW mode in the slit. The optical resolution is similar to the slit width, 300nm. Nevertheless, in the same slit but changing the incident angle to 30°, light is emitted just from right edge of the slit as shown in the blue line. The FWHM of the emission light is greatly reduced to ~44nm. In addition, the peak intensity is twice that the peak intensity at normal incidence.
The off-angle illumination takes advantages of high spatial resolution and high transmission intensity through a subwavelength slit. Therefore, it can be used for massively manufacturing nano structures by using the photolithographic process. Previous studies  using the interference pattern of SPWs on the surface of multiple metallic slits to make periodic nano structures. Using off-angle illumination, it is possible to make a single nanometer line or non-periodic structures. For example, the green line in the middle of Fig. 7 shows a calculation result for exposing a single line on photoresist. The calculation is performed by putting a photoresist layer (n=1.7+0.001i) on the slit surface. The result demonstrates a single 37nm-wide line on the photoresist.
The off-angle illumination can also be used for generating a nanometer excitation source for scanning biomolecues in water environment. The red line in the bottom of Fig. 7 shows a calculation result for a slit in water. Note that the coupling length in water is longer than that in air. The slit thickness in this calculation is changed to 600nm. The resultant emissive light has a high optical resolution (~50nm). Although this resolution can be achieved by using a 50nm-wide slit. The off-angle illumination has much higher optical intensity than that in a 50nm-wide slit.
The single-side emission can provide a nanometer light source at the slit edge. It is noted that such light source has very large near-field components. Its intensity will decay exponentially and FWHM of the light increases very quickly with the propagation distance. Figure 8(a) shows the calculated peak intensity distribution as a function of height for different surroundings (air and water). The metal is Al. The slit width is 300nm and thickness is equal to a coupling length (300nm for air, 600nm for water). The incident angle is 30°. The intensity plots show the exponentially decays for the nanometer light sources. They decay to one-tenth of the original intensity at ~40nm propagation distance. Figure 8(b) shows the calculation results for the FWHM as a function of height. The FWHM increases from several nanometer to 100nm when light propagates ~20nm distance. The water has a larger refractive index than the air, hence the increase of FHWM is slower at a longer propagation distance. From the intensity and FWHM plots, it is suggested that interactions in close proximity to the metallic slit is very important for nanometer photolithography and biological imaging.
FDTD simulations were performed to explore the propagation behavior of obliquely incident light in a subwavelength slit. The propagation can be described by the directional coupling of two SPW modes. An appropriate slit width, ~ λ/2 to λ, is required to generate both SPW modes without the other propagation waves in the slit. An incident angle between 30° to 40° causes most of the optical energy to be transferred from one side to the other side. If the thickness of the slit equals the coupling length, then the optical energy can be emitted from only one edge of the slit. An oblique incident angle provides a near-field light source with a higher intensity and a much narrower width than are obtained with normal incidence. The light source can be used in a near-field bio-scanner to increase optical resolution. It can also be used in the photolithographic process to make nanometer structures.
The authors would like to thank the National Science Council of Taiwan, for financially supporting this research under Contract No. NSC_93-2215-E-001-001, and the support of the Nano Program of Academia Sinica, Taiwan.
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