1. Introduction

The analysis of light transmission through a slit with a subwavelength width in a thin plate has started in early days, dating back to Lord Rayleigh [1,2]. Recently, Ebbesen et al. observed extraordinary optical transmission through sub-wavelength hole arrays [3]. Because of these unexpected demonstrations, light propagations in periodic arrays of nano-slits and nano-apertures have received much attention due to their extraordinary behavior and potential applications in biologic and photonic devices [4–10]. In addition to the researches on the periodic arrays, a few studies on single nano-slit have also been performed [11–14].

To our best knowledge, however, there has not been an intensive study on the near-field properties of nano-metallic double-slit, specifically on the interference of evanescent waves in the near-field region. Considering that interference patterns of a double-slit in the far-field region is distinct from diffraction patterns of single and periodic slits, interference patterns of a double-slit in the near-field region is expected to be salient. A few authors have already studied interference of evanescent waves or surface plasmons. Bainier et al. reported the interaction of an evanescent standing wave with nanometer-sized objects [15] and Novotny et al. investigated surface plasmon interactions on a finite silver layer [16]. However, the interference of evanescent waves in nano-metallic double-slit has not been reported. In this letter, therefore, we present the characteristics of nano-metallic double-slit in near- and far-field regions using two-dimensional finite-difference time-domain (FDTD) method. With this analysis, we have distinguished several differences between the two interference patterns of nano-metallic double-slit in the near- and far-field region. We believe that the differences originate from the fact that evanescent waves are dominant in the near-field region whereas propagating waves are dominant in the far-field region.

 

Fig.1. Schematics of modeled space of a nano-metallic double-slit. Soft source is used for the illumination.

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Figure 1 shows the modeled space of a nano-metallic double-slit. The PEC in the figure describes a material whose electric conductivity is infinite, where the initials stand for the perfect electric conductor, i.e., a perfect metal. The nano-metallic slit is assumed to be composed of perfect metal and illuminated by a Gaussian beam of λ=500 nm. Perfect metal means that both skin depth and the amplitude of electric field are zero at the metal surface. In addition to the analysis of nano-metallic double-slit with no cladding, the slit with a dielectric cladding was also analyzed to assure the possibility that evanescent waves can be enhanced and confined with the dielectric cladding. Besides, numerous simulations were performed varying the distance between the two slits, the slit thickness, and the thickness of the dielectric cladding, i.e., the values of d, t, l. It is assumed that the width of both slits, i.e., the value of a, is 50 nm and the refractive index of the glass substrate and the dielectric cladding is 1.5. The origin of the modeled space is the center of the double-slit on the line just one sampling point right from the metal/air interface, as shown in Fig. 1.

2. Field distribution due to polarization direction

When both the structure to be modeled and the incident wave are uniform in the z-direction, we can split Maxwell’s curl equations into two groups. One group consists of only H_x, H_y, and E_z, and the other only E_x, E_y, and H_z. The field components of the first group are called transverse-magnetic modes with respect to z (TM_z) in two dimensions and the other transverse electric modes with respect to z (TE_z) [17]. We can notice that the polarization of incident light in the TM_z mode is normal to the plane, i.e., x-y plane and that the TE_z mode is parallel to the plane. With the assumption of uniformity in the z-direction, therefore, we can calculate the near- and far-field properties of nano-metallic double-slits using two-dimensional FDTD simulation for two polarization directions. There are a few differences between the TM_z modes and the TE_z modes for the different polarization direction. The first is the difference of transmission. For the TM_z modes, because electric field is tangential to the metal surface in the slit, light cannot transmit through the slit unless the slit is thin enough. Therefore, transmission decreases exponentially as the thickness of the slit increases. For TE_z modes, however, high transmission can be acquired since light can be guided well by metal-slits.

Another difference is the position and shape of the peak at the exits of the two slits, as shown in Fig. 2, which shows the intensity distribution in the region after the slit/air interface for the TM_z and TE_z modes. In the result for the TM_z mode, light is confined to the middle of the slit since the amplitude of electric field is diminished to approximately zero in the region near the metal surface. For the TE_z mode, however, we can see that two sharp peaks appear at the edges of slits. This is due to the fact that light propagates via metal surface at the edges, and not between the slits. This result is consistent with the result of Wei et al. [11].

In Figs. 2(c) and (d), we can see that the double-slit forms interference fringes in the near-field region as well as in the far-field region and the interval between fringe maxima is half of wavelength as expected. The interference pattern is also different for the two modes. Interestingly, in the near-field region, interference patterns of TE_z modes seem to have a phase difference of π with respect to that in far-field region. We can easily discern the phase difference from the different line profiles of Fig. 2(d). The dashed line, i.e., the intensity profile in the near-field region, reaches a valley at the center while the intensity of the far-field reaches a peak at that point. From the difference, we can say that the two interference patterns in near- and far-field regions have phase difference of π for TE_z modes. In addition, the intensity variation of the near-field interference pattern is greater than that of the far-field interference pattern. In contrast to TE_z modes, it is hard to find any distinct difference between near- and far-field interference patterns for TM_z modes. Actually, it is even difficult to identify near-field interference for TM_z modes. From these results, we can conclude that the interference patterns of TE_z modes change their characteristics as light propagates from the near-field region to the far-field region. The differences of the interference patterns in near- and far-field regions will be further discussed in the next section. Since, the distinction appears only for TE_z modes, we will discuss the results of TE_z modes simulation from now on.

 

Fig. 2. Intensity distribution of the two-dimensional simulation when the slit width is a=50nm and slit thickness is t=20 nm. (a) Shaded display of TM_z mode intensity distribution when the distance between two slits; d=1µm. (b) Line profile of (a). (c) Shaded display of TE_z mode intensity distribution when the distance between two slits is d=1µm. (d) Line profile of (c). Solid line represents the line profile just above the metal/air interface, dashed line is the line profile of x=50 nm, and dotted line represents the line profile of x=1µm. The insets of (b) and (d) are the magnifications of dotted and dashed line profiles.

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3. Interference of nano-metallic double-slits in near- and far-field regions

In the far-field region, propagating waves generate interference patterns. In the near-field region, however, interference patterns are principally shaped by evanescent waves since evanescent waves are stronger than propagating waves in near-field region. Therefore, interference patterns may have different characteristics from that of the far-field region. Figure 3 shows other displays for the results of Figs. 2(c) and (d), featuring the differences between two interference patterns in near- and far-field regions. As expected, in the far-field region, constructive interference occurs at the line of y=0, which is illustrated in Fig. 3(d). In contrast to the interference pattern of the far-field region, that of the near-field region is destructive at the points of equal distance from two slits. As pointed out above, we believe that the difference is originated from the fact that the evanescent wave, appearing as an electric field normal to the metal/air interface, is dominant in the near-field region and out of phase at the exits of two slits whereas the propagating wave is dominant in the far-field region and in phase at the exits.

 

Fig. 3. Interference patterns in near-field and far-field regions when d=1µm and λ=620 nm: (a) Intensity distribution from the entrance of the two slits, (b) Magnified intensity distribution of the region between the two slits after the metal/air interface, (c) contour display of (b), (d) far-field interference pattern.

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Because both waves in the near- and far-field regions emanate from the same slits, it seems to be inconsistent with the fact that the electric field at the exits of the two slits is out-of-phase in one region, while they are in phase in another. However, considering the superposition principle of electromagnetic waves and the fact that evanescent waves do not propagate to the far-field region, this paradox can be easily explained. If there are mixtures of two modes at the exits of double-slit, one corresponding to the evanescent wave, having phase difference at the exits of slits and appearing as an x component of the electric field, and the other a propagating wave with no phase difference, appearing as a y component, the difference between near- and far-field interferences will arise.

Figure 4 confirms this explanation. The x component of the electric field distribution at the slit edges appears anti-symmetric about the center of the double-slit whereas the y component is symmetric. Evanescent waves are assumed to be induced by E_x, the x-component of the electric field, at the slit exits since the x-axis is normal to the metal/air interface, and the propagating waves have the same polarization direction at the exit of slit as the incident waves, i.e., the direction of y-axis. We think that the phase difference originates from the fact that charge oscillations in the region between the two slits give opposite effects to each slit. Considering the differences between interference patterns in both regions, it is legitimate to say that the interference patterns evolve from the near-field region, where destructive interference occurs at the places where distances from the two slits are equal, to the far-field region, where constructive interference occurs.

 

Fig. 4. Electric field distribution when d=1µm and t=200 nm: (a) amplitude distribution of E_x (x component of electric field), (b) line profile of (a) at the metal/air interface, (c) E_y distribution, and (d) line profile of (c).

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In the case of a nano-metallic single-slit, the intensity distribution decays exponentially in the lateral direction as well as the longitudinal direction as light propagates. In a nano-metallic double-slit, however, the intensity distribution in the region between the two slits appears as standing waves. It is also found that the interval of nodes is half of the wavelength λ. We suppose that the intensified standing wave in the region between the two slits is also obtained with the help of evanescent waves, whose intensity in the near-field region is much greater than that of propagating waves in the far-field region.

The interference pattern varies as the distance d between two slits increases. Figure 5 shows the results of simulation with various values of d. When d <λ, the interference fringe in the near- and far-field regions is not clear. As d grows to be greater than λ, clear interference patterns emerge. Considering that two point sources cannot be easily identified as two separate light sources when the distance between them is smaller than a wavelength, this phenomenon may be easily explained. Since the two slits act as a single light source at the far-field region, no interference pattern is constructed in that region. Similarly, it is hard to observe clear fringes in the near-field region because two or more distinct sources are required to shape clear fringes. It would be better to explain the phenomenon with diffraction when d<λ as follows.

 

Fig. 5. Intensity distribution with various distances between two slits: (a) d=250 nm, (b) d=2 µm, (c) magnified display of the region between two slits when d=2 µm, and (d) magnified display of the region between two slits when d=5 µm.

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Because near-field interference patterns have phase differences of π from far-field interference patterns, there should be a boundary that separates the interference patterns of the near-field region from that of the far-field region. In Figs. 5(c) and (d), we can easily discern the boundary circle that separates the two interference patterns. We can also see that the radius of the circle increases linearly as the distance between two slits increase. Specifically, the distance from the center of the double slit to the boundary circle equals to a half of the distance d between the two slits. Because light decays as it propagates along the metal/air interface in real metal, the assumption of perfect metal is necessary for these ideal results. However, we expect that the difference between interference patterns in both regions will affect the field distribution in the case of real metals.

We can shift the near-field interference pattern as well as the far-field interference pattern by filling the slits with different dielectric materials. For example, a constructive interference pattern will be obtained at the center of the double-slit when the slit thickness is half of the wavelength and one of the slits is filled with Si₃N₄ whose refractive index is 2. Figures 6(a) and (b) show examples of constructive interference. Figures 6(c) and (d) illustrate other results corresponding to interference of three slits and interference between slits of different widths, respectively. With the simulation where three slits are employed, we can see that each of the two pairs of slits show independent and identical interference patterns, which is equivalent to that of the double-slit. The intensity difference of the peaks originates from the employment of a Gaussian beam for the illumination. From Fig. 6(d), we can verify that the destructive interference occurs at the center of the double-slit in spite of the different widths.

 

Fig. 6. Interference patterns with various conditions are presented. (a) and (b) correspond to the results when one of the slits is filled with silicon nitride (n=2) to introduce phase difference resulting in a phase shift of π. (c) Interference pattern with three slits. (d) Interference between slits of different widths.

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4. Evanescent wave enhancement by dielectric cladding

If a dielectric cladding is added to the double slit, more evanescent waves will be guided along the cladding/metal interface. Figure 7 shows interference patterns generated by the nano-metallic double-slit with a dielectric cladding of refractive index 1.5. Comparing Figs. 5(c) and 7(b), we can easily verify the enhancement of evanescent waves. In addition, evanescent waves decay more rapidly along the x-axis direction. In other words, evanescent waves are confined more effectively to the cladding/air interface.

 

Fig. 7. Intensity distribution with various slit intervals for a cladding thickness of 50 nm: (a) d=2 µm, (b) magnified display of the region between the two slits.

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Fig. 8. Line profiles of the intensity distribution along the y direction when (a) l=20 nm, (b) l=50 nm, (c) l=100 nm, and (d) l=200 nm.

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Figure 8 shows the intensity distributions with various cladding thicknesses l when the distance d between the two slits is 750 nm. The line profiles are obtained at the planes a few nano-meters apart from the cladding/air interface because the height of line profile reaches its peak at that position. In comparison with the case with no cladding, the interval between the nodes of the standing wave decreases due to the increase of the refractive index. We can also see that the intensity distribution at the cladding/air interface varies as the cladding thickness is changed. When the cladding is too thin, little enhancement occurs. However, as the cladding thickness l increases, evanescent waves in the region between two slits grow stronger while those outside this region decay exponentially identical to the case of no cladding. The enhancement of the evanescent waves in this region continues until l is approximately 50 nm. As l grows to be greater than 50 nm, the intensity distribution in the outer regions between the slits becomes stronger. When l is 100 nm, the height of the intensity distribution in the outside regions equals to that in the region between the two slits, as shown in Fig. 5(c). The intensity distribution reduces as the slit thickness grows to be greater than 100 nm. With the results of the simulations, we found that the intensity distribution in the region between the two slits is maximally enhanced when l is approximately 50 nm, and approximately 100 nm for the case of the outside region.

It can also be noticed that the interval between the nodes of the standing wave decreases slightly as the thickness increases. Since the cladding can be considered as a waveguide, we suppose that the change of the dominant spatial mode leads to the variation of the interval between nodes. In other words, we suppose that waves propagate along the direction which is more inclined to the x-axis as the cladding grows thicker.

5. Conclusion

The characteristics of nano-metallic double-slits, assuming a perfect metal, in both near and far-field regions were analyzed. With the finite-difference time-domain (FDTD) simulation of a nano-metallic double-slit, we can see that the interference patterns in the near-field region have many characteristics different from those in the far-field region. Specifically, the interference patterns in both regions have a phase difference of π with respect to each other. In addition, a boundary separating the near-field interference from the far-field interference appears as a half circle, whose radius is half of the distance between the two slits. We think that this difference originates from the fact that the near-field interference is determined by evanescent waves whereas in the far-field propagating waves are dominant. We also show that filling the slits with different dielectric materials can shift fringes in the near-field region as well as the far-field region. In addition, simulations for three slits and double slits of different widths are also performed.

It is also found that when a dielectric cladding is added to the double-slit, evanescent waves are enhanced and confined more effectively to the cladding/air interface. As the cladding grows thicker, the intensity distribution varies as in the following orders. 1) Intensity in the region between two slits increases. 2) Light easily propagates laterally through the cladding and the light intensity propagating to the outer region of the two slits becomes equal to that of the region between the two slits. 3) Light propagation to the outside region becomes weaker. We think that a resonator of evanescent waves can be constructed by placing a pair of reflectors using the fact that light easily propagates laterally along the metal/cladding interface.