1. Introduction

Optical transmission through subwavelength apertures in periodically structured, optically opaque metal films that exceeds classically predicted values [1] is of potential technological importance in the areas of near-field microscopy, nanolithography, optical data storage, and display technology [2, 3]. This effect has been displayed by a variety of periodic structures including hole arrays [1], gratings [4], bull’s eye structures [5], and single linear apertures flanked by periodic grooves [6]. Since the first observation of enhanced transmission through small apertures [1, 7] there has not, however, been a consensus in the literature as to the specific mechanisms involved in the phenomenon.

Initially, this effect was attributed to a resonant grating excitation of a surface plasmon polariton (SPP) [8] due to the periodicity of the surface structure. In this view, the SPP mode interacts with the aperture giving enhanced transmission. Using this theory, the wavelengths of transmission resonances, λ_max, for normal incidence are predicted to be [3]

where a ₀ is the periodicity of the surface features, n_SP is the effective index of refraction of the surface plasmon, m is an integer, and ε_m and ε_d are the wavelength-dependant complex dielectric constants for the metal and dielectric media, respectively, that form the interface. There has been disagreement, however, about how effective the resonant SPP model is at predicting the wavelength of maximum transmission [4, 9, 10, 11]. Equation 1 predicts that transmission maxima will be confined to a narrow spectral range defined solely by the periodicity. In fact, there have been observations that changing the separation between the grating and aperture while leaving the period fixed has a dramatic effect on peak position as shown in Ref. [12] for a bull’s eye grating structure. Below, we show the same behavior for a linear grating structure.

Recently, because of the differences between the transmission enhancement wavelengths predicted by Eq. 1 and experimental observations, it has been suggested that the effect is not connected with SPPs at all [12], but, rather, that surface waves distinct from SPPs denoted as composite diffracted evanescent waves (CDEWs) are responsible. Ref. [12] claims that peak location is determined by an interference effect between the CDEWs and the incident wave. The CDEW either constructively or destructively interfere with the incident field at the aperture, causing either enhancement or suppression of transmission. Although the peak location is well predicted by an interference model, in other recent experiments, the surface wave has been reported to have both CDEW and SPP-like properties [13].

In this paper, we present results obtained experimentally and theoretically from Au film structures consisting of a subwavelength, linear aperture centered symmetrically in a smooth cavity between two finite grating structures. Transmission is measured as the cavity width (the spacing between the grating structures) is varied. We show that the wavelengths where the maximum transmission enhancement is observed are not related to the periodicity of the grating structure, but rather the wavelength of maximum transmission can be tuned by changing the location of the aperture with respect to the edge of the grating structure. We use these results to confirm that the enhanced transmission does arise from interference between a surface wave and the incident field in agreement with Lezec and Thio [12]. However, in contradiction to their results, testing the properties of the surface wave, we find that the surface wave is an SPP, and not a CDEW. In addition, we find that because the two sets of gratings behave as mirrors for the SPP, resonant surface cavity modes also exist and make a significant contribution to the transmission maxima and minima.

2. Experimental and modeling details

 

Fig. 1. (a) Schematic cross-section of structure geometry. Cavity width, w_C, was varied from 800 – 2000 nm; t_Au = 200 nm; t_SiN = 200 nm; depth of grooves cut into SiN was 75 nm; P = 400 or 450 nm; and aperture width was 150 nm at substrate and 450 nm at air side. Light is normally incident on the glass side of the film. (b) Experimental absolute transmission spectra (top) and FEM transmission spectra (bottom), for varying cavity widths. All curves are for P = 400 nm. Note the red shift as w_C is increased.

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Figure 1(a) presents a schematic cross-section of the structure used in this study. A layer of SiN was deposited on a glass slide followed by a 10 nm Au charge dissipation layer. Fifty micron long grooves were cut into the SiN using a combination of electron-beam lithography and broad-beam Ar-ion milling. A 2.5 nm Ti adhesion layer followed by an opaque Au layer were then sequentially evaporated onto the substrate. We note Ti only contacts the SiN at the bottom of the grooves because the Au charge dissipation layer is still present elsewhere. This process produces gratings on both the SiN and air side of the film. Prior studies have shown that the transmission spectrum is primarily influenced by the grating on the input side [14], which in this case, takes the form of raised ridges in the Au at the SiN/Au interface as shown in Fig. 1(a). Finally, a linear aperture also fifty microns long was milled through the Au film. This process results in naturally sloped side walls. All parameters were confirmed using a combination of scanning electron microscopy (SEM) and atomic force microscopy (AFM).

Transmission measurements were made using a micro-transmission system with an effective 20 μm diameter collection area. Light was normally incident on the glass side of the structures. Sample transmission was normalized to the transmission from a blank substrate supporting only the SiN layer. Rotatable polarizers allowed the selection of TM polarized light (electric field perpendicular to grooves) or TE polarized light (electric field parallel to grooves). Consistent with prior studies, TE polarization shows an essentially featureless spectrum.

Modeling was performed using a commercial, finite element model (FEM), Comsol Multi-physics, of the harmonic Maxwell’s equations. Because the length of the grooves and aperture are much greater than characteristic lateral geometric parameters such as cavity width, a 2-dimensional model was used. The FEM assumed a TM-polarized plane wave. The dielectric constant used for the glass was 2.25 as quoted by the manufacturer, while measured values obtained by ellipsometry were used for SiN and Au. Perfectly matched layers [15] were used to truncate open edges of the FEM. The modeled transmission was calculated by integrating the flux through a line 0.5 μm below the aperture. The FEM was also used to look at field behavior, including phase information, around the structure. The time-averaged power flow (Poynting vector) was found to be the most effective field to display interference and other important phenomena [16].

3. Discussion of results

The transmission spectra for TM polarization from measurement and the FEM for several cavity widths are shown in Fig. 1(b). In both measurement and FEM, the maxima are asymmetric and broadened suggesting the presence of more than one peak. Minima are similarly asymmetric with modeled minimum transmission approaching zero in some cases. The measured minimum transmission is less than the transmission through a plain aperture with no surrounding grooves (not shown) [17]. In fact, two effects have clearly been identified in the FEM, as will be discussed below.

To assist in interpreting the measurement, the maxima in the measured spectra were analyzed as the sum of two Gaussians which provided a good fit. The Gaussian peak locations for all the experimental structures studied are shown in Fig. 2 as a function of cavity width, w_C. The gray scale background in Fig. 2 is a density plot of the calculated transmission from the FEM. In this representation, transmission enhancement and suppression bands are clearly visible, and the experimentally determined peak locations are in good agreement with the calculated regions of enhanced transmission. We also note that the peak locations systematically shift up in wavelength as the cavity width is increased, and that changing the periodicity of the grooves from 400 nm to 450 nm for a fixed cavity width has little or no systematic effect on peak location. As a matter of fact, Eq. 1 predicts very different enhancement locations for the two wavelengths: λ_max = 768 nm for 400 nm period, λ_max = 844 nm for 450 nm period. Clearly, Fig. 2 shows Eq. 1 does not accurately predict enhancement wavelengths. The curves in Fig. 2 are explained in Section 5 of the paper. The agreement between experimental and FEM spectral shape in Fig. 1(b) and peak position in Fig. 2 lends confidence in the FEM and allows us to use the model to investigate the microscopic behavior of the electromagnetic fields in the vicinity of the structure.

 

Fig. 2. Peak location as a function of cavity width, w_C. Gray scale background is a linearly-scaled density plot of transmission generated by the FEM; white represents highest transmission and black represents lowest transmission. Experimental peak positions are shown for P = 400 nm and P = 450 nm as triangles and boxes, respectively. Analytically predicted maxima (white lines) and minima (grey lines) taken from a modification of the interference theory by Lezec and Thio [12] and a resonant cavity theory are displayed as dashed and solid lines, respectively.

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4. Determining the nature of the surface wave

As a starting point, we used the FEM to model a simpler structure to help isolate effects giving rise to transmission enhancement: an Au film on glass with five grooves cut into the glass and an aperture cut into the Au to one side of the grooves (see Fig. 3(a)). Figures 3(b) and (c) show the time-averaged power flow for two incident wavelengths. For both wavelengths, a surface wave emanates from the grating structure. The wave exhibits undulations in power flow. Based on insight gained from field visualization in the FEM, this undulation in power flow has been found to arise from interference between the surface wave and the vertically propagating waves (normally incident and reflected light that forms a standing wave pattern above the smooth surface). Positions of large power flow are where the incident wave and reflected wave are consistently in phase with the surface wave; positions of small power flow are where the surface wave is consistently out of phase with the incident wave and reflected wave. The height of the undulation is half the wavelength of the incident wave [17].

Large transmission occurs when a region of high power flow at the surface occurs at the location of the aperture which is the case for λ ₀ = 660 nm (Fig. 3(b)). As seen from Fig. 3(c), a strong surface wave is still present at λ ₀ = 730 nm, but almost no transmission is observed. We conclude that minima and maxima in transmission arise from interference of a surface wave with the propagating incident and reflected fields, and the wavelengths of the maxima and minima are determined by this interference effect.

Lezec and Thio previously argued that interference of a surface wave with the incident and reflected fields could lead to transmission enhancement and suppression. They concluded, however, that the wave was a CDEW [12]. Careful analysis of the surface wave in our model shows, instead, that the surface wave is indeed an SPP. Below we summarize arguments that have been quoted in the literature against SPPs having a role in enhanced transmission. We further resolve these issues and show how the results from our experiment and FEM are consistent with SPP generation.

 

Fig. 3. (a) Modeled geometry consisting of an Au film on glass with a single set of five 50 nm tall grooves cut into the glass forming raised ridges in the Au with 400 nm period at the glass/Au interface and a 100 nm wide aperture in the Au located 1.1 μm from the edge of the grooves. A TM-polarized plane wave is normally incident on the Au from above (through the glass). The power flow within the dashed box is shown in (b) and (c) for free space wavelengths of 660 nm and 730 nm, respectively. Gray-scale is the magnitude of time-average power flow and streamlines show the direction of the power flow.

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Four main arguments have been given in the literature against associating the effect with SPP creation: (1) the range of wavelengths that support transmission enhancement is too broad to be the result of an SPP resonance in the grating structure as described in Eq. 1; (2) recent fits using the CDEW model yielded effective refractive indices larger than expected for SPPs; (3) propagation lengths are reported to have a different behavior than that predicted for SPPs; and (4) phase shifts in the surface wave have been observed which are thought to be contradictory to SPP phase shifts (see Ref. [12] and references therein for detailed discussions).

Concerning argument (1), it is incorrect to use the fact that Eq. 1 does not predict peak location as an argument against SPPs being the surface wave responsible for the enhanced transmission. Equation 1 is theoretically based on there being a large periodic array of scatterers, and is only valid far from the edges of that array. In our case, as in many cases studied, we are concerned with the edge of a finite array, and we therefore expect either broadening of the peak location given by Eq. 1 or Eq. 1 to be invalid altogether. Modeling of grating structures similar to our structure using the FEM demonstrated creation of a substantial surface wave for a broad range of wavelengths, roughly 600–1000nm. Changing the grating periodicity would shift this range of wavelengths, but the range of wavelengths was always substantially broader than experimentally observed enhancement peaks. In fact, all modeling results showed that almost any grating structure would generate SPPs over a broad wavelength band, and the governing factors that determine the generation of an SPP by some structure (periodic or otherwise) are (a) whether the metal is good enough to support an SPP, and (b) if the geometric parameters near the edge of the structure are of the right size compared to the wavelength to effectively scatter the light.

One set of models of interest consisted of 250 nm wide 75 nm tall ridges in a gold film to match the experimental structures, varying the number of ridges from 1 to 20. It was found that for all the structures, a sizeable surface wave was created to the side of the ridges in the wavelength range from 600 nm to 1000 nm. It was also found that the amplitude of the surface wave increased moderately as the number of ridges was increased from one to three, but additional ridges did not increase the amplitude noticeably. The result of this is that Eq. 1 really cannot be expected to predict large scale coherent scattering from a periodic structure like ours into an SPP at the edge of that array.

Apropos argument (2), we examined an FEM with 200 nm of SiN on glass, a thick Au film on the SiN, and only a single groove in the SiN forming a single ridge in the Au. A normally incident TM wave was used to generate a surface wave and the surface wave was fit over 5 μm to the form

where A, b, k_SW, and ε were fit parameters. From the fit, we extracted the dispersion relation (k_SW versus energy, h̄ω) and propagation constant, L = (2b)^-1, of the surface wave. Figure 4(top) shows the dispersion relation compared to the analytical dispersion relation of an SPP [8]. The dispersion relation is in agreement to within less than 1% at all points. This tells us the surface wave observed in the FEM is an SPP. The excellent agreement between experimental and FEM peak location confirms that the FEM is correctly modeling the effect, and therefore, transmission enhancement is indeed caused by an SPP interfering with the incident and reflected waves.

 

Fig. 4. Comparison of the dispersion relation (top) and propagation length (bottom) for surface waves observed in the FEM (shown as boxes) and the analytical expressions for SPPs [8] (shown as lines).

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With regard to argument (3), Fig. 4 (bottom) shows the propagation length obtained from the FEM compared to the analytical SPP result [8]. These are also in very good agreement. Refs. [12] and [13] reported that the measured decay is inconsistent with SPPs because it exhibits an inverse distance dependence. A possible explanation for their observation is provided by the geometry of their groove/aperture structure. The relative lengths of their structures are much shorter than the structures studied here. Their geometry is, therefore, less compatible with a 2-dimensional treatment, which could cause systematic errors in determining decay characteristics [18].

In reference to argument (4), there are reports in the literature that the experimentally measured phase shift between surface waves launched by holes, slits or grooves and the incident field is π/2 [12, 13]. In these studies, the expected phase shift for an SPP was quoted to be 0, and the experimental result was taken as strong evidence against the surface wave being an SPP. Theoretically, a good conducting structure, such as a ridge, scatters TE waves (electric field parallel to the surface) nearly π out of phase and TM waves (magnetic field parallel to the surface) in phase. Therefore, as SPPs are only generated for TM waves, we expect SPPs created by a ridge to exhibit phase shifts close to zero. Consistent with this, the surface waves observed in the FEM generated by small, raised ridges show nearly zero phase shifts with respect to the center of the ridge. For a hole or linear aperture, which extends through the film, and is smaller than the wavelength of the light, one expects the SPP to be generated in antiphase with the incident field since scattering from an aperture is effectively the absence of a reflector. This has been confirmed using a finite element model in Ref. [19]. For grooves or troughs of finite depth in a metal, one expects the phase shift to depend on groove depth and width as an organ pipe cavity mode is set up in the groove [18]. Using the FEM, we have found that the phase shift of the SPP does indeed change as the dimensions of the groove are changed. Therefore, depending on the geometry of the groove, an SPP can be generated with a broad range of phases including the π/2 reported in Refs. [12] and [13].

We conclude that the surface wave responsible for transmission enhancement is an SPP. The logic is as follows: (1) there is very good agreement between the FEM and the experiment leading us to conclude the FEM is modeling the transmission enhancement correctly, (2) there is a near perfect match between the dispersion relation and propagation length of the surface wave from the FEM and that of an SPP and (3) the dispersion relation and decay properties of any guided mode, including surface waves, uniquely define that mode, and, in the present case, are those of an SPP. In other words, we have shown that the experimental results are in agreement with the surface wave being an SPP, and an SPP-based theory may be used to predict transmission enhancement.

5. An analytical prediction

To show the predictive power of the conclusions from the last section, we now develop an analytical prediction for wavelengths of transmission enhancement and suppression. We begin by taking into account interference between the surface wave and the incident and reflected waves. This idea was first introduced by Lezec et al. [12] and used later by Janssen et al. [19]. In contrast to Lezec, however, and in agreement with Janssen, note that we are now treating the surface wave as an SPP.

At this point, it is also important to note that while Lezec et al. did their development in terms of the electric field, the appropriate field to use when discussing phase shifts and field interference for TM polarization is actually the magnetic field, since it can be treated as a scalar in this polarization while the electric field cannot. In this case, however, it turns out that using the magnetic field results in the same equations as reported by Lezec et al. We treat the center of the first raised ridge at the edge of the cavity as the primary source location of the SPP (in agreement with observations from the FEM). Therefore, for the structures with 400 nm period grooves, the effective distance between the SPP source and the aperture is (w_C + 200 nm)/2. This yields the following expression for predicting constructive or destructive interference between the SPP and the incident field at the aperture.

When m is an integer, the transmission should be enhanced and when m is a half integer, the transmission should be suppressed. Here n_SP is the effective index for the SPP, and ε is the intrinsic phase shift of the SPP.

As noted above, ε should be close to zero in our case because the scatterer is a good metal. Also, the height of the ridge is small compared to the wavelength so one does not expect a large phase shift due to coupling between the top and bottom of the ridge. The dashed lines in Fig. 2 show the predicted locations for minima and maxima using ε = 0. These line up well with the upper peak position in the enhancement bands and the upper minima in the suppression bands for cavity widths substantially larger than the wavelength. The simplicity of the theory, and the fact that the theory fits the progression of the experimental and FEM data, adds additional support for this interference model.

However, as mentioned before, another effect was noticed in the FEM, namely a resonant cavity effect. The walls of the ridges closest to the aperture behave as mirrors reflecting SPPs back and forth between the mirrors and forming a resonant cavity. This is evident in Fig. 5 which shows FEM calculations of the integrated energy density along a line between the walls of the cavity. Two calculations are shown, one which uses the model structure in Fig. 1(a) with a cavity width of 1450 nm and another which uses the same geometry but with the aperture removed. The observed peaks are the expected behavior of a resonant cavity. The vertical lines are the predicted positions of cavity resonances using the condition for standing waves between the mirrors given by

where w_C is set to 1450 nm, m is again an integer and n_SP is calculated from Ref. [8]. Half integer values corresponding to antisymmetric modes are excluded due to the symmetry of the structure and incident field.

 

Fig. 5. Calculated integrated energy density inside the cavity of the structure in Fig. 1(a) with a cavity width of 1450 nm are shown as a function of wavelength. The solid curve is without an aperture present. The dashed curve is with the aperture present. The vertical dotted lines are predicted cavity resonant wavelengths using Eq. 4 for m = 3, 4, and 5.

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Focusing first on the solid curve in Fig. 5, for the structure without an aperture, the peaks are close to the expected wavelengths. However, for shorter wavelengths, the resonances are red shifted relative to the analytical expression. This is because the dielectric properties of Au become less metallic at shorter wavelengths and there is some penetration of the SPP into the mirrors formed at the cavity edges. Using the FEM without the aperture, the penetration into the cavity wall mirrors was explored for various wavelengths and cavity widths. The effective penetration consistently decreased as wavelength was increased. This is because Au becomes a better conductor at longer wavelengths. There were also small effects from changing cavity width that are presumably from secondary causes such as internal ridges or interference of the cavity mode with the normally incident and reflected fields at the cavity mirrors.

On average, the penetration into the walls of the cavity was roughly 25 nm into the wall of the leading ridge. However, note that the dashed curve in Fig. 5 (for the energy density in the cavity with the aperture present) indicates that the aperture perturbs the resonant effect. This perturbation is complicated because the aperture does a number of things to the cavity: it reduces the quality of the cavity; it also provides a way to couple energy out of the cavity; finally, the presence of the aperture can change the effective cavity width. For the m = 3 curve, it can clearly be seen that the cavity resonant wavelength is blue shifted by the presence of the aperture. Consistently, the presence of an aperture blue shifted the resonance counteracting the red shifting from penetration into the cavity walls. For longer wavelengths, where the penetration into the cavity walls is small, the blue shift from the aperture dominates and the resonant wavelengths are significantly blue shifted to wavelengths shorter than the prediction from Eq. 4.

For resonant wavelengths, the magnetic field of the SPP is always a maximum at the center of the cavity. However, once again taking into account interference with the incident field, we expect constructive interference at the aperture (enhancement) for even m and destructive interference (suppression) for odd m. The predictions this makes for enhancement and suppression bands are shown as solid lines in Fig. 2. The results are in good agreement with the lower enhancement peaks (or suppressions) in the enhancement bands (or suppression bands). Note the deviation from the theory for wavelengths above about 900 nm (see the top left of Fig. 2). This is attributable to the blue shifting of Eq. 4 for longer wavelengths as discussed above. It is possible for Eq. 4 to be modified to include a wavelength dependent cavity width arising from the penetration into the cavity walls and the presence of the aperture; in this paper, however, we choose to use the expression from Eq. 4 to show the predictive power of that simple theory.

These observations, along with the good agreement of the cavity resonance model with the lower peak and minima positions, lead us to conclude that, in addition to the interference effect discussed above, resonant cavity modes play a significant role in transmission enhancement for this geometry.

6. Conclusion

We have examined transmission through a subwavelength linear aperture in an Au film centered symmetrically in a smooth cavity between two linear grating structures. We find the transmission enhancement wavelengths of this structure are determined by two effects: interference between an SPP and the normally incident and reflected fields, and resonant cavity modes. The separation of the SPP source and the aperture, in our case roughly half of the cavity width, governs when transmission enhancement occurs due to the interference effect. The cavity width, of course, also governs the resonant cavity effect. In addition to this, we found that for a given grating periodicity, a sizable SPP was generated for a much broader range of wavelengths than the peak widths for the two effects mentioned above. Therefore, the grating periodicity is not a controlling factor in determining enhancement wavelengths for these structures. In short, using the SPP dispersion relation, interference theory, and resonant cavity theory, we have developed simple analytical expressions that predict transmission enhancement and suppression wavelengths for structures consisting of a set of scatterers (ridges in our case), and a single aperture.