The transmission of ultrashort (7 fs) broadband laser pulses through periodic gold nano-structures is studied. The distribution of the transmitted light intensity over wavelength and angle shows an efficient coupling of the incident p-polarized light to two counter-propagating surface plasmon (SP) modes. As a result of the mode interaction, the avoided crossing patterns exhibit energy and momentum gaps, which depend on the configuration of the nano-structure and the wavelength. Variations of the widths of the SP resonances and an abrupt change of the mode interaction in the vicinity of the avoided crossing region are observed. These features are explained by the model of two coupled modes and a coupling change due to switching from the higher frequency dark mode to the lower frequency bright mode for increasing wavelength of the excitation light.
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The interaction of light with periodic metal structures, allowing for efficient coupling of light to surface plasmons (SPs) with their remarkable properties continues attracting considerable interest. In particular, sharp resonances [1–4], localization and enhancement of the electromagnetic field [5,6], and wave guiding along metal surfaces  were demonstrated. Corrugated metal surfaces and arrays of nanowires find applications in sensing  and enhancing light absorption [9,10]. The interaction of light with periodic structures is also of interest, because excited SPs exhibit gaps in the energy spectrum (ω-gaps) [3,5,7,11,12], rendering these structures as simple plasmonic crystals. The gaps appear due to mode interaction near the crossing of their unperturbed dispersion curves and can be employed for wave guiding , developing sensors  and photonic notch filters . The existence of momentum gaps (or k-gaps) was also proposed and confirmed in experimental [11,12] and theoretical works [16–18]. However, it has also been suggested that a k-gap is an artifact, resulting from the mode over coupling . Recently, after the discovery of the extraordinary transmission (EOT) of light through arrays of small holes  the investigation of the role of SPs in this phenomenon was extensively investigated, revealing effects of the diffraction, resonance excitations, and interplay between localized and delocalized surface plasmons on the enhanced light transmission (see recent reviews [21,22]).
In this paper we investigate the interaction of SP modes observed with the transmission far-field spectroscopy [23,24] of broadband laser pulses (transform limited duration 7 fs) through gold nano-structures with gratings. We study the interaction of light with gold gratings at parameters that are quite different from those used for the observation of the EOT. The latter is observed in optically thick (150-300nm) metal films with small holes (surface fraction of openings less than 20%). The thickness of the gold layer in our samples is about 50 nm, so that the film is partially transparent (the absorption length of light around 700 nm, calculated from the optical constants of gold  is about 14 nm). If the slits (through or partial) are also taken into account the transmittance of our samples is much higher than in EOT observations. Consequently, at such conditions the positions of the SP resonances are close to the minima in the transmission [26,27], while for typical conditions of the EOT the loci of the SP resonances correspond to the transmission enhancement . We observe efficient coupling of the incident light to two counter-propagating SP modes, exhibiting avoided crossing with features that were not reported previously. The experimentally measured minima in the transmitted light form patterns, indicating the existence of ω- and k-gaps, confirmed also by calculations with the coupled mode model [28,29].
2. Experimental section
We studied two types of metallic structures: a periodic array of gold nanowires on a glass substrate and a similar array with an additional gold sub-layer. The choice of the samples was aimed at observing the differences in the dispersion relations and the interaction of the SP modes in a sample with and without slits, since these configurations can be expected to have different radiative damping and mode coupling. The periods of the arrays were chosen to provide SP resonances at normal incidence close to the middle of the spectral range (from 650 to 850 nm) of the laser pulses. The configurations of the two samples produced by electron beam lithography and thermal vapor deposition are shown in Fig. 1(a) . Both samples had a gold grating structure with a profile close to rectangular. For the first sample the grating was thinner, and it was deposited onto an underlying gold film. The thickness of the films during the deposition was monitored with a quartz crystal monitor. The samples were also characterized with an atomic force microscope (AFM) and by scanning electron microscopy (SEM). A SEM image of the surface of the first sample is shown in Fig. 1(b).
Measurements of the transmitted light for a set of incidence angles were performed. A portion of the beam from a broadband pulsed laser (Rainbow, Femtolasers) was weakly focused and front-illuminated the sample, as shown in Fig. 1(a). The sample was mounted on a rotation stage allowing for variation of the incidence angle from −6° to + 6° with small steps and accuracy better than 0.05°. The light that interacted with the grating on the sample was selected by a small aperture and directed into an Ocean Optics spectrometer, registering the distribution of the spectral intensity.
SPs can be efficiently excited when the incident light interacting with the grating produces a polarization wave propagating along the metal surface with the phase velocity of the SP wave.
This condition, following also from the conservation of energy and the momentum component along the surface, can be realized by adjusting the wavelength, the incidence angle of the light wave, the period of the grating [30–32] or the refractive index of the dielectric medium adjacent to the metal . Thus, the resonant coupling takes place, when the wave vector of light after interaction with the grating has a component in the plane of the grating equal to the wave vector of the SP,
Normalized transmission spectra for two samples are shown as density plots in Fig. 2(a,c) . The intensity distribution over incidence angle and wavelength displays two diagonal valleys corresponding to orders of the SP resonance excitation, producing a decrease in the transmitted intensity. The higher frequency branch demonstrates a narrowing for sample 1 and a slight broadening for sample 2, when approaching the crossing region. This can be seen by comparing the shorter-wavelength dips of the spectral intensity profiles shown for a set of angles in Fig. 2(b,d), which exhibit asymmetric Fano-type resonances . Such changes indicate damping variations of the SP modes. In particular, radiative damping can change due to a shift of the mode intensity distribution relative to the grooves of the grating, as the wavelength changes [35,36], and also the effect of the mode interaction due to Bragg scattering  increases closer to the normal incidence, corresponding in -space to the boundary of the Brillouin zone.
We note that the coupling of the incident light to SPs at a grating surface is highly dependent on the polarization. The maximum coupling is achieved, when the projection of the electric field vector on the sample surface is perpendicular to the grooves. We measured SP coupling with light at normal incidence for varied polarizations, and the normalized data (Fig. 3 ) suggests a dependence on the polarization angle ϕ, i.e. only the component with the polarization perpendicular to the grooves couples to SPs.
As was shown previously [30–32], in the optical spectral region the coupling reaches maximum values for the grating thickness h from 30 to 50 nm, close to the values used for our samples. The relative strength of coupling for the lower and higher frequency branches depends also on the relative phase of the fundamental and the next higher (second) harmonic of the grating spatial profile . The coupling was inferred from the normalized intensity distributions and characterized by the ratio of the depth of the intensity minimum measured from the height of the nearby maximum to this height for observed Fano-type resonances as a function of the incidence angle (see Fig. 4 ). The coupling for the lower frequency branch (“bright” mode) is substantially stronger than the coupling for the higher frequency branch (“dark” mode). Note that the coupling of the former increases, and the coupling of the latter abruptly decreases, as the incidence angle approaches zero. For sample 2 the coupling dependences were qualitatively similar, but the decrease for the higher frequency branch near was even steeper.
The plots of the minima loci in the plane of the experimentally measured transmission show qualitatively different patterns of the dispersion relations, depicted in Fig. 5(a,d) . For sample 1 an ω-gap of about 30 meV is observed (Fig. 5(a)), which appears as a result of the interaction of modes and is mainly due to the presence of the second harmonic in the spatial profile of the nano-structure [19,32]. The lower and higher frequency branches can be related to the symmetric and antisymmetric modes [35–38]. The edges of the gap at have wavelength values nm and nm and two different SP phase velocities and of the higher and lower frequency SP branches, which have the following ratios to the speed of light c in air, and , inferred from Eq. (1) for normal incidence.
For sample 2, the mode pattern in the band-gap region is qualitatively different for higher and lower frequency branches. For the higher frequency branch, the dispersion curves with do not tend to merge when they approach the avoided crossing region, but rather form a gap in θ values near (k-gap, 1.91 × 103 cm- 1).
Figures 5(b,e) show the evolution of the transmission angular dependence for the two samples in the avoided crossing regions, indicated by dashed boxes in Fig. 5(a,d), as the wavelength is changed in small steps. With increasing wavelength the two side minima in the angular intensity distribution (Fig. 5(b), sample 1), which are seen clearly for nm, initially merge into a valley and then re-appear again. In Fig. 5(e) (sample 2) with increasing wavelength the two side minima stay separated until they disappear (four lower curves), and then two shallow minima in the central part of the plot are formed.
To describe the observed behavior we used a model of two coupled modes [28,29,39] with slowly varying complex amplitudes (mode with n = + 1) and (n = −1) and propagation constants (without accounting for interaction) and , counter-propagating along z-axis. The main mechanism of the interaction of these modes is their Bragg scattering with [5,19,23]. The wave numbers of re-scattered waves can be shifted by and create a polarization wave with a wave vector value . Then and considered as functions of the incident light frequency cross at some frequency value , such that . The crossing of the modes means that they propagate with the same phase velocity and therefore their coupling can take place, modifying also the dispersion relations. To find these new relations we use a system of two coupled equations :Eqs. (2) that describe dispersion relations. To have the correct choice of signs we note that far from the crossing point new solutions should transform into unperturbed values, .Consequently, we find the new propagation constants as functions of ω:29]. When attenuation of the SP modes takes place or the coupling constants are complex, the values of also become complex, so that the real parts determine the phase constants of the modes, and the imaginary parts determine their attenuation or amplification, depending on the sign.
The results of the dispersion relation calculations with the model of two coupled SP modes (Eq. (2)) are shown in Figs. 5(c,f) as dependences to enable direct comparison with the experimental data. The unperturbed dispersion relations were calculated using the three layer model , taking into account the grating as an additional gold layer with effective thicknesses nm and nm for samples 1 and 2, respectively. For sample 1, agreement with the experiment is obtained when the value of G has a negative real part and a relatively smaller imaginary part, namely (meV)2, so that in the assumption we obtain for the magnitude of the coupling constant meV. A significant negative real part is required for a large ω-gap with a strong attenuation of SP modes within the respective frequency interval. When is positive, a k-gap (or a gap in θvalues) and two almost vertical portions of the dispersion curves appear. Note that the presence of the imaginary part can also lead to the formation of a k-gap. This can be the reason for the appearance of a relatively narrow k-gap in calculated dispersion curves of Fig. 5(c). When this k-gap is small and damping is present, as in our case, the vertical portions of the dispersion curves can merge, forming a vertical line of minima in the transmission, similar to the observed in the experiment (Fig. 5(a), the data points shown by crosses). Thus, for and features of both ω- and k-gaps can be present as in Fig. 5(c), where strong attenuation of the SP modes takes place for the vertical portions of the dispersion curves. This strong attenuation leads to the broadening of the central portion of the SP modes seen in Fig. 2(a).
For sample 2, the lower frequency branch looks similar to that of sample 1. However, the higher frequency branch shows the divergence of the dispersion curves for and in the vicinity of the avoided crossing region (nm), while in the region the portions of the dispersion curve tend to merge as the wavelength is approaching the central wavelength . The different behavior in these two wavelength regions provides evidence that the sign of the real part of the product G changes, as the wavelength changes from to. The solid lines in Fig. 5(f) are calculated with (meV)2 for and with (meV)2 for and give good agreement with experimental points.
The change of the coupling constant can be related to a spatial shift of the intensity distribution relative to the grating slits with increasing wavelength, as the excitation switches from the higher frequency dark mode to the lower frequency bright mode, which was recently observed in grating structures [35,36]. The coupled mode model directly calculates dispersion dependences of the modes, and thus, plots of Fig. 5(c,f) clearly demonstrate the possibility for existence of k-gaps. The experiment shows the presence of the transmission minima also in the gap region (in Fig. 5(d) these points are shown by crosses), which are not reproduced by the presented theory. These minima gradually disappear farther away from the respective dispersion branch, which indicates their transitional nature.
We investigated the SP modes related to the air-metal interface. The SP resonance corresponding to the interface metal/glass has at normal incidence a typical resonance wavelength in the infrared region  (around 1.1 µm for our samples), which is beyond the spectral range of the laser pulses that were used, and therefore it was not observed. As was expected from their configurations, our samples 1 and 2 have shown different behavior in the avoided crossing region, exhibiting differences in the radiative damping, mode coupling and also the degree of asymmetry above and below of the crossing region. It is known (see, for instance , ) that in a periodic metallic structure two modes can exist, such that for one of them the field is preferentially concentrated in the region with abundance of the metal (ridges), while for the other the field is preferentially concentrated in the regions with deficiency of the metal (troughs). Previously it was shown that switching between the preferential excitation of one or another of these two modes can take place when the wavelength passes through the central wavelength of the crossing region [19,36,37]. For the sample 1 the higher frequency mode shows a reduction of both the radiative damping and coupling as the wavelength approaches , similar to what was observed in Refs. [26,36], and the two SP modes tend to merge above and below the central wavelength, forming an ω-gap.
For sample 2 a qualitatively different mode behavior is observed above and below the central wavelength. Above modes merge similar to sample 1, however below the modes tend to diverge, forming a k-gap. Thus, for both samples the higher frequency mode disappears with , as it also follows from the symmetry considerations , but realizations of this requirement are different. Also the width of the higher frequency branch somewhat increases with λ approaching, unlike sample 1. In previous modeling of the spectral gaps only one type of gaps per avoided crossing event was discussed. Here we indicate a possibility of a mixed case, where features characteristic of both types of gaps can be present in one crossing region. The origin of the observed k-gap cannot be attributed to a partial overlap of the approaching modes in the avoided crossing region, as was suggested , since the higher frequency branch vanishes in this region.
We also would like to compare our results with those obtained in the observation of the EOT effect, which stimulated an extensive discussion of the role of SPs in this phenomenon. For the normal incidence the transmission minima take place when the period of the grating is close to a multiple of the wavelength of the surface plasmon wave . This is consistent with our results, since at the normal incidence we observe a strong minimum in the transmission at the , which corresponds to , i.e. (d is the grating period). We determine the position of the SP mode from the transmission measurements. The transmitted intensity results from the interference of the direct wave, which is slowly varying, and contributions from scattered SPs that are re-emitted as bulk waves [35,36]. Since at resonance SPs and the waves they re-emit experience a sharp change of phase, SP resonances are often exhibited in the transmission as Fano-type profiles . The effect of the EOT is observed in optically thick metal films with small holes. Recent theoretical calculations have shown that for such conditions the positions of the surface plasmon resonances for a plane metal film correspond to minima in the transmission [42,43], and transmission maxima match the positions of SP resonances, taking into account their displacement due to collective and localized interactions with the holes .
We study interaction of light with gold gratings at parameters which are quite different from EOT observations. At such conditions the positions of the SP resonances are close to the minima in the transmission [26,27] with a small uncertainty due to asymmetry of the Fano profiles. This was also confirmed by calculations and measurements [27,45], showing that the transmission minima coincide with the maxima of the power losses, arising from enhancement of the local field at the SP resonance and also demonstrating the existence of k-gaps. Our observations of SP resonances at the positions of the minima in the transmission are in agreement with these previous results. Thus, in our case with a significant nonresonant transmission, the excitation of SPs takes some energy from the incident light beam and therefore leads to a minimum in the transmission.
A k-gap in our experiment was observed for a dark higher frequency mode, which corresponds to the maxima of the field on the metal ridges [35,36]. For such a spatial configuration of the mode the scattering should be reduced and the coupling is also reduced accordingly. Consequently, also for this reason over coupling cannot be responsible for the observed k-gap effect. This is consistent with the results of Ref. 18, showing that a k-gap appears for a small direct coupling of the SP modes. From calculations based on the Rayleigh expansion and transfer-matrix theory it was shown  that k-gaps exist only up to a certain value of the ratio of the real and imaginary parts of the coupling constant. The presence of a significant imaginary part of the coupling constant is also a prerequisite of appearance of a k-gap according to presented in our paper model of two coupled modes. We note that recent theoretical analysis of a grating-coupled multimode waveguide based on a study of the trajectories of the poles and zeroes of the reflection and transmission coefficients also revealed the existence of significant k-gaps .
In conclusion, we experimentally investigated the interaction of light with periodic gold nano-structures on glass substrates and observed qualitatively different behavior near the avoided crossing region, depending on the configuration of the sample. The sample with a grating deposited on top of a gold film exhibited an energy (or ω-) gap between the two branches of the dispersion relation. The valley in the intensity distribution of the transmitted light in the vicinity of normal incidence and the calculation with the coupled mode model indicate possible presence of a small momentum (or k-) gap as well. Additionally, the higher frequency branch experienced narrowing of the width of the transmission minimum and a strong reduction of coupling to the incident light, which indicate a decrease of radiative damping in the avoided crossing region. For the sample with only a gold grating deposited on a glass substrate the two branches of the dispersion relation displayed qualitatively different patterns: while the lower frequency branch exhibited behavior typical for an ω-gap, the higher frequency branch was split into two portions showing a significant k-gap. Therefore, our observations indicate that interacting modes in one avoided crossing region can exhibit features characteristic of both a k-gap and an ω-gap; this behavior we explain by switching from the higher frequency dark mode to the lower frequency bright mode as the wavelength increases. The coupled mode model shows that ω-gaps are formed, when the coupling constants are predominantly real, and k-gaps appear for predominantly imaginary coupling constants. Thus, when the latter have both real and imaginary parts, features characteristic for gaps of both types are possible.
We thank Wonmuk Hwang for providing the AFM for sample profile measurements and Norbert Kroo for a stimulating discussion. This work was partially supported by the Robert A. Welch Foundation (grants Nos. A1546 and A1585), the National Science Foundation (NSF) (grants Nos. 0722800 and 0555568), the Qatar National Research Fund (grant NPRP30-6-7-35), and the United States Air Force Office of Scientific Research (USAFOSR) (grant FA9550-07-1-0069).
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