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Surface plasmon polaritons assisted diffraction in periodic subwavelength holes of metal films with reduced interplane coupling

Metal films grown on Si wafer perforated with a periodic array of subwavelength holes have been fabricated and anomalous enhanced transmission in the mid-infrared regime has been observed. High order transmission peaks up to Si(2,2) are clearly revealed due to the large dielectric constant contrast of the dielectrics at the opposite interfaces. Si(1,1) peak splits at oblique incidence both in TE and TM polarization, which confirms that anomalous enhanced transmission is a surface plasmon polaritons (SPPs) assisted diffraction phenomenon. Theoretical transmission spectra agree excellently with the experimental results and confirm the role of SPPs diffraction by the lattice.

Since the discovery of anomalous enhanced transmission of subwavelength metallic hole arrays in 1998 [1], enormous interest has been sparked among world wide researchers due to its potential application in subwavelength optoelectronic devices as well as the underlying physics. Although the exact origin of the enhanced transmission remains under controversial [2,3], it is generally acknowledged that surface plasmon polaritons (SPPs) are playing a crucial role. Recent researches indicate that this phenomenon could be a SPPs assisted diffraction process [4,5]. In this process, the coupling of SPPs on the opposite interfaces of the metal film helps the electromagnetic energy to tunnel through the subwavelength holes [6]. This interplane coupling is substantial when the dielectrics at the interfaces are similar, and is the strongest when they are identical [7]. To achieve high transmittance, most previous work adopts either free standing films or films on substrates with small dielectric constants. However, the strong interplane interference in this kind of samples will hinder the mechanism from getting fully revealed [8]. For thin films with identical interfaces, this interference could even lift the degeneracy of SPPs on the two interfaces and alter their modes [9,10]. So it is of great interest to reduce the interference in order to study the underlying physics of the SPPs at each individual interface. For this purpose, we adopt the Air/Au/Si system in which air and Si have large difference in their dielectric constants in the mid-infrared. The thickness of the metal films is an order larger than the skin depth (about 20 nm for Au at the mid-infrared). It is expected that in such system the interference of the SPP modes at two interfaces is greatly reduced and the characteristics of SPPs at a single interface are preserved.
Metal films were deposited on Si wafer by magnetic sputtering. The hole arrays were fabricated by conventional ultraviolet photolithography and reactive ion etching. All the patterns are square arrays of circular holes, occupying an area of 7×7 mm 2 respectively. The hole number for each pattern is over one million and the finite size effect is assumed to be well avoided [11]. The infrared transmittance spectra were obtained with an ABB Bomem DA8 Fourier transform infrared spectrometer. Fig. 1(a) is the zeroth-order transmittivity (transmittance normalized by the porosity of the film) of three Au films under normal incidence. All three films have a thickness of 320 nm and a hole diameter of 3 µm. The lattice constants are 5 (sample A), 6 (sample B) and 7 (sample C) µm respectively. Three salient features can clearly be discerned. (i) Because the dielectric constant of Si (ε=11.7) is much larger than that of air in the mid-infrared regime, the distance between Air(1,0) and Si(1,0) is greatly expanded in comparison with previous reports (Si (1,0) are at 21.02 µm and 24.41 µm for samples B and C respectively, which are not shown in the figure). (ii) High order peaks such as Si(1,1), (2,0) and (2,1) can be clearly observed between Air(1,0) and Si(1,0). (iii) Peaks with the same order of the three samples shift according to the periodicities, and they have larger wavelengths for larger periodicities. Numerical calculations on the transmission spectra have been performed by means of a transfer-matrix method combined with an analytical modal solution [12], using the same material parameters as in the experiments, and the results are displayed in Fig. 1(b). The circular holes are treated as square ones with the same area, considering the convenience in modeling. A series of resonant peaks can be clearly found and their positions agree well with the experimental data.

Shown in
We attribute the above observation to be a result of SPPs assisted diffraction process.
SPPs have different momenta from light of the same energy [13]. The gap between the momenta can be compensated with the presence of a periodic hole array. The momentum matching condition is where k sp is the wave vector of SPPs, k is the inplane wave vector component of the incident light (zero for normal incidence), i and j are integers, G x and G y are the reciprocal vectors [14]. These two indices have been used to designate the resonant peaks in Fig. 1(a). We have extracted the experimental SPPs dispersion from the data shown in Fig. 1

(a) through
Eq. (1) and compared with the theoretical one which adopts the material parameters in Ref. [15,16], and found that these two curves fit very well.
To further confirm the view above, we measured the angular dependent transmittivity The discussion above also applies well to the high order peaks of Si(2,0), (2,1) and (2,2).
The change for high order peaks is generally not as significant as the lower order ones.
The reason is that the reciprocal vectors for high order peaks are large, making them less sensitive to the change of the inplane vector of the incident light.
To verify that the interplane coupling is indeed very weak in the present system, we fabricated a series of Pt films with different thickness and did the same measurements. All the peaks stay unchanged in wavelength until being overlapped by other peaks or merged into the background as the thickness decreases. Further numerical calculations on sample D with various thicknesses are made. The calculated transmission spectra of these samples under TE-polarized illumination at an incident angle of 16 • are displayed in Fig. 4. The resonant peaks stay unchanged in wavelength while the thickness changes over 7 times from 120 nm to 920 nm, which clearly manifests the character of decoupled SPPs [19,20]. The splitting behavior does not change with the thickness, as long as the interplane coupling is weak.
We notice that although most existing relevant theories treat the holes in the array as point scattering centers, it is expected that the finite size of the hole diameter will modify the exact behavior of the diffraction [21,22,23,24]. It is found that when the diameter of the holes is larger than half of the periodicity, the (1,0) peak will split into two [25]. The Since SPPs on the opposite interfaces overlap greater for larger holes, one may attribute this phenomenon to the enhancement of the interplane coupling. We believe it is not the real reason. First, the wave vectors of SPPs corresponding to Si(2,0) and (2,1) are quite different from those of Air(1,0). There could not be any strong coupling. Second, we have done the angular dependence experiments of sample D and obtain similar results. As discussed previously, this verifies that interplane coupling is weak in these samples.
We attribute this change to the inplane Bragg scattering of SPPs. When the hole diameter exceeds half of the periodicity, Bragg scattering will introduce band gaps to the dispersion relation, which correspond to different distributions of charges and electromagnetic field in space [26]. SPPs corresponding to the high order peaks have small wavelengths, so they are more sensitive to the actual size of the holes. We also find through numerical calculation that high order peaks are usually sharper and more pronounced when the hole diameter is