## Abstract

We have systematically measured *ε*(*ω*) of subwavelength aperture arrays fabricated in metal films as a function of aperture size and incidence angle using terahertz time-domain spectroscopy. This approach simultaneously yields both the real and imaginary *ε*(*ω*) components, enabling deeper insight into the underlying mechanism of the ‘enhanced optical transmission’ (EOT) phenomenon. For random aperture arrays we find that *ε*(*ω*) has a plasma response, with an *effective* plasma frequency that is determined by the waveguide mode cutoff frequency of the individual apertures. However *ε*(*ω*) in plasmonic lattices is strongly modulated at discrete resonant frequencies that correspond to the *reciprocal vectors in the structure factor* that are *superposed* on the plasma envelope response and appear as *dips* in the EOT spectrum. The existence of a sum rule for the discrete resonance oscillator strengths when the aperture size or incidence angle are changed validates our approach and allows for engineering of the individual resonances in the EOT spectrum.

©2008 Optical Society of America

## 1. Introduction

The complex dielectric response of a medium is of fundamental importance in determining the propagation properties of electromagnetic waves. Hence, the ability to change the dielectric response of a medium in an unexpected manner, and thereby create unusual electromagnetic properties across a range of artificially structured materials, has attracted significant interest in recent years. Examples include enhanced optical transmission (EOT) through periodic and aperiodic subwavelength aperture arrays [1,2] and ‘left-handed materials’ [3–5]. While significant insight has been gained on the EOT phenomenon in aperture arrays (also referred to as ‘plasmonic lattices’) as mediated by surface plasmon polaritons (SPP) [1,2,6–13], the prediction and consequent engineering of both the EOT resonant frequencies and their corresponding strengths simultaneously in the transmission spectrum have not been possible to date. This is because fundamental information, such as the effective dielectric constant of the perforated metallic medium is still not known. Based on the approximation that metals are perfect conductors, Pendry and coworkers [14,15] recently proposed a theoretical framework hypothesizing that metallic films perforated with periodically spaced apertures exhibit an *effective* plasma response (‘spoof plasmons’), which is approximated by the cut-off frequency of the waveguide mode in the individual apertures. If this model were true, then the effective plasma frequency can, in principle, be designed to occur at almost any frequency below the natural surface plasma frequency of the unperforated metal film, providing great flexibility in designing a variety of unique plasmonic lattices. Therefore further insight into the *complex* dielectric response of plasmonic lattices and its subsequent engineering is important to advance of this relatively young field.

In this submission, we describe measurements of the complex dielectric function, *ε*(*ω*), for a systematic group of random and corresponding periodic arrays of subwavelength apertures fabricated on free-standing metal films using THz time-domain spectroscopy (THz-TDS). This experimental technique is unique in that it allows for direct measurement of the transient THz electric field transmitted through the structures, yielding both amplitude and phase information [10–13]. This information is used to extract both the real and imaginary components of *ε*(*ω*) for the effective array medium without resorting to traditional Kramers-Kronig transformations, where somewhat arbitrary assumptions about asymptotic behavior are usually made. By examining and fitting the obtained *ε*(*ω*) response of the group of perforated metal films, we deepen the understanding of the EOT phenomenon in plasmonic lattices.

First, we demonstrate that the effective dielectric constant of the perforated metallic medium is *much* “*richer*” than was predicted by the ‘spoof plasmon’ model [14,15], and is valid when a sizable attenuation or loss mechanism is involved. Second, we find that *ε*(*ω*) in plasmonic lattices is a superposition of *ε*(*ω*) of the random aperture structures describing the individual aperture response, and a response due to the interference of surface waves on the metallic films having resonant frequencies that are given by the reciprocal vectors of the underlying structure factor. These resonant frequencies correspond to *dips* rather than peaks in the EOT spectrum [16–19]. Finally, we find an oscillator strength ‘sum rule’ for the discrete resonances in the dielectric response upon changing the aperture size and/or the impinging angle, θ, of the incident beam. This leads to greater control over the individual resonances in the optical transmission, and consequently to a tuning method.

## 2. Experimental details

We fabricated periodic and random arrays of subwavelength apertures on 75 µm thick free-standing stainless steel metallic films. The apertures in the periodic array were arranged in a square lattice with various lattice constants, *a*, and aperture diameters, *D*. The samples were placed in the THz beam at various angles, θ, with respect to the incident electromagnetic radiation. The random arrays were designed to have the same average spacing, *a*, and *D* (hence, the same fractional aperture area) as the corresponding periodic array. Two-dimensional (2D) Fourier transforms applied to the random aperture arrays pattern did not yield any discrete peaks in the reciprocal space, confirming that the aperture distribution in these samples was truly random.

We used a conventional THz time-domain spectroscopy setup [10] to characterize the 2D aperture arrays. Photoconductive devices were used for both emission and coherent detection. An off-axis paraboloidal mirror was used to collect and collimate the THz radiation from the emitter to the samples. The THz beam was incident on the aperture array structures at different incident angles *θ*. Samples were attached to a solid metal plate with a 5 cm×5 cm opening that was placed in the path of the collimated THz beam. The 1/e THz beam diameter was smaller than the aperture opening in the metal holder and the spatial extent of the aperture array structures, thereby minimizing edge effects due to the finite size of the array sample. Reference transmission spectra were taken with just the metal holder in the THz beam path using the same setup. The metal foils were completely opaque prior to fabrication of the arrays; therefore transmission through the perforated metal films was uniquely due to the apertures. The detected transient photocurrent was then Fourier transformed and normalized to the reference transmission, yielding the electric field transmission spectrum, *t*(*ω*) in the range ~0.05 to 0.6 THz.

Furthermore, THz time-domain spectroscopy allows for the direct measurement of the THz electric field, yielding both amplitude and phase information. By transforming the time-domain data to the frequency domain, we are able to determine independently both the magnitude and phase of the amplitude transmission coefficient, *t*(*ω*), using the relation

In this expression, E_{incident} and E_{transmitted} are the incident and transmitted THz fields, respectively, |*t*(*ω*)| (labeled simply as *t*(*ω*) hereafter) and φ(*ω*) are the magnitude and phase of the amplitude transmission coefficient, respectively, and *ω*/*2π* is the THz frequency.

## 3. Experimental results and discussion

The amplitude transmittance, *t*(*ω*), and phase, *φ*(*ω*), spectra for a random aperture array medium with *a*=2 mm and 12% fractional aperture area at normal incidence (θ=0) are shown in Fig. 1(a). From these spectra we directly calculated *ε*(*ω*) for the array medium, as shown in Fig. 1(b) for the real and imaginary components. *ε*(*ω*) exhibits the *dispersion properties of a plasma*, similar to that of a metallic medium, but with an *effective* plasma frequency approximately equal to the waveguide mode cutoff frequency, *f _{c}* of a cylindrical aperture.

*f*here is given by the relation

_{c}*f*=

_{c}*ca*/

_{m}*πD*, where

*a*is a mode-dependent constant equal to 1.841 for the lowest-order mode (TE

_{m}_{11}), and

*c*is the speed of light [20]. To demonstrate this, we modeled

*ε*(

*ω*) for the random array using the form of a ‘lossy plasma’:

where $\tilde{\omega}$
_{P} is the *effective* plasma frequency, ε_{∞} is the high frequency dielectric constant, and *γ* is the plasma relaxation rate. Using Eq. (2) with $\tilde{\omega}$
_{P}/2π=0.19 THz, ε_{∞}=927, and *γ*=0.45 rad/ps, we obtained an excellent fit to the real and imaginary components of *ε*(*ω*), as shown in Fig. 1(b) (blue lines).

The value of f_{c}=$\tilde{\omega}$
_{P}/2π used here (where the real component of *ε*(*ω*)~0) is in agreement with the calculated *f _{c}*=0.22 THz for an aperture with

*D*=800 µm [20], and is consistent with the predictions of the “spoof plasmon” model [14,15,21]. We further measured and modeled the effective

*ε*(

*ω*) for a variety of random aperture arrays with the same average spacing,

*a*=2 mm and different aperture diameters,

*D*using Eq. (2). The corresponding “best fit” parameters obtained from our measurements are summarized in Table I. It is apparent that γ does not vary in a systematic manner with D. This may be due to sample-to-sample fabrication variation, but further investigation regarding this point is necessary. In contrast, ε

_{∞}monotonically decreases with

*D*. This may result from the fact that the fractional aperture area (where ε

_{∞}=1 in the apertures) increases with

*D*.

In contrast to the random aperture array in which *t*(*ω*) consists of a broad continuous transmission band (Fig. 1(a)) that results from transmission through the individual uncorrelated apertures; *t*(*ω*) associated with a *periodic aperture array* (plasmonic lattice) is characterized by *resonant* transmission bands. For example, Figs. 2(a) and 2(b) show *t*(*ω*) and *φ*(*ω*) spectra, respectively, for a plasmonic lattice with *a*=2 mm and *D*=800 µm measured at normal incidence. We note that in addition to the resonant EOT peaks (R_{i}), *t*(*ω*) also shows sharp anti-resonance (AR_{i}) dips on the high frequency side of each R_{i}. Using the two measured spectra, we obtained the dielectric response shown in Figs. 2(c) and 2(d), respectively, for the real and imaginary *ε*(*ω*) components. As is apparent from the figure, rather than a smooth ‘plasma-like’ response expected from a metallic medium, *ε*(*ω*) of the plasmonic lattice is strongly modulated in the vicinity of the resonant frequencies.

In our previous work, we have discussed how Fano-type interference between a broad continuum and discrete resonances qualitatively describes the observed EOT spectra [2,17]. Obtaining the dielectric response *ε*(*ω*) enables us now to discuss the Fano-type interference in more quantitative terms. It is easy to verify that the sharp modulation in *ε*(*ω*) is *superposed* on an envelope that has a ‘plasma-like’ response; in fact very similar to that of the corresponding random array medium. As shown in previous studies [2] the superposed modulation can be directly attributed to the EOT resonances associated with the inherent reciprocal vectors of the periodic aperture arrays having periodicity *G*=2π/*a*. We therefore modeled *ε*(*ω*) of the plasmonic lattice by adding the contributions from the plasma (the smooth envelope) and several discrete resonances. Being discrete, the resonant contribution to *ε*(*ω*) may be written using longitudinal, ω_{L} and transverse, ω_{T}
*effective* frequencies, similar to phonon or exciton contributions to *ε*(*ω*) [22,23]. Hence, the resulting dielectric response has the form:

where *γ _{j}* is the relaxation rate (linewidth) of the

*j*th resonance. The first two terms in Eq. (3) correspond to the continuous broad response associated with the individual apertures as in Eq. (2) above, while the last term in Eq. (3) corresponds to the

*discrete*resonances. We note that this functional form is consistent with a Fano-type interference between a continuum band and discrete lines [2,7,24]. Importantly, the maxima obtained in the EOT spectrum typically occur at the longitudinal resonant frequencies, and thus should not be considered as the real eignfrequencies here. In contrast, the dips occur at the transverse resonant frequencies, and are thus the principal eignfrequencies that describe the underlying physical properties of the medium. In fact, ω

_{Tj}/2π in Eq. (3) are the AR

_{j}frequencies in the EOT spectrum that correspond to the various SPP branches. Specifically for normal incidence the transverse component of the propagation wave vector is zero, k

_{x}=0; hence the AR frequencies in the EOT spectrum are given by the standard SPP model [1]

where *c* is the speed of light in vacuum, *a* is the lattice periodicity and *m* and *n* are integers, and we assume *n*
_{eff}=1 for the SPP refraction index in the THz spectral range [2]. In fitting the data for the plasmonic lattice, we used the same effective plasma frequency ($\tilde{\omega}$
_{P}) that was used above for describing *ε*(*ω*) of the corresponding random aperture array (Fig. 1); while the transverse frequencies (ω_{Tj}) in the resonant contribution of Eq. (3) correspond to the (±1, 0) and the (±1, ±1) plasmonic branches derived from the standard SPP model (given by Eq. (4)) [1]. The very good fit obtained for the two experimental *ε*(*ω*) components using Eq. (3) is shown in Figs. 2(c) and 2(d) (blue lines), and the corresponding ‘best fit’ parameters are summarized for various aperture arrays in Table II. The good agreement obtained between theory and data validates the model used for *ε*(*ω*).

Given the measured *ε*(*ω*), we directly obtained the amplitude reflection, *r*(*ω*), and field absorption, *α*(*ω*), spectra for this aperture array, as plotted in Figs. 2(e) and 2(f), respectively. It is interesting to note that *r*(*ω*) spectrum is *opposite* in phase to the EOT spectrum; namely in contrast to the EOT spectrum, *r*(*ω*) is characterized by *dips* (peaks) at the *resonant* (AR) frequencies. Surprisingly, *α*(*ω*) resonant spectrum also exhibits an *opposite* phase with respect to that in the EOT spectrum, showing *peaks* at the AR frequencies. These observations at THz frequencies are inconsistent with *α*(*ω*) response obtained for plasmonic lattices at optical frequencies [8]. We note, however, that α(ω) obtained in that case was not directly extracted from the complex dielectric response, as is done here in the THz range.

We now consider the effect on the EOT spectrum upon changing the aperture diameter, *D*. The transmittance spectra for four representative plasmonic lattices with 2 mm periodicity and a systematic increase in the aperture diameters, along with spectra for the corresponding random arrays, are shown in Figs. 3(a)–3(d). *t*(*ω*) of the random arrays is again characterized by a broad continuous spectrum lacking any sharp resonance. Furthermore, *t*(*ω*) increases with frequency reaching a broad maximum at approximately the cutoff frequency (*f _{c}*) of the cylindrical apertures; and as

*D*increases this continuous band is red-shifted. In contrast,

*t*(

*ω*) associated with the plasmonic lattices is characterized by two (or more) sharp resonances at frequencies corresponding to R

_{1}(±1, 0) and R

_{2}(±1, ±1) SPP resonances. Also, it can be readily recognized that the discrete resonances in

*t*(

*ω*) (Fig. 3 red curves) are superposed on the continuum spectrum of the individual apertures (Fig. 3 blue curves). We also note that the relative EOT amplitudes of these two resonances

*vary dramatically*with

*D*. Specifically; R

_{1}becomes stronger as

*D*increases, while R

_{2}is increasingly suppressed. These spectral features arise from the interference between discrete resonances caused by the SPP diffraction from the aperture array, and the broad (continuum) transmission spectrum associated with the individual apertures on the metal film and is well described by the Fano interference mechanism [7] that results from the complex dielectric response given by Eq. (3) above. For small

*D*, the continuum band blue-shifts to higher frequencies and, thus, discrete resonances having

*higher*frequencies (R

_{3}and R

_{4}) become more apparent. The reverse situation occurs at large

*D*, in which case the continuum band red-shifts, and thus the discrete resonances with

*lower*frequencies (R

_{1}) are better resolved. We note that the changes occurring in the resonance

*amplitudes*are

*independent*of the resonance

*frequencies*(ω

_{Ti}), as seen from the robust AR frequencies in Fig. 3. In contrast, the resonance

*maxima*frequencies

*change*with

*D*, again emphasizing that the EOT maxima do not represent the important underlying EOT mechanism.

In the inset of Figure 3(d), we plot the ratio of the EOT amplitude, *t*
_{R1} of the R_{1} (±1,0) resonant mode for the various plasmonic lattices to the peak amplitude transmission, *t*
_{NR} of the non-resonant background from the corresponding random array, as a function of *D*/*a*. It is seen that the optimum coupling of the R_{1} resonance occurs at *D*/*a*≈0.5. Thus, by simply varying *D* it is possible to *engineer* the relative amplitudes of the EOT resonances. It is also seen from Fig. 3(a) that for plasmonic lattices with small *D*, the continuum band can be blue-shifted to an extent that it becomes possible to observe the higher order resonances (R_{3} and R_{4}), while, at the same time suppressing the EOT of the lower order resonances (R_{1} and R_{2}).

Using the EOT spectra associated with these structures (Fig. 3), we calculated the real and imaginary components of *ε*(*ω*) for all four plasmonic lattices in this series with various *D*, assuming that the effective permeability is frequency independent and of the order of 1 [14,25]. Subsequently we fitted the obtained dielectric functions for the various random and periodic aperture arrays using the modeled *ε*(*ω*) given by Eqs. (2) and (3), respectively. From the fits (Table II) we calculated the associated *t*(*ω*) spectra as shown in Figs. 4(a)–4(d) that correspond to the measured *t*(*ω*) spectra shown in Fig.3. The good agreement between the calculated and the measured *t*(*ω*) spectra validates the model used for *ε*(*ω*). The corresponding ‘best fit’ parameters are summarized in Tables I and II.

The ‘best fit’ values for $\tilde{\omega}$
_{P} and ε_{∞} vs. *D* (Table II) are specifically shown in Fig. 5. In the case of ε_{∞}, we found that the data fit best to an offset exponential of the form: ε_{∞}=ε^{o}
_{∞}+Aexp [(D-D_{o})/ℓ_{eff}], where ε^{o}
_{∞}=50, D_{o}=500 µm, and ℓ_{eff}=174 µm. This is somewhat surprising, since one would expect a quadratic decrease with *D* based on the fractional aperture area. Furthermore, we found that $\tilde{\omega}$
_{P}~1/*D* (Fig. 5 inset), in agreement with the cylindrical cut-off model mentioned above. Analogous to the conventional description of discrete resonances in the dielectric response of semiconductors, such as polar phonons or excitons [22,23], we define the oscillator strength, S_{j} of the individual resonances ω_{j} by the relation S_{j}=(ω^{2}
_{Lj}-ω^{2}
_{Tj})/ω^{2}
_{Tj} with parameters given in the fit for *ε*(*ω*). Based on this definition, we found from the fits (Table II) that the strength of individual resonances (*S _{j}*) depends primarily on

*D*/

*a*. From the existence of the Fano-type interaction between the resonant and continuous contributions, R

_{1}strengthens (S

_{1}increases) whereas R

_{2}weakens (S

_{2}decreases) as

*D*increases. Therefore a sum rule for their combined oscillator strengths, Σ

*S*in the EOT spectra appears to be in order. As shown in Table II and Fig. 5, when

_{j}*all*resonant modes are taken into account, then Σ

*S*

_{j}*remains constant*at ~2.45, demonstrating that a

*sum rule for the oscillator strengths*is indeed viable here. The numerical value of Σ

*S*is likely related to the aperture shape, film thickness, and properties of the underlying Bravais lattice. However, the existence of a sum rule strongly validates the model used for the discrete resonances in

_{j}*ε*(

*ω*), and allows for further EOT engineering.

We further analyzed the change in *t*(*ω*) and corresponding change in *ε*(*ω*) by measuring the optical transmission as a function of the impinging angle, θ of the incident beam with respect to the film surface. In Fig. 6(a), we show the *t*(*ω*) spectra of a random aperture array with *D*=750 µm at six different incident angles. It is apparent that the transmission spectrum does not change with θ. In Fig. 6(b) we show *t*(*ω*) spectra of a plasmonic lattice with *a*=1.5 mm and *D*=750 µm at four different incident angles. In contrast to the case of the random hole structure, and as observed previously with other plasmonic lattices structures in the visible range [1,17], the fundamental (1,0) SP resonance (at ~0.19 THz] splits into two resonances at frequencies ω_{T1}′ and ω_{T1}″ for θ>0. Specifically, the ω_{T1}′ branch frequency increases with θ, whereas the ω_{T1}″ branch frequency decreases in a corresponding manner. The splitting in the primary transmission peak occurs because the dynamical diffraction process acquires a vector component **k**
_{‖} parallel to the surface, which results in a symmetry breaking between reciprocal vectors in and against the **k**
_{‖} direction; specifically, ω_{T1}′~*c*(G_{1,0}-**k**
_{‖}); whereas ω_{T1}″~*c*(G_{1,0}+**k**
_{‖}). Therefore, the split frequencies yield little information regarding the SPP dispersion relation on the metallic surface, as was claimed in [1].

In Fig. 6(c) we show the corresponding real and imaginary components of *ε*(*ω*) at various incident angles directly obtained from the measured *t*(*ω*) and *φ*(*ω*) spectra as done for the case of θ=0 discussed above. We note that the “plasma” like envelope in *ε*(*ω*) *remains unaltered* at different θ’s, whereas the superposed discrete resonances split with θ in accordance with the split in the major resonance in *t*(*ω*) (shown in Fig 6(c) inset). As before, the “plasma” like behavior depends on the shape and cutoff properties of the individual apertures, but not on their distribution; hence the envelope is expected to be unaltered as θ is changed; this is apparent in Fig. 6(a) since no change in the *t*(*ω*) and *φ*(*ω*) spectra as a function of θ is observed for the random arrays. It is also apparent (Fig. 6(c)) that the strengths of the two split resonances at frequencies ω_{T1}′ and ω_{T1}″ also vary as a function of θ. Specifically the oscillator strength associated with the resonance at ω_{T1}′ decreases, while the oscillator strength associated with the resonance at ω_{T1}″ increases (Fig. 6(c)), in a manner similar to that described above when tuning the individual resonant oscillator strength by changing the aperture size. In this case, however, the *resonant frequency* changes rather than the *plasma frequency* associated with the individual apertures.

From our discussion above, the induced changes with θ suggests that a sum rule for the split resonance is in order in this case too. In fact, as we show in Table III and Fig. 6, we found that the oscillator strengths sum of these two split resonances in *ε*(*ω*) indeed remains constant, ∑*S _{j}*≅1.00. This further justifies and validates the model used here for describing

*ε*(

*ω*).

## 4. Conclusion

In summary, we have measured the complex dielectric function, *ε*(*ω*) at THz frequencies for periodic and corresponding random aperture arrays perforated in metallic films, at various incident angles. The dielectric function, *ε*(*ω*) is fundamental for understanding the complete linear propagation properties of electromagnetic radiation in plasmonic lattices, as well as deepening our understanding of the EOT underlying mechanism. We found that *ε*(*ω*) consists of a superposition of an effective ‘plasma-like’ response that is associated with the uncorrelated individual apertures, and discrete resonances response that is associated with the launched SPP interference related to the reciprocal vectors of the underlying aperture arrays in the Fourier space. The application of this approach to other spectral ranges, where the attenuation is larger should be interesting. The existence of a sum rule for the resonant mode oscillator strengths upon changing the aperture size and incident angle allows for tuning of the individual resonance amplitudes in the EOT spectrum. In addition, we also obtained the reflectivity and absorption spectra of the aperture arrays, which are important in understanding the thermal emission properties of such devices. The obtained dielectric response and model are important for designing novel plasmonic structures with almost arbitrary dispersion relations, opening up new and exciting opportunities in the development of useful passive and active optoelectronic devices. This includes sensors, switches, filters, and unique waveguide architectures for the THz spectral range, which is otherwise largely devoid of useful devices. Since Maxwell’s equations are scale invariant, our approach may be easily extended to other spectral regions of the electromagnetic spectrum.

## Acknowledgment

We gratefully acknowledge support from the National Science Foundation through grant #ECCS-0801965.

## References and links

**1. **T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature **391**, 667–669 (1998). [CrossRef]

**2. **T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature **446**, 517–521 (2007). [CrossRef] [PubMed]

**3. **V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. **10**, 509–514 (1968). [CrossRef]

**4. **J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. **76**, 4773–4776 (1996). [CrossRef] [PubMed]

**5. **D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. **84**, 4184–4187 (2000). [CrossRef] [PubMed]

**6. **L. Martin-Moreno, F. J. Garcia-Videl, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of Extraordinary Optical Transmission through Subwavelength Hole Arrays,” Phys. Rev. Lett. **86**, 1114–1117 (2001). [CrossRef] [PubMed]

**7. **C. Genet, M. P. van Exter, and J. P. Woerdman, “Fano-type interpretation of red shifts and red tails in hole array transmission spectra,” Opt. Commun. **225**, 331–336 (2003). [CrossRef]

**8. **W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. **92**, 107401 (2004). [CrossRef] [PubMed]

**9. **H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature **452**, 728–731 (2008). [CrossRef] [PubMed]

**10. **H. Cao and A. Nahata, “Resonantly enhanced transmission of terahertz radiation through a periodic array of subwavelength apertures,” Opt. Express **12**, 10041–1010 (2004). [CrossRef]

**11. **J. Gomez-Rivas, C. Schotsch, P. Haring-Bolivar, and H. Kurz, “Enhanced transmission of THz radiation through subwavelength holes,” Phys. Rev. B **68**, 201306 (2003). [CrossRef]

**12. **D. Qu, D. Grischkowsky, and W. Zhang, “Terahertz transmission properties of thin, subwavelength metallic hole arrays,” Opt. Lett. **29**, 896–898 (2004). [CrossRef] [PubMed]

**13. **F. Miyamaru, M. Tanaka, and M. Hangyo, “Effect of hole diameter on terahertz surface-wave excitation in metal-hole arrays” Phys. Rev. B **74**, 153416 (2006). [CrossRef]

**14. **J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science **305**, 847–848 (2004). [CrossRef] [PubMed]

**15. **F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. Pure Appl. Opt. **7**, S97–S101 (2005). [CrossRef]

**16. **M. Sarrazin, J.-P. Vigneron, and J-M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B **67**, 085415 (2003). [CrossRef]

**17. **A. Agrawal, T. Matsui, Z. V. Vardeny, and A. Nahata, “Terahertz transmission properties of quasiperiodic and aperiodic aperture arrays,” J. Opt. Soc. Am. B **24**, 2545–2555 (2007). [CrossRef]

**18. **C. Huang, Q. Wang, and Y. Zhu, “Dual effect of surface plasmons in light transmission through perforated metal films,” Phys. Rev. B **75**, 245421 (2007) [CrossRef]

**19. **P. B. Catrysse and S. Fan, “Propagating plasmonic mode in nanoscale apertures and its implications for extraordinary transmission,” J. Nanophotonics **2**, 021790 (2008). [CrossRef]

**20. **N. Marcuvitz, *Waveguide Handbook* (McGraw-Hill, 1951).

**21. **A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science **308**, 670–672 (2005). [CrossRef] [PubMed]

**22. **N. W. Ashcroft and N. D. Mermin, *Solid State Physics* (Saunders College, 1976).

**23. **E. L. Albuquerque and M. G. Cottam, *Polaritons in Periodic and Quasiperiodic Structures* (Elsevier B.V., 2004).

**24. **U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. **124**, 1866–1878 (1961). [CrossRef]

**25. **A. Pimenov and A. Loidl, “Conductivity and permittivity of two-dimensional metallic photonic crystals,” Phys. Rev. Lett. **96**, 063903 (2006). [CrossRef] [PubMed]