We demonstrate resonantly enhanced transmission of freely propagating coherent terahertz radiation through free-standing metal foils perforated with periodic arrays of sub-wavelength apertures. These arrays consist of 400 µm diameter apertures periodically spaced by 1 mm and 600 µm diameter apertures periodically spaced by 1.5 mm. We measure absolute amplitude transmission coefficients of ~0.6 at the resonance wavelength. Correspondingly, the ratio of the absolute amplitude transmission coefficient to the fractional aperture area at these resonance frequencies is ~5. This value at terahertz frequencies is significantly larger than equivalent values measured at optical frequencies.
©2004 Optical Society of America
The recent observation of enhanced optical transmission at through subwavelength apertures in metallic films and the subsequent association of this phenomenon with a resonance supported by periodic surface features raises the prospect of further applications of these phenomena. It is now well established that the optical transmission through subwavelength holes in metal films can be enhanced by orders of magnitude when the holes are arranged in a periodic array . There is a growing consensus that the mechanism thought to be responsible for these observations is a resonance supported by the periodic corrugations in which surface waves, or surface plasmon-polaritons (SPPs), which are the dominant mode of propagating electromagnetic radiation along the metal surface . The electromagnetic fields in the subwavelength apertures are evanescent and are thus highly attenuated. However, the fields at the entrance to the hole and consequently the intensity of the light radiated from the hole, are enhanced by the resonance and are much greater than would be the case without the surface resonance .
To date, both theoretical [4–7] and experimental investigations [3, 8–12] of this phenomenon have concentrated on the elucidation of this phenomenon at optical frequencies. This may be attributed largely to the fact that many of the proposed applications are perceived to have the greatest impact at optical frequencies. However, this emphasis has also been bolstered by conclusions drawn by experimental findings of Grupp et al. , who found phenomenologically that at optical frequencies, the magnitude of the real component of the dielectric constant of the metal must be much larger than the corresponding imaginary component of the dielectric constant. Therefore, at optical frequencies, periodic structures made from metals such as silver (ε=-29.7+0.981i at 800 nm) show more pronounced effects than metals such as nickel (ε=-13+21.7i at 800 nm) or iron (ε=-4.91+23i at 827 nm) . At THz frequencies, the imaginary component of dielectric constant is typically much larger than the real component for most metals . Thus, it is not clear that the use of metals is appropriate in this frequency range.
In this submission, we demonstrate resonantly enhanced transmission of THz radiation through a periodic array of sub-wavelength apertures fabricated in a metal sheet. The apertures are formed in a free-standing metal foil using two different metals for the top surface layer. We use THz time-domain spectroscopy to measure simultaneously the magnitude and phase of the absolute amplitude transmission coefficient and observe strong enhancement at the resonant frequencies. The locations of these frequencies agree well with established theories. To our knowledge, the observed transmissivity is the largest measured for such structures.
2. Description of the Phenomenon
Before moving to the experimental details, we discuss properties related to the interaction of electromagnetic radiation with the aperture array necessary to determine the location of the transmission peaks. It should be noted that several models have been put forward to explain the spectral modulation in the optical transmission and there continues to be disagreement over which of these most accurately describes the situation. Despite this, all of the theories concur one basic feature of the experimentally observed transmission spectra. Specifically, the spectra exhibit maxima that are directly related to the physical periodicity of the surface corrugation and the dielectric properties of the interfacial media (i.e., the metal and the surrounding dielectric).
The scattering of light is inherently diffractive in nature. The diffracted wave field from a structured metal surface is associated with a polarization charge, which corresponds to the induced surface plasmon-polaritons. Therefore, it is convenient to view the properties of the wave field propagating on and through the perforated metal foil in terms of the properties of SPPs. On a plane metal-dielectric interface, electromagnetic radiation incident on the surface cannot couple to the SPP waves because conservation of energy and momentum laws are not obeyed. However, when the metal film is perforated with a two-dimensional periodic array of apertures, it acts as a grating. The additional degree of freedom associated with the grating momentum allows for coupling between the incident radiation and SPP modes. The corresponding relation for conservation of momentum is then given by [1,2]
where k sp is the wave vector associated with the SPP, kx is the transverse wave vector component of the incident radiation, Gx and Gy are the wave vector components associated with the two-dimensional array, and i and j are integers. In the experiments described below, the apertures were formed on a square lattice. Therefore, |Gx |=|Gy |=2π/P, where P is the period spacing between apertures. We use bold face, kx , to symbolize vectors and plain face, kx, to symbolize its magnitude.
The dispersion relation for SPP waves associated with these perforated metal films is not well established. However, it has been shown experimentally that the SPP dispersion relation for a plane metal-dielectric interface allows one to determine the locations of the transmission peaks at optical frequencies to good approximation. We assume that such an approximation is equally valid at THz frequencies. This dispersion relationship is given by 
where ω is the frequency of the incident electromagnetic radiation, c is the speed of light in vacuum, εd is the dielectric constant of the dielectric interface medium and εm is the (complex) dielectric constant of the metal. This last quantity may be expressed as εm=εmr+iεmi, where εmr and εmi are the real and imaginary components of the dielectric constant of the metal, respectively. The complex propagation constant for the SPP wave can be expressed as ksp=kspr+i kspi, where the two individual components can be written as 
with =++εdεmr. In the limit where εmi > |εmr| and |εmr|, εmi≫εd, which is true for most metals at THz frequencies, the real and imaginary components of the SPP wave vector can be greatly simplified. The real component of the SPP propagation constant may be approximated as
and the imaginary component of the SPP propagation constant may be approximated as
For radiation normally incident of the perforated metal, the transverse component of the propagation wave vector is zero, kx=0. Therefore, using equations (1) and (5), we find that the locations of the transmission peaks is given by 
Note that this is essentially the same result as obtained at optical frequencies. In principle, both metal-dielectric interfaces will contribute to the observed transmission spectra. The imaginary component of the SPP propagation constant, kspi, corresponds to the 1/e propagation distance of the SPP on the metal-dielectric interface.
3. Experimental details
Krishnan et al.  found that maximum transmission enhancement was observed when the two dielectric materials, on either side of the metal film, had the same dielectric constant. Based on this observation, we fabricated square arrays of subwavelength apertures in a free-standing 75 µm thick stainless steel metal foil. Circular apertures of 400 µm diameter with a periodic spacing of 1 mm (Sample A) or 600 µm diameter with a periodic spacing of 1.5 mm (Sample B) were formed by laser cutting. Each array measured 5 cm×5 cm. It is worth noting that the ratio of the aperture diameter to the period spacing for both samples is identical to that used in many of the periodic aperture arrays developed for optical transmission studies. Two separate sets of these arrays were fabricated. The first set consisted simply of the apertures arrays in the stainless steel foil. With the second set, we sputter deposited approximately 3 µm of silver onto both surfaces of the stainless steel foil. We expect that this deposition approach will also deposit silver onto the sidewalls of the apertures. This second set was fabricated primarily because we do not know the dielectric properties of stainless steel at THz frequencies. Deposition of a metal with thickness greater than one skin depth at the relevant frequency allows us to straightforwardly test the transmission properties of the arrays using the dielectric constant of the top metal layer without the need to fabricate completely new aperture arrays. For frequencies between 0.1 and 1 THz, the skin depth in silver is less than 1 µm.
We used a conventional time-domain THz spectroscopy setup  to characterize these aperture arrays. Photoconductive devices were used for both generation and coherent detection. The arrays were attached to a solid metal plate with a 5 cm×5 cm opening that was placed at the center of the two off-axis parabolic mirrors in the spectroscopy system. The 1/e THz beam diameter was made slightly less than the aperture opening in the metal holder, and therefore less than the spatial extent of the array, thereby minimizing edge effects due to the finite size of the array. The THz beam was normally incident of the aperture array and horizontally polarized, parallel to the aperture rows. Reference transmission spectra were taken in the absence of the perforated metal foil, with the metal holder in place.
These experimental parameters allow us to directly determine the absolute amplitude transmission coefficients. 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(f), using the relation
In this expression, Eincident and Etransmitted are the incident and transmitted THz fields, respectively, |t(f)| and φ(f) are the magnitude and phase of the amplitude transmission coefficient, respectively, and f is the THz frequency.
4. Experimental results and discussion
The temporal waveforms of the transmitted THz pulses through the stainless steel aperture arrays and the silver coated aperture arrays are shown in Figs. 1(a) and 1(b), respectively. The waveforms are offset from the origin for clarity and the reference waveform is shown in both figures for the sake of comparison. From the figure, it is apparent that each waveform corresponding to Samples A and B encounters a sign reversal relative to the reference waveform. The cause of this inversion is not clear, but my simply arise from the induced phase shift caused by the resonance. Further investigation is required. Furthermore, there appears to be little difference between the measured waveforms for each sample, regardless of which of the two metals is used to form the aperture. We discuss below how this may be a simple consequence of the fact that many metals exhibit a large imaginary component of the dielectric constant at THz frequencies.
Figure 2 shows the magnitude and phase of the amplitude transmission coefficient versus THz frequency for the aperture arrays fabricated in the stainless steel foil. We show only the spectra up to 0.5 THz, since higher frequency resonances exhibited reduced signal-to-noise characteristics. For sample A, which has a periodic aperture spacing of 1 mm and air (εd=1) as the adjacent dielectric medium on both metal interfaces, we would expect to see only two SPP resonances in this frequency window. From Eq. (7), these resonances occur at ~0.33 THz and 0.46 THz, corresponding to indices (i,j) equal to (±1,0) and (±1,±1), respectively. From Fig. 2, it is apparent that these resonances occur at slightly lower frequencies. This is not surprising, given that Eq. (2) is not strictly valid for a periodically perforated metal film and is consistent with observations made at optical frequencies. For sample B, which has a periodic aperture spacing of 1.5 mm and air as the adjacent dielectric medium on both metal interfaces, we would expect to see four SPP resonances in this frequency window. From Eq. (7), these resonances occur at ~0.2 THz, ~0.28 THz, ~0.4 THz, and ~0.45 THz corresponding to indices (i,j) equal to (±1,0), (±1,±1), (±2,0), and (±1,±2)=(±2,±1) respectively. The last two of the four resonances is slightly obscured in the magnitude spectrum, Fig. 2(a), but are evident in the corresponding phase spectrum. Figure 3 shows the magnitude and phase of the amplitude transmission coefficient versus THz frequency for the aperture arrays fabricated in the silver-coated stainless steel foil. Once again, the SPP resonances for sample A occur at ~0.33 THz and ~0.46 THz and the SPP resonances for sample B occur at ~0.2 THz, ~0.28 THz, ~0.4 THz, and ~0.45 THz. The magnitude of the transmission coefficients of the lowest frequency resonances are ~0.6. The ratio of this value to the fractional aperture area is ~5 for both samples A and B. In comparing this value to a similarly defined ratio at optical frequencies , it is important to note that the current measurements relate to the THz electric field, while the optical measurements pertain to the optical intensity.
The measured full-width at half-maximum (FWHM) linewidth for each of the lowest frequency resonances in the spectra shown in Figs. 2 and 3 is ~10 GHz, which corresponds directly to the inverse of the temporal length of the waveforms in Fig. 1 (100 ps). Since the oscillations appearing after the main bipolar pulse in the corresponding temporal waveforms are not completely damped, we believe that experimental parameters (the temporal scan window) place an upper limit on the linewidth. While simply increasing the time delay in these measurements will yield improved frequency resolution, we are exploring an alternate approach  that will allow for significantly increased frequency resolution, in order to more accurately measure the lineshape of this resonance feature.
The observed linewidth relative to the (lowest) resonance frequency is much smaller than the equivalent value at observed with samples designed for optical frequency studies. This difference may arise, in part, from the longer propagation lengths for SPP waves at THz frequencies. The dielectric constant for silver at 0.2 THz is ε=-2.5×105+i5.5×106 . Thus, the imaginary component of the propagation constant, kspi=1.6×10-6 cm-1. While this value is significantly smaller than corresponding values at optical frequencies, one cannot expect a commensurate increase in the quality factor, since radiative losses from the grating structure are believed to be the dominant loss mechanism. For this reason, the observation that the stainless steel and silver-coated aperture arrays exhibit very similar transmission enhancement properties is not surprising. In fact, we would expect that aperture arrays fabricated from other good metallic conductors at THz frequencies would exhibit similar characteristics.
In conclusion, we have presented the first demonstration of resonantly enhanced transmission through periodic arrays of sub-wavelength apertures at THz frequencies. The arrays were fabricated in free-standing stainless steel metal foils, with and without a silver surface layer. We observed strong enhancements in the transmission at frequencies related to the aperture periodicity and the refractive index of the SPP waves at the metal dielectric interface. The peak transmission at the lowest frequency resonance was ~0.6 for each aperture array, which was a factor of ~5 larger than the fractional area occupied by the apertures. In contrast to studies at optical frequencies, where only silver and gold have been demonstrated to yield strong resonant transmission enhancement, the similar results from the two different metal foils presented above suggest that a broader range of metals may be used to investigate this phenomenon at THz frequencies.
We thank S. Blair and R.A. Linke for helpful discussions. We would also like to thank the reviewers for helpful suggestions.
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