1. Introduction

The demonstration of enhanced optical transmission through periodic arrays of subwavelength apertures [1] has elicited significant interest in recent years. While much of the initial work in this topic concentrated on understanding the underlying physical principles behind this phenomenon, there has been significant expansion recently in the range of investigations. For example, several studies have explored and demonstrated this phenomenon at mid-infrared [2] and far-infrared [3–6] frequencies. In this latter spectral range, we have shown that periodic arrays fabricated in metal films allow for larger transmission coefficients and narrower resonance linewidths than have been observed at visible frequencies [4]. There have also been several examples demonstrating the broader range of potential applications that may arise from the use of these structures. These include the demonstration of enhanced fluorescence from molecules attached to the metal surface [7], highly directional radiation from the structured metal [8], and the enhancement of nonlinear optical processes [9].

While this broadened range of exploration is of critical importance for developing successful applications, a number of unexplored properties of this process remain. As an example, Gordon et al. recently showed that the transmission properties of a periodic array of elliptical apertures exhibit strong polarization dependence [10]. In fact, the anisotropy in the polarization properties increases with increasing ellipticity. More generally, the details of the aperture shape should have significant impact on a wider range of optical properties related to the structure. This is particularly true when the aperture is only slightly subwavelength.

In this submission, we demonstrate that the resonantly enhanced transmission spectrum associated with a periodic array of subwavelength apertures is dependent upon the shape of the apertures. This is demonstrated using terahertz (THz) time-domain spectroscopy in aperture arrays fabricated in metal foils. Periodic and aperiodic arrays were fabricated using four different aperture shapes: three rectangular apertures with different aspect ratios and a circular aperture. Resonance features are evident only in the periodically spaced arrays. We show that the ratio of the transmission coefficients of the two lowest order resonances can be directly related to the geometrical properties of the rectangular apertures. We observe two independent, but phase-coherent, transmission processes: non-resonant transmission of the broadband THz pulse with no time delay related to the simple transmission through subwavelength apertures and a time-delayed resonant transmission related to the interaction of the THz pulse with the periodic aperture array. The former contribution is a bipolar waveform that exhibits a sign reversal relative to the reference waveform. This phenomenon is related simply to the diffractive properties of the subwavelength apertures and does not rely on resonances in the transmission spectrum.

2. Experimental details

The aperture arrays were fabricated in free-standing 75 µm thick stainless steel foils. We have previously shown that this substrate medium works well for the observation of resonantly enhanced transmission at THz frequencies [4]. We first fabricated aperture arrays in a periodic square lattice with a center-to-center spacing of 1 mm. Four separate arrays, shown schematically in Fig. 1, were fabricated, each with a different aperture shape: Array A consisted of 400 µm diameter circular apertures, Array B consisted of 400 µm×400 µm square apertures, Array C consisted of 400 µm×300 µm rectangular apertures, and Array D consisted of 400 µm×200 µm rectangular apertures. We also fabricated a second set of apertures arrays utilizing the four aperture shapes described above. However, in this case, the aperture spacing was aperiodic and designed to yield non-resonant transmission behavior. It should be noted that the aperture-to-aperture spacing in these latter four arrays was not completely random, since a minimum spacing was enforced to eliminate the possibility of overlapping apertures [11]. Nevertheless, these four aperiodic arrays exhibited no resonance features and demonstrated qualitatively identical transmission properties. Therefore, we only show the observations of the 400 µm×400 µm square aperture array (Array E) that was fabricated to yield non-resonant behavior. In all cases, the aperture arrays measured 5 cm×5 cm.

The arrays were characterized using a standard THz time-domain spectroscopy system [12]. Conventional photoconductive devices were used for both emission and detection. As is common in such spectroscopy systems, two off-axis paraboloidal mirrors were used to collect, collimate, and refocus the THz radiation from the emitter to the detector. The arrays were attached to a solid metal plate with a 5 cm×5 cm opening that was placed at the center of these two mirrors in the spectroscopy system. The 1/e THz beam diameter was smaller than the aperture opening in the metal holder, and therefore less than the spatial extent of the array. This was designed to minimize edge effects due to the finite size of the array. Reference spectra were obtained with the bare metal holder placed in the system. The THz radiation was horizontally polarized and normally incident on the aperture array. The orientation of the apertures with respect to the polarization direction is shown in Fig. 1.

 

Fig. 1. The four different aperture shapes used in this investigation and the polarization direction of the normally incident THz pulses. Array A consists of 400 µm diameter circular apertures, Array B consists of 400 µm×400 µm square apertures, Array C consists of 400 µm×300 µm rectangular apertures, Array D consists of 400 µm×200 µm rectangular apertures, and Array E consists of 400 µm×400 µm square apertures. In Arrays A-D, the apertures are periodically spaced by 1 mm. In Array E, the spacing is designed to yield a non-resonant transmission behavior. The dashed lines correspond to the aperture dimension at 45° with respect to the polarization direction, along the (+1, +1) axis. This last dimension is necessary for Fig. 5.

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Time-domain THz spectroscopy allows for the measurement of the transmitted THz electric field [12]. In order to compare the spectral transmission characteristics of the different arrays, we normalized the time-domain waveforms to the aperture fill fraction of Array A. Specifically, the aperture fill fraction of Array A is 0.1257. The waveforms of the other four arrays are corrected to match this fill fraction. Thus, the time-domain waveforms for Arrays B, C, D, and E are divided by 1.273, 0.955, 0.637, and 0.159, respectively. We then transformed the time-domain data to the frequency domain, allowing us to determine independently both the magnitude and phase of the normalized amplitude transmission coefficient, t_N (f), using the relation

In this expression, E_reference and E_transmitted are the reference and normalized transmitted THz fields, respectively, |t_N(f)| and φ_N(f) are the magnitude and phase of the normalized amplitude transmission coefficient, respectively, and f is the THz frequency. Using this procedure, the normalized amplitude transmission coefficient of Array A is the same as the absolute amplitude transmission coefficient. Absolute coefficients for the other four arrays require an inverse correction for the aperture fill fraction.

3. Experimental results and discussion

The time-domain waveforms corresponding to the transmitted THz pulses through the five different aperture arrays, as well as the reference waveform, are shown in Fig. 2. The waveforms are offset from the origin for clarity. Expanded versions of these time-domain waveforms are given in the accompanying multimedia file. In contrast to our earlier demonstration of resonantly enhanced THz transmission [4], the temporal scan window has been extended to more accurately obtain the linewidths of the resonance features. While each waveform was initially measured over a 320 ps temporal window, the waveforms in Fig. 2 were all truncated at 267 ps, since no useful signal information was present beyond this time delay value. There are several interesting features to note in these time-domain traces. If we compare the waveforms of Array A (circular apertures) and Array B (square apertures), there are some apparent differences. The oscillations after the main bipolar pulse for Array A exhibit a smaller peak-to-peak value than those of Array B, but extend for much longer in time. In general, the magnitude of the oscillations corresponds to the magnitude of the resonance feature, while the oscillation duration corresponds to the linewidth of that feature. As the aspect ratio of the square aperture increases, moving from Array B to D, the peak-to-peak value of the oscillations decreases, but the duration of the oscillations remains largely unchanged.

In comparing the reference waveform and the waveform associated with Array E to the waveforms obtained for the four periodic arrays, it appears that there are two independent, yet phase-coherent, contributions to the waveforms for Arrays A-D. The first contribution is a bipolar pulse that is similar in form to the reference waveform. This bipolar waveform does not exhibit any additional time delay relative to the reference pulse, within experimental error. We attribute this feature to the nonresonant transmission of the broadband THz pulses through the subwavelength aperture arrays, since it is present in all of the waveforms associated with the periodic and aperiodic arrays. The second contribution present in the waveforms associated with Arrays A-D is a damped oscillatory waveform. These oscillations contain the spectral features of the transmission resonances, as shown below. We attribute this latter contribution to the resonant interaction of the THz pulse with the periodically perforated metal film. It is important to note that the waveforms associated with Arrays A-E in Fig. 2 are multiplied by a factor of 10 for clarity. Thus, the nonresonant transmission (first contribution mentioned above) is strongly attenuated. The amplitude of the damped oscillations, though small relative to the reference, corresponds to a large amplitude transmission coefficient with a narrow resonance linewidth. We discuss this in detail below.

It is also apparent from Fig. 2 that the bipolar pulse feature (first contribution in previous paragraph) in the waveforms associated with Arrays A-E exhibits a sign reversal relative to the reference waveform. This characteristic was also observed in earlier measurements with similar array structures [4]. Since Array E exhibits non-resonant transmission behavior, as demonstrated below, this behavior cannot be related to the resonance phenomenon. We have previously shown that the traversal of a THz pulse through a single subwavelength slit will cause significant reshaping of the incident THz pulse due to diffraction [13]. One consequence of this reshaping process is that in the low frequency limit, the transmitted THz pulse would exhibit an additional ±π/2 phase shift, depending upon whether the polarization direction is parallel or perpendicular to the slit axis [13,14]. The phase imparted to a THz pulse transmitted through a subwavelength aperture (or an array of subwavelength apertures) in a conductive screen will also be frequency dependent. A full description of this dependence is beyond the scope of this contribution. Nevertheless, a sign reversal of the transmitted time-domain waveform would correspond to a constant phase of π superposed on the frequency dependent phase function. This low frequency constant phase contribution of π is consistent with conventional diffraction theory [14–16].

 

Fig. 2. Measured time-domain THz waveforms transmitted through five different aperture arrays fabricated in 75 µm thick free-standing stainless steel foils. [Expanded Fig. 2 (a), (b), (c)]

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Figure 3 shows the magnitude and Fig. 4 shows the phase of the normalized amplitude transmission coefficient, tN (f), versus THz frequency for the five aperture arrays. We show only the spectra up to 0.5 THz, since higher frequency resonances exhibited reduced signal-to-noise characteristics. As expected, the amplitude and phase spectrum for Array E shows non-resonant behavior. As we noted above, we fabricated four separate aperiodic arrays, one for each aperture shape. In each case we observed qualitatively similar non-resonant behavior. Thus, there is no observed resonance associated with the aperture shape. For the four periodic arrays, the lowest order transmission resonance occurred at ~0.29 THz and the next higher order resonance occurred at ~0.4 THz. It is apparent that the resonance frequencies vary slightly between Arrays A-D, even though the physical periodicity was fixed at 1 mm in each case.

As we have noted before, this enhanced transmission phenomenon is inherently a diffractive process. The diffracted wave field from a structured metal surface is associated with a polarization charge, which corresponds to the induced surface plasmon-polaritons (SPPs) [17]. 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. It has been shown that the spectra exhibit transmission maxima that are directly related to the physical periodicity of the surface corrugation and the relevant material properties of the interfacial media [18]. In the far-infrared, where the magnitudes of the real and imaginary components of the metal dielectric constant are large, the approximate locations of the transmission peaks are given by

Here, P is the physical periodicity, ε_d is the dielectric constant of the interfacial dielectric media (ε_d=1 in our case), and i and j are indices corresponding to the resonance order. Thus, we would expect to see only two SPP resonances in this frequency window at ~0.33 THz and 0.46 THz, corresponding to indices (i,j) equal to (±1,0) and (±1,±1), respectively. The fact that we observe resonance frequencies at slightly lower frequencies than predicted by Eq. (2) and that these frequencies vary slightly between Arrays A-D is not surprising, since Eq. (2) is only strictly valid for a plane metal film. A more correct description is expected to depend upon the shape, spatial distribution, and fill fraction of the apertures.

From Fig. 3, it is apparent that the magnitude of the normalized amplitude transmission coefficients of the lowest frequency resonances varies with aperture shape. As expected from the time-domain waveforms, the normalized transmission coefficient of the lowest order resonance is largest for the square apertures. The corresponding absolute transmission coefficient of the (±1,0) resonance for that aperture shape is ~0.8. This represents the largest transmission coefficient, to our knowledge, for such structures. The measured 3-dB linewidth for Array A was ~8 GHz, while the 3-dB linewidths for Arrays B-D varied between 9.5 GHz and 10.5 GHz. This is consistent with our earlier discussion regarding the duration of the oscillations in the time-domain waveforms. In contrast to our earlier measurements, we do not believe that these values are limited by the measurement technique. Using the phase spectra for each aperture, we can calculate the group delay for the transmitted time-domain waveform. Dogariu et al. have shown that this calculation matches experimentally observed values at optical frequencies [11]. It is also worth noting in the phase spectra that the low frequency phase shift for all five arrays is ~±, as expected from the discussion above.

One of the more interesting aspects of Fig. 3 is the observation that the ratio of the normalized amplitude transmission coefficient of the (±1,0) peak to the normalized amplitude transmission coefficient of the (±1,±1) peak varies with aperture shape. We can understand this variation based on the geometrical differences between the apertures. We demonstrate this quantitatively. For each aperture shape we subtract a scaled version of the magnitude spectrum for the aperiodic array from the magnitude spectrum of the corresponding periodic array. Thus, for example, we subtracted a scaled version of the magnitude spectrum of Array E from the magnitude spectrum of Array B. To first order, this removes the background transmission spectrum leaving only the transmission resonances. We use these modified magnitude spectra to demonstrate the aperture shape dependence of the transmission spectra.

The ordinate values of Fig. 5 correspond to the ratios of the magnitude of the (±1,0) peaks to the magnitude of the (±1,±1) peaks for the four periodic arrays. The abscissa values of Fig. 5 correspond to inverse ratios of the aperture lengths in the direction perpendicular to the vector associated with the resonance order. Thus the abscissa values correspond to the ratio of the aperture length along the (±1,±1) axes to the aperture length along the (0, ±1) axes. These dimensions are shown in Fig. 1. Numerically, the abscissa value for Array A is 400µm/400 µm, for Array B is 400√2 µm/400 µm, for Array C is 300√2 µm/400 µm, and for Array D is 200 √2 µm/400 µm. Fig. 5 shows the resulting data for the four periodic arrays along with a least squares linear fit to the data for Arrays B, C, and D. The fact that the data point for Array A does not fall on this line is believed to reflect the fact that the transmission characteristics of rectangular apertures differ from that of circular apertures. This difference is also related to the variation in the resonance peak between Arrays A and B (Fig. 3). It is worth noting that as the ratio of the aperture dimensions approaches zero, this does not correspond to a periodic array of infinitely long subwavelength slits. The data in Fig. 5 pertains only to periodic arrays that contain (rectangular) apertures that are subwavelength in both dimensions. The electric field distribution within one-dimensional and two-dimensional subwavelength apertures is fundamentally different.

 

Fig. 3. Magnitude of the normalized amplitude transmission spectra for (upper) Arrays A and B and (lower) Arrays B-E.

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Fig. 4. Phase of the normalized amplitude transmission spectra for (upper) Arrays A and B and (lower) Arrays B-E.

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In conclusion, we have shown that the aperture shape influences the transmission spectrum associated with a periodic array of subwavelength apertures. Specifically, we have shown that the ratio of the transmission coefficients for the two lowest order resonances can be related to the geometrical properties of the aperture. We expect that similar considerations may be used to predict ratios involving higher order resonances. We found two independent, yet phase-coherent, transmission processes: non-resonant transmission related to the simple transmission through subwavelength apertures and a time-delayed resonant transmission related to the interaction of the THz pulse with the periodic aperture array. We have also shown that the sign inversion observed in the main bipolar waveform for each of the transmitted THz time-domain waveforms can be attributed to diffraction and does not rely on the existence of any resonances. Of the four aperture shapes used, the array containing 400 µm×400 µm square apertures fabricated with a periodic spacing of 1 mm exhibits an absolute amplitude transmission coefficient of 0.8. Also, the 3-dB linewidths of the lowest order resonance features for the four periodic arrays are between 8 and 10.5 GHz. To our knowledge, these are the narrowest linewidths, relative to the resonance frequency, that have been observed in such structures.

 

Fig. 5. The ratio of the normalized amplitude transmission coefficients versus the ratio of relevant aperture dimensions. See text for details of the definitions of these ratios. The filled markers correspond to data points for the four periodic arrays. The dashed line is a linear least squares fit to the data for Arrays B-D.

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