We present a comprehensive experimental study comparing the propagation characteristics of the virtually unknown TE1 mode to the well-known TEM mode of the parallel-plate waveguide (PPWG), for THz pulse applications. We demonstrate that it is possible to overcome the undesirable effects caused by the TE1 mode’s inherent low-frequency cutoff, making it a viable THz wave-guiding option, and that for certain applications, the TE1 mode may even be more desirable than the TEM mode. This study presents a whole new dimension to the THz technological capabilities offered by the PPWG, via the possible use of the TE1 mode.
© 2009 OSA
Ever since the first demonstration of the parallel-plate waveguide (PPWG) for undistorted terahertz (THz) pulse propagation , the PPWG geometry has proven to be a breakthrough technology, enabling numerous THz applications. These include THz interconnects , pulse generation [3,4], spectroscopy [5–8], sensing [9,10], imaging , and signal processing . The PPWG has also been employed as a convenient two-dimensional environment for many THz experiments, such as to study photonic crystals [13,14], photonic waveguides , the super-prism effect , and Bragg resonances . Interestingly, all of these THz applications have exploited the lowest-order transverse-magnetic (TM0) mode, which is also the transverse-electromagnetic (TEM) mode of the PPWG. The TEM mode has been the obvious choice due to its low loss and ease of quasi-optic coupling, and perhaps most importantly, due to its negligible group-velocity-dispersion (GVD), as a result of having no low-frequency cutoff.
The lowest-order transverse-electric (TE1) mode of the PPWG has not been considered for THz pulse applications, mainly due to the presence of a low-frequency cutoff at fc = c/(2nb), where b is the plate separation, and n is the refractive index of the medium between the plates. This cutoff causes spectral filtering and introduces high GVD that results in broadening and reshaping of input broadband THz pulses, which has discouraged the use of the TE1 mode in the past. In this work, we compare the propagation characteristics of the TE1 mode with those of the TEM mode for possible THz pulse applications. We demonstrate how to negate the undesirable properties of the TE1 mode, making it a viable option for wave guiding, and further demonstrate that for certain THz applications, the TE1 mode may be more desirable than the TEM mode. Specifically, we show that one could achieve undistorted THz pulse propagation using the TE1 mode, similar to the TEM mode, but with the added potential for ultra-low ohmic losses in the dB/km range. We also show that it is possible to excite a practically simple, but highly effective, resonant cavity that is integrated with a PPWG, via the TE1 mode, and that this is not possible via the TEM mode.
2. Short path-length experiment
For the first experiment, we employ PPWGs with relatively short propagation path lengths. We fabricate these using highly polished aluminum plates, all having a transverse width larger than the ≈10 mm 1/e-diameter input Gaussian beam. As in previous experiments [1,2], the THz pulses, generated and detected using a commercial THz time-domain-spectroscopy (THz-TDS) system, are coupled into and out of the PPWGs using two high-resistivity silicon plano-cylindrical lenses positioned at the input and output. The input reference pulse, obtained using the cylindrical lenses in their confocal configuration with no waveguide in place, is
shown in Fig. 1(a) . The positive peak of the main transient has a FWHM of 0.8 ps, and the observed secondary features are inherent to the THz-TDS system. Figure 1(b) shows the pulse after propagating through a 2.5 cm long, air-filled PPWG having b = 0.5 mm, with the input beam polarized perpendicular to the plates [inset of Fig. 1(b)] to excite the TEM mode. As expected, there is no change in the shape of the pulse compared to the reference, indicating single TEM mode propagation. This is confirmed by the corresponding amplitude spectrum shown in Fig. 2(a) by the open circles, derived by Fourier transforming the truncated pulse, which gives no indication of a low-frequency cutoff. Figure 1(c) shows the pulse after propagating through the same PPWG, but with the input beam polarized parallel to the plates
[inset of Figs. 1(c) and 1(d)] to excite the TE1 mode. In sharp contrast to the TEM case, the picosecond-scale input pulse is broadened to more than 150 ps, strongly reshaped, and exhibits a negative chirp. Its amplitude spectrum shown by the dotted curve in Fig. 2(a) indicates a cutoff at fc = 0.3 THz, as expected for the TE1 mode with b = 0.5 mm. Since the next possible even-symmetric higher-order mode [1,2], the TE3 mode, has its cutoff at 0.9 THz, almost at the high-end of the input spectrum, we can conclude that the pulse has propagated via the single TE1 mode.
In order to analyze the propagation behavior, we can write the frequency-domain input-output relationship for the single-mode waveguide asFig. 1 (squared, since it is the same at the input and output), and Cx is the coupling coefficient for the x direction (not squared, since it is 100% at the input). The coupling coefficients are analyzed using the standard overlap-integral method , where Cy takes into account the similarity of the input/output Gaussian beam to the guided mode in the y direction, while Cx takes into account the spreading of the beam in the x direction due to diffraction. L is the propagation length, α is the attenuation constant, βz is the phase constant, and βo = 2π/λo, where λo is the free-space wavelength. The βo-term accounts for the fixed THz transmitter and receiver positions during the experiment.
Applying Eq. (1) to the data corresponding to two different propagation lengths, taking the complex ratio, and extracting the phase and amplitude information, we can derive the experimental βz and α for the propagating mode. This assumes that Cx is the same for the two lengths, which is justified since both lengths are relatively short. Using βz, we can derive the phase velocity υp ( = ω/βz) and the group velocity υg ( = ∂ω/∂βz), where the TE1-mode values are plotted in Fig. 2(b) by the dots and open circles, respectively. These show excellent agreement with the theoretical (thick red) curves calculated using classical guided-wave theory [18,19] for the TE1 mode of the PPWG with b = 0.5 mm. The curves clearly demonstrate the highly dispersive nature of propagation due to the cutoff at 0.3 THz, where the high-frequency components travel faster, resulting in the observed negative chirp.
The experimentally derived α, for both the TE1 and TEM modes, are plotted in Fig. 3(a) by the dots and open circles, respectively. Also plotted are the corresponding theoretical (red and blue) ohmic loss curves, computed using the expressions [18,19]
Here, Zo is the free-space impedance and Rs = [π fμ /σ]1/2 is the surface resistance, where μ is the permeability and σ is the DC conductivity. The experimental and theoretical curves agree reasonably well, and reveal a remarkable counter-intuitive property of the TE1 mode, where the attenuation actually decreases with increasing frequency for all frequencies above cutoff. This dependence is in direct contrast to that of the TEM mode, which increases with frequency, and has not been observed with any other THz waveguide to date. The frequency dependence in the TEM-mode’s attenuation can be physically attributed to the decrease in skin depth, resulting in an increased ohmic loss as the frequency increases, and is the reason the observed TE1-mode’s dependence is counterintuitive, since one would expect the same skin-depth dependence. However, it turns out that in the TE1 mode, this physical effect is
offset by a more dominant geometric effect. An in-depth discussion of this remarkable phenomenon is presented elsewhere .
This frequency dependence suggests that we should be able to reduce the TE1-mode’s attenuation by pushing fc to lower frequencies. We can lower fc by increasing b; for example, for an air-filled PPWG, when b = 5 mm, fc = 30 GHz, which is at the low end of the input spectrum. The corresponding αTE is plotted in Fig. 3(b) by the thick red curve, along with the analogous (same b) αTEM by the thick blue curve. For clarity, a close-up of the baseline of the αTE curve is shown in Fig. 3(c). We note that when b increases from 0.5 mm to 5 mm, by a factor of 10, αTE varies in a highly non-linear fashion, reducing from 2.7 × 10−2 dB/cm to 2.6 × 10−5 dB/cm (@ 1 THz), by a factor of more than 1000. In contrast, αTEM varies in a linear fashion, reducing from 1.5 × 10−1 dB/cm to 1.5 × 10−2 dB/cm (@ 1 THz), by the same factor of 10. Therefore, increasing b actually decreases both αTE and αTEM, although the effective reduction can be several orders of magnitude stronger in αTE, which is a clear advantage. The larger b predicts an extraordinarily low attenuation for the TE1 mode, where the calculated value of 2.6 × 10−5 dB/cm ( = 2.6 dB/km) is only an order of magnitude higher than that of telecommunications-grade optical fiber operating at 1550 nm.
It is also interesting to consider a situation where the PPWG is filled with a dielectric medium other than air. For example, when the above PPWG is filled with a 5 mm thick high-resistivity silicon (n = 3.42) slab, this would affect αTE and αTEM according to Eqs. (2) and (3). In fact, αTEM would proportionately increase, which is attributed to an increase in the magnetic field, generating higher conduction currents, and therefore, a higher ohmic loss in the metal plates , as indicated by the thin blue curve in Fig. 3(b). Although, at first glance of Eq. (2), one might predict the same dependence for αTE, the fc terms bring about a more complex n dependence. Interestingly, αTE is actually reduced due to the high-index dielectric, as shown by the thin red curve in Fig. 3(c). One way to understand this is to realize that fc is being pushed to lower frequencies by the high-index medium, thereby naturally reducing the loss, similar to that observed when increasing b. However, although it is possible to reduce the ohmic loss αTE via the use of a dielectric filling, one need to be aware that an additional loss would be introduced by the dielectric medium (the dielectric loss), which would contribute to the overall loss. In fact, it is likely that the dielectric loss may turn out to be the more dominant loss in the TE1 case.
There are other important advantages in lowering the cutoff of the TE1 mode. This would assist to maintain the spectral integrity of the input pulses, while reducing the GVD to almost negligible values, as seen by the (thin red) velocity curves shown in Fig. 2(b) for an air-filled PPWG with b = 5 mm. Except for the low end of the spectrum, these curves indicate virtually zero dispersion. However, this method of lowering the cutoff by increasing the plate separation has the obvious disadvantage of permitting multimode propagation, and is also a problem even for the TEM case. Since the cutoff frequencies of many higher-order modes now fall within the input spectrum, multiple modes could be simultaneously excited, leading to excessive loss and dispersion. Nevertheless, this problem can be overcome if the input coupling is optimized to selectively excite only the mode of interest via mode-matching. We note that the electric field of the TE1 mode has a ‘sin(πy/b)’ spatial dependence (where y = 0 is at one of the plate surfaces), whereas the TEM mode has a flat-top profile, independent of y. Therefore, the TE1 profile is better matched to a Gaussian profile, and should enable better coupling and selectivity. To quantify this, we calculate the power coupling efficiency (η) from an input Gaussian beam to the TE1 and TEM modes, as a function of b/D, where D is the 1/e beam size of the Gaussian. This result, shown inset of Fig. 3(a), predicts that we can couple to the TE1 mode with a maximum possible η of 99%, achieved when b/D = 1.42, whereas to the TEM mode, the maximum possible η is only 89%, achieved when b ≈D. Additionally, the η curve for the TE1 mode is much broader than that for the TEM mode, which implies that we can achieve single-mode propagation for a larger range of input beam sizes when exciting the TE1 mode, compared to the TEM mode. This would mean that, when utilizing a possible multimode (or over-moded) PPWG configuration (having a large b), it would be easier to achieve single-TE1 mode propagation than to achieve single-TEM mode propagation.
To test an over-moded configuration for the TE1 case, we used the same aluminum plates as in our earlier experiment, and constructed PPWGs with a large b = 5 mm. We directly excited the guided-wave with a weakly focused (≈10 mm 1/e-diameter) THz input beam polarized parallel to the plates, without using cylindrical lenses. In practice, when using a large b, it is important to ensure that the input beam size is sufficiently large, so that it interacts with the (inside) surfaces of the plates, right at the input plane, to setup the guided mode. Figure 1(d) shows the resulting pulse after propagating though a 2.5 cm long PPWG, which indicates only a mild reshaping, in sharp contrast to the severe pulse distortion seen in Fig. 1(c). The experimentally derived phase velocity for this larger value of b, shown by the dots in Fig. 2(c), is in excellent agreement with the theoretical prediction (red curve) for the TE1 mode and indicates negligible dispersion. This clearly demonstrates that we have achieved single TE1 mode propagation, confirming the high selectivity of the input coupling, and moreover, demonstrates that we have overcome the problems due to the cutoff. Determining the experimental attenuation is not possible here, because the ohmic losses [Fig. 3(c)] are far too small to measure with these propagation lengths.
3. Long path-length experiment
An important consideration when using relatively long PPWGs is the energy leakage caused by wave diffraction in the unconfined (transverse) direction . This would result in an additional loss process termed as the “diffraction loss”. In the case of the TEM mode, the diffraction is identical to that of a freely propagating wave in space, and the loss can be quantified accordingly . However, in the case of the TE1 mode, there is a subtle difference, as demonstrated in the following experiment and subsequent analysis.
In the experiment, to accommodate the long (and later, curved) path lengths, we used a fiber-coupled THz-TDS system, different to that used previously. Beam shaping optics were employed to increase the input beam (waist) diameter to a frequency-independent size of 2 cm. This size was chosen to reduce the diffractive spreading in the x direction. To match this input beam size, all the PPWGs were fabricated with b = 10 mm (fc = 15 GHz), using polished aluminum plates. The input beam was polarized parallel to the plates to excite the TE1 mode. Figure 4(a) shows the pulse after propagating through a 2.5 cm long PPWG, used as the reference waveform in this experiment. The 1.05 ps FWHM of the positive peak is slightly larger than that of the reference pulse in the previous experiment, since the bandwidth was
slightly less, as seen in the amplitude spectrum shown by the open circles in Fig. 5(a) . In fact, compared to the previous system, there is more low-frequency content in the input pulse. Figure 4(b) shows the pulse after propagating through a 25 cm long PPWG, which was fabricated using plates having a transverse-width of 10 cm to accommodate unbounded diffractive spreading in the x direction. Although the pulse distortion is clearly not as significant as that observed in the previous experiment [Fig. 1(c)], there is some reshaping, and clear evidence of a negative chirp, where the low frequencies have been pulled out of the main pulse. This is a direct consequence of the increased low-frequency content in the input pulse. Based on this time dependent behavior and the associated clean spectrum (dotted curve) shown in Fig. 5(a), we can conclude that the pulse propagates via the single TE1 mode. The spectra for the 2.5 cm and 25 cm long PPWGs in Fig. 5(a) indicate a loss in the low-frequency components of the propagated signal, but no discernible loss in the high-frequency components. We will later show that this loss behavior can be explained by the diffraction loss, which is the dominant, and in fact, the only appreciable loss mechanism here.
We also fabricated a long, curved PPWG (with b = 10 mm), to investigate the effect of bends on the propagation behavior of the TE1 mode. This was fabricated using polished aluminum adhesive tape (from 3M) and consisted of a 3 cm long straight section followed by a 23.6 cm long semi-circular section, as illustrated in Fig. 4(d). Again, to accommodate unbounded diffractive spreading of the 2 cm diameter input beam, the transverse-width was chosen to be 10 cm. This composite PPWG was excited at the input-end near the 3 cm section, polarized parallel to the plates to excite the TE1 mode. The propagated pulse shown in Fig. 4(c) show remarkably low dispersion and very mild reshaping, in contrast to the pulse propagated through the 25 cm straight PPWG. In fact, the FWHM of the positive peak is 0.95 ps, even slightly smaller than that for the 2.5 cm long PPWG. Although not presented here, we see the same low-frequency loss in the amplitude spectrum as seen before [in Fig. 5(a)], apparently caused by the diffraction loss. This low-frequency loss improves the “effective bandwidth” of the signal, and along with the negligible dispersion, results in the observed narrowing of the pulse. We believe that this interesting phenomenon of reduced dispersion is related to whispering-gallery-mode excitation in the curved section of the
composite PPWG .
In order to investigate the loss behavior observed in the long path-length PPWG [Fig. 5(a)], we again resort to the fundamental input-output expression given in Eq. (1). As mentioned before, the one-dimensional amplitude-coupling-coefficient Cx takes into account the spreading of the beam in the unconfined x direction, and was assumed to be the same for the considered (two) path lengths in the first experiment, due to the relatively short lengths used. However, since the current experiment deals with longer path lengths, Cx plays a major role. The longer path lengths would result in much higher lateral spreading of the beam, and in the case of the TEM mode, this spreading would be identical to that of a freely propagating beam in space, allowing the direct application of Gaussian-beam optics in one dimension to quantify this effect . In the case of the TE1 mode, however, this spreading cannot be immediately generalized to be identical to a freely-propagating beam, for the simple reason that it is not a TEM wave. Nevertheless, if we could assign an “effective refractive index” for the wave propagating in the TE1 mode, then mathematically, it would still be possible to apply Gaussian-beam optics, using this equivalent index, to determine the lateral spreading in the air-filled PPWG. In fact, analogous to a conventional dielectric medium, we can define an effective index as neff = c/υp, based on the phase-velocity curves presented in Fig. 2(b). These curves predict an neff that varies between zero and unity, since υp varies from ∞ to c, as the frequency increases from cutoff. This plasma-like behavior mimicked by the TE1 mode actually opens up the possibility for a whole new class of artificial dielectric media having a refractive index less than unity . Therefore, neff = 0 at fc, and increases towards unity as the frequency increases.
An interesting consequence of this result is that the lateral spreading for a wave propagating in the TE1 mode would be generally more than if it were to freely propagate, starting from the same Gaussian beam (waist) size, due to the fact that the index is lower than that of free space. This spreading is more severe for frequencies near the cutoff, but becomes negligible at high frequencies, as shown in Fig. 5(c). This figure shows the frequency-dependent lateral output beam size after propagating through a 25 cm long PPWG, starting from a frequency-independent 2 cm diameter input beam, for b = 0.5 mm (thick red curve) and b = 10 mm (thin red curve), in the TE1 mode. Also shown (blue curve, which is almost perfectly overlapping with the thin red curve) is the frequency-dependent output beam size for the TEM mode, starting from the same input beam size, and after propagating 25 cm. Note that in the case of the TEM mode, the value of b does not affect the spreading. This demonstrates that for large values of b, the beam spreading associated with the TE1 mode is very similar to that of the TEM mode, or to a freely propagating beam, except for the very low frequency end (near fc), where it is slightly higher.
Once the diffractive spreading is known, we can determine Cx for the short (2.5 cm) and long (25.0 cm) PPWGs in the current experiment, using the standard overlap-integral method . These are plotted in Fig. 5(d), assuming a collecting-aperture size of 6 mm for the silicon-lens-coupled THz receiver. Now, applying Eq. (1) to the short and long path-lengths separately, taking the complex ratio, and extracting the amplitude information, we can writeEquation (4) would allow us to determine the experimental α due to the ohmic loss, provided there is a measurable change between |Eoutl| and |Eouts| × [Cxl /Cxs], where the latter term is the adjusted spectrum of the short waveguide (mathematically) accounting for the diffraction. These two spectra are plotted in Fig. 5(b) by the dots and open circles, respectively, and are experimentally indistinguishable. This implies that the experiment does not allow a meaningful measurement of the ohmic loss. However, this does demonstrate that the ohmic loss is virtually negligible, which is also confirmed by the theoretical loss, computed using Eq. (2) for b = 10 mm, where it is found to be 9.3 × 10−6 dB/cm at 0.5 THz and 3.3 × 10−6 dB/cm at 1 THz. To meaningfully measure these extraordinarily low ohmic losses, we would need to employ PPWGs that are several tens of meters in length, while overcoming the diffraction losses. This analysis clearly demonstrates that for a PPWG with a large enough b, the only appreciable loss for the TE1 mode is caused by diffraction. Therefore, mitigating this diffraction loss is an important consideration towards the practical realization of an ultra-low loss THz waveguide .
4. Resonant-cavity experiment
In this experiment, we investigate the feasibility of exciting a resonant cavity integrated with a PPWG, via the TE1 and TEM modes. We note that there have been several interesting experimental THz studies demonstrating resonant spectral features using (modified) PPWG structures, fabricated using advanced lithographic techniques [10,13–15,17]. Here, we study a very simple cavity, which can be easily integrated with a PPWG, fabricated using conventional machining.
Figure 6(a) shows a 10,000-scan average of a THz pulse after propagating through a 6.4 mm long (reference) PPWG with b = 1 mm in the single TE1 mode. As expected, we observe a negative chirp with pulse broadening due to the cutoff. The corresponding amplitude spectrum derived by Fourier-transforming the original 320 ps time-scan, zero-padded to 5120 ps, is shown in Fig. 6(c) on a logarithmic scale. This exhibits a cutoff fc = 0.15 THz and two strong water-vapor absorption lines (green arrows) at 0.557 THz and 0.752 THz. Figure 6(b) shows the pulse after propagating through a PPWG with the same b, but where the top plate has a square groove with side d, situated perpendicular to the direction of propagation, and centered between the input and output planes of the waveguide. The longitudinal cross-section of the fully integrated cavity is shown schematically in the inset of Fig. 6(b), and a photograph of the top plate is shown in Fig. 7(b) . When comparing the two time pulses, we can see the presence of a low-frequency envelope for the one with the cavity, with a more dramatic, but localized effect in the corresponding amplitude spectrum (derived as before) shown in Fig. 6(d). This
figure shows a very strong, narrow extinction feature (red arrow), in addition to the two water-vapor lines.
By comparing the spectra of the propagated pulses, with and without the cavity, we can derive the power transmission for the integrated device, which is plotted in Fig. 7(a) by the dots, in the vicinity of the extinction feature. This is fit with a Lorentzian-line shape (red curve), giving a center frequency fo = 0.280 THz, a 3-dB linewidth Δf = 5 GHz, and a peak extinction coefficient of almost 30 dB. Although the derived quality-factor Q = fo/Δf = 56, is not so impressive due to the relatively lower fo, these values of Δf and extinction-coefficient are the best ever measured in the THz regime for a PPWG-based device, to the best of our knowledge. In order to theoretically model the cavity, we resort to the well-known resonance-
frequency expression for an air-filled, generalized 3-D rectangular cavity, given by Eq. (5). This calculated value of fr is in excellent agreement with the experimental fo, and suggests that this groove behaves as a 1-D cavity, where standing waves are setup between the two vertical sidewalls of the groove, similar to a Fabry-Perot cavity.
Next, we conduct the same experiment, but using the TEM mode of the same PPWG instead of the TE1 mode. The propagated pulses, without and with the cavity, are given in Figs. 8(a) and 8(b), with their corresponding spectra in Figs. 8(c) and 8(d). Unlike the case of the TE1 mode, we do not see any strong, localized resonance dips in this case, although there is some additional structure in the propagated pulse and its spectrum, when the cavity is present. This means the cavity does perturb the propagating mode, but not in any meaningful manner. This is clear evidence that this particular cavity configuration cannot be excited by the TEM mode. This contrasting behavior, where the TE1 mode can excite the integrated cavity, and the TEM mode cannot, is a direct consequence of the better matched field orientation of the TE1 mode to that of the resonant cavity. The cavity is more efficiently excited via open-aperture-coupling of the electric field oriented parallel to the x direction. Additionally, this experiment also reveals that, although there is significant pulse broadening due to the TE1 mode’s inherent cutoff, this does not inhibit the use of this mode for resonant-cavity-based THz applications. As demonstrated in the previous experiments, we could minimize the dispersion by increasing b. However, we have observed that this weakens the strength of the resonance dip, decreasing the sensitivity, implying less coupling to the cavity,
due to the reduced energy density inside the PPWG. Nevertheless, the dispersive broadening due to a smaller b can be readily ignored in this type of application.
In this comprehensive study, we have compared the propagation characteristics of the virtually unknown (in the THz regime) TE1 mode of the PPWG with those of the well-known TEM mode. We demonstrate that by the proper choice of the plate separation and input excitation, we could negate the dispersive pulse broadening that has discouraged the use of the TE1 mode for THz-pulse applications, making it a viable wave-guiding option. We find that it is possible to achieve extraordinarily low ohmic losses with the TE1 mode, which would make this an ideal candidate for long-path-length applications, provided we could mitigate the diffraction losses. We also demonstrate that it is possible to excite a simple resonant cavity integrated with a PPWG using the TE1 mode, but not with the TEM mode, implying that the TE1 mode may be more advantageous than the TEM mode for certain THz applications. As a closing remark, we point out that this study in no way lessens the value of the plethora of unique THz applications made possible by the TEM mode of the PPWG, as presented in Refs. [1–17], but rather, presents a whole new dimension to the technological capabilities offered by the PPWG, for many other possible THz applications via the use of its TE1 mode.
This work was supported in part by the National Science Foundation (NSF) and by the United States Air Force through the CONTACT program.
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