2. Experiment and Analysis
The experimental setup, shown in Fig. 1
(a), is a standard THz-TDS system modified for reflection-mode measurements. THz radiation is generated and detected using photoconductively switched dipole antennas backed by hyper-hemispherical silicon lenses. The p
-polarized THz beam travels from the emitter to a paraboloidal collimating mirror, passes through a Teflon lens, and forms a planar-wavefront focus with a 1/e
amplitude diameter of ~6 mm on the grating. All of the optics are confocal and detection is reciprocal to maximize power transfer. The bistatic angle between the emitter and detector assemblies is approximately 37°. To examine the gratings in transmission mode, the system was also re-configured as shown in Fig. 1
(b). The detector-emitter separation was increased to maintain confocality of the Teflon lenses and paraboloids. In both reflection and transmission modes, polarization effects were examined by rotating each grating about its surface normal axis.
(a) shows the energy spectral density (ESD) obtained from these data along with a reference spectrum (thick dashed curve) obtained by replacing the rods with a flat polished aluminum plate. The data for the parallel grating configuration, in which SPPs cannot couple, show a smooth spectrum and broadband loss due predominately to scattering. In the perpendicular case, two spectra were generated by computing the ESD on windowed portions of the time-domain data. The window designated by the ‘Window’ label in Fig. 2
(a) encompasses only the ringing portion of the time-domain data, and its spectrum reveals a sharp peak at 450 GHz. We believe this feature corresponds to the coupling and propagation of a SPP for three main reasons: 1) SPP dispersion and grating calculations predict coupling on the m
=-1 diffraction order at 455 GHz for our experimental setup, 2) the ringing is present only in the perpendicular configuration, and 3) the extended duration and gradual decay of the narrowband ringing is consistent with the continual decoupling of a SPP as it travels along the grating surface. While SPP calculations also predict SPP coupling on the m
=+1 diffraction order at 879 GHz, we do not observe this mode. We attribute this to a blazed grating effect wherein the THz beam is preferentially diffracted along the m
=-1 order due to both the shape of the grating rods and the angle of incidence of the THz beam. The onset of this effect should be observable when the THz beam is nearly normally incident upon the grating. For this situation, SPPs in both diffraction orders should have roughly equal strength but their coupling frequencies should be starting to diverge from equality. This effect may be occurring for SPPs transmitting through the 0.5 mm rod grating at near normal incidence and is discussed in more detail below.
Time-domain THz signals. (a) and (c) are reflected signals from brass rod and Al plate gratings, respectively. (b) Transmission through 0.5 mm rod grating. Thin and thick curves in (a)–(c) correspond to parallel and perpendicular grating configurations, respectively. Signals are shifted for clarity. All signals in (b) are scaled up in amplitude by three times. Window boundaries for spectra in Fig. 3
are indicated by boxes in. (d) Plan view of geometric reflection effects accounting for additional scattering loss in perpendicular configuration.
The duration of the narrowband ringing provides an estimate of the maximum distance traveled by the SPP along the grating. Assuming a speed no greater than c (speed of light in free-space), the ringing indicates that the SPP travels <8 mm before it is decoupled, implying that SPP coupling, travel, and decoupling all occur in a span of <16 grating periods. This distance is consistent with the full-width at half-max (FWHM) bandwidth of the SPP signal, which was found to be 0.055 THz. Assuming a SPP propagation speed of c, the SPP bandwidth indicates a travel distance on the order of 6 mm. Though the SPP propagation distance is slightly greater than the diameter of the receiver’s maximum sensitivity spot, most of the energy of the SPP has already decoupled before even 6 mm of travel. Therefore, the limited spot size has only a minor effect, if any, on the received waveform. This very rapid decoupling greatly limits the utility of the grating coupling method for launching usable SPPs on guiding structures.
An estimate of the SPP coupling efficiency can be obtained by first considering the parallel and perpendicular time-domain data. Since the energy of the SPP is delayed in time, due to traveling along the grating prior to decoupling, it is possible to temporally window SPP data from the portion of the THz pulse that is specularly reflected. The ESDs of the total parallel waveform and the ringing (windowed) perpendicular waveform are then computed. These ESDs are shown in Fig. 3
(a), as previously described. We sum over these ESDs to compute the energy contained in both waveforms. However, to confine our analysis to energy in the SPP
, this computation is limited to a narrow band around which the SPP spectrum peaks. The ratio of this narrowband energy in the ringing perpendicular data versus that in the parallel data provides an estimate of the coupling efficiency. By comparing the ringing energy with the energy in the parallel grating configuration, this estimate automatically normalizes out the scattering losses common to both grating configurations. One additional normalization must be performed to adjust for scattering losses unique to the perpendicular configuration. A more detailed account of this analysis follows.
For discrete-time data the energy contained in a particular spectral band, ESB, can be determined by
] is the discrete Fourier transform of the time-domain data and is a function of discrete frequency f
is the total number of data points, and f
represent the frequencies bounding the band of interest. The full-width at half-max bandwidth of the SPP was found to be f
=0.475-0.42=0.055 THz by inspection of the ESD of the windowed ringing data shown in Fig. 3
(a). By the above formula, the energy in this band was then found for both the SPP data and the parallel configuration data. The ratio of these energies indicates that 48% of the incident energy in the SPP band (that is not specularly scattered) is returned to the detector but delayed in time. This ratio was then normalized to adjust for scattering losses unique to the perpendicular grating configurations. These losses arise mainly from geometric multiple-reflection effects, one of which is illustrated in Fig. 2
(d) where the rods behave like retro-reflectors scattering a fraction of the incident energy into undetectable directions. The parallel configuration does not suffer this retro effect so more of the incident energy is detected. This loss mechanism can be quantified by first assuming that, in the absence of this geometric effect, both configurations should return the same overall energies. By “overall”, we mean the energies of the entire waveforms, not just windowed portions. Assuming we collect all of the decoupled SPP energy we can also neglect SPP losses, including joule losses (valid given the high conductivity of brass at THz frequencies). We thus attribute all of the measured
loss between configurations to geometric reflection effects. This loss must be normalized out of our efficiency estimate. The total waveform energies in both configurations can be computed by applying the previous formula to the entire bandwidth (~0.1-1.3 THz) of the signals. We emphasize that in both configurations the argument in the formula is now the DFT of the entire
time-domain scan, not just a windowed portion. The resulting energy ratio (perpendicular/ parallel) is 0.68. Normalizing this loss out of the previous SPP detected energy we obtain an adjusted result of 0.48/0.68=71%. Finally, the nearly complete decay of the SPP ringing in Fig. 2
(a) indicates that essentially 100% of the energy coupled into the SPP mode is decoupled. Therefore, assuming we fully detect the decoupled SPP and that the decoupling efficiency is ~100%, the coupling efficiency is ~71%.
Energy spectra of collected waveforms. The thick dashed curves are reference spectra from flat mirror targets in (a) and (c) or from free-space transmission (no target) in (b). Thin curves in (a) and (c) correspond to the parallel grating configurations. Thick curves in (a) and (c) are spectra of the windowed ringing data labeled in Fig. 2
. Both solid curves in (b) are spectra of windowed ringing data labeled in Fig. 2
. All spectra were normalized with the same value and, for clarity, the spectra in (b) were scaled up as indicated. (d) Illustration of rod imperfections.
In addition, transmission data were collected for normal and 19° incidence on the free-standing 0.5 mm rod grating. The corresponding data are shown in Figs. 2
(b) and 3
(b). The 19° data in Fig. 2
(b) were shifted up +1 nA and the 0° data were shifted -1 nA and -5 ps for visual clarity. These time-domain data show that some energy was transmitted through the grating in the perpendicular configuration. The thin curve in Fig. 2
(b) indicates that essentially no energy was transmitted when the grating was in the parallel configuration, consistent with SPP behavior. The spectra of the transmission data reveal the expected peaks corresponding to SPP modes. For example, the spectral peak for the 19° incidence data is at 418 GHz, which is quite close to the spectral peak (at 450 GHz) obtained from the bistatic reflection-mode data, wherein the incidence angle was approximately 18.5°. The discrepancy is partially due to the limited accuracy in setting the angle of incidence for each configuration. According to the energy spectra in Fig. 3
(b), the SPP energy transmitted through the grating accounts for only about 1–3% of the total SPP bandwidth energy incident on the grating, for both 19° and normal incidence. This indicates that the SPP is largely bound to a single side of the grating, but also leaks energy to the other side. The duration of ringing is very similar in both the transmission and reflection data, again suggesting that the lifetime of the SPP on the grating structure is short.
Reflection data were also collected for the target shown in Fig. 1
(d): an aluminum plate with v-shaped grooves. The grooves were about 0.25 mm deep and were separated by 0.5 mm. As shown in Figs. 2
(c) and 3
(c), this structure produced very similar data to the 0.5 mm rods. Again, the time-domain SPP signal (thick curve) is shifted +1 nA and +5 ps for clarity. Though similar to the time-domain rod grating data, this data reveals a slight difference in the shape and magnitude of the initial peak. This is also likely due to geometric reflection effects since the curved surfaces of the rods should scatter differently than the thin ridges on the grooved plate. The blazed grating effect is also present in this grating, so it is not surprising that we still did not observe the SPP corresponding to the m
=+1 diffraction order. Once coupled, however, the SPP appeared mostly unaffected by the grating profile and we estimated the coupling efficiency of the SPP on this grating to be approximately 72%, after accounting for specular scattering. Again, the total
returned energy from the perpendicular grating configuration was lower than that of the parallel configuration by a factor of 0.66 and the estimated efficiency was normalized with this value. Transmission measurements were clearly not possible with this grating.
Assuming a metal dielectric constant equal to that of gold, simple calculations of dispersion and grating diffraction [8
8. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988).
] reveal that coupling to SPPs in the reflection arrangements should occur via the m
=-1 diffraction order at 455 GHz, for the given 37° bistatic angle. This calculation also applies to a transmission arrangement at 18.5° incidence. For transmission at normal incidence both the m
=1 and m
=-1 diffraction order SPPs should occur at the same frequency, 600 GHz, assuming the rod spacing is exactly 0.5 mm and that the incident beam is perfectly normal. Our data indicates a peak at 526 GHz, but an increase in rod spacing, due to rod diameter variations, slight bends in the rods, as shown in Fig. 3
(d), or mounting imperfections, would account for some of this discrepancy. Small variations from normal incidence can also alter the predicted SPP frequency. For example, at only 2° incidence and 0.55 mm average rod spacing, the predicted m
=-1 order SPP frequency is 527 GHz, while the m
=+1 order SPP frequency is 565 GHz. As stated earlier, these two SPPs should have similar strengths but, given a 55 GHz FWHM bandwidth for each SPP, they would not be spectrally resolvable. However, this effect may explain the anomalously high FWHM bandwidth of our 0° incident SPP data in transmission. We measured this bandwidth to be ~104 GHz and the theory predicts an overall bandwidth of 93 GHz for the two blended spectral peaks. Imperfections in our physical gratings and their current mounting structures forbid us from verifying a perfect match to theory. Nevertheless, within our measurement tolerances, our data exhibit good agreement with theoretical predictions. Additional measurements with smaller tolerances should remove this ambiguity. Other experiments, in which the detector and emitter polarizations are rotated instead of the grating, are also planned to verify the geometric loss effects.