We demonstrate the ability to characterize independently the vector components of the electric field associated with terahertz surface plasmons. This is accomplished via electro-optic sampling, using an electro-optic crystal placed in close proximity to a corrugated metal foil. The individual electric field vector components are measured using two separate ZnTe crystals. Since ZnTe exhibits isotropic dielectric properties, all of the detection configurations obey identical phase-matching constraints. Furthermore, since ZnTe is characterized by a single independent electro-optic tensor component, the field measurements may be directly compared against one another.
©2007 Optical Society of America
Although surface plasmon-polaritons have been studied extensively for many decades , there has been a resurgence of interest in this topic. Much of the appeal for this topic is based on the broad range of potential applications, including the miniaturization of photonic devices and circuits , sensing for biomedical applications , and the development of nanolithography . In these and associated applications, detailed knowledge of the properties of the surface electromagnetic field is of fundamental importance. There has been significant work in characterizing and optimizing the excitation and propagation parameters of surface plasmon-polaritons for various frequencies across much of the electromagnetic spectrum. For example, at optical frequencies, numerous studies have examined the excitation of surface plasmon-polaritons using near-field scanning optical microscope (NSOM) probes, gratings, and prisms . The subsequent propagation, interference, and scattering effects are commonly determined using far-field measurements [1, 5]. In contrast, there has been very little work on characterizing the vector nature of the surface electromagnetic field.
Scanning probe techniques offer a convenient means for investigating surface plasmon-polariton properties . Among these techniques, near-field scanning optical microscopy has been the most widely utilized. Conventional NSOM probes, based on metallized tapered fibers with a subwavelength aperture at the apex, are typically sensitive to the in-plane component of the electric field , while apertureless NSOM probes are typically sensitive to the out-of-plane component of the electric field [7,8]. Recently, Lee et al. used an NSOM probe functionalized with a gold nanoparticle at the tip that allows one to measure both the in-plane and out-of-plane electric field components . By analyzing the polarization of the far-field scattered radiation, the authors were able to map the two-dimensional vector electric field distribution at the surface of an aperture array structure designed for optical frequencies. While near-field optical techniques can be extended to longer wavelengths, such as the terahertz (THz) spectral range, scaling in the wavelength and the dimensions of the corresponding structures creates the opportunity to utilize approaches not easily amenable to measurements at optical frequencies.
In this submission, we demonstrate the capability for mapping the complete three-dimensional vector nature of the surface electromagnetic field associated with a propagating SPP field using electro-optic sampling in a terahertz time-domain spectroscopy setup. In this approach, by appropriately choosing the experimental configuration, all three vector components of the surface propagating electric field can be measured independently. These measurements are performed near the center of a metal foil that contains periodically spaced concentric annular grooves and can be straightforwardly extended to map the field properties across the entire sample. Such a structure has previously been shown to allow for the excitation of narrowband terahertz surface plasmons (TSPs), where the center frequency and linewidth are directly related to the groove periodicity and number of grooves, respectively . Using two separate ZnTe detection crystals, we are able to independently measure all three vector components of the TSP electric field. It is worth noting that similar vector field characterization based on electro-optic sampling has been used previously for circuit characterization [11, 12]. However, in sharp contrast to those measurements, phase-matching considerations must be carefully considered in the present measurements in order to optimize the conversion efficiency. ZnTe is a cubic nonlinear optical medium that exhibits isotropic dielectric properties. Thus, the linear dielectric properties are the same regardless of the propagation direction and polarization of the optical and THz beams, ensuring that identical phase-matching properties are observed for all of the measurements. Furthermore, since the crystal is characterized by only one independent electro-optic tensor component, the magnitudes of the three electric field components can be directly compared to one another.
2. Experimental details
We fabricated bullseye patterns to measure the vector nature of the surface propagating THz electric field. The bullseye pattern was fabricated by chemical etching on freestanding 150μm thick stainless steel foils. In order to produce the required structures, photoresist was initially patterned on the stainless steel foil using conventional photolithographic processes. The exposed metal was then etched in a ferric chloride solution at 110°C, resulting in an etch rate of ~0.6 μm/minute. Finally, the photoresist and residual ferric chloride was stripped in a potash solution heated to 130°C. The resulting bullseye pattern consisted of 25 concentric annular grooves, in which each groove had a rectangular cross-section with a width of 500 μm and a depth of 100 μm. The center-to-center groove spacing was 1 mm, with a total spatial extent of 50 mm. The 1 mm annular groove periodicity corresponds to the excitation of a narrowband TSP centered at a frequency of ~0.3 THz .
The experimental setup is shown schematically in Fig. 1. The overall experimental configuration, shown in Fig. 1(a), included an amplified Ti:Sapphire laser with an average power of ~1W, repetition rate of 1kHz, and temporal pulse duration of ~40fs that was used as the optical source in the experiment. This optical radiation was split 80:20 for use as the optical pump and probe beams, respectively. The p-polarized optical pump beam was incident on a 1 mm thick (110) ZnTe emitter at normal incidence. The c-axis of the crystal was rotated at an angle of 54° with respect to the pump beam polarization in order to maximize the nonlinear response . The p-polarized THz radiation produced by optical rectification in the ZnTe crystal was collimated by a parabolic mirror and normally incident on the bullseye sample.
Figure 1(b) shows the bullseye structure and the TSP detection geometry. We used a 10 mm × 10 mm × 1mm thick ZnTe crystal as the detection medium. The diameter of the optical probe beam is approximately 500 μm within this crystal. With respect to the laboratory coordinate system (xyz coordinates), shown in Fig. 1(b), the polarization of the incident THz radiation is parallel to the x-axis. Therefore, TSPs generated at the metal-dielectric interface will propagate along the ±x axes. Specific vector components of the TSP are detected using either a (100) or (110) ZnTe crystal via electro-optic sampling, with the crystal face oriented orthogonal to the x-axis. The detection system utilizes differential detection for improved sensitivity. In order to measure the electric field of the TSP at different heights about the sample surface, the optical probe beam and electro-optic crystal are moved relative to the surface of the bullseye structure.
As we mentioned earlier, phase-matching is an important consideration in the present measurements. It has previously been shown that for free-space electro-optic detection of THz radiation, non-critical phase-matching is possible when the terahertz wave and the optical probe beam co-propagate. This arises from the fact that the optical group velocity of the probe beam approximately matches the phase velocity of the freely propagating THz beam . The effective refractive index for TSPs are known to very nearly match the refractive index of the dielectric adjacent to the metal surface, which in the present configuration is air . Thus, the phase-matching constraint for freely-propagating THz radiation is equally valid for detection of TSPs. Since ZnTe exhibits isotropic linear dielectric properties, the phase-matching considerations are identical for (110) and (100) crystals. By selecting the crystal orientation, we can selectively measure each of the TSP electric field vector components. Finally, since ZnTe exhibits only one independent electro-optic tensor component, r41, measurements of the different vector components may be directly compared to one another.
3. Theory of electro-optic sampling
Before moving to the experimental results, it is important to discuss the use of electro-optic sampling for the characterization of arbitrarily polarized terahertz electric fields. As mentioned above, we define the laboratory coordinate frame using the xyz coordinate system, as shown in Fig. 1(b). The terahertz field along each axis in this coordinate frame is defined as Ex, Ey, and Ez. We also define a crystalline coordinate frame related to the crystallographic axes of the detection crystal. These coordinates are designated x′y′z′, shown in Fig. 2. The corresponding terahertz electric field component along each crystallographic axis is defined as Ex′, Ey′ and Ez′. The relationship between these two coordinate systems is shown in Fig. 2 for the (100) and (110) crystals. Because of the electro-optic effect, the electric field associated with the TSP alters the polarization of the optical probe beam from linear to elliptical. Zincblende crystals, such as ZnTe, possess isotropic dielectric properties and are described by the point group 4̄3m. The corresponding index ellipsoid, written in terms of the crystallographic coordinate system, is given by 
where n is the refractive index of the ZnTe crystal and r41 is the only nonzero component of the electro-optic tensor.
After passing through the λ/4 plate and Wollaston prism, the terahertz field is measured by detecting the change in the optical probe beam power, which is given by 
where s′ = (sx',sy',sz') represents the propagation direction of the probe beam in x′y′z′ coordinate system, and δ is the polarization state of the probe beam in xyz coordinate system, as shown in Fig. 2, ω is the frequency of the optical probe beam, and L is the thickness of the crystal.
The coherent detection of the longitudinal component Ex of the TSP is achieved by using a (100) oriented ZnTe crystal. We emphasize that this is different from the (110) ZnTe that is typically used in detecting freely propagating THz radiation in conventional THz-TDS systems. The (100) zincblende crystal is known to be sensitive only to the electric filed component that is normal to the crystal surface [8, 11, 12]. Figure 2(a) shows the configuration of the probe beam and the (100) ZnTe. In this configuration, the crystalline coordinates coincide with the laboratory coordinates. Therefore, with s′ = (1, 0, 0), (Ex′, Ey′, Ez′) = (Ex, Ey, Ez). Using Eq. (2), the change in the detected optical probe power is given by
Note here that Ex is in the laboratory coordinate frame, corresponding to the longitudinal component of the TSP on the bullseye structure. For maximum change in the detected optical probe beam power, either a p-polarized (δ = 0) or s-polarized (δ = π/2) probe beam can be used.
In order to measure the transverse electric field components of the TSP, we use the same setup but replace the (100) ZnTe detection crystal with a conventional (110) ZnTe crystal. Because of the curved nature of the grooves on the bullseye sample, the polarization direction of the TSP exhibits a small in-plane (with respect to the sample) terahertz component Ey along the y-axis, as well as an out-of-plane field, Ez. It should be noted that it is possible to separate these two orthogonal fields Ey and Ez by appropriately choosing the polarization direction of the probe beam. This can be done even in the presence of a longitudinal electric field. Figures 2(a) and 2(b) show the beam configurations for the (100) ZnTe and (110) ZnTe, respectively. Although the direction of s′ is shown differently in Figs. 2(a) and 2(b), they are identical in xyz coordinates. However, the crystalline coordinate changes in this configuration, with s′= (1, 1, 0) and (Ex′, Ey′, Ez′) = (Excos(π/4)-Eysin(π/4), Exsin(π/4)+Eycos(π/4), Ez). Therefore, using Eq. (2), the observed change in the optical probe beam power is given by
In Eq. (4), Ey and Ez are in the laboratory coordinates, corresponding to the transverse components of the TSP on the sample. By using either a p-polarized (δ = 0) or s-polarized (δ = π/2) probe beam in the laboratory coordinates, the change in the optical probe beam power is determined only by Ey, which is parallel to the [-110] direction of the crystal. On the other hand, by using a 45°-polarized (δ = π/4) probe beam, only Ez is present in Eq. (4). In our measurements, we choose to rotate the crystal by 90° to make either Ey or Ez parallel to the [-110] direction of the crystal while keeping the other experimental conditions unchanged. This leads to a measured signal that is √2 times larger than that obtained if we were to change the polarization of the probe beam.
4. Experimental results and discussion
We began by measuring the temporal and spectral properties of the THz pulse incident on the bullseye structure. This measurement was performed by removing the bullseye structure and placing a second parabolic mirror in the THz beam path. A (110) ZnTe detection crystal was then positioned near the focus of this latter mirror to allow for detection of the reference. The resulting time-domain waveform (top black) is shown in Fig. 3(a) and the corresponding amplitude spectrum (black trace) is shown in Fig. 3(b). The nominal frequency content of the incident beam extends from ~0.1 THz (λ = 3000 μm) to over ~1.5 THz (λ = 200 μm), with a center frequency of ~0.3 THz (λ = 1000 μm). The sharp dips in the amplitude spectrum are due to absorption from ambient water vapor .
We then measured all three vector components of the TSP at the center of the bullseye structure for 11 different heights above the surface. The lower red waveform in Fig. 3(a) shows a typical measured time-domain waveform for the out-of-plane field component of the TSP. It should be noted that all of the time-domain waveforms for the three field components at different heights above the surface were nearly identical and differed only in magnitude. The waveform consists of approximately 25 oscillations, which is equal to the number of grooves. We expect that the magnitude of the measured THz waveform (within the crystal) is smaller than that incident of the detection crystal. However, because of the isotropic linear dielectric properties of ZnTe, this effect is polarization independent. No resonances arising from multiple internal reflections within the crystal are observed.
It is worth noting that the observed waveform is similar to the transmitted THz waveform obtained using bullseye structures with a subwavelength aperture at the center , demonstrating that the surface field is coupled to the radiated field through the aperture. It is apparent from Fig. 1(a) that the entire bullseye structure is illuminated by the incident THz beam. Therefore, identical narrowband TSPs will be excited from the two halves of the structure and will propagate towards the center of the pattern . However, we only observe the TSP that co-propagates with the optical probe beam, since that interaction is phase-matched. The case of the counter-propagating optical probe beam and TSP is not phase-matched and the oscillatory nature of the corresponding time-domain waveform ensures that it cannot contribute to the observed signal. We have separately verified that this is true experimentally.
Figure 3(b) shows the amplitude spectrum (red trace) associated with the Ez component of the TSP electric field and the spectrum of the incident THz pulse. The amplitude spectrum shows a sharp resonance at 0.3 THz, corresponding to the wavelength of 1mm. This is as expected given the 1 mm groove periodicity. Since the incident THz pulses are broadband, we also observe resonances at harmonics of this frequency (i.e., 0.6 THz, 0.9THz, and 1.2THz), although with increasing frequency, these resonances are diminished in amplitude. This may arise, in part, from reduced coupling to TSPs on the bullseye structure. Higher order harmonic resonances are below the noise level. It should be pointed out that based on simulations using bullseye patterns, the magnitude of Ez component is expected to be zero at the very center of the bullseye pattern . However, the finite thickness of our detection crystal yields a spatial average over the thickness of the crystal, yielding non-zero values for the out-of-plane field component.
We measured the vector components of the THz electric field at various points above the center of the bullseye pattern by moving the electro-optic crystal away from the metal surface. Specifically, for each distance, the ZnTe crystal was moved along the z-axis, from 0mm (nominally ~500 μm) to 10mm in steps of 1mm, with appropriate translation of the optical probe beam. For each z position, we measured the time-domain waveform for each of the three vector components. These temporal waveforms were Fourier transformed and the magnitude of the resonance at 0.3 THz was determined. We note that simply using the magnitude is appropriate, since the line shape of the low frequency spectral resonance at 0.3 THz is nearly identical in each measurement. Figure 4 summarizes the relative magnitudes of the in-plane and out-of-plane electric field components. Each vector field component was calibrated according to the theory in Section 3. It should be noted that the
The surface propagating electric field can be expressed as 
where E⃗o includes the spatial, temporal, and vector properties of the TSP electric field, kx and kz are the propagation constants of the surface wave along the x and z directions, respectively; and ω is the frequency of the TSP. For a TSP, kx is complex, corresponding to a damped propagating wave along the surface, while kz is imaginary, corresponding to an evanescent field bound to the metal surface. Here, we focus on the evanescent nature of TSPs along the out-of-plane axis. We fit each vector field component in Fig. 4 to an exponential decay. The 1/e decay length along the z-axis is approximately 4.3 mm and is generally consistent with recent measurements using broadband THz radiation on planar metal films . This value is significantly less than one would expect using the dielectric properties of metals at THz frequencies [1, 21]. We do not know the dielectric properties of stainless steel at THz frequencies. However, the imaginary component of the dielectric constant for typical metals in this frequency range is in the range of 105 – 106 . Therefore, the surface wave would be expected to extend ~3–11 cm into the dielectric (air) half-space. We note that from Fig. 4, at any given value of z, Ex ≈ Ey and Ez ≈ 10∙Ex. It should be noted that at optical frequencies, this latter relationship showed a magnitude difference of ~3 rather than 10 . This discrepancy may arise, in part, from the fact that, in both cases, a dielectric medium is brought in close proximity to a metal surface, which can perturb the surface electromagnetic field distribution. In the aforementioned optical measurements, this resulted from a NSOM probe brought in close proximity to a metal surface [19 ].
In conclusion, we have demonstrated the ability to measure the complete three-dimensional vector nature of TSPs using electro-optic sampling. This is accomplished using only two ZnTe crystals and can be extended to map completely the field orientation at all points on a sample. Because of linear isotropic dielectric properties and the fact that ZnTe exhibits only one independent electro-optic tensor component, the magnitudes of the three electric field components can be directly compared to one another. This general approach may be used to measure surface field properties on a broad range of materials and metamaterials.
We thank Amit Agrawal for helpful comments regarding the manuscript. We acknowledge support from the National Science Foundation through grant #DMR-0415228.
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