1. Introduction

Surface plasmon-polaritons (SPPs), which are composed of electromagnetic waves coupled to a collective excitation of the electrons at the interface between a conductor and a dielectric, have elicited renewed attention in recent years, since they offer significant promise in a range of applications [1,2]. One specific area of applications, for which there is significant ongoing activity, is in the field of photonics. SPPs may allow for the miniaturization of discrete optical elements and guided wave devices [1]. In order to fully realize a range of device functionalities, it is important to be able to manipulate and fully understand the propagation properties of these surface waves. One key capability is the ability to focus this radiation. A simple curved groove, slit or protrusion is able to accomplish this. In the simplest embodiment, incident freely propagating radiation is coupled to SPPs by the curved element, whose geometry subsequently focuses the propagating SPP waves. This focusing phenomenon has been observed using circular [3–5], parabolic [6,7] and elliptical [8] curved elements. In fact, such structures have been examined previously in the context of enhanced transmission through a subwavelength aperture. In that application, a single aperture fabricated in a metal film is surrounded by periodically spaced concentric grooves [9–12]. This bullseye pattern has been shown to increase the transmission through the aperture by more than two orders of magnitude relative to an identical bare aperture [11].

The question, then, is how do these surface waves evolve as they propagate through the focus. It is well known that when a freely propagating wave propagates through its focus from -∞ to +∞, it experiences an additional axial phase change relative to a plane wave [13]. The effect, first described by Gouy in 1890, was based on observing interference fringes between a plane wave and a spherical wave using a white light source [13]. This phase shift is most commonly discussed in the context of the propagation properties of Gaussian beams, either in free space or within curved mirror resonators [13,14]. In recent years, there have been a number of experimental studies that have tracked the temporal evolution of the electric field component of electromagnetic transients, thereby allowing for a direct, non-interferometric measurement of this phase anomaly for free-space [15] and cylindrical [16] Gaussian beams. However, the Gouy phase shift is not limited to electromagnetic waves and has been observed using phonon polaritons pulses [17] and acoustic pulses [18,19]. Recently, Feng and Winful demonstrated that the Gouy phase shift is a consequence of transverse spatial beam confinement [20]. Their model theoretically confirms both the arctangent-like phase transition that a focused Gaussian wave experiences as it passes through its focus and the numerical value for the cumulative phase shift. This interpretation further demonstrates that any converging wave will experience an additional geometrical phase change as it passes through its focus.

In this submission, we directly measure the Gouy phase shift for focused SPPs on metal films using THz time-domain spectroscopy, by measuring the temporal properties of the outof- plane electric field component of the SPP as it propagates through its focus. It is important to note that since the conductivity of metals is finite at THz frequencies [21], the metaldielectric interface can support SPPs, although in this frequency range they have also been referred to as surface electromagnetic waves, Zenneck waves, and THz surface plasmons. We inscribe a curved groove in a metal foil that allows for coupling of the incident freely propagating terahertz radiation and focuses the SPP propagating along the metal-dielectric interface at the center of the groove [22,23]. Since THz time-domain spectroscopy is a phase-sensitive measurement technique, the axial phase change as the beam moves through its focus is revealed by measuring the waveforms at different positions with respect to the focus. However, since the two-dimensional (2D) SPP beam does not exhibit a Gaussian spatial distribution, conventional analytic expressions for the Gouy phase shift cannot be used. Therefore, we rely on numerical simulations that yield the spatial electric field distribution on the metal foil and use the aforementioned model to compute the (axial) Gouy phase shift. These numerical results confirm our experimental observations.

2. Experimental details

To generate a focused SPP beam, we fabricated a semicircular groove in a freestanding 150µm-thick stainless steel foil by chemical etching. The 40 mm diameter groove had a rectangular cross-section with a width of 300µm and a depth of 100µm. In order to produce the groove, 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.

 

Fig. 1. Schematic diagram of the experimental setup for measuring the Gouy phase shift (a) a (110) ZnTe crystal is used as the detection medium, λ/4 corresponds to a quarter-wave plate, WP corresponds to a Wollaston prism, and differential detection is used for improved sensitivity. (b) the optical probe beam and the SPP direction co-propagate. The position of the ZnTe crystal can be changed with respect to the center of the semicircular groove to measure the relative phase change or spatial properties of the SPP beam.

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The basic experimental setup is shown schematically in Fig. 1, with the overall experimental configuration shown in Fig. 1(a). An amplified Ti:Sapphire laser with an (C) 2007 OSA 6 August 2007/Vol. 15, No. 16/OPTICS EXPRESS 9997 average power of ~1W, repetition rate of 1kHz, and temporal pulse duration of ~40fs was used as the optical source in the experiment. Broadband terahertz radiation was generated via optical rectification in a (110) ZnTe crystal [24]. The resulting free-space terahertz pulses were collimated and normally incident on the metal film. The incident terahertz beam, polarized along the x-axis, had a 1/e beam diameter of approximately 50mm and was positioned at the center of the groove, so that the entire groove was equally excited by the incident radiation. The groove acts as a scattering element that allows for coupling of incident radiation to SPPs and vice versa [22,23,25]. The efficiency of this coupling is dependent upon the spatial properties of element. Since the incident THz beam is polarized along the x-axis, it couples to SPPs that propagate along ±x-axes.

Figure 1(b) shows the semicircular groove and the SPP detection geometry. We used a 10 mm x 10 mm x 1mm thick (110) ZnTe crystal to measure the out-of-plane component of the SPP time-domain waveform via electro-optic sampling in a phase-matched geometry [24]. In this configuration [24,26], the probe beam co-propagates with the SPP, since the phase velocity of the SPP matches the group velocity the optical probe beam. The ZnTe detection crystal was placed in contact with the sample surface with the probe beam positioned approximately 400µm from the surface, which is well within the 1/e field extension of the SPP from the metal surface (~5 cm into the air dielectric) [26,27]. For phase-sensitive measurements, moving the detection crystal along the x-axis does not result in timing changes (i.e. no additional phase change occurs from simply moving the crystal).

3. Experimental results and discussion

We first measured the time-domain waveforms at 17 different positions along the x-axis. Fig. 2(a) shows three specific time-domain waveforms measured with the detection crystal positioned at x=-10 mm, x=0 mm, and x=+10 mm on the x-axis. The center of rotation for the semicircular groove is defined as the origin of the xyz coordinates in our discussion, as shown in Fig. 1(b). The upper curve shown in Fig. 2(a) shows the temporal waveform before the SPP is focused, while the lower curve shows the temporal waveform after the focus. Because of the curved nature of the groove, the SPP possesses a curved wavefront and is focused at the origin (middle curve). The peak-to-peak value for the middle waveform is ~3 times larger than that of the upper or lower waveforms. It is clear that propagation of the SPP in this geometry causes reshaping of the temporal waveforms. We Fourier transformed these three waveforms and used the phase spectrum associated with waveform at the focus (groove origin) as the reference. The relative phase difference between the x=-10 mm waveform and the x=+10 mm waveform is ~1.6 rad below 0.5 THz, as shown in Fig. 2(b). We have also measured the phase shift associated with a linear groove. Over the same propagation distance (20 mm), we observe no phase shift (not shown). Thus, the phase shift shown in Fig. 2(b) is directly related to the properties of the curved groove.

As noted earlier, it has been demonstrated that Gouy phase shift originates from the transverse spatial confinement of a wave [8]. We now discuss why we attribute this phase shift (from x=-10 mm to x=+10 mm) to the Gouy phase shift. The electric field distribution of the SPP beam coupled to the metal surface by the semicircular groove can be modeled by assuming a superposition of point sources along the groove. The resulting electric field distribution does not have a Gaussian spatial distribution, nor can it be expressed by a closed form analytical expression. Therefore, we relied on numerical techniques to compute the electric field distribution on the metal surface. For the purposes of exposition, we show results at 0.3 THz, although we have determined that the basic results are valid at other frequencies within the THz bandwidth of the SPP. We used finite differential time-domain (FDTD) simulations to calculate the field pattern excited by the semicircular groove, assuming a Drude model for the dielectric properties of the metal [21]. An x-polarized monochromatic plane wave was assumed to be normally incident on the sample and, as discussed above, the groove acts to couple the incident radiation to SPPs [22,23,25]. The steady-state field distribution of the z-component of the SPP electric field at 0.3 THz is shown in Fig. 3(a). This field pattern shows a tightly focused beam at the origin of the groove.

 

Fig. 2. (a) Time-domain waveforms of the SPP measured at x=-10mm (upper blue curve) and x=0 mm (middle black curve), and x=+10 mm (lower red curve) for a semicircular groove. (b) The relative phase shift difference between the waveform obtain for x=-10 mm and for x=+10 mm. When a linear groove is used, we observe no phase shift over the same propagation distance.

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To confirm the predictions of the simulations, we measured the z-component of the surface electric field along y-axis at different positions on x-axis. Figs 3(b)–3(d) summarize the measured and calculated z-component of the electric field distribution along y-axis for x=0 mm, x=1 mm, and x=2 mm. The magnitude of the electric field at 0.3 THz for each position is obtained from the Fourier transform of the corresponding time-domain waveform. The FDTD simulations are in good agreement with the measured electric field distribution. We are not able to resolve the oscillations obtained in the simulations, because of the finite optical probe beam diameter.

Based on good agreement between experiment and simulation in Fig. 3, we can now use the calculated electric field distribution from the FDTD simulations to numerically calculate the Gouy phase shift for the SPPs. While this phase shift can be directly taken from the FDTD results, we use the theory developed by Feng and Winful [20], which shows that the Gouy phase shift of a three-dimensional (3D) Gaussian beam can be computed using the expectation values of the transverse momentum. This approach shows clearly the role of spatial confinement in creating this phase shift. Using this general process, the Gouy phase shift for the 2D SPPs can be obtained by converting the 3D approach to 2D. We begin by considering the z-component of the surface electric field distribution, E_z(x,y), defined across the entire metal sheet The distribution of the transverse momentum k_y is given by the spatial Fourier transform:

The expectation value of the square of the transverse momentum is then given by

Note that 〈k_y²〉 is a function of the parameter x, since the integrations in Eqs. (1) and (2) are only on the parameter y.

 

Fig. 3. (a) FDTD simulated steady-state field distribution of the z-component of the SPP electric field at 0.3 THz shown on a log scale, with the corresponding color map shown on the right hand side. The measured (filled circles) and simulated (solid line) cross-section of the electric field distribution is shown for (b) x=0 mm; (c) x=1 mm; and (d) x=2 mm. The measured field magnitude values at 0.3 THz are taken from Fourier transforms of the corresponding time-domain waveforms.

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Based on this expectation value, the Gouy phase shift for this focused beam as a function of the x-coordinate is given by [20]

where λ is the wavelength of the SPP. Using the 2D spatial distribution of the SPP electric field at 0.3 THz obtained from FDTD simulations, we calculate φ_G(x). The resulting phase shift is shown by the solid curve in Fig. 4. We have also computed the relative phase shift of the focused SPP from the measured time-domain waveforms at different distances from the focus. As with the data in Fig. 2(b), we used the phase spectra associated with the timedomain waveform measured at the focus as the reference. The experimentally measured relative phase difference at 0.3 THz from −10 mm to +10 mm is found to be ~π/2, as shown in Fig. 4, in excellent agreement with the simulations. Just as with Gaussian beams [13,14], the primary change in phase occurs within several wavelengths of the beam focus. As noted earlier, we have performed similar comparisons between simulations and experiment at several other frequencies within the bandwidth of the SPP and found equally good agreement. To our knowledge, this is the first demonstration applying the theory of Feng and Winful [20] to beams that do not exhibit Gaussian spatial cross-sections. Finally, it is worth commenting on the numerical value of the total accumulated (Gouy) phase shift. It is well known that for spherical Gaussian beams, this phase change takes on a value of π, while for cylindrical Gaussian beams its value is π/2, as expected from the reduced dimensionality of the beam. Based on the properties of Gaussian beams, we believe that the measured phase shift in Fig. 4 confirms the surface nature (reduced dimensionality) of the propagating waves.

 

Fig. 4. The measured and simulated phase shift at 0.3 THz for a SPP focused using a semicircular groove. The total phase shift is ~π/2. The filled triangles represent the measured phase shift at 0.3 THz retrieved by the Fourier transforms of time-domain waveforms measured at different positions. The solid curve represents the simulated phase transition based on the spatial field distribution calculated from FDTD simulations.

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4. Conclusion

In conclusion, we have directly measured the Gouy phase shift for a SPP that was excited by a semicircular groove. Since there is no closed form analytical expression for the surface electric field, we performed numerical simulations to determine the spatial properties of the focused surface wave. This spatial distribution was used to calculate the axial Gouy phase shift along the propagation direction. This general approach may be used to calculate the associated axial phase shift for other types of focusing elements. While the current measurements were performed at THz frequencies, we believe that essentially identical results would be obtained in other regions of the electromagnetic spectrum. Such information is critically important in understanding the propagation properties of SPP beams and in designing optical devices and resonator cavities on metal surfaces to manipulate SPPs.