1. Introduction

Spatially oscillating electromagnetic (EM) fields are completely defined only when both amplitude and phase are simultaneously known. The knowledge on spatial distribution of optical field is especially important in optical phenomena of subwavelength structures where the behavior of the optical polarization and the phase is not simply estimated. Complex diffraction phenomena caused by interaction between nanostructures are being exploited to efficiently manipulate EM fields, for instances, focussing or guiding of light using nanoslits [1], grooves [2], and holes [3, 4]. For the design of optical antenna, spatial mapping of amplitude and phase performed in near-field regime is also crucial [5].

Measurement of optical amplitudes and phases on a few tens of nanometer resolution can be achieved by means of near-field scanning optical microscope (NSOM) combined with an interferometer [6–8]. Among different types of phase-sensitive NSOM, heterodyne detection scheme is frequently used due to relatively low background noise. In this scheme the scattered light from the near-field probe is measured through lock-in detection at the beating frequency given by the probe and reference beam modulations [9, 10]. However, heterodyne technique is not useful for measuring optical fields far away from surfaces, especially, where near-to-far field conversion occurs, because the optical signal modulated by the probe is mainly sensitive to the surface-probe optical interaction.

In this letter, we demonstrate phase-sensitive imaging of diffracted light by a single nanoslit using homodyne detection technique to perform height dependent measurement of optical amplitude and phase. We choose a single nanoslit as a test bed for our homodyne phase-sensitive NSOM measurement, because it is one of fundamental structures launching surface plasmon polariton (SPP) [11–17]. Cylindrical wavefronts emanating from the slit and propagating into free space up to a few wavelengths away from the slit are successfully reconstructed. Our phase-sensitive measurement performed on single slits provides information about propagation directions of surface waves and phase difference evolution between the horizontal and the vertical polarization components for increasing distance from the slit.

2. Experiments and Discussion

Figure 1 is a schematic of our homodyne phase-sensitive NSOM. Incident pulse laser (Mira900, Coherent) with spectral bandwidth less than 5 nm at 780 nm center wavelength is split into an excitation and a reference beam by the non-polarizing beam splitter (BS). The excitation beam illuminates a slit sample at the normal incidence from the bottom with the vertical polarization to the slit axis. A single slit with 300 nm width is perforated in a 150 nm thick gold film. The diffracted light from the slit is scattered at the apex of a commercial metal coated NSOM probe (Nanonics) with aperture diameter of 100 nm. A sufficiently blunt metal coated probe is used to suppress the image dipole effects [18] which strongly modify the scattering signal on metal surfaces [19]. The scattered light from the probe apex is guided toward the second non-polarizing BS to be mixed with the reference beam, which is retro reflected by the moving mirror (Fig. 1).

The moving mirror driven by a saw-tooth waveform with modulation frequency of Ω (500 Hz) provides an additional optical phase to the reference beam. We note that the saw-tooth waveform gives an ideal sinusoidal interference signal between the scattered light and the reference beam, while sinusoidal or triangular waveforms result in distorted interference signals. This is because sinusoidal or triangle waveform needs both of its amplitude and phase to be controlled simultaneously to give a sinusoidal interference signal, which is ideal for lock-in detection operated in a single frequency. However, controlling the phase of the waveform is difficult because the phase of the target signal ϕ_sig [Eq. (1)] is not constant but dependent on the probe position. On the other hands, saw-tooth waveform only needs its amplitude controlled to calibrate the modulation amplitude of the moving mirror to be equal to the half wavelength of the incident light, which gives sinusoidal interference signal.

Co-linearly recombined reference beam with the scattered light generates an interference signal which is described by

where E _sig(ref) and ϕ _sig(ref) are electric field amplitudes and phases of signal (reference) beam, respectively. Time varying optical phase of the reference beam makes the simultaneous detection of E_sig and ϕ_sig feasible through the lock-in detection at the modulation frequency of ϕ_ref (Ω). The interference signal is split again by the polarizing BS for polarization resolved detections.

 

Fig. 1. The experimental setup: scattering type NSOM is combined with an optical interferometer, which is comprised of 1,5:non-polarizing BS, 2: linear polarizer, 3: half wave plate, 4: NSOM probe, 6: polarizing BS, 7: moving mirror modulated by saw-tooth waveform. Arrowed lines among the optical components represents beam path of signal excitation beam (solid) and reference beam (dashed), respectively.

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Fig. 2. Visualized optical wavefronts of light diffracted by the single slit. (a, d) Spatial mappings of the measured amplitudes multiplied by sine functions of phases. (b, e) Spatial phase mappings. (c, f) Linear profiles along the arrowed white lines in (b) and (e), respectively for the horizontal (left panel) and the vertical (right panel) polarization component. Dashed lines are guiding for eyes.

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Height dependent phase-sensitive measurements of the diffracted light from the slit are performed through sample scanning geometry at a constant probe height ranging from 0.1 µm up to 2.8 µm. We visualize diffracted optical wavefronts in Fig. 2(a) and 2(d) by mapping the measured optical amplitudes multiplied by sine function of measured ϕ_sig.

While the wavefronts of the horizontal polarization component propagate into far-field with quasi cylindrical symmetry, those of the vertical polarization components propagate mostly in both side directions along the surface, as expected from theory [14, 20]. Figure 2(b) and 2(e) where measured phases are mapped as a function of position more clearly explain different wavefront propagation characteristics of two polarization components: while only radial equiphase lines are shown in the horizonal component (Fig. 2), a π-phase difference between the left and the right side of the slit and a phase-disjunction above the slit aperture can be seen in the vertical component (Fig. 2).

Cross sections in phase maps along the arrowed lines show two linear curves with opposite slopes as shown in Fig. 2(c) and 2(f). The absolute value of the slopes are obtained as (2.53± 0.03)π [µm ⁻¹], which shows good agreement with the wavenumber of the incident light (k ₀ = 2.56π[µm ⁻¹]). The presented results indicate that our phase-sensitive NSOM can successfully measure propagating wavefronts in the near-field as well as far away from the surface.

Next, we investigate evolution of the phase relation between two polarization components in a close proximity of the metal surface. The phase relation is especially important for discriminating the non-confined cylindrical wave from SPP [20]. As presented in Fig. 3, hardly distinguishable mixture of SPP and the surface diffracted wave at distance of 1 µm from the slit [Fig. 3(a) and 3(b)] are clearly separated at 30 µm away from the slit [Fig. 3(e) and 3(f)]. By comparing field profiles of measured optical wavefronts along the horizontal line at a fixed height (0.1 µm), we can read phase differences of π and π/2 between the horizontal and the vertical polarization components at 1 µm [Fig. 3 (c)] and at 30 µm [Fig. 3(g)] from the slit, respectively.

The varying phase relation between two components for the increasing distances from the slit can be explained by a surface wave model composed of SPP and a cylindrical wave. In magnetic field representation two waves on the right side of the slit are written by

where $k_0=\sqrt{k_x^2+k_z^2}$ and k_spp are the wavenumbers of the cylindrical wave and SPP, $\kappa =\sqrt{k_{\mathrm{spp}}^2-k_0^2}$ the decay rate of SPP along the vertical axis. H ₀ is the magnetic field strength of the cylindrical wave with 1/√x slit distance dependency [21–23], and H ₁ is that of SPP with $e^{-\mathrm{Im}\left(k_{\mathrm{spp}}\right)}$ decay behavior [18]. Note that the time-dependent term e^iωt is suppressed throughout. The electric fields of SPP and the cylindrical wave are derived from Eq. (2) through Maxwell’s equations. Then, the horizontal and the vertical polarization component E_x and E_z of the electric field are obtained as

When the cylindrical wave is predominant over SPP near the slit, the surface wave can be approximated by the first terms of the right hand side of Eq. (3) and (4): there exists a π phase difference between two polarization components. As the distance from the slit increases to about 30 µm, SPP begins to predominate the cylindrical wave due to the fact that SPP has a negligible decay constant along the metal surface whereas the non-confined cylindrical wave decays with dependence on the reciprocal distance [20]. Therefore, the surface wave in this regime is decided by the second term in each equation, and a π/2 phase difference between two components is expected as our experimental result presents. When the phase difference would be measured on the left side of the slit, zero phase difference is observed in the proximity of the slit, because in this case k_x in Eq. (4) should have a negative value due to the reversed propagation direction of SPP.

 

Fig. 3. Optical surface wavefronts measured at 1 µm (left panel) and 30 µm (right panel) away from the right side of the slit. (a, e) The horizontal (x-pol) polarization components. (b, f) The vertical (z-pol) polarization components. (c, g) Cross sections of the wavefronts along the scan line at a constant height (0.1 µm) from the surface. (d, h) FDTD simulation results.

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We note that the observation of the phase difference near the slit is possible only when the height of NSOM probe position is large enough to detect the vertical wavevector component k_z [Eq. (3)]. FDTD simulation results at 0.3 µm height [Fig. 3(d) and 3(h)] show good agreement with our experimental results [Fig. 3(c) and 3(g)] at 0.1 µm, while only π/2 phase difference is observed with less than 0.1 µm height. The discrepancy of scanning height between FDTD and experimental results can be attributed to the scattering signal from the probe shaft which mainly detects the cylindrical wave in the free space.

 

Fig. 4. Phase difference between horizontally and vertically polarized surface waves as a function of distance from the slit. NSOM measurements (open circles) and FDTD calculations (red dashed lines) are compared.

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As the next step, we proceed to measure the phase difference between the horizontally and the vertically polarized surface waves as a function of distance from the slit. We increase the detection position from the slit with a step of 5 µm and raster scanned the sample within 2 µm range to measure the phase differences of surface waves. The π-phase difference in the proximity of the slit changes to π/2, showing good consistency between the experimental measurement and FDTD calculation, as shown in Fig. 4. However, there exists an abrupt decrease of the phase difference below π/2 in the NSOM result which can not be explained by the theoretical model discussed above. Although origin of the discrepancy has not been cleared yet, it could be attributed to a defect on the surface which was not considered in the FDTD simulation. Defect on the surface can contribute to conversion between SPP and the diffracted wave, as a consequence, a clear discrimination of the phase difference between two polarization components is difficult. Another possible reason is the interference between SPP and the diffracted light at the probe shaft which hinders separating one from the other in the intermediate region. To clarify this issue, further investigation on a defect controlled nanoslit structure with high flatness is needed.

3. Conclusion

In conclusion, we demonstrate phase-sensitive near-field imaging of diffracted light from a single nanoslit. Homodyne detection technique is used for height dependent phase-sensitive measurement to visualize the cylindrical wave and SPP emanating from the slit. Analysis of the measured phase map determines the propagation direction of the surface wavefronts. Furthermore, varying phase difference between the horizontal and the vertical polarization component of the electric field is observed for increasing distance from the slit. We believe that our phase-sensitive measurement technique will be beneficial for investigating complex diffraction phenomena caused by nano structures and interactions among them.