1. Introduction

Nanoscale near-field imaging in the terahertz (THz) spectral range is of great importance for studying intriguing phenomena such as biomolecular vibrations and carrier dynamics in quantum-confined nanostructures. Conventional THz time-domain spectroscopy can provide macroscopic imaging averaged over an ensemble of such nanostructures. Their spatial resolutions are, however, limited to ~λ/2 by diffraction. Therefore, several types of THz pulse scanning near-field optical microscopes (SNOMs) have been developed to achieve sub-wavelength resolutions [1–9]. In contrast to visible or IR SNOMs, most THz SNOMs have been based on THz pulse TDS systems [1–8], making it possible to perform ultra-broadband THz spectroscopy. Among the THz SNOM systems, the scattering-type SNOM (s-SNOM) has been the most successful technique so far [6–13].

In the THz s-SNOM, the scattered field from the tip apex is measured in the far-field region. Sub-micrometer resolutions are enabled by the strongly localized near-field around the probe tip [14]. Thus, it is essential to understand the near-field interaction in the tip-substrate system, and there have been several analytic models [15–18] and also numerical simulations [19,20] to solve the problems. The most popular approach has been the point dipole image method (PDIM) [8–12] where the probe tip is replaced by a polarizable point dipole [10]. Because of its simplicity, the PDIM has been widely used to analyze experimental data [8–12], and has provided qualitative understanding on several important aspects of the s-SNOM, including resolution [10] and optical phase contrast [11]. However, because the boundary conditions are not matched on the surface of the probe sphere, the PDIM becomes incorrect as the sphere approaches the substrate [15–18].

In this work, we present a self-consistent line dipole image method (LDIM) based on an exact quasi-electrostatic image theory [21–24] for the analysis of THz s-SNOMs. The accuracy of the LDIM was verified by a quantitative comparison with numerical simulations based on the finite element method (FEM). The experimental approach curves and contrasts of THz s-SNOMs were in excellent agreement with the LDIM calculations.

2. Self-consistent image theory

For a theoretical study of s-SNOMs, the probe tip is usually replaced by a dielectric sphere of radius a and complex relative permittivity ε_p, as shown in Fig. 1(a) , where g is the tip-substrate gap distance. The near-field interactions and scattering in s-SNOMs have been studied by solving this simplified sphere-substrate system. Because the probe sphere is much smaller than the wavelength of the incident field E _in, quasi-static image methods are applicable to the analysis of the near-field interaction. In the quasi-static image methods, the near-field interaction and the subsequent scattering field E _sc can be calculated by the dipoles induced in the probe and substrate. In general, the analytic theory for those image dipoles becomes very involved to match the exact boundary conditions [16,17].

 

Fig. 1 Schematics of a THz s-SNOM system. (a) the probe-substrate system, (b) the point dipole image method (PDIM), and (c) the self-consistent line dipole image method (LDIM).

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In the conventional PDIM, the incident field polarizes the probe sphere, resulting in a point image dipole d _p^PD = α E _in at the center of the sphere, as shown in Fig. 1(b), where ε_s is the complex relative permittivities of the substrate, ε ₀ is the permittivity in free space, and the polarizability of the probe sphere is given by α = 4πε ₀ a ³(ε_p−1)/(ε_p + 2). The probe dipole induces a substrate image dipole, d _s^PD = β _T∙d _p^PD at z = −h. Here, β _T = [(ε_s−1)/(ε_s + 1)](I−2u _x u _x) is the probe-substrate image generation dyadic, where u _x and u _z are the unit vectors along the x and z axes, respectively, and I = u _x u _x + u _z u _z. In principle, the electric field from the substrate dipole induces not only a single point dipole but also a continuous dipole distribution in the probe sphere, as we shall see in the LDIM. In the PDIM, the continuous dipole distributions are usually replaced by point image dipoles. This point-dipole approximation, however, works well only for a large tip-substrate distance. By taking into account the recursive generation of the point image dipoles [10,11], the total image dipole d ^PD in the probe-substrate system is given by

In contrast to the PDIM, the LDIM applies exact electrostatic boundary conditions to all dielectric surfaces of the sphere-substrate system through iteration processes. In the first iteration, the initial dipole moment at the probe sphere is induced by the sum of the incident field and the reflected field from the substrate, resulting in the probe dipole d _{p (1)} ^LD = α(1 + r)E _in, where we include the reflection coefficient r at the substrate surface to match the exact boundary conditions at the substrate surface because there is an image of the radiation source for the incident field. The probe dipole then induces the substrate image dipole d _{s (1)} ^LD = β _T∙d _{p (1)} ^LD at z = − h. In the second iteration, the substrate point dipole d _{s (1)} ^LD induces a line dipole density p _{s (2)} ^LD(z, h) = γ(z, h _, 2h)∙d _{s (1)} ^LD. From the exact image theory [21–24], we can derive the substrate-probe image generation dyadic γ(z, h, L) given by

 

Fig. 2 Near-field distributions (80 nm × 80 nm) of the sphere-substrate system. LDIM calculations on (a) Au substrate and (b) FZ-Si substrate, and FEM simulations on (c) Au substrate and (d) FZ-Si substrate. The diameter of the tungsten sphere is 40 nm and the gap distance is 5 nm.

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The sphere-substrate problem has been studied by other self-consistent methods such as multipole expansion method (MEM) [15,16] using bispherical coordinates where the Laplace equation becomes separable and the field solution can be obtained in a closed form of series. The LDIM also starts from the Laplace equation, and make use of the Mellin transform [21] to directly calculate the line dipoles, providing more explicit picture of the near-field interaction. Moreover, the LDIM can easily be applied to multiple spheres, and has much better convergence than the MEM [24].

3. Measurements, calculations, and discussion

The THz pulse s-SNOM consists of a THz time-domain spectroscopy (TDS) system and an atomic force microscope (AFM). We used a FZ-Si wafer with high resistivity (>10 kΩ·cm) partially coated by a 30 nm gold film, and the p-polarized incident THz pulse was focused upon a tungsten AFM tip glued to a quartz tuning fork driven at a mechanical resonance frequency Ω. The tip-sample gap distance is given by g(t) = g ₀ + g ₁ cos Ωt = h(t) − _. a, where t denotes the time coordinate for the tip oscillation. When scanning the time delay of the TDS system, the scattered field E _sc was measured using a photoconductive antenna in the far-field region. The near field is a highly nonlinear function of g(t), and the scattered field can be expressed as

Figure 3 shows the LDIM and PDIM calculations of the dipole moments and their responses to the tip modulation on Au and FZ-Si substrates for a = 1100 nm and g ₁ = 110 nm, where the total dipole moments are normalized to the maximum of d_z^LD on the Au substrate. We used the Drude model for the frequency-dependent permittivities of gold and tungsten, where the plasma (damping) frequencies of gold and tungsten were 2184 THz (1551 THz) and 6.45 THz (14.61 THz), respectively [26]. For the FZ-Si wafer, we used a frequency-independent refractive index of 3.42 [27]. The LDIM calculations result in larger induced dipole moments and also steeper increase of the dipole moments than the PDIM as the tip approaches the substrate. Consequently, the LDIM has larger harmonic components than the PDIM, as shown in Fig. 3(b). As expected from the standard electrostatic image theory, the vertical component (d_z) is much larger than the tangential component (d_x) for both LDIM and PDIM, and also the Au substrate shows stronger enhancements for the harmonic modulations than the Si substrate.

 

Fig. 3 LDIM and PDIM calculations for Au and FZ-Si substrates. (a) x- and z-components of total induced dipole moments, and (b) tip-modulation responses of x- and z-components of the total induced dipole moments. The tungsten tip has a radius of a = 1100 nm, and the dithering amplitude is g ₁ = 110 nm. The solid and dashed curves stand for LDIM and PDIM calculations, respectively. The red and blue curves represent calculations for gold and Si substrates, respectively.

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The measured and calculated approach curves for E _{1 z} on the Au and Si surfaces are shown in Fig. 4(a) , where the inset shows the scattering THz pulse signals, and the approach curves are normalized to the maximum of E _{1 z} on the Au substrate. During measurements, the dithering amplitude was g ₁ = 45 nm, maintained by a proportional-integral-derivative (PID) controller, and the mechanical delay line was fixed at the positive peaks of the THz scattering pulses. All of the approach curves were measured by placing the probe tip a few millimeters away from the edge of the gold film to avoid the possible effects of fringing fields. We used the tip radius as a fitting parameter to compare the measurements with the calculations. For the best-fit curves, the effective radii of the probe tip were found to be 1100 nm and 300 nm for the LDIM and PDIM, respectively. The measurements were in excellent agreement with the LDIM calculations even with a single fitting parameter for both gold and Si substrates. The PDIM calculations, however, show significant deviations from the measurements, especially for the Si substrates. Another effective radius for the Si substrate can be defined for better fits, but this is obviously not consistent with the PDIM.

 

Fig. 4 Approach curves and relative contrasts of a THz pulse s-SNOM. (a) Approach curves on gold (red) and FZ-Si (blue) substrates and (b) calculated relative contrasts. The inset shows the THz scattering pulses on gold and Si substrates. The closed circles represent experimental results, the sold (dashed) lines stand for the LDIM (PDIM) calculations, respectively.

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The steep increase of the measured THz scattering signals for small gaps is convincing experimental evidence for the strong tip-substrate THz near-field interaction. Moreover, no undulation was observed in the measured approach curves, which means that the background artifact was negligible in our THz pulse SNOM. This is because the THz pulse s-SNOMs directly measure the instantaneous THz electric field, not the power [10,25]. Therefore, the THz pulse s-SNOMs can obtain artifact-free near-field signals without using higher harmonic demodulation, in contrast to visible and infrared s-SNOMs using optical power detectors [4–7,9–13]. As shown in Fig. 4(b), the measured relative contrast, E _{1 z} (Si)/E _{1 z} (Au) decreases as the tip approaches the substrate. The measurements were in good agreement with the LDIM calculation whereas the PDIM shows a rather large offset. These results demonstrate that THz pulse s-SNOMs with accurate near-field analyses can provide quantitative spectroscopic information at THz frequencies for a variety of nanomaterials such as quantum-confined semiconductors and biomolecular nanostructures.

4. Conclusion

Based on the LDIM, we have developed a new analytic theory for near-field interactions in THz pulse s-SNOMs. The measurements of approach curves and relative contrasts on Au and FZ-Si substrates were in excellent agreement with the LDIM calculations. We believe that this is an important step toward solving the inverse scattering problems for quantitative spectroscopic imaging in THz near-field microscopy.