## Abstract

This study constructs interference-based model of the apertureless scanning near-field optical microscopy (A-SNOM) heterodyne detection signal which takes account of both the tip enhancement phenomena and the tip reflective background electric field. The analytical model not only provides a meaningful explanation of the image artifacts and errors, but also suggests methods for reducing these effects. It is shown that the detection signal obtained in the heterodyne A-SNOM method has a significantly higher signal-to-background (S/B) ratio than in the homodyne method. It is also shown that the S/B ratio increases as the wavelength of the illuminating light source is increased or the incident angle is reduced. Finally, an inspection reveals two fundamental phenomena which may potentially be exploited to obtain further significant improvements, namely (1) the modulation depth parameter has certain specific values greater than 1; and (2) the AFM tip apparatus using a ramp function.

©2008 Optical Society of America

## 1. Introduction

This paper develops a detailed analytical model of the detected A-SNOM signal and investigates the variation in the signal contrast and intensity as a function of the phase modulation depth, the wavelength and angle of the incident light, and the amplitude of the AFM tip vibration. The analytical results are intended to clarify the factors determining the detection signal contrast such that the imaging capabilities of A-SNOM can be further improved. As comparison with the experimental results in [9], the authors adopted higher order harmonic radian frequency in order to improve signal contrast, and it consists with ones of our major findings. Aperture scanning near-field optical microscopes (SNOM) are one of the most commonly-employed instruments for obtaining optical resolutions below the diffractive limit [1-2]. In such systems, the tapered metal-coated optical fiber aperture confines the ranges of the illumination and detection electric fields to the near-field regime, and thus effectively eliminates background noise. However, SNOM has a number of fundamental drawbacks. For example, the maximum attainable resolution in the visible light range is limited to just 20 nm due to the finite skin depth of the metal coating used to define the aperture. Furthermore, the very small apertures typical of SNOM devices severely restrict the light throughput, and the corresponding loss in intensity of the detection signal cannot be compensated simply by increasing the incident power level due to the risk of damaging the probe via localized heating effects [3].

Compared to the conventional SNOM method, apertureless SNOM (A-SNOM), in which a sharp vibrating tip supporting a sphere with a nanometer-scale radius achieves a local enhancement of the electric field and makes possible an optical resolution at the sub-10 nm scale [4–6]. However, the A-SNOM detection signal is contaminated by a complex interference between the background electric field and the near-field electric field. As a result, developing effective techniques for suppressing the contribution of background-scattering effects within the detection signal is essential in improving the precision and reliability of the A-SNOM results. In A-SNOM, the detection process is performed using either a heterodyne technique [7–8] or a homodyne method [8–9]. Both techniques have attracted considerable attention in the literature, and various analytical formulae have been proposed to model the respective characteristics of the two detection signals. The present study makes an improvement of refs. [10–12, 17] by analyzing both tip enhancement field and modulated background field components comprehensively in the total A-SNOM signal. These background components play a crucial role in back-scattering configuration and hard to eliminate since the probe is directly illuminated by the incident wave. Furthermore, whilst other studies [13-14] investigated the problem of image artifacts and errors in A-SNOM, these studies not only ignored the near-field enhancement signal, but also utilized models which were overly complex. In a recent study, the current group conducted an analytical investigation into the modulation of A-SNOM homodyne signals and proposed new methods to improved signal contrast [15]. Compared to homodyne detection techniques, heterodyne detection enables a magnification of the near field signal [7, 16], and is therefore an ideal solution for measuring the near-field complex dielectric constant. However, the literature contains very little information regarding the respective effects of the principal A-SNOM parameters (i.e. the wavelength and incident angle of the illumination light and the amplitude of the AFM tip vibration) on the contrast and intensity of the heterodyne detection signal [7–9].

Accordingly, the current study develops a comprehensive interference-based model with which to analyze the amplitude and phase of the heterodyne detection signal at different harmonics of the tip vibration frequency. In constructing this model, the present study expands the model presented in [15] to take account not only of the electric field scattered directly from the AFM tip, but also the tip scattering field reflected from the sample. The latter field is highly important in A-SNOM detection since it is has a very high intensity and a double value of the phase modulation depth. The current analysis deliberately considers the reflective-type A-SNOM technique [7–9] since this technique is invariably the method of choice when measuring the surface properties of materials at the nanoscale. Compared to the models presented in [7–8, 13–14], the model developed in this study provides clearer and more concise explanations for the various phenomena observed in the A-SNOM detection technique. Moreover the interference-based formulation developed for the S/B ratio in the current study has no points of discontinuity, and therefore represents a more suitable means of analyzing the detection signal than the signal contrast formula presented by the current group in [15]. The dependency of the S/B ratio on the wavelength and incident angle of the illuminating light source is systematically examined and discussed. Finally, various means of improving the S/B ratio are introduced and discussed.

## 2. Electric fields in heterodyne A-SNOM

Figure 1 presents a schematic illustration of a Mach-Zehnder interferometer-type A-SNOM. As shown, a radian frequency shift Δ*ω* is added to the reference beam by a generic frequency shifting device such as an acousto-optic modulator (AOM). Therefore, the reference beam can be expressed analytically as

where *E _{R}* is the amplitude of the reference beam and

*ω*and

*Φ*are the radian frequency and initial phase of the incident light, respectively. As shown in Fig. 1, the measurement beam,

_{R}*Ē*, is focused on a vibrating AFM tip by an objective lens. Figure 2 presents an enlarged view of the near-field region. It can be seen that the incident electric field,

_{i}*Ē*, strikes the sample with an angle

_{i}*θ*and produces four discrete electromagnetic waves, namely (1) an interaction signal,

*Ē*-

_{T}*, between the AFM tip and the sample; (2) an electric field,*

_{S}*Ē*, scattered directly from the AFM tip; (3) an electric field,

_{Tip}*Ē*

_{Tip}_{_Reflective}, scattered from the AFM tip and then reflected from the sample surface; and (4) an electric field,

*Ē*, scattered directly from the sample. As shown, an assumption is made in the current analysis that all the incident light and detected light passes through the objective lens.

_{Sample}The first field of interest in A-SNOM detection is that produced by the interaction between the AFM tip and the sample. According to the general model of quasi-electrostatic theory [7–9], the interaction (or tip enhancement) electric field can be formulated as

where *α _{eff}* is the effective polarizability, and it can be expressed as

*α*=

_{eff}*α*(1+

*β*)/(1-

*αβ*/16

*πr*

^{3}) with

*α*=4

*πa*

^{3}(

*ε*-1)/(

_{p}*ε*+2) and

_{p}*β*=(

*ε*-1) (

*ε*+1).

*a*is radius of probe dipole, and

*r*is distance between dipole and sample surface.

*ε*and

_{p}*ε*are complex dielectric numbers of probe and sample, respectively.

*E*is the amplitude of the incident electric field, and

_{i}*ω*and

*Φ*are the radian frequency and initial phase of the interaction light, respectively. Note that

_{TS}*α*is a highly important parameter since it contains everything necessary to predict the relative constants observable in the A-SNOM technique. In practice, its value is determined by the tip radius, the dielectric constants of the AFM tip and the sample, respectively, and the tip-to-sample distance [7–9].

_{eff}In the present analysis, it is assumed that the AFM tip does not perturb the near-field region as the model in [17]. Consequently, the electric field scattered from the probe can be formulated as

where *E _{P}* and

*Φ*are the amplitude and initial phase of the scattering electric field,

_{P}*ω*is the radian frequency of the incident light, and

*K*is the wave number of the incident light and is given by 2

*π*/

*λ*. In addition,

*e*(

^{i}^{2Ksin}

^{(θ)Z(t))}represents the phase vibration caused by the probe’s vertical dither (Note that A-SNOM is generally performed using an AFM operating in tapping mode). Also, it should be noticed that probe scattering electric field has two components. One is directly scattering from the incident light, and another scattering from the light reflected from the sample. However, they have the same features in 2

*K*sin(

*θ*)

*Z*(

*t*) and light frequency

*ω*. Therefore, the two electric fields scattering from the probe can be merged into one probe scattering electric field

*Ē*for the sake of simplicity.

_{Probe}In the A-SNOM scanning procedure, the AFM drives the probe with a vertical cosine vibration around a mean position *Z*
_{0} (see Fig. 2). Assuming that the amplitude and radian frequency of the vibration of the probe are denoted by *A* and *ω*
_{0}, respectively, the dynamic variation of the tip position over time can be written as

The third electric field of interest in A-SNOM detection is the AFM probe scattering field reflected from the sample. From Fig. 2, it can be seen that the optical path difference between the AFM direct scattering electric field and the probe scattering field reflected from the sample surface is equivalent to 2*K*sin(*θ*)*Z*(*t*). Therefore, the reflected probe scattering field can be formulated as

$$={E}_{P\_R}{e}^{i\left(\omega t+{\varphi}_{P}\right)}{e}^{i\left(4K\mathrm{sin}\left(\theta \right)Z\left(t\right)\right)}.$$

The final electric field in the near-field region is that of the light scattered directly from the sample surface. Since this electric field is not modulated by the AFM tip motion, it can be expressed simply as

where *E _{S}* and

*Φ*are the amplitude and initial phase of the scattering light, respectively.

_{S}## 3. Analysis of heterodyne A-SNOM detection signal

The total electric field entering the A-SNOM detector is equivalent to the sum of the reference beam and the four electric fields in the near-field region, respectively, i.e.

Therefore, the corresponding intensity signal, *I*(*t*), is given by

in which the homodyne intensity component may be further derived as

$$+\mathit{2}{E}_{T-S}{E}_{S}\mathrm{cos}\left({\varphi}_{\mathrm{TS}}-{\varphi}_{S}\right)\text{}$$

$$+\mathit{2}{E}_{P}{E}_{S}\mathrm{cos}[{\varphi}_{T}-{\varphi}_{S}+\mathit{2}K\mathrm{sin}\left(\theta \right){Z}_{\mathit{0}}+\mathit{2}K\mathrm{sin}\left(\theta \right)A\mathrm{cos}\left({\omega}_{\mathit{0}}t\right)]$$

$$+\mathit{2}{E}_{T-S}{E}_{P}\mathrm{cos}[{\varphi}_{T}-{\varphi}_{\mathrm{TS}}+\mathit{2}K\mathrm{sin}\left(\theta \right){Z}_{\mathit{0}}+\mathit{2}K\mathrm{sin}\left(\theta \right)A\mathrm{cos}\left({\omega}_{\mathit{0}}t\right)]$$

$$+\mathit{2}{E}_{P\_R}{E}_{S}\mathrm{cos}[{\varphi}_{T}-{\varphi}_{S}+\mathit{4}K\mathrm{sin}\left(\theta \right){Z}_{\mathit{0}}+\mathit{4}K\mathrm{sin}\left(\theta \right)A\mathrm{cos}\left({\omega}_{\mathit{0}}t\right)]$$

$$+\mathit{2}{E}_{T-S}{E}_{P\_R}\mathrm{cos}[{\varphi}_{T}-{\varphi}_{\mathrm{TS}}+\mathit{4}K\mathrm{sin}\left(\theta \right){Z}_{\mathit{0}}+\mathit{4}K\mathrm{sin}\left(\theta \right)A\mathrm{cos}\left({\omega}_{\mathit{0}}t\right)]$$

$$+\mathit{2}{E}_{P}{E}_{P\_R}\mathrm{cos}[\mathit{2}K\mathrm{sin}\left(\theta \right){Z}_{\mathit{0}}+\mathit{2}K\mathrm{sin}\left(\theta \right)A\mathrm{cos}\left({\omega}_{\mathit{0}}t\right)]$$

It is noted here that the last three terms with *E _{P}*

_{_}

*in Eq.(9) are modified from their respective forms in the homodyne model presented in [15] to take account of the inclusion of the electric field*

_{R}*Ē*in the current analysis. Meanwhile, the heterodyne intensity component has the form

_{Probe _ Reflective}$$+\mathit{2}{E}_{T-S}{E}_{R}\mathrm{cos}\left({\Delta \omega t+\varphi}_{R}-{\varphi}_{\mathrm{TS}}\right)\text{}$$

$$+\mathit{2}{E}_{S}{E}_{R}\mathrm{cos}\left({\Delta \omega t+\varphi}_{R}-{\varphi}_{S}\right)\text{}$$

$$+\mathit{2}{E}_{R}{E}_{P}\mathrm{cos}(\Delta \omega t{+\varphi}_{R}-{\varphi}_{T}-\mathit{2}K\mathrm{sin}\left(\theta \right){Z}_{\mathit{0}}-\mathit{2}K\mathrm{sin}\left(\theta \right)A\mathrm{cos}\left({\omega}_{\mathit{0}}t\right))$$

$$+\mathit{2}{E}_{R}{E}_{P\_R}\mathrm{cos}(\Delta \omega t+{\varphi}_{R}-{\varphi}_{T}-\mathit{4}K\mathrm{sin}\left(\theta \right){Z}_{\mathit{0}}-\mathit{4}K\mathrm{sin}\left(\theta \right)A\mathrm{cos}\left({\omega}_{\mathit{0}}t\right))$$

In the current analysis, the component of interest in the detected intensity signal *I*(*t*) is the heterodyne intensity, *I _{het}*(

*t*). Applying the Fourier Bessel series expansion and introducing the phase differences

*ψ*

_{1}=

*Φ*-

_{R}*Φ*-2

_{P}*K*sin(

*θ*)

*Z*

_{0}and

*ψ*

_{2}=

*Φ*-

_{R}*Φ*-4

_{P}*K*sin(

*)*

*Z*_{0}, and the phase modulation depth*ψ*_{3}=2*K*sin(*θ*)*A*, Eq. (10) can be rewritten as*$${I}_{\mathrm{het}}\left(t\right)={E}_{R}^{\mathit{2}}$$*

$$+\mathit{2}{E}_{T-S}{E}_{R}\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{\mathrm{TS}}+\mathrm{\Delta \omega}t\right)\text{}$$

$$+\mathit{2}{E}_{S}{E}_{R}\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{S}+\Delta \omega t\right)\text{}$$

$$+\mathit{2}{E}_{R}{E}_{P}\left\{\right[{J}_{\mathit{0}}\left({\psi}_{\mathit{3}}\right)+\mathit{2}\sum _{j=\mathit{1}}^{\infty}{\left(-\mathit{1}\right)}^{j}{J}_{\mathit{2}j}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}\left(\mathit{2}j{\omega}_{\mathit{0}}t\right)]\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{1}})$$

$$+\mathit{2}\sum _{j=\mathit{0}}^{\infty}{\left(-\mathit{1}\right)}^{j}{J}_{\mathit{2}j+\mathit{1}}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}\left[\right(\mathit{2}j{+\mathit{1})\omega}_{\mathit{0}}t]\mathrm{sin}(\Delta \omega t+{\psi}_{\mathit{1}})\}$$

$$+\mathit{2}{E}_{R}{E}_{P\_R}\left\{\right[{J}_{\mathit{0}}\left({\mathit{2}\psi}_{\mathit{3}}\right)+\mathit{2}\sum _{j=\mathit{1}}^{\infty}{\left(-\mathit{1}\right)}^{j}{J}_{\mathit{2}j}\left({\mathit{2}\psi}_{\mathit{3}}\right)\mathrm{cos}\left(\mathit{2}j{\omega}_{\mathit{0}}t\right)]\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{2}})$$

$$+\mathit{2}\sum _{j=\mathit{0}}^{\infty}{\left(-\mathit{1}\right)}^{j}{J}_{\mathit{2}j+\mathit{1}}\left(\mathit{2}{\psi}_{\mathit{3}}\right)\mathrm{cos}\left[\right(\mathit{2}j{+\mathit{1})\omega}_{\mathit{0}}t]\mathrm{sin}(\Delta \omega t+{\psi}_{\mathit{2}})\}$$

$$+\mathit{2}{E}_{T-S}{E}_{R}\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{\mathrm{TS}}+\mathrm{\Delta \omega}t\right)\text{}$$

$$+\mathit{2}{E}_{S}{E}_{R}\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{S}+\Delta \omega t\right)\text{}$$

$$+\mathit{2}{E}_{R}{E}_{P}\left\{\right[{J}_{\mathit{0}}\left({\psi}_{\mathit{3}}\right)+\mathit{2}\sum _{j=\mathit{1}}^{\infty}{\left(-\mathit{1}\right)}^{j}{J}_{\mathit{2}j}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}\left(\mathit{2}j{\omega}_{\mathit{0}}t\right)]\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{1}})$$

$$+\mathit{2}\sum _{j=\mathit{0}}^{\infty}{\left(-\mathit{1}\right)}^{j}{J}_{\mathit{2}j+\mathit{1}}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}\left[\right(\mathit{2}j{+\mathit{1})\omega}_{\mathit{0}}t]\mathrm{sin}(\Delta \omega t+{\psi}_{\mathit{1}})\}$$

$$+\mathit{2}{E}_{R}{E}_{P\_R}\left\{\right[{J}_{\mathit{0}}\left({\mathit{2}\psi}_{\mathit{3}}\right)+\mathit{2}\sum _{j=\mathit{1}}^{\infty}{\left(-\mathit{1}\right)}^{j}{J}_{\mathit{2}j}\left({\mathit{2}\psi}_{\mathit{3}}\right)\mathrm{cos}\left(\mathit{2}j{\omega}_{\mathit{0}}t\right)]\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{2}})$$

$$+\mathit{2}\sum _{j=\mathit{0}}^{\infty}{\left(-\mathit{1}\right)}^{j}{J}_{\mathit{2}j+\mathit{1}}\left(\mathit{2}{\psi}_{\mathit{3}}\right)\mathrm{cos}\left[\right(\mathit{2}j{+\mathit{1})\omega}_{\mathit{0}}t]\mathrm{sin}(\Delta \omega t+{\psi}_{\mathit{2}})\}$$

*where J_{n}(ψ
_{3}) and J_{n}(2ψ
_{3}) are n-th order Bessel functions of the first kind at the phase modulation depths ψ
_{3} and 2ψ
3, respectively. In Eq.(11), it can be seen that the higher order background electric field signals in E_{R}E_{P} and E_{R}E_{P}
_{_}
_{R} decay more rapidly as the modulation depth, ψ
_{3}, approaches zero [15].*

*Since the amplitude of the interaction electric field between the tip and the sample is nonlinear, an assumption is made that E_{T}
_{-}
_{S} can be expressed as the sum of the individual components oscillating at different harmonics of the AFM probe modulation radian frequency [18], i.e.*

*$${E}_{T-S}={E}_{T-S}^{0{\omega}_{0}}+{E}_{T-S}^{1{\omega}_{0}}\mathrm{cos}\left({\omega}_{0}t\right){+E}_{T-S}^{2{\omega}_{0}}\mathrm{cos}\left(2{\omega}_{0}t\right)+{E}_{T-S}^{3{\omega}_{0}}+\mathrm{cos}\left(3{\omega}_{0}t\right)+\dots $$*

*Note that the series coefficients ${E}_{T-S}^{n{\omega}_{0}}$ in Eq. (12) can be obtained from the Fourier components of E_{i}
α_{eff}. Substituting Eq.(12) into Eq.(11), the heterodyne intensity I_{het}(t) signal can be reformulated as*

*$${I}_{\mathrm{het}}\left(t\right)={E}_{R}^{\mathit{2}}$$*

$$+\mathit{2}\sum _{n=\mathit{0}}^{\infty}{E}_{T-S}^{n{\omega}_{\mathit{0}}}\mathrm{cos}\left(n{\omega}_{\mathit{0}}t\right){E}_{R}\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{\mathrm{TS}}+\mathrm{\Delta \omega}t\right)\text{}$$

$$+\mathit{2}{E}_{S}{E}_{R}\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{S}+\Delta \omega t\right)\text{}$$

$$+\mathit{2}{E}_{R}{E}_{P}\left\{\right[{J}_{\mathit{0}}\left({\psi}_{\mathit{3}}\right)+\mathit{2}\sum _{j=\mathit{1}}^{\infty}{\left(-\mathit{1}\right)}^{j}{J}_{\mathit{2}j}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}\left(\mathit{2}j{\omega}_{\mathit{0}}t\right)]\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{1}})$$

$$+\mathit{2}\sum _{j=\mathit{0}}^{\infty}{\left(-\mathit{1}\right)}^{j}{J}_{\mathit{2}j+\mathit{1}}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}\left[\right(\mathit{2}j{+\mathit{1})\omega}_{\mathit{0}}t]\mathrm{sin}(\Delta \omega t+{\psi}_{\mathit{1}})\}$$

$$+\mathit{2}{E}_{R}{E}_{P\_R}\left\{\right[{J}_{\mathit{0}}\left({\mathit{2}\psi}_{\mathit{3}}\right)+\mathit{2}\sum _{j=\mathit{1}}^{\infty}{\left(-\mathit{1}\right)}^{j}{J}_{\mathit{2}j}\left({\mathit{2}\psi}_{\mathit{3}}\right)\mathrm{cos}\left(\mathit{2}j{\omega}_{\mathit{0}}t\right)]\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{2}})$$

$$+\mathit{2}\sum _{j=\mathit{0}}^{\infty}{\left(-\mathit{1}\right)}^{j}{J}_{\mathit{2}j+\mathit{1}}\left(\mathit{2}{\psi}_{\mathit{3}}\right)\mathrm{cos}\left[\right(\mathit{2}j{+\mathit{1})\omega}_{\mathit{0}}t]\mathrm{sin}(\Delta \omega t+{\psi}_{\mathit{2}})\}$$

$$+\mathit{2}\sum _{n=\mathit{0}}^{\infty}{E}_{T-S}^{n{\omega}_{\mathit{0}}}\mathrm{cos}\left(n{\omega}_{\mathit{0}}t\right){E}_{R}\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{\mathrm{TS}}+\mathrm{\Delta \omega}t\right)\text{}$$

$$+\mathit{2}{E}_{S}{E}_{R}\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{S}+\Delta \omega t\right)\text{}$$

$$+\mathit{2}{E}_{R}{E}_{P}\left\{\right[{J}_{\mathit{0}}\left({\psi}_{\mathit{3}}\right)+\mathit{2}\sum _{j=\mathit{1}}^{\infty}{\left(-\mathit{1}\right)}^{j}{J}_{\mathit{2}j}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}\left(\mathit{2}j{\omega}_{\mathit{0}}t\right)]\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{1}})$$

$$+\mathit{2}\sum _{j=\mathit{0}}^{\infty}{\left(-\mathit{1}\right)}^{j}{J}_{\mathit{2}j+\mathit{1}}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}\left[\right(\mathit{2}j{+\mathit{1})\omega}_{\mathit{0}}t]\mathrm{sin}(\Delta \omega t+{\psi}_{\mathit{1}})\}$$

$$+\mathit{2}{E}_{R}{E}_{P\_R}\left\{\right[{J}_{\mathit{0}}\left({\mathit{2}\psi}_{\mathit{3}}\right)+\mathit{2}\sum _{j=\mathit{1}}^{\infty}{\left(-\mathit{1}\right)}^{j}{J}_{\mathit{2}j}\left({\mathit{2}\psi}_{\mathit{3}}\right)\mathrm{cos}\left(\mathit{2}j{\omega}_{\mathit{0}}t\right)]\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{2}})$$

$$+\mathit{2}\sum _{j=\mathit{0}}^{\infty}{\left(-\mathit{1}\right)}^{j}{J}_{\mathit{2}j+\mathit{1}}\left(\mathit{2}{\psi}_{\mathit{3}}\right)\mathrm{cos}\left[\right(\mathit{2}j{+\mathit{1})\omega}_{\mathit{0}}t]\mathrm{sin}(\Delta \omega t+{\psi}_{\mathit{2}})\}$$

*Rearranging Eq. (13) in order of the modulation radian frequency, i.e. Δ ω+nω
_{0}, the heterodyne intensity signal can be expressed in terms of the following components:*

*$${I}_{\mathrm{het}}\left(t\right)={E}_{R}^{\mathit{2}}........................................................................\mathrm{DC}$$*

$$+\mathit{2}{E}_{R}{E}_{P}{J}_{\mathit{0}}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{1}})+\mathit{2}{E}_{T-S}^{\mathit{0}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{\mathrm{TS}}+\mathrm{\Delta \omega}t\right)$$

$$+\mathit{2}{E}_{S}{E}_{R}\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{S}+\Delta \omega t\right)\text{}+\mathit{2}{E}_{R}{E}_{P\_R}{J}_{\mathit{0}}\left(\mathit{2}{\psi}_{\mathit{3}}\right)\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{2}})..................\Delta \omega t$$

$$+\mathit{4}{E}_{R}{E}_{P}{J}_{\mathit{1}}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}\left({\omega}_{\mathit{0}}t\right)\mathrm{sin}(\Delta \omega t+{\psi}_{\mathit{1}})$$

$$+\mathit{4}{E}_{R}{E}_{P\_R}{J}_{\mathit{1}}\left({\mathit{2}\psi}_{\mathit{3}}\right)\mathrm{cos}\left({\omega}_{\mathit{0}}t\right)\mathrm{sin}(\Delta \omega t+{\psi}_{\mathit{2}})$$

$$+\mathit{2}{E}_{T-S}^{\mathit{1}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left({\omega}_{\mathit{0}}t\right)\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{\mathrm{TS}}+\mathrm{\Delta \omega}t\right).................................................(\Delta \omega +{\mathit{1}\omega}_{\mathit{0}})t$$

$$-\mathit{4}{E}_{R}{E}_{P}{J}_{\mathit{2}}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}\left(\mathit{2}{\omega}_{\mathit{0}}t\right)\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{1}})$$

$$-\mathit{4}{E}_{R}{E}_{P\_R}{J}_{\mathit{2}}\left({\mathit{2}\psi}_{\mathit{3}}\right)\mathrm{cos}\left({\mathit{2}\omega}_{\mathit{0}}t\right)\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{2}})$$

$$+\mathit{2}{E}_{T-S}^{\mathit{2}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left({\mathit{2}\omega}_{\mathit{0}}t\right)\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{\mathrm{TS}}+\mathrm{\Delta \omega}t\right).................................................(\Delta \omega +{\mathit{2}\omega}_{\mathit{0}})t$$

$$-\mathit{4}{E}_{R}{E}_{P}{J}_{\mathit{3}}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}\left(\mathit{3}{\omega}_{\mathit{0}}t\right)\mathrm{sin}(\Delta \omega t+{\psi}_{\mathit{1}})$$

$$-\mathit{4}{E}_{R}{E}_{P\_R}{J}_{\mathit{3}}\left({\mathit{2}\psi}_{\mathit{3}}\right)\mathrm{cos}\left({\mathit{3}\omega}_{\mathit{0}}t\right)\mathrm{sin}(\Delta \omega t+{\psi}_{\mathit{2}})$$

$$+\mathit{2}{E}_{T-S}^{\mathit{3}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left({\mathit{3}\omega}_{\mathit{0}}t\right)\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{\mathrm{TS}}+\mathrm{\Delta \omega}t\right).................................................(\Delta \omega +{\mathit{3}\omega}_{\mathit{0}})t$$

$$+\mathit{4}{E}_{R}{E}_{P}{J}_{\mathit{4}}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}\left(\mathit{4}{\omega}_{\mathit{0}}t\right)\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{1}})$$

$$+\mathit{4}{E}_{R}{E}_{P\_R}{J}_{\mathit{4}}\left({\mathit{2}\psi}_{\mathit{3}}\right)\mathrm{cos}\left({\mathit{4}\omega}_{\mathit{0}}t\right)\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{2}})$$

$$+\mathit{2}{E}_{T-S}^{\mathit{4}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left({\mathit{4}\omega}_{\mathit{0}}t\right)\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{\mathrm{TS}}+\mathrm{\Delta \omega}t\right).................................................(\Delta \omega +{\mathit{4}\omega}_{\mathit{0}})t$$

$$+\mathrm{Higher}\phantom{\rule{.2em}{0ex}}\mathrm{Order}\phantom{\rule{.2em}{0ex}}\mathrm{Heterodyne}\phantom{\rule{.2em}{0ex}}\mathrm{Modulation}\phantom{\rule{.2em}{0ex}}\mathrm{Radian}\phantom{\rule{.2em}{0ex}}\mathrm{Frequencym}\phantom{\rule{.2em}{0ex}}\mathrm{Terms}$$

$$+\mathit{2}{E}_{R}{E}_{P}{J}_{\mathit{0}}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{1}})+\mathit{2}{E}_{T-S}^{\mathit{0}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{\mathrm{TS}}+\mathrm{\Delta \omega}t\right)$$

$$+\mathit{2}{E}_{S}{E}_{R}\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{S}+\Delta \omega t\right)\text{}+\mathit{2}{E}_{R}{E}_{P\_R}{J}_{\mathit{0}}\left(\mathit{2}{\psi}_{\mathit{3}}\right)\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{2}})..................\Delta \omega t$$

$$+\mathit{4}{E}_{R}{E}_{P}{J}_{\mathit{1}}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}\left({\omega}_{\mathit{0}}t\right)\mathrm{sin}(\Delta \omega t+{\psi}_{\mathit{1}})$$

$$+\mathit{4}{E}_{R}{E}_{P\_R}{J}_{\mathit{1}}\left({\mathit{2}\psi}_{\mathit{3}}\right)\mathrm{cos}\left({\omega}_{\mathit{0}}t\right)\mathrm{sin}(\Delta \omega t+{\psi}_{\mathit{2}})$$

$$+\mathit{2}{E}_{T-S}^{\mathit{1}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left({\omega}_{\mathit{0}}t\right)\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{\mathrm{TS}}+\mathrm{\Delta \omega}t\right).................................................(\Delta \omega +{\mathit{1}\omega}_{\mathit{0}})t$$

$$-\mathit{4}{E}_{R}{E}_{P}{J}_{\mathit{2}}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}\left(\mathit{2}{\omega}_{\mathit{0}}t\right)\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{1}})$$

$$-\mathit{4}{E}_{R}{E}_{P\_R}{J}_{\mathit{2}}\left({\mathit{2}\psi}_{\mathit{3}}\right)\mathrm{cos}\left({\mathit{2}\omega}_{\mathit{0}}t\right)\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{2}})$$

$$+\mathit{2}{E}_{T-S}^{\mathit{2}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left({\mathit{2}\omega}_{\mathit{0}}t\right)\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{\mathrm{TS}}+\mathrm{\Delta \omega}t\right).................................................(\Delta \omega +{\mathit{2}\omega}_{\mathit{0}})t$$

$$-\mathit{4}{E}_{R}{E}_{P}{J}_{\mathit{3}}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}\left(\mathit{3}{\omega}_{\mathit{0}}t\right)\mathrm{sin}(\Delta \omega t+{\psi}_{\mathit{1}})$$

$$-\mathit{4}{E}_{R}{E}_{P\_R}{J}_{\mathit{3}}\left({\mathit{2}\psi}_{\mathit{3}}\right)\mathrm{cos}\left({\mathit{3}\omega}_{\mathit{0}}t\right)\mathrm{sin}(\Delta \omega t+{\psi}_{\mathit{2}})$$

$$+\mathit{2}{E}_{T-S}^{\mathit{3}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left({\mathit{3}\omega}_{\mathit{0}}t\right)\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{\mathrm{TS}}+\mathrm{\Delta \omega}t\right).................................................(\Delta \omega +{\mathit{3}\omega}_{\mathit{0}})t$$

$$+\mathit{4}{E}_{R}{E}_{P}{J}_{\mathit{4}}\left({\psi}_{\mathit{3}}\right)\mathrm{cos}\left(\mathit{4}{\omega}_{\mathit{0}}t\right)\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{1}})$$

$$+\mathit{4}{E}_{R}{E}_{P\_R}{J}_{\mathit{4}}\left({\mathit{2}\psi}_{\mathit{3}}\right)\mathrm{cos}\left({\mathit{4}\omega}_{\mathit{0}}t\right)\mathrm{cos}(\Delta \omega t+{\psi}_{\mathit{2}})$$

$$+\mathit{2}{E}_{T-S}^{\mathit{4}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left({\mathit{4}\omega}_{\mathit{0}}t\right)\mathrm{cos}\left({\varphi}_{R}-{\varphi}_{\mathrm{TS}}+\mathrm{\Delta \omega}t\right).................................................(\Delta \omega +{\mathit{4}\omega}_{\mathit{0}})t$$

$$+\mathrm{Higher}\phantom{\rule{.2em}{0ex}}\mathrm{Order}\phantom{\rule{.2em}{0ex}}\mathrm{Heterodyne}\phantom{\rule{.2em}{0ex}}\mathrm{Modulation}\phantom{\rule{.2em}{0ex}}\mathrm{Radian}\phantom{\rule{.2em}{0ex}}\mathrm{Frequencym}\phantom{\rule{.2em}{0ex}}\mathrm{Terms}$$

*Equation (14) provides a number of basic insights into the fundamental characteristics of the heterodyne intensity signal. For example, it can be seen that it is impossible to acquire the absolute interaction electric field Ē_{T-S} in A-SNOM. However, the intensities of the background electric fields E_{R}E_{P} and E_{R}E_{P_R} have coefficients of J_{n}(ψ
_{3}) and J_{n}(2ψ
_{3}), respectively, and thus if the higher-order coefficients of these fields decay more rapidly than those of the E^{nω}
_{T-S} signal, the lock-in detection signal will exhibit an improved signal contrast when applied to different samples at higher orders of the harmonic modulation radian frequency.*

*4. Simulation of amplitude and phase of detection signal at (Δω+1ω0)-order harmonics of modulation radian frequency*

*The analytical formulation given in Eq. (14) provides a complete description of the lock-in heterodyne A-SNOM detection signal from its basic DC component to its (Δ ω+4ω
_{0})-order harmonic radian frequency component. To confirm the validity of this formulation, this section of the paper simulates the (Δω+1ω
_{0})-order harmonics of the heterodyne detection signal as a function of the tip-to-sample distance and compares the analytical results with the experimental results presented in the literature [7–8].*

*In extracting the amplitude and phase of the detection signal, the simulations make the assumption that the amplitude of the 1ω _{0} component of the tip enhancement electric field is given by ${E}_{T-S}^{1{\omega}_{0}}\propto {\stackrel{\rightharpoonup}{E}}_{T-S}\propto {\alpha}_{\mathrm{eff}}{\stackrel{\rightharpoonup}{E}}_{i}$. Therefore, the variation of ${E}_{T-S}^{l{\omega}_{0}}$ with the tip-to-sample distance, Z
_{0}, can be simulated using the effective polarizability formula given in [7–9]. Moreover, the simulations make the following additional assumptions: (1) the illuminating light source is a He-Ne laser with a wavelength of 633 nm, (2) the sample is made of Si and has a dielectric constant of 15 [7–8,19], and (3) the tip-mounted sphere is made of Au and has a radius of 20 nm and a dielectric value of -10+2 i [7–8, 19]. The normalized simulation results presented in Fig. 3 show that the amplitude of the tip enhancement electric field decays steeply at tip-to-sample separation distances of less than 40 nm, but then converges to a value of approximately 0.2 as the separation distance increases. This tendency reflects the fact that the electric dipole enhancement effect is very high in the near-field region, but has a low, constant value at greater tip-to-sample distances [7–9].*

*In order to achieve a level playing field when comparing the present analytical results with the experimental results presented in [7–8], the parameters in the (Δ ω+1ω
_{0})-order components of Eq. (14) must be assigned reasonable values. Thus, the phase difference (Φ_{R}-Φ_{TS}) is set to π/3. Note that the phase Φ_{TS} of the near-field electric field is assumed to be independent of the tip-to-sample distance Z
_{0} because it is related only to the effective polarizability parameter α_{eff} [7]. Furthermore, the relative amplitudes E_{P} and E_{P_R}, are assigned values of 0.05 and 0.5, respectively. In addition, the relative amplitude of the reference electric field, E_{R}, is specified as 1000 referring to [7–8]. The phase differences between the two fields can be written as ψ
_{1}=(Φ_{R}-Φ_{P})+2Ksin(θ)Z
_{0} and ψ
_{2}=(Φ_{R}-Φ_{P})+4Ksin(θ)Z
_{0}, respectively, where K is the wave number and the initial phase (Φ_{R}-Φ_{P}) is assumed to be -π. Finally, the incident angle of the illuminating light, θ, is specified as π/6 and the amplitude of the tip vibration is assigned a value of A=20 nm [7-8]. As a result, the phase modulation depth has a value of ψ
_{3}=2Ksin(θ)A=0.199. The variations of the amplitude and phase of the (Δω+1ω
_{0})-order frequency component of the interaction electric field signal can then be obtained as a function of the tip-to- sample distance, Z
_{0}, directly from Eq. (14), as illustrated in Fig. 4.*

*In general, Fig. 4 shows that the amplitude and phase of the heterodyne signal are both highly sensitive to the tip-to-sample distance. In the near-field region, the interaction electric field is the largest of the four fields generated by the illuminating light source (see Fig. 2). However, its intensity decays rapidly with an increasing tip-to-sample distance [7–8]. In the far-field region, the tip scattering field reflected from the sample dominates the detection signal since the periods of the amplitude and phase of the interaction electric field shown in Fig.4 are equal to that of the variation of the phase difference ψ
_{2}, i.e. λ/2, which implies that the electric field reflected from the sample surface, E_{P_R}, is much stronger than the tip scattering electric field, E_{P}. This far-field phenomenon demonstrates the importance of modifying the near-field homodyne model presented in [15] to take account of the tip scattering electric field reflected from the sample. Comparing the simulation results presented in Fig. 4 with the experimental results presented in Fig. 3 in [7], it is found that a good general agreement exists between the two sets of results. Thus, the validity of the analytical model presented in Eq. (14) is confirmed.*

*5. Analysis of A-SNOM image artifacts and errors using analytical model*

*From Section 3, it is clear that the A-SNOM detection signal (either homodyne or heterodyne) is highly complex. As a result, the quality of the A-SNOM image obtained from the AFM tip scanning operation is subject to many influencing factors. Therefore, to improve the A-SNOM image quality, it is essential to identify the various sources of artifacts and errors, and to take appropriate remedial actions to suppress their effects [13–14].*

*5.1 Image artifacts*

*Image artifacts are induced by variations in the topography of the scanned sample surface and are essentially fake near-field signals. As shown in Fig 2, the sample topography is not flat, and thus the height of the tip must be dynamically adjusted in accordance with the atomic force between the tip and the sample in order to maintain a constant tip-to-sample distance Z
_{0} during the tip scanning process. The initial phase Φ_{P} of the tip scattering electric field Ē_{Probe}should therefore be modified to the form Φ_{P}+2Ksin(θ)ΔZ, where ΔZ is the variation of the sample surface from its initial position. Consequently, the phase differences ψ
_{1} and ψ
_{2} should be reformulated as follows:*

*$${\psi}_{\mathit{1}}={\varphi}_{R}-{\varphi}_{P}-\mathit{2}K\phantom{\rule{.2em}{0ex}}\mathrm{sin}\left(\theta \right){Z}_{\mathit{0}}-\mathit{2}K\phantom{\rule{.2em}{0ex}}\mathrm{sin}\left(\theta \right)\mathrm{sin}\phantom{\rule{.2em}{0ex}}\mathrm{\Delta Z}$$*

$${\psi}_{\mathit{2}}={\varphi}_{R}-{\varphi}_{P}-\mathit{4}K\phantom{\rule{.2em}{0ex}}\mathrm{sin}\left(\theta \right){Z}_{\mathit{0}}-\mathit{2}K\phantom{\rule{.2em}{0ex}}\mathrm{sin}\left(\theta \right)\mathrm{sin}\phantom{\rule{.2em}{0ex}}\mathrm{\Delta Z}$$

$${\psi}_{\mathit{2}}={\varphi}_{R}-{\varphi}_{P}-\mathit{4}K\phantom{\rule{.2em}{0ex}}\mathrm{sin}\left(\theta \right){Z}_{\mathit{0}}-\mathit{2}K\phantom{\rule{.2em}{0ex}}\mathrm{sin}\left(\theta \right)\mathrm{sin}\phantom{\rule{.2em}{0ex}}\mathrm{\Delta Z}$$

*From Eq. (15), it can be seen that the image artifacts induced by the sample topography have the extra form 2 Ksin(θ)ΔZ. The term, 2Ksin(θ)ΔZ in Eq. (15) is consistent with that in Eq. (4) presented in [13]. Significantly, however, compared to [13], the derivation of the image artifact formulation presented in this study does not require the establishment of a complex dipole-dipole theoretical model and, moreover, takes explicit account of the near-field interaction effect. In other words, the model developed in this study provides both a simpler and a more comprehensive explanation of the A-SNOM artifact phenomenon than that proposed in [13].*

*From the derivations and discussions above, it is clear that the problem of image artifacts in heterodyne A-SNOM detection can be easily resolved by performing a sample stage scanning operation rather than a tip scanning operation. In this way, the initial phase Φ_{P} of the tip scattering electric field, Ē_{Probe}, remains constant throughout the sample stage scanning operation. Furthermore, even though the initial phase of the sample scattering field is modified to Φ_{S}+2Ksin(θ)ΔZ, the heterodyne detection signal intensities I
_{Δω}+_{nω}
_{0},(n≧ 1) in Eq.(14) are not affected, and thus the image artifacts induced by the sample topography effectively disappear by operating a sample stage scanning. It should be noticed that the stage scanning does not affect the optical paths of the probe direct scattering field Ē_{Probe} and the probe scattered field Ē _{Probe _Reflective} reflected from the sample. It is because that the tip-tosample distance keeps constant with AFM feedback control.*

*5.2 Image errors*

*When an A-SNOM tip scans a sample with discontinuous boundaries, the amplitude of the tip vibration, A, is affected by changes in the discontinuous geometry and/or material as a result of the corresponding variations in the attractive Van der Waals forces between the tip and the sample surface. Thus, even though an AFM feedback control scheme is employed to trace the tip amplitude vibration variation ΔA, the speed of the scanning process is usually such that the vibration of the tip amplitude can not be maintained as a perfect constant, and thus the A-SNOM detection signal contains fake signals induced by variations in the tip vibration amplitude [14]. Furthermore, the amplitude of the tip vibration is related to the modulation depth, ψ
_{3}, and the Fourier expansion coefficients ${E}_{T-S}^{n{\omega}_{0}}$ in Eq.(14), and therefore the intensity of the detected signal when performing scanning in the forward direction varies inversely with that detected when scanning in the backward direction at the same boundary [14].*

*In order to correct these image errors, the tip scanning rate should be carefully controlled in such a way as to allow the AFM feedback control scheme sufficient time to trace the variations in the amplitude of the tip vibration and to take appropriate remedial actions to maintain a constant amplitude, A. Clearly, this is particularly important when the sample surface is characterized by many steep boundaries and ridges.*

*6. Comparison of signal-to-background ratio and signal contrast of homodyne and heterodyne A-SNOM detection signals in visible light region*

*Homodyne detection is challenging in the visible light region since the detection signal has a low intensity, and thus heterodyne detection is generally preferred [8]. In order to clarify this phenomenon in a quantitative manner, this section of the paper compares the S/B ratios of the two detection schemes at different harmonic orders of the modulation radian frequency. In general, the S/B ratio can be defined as*

*$${\left(\frac{S}{B}\right)}_{\mathrm{Radian}\phantom{\rule{.2em}{0ex}}\mathrm{Frequency}\phantom{\rule{.2em}{0ex}}\mathrm{Order}}=\frac{\mid \mathrm{Signal}\phantom{\rule{.2em}{0ex}}\mathrm{Intensity}\mid}{{\mid \mathrm{Background}\phantom{\rule{.2em}{0ex}}\mathrm{Intensity}\mid}_{\mathrm{Radian}\phantom{\rule{.2em}{0ex}}\mathrm{Frequency}\phantom{\rule{.2em}{0ex}}\mathrm{Order}}}$$*

*where the signal intensity term is given by the sum of all the terms relating to the near-field interaction electric field ${E}_{T-S}^{n{\omega}_{0}}$ in Eq.(14), and the noise intensity term is the sum of the background noise terms. The signal intensity contrast ( S1/S2) [15] can be defined as*

*$${\mid S1/S2\mid}_{\mathrm{Radian}\phantom{\rule{.4em}{0ex}}\mathrm{Frequency}\phantom{\rule{.4em}{0ex}}\mathrm{Order}}={\mid {I}_{n{\omega}_{0}\left(S1\right)}/{I}_{n{\omega}_{0}\left(S2\right)}\mid}_{\mathrm{Radian}\phantom{\rule{.4em}{0ex}}\mathrm{Frequency}\phantom{\rule{.4em}{0ex}}\mathrm{Order}}$$*

*where S1 and S2 represent sample 1 and sample 2, respectively. It is assumed that S1 and S2 represent two distinct signals with different near-field interaction amplitude but same phase difference relation and background contribution.*

*6.1 S/B ratio and signal contrast of homodyne detection signal*

*In the following analysis, an assumption is made that the amplitude of the near-field DC term, ${E}_{T-S}^{0{\omega}_{0}}$, has a value of 1.2 while the amplitude of the tip vibration, A, is equal to 60 nm. Furthermore, the higher-order amplitudes are given ideally by ${E}_{T-S}^{n{\omega}_{0}}=\frac{{E}_{T-S}^{0{\omega}_{0}}}{2n}$ as determined from a Fourier analysis of E_{i}
α_{eff}. In investigating the effect of the incident wavelength on the S/B ratio in homodyne detection, the incident angle is assumed to be θ=π/6 and the background electric field is assigned a value of E_{S}=10. As in the simulations performed in Section 3, the amplitudes of the direct tip scattering field, E_{P}, and the reflected field, E_{P_R}, are specified as 0.05 and 0.5, respectively. In addition, the phase differences Φ_{TS}-Φ_{S}, Φ_{P}-Φ_{S}, and Φ_{P}-Φ_{TS} are all assumed to be π/4. Finally, Ksin(θ)Z
_{0} is assigned a value of 0.1.*

*Figure 5 presents the results obtained for the variation of the S/B ratio with the wavelength of the illuminating light source when substituting the parameter values given above into the homodyne signal formula (derived from Eq.(9)). Clearly, in performing A-SNOM detection (either homodyne or heterodyne), it is desirable to maximize the S/B ratio in order to improve the quality of the imaging results. Figure 5 shows that the S/B ratio of the I_{DC} component of the detected signal has a value of approximately 0 at all values of the incident wavelength. However, for the higher-order harmonic radian frequency components, an improved S/B ratio is obtained as the wavelength of the illuminating light source increases. The improvement in the S/B ratio with an increasing wavelength is particularly pronounced at higher harmonic orders. However, whilst the I
_{4ω0} order has the best S/B ratio of the various harmonic components illustrated in Fig. 5, its intensity is still relatively low (see Fig. 6), and thus its suitability for practical high-resolution scanning applications is limited.*

*Figure 7 illustrates the variation of the signal intensity contrast ( S1/S2) with the wavelength of the illuminating light source as computed using the original homodyne model presented in [15]. Note that in deriving these results, it is assumed that the near-field amplitude of sample (S1) is 1.2 times larger than that of (S2) in every radian frequency order, ${E}_{T-S\left(S1\right)}^{n{\omega}_{0}}=1.2{E}_{T-S\left(S2\right)}^{n{\omega}_{0}}$. In other words, the ideal signal intensity contrast has a value of 1.2, i.e. ${\mid \frac{S1}{S2}\mid}_{n}=\mid \frac{{I}_{n{\omega}_{0}\left(S1\right)}}{{I}_{n{\omega}_{0}\left(S2\right)}}\mid \cong \mid \frac{{E}_{T-S\left(S1\right)}^{n{\omega}_{0}}}{{E}_{T-S\left(S2\right)}^{n{\omega}_{0}}}\mid $. The results indicate that the I
_{4ω0} order component of the detected signal yields a contrast ratio close to the ideal value of 1.2. The signal contrast diagram may have the discontinuous profile for the order I
_{1ω0} in Fig. 7, which is caused by the deconstructive interference between the near-field and background noise in Fig. 6.*

*6.2 S/B ratio and signal contrast of heterodyne detection signal*

*Due to the weak intensity of the homodyne detection signal (see Fig. 6), attempting to improve the S/B ratio by utilizing conventional A-SNOM techniques such as raising the order of the radian frequency used for detection purposes, decreasing the amplitude of the tip vibration, A, or increasing the wavelength of the illuminating light source is bound to meet with limited success. However, as shown in Fig. 8, the intensity of the heterodyne detection signal is far greater than that of the homodyne detection signal. (Note that in computing the results presented in Fig. 8, the amplitude of the reference beam is specified as E_{R}=1000 in the visible light region). Thus, even if the amplitudes of the background electric fields E_{P} and E_{P_R} are amplified by the reference electric field Ē_{Referenc} to the extent that they are equivalent to the near-field amplitude ${E}_{T-S}^{n{\omega}_{0}}$, the S/B ratio can still be improved by increasing the radian frequency order used for detection purposes or decreasing the amplitude of the tip vibration, A.*

*The following discussions consider the case where the S/B ratio is improved by reducing the tip vibration amplitude, A, from 60 nm to 20 nm. By doing so, the amplitude of the DC term, ${E}_{T-S}^{0{\omega}_{0}}$, in the tip enhancement can be assigned a reasonable value of 1.8, while the higher-order near-field interaction amplitudes are assigned ideally as ${E}_{T-S}^{n{\omega}_{0}}=\frac{{E}_{T-S}^{0{\omega}_{0}}}{3n}$ since a Fourier analysis reveals that these terms decay more rapidly at smaller values of the tip vibration. Note that all the other parameters in Eq. (14) are assumed to have the same values as those assigned in subsection 6.1. Figure 9 shows that the S/B ratio of the heterodyne detection signal is an order of magnitude higher than that of the homodyne detection signal in the visible wavelength region (see Fig. 5). In fact, the S/B ratio obtained from the I_{Δω}
_{+2ω0} harmonic component is already sufficiently high for detection purposes without the need to consider higher order harmonics. Even though the near-field interaction amplitudes ${E}_{T-S}^{n{\omega}_{0}}$ of the heterodyne detection signal decay more rapidly than their homodyne counterparts, the background noise is also reduced due to the lower modulation depth ψ
_{3} achieved by decreasing the amplitude of the tip vibration A vibration (see Eq. (14).) Significantly, the use of a reference electric field Ē_{Reference} in the heterodyne detection method improves both the S/B ratio and the signal intensity compared to the homodyne technique. Thus, as shown in Fig. 10, an acceptable signal contrast (i.e. a value close to the ideal value of 1.2) can be obtained at harmonic orders as low as the second order, i.e. I_{Δω+2ω0}, in the visible region (compared to the fourth order in the homodyne method).*

*7. Potential methods for improving S/B ratio of A-SNOM detection signal*

*This section discusses various additional techniques for improving the S/B ratio of the heterodyne A-SNOM detection signal as suggested by an inspection of the analytical formulae presented in Section 3. Subsections 7.1 and 7.2 investigate the correlation between the S/B ratio of the heterodyne detection signal and the modulation depth parameter, ψ
_{3}, in two different regimes, i.e. ψ
_{3} < 1 and ψ
_{3} > 1, respectively. (Note that in performing the analyses, all of the parameter values are identical to those considered in subsection 6.2.) Finally, subsection 7.3 considers the feasibility of improving the S/B ratio of the heterodyne A-SNOM detection signal by replacing the conventional sinusoidal tip vibration modulation function by a ramp function.*

*7.1 Correlation between S/B ratio and phase modulation depth parameter ψ3 for ψ3 <1*

*Equation (14) shows that the phase modulation depth ψ _{3} is a highly important parameter since its value has a direct effect upon the intensity of the background noise. Thus, intuitively, the potential exists for improving the S/B ratio by specifying the value of the modulation depth in such a way that the intensity of the background noise is reduced, or even eliminated entirely. Figure 11 illustrates the variation of the S/B ratio of the detection signal components I_{Δω} to I_{Δω+4ω0} with the modulation depth, ψ_{3}, for ψ_{3} < 1. It is observed that the S/B ratios tend toward infinity as the modulation depth approaches 0 for all harmonic orders other than zero. Furthermore, it can be seen that for a given harmonic order, the S/B ratio decays rapidly with an increasing modulation depth. However, comparing the various profiles presented in the figure, it is apparent that the S/B ratios of the higher-order radian frequency components decay more slowly with increasing ψ_{3} than the lower-order harmonic order components. Thus, it is inferred that higher-order intensity signals represent a more suitable choice for heterodyne A-SNOM detection purposes.*

*As described in Section 3, the phase modulation depth, ψ
_{3}, has the form ψ
_{3}=2Ksin(θ)A. In other words, ψ
_{3} is governed by three experimental parameters, namely the incident wavelength λ (K=2π/λ), the incident angle θ, and the tip vibration amplitude A, respectively. Since Fig. 11 shows that the S/B ratio improves as the value of ψ
_{3} reduces, it can be inferred that for constant θ and A, the S/B ratio improves as the wavelength of the illuminating light source increases. (Note that this observation is consistent with the discussions presented in subsection 6.2.)*

*Many researchers have investigated the correlation between the incident angle of the illuminating light source, θ, and the tip-sample enhancement phenomenon in the A-SNOM technique [7–9, 20]. Irrespective of the method used to analyze this correlation (e.g. quasi-electrostatic, Finite-Difference Time-Domain (FDTD), Finite Element Method (FEM)), these studies all concluded that if the polarized direction of the incident electric field is perpendicular to the sample surface (as shown in Fig. 2), a smaller value of the incident angle induces a stronger enhancement scattering field. The following analysis examines the effect of the incident angle on the heterodyne A-SNOM detection signal. Note that in conducting the analysis, the incident wavelength is specified as 633 nm and the remaining parameters are assigned the same values as those given in Subsection 6.2. Substituting these parameter values into Eq. (14) yields the S/B vs. θ profiles shown in Fig. 12. It can be seen that the S/B ratio of the zero-order harmonic intensity signal has a value of approximately 0 at all values of the incident angle. However, the S/B ratios associated with the higher-order intensity signals tend to a high value at small values of the incident angle, but reduce as the incident angle increases. From inspection, it is found that the rate of decay of the S/B ratio with an increasing incident angle slows as the harmonic order of the radian modulation frequency increases. Thus, the results once again indicate that higher-order intensity signals represent a more suitable choice for heterodyne A-SNOM detection purposes.*

*Regarding the effect of the AFM tip vibration amplitude A on the value of the S/B ratio, from a lock-in detection perspective, a S/B vs A figure similar to those presented in Figs. 11 and 12 can be derived. However, to do so, it is necessary to abandon the assumption that the n-th order electric field amplitude is given by ${E}_{T-S}^{n{\omega}_{0}}=\frac{{E}_{T-S}^{0{\omega}_{0}}}{3n}$. In practice, however, this assumption can not be ignored since it has a significant effect on the interaction electric field Ē
_{T-S}. Accordingly, in optimizing the tip vibration amplitude, a Fourier transform analysis of Ē
_{T-S} should first be performed (as in [9]) since from the definition of the phase modulation depth (ψ
_{3}=2Ksin(θ)A), it is known that a smaller tip vibration amplitude A yields an improved contrast. However, a smaller value of A also reduces the amplitude of the interaction signal since the high-order Fourier components decay more rapidly. If the Fourier components decay more rapidly than the Bessel function of the same order or the amplitude of the high-order electric field ${E}_{T-S}^{n{\omega}_{0}}$ is lower than the noise floor, A-SNOM measurement cannot be performed.*

*7.2 Correlation between S/B ratio and modulation depth parameter ψ3 for ψ3 >1*

*Figure 11 shows that the S/B ratios of all the harmonic-order intensity signals converge toward a low value as the modulation depth approaches 1. Since all orders of the Bessel function of the first kind J_{n}(ψ
_{3}) have zero points when the phase modulation depth ψ
_{3} is assigned certain values greater than one, the opportunity arises to increase the S/B ratio by specifying the value of the modulation depth such that these zero points are obtained, thereby causing the background noise to disappear in Eq. (14). Figure 13 illustrates the S/B ratios associated with detection signal components I
_{Δω} to I
_{Δω+4ω0} over the modulation depth range of ψ
_{3}=1~4. It can be seen that the I
_{Δω+ω0} component exhibits two extremely high S/B ratio points at ψ
_{3}=2 and 3.5, respectively; while I
_{Δω+2ω0} has a high S/B ratio at ψ
_{3}=2.65; I
_{Δω+3ω0} at ψ
_{3}=3.25; and I
_{Δω+4ω0} at ψ
_{3}=3.85, respectively. However, despite these ultra-high S/B ratio points, to the best of the current authors’ knowledge, no attempt has been made in the literature to exploit this characteristic of the modulation depth parameter to enhance the S/B ratio of the A-SNOM detection signal due to the practical difficulties involved in establishing the necessary experimental conditions. For example, to obtain a modulation depth of ψ
_{3}=2, corresponding to the first ultra-high S/B point in the I
_{Δω+ω0} profile, the incident wavelength should have a value of 62.8 nm (assuming that the incident angle and tip vibration amplitude are π/6 and 20 nm, respectively). However, this wavelength lies deep within the UV region of the light spectrum, and thus the sample may be damaged during the scanning process. In addition, Fig. 13 shows that the high S/B ratio points associated with the different harmonic-order intensity signals not only occur at different modulation depth values, but are also restricted to very narrow ranges of the modulation depth. Thus, it is difficult to regulate the three experimental parameters affecting the phase modulation depth (i.e. the wavelength, incident angle, and AFM tip vibration amplitude) with a sufficient precision to obtain the precise values of ψ
_{3} required to generate the ultra-high S/B ratio.*

*7.3 Modulation of AFM tip vibration utilizing ramp function*

*The sections above have assumed the AFM tip vibration to be modulated by a sinusoidal function. This section of the paper investigates the feasibility of utilizing a ramp function in place of this sinusoidal function as a means of eliminating the background noise, thereby enhancing the S/B ratio. Accordingly, the tip-to-sample distance is reformulated as follows:*

*where A and T are the amplitude and period of the ramp function, respectively, and mT ≤t≤(m+1)T. m is a positive integer. Substituting Z(t) into the heterodyne intensity formulation I
_{het}(t) given in Eq. (10), it can be shown that*

*$${I}_{\mathrm{het}}\left(t\right)={E}_{R}^{\mathit{2}}$$*

$$+\mathit{2}{E}_{T-S}{E}_{R}\mathrm{cos}\left(\Delta \omega t+{\varphi}_{R}-{\varphi}_{\mathrm{TS}}\right)$$

$$+\mathit{2}{E}_{S}{E}_{R}\mathrm{cos}\left(\Delta \omega t+{\varphi}_{R}-{\varphi}_{S}\right)$$

$$+\mathit{2}{E}_{R}{E}_{P}\mathrm{cos}\left(\Delta \omega t+{\psi}_{\mathit{1}}-{\psi}_{\mathit{3}}{\omega}_{\mathit{0}}t\right)$$

$$+\mathit{2}{E}_{R}{E}_{{P}_{-}R}\mathrm{cos}\left(\Delta \omega t+{\psi}_{\mathit{2}}-\mathit{2}{\psi}_{\mathit{3}}{\omega}_{\mathit{0}}t\right)$$

$$+\mathit{2}{E}_{T-S}{E}_{R}\mathrm{cos}\left(\Delta \omega t+{\varphi}_{R}-{\varphi}_{\mathrm{TS}}\right)$$

$$+\mathit{2}{E}_{S}{E}_{R}\mathrm{cos}\left(\Delta \omega t+{\varphi}_{R}-{\varphi}_{S}\right)$$

$$+\mathit{2}{E}_{R}{E}_{P}\mathrm{cos}\left(\Delta \omega t+{\psi}_{\mathit{1}}-{\psi}_{\mathit{3}}{\omega}_{\mathit{0}}t\right)$$

$$+\mathit{2}{E}_{R}{E}_{{P}_{-}R}\mathrm{cos}\left(\Delta \omega t+{\psi}_{\mathit{2}}-\mathit{2}{\psi}_{\mathit{3}}{\omega}_{\mathit{0}}t\right)$$

*When the modulation depth term in Eq. (17), i.e. ψ
_{3}=2sin(θ)A/λ, has a value equal to 1, the heterodyne intensity signal can be written as*

*$${I}_{\mathrm{het}}\left(t\right)={E}_{R}^{\mathit{2}}...................................................................................................................................\mathrm{DC}$$*

$$+\mathit{2}{E}_{S}{E}_{R}\mathrm{cos}\left(\Delta \omega t+{\varphi}_{R}-{\varphi}_{S}\right)+\mathit{2}{E}_{T-S}^{\text{'}\mathit{0}{\omega}_{\mathit{0}}}{E}_{S}\mathrm{cos}\left(\Delta \omega t+{\varphi}_{R}-{\varphi}_{\mathrm{TS}}\right).......................\Delta \omega t$$

$$+\mathit{2}{E}_{R}{E}_{P}\mathrm{cos}\left(\Delta \omega t+{\psi}_{\mathit{1}}-\mathit{1}{\omega}_{\mathit{0}}t\right)$$

$$+\mathit{2}{E}_{T-S}^{\text{'}\mathit{1}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left(\mathit{1}{\omega}_{\mathit{0}}\right)\mathrm{cos}\left(\Delta \omega t+{\varphi}_{R}-{\varphi}_{\mathrm{TS}}\right)............................................(\Delta \omega \pm \mathit{1}{\omega}_{\mathit{0}})t$$

$$+\mathit{2}{E}_{R}{E}_{{P}_{-}R}\mathrm{cos}\left(\Delta \omega t+{\psi}_{\mathit{2}}-\mathit{2}{\omega}_{\mathit{0}}t\right)$$

$$+\mathit{2}{E}_{T-S}^{\text{'}\mathit{2}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left(\mathit{2}{\omega}_{\mathit{0}}\right)\mathrm{cos}\left(\Delta \omega t+{\varphi}_{R}-{\varphi}_{\mathrm{TS}}\right)............................................(\Delta \omega \pm \mathit{2}{\omega}_{\mathit{0}})t$$

$$+\mathit{2}{E}_{T-S}^{\text{'}\mathit{3}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left(\mathit{3}{\omega}_{\mathit{0}}\right)\mathrm{cos}\left(\Delta \omega t+{\varphi}_{R}-{\varphi}_{\mathrm{TS}}\right)............................................(\Delta \omega \pm \mathit{3}{\omega}_{\mathit{0}})t$$

$$+\mathit{2}{E}_{T-S}^{\text{'}\mathit{4}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left(\mathit{4}{\omega}_{\mathit{0}}\right)\mathrm{cos}\left(\Delta \omega t+{\varphi}_{R}-{\varphi}_{\mathrm{TS}}\right)............................................(\Delta \omega \pm \mathit{4}{\omega}_{\mathit{0}})t$$

$$+\mathit{2}{E}_{S}{E}_{R}\mathrm{cos}\left(\Delta \omega t+{\varphi}_{R}-{\varphi}_{S}\right)+\mathit{2}{E}_{T-S}^{\text{'}\mathit{0}{\omega}_{\mathit{0}}}{E}_{S}\mathrm{cos}\left(\Delta \omega t+{\varphi}_{R}-{\varphi}_{\mathrm{TS}}\right).......................\Delta \omega t$$

$$+\mathit{2}{E}_{R}{E}_{P}\mathrm{cos}\left(\Delta \omega t+{\psi}_{\mathit{1}}-\mathit{1}{\omega}_{\mathit{0}}t\right)$$

$$+\mathit{2}{E}_{T-S}^{\text{'}\mathit{1}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left(\mathit{1}{\omega}_{\mathit{0}}\right)\mathrm{cos}\left(\Delta \omega t+{\varphi}_{R}-{\varphi}_{\mathrm{TS}}\right)............................................(\Delta \omega \pm \mathit{1}{\omega}_{\mathit{0}})t$$

$$+\mathit{2}{E}_{R}{E}_{{P}_{-}R}\mathrm{cos}\left(\Delta \omega t+{\psi}_{\mathit{2}}-\mathit{2}{\omega}_{\mathit{0}}t\right)$$

$$+\mathit{2}{E}_{T-S}^{\text{'}\mathit{2}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left(\mathit{2}{\omega}_{\mathit{0}}\right)\mathrm{cos}\left(\Delta \omega t+{\varphi}_{R}-{\varphi}_{\mathrm{TS}}\right)............................................(\Delta \omega \pm \mathit{2}{\omega}_{\mathit{0}})t$$

$$+\mathit{2}{E}_{T-S}^{\text{'}\mathit{3}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left(\mathit{3}{\omega}_{\mathit{0}}\right)\mathrm{cos}\left(\Delta \omega t+{\varphi}_{R}-{\varphi}_{\mathrm{TS}}\right)............................................(\Delta \omega \pm \mathit{3}{\omega}_{\mathit{0}})t$$

$$+\mathit{2}{E}_{T-S}^{\text{'}\mathit{4}{\omega}_{\mathit{0}}}{E}_{R}\mathrm{cos}\left(\mathit{4}{\omega}_{\mathit{0}}\right)\mathrm{cos}\left(\Delta \omega t+{\varphi}_{R}-{\varphi}_{\mathrm{TS}}\right)............................................(\Delta \omega \pm \mathit{4}{\omega}_{\mathit{0}})t$$

*Note that in Eq. (20), ${E}_{T-S}^{\text{'}n{\omega}_{0}}$ is the Fourier component of the nω
_{0} ramp function used to modulate the AFM tip. In contrast to Eq.(14), it can be seen in Eq. (20), that a pure near-field intensity signal can theoretically be obtained at radian frequency orders greater than 3 since the background noise term corresponding to the scattered tip field reflected from the sample surface is eliminated. Therefore, the use of a ramp modulation function appears to offer the potential for enhancing the S/B ratio in near-field optical measurement applications; particularly when using short illuminating wavelengths, i.e. λ=A according to ψ
_{3}=2sin(θ)A/λ.*

*It should be also noted that operating with a ramp function also eliminates the topography artifact due to the Z-dependant phase modulation when lock-in detection is set at radian frequency orders greater than 3.*

*8. Conclusions*

*This study has developed a robust analytical interference-based model to investigate the detection signals obtained in the reflective-type A-SNOM technique at various harmonics of the AFM tip vibration frequency. The validity of the proposed model has been confirmed by comparing the analytical results obtained for the intensity and phase of the heterodyne A-SNOM detection signal with the experimental results presented in [7–8, 13–14]. It has been shown that the analytical model provides satisfactory explanations for the image artifacts and errors observed in typical A-SNOM applications. Furthermore, it has been confirmed that the S/B ratio obtained in the visible region using the heterodyne detection technique is far higher than that generated in the homodyne technique.*

*In general, the results have shown that the heterodyne S/B ratio can be improved by increasing the wavelength of the incident electric field and reducing its incident angle. Furthermore, the analytical model has suggested two potential techniques for obtaining further dramatic improvements in the S/B ratio, namely (1) setting the modulation depth to specific values in the range ψ
_{3} > 1 such that the Bessel function of the first kind J_{n}(ψ
_{3}) has a zero point and therefore causes the contribution of the background noise to the detection signal to disappear; and (2) replacing the conventional sinusoidal function used to modulate the AFM tip with a ramp function and increasing the order of the radian frequency used for detection purposes.*

*Acknowledgments*

*The authors gratefully acknowledge the financial support provided to this study by the National Science Council of Taiwan under Grant No. NSC 95-2221-E-006-049-MY3. C. H. Chuang was supported by a fellowship within Postdoc-Program of the National Science Council of Taiwan under Grant No. NSC 97-2811-E-006-021.*

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