1. Introduction

For many years, the resolution of optical microscopy was limited to the order of approximately 1/2λ as a result of the far-field diffraction effect. Although the concept of near-field microscopy was first proposed by Synge as early as 1928 [1], it only became a reality with the advent of scanning probe techniques such as scanning tunneling microscopy (STM) [2] and atomic force microscopy (AFM) [3] in the 1980’s. Aperture scanning near-field optical microscopy (SNOM) was first introduced by Pohl et al. in 1984 [4]. In SNOM, a metallic aperture is used to confine the near-field light emanating from or entering the probe tip. However, the resolution is limited to approximately 50 nm since the tapered glass fiber tip causes a waveguide cut-off effect [5]. Accordingly, an alternative SNOM configuration was proposed in which the optical fiber was replaced with small scatter, yielding an enhanced resolution of approximately 10 nm [6–8] depending on the tip diameter. In this configuration, the incident light illuminates the small scatter and induces an enhanced electric field between the tip and the sample whose magnitude depends on the dipole effect. Measuring the near-field interaction electric field is the operating principle. This device is conventionally referred to as the apertureless scanning near-field optical microscope (A-SNOM) or the scattering-type scanning near-field optical microscope (s-SNOM). However, in A-SNOM, the near-field electric field is seriously affected by a background interference electric field and therefore it is necessary to develop techniques for eliminating the background-scattering noise from the detected signal in order to improve the imaging resolution.

The detected signals in A-SNOM are highly complex, and are therefore most commonly acquired using a lock-in amplifier [9–12]. In analyzing the detected signal, most previous studies applied a simple mathematic approach to derive the near-field signal and background noise, respectively. For example, in [9] and [11], the authors simply assumed that the lock-in harmonic electric fields were given by the sum of the near-field electric field and the background electric field, respectively. As a result of this crude approximation, the authors were able only to derive a simple mathematical formula for the intensity signal. The authors reasoned that the un-modulated background electric field would disappear at high-order frequencies. In previous A-SNOM investigations [9, 11, 13], researchers generally claimed that the resolution is independent of the wavelength of the incident light and is determined primarily by the tip radius. By contrast, the current study examines the hypothesis that a longer wavelength improves the signal contrast for a constant order of modulation frequency. Recently, some studies has analyzed the complicated background noise in transmission or total reflection type of A-SNOM [14–16] and got the result that the background signal decays with a decreasing value of the Bessel function J _n. This conclusion also could be found and confirmed in our study. However, they didn’t analyze the signal by a lock-in detection technique in different harmonics of tip vibration frequency, nor the signal contrast relative to wavelength, incident angle and tip vibration amplitude.

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.

2. Analytical model of A-SNOM

Figure 1 presents a schematic illustration of the A-SNOM near-field region. Note that to simplify the analytical model used in this study, an assumption is made that both the incident light and the detection light pass through the same objective lens. As shown, the incident angle of the electric field $\stackrel{\rightharpoonup }{\mathit{E}}$ _i is denoted by θ. Three major electromagnetic wave sources exist in the near-field region and are subsequently detected in the far-field region, namely the electromagnetic interaction signal $\stackrel{\rightharpoonup }{\mathit{E}}$ ̄_T-S between the AFM tip and the sample, the scattering electric field $\stackrel{\rightharpoonup }{\mathit{E}}$ _Tip from the AFM tip, and the scattering electric field $\stackrel{\rightharpoonup }{\mathit{E}}$ _Sample from the sample. Of these three signals, the most important, yet the weakest, is the interaction signal between the AFM tip and the sample. This interaction (or enhancement) effect can be described using the general model of quasi-electrostatic theory [9–11], i.e.

where $\stackrel{\rightharpoonup }{\mathit{E}}$ _T-S is the interaction electric field, α _eff is the effective polarizability, E _i is the amplitude of the incident electric field, and ω and ϕ _TS are the frequency and initial phase of the interaction electric field, respectively. Of these parameters, α _eff is a critically important factor in A-SNOM since it contains all the necessaries to predict the relative contrasts observable in A-SNOM. The value of α _eff is determined by the tip radius, the dielectric constants of the AFM tip and the sample, respectively, and the distance between them.

During the imaging process, the AFM drives the probe with a vertical cosine displacement around a mean position Z ₀, with an amplitude A and radian frequency of ω ₀, respectively. Therefore, the position of the probe at any time t can be written as

An assumption is made that the AFM tip does not perturb the near-field region. As a result, the scattering electric field from the tip (See Fig. 1) can be expressed as

where E _T and ϕ _T are the amplitude and initial phase of the scattering electric field, respectively, ω is the radian frequency, and K is the wave number of the incident light (given by 2π/λ). Finally, the term e ^{i(2Ksin(θ)Z(t))} is the phase vibration caused by the vertical dither of the probe, and has a factor K sin(θ).

The third electric field in the near-field region is the scattering light from the sample surface. Since this light is not modulated by the AFM tip, it can be described simply as

where E _S and ϕ _s are the amplitude and initial phase of the scattering light from the sample surface.

 

Fig. 1. Model of A-SNOM near-field region.

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3. Modulation signals of A-SNOM

As described above, the incident electric field E⇀_i with an incident angle θ generates three major electric fields in the near-field and far-field detection regions, namely the electromagnetic interaction electric field between the AFM tip and the sample, the scattering electric field from the AFM tip, and the scattering electric field from the sample. Thus, the total electric field coupled into the objective lens is equal to the sum of the three individual electric fields, i.e.

Therefore, the intensity of the detected signal, I(t), can be expressed from Eq. (5) as

Applying the Fourier-Bessel series expansion and assuming that Ψ ₁=ϕ _T-ϕ _S+2Ksin(θ)Z₀, Ψ ₂=ϕ _T-ϕ _TS+2Ksin(θ)Z₀ and Ψ ₃=2Ksin(θ)A, Eq. (6) can be rewritten as [12,17]

where J _n(ψ ₃) is a n-th order Bessel function of the first kind at ψ ₃, and ψ ₃ can be defined as the phase modulation depth. From Eq. (7), it can be seen that the electric field intensities E _T E _S and E _T-S E _T at the higher-order harmonics of the probe vibration frequency have coefficient of higher-order Bessel function, J _n(ψ ₃). As ψ ₃ approaches zero, these higher-order Bessel functions rapidly decay.

Since the amplitude of the interaction electric field is nonlinear, it is assumed that E _T-S can be written as the sum of the components oscillating at different harmonics of the AFM probe modulation frequency [14], i.e.

The series coefficient $E_{T-S}^{n{\omega }_0}$ can be obtained from the Fourier components of $E_i{\alpha }_{\mathrm{eff}}.$ Substituting Eq. (8) into Eq. (7), the intensity I (t) becomes

Clearly this formulation describing the detection signal I (t) is highly complicated. Furthermore, the electric field of interest, i.e. the interaction electric field $\stackrel{\rightharpoonup }{\mathit{E}}$ _T-S, yields the weakest signal. Therefore, it is difficult to distinguish among the different signals in the near-field region. To overcome this problem, the interaction signal is generally extracted using a lock-in detection technique. Eq. (9) can be rearranged in order of modulation radian frequency nω ₀ using the formula

I (t) can then be decomposed into the following major terms:

Although Eq. (11) still appears complicated, it provides some indications as to how to deal with the three different electric fields, $\stackrel{\rightharpoonup }{\mathit{E}}$ _T-S, $\stackrel{\rightharpoonup }{\mathit{E}}$ _Tip, and $\stackrel{\rightharpoonup }{\mathit{E}}$ _Sample, within the detected signal. Firstly, the absolute interaction electric field, $\stackrel{\rightharpoonup }{\mathit{E}}$ _T-S, cannot be obtained easily from the lock-in detection because those different order modulation radian frequencies mix with three electric fields. Secondly, the intensity of the background electric field, E _T E _S, has a coefficient of J _n(ψ ₃), and if the higher-order Bessel function decays more rapidly than the same order Fourier component E ^nω _T-S, the lock-in detection signal will have an enhanced contrast at higher-order harmonic radian frequencies. (Note that it is for this reason that researchers generally adopt higher-order harmonics to enhance the S/N ratio in A-SNOM [9–12]). Thirdly, it is known that the phase modulation depth, ψ ₃=2Ksin(θ)A, and the signal contrast between the different samples can be analyzed and optimized by adjusting the phase modulation depth ψ ₃, as discussed in the following section.

4. Effect of phase modulation depth ψ₃ in A-SNOM lock-in detection

Equation (11) gives the lock-in detection signal from DC to the 4^th-order harmonic radian frequency. This section investigates the effect on the signal contrast of varying the phase modulation depth ψ ₃ at different orders of harmonic radian frequency. In conducting this investigation, it is necessary to make a number of assumptions and to assign certain values. Firstly, the phase differences of the three electric fields are specified as ϕ _TS-ϕ _S=0, ψ ₁=π/4 and ψ ₂=π/4, respectively. It is in order to ensure that all the signal sources survive in lock-in detection at any order of the harmonic radian frequency. Secondly, it is assumed that there are two different measurement samples S1 and S2 with the DC interaction amplitudes $E_{T-S}^{0{\omega }_0}$ are 1.2 and 1, respectively. The n-th order amplitude is given ideally as $E_{T-S}^{n{\omega }_0}=E_{T-S}^{0{\omega }_0}⁄3n$ that approximates to the result of n-th order Fourier components with distance modulation Z(t)=0,…, 0.5a in [9] where a is radius of a polarizable sphere. As a result, the series $\sum _{n=1}^{\infty }{E}_{T-S}^{n{\omega }_0}{E}_{T-S}^{\left(n+1\right){\omega }_0^{*}}$ converge to zero rapidly as the strength of the successive amplitude harmonics decreases. Thirdly, E _S and E _T are assigned amplitudes of 20 and 15, respectively, to satisfy the homodyne field amplification factors $\sum _{n=1}^{\infty }{E}_{T-S}^{n{\omega }_0}{E}_{T-S}^{\left(n+2\right){\omega }_0^{*}}$ and $\sum _{n=1}^{\infty }{E}_{T-S}^{n{\omega }_0}{E}_{T-S}^{\left(n+3\right){\omega }_0^{*}}$ described in [9]. I _DC is the DC term of the signal intensity in Eq. (11), and $g_1=I_{\mathrm{DC}}⁄I_{1{\omega }_0}\approx 100$ and $g_2=I_{\mathrm{DC}}⁄I_{2{\omega }_0}\approx 1000$ are the first- and the second- order term of the signal intensity in Eq. (11). The signal contrast between sample S1 and sample S2 in different order term of the signal intensity in Eq. (11) can be defined as

where $I_{n{\omega }_0(S1,S2)}$ are the n-order term of the signal intensity in Eq. (11) for samples S1 and S2. Applying these assumptions and values, Eq. (12) is then used to plot Fig. 2, which illustrates the relationship between the phase modulation depth ψ ₃ and the contrast ratio |S1/S2| at different harmonic orders. The objective is to achieve the correct contrast of the electric field intensity |S1/S2|_n=1.2 of two different material samples S1 and S2. It is found in Fig. 2 that with $\underset{{\psi }_3\to 0}{lim}{J}_n\left({\psi }_3\right)=0,n=1,2,3\dots ,$ all the background signal intensities in the same order modulation frequency approach to zero, thus the signal contrast in Eq. (12) becomes ${\mid S1⁄S2\mid }_n=\mid I_{n{\omega }_0\left(S1\right)}⁄I_{n{\omega }_0\left(S2\right)}\mid \cong \mid E_{T-S\left(S1\right)}^{n{\omega }_0}⁄E_{T-S\left(S2\right)}^{n{\omega }_0}\mid$. Therefore, the intensity contrast can be described by the field amplitude ratio in this case.

An observation of Fig. 2 reveals a number of interesting findings. For example, it can be seen that irrespective of the value of ψ ₃, the contrast of I _DC remains constant at I _DC≈1. Furthermore, as the phase modulation depth ψ ₃ approaches zero, the contrasts of $I_{{1\omega }_0}$, $I_{{2\omega }_0}$, $I_{{3\omega }_0}$, and all converge to a value of 1.2 and the high-order background noise electric fields are zero. Also, it can be seen that all of the contrast values approach a value of 1 when ψ ₃ is increased. As illustrated in Fig. 3(a), this result arises because the background signal intensity dominates the intensity of the detected signal at higher values of the phase modulation depth. Comparing the contrast profiles of $I_{{1\omega }_0}$, $I_{{2\omega }_0}$, $I_{{3\omega }_0}$, and $I_{{4\omega }_0}$, it is apparent that a higher-order frequency lock-in signal yields a more stable contrast with a value closer to the required value of 1.2. Finally, it can be seen that $I_{{1\omega }_0}$ and $I_{{2\omega }_0}$ have discontinuous contrast profiles. While this may initially be thought of as advantageous in terms of an increased contrast, Figs. 3(a) and 3(b) indicate that these discontinuities correspond to points of zero intensity. As a result, it is difficult to measure and distinguish between the different samples by extracting $I_{{1\omega }_0}$ and $I_{{2\omega }_0}$ because the contrast is unstable and incorrect. At this singularity point, it can be explained that the interaction electric field amplified by a lock-in technique and the background electric field effectively have destruction interference. This situation can be confirmed by the fore experimental results in [9]. However, the divergence in signal contrast figure would not be observed if ψ ₁=ψ ₂=5π/4 is assumed. In this case, the interaction electric field amplified by a lock-in technique and background electric field will have constructive interference.

 

Fig. 2. Variation of signal contrast |S1/S2 in different order of modulated frequency with phase modulation depth ψ ₃ in A-SNOM lock-in detection.

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Fig. 3. (a). Variation of signal intensity |S1| in different order of modulated frequency with phase modulation depth ψ3 in A-SNOM lock-in detection.

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Fig. 3. (b). Partial enlargement of Fig. 3. (a).

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From the discussions above, it is seen that a higher-order lock-in detection frequency and a lower phase modulation depth ψ ₃ result in a higher and more stable signal contrast. However, a higher-order lock-in detection radian frequency also results in the weaker signal intensity with the higher-order Fourier component decay, i.e. $E_{T-S}^{n{\omega }_0}=E_{T-S}^{0{\omega }_0}/3n$. Therefore, it is necessary to analyze the individual components of ψ ₃=2Ksin(θ)A in order to optimize the near-field signal contrast. Accordingly, the following subsections examine the respective effects of the wavelength and incident angle of the incident light, and the amplitude of the tip vibration on the signal contrast and signal intensity. (Note that the discussions are based on the same set of assumptions and assigned values as those considered above).

4.1 Wavelength of incident electric field

In classical idea of microscopy, the resolution on is R≥1/2λ. In optical systems, the longer wavelength results in the poorer resolution, and thus electron microscopes provide a far better resolution. In previous A-SNOM investigations, researchers have claimed that the resolution is independent of the wavelength of the incident light [11,14] and have suggested that the key factors determining the resolution are actually the aperture and tip size in SNOM and A-SNOM, respectively. In [9], the qualitative observation that using the longer wavelength improves the signal contrast for a constant order of modulation radian frequency is found. In our study, a quantitative model is presented and given for more detail discussion in wavelength influence. To verify this claim, this study examines the effect of the wavelength on the contrast and intensity of the various components in the A-SNOM lock-in detected signal. In performing the analysis, the effect of the wavelength on the tip and sample resonance [5] is ignored. The relationships ψ ₃=2Ksin(θ)A and K=2π/λ are substituted into Eq. (11) in order to get the relation between wavelength and signal contrast. Note that in accordance with the experimental results presented in [9], the incident angle θ is specified as π/4 and the amplitude A of the tip vibration is set to 60 nm. Therefore, Fig. 4 plots the results obtained for the variation of the signal contrast (S1/S2) with the wavelength of the incident light.

Although the effect of the wavelength on the tip and sample resonance is ignored, Fig. 4 contains several features of note from a signal analysis perspective. For example, it can be seen that the contrast of I _DC is always equal to 1 irrespective of the wavelength. Furthermore, in the short wavelength region, there is no contrast between S1 and S2. However, in the long wavelength region, the high-order contrasts in $I_{{2\omega }_0}$, $I_{{3\omega }_0}$, and $I_{{4\omega }_0}$ approach the desired value of 1.2. It is observed that the contrast profiles of $I_{{1\omega }_0}$ and $I_{2{\omega }_0}$ have points of discontinuity because, as shown in Fig. 5, their intensities fall to zero as a result of destructive interference between the interaction electric field and the background electric field. Finally, it is seen that the contrast profiles of $I_{3{\omega }_0}$ and $I_{4{\omega }_0}$ have a smooth appearances because the background electric field decays rapidly with increasing wavelength at a high-order modulation frequency, as shown in Fig. 5. A-SNOM in the visible wavelength can be designed if the higher order mode term of modulation frequency can be extracted by a lock-in technique. However, in reality the higher order modulated frequency signal will induce lower signal intensity. From this reason, by adopting the heterodyne technique [10] in the visible wavelength is recommended on A-SNOM.

 

Fig. 4. Variation of signal contrast |S1/S2| in different order of modulated frequency with wavelength of incident electric field in A-SNOM lock-in detection.

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Fig. 5. Variation of signal intensity |S1| in different order of modulated frequency with wavelength of incident electric field in A-SNOM lock-in detection.

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4.2 Incident angle of incident electric field

Many studies have investigated the effect of the incident angle of the incident electric field on the enhancement of the tip and sample electric fields [9–11, 18]. Irrespective of the analytical model employed, e.g. quasi-electrostatic, Finite-Difference Time-Domain Method (FDTD), Finite Element Method (FEM), and so forth, the results indicated that provided the polarization direction of the incident electric field was perpendicular to the sample surface, a smaller incident angle induced a stronger signal enhancement. In this study, the effect of the incident angle is examined from a lock-in signal analysis perspective to determine whether this holds true. In performing the analysis, the wavelength of the incident light and the amplitude of the tip vibration are specified as 10 µm and 60 nm (as in [9]), respectively. Equation (11) is then used to derive the relationship between the signal contrast and the incident angle θ, as shown in Fig. 6.

Figure 6 shows that irrespective of the incident angle, the contrast of I _DC remains constant and has a value of approximately 1. Furthermore, it can be seen that in the small incident angle region, the $I_{2{\omega }_0}$, $I_{3{\omega }_0}$, and $I_{4{\omega }_0}$ contrasts are close to a value of 1.2. Finally, in the high incident angle region, the contrasts of $I_{2{\omega }_0}$, $I_{3{\omega }_0}$ and $I_{4{\omega }_0}$ re still close to 1.2, but that of $I_{1{\omega }_0}$ become discontinuously. Figure 7 illustrates the variation of the intensity profile in |S1| with the incident angle for I _DC $I_{1{\omega }_0}$, $I_{2{\omega }_0}$, $I_{3{\omega }_0}$, and $I_{4{\omega }_0}$, and shows that the intensities $I_{2{\omega }_0}$, $I_{3{\omega }_0}$ remain stable as the incident angle is increased.

From the results above, it is clear that the angle of the incident light is not as influential as the wavelength in the lock-in detection technique, i.e. since $I_{2{\omega }_0}$, $I_{3{\omega }_0}$ and $I_{4{\omega }_0}$ have very similar signal contrasts. However, the incident angle is known to have a key effect on the tip enhancement in A-SNOM [9]. Overall, combining the results presented here with those reported in the literature [9–11, 17], it can be inferred that a smaller incident angle provides both a better tip enhancement and an improved intensity contrast.

 

Fig. 6. Variation of signal contrast |S1/S2| in different order of modulated frequency with incident angle of $\stackrel{\rightharpoonup }{\mathit{E}}$ _i in A-SNOM lock-in detection.

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Fig. 7. Variation of signal intensity |S1| in different order of modulated frequency with incident angle of $\stackrel{\rightharpoonup }{\mathit{E}}$ _i in A-SNOM lock-in detection.

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4.3 Tip vibration amplitude

The final factor to be considered in ψ ₃ is the tip vibration amplitude. Although from a lock-in detection perspective, signal contrast and intensity figures similar to those presented in the sections above can again be derived, doing so requires a departure from the assumption of an n-th order electric field amplitude of $E_{T-S}^{n{\omega }_0}=E_{T-S}^{0{\omega }_0}⁄3n$. Accordingly, in optimizing the tip vibration amplitude, Fourier transform analysis of $\stackrel{\rightharpoonup }{\mathit{E}}$ _T-S should first be performed as in [9] since it is known that a smaller tip vibration amplitude A can yield an improved contrast as illustrated in Fig. 2 according to ψ ₃=2Ksin(θ)A. However, a smaller value of A also rapidly decays the high-order Fourier components of the amplitude of the interaction field in Eq. (8). If the Fourier components decay more rapidly than the Bessel function or the amplitude of the high-order electric field $E_{T-S}^{n{\omega }_0}$ is less than that of the background noise, ASNOM measurement cannot be performed.

5. Relative merits of tip scan versus sample stage scan

Although the phase differences ψ ₁=ϕ _T-ϕ _S+2K(θ)Z ₀ and ψ ₂=ϕ _T-ϕ _TS+2K(θ)Z ₀ can be arbitrarily adjusted during the analysis by changing the positions of the incident light spot and the tip vibration mean position Z ₀, respectively, it is necessary to consider the roles of ψ ₁ and ψ ₂ in scanning type. In the sample stage scanning operation, this signal intensity variation does not arise because the phase differences are fixed during scanning. However, commercial AFMs generally perform a tip scan since the cost is lower and the control system is more straightforward. It’s not a problem in AFM measurement, but it is an important issue in A- SNOM measurement, because the A-SNOM is an electric field measurement microscope. Figure 8 illustrates the variation of the signal intensity |S1| with the phase difference with ϕ _TS-ϕ _S=00 and ψ ₂=ψ ₁ in the A-SNOM lock-in detection process. The intensity profiles of $I_{1{\omega }_0}$ and $I_{2{\omega }_0}$ in Fig. 8 are similar to the measurement results presented in Fig. 6 in [9]. Furthermore, the intensity $I_{2{\omega }_0}$, $I_{3{\omega }_0}$, and $I_{4{\omega }_0}$ are maximum when ψ ₂=ψ ₁=0° in Fig. 8. Therefore, it can help us how to choose the maximum signal intensity for improving ASNOM measurements. Figure 9 shows that the contrast of I _DC is approximately 1. Furthermore, the contrasts of $I_{2{\omega }_0}$, $I_{3{\omega }_0}$ and $I_{4{\omega }_0}$ are approximately 1.2 and are independent of the phase difference. Finally, it can be seen that the contrast of $I_{1{\omega }_0}$ is unstable during tip scanning.

In general, the results presented above imply that sample stage scanning provides a better imaging performance than tip scanning. Nevertheless, the signal intensity and contrast are not greatly variation at $I_{2{\omega }_0}$, $I_{3{\omega }_0}$, and $I_{4{\omega }_0}$ in Figs. 8 and Fig. 9, and hence it can be inferred that tip scanning provides a satisfactory low-cost and faster imaging alternative provided that a higher than 2nd order harmonic frequency and a small scan area is used. The difference between the tip scan and the stage scan had been discussed in 2004 with the experiment [19], the author found that it is possible to do imaging by tip scan and still acquire optical data-modulated, of course with the envelope of the local amplitude and phase of the exciting beam focus as above discussions.

If the illumination is highly focused in A-SNOM, the quasi-electrostatic theory is still practical because the near-field area is much smaller than focus point. However, the tip scan area needs to be limited in local incident electric field $\stackrel{\rightharpoonup }{E}$ _i as in Eq. (1). Therefore, the envelope of the local amplitude and phase of the exciting beam focus [19] should be considered in a larger scan area by using the tip scan.

 

Fig. 8. Variation of signal intensity |S1| in different order of modulated frequency with phase difference with ψ ₁=ψ ₂ in A-SNOM lock-in detection.

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Fig. 9. Variation of signal contrast |S1/S2| in different order of modulated frequency with phase difference with ψ ₁=ψ ₂ in A-SNOM lock-in detection.

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6. Conclusions and discussions

This study has presented a comprehensive modulation analysis of the detected signal in A-SNOM. To the best of the current authors’ knowledge, the study represents the first reported attempt to clarify the complicated physical phenomena of A-SNOM from the perspective of the lock-in detection signal. A mathematical model has been constructed to describe the interference among the electromagnetic interaction field between the AFM tip and the sample, the scattering electric field from the AFM tip, and the scattering electric field from the sample surface. The model has then been used to perform a systematic investigation into the respective effects of the phase modulation depth, the wavelength and angle of the incident light, and the tip vibration amplitude on the signal contrast and intensity. The results support the following major findings:

1. The background signal decays with a decreasing value of the Bessel function J _n(ψ ₃) [14–16].

2. A longer incident light wavelength improves the signal contrast for a constant order of modulation radian frequency.

3. A smaller incident angle yields an improved signal contrast and enhancement effect.

4. Sample stage scanning produces improved imaging results. However, tip scan provides a reasonable lower-cost and faster solution provided that the scan operation is performed in a smaller scan area using a higher 2^nd order harmonic signal.

As comparison with the experimental results in [9], the authors adopt higher order harmonic frequency to improve signal contrast, and it consists with our major finding. Besides, the intensity profiles of $I_{1{\omega }_0}$ and $I_{2{\omega }_0}$ in Fig. 8 also consist with the measurement results presented in Fig. 6 in [9].

In conclusion, the analytical results presented in this study provide many insights into the complex phenomena of A-SNOM and indicate potential techniques for improving the signal resolution of A-SNOM measurement systems.