Analysis on heterodyne signals in apertureless scanning near-field optical microscopy

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. 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. 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.


Introduction
Apertureless SNOM (A-SNOM) adopts 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 [1][2].However, the A-SNOM detection signal is contaminated by a complex interference between the background electric field and the nearfield 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.
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 [3] 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 [4][5] since this technique is invariably the method of choice when measuring the surface properties of materials at the nanoscale.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 [3].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.

Electric fields and signals in heterodyne A-SNOM
Figure 1 presents a schematic illustration of a Mach-Zehnder interferometer-type A-SNOM.As shown, a radian frequency shift Δω 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 1- where E R is the amplitude of the reference beam and ω and Φ R are the radian frequency and initial phase of the incident light, respectively.As shown in Fig. 1, the measurement beam, i E   , 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 E   , strikes the sample with an angle θ and produces four discrete electromagnetic waves.
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 [4][5], the interaction (or tip enhancement) electric field can be formulated as where α eff is the effective polarizability.Ei is the amplitude of the incident electric field, and ω and Φ TS are the radian frequency and initial phase of the interaction light, respectively.Note that α eff is a highly important parameter since it contains everything necessary to predict the relative constants observable in the A-SNOM technique.
In the present analysis, it is assumed that the AFM tip does not perturb the near-field region as the model in [6].Consequently, the electric field scattered from the probe can be formulated as where E P and Φ P are the amplitude and initial phase of the scattering electric field, ω is the radian frequency of the incident light, and K is the wave number of the incident light and is given by 2π/λ.In addition, e i(2Ksin(θ)Z(t)) represents the phase vibration caused by the probe's vertical dither.
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 2Ksin(θ)Z(t).Therefore, the reflected probe scattering field can be formulated as 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 S are the amplitude and initial phase of the scattering light, respectively.1- 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 Applying the Fourier Bessel series expansion and introducing the phase differences ψ 1 =Φ R -Φ P -2Ksin(θ)Z 0 and ψ 2 =Φ R -Φ P -4Ksin(θ)Z 0 , and the phase modulation depth ψ 3 = 2Ksin(θ)A, I het (t) can be rewritten in order of the modulation radian frequency, i.e.Δω+nω 0 , the heterodyne intensity signal can be expressed in terms of the following components:

Potential methods for improving S/B ratio of A-SNOM detection signal
The S/B ratio can be defined as where the signal intensity term is given by the sum of all the terms relating to the near-field interaction electric field 0 ω n TS E  in Eq.( 9), and the noise intensity term is the sum of the background noise terms.

Correlation between S/B ratio and modulation depth parameter ψ 3 for ψ 3 >1
Figure 3 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. ( 9).In addition, Fig. 3 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.

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 ( 1) mT t m T    .m is a positive integer.Substituting Z(t) into the heterodyne intensity formulation I het (t).When the modulation depth term, i.e. ψ 3 = 2sin(θ)A/λ, has a value equal to 1, the heterodyne intensity signal can be written as is the Fourier component of the nω 0 ramp function used to modulate the AFM tip.In contrast to Eq.( 9), it can be seen in Eq. ( 12), 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.

Conclusion
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.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.

Fig. 3 .
Fig. 3. Simulation results obtained for variation of heterodyne S/B ratio with modulation depth ψ 3 in range ψ 3 >1 at various values of radian frequency order, I Δω+nω0 .