1. Introduction

The advent of the near-field optical microscope successfully showed, in the early eighties, the possibility of beating the classical diffraction limit of conventional imaging systems [1]. This new kind of microscopy is aimed at acquiring information about a nanostructure through the employment of a probe, located in the vicinity of the sample, to detect a non-propagating field through its conversion into an homogeneous one. The lateral resolution (or resolving power) of the Scanning Near-Field Optical Microscope (SNOM) is closely related to the size of the probe-end. Nevertheless, if it is too small, the detected signal would be buried in the noise [2, 3, 4]. Consequently, to increase the signal-to-noise ratio, a lock-in detection at the harmonics of the frequency of the vertical vibration of the probe must be used. The amplitude of the vertical vibration is a critical parameter as illustrated theoretically [5] and experimentally [6]. Walford et al have demonstrated that a large modulation amplitude should be used to produce a realistic image of the field [5]. On the other hand, a small modulation amplitude should be used if information on specific small details of the object are of interest. Moreover, to reproduce the experimental data with numerical computations, the vibration of the probe must be also included in the model of an a-SNOM [7, 8]. For a given amplitude of vibration, the data obtained through the lock-in detection are the Fourier harmonics of the signal diffracted by the probe. Therefore, this signal must be reconstructed from the lock-in data before physical interpretation.

In this study, we propose a general method for the reconstruction of this “ideal” signal as a function of the probe vertical position. In Sec. 2 we describe a general method for the demodulation of the lock-in detected signal. Then, in Sec. 3, the proposed method is applied to simulated experimental data, in the case of approach curves. We present our main conclusions in Sec. 4.

2. The lock-in demodulation

In the following paragraphs we expose the principles of the method that will be employed for the demodulation of the simulation of the detected signals in Sec. 3.

2.1. The method

The purpose of the experiment is to measure the variations of the detected signal along an approach curve. As the signal-to-noise ratio is too weak, a lock-in amplifier is necessary to get significative data. The lock-in amplifier is tuned on the frequency of the vertical vibration of the probe and gives therefore the harmonics of the signal. A physical interpretation cannot be achieved directly from these raw lock-in data without a comparison with the “ideal signal” that would be measured with a non-vibrating probe. In the following, we will introduce the notations used for the various parameters and signals involved in this study.

The ideal signal, as shown in Fig. 1(a), corresponds to the data that may be obtained without any probe vibration and without lock-in. Therefore it can be considered as a function of the mean position of the probe z ₀: s_i(z ₀). The corresponding experimental case is shown in Fig. 1(b), where the detected signal is obtained experimentally through the lock-in amplifier. It is in fact the harmonics H_n of the signal measured along a probe vibration period. In order to extract the physical information from the recorded data, it is necessary to reconstruct s_i from these harmonics H_n through a demodulation scheme [5].

Let us consider the experimental configuration shown in Fig 1(b). We will investigate the signal s_i(z _tip(t)) that results from the interaction between the illuminating field and the involved materials, including those of the nanostructures and the time vibrating probe. If the probe vibrates at a frequency f, the detected signal varies periodically in time and, if the vibration is harmonic, the vertical position of the probe can be expressed as z _tip(t) = z ₀+A(z ₀)(cos(2πft)), where A(z ₀) is the amplitude of vibration of the probe about its mean position z ₀, A(z ₀) = A if z ₀ > A and A(z ₀) = z ₀ if z ₀ < A. With reference to Fig. 1(b), the position of the probe varies within the range [z ₀-A, z ₀+A] if z ₀ > A and [0, 2z ₀] elsewhere. The last case corresponds to the tapping mode [6]. The harmonics H ₀, H ₁, …, H_N of the signal s_i(z _tip(t)) are provided by the lock-in detection and can be written as:

 

Fig. 1. (a) Schematic of the “ideal” ASNOM experimental setup without lock-in detection. The corresponding ideal signal s_i(z ₀) is the near-field data of interest. (b) Schematic of the ASNOM experimental setup with lock-in detection, used to increase the signal to noise ratio. The recorded data are Fourier harmonics of the near-field signal H ₁, …, H_N.

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The signal s_i(z _tip(t)) in Eq. (1) can be considered as an even function of time. Indeed it is dependent on the probe position but not on the fact that it is approached or removed from the surface of the sample. In the case of the approach curves it is convenient to consider z _tip as the variable of interest instead of time t. Therefore, the integral in Eq. (1) can be written as a function of the probe vertical position z _tip = z ₀+A(z ₀)cos(2πft):

where ζ= (z _tip-z ₀)/A(z ₀) and T_n(ζ) = cos(narccos(ζ)) is the Chebyshev’s polynomial of the first kind (ζ ∈ [-1,1]) [9]. Thus the ideal signal can be written:

From Eq. (4), it is clear that the detection of several harmonics of the signal would make possible its reconstruction. However, in an experimental situation, the number of harmonics is limited. Thus the reconstructed signal is given by:

The signal R_N reconstructed from the harmonics H ₀,…,H_N is a function of the amplitude of the vibration A and the mean position z ₀ of the probe. Also R_N can be expressed as a function of the normalized position of the probe ζ = (z _tip-z ₀)/A(z ₀). The relationship between s_i and R_N is therefore straightforwardly established as:

This equation enables to reconstruct the “ideal” signal for each position of the probe in the range [z ₀-A(z ₀),z ₀+A(z ₀)] but in the following, we focuss our attention on approach curves and therefore, all the computations will be done for ζ = 0.

In the next subsection, we illustrate the reconstruction in the case of a pure exponential decay.

2.2. The example of the pure exponential decay

To illustrate the foundations of our method, we will consider the case of the study of the evanescent decay D_p of a Fresnel wave, generated in total internal reflection of a laser beam, above a glass prism. For this purpose, the ideal signal should be the square modulus of an evanescent Fresnel field [10]:

To clarify the interpretation of our results, the underlying hypothesis is that the probe is passive. Nevertheless, the method previously described would be also valid for the case when the probe perturbs the near-field. The measured data is the signal diffracted by the probe and recorded by the detector. In this numerical simulation, we have chosen E_t = 2.4, D_p = 0.1874μm in order to be close to the experimental conditions described in [6]. The Fourier harmonics given by the lock-in can be calculated analytically:

where I_n(x) is the modified Bessel function of the first kind and A(z ₀) = A is the amplitude of vibration of the probe if z ₀ > A. For z ₀ < A, in tapping mode, A(z ₀) = z ₀. Figure 2 shows the influence of the amplitude of vibration A(z ₀) on the first harmonic |H ₁| comparatively to the ideal signal s_i(z ₀). For a small amplitude of vibration, the first harmonic of the signal is weak. When the probe approaches the sample, in the case of the experimental “tapping” mode, the amplitude of vibration is equal to z ₀. In this zone, the first harmonic |H ₁| decreases toward 0 therefore the variations of |H ₁| are far from the ideal signal (Eq. (7)).

 

Fig. 2. Influence of the amplitude of vibration A(z ₀) on the first harmonic |H ₁| and the ideal signal s_i(z ₀).

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In Fig. 3, we show the ideal signal s_i(z ₀), the modulus of the first harmonic |H ₁| that is directly obtained from the lock-in amplifier, the reconstructions R ₁(z ₀) and R ₂(z ₀) for A = 70nm (close to the experimental value [6]). In this case, only two harmonics are necessary to reconstruct the ideal signal s_i(z ₀) along the approach curve but the shape of the first harmonic |H ₁| differs strongly from the s_i(z ₀). This example shows that the reconstruction is necessary to get a signal close to the ideal one that should be obtained without the lock-in. It is clear that in a more general case of experiments, if we have no “ab initio” information on the detected signal, its interpretation may be hazardous before reconstruction.

 

Fig. 3. Reconstitution of the approach curve in the case of an evanescent field. The ideal s_i(z ₀) and the detected |H ₁| signals are plotted with plus and star symbols, respectively.

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In the Sec. 3, we illustrate the behavior of the lock-in detection by considering simulated near-field signals.

3. Reconstruction of the ASNOM signal from harmonics

This section is devoted to the application of the reconstruction scheme proposed in this work to the case of an approach of the probe above an interference pattern. The ideal signal s_i(z) results from the interferences between an evanescent wave with amplitude E_t and a background homogeneous wave E_g. The corresponding experiment has been described by Aubert et al [6]. In that paper, the detected signal is supposed to be independent of the characteristics of the probe, which also is supposed to be passive. Therefore it does not modify the interference fringes that would exist without its presence in the experiment. This assumption seems to be valid in this particular case; however, in the general case, the method of reconstruction of the ideal signal takes into account the interaction with the probe.

The ideal signal along the approach curve can be written as:

where D_p is the decay of the evanescent wave [6, 10] and ϕ the phase between the two waves. The purpose of the experiment is to measure this ideal signal. The lock-in furnishes harmonics of this signal deduced from the vertical vibration of the probe around z ₀. The amplitude of vibration is A(z ₀). In this example, we can calculate analytically the harmonics of the ideal signal as:

where δ _n0 is the Kronecker delta. Equation (11) shows that the two terms with exponential decay are modulated by modified Bessel functions of the first kind I_n. To illustrate the difference between the first harmonic data |H ₁| (given by Aubert et al [6]) and the ideal signal s_i(z ₀) we show in Fig. 4 the associated maps as functions of the approach distance z ₀ and the phase ϕ. From this map, it appears that it may be hazardous to do physical interpretation from the first harmonic as it can differ strongly from the ideal signal. At this stage, we may conclude that the discrepancy between the recorded signal |H ₁| and the ideal signal S_i may be mainly due to the amplitude of the vibration of the probe A = 70nm.

 

Fig. 4. Approach curves map on a period of the fringes: ϕ ∈ [0,180°]. The maps |H ₁| and S_i correspond respectively to the first harmonic and to the ideal signal.

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Consequently, to complete the discussion, we show in Figures 5, the influence of the amplitude of vibration on the discrepancy between the recorded signal |H ₁|, the reconstructed signal R ₁ and the ideal signal s_i. Figures 5(a-b) show the case of constructive interference (ϕ = 0) and Figs. 5(c-d) show the case of destructive interference (ϕ = π) for various amplitudes of vibration A ∈ [40nm, 200nm]. As expected, the influence of the amplitude of vibration is visible on the reconstructed signal even if A is much smaller than D_p. Moreover, the first harmonic amplitude differs strongly from the useful ideal signal. Therefore, the aim of the following is to reconstruct the ideal signal when the probe is above a bright fringe and a dark fringe.

 

Fig. 5. Ideal signal and simulation of the first harmonic amplitude given by the lock-in amplifier for various amplitudes of vibration A for ϕ = 0 (a) and ϕ = π (c). Ideal signal and simulation of the reconstructed signal from the first harmonics for various amplitudes of vibration A for ϕ= 0 (b) and ϕ=π(d).

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Fig. 6. Computed first harmonic of the detected signal |H ₁|, the reconstructed signals R ₀(z ₀), R ₁(z ₀) and R ₂(z ₀) and the ideal signal s_i(z ₀) above (a) a bright fringe (ϕ = 0) and (b) a dark fringe (ϕ = π).

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Finally, we show in Figs. 6(a) and 6(b) the simulated first harmonic |H ₁| (Eq. (11)), the reconstructed signals R ₀(z), R ₁(z), R ₂(z) from computed harmonics (Eq. (5)) and the ideal signal that should be measured without probe vibration along the approach curve s_i(z ₀) (Eq. (9)), for ϕ= 0 (bright fringe) and ϕ=π(dark fringe), respectively. The other parameters are A = 70 nm, D_p = 187.4 nm, E_t = 2.4 and E_g = 1.28. They are either deduced from the experimental setup [6] or from a non linear fit using evolutionary algorithm [11]. The field amplitude are in arbitrary units because they are deduced from the experimental curve, which is measured in arbitrary units. Again, a first visual inspection reveals that |H ₁| is far from the expected (or ideal) signal s_i(z ₀). For this example, we may conclude that the reconstruction of the signal may be considered as satisfactory by using only three harmonics H ₀, H ₁ and H ₂.

4. Conclusions

We have demonstrated that the SNOM recorded signal, if a lock-in detection is used, cannot be interpreted directly. In order to get information on the electromagnetic interaction between the light, the nanostructures and the probe along an approach curve, we have presented a method to reconstruct the “ideal” signal, from the harmonics of the lock-in detection. The use of the Chebyshev polynomials enables a versatile reconstruction that could permit to deduce a physically meaningful signal from the lock-in experimental data. In principle, there are no visible restrictions to extend the application of the proposed method to the case of scanned maps. In the future, the use of near-field intensity data obtained from heterodyne setups will open the possibility of using this method for a more general application than for approach curves.