Measurement of decay lengths of evanescent waves: the lock-in nonlinear filtering

We study the influence of the lock-in amplifier and probe vibration on the measurement of decay lengths of evanescent waves after reconstruction of the Scanning Near-field Optical Microscopy signal. Thanks to the reconstruction which gives a tomography-like 3D map of the detected signal, the vertical decay lengths can be measured directly, from only one lateral scan. A nonlinear fit is applied to recover the exponential decays in simulated signals and experimental data.


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Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT

Introduction
The characterization and imaging of nanostructures and fields are becoming a domain of interest in several fields such as physics, chemistry, biology and data storage [1]- [4]. The Scanning Near-Field Optical Microscopes (SNOM) allow the detection of the optical field at a few nanometres above nanostructures. The SNOM techniques are based on the detection of an electromagnetic signal scattered by a subwavelength-sized probe scanning the investigated sample. Due to the nanometric size of the probe end, the nanostructure and the confinement of the light, the signalto-noise obtained from such a configuration is low. In order to increase the signal-to-noise ratio and to decrease the useless spatially slow varying signals, a lock-in amplifier is often used with homodyne [5] or heterodyne [6,7] detection techniques. The detection is locked in the vertical vibration of the probe and the amplifier gives the first Fourier harmonics of the detected signal. Due to the filtering character of the lock-in, a direct and general physical discussion on the contrast in near-field optics, based on a given harmonic, is obviously difficult [8]- [10]. A reconstruction of the 'real' optical signal from all the available harmonics is necessary to discuss the contrast of the data [11,12]. Moreover, the reconstruction of such a signal gives a tomography-like map of the near-field along the vertical vibration of the probe. Therefore, from a unique xy-scan, an approximation of the exponential decays of evanescent waves can be computed and the decays can be deduced from an appropriate fit.
In this paper, we discuss the filtering effect of the homodyne detection mode and its influence on the measurement of the decay of evanescent waves. Experimental data are processed and it is shown that the signal detected above the nanostructures is a superposition of exponential decays.

The lock-in demodulation
The basic principle of the lock-in detection is the use of the periodic vibration of a metallic or dielectric probe, in the vicinity of an investigated sample, in order to record their diffraction patterns. Assuming an harmonic vibration of the probe, working in tapping mode, with amplitude A: z(t) = A(1 + cos(2πft)), f being the frequency of the probe vibration, the recorded signals through the lock-in detection are the first Fourier series amplitudes and phases of the harmonics H i of the field diffracted by the probe. The harmonics H 0 , H 1 , . . . , H N of the signal S(x, y, z) are provided through the lock-in amplifier and can be written as a function of the probe position in the scanning xy-plane [11,12] where T n (z/A − 1) = cos(n arccos(z/A − 1)) is the Chebyshev polynomial of the first kind (z ∈ [0, 2A]) [13]. The physical information on the near-field is present in all harmonics given by the lock-in. We investigate the filtering properties of the lock-in and a method to measure the characteristics of the near-field signal (i.e. the decay lengths).

The direct measurement of a unique decay length
The contrast of the high harmonics images strongly depends on the amplitude of vibration A of the probe and the lock-in acts like a z-filter. This filtering effect and the contrast dependance are intrinsic to the lock-in detection, to the vertical vibration of the probe and to the probe properties [14].
As an example, we focus on the measurement of a a pure evanescent decay length D p . In the general case, the detected signal results from a superposition of evanescent waves but can reveal only one decay above a flat surface S(x, y, z) = S(z) = exp(−z/D p ). ( The investigated signal is recorded through the lock-in and therefore takes into account the interaction between the probe and the sample. No a priori hypothesis is necessary to discuss this decay length. To measure such a decay length, the classical method consists of recording approach curves [8,10]. The drawback of this method is the time requirement to record the approach curves at each scanning step in the xy-plane. Another approach could consist of taking advantage of the filtering effect of the lock-in to measure D p . We show in figure 1 the computation of the amplitude of the four first harmonics of the signal (see equation (2)) as a function of the relative decay length D p /A. The probe is supposed to vibrate in tapping mode and therefore its vertical position varies in [0, 2A]. Each harmonic exhibits a maximum of its amplitude for a fixed ratio D p /A. Therefore, by varying A and detecting this maximum, the measurement of D p becomes possible. Table 1 helps to deduce D p from the maximum of the harmonic amplitudes. The drawbacks of this method are the detection of the maximum, which could be tedious and difficult due to the experimental noise, the time needed to vary A with small steps and its accurate control. Moreover, A depends on the mechanical properties of the probe and of the cantilever themselves. In order to avoid such difficulties, a reconstruction of the signal along the probe vibration direction can be used.

Principle of reconstruction
In order to ensure the pertinence of the physical interpretation and to study the real contrast of the field, the ASNOM real signal must be at least partially reconstructed from the first Fourier harmonics [12]. This reconstruction is a tomography-like method which enables the computation of an approximation of the approach curve directly from the lock-in data [15]. The reconstructed ASNOM signal R N from N recorded harmonics can be expressed as a function of the position of the probe z and is given by: The reconstruction accuracy increases with the number of recorded harmonics and it enables a z-tomography of the near-field for z ∈ [0, 2A], without any approach curve. Nevertheless, the limitation of the number of detected harmonics N and the amplitude of vibration A induce a filtering of the detected signal that is characterized in the following subsection.

The lock-in filtering of one evanescent wave
The lock-in is used to remove the useless background and to increase the signal to noise ratio. Therefore the expected behaviour of the detection is high-pass filtering in terms of diffracted waves. The low diffracted orders (background) are partially removed from the signal and the relative magnitude of the evanescent waves depends on the amplitude of vibration of the probe [16]. Theoretically, an infinite number of harmonics is necessary to accurately measure any decay length. Experimentally, only a few number of harmonics are detected. Nevertheless, it is possible to compute an approximation of the decay length measurement m(D p ). The filter associated to the measurement of one decay length D p can be expressed as: where the measured decay length m(D p ) is computed from an exponential nonlinear mean square fit (Prony's method [17]) of the reconstructed data. This exponential fit could reveal the existence of more than only one decay in the reconstructed signal. The signal is given in equation (2), the lock-in is modelled with equation (1) and the reconstruction is computed with equation (3). Figure 2 shows the filter F evolution as a function of the number of harmonics N used in the reconstruction. The measurement of D p when the amplitude of vibration of the probe A  varying, is illustrated in figure 2(a). The measured decay length m(D p ) is greater than the real D p (m(D p ) > D p ) for small N and A increasing. Figure 2(b) shows the filter effect for a given A. For D p lower than the amplitude of the probe vibration A, m(D p ) > D p . A being fixed, the greater D p is, the better is its measurement. Therefore, the probe vibration induces a cut-off frequency for the measurement of evanescent waves with small D p (high confinement). The amplitude of vibration A limits the accurate measurement of small D p . Table 2 shows the evolution of the limit of accuracy on the measurement of D p as a function of N. It becomes possible to deduce the lower limit of the measurement of D p with an accuracy of 1, 5 and 10% respectively. For example, for N = 3 and an amplitude of vibration A = 18 nm, min(D p ) is equal to 12.8, 8.5 and 7 nm with accuracy of 1, 5 and 10% respectively. The measurement of such a decay with the technique of approach curve would be more tedious [11]. Actually, the probe vibration is necessary to prevent tip crash on the sample. Therefore, the reconstruction must be also applied to the approach curve and this measurement is limited by the signal-to-noise ratio. In that case, the reconstruction of approach curves will enable the recovering of exponential decays with high accuracy but the experimental approach curve is more time consuming (especially if it is achieved at each scanning step) than the scan in tapping mode presented in this paper. In the following section, the presence of one or more evanescent decays is studied by considering the experimental signal obtained through a lock-in amplifier in the xy-scanning plane and its reconstruction.

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Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT

The decays of experimental data
We consider the experimental optical images of latex nanoparticles obtained with ASNOM. That consists of illuminating by a p-polarized λ 0 = 488 nm argon laser beam, with an angle of incidence θ = 40 • , focused on the surface of a sample made of 80-100 nm diameter latex particles deposited on a glass substrate [12]. The optical signals, produced by a vertical vibrating AFM commercial tetrahedral silicon probe (AC Series, Olympus) at frequency f 300 kHz and with amplitude A = 18 nm (this parameter is not restrictive), is recorded in the xy-scanning plane. The estimated shape of the end of the probe is spherical with radius lower than 10 nm. The tip angle is about 15-25 • . The dc-term, corresponding to the zeroth order of the Fourier harmonics H 0 and the three first harmonics (H 1 , H 2 and H 3 ) are used to compute the reconstruction R 3 . The detection of higher harmonics is impossible in the present case due to the signal-to-noise ratio. If finer information is needed, an heterodyne detection should be used. Actually, this technique enables the detection of more than three harmonics as it is less sensitive to the noise. Moreover, the various harmonics are not acquired simultaneously with the lock-in amplifier (Model 5302 EG&G Instruments Corporation). Therefore, before reconstruction, a signal preprocessing using the maximum of correlation of AFM data is applied to the optical signal to compensate the possible lateral shifts. To measure the decay lengths we use the above mentioned exponential nonlinear mean square fit with a number of exponential terms that is optimized with regards to the rank of the Prony's matrices [17]. In these computations, the fitting function of the reconstruction with three harmonics and H 0 at each scanning step (x, y) (see equation 3) is given by where θ is the illumination angle and λ 0 the wavelength in vacuum. From this equation (6), the value of m(D s ) can be deduced from the measured m(D p ) by: Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Table 3. Computed parameters of the fitting function (see equation (5)) for the reconstructions at the centre of zones shown in figure 3. The evanescent decays correspond to positive values of m(D p ). The corresponding values of m(D s ) are also presented. The rms error of the fit is also indicated. Each column corresponds to the result of the fit with height terms (rank = 8). For exponential decreasing terms, a first approximation of the size m(D s ) of the involved diffracting structures computed from equation (7) is also indicated.
Zone1   In the particular case of m(D s ) λ 0 an approximation of equation (7) can be easily deduced: m(D s ) ≈ 2πm(D p ) [18]. Table 3 shows the values of m(D s ) deduced from equation (7) table 3 if m(D p ) < 0. The zones 1 and 2 are above nanoparticles and the zones 3, 4 and 5 are above the glass substrate but correspond to different intensity patterns. Actually, zone 3 is above a dark fringe and therefore, no decay is expected above the sample. In contrast, zone 4 is above a bright fringe and zone 5 is above more complex interference pattern.
The examination of the results in table 3 (i.e. amplitudes c n and m(D p ) values) leads to show that the usual hypothesis of a passive probe is erroneous. Indeed, under such an hypothesis of passive probe, in the dark-zone (i.e. zone 3, far from any particle) the major contribution would be due to a constant background and no evanescent wave would be revealed. We can remark that, in contrast to this hypothesis, the zone 3 is characterized by evanescent decays (m(D p ) = 9.31, 1.32 and 0.64 nm) times a cosine and sine oscillation of periods 18.04, 13.77 and 3.64 nm, respectively. We can note that the value m(D p ) = 1.32 and 0.64 nm are not significantly in agreement with the accuracy limit of measurement of D p . This reveals a contribution due to the probe hidden in the global flat decreasing signal (see figure 4). Therefore, even in the dark fringes, where the signal is expected to be negligible, one evanescent decreases can be measured: m(D p ) = 9.31 nm. This measured decay can only be associated to the presence of the probe under illumination. Moreover, under the hypothesis of passive probe, far from any particle, the differences between the dark zone (zone 3) and bright zones (zones 4 and 5) would come from a phase effect between the diffracted fields and the background field. In contrast to this, the zones 4 and 5 reveal not only a phase effect but also evanescent decay lengths m(D p ) equal to 25.32 and 8.07 nm (8.07 nm may be doubtful (table 2)) in zone 4 and 19.06 nm in zone 5 (the value 3.58 nm is out of the accuracy limit). The values 19.06 and 25.32 nm differ from more than 30% and show that the exponential decay differs in the complex intensity pattern (zone 5) from the bright fringe (zone 4). The obvious trace of the probe can be observed in some zones of the sample. This fact may lead to the following conclusion: the image formation cannot be considered as a superposition of evanescent decays and the hypothesis of passive probe is not relevant.
Finally, the zones just above nanoparticles reveal evanescent decay lengths m(D p ) equal to 38.09 and 14.50 nm in zone 1 and 29.24 and 9.99 nm in zone 2. Such decay lengths m(D p ) cannot be directly related to the sizes of the nanoparticles and/or of the probe, as noted in table 3. This analysis of the parameters of fit clearly shows that a complete model of detection should be used to recover the physical (optical properties and geometry) parameters, including the detection [19].

Conclusion
We have studied the influence of the lock-in and the probe vibration on the measurement of the decay length of evanescent waves after reconstruction of the signal from the detected harmonics. We have applied a nonlinear method to fit the exponential decays in theoretically and experimentally reconstructed signals. The number of exponentials in the series has been chosen to get the best accuracy. This method could be applied to any near-field optical signal, recorded through homodyne or heterodyne lock-in detection. Thanks to the reconstruction, no approach curve is necessary and the decay lengths along the vertical direction can be measured directly, from only one lateral scan. In contrast to the usual hypothesis of passive probe and interferometric description of the electromagnetic field, this study reveals that the probe cannot be considered as passive and that the measured decay lengths cannot be directly related to the sizes of the nanostructures and of the probe. Therefore, the only way to interpret the decays is the resolution