A study on the image contrast of pseudo-heterodyned scattering scanning near-field optical microscopy

D. E. Tranca; C. Stoichita; R. Hristu; S. G. Stanciu; G. A. Stanciu

doi:10.1364/OE.22.001687

1. Introduction

In the past decade, scanning near-field optical microscopy (SNOM) has attracted a considerable amount of interest as nano-scale research is becoming more important with each day. The scattering-SNOM (s-SNOM) variants have been shown to provide very promising high resolution imaging capabilities in more than several experiments [1–3]. The working principle of the s-SNOM technique relies on a metallic probe being brought up in the close proximity of a sample surface and scanned across it, while a laser beam is focused on the interaction region between the probe and the sample. The electric field intensity of the laser beam causes the formation of an oscillating dipole [4–9] located in the near-field region, which radiates in the far-field range. This scattered light contains information from the near-field region and its detection is exploited so as to provide the s-SNOM image.

The main drawback of this technique consists in the presence of an intense background light (reflected from outside the interaction area between the probe and the sample), which makes the detection of the near-field scattered light very difficult. Reported solutions to this problem are based on the non-linearity of the scattered light signal on the tip-sample distance: when driving the probe into oscillation close above the sample with frequency f_o and amplitude A_probe, the modulated near-field signal will contain several higher harmonics of the oscillation frequency, while the background light signal will be constrained to the DC and fundamental frequency [8,10]. Thus, demodulating the backscattered light at higher harmonics, the influence of the background light will be diminished. It has been shown however that this method is not sufficient for an efficient background light suppression and, along with higher harmonics demodulation, three interferential detection methods are widely used: homodyne [8,11–13], heterodyne [1,11,14] and pseudo-heterodyne detection [15–17]. Among these three methods, the pseudo-heterodyne detection proved to be the most efficient to suppress the background-light and to lead to a good image contrast in s-SNOM imaging [15,18]. This detection configuration consists in a Michelson interferometer with one interferometer arm focused onto the tip (which will determine the formation of the oscillating dipole) and the other one (the reference beam) reflected off a harmonic oscillating reference mirror. The reference beam interferes with the scattered light from the near-field of the sample and the interference signal will contain the near-field information at frequencies $n \cdot f_{o} \pm m \cdot M$ [15], where f_o is the probe oscillation frequency, M is the mirror oscillation frequency and n, m are integers. By using a lock-in amplifier locked at the $n \cdot f_{o} \pm m \cdot M$ spectral harmonics, the near-field signal is straightforward collected.

Although the pseudo-heterodyne method is considered at this time to be one of the best choices for s-SNOM detection, the influence of the involved parameters on the resulting image contrast is still not completely understood. The present work contributes to this area by providing theoretical and experimental evidence that the oscillation amplitude of the reference mirror has a strong influence on s-SNOM image contrast.

2. Materials and methods

Computer simulations have been performed based on the characteristics of the calibration sample that has been used in the experimental work. This sample contains 5x5 μm² rectangular Platinum elements deposited on a Sapphire substrate. The Platinum domains have a thickness of 5 nm. The s-SNOM contrast corresponds to different intensities of the scattered light generated by the metal (Platinum, ε_Platinum = −11.275-18.722i at wavelength 635 nm [19]) and by the dielectric material (Sapphire, ε_Sapphire = 3.139 at wavelength 635 nm [19]).

By using the MATLAB software platform, we have performed simulations that reflect the influence of the reference mirror’s oscillation amplitude on the pseudo-heterodyne s-SNOM image contrast.

The parameters involved in all calculations and simulations are based on the configuration of our experimental homemade pseudo-heterodyne s-SNOM setup [20]: beam wavelength, λ = 635 nm; probe’s oscillation frequency, f_o = 50 kHz; oscillation amplitude of the probe, A_probe = 75 nm; probe’s tip diameter, a = 30 nm; reference mirror oscillation frequency, M = 800 Hz. The long working distance objective required by the s-SNOM system had a numerical aperture NA = 0.42. The oscillation amplitude of the reference mirror was varied between 10 – 800 nm to study its influence in the image contrast. The scanning probe has a Platinum coated tip.

In s-SNOM, the image is generated by collecting the light that is scattered from the tip – sample’s surface interaction region. The intensity of the electric field scattered from the near-field region of the sample (E_nf) can be represented as [4,5,21]:

E_{n f} = C t \cdot α_{e f f} \cdot E_{i} .

In Eq. (1), E_i is the intensity of the incident electric field, α_eff is the effective polarizability and Ct is a proportional constant (considered to have the value of 10¹⁹ for a reasonable interference [21]). The effective polarizability is given by [4,5]:

α_{e f f} = α \cdot \frac{β (ε_{s}) + 1}{1 - \frac{α \cdot β (ε_{s})}{16 \cdot {(a + \tilde{z})}^{3}}},

where

α

is the polarizability of a sphere with its diameter a equal to the diameter of the tip, β(ε_s) is a parameter which depends on the electric permittivity of the sample

β (ε_{s}) = (ε_{s} - 1) / (ε_{s} + 1)

, and

\tilde{z}

is the instantaneous distance from the tip of the probe to the sample’s surface. As the probe is oscillating above the sample’s surface, the distance

\tilde{z}

will be a periodic function with frequency f_o:

\tilde{z} = z (f_{o} \cdot t) + d

[22] (with d being the minimal separation tip-sample during probe’s oscillation above the sample and t is time). Thus,

α_{e f f} = α_{e f f} (f_{o} \cdot t)

, which implies that

E_{n f} = E_{n f} (f_{o} \cdot t)

.

The reference beam (phase modulated by means of the reference mirror’s oscillation) can be written as:

E_{r e f} = E_{i} \cdot \exp [i \cdot \frac{2 π}{λ} \cdot A \sin (2 π \cdot M \cdot t)] .

Here, $i = \sqrt{- 1}$ , λ is the wavelength of the laser beam, A is the oscillation amplitude of the mirror, M is the oscillation frequency of the mirror, and t is time.

The intensity of the background electric field (amplitude modulated by the probe’s oscillation) can be expressed as [4]:

E_{b k g} = E_{i} \cdot [1 + \frac{1}{2} \cdot \sin (2 π \cdot f_{o} \cdot t)] .

Because the background field has no influence on the spectral components situated at $n \cdot f_{o} \pm m \cdot M$ (where n,m ≠ 0) [15], in a first approximation we neglect the background light given by Eq. (4). Thus, in the pseudo-heterodyne scheme, the waves E_nf and E_ref will interfere, and the interference signal (which represents the detected near-field signal), I, will be given by:

I = {| E_{r e f} |}^{2} + {| E_{n f} |}^{2} + 2 \cdot | E_{r e f} | | E_{n f} | \cos (ϕ_{r e f} - ϕ_{n f}),

where

φ_{r e f}

and

φ_{n f}

are the phases of the reference wave and near-field scattered wave, respectively, with

φ_{r e f} = \frac{2 π}{λ} \cdot A \cdot \sin (2 π \cdot M \cdot t)

. If we assume the situation when the probe is in close proximity with the sample’s surface, but not driven into oscillation, it can be noticed that in the interference signal in Eq. (5) only the interference term is relevant, since the other terms will be constant. Therefore, I is proportional with the cosine of phase difference between the two waves:

I \propto \cos (ϕ_{r e f} - ϕ_{n f}) .

3. Results

3.1 Magnitude of the spectral components vs. oscillation amplitude of the reference mirror

For a better understanding of the reference wave’s influence on the interference signal, the phase of the near-field scattered wave can be kept constant (which is equivalent to the situation when the probe tip is situated at a constant distance from the sample) and the cosine in Eq. (6) can be viewed as the real part of an exponential complex number. In this scenario, the interference term in Eq. (5) can be written as:

I \propto \exp (i \cdot ϕ_{r e f}) = \exp (i \cdot \frac{2 π}{λ} A \cdot \sin (2 π \cdot M \cdot t)) .

Transforming this signal to express it on the basis of its harmonics using Jacobi – Anger expansion [4,23,24], the corresponding signal can be described as:

I = \sum_{m} k_{m} \cdot \exp (i \cdot 2 π \cdot m M \cdot t),

with k_{m} = J_{m} (\frac{2 π}{λ} \cdot A) .

The coefficients k_m are given by the Jacobi – Anger expansion and J_m is the Bessel function of order m. Thus, Eq. (9) shows that each spectral component at frequencies $n \cdot f_{o} \pm m \cdot M$ is strongly dependent on the oscillation amplitude A of the reference mirror.

This dependency is shown graphically computing the magnitude of k_m from Eq. (9) as a function of the mirror’s oscillation amplitude, A, which is varied between 10 nm and 800 nm. The result is plotted in Fig. 1, with continuous curves, for m = 1 and m = 2 (the first two harmonics), corresponding to the fundamental mirror frequency and second harmonic frequency, respectively.

Fig. 1 Normalized near-field signal magnitude dependence on the oscillation amplitude of the reference mirror: a) frequency f_o + M; b) frequency f_o + 2M. Continuous curves: Jacobi–Anger expansion coefficients; circled curves: Fast Fourier Transform; Black dots: experimental determinations.

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For a second confirmation of the s-SNOM signal’s spectral components dependency on the mirror’s oscillation amplitude we compute the Fast Fourier Transform (FFT) of the interference signal given by Eq. (5). Using the dedicated function for FFT in MATLAB, the magnitude of any spectral component of the signal can be evaluated. The result is plotted in Fig. 1, with circled curves, for m = 1 and m = 2, corresponding to the fundamental mirror frequency and second harmonic frequency, respectively.

The differences between the two types of curves (Fig. 1) are caused by the variation of near-field wave phase during probe vibration (not taken into consideration in Eq. (9)) and by the fact that the calculation of the circled curves takes into consideration the optical properties of the materials involved.

An experimental confirmation of the s-SNOM signal dependency on the mirror’s oscillation amplitude is represented in Fig. 1 with black dots for values of the amplitude A in the interval 30 – 800 nm. The experiment was conducted by varying the oscillation amplitude of the mirror with a constant step while reading the output signal magnitude of the lock-in amplifier, which was locked on f_o + mM, with m = 1 for a) and m = 2 for b). The probe was maintained in a close proximity to the Platinum sample surface, oscillating with frequency f_o, at a fixed x, y position.

3.2 Platinum – Sapphire contrast vs. oscillation amplitude of the reference mirror

In the following section we take into consideration the fact that an s-SNOM instrument is capable of contrast differentiation between two different materials [5,8,25]. In the case of the presented experiment, a sample containing thin Platinum domains (ε_Platinum = −11.275-18.722i [19] at wavelength 635 nm [19]) deposited on a Sapphire substrate (ε_Sapphire = 3.139 at wavelength 635 nm [19]) was imaged, and the s-SNOM contrast between the two materials was computationally evaluated. For this, Eq. (1)–(5) and FFT of the interference intensity, I, have been used for both materials to represent the magnitude of the signal at f_o + M and f_o + 2M harmonic frequencies. For each of the amplitude values of the oscillating mirror, a pair of two magnitude values corresponding to the two materials, |k_i|_Platinum and |k_i|_Sapphire is obtained. For the image contrast calculation between the two materials the Michelson contrast formula [26] is used:

C o n t r a s t (A) = \frac{| k_{m} (A) |_{P l a t i n u m} - | k_{m} (A) |_{S a p p h i r e}}{| k_{m} (A) |_{P l a t i n u m} + | k_{m} (A) |_{S a p p h i r e}} \cdot 100 %

For a more comprehensive evaluation, the contrast is computed for four different minimal separations between the tip and the sample: d = 0.1 nm (tip is nearly touching the sample), d = 10 nm, d = 20 nm and d = 30 nm.

Figure 2 contains the results representing the variation of s-SNOM contrast between Platinum and Sapphire at (a) f_o + M and (b) f_o + 2M harmonic frequencies with the oscillation amplitude of the reference mirror, A.

Fig. 2 Image contrast vs. oscillation amplitude of the reference mirror for different tip-sample minimal separations, d: a) frequency f_o + M; b) frequency f_o + 2M.

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Figure 2 shows that the s-SNOM material contrast obtained in the pseudo-heterodyne configuration is highly dependent on the oscillation amplitude of the reference mirror, especially when the detection is taken on the fundamental frequency of the mirror. Also, it can be noticed that the minimal separation between tip and sample has a major influence in the image contrast, but again the highest influence is observed in the case of the fundamental frequency of the mirror.

This simulation shows that in the proximity of mirror’s oscillation amplitude values for which the magnitude of spectral components |k_m| is zero (Fig. 1), the material contrast is decreasing, closing in to zero (Fig. 2). At the same time, it can be observed that the amplitude domains for which this effect occurs are getting larger when the distance d is increased between 0.1 – 30 nm. This increase of distance d affects as well the mean value of the contrast, yielding smaller values for the material contrast.

Although for the case m = 2 the contrast quickly drops to a low value when distance d is increased, it is preferred to use detection on m = 2 because it is less sensitive (than m = 1) to oscillation amplitude variations and the real variations of the parameter d are extremely small as the mechanical feedback system of the AFM keeps the minimal separation d to a constant value.

3.3 Influence of the cantilever instability on the image contrast

Next we take into consideration a more realistic case when, during a scanning session, the minimal position of the cantilever relative to the sample’s surface beneath it is not constant, but varies randomly in the small interval 0.1 – 10 nm [27]. A computational evaluation of the contrast between the two materials as a function of the reference mirror’s oscillation amplitude was performed, using as imaging support the same sample as in previous experiments. For each value of the oscillation amplitude A, the minimal separation d between the tip and the sample is randomly modified in the interval 0.1 – 10 nm and the signal magnitude values corresponding to the two materials, |k_i|_Platinum and |k_i|_Sapphire are calculated. The corresponding image contrast is presented in Fig. 3 as a function of the oscillation amplitude, along with the contrast that results in an idealistic case with a perfectly oscillating cantilever, for a constant minimal tip-sample separation d = 0.1 nm. Figure 3 contains the results for two different frequency detection, (a) f_o + M and (b) f_o + 2M.

Fig. 3 Image contrast vs. oscillation amplitude of the reference mirror for random 0.1 – 10 nm tip-sample separations, d: a) frequency f_o + M; b) frequency f_o + 2M.

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The effects of a mechanical unstable system can be noticed in Fig. 3. Figure 3(a) shows that if the oscillation of the reference mirror is also unstable and oscillates with an amplitude that randomly varies between 180 nm and 190 nm, the contrast variations will be consistent (between 31.9% and 63.8%).

We have used our s-SNOM setup to experimentally confirm this effect. During a scanning session of the sample containing Platinum domains on Sapphire substrate, a small external mechanical perturbation that caused a temporary instability of the probe vibration was deliberately introduced, resulting in a severe changing of the s-SNOM image contrast, as can be observed in Fig. 4. Figure 4(b) presents also an image contrast outside the Pt square boundaries – a scanning artifact that is due to the external mechanical perturbation.

Fig. 4 5x5 µm rectangular domain of a thin Platinum film deposited on a Sapphire substrate. (a) Topographic image; (b) Experimental pseudo-heterodyne s-SNOM image obtained at f_o + M harmonic frequency.

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3.4 Simulated and experimental s-SNOM images

A further step towards reproducing a realistic case was placed by taking into consideration the background light given in Eq. (4). A topographic image of Platinum rectangular domains deposited on the Sapphire substrate sample was collected by Atomic Force Microscopy and was used in our simulations to define the regions where the Platinum film is deposited (Fig. 5(a)). We choose three different amplitude values around the first point where the contrast in Fig. 2(a) is zero to evidence the inversion of the image contrast in the case of f_o + M frequency detection. The three reference mirror’s oscillation amplitudes are A₁ = 365 nm (Contrast ≈36%), A₂ = 385 nm (Contrast ≈0%) and A₃ = 405 nm (Contrast ≈-5%). In the frame of the performed simulation, the topographic image was computationally “scanned” pixel-by-pixel and line-by-line taking into accounts the identified Platinum regions. If a pixel is contained in a Platinum region, the complex electric permittivity will be ε_Platinum; otherwise the complex electric permittivity will have the value of ε_Sapphire. For each image pixel Eqs. (1)–(5) are applied, but the interference signal will contain the contribution of E_bkg given by Eq. (4):

Fig. 5 5x5 µm thin rectangular domains of Platinum deposited on a Sapphire substrate. (a) Topographic image; (b) – (d) Simulated pseudo-heterodyne s-SNOM image obtained on fo + M with respectively A1 = 365 nm, A2 = 385 nm, A3 = 405 nm; (e) – (g) Experimental pseudo-heterodyne s-SNOM image obtained on fo + M with respectively A1 = 365 nm, A2 = 385 nm, A3 = 405 nm.

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\begin{array}{l} I = {| E_{r e f} |}^{2} + {| E_{n f} |}^{2} + {| E_{b k g} |}^{2} + 2 \cdot | E_{r e f} | | E_{n f} | \cos (ϕ_{r e f} - ϕ_{n f}) + \\ + 2 \cdot | E_{r e f} | | E_{b k g} | \cos (ϕ_{r e f} - ϕ_{b k g}) + 2 \cdot | E_{n f} | | E_{b k g} | \cos (ϕ_{n f} - ϕ_{b k g}) . \end{array}

For this signal the FFT is applied and the signal magnitude at f_o + M is retained. When all pixels have been “scanned”, the values of all retained magnitudes are scaled to obtain a grayscale s-SNOM image, and the simulation is complete. Figures 5(b)–5(d) present the resulting simulated pseudo-heterodyne s-SNOM image. We considered a minimal separation d = 0.1 nm between the tip and the sample during the simulated scanning session. Also for the scanning experiments the distance d was estimated to a value of 0.1 nm using the AFM scanning software. The resulting simulated images are illustrated in Fig. 5 along with experimental s-SNOM images collected while using the same parameter values.

Figures 5(b)–5(d) and 5(e)–5(g) illustrate how the contrast between the Platinum rectangular domain and the Sapphire substrate varies as a function of the oscillation amplitude of the reference mirror, in the case of f + M frequency detection. In the simulated images topographic features are visible due to small variations of the cantilever’s oscillation amplitude, deliberately introduced in the simulation for a more realistic simulation. Figure 5 shows as well that the simulation and the experimental results correspond in a consistent proportion. The contrast values between Platinum and Sapphire for the three different mirror amplitudes, for both simulated and experimental images are presented in Table 1.

Table 1. Image contrast for simulated and experimental images, in the case of three different mirror oscillation amplitudes; detection on f + M frequency.

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Similar simulations and experiments were done for signal detection on f_o + 2M frequency. In this case, the oscillation amplitudes for which the determinations were made were chosen in a region where the contrast variation with oscillation amplitude is most evident from Fig. 2(b): A4 = 480 nm, A5 = 500 nm and A6 = 520 nm.

Figures 6(b)–6(d) and 6(e)–6(g) illustrate how the contrast between Platinum and Sapphire domains varies as a function of the oscillation amplitude of the reference mirror, when signal demodulation is performed on f + 2M frequency. Figure 6 as well shows that the simulation and the experimental results correspond in a consistent proportion. The contrast values between Platinum and Sapphire for the three mirror amplitudes, for both simulated and experimental images are presented in Table 2.

Fig. 6 5x5 µm thin rectangular domains of Platinum deposited on a Sapphire substrate. (a) Topographic image; (b) – (d) Simulated pseudo-heterodyne s-SNOM image obtained on f_o + 2M with respectively A₁ = 480 nm, A₂ = 500 nm, A₃ = 520 nm; (e) – (g) Experimental pseudo-heterodyne s-SNOM image obtained on f_o + 2M with respectively A₁ = 480 nm, A₂ = 500 nm, A₃ = 520 nm.

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Table 2. Image contrast for simulated and experimental images, in the case of three different mirror oscillation amplitudes; detection on f + 2M.

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Comparing the two detection modes (on frequencies f + M and f + 2M) it can be clearly stated that detection in the second harmonic frequency of the mirror is preferred to detection on the fundamental frequency, as the tip-sample separation, d, is kept constant by the mechanical system of the AFM. Furthermore, in the region where detection on the fundamental frequency of the mirror gives a contrast that is highly dependent on the oscillation amplitude (e.g. 365 – 405 nm), for detection on the second harmonic of the mirror’s oscillation frequency a constant value of the contrast (~-41%) is obtained.

4. Discussion

Our experiment demonstrates that in a pseudo-heterodyne s-SNOM configuration the image contrast between two different materials is highly dependent on the oscillation amplitude of the reference mirror, A. The contrast may take values of opposite signs for different amplitude values, and even zero values in some cases. The performed simulations show that it is advisable to avoid the values of the mirror’s oscillation amplitudes at which |k_m| (given by Eq. (9) and plotted in Fig. 1) is zero due to the fact that around these points the material contrast is the lowest. For future developments of a consistent method for material-specific identification using the pseudo-heterodyne s-SNOM scheme, this influence of the reference mirror must be taken into consideration.

The results show that the contrast between two different materials has a periodical behavior when varying the mirror’s oscillation amplitude (see Fig. 2), and its period is larger for higher harmonic demodulation of the mirror’s oscillation frequency. Figure 2 also reveals that for the |k₂| (m = 2) spectral component the image contrast tends to maintain a constant value for wider ranges of the mirror’s oscillation amplitude than for the |k₁| (m = 1) component. This leads to the conclusion that is better to detect the near-field signal based on higher harmonics of the mirror oscillation frequency, as small variations of the oscillation amplitude of the mirror will have a lower effect on image contrast.

Our experiment shows as well that the mechanical stability of the probe above the sample during scanning plays a key role when performing s-SNOM imaging. As Figs. 2 and 3 show, the contrast is highly dependent on tip-sample minimal separation and the contrast can drastically change during a scanning session if the minimal separation varies within just a few nanometers.

5. Conclusion

This theoretical and experimental study on image contrast for pseudo-heterodyned s-SNOM configuration clearly demonstrates that the oscillation amplitude of the reference mirror has an important influence on image contrast between different materials. The image contrast has a periodical behavior when varying the oscillation amplitude of the reference mirror. For higher harmonic demodulation, its period is larger and tends to maintain a constant value for wider ranges of the mirror’s oscillation amplitude. Our experiments reveal as well that the instabilities in the probe’s vibration have a severe influence on s-SNOM image contrast.

Acknowledgments

The presented work was supported by the UEFISCDI PN-II-PT-PCCA-2011-3.2-1162 Research Grant and by the EC 7th Framework Programme under grant agreement n° 280804 (LANIR).

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	A₁ = 365 nm	A₂ = 385 nm	A₃ = 405 nm
Contrast (Simulation)	34.48%	0.63%	−4.63%
Contrast (Experiment)	36.56%	−0.55%	−6.6%

	A₁ = 480 nm	A₂ = 500 nm	A₃ = 520 nm
Contrast (Simulation)	−40.42%	−34.34%	−6.22%
Contrast (Experiment)	−38.88%	−33.41%	−5.65%

	A₁ = 365 nm	A₂ = 385 nm	A₃ = 405 nm
Contrast (Simulation)	34.48%	0.63%	−4.63%
Contrast (Experiment)	36.56%	−0.55%	−6.6%

	A₁ = 480 nm	A₂ = 500 nm	A₃ = 520 nm
Contrast (Simulation)	−40.42%	−34.34%	−6.22%
Contrast (Experiment)	−38.88%	−33.41%	−5.65%

A study on the image contrast of pseudo-heterodyned scattering scanning near-field optical microscopy

Abstract

1. Introduction

2. Materials and methods

3. Results

3.1 Magnitude of the spectral components vs. oscillation amplitude of the reference mirror

3.2 Platinum – Sapphire contrast vs. oscillation amplitude of the reference mirror

3.3 Influence of the cantilever instability on the image contrast

3.4 Simulated and experimental s-SNOM images

4. Discussion

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Tables (2)

Equations (11)

Optics Express