1. Introduction

Fano resonance can be characterized as an asymmetric resonance in the excitation spectra of atoms, molecules, and single quantum dots [1,2]. The existence of two scattering channels, one from a discrete state and the other from a continuum state, modifies the Lorentzian resonant profile into a peak-and-trough resonance curve. This phenomenon is not limited to the field of atomic physics, but also extends to optical microcavity systems where the discrete state in the microcavity interferes with continuous uncoupled light [3,4]. It has been shown that a resonance analogous to Fano resonance can also appear in a pair of classical oscillators coupled by a weak spring [5]. These show that Fano resonance is universal in physical systems.

The microcantilever structure is becoming popular as the main element in microactuators and microsensors [6–8]. For atomic force microscopy (AFM) in dynamic mode, the cantilever probe is excited into mechanical vibration; for example, by means of piezo or magnetic actuation. Photothermal excitation provides an alternative actuation method in extreme conditions, such as in liquids or remote probing of samples [9]. In general, photothermally induced vibration of cantilevers is based on the conversion of absorbed optical energy into mechanical motion through an intermediate thermal stage [10]. The temperature gradient established inside as well as on the surface of the cantilever may cause interference between various mechanisms of thermally related forces (i.e. radiometric or photothermal forces) and radiation pressure [11–14]. If the cantilever is made of two different layers of materials, then the difference in expansion coefficients causes the structure to bend [15–17].

Although photothermally induced vibration has been reported for various cantilever types and excitation conditions, each case exhibits vibration spectra with distinct features [18–20]. In this paper, we demonstrate the appearance of Fano-like resonances in the vibration spectrum of an optically driven, triangular-shaped AFM cantilever system. By investigating the general characteristics of the complex amplitude of the vibration spectrum, we construct a simple physical model to extract the main spectral components that contribute to the asymmetric behavior at resonance. We find that the asymmetry of the resonance changes as the excitation spot position moves along the central axis of the cantilever. Furthermore, when the position of the excitation spot is correctly selected, a very high resonance phase gradient can be achieved. This phenomenon might find applications in high-sensitivity AFMs, microsensors, and microactuators.

2. Experiment

The experimental setup is shown in Fig. 1(a) . We automated the control of the signal generator (NF Corporation, DF1906) and acquired the vibration signal via a lock-in amplifier (Stanford Research, SR 830). A sinusoidal voltage waveform from a signal generator drove a laser diode at 780 nm (Sacher Lasertechnik, LION) to produce an 8-μW-amplitude intensity modulated beam. By using an objective lens (Olympus, 60× , NA 0.9), the laser beam was focused to a 2-μm diameter spot on the cantilever surface. We used a commercially available triangular AFM cantilever (Nanoworld, PNP-TR) with length, and arm width of 208-µm, and 24-µm, respectively (Fig. 1(b)). The main layer consists of 535-nm silicon nitride and a 65-nm gold layer on the backside. An optical lever monitors the vertical vibration amplitude and phase of the cantilever. We modified the AFM system (Veeco, Caliber) to monitor the induced vibration of the cantilever. A 670-nm laser diode beam was focused at 21 μm from the free end of the cantilever, and the reflected beam was directed onto a position-sensitive detector (PSD), which was read by a lock-in amplifier. 700-nm short-pass filter (optical density 5 at 780 nm) is placed behind the cantilever to block any spurious disturbances from the excitation beam. We confirmed that after inserting the filter, no significant signal was detected by the lock-in amplifier when the excitation laser illuminated the cantilever with the probe laser being switched off. The AFM head was placed on a movable stage to change the location of the excitation spot relative to the free end of the cantilever.

 

Fig. 1 (a) Experimental setup and (b) scanning electron microscope image of the AFM cantilever.

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At the peak laser intensity, the cantilever bends downward to position A (Fig. 2 ) because of the difference in the thermal expansion coefficients of the gold and silicon nitride layers (the thermal expansion coefficients for gold and silicon nitride are 14.2 × 10⁻⁶ K⁻¹ and 0.8 × 10⁻⁶ K⁻¹, respectively). During a minimum in the laser intensity cycle, the cantilever relaxes back to its initial condition (position C). The measurement point oscillates about the equilibrium point B. Note that this explanation is valid only for the fundamental flexural mode of the cantilever; a higher flexural mode would follow a different bending mechanism [21,22].

 

Fig. 2 (a) Modulation intensity profile and (b) cantilever bending mechanism. Top and bottom layers of the cantilever are gold and silicon nitride, respectively

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3. Results and discussion

All measurements were performed at room temperature. We measured the vibration spectra for various positions of the excitation spot along the central axis of the AFM cantilever. Because the cantilever has a hollow section, the measurement area is limited within 94 μm from the free end. Figure 3 shows the thermal noise spectrum of the cantilever in the absence of external excitation. The result was acquired by a Fourier transform of the PSD output. The spectrum shows that the fundamental resonance frequency of the cantilever under thermal fluctuation is 12.2 kHz.

 

Fig. 3 Thermal noise of the cantilever in the absence of excitation laser

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Figure 4 shows the spectra of the vibration amplitude and phase for modulation frequencies from 6 to 18 kHz at excitation spot positions of 23, 36, 48, 61, 74, and 87 μm from the free end of the cantilever. All the amplitude spectra share a common declining baseline characteristic, which is attributed to the thermal response of the cantilever to optical excitation. The induced photothermal effect in the cantilever decreases as modulation frequency increases. Because of slow thermal response, the temperature change of the cantilever cannot follow high-frequency excitation. Thus, the structural bending of the cantilever, which depends primarily on thermal stress, decreases because of the small temperature gradient

 

Fig. 4 Vibration spectra in terms of amplitude (right column) and phase (left column) show the measurement data (black) and fit (red). Measured from the free end of the cantilever, the position of the excitation spot was 23 µm for (a) and (g), 36 µm for (b) and (h), 48 µm for (c) and (i), 61 µm for (d) and (j), 74 µm for (e) and (k), and 87 µm (f) and (l).

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All amplitude spectra clearly exhibit asymmetric resonance profiles, which are analogous to Fano resonance. The dip moves from higher frequencies at 87 μm to lower frequencies at 61 μm relative to the resonance (~12.2 kHz). The deepest dip is observed when the excitation spot is 61 μm from the free end of the cantilever. Furthermore, the phase spectra for positions of 61, 74, and 87 μm clearly show 2π shifts at the resonance. We find that the phase gradients of these spectra, determined from the range of 12.0 to 12.1 kHz, are 0.210 rad/Hz, 0.058 rad/Hz, and 0.048 rad/Hz. This indicates that the resonance phase becomes very sensitive to frequency modulations when the excitation spot is at 61 μm. In contrast, the phase spectrum when the excitation spot is at 23, 36, and 48 μm shows no phase shift before and after the resonance. These observations imply that a continuous component exists, which interferes with the resonance component of the vibration spectra.

We construct a simple analytical model from the standard solution to a forced harmonic oscillator with damping [23]. The amplitude |A(ω)| of the oscillator exhibits Lorentzian profile with a maximum at resonance. Because the cantilever exhibits a thermal response to the photothermal driving force, the amplitude spectrum is multiplied by a declining factor. The appearance of the asymmetric resonance curve can be explained by considering Fano resonance approach. In Fano resonance, interference between a resonance component and a continuous component causes the resonance curve to become asymmetrical [1]. These phenomena can be expressed using the complex amplitude of the vibration as [3,12]:

where τ is the thermal response, A _r is the complex resonance amplitude, A _c is the complex amplitude of the continuous component, ω = 2πf where f is the modulation frequency, ω₀ = 2πf₀ where f₀ is the resonance frequency, and β is the damping factor. The resulting curves are shown as fits in Figs. 4 and 5 . The two components in Eq. (1) are called the resonance component and continuous component. Both components share a common term τ, which characterizes a decrease in the photothermal effect with an increase in the modulating frequency. The term A _c causes the Lorentzian profile of the resonance component to become asymmetric and modifies the phase shift such that it is either larger or smaller than π radians.

 

Fig. 5 Nyquist plot for vibration spectrum at various excitation-spot positions; measurement data (black) and fit (red). The excitation-spot positions are (a) 23 µm, (b) 36 µm, (c) 48 µm, (d) 61 µm, (e) 74 µm, and (f) 87 µm from the free end of the cantilever.

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There are five unknowns in Eq. (1): τ, A _r, A _c, f₀ and β. We first evaluate data set for the excitation-spot position at 87 µm because its amplitude spectrum closely resembles that of a standard Lorentzian profile, which helps the fitting algorithm to fit the data more accurately. The cantilever experiences under-damped oscillations (ω_o >>β) with β = 1574 s⁻¹. The thermal response of the cantilever τ is 0.209 ms, which is in the sub millisecond range, as predicted from a published model [8]. Fit parameters from this first evaluation are used as initial values for the next data set. However, values of the quantities τ, f₀ and β are fixed at 0.209 ms, 12.18 kHz, and 1574 s⁻¹, respectively, because these quantities should not vary dramatically between data sets. By reducing the unknown parameters to only two (A _r and A _c), the speed and accuracy of the fit are improved. The procedure is repeated for consecutive data sets in the order of decreasing excitation-spot position, and the result obtained from the previous fit of data set is used as initial values for the subsequent data set.

We can obtain detailed explanation by projecting the cantilever response A(ω) in Nyquist plots, which are shown in Fig. 5. In a Nyquist plot, the complex amplitude of the vibration is resolved into a real component on the x-axis and an imaginary component on the y-axis. Each point on the plot represents the vibration vector at the corresponding modulation frequency. In other words, the Nyquist plot is a parametric plot of the complex amplitude as a function of modulation frequency. Figure 5 shows that Eq. (1) fits well to the experimental data. The circle in each graph represents the resonance behavior of the cantilever vibration. The resonance circle radius (which represents the resonance amplitude) becomes smaller as the excitation-spot position approaches the free end of the cantilever.

The relatively small distance between the plot origin and the perimeter of the resonance circle in Fig. 5(d) requires special attention. It shows a deep resonance trough in the amplitude spectrum and a high, steep phase gradient. The measurement point labeled D in Fig. 5(d) is very close to the deepest dip in Figs. 4(d). The vibration amplitude is suppressed near this position, which can be understood by decomposing the vibration spectrum into resonance and continuous components. This is done by inserting the fit parameters into Eq. (1), and plotting the resonance and continuous components in amplitude and phase plots, respectively, as shown in Fig. 6 . In Fig. 6(b), only π shift is observed if the continuous component is absent, and the phase gradient is approximately 2.5 × 10⁻⁴ rad/Hz. However, interference between two components introduces a 2π shift and causes the phase gradient to increase to 0.210 rad/Hz (Fig. 4(j)). At resonance, the amplitude of both components is almost equivalent and their relative phase is approximately π. The sum of these two components results in destructive interference. In other words, mechanical vibration is minimized at resonance. The observation of this Fano-like resonance phenomenon reveals two useful features: (1) mechanical vibration suppression, and (2) high phase gradient at resonance.

 

Fig. 6 Decomposition of resonance (solid line) and continuous (dashed line) components in the vibration spectrum for an excitation-spot position of 61μm.

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The fit results (| A _r| and |A _c|) are plotted in Fig. 7 . The resonance amplitude | A _r| decreases as the excitation spot approaches the free end of the cantilever. This behavior follows from the fact that the deflection at the measurement point is approximately proportional to the flexural mode curvature at the excitation-spot position On the other hand, the quantity |A _c| for the continuous component does not tend to zero with decreasing the distance from the free end of the cantilever. If the continuous component is part of the higher resonance modes, its value should be a minimum at the free end [24]. One possible origin of this continuous component is the thermal expansion without force being exerted on the cantilever. Both resonance and continuous components share the same thermal response characteristic against the optical excitation. The local heating by the laser irradiation causes the bending deformation at the excitation spot-position. The bending moment induces the force exerted on the cantilever that excites the resonance vibration, while a part of the bending deformation, so called uncoupled deformation, remains without coupling into the mechanical resonance mode. For example, the bending deformation near the free end of the cantilever barely initiates the resonance vibration, as discussed above, but the uncoupled deformation can still be observed as the deflection. We calculated the uncoupled deformation under the present experimental conditions, based on references [24,25]. The temperature change and the diffusion length were determined as ~0.1 K and ~20 μm, respectively, and the observed deflection amplitude caused by the uncoupled deformation was estimated to be ~0.5 nm, which well agreed with the experimental result (Fig. 6(a)). The non-resonant vibration due to the uncoupled deformation might be related to the continuous component, which is responsible for the appearance of Fano-like resonance in the amplitude spectra. The mechanism details of the continuous component would be of the great interest in the next research step.

 

Fig. 7 Result of fit for resonating component |A _r| (filled circles), and continuous component |A _c| (filled squares) as a function of the excitation-spot position.

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In practical applications, one may select a specific excitation-spot position along the central axis of the structure i.e., the rectangular-shaped cantilever. For a triangular shape similar to the cantilever used in the present study, the range of excitation-spot position or the active area is limited by the inner hole and its geometry. In addition, it would be useful to select the excitation-spot position based on the Nyquist plot rather than only on amplitude spectrum because the resonance vector in the former gives information on amplitude amplification and phase. The decreasing trend observed in the amplitude of the resonance component is not seen clearly in the amplitude spectra because of interference with the continuous component. However, in Nyquist plots, the amplitude of the complex value of the resonance component or the resonance circle radius decreases as the excitation-spot position approaches the free end of the cantilever.

4. Conclusions

We demonstrate dependence of photothermally induced vibration on excitation-spot position in an optically driven AFM cantilever. The vibration spectrum exhibits Fano-like resonance. The asymmetric resonance profile strongly depends on the excitation spot-position along the central axis of the AFM cantilever. We find that the cantilever vibration is suppressed with a phase gradient of 0.210 rad/Hz at resonance when the excitation-spot position is 61 μm from the free end of the cantilever. From our simple analytical model, we attribute this phenomenon to the interference between resonance and continuous components. The observed Fano-like resonance might find useful applications in high-sensitivity AFM systems; for example, in constructing phase sensitive mode AFMs, where a high phase gradient is required at resonance. In addition, the variation in cantilever response for different excitation-spot positions provides more options to design cantilever-based devices; in particular, because it is possible to maximize or minimize the resonance amplitude by selecting the correct excitation- spot position.