Open Access
Add to BibSonomy
Abstract
This study presents a simulation-based analysis on the excitation of microcantilever in air using pulsed-laser-induced photoacoustic waves. A model was designed and coded to investigate the effects of consecutive photoacoustic waves, arising from a spherical light absorber illuminated by short laser pulses. The consecutiveness of the waves were adjusted with respect to the pulse repetition frequency of the laser to examine their cumulative effects on the oscillation of microcantilever. Using this approach, oscillation characteristics of two rectangular cantilevers with different resonant frequencies (16.9 kHz and 505.7 kHz) were investigated in the presence of the random oscillations. The results show that the effective responses of the microcantilevers to the consecutive photoacoustic waves provide steady-state oscillations, when the pulse repetition frequency matches to the fundamental resonant frequency or its lower harmonics. Another major finding is that being driven by the same photoacoustic pressure value, the high frequency cantilever tend to oscillate at higher amplitudes. Some of the issues emerging from these findings may find application area in atomic force microscopy actuation and photoacoustic signal detection.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Excitations of microcantilevers by laser-induced photoacoustic (PA) waves have recently been reported [1]. This excitation technique typically relies on the generation of PA waves from an optical absorber, followed by effective delivery of these propagating PA waves on the lever surface through the medium. An effective response of the microcantilever to the PA waves holds promise to actuate and characterize atomic force microscopy (AFM) cantilevers. Moreover, it may enable microcantilevers to be used as photoacoustic sensors, a substitute method for detecting small signals. Nonetheless, these potential applications call on a comprehensive understanding of the microcantilever response to the laser-induced PA waves.
Piezoelectric actuation [2] is a commonly used base-driven AFM actuation model in vacuum and ambient conditions. However, its applications are confounded by its peculiar limitations such as utilization of a bulky piezo-actuator mounted to the cantilever holder. This may cause spurious resonance vibrations [3] especially in liquid environment, mostly due to the resultant motion of the surrounding fluid. Although there are rigorous attempts [4] to eliminate or reduce these spurious peaks, it still remains as a difficult problem to be dealt with. Another common AFM actuation technique is the photothermal excitation [5], which is based on the direct illumination of the lever surface. However, this approach causes undesired thermal effects to occur on the microcantilever oscillation [1] and on the samples [6], making them be exposed to the laser beams and to the heat as well. Several attempts [7] have been done to eliminate this problem by situating the laser focus on a substrate at a distance from the clamped end, which is not possible with standard cantilevers. Electrostatic [8] and magnetic excitations [9] are other demonstrated techniques, but they require surface modifications, sophisticated microfabrication processes, and include bulky units, hampering the cost-effectiveness and practical utilization. Furthermore, an efficient alternative is the base-integrated piezoelectric actuator, but it requires high voltages for actuation [10] and may suffer from repeatability and passivation problems [11]. In addition, acoustic actuation method either by a piezoelectric transducer [12] or an acoustic radiation pressure actuator [13] to generate acoustic waves is of particular interest, providing the advantage of non-contact interactions. Although constituting a promising approach, this technique is also hinged on the utilization of bulky units. In view of all that has been mentioned so far, one may appreciate why microcantilever actuation by laser-induced PA waves has taken its own place as an alternative technique. This method does not impose direct laser illumination onto the lever surface without sacrificing its non-contact interaction property that is provided by generation of the PA waves from an optical absorber.
On the other hand, the present literature shows the usage of cantilevers (with few millimeters in length and width, 5 to 10 microns in thickness) as PA sensors particularly in the photoacoustic trace gas analysis [14–16]. In these studies, photoacoustic generation is commonly based on a modulated laser light in which the source term for the excitation of cantilever is simply defined by a sinusoidal force [14]. However, when using pulse laser illumination to generate PA waves to be detected by cantilevers (for AFM actuation or photoacoustic detection), there remains a lack of knowledge about the response of microcantilever to the consecutive PA waves to achieve a more realistic model.
In this work, we investigate the excitations of two rectangular microcantilevers with different resonant frequencies (16.9 kHz and 505.7 kHz) in air medium using PA waves induced by a pulsed laser. Instead of using a sinusoidal force for the excitation source, we use a purely N-shaped PA wave from an ideal spherical light absorber. A specific objective of the present study is to analyze the cumulative effect of these waves on the microcantilever oscillation characteristics by combining them consecutively with respect to pulse repetition frequency (PRF) of the excitation laser. This novel approach is expected to yield a better understanding of the response of microcantilever to the pulse-laser-induced PA waves.
2. Theory and method
2.1. Generation of the PA wave by the Dirac delta pulse illumination
PA generation can be enabled by either intensity modulated continuous wave lasers [17, 18] or pulsed lasers [19, 20]. In the latter case, which is a profoundly efficient [18, 21–23] method in PA generation, a short-pulsed laser is used to irradiate an optical absorber. Once the light is absorbed by the absorber, an initial temperature rise occurs, resulting in an adiabatic volume expansion. This expansion triggers a pressure wave out of the irradiated absorber, that is called the photoacoustic effect. The propagation of PA waves is described by the following wave equation [19]:
Here, p(r, t) is the PA wave at position r and time t, vs the speed of sound, β the thermal coefficient of volume expansion, CP the specific heat capacity at constant pressure, and H the heating function defined as the amount of heat generated by light absorption per unit volume and unit time. In PA effect, H is proportional to the time derivative of the increase in temperature, and can be represented by the product of the optical absorption coefficient μa and the optical fluence rate F (H = μaF), since the absorption is dominant over scattering. It is important to note that PA wave is produced by only time-variant heating as can be seen in Eq. (1).PA effect requires that the pulse duration of the laser is shorter than the thermal and stress relaxation time of the tissue, in which the heat caused by optical absorption does not have enough time to diffuse inside the absorber. The resultant initial pressure rise can be defined as [19]
Here, Gruneisen parameter, Γ, is a unique and dimensionless parameter for the sample [24], and ηth is the percentage of absorbed energy that is converted to heat which is generally assumed as 1 when laser pulse is short enough [19]. Therefore, the light absorption and the optical fluence are the primary parameters for the effective generation of PA waves.Solutions of the PA wave equation have been presented for both point source described by the Dirac delta function [19, 25, 26] and a source which has Gaussian spatiotemporal profile [27]. The time domain solution for a delta-pulse excitation of a spherical light absorber is obtained by the Greens’ function method [19]. When a spherical object of radius R is illuminated with a laser pulse, the resultant initial pressure p0 occurs inside the object. Three cases need to be considered based on the propagation time if the observation point is outside the object. The first case is that the spherical object does not intersect with the spherical shell of radius vst centered at the observation point so that the photoacoustic signal is zero if r − R > vst. The second case is that upon the heating, the spherical object touches the spherical shell of radius vst so that the pressure is if vst is inside [r − R, r + R]. The third case is that the spherical object cannot intersect with the spherical shell and hence p(r, t) is zero if r + R < vst. On the other hand, the other three cases need to be considered if the observation point is inside the spherical object (r < R). The first case is that when vst < R − r, the spherical shell is surrounded completely by the excited spherical object so that p(r, t) = p0. The second case is that when vst is inside [R − r,R + r], the spherical shell comes out of the excited object and the pressure wave is . The final case is that when R + r < vst, spherical shell surrounds the spherical object and p(r, t) becomes zero [19].
By introducing the Heaviside step function, U, and taking into consideration the aforementioned cases, the PA signal is expressed by the following equation
If the initial pressure distribution can be described by p0(r) = p0U(r)U(−r + R) for r is inside [0, R], the following PA pressure wave expression is obtained In Eq. (4), the first term on the right-hand side describes a converging spherical wave while the second term describes a diverging spherical wave, which is resulted from the initially converging wave propagating through the center. The last term on the right-hand side in Eq. (4) also describes a diverging spherical wave. Upon the delta heating, a constant initial pressure wave through the entire sphere is created. This pressure wave consists of two spherical waves with the same magnitude. One of them moves inward (as a converging spherical wave, the first term in Eq. (4) as the other one moves outward (as a diverging spherical wave the second term in Eq. (4)). The convergent spherical wave turns to a divergent spherical wave when it reaches the center of the object (the last term in Eq. (4)). Therefore, the PA pressure wave resulted from a delta heating has an N-shaped form as can also be seen in Fig. 1(a), where a characteristic N-shaped waveform (bipolar; positive followed by negative) is generated from a spherical light absorber with a radius of R = 250 μm as a function of time in the forward direction that is relative to the pulsed laser incident. It is also observed that the amplitude of PA wave is decreasing with the distance from the absorber; thus, the distance between the absorber and the microcantilever is an important parameter for the oscillation amplitude.
Fig. 1 (a) The characteristic N-shaped photoacoustic waves generated from a spherical light absorber of R = 250 μm for p0 = 75 Pa. The amplitude of the photoacoustic wave decays with distance with respect to inverse distance law [Eq. (4)], (b) Normalized power spectral density of this photoacoustic wave.
Wavelength of the generated PA wave is an important parameter to adjust the gap between the absorber and the cantilever. Laser-induced PA waves, especially for short pulses, have characteristically broadband frequency spectrum. In other words, the wave packet contains relatively low frequency components with dominant amplitudes together with the high frequency components with decreasing amplitudes. Particularly, a spherical absorber with a radius of 250 μm generates a PA wave that has a frequency of 0.455 MHz (with a peak amplitude) and other high frequency components up to several tens of MHz as shown in Fig. 1(b). A sound wave with a frequency of 0.455 MHz has a wavelength of about 760 μm in air, and the higher frequency components inside the wave packet has shorter wavelengths. Therefore, the distance between the absorber and the cantilever should be sufficiently large compared to the acoustic wavelength, so that its shape can be formed efficiently and the propagation along the gap is permitted.
2.2. The creation of consecutive PA waves
Investigating cumulative effects of the consecutive PA waves acting on the cantilever surface provides valuable information for characterizing the cantilever oscillation, which cannot be obtained by regarding a single wave. To achieve such approach, the single N-shaped PA wave was repeated numerically. The care was taken to present a realistic PRP by defining each period with respect to the time spanned between ascending peaks of the successive photoacoustic pulses. Figures 2(a)–2(b) show two examples of consecutive PA waves generated by a pulsed laser with PRF of 100 kHz for the values of p0 = 75 Pa, R = 250 μm, and r = 2R. Note that the characteristic N-shaped PA wave is not perceptible in Fig. 2(b) since the duration of the PA wave is very small compared to the PRP of the PA wave train. While the latter is simply the inverse of the PRF of the excitation laser, the former varies with respect to the radius of the light absorber and the speed of sound in the medium.
Fig. 2 Two examples of numerically created consecutive PA waves comprising (a) 5 and (b) 50 waves, which are generated by a 100 kHz excitation laser for the values of p0 = 75 Pa, R = 250 μm, and r = 2R.
2.3. Photoacoustically driven microcantilevers
The oscillation dynamics of the microcantilever is well described by the Euler-Bernoulli beam equation [28]. Here, the most straightforward approximation to model the dynamics is the one-dimensional simple harmonic oscillator under external PA excitation within a noisy environment as follows [14, 29]
where meff, x, γ, and keff are the effective mass, the vertical displacement of the free end, the damping coefficient, and the effective spring constant of the cantilever, respectively. The right hand side terms FPA(t) and are the driving forces which represent the force due to the PA pressure and the Gaussian white noise, respectively. The noise term is incorporated into the model to account for the main sources of noise such as shot noise of the photodetector, thermal noise of the cantilever, thermal drift in the system, and other ambient noise sources [30]. Energy transfer at the air-cantilever interface is based on the average force acting on the lever surface due to the PA wave packet. The approximation made here can be fairly regarded as realistic, since the duration of the PA wave is short, and the size of the microcantilever is small compared to the width of the photoacoustic wavefront. In addition, acoustic attenuation can also be taken into account. However, it is negligible for small distances (particularly for such frequencies). Thus, plane wave approximation for the interaction between PA pressure wave and cantilever surface leads to a uniform pressure load on the cantilever surface such that the force is simply FPA(t) = P(t)Alever where P(t) is the PA wave as a function of time t at a certain distance r, and Alever is the one-side surface area of cantilever. Hence, the related force term due to the consecutive PA waves as a function of time can be defined as where PRP = 1/PRF. Note that for a conventional dynamic AFM model, in addition to the driving force, the interaction force between tip and sample surface is considered. Nevertheless, we use a cantilever clamped at one end and has no interaction with any sample on its free end. However, it is driven only by the random noise and the force due to external consecutive PA pressure waves separated by the PRP as shown in Fig. 3.Fig. 3 Schematic illustration of the setup for the microcantilever excitation by pulsed-laser-induced photoacoustic waves generated from a spherical light absorber.
The effective mass of cantilever for this case is defined as meff = 0.647mlever [15] such that mlever = ρlever Lwh where L, w, and h are the length, width, and thickness of cantilever, respectively; and ρlever the density of cantilever material. Furthermore, the effective spring constant keff for the uniform pressure distribution is [15, 31]
where E is the Young’s modulus for the cantilever material. The resonance frequency of the cantilever is then described as It can be shown the above definition of the resonance frequency yields an equivalent result when compared to the Euler-Bernolli beam theory for the first mode [1, 29].2.4. Oscillating cantilever in a viscous fluid
We need to analyze the system in a medium due to excitation of microcantilever by PA waves. Besides, the light absorber and cantilever should be located in the same medium to avoid acoustic impedance mismatch between boundaries, where acoustic wave can be attenuated considerably due to the strong reflection. When oscillating in a viscous fluid, the resonant frequency of cantilever, ffluid, decreases compared to the value in vacuum, fvacuum, due to the drag force of the surrounding fluid acting on cantilever surface. The shift in the resonant frequency in a fluid is defined as [32, 33]
where ρfluid is the density of fluid. The resultant motion of the surrounding fluid due to the vibration of cantilever has considerable effects on the oscillation behavior, particularly in liquids. The effect is generally modeled considering an additional mass, madded, to the cantilever’s original mass. Hence, this additional mass results in an increase in the damping coefficient of the oscillation. While this effect in air can be negligible [34], there is value in considering that in the fluid particularly for high frequency cantilevers. For a rectangular cantilever beam, the additional mass term madded and the resultant damping coefficient γ can be derived by the following expressions [34] where ωfluid is the pulsation in fluid, and the parameters Γ′ and Γ″ are defined by forming a hydrodynamic function Γ = Γ′ + iΓ″ for the oscillating cantilever. For an infinitely thin rectangular beam, the above parameters are approximated [34] for a1 = 1.0553, a2 = 3.7997, b1 = 3.8018, and b2 = 2.7364. The length parameter δ corresponding to the fluid is defined by that gives the thickness of the thin viscous layer surrounding the cantilever in which the velocity decreases by a factor of 1/e. The parameter η is the fluid viscosity. More detailed information about the additional mass and the resulting damping including the hydrodynamic function parameters can be found elsewhere [33, 34].3. Results
To simulate the oscillation behavior of the microcantilever, the related second-order differential equation [Eq. (5)] was numerically solved by using the finite-difference method in MATLAB (Mathworks Inc., Natick, MA). The position and the velocity of the cantilever were assumed to be zero as an initial condition where the first PA wave reaches the cantilever surface. The simulations were performed in air environment at a temperature of 25 °C for two rectangular shaped silicon cantilevers C1 and C2 with different sizes and resonant frequencies listed in Table 1. The density, viscosity of air, and the speed of sound were taken as 1.184 kg m−3, 1.844 × 10−5 Pa s, and 346 m s−1, respectively. Also, the Young’s modulus and density of silicon were taken as 169 GPa [35] and 2329 kg m−3 [36], respectively.
Table 1. Cantilevers used for simulations
First, the random oscillations of the microcantilevers in the absence of the external PA waves (PA off) were analyzed. The root mean square (RMS) of the white noise including the all noise sources mentioned before was adjusted around 1 nm for both cantilevers. Here, the effects of the thermal noise of the cantilevers were ignored by considering the shot noise as the dominant noise source in our system. The simulated random oscillations with respect to time up to 4 ms are shown in Figs. 4(a)–4(b). Next, the fast Fourier transform (FFT) of the random oscillations was taken, which reveals the power spectral density shown in Figs. 4(c)–4(d) for C1 and C2, respectively. Notice that C2 has a higher resonant frequency than C1, leading to a resonance curve with higher quality as expectedly.
Fig. 4 (a)–(b) Random oscillations of the cantilevers in time domain modeled by the Gaussian white noise with an RMS value of approximately 1 nm, (c)–(d) the corresponding normalized power spectral densities calculated by taking the FFT of these time domain random oscillations.
The oscillation behaviors of the microcantilevers in the presence of the external PA waves (PA on) were investigated. Figures 5(a)–5(b) show the oscillations with respect to time (again up to 4 ms) response to the PRF of the excitation laser that is equal to the natural resonance frequency values fC1 and fC2 of C1 and C2, respectively. As can be seen, when the PRF matches to the resonant frequency, the oscillation amplitude maximizes in a short time and then exhibits a steady-state behavior. Here, the radius of the light absorber was taken as 250 μm, and the initial PA pressure as 75 Pa on the surface of the absorber. The distance of the cantilever to the center of the spherical absorber was considered as 2000 μm which is larger than the acoustic wavelength, and ensures plane-wave approximation (e.g., after being emitted from an absorber with a radius of 250 μm and propagated through the medium for 1750 μm, the photoacoustic wavefront’s interaction with the surface of the cantilevers considered in the present study can be fairly presumed as planar). Each single PA pressure wave has an amplitude of 4.7 Pa on the cantilever located at this distance [Fig. 1(a)]. Furthermore, Figs. 5(c)–5(d) show the FFTs of the oscillations at the resonance in which the clear maximums of the spectral densities correspond to the PRFs that are equal to the resonant frequencies.
Fig. 5 (a)–(b) The responses of the cantilevers to the consecutive photoacoustic waves when the PRF of the excitation laser matches to the fundamental resonant frequency, (c)–(d) the corresponding normalized power spectral densities calculated by taking the FFT of these time domain oscillations.
Cantilevers can be excited to oscillate at relatively higher amplitudes by setting PRF of the laser to the integer fractions of the cantilever’s fundamental frequency f0. Each division of fundamental frequency by nth integer fraction gives nth lower harmonic frequency fn. Figures 6(a)–6(b) show the maximum oscillation amplitudes obtained for each PRF within a range that encompasses driving frequencies up to 3rd order of C1 and C2, respectively. Note that to obtain each of these data points, driven oscillations were computed with a high sampling rate, which imposes a considerable computational overhead. Regarding the memory limits of the hardware available and the encompassed range of driving frequencies, PRF step sizes of 25 Hz and 50 Hz were selected to extract resonance curves for C1 and C2, respectively. The amplitudes of the curves decreases as the harmonic value increases as expected. For a better understanding of the physical origin of the effective response of the cantilever to the lower harmonic frequencies emerged in Figs. 6(a)–6(b), it is important to observe time domain oscillations of the cantilever by applying PRF at these lower harmonic frequencies. The consecutiveness of the PA waves (PRP) plays an important role in determining the oscillation amplitude [Fig. 6(c)]. When the cantilever is driven by f0, the magnitude of the oscillation is elevated at its each period. Whereas it is driven by f1, which equals to the half of f0, the increment of the oscillation magnitude occurs once in two oscillation periods.
Fig. 6 (a)–(b) Frequency response curves corresponding to the fundamental resonant frequencies and their lower harmonics up to 3rd order, obtained by iterating the PRF of the excitation laser with a step size of 25 Hz and 50 Hz for C1 and C2, respectively, (c) the time domain oscillations (without noise) of C1 up to 0.4 ms driven by f0, f1, f2, and f3, in which the colored arrows show the inclusion of the PA waves.
Off-resonance excitation of the cantilever leads to an oscillation at very low amplitudes. For example, the oscillations driven by the PRF of 150 kHz which is above of the resonance value of C1, below of the resonance value of C2, and does not correspond to any harmonic values for both, are shown in Figs. 7(a)–7(b). Notice that the oscillations with respect to time behave like a random noise for both cantilevers in the manner of amplitude and characteristic. Besides, the normalized spectral densities shown in Figs. 7(c)–7(d) for C1 and C2, respectively, reveal the characteristics of the oscillations. One can see that the densities corresponding to the driving frequency (PRF) and its resultant harmonics remain low compared to the resonance curve which indicates the random noise level. Therefore, the repetition rate of the PA waves, or the PRF of the excitation laser, has remarkable effects to drive the microcantilever effectively.
Fig. 7 (a)–(b) The responses of the cantilevers to the consecutive photoacoustic waves with a repetition frequency of 150 kHz that does not match to any harmonic values of both cantilevers, (c)–(d) the corresponding normalized power spectral densities calculated by taking the FFT of these time domain oscillations.
To obtain the quality factors of the cantilevers for the fundamental resonance frequencies, the frequency response curves (with and without noise) were utilized shown in Figs. 8(a)–8(b). These resonance curves were obtained by iterating PRF with a step size of 25 Hz within the respective ranges and recording the maximum oscillation amplitude for each iteration. The resonance values were determined to be 16.8 kHz and 505.5 kHz, which agree well with the theoretical calculations (Table 1), with quality factors of 40 and 581 for C1 and C2, respectively.
Fig. 8 Frequency response curves of the microcantilevers simulated with and without noise. To find f0, f1, and f2 values for each curve, the correct positions for the vertical intersections for the data with noise were obtained by the data without noise. Then, the horizontal red lines correspond to of the peak amplitude values of the curves.
The resonance amplitudes as a function of the distance between the optical absorber and the microcantilever were also investigated as shown in Figs. 9(a)–9(b). The oscillation amplitudes decay according to inverse distance law due to the fact that PA wave loses its power with distance as expected [Fig. 1(a)]. It should be indicated here that when the distance is equal to an integer multiple of the half of the acoustic wavelength, a standing wave can be generated which may provide an easier energy transfer on the microcantilever, although the resultant increase in the oscillation amplitude is not remarkably high in this particular condition [1]. However, an analysis that examines the reflected sound wave from the lever surface should be performed to observe this effect which is not covered in our model that includes a wave propagation in one direction onto the microcantilever.
Fig. 9 Oscillation amplitudes of the microcantilevers as a function of the distance between cantilever and the surface of the spherical absorber.
4. Discussion
The present study was designed to investigate microcantilever excitation by simulating pulsed-laser-induced PA waves generated from a spherical light absorber. The analysis of the cumulative effects of the consecutive waves on microcantilever oscillation undertaken here compromises a novel approach to such investigation. Both time and frequency domain results indicate that various standard AFM cantilevers can be effectively excited (e.g., driving at resonant or any other desired frequency) with steady-state oscillations and can be characterized (e.g., determining resonant frequency and quality factor from PRF response curves). Given that the most common types of microcantilever materials, e.g., silicon, silicon-nitride, gold, aluminum, etc., are profoundly reflective to acoustic waves, they can be utilized for actuation and characterization of AFM cantilevers or photoacoustic signal detection without the need of special coating on cantilever, or complex microfabrication processes.
We also indicate that microcantilevers can be used to detect small photoacoustic signals with the help of an increase in the oscillation amplitude by the cumulative effects of consecutive photoacoustic waves when working at the resonant frequency. While a single photoacoustic wave yields a deflection of about 0.43 nm for C1 and 0.07 nm for C2, which are not easily measurable levels within a noisy environment (e.g., by a four quadrant photodetector), the consecutive photoacoustic waves provide deflections of about 7 nm for C1 and 24 nm for C2 at the resonance [Figs. 5(a)–5(b)]. This enhancement in the oscillation amplitude may be favorable for the detection of small PA signals which is not possible by the traditional detection tools.
Differential acoustic impedance of the absorber and the surrounding medium distorts the N-shape of the PA waves, manipulating their propagation characteristics [37]. This condition has been commonly reported to challenge non-contact PA detection in air medium [38–41]. In this regard, air-coupled materials such as aerogels [42, 43] can be promising for utilizing in the medium considered in the present study, given that they have an acoustic impedance value around 0.01 MRayl [44, 45], which is comparable to that of the air. Apart from the excitation in air medium, excitation in another viscous fluid such as water, which may possibly be necessary for AFM actuation or photoacoustic measurements, includes similar calculations mentioned in Section 2.4. However, the shift in the resonant frequency in water will be high due to strong hydrodynamic damping effect. Hence, oscillation amplitude will remain low even at the resonance, and may be below the measurable level. Besides, it may be possible to obtain oscillations at sufficiently high amplitudes in water as well. This can be achieved when cantilevers with frequencies in the MHz level are used, or photoacoustic waves are generated with higher pressure values by increasing the excitation laser energy and/or light absorption. It is worth noting that, the noise level was set around 1 nm in the present study. Therefore, there is still room for the improvement of minimum detectable level by the utilization of an experimental setup with a reduced noise level [46] that allows measuring smaller displacements and pressure values.
Possible exposure of the microcantilever to the pulsed-laser light, which may not be completely shadowed by the photoacoustic absorber can be problematic. This is because it can manipulate the oscillation of microcantilever. Nevertheless, this effect can be eliminated with the precise alignment of the laser. On the other hand, such experimental setups cannot be easily obtained using commercial AFMs since they do not have enough space around microcantilever for the photoacoustic instrumentation. Custom-made portable AFM heads [47] can permit configurations in which the excitation setup can be integrated to the system. In addition, tunability of the laser PRF plays an important role in the efficiency of the photoacoustic excitation. While this property is commonly not available in commercial lasers, custom developed fiber lasers can be equipped with it, permitting tunability of the PRF even up to MHz levels [20]. In conclusion, these needs highlight the importance of the advancements in the custom made AFM instrumentations and lasers, indicating a new potential application field.
The energy transfer at the air-cantilever interface in the present study is merely based on a uniform pressure distribution, which allowed us to use plane-wave approximation for the interaction between PA wavefront and lever surface. This approach can be considered as realistic for this scenario in which the width of the wavefront at the interface is large enough compared to the length of the cantilever. A limitation of this study is that this assumption would be very rough for relatively small absorbers in close regions, where both the energy transfer and oscillation parameters should be modified.
Funding
Türkiye Bilimsel ve Teknolojik Araştirma Kurumu (TUBITAK) Grant No. 213E033; Bogazici University Research Fund Grant No. BAP 15B03TUG3; Republic of Turkey Ministry of Development Grant No. 2009K120520.
References and links
1. N. Gao, D. Zhao, R. Jia, and D. Liu, “Microcantilever actuation by laser induced photoacoustic waves,” Sci. Rep. 6, 19935 (2016). [CrossRef] [PubMed]
2. M. Huefner, A. Pivonka, J. Kim, C. Ye, M. A. Blood-Forsythe, M. Zech, and Jennifer E. Hoffman, “Microcantilever Q control via capacitive coupling,” Appl. Phys. Lett. 101, 173110 (2012). [CrossRef]
3. T. E. Schaffer, J. P. Cleveland, F. Ohnesorge, D. A. Walters, and P. K. Hansma, “Studies of vibrating atomic force microscope cantilevers in liquid,” J. Appl. Phys. 80(7), 3622–3627 (1996). [CrossRef]
4. C. Carrasco, P. Ares, P. J. de Pablo, and J. Gómez-Herrero, “Cutting down the forest of peaks in acoustic dynamic atomic force microscopy in liquid,” Rev. Sci. Instrum. 79, 126106 (2008). [CrossRef]
5. V. Pini, B. Tiribilli, C. M. C. Gambi, and M. Vassalli, “Dynamical characterization of vibrating AFM cantilevers forced by photothermal excitation,” Phys. Rev. B 81(5), 054302 (2010). [CrossRef]
6. H. Yamashita, N. Kodera, A. Miyagi, T. Uchihashi, D. Yamamoto, and T. Ando, “Dynamical characterization of vibrating AFM cantilevers forced by photothermal excitation,” Rev. Sci. Instrum. 78, 083702 (2007). [CrossRef]
7. B. Ilic, S. Krylov, and H. G. Craighead, “Theoretical and experimental investigation of optically driven nanoelectromechanical oscillators,” J. Appl. Phys. 107, 034311 (2010). [CrossRef]
8. K. Umeda, N. Oyabu, K. Kobayashi, Y. Hirata, K. Matsushige, and H. Yamada, “High-Resolution Frequency-Modulation Atomic Force Microscopy in Liquids Using Electrostatic Excitation Method,” Appl. Phys. Express 3, 065205 (2010). [CrossRef]
9. E. T. Herruzo and R. Garcia, “Frequency response of an atomic force microscope in liquids and air: Magnetic versus acoustic excitation,” Appl. Phys. Lett. 91, 143113 (2007). [CrossRef]
10. T. Michels and I. W. Rangelow, “Review of scanning probe micromachining and its applications within nanoscience,” Microelectron. Eng. 126, 191–203 (2014). [CrossRef]
11. A. G. Onaran, F. L. Degertekin, B. Hadimioglu, T. Sulchek, and C. F. Quate, “Actuation of atomic force microscope cantilevers by acoustic radiation pressure,” in The 15th IEEE International Conference on Micro Electro Mechanical Systems, Las Vegas, NV (IEEE, 2002), pp 356–359.
12. T. M. Huber, B. C. Abell, D. C. Mellema, M. Spletzer, and A. Raman, “Mode-selective noncontact excitation of microcantilevers and microcantilever arrays in air using the ultrasound radiation force,” Appl. Phys. Lett. 97, 214101 (2010). [CrossRef]
13. F. L. Degertekin, B. Hadimioglu, T. Sulchek, and C. F. Quate, “Actuation and characterization of atomic force microscope cantilevers in fluids by acoustic radiation pressure,” Appl. Phys. Lett. 78, 1628 (2001). [CrossRef]
14. T. Kuusela and J. Kauppinen, “Photoacoustic Gas Analysis Using Interferometric Cantilever Microphone,” Appl. Spectrosc. Rev. 42(5), 443–474 (2007). [CrossRef]
15. V. Koskinen, J. Fonsen, K. Roth, and J. Kauppinen, “Progress in cantilever enhanced photoacoustic spectroscopy,” Appl. Spectrosc. Rev. 48, 16–21 (2008).
16. J. Rakovsky and O. Votava, “A simple photoacoustic detector for highly corrosive gases,” Rev. Sci. Instrum. 88, 013103 (2017). [CrossRef] [PubMed]
17. A. G. Bell, “On the production and reproduction of sound by light,” Am. J. Sci. 20, 305–324 (1880). [CrossRef]
18. K. Maslov and L. V. Wang, “Photoacoustic imaging of biological tissue with intensity-modulated continuous-wave laser,” J. Biomed. Opt. 13, 024006 (2008). [CrossRef] [PubMed]
19. L. V. Wang and H. Wu, Biomedical Optics: Principles and Imaging, (Wiley, 2007).
20. E. Aytac-Kipergil, A. Demirkiran, N. Uluc, S. Yavas, T. Kayikcioglu, S. Salman, S. G. Karamuk, F. O. Ilday, and M. B. Unlu, “Development of a fiber laser with independently adjustable properties for optical resolution photoacoustic microscopy,” Sci. Rep. 6, 38674 (2016). [CrossRef] [PubMed]
21. A. C. Tam, “Applications of photoacoustic sensing techniques,” Rev. Mod. Phys. 58, 381–431 (1986). [CrossRef]
22. A Petschke and P. J. La Riviere, “Comparison of intensity-modulated continuous-wave lasers with a chirped modulation frequency to pulsed lasers for photoacoustic imaging applications,” Biomed. Opt. Exp. 1(4), 1188–1195 (2010). [CrossRef]
23. J. Yao and L. V. Wang, “Sensitivity of photoacoustic microscopy,” Photoacoustics 2(2), 87–101 (2014). [CrossRef] [PubMed]
24. D. K. Yao, C. Zhang, K. Maslov, and L. V. Wang, “Photoacoustic measurement of the Grüneisen parameter of tissue,” J. Biomed. Opt. 19, 017007 (2014). [CrossRef]
25. G. J. Diebold, T. Sun, and M. I. Khan, “Photoacoustic monopole radiation in one, two, and three dimensions,” Phys. Rev. Lett. 67, 3384–3387 (1991). [CrossRef] [PubMed]
26. H. Erkol and M. B. Unlu, “Spectral power density of the random excitation for the photoacoustic wave equation,” AIP Adv. 4,097103 (2014). [CrossRef]
27. H. Erkol, E. Aytac-Kipergil, and M. B. Unlu, “Photoacoustic radiation force on a microbubble,” Phys. Rev. E 90, 023001 (2014). [CrossRef]
28. R. W. Stark and W. M. Heckl, “Fourier transformed atomic force microscopy: tapping mode atomic force microscopy beyond the Hookian approximation,” Surf. Sci. 457, 219–228 (2000). [CrossRef]
29. R. Garcia and R. Perez, “Dynamic atomic force microscopy methods,” Surf. Sci. 47, 197–301 (2002). [CrossRef]
30. G. G. Yaralioglu, A. Atalar, S. R. Manalis, and C. F. Quate, “Analysis and design of an interdigital cantilever as a displacement sensor,” J. Appl. Phys. 83, 7405 (1998). [CrossRef]
31. U. Rabe, K. Janser, and W. Arnold, “Vibrations of free and surface-coupled atomic force microscope cantilevers: Theory and experiment,” Rev. Sci. Instrum. 67, 3281 (1996). [CrossRef]
32. U. S. Lindholm, D. D. Kana, W. H. Chu, and H. N. Abramson, “Elastic vibration characteristics of cantilever plates in water,” J. Ship Res. 9, 11 (1965).
33. J. E. Sader, “Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope,” J. Appl. Phys. 84, 64 (1998). [CrossRef]
34. A. Maali, C. Hurth, R. Boisgard, C. Jai, T. Cohen-Bouhacina, G. Couturier, and J. P. Aime, “Hydrodynamics of oscillating atomic force microscopy cantilevers in viscous fluids,” J. Appl. Phys. 97, 074907 (2005). [CrossRef]
35. M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What is the Young’s Modulus of Silicon?” J. Microelectromech. Syst. 19, 229–238 (2010). [CrossRef]
36. O. N. Pierron, C. C. Abnet, and C. L. Muhlstein, “Methodology for low- and high-cycle fatigue characterization with kHz-frequency resonators,” Sens. Actuat. A Phys. 128, 140–150 (2006). [CrossRef]
37. J. M. Sun and B. S. Gerstman, “Photoacoustic generation for a spherical absorber with impedance mismatch with the surrounding media,” Phys. Rev. E 59, 5772 (1999). [CrossRef]
38. R. G. M. Kolkman, E. Blomme, T. Cool, M. Bilcke, T. G. van Leeuwen, W. Steenbergen, K. A. Grimbergen, and G. J. den Heeten, “Feasibility of noncontact piezoelectric detection of photoacoustic signals in tissue-mimicking phantoms,” J. Biomed. Opt. 15(5), 055011 (2010). [CrossRef]
39. A. Hochreiner, J. Bauer-Marschallinger, P. Burgholzer, B. Jakoby, and T. Berer, “Non-contact photoacoustic imaging using a fiber based interferometer with optical amplification,” Biomed. Opt. Express 4(11), 2322–2331 (2013). [CrossRef] [PubMed]
40. G. Rousseau, A. Blouin, and J.-P Monchalin, “Non-contact photoacoustic tomography and ultrasonography for tissue imaging,” Biomed. Opt. Express 3(1), 16–25 (2012). [CrossRef] [PubMed]
41. K. H. Kim, W. Luo, C. Zhang, C. Tian, L. J. Guo, X. Wang, and X. Fan, “Air-coupled ultrasound detection using capillary-based optical ring resonators,” Sci. Rep. 7, 109 (2017). [CrossRef] [PubMed]
42. S. R. Meier, M. L. Korwin, and C. I. Merzbacher, “Carbon aerogel: a new nonreflective material for the infrared,” Appl. Opt. 39(22), 3940–3944 (2010). [CrossRef]
43. T. Fu, J. Tang, K. Chen, and F. Zhang, “Visible, near-infrared and infrared optical properties of silica aerogels,” Appl. Opt. 55(4), 705–711 (2016). [CrossRef] [PubMed]
44. R. Gerlach, O. Kraus, J. Fricke, P.-Ch. Eccardt, N. Kroemer, and V. Magori, “Modified SiO2 aerogels as acoustic impedance matching layers in ultrasonic devices,” J. Non-Cryst. Solids 145, 227–232 (1992). [CrossRef]
45. T. E. G. Álvarez-Arenas, “Acoustic impedance matching of piezoelectric transducers to the air,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control 51(5), 624–633 (2004). [CrossRef]
46. Y. Liu, H. Miao, V. Aksyuk, and K. Srinivasan, “Wide cantilever stiffness range cavity optomechanical sensors for atomic force microscopy,” Opt. Express 20(16), 18268–18280 (2012). [CrossRef] [PubMed]
47. S. Sevim, N. Shamsudhin, S. Ozer, L. Feng, A. Fakhraee, O. Ergeneman, S. Pane, B. J. Nelson, and H. Torun, “An Atomic Force Microscope with Dual Actuation Capability for Biomolecular Experiments,” Sci. Rep. 6, 27567 (2016). [CrossRef] [PubMed]
