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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 3 — Jan. 31, 2011
  • pp: 2317–2324
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Fano-like resonance in an optically driven atomic force microscope cantilever

Shahrul Kadri, Hideki Fujiwara, and Keiji Sasaki  »View Author Affiliations


Optics Express, Vol. 19, Issue 3, pp. 2317-2324 (2011)
http://dx.doi.org/10.1364/OE.19.002317


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Abstract

We observe Fano-like resonance in the vibration spectrum of an optically driven atomic force microscope cantilever system. The vibration of the cantilever is photothermally induced by exciting it with a 780-nm laser diode. The asymmetry of the resonance curve strongly depends on the position of the excitation spot along the central axis of the cantilever. By using a simple physical model, we could extract and analyze the hidden resonance and continuous components in the vibration spectrum.

© 2011 OSA

1. Introduction

Fano resonance can be characterized as an asymmetric resonance in the excitation spectra of atoms, molecules, and single quantum dots [1

1. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]

,2

2. M. Kroner, A. O. Govorov, S. Remi, B. Biedermann, S. Seidl, A. Badolato, P. M. Petroff, W. Zhang, R. Barbour, B. D. Gerardot, R. J. Warburton, and K. Karrai, “The nonlinear Fano effect,” Nature 451(7176), 311–314 (2008). [CrossRef] [PubMed]

]. 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

3. A. Chiba, H. Fujiwara, J. Hotta, S. Takeuchi, and K. Sasaki, “Fano resonance in a multimode tapered fiber coupled with a microspherical cavity,” Appl. Phys. Lett. 86(26), 261106 (2005). [CrossRef]

,4

4. Y. Lu, J. Yao, X. Li, and P. Wang, “Tunable asymmetrical Fano resonance and bistability in a microcavity-resonator-coupled Mach-Zehnder interferometer,” Opt. Lett. 30(22), 3069–3071 (2005). [CrossRef] [PubMed]

]. 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

5. Y. S. Joe, A. M. Satanin, and C. S. Kim, “Classical analogy of Fano resonances,” Phys. Scr. 74(2), 259–266 (2006). [CrossRef]

]. These show that Fano resonance is universal in physical systems.

Although photothermally induced vibration has been reported for various cantilever types and excitation conditions, each case exhibits vibration spectra with distinct features [18

18. Y. Song, B. Cretin, D. M. Todorovic, and P. Vairac, “Study of photothermal vibrations of semiconductor cantilevers near the resonant frequency,” J. Phys. D Appl. Phys. 41(15), 155106 (2008). [CrossRef]

20

20. G. C. Ratcliff, D. A. Erie, and R. Superfine, “Photothermal modulation for oscillating mode atomic force microscopy in solution,” Appl. Phys. Lett. 72(15), 1911–1913 (1998). [CrossRef]

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

At the peak laser intensity, the cantilever bends downward to position A (Fig. 2
Fig. 2 (a) Modulation intensity profile and (b) cantilever bending mechanism. Top and bottom layers of the cantilever are gold and silicon nitride, respectively
) 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

21. R. W. Stark, T. Drobek, and W. M. Heckl, “Thermomechanical noise of a free v-shaped cantilever for atomic-force microscopy,” Ultramicroscopy 86(1-2), 207–215 (2001). [CrossRef] [PubMed]

,22

22. G. Jourdan, F. Comin, and J. Chevrier, “Mechanical mode dependence of bolometric backaction in an atomic force microscopy microlever,” Phys. Rev. Lett. 101(13), 133904 (2008). [CrossRef] [PubMed]

].

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

Figure 4
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).
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

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

23. D. W. Jordan and P. Smith, Mathematical Techniques, 3rd ed. (Oxford, New York, 2002), Chap. 20.

]. 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

1. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]

]. These phenomena can be expressed using the complex amplitude of the vibration as [3

3. A. Chiba, H. Fujiwara, J. Hotta, S. Takeuchi, and K. Sasaki, “Fano resonance in a multimode tapered fiber coupled with a microspherical cavity,” Appl. Phys. Lett. 86(26), 261106 (2005). [CrossRef]

,12

12. D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444(7115), 75–78 (2006). [CrossRef] [PubMed]

]:

A(ω)=11+iωτ(Ar1(ω/ω0)2+i2βω/ω02+Ac),
(1)

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
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.
. 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.

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

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.

This research was supported by Grant-in-Aid for Scientific Research in the Priority Area “Strong Photon-Molecule Coupling Fields” from the Ministry of Education, Culture, Sports, Science and Technology of Japan. Shahrul Kadri gratefully acknowledges support from the Malaysian Ministry of Higher Education.

References and links

1.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]

2.

M. Kroner, A. O. Govorov, S. Remi, B. Biedermann, S. Seidl, A. Badolato, P. M. Petroff, W. Zhang, R. Barbour, B. D. Gerardot, R. J. Warburton, and K. Karrai, “The nonlinear Fano effect,” Nature 451(7176), 311–314 (2008). [CrossRef] [PubMed]

3.

A. Chiba, H. Fujiwara, J. Hotta, S. Takeuchi, and K. Sasaki, “Fano resonance in a multimode tapered fiber coupled with a microspherical cavity,” Appl. Phys. Lett. 86(26), 261106 (2005). [CrossRef]

4.

Y. Lu, J. Yao, X. Li, and P. Wang, “Tunable asymmetrical Fano resonance and bistability in a microcavity-resonator-coupled Mach-Zehnder interferometer,” Opt. Lett. 30(22), 3069–3071 (2005). [CrossRef] [PubMed]

5.

Y. S. Joe, A. M. Satanin, and C. S. Kim, “Classical analogy of Fano resonances,” Phys. Scr. 74(2), 259–266 (2006). [CrossRef]

6.

M. Z. Ansari and C. Cho, “Deflection, frequeny, and stress characteristics of rectangular, triangular, and step profile microcantilevers for biosensors,” Sensors (Basel Switzerland) 9(8), 6046–6057 (2009).

7.

S. W. Stahl, E. M. Puchner, and H. E. Gaub, “Photothermal cantilever actuation for fast single-molecule force spectroscopy,” Rev. Sci. Instrum. 80(7), 073702 (2009). [CrossRef] [PubMed]

8.

J. R. Barnes, R. J. Stephenson, C. N. Woodburn, S. J. O’Shea, M. E. Welland, T. Rayment, J. K. Gimzewski, and C. Gerber, “A femtojoule calorimeter using micromechanical sensors,” Rev. Sci. Instrum. 65(12), 3793–3798 (1994). [CrossRef]

9.

S. Nishida, D. Kobayashi, T. Sakurada, T. Nakazawa, Y. Hoshi, and H. Kawakatsu, “Photothermal excitation and laser Doppler velocimetry of higher cantilever vibration modes for dynamic atomic force microscopy in liquid,” Rev. Sci. Instrum. 79(12), 123703 (2008). [CrossRef]

10.

R. M. A. Fatah, “Mechanisms of optical of micromechanical resonators,” Sens. Actuators A Phys. 33(3), 229–236 (1992). [CrossRef]

11.

C. H. Metzger and K. Karrai, “Cavity cooling of a microlever,” Nature 432(7020), 1002–1005 (2004). [CrossRef] [PubMed]

12.

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444(7115), 75–78 (2006). [CrossRef] [PubMed]

13.

N. Selden, C. Ngalande, S. Gimelshein, E. P. Muntz, A. Alexeenko, and A. Ketsdever, “Area and edge effects in radiometric forces,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(4), 041201 (2009). [CrossRef] [PubMed]

14.

C. Metzger, I. Favero, A. Ortlieb, and K. Karrai, “Optical self cooling of a deformable Fabry-Perot cavity in the classical limit,” Phys. Rev. B 78(3), 035309 (2008). [CrossRef]

15.

D. Ramos, J. Mertens, M. Calleja, and J. Tamayo, “Study of the origin of bending induced by bimetallic effect on microcantilever,” Sensors (Basel Switzerland) 7(9), 1757–1765 (2007).

16.

S. Kadri, H. Fujiwara, and K. Sasaki, “Analysis of photothermally induced vibration in metal coated AFM cantilever,” Proc. SPIE 7743, 774307, 774307-6 (2010). [CrossRef]

17.

A. Wig, A. Passian, E. Arakawa, T. L. Ferrell, and T. Thundat, “Optical thin-film interference effects in microcantilevers,” J. Appl. Phys. 95(3), 1162–1165 (2004). [CrossRef]

18.

Y. Song, B. Cretin, D. M. Todorovic, and P. Vairac, “Study of photothermal vibrations of semiconductor cantilevers near the resonant frequency,” J. Phys. D Appl. Phys. 41(15), 155106 (2008). [CrossRef]

19.

K. Hane, T. Iwatuki, S. Inaba, and S. Okuma, “Frequency shift on a micromachined resonator excited photothermally in vacuum,” Rev. Sci. Instrum. 63(7), 3781–3782 (1992). [CrossRef]

20.

G. C. Ratcliff, D. A. Erie, and R. Superfine, “Photothermal modulation for oscillating mode atomic force microscopy in solution,” Appl. Phys. Lett. 72(15), 1911–1913 (1998). [CrossRef]

21.

R. W. Stark, T. Drobek, and W. M. Heckl, “Thermomechanical noise of a free v-shaped cantilever for atomic-force microscopy,” Ultramicroscopy 86(1-2), 207–215 (2001). [CrossRef] [PubMed]

22.

G. Jourdan, F. Comin, and J. Chevrier, “Mechanical mode dependence of bolometric backaction in an atomic force microscopy microlever,” Phys. Rev. Lett. 101(13), 133904 (2008). [CrossRef] [PubMed]

23.

D. W. Jordan and P. Smith, Mathematical Techniques, 3rd ed. (Oxford, New York, 2002), Chap. 20.

24.

D. Ramos, J. Tamayo, J. Mertens, and M. Calleja, “Photothermal excitation of microcantilevers in liquids,” J. Appl. Phys. 99(12), 124904 (2006). [CrossRef]

25.

E. Finot, A. Passian, and T. Thundat, “Measurement of mechanical properties of cantilever shaped materials,” Sensors (Basel Switzerland) 8(5), 3497–3541 (2008).

OCIS Codes
(120.7280) Instrumentation, measurement, and metrology : Vibration analysis
(190.4870) Nonlinear optics : Photothermal effects
(260.5740) Physical optics : Resonance

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: November 22, 2010
Revised Manuscript: January 14, 2011
Manuscript Accepted: January 14, 2011
Published: January 24, 2011

Virtual Issues
Vol. 6, Iss. 2 Virtual Journal for Biomedical Optics

Citation
Shahrul Kadri, Hideki Fujiwara, and Keiji Sasaki, "Fano-like resonance in an optically driven atomic force microscope cantilever," Opt. Express 19, 2317-2324 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-2317


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References

  1. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]
  2. M. Kroner, A. O. Govorov, S. Remi, B. Biedermann, S. Seidl, A. Badolato, P. M. Petroff, W. Zhang, R. Barbour, B. D. Gerardot, R. J. Warburton, and K. Karrai, “The nonlinear Fano effect,” Nature 451(7176), 311–314 (2008). [CrossRef] [PubMed]
  3. A. Chiba, H. Fujiwara, J. Hotta, S. Takeuchi, and K. Sasaki, “Fano resonance in a multimode tapered fiber coupled with a microspherical cavity,” Appl. Phys. Lett. 86(26), 261106 (2005). [CrossRef]
  4. Y. Lu, J. Yao, X. Li, and P. Wang, “Tunable asymmetrical Fano resonance and bistability in a microcavity-resonator-coupled Mach-Zehnder interferometer,” Opt. Lett. 30(22), 3069–3071 (2005). [CrossRef] [PubMed]
  5. Y. S. Joe, A. M. Satanin, and C. S. Kim, “Classical analogy of Fano resonances,” Phys. Scr. 74(2), 259–266 (2006). [CrossRef]
  6. M. Z. Ansari and C. Cho, “Deflection, frequeny, and stress characteristics of rectangular, triangular, and step profile microcantilevers for biosensors,” Sensors (Basel Switzerland) 9(8), 6046–6057 (2009).
  7. S. W. Stahl, E. M. Puchner, and H. E. Gaub, “Photothermal cantilever actuation for fast single-molecule force spectroscopy,” Rev. Sci. Instrum. 80(7), 073702 (2009). [CrossRef] [PubMed]
  8. J. R. Barnes, R. J. Stephenson, C. N. Woodburn, S. J. O’Shea, M. E. Welland, T. Rayment, J. K. Gimzewski, and C. Gerber, “A femtojoule calorimeter using micromechanical sensors,” Rev. Sci. Instrum. 65(12), 3793–3798 (1994). [CrossRef]
  9. S. Nishida, D. Kobayashi, T. Sakurada, T. Nakazawa, Y. Hoshi, and H. Kawakatsu, “Photothermal excitation and laser Doppler velocimetry of higher cantilever vibration modes for dynamic atomic force microscopy in liquid,” Rev. Sci. Instrum. 79(12), 123703 (2008). [CrossRef]
  10. R. M. A. Fatah, “Mechanisms of optical of micromechanical resonators,” Sens. Actuators A Phys. 33(3), 229–236 (1992). [CrossRef]
  11. C. H. Metzger and K. Karrai, “Cavity cooling of a microlever,” Nature 432(7020), 1002–1005 (2004). [CrossRef] [PubMed]
  12. D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444(7115), 75–78 (2006). [CrossRef] [PubMed]
  13. N. Selden, C. Ngalande, S. Gimelshein, E. P. Muntz, A. Alexeenko, and A. Ketsdever, “Area and edge effects in radiometric forces,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(4), 041201 (2009). [CrossRef] [PubMed]
  14. C. Metzger, I. Favero, A. Ortlieb, and K. Karrai, “Optical self cooling of a deformable Fabry-Perot cavity in the classical limit,” Phys. Rev. B 78(3), 035309 (2008). [CrossRef]
  15. D. Ramos, J. Mertens, M. Calleja, and J. Tamayo, “Study of the origin of bending induced by bimetallic effect on microcantilever,” Sensors (Basel Switzerland) 7(9), 1757–1765 (2007).
  16. S. Kadri, H. Fujiwara, and K. Sasaki, “Analysis of photothermally induced vibration in metal coated AFM cantilever,” Proc. SPIE 7743, 774307, 774307-6 (2010). [CrossRef]
  17. A. Wig, A. Passian, E. Arakawa, T. L. Ferrell, and T. Thundat, “Optical thin-film interference effects in microcantilevers,” J. Appl. Phys. 95(3), 1162–1165 (2004). [CrossRef]
  18. Y. Song, B. Cretin, D. M. Todorovic, and P. Vairac, “Study of photothermal vibrations of semiconductor cantilevers near the resonant frequency,” J. Phys. D Appl. Phys. 41(15), 155106 (2008). [CrossRef]
  19. K. Hane, T. Iwatuki, S. Inaba, and S. Okuma, “Frequency shift on a micromachined resonator excited photothermally in vacuum,” Rev. Sci. Instrum. 63(7), 3781–3782 (1992). [CrossRef]
  20. G. C. Ratcliff, D. A. Erie, and R. Superfine, “Photothermal modulation for oscillating mode atomic force microscopy in solution,” Appl. Phys. Lett. 72(15), 1911–1913 (1998). [CrossRef]
  21. R. W. Stark, T. Drobek, and W. M. Heckl, “Thermomechanical noise of a free v-shaped cantilever for atomic-force microscopy,” Ultramicroscopy 86(1-2), 207–215 (2001). [CrossRef] [PubMed]
  22. G. Jourdan, F. Comin, and J. Chevrier, “Mechanical mode dependence of bolometric backaction in an atomic force microscopy microlever,” Phys. Rev. Lett. 101(13), 133904 (2008). [CrossRef] [PubMed]
  23. D. W. Jordan and P. Smith, Mathematical Techniques, 3rd ed. (Oxford, New York, 2002), Chap. 20.
  24. D. Ramos, J. Tamayo, J. Mertens, and M. Calleja, “Photothermal excitation of microcantilevers in liquids,” J. Appl. Phys. 99(12), 124904 (2006). [CrossRef]
  25. E. Finot, A. Passian, and T. Thundat, “Measurement of mechanical properties of cantilever shaped materials,” Sensors (Basel Switzerland) 8(5), 3497–3541 (2008).

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