## Fano-like resonance in an optically driven atomic force microscope cantilever |

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

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]

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. 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]

## 2. Experiment

^{−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. 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

**(**

*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]

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

*τ,*

*A**,*

_{r}

*A**,*

_{c}*f*and

_{0}*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

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]

*τ, f*and

_{0}*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**and*

_{r}

*A**), 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.*

_{c}**(**

*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.

^{−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.

*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

## References and links

1. | U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. |

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 |

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. |

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. |

5. | Y. S. Joe, A. M. Satanin, and C. S. Kim, “Classical analogy of Fano resonances,” Phys. Scr. |

6. | M. Z. Ansari and C. Cho, “Deflection, frequeny, and stress characteristics of rectangular, triangular, and step profile microcantilevers for biosensors,” Sensors (Basel Switzerland) |

7. | S. W. Stahl, E. M. Puchner, and H. E. Gaub, “Photothermal cantilever actuation for fast single-molecule force spectroscopy,” Rev. Sci. Instrum. |

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. |

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. |

10. | R. M. A. Fatah, “Mechanisms of optical of micromechanical resonators,” Sens. Actuators A Phys. |

11. | C. H. Metzger and K. Karrai, “Cavity cooling of a microlever,” Nature |

12. | D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature |

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. |

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 |

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) |

16. | S. Kadri, H. Fujiwara, and K. Sasaki, “Analysis of photothermally induced vibration in metal coated AFM cantilever,” Proc. SPIE |

17. | A. Wig, A. Passian, E. Arakawa, T. L. Ferrell, and T. Thundat, “Optical thin-film interference effects in microcantilevers,” J. Appl. Phys. |

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. |

19. | K. Hane, T. Iwatuki, S. Inaba, and S. Okuma, “Frequency shift on a micromachined resonator excited photothermally in vacuum,” Rev. Sci. Instrum. |

20. | G. C. Ratcliff, D. A. Erie, and R. Superfine, “Photothermal modulation for oscillating mode atomic force microscopy in solution,” Appl. Phys. Lett. |

21. | R. W. Stark, T. Drobek, and W. M. Heckl, “Thermomechanical noise of a free v-shaped cantilever for atomic-force microscopy,” Ultramicroscopy |

22. | G. Jourdan, F. Comin, and J. Chevrier, “Mechanical mode dependence of bolometric backaction in an atomic force microscopy microlever,” Phys. Rev. Lett. |

23. | D. W. Jordan and P. Smith, |

24. | D. Ramos, J. Tamayo, J. Mertens, and M. Calleja, “Photothermal excitation of microcantilevers in liquids,” J. Appl. Phys. |

25. | E. Finot, A. Passian, and T. Thundat, “Measurement of mechanical properties of cantilever shaped materials,” Sensors (Basel Switzerland) |

**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

- U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]
- 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]
- 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]
- 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]
- Y. S. Joe, A. M. Satanin, and C. S. Kim, “Classical analogy of Fano resonances,” Phys. Scr. 74(2), 259–266 (2006). [CrossRef]
- 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).
- 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]
- 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]
- 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]
- R. M. A. Fatah, “Mechanisms of optical of micromechanical resonators,” Sens. Actuators A Phys. 33(3), 229–236 (1992). [CrossRef]
- C. H. Metzger and K. Karrai, “Cavity cooling of a microlever,” Nature 432(7020), 1002–1005 (2004). [CrossRef] [PubMed]
- D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444(7115), 75–78 (2006). [CrossRef] [PubMed]
- 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]
- 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]
- 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).
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- D. W. Jordan and P. Smith, Mathematical Techniques, 3rd ed. (Oxford, New York, 2002), Chap. 20.
- D. Ramos, J. Tamayo, J. Mertens, and M. Calleja, “Photothermal excitation of microcantilevers in liquids,” J. Appl. Phys. 99(12), 124904 (2006). [CrossRef]
- 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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