本論文在探討掃描式近場光學顯微鏡，探針掃描樣品時之振動頻率(frequency)、靈敏度(sensitivity)與接觸力(interactive force)等三部分。在振動頻率方面，以線性彈簧及黏滯阻尼模擬探針與樣品之接觸力，並考慮探針之旋轉慣性(rotary inertia)與剪應變(shear deformation)之效應，即應用Timoshenko beam理論，推導出在阻尼振動下，探針在不同接觸勁度(contact stiffness) 、阻尼與長細比之撓曲(damping flexural vibration)頻率閉合(closed form)解。經分析顯示，探針前五個模態之振動頻率，除隨探針與樣品之接觸勁性變大而增加外，亦隨探針之長細比增加而增加。而且經比較以Bernoulli-Euler beam理論推導出，具阻尼振動之撓曲頻率結果顯示，在高頻、高接觸勁度與低長細比之探針，其旋轉慣性與剪應變之效應較顯著。
在靈敏度方面，對非均勻形狀之圓錐型探針，以Bernoulli-Euler beam理論，應用Rayleigh-Ritz method推導出，具阻尼振動之撓曲與軸向之靈敏度。經分析顯示，在低接觸勁度時，無論撓曲或軸向振動靈敏度，隨著阻尼增加而降低。而且，在接觸勁度低時，除第一模態外，錐形探針之撓曲振動零敏度，隨著錐度增加而降低，其餘模態或高接觸勁度時則相反。但軸向振動零敏度，無論高或低的接觸勁度，隨著錐度增加而皆增高。
在探針與樣品之接觸力，以逆運算之共軛梯度法(conjugate gradient method)，及伴隨方程式(adjoint equation)，計算出錐形探針於掃描過程中，探針之側向接觸剪力或軸向接觸力。即在無法預知探針掃描之接觸力過程函數，僅以探針尖端之側向或軸向撓度值，得準確的估算探針與樣品間之接觸力。經數值驗證顯示：縱使探針尖端之量測撓度值含有誤差，或迭代之應力始猜值與真實值相差大，亦能計算出準確之接觸力。

The objective of this dissertation is to study the vibration of Scanning Near-Field Optical Microscope (SNOM) probe including frequency, sensitivity, and interactive force, respectively, while the probe scans samples.
In the frequency analysis, the effect of interactive damping on the flexural vibration frequency for the SNOM’s probe based on the Timoshenko beam theory, with the effects of shear deformation and rotary inertia, has been analyzed. Besides, the effects of the contact stiffness, damping factor and the ratio of different probe dimensions on the damping vibration frequency were studied. The results show that increasing the ratio of probe length to radius increases the vibration frequency of first five modes. In addition, the resonant frequencies based on the Bernoulli-Euler beam theory and the Timoshenko beam theory are compared. When the contact stiffness is very large for the higher modes, the effects of shear deformation and rotary inertia on the frequency become significant. This observation that the Timoshenko beam theory is able to predict the frequencies of flexural vibrations of the higher modes with higher contact stiffness for the SNOM fiber probe.
In the sensitivity analysis, the effect of interactive damping on the sensitivity of flexural and axial vibration modes of SNOM with a tapered optical fiber probe has been analyzed. The interaction of the SNOM probe with the sample surface is modeled by a combination of a spring and a dashed pot in the lateral direction and a similar combination in the axial direction. An approximate form for the sensitivities of both modes was derived by using the Rayleigh-Ritz method. The results show that the interactive damping will decrease the sensitivities of both flexural and axial vibration modes when the contact stiffness is low. The more the damping effect, the lower the sensitivities are. In addition, when the contact stiffness was low, the mode 1 of flexural sensitivity slightly increased as the tapered angle decreased, but the flexural sensitivity increased as the tapered angle increased in high modes. However, the axial sensitivity apparently decreased as the tapered angle decreased. When the contact stiffness became higher, the sensitivities of both flexural and axial vibration modes increased as the tapered angle increased.
In the interactive force, the conjugate gradient method of minimization with an adjoint equation is successfully applied to solve the inverse problem in estimating the shear or axial force between the tapered probe and sample during the scanning process SNOM. While knowing the available deflection at the tapered probe tip, the determination of the interaction force is regarded as an inverse vibration problem. In the estimating processes, no prior information on the functional form of the unknown quantity is required. The accuracy of the inverse analysis is examined by using the simulated exact and inexact measurements of deflection at the tapered probe tip. Numerical results show that good estimations on the interaction shear force can be obtained for all the test cases considered in this study.

[1] 蔡定平, “近場光學顯微術”, 自然科學簡訊, 第7卷, 第3期, P110, 1995.
[2] 蔡定平, “近場光學顯微術及其應用”, 科儀新知, 第17卷, 第5期, P4, 1996.
[3] 蔡定平, “掃描近場光學顯微儀”, 科儀新知, 第21卷, 第5期, P17, 2000.
[4] 吳靖宙,張憲章, “掃描式近場光學顯微鏡於生物樣本的量測與應用” , 科儀新知, 第23卷, 第5期, P88, 2002.
[5] 王振宇,潘善鵬, “奈米粒徑量測技術及先期能力試驗”, 量測資訊,第99期, P 9, 2004.
[6] R. Toledo-Crow, P. Yang, Y. Chen, and M. Vaez-Iravani, ” Near-field differential scanning optical microscope with atomic force regulation”, Appl. Phys. Lett. 60, 2957, 1992.
[7] E. Betzig, P. L. Finn, and J. S. Weiner, “Combined shear force and near-field scanning optical microscopy”, Appl. Phys. Lett. 60, 2484, 1992.
[8] B. Vohnsen, S. Bozhevolnyi, and R. Olesen , “Study of shear force technique for near-field microscopy with an uncoated fiber tip”, Ultramicroscopy 61, 207, 1995.
[9] R. Bachelot, P. Glryzes, and A. G. Boccara,”Near-field optical microscope based on local perturbation of a diffraction spot”, Opt. Lett. 20, 18, 1924, 1995.
[10] U. Durig, D. W. Pohl, and F. Rohner, “Near-field optical-scanning microscopy”, J. Appl. Phys. 59, 3318, 1986.
[11] K. Karrai, and R. D. Grober,” Piezoelectric tip-sample distance control for near field optical microscopes”, Appl. Phys. Lett. 66, 1842, 1995.
[12] K. Karrai, and R. D. Grober, “Piezo-electric tuning fork tip-sample distance control for near field optical microscopes”, Ultramicroscopy 61, 197, 1995.
[13] O. Hollricher, R. Brunner, and O. Marti “Piezoelectrial shear-force distance control in near-field optical microscopy for biological applications”, Ultramicroscopy 71, 143, 1998.
[14] D. P. Tsai, and Y. Y. Lu, ”Tapping-mode tuning fork force sensing for near-field scanning optical microscopy”, Appl. Phys. Lett. 73, 2724, 1998.
[15] D. Sard, Scanning Force Microscopy, Oxford University Press, New York, 1991.
[16] A. G. T. Ruiter, K. O. van der Werf, J. A. Veerman, M. F. Garica-Parajo, W. H. J. Rensen, N. F., and Hulst, ”Tuning fork shear-force feedback”, Ultramicroscopy 71, 149, 1998.
[17] J. W. P. Hsu, Mark Lee, and B. S. Deaver, “A nonoptical tip–sample distance control method for near-field scanning optical microscopy using impedance changes in an electromechanical system”, Rev. Sci. Instrum. 66, 3177, 1995.
[18] J. Barenz, O. Hollricher, and O. Marti, “An easy-to-use non-optical shear-force distance control for near-field optical microscopes”, Rev. Sci. Instrum. 67, 1912, 1996.
[19] A. Drabenstedt, J. Wrachtrup, and C. von Borczyskowski, “A distance regulation scheme for scanning near-field optical microscopy”, Appl. Phys. Lett. 68, 3497, 1996.
[20] Y. H. Chuang, C. J. Wang, J. Y. Huang, and C. L. Pan, “Nonoptical tip–sample distance control for scanning near-field optical microscopy”, Appl. Phys. Lett. 69, 3312, 1996.
[21] J. A. Turner and J. S. Wiehn, ”Sensitivity of flexural and torsional vibration modes of atomic force microscope cantilevers to surface stiffness variations”, Nanotechnology 12, 322, 2001.
[22] H. L. Lee, W. J. Chang and Y. C Yang, ” Flexural sensitivity of a V-shaped cantilever of an atomic force microscope”, Mater. Chem. Phys. 92, 438, 2005.
[23] J. C. Hsu, H. L. Lee, and W. J. Chang, ”Flexural Vibration Frequency of Atomic Force Microscope Cantilevers Using the Timoshenko Beam Model”, Nanotechnology, 18, 285503, 2007
[24] T. H. Fang and W. J. Chang, “Sensitivity Analysis of Scanning Near-Field Optical Microscope Probe”, Opt. Laser Technol. 35, 267, 2003.
[25] W. J.Chang, T. H. Fang, H. L. Lee and Y. C, Yang ”Vibration sensitivity of the scanning near-field optical microscope with a tapered optical fiber probe”, Ultramicroscopy. 102, 85, 2005.
[26] H. L. Lee and T. Y. Chen,” Effect of Interactive Damping on Vibration Sensitivities of Scanning Near-Field Optical Microscope Probe”, Appl. Phys. B-Lasers Opt. 88, 179, 2007.
[27] W. J. Chang, T. Y. Chen, and H. L. Lee, “Vibration Analysis of Scanning Near-Field Optical Microscope Probe Using the Timoshenko Beam Model”, Jpn. J. Appl. Phys. 47, 3657, 2008.
[28] H. L. Lee, W. J. Chang, W. L. Chen, and Y. C. Yang, “An inverse problem of estimating the heat source in tapered optical fibers for scanning near-field optical microscopy”, Ultramicroscopy. 107, 656, 2007.
[29] W. L. Chen, Y. C. Yang, and H. L. Lee, “Inverse problem in determining convection heat transfer coefficient of annular fin”, Energy Conv. Manag. 48, 1081, 2007
[30] G.M.L.Gladwell Inverse problem in Vibration, The Netherlands, Kluwer Academic Pubishers, 1986.
[31] L. Starek and D. J. INMAN, ”On the inverse vibration problem with rigid-body modes”, J. Appl. Mech.-Trans. ASME 58, 1101, 1991.
[32] L. STAREK and D. J. INMAN and A. KRESS, “A symmetric inverse vibration problem”, J. Vib. Acoust.-Trans. ASME 114, 565, 1992.
[33] L. STAREK and D. J. INMAN, ”A symmetric inverse vibration problem with overdamped modes”, J. Sound Vibr. 181, 893, 1995.
[34] C. H. Huang, “An Inverse Nonlinear Force Vibration Problem of Estimating the External Forces in a Damped System with Time-Dependent System Parameters”, J. Sound Vibr. 242, 749, 2001.
[35] C. H. Huang, ”A Generalized Inverse Force Vibration Problem for Simultaneously Estimating the Time-Dependent External Forces”, Appl. Math. Model. 29, 1022, 2005.
[36] W. J. Chang, J. C. Hsu, and T. H. Lai, “Inverse Calculation of the Tip-Sample Interaction Force in Atomic Force Microscopy by the Conjugate Gradient Method”, J. Phys. D-Appl. Phys. 37, 1123, 2004.
[37] W. J. Chang, T. H. Fang, and C. I, Weng, “Inverse Determination of the Cutting Force on Nanoscale Processing Using Atomic Force Microscopy”, Nanotechnology, 15, 427, 2004.
[38] W. J. Chang, C. M. Lin, J. F. Lee, and S. L. Lin, “Inverse Determination of Tip-Sample Damping Force in Atomic Force Microscopy”, Phys. Lett. A 343, 79, 2005.
[39] W. J. Chang, and T. H. Fang, “An Inverse Method for Determining the Interaction Force Between the Probe and Sample Using Scanning Near-Field Optical Microscopy”, Phys. Lett. A 348, 260, 2006.
[40] H. L. Lee, “Inverse estimation of the tapered pro-sample shear force of scanning near-field optical microscope”, Ultramicroscopy 106, 547, 2006.
[41] L. Meirovuth, Analytical Methods in Vibrations, Macmillan, New York, 1967.
[42] .C. R. Cowper, ”The shear coefficient in Timoshenko’s Beam Theory”, J. Appl. Mech., 33, 335, 1996.
[43] W. Weaver, JR., S. P. Timoshenko, and D. H. Young, Vibration problems in engineering, 5th ed., John Wiley & Sons, New York, 1990.
[44] O.M. Alifanov, Inverse Heat Transfer Problem, Springer-Verlag, New York, 1994
[45] T. Y. Chen and H. L. Lee,” Damping Vibration of Scanning Near-Field Optical Microscope Probe Using the Timoshenko Beam Model”, Microelectron. J. June, 2008, (in press).