本篇論文考量三維幾何結構對光作用力與光力矩變化。我們選用不同波長的圓偏振入射光源，透過有限元素法得到介電質表面電磁場分布，代入馬克斯威爾應力方程得到每一網格點之的光應力，將每網格點光作用力，對位置向量進行外積，接著加總每網格的光力矩，得到全體三維結構之光力矩。接著分析局部結構電場分布與光力矩振盪表現，透過篩選不同入射波長，達到光操作物體旋轉方向的成效，並以圓盤結構為基礎，延伸設計為各樣結構進行分析，觀察其光力矩振盪行為變化。

最後給出我們自行提出的5-layer結構，此設計於1.66 um ~0.62 um處皆為正光力矩，而0.62 um ~0.51 um皆為負光力矩，在特定波長達到正負力矩一分為二的效果，除此之外，y分量與z分量之光力矩於1.66 um~0.51 um波長內皆趨於零，成功降低其正負光力矩對波長之振盪行為，其設計優勢將光操作物體旋轉領域，帶來全新穩定操作微米物質旋轉的成果。

In this thesis, we discuss the optical force and optical torque of circularly polarized incident light on three-dimensional objects. We use circularly polarized light as incident light in different wavelengths. The electromagnetic field distribution on the grid surface is obtained by the finite element method, and the light stress on each grid point is obtained from Maxwell's stress equation, and the overall optical force is evaluated by summing over the contributions from all grid points. Similarly, to calculate the optical torque of a three-dimensional object, we first calculate at each grid point the cross product of the position vector and the optical force, then sum over the surface contributions from the whole geometric structure. We analyzed the surface distribution of the electric field and the local oscillation behavior of the optical torque density on the surfaces of various three-dimensional structures. These results provide us with the information on how to change the wavelength of incident light to control the direction of rotation. Based on the knowledge obtained in the study of disc structures, we further designed various structures and analyzed their optical torque oscillation characteristics.

Finally, we proposed a 5-layer structure. This design has positive optical torque in 1.66 um ~ 0.62 um, and negative optical torque in 0.62 um ~ 0.51 um. In addition, the optical torque of the y and z components approach zero within the wavelength of 1.66 um ~ 0.51 um. This structure successfully reduces the unwanted oscillation behavior of the optical torque appearing in the positive or negative region, and thus has the advantage of providing the necessary stability in controlling the rotation direction of microscopic structures.

[1] Ashkin, A., "Acceleration and Trapping of Particles by Radiation Pressure," Physical Review Letters 24, 156 (1970).
[2] Ashkin, A. and J. M. Dziedzic, "Observation of Resonances in the Radiation Pressure on Dielectric Spheres," Physical Review Letters 38, 1351 (1977).
[3] Ashkin, A., J. M. Dziedzic, J. E. Bjorkholm and S. Chu., "Observation of a single-beam gradient force optical trap for dielectric particles," Optics Letters 11, 288 (1986).
[4] Phillips, W. D., "Nobel Lecture: Laser cooling and trapping of neutral atoms," Reviews of Modern Physics 70, 721 (1998).
[5] Steane, A. M., M. Chowdhury and C. J. Foot., "Radiation force in the magneto-optical trap," Journal of the Optical Society of America B 9, 2142 (1992).
[6] Zhang, H. and K.-K. Liu., "Optical tweezers for single cells," Journal of the Royal Society, Interface 5, 671 (2008).
[7] Zhong, M. C., X. B. Wei, J. H. Zhou, Z. Q. Wang and Y. M. Li., "Trapping red blood cells in living animals using optical tweezers," Nature Communications 4, 1768 (2013).
[8] Zhang, Z., T. E. P. Kimkes and M. Heinemann., "Manipulating rod-shaped bacteria with optical tweezers," Scientific Reports 9, 19086 (2019).
[9] Povinelli, M. L., M. Lončar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso and J. D. Joannopoulos., "Evanescent-wave bonding between optical waveguides," Optics Letters 30, 3042 (2005).
[10] Ma, J. and M. L. Povinelli., "Large tuning of birefringence in two strip silicon waveguides via optomechanical motion." Optics Express 17, 17818 (2009).
[11] 欒丕綱、陳啟昌--光子晶體 (從蝴蝶翅膀到奈米光子學)，第二版，五南出版社 (2010).
[12] Pernice, W. H. P., M. Li and H. X. Tang., "Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate," Optics Express 17, 1806 (2009).
[13] Eichenfield, M., J. Chan, R. M. Camacho, K. J. Vahala and O. Painter., "Optomechanical crystals," Nature 462, 78 (2009).
[14] Chan, J., M. Eichenfield, R. Camacho and O. Painter., "Optical and mechanical design of a “zipper” photonic crystal optomechanical cavity," Optics Express 17, 3802 (2009).
[15] Wiederhecker, G. S., L. Chen, A. Gondarenko and M. Lipson., "Controlling photonic structures using optical forces," Nature 462, 633 (2009).
[16] Van Thourhout, D. and J. Roels., "Optomechanical device actuation through the optical gradient force," Nature Photonics 4, 211 (2010).
[17] Einat, A. and U. Levy., "Analysis of the optical force in the Micro Ring Resonator," Optics Express 19, 20405 (2011).
[18] Li, M., S. Yan, B. Yao, Y. Liang and P. Zhang., "Spinning and orbiting motion of particles in vortex beams with circular or radial polarizations," Optics Express 24, 20604 (2016).
[19] Diniz, K., R. S. Dutra, L. B. Pires, N. B. Viana, H. M. Nussenzveig and P. A. Maia Neto., "Negative optical torque on a microsphere in optical tweezers," Optics Express 27, 5905 (2019).
[20] Chen, J., J. Ng, K. Ding, K. H. Fung, Z. Lin and C. T. Chan., "Negative optical torque," Scientific Reports 4, 6386 (2014).
[21] Han, F., J. A. Parker, Y. Yifat, C. Peterson, S. K. Gray, N. F. Scherer and Z. Yan ., "Crossover from positive to negative optical torque in mesoscale optical matter," Nature Communications 9, 4897 (2018).
[22] David J. Griffiths, “Introduction to Electrodynamics,” 3rd Edition, ‎Cambridge University Press (1999).
[23] Coggon, J. H.., "ELECTROMAGNETIC AND ELECTRICAL MODELING BY THE FINITE ELEMENT METHOD," GEOPHYSICS 36, 132 (1971).
[24] 皮托科技, COMSOL Multiphysics 電磁模擬有限元素分析