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在光镊的射线模型中，追迹光线在界面的折射反射光是较基本的也是较复杂的问题.传统的几何光学法计算光线的方位，对于一些不规则的物体来说，存在一定的困难度.本文提出使用四元数简化空间光线追迹，从而可计算非球形颗粒的光阱力.以入射光线和界面法线的外积确定入射面的法线为旋转轴；根据折射定律确定由入射光线旋转到反射光线和折射光线的角度.将入射光线以四元数表示，根据四元数旋转，可获得反射光线和折射光线的空间矢量.根据光子的动量变化，求得光阱对微粒的作用力.本文以球差影响下的不同形变的椭球为计算示例，模拟了椭球在光阱下的动力学行为，结果表明椭球的横向和轴向俘获效率受到各方向变形系数的影响；球差增大降低了轴向的最大俘获效率，稳定俘获位置也随之朝负轴偏离中心更远；在固定球差作用下，最大轴向俘获效率与轴向变形系数相关，在特定的变形下轴向俘获效率变得较大.由此验证了四元数方法的正确性、实用性与普遍性.In the ray-optics (RO) model of optical tweezers, tracing refractive and reflected rays with vectors play important roles in calculating the trapping forces. Traditional ray-tracing method with solid geometry, to some extent, is complicated in determining the orientations of those refractive and reflected rays according to spatial incident rays. It is difficult to calculate the trapping forces for irregular particles. In this paper, quaternion is proposed to rotate ray vectors for simplifying the traces of all kinds of spatial rays. Then, it is appropriate to calculate the trapping force of an ellipsoid bead. Based on the algorithm of quaternion and the convention between the interface normal and angular directions, the direction of normal always points from optically denser medium to thinner medium. The rotation axis is the cross product of the incident ray and the interface normal. And the positive angular direction can be determined by right-hand rule based on the orientation of the rotation axis. According to Snell' law, the rotation angle between the incident ray and refractive/reflected ray can be determined. The quaternion for rotation consists of rotation axis and angle. So the refractive and reflected rays are both determined by quaternions of incident ray and rotation based on rotation rules. Furthermore, the force on interface can also be calculated according to momentum changes of the photon before and after the interface refraction and reflection. The quaternion method is used to analyze the effects of coverslip position and deformation ratio on the trapping efficiency of ellipsoid particles. Our simulative results show that the lateral and axial trapping efficiencies are obviously affected by the deformation of the ellipsoid itself. No matter whether the bead deforms transversely or axially, the transverse and axial trapping efficiencies both become larger at a specific deformation. Meantime, the increase of the spherical aberration reduces the maximum axial trapping efficiency, and the equilibrium position of the bead becomes farther away from the center. Using quaternion method, the calculation of refractive lightvector can be simplified in comparison with by using the method of Euclidean geometry or transformation matrix. Theoretically, this quaternion can be used to trace rays on any irregular geometric surfaces. In conclusion, the method of quaternion can make ray tracing easier and extend the applications of RO model.
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Keywords:
- quaternion /
- ray tracing /
- ellipsoidal bead /
- trapping force
[1] Schnitzer M J, Block S M 1997 Nature 388 386
[2] Qian H, Chen H, Yan J 2016 Acta Phys. Sin. 65 188706 (in Chinese)[钱辉, 陈虎, 严洁 2016 65 188706]
[3] Fazal F M, Block S M 2011 Nature Photon. 5 318
[4] Xia P, Zhou J H, Song X Y, Wu B, Liu X, Li D, Zhang S Y, Wang Z K, Yu H J, Ward T, Zhang J C, Li Y M, Wang X N, Chen Y, Guo Z, Yao X B 2014 J. Mol. Cell Biol. 6 240
[5] Ouyang H D, Wei M T 2010 Annu. Rev. Phys. Chem. 61 421
[6] Zhou J H, Ren H L, Cai J, Li Y M 2008 Appl. Opt. 47 6307
[7] Xu S H, Li Y M, Lou L R 2006 Chin. Phys. 15 1391
[8] Wright W H, Sonek GJ, Berns M W 1994 Appl. Opt. 33 1735
[9] Stilgoe A B, Nieminen T A, Knoner G, Heckenberg N R, Rubinsztein-Dunlop H 2008 Opt. Express 16 15039
[10] Gu Y Q, Gong Z, Lou L R, Li Y M 2007 Appl. Laser 27 98 (in Chinese)[谷勇强, 龚錾, 楼立人, 李银妹 2007 应用激光 27 98]
[11] Ashkin A 1992 Biophys. J. 61 569
[12] Fällman E, Axner O 2003 Appl. Opt. 42 3915
[13] Reihani S N S, Oddershede L B 2007 Opt. Lett. 32 1998
[14] Sidick E, Collins S D, Knoesen A 1997 Appl. Opt. 36 6423
[15] Bareil P B, Sheng Y, Chiou A 2006 Opt. Express 14 12503
[16] Zhou J H, Zhong M C, Wang Z Q, Li Y M 2012 Opt. Express 20 14928
[17] Kuipers J B 1999 Geometry, Intergrability and Quantization (New York:Coral Press) pp127-143
[18] Xu F G 2012 Physics with Quaternions (Beijing:Peking University Press) pp16-186 (in Chinese)[许方官 2012 四元数物理学 (北京:北京大学出版社) 第16–186页]
[19] Pletinckx D 1989 Visual Comput. 5 2
[20] Zhang R H, Jia H G, Chen T, Zhang Y 2008 Opt. Precis. Eng. 16 1965 (in Chinese)[张荣辉, 贾宏光, 陈涛, 张跃 2008 光学精密工程 16 1965]
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[1] Schnitzer M J, Block S M 1997 Nature 388 386
[2] Qian H, Chen H, Yan J 2016 Acta Phys. Sin. 65 188706 (in Chinese)[钱辉, 陈虎, 严洁 2016 65 188706]
[3] Fazal F M, Block S M 2011 Nature Photon. 5 318
[4] Xia P, Zhou J H, Song X Y, Wu B, Liu X, Li D, Zhang S Y, Wang Z K, Yu H J, Ward T, Zhang J C, Li Y M, Wang X N, Chen Y, Guo Z, Yao X B 2014 J. Mol. Cell Biol. 6 240
[5] Ouyang H D, Wei M T 2010 Annu. Rev. Phys. Chem. 61 421
[6] Zhou J H, Ren H L, Cai J, Li Y M 2008 Appl. Opt. 47 6307
[7] Xu S H, Li Y M, Lou L R 2006 Chin. Phys. 15 1391
[8] Wright W H, Sonek GJ, Berns M W 1994 Appl. Opt. 33 1735
[9] Stilgoe A B, Nieminen T A, Knoner G, Heckenberg N R, Rubinsztein-Dunlop H 2008 Opt. Express 16 15039
[10] Gu Y Q, Gong Z, Lou L R, Li Y M 2007 Appl. Laser 27 98 (in Chinese)[谷勇强, 龚錾, 楼立人, 李银妹 2007 应用激光 27 98]
[11] Ashkin A 1992 Biophys. J. 61 569
[12] Fällman E, Axner O 2003 Appl. Opt. 42 3915
[13] Reihani S N S, Oddershede L B 2007 Opt. Lett. 32 1998
[14] Sidick E, Collins S D, Knoesen A 1997 Appl. Opt. 36 6423
[15] Bareil P B, Sheng Y, Chiou A 2006 Opt. Express 14 12503
[16] Zhou J H, Zhong M C, Wang Z Q, Li Y M 2012 Opt. Express 20 14928
[17] Kuipers J B 1999 Geometry, Intergrability and Quantization (New York:Coral Press) pp127-143
[18] Xu F G 2012 Physics with Quaternions (Beijing:Peking University Press) pp16-186 (in Chinese)[许方官 2012 四元数物理学 (北京:北京大学出版社) 第16–186页]
[19] Pletinckx D 1989 Visual Comput. 5 2
[20] Zhang R H, Jia H G, Chen T, Zhang Y 2008 Opt. Precis. Eng. 16 1965 (in Chinese)[张荣辉, 贾宏光, 陈涛, 张跃 2008 光学精密工程 16 1965]
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