偏振光学的四元数方法
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摘要
在偏振光学中系统地引入四元数方法。分别给出了在Poincare球和斯托克斯参数基础上建立的偏振光四元数表示,并证明这两种表示是等价的。讨论了偏振光四元数表示的多样性。导出了偏振器件和系统的四元数表示。利用四元数表示证明了偏振系统的等效定理,导出了等价简化系统的组成和四元数表示。提出了偏振系统的四元数矩阵计算方法,得到了四元数表示和Mueller矩阵之间的变换关系。讨论了根据四元数矩阵乘法有条件的交换性优化偏振系统的四元数矩阵算法的途径,并指出了这种算法的应用前景。
The quaternion method is introduced into the polarization optics systematically.Two quaternion representations of polarized light are presented which are based on the Poincare sphere representation and the Stokes parameters,respectively,and their equivalence relation is proved.The diversity of the quaternion representations of the polarized light is discussed.The quaternion representations of polarizing devices and systems are obtained.By using the quaternion representation,the equivalent theorem of polarizing system is proved,and the composition and the quaternion representations of the reduced equivalent system are deduced.The quaternion matrix calculation method for polarizing systems is presented,and the transform relations between the quaternion representations and the Mueller matrixes are obtained.Based on the conditional commutative low of the quaternion matrix multiplication,the optimization way of the quaternion matrix calculation method are discussed,and the application prospects of the algorithm are pointed.
The quaternion method is introduced into the polarization optics systematically.Two quaternion representations of polarized light are presented which are based on the Poincare sphere representation and the Stokes parameters,respectively,and their equivalence relation is proved.The diversity of the quaternion representations of the polarized light is discussed.The quaternion representations of polarizing devices and systems are obtained.By using the quaternion representation,the equivalent theorem of polarizing system is proved,and the composition and the quaternion representations of the reduced equivalent system are deduced.The quaternion matrix calculation method for polarizing systems is presented,and the transform relations between the quaternion representations and the Mueller matrixes are obtained.Based on the conditional commutative low of the quaternion matrix multiplication,the optimization way of the quaternion matrix calculation method are discussed,and the application prospects of the algorithm are pointed.
引文
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3SDe Jeo,G Ducati.Quaternionic groups in physics:apanoramicreview[J].Int J Theor Phys,1999,38:2179-2220.
4A P Yefremov.Quaternions:algebra,geometry and physicaltheories[J].Hypercomplex Numbers in Geometry and Physics,2004,1(1):104-119.
5Ding Guangtao.Bi-quaternion form of electromagnetism thoery[J].Scientia Sinica Physica,Mechanic&Astronomica,2012,42(10):1029-1039.丁光涛.双四元数形式的电磁理论[J].中国科学:物理学力学天文学,2012,42(10):1029-1039.
6Xu Fangguan.Quaternion Physics[M].Beijing:BeijingUniversity Press,2012.许方官.四元数物理学[M].北京:北京大学出版社,2012.
7Xiao Shangbin.The method of quaternion and its application[J].Advances in Mechanics,1993,23(2):249-260.肖尚彬.四元数方法及其应用[J].力学进展,1993,23(2):249-260.
8BP Ickes.A new method for performing digital control systemattitude computations[J].AIAA J,1970,8(1):13-17.
9Liao Yanbiao.Polarization Optics[M].Beijing:Science Press,2003.45-63.廖延彪.偏振光学[M].北京:科学出版社,2003.45-53.
10M Richartz,H Y Hsu.Analysis of elliptical polarization[J].JOpt Soc Am,1949,39(2):136-157.
11JCernosec Simple geometrical method for analysis of ellipticalpolarization[J].J Opt Soc Am,1971,61(3):324-327.
12Ding Guangtao.Quaternion representation of polarized light[J].Huanghuai Journal,1993,9(1):43-48.丁光涛.偏振光的四元数表示[J].黄淮学刊,1993,9(1):43-48.
13Ding Guangtao.Quaternion representations of polarizing devices[J].Huanghuai Journal,1993,9(2):23-30.丁光涛.偏振器件的四元数表示[J].黄淮学刊,1993,9(2):23-30.
14Ding Guangtao.A quaternion representation of polarization optics[J].Journal of Anhui Normal University(Natural Science),1993,16(1):31-38.丁光涛.偏振光的四元数表示[J].安徽大学学报(自然科学版),1993,16(1):31-38.
15Wang Haixia,Ding Chaoliang,Zhang Yongtao,et al..Polarization properties of spatially and spectrally partiallycoherent electromagnetic Hermite-Gaussian pulsed beams[J].Acta Optica Sinica,2011,31(3):0326001.王海霞,丁超亮,张永涛,等.部分空间相干部分光谱相干厄米-高斯脉冲电磁光束的偏振特性[J].光学学报,2011,31(3):0326001.
16Leng Mei,Yang Yanfang,He Ying,et al..Effect actors of thefocal shift in spatial-variant polarized vector beams[J].ActaOptica Sinica,2012,32(5):0526001.冷梅,杨艳芳,何英,等.影响空间变化偏振矢量光束强聚焦后焦斑移位的因素[J].光学学报,2012,32(5):0526001.
17Chang Qiang,Yang Yanfang,He Ying,et al..Focusing featuresof concentric three-ring non-uniform mixing polarization vectorbeams[J].Acta Optica Sinica,2012,32(6):0626001.常亮,杨艳芳,何英,等.三环非均匀混合偏振同轴矢量光束的聚焦特性[J].光学学报,2012,32(6):0626001.
2A Waser.Application of bi-quaternions in physics[J/OL].2007,www.Andre-waser.ch/pulications/Application of Bi-quaternions in physics.
3SDe Jeo,G Ducati.Quaternionic groups in physics:apanoramicreview[J].Int J Theor Phys,1999,38:2179-2220.
4A P Yefremov.Quaternions:algebra,geometry and physicaltheories[J].Hypercomplex Numbers in Geometry and Physics,2004,1(1):104-119.
5Ding Guangtao.Bi-quaternion form of electromagnetism thoery[J].Scientia Sinica Physica,Mechanic&Astronomica,2012,42(10):1029-1039.丁光涛.双四元数形式的电磁理论[J].中国科学:物理学力学天文学,2012,42(10):1029-1039.
6Xu Fangguan.Quaternion Physics[M].Beijing:BeijingUniversity Press,2012.许方官.四元数物理学[M].北京:北京大学出版社,2012.
7Xiao Shangbin.The method of quaternion and its application[J].Advances in Mechanics,1993,23(2):249-260.肖尚彬.四元数方法及其应用[J].力学进展,1993,23(2):249-260.
8BP Ickes.A new method for performing digital control systemattitude computations[J].AIAA J,1970,8(1):13-17.
9Liao Yanbiao.Polarization Optics[M].Beijing:Science Press,2003.45-63.廖延彪.偏振光学[M].北京:科学出版社,2003.45-53.
10M Richartz,H Y Hsu.Analysis of elliptical polarization[J].JOpt Soc Am,1949,39(2):136-157.
11JCernosec Simple geometrical method for analysis of ellipticalpolarization[J].J Opt Soc Am,1971,61(3):324-327.
12Ding Guangtao.Quaternion representation of polarized light[J].Huanghuai Journal,1993,9(1):43-48.丁光涛.偏振光的四元数表示[J].黄淮学刊,1993,9(1):43-48.
13Ding Guangtao.Quaternion representations of polarizing devices[J].Huanghuai Journal,1993,9(2):23-30.丁光涛.偏振器件的四元数表示[J].黄淮学刊,1993,9(2):23-30.
14Ding Guangtao.A quaternion representation of polarization optics[J].Journal of Anhui Normal University(Natural Science),1993,16(1):31-38.丁光涛.偏振光的四元数表示[J].安徽大学学报(自然科学版),1993,16(1):31-38.
15Wang Haixia,Ding Chaoliang,Zhang Yongtao,et al..Polarization properties of spatially and spectrally partiallycoherent electromagnetic Hermite-Gaussian pulsed beams[J].Acta Optica Sinica,2011,31(3):0326001.王海霞,丁超亮,张永涛,等.部分空间相干部分光谱相干厄米-高斯脉冲电磁光束的偏振特性[J].光学学报,2011,31(3):0326001.
16Leng Mei,Yang Yanfang,He Ying,et al..Effect actors of thefocal shift in spatial-variant polarized vector beams[J].ActaOptica Sinica,2012,32(5):0526001.冷梅,杨艳芳,何英,等.影响空间变化偏振矢量光束强聚焦后焦斑移位的因素[J].光学学报,2012,32(5):0526001.
17Chang Qiang,Yang Yanfang,He Ying,et al..Focusing featuresof concentric three-ring non-uniform mixing polarization vectorbeams[J].Acta Optica Sinica,2012,32(6):0626001.常亮,杨艳芳,何英,等.三环非均匀混合偏振同轴矢量光束的聚焦特性[J].光学学报,2012,32(6):0626001.