矢量光束的表征及无衍射矢量光束
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摘要
本论文主要研究了当今光学领域的一个热点问题—矢量光束的表征问题。首先，基于光束的精确表征，分析了Jones矢量在任意矢量光束中应具有的形式，在此基础上研究了具有螺旋相位的柱矢量光束的性质；其次，研究了一般无衍射矢量光束的表征方法及其性质，并探讨了将它们作为表示任意矢量光束的正交完备基的条件；另外，还改进了一个描述光束近轴度的参数；最后研究了矢量光束的级数表示。本文的主要工作和创新点如下：
一，仔细分析了李的表征方法中的广义Jones矢量，指出Jones矢量在任意矢量光束中应具有的形式，并纠正了文献中对偏振椭圆度的误解。另外，研究发现，具有螺旋相位的柱矢量光束是总角动量算符沿着传播方向分量的本征态。
二，研究了一般无衍射矢量光束的表征方法及其性质，并探讨了将它们作为表示任意矢量光束的正交完备基的条件。在用它们作为一组正交完备基来表征任意矢量光束的时候，分析了我们提出的一个特征单位矢量I的作用。另外，讨论了和I相关的电场的纵向场分量，纵向场和总场的相对大小可以体现光束的矢量性。最后，推导了无衍射光束的横向位移和角动量。
三，改进了Gawhary和Severini提出的光束的近轴度，通过比较光束能量的纵向分量和横向分量来定义描述光束近轴度的参数。它的值只和矢量角谱的绝对值有关,和其矢量性无关。该值的范围从0到1，值越大，光束的近轴性越好。文中通过分析无衍射矢量光束和角谱分布为高斯函数的矢量光束的近轴性，来论证该定义的正确性。通过分析，我们发现该定义可以很方便地推广到多色场情况。
四，从光束的表征理论出发，通过选择不同的参数I,和标量角谱函数f,可以得到不同矢量光束关于一个参量s的级数表示。当特征单位矢量和传播轴垂直，得到和文献中自洽的结果,电场的横向场仅仅是s的偶阶项，而纵向场仅仅是s的奇阶项；当特征单位矢量和传播轴平行，电场的横向场是s的偶阶项，纵向场没有高阶修正，角向偏振光束的电场没有纵向场；当特征单位矢量和传播轴既不垂直也不平行，如果采用一阶近似，发现电场的横向场不仅包括s的偶阶项也包括其奇阶项。
This thesis is concerned mainly with one of the hot topics, the characterization ofvector light beams. First of all, the form of Jones vector for arbitrary vector beams isstudied on the basis of the accurate representation theory for the light beams and thecylindrical vector beams with a helical phase structure are analyzed carefully; Then,the characterization and properties of general vector diffraction-free beams arestudied and the condition of the diffraction-free beams to expand a finite vectorbeam as a complete orthonormal set is discussed here; In addition, a recentlyadvanced parameter, called degree of paraxiality, is refined; Finally, the power-seriesexpansion of the electromagnetic beams is investigated. The main results andinnovations are as follows:
First, the generalized Jones vector in Li’s representation theory is analyzedcarefully and the form of Jones vector for arbitrary vector beams is studied. Also, themisunderstanding of the polarization ellipticity in the previous work is corrected. Itshows that cylindrical vector beams with a helical phase structure are the eigenstatesof total angular momentum in the propagation direction.
Second, the characterization and properties of general vector diffraction-freebeams are studied here. And the condition of the diffraction-free beams to expand afinite vector beam as a complete orthonormal set is discussed here. The importanceof the so-called characteristic unit vector I for the complete orthonormal set isinvestigated. Also, the I-dependence of the longitudinal component of thediffraction-free beam is analyzed. The relative strength of the longitudinalcomponent with the total component for the electric field can reflect its vectorialproperty. At last, the transverse displacement and angular momentum for thenon-diffracting beams are given.
Third, a recently advanced parameter given by Gawhary and Severini, calleddegree of paraxiality, is refined. By examining the property of the energies originatingfrom the longitudinal as well as the transverse components, we can define the parameter. It is shown that this parameter is determined by the magnitude of theangular spectrum, without depending on the vectorial feature of it. Its value rangesfrom0to1. A larger value stands for a more paraxial beam. As an application, twoexamples, the family of diffraction-free beams and a class of beams that has aGaussian-like angular spectrum are examined. Also, this definition can be generalizedto polychromatic beams without any difficulty.
Fourth, from the representation theory, power-series expansion according to aparameter s for all kinds of beams can be obtained by choosing differentparameters I, and the scalar angular spectrum f. When the characteristic unitvector is vertical to the propagation direction, the consistent result is achieved withthe previous work. The transverse component of the electric field involves only evenpowers of s, while the longitudinal component of it involves only odd powers of s.When the characteristic unit vector is parallel to the propagation direction, there isno high-order correction to the longitudinal component of the electric field, still thetransverse component of it involves only even powers of s. It is found that theelectric field of the azimuthally polarized beam has no longitudinal component.When the characteristic unit vector is neither vertical nor parallel to the propagationdirection, in the first-order approximation, it is noticed that the transverse componentof it contains not only the even powers but also the odd powers of s.
一，仔细分析了李的表征方法中的广义Jones矢量，指出Jones矢量在任意矢量光束中应具有的形式，并纠正了文献中对偏振椭圆度的误解。另外，研究发现，具有螺旋相位的柱矢量光束是总角动量算符沿着传播方向分量的本征态。
二，研究了一般无衍射矢量光束的表征方法及其性质，并探讨了将它们作为表示任意矢量光束的正交完备基的条件。在用它们作为一组正交完备基来表征任意矢量光束的时候，分析了我们提出的一个特征单位矢量I的作用。另外，讨论了和I相关的电场的纵向场分量，纵向场和总场的相对大小可以体现光束的矢量性。最后，推导了无衍射光束的横向位移和角动量。
三，改进了Gawhary和Severini提出的光束的近轴度，通过比较光束能量的纵向分量和横向分量来定义描述光束近轴度的参数。它的值只和矢量角谱的绝对值有关,和其矢量性无关。该值的范围从0到1，值越大，光束的近轴性越好。文中通过分析无衍射矢量光束和角谱分布为高斯函数的矢量光束的近轴性，来论证该定义的正确性。通过分析，我们发现该定义可以很方便地推广到多色场情况。
四，从光束的表征理论出发，通过选择不同的参数I,和标量角谱函数f,可以得到不同矢量光束关于一个参量s的级数表示。当特征单位矢量和传播轴垂直，得到和文献中自洽的结果,电场的横向场仅仅是s的偶阶项，而纵向场仅仅是s的奇阶项；当特征单位矢量和传播轴平行，电场的横向场是s的偶阶项，纵向场没有高阶修正，角向偏振光束的电场没有纵向场；当特征单位矢量和传播轴既不垂直也不平行，如果采用一阶近似，发现电场的横向场不仅包括s的偶阶项也包括其奇阶项。
This thesis is concerned mainly with one of the hot topics, the characterization ofvector light beams. First of all, the form of Jones vector for arbitrary vector beams isstudied on the basis of the accurate representation theory for the light beams and thecylindrical vector beams with a helical phase structure are analyzed carefully; Then,the characterization and properties of general vector diffraction-free beams arestudied and the condition of the diffraction-free beams to expand a finite vectorbeam as a complete orthonormal set is discussed here; In addition, a recentlyadvanced parameter, called degree of paraxiality, is refined; Finally, the power-seriesexpansion of the electromagnetic beams is investigated. The main results andinnovations are as follows:
First, the generalized Jones vector in Li’s representation theory is analyzedcarefully and the form of Jones vector for arbitrary vector beams is studied. Also, themisunderstanding of the polarization ellipticity in the previous work is corrected. Itshows that cylindrical vector beams with a helical phase structure are the eigenstatesof total angular momentum in the propagation direction.
Second, the characterization and properties of general vector diffraction-freebeams are studied here. And the condition of the diffraction-free beams to expand afinite vector beam as a complete orthonormal set is discussed here. The importanceof the so-called characteristic unit vector I for the complete orthonormal set isinvestigated. Also, the I-dependence of the longitudinal component of thediffraction-free beam is analyzed. The relative strength of the longitudinalcomponent with the total component for the electric field can reflect its vectorialproperty. At last, the transverse displacement and angular momentum for thenon-diffracting beams are given.
Third, a recently advanced parameter given by Gawhary and Severini, calleddegree of paraxiality, is refined. By examining the property of the energies originatingfrom the longitudinal as well as the transverse components, we can define the parameter. It is shown that this parameter is determined by the magnitude of theangular spectrum, without depending on the vectorial feature of it. Its value rangesfrom0to1. A larger value stands for a more paraxial beam. As an application, twoexamples, the family of diffraction-free beams and a class of beams that has aGaussian-like angular spectrum are examined. Also, this definition can be generalizedto polychromatic beams without any difficulty.
Fourth, from the representation theory, power-series expansion according to aparameter s for all kinds of beams can be obtained by choosing differentparameters I, and the scalar angular spectrum f. When the characteristic unitvector is vertical to the propagation direction, the consistent result is achieved withthe previous work. The transverse component of the electric field involves only evenpowers of s, while the longitudinal component of it involves only odd powers of s.When the characteristic unit vector is parallel to the propagation direction, there isno high-order correction to the longitudinal component of the electric field, still thetransverse component of it involves only even powers of s. It is found that theelectric field of the azimuthally polarized beam has no longitudinal component.When the characteristic unit vector is neither vertical nor parallel to the propagationdirection, in the first-order approximation, it is noticed that the transverse componentof it contains not only the even powers but also the odd powers of s.
引文
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