菲涅耳-基尔霍夫积分与半波带法分析衍射光谱

惠更斯菲 涅耳原理指出光的衍射本质是无穷多次波的相干叠加.在研究衍射光谱的强度时可使用菲涅耳-基尔霍夫衍射积分进行计算,但该积分的计算较为复杂.为了简化计算而引入菲涅耳半波带的概念,把积分运算简化为振幅矢量的叠加,从而在研究衍射时由菲涅耳-基尔霍夫衍射积分过渡到菲涅耳半波带法.在菲涅耳半波带法中波前上相邻两个半波带到达叠加点的光程差为半个波长,从而导致相位差π.因此相邻两个半波带引起的振动在叠加点发生相消,最终叠加点的光强由发出次波的半波带数目的奇偶性决定.奇数个半波带在叠加点的光强得到加强,偶数个半波带在叠加点的光强发生相消.     

菲涅耳双面镜测波长

通过用菲涅耳双棱镜测波长实验的研究,得出用菲涅耳双面镜测波长的方法,实验简便易行.设计了两平面镜夹角的测量方法,减小了实验误差.     

X射线波带片

综述了用于Ｘ射线聚焦、成像的光学元件———Ｘ射线菲涅耳波带片及近几年才出现的布拉格－菲涅耳波带片的历史背景、基本概念、基本特征和主要类型．同时介绍了当前国际上Ｘ射线波带片的主要制作方法及最新进展．     

菲涅耳衍射公式误差相位的讨论

根据波动光学的理论,对菲涅耳近场衍射公式误差相位的影响进行了分析。以单缝衍射为例,讨论了误差相位与计算精度的联系。结合菲涅耳线波带片焦面光强分布的计算,对比了菲涅耳衍射公式与基尔霍夫衍射公式的计算结果,说明在不满足误差相位近似条件下使用菲涅耳衍射公式带来的影响。最后对影响波带片焦面光强分布的几个因素进行了讨论。     

关于菲涅耳全息滤波器的实验与分析

本文对菲涅耳全息滤波器作为强度相关器的功能作了详细的实验分析研究,结果表明菲涅耳全息滤波器不适宜于用作相关识别检测。     

验证菲涅耳圆孔衍射中的一个重要结论

验证菲涅耳圆孔衍射中的一个重要结论温淑珍（黑龙江绥化师专）在菲涅耳圆孔衍射中，如图１，圆孔ＣＣ’露出的波阵面，在与点光源Ｐ０处同一轴上的Ｐ点引起的光振动的振幅”’为其中山和山分别为波阵面上相对于ＰＯ和Ｐ所作的第一个和最后一个菲涅耳半波带单独对该点引起...     

菲涅耳衍射和分数傅里叶变换

介绍了衍射孔的菲涅耳衍射和分数傅里叶变换的对应关系,使得可以用分数傅里叶变换来描述光由原始光场经过菲涅耳衍射区一直到无穷远处夫琅禾费衍射区的自由空间标量衍射传播全过程。     

“四分之一光强”结论的实验验证

在菲涅耳圆孔衍射(图1)中,根据圆形菲涅耳半波带的划分,可知没有遮蔽的整个波阵面在P_0点引起的光强只有使圆孔露出第一个半波带的那一小部分波阵面在同一点引起光强的四分之一。笔者对此“四分之一光强”的结论作了实验验证。该实验的主要方法是,设计一个能露出第一个半波带的菲涅耳圆孔衍射系统,以线性光电探头测出被测点光强引起的光电流,再与     

环形光栅的菲涅耳衍射

在圆孔的菲涅耳衍射的基础上 ,讨论环形光栅的菲涅耳衍射 ,并给出了计算机仿真结果     

菲涅耳衍射解析表达式的物理意义

文中讨论了圆孔菲涅耳衍射解析表达式的物理意义,指出了精确解(在菲湿耳近似下)与近似解的区别,并与用数值积分法计算的结果进行比较,证明了圆孔菲涅耳衍射解析表达式的正确性。     

Quantum mechanical version of the classical Liouville theorem

In terms of the coherent state evolution in phase space,we present a quantum mechanical version of the classical Liouville theorem.The evolution of the coherent state from |z>to|sz-rz*> corresponds to the motion from a point z(q,p) to another point sz-rz* with |s|2-|r|2=1.The evolution is governed by the so-called Fresnel operator U(s,r) that was recently proposed in quantum optics theory,which classically corresponds to the matrix optics law and the optical Fresnel transformation,and obeys group product rules.In other words,we can recapitulate the Liouville theorem in the context of quantum mechanics by virtue of coherent state evolution in phase space,which seems to be a combination of quantum statistics and quantum optics.     

New approach to Q-P （P-Q） ordering of quantum mechanical operators and its applications

In this paper, we introduce a new way to obtain the Q-P （P-Q） ordering of quantum mechanical operators, i.e., from the classical correspondence of Q-P （P-Q） ordered operators by replacing q and p with coordinate and momentum operators, respectively. Some operator identities are derived concisely. As for its applications, the single （two-） mode squeezed operators and Fresnel operator are examined. It is shown that the classical correspondence of Fresnel operator’s Q-P （P-Q） ordering is just the integration kernel of Fresnel transformation. In addition, a new photo-counting formula is constructed by the Q-P ordering of operators.     

A new theorem relating quantum tomogram to the Fresnel operator

According to Fan-Hu's formalism (Fan Hong-Yi and Hu Li-Yun 2009 Opt. Commun. bf282 3734) that the tomogram of quantum states can be considered as the module-square of the state wave function in the intermediate coordinate-momentum representation which is just the eigenvector of the Fresnel quadrature phase, we derive a new theorem for calculating quantum tomogram of density operator, i.e., the tomogram of a density operator ρ is equal to the marginal integration of the classical Weyl correspondence function of F⁺ρF, where F is the Fresnel operator. Applications of this theorem to evaluating the tomogram of optical chaotic field and squeezed chaotic optical field are presented.     

Optical Fresnel Transform as a Correspondence of the SU(1,1) Squeezing Operator Composed of Quadratic Combination of Canonical Operators and the Entangled State

We study optical Fresnel transforms by finding the appropriate quantum mechanical SU(1,1) squeezing operators which are composed of quadratic combination of canonical operators. In one-mode case, the squeezing operator's matrix element in the coordinate basis is just the kernel of one-dimensional generalized Fresnel transform (GFT); while in two-mode case, the matrix element of the squeezing operator in the entangled state basis leads to the two-dimensional GFT kernel. The work links optical transforms in wave optics to generalized squeezing transforms in quantum optics.     

Quantum optical ABCD theorem in two-mode case

By introducing the entangled Fresnel operator （EFO） this paper demonstrates that there exists ABCD theorem for two-mode entangled case in quantum optics. The canonical operator method as mapping of ray-transfer ABCD matrix is explicitly shown by EFO＇s normally ordered expansion through the coherent state representation and the technique of integration within an ordered product of operators.     

New theorem relating two-mode entangled tomography to two-mode Fresnel operator

Based on the Fan-Hu's formalism, i.e., the tomogram of two-mode quantum states can be considered as the module square of the states' wave function in the intermediate representation, which is just the eigenvector of the Fresnel quadrature phase, we derive a new theorem for calculating the quantum tomogram of two-mode density operators, i.e., the tomogram of a two-mode density operator is equal to the marginal integration of the classical Weyl correspondence function of F₂⁺ρF₂, where F₂ is the two-mode Fresnel operator. An application of the theorem in evaluating the tomogram of an optical chaotic field is also presented.     

Quantum mechanical version of classical Liouville theorem

In terms of the coherent state evolution in phase space, we present a quantum mechanical version of the classical Liouville theorem. The evolution of coherent state from |z〉to |sz-rz^*〉angle corresponds to the motion from a point z(q,p) to another point sz-rz^* with |s|²-|r|²=1. The evolution is governed by the so-called Fresnel operator U(s,r) recently proposed in quantum optics theory, which classically corresponds to the matrix optics law and the optical Fresnel transformation and obeys the group product rules. In another word, we can recapitulate the Liouville theorem in the context of quantum mechanics by virtue of coherent state evolution in phase space, which seems to be a combination of quantum statistics and quantum optics.     

The two-mode quantum Fresnel operator and the multiplication rule of 2D Collins diffraction formula

By using the two-mode Fresnel operator we derive a multiplication rule of two-dimensional (2D) Collins diffraction formula, the inverse of 2D Collins diffraction integration can also be conveniently derived in this way in the context of quantum optics theory.     

Fresnel diffraction of truncated Gaussian beam

We study the Fresnel diffraction of Gaussian beam truncated by one circular aperture, and give the general analytic expression of the Fresnel diffraction of truncated Gaussian beam denoted by Bessel functions. Then the characteristic of the axial diffraction fluctuation and the influence of the caliber of the circular aperture and the wave waist of Gaussian beam on the diffraction distributions are discussed, respectively. Through the numerical calculations, the characteristics of the transverse diffraction are presented and the relationship of the fluctuation of the transverse diffraction profile and the position of the axial point is shown. The physical origin of the fluctuation of Fresnel diffraction intensities of truncated Gaussian beam is expressed in terms of Fresnel half-zone theory. These phenomena and the conclusions are important for the measurement of the parameters of the beam and its applications.     

Correspondence between quantum-optical transform and classical-optical transform explored by developing Dirac’s symbolic method

By virtue of the new technique of performing integration over Dirac’s ket–bra operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel–Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel operator (GFO) to correspond to the generalized Fresnel transform (GFT) in classical optics. We derive GFO’s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used. All these have confirmed Dirac’s assertion: “...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory”.