On the Entangled Fractional Fourier Transform in Tripartite Entangled State Representation

Based on the newly constructed two mutually conjugate 3-mode entangled states of continuum variables in three-mode Fock space we introduce entangled fractional Fourier transform (EFFT) for the tripartite entangled state representations, which are not a direct product of three 1-dimensional FFTs. The eigenmodes of EFFT are obtained,which is different from the usual Hermite polynomials. The EFFT of the three-mode squeezed state is derived.     

Entangled Fractional Fourier Transform for the Multipartite Entangled State Representation

We deduce entangled fractional Fourier transformation (EFFT) for the multipartite entangled state representation, which was newly constructed with two mutually conjugate n-mode entangled states of continuum variables in n-mode Fock space. We establish a formalism of EFFT for quantum mechanical wave functions, which provides us a convenient way to derive some wave functions. We find that the eigenmode of EFFT is different from the usual Hermite Polynomials. We also derive the EFFT of the n-mode squeezed state.     

New Tripartite Nonlinear Entangled State Representation in Quantum Mechanics

Based on the technique of integral within an ordered product of nonlinear bosonic operators, we construct a new kind of tripartite nonlinear entangled state ｜α,γ）λ in 3-mode Fock space, which can make up a complete set. We also simply discuss its properties and application.     

New Tripartite Nonlinear Entangled State Representation in Quantum Mechanics

Based on the technique of integral within an ordered product of nonlinear bosonic operators, we construct a new kind of tripartite nonlinear entangled state |α,γ〉_λ in 3-mode Fock space, which can make up a complete set. We also simply discuss its properties and application.     

From Complex Fractional Fourier Transform to Complex Fractional Radon Transform

We show that for n-dimensional complex fractional Fourier transform the corresponding complex fractional Radon transform can also be derived, however, it is different from the direct product of two n-dimensional real fractional Radon transforms. The complex fractional Radon transform of two-mode Wigner operator is calculated.     

分数傅立叶变换的进展与展望

在综合国内外有关文献的基础上，评述分数傅立叶变换在信息光学中的新进展。概述分数傅立叶变换的发展过程，并对其研究现状进行了简略总结，同时对其未来的发展作出展望。     

From Complex Fractional Fourier Transform to Complex Fractional Radon Transform

We show that for n-dimensional complex fractional Fourier transform the corresponding complex fractional Radon transform can also be derived, however, it is different from the direct product of two n-dimensional real fractional Radon transforms. The complex fractional Radon transform of two-mode Wigner operator is calculated.     

分数傅里叶变换全息图及其在防伪中的应用

提出分数傅里叶变换全息图,讨论了它的性质。拍摄分数傅里叶变换彩虹全息图。基于其再现条件的特殊性,可建立一种新型的伪全息术。     

New Three-Mode Squeezing Operators Gained via Tripartite Entangled State Representation

We show that the Agarwal-Simon representation of single-mode squeezed states can be generalized to find new form of three-mode squeezed states. We use the tripartite entangled state representations ｜p, y, z） and ｜x, u, v） to realize this goal.     

基于联合广义分数相关器的相关峰特性分析

分析了联合广义分数傅里叶变换相关器相关峰的特性,得到通过改变广义分数傅里叶变换的系统参量可以提高广义分数相关峰性能的结论.进行了数值模拟和光学实验,并根据两者的结果对四个相关峰的性能指标相关峰强度最大值、峰能比、识别能力、信噪比进行了比较分析,说明只要适当控制系统参量,联合广义分数傅里叶变换相关器比联合分数傅里叶变换相关器具有更好的相关性能,有助于提高光学相关器识别的准确率.     

Quantum Optical Squeezing Transform for Generalizing Fractional Fourier Transform

By establishing the relation between the optical scaled fractional Fourier transform （FFT） and quantum mechanical squeezing-rotating operator transform, we employ the bipartite entangled state representation of two-mode squeezing operator to extend the scaled FFT to more general cases, such as scaled complex FFT and entangled scaled FFT. The additiyity and eigenmodes are presented in quantum version. The relation between the scaled FFT and squeezing-rotating Wigner operator is studied.     

On Fermionic Entangled State Representation and Fermionic Entangled Wigner Operator

By analogy with the bosonic bipartite entangled state we construct fermionic entangled state with the Grassmann numbers. The Wigner operator in the fermionic entangled state representation is introduced, whose marginal distributions are understood in an entangled way. The technique of integration within an ordered product (IWOP) of Fermi operators is used in our discussion.     

On the Entangled State Representation with Continuum Variables

we propose the conception of entangled state representation with continuum variables. We analyze the non-factorizable properties of the |η> state, the common eigenvector of two-particle relative position and total moment um (Fan Hongyi and J.R. Klauder, Phys. Rev. A49 (1994) 704), and recast it into the standard form as the state prepared in a parametric down conversion process which involves the entanglement of idle and signal photons. We name the set of |η> as entangled state representation, as it is orthonormal and complete     

On Fermionic Entangled State Representation and Fermionic Entangled Wigner Operator

By analogy with the bosonic bipartite entangled state we construct fermionic entangled state with the Grassmann numbers. The Wigner operator in the fermionic entangled state representation is introduced, whose marginal distributions are understood in an entangled way. The technique of integration within an ordered product （IWOP） of Fermi operators is used in our discussion.     

Tripartite Nonlinear Entangled State Representations in Quantum Mechanics

Based on the technique of integral within an ordered product of nonlinear bosonic operators we construct a kind of tripartite nonlinear entangled states, which can make up a complete set. As its application, we also derive nonlinear 3-mode charge-related entangled state. The essential point for constructing these states lies in choosing the appropriate charge operator.     

Tripartite Nonlinear Entangled State Representations in Quantum Mechanics

Based on the technique of integral within an ordered product of nonlinear bosonic operators we construct a kind of tripartite nonlinear entangled states, which can make up a complete set. As its application, we also derive nonlinear 3-mode charge-related entangled state. The essential point for constructing these states lies in choosing the appropriate charge operator.     

联合分数变换相关器的分数相关特性分析

从分数傅里叶变换（FRFT）的定义出发，理论分析了联合分数变换相关器（JFRTC）的分数相关特性.从所得JFRTC的数学表达式中可以看出，将FRFT应用到联合变换相关器（JTC）中得到的JFRTC具有与传统JTC不同的性质.对于传统JTC，一旦输入平面上参考图像与目标图像之间的距离给定，相关输出峰的位置即确定，而JFRTC的相关输出峰的位置则可以由分数级次p₁和p₂来自由调节，这个特性在实际模式识别中非常有用.另一方面，JFRTC的相关输出峰值在大多数情况下低于传统JTC的相关峰值，却是JFRTC的一大缺点.最后，从FRFT的比例性质出发，给出了FRFT谱畸变不变的实现条件，并由此预言了JFRTC畸变不变模式识别的功能.     

用全息透镜记录多重分数傅里叶变换全息图

提出一种全息透镜记录多重分数傅里叶变换全息图的新方法，它能在三维空间不同位置和不同方向上分别再现所记录的多个物体的图象，分析了利用全息透镜记录多重分数傅里叶变换全息图的原理及特点，制作了多重分数傅里叶变换全息图，并获得了满意的再现结果。     

Recording Fractional Fourier Transform Hologram Using Holographic Zone Plate

1　Ｉｎｔｒｏｄｕｃｔｉｏｎ　　ＦＲＴ(ＦｒａｃｔｉｏｎａｌＦｏｕｒｉｅｒＴｒａｎｓｆｏｒｍ )ｗａｓｆｉｒｓｔｌｙｉｎｔｒｏｄｕｃｅｄｂｙＮａｍｉａｓｉｎ 1 980ａｓａｍａｔｈｅｍａｔｉｃｔｏｏｌｉｎｑｕａｎｔｕｍｍｅｃｈａｎｉｃｓ[1] .Ｉｎ1 993,ＭｅｎｄｌｏｖｉｃａｎｄＯｚａｋｔａｓｉｍｐｌｅｍｅｎｔｅｄｔｈｅＦＲＴａｔｏｐｔｉｃａｌｆｉｅｌｄｗｈｅｎｔｈｅｙｓｔｕｄｉｅｄｇｒａｄｉｅｎｔｉｎｄｅｘ (ＧＲＩＮ )ｆｉｂｅｒ[2 ] ａｎｄａｐｐｌｉｅｄｔｈｅｐｕｒｅｍａｔｈｅｍａｔｉｃｆｏｒｍｕｌａｔｉｏ…     

Recording Fractional Fourier Transform Hologram Using Holographic Zone Plate

FRTH(fractional Fourier transform hologram) is a new kind of hologram that differs from common Fresnel holograms and Fourier transform holograms. Due to the flexibility of zone plate. A method that uses the -1 order diffraction wave of zone plate as the object wave and the 0 order diffraction wave as the reference wave to record FRTH is presented. It provides a new simple way to record FRTH. In this paper, the theory of achieving FRT and recording FRTH using holographic zone plate is presented and experimental results are given.