In this Letter, a new fractional entangling transformation (FrET) is proposed, which is generated in the entangled state representation by a unitary operator exp{iθ(ab +a b)} where a(b) is the Bosonic annihilate operator. The operator is actually an entangled one in quantum optics and differs evidently from the separable operator, exp{iθ(a a+b b)}, of complex fractional Fourier transformation. The additivity property is proved by employing the entangled state representation and quantum mechanical version of the FrET. As an application, the FrET of a two-mode number state is derived directly by using the quantum version of the FrET, which is related to Hermite polynomials.

The phase aberrations caused by thermal deformations is one of the main limitations in a beam control system. If left uncorrected, it would significantly reduce the effectiveness of high-power lasers as devices for beaming power over long paths through the atmosphere. For the purpose of overcoming the effects of phase aberrations, the method of conjugation wavefront pre-compensation of incident laser is introduced and theoretically studied. Based on the simulated results of thermal deformations at different irradiations time, choosing the power in the bucket (PIB), Strehl ratio, and the root-mean-square of phase as the characteristic parameters, the beam quality of emitted laser with and without wavefront pre-compensation is calculated respectively and compared. The result shows that under the condition that the conjugation compensation phase is placed on the incident beam, the beam quality of emitted laser is ameliorated prominently especially for a beam control system with serious thermal deformations.

A novel fully phase color image encryption/decryption scheme based on joint fractional Fourier transform correlator (JFRTC) and phase retrieval algorithm (PRA) is proposed. The security of the system is enhanced by the fractional order as a new added key. This method takes full advantage of the parallel processing features of the optical system and could optically realize single-channel color image encryption. The system and operation procedures are simplified. The simulation results of a color image indicate that the new method provides efficient solutions with a strong sense of security.

基于柯林斯（Collins）积分公式，导出了高斯-谢尔模型(GSM)光束在有色差的分数傅里叶变换(FRFT)系统中的光谱传输表达式，并以Lohmann Ⅱ型系统为例，通过数值计算与分析，研究了色差大小、分数傅里叶变换阶数、以及相对横向坐标对输出面归一化光谱和相对谱移动的影响。研究表明，相对谱移动随FRFT阶数的变化而改变，色差的大小会影响相对谱移动的大小、光谱跃变发生的位置以及光谱跃变量的大小；相对谱移动随色差的变化而改变，在不同阶数的FRFT系统中，相对谱移动随色差的变化规律各不相同。此外，归一化的光谱和相对谱移动还随相对横坐标的变化而改变。

In order to realize holographic display of three-dimensional (3D) objects and suppress zero-order light, conjugate image, and speckle noise, a novel method is proposed based on multiple fractional Fourier transform (M-FrFT) for calculating holograms of 3D objects. A series of kinoforms are generated by adding pseudorandom phase factor (PPF) to object planes in calculating each kinoform, and generating the PPF randomly again in the next kinoform calculation. The reconstructed images from kinoform sequence are superposed together in order to suppress the speckle noise of reconstructed image and improve the contrast and detail resolution of the reconstructed images. The qualities of reconstructed images from single amplitude hologram, single kinoform, and kinoform sequence calculated by M-FrFT are compared. The effects of suppressing speckle noise are analyzed by calculating the speckle index of numerical reconstructed images. The analytical results illustrate that, with the proposed method for 3D holographic display, the zero-order light, conjugate image, and speckle noise can be suppressed, and the qualities of reconstructed images can be improved significantly.

基于分数傅里叶变换(FRFT)与维格纳分布函数(WDF)旋转等效的性质,推导出了双曲余弦-高斯(ChG)光束在分数傅里叶变换面上光强分布和束宽的解析公式,研究了双曲余弦-高斯光束光强和束宽随分数傅里叶变换阶数的变化规律,分析了偏心参量对双曲余弦-高斯光束分数傅里叶变换特性的影响,并对数值计算结果进行了分析和讨论。结果表明,对给定偏心参量的双曲余弦-高斯光束,可以使其经过一定阶数的分数傅里叶变换来获得平顶光束。