The wigner distribution function and Hamilton''s characteristics of a geometric-optical system

Four system functions have been defined for an optical system; each of these functions describes the system completely in terms of Fourier optics. From the system functions the Wigner distribution function of an optical system has been defined; although derived from Fourier optics, this Wigner distribution function can directly be interpreted in terms of geometrical optics. In the special case of a geometric-optical system, the Wigner distribution function clearly demonstrates the relations that exist between the four system functions and the four Hamilton's characteristics of the system.     

HfO2/SiO2高反射膜的缺陷及其激光损伤

用原子力、Normaski和扫描电子显微镜等分析仪器,对高损伤阈值薄膜常采用的HfO2光损伤所形成的孔洞,与镀制过程中形成的孔洞形貌相似,激光再损伤能力也相似。低能量密度的激光把节瘤缺陷变为孔洞缺陷是激光预处理提高薄膜损伤阈值的原因之一。     

Quantum coherent versus classical coherent light

The paper shows that the Wigner distribution function of quantum optical coherent states, or of a superposition of such states, can be produced and measured with a classical optical set-up using classical coherent light fields. This measurement cannot be done directly in quantum optics since the quantum phase space variables correspond to non-commuting operators. As an example, the Wigner distribution function of Schrödinger cat states of light has been measured. It is also shown that the possibility of measuring the Wigner distribution function of quantum coherent states with classical coherent fields is unique in the sense that it cannot be extended to other quantum states, not even to the incoherent limit of the superposition of coherent states.     

Linear canonical Wigner distribution and its application

Linear canonical transform (LCT) form a three-parameter family of intergral transforms with wide application in optics. In this paper, we investigate the linear canonical Wigner distribution (LCWD) which is based on the LCT and the classical Wigner distribution (WD). Firstly, the definition of LCWD is discussed. Moreover, the transformation law for the LCWD through a first-order optical system is derived. This new phase-space distribution provides analysis of signals in both space and LCT domains simultaneously. Then, the main properties of LCWD are investigated in detail. Finally, the application of the LCWD is presented. The LCWD is found to be the appropriate phase-space distribution function for light-beam characterization in first-order optical system. Moreover, the moment matrix formalism for beam characterization is studied.     

Some Evolution Formulas on the Optical Fields Propagation in Realistic Environments

Evolution formulas of the density operator, the photon number distribution, and the Wigner function are derived for the problem on the optical fields propagation in realistic environments. Using the idea “reservoir modeled by beam splitter (BS)” and the Weyl expansion of the density operator, we obtain these formulas cleverly, which are very useful for quantum optics and quantum statistics. As an application, we study the time evolution of the photon number distribution and the Wigner function for single-photon-added coherent state in thermal environment.     

在量子光学框架中研究魏格纳-维利分布

我们在量子光学框架中研究光信号的魏格纳-维利分布,指出利用魏格纳算符和纠缠魏格纳算符的显示正规乘积形式以及压缩算符的纠缠态表象,这方面的研究就可做到数学上简明和物理上有吸引力.     

Characterization of freeform optics in automotive lighting systems using an Optical-Geometrical Feature Based Method

The use of freeform optics has been more widespread. It is difficult to evaluate the optical performance of the freeform optics due to its geometrical complexity. This paper presents a novel method named Optical-Geometrical Feature Based Method (OGFM) for the characterization of freeform optics. The method attempts to establish the relationship between the geometrical features of freeform surfaces (e.g. the surface roughness and form error) and their corresponding optical performance. Hence, the optical performance of the freeform optics can be evaluated by the measurement of the geometrical features which is relative easy to be carried out as compared to the optical performance tests conducted in automotive lighting systems. A series of simulation experiments have been conducted on the freeform optics in an automotive lighting system. The results show that the geometrical errors in micrometer scale appear not to have significant effect on the optical performance of the freeform optics. However, the optical performance is adversely affected when the geometrical form error increases from more than tens of micrometers scale to sub-millimeters scale. The successful development of the proposed OGFM provides an important means for the optimization of manufacturing tolerance, the selection of appropriate optics manufacturing process, and the evaluation of the optical performance evaluation of the freeform optics.     

General Wigner Transforms Studied by Virtue of Weyl Ordering of the Wigner Operator

By virtue of the property that Weyl ordering is invariant under similar transformations we show that the Weyl ordered form of the Wigner operator, a Dirac δ-operator function, brings much convenience for deriving miscellaneous Wigner transforms. The operators which engender various transforms of the Wigner operator, can also be easily deduced by virtue of the Weyl ordering technique. The correspondence between the optical Wigner transforms and the squeezing transforms in quantum optics is investigated.     

General Wigner Transforms Studied by Virtue of Weyl Ordering of the Wigner Operator

By virtue of the property that Weyl ordering is invariant under similar transformations we show that the Weyl ordered form of the Wigner operator, a Dirac δ-operator function, brings much convenience for derivingmiscellaneous Wigner transforms. The operators which engender various transforms of the Wigner operator, can alsobe easily deduced by virtue of the Weyl ordering technique. The correspondence between the optical Wigner transformsand the squeezing transforms in quantum optics is investigated.     

White-Noise and Geometrical OpticsLimits of Wigner–Moyal Equation for Beam Waves in Turbulent Media II: Two-Frequency Formulation

We introduce two-frequency Wigner distribution in the setting of parabolic approximation to study the scaling limits of the wave propagation in a turbulent medium at two different frequencies. We show that the two-frequency Wigner distribution satisfies a closed-form equation (the two-frequency Wigner–Moyal equation). In the white-noise limit we show the convergence of weak solutions of the two-frequency Wigner–Moyal equation to a Markovian model and thus prove rigorously the Markovian approximation with power-spectral densities widely used in the physics literature. We also prove the convergence of the simultaneous geometrical optics limit whose mean field equation has a simple, universal form and is exactly solvable     

Wigner distribution function matrix of electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis

The Wigner distribution function matrix of electromagnetic beams propagating in uniaxial crystals is introduced, and its analytical propagation expressions through uniaxial crystals orthogonal to the optical axis are derived. The application of the Wigner distribution function matrix of the electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis is illustrated with the stochastic electromagnetic beams.     

Wigner分布函数及非整数傅里叶变换

本文研究了Wigner分布函数的特性,利用Wigner分布函数的旋转,将整数域傅里叶变换推广到了非整数域,描述了自由空间光场的演变过程.     

Approximate analytical representation of Wigner distribution function for Gaussian beams passing through ABCD optical systems with hard aperture

Based on the treatment that a rectangular function can be expanded as an approximate sum of complex Gaussian functions with finite numbers, the analytical expression of the Wigner distribution function for a Gaussian beam passing through a paraxial ABCD optical system with hard-edged aperture is obtained. By numerical simulation, it is shown that the effect of the aperture on the Wigner distribution function is prominent. By comparing the analytical results with the numerical integral results, it is shown that this method of expanding hard-edged aperture into Gaussian functions with finite numbers is proper and ascendant. This method could be extended to study the Wigner distribution functions of other light beams passing through a paraxial ABCD optical system with hard-edged aperture.     

Analysis of eccentric photorefraction by Fourier optics

Eccentric photorefraction usually is used as early eyesight diagnostic test of infants and small children. Unlike currently approved geometrical optical model of eccentric photorefractometer, the crescent formation and the light-intensity distribution in the pupil image of a myopic eye are analyzed by Fourier optics with the assumption of an isotropic scattering retina. In the case of little circular light source and rectangular slit, the simulation results of different myopic diopters are obtained by geometrical optical theory and Fourier optics respectively. It is found that the simulation results by Fourier optics are similar as those obtained by geometrical optics, and all simulations are almost corresponding to the experimental result.The result demonstrates that the new method presented here is feasible.     

基于Wigner分布函数的光学元件评价方法

为了有效地表征和评价光学元件局部小尺度波前畸变,提出了能同时反映空间频率和空间位置的Wigner分布函数评价方法。通过理论分析Wigner分布函数与功率谱密度之间的关系,得到了局部波前畸变的评价方法和指标。模拟计算和实验结果验证了该方法的有效性,并表明:该方法不但能判断光学元件的中频误差是否满足惯性约束聚变系统要求,而且还能确定不满足要求的特定区域,从而有效地指导光学元件的返修。     

An Approach to Optical Reflection Tomography along the Geometrical Thickness

We propose and demonstrate a novel optical reflection tomography along the geometrical thickness. This technique is based on simultaneous measurement of refractive index n and thickness t of a sample using the combination of a low coherence interferometer and confocal optics. The interferometer provides optical coherence tomography (OCT) of the dimension of the optical thickness (=n × t) along the optical axis, while the confocal optics gives us another type of reflection tomography, having the thickness dimension of nearly t/n along the optical axis. This sort of tomography can be called confocal reflection tomography (CRT) and has not yet been demonstrated, to our knowledge. Simple image processing of OCT and CRT results in the desired reflection tomographic image, showing two-dimensional refractive index distribution along the geometrical thickness.     

Third- and fourth-order parametric characterization of partially coherent beams propagating throughABCD optical systems

Within the formalism of the Wigner distribution function, third- and fourth-order measurable beam parameters are proposed and their physical meaning is investigated. The general propagation law of such characteristic quantities inABCD optical systems is derived and a number of paramters are shown to be invariant under propagation inABCD devices.