Spectral shift of partially coherent flattened multi-Gaussian beams with a hole passing through ABCD optical systems

Based on flattened multi-Gaussian beam model and partially coherent theory, the expression for cross-spectral density of partially coherent flattened multi-Gaussian beam with a hole was given. The analytical expression for on-axis spectrum of partially coherent flattened multi-Gaussian beam with a hole passing through a paraxial ABCD optical system was derived. The spectral shifts of the partially coherent flattened multi-Gaussian beam with a hole propagating in free space and passing through the lens have been analyzed. The effects of the beam profile, spatial coherence parameter and the system parameters on the relative spectral shift have been discussed. Our results show that the inner radius of flattened multi-Gaussian beam with a hole determines its relative spectral shift in near field and its outer radius determines that in far field. For the flattened multi-Gaussian beam with a hole passing through a lens, the on-axis relative spectral shift changes sharply near the focal plane with the increase in Fresnel number and the spatial coherence parameter.     

A new definition of the degree of paraxiality for vector light beams

A parameter, called the degree of paraxiality, is introduced upon examining the property of energies originating from the longitudinal as well as the transverse components of both the electric and magnetic fields. It is shown that this degree of paraxiality is determined by the magnitude of the plane-wave spectrum, without depending on the vectorial feature of the plane-wave spectrum. Its value ranges from 0 to 1. A larger value stands for a more paraxial beam. As an application, two examples are considered.     

Degree of paraxiality for monochromatic light beams

We introduce the degree of paraxiality as a paraxiality measure of a monochromatic light beam. Computation of this parameter is possible once the plane-wave spectrum of the field being analyzed is given for a starting source plane. On the basis of this definition, quantitative comparisons of the paraxiality of different types of fields are possible. In addition, the recently introduced paraxial estimator [Opt. Lett.32, 927 (2007)] is identified as the limiting case of the degree of paraxiality.     

Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam

The concept of the degree of paraxiality introduced recently for monochromatic fields is extended to the domain of stochastic electromagnetic fields. Analytical expression for the degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam is derived. Numerical results show that the degree of paraxiality of an EGSM beam is determined by the degree of polarization, r.m.s. widths of the spectral densities and of the correlation functions of its source. Degree of paraxiality of an EGSM beam after passing through a linear polarizer is also analyzed. Our results show that one can modulate the degree of paraxiality of an EGSM beam by a linear polarizer.     

拉盖尔-高斯光束傍轴度的变化

以拉盖尔-高斯(L-G)光束为例,详细研究了单色光束傍轴度的变化.结果表明,模指数、相对束腰宽度等光束参数的变化,通过近轴ABCD光学系统的传输,以及光阑衍射都会引起光束傍轴度的变化.但是,在自由空间中传输时,光束的傍轴度不变.对上述结果以数值计算例加以说明,并用光束傍轴度与远场发散角间的联系做了物理解释. 关键词： 傍轴度 远场发散角 拉盖尔-高斯光束     

Spectral intensity characteristics of partially coherent flattened multi-Gaussian beams with an axial shadow passing though paraxial ABCD optical systems

Based on the beam model for the flattened multi-Gaussian beams (FMGBs) with an axial shadow and partially coherent theory, the expression for cross-spectral density of partially coherent FMGBs with an axial shadow has been given, and the analytical expression for the spectral intensity of partially coherent FMGBs with an axial shadow passing through paraxial ABCD optical systems has been derived. The spectral shift of partially coherent FMGBs with an axial shadow focused by a thin lens has been analyzed quantitatively. The effects of the beam profile, spatial coherence parameter and the system parameters on the relative spectral shift have been discussed in detail. Our results show that the spectral intensity and the spectral shift of partially coherent FMGBs with an axial shadow passing through paraxial ABCD optical systems depend on the source spectral density, the generalized Fresnel number of the system, the transverse coordinate, the spatially coherence parameter, as well as the beam orders.     

General formulation of the edge-diffracted paraxial waves

A general formula for edge-diffracted paraxial waves is derived in terms of the Fourier integral transform. It is observed that the scattered paraxial wave can be divided into components of the geometrical optics and diffracted fields in the spectral domain. An edge-diffracted Gaussian beam is examined numerically as an application of the formula.     

几种描述光束傍轴性定义的比较研究

以平顶高斯光束为例,对光束傍轴度3种定义,即傍轴估计量、傍轴度和傍轴度修正式做了解析研究和计算分析。结果表明,当束腰宽度与波长之比很小时,三者存在显著差异,傍轴估计量会导致非物理结果。而对实际应用中感兴趣的束腰宽度与波长之比取中间值情况,傍轴度和傍轴度修正式得出类似的结果。因此,傍轴度和傍轴度修正值都可以用来描述光束的傍轴性。     

Complex modes and instability of full-vectorial beam propagation methods

Full-vectorial beam propagation methods (FVBPMs) are widely used to model light waves propagating in high-index-contrast optical waveguides. We show that the paraxial FVBPM and wide-angle FVBPMs based on diagonal Padé approximants are analytically unstable for waveguides with complex modes. The instability cannot be removed by enlarging the computational domain, increasing the numerical resolution, or using perfectly matched layers, because the complex modes are highly confined around the waveguide core.     

Spectral shifts and spectral switches of partially coherent vector beams propagating through an astigmatic aperture lens

By using the polarization matrix and propagation equation of the cross-spectral density function matrix, the spectral shifts and spectral switches of partially coherent vector beams propagating through an astigmatic aperture lens are studied, where the effect of lens astigmatism and beam cross-correlation on the spectral switches is stressed. It is shown that the critical position, spectral transition height and spectral minimum at spectral switches depend on the astigmatism. The beam cross-correlation plays a role in spectral switches under the condition that a polarizer is inserted. The spectral switch vanishes when the astigmatic coefficient is larger than a certain value, or the cross-correlation length is less than a certain value.     

Effect of chromatic aberration on Gaussian beams: non-dispersive laser resonators

The propagation of a non-monochromatic (polychromatic) TEM₀₀ Gaussian beam in vacuum, its passage through a thin plate and its transformation by a thin lens are studied in the case of a non-dispersive laser resonator. The basic assumptions of the model are as follows: optical fields are stationary and plane-polarized, the paraxial wave equation is valid, an equivalent non-dispersive hemiconfocal resonator represents the lasing medium and its stable resonator, the laser emits in a single mode. It is also assumed that the plate and the lens have large transverse dimensions. Mathematical expressions, for beam radius, divergence, radius of curvature and beam parameter product, are obtained. A beam quality factor for polychromatic Gaussian beams is defined and its value calculated in each case of interest. It is proposed to simulate a dispersive laser resonator by a non-dispersive resonator complemented with a plate and/or a thin lens.     

Virtual source for rotational symmetric Lorentz-Gaussian beam

The virtual source for generation of rotational symmetric Lorentz-Gaussian (RLG) wave whose propagating dynamics present the rotational symmetry is identified. Closed-form expressions, including integral and differential representations, are derived for this kind of Lorentz-Gaussian (LG) wave, thereby yieldingparaxial approximation of the RLG beam in the appropriate regime. From the spectral representation of this wave, the first three order corrections of nonparaxial approximations are determined for a corresponding paraxial RLG beam. Moreover, the relationship between the RLG beam and the Hermite-Gaussian beam is revealed.     

Propagation of paraxial circular symmetric beams in a general optical system

Starting from flattened Gaussian beams, we deduce analytical expressions for the propagation of paraxial circular symmetric beams through a general first-order optical system. It is shown that the propagation of any scalar beam in a paraxial domain takes the same form as that in free space. Three parameters were introduced to characterize the evolution of the propagating beam. Numerical examples are also presented to illustrate the effectiveness of this method.     

复宗量双曲余弦高斯光束的束腰及其位置

 由复宗量双曲余弦高斯光束通过近轴-ABCD-光学系统传输公式，推导出了基于二阶矩定义的光斑尺寸的解析表达式，给出了复宗量双曲余弦高斯光束在自由空间传输和通过薄透镜时的光斑尺寸、束腰宽度及其位置的解析表达式，并进行了数值计算，对计算结果进行了分析和讨论，该方法可用于其它光束通过近轴ABCD光学系统的类似计算。     

复宗量双曲余弦高斯光束的束腰及其位置

由复宗量双曲余弦高斯光束通过近轴-ABCD-光学系统传输公式,推导出了基于二阶矩定义的光斑尺寸的解析表达式,给出了复宗量双曲余弦高斯光束在自由空间传输和通过薄透镜时的光斑尺寸、束腰宽度及其位置的解析表达式,并进行了数值计算,对计算结果进行了分析和讨论,该方法可用于其它光束通过近轴ABCD光学系统的类似计算。     

Nonparaxial propagation of phase-flipped Gaussian beams

This paper derives the closed-form expressions for nonparaxial phase flipped Gaussian （PFG） beams propagating in free space, through a knife edge and an aperture, which enable us to study nonparaxial propagation properties of PFG beams and to compare nonparaxial results with paraxial ones. It is found that the f parameter, offsetting distance of the knife edge and truncation parameter affect the nonparaxial beam propagation properties. Only under certain conditions the paraxial approximation is applicable. The results are illustrated by numerical examples.     

一种自聚焦透镜的成像解

自聚焦锥透镜与圆柱透镜相比较，有其特殊之处。柱透镜在近轴平行光入射时，光线的振幅、相位、周期保持不变，而锥透镜的三者均发生变化，而且求解锥透镜的光线方程也比圆柱透镜复杂得多。本文推导了一个锥透镜的光线方程。可据此设计经过锥透镜后的光源大小，这在微系统照明中非常有用。     

Determination of the final equilibrium radius of a relativistic electron beam mismatched to the equilibrium conditions in transportation in the ion focusing regime

We derive the equation connecting the final equilibrium radius of a paraxial azimuth-symmetrical relativistic electron beam propagating in the ion focusing regime with its initial nonequilibrium value. The dependence of the final equilibrium radius and the increment of the rms emittance on the density of the scattering background medium is analyzed.     

超短脉冲经过小数值口径透镜的聚焦特性

 从亥姆霍兹方程出发，考虑透镜的色差和球差，得到了超短脉冲经过小数值口径透镜后光场的解析表达式；利用数值模拟得到了旁轴焦平面和边缘焦平面上光场强度的时空分布，以及聚焦区域内轴上点的强度分布图，并分析了色差和球差的大小对它们的影响。由于通过透镜中心和边缘之间的大部分脉冲产生了干涉相消，因此在旁轴焦平面与边缘焦平面上都出现前、后两个峰值相差很大的脉冲，并且随着球差的增大，两峰值差更大了；聚焦区域内轴上点的强度分布基本是对称的，强度最大点不是出现在旁轴和边缘焦平面上，而是在它们之间的某个位置，并且随着球差的增加，该位置将向聚焦区域的边缘靠近。     

Bragg-induced oscillations in non-PT complex photonic lattices

We investigate Bragg-induced oscillations in complex non-PT periodic structures, that is, a periodic medium with gain and loss and not necessarily symmetric under the combined action of parity and time reversal operations. We compare our analytic results based on the expansion of the optical field in Bragg-resonant plane waves with a direct numerical integration of the paraxial wave equation beyond the shallow potential approximation and using a wide Gaussian beam as initial condition. In particular, we study under which conditions a mode trapping phenomenon may still be observed and to inspect how the energy exchange between the spectral modes takes place during propagation in this more general class of asymmetric complex potentials.