Focal shift and focal switch of partially polarized Gaussian Schell-model beams passing through a system with the aperture and spherically aberrated lens separated

Based on the beam coherent-polarization (BCP) matrix approach and propagation law of partially coherent beams, the focal shift and focal switch of partially polarized Gaussian Schell-model (PGSM) beams passing through a system with the aperture and spherically aberrated lens separated is studied in detail. Our main attention is focused on the effect of spherical aberration and partial coherence on the focal shift and focal switch of PGSM beams. It is shown that for polarizer-free case there is no focal switch of PGSM beams, the focal shift of PGSM beams is closely related with spherical aberration coefficient C₄, auto-coherence length σ_a, truncation parameter δ and relative position s/f between the aperture and lens in general, and is independent of the cross-coherence length σ_c. After inserting a polarizer the focal switch can take place. Numerical calculation results are given to illustrate how the spherical aberration and partial coherence affect the focal shift and focal switch of PGSM beams.     

Focal shift and focal switch of Bessel–Gaussian beams passing through a lens system with or without aperture

The focal shift and focal switch of Bessel–Gaussian (B–G) beams passing through a lens system with or without aperture is studied in detail. For the unapertured case, the necessary condition for the focal switch and the expression for the amplitude of the focal switch are derived. It is shown that if the truncation parameter δδ_c or beam parameter ββ_c (δ_c, β_c are the corresponding critical values), there exits only one axial intensity maximum, and the focusing without focal shift can be achieved for u/f=1 (u is the separation between the aperture and lens, f is the focal length of the lens); if δ>δ_c and β>β_c, there exist two axial intensity maxima, and the focal switch can take place at the turning point u/f=1 for both apertured and unapertured cases. The dependence of the amplitude of focal switch and normalized axial intensity minimum on the truncation parameter, beam parameter and Fresnel number is also studied.     

Focal shift of hollow Gaussian beams through a thin lens

In this paper, the focal shift of hollow Gaussian beams (HGBs) passing through a thin lens is investigated in detail. An analytic expression of the location of the axial maximum intensity for the HGBs is derived. It is found that, both the relative focal shift and the relative location of the axial maximal intensity of the HGBs is strongly affected by the ratio α (where α=s/f, here s is the axial distance from the input plane to the lens plane and f is the focus length of the lens), and it is also greatly affected by the changes of both the effective Fresnel number N_w and the order n of HGBs.     

Propagation of flat-topped beams

The propagation of flat-topped beams passing through paraxial ABCD optical system is investigated based on the propagation formulas of Gaussian beam. The focal shift of focused coherent flat-topped beam is also studied in detail. Analytical expressions of the M² factor and the far-field intensity distribution for flat-topped beams are derived on the basis of second-order moments.     

拉盖尔-高斯光束通过光阑-透镜分离系统的聚焦特性和焦移

 使用Collins公式推导了拉盖尔-高斯光束通过光阑-透镜分离系统的光强分布。利用数值计算详细分析和说明了其聚焦特性。分析得出光阑-透镜分离程度的大小影响相对焦移的大小和光强的分布，并引起焦移可能相反或消失。而且发现满足一定条件时在一段较大的轴向距离上光强几乎等于光强极大值，从而可获得一个较大的光强极大范围。     

拉盖尔-高斯光束通过光阑-透镜分离系统的聚焦特性和焦移

使用Collins公式推导了拉盖尔-高斯光束通过光阑-透镜分离系统的光强分布。利用数值计算详细分析和说明了其聚焦特性。分析得出光阑-透镜分离程度的大小影响相对焦移的大小和光强的分布,并引起焦移可能相反或消失。而且发现满足一定条件时在一段较大的轴向距离上光强几乎等于光强极大值,从而可获得一个较大的光强极大范围。     

Gradient force pattern, focal shift, and focal switch in an apodized optical system

In this paper, the evolution of the gradient force pattern, focal shift, and focal switch induced by a three-portion pure phase-shifting apodizer is numerically investigated in detail. The results show that the proposed apodizer may induce tunable gradient force on the particles in the focal region, focal shift, and focal switch. By adjusting the geometrical parameters of the phase-shifting apodizer, multiple traps may occur with changeable distance between them, and the shape of the optical trap also evolves evidently. More interestingly, for certain geometrical parameters of the proposed apodizer, by changing the phase shift of inner annular portion, the considerable focal shift may occur with focal switch accompanying, which is discussed to show that this kind of apodizer may be a very promising method of transporting trapped particles.     

Focal shift of flat-topped beams passing through a lens system with or without aperture

We theoretically investigated on the focal shift of flat-topped beams (FTBs) passing through a lens system with or without aperture. We find that the position of on-axis peak intensity of focused FTBs strongly depends on the Fresnel number, the flatness order of FTBs, the truncation of the aperture, as well as the relative incident distance. The results indicate that the focal shift away from the geometrical focus of the focused FTBs appears when the relative incident distance does not equal to 1 and increases with the decreasing of the Fresnel number or the beam flatness order for both the apertured and unapertured lens systems. In the lens system with aperture, the focal shift effect decreases with the increase of the truncation parameter.     

厄米-高斯光束的焦开关

考虑了由光阑和透镜组成的简单光学系统。研究表明，当厄米-高斯光束入射这一系统，在系统和光束参数满足一定条件时会出现称为焦开关的新现象。详细的数值计算和物理分析说明了厄米-高斯光束的焦开关特性。与已有的工作做了比较，并指出了焦开关可能的应用。     

A detailed study of Mathieu-Gauss beams propagation through an apertured ABCD optical system

Using Collins formula and the expansion of Mathieu beams in terms of Bessel beams we derive the exact propagation equations of Mathieu-Gauss beams through an apertured paraxial ABCD optical system. A comparison between the exact propagation equations and the approximated ones, which are derived by expanding the circ function into a finite sum of Gaussian functions, is presented. The propagation characteristics of zeroth-order Mathieu-Gauss beams in (y-z) and (x-z) planes are analyzed with detailed numerical calculations. It is found that the profile of the propagated Mathieu-Gauss beam is similar to that of Bessel-Gauss beam. Furthermore, the focalization of the Mathieu-Gauss beams through a thin lens is illustrated and analyzed with numerical results.     

Focal switch in unapertured converging Hermite-cosh-Gaussian beams

Based on the Huygens–Fresnel diffraction integral, analytical representation of unapertured converging Hermite-cosh-Gaussian beams is derived. Focal switch of Hermite-cosh-Gaussian beams is studied detailedly with numerical calculation examples and a physical interpretation of focal switch is presented. It is found that decentered parameter is the dominant factor for the emergence of focal switch, and Fresnel number affects the amplitude of focal switch and the value of critical decentered parameter to determine emergence of focal switch. Physically, the emergence of focal switch of Hermite-cosh-Gaussian beams is resulted from competition between two major maximum intensities and switch of the absolute maximum intensity from a point to another when decentered parameter increases.     

Focal shift of focused truncated random electromagnetic beams

The focal shifts of focused truncated random electromagnetic beams are investigated. Based on the complex Gaussian expansion method for a hard-edged aperture function, the analytical propagation formula of cross-spectral density matrix for a random electromagnetic beam focused by an optical system with a thin lens and a circular aperture is derived. The Fresnel numbers related with the beam and system parameters are defined and used to examine focal shifts. The dependence of the focal shifts on the different Fresnel numbers and polarization distribution are discussed in detail with numerical examples.     

Focal shift for off-axial partially coherent flat topped beams passing a thin lens

Focal shift is evaluated for partially coherent off-axis flat topped beams propagated through a thin lens. We have shown that the focal shift decreases as the off-axis parameter (x₀) increases and, for fully coherent flat topped beams, focal shift is enhanced as the order of flat topped beam decreases, but the partially coherent one does not show this behavior. Also we have shown that there is a critical point for partially coherent flat topped beams. We investigated the dependency of focal shift in thin lens on correlation length and focal length.     

Focal shift in focused off-axial Hermite-cosh-Gaussian beams

Based on the encircled-power criterion, the focal shift of off-axial Hermite-cosh-Gaussian beams passing through an apertured lens is studied and illustrated with detailed numerical examples. The dependence of focal shift on the beam and optical system parameters is discussed.     

聚焦贝塞尔-高斯光束的焦开关效应

 详细研究了贝塞尔-高斯光束被无光阑透镜聚焦后的焦开关效应。推导出了轴上最大光强位置满足的一元三次方程和相对跃迁量公式,并用数值计算对理论结果作了说明。研究表明,当光束参数大于2时,聚焦贝塞尔-高斯光束会有焦开关现象出现。相对跃迁量随着光束参数的增大而增大,轴上归一化光强最小值随着光束参数的增大而减小,并且当光束参数大于3.9时,轴上归一化光强最小值为零。     

Propagation of off-axial cosine-Gaussian beams passing through an apertured and misaligned optical system

A closed-form expression for the one-dimensional off-axial cosine-Gaussian beams passing through an apertured and misaligned paraxially ABCD optical system is derived. As special cases, the corresponding closed-forms for the off-axial or non off-axial cosine-Gaussian beams passing through apertured or unapertured and misaligned or aligned optical systems are also given. The obtained results could be straightforward to the two-dimensional case or sine-Gaussian beams.     

Focal shift of the partially coherent vortex beams focused by an aperture lens

Based on the propagation law of partially coherent beams, the propagation equations of the partially coherent vortex beams focused by an aperture lens are derived. Based on these equations, the focal shift of partially coherent vortex beams is studied, and illustrative numerical results are given. It is found that the focal shift of the partially coherent vortex is dependent not only on the Fresnel number but also on the relative coherent length and the topological charge of the vortex.