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 switch of spherically aberrated Laguerre–Gaussian beams

Based on the Collins formula, we study the focal switch of spherically aberrated Laguerre–Gaussian (L–G) beams passing through an optical system with the aperture and lens separated. Our results demonstrate that the behavior of the focal switch of spherically aberrated L–G beams is generally dependent on the spherical aberration coefficient, truncation parameter, Fresnel number and mode indices. The spherical aberration affects the relative focal shift, critical position of the focal switch and the relative transition. We also illustrate that there exists a minimum in addition to a maximum in the truncation parameter. Only inside the region between the minimum and the maximum the focal switch effect can be expected. The region depends on the spherical aberration coefficient as well as the Fresnel number.     

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 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.     

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.     

多色矢量高斯-谢尔模型光束的焦移和焦开关

从交叉谱密度矩阵的传输公式出发，对多色矢量高斯-谢尔模型（GSM）光束的焦移和焦开关作了详细的研究.插入偏振片之前，多色矢量高斯-谢尔模型（GSM）光束通过硬边光阑透镜分离光学系统后，有焦移，但无焦开关；而插入偏振片之后，会出现焦开关.改变偏振片的旋转角度可以控制焦开关的特性.     

Focal switching of partially coherent modified Bessel–Gaussian beams passing through an astigmatic lens with circular aperture

The axial intensity distribution of partially coherent modified Bessel–Gaussian (PCMBG) beams passing through an astigmatic lens with circular aperture is derived on the basis of Collins formula. The influences of truncation parameter δ and astigmatic coefficient α on the axial intensity distribution are presented. The research shows that the axial intensity changes with the variation of α, and twice focal switches have been found. Furthermore, the physical interpretation of the twice focal switches is given and is also illustrated by numerical calculations. It has been revealed that the existence of astigmatism and the effect of the truncation parameter are the two main reasons for the twice focal switches of PCMBG beams passing through an astigmatic lens with circular aperture.     

部分相干修正贝塞尔高斯光束通过光阑透镜的光谱特性

基于部分相干光的传输理论，研究了部分相干修正贝塞尔高斯光束（MBGB）通过光阑透镜系统后,轴上光谱位移及光谱开关.数值计算结果表明，光谱开关主要与截断参量、光束菲涅耳数以及光谱相干度有关.光谱开关的个数随着光束菲涅耳数以及截断参量的增大将增多，而随光谱相干度的增大而减少.但当截断参量增大到一定值时，光谱开关将消失.例如，当光谱相干度取0.5时，截断参量大于2.2，光谱开关将消失，仅表现出蓝移.     

Focal switch of cosine-Gaussian beams

The focal switch of cosine-Gaussian (CsG) beams passing through a system with the aperture and lens separated is studied analytically and numerically. It is shown that the focal switch of CsG beams can appear not only for the apertured case, but also for the unapertured case. The necessary condition for the focal switch is that truncation parameter α > α_c and the beam parameter β > β_c, α_c, β_c being the corresponding critical values. There exists a maximum of the relative transition height Δz _sw as α varies, and Δz _sw increases with increasing β and decreasing N _w. The normalized axial intensity minimum I _min / I _max decreases with an increase of α and β, and I _min / I _max remains unchanged as N _w varies.     

Propagation characteristics of flattened Gaussian beams through a misaligned optical system with a misaligned annular aperture

The approximate analytical formula for flattened Gaussian beams through a misaligned optical system with a misaligned annular aperture was derived by the extended Huygens–Fresnel principle. Some numerical simulations are illustrated to the effects on the propagation of flattened Gaussian beams by the misaligned annular aperture. To compare the difference between annular apertured system and circular apertured system, the circular apertured system is also studied. The results show that angle misalignments and lateral displacements of aperture create asymmetrical average intensity distribution at receiving plane z = 500. The effects on intensity distribution by angle misalignments of annular aperture were small. In annular aperture case, the smooth of intensity distribution was worse with escalating obscure ratio ? in near-field; the side-lobes increased and the central lobe decreased with escalating obscure ratio ? in far-field. At receiving plane z = 500: for circular aperture, the side-lobes decreased, even to be neglected, with the increasing of truncation parameter δ; for annular aperture, the side-lobes increased with the increasing of truncation parameter δ. In addition, it is found that the aligned thin lens can fix asymmetry of intensity distribution which was caused by the misaligned annular aperture.     

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。     

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.     

Effect of quartic-phase aberrations on the focal switch of Hermite–Gaussian beams

The focal switch of Hermite–Gaussian beams diffracted at an aperture and subsequently focused by a spherically aberrated lens is studied. Our main attention is focused on the effect of quartic-phase aberrations on the behavior of the focal switch. It is shown that quartic-phase aberrations affect the relative focal shift Δz_f, turning position s_1,t, and relative transition height Δz_sw. Apart from a critical maximum truncation parameter α_c,max, there is a critical minimum truncation parameter α_c,min. Within the region α_c,min<α<α_c,max the focal switch can take place, but quartic-phase aberrations give rise to a decrease of α_c,max-α_c,min in comparison with the aberration-free case.     

Tightly focusing of spirally polarized Quadratic Bessel Gaussian beam through a dielectric interface

Based on vector diffraction theory, the tight focusing properties of spirally polarized axisymmetric Bessel-modulated Gaussian beam through a dielectric interface on high numerical aperture (NA) are investigated theoretically. The optical intensity distribution in the focal region of high NA objective lens is investigated in detail by numerical calculations. The results show that the focal shift induced in the focal region is by mismatch of refractive indices across the dielectric interface. It is also found that the optical intensity in focal region of spirally polarized Quadratic Bessel Gaussian (QBG) beam can be altered considerably by changing spiral parameter C that indicates the polarization spiral degree of the incident beam.     

Focusing of flattened Gaussian beams by an annular lens

Focusing properties of flattened Gaussian beams (FGBs) passing through an annular lens is studied based on the Collins formula. It is found that the on-axis irradiance distributions of focused FGBs are unsymmetrical with respect to the geometrical focal plane even for large values of Fresnel number F_w associated with the beam, so that there exist focal shifts in general. Detailed numerical results show the dependence of focal shifts on the beam and system parameters. Focal shifts of FGBs by a lens without central obscuration and focal shifts of Gaussian beams by an annular lens can be treated as special cases of the obscure ratio =0 and beam order N=0, respectively. Furthermore, focal shifts of plane waves by an annular lens can also be treated as a special case of N=0 and F_w→∞.     

Quasi Bessel beam by Billet's N-split lens

This study utilizes the focal property of a classical Billet's split lens to create more focal points by splitting the lens. This approach distributes the focal points circularly on the focal plane. This study explores the characteristics of beam propagation and analytically derives the asymptotic characteristics of beam propagation based on the stationary phase approximation and the moment-free Filon-type method. Results show that the unique Billet's N-split lens can generate a quasi Bessel beam if the number of splitting N is large enough, e.g., N ≧ 24. This study also explores the diffraction efficiency of corresponding quasi Bessel beam and the influence of aperture size. The potential advantage of proposed split lens approach is that, unlike the classical means of annular aperture, this simple lens approach allows a much larger throughput in creating the Bessel beam and hence the Bessel beam could have more optical energy.     

Nonlinear change of on-axis pressure and intensity maxima positions and its relation with the linear focal shift effect

A comprehensive experimental, analytical and numerical study of the true focal region drift relative to the geometrical focus (focal shift effect) in acoustic focused beams and its nonlinear evolution is presented. For this aim, the concept of Fresnel number, proportional to the linear gain, is introduced as a convenient parameter for characterizing focused sources. It is shown that the magnitude of the shift is strongly dependent on the Fresnel number of the source, being larger for weakly focused systems where a large initial shift occurs. Analytical expressions for axial pressure distributions in linear regime are presented for the general case of truncated Gaussian beams. The main new contribution of this work is the examination of the connection between the linear and nonlinear stages of the focal shift effect, and its use for the estimation of the more complicated nonlinear stage. Experiments were carried out using a continuous-wave ultrasonic beam in water, radiated by a focused source with nominal frequency f = 1 MHz, aperture radius a = 1.5 cm and geometrical focal distance R = 11.7 cm, corresponding to a Fresnel number N_F = 1.28. The maximum measured shifts for peak pressure and intensity were 4.4 and 1.1 cm, respectively. The evolution of the different maxima with the source amplitude, and the disparity in their axial positions, is interpreted in terms of the dynamics of the nonlinear distortion process. Analytical results for the particular case of a sound beam with initial Gaussian distribution are also presented, demonstrating that the motion of peak pressure and peak intensity may occur in opposite directions.     

高功率激光通过有中心遮拦透镜的聚焦

用广义惠更斯－菲涅耳衍射积分对有振幅调制和位相畸变的高功率激光束通过有中心遮拦透镜的聚焦特性作了详细研究，并对焦面处光束的可聚焦性进行了讨论。数值计算结果表明，功率集中度与光阑的遮拦比，光束截断参数以及振幅调制和位要畸变参数等有关。     

会聚厄米-双曲余弦高斯光束的光强分布和焦移

 基于厄米-双曲余弦高斯光束通过无光阑限制薄透镜聚焦的解析传输公式，研究了厄米-双曲余弦高斯光束聚焦区域的光强分布，并对光束的焦移进行了分析，讨论了偏心参数对光强主极大位置的影响。结果表明：TEM₁₁模厄米-双曲余弦高斯光束的相对焦移（绝对值）随偏心参数和菲涅尔数的减小而增大，菲涅尔数较大时相对焦移趋于零。TEM₂₂模光束在偏心参数小于0.54时，轴外与轴上光强极大值的比值大于1，此时光强主极大在轴外，偏心参数大于0.54时则相反；在偏心参数等于0.54时比值为1，此时光束有两个主极大，偏心参数愈大光强愈集中于轴上。使用LW法和GH法得到的TEM₂₂模光束的相对焦移（绝对值）随偏心参数和菲涅尔数的变化规律与TEM₁₁模光束一致，但相同参数下使用这两种方法得出的具体结果不同。     

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.