部分相干电磁光束的焦移

 基于有效菲涅耳数的概念,给出一种简单的表达式来评估部分相干电磁光束经圆形光阑透镜后所产生的相对焦移,对部分相干电磁光束经透镜聚焦的情况进行了研究。结果表明:当交叉谱密度函数的非对角元素为零时,有效菲涅耳数不仅依赖于真实的菲涅耳数大小,而且依赖于光束的偏振分布和入射部分相干电磁光束两个偏振分量的空间相干长度;相对焦移量仅由等效菲涅耳数决定。将此方法与数值积分法进行的比较表明,等效菲涅耳数可以简单、准确地评估焦移。     

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.     

Spectral changes of partially coherent flat topped beam in turbulent atmosphere

Spectral characteristics of partially coherent flat-topped (PCFT) beams propagated in a turbulent atmosphere are investigated and an analytical formula for the spectrum S(ω) is derived. It is shown that, if the scaling law is valid, the normalized spectrum S(ω) of PCFT beams propagating through turbulent atmosphere is the same as the normalized source spectrum S⁽⁰⁾(ω), this similarity is valid for any arbitrary order of the circular flat-topped “N”. It is proved that if the scaling law is not valid, S(ω) of PCFT is different from S⁽⁰⁾(ω). Also spectral behavior of PCFT beams in free space and turbulence media is investigated in detail.     

部分相干光被环形透镜聚焦而产生的焦移

 从理论上研究部分相干光被一环形透镜聚焦,在焦点附近的轴上点的光强分布。研究结果表明,当部分相干光被一环形透镜聚焦时,最大聚焦光强不在几何焦点,而是位于透镜与几何焦点之间,出现焦移现象。并且,焦移量不仅依赖于透镜外半径的菲涅耳数,还依赖于部分相干光的空间相干度和中心拦截比。透镜的菲涅耳数越小,焦移越大;部分相干光的空间相干度越低,焦移越大。当菲涅耳数一定时,环形透镜的中心拦截比越大,焦移越大;当空间相干度很小时,情况就变得比较复杂。     

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.     

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.     

Transformation and spatial shaping of partially coherent cosh-Gaussian beams through an astigmatic lens

Based on the propagation law of partially coherent beams, the closed-form propagation expression for partially coherent cosh-Gaussian (ChG) beams through an astigmatic lens is derived. The transformation and spatial shaping of partially coherent ChG beams through the astigmatic lens are studied and illustrated by numerical examples. It is shown that a suitable choice of the spatial coherence parameter and/or astigmatic coefficient, different beam profiles, such as Gaussian-like, flat-topped and bottle beam profiles, and beam profile with a central dip, at the geometric focal plane and at a certain plane are realizable.     

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.     

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.     

Focal shift of partially polarized Gaussian Schell-model beams

A detailed study of the focal shift of partially polarized Gaussian Schell-model (PGSM) beams is made on the basis of the beam coherence-polarization (BCP) matrix approach. It is shown that, apart from the Fresnel number and truncation parameter, the focal shift of PGSM beams depends on the angle of polarizer and the ratio of degree of spatial coherence.     

Focal shift and focal switch of flat-topped Mathieu-Gaussian beams passing through an apertured lens system

By introducing a hard aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression predominating the distribution of axial intensity for the flat-topped Mathieu-Gauss (FTMG) beams passing through a system with the aperture and lens separated has been derived. The focal shift and the focal switch effect of FTMG beams passing through the system is studied in detail. Numerical calculations have shown that the position of real focal plane is not coincident with the geometrical focus but is somewhat shifted toward the lens. The focal shift and focal switch of FTMG beams take place when the relative separation s/f = 1 by a suitable choice of beam parameter and truncation parameter, for example, the beam parameter is smaller than its corresponding critical value or the truncation parameter is between its two corresponding critical values.     

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.     

湍流大气中聚焦平台光束焦移及光斑尺度

通过理论分析和数值模拟,对聚焦平台光束大气传输的焦移问题进行了研究。聚焦平台光束在湍流大气中传输时,湍流将导致光束的束腰向发射点移动,且湍流越强移动的幅度越大;大气湍流强度相同时,聚焦平台光束发射口径越小,其束腰移动的幅度越大。另外,通过调节发射系统的焦距,使聚焦平台光束的束腰正好位于接收探测器处时,接收探测器上的激光功率密度并非最大,只有当发射系统的焦距等于激光的传输距离时,接收探测器上的激光功率密度才会最大。     

Coherent and incoherent combinations of partially coherent beams beyond the paraxial approximation

The concept of coherent and incoherent combinations of partially coherent beams is described and interpreted physically. Taking the Gaussian Schell-model (GSM) beam as a typical example of a partially coherent beam, the analytical propagation expressions for the coherent and incoherent combinations of GSM beams beyond the paraxial approximation are derived and illustrated numerically. Some special cases of the general results are analyzed.     

Propagation of the off-axis superposition of partially coherent beams through atmospheric turbulence

The propagation properties of the off-axis superposition of partially coherent beams through atmospheric turbulence and their beam quality in terms of the mean-squared beam width w(z) and the power in the bucket (PIB) are studied in detail, where the effects of partial coherence, off-axis beam superposition and atmospheric turbulence are considered. The analytical expressions for the intensity, the beam width and the PIB are derived, and illustrative examples are given numerically. It is shown that the maximum intensity I_max and the PIB decrease and w(z) increases as the refraction index structure constant C_n² increases. Therefore, the turbulence results in a degradation of the beam quality. However, the resulting partially coherent beam with a smaller value of spatial correlation parameter γ and larger values of separate distance x_d and beam number M is less affected by the turbulence than that with a larger value of γ and smaller values of x_d and M. The main results obtained in this paper are explained physically.     

湍流大气中聚焦平台光束焦移及光斑尺度

 通过理论分析和数值模拟,对聚焦平台光束大气传输的焦移问题进行了研究。聚焦平台光束在湍流大气中传输时,湍流将导致光束的束腰向发射点移动,且湍流越强移动的幅度越大;大气湍流强度相同时,聚焦平台光束发射口径越小,其束腰移动的幅度越大。另外,通过调节发射系统的焦距,使聚焦平台光束的束腰正好位于接收探测器处时,接收探测器上的激光功率密度并非最大,只有当发射系统的焦距等于激光的传输距离时,接收探测器上的激光功率密度才会最大。     

Spectral shifts of partially coherent radial array beams passing through ABCD optical systems

The expressions for the spectral intensity of partially coherent Gaussian Schell-model (GSM) radial array beams for both the correlated and uncorrelated superpositions passing through ABCD optical systems have been derived by using the extended Huygens-Fresnel diffraction integral. The effects of the normalized radius R, the number of beamlets N, the spatial coherent parameter of array beamlets β and the optical system parameters on the on-axis and off-axis relative spectral shifts for the two types of superposition have been discussed in detail. The results show that for the correlated superposition, the on-axis spectral intensity in free space and the off-axis spectral intensity on the geometrical focal plane depends on the source spectral density S⁰(ω), the spatial coherent parameter of array beamlets β, the generalized Fresnel number of the system F, the normalized radius R and the number of beamlets N, whereas the spectral intensity for the uncorrelated superposition is independent of the number of beamlets N. Furthermore, as for on the actual focal plane, the off-axis spectral intensity for the two types of superposition is closely related to N.     

Spectral properties of spatially and spectrally partially coherent cosh-Gaussian pulsed beams focused by an aperture lens

The analytical expression for the spectrum of spatially and spectrally partially coherent cosh-Gaussian pulsed (ChGP) beams focused by an aperture lens is derived and used to study their spectral properties at the focal plane and on the axis. The effect of the truncation parameter, spatial correlation parameter, decentered parameter and temporal coherence length on the spectral shifts and spectral switches of spatially and spectrally partially coherent ChGP beams is stressed. The results are illustrated numerically and interpreted physically.     

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.