Spectral shifts and spectral switches of a pulsed Gaussian beam from a rectangular aperture in the far field

The far-field anomalous spectral behaviours of a space–time-dependent Gaussian pulsed beam passing through a rectangular aperture are studied. By expanding a hard aperture function into a finite sum of complex Gaussian functions and starting from the Fresnel diffraction integral, the approximate analytical expression for the spectral intensity of a space–time-dependent Gaussian pulsed beam passing through a rectangular aperture is derived. Meanwhile, the corresponding closed-forms for the slit and the unapertured cases are also given as special cases of the general results. The red and blue shifts and the spectral intensity distribution are studied and illustrated with numerical calculations. Specifically, it is shown that the spectral switch takes place when the truncation parameter is equal to particular values or the observation position is at the critical diffraction angle. The possibility of tunable spectral switching in the far field with an apertured pulsed beam by varying the size of the rectangular aperture is highlighted.     

An intuitive model for on-axis pulse evolution of ultrashort pulsed Gaussian beams diffracted from a circular aperture

The diffraction of ultrashort pulsed Gaussian beams from a circular aperture is studied by means of Fresnel diffraction integral and Fourier transform method. A uniform analytical expression is derived for temporal pulse form of ultrashort pulsed Gaussian beams in two cases, i.e. with constant beam waist and with constant diffraction length. It is shown that the on-axis pulse can be formulated as a superposition of an unapertured pulse and an aperture-induced pulse. The superposition of these two pulses leads to an enhanced pulse intensity for small truncation parameters at certain distances in the near field. Our results may find applications in high-intensity laser waveform control.     

Phase singularities and spectral switches of focused higher-order Bessel–Gauss pulsed beams

The phase singularities and spectral switches of focused higher-order Bessel–Gauss pulsed beams are studied. Numerical calculation results are given to illustrate the dependence of phase singularities and spectral switches of focused higher-order Bessel–Gauss pulsed beams on the truncation parameter, topological charge, spatial parameter and propagation distance. It is shown that there always exists an optical vortex at the center of focused higher-order Bessel–Gauss pulsed beams and the topological charge is conserved during the propagation. The spectral switch appears in the neighborhood of the zero- or minimum-intensity position. With increasing topological charge or spatial parameter, the size of the vortex core increases and the spectral transition height decreases.     

Optimum truncation of a Gaussian beam for propagation through atmospheric turbulence

The mean on-axis far-field (or focal-plane) irradiance of a Gaussian beam that is truncated by a circular aperture in the presence of atmospheric turbulence is considered. In the absence of turbulence, an accurate analytic approximation for the irradiance distribution that is valid within the main central lobe of the beam is presented. Based on this approximation, the mean on-axis far-field irradiance and the corresponding turbulence Strehl ratio for the truncated Gaussian beam are then obtained. By maximization of the on-axis irradiance, the optimum ratio of the beam diameter to the aperture diameter in the presence of turbulence is obtained, and the results for the corresponding maximum on-axis irradiance as a function of the strength of turbulence are presented. In particular, for D/r(0) > 1, where D is the aperture diameter and r(0) is Fried's coherence length, optimum truncation of a Gaussian beam and uniform illumination of a circular aperture (where the same total power isuniformly distributed over the aperture) result in the same on-axis irradiance in the presence of uncompensated turbulence.     

Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture

Based on the vectorial Rayleigh-Sommerfeld diffraction integrals, an analytical propagation equation of vectorial, nonparaxial, elliptical Gaussian beams through a rectangular aperture is derived. Unlike in previous work, the aperture effect and nonrotational symmetry of the beam and aperture are considered in our theoretical model. The results of the far-field and paraxial approximation for the apertured case are treated as special cases of our general expression. It is found that two f parameters fx, fy and two truncation parameters deltax, deltay in the x and y directions, respectively, have to be introduced that affect the beam nonparaxial evolution behavior in both the near field and the far field.     

Spatial correlation properties and correlation vortices of partially coherent vortex beams diffracted by an aperture

Taking the Gaussian–Schell model vortex beam as a typical example of partially coherent vortex beams, the spatial correlation properties and correlation vortices of partially coherent vortex beams diffracted by an aperture are studied. It is shown that the off-axis displacement and spatial coherence affects the spectral degree of coherence. The number and position of correlation vortices depend on the off-axis displacement, spatial coherence, aperture truncation and propagation distance, where the effect of aperture diffraction on the correlation vortices is stressed. The number of correlation vortices decrease as the truncation parameter increases. The correlation vortices in the diffracted field result from the vortex embedded in partially coherent beams at the source plane rather than from the aperture diffraction. The correlation vortices in the diffracted field appear even when the vortex core is stopped by the aperture.     

Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam

The truncated fractional Fourier transform (FRT) is applied to a partially coherent Gaussian Schell-model (GSM) beam. The analytical propagation formula for a partially coherent GSM beam propagating through a truncated FRT optical system is derived by using a tensor method. Furthermore, we report the experimental observation of the truncated FRT for a partially coherent GSM beam. The experimental results are consistent with the theoretical results. Our results show that initial source coherence, fractional order, and aperture width (i.e., truncation parameter) have strong influences on the intensity and coherence properties of the partially coherent beam in the FRT plane. When the aperture width is large, both the intensity and the spectral degree of coherence in the FRT plane are of Gaussian distribution. As the aperture width decreases, the diffraction pattern gradually appears in the FRT plane, and the spectral degree of coherence becomes of non-Gaussian distribution. As the coherence of the initial GSM beam decreases, the diffraction pattern for the case of small aperture widths gradually disappears.     

Topological phase in a nonparaxial Gaussian beam

An analysis is made of the structure and evolution of the singularities of a nonparaxial Gaussian beam. It is shown that a Gaussian beam may be represented by a family of straight lines lying on the surface of a hyperboloid and that the wavefront of this beam is a function of a point source situated at a point on the z axis with the imaginary coordinate iz ₀. The argument of this complex function is the topological phase of the beam which characterizes the rotation of the wavefront. The singularities of a nonparaxial Gaussian beam are located in the focal plane and are annular edge dislocations. Dislocation processes near the constriction of the Gaussian beam only occur as a result of aperture diffraction. Pis’ma Zh. Tekh. Fiz. 25, 14–20 (November 26, 1999)     

Propagation characteristics of Gaussian beams through a paraxial ABCD optical system with an annular aperture

By introducing a hard aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression for a Gaussian beam passing through a paraxial ABCD optical system with an annular aperture has been derived. The results could be reduced to the case of circular black screen or circular aperture. Some numerical simulations are also performed and illustrated for the propagation characteristics of a Gaussian beam through a paraxial ABCD optical system with an annular aperture, a circular black screen or a circular aperture.     

Power deposited by a Gaussian beam on a decentered circular aperture

An expression for the energy or power deposited by a Gaussian beam on a decentered circular aperture is derived. It represents a generalization of the classic laser-beam truncation problem, with applications in the areas of laser scanning, detection theory, lidar, free-space communications, and so on. In addition, it can be used to quantify the effect of alignment errors on laser systems.     

Generation of dark hollow femtosecond pulsed beam by phase-only liquid crystal spatial light modulator

Based on the refractive laser beam shaping system, the dark hollow femtosecond pulse beam shaping technique with a phase-only liquid crystal spatial light modulator (LC-SLM) is demonstrated. The phase distribution of the LC-SLM is derived by the energy conservation and constant optical path principle. The effects of the shaping system on the temporal properties, including spectral phase distribution and bandwidth of the femtosecond pulse, are analyzed in detail. Experimental results show that the hollow intensity distribution of the output pulsed beam can be maintained much at more than 1200 mm. The spectral phase of the pulse is changed, and the pulse width is expanded from 199 to 230 fs, which is caused by the spatial-temporal coupling effect. The coupling effect mainly depends on the phase-only LC-SLM itself, not on its loaded phase distribution. The experimental results indicate that the proposed shaping setup can generate a dark hollow femtosecond pulsed beam effectively, because the temporal Gaussian waveform is unchanged.     

Focal shift of focused truncated Lorentz-Gauss beam

Based on the Collins diffraction integral formula and the complex Gaussian expansion of the aperture function, an analytical expression for a Lorentz-Gauss beam focused by an optical system with a thin lens and a circular aperture has been derived. The focal shift of the focused truncated Lorentz-Gauss beam is investigated with numerical examples, and the dependence of the focal shift on the different parameters of the focused truncated Lorentz-Gauss beam is discussed in detail. This research is useful to the applications of highly divergent laser beams.     

Variational analysis of z scan of thick medium with an elliptic Gaussian beam

By variational approach, we analyze the characteristics of beam propagation through a cubic optical nonlinear medium using a laser beam that has a transverse elliptic Gaussian profile. The analytic solution to the normalized transmittance at the center of the far field as a function of medium position and the beam characteristics is obtained and compared with the numerical simulation, which is realized by a combination of algorithms. We also analyze the peak-valley transmittance difference as a function of medium length, ellipticity, and a stigmatism. The relationship between peak-valley normalized transmittance difference of the z-scan trace and aperture size or the slit width are obtained. Meanwhile, the comparison of z-scan characteristics with an elliptic Gaussian beam with those using a circular symmetric Gaussian beam is made.     

Spectral changes and spectral switches of partially coherent beams focused by an aperture lens

Starting from the propagation law of partially coherent polychromatic light in the space-frequency domain, detailed numerical results and physical analysis are given to elucidate spectral changes and spectral switches at the geometrical focal plane of Gaussian Schell-model beams focused by an aperture lens. It is found that, in contrast to the aperture-induced spectral anomalies of spatially fully coherent polychromatic light, for partially coherent polychromatic light aperture diffraction plays an important role in spectral switching, but the truncation parameter, spectral correlation, and bandwidth all affect its spectral behavior.     

环形孔径高斯光束的远场特性

讨论环形孔径高斯光束的远场光斑能量分布，并给出拟合表达式。首先推导光束没有抖动时远场光斑的光能分布表达式，然后研究各种不同中心遮拦和不同光束截断比，在有以及没有激光束抖动的情况下，对高斯光束的远场环围能量分布变化的影响。     

Analytical far-field divergence angle of a truncated gaussian beam

Approximate, but accurate, analytical expressions for the far-field divergence angle of a Gaussian beam normally incident on a circular aperture are derived. A first equation is obtained based on the concept of Gaussian transform, in which the Bessel function present in the far-field diffraction integral is approximated by a Gaussian function. Refining this approach yields another simple, practical closed-form formula with such a level of accuracy that we propose that it can be used as an exact reference. All approximations hold for any combination of Gaussian beam width and aperture radius.     

Target-plane intensity approximation for apertured Gaussian beams applied to heterodyne backscatter lidar systems

With an application in lidar systems in mind, we investigate the effects of transmit-aperture truncation of Gaussian beams by employing the extended Huygens-Fresnel principle. We derive an approximation to the top-hat aperture-transmission function by defining an abstract Gaussian aperture-transmission function. The two fitting parameters of the latter are found when the beam radius and the on-axis intensity for both aperture cases are equated in the observation plane. Bounds for the applicability of the approximation are established, and its accuracy and usefulness is demonstrated through application to the calculation of the return signal of a heterodyne lidar system.     

Effective schema for the rigorous modeling of grating diffraction with focused beams

Most modal diffraction methods are formulated for incident plane waves. In practical applications, the probing beam is focused. Usually, this is simulated by means of numerical integration where Gaussian quadrature formulas are most effective. These formulas require smooth integrands, which is not fulfilled for gratings due to Rayleigh singularities and physical resonances. The violation of this condition entails inaccurate integration results, such as kinks and other artifacts. In this paper, a methodology for the efficient treatment of the numerical integration with improved accuracy is presented. It is based on the subdivision of the aperture along the lines of Rayleigh singularities, mapping of these subapertures into unit squares, and separate application of the Gaussian cubature formulas for each subarea.     

Some Effects of Temporal Coherence on the First Order Statistics of Speckle

Moments and the first order probability density function of the intensity in a speckle pattern are considered as a function of the spectral bandwidth of the incident light. It is shown that the standard deviation of the intensity generally depends on the surface roughness of the diffuser, and on the position of measurement in the far field. Numerical results are given for a diffuser with a Gaussian distribution of surface height limited by a circular aperture. Exact and approximate forms of the probability density function are discussed. The analysis applies when the fluctuations in optical path are greater than the maximum wavelength incident.     

Generalized M* factor of truncated partially coherent controllable dark-hollow beams

On the basis of the fact that a hard-edged aperture function can be expanded into an approximate sum of complex Gaussian functions with finite numbers and the method of truncated second-order moments, the generalized beam propagation factor of truncated partially coherent controllable dark-hollow beams is derived. Some typical numerical simulations are given to illustrate the relations of the generalized beam propagation factor to four parameters: beam parameter ε, beam order N, truncation parameter F and coherence parameter T.