Experimental investigation of the intrinsic properties of the optical coupling between axisymmetric Gaussian beams

Continuing a previous analytical and numerical work, an experimental investigation of seven intrinsic properties of the optical coupling between axisymmetric Gaussian beams is presented. In this study, two single-mode fibers are used as the receiver and the emitter and a five-axis nanopositioning system is used to investigate optical coupling properties by moving one fiber relative to the other. Experiments demonstrate the existence of sufficiently accurate hyperbolic, parabolic, and linear trends for the optical coupling phenomenon, which can be useful for developing model-based alignment algorithms.     

Gaussian content as a laser beam quality parameter

We propose the Gaussian content (GC) as an optional quality parameter for the characterization of laser beams. It is defined as the overlap integral of a given field with an optimally defined Gaussian. The definition is especially suited for applications where coherence properties are targeted. Mathematical definitions and basic calculation procedures are given along with results for basic beam profiles. The coherent combination of an array of laser beams and the optimal coupling between a diode laser and a single-mode fiber are elaborated as application examples. The measurement of the GC and its conservation upon propagation are experimentally confirmed.     

Propagation of a decentered elliptical Gaussian beam through apertured aligned and misaligned paraxial optical systems

By expanding the hard aperture function into a finite sum of complex Gaussian functions, approximate analytical formulas for a decentered Gaussian beam (DEGB) passing through apertured aligned and misaligned paraxial apertured paraxial optical systems are derived in terms of a tensor method. The results obtained by using the approximate analytical expression are in good agreement with those obtained by using the numerical integral calculation. Furthermore, approximate analytical formulas for a decentered elliptical Hermite-Gaussian beam (DEHGB) through apertured paraxial optical systems are derived. As an application example, approximate analytical formulas for a decentered elliptical flattened Gaussian beam through apertured paraxial optical systems are derived. Our results provide a convenient way for studying the propagation and transformation of a DEGB and a DEHGB through apertured paraxial optical systems.     

Model-based far-field alignment algorithm for Gaussian beamlike single-mode optical devices

Single-mode device-to-fiber alignment automation is usually achieved with a classical mathematical optimization approach. We present a different approach, which is based on the identification of particular intrinsic characteristics of the coupled optical power and on estimating residual axial, transverse, and angular misalignments in the far field. Such a model-based approach is based on the physical nature of the optical coupling phenomenon and can replace or be complementary to already known automated alignment methods. An alignment algorithm is described and validated experimentally using two single-mode fibers as Gaussian beam emitter and receiver.     

Propagation of anomalous hollow beams through a circular-apertured optical system

In this paper, we consider the propagation of anomalous hollow beams through a circular-apertured paraxial ABCD optical system. Based on the Huygens–Fresnel integral and the expansion of the circular aperture function into a finite sum of Gaussian functions, the approximation analytical expression of an anomalous hollow beam passing through a circular-apertured paraxial ABCD optical system is derived. Based on the approximation analytical expression, the propagation properties in free space are numerically illustrated. This method provides a convenient tool for studying the propagation and transformation properties of anomalous hollow beams passing through a circular-apertured paraxial ABCD optical system.     

Coupling plane wave received by an annular aperture into a single-mode fiber in the presence of atmospheric turbulence

The efficiency of coupling a plane wave into a single-mode fiber can be reduced by both the aperture obstruction of receivers and the turbulence-induced degradation of optical coherence. Using the Gaussian approximation to the mutual coherence function of the incident optical field, we derived an analytical solution for the fiber-coupling efficiency when a plane wave, propagating through atmospheric turbulence, is received by an annular-aperture receiver and coupled into a single-mode fiber. It is a function of the coupling geometry, the aperture-radius-to-coherence-radius ratio (ARCRR), and the aperture-obstruction parameter. It is found by the numerical optimization method that the optimal coupling geometry depends on both the ARCRR and the aperture-obstruction parameter. The results obtained are useful for analyzing and designing a fiber-coupling system influenced by atmospheric turbulence.     

Propagation of decentered elliptical Gaussian beams in apertured and nonsymmetrical optical systems

A hard-edged elliptical aperture is described approximately by a tensor form, which can be expanded as a finite sum of complex Gaussian functions. An analytical propagation expression for a decentered elliptical Gaussian beam (DEGB) through an axially nonsymmetrical optical system with an elliptical aperture is derived by using vector integration. The approximate analytical results are compared with numerically integral ones, and it is shown that this method can significantly improve the efficiency of numerical calculation. Some numerical simulations are illustrated for the propagation properties of DEGBs through apertured and nonsymmetrical optical transforming systems.     

Fast simulation of the propagation of a spatially non-uniform laser beam through apertured ABCD systems

Based on the treatment that the rectangular function can be expanded into an approximate sum of complex Gaussian functions with finite numbers, the analytical expressions for the propagation of partially non-uniform laser beams through apertured ABCD optical systems were derived. The typical numerical examples were given and were compared with those obtained from numerically integral calculations. It can be shown from the results of our study that our method can significantly improve the numerical calculation ejciency.     

Propagation of elliptical Gaussian beams in apertured and misaligned optical systems

On the basis of the fact that a hard-edged elliptical aperture can be expanded approximately as a finite sum of complex Gaussian functions in tensor form, an analytical propagation expression for an elliptical Gaussian beam (EGB) through a misaligned optical system with an elliptical aperture is derived by use of vector integration. The approximate analytical results provide more convenience for studying the propagation and transformation of EGBs than the usual way by using a diffraction integral directly, and the efficiency of numerical calculation is improved. Some numerical simulations are illustrated for the propagation properties of EGBs through apertured optical transforming systems with misaligned thin lenses.     

Generalized huygens-fresnel diffraction integral for misaligned asymmetric first-order optical systems and decentered anisotropic Gaussian Schell-model beams

The generalized Huygens-Fresnel diffraction integral for misaligned asymmetric first-order optical systems is derived by using the canonical operator method, which enables us to study propagation properties of anisotropic Gaussian Schell-model (AGSM) beams through misaligned asymmetric first-order optical systems. It is shown that under the action of misaligned asymmetric first-order optical systems AGSM beams do not preserve the closed property. Therefore generalized partially coherent anisotropic Gaussian Schell-model beams called decentered anisotropic Gaussian Schell-model (DAGSM) beams are introduced, and AGSM beams can be regarded as a special case of DAGSM beams.     

Propagation of hollow Gaussian beams through apertured paraxial optical systems

On the basis of the generalized Collins formula and the expansion of the hard-aperture function into a finite sum of complex Gaussian functions, an approximate analytical formula for a hollow Gaussian beam propagating through an apertured paraxial stigmatic (ST) ABCD optical system is derived. Some numerical examples are given. Furthermore, by using a tensor method, we derive approximate analytical formulas for a hollow elliptical Gaussian beam propagating through an apertured paraxial general astigmatic ABCD optical system and an apertured paraxial misaligned ST ABCD optical system. Our results provide a convenient way for studying the propagation and transformation of a hollow Gaussian beam and a hollow elliptical Gaussian beam through an apertured general optical system.     

Hollow Gaussian beams with the power-exponent-phase vortex

A kind of hollow Gaussian beams with the power-exponent-phase vortex is introduced. Based on the Collins integral, an analytical expression of a hollow Gaussian beam with the power-exponent-phase vortex passing through a paraxial optical system described by the ABCD matrix approach is derived. The analytical expressions for the beam propagation factors and the orbital angular momentum density of such hollow vortex Gaussian beam passing through a paraxial optical system described by the ABCD matrix approach are also derived, respectively. As a numerical example, the propagation properties of a hollow Gaussian beam with the power-exponent-phase vortex are demonstrated in free space. The evolutions of the normalized intensity, the phase and the orbital angular momentum density distributions are investigated, respectively. The influences of the power order and the topological charge on the beam propagation factors in the x- and y-directions are analysed. The introduced hollow Gaussian beam has potential applications in the atom manipulation and the optical trapping.     

Fractional Fourier transform for beams generated by Gaussian mirror resonator

Based on the definition of the fractional Fourier transform (FRFT) in the cylindrical coordinate system and the fact that a hard-edged-aperture function can be expanded into a finite sum of complex Gaussian functions, the propagation properties of beams generated by Gaussian mirror resonator passing through FRFT optical systems have been studied in detail, and some typical numerical examples are given to compare the results obtained by the approximate analytical method with those by the numerical integral method, and it is shown that our method can significantly improve the numerical calculation efficiency. Further, the results indicate that the normalized intensity distributions in the FRFT plane are closely related to the fractional order, truncation parameter and initial beam parameters. The variation period of normalized intensity distributions with p is 2 whatever the value of the truncation parameter δ is.     

Single-mode fiber coupling efficiency with graded-index rod lenses

Losses are calculated for the single-mode graded-index- (GRIN-) lens coupler. The main advantage of this coupling system is large separation between fibers with small power loss. The excess loss of the GRIN-lens coupler is due primarily to the misalignments of the GRIN lenses and is most sensitive to angular tilt rather than lateral offset or end separation. The excess loss is calculated from the overlap of two differently sized or misaligned Gaussian beams. Experimental results show that this method is adequate for predicting losses that are due to misalignments of the GRIN lenses.     

Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane

An analytical and concise formula is derived for the fractional Fourier transform (FRT) of partially coherent beams that is based on the tensorial propagation formula of the cross-spectral density of partially coherent twisted anisotropic Gaussian-Schell-model (GSM) beams. The corresponding tensor ABCD law performing the FRT is obtained. The connections between the FRT formula and the generalized diffraction integral formulas for partially coherent beams passing through aligned optical systems and misaligned optical systems are discussed. With use of the derived formula, the transformation and spectrum properties of partially coherent GSM beams in the FRT plane are studied in detail. The results show that the fractional order of the FRT has strong effects on the transformation properties and the spectrum properties of partially coherent GSM beams. Our method provides a simple and convenient way to study the FRT of twisted anisotropic GSM beams.     

Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems

On the basis of the fact that a hard-edged-aperture function can be expanded into a finite sum of complex Gaussian functions, approximate analytical expressions for the output field distribution of a Laguerre-Gaussian beam and an elegant Laguerre-Gaussian beam passing through apertured fractional Hankel transform systems are derived. Some numerical simulation comparisons are done, by using the approximate analytical formulas and diffraction integral formulas, and it is shown that our method can significantly improve the numerical calculation efficiency.     

Diffraction of cylinders with longitudinal surface structures

We present a model to determine the light scattered by a metallic cylinder with longitudinal structures when the cylinder is illuminated by a Gaussian light beam in oblique incidence. The model is based on an approximate solution to the Helmholtz-Kirchhoff integral by means of the stationary-phase method. We have studied the variations of the diffraction pattern in terms of the size of the defect and other geometrical parameters. The width of the beam and the misalignment between the beam and the cylinder have also been considered, as well as the optical properties of the surface.     

Approximate method for the generalized M2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams

On the basis of the truncated second-order moments method in the cylindrical coordinate systems and the expansion of the hard-edged aperture function into a finite sum of complex Gaussian functions, an approximate method used to calculate the generalized beam propagation factor (M2 factor) is proposed. The approximate analytical expressions of the generalized M2 factor for rotationally symmetric hard-edged diffracted flattened Gaussian beams defined by Gori [Opt. Commun. 107, 335 (1994)] and Li [Opt. Lett. 27, 1007 (2002)] are derived, respectively; we show that it depends on the beam order N and the beam truncation parameter delta. Some typical numerical examples are given to illustrate its applications that we compare by using the obtained analytical method and the numerical integration method.     

光纤耦合效率与接收光强计算研究

由于用近似方法计算出的光纤耦合效率和接收光功率与实验测量之间存在差异,有必要探讨更准确的计算方法以满足应用的需要。通过对径向变量的积分,导出了在高斯光强分布下对接收光纤端面进行面积分计算的一元积分式,从而可以快速准确地对光纤的耦合效率和接收光功率进行数值计算。对62.5/125多模光纤的耦合效率和接收光功率进行的数值计算和实验测量说明,光纤的耦合效率只有用准确的数值计算才能给出与实际相符合的计算结果。使用芯径较大的接收光纤测量出射光束的近场光强分布时,光纤接收光强分布较光束的光强分布有较大的展宽,其相对误差甚至可达60%。     

Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems

A new mathematical model called hollow elliptical Gaussian beam (HEGB) is proposed to describe a dark-hollow laser beam with noncircular symmetry in terms of a tensor method. The HEGB can be expressed as a superposition of a series of elliptical Hermite-Gaussian modes. By using the generalized diffraction integral formulas for light passing through paraxial optical systems, analytical propagation formulas for HEGBs passing through paraxial aligned and misaligned optical systems are obtained through vector integration. As examples of applications, evolution properties of the intensity distribution of HEGBs in free-space propagation were studied. Propagation properties of HEGBs through a misaligned thin lens were also studied. The HEGB provides a convenient way to describe elliptical dark-hollow laser beams and can be used conveniently to study the motion of atoms in a dark-hollow laser beam.