Analytical structure of an apertured vector Gaussian beam in the far field

Based on the angular spectrum representation of the Maxwell’s equations and the complex Gaussian expansion of the aperture function, the structure of an apertured vector Gaussian beam in the far field is presented in the integral form. By means of the method of stationary phase, the analytical vectorial structures are obtained. According to the analytical expressions, the characteristics of vectorial structure of an apertured Gaussian beam are investigated in the far field. The influence of a linearly polarized angle on the vectorial structure is also studied in the far field. This research provides a novel approach to further comprehend the vectorial property of an apertured Gaussian beam.     

Transformation of a hollow Gaussian beam into a ring-shaped beam using a phase aperture

Propagation of a hollow Gaussian beam diffracted by a circular phase aperture is studied without making the paraxial approximation. The analytical expression of the intensity of the apertured hollow Gaussian beam is presented in the far field. The influences of the truncation parameter and the order of hollow Gaussian beam on the intensity distributions are discussed. It is shown that a circular ?-phase aperture can be used to transform a hollow Gaussian beam into a ring-shaped beam in the far field with the appropriate parameters.     

Vectorial structure of the far-field of an apertured four-petal Gaussian beam

Based on the theorem of vectorial structure and the method of stationary phase, an analytical vectorial structure of the far-field of an apertured four-petal Gaussian beam has been derived without any approximation. The analytical expressions of the energy flux of the TE term, the TM term, and the apertured four-petal Gaussian beam are also presented in the far-field reference plane, respectively. The energy flux distributions of the TE term, the TM term, and the apertured four-petal Gaussian beam are graphically demonstrated in the far-field plane. The dependences of the energy flux distributions of the TE term, the TM term, and the apertured four-petal Gaussian beam on the f-parameter, the truncation parameter, and the beam order are also examined.     

Far-field structural property of a Gaussian beam diffracted by a phase aperture

Based on the angular spectrum representation of an arbitrary electromagnetic beam and the method of stationary phase, an analytically vectorial structure of a Gaussian beam diffracted by a phase aperture has been derived in the far-field. Moreover, the derivation is performed without any approximation. The analytical expressions of the energy flux of the TE term, the TM term, and the apertured Gaussian beam are also presented in the far-field, respectively. The influence of the phase delay on the energy flux distributions of the TE term, the TM term, and the apertured Gaussian beam is discussed in the far-field.     

Vectorial structure of the far field of an elegant Hermite–Gaussian beam

Based on the vector angular spectrum method and the method of stationary phase, an analytical expression for the vectorial structure of the far field of an elegant Hermite–Gaussian beam is derived. The analytical formulae of the energy flux distributions of the TE term, the TM term, and the whole beam are presented in the far field. Analytical expressions for the ratios of the powers of the TE and TM terms to those of the elegant Hermite–Gaussian beam are obtained without any approximation. The physical pictures of the far field of an elegant Hermite–Gaussian beam are demonstrated and compared with those of the far field of the corresponding standard Hermite–Gaussian beam. This research reveals the internal vectorial structure of the far field of an elegant Hermite–Gaussian beam from an alternative viewpoint.     

Analytical solution for an anomalous hollow beam in a fractional Fourier transforming optical system with a hard aperture

Based on the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, the approximate analytical expressions for the output field distribution of an anomalous hollow beam (AHB) passing through an apertured fractional Fourier transform (FRT) system are derived. By using the approximate analytical formulae and diffraction integral formulae, the propagation properties of an AHB in circular and rectangular apertured FRT system are studied numerically. The results show that this method provides a convenient tool for studying the propagation properties of an AHB passing through apertured FRT system, and the apertured FRT system can be applied to laser beam shaping conveniently.     

Propagation of a hollow Gaussian beam through a paraxial misaligned optical system

Based on the generalized diffraction integral formula for treating the propagation of a laser beam through a paraxial misaligned optical system in the cylindrical coordinate system, we obtain an analytical formula for a hollow Gaussian beam passing through a paraxial misaligned optical system. Furthermore, we also obtain the approximate analytical formula for a hollow Gaussian beam passing through a paraxial circularly apertured misaligned optical system by expanding the hard aperture function into a finite sum of complex Gaussian functions. As a numerical example, the propagation properties a hollow Gaussian beam through a misaligned thin lens are studied numerically.     

Propagation of a vectorial Laguerre–Gaussian beam beyond the paraxial approximation

Based on the vectorial Rayleigh–Sommerfeld integrals, the analytical propagation expression of a vectorial Laguerre–Gaussian beam beyond paraxial approximation is presented. The far field expression and the scalar paraxial result are obtained as special cases of the general formulae. According to the analytical representation, the light intensity distribution of the vectorial Laguerre–Gaussian beam is depicted in the reference plane. The light intensity distribution of a vectorial Laguerre–Gaussian beam with cos m is also compared with that of a vectorial Laguerre–Gaussian beam with sin m.     

Analytical structure of Hermite Gaussian beam in far field

Based on the vectorial structure of electromagnetic beam and the method of stationary phase, the analytical structure of Hermite Gaussian beam in far field is presented. The structural energy flux distributions are also investigated in the far field. The structural pictures of some Hermite Gaussian beams are depicted in the far field. As the structure of Hermite Gaussian beam is dominated by the transverse mode numbers and the initial transverse Gaussian half width, it is more complex than that of Gaussian beam. The ratios of the structural energy fluxes to the whole energy flux are independent of the transverse mode numbers and the initial transverse Gaussian half width. The present research reveals the internal vectorial structure of Hermite Gaussian beam from other viewpoint.\     

横向激发线偏振高斯光束的非傍轴传输

文章提出一种横向激发的任意线偏振高斯光束,该光束场量的模与传统的任意线偏振高斯光束具有相同的模,在源场区,其满足自由空间无源场的场方程。同时利用瑞利—索末菲衍射积分对该高斯光束的非傍轴传输特性进行了解析研究,给出了空间光强分布、束腰和远场发散角的解析通式,发现该类光束的远场光强呈一平顶空心光束。     

Vectorial structure of helical hollow Gaussian beams in the far field

The analytical vectorial structure of helical hollow Gaussian beam (HHGB) is investigated in the far field based on the vector plane wave spectrum and the method of stationary phase. The energy flux distributions of HHGB in the far-field, which is composed of TE term with the electric field transverse to the propagation axis, and TM term with the magnetic field transverse to the propagation axis, are demonstrated. The physics pictures of HHGB is illustrated from the vectorial structure, which is important to understand the theoretical aspects of both scalar and vector HHGB propagation.     

Approximate analytical expressions of apertured broadband beams in the far field

The approximate analytical expressions of the apertured broadband beams in the far field with Gaussian and Laguerre-Gaussian spatial modes are presented.For the radially polarized Laguerre-Gaussian beam,the result reveals that the electromagnetic field in the far field is transverse magnetic.The influences of bandwidth(Γ) and truncation parameter(C 0) on the transverse intensity distribution of the Gaussian beam and on the energy flux distribution of radially polarized Laguerre-Gaussian beam are analysed.     

Analytical vectorial structure of the anomalous hollow beam in the far field

On the basis of the vector angular spectrum representation of the Maxwell equations and the method of stationary phase, the analytical vectorial structure of an anomalous hollow beam in free space is presented. The energy flux distributions of the anomalous hollow beam, its TE and TM terms in the far field are investigated. The analytical results are analyzed with numerical examples.     

Analytical vectorial structure of non-paraxial nonsymmetrical vector Gaussian beam in the far field

Based on the vectorial structure of non-paraxial electromagnetic beam and the method of stationary phase, the analytical TE and TM terms of non-paraxial nonsymmetrical vector Gaussian beam in the far field are presented. According to the analytical electromagnetic fields of the TE and TM terms, the energy flux distributions of the whole beam, its TE and TM terms are investigated. Moreover, the influences of non-symmetries on the energy flux distributions of the whole beam, its TE and TM terms are also analyzed, respectively. This research reveals the internal vectorial structure of non-paraxial laser beam in the far field and is useful to the propagation of non-paraxial laser beam.     

Vectorial structure of far field of cylindrically polarized beams diffracted at a circular aperture

Based on the vector angular spectrum representation and the method of stationary phase, the analytical vectorial structure of the cylindrically polarized Laguerre-Gaussian beam diffracted at a circular aperture is derived in the far field, which provides an approach to further comprehend the vectorial properties of the apertured cylindrically polarized beams. The radially polarized, azimuthally polarized, and unapertured cases can be viewed as the special cases of our general result. The analyses show that the far-field energy flux distributions of the entire beam, the TE term, and the TM term depend on the beam order, the ratio of the waist width to the wavelength, the truncation parameter, and the angle between the electric field vector and the radial direction.     

Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture

Based on the vectorial Rayleigh diffraction integral, the nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture is studied. The far-field and paraxial cases are treated as special cases of our general result. It is shown that for the apertured case the f parameter still plays an important role in determining the nonparaxiality of vectorial diffracted Gaussian beams, but both the f parameter and truncation affect the beam evolution behavior.     

Vectorial structure of Hermite–Laguerre–Gaussian beam in the far field

Starting from Maxwell's equations, a Hermite–Laguerre–Gaussian (HLG) beam is decomposed into the TE and TM terms by using the vector angular spectrum representation. By means of the method of stationary phase, the analytical TE and TM terms are presented in the far field. The energy flux distributions of the TE and TM terms are also investigated and depicted in the far field. The influences of the additional angle parameter and Gaussian waist width on the vectorial structure and energy flux pattern of HLG beam are also investigated. This research reveals the internal vectorial structure of HLG beam and may provide a new approach to the manipulation of laser beams.     

Nonparaxial propagation of a vectorial apertured off-axis Lorentz beam

Based on the vectorial Rayleigh--Sommerfeld integral formula and the complex Gaussian expansion of the hard-edge aperture function, an analytical propagation expression for a nonparaxial vectorial off-axis Lorentz beam passing through a rectangular aperture is derived. The unapertured case, the far field expression and the scalar paraxial result are also presented as special cases of the general formulae, respectively. Some numerical examples are also given to show the propagation characteristics of a nonparaxial vectorial off-axis Lorentz beam through a rectangular aperture. It is indicated that the f parameter, the off-axis displacement and the truncation parameter all play an important role in determining nonparaxial propagation behaviour.