Analytical propagation equation of astigmatic Hermite–Gaussian beams through a 4×4 paraxial optical system and their symmetrizing transformation

Starting from the Collins formula, a closed-form propagation equation of astigmatic Hermite–Gaussian (H–G) beams through a 4×4 paraxial optical system is derived, which permits us to calculate the irradiance distribution at any propagation plane and to study the symmetrization of astigmatic standard and elegant H–G beams. A detailed symmetrizing transformation of astigmatic H–G beams through a three-cylindrical-lens mode converter is illustrated both analytically and numerically. It is found that in accordance with the second-order moments characterization, there are two types of beam symmetrization. The transformation of standard H–G beams through the three-cylindrical-lens mode converter belongs to the perfect symmetrization, whereas the transformation of elegant H–G beams is the imperfect one.     

The elliptical elegant Laguerre–Gaussian beam and its propagation through aligned and misaligned paraxial optical systems

A new kind of laser beam called the elliptical elegant Laguerre–Gaussian beam (EELGB) is defined by using tensor method. By using the generalized diffraction integral formulas for light beam passing through paraxial optical system, the analytical propagation formulas for EELGB passing through paraxial aligned and misaligned optical systems are obtained through vector integration. As examples of applications, the propagation properties of EELGBs in free space propagation and through a misaligned thin lens are studied.     

Novel unitary optical mode converter

A new unitary optical mode converter for transforming Hermite–Gaussian beams into Laguerre–Gaussian beams is proposed, which is made up of gradient index medium. Being unitary cylinder with only two reflecting surfaces, the mode converter is easily adjusted and has very low reflecting loss.     

Rectangular hard-edged aperture diffracted Laguerre–Gaussian beams

Based on the relations between Laguerre–Gaussian (LG) and Hermite–Gaussian (HG) modes and by introduced the complex Gaussian expansion method for two dimensional rectangular aperture, the approximate analytical propagation expressions of the rotational symmetrical LG beams along with their even and odd modes through a paraxial ABCD optical system with rectangular hard-edged aperture are derived. As special cases of the results, the corresponding closed-forms of the circular aperture diffracted LG beams and non-truncated LG beams are also given. Numerical examples are given to prove the validity of this approximate analytical method and illustrate the propagation properties of the rectangular hard-edged aperture diffracted LG beams.     

Application of the Wigner distribution function to complex-argument Hermite– and Laguerre–Gaussian beams beyond the paraxial approximation

The Wigner distribution function (WDF) is applied to study the propagation of complex-argument Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) beams beyond the paraxial approximation. The analytical expressions for their intensity distributions in free-space propagation are derived, which are expressed in terms of Hermite polynomials for nonparaxial complex-argument HG beams and in terms of the sum of finite Hermite polynomials for nonparaxial complex-argument LG beams. A detailed comparison of the WDF approach, series expansion method and paraxial expressions is made, which shows that in the paraxial regime the WDF approach and series expansion method deliver consistent results with that of paraxial expressions. Beyond the paraxial approximation, the WDF approach offers convergent results, whereas the series expansion method has a limited applicable range, within which it gives consistent results with that of WDF approach but beyond which it gives unrealistic and divergent results.     

Generalized beam propagation factor of hard-edged circular apertured diffracted Bessel–Gaussian beams

Based on the truncated second-order moments method on the cylindrical coordinate systems and the incomplete gamma function, an analytical expression of the generalized beam propagation factor (M_G² factor) of hard-edged circular apertured diffracted Bessel–Gaussian beams is derived and illustrated numerically. It is shown that the M_G² factor of hard-edged diffracted BGBs mainly depends on the truncation parameter δ and the beam parameters m and η. The results can be reduced to that for the non-truncated Bessel–Gaussian beams case and that for the truncated fundamental Gaussian beams case under certain conditions, respectively. The power fraction is also discussed analytically and numerically.     

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.     

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.     

Propagation of elegant Laguerre–Gaussian beams through an annular apertured paraxial ABCD optical system

Based on the generalized Huygens–Fresnel diffraction integral and the expansion of the hard aperture function into a finite sum of complex Gaussian functions, the approximate analytical expression of elegant Laguerre–Gaussian beams passing through a paraxial ABCD optical system with an annular aperture is derived. Meanwhile, the corresponding closed-forms for the unapertured, circular apertured or circular black screen cases are also given. The obtained results provide more convenience for studying their propagation and transformation than the usual way by using diffraction integral formula directly. Some numerical examples are given to illustrate the propagation properties of elegant Laguerre–Gaussian beams.     

Generation of multi-axis Laguerre–Gaussian beams from?geometric modes of a hemiconfocal cavity

We study the transformation of multi-axis Laguerre–Gaussian beams from optical beams comprising Hermite–Gaussian modes with ray–wave duality. By use of cylindrical lenses, Hermite–Gaussian modes can be transformed into Laguerre–Gaussian modes possessing optical orbital angular momentum. The superposed Hermite–Gaussian modes localized on geometric trajectories are generated from a degenerate hemiconfocal cavity and transformed into multi-axis Laguerre–Gaussian beams. Experimental results of the structured beams are systematically manipulated and in good agreement with theoretical analyses.     

Far-field properties and beam quality of vectorial Hermite–Laguerre–Gaussian beams beyond the paraxial approximation

The far-field properties and beam quality of vectorial nonparaxial Hermite–Laguerre–Gaussian (HLG) beams are studied in detail, where, instead of the second-order-moments-based M² factor, the extended power in the bucket (PIB) and βparameter are used to characterize the beam quality in the far field and the intensity in the formulae is replaced by the z component of the time-averaged Poynting vector S_z. It is found that the S_z PIB and βparameter of vectorial nonparaxial HLG beams depend on the mode indices n, m, αparameter and waist-width-to-wavelength ratio w₀/λ and the PIB and βparameter are additionally dependent on the bucket's size taken.     

Propagation of a Laguerre–Gaussian beam through a slightly misaligned paraxial optical system

Based on the generalized diffraction integral formula for treating the propagation of a laser beam through a slightly misaligned optical system in a cylindrical coordinate system, an analytical formula for a Laguerre–Gaussian beam passing through such an optical system is derived. Furthermore, an approximate analytical formula is derived for a Laguerre–Gaussian beam passing through an apertured slightly misaligned optical system by expanding the hard aperture function as a finite sum of complex Gaussian functions. Some analytical formulas are also given for a flattened Gaussian beam by expanding its field as a superposition of a finite series of Laguerre–Gaussian beams. PACS 42.25.Bs; 41.85.Ew; 41.85.Ct     

M factor and kurtosis parameter of super-Gaussian beams passing through an axicon

The effects of axicons on the M² factor and kurtosis parameter of super Gaussian beams(SGBs) are studied in detail. The closed-form expression for the M² factor of SGBs after passing through an axicon is derived, and the reason why the convergent and divergent axicons give rise to the same increase of the M² factor is explained physically from the similarity of the resulting irradiance distributions. The analytical propagation equation of the K parameter of SGBs passing through an axicon followed by a paraxial optical ABCD system is given, some interesting special cases are discussed. In particular, it is found that even for the Gaussian special case theK parameter is no longer a constant due to the effect of the axicon. Numerical examples are also given to illustrate our analytical results.     

Beam wander of J 0- and I 0-Bessel Gaussian beams propagating in turbulent atmosphere

Root mean square (rms) beam wander of J ₀-Bessel Gaussian and I ₀-Bessel Gaussian beams, normalized by the rms beam wander of the fundamental Gaussian beam, is evaluated in atmospheric turbulence. Our formulation is based on the first and the second statistical moments obtained from the Rytov series. It is found that after propagating in atmospheric turbulence, the collimated J ₀-Bessel Gaussian and the I ₀-Bessel Gaussian beams have smaller rms beam wander than that of the Gaussian beam, regardless of the choice of Bessel width parameter. However, the extent of such an advantage depends on the chosen width parameter, Gaussian source size, propagation distance and the wavelength. Focusing at finite distances of the considered beams causes the rms beam wander to decrease sharply at the propagation distances equal to the focusing parameter.     

High-order mode Laguerre–Gaussian beam transformation using a 127-actuator deformable mirror: numerical simulations

Transforming high-order mode Laguerre–Gaussian beams in the far-field using a 127-actuator deformable mirror controlled by a stochastic parallel gradient descent algorithm is presented. As a phase shift of half wave exists between every neighboring lobes of high-order mode Laguerre–Gaussian beams, there are multiple lobes in the far-field. The suitable beam radius related to the aperture size of the deformable mirror is discussed. Three system performance metrics are evaluated, and encircled energy is preferred. Simulation results show that it is possible to compensate for the phase shifts and other phase aberrations of a high-order mode Laguerre–Gaussian beam and achieve a single bright lobe with this approach. Transforming the far-field intensity distribution of high-order mode Laguerre–Gaussian beams into Gaussian and super Gaussian distributions are also investigated.     

Propagation and the kurtosis parameter of Gaussian flat-topped beams in uniaxial crystals orthogonal to the optical axis

The analytical formulae for the Gaussian flat-topped beams propagating in uniaxial crystals orthogonal to the optical axis are derived. The numerical results show that the Gaussian flat-topped beams spread at different rates in the directions parallel and orthogonal to the optical axis due to anisotropic crystals. An analytical expression for the kurtosis parameter of the Gaussian flat-topped beams propagating in uniaxial crystals is derived and illustrated with numerical examples. It is shown that the evolution of the kurtosis parameters K_x and K_y depend on the ratio of extraordinary to ordinary refractive indices.     

Studies of propagation characteristics of elliptical Gaussian beams through circular apertured optical systems

An approximate analytical propagation formula of the elliptical Gaussian beams through a paraxial ABCD optical system with a circular aperture is obtained on the basis of the expansion of a hard-edged aperture into a finite sum of complex Gaussian functions. And the parameter characteristics of the truncated beams including the beam propagation factor and the kurtosis parameter are also studied in detail by using the second-order-moments method.     

Roughness and gradient parameters for characterising shape uniformity of laser beams

We define six parameters, the roughness parameter, the local gradient parameter, the directionless gradient parameter, the integral gradient parameter, the directionless integral gradient parameter and the absolute gradient parameter for characterising the uniformity of a laser beam in its transversal section. For shape-invariant beams the roughness parameter is propagation invariant. The absolute gradient parameter is propagation invariant for symmetric shape invariant beams. As an example, five of the defined parameters are calculated for a dummy irradiance. The values of the roughness parameter and the absolute parameter are calculated and tabulated for symmetric Hermite–Gaussian, Laguerre–Gaussian, spherical Gaussian, flattened Gaussian (at the waist plane) and super Gaussian beams (at the waist plane). The results are discussed.     

Fresnel tomography and interferometric technique for characterizing Laguerre–Gaussian beams

The Fresnel tomograms of Laguerre–Gaussian beams are found and the relation of the two-point spatial correlation function with the tomographic map is discussed. The spatial distribution and signature of vorticity of Laguerre–Gaussian modes is analyzed by measuring the two-point spatial correlation function of the one-dimensional projection of the vortex field. An interferometric approach, based on a Fourier projection algorithm, for the experimental determination of the two-point correlation function of the full complex vortex field at the cross-section plane of the recorded interference pattern is also described; this approach allows one to characterize the vortex structure and spatial coherence along the direction of propagation.