Closed-form Propagation Expressions Research Articles

Propagation Property of an Astigmatic sin–Gaussian Beam in a Strongly Nonlocal Nonlinear Media

Based on the Snyder and Mitchell model, a closed-form propagation expression of astigmatic sin-Gaussian beams through strongly nonlocal nonlinear media (SNNM) is derived. The evolutions of the intensity distributions and the corresponding wave front dislocations are discussed analytically and numerically. It is generally proved that the light field distribution varies periodically with the propagation distance. Furthermore, it is demonstrated that the astigmatism and edge dislocation nested in the initial sin-Gaussian beams greatly influence the pattern configurations and phase singularities during propagation. In particular, it is found that, when the beam parameters are properly selected, a vortex beam with perfect doughnut-shaped profile can be obtained for astigmatic sin-Gaussian beams with two-lobe pattern propagating in SNNM.

Propagation of truncated modified Laguerre–Gaussian beams

By expanding the circ function into a finite sum of complex Gaussian functions and applying the Collins formula, the propagation of hard-edge diffracted modified Laguerre–Gaussian beams (MLGBs) through a paraxial ABCD system is studied, and the approximate closed-form propagation expression of hard-edge diffracted MLGBs is obtained. The transverse intensity distribution of the MLGB carrying finite power can be characterized by a single bright and symmetric ring during propagation when the aperture radius is very large. Starting from the definition of the generalized truncated second-order moments, the beam quality factor of MLGBs through a hard-edged circular aperture is investigated in a cylindrical coordinate system, which turns out to be dependent on the truncated radius and the beam orders.

Dynamic evolution of an edge dislocation through aligned and misaligned paraxial optical ABCD systems

The closed-form propagation expressions of an edge dislocation nested in a general elliptical Gaussian beam through aligned and misaligned paraxial optical ABCD systems are derived and used to study the dynamic evolution behavior of the edge dislocation propagating in free space and through a misaligned lens. It is shown that the noncanonical vortex and dynamic inversion of the topological charge, as well as a new edge dislocation may appear under certain conditions. The results are illustrated analytically and numerically.

Propagation of partially polarized, partially coherent beams

The propagation of partially polarized and partially coherent beams in free space is investigated. Using the partially polarized Gaussian Schell model beam as an example, closed-form propagation expressions are derived for the intensity distribution and the degree of polarization of the beam. The effect of propagation parameters on the intensity and the polarization property of the beam is illustrated and analyzed when the components of the incident field change. To our knowledge it is the first time a recent definition of the degree of polarization for general three-dimensional fields has been used to evaluate the propagation of partially polarized and partially coherent beams.

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.

Vectorial Nonparaxial Four-Petal Gaussian Beams and TheirPropagation in Free Space

The vectorial nonparaxial four-petal Gaussian beam (FPGB) isintroduced. The closed-form propagation expressions for the free-spacepropagation of FPGBs are derived and their more general applicableadvantages are illustrated analytically and numerically. Some specialinteresting cases, in particular the paraxial one, are discussed. Itis found that the parameter f = 1/kw0 with the kbeing the wave number and w0 being the waist width playsa crucial role in determining the nonparaxiallity of FPGBs. For smallvalues of the f parameter the paraxial approximation isallowable. In the nonparaxial regime the beam order nadditionally affects the vectorial and nonparaxial behaviour of FPGBs.

Kurtosis parameter of truncated standard and elegant Laguerre–Gaussian beams propagating through ABCD optical systems

On the basis of the propagation equation of truncated standard and elegant Laguerre–Gaussian beams, the closed-form propagation expressions for the kurtosis parameter K of standard and elegant Laguerre–Gaussian beams passing through apertured optical systems are derived, it is shown that the kutosis parameter K of standard and elegant Laguerre–Gaussian beams depend on order p and index m of Laguerre polynomial, the beam-truncated parameter δ and the generalized Fresnel number F. Detailed numerical examples are given to illustrate the analytical results.

Propagation of the Wigner distribution function for partially coherent nonparaxial beams

On the basis of the Rayleigh-Sommerfeld diffraction integral, a closed-form propagation expression for the Wigner distribution function of partially coherent nonparaxial beams in free space is derived for what is to our knowledge the first time. The propagation of spatially fully coherent nonparaxial beams is treated as a special case of our general result. Application of the result is illustrated with the nonparaxial propagation of partially coherent anisotropic Gaussian-Schell-model beams and TEM11-mode Hermite-Gaussian beams.

Propagation properties of Hermite-cosine-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture

Based on the generalized Huygens–Fresnel diffraction integral, a recurrence propagation equation of Hermite-cosine-Gaussian (HcoG) beams through a paraxial optical ABCD system with hard-edge aperture is derived, which permits us to obtain the analytical propagation expression for HcoG beams of any order. Furthermore, the closed-form propagation expressions for the Gaussian, Hermite–Gaussian, cosine (sine)-Gaussian, cosh (sinh) and cosine (sine) beams passing through a paraxial optical ABCD system with hard-edge aperture can be obtained by treating them as special cases of the analytical propagation expression for HcoG beams. The advantages of our analytical results are analyzed, and the applications are illustrated with numerical examples.

Analytical expression for the kurtosis parameter of flattened Gaussian beams propagating through ABCD optical systems

The closed-form propagation expression for the kurtosis parameter of flattened Gaussian beams (FGBs) in passage through paraxial ABCD optical systems is derived on the basis of the propagation equation of FGBs and analysed with numerical examples. A comparison of our results with those obtained by using the propagation law of irradiance moments is made, showing the advantage of the method used in this paper.

Propagation of the kurtosis parameter of elegant Hermite-Gaussian and Laguerre-Gaussian beams passing through ABCD systems

By using the propagation formulae of elegant Hermite-Gaussian beams (EHGBs) and Laguerre-Gaussian beams (ELGBs), the closed-form propagation expressions for the kurtosis parameter K of EHGBs and ELGBs in passage through paraxial ABCD optical systems are derived, which enable us to calculate the K parameter of EHGBs and ELGBs at any propagation plane. Detailed numerical examples are given to illustrate our analytical results

Closed-form propagation expressions of flattened Gaussian beams through an apertured ABCD optical system

Within the framework of diffraction theory, we investigate some closed-form propagation expressions of flattened Gaussian beams through a paraxial optical ABCD system. By studying the propagation through an aperture, we have derived the closed-form propagation expressions through an ABCD system as a limit when the aperture width is infinite. The closed-form propagation expressions are derived, as a function of the Gaussian probability function, which relates the electromagnetic fields on the input plane to those on its output plane. Some numerical examples are also illustrated.

PROPAGATION OF QBG BEAMS THROUGH AN APERTUREDABCD OPTICAL SYSTEM

Based on the generalized Huygens-Fresnel diffraction integral, the propagation of the Bessel-modulated Gaussian beams with quadratic radial dependence (QBG beams) through a circle-apertured ABCD optical system has been studied.-In our treatment the circle function is expanded into a sum of complex Gaussian functions, thus a closed-form propagation expression of QBG beams has been derived. As an application example, numerical calculations have been performed for the focusing of QBG beams by an apertured lens. The calculatied results have shown the advantage of our method, and are in good agreement with those obtained by the straightforward numerical integration of the generalized diffraction formula.

Propagation properties of Hermite-cosh-Gaussian laser beams

In this paper, we investigate theoretically some closed-form propagation expressions of Hermite-cosh-Gaussian laser beams through a paraxial optical ABCD system. By studying the propagation through an aperture, we have derived the closed-form propagation expressions through an ABCD system as a limit when the aperture width is infinite. The closed-form propagation expressions are derived, as a function of the probability function, which relates the electromagnetic fields on the input plane to those on its output plane. Some numerical examples are also illustrated.

Beam propagation properties of radial laser arrays.

A detailed study of phase-locked and non-phase-locked radial laser arrays is presented. The closed-form propagation expressions for the beamlets and the resulting beam are given, which enable us to study beam propagation properties of radial laser arrays for both phase-locked and non-phase-locked cases. Numerical calculation examples are given to illustrate the application of our analytical results and the differences between phase-locked and non-phase-locked radial arrays.

Propagation of flattened Gaussian beams with rectangular symmetry passing through a paraxial optical ABCD system with and without aperture

The propagation of flattened Gaussian beams (FGBs) passing through a paraxial optical ABCD system has been studied in detail, where both unapertured and apertured cases have been considered. For the unapertured case, a closed-form propagation expression has been derived, and for the apertured case typical numerical examples have been given to illustrate our results and method.