Propagation properties of vectorial Hermite–cosine–Gaussian beams beyond the paraxial approximation

Starting from the vectorial Rayleigh–Sommerfeld diffraction integrals, the analytical propagation expressions of vectorial nonparaxial Hermite–cosine–Gaussian (HCosG) beams in free space are derived. The on-axis intensity, far-field equation and, in particular, paraxial expressions are given and treated as special cases of our result. It is shown that for vectorial nonparaxial HCosG beams, the parameter f = 1/kw ₀, with k being the wave number and w ₀ being the waist width, determines their nonparaxiality. What is more, the decentered parameter also affects their intensity distribution and nonparaxial behavior. The calculation results indicate that the position of maximum on-axis intensity changes with the mode indices.     

The vertical beam quality of GaInP/AlGaInP strained multiple quantum well laser

Abstract

The waveguided mode of the visible GaInP/AlGaInP compressive strained multiple quantum well laser is calculated by using transfer matrix method. On the basis of the nonparaxial vectorial moment theory of light beam propagation, the vertical (perpendicular to junction plane) optical beam quality factor M ²? of the waveguided mode is shown to be smaller than unity. This result may be useful for the design of semiconductor lasers and analysis of nonparaxial beam propagating characteristics.     

Fully vectorial highly nonparaxial beam close to the waist

I use the angular spectrum representation to compute exactly the Gaussian beam close to the waist (w(0)) in the case of a highly nonparaxial field (w(0)相似文献   

Nonparaxial description of reflection and transmission at the interface between an isotropic medium and a uniaxial crystal

Angular spectra of reflected and transmitted fields, induced by an arbitrary electromagnetic beam passing through the planar interface between a homogeneous medium and a uniaxially anisotropic medium, are derived and related to the incident medium. By using these formulas, we obtain the expressions for paraxial and slightly nonparaxial fields. The reflected paraxial field is related to the incident one by means of Fresnel relations; the transmitted paraxial field is the superposition of an ordinary and an extraordinary beam, multiplied by the Fresnel coefficient. We find that the nonparaxial corrections, owing to the medium discontinuity, are larger than their free-propagation counterparts and that they are very simply related to the paraxial solutions of the incident beam. The case of two homogeneous media with different refractive indices is also discussed. The general expressions obtained are applied to the case of a nonparaxial Gaussian beam.     

Nonparaxial propagation properties of an anomalous hollow beam with orbital angular momentum

The analytical nonparaxial propagation formula of an anomalous hollow beam (AHB) with orbital angular momentum (OAM) in free space is derived based on the generalized Raleigh–Sommerfeld diffraction integral. The nonparaxial properties of AHB with OAM such as intensity, phase and OAM density distributions are studied in detail, using the pertinent nonparaxial propagation formula. The comparison between the paraxial and nonparaxial results is also carried out. The results show that the nonparaxial properties of an AHB with OAM are determined by the initial beam parameters, such as beam waist size and topological charge and propagation distance.     

Nonparaxial gaussian beams: 1. Vector fields

Rigorous analytical solutions were obtained for the vector field of a propagating nonparaxial Gaussian beam in the vicinity of the beam focus. It is demonstrated that a nonparaxial beam of the lowest order has six modes. Upon the limiting transition _{kz 0} ≫ 1, four of these modes exhibit degeneracy and convert into a paraxial Gaussian beam, while the other two modes form the azimuth-symmetric TE and TM fields. A characteristic feature of the nonparaxial modal beam is the absence of symmetry between (still mutually orthogonal) electric and magnetic fields. The deviation from symmetry results in the appearance of negative energy fluxes in the vicinity of the phase singularity manifested by Airy’s fringes, “light drops” (special states of the light field), and “singularity islands”. The obtained results are in good agreement with other relevant data published.     

Propagating free-space nonparaxial beams

We use a method based on the simultaneous combination of the propagation operator and the Fourier transform with arbitrary index in propagating the transverse component of a nonparaxial beam in free space from an arbitrary initial transverse field structure. Being an iterative method, this approach can easily be implemented computationally. As an example of its efficiency, we derive the closed-form nonparaxial corrections to a Bessel-Gaussian beam, showing that our results differ strongly from those reported previously. The validity of our approach is supported by an analysis of the paraxiality estimator recently introduced in the literature.     

Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond

We describe monochromatic light propagation in uniaxial crystals by means of an exact solution of Maxwell's equations. We subsequently develop a paraxial scheme for describing a beam traveling orthogonal to the optical axis. We show that the Cartesian field components parallel and orthogonal to the optical axis are extraordinary and ordinary, respectively, and hence uncoupled. The ordinary component exhibits a standard Fresnel behavior, whereas the extraordinary one exhibits interesting anisotropic diffraction dynamics. We interpret the anisotropic diffraction as a composition of two spatial geometrical affinities and a single Fresnel propagation step. As an application, we obtain the analytical expression of the extraordinary Gaussian beam. We then derive the first nonparaxial correction to the paraxial beam, thus giving a scheme for describing slightly nonparaxial fields. We find that nonparaxiality couples the Cartesian components of the field and that the resultant longitudinal component is greater than the correction to the transverse component orthogonal to the optical axis. Finally, we derive the analytical expression for the nonparaxial correction to the paraxial Gaussian beam.     

Bessel-Gauss beams as rigorous solutions of the Helmholtz equation

The study of the nonparaxial propagation of optical beams has received considerable attention. In particular, the so-called complex-source/sink model can be used to describe strongly focused beams near the beam waist, but this method has not yet been applied to the Bessel-Gauss (BG) beam. In this paper, the complex-source/sink solution for the nonparaxial BG beam is expressed as a superposition of nonparaxial elegant Laguerre-Gaussian beams. This provides a direct way to write the explicit expression for a tightly focused BG beam that is an exact solution of the Helmholtz equation. It reduces correctly to the paraxial BG beam, the nonparaxial Gaussian beam, and the Bessel beam in the appropriate limits. The analytical expression can be used to calculate the field of a BG beam near its waist, and it may be useful in investigating the features of BG beams under tight focusing conditions.     

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)     

Diffraction by a circular aperture: an application of the vectorial theory of Huygens's principle in the near field

A vectorial theory that brings new insight into the nature of diffraction is used to obtain mathematical expressions that evaluate diffraction patterns in the near field. The equations allow us to discriminate between the contributions of the vectorial and the scalar approaches. In the near field we studied the pattern of light diffracted through a circular aperture, and it was proved that the vectorial approach is significant in a region very near the circular aperture. In spite of the obvious differences between the circular aperture and other obstacles, the present theory may also be used with other geometries.     

Nonparaxial propagation of spirally polarized optical beams

The free-propagation features of light beams whose transverse electric field lines are logarithmic spirals (namely, spirally polarized beams) are investigated in both the paraxial and the nonparaxial regime. The complete propagated electric field is considered, and some general properties are obtained regardless of the specific transverse distribution. Simple and significant analytical results are obtained when the transverse intensity profile is chosen as that pertinent to an axially symmetric Laguerre-Gaussian beam of order 1 (namely, spirally polarized donut beams). In particular, it is found that for such beams, the propagated longitudinal electric field can be expressed as a simple superposition of elegant Laguerre-Gaussian beams. Numerical results are presented for different values of the beam parameters and are compared with recently obtained experimental results.     

Response to "Comments on 'Quality of paraxial electromagnetic beams'"

The nonparaxial wave obtained by Seshadri [Appl. Opt.45, 5335 (2006)]APOPAI0003-693510.1364/AO.45.005335 is correct. The difference in the input field distributions of the nonparaxial wave and the corresponding paraxial beam is correct and is not caused by any error in the treatment. As is to be expected, in the appropriate limit, the nonparaxial wave reduces to the paraxial beam for 0相似文献   

Light intensity distributions of the Gaussian laser beam through an aperture team

Based on the propagating theory of the laser beam, the propagating characteristics of the Gaussian beam through an aperture team that comprises two apertures and a convergent lens, are studied. The approximate expressions for the field distribution are derived by the diffracted integral equation in detail under the condition of approximations. In comparison with the approximate expression and the precise expression, we know that there are the approximate same results for the two expressions if the radius of the second aperture is not too large. The numerical examples are given to confirm the correctness of our calculated results.     

Fundamental electromagnetic Gaussian beam beyond the paraxial approximation

The fundamental electromagnetic Gaussian beam is constructed from a single component of the electric vector potential oriented normal to the propagation direction. The potential is cylindrically symmetrical about the propagation direction. The paraxial beam and the first-order nonparaxial beam are obtained. In solving the inhomogeneous paraxial wave equation governing the evolution of the nonparaxial beam, both the particular integral and the complementary function are included. A procedure for deducing the proper asymptotic state of the nonparaxial beam is summarized. The amplitude coefficients of the cylindrically symmetric complex-argument Laguerre-Gauss beams, which constitute the complementary function are determined by requiring the potential to have the proper behavior asymptotically at infinity and near the input plane. From the potential function, the electromagnetic fields are developed and the electrodynamics of the fundamental electromagnetic Gaussian beam beyond the paraxial approximation is investigated. The role of the first-order nonparaxial beam in determining the average beam characteristics is examined.     

Beam shaping in the nonparaxial domain of diffractive optics

We address the problem of shaping the radiant intensity distribution of a highly nonparaxial coherent field by means of a diffractive element located in the plane of the beam waist. To be capable of wide-angle energy redistribution the element must necessarily contain wavelength-scale transverse features, and consequently it must be designed on the basis of rigorous diffraction theory. We consider, in particular, wide-angle Gaussian to flat-top beam shaping in one dimension. Scalar designs are provided and their validity is evaluated by rigorous diffraction theory, which is also used for optimization deep inside the nonparaxial domain, where the scalar designs fail. Experimental verification is provided by means of electron-beam lithography.     

Second-order moments representation of nonparaxial TM flattened Gaussian beams

By using the angular spectrum representation of the Hertz vector and the theory of nonparaxial vectorial moments of light beam propagation, expressions for the width matrix, curvature and divergence matrices and M ² factor of nonparaxial TM flattened Gausssian beams (FGBs) are derived and their matrix elements are expressed in terms of a series sum. Generally, nonparaxial TM FGBs possess different beam widths, divergence angles and M ² factors in the x and y directions, which are dependent on the beam order N and f parameter. For 1/f → 0 the divergence angles θ _x and θ _y in the x and y directions approach their maximum values θ _x = 54.736° and θ _y = 63.435°, respectively. The M ² factor can be less than 1 and take arbitrarily small values if 1/f is small enough.     

Propagation of Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation

On the basis of the vectorial Rayleigh-Sommerfeld formulas and by means of the relation between Hermite and Laguerre polynomials, the analytical expressions for the propagation of the Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) beams beyond the paraxial approximation are derived, with the corresponding far-field propagation expressions and that for the Gaussian beams being given as special cases of the results. Some detailed comparisons of our results with the expansion series and paraxial expressions are made, which show the advantages of our results over the expansion series. With the results obtained, some typical intensity patterns of nonparaxial HG and LG beams are shown.