Vectorial Rayleigh Sommerfeld Diffraction Integrals Research Articles

Concise and explicit expressions for typical spatial-structured light beams beyond the paraxial approximation.

Structured light beams with distinct spatial inhomogeneity of amplitude, phase, and polarization have garnered tremendous attention in recent years. A better understanding of the vectorial structure of such beams is helpful to reveal their important and interesting features for further applications. In this paper, explicit analytical expressions for the electric field components of typical spatial-structured light beams, including fundamental Gaussian beams, Hermite-Gaussian beams, Laguerre-Gaussian beams, Bessel/Bessel-Gaussian beams, and Airy beams, beyond the paraxial approximation are derived on the basis of the vectorial Rayleigh-Sommerfeld diffraction integrals. Compared with the existing expressions in the literature, the expressions given in this paper are very concise. Using the derived analytical expressions, distributions of the electric field components of these typical structured light beams are displayed and analyzed.

A unifying approach for the vectorial Rayleigh–Sommerfeld diffraction integrals

A unique approximation to the vectorial Rayleigh–Sommerfeld diffraction integrals (RSDI) is developed to study the diffraction properties of apertures illuminated with collimated, converging or diverging polarized waves. The apertures can be bigger or smaller than the wavelength of the illumination light. For circular apertures, it is shown that the approach can be expressed as a simple set of equations containing one-dimensional integrals.

Alternative approach to evaluate the Rayleigh-Sommerfeld diffraction integrals using tilted spherical waves.

It is reported an approximation to the scalar and vectorial Rayleigh-Sommerfeld diffraction integrals performed with a Fourier transform operation, as done in Fresnel and Fraunhoffer approximations, but suitable for paraxial, non-paraxial and off-axis regimes. High accuracy is obtained even for waves with low f numbers. The approximation becomes exact on-axis.

Far-Field Properties of Vectorial Cosh-Squared-Gaussian Beams beyond the Paraxial Approximation

The concept of vectorial nonparaxial cosh-squared-Gaussian (ChsG) beams is introduced and their free space propagation equations are derived analytically by using the vectorial Rayleigh-Sommerfeld diffraction integrals, and far-field expressions as well as paraxial propagation formula are obtained in free space. Numerical calculations and analysis show that, the f-parameter and decentered parameter plays an important role in determining the far-field properties of the vectorial nonparaxial ChsG beams.

Polarisation singularities of non-paraxial Gaussian vortex beams diffracted by an annular aperture

Based on the vectorial Rayleigh–Sommerfeld diffraction integrals, the propagation expressions for non-paraxial Gaussian vortex beams diffracted by an annular aperture are derived, and are used to analyse polarisation singularities of non-paraxial Gaussian vortex beams in the diffraction field. It is shown that there exist polarisation singularities including C-points (S 12 vortices), S 23 vortices, and S 31 vortices. The number and position of polarisation singularities depend on the waist width of the beam, the off-axis distance of the vortex and the aperture parameter. The creation and annihilation of polarisation singularities appear as a controlling parameter – including the waist width, off-axis distance and aperture parameter – varies. When compared with the free-space propagation, more polarisation singularities may take place in the diffraction field.

Airy beams beyond the paraxial approximation

The behavior of the vectorial Airy beams beyond the paraxial approximation is investigated. Indeed, closed-form (even though non exact) expressions for the electric components of the fields generated by the same boundary conditions, which should pertain to the scalar Airy beams, are obtained on the basis of the vectorial Rayleigh–Sommerfeld diffraction integrals under suitable approximations. Such expressions may accompany more complete approaches, like that in Opt. Expr. 17, 22432 (2009), where a fully numerical analysis of the propagation of exponentially smoothed Airy beams has been presented, faithfully reproducing the conditions of their experimental demonstration as reported in Phys. Rev. Lett. 99, 213907 (2007). Comments on other well known approaches to the investigation of the nonparaxial propagation of definite paraxial beams are also given.

Effect of vectorial nature on spectral anomalies of ultrashort pulsed beams from a hard-edged aperture

Based on the vectorial Rayleigh–Sommerfeld diffraction integrals, the analytical expression for the spectral intensity of a vectorial nonparaxial ultrashort pulsed Gaussian beam diffracted at a rectangular hard-aperture is derived. The results of the far-field and paraxial cases can be regarded as special cases of the general expression. Effect of vectorial nature on spectral anomalies of ultrashort pulsed beams passing through a hard-edged aperture is analyzed in detail. It is shown that the spectral switch near the phase singularity, which is predicated under the condition of the scalar paraxial theory, disappears with the increasing effect of the vectorial and nonparaxial nature.

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 beam quality of vectorial nonparaxial Hermite-Laguerre-Gaussian beams

Based on the vectorial Rayleigh-Sommerfeld diffraction integrals, the closed-form expressions of vectorial nonparaxial circularly polarized Hermite-Laguerre-Gaussian (HLG) beams in the far-field are derived. The vectorial nonparaxial circularly polarized Gaussian beam is treated as a special case of our general formulae. The power in the bucket (PIB) is extended to the nonparaxial regime, where the intensity is replaced by the z component of the time-averaged Poynting vector, and used to characterize the beam quality of nonparaxial beams in the far field. Numerical calculation and analysis show the dependence of the PIB of vectorial nonparaxial HLG beams on the waist width-to-wavelength ratio w0/λ, as well as on the parameter α, mode indexes n and m, and bucket's size chosen.

Focusing and diffraction by an optical lens and a small circular aperture

Based on the vectorial Rayleigh–Sommerfeld diffraction integrals, analytical expressions for the transversal and axial field distribution of plane waves propagating through a thin lens followed by a small circular aperture are derived and used to study the focusing and diffraction properties of plane waves. Some special cases of our general result are discussed, and illustrative numerical calculation results are given. It is found that the vectorial nonparaxial approach should be applied if the aperture dimension is comparable with the wavelength or the focusing is strong.

Analytical description of propagation of vectorial apertured off-axis Gaussian beams beyond the paraxial approximation

Starting from the vectorial Rayleigh–Sommerfeld diffraction integrals, an analytical description of propagation of vectorial nonparaxial off-axis Gaussian beams propagating through a rectangular aperture is presented. The on-axis, far-field and paraxial cases are dealt with as special cases of our general result and the corresponding mathematical formulations are given. It is shown that for the apertured off-axis cases, besides the f parameter, which determines the beam nonparaxiality, the truncation and off-axis parameters have to be introduced, which additionally affect the beam nonparaxial evolution behavior.

Propagation of vectorial apertured cosh-Gaussian beams beyond the paraxial approximation

By using the vectorial Rayleigh–Sommerfeld diffraction integrals, the propagation equation of vectorial nonparaxial cosh-Gaussian (ChG) beams in the presence of an aperture is derived and expressed in a closed form. The on-axis, far-field and paraxial cases are studied as special cases. Analytical results are illustrated with numerical examples.

Partially coherent vectorial nonparaxial beams

Generalized vectorial Rayleigh-Sommerfeld diffraction integrals are developed for the cross-spectral-density matrices of spatially partially coherent beams. Using the Gaussian Schell-model (GSM) beam as an example, we derive the expressions for the propagation of cross-spectral-density matrices and intensity of partially coherent vectorial nonparaxial beams, and the corresponding far-field asymptotic forms, beyond the paraxial approximation. The propagation of the vectorial nonparaxial GSM beams are evaluated and analyzed. It is shown that a 3 x 3 cross-spectral-density matrix or a vector theory is required for the exact description of nonparaxial GSM beams.

Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture.

Based on the vectorial Rayleigh-Sommerfeld diffraction integrals, an analytical propagation equation of vectorial, nonparaxial, elliptical Gaussian beams through a rectangular aperture is derived. Unlike in previous work, the aperture effect and nonrotational symmetry of the beam and aperture are considered in our theoretical model. The results of the far-field and paraxial approximation for the apertured case are treated as special cases of our general expression. It is found that two f parameters fx, fy and two truncation parameters deltax, deltay in the x and y directions, respectively, have to be introduced that affect the beam nonparaxial evolution behavior in both the near field and the far field.