Nonparaxial Gaussian Research Articles

Complex source point theory of paraxial and nonparaxial elliptical Gaussian beams.

Using the operator transformation technology, we extend the circular Gaussian beam based virtual (complex) sources method to investigate the paraxial and nonparaxial propagation properties of the elliptical Gaussian beams (EGBs) with planar or cylindrical wavefronts travelling in free space. The paraxial approximation analysis reveals the self-reappearance and self-focusing propagation features for the EGBs with cylindrical wavefront under proper parameter conditions. We further introduce the nonparaxial theory to derive the analytical expressions for the field distribution of an EGB in free space, and confirm that these intriguing propagation features can still be observed with added nonparaxial correction. Comparing with the paraxial approximation results, it is worth noting that there is a clear deviation of the on-axial intensity and phase distributions near the self-focusing position on the basis of nonparaxial correction solution. Our results reveal that the anisotropic diffraction of light propagating through homogeneous medium or free space is possible. The approach in this work can easily be generalized to other beam models with elliptical geometry, which allows us to correctly predict some important information about their near field propagation characteristics for various applications.

Characteristics of Light Field Near Focal Line of One-Dimensional Nonparaxial Convergent Gaussian Beam

Characteristics of Light Field Near Focal Line of One-Dimensional Nonparaxial Convergent Gaussian Beam

Nonparaxial Gaussian beam

A new analytical solution of the nonparaxial Helmholtz equation for the Gaussian beam has been obtained. It is shown that the beam retains the Gaussian distribution of the amplitude at propagation in space. The scale transformation of the beam has been determined. The Kogelnik–Li law applies to a nonparaxial Gaussian beam.

Splitting of an edge dislocation by a vortex emergent from a nonparaxial beam

The splitting of a linear or circular edge dislocation by a vortex nested in a nonparaxial Gaussian beam is studied. It is found that the edge dislocations are unstable and vanish, while new vortices may take place and the total topological charge may change with propagation. The location of the vortex, the intercept and slope of the edge dislocation, or the location and radius of the circular edge dislocation in the initial beam may affect the number and positions of vortices occurring in the field. The results are compared with previous works.

Skew line ray model of nonparaxial Gaussian beam.

Many ray-optics models have been proposed to describe the propagation of paraxial Gaussian beam. However, those paraxial ray-optics models are inapplicable to the beams that violate the paraxial condition. In this paper, we present a skew line ray (SLR) based model to represent the propagation properties of nonparaxial Gaussian beam under the oblate spheroidal coordinates. The free-space evolution of complex wavefront of the light beam including amplitude and phase is derived via this model. Our analysis demonstrates that the SLR model is available for both nonparaxial and paraxial conditions, and can be used to precisely describe the propagation of complex wavefront. Furthermore, this model changes the transverse density of rays while propagating. The behavior influences the transverse intensity distribution and makes the optical rays become concentrated towards the center. We believe that this ray-optics model can be further developed to describe other kind of structured beams such as Laguerre-Gauss and Bessel-Gauss beams.

Ultrasonic superlensing jets and acoustic-fork sheets

Focusing acoustical (and optical) beams beyond the diffraction limit has remained a major challenge in imaging instruments and systems, until recent advances on “hyper” or “super” lensing and higher-resolution imaging techniques have shown the counterintuitive violation of this rule under certain circumstances. Nonetheless, the proposed technologies of super-resolution acoustical focusing beyond the diffraction barrier require complex tools such as artificially engineered metamaterials, and other hardware equipment that may not be easily synthesized or manufactured. The present contribution therefore suggests a simple and reliable method of using a sound-penetrable circular cylinder lens illuminated by a nonparaxial Gaussian acoustical sheet (i.e. finite beam in 2D) to produce non-evanescent ultrasonic superlensing jets (or bullets) and acoustical ‘snail-fork’ shaped wavefronts with limited diffraction. The generalized (near-field) scattering theory for acoustical sheets of arbitrary wavefronts and incidence is utilized to synthesize the incident beam based upon the angular spectrum decomposition method and the multipole expansion method in cylindrical wave functions to compute the scattered pressure around the cylinder with particular emphasis on its physical properties. The results show that depending on the beam and lens parameters, a tight focusing (with dimensions much smaller than the beam waist) can be achieved. Subwavelength resolution can be also achieved by selecting a lens material with a speed of sound exceeding that of the host fluid medium. The ultrasonic superlensing jets provide the impetus to develop improved subwavelength microscopy and acoustical image-slicing systems, cell lysis and surgery, and photoacoustic imaging to name a few examples. Moreover, an acoustical fork-sheet generation may open innovative avenues in reconfigurable on-chip micro/nanoparticle tweezers and surface acoustic waves devices.

Extinction efficiency of “elastic–sheet” beams by a cylindrical (viscous) fluid inclusion embedded in an elastic medium and mode conversion—Examples of nonparaxial Gaussian and Airy beams

Stemming from the law of the conservation of energy in an elastic medium, this work extends the scope of the previous analysis for a scatterer immersed in a nonviscous liquid [F. G. Mitri, Ultrasonics 62, 20–26 (2015)] to the case of a (viscous) fluid circular cylinder cross-section encased in a homogeneous, isotropic, elastic matrix. Analytical expressions for the absorption, scattering, and extinction efficiencies (or cross-sections) are derived for “elastic-sheets” (i.e., finite beams in 2D propagating in elastic media) of arbitrary wavefront, in contrast to the ideal case of plane waves of infinite extent. The mathematical expressions are formulated in generalized partial-wave series expansions in cylindrical coordinates involving the beam-shape coefficients of finite elastic-sheet beams with arbitrary wavefront, and the scattering coefficients of the fluid cylinder encased in the elastic matrix. The analysis shows that in elastodynamic scattering, both the scattered L-wave as well as the scattered T-wave contribute to the time-averaged scattered efficiency (or power). However, the extinction efficiency only depends on the scattering coefficients characterizing the same type (L or T) as the incident wave. Numerical computations for the (non-dimensional energy) efficiency factors such as the absorption, scattering, and extinction efficiencies of a circular cylindrical viscous fluid cavity embedded in an elastic aluminum matrix are performed for nonparaxial focused Gaussian and Airy elastic-sheet beams with arbitrary longitudinal and transverse normally-polarized (shear) wave incidences in the Rayleigh and resonance regimes. A series of elastic resonances are manifested in the plots of the efficiencies as the non-dimensional size parameters for the L- and T-waves are varied. As the beam waist for the nonparaxial Gaussian beam increases, the plane wave result is recovered, while for a tightly focused wavefront, some of the elastic resonances can be suppressed. Moreover, the efficiencies for the embedded circular viscous fluid cylinder in the field of an Airy elastic-sheet display a spatial parabolic (nonlinear) type of absorption, scattering, and extinction, representative of the intrinsic property of the curved accelerating beam in the elastic matrix. The present analysis provides an improved method for the computations of energy efficiency factors in elastodynamics for finite beams in 2D, which can be used as a priori information in the direct or inverse characterization of the mechanical properties of cylindrical fiber-reinforced materials, pipes, vessels, etc., embedded in an elastic medium.

Propagation of a Laguerre–Gaussian correlated Schell-model beam beyond the paraxial approximation

Recently, Laguerre–Gaussian correlated Schell-model (LGCSM) beam was introduced theoretically [Mei and Korotkova, Opt. Lett. 38(2), (2013), 91–93] and generated experimentally [Wang et al., Opt. Lett. 38(11), (2013), 1814–1816]. In this paper, we treat the propagation of a LGCSM beam beyond the paraxial approximation. Based on the generalized Rayleigh–Sommerfeld diffraction integral, analytical expressions for the intensity and spectral degree of coherence of a nonparaxial LGCSM beam propagating in free space are obtained, and the corresponding results of a paraxial LGCSM beam are also derived as a special case. Our numerical results show that the nonparaxial propagation properties of a LGCSM beam are closely related to the initial beam parameters, such as the beam waist width, the coherence width and the mode order. Furthermore, in the far field, flat-topped, hollow intensity profiles can be formed through varying the initial beam parameters, which is different from that of a nonparaxial Gaussian Schell-model beam.

Elliptic cylindrical pseudo-optical black hole for omnidirectional light absorber: comment

We prove that Lee et al.’s theoretical analysis and final formula for ray trajectories in an elliptic cylindrical optical “black hole” are incorrect. By utilizing a full-wave finite-element simulation, we show that a nonparaxial Gaussian beam does not follow a trajectory obtained using their formula.

The Far Properties of Nonparaxial Cosine-Squared Gaussian Beams

Based on the Rayleigh-Sommerfeld diffraction integrals, the closed-form expression of nonparaxial cosine-squared Gaussian beams in free space is derived. The power in the bucket (PIB) of nonparaxial regime is introduced to characterize the beam quality of nonparaxial beams in the far field. It is found that the intensity and PIB of nonparaxial cosine-squared Gaussian beams depends onfparameter, decentered parameter, and the PIB is additionally depends on the bucket’s size taken. For a given bucket’s size, the smaller decentered parameter andfparameter result in the better power focusability in the far field and the better beam quality.

Vortex generation and inhomogeneous Faraday rotation of a nonparaxial Gaussian beam in isotropic magneto-optic crystals.

We study the propagation of a nonparaxial Gaussian beam in the isotropic Bi₄Ge₃O₁₂ crystal under an external magnetic field. The analytical solutions in the paraxial limit are derived, which show that a vortex with topological charge m=2 (m=1) generates in the E_ (E(z)) component. Numerical calculations in the nonparaxial regime confirm that, with the assistance of an external magnetic field, the spin-to-orbital angular momentum coupling effect can occur even in the isotropic crystal. We also study the Faraday rotation of the nonparaxial Gaussian beam. At the center of the transverse plane, the rotation of the plane of polarization is linearly proportional to the magnetic field. However, the rotation is not uniform on the whole transverse plane, manifesting the nonparaxial effect.

Resonant laser focus light by uniformity dielectric microcylinder

In this paper we provide a numerically obtained result for analytical solution for the problem of monochromatic nonparaxial Gaussian beam diffraction by uniform dielectric cylinder, which yields a focal spot outside the cylinder with linear sizes twice less than the diffraction limit in the free space in case of cylinder radius comparable with the wavelength. Although the radius of cylinder is only two times larger than the wavelength, the produced intensity in focal spot is 48 times larger than the maximal intensity of incident Gaussian beam. Two percent deviation from the resonant radius (for a fixed wavelength) causes the focal spot increase in half, and the resulted intensity in focal spot drop to value of 12 times greater than the incident beam intensity. Obtained solution corresponds to the growth of the Bessel function of order 18 coefficient in E-field amplitude Bessel series expansion.

Spiral light fields in the form of closed curves as generalization of laguerre-gaussian beams

In this paper we provide a numerically obtained result for analytical solution for the problem of monochromatic nonparaxial Gaussian beam diffraction by uniform dielectric cylinder, which yields a focal spot outside the cylinder with linear sizes twice less than the diffraction limit in the free space in case of cylinder radius comparable with the wavelength. Although the radius of cylinder is only two times larger than the wavelength, the produced intensity in focal spot is 48 times larger than the maximal intensity of incident Gaussian beam. Two percent deviation from the resonant radius (for a fixed wavelength) causes the focal spot increase in half, and the resulted intensity in focal spot drop to value of 12 times greater than the incident beam intensity. Obtained solution corresponds to the growth of the Bessel function of order 18 coefficient in E-field amplitude Bessel series expansion.

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.

Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal

Nonparaxial propagation theory of coherent beams in a uniaxial crystal is extended to the partially coherent case. An analytical formula for the 3×3 cross-spectral density matrix of a nonparaxial Gaussian Schell-model (GSM) beam propagating in a uniaxial crystal orthogonal to the optical axis is derived. Statistical properties, such as the spectral intensity and the degree of polarization, of a nonparaxial GSM beam in a uniaxial crystal are studied numerically. It is found that the statistical properties of a nonparaxial GSM beam are closely determined by its initial beam parameters and the parameters of the crystal. Uniaxial crystal can be used to modulate the spectral density and degree of polarization of a nonparaxial partially coherent beam. Our results may be useful in some applications, such as optical trapping and nonlinear optics, where a light beam with special beam profile and polarization is required.

Propagation characteristics of nonparaxial Gaussian beams

Based on the rigorous definition of light intensity of nonparaxial vectorial beams in the transverse plane, beam width, far-field divergence angles and M2 factors are defined by the power in the bucket. Taking nonparaxial vectorial Gaussian beam as an example, the detailed theoretical analyses and numerical calculations about the propagation characteristics are performed, and the results are compared with those from the nonparaxial scalar theory. Showing that when the half width of light source ω0 is less than Λ, there are obvious differences between the propagation characteristics described by the nonparaxial scalar and the vectorial theory, and their propagation characteristics tend to be consistent with ω0 increasing, change to that obtained in the paraxial scalar approximation gradually.

Phase and polarization singularities of nonparaxial Gaussian vortex beams

The analytical expression for nonparaxial Gaussian vortex beams propagating in free space is derived, which enables us to study phase and polarization singularities in nonparaxial vector wavefields. Differing from the polarization singularities formed by two transverse electric-field components in the paraxial regime, the polarization singularities can be formed by the transverse and longitudinal electric-field components of nonparaxial beams, and there exist C-points and L-lines. The variation of the beam parameters and propagation distance will result in a shift of phase and polarization singularities, but their position relation remains unchanged and the topological relationship holds true.

Analytical non-paraxial TM polarized Gaussian beam in the source region

The non-paraxial TM polarized Gaussian beam, which approximately reduces to be a Gaussian TEM00 mode at the paraxial case, is the rigorous solution of Maxwell's equations for a confocal resonator. By means of mathematical techniques, the analytical expression in the source region has been derived without any approximation and expressed as a sum of the propagating part and the evanescent part. With some certain given values of the initial Gaussian half width w 0, the distributions of the squared modulus of the propagating part, the evanescent part, and the whole beam are depicted in the reference plane close to the source. When w 0 is small enough, the squared modulus of the non-paraxial TM polarized Gaussian beam is composed of two lobes, which results from the contribution of the evanescent wave. This research further reveals the difference of a non-paraxial TM polarized Gaussian beam from a non-paraxial Gaussian beam.

Propagation properties of a non-paraxial hollow Gaussian beam

Based on the non-paraxial vectorial moment theory of beam propagation, the analytical expressions of the M 2 factors for a non-paraxial hollow Gaussian beam (HGB) have been derived. The analytical formulae are further simplified for paraxial and highly non-paraxial cases. The beam waists, the divergence angles and the beam propagation factors are also depicted as functions of the parameter w 0. The divergence angles will not exceed the maximum value of 90°. When w 0 is within the scale of one time of light wavelength, the TE polarization results in the different beam propagation factors in the two transverse directions and the beam propagation factors first increase and then decrease. When w 0 is large enough, the beam propagation factors are determined only by the beam order.

The analytical vectorial structure of a nonparaxial Gaussian beam close to the source

The description of a nonparaxial Gaussian beam is made directly staring with the Maxwell's equations. The vector angular spectrum method is used to resolve the Maxwell's equations. As the vector angular spectrum can be decomposed into the two terms in the frequency domain, the nonparaxial Gaussian beam is also expressed as a sum of two terms. One term is the electric field transverse to the propagation axis, and the other term is the associated magnetic field transverse to the propagation axis. By means of mathematical techniques, the analytical expressions for the two terms in the source region have been derived without any approximation. The influence of the evanescent plane wave on the vectorial structure is also investigated. The results are analyzed with numerical example. This research is useful to the optical trapping and the optical manipulation.