Tight Focusing Properties Research Articles

Vector beams excited nonlinear optical effects

Vector beams with the desired polarization structure interacting with matter result in many novel nonlinear optical effects. Herein, we review the tight focusing and weak focusing properties of three types of light beams, namely, linearly polarized beams, cylindrical vector beams, and hybridly polarized beams. In particular, we revisit the second- and third-order nonlinear optical effects in nonlinear optical media under the excitation of several types of vector beams with the desired polarization distribution. The prospects of their applications in nonlinear optical microscopy and materials characterization are also briefly discussed.

Tight focusing properties of a circular partially coherent Gaussian beam.

Tight focusing properties of a circular partially coherent Gaussian (CPCG) beam with linear polarization have been studied based on vectorial Debye theory. Expressions for the intensity distribution and degree of coherence near the focus are derived. Numerical calculations are performed to show the intensity distribution and degree of coherence of the CPCG beam in the focal region. It is interesting to find that after focusing the CPCG beam through a high numerical-aperture objective we can obtain a super-length optical needle (>12λ) with homogeneous intensity along the propagation axis and wavelength beam size (∼λ). Moreover, the numerical calculations of coherence illustrate that, in the range of full width at half-maximum of the optical needle, for any two of the parallel electric field components of the optical needle the coherence is close to 1, but for any two of orthometric electric field components the value of coherence is between 0.4 and 0.9. Such a non-diffracting optical needle may have potential applications in atom optical experiments, such as in atom traps and atom switches.

Focusing properties of spirally polarized sinh Gaussian beam

The tight focusing properties of spirally polarized sinh-Gaussian beam is investigated numerically by vector diffraction theory. Results show that the optical intensity in focal region of spirally polarized sinh Gaussian beam can be altered considerably by the beam order, relative waist width and the spiral parameter that indicates the polarization spiral degree of the spirally polarized sinh Gaussian beam. Many novel focal patterns including flattop profile, focal hole axially separated focal spots and focal spot with long focal depth are evolved considerably for the suitable beam parameters. We expect such a tunable focal patterns are useful for optical manipulation of micro particles.

Focus shaping of the radially polarized Laguerre-Gaussian-correlated Schell-model vortex beams.

In this paper, we introduce a new kind of partially coherent vector beam with special correlation function and vortex phase named radially polarized Laguerre-Gaussian-correlated Schell-model (LGCSM) vortex beam as a natural extension of scalar LGCSM vortex beam. The realizability conditions for such beam are derived. The tight focusing properties of a radially polarized LGCSM vortex beam passing through a high numerical aperture (NA) objective lens are investigated numerically based on the vectorial diffraction theory. We find that not only the transverse component but also the longitudinal component of the focal field distributions can be shaped by regulating the structures of the correlation functions, which is quite different from that of the conventional radially polarized partially coherent beam. Moreover, a series of wildly used focal field with novel structure, e.g., focal spot, flat-topped or doughnut beam profiles, needle-like focal field and controllable three-dimensional (3D) optical cage, were obtained. These results indicate that the focus shaping can be achieved by combining the regulation of the structures of the correlation functions with the regulation of beam parameters effectively. Our results may be useful for potential applications in optical trapping, optical high-resolution microscopy and optical data storage.

Focusing properties of arbitrary optical fields combining spiral phase and cylindrically symmetric state of polarization.

The tight focusing properties of optical fields combining a spiral phase and cylindrically symmetric state of polarization are presented. First, we theoretically analyze the mathematical characterization, Stokes parameters, and Poincaré sphere representations of arbitrary cylindrical vector (CV) vortex beams. Then, based on the vector diffraction theory, we derive and build an integrated analytical model to calculate the electromagnetic field and Poynting vector distributions of the input CV vortex beams. The calculations reveal that a generalized CV vortex beam can generate a sharper focal spot than that of a radially polarized (RP) plane beam in the focal plane. Besides, the focal size decrease accompanies its elongation along the optical axis. Hence, it seems that there is a trade-off between the transverse and axial resolutions. In addition, under the precondition that the absolute values between polarization order and topological charge are equal, a higher-order CV vortex can also achieve a smaller focal size than an RP plane beam. Further, the intensity for the sidelobe admits a significant suppression. To give a deep understanding of the peculiar focusing properties, the magnetic field and Poynting vector distributions are also demonstrated in detail. These properties may be helpful in applications such as optical trapping and manipulation of particles and superresolution microscopy imaging.

Tight focusing properties of spirally polarized LG (1,1)* beam with High NA Parabolic mirror

The tight focusing of double ring shaped spirally polarized beam with high NA Parabolic mirror system is analyzed numerically based on the vector diffraction theory. Calculation results show that intensity distribution in focal region can be altered considerably on changing the spiral parameter (C) and pupil to beam ratio (β) of the incident double ring shaped beam. Many novel focal patterns such as focal spot, dark hollow focus and flat top profile of sub wavelength scale are achieved by proper tuning β and C.

Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties

In this paper, we introduce a new kind of vector multi-Gaussian Schell-model (MGSM) beam named radially polarized MGSM beam as an extension of recently introduced scalar MGSM beam [Opt. Lett.37, 2970 (2012)], and we obtain the realizability conditions of such beam. The tight focusing properties of a radially polarized MGSM beam passing through a high numerical aperture objective lens are investigated numerically based on the vectorial diffraction theory. It is interesting to find that the intensity distributions of both transverse and longitudinal fields near the focus display flat-top beam profiles due to the special correlation functions of the incident beam. Our results clearly show that engineering the structures of the correlation functions of a radially polarized partially coherent beam provides a novel way for shaping both transverse and longitudinal fields distributions, which have potential applications in particle acceleration and laser machining. Finally, we report experimental generation of a radially polarized MGSM beam with the help of a spatial light modulator and a radial polarization converter.

Tight focusing properties of anomalous vortex beams

Tight focusing properties of an anomalous vortex beam (AVB) passing through a high numerical aperture (NA) lens system are investigated, based on vector Debye integral. The numerical examples show that intensity distribution for the AVBs in focal region depend on the beam parameters including the topological charge m, the order of the AVBs n. Furthermore, the phase distribution and the intensity distribution at the different propagation distance z are discussed in detail. Our results will be useful to find the potential applications of AVBs.

Tight focusing of radially polarized circular Airy vortex beams

Tight focusing properties of radially polarized circular Airy vortex beams (CAVB) are studied numerically. The light field expressions for the focused fields are derived based on vectorial Debye theory. We also study the relationship between focal profiles, such as light intensity distribution, radius of focal spot and focal length, and the parameters of CAVB. Numerical results demonstrate that we can generate a radially polarized CAVB with super-long focal length, super-strong longitudinal intensity or subwavelength focused spot at the focal plane by properly choosing the parameters of incident light and high numerical aperture (NA) lens. These results have potential applications for optical trapping, optical storage and particle acceleration.

Tight Focusing Properties of Azimuthally Polarized Pair of Vortex Beams through a Dielectric Interface

Tight focusing properties of an azimuthally polarized Gaussian beam with a pair of vortices through a dielectric interface is theoretically investigated by vector diffraction theory. For the incident beam with a pair of vortices of opposite topological charges, the vortices move toward each other, annihilate and revive in the vicinity of focal plane, which results in the generation of many novel focal patterns. The usable focal structures generated through the tight focusing of the double-vortex beams may find applications in micro-particle trapping, manipulation, and material processing, etc.

Tight focusing properties of phase modulated transversely polarized sinh Gaussian beam

We studied the tight focusing properties of an azimuthally polarized sinh-Gaussian beam phase modulated with multi belt vortex phase filter on the basis of the vector diffraction theory. The sinh-Gaussian beam can be considered as superposition of a series of eccentric Gaussian beams and it is observed that focusing such a beam with a properly designed dedicated vortex phase filter generates many novel focal patterns including transversely polarized optical needle of extended focal depth with sub wavelength scale lateral resolution and mutiple focal spots with different axial seperation distance. We expect that such a focal patterns are usefull in applications such as semiconductor inspection and multiple optical trapping.

Tight Focusing Properties of Radially Polarized Gaussian Beams with Pair of Vortices

The tight focusing properties of a radially polarized Gaussian beam with a nested pair of vortices having a radial wave front distribution are investigated theoretically by the vector diffraction theory. The results show that the optical intensity in the focal region can be altered considerably by changing the location of the vortices nested in a radially polarized Gaussian beam. It is noted that focal evolution from one annular focal pattern to a highly confined focal spot in the transverse direction is observed corresponding to the change in the location of the optical vortices in the input plane. It is also observed that the generated focal hole or spot lead to a focal shift along the optical axis remarkably under proper radial phase modulation. Hence the proposed system may be applied to construct tunable optical traps for both high and low refractive index particles.

Tight focusing properties of phase modulated azimuthally polarized doughnut Gaussian beam

The intensity distribution in the focal region of a high NA lens for the incident azimuthally polarized doughnut Gaussian beam transmitted through a multi belt spiral phase hologram (MBSPH) is studied on the basis of the vector diffraction theory. Here we report a new method that generates a needle of transversely polarized light beam with sub diffraction beam size of 0.48λ that propagates without divergence over a long distance of about 49λ in free space and designed a dedicated MBSPH to generate multiple focal spot of transversely polarized. The authors also expect such a light needle of transversely polarized beam may find its application when using optical materials or instruments responsive to the transversal field only and to use in multiple optical trapping.

Focusing properties of radially polarized modified Bessel-modulated Gaussian beam by a high numerical aperture objective

In this paper, the focusing properties of radially polarized modified Bessel-modulated Gaussian (RPMQBG) beam with quadratic radial dependence through a high numerical aperture (NA) lens are investigated, theoretically and numerically, using the vector diffraction theory model. The tight focusing properties are studied in great detail. The numerical simulation results show that the intensity distribution of the RPMQBG beam at the focal region of the numerical aperture can be changed considerably by the variation of the optical vortex beam topological charge l and by the high numerical aperture angle (NA).

Tight focusing properties of cylindrically polarized annular multi-Gaussian beam

Focusing properties of cylindrically polarized annular multi gaussian beam with high numerical aperture lens system is investigated theoretically by vector diffraction theory. Results show that intensity distribution in focal region can be tailored considerably by appropriately adjusting the polarization angle. Peak-centered, donut and flattop focal shapes with extended focal depth, potentially useful in optical tweezers, material processing and laser printing can be obtained using this technique.

The tight focusing properties of Laguerre-Gaussian-correlated Schell-model beams

Based on the Richards–Wolf vectorial diffraction theory, the tight focusing properties, including the intensity distribution, the degree of polarization and the degree of coherence, of the Laguerre–Gaussian-correlated Schell-model (LGSM) beams through a high-numerical-aperture (NA) focusing system are investigated in detail. It is found that the LGSM beam exhibits some extraordinary focusing properties, which is quite different from that of the GSM beam, and the tight focusing properties are closely related to the initial spatial coherence and the mode order n. The LGSM beam can form an elliptical focal spot, a circular focal spot or a doughnut-shaped dark hollow beam at the focal plane by choosing a suitable value of the initial spatial coherence , and the central dark size of the dark hollow beam increases with the increase of the mode order n. In addition, the influences of the initial spatial coherence and the mode order n on the degree of polarization and the degree of coherence are also analysed in detail, respectively. Our results may find applications in optical trapping.

Generation of arbitrary vector fields based on a pair of orthogonal elliptically polarized base vectors.

We present an arbitrary vector field with hybrid polarization based on the combination of a pair of orthogonal elliptically polarized base vectors on the Poincaré sphere. It is shown that the created vector field is only dependent on the latitude angle 2χ but is independent on the longitude angle 2ψ on the Poincaré sphere. By adjusting the latitude angle 2χ, which is related to two identical waveplates in a common path interferometric arrangement, one could obtain arbitrary type of vector fields. Experimentally, we demonstrate the generation of such kind of vector fields and confirm the distribution of state of polarization by the measurement of Stokes parameters. Besides, we investigate the tight focusing properties of these vector fields. It is found that the additional degree of freedom 2χ provided by arbitrary vector field with hybrid polarization allows one to control the spatial structure of polarization and to engineer the focusing field.

Tight focusing of axially symmetric polarized beams with fractional orders

Tight focusing properties of axially symmetric polarized beams with fractional orders are studied theoretically based on vector diffraction theories. The mathematical expressions of focused fields are derived, which can be used to calculate the amplitude, phase and polarization distributions of focused fields. Simulation results are further shown, and many interesting patterns of focused fields are presented, which may find some potential applications in optical trapping, microscopy, near-field optics and so on. In addition, the influences of system parameters including numerical-aperture of objective lens, the beam filling factor and the beam waist factor on focused fields are analyzed, which provide more flexible modulation of focused field patterns.

Radiation forces of highly focused radially polarized hollow sinh-Gaussian beams on a Rayleigh metallic particle

Based on the vector diffraction theory, the tight focusing properties of radially polarized hollow sinh-Gaussian (HsG) beams are theoretically studied. It is found that the radially polarized HsG beams can form a longitudinally polarized sub-wavelength focal spot. Moreover, the radiation forces acting on a Rayleigh metallic particle are calculated for the case where the radially polarized HsG beams are applied. Compared with the use of conventional Gaussian beams, the high-order radially polarized HsG beams can largely enhance the radial trap stiffness and broaden the axial trap distance. The influence of the beam order m on the focusing properties and trap stiffness is investigated in detail.

Degree of polarization of a tightly focused, partially coherent anomalous hollow beam.

By using the Richards-Wolf vector diffraction theory, the formulas of the propagation of a linearly polarized, partially coherent anomalous hollow beam (AHB) through a high numerical aperture (NA) thin lens are derived. The tight focusing properties of a partially coherent AHB, including the degree of polarization, the intensity distribution, and the degree of coherence, are studied in detail. It is found that the focusing properties of a partially coherent AHB are closely determined by the initial coherence, initial beam parameters, and NA of the thin lens. However, no matter how high the value of eccentricity, the initial beam will reduce to a Gaussian beam of quasi-circular symmetry at the focal plane. No matter the value of NA, the size of the beam will remain invariant, which is an interesting phenomenon, and the beam has good stability. The results may find application in optical trapping with a partially coherent field.