Vectorial Rayleigh Diffraction Integral Research Articles

Vectorial diffraction properties of THz vortex Bessel beams.

A vortex Bessel beam combines the merits of an optical vortex and a Bessel beam, including a spiral wave front and a non-diffractive feature, which has immense application potentials in optical trapping, optical fabrication, optical communications, and so on. Here, linearly and circularly polarized vortex Bessel beams in the terahertz (THz) frequency range are generated by utilizing a THz quarter wave plate, a spiral phase plate, and Teflon axicons with different opening angles. Taking advantage of a THz focal-plane imaging system, vectorial diffraction properties of the THz vortex Bessel beams are comprehensively characterized and discussed, including the transverse (Ex, Ey) and longitudinal (Ez) polarization components. The experimental phenomena are accurately simulated by adopting the vectorial Rayleigh diffraction integral. By varying the opening angle of the axicon, the characteristic parameters of these THz vortex Bessel beams are exhibited and compared, including the light spot size, the diffraction-free range, and the phase evolution process. This work provides the precise experimental and theoretical bases for the comprehension and application of a THz vortex Bessel beam.

Vector characterization of zero-order terahertz Bessel beams with linear and circular polarizations

As a kind of special beams, Bessel beams are always a research hot spot in optics due to its non-diffractive and self-healing properties. Here, zero-order terahertz (THz) Bessel beams with linear and circular polarizations are generated by using a THz quarter wave plate and Teflon axicons with different opening angles. By applying a THz digital holographic imaging system, the evolutions of the transverse (Ex, Ey) and longitudinal (Ez) electric fields are coherently measured and analyzed during the propagation processes of the THz Bessel beams. The vectorial Rayleigh diffraction integral is used to accurately reproduce the amplitude, phase, and non-diffractive feature of each polarization component for the THz Bessel beams. With varying opening angles of the axicons, the focal spots, diffraction-free ranges, and Gouy phase shifts of the THz Bessel beams are compared and discussed. The experiment and simulation results provide a comprehensive view for exactly understanding peculiar features of THz Bessel beams.

Topological charges shift of vectorial nonparaxial vortex beam

Based on the vectorial Rayleigh diffraction integral, the analytical propagation expression of a nonparaxial vectorial vortex beam carrying orbital angular momentum (OAM), diffracted at a circular aperture, is presented. The effects of both f parameter and topological charge (m) on the nonparaxiality of apertured vectorial vortex beams are studied. Simulation results show that the topological charge of transverse field component m will shift to m-1, for that of longitudinal field component.

Non-Paraxial Propagation of Cylindrical Vector Vortex Beams in the Far-Field

On the basis of the vectorial Rayleigh diffraction integral, the analytical expressions for the electric fields of cylindrical vector vortex beams (CVVBs) with an arbitrary polarization order in the far-field are derived, which helps us investigate the far-field properties of the CVVBs in the non-paraxial and paraxial regimes. Some detailed comparisons of the non-paraxial results with the paraxial results are made, which show that the polarization order and the ratio of beam waist width to wavelength play an important role in determining the non-paraxiality of CVVBs. In addition, the polarization order and the ratio of beam waist width to wavelength have a great impact on the diffraction effect and the divergent behavior of CVVBs in the far-field.

Spectral characteristic of a pulsed hollow Gaussian beam passing through a circular aperture

The spectral characteristics of a pulsed hollow Gaussian beam passing through a circular aperture are studied. Based on the vectorial Rayleigh diffraction integrals, the analytical expressions of the spectra for the complex analytic signal representation and for the complex amplitude envelope representation are derived, respectively, and the comparison between them is made. The influences of the truncation parameter and the beam order on the spectral shifts and on the spectral switches are illustrated. It is shown that the spectrum for the complex analytic signal representation and that for the complex amplitude envelope representation are obvious different as the pulse shorter than an optical period.

Focusing of Optical Vector-vortex Beams

Theoretical formalism using vectorial Rayleigh diffraction integrals is developed to calculate the electric field components   z y x E E E , of generalized vector-vortex (VV) beams of different phase and polarization characteristics as a function of propagation distance ‘z’ in the focal region of an axicon. This formalism is used to generate sub-wavelength spot-size (0.43λ) ultra-long length (80λ) longitudinally-polarized optical needle beam by appropriately selecting the phase and polarization characteristics of the input VV beam. The formalism is further extended to also generate purely transverse polarized beam with similar characteristics. The focusing process leads to interference between different field components of the beam resulting in the formation of C-point polarization singularities of index Ic = ±1 whose transverse characteristics evolve with propagation distance. Experimental results to support our theoretical calculations are presented along with lens focus comparison results.

Nonparaxial Propagation of a Radially Polarized Beam Diffracted by an Annular Aperture

Based on the vectorial Rayleigh diffraction integral, the integral formulae of three electromagnetic field components for radially polarized Laguerre—Gaussian beams after diffraction by an annular aperture and the propagation distance in nonparaxial regimes are presented. The diffraction by a circular aperture or a circular disk or propagation in free space can be treated as the special cases of this general result. The numerical simulation shows that the electric field intensity distributions closely depend on both the inner truncation parameters and the outer truncation parameters of the annular aperture, as well as on the ratio of the waist of the incident Laguerre-Gaussian beam to the wavelength.

Nonparaxial analyses of radially polarized beams diffracted at a circular aperture

On the basis of the vectorial Rayleigh diffraction integral, the analytical expressions for the electromagnetic fields of the radially polarized beams diffracted at a circular aperture are derived, which helps us investigate the propagation properties of the apertured radially polarized beams in the nonparaxial and paraxial regimes. The unapertured and paraxial cases can be viewed as the special cases of the general result obtained in this paper. The analyses indicate that the nonparaxiality of the apertured radially polarized beams depends on the ratio of the waist width to the wavelength and the truncation parameter. In addition, the truncation parameter and the beam order have a great impact on the beam diffraction effect and the beam evolution behavior.

Anomalous spectral behavior of vectorial apertured ultrashort pulsed beams beyond the paraxial approximation

Based on the vectorial Rayleigh diffraction integrals, the analytical expression for the spectral intensity of a vectorial nonparaxial ultrashort pulsed Gaussian beam diffracted at a circular hard-aperture is derived. The effect of f-parameter ( f = 1/ k 0 w 0) on the spectral anomalies near phase singularities of the vectorial nonparaxial ultrashort pulsed beams is studied. It is shown that the spectral switch near the phase singularity of diffracted vectorial nonparaxial ultrashort pulsed beam still exists beyond paraxial regime, but disappears when the f-parameter is larger than a certain value.

Partially coherent vectorial cosh-Gaussian beams beyond the paraxial approximation

The concept of partially coherent vectorial nonparaxial cosh-Gaussian (ChG) beams is introduced, and their analytical propagation expressions for the cross-spectral density matrix in free space are derived by using the generalized vectorial Rayleigh diffraction integrals. Some interesting cases, in particular, the vectorial nonparaxial Gaussian–Schell-model (GSM) beams are discussed and treated as special cases of our general expressions. It is shown that the f and f σ parameters play a crucial role in determining the vectorial property and nonparaxiality of partially coherent ChG beams, but the decentered parameter additionally affects their behavior.

Propagation of vectorial Gaussian beams behind a circular aperture

Based on the vectorial Rayleigh diffraction integral and the hard-edge aperture function expanded as the sum of finite-term complex Gaussian functions, an approximate analytical expression for the propagation equation of vectorial Gaussian beams diffracted at a circular aperture is derived and some special cases are discussed. By using the approximate analytical formula and diffraction integral formula, some numerical simulation comparisons are done, and some special cases are discussed. We find that a circular aperture can produce the focusing effect but the beam becomes the shape of ellipse in the Fresnel region. When the Fresnel number is equal to unity, the beam is circular and the focused spot reaches a minimum.

Vector propagation of radially polarized Gaussian beams diffracted by an axicon

On the basis of the vectorial Rayleigh diffraction integrals and stationary-phase method, the analytic expression describing the vectorial field distribution of radially polarized Gaussian beams diffracted by an axicon is derived. The theoretical analysis and simulation calculation show that the radial component of the diffraction field is the propagation-invariant first-order Bessel beam when the radially polarized Gaussian beam illuminates the axicon. However, the longitudinal component possesses no such behavior because of its intrinsic r dependence, and its central intensity is the maximum. The longitudinal component is related to the open angle and index of the axicon, which has to be considered when the open angle and index are large. For a small open angle and index, the longitudinal component can be neglected, and the scalar approximation is valid.

Propagation of vectorial nonparaxial Gaussian beams through an annular aperture

Starting from the vectorial Rayleigh diffraction integrals, the nonparaxial propagation of vectorial Gaussian beams through an annular aperture is studied. The analytical propagation expressions are derived, which permit us to treat the on-axis field and far field of vectorial nonparaxial Gaussian beams diffracted at the annular aperture, the nonparaxial diffraction at a circular aperture and a circular disc as our special cases in a unified way. The validity of our treatment is confirmed by direct numerical integration of the Rayleigh formulae. It is shown that the f -parameter and annular obscuration affect the beam nonparaxiality in the case of diffraction at the annular aperture.

Nonparaxial diffraction of vectorial plane waves at a small aperture

By using the vectorial Rayleigh diffraction integrals, a nonparaxial propagation equation of vectorial plane waves diffracted at a small rectangular aperture is derived analytically and some special cases are discussed. Numerical calculation results are given to illustrate the applicability and validity of our theoretical formulae. It is shown that for the apertured case the ratio of the aperture width and wavelength affects the beam nonparaxiality. The nonparaxial approach presented in this paper has to be used for diffracted plane waves if the aperture width is comparable with or less than the wavelength.

A comparative study of the propagation of vectorial nonparaxial beams in the use of different approaches

Starting from the vectorial Rayleigh diffraction integral formula and using a simple expansion of the nucleus in the Rayleigh formula, an analytical propagation equation of vectorial nonparaxial Gaussian beams in free space is derived, which permits us to perform numerical calculations in comparison with the expression derived by Ciattoni et al. and with the direct numerical integration of the Rayleigh formula. It is found that as usual the use of expansion of vectorial Rayleigh diffraction integral is sufficient to provide satisfactory numerical results as compared with the direct integration of the Rayleigh formula. The above two analytical expressions are valid under certain conditions, however both are applicable in the far field.

A comparison of the vectorial nonparaxial approach with Fresnel and Fraunhofer approximations

Based on the vectorial Rayleigh diffraction integrals, a nonparaxial propagation equation of vectorial plane waves diffracted at a circular aperture is derived. The nonparaxial far-field expression, Fresnel and Fraunhofer diffraction formulae are given and treated as special cases of our general expression. The theoretical formulation permits us to study and compare the transversal and axial intensity distributions of diffracted plane waves both analytically and numerically. Illustrative numerical examples are given. It is shown that the vectorial nonparaxial approach has to be used if the aperture size is comparable with or less than the wavelength, and the knowledge of both transversal and axial intensity distributions is required to provide a comprehensive comparison of the paraxial and nonparaxial results.

Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture.

Based on the vectorial Rayleigh diffraction integral, the nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture is studied. The far-field and paraxial cases are treated as special cases of our general result. It is shown that for the apertured case the f parameter still plays an important role in determining the nonparaxiality of vectorial diffracted Gaussian beams, but both the f parameter and truncation affect the beam evolution behavior.