Rayleigh Diffraction Integral Research Articles

Vector Rayleigh Diffraction of High-Power Laser Diode Beam in Optical Communication

Laser diodes (LDs) are widely used in optical wireless communication (OWC) and optical networks, and proper theoretical models are needed to precisely describe the complicated beam field of LDs. A novel mathematical model is proposed to describe the vectorial field of nonparaxial LD beams. Laser beam propagation is studied using the vector Rayleigh diffraction integrals, and the stationary phase method is used to find the asymptotic expansion of diffraction integral. The far-field distribution of the LD beam in the plane parallel and perpendicular to the junction is considered in detail, and the computed intensity distributions of the theory are compared with the corresponding measurements. This model is precise for single transverse model beam of LDs and can be applied to describe the LD beams in OWC and optical networks.

Numerical study of the reflected elastic waves using Rayleigh diffraction integral

In this work, the transient ultrasonic waves radiated by a linear phased array transducer in a liquid then reflected at a liquid solid interface is studied. A model based on the Rayleigh integral is used where the reflection at the plane interface is considered by using the reflection coefficients for harmonic plane waves. The transient field is obtained by an inverse Fourier transform of the harmonic field. The obtained results highlight the different components of the ultrasonic field: the direct and edge waves as well as the longitudinal head waves or leaky Rayleigh waves. The temporal representation of these waves has been analyzed and discussed by the rays’ model. Instantaneous cartographies allowed a clear description of all the waves which appear at the liquid-solid interface. The results have been compared to those obtained by using a finite element method.

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.

Spectral switches of partially coherent nonparaxial beams diffracted at an aperture

Based on the generalized Rayleigh diffraction integral, the spectral shifts and spectral switches of nonparaxial partially coherent beams diffracted at an aperture are studied. Our main attention is paid to the effect of beam nonparaxiality on the spectral switches of partially coherent beams. It is shown that the spatial coherent parameter β, trunction parameter δ and waist width-to-central wavelength ratio w 0/ λ 0 affect the behavior of spectral switches. Only when β, δ and w 0/ λ 0 satisfy certain conditions, can the difference between the nonparaxial and paraxial results be negligible. The discrepancy arises physically from the change of the spectral modifier by using the paraxial approximation.

Analytic expressions and approximations for the on-axis, aberration-free Rayleigh and Debye integral in the case of focusing fields on a circular aperture

We present a derivation of the analytic result for on-axis field values of the Rayleigh diffraction integral, a result that was originally presented in a paper by Osterberg and Smith (1961). The method on which our derivation is based is then applied to other diffraction integrals used in acoustics and optics, e.g., the far-field Rayleigh integral, the Debye integral and the separate near-field part of the Rayleigh integral. Having available our on-axis analytic or semi-analytic solutions for these various cases, we compare the various integrals for wave numbers k pertaining to low-frequency acoustic applications all the way up to high-frequency optical applications. Our analytic results are compared to numerical results presented in the literature.

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.

Spectral behavior of partially coherent beam passing through an aperture

Based on the Rayleigh diffraction integral and the expansion of the hard-aperture function into a finite series of complex Gaussian functions, an approximate analytical spectrum expression for a Gaussian schell-model (GSM) beam passing through an aperture is given. The spectral shifts and spectral switches of near-zone off-axis are studied. The effect of the coherence and hard-edge on the spectral behavior are also presented.

Effect of nonparaxiality of optical field on spectral shift and spectral switching

Based on the Rayleigh diffraction integral, an analytical expression for the spectrum of polychromatic Gaussian beams diffracted at an aperture is derived， and the paraxial results are treated as a special case of our general expression. The main attention is focused on the effect of ratio of the waist width to the central wavelength w0/λ0 and truncation parameter δ on spectral shift and spectral switching. Only when w0/λ0 and δ satisfy certain conditions are the paraxial approximation results consistent with the nonparaxial ones. The nonparaxiality of field results in different spectral shifts and changes of the critical position where the spectral switching takes place.

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 properties of ultrashort pulsed beams with constant waist width in free space

Based on the Rayleigh diffraction integral and complex analytical signal representation, the strict propagation expression and its Fourier spectrum for ultrashort pulsed Gaussian beams with constant waist width are derived, which have the general applicable advantage, because the paraxial approximation, pulsed envelope representation and far-field approximation are not invoked. The far-field and paraxial expressions are obtained as the special cases of our general results. Numerical calculation results are given to illustrate the spatiotemporal propagation properties of few- and single-cycle pulsed Gaussian beams with constant waist width. A comparison with the paraxial results and previous work is also made.

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.

Propagation properties of ultrashort chirped pulsed Gaussian beams in the free space

Based on the Rayleigh diffraction integral and complex analytical signal representation, the free-space analytical propagation equation and its Fourier spectrum for ultrashort chirped pulsed Gaussian beams with constant diffraction length are derived. The effect of chirp parameter on the spatiotemporal and spectral properties is illustrated with analytical formulas and numerical calculation results. It is shown that the axial spectra of ultrashort chirped pulsed Gaussian beams become broadened with increasing chirp parameter. For single optical cycle, the transversal intensity distribution is affected by increasing chirp parameter, but almost not affected for several optical cycles. Moreover, the positive or negative sign of the chirp parameter has no effect on the spectral distribution and intensity distribution.

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.