The influence of nonparaxiality on the spectral behavior in Young's experiment illuminated by partially coherent light

Starting from the Rayleigh-Sommerfeld diffraction integral, this paper studies the spectral behavior in Young＇s experiment illuminated by nonparaxial partially coherent light and compares with the paraxial case, where the influence of nonparaxiality of partially coherent light on the spectral shifts and spectral switches is stressed. It is shown that there is a spectral shift in the nonparaxial case relative to the paraxial one and the critical position changes, at which the spectral switch occurs. The ratio of the waist width to the central wavelength ω0/λ0 and relative spatial correlation length △ affect the spectral difference. The smaller ω0/λ0 is, the larger the difference between the nonparaxial and paraxial results appears. The effect of relative spatial correlation length △ is relatively small.

Nonparaxial propagation properties of the chirped Airy Gaussian vortex beams in uniaxial crystals orthogonal to the optical axis

In this article, we investigate the nonparaxial propagation properties of the chirped Airy Gaussian vortex（CAiGV）beams in uniaxial crystals orthogonal to the optical axis analytically and numerically. We discuss how the linear chirp parameters, the quadratic chirp parameters, and the Gaussian factors influence the nonparaxial propagation dynamics of the CAiGV beams. The intensity, the energy flow, the beam center, and the angular momentum of the CAiGV beams are deeply investigated. It is shown that the Gaussian factors have a great effect on the intensity and the centroid positions of the CAiGV beams. With the Gaussian factors increasing, the intensity of CAiGV beams decreases rapidly. The main lobes of the transverse intensity distribution of the CAiGV beams are similar to triangles.

Partially coherent nonparaxial modified Bessel-Gauss beams

The concept of partially coherent nonparaxial modified Bessel Gauss （MBG） beams is proposed. Based on the generalized Rayleigh-Sommerfeld diffraction integral, the analytical propagation equations of nonparaxial MBG beams in free space are derived and analysed, and some special cases are discussed. In particular, under the paraxial approximation our results reduce to the corresponding paraxial ones. Numerical calculation examples are given to illustrate the dependence of intensity and spectral degree of coherence on the beam order m, ζ and f parameters, and to compare the difference between the paraxial and nonparaxial results.

Nonparaxial propagation of radially polarized chirped Airy beams in uniaxial crystal orthogonal to the optical axis

The nonparaxial propagation of radially polarized chirped Airy beams(RPCAiBs) in uniaxial crystal orthogonal to the optical axis is analytically studied. The effects of the first and the second order chirp factors, and the ratio of the extraordinary refractive index to the ordinary refractive index on the nonparaxial evolution of RPCAi Bs in different observation planes are investigated in detail. The results show that when one parameter changes, different components behave differently, and even the same component has various behavior in different directions. The initial slope of RPCAi Bs in the x-direction varies more with the first order chirp factor than that in the y-direction. Meanwhile, with the second order chirp factor becoming larger, the depth of the focus of the y-component in the x-direction decreases while that in the y-direction has no difference. In addition, the different ratios of the extraordinary refractive index to the ordinary refractive index change the difference of the depth of the focus between the x-and the y-components.

Nonparaxial propagation of phase-flipped Gaussian beams

This paper derives the closed-form expressions for nonparaxial phase flipped Gaussian （PFG） beams propagating in free space, through a knife edge and an aperture, which enable us to study nonparaxial propagation properties of PFG beams and to compare nonparaxial results with paraxial ones. It is found that the f parameter, offsetting distance of the knife edge and truncation parameter affect the nonparaxial beam propagation properties. Only under certain conditions the paraxial approximation is applicable. The results are illustrated by numerical examples.

One-dimensional nonparaxial bright solitons and their coherent interactions in Kerr media

The existence of one-dimensional bright Kerr solitons is investigated in Kerr media beyond the paraxial approximation. It is found that a nonparaxial soliton with no less than a minimum dimensionless width of about 0.76 can exist, which corresponds to the real width about a wavelength. Besides, the coherent interactions between two nonparaxial bright solitons in Kerr media are also investigated in detail. It is found that their separation and intensity ratio have great influence on the coherent interaction between these two solitons. Furthermore, the effect of the relative phase difference on the nonparaxial interaction is quite different from that on the paraxial interaction. Periodical breath, merging, repulsion, and energy transferring can be realized separately by choosing an appropriate initial relative phase between the coherent solitons.

Nonparaxial propagation of Hermite-Laguerre-Gaussian beams in uniaxial crystal orthogonal to the optical axis

Analytical expressions for the three components of the nonparaxial propagation of a Hermite-Laguerre-Gaussian （HLG） beam in uniaxial crystal orthogonal to the optical axis are derived. The intensity distribution of an HLG beam and its three components propagating in a uniaxial crystal orthogonal to the optical axis are demonstrated by numerical examples. Although the y and z components of an HLG beam in the incident plane are both equal to zero, they emerge upon propagation inside the uniaxial crystal. Moreover, the beam profile of the x component is relatively stable and the beam profiles of the y and z components have the same evolution law. If the ratio of the extraordinary refractive index to the ordinary refractive index is larger than unity, the beam profile of the HLG beam is elongated in the x direction and generally rotates clockwise. Otherwise, the beam profile of the HLG beam is elongated in the y direction and generally rotates anticlockwise. This research is beneficial to the optical trapping and nonlinear optics involved in the rotation of a beam profile.

Nonparaxial propagation of a super-Lorentz-Gauss SLGmode beam

Based on the vectorial Rayleigh-Sommerfeld integral formulae, this paper derives the analytical nonparaxial propagation equation of a super-Lorentz-Gauss （SLG） SLGo1 mode beam in free space. The far field expression and the scalar paraxial result are treated with special cases of the general formulae. According to the obtained analytical representation, the nonparaxial propagation properties of the SLG01 mode beam are illustrated and analysed with numerical examples. This research provides an approach to investigate the propagation of the SLG01 mode beam within the framework of the nonparaxial regime.

Nonparaxial propagation of a vectorial apertured off-axis Lorentz beam

Based on the vectorial Ftayleigh-Sommerfeld integral formula and the complex Gaussian expansion of the hardedge aperture function, an analytical propagation expression for a nonparaxial vectorial off-axis Lorentz beam passing through a rectangular aperture is derived. The unapertured case, the far field expression and the scalar paraxial result are also presented as special cases of the general formulae, respectively. Some numerical examples are also given to show the propagation characteristics of a nonparaxial vectorial off-axis Lorentz beam through a rectangular aperture. It is indicated that the f parameter, the off-axis displacement and the truncation parameter all play an important role in determining nonparaxial propagation behaviour.

Talbot effect in nonparaxial self- accelerating beams with electromagnetically induced transparency

In this study,we report on the fractional Talbot ffect of nonpar axial self-accelerating beams in a multilevel electromagnetically induced transparency(EIT)atomic configuration,which,to the best of our knowledge,is the first study on this subject.The Tallbot ffect originates from superposed eigenmodes of the Helmholtz equation and forms in the EIT window in the presence of both linear and cubic susceptibilities.The Talbot ffect can be realized by appropriately selecting the cofficients of the beam components.Our results indicate that the larger the radial difference between beam components,the stronger the interference between them,the smaller the Tallbot angle is.The results of this study can be useful when studying optical imaging.optical measurements,and optical computing.

The nonparaxial property of chirped pulsed beam

The nonparaxial property of the chirped pulsed beam is analyzed both quantitatively and qualitatively. Through the qualitative investigation of the paraxial approximation condition, we show there are chirp-induced changes in the nonparaxial propagation of the chirped pulsed beam. A quantitative nonparaxial correction was developed by use of the perturbational technic and the Fourier transform for a few-cycle chirped pulsed beam with relative small chirp parameter. It was shown that the nonparaxial corrections were enhanced near the leading or trailing edge of pulse depending on weather the chirp parameter is positive or negative. An example for pulsed Gaussian beam driven by a chirped Gaussian pulse is shown in the numerical result to confirm our analysis.

Multimode Interference Optical Power Splitter in Proton-Exchange LiNbO_3 Waveguides

The self imaging effect in graded index waveguides using annealed proton exchange (APE) technique in lithium niobate (LiNbO 3) waveguides is analyzed and simulated using the three dimensional nonparaxial beam propagation method (BPM).On this basis,a 1×8 multimode interference (MMI) optical power splitter by APE technique in X cut LiNibO 3 with Y propagation substrate is fabricated.Measurements show that the device has realized eight powers splittings.

Vectorial Hermite-Laguerre-Gaussian beams beyond the paraxial approximation

Starting from the vectorial Rayleigh-Sommerfeld integrals, the free-space propagation expressions for vectorial Hermit-Lagucrre-Gaussian （HLG） beams beyond the paraxial approximation are derived. The far-field expressions and the scalar paraxial results are given as special cases of our general expressions. The intensity distributions of vectorial nonparaxial HLG beams are studied and illustrated with numerical examples.