Elegant Hermite-Gaussian Beams Research Articles

Propagation properties of elegant modified Bessel Gaussian beams.

A kind of optical beam with a radially parabolic propagating manner and intensity decay inversely proportional to propagating distance in the far field is investigated. The initial complex amplitudes of this kind of beam have the form of a Gaussian function multiplied by a m/2-order modified Bessel function and a helical phase factor with topological charge m. The arguments for Bessel and Gauss parts in the propagating solutions of these beams are complex and symmetric as elegant Laguerre and Hermite Gaussian beams. As a result, the beams can be referred to as elegant modified Bessel Gauss (EMBG) beams. Similar to non-diffractive beams such as Bessel and Airy beams, the EMBG beams also carry infinite power due to a transversely slowly decaying tail of complex amplitude. The EMBG beams demonstrate intermediate propagating properties between non-diffractive and finite-power beams. Unlike non-diffractive beams that never spread their power and finite-power beams that always diverge in a linear manner and spread their power by inversely square law in the far field, the EMBG beams demonstrate a far-field parabolic propagating manner and decay their power by inversely linear law. In addition, the EMBG beams have total Gouy phase, which is only half of that of elegant Laguerre Gauss beams with the same topological charge, and have far-field intensity distributions regardless of the beam waist radius in the initial plane. The propagating and focusing properties of EMBG beams represent an intermediate status between the non-diffractive and finite-power beams.

Properties of Airy transform of elegant Hermite-Gaussian beams

Airy transform of elegant Hermite-Gaussian beams is theoretically and experimentally investigated. The analytically general formula of the Airy transform of elegant Hermite-Gaussian beams is presented. The analytically general expressions of the centroid and the beam spot size of an elegant Hermite-Gaussian beam passing through an Airy transform optical system are derived. The effects of the Airy control parameters and the transverse mode numbers on the intensity distribution, the centroid, and the beam spot size of elegant Hermite-Gaussian beams passing through Airy transform optical systems are investigated, respectively. The experimental results coincide with the theoretical results. Also, the Airy transform of the elegant and the standard Hermite-Gaussian beams is compared. The analytical expression of the Airy transform of the elegant Hermite-Gaussian beam is more concise than that of the standard Hermite-Gaussian beam. Although the optical fields in the x-direction of the Airy transform of the elegant and the standard Hermite-Gaussian beams with transverse mode number m > 1 are both the sum of the zero-order derivative up to mth-order derivative of the Airy function, the weight coefficient of the same order derivative of the Airy function is different for the elegant and the standard transformed Hermite-Gaussian beams. The centroids of the elegant and the standard Hermite-Gaussian beams passing through the same Airy transform optical system are the same. However, the beam spot size of the elegant transformed Hermite-Gaussian beam is less than or equal to that of the standard transformed Hermite-Gaussian beam. The meaning of elegance is also embodied in the Airy transform of the elegant optical beams. This research reveals the difference between the Airy transform of the elegant and the standard Hermite-Gaussian beams.

Evolution properties of partially coherent standard and elegant Hermite-Gaussian beams in uniaxial crystals.

The propagation properties of a partially coherent elegant Hermite-Gaussian beam (EHGB) and a partially coherent standard Hermite-Gaussian beam (SHGB) in uniaxial crystals orthogonal to the optical axis are studied. The propagation expression for the cross-spectral density of the partially coherent EHGB and SHGB propagating in a uniaxial crystal is derived. With the help of the derived formula, some analyses are illustrated by numerical examples related to the evolution properties of the spectral density, the spectral degree of coherence, and the effective beam width of the partially coherent EHGB and SHGB in a uniaxial crystal. It is shown that propagation properties of the partially coherent EHGB and SHGB in an isotropic crystal are much different from the evolution properties in an anisotropic crystal. The evolution behavior of spectral density and the spectral degree of coherence depends on the parameters of the uniaxial crystals (the ratio of extraordinary and ordinary refractive indices), and beam parameters including the coherence length and the beam order, as well as the propagation distance.

Comparative study of propagation properties of partially coherent standard and elegant Hermite-Gaussian beams in inhomogeneous atmospheric turbulence

Based on the extended Huygens-Fresnel principle (EHFP) and the definition of the Wigner distribution function (WDF), the propagation properties of partially coherent standard Hermite-Gaussian beams (SHGBs) and elegant Hermite-Gaussian beams (EHGBs) in inhomogeneous atmospheric turbulence have been investigated. The M2-factor, the curvature radius and the root-mean-square (rms) angular width of partially coherent SHGBs and EHGBs have been explored numerically and comparatively. It can be found that the saturation propagation distances (SPDs) of the M2-factor and the rms angular width of EHGB and SHGB in inhomogeneous turbulence with a small zenith angle of π/6 are about 20km and 0.3km, respectively. What’s more, the M2-factor and the curvature radius of the partially coherent EHGB are smaller than that of the partially coherent SHGB in inhomogeneous turbulence. It can be shown that the M2-factor and curvature radius depend on the zenith angle, coherence width, inner scale of the turbulence, beam order, wave number and waist width in inhomogeneous turbulence. Furthermore, it is found that the partially coherent SHGB and EHGB have the same rms angular width in inhomogeneous turbulence.

Elegant Hermite–Airy beams

As can be approximated by , a kind of elegant Hermite–Airy (EHA) beam that is similar to the elegant Hermite–Gaussian (EHG) beam is introduced in this paper. Analytical expression of the EHA beams passing through an ABCD paraxial optical system is derived. By using the method of numerical fitting, the approximate expressions of , , , , and for an EHA beam are presented, respectively. When the transverse mode number is larger than 2, , , , , and of an EHA beam are all larger than those of the EHG beam. Based on the higher-order intensity moments, one can calculate the beam propagation factor, the beam half width, and the kurtosis parameter of the EHA beam passing through an ABCD paraxial optical system. As a numerical example, the propagation characteristics of the EHA beam are demonstrated in free space. Moreover, the propagation properties of the EHA beam are compared with those of the corresponding EHG beam. The evolutionary process of the EHA beam is far slower than that of the corresponding EHG beam. The research denotes that the EHA beams can be used to describe specially distributed optical beams that can not be characterized by the existing EHG beam model. The EHA beam model enriches and replenishes the existing beam model.

Two kinds of Airy-related beams

Two kinds of Airy-related beams are introduced in this manuscript. The normalized intensity distribution in the x-direction of the two kinds of Airy-related beams is close to that of the Gaussian beam. The normalized intensity distribution in the y-direction of the two kinds of Airy-related beams is close to that of the second-order and the third-order elegant Hermite–Gaussian beams, respectively. Analytical expressions of the two kinds of Airy-related beams passing through an ABCD paraxial optical system are derived. The beam propagation factors for the two kinds of Airy-related beams are 1.933 and 2.125, respectively. Analytical expressions of the beam half widths and the kurtosis parameters of the two kinds of Airy-related beams passing through an ABCD paraxial optical system are also presented. As a numerical example, the propagation properties of the two kinds of Airy-related beams are demonstrated in free space. Moreover, the comparison between the two kinds of Airy-related beams and their corresponding elegant Hermite–Gaussian beams along the two transverse directions are performed in detail. Upon propagation, the former kind of Airy-related beam will evolve from the central bright beam into the dark hollow beam. Contrarily, the latter kind of Airy-related beam will evolve from the dark hollow beam into the central bright beam. These two kinds of Airy-related beams can be used to describe specially distributed beams.

Airyprime beams and their propagation characteristics

A type of Airyprime beam is introduced in this document. An analytical expression of Airyprime beams passing through a separable ABCD paraxial optical system is derived. The beam propagation factor of the Airyprime beam is proved to be 3.676. An analytical expression of the kurtosis parameter of an Airyprime beam passing through a separable ABCD paraxial optical system is also presented. The kurtosis parameter of the Airyprime beam passing through a separable ABCD paraxial optical system depends on the two ratios B/(Azrx) and B/(Azry). As a numerical example, the propagation characteristics of an Airyprime beam is demonstrated in free space. In the source plane, the Airyprime beam has nine lobes, one of which is the central dominant lobe. In the far field, the Airyprime beam becomes a dark-hollow beam with four uniform lobes. The evolvement of an Airyprime beam propagating in free space is well exhibited. Upon propagation, the intensity distribution of the Airyprime beam becomes flatter and the kurtosis parameter decreases from the maximum value 2.973 to a saturated value 1.302. The Airyprime beam is also compared with the second-order elegant Hermite–Gaussian beam. The novel propagation characteristics of Airyprime beams denote that they could have potential application prospects such as optical trapping.

部分相干复宗量厄米高斯光束捕获两种类型粒子

部分相干复宗量厄米高斯光束捕获两种类型粒子

Orbital angular momentum of superposition of two generalized hermite–gaussian laser beams

We have obtained an expression for complex amplitude of the generalized paraxial Hermite-Gaussian (HG) beams. At certain parameters these beams become well-known HG modes and elegant HG beams. We calculated an orbital angular momentum (OAM) of linear combination of two generalized HG beams with transposed adjacent indices and with phase delay of p/2. It has been shown that for a sum of two HG modes the OAM is integer, while for two elegant HG beams it is always unitary and for two hybrid HG beams it is fractional.

A comparative study of standard and elegant Hermite–Gaussian beams propagating through turbulent atmosphere

Based on the extended Huygens–Fresnel principle and the definition of the second-order moments of the Wigner distribution function (WDF), the analytical expressions for the root-mean-square (rms) beam width and far-field divergence angle, curvature radius and M 2-factor of standard Hermite–Gaussian (SHG) and elegant Hermite–Gaussian (EHG) beams passing through turbulent atmosphere are derived and compared. It is shown that in turbulent atmosphere the far-field divergence angle of SHG and EHG beams is equal under the same conditions, but the rms beam width, curvature radius and the M 2-factor of SHG and EHG beams are different except for beam orders m = 0 and m = 1. The relative rms beam width, relative curvature radius and relative M 2-factor of SHG beams are less than those of EHG beams. Therefore, the conclusion that SHG beams are less influenced by turbulence than EHG beams can be drawn if we examine one of the above three relative beam parameters.

Vectorial structure of the far field of an elegant Hermite–Gaussian beam

Based on the vector angular spectrum method and the method of stationary phase, an analytical expression for the vectorial structure of the far field of an elegant Hermite–Gaussian beam is derived. The analytical formulae of the energy flux distributions of the TE term, the TM term, and the whole beam are presented in the far field. Analytical expressions for the ratios of the powers of the TE and TM terms to those of the elegant Hermite–Gaussian beam are obtained without any approximation. The physical pictures of the far field of an elegant Hermite–Gaussian beam are demonstrated and compared with those of the far field of the corresponding standard Hermite–Gaussian beam. This research reveals the internal vectorial structure of the far field of an elegant Hermite–Gaussian beam from an alternative viewpoint.

Spreading and M 2-factor of elegant Hermite–Gaussian beams through non-Kolmogorov turbulence

Using a non-Kolmogorov power spectrum, analytical expressions for the beam width, angular spread and M 2-factor of elegant Hermite–Gaussian (EHG) beams through non-Kolmogorov turbulence are derived. It is shown that the beam-width spreading, angular spread and M 2-factor of EHG beams through non-Kolmogorov turbulence depend on the non-Kolmogorov turbulence, i.e. on the generalised exponent parameter, , the inner scale, l 0, and the outer scale, L 0, at a sufficient propagation distance. They increase with increasing outer scale for 3.6<<4, and with decreasing inner scale. Only at a shorter propagation distance is the effect of turbulence on the beam-width spreading and M 2-factor negligible. The results are interpreted physically.

强非局域非线性介质中复宗量厄米高斯光束的传输

强非局域非线性介质中复宗量厄米高斯光束的传输

Enhancement of cross-polarized beam components at a metamaterial surface

Beam-interface interactions under incidence of a TE polarized elegant Hermite–Gaussian beam are considered in order to obtain efficient excitation of higher-order beam modes at a homogeneous, isotropic metamaterial surface. It is shown by numerical simulations that two transmitted TE and TM polarization beam components are enhanced more efficiently at a metamaterial surface than in the case of a dielectric interface. The mixed Poynting vector contribution to the incident energy flux is evaluated to explain this enhancement. The numerical results obtained are verified against energy balance at the interface.

Partially coherent elegant Hermite–Gaussian beam in turbulent atmosphere

Based on the extended Huygens–Fresnel integral, analytical formulas for the cross-spectral density, mean-squared beam width and angular spread of a partially coherent elegant Hermite–Gaussian (HG) beam in turbulent atmosphere are derived. The evolution properties of the average intensity, spreading and directionality of a partially coherent elegant HG beam in turbulent atmosphere are studied numerically. It is found that the partially coherent elegant HG beam with smaller initial coherence width, larger beam order and longer wavelength is less affected by the atmospheric turbulence. Compared to the partially coherent standard HG beam, the partially coherent elegant HG beam is less affected by turbulence under the same condition. Furthermore, it is found that there exist equivalent partially coherent standard and elegant HG beams, equivalent fully coherent standard and elegant HG beams, and an equivalent Gaussian–Schell-model beam may have the same directionality as a fully coherent Gaussian beam whether in free space or in turbulent atmosphere. Our results can be utilized in short and long atmospheric optical communication systems.

Partially coherent elegant Hermite–Gaussian beams

Elegant Hermite–Gaussian beams (EHGBs) are extended to the partially coherent case. An explicit and analytical formula is derived for the cross-spectral density of a partially coherent EHGB propagating through an aligned or misaligned paraxial ABCD optical system. The propagation properties of a partially coherent EHGB in free space and its focusing properties through a thin lens are studied numerically, and are compared to those of a partially coherent standard Hermite–Gaussian beam (SHGB). It is found that the propagation and focusing properties of a partially coherent EHGB are closely related to its initial coherence. A partially coherent EHGB spreads slower than a partially coherent SHGB in free-space propagation. A partially coherent EHGB can be focused more tightly than a partially coherent SHGB.

复宗量厄米高斯光束在大气湍流中的传输特性

复宗量厄米高斯光束在大气湍流中的传输特性

Average intensity and spreading of an elegant Hermite–Gaussian beam in turbulent atmosphere

The propagation of an elegant Hermite-Gaussian beam (EHGB) in turbulent atmosphere is investigated. Analytical propagation formulae for the average intensity and effective beam size of an EHGB in turbulent atmosphere are derived based on the extended Huygens-Fresnel integral. The corresponding results of a standard Hermite-Gaussian beam (SHGB) in turbulent atmosphere are also derived for the convenience of comparison. The intensity and spreading properties of EHGBs and SHGBs in turbulent atmosphere are studied numerically and comparatively. It is found that the propagation properties of EHGBs and SHGBs are much different from their properties in free space, and the EHGB and SHGB with higher orders are less affected by the turbulence. What's more, the SHGB spreads more rapidly than the EHGB in turbulent atmosphere under the same conditions. Our results will be useful in long-distance free-space optical communications.

Elegant Hermite–Laguerre–Gaussian beams

A novel family of paraxial laser beams called elegant Hermite-Laguerre-Gaussian beams (EHLGBs) are presented. The EHLGBs, a unity of elegant Hermite-Gaussian beams (EHGBs) and elegant Laguerre-Gaussian beams (ELGBs), constitute the exact and continuous transition modes between EHGBs and ELGBs when an additional parameter continuously changes. The virtual source for generation of the EHLGBs is identified. From the spectral representation of the elegant Hermite-Lauguerre-Gaussian waves, we derive the first three orders of nonparaxial corrections for the corresponding paraxial EHLGBs.

Spatiotemporal behaviors and singularity of ultrashort pulsed Elegant Hermite-Gaussian beams

Analytical propagation expressions of ultrashort pulsed Elegant Hermite-Gaussian beams are derived and spatiotemporal properties of the pulses with different transverse modes are studied. Singularity of the complex amplitude envelope solution of the pulses obtained under slowly varying envelope approximation is analyzed in detail. The rigorous analytical solution of the pulse is deduced and no singularity emerges in the solution. The obtained results indicate that the transverse mode affects not only the spatiotemporal properties but also the singularity of the pulses. Time delay of the off-axis maximum intensity is more obvious and the singularity is located nearer to the z-axis for the pulse with higher transverse modes.