Series representation of the Gaussian beam far-field diffracted by annular aperture

Accurate, analytical series expressions for the far-field diffraction of a Gaussian beam normally incident on a circular and central obscured aperture are derived with the help of the integration of parts method. With this expression, the far-field intensity distribution pattern can be obtained and the divergence angle is deduced too. Using the first five items of the series, the accuracy can satisfy most laser application fields. Compared with the conventional numerical integral method, the series representation is very convenient for understanding the physical meanings.     

Enhanced transmission through nano-aperture on a tapered metallic substrate

The optical transmission and distribution through a subwavelength slit on a tapered metallic substrate was investigated. By using a 45° tapered structure rather than a traditional metallic plate, a 6-fold transmission enhancement could be achieved. This is due to the asymmetrical excitation of surface waves and the matching of propagation constants between the surface waves and slit waveguide. In addition, by patterning surface corrugations in the exit plane, the beam could be focused. By tuning the period of the surface corrugations, we were able to adjust the focal length. For an input wavelength of 0.5 μm, the focal point could be kept within 0.6 μm with a focal length extending from 0.5 μm to 2.5 μm and a grating period ranging from 0.5 μm to 0.6 μm.     

Transmission enhancement through a metallic slit assisted by low scattering loss corrugations

A method to obtain large transmission of light through a nano-metallic slit bordered by nano trenches is demonstrated theoretically. The model is based on the composite diffraction of evanescent waves (CDEW) theory and utilizes a single trench design to effectively convert free-space light into surface plasma waves (SPW). The effectiveness is confirmed by the finite difference time domain (FDTD) method. The SPW loss due to scattering by the trench is also analyzed. It is found that when the width of the trench is slightly larger than λ_spp/2, the SPW becomes localized and resonates at the bottom edges performing as a dipole-like radiation source. At this time, the propagating SPW suffers from high scattering loss. Trenches can be used to enhance the transmission of a nano-scaled slit while simultaneously preserving high conversion efficiency and low scattering loss. A nano-scaled slit bordered by 10 pairs of trenches can be enhanced by a factor as high as 16. It is determined that a device designed to have a large clear aperture of 15 μm would have an advantage in terms of alignment.     

Focal shift and focal switch of flat-topped Mathieu-Gaussian beams passing through an apertured lens system

By introducing a hard aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression predominating the distribution of axial intensity for the flat-topped Mathieu-Gauss (FTMG) beams passing through a system with the aperture and lens separated has been derived. The focal shift and the focal switch effect of FTMG beams passing through the system is studied in detail. Numerical calculations have shown that the position of real focal plane is not coincident with the geometrical focus but is somewhat shifted toward the lens. The focal shift and focal switch of FTMG beams take place when the relative separation s/f = 1 by a suitable choice of beam parameter and truncation parameter, for example, the beam parameter is smaller than its corresponding critical value or the truncation parameter is between its two corresponding critical values.     

Phase anomaly and phase singularities of the field in the focal region of high-numerical aperture systems

The phase behavior of the three Cartesian components of the electric field in the focal region of a high-numerical aperture focusing system is studied. The Gouy phase anomaly and the occurence of phase singularities are examined in detail. It is found that the three field components exhibit different behaviors.     

Approximate analytical expressions of apertured broadband beams in the far field

The approximate analytical expressions of the apertured broadband beams in the far field with Gaussian and Laguerre-Gaussian spatial modes are presented.For the radially polarized Laguerre-Gaussian beam,the result reveals that the electromagnetic field in the far field is transverse magnetic.The influences of bandwidth(Γ) and truncation parameter(C 0) on the transverse intensity distribution of the Gaussian beam and on the energy flux distribution of radially polarized Laguerre-Gaussian beam are analysed.     

Focusing of stochastic electromagnetic Gaussian Schell-model beams through a high numerical aperture objective

Based on vectorial Debye diffraction theory, the focusing properties of stochastic electromagnetic Gaussian Schell-model beams in the focal region of high numerical aperture objective are investigated. Expressions for the intensity distribution and the degree of polarization are derived near the focus. Numerical calculations are performed to analyze the influences of varying corresponding parameters on the changes in the intensity distribution and in the degree of polarization in the focal region.     

离子通道中环形束流分布与场解的自洽理论

考虑环形束流的空间电荷效应与相对论状态下的自磁效应对离子通道——环形束流系统的影响，研究了通道中环形束流的轨道与场解的自洽平衡态.利用动力学理论得到了电子平衡态分布函数以及环形束流的径向位置和通道内电磁场的自洽解.研究发现，相对论束流的自场对系统平衡态有很大影响，并证实了系统自洽平衡态的存在.数值计算给出了束流位置与通道半径间随能量的分布关系以及通道内场随半径分布的规律，为相关器件的设计提供了重要的理论数据. 关键词： 离子通道 环形束流 动力学理论 自洽平衡态     

Coherence vortex evolution of partially coherent vortex beams in the focal region

Taking Gaussian Schell-model vortex beams as an example of partially coherent vortex beams, we study the evolution and subwavelength structures of coherence vortices in the focal region. The dependence of coherence vortices on the truncation parameter δ, the normalized coherence length ε and the topological charges m is illustrated numerically. It is found that during the evolution process the innermost m coherence vortices and the saddle points near the coherence vortices always remain in the focal plane.     

The creation-annihilation process of phase singularities of flat-topped beams in the focal plane

The creation-annihilation process of phase singularities of flat-topped beams in the focal plane is studied within the framework of the scalar paraxial approximation, where the new expression for flat-topped beams introduced by Li is taken as the beam model. It is shown that by suitably varying the truncation parameter, the phase singularities reorganize themselves, which leads to the annihilation of singularities in the focal plane, the creation of singularities outside the focal plane and subwavelength structures. Apart from the truncation parameter, the beam order of flat-topped beams additionally affects the spatial distribution of phase singularities.     

Beam propagation factor of partially coherent beams in gain or absorbing media

The definition of second-order intensity moments in the spatial domain and spatial frequency domain has been generalized for the case that the linear gain or absorbing media are included, where the wave number is generally complex. The formula for beam propagation M²-factor of partially coherent beams in linear gain or absorbing media has been given. The partially coherent flat-topped Schell-model beam is taken as an example. The closed-form expression for the beam propagation M²-factor of partially coherent flat-topped beam in gain or absorbing media has been derived. The changes of the M²-factor in media have been discussed with numerical examples. It can be shown that the M²-factor of flat-topped Schell-model beams in gain or absorbing media depends on the coherent parameter β, the coherent length σ₀, the beam order M, the propagation distance B, the imaginary part of the wave number K_i, as well as the real part of the wave number K_r.     

Spatial correlation properties of apertured partially coherent radially polarized beams in turbulent atmosphere

Based on the extended Huygens–Fresnel integral, the analytical formulae for the cross-spectral density matrix of the partially coherent radially polarized beams diffracted at a circular aperture in turbulent atmosphere are derived. The unapertured and free-space cases can be viewed as the special cases of our general result. By using the degree of coherence formula, the spatial correlation properties of the apertured partially coherent radially polarized beams in turbulent atmosphere are studied. The analyses indicate that the spatial correlation of the apertured partially coherent radially polarized beams are more affected by the atmospheric turbulence with the larger structure constant, the smaller truncation parameter, the larger coherence length, and the farther propagation distance.     

Paraxial waves in the far-field region

By investigating the changes suffered by a paraxial beam propagating in the near-field and in the far-field regions, it has been found a set of wave equations valid for points gradually closer to the near field. A relevant expression for the validity of the far-field approximation is given from the paraxial Helmholtz equation. It is pointed out that the well-known Fresnel number associated with every transverse diffraction pattern can be interpreted as a magnitude that measures the relative standard deviation of the Fraunhofer pattern and a first-order field, thus reporting on an integral expression suitable for a general case. Finally, the Rayleigh range of the optical beam is deduced from the previously inferred Fresnel number, what has been applied for the cases of a spherical Gaussian beam and a uniform-illuminated circular aperture.     

Bessel–Gauss beam optical resonator

In a simple picture, a Bessel beam is viewed as a transverse standing wave formed in the interference region between incoming and outgoing conical waves. Based on this interpretation we propose an optical resonator that supports modes that are approximations to Bessel–Gauss beams. The Fox–Li algorithm in two transverse dimensions is applied to confirm the conclusion.     

Whispering gallery mode propagation in photonic crystals in front of subwavelength slit arrays: Interplay with extraordinary transmission

We numerically study the propagation of light in 2D photonic crystals (PC) made of infinite cylinders of high refractive index, via the excitation of their morphological dependent resonances, in the presence of metallic arrays of subwavelength slits, either corrugated or not, that may produce extraordinary transmission. In this way, we confirm and illustrate previous theoretical findings on PCs and show new effects when combined with slits. Among them, we show that the whispering gallery mode excitation in the PC couples dominates the transmission of the slit arrays alone. Appropriate design of the system parameters and the illumination conditions, selects the transport and confinement of energy.     

平顶光束K参数的解析传输方程

 从Li提出的平顶光束模型和Collins公式出发,推导出平顶光束通过近轴ABCD光学系统K参数的解析传输公式。平顶光束的K参数与光束阶数M、瑞利长度z_R和传输矩阵元A/B有关。另一方面,基于强度矩的传输方程,通过繁冗的积分,也推导出K参数的解析传输方程。大量数值计算结果证明二者所得结果相同。对平顶光束K参数的传输特性用数值计算例作了说明。     

More precise determination of thermal lens focal length for end-pumped solid-state lasers

A method of more precise determination of the focal length of equivalent thermal lens (TL) is presented in this paper. The method is based on the diffraction theory of aberrations. By numerically calculating the optical path difference (OPD) distribution and the Strehl ratio, the focal lengths for “top-hat” pumping and Gaussian pumping are obtained and the equations for the focal lengths are presented by curve fitting. Also the equations for average pump radii are presented. The experiment on a fiber-coupled LD pumped Nd:YAG laser is carried out. The experimental results verify the validity of the method.     

Formation of sub-half-wavelength focal spot with ultra long depth of focus

We theoretically demonstrate the creation of sub-wavelength focal spot with ultra long depth of focus (DOF) non-diffracting. To achieve a sub-half-wavelength focal spot (0.4λ) with modulation, extended DOF (over 9.5λ) in vacuum, an integration of amplitude phase encoding, cylindrical polarization and the combination of dual-beams is utilized. Further analysis also predicts that a flattop intensity distribution along z axis can be observed around the ideal focus point.     

Intensity distribution in the focal region of a “doughnot” mode

We study the intensity distribution of a TEM₀₁₊₁₀ mode propagating through a single diffraction limited lens. The theoretical intensity distributions along the optical path remain similar until the truncation effects, due to the lens diameter, are negligible. The focal distribution is however very sensitive to truncation, and this is relevant for industrial applications of high power lasers. An experimental system to measure the intensity distribution at the focal plane is described and its results are compared with the theory.