J. Rogel-Salazar, G. H. C. New, S. Chavez-Cerda
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 dimension is applied to confirm the conlcusion
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Bessel beams, propagationally invariant beams, and axicons: An historical and tutorial review
本文综述贝塞尔光束的历史发展，阐述其与阿拉戈光斑、轴锥镜等的关系，并涵盖高阶、偏振、波导、脉冲、艾里光束、扩展焦深及光片和光子纳米射流等主题。
We present a study of radially and azimuthally polarized Bessel-Gauss beams in both the paraxial and nonparaxial regimes. We discuss the validity of the paraxial approximation and the form of the nonparaxial corrections for Bessel-Gauss beams. We show that, independently from the ratio between the Bessel aperture cone angle $\vartheta_0$ and the Gauss beam divergence $θ_0$, the nonparaxial corrections are always very small and therefore negligible. Explicit expressions for the nonparaxial vector electric field components are also reported.
Collapse of a Gaussian beam in self-focusing Kerr media arrested by nonlinear losses may lead to the spontaneous formation of a quasi-stationary nonlinear unbalanced Bessel beam with finite energy, which can propagate without significant distortion over tens of diffraction lengths, and without peak intensity attenuation while the beam power is drastically diminishing.
Constructing various paraxial beams out of regular and modified Bessel-Gaussian modes
本文提出利用正则与修正贝塞尔-高斯模式，通过调节拓扑指标和伸缩参数并选取适当权重因子，系统构造出高斯光束、Kummer-Gaussian光束等多种已知及新型傍轴光束解。
A unified description of the free-space cylindrical vector beams is presented, which is an integral transformation solution to the vector Helmholtz equation and the transversality condition. The amplitude 2-form of the angular spectrum involved in this solution can be arbitrarily chosen. When one of the two elements is zero, we arrive at either transverse-electric or transverse-magnetic beam mode. In the paraxial condition, this solution not only includes the known $J_1$ Bessel-Gaussian vector beam and the axisymmetric Laguerre-Gaussian vector beam that were obtained by solving the paraxial wa…
The generation of isolated attosecond pulses with high efficiency and high beam quality is essential for attosec- ond spectroscopy. We numerically investigate the supercontinuum generation in a neutral rare-gas medium driven by a two-color Bessel-Gauss beam. The results show that an efficient smooth supercontinuum in the plateau is obtained after propagation, and the spatial profile of the generated attosecond pulse is Gaussian-like with the divergence angle of 0.1 degree in the far field. This bright source with high beam quality is beneficial for detecting and controlling the microscopic pro…
We describe a new class of propagation-invariant light beams with Fourier transform given by an eigenfunction of the quantum mechanical pendulum. These beams, whose spectra (restricted to a circle) are doubly-periodic Mathieu functions in azimuth, depend on a field strength parameter. When the parameter is zero, pendulum beams are Bessel beams, and as the parameter approaches infinity, they resemble transversely propagating one-dimensional Gaussian wavepackets (Gaussian beam-beams). Pendulum beams are the eigenfunctions of an operator which interpolates between the squared angular momentum ope…
A simple analytical way of creating superpositions of Bessel-Gaussian light beams with knotted nodal lines is proposed. It is based on the equivalence between the paraxial wave equation and the two-dimensional Schrödinger equation for a free particle. The $2D$ Schrödinger propagator is expressed in terms of Bessel functions, which allows to obtain directly superpositions of beams with a desired topology of nodal lines. Four types of knots are constructed in the explicit way: the unknot, the Hopf link, the Borromean rings and the trefoil. It is also shown, using the example of the figure-eight …
The normalization of energy divergent Weber waves and finite energy Weber-Gauss beams is reported. The well-known Bessel and Mathieu waves are used to derive the integral relations between circular, elliptic, and parabolic waves and to present the Bessel and Mathieu wave decomposition of the Weber waves. The efficiency to approximate a Weber-Gauss beam as a finite superposition of Bessel-Gauss beams is also given.
Multi-modes Bessel-Gaussian-Orbital Angular Momentum Beams Quantum Holography
本文提出基于多模式贝塞尔-高斯光束的轨道角动量量子全息方案，利用纠缠光子对构建量子选择全息图，并通过信号光子的模式参数实现信息重建，提升多路复用维度和编码能力。
We report, in theory and experiment, on a novel class of controlled light capsules with nearly perfect darkness, directly employing intrinsic properties of modified Bessel-Gauss beams. These beams are able to naturally create three-dimensional bottle-shaped region during propagation as long as the parameters are properly chosen. Remarkably, the optical bottle can be controlled to demonstrate various geometries through tuning the beam parameters, thereby leading to an adjustable light capsule. We provide a detailed insight into the theoretical origin and characteristics of the light capsule der…
Goos-Haenchen and Imbert-Fedorov shifts are diffractive corrections to geometrical optics that have been extensively studied for a Gaussian beam that is reflected or transmitted by a dielectric interface. Propagating in free space before and after reflection or transmission, such a Gaussian beam spreads due to diffraction. We address here the question how the Goos-Haenchen and Imbert-Fedorov shifts behave for a ``nondiffracting'' Bessel beam.
We propose the braiding of optical vortices in a laser beam with more than 2π rotation by superposing Bessel modes with a plane wave. We experimentally demonstrate this by using a Bessel-Gaussian beam and a coaxial Gaussian, and we present measurement of three complete braids. The amount of braiding is fundamentally limited only by the numerical aperture of the system and we discuss how braiding can be controlled experimentally for any number of vortices.
Nonlinear Bessel beams in self-defocusing media are found to be the natural, non-diffracting background where vortex solitons can be nested, interact and survive for propagation distances that are one order of magnitude larger than in the usual Gaussian or super-Gaussian backgrounds. The dynamics of the vortex solitons approaches that in the ideal, uniform background, preventing vortex spiraling and decay, which eases vortex steering for applications.
Bessel-Gauss beams of arbitrary integer order: propagation profile, coherence properties and quality factor
本文提出在梯度指数介质中生成任意整数阶贝塞尔-高斯光束的方法，并揭示了其与SU(1,1)群的关联。
Super-resolution imaging enabled by higher-order Bessel Gaussian beams in nanoscale optical structures
研究旨在突破传统光学显微镜阿贝衍射极限以获取亚波长信息，通过利用高折射率纳米级结构和高阶贝塞尔光束实现超分辨率成像，展示了（λ/53）的分辨率，阐明了相关因素，凸显该方法在多领域应用的潜力。
Propagation of Laguerre-Gaussian and Bessel-Gaussian scalar beams in an effective anisotropic background
该研究探讨有效各向异性背景下结构化标量光束的传播，采用格林函数方法，以拉盖尔-高斯、贝塞尔-高斯光束为对象，揭示各向异性对径向强度分布等的调控作用。
Non-separable superpositions of polarization and spatial modes in Gaussian beams have shown rich patterns of spatially-variable polarization encoded in the transverse mode of the light, with the potential for many applications. In this work we investigate encoding similar types of superpositions in Bessel beams. We find that analogously to what is observed in Poincaré-Gaussian beams, Poincaré-Bessel beams can be produced by Bessel-mode superpositions. They show polarization patterns with disclinations in the orientation of the polarization ellipse and mappings of the Poincaré sphere onto the t…
Monochromatic high-harmonic generation by Bessel-Gauss beam in periodically modulated media
针对高次谐波产生宽光谱限制单波长应用的问题，提出利用Bessel-Gauss光束与周期性调制气体介质直接产生单色高次谐波的方法，经解析计算与数值模拟验证，无需额外单色化光学元件，显著提升了通量并简化了装置。
In this paper, we present the theoretical studies of a refractive index map to implement a Gauss to J0-Bessel-Gauss convertor. We theoretically demonstrate the viability of such device by solving the inverse electromagnetic problem. The computed conversion efficiency is 90%. The theoretical results, obtained from the beam conversion efficiency, self-regeneration, and propagation through an opaque obstruction; demonstrate that a 2D graded index map of the refractive index can be used to transform a Gauss beam into a J0-Bessel-Gauss beam. To the best of our knowledge, this is the first demonstra…
Bessel Gaussian Beam Propagation in a Thermally Induced Axially Varying GRIN Medium
研究热诱导轴向变化GRIN介质中Bessel高斯束传播，推导了考虑轴向吸收的闭式ABCD传输矩阵，验证其在均匀介质和常梯度GRIN极限下的有效性。
The properties of electron round lenses produced by the ponderomotive potential are investigated in geometrical optics. The potential proportional to the intensity distribution of a focused first-order Bessel or Laguerre-Gaussian beam is exploited to produce an electron round lens and a third-order spherical aberration corrector. Several formulas for the focal length and spherical aberration coefficients in the thin-lens approximation are derived to set the lens properties and associated light beam parameters. When the mode field of the optical beam is small, the electron trajectory calculatio…
Contrarily to a common belief, any beam of light possesses to a some extent the ability to "reconstruct itself" after hitting an obstacle. The celebrated Arago spot phenomenon is nothing but a manifestation of this property. In this work we analyze the self-healing mechanism from both a mathematical and a physical point of view, eventually finding a new expression for the minimum reconstruction distance, which is valid for \emph{any} kind of beam, including Gaussian ones. Finally, a witness function that quantify the self-reconstruction capability of a beam is proposed and tested. The results …
A promising alternative to Gaussian beams for use in strong field science is Bessel-Gauss (BG or Bessel-like) laser beams as they are easily produced with readily available optics and provide more flexibility of the spot size and working distances. Here we use BG beams produced with a lens-axicon optical system for higher order harmonic generation (HHG) in a thin gas jet. The finite size of the interaction region allows for scans of the HHG yield along the propagation axis. Further, by measuring the ionization yield in unison with the extreme ultraviolet (XUV) we are able to distinguish region…
In this paper we present a simple and effective method, based on appropriate superpositions of Bessel-Gauss beams, which in the Fresnel regime is able to describe in analytic form the 3D evolution of important waves as Bessel beams, plane waves, gaussian beams, Bessel-Gauss beams, when truncated by finite apertures. One of the byproducts of our mathematical method is that one can get in few seconds, or minutes, high-precision results which normally require quite long times of numerical simulation. The method works in Electromagnetism (Optics, Microwaves,...), as well as in Acoustics. OCIS code…
Properties of "special'' hyperbolic Bessel-Gaussian optical beams
本文推导了“特殊”双曲贝塞尔-高斯（SHBG）光束的显式公式，分析其基本特性，发现参数γ可调整形状，光束呈环形且传播中分裂，相位具涡旋模式特征。
The first part of the paper is devoted to diffraction phenomena that can be expressed by fractional Fourier transforms whose orders are real numbers. According to a scalar theory, diffraction acts on the amplitude of the electric field as well as on its spherical angular spectrum, and Wigner distributions can be defined on a space-frequency phase space. The phase space is equipped with an Euclidean structure, so that the effects of diffraction are rotations of Wigner distributions associated with optical fields. Such a rotation is shown to split into two specific elliptical rotations. Wigner d…
The electromagnetic field of optical vortices is in most cases derived from vector and scalar potentials using either a procedure based on the Lorenz or the Coulomb gauge. The former procedure has been typically used to derive paraxial solutions with Laguerre-Gauss radial profiles, while the latter procedure has been used to derive full solutions of the wave equation with Bessel radial profiles. We investigate the differences in the derivation procedures applying each one to both Bessel and Laguerre-Gauss profiles. We show that the electromagnetic fields thus derived differ in the relative str…
We experimentally studied intense femtosecond pulse filamentation and propagation in water for Gaussian, Laguerre-Gaussian, and Bessel-Gaussian incident beams. These different transverse modes for incident laser pulses were created from an initial Gaussian beam by using a computer generated hologram technique. We found that the length of the filament induced by the Bessel-Gaussian incident beam was longer than that for the other transverse modes under the conditions of the same peak intensity, pulse duration, and the size of the central part of the beam. To better understand the Bessel-Gaussia…
Orbital angular momentum (OAM) entanglement is investigated in the Bessel-Gauss (BG) basis. Having a readily adjustable radial scale, BG modes provide a more favourable basis for OAM entanglement over Laguerre-Gaussian (LG) modes. The OAM bandwidth in terms of BG modes can be increased by selection of particular radial modes and leads to a flattening of the spectrum. The flattening of the spectrum allows for higher entanglement. We demonstrate increased entanglement in terms of BG modes by performing a Bell-type experiment and violating the appropriate Clauser Horne Shimony Holt (CHSH) inequal…
In this paper, an analytical theory for the diffraction of a Bessel beam of arbitrary order $J_l(κr)$ on a 2D amplitude grating is presented. The diffraction pattern in the main and fractional Talbot planes under certain conditions is a lattice of annular microbeams, the diameters of which depend on the grating period, the illuminating beam diameter, the number of the Talbot plane, and the topological charge $l$. For the rings near the optical axis, the latter reproduces $l$ of the illuminating beam. Experiments carried out on the Novosibirsk free electron laser at a wavelength $λ= 141$$μ$m us…
We show that in addition to well known Bessel, Hermite-Gauss, and Laguerre-Gauss beams of electromagnetic radiation, one may also construct exponential beams. These beams are characterized by a fall-off in the transverse direction described by an exponential function of rho. Exponential beams, like Bessel beams, carry definite angular momentum and are periodic along the direction of propagation, but unlike Bessel beams they have a finite energy per unit beam length. The analysis of these beams is greatly simplified by an extensive use of the Riemann-Silberstein vector and the Whittaker represe…
A concise method of deriving expressions for Gaussian-like solutions of the paraxial and d'Alembert equations is presented. This method is based on the Hankel transform. Choosing some Gaussian base functions with slight modifications of the prefactor all basic beams of cylindrical character can be easily obtained. This refers to Gaussian, Bessel-Gaussian, modified Bessel-Gaussian, Laguerre-Gaussian and Kummer-Gaussian (i.e., Hypergeometric-Gaussian) beams although potentially other beams can come into play as well. For instance a new type of a beam that can be derived in this way is described …
The effect of the Kerr nonlinearity on linear non-diffractive Bessel beams is investigated analytically and numerically using the nonlinear Schrödinger equation. The nonlinearity is shown to primarily affect the central parts of the Bessel beam, giving rise to radial compression or decompression depending on whether the nonlinearity is focusing or defocusing, respectively. The dynamical properties of Gaussian-truncated Bessel beams are also analysed in the presence of a Kerr nonlinearity. It is found that although a condition for width balance in the root-mean-square sense exists, the beam pro…
Visualization of Gaussian Mode Profile in Gigahertz Surface-Acoustic-Wave Resonators
本研究通过传输模式微波阻抗显微镜可视化吉赫兹SAW谐振器中的高斯模式剖面，实现了谐振器线宽提取及故障分析，为优化经典和量子声学应用提供了新路径。
Generalized Bessel beams with two indices
该研究通过设计 Bessel 光束的角谱，提出一类具有两个指数的标量 Helmholtz 方程精确解，并分析其在近轴条件下的结构与传播特性。
Planar Waveguide Circulating Gaussian Beam Resonators on a Silicon Photonic Chip
本研究提出一种硅基平板波导循环高斯光束谐振器设计，采用折叠腔结构与缝隙耦合方式，实现了高Q值与宽带工作，降低了侧壁粗糙度等带来的损耗和模式分裂影响。
We present here a nanosecond laser ablation of Si(111) wafer with diffraction-free (Bessel-$J_0$) beams. First, the Axicon a conical shaped optical element for beam shaping is characterized with a visible and infrared light sources. Then, ablation profiles are obtained with Bessel beams generated for $α=1^{\circ}$ and $α=20^{\circ}$ base angles, then they are characterized via a Scanning Electron, an Atomic Force and an Optical Interferometric Microscopes. Experimental results compared with theoretical predictions obtained by using Bessel functions. Results show that Bessel beams give possibil…
Passive Cross-Basis Mode Transitions Along a Single Freely Propagating Bessel Beam
该研究展示了如何通过单个自由传播的贝塞尔光束中的径向位置与轴向重建距离的一一映射，将静态相位-only空间光调制器的径向孔径转换为可编程的纵向模式寄存器，从而实现离散的横模编码。
Bessel beams (BBs) appear immune to diffraction over finite propagation distances due to the conical nature of light propagation along the optical axis. This offers promising advantages in laser fabrication. However, BBs exhibit a significant intensity variation along the direction of propagation. We present a simple technique to engineer the axial intensity of the BB over centimeters-long propagation distances without expansion of the incoming laser beam. This method uses two diffractive optical elements (DOEs), one converts the input Gaussian intensity profile to an intermediate intensity di…
Speckle patterns are formed by random interferences of mutually coherent beams. While speckles are often considered as an unwanted noise in many areas, they also formed the foundation for the development of numerous speckle-based imaging, holography and sensing technologies. In the recent years, artificial speckle patterns have been generated with spatially incoherent sources using static and dynamic optical modulators for advanced imaging applications. In this report, a fundamental study has been carried out with Bessel distribution as the fundamental building block of the speckle pattern: sp…
Ablation of black-Si by (Gauss-)Bessel femtosecond laser beams
本文比较了高斯与高斯-贝塞尔飞秒激光对黑硅的改性效果，发现贝塞尔光束可在低50倍能量密度下实现更窄改性并获更高深宽比沟槽，且双面织构黑硅近红外透射率超95%。
We propose and experimentally demonstrate a novel experimental scheme to generate high peak power, segmented, smooth, zero-order Bessel beams with tunable range. Illuminating the axicon with hollow Gaussian beams (HGBs) of different orders we have generated Bessel beams of varying range at different positions away from the axicon. The presence of dark core at the center of the HGBs removes the effect of imperfection in the axicon tip. As a result, the entire power of the input beam is transformed into zero-order Bessel beam without any on-axis intensity modulation. We observe the decrease in r…
Non-diffracting chirped Bessel waves in optical antiguides
本文提出光学反波导中近轴波动方程的无衍射啁啾贝塞尔波解，并通过数值模拟证明其有限能量形式可抵抗散焦、弯曲和偏心发射影响，且具有自愈能力。
We demonstrate scattering control of Gaussian-like wave packets propagating with constant envelope velocity and invariant waist through coupled resonator optical waveguides (CROW) via an external resonator coupled to multiple sites of the CROW. We calculate the analytical reflectance and transmittance using standard scattering methods from waveguide quantum electrodynamics and show it is possible to approximate them for an external resonator detuned to the CROW. Our analytical and approximate results are in good agreement with numerical simulations. We engineer various configurations using an …
Bessel beams are becoming a very useful tool in many areas of optics and photonics, because of the invariance of their intensity profile over an extended propagation range. Finite-Difference-Time-Domain (FDTD) approach is widely used for the modeling of the beam interaction with nanostructures. However, the generation of the Bessel beam in this approach is a computationally challenging problem. In this work, we report an approach for the generation of the infinite Bessel beams in three-dimensional FDTD. It is based on the injection of the Bessel solutions of Maxwell's equations from a cylindri…
The motion of an electromagnetic wave, through a classically-forbidden region, has recently attracted renewed interest because of its implication with regard to the theoretical and experimental problems of superluminality. From an experimental point of view, many papers provide an evidence of superluminality in different physical systems. Theoretically, the problem of a passage through a forbidden gap has been treated by considering plane waves at oblique incidence into a plane parallel layer of a medium with a refractive index smaller than the index of the surrounding medium, and also confine…
Current optoacoustic microscopy configurations often have narrow focal ranges that limit their use for fast volumetric imaging applications. Here, the focal range of optoacoustic microscopes is extended by matching the elongated optical illumination profile of a Bessel beam with the pencil beam acoustic sensitivity profile of a broadband axicon detector. An inverted optoacoustic microscope was developed with interchangeable optical illumination and acoustic detection units to assess the imaging depths and resolutions retained with several combinations of illumination and detection profiles. Ma…
Systems of coupled cavity modes have the potential to provide bright quantum optical states of light in a highly versatile manner. Microring resonators for instance are highly scalable candidates for photon sources thanks to CMOS fabrication techniques, their small footprint and the relative ease of coupling many such microrings together, however, surface roughness of the wave-guides, and defects in the coupler geometry routinely induce splitting of the cavity modes due to backscattering and backcoupling. The parasitic back-propagating mode in the microring leads to hybridisation of the modes,…
Stimulated emission depletion (STED) microscopy has become a powerful imaging and localized excitation method beating the diffraction barrier for improved lateral spatial resolution in cellular imaging, lithography, etc. Due to specimen-induced aberrations and scattering distortion, it has been a great challenge for STED to maintain consistent lateral resolution deeply inside the specimens. Here we report on a deep imaging STED microscopy by using Gaussian beam for excitation and hollow Bessel beam for depletion (GB-STED). The proposed scheme shows the improved imaging depth up to ~155μm in so…
We report a theoretical study showing that rogue waves can emerge in whispering gallery mode resonators as the result of the chaotic interplay between Kerr nonlinearity and anomalous group-velocity dispersion. The nonlinear dynamics of the propagation of light in a whispering gallery-mode resonator is investigated using the Lugiato-Lefever equation, and we evidence a range of parameters where rare and extreme events associated with a non-gaussian statistics of the field maxima are observed.
The paper aims at presenting a didactic and self-contained overview of Gauss-Hermite and Gauss-Laguerre laser beam modes. The usual textbook approach for deriving these modes is to solve the Helmoltz electromagnetic wave equation within the paraxial approximation. Here, a different technique is presented: Using the plane wave representation of the fundamental Gaussian mode as seed function, all higher-order beam modes are derived by acting with differential operators on this fundamental solution. Even special beam modes as the recently introduced Bessel beams are easily described within that f…
Shaping the angular spectrum of a Bessel beam to enhance light transfer through dynamic strongly-scattering media
本研究通过空间光调制器制备环形角谱贝塞尔光束，发现横向波矢与分段环方位频率匹配时透射强度最优，且仅用64段即可优于二维网格光束，并支持动态散射体测量。
In this paper, we develop a theoretical analysis to efficiently handle superpositions of waves with concentrated wavevector and frequency spectra, allowing an easy analytical description of fields with interesting transverse profiles. First, we analyze an extension of the paraxial formalism that is more suitable for superposing these types of waves, as it does not rely on the use of coordinate rotations combined with paraxial assumptions. Second, and most importantly, we leverage the obtained results to describe azimuthally symmetric waves composed of superpositions of zero-order Bessel beams …
Recently, a series of theoretical and experimental papers on free-space propagation of pulsed Laguerre-Gaussian and Bessel beams was published, which reached contradictory and controversial results about group velocities of such pulses. Depending on themeasurement scheme, the group velocity can be defined differently.We analyze how different versions of group velocity are related to the measurable travel time (time of flight) of the pulse between input (source) and output (detecting) planes. The analysis is tested on a theoretical model---the Bessel-Gauss pulse whose propagation path exhibits …
We analyse the polarisation properties of full Poincaré beams. We consider different configurations, such as Laguerre-Poincaré, Bessel-Poincaré, and Lambert-Poincaré beams. The former is the original Poincaré beam produced by a collinear superposition of two Laguerre-Gauss beams with orthogonal polarisations. For this configuration, we describe the Stokes statistics and overall invariant parameters. Similarly, Bessel-Poincaré beams are produced by the collinear superposition of Bessel beams with orthogonal polarisations. We describe their properties under propagation and show that they behave …
Operator based propagation of Whittaker and Helmholtz Gauss beams
本文提出一种基于运算符的高效方法，用于传播Whittaker和Helmholtz高斯光束，通过闭合形式表达保留高斯宽度，实现有限能量光束的描述，并验证了其在非衍射结构中的应用效果。
It has been proven that the intense tightly focused Gauss beam (GB) generates pressures in excess of a few TPa creating the novel super-dense phases of Aluminium and silicon [1-5]. Recently it was demonstrated that the Bessel beam (BB) focused inside sapphire produced the cylindrical void being two orders of magnitude larger than that generated by the GB [6-8]. Analysis of the experimental data presented below allows making the remarkable conclusions based solely on the void size measurements without any ad hoc assumptions about the interaction process. First, the void size is direct evidence …
It was noticed in the past that, to avoid physical inconsistencies, in Marcatili's lossless tapers through-flowing waves must be drastically different from standing waves. First, we reconfirm this by means of numerical results based on an extended BPM algorithm. Next, we show that this apparently surprising behavior is a straightforward fallout of Maxwell's equations. Very similar remarks apply to Gaussian beams in a homogeneous medium. As a consequence, Gaussian beams are shown to carry reactive powers, and their active power distributions depart slightly from their standard pictures. Similar…
Quantum limited imaging of a nanomechanical resonator with a spatial mode sorter
本文提出用空间模式解复用器成像纳米机械谐振器，通过HG模式转换读出扭转振动，实现接近量子极限的角位移测量。
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