Focusing and phase compensation of paraxial beams by a left-handed material slab

On the basis of angular spectrum representation, a formalism describing paraxial beams propagating through an isotropic left-handed material (LHM) slab is presented. The treatment allows us to introduce the ideas of beam focusing and phase compensation by LHM slab. Because of the negative refractive index of LHM slab, the inverse Gouy phase shift and the negative Rayleigh length of paraxial Gaussian beam are proposed. It is shown that the phase difference caused by the Gouy phase shift in right-handed material (RHM) can be compensated by that caused by the inverse Gouy phase shift in LHM. If certain matching conditions are satisfied, the intensity and phase distributions at object plane can be completely reconstructed at the image plane.     

左手材料平板对傍轴高斯光束聚焦特性分析

在傍轴近似下,推导出了左手材料平板聚焦系统的传输矩阵,并利用ABCD定律得到了高斯光束在左手材料中和经过平板透镜聚焦后的传输公式。高斯光束在左手材料内部和像空间的传输公式的研究表明:像高斯光束和物高斯光束束腰大小一致,即左手材料平板透镜实际上对高斯光束没有聚焦作用。研究同时表明左手材料平板对高斯光束的聚焦与几何光学成像规律完全一致,而不存在一般透镜聚焦时的焦移效应。     

Propagation of Gaussian beams family through a Kerr-type left-handed metamaterial

In this paper the propagation of elegant Hermite-cosh-Gaussian, elegant Laguerre Gaussian, and Bessel Gaussian beams through a Kerr left-handed metamaterial(LHM) slab have been studied. A split-step Fourier method is used to investigate the propagation of laser beams through this media. Numerical simulation shows that Gaussian beams have different focusing behaviors in a Kerr LHM slab with positive or negative nonlinearity. Beam focusing happens in slabs with positive nonlinearity and not in slabs with negative nonlinearity; however, negative nonlinearity is required for a Kerr LHM slab to act like a lens. Additionally, the focusing properties of beams can be controlled by controlling the thickness of the slab or the input power of the incident beam. A multilayer structure is also proposed to have beam focusing by thinner slabs and passing longer distances.     

Propagation of Airy beams from right-handed material to left-handed material

Based on the ABCD matrix formalism,the propagation property of an Airy beam from right-handed material(RHM) to left-handed material(LHM) is investigated.The result shows that when the Airy beam propagates in the LHM,the intensity self-bending due to its propagation in the RHM can be compensated.In particular,if the propagation distance in the RHM is equal to that in the LHM and the refractive index of the LHM is n L =-1,the transverse intensity distribution of the Airy beam can return to its original state.     

Partially coherent matter wave and its evolution

The evolution and propagation of a partially coherent matter wave (PCMW) is investigated theoretically by the correlation function method. The ABCD matrix formalism previously used for a fully coherent matter wave is extended to make it applicable for the PCMW. A new tensor ABCD law is derived, which makes the propagation and evolution of a PCMW very simple and clear. As an example, the evolution of the coherence of a PCMW in a gravitational field is calculated.     

梯度负折射率介质中高斯光束传输特性的研究

导出了高斯光束在梯度负折射率介质中的ABCD矩阵,据此得到光束在此介质中的传输模型.并利用此模型分析了高斯光束在梯度负折射率介质中的传输特性,发现它能产生空间孤子及呼吸子形式的传输,并发现光束的束腰半径不一定是最小束宽半径.还研究了梯度系数对介质聚焦能力的影响,据此可以设计出相应聚焦能力所需要的折射率分布.最后分析了传输时高斯光束曲率半径的变化情况,与光束束宽半径的变化显著不同,曲率半径始终从无穷大开始,然后产生一个个周期性的变换.     

Generalized tensor ABCD law for an elliptical Gaussian beam passing through an astigmatic optical system in turbulent atmosphere

The propagation of an elliptical Gaussian beam (EGB) through an astigmatic ABCD optical system in a turbulent atmosphere is investigated. An analytical formula for the average intensity of an EGB and a generalized tensor ABCD law for the generalized complex curvature tensor are derived. As an application example, we derived an analytical formula for the average intensity of an elliptical flat-topped beam propagating through an astigmatic ABCD optical system in a turbulent atmosphere. As a numerical example, the focusing properties of an EGB focused by a thin lens in a turbulent atmosphere are studied. It is found that the focused beam at the focal plane becomes a circular Gaussian beam when the atmospheric turbulence is strong enough, and the beam width of the circular Gaussian beam is determined by atmospheric turbulence strength, focal length of the thin lens, and wavelength of the initial beam but is independent of the initial beam widths (i.e., initial intensity distribution).     

NOVEL ANALYTIC THEORY FOR ACTIVELY MODE-LOCKED FIBER LASER

The time-domain ABCD matrix formalism is based on the propagation of a Gaussian pulse which is characterized by the pulsewidth and chirp. In this paper, the time-domain ABCD matrix of th SPM (self phase modulation) is perfected so as to acquire more information. With the time-domain ABCD matrices of amplitude modulation, filtration from the optical bandpass, group velocity dispersion and SPM, the detailed characteristics of actively mode-locked fiber laser are described and the steady-state solutions are derived for two simple cases. The application of the formalism is also helpful to understand the mechanism of mode-locking in an actively mode-locked fiber laser.     

Gaussian wave formalism model for propagation of charged-particle beam through a first-order optical systems

An elliptical Gaussian wave formalism model of a charged-particle beam is proposed by analogy with an elliptical Gaussian light beam. In the paraxial approximation, the charged-particle beam can be described as a whole by a complex radius of curvature in the real space domains. Therefore, the propagation and transform of charged-particle beam passing through a first-order optical system is represented by the \textit{ABCD}-like law.As an example of the application of this model, the relation between the beam waist and the minimum beam spot at a fixed target is discussed. The result, well matches that from conventional phase space model, and proves that the Gaussian wave formalism model is highly effective and reasonable.     

Modeling of astigmatic-elliptical beam shaping during fs-laser waveguide writing including beam truncation and diffraction effects

In this work, we report a model for accurately calculating the focal volumes corresponding to astigmatic elliptical beams used in fs-laser waveguide writing. The model is based on the use of the ABCD matrix formalism for the propagation of a Gaussian beam. The code includes the effects of propagation on the astigmatic elliptical beam, and the effects of beam truncation and diffraction at the entrance pupil of the focusing objective due to beam clipping when overfilling the pupil. The results predict that for a given astigmatism value and propagation distance it is possible to efficiently suppress the astigmatic focus closer to the surface. This explains previous experimental results where single structure waveguides with controllable aspect-ratio were fabricated using astigmatic-elliptical beams. Furthermore, we investigate the respective roles of astigmatism and beam propagation, as well as the strong impact of truncation and diffraction effects caused by clipping the beam at the pupil of the focusing optics. Finally, based on the results from our model, we present some practical considerations in terms of beam propagation and phase wrapping constraints.     

Simple ABCD matrix treatment for transversely varying saturable gain

We have developed an ABCD matrix that, for the first time to our knowledge, accurately describes the transformation of a Gaussian beam by a medium with transversely varying saturable gain. In contrast with the conventional ABCD matrix, the newly developed matrix is shown to be in excellent agreement with a full beam propagation code over a wide parameter range. Accurate treatment of transversely varying saturable gain in laser resonators is important for the optimization of end-pumped lasers, particularly for efficient diode-pumped solid-state and Kerr-lens mode-locked systems.     

Minimization of scintillation index against displacement parameters

For sinusoidal beams, minimization of scintillation index is carried out against the displacements parameters. It is found that x-y asymmetric cosh-Gaussian beam fulfills the requirements of such optimum beam. Our minimization procedure reveals that the optimum beam is achieved by continually focusing it at the chosen propagation length and by further adjusting displacements parameters to be propagation distance dependent. Scintillation index of thus constructed optimum beam is formulated and numerically evaluated. Our graphical comparisons entailing collimated and focused versions of cos-, cosh-Gaussian, annular-Gaussian and Gaussian beams show that the optimum beam yields the lowest scintillations provided that propagation range is less than or equal to the focusing distance.     

Propagation of Gaussian beams in negative-index metamaterials with cubic nonlinearity

Propagation of Gaussian beams in the negative-index metamaterials (NIMs) with cubic nonlinearities is investigated, both theoretically and numerically. The role of the status of the incident Gaussian beam, which is scaled by a converging parameter in this paper, in beam self-focusing and self-defocusing in NIMs is specially identified. The expressions for beam self-focusing and self-defocusing for different converging parameter cases, and the dependence of the critical power and the focus location of self-focusing in NIMs on the converging parameter are obtained. It is found that it is the divergent rather than convergent incident beams which are self-focused more quickly in NIMs with defocusing nonlinearities, in sharp contrast with the propagation property of Gaussian beams in conventional Kerr media, in which beam self-focusing only occurs in the media with focusing nonlinearities and a convergent incident beam self-focuses more quickly than a divergent one. By adjusting the converging parameter of incident Gaussian beam or the controllable magnetic permeability of NIM, or both, one can manipulate the beam self-focusing in NIMs at will.     

Paraxial propagation of a partially coherent flattened Gaussian beam through apertured ABCD optical systems

By expanding the hard aperture function into a finite sum of complex Gaussian functions, some approximate analytical formulae for the cross-spectral density of a partially coherent flattened Gaussian beam (FGB) propagating through apertured aligned and misaligned ABCD optical systems are derived based on the generalized Collins formula. The results obtained by using the approximate analytical formula are in good agreement with those obtained by using the numerical integral calculation. As a numerical example, the focusing properties (including average irradiance distribution and focal shift) of a partially coherent FGB focused by an apertured thin lens are studied, and it is found that the focusing properties of a partially coherent FGB are closely related to its initial coherence and the aperture width. Our results provide an effective and fast way for studying the paraxial propagation of a partially coherent FGB through apertured ABCD optical systems.     

Novel Analytic Theory for Laser Mode-Locking

The time-domain ABCD matrix formalism is based on the propagation of a Gaussian pulse which is characterized by the pulsewidth and chirp. In this paper, we will perfect this method and derive time-domain ABCD matrices for more optical devices. Especially, we extend this perfected theory to analyze the principle of passive mode-locking by reasonable approximation though, as we know, the output pulse generated from passive mode-locking fiber laser is of sech-shape. It is attractive because it can present a fairly apparent view to understand the mechanism of fiber lasers as well as analytic results to understand the pulse characteristics.     

Waveguide lenses with multimode interference for low-loss slab propagation

It is shown that a nearly ideal two-dimensional focusing Gaussian beam can be synthesized by use of a linear combination of the two lowest-order even modes of an optical waveguide. This property can be used to couple laterally guided modes across slab waveguide regions with low loss. The technique is illustrated by use of a conventional multimode interference (MMI) geometry, in which the MMI coupler transforms the fundamental mode of an initial waveguide into a focusing Gaussian beam, which is then fed to a slab region. Two-dimensional beam propagation simulations show that the beam does not initially diverge in the slab region, but rather comes to a focus. A second MMI coupler then transforms the diverging beam back to the initial mode. A structure is designed that can couple the fundamental mode of a 9-microm-wide waveguide across an 88-microm-long slab region with only a 0.036-dB loss. This technique can be applied to improve the performance of small-angle waveguide crossings and integrated turning mirrors.     

Focusing properties of a general-type beam in turbulent atmosphere

Based on the generalized Huygens–Fresnel integral, analytical propagation formulas for a general-type beam propagating through aligned or misaligned ABCD optical systems in turbulent atmosphere are derived. The derived formulas provide a convenient way for studying the focusing properties of a variety of laser beams, such as Gaussian, cos-Gaussian, cosh-Gaussian, sine-Gaussian, sinh-Gaussian, flat-topped, Hermite-cosh-Gaussian, Hermite-sine-Gaussian, higher-order annular Gaussian, Hermite-sinh-Gaussian and Hermite-cos-Gaussian beams in turbulent atmosphere. As an application example, the focused intensities of cos-Gaussian, Hermite-sine-Gaussian and flat-topped beams in turbulent atmosphere are studied numerically. Focal shift of a flat-topped beam in turbulent atmosphere is investigated. Effect of the misalignment of the thin lens on the focusing properties of a cos-Gaussian beam is also explored. Our results will be useful for the applications of the general-type beam in LIDAR systems and remote sensing operating in turbulent atmosphere, where optical elements such as aligned or misaligned thin lens are commonly encountered.     

Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams

An active bistatic LIDAR system operating through atmospheric turbulence is considered. Illumination field is assumed to be an electromagnetic Gaussian-Schell model beam. Target surface is modeled as a combination of isotropic phase screen governed by Gaussian statistics, to account for its roughness, and a Gaussian lens to account for its size and radius of curvature. With the help of a recently developed tensor method for propagation of stochastic electromagnetic beams through ABCD systems and random media we examine the evolution of states of coherence and polarization of the beam. In the case of unresolved flat (planar) target we show that by comparing coherence and polarization properties of the illumination beam and of the return beam it is possible to predict the typical roughness of the target surface.     

Gaussian laser beam tailoring using acoustooptic cell

Profile shaping of a Gaussian laser beam by an acoustic wave is well described using Collins integral and ABCD matrix formalism. It is shown by a numerical simulation that the relative width of the laser beam to the ultrasonic wavelength and the acoustic pressure inside the acoustooptic cell act on the light intensity diffraction pattern.Obtained results show that the output intensity profile differs from the incident Gaussian beam shape, and it is more broadened with an increase in the acoustic pressure. The intensity of a focused laser beam is transformed in a flat form in the central region if the acoustic pressure is proprely controlled.On the other hand the intensity longitudinal range (ILR) of the flat shape is discussed along the propagation axes, we have found the ILR is about 2 mm for a focal length distance f=100 mm.     

Propagation of Single-Mode Fibre Laser Beams through an Optical ABCD System with Circular Aperture at the Fibre Output End

Based on the expansion expression of the fundamental mode of a single-mode fibre in terms of Laguerr-Gauss modes, the propagation of a beam of a weakly guiding fibre laser through an opticM ABCD system with a circular aperture at the fibre end is studied. The results show that there is much difference between the propagation of the laser beam described by the expansion expression and by the Gaussian mode approximation. The depth of focus of the laser beam is longer than that of the Gaussian modes.