Partially coherent anomalous hollow beam and its paraxial propagation

Anomalous hollow beam is extended to the partially coherent case. Analytical propagation formulae for a partially coherent anomalous hollow beam passing through a paraxial ABCD optical system are derived. The propagation properties of a partially coherent anomalous hollow beam in free space and the focusing properties of a partially coherent anomalous hollow beam are studied numerically. It is found that the propagation and focusing properties of the partially coherent anomalous hollow beam are closely related to its initial coherence.     

Analytical formulas for a circular or non-circular flat-topped beam propagating through an apertured paraxial optical system

Propagation of a flat-topped beam of circular or non-circular (rectangular or elliptical) symmetry through an apertured optical system is investigated. By expanding the hard aperture function as a finite sum of complex Gaussian functions, some approximate analytical propagation formulas are derived for a flat-topped beam of circular or non-circular (rectangular or elliptical) symmetry propagating through an apertured paraxial general astigmatic (GA) optical system or an apertured paraxial misaligned stigmatic (ST) optical system. The derived formulas are very fast to compute. The results obtained by using the approximate analytical expressions are in a good agreement with those obtained by direct numerical integration. The present analytical formulas provide a convenient and effective way for studying the propagation and transformation of a circular or non-circular flat-topped beam through an apertured general optical system.     

Modified hollow Gaussian beam and its paraxial propagation

A model named modified hollow Gaussian beam (HGB) is proposed to describe a dark hollow beam with adjustable beam spot size, central dark size and darkness factor. In this modified model, both the beam spot size and the central dark size will be convergent to finite constants as the beam order approaches infinity, which are much different from that of the previous unmodified model, where the beam spot size and the central dark size will not be convergent as the beam order approaches infinity. The dependences of the propagation factor of modified and unmodified HGBs on the beam order are found to be the same. Based on the Collins integral, analytical formulas for the modified HGB propagating through aligned and misaligned optical system are derived. Some numerical examples are given.     

Generalized beam propagation factors of truncated partially coherent cosine-Gaussian and cosh-Gaussian beams

Based on the method of truncated second-order moments, the analytical expressions of the generalized beam propagation factor for truncated partially coherent cosine-Gaussian and cosh-Gaussian beams are derived. The beam propagation factors of the truncated partially coherent cosine-Gaussian and cosh-Gaussian beams depend on the acentric parameter, the truncation parameter, and the coherent parameter. According to the derived expressions, the beam propagation factors are illustrated and analyzed with numerical examples, and the influences of the different parameters on the beam propagation factors are also discussed in detail.     

Camera calibration based on paraxial raytracing

Traditional camera calibration is based on the pinhole model, which is an approximation algorithm using untrue geometrical assumptions and giving a single lumped result for the multiple optical elements in a camera. To provide an alternative method of camera calibration, we extend the traditional 2×2 matrix-based paraxial raytracing method to 6×6 in order to trace paraxial rays by using the first-order Taylor series expansion of Snell’s laws. Then we establish the geometric relationship between images and objects. Compared with the Snell’s Law camera calibration model of our previous work, the paraxial model offers explicit analytical sensitivity analysis for the mathematical manipulation of problematical conditions. Compared with the existing pinhole model, the proposed method, in addition to five intrinsic and six extrinsic parameters, gives the position parameters of each optical element of the camera system.     

Propagation characteristics of the rectangular flattened Gaussian beams through circular apertured and misaligned optical systems

Based on the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, the approximate analytical expression for the output field distribution of a rectangular flattened Gaussian beam passing through a circular apertured and misaligned paraxial ABCD system is derived. The result brings more convenient for studying its propagation than the usual way by using diffraction integral directly. Some numerical simulations are also given for illustrating the propagation properties of a rectangular flattened Gaussian beam through a circular apertured and misaligned optical system.     

Generalized beams in ABCDGH systems

For a generalized beam at the source plane passing through co-located aperture and a propagation path consisting of an off-axis x-y asymmetric ABCDGH system, the receiver plane irradiance expression is derived using the Collins integral. A collection of source beam profiles that are obtainable from the generalized beam formulation are illustrated. Plots are given for viewing the progress of selected generalized beam types along the propagation axis, containing a single thin lens, co-centric and misaligned in the x-direction. The received power falling onto a finite aperture surface is calculated for various misalignment situations.     

Geometrical optical modeling of femtosecond setups having angular dispersion

A reliable and efficient method based on a geometrical optical approach is presented to model the propagation of ultrashort pulses in optical systems. It is shown that the commonly used method to determine the group delay of the spectral components of a pulse from their geometrical optical path lengths is only correct for aberration-free optical systems. In the case of systems with angular dispersion and imaging errors, a correction to the path values obtained from ray-tracing calculations must be applied, since for specific systems neglect of the correction causes significant errors. A technique for performing this correction is presented. Two optical arrangements used for the generation and detection of tunable THz radiation by the femtosecond tilted-pulse-front excitation technique are analyzed to demonstrate the indispensability of the correction. Received: 29 July 2002 / Revised version: 22 October 2002 / Published online: 29 January 2003 RID="*" ID="*"Corresponding author. Fax: +36-72/501-571, E-mail: kozma@fizika.ttk.pte.hu     

Performance of a zoom homogenizer for reshaping coherent laser beams

We have reshaped the TEM₀₀ beam emitted by a diode-pumped Nd-Yag laser by means of an optical homogenizer with zoom. The laser beam, first enlarged up to a size of (100 × 37.5) mm², is homogenized and resized to a final dimension continuously adjustable from (130 × 4.5) mm² up to (130 × 52) mm². We measured the plateau uniformity, the root mean square fluctuation and the edge steepness of the beam according to the ISO standard definitions, showing a poor reliability of the above parameter values due to the definitions themselves, and propose an amendment to overcome this problem. We obtained a very sharp edge steepness, but the spatial coherence of the laser beam put a lower limit to the high-frequency intensity fluctuations on the plateau region of the homogenized beam. Finally, we discuss the optimum homogenizer design for spatially coherent laser beams, including the depth of focus issue.     

Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere

The propagation of an off-axis Gaussian Schell-model (GSM) beam in a turbulent atmosphere is investigated based on the extended Huygens-Fresnel integral formula. Analytical formulae for the cross-spectral density and corresponding partially coherent complex curvature tensor of an off-axis GSM beam propagating in a turbulent atmosphere are derived. Based on these formulae, the propagation properties of such kind of beam in a turbulent atmosphere are investigated in detail. Furthermore, the methods are extended to investigate the propagation properties of a partially coherent laser array beam in a turbulent atmosphere. The properties of an off-axis GSM beam and a partially coherent laser array beam in a turbulent atmosphere are closely related with the beam parameters and the structure constant of the turbulent atmosphere.     

Study on preparation and fluorescence characteristic of coordinated Eu2O3/polymer hybrid films

The cyclohexane solution of TTA (trifluorothenoyl-acetone), phen (8-hydroxylquinoline) and PS (polystyrene), the ethyl acetate solution of TTA, phen and PMMA (polymethyl methacrylate) were used as flowing liquid, the coordinated Eu₂O₃/polymer hybrid colloids were successively produced by focused pulsed laser ablation of Eu₂O₃ target in interface of solid and flowing liquid. As solvent in the hybrid colloids has volatilized, the coordinated Eu₂O₃/polymer hybrid films were obtained. The hybrid colloids and films were characterized by TEM, UV-vis spectrum, fluorescence spectrum, TG-FTIR and X-ray photoelectron spectrum. The results show the coordinated Eu₂O₃ nanoparticles with average size of less than 20 nm are surrounded by the three-dimensional network and are properly incorporated into the PMMA and PS matrix, the hybrid films can emit intense red light under ultraviolet radiation, and their emission fluorescence spectra display same characteristic emission peaks of Eu³⁺ ions. The Eu₂O₃ hybrid films have better thermo stability than the related pure polymers because of strong interaction between surface europium ions of the nanoparticles and polymer. Because the coordinated Eu₂O₃ nanoparticles were wrapped by polymer, they have higher chemical stability than the related europium complex.     

Modelling of angular effects in nonlinear optical processes

Two different methods for studying transverse effects in nonlinear optical interactions are compared, and their strengths and weaknesses discussed. The simpler split-step approach is found to be less restricted in its capabilities and computationally more efficient. A number of interesting results obtained using this technique are presented.     

Successive preparation of decorated zinc oxide organic sol by pulsed laser ablation and their luminescence characteristics

Nano-ZnO organic sols which were modified in situ were successively produced through focused pulsed laser ablation of ZnO target in interface of solid and flowing liquid which contained modification agents or polymer. It is found that the ZnO ethanol sol decorated by Q (8-hydroxylquinoline) radiates intense green light under ultraviolet radiation and has a broad emission band centered at 555 nm in the emission fluorescence spectrum. The influence of factors including different modification agents and their added methods, laser fluence, aging time after preparation, compositions of flowing liquid and their velocity on luminescence characteristics of nano-ZnO organic sol was characterized by TEM, UV-vis and fluorescence spectrum.     

Optimization of phase objects in 4f coherent imaging system for nonlinear refraction measurements

We numerically analyze the effects of radius and phase shift of phase objects on the diffraction image of the 4f coherent imaging system, a system used for measuring the third-order nonlinear refractive index. The selection of the aperture radius is discussed. We prove that when the phase object radius is 0.1 time of the aperture radius and the phase change of the phase object is 0.57π, one can get the highest sensitivity for nonlinear refraction measurement.     

Equiangular-spiral bent lightpipes with arbitrary bent angle

A recent study proposed the concept of a leakage-free bent lightpipe with an equiangular-spiral shape [S.-C. Chu, J.-L. Chern, Opt. Lett. 30 (2005) 3006]. This paper extends the design of a leakage-free equiangular-spiral bent lightpipe with arbitrary-bend-angle and addresses in detail the mathematical formalism required to build such an equiangular-spiral bent lightpipe. Furthermore, a comparison of the proposed equiangular-spiral bent lightpipe and a conventional circular bent lightpipe is provided. Numerical verifications are discussed, and experimental explorations with different bent shapes and angles are carried out for comparison. Results show that equiangular-spiral bent lightpipes with different bent angles exhibit a theoretical 100% transmission with more than 76% efficiency when practically propagating, which is much better than conventional circular bent lightpipes. The output irradiance distributions of bent lightpipes with different bent shapes are also investigated.     

Low order adaptive optics on Z-Beamlet using a single actuator deformable mirror

The Z-Beamlet laser at Sandia National Laboratories can perform a full system shot every 3-4 h. This shot rate is limited by thermal aberrations that result from the flashlamp pumped Nd:phosphate amplifier slabs. The lowest order as well as the strongest aberration is of cylindrical shape. Therefore, a single actuator deformable mirror assembly for correction of cylindrical aberration was developed. Mirror performance was modeled using finite element analysis and showed good agreement with derived analytical expressions. Quantitative measurements were performed with an interferometer and thermal lens compensation was achieved in the Z-Beamlet laser system leading to an increased shot rate of one in every 2 h.     

Analytical vectorial structure of non-paraxial nonsymmetrical vector Gaussian beam in the far field

Based on the vectorial structure of non-paraxial electromagnetic beam and the method of stationary phase, the analytical TE and TM terms of non-paraxial nonsymmetrical vector Gaussian beam in the far field are presented. According to the analytical electromagnetic fields of the TE and TM terms, the energy flux distributions of the whole beam, its TE and TM terms are investigated. Moreover, the influences of non-symmetries on the energy flux distributions of the whole beam, its TE and TM terms are also analyzed, respectively. This research reveals the internal vectorial structure of non-paraxial laser beam in the far field and is useful to the propagation of non-paraxial laser beam.     

Influence of Misalignment on High-Order Aberration Correction for Normal Human Eyes

Although a compensation device can correct aberrations of human eyes, the effect will be degraded by its misalignment, especially for high-order aberration correction. We calculate the positioning tolerance of correction device for high-order aberrations, and within what degree the correcting effect is better than low-order aberration （defocus and astigmatism） correction. With fixed certain misalignment within the positioning tolerance, we calculate the residual wavefront rms aberration of the first-6 to first-35 terms along with the 3rd-Sth terms of aberrations corrected, and the combined first-13 terms of aberrations are also studied under the same quantity of misalignment. However, the correction effect of high-order aberrations does not meliorate along with the increase of the high-order terms under some misalignment, moreover, some simple combined terms correction can achieve similar result as complex combinations. These results suggest that it is unnecessary to correct too much the terms of high-order aberrations which are difficult to accomplish in practice, and gives confidence to correct high-order aberrations out of the laboratory.     

Propagation properties of anomalous hollow beams in a turbulent atmosphere

Propagation of circular and elliptical anomalous hollow beams in a turbulent atmosphere is investigated in detail. Based on the extended Huygens–Fresnel integral, analytical formulae for the average irradiance of circular and elliptical anomalous hollow beams propagating in a turbulent atmosphere are derived. The irradiance and spreading properties of circular and elliptical anomalous hollow beams in a turbulent atmosphere and in free space are studied numerically. It is found that circular and elliptical anomalous hollow beams at short propagation distance in turbulent atmosphere have similar propagation properties to those of free space, while at long propagation distance, circular and elliptical anomalous hollow beams eventually become circular Gaussian beams in a turbulent atmosphere, which is much different from their propagation properties in free space. The conversion from an anomalous hollow beam to a circular Gaussian beam becomes quicker and the beam spot spreads more rapidly for a larger structure constant, a shorter wavelength and a smaller waist size of the initial beam.