Two-dimensional simulation of high-order laterally-coupled GaAs-AlGaAs DFB laser diodes

Laterally-coupled distributed feedback (LC-DFB) laser diodes made without an epitaxial re-growth process have the advantage of a simple fabrication process. In this paper, two-dimensional optical field distribution of the fundamental quasi TE (transverse electric) mode is calculated by means of a semivectorial finite-difference method (SV-FDM). The dependence of the effective coupling coefficient (kappa(eff)) on the dutycycle of first-, second- and third-order LC-DFB LDs is investigated using modified coupled wave equations.

A multi-moment finite volume formulation for shallow water equations on unstructured mesh

A novel and accurate finite volume method has been presented to solve the shallow water equations on unstructured grid in plane geometry. In addition to the volume integrated average (VIA moment) for each mesh cell, the point values (PV moment) defined on cell boundary are also treated as the model variables. The volume integrated average is updated via a finite volume formulation, and thus is numerically conserved, while the point value is computed by a point-wise Riemann solver. The cell-wise local interpolation reconstruction is built based on both the VIA and the PV moments, which results in a scheme of almost third order accuracy. Efforts have also been made to formulate the source term of the bottom topography in a way to balance the numerical flux function to satisfy the so-called C-property. The proposed numerical model is validated by numerical tests in comparison with other methods reported in the literature. (C) 2010 Elsevier Inc. All rights reserved.

Statistical distribution of wave-surface elevation for second-order random directional ocean waves in finite water depth

Based on the second-order random wave solutions of water wave equations in finite water depth, a statistical distribution of the wave-surface elevation is derived by using the characteristic function expansion method. It is found that the distribution, after normalization of the wave-surface elevation, depends only on two parameters. One parameter describes the small mean bias of the surface produced by the second-order wave-wave interactions. Another one is approximately proportional to the skewness of the distribution. Both of these two parameters can be determined by the water depth and the wave-number spectrum of ocean waves. As an illustrative example, we consider a fully developed wind-generated sea and the parameters are calculated for various wind speeds and water depths by using Donelan and Pierson spectrum. It is also found that, for deep water, the dimensionless distribution reduces to the third-order Gram-Charlier series obtained by Longuet-Higgins [J. Fluid Mech. 17 (1963) 459]. The newly proposed distribution is compared with the data of Bitner [Appl. Ocean Res. 2 (1980) 63], Gaussian distribution and the fourth-order Gram-Charlier series, and found our distribution gives a more reasonable fit to the data. (C) 2002 Elsevier Science B.V. All rights reserved.

准噶尔盆地西北缘三叠系层序地层及有利勘探目标选择

Based on outcrop, borehole, seismic and regional geological data, the sequence stratigraphy, sedimentary facies of the Triassic in the western margin of the Zhugaer basin was studied, and favorable exploration target was forecasted. The major achievements include: (1) the Triassic in the western margin of the Zhugaer basin can be divided into 1 second-order sequence and 5 third-order sequences, which are, in ascending order, TSQ1, TSQ2, TSQ3, TSQ4, and TSQ5. TSQ1 is equivolent to Baikouquan formation, TSQ2 is equivolent to lower Kelamayi formation, TSQ3 is equivolent to upper Kelamai formation, TSQ4 is equivolent to lower and middle Baijiantan formation, and TSQ5 is equivolent to upper Baijiantan formation. Each sequence is divided into transgressive and regressive system tracts. Thus the sequence correlation framework is established. (2) The factors controlling development of sequences are analyzed, and it is believed that tectonic is the major controlling factor. Model of sequence development is summarized. (3)Through study on sedimentary facies, 6 types of facies are recognized: alluvial fan, fan delta, braided river, braided delta, delta and lake. Their microfacies are also recognized. In this study, it is proposed that the upper and lower Kelamayi formation（TSQ2、 TSQ3）is deposited by braided river instead of alluvial fan. This conclusion is of important theoretical and practical significance.(4) The sedimentary facies map of each sequence is compiled, and the sedimentary facies developed in each sequence is determined. In TSQ1, the sedimentary facies developed is alluvial fan and fan delta. In TSQ2, the sedimentary facies developed is mainly alluvial fan and fan delta in the north, and braided river and braided delta in the south. In TSQ3, the sedimentary facies developed is mainly braided river and braided delta. In TSQ4, the sedimentary facies developed is mainly braided delta in the north, and meandering delta in the south. In TSQ5, the sedimentary facies developed is mainly braided river and braided delta. (5) In the framework of sequence stratigrahpy, favorable areas for concealed traps are forecasted, and different types of traps are developed in different system tracts. (6) Favorable areas for future exploration are predicted.

砂砾岩油藏精细油藏描述–以准噶尔盆地西北缘八区下乌尔禾组油藏为例

Glutenite reservoir is one of the most important reservoir types in china. Because of its particularity of rock structure and pore structure, it is usually difficult in development, especially for its serious heterogeneity. On the basis of seismic, well logs, core data and production performance, the lower Wuerhe group can be divided into one second-order sequences, two third-order sequences and twenty two subsequences, corresponding to the five stages and twenty two minlayers. In addition, the fault systems are interpreted and the control action of fault systems to reservoir development is also described. The lower Wuerhe formation of 8th district belongs to fluvial-dominated fan delta sedimentation, according to the analysis of well logs, logging data and core data. It can be subdivided into two kinds of subfacies and nine kinds of microfacies. The fan delta plain subfacies mainly consist of braided channel, unconcentrated flow, mud flow and sieve deposit microfacies. The fan delta front subfacies include subaqueous distributary channel, subaqueous interdistributary channel, debris flow, subaqueous barrier and grain flow microfacies. Combined with the regional geological characteristics, the porosity model of lower Wuerhe formation is performed using core data. A permeability model based on the flow zone index is also formed according to the pore throat characteristics and flow property. Finally, the heterogeneity is analyzed. The result shows that the lower Wuerhe formation has a feature of middle-high heterogeneity, and it is controlled by material sources and sedimentary facies belt.

柴西南区下干柴沟组下段层序地层格架内沉积相展布及有利储集砂体分布研究

Lower member of the lower Ganchaigou Formation in the southwestern of Qaidam Basin is one of the main targeted exploration zones. With the advancement of exploration, the targets are gradually switching into the lithologic reservoirs and it is urgent to gain the more precise research results in distribution of sedimentary facies and sandstones. Guided by the theory of sequence stratigraphy and sedimentology as well as on the basis of many logging data, drillings, seismic data and chemical tests, the paper comprehensively analyzes the sedimentary facies and sandstones in the lower member of lower Ganchaigou Formation in the southern of Chaixi. According to the identification marks of the key interface in sequence stratigraphy, the key interfaces in lower member of lower Ganchaigou Formation in the southwestern of Qaidam Basin are identified as two third-order sequences SQ1、SQ2. By calibrating the synthetic seismogram, the seismic sequence, well drilling and logging sequences are united. Based on the works above, this paper chooses seven primary cross-sections and builds connecting-well stratigraphic correlation of seven main connecting-well sections. Ultimately, the high-resolution sequence stratigraphic frameworks in the lower member of the lower Ganchaigou Formation, which are uniform to logging and seismic data, are figured out. In terms of study on each sequence features, the main style of the base-level cycle overlay which forms the third-order sequence is confirmed. It contains asymmetric “becoming deep upward” style and symmetry style. Researching on the spreading characters of sequence stratigraphy indicates that SQ1 and SQ2 are rather thicker near northwest well Shashen 20 and Shaxin1 while they are quite thiner near Hongcan 1, Yuejin, Qie 4 and Dong8-Wu3, and the thickness of SQ1 is thicker than SQ2.Based on the deep analysis of the marks for depositional facies, it is proposed that the lake facies and braid river deltas facies mainly occurred in study areas. Besides, the sorts of sub-facies and micro-facies model are divided and described. Under the control of high-resolution sequence stratigraphic framework, three source directions from Arlarer Mountain、Qimantage Mountain and Dongchai Mountain are identified by using the features of heavy mineral assemblage and paleogeomorphy. In addition, regularities of distribution sedimentary facies in sequence stratigraphic framework are studied in accordance with research thinking of the "point" (single well) "line" (section) "face" (plane). In the stage of lower member in the lower Ganchaigou Formation in the southwestern of Qaidam Basin, it is at the early phrase of evolution of the lake basin with the gradual outspread and the rise of the lake level. Combined with physical analysis of reservoir sands formed in different sedimentary environment, the paper studies the style of favorable sandstone bodies that are underwater distributary channel of braided rive delta front, coarse sand in mouth bar and the sand body in sand flat of shore-shallow lacustrine facies. Finally, this article comprehensively analyzes the distribution relationship between sedimentary facies and favorable sandstone body and proposes the ideas that sequence SQ1 Yuejin area, well east 8-wu3 area, well qie4-qie1 area and well hongcan2 area are distributed areas of favorable sandstone.

断层封闭性及其与油气成藏的关系研究-以沾化凹陷孤岛地区为例

Godao area, located in the east of the Zhanhua depression of Jirang sag in Bohai Bay Basin, is the studying area in my dissertation. It is first time that fault sealing properties and the related relationship with the pool forming are studied in Gudao area. On the base of the analysis of the regional tectonics, the author has studied the tertiary structural evolution of the Gudao area and distinguished the fault's level and put forward the distinguishing principle. The geometrical feature, mechanical characters, developmental mechanism and history of the boundary faults in the tectonic unit of this area are all studied and emphasized especially. The buried history of oil-generating depression (that is Gudao depression) and the history of oil and gas migration simultaneously are discussed, the juxtaposition relationship between boundary fault evolutionary history and oil and gas migrated history are expatiated. To the geological condition of the Gudao area, three level faults sealing properties of this area were discussed in detail. Their characteristics of behavior and the intrinsic relationship between their sealing and oil and gas migrated reservoir are elucidated. The pool-forming models related to fault seal are exposed. The author has studied the lithologies of different order of faults, the relationship of occurrence assemblage analysis, normal stress of fault plane in different depth and shale smear factor faults. Then analysis their role in the various faults sealing and confirms fault sealing marks of three different orders faults and exposes the mechanism of fault sealing. Shale smear zone formed by first order fault in lasting activities is one different type of fault breccia and mainly controlled factor to its entrapment of petroleum. Effective sealing threshold value and fault displacement is ascertained. Mainly controlled factor of second fault sealing is bigger compressive stress loaded on fault plane. According to this, quantitatively evaluated index is given. Shale smear zone is necessary condition for second fault stress entrapment. The juxtaposed relationship between the different lithologies within third order faults is most important controlled factor for its sealing. Based on various order of fault sealing features and mechanism in Gudao area, the author proposed three orders of fault sealing models. Shale smear zone sealing model, normal stress sealing model and lithologies juxtaposed sealing model are suggested to first, second and third order fault respectively. The conclusion of this studying has not only the very important theoretical significance and practical value in Gudo area but also the very important guiding role for other areas of related aspects.

孤北洼陷沙三-沙四段隐蔽油藏形成机制及其预测

Based on the study of sequence stratigraphy, modern sedimentary, basin analysis, and petroleum system in Gubei depression, this paper builds high resolution sequence stratigraphic structure, sedimentary system, sandbody distribution, the effect of tectonic in sequence and sedimentary system evolution and model of tectonic-lithofacies. The pool formation mechanism of subtle trap is developed. There are some conclusions and views as follows. 1.With the synthetic sequence analysis of drilling, seismic, and well log, the highly resolution sequence structure is build in Gubei depression. They are divided two secondary sequences and seven three-order sequences in Shahejie formation. They are include 4 kinds of system traces and 7 kinds of sedimentary systems which are alluvial fan, under water fan, alluvial fan and fan-delta, fan-delta, lacustrine-fan, fluvial-delta-turbidite, lakeshore beach and bar, and deep lake system. Sandbody distribution is show base on third order sequence. 2.Based on a lot of experiment and well log, it is point out that there are many types of pore in reservoir with the styles of corrosion pore, weak cementing, matrix cementing, impure filling, and 7 kinds of diagenetic facies. These reservoirs are evaluated by lateral and profile characteristics of diagenetic facies and reservoir properties. 3.The effect of simultaneous faulting on sediment process is analyzed from abrupt slope, gentle slope, and hollow zone. The 4 kinds of tectonic lithofacies models are developed in several periods in Gubei depression; the regional distribution of subtle trap is predicted by hydro accumulation characteristics of different tectonic lithofacies. 4.There are 4 types of compacting process, which are normal compaction, abnormal high pressure, abnormal low pressure and complex abnormal pressure. The domain type is normal compaction that locates any area of depression, but normal high pressure is located only deep hollow zone (depth more than 3000m), abnormal low pressures are located gentle slope and faulted abrupt slope (depth between 1200~2500m). 5.Two types dynamic systems of pool formation (enclosed and partly enclosed system) are recognized. They are composed by which source rocks are from Es3 and Es4, cap rocks are deep lacustrine shale of Esl and Es3, and sandstone reservoirs are 7 kinds of sedimentary system in Es3 and Es4. According to theory of petroleum system, two petroleum systems are divided in Es3 and Es4 of Gubei depression, which are high or normal pressure self-source system and normal or low pressure external-source system. 6.There are 3 kinds of combination model of pool formation, the first is litholgical pool of inner depression (high or normal pressure self-source type), the second is fault block or fault nose pool in marginal of depression (normal type), the third is fault block-lithological pool of central low lifted block (high or normal pressure type). The lithological pool is located central of depression, other pool are located gentle or abrupt slope that are controlled by lithological, faulting, unconfirmed. 7.This paper raise a new technique and process of exploration subtle trap which include geological modeling, coring description and logging recognition, and well log constrained inversion. These are composed to method and theory of predicting subtle trap. Application these methods and techniques, 6 hydro objects are predicted in three zone of depression.

东营凹陷成藏组合体研究与勘探实践

With the development of petroleum exploration, subtle reservoir has become the main exploration object in Dongying Depression, which requires some new technologies and methods to further reveal the geological characteristics in step with the mature exploration stage. In this paper, on the references to the studies of petroleum system and multiple oil-gas accumulation belt with flexible maneuverability, and the application of systematic theory, the concept of reservoir assemblage is initially defined as "the association of active source rock(s) and hydrocarbon reservoir(s) that are genetically related, with the bridge of pathway system in an oil and gas bearing basin". Compared with the theories of petroleum system and multiple oil-gas accumulation belts, it emphasizes on the processes of petroleum migration and accumulation and the correlation among active source rock, trapped hydrocarbon and migration pathway, and has been confirmed to be more suitably applied to high maturely explored basin. In the first study of this paper, sequence stratigraphy and subtle analytical technology of source rock have been employed to find that two categories of source rock with their characteristic types of organic matter and substantial states occurred in Dongying Depression. The first category, consisting of the oil shales within the third-order sequences of lacustrine expanding system tracts in the upper interval of the fourth Member of Shahejie Formation and both in the middle and lower intervals of the third Member of Shahejie Formation, is featured with the highest abundance of total organic matter (TOC) and the strongest abilities of hydrocarbon generation and expulsion, which is classified into the standard of good hydrocarbon source rock. Exploration assessment confirmed that about 70-80% of hydrocarbon in Dongying Depression came from this set of source rock for which the low sedimentary rate and strong oxygen-free environment would play the key role during its generation. The second category, composed of organic matter of dark mudstone in high stand system tracts in the upper and middle intervals of the third Member of Shahejie Formation, has been characterized by low content of total organic matter which mostly dispersedly distributes, and formed in the pre-delta to delta front environments. In classification, it belongs to the ordinary standard of source rocks. In the second research part, through the studies of high frequency sequence stratigraphy, fault geometry and active history combining with geochemistry of fluid inclusion and nitrogen compound and simulation test of hydrocarbon migration and accumulation, the faults have been thought to be the principal conduits, and the sandy bodies and unconformities might played the complementary pathways for hydrocarbon migration and accumulation in Dongying Depression of the continental faulted basin. Therefore, the fault activities may mainly constrain on the development of hydrocarbon pathways in space and time. Even more, using homogenization temperatures of fluid inclusion in digenetic minerals, three critical moments for hydrocarbon accumulation have been determined as well in Dongying Depression, which happened during the late stage of Dongying Formation (Ed), the early stage of Guantao Formation (Nig) and the early stage of Minghuazhen Formation (Nim), respectively. Comparatively, the last stage is looked as the main forming-reservoir period, which has also been supported by the results of geochemical analysis and simulation experiments of hydrocarbon generation and expulsion. Clearly, the times of hydrocarbon migration and accumulation are consistent with those of the fault activities in Dongying Depression, which indicate that tectonic activities would control the forming-reservoir. A conceptual model of faulting-episodic expulsion coupled with episodic forming-reservoir has then been established in this study. In the third part of this paper, some focusing areas were selected for the fine descriptions of pathway distribution and forming-reservoir, which has given four types of reservoir assemblage in terms of the main pathway and its correlation with the reservoir and trap: (1) mainly consisted of sandy bodies; (2) mainly consisted of faults; (3) mainly consisted of unconformities; and (4) their complex with two or three types of pathways. This classified criteria has also been applied to access the risk of some prospected traps in Dongying Depression. Finally, through the application of reservoir assemblage integrated with pathway distribution to all the prospective targets in Dongying Depression, the new favorably hydrocarbon accumulated belts have been figured out, and more subtle reservoirs have also been found. For examples, during 2000 and 2002, in the mature exploration areas, such as Liangjialou and Shengtuo structural closures etc., newly proved reserves were 2274 * 104t, and forecasted oil reserves 5660-5860xl04t; and in the predicted favorable areas, newly additional controlled oil reserves was 3355xl04t. Besides those, many other favorable exploration areas need to be further appraised.

各向异性与非线性介质中弹性波场传播特征研究

The topic of this study is about the propagation features of elastic waves in the anisotropic and nonlinear media by numerical methods with high accuracy and stability. The main achievements of this paper are as followings: Firstly, basing on the third order elastic energy formula, principle of energy conservation and circumvolved matrix method, we firstly reported the equations of non-linear elastic waves with two dimensions and three components in VTI media. Secondly, several conclusions about some numerical methods have been obtained in this paper. Namely, the minimum suitable sample stepth in space is about 1/8-1/12 of the main wavelength in order to distinctly reduce the numerical dispersion resulted from the numerical mehtod, at the same time, the higher order conventional finite difference (CFD) schemes will give little contribution to avoid the numerical solutions error accumulating with time. To get the similar accuracy with the fourth order center finite difference method, the half truncation length of SFFT should be no less than 7. The FDFCT method can present with the numerical solutions without obvious dispersion when the paprameters of FCT is suitable (we think they should be in the scope from 0.0001 to 0.07). Fortunately, the NADM method not only can reported us with the higher order accuracy solutions (higher than that of the fourth order finite difference method and lower than that of the sixth order finite difference method), but also can distinctly reduce the numerical dispersion. Thirdly, basing on the numerial and theoretical analysis, we reported such nonlinear response accumulating with time as waveform aberration, harmonic generation and resonant peak shift shown by the propagation of one- and two-dimensional non-linear elasticwaves in this paper. And then, we drew the conclusion that these nonlinear responses are controlled by the product between nonlinear strength (SN) and the amplitude of the source. At last, the modified FDFCT numerical method presented by this paper is used to model the two-dimensional non-linear elastic waves propagating in VTI media. Subsequently, the wavelet analysis and polarization are adopted to investigate and understand the numerical results. And then, we found the following principles (attention: the nonlinear strength presented by this paper is weak, the thickness of the -nonlinear media is thin (200m), the initial energy of the source is weak and the anisotropy of the media is weak too): The non-linear response shown by the elastic waves in VTI media is anisotropic too; The instantaneous main frequency sections of seismic records resulted from the media with a non-linear layer have about 1/4 to 1/2 changes of the initial main frequency of source with that resulted from the media without non-linear layer; The responses shown by the elasic waves about the anisotropy and nonlinearity have obvious mutual reformation, namely, the non-linear response will be stronger in some directions because of the anisotropy and the anisotropic strength shown by the elastic waves will be stronger when the media is nonlinear.

Numerical Investigation of Richtmyer-Meshkov Instability Driven by Cylindrical Shocks

In this paper, a numerical method with high order accuracy and high resolution was developed to simulate the Richtmyer-Meshkov(RM) instability driven by cylindrical shock waves. Compressible Euler equations in cylindrical coordinate were adopted for the cylindrical geometry and a third order accurate group control scheme was adopted to discretize the equations. Moreover, an adaptive grid technique was developed to refine the grid near the moving interface to improve the resolution of numerical solutions. The results of simulation exhibited the evolution process of RM instability, and the effect of Atwood number was studied. The larger the absolute value of Atwood number, the larger the perturbation amplitude. The nonlinear effect manifests more evidently in cylindrical geometry. The shock reflected from the pole center accelerates the interface for the second time, considerably complicating the interface evolution process, and such phenomena of reshock and secondary shock were studied.

A Study of Composite Beam with Shape Memory Alloy Arbitrarily Embedded under Thermal and Mechanical Loadings

The constitutive relations and kinematic assumptions on the composite beam with shape memory alloy (SMA) arbitrarily embedded are discussed and the results related to the different kinematic assumptions are compared. As the approach of mechanics of materials is to study the composite beam with the SMA layer embedded, the kinematic assumption is vital. In this paper, we systematically study the kinematic assumptions influence on the composite beam deflection and vibration characteristics. Based on the different kinematic assumptions, the equations of equilibrium/motion are different. Here three widely used kinematic assumptions are presented and the equations of equilibrium/motion are derived accordingly. As the three kinematic assumptions change from the simple to the complex one, the governing equations evolve from the linear to the nonlinear ones. For the nonlinear equations of equilibrium, the numerical solution is obtained by using Galerkin discretization method and Newton-Rhapson iteration method. The analysis on the numerical difficulty of using Galerkin method on the post-buckling analysis is presented. For the post-buckling analysis, finite element method is applied to avoid the difficulty due to the singularity occurred in Galerkin method. The natural frequencies of the composite beam with the nonlinear governing equation, which are obtained by directly linearizing the equations and locally linearizing the equations around each equilibrium, are compared. The influences of the SMA layer thickness and the shift from neutral axis on the deflection, buckling and post-buckling are also investigated. This paper presents a very general way to treat thermo-mechanical properties of the composite beam with SMA arbitrarily embedded. The governing equations for each kinematic assumption consist of a third order and a fourth order differential equation with a total of seven boundary conditions. Some previous studies on the SMA layer either ignore the thermal constraint effect or implicitly assume that the SMA is symmetrically embedded. The composite beam with the SMA layer asymmetrically embedded is studied here, in which symmetric embedding is a special case. Based on the different kinematic assumptions, the results are different depending on the deflection magnitude because of the nonlinear hardening effect due to the (large) deflection. And this difference is systematically compared for both the deflection and the natural frequencies. For simple kinematic assumption, the governing equations are linear and analytical solution is available. But as the deflection increases to the large magnitude, the simple kinematic assumption does not really reflect the structural deflection and the complex one must be used. During the systematic comparison of computational results due to the different kinematic assumptions, the application range of the simple kinematic assumption is also evaluated. Besides the equilibrium study of the composite laminate with SMA embedded, the buckling, post-buckling, free and forced vibrations of the composite beam with the different configurations are also studied and compared.

High-Resolution Finite Compact Difference Schemes for Hyperbolic Conservation Laws

A finite compact (FC) difference scheme requiring only bi-diagonal matrix inversion is proposed by using the known high-resolution flux. Introducing TVD or ENO limiters in the numerical flux, several high-resolution FC-schemes of hyperbolic conservation law are developed, including the FC-TVD, third-order FC-ENO and fifth-order FC-ENO schemes. Boundary conditions formulated need only one unknown variable for third-order FC-ENO scheme and two unknown variables for fifth-order FC-ENO scheme. Numerical test results of the proposed FC-scheme were compared with traditional TVD, ENO and WENO schemes to demonstrate its high-order accuracy and high-resolution.

Saltation in wind blown sand

The Boltzmann equation of the sand particle velocity distribution function in wind-blown sand two-phase flow is established based on the motion equation of single particle in air. And then, the generalized balance law of particle property in single phase granular flow is extended to gas-particle two-phase flow. The velocity distribution function of particle phase is expanded into an infinite series by means of Grad's method and the Gauss distribution is used to replace Maxwell distribution. In the case of truncation at the third-order terms, a closed third-order moment dynamical equation system is constructed. The theory is further simplified according to the measurement results obtained by stroboscopic photography in wind tunnel tests.

Quasi-Dammann Grating With Proportional Intensity Array Spots

A quasi-Dammann grating is proposed to generate array spots with proportional-intensity orders in the far field. To describe the performance of the grating, the uniformities of the array spots are redefined. A two-dimensional even-sampling encode scheme is adopted to design the quasi-Dammann grating. Numerical solutions of the binary-phase quasi-Dammann grating with proportional-intensity orders are given. The experimental results with a third-order quasi-Dammann grating, which has an intensity proportion of 3:2:1 from zero order to second order, are presented. (C) 2008 Optical Society of America