Article(id=1315741687209639985, tenantId=1146029695717560320, journalId=1315121910733963395, issueId=1315741514865689424, articleNumber=null, orderNo=null, doi=10.3788/gzxb20265503.0326002, pmid=null, cstr=32255.14.gzxb20265503.0326002, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1760630400000, receivedDateStr=2025-10-17, revisedDate=null, revisedDateStr=null, acceptedDate=1763395200000, acceptedDateStr=2025-11-18, onlineDate=1791626863528, onlineDateStr=2026-10-10, pubDate=1774368000000, pubDateStr=2026-03-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1791626863528, onlineIssueDateStr=2026-10-10, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1791626863528, creator=13701087609, updateTime=1791626863528, updator=13701087609, issue=Issue{id=1315741514865689424, tenantId=1146029695717560320, journalId=1315121910733963395, year='2026', volume='55', issue='3', pageStart='1', pageEnd='378', issueExtLink='null', onlineDate='null', pubDate='1774368000000', pubDateStr='2026-03-25', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=1, specialIssue=null, createTime=1791626822438, creator='13701087609', updateTime=1791626932547, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext={EN=IssueExt(id=1315741977111540218, tenantId=1146029695717560320, journalId=1315121910733963395, issueId=1315741514865689424, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1315741977111540219, tenantId=1146029695717560320, journalId=1315121910733963395, issueId=1315741514865689424, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null, downloadFileDto=null}, startPage=230, endPage=243, ext={EN=ArticleExt(id=1315741688899944498, articleId=1315741687209639985, tenantId=1146029695717560320, journalId=1315121910733963395, language=EN, title=Remodeling of the Localized Approximation： Beam Shape Coefficient Calculation for the Gaussian Beam by Using Scalar Translation Addition Theorem, columnId=1315741623712067945, journalTitle=Acta Photonica Sinica, columnName=Physical Optics, runingTitle=null, highlight=null, articleAbstract=

The accurate and efficient calculation of the Beam Shape Coefficients （BSCs） is one of the key issues in researching the interaction between a structured or shaped beam and the particles. For this reason， different techniques have been established， such as the quadrature method， the Localized Approximation （LA） or its variants， the Finite Series （FS） method， the Angular Spectrum Decomposition （ASD）， and the others. Among these methods， the Localized approximation has been widely used due to its advantages including mainly the efficiency in numerical calculation， the conciseness of the expressions and the straightforwardness in deducing the BSCs for the beams in both the on-axis and off-axis scenarios. However， this method， just as it is named， is an approximate one. The beam remodeling takes place when the beam field is reconstructed from the BSCs， i.e. the reconstructed beam field deviates from the originally given field. Up to date， the remodeling of the beam field has not been studied systematically.

In this work， the two-step indirect method is employed to study the remodeling effects of the localized approximation method for the Gaussian beams. In the first step， the expression of the BSCs is obtained， by using the localized approximation， for the Gaussian beam which is centered in the coordinate system （called beam system） and is described by using a scalar potential function （therefore the BSCs are called scalar BSCs）. Subsequently， the scalar translational addition theorem is used to transform the BSCs into another coordinate system whose axes are parallel to those of the beam system. The second coordinate system in which the spherical particle is centered is called particle system. In the second step of the indirect method， the electric and magnetic （EM） fields of the Gaussian beam are described by using the scalar potential function together with a polarization parameter. Based on this relation， the BSCs of the EM field （called vector BSCs or EM BSCs） are expressed as a linear combination of the scalar BSCs. In this way， the EM BSCs for the off-axis located Gaussian beam are obtained indirectly from the scalar BSCs of the centered beam. The two-step indirect method simplifies the analytical derivation of the BSCs. The expressions of the EM BSCs which are directly deduced in the localized approximation for the off-axis located Gaussian beam are used in the numerical calculations for making a comparison. It is found that the indirect method is at least eight times faster than the direct method in numerical calculation of the BSCs.

The remodeling of the localized approximation is studied， based on a comparison between the originally given field of the Gaussian beam and the fields which are reconstructed from the directly and/or indirectly calculated EM BSCs. The numerical results reveal that， in the direct LA method， the axisymmetric structure of the Gaussian beam is broken in the reconstructed beam field， exhibiting spurious-peaks in the electric field. When the off-axis distance becomes sufficiently large， a ring structure is produced in the field. The deviation of the reconstructed beam field from the given one increases gradually along the increase of the off-axis distance. However， in the indirect method， the reconstructed field is always axisymmetric and the discrepancy between the reconstructed and given fields is independent of the off-axis distance. In both the direct and indirect methods， the reconstructed beam fields deviate from the given one， showing a close dependence on the beam waist radius. Namely， the reconstructed beam fields agree much better with the given field when the beam waist radius is large， but the quality of the remodeled fields become poorer when the beam waist radius decreases， especially when the beam waist radius is close to or less than the wavelength of the light beam.

It is concluded that， compared with the direct localized approximation method， the two-step indirect method has advantages in analytical derivation of the BSCs， in conciseness of the expressions， in efficiency of numerical calculation， and in quality of reconstructed beam field. Besides， the work presented here suggests a warning against the use of the direct/indirect LA methods for strongly focused Gaussian beams and the use of the direct LA for large off-axis located beams.

, authors=Yu WANG, Yiqian TIAN, Haoyu JIANG, Jianqi SHEN, authorsList=Yu WANG, Yiqian TIAN, Haoyu JIANG, Jianqi SHEN, authorCompany=null, correspAuthors=Jianqi SHEN, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, fund=null), CN=ArticleExt(id=1315741693358489671, articleId=1315741687209639985, tenantId=1146029695717560320, journalId=1315121910733963395, language=CN, title=局域近似法的光束重构：高斯光束形状系数的标量平移计算方法, columnId=1315741623934366059, journalTitle=光子学报, columnName=物理光学, runingTitle=null, highlight=null, articleAbstract=

采用两步间接法结合局域近似法计算有形光束电磁场的光束形状系数，并讨论了两步间接法结合局域近似法和直接采用局域近似法重构高斯光束之间的差异。该方法首先采用标量势函数描述在轴时的结构光束，利用标量平移定理及标量势函数与电磁场之间的关系得到离轴时的矢量光束形状系数。采用两步间接法计算有形光束的光束形状系数，不仅简化了光束形状系数的表述，还加快了数值计算的速度。高斯光束的数值计算结果验证了该方法在计算光束形状系数和进行场重构上有较好的精度和效率。

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Journal of the Optical Society of America A， 2024， 41（7）： 1403-1412., articleTitle=Beam shape coefficients of hollow vortex Gaussian beam and near-field scattering, refAbstract=null), Reference(id=1315741704905408683, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, doi=null, pmid=null, pmcid=null, year=1994, volume=11, issue=9, pageStart=2503, pageEnd=2515, url=null, language=null, rfNumber=40, rfOrder=39, authorNames=LOCK J A, GOUESBET G, journalName=Journal of the Optical Society of America A, refType=null, unstructuredReference=LOCK J A， GOUESBET G. Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams［J］. Journal of the Optical Society of America A， 1994， 11（9）： 2503-2515., articleTitle=Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams, refAbstract=null)], funds=null, companyList=[AuthorCompany(id=1315741693668868168, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, xref=1, ext=[AuthorCompanyExt(id=1315741693677256777, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, companyId=1315741693668868168, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=¹College of Science， University of Shanghai for Science and Technology， Shanghai 200093， China), AuthorCompanyExt(id=1315741693689839690, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, companyId=1315741693668868168, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=¹上海理工大学 理学院，上海 200093)])], figs=[ArticleFig(id=1315741698072887402, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=EN, label=Fig.1, caption=Geometric relation between the beam and particle systems, figureFileSmall=gdzmi5hC2+Ajx9pOJiRBEg==, figureFileBig=k7Wy9FobcpxcPbr92b4haA==, tableContent=null), ArticleFig(id=1315741698135801963, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=CN, label=图1, caption=光束系和颗粒系的几何关系, figureFileSmall=gdzmi5hC2+Ajx9pOJiRBEg==, figureFileBig=k7Wy9FobcpxcPbr92b4haA==, tableContent=null), ArticleFig(id=1315741698353905772, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=EN, label=Fig.2, caption=Numerical results of the vector BSCs G_{\mathrm{n}\mathrm{m}}^{\mathrm{T}\mathrm{M},\mathrm{2}} and the comparison. （a） Direct method； （b） Indirect method with DP； （c） Indirect method with HP； （d） Difference between direct and DP indirect method； （e） Difference between direct and HP indirect method； （f） Difference between DP and HP 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tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=CN, label=图3, caption=直接法和间接法光束形状系数G_{nm}^{\mathrm{T}\mathrm{M},\mathrm{2}}之间的偏差。 （a）不同分波n； （b） 不同方位角模式m, figureFileSmall=yruxLvTaJQXHhn+rszx1BQ==, figureFileBig=2vwrs4G5YtPrqMDojyEOiQ==, tableContent=null), ArticleFig(id=1315741698672672880, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=EN, label=Fig.4, caption=Reconstructed electric fields of the Gaussian beam centered at .\left(x_{\mathrm{0}},y_{\mathrm{0}},z_{\mathrm{0}}\right)=\left(\mathrm{2,2},\mathrm{0}\right)\mathrm{\mu }\mathrm{m}. （a） Electric field reconstructed in direct method； （b） Electric field reconstructed in indirect method； （c） Difference between the original and directly reconstructed fields； （d） Difference between the original and indirectly reconstructed fields, figureFileSmall=6B2XZWbyJvnTPg5W+eY59g==, figureFileBig=2WaQHjSdhKFMEXZc6zmxzw==, tableContent=null), ArticleFig(id=1315741698748170353, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=CN, label=图4, caption=高斯光束的重构场及其偏差，光束中心位于\left(x_{\mathrm{0}},y_{\mathrm{0}},z_{\mathrm{0}}\right)=\left(\mathrm{2,2},\mathrm{0}\right)\mathrm{\mu }\mathrm{m}。（a）直接法重构电场；（b）间接法重构电场；（c）直接法重构误差；（d）间接法重构误差, figureFileSmall=6B2XZWbyJvnTPg5W+eY59g==, figureFileBig=2WaQHjSdhKFMEXZc6zmxzw==, tableContent=null), ArticleFig(id=1315741698827862130, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=EN, label=Fig.5, caption=Reconstructed electric fields of the Gaussian beam centered at \left(x_{\mathrm{0}},y_{\mathrm{0}},z_{\mathrm{0}}\right)=\left(\mathrm{4,2},\mathrm{0}\right)\mathrm{\mu }\mathrm{m}. （a） Electric field reconstructed in direct method； （b） Electric field reconstructed in indirect method； （c） Difference between the original and directly reconstructed fields； （d） Difference between the original and indirectly reconstructed fields, figureFileSmall=3YRmJ/uQUnorZHX9MyftIw==, figureFileBig=w2Rj8VPzEAQ5bEz9anlsXg==, tableContent=null), ArticleFig(id=1315741698911748211, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=CN, label=图5, caption=高斯光束的重构场及其偏差，光束中心位于\left(x_{\mathrm{0}},y_{\mathrm{0}},z_{\mathrm{0}}\right)=\left(\mathrm{4,2},\mathrm{0}\right)\mathrm{\mu }\mathrm{m}。（a）直接法重构电场；（b）间接法重构电场；（c）直接法重构误差；（d）间接法重构误差, figureFileSmall=3YRmJ/uQUnorZHX9MyftIw==, figureFileBig=w2Rj8VPzEAQ5bEz9anlsXg==, tableContent=null), ArticleFig(id=1315741698978857076, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=EN, label=Fig.6, caption=Reconstructed electric fields of the Gaussian beam centered at \left(x_{\mathrm{0}},y_{\mathrm{0}},z_{\mathrm{0}}\right)=\left(\mathrm{6,2},\mathrm{0}\right)\mathrm{\mu }\mathrm{m}. （a） Electric field reconstructed in direct method； （b） Electric field reconstructed in indirect method； （c） Difference between the original and directly reconstructed fields； （d） Difference between the original and indirectly reconstructed fields, figureFileSmall=r7AcvY45QHjyTzCmBRuT2A==, figureFileBig=eOYgLvvVzqZD5v1yKjm/aw==, tableContent=null), ArticleFig(id=1315741699045965941, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=CN, label=图6, caption=高斯光束的重构场及其偏差，光束中心位于\left(x_{\mathrm{0}},y_{\mathrm{0}},z_{\mathrm{0}}\right)=\left(\mathrm{6,2},\mathrm{0}\right)\mathrm{\mu }\mathrm{m}。（a）直接法重构电场；（b）间接法重构电场；（c）直接法重构误差；（d）间接法重构误差, figureFileSmall=r7AcvY45QHjyTzCmBRuT2A==, figureFileBig=eOYgLvvVzqZD5v1yKjm/aw==, tableContent=null), ArticleFig(id=1315741699146629238, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=EN, label=Fig.7, caption=Reconstructed electric fields of the Gaussian beam centered at \left(x_{\mathrm{0}},y_{\mathrm{0}},z_{\mathrm{0}}\right)=\left(\mathrm{8,2},\mathrm{0}\right)\mathrm{\mu }\mathrm{m}. （a） Electric field reconstructed in direct method； （b） Electric field reconstructed in indirect method； （c） Difference between the original and directly reconstructed fields； （d） Difference between the original and indirectly reconstructed fields, figureFileSmall=CmS2GWIgULd1ltLmfeDJtA==, figureFileBig=1LtQryknIMXkAOwfyXfgYQ==, tableContent=null), ArticleFig(id=1315741699222126711, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=CN, label=图7, caption=高斯光束的重构场及其偏差，光束中心位于\left(x_{\mathrm{0}},y_{\mathrm{0}},z_{\mathrm{0}}\right)=\left(\mathrm{8,2},\mathrm{0}\right)\mathrm{\mu }\mathrm{m}。（a）直接法重构电场；（b）间接法重构电场；（c）直接法重构误差； （d）间接法重构误差, figureFileSmall=CmS2GWIgULd1ltLmfeDJtA==, figureFileBig=1LtQryknIMXkAOwfyXfgYQ==, tableContent=null), ArticleFig(id=1315741699289235576, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=EN, label=Fig.8, caption=Reconstructed electric fields of the Gaussian beam with a waist radius of w_{\mathrm{0}}=\mathrm{3}\mathrm{\mu }\mathrm{m}. （a） Electric field reconstructed in direct method； （b） Electric field reconstructed in indirect method； （c） Difference between the original and directly reconstructed fields； （d） Difference between the original and indirectly reconstructed fields, figureFileSmall=UQAlv6cZiS6crVNV8LK5FA==, figureFileBig=R9scmXYYyCrUVegCJRWeUQ==, tableContent=null), ArticleFig(id=1315741699364733049, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=CN, label=图8, caption=高斯光束的重构场及其偏差，光束束腰半径为w_{\mathrm{0}}=\mathrm{3}\mathrm{\mu }\mathrm{m}。（a）直接法重构电场；（b）间接法重构电场；（c）直接法重构误差；（d）间接法重构误差, figureFileSmall=UQAlv6cZiS6crVNV8LK5FA==, figureFileBig=R9scmXYYyCrUVegCJRWeUQ==, tableContent=null), ArticleFig(id=1315741699440230522, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=EN, label=Fig.9, caption=Reconstructed electric fields of the Gaussian beam with a waist radius of w_{\mathrm{0}}=\mathrm{2}\mathrm{\mu }\mathrm{m}. （a） Electric field reconstructed in direct method； （b） Electric field reconstructed in indirect method； （c） Difference between the original and directly reconstructed fields； （d） Difference between the original and indirectly reconstructed fields, figureFileSmall=0Fi7pw3zfGDAH05W7vrs4w==, figureFileBig=L8/pXaawi9m1eeUb34rTng==, tableContent=null), ArticleFig(id=1315741699519922299, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=CN, label=图9, caption=高斯光束的重构场及其偏差，光束束腰半径为w_{\mathrm{0}}=\mathrm{2}\mathrm{\mu }\mathrm{m}。（a）直接法重构电场；（b）间接法重构电场；（c）直接法重构误差； （d）间接法重构误差, figureFileSmall=0Fi7pw3zfGDAH05W7vrs4w==, figureFileBig=L8/pXaawi9m1eeUb34rTng==, tableContent=null), 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tableContent=null), ArticleFig(id=1315741699767386238, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=EN, label=Table1, caption=

Comparison of CPU time for BSC calculation

, figureFileSmall=null, figureFileBig=null, tableContent=

{\mathit{r}}_{12}\left(x_{12},y_{12},z_{12}\right)	g_{n,0}^{\left(1\right)}	B-to-P	S-to-V	Indirect method	Direct method
					g_{nm}^{\left(2\right)}	\left(G_{nm}^{\mathrm{T}\mathrm{E},2},G_{nm}^{\mathrm{T}\mathrm{M},2}\right)
（0， 2， 0）	0.224	0.097	0.017	0.338	1.338	7.896
（3， 0， 0）	0.242	0.098	0.017	0.357	1.359	8.783
（2， 2， 5）	0.237	0.094	0.017	0.347	1.350	9.418
（3， 4， -10）	0.381	0.138	0.022	0.541	1.583	12.69

), ArticleFig(id=1315741699826106495, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=CN, label=表1, caption=

比较光束形状系数的计算时间

, figureFileSmall=null, figureFileBig=null, tableContent=

{\mathit{r}}_{12}\left(x_{12},y_{12},z_{12}\right)	g_{n,0}^{\left(1\right)}	B-to-P	S-to-V	Indirect method	Direct method
					g_{nm}^{\left(2\right)}	\left(G_{nm}^{\mathrm{T}\mathrm{E},2},G_{nm}^{\mathrm{T}\mathrm{M},2}\right)
（0， 2， 0）	0.224	0.097	0.017	0.338	1.338	7.896
（3， 0， 0）	0.242	0.098	0.017	0.357	1.359	8.783
（2， 2， 5）	0.237	0.094	0.017	0.347	1.350	9.418
（3， 4， -10）	0.381	0.138	0.022	0.541	1.583	12.69

), ArticleFig(id=1315741699901603968, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=EN, label=Table 2, caption=

Dependence of the reconstructed field on the off-axis distance

, figureFileSmall=null, figureFileBig=null, tableContent=

{\mathit{r}}_0\left(x_0,y_0,z_0\right)	Direct method		Indirect method		N
	{\epsilon }_{\mathrm{a}\mathrm{v}\mathrm{e}}^{\mathrm{d}\mathrm{i}\mathrm{r}}	{\epsilon }_{\mathrm{r}\mathrm{e}\mathrm{l}}^{\mathrm{d}\mathrm{i}\mathrm{r}}	{\epsilon }_{\mathrm{a}\mathrm{v}\mathrm{e}}^{\mathrm{i}\mathrm{n}\mathrm{d}}	{\epsilon }_{\mathrm{r}\mathrm{e}\mathrm{l}}^{\mathrm{i}\mathrm{n}\mathrm{d}}
（2， 2， 0）	9.47×10-5	0.285	9.42×10-5	9.69×10-2	11 433
（4， 2， 0）	1.12×10-4	0.522
（6， 2， 0）	1.44×10-4	1.928
（8， 2， 0）	1.82×10-4	1.785

), ArticleFig(id=1315741699972907137, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=CN, label=表2, caption=

重构场效应与离轴距离的关系

, figureFileSmall=null, figureFileBig=null, tableContent=

{\mathit{r}}_0\left(x_0,y_0,z_0\right)	Direct method		Indirect method		N
	{\epsilon }_{\mathrm{a}\mathrm{v}\mathrm{e}}^{\mathrm{d}\mathrm{i}\mathrm{r}}	{\epsilon }_{\mathrm{r}\mathrm{e}\mathrm{l}}^{\mathrm{d}\mathrm{i}\mathrm{r}}	{\epsilon }_{\mathrm{a}\mathrm{v}\mathrm{e}}^{\mathrm{i}\mathrm{n}\mathrm{d}}	{\epsilon }_{\mathrm{r}\mathrm{e}\mathrm{l}}^{\mathrm{i}\mathrm{n}\mathrm{d}}
（2， 2， 0）	9.47×10-5	0.285	9.42×10-5	9.69×10-2	11 433
（4， 2， 0）	1.12×10-4	0.522
（6， 2， 0）	1.44×10-4	1.928
（8， 2， 0）	1.82×10-4	1.785

), ArticleFig(id=1315741700044210306, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=EN, label=Table 3, caption=

Dependence of the reconstructed field on the beam waist radius.

, figureFileSmall=null, figureFileBig=null, tableContent=

w_0	Direct method		Indirect method		N
	{\epsilon }_{\mathrm{a}\mathrm{v}\mathrm{e}}^{\mathrm{d}\mathrm{i}\mathrm{r}}	{\epsilon }_{\mathrm{r}\mathrm{e}\mathrm{l}}^{\mathrm{d}\mathrm{i}\mathrm{r}}	{\epsilon }_{\mathrm{a}\mathrm{v}\mathrm{e}}^{\mathrm{i}\mathrm{n}\mathrm{d}}	{\epsilon }_{\mathrm{r}\mathrm{e}\mathrm{l}}^{\mathrm{i}\mathrm{n}\mathrm{d}}
1	1.33×10-3	19.61	8.22×10-4	0.525	1 101
2	2.20×10-4	1.130	2.11×10-4	0.201	4 381
3	9.29×10-5	0.248	9.42×10-5	9.69×10-2	9 777

), ArticleFig(id=1315741700123902083, tenantId=1146029695717560320, journalId=1315121910733963395, articleId=1315741687209639985, language=CN, label=表3, caption=

重构场效应与光束束腰的关系

, figureFileSmall=null, figureFileBig=null, tableContent=

w_0	Direct method		Indirect method		N
	{\epsilon }_{\mathrm{a}\mathrm{v}\mathrm{e}}^{\mathrm{d}\mathrm{i}\mathrm{r}}	{\epsilon }_{\mathrm{r}\mathrm{e}\mathrm{l}}^{\mathrm{d}\mathrm{i}\mathrm{r}}	{\epsilon }_{\mathrm{a}\mathrm{v}\mathrm{e}}^{\mathrm{i}\mathrm{n}\mathrm{d}}	{\epsilon }_{\mathrm{r}\mathrm{e}\mathrm{l}}^{\mathrm{i}\mathrm{n}\mathrm{d}}
1	1.33×10-3	19.61	8.22×10-4	0.525	1 101
2	2.20×10-4	1.130	2.11×10-4	0.201	4 381
3	9.29×10-5	0.248	9.42×10-5	9.69×10-2	9 777

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