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Kramers-Kronig关系的研究与发展

阎春生

阎春生. Kramers-Kronig关系的研究与发展[J]. 中国光学(中英文), 2019, 12(2): 179-198. doi: 10.3788/CO.20191202.0179
引用本文: 阎春生. Kramers-Kronig关系的研究与发展[J]. 中国光学(中英文), 2019, 12(2): 179-198. doi: 10.3788/CO.20191202.0179
YAN Chun-sheng. Research and development on Kramers-Kronig relationship[J]. Chinese Optics, 2019, 12(2): 179-198. doi: 10.3788/CO.20191202.0179
Citation: YAN Chun-sheng. Research and development on Kramers-Kronig relationship[J]. Chinese Optics, 2019, 12(2): 179-198. doi: 10.3788/CO.20191202.0179

Kramers-Kronig关系的研究与发展

doi: 10.3788/CO.20191202.0179
基金项目: 

国家自然科学基金 11621101

国家自然科学基金 91233208

浙江省科技部中央高校基础研究经费 2017FZA5001

详细信息
    作者简介:

    阎春生(1973-), 男, 山西文水人, 博士, 副教授, 硕士生导师, 1994年、1999年于电子科技大学分别获得学士、硕士学位, 2003年于清华大学获得博士学位, 主要从事光传感、光层析成像技术及近场光学等方面的研究。E-mail:yancs@zju.edu.cn

  • 中图分类号: O174.5

Research and development on Kramers-Kronig relationship

Funds: 

National Natural Science Foundation of China 11621101

National Natural Science Foundation of China 91233208

Fundamental Research Funds for the Central Universities from the Science and Technology Department of Zhejiang Province 2017FZA5001

More Information
  • 摘要: Kramers-Kronig关系(简称KK关系)是希尔伯特变换的一个特例,描述了具有因果性的平方可积函数实部与虚部之间的数学联系,具有普适的物理背景。本文介绍了KK关系的历史及数学物理本质,详细阐述了其在电学、磁学、声学、光学、人工介质以及光通信中的具体形式、涵义及应用,包括反射和透射响应函数、电极化率、介电常数、折射率、电导率、电阻抗、磁导率、原子散射因子、绝热压缩系数、声折射率、单边带时域信号、空间隐身介质还有各种非线性介质等。分析了截断误差在实际应用中对KK积分计算结果的影响,总结了各种积分限外推方法以及各种基于锚点的减法KK关系,包括单减KK关系、多减KK关系及差分多减KK关系等。

     

  • 图 1  公式(1)的积分区域(R→∞, r→0)

    Figure 1.  Integral region of formula 1(R→∞, r→0)

    图 2  κ(黑点)和n(灰线)的雅克比谱分布[60]

    Figure 2.  Jacobian spectrum distribution of the κ(black dot) and n(grey line)[60]

    图 3  消光谱:×为理论值;○为KK计算值[63]

    Figure 3.  Extinction spectrum:× is theoretical value; ○ is KK calculation value[63]

    图 4  (a) 和(b)分别是先验和重建的κn[60]

    Figure 4.  (a) and (b) are priori and retrieved values of κ and n, respectively [60]

    图 5  色散曲线[58]

    Figure 5.  Dispersion curve [58]

    图 6  χ(3)(3ω; ω, ω, ω)实部(a)和虚部(b)的测量值,KK和SKK计算值[67]

    Figure 6.  Measured, KK and SKK values of χ(3)(3ω; ω, ω, ω):(a)the real part; (b)the imaginary part[67]

    图 7  一维周期介质散射谱相位差重建[69]:实验测量SPEBI方法(实线),KK关系(点画线),DSSKK关系(点),DMSKK(线段)

    Figure 7.  Phase difference reconstruction of one-dimensional periodic dielectric scattering spectrum[69]:experimental measurement SPEBI method(solid line); KK relationship(dash dot); DSSKK relationship(dot); DMSKK relationship(line segment)

    图 8  双向隐身KK平面介质[74]:(a)复介电常数ε(x)的谱,(b)和(c)分别是TE和TM极化波传输及反射系数谱

    Figure 8.  Bi-directional stealth KK plane medium [74]: (a) is the spectrum of the complex permittivity ε(x); (b) and (c) are TE and TM polarized wave propagation and reflection coefficient spectra, respectively

    图 9  一维非均匀空间KK介质[77]。(a)印制卷绕金属丝制成的二维人工介质及几何参数;(b)具有91个单元的沿x方向的周期性条形结构;(c)利用全方位单极探针测量电场的实验系统;(d)测量及(e)仿真得到的2.4 GHz的电场分布;(f)y=0时,沿x方向的电场|Ez|的分布曲线

    Figure 9.  1-dimensional non-uniform space KK medium[77]: (a)2-dimensional artificial medium made of printed rolled-up wire and its geometric parameters; (b)a periodic bar structure with 91 units along the x direction; (c)an experimental system for measuring electric field by omnidirectional monopole probe; (d) and (e) are the electric field distributions of 2.4 GHz for measurement and simulation, respectively; (f)the distribution curve of electric field |Ez| along the x direction as y=0

    图 10  (a) 220 Gb/s单波长、单偏振、单探测器的基于KK关系的直接探测系统;(b)测量结果[80]

    Figure 10.  (a)220 Gb/s direct detection system based on KK relationship with single wavelength, single polarization and single detector and (b)detection results[80]

    图 11  偏振复用KK收发机[79]:(a)原理图;(b)实验结果

    Figure 11.  (a)Schematic diagram and (b)experimental results for polarization multiplexing KK transceiver [79]

    图 12  斯托克斯向量KK收发机[80]

    Figure 12.  Stokes vector KK transceiver [80]

    表  1  非线性效应及数学形式

    Table  1.   Nonlinear effects and mathematical forms

    名称 数学形式
    二倍频上半,ω σijk(2)(2ω, ω)
    三倍频上半,ω σijkl(3)(3ω, 2ω, ω)
    和频上半,ω1ω2 σijk(2)(ω1+ω2, ω1)
    σijkl(3)(ω1+2ω2, 2ω2, ω2)
    差频 σijk(2)(ω1-ω2, ω1)
    上半,ω1; 下半,ω2 σijkl(3)(2ω1-ω2, 2ω1, ω1)
    ω2激发对ω1影响上半,ω1 σijkl(3)(ω1, 0, ω2)
    整流效应不解析 σijk(2)(0, ω)
    “自作用”效应不解析 σijkl(3)(ω, 0, ω)
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    WANG X J, ZHU R CH, HONG L. Kramers-Kronig relations and frequency to time-domain transformation method for time domain calculation of floating body with forward speed[J]. Shipbuilding of China, 2018, 59(2):9-23.(in Chinese) doi: 10.3969/j.issn.1000-4882.2018.02.002
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出版历程
  • 收稿日期:  2018-06-11
  • 修回日期:  2018-07-13
  • 刊出日期:  2019-04-01

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