Kramers-Kronig关系的研究与发展 -

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Kramers-Kronig关系的研究与发展

doi:10.3788/CO.20191202.0179

阎春生^{1, 2,,}

1.
浙江大学图书馆, 浙江杭州 310058
2.
浙江大学光电科学与工程学院, 浙江杭州 310058

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
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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

Research and development on Kramers-Kronig relationship

YAN Chun-sheng^{1, 2,,}

1.
Library of Zhejiang University, Hangzhou 310058, China
2.
College of Optical Science and Engineering, Zhejiang University, Hangzhou 310058, China

Funds:

National Natural Science Foundation of China11621101

National Natural Science Foundation of China91233208

Fundamental Research Funds for the Central Universities from the Science and Technology Department of Zhejiang Province2017FZA5001

More Information

Corresponding author:YAN Chun-sheng: E-mail:yancs@zju.edu.cn

摘要

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。
- Kramers-Kronig关系/
- 希尔伯特变换/
- 因果关系/
- 空间KK隐身介质/
- KK光通信收发机
Abstract:The Kramers-Kronig(KK for short) relationship is a special case of Hilbert transformation, which describes the mathematical connection of square integrable function with causality between its real and imaginary parts. In this paper, the history, mathematics and physics essence of KK relationship are introduced. Its concrete form, meaning and application in electricity, magnetics, acoustics, optics, artificial medium and optical communication are presented, including its reflection and transmission functions, electric susceptibility, dielectric constant, refractive index, electrical conductivity, electrical impedance, magnetic permeability, atomic scattering factor, adiabatic compression coefficient, acoustic refractive index, single band time domain signal, space stealth medium and various nonlinear media and so on. The influence of truncation error on the calculation results of KK integral in practical applications is analyzed. Various integral limit extrapolation methods and various subtractive KK relationships based on anchor point are summarized, including single, multiple subtractive and differential multiple subtractive KK relationships.
- Kramers-Kronig relationship/
- Hilbert transformation/
- causality/
- space KK stealth medium/
- KK optical transceiver

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图 1公式(1)的积分区域(R→∞,r→0)

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

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图 2κ(黑点)和n(灰线)的雅克比谱分布^[60]

Figure 2.Jacobian spectrum distribution of theκ(black dot) andn(grey line)^[60]

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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]

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图 4(a) 和(b)分别是先验和重建的κ和n^[60]

Figure 4.(a) and (b) are priori and retrieved values ofκandn, respectively^[60]

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图 5色散曲线^[58]

Figure 5.Dispersion curve^[58]

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图 6χ⁽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 ofχ⁽³⁾(3ω;ω,ω,ω):(a)the real part; (b)the imaginary part^[67]

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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)

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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 permittivityε(x); (b) and (c) are TE and TM polarized wave propagation and reflection coefficient spectra, respectively

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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|E_z|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 thexdirection; (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 |E_z| along thexdirection asy=0

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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]

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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]

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图 12斯托克斯向量KK收发机^[80]

Figure 12.Stokes vector KK transceiver^[80]

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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	σ_ijk⁽²⁾(2ω,ω)
三倍频上半，ω	σ_ijkl⁽³⁾(3ω, 2ω,ω)
和频上半，ω₁，ω₂	σ_ijk⁽²⁾(ω₁+ω₂,ω₁)
	σ_ijkl⁽³⁾(ω₁+2ω₂, 2ω₂,ω₂)
差频	σ_ijk⁽²⁾(ω₁-ω₂,ω₁)
上半，ω₁; 下半，ω₂	σ_ijkl⁽³⁾(2ω₁-ω₂, 2ω₁,ω₁)
ω₂激发对ω₁影响上半，ω₁	σ_ijkl⁽³⁾(ω₁, 0,ω₂)
整流效应不解析	σ_ijk⁽²⁾(0,ω)
“自作用”效应不解析	σ_ijkl⁽³⁾(ω, 0,ω)

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参考文献 (82)

[1]	KRONIG R D L. On the theory of dispersion of X-rays[J]. Journal of the Optical Society of America, 1926, 12(6):547-557.doi:10.1364/JOSA.12.000547
[2]	KRAMERS H A. La diffusion de la lumiere par les atomes[J]. Atti del Congresso Internazionale dei Fisici, 1927, 2:545-557.http://d.old.wanfangdata.com.cn/OAPaper/oai_arXiv.org_astro-ph%2f0205136
[3]	BOHREN C F. What did Kramers and Kronig do and how did they do it?[J]. European Journal of Physics, 2010, 31(3):573-577.doi:10.1088/0143-0807/31/3/014
[4]	GRAF U. Introduction to Hyperfunctions and Their Integral Transforms:An Applied and Computational Approach[M]. Basel:Birkhäuser, 2010.
[5]	汪璇, 曹万强.Hilbert变换及其基本性质分析[J].湖北大学学报(自然科学版), 2008, 30(1):53-55.doi:10.3969/j.issn.1000-2375.2008.01.014 WANG X, CAO W Q. The Hilbert transform and its characters[J]. Journal of Hubei University(Nature Science), 2008, 30(1):53-55.(in Chinese)doi:10.3969/j.issn.1000-2375.2008.01.014
[6]	LUCARINI V, PEIPONEN K E, SAARINEN J J, et al.. Kramers-Kronig Relations in Optical Materials Research[M]. Berlin, Heidelberg:Springer-Verlag, 2005.
[7]	ARFKEN G B, WEBER H J. Mathematical Methods for Physicists[M]. 6th ed. Amsterdam:Elsevier Academic Press, 2005.
[8]	NUSSENZVEIG H M. Causality and Dispersion Relations[M]. Amsterdam:Elsevier Science, 1972.
[9]	ALTARELLI M, DEXTER D L, NUSSENZVEIG H M, et al.. Superconvergence and sum rules for the optical constants[J]. Physical Review B, 1972, 6(12):4502-4509.doi:10.1103/PhysRevB.6.4502
[10]	ALTARELLI M, SMITH D Y. Superconvergence and sum rules for the optical constants:physical meaning, comparison with experiment, and generalization[J]. Physical Review B, 1974, 9(4):1290-1298.doi:10.1103/PhysRevB.9.1290
[11]	LANDAU L D, LIFSHITZ E M. Electrodynamics of Continuous Media[M]. Oxford:Pergamon, 1960.
[12]	JAHODA F C. Fundamental absorption of barium oxide from its reflectivity spectrum[J]. Physical Review, 1957, 107(5):1261-1265.doi:10.1103/PhysRev.107.1261
[13]	BODE H W.Network Analysis and Feedback Amplifier Design[M]. New York: D. Van Nostrand Company, Inc., 1945.
[14]	PHILIPP H R, TAFT E A. Optical constants of germanium in the region 1 to 10 eV[J]. Physical Review, 1959, 113(4):1002-1005.doi:10.1103/PhysRev.113.1002
[15]	PHILIPP H R, TAFT E A. Kramers-Kronig analysis of reflectance data for diamond[J]. Physical Review, 1964, 136(5A):A1445-A1448.doi:10.1103/PhysRev.136.A1445
[16]	ANDERMANN G, CARON A, DOWS D A. Kramers-Kronig dispersion analysis of infrared reflectance bands[J]. Journal of the Optical Society of America, 1965, 55(10):1211-1216.
[17]	牟媛, 吴振森, 张耿, 等.基于Kramers-Kronig关系建立金属太赫兹色散模型[J].物理学报, 2017, 66(12):120202.doi:10.7498/aps.66.120202 MOU Y, WU Z S, ZHANG G, et al.. Establishment of THz dispersion model of metals based on Kramers-Kronig relation[J]. Acta Physica Sinica, 2017, 66(12):120202.(in Chinese)doi:10.7498/aps.66.120202
[18]	KOZIMA K, SUËTAKA W, SCHATZ P N. Optical constants of thin films by a Kramers-Kronig method[J]. Journal of the Optical Society of America, 1966, 56(2):181-184.doi:10.1364/JOSA.56.000181
[19]	陈金金.Kramers-Kronig关系在光学中的应用[D].天津: 南开大学, 2011. CHEN J J. The applications of Kramers-Kronig relations in the optics[D]. Tianjin: Nankai University, 2011.(in Chinese)
[20]	PEIPONEN K E, ASAKURA T. Dispersion theory for two-phase layered-geometry nanocomposites[J]. Optical Review, 1999, 6(5):410-414.doi:10.1007/s10043-999-0410-z
[21]	PEIPONEN K E, MARTTI O A, SAARINEN J, et al.. Dispersion theory of liquids containing optically linear and nonlinear Maxwell Garnett nanoparticles[J]. Optical Review, 2001, 8(1):9-17.doi:10.1007/s10043-001-0009-5
[22]	BODE H W, .网络分析和反馈放大器设计[M].陈志刚, 译.北京: 人民邮电出版社, 1958. BODE H W.Network Analysis and Feedback Amplifier Design[M]. CHEN ZH G, trans. Beijing: People's Post and Telecommunications Press, 1958.(in Chinese)
[23]	GINER-SANZ J J, ORTEGA E M, PÉREZ-HERRANZ V. Monte carlo based quantitative Kramers-Kronig test for PEMFC impedance spectrum validation[J]. International Journal of Hydrogen Energy, 2015, 40(34):11279-11293.doi:10.1016/j.ijhydene.2015.03.135
[24]	VAN MEIRHAEGHE R L, DUTOIT E C, CARDON F, et al, . On the application of the Kramers-Kronig relations to problems concerning the frequency dependence of electrode impedance[J]. Electrochimica Acta, 1975, 20:995-999.doi:10.1016/0013-4686(75)85062-6
[25]	FANO W G, BOGGI S, RAZZITTE A C. Causality study and numerical response of the magnetic permeability as a function of the frequency of ferrites using Kramers-Kronig relations[J]. Physica B:Condensed Matter, 2008, 403(4):526-530.doi:10.1016/j.physb.2007.08.218
[26]	GREINER F. Classical Electrodynamics[M]. New York:Springer, 1998.
[27]	PEIPONEN K E, LUCARINI V, VARTIAINEN E M, et al.. Kramers-Kronig relations and sum rules of negative refractive index media[J]. The European Physical Journal B-Condensed Matter and Complex Systems, 2004, 41(1):61-65.doi:10.1140/epjb/e2004-00294-6
[28]	SZABÒ Z, PARK G H, HEDGE R, et al.. A unique extraction of metamaterial parameters based on Kramers-Kronig relationship[J]. IEEE Transactions on Microwave Theory and Techniques, 2010, 58(10):2646-2653.doi:10.1109/TMTT.2010.2065310
[29]	DING W W, SUN L Q, YI L Y, et al.. Dual-sideband heterodyne of dispersion spectroscopy based on phase-sensitive detection[J]. Applied Optics, 2016, 55(31):8698-8704.doi:10.1364/AO.55.008698
[30]	STOCKMAN M I. Criterion for negative refraction with low optical losses from a fundamental principle of causality[J]. Physical Review Letters, 2007, 98(17):177404.doi:10.1103/PhysRevLett.98.177404
[31]	MACKAY T G, LAKHTAKIA A. Comment on "criterion for negative refraction with low optical losses from a fundamental principle of causality"[J]. Physical Review Letters, 2007, 99(18):189701.doi:10.1103/PhysRevLett.99.189701
[32]	彭文胜, 王建中.光电导效应及其应用探究[J].高等函授学报(自然科学版), 2007, 21(6):32-35.doi:10.3969/j.issn.1006-7353.2007.06.012 PENG W SH, WANG J ZH. Photoconductivity effect and its application[J]. Journal of Higher Correspondence Education(Natural Sciences), 2007, 21(6):32-35.(in Chinese)doi:10.3969/j.issn.1006-7353.2007.06.012
[33]	OPPENEER P M, MAURER T, STICHT J, et al.. Ab initio calculated magneto-optical Kerr effect of ferromagnetic metals:Fe and Ni[J]. Physical Review B, 1992, 45(19):10924-10933.doi:10.1103/PhysRevB.45.10924
[34]	WANG C S, CALLAWAY J. Band structure of nickel:Spin-orbit coupling, the Fermi surface, and the optical conductivity[J]. Physical Review B, 1974, 9(11):4897-4907.doi:10.1103/PhysRevB.9.4897
[35]	RATHGEN H, KATSNELSON M I. Symmetry assumptions, kramers kronig transformation and analytical continuation in Ab initio calculations of optical conductivities[J]. Physica Scripta, 2004, T109:170-174.doi:10.1238/Physica.Topical.109a00170
[36]	(美)阿特伍德D T.软X射线与极紫外辐射的原理和应用[M].张杰, 译.北京: 科学出版社, 2003. ATTWOOD D T.Soft X-rays and Extreme Ultraviolet Radiation:Principles and Applications[M]. ZHANG J, trans. Beijing: Science Press, 2003.(in Chinese)
[37]	STONE K H, VALVIDARES S M, KORTRIGHT J B. Kramers-Kronig constrained modeling of soft X-ray reflectivity spectra:Obtaining depth resolution of electronic and chemical structure[J]. Physical Review B, 2012, 86(2):024102.doi:10.1103/PhysRevB.86.024102
[38]	MANGULIS V. Kramers-Kronig or Dispersion Relations in Acoustics relationship between ultrasonic attenuation and phase velocity[J]. The Journal of the Acoustical Society of America, 1964, 36(1):211-212.doi:10.1121/1.1918936
[39]	O'DONNELL M, JAYNES E T, MILLER J G. Kramers-Kronig relationship between ultrasonic attenuation and phase velocity[J]. The Journal of the Acoustical Society of America, 1981, 69(3):696-701.doi:10.1121/1.385566
[40]	ÁLVAREZ F J, KUC R. Dispersion relation for air via Kramers-Kronig analysis[J]. The Journal of the Acoustical Society of America, 2008, 124(2):EL57-EL61.doi:10.1121/1.2947631
[41]	YE ZH. Acoustic dispersion and attenuation in many spherical scatterer systems and the Kramers-Kronig relations[J]. Journal of the Acoustical Society of AmericaJ, 1997, 101(6):3299-3305.doi:10.1121/1.418311
[42]	BOYD R W.Nonlinear Optics[M]. 2nd ed. London: Academic, 2003.
[43]	LI CH. Nonlinear Optics:Principles and Applications[M]. Singapore:Springer, 2017.
[44]	HUTCHINGS D C, SHEIK-BAHAE M, HAGAN D J, et al.. Kramers-Kronig relations in nonlinear optics[J]. Optical and Quantum Electronics, 1992, 24(1):1-30.http://d.old.wanfangdata.com.cn/OAPaper/oai_arXiv.org_physics%2f0302080
[45]	BASSANI F, SCANDOLO S. Dispersion relations and sum rules in nonlinear optics[J]. Physical Review B, 1991, 44(16):8446-8453.doi:10.1103/PhysRevB.44.8446
[46]	PEIPONEN K E. Sum rules for the nonlinear susceptibilities in the case of sum frequency generation[J]. Physical Review B, 1987, 35(8):4116-4117.doi:10.1103/PhysRevB.35.4116
[47]	BASSANI F, LUCARINI V. General properties of optical harmonic generation from a simple oscillator model[J]. Nuovo Cimento D, 1998, 20(7-8):1117-1125.doi:10.1007/BF03185520
[48]	BASSANI F, LUCARINI V. Asymptotic behaviour and general properties of harmonic generation susceptibilities[J]. The European Physical Journal B-Condensed Matter and Complex Systems, 2000, 17(4):567-573.doi:10.1007/PL00011069
[49]	SAARINEN J J. Sum rules for arbitrary-order harmonic generation susceptibilities[J]. The European Physical Journal B-Condensed Matter and Complex Systems, 2002, 30(4):551-557.doi:10.1140/epjb/e2002-00413-5
[50]	SHEIKBAHAE M, HUTCHINGS D C, HAGAN D J, et al.. Dispersion of bound electronic nonlinear refraction in solids[J]. IEEE Journal of Quantum Electronics, 1991, 27(6):1296-1309.doi:10.1109/3.89946
[51]	MILLER D A B, SEATON C T, PRISE M E, et al.. Band-gap-resonant nonlinear refraction in Ⅲ-V semiconductors[J]. Physical Review Letters, 1981, 47(3):197-200.doi:10.1103/PhysRevLett.47.197
[52]	CHEMLA D S, MILLER D A B, SMITH P W, et al.. Room temperature excitonic nonlinear absorption and refraction in GaAs/AlGaAs multiple quantum well structures[J]. IEEE Journal of Quantum Electronics, 1984, QE-20(3):265-275.
[53]	LEE Y H, CHAVEZ-PIRSON A, KOCH S W, et al.. Room-temperature optical nonlinearities in GaAs[J]. Physical Review Letters, 1986, 57(19):2446-2449.doi:10.1103/PhysRevLett.57.2446
[54]	RIDENER JR F L, GOOD JR R H. Dispersion-relations for third-degree nonlinear-systems[J]. Physical Review B, 1974, 10(12):4980-4987.doi:10.1103/PhysRevB.10.4980
[55]	RIDENER JR F L, GOOD JR R H. Dispersion-relations for nonlinear-systems of arbitrary degree[J]. Physical Review B, 1975, 11(8):2768-2770.doi:10.1103/PhysRevB.11.2768
[56]	KADOR L. Kramers-Kronig relations in nonlinear optics[J]. Applied Physics Letters, 1995, 66(22):2938-2939.doi:10.1063/1.114235
[57]	KOGAN S M. On the electrodynamics of weakly nonlinear media[J]. Soviet Physics Jetp, 1963, 16(1):217-219.
[58]	BOWLDEN H J, WILMSHURST J K. Evaluation of the one-angle reflection technique for the determination of optical constants[J]. Journal of the Optical Society of America, 1963, 53(9):1073-1078.doi:10.1364/JOSA.53.001073
[59]	MYHRE C E L, CHRISTENSEN D H, NICOLAISEN F M, et al.. Spectroscopic study of aqueous H₂SO₄at different temperatures and compositions:variations in dissociation and optical properties[J]. The Journal of Physical Chemistry A, , 2003, 107(12):1979-1991.doi:10.1021/jp026576n
[60]	HERBIN H, PUJOL O, HUBERT P, et al.. New approach for the determination of aerosol refractive indices-Part I:theoretical bases and numerical methodology[J]. Journal of Quantitative Spectroscopy and Radiative Transfer, 2017, 200:311-319.doi:10.1016/j.jqsrt.2017.03.005
[61]	GOTTLIEB M. Optical properties of lithium fluoride in the infrared[J]. Journal of the Optical Society of America, 1960, 50(4):343-349.doi:10.1364/JOSA.50.000343
[62]	THOMAS D G, HOPFIELD J J. Exciton spectrum of cadmium sulfide[J]. Physical Review, 1959, 116(3):573-582.doi:10.1103/PhysRev.116.573
[63]	SPITZER W G, KLEINMAN D A. Infrared lattice bands of quartz[J]. Physical Review, 1961, 121(5):1324-1335.doi:10.1103/PhysRev.121.1324
[64]	SON H J, CHOI D H, PARK J S. Improved thickness estimation of liquid water using Kramers-Kronig relations for determination of precise optical parameters in terahertz transmission spectroscopy[J]. Optics Express, 2017, 25(4):4509-4518.doi:10.1364/OE.25.004509
[65]	BACHRACH R Z, BROWN F C. Exciton-optical properties of tlbr and tlcl[J]. Physical Review B, 1970, 1(2):818-831.doi:10.1103/PhysRevB.1.818
[66]	AHRENKIE R K. Modified Kramers-Kronig analysis of optical spectra[J]. Journal of the Optical Society of America, 1971, 61(12):1651-1655.doi:10.1364/JOSA.61.001651
[67]	LUCARINI V, SAARINEN J J, PEIPONEN K E. Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities[J]. Optics Communications, 2003, 218(4-6):409-414.doi:10.1016/S0030-4018(03)01259-8
[68]	PALMER K F, WILLIAMS M Z, BUDDE B A. Multiply subtractive Kramers-Kronig analysis of optical data[J]. Applied Optics, 1998, 37(13):2660-2673.doi:10.1364/AO.37.002660
[69]	GRANOT E, BEN-ADERET Y, STERNKLAR S. Differential multiply subtractive Kramers-Kronig relations[J]. Journal of the Optical Society of America B, 2008, 25(4):609-613.doi:10.1364/JOSAB.25.000609
[70]	BEN-ADERET Y, GRANOT E, STERNKLAR S, et al.. Spectral analysis of a one-dimensional scattering medium with the differential multiply subtractive Kramers-Kronig method[J]. Journal of the Optical Society of America B, 2009, 26(1):125-128.doi:10.1364/JOSAB.26.000125
[71]	HORSLEY S A R, ARTONI M, LA ROCCA G C. Spatial Kramers-Kronig relations and the reflection of waves[J]. Nature Photonics, 2015, 9(7):436-439.doi:10.1038/nphoton.2015.106
[72]	LONGHI S. Wave reflection in dielectric media obeying spatial Kramers-Kronig relations[J]. Europhysics Letters, 2015, 112(6):64001.doi:10.1209/0295-5075/112/64001
[73]	KOBER H. A note on Hilbert's operator[J]. Bulletin of the American Mathematical Society, 1942, 48(6):421-427.doi:10.1090/S0002-9904-1942-07688-9
[74]	LONGHI S. Bidirectional invisibility in Kramers-Kronig optical media[J]. Optics Letters, 2016, 41(16):3727-3730.doi:10.1364/OL.41.003727
[75]	PHILBIN T G. All-frequency reflectionlessness[J]. Journal of Optics, 2016, 18(1):01LT01.http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=30019534296ebe105e4635bcfaa2d372
[76]	KING C G, HORSLEY S A R, PHILBIN T G. Zero reflection and transmission in graded index media[J]. Journal of Optics, 2017, 19(8):085603.doi:10.1088/2040-8986/aa7783
[77]	YE D X, CAO C, ZHOU T Y, et al.. Observation of reflectionless absorption due to spatial Kramers-Kronig profile[J]. Nature Communications, 2017, 8(1):51.doi:10.1038/s41467-017-00123-4
[78]	MECOZZI A, ANTONELLI C, SHTAIF M. Kramers-Kronig coherent receiver[J]. Optica, 2016, 3(11):1220-1227.doi:10.1364/OPTICA.3.001220
[79]	ANTONELLI C, MECOZZI A, SHTAIF M. Kramers-Kronig PAM transceiver and two-sided polarization-multiplexed Kramers-Kronig transceiver[J]. Journal of Lightwave Technology, 2018, 36(2):468-475.doi:10.1109/JLT.2018.2796306
[80]	CHEN X, ANTONELLI C, CHANDRASEKHAR S, et al.. Kramers-Kronig receivers for 100-km datacenter interconnects[J]. Journal of Lightwave Technology, 2018, 36(1):79-89.doi:10.1109/JLT.2018.2793460
[81]	HOANG T M, SOWAILEM M Y S, ZHUGE Q B, et al.. Single wavelength 480 Gb/s direct detection over 80 km SSMF enabled by Stokes vector Kramers -Kronig transceiver[J]. Optics Express, 2017, 25(26):33534-33542.doi:10.1364/OE.25.033534
[82]	汪小佳, 朱仁传, 洪亮.有航速Kramers-Kronig关系及浮体运动的间接时域法[J].中国造船, 2018, 59(2):9-23.doi:10.3969/j.issn.1000-4882.2018.02.002 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