摘要
空间中波动的基本类型一般有平面波、柱面波和球面波等几类。传统的非线性光学一般是建立在平面电磁波在非线性介质中的传播的问题上。柱面非线性光学主要研究柱面电磁波在各种非线性非均匀介质中的传播,在理解柱面电磁波和物质相互作用领域起着重要作用。作为几种基本的电磁波之一,柱面电磁波在各种非线性非均匀介质中的传播是一个古老、有趣而困难的话题,特别是对柱面电磁波在各种非线性非均匀受限介质中的传播进行精确求解,更是一个新兴的领域。柱面电磁波的偏振、传播等特性都和平而电磁波很不相同,例如：柱而腔中的电磁场的本征模式依赖于电磁场的偏振方向；对于非线性介质中的柱而电磁波来说,不同的偏振对应着不同的非线性效应；充满非线性介质的柱而腔中会出现一种全新的电磁模式等等。由于柱面电磁波的新奇特性,对于其在各种非线性非均匀介质中的研究具有重要的理论意义。不仅如此,对柱面受限体系非线性光学现象的研究能够揭示出一些新的物理内容,在高精度光学测量技术、光学成像、全光信息处理器件等高新技术领域有潜在的重要应用前景。
本论文研究柱面电磁波在非线性非均匀介质中传播的各种性质,主要工作包括以下几个方面：
1.研究了求解非线性非均匀受限介质中柱面电磁波传播精确解的方法,给出了一系列非线性非均匀介质中轴对称电磁场的精确解。这些精确解不但可以用来描述弱非线性介质中的柱面电磁波传播,而且适用于强非线性情况,可以用来处理自陡峭效应以及电磁冲击波等非微扰现象,对非线性非均匀介质中的柱面电磁波由小振幅弱非线性过渡到大振幅强非线性以至电磁冲击波的形成,都可以给出完整清晰的物理图像和准确的数学描述。
2.我们提出了一整套有效的数学方法对这些精确解进行系统地分析。利用这些数学方法,我们发现二次谐波产生、和差频产生、光克尔效应、电光效应等非线性效应可以很简单自然地从精确解中推导出来,而且可以给出相应的解析表达式。这些解析表达式和数值求解耦合波方程组得到的结果定量地符合。
3.理论上讨论了受限介质中的非线性光学效应。在空间受到限制时,光的存在形态和传播传播都与无边界情况有显著不同,特别是处于受限的非线性介质中。对受限非线性甚至非均匀介质中的强光光场的演化和传播的研究较之自由空间的情况要困难的多。对于微腔或者波导中的非线性光学问题,传统的观点认为：由于边界条件对二次谐波的禁戒,理想的柱面腔中不存在二次谐波等非线性过程。但是我们利用精确解得出的结论是：理想的柱面腔中可以存在二次谐波等非线性过程。我们分析了精确解中的二次谐波成分,指出满足边界条件并不意味着禁止二次谐波。我们的研究揭示出一种新的满足边界条件的机制。基于这种满足边界条件的机制,我们提出了振幅模式的概念,以及相应的态密度。不仅仅局限于光学领域,在非线性声学中对大振幅声驻波进行系统地、可靠地理论描述仍然是一个悬而未解的难题,我们的工作为系统可靠地研究大振幅声驻波提供了思路。
4.讨论了z偏振和φ偏振的柱面电磁波在介质中传播的情况。柱面电磁波存在一个特定的对称轴z轴,这使得z偏振和φ偏振这两种电场方向并不处于完全相同的地位。我们发现不同偏振的柱面电磁波满足的麦克斯韦方程组的形式有很大的不同,这将带来一系列新的电磁特性：对于柱面腔中的电磁模式来说,不同的偏振对应着完全不同的本征频率谱；对于非线性介质中的柱面电磁波来说,不同的偏振对应着不同的非线性效应。
总之,本论文系统地研究了柱面电磁波在非线性非均匀介质中的传播问题。这些结果对于了解柱面电磁波的传播特性、理解和掌握受限非线性介质中光学现象的规律和特性、探讨非线性偏微分方程的求解甚至处理非线性声学中的大振幅声驻波的问题都有启发意义。
Cylindrical nonlinear optics, which describes cylindrical electromagnetic waves'propagation in media with different types of nonlinearity and inhomo-geneity, is a fundamental branch of nonlinear optics and becomes a burgeoning research area recently. For the features of propagation and polarization, there are many differences between cylindrical electromagnetic waves and traditional plane electromagnetic waves. For example, electromagnetic modes in a cylin-drical cavity depends on the polarization direction of the electromagnetic field; The propagation features of cylindrical electromagnetic waves in a nonlinear medium also depends on the polarization direction of the electromagnetic field; There is a new type of electromagnetic modes:amplitude modes, in which the amplitudes, rather than the frequencies, of the electromagnetic waves are lim-ited by the boundary condition. Due to the novel characteristics of cylindrical electromagnetic wave, the researches of cylindrical electromagnetic waves in inhomogeneous and nonlinear media are of great interest and meaningful for theory and application, and the studies of cylindrical nonlinear optics can al-so deepen the understanding of traditional nonlinear optical effects with plane waves, even provide simple methods of dealing traditional nonlinear optical ef-fects.
In this thesis, we study the features of cylindrical electromagnetic waves' propagation in nonlinear and inhomogeneous media, including the following several aspects:
1. We study the method of solving cylindrical electromagnetic waves prop-agation in nonlinear and inhomogeneous media, and obtain a series of exact so-lutions which describe cylindrical electromagnetic fields in media with different types of nonlinearity and inhomogeneity.
2. We develop a set of effective mathematical methods to analyze the exact solution, and found that most of the nonlinear optical effects, including second-harmonic generation, sum-and difference-frequency generation, op-tical Kerr effect, and linear electrooptic effect, come out quite naturally from the exact solution, and the analytical expressions for these effects can be ob-tained. We also deduce coupled-wave equations which describe the interaction between cylindrical electromagnetic waves and nonlinear inhomogeneous me-dia. Using the coupled-wave equations, we analyze some nonlinear effects, and compare the results obtained from the two different methods, the method of us-ing exact solutions and the method of using coupled-wave equations, and find that descriptions of nonlinear effects by the coupled-wave equations are in good agreement with the exact solutions. We also show that the traditional slowly-varying-envelope approximation, which is widely used in nonlinear optics of plane waves, is inapplicable in cylindrical nonlinear optics.
3. We theoretically analyze nonlinear optics in a cylindrical ideal cavi-ty. Nonlinear optical processes under boundary conditions (such as cavity and waveguide) is a fundamental problem in optics, and the traditional view of such processes is that the electromagnetic modes with frequencies of second-and higher-order harmonics are not allowed by the boundary conditions, regardless of the function of polarizability P depends on E. However, both the numer-ical and analytical methods of using the exact solution suggest that there are second-and higher-order harmonics generation in such a cylindrical cavity, and it can be easily verified that the exact solution satisfies the boundary conditions. These electromagnetic oscillations obey a new type of electromagnetic modes: amplitude modes, in which the amplitudes, rather than the frequencies, of the electromagnetic waves are limited by the boundary condition. We furthermore show that the photon state density of the system is needed to be modified when taking account of the amplitude modes.
4. Starting from the Maxwell equations, the propagation of cylindrical electromagnetic waves with z polarization and (?) polarization in media are dis-cussed. We find that the equations of z-polarized cylindrical electromagnetic wave are very different from (?)-polarized, which may lead to a series of new properties. For electromagnetic modes of a cylindrical cavity, eigenfrequencies are related to the polarization; For cylindrical electromagnetic waves in nonlin-ear media, nonlinear effects are also related to the polarization.
In a word, this thesis is the summary of preliminary exploration of cylin-drical nonlinear optics. These findings may shed light on solving nonlinear partial differential equation and understanding the propagation features of cylindrical electromagnetic wave.
引文
[1]Petrov E Y and Kudrin A V. Exact axisymmetric solutions of the Maxwell equations in a nonlinear nondispersive medium. Phys. Rev. Lett.,2010, 104(19):190404(1-4)
[2]Whitham G B. Linear and Nonlinear Waves. Wiley, New York,1974
[3]Fokas A S. Integrable Nonlinear Evolution Partial Differential Equations in 4+2 and 3+1 Dimensions. Phys. Rev. Lett.,2006,96(19):190201(1-4)
[4]郭伯灵.非线性演化方程：上海：上海科技教育出版社,1993
[5]Nozaki K and Bekki N. Chaos in a perturbed nonlinear Schrodinger equation. Phys. Rev. Lett,1983,50(17):1226-1269
[6]郭柏灵和庞小峰.孤立子.上海：科学出版社,1987
[7]谷超豪等.孤子理论与应用.浙江：浙江科技出版社,1987
[8]Hasegawa A and Kodama Y. Solitons in Optical Communications. Oxford: Oxford University Press,1995
[9]Zabusky N J and Kruskal M D. Interaction of " Solitons " in a Collisionless Plas-ma and the Recurrence of Initial States. Phys. Rev. Lett.,1965,15(6):240-243
[10]Liang Z X, Zhang Z D, and Liu W M. Dynamics of a Bright Soliton in Bose-Einstein Condensates with Time-Dependent Atomic Scattering Length in an Ex-pulsive Parabolic Potential. Phys. Rev. Lett.,2005,94(5):050402(1-4)
[11]Segev M and Stegeman G I. Self-Trapping of Optical Beams:Spatial Solitons. Phys. Today,1998,51(1):43-48
[12]Segev M, et al. Steady-State Spatial Screening Solitons in Photorefractive Ma-terials with External Applied Field. Phys. Rev. Lett.,1994,73(24):3211-3214
[13]Wu Y and Deng L. Ultraslow bright and dark optical solitons in a cold three-state medium. Opt. Lett.,2004,29:2064-2066
[14]Wu Y and Deng L. Ultraslow Optical Solitons in a Cold Four-State Medium. Phys. Rev. Lett.,2004,93:143904(1-4)
[15]Kudrin A V and Petrov E Y. Cylindrical electromagnetic waves in a nonlinear nondispersive medium:Exact solutions of the Maxwell equations. Journal of Experimental and Theoretical Physics,2010,110(3):537-548
[16]Armstrong J A. Interactions between light waves in a nonlinear dielectric. Phys. Rev.,1962,127(6):1918-1939
[17]Bloembergen N. Light Waves at the Boundary of Nonlinear Media. Phys. Rev., 1962,128(2):606-622
[18]Bloembergen N. Nonlinear optics and spectroscopy. Rev. Mod. Phys.,1982, 54(3):685-695
[19]叶佩弦.北京大学物理学丛书·理论物理专辑L:非线性光学物理.北京：北京大学出版社,2007：1-53
[20]钱士雄,王恭明.非线性光学：原理与进展.上海：复旦大学出版社,2007：10-53
[21]Franken P A, Hill A E, Peters C W, and Weinreich G. Nonlinear optics and spectroscopy. Phys. Rev. Lett.,1961,7(4):118-119
[22]Chiao R Y, Garmire E, and Townes C H. Self-Trapping of Optical Beams. Phys. Rev. Lett.,1964,13(15):479-482
[23]Bjorkholm J E and Ashkin A A. cw Self-Focusing and Self-Trapping of Light in Sodium Vapor. Phys. Rev. Lett.,1974,32(4):129-132
[24]Duree G C, et al. Observation of self-trapping of an optical beam due to the photorefractive effect. Phys. Rev. Lett.,1993,71(4):533-536
[25]Harris S E. Electromagnetically induced transparency. Phys. Today,1997, 50(7):36-42
[26]Harris S E and Hau L V. Nonlinear Optics at Low Light Levels. Phys. Rev. Lett., 1999,82(23):4611-4614
[27]Schmidt H and Imamoglu A. Giant Kerr nonlinearities obtained by electromag-netically induced transparency. Opt. Lett.,1996,21:1936-1938
[28]Lukin M D and Imamoglu A. Nonlinear Optics and Quantum Entanglement of Ultraslow Single Photons. Phys. Rev. Lett.,2000,84:1419-1422
[29]Agarwal GP. Lasers with three-level absorbors. Phys. Rev. A,1981,24(3):1399-1403
[30]Wall D F and Zoller P A Coherent nonlinear mechanism for optical bistability from three level atoms. Opt. Commun.,1980,34(2),260-264
[31]Harshawardhan W and Agarwal G S Controlling optical bistability using elec-tromagnetic field induced transparency and quantum interferences. Phys. Rev. A,1996,53(3):1812-1817
[32]Gong S, Du S, Xu Z, et al. Optical bistability via a phase fluctuation effect of the control field. Phys. Lett. A,1996,222(4):237-240
[33]Hu X M and Xu Z Phase control of amplitude-fluctuation-induced bistability. J. Opt B:Quantum Semiclass. Opt.,2001,3(2):35-38
[34]Chen Z, Du C, Gong S, et al. Optical bistability via squeezed vacuum input. Phys. Lett. A,1999,259(1):15-19
[35]Brown A, Joshi A and Xiao M Controlled steady-state switching in optical bista-bility. Appl. Phys. Lett.,2003,83(7):1301-1303
[36]Schmidt H, Campman K L, Gossard A C, et al. Tunneling induced trans-parency:Fano interference in intersubband transitions. Appl. Phys. Lett.,1997, 70(25):3455-3457
[37]Paspalakis E, Kylstra N J and Knight P L. Transparency induced via decay inter-ference. Phys. Rev. Lett.,1999,82(10):2079-2082
[38]Field J E and Imagoglu A. Spontaneous emission into an electromagnetically induced transparency. Phys. Rev. A,1993,48(3):2486-2489
[39]Li Y Q and Xiao M. Observation of quantum interference between dressed states in an electromagnetically induced transparency. Phys. Rev. A,1995,51(6):4959-4962
[40]Zhang G Z, Katsuragawa M, Hakuta K, et al. Sum-frequency generation using srtong-field and induced transparency in atomic hydrogen. Phys. Rev. A,1995, 52(2):1584-1593
[41]Harris S E. Electromagnetically induced transparency with matched pulses. Phys. Rev. Lett.,1993,72(5):552-555
[42]Braje D A, Balic V, Goda S, et al. Frequency mixing using electromagnetically induced transparency in cold atoms. Phys. Rev. Lett.,2004,93(18):183601(1-4)
[43]Xiong H, Si L G, Huang P, and Yang X. Analytic description of cylindrical elec-tromagnetic wave propagation in an inhomogeneous nonlinear and nondispersive medium. Phys. Rev. E,2010,82(5):057602(1-4)
[44]Es'kin V A, Kudrin A V, and Petrov E Y. Exact solutions for the source-excited cylindrical electromagnetic waves in a nonlinear nondispersive medium. Phys. Rev. E,2011,83(6):067602(1-4)
[45]Xiong H, Si L G, Guo J F, Lii X Y, and Yang X. Classical theory of cylin-drical nonlinear optics:Second-harmonic generation. Phys, Rev. E,2011, 83(6):063845(1-8)
[46]Xiong H, Si L G, Ding C, Yang X, and Wu Y. Classical theory of cylindrical nonlinear optics:Sum-and difference-frequency generation. Phys. Rev. A,2011, 84(4):043841(1-6)
[47]Xiong H, Si L G, Ding C, Lu X Y, Yang X, and Wu Y. Solutions of the cylindrical nonlinear Maxwell equations. Phys. Rev. E,2012,85(1):016602(1-5)
[48]Xiong H, Si L G, Ding C, Yang X, and Wu Y. Second-harmonic generation of cylindrical electromagnetic waves propagating in an inhomogeneous and nonlin-ear medium. Phys. Rev. E,2012,85(1):016606(1-8)
[49]Chew W C. Waves and Fields in Inhomogeneous Media. Van Nostrand, New York,1990
[50]钱祖文.非线性声学(第二版).北京：科学出版社,2009
[51]张海澜.理论声学(修订版).北京：高等教育出版社,2012
[52](俄罗斯)古尔巴托夫(S.N.Gurbatov).非线性非分散介质中的波与结构：非线性声学的一般理论及应用.北京：高等教育出版社,2011
[53]Solli D R, Ropers C, Koonath P, and Jalali B. Optical rogue waves. Nature, 450(7172):1054-1057
[54]Soto-Crespo J M, Grelu P, and Akhmediev N. Dissipative rogue waves:Ex-treme pulses generated by passively mode-locked lasers. Phys. Rev. E,2011, 84(1):016604(1-7)
[55]Pisarchik A N, et al. RogueWaves in a Multistable System. Phys. Rev. Lett., 2011,107(27):274101(1-5)
[56]Boyd R W. Nonlinear optics(Third Edition). Elsevier, Singapore,2010
[57]Jackson J D. Classical Electrodynamics. Wiley, New York,1998
[58]Zeng Y, Hoyer W, Liu J, Koch S W, and Moloney J V. Classical theory for second-harmonic generation from metallic nanoparticles. Phys. Rev. B,2009, 79(23):235109(1-9)
[59]彭金生,李高翔.近代量子光学导论(第一版).北京：科学出版社,1996：295-305.
[60]Courant R and Hilbert D. Methods of Mathematical physics. Wiley, New York, 1989
[61]Shen Y R. Principles of Nonlinear Optics. Wiley, New York,1984
[62]Yang S A, Li X, Bristow A D, and Sipe J E. Second harmonic generation from tetragonal centrosymmetric crystals. Phys. Rev. B,2009,80(16):165306(1-9)
[63]Hewageegana P and Apalkov V. Second harmonic generation in disordered me-dia:Random resonators. Phys. Rev. B,2008,77(7):075132(1-8)
[64]Saltiel S M, et al. Spatiotemporal toroidal waves from the transverse second-harmonic generation. Opt Lett.,2008,33(5):527-529
[65]Saltiel S M, et al. Generation of Second-Harmonic Conical Waves via Nonlinear Bragg Diffraction. Phys. Rev. Lett.,2008,100(10):103902(1-4)
[66]Lin H B and Campillo A J. cw Nonlinear Optics in Droplet Microcavities Dis-playing Enhanced Gain. Phys. Rev. Lett.,1994,73(18):2440-2443
[67]Nakayama Y, Pauzauskie P J, Radenovic A, et al. Tunable nanowire nonlinear optical probe. Nature,2007,447(7148):1098-1101
[68]Poutrina E, Huang D, and Smith D R. Analysis of nonlinear electromagnetic metamaterials. NewJ. Phys.,2010,12(9):093010
[69]Petschulat J, Chipouline A, TUnnermann A, et al. Multipole nonlinearity of meta-materials. Phys. Rev. A,2009,80(6):063828(1-7)
[70]Rose A and Smith D R. Broadly tunable quasi-phase-matching in nonlinear metamaterials. Phys. Rev. A,2011,84(1):013823(1-5)
[71]Powell D A, Shadrivov I V, and Kivshar Y S. Nonlinear electric metamaterials. Appl. Phys. Lett.,2009,95(8):084102(1-3)
[72]Lapine M and Gorkunov M. Three-wave coupling of microwaves in meta-material with nonlinear resonant conductive elements. Phys. Rev. E,2004, 70(6):066601(1-8)
[73]Rose A, Huang D, and Smith D R. Controlling the Second Harmonic in a Phase-Matched Negative-Index Metamaterial. Phys. Rev. Lett.,2011,107(6):063902(l-4)
[74]Belkin M A, Capasso F, Belyanin A, et al. Terahertz quantum-cascade-laser source based on intracavity difference-frequency generation. Nature Photonics, 2007, 1(5):288-292
[75]Fu L, Ma G, and Yan E C Y. In Situ Misfolding of Human Islet Amyloid Polypep-tide at Interfaces Probed by Vibrational Sum Frequency Generation. J. Am. Chem. Soc.,2010,132(15):5405-5412
[76]Ji N, Ostroverkhov V, Tian C S, and Shen Y R. Characterization of Vibrational Resonances of Water-Vapor Interfaces by Phase-Sensitive Sum-Frequency Spec-troscopy. Phys. Rev. Lett.,2008,100(9):096102(1-4)
[77]Shen Y R. Surface properties probed by second-harmonic and sum-frequency generation. Nature,1989,447(6207):519-525
[78]Anglin T C and Conboy J C. Kinetics and Thermodynamics of Flip-Flop in Bi-nary Phospholipid Membranes Measured by Sum-Frequency Vibrational Spec-troscopy. Biochemistry,2009,48(43):10220-10234
[79]Fu Y,Wang H, Shi R, and Cheng J X. Second Harmonic and Sum Frequency Generation Imaging of Fibrous Astroglial Filaments in Ex Vivo Spinal Tissues. Biophys. J.,2007,92(9):3251-3259
[80]Lu X, Shephard N, Han J, et al. Probing Molecular Structures of Polymer/Metal Interfaces by Sum Frequency Generation Vibrational Spectroscopy. Macro-molecules,2008,41(22):8770-8777
[81]Hagenmuller D and Ciuti C. Cavity QED of the Graphene Cyclotron Transition. Phys. Rev. Lett.,2012,109(26):267403(1-5)
[82]Brierley R T, Creatore C, Littlewood P B, and Eastham P R. Adiabatic S-tate Preparation of Interacting Two-Level Systems. Phys. Rev. Lett.,2012, 109(4):043002(1-5)
[83]Hughes S and Carmichael H J. Stationary Inversion of a Two Level System Coupled to an Off-Resonant Cavity with Strong Dissipation. Phys. Rev. Lett., 2011,107(19):193601(1-4)
[84]Wu Y. Simple algebraic method to solve a coupled-channel cavity QED model. Phys. Rev. A,1996,54(5):4534-4543
[85]Wu Y and Yang X. Jaynes-Cummings Model for a Trapped Ion in Any Position of a Standing Wave. Phys. Rev. Lett.,1997,78(16):3086-3088
[86]Schmidt S and Blatter G. Strong Coupling Theory for the Jaynes-Cummings-Hubbard Model. Phys. Rev. Lett.,2009,103(8):086403(1-4)
[87]Hong H G, et al. Spectrum of the Cavity-QED Microlaser:Strong Coupling Effects in the Frequency Pulling at Off Resonance. Phys. Rev. Lett.,2012, 109(24):243601(1-5)
[88]Madsen K H, et al. Observation of Non-Markovian Dynamics of a Single Quan-tum Dot in a Micropillar Cavity. Phys. Rev. Lett.,2011,106(23):233601(1-4)
[89]Eto Y, et al. Projective Measurement of a Single Nuclear Spin Qubit by Using Two-Mode Cavity QED. Phys. Rev. Lett.,2011,106(16):160501 (1-4)
[90]Cary J R and Xiang N. Wave excitation in inhomogeneous dielectric media. Phys. Rev. E,2007,76(5):055401(R)(1-4)
[91]Jiang W X, et al. Suppression of spirals and turbulence in inhomogeneous ex-citable media. Phys. Rev. E,2008,78(6):066607(1-4)
[92]Choubani M, et al. Analysis and design of electromagnetic band gap structures with stratified and inhomogeneous media. Opt. Commun.,2010, 283(22):4499-4504
[93]Cui T J, et al. Metamaterials:Theory, Design, and Applications. Springer, New York,2010
[94]Ruan Z C, et al. Ideal cylindrical cloak:perfect but sensitive to tiny perturbations. Phys. Rev. Lett.,2007,99(11):113903(1-4)
[95]Yan M, Ruan Z, and Qiu M. Cylindrical Invisibility Cloak with Simplified Mate-rial Parameters is Inherently Visible. Phys. Rev. Lett,2007,99(23):233901(1-4)
[96]Lai Y. Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell. Phys. Rev. Lett,2009,102(9):093901(1-4)
[97]You Y, Kattawar G W, and Yang P. Invisibility cloaks for toroids. Opt. Express, 2009,17(8):6591-6599
[98]Chen J X, et al. Suppression of spirals and turbulence in inhomogeneous ex-citable media. Phys. Rev. E,2009,79(6):066209(1-8)
[99]Levy U, et al. Inhomogenous Dielectric Metamaterials with Space-Variant Po-larizability. Phys. Rev. Lett,2007,98(24):243901(1-4)
[100]刘式适.物理学中的非线性方程.北京：北京大学出版社,2000
[101]梁昆淼.数学物理方法(第三版).北京：高等教育出版社,1998