不同媒质模型左手材料的时域FDTD分析及方法研究
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摘要
左手材料是当前科学研究的热点之一,在电磁学、光学和材料学领域都有着巨大的应用前景。对于其特性的研究,在理论分析、数值计算和实际应用等方面还需要不断地完善和深入发展。特别是利用数值计算方法对左手材料电磁特性进行分析,避开了解析方法中繁琐的推导和复杂的数学计算,在一定的精度范围内,可以灵活地分析目标的电磁性能,大大提高分析的效率。同时,通过现代先进的作图手段,可以很直观地了解电磁场在目标周围空间和时间上的变化过程和趋势,提高数值分析的有效性。时域有限差分(FDTD)算法从麦克斯韦(Maxwell)方程组出发,不需要任何导出方程,避免了使用更多的数学工具,具有简单、直观、易于矢量化和并行化的特点,被广泛的应用于求解各种复杂情况的电磁问题中。但是,由于左手材料具有介电常数和磁导率同时为负数的媒质特性,应用传统的时域有限差分法直接分析左手材料中的电磁波,将会造成计算结果发散,使数值计算无法进行。
本文主要对不同数学模型左手材料中传统的时域有限差分算法进行改进,利用计算机编程技术,成功实现了对它们电磁特性的准确数值分析。同时,通过大量研究工作,讨论了各模型左手材料的负折射和电磁平板聚焦等现象。提出了一些具有特殊性质的左手材料系统结构,探索了其在微波中的一些潜在应用。本文的主要工作可以概括为:
1.本文简要分析了左手材料的特殊性质及其研究的理论基础。掌握并熟练应用时域有限差分法对典型的电磁场问题进行时域的数值分析。
2.本文对传统的时域有限差分法进行了改进,成功将改进后的时域有限差分法应用到左手材料的数值计算中去。解决了直接修改传统时域有限差分法中的媒质等效介电常数和等效磁导率所产生的数值发散问题。
3.本文通过数学推导,分别给出了TM模和TE模电磁波在冷等离子体媒质模型下的左手材料中的改进的时域有限差分法数值计算公式以及该左手材料中的完美匹配层(PML)的数值计算表达式。引入背景媒质参数ε′_r和μ′_r,对二维有限长和无限长左手材料平板中的电磁波的传播特性进行了数值分析。同时,分别计算了不同负折射率边界条件下,左手材料平板的电磁波折射规律,数值验证了电磁波在二维左手材料中的传播满足推广的斯涅尔(Snell)定律。
4.本文进一步引入了更接近实际材料的Drude媒质模型下的左手材料;详细推导了Drude模型下左手材料中的时域有限差分法数值计算公式。讨论了有损耗左手材料平板的电磁特性;分析了不同损耗因子y的有损耗左手材料平板中电磁波的分布变化。同时,计算了不同长度的有损耗左手材料平板中的电磁场分布。数值分析了一种利用有损耗左手材料平板来实现电磁能量聚焦的系统结构。数值分析结果,验证了该结构的能量局域化特性。
5.本文详细讨论了具有更普适特性的洛伦兹(Lorentz)模型下的左手材料的特性。以第三章、第四章和第五章中的FDTD分析方法为基础,推导了洛伦兹模型下左手材料中的电磁场FDTD计算公式。数值模拟了该模型下不同厚度的左手材料平板结构的电磁场分布情况。根据左手材料平板的聚焦性质,提出了一种基于左手材料平板的电磁波多次聚焦结构,并对其作了数值分析,得到了较一致的结果。最后,利用该FDTD数值方法,模拟分析了左手材料平板的分层聚焦特性。采用电磁场数值仿真软件Ansoft,设计了具有折射率近似等于-1的实际金属阵列结构。同时,利用该结构进行了多层平板的Ansoft数值仿真实验,实验结果与FDTD数值仿真结果得到较好的吻合。
6.本文最后结合前面几章对线性左手材料的分析结果,初步讨论了非线性左手材料的特性。推导了克尔(Kerr)非线性左手材料中的FDTD计算式。运用该方法数值模拟了具有该非线性效应左手材料平板中的电磁波,得到了稳定的数值结果。与线性左手材料平板结果进行了对比,数值计算和分析了波源到该平板不同距离情况下,该非线性左手材料平板对电磁波聚焦点的影响。
Presently,left-handed materials(LHMs)is one of the hot research interests for it has intensively potential application in the fields of Electromagnetics,Optics and Materials.However,for the newly artificial electromagnetic materials,these traditional theories,numerical methods and applications of electromagnetic fields need to be developed.Especially,by using the numerical methods to analyze the characteristics of the LHMs and the distributions of electromagnetic fields in the LHMs would greatly enhance the people's right understanding for these newly man-made materials and reduce the research time and costs.Besides,the variation of electromagnetic fields with time could be easily and clearly illustrated by recent drawing tools. Finite-difference time-domain(FDTD)method is directly derived by Maxwell's Equations which is widely used in many complex conditions.However,the traditional FDTD method would make the numerical simulation of LHMs divergence.
In this thesis,finite-difference time-domain(FDTD)methods in different mathematic modeling LHMs are developed.Programs based on these modified algorithms are performed in order to analyze the electromagnetic fields in such LHMs. According a great quantity of calculations,the characteristics of different modeling LHMs are discussed.Some special LHMs structures or systems are proposed.Their potential applications are explored.The author's work is mainly focused on:
1.The special characteristics and the basic theory of LHMs are introduced. Finite-difference time-domain(FDTD)method is grasped and applied masterly to the numerical analysis of electromagnetic problems.
2.Traditional FDTD method is modified.Such modified FDTD method is successfully applied in the numerical calculation of LHMs.It could solve the numerical problem of traditional FDTD in LHMs simulation.
3.TM and TE mode electromagnetic waves are separately concerned.The numerical expressions of the perfect-matched layer(PML)and such modified FDTD method in LHMs modeled by cold plasma model are derived.The permittivityε'_r and permeabilityμ'_r of back ground media are introduced.Two dimensional finite LHMs slab and infinite LHMs slab are numerically analyzed by such modified method.Besides,the refractional phenomena of electromagnetic waves in LHMs slab surface with different refraction-index are simulated.The extended Shell's law which electromagnetic waves satisfy in the LHMs could be verified by the numerical results.
4.The much more real media model,Drude model,compared to the cold plasma model is introduced into the LHMs.The numerical expressions of the modified FDTD method in Drude model LHMs are derived in detail.The lossy LHMs slab is studied.The distributions of electromagnetic waves around such LHMs slab with different lossy factorγare calculated.An electromagnetic energy localizing system with such lossy LHMs slab is numerically analyzed.The characteristic of electromagnetic energy localization in such system could be proved by the numerical results.
5.The most general model,Lorentz model,is introduced in LHMs.Based on the modified FDTD theory of chapter 3,chapter 4 and chapter 5,the numerical expressions of the modified FDTD method in Lorentz model LHMs are derived. The distributions of electromagnetic waves around such LHMs slab with different thickness are calculated.A multipoint focusing system with LHMs is discussed. According to the numerical results,these multipoint focuses certainly agree with the focusing theory of LHMs.Besides,a multi-layer LHMs system with multi-layer focusing characters is also discussed.A real metal-ring LHMs structure with the refraction-index approximately to -1 is designed which is applied to the multi-layer LHMs system.Ansoft is used to simulate the electromagnetic waves around the real LHMs.Great agreement could be seen between the FDTD results and the Ansoft results.
6.Nonlinear LHMs is primarily discussed.The numerical expressions of the modified FDTD method in Nonlinear LHMs with Kerr effects are derived.The distribution of electromagnetic waves around such nonlinear LHMs slab is calculated steadily. Compare to the linear LHMs,the electromagnetic focus could also be observed in this nonlinear LHMs slab.The location of the focus would vary with the distance between the source and the nonlinear LHMs slab.
本文主要对不同数学模型左手材料中传统的时域有限差分算法进行改进,利用计算机编程技术,成功实现了对它们电磁特性的准确数值分析。同时,通过大量研究工作,讨论了各模型左手材料的负折射和电磁平板聚焦等现象。提出了一些具有特殊性质的左手材料系统结构,探索了其在微波中的一些潜在应用。本文的主要工作可以概括为:
1.本文简要分析了左手材料的特殊性质及其研究的理论基础。掌握并熟练应用时域有限差分法对典型的电磁场问题进行时域的数值分析。
2.本文对传统的时域有限差分法进行了改进,成功将改进后的时域有限差分法应用到左手材料的数值计算中去。解决了直接修改传统时域有限差分法中的媒质等效介电常数和等效磁导率所产生的数值发散问题。
3.本文通过数学推导,分别给出了TM模和TE模电磁波在冷等离子体媒质模型下的左手材料中的改进的时域有限差分法数值计算公式以及该左手材料中的完美匹配层(PML)的数值计算表达式。引入背景媒质参数ε′_r和μ′_r,对二维有限长和无限长左手材料平板中的电磁波的传播特性进行了数值分析。同时,分别计算了不同负折射率边界条件下,左手材料平板的电磁波折射规律,数值验证了电磁波在二维左手材料中的传播满足推广的斯涅尔(Snell)定律。
4.本文进一步引入了更接近实际材料的Drude媒质模型下的左手材料;详细推导了Drude模型下左手材料中的时域有限差分法数值计算公式。讨论了有损耗左手材料平板的电磁特性;分析了不同损耗因子y的有损耗左手材料平板中电磁波的分布变化。同时,计算了不同长度的有损耗左手材料平板中的电磁场分布。数值分析了一种利用有损耗左手材料平板来实现电磁能量聚焦的系统结构。数值分析结果,验证了该结构的能量局域化特性。
5.本文详细讨论了具有更普适特性的洛伦兹(Lorentz)模型下的左手材料的特性。以第三章、第四章和第五章中的FDTD分析方法为基础,推导了洛伦兹模型下左手材料中的电磁场FDTD计算公式。数值模拟了该模型下不同厚度的左手材料平板结构的电磁场分布情况。根据左手材料平板的聚焦性质,提出了一种基于左手材料平板的电磁波多次聚焦结构,并对其作了数值分析,得到了较一致的结果。最后,利用该FDTD数值方法,模拟分析了左手材料平板的分层聚焦特性。采用电磁场数值仿真软件Ansoft,设计了具有折射率近似等于-1的实际金属阵列结构。同时,利用该结构进行了多层平板的Ansoft数值仿真实验,实验结果与FDTD数值仿真结果得到较好的吻合。
6.本文最后结合前面几章对线性左手材料的分析结果,初步讨论了非线性左手材料的特性。推导了克尔(Kerr)非线性左手材料中的FDTD计算式。运用该方法数值模拟了具有该非线性效应左手材料平板中的电磁波,得到了稳定的数值结果。与线性左手材料平板结果进行了对比,数值计算和分析了波源到该平板不同距离情况下,该非线性左手材料平板对电磁波聚焦点的影响。
Presently,left-handed materials(LHMs)is one of the hot research interests for it has intensively potential application in the fields of Electromagnetics,Optics and Materials.However,for the newly artificial electromagnetic materials,these traditional theories,numerical methods and applications of electromagnetic fields need to be developed.Especially,by using the numerical methods to analyze the characteristics of the LHMs and the distributions of electromagnetic fields in the LHMs would greatly enhance the people's right understanding for these newly man-made materials and reduce the research time and costs.Besides,the variation of electromagnetic fields with time could be easily and clearly illustrated by recent drawing tools. Finite-difference time-domain(FDTD)method is directly derived by Maxwell's Equations which is widely used in many complex conditions.However,the traditional FDTD method would make the numerical simulation of LHMs divergence.
In this thesis,finite-difference time-domain(FDTD)methods in different mathematic modeling LHMs are developed.Programs based on these modified algorithms are performed in order to analyze the electromagnetic fields in such LHMs. According a great quantity of calculations,the characteristics of different modeling LHMs are discussed.Some special LHMs structures or systems are proposed.Their potential applications are explored.The author's work is mainly focused on:
1.The special characteristics and the basic theory of LHMs are introduced. Finite-difference time-domain(FDTD)method is grasped and applied masterly to the numerical analysis of electromagnetic problems.
2.Traditional FDTD method is modified.Such modified FDTD method is successfully applied in the numerical calculation of LHMs.It could solve the numerical problem of traditional FDTD in LHMs simulation.
3.TM and TE mode electromagnetic waves are separately concerned.The numerical expressions of the perfect-matched layer(PML)and such modified FDTD method in LHMs modeled by cold plasma model are derived.The permittivityε'_r and permeabilityμ'_r of back ground media are introduced.Two dimensional finite LHMs slab and infinite LHMs slab are numerically analyzed by such modified method.Besides,the refractional phenomena of electromagnetic waves in LHMs slab surface with different refraction-index are simulated.The extended Shell's law which electromagnetic waves satisfy in the LHMs could be verified by the numerical results.
4.The much more real media model,Drude model,compared to the cold plasma model is introduced into the LHMs.The numerical expressions of the modified FDTD method in Drude model LHMs are derived in detail.The lossy LHMs slab is studied.The distributions of electromagnetic waves around such LHMs slab with different lossy factorγare calculated.An electromagnetic energy localizing system with such lossy LHMs slab is numerically analyzed.The characteristic of electromagnetic energy localization in such system could be proved by the numerical results.
5.The most general model,Lorentz model,is introduced in LHMs.Based on the modified FDTD theory of chapter 3,chapter 4 and chapter 5,the numerical expressions of the modified FDTD method in Lorentz model LHMs are derived. The distributions of electromagnetic waves around such LHMs slab with different thickness are calculated.A multipoint focusing system with LHMs is discussed. According to the numerical results,these multipoint focuses certainly agree with the focusing theory of LHMs.Besides,a multi-layer LHMs system with multi-layer focusing characters is also discussed.A real metal-ring LHMs structure with the refraction-index approximately to -1 is designed which is applied to the multi-layer LHMs system.Ansoft is used to simulate the electromagnetic waves around the real LHMs.Great agreement could be seen between the FDTD results and the Ansoft results.
6.Nonlinear LHMs is primarily discussed.The numerical expressions of the modified FDTD method in Nonlinear LHMs with Kerr effects are derived.The distribution of electromagnetic waves around such nonlinear LHMs slab is calculated steadily. Compare to the linear LHMs,the electromagnetic focus could also be observed in this nonlinear LHMs slab.The location of the focus would vary with the distance between the source and the nonlinear LHMs slab.
引文
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