【作者】 冯传胜；

【导师】 梅良模；

【作者基本信息】 山东大学 ， 光学工程， 2007， 博士

【摘要】 近二十年来,“光子晶体”引起了越来越多人的注意。光子晶体的概念是在1987年由E.Yablonovitch和S.John同时提出的,它是一种介电常数周期性分布的人工“带隙”材料。与固体理论中研究的半导体类似,光子晶体被称为“光子”的半导体。它最突出的特点是具有“光子带隙”,频率处于光子带隙中的电磁波是禁止传播的。在光子晶体中精心设计缺陷可以在光子带隙中引入“缺陷态”,而这些缺陷态使光子晶体具有更加丰富的应用,如光子晶体光纤、微谐振腔等。另外,光子晶体也可以作为光子的导体。某些特殊结构的光子晶体在特定的频率范围内具有异常色散特性,如超棱镜效应、负折射现象等。总之,光子晶体使成功地控制光子成为可能,将为光集成的实现铺平道路。用来描述光子晶体特性的主要手段有两个,一个是能带结构,另一个是复能带结构。与传统半导体中电子的能带结构类似,光子晶体中光子的能带结构给出了光子晶体中传播模的色散关系。通过光子晶体的能带结构,我们可以知道光子带隙的频率范围,但不知道光子带隙中消散模的任何信息,如衰减系数等。传统的平面波展开法是用于计算光子晶体能带结构的最经典的方法,由于它基于“无限”、“周期性”的假设,通过它不能得到关于光子晶体中消散模的任何信息,所以不能用于解释半无限和有限光子晶体中光的传输特性。光子晶体的复能带结构可以看作能带结构的扩展,它除了给出光子晶体中传播模的色散关系外,还给出了光子带隙中消散模的衰减系数。由于它的本征模既包含传播模又包含消散模,所以能够用于半无限和有限厚度光子晶体的研究。本论文的主要目的是给出一种计算光子晶体复能带结构方法,并系统地讨论它在半无限和有限厚度光子晶体中的应用。论文的主要内容如下;1.在论文的第一章对光子晶体的研究进展做了简单介绍。介绍了光子晶体的概念、制备及应用、常用研究方法等。2.在论文的第二章详细描述了本论文中用到的两种方法,即平面波展开法和基于平面波的传输矩阵法。在复能带结构的计算中,我们用平面波展开法来验证我们得到的复能带结构中传播模的色散关系是否正确,用基于平面波的传输矩阵法来验证复能带结构中衰减系数的正确性。在第五章中,用基于平面波的传输矩阵法来验证复能带结构的本征模在有限厚度光子晶体中的应用是否正确。我们用Matlab语言对这两种方法进行了编程,用文献上的数据证明了程序的正确性。3.在论文的第三章中我们提出了一种基于平面波的用于光子晶体复能带结构计算的方法。从Maxwell方程组出发,推导出了二维光子晶体TE模和TM模的复能带结构的本征方程。分析了本征方程的特点,指出该本征方程除了可以得到光子晶体的复能带结构之处还有一些其它的优点,如可以计算色散材料构成的光子晶体的复能带结构及有利于计算某一特定频率的等频线等。计算了二维光子晶体(包含正方晶格结构和三角晶格结构)的复能带结构,提出采用坐标变换的方法来解决计算沿Γ-M方向复能带结构遇到的困难。用平面波展开法和基于平面波的传输矩阵法验证了计算得到的复能带结构。我们还以二维正方晶格结构光子晶体为例,计算了光子晶体中特定频率的等频线,分析了复能带结构本征方程的收敛特性。我们还推导出了三维光子晶体复能带结构的本征方程。以空气中的介质球形成的简立方晶格结构三维光子晶体为例,计算了三维光子晶体的复能带结构,并用平面波展开法和基于平面波的传输矩阵法验证了复能带结构的正确性。4.在论文的第四章中我们用复能带结构本征方程的本征模对半无限光子晶体中光的传输进行了分析。我们利用传播模的群速度对光子晶体界面处的折射现象进行了分析,并用Notomi提出的等频面(线)法和有限时域差分法(FDTD)进行了验证。我们发现利用传播模的群速度可以很方便地研究光子晶体中的异常色散现象。相对于Notomi的方法和FDTD,该方法更容易获得规律性的结论。我们用复能带结构本征方程的本征模给出了光子晶体界面处的边界条件,并利用它对电磁波穿过半无限光子晶体界面的传输谱进行了分析。我们分析的半无限光子晶体系统有两种,一种由均匀介质和半无限光子晶体构成,另一种由两种不同的半无限光子晶体构成。我们将计算结果与文献中的数据进行了比较,证明了数据的正确性。对于由均匀介质和半无限光子晶体构成的系统,我们还给出了一种“单模近似模型”来计算光从均匀介质中入射时的反射系数及反射光相对于入射光的相位变化。5.在论文的第五章我们用复能带结构本征方程的本征模对有限厚度光子晶体中光的传输进行了分析。在由本征模表示的边界条件的基础上,给出了用于单层有限厚度光子晶体的传输矩阵。我们针对三种结构讨论了电磁波穿过单层光子晶体时的传输谱,第一种是单层光子晶体的两侧为均匀介质的结构,第二种是单层光子晶体的一侧为均匀介质而另一侧为半无限光子晶体的结构,第三种是单层光子晶体的两侧均为不同的半无限光子晶体的结构。我们将计算结果与基于平面波的传输矩阵法得到的结果进行了比较,证明了该传输矩阵法的正确性。在单层有限厚度光子晶体传输矩阵的基础上,给出了用于多层有限厚度光子晶体的传输矩阵法。用该传输矩阵法计算了电磁波穿过多层有限厚度光子晶体时的传输谱,并将其与基于平面波的传输矩阵法得到的结果进行了比较,证明了该方法的正确性。光子晶体量子阱结构是一种特殊的光子晶体异质结结构,我们讨论了光子晶体量子阱结构中的谐振模。对于阱层为有限厚度的量子阱结构,我们可以用多层有限厚度光子晶体的传输矩阵法来计算其中的谐振模。当阱层为无限厚度时,我们不得不采用另外的方法。在这里我们利用谐振模的驻波特性,推导了谐振模对应的本征方程。利用谐振模的本征方程,对多种不同的量子阱结构中的谐振模进行了计算与分析,并用多层有限厚度光子晶体的传输矩阵法进行了验证。对于对称的量子阱结构,我们给出了判断谐振模奇偶性的方法。对于阱层为均匀介质的光子晶体量子阱结构,我们给出了根据反射波相对于入射波相位差来计算其谐振模的公式,并用它对垒层为不同晶格常数光子晶体的量子阱结构中的谐振模进行了计算。总之,我们在本论文中给出了一种基于平面波的方法来计算光子晶体的复能带结构,并讨论了它在半无限和有限厚度光子晶体中的应用。通过分析发现,虽然复能带结构本征方程的得到也是基于“无限”、“周期性”的假设,但复能带结构及其本征模包含了足够的信息来描述光在半无限或有限厚度光子晶体中的行为。更多还原

【Abstract】 In the past two decades, "photonic crystals" have attracted more and more people’s attention. A photonic crystal, proposed simultaneously by E. Yablonovitch and S. John, is an artificial "band gap" material in which the dielectric constant is arranged periodically. Analogous to the semiconductors considered in solid state theory, photonic crystals are called semiconductors of photons. A photonic crystal is characteristic of "photonic band gap", in which the propagation of electromagnetic wave is highly prohibited. The defects modes can be introduced into the photonic band gap by fabricating carefully designed defects in perfect photonic crystal, and these defects modes are of more plentiful applications, such as photonic crystal fiber, micro resonant cavity etc.. Additionally, a photonic crystal can act as a conductor of photons. Some abnormal dispersive phenomena have been found in certain frequency range in some photonic crystals of special structures, such as super prism effect and negative index phenomena etc. In a word, photonic crystals give us a possibility to successfully control photons, and will pave the road for optical integration.There are two main ways to describe the property of a photonic crystal, the photonic band structure and the complex photonic band structure. Analogous to the band structure of electrons in traditional semiconductor, the photonic band structure gives the dispersive relation of propagating modes in a photonic crystal. From the band structure, we can see the frequency scope of photonic band gaps, but we know nothing about the evanescent modes in the band gap, such as the decay constant of an evanescent mode. The traditional plane wave expansion method (PWM) is the most classical way to calculate the band structure of a photonic crystal. Because PWM is based on the infinite and periodic assumptions and it gives nothing about the evanescent modes, it can not be used to describe the light behavior in semi-infinite and finite systems of photonic crystals. The complex photonic band structure can be regarded as an extension of traditional photonic band structure, which gives the decay constants of evanescent modes in the photonic band gaps in addition to the dispersive relation of the propagating modes. Because the eigen modes contain both the propagating modes and the evanescent modes, the complex photonic band structure and its eigen modes can be applied to semi-infinite and finite systems of photonic crystals.The aim of my dissertation is to give a method to calculate the complex photonic band structure of a photonic crystal and discuss its applications to semi-infinite and finite systems of photonic crystals systematically. The main content of the dissertation is given below:1. A brief introduction about the research progress of photonic crystal is given in chapter 1.In this chapter, we give a brief introduction about photonic crystals, such as concept, fabrication methods and applications, research methods, etc.2. A detailed description of two methods used in my dissertation, plane wave expansion method (PWM or PWEM) and plane-wave-based transfer matrix method (TMM), is given in chapter 2.In the calculation of complex photonic band structure, PWM is used to verify the dispersive relation of propagating modes and the plane-wave-based TMM is used to verify the decay constants of evanescent modes in photonic band gap. In chapter 5, the plane-wave-based TMM is used to verify the validity of our TMM for finite system of photonic crystals. We programmed for these two methods in Matlab language, and checked the programs with the data from references.3. A plane-wave-based approach for complex photonic band structure is given in chapter 3.Start from Maxwell’s equations, we deduced the eigen equations of TE mode and TM mode for complex photonic band structure of two-dimensional (2D) photonic crystals. We have discussed the properties of eigen equations, and found that the eigen equations have some other virtues in addition to the calculation of complex photonic crystals, such as its application to photonic crystals composed of dispersive material, its convenience to get the equi-frequency curve (surface for three-dimensional (3D) photonic crystals) for a fixed frequency value. We have calculated the complex photonic band structures of 2D photonic crystals (including square lattice and triangular lattice), and applied coordinates transformation method to overcome the difficulty encountered in the calculation of complex photonic band structure alongΓ- M direction. The calculated results have been verified with PWM and plane-wave-based TMM. For 2D photonic crystal of square lattice, the equi-frequency curves have been calculated, and the convergence property of present approach has been discussed. We have also deduced the eigen equation for complex photonic band structure of 3D photonic crystals. The complex photonic band structure of a 3D photonic crystal of simple cubic lattice (dielectric spheres in air) has been calculated and verified with PWM and plane-wave-based TMM.4. We have applied the eigen modes for complex photonic band structure to semi-infinite systems of photonic crystals in chapter 4.Based on the group velocity of propagating mode, the refraction phenomena at the interface of photonic crystal have been discussed, and the results have been verified with Notomi’s method and finite difference time domain (FDTD) simulation. We have found that it is very convenient to analyze the abnormal dispersive phenomena with group velocity of propagating mode. Compared with Notomi’s method and FDTD simulations, present method is convenient to get generalized conclusion.We have described the boundary conditions at the interface of a photonic crystal with the eigen modes from eigen equation for complex photonic band structure, and applied the boundary conditions to get the spectra for electromagnetic wave propagating through the interface of semi-infinite photonic crystal. Two different systems are considered, one is composed of homogeneous media and semi-infinite photonic crystal, and the other is a heterostructure composed of two different semi-infinite photonic crystals. The calculated results are verified with the data from references. For the system composed of homogeneous media and semi-infinite photonic crystal, a "single mode approximation model" is given to get the reflection coefficients and the phase shift of reflected wave relative to the incident wave.5. We have applied the eigen modes for complex photonic band structure to finite systems of photonic crystals in chapter 5.Based on the boundary conditions described with eigen modes, a TMM for single slab of photonic crystal is proposed. We have discussed the spectra for electromagnetic wave propagating through a single slab of photonic crystal in three different structures. The first considered structure is composed of a single slab of photonic crystal sandwiched by homogeneous media, the second one is composed of a single slab sandwiched by homogeneous media and a different semi-infinite photonic crystal, and the third one is composed of a single slab sandwiched by two different semi-infinite photonic crystals. The calculated results for the first structure are verified with plane-wave-based TMM.Based on the transfer matrix for single slab of photonic crystal, an extended transfer matrix method (ETMM) for multiple slabs of photonic crystals is given. The calculated spectra for multiple slabs of photonic crystals are verified with plane-wave-based TMM.A quantum well structure of photonic crystals is a special double-heterostructure of photonic crystals, in which we have discussed the resonant modes. For the quantum well structure with finite barrier thickness, ETMM for multiple slabs of photonic crystals can be used to calculate the resonant modes. For the quantum well structure with infinite barrier thickness, we have to resort to other method. Here, based on the standing wave property of resonant modes, the eigen equation for resonant modes is deduced. Using the eigen equation, we have discussed the resonant modes in some different quantum well structures, and the results are verified by ETMM for multiple slabs of photonic crystals. For symmetric quantum well structure, we give a method to distinguish the parity of the resonant modes. For quantum well structure with homogeneous media well slab, a simple method is given to calculate the resonant modes, which is based on the phase shift of reflected wave relative to incident wave, and this method is very convenient to calculate resonant modes in a quantum well structure with barriers of different lattice constants.In brief, we have proposed a plane-wave-based approach to calculate the complex photonic band structure of photonic crystals, and applied it to semi-infinite and finite systems of photonic crystals. From the discussion, we can see that although the eigen equations are dependent on the infinite and periodic assumptions, the complex photonic band structure and its eigen modes contain enough information to describe the light behavior in semi-infinite and finite systems of photonic crystals.更多还原