几类典型光波导中泄漏模与Berenger模的渐近解
发布时间：2018-11-24 12:27
【摘要】：本文主要研究几类典型光波导经完美匹配层截断后其泄漏模和Berenger模的求解问题。由于常见的数值求解方法,比如有限差分法和有限元方法,在求解模较大的传播常数时会遇到算不好的困难,因此采用渐近展开方法来推导这些传播常数的渐近解。只要传播常数的模足够大,这些渐近解均具有不错的精度,而且当传播常数的模变大时,其精度还会进一步提高。在第二章,我们通过一种系统的推导方法重新得到了各向同性圆柱形光波导中泄漏模和Berengeir模的渐近解,这些新的渐近结果将现有文献中的零阶精度提高到了 1阶和2阶。在推导中,我们先利用大参数Bessel函数的渐近表达式,并结合泄漏模和Berenger模对应传播常数幅角的渐近特点,对色散关系进行渐近处理;接着利用逆幂级数的渐近展开根据波导有没磁性差别对HE泄漏模进行推导;然后就HE和EH Berenger模的渐近解进行了类似的推导,同时给出了其与对泄漏模的推导的主要区别。后面给出两个例子将各阶渐近解与精确解做了比较。第三章则是将第二章的推导方法应用到三层各向异性平板波导中泄漏模和PML模的渐近求解。由于考虑的各向异性平板波导仅支持TE模式和TM模式,而TE模又与各向同性平板波导中的TE模类似,于是仅需考虑TM模式下渐近解的推导。由于各向异性的存在,在求TM泄漏模的渐近解时需要分三种情形进行推导,其中前面两种情形分别类似于各向同性波导中TE模和TM模的推导,而第三种情形则需综合利用在前面两种情形下所采用的推导方法。同样,各向异性的存在也导致在推导两列Berenger模时均需要分两种情形进行。最终推导得到的TM泄漏模和Berenger模的渐近解的最高阶数均不低于4阶。本章后面给出了三个数值例子以验证推导得到的渐近解的有效性以及高精度性。第四章继续将第二章的推导方法进行推广,考虑带完美匹配层的各向异性圆柱形光纤波导。首先根据引入完美匹配层的完美导电条件以及包层与芯层间的连续性条件,较为详细地推导了色散关系。然后根据渐近处理后的色散关系,推导了泄漏模和Berenger模的零阶和1阶的渐近解。需要注意的是,在推导EH模时,由于各向异性的存在,我们需要分两种情形进行推导;其中一种情形的推导与各向同性的类似,而另外一种情形的推导则与各向同性的完全不同。后面给出了两个用来验证推导结果的数值例子。
[Abstract]:In this paper, the problem of solving the leakage mode and Berenger mode of several typical optical waveguides after truncated by perfectly matched layer is studied. Because common numerical methods, such as finite difference method and finite element method, are difficult to solve the propagation constants with large modulus, the asymptotic expansion method is used to deduce the asymptotic solutions of these propagation constants. As long as the modulus of the propagation constant is large enough, these asymptotic solutions have good accuracy, and the accuracy will be further improved when the modulus of the propagation constant becomes larger. In chapter 2, we obtain the asymptotic solutions of leaky modes and Berengeir modes in isotropic cylindrical optical waveguides by a systematic derivation method. These new asymptotic results improve the zero order accuracy of the existing literatures to order 1 and order 2. In the derivation, we use the asymptotic expression of the large parameter Bessel function, and combine the asymptotic characteristic of the leaky mode and the Berenger mode corresponding to the amplitude angle of the propagation constant to deal with the dispersion relation asymptotically. Then the asymptotic expansion of the inverse power series is used to deduce the HE leaky mode according to the magnetic difference of the waveguide, and the asymptotic solution of the HE and EH Berenger modes is similarly derived, and the main differences between the asymptotic solution and the leaky mode are given. Two examples are given to compare the asymptotic solutions of each order with the exact solutions. In the third chapter, the derivation of the second chapter is applied to the asymptotic solution of the leaky mode and the PML mode in a three-layer anisotropic planar waveguide. Because the anisotropic planar waveguide only supports the TE mode and the TM mode, and the TE mode is similar to the TE mode in the isotropic planar waveguide, we only need to consider the derivation of asymptotic solution in the TM mode. Due to the existence of anisotropy, the asymptotic solutions of TM leakage modes need to be deduced in three cases. The first two cases are similar to the derivation of TE modes and TM modes in isotropic waveguides, respectively. In the third case, the derivation method used in the first two cases should be used synthetically. Similarly, the existence of anisotropy results in the derivation of two Berenger modules in two cases. The highest order of asymptotic solutions of TM leaky mode and Berenger mode obtained from the final derivation is not less than 4 order. At the end of this chapter, three numerical examples are given to verify the validity and high accuracy of the derived asymptotic solution. In chapter 4, the derivation method of the second chapter is extended to consider the anisotropic cylindrical fiber waveguide with perfectly matched layer. Firstly, the dispersion relation is deduced in detail according to the perfect conduction condition of the perfectly matched layer and the continuity condition between the cladding layer and the core layer. Then, according to the dispersion relation after asymptotic treatment, the asymptotic solutions of zero order and first order of leaky mode and Berenger mode are derived. It is important to note that we need to deduce the EH mode in two cases because of the existence of anisotropy. The derivation of one case is similar to that of isotropy, while the derivation of the other case is completely different from that of isotropy. Two numerical examples are given to verify the derivation.
【学位授予单位】：浙江大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175
本文编号：2353713
[Abstract]:In this paper, the problem of solving the leakage mode and Berenger mode of several typical optical waveguides after truncated by perfectly matched layer is studied. Because common numerical methods, such as finite difference method and finite element method, are difficult to solve the propagation constants with large modulus, the asymptotic expansion method is used to deduce the asymptotic solutions of these propagation constants. As long as the modulus of the propagation constant is large enough, these asymptotic solutions have good accuracy, and the accuracy will be further improved when the modulus of the propagation constant becomes larger. In chapter 2, we obtain the asymptotic solutions of leaky modes and Berengeir modes in isotropic cylindrical optical waveguides by a systematic derivation method. These new asymptotic results improve the zero order accuracy of the existing literatures to order 1 and order 2. In the derivation, we use the asymptotic expression of the large parameter Bessel function, and combine the asymptotic characteristic of the leaky mode and the Berenger mode corresponding to the amplitude angle of the propagation constant to deal with the dispersion relation asymptotically. Then the asymptotic expansion of the inverse power series is used to deduce the HE leaky mode according to the magnetic difference of the waveguide, and the asymptotic solution of the HE and EH Berenger modes is similarly derived, and the main differences between the asymptotic solution and the leaky mode are given. Two examples are given to compare the asymptotic solutions of each order with the exact solutions. In the third chapter, the derivation of the second chapter is applied to the asymptotic solution of the leaky mode and the PML mode in a three-layer anisotropic planar waveguide. Because the anisotropic planar waveguide only supports the TE mode and the TM mode, and the TE mode is similar to the TE mode in the isotropic planar waveguide, we only need to consider the derivation of asymptotic solution in the TM mode. Due to the existence of anisotropy, the asymptotic solutions of TM leakage modes need to be deduced in three cases. The first two cases are similar to the derivation of TE modes and TM modes in isotropic waveguides, respectively. In the third case, the derivation method used in the first two cases should be used synthetically. Similarly, the existence of anisotropy results in the derivation of two Berenger modules in two cases. The highest order of asymptotic solutions of TM leaky mode and Berenger mode obtained from the final derivation is not less than 4 order. At the end of this chapter, three numerical examples are given to verify the validity and high accuracy of the derived asymptotic solution. In chapter 4, the derivation method of the second chapter is extended to consider the anisotropic cylindrical fiber waveguide with perfectly matched layer. Firstly, the dispersion relation is deduced in detail according to the perfect conduction condition of the perfectly matched layer and the continuity condition between the cladding layer and the core layer. Then, according to the dispersion relation after asymptotic treatment, the asymptotic solutions of zero order and first order of leaky mode and Berenger mode are derived. It is important to note that we need to deduce the EH mode in two cases because of the existence of anisotropy. The derivation of one case is similar to that of isotropy, while the derivation of the other case is completely different from that of isotropy. Two numerical examples are given to verify the derivation.
【学位授予单位】：浙江大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175
【参考文献】
相关博士学位论文 前1条
1 张学仓;Sturm-Liouville算子的矩阵逼近及其应用[D];浙江大学;2011年
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