【摘要】：本文主要研究幾類典型光波導經(jīng)完美匹配層截斷后其泄漏模和Berenger模的求解問題。由于常見的數(shù)值求解方法,比如有限差分法和有限元方法,在求解模較大的傳播常數(shù)時會遇到算不好的困難,因此采用漸近展開方法來推導這些傳播常數(shù)的漸近解。只要傳播常數(shù)的模足夠大,這些漸近解均具有不錯的精度,而且當傳播常數(shù)的模變大時,其精度還會進一步提高。在第二章,我們通過一種系統(tǒng)的推導方法重新得到了各向同性圓柱形光波導中泄漏模和Berengeir模的漸近解,這些新的漸近結(jié)果將現(xiàn)有文獻中的零階精度提高到了 1階和2階。在推導中,我們先利用大參數(shù)Bessel函數(shù)的漸近表達式,并結(jié)合泄漏模和Berenger模對應傳播常數(shù)幅角的漸近特點,對色散關(guān)系進行漸近處理;接著利用逆冪級數(shù)的漸近展開根據(jù)波導有沒磁性差別對HE泄漏模進行推導;然后就HE和EH Berenger模的漸近解進行了類似的推導,同時給出了其與對泄漏模的推導的主要區(qū)別。后面給出兩個例子將各階漸近解與精確解做了比較。第三章則是將第二章的推導方法應用到三層各向異性平板波導中泄漏模和PML模的漸近求解。由于考慮的各向異性平板波導僅支持TE模式和TM模式,而TE模又與各向同性平板波導中的TE模類似,于是僅需考慮TM模式下漸近解的推導。由于各向異性的存在,在求TM泄漏模的漸近解時需要分三種情形進行推導,其中前面兩種情形分別類似于各向同性波導中TE模和TM模的推導,而第三種情形則需綜合利用在前面兩種情形下所采用的推導方法。同樣,各向異性的存在也導致在推導兩列Berenger模時均需要分兩種情形進行。最終推導得到的TM泄漏模和Berenger模的漸近解的最高階數(shù)均不低于4階。本章后面給出了三個數(shù)值例子以驗證推導得到的漸近解的有效性以及高精度性。第四章繼續(xù)將第二章的推導方法進行推廣,考慮帶完美匹配層的各向異性圓柱形光纖波導。首先根據(jù)引入完美匹配層的完美導電條件以及包層與芯層間的連續(xù)性條件,較為詳細地推導了色散關(guān)系。然后根據(jù)漸近處理后的色散關(guān)系,推導了泄漏模和Berenger模的零階和1階的漸近解。需要注意的是,在推導EH模時,由于各向異性的存在,我們需要分兩種情形進行推導;其中一種情形的推導與各向同性的類似,而另外一種情形的推導則與各向同性的完全不同。后面給出了兩個用來驗證推導結(jié)果的數(shù)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.
【學位授予單位】：浙江大學
【學位級別】：博士
【學位授予年份】：2017
【分類號】：O175

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本文編號：2353713