上半平面的規範化解析延拓

A Regularized Analytic Extension on the Upper Half Plane

羅彥博(Luo, Yen Po)

指導教授 ： 王偉成

國立清華大學/理學院/數學系/碩士(2016年)

摘要

在電磁學中，我們以 D=ε_0 E 來表示電場和電位移的關係，ε_0 為電容率。 而電極化率和相對電容率有 χ=ε_r-1 的關係。 電極化率本身滿足 KK 關係式，且可以從波方程中得到對應折射率以及衰退係數的關係 χ=〖(n+iq)〗^2。 在真實生活中，我們只能在有限的範圍下測量以及得知n及q的資料，因此我們利用電極化率會滿足KK關係式的條件，以及假設電極化率在可測量資料的範圍外部會有光滑的行為，使用規範化的方法對 χ 在無法測量的範圍延拓。然而金屬導體的電極化率在0點有瑕點 χ，要先去除奇異的部分才能夠使 χ 滿足KK關係式。 在第1節中，我們會討論如何處理瑕點的細節。

關鍵字

解析延拓

並列摘要

In electromagnetism, the electric displacement field D represents how an electric field E affects in a given medium. The actual permittivity ε is calculated by =ε_r ε_0=(1+χ)ε_0 , where χ is the electric susceptibility of given material. The electric susceptibility χ satisfies the Kramers-Krönig relation. From the wave equation, electric susceptibility χ , the refractive index n and attenuation coefficient q have the relation χ=〖(n+iq)〗^2. However we can only measure n and q within a finite bandwidth. In this paper, we will find out a subset of the kernel of the Kramer-Krönig relation, and then use a minimization with a regularized extension to recover χ outside the measured bandwidth. We will recover the data of silicon and silver. For silver, we observe the singularity of its data χ_s and apply the KK-relation on χ-χ_(s ). For the singularity of conductors, we put the detail in the section 1.3.

並列關鍵字

regularized analytic extension

參考文獻

[5] Rainer Kress Linear Integral Equations Appl. Math. Sci. (1989) v. 82 Springer-Verlag

[6] Mark Fox Optical Properties of Solids (2010) Oxford University Press

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