臺灣博碩士論文加值系統

本文主要利用有限時域差分法(finite-difference time-domain method)分析消散波(evanescent wave)與繞射極限(diffraction limit)之間的關係。我們用二維有限時域差分法模擬次波長單狹縫繞射(sub-wavelength single slit diffraction)，觀察其穩態下的瞬時波印亭向量(instantaneous Poynting vector)後發現，瞬時波印亭向量的大小會隨週期改變，但方向恆定。與全反射(total internal reflection)消散波的瞬時波印亭向量比較後，我們認為，次波長單狹縫繞射波全部都是傳輸波(propagating wave)，而不存在消散波。我們利用三面的光學相位共軛鏡(phase conjugate mirror)，回溯狹縫寬為次波長大小(2λ/5)的單狹縫繞射，並重新聚焦於維度小於繞射極限的點上。進一步地，我們改變狹縫的截面形狀為高斯函數(Gaussian function)，在狹縫寬為2λ的情況下，得到完美的回溯剖面場型。

In this thesis, the finite-difference time-domain (FDTD) technique is applied to analyze the relationship of evanescent wave and diffraction limit. We employ 2D FDTD to simulate sub-wavelength single slit diffraction and observe the instantaneous Poynting vector in steady-state. We find that the magnitude of the instantaneous Poynting vector varies periodically, while the direction remains unchanged. By comparing the result with the instantaneous Poynting vector of total internal reflection evanescent wave, we think that the diffracted waves of sub-wavelength single slit diffraction are all propagating waves rather than evanescent waves. We simulate the playback of the sub-wavelength (slit width is 2λ/5) single slit diffraction via 3-sided phase conjugate mirror and the phase conjugate waves re-focus back onto the spot with dimension below the diffraction limit. Furthermore, the cross-sectional shape of slit is modified from a rectangular to a Gaussian function. For the slit width is 2λ, a perfect re-focusing profile is generated.

Thesis Certification...........................................................................i

Acknowledgement...........................................................................ii

Traditional Chinese Abstract.............................................................iii

English Abstract.............................................................................iv

Contents........................................................................................v

List of Figures................................................................................viii

Chapter 1 Introduction......................................................................1

1.1 Preface.. ...................................................................................1
1.2 Chapter Outline..........................................................................4

Chapter 2 Theory.............................................................................5

2.1 Introduction................................................................................5
2.2 Diffraction Theory.......................................................................6
2.2.1 Huygens-Fresnel Principle........................................................6
2.2.2 Fresnel and Fraunhofer Approximation.......................................7
2.2.3 Diffraction Limit.......................................................................9
2.3 Evanescent Field......................................................................11
2.3.1 Introduction............................................................................11
2.3.2 Total Internal Reflection...........................................................12
2.4 Optical Phase Conjugation.........................................................15
2.4.1 Introduction............................................................................15
2.4.2 Basic Mathematical Formulation..............................................16
2.5 Chapter Summary.....................................................................18

Chapter 3 The Finite-Difference Time-Domain Method........................19

3.1 Introduction..............................................................................19
3.2 Maxwell’s Equations.................................................................20
3.3 Central Difference.....................................................................24
3.4 The Yee Algorithm....................................................................27
3.5 Numerical Dispersion and Stability of FDTD................................34
3.6 Perfectly Matched Layer Absorbing Boundary Condition..............37
3.7 Total-Field / Scattered-Field Technique....................................42
3.8 Near-to-Far-Field Transformation.............................................44
3.9 Chapter Summary...............................................................47

Chapter 4 Simulation Results and Discussion................................49

4.1 Introduction........................................................................49
4.2 Validation..........................................................................50
4.3 Sub-Wavelength Single Slit Diffraction Simulation......................53
4.3.1 Simulation Results............................................................53
4.3.2 Summary........................................................................57
4.4 Re-focus Diffracted Wave via Phase Conjugate Mirror.................58
4.4.1 Conventional Single Slit Diffraction.......................................59
4.4.2 Gaussian Function Shape Single Slit Diffraction.....................66
4.4.3 Summary.......................................................................70
4.5 Chapter Summary..............................................................71

Chapter 5 Conclusion and Future Prospect..................................72

5.1 Conclusion.......................................................................72
5.2 Future Prospect................................................................73

Bibliography..........................................................................74

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