||Reverse ray tracing for transformation optics
||Department of Photonics
因為因為超材料(Metamaterial)技術的進步，轉換光學(Transformation optics)被廣泛的討論，轉換光學可以利用空間中座標的轉換來控制電磁波的行進方向。幾何光學中，光束追跡是模擬光線軌跡和評估光學系統效率的工具。在轉換光學領域中，哈密頓運動方程式(Hamiltonian equations of motion)是一組微分方程式(Ordinary differential equations)，可求解光線軌跡。轉換光學的材料特性因為經過座標變換，會有非線性，不均勻，非等向性的材料分佈，提高微分求解器解哈密頓運動方程式的難度。在過去相關研究中，仍無探討在微分方程式中可能產生的奇點，與在轉換座標空間與原始空間的界面可能發生的衰減波造成哈密頓運動方程式無法求解的情況。本論文提出逆向光束追跡，結合掃描求解(Sweeping method)哈密頓-雅可比方程(Hamiltonian-Jacobi)的方法與反向傳播(Back propagation)，從感興趣的像面點計算光束線軌跡到光源。求解的步驟為，先掃描求解哈密頓-雅可比方程，得到感興趣元件的光程函數分佈(Eikonal map)。對光程函數分佈微分後求得波向量(Wave vector)，波向量可以計算像面上的光束能量分佈。逆向光束追跡法適用在材料特性(Material tensor)滿足對稱的正定矩陣時(Positive definite)的任何轉換光學元件。此方法可避開求解哈密頓微分方程式的解。第一個實現例為一個朗伯光源(Lambertian source)到像面後光束可以垂直出射的元件，採用與解析解的光束線軌跡比較與觀測面的能量分佈，驗証逆向光束追跡的準確度。第二與第三個光學實現例分別為可調整出射光角度，與一個二次光學元件可以旋轉偏振方向90與使出射光垂直出射。本文提出的逆向光束追跡成功解決哈密頓運動方程式無法順利求解的情形。
Transformation optics have been widely discussed in the area of advanced metamaterials since it allows spatial coordinate transformations of electromagnetic fields. Ray tracing is a method for the simulation of ray propagation in geometrical optics allowing for calculation of the optical system efficiency. The Hamiltonian equations of motion are based on ordinary differential equations (ODEs) and are used for ray tracing. However, the full solution to ordinary differential equations is may not be easily found because of the complexities of the inhomogeneous and anisotropic indices of the optical device. The failure of ray tracing due to singularity and complex wave vectors at the interface between air and transformed spaces is not well studied. To resolve this deficiency, we provide a 3D reverse ray tracing method for these situations which combines the sweeping method for Hamilton–Jacobi equations and ray trajectory. The sweeping method provides the eikonal function (time map) of the interested domain and back-propagation from the location of interest to the source gives the ray trajectory. Wave vectors, which represent illuminance, are obtained from the gradient of the eikonal function map in the transformed space.
This approach is applicable in any form of transformation optics where the material property tensor is a symmetric positive definite matrix. This method is not dependent on finding solutions to the Hamiltonian motion equations and also avoids the problems of a singularity or complex wave vector arising from the evanescent wave for the initial Hamiltonian motion equation conditions. In this thesis, the idea of transformation optics with function of directivity emitting and polarization rotation as secondary optics are explored. The accuracy of ray trajectories and illuminances are demonstratively solved by the proposed reverse ray tracing method for a number of example instances.
Chapter 1 Background 1
1 Background 1
1.1 Introduction 1
1.2 Maxwell equations in transformation space 3
1.3 Transformation optics 9
1.4 Hamiltonian ray tracing in transformation optics 26
1.5 Fresnel coefficients in transformation optics 41
1.6 Illuminance 45
Chapter 2 Reverse ray tracing 48
2 Reverse ray tracing 48
2.1 Motivation of reverse ray tracing 48
2.2 Methods 61
2.3 Geodesic distance on manifolds 72
Chapter 3 Numerical Examples 77
3 Numerical examples 77
3.1 Reverse ray tracing for transformation optics 77
3.2 Simulation accuracy 84
3.3 Wave vector and illuminance 90
3.4 Example 1: A 3D highly directive emitting device 93
3.5 Example 2: A 3D highly directive emitting device with polarization conversion 97
Chapter 4 Conclusion 101
4 Conclusion 101
Appendix A 105
Appendix B 110
Appendix C 119
Appendix D 136
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