本文主要利用歪斜光線追縱法，探討像差的産生，並利用齊次座標
轉換矩陣，將光學系統模型以矩陣的型式，建構出完整的光線追縱理論。利用光線的一次微分(Jacobin Matrices)，求得焦散面産生的位置。最後將計算出的結果繪製成圖，並加以探討。
不同以往以向量方式計算光線追蹤方程式，本論文將光線以矩陣的型式表現，並對光線作一次微分，求得光線在像平面上的位置，再求得各個像差所需要之數值，將像差數值化，以應變數為縱座標，自變數為橫座標，繪製各個像差的座標圖。
利用點分散函數(Point Spread Function)，得知像平面上輻照度(Irradiance)的分佈情形，以點分散函數的一次微分與Irradiance-Based Method，求得輻照度無限大所在之位置，即為焦散面之位置。並將光源分成點光源與平行光，系統為球面反射鏡與一組光學透鏡組兩種，經過牛頓逼近法(Newton-Raphson Method)求得焦散面的位置與形狀，並加以探討。

關鍵字：歪斜光線追蹤、初級像差、焦散面

SUMMARY

Aberrations is always an optic designs problem and cause the unpredictable of images. So the research of aberrations of optics is an important thing to reduce aberrations.
A caustic surface is the locus of singularities in the flux density. This regions are special brightness. Caustic surface is a special case. They may also cause optic designs unpredictable. So the research of caustic surfaces is important, too.
In this thesis, we use skew ray tracing, homogeneous coordinate notation, point spread function, irradiance-base method, Newton-Raphson method and Jacobin matrices to calculate first order aberrations and caustic surfaces. Then making coordinate diagram to discuss the results of aberrations and caustic surfaces.

Key words: skew ray tracing, aberrations, caustic surfaces

INTRODUCTION
The purpose of this thesis is to research Aberrations and find real data by skew ray tracing and Jacobin matrices. Then making coordinate diagram with this data to discuss the results.
A caustic surface can be defined as either the envelope of the reflected/refracted rays, or as the loci of the centers of curvature of the wavefront. There are three approaches to determine the caustic surfaces of optical systems. (1) From the solution of eikonal equation. (2) From the formula of illuminance(or flux density). (3) From the expression of PSF on image plane. The first approach is probably first presented by Stavroudis. The second method could be first reported by Shealy. The third method utilizes a moving virtual image plane to collect the points possessing infinite PSF. This thesis is to use the third method to determine the caustic surfaces position in image plane and a virtual image plane.

MATERIALS AND METHODS

1. The Jacobian Matrices of a Ray with respect to Its Incoming Ray and Boundary Variable Vector
In the thesis, Jacobian matrices is the major method. There are two boundary surfaces: (1) flat boundary surface (2) spherical boundary surface.
(1). Flat Boundary Surface
The Jacobian matrix of the incident point:

The Jacobian matrix of the reflective unit vector:
The Jacobian matrix of the refractive unit vector:

(2). Spherical Boundary Surface
The Jacobian matrix of the incident point:

The Jacobian matrix of the reflective unit vector:

The Jacobian matrix of the refractive unit vector:

If there is not any changes of boundary variables , can be obtained by successively using the matrices of above equations.

2. The Point Spread Function Based on Irradiance Method
The PSF in an optical system describes the irradiance distribution on the image plane associated with a point source . The energy flux emitted from into a solid angle along the ray tube is given by:
.
Defining as the irradiance on an infinitesimal area is given by .Assuming no transmission losses, the following equation is obtained by applying the principle of energy flux conservation along this ray then . So PSF can be written as:

RESULTS AND DISCUSSIONS
The main purpose of this thesis is to make coordinate diagram with data which we have calculated. It is the results of this thesis in figure1 to figure9 and table1.
Figure 1. Longitudinal spherical aberration and traverse spherical aberration coordinate diagram

Figure 2. Sagittal coma and meridional coma coordinate diagram.

Figure 3. Astigmatism coordinate diagram at and

Figure 4. Field curvature coordinate diagram at and

Figure 5. Distortion coordinate diagram at and
Table 1. Longitudinal chromatic aberration and traverse chromatic aberration
Wavelength Paraxial image Ray angle =

Chromatic Aberration LCA=-0.24588mm

Figure 6. The cross-sectional curves of caustic surfaces on plane generated by rays originated from different point source located at , and

Figure 7. The cross-sectional of caustic surfaces on plane formed by refracted rays originated from on-axis point source located at and

Figure 8. The cross-sectional curves of primary and secondary caustic surfaces on plane generated by collimated rays parallel to the optical axis of system containing only a spherical mirror.

Figure 9. The caustic surfaces formed by collimated rays with to optical axis.

CONCLUSION

(1) It is convenience and easy to build the exact and complete optical systems by using homogeneous coordinate notation, skew ray tracing, law of reflection (/refraction) and the determined optical elements coordinate by math method.
(2) Since only a single traced ray suffices for finding the position of source ray in image plane. It is efficient to calculating aberrations by Jacobian matrix and skew ray tracing.
(3) Only a single traced ray suffices for the PSF at the incidence point on the image plane for a given source ray . Thus, the proposed PSF method is far more computationally efficent than the ray-counting method which sufficient rays must be traced.
(4) It is possible to calculate all of system variable in the same time by Jacobian matrix and skew ray tracing. In the future, we could use these methods to design optical systems to achieve the designer purpose.

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