The Calculation of Caustic Surfaces and First Order Aberrations.

• Main Authors: Zhen-XiangFan, 范振翔
• Other Authors: Psang-Dain Lin
• Format: Others
• Language: zh-TW
• Published: 2014
• Online Access: http://ndltd.ncl.edu.tw/handle/35460743520836660022

碩士 === 國立成功大學 === 機械工程學系 === 102 === 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.