| Summary: | 碩士 === 國立成功大學 === 機械工程學系 === 102 === The Analysis of Dispersive Prisms and Image Orientation Change of Prisms
Chun-Wei Huang
Psang Dain Lin
Department of Mechanical Engineering,
National Cheng Kung University
SUMMARY
Prism is a common object in optic, and has two important function which are changing image orientation and dispersion. This thesis presents a method to analyze image orientation change and dispersive prisms with Jacobian matrices and skew-ray tracing. In this thesis, lots of prisms are analyzed so that all kinds of prisms’ image orientation change have been known. Besides, a mathematic model are build on dispersive prisms, including triangular prism and compound prisms.
Key word : prism、Jacobain matrices、dispersive prisms
INTRODUCTION
Image orientation change and dispersion are two important function of prisms. But there was not a appropriate method to calculate prism’s image orientation change conveniently, and there wasn’t a good way to calculate the dispersion of dispersive prisms either. W. J. Smith presented a method to analyze image orientation change is that imaging a pencil, which is bounced off the reflecting part of the prism, to determine the orientation change, but this method has a restriction that the inner light and the outer light have to be perpendicular or parallel. R. E. Hopkins build a mathematic method, which is called reflecting matrix. When these matrices are multiplied, the image orientation can be analyzed. P. D. Lin and T. T. Liao analyzed the light’s direction after light is reflected or refracted by Snell’s law in their paper. G. J. Zissis analyzed the triangular dispersive prism, and used the geometric method to present the situation of dispersive light in the triangular prism. In this thesis, Jacobian matrix are used to analyze non-dispersive prisms, and dispersive prism. By this method, not only image orientation change can be presented as a matrix, but also the mathematic model is build. To analyze dispersive prisms, the relative of wavelength and refractive index have to be discussed. Because of this reason, 4 kinds of equation about the refractive index and wavelength are discussed in the thesis.
MATERIALS AND METHODS
Jacobian matrix of the flat surface boundary is the major method of this thesis. There are three kinds of situations should be discussed:
1. the Jacobian matrix of the incident point
2. the Jacobian matrix of the reflective unit vector
3. the Jacobian matrix of the refractive unit vector
Combing theses equations cane get the result:
So the reflective and refractive equations are:
This equation is at the reflective boundary
= =
And this equation is at the refractive boundary
RESULTS AND DISCUSSIONS
Table 1. the form of image orientation change
Prism Image orientation change
Right angle prism
Amici prism
Porro prims I
Porro prism II
Dove prism
Abbe-Koening prism
Schmidt prism
Leman prism
Goerz prism
Pechan-Schmidt prism
Roffe Pechan-Schmidt prism
Delta prism
Roofed delta prism
Rhomboid prism
Penta prism
This table is the table of image orientation change, which is made by the equations above. Through this table, image orientation change of many kinds of prisms can be known and used.
CONCLUSION
Firstly, by Jacobian matrix, image orientation change is presented in a complete theory. This method is more convenient and easier to calculate. Secondly, by Jacobian matrix and skew-ray tracing, the mathematic model of triangular prism and compound prism are build, and this model has the equal value with geometric method. Thirdly, the equations of wavelength and refractive index are compared in the thesis while building the mathematic model of dispersive prisms, and this thesis provides a table of these equations which describes their advantages and disadvantages.
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