本文以齊次座標轉換和光線追蹤法在平面邊界上的應用，來計算光線在經過非色散稜鏡後成像位姿的變化，文中探討了各式稜鏡並將其成像位姿變化以矩陣的方式表示，這有利於稜鏡設計和稜鏡組的配合;另外也針對了波長和折射係數的關係式進行分析，本文將各關係式於三種不同的材質比較，再從中挑選出最易於計算且不失其準確度的關係式，我們也將各關係式在三種材質中微分，並且比較其微分之後的結果，最後找出在計算色散時最簡易的式子；對於色散稜鏡，我們針對三稜鏡、貝林-布洛卡稜鏡、阿貝稜鏡和複合式稜鏡這幾種稜鏡進行分析，在三稜鏡中我們以光線追蹤的方法將兩種計算色散的式子交互比對並驗證，對於貝林-布洛卡稜鏡和阿貝稜鏡我們以簡單的幾何方式證明其分散光線的原理，而對於複合式稜鏡我們提出了一套方法可以不受限制地計算出複合稜鏡所需的頂角角度和折射係數，以達到我們想要的色散角度和值。

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.

1.Isaac Newton, Hydrostatics, Optics, Sound and Heat. Retrieved 10 January 2012: Cambridge University Digital Library.
2.Psang Dain Lin, Analysis and Modeling of Optical elements and Systems. ASME Journal of Engineering for Industry, 1994. 116(1): p. 101-107.
3.W.J.Smith, Modern Optical Engineering. Vol. 3. 2001, Barrington,NJ: Edmund Industrial Optics.
4.Optical Design. Vol. MIL-HDBK-141. 1962, Washington, D.C.: U.S. Defense Supply Agency Rep . 13-113-52.
5.Te-Tan Liao and Psang Dain Lin, Analysis of optical elements with flat boundary surfaces. Applied Optics, 2003. 42(7): p. 1191-1202.
6.D. W. Swift, Image rotation devices—a comparative study. Opt. Laser Technol. , 1972: p. 175-188.
7.R. E. Hopkins, Mirror and prism systems in Optical Design. 1962, U.S. Defense Supply Agency: Washington, D.C.
8.R. E. Hopkins, Mirror and prism systems. Appl. Opt. Opt., 1965. 3: p. 269-308.
9.William L . Wolfe, NONDISPERSIVE PRISMS, in hanbook of optics. 1995, McGraw-Hill , Inc: United States of America.
10.George J. Zissis, Dispersive Prisms and Gratings, in HANDBOOK OF OPTICS. 1995, McGraw-Hill , Inc: United States of America.
11.David A. Atchison and George Smith, Chromatic dispersions of the ocular media of human eyes. J. Opt. Soc. Am. A, 2005. 22(1): p. 29-37.
12.M. Born and E. Wolf, Principles of Optics. 1985, New York: Pergamon.
13.J. D. Foley, et al., Computer Graphics Principles and Practices. 2 ed. 1981: Addision-Wesley Publishing Company.
14.C. H. Wu, Robot Accuracy Analysis Based on Kinematics. IEEE Journal of Robotics and Automation, 1986. RA-2/3: p. 171-179.
15.R. P. Paul, Programming and Control, in Robot Manipulators-Mathematics. 1982, MIT press, Cambridge, Mass.
16.J. Denavit and R. S. Hartenberg, A Kinematic Notation for Lower Pair Mechanisms Based on Matrices. Transactions of the ASME Journal of applied mechanics, 1955. 77: p. 215-221.
17.J.J. Uicker, On the Dynamic Analysis of Spatial Linkages Using 4x4 Matrices, in Dissertation for Doctor of Philosophy. 1965, Northwestern University: Evanston ILL.
18.P.D. Lin, New Computation Methods for Geometrical Optics. 2013: Springer Singapore.
19.Paul R. Yoder, Design and Mounting of Prisms and Small Mirrors in Optical Instruments. 1998: SPIE Press.
20.N. Shojiro and K. Jun, Retroreflector Using Gradient-index Rods. Appl. Opt. , 1991. 30: p. 815-822.
21.M. Atsushi and S. Nobuo, Design of a Four-element, Hollow-cube Corner Retroreflector for Satellities by Use of a Genetic Algorithm. Appl. Opt. , 1998. 37: p. 438-442.
22.L. Z. James, Cube Corner Retroreflector Test and Analysis. Appl. Opt., 1976. 15: p. 445-452.
23.M. Born and E. Wolf, Principles of Optics. 6 ed. 1989, Oxford, UK: Pergamon.
24.M.Herzberger Colour correction in optical systems and a new dispersion formula. Opt. Acta, 1959. 6: p. 197-215.
25.Bernard H. W. S. De Jong, Ruud G. C.Beerkens, and Peter A.Van Nijnatten, Glass, in Ullmann's Encyclopedia of Industrial Chemistry. 2000.
26.Y. Le Grand, Form and Space Vision. 1967, Indiana University Press: Bloomington, Ind.
27.A. L. Cauchy, Me´moire sur la dispersion de la lumie`re. Nouveaux Exercices de Mathe´matiques, ed. A. Cauchy. Vol. 10. 1895, Paris: Gauthier-Villars et Fils.
28.F.A. Jenkins and H.E. White, Fundamentals of Optics. 4 ed. 1981: McGraw-Hill, Inc.
29.Rüdiger Paschotta. Chromatic Dispersion.
30.P. Pellin, A.B., A Spectroscope of Fixed Deviation. Astrophysical Journal 1899. 10(337-342).
31.Eugene Hecht, Optics. 4 ed. 2001, Pearson Education.
32.Nathan Hagen and Tomasz S. Tkaczyk, Compound prism design principles I. Appl. Opt., 2011. 50: p. 4998-5011.
33.John Browning, Note on the use of compound prisms. MNRAS, 1871. 31: p. 203-205.
34.蔡忠佑 Chuang-Yu Tsai, 稜鏡成像位姿變化之分析與設計 Prisms Analisis and Design Base onImage Orientation Change, in 機械工程學系 ME. 2007, 國立成功大學 National Cheng Kung University.