رهگیری پرتو چوله و تجزیه و تحلیل حساسیت مرزی نوری
|کد مقاله||سال انتشار||مقاله انگلیسی||ترجمه فارسی||تعداد کلمات|
|26736||2013||11 صفحه PDF||سفارش دهید||محاسبه نشده|
Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : Optik - International Journal for Light and Electron Optics, Volume 124, Issue 12, June 2013, Pages 1159–1169
One of the most popular mathematical tools in fields of robotics, mechanisms and computer graphics is the 4 × 4 homogeneous transformation matrix. Our group's previous application of the homogeneous transformation matrix to flat and spherical optical boundaries has been extended to hyperboloid surfaces for: (1) skew ray tracing to determine the paths of reflected/refracted skew rays; (2) sensitivity analysis for direct mathematical expression of differential changes of incident points and reflected/refracted vectors with respect to changes in incident light sources and boundary geometric parameters; (3) a sensitivity analysis-based merit function derived directly from mathematical expression of catadioptric imaging system components. The presented methodology is highly suited to digital implementation and offers direct and rapid analytical statement of ray path, chief ray, marginal rays and merit functions of optical systems.
Optical system design requires accurate determination of the paths of light rays interacting with reflective and refractive system components. This may be done by successive application of Snell's well-known laws of reflection and refraction, an extensively employed method known as ray tracing. The generalized version is known as skew ray tracing and is more difficult to perform, but analytical modeling and evaluation of optical systems is presently impossible without this technique. Traditional skew ray tracing, even with digital computers, is computationally very expensive. Our previous work expedited skew ray tracing by formulating it in terms of homogeneous transformation matrices and applied it to the easily-manufactured flat  and spherical  optical components commonly used by industry. However, aspherical boundary surfaces such as hyperboloid surfaces sometimes have significant advantages over spherical boundary surfaces. For example, it is known that parallel incident rays reflected by a spherical concave mirror undergo spherical aberration and do not converge at the focal point. However, a convex/concave hyperboloid reflecting mirror can converge these rays to form a virtual/real imaged point. Therefore, hyperboloids are used in a great variety of applications such as flashlights, automobile headlight reflectors, radiotelescope antennas, microwave horns, acoustical dishes and optical telescope mirrors. Ray tracing though an aspherical surface is difficult since the intersection of a ray and an aspherical surface cannot be determined directly. Smith  performed aspherical-boundary skew ray tracing by a series of approximations which continued until approximation error became negligible, a computationally expensive procedure. This present work achieves more efficient aspherical-boundary skew ray tracing by use of Snell's laws formulated as the homogeneous transformation matrices shown in Section 2. Sensitivity analysis is presented in Section 3. The discussion in Sections 2 and 3 is limited to monochromatic light. However, most light sources are polychromatic. When polychromatic light is refracted, each monochromatic component has its own unique interaction with the refractive components of the optical system. Each monochromatic component thus takes a different ray path through the system and each arrives at a slightly different position. The resulting image is different for different colors, an effect called chromatic aberration. Welford  pointed out that exact formulae for chromatic aberration are cumbersome, but Welford's numerical ray-tracing technique remains the universally adopted method for detailed analysis of optical systems. Even with computers and commercial software, the process is slow. Therefore, Section 4 provides algebraic expressions for chromatic aberration suitable for application in rapid computerized evaluation of optical system quality. Conclusions are presented in Section 5. The 4 × 4 homogeneous transformation matrix is one of the most efficient and useful tools in robotics  and , mechanisms  and  and computer graphics . In homogeneous coordinate representation, a position vector View the MathML sourcePixi+Pjiy+Pizk is written as a column matrix View the MathML sourcePij=[PixPiyPiz1]T. In the following, the pre-superscript “j ” of the leading symbol jPi means this vector is referred with respect to coordinate frame (xyz )j. Given a point jPi, its transformation kPi is represented by the matrix product View the MathML sourcePik=AjkPij, where kAj is a 4 × 4 matrix defining the position and orientation (referred to as pose matrix hereafter) of a frame (xyz)j with respect to another frame (xyz)k . If the ith vector is referred with respect to the ith frame (xyz)i (i.e. iPi uses the same number as both its pre-superscript and post-subscript), then its pre-superscript “i” will be omitted for reasons of simplicity. These notational rules are also applied to unit directional vector jℓi = [ℓix ℓiy ℓiz 0]T.
نتیجه گیری انگلیسی
Our group's previous application of the homogeneous transformation matrix to flat and spherical optical boundaries has been extended to hyperboloid surfaces for: (1) skew ray tracing to determine the paths of skew rays being reflected/refracted; (2) sensitivity analysis for direct mathematical expression of differential changes of incident points and reflected/refracted vectors with respect to changes in incident light sources and boundary geometric parameters; (3) a sensitivity analysis-based merit function derived directly from mathematical expression of catadioptric imaging system components. The results of this paper offer direct and rapid analytical expression of ray path, chief ray, marginal rays and merit functions of optical systems, making optimization theory applicable in the early optical design stage. The proposed methodology is particularly suited to digital implementation. We also feel that this presentation gives a more complete and robust approach to geometrical optics than is available in the contemporary literature. Future work will extend application of this methodology to more complex surfaces.