By Psang Dain Lin (auth.)
This ebook computes the 1st- and second-order spinoff matrices of skew ray and optical direction size, whereas additionally offering a massive mathematical instrument for computerized optical layout. This booklet contains 3 elements. half One stories the elemental theories of skew-ray tracing, paraxial optics and first aberrations – crucial studying that lays the root for the modeling paintings provided within the remainder of this publication. half derives the Jacobian matrices of a ray and its optical course size. even if this factor can also be addressed in different guides, they typically fail to think about all the variables of a non-axially symmetrical approach. The modeling paintings hence presents a much better framework for the research and layout of non-axially symmetrical platforms corresponding to prisms and head-up screens. finally, half 3 proposes a computational scheme for deriving the Hessian matrices of a ray and its optical direction size, supplying a good technique of opting for a suitable seek path whilst tuning the process variables within the process layout process.
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Extra info for Advanced Geometrical Optics
18 1 Mathematical Background Fig. 16 Deﬁnition of a flat boundary surface in 3-D space by means of four parameters Given a flat boundary surface gr and two coordinate frames ðxyz)g and ðxyz)h , the transformation of flat boundary surface hr can be computed via the following matrix product: 3 2g 3 nx nx g h T 7 6 7 6 ny 7 h h À1 g 0 6 g ny 7 ; r ¼ 6 r ¼ h A g4 n 5 4 h nz 5 ¼ ð A g Þ z g h e e 2h ð1:34Þ g describes the pose of coordinate frame (xyz)g with respect to coordinate where h A 0 is the frame ðxyzÞh , and is given by Eq.
Again, the pre-superscript “g” of the leading symbol gr in Eq. 33) indicates that the flat boundary surface is deﬁned with respect to coor lying on flat boundary surface gr, the matrix dinate frame ðxyz)g . For any point g P g g product r Á P ¼ 0 is fulﬁlled. 18 1 Mathematical Background Fig. 16 Deﬁnition of a flat boundary surface in 3-D space by means of four parameters Given a flat boundary surface gr and two coordinate frames ðxyz)g and ðxyz)h , the transformation of flat boundary surface hr can be computed via the following matrix product: 3 2g 3 nx nx g h T 7 6 7 6 ny 7 h h À1 g 0 6 g ny 7 ; r ¼ 6 r ¼ h A g4 n 5 4 h nz 5 ¼ ð A g Þ z g h e e 2h ð1:34Þ g describes the pose of coordinate frame (xyz)g with respect to coordinate where h A 0 is the frame ðxyzÞh , and is given by Eq.
1). , the z0 axis direction) of the unit sphere. For convenience, the cone shown in Fig. 1 is referred to hereafter as the a0 ðb0 Þ cone, and is generated by sweeping ‘0 with a constant value of b0 around the zenith direction of the unit sphere. It is noted that ‘0 is parallel with the y0 axis when 0 of the a0 ¼ 0 and b0 ¼ 0 . From Eqs. 3), the variable vector X source ray R0 is obtained as 0 ¼ ½ P0x X P0y P0z a0 b0 T : ð2:4Þ 0 of an axis-symmetrical system is conﬁned to the y0 z0 When a point source P plane (where y0 points along the optical axis of the system), then a meridional ray (or tangential ray) is the ray lying on that plane.
Advanced Geometrical Optics by Psang Dain Lin (auth.)