By Psang Dain Lin (auth.)
This e-book computes the 1st- and second-order by-product matrices of skew ray and optical course size, whereas additionally supplying an immense mathematical software for automated optical layout. This ebook includes 3 components. half One studies the fundamental theories of skew-ray tracing, paraxial optics and first aberrations – crucial analyzing that lays the root for the modeling paintings provided within the remainder of this ebook. half derives the Jacobian matrices of a ray and its optical direction size. even supposing this factor is usually addressed in different guides, they typically fail to think about the entire variables of a non-axially symmetrical method. The modeling paintings therefore offers a much better framework for the research and layout of non-axially symmetrical structures reminiscent of 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, providing a good technique of selecting a suitable seek course while tuning the process variables within the method layout process.
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Additional resources for Advanced Geometrical Optics
13): h Fig. 12 Roll, pitch and yaw angles 14 1 Mathematical Background Fig. 13 Euler angles hA g ¼ Eulerðxz ; xy ; hz Þ ¼ rotðz; xz Þrotðy; xy Þrotðz; hz Þ 2 Cxz Cxy Chz À Sxz Shz ÀCxz Cxy Shz À Sxz Chz 6 Sx Cx Ch þ Cx Sh z z ÀSxz Cxy Shz þ Cxz Chz 6 z y z ¼6 4 ÀSxy Chz Sxy Shz 0 0 Cxz Sxy Sxz Sxy Cxy 0 0 3 07 7 7: 05 1 ð1:27Þ However, it is noted from Eq. 27) that the Euler transformation involves two rotations about two different z axes. Consequently, it is not used in this book in order to avoid confusion.
E A ffA g; Ag ¼ h A ð1:11Þ where Eq. 11) shows successive coordinate transformation from ðxyz)h to ðxyzÞa , (xyz)a to (xyz)b ,…, and (xyz)f to (xyz)g . 5 of the two coordinate frames shown in Fig. 4 Given a vector 5 P 55P 1 is obtained as 4 P 1 ¼ 4 A 1 ¼ ½ 7 0 À2 1T . 4 Basic Translation and Rotation Matrices 9 Fig. 4 Basic Translation and Rotation Matrices The transformation matrices corresponding to translations along vectors txi, tyj and tz k with respect to coordinate frame ðxyz)h are given respectively by (see Figs.
10 Schematic illustration of rotðz; hÞ 11 12 1 Mathematical Background axes, not of the xg , yg and zg axes. Furthermore, the transformation matrices have a simple geometric interpretation. For example, in the transformation matrix h Ag ¼ rotðz; hÞ, the third column represents the z axis and remains constant during rotation, while the ﬁrst and second columns represent the x and y axes, and vary as shown in Fig. 10. When performing matrix manipulations, the following properties are of great practical use: tranðtx ; ty ; tz Þtranðpx ; py ; pz Þ ¼ tranðtx þ px ; ty þ py ; tz þ pz Þ; ð1:18Þ rotðx; hÞrotðx; UÞ ¼ rotðx; h þ UÞ; ð1:19Þ rotðy; hÞrotðy; UÞ ¼ rotðy; h þ UÞ; ð1:20Þ rotðz; hÞrotðz; UÞ ¼ rotðz; h þ UÞ ð1:21Þ rotðx; hÞtranðtx ; 0; 0Þ ¼ tranðtx ; 0; 0Þrotðx; hÞ; ð1:22Þ rotðy; hÞtranð0; ty ; 0Þ ¼ tranð0; ty ; 0Þrotðy; hÞ; ð1:23Þ rotðz; hÞtranð0; 0; tz Þ ¼ tranð0; 0; tz Þrotðz; hÞ: ð1:24Þ The transformation matrix representing rotation around an arbitrary unit vector Â ÃT j ¼ jx jy jz 0 , where j2x þ j2y þ j2z ¼ 1, located at the origin of coordinate frame ðxyz)h (see Fig.