By Psang Dain Lin (auth.)
This publication computes the 1st- and second-order by-product matrices of skew ray and optical direction size, whereas additionally supplying a massive mathematical device for computerized optical layout. This ebook contains 3 components. half One reports the elemental theories of skew-ray tracing, paraxial optics and first aberrations – crucial studying that lays the basis for the modeling paintings awarded within the remainder of this publication. half derives the Jacobian matrices of a ray and its optical direction size. even supposing this factor is additionally addressed in different guides, they typically fail to contemplate the entire variables of a non-axially symmetrical approach. The modeling paintings hence offers a much better framework for the research and layout of non-axially symmetrical platforms similar 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, providing a good technique of deciding upon a suitable seek course whilst tuning the process variables within the approach layout process.
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Extra info for Advanced Geometrical Optics
However, many derivations in this book are built relative to the world coordinate frame ðxyz)i . The following pose matrix of ðxyz)i with respect to ðxyz)0 is thus required: 34 2 Skew-Ray Tracing of Geometrical Optics 0 Ai ¼ tranðtix ; 0; 0Þtranð0; tiy ; 0Þtranð0; 0; tiz Þrotðz; xiz Þrotðy; xiy Þrotðx; xix Þ 2 Cxiz Cxiy Cxiz Sxiy Sxix À Sxiz Cxix Cxiz Sxiy Cxix þ Sxiz S xix 6 Sx Cx 6 iz iy Sxiz Sxiy Sxix + Cxiz Cxix Sxiz Sxiy Cxix À Cxiz Sxix ¼6 4 ÀSxiy Cxiy Sxix Cxiy Cxix 2 Iix 6I 6 iy ¼6 4 Iiz 0 0 Jix Jiy Kix Kiy Jiz 0 Kiz 0 3 0 0 3 tix tiy 7 7 7 tiz 5 1 tix tiy 7 7 7; tiz 5 1 ð2:9Þ where tix ; tiy ; tiz ; xix ; xiy and xiz are the pose variables of the spherical boundary surface.
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. 7). , the order of multiplication does not change the result. Fig. 5 Translation along xh axis by distance tx 10 1 Mathematical Background Fig. 6 Translation along yh axis by distance ty Fig. 7 Translation along zh axis by distance tz The transformation matrices corresponding to rotations about the xh , yh and zh axes of coordinate frame ðxyz)h through an angle h each time are given respectively by (see Figs.
70) can be mistakenly rewritten as 2 @2G @2F @ H þ 2 @G @H : þG ¼ H 2 2 2 @ Xi @ X @ Xi @ Xi @ Xi i ð1:71Þ X 2 This mathematical error can be observed from the alternative form of @ 2 F=@ i obtained by directly differentiating Eq. 67) to give ! M X @ 2 gpm @gpm @hmq @gpm @hmq @ 2 f pq @ 2 hmq : ¼ hmq þ þ gpm þ @xiw @xiv m¼1 @xiw @xiv @xiv @xiw @xiw @xiv @xiw @xiv ð1:72Þ In Eq. 71) is valid only if p ¼ q and w ¼ v). In order to prevent confusion, the following equation is thus used instead of Eq.
Advanced Geometrical Optics by Psang Dain Lin (auth.)