By Psang Dain Lin (auth.)
This booklet computes the 1st- and second-order spinoff matrices of skew ray and optical course size, whereas additionally supplying an immense mathematical device for computerized optical layout. This ebook contains 3 elements. half One reports the elemental theories of skew-ray tracing, paraxial optics and first aberrations – crucial analyzing that lays the basis for the modeling paintings awarded within the remainder of this e-book. half derives the Jacobian matrices of a ray and its optical course size. even supposing this factor can be addressed in different guides, they typically fail to think about all the variables of a non-axially symmetrical procedure. The modeling paintings therefore presents a much better framework for the research and layout of non-axially symmetrical platforms equivalent to prisms and head-up screens. finally, half 3 proposes a computational scheme for deriving the Hessian matrices of a ray and its optical course size, supplying a good technique of picking out a suitable seek path while tuning the process variables within the process layout process.
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Extra resources for Advanced Geometrical Optics
Given the pose matrix described in Eq. , hA g ¼ tranðtx ; 0; 0Þtranð0; ty ; 0Þtranð0; 0; tz Þrotðz; x Þrotðy; xy Þrotðx; xx ÞÞ; z then, from basic matrix theory, the inverse matrix is given as hA g À1 ¼ rotðx; Àxx Þrotðy; Àxy Þrotðz; Àxz ÞtranðÀtx ; Àty ; Àtz Þ: ¼ gA h g by g A . 3. 7 0 0 1 Flat Boundary Surface In general, any flat boundary surface can be speciﬁed uniquely in 3-D space in a Â ÃT coordinate frame ðxyzÞg using its unit normal vector g n ¼ g nx g ny g nz 0 and a point by ðg nx Þx þ ðg ny Þy þ ðg nz Þz þ g e ¼ 0.
Any intermediate point P iÀ1 iÀ1 þ k‘iÀ1 . The parameter k ¼ ki , for which the incoming 0 ¼ P ray is given by P iÀ1 iÀ1 hits the current boundary surface ri at incidence point ray R 3 2 3 Pix PiÀ1x þ ki ‘iÀ1x 7 6 7 6 i ¼ 6 Piy 7 ¼ 6 PiÀ1y þ ki ‘iÀ1y 7 ¼ P iÀ1 þ ki ‘iÀ1 ; P 4 Piz 5 4 PiÀ1z þ ki ‘iÀ1z 5 1 1 2 0 P i ÞÀ1 P i ¼ i A i ¼ ð0 A i . That is, can be obtained by equating Eq. 2 Spherical Boundary Surfaces 35 Fig. 4 Raytracing at spherical boundary surface ri 2 Iix Iiz Jiy Jiz ÀðIix tix þ Iiy tiy þ Iiz tiz Þ 32 PiÀ1x þ ki ‘iÀ1x 3 7 6 ÀðJix tix þ Jiy tiy þ Jiz tiz Þ 7 76 PiÀ1y þ ki ‘iÀ1y 7 76 7 Kiy Kiz ÀðKix tix þ Kiy tiy þ Kiz tiz Þ 54 PiÀ1z þ ki ‘iÀ1z 5 0 0 0 1 1 2 3 2 3 ri jRi j Cbi Cai 6q 7 6 jR j Cb Sa 7 i7 6 i7 6 i i ¼ 6 7 ¼ iri ¼ 6 7; 4 si 5 4 jRi j Sbi 5 6J 6 ix Pi ¼ 6 4 Kix i Iiy 1 ð2:12Þ 1 where ri , qi , and si are coordinates of incidence point expressed in boundary coordinate frame ðxyzÞi , given by À Á ri ¼ Iix ðPiÀ1x þ ‘iÀ1x ki Þ þ Iiy PiÀ1y þ ‘iÀ1y ki þ Iiz ðPiÀ1z þ ‘iÀ1z ki Þ À ðIix tix þ Iiy tiy þ Iiz tiz Þ ð2:13Þ ¼ Iix Pix þ Iiy Piy þ Iiz Piz À ðIix tix þ Iiy tiy þ Iiz tiz Þ; À Á qi ¼ Jix ðPiÀ1x þ ‘iÀ1x ki Þ þ Jiy PiÀ1y þ ‘iÀ1y ki þ Jiz ðPiÀ1z þ ‘iÀ1z ki Þ À ðJix tix þ Jiy tiy þ Jiz tiz Þ ¼ Jix Pix þ Jiy Piy þ Jiz Piz À ðJix tix þ Jiy tiy þ Jiz tiz Þ; ð2:14Þ 36 2 Skew-Ray Tracing of Geometrical Optics À Á si ¼ Kix ðPiÀ1x þ ‘iÀ1x ki Þ þ Kiy PiÀ1y þ ‘iÀ1y ki þ Kiz ðPiÀ1z þ ‘iÀ1z ki Þ À ðKix tix þ Kiy tiy þ Kiz tiz Þ ¼ Kix Pix þ Kiy Piy þ Kiz Piz À ðKix tix þ Kiy tiy þ Kiz tiz Þ: ð2:15Þ From the sum of r2i , q2i and s2i , ki is obtained as ki ¼ ÀDi Æ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D2i À Ei ; ð2:16Þ where Di ¼ ‘iÀ1x ðPiÀ1x À tix Þ þ ‘iÀ1y ðPiÀ1y À tiy Þ þ ‘iÀ1z ðPiÀ1z À tiz Þ; ð2:17Þ Ei ¼ P2iÀ1x þ P2iÀ1y þ P2iÀ1z À R2i þ t2ix þ t2iy þ t2iz À 2ðtix PiÀ1x þ tiy PiÀ1y þ tiz PiÀ1z Þ: ð2:18Þ The non-negative parameter ki represents the geometrical path length from point iÀ1 to point P i .
7, designate i ¼ ½ tix X tiy tiz xix xiy xiz niÀ1 ni R i T ð2:30Þ as the boundary variable vector of a spherical boundary surface, where this vector comprises the six pose variables of Eq. 9), the refractive indices niÀ1 and ni , and the radius Ri of the boundary surface. i is a function of the It is noted from Eqs. 29) that the ray R iÀ1 with the given source ray R 0 (Fig. 8). , a function which operates in turn on another function (or functions). It is like a Russian nesting doll. Each doll has a smaller and smaller doll inside it.
Advanced Geometrical Optics by Psang Dain Lin (auth.)