Introduction to Geometrical Optics - a 2D ray tracing Excel model for spherical mirrors - Part 2
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1 Introducton to Geometrcal Optcs - a D ra tracng Ecel model for sphercal mrrors - Part b George ungu - Ths s a tutoral eplanng the creaton of an eact D ra tracng model for both sphercal concave and sphercal conve mrrors. - Whle the prevous secton dealt wth creatng and plottng the mrror, ths secton eplans the geometr of the 5-ra lght beam emergng from an artfcal star. - Ths s an eact model n the sense that no geometrcal appromatons are used, however the model does not take nto consderaton dffracton effects. The nput beam parameters: - There are two varable beam parameters set b the user: and alpha. Parameter wll be calculated from, and alpha. - Essentall we have two ponts and and we wrte the slope formula of the segment connectng them (see the dagram n the net page: alpha and from here we can derve the formula for : alpha < The European Space Agenc's Herschel telescope mrror (Far Infraed and Subllmeter telescope - FIST 1
2 Create spreadsheet entres for the nput beam parameters n the spreadsheet: - ange A9:A1 contans labels - Name call B9, name cell B1 and cell B1 alpha - B1: =(-*TAN(alpha*PI(/18+ - and alpha are nput parameters and ou can tpe some constants n those cells. Calculatng the ncdence parameters: -et s create 5 ras startng from pont wth the condton that all ht the mrror and are unforml spread. - The frst (lowest ra wll ht the mrror at the lowest etremt (A and the last one at the hghest etremt (B - Based on the above descrpton we can wrte: O(, alpha B( B, B (, d mn ma atan atan We wll retreve the coordnates of ponts A and B from the mrror table where the were alread calculated B4:C4 and B6:C6 A A B B < (, et s epress the angle dfference between two consecutve ras delta A( A, A ma mn 4
3 Spreadsheet mplementaton of the ncdence parameters: - ange A13:A15 contans just labels. - Name B13 alpha_mn, B14 alpha_ma and B15 delta_alpha - B13: =ATAN((C4-/(B4- - B14: =ATAN((C6-/(B6- - B15: =(alpha_ma-alpha_mn/4 Wrtng the equatons of the ncdent ras: - We have 5 ncdent ras startng from the artfcal star. In the prevous two pages we derved constrants for the angle of these ras so that the all end up meetng the mrror surface and coverng t unforml. - Knowng one pont through whch all the ras pass ( pont of coordnates (, and knowng the angle of each ra wth respect to the horzontal -as we can wrte the followng 5 equatons: for =,1,,.., 4 we can wrte the prevous epresson as: ( And we can fnall brng t to a standard lne equaton n the - plane: < 3 for =,1,,.., 4
4 Wrtng the equaton of the mrror surface (t s actuall a curve snce t s all n D: - We need the mrror surface equaton so that formng a sstem wth the prevousl derved ncdent ra equatons we can calculate the eact coordnates of the ponts where the ncdence ras meet the mrror. Wth that nformaton we can also calculate the slope of the reflected ras hence beng able to wrte the equatons of the reflected ras and chart them. - A standard crcle equaton n the - plane looks lke ths: C C where ( C, C are the Cartesan coordnates of the orgn In our case the mrror equaton becomes: Whch can be rewrtten as: Combnng the ncdent ra equatons wth the mrror equaton we get the followng sstem to solve to fnd the ncdence coordnates (ntersecton ponts between the ras and the mrror: for =,1,,.., 4 < 4
5 < 5 Solvng the sstem of equatons to fnd the ncdence ponts: We have the followng sstem: for =,1,,.., 4 et s substtute the from the latter equaton nto the former equaton and for now carr just the former equaton: ( tan 1 c b a ( cos 1 a b c The equaton becomes: where a, b, c are gven here:
6 a b c Ths s a quadratc equaton whose soluton s: 1, b b a 4 a c If the determnant: b 4 a c the equaton has no soluton In most of cases there are two solutons snce a lne usuall ntersects a crcle n two ponts. Judgng b the dagram to the left we can have three condtons: (a the ra msses the mrror (no soluton to the quadratc equaton - the determnant s negatve (b the ra s tangent to the mrror (the determnant s null. (c the ra ntersects the mrror twce (the determnant s postve and we need to choose the hgher value (wth + for a concave mrror ( < and choose the lower value (wth mnus n front of the square root for conve mrrors ( >. (a (c (b mrror Of course n the spreadsheet we need to ntroduce one more condton n the functon, namel when the ra goes hgher than the upper end of the mrror or lower than the lower end of the mrror the ra msses the mrror and t passes undevated. to be contnued < 6
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