PROCEEDINGS OF SPIE. Use of computer graphics methods for efficient stray light analysis in optical design

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1 PROCEEDINGS OF SPIE SPIEDigitalLibrary.rg/cnference-prceedings-f-spie Use f cmputer graphics methds fr efficient stray light analysis in ptical design Dmitry Zhdanv, Igr S. Ptemin, Andrey D. Zhdanv, Alexey G. Vlby Dmitry Zhdanv, Igr S. Ptemin, Andrey D. Zhdanv, Alexey G. Vlby, "Use f cmputer graphics methds fr efficient stray light analysis in ptical design," Prc. SPIE 10690, Optical Design and Engineering VII, H (5 June 2018); di: / Event: SPIE Optical Systems Design, 2018, Frankfurt, Germany

2 Use f cmputer graphics methds fr efficient stray light analysis in ptical design Dmitry D. Zhdanv* a, Igr S. Ptemin a, Andrey D. Zhdanv a, Alexey G. Vlby b a ITMO University, 49 Krnverksky Pr., St. Petersburg, , Russia; b Keldysh Institute f Applied Mathematics, Miusskaya sq., 4, Mscw, , Russia ABSTRACT The prblems f using stray light visualizatin fr the effective analysis and design f cmplex ptical systems are cnsidered. Examples f real applicatins are given where the use f the light prpagatin criterin in cnjunctin with the visual representatin f the ray path makes it pssible t effectively analyze cmplex ptical design prblems. The suggested slutin allws nt nly t visualize surce f the stray light in the ptical system but alt t render the image n the detectr taking int accunt diffuse scattering n all illuminated surfaces. Keywrds: stray light analysis and preventin, mdeling, simulatin, stray light, bidirectinal stchastic ray tracing, image frmatin, ptical design, cmputer graphics, rendering. 1 INTRODUCTION Stray light is defined as unwanted light that reaches the fcal plane f an ptical system. There are a number f reasns when the stray light can ccur: a specular r Fresnel reflectin frm ptical surfaces resulted in ghsts, diffractin n diaphragms r gratings, diffuse scattering n the unplished lens surface, diffuse scattering n the surfaces f supprting structures within the ptical system (baffles, munts, struts, vanes), diffuse scattering n the surface defects (scratches and digs) and dust. The task f stray light simulatin is very imprtant and has different engineering slutins 1,2. The idea f these slutins is t trace visual rays (either frward r backward), find stray light paths reaching the detectr and define the pwer f such rays. In case f diffuse scattering n unplished parts f lens and mechanical elements the prcess f simulatin becmes very time cnsuming because f large number f inter-reflectins f stray light rays. The results f cmputer ptical mdeling are mainly presented as graphs, tables r images f distributins f such utput light characteristics as brightness, illuminatin r intensity n radiatin receivers 3. Output ptical characteristics are the direct result f ptical mdeling, bserved at the radiatin receiver. This frm f representatin f the result is cnvenient when we are primarily interested in the result f mdeling itself, and nt in hw it was btained, fr example when mdeling the brightness distributin n the surface f a liquid crystal display r the image frmed by a phtgraphic lens. Hwever, when designing ptical systems, it is ften necessary t understand hw the utput was btained, that is, hw the light frm the light surces hit the radiatin receiver. Fr example, when analyzing scattered light in a lens ptical system, yu need t knw which surface and which lens creates a glare in the image. T btain infrmatin n hw the light prpagates in an ptical system, it is mst cnvenient t use the visualizatin f ray paths in that ptical system. In additin, visual representatin f the paths f light rays in the ptical system is als useful fr the sftware develper, fr the mean f debugging and ptimizing ptical simulatin algrithms 4. Fr physically crrect and effective mdeling f the light prpagatin in ptical systems, varius ray methds are used. The mst efficient methd f simulatin the light prpagatin in the ptical system, which allws slving cmplex prblems f light prpagatin in light-cnducting ptical systems with surface and vlume scattering elements, is the methd f frward Mnte Carl ray tracing. This algrithm was first prpsed in 5 and then develped widely in ptical mdeling systems 6. Fr ptical mdeling, the preferred frm f the Mnte Carl methd is the "Russian rulette" 1. This methd, n the ne hand, has a simple prgram interface, and n the ther hand it has quite high perfrmance in mdeling high-efficient ptical systems. Frward Mnte Carl ray tracing methd simulates the prpagatin f light rays frm the light surce t the radiatin receiver and thereby statistically reprduces the distributin f illuminatin, intensity r brightness at the radiatin receiver. The Mnte Carl methd allws t simulate all the physical effects f ray prpagatin (diffuse scattering n surfaces, mirrr reflectin, refractin, plarizatin state change, birefringence, etc.). *ddzhdanv@mail.ru; phne Optical Design and Engineering VII, edited by Laurent Mazuray, Rlf Wartmann, Andrew P. Wd, Prc. f SPIE Vl , H 2018 SPIE CCC cde: X/18/$18 di: / Prc. f SPIE Vl H-1

3 When designing ptical systems, it is ften necessary t determine the effect f the specified cmpnents f the ptical system n the quality f the image r hw these cmpnents affect the prpagatin f light. Fr example, when analyzing an illuminating system, it may be necessary t knw what cntributin t the utput light distributin is made by light reflected frm a particular surface f the system, r, fr example, it is necessary t determine hw the highlight between tw surfaces f the lens ptical system is frmed. The use f the Mnte Carl ray tracing methd t simulate the prpagatin f light in ptical systems is in gd agreement with the pssibility t analyze the influence f individual cmpnents f the system n the light prpagatin. When tracing rays, it is necessary t select nly rays that satisfy a certain criterin. This criterin can be either simple, fr example, t islate all rays that have undergne a duble Fresnel reflectin, r mre cmplicated, taking int accunt the rder f events ccurring with the ray in time, fr example, first the ray must pass thrugh a certain surface, and then underg a duble Fresnel reflectin. The ray prpagatin criterin in the ptical system can be used bth fr visualizatin f ray trajectries and fr calculating the distributin f light characteristics n radiatin receivers. The use f ray path criteria makes it pssible t clearly demnstrate the effect f individual effects and cmpnents f the ptical system n the prpagatin f light in an ptical system. Even if the frward Mnte Carl ray tracing methd is rather gd fr light mdeling it is nt very effective slutin fr simulatin f light scattering n diffuse surfaces. T effectively slve this prblem, we designed the slutins which allws nt nly visualize stray light paths but als t render the virtual image which is frmed n the detectr. It takes int accunt a diffuse scattering n all illuminated surfaces. The article presents results f the stray light analysis and simulatin in the typical lens systems. 2 RAY VISUALIZATION When designing an ptical system, it is required t analyze hw light ges thrugh this system, hw rays frms an image n the image sensr r what path d rays have that frm sme effect visible n the final image. Figure 1a shws paths f first 1000 randm rays emitted by sun prpagating thrugh an ptical system. Hwever, visualizatin f all rays traced thrugh the ptical system d nt have t much sense. Frm the figure abve, ne can see that all rays emitted by sun that gets in ptical system d nt frm glares n the luminance sensr. But if criterin f at least ne reflectin n the ray path and a hit with image sensr is specified a different picture shwn n figure 1b wuld be seen. N, \ latha, I"-,l r Figure 1. Visualizatin f a) first 1000 rays paths emitted by sun and traced thrugh the ptical system; b) first 1000 rays paths emitted by sun and traced thrugh the ptical system qualifying the criterin f tw specular reflectins Nw we can see that sme f rays emitted by Sun hits the image sensr. The percentage f such rays is quite small, hwever, taking int the accunt that the Sun luminance is high and as result its influence n the final image frmed n the image sensr might be clearly visible. T analyze light prpagatin in the ptical system a criterin shuld be designed. 2.1 Ray selectin criterin When designing and analyzing the designed ptical systems it is required t include in the special criteria int the ptical mdeling system that allws selectin f rays that have certain prperties and are the subject f analysis f the light prpagatin in that ptical system. The light prpagatin criterin is a special prgram bject that analyzes the histry f beam prpagatin in the ptical system and decides whether the specific ray satisfies the given criterin r nt. If the ray Prc. f SPIE Vl H-2

4 meets the criterin, it is accepted by the ptical mdeling system and its path can be visualized n tp f the ptical system image. The light prpagatin criterin is clsely related t the histry f ray path in the ptical system. The histry f the ray path captures all the events that ccur n the path f the ray frm the light surce t the end: e i Ray is emitted by LED light surce emissin the ray frm the light surce; ray path as a set f linear segments; clr f the ray at each segment; an event that has ccurred at the end f each segment: ray emissin; ray reflectin and its type; ray refractin and its type; vlume scattering; absrptin n the surface r in media; ray leaving the scene dmain; etc. scene bjects crrespnding t the ccurred event; luminance r illuminance detectrs that accumulated the ray energy. Figure 2 shws tw examples f ray prpagatin path in a reflectr light surce ptical system. Ray htst "Reflectr" surface Specular Ray hits the Specular -i. reflectin "Lens" surface refractin Ray enters the "Glass" medium Medium absrptin Ray is emitted by LED light surce Ray hits the "Lens" surface Specular refractin Ray enters the "Glass" medium Ray hits the "Lens" surface Specular refractin Ray enters the "Envirnment" medium Ray leaves the scene dmain Figure 2. Ray prpagatin histry examples. T acquire the ptimal interactin between the ray prpagatin criterin and its prpagatin histry, the criterin is implemented as a tree which leaves are elementary events that shuld ccur with the ray n its path and ndes f this tree are lgical peratins (AND, OR and NOT) that determine the lgical cnnectin between elementary events. Pssible elementary events are based n the events that can ccur in histry f ray prpagatin in the ptical system that simplifies the applicatin f the ray path criterin t its histry. The main events that criterin can analyze are: the ray is emitted frm the specified light surce; the ray hits the specified gemetrical bject; the ray hits the specified luminance r illuminance detectr; the ray hits a surface with specified ptical prperties r a medium; the specified event ccurs with ray. Fr each elementary event a cunter can be specified that adds a requirement n number f ccurrence f this event in ray path histry. In additin, a special sequence nde can be added t a tree that requires that all its children ccur in a specified rder. Fr example, this allws t select rays that befre hitting the lens surface either secularly refracts r hits the glass medium. The crrespnding tree is shwn n figure 3. Prc. f SPIE Vl H-3

5 Sequence Ray hits the "Lens" surface Specular refractin Ray enters the "Glass" medium Figure 3. Ray selectin criterin example. Secnd sequence frm figure 3 qualifies this criterin while the first ne des nt. Fr quick executin f the criterin check, the regular expressin is cmmnly transfrmed int a finite state machine. Hwever, the number f states f that finite state machine increases expnentially with the number f ndes in the criterin tree. Using this apprach n real ray path criteria, the number f states f the finite state machine becmes extremely large. S, we have develped an riginal methd fr checking if the ray path histry qualifies the criterin. The develped methd must have high perfrmance and lw memry requirements. The fllwing testing methd is prpsed: The sequence f events is checked using the criterin tree described abve. Each tree nde can be in ne f three pssible states: "true", "false" r "undefined ". As sn as the state f the rt nde f the tree changes frm "undefined" t "true" r "false," the calculatin terminates, and the state f the rt nde is taken as the result f the criterin checker wrk. The elements f the sequence f events are applied alternately t the ray prpagatin criterin tree. Each element f this sequence sets "true" values in the crrespnding "undefined" tree leaves when the fllwing cnditins are met: 1. The event cnditin specified in the tree leaf fulfils the next element f the sequence f events. 2. All parents f this leaf are in the "undefined" state. 3. If amng the parents f this leaf there is a time sequence cnditin, then all direct children f the time sequence cnditin lcated "t the left" f direct leaf parent shuld be in the "true" state. When the value f any nde in the tree is changed, the value f its parent nde is recalculated. As a result, the parent nde can either change its value frm "undefined" t "true" r "false" r leave it untuched. In the case when the criterin remains in the state f "undefined" at the end f the sequence f events, all leaves f the criterin tree which are still in the "undefined" state are assigned the "false" value and the result f the criterin calculated. 2.2 Simulatin results One f the basic requirements fr image quality frmed by lens ptical systems is absence f the parasitic lighting. Mre ften the parasitic lighting in ptical systems is frmed either as a result f light interreflectins between lens surfaces, frming ghsts, r as a result f diffuse scattering f ptical elements like fr example lens munts r butt ends. T analyze the ghsts in the lens ptical system a special criterin fr the ray path in the ptical system shuld be used. The criterin cnsisted f three successive events-the reflectin f the ray frm the back surface f the furth lens, then the reflectin frm the frnt surface f the furth lens and, finally, the hit with the image sensr. T analyze the diffuse scattering n the lens munts, a criterin cnsisting f tw events was used: diffuse scattering n the munts f lenses and hit with the image sensr. Figure 4 demnstrates simulatin results with pssible paths f the parasitic rays in the lens ptical system with the crrespnding utput distributin f parasitic illuminatin that was added t the final image due t light scattering n the lens surfaces and munts. Prc. f SPIE Vl H-4

6 Stray light frm primary light surce Stray light frm secndary light surce Light surce Image plane Object Scattering \111% - Lens with case Figure 4. Stray light in lens ptical systems frmed by the interreflectins frm the lens surfaces and the scattering n the lens munts. These examples shw the simplest cases f parasitic light analysis. Figure 5 demnstrate the flexibility and simplicity f the criterin fr the paths f the ray that makes it pssible t analyze any surces f parasitic light, bth cmpsite (all ghsts f all rders) and individual (ghst f a given rder between given surfaces). 11W"'M TOWill Figure 5. Stray light in lens ptical systems frmed by a) the interreflectin frm lenses; b) the interreflectin frm the frth lens nly In additin, this flexibility allws specifying the surce f light scattering fr investigatin. The fllwing figure 6 shws hw surce f scattering can be investigated with ray path criterin specificatin. Prc. f SPIE Vl H-5

7 Figure 6. Stray light in lens ptical systems frmed by a) the scattering n the lens butt ends; b) the scattering n the lens case. 3 STRAY LIGHT SIMULATION Fr stray light analysis caused by surface scattering the mst imprtant unwanted luminance cmpnent is the secndary and caustic luminance. Physically accurate calculatin f the light scattering n diffuse surfaces is prvided by the rendering equatin 7. Fr static scenes, the luminance f the clr cmpnent f the bject c at the pint f the surface, with the lcal nrmal and in the directin, can be written as fllwing: ( ) ( ) ( ( ) ( ) ( )( ) ) (1) where ( ) is the wn bject luminance in an bservatin pint, ( ) the transmittance (transparency) f the medium between the bserver and the bservatin pint ( ) ( )( ) the luminance frmed by primary and secndary illuminatin f the bserved bject, where ( ) is the luminance factr f the surface (r Bidirectinal Scattering Distributin Functin (BSDF)) frm the surce in directin t the bserver, ( ) the luminance f the light surces in a slid angle dω in the directin t the bservatin pint. T slve rendering equatin, the fllwing three basic methds f stchastic ray tracing are applied: frward stchastic ray tracing 8 ; backward stchastic ray tracing 9 ; bidirectinal stchastic ray tracing 10. The chice f the ray tracing methd depends n the regin in which the integratin can be perfrmed mst efficiently; i.e., the effective regin f integratin is maximal. In turn, the effective regin f integratin is determined by parameters f the bserver, light surces, and ptical prperties f the ptical system elements. If cntributin f secndary lighting t the integral luminance value is essential then the mst apprpriate methd fr slutin f the rendering equatin is the bidirectinal stchastic ray tracing. The bidirectinal stchastic ray tracing is a sequential stchastic tracing f backward and frward rays and special prcessing f saved paths. In the result f the ray paths prcessing the fllwing fur surces f luminance are frmed: visible luminance f light surces; i.e. the luminance frmed by traces f backward rays withut any diffuse scattering; primary r direct luminance; i.e. the luminance frmed by lights directly illuminating the pint f bservatin; Prc. f SPIE Vl H-6

8 caustic luminance; i.e. the luminance f directly viewed diffuse bjects frmed by light surces which illuminate the pint f bservatin withut scattering; secndary luminance; i.e. the luminance frmed by diffusively scattered rays with tw r mre scatterings n the traces frm light surce t the pint f bservatin. Bidirectinal stchastic ray tracing cmbines all the main advantages f the frward and backward ray tracing and, in additin, allws t find ut the mst suitable pint fr the integratin f secndary luminance. Fr effective calculatin f caustic luminance cmpnent this methd shuld be extended with pssibility t frm secndary and caustic illuminances n the phtn maps and recalculate them int the visible luminance. We elabrated the algrithm f bidirectinal stchastic ray tracing which includes tw stages (see Figure 7). On the first stage backward rays are traced and the direct luminance and the luminance f directly visible light surces are calculated. Als, the phtn maps as an array f spheres f integratin with centers in the pint f intersectin f backward rays with diffuse surfaces f the ptical system are stred. On the secnd stage, the frward rays are traced and the secndary and caustic luminances are calculated in the areas f intersectin f the frward rays with the phtn map that were stred n the first stage as the spheres f integratin. Backward stchastic ray tracing, calculatin f direct luminance, luminance f directly visible light surces and generatin f view maps Light surce Lens munt Lens Detectr í. Frward stchastic ray tracing, calculatin f indirect and caustic luminances Sphere f integratin Frward ray Backward ray Figure 7. Tw stages f bidirectinal stchastic ray tracing with phtn maps algrithm. Integratin f the calculated luminance f detectr pint ver whle exit aperture allws calculating detectr illuminance and repeating the illuminance calculatin fr all pints f detectr. It allws t render image created frm stray light illuminatin. Generally, bidirectinal stchastic ray tracing with phtn maps allws t slve the rendering equatin in a physically accurate way. But taking limited cmputer resurces int accunt equatins can be slved nly fr limited number f frward and backward rays (usually up t tens millins). Usually this number is nt sufficient fr accurate slutin s the simulatin prcess (backward ray tracing, phtn map creatin, frward rays tracing, and luminance Prc. f SPIE Vl H-7

9 calculatins) is repeated and temprary results are accumulated while required simulatin accuracy is reached. Stchastic apprach f simulatin prcess has a number f advantages. At first, average detectr illuminance gives crrect value frm the first accumulatin steps that allws t estimate general parameters f stray light (transmittance f the ptical system fr stray light illuminatin) frm the beginning. At secnd, the simulatin prcess can be cntinued if accuracy f calculated illuminance distributin n light detectr is nt sufficient. Fr stray light analysis f the ptical system, the main cmpnents f parasitic illuminatin are caustic and indirect ne. The main factrs t influence n accuracy f indirect and caustic illuminance calculatins are prbability f intersectin f paths f frward rays with integratin spheres (phtn maps f backward rays). S in the case f unifrm BSDF the methd f bidirectinal ray tracing allws calculatin f stray light illuminance frm very dark surfaces with the same efficiency as light nes. Fr example, cmputatin time t reach the same accuracy f the stray light illuminance distributin fr ptical system with specially cvered lens munt and reflectance abut 0.1%, and nt cvered cmpnents f the lens munt and reflectance abut 30% will be the same. The bidirectinal stchastic ray tracing slutin was integrated in Lumicept sftware package 10 and efficiently used fr stray light analysis f different kinds f ptical systems. 3.1 Simulatin results The current article presents the results f the stray light mdeling with the help f the develped methds and algrithms fr tw different ptical systems: 6-lenses lng-fcus lens system; 2-lens 2-mirrr ultra-lng-fcus catadiptric lens system. First lens is 6-lenses ptical system with 360mm fcal length (f/7) and ±5 rectangular field f view (FOV). The light surce is the Sun that illuminates the lens entrance pupil frm an ut f the field f view under an incident angle ω Sun = 7. As the field f view f the simulated lens (ω FO V = 5 ) is less than ω Sun s the detectr des nt see the Sun directly. General setup design is shwn n figure 8. Diaphragm 50um width Detectr WSun Figure 8. 6-lensel lng-fcus lens general design with an ut f view field light surce. The light frm sun can reach the detectr nly if it is scattered n sme element f the lens system. In given simulatin we tk int accunt nly stray light scattered n diffuse surfaces f lens system cmpnents (munts and unplished lens butt end surfaces). All image frming ptical surfaces are assumed t be cvered with perfect antireflectin layer s n reflectin shuld ccur n the clear lens surfaces. Figure 9 shws simulatin results f the caustic illuminance distributin frm the Sun light scattering n the edge f aperture diaphragm. This edge has 50 um width and 10% Lambertian scattering. Excepting the diaphragm edge all ther diffuse surfaces were cnsidered as abslutely absrbing. Calculatin time was 6 hurs and 33 minutes. Prc. f SPIE Vl H-8

10 Figure 9. Stray light illuminance distributin n the detectr in case f Lambertian scattering n the aperture diaphragm. Figure 10a shws simulatin results f the caustic and indirect illuminance distributin frm the Sun light scattering n the ptical system lens munts and lens butt ends. In this simulatin all lens munts and butt ends scatterings are set t 5% Lambertian. Diaphragm edge was cnsidered as an abslutely absrbing. Calculatin time was 18 hurs and 30 minutes. Figure 10b shws simulatin results f the caustic and indirect illuminance distributin frm the Sun light scattering n the ptical system lens munts and lens butt ends. In this simulatin all lens munts and butt ends scatterings are set t 5% Gaussian with 5 half-width diagram. Diaphragm edge was cnsidered as an abslutely absrbing. As in the previus case, the calculatin time was 18 hurs and 30 minutes..0500bc óc óc óc k óc óc óc bc k bc k óc bc c 0.00óc a b à óc óc bc óc óc óc óc óc óc óc óc óc óc óc b Figure 10. Stray light illuminance distributin n the detectr in case f a) Lambertian scattering n the lens munts and butt ends; b) Gaussian scattering n the lens munts and butt ends. Frm simulatin results it can be clearly seen that the images frmed by stray light are mre r less the same in bth cases f scattering n lens munts and butt ends. But in spite f such a similarity in case f Gaussian reflectance the illuminance f stray light n the detectr is essentially higher than in case f Lambertian ne. The secnd lens is a tw-mirrr Catadiptric ptical system with 2000mm fcal length (f/10) and 1 15 field f view. The entrance windw f the lens system is unifrmly illuminated inside f 3.25 cne. Ttal flux f the illuminatin is lm. Its general design with illuminatin setup is shwn n figure 11. Prc. f SPIE Vl H-9

11 Illuminatin cne wfov Figure 11. Catadiptric tw-mirrr lens general design and illuminatin setup. Lens munt The simulatin gal was t simulate indirect and caustic cmpnents f stray light illuminatin f detectr caused by diffuse scattering n the lens butt ends and munts. Tw independent simulatins were perfrmed t find ut the stray light cause: simulatin f light scattering n the lens butt ends and munts; simulatin f light scattering n the lens butt ends nly. Figure 12 shws simulatin results f the light scattering n the ptical system lens munts and lens butt ends. In this simulatin all lens butt ends are diffuse surfaces with 50% diffuse transmittance and 50% diffuse reflectance; lens munts are almst abslutely absrbing with nly 1% diffuse reflectance; lens surfaces are clear with 1% specular reflectance. The calculatin time was 8 hurs and 33 minutes óc e-005óc I óc e-005óc óc e-005óc óc e-0051c óc e-0051x óc e-005k óc e-005óc Caustic illuminatin óc In d itect 'situ min atin óc Figure 12. Stray light illuminance distributin n the detectr in case f scattering n the lens munts and butt ends. Figure 13 shws simulatin results f the light scattering n the ptical system lens butt ends nly. In this simulatin all lens butt ends are diffuse surfaces with 50% diffuse transmittance and 50% diffuse reflectance; lens munts are abslutely absrbing; lens surfaces are clear with 1% specular reflectance. The calculatin time was 22 hurs and 57 minutes. Prc. f SPIE Vl H-10

12 c b óc e-0066c e-006bc e-006óc óc e-0061x óc e-006óc % e-006c óc e-006óc 5.150e-005óc e-006bc Caustic illuminatin c Indirect illuminatin óc Figure 13. Stray light illuminance distributin n the detectr in case f scattering n the lens butt ends. Frm these simulatin results we can see that level f light scattered n the lens butt ends that reached the detectr is essentially lwer than stray light level frm munt parts f the lens ptical system. 4 CONCLUSION The cmputer graphics technlgy methds prvide quite gd results in visualizatin f stray light in cmplex ptical systems. The physically crrect methd f stchastic bidirectinal ray tracing with phtn maps was elabrated that allws simulating stray light caused by scattering n the diffuse surfaces f ptical systems. The elabrated slutin prvides physically accurate and high-efficient rendering f the parasitic light (sky and sun, fr example) scattered n lens munt and ther scattering elements f the ptical system. All f the elabrated slutins were integrated in the Lumicept sftware package 11 and can be successfully used fr stray light analysis and simulatin. ACKNOWLEDGMENTS The research was partially supprted by RFBR grants N and , as well as by Integra Inc. REFERENCES [1] Fest, E. C., "Stray light Analysis and Cntrl," SPIE press, Vl. N.: PM229 (2013). [2] Kleiner. S., "StrayLight and Ghst Images," Laser and Phtnik, 1, 46-49, (2007). [3] Kpylv, E. A. and Dmitriev. K. A., "Light prpagatin visualizatin as a tl fr 3D scene analysis in lighting design," Cmputers & Graphics, vl.24, n.1, 31-39, (2000). [4] Wernert, E., "A unified envirnment fr presenting, develping and analyzing graphics algrithms," Cmputer Graphics, vl. 31, n. 3, (1997). [5] Ck, R., Prter, T. and Carpenter, L., "Distributed ray tracing," Prc. SIGGRAPH 84, Cmputer Graphics, (1984). [6] Pharr, M. and Humphreys, G., "Physically Based Rendering. Frm thery t implementatin," Mrgan Kaufmann (2004). [7] Kajiya, J. T., "The rendering equatin," Prc. SIGGRAPH '86, Cmputer Graphics, vl. 20, (1986). [8] Tshiya, H. and Wann, J. H., "Stchastic prgressive phtn mapping," ACM Trans. Graph., Vl. 28, n. 5. P. 141:1 141:8 (2009). [9] Veach E. Rbust mnte carl methds fr light transprt simulatin," Ph. D. thesis. Stanfrd, CA, USA: Stanfrd University, AAI (1998). Prc. f SPIE Vl H-11

13 [10] Zhdanv, D., Garbul, A., Mayrv, V., Sklv, V., Ptemin, I., Hyd, T., Vlby, A. and Galaktinv, V., "Autmatic Design f Illuminatin Systems", Optical Review, vl. 20, n.2, p (2013). [11] "Lumicept Hybrid Light Simulatin Sftware," Integra Inc., 2018, < (9 April 2018). Prc. f SPIE Vl H-12