. Perform a geometric (ray-optics) construction (i.e., draw in the rays on the diagram) to show where the final image is formed.

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1 MASSACHUSETTS INSTITUTE of TECHNOLOGY Departmet of Electrical Egieerig ad Computer Sciece Moder Optics Project Laboratory Optical Sigals, Devices & Systems Problem Set No. 1 Geometric optics Issued Tues. 9/13/2016 Fall Term, Due Tues. 9/20/2016 Readig recommedatio: Class Notes, Chapter 1. Be eat i your work! Problem 1.1 The two-mirror imagig system show o the ext page cosists of a large primary mirror, M 1, with radius of curvature, R 1, ad a small secodary mirror, M 2, with a radius of curvature, R 2. Both mirrors are cocave. I the system, d 1 is the distace of the object from the primary mirror, d 2 is the separatio betwee the mirrors, ad d 3 (ot show) is the distace of the fial image from M 2. (a) I the figure, you are give the special case where d 2 = R 1 2. Perform a geometric (ray-optics) costructio (i.e., draw i the rays o the diagram) to show where the fial image is formed. (b) Is the fial image real or virtual? (c) Show the positio ad orietatio of itermediate images, if ay, ad label them as real or virtual o the diagram. (d) For the case where both the mirror separatio, d 2, is arbitrary ad d 1 {d 2,R 1,R 2 }, ad with the help of the class otes, write dow ad simplify a expressio for the fial image distace, d 3, i terms of d 1, d 2, R 1, ad R 2. 1

2 Problem 1.1, Cotiued... object R 1 R 2 M 2 d 1 d 2 = R 1 /2 M 1 2

3 Problem 1.2 P d P.. h F * F* BFL R 2 R 1 z (a) A ray of light eters a cylidrically symmetric bead of glass ( thick les ) of refractive idex at a height h above its pricipal axis. The les has etrace ad exit faces with radii of curvature R 1 ad R 2 as show. Assumig the usual small-agle ad thi-les approximatios hold for each compoet of the thick les, use the ray-matrix approach to determie the approximate distace F at which the ray crosses the z-axis. (b) Now ru the ray backwards through the bead, keepig it still at a height h above the pricipal axis, so it is icidet of the R 2 facet first. What is the ew value F of F? (c) Does your result deped o h?. (d) Whe d = 0, show that your result for F is i agreemet with the les maker s formula. (e) Whe d = 0, do you get the expected result for two leses i series? (f) Based o your results for (a)-(e), commet o the use of such a bead as a les. For example, does it have a well-defied focal legth? What are its imagig properties? The plaes P ad P are ofte called pricipal plaes. F ad F are called the Effective Focal Legths (EFL). However, i practical use, the tedecy is to measure the Frot Focal Legth (FFL), sometimes called the frot focal distace (FFD), which is the distace from the vertex of the first optical surface to the frot focal poit. A similar defiitio holds for the Back Focal Legth (BFL). The FFL ad the BFL ca be calculated from the EFLs ad R 1 ad R 2 ad d. 3

4 Problem 1.3 Matchig FOV ad etrace pupil of cameras Whe creatig sythetic scees, projectio optics are ofte used to maipulate the source image (typically formed o a real-time display). The figure below shows a 3-les system that is desiged to couple the image o a display ito a camera. The display has a diameter 2D m ad emits light over a full coe agle of 2θ m. The camera has a fixed full agular field of view (FOV)of 2θ s ad a fixed etrace pupil of diameter 2D s. It is assumed the camera has its ow les F 4 (show as a dashed outlie) ad ca therefore form a image of the object geerated by the display. iput plae with display (ρi,ρ i ') F 2 F 3 output plae (ρ o,ρ o ') 2D m 2D s d 1 d 2 d 3 Camera d 4 The goal is to desig the 3-les projectio optics so that o light leavig the display is wasted, ad that maximum image size is achieved iside the camera. That is, we wat to icomig light to match the FOV ad the pupil diameter of the camera. (a) Derive A, B, C, ad D of the ABCD ray-optics matrix (i terms of the focal legths of the leses ad d 1, d 2, d 3 ad d 4 ) for the system bouded by the give object ad the iput plae to the camera. To help elimiate algebraic errors, you may wat to use Mathematica, Maple or Matlab for this exercise. (b) Cosider the special imagig case where we wat to make the image o the display appear to the camera as if the object was ifiitely far away. That is, whe the camera is focused at ifiity, the camera output image must be sharp. I this case, the regio betwee F 3 ad the camera is ofte referred to as collimated space. What coditio o the matrix elemets A, B, C, ad D would have to hold so that this is the case? (c) To match the FOV ad the iput aperture dimesios of the camera, what costraits o the matrix elemets A, B, C, ad D would have to hold to realize these two coditios? (d) For the special case where d 1 =, ad ad F 2 are idetical, draw the ray-optics system correspodig to the matched projectio system. (e) Assumig Case (d), d 2, d 3 ad d 4 are ucostraied. What are the coditios o these 3 variables so that matchig FOV ad etrace pupil occurs simultaeously? 4

5 Problem oly Cosider a microscope with the geometry show below. eyepiece objective y. g. δ F 2 F 2 eye α' + ε itermediate image <F 2 (a) Use the ABCD matrix method to show that the effective focal legth of the two-les combiatio (the distace behid L 2 that collimated iput light comes to a focus)is approximately d 2 F 2 /g, where d 2 is the separatio betwee the leses. For positive values of g, what is your iterpretatio of this result? (b) Use the M system equatio i the otes for the two-les system to calculate the exact (o approximatios) agular magificatio of the microscope. That is, assume d 1 = +ǫ, the itermediate image is placed at a distace a little less tha F 2 from the eyepiece, ad d 3 = -d mi. 5

6 Problem oly A illumiatio poit source is located at the back of a short cylidrical glass slab (coupler) which has a radius of curvature R at its other ed as show. The coupler has a legth d ad a refractive idex. Source Coupler R * d (a) What is the legth, d 1, of the coupler which produces a collimated exitig beam (i air)? (b) What is the legth, d 2, of the coupler so that the poit source is imaged at a equal distace d 2 i air away from the the covex ed of the coupler? The coupler is ow butted axially agaist a log glass rod (light pipe)of the same material ad of the same diameter. The goal is to efficietly trasfer light from the source ito the light pipe, but it turs out that the cotiguous ed of the light pipe also has a covex surface of radius of curvature R as show. Source Coupler R R Light Pipe Plae of iterest * d L (c) Igorig the outer boudaries of the coupler ad the light pipe, what should be the legth, d 3,of the coupler so that the light is collimated withi the light pipe? (d) Agai igorig the outer boudaries of the coupler ad the light pipe, what should be the legth, d 4,of the coupler so that the poit source is is imaged at a equal distace L = 2d 4 i the light pipe away from the covex side of the iterface betwee the coupler ad the light pipe? 6