Section 2. Mirrors and Prism Systems

Transcription

1 Secion 2 Mirrors and Prism Sysems 2-1 Plane Mirrors Plane mirrors are used o: Produce a deviaion Fold he opical pah Change he image pariy Each ray from he objec poin obeys he law of reflecion a he mirror surface. A virual image of he objec poin is produced. Objec M 2-2 Image The rules of plane mirrors: The line connecing an objec poin and is image is perpendicular o he mirror and is biseced by he mirror. Any poin on he mirror surface is equidisan from a given objec poin and is image poin. The image pariy is changed on reflecion.

2 Pariy Change on Reflecion In addiion o bending or folding he ligh pah, reflecion from a plane mirror inroduces a pariy change in he image. Righ Handed Pariy (RH) Lef Handed Pariy (LH) Each ray from an objec obeys he law of reflecion a a plane mirror surface, and a virual image of he objec is produced. B A Observer 2-3 B' M A' Image Pariy and Orienaion Terminology: Image Roaion he image is roaed abou a line normal o he image (opical axis). An image roaion does no produce a pariy change. Image Pariy Changes Inver Pariy change abou horional line; verical flip. Rever Pariy change abou a verical line; horional flip. An inversion and a reversion is equivalen o a 180 roaion. Inversion: Reversion: 180 Roaion: 2-4 An image seen by an even number of reflecions mainains is pariy. An odd number of reflecions changes he pariy. Mirror Normal (LH) Pariy is deermined by looking back agains he propagaion direcion owards he objec or image in ha opical space; le he ligh from he objec or image come o you. (RH)

3 Pariy and Imaging The objec and he image formed by a lens are flipped op and boom flipped lef and righ 2-5 This consiues an inversion and a reversion. The image is roaed 180 abou he opical axis. There is no pariy change. Sysems of Plane Mirrors The rules of plane mirrors are used sequenially a each mirror in a sysem of plane mirrors. Two Parallel Mirrors Two parallel plane mirrors ac as a periscope and displace he line of sigh. There is no pariy change. All image rays are parallel o he corresponding objec rays. The image is displaced by wice he perpendicular separaion of he mirrors d. O'' O 2d d O' M 1 Proof: Use a single ray. Two idenical riangles are formed. 2d M 2 M M 2

4 Two Non-Parallel Plane Mirrors The dihedral line is he line of inersecion of wo nonparallel plane mirrors. A principal secion is a plane perpendicular o he dihedral line. In a plane perpendicular o he dihedral line (a principal secion), he projeced ray pah is deviaed by wice he angle beween he mirrors (he dihedral angle ). 2 This deviaion is independen of he inpu angle. Mirror 1 Mirror Dihedral Line < 90 : > 90 : = 90 : The inpu and oupu rays cross. The inpu and oupu rays diverge. (virual rays cross behind he mirrors) The inpu and oupu rays are ani-parallel. When he dihedral angle is 90, he inpu and oupu rays are ani-parallel in he principal secion. This combinaion of wo mirrors is called a roof mirror. The projecion of he ray pahs ino a plane conaining he dihedral line shows a simple reflecion a he dihedral line. Two Non-Parallel Plane Mirrors - Derivaion Negaive quaniies: Triangle 1: and M 1 M Triangle 2:

5 Two Non-Parallel Plane Mirrors Image Locaion Dihedral Line 2-9 Rules: 1. Every objec poin and is image lie in he same principal secion. 2. Every objec poin and is image are equidisan from he dihedral line. 3. Every image poin is displaced from is objec poin by roaion abou he dihedral line hrough wice he dihedral angle beween he mirrors in he same sense as he sequence of reflecions. 4. Every image ray is deviaed from is objec ray by wice he dihedral angle beween he mirrors in he same sense as he sequence of reflecions. 5. Pariy is unchanged A principal secion is defined o be a plane perpendicular o he dihedral line. Roof Mirror A roof mirror is wo plane mirrors wih a dihedral angle of 90, and he inpu and oupu rays are ani-parallel. This roof mirror can replace any fla mirror o inser an addiional reflecion or pariy change. I preserves pariy on reflecion. (LH) (RH 180 ) 2-10 (RH) (RH) Plane Mirror Roof Mirror The dihedral line is ofen in he plane of he drawing, and he presence of a roof mirror is indicaed by a V a he equivalen mirror or dihedral line. Roof Symbol

6 Roof Mirror The addiion of a roof mirror does no add any exra opical pah. An equivalen plane mirror is formed a he dihedral line. Equivalen Mirror The projecion of he ray pahs ino a plane conaining he dihedral line shows a simple reflecion a he dihedral line Dihedral Line V End View Side View Prism Sysems Prism sysems can be considered sysems of plane mirrors. An excellen reference for prisms is MIL_HDBK-141 Secion If he angles of incidence allow, he reflecion is due o TIR. Prisms fold he opical pah and change or correc he image pariy. Surfaces where TIR fails mus have a reflecive coaing. A unnel diagram unfolds he opical pah a each reflecion hrough he prism and shows he oal lengh of he pah hrough he prism. The prism is represened as a block of glass of he same hickness. A ray propagaes sraigh hrough he diagram. The unnel diagram aids in deermining Field of View (FOV), clear aperure, and vigneing. The addiion of a roof mirror o he prism does no change he unnel diagram Prisms are classified by he overall ray deviaion angle and he number of reflecions (# of R s).

7 90 Deviaion Prisms Righ angle prism (1 R) he acual deviaion depends on he inpu angle and prism orienaion. Image is invered or revered depending on prism orienaion. Amici or Roof prism (2 R) a righ angle prism wih a roof mirror. The image is roaed 180. No pariy change The unnel diagram is idenical o he righ angle prism. More 90 Deviaion Prisms Penaprism (2 R) wo surfaces a 45 produce a 90 deviaion independen of he inpu angle. I is he sandard opical merology ool for defining a righ angle. The wo reflecing surfaces mus be coaed. No pariy change Wollason Prism (2 R) wo surfaces a 135 produce a deviaion of 270 (or 90 ).

8 More 90 Deviaion Prisms Reflex Prism Reflex prism (3 R) a penaprism wih an added roof mirror. Used in single lens reflex (SLR) camera viewfinders o provide an erec image of he proper pariy. The roof surfaces mus also be coaed SLR Camera he flip mirror roaes up during he exposure. The film and viewing screen planes are opically coinciden. Viewing Screen Deecor Camera Lens Flip Mirror SLR Camera 2-16 Wikipedia LoneTreeImages

9 180 Deviaion Prisms Porro prism (2 R) a righ angle prism using he hypoenuse as he enrance face. I conrols he deviaion in only one dimension. or V Dihedral line perpendicular o he plane of he paper Dihedral line in he plane of he paper. Noe ha boh represenaions of he unnel diagram have he same lengh. Corner cube (3 R) hree surfaces a 90. The oupu ray of his reroreflecor is ruly ani-parallel o he inpu ray. The deviaion is conrolled in boh direcions, and ligh enering he corner cube reurns o is source These figures appear skewed due o he compound angles needed o represen a prism face and a roof edge when all hree prism faces have equal angles wih he opical axis. Reroreflecors Taillighs and Bicycle Reflecors: Three reflecions are made and he ligh reurns o he source bannerengineering.com

10 Applicaions of Corner Cubes Time-of-Fligh Measuremens Surveying Prism Lunar Laser Ranging Reroreflecor Arrays Apollo 11 and 14: 100 prisms Apollo 15: 300 prisms 2-19 Laser Geodynamics Saellie (Orbi a 5,900 km): The LAGEOS mission goals: - Provide an accurae measuremen of he saellie's posiion wih respec o Earh, - Deermine he plane's shape (geoid), and - Deermine econic plae movemens associaed wih coninenal drif. 45 Deviaion Prisms 45 Prism (2 R) half a penaprism. Schmid prism (4 R) a 3 R version wihou a roof also exiss. No coaed surfaces 2-20

11 Image Roaion Prisms Image Roaion Prisms as he prism is roaed by abou he opical axis, he image roaes by wice ha amoun (2). In all of hese prisms, he inpu and oupu rays are co-linear (he ligh appears o go sraigh hrough) and here are an odd number of reflecions (and herefore a pariy change). Symmery and he pariy change explains he image roaion. Each prism has an inversion direcion and flip axis associaed wih i. As he prism roaes relaive o he objec, his flip axis roaes, and he objec is invered abou his axis. By symmery, he objec mus roae wice as fas (he prism a 0 and 180 mus give he same oupu) Objec wih Flip Axis: Prism wih Inversion Direcion: Roaed Image: R R R R R R Prism Angle: Image Roaion Angle: Image Roaion Prisms Dove prism (1 R) because of he iled enrance and exi faces of he prism, i mus be used in collimaed ligh. Laeral chromaic aberraion is inroduced. Reversion or K-prism (3 R) he upper face mus be coaed Pechan prism (5 R) a small air gap provides a TIR surface inside he prism. This compac prism suppors a wide FOV. The wo exerior surfaces mus be coaed. The Pechan prism is a combinaion of a 45 prism (2R) and a 45 equivalen of a non-roof Schmid prism (3 R). Air Gap

12 Image Erecion Prisms Image erecion prisms are insered in an opical sysem o provide a fixed 180 image roaion. They are commonly used in elescopes and binoculars o provide an uprigh image orienaion. No pariy change Porro Prism Sysem Wikipedia Image Erecion Prisms Porro sysem (4 R) wo Porro prisms. The firs prism flips he image in one plane and he second prism flips he image in he oher plane. This prism accouns for he displacemen beween he objecive lenses and he eyepieces in binoculars. Invened in 1854 by Ignaio Porro he firs pracical implemenaion was by Zeiss in The unnel diagram shows wo Porro prisms. In he second represenaion, only he available enrance face is shown. or 2-24 Porro-Abbe sysem (4 R) a variaion of he Porro sysem where he sequence of reflecions is changed.

13 Image Erecion Prisms Pechan-roof prism (6 R) a roof is added o a Pechan prism. This prism is used in compac binoculars and provides a sraigh-hrough line of sigh. I is a combinaion of a 45 prism and a Schmid prism. Noe ha he roof surface does no need o be coaed. This prism sysem is also known as a Schmid-Pechan prism as i is he combinaion of a 45º prism (2 R) and a Schmid prism. I appears ha his prism was firs used in The unnel diagram is he same as ha of he Pechan prism This prism sysem is used in wha are markeed as roof-prism binoculars. Image Erecion Prisms Lehman prism or Sprenger-Lehman prism (4 R) he usable aperure is only a small porion of he enrance and exi faces Ligh Pah Roof Surface Theais 3½X ; J.D. Möller, Wedel, Germany, lae 1920s

14 Image Erecing Prisms Abbe-Köenig erecing prism (4 R) his prism sysem is a reversion or K-prism wih a roof surface. This prism sysem appears in he early 1900s, and image erecion is obained wihou a displacemen of he opical axis. Roof Surface 2-27 Hensold, Welar 1905 Some asymmery can be buil ino he Abbe- Köenig prism o provide an offse of he opical axis. This is useful for large diameer objecive lenses. Abbe-Köenig Prism Binoculars 2-28 Jagd-Dialy 7x44 S/N Hensold, Welar

15 Cube Beamsplier A beamsplier splis one inpu beam ino wo oupu beams. The cube beamsplier is a combinaion of wo righ angle prisms. A parially reflecing coaing is applied o one hypoenuse before he wo prisms are glued ogeher. There is no TIR a he inerface. The beamsplier usually provides a 50/50 spli of he beam ino he resuling wo beams. The densiy of he coaing can be varied o produce differen raios (such as 70/30). The coaing can be meallic or a dielecric (hin film) inerference coaing. Coaings can also be applied so ha he oupu beams are polaried. One beam would be verically polaried and he oher is horionally polaried. This device is called a polariing beamsplier (PBS) Prism Commens Prisms wih enrance and exi faces normal o he opical axis can be used in converging or diverging ligh. They will, however, inroduce he same aberraions as an equivalen hickness plane parallel plae. Spherical aberraion and longiudinal chromaic aberraion are inroduced ino an on-axis beam. TIR ofen fails when prisms are used wih converging or diverging beams. In laser or polaried ligh applicaions, TIR a he prism surfaces will change he polariaion sae of he ligh. In boh of hese siuaions, silvered or coaed prisms mus be used. This polariaion siuaion occurs frequenly wih corner cubes used wih laser-based disance measuring inerferomeers. 2-30

16 TIR Limi in Prism Sysems When using a prism wih a converging beam, he angle of incidence on he hypoenuse surface will vary, and rays below he criical angle can occur. TIR is hen los. In hese cases, he surface mus be coaed wih a reflecive coaing. Consider a righ angle prism in air used wih a beam ha has a convergence of ±: n L ' ' º U º 1 C n U U L 1 sin 41.8 n sin nsin 0 0 The lower ray is a an angle greaer han he criical angle, and i will TIR. As and ' increase, he angle of he upper ray wih respec o he surface normal of he hypoenuse will decrease. A some value of, he ray will lose TIR. Plane Parallel Plae A ray passing hrough a plane parallel plae is displaced bu no deviaed; he inpu and oupu rays are parallel. n D (in air) D sin 1 Small angle approximaions: n 1 D n 2 1 sin 2 2 n sin 2-32 An image formed hrough a plane parallel plae is longiudinally displaced, bu is magnificaion or sie is unchanged. n (in air) n 1 d n d d 3 for n 1.5

17 Ray Displacemen Derivaion x1 an x x an x an an 90 D sin sin 90 cos x D x cos cos an an 2 cos sin D sin cos sin n sin cos D sin 1 ncos 2 cos 1 sin n 1 x 2 D x 1 (in air) 2 1 sin D sin n sin Small angle approximaions: 1 2 cos 1 sin 2 n 1 n 1 D 1 n n ncos 2 2 n sin Image Displacemen Derivaion 1 n h A b d B 1 A: The original image locaion (no plae) B: The displaced image posiion Se he problem up so ha B is locaed a he second surface of he plae (i.e. ranslae he plae o pu he image or ray jus a is rear surface). Use small angle approximaions A: B: h an b h an Refracion: n h nh b b n Reduced Thickness Image Shif: d d n n 1 d n

18 Reduced Thickness The reduced hickness gives he air-equivalen hickness of he glass plae. A reduced diagram shows he amoun of air pah needed o fi he plae in he sysem, and no refracion is shown a he faces of he plae. Reduced diagrams can be placed direcly ono sysem layou drawings o deermine he required prism aperure sies for a given FOV. n Acual d n 1 d n n ReducedThickness n 2-35 Reduced The reduced hickness is useful o deermine wheher a cerain sie plae or prism will fi ino he available airspace in an opical sysem (beween elemens or beween he final elemen and he image plane). Since he plae makes some exra room for iself by pushing back he image plane, he required space is less han he acual plae hickness. Reduced Thickness vs. Opical Pah Lengh Reduced Thickness (/n) is a geomerical concep. I relaes o he amoun of air space ha is used when insering a plae of index n ino a sysem. I is he air-equivalen hickness of he glass plae. A reduced diagram shows he amoun of air pah needed o fi he plae in he sysem. The reduced drawing is an opical represenaion of he sysem masks he refracions a he surfaces and provides a simplified view of he sysem. This opical model is no a mechanical represenaion of he sysem. Opical Pah Lengh (nl) is a physical propery of he opical sysem ha is proporional o he propagaion ime, and i can be hough of as he air-equivalen propagaion disance. The OPL of he opical sysem increases grealy when a prism or glass plae is insered in he sysem. - The acual propagaion disance increases by d - The index of refracion slows down he propagaion velociy 2-36 Acual or Physical Drawing n Opical or Reduced Drawing d

19 Reduced Tunnel Diagram A reduced unnel diagram shorens he lengh of a unnel diagram by 1/n o show he airequivalen lengh of he prism. The diagram is foreshorened only along he direcion of propagaion (he opical axis). L 2-37 L/n When reduced diagrams are used, no refracion is shown a he surfaces. Example of he Use of he Reduced Tunnel Diagram Deermine he required sie of a righ angle prism ha is o be placed behind a lens. Tunnel a Reduced a/n Diagram a Diagram a The reduced unnel diagram can be overlaid scaled on he sysem drawing o deermine he required prism sie a for differen prism locaions. The refracions a he surfaces are no shown in hese opical represenaions. Opical Sysem Used wih a Deecor 2-38 a 1 /n a 2 /n Prism Near Image Plane Prism Near Lens a 2 > a 1 Remember ha hese are opical represenaions and he unfolded mechanical drawing would use he unnel diagram and would show he refracions a he surface.

20 2-39 Opical Drawing and he Mechanical Drawing Opical drawing wih he reduced unnel diagram: a 1 /n Mechanical drawing of he unfolded sysem wih he unnel diagram refracions are included: a 1 The full mechanical drawing of he sysem can be obained by folding he ligh pahs and showing he reflecions from he prism: a 1