Section 2 Mirror and Prism Systems OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E

OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-1 Section 2 Mirror and Prism Systems OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-2 M reflection at the mirror virtual surface. A equidistant from a given object equidistant from a given Object Image the mirror and is bisected by the mirror. is bisected by the mirror and point and its point and image point. The rules of plane mirrors: • line The connecting an object point and its image is perpendicular to •the mirror point on surface is Any •reflection. on image parity is changed The Plane Mirrors Plane mirrors are used to: • a deviation Produce • the optical Fold path •the image parity Change Each ray from the object point obeys law of image of the object point is produced. OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-3 M Observer Left Handed Parity (LH) Parity Left Handed ection at a plane mirror surface, and a virtual path, reflection from a plane mirror introduces A' A B' B Right Handed Parity (RH) Each ray from an object obeys the law of refl image of the object is produced. a parity change in the image. a parity change In addition to bending or folding the light In addition to bending Parity Change on Reflection on Parity Change OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-4 Rotation: (LH) ° Inversion: Reversion: 180 (RH) Mirror Normal equivalent to a 180° rotation. An image seen by an even number of An reflections maintains its parity. odd number of reflections changes the parity. back looking Parity is determined by direction against the propagation towards the object or image in that optical space; let the light from the object or image come to you. Image Parity and Orientation Terminology: Image Rotation –a line normal to image is rotated about the not produce image rotation does An the image (optical axis). a parity change. Image Parity Changes –Invert line; vertical flip. horizontal about change Parity Revert –about a vertical Parity change line; horizontal flip. An inversion and a reversion is OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-5 flipped top and bottom flipped top and flipped left right and This constitutes an inversion and a reversion. image is rotated 180°The about the optical axis. There is no parity change. The object and the image formed by a lens are object and The Parity and Imaging Parity and OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-6 1 M 2 M 2d Proof: a single ray. Use identical triangles are formed. Two tially at each mirror in a system of plane scope and displace the line of sight. scope and 1 M 2 M O' d O 2d O'' Systems of Plane Mirrors The rules of plane mirrors are used sequen peri a as mirrors act parallel plane Two mirrors. Two Parallel Mirrors There is no parity change. object rays. All image rays are parallel to the corresponding separation of the mirrors twice the perpendicular d. image is displaced by The OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-7 Line Dihedral  Mirror 1 Mirror 2  and output rays are anti-parallel and in the ane containing the dihedral line shows a ane containing the dihedral line shows of two mirrors of two is called a roof mirror. ed ray path is deviated by nt of the input angle. of the input angle. nt

 2  (virtual the mirrors) rays cross behind

 ).

 < 90°: The input and output rays cross. > 90°: The input and output rays diverge. = 90°: The input and output rays are anti-parallel. Two Non-Parallel Plane Mirrors principal section. This combination principal section. projection of the ray paths into a pl The simple reflection at the dihedral line. When the dihedral angle is 90°, the input the dihedral angle is 90°, When The dihedral line is the line of intersection of two non- line of intersection two dihedral line is the The parallel plane mirrors. A principal section is a plane perpendicular to the dihedral line. to the dihedral line (a In a plane perpendicular principal section), the project twice the angle between the mirrors between (the dihedral twice the angle angle This deviation is independe    OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-8 1  2 M M 1   1   2    1 2   1  2   2  

   180 

 

  180    21  22

 90    12 11   22 22  

21  90 90 180 2180 22 2 180 2 180 90 180 2     2 and    21 1         

           

   90 112  

  11       11 11 Triangle 2: Triangle Triangle 1: Triangle 90 Negative quantities:   Two Non-Parallel Plane Mirrors - Derivation OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-9 Roof Symbol (RH 180°) Roof Mirror (RH) dral angle of 90°, and the input and output the input and and dral angle of 90°, replace any flat mirror to insert an additional (LH) Plane Mirror (RH) The dihedral line is often in the plane of drawing, The of a roof mirror the presence and is indicated by a at the equivalent mirror “V” or dihedral line. rays are anti-parallel. This roof mirror can reflection It or parity preserves parity on reflection. change. A roof mirror is two plane mirrors with a dihe Roof Mirror OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-10 V Mirror ny extra optical path. An equivalent plane ane containing the dihedral line shows a ane containing the dihedral line shows Equivalent Dihedral Line End View Side View simple reflection at the dihedral line. The projection of the ray paths into a pl The mirror is formed at the dihedral line. The addition of a roof mirror does not add a addition of a roof mirror not add The does Roof Mirror OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-11 plane mirrors. An excellent reference for plane mirrors. An e prism. The prism is represented as a prism is represented prism. The e ation angle and the number of reflections on is due to TIR. Prisms fold the optical on is due eld of View (FOV), clear aperture, and (FOV), eld of View at each reflection through the prism and to the prism does not change the tunnel not change to the prism does ity. Surfaces where TIR fails a ity. Surfaces where must have prisms is MIL_HDBK-141 Section 13.10 prisms is MIL_HDBK-141 If the angles of incidence allow, reflecti or correct the image par change path and reflective coating. the optical path tunnel diagram unfolds A the total th length of the path through shows the diagram. straight through ray propagates of glass the same thickness. A block tunnel diagram aids in determining Fi The vignetting. The addition of a roof mirror diagram. Prisms the overall ray devi are classified by (# of R’s). Prism systems can be considered of Prism Systems OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-12 The tunnel diagram is identical to the right angle prism. prism with a roof mirror. The image is The prism with a roof mirror. ation depends on the input angle and prism ation depends verted depending on prism orientation. on verted depending Amici or Roof prism (2 R) –Amici or Roof a right angle parity change. No rotated 180°. Right angle prism (1 R) – actual devi the orientation. Image is inverted or re Image is inverted orientation. 90° Deviation Prisms OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-13 trology tool for defining a right angle. The trology tool for defining a right angle. produce a 90°produce of the deviation independent More 90° Deviation Prisms input angle. It is the standard optical me input angle. Pentaprism (2 R) –surfaces at 45° two parity change. No coated. reflecting surfaces must be two OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-14 Detector Flip Mirror Screen Viewing d roof mirror. in single lens reflex Used d image of the proper parity. The roof Camera Lens SLR Camera – the flip mirror the exposure. during rotates up The film and viewing screen planes are optically coincident. (SLR) camera viewfinders to provide an erect surfaces must also be coated. Reflex prism (3 R) – a pentaprismadde with an More 90° Deviation Prisms – Prism Reflex OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp

Wikipedia 2-15

LoneTreeImages SLR Camera OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-16 Dihedral line perpendicular to the plane of the paper Dihedral line in the Note plane of the paper. representations that both of the tunnel diagram have the same length. have equal angles with the optical axis. e output ray of this retroreflector is truly mpound angles needed to represent a prism on is controlled in both directions, and light on is controlled in both directions, and

V or Corner cube (3 R) –cube Corner Th surfaces at 90°. three deviati The anti-parallel to the input ray. returns to its source. entering the corner cube due to These figures the co appear skewed all three prism faces when a roof edge face and Porro prism (2 R) – a right as the entrance face. angle prism using the hypotenuse dimension. one It controls the deviation in only 180° Deviation Prisms OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-17

bannerengineering.com Three reflections are made and source. the light returns to Taillights and Bicycle Reflectors: Bicycle and Taillights Retroreflectors OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-18 Lunar Laser RetroreflectorRanging Arrays Apollo 15: 300 prisms and 14: 100 prisms Apollo 11 Laser Geodynamics Satellite km): (Orbit at 5,900 mission The LAGEOS goals: - an accurate measurement of the Provide satellite's position with respect to Earth, -(geoid), and the planet's shape Determine - tectonic plate movements associated Determine with continental drift. Surveying Prism Applications of Corner Cubes – Time-of-Flight Measurements OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-19 A 3 R version without a roof also can be made. A Schmidt prism (4 R) –coated surfaces. No 45° Deviation Prisms 45°– Prism (2 R) pentaprism. half a OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-20 R 90° 225° 0° R 180° 135° 270° about the optical axis, image about R 90°  180° flip axis rotates, and the object is inverted flip axis rotates, and rotate twice as fast (the prism at 0° rotate twice as and RR 45° 90° ght through) and there are an odd number of ). In all of these prisms, the input and output rays are ). In all of these prisms, the input and  R 0° 0° co-linear (the light appears to go strai to go co-linear (the light appears reflections (and therefore a parity change). rotates by twice that amount (2 Image Rotation Prisms – as the prism is rotated by Image Rotation Prisms Symmetry and the parity change explains the image rotation. Each prism has an explains the image rotation. Each the parity change Symmetry and flip inversion direction and axis associated with it. As the prism rotates relative to the object, this about this axis. By symmetry, the object must this symmetry, about axis. By 180° must give the same output). Object with Flip Axis: Object with Flip Prism with Inversion Direction: Rotated Image: Prism Angle: Angle: Image Rotation OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-21 TIR surface inside the prism. This compact TIR surface inside the prism. rfaces must be coated. The PechanThe rfaces must be coated. prism is a valent of a non-roof Schmidt prism (3 R). ce and exit faces of the prism, it ce and must be Air Gap Dove prism (1 R) –Dove of the tilted entran because in collimated light. Lateral chromatic aberration is introduced. used Pechan prism (5 R) – a small air gap provides a prism supports a wide FOV. The two exterior su prism supports a wide FOV. combination of a 45° prism (2R) and a 45° equi Image Rotation Prisms OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-22 Wikipedia System Porro Prism scopes and binoculars to provide an upright binoculars to provide an scopes and optical system to provide a fixed 180° image rotation. They are commonly used in tele parity change. No image orientation. Image erection prisms are inserted in an Image Erection Prisms OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-23 the second representation, only the available the second or Image Erection Prisms Porro-Abbe system (4 R) –Porro-Abbe a variation of the Porro system of reflections is changed. the sequence where entrance face is shown. The tunnel diagram shows two Porro prisms. In PorroR) – system (4 Porro two first prism flips The prisms. prism flips the second the image plane and the image in one This prism for the displacement accounts in the other plane. in binoculars. the eyepieces the objective lenses and between Invented in 1854 by Ignatio Porro – the first practical implementation by Zeiss in 1894. was OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-24 are marketed as “roof-prism” binoculars. is added to a Pechan is added prism. It is a combination of a 45° combination of It is a This prism system is used in what This prism system is used Pechan-roof prism (6 R) – a roof This prism is used in compact binoculars and provides a straight-through line of sight. prism and a Schmidt prism. Note that the roof surface does as a This prism system is also known to be coated. not need Schmidt-Pechan prism as it is the combination of a 45º prism that this prism was It appears Schmidt prism. a and (2 R) 1964. in first used tunnel diagram is the same as that of PechanThe prism. Image Erection Prisms OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-25 the beam into the resulting two beams. the beam into resulting two beams. e two prisms are glued together. There is together. There e two prisms are glued put beams are polarized. One beam would be put beams are polarized. One beam would ontally polarized. This device is called a to produce different ratios (such as 70/30). to produce ectric (thin film) interference coating. no TIR at the interface. The beamsplitter usually provides a 50/50 split of The density of the coating can be varied The coating can be metallic or a diel Coatings can also be applied so that the out vertically the other is horiz polarized and polarizing beamsplitter (PBS). A beamsplitter splits one input beam into two output beams. The cube beamsplitter is a combination of two right angle prisms. A partially reflecting before th coating is applied to one hypotenuse Cube Beamsplitter OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-26 the optical axis can be used in converging surfaces will change the polarization state of surfaces will change with corner cubes used with laser-based onverging or diverging beams. In laser or or diverging onverging oduce the same aberrations as an equivalent erration and longitudinal chromatic aberration erration and ered or coated prisms must be used. used. prisms must be ered or coated or diverging light. They will, however, intr will, however, light. They or diverging parallel plate. Spherical ab thickness plane are introduced into an on-axis beam. prisms TIR often fails are used with c when polarized light applications, TIR at the prism the light. In both of these situations, silv frequently This polarization situation occurs distance measuring interferometers. Prisms and exit faces normal to with entrance Prism Comments OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-27 n :  , the ray will lose TIR. , the ray will lose    n 1    45 1   45 45 45 45 sin 41.8 1.5     U U L C     ' increase, the angle of the upper ray with ' increase, the angle of upper  e can occur. TIR is then lost. In these º and  , the angle of incidence on the hypotenuse of incidence on , the angle  The lower ray is at an angle greater than the critical angle, and it will TIR. As respect to the surface normal of the hypotenuse will respect to the surface normal of hypotenuse At some value of decrease. with a beam that has a convergence of ± with a beam that has convergence U  º '   '  L   n   n  00    sin sin   surface will vary, and rays below the critical angl cases, the surface must be coated with a reflective coating. Consider a right angle prism in air used When using a prism with a converging beam When using a prism with a converging TIR Limit in Prism Systems Limit in TIR OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-28

  2 sin    22 1sin n 1  n for for 1.5 n t 3    1        n dn dt

 n   

sin 1     Dt Small angle approximations: Small angle Dt z displaced but not deviated; the input and ate is longitudinally displaced, but its but ate is longitudinally displaced, d D (in air) (in air)  t n n t  An image formed through a plane parallel pl magnification or size is unchanged. output rays are parallel. A ray passing through a plane parallel plate is A Plane Parallel Plate OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-29    D 2 1 n sin (in air)   22 1sin  n    1 nn 11 2     n x x

  1      

sin 1          t Dt Dt t Small angle approximations: Small angle 1      2 2 2 D 2 x 1  n  22      n         

  

     cos cos   cos n cos sin cos sin nn coscos 1 sin 1 sin sin sin      t     12 1 xx t cos cos tan tan sin x     sin 1 2 tan tan 90        

          2 xt tan tan sin sin 90 cos Dx t Dt Dt Ray Displacement Derivation OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-30 is located at the Reduced Thickness B t  n b 1  n  n          n dt dt dt  : The original image location (no plate) The : : The displaced image position  bt n hnh t second surface of the plate (i.e. translate the second plate to put the image or ray just at its rear surface). A B Set the problem up so that Use small angle approximations.   1 B Refraction: Image Shift:   d n t A  

 t n

  1  nn nt        ReducedThickness tdt t red prism aperture sizes for a given FOV. red prism aperture sizes for a given FOV. whether a certain size plate or prism will fit e makes some extra room for itself by pushing to fit the plate in the system, and no to fit the plate in system, and em (between elements or between the final Reduced diagrams can be placed directly onto diagrams can Reduced z z is less than the actual plate thickness. is less than d Actual Reduced gives the air-equivalent thickness of the glass plate. A reduced gives the air-equivalent thickness of glass plate. A t

 n  The reduced thickness is useful to determine into the available airspace in an optical syst element and the image plane). Since plat back the image plane, required space diagram shows the amount of air path needed diagram shows refraction is shown at the faces of plate. refraction is shown to determine the requi system layout drawings The reduced thickness Reduced Thickness OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp z 2-32  Optical or Drawing Reduced tical system that is proportional es greatly when a prism or glass plate is es greatly when e amount of air path needed to fit e amount of air path needed the plate lified of the system. This optical view a system. It is the “air-equivalent thickness” tical representation of the system masks z d ) is a physical property of the op ) is a geometrical It concept. relates to the amount of air space that nL t n t/n Actual or Physical Drawing Actual or Physical - actual propagation distance increases by d The -the propagation velocity index of refraction The slows down to the propagation time, and it can be thought of as the “air-equivalent propagation of the optical system increas OPL distance.” The system. inserted in the Reduced Thickness ( is used when inserting a plate of index n into n plate of index inserting a when is used th diagram shows of the glass plate. A reduced is an op drawing reduced in the system. The refractions at the surfaces and provides a simp model is not a mechanical representation of the system. ( Optical Path Length Reduced Thickness vs. Optical Path Length Thickness Reduced OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-33 to show the air- to show n m is foreshortened only along the direction of L L/n When reduced diagrams are used, no refraction at is the surfaces. shown equivalent length of the prism. The diagra propagation (the optical axis). A reduced tunnel diagram shortens the length of a tunnel diagram by 1/ tunnel diagram shortens the length of a by reduced A Reduced Tunnel Diagram OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1 2-34 z > a 2 a/n a Prism Near Lens a Reduced Diagram /n 2 a ed on the system drawing to determine the a Optical System Used with a Detector ons. The refractions at the surfaces are not ons. The prism that is to be placed behind a lens. a behind placed prism that is to be tions and the unfolded mechanical drawing the unfolded tions and z z show the refractions at surface. show a /n 1 a Tunnel Diagram Prism Near Image Plane required prism size a for different prism locati prism size a required shown in these optical representations. Remember that these are optical representa use the tunnel diagram and would would Determine the required size of a right angle The reduced tunnel diagram can be overlaid scal Example of the Use of the Reduced Tunnel Diagram Tunnel of the Reduced Example of the Use OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2-35 z z the tunnel diagram – refractions are included: be obtained by folding the light paths and z 1 1 /n a a 1 a showing the reflections from the prism: showing Optical drawing with the reduced tunnel diagram: with the reduced Optical drawing The full mechanical drawing of the system can Mechanical drawing of the unfolded system with of the unfolded Mechanical drawing Optical Drawing and the Mechanical Drawing and Optical Drawing