Section 2 Mirrors and Prism Systems Mirrors and Prism Object Image the Mirror the Mirror Is Bisected the Mirror

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OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 2-2 2-1 M and its image its image and is perpendicular to Section 2 Mirrors and Prism Systems Mirrors and Prism Object Image the mirror the mirror is bisected the mirror. and by point and its image point. • the mirror surface point on is equidistant given object Any from a • reflection. on image parity The is changed The rules ofplane mirrors: •object line connecting an point The Plane Mirrors Plane mirrors Plane mirrors are to: used • a deviation Produce • the Fold optical path • Change the image parity Each ray reflectionthe law offrom the object point obeys at the mirror surface. virtual A image of the object point is produced. OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 2-4 2-3 M Rotation: (LH) ° Observer Inversion: Reversion: 180 (RH) Left Handed Parity (LH) Left Handed Parity th, reflection from a plane mirror mirror reflection th, introduces a plane from Mirror Normal equivalent a 180°to rotation. image image rotation does not produce A' A B' B Right Handed Parity Right Handed Parity (RH) Each ray from Each ray reflectionobject law of from an obeys the at a plane mirror surface, virtual a and image of the object is produced. An image seen by an even number of even number image seen by an An An parity. its reflections maintains odd number of reflections changes the parity. back Parity looking is determined by direction against the propagation towards the object image that or in optical space; let the light from the object or image you. come to a parity change in the image. In addition to bending or folding the light pa the light folding or bending addition to In Image Parity and Orientation and Parity Image Parity Change on Reflection on Change Parity Terminology: –Image Rotation image the a line normal is rotated to about An the image (optical axis). a parity change. Changes Parity Image –Invert line; vertical flip. horizontal about change Parity Revert – Parity change about a vertical line; horizontal flip. An inversion and a reversion is OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 2-6 2-5 1 M 2 M 2d Proof: Proof: single ray. Use a identical triangles are formed. Two tially at each mirror a system in plane of rpendicular separation of the mirrors d. mirrors the of rpendicular separation 1 M 2 M O' flipped top and bottom flipped left and right d O 2d This constitutes an inversion and a reversion. image The is rotated 180°the optical about axis. change. parity no is There O'' The object and the image the image formed The object and are by a lens Systems of Plane Mirrors Plane of Systems Parity and Imaging Parity and The rules of plane mirrors mirrors plane The rules of are sequen used parallel Two plane mirrors sight. act as a periscope and displace the line of change. parity no is There object rays. corresponding the parallel to are image rays All The image is displaced twice by the pe mirrors. mirrors. Two Parallel Mirrors OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 2-8 2-7 Line 1 2 Dihedral M M  Mirror 1 Mirror 2    180  ane containing the dihedral a containing the dihedral line shows ane two mirrors two mirrors is called a mirror. roof     180    21  22 nt of the input angle.  90 to the dihedral line (a    12 11   22 22    21 90 90 180  2180 22180 22 180 90 180 2      2  and        21 1    2           (virtual rays cross behind the mirrors) the behind cross (virtual rays     90 112     11 ).         11 11 Triangle 2: Triangle Triangle 1: Triangle < 90°: < 90°: The input and output rays cross. > 90°: The rays input and output diverge. = 90°: The input and output rays are anti-parallel. 90 Negative quantities:  Two Non-Parallel Plane Mirrors -Non-Parallel Plane Two Derivation Two Non-Parallel Plane Mirrors When the dihedral angle is 90°, the input and output rays are anti-parallel rays output and in the the input the dihedral angle is 90°, When of combination This principal section. into a pl the ray paths projection The of line. simple dihedral reflection at the    The dihedral line is the line of intersection non- line intersection two dihedral The is the line of of principal parallel A plane mirrors. section is a plane line. dihedral the to perpendicular plane perpendicular In a principal section), path is deviated by the projected ray twice the angle between the mirrors (the dihedral angle This deviation is independe This OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 2-9 2-10 Roof Symbol V (RH 180°) and the input and output output the input and and Roof Mirror (RH) place any flat mirror to insert additional an Mirror y extra optical path. An equivalent plane An optical path. extra y hedral hedral angle of 90°, Equivalent Dihedral Line preserves parity on reflection. on parity preserves (LH) End ViewView Side Plane Mirror (RH) The dihedral line is often in the plane of the drawing, the drawing, of line in the plane dihedral The is often and the presence of a roof mirror is indicated by a at the equivalent dihedral line. “V” mirror or The projection of the ray paths into a plane containing the dihedral a containing the dihedral line shows into a plane the ray paths projection The of line. simple dihedral reflection at the mirror mirror is formed at the dihedral line. The addition of a roof mirror The addition of a mirror roof does not add an A roof mirror is two plane mirrors di is two plane mirrors mirror with a roof A rays are anti-parallel. This roof mirror mirror rays are anti-parallel. re roof can This It change. parity or reflection Roof Mirror Roof Mirror OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 2-11 2-12 number of reflections reflections of number The tunnel diagram diagram is identical tunnel The the right angle prism. to plane mirrors. plane mirrors. excellent An reference for on is due to TIR. Prisms on TIR. Prisms the optical to fold is due y propagates straight through the diagram. diagram. the through straight propagates y viation angle and the and viation angle at each reflection through the prism and to the prism to the prism does not change the tunnel ation depends on the input angle and prism ation depends on the input angle and prism ity. Surfaces where TIR fails must have a reverted depending on prism reverted depending on prism orientation. Amici or Roof prism –prism R) (2 Roof Amici or image The is mirror. right with a roof angle prism a rotated No parity 180°. change. (# ofR’s). path and change or correct the image correct or change path and the image par represented prism The as a is prism. the through the path of the total length shows block of glass of the same thickness. A ra clear diagram View (FOV), aids in determining tunnel The aperture, Field of and mirror a roof addition of The vignetting. Prism systems can be considered systems of systems considered be can systems Prism 13.10 Section MIL_HDBK-141 is prisms Ifincidence allow, the angles of the reflecti reflective coating. optical the tunnel diagram path A unfolds diagram. overall ray de Prisms the are classified by orientation. Image is inverted or or inverted is Image orientation. Right angle prism –R) angle prism (1 Right actual the devi 90° Deviation Prisms Prism Systems Prism OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 2-14 2-13 Detector Flip Mirror Flip independent of the independent of Screen Viewing image image of the proper parity. The roof Camera Lens dded roof mirror. mirror. dded roof Used in single lens reflex 135°°270 a deviation of produce (or 90°). trology tool for defining a right angle. The The trology a right angle. defining tool for produce a 90°produce a deviation SLR Camera Camera SLR – the flip mirror exposure. the during rotates up viewing screen film The and planes are optically coincident. Wollaston Prism Prism –(2 R) Wollaston surfaces at two Reflex prism Reflex prism –(3 R) pentaprism a with an a erect an (SLR) camera provide viewfinders to surfaces must also be coated. be surfacesalso must More More 90° Deviation Prisms – Reflex Prism More 90°More Prisms Deviation input angle. It is the standard optical me standard the is It angle. input two reflecting surfaces must be coated. No parity change. Pentaprism Pentaprism (2 R) –surfaces at 45° two 2-15 OPTI-502Optical Designand Instrumentation I

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Porro prism (2 R) – a right angle prism using the hypotenuse as the entrance face. It controls the deviation in only one dimension. Dihedral line perpendicular to the plane of the paper

or Dihedral line in the

V plane of the paper. Note that both representations of the tunnel diagram have the same length.

Corner cube (3 R) – three surfaces at 90°. The output ray of this retroreflector is truly anti-parallel to the input ray. The deviation is controlled in both directions, and light entering the corner cube returns to its source.

These figures appear skewed due to the compound angles needed to represent a prism face and a roof edge when all three prism faces have equal angles with the optical axis. 2-17 OPTI-502Optical Designand Instrumentation I

Taillights and Bicycle Reflectors: bannerengineering.com

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Surveying Lunar Laser Ranging Retroreflector Arrays Prism Apollo 11 and 14: 100 prisms Apollo 15: 300 prisms

Laser Geodynamics Satellite (Orbit at 5,900 km): The LAGEOS mission goals: - Provide an accurate measurement of the satellite's position with respect to Earth, - Determine the planet's shape (geoid), and - Determine tectonic plate movements associated with continental drift. OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 2-20 2-19 90° R 225° 0° R 180° 135° 270° . No coated surfaces coated . No s, s, the object is inverted and about the optical axis, the image the optical the image about axis, R 90°  180° rotate twice as fast (the prism at 0° prism (the fast twice as rotate and a roof also exists RR 45° 90° ). ). In all these of prisms, the input and output rays are the object, this flip axis rotate  R 0° 0° axis associated with it. Schmidt Schmidt prism (4 R) – a version 3 R without rotates by twice that amount (2 amount twice that rotates by Image Rotation Prisms –Prisms Rotation Image by rotated is prism the as of number odd an are there and through) straight go to appears light (the co-linear reflections (and therefore a parity change). Image Rotation Prisms Rotation Image 45° Deviation Prisms 45° –R) Prism (2 pentaprism. half a Symmetry an image change explains the prism the parity rotation. Each has Symmetry and direction flip inversion and As the prism rotates relative to about this axis. By symmetry, the object must about this axis. symmetry, By 180° must output). give the same Object with Flip Axis: Flip with Object Prism with Inversion Direction: Rotated Image: Prism Angle: Angle: Image Rotation OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 2-22 2-21 Wikipedia System The Pechan prism The Pechan prism is a Porro Prism provide provide an upright rfaces must coated. be rfaces must valent of a non-roof Schmidt prism R). (3 Schmidt prism a non-roof valent of ce and exit faces of the prism, ce exit facesthe prism, ofand it must be a TIR surface inside the prism. This compact This compact surface TIR inside the prism. a scopes and binoculars to binoculars scopes and chromatic aberration is introduced. is aberration chromatic he upper face must coated. he upper face must be Air Gap used in collimated in collimated used Lateral light. Dove prism prism R) –(1 Dove the tilted because of entran Reversion or K-prism Reversion or K-prism –(3 R) t combination a 45°combination of a 45° and prism (2R) equi prism supports a wide FOV. The two exterior su prism supports a wide FOV. Pechan prism Pechan prism (5 R) – small a air provides gap image orientation. No parity change. change. parity No orientation. image Image erection prisms are inserted in an optical system to provide a fixed 180°fixed a provide to optical system an in inserted are prisms erection Image image rotation. They are in tele commonly used Image Erection Prisms Erection Image Image Rotation Prisms Rotation Image OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 2-24 2-23 the second representation, the second representation, the only available or are marketed as “roof-prism” binoculars. “roof-prism” as marketed are e eyepieces in binoculars. e second prism flips the image prism e second ism ism accounts for the displacement This prism This prism system what in is used Pechan-roof prism – prism R) (6 Pechan-roof to a Pechanis added roof a prism. a provides in compact binoculars and is used prism This 45° a It is a combination of sight. line of straight-through prism and a Schmidt prism. Note that the roof surface does not need This to be coated. prism system is as also known a Schmidt-Pechan prism prism 45º a as it is the combination of a Schmidt prism. and It appears (2 R) this prism that was 1964. in used first the Pechandiagram is the same as that tunnel The of prism. Image Erection Prisms Erection Image Image Erection Prisms Erection Image Porro-Abbe system –(4 R) a variation ofthe Porro system reflectionswhere the sequence of is changed. entrance face is shown. shown. entrance face is The tunnel diagram The tunnel diagram shows two Porro prisms. In Porro system (4 R) –Porro two The first prisms. prism flips th plane and the image in one This pr the other in plane. th between the objective and lenses Ignatio by in 1854 Invented Porro – first the practical implementation was Zeiss by in 1894. OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 2-26 2-25 Roof Surface Path Light Hensoldt, Wetzlar 1905 Hensoldt, Wetzlar system system is a reversion K-prism or a with Theatis 3½X Theatis 1920s late Germany, Wedel, ; J.D. Möller, Roof Surface s in the early 1900s, and image erection is built into the Abbe- built into the Some asymmetrybe Some can Köenig prism of the to provide an offset fordiameter This is useful large optical axis. objective lenses. Abbe-Köenig erecting prism erecting Abbe-Köenig prism R) (4 – this prism surface. roof This prism system appear obtained without a displacement of the optical axis. Lehman prism or Sprenger-Lehman –R) prism Sprenger-Lehman (4 or prism Lehman small of a portion usable aperture the is only exit faces. entrance and the Image Erecting Prisms Erecting Image Image Erection Prisms Erection Image OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 2-27 2-28 Hensoldt, Wetzlar Jagd-Dialyt 7x44 S/N 13486 13486 S/N 7x44 Jagd-Dialyt the beam beam into the resulting beams. the two right angle prisms. right angle prisms. A partially reflecting e two prisms together. There are e two prisms is glued put beams are polarized. One beam would be would beam One are polarized. beams put ontally polarized. This device is called a to produce different as 70/30). different ratios (such to produce The beamsplittersplit usually of provides a 50/50 varied the coating can be of density The A beamsplitter splits one input beam into two output beams. beamsplitterThe cube two is a combination of th before hypotenuse coating is applied to one TIR at the interface. no be metallic The coating can or a dielectric (thin film) interference coating. applied that the out can also be so Coatings vertically vertically is horiz other the polarized and polarizing beamsplitter (PBS). Abbe-Köenig Prism Prism Abbe-Köenig Binoculars Cube Beamsplitter 2-29 OPTI-502Optical Designand Instrumentation I

Prisms with entrance and exit faces normal to the optical axis can be used in converging or diverging light. They will, however, introduce the same aberrations as an equivalent thickness plane parallel plate. Spherical aberration and longitudinal chromatic aberration are introduced into an on-axis beam.

TIR often fails when prisms are used with converging or diverging beams. In laser or polarized light applications, TIR at the prism surfaces will change the polarization state of the light. In both of these situations, silvered or coated prisms must be used.

This polarization situation occurs frequently with corner cubes used with laser-based distance measuring interferometers.

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When using a prism with a converging beam, the angle of incidence on the hypotenuse surface will vary, and rays below the critical angle can occur. TIR is then lost. In these cases, the surface must be coated with a reflective coating.

Consider a right angle prism in air used with a beam that has a convergence of ±:

1 1 C sin 41.8n 1.5 n  '  U º  45 ' U n 45 45  U L º

 L 45 45

The lower ray is at an angle greater than the critical sin  n sin angle, and it will TIR.

As  and ' increase, the angle of the upper ray with 00 respect to the surface normal of the hypotenuse will decrease. At some value of , the ray will lose TIR. 2-31 OPTI-502Optical Designand Instrumentation I

A ray passing through a plane parallel plate is displaced but not deviated; the input and output rays are parallel.

n 1sin 2  Dtsin  1  22  D n  sin 

 Small angle approximations:

t n 1 Dt   (in air) n

An image formed through a plane parallel plate is longitudinally displaced, but its magnification or size is unchanged.

n 1 n (in air) dt  n

z t dn for  1.5 3 t d

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x1 tan   1 n 1 t xx 12  tan    x2 D t   xt2 tan tan  90   D x  sin  sin 90 cos 1  t x2 (in air) Dx2 cos  t cos tan  tan  cos sin  Dtsin   cos  1sin 2  sinn sin  Dtsin  1   22 n  sin  cos  Dtsin  1  ncos  Small angle approximations: cos 1 sin2

1 2 11n  cos  1 sin    2 Dt1   t  n nn  nncos 22 sin  2-33 OPTI-502Optical Designand Instrumentation I

1 n 1 A: The original image location (no plate)

 B: The displaced image position  h  Set the problem up so that B is located at the  second surface of the plate (i.e. translate the A B plate to put the image or ray just at its rear b surface). t d Use small angle approximations.

h A: tan  b Refraction:   n h  hnh t B: tan  b  Reduced Thickness t bt n

t Image Shift: dt  dt n n 1 dt  n

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The reduced thickness  gives the air-equivalent thickness of the glass plate. A reduced diagram shows the amount of air path needed to fit the plate in the system, and no refraction is shown at the faces of the plate. Reduced diagrams can be placed directly onto system layout drawings to determine the required prism aperture sizes for a given FOV.

n Actual nt 1 tdt  t   z nn t d t ReducedThickness  n

z  Reduced

The reduced thickness is useful to determine whether a certain size plate or prism will fit into the available airspace in an optical system (between elements or between the final element and the image plane). Since the plate makes some extra room for itself by pushing back the image plane, the required space is less than the actual plate thickness. 2-35 OPTI-502Optical Designand Instrumentation I

Reduced Thickness (t/n) is a geometrical concept. It relates to the amount of air space that is used when inserting a plate of index n into a system. It is the “air-equivalent thickness” of the glass plate. A reduced diagram shows the amount of air path needed to fit the plate in the system. The reduced drawing is an optical representation of the system masks the refractions at the surfaces and provides a simplified view of the system. This optical model is not a mechanical representation of the system.

Optical Path Length (nL) is a physical property of the optical system that is proportional to the propagation time, and it can be thought of as the “air-equivalent propagation distance.” The OPL of the optical system increases greatly when a prism or glass plate is inserted in the system. - The actual propagation distance increases by d - The index of refraction slows down the propagation velocity

Actual or Physical Drawing Optical or Reduced Drawing n

z z

t d 

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A reduced tunnel diagram shortens the length of a tunnel diagram by 1/n to show the air- equivalent length of the prism. The diagram is foreshortened only along the direction of propagation (the optical axis).

L/n

When reduced diagrams are used, no refraction is shown at the surfaces. 2-37 OPTI-502Optical Designand Instrumentation I

Determine the required size of a right angle prism that is to be placed behind a lens. Tunnel a Reduced a/n Diagram a Diagram a

The reduced tunnel diagram can be overlaid scaled on the system drawing to determine the required prism size a for different prism locations. The refractions at the surfaces are not shown in these optical representations.

Optical System Used z with a Detector

a2/n a1/n

z z

Prism Near Image Plane Prism Near Lens a2 > a1 Remember that these are optical representations and the unfolded mechanical drawing would use the tunnel diagram and would show the refractions at the surface.

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Optical drawing with the reduced tunnel diagram:

a1/n

Mechanical drawing of the unfolded system with the tunnel diagram – refractions are included:

The full mechanical drawing of the system can be obtained by folding the light paths and showing the reflections from the prism: