OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-1 Section 5 Gaussian Imagery OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-2 system analysis that treats imaging as a ogy to determine ogy ray paths through optical pal optical configuration Planes for a given . This has also allowed the Focal Length of a tion for a general system can be calculated tion for a general system can race can be used to answer these questions. not yet complete. The details of determining of optical elements is also not clear. optical elements is also not of the system’s overall refractive properties. systems. Using this the image loca systems. Using method, relative to the Principal Planes of system general system to be defined in terms of the physical locations of Princi However, still derived, so the analysis in to be need the system focal length of a combination In addition, the image size is as of yet unknown. As will be demonstrated later, paraxial rayt Gaussian optics provides an alternative method of mapping from object space into image space. Paraxial optics provides a simplified methodol Imaging OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-3 tion is the ratio of the image point height em to its image point (perfect conjugate m object space into image space. It is a m object space into image space.  ndently radiating point point sources. Each lied to rotationally symmetric systems, and e superposition of all of the point images. - usually the y-z plane h h ts are called conjugate elements. called conjugate ts are nce locations will be applied as usual.  optics holds: any three-dimensional object optics holds: any m lines and planes to planes. lines and Meridional Plane - contains the axis of symmetry Sign Conventions and definitions of refere The transverse magnification or lateral magnifica object point height: from the axis to conjugate The fundamental principle of geometrical scene is represented by a collection of indepe source is independently imaged through the syst three-dimensional image is th imagery). The The corresponding object and image elemen special case of a collinear transformation app it maps points to points, lines Assumptions: Axial Symmetry Gaussian optics treats fro Gaussian imaging as a mapping Gaussian Optics OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-4   z z map to lines and are conjugate elements. are conjugate to lines and map system), or are also parallel to the axis to conjugate lines in the other space that to conjugate ace are mapped to planes perpendicular the z z In a Gaussian or collinear lines mapping, must Afocal System: Focal System: axis in the other space. - parallel Lines space map to the axis in one either intersect point (focal the axis at a common (afocal system). Theorems (all Theorems derive from rotational symmetry of the system): -sp to the axis in one perpendicular Planes Gaussian Optics Theorems OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-5  z conjugate planes perpendicular to the axis. tant, the images of grid lines would z If the magnification in a plane were not cons If the magnification in a plane were become curved or distorted. -in transverse magnification is constant The Gaussian Optics Theorems - Continued OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-6  h h  m  z  z     F   h0 m0 ∞ hH  R f ribe the focal mapping. They are defined They ribe the focal mapping.    z P   hH m1 ) are defined as the directed distances from f  P  and hH m1 FR z f F f   F  hH h0 m FF'PP' focal point/plane Front Rear focal point/plane principal plane Front Rear principal plane m = m = 0 m = 1 m = 1 The front and rear focal lengths ( front and The respective focal points. to the rear principal planes front and Cardinal Points and Planes Cardinal Points and Consider conjugate line elements in object and image space: conjugate line elements in object and Consider The cardinal points and planes completely desc cardinal points and The by specific magnifications: OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-7 . . z focal planes principal planes θ  N location is related to its conjugate image location is related to its conjugate fication associated with those planes. N θ For a focal imaging system, an object plane plane location through the transverse magni plane location through image distances from the measure object and equations Newtonian image distances from the equations measure object and Gaussian Unprimed variables are in object space. Unprimed Primed variables are in image space. APAP' Front antiprincipal plane/pointN antiprincipal Rear plane/pointN'The two nodal points of a system are conjugate points. Front nodal point Rear nodal point m = -1 m = -1 Angular Mag = 1 Angular Mag = 1 Cardinal Points and Planes Cardinal Points and OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-8 z  h  F  R f will emerge from the rear principal plane at from the rear principal plane will emerge  set of Principal Planes and a set of Focal set of Principal Planes and P of effective object space refraction between of effective ect space passes through the rear focal point. passes through ect space emerges parallel to the optical axis. P fines an object or image point. F f F h The Principal Planes serve as the planes and image space. Points (with the respective focal lengths). Points (with the -the front principal plane ray incident on A the same height. -ray parallel to the optical axis in obj A - ray A through the front focal point - intersection The of two rays de The optical system can be represented as a Locating an Image with the Cardinal Points Locating an OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-9 z z  h   F F   R R f f   P P P P F F f f F F h h Positive System – object to the left Real of the front focal point F Locating an Image with the Cardinal Points –Locating an 1 Example OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-10 z z   F F   R R f f focal point F and the front principal plane focal point F and rtual crossing. An enlarged, erect virtual enlarged, An rtual crossing.   P P P P F F f f h h F F  h Positive System – object between the front Real image is produced. The image is in space. The image is produced. The two image space rays diverge and have a vi The two image space rays diverge Locating an Image with the Cardinal Points –Locating an 2 Example OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-11 z z F F f f have a virtual crossing. A minified, erect A have a virtual crossing. the system object space. Similarly, the Rear Similarly, the system object space.   both in the system image space. The same The both in the system image space. P P P P    R R f f h   FF FF h h Negative SystemNegative – object – Real Front Focal The the locations of Focal Points. Note are both in Front Principal Plane P Point F and Focal Point F' and Rear Principal Plane P' are image formation rules apply. virtual image is produced. The image is in space. virtual image is produced. Once again, the two image space rays diverge and again, the two image space rays diverge Once Locating an Image with the Cardinal Points –Locating an 3 Example OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp F F  5-12 . 0 0 F   z   F  Focal Points z  h z Image DistanceImage = z Object Distance = z  h  n height distance object distance Magnification: - of the Independent object - to the image Proportional - proportional to the Inversely  F  z   z  F   R  F z   R  f m ff tan  ussian mapping when the axial locations of  P  hf h    hs continue to be measured relative to the  RF  f hh FF FR  m zz z F f   1  F F  FP f F 1 z  

    n  tan h hz h FR FF  fm f zz FR FF  ff zz FF hh zf    m Newtonian Equations The Newtonian equations characterize this Ga the conjugate object and image planes are measured relative image planes are measured to the respective object and the conjugate By definition, the front and rear focal lengt definition, the front and By principal planes. Use similar triangles. OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-13  h h . 2  E f  2 m  f  ff       Focal Points FF E FF FF FR nn zz z zz zz     h  n  F  E z mf      F  F n z  R f ussian mapping when the axial locations of  P F f m E f m     F f FE FFR nm FFE zf zzmf zzmf FP F z n h 1   nf nn In air: nf

n  

n      RE FE f f Newtonian Equations Applied to a System The Newtonian equations characterize this Ga the conjugate object and image planes are measured relative image planes are measured to the respective object and the conjugate OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp  5-14 0 0   z/z 

  h  z Image Distance = z Image Object Distance = z Principal Planes   h n height the image distance to object distance Magnification: - of the Independent object - to the ratio of Proportional  z f    z R 1  m R  F     zf  R f  tan      hf h P of the conjugate object and image planes. object and of the conjugate  RR  f hh m  m  R F z f e focal mapping when the respective when e focal mapping    1 P   F zf z  f 11    m F F z F f  

    11 Ratio: n h tan          FR FR hzf h FF 11 zzm m zz ff ff     hh  FR zf f zm z m fmm f Add: The Gaussian equations describe th are the references for measuring locations the same similar Use triangles. Gaussian Equations OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp  h h 5-15  m z  1 E E 1 h   n   Principal Planes        FR zz 111 ff zzf zzf nn z  m F   R f  R  F f z / / z  zn zn       P m  m  zf z of the conjugate object and image planes. object and of the conjugate E E F  f m mf   1 e focal mapping when the respective when e focal mapping FP 1 1    z   R   z f    zmf z n n h E E f f 1 m m m 1 m    1 1   F z f   z z nm nf  n nf 1 n

    R F

n  fn f       nn The Gaussian equations describe th are the references for measuring locations Gaussian Equations Applied to a System RE In air: FE f f OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-16 ∞ ∞ are conjugate. ∞ and F' are conjugate. F and P and P' and P are conjugate. ∞ Image plane located at Image plane located at P´ Image plane located at F' Object plane located at F Object plane located at P Object plane located at m m m  0 R      z   z zf       ∞ F 1 1 1 0             z zf 11 z 11 11       FR FR FR zz zz zz fm f fm f fm f ∞ m = 1 m = m = 0 Use the Gaussian equations. Conjugate planes for m = 1, 0 and OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-17     F F F F f f

0 

0     tan Same result!  tan hz

 h hz

 h     FF hh FF zf hh m zf m are negative. is negative. F F and f ′ , h z z F  h  h the right of F: now only f the right of F: now h F F f f  F  z FP FP F z  n h n New Configuration – Object to Newtonian Equations Derivation Sign Conventions Revisited Original Configuration – to the left Object of F; z OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-18   F F f Same result!

0    tan hz

 h   FF hh zf m are negative. F and f ′ z te in the equations. The net result The is in the equations. te  h ith the figure, the sign conventions will ith the figure, sign conventions ese figures, the reference location used to ese figures, the reference location used h F f goes positive to place the object to the right of F, positive to place the object right of F, goes the F F z  FP n object height h becomes negative to compensa the same. Note that in all of th always define the quantities not changed. has to be consistent w Set the equations up configurations. valid for different allow the equation to be As the object distance z Original Figure –h, h to the right of F; now, Object Sign Conventions Revisited OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp  5-19  and zz  m     z The thicknesses of the are independent origins used. 2  h  z  1  h  F2 z 1   F1 n  z   F 12 mm FR FF fm f zz   R f    21 21 211212 P 1111 mmmm mmmmmm       21 z 12 mm  mm 2112  P  R F f  F f 21 21 21 21 21   FF FF FF FF zz    zz mm R F   F        / //  F2 n   zf  z zz z z z z zf m m mm z RRR z zf FFF z 2 F1 fff fff            z h z 1  h The thickness magnification relates the distances between pairs of conjugate planes. thickness magnification relates pairs of conjugate The the distances between Use Newtonian equations: Distances Between Pairs Distances Between of Conjugate Planes – Magnification Thickness OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-20 nf   n nf 

n    R F n

  fn f     RE FE f f / / r widely separated planes. Since it is a cal longitudinal or axial magnification is  z 22 ndependent of the choice origins. and . and z 12    mm 12  R F 12 mm   f zn zn mm zn zn    0       z   lim / /     zn zf zn   zn zn z       mm m varies with position, the longitudinal magnification the thickness and m Since magnification are a function of As the plane separation approaches zero, the lo As the plane separation approaches difference in position, the result is i obtained. The thickness magnification equations are valid fo Thickness Magnification and Longitudinal Magnification OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-21   2 2 1 1 h h h h   1 2 m m 2 12 mm    mm z z   z  12 2  mmm h  z  1  h In air: In air: 2  z   1 n z for widely separated planes. As the plane for widely separated planes. nal or axial magnification is obtained: axial magnification is obtained: nal or 2  m   z     zn zn and . and P   z 12 P mm  2  zn zn 21 21    2 zn n zn z 2 1 zz z z z z   h z      0 z lim z  1 varies with position, the longitudinal magnification the thickness and  h  m mm Since magnification are a function of The thickness magnification equations are valid separation approaches zero, the local longitudi separation approaches The thickness magnification relates the distances between pairs of conjugate planes. thickness magnification relates pairs of conjugate The the distances between Thickness Magnification and Longitudinal Magnification – System OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-22 1 Ray 2 Ray 1  P  z  m n  n n   F N   R f m of the principal planes and nodal points): of the principal planes and nodal N  PN z  P N  mm  since their conjugate rays cross in the front since their conjugate  P   nd rear nodal points (N and N') that define r a focal system. A ray passing through one ff    axis. The nodal points are conjugate points. points are conjugate nodal axis. The z y passing through the other nodal point having the other nodal through y passing   N NPNFR m NF R   P P zn PN zz ff z are not only similar, but identical. are not only similar, z   N n n     1 F f  FP    n NPPN NPPN  NPN zzz zzz z n  P zz Using the thickness magnification (and the locations Rays 1 and 2 must be parallel in image space, focal plane. The indicated triangles The focal plane. the location of unit angular magnification fo to a ra point of a system is mapped nodal the same angle with respect to optical Two additional cardinal points are the front a Two Nodal Points OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-23 incipal planes if the image space index of index if the image space incipal planes 0  1   N PN PN m zz  f   n n     NPNFR N FR  P f nn m zz ff The nodal points are located at the respective pr The refraction equals the object space index of refraction. If the same index occurs in object space and image space: Nodal Points of a System Nodal OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-24  n n  N m  z  h N N  z N  N mm z   hz h u=u   n     N m  measured from the same origins same the from measured relative to the Nodal points, an interesting relative to the Nodal mm from the rear nodal point equals the angular from the rear nodal z         N nn     NN NN   zzz n zzz   u N N N  z z znn een from the front nodal point. een from the front nodal NN hh zz  uu h 12  mm     and and point positions nodal the are and are the object and positions image origins measured some from  NN    zz zz zn zn   The angular subtense of an image as seen subtense of the object as s and important result is obtained. Use the thickness magnification relationship: If the object and image locations are measured Origins at the Nodal Points Origins at the Nodal OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-25   f ′ z FR nn f    F E   ff n BFD  R f from the system back vertex V vertex from the system back ′   d V  P stance from the system front vertex stance from the stance from the system back vertex system back stance from the pal Plane P from the system front vertex V system front vertex from the P pal Plane ′ ′ P Optical System V d F f FFD n to the front focal point F to the back focal point F to the back F is the shift of the system rear Principal Plane P ′ The Front Focal Distance FFD is the di Front Focal Distance FFD The The Back Focal Distance BFD is the di d is the shift of system front Princi d Cardinal Points and Planes of a System Cardinal Points and The system focal length is f system focal length The The object space index of refraction is n The image space index of refraction is n OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-26 z  F  P,N P,N z cal length to be experimentally determined. incide with the principal points. The use of a nt, the rays will converge to the same point. nt, the rays will converge  F F points (N, F, P) can also be located. P) can points (N, F,  V   P,N P,N F nodal slide allows the principal planes and fo nodal poi the lens is rotated about its rear nodal When F'and forming the image is skewed, the ray bundle though even image will not move The is side. shifted to one When a lens system is in air, the nodal points co When Nodal Slide By inverting the lens, front cardinal By OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp z 5-27  F   P,N P, N F ns with the translation stage so that distance from the rear vertex to the  is rotated. The rear nodal point (and the The rear nodal is rotated. measure the distance between the rear measure the distance between on axis. When properly positioned, the on axis. z on stage which is on a rotation stage.  F  ER ffBFDd  R BFD  f   V f  d  P,N P, N vertex V' and the focus (rear focal vertex point F'). •–Focal Distance BFD is the Back This the rear focal point. •the image, reposition le observing While the lens not translate the image does when over the rotation point. rear principal point) are now •is the separation d'moved amount the lens was The the rear vertex and between the rear principal plane. •by is found system focal length The • the rear Position vertex over the rotati vertex will the lens is rotated. not translate when • Use collimated illumination. • Use a microscope (with a micrometer) to •a translati the lens system on Mount Nodal Slide – Procedure F OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp z 5-28  n  CC N, N R nf V n

n    FE f  will emerge from that surface at the same nf cident and located at the surface vertex V. cident and 

n    RE f e measured from the surface vertex. e measured  nnC  

 Nodal Points: nodal points are located at the center of Both curvature CC of the optical surface. towards the center of curvature is normal ray heading A ray has is not refracted. The to the refracting surface and the same angle in object space and image (angular the ray’s axial intercept at CC magnification = 1), and nodal points. defines both Gaussian Properties of a Single Refracting Surface Gaussian Principal Planes: The front and rear principal planes are coin rear focal lengths ar front and The A ray incident on the surface at some height ray incident on A height. Unit magnification occurs at the surface. Surface Power and Focal Points: OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-29 nf  nf

n   n

     RE FE f f  n n  N m    nnC  

    

nn nn  1 C       NPNFR PN PN P PN PN zz R zz ff zz The principal planes are at the surface vertex The The nodal points are at the surface center of curvature. Nodal point magnification for a single refracting surface: Verify the Nodal Point –Verify the Nodal Principal Plane separation: Nodal Points of a Single Refracting Surface Nodal OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-30  > 0 n>n z   R > 0 R >   F n  n n   R f N CC m R 1 C V  n  

 PN PN zz R  f  n  from the surface vertex (principal planes).    zz nn

n       RE f f  R  nn r of curvature the surface. FR  nn f E  1 / /  1

 zn zn f   n  E   nnC  zzf nn m    ff  n    FE f Magnification: Imaging: Principal plane -point separation: Nodal Principal planes/points –surface vertex. at the located Nodal points – at located the cente The front and rear focal lengths are measured front and The Focal Lengths: Power: Gaussian Properties of a Refracting Surface –Gaussian Summary OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-31   f/n

1  ER yy f   R f n the refracting surface is u', the refracting surface 

    thin lens equivalent. Both have the same have thin lens equivalent. Both  nu nu y nu nu  RE E E fnff f allows any system to be represented as an allows any ce for the thin lens becomes the reduced image The reduced focal length of the surface equals reduced The length of the thin lens. If the ray angle or slope for for the thin lens. then the ray angle is n'u' ray angle multiplied by the refractive index of its A optical angle: nu. is called an optical space is in terms of optical paraxial raytrace equation The focal length of the (or reduced) the effective angles and system: z  F z   u n  F  R  f  u=nu   ER f = f/n f =  .  n = 1 n = 1 air-equivalent system comprised of thin lenses. of air-equivalent system comprised If the object is not at infinity, the image distan distance of the refracting surface. The use of reduced distances and optical angles distances and use of reduced The power Consider the example of a refracting surface and its the example of a refracting surface and Consider Reduced Distance Equivalence OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-32 z z z    F F F   R R f f age space for a focal system. Gaussian stem, regardless of the number surfaces,   ed focal completely lengths and power, P P P P F F f f F F F Gaussian Imagery specify the mapping from object space into im from object space specify the mapping focal imaging sy any imagery aims to reduce set of cardinal points. to a single, unique The cardinal points, along with the associat cardinal points, along The OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-33 z  h  F z F'  R f .50 .50 100 50 P'   .50 10 5.0 P 200 100  mm 1.0 F 200 f E F  f  z  100      10         FE   F E zmf mmmm m hmh mm mm zmm hmm f nn F E mf   FE F nm  zf F  z n z h = 10 mm h Note: The physical separation between P and P' is not known. P The physical separation between Note: Object:mm to left 200 of F Use Newtonian equations: Imaging Example 1 Positive Focal System OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-34 z mm 1.0   100    E h f nn .50 F'  z  R f 100 P' 100 300 P  E f E F  f   F E EE fm .50 10 5.0 50 mm to the right of F' 50 F m mf z f m m mm  1 z 1 1 100 1.50 150                 zmm n m f z mm mm z n hmh mm mm 200 300 10     zmmfmm hmm h Use Gaussian equations: Distances from Principal Gaussian planes Use Imaging Example 1 – Equations Gaussian OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-35 z 2.5 2.5 100 250    2.5 10 25 40 F' 100 mm 1.0  E F  40 100  f R z f 10             E F   FE f zmm nn hmm zmf mmmm m hmh mmmm P' F  z E P mf   FE h nm F  zf  F f n z F z F h = 10 mm  h Object: F mm to right of 40 Use Newtonian equations: Imaging Example 2 Same Positive Focal System OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-36 z  A F'  R f P' P F f B anes are shown as coincident. anes are shown F  B A For convenience, the Principal Pl For convenience, Example Summary OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp  5-37 2  E f   // 12 12  12 FF zn z n        /// / ///  / zn z n z n zn  zn z n z n zn     2 2 F F 2 2  f   nz n nz nz       l to the Newtonian object distance squared. l to the Newtonian   m m / / 2 12 f magnification is obtained Newtonian: Gaussian:  or / /  zn   are the lateral magnifications planes for the two  12 12 12 12 2 zz mm mm 2  FF / / FFF       and m f   zznzz zn n   1 n zn       m znzz zn zn  zn zn zn        z is small, the longitudinal z m  12 m mm Δ mmm Newtonian Equations (distances measured from F, F') Equations (distances measured from F, Newtonian When The image space spacing is inversely proportiona Gaussian Equations (distances measured from P, P') Gaussian Equations (distances measured from P, Thickness and Longitudinal Magnification – Focal Systems OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-38 21 21 FF  FF 405      zz z zz z  F  zmm 2 F 0.061  0.61 2 2    10 zmm   mm mm zmm  405 100 2   EF fz mm   1.0 0.61    2  2 F 100 F f     E 0.061  25.0 410 400 24.39 f nz nn n           2 1 z z  FF E    F  F 2 1 nn z zmz mm zz f zmm zmm  F F  zmm zmm m      400 mm to left 400 of F Exact thickness magnification: Use Newtonian equations: Objects:mm to left 410 of F Imaging Example 3 – Magnification Longitudinal Same Positive Focal System OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-39 21  21 FF    45 zzz zz z     F zmm 2 F  4.94 50.0 2 2    10 zmm  mm  mm zmm  45 2 100  EF  fz   mm 1.0 49.4    2 2 F F  100 f     5.00  50 40 250.0 E 200.0 nz f n  nn          2 1 z FF E z  nn z   2  F  F 1 zz f 40 mm to left of F zmz mm zmm zmm F F      zmm zmm m   Exact thickness magnification: Objects closer to the system: 50 mm to left of F Use Newtonian equations: Imaging Example 3A – Magnification Longitudinal Same Positive Focal System OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-40 2 f E  f n   

n  E  . As the course   E mf  f RE f   FF nn  FE zz F  nm n zf z    f her in terms of the front and rear focal her in terms of the front and n

n    FE f opped, and the more common expression for the licitly note the focal length as ve focal length and indices of refraction. ve focal length and

1  E  ff 1 m m  will be used. f     F R F FF FR F  f f z zz ff z Newtonian Equations (Origins at F, F'): Newtonian Equations (Origins at F, progresses, the “E” subscript will be dr focal length lengths or in terms of the effecti For clarity, the expressions will exp The general imagery relationships can be written eit Object Image Relationships OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 2 m 5-41    zn zn   E E f E 12  1 2 mf mm m / /     n zn  n zn 1 1 n n         zn  N zn     m  z z zzf n nm  nn mm m    12 12 mm mm 2 m 1   R F R f F f 1  m  m R F R F  f f  f f    1 1               FR    F  R zf n zz zn ff z N zf f z f z   mm m Object Image Relationships –2 Page Gaussian Equations (Origins at P, P'): Gaussian Equations (Origins at P, Thickness and Longitudinal Magnification – Focal System: Points: Magnification of the Nodal OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-42

 : t or or E f the EFL or power ( ), or power the EFL focal length or minus the its optical space is called an optical angle: is called an its optical space he physical distance to the index of refraction in he physical distance to the index called a reduced distance and is usually denoted by a by is usually denoted distance and called a reduced nu t n ength: it equals the reduced rear the reduced ength: it equals   represents the reduced distance associated with the thickness represents the reduced    Reduced Distances and Optical Angles reduced front focal length. ray angle multiplied by the refractive index of A The EFL is the reduced focal l is the reduced EFL The When the Gaussian imagery equations are expressed in terms of all of the axial distances appear as a ratio all of the axial distances appear of t This ratio is optical space. the same For example letter. Greek OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 5-43    111 zzf 2 1   z z     m mm m FF zz f 1  N  m 

1    2 2  ERF fff  1         F f mm mm zmf zmf f m 1 12 2 1 m f  f mm f 1  m       z  z F z   nn z m Thickness and Longitudinal Magnification – Focal System: Magnification of the Nodal Points: Magnification of the Nodal Newtonian Equations (Origins at F, F'): Newtonian Equations (Origins at F, Afocal Systems: Gaussian Equations (Origins at P, P'): Gaussian Equations (Origins at P, Imaging Equations in Air