502-07 Gaussian Reduction

OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 7-2 7-1 ' t z  y t n   within an optical space (including  t  u  n  two optical spaces. The transfer distance The optical spaces. two   y nnC   Section 7   t n   P    y  P u  Gaussian Reduction           yytu yy yynu u nu nu y nu  ' to be determined be determined ' to at any plane n y   Transfer: Refraction: Refraction occurs at an interface between allows the ray height segments). virtual Paraxial Raytrace Equations OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 7-4 7-3 z z  1  y    2 1 t y n     2       2   F    F   32   32 u = u u = n= n = n = n n= u = u u = n= n = n = n n=     2 2  2 2 y P y P 2 2      2 0 2      P P  rties (power, focal lengths, and the location rties and focal lengths, (power,  es multiple time at a two into a components     ect space. This ray must the rear through ect space. This ray must go P     P       t = t t =  t = t t = 21   the refraction equation to the system: 21 n=n  12 yynutnyy nu nu y y y n=n   yy 12 u = u u = 1  u = u u = 1 P P  1 1       1 1   1 1 11 2 12 1 y  P y 11 111 1 1 2 2 11 2  yy P y yy yy y 2222 1212                           2111111 211 Transfer yy  21      1 1 n=n 1  n=n 1   u = u= u 0 = u = u= u 0 = System power: System Trace the ray: the Trace Define the system applying by Define the system power Paraxial raytrace: Refraction single equivalent system. Gaussian prope single equivalent system. The the cardinal are points) determined. of –system component Two Power: System Trace a ray parallel the optical to axis in obj focal point of the system. Gaussian reduction is the process Gaussian reduction is the that combin Two Component System System Component Two – System Power Gaussian Reduction OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp y 7-6 7-5     n    1   y  R  f      1   2  12 11 211 PF yy    2   t  n  2 1 E P  f   z If is the rear vertex the system, of then the distance is the BFD.    1  1  z 2  2    y u  32 d   n= n = n = n n=         u = u = 0 u= u =  yy yy 11    2             2  ny F 221111 2     2 122212 21 2 12 2 1 PF y y y PF PF d 1212 yy    32      2 2   u = u u = n= n = n = n n=   y P        R 1111211 R R f n fPF nnn f   System System power: same result as forforward ray.  2   2  0 PF  R  nal Start points. with a ray the system at f  2 2 2  y P P   2    12  21  u = u u = d 2 n=n t = t t = P P 0  the plane of unit system system unit magnification. the plane of   1  P  P  2 y  1   t = t t = 1 21  1 1  2  n=n  12 y P P   u = u u = 1 1 2 P  11  22 1 2 1 y  y             1 1 y y  P  y  of the system of the system rear uu F   1  n=n y    1  11             1111  2222 222 212 122 2 2  u  yy y  yy     1 1 21 u 1 n=n yy      nn ny dt d     u = u= u 0 =     System:   Work the ray backwards from the system. the ray backwards from image space through Work d d The system The system rear principal plane is Repeat the process to determine Repeat the process to determine cardi the front front focal point F. It will emerge It F. will emerge front focal point from the system parallel the optical to axis. Two Component System – System Component Two Points Cardinal Rear Two Component System System Component –Two Front Properties Note that the shift d' principal plane P' the rear from principal in the occurs element the second plane of n'. space image system OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 1   n 7-8 7-7   E        F  2 f ff y  y are not    2  1 1 222 212 PF yy   z nf E  is the front vertex vertex is the front f  1 nf    2  n  of the system, the system, of then the distance is the FFD. If P   1 1  n 3   F n= n n= 22   y  R F f f z  R . these Both of principal  f  2 the rear principal the plane of P 1  u y    2 22  d  P 2  2         2 32  ny 1 11 2  1 n= n = n = n n=  1 PF y 1 PF  d  u = u = 0 u= u = PF d 2  P   incipal plane from from the rear incipal principal plane plane F F F n  f nnn f fPF principal from the front principal plane P plane  P 11  2 2 cation (effective refraction for the system). cationsystem). (effectivethe refraction for 2  t y  P n P ents and the intermediate optical space n space optical intermediate the ents and 2     2 1 nn dt t 2 n P d P 1        1  P  12 21  u = u u = n=n t = t t = e intermediate e intermediate optical space from F f P  1 2 1 P  P d occurs in the 2 1 F 2 1  y n=n   2 1 1 F f P y P  1  P 222  22  1 1   PF  22   u u=u   1 2 21 u F nn ny d P dt yy  1 1212     nn n=n dt         d d    Two Component System – System Component Two Points Cardinal Front Gaussian Reduction -Gaussian Reduction Summary Note that the shift d of the system the system of front Note that the shift d principal the front from principal plane P the firstplane of element n. space object system planes must be in the same in the same optical planes must be space. • reduction, two original Following the elem •P' and P unit system are magnifi the planes of •object is the shiftthe front system in space of d of the first system . • d' image the shift is in the rear space of system pr system P the second principal plane of the front to first system of the second system of the system second . • is the directed t distance in th needed or used. OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp  z 7-9 7-10    R z 2      z z 12   12   FD f d   F  B  P s F z  d 1   R t n V = 1  R BFD 1 V  f  3  t  n f=f 2 V n    BFD 2 d  C     F  V f    P PP n d  FFD  P PP t d d t P  t  = n  d  d 2 dVP dVP n  P F P V d 1 V C = 1 s z in a system and are often the reference 1 n 1    3 2 tt  n nn    R 1   f  RF 1 12 12     nC n C 1 ff f   11/ 1  The nodal points are are located points at the respective nodal The principal planes.  R n F nC 11 nCCnCCtn            zd  zd   22 2 11 F  f     FD f d FD f d  s s B F nnC 1212 nnC       2322 1211 Back focal distance BFD: Front focal distance FFD: vertex distances are image Object and determined Gaussian distances: the using  A thick lens is the combination thick lens is the combination of A two refracting surfaces.  The surface vertices are the mechanical datums cardinal points. the locations for Thickin Lens Air Vertex Distances  OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 7-11 7-12 z z  F 2    12 F    R f=BFD FD f BFD 2 2 B   f=f   1 d      P  Ptdd t t P  P 0 d   1   dd 2    dt 1    1   t R 1 1 t    1 R   nC  12  nC     FD f d ff 1 dt ER F   B   1 11 22 2    00 1212 n nCC 1212  12 ff f f        The The nodal points are coincident with the principal planes.   t     The principal planes and nodal points are points are located at lens. nodal principal the The planes and Two Separated Two Thin Lenses in Air ThinLens in Air OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 7-14 7-13 -1 25  1 1     mm mm R 0.02 mm 16.667   0.03 0.02 FD mm 12 z B BFD f d   1  16.667     F   dtdmm mm BFD   R V 2  f = f f =  d  P 2 Separated Thin Lenses in Air Lenses in Separated Thin  t25mm . Focal lengths do not add. add. not . Focal lengths do 1  nses, the total power is approximately nses, total the power is approximately the of the sum  -1 E ( in m) ( in (in m ) E 1 f  t   1 11 1 1 D Df     m 33.333 mm mm mm mm mm   R 1212 0.02 0.020.03 0.02 25       With closely spaced thin le thin closely spaced With powers powers of the individual lenses ff mm Two 50 mm mm. 25 mm focal length lenses are separated by 50 Two Lens power is often quoted in diopters diopters D. in is often quoted power Lens Units are Gaussian Reduction Example Example – Reduction Gaussian Two Diopters  OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 7-16 7-15    4   n   4 nn 4 V P 4 C 44 34    P P d  n  3  3  34  t t  34  pal the last plane of nn  34 34  1234 34 P P d  d 3 3 33 C  P P 3  12 2  2   t  n t t  23  nn 2 P P 2 C 22  12   P P d 1234 d dd  1  1 12  t t   12 nn n  ed relative rear and to the front vertices 12 12 12 P P d 1 lative to the rear princi rear the lative to relative the front principal the to plane of 1 1 C 11 V  P P P n 1 and other cardinal points can be found. other cardinal can be and points  nn   1 2 3 4 (34) (12) 1 2 3 4 (1234) 3 4 (12) 4 (123) (1234) (1234) 34 1234 12 1234 (12) (34) (34) (12)          12 2 12 34 dd d dd d ttdd 1 2 3 4 3 2 1 surface or element. surface or of the systems: - system The front principal plane is located firstelement. surface or - system The rear principal plane is located re The system are usually measur system principal planes The There are several reduction strategies possible for multiple strategies forThere are several reduction multiple surfaces.elements possible or Multiple element systems are reduced two elements at a time. a at elements two reduced are systems element Multiple principal planes results. pair of and system single power A lengths focal the these quantities, Given Reduction in Pairs Multi-Element Reduction OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 7-18 7-17      4    n  n nn  4 4 P 4  44 44 44   P P P P P P  nn 1.0 n 4  3  3  3   n t t  1234 V C 4 34  d n  123 nn 2.43 t    3  3 2 33 33 2  t n   t P P P P  123 4.0 d 2     t   2 3   23 2   3 2  12 C n t nn t  n 1.649 123 P P  1234 2 22  12  d  P P d dd 123 123  1 123 12  t P P d   12  1 nn n t  12 12 12  P P d 6.92 1 2 1 11  P n1.517 P P n n 1 2  1 t n 1 10.5   C  V nn  1  1 t  73.8950 51.7840 162.2252 .0135327 .0193110 .00616427 1 .00700 .00255 .00400     12 3 12 3 12 3 RR R nn 1.0 CC C   (1234) (123) 4 (123)   (12) 3 4 3 (12) 1234 12 123 1234          1 2 3 4 12 2123 12 3 123 dd d d dd ttd ttd Cemented Doublet Cemented Gaussian Reduction –Gaussian Reduction Example Reduction – at One a Time OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 7-20 7-19  12 P z z 12 P 3 3 l planes and . V V  d  123 d  2  P t     2 12 n n t V  12 t 12  17.48  12 d   12 12 P P   13.03 RF   Image space space Image 1  21.48 Object space 1    12 3 12 12 12 dn d represented by and the principa 12 2.40  P 3  12 n t    .00457 d 7.15 1 1 .00833 V   V 12 2 12 12 2 12 ttd   E 123 6.26 fff d  12 12 10.60 d d 3.86 7.15   12  3  12 12   P P 1 12   123   2 12  12 123 12 123           12 3 12 3 12       12 1 2 1 2 1  12 1 .008333 120.0 120.0 12 1               123 123 12  123 123 12   d d ddd d  Add the third surface: First, reduce the first two surfaces: first two First, reduce the Gaussian Reduction – Gaussian Reduction Example – Continued Gaussian Reduction –Gaussian Reduction Example – Continued At this point, the first At this point, two surfaces are OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 7-22 7-21 z  F BFD  R f  V  d  Back Focal Distance BFD Front Focal Distance FFD P  RF  112.85 PP 117.6 12 2.40 R F P E fff d 3 7.15 1 V 4.95        F f     123 PP PVdt t dd PPVVdd VF PF f d VF PF f d  12 123 12 123 Principal plane separation: plane Principal  FFD   .008333 120.0 120.0          F ddd d  Real Lens to Thin Lens Model Lens Thin to Real Lens Gaussian Reduction – Reduction Gaussian Example – Summary 7-23 OPTI-502Optical Designand Instrumentation I

Consider a single reflecting surface with a radius of curvature of R. The rays propagate in an index of refraction of n.

The angles of incidence and refraction (I and I') are measured with respect to the surface normal.

The ray angles U and U', as well as the elevation angle A of the surface normal at the ray intersection are measured with respect to the optical axis. The usual sign conventions apply.

Reflecting I' Surface n I

y U V U' A z O O' CC Sag 1 Center of Curvature C R R = 1/C Curvature

7-24 OPTI-502Optical Designand Instrumentation I

Relating the angles at the ray intersection with the surface:

UAI    I  UA IU  A

Apply the Law of Reflection: I  I

UA   UA  7-25 OPTI-502Optical Designand Instrumentation I

UA  UA

sinUA  sin  UA

sinUA cos cos UA sin sin UAUA cos cos sin

sinAA sin sinUU cos sin UU cos cosAA cos

Approximation #1 implies that sinUUA cos tan sin UUA cos tan magnitude of the ray angle is approximately constant. Approximation #1: cosUU cos  UU 

sinU  sinU tan A  tan A cosU cosU U' U z tanUA  tan tan UA tan

7-26 OPTI-502Optical Designand Instrumentation I

tanUA  tan  tan UA tan 

tanUUA  tan  2tan

Reformulate in terms of the paraxial angles or ray slopes:

uUuUtan tan  tan A

uu  2 n

y  tan A R  Sag y V A Approximation #2: Sag R CC z y Sag   R R = 1/C y uu  2 R Approximation #2 implies that the sag of the surface at the ray This is the Paraxial intersection is much less than the uuyC  2 Reflection Equation. radius of curvature of the surface. OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp z 7-28 7-27  nC  2 nnC      nn    C 1 22  yy   yy   nn  z Sag z Sgz zSag  CC  nR          F  f tan tan  A     R     f FR E The front and rear and The front focal lengths are curvature. of half the radius equal to ff nf  uU uU   nu nu y nn  U' Surface O' 2 also both measured measured also both from the surface vertex.  Reflecting z'  y Sag nC F 2 f     C   1 RR   22 ff V  2  I' nn I nR  F nn n  f if    nn           REFLECTION     F   R uuyC f nu nu y nu ynu n n C nu y nu nyC f 1  z ReflectionRefraction Equals with    U   E ff n Sag z Sag z Note that a reflector with a positive curvature has a negative power. Note that a reflector with a positive curvature a negative power. has Reflection: Approximation #3: The object and image image object and The #3: Approximation of sag greater the than distances are much the surface at the ray intersection. O The object and image image The object and z') are distances (z and Refraction: Object and Image Object Image distances and for a Single Reflecting Surface Reflection and Refraction Reflection and 7-29 OPTI-502Optical Designand Instrumentation I

The same set of approximations hold for paraxial analysis of a reflecting surface as for a refraction surface:

The surface sag is ignored and paraxial reflection occurs at the surface vertex. The ray bending at each surface is small.

By ignoring the surface sag in paraxial optics, the planes of effective refraction for the single reflecting surface are located at the surface vertex plane V. The Front and Rear Principal Planes (P and P') of the surface are both located at the surface.

The nodal points of a reflecting surface are located at its center of curvature as a ray perpendicular to the surface is reflected back on itself.

Vertex Plane Reflecting Surface

V P P' CC z N, N'

7-30 OPTI-502Optical Designand Instrumentation I

cosU cosUU cos or 1 or UU cosU

Consider a reflecting surface perpendicular to the optical axis: nn1.0 1.0 A  0 Surface I  U Normal 1.04 I U 1.03 n 1.02 U' U z 1.01 cosU'/cosU 1

0.99 -30 -20 -10 0 10 20 30 For a plane mirror perpendicular to U (degrees) the axis, there is no error associated with this approximation. 7-31 OPTI-502Optical Designand Instrumentation I

With curved surfaces, this approximation is more difficult to interpret as it relates to the ray angle with respect to the optical axis, not with respect to the surface normal.

Consider a surface with a radius of curvature R = 200 mm with a ray height of 10 mm. The surface normal at the ray intersection is tilted at about 2.9 degrees.

n IUA  I U A U' A U

y V A z R CC

A ray will be incident perpendicular to the surface when U = A.

7-32 OPTI-502Optical Designand Instrumentation I

cosU cosUU cos or 1 or UU cosU

Consider: R 200mm y 10 mm A 2.9 deg nn1.0  1.0

Surface Normal The approximation error is 1.04 approximately linear with the ray 1.02 angle relative to the surface normal. 1 At normal incidence, there is no 0.98 approximation error. cosU'/cosU 0.96 For rays incident within about 10 deg of the surface normal, the 0.94 error is about 2%. -30 -20 -10 0 10 20 30 U (degrees) The approximation is in terms of the ray angle with respect to the optical axis, not the angle of incidence. OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 7-34 7-33 z V        R  t   f  0 C 1 yy    F  E     t R       n yytu yy nf tu y y u 22  1 R   CC R F  n fn f    C   R > 0 f n=-n  R < 0 z nR    as fora refractive system, except that the e refractive or reflective surface are located   nn ual sign convention. A distance to the left A ual sign convention. E tical surface are coincident and located at the located at are coincident and tical surface y   R cal angles will unfold the mirror system mirror the system will unfold cal angles  f nf z FR E ff nf F nn f are opposite those of the corresponding u and t. A A t. and u the corresponding those of opposite are n      F : n  tion following a reflection are reversed.  u   F and E f f  R   nn ()  n R CC R f  2 t 1 R   nC  E 2 nnC V  ()  n     ff   > 0 n > n  R > 0  y A reflective surface is a special case with The front and rear principal planes of an op a singl points of nodal surface Both vertex V. at the center curvature of the surface.of Sign conventions and reflection: Sign conventions and reflection: - the us directed Use by distances as defined is negative, and a distance distance is negative, a to the right is positive. and -refrac of all indices of signs The drawing done in reduced distances and opti and show a thin lens equivalent system. The transfer equation is independent of the direction of transfer. the direction of of equation is independent transfer The of signs the reflection, After Optical Surfaces Transfer After Reflection distance the next surface to (to the left) is negative. Transfer after reflection works exactly the same exactly the afterTransfer reflection works 7-35 OPTI-502Optical Designand Instrumentation I

Power of a refractive surface:   ()nnC 

Assume nn 1 and  1.5

 (0.5)CC / 2

Power of a reflective surface:   ()nnC  2 nC

Assume n  1

 2C

For the same optical power, a reflective surface requires approximately one quarter of the curvature of a refractive surface. However the signs of the powers are opposite.

This is one advantage of using reflective surfaces.

7-36 OPTI-502Optical Designand Instrumentation I

()nn  1  ()nnC  C  R R PP, at surface vertex n n  f  nf f  nf NN, at center of curvature F  E R  E

Positive Refracting: n n   0  N,N nn  PP F CC F z C  0 R

fF fR

n n   0 N,N  nn  PP F CC F z C  0 R  fF fR OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 7-38 7-37 z z z z     F n n  PP CC N,N F  d rear focal lengths. 2 R   FR  ff F F  F  F F F f f  CC R R N,N    FR   ff R  front and rear front and focal lengths. FR   CC  N,N PP ff  , at centerof curvature , at surfacevertex NN PP , at centerof curvature , at surfacevertex PP PP NN PP   n=-n n=-n R    R R f f h, but negative front an CC N,N  E n n 1 R  1 F R   nf RE    fnf F C C    n F f   R f and focal length, but positive focal length, but and  R  F nn f () n 0 0 0 0 0 0 E 0 0 0 0 2 R            nf     n nn nn C C C  n   C  n  nC   2  nn nnC ()         F R f f     Optical Surfaces – Cardinal points Optical Surfaces – Cardinal points nn Negative Refracting: Positive Reflecting: Negative Reflecting: A convex mirror convex mirror has a negative power A A concave mirror concave mirror focal lengt a positive power and has A OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 7-40 7-39 z z :  X F F  12   fference in curvatures in fference  F F 1  and shape factor   nCC    2  2 1  X  X   1    11 0 0 Plano Spheric Lens Spheric Plano Lens wards the image. 1Meniscus Lens 1Meniscus     0 Symmetric Bi-SphericLens         111 X away from the image. The rays must be must rays The image. the from away XC XC X The distribution ofeach other in shape. Thin Lens –Lens Thin it is the di that determines the lens power:  2  z z   with a specified power 1212 1212  CC CC  or or t  X  F image (virtual ray segments). ray (virtual image image space head to head space image 2   X F   X   1    111 : X 1 independent of X    1 F 1   or or 1 1 1  2 1 F   t  2 1 2 1 2 1           2 1 PP n X dX dX  For small For small projected the backwards to find Virtual –Virtual head image space actual the rays in For a thick lens in air thickness For a of Lenses having the same the same power can differLenses having from surfaces power between the two can vary. Factor: Shape Symmetrical Real images can be projected and made a screen; virtual made visible Real images be projected images and on cannot. can Real images– the actual rays in Shape Factor and Bending Real and Virtual Real and Images OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 7-42 7-41 z z z d = 0 d =  2 V     2 2 P P P P 2 2 2 are the mechanical are the mechanical datums for the system. P P P P 2 V  1  V d = 0 d =   1 1 P P 1 1 1 P P 1 V -3 -1 0 1 3 solution to the problem: solution to X = The vertices and vertex-to-vertex vertices The vertex-to-vertex and separations Bending does not change the element does the Bending not change element power. 3) Locate Locate the real 3) the vertices components: of 2) Include the principal plane separations real of elements: 1) Obtain the thin lens 1) Obtain the thin System System Design Lenses Thin Using Shape Factor OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp

7-44 7-43

Magnification, etc. Magnification,

Object and Inage Locations, Locations, Inage and Object Imaging Properties Imaging z z   F F Imagery Raytrace Gaussian BFD the imaging the imaging properties of  R  f  V V  P represented by a system represented a system by focal or power of focal points. In initial the P-P' focal points. design, of  Raytrace P Gaussian System Gaussian System Cardinal Points Cardinal System and Gaussian reduction is that and V V F f FFD Gaussian Raytrace Reduction

F F

Element Focal Lengths, etc. Lengths, Focal Element

Radii, Indices, Thicknesses, Spacings, Spacings, Thicknesses, Indices, Radii, System Specification System length, a pair of principal planes and a pair principal a length, a pair of planes and lens model). (i.e. a thin separation is often ignored any combination any combination optical of elements can be The utility optics Gaussian of Methods of System System Analysis of Methods Gaussian Imagery Gaussian Imagery Reduction Gaussian and OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 7-46 7-45 m 1 z m E h' E f  2  1  m E E E m  2   E 1   '   m   z 1 h 1 h  m L m 1 m      1 conjugates: and Reciprocal magnifications. zmf m 14    0    E 2 mzfzf mLf 1  f E L f m  E 2 1  1 E    m m E L m fm m    1 zz mf f        z   111 zzf L L zf h Real object and real image: object and Real Minimum Minimum object to image distance: Overall object-to-image Overall object-to-image distance: For each L, there are possible magnifications two Thin Lens Design –Thin Overall Object-to-Image Distance Reciprocal Magnifications Reciprocal OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 7-48 7-47 Image Lens Location m To Object m Position To Object Position Relative Position Relative Position 4 -4 Image Location Real Image Real Object 1/4  -1/4 Real Image z Virtual Object Virtual 3 -3 -1/3 1/3 Lens 2 z Position Object -2 Location E 1/2 -1/2 4f 1 -1  z E E f 3f z  L E E 2f 2f  z Real Object Virtual Image Virtual E E f Real Image L  Virtual Object Virtual 3f z  Object Location 1 -1 E 4f  1/2 2 -1/2 z -2 1/3 3 -1/3  -3 z   E f0 1/4 4 E -1/4 Real Object f0 -4 Virtual Image Virtual m m E     z0 zf z0 E z0 Virtual Image Virtual Virtual Object Virtual     z0 z0 z0 zf Negative Lens Positive Lens OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 7-50 7-49  z   z mt t t h n    123 13 tttm tt 1 u   z 123  P  mt t t m P P 2 t 3 tm h      13  13 ttm  P ttm P t       3 P 1 t  z   u  P nu 23 23 P nu    n 11 12 htmtt h    ht tt 23 123 12 2 12122   2  h     h L2     2   1     P P m  hh hh zn z n   P zz ys ys on-axis conjugate passing through points.   hh hznh hzn    2 2  t   m zn z n uu    nu n u 1 L1 h  13 1 ttm 1  t 12 1 u   ,,, ,, 123  tttm 1 u  11 mm 22   tt  21   hh hzn  hzn  1        2 2333 31 1111 httm htt m n  Let:  This angle relationship holds for all for ra angle relationship holds This Given: Determine: The Gaussian Magnification may The Gaussian Magnification may also be determined from the object and angles. ray image Thin Lens Design –Two Lenses at Fixed Separation Separation Lenses at Fixed Design –Two Lens Thin Magnification Properties Magnification OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 7-52 7-51 z  z P' t 2  m m    1  tm 2f tm   12 22 fL F fL        12122 1 11  11 2 2 F  M   3 3 1 t t t  f 2 2 2  Define:   L2 -2f 12  nal points of a system nal points of a system completely define the   L Pt ng system consisting of cascaded 2f-2f systems. systems. 2f-2f cascaded of consisting ng system  2 P t 0 0 with the object and image planes. image object and the with 2f 1 L1  2 t   122   f    2 12 12  1   t 122 1 2 122 morff ,,, ,,, ,     123    ttt 2 L -2f MtL t 114  12 2 1 2 2 2 L ff t L f f Lt      MM 2 22 1 2 12 L     tF LMFF ttLFF   M  2 22 12 12 21212 P 1 L LLfMff tLtLff tFFMFF    LPP Given: Determine: Where Where are the focal points? Because the object and image image magnification, Because the object and planes are unit planes of the system and front rear are coincident principal planes The power and the relative and The power the cardi locations of mapping. imaging elements Different interesting combinations of can produce situations. an example imagi consider As this true 1:1 An inverted intermediate image is formed. Cardinal Points Example Cardinal Points Thin Lens Lens Design Thin – Given Lenses Two 7-53 OPTI-502Optical Designand Instrumentation I

To find the rear focal point of the system, launch a ray parallel to the axis: f f

d

z P F1 F' P'

-2f f -3f 1.5f -.5f

The system rear focal point is to the left of the system rear principal plane, and the system power is negative! This infinity ray diverges from the rear principal plane.

1 In a simplified Gaussian model that ignores the tf 4 12f P-P' separation, this system looks just like a negative thin lens: 24f 2  t   1212 f ff2 P P' z fSYSTEMff R 0.5

 1/ f By symmetry, the front focal point of the system dt 1 42 ff   2/ f is to the right of the system front principal plane.

7-54 OPTI-502Optical Designand Instrumentation I

Two 100 mm focal length thin lenses are separated by 50 mm. What is the focal length of this combination of lenses? [ ] a. 66.67 mm [ ] b. 200 mm [ ] c. 50 mm [ ] d. It is an afocal system 7-55 OPTI-502Optical Designand Instrumentation I

Two 100 mm focal length thin lenses are separated by 50 mm. What is the focal length of this combination of lenses? [X] a. 66.67 mm  [ ] b. 200 mm 1 2 [ ] c. 50 mm [ ] d. It is an afocal system F z

1 f fmm100   0.01 mm1 12f 12

tmm 50

  1212 t

  0.015mm1

fmm 66.67