7-1 I and Instrumentation Design Optical OPTI-502 © Copyright 2019 John E. Greivenkamp E. John 2019 © Copyright Section 7 Gaussian Reduction 7-2 I and Instrumentation Design Optical OPTI-502 Paraxial Raytrace Equations Greivenkamp E. John 2019 © Copyright Refraction occurs at an interface between two optical spaces. The transfer distance t' allows the ray height y' to be determined at any plane within an optical space (including virtual segments). t nu nnC n Refraction: nu nu y y n n u y u y u P P z t Transfer: yytu t yynu n yy 7-3 I and Instrumentation Design Optical OPTI-502 Gaussian Reduction Greivenkamp E. John 2019 © Copyright Gaussian reduction is the process that combines multiple components two at a time into a single equivalent system. The Gaussian properties (power, focal lengths, and the location of the cardinal points) are determined. Two component system – System Power: Trace a ray parallel to the optical axis in object space. This ray must go through the rear focal point of the system. Paraxial raytrace: Refraction nu nu y y Transfer yynutnyy 1 2 u = u1 = 0 u12 = u y1 u = u2 y2 z P1 P1 P P2 P2 F n=n1 n=n21 n=32 n = n t = t Define the system power by applying the refraction equation to the system: y 21 0 y1 7-4 I and Instrumentation Design Optical OPTI-502 Two Component System – System Power Greivenkamp E. John 2019 © Copyright 1 2 u = u1 = 0 u12 = u y1 u = u2 y2 z P1 P1 P P2 P2 F n=n1 n=n21 n=32 n = n t = t Trace the ray: 2111111 yy yy211 t 2222y n2 yy11 1 1 2 yy y 11 1 2 11 2 y1 yy11 2 12 1 System power: 1212 7-5 I and Instrumentation Design Optical OPTI-502 Two Component System – Rear Cardinal Points Greivenkamp E. John 2019 © Copyright The system rear principal plane is the plane of unit system magnification. 1 2 yy211 u = u1 = 0 u12 = u y1 u = u2 y2 12 y 11 z P1 P1 P P2 P2 F y1 n=n1 n=n21 n=32 n = n t = t PF d 2 If P 2 is the rear vertex yy d 21 of the system, then the fR u distance PF 2 is the BFD. 1 y d PF 2 u 2 u d y PF y y y1 1 11 221111 ny 1 ny 1 dt 11 fR PF2 d nn 2 fPFd R 2 Note that the shift d' of the system rear nnn principal plane P' from the rear principal f 1 1 n plane of the second element P occurs in the R 11 2 fE fR system image space n'. n 7-6 I and Instrumentation Design Optical OPTI-502 Two Component System – Front Properties Greivenkamp E. John 2019 © Copyright Repeat the process to determine the front cardinal points. Start with a ray at the system front focal point F. It will emerge from the system parallel to the optical axis. 1 2 u = u2 = 0 uu 1 u = u y2 y1 12 z F P1 P1 P P2 P2 n=n 1 n=n21 n=32 n = n t = t System: 22y 0 t y2 n2 Work the ray backwards from image space through the system. 2222y 2 0 1111211 yy 222 y yy212 122212yy1 yy122 y 21 2 yy21 2 12 2 1111 y 1212 System power: same result as for forward ray. 7-7 I and Instrumentation Design Optical OPTI-502 Two Component System – Front Cardinal Points Greivenkamp E. John 2019 © Copyright 1 2 u = u = 0 yy212 u=u1 2 u = u y2 y1 12 z 222 y F P1 P1 P P2 P2 n=n 1 n=n21 n=32 n = n y2 t = t PF1 d If P1 is the front vertex of the system, then the fF distance PF 1 is the FFD. yy21 d y u 1 PF1 u d 2 PF y y 1 1 u 11 22 2 ny 2 d y 222 ny 2 fF PF1 d dt22 fPFF 1 d nn 2 nnn Note that the shift d of the system front f 1 1 n F 22 f f principal plane P from the front principal n E F plane of the first element P 1 occurs in the system object space n. 7-8 I and Instrumentation Design Optical OPTI-502 Gaussian Reduction - Summary Greivenkamp E. John 2019 © Copyright 1 2 n= n n=n1 n2 3 z F P1 P1 P P P2 P2 F fR fF d t d t n 1212 fR nf n2 1 ff E dt22 dt 11 n fF nf nn2 nn 2 • P and P' are the planes of unit system magnification (effective refraction for the system). • d is the shift in object space of the front system principal plane P from the front principal plane of the first system P 1 . • d' is the shift in image space of the rear system principal plane P from the rear principal plane of the second system P 2 . • t is the directed distance in the intermediate optical space from the rear principal plane of the first system P 1 to the front principal plane of the second system P2. Both of these principal planes must be in the same optical space. • Following reduction, the two original elements and the intermediate optical space n2 are not needed or used. 7-9 I and Instrumentation Design Optical OPTI-502 Vertex Distances Greivenkamp E. John 2019 © Copyright The surface vertices are the mechanical datums in a system and are often the reference locations for the cardinal points. d BFD Back focal distance BFD: P V F z BFD f d R fR Front focal distance FFD: FFD d FFD f d F F V P z fF Object and image vertex distances are determined using the Gaussian distances: s zd s d d s szd V P P V z z z 7-10 I and Instrumentation Design Optical OPTI-502 Thick Lens in Air Greivenkamp E. John 2019 © Copyright n1 = 1 n2 = n n3 = 1 A thick lens is the combination of C two refracting surfaces. C1 2 1211nnC V P P V F z 11nC1 f=fR d d BFD 2322nnC t 2211nC n C 2 t dVP1 tt n 1212 nn2 BFD fR d 2 t nCCnCCtn11/ dVP 12 12 n n1 1 n3 1 f f 12 F R PP t d d t 1 12 2 ffRF f PP n 1 The nodal points are located at the respective principal planes. 7-11 I and Instrumentation Design Optical OPTI-502 Thin Lens in Air Greivenkamp E. John 2019 © Copyright t 00 1212 1 2 12 F z 11nC1 f=BFD 22nC 1 nCC1 12 1 ff f f dd 0 BFD f ER F The principal planes and nodal points are located at the lens. 7-12 I and Instrumentation Design Optical OPTI-502 Two Separated Thin Lenses in Air Greivenkamp E. John 2019 © Copyright n 1 t 2 1 2 1212t P P F z d d BFD 1 t ff f=f R R dt 1 dt 2 BFD f d PPtdd 12 t2 R The nodal points are coincident with the principal planes. 7-13 I and Instrumentation Design Optical OPTI-502 Gaussian Reduction Example – Two Separated Thin Lenses in Air Greivenkamp E. John 2019 © Copyright Two 50 mm focal length lenses are separated by 25 mm. 1 2 -1 120.02 mm P V F z d t25mm BFD f = fR 0.02mm1 1 t dt 1 25 mm 1212 0.03mm 2 0.02mm11 0.02 mm 0.02 mm 1 25 mm dmm 16.667 1 0.03mm BFD fR d ffR 33.333 mm BFD16.667 mm 7-14 I and Instrumentation Design Optical OPTI-502 Diopters Greivenkamp E. John 2019 © Copyright Lens power is often quoted in diopters D. Units are m1 D (in m-1 ) 1 Df (E in m) fE With closely spaced thin lenses, the total power is approximately the sum of the powers of the individual lenses. Focal lengths do not add. 7-15 I and Instrumentation Design Optical OPTI-502 Multi-Element Reduction Greivenkamp E. John 2019 © Copyright Multiple element systems are reduced two elements at a time. A single system power and pair of principal planes results. Given these quantities, the focal lengths and other cardinal points can be found. There are several reduction strategies possible for multiple elements or surfaces. 1 2 3 4 (12) (34) (1234) 1 2 3 4 (12) 3 4 (123) 4 (1234) The system principal planes are usually measured relative to the front and rear vertices of the systems: - The system front principal plane is located relative to the front principal plane of the first surface or element. - The system rear principal plane is located relative to the rear principal plane of the last surface or element. 7-16 I and Instrumentation Design Optical OPTI-502 C C C C Reduction in Pairs 1 2 3 4 Greivenkamp E. John 2019 © Copyright nn 1 nn12 nn23 nn34 nn4 V t1 t2 t3 V 1 2 3 4 (12) (34) (1234) 1 2 3 4 P11 P P22 P P33 P P44 P t1 t2 t3 d12 d12 d34 d34 12 34 P12 P 12 P34 P 34 t12 n n3 n ttdd12 2 12 34 d1234 d1234 dd12 d 1234 P P P P dd34 d 1234 1 4 dd n n 7-17 I and Instrumentation Design Optical OPTI-502 nn 1 nn12 nn23 nn34 nn4 Reduction – One at a Time Greivenkamp E.