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ECEG105 & ECEU646 Optics for Engineers Course Notes Part 4: Apertures, Aberrations Prof. Charles A. DiMarzio Northeastern University Fall 2003 July 2003 Chuck DiMarzio, Northeastern University 10351-4-1 Advanced Geometric Optics • Introduction • Stops, Pupils, and Windows • f-Number • Examples • Design Process – From Concept through Ray Tracing – Finalizing the Design – Fabrication and Alignment – Magnifier – Microscope • Aberrations July 2003 Chuck DiMarzio, Northeastern University 10351-4-2 Some Assumptions We Made • All lenses are infinite in diameter – Every ray from every part of the object reaches the image • Angles are Small: – sin(q)=tan(q)=q – cos(q)=1 July 2003 Chuck DiMarzio, Northeastern University 10351-4-3 What We Have Developed • Description of an Optical System in terms of Principal Planes, Focal Length, and Indices of Refraction s, s’ are z coordinates • These equations describe a mapping – from image space (x,y,z) – to object space (x’,y’,z’) July 2003 B Chuck DiMarzio, Northeastern University H V V’ H’ B’ 10351-4-4 Lens Equation as Mapping f = 10 cm. 50 July 2003 30 20 s', Image Dist., cm. • The mapping can be applied to all ranges of z. (not just on the appropriate side of the lens) • We can consider the whole system or any part. • The object can be another lens 40 10 0 -10 -20 -30 -40 -50 -60 -40 L4’ Chuck DiMarzio, Northeastern University -20 0 20 s, Object Dist., cm. L1 L2 L3 40 60 L4 10351-4-5 Stops, Pupils, and Windows (1) • Intuitive Description – Pupil Limits Amount of Light Collected – Window Limits What Can Be Seen Window July 2003 Chuck DiMarzio, Northeastern University Pupil 10351-4-6 Stops, Pupils and Windows (2) Images in Object Space Entrance Pupil Limits Cone of Rays from Object Entrance Window Limits Cone of Rays From Entrance Pupil July 2003 Physical Components Aperture Stop Limits Cone of Rays from Object which Can Pass Through the System Field Stop Limits Locations of Points in Object which Can Pass Through System Chuck DiMarzio, Northeastern University Images in Image Space Exit Pupil Limits Cone of Rays from Image Exit Window Limits Cone of Rays From Exit Pupil 10351-4-7 Finding the Entrance Pupil • Find all apertures in object space L4’ is L4 seen through L1-L3 L1 L2 L3 L4 • Entrance Pupil Subtends Smallest Angle from Object L3’ L4’ L1 L2’ L3’ is L3 seen through L1-L2 July 2003 Chuck DiMarzio, Northeastern University 10351-4-8 Finding the Entrance Window • Entrance Window Subtends Smallest Angle from Entrance Pupil L3’ L4’ July 2003 L1 L2’ • Aperture Stop is the physical object conjugate to the entrance pupil • Field Stop is the physical object conjugate to the entrance window • All other apertures are irrelevant Chuck DiMarzio, Northeastern University 10351-4-9 f=28 mm Field of View f=55 mm f=200 mm July 2003 Film= Exit Window Chuck DiMarzio, Northeastern University 10351-4-10 Example: The Telescope Aperture Stop July 2003 Chuck DiMarzio, Northeastern University Field Stop 10351-4-11 The Telescope in Object Space Secondary’ Primary Secondary Entrance Window July 2003 Entrance Pupil Chuck DiMarzio, Northeastern University 10351-4-12 The Telescope in Image Space July 2003 Primary Primary’’ Secondary Exit Pupil Chuck DiMarzio, Northeastern University Exit Window Stopped 26 Sep 03 10351-4-13 Chief Ray Aperture Stop Exit Pupil Field Stop • Chief Ray passes through the center of every pupil July 2003 Chuck DiMarzio, Northeastern University 10351-4-14 Hints on Designing A Scanner • Place the mirrors at pupils July 2003 Chuck DiMarzio, Northeastern University Put Mirrors Here 10351-4-15 f-Number & Numerical Aperture Numerical Aperture f-Number f q A F’ A’ F D is Lens Diameter 5 f#, f-number 4 3 2 1 0 0 0.2 0.4 0.6 NA, Numerical Aperture July 2003 0.8 1 Chuck DiMarzio, Northeastern University 10351-4-16 Importance of Aperture • ``Fast’’ System – – – – Low f-number, High NA (NA1, f# 1) Good Light Collection (can use short exposure) Small Diffraction Limit (l/D) Propensity for Aberrations (sin q q) • Corrections may require multiple elements – Big Diameter • Big Thickness Weight, Cost • Tight Tolerance over Large Area July 2003 Chuck DiMarzio, Northeastern University 10351-4-17 The Simple Magnifier F A’ July 2003 A F’ Chuck DiMarzio, Northeastern University 10351-4-18 The Simple Magnifier (2) • Image Size on Retina Determined by x’/s’ • No Reason to go beyond s’ = 250 mm • Magnification Defined as • No Reason to go beyond D=10 mm • f# 1 Means f=10 mm • Maximum Mm=25 For the Interested Student: What if s>f ? July 2003 Chuck DiMarzio, Northeastern University 10351-4-19 Microscope F’ F F’ F A’ A • Two-Step Magnification – Objective Makes a Real Image – Eyepiece Used as a Simple Magnifier July 2003 Chuck DiMarzio, Northeastern University 10351-4-20 Microscope Objective F F’ F’ F A’ A July 2003 Chuck DiMarzio, Northeastern University 10351-4-21 Microscope Eyepiece F F’ A2 A A2’ July 2003 F’ F Chuck DiMarzio, Northeastern University 10351-4-22 Microscope Effective Lens H H’ 192 mm Barrel Length = 160 mm FA f2=16mm f1=16mm D 19.2 mm Effective Lens: f = -1.6 mm H’ H A F’ D’ 19.2 mm F’ F July 2003 Chuck DiMarzio, Northeastern University 10351-4-23 Microscope Aperture Stop Analysis in Image Space F’ F Exit Pupil Image Aperture Stop =Entrance Pupil Put the Entrance Pupil of your eye at the Exit Pupil of the System, Not at the Eyepiece, because 1) It tickles (and more if it’s a rifle scope) 2) The Pupil begins to act like a window July 2003 Chuck DiMarzio, Northeastern University 10351-4-24 Stopped here 30 Sep 03 Microscope Field Stop F F’ Entrance Window Field Stop = Exit Window July 2003 Chuck DiMarzio, Northeastern University 10351-4-25 Microscope Effective Lens July 2003 Chuck DiMarzio, Northeastern University 10351-4-26 Aberrations • Failure of Paraxial Optics Assumptions – Ray Optics Based On sin(q)=tan(q)=q – Spherical Waves f=f0+2px2/rl • Next Level of Complexity – Ray Approach: sin(q)=q+q3/3! – Wave Approach: f=f0+2px2/rl+cr4+... • A Further Level of Complexity – Ray Tracing July 2003 Chuck DiMarzio, Northeastern University 10351-4-27 Examples of Aberrations (1) 1 Paraxial Imaging 0.5 0 -0.5 -1 -10 m4061_3 July 2003 -5 0 5 10 Chuck DiMarzio, Northeastern University R = 2, n=1.00, n’=1.50 s=10, s’=10 In this example for a ray having height h at the surface, s’(h)<s’(0). 10351-4-28 Example of Aberrations (2) 0.2 D z(h=1.0) Longitudinal Aberration = D z D z(h=0.6) 0.15 0.1 0.05 Transverse Aberration =D x 0 -0.05 Where Exactly is the image? -0.1 -0.15 2D x(h=1.0) m4061_3 -0.2 8.5 July 2003 9 9.5 10 10.5 Chuck DiMarzio, Northeastern University What is its diameter? 10351-4-29 Spherical Aberration Thin Lens in Air July 2003 Chuck DiMarzio, Northeastern University 10351-4-30 Transverse Spherical Aberration h Dx Ds s(0) July 2003 Chuck DiMarzio, Northeastern University 10351-4-31 Evaluating Transverse SA July 2003 Chuck DiMarzio, Northeastern University 10351-4-32 Coddington Shape Factors -1 -1 July 2003 +1 p=0 q=0 Chuck DiMarzio, Northeastern University +1 10351-4-33 Numerical Examples Beam Size, m -2 10 5 s=1m, s’=4cm q, Shape Factor 10 n=1.5 0 n=2.4 n=4 -5 -1 -0.5 0 p, Position Factor 0.5 10 10 -3 n=2.4 -4 -5 1 -6 10 -5 n=1.5 n=4 DL at 10 mm DL at 1.06 mm 500 nm 0 5 q, Shape Factor July 2003 Chuck DiMarzio, Northeastern University 10351-4-34 Phase Description of Aberrations v y Image Object z Entrance Pupil Exit Pupil • Mapping from object space to image space • Phase changes introduced in pupil plane – Different in different parts of plane – Can change mapping or blur images July 2003 Chuck DiMarzio, Northeastern University 10351-4-35 Coordinates for Phase Analysis v y v D z phase v 2a ra Solid Line is phase of a spherical wave toward the image point. Dotted line is actual phase. Our goal is to find D(r,f,h). y u h 0<r<1 July 2003 Chuck DiMarzio, Northeastern University x 10351-4-36 Aberration Terms Odd Terms involve tilt, not considered here. D r cos f July 2003 Chuck DiMarzio, Northeastern University 10351-4-37 Second Order Image Position Terms: The spherical wave is approximated by a second-order phase term, so this error is simply a change in focal length. July 2003 Chuck DiMarzio, Northeastern University 10351-4-38 Fourth Order (1) r2 is focus: depends on h2 and h2cosf Spherical Aberration Astigmatism and Field Curvature Tangential Plane S T Sagittal Plane Sample Images At T At S July 2003 Chuck DiMarzio, Northeastern University 10351-4-39 Fourth Order (2) r2cosf is Tilt: Depends on Coma h3 y Object Barrel Distortion Pincushion Distortion v x u July 2003 Chuck DiMarzio, Northeastern University 10351-4-40 Example: Mirrors By Fermat’s Principle, r1+r2 = constant v r1 r2 z Result is an Ellipse with these foci. One focus at infinity requires a parabola Foci co-located requires a sphere. July 2003 Chuck DiMarzio, Northeastern University 10351-4-41 Mirrors for Focussing v, cm. 0.1 0.05 Sphere Parabola Ellipse 0 Note: These Mirrors have a very large NA to illustrate the errors. In most mirrors, the errors could not be seen on this scale. s=30 cm, s’=6cm -0.05 -0.1 -0.1 July 2003 -0.08 -0.06 -0.04 -0.02 0 Chuck DiMarzio, Northeastern University z, cm. 10351-4-42 Numerical Example July 2003 Chuck DiMarzio, Northeastern University 10351-4-43 Optical Design Process July 2003 Chuck DiMarzio, Northeastern University 10351-4-44 Ray Tracing Fundamentals July 2003 Chuck DiMarzio, Northeastern University 10351-4-45 Ray Tracing (1) July 2003 Chuck DiMarzio, Northeastern University 10351-4-46 Ray Tracing (2) July 2003 Chuck DiMarzio, Northeastern University 10351-4-47 If One Element Doesn’t Work... “Let George Do It” Add Another Lens Different Index? Smaller angles with higher index. Thus germanium is better than ZnSe in IR. Not much hope in the visible. July 2003 Aspherics Chuck DiMarzio, Northeastern University 10351-4-48 Summary of Concepts So Far • • • • Paraxial Optics with Thin Lenses Thick Lenses (Principal Planes) Apertures: Pupils and Windows Aberration Correction – Analytical – Ray Tracing • What’s Missing? Wave Optics July 2003 Chuck DiMarzio, Northeastern University 10351-4-49