July © Chuck DiMarzio, Northeastern University ECEG105/ECEU646 Optics for Engineers Course Notes Part 4: Apertures, Aberrations Prof. Charles A. DiMarzio Northeastern University Fall 2008 Aug 2007Sept 2008

July © Chuck DiMarzio, Northeastern University Advanced Geometric Optics Introduction Stops, Pupils, and Windows f-Number Examples –Magnifier – Microscope Aberrations Design Process –From Concept through Ray Tracing –Finalizing the Design –Fabrication and Alignment

July © Chuck DiMarzio, Northeastern University 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(  )=tan(  )=  –cos(  )=1

July © Chuck DiMarzio, Northeastern University What We Have Developed Description of an Optical System in terms of Principal Planes, Focal Length, and Indices of Refraction These equations describe a mapping –from object space (x,y,z) –to image space (x’,y’,z’) s, s’ are z coordinates H H’ VV’ B B’ Sept 2007Sept 2008

July © Chuck DiMarzio, Northeastern University Lens Equation as Mapping 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 L1L1 L2L2 L3L3 L4L4 L4’L4’ Sept 2008

July © Chuck DiMarzio, Northeastern University Stops, Pupils, and Windows (1) Intuitive Description –Pupil Limits Amount of Light Collected –Window Limits What Can Be Seen Pupil Window

July © Chuck DiMarzio, Northeastern University Stops, Pupils and Windows (2) 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 Physical Components Images in Object Space Entrance Pupil Limits Cone of Rays from Object Entrance Window Limits Cone of Rays From Entrance Pupil Images in Image Space Exit Pupil Limits Cone of Rays from Image Exit Window Limits Cone of Rays From Exit Pupil Sept 2008

July © Chuck DiMarzio, Northeastern University Finding the Entrance Pupil Find all apertures in object space L1L1 L2L2 L3L3 L4L4 L 4 ’ is L 4 seen through L 1 -L 3 L 3 ’ is L 3 seen through L 1 -L 2 Entrance Pupil Subtends Smallest Angle from Object L1L1 L2’L2’ L4’L4’L3’L3’

July © Chuck DiMarzio, Northeastern University Finding the Entrance Window Entrance Window Subtends Smallest Angle from Entrance Pupil 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 L1L1 L2’L2’ L4’L4’L3’L3’ Sept 2008

July © Chuck DiMarzio, Northeastern University Field of View Film= Exit Window f=28 mm f=55 mm f=200 mm

July © Chuck DiMarzio, Northeastern University Example: The Telescope Aperture Stop Field Stop

July © Chuck DiMarzio, Northeastern University The Telescope in Object Space Primary Secondary Secondary’ Entrance Pupil Entrance Window

July © Chuck DiMarzio, Northeastern University The Telescope in Image Space Primary Secondary Primary’’ Exit Pupil Exit Window Stopped 26 Sep 03

July © Chuck DiMarzio, Northeastern University f-Number & Numerical Aperture F’ F A A’ f D is Lens Diameter  NA, Numerical Aperture f#, f-number f-Number Numerical Aperture

July © Chuck DiMarzio, Northeastern University Importance of Aperture ``Fast’’ System –Low f-number, High NA (NA  1, f#  1) –Good Light Collection (can use short exposure) –Small Diffraction Limit ( /D) –Propensity for Aberrations (sin  Corrections may require multiple elements –Big Diameter  Big Thickness  Weight, Cost Tight Tolerance over Large Area Sept 2008

July © Chuck DiMarzio, Northeastern University The Simple Magnifier F F’ AA’

July © Chuck DiMarzio, Northeastern University 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#  Means f=10 mm Maximum M m =25 For the Interested Student: What if s>f ? Sept 2008

July © Chuck DiMarzio, Northeastern University Microscope Two-Step Magnification –Objective Makes a Real Image –Eyepiece Used as a Simple Magnifier F’ F A’ A F’ F

July © Chuck DiMarzio, Northeastern University Microscope Objective F’ F A’ A F’ F

July © Chuck DiMarzio, Northeastern University Microscope Eyepiece F’ F A2A2 A F A2’A2’

July © Chuck DiMarzio, Northeastern University Microscope Effective Lens A D’D F F’ H H’ A F H F’ H’ 192 mm 19.2 mm f 1 =16mm f 2 =16mm Barrel Length = 160 mm Effective Lens: f = -1.6 mm

July © Chuck DiMarzio, Northeastern University Microscope Aperture Stop F’ F Aperture Stop =Entrance Pupil Exit 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 Image Analysis in Image Space Stopped here 30 Sep 03 Sept 2008

July © Chuck DiMarzio, Northeastern University Microscope Field Stop F’ F Field Stop = Exit Window Entrance Window

July © Chuck DiMarzio, Northeastern University Microscope Effective Lens

July © Chuck DiMarzio, Northeastern University Apertures Summary Object and Image Space Locating All the Elements Finding the Pupil –Computing the Pupil Size and NA or f# Finding the Window –Computing the Field of View Dec 2004

July © Chuck DiMarzio, Northeastern University Aberrations Failure of Paraxial Optics Assumptions –Ray Optics Based On sin(  )=tan(  )=  –Spherical Waves  =  0 +2  x 2 /  Next Level of Complexity –Ray Approach: sin(  )=    –Wave Approach:  =  0 +2  x 2 /  c    A Further Level of Complexity –Ray Tracing

July © Chuck DiMarzio, Northeastern University Examples of Aberrations (1) R = 2, n=1.00, n’=1.50 s=10, s’=10 Paraxial Imaging In this example for a ray having height h at the surface, s’(h)<s’(0). m4061_3

July © Chuck DiMarzio, Northeastern University Example of Aberrations (2) m4061_3  z(h=0.6)  z(h=1.0) Longitudinal Aberration =  z Transverse Aberration =  x  x(h=1.0) Where Exactly is the image? What is its diameter?

July © Chuck DiMarzio, Northeastern University Spherical Aberration Thin Lens in Air

July © Chuck DiMarzio, Northeastern University Transverse Spherical Aberration ss xx h s(0)

July © Chuck DiMarzio, Northeastern University Evaluating Transverse SA

July © Chuck DiMarzio, Northeastern University Coddington Shape Factors q=0 +1 p=0 +1

July © Chuck DiMarzio, Northeastern University Numerical Examples Beam Size, m q, Shape Factor s=1m, s’=4cm DL at 10  m n=2.4n=1.5 n=4 DL at 1.06  m 500 nm p, Position Factor q, Shape Factor n=4 n=2.4 n=1.5

July © Chuck DiMarzio, Northeastern University Phase Description of Aberrations Entrance Pupil Exit Pupil Object Image z y v 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 Sept 2008

July © Chuck DiMarzio, Northeastern University Coordinates for Phase Analysis 2a aa 0<  <1 z y v v u y x h phase v  Solid Line is phase of a spherical wave toward the image point. Dotted line is actual phase. Our goal is to find  ( ,h).

July © Chuck DiMarzio, Northeastern University Aberration Terms Odd Terms involve tilt, not considered here.

July © Chuck DiMarzio, Northeastern University 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 © Chuck DiMarzio, Northeastern University Fourth Order (1) Spherical Aberration Astigmatism and Field Curvature S Sagittal Plane Tangential Plane T Sample Images At T At S  2 is focus: depends on h 2 and h 2 cos  Sept 2008

July © Chuck DiMarzio, Northeastern University Fourth Order (2) Barrel Distortion Pincushion Distortion y x Coma u v   cos  is Tilt: Depends on h 3 Sept 2008

July © Chuck DiMarzio, Northeastern University Optical Design Process

July © Chuck DiMarzio, Northeastern University Ray Tracing Fundamentals

July © Chuck DiMarzio, Northeastern University If One Element Doesn’t Work... Add Another Lens “Let George Do It” Aspherics Different Index? Smaller angles with higher index. Thus germanium is better than ZnSe in IR. Not much hope in the visible.

July © Chuck DiMarzio, Northeastern University Aberrations Summary Origin of Aberrations On-Axis Aberrations –Change of Focus –Spherical Aberration Off-Axis Aberrations –Additional Blurring Effects –Distorting Effects Dec 2004

July © Chuck DiMarzio, Northeastern University 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