optical systems images and pupils rays wavefronts...

Lecture 4: Geometrical Optics 2

Outline

1 Optical Systems2 Images and Pupils3 Rays4 Wavefronts5 Aberrations

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 1

Optical Systems

Overview

combinations of several optical elements (lenses, mirrors, stops)examples: camera “lens”, microscope, telescopes, instrumentsthin-lens combinations can be treated analyticallyeffective focal length: 1

f = 1f1+ 1

f2

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 2

Simple Thin-Lens Combinations

distance > sum of focal lengths⇒ real image between lensesapply single-lens equation successively

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 3

Thin-Lens Combinations 1

construct image formed by lens 1 using rays 2 and 3ray 2 passes through focal point Fi1

ray 3 passes through focal point Fo1

ray 4 passes backwards through center of lens 2

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 4

Thin-Lens Combinations 2

adding lens 2 does not refract ray 4ray 3 is refracted to image focus Fi2

intersection of rays 3 and 4 determine image locationlens 2 adds convergence or divergence

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 5

Second Lens Adds Convergence or Divergence

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 6

F-number and Numerical Aperture

Aperture

all optical systems have a place where ’aperture’ is limitedmain mirror of telescopesaperture stop in photographic lensesaperture typically has a maximum diameteraperture size is important for diffraction effects

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 7

F-number

f/2: f/4:

describes the light-gathering ability of the lensf-number given by F = f/Dalso called focal ratio or f-ratio, written as: f/Fthe bigger F , the better the paraxial approximation worksfast system for F < 2, slow system for F > 2

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 8

F-number on Camera Lens

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 9

Numerical Aperture

en.wikipedia.org/wiki/File:Numerical_aperture.svg

numerical aperture (NA): n sin θn index of refraction of working mediumθ half-angle of maximum cone of light that can enter or exit lensimportant for microscope objectives (n often not 1)

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 10

Numerical Aperture in Fibers

en.wikipedia.org/wiki/File:OF-na.svg

acceptance cone of the fiber determined by materials

NA = n sin θ =√

n21 − n2

2

n index of refraction of working medium

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 11

Ray Definitions

Planes and Raysmeridional plane definedby optical axis and chiefray going through center ofoptical systemsagittal plane isperpendicular to it

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 12

Meridional (or Tangential) Ray

confined to plane containingoptical axis and object pointfrom which ray originates

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 13

Chief (or Principal) Raygoes through center ofaperturemeridional ray that starts atedge of object, and passesthrough center of aperturestopcrosses optical axis atlocations of pupilschief rays are equivalent to therays in pinhole cameradistance between chief rayand optical axis at an imagelocation defines size of image

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 14

Skew Raydoes not propagate in planethat contains both object pointand optical axisdoes not cross optical axisanywhere, and not parallel to it

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 15

Marginal Rayis meridional ray that starts atpoint where object crossesoptical axis and touches edgeof aperture stopuseful because it crossesoptical axis again at locationswhere image is formeddistance of marginal ray fromoptical axis at entrance andexit pupils defines their sizes

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 16

Sagittal (or Transverse) Ray

comes from off-axis objectpoint, propagates in planeperpendicular to meridionalplaneintersects the pupil along aline that is perpendicular tomeridional planechief ray is both sagittal andmeridionalall other sagittal rays are skewrays

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 17

Paraxial Raymakes a small angle to theoptical axis of the systemlies close to the axisthroughout the systemcan be modeled reasonablywell by using the paraxialapproximation.

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 18

Images and Pupils

Converging, Diverging and Collimated beams

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 19

Images and Pupilsimage

every object point comes to a focus in an image planelight in one image point comes from pupil positionsobject information is encoded in position, not in angle

pupilall object rays are smeared out over complete aperturelight in one pupil point comes from different object positionsobject information is encoded in angle, not in position

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 20

Aperture and Field Stops

aperture stop limits the amount of light reaching the imageaperture stop determines light-gathering ability of optical systemfield stop limits the image size or angle

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 21

Entrance and Exit Pupils

pupil is an image of the aperture stopentrance pupil: image of the aperture stop as seen from a pointon the optical axis and on the object through optical elementspreceeding the aperture stopexit pupil: image of the aperture stop as seen from a point on theoptical axis and in the image through optical elements after theaperture stop

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 22

Entrance and Exit Pupils

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 23

Entrance and Exit Pupils

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 24

Vignetting

effective aperture stop depends on position in objectimage fades toward its edges

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 25

Telecentric Arrangement

Made with Touch Optical Design

Made with Touch Optical Design

as seen from image, pupil is at infininityeasy: lens is its focal length away from pupil (image)magnification does not change with focus positionsray cones for all image points have the same orientation

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 26

Aberrations

Spot Diagrams and Wavefronts

plane of least confusion is location where image of point sourcehas smallest diameterspot diagram: shows ray locations in plane of least confusionspot diagrams are closely connected with wavefrontsaberrations are deviations from spherical wavefront

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 27

Made with Touch Optical Design

Spherical Aberrationsdifferent focal lengthsof paraxial andmarginal rayslongitudinal sphericalaberration along opticalaxistransverse (or lateral)spherical aberration inimage planemuch morepronounced for shortfocal ratios

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 28

Minimizing Spherical Aberrations

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 29

Spherical Aberration of Spherical Lens

Made with Touch Optical Design

foci from paraxial beams are further away than marginal raysspot diagram shows central area with fainter disk around it

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 30

Spherical Aberration Spots and Waves

spot diagram shows central area with fainter disk around itwavefront has peak and turned-up edges

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 31

Aspheric Lens

conic constant K = −1−√

n makes perfect lensdifficult to manufacturebut possible these days

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 32

HST

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 33

Coma

typically seen for object points away from optical axisleads to ’tails’ on stars

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 34

Positive Coma

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 35

Coma

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 36

Coma Spots and Waves

parabolic mirror with perfect on-axis performancespots and wavefront for off-axis image pointswavefront is tilted in inner part

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 37

Astigmatism

image of a point forms focal lines at the sagittal and tangental fociin between an elliptical shape

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 38

Tilted Glass Plate in Converging Beam

astigmatism and spherical aberrationnote beam shifttilted plates: beam shifters, filters, beamsplitters

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 39

Astigmatism Spots and Waves

focus in two orthogonal directions, but not in both at the same timedifference of two parabolae with different curvatureswavefront has saddle shape

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 40

Field Curvature

field (Petzval) curvature: image lies on curved surfaceproblems with flat detectors (e.g. CCDs)solution: field flattening lens close to focus

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 41

Distortion

image is sharp but geometrically distorted(a) object(b) positive (or pincushion) distortion(c) negative (or barrel) distortion

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 42

Aperture Stop Creates Distortion

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 43

Aberration Descriptions

Seidel AberrationsLudwig von Seidel (1857)Taylor expansion of sinφ

sinφ = φ− φ3

3! +φ5

5! − ...paraxial: first-order opticsSeidel optics: third-order opticsSeidel aberrations: spherical, astigmatism, coma, field curvature,distortion

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 44

Zernike Polynomials

tip

coma (0deg)

tilt

coma (90deg)

focus

trefoil (0deg)

astigmatism(45 deg)

trefoil (30deg)

astigmatism0 deg

third-orderspherical

low orders equal Seidel aberrationsform orthonormal basis on unit circle

Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl Lecture 4: Geometrical Optics 2 45