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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
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