4-1 Sectio 4 Imagig ad Paraxial Optics
Optical Sstems A optical sstem is a collectio of optical elemets (leses ad mirrors). While the optical sstem ca cotai multiple optical elemets, the first order properties of the sstem are characteried b a sigle focal legth or magificatio. 4-2 leses.eiss.com
4-3 tf.ui-kiel.de oddstuffmagaie.com Optical Sstems base24.com
Mappigs A mappig is a coversio or trasformatio from oe represetatio to aother represetatio. 4-4 progoos.com
Color Mappigs Color to Grascale: False-color: I mapped to a visible color Wikipedia Pseudo-color: stackoverflow.com 4-5 Differet tissues are displaed as differet colors.
Imagig as a Mappig First-order optics is the optics of perfect imagig sstems. Aberratios are igored. The object is mapped or trasformed to its image. The process ca be aaled without kowig the details of the optical sstem. A small umber of sstem properties will completel defie ad determie the mappig or first-order imagig properties. These are kow as the cardial poits of the imagig sstem. 4-6 This is a mappig from object space to image space. Image from: supercolorig.com
Imagig Goal: Determie the sie, orietatio ad locatio of a image for a give object. This is the optics of perfect imagig sstems; aberratios are igored. Colliear Trasformatio A geeralied mappig from oe space ito aother i which poits map to poits, lies map to lies, ad plaes map to plaes. Gaussia Imager A specific colliear trasformatio usig assumptios that are appropriate for optical sstems. The cardial poits result. First-Order Optics The actual ra paths through a sstem ca be expaded i a power series of heights ad agles. A axiall smmetric sstem will have ol odd power terms, ad the first-order terms give the positio ad sie of the image. First-order optics is the optics of perfect optical sstems. The deviatios from this perfectio are the sstem aberratios. Paraxial Optics A method of determiig the first-order properties of a optical sstem b tracig ras usig the slopes of the ras istead of the ra agles. The agles of icidece ad refractio or reflectio at surfaces are assumed to be small. The sag* of the refractig or reflectig surface is igored or is cosidered to be egligible compared to other distaces. 4-7 Paraxial aalsis is also useful for relatig the phsical properties of a refractig or reflectig surface (curvature ad idex) to its Gaussia properties (focal legth ad cardial poits). Fortuatel, all of these theories are cosistet ad give the same results. *The sag of a surface is the separatio of the surface from a plae taget to the surface vertex.
Optical Spaces Each time a refractig or reflectig surface is ecoutered, a ew optical space is etered. Each space exteds from to ad has a associated idex. If a sstem has N surfaces, there will be N+1 optical spaces. The first space is usuall called object space ad the last is called image space. Surface: Object Space 1 1 2 3 4 5 2 3 4 5 6 1 1' 2 2' 3 3' 4 4' 5 5' 0 0' = I Image Space 4-8 A sigle object or image exists i each space. Each surface has a object space ad image space of its ow. These itermediate optical spaces serve as the image space for the precedig surface ad the object space for the followig surface. Each surface will form a image of the object i the precedig space. There are real ad virtual segmets of each optical space. The real segmet of a optical space is the volume betwee surfaces defiig etr ad exit ito that space.
eal ad Virtual as ca be traced from optical space to optical space. Withi a optical space, a ra is straight ad exteds from to with real ad virtual segmets. as from adjoiig spaces meet at the commo optical surface. 1 2 A real object is to the left of the surface; a virtual object is to the right of the surface. A real image is to the right of the surface; a virtual image is to the left of the surface. I a optical space with a egative idex (light propagates from right to left), left ad right are reversed i these descriptios of real ad virtual. 4-9 Alterate descriptio: If a object is upstream of the surface or sstem, it is cosidered to be real. A dowstream object is virtual. Images dowstream of the surface or sstem are real; images upstream are virtual. It is also commo to combie multiple optical surfaces ito a sigle sstem elemet ad ol cosider the object ad image spaces of the elemet; the itermediate spaces withi the elemet are the igored.
eal ad Virtual Cofusio The distictios betwee real ad virtual ma become cofused whe discussig itermediate optical spaces. eal ad virtual ca be discussed either from the perspective of a sigle surface or from the perspective of the sstem. For example, it is commo for the real image created b oe surface to serve as a virtual object for the ext surface. 1 The real image formed b Surface 1 becomes virtual due to the presece of Surface 2, ad this image serves as the virtual object for Surface 2. 4-10 1 2 3 4 I a similar maer, the virtual image produced b Surface 3 ca be cosidered to be a real object for Surface 4.
Geeral Sstem Cosider a black-box model of a optical sstem. A optical sstem is a collectio of optical elemets (leses, mirrors, etc.) comprisig a rotatioall smmetric optical sstem. A ra from a object at ifiit will emerge from the sstem ad go through the ear Focal Poit of the sstem F': F 4-11
Plaes of Effective efractio B extedig the image space ra back to the height of the object space ra, the plae of effective refractio ito image space for the sstem is foud. All of the refractios at the idividual elemets withi the sstem are combied ito a sigle effective refractio. P F 4-12 f This plae of effective refractio ito image space is called the ear Pricipal Plae P'. The distace from the ear Pricipal Plae to the ear Focal Poit is the ear Focal Legth of the sstem f.
Plaes of Effective efractio I a similar maer, the plae of effective refractio out of object space ca be foud b usig a ra startig at the Frot Focal Poit of the sstem F. The image ra is parallel to the axis. F f F P 4-13 This plae of effective refractio out of object space is called the Frot Pricipal Plae P. The distace from the Frot Pricipal Plae to the Frot Focal Poit is the Frot Focal Legth of the sstem. f F
Plaes of Effective efractio For the sstem used at fiite cojugates, the ra from the object poit will appear to refract at the Frot Priciple Plae ad emerge from the ear Pricipal Plae at the same height. The image poit is located where this ra crosses the axis. O P P O 4-14 The object ad image distaces ad ' are measured from the respective Pricipal Plaes. This treatmet allows the sstem to be cosidered to be just like a thi les except the refractio occurs at separated plaes of effective refractio.
Paraxial Optics I order to relate the phsical parameters of a optical sstem (radii of curvature of surfaces, spacigs ad thickesses) to its imagig properties, ras must be traced through the sstem usig Sell s law or the law of reflectio. While exact ratracig ca be used, the first-order or imagig properties of the sstem ca be foud usig the approximate ra paths computed b paraxial optics. as are traced usig the slopes of the ras istead of the ra agles. The amout a ra is bet at the refractig or reflectig surface is assumed to be small. The sag of the refractig or reflectig surface is igored or is cosidered to be egligible compared to other distaces. The radius of curvature of a surface is defied to be the distace from its vertex to its ceter of curvature CC. The sag of a surface is measured relative to a plae taget to the surface vertex V. The sag will var with the radial positio o the surface. Sag V Surface CC 4-15 Ceter of Curvature
Sigle efractig Surface Cosider a sigle refractig surface with a radius of curvature of that separates a idex of refractio from ad idex of refractio '. The agles of icidece ad refractio (I ad I') are measured with respect to the surface ormal. The ra agles U ad U', as well as the elevatio agle A of the surface ormal at the ra itersectio are measured with respect to the optical axis. The usual sig covetios appl. O U 1 Curvature C I V Sag efractig Surface I' A = 1/C CC U' ' Ceter of Curvature O' 4-16
Sigle efractig Surface U U O A V I A Sag efractig Surface I' A = 1/C U' CC U' 1 Curvature C ' Ceter of Curvature O' 4-17 elatig the agles at the ra itersectio with the surface: A U I I U A I U A Appl Sell s Law: si I si I si si U A U A
Sigle efractig Surface ad Sell s Law si I si I si si U A U A si cos cos si si cos cos si U A U A U A U A si A si A siu cosu siucosu cos A cos A si cos ta si cos ta U U A U U A 4-18 Approximatio #1: cosu cosu siu siu ta A ta A cosu cosu ta ta ta ta U A U A Approximatio #1 implies that the ra bedig at the surface is small.
Paraxial Agles ta ta ta ta U A U A Defie the paraxial agles: u tau utau ta A u u u u 4-19 Paraxial agles are the tagets of the real ra agles ad are the slopes of the ras. Paraxial agles are ot agles ad are icorrectl called agles ol because of traditio. Paraxial agles are ra slopes. Paraxial agles are uitless.
Power of a Surface ad Paraxial atrace Equatio u u ta A Sag Approximatio #2: Sag u u uc V Sag ' A = 1/C CC Approximatio #2 implies that the sag of the surface at the ra itersectio is much less tha the radius of curvature of the surface. 4-20 Defie the Power of the surface: u u C This is the Paraxial atrace Equatio. Note that the power of the surface depeds ol o the costructio parameters (,, ') of the surface. It is ra idepedet.
Object ad Image distaces for a Sigle efractig Surface The object ad image distaces ( ad ') are also both measured from the surface vertex. O U Vertex Plae V Sag efractig Surface A CC U' ' O' 4-21 ' Approximatio #3: The object ad image distaces are much greater tha the sag of the surface at the ra itersectio. Sag Sag u tau Sag u tau Sag
Surface Vertex Plae ad Pricipal Plaes Approximatios #2 ad #3 state that the sag is cosidered to be small with respect to the radius of curvature of the surface ad with respect to the object ad image distaces. Sice these coditios are ol trul met for small ra heights o the surface, this method of aalsis is called paraxial optics or paraxial ratracig. Paraxial meas ear axis. B igorig the surface sag i paraxial optics, the plaes of effective refractio for the sigle refractig surface are located at the surface vertex plae V. The Frot ad ear Pricipal Plaes (P ad P') of the surface are both located at the surface. Here, the paraxial agles u ad u' are show with the refractio at the vertex plae. The paraxial ras show approximate the actual ras as the surface sag is igored. efractio is cosidered to occur at the surface vertex. Vertex Plae efractig Surface ' 4-22 O u u u P V P' ' u' O' This is the paraxial model of refractio at a surface with power.
Imagig ad Focal Legths of a Sigle efractig Surface u u- Object at Ifiit Image at the ear Focal Poit u u ad ' are measured from the surface vertex or the Pricipal Plaes. Image at Ifiit Object at the Frot Focal Poit 4-23 f f F f f f f F F f f F f f F
Object-Image elatioship ad the Focal legth f f F Defie the THE Focal Legth, also called the Effective (or Equivalet) Focal Legth EFL: 1 ff f fe 1 f E 1 f f Note that the focal legth f is the reduced frot ad rear focal legths. 4-24 The effective or equivalet i EFL is actuall uecessar. There is a sigle focal legth f. The frot ad rear focal legths are related to the focal legth through the idices of refractio: ff fe f f fe f
Surface Sag Sag Circle (or Sphere): 2 Sag 2 2 The surface sag is the measure of the surface shape relative to a plae. CC 4-25 2 Sag Sag 2 2 2 2 2 Sag 2 2 2 Sag 0 2 Sag 2 This is the parabolic approximatio for a circle or sphere.
adius of Curvature Measuremet The surface sag over a fixed baselie is a commo method of measurig radius of curvature i the optics shop. The istrumet is kow as a Les Clock or a Geeva Gauge. It cosists of three pis that cotact the surface. The outside two pis are fixed, separated b a distace D, ad defie a referece lie. The middle pi is sprig loaded ad coected to a displacemet gauge. Sag D = Separatio of Fixed Pis Displacemet Gauge Sag 2 D 8 2 D 8Sag 4-26 Surface eferece Lie Defied b Fixed Pis The gauge ofte reads i Diopters ad assumes a specific idex of refractio. A covex surface reads a positive power ad a cocave surface reads a egative power. A spherometer is a more precise istrumet for measurig adius of Curvature. The surface is cotacted with three fixed pis or a large rig. A micrometer i the ceter reads the sag of the surface relative to the plae defied b the three pis or the rig.
Approximatio Summar for Paraxial Optics Approximatio #1: Cosie Coditio cosu cosu cosu or 1 cosu This coditio is met if the ra bedig at a surface is small. This will occur whe the icidet ra is approximatel perpedicular to the surface at the ra itersectio. Note that this approximatio does ot require that the ra agles U ad U' are small. Approximatio #2: Sag This coditio requires that the sag of the surface at the ra itersectio is much less tha the radius of curvature of the surface. 4-27 Approximatio #3: Sag Sag The object ad image distaces are much greater tha the sag of the surface at the ra itersectio. The surface sag is igored ad paraxial refractio occurs at the surface vertex. The ra bedig at each surface is small.
Approximatio Summar for Paraxial Optics Approximatio #1: Cosie Coditio Cosider a surface perpedicular to the optical axis: 1.0 1.5 U For For U 10 deg ' U' A 0 I U The error is about 1% U 20 deg cosu cosu cosu or 1 cosu The error is about 3.5% cosu'/cosu 1.04 1.03 1.02 1.01 1 Surface Normal 4-28 0.99-30 -20-10 0 10 20 30 U (degrees)
Approximatio Summar for Paraxial Optics Cosie Coditio: The situatio of small ra bedig at each surface is commo i welldesig sstems. The ras are getl guided through the sstem. Double Gauss photographic objective with 6 elemets 4-29 Lithograph les with 30 elemets US Patet 5,835,285 Niko Corporatio
Approximatio Summar for Paraxial Optics Approximatio #2 states that the sag of the surface at the ra itersectio is much less tha the radius of curvature of the surface. Approximatio #3 states that the object ad image distaces are much greater tha the sag of the surface at the ra itersectio. The object ad image distaces for a ra are the distaces to the ra crossigs at the optical axis. Cosider: Sag Sag Sag 100 mm 10 mm 2 Sag 0.5 mm 2 1.0 1.5 ' 4-30 For a icidet ra parallel to the axis: f 300 mmsag V Sag A CC = 1/C
Paraxial atrace Thi Les h u u u 1 u 1 u f Appl the imagig equatio for a thi les: u h 1 1 1 f u u 1 f 4-31 Let f f E 1 u u f u u This is the paraxial ratrace equatio of a refractig surface where = = 1 u u
Geeral Sstem The refractive properties of a geeral sstem ca be used to defie its sstem focal legth f or sstem power. For a arbitrar optical sstem, the sstem power or sstem focal legth f is determied so that the sstem obes the paraxial refractio equatio. The sstem is treated as if it were a sigle refractig surface where the pricipal plaes of the sstem are the plaes of effective refractio. 4-32 h u u F P P ff f F h u u 1 f f E The paraxial ratrace equatio becomes the defiitio of the sstem power ad focal legth
ear Focal Legth of a Geeral Sstem Trace a ra parallel to the axis. This correspods to a object at ifiit located o the optical axis. The cojugate ra crosses the axis at the rear focal poit. u 0 u Appl the paraxial refractio equatio to the sstem: P P f F 4-33 u u u 0 u u u u f f Equate: f f 1 f fe This is exactl the same relatioship as for a sigle refractig surface.
Frot Focal Legth of a Geeral Sstem Trace a ra from a object at the frot focal poit. The cojugate ra is parallel to the optical axis i image space. This correspods to a image at ifiit located o the optical axis. F u f F Appl the paraxial refractio equatio to the sstem: P P u 0 4-34 u u u 0 u u u u f f F F Equate: f F f 1 f f F E This is also exactl the same relatioship as for a sigle refractig surface.
Paraxial atrace ad Imagig for a Geeral Sstem h u u F P P u Appl the paraxial refractio equatio that defies the sstem power:: ff f F u h 4-35 u u f f E 1 1 f E This is the same relatioship as was determied for a sigle refractig surface.
Focal Legths of a Geeral Sstem epeatig the results for the frot ad rear focal legths: f f ff 1 f 1 f ff f fe f The focal legth is the reduced rear focal legth ad mius the reduced frot focal legth. I geeral, the focal legth is ot a phsical distace. The frot ad rear focal legths are phsical distaces. The are the directed or siged distaces from the Pricipal Plaes to the respective Focal Poits. f f F 4-36 These are exactl the same relatioships betwee power ad focal legth as foud for a sigle refractig surface, ad the ca be applied to a optical sstem. The pricipal plaes are the effective plaes of refractio i object space ad image space. For a refractive sstem i air: 1 f ff f 1
Commets o Paraxial Optics There is a oe-to-oe correspodece betwee object ad image poits i paraxial optics. This is just a wa of saig that there are o aberratios i paraxial or first-order optics it is the optics of perfect imagig sstems. A paraxial ratrace is liear with respect to ra agles ad heights sice all paraxial agles are defied to be the taget of the actual ra agle the are actuall the slope of the ra. I a paraxial aalsis, these approximatios (ad the resultig liearit) are maitaied eve for large object ad image heights ad large ra agles. 4-37