Lenses and Imaging (Part I) Parabloid mirror: perfect focusing

Transcription

1 Leses ad Imagig (Part I) eview: paraboloid reflector, focusig Why is imagig ecessary: Huyges priciple Spherical & parallel ray budles, poits at ifiity efractio at spherical surfaces (paraial approimatio) Optical power of surfaces: positive, egative surfaces Paraial ray-tracig atri formulatio of paraial geometrical optics Imagig coditio Object/image at ifiity 9/4/5 wk-b- Parabloid mirror: perfect focusig (e.g. satellite dish) s() f: focal legth z f s ( ) 4 f 9/4/5 wk-b-

2 Les: mai istrumet of image formatio air glass air optical ais Poit source (object) Poit image 9/4/5 wk-b-3 The curved surface makes the rays bed proportioally to their distace from the optical ais, accordig to Sell s law. Therefore, the diverget wavefrot becomes coverget at the right-had (output) side. Why are focusig istrumets ecessary? ay budles: spherical waves ad plae waves Poit sources ad poit images Huyges priciple ad why we ca see aroud us The role of classical imagig systems 9/4/5 wk-b-4

3 ay budles poit source spherical wave (divergig) wave-frot rays poit source very-very far away wave-frot rays plae wave 9/4/5 wk-b-5 Huyges priciple Each poit o the wavefrot acts as a secodary light source emittig a spherical wave The wavefrot after a short propagatio distace is the result of superimposig all these spherical wavelets 9/4/5 wk-b-6 optical wavefrots 3

4 Why are focusig istrumets ecessary? icidet light... is scattered by the object object 9/4/5 wk-b-7 Why are focusig istrumets ecessary? icidet light... is scattered by the object object eed optics to udo the effects of scatterig, i.e. focus the light image 9/4/5 wk-b-8 4

5 Ideal les air glass air optical ais Poit sources (object) Poit images Each poit source from the object plae focuses oto a poit image at the image plae; NOTE the image iversio 9/4/5 wk-b-9 Summary: Why are imagig systems eeded? Each poit i a object scatters the icidet illumiatio ito a spherical wave, accordig to the Huyges priciple. A few micros away from the object surface, the rays emaatig from all object poits become etagled, delocalizig object details. To relocalize object details, a method must be foud to reassig ( focus ) all the rays that emaated from a sigle poit object ito aother poit i space (the image. ) The latter fuctio is the topic of the disciplie of Optical Imagig.? 9/4/5 wk-b- 5

6 The ideal optical imagig system optical elemets 9/4/5 wk-b- object Ideal imagig system: each poit i the object is mapped oto a sigle poit i the image eal imagig systems itroduce blur... image Focus, defocus ad blur Perfect focus efocus 9/4/5 wk-b- 6

7 Focus, defocus ad blur Perfect focus Imperfect focus spherical aberratio 9/4/5 wk-b-3 Why optical systems do ot focus perfectly iffractio Aberratios However, i the paraial approimatio to Geometrical Optics that we are about to embark upo, optical systems do focus perfectly To deal with aberratios, we eed o-paraial Geometrical Optics (higher order approimatios) To deal with diffractio, we eed Wave Optics 9/4/5 wk-b-4 7

8 Ideal les air glass air optical ais Poit sources (object) Poit images Each poit source from the object plae focuses oto a poit image at the image plae 9/4/5 wk-b-5 efractio at sigle spherical surface for each ray, must calculate poit of itersectio with sphere agle betwee ray ad ormal to surface apply Sell s law to fid directio of propagatio of refracted ray : radius of spherical surface medium ide, e.g. air ceter of spherical surface medium ide, e.g. glass.5 9/4/5 wk-b-6 8

9 9/4/5 wk-b-7 Paraial approimatio / I paraial optics, we make heavy use of the followig approimate ( st order Taylor) epressios: si ε ε taε ε ε cosε where ε is the agle betwee a ray ad the optical ais, ad is a small umber (ε rad). The rage of validity of this approimatio typically eteds up to ~-3 degrees, depedig o the desired degree of accuracy. This regime is also kow as Gaussia optics or paraial optics. Note the assumptio of eistece of a optical ais (i.e., perfect aligmet!) Paraial approimatio / Igore the distace betwee the locatio of the aial ray itersectio ad the actual off-ais ray itersectio Apply Sell s law as if ray bedig occurred at the itersectio of the aial ray with the les off-ais ray Valid for small curvatures & thi optical elemets aial ray 9/4/5 wk-b-8 9

10 efractio at spherical surface efractio at positive spherical surface: ( ) : ray height : ray directio : radius of curvature off-ais ray optical ais 9/4/5 wk-b-9 efractio at spherical surface efractio at positive spherical surface: ( ) : ray height : ray directio : radius of curvature off-ais ray Power optical ais 9/4/5 wk-b-

11 Propagatio i uiform space Propagatio through distace : : ray height : ray directio off-ais ray optical ais 9/4/5 wk-b- Sig covetios for refractio Light travels from left to right A radius of curvature is positive if the surface is cove towards the left Logitudial distaces are positive if poitig to the right Lateral distaces are positive if poitig up ay agles are positive if the ray directio is obtaied by rotatig the z ais couterclockwise through a acute agle optical ais z 9/4/5 wk-b-

12 The power of surfaces Positive power beds rays iwards (covergig budle) > Simple spherical refractor (positive) P ( right left ) ( ) ( ) > Negative power beds rays outwards (divergig budle) 9/4/5 wk-b-3 < Simple spherical refractor (egative) P ( right left ) ( ) ( ) < The power of surfaces Positive power beds rays iwards (covergig budle) < Simple spherical refractor (positive) P ( right left ) ( ) ( ) > Negative power beds rays outwards (divergig budle) 9/4/5 wk-b-4 > Simple spherical refractor (egative) P ( right left ) ( ) ( ) <

13 Paraial ray-tracig air glass air Free-space propagatio Free-space propagatio 9/4/5 wk-b-5 efractio at air-glass iterface Free-space propagatio efractio at glass-air iterface Eample: oe spherical surface, traslatiorefractiotraslatio : radius of spherical surface 9/4/5 wk-b-6 No-paraial ray (approimatio gives large error) medium ide, e.g. air Paraial rays (approimatio valid) ceter of spherical surface medium ide, e.g. glass.5 3

14 Traslatiorefractiotraslatio / Startig ray: locatio Traslatio through distace 9/4/5 wk-b-7 directio ( directio): efractio at positive spherical surface: ( ) Traslatiorefractiotraslatio / Traslatio through distace Put together: ( directio): 9/4/5 wk-b-8 4

15 Traslatiorefractiotraslatio /3 ( ) ( ) ( ) 9/4/5 wk-b-9 (Paraial) ay-tracig i geeral i out out i i OPTICAL SYSTE out 9/4/5 wk-b-3 For a arbitrary ray eterig with lateral departure displacemet i wrt the optical ais agle of departure i wrt the optical ais i a departure space of refractive ide i determie the ray s lateral arrival displacemet out wrt the optical ais agle of arrival out wrt the optical ais i a arrival space of refractive ide out upo eitig the optical system 5

16 6 9/4/5 wk-b-3 atri formulatio / ( ) m m f refractio by surface with radius of curvature ray-tracig object-image trasformatio traslatio by distace form commo to all i i out i i out 9/4/5 wk-b-3 atri formulatio / i i out i i out i i i out out out ( ) efractio by spherical surface Traslatio through uiform medium Power

17 Traslatiorefractiotraslatio traslatio refractio traslatio by by r.curv. by 9/4/5 wk-b-33 result ( ) ( ) ( ) atri ray-tracig i geeral i out out i i OPTICAL SYSTE out out out out ii i 9/4/5 wk-b-34 is the system matri, foud as the product, i reverse order, of the matrices of the elemets (free space, spherical surfaces) comprisig the system 7

18 O-ais image formatio Poit object Poit image 9/4/5 wk-b-35 O-ais image formatio All rays emaatig at arrive at irrespective of departure agle ( ) [ L] Imagig coditio 9/4/5 wk-b-36 8

19 O-ais image formatio All rays emaatig at arrive at irrespective of departure agle Imagig coditio Power of the spherical - surface [uits: diopters, m ] 9/4/5 wk-b-37 Off-ais image formatio Poit object (off-ais) optical ais Poit image 9/4/5 wk-b-38 9

20 agificatio: lateral (off-ais), agle Lateral m... ' Agle m 9/4/5 wk-b-39 Object-image trasformatio m Paraial ray-tracig f trasformatio betwee m object ad image poits subject to (imagig coditio) 9/4/5 wk-b-4

21 atri ray-tracig: summary i out out i i OPTICAL SYSTE out 9/4/5 wk-b-4 out out Power out ii i Imagig coditio Lateral magificatio Agular magificatio Image of poit object at ifiity Poit image 9/4/5 wk-b-4

22 object at 9/4/5 wk-b-43 Image of poit object at ifiity Note: f ambiet refractive ide at space of poit image image f :image focal legth /Power Poit object imaged at ifiity Poit object 9/4/5 wk-b-44

23 Poit object imaged at ifiity image at object 9/4/5 wk-b-45 Note: f ambiet refractive ide at space of poit object f : object focal legth /Power Image / object focal legths Poit image Poit object 9/4/5 wk-b-46 Object at f Power f Power Image at 3