Lenses and Imaging (Part I)

Transcription

1 Leses ad Imagig (Part I) Why is imagig ecessary: Huyge s priciple Spherical & parallel ray budles, poits at ifiity efractio at spherical surfaces (paraial approimatio) Optical power ad imagig coditio Matri formulatio of geometrical optics The thi les Surfaces of positive/egative power eal ad virtual images 9/5/4 wk2-b-

2 Parabloid mirror: perfect focusig (e.g. satellite dish) s() f: focal legth z f s ( ) 2 4 f 9/5/4 wk2-b-2

3 Les: mai istrumet for image formatio air glass air optical ais Poit source (object) Poit image 9/5/4 wk2-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 (put) side.

4 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 9/5/4 wk2-b-4

5 ay budles poit source spherical wave (divergig) wave-frot rays poit source very-very far away wave-frot rays plae wave 9/5/4 wk2-b-5

6 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 9/5/4 wk2-b-6 optical wavefrots

7 Why are focusig istrumets ecessary? icidet light... is scattered by the object object 9/5/4 wk2-b-7

8 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 9/5/4 wk2-b-8

9 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 9/5/4 wk2-b-9

10 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.? 9/5/4 wk2-b-

11 The ideal optical imagig system optical elemets 9/5/4 wk2-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

12 Focus, defocus ad blur Perfect focus Defocus 9/5/4 wk2-b-2

13 Focus, defocus ad blur Perfect focus Imperfect focus spherical aberratio 9/5/4 wk2-b-3

14 Why optical systems do ot focus Diffractio Aberratios perfectly However, i the paraial approimatio to Geometrical Optics that we are ab 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 9/5/4 wk2-b-4

15 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 9/5/4 wk2-b-5

16 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 D D 2 medium 2 ide, e.g. glass.5 ceter of spherical surface 9/5/4 wk2-b-6

17 9/5/4 wk2-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 2 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!)

18 Paraial approimatio /2 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 9/5/4 wk2-b-8

19 efractio at spherical surface efractio at positive spherical surface: ( ) : ray height : ray directio : radius of curvature off-ais ray Power optical ais 9/5/4 wk2-b-9

20 Propagatio i uiform space Propagatio through distace D: + D : ray height : ray directio off-ais ray D optical ais 9/5/4 wk2-b-2

21 Paraial ray-tracig air glass air Free-space propagatio Free-space propagatio 9/5/4 wk2-b-2 efractio at air-glass iterface Free-space propagatio efractio at glass-air iterface

22 Eample: oe spherical surface, traslatio+refractio+traslatio : radius of spherical surface 9/5/4 wk2-b-22 No-paraial ray (approimatio gives large error) D 2 medium ide, e.g. air D medium 2 ide, e.g. glass.5 Paraial rays (approimatio valid) ceter of spherical surface

23 Traslatio+refractio+traslatio / 2 2 D D2 Startig ray: locatio directio Traslatio through distace (+ directio): D 9/5/4 wk2-b-23 + D efractio at positive spherical surface: ( )

24 Traslatio+refractio+traslatio /2 2 2 D D2 Traslatio through distace (+ directio): D 2 Put together: 2 + D 2 2 9/5/4 wk2-b-24

25 Traslatio+refractio+traslatio /3 2 2 ( ) D D ( ) D D + + D D2 + + ( ) D 2 D 9/5/4 wk2-b-25

26 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 cerclockwise through a acute agle optical ais +z 9/5/4 wk2-b-26

27 O-ais image formatio Poit object Poit image 9/5/4 wk2-b-27

28 O-ais image formatio 2 2 D D2 All rays emaatig at arrive at irrespective of departure agle 2 2 [ ] D ( ) D D + D 9/5/4 wk2-b-28

29 O-ais image formatio 2 2 D D2 All rays emaatig at arrive at irrespective of departure agle D 2 + D 2 2 Power of the spherical - surface [uits: diopters, D m ] 9/5/4 wk2-b-29

30 Off-ais image formatio Poit object (off-ais) optical ais Poit image 9/5/4 wk2-b-3

31 Magificatio: lateral (off-ais), agle 2 2 D D2 2 Lateral D 2 2 m +... ' Agle m 2 D D 2 D D 2 9/5/4 wk2-b-3

32 Object-image trasformatio 2 2 D D2 2 m m f ay-tracig trasformatio (paraial) betwee object ad image poits 9/5/4 wk2-b-32

33 Image of poit object at ifiity Poit image 9/5/4 wk2-b-33

34 object at Image of poit object at ifiity 2 D D2 image 2 9/5/4 wk2-b-34 D 2 Note: f D 2 ambiet refractive ide at space of poit image f :image focal legth /Power

35 Poit object imaged at ifiity Poit object 9/5/4 wk2-b-35

36 Poit object imaged at ifiity 2 image at object 2 D D 2 9/5/4 wk2-b-36 D Note: f D ambiet refractive ide at space of poit object f : object focal legth /Power

37 Image / object focal legths Poit image Poit object 9/5/4 wk2-b-37 Object at f Power f Power Image at

38 Matri formulatio / 9/5/4 wk2-b-38 + D traslatio by distace D M M 2 i i form commo to all + M + M refractio by surface with radius of curvature i i ( ) m m f ray-tracig object-image trasformatio

39 9/5/4 wk2-b-39 Matri formulatio /2 i 22 i 2 i 2 i M M M M + + i i M M M M ( ) + + D D efractio by spherical surface Traslatio through uiform medium Power

40 Traslatio+refractio+traslatio D D2 traslatio refractio traslatio by D2 by r.curv. by D 9/5/4 wk2-b-4 result ( ) D D ( ) + + D D ( ) D D

41 Thi les i air i i ()??? i? i Objective: specify iput-put relatioship 9/5/4 wk2-b-4

42 Thi les i air + () Model: refractio from first (positive) surface + refractio from secod (egative) surface Igore space i-betwee (thi les appro.) 9/5/4 wk2-b-42

43 9/5/4 wk2-b-43 Thi les i air () + i i i i

44 9/5/4 wk2-b-44 Thi les i air () i i ( ) i i i i

45 Thi les i air i i () Power of the first surface P P Power of the secod surface 9/5/4 wk2-b-45 ( ) i i

46 9/5/4 wk2-b-46 Thi les i air P P i i i i thi les P ( ) + + P P P thi les Les-maker s formula

47 Thi les i air P i i P P thi les i i 9/5/4 wk2-b-47 i i P thi les i ay bedig is proportioal to the distace from the ais

48 The power of surfaces Positive power beds rays iwards + > Simple spherical refractor (positive) + > N Plao-cove les Negative power beds rays wards > + < + Bi-cove les 9/5/4 wk2-b-48 < Simple spherical refractor (egative) < N Plao-cocave les < Bi-cocave les >