Lenses and imaging. MIT 2.71/ /10/01 wk2-a-1

Transcription

1 Leses ad imagig Huyges priciple ad why we eed imagig istrumets A simple imagig istrumet: the pihole camera Priciple of image formatio usig leses Quatifyig leses: paraial approimatio & matri approach Focusig a les: Imagig coditio Magificatio Aalyzig more complicated (multi-elemet) optical systems: Pricipal poits/surfaces Geeralized imagig coditios from matri formulae MIT 2.7/2.7 9// wk2-a-

2 The miimum path priciple P o-uiform object Γ light ray Γ (, P y, z) dl Γ is chose to miimize this path itegral, compared to alterative paths (aka Fermat s priciple) Cosequeces: law of reflectio, law of refractio MIT 2.7/2.7 9// wk2-a-2

3 The law of refractio reflected θ refracted θ θ icidet siθ siθ Sell s Law of Refractio MIT 2.7/2.7 9// wk2-a-3

4 Ray budles poit source spherical wave (divergig) poit source very-very far away plae wave MIT 2.7/2.7 9// wk2-a-4

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 MIT 2.7/2.7 9// wk2-a-5 optical wavefrots

6 Why imagig systems are 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. MIT 2.7/2.7 9// wk2-a-6

7 The pihole camera opaque scree image pihole MIT 2.7/2.7 9// wk2-a-7 object The pihole camera blocks all but oe ray per object poit from reachig the image space a image is formed (i.e., each poit i image space correspods to a sigle poit from the object space). Ufortuately, most of the light is wasted i this istrumet. Besides, light diffracts if it has to go through small piholes as we will see later; diffractio itroduces artifacts that we do ot yet have the tools to quatify.

8 Les: mai istrumet for image formatio air glass air optical ais Poit source (object) Poit image MIT 2.7/2.7 9// wk2-a-8 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.

9 Aalyzig leses: paraial ray-tracig air glass air Free-space propagatio Free-space propagatio MIT 2.7/2.7 9// wk2-a-9 Refractio at air-glass iterface Free-space propagatio Refractio at glass-air iterface

10 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. Note the assumptio of eistece of a optical ais (i.e., perfect aligmet!) MIT 2.7/2.7 9// wk2-a-

11 Paraial approimatio /2 Igore the distace betwee the locatio of the aial ray itersectio ad the actuall 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 MIT 2.7/2.7 9// wk2-a-

12 Eample: oe spherical surface, traslatiorefractiotraslatio R: radius of spherical surface MIT 2.7/2.7 9// wk2-a-2 No-paraial ray (approimatio gives large error) 2 medium ide, e.g. air R medium 2 ide, e.g. glass.5 Paraial rays (approimatio valid) ceter of spherical surface

13 Traslatiorefractiotraslatio / Startig ray: locatio directio Traslatio through distace ( directio): MIT 2.7/2.7 9// wk2-a-3 Refractio at positive spherical surface: ( ) R

14 Traslatiorefractiotraslatio / Traslatio through distace ( directio): 2 Put together: MIT 2.7/2.7 9// wk2-a-4

15 Traslatiorefractiotraslatio /3 2 2 ( ) ( ) R R R 2 ( ) R 2 MIT 2.7/2.7 9// wk2-a-5

16 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 Ray agles are positive if the ray directio is obtaied by rotatig the z ais couterclockwise through a acute agle optical ais z MIT 2.7/2.7 9// wk2-a-6

17 O-ais image formatio All rays emaatig at arrive at irrespective of departure agle 2 R 2 2 Power of the spherical - surface [uits: diopters, m ] MIT 2.7/2.7 9// wk2-a-7

18 Magificatio: lateral (off-ais), agle Lateral 2 2 m... R ' Agle m MIT 2.7/2.7 9// wk2-a-8

19 Object-image trasformatio m 2 2 m f Ray-tracig trasformatio (paraial) betwee object ad image poits MIT 2.7/2.7 9// wk2-a-9

20 Image of poit object at ifiity R 2 R f :image focal legth MIT 2.7/2.7 9// wk2-a-2

21 Poit object imaged at ifiity R R f : object focal legth MIT 2.7/2.7 9// wk2-a-2

22 Matri formulatio / MIT 2.7/2.7 9// wk2-a-22 traslatio by distace out out M M 2 i i form commo to all M M 2 22 refractio by surface with radius of curvature R i i ( ) R m 2 2 m f ray-tracig object-image trasformatio

23 MIT 2.7/2.7 9// wk2-a-23 Matri formulatio /2 i 22 i 2 out i 2 i out M M M M i i out out out M M M M R ( ) R Refractio by spherical surface Traslatio through uiform medium Power

24 Traslatiorefractiotraslatio traslatio refractio traslatio by 2 by r.curv. R by MIT 2.7/2.7 9// wk2-a-24 result ( ) ( ) R R R ( ) R 2

25 MIT 2.7/2.7 9// wk2-a-25 Thi les P ( )P i i out out [ ] i i out out P P ( ) R R R R P thi les Les-maker s formula

26 The power of surfaces Positive power beds rays iwards R> Simple spherical refractor (positive) R> N Plao-cove les Negative power beds rays outwards R> R< Bi-cove les MIT 2.7/2.7 9// wk2-a-26 R< Simple spherical refractor (egative) R< N Plao-cocave les R< Bi-cocave les R>

27 The power i matri formulatio out out out M M 2 M M 2 22 i i i i M out < 2i i M2 out > i R> Simple spherical refractor (positive) M 2 R R< Simple spherical refractor (egative) M 2 R ( Ray bedig) ( Power) ( Lateral coordiate) MIT 2.7/2.7 9// wk2-a-27 ( Power) M2

28 i Power ad focal legth out M2 < i Simple spherical refractor (positive) i ( > out i out > ) M2 M i 2 ( Focal legth) ( Power) M2 Simple spherical refractor (egative) MIT 2.7/2.7 9// wk2-a-28 i out M ( < ) 2

29 Thick/compoud elemets: focal & pricipal poits (surfaces) st Pricipal Poit/Surface 2 d Pricipal Poit/Surface 2 d Focal Poit/Surface Optical Ais st Focal Poit/Surface geeralized optical system MIT 2.7/2.7 9// wk2-a-29 Note: i the paraial approimatio, the focal & pricipal surfaces are flat (i.e., plaar). I reality, they are curved (but ot spherical!!).the eact calculatio is very complicated.

30 Focal Legths for thick/compoud elemets geeralized optical system st FP FFL EFL st PS 2 d PS BFL EFL 2 d FP EFL: Effective Focal Legth (or simply focal legth ) FFL: Frot Focal Legth BFL: Back Focal Legth MIT 2.7/2.7 9// wk2-a-3

31 PSs ad FLs for thi leses glass, ide P P2 l (EFL) P P P 2 (BFL) (EFL) (FFL) The pricipal plaes coicide with the (collocated) glass surfaces The rays bed precisely at the thi les plae (collocated glass surfaces & PP) MIT 2.7/2.7 9// wk2-a-3

32 The sigificace of pricipal plaes / st FP st PS 2 d FP geeralized optical system 2 d PS thi les of the same power located at the 2 d PS for rays passig through 2 d FP MIT 2.7/2.7 9// wk2-a-32

33 The sigificace of pricipal plaes /2 st FP st PS 2 d FP geeralized optical system 2 d PS thi les of the same power located at the st PP for rays passig through st FP MIT 2.7/2.7 9// wk2-a-33

34 Imagig coditio: ray-tracig st PS 2 d PS object st FP chief ray 2 d FP image Image poit is located at the commo itersectio of all rays which emaate from the correspodig object poit The two rays passig through the two focal poits ad the chief ray ca be ray-traced directly MIT 2.7/2.7 9// wk2-a-34

35 MIT 2.7/2.7 9// wk2-a-35 Imagig coditio: matri form / 2 d PS st PS 2 d FP st FP object image S S PS PSS S S P PS S P S system matri

36 Imagig coditio: matri form /2 st PS 2 d PS S st FP object 2 d FP image S Imagig coditio: Output coordiate must ot deped o etrace agle γ MIT 2.7/2.7 9// wk2-a-36 γ S S PS PSS P PS γ

37 Imagig coditio: matri form /3 st PS 2 d PS S st FP object 2 d FP image S Imagig coditio: S S PSS S S P system immersed i air, ; power P/f S S f MIT 2.7/2.7 9// wk2-a-37

38 Lateral magificatio st PS 2 d PS st FP object S S 2 d FP image γ (assume imagig coditio is satisfied) PS P γ PS m PS MIT 2.7/2.7 9// wk2-a-38

39 Agular magificatio st PS 2 d PS γ S st FP object 2 d FP image S γ γ (assume imagig coditio is satisfied) PS P PS γ m a γ γ PS MIT 2.7/2.7 9// wk2-a-39

40 Geeralized imagig coditios M M 2 M M 2 22 image system matri object Power: P M 2 Imagig coditio: Lateral magificatio: Agular magificatio: M 2 m M 22 m a M MIT 2.7/2.7 9// wk2-a-4