GeometricOptics

LearningGoals

Lookingforwardat…• howaplanemirrorformsanimage,andwhyconcaveandconvexmirrorsformimagesofdifferentkinds.• howimagescanbeformedbyacurvedinterfacebetweentwotransparentmaterials.• whataspectsofalensdeterminethetypeofimagethatitproduces.• whatcausesvariousdefectsinhumanvision,andhowtheycanbecorrected.• howmicroscopesandtelescopeswork.

Introduction

• Thissurgeonperformingmicrosurgeryneedsasharp,magnifiedviewofthesurgicalsite.• Toobtainthis,she’swearingglasseswithmagnifyinglenses.• Howdomagnifyinglenseswork?• Howdolensesandmirrorsformimages?• Weshalluselightraystounderstandtheprinciplesbehindopticaldevicessuchascameralenses,theeye,microscopes,andtelescopes.

Reflectionataplanesurface

• LightraysfromtheobjectatpointP arereflectedfromaplanemirror.• ThereflectedraysenteringtheeyelookasthoughtheyhadcomefromimagepointP’

Imageformationbyaplanemirror

Imageformationbyaplanemirror:Signrules

Characteristicsoftheimagefromaplanemirror

• Inaplanemirror,theimageisvirtual,erect,reversed,andthesamesizeastheobject.

Theimageisreversed• Theimageformedbyaplanemirrorisreversed;theimageofarighthandisalefthand,andsoon.

Sphericalmirrorwithapointobject

• AsphericalmirrorwithradiusofcurvatureR formsarealimageP’ ofthepointobjectP.

GeneralSignconventions(mirror&lens)

• IftheobjectpointP isonthesamesideastheincidentlight,thens ispositive.• IftheimagepointP’ isonthesamesideasthereflectedlight,thens’ ispositive.• IfthecenterofcurvatureCisonthesamesideastheoutgoing light,thenR ispositive.

Focalpointandfocallength• Whentheobjectisveryfarfromthesphericalmirror,theincomingraysareparallel.• Thebeamofincidentparallelraysconverges,afterreflectionfromthemirror,toafocal point,pointF.• Thedistancefromthevertextothefocalpoint,denotedbyf,iscalledthefocal length.

Focalpointandfocallength

• Withtheobjectatthefocalpoint,thereflectedraysareparalleltotheopticaxis.• Thereflectedraysmeetonlyatapointinfinitelyfarfromthemirror,sotheimageisatinfinity.

Imageofanextendedobject:Sphericalmirror

• Shownishowtodeterminetheposition,orientation,andheightofanimageformedbyaconcavesphericalmirror.

Imageformationbyaconvexmirror• Ifthemirrorisconvex,sothatR isnegative,theresultingimageisvirtual(thatis,theimagepointisontheoppositesideofthemirrorfromtheobject),erect,andsmallerthantheobject.

Focalpointandfocallengthofaconvexmirror

• Whenincomingraysthatareparalleltotheopticaxisarereflectedfromaconvexmirror,theydivergeasthoughtheyhadcomefromthevirtual focalpoint F atadistancefbehindthemirror.• Thecorrespondingimagedistances isnegative.

Focalpointandfocallengthofaconvexmirror

• WhentheincomingraysareconvergingasthoughtheywouldmeetatthevirtualfocalpointF,thentheyarereflectedparalleltotheopticaxis.

Graphicalmethodoflocatingimages

Graphicalmethodoflocatingimages

Imageofapointobjectatasphericalsurface

Apparentdepthofaswimmingpool

• Whenlighttravelsthroughaplanesurfacebetweentwoopticalmaterials,theimagehasthesamelateralsize(m =1)andisalwayserect.• s’=-s/1.33• Theapparentdepthofapoolislessthanitsactualdepth.

R→∞

Lensmaker’s equation

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Magnification

m = y 'y= −s 's

Typesoflenses• Shownbelowarevarioustypesoflenses,bothconverginganddiverging.

• Anylensthatisthickeratitscenterthanatitsedgesisaconverging lenswithpositivef ;andanylensthatisthickeratitsedgesthanatitscenterisadiverging lenswithnegativef.

1f= (n−1) 1

R1−1R2

"

#$$

%

&''

R1 > 0 > R2R1 =∞0 > R2

0 > R2 > R1 R2 > 0 > R1

f > 0 f < 0

R1 > R2 > 0R1 =∞R2 > 0

Thinconverginglens

• Alens isanopticalsystemwithtworefractingsurfaces.• Thesimplestlenshastwosphericalsurfacescloseenoughtogetherthatwecanignorethedistancebetweenthem;wecallthisathin lens.

Thinconverginglens• RayspassingthroughthefirstfocalpointF1 emergefromaconverginglensasabeamofparallelrays.

• fiscalledthefocal length.

Imageformedbyathinconverginglens

Thindiverginglens• Whenabeamofparallelraysisincidentonadiverging lens,theraysdivergeafterrefraction.• Thefocallengthofadiverginglensisanegativequantity,andthelensisalsocalledanegative lens.

Thindiverginglens

• Incident raysconvergingtowardthefirstfocalpointF1ofadiverginglensemergefromthelensparallel toitsaxis.

Graphicalmethodsforlenses

• Shownbelowisthemethodfordrawingthethreeprincipal rays forarealimageformedbyaconverginglens.

Graphicalmethodsforadiverginglens

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Example

Cameras

• Whenacamera isinproperfocus,thepositionoftheelectronicsensorcoincideswiththepositionoftherealimageformedbythelens.

Cameralensbasics

• Thefocallengthf ofacameralensisthedistancefromthelenstotheimagewhentheobjectisinfinitelyfaraway.

• Theeffectiveareaofthelensiscontrolledbymeansofanadjustablelensaperture,anearlycircularholewithdiameterD.

• Photographerscommonlyexpressthelight-gatheringcapabilityofalensintermsoftheratiof/D,calledthef-numberofthelens:

Theeye• Theopticalbehavioroftheeyeissimilartothatofacamera.

Defectsofvision

• Anormaleyeformsanimageontheretinaofanobjectatinfinitywhentheeyeisrelaxed.• Inthemyopic (nearsighted)eye,theeyeballistoolongfromfronttobackincomparisonwiththeradiusofcurvatureofthecornea,andraysfromanobjectatinfinityarefocusedinfrontoftheretina.

Nearsightedcorrection

• Thefarpointofacertainmyopiceyeis50cminfrontoftheeye.• Whenadiverginglensoffocallengthf =−48cmisworn2cminfrontoftheeye,itcreatesavirtualimageat50cmthatpermitsthewearertoseeclearly.

Farsightedcorrection• Aconverginglenscanbeusedtocreateanimagefarenoughawayfromthehyperopiceyeatapointwherethewearercanseeitclearly.

Angularsize

• Themaximumangular sizeofanobjectviewedatacomfortabledistanceistheangleitsubtendsatadistanceof25cm.

𝜃 ≈ tan 𝜃 =𝑦

25𝑐𝑚

Themagnifier• Theangular magnification ofasimplemagnifieris:

Thecompoundmicroscope

Theastronomicaltelescope

• Thefigurebelowshowstheopticalsystemofanastronomicalrefracting telescope.

Thereflectingtelescope

• TheGeminiNorthtelescopeusesanobjectivemirror8metersindiameter.

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Example:Inatelescope,theobjectivelensformsareal,reducedimageIoftheobject.Thisimageistheobjectfortheeyepiece,whichformsanenlarged,virtualimageofIwhichisatinfinity(formastcomfortableviewingbyaeye).

ϑ =−y 'f1

and ϑ ' = y 'f2

angular magnification M =ϑ 'ϑ= −

f1f2

(-ve for the inverted final image)