Department of Physics & Astronomy Undergraduate Labs

11/15/2016

ThinLenses

ReflectionandRefraction

Whenlightpassesfromonemediumtoanother,partofthelightisreflectedandtherestistransmitted.Light rays that are transmitted undergo refraction (bending) at the interface of the twomedia. Thedirections of light rays at the interface of two media are defined with respect to a line normal(perpendicular)totheinterfaceasshownbelow.

The angles that the incident (𝑖) and reflected (𝑟) rays make with the normal are the same. That is,𝜃! = 𝜃!.Theanglesthattheincident(𝑖)andtransmitted(𝑡)raysmakearerelatedbySnell’sLaw:

𝑛! sin 𝜃! = 𝑛! sin 𝜃! Here,𝑛! and𝑛!aretheindexofrefractionforthemediumthathoststheincidentandtransmittedray,respectively.Lenses

RefractionandSnell’sLawisthebasisforhowalensworks.Alensisanopticaldevicewhichtransmitsandrefractslight.Lensescanbecategorizedbyhowtheytransmitlightrays.Aconverginglenstransmitsparallellightraystoasinglepointcalledthefocalpoint𝐹.Ontheotherhand,parallellightraysdivergeafterpassingthroughadiverginglens.Thediverginglightraysaretransmittedsuchthattheyappeartoemergefromafocalpoint𝐹locatedbehindthelens.

ConvergingLens DivergingLens

𝜃! 𝜃!

𝜃!

𝑛!

𝑛!

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Alenscanformanimageofanextendedobjectasshownbelowforaconverginglens.Thefocalpoint𝐹islocatedatadistance𝑓,knownasthefocallength,fromthecenterofthelens.Theobjectislocatedatadistance𝑠!,knownastheobjectdistance,fromthelens.Arayemittedparalleltotheopticalaxisofthe lenswillpassthroughthefocalpoint𝐹.Araythatpassesthroughthecenterofthe lenswillpassthrough the lensundeflected.Thepointatwhich these tworayscrossdetermines thepositionof theimage,whichislocatedatadistance𝑠!,knownastheimagedistance,fromthelens.

TherelationshipbetweenallthesedistancesisgivenbytheThinLensFormula:

1𝑠!+1𝑠!=1𝑓

Themagnification𝑚ofthelensisgivenby

𝑚 = −𝑠!𝑠!

Thenegativesigntellsusthatwhen𝑠! and𝑠!arepositive,theimagewillbeinverted.

PrelabExerciseYouhaveaconverging lens thathasa focal lengthof𝑓 = 30cm.Calculate the imagedistance𝑠!, themagnification𝑚andindicatewhethertheimageisuprightorinvertedforeachofthefollowingobjectdistances.

1. 𝑠! = 1𝑓2. 𝑠! = 3𝑓3. 𝑠! = 10𝑓4. 𝑠! = 100𝑓5. 𝑠! = ∞

6. What is the percent difference in𝑠! for an object located at 𝑠! = 10𝑓 and𝑠! = ∞?What about𝑠! = 100𝑓and𝑠! = ∞?Whenisitjustifiedtoconsideratobjecttobelocatedat“opticalinfinity”?

𝑠!

𝑠!

𝐹

𝑓

Object

Image

OpticalAxis

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31/15/2016

Lensmaker’sFormula

TheLensmaker’sFormulaprovidesarecipeforengineeringalenstohaveaparticularfocallength.Thisformularelatesthefocallength𝑓totheindexofrefractionofthelens𝑛andtotheradiiofcurvature𝑅!and𝑅!foreachsideofthelens.TheLensmaker’sFormulaisgivenby

1𝑓= 𝑛 − 1

1𝑅!+1𝑅!

Theradiusofcurvature𝑅canbefoundusingthefiguretotheleft.UsingthePythagoreanTheorem𝑅! = 𝑅 − 𝑎 ! + 𝑟!.Solvingfor𝑅gives

𝑅 =𝑟! + 𝑎!

2𝑎

TypesofLenses

Lensesarenamedaccordingtotheradiiofcurvatureofeachside.

PrelabExercise7. You have a plano-convex lens 1 cm high with adiameter of 15 cm as shown to the left. Its index ofrefraction𝑛 = 1.5. What is the radius of curvature of thelens?Whatisthefocallength?

𝑅𝑅 − 𝑎

𝑎

𝑟

15 cm

1 cm

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41/15/2016

Telescopes

A simple telescope consists of only two lenses: an objective lens, which focuses the image, and theeyepiece,whichmagnifies it.Belowaretwotypesofsimpletelescopes.TheGalileantelescopeusesaconvergent lens for the objective and a divergent lens for the eyepiece. The lenses are located adistanceaway fromeachotherequal to thedifference in their focal lengths. TheKeplerian telescopeusesaconvergentlensforboththeobjectiveandeyepiece.Thelensesarelocatedadistanceawayfromeachotherequaltothesumoftheir focal lengths.TheadvantageoftheKepleriantelescope is that itprovides a wider field of view compared to the Galilean telescope. However, unlike the GalileantelescopetheimageproducedbyaKepleriantelescopeisinverted.

LensAberrations

Lens aberrations are deformations of the image formed when light travels through a lens. Sphericalaberrationisacommonlensaberrationthatoccurswithsphericallenssurfaces.Theaboveformulasarevalidforraysthatmakesmallangleswiththelenssurfaces.Theseraysarereferredtoasparaxialrays.The behavior of light rays that make larger angles with the lens surface do not follow the aboveformulas.Asaresult,non-paraxialrayswillfocusatadifferentpointandresultinablurredimage.

GoalsofthisLab

1. Observeandunderstandhowlensesrefractlight

2. Measurethefocallengthofalensusingseveraltechniques

3. Measuretheindexofrefractionofalens

4. Constructatelescope

fo

fofe

fe

Keplerian Telescope

Galilean Telescpoe

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51/15/2016

LabMaterials

LaserRayBoxandLensesA laser ray box can produce 1, 3 or 5 laser beams that can berefractedbydifferentlenses.

TelescopeKitA large converging objective lens and two eyepieces, converginganddiverging,inmountsonameterstick.

LabProcedure

IndexofRefraction

Youwillmeasuretheindexofrefractionofaplano-convexlens.

1. Unrollthescaleandplaceitonthetable.

2. Turnthelaserrayboxonandsetitforasingleray.

3. Positionthelarge,rectangularplano-convexlenssuchthattheplanarsidefacestheraybox.

4. Orientthelenssothatyoucanobservetheincidentandtransmitted(refracted)ray.

5. Recordtheanglesofthe incidentandtransmittedrayandcalculatethe indexofrefractionforthelensmaterialusingSnell’slaw.

Questions

1. What is the index of refraction for your lens? The lens is made of acrylic. Look up the index ofrefractionforacrylicandcomparethiswithyourmeasuredvalue.

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61/15/2016

RayTracing

Youwillobservehowlightraysarebentbylenses.

1. Usethelaserrayboxtoshine3-5lightraysthroughtheplano-convex,biconcaveandbiconvexlensseparately.Observehoweachrayisbentbythelens.

2. Measurethefocallengthofeachlensusingraytracing.

3. Construct theGalileanandKeplerian telescopesusing theappropriate lensesandobserve thepathoftheraysastheyshinethrougheach.

PrelabExercise8. Anextendedobjectemitslightraysinalldirections.Whatistheobjectdistance𝑠!ifalllightraysare

parallel to each other when they arrive at a lens? How can these parallel light rays be used todeterminethefocallengthofalens?

Questions

2. Whenshining5lightrays,aretheoutermost2raysfocusedtothesamepointastheothers?Why?

3. Whatisthefocallengthofeachlens?

Lensmaker’sFormula

1. Traceanoutlineofthelargeplano-convexlensonapieceofpaper.

2. Use the Lensmaker’s Formula and any measurements necessary to calculate its index ofrefraction.

Questions

4. Howdoes yourmeasured index of refraction compare to the value you obtained in the previousstep?Whatarepossiblesourcesoferror?

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Telescopes

1. Gooutintothehallwayandarrangethetwoconverginglensessothattheobjectivelens,(thebigone)isinfrontoftheotherandtheysharethesameopticalaxis(similartothediagramfortheKepleriantelescopeonp.3).

2. Positionan indexcardbetween the lenses.Move the lensesalong theiropticalaxisuntil lightfromthewindowateitherendofthehallwayisfocusedoneachsideoftheindexcard.Whenthisoccurs,youcanthenbesurethatthelensesareseparatedbythesumoftheirfocallengths.

3. Removethecardandusethetelescopetoobserveadistantobject.

4. NowconstructaGalileantelescopebyusingadivergentlensfortheeyepiece.Youalreadyknowthe focal length of your objective lens. You can’t directly measure the focal length of thedivergent lens. However, you can move the eyepiece along the common optical axis of thetelescopeuntilyouobtainafocusedimage.

Questions

5. Whatisthefocallengthofyourobjectivelensandbotheyepieces?

6. Whichlenswasthemosteffectiveeyepiece?Why?

7. WhydoestheGalileantelescopehavesuchanarrowfieldofview?

Feedback

Do you have any comments, complaints, or suggestions about this lab? Feel free to email our LabManagerdirectlyatPharnish@physics.upenn.edu.