PHYS 2212K : Ray Tracing - Convex/Concave Waves
PHYS 2212K : Ray Tracing - Convex/Concave Waves
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Date Created: 10/12/15
40quot P12 ijCS 2212K Chapter 236 238 39 PHYsiC S Hm SCIENTIS39I S AND ENGINEERS x n mm mm i w A STRATEGIC APPROACH quot1 RANDALL DKNIGHT 236 The Art Of Tracing A Ray And Other Aspects of Refraction Images can be seen in a variety of ways clunky or unclear unfocused or clear The way we get a well focused image is by introducing a lens A lens is a transparent material that uses refraction at curved surfaces to form an image from diverging rays 0 This basically means that lenses are used to look at an image more clearly by spacing them out and essentially concentrating its details across the lens The way that it does this is by using different types of rays which we will discuss in a minute The pictorial representation of image formation is called ray tracing o It is utilized by considering different ways a light ray diffracts through the lens Let39s look at the diagram above which is an example of a converging lens 0 We see that after light passes through the lens it causes the rays to refract or come together at a point towards the optical axis or the base line that goes through the center of the lens 0 We say that the above converging lens is called a convex lens o It is important to note that all three rays that are being refracted come to a point known as the focal point which is again the intersection of the refracted rays The distance from the focal point to the lens is called the focal length 6 Now let39s look at another diagram the one below is called a diverging lens and it is called so because the rays refract away from each other to a point far away 0 They refract parallel rays away from the optical axis 0 Note These rays do have a focal point but it is not as obvious 0 They are also known as concave lenses get it the concave lenses go inward like a cave and the convex lenses go outward like uh not a cave Let39s remember each by the horrible analogy I gave earlier it39s not a good one but hopefully it is so bad that you remember it Note The converging lens is thicker in the center than the edges while a diverging lens is thicker at the edges than at the center Let39s look at the focal lengths for a converging and diverging lens 0 As you can see the focal length is relative to where a bundle of rays converge or diverge and with this knowledge we can either retrace or draw forward the focal points Note The left is diverging the right is converging Focal length f 1 F v gthf Parallel rays I 3 g Paralll ll rays f 395 139 39t a llquot 1395 1 39l H 391f 3911 1 r n f li39i hi it 4 ii ll l39 i f Ellipticall axis p lmcal IlE I quotI Ff l 1 V g This E 1hquot f filmm Tltis is the focal point Converging lens Rays actuallyquot pomerge Rays appear to diverge at this pninL frown this point Diverging lens Note The focal length is the distance from the lens at which rays parallel to the optical axis converge or from which they diverge The focal length if is a property of the lens which is independent of how the lens is used Converging Lenses Let39s first introduce the easier in my opinion of the two lenses the converging lens Let39s first start of by talking about different aspects of lenses in general 0 A thin lens is a lens whose thickness is very small in comparison to its focal length We assume the thickness of a thin lens is zero 0 This lens lies in a plane called the lens plane not like the flying one o It is important to note that all refraction occurs as the rays cross the lens plane and all distances are measured from the lens plane Now that we got that covered we can talk about the three most important situations of light rays passing through a thin converging lens First up we have what is commonly known as the a Lens plane 7 Farf alwlm PRay or the ray that is initially parallel to the a ll 1 optical axis bx 0 From the diagram we see that these types of quot rays run parallel to the optical axis which is not Parallel rays shown in the diagram sorry and meet at a point past the lens known as the far focal point this is due to the fact that it is after the lens plane Next up we have the NRay which is lb LENS llama called this because of the initial NEW f l Pill quotif 39lllquot convergence at the near focal point 7 39 E 39IIF 0 After they converge they pass HF 391 through a lens and appear to be P M H am lie rays parallel to the optical axis which they are 0 This is an important contrast to the way light converge and is a great contrast to the PRay since they two almost mirror each other in the way light reacts to the lens Note It is important to note how the focal length differ in their relation to the focal point for the P and N Ray Next we have the CRay which is the most basic of the three oThe rays basically pass through the center by using a straight line nothing to challenging about that Rays are oThey are not bent and are continuously straight Center ef liens net bent These three situations are the basis for ray tracing Real Images Real images are images that come from a source get refracted converge and meet at a similar side on in the image plane A virtual image is a point from which rays never meet and appear to diverge In the diagram below P39 is a real image from point P d All the tays leaving nne paint in the Sll i lIF i S Lens lll ll 39quotf abject plane P are refraeted by the lens and P 39 If I ennaerge tn nne paint in the image plane Pi Ubjeetk 39 r v I V I Q Far fneal point V R quotat 77 I Near fecal point quot V39GGSSSSSQ v I quotI Iterrsiih i I Image Optical axis i 1 i I 739 I if c cf I la Fifi u HquotquotCllajeet plane I4 f M f ltnage plane I S 5quot All points on the object that are on the same plane are said to be on the object plane They converge in the image plane As you can see points PQ and R translate over past the lens and make an image e these planes A sharp well fecusecl image is seen an a screen placed in the image plane The rays clen t step unless tl rep re 39bleclcecl by a screen The image will be blurry and cut at fDCLIS en a screen in The image seen is called an inverted image or an upside down image with respect to the subject olt is a standard characteristic of realimage formation with a converging lens oNote A larger lens will quotcollectquot more rays and thus make a brighter image also rays don39tjust stop at the point of convergence the continue until being opposed by a screen The diagram shows a closer look at rays near the image plane The book provides a Chart to list the steps out for ray tracing for diverging lens use at your own discretion hay tracing fer a converging lens ill Ilraar an nptical axis Use graph p aper er a ruler Establish an apprepriate scale 93 Center the lens an the axis Mark anti label the fecal paints at distance f en either side 83 Represent the abject tsith an upright arrest at distance 539 lt s usually best te place the base at the anew en the axis and te draw the anew abeut halt the radius at the lens Illranr the three special rays lirem the tip at the arrest Use a straight edge a A ray parallel tn the axis retracts threngh the far lineal paint I A ray that enters the lens alen g a line threngh the near fecal paint emerges parallel tn the axis c A ray thrnugh the center at the lens tines net bendt t3 Extend the rays until they cenrerge This is the image paint Draw the rest at the image in the image plane lithe base at the abject is en the axis then the base at the image will alse be en the axis Measure the image distance squot Alse if needed measure the image height relative tn the abject height Lateral Magnification Lateral Magnification is the combined size orientation information of an image denoted by a single number 0 Its orientation is relative to the object o Is it large or small 0 What is the focal length Location We use an equation to denote it as 57 m S 0 We refer to the imagetoobject height ratio as h 7 575 We can interpret the lateral magnification by 0 A positive value of m indicates that the image is upright relative to the object A negative value of m indicates that the image is inverted relative to the object o The absolute value of m gives the size ratio of the image and object h39h m Virtual Images In this section we basically reviewed converging lenses which have a focal point that is relative to a certain distance that is the focal point covers less ground than the objects distance from the lens What happens when the object is inside the focal point o What if the distance of the object from the lens is less than the focal length In this case all three rays appear to let The refracted rays are diverging P peltrl appear to eeme frem F of the height of the image in the diverge from point P39 or the position image plane Let39s look at the diagram Virtual eejee 0 First off we can see our claSSIc image ray patterns The PRay NRay I J r Ye ur eye eeee the and CRay virtual ittnege at F o The only distinct difference is the orientation of the virtual image as it relates to the object 0 These rays follow a diverging pattern so its destination is to be expected 0 When it comes to the virtual image however we have to backtrack a bit from the rays we created which is indicated by the dotted lines o The virtual image is not a real one you can think of it as a result of confusing the regular orientation of lens diagram 0 The virtual image is created as if on object was there in the first place but it39s not 0 This example relates to the study of a mirror where a reflection is just the ray s quotinterpretationquot of how an object would look if it were behind the mirror Note Dashed lines on a diagram typically signal a virtual image 0 To further explain this think of a virtual image as an image where you look through the lens Essentially this leads do magnification for your eye collects and focuses bundles of diverging rays and views them all at once o In this example we get a virtual upright image which means it is not real and it retains the same orientation as the object o The image distance 51 is seen as a negative number which indicates the image is on the opposite side of the lens from the real image Note The light rays through a lens create a real image if the object distance is larger than the focal length 5 gt 6 and a virtual image if the focal length is greater than the object distance fgt 5 Diverging Lenses When a lens is thicker in the middle than it is on the edges it is called a converging lens When a lens is thicker on the edges than it is in the middle it is called a diverging lens Like with converging lenses there are also three important sets of rays that pass through the diverging lens Parallel rays H 2 Parallel rays Center ef le e Rays are net hem w ll any ray lllllll1llli parallel ta the Amy ray directed alerrg a line reward Arr ray directed at the reenter rptleal axis dlrergyes alerrg a line the far feeal palrrt emergea free the of the lens passes tlrraagh in a lireuglr the near ideal radial lens parallel la the optical axis Straight llrre x77 3 k h l l 6 00 Pm esac ar 0 TACTICS list the steps out for ray tracing 3 233 Ray tracing far a dwergmg liens for diverging ens use at your 099 hulluw steps ll thruugh 3 uf39Taeties has 232 raw the three slpeeial rays frum the tip at the arruw Use a straights erlge own discretion Diverging lenses always make virtual images A my parallel hr the aaie tliuerges along a line through the near lueal palm h A ray along a lime tmaarel the tar tueal paint emerges parallel to the axis e A ray through the center at the lens dues nut hemi aso pro video a few more tabes just for your we Wing Theme 233 Sign vcnaentien llr retraet quot 9 Mamquot t3 Trace the elirergiug raga haehwarrl The paint hum whieh they are tlieerg mm Nail lag is the image point which is always a virtual image a Camera tetearti Cemeaee tewarri Measure the image distauee hi this will he a negative number the ehjeet the ehjeet 339quot Real image lliirtual image Eili FCAE 231 epgeaite aisle saute aisle tram ehjeet ehjeet experience TALE 234 Sign eeneentien tir thin lenses Peeiti ee R R2 Creamert tewerei the ehjeet Cemeeee toward the object f Cereaerging leme thiekee in center Diverging leme thinner ire eemter s Real irmege eppeeite side item ehjeet Virtue image same side ale ehjeet 237 238 Making Images With Spheres And A Quick Little Recap Of Concavevex Lenses Along with all the lenses we have seen curved mirrors can also be used to form images 0 They can also be analyzed with ray diagrams Yah To analyze this we only take into account the most important cases of spherical mirrors which are mirrors whose surface is a section of a sphere Concave Mirrors A concave mirror is a mirror in which the edges curle toward the light source Rays parallel to the optical axis reflect from the surface of the mirror they pass through a single point on the optical axis which is the focal point 0 The focal length is the distance from the mirror surface to the focal point If we consider the case that the distance from the mirror is greater than the focal length 5 gt f we see o The image is real 0 The image is inverted FIGTUlFlE 2352 A real We eformed b a concave mirror g F Although there are an IannIte P Special rays Minor plane Iquot a number of rays each ray obeys the law of reflection 0 You can see the three quotspecial raysquot which determine the position and size of the image A ray parallel to the axis reflects through the focal point O A ray through the focal point reflects parallel to the axis A ray striking the center of the mirror reflects at an equal angle on the opposite side of the axis Note The three rays also locate the image if the focal length is greater than the object length which would be the case of a virtual image behind the mirror Convex Mirrors A convex mirror shows parallel light rays approaching a mirror in which the edges curve away from the light source O In this case the reflected rays appear to come from a point behind the mirror The diagram shows an object in front of a convex mirror O In this case the reflected rays each obeying the law of reflection create an upright image of reduced height quotbehindquot the mirror We say the image is virtual because no rays actually converge FIGURE 2355 A sin lual image formed 3 a convex mirror This tag was heading for the foeal paint and thus emerges parallel te the optical axis 4 r This my entered parallel to the optical axis and thus appeals to m a have eeme float the focal peint 394 Special rays Optical axis image f t Mirmr plane at the image point P39 They instead appear to come from this point Once again the three special rays are enough to find the image The book provides a chart to list the TACTICS Ra tracin tier 3 s herieal mirrer r m sea 234 y g p steps out for ray ill Draw an eptieal axis Use graph paper er a ruler Establish an appropriate tracing for Scales 393 Center the mirrer en the axis Mark and label the fecal peint at distance f 51079 r 53 mrrOrS frem the miner s surface 33 Represent the ehjeet with an upright arrew at distanee 5 it s usually best te place the base ef the arrew en the axis and te draw the anew abeut halt the dScretbn radius at the miner E Draw the three speeial rays tram the tip ef the arrew Use a straight ed get use at your own a A parallel to the axis refleets threugh eeneare er away here can sex the teeal paint I An ineerning ray passing threagh eeneare er heading teward CDIIWEK the fecal peiht re ects parallel te the artist e A ray that strikes the center ef the miner re ects at an equal angle en the eppesite side at the eptieal artist 3 Extend the rays ferwartl er haetsward until they enmrerge This is the image peinti llDraw the rest ef the image in the image planet It the base ef the ebjieet is en the asisa then the base at the image will else be en the axis Measure the image distance 3 Alan it needed measure the image height relative tn the abject heights Exereses 32 33 The Mirror Equation The thinlens equation assumes that lenses have negligible thickness 0 This causes single refraction in the lens plane 0 The rays are nearly parallel to the optical axis We can make the same assumptions about spherical mirrors 0 So we relate object and image distances by the equation for thin lenses 4 l miner equatien The focal length is related to the mirror39s radius of curvature Iuq A concave mirror converging lens has a positive focal length while a convex mirror diverging lens has a negative focal length The lateral magnification of a spherical mirror is a computed exactly as it is for a lens m 5 TABLE 235 Sign convention for spherical mirrors Pooitivo Negative R Concave toward Convex toward t lho objoot lilo ohjoot 5 Real imago Virtual imago salno aido oppoaito Sirlo as ohjoot from ohjoot lmpmrtnt E l lEEtE The my midEll ll light Hams ung nial Hm lhill In 31 39 J v 3 arm A light my EMiIIJJEE Fm minim an intermim will minim 1 ii I lili39l39ln ill 5 HTML rillJr iii if m rays Em mm whim F ilil an IE my mum raw inn all Minna The Ele m an mg m an mm In 7 em by In pupil and are J m litE Mina F Illajr n im ii lial lm lam minim and mums 311E law rm a l mm Aplcamns Hay farming 3 51mm rays in ii basil Eil lfhl EmuEm his IE IE HE Hill in mum Mlagni icatin m E mislmanup g timg5 rmr The is 39m Imll Ephraim nmilrmrs ImElQiE l irmati n lfmjrss iw fmFm ini ra n 1 n wlmal mummmmrm mm my Emmy at l Hm Mr Fquot is a IFEEII image of F I quotaquot arrays dilf lg 1mm 311 an mini a lawn Mm m mm r m m w rg m Fquot M l l I m E mm Em E39Esaiiirl zuall imagine HEP E l l mm EIth mil image dim 311E mm by H n HE n 3 squot 2 E 2u1m Flame mm RH 91 EE If Th i i Winingamhjm WWWth 1 11 s 539f Wm IE nal my 5E by litE l m maker s aquarium 1 mi 1 r K i E E Inr5url msmmmmanhj 427me f Inramn rgingl m Inniiir gim 339 Inramlimag Inn inua Elli f for mm mirmr linrr 39 Inramlimag ltrrquotrirlural WWFHH