Wave Phenmena Phsics 5c Lecture Gemetrical Optics (H&L Chapter )
Tw Mre Lectures T G!! Will inish gemetrical ptics tda! Next week will cver less serius material! Laser and hlgraph! Quantum Mechanics Hw much can we d in a lecture?! Final Exam! Ma 7 (Frida) 9:5 :5. Sever 0! Prepare a cheat sheet! letter-size sheet paper. Use bth sides i u wish! Bring a calculatr with resh batteries
What We Did Last Time! Intrduced gemetrical ptics! Size ptical elements much larger than the wavelength " Intererence and diractin can be ignred! Discussed lenses! Fcal pint and cal length! Real lenses are imperect aberratins! Spherical aberratin when aperture is large! Chrmatic aberratin due t dispersin the glass! There are slutins, but nthing is perect! Als talked abut rainbws and Hubble Space Telescpe R = n
Gals r Tda! Stud ptical devices made lenses! I assume that ur lenses are perect N aberratin! Start b deining what such a lens wuld d! Intrduce lens rmula r the lcatin the image! Trace light ras t analze a ew simple devices! Intrduce matrix technique t simpli the analsis! Describe an ptical device b a prduct matrices! Derive a undamental rule related t uncertaint and diractin, as well as t statistical mechanics
Ideal Lens! An ideal lens wuld use ver-high-index, nndispersive material We can dream R! Since = such a lens have ver large R n! It can be made ver thin, with n spherical aberratin! In the n limit, we ind an ininitel thin ilm! Light entering the ilm magicall bends b θ = θ () r that satisies tanθ = r r θ F
Ideal Lens! What abut a cncave lens?! Eas: tanθ = r! Negative signs n θ and cancel each ther! An ideal lens (cncave r cnvex) bends the light that passes at radius r b θ that satisies tanθ = r! We use this idealized rmula r analzing ptical devices tda F r θ
Lens Frmula! First, we trace ras light rm a pint thrugh a lens! We assume ideal lens with n aberratin! = distance rm the bject r! i = distance t the image! Assuming small angles r r θ = θ+ θ + i gemetr! This is mre useul that it lks θ tanθ = + = General lens rmula i r θ θ i θ ideal lens
Lens Frmula + = i! The rmula wrks in all cmbinatins, i, and i > i = 0 > i! Negative i " Image is virtual, i.e. light des nt actuall cus in a pint < i = 0 <
Lens Frmula + = i! It wrks with cncave lenses as well i! Since is negative, i is alwas negative " Image is virtual! What d we mean b images?! S ar ur bject is a pint < 0 i = 0 <! Hw des a real bject (with size) lk thrugh lenses?
Magniing Glass! Simplest ptical device: a magniing glass + = i! We knw image distance rm the lens rmula! Ras passing the middle the lens dn t bend i! We can trace ras rm varius pints the bject
Magniicatin! Magniicatin pwer is m i =! Using the lens rmula + = i m =! I bject is utside the cal pint! Image is inverted! Magniicatin is again i m = = Negative = inverted i Object shuld be placed slightl inside the cal pint i
Practical Issues! Have u ever seen a 00x magniing glass?! What s wrng with it?! Tr t make small i =! Image is bigger, but it s ar awa! Desn t help seeing mre details! Usuall i = 5 cm r eas viewing! Hw abut making small?! Object must it between and the lens! i i m = m = 5 cm
Micrscpes! Cmbine the tw cases t make a micrscpe! Object just utside! Intermediate image is inverted and just inside! Final image is inverted! Ttal magniicatin is i i m= mm =! This is getting tedius i! We want a simpler mathematical rmalism t deal with mre cmplex ptical devices i
Tracing a Ra Light! Natural wa representing a ra light is = ( x) x ( )! Mst the time, it s just a linear unctin: ( x) = 0 + x! Slpe changes at each lens: θ! Fr ideal lens, tanθ =! All that happens t a ra light is expressed b a linear transrmatin (, )! Hw abut using linear algebra here? θ Because we lve matrices small angle! x
Matrix Technique! Hw des the vectr [, ] change?! B ling a distance L + L L = = 0! B passing a lens cus length 0 = =! Lenses and space between them are represented b matrices! We can build an ptical device b multipling them L x x
Magniing Glass! First, an eas example! Ras leaving the head the duck are, [ ] height randm! The ras travel distance and thrugh the lens 0 = 0 =! Beware the s!!!!! OK, hw des this equatin relate t the image?! Hw des it lk t ur ees? i
Interpreting Slutin =! What ur ees see is [, ]! Image is created b ur brains, assuming that the i light traveled straight! Suppse that light riginated as, at i let the lens i 3 = 0 3 [ ] 3 3 3 i = 3 0 + i i i = This is dne in ur brain
Interpreting Slutin i i 3 + i = 3! All ras must lk like cming rm the image i r an value i + i = 0 i = Equivalent t lens rmula! Nw we can calculate the height the image as 3! Magniicatin is m i = = = = ( ) i 3
Micrscpes L! Appl the same methd 3 i 0 L 0 = 3 0 0 0 = m! Tp-let is magniicatin m = { } ( L) + i( L ) ( L+ L) + i ( L ) ( L ) L ( L ) ( L ) =! Tp-right = 0 ( L+ L) i = L ( ) ( L ) i = 0 ( L) + ( L )
Diractin! Light entering the telescpe suers diractin θ! Can we crrect r this?! What wuld be the real limit reslutin i we are allwed t use unlimited number ptical elements?! Impssible t answer this withut speciing hw we arrange the lenses r is it?! Matrix technique has an answer a θ =. a λ
Light in the - Plane! A ra light [, ] is a pint in the - plane! Multipling a matrix mves the pint! Because diractin, input ra cannt be a single pint! We must deal with a cllectin ras " an area in the - plane! Light rm a distant star entering the telescpe ccupies a rectangle [, ] = M M θ [, ] θ a θ θ a
Linear Algebra! Hw des the rectangle mve?! Multipling b M is a linear transrmatin! Linear algebra tells us that the result must be a parallelgram! Hw abut the areas the rectangle/parallelgram?! Cnsider vectrs that lie n tw edges the area v u v u v = v u = u A= vu vu= det v u v u v u v u A = = M = M = det det det( )det Adet( M ) v u v u v u u A v u v A
Cnservatin Area! S, what is det(m)? 0! Fr a lens and a space det =! Generall det( AB) = det( A)det( B)! Fr an ptical device det( M ) = L det = 0 Area in the - plane ccupied b a set light ras is cnserved as it ges thrugh an ptical device! I u shrink the light in space, it diverges in angle! I u tr t make the light parallel, it blws up in width! We need t see an example r tw
Fcusing Light! Let s tr t cus light int a pint! Incming light is parallel! Diractin makes it slightl nn-parallel! As the light enters the lens a a < < θ < < θ λ θ =. a! Light passes the lens and travels 0 = 0 = +! Light des nt cus n a spt, but a θ < < θ < < a Prduct didn t change
Shrinking Aperture! Tr t shrink the aperture using tw lenses! Incming light is, again, < < θ < < θ a a θ =! Light is transrmed as. a λ 0 + 0 + ( + ) = 0 =! Distributin ater shrinking is a a < < θ < < θ Smaller in aperture, larger in angular spread
Liuville s Therem! Cnservatin area in the - space is a special case Liuville s therem in statistical mechanics:! The area in the phase space ccupied b an ensemble particles is cnserved! Phase space = (space, mmentum) plane! We are dealing with light made phtns! Each phtn has a mmentum p =! k = h λ h! -cmpnent is p = psinθ λ! - plane = phase space times cnstant! I we start rm a bunch phtns ccuping certain area in the phase space, the area shuld never change
Uncertaint Principle! A phtn ging thrugh an aperture a ma have! Psitin anwhere inside a! Mmentum uncertain within h λ = λ α! We culd call these quantities as uncertainties in space and mmentum, and p p = h! Heisenberg s uncertaint principle! The act that this prduct cannt be changed is critical r the validit the principle under all circumstances h a Ignring.
Summar! Discussed ideal lens and lens rmula! Ideal lens bends light b! Such a lens create an image at distance given b! Analzed magniing glass and micrscpe! Intrduced matrix technique! Light ra = vectr [, ]! Lenses and the space between them! Optical device is a prduct matrices! Area ccupied b light in the - plane is cnserved! Liuville s therem tanθ = r! Heisenberg s uncertaint principle 0 + = i L 0