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by: Ubaldo Jacobson

Optics PHYS 5310

Ubaldo Jacobson
GPA 3.93

Charles Sackett

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Charles Sackett
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This 37 page Class Notes was uploaded by Ubaldo Jacobson on Monday September 21, 2015. The Class Notes belongs to PHYS 5310 at University of Virginia taught by Charles Sackett in Fall. Since its upload, it has received 39 views. For similar materials see /class/209743/phys-5310-university-of-virginia in Physics 2 at University of Virginia.


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Date Created: 09/21/15
Phys 531 Lecture 26 29 November 2005 Photons Last time finished polarization Learned about retarders Jones matrices Today introduce quantum optics Whole new way to look at light Outline 0 Photon optics o Photon detectors 0 Quantum noise 0 Quantum states Photon optics not too hard Helpful to know quantum mechanics not totally necessary Next time Quantum field theory of light Photon Optics Hecht 333 Simple version of quantum theory Say that light is really composed of particles particles called photons Energy of photon 2 had it Planck39s constant 2 1054gtlt1O 34 J s w oscillation frequency of light Don39t worry about what is oscillating for now Polychromatic light many photons with different w39S 0 Pulse of light with energy U in J U contains N photons hw N photon number o Beam with power P in W Js P delivers CD 7 photonss hw CD photon flux 0 Beam with irradiance r in Wm2 I photonss at I ha m2 5 photon flux density haqu Also light with energy density ur in Jm3 hotons has photon denSIty i ha m Example Sunlight has an irradiance of about 250 Wm2 Assume average wavelength 2 500 nm Then average ha 2 4 x 10 19 J Photon flux density 6 x 1020 photonss m2 Looking at sun pupil area m 10 6 m 2 collect about 6 X 1014 photons in 1 s Photons are not classical particles gt propagate according to wave equation not Newton39s laws Procedure use wave techniques to calculate r Then I gives flux density Imagine photons follow wave like surfers in ocean But photons and wave are inseparable Best to interpret probabilistically Average number of photons in volume d3quot ur d3r r d3r N hw have If N ltlt 1 interpret as probability to find one photon Important No definite trajectory for individual photons Picture surfers scrambling back and forth on wave Avoids two slitquot paradox In two slit interferometer which slit does photon pass through Correct interpretation It doesn39t matter The wave passes through both Interference pattern says where photons can go Question What ifa wave passes through beam spitter and the outputs are separated by a large distance Does it make sense to ask where one of the photon is If wave determines photon distribution why use photons Because detectors see photons not waves Or light can only transfer energy in units of had So photons important whenever light is emitted or absorbed Example photoelectric effect Shine light on metal Electrons absorb energy from light and escape 39 electrons 39 Find that maximum electron energy 2 ha 2 max energy absorbed from photon Doesn39t depend on irradiance just a Another example photomultiplier tube Light hits metal plate detaches single electron of K 1000 V 800 V 600 V Accelerate electron to second plate detach more electrons Cascade many plates get big current pulse See blip on output detection of one photon Other photon properties 0 Momentum p hk Hecht 334 Interesting effect suppose atom absorbs photon makes internal transition U0 gt U1 Say atom velocity before absorption 2 v Velocity after absorption 2 v l hkM gets kick from photon v H atom mass M k 1 Initial energy Utot 2 U0 l 5M1 l ha 2 1 k Final energy Utot 2 U1 l EM v l Energy conserved so U1 U0 E First term standard QM mg2 7 k w 2 V Second term radiative correction Third term Doppler shift Derive Doppler effect from QM o Angular momentum Hecht 815 Circular polarization states have definite spin Right circular Lphoton 4112 Left circular Lphoton hR Linear polarized states 2 superposition of ER and A35 Photon equally likelty to be spin upquot and spin down Important for transition selection rules Detecting Photons Already mentioned one way to see single photons photo mulitplier tube PMT Are there other methods Can characterize light detectors by three parame ters quantum efficiency background rate saturation rate Quantum efficiency 5 probability that incident photon is detected For PMT typically about 5 limited by emission of first electron Background rate R0 2 count rate observed with no incident light For good PMT as low as N 1 counts due to thermal emission of electrons Saturation rate Rsat s max measurable count rate For PMT about 106 countss takes about 1 us to recover between clicks39 To see single bhotons need incident flux ltlgt with R0 lt 64 lt Rsat Impossible if R0 2 Rsat Some detectors Device l a l R0 countss l Rsat countss PMT 1 20 lt 1 to 105 1o6 1o9 Photodiode 20 90 104 lt R0 Avalanche PD 30 70 10 to 5000 106 107 Note PMTs have much larger area than APDs Quantum Noise Important effect of photon theory light is noisy Suppose laser beam with power P measure with ideal photon counter In time T detect N PThw photons But photons are randomly distributed in wave Don39t expect to get exactly N every time Guess that photons are distributed independently Probablity to detect a photon doesn39t depend on when previous photon was detected Why Photons emitted by independent atoms Don39t expect correlations in emission times Reconsider this assumption next time 20 Then photons obey Poisson statistics If on average you detect N photons in time T then N fluctuates by AN N12 Same as fluctuations in flipping coins most other statistical noise Sometimes called shot noise comes from photons being discrete shots Better name is quantum noise 21 So no such thing as light with constant P Always detect some fluctuations Size of fluctuations depends on time scale Measure for time T get relative noise APAN 1 hw 12 P N N12 PT Noise goes up as T goes down just like normal noise goes down with averaging 22 Question If your detector has a quantum efficiency less than one will the quantum noise scale as the square root of the number of incident photons or the number of mea sured photons All real detectors also have technical noise 2 noise sources besides quantum mechanics For instance noise in background signal electrical pick up vibration induced noise noise from temperature drifts etc 23 Most often technical noise dominates Quantum noise typically important for Photon counting applications N is low so ANN is large High speed applications T is small gt need high power to measure fast signals Can distinguish quantum and technical noise Quantum noise oc P12 Technical noise oc P or indep of P 24 Quanturn States So far haven39t really needed quantum mechanics just idea that light energy is discrete But quantum mechanics of light is interesting too Easiest to see using polarization Suppose beam with j c0505 l sin a 25 Count number of photons polarized along 96 y For instance PMT Imm PMT polarizing beam splitter 26 If average photon number N expect N36 2 cos2a N Ny sin2 oz N on average If N ltlt 1 often see no photons but when one is detected have prob cos2a to be along 90 sin2a along y Similar to QM for particle in superposition state 27 Natural to describe photon with wave function w COSaX l sin a where 2 state polarized along a 2 state polarized along g Q R Note 1p Ni for right interpretation of gt2 37 More generally have Mr N Er Electric field of wave m wave function of photons Optics and quantum mechanics very similar 28 But quantum mechanics has more possibilities Imagine light pulse with two photons QM allows polarization state 12 E ii X1372 Y1X2 Always measure one photon polarized along 90 and one along y 29 But light not polarized in any usual way Compare light polarized at 450 has m 221 2 X2 2 X1922 l X1372 l 371922 l Y1Y2gt Have 50 Chance to measure both photons in same state Not the same as state X 30 Phys 531 Lecture 13 6 October 2004 Superposition and Interference Last time wrapped up ray optics o aberrations 0 practical considerations Today return to wave optics Start developing tools for real calculations Outline 0 Interference o Interference examples o Interference in time o Pulse propagation group and phase 11 Next time see how to construct pulses from fre quency components Fourier transforms Interference HeCht 71 Consider two waves E1 A1 ikz wt E2 A2 ikz wt Note for next six lectures ignore polarization If you want assume all fields polarized along gt2 Also remember really E1 lA1lcoslltz wt l 51 for A1 lAllei 1 Total field is sum of individual waves Etot E1 E2 A1 A2 ikzw What is the real field Need magnitude and phase of Amt A1 l A2 Work out lAtotl2 Atot Atot lAlleiqbll lA2l i 2lA1l i 1llA2l i 2 lA1l2 r42 lA1llA2le 1 2 lA1llA2le l 1 2 l141l2 lA2l2 2lA1llA2l COSlt51 52 4 and Im Atot Re Atot Im A1Im A2 ReA1 l Re A2 l141l5in 51 lA2l Si 52 lA1lCOSlt151 lA2lCOS 2 So can find amplitude and phase of total wave tan lt15tot Usually amplitude is more interesting lAtotl2 0 tot l lAtotl2 r41 mg 2lA1llA2l cosltqgt1 2 Lets us be quantitative about interference Constructive interference when 51 s 52 lAtotl2 lA1l lA2l2 Destructive interference when 51 s q52 lAtotl2 lA1l l142l2 Given any 6 E 51 52 we have formula Note if 6 7r2 then lAtotl2 l141l2 l142l2 Irradiances add Irradiances also add if 6 fluctuates randomly 1 tot TUE 0C lAtotl2gt 770 so interference term gt 0 if cos 6 0 Then say waves are incoherent Question Can two harmonic waves be incoherent Generalizes to N waves Atot Z An TL lAtotl2 ZAnZAii n m E lAnllAmleWn fm nm N Z lAnl2 Z lAnllAmleW rm n21 n m Second term gt O for incoherent fields Also get tanqb anAnlsmqbn tot Zn lAnlCOS qbn For incoherent fields qbtot fluctuates randomly More about incoherent fields at end of course generally assume coherent now Look closer at phases Say waves generated by two point sources 1 0 0 1391 r2 Field from 1 AleiCklr rll wt Field from 2 AQeiCklr r2l wt Total eld lEtotl2 lE1 E2l2 l141l2 lA2l2 2lA1llA2l COSltgt1 52 kl r1l kl r2l Interference term depends on position get interference pattern in space In medium k nkio Optical path length difference 81 82 lrl Tl lr2 Tl Write interference phase 6 151 152 l koAr Like in scattering picture 8 determines interference effects Example Two sources at 00 Hecht 714 olll eHilu Z Then fields gt plane waves Equivalent to plane wave reflected by mirror ill What is Em EM 2 A1 ikz wt A2 i kz wt Suppose A1 A2 Then Etot Ale iwteikz 6 ik2 21416 th coskz 50 lEtotl 2lA1ll COSkZl and real field Etot 2lA1l cosq51 wt coskz Not a travelling wave Called standing wave Fixed pattern oscillating in time Nodes points where Etot O Antinodes points where lEtotl max 2lA1l 14 Standing wave is one interference pattern Another example Hecht 91 Two point sources separated by a Observation point gt 00 at angle 6 Here dasin0a0 If A1 A2 then iEtoti2 iA1i2 iA2i2 2iA1HA2i COSlt51 52 koA 2iA1i21 COSkOA 4iA1i cos2 s 4iA1i cos2 koge 0 6 Nodes where koA2 n l 7r 1 koaG n E 27r A l 27r Qm s ncek or altn2 0 A Demo two slit interference Setup I Laser I gt gt Due to diffraction slits m point sources For large 6 more complicated Can39t use sin 6 m 6 Can39t treat slit as point source For now gives idea Basic points 0 Calculate field at position r by algebraically sum ming incident waves o Phase of each field depends on source phase optical path length between source and r Question If the phase of the two point sources were dif ferent how would the two slit pattern change Question What if the amplitudes of the sources were dif ferent Temporal Interference Hecht 72 Can also get interference in time Say E1 Aleiklzw1t E2 A2 ik22 w2t Wiwi With kil39 and W1 3k 412 What is total field Etot E1 l E2 20 Define 01239 kiz wit lEtotl2 lAlleml lA2lem2 lAlle m1lA2le m2 l141l2 l142l2 2lA1llA2lCOSa1 012 Intererence term 2 COSIlt1 k2Z W1 W2t 51 2 At fixed 2 irradiance oscillates at M W E beat note Identical effect with sound waves demo 21 Often useful to think of Etot as single plane wave with changing amplitude Say lA1l lA2l3 Etot A1 ik12 w1t 1 lAlleiCk2Z W2t 2 Define J w I w and wmw1 w2 Then cal 252 El l l W2 Define similarly for E km lt13 1 22 Then write Etot lA1l iEzQt X egkmz wmt m kmz wmt m 2lA1leiEZ t cos kmz wmt pm 2 Looks like plane wave Jab Qt with amplitude kimZ wmt ban AtotCZat 2lA1l i C05 2 Most useful if Um ltlt J Then Amt slowly varying Field m plane wave at any z t 23 Picture at fixed t cos 2kawmtl coslzat tot Z This is example of a modulated wave like a plane wave but amplitude andor phase slowly varies 24 Call 610324 2 carrier wave Call At0tzt envelope function contains carrier wave Might ask how fast does wave propagate Important envelope used to transmit information Define phase velocity 2 speed of carrier wave 2 speed of individual peaks I W1 412 Uphase E Cnlwl ngwg 25 If cal tug then n1 n2 Uphase gt Cn Makes sense Define group velocity 2 speed of envelope cum cal Lug cal Lug U rou g p km k1 k2 nlwl n2w2 If cum ltlt 5 then get dd 72 U rou gt U 9 D dk Q phase 26 So envelope travels at entirely different speed Generalizes to any modulated wave Complicated envelopes more frequency components gt single pulse of light travels at Ugroup Usually easiest to evaluate ii 1dn vgdwdw c c dw C So 11 g nDj Z 27 In vacuum n 1 and dndw O 50 vgroup phase 0 Only matters in medium Recall for normal dispersion dndw gt O C lUQI OUD lt g 39Uphase Example In BK glass n 546 nm 2 151872 and n588 nm 2 15168 What is the difference between the group and phase velocities near these wavelengths 28 Solution Average n 15178 so phase velocity z 06588c dn dn dA For 129 need 4 w dw dA dw dn 151872 15168 5 1 Have 46 x 10 nm dA 546 588 2 dA 2 Also A 3 so 4 w A Use A 567 nm to w 42 dn dn 75 71 So 4 A 576 nm x 46 x 10 nm 2 00265 dw dA 12 c c 064750 n 6 15178 00265 Difference up v9 0011c z 2 29 Usually a small effect dn But not always Near resonance cum gtgt 1 Generally uphase and Ugroup both arbitrary Can get n lt 1 near resonance gt up gt c Get 119 gt c for large anomalous dispersion Didn39t worry about up gt c carrier has no information Should we worry about 119 gt c 30 Remember in scattering picture fields travel at c gt c remains ultimate speed limit What does 119 gt c mean 8 Z1 Consider pulse of light Li D Q 2 1 E 22 Two features J o Peak at some position Z1 Z o Leading edge at position Z2 Leading edge indicates time you initiated pulse physical discontinuity 31 Group velocity gives speed of peak Can be larger than 0 Simple example Suppose medium transmits for time ltlt Atpulse Only front part of pulse transmitted 121 Peak Z1 has advanced 39UggtC 32 Phys 531 Lecture 9 30 September 2004 Ray Optics II Last time developed idea of ray optics approximation to wave theory Introduced paraxial approximation rays with 6 ltlt 1 Will continue to use Started disussing imaging and lenses 1 1 52 f Basic equation of paraxial optics 1 Thin lens equation l 50 Example Suppose a thin lens of focal length f 100 cm is placed 50 cm in front of a small light bulb Where will the image of the bulb be located focal point object W l l 9so 50 cm f100 cm Solution Real object upstream of lens so so 50 cm 1 1 1 1 1 Then Cmil s f so 100 cm 50 cm So 3 333 cm Negative so image located 333 cm before lens 2 167 cm from object Today continue with imaging 0 Imaging extended objects 0 Multiple lenses 0 Mirrors 0 Apertures Next time Techniques for dealing with multi lens systems Finite Imaging Hecht 523 Before all pictures showed point to point imaging Points are on optic axis 2 symmetry axis of lens Finite object 2 collection of points must deal with points that are off axis Picture f Construct image using ray diagram Three simple rays ray through lens center undeviated ray through front focal point becomes horizontal horizontal ray hits back focal point NW Image where rays intersect Each point in object plane gt point in image plane Here image inverted object height go gt 0 maps to image height yi lt 0 Define magnification m amp yo Get magnification from diagram Yo r Si so vh 8239 80 Triangles are Similar so 7 7 9239 yo 8 Then m g Z 0 50 with 3239 determined by thin lens equation For negative parameters follow sign conventions Since f lt 0 front and back focal points reversed See that silt0 and mgtO Ray diagrams good tool for understanding Lens systems If more than one lens apply thin lens law in succession Consider two lenses f1 f2 separated by d will v v SO u 3 Find image distance si First lens image distance sil 1 1 1 821 f1 80 Second lens object distance 302 d sil 1 1 Then final image at 5239 8239 f2 802 Combine get 8 f2d80 f1f2d f1f280 Z Sod Sofl Son dfl l39flfQ Not very enlightening


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