Optics and Non-Linear Optics I Non-linear Optics Tutorial Sheet November 2007
|
|
- Cynthia Henry
- 5 years ago
- Views:
Transcription
1 Optics and Non-Linar Optics I Non-linar Optics Tutorial Sht Novmbr An altrnativ xponntial notion somtims usd in NLO is to writ Acos (") # 1 ( Ai" + A * $i" ). By using this notation and substituting into th inducd non-linar polarisation quation (us only th χ (1) and χ () trms), show that th trms gnratd ar th sam as thos w obtaind in lcturs..a. Explain what is mant by cohrnc lngth. b. Calculat th cohrnc lngth for a fundamntal wav of 1064nm gnrating a scond harmonic wav of 53nm in a KTP crystal. (n ω = 1.74, n ω =1.78) c. Sktch how th nd harmonic convrsion fficincy varis with distanc along th crystal with th cohrnc lngths markd on th x-axis. d. Calculat th maximum convrsion fficincy for SHG for a mm long non-phas matchd crystal (d ff =3.18pm/V) for th following pump powrs (P ω = 50mW, 5W, 100W, 1000W) focussd to a spot siz of 0µm.. What input powr is rquird to obtain an SHG fficincy of 10% aftr 1 cohrnc lngth in th scnario outlind abov? 3. a. Sktch th rfractiv indx as a function of angl for th ordinary and xtraordinary rays in a ngativ uniaxial crystal. Mark on your diagram th valus of n o and n. b. Starting from th llipsoidal quation: # 1 & % ( $ n (")' = 1 # & # % ( cos " + % $ n o ' $ 1 & ( sin " n ' Show that th phas matching angl for scond harmonic gnration from frquncy ω to ω, θ m, is givn by: o n " m = sin #1 $ ( ) # o # ( n $ ) # ( n $ ) # o # ( n $ ) # c. In ordr to work out th rfractiv indx for a particular wavlngth, w mak us of formula known as th Sllmir Equations. For th common non-linar matrial Bta-Barium Borat (BBO) w hav: n o = (" ) " and n = (" ) " Whr th wavlngth, λ, is writtn in µm. Calculat n o and n for wavlngths of 1064nm and 53nm and hnc work out th phas matching angl for scond harmonic gnration.
2 d. It is now dcidd to frquncy doubl th 53nm radiation in a scond crystal. What is th nw phas matching angl? Will th fficincy of gnration b th sam?. What limits th rang of wavlngths that a particular non-linar optical can b usd for SHG? 4. In our labs w hav grown a nw ngativ uniaxial crystal. Th crystal mlts at 140 o C is it suitabl to us this crystal in a non-critical phas matching gomtry for scond harmonic gnration from a fundamntal wavlngth of 800nm? Data: n o 800nm = and n 400nm = (Both masurd at 0 o C) dn o /dt = 0.001K -1 dn /dt = 0.00K a. For a rang from 0 to 10xl c, sktch th scond harmonic output powr vrsus distanc along th crystal for unphasmatchd LiNbO 3. Mark th x axis in units of l c. b. In ordr to obtain quasi-phasmatching, w now priodically pol th crystal by invrting a distanc of with = l c with a grating priod = l c, sktch on you diagram th gnratd scond harmonic vrsus distanc. c. For som wavlngths poling vry cohrnc lngth is impossibl. Instad w hav to mov to highr ordr priods. Sktch on your graph th gnratd scond harmonic vrsus distanc for third ordr (i width of pold rgion = 3l c, priod = 6l c ) priodic poling. Commnt on th fficincy of this mthod of SHG. d. Commnt on th proprtis rquird from a crystal to allow lctric fild poling for fficint SHG.. Using th data providd in lcturs, calculat th grating priod rquird for first-ordr QPM to obtain SHG from λ incidnt =946nm. 6.a. Show that whn considring χ (3) intractions and starting from E 3 0 χ (3) cos 3 (ωt) th gnratd trms can b writtn in th form: " (3) ( ( ) + cos(3#t) ). 4 E 3 0 3cos #t b. Starting from n = ((1 + χ (1) + χ (3) I(ω)), show that th rfractiv indx can b approximatd such that: n " n 0 + n I(#) whr n = " (3). n 0 7. a. Dscrib th origin of th stoks and antistoks photons. b. An 800nm lasr is focussd onto a sampl of intrst and stoks lins ar obsrvd at 847nm, 876nm, 947nm and 1030nm. Calculat υ R for ach transition in units of wavnumbrs. c. Using th simpl diagram shown in lcturs for th origin of antistoks radiation, calculat th xpctd antistoks wavlngths from this sampl. d. Assuming a 1064nm lasr was usd to xcit th sampl, calculat th stoks wavlngths you would xpct to obsrv.
3 1. An altrnativ xponntial notion som tim usd in NLO is to writ Acos (") # 1 ( Ai" + A * $i" ). By using this notation and substituting into th inducd non-linar polarisation quation (us only th χ (1) and χ () trms), show that th sam gnratd frquncy trms ar obtaind as obtaind in lcturs. So w now hav: E = E " cos("t) = 1 (E " i"t + E " * #i"t ) Substituting into: P = ε 0 χ (1) E +ε 0 χ (1) E givs: P = " 0# (1) ( E $ i$t + E * $ %i$t ) + " 0# () ( E $ i$t + E * $ %i$t )( E $ i$t + E * $ %i$t ) 4 = " 0# (1) ( E $ i$t + E * $ %i$t ) + " 0# () E $ i$t + E $ E * $ + E * * $ E $ + E $ %i$t 4 Now : EE * = EE * = E ( ) ( ) So : P = " 0# () E $ + " 0# (1) ( E $ i$t + E * $ %i$t ) + " 0# () 1 E $ i$t * + E $ %i$t & " 0# () E $ + " 0 # (1) cos($t) + " 0# () cos($t) Giving th sam trms as gnratd in lcturs (without nding to know th trig id s!).a. Explain what is mant by cohrnc lngth. Th cohrnc lngth is th ffctiv intraction lngth in non-linar optics whn k ω k ω. Th gnratd output rachs a maxmimum aftr on cohrnc lngth bfor dstruciv intrfrnc rducs it to 0 aftr anothr cohrnc lngth. b. Calculat th cohrnc lngth for a fundamntal wav of 1064nm gnrating a scond harmonic wav of 53nm in a KTP crystal. (n ω = 1.74, n ω =1.78) l 0 = " #k = $ fundamntal 4[n % & n % ] = 1.064x10-6 / (4 ( )) = 6.65µm c. Sktch how th nd harmonic convrsion fficincy varis with distanc along th crystal with th cohrnc lngths markd on th x-axis. d. Calculat th maximum convrsion fficincy for SHG (1.064µm 53nm) for a mm long non-phas matchd crystal (d ff =3.18pm/V) for th following pump powrs focussd to a spot siz of 0µm. P ω = 50mW, 5W, 100W, 1kW Maximum convrsion fficincy occurs aftr ach odd intgr cohrnc lngth. Tak xprssion from lcturs and substitut Δk = π/l 0 : " SHG = C L I # sinc ($kl /) = " 0 sinc (%L /l 0 )
4 So aftr on cohrnc lngth: sinc (π/) = (/π) = So η SHG = x η 0 Powr/W Intnsity = η 0 η SHG powr / πω 0 [W/m ] E E % E E % E E % E E %. What input powr is rquird to obtain an SHG fficincy of 10% aftr 1 cohrnc lngth in th scnario outlind abov? In this cas w want η SHG = 0.1 η 0 = 0.1/0.405 = = C L I ω I ω =0.47/C L = 0.47 / (5.036x10-8 x 4x10-6 ) = 1.x10 1 Wm - P ω = I ω A = I ω x πω 0 = 1533W 3. i. Sktch th rfractiv indx as a function of angl for th ordinary and xtraordinary rays in a ngativ uniaxial crystal. Mark on your diagram th valus of n o and n. From lctur nots: Optical Axis n o (θ) n o n n (θ) ii. Starting from th llipsoidal quation: # 1 & % ( = 1 # & % ( cos " + 1 # & % ( sin " $ n (")' $ n o ' $ n ' Show that th phas matching angl for scond harmonic gnration from frquncy ω to ω, θ m, is givn by:
5 o n " m = sin #1 $ ( ) # o # ( n $ ) # ( n $ ) # o # ( n $ ) # For scond harmonic gnration w hav to st n ω(θ) = n o ω. Substituting into th abov qution: # 1 & # 1 & # % o ( = % o ( cos 1 & # ) + % ( sin 1 & ) = % o ( 1* sin ) $ n " ' $ n " ' $ n " ' $ n " ' Thrfor : ( ) sin ) o n * o " * n * " = n * o " * n * " So : sin ) = n o * o " * n * " n * o " * n * " o * o n Thrfor :) = sin *1 " * n * " n * o " * n * " ( ) + # 1 & % ( sin ) $ ' n " iii. In ordr to work out th rfractiv indx for a particular wavlngth, w mak us of formula known as th Sllmir Equations. For th common non-linar matrial Bta-Barium Borat (BBO) w hav: n o = (" ) " and n = (" ) " Whr th wavlngth, λ, is writtn in µm. Calculat n o and n for wavlngths of 1064nm and 53nm and hnc work out th phas matching angl for scond harmonic gnration. n o 1064 = 1.655, n o 53 = 1.674, n 1064 = 1.539, n o 53 = Substitut back into th answr drivd in part ii to obtain θ m =.8 o iv. It is now dcidd to frquncy doubl th 53nm radiation in a scond crystal. What is th nw phas matching angl? Will th fficincy of gnration b th sam? Scond harmonic of 53nm is 66nm. n o 53 =1.674, n o 66 = 1.759, n 53=1.555, n 66=1.613 Givs phas matching angl of 47.6 o. Efficincy will not b th sam as χ cofficint dpnds on dirction. v. What limits th rang of wavlngths that a particular non-linar optical can b usd for SHG? Limitd by crystal transparncy rang. For BBO 190nm to 3500nm. 4. In our labs w hav grown a nw ngativ uniaxial crystal. Th crystal mlts at 140 o C is it suitabl to us this crystal in a non-critical phas matching gomtry for scond harmonic gnration from a fundamntal wavlngth of 800nm? Data: n o 800nm = and n 400nm = (Both masurd at 0 o C) dn o /dt = 0.001K -1 dn /dt = 0.00K -1. For NCPM, us 90 o phas matching and us tmpratur to tun n ω = n o ω. So: n o 800+ dn o /dt x ΔT = n dn /dt x ΔT
6 Givs ΔT = 86K so crystal should b hatd to 106 o C and things should b ok! 5. a. For a rang from 0 to 10xl c, sktch th scond harmonic output powr vrsus distanc along th crystal for unphasmatchd LiNbO 3. Mark th x axis in units of l c. b. In ordr to obtain quasi-phasmatching, w now priodically pol th crystal by invrting a distanc of with = l c with a grating priod = l c, sktch on you diagram th gnratd scond harmonic vrsus distanc. c. For som wavlngths poling vry cohrnc lngth is impossibl. Instad w hav to mov to highr ordr priods. Sktch on your graph th gnratd scond harmonic vrsus distanc for third ordr (i width of pold rgion = 3l c, priod = 6l c ) priodic poling. Commnt on th fficincy of this mthod of SHG. Sktch is shown on nxt pag. Th procss of priodic poling is obviously lss fficint that BPM. In gnral th fficincy of QPM can b shown to b (s Kochnr p.65) /(πw ) x E bpm whr w is th numbr of cohrnc lngths pold and E bpm is th fficincy of BPM. So whn w pol 3l c w hav a factor 9 rduction in fficincy. d. Commnt on th proprtis rquird from a crystal to allow lctric fild poling for fficint SHG. In ordr for lctric fild poling to b succssful, th crystal must b frrolctric (i, a prmannt chang in polarisation must b causd by th incidnt lctric fild.) Th crystal must also hav a good non-linar cofficint in th dsird propagation dirction and also b transparnt for th fundamntal and SHG wavlngths.. Using th data providd in lcturs, calculat th grating priod rquird for first-ordr QPM to obtain SHG from λ incidnt =946nm. So λ incidnt =946nm λ SHG = 473nm From lcturs: ( n o ) = " " # So: n 946 =.404 n 473 =.3593 (rmmbr to us λ in µm in th quation!)
7 So following th quations givn: " = 4" [ # g $ n % & n % ] Thrfor : $ % # g = (n % & n % ) W can obtain Λ g = / ( ) = (-) 3.98µm this is vry hard to do! 6.a. Show that whn considring χ (3) intractions and starting from E 3 0 χ (3) cos 3 (ωt) th gnratd trms can b writtn in th form: " (3) 3 ( 3cos (#t) + cos(3#t) ). 4 E 0 W hav trms arising from: E 0 3 χ (3) cos 3 (ωt) E 0 3 χ (3) cos 3 (ωt) = E 0 3 χ (3) cos(ωt)cos (ωt) Using th idntity: cos x = ½ + ½ cos(x) E 3 0 χ (3) cos 3 (ωt) = E 3 0 χ (3) cos(ωt)( ½ + ½ cos(ωt)) Using th idntity: = E 0 3 χ (3) ( ½cos(ωt) + ½cos(ωt)cos(ωt)) cosacosb = ½ (cos(a-b) + cos(a+b)) E 0 3 χ (3) cos 3 (ωt) = E 0 3 χ (3) ( ½cos(ωt) + ¼ (cos(-ωt)+cos(3ωt))) Now cos(-x) = cos(x) so: E 0 3 χ (3) cos 3 (ωt) = E 0 3 χ (3) ( ¾ cos(ωt) + ¼cos(3ωt)) = " (3) 4 E 3 0 3cos #t ( ( ) + cos(3#t) ) QED b. Starting from n = ((1 + χ (1) + χ (3) I(ω)), show that th rfractiv indx can b approximatd such that: n " n 0 + n I(#) whr n = " (3) n 0. n = = 1+ " (1) 1+ " (1) + " (3) I(#) $ ( ) 1 1+ " (3) I(#) & % $ = n 0 1+ " (3) I(#) ' & ) % n 0 ( $ * n 0 1+ " (3) I(#) ' & ) % n 0 ( 1+ " (1) 1 * n 0 + " (3) I(#) n 0 = n 0 + n I(#) Q.E.D ' ) ( 1 Making us of th binomial xpansion of (1+x) 1/ 1+x/+. 7. a. Dscrib th origin of th stoks and antistoks photons.
8 Stoks and antistoks radiation arrivs as a consqunc of th Raman ffct. In th cas of stoks light, som of th incidnt radiation is coupld into a vibrational mod rducing th nrgy of th photon and gnrating a longr wavlngth photon. In th cas of antistoks, an xcitd vibrational mod givs up som nrgy to th incoming photon gnrating radiation of a shortr wavlngth than th pump. b. An 800nm lasr is focussd onto a sampl of intrst and stoks lins ar obsrvd at 847nm, 876nm, 947nm and 1030nm. Calculat υ R for ach transition in units of wavnumbrs. In this cas, w nd to calculat th frquncy of th radiations in wavnumbr units. So: 800nm 800x10-7 cm 1500cm -1 Similarly 847nm 11806cm -1, 876nm 11415cm -1, 947nm 10560cm -1, 1030nm 9708cm -1 In ach cas w can obtain ν R from ν P - ν S : 847nm ν R = = 694cm -1, 876nm ν R = 1085cm -1, 947nm ν R = 1940cm -1, 1030nm ν R = 79cm -1. c. Using th simpl diagram shown in lcturs for th origin of antistoks radiation, calculat th xpctd antistoks wavlngths from this sampl. From th simpl diagram ν AS = ν R +ν P, so w hav antistoks wavs at frquncis of: = 13194cm -1 λ AS =(1/1319)cm = 757nm = 13585cm -1 λ = 736nm = 14440cm -1 λ = 69nm λ = 654nm d. Assuming a 1064nm lasr was usd to xcit th sampl, calculat th Stoks wavlngths you would xpct to obsrv. λ p =1064nm 9398cm -1, thrfor w ll hav stoks wavs at: cm nm, nm, nm and nm.
surface of a dielectric-metal interface. It is commonly used today for discovering the ways in
Surfac plasmon rsonanc is snsitiv mchanism for obsrving slight changs nar th surfac of a dilctric-mtal intrfac. It is commonl usd toda for discovring th was in which protins intract with thir nvironmnt,
More informationPhys 402: Nonlinear Spectroscopy: SHG and Raman Scattering
Rquirmnts: Polariation of Elctromagntic Wavs Phys : Nonlinar Spctroscopy: SHG and Scattring Gnral considration of polariation How Polarirs work Rprsntation of Polariation: Jons Formalism Polariation of
More information2. Background Material
S. Blair Sptmbr 3, 003 4. Background Matrial Th rst of this cours dals with th gnration, modulation, propagation, and ction of optical radiation. As such, bic background in lctromagntics and optics nds
More information2. Laser physics - basics
. Lasr physics - basics Spontanous and stimulatd procsss Einstin A and B cofficints Rat quation analysis Gain saturation What is a lasr? LASER: Light Amplification by Stimulatd Emission of Radiation "light"
More information6. The Interaction of Light and Matter
6. Th Intraction of Light and Mattr - Th intraction of light and mattr is what maks lif intrsting. - Light causs mattr to vibrat. Mattr in turn mits light, which intrfrs with th original light. - Excitd
More information22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.
Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M
More informationChapter 6: Polarization and Crystal Optics
Chaptr 6: Polarization and Crystal Optics * P6-1. Cascadd Wav Rtardrs. Show that two cascadd quartr-wav rtardrs with paralll fast axs ar quivalnt to a half-wav rtardr. What is th rsult if th fast axs ar
More informationorbiting electron turns out to be wrong even though it Unfortunately, the classical visualization of the
Lctur 22-1 Byond Bohr Modl Unfortunatly, th classical visualization of th orbiting lctron turns out to b wrong vn though it still givs us a simpl way to think of th atom. Quantum Mchanics is ndd to truly
More informationTitle: Vibrational structure of electronic transition
Titl: Vibrational structur of lctronic transition Pag- Th band spctrum sn in th Ultra-Violt (UV) and visibl (VIS) rgions of th lctromagntic spctrum can not intrprtd as vibrational and rotational spctrum
More informationLecture Outline. Skin Depth Power Flow 8/7/2018. EE 4347 Applied Electromagnetics. Topic 3e
8/7/018 Cours Instructor Dr. Raymond C. Rumpf Offic: A 337 Phon: (915) 747 6958 E Mail: rcrumpf@utp.du EE 4347 Applid Elctromagntics Topic 3 Skin Dpth & Powr Flow Skin Dpth Ths & Powr nots Flow may contain
More informationThe pn junction: 2 Current vs Voltage (IV) characteristics
Th pn junction: Currnt vs Voltag (V) charactristics Considr a pn junction in quilibrium with no applid xtrnal voltag: o th V E F E F V p-typ Dpltion rgion n-typ Elctron movmnt across th junction: 1. n
More informationSAFE HANDS & IIT-ian's PACE EDT-15 (JEE) SOLUTIONS
It is not possibl to find flu through biggr loop dirctly So w will find cofficint of mutual inductanc btwn two loops and thn find th flu through biggr loop Also rmmbr M = M ( ) ( ) EDT- (JEE) SOLUTIONS
More informationA Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
More informationPrinciples of active remote sensing: Lidars. 1. Optical interactions of relevance to lasers. Lecture 22
Lctur 22 Principls of activ rmot snsing: Lidars Ojctivs: 1. Optical intractions of rlvanc to lasrs. 2. Gnral principls of lidars. 3. Lidar quation. quird rading: G: 8.4.1, 8.4.2 Additional/advancd rading:.m.
More information10. The Discrete-Time Fourier Transform (DTFT)
Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w
More informationSection 11.6: Directional Derivatives and the Gradient Vector
Sction.6: Dirctional Drivativs and th Gradint Vctor Practic HW rom Stwart Ttbook not to hand in p. 778 # -4 p. 799 # 4-5 7 9 9 35 37 odd Th Dirctional Drivativ Rcall that a b Slop o th tangnt lin to th
More informationWhy is a E&M nature of light not sufficient to explain experiments?
1 Th wird world of photons Why is a E&M natur of light not sufficint to xplain xprimnts? Do photons xist? Som quantum proprtis of photons 2 Black body radiation Stfan s law: Enrgy/ ara/ tim = Win s displacmnt
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationRandom Process Part 1
Random Procss Part A random procss t (, ζ is a signal or wavform in tim. t : tim ζ : outcom in th sampl spac Each tim w rapat th xprimnt, a nw wavform is gnratd. ( W will adopt t for short. Tim sampls
More informationQuasi-Classical States of the Simple Harmonic Oscillator
Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More information5.80 Small-Molecule Spectroscopy and Dynamics
MIT OpnCoursWar http://ocw.mit.du 5.80 Small-Molcul Spctroscopy and Dynamics Fall 008 For information about citing ths matrials or our Trms of Us, visit: http://ocw.mit.du/trms. Lctur # 3 Supplmnt Contnts
More informationAtomic and Laser Spectroscopy
L-E B, OL, MOV 83 Atomic and Lasr Spctroscopy Th aim of this xrcis is to giv an ovrviw of th fild of lasr spctroscopy and to show modrn spctroscopic mthods usd in atomic, molcular and chmical physics.
More informationCollisions between electrons and ions
DRAFT 1 Collisions btwn lctrons and ions Flix I. Parra Rudolf Pirls Cntr for Thortical Physics, Unirsity of Oxford, Oxford OX1 NP, UK This rsion is of 8 May 217 1. Introduction Th Fokkr-Planck collision
More informationForces. Quantum ElectroDynamics. α = = We have now:
W hav now: Forcs Considrd th gnral proprtis of forcs mdiatd by xchang (Yukawa potntial); Examind consrvation laws which ar obyd by (som) forcs. W will nxt look at thr forcs in mor dtail: Elctromagntic
More informationOn the Hamiltonian of a Multi-Electron Atom
On th Hamiltonian of a Multi-Elctron Atom Austn Gronr Drxl Univrsity Philadlphia, PA Octobr 29, 2010 1 Introduction In this papr, w will xhibit th procss of achiving th Hamiltonian for an lctron gas. Making
More informationThe graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the
Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th
More informationu 3 = u 3 (x 1, x 2, x 3 )
Lctur 23: Curvilinar Coordinats (RHB 8.0 It is oftn convnint to work with variabls othr than th Cartsian coordinats x i ( = x, y, z. For xampl in Lctur 5 w mt sphrical polar and cylindrical polar coordinats.
More informationAdditional Math (4047) Paper 2 (100 marks) y x. 2 d. d d
Aitional Math (07) Prpar b Mr Ang, Nov 07 Fin th valu of th constant k for which is a solution of th quation k. [7] Givn that, Givn that k, Thrfor, k Topic : Papr (00 marks) Tim : hours 0 mins Nam : Aitional
More informationCOMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.
C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH
More informationIntroduction to the quantum theory of matter and Schrödinger s equation
Introduction to th quantum thory of mattr and Schrödingr s quation Th quantum thory of mattr assums that mattr has two naturs: a particl natur and a wa natur. Th particl natur is dscribd by classical physics
More informationGeneral Notes About 2007 AP Physics Scoring Guidelines
AP PHYSICS C: ELECTRICITY AND MAGNETISM 2007 SCORING GUIDELINES Gnral Nots About 2007 AP Physics Scoring Guidlins 1. Th solutions contain th most common mthod of solving th fr-rspons qustions and th allocation
More informationLinear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let
It is impossibl to dsign an IIR transfr function with an xact linar-phas It is always possibl to dsign an FIR transfr function with an xact linar-phas rspons W now dvlop th forms of th linarphas FIR transfr
More information1973 AP Calculus AB: Section I
97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=
More informationMath 34A. Final Review
Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right
More informationLecture 28 Title: Diatomic Molecule : Vibrational and Rotational spectra
Lctur 8 Titl: Diatomic Molcul : Vibrational and otational spctra Pag- In this lctur w will undrstand th molcular vibrational and rotational spctra of diatomic molcul W will start with th Hamiltonian for
More informationCoupled Pendulums. Two normal modes.
Tim Dpndnt Two Stat Problm Coupld Pndulums Wak spring Two normal mods. No friction. No air rsistanc. Prfct Spring Start Swinging Som tim latr - swings with full amplitud. stationary M +n L M +m Elctron
More informationProblem Set 6 Solutions
6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More informationNumerical Problem Set for Atomic and Molecular Spectroscopy. Yr 2 HT SRM
Numrical Problm St for Atomic and Molcular Spctroscopy Yr HT SRM Sction 1: Atomic Spctra 1. For ach of th atomic trm symbols 1 S, P, 3 P, 3 D, 4 D, writ down: a) Th associatd valus of th total spin and
More informationAlpha and beta decay equation practice
Alpha and bta dcay quation practic Introduction Alpha and bta particls may b rprsntd in quations in svral diffrnt ways. Diffrnt xam boards hav thir own prfrnc. For xampl: Alpha Bta α β alpha bta Dspit
More informationBifurcation Theory. , a stationary point, depends on the value of α. At certain values
Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local
More informationChapter 1. Chapter 10. Chapter 2. Chapter 11. Chapter 3. Chapter 12. Chapter 4. Chapter 13. Chapter 5. Chapter 14. Chapter 6. Chapter 7.
Chaptr Binomial Epansion Chaptr 0 Furthr Probability Chaptr Limits and Drivativs Chaptr Discrt Random Variabls Chaptr Diffrntiation Chaptr Discrt Probability Distributions Chaptr Applications of Diffrntiation
More informationHigher order derivatives
Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of
More informationHydrogen Atom and One Electron Ions
Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial
More informationde/dx Effectively all charged particles except electrons
de/dx Lt s nxt turn our attntion to how chargd particls los nrgy in mattr To start with w ll considr only havy chargd particls lik muons, pions, protons, alphas, havy ions, Effctivly all chargd particls
More informationStatistical Thermodynamics: Sublimation of Solid Iodine
c:374-7-ivap-statmch.docx mar7 Statistical Thrmodynamics: Sublimation of Solid Iodin Chm 374 For March 3, 7 Prof. Patrik Callis Purpos:. To rviw basic fundamntals idas of Statistical Mchanics as applid
More informationThere is an arbitrary overall complex phase that could be added to A, but since this makes no difference we set it to zero and choose A real.
Midtrm #, Physics 37A, Spring 07. Writ your rsponss blow or on xtra pags. Show your work, and tak car to xplain what you ar doing; partial crdit will b givn for incomplt answrs that dmonstrat som concptual
More informationHow can I control light? (and rule the world?)
How can I control light? (and rul th world?) "You know, I hav on simpl rqust. And that is to hav sharks with frickin' lasr bams attachd to thir hads! - Dr. Evil Phys 230, Day 35 Qustions? Spctra (colors
More informationDefinition1: The ratio of the radiation intensity in a given direction from the antenna to the radiation intensity averaged over all directions.
Dirctivity or Dirctiv Gain. 1 Dfinition1: Dirctivity Th ratio of th radiation intnsity in a givn dirction from th antnna to th radiation intnsity avragd ovr all dirctions. Dfinition2: Th avg U is obtaind
More informationContent Skills Assessments Lessons. Identify, classify, and apply properties of negative and positive angles.
Tachr: CORE TRIGONOMETRY Yar: 2012-13 Cours: TRIGONOMETRY Month: All Months S p t m b r Angls Essntial Qustions Can I idntify draw ngativ positiv angls in stard position? Do I hav a working knowldg of
More informationPHA 5127 Answers Homework 2 Fall 2001
PH 5127 nswrs Homwork 2 Fall 2001 OK, bfor you rad th answrs, many of you spnt a lot of tim on this homwork. Plas, nxt tim if you hav qustions plas com talk/ask us. Thr is no nd to suffr (wll a littl suffring
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationStudies of Turbulence and Transport in Alcator C-Mod Ohmic Plasmas with Phase Contrast Imaging and Comparisons with GYRO*
Studis of Turbulnc and Transport in Ohmic Plasmas with Phas Contrast Imaging and Comparisons with GYRO* L. Lin 1, M. Porkolab 1, E.M. Edlund 1, J.C. Rost 1, M. Grnwald 1, D.R. Mikklsn 2, N. Tsujii 1 1
More informationChapter 6. The Discrete Fourier Transform and The Fast Fourier Transform
Pusan ational Univrsity Chaptr 6. Th Discrt Fourir Transform and Th Fast Fourir Transform 6. Introduction Frquncy rsponss of discrt linar tim invariant systms ar rprsntd by Fourir transform or z-transforms.
More informationLecture 2: Discrete-Time Signals & Systems. Reza Mohammadkhani, Digital Signal Processing, 2015 University of Kurdistan eng.uok.ac.
Lctur 2: Discrt-Tim Signals & Systms Rza Mohammadkhani, Digital Signal Procssing, 2015 Univrsity of Kurdistan ng.uok.ac.ir/mohammadkhani 1 Signal Dfinition and Exampls 2 Signal: any physical quantity that
More information1.2 Faraday s law A changing magnetic field induces an electric field. Their relation is given by:
Elctromagntic Induction. Lorntz forc on moving charg Point charg moving at vlocity v, F qv B () For a sction of lctric currnt I in a thin wir dl is Idl, th forc is df Idl B () Elctromotiv forc f s any
More informationBackground: We have discussed the PIB, HO, and the energy of the RR model. In this chapter, the H-atom, and atomic orbitals.
Chaptr 7 Th Hydrogn Atom Background: W hav discussd th PIB HO and th nrgy of th RR modl. In this chaptr th H-atom and atomic orbitals. * A singl particl moving undr a cntral forc adoptd from Scott Kirby
More informationSundials and Linear Algebra
Sundials and Linar Algbra M. Scot Swan July 2, 25 Most txts on crating sundials ar dirctd towards thos who ar solly intrstd in making and using sundials and usually assums minimal mathmatical background.
More informationThe van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012
Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor
More informationSec 2.3 Modeling with First Order Equations
Sc.3 Modling with First Ordr Equations Mathmatical modls charactriz physical systms, oftn using diffrntial quations. Modl Construction: Translating physical situation into mathmatical trms. Clarly stat
More information4. (5a + b) 7 & x 1 = (3x 1)log 10 4 = log (M1) [4] d = 3 [4] T 2 = 5 + = 16 or or 16.
. 7 7 7... 7 7 (n )0 7 (M) 0(n ) 00 n (A) S ((7) 0(0)) (M) (7 00) 8897 (A). (5a b) 7 7... (5a)... (M) 7 5 5 (a b ) 5 5 a b (M)(A) So th cofficint is 75 (A) (C) [] S (7 7) (M) () 8897 (A) (C) [] 5. x.55
More informationRadiation Physics Laboratory - Complementary Exercise Set MeBiom 2016/2017
Th following qustions ar to b answrd individually. Usful information such as tabls with dtctor charactristics and graphs with th proprtis of matrials ar availabl in th cours wb sit: http://www.lip.pt/~patricia/fisicadaradiacao.
More informationNumbering Systems Basic Building Blocks Scaling and Round-off Noise. Number Representation. Floating vs. Fixed point. DSP Design.
Numbring Systms Basic Building Blocks Scaling and Round-off Nois Numbr Rprsntation Viktor Öwall viktor.owall@it.lth.s Floating vs. Fixd point In floating point a valu is rprsntd by mantissa dtrmining th
More informationPHYSICS 489/1489 LECTURE 7: QUANTUM ELECTRODYNAMICS
PHYSICS 489/489 LECTURE 7: QUANTUM ELECTRODYNAMICS REMINDER Problm st du today 700 in Box F TODAY: W invstigatd th Dirac quation it dscribs a rlativistic spin /2 particl implis th xistnc of antiparticl
More informationClassical Magnetic Dipole
Lctur 18 1 Classical Magntic Dipol In gnral, a particl of mass m and charg q (not ncssarily a point charg), w hav q g L m whr g is calld th gyromagntic ratio, which accounts for th ffcts of non-point charg
More informationElectromagnetic scattering. Graduate Course Electrical Engineering (Communications) 1 st Semester, Sharif University of Technology
Elctromagntic scattring Graduat Cours Elctrical Enginring (Communications) 1 st Smstr, 1388-1389 Sharif Univrsity of Tchnology Contnts of lctur 8 Contnts of lctur 8: Scattring from small dilctric objcts
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More informationREADING ASSIGNMENTS. Signal Processing First. Problem Solving Skills LECTURE OBJECTIVES. x(t) = cos(αt 2 ) Fourier Series ANALYSIS.
Signal Procssing First Lctur 5 Priodic Signals, Harmonics & im-varying Sinusoids READING ASSIGNMENS his Lctur: Chaptr 3, Sctions 3- and 3-3 Chaptr 3, Sctions 3-7 and 3-8 Nxt Lctur: Fourir Sris ANALYSIS
More informationFinite element discretization of Laplace and Poisson equations
Finit lmnt discrtization of Laplac and Poisson quations Yashwanth Tummala Tutor: Prof S.Mittal 1 Outlin Finit Elmnt Mthod for 1D Introduction to Poisson s and Laplac s Equations Finit Elmnt Mthod for 2D-Discrtization
More informationCh. 24 Molecular Reaction Dynamics 1. Collision Theory
Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic
More informationMiddle East Technical University Department of Mechanical Engineering ME 413 Introduction to Finite Element Analysis
Middl East Tchnical Univrsity Dpartmnt of Mchanical Enginring ME 43 Introduction to Finit Elmnt Analysis Chaptr 3 Computr Implmntation of D FEM Ths nots ar prpard by Dr. Cünyt Srt http://www.m.mtu.du.tr/popl/cunyt
More informationExercise 1. Sketch the graph of the following function. (x 2
Writtn tst: Fbruary 9th, 06 Exrcis. Sktch th graph of th following function fx = x + x, spcifying: domain, possibl asymptots, monotonicity, continuity, local and global maxima or minima, and non-drivability
More information4. Money cannot be neutral in the short-run the neutrality of money is exclusively a medium run phenomenon.
PART I TRUE/FALSE/UNCERTAIN (5 points ach) 1. Lik xpansionary montary policy, xpansionary fiscal policy rturns output in th mdium run to its natural lvl, and incrass prics. Thrfor, fiscal policy is also
More informationPair (and Triplet) Production Effect:
Pair (and riplt Production Effct: In both Pair and riplt production, a positron (anti-lctron and an lctron (or ngatron ar producd spontanously as a photon intracts with a strong lctric fild from ithr a
More informationDSP-First, 2/e. LECTURE # CH2-3 Complex Exponentials & Complex Numbers TLH MODIFIED. Aug , JH McClellan & RW Schafer
DSP-First, / TLH MODIFIED LECTURE # CH-3 Complx Exponntials & Complx Numbrs Aug 016 1 READING ASSIGNMENTS This Lctur: Chaptr, Scts. -3 to -5 Appndix A: Complx Numbrs Complx Exponntials Aug 016 LECTURE
More informationExam 2 Thursday (7:30-9pm) It will cover material through HW 7, but no material that was on the 1 st exam.
Exam 2 Thursday (7:30-9pm) It will covr matrial through HW 7, but no matrial that was on th 1 st xam. What happns if w bash atoms with lctrons? In atomic discharg lamps, lots of lctrons ar givn kintic
More informationGive the letter that represents an atom (6) (b) Atoms of A and D combine to form a compound containing covalent bonds.
1 Th diagram shows th lctronic configurations of six diffrnt atoms. A B C D E F (a) You may us th Priodic Tabl on pag 2 to hlp you answr this qustion. Answr ach part by writing on of th lttrs A, B, C,
More informationECE 344 Microwave Fundamentals
ECE 44 Microwav Fundamntals Lctur 08: Powr Dividrs and Couplrs Part Prpard By Dr. hrif Hkal 4/0/08 Microwav Dvics 4/0/08 Microwav Dvics 4/0/08 Powr Dividrs and Couplrs Powr dividrs, combinrs and dirctional
More informationSolution: APPM 1360 Final (150 pts) Spring (60 pts total) The following parts are not related, justify your answers:
APPM 6 Final 5 pts) Spring 4. 6 pts total) Th following parts ar not rlatd, justify your answrs: a) Considr th curv rprsntd by th paramtric quations, t and y t + for t. i) 6 pts) Writ down th corrsponding
More informationBrief Introduction to Statistical Mechanics
Brif Introduction to Statistical Mchanics. Purpos: Ths nots ar intndd to provid a vry quick introduction to Statistical Mchanics. Th fild is of cours far mor vast than could b containd in ths fw pags.
More informationMATH 319, WEEK 15: The Fundamental Matrix, Non-Homogeneous Systems of Differential Equations
MATH 39, WEEK 5: Th Fundamntal Matrix, Non-Homognous Systms of Diffrntial Equations Fundamntal Matrics Considr th problm of dtrmining th particular solution for an nsmbl of initial conditions For instanc,
More informationMolecular Orbitals in Inorganic Chemistry
Outlin olcular Orbitals in Inorganic Chmistry Dr. P. Hunt p.hunt@imprial.ac.uk Rm 167 (Chmistry) http://www.ch.ic.ac.uk/hunt/ octahdral complxs forming th O diagram for Oh colour, slction ruls Δoct, spctrochmical
More informationINTEGRATION BY PARTS
Mathmatics Rvision Guids Intgration by Parts Pag of 7 MK HOME TUITION Mathmatics Rvision Guids Lvl: AS / A Lvl AQA : C Edcl: C OCR: C OCR MEI: C INTEGRATION BY PARTS Vrsion : Dat: --5 Eampls - 6 ar copyrightd
More informationThat is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More informationThe failure of the classical mechanics
h failur of th classical mchanics W rviw som xprimntal vidncs showing that svral concpts of classical mchanics cannot b applid. - h blac-body radiation. - Atomic and molcular spctra. - h particl-li charactr
More informationUnfired pressure vessels- Part 3: Design
Unfird prssur vssls- Part 3: Dsign Analysis prformd by: Analysis prformd by: Analysis vrsion: According to procdur: Calculation cas: Unfird prssur vssls EDMS Rfrnc: EF EN 13445-3 V1 Introduction: This
More informationCHAPTER 10. Consider the transmission lines for voltage and current as developed in Chapter 9 from the distributed equivalent circuit shown below.
CHAPTER 1 1. Sinusoidal Stady Stat in Transmission ins 1.1 Phasor Rprsntation of olta and Currnt Wavs Considr th transmission lins for volta and currnt as dvlopd in Chaptr 9 from th distributd quivalnt
More informationA 1 A 2. a) Find the wavelength of the radio waves. Since c = f, then = c/f = (3x10 8 m/s) / (30x10 6 Hz) = 10m.
1. Young s doubl-slit xprint undrlis th instrunt landing syst at ost airports and is usd to guid aircraft to saf landings whn th visibility is poor. Suppos that a pilot is trying to align hr plan with
More informationThe Transmission Line Wave Equation
1//5 Th Transmission Lin Wav Equation.doc 1/6 Th Transmission Lin Wav Equation Q: So, what functions I (z) and V (z) do satisfy both tlgraphr s quations?? A: To mak this asir, w will combin th tlgraphr
More informationExam 1. It is important that you clearly show your work and mark the final answer clearly, closed book, closed notes, no calculator.
Exam N a m : _ S O L U T I O N P U I D : I n s t r u c t i o n s : It is important that you clarly show your work and mark th final answr clarly, closd book, closd nots, no calculator. T i m : h o u r
More informationTypes of Transfer Functions. Types of Transfer Functions. Types of Transfer Functions. Ideal Filters. Ideal Filters
Typs of Transfr Typs of Transfr x[n] X( LTI h[n] H( y[n] Y( y [ n] h[ k] x[ n k] k Y ( H ( X ( Th tim-domain classification of an LTI digital transfr function is basd on th lngth of its impuls rspons h[n]:
More informationMor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration
More informationSection 6.1. Question: 2. Let H be a subgroup of a group G. Then H operates on G by left multiplication. Describe the orbits for this operation.
MAT 444 H Barclo Spring 004 Homwork 6 Solutions Sction 6 Lt H b a subgroup of a group G Thn H oprats on G by lft multiplication Dscrib th orbits for this opration Th orbits of G ar th right costs of H
More informationcycle that does not cross any edges (including its own), then it has at least
W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th
More informationPropositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018
Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric Rodríguz-Carbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs
More informationNEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA
NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals
More informationIntroduction to Condensed Matter Physics
Introduction to Condnsd Mattr Physics pcific hat M.P. Vaughan Ovrviw Ovrviw of spcific hat Hat capacity Dulong-Ptit Law Einstin modl Dby modl h Hat Capacity Hat capacity h hat capacity of a systm hld at
More informationEXST Regression Techniques Page 1
EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy
More information