Time: Apil, 006, -3:30pm MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Electical Engineeing and Compute Science 6.097 UG Fundamentals of Photonics 6.974 G Quantum Electonics Sping 006 Qui II Poblems maked with Gad ae fo gaduate students only. This is a closed book exam, but two 8/ x sheets both sides is allowed. At the end of the booklet thee is a collection of equations you might find helpful fo the exam. Eveything on the notes must be in you oiginal handwiting i.e. mateial cannot be Xeoxed. You have 90 minutes fo this exam. Thee ae 5 poblems fo undegaduate and 6 poblems fo gaduate students on the exam with the numbe of points fo each pat and the total points fo each poblem as indicated. Note, that the poblems do not all have the same total numbe of points. Some of the poblems have pats fo gaduate students only. Undegaduate students solving these poblems can make these additional points and compensate eventually fo points lost on othe poblems. Make sue that you have seen all numbeed sides of this answe booklet. The poblems ae not in ode of difficulty. We ecommend that you ead though all the poblems, then do the poblems in whateve ode suits you best. We tied to povide ample space fo you to wite in. Howeve, the space povided is not an indication of the length of the explanation equied. Shot, to the point, explanations ae pefeed to long ones that show no undestanding. Please be neat-we cannot gade what we cannot deciphe. All wok and answes must be in the space povided on the exam booklet. You ae welcome to use scatch pages that we povide but when you hand in the exam we will not accept any pages othe than the exam booklet. Exam Gading In gading of the exams we will be focusing on you level of undestanding of the mateial associated with each poblem. When we gade each pat of a poblem we will do ou best to assess, fom you wok, you level of undestanding. On each pat of an exam question we will also indicate the pecentage of the total exam gade epesented by that pat, and you numeical scoe on the exam will then be calculated accodingly. Ou assessment of you level of undestanding will be based upon what is given in you solution. A coect answe with no explanation will not eceive full cedit, and may not eceive much-if any. An incoect final answe having a solution and explanation that shows excellent undestanding quite likely will eceive full o close to full cedit.
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Time: Apil, 006, -3:30pm MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Electical Engineeing and Compute Science 6.097 UG Fundamentals of Photonics 6.974 G Quantum Electonics Sping 006 Qui II Full Name: Ae you taking 6.974 o 6.097? Poblem Max points Max points undegad. gad.. Gaussian Beams and Resonatos 36. Coupled Waveguide Modes 0 0 3. Polaiation of Light 4 4 4. Themal Radiation 8 8 5. Schödinge s Equation 7 7 6. Supeposition State - 5 Total 90 0 You Points 3
Poblem : Gaussian Beams and Resonatos points undegad/36 points gad a 4 points What ae the chaacteistics of the Gaussian beam at the waist? b 3 points What is the physical meaning of the Rayleigh ange R of a Gaussian beam? c 5 points In the esonato shown below the spot sie of the mode is lage at mio. Is the waist of the beam situated close to mio o mio? Why? 4
d 9 points Conside the stable plano-concave esonato shown below. The fundamental mode of the esonato at wavelength λ μm is a Gaussian beam with the waist adius w 0. How will the waist adius of the mode change if λ is inceased to 0 μm? Show you easoning. How will the divegence angle of the beam change if λ is inceased to 0 μm? 5
e Gad 5 points Suppose you need to design a two-mio esonato fo a high-powe lase opeating at λ μm. You have at you disposal 3 pais of mios: two spheical mios with ROC adius of cuvatue of m, two spheical mios with ROC m, and two plane mios ROC. Thee ae two equiements that must be satisfied: i fo a given powe, the intensity at the mios must be as small as possible to avoid mio damage; ii to achieve most effective amplification, the gain medium with adius 0. 7mm is placed into the waist egion of the Gaussian beam. The waist sie must not be too small, othewise the beam will fill just a pat of the gain medium, and it also must not be too lage, othewise only a faction of the beam will be amplified. Suppose that the optimum waist adius is half the gain medium adius, w / 0. 35mm. Thus the waist adius of the beam is fixed by the gain 0 medium and cannot be vaied in esonato design. Which two of the available mios would you choose fo the esonato? Why? Appoximately what length of esonato would you choose? 6
Poblem : Coupled Waveguide Modes 0 points Two waveguides ae spaced closely to each othe. In the absence of the othe waveguide, the unpetubed waveguides each have two guided modes with the popagation constants sketched below note that the subscipts denote the waveguide numbe and supescipts denote the mode numbe. β, μm 4 β I β II 3 β I β II Waveguide I Waveguide II a 5 points Which pai of modes can exchange full powe upon abitaily small coupling between these waveguides? Give a bief explanation. 8
b 0 points Now conside that in the absence of the othe waveguide, the unpetubed waveguides each have only one guided mode with the popagation constants sketched below. Powe is launched in waveguide I with wave numbe β Ι and thee is coupling to the guided mode in waiveguide II with the wave numbe β ΙΙ. The coupling is descibed by the coupling coefficient κ I II 0.5μm waveguide II?. What will be the maximum powe coupling fom waveguide I to β, μm β I β II Waveguide I Waveguide II c 5 points At what length L of the two waveguides does the maximum powe tansfe fom waveguide I to waveguide II occu accoding to pat b? 9
Poblem 3: Polaiation of Light 4 points A plane electomagnetic wave popagates in fee space along the positive -axis. The electic field vecto of the wave is given as E e, t E0x cos ω t k ex + E0 y cos ω t k + ϕ y a 4 points What is the polaiation in the following cases, i.e. is the light linealy polaied, ciculaly polaied, o elliptically polaied? ϕ 0, E0 x 0, and E 0 ; 0 y ϕ π / and E. E0 x 0 y b points Detemine a Jones vecto fo each of the two cases in pat a. 0
c points Suppose we have a half-wave plate with a phase etadation of Γ π and the pincipal axes otated by an angle ψ 45 with espect to the x- and y- axes. What is the Jones Matix of this half-wave plate? d 6 points What is the final polaiation of the two cases in pat a afte popagation though the half-wave plate of pat c?
Poblem 4: Themal Radiation 8 points We ae given a system in an Ulbicht sphee of atoms and photons in themal equilibium at a tempeatue T300K. We will examine the photons that ae emitted and absobed at a wavelength of 60nm. a 6 points What kind of adiato is descibed above? Why? b 4 points Is 60nm the pat of the enegy spectum whee the atoms emit the most electomagnetic enegy? 3
c 4 points What is the aveage numbe of photons you expect to be stoed in a adiation mode of this system at 60nm? d 4 points Now assume that ou system only had two enegy levels. Show that at high tempeatues the population densities and n of the two states tend to become equal. n 4
Poblem 5: Schödinge s Equation 7 points A paticle of mass m that moves in a potential Vx is in the state Ψ x, t whee a and b ae positive eal constants. 4 a 5 points Find the uncetainty in paticle position. a e π ax jbt, b 7 points Is this a stationay state? Why o why not? If this is a stationay state, find the enegy of the system. c 5 points Find the pobability cuent, Jx, t. 5
Poblem 6: Supeposition State Gaduate poblem, 5 points Conside a paticle in an infinite -D squae well with potential 0, x a / V x V0, x > a / The paticle is initially in a supeposition state of the fist two stationay states, i.e. Ψ x, t 0 [ ψ x + ψ x ], π π whee the wave functions ae ψ x cos x and ψ x sin x inside the well and a a a a eo outside the well. Fo a given state, n, the coesponding enegy eigenvalue is E n n π h. ma What is the expected value of the paticle position, x, fo t > 0? Note that the equation sheet at the back of this booklet contains some fomulas you may find useful fo this poblem. 6
Maxwell s Equations Mateial Equations Index of Refaction Poynting Vecto Enegy density Qui I Equation Sheet B D E H + J t t D ρ B 0 D ε 0 E + P ε E P ε 0 χee ε ε 0 + χe B μ H + μ M μ H M χ μ μ + χ n 0 0 mh + χ, fo χ << : n + χ / S E H we ε E T E H * 0 m w m μ H w w e + wm Snell s Law nsinθ nsinθ Bewste s Angle tan θ B n n Reflectivity TE TE TE TM TM Z Z TM Z Z TE TE TM TM Z + Z Z + Z μ TE / TM Z / Z / / cosθ/ ε/ cosθ/ ε/ n cosθ n n TE n cosθ + cosθ cosθ μ n n cosθ cosθ n cosθ cosθ TM n + Tansmitivity t TE TM TE Z TM Z t TE TE TM TM Z + Z Z + Z t n cos θ TE n cos θ + n cos θ n cos θ t TM n n + cos θ cos θ Powe Refl. Coef. TE R TE TM R TM TE TE TE Z 4Z Z Powe Tansm. Coef. T t TE TE TE Z Z + Z TE TE TM TM TM TM TM Z 4Z Z T t TM TM TM Z Z + Z continued on the next page 8
A, t k A, t Pulse Dispesion j t Gaussian Pulse Faby Peot Beam Splitte S-matix k L τ L τ + τ FWHM ln τ τ S R π f whee φ kl, k n R + 4Rsin φ / c jt S, with jt + t 0 Constants ε o 8.85 0 - C /Nm μ ο 4π 0-7 N/A m 9. 0-3 kg e.60 0-9 C pemittivity of fee space pemeability of fee space mass of an electon chage of an electon continued on the next page 9
Qui II Equation Sheet Gaussian Beams + exp, 0 j R jk w w P E ζ π 0 + R w w + R R R actan ζ exp, 0 0 w w w I I λ π 0 0 w k w R w 0 π λ θ q-paamete R j q + w j R q π λ exp, 0 q jk q E Resonato Stability 0 g g R L g R L g Waveguide Coupling Δ + P P γ γ β γ sin cos 0 sin 0 P P γ γ κ κ β γ + Δ β β β Δ 0 β β β + Polaiation, Retadation Plate + Γ Γ + Γ Γ Γ Γ ψ ψ ψ ψ ψ ψ / / / / cos sin sin sin sin sin sin cos j j j j e e j j e e W ψ is the otation angle of the pincipal axes with espect to the x- and y- axes continued on the next page 0
cosψ sinψ if Γ π : W j sinψ cosψ if π Γ : W [ j cosψ ] j sinψ j sinψ [ + j cosψ ] Wien s Law Wien Displacement Law Rayleigh-Jeans Law 8π hf w f 3 c λ max w f 3 e hc 4.965kT 8π 3 c f kt hf kt Planck s Law 8π f w f 3 c hf hf exp kt Einstein s A and B Coefficients Schödinge s Equation Heisenbeg s Uncetainty Pinciple A 8π f c 3 B 3 B B dψ, t h jh ΔΨ, t + V Ψ, t dt m ΔxΔp h de Boglie s Fomula p h k Einstein s Enegy/Fequency Relation E h ω Tigonometic Identities sin α sinα cosα cosα cos α sin α sinα sin β cos α β cos α + β cosα cos β cos α β + cos α + β sinα cos β sin α + β + sin α β cosα sin β sin α + β sin α β continued on the next page
Helpful Integals e a x x e dx a x π a dx π 3 a xe sin α x α x cos α x x sin α x dx + C α cos α x + α xsin α x x cos α x dα + C α a x dx 0 Constants k.380650 0-3 J/K Boltmann's constant c 0.99795 0 8 m/s - h 6. 66068 0 - h 05457. 0 34 34 J s J s Speed of light in fee space Planck's constant