MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.097 (UG) Fundamentals of Photonics 6.974 (G) Quantum Electronics Spring 2006 Final Exam Time: May 23, 2006, 1:30-4:30pm Problems marked with (Grad) are for graduate students only. • This is a closed book exam, but three 81/2”x11” sheets (both sides) are allowed. • At the end of the booklet there is a collection of equations you might find helpful for the exam. • Everything on the notes must be in your original handwriting (i.e. material cannot be Xeroxed). • You have 3 hours for this exam. • There are 8 problems for undergraduate and 9 problems for graduate students on the exam with the number of points for each part and the total points for each problem as indicated. Note, that the problems do not all have the same total number of points. • Some of the problems have parts for graduate students only. Undergraduate students solving these problems can make these additional points and compensate eventually for points lost on other problems. • Make sure that you have seen all 29 numbered sides of this answer booklet. • The problems are not in order of difficulty. We recommend that you read through all the problems, then do the problems in whatever order suits you best. • We tried to provide ample space for you to write in. However, the space provided is not an indication of the length of the explanation required. Short, to the point, explanations are preferred to long ones that show no understanding. • Please be neat-we cannot grade what we cannot decipher. All work and answers must be in the space provided on the exam booklet. You are welcome to use scratch pages that we provide but when you hand in the exam we will not accept any pages other than the exam booklet. Exam Grading In grading of the exams we will be focusing on your level of understanding of the material associated with each problem. When we grade each part of a problem we will do our best to assess, from your work, your level of understanding. On each part of an exam question we will also indicate the percentage of the total exam grade represented by that part, and your numerical score on the exam will then be calculated accordingly. Our assessment of your level of understanding will be based upon what is given in your solution. A correct answer with no explanation will not receive full credit, and may not receive much-if any. An incorrect final answer having a solution and explanation that shows excellent understanding quite likely will receive full (or close to full) credit. 1This page is intentionally left blank. Use it as scratch paper. No work on this page will be evaluated. 2MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.097 (UG) Fundamentals of Photonics 6.974 (G) Quantum Electronics Spring 2006 Final Exam Time: May 23, 2006, 1:30-4:30pm Full Name: ______________________________________________ Are you taking 6.974 ____ or 6.097 ____ ? Your Points Max points (undergrad.)Max points (grad.) 1 Fabry-Perot Filter 20 20 2 ABCD Matrices 12 12 3 Gaussian Beams/Resonators 5 5 4 Polarization of Light 18 18 5 Harmonic Oscillator 20 20 6 Operators/Commutators 10 10 7 Two-Level Atom 15 15 8 Laser 25 25 9 Three-Level System - 25 Total 125 150 3Problem 1: Fabry-Perot Filter (20 points) In most fiber optics transmission systems a single fiber is used for transmission of multiple channels, each of which has a different carrier frequency. Suppose that you have a system with 40 channels; each channel has 40 GHz bandwidth and these channels are spaced by 100 GHz. A part of this spectrum is illustrated below. At the output of the fiber, each of these channels must be separated from the others before information carried by this channel can be used. Suppose we want to use a Fabry-Perot resonator as a filter to select the single channel at 194 THz. The two mirrors of the Fabry-Perot are identical and are separated by material with a refractive index of . 5.1=n(a) (5 points) What is the requirement for the free spectral range of the Fabry-Perot filter to be used for selecting 1 of these 40 channels? 4(b) (15 point) Choose the length Lof the Fabry-Perot cavity and the reflectivity of the mirrors,R, that can be used for extracting the channel at 194 THz. We have the constraint thatR cannot exceed 99%. If you think that the answer to this problem is not unique, give a combination of L and R that will work. 5(b) continued 6Problem 2: ABCD Matrices (12 points) In each figure below, we have an input plane and an output plane separated by some unknown optical elements. The arrows in these figures denote optical rays. What can you say about the matrix elements of each optical system’s ABCD matrix? (a) (3 points) Input plane Output plane (b) (3 points) Input plane Output plane 7(c) (3 points) Input plane Output plane (d) (3 points) Input plane Output plane 8Problem 3: Gaussian Beams and Resonators (5 points) Suppose you have a resonator with identical mirrors as shown below. If we increase the radius of curvature of both mirrors but don’t modify the resonator length, will the waist radius of the resonator mode increase or decrease? Why? 9Problem 4: Polarization of Light (8 points) A plane electromagnetic wave propagates in free space along the positive z-axis. The electric field vector of the wave is given as yyxxezktEezktEtzErrr)cos()cos(),(00ϕωω+−+−= (1) with 2/πϕ= , xyEE003= (a) (3 point) What is the polarization of the wave i.e. is the light linearly polarized, circularly polarized, or elliptically polarized? (b) (15 points) We would like to transform this wave into a circularly polarized wave using one or several half-wave and/or quarter-wave plates. Give a sequence of half-wave and quarter-wave plates and their rotation angles with respect to the x-axis such that the output is circularly polarized. If the answer is not unique, please give at least one combination that works. 10(b) continued 11Problem 5: Harmonic Oscillator (20 points) In class, we determined the wave functions, ()nxψ, of a one dimensional harmonic oscillator with 221()2Vx m xω=. Below are the solutions for the first 4 modes: (note n indicates the state number) Now