Phys4510 - Optics, Midterm Take-home Exam Tuesday, October 29, 2011 Name: This is a take-home exam. You can discuss and work together, but you have to submit your own original solution. The exam is due on Thursday, November 7, in class. The exam consists of three parts. In part I you solve general optics and spectroscopy problems. In Part II you will perform some optics experiments, record your observations, analyze and discuss the results. In Part III you will research the assigned topic and write a 2 page essay about it. There will be no class this Th, but regular class Tu and Th next week. The due date for HW 7 is next Tu, Nov. 5. During the exam days we have extended office hours. Here are is office hours schedule: Wednesday: 3pm-4pm (Sven) Thursday: 2pm-3pm (Will, in his office; no class that day) Friday: 9am-10am (Sven), 2pm-3pm (Will) Saturday/ Sunday: Will, Sven, Markus by appointment Monday: 9am-10am (Sven), 1 pm-2pm (Markus) Tuesday: 9am-11am (Sven), 11am-12pm (Will) Wednesday: 1pm-2pm (Will), 3pm-4pm (Markus) Thursday: NO office hours, exam due in class. Part I: General problems Problem I.1: Fundmentals Please provide a one to two sentence answer to each of the following questions. You may, in addition, supply an equation or a drawing with your response. Note: some of the questions ask you to provide a graph. Be precise in your drawing, since even a simple graph can cover a lot of physics in different regions; pay attention to slopes and curvatures. a. What are the relationships between optical susceptibility, index of refraction, and dielectric function in linear optics? b. Sketch for a typical solid the frequency dependence of the real and imaginary part of the dielectric function. Label and indicate the origin of the main spectral features. 1
c. A continuous wave monochromatic optical field E(t) interacts with a polarizable linear medium. Plot the time trace of the induced optical polarization P (t) in relation to E(t) for i) far off-resonance interaction, and ii) on resonance excitation, for a few optical cycles and clearly indicate the differences between the two scenarios. d. For both cases in (c) at some time turn the field E(t) suddenly off to zero. How does P (t) progress for both cases? Sketch and explain. e. Explain the classical harmonic oscillator model to describe the resonant light-matter interaction. Discuss why the resulting spectral lineshapes are Lorentzians. Problem I.2: Silicon carbide A material with a strong phonon resonance in the infrared is SiC. On the class website you find a table with its complex dielectric function ɛ = ɛ 1 + iɛ 2. a. Plot ɛ 1 (ω) and ɛ 2 (ω) as a function of frequency in units of photon energy in ev. Identify regions of negative ɛ 1. Discuss the similarity in the spectral behavior compared to what you learned for the dielectric function of metals? Discuss the excitation condition for a corresponding surface polariton (here termed surface phonon polariton, as opposed to surface plasmon polariton in the case of metals). b. Derive and plot the real and imaginary part of the index of refraction ñ = n + iκ. Where do you expect the material to exhibit strong absorption or reflection? What are the regions of normal and anomalous dispersion? Discuss the implications of the wildly changing n on the speed of light traveling through the material? c. Derive and plot the phase velocity v ph and group velocity v g. Indicate points of maxima and minima in phase and group velocity? Are there regions of negative phase and group velocity? Discuss their meaning and the physical connection of these points/regions in terms of the interaction of light with the vibrational oscillator. d. Assuming a semi-infinite SiC medium, derive and plot the reflectivity at normal incidence. e. Still assuming a semi-infinite SiC medium, derive and plot the depth of material at which you obtain half intensity transmission accounting for the single reflection upon entering the medium and absorption by the medium. 2
Problem I.3: Distributed Bragg Reflector A Distributed Bragg Reflector is a layered thin film structure designed to reflect certain wavelengths very effectively. a. Suppose we have a periodic dielectric structure with 20 total layers, 10 each of n H = 1.7 and n L = 1.38 alternating from one to the other starting with n H i.e. {H, L, H, L, H, L, H, L, H, L, H, L, H, L, H, L, H, L, H, L}. Calculate and plot the reflectance at normal incidence for this structure. For definiteness, you may take each layer to have thickness 158 nm. Hint, use the transfer matrix formalism described in section 4.4 of Fowles. b. Now, suppose we have a smooth transition between the layers i.e. approximate the structure by the first say 10 terms of its Fourier series to make the calculation easier. Calculate the Fourier transform of n(z). Where is the Fourier transform peaked? Is this expected? c. Is the reflectance narrow or broadband? What can we do to make it broadband if it is narrow, or vice versa? How does this relate to what you found in terms of the Fourier transform? d. Now suppose we arrange the layers in the following order {H, H, H, L, H, L, L, H, L, L, H, L, L, H, H, H, L, H, L, L}. Calculate the reflectance versus wavelength with the transfer matrix formalism. Plot the result. What changed? Part II: Experiment Note: for the experimental tasks we expect you to i) briefly describe your experimental approach and physical principles you take advantage of, ii) provide a schematic of the experiment, iii) describe your observation, and iv) describe the conclusions. Problem II.1 You were handed out two sheets of polarizers in class. Their polarization direction is not indicated. You goal is to experimentally determine the polarizing axis, in two different ways, based on different optical principles. Use your knowledge from class and identify different polarization sensitive processes you can use for that purpose. Problem II.2 With this exam you have been handed out two more sheets of an optically active element. Determine the characteristics of the two sheets with their unknown optical properties. Hint: Once you have a clue from playing around with the sheets what they are, read and describe the specific properties of such a component. Then device a specific experiment with your polarizers how you can uniquely verify your hypothesis. 3
Problem II.3 Investigate the polarization dependence of reflection from a metal surface (e.g., bathroom mirror). Why does it behave differently compared to a dielectric? Describe in terms of the model we developed in class concerning dipole radiation (hint: dipole orientation). Problem II.4 Investigate the polarization dependence of reflection from a water surface. Is the behavior the same or different compared to a glass surface or any ordinary dielectric for that matter? Problem II.5 Observe a LCD display from a pocket calculator or watch and determine its polarization orientation. Do some literature research and describe how an LCD display works, and what the purpose of the polarizer is (ca. 1/2 page text including some schematics). Part III: Essay The essay is an opportunity for you to explore in greater depth an area you find particularly interesting. Your write up should be 2-3 pages, including graphics, and up to 5 references. Here is a list of topics you can choose from, and we will make the assignments in class: (1) Frequency mixing (and phasematching) (2) Kerr effect (and self-focusing) (3) Vector Babinet s principle (4) Birefringence and Dichroism (5) Fiber optics in telecommunications (6) Optical Trapping (7) Fourier Transform Infrared Spectroscopy (8) Stable Laser Cavity Design (9) Gradient Index Materials (10) Metamaterials (11) Raman Spectroscopy (12) Multilayer Films and Anti-Reflection Coatings (13) Fermat s Principle (14) Quantization of Light and the Ultraviolet Catastrophe (15) Symmetry Selection Rules for Radiation from Atoms (16) Transverse Modes of Laser Cavities (17) Mode-Locking in Ultrafast Laser Sources (18) Slow Light 4
(19) Photonic Crystals (20) Optics of the Human Eye and Vision (21) Physics Behind the Depth of Field in Photography and Cinematography (22) Radial Polarization of Beams (23) Rainbow (24) Principles of Holography (25) Mie Scattering from Spherical Particles (26) Angular Momentum of Light (27) Quantum Mechanical Nature of Light (28) X-Ray Crystallography (29) Diffraction Gratings (30) Cherenkov Radiation (31) Spectroscopic Ellipsometry (32) LIDAR Requirements for the writeup: Your writing will be graded on: (1) its scientific merit; (2) organization and content; (3) clear and accurate writing; and, (4) your level of understanding. We aim for a short, but well-thought-out, well-written, and well-conceived report. Some basic suggestions: 1. Use diagrams and pictures to simplify your work and to enable the reader to come to an easier understanding of the material. They should be simple and well labeled. The content or meaning of a graph or picture with its figure caption should be selfexplanatory. 2. Footnote and endnote your sources. Advances in scientific knowledge and understanding often come from rephrasing and reorganizing ideas presented by other authors. Use your own words, based on your own understanding, credited to the appropriate source. 3. Structure your report well: start with an introductory statement related to the effect/process. Then describe the physical basis. This shall consume most of your attention. Do not simply write encyclopedically about different phenomena or applications. 5