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Nonlinear Optics Nonlinear Optics Instructor: Mark G. Kuzyk Students: Benjamin R. Anderson Nathan J. Dawson Sheng-Ting Hung Wei Lu Shiva Ramini Jennifer L. Schei Shoresh Shafei Julian H. Smith Afsoon Soudi Szymon Steplewski Xianjun Ye August 4, 2010 2 Dedication We dedicate this book to everyone who is trying to learn nonlinear optics. i ii Preface This book grew out of lecture notes from the graduate course in Nonlinear Optics taught in spring 2010 at Washington State University. In its present form, this book is a first cut that is still in need of editing. The homework assignments and solutions are incomplete and will be added at a later time. Mark G. Kuzyk July, 2010 Pullman, WA iii iv Contents Dedication i Preface iii 1 Introduction to Nonlinear Optics 1 1.1 History . 1 1.1.1 Kerr Effect . 1 1.1.2 Two-Photon Absorption . 3 1.1.3 Second Harmonic Generation . 4 1.1.4 Optical Kerr Effect . 5 1.2 Units . 7 1.3 Example: Second Order Susceptibility . 8 1.4 Maxwell's Equations . 10 1.4.1 Electric displacement . 11 1.4.2 The Polarization . 13 1.5 Interaction of Light with Matter . 14 1.6 Goals . 19 2 Models of the NLO Response 21 2.1 Harmonic Oscillator . 21 2.1.1 Linear Harmonic Oscillator . 21 2.1.2 Nonlinear Harmonic Oscillator . 23 2.1.3 Non-Static Harmonic Oscillator . 26 2.2 Macroscopic Propagation . 30 2.3 Response Functions . 38 2.3.1 Time Invariance . 39 2.3.2 Fourier Transforms of Response Functions: Electric Susceptibilities . 40 2.3.3 A Note on Notation and Energy Conservation . 41 v vi 2.3.4 Second Order Polarization and Susceptibility . 42 2.3.5 nth Order Polarization and Susceptibility . 43 2.3.6 Properties of Response Functions . 43 2.4 Properties of the Response Function . 46 2.4.1 Kramers-Kronig . 47 2.4.2 Permutation Symmetry . 51 3 Nonlinear Wave Equation 55 3.1 General Technique . 55 3.2 Sum Frequency Generation - Non-Depletion Regime . 58 3.3 Sum Frequency Generation - Small Depletion Regime . 62 3.4 Aside - Physical Interpretation of the Manley-Rowe Equation 66 3.5 Sum Frequency Generation with Depletion of One Input Beam 68 3.6 Difference Frequency Generation . 72 3.7 Second Harmonic Generation . 75 4 Quantum Theory of Nonlinear Optics 87 4.1 A Hand-Waving Introduction to Quantum Field Theory . 87 4.1.1 Continuous Theory . 87 4.1.2 Second Quantization . 91 4.1.3 Photon-Molecule Interactions . 93 4.1.4 Transition amplitude in terms of E~ ........... 96 4.1.5 Stimulated Emission . 97 4.2 Time-Dependent Perturbation Theory . 98 4.2.1 Time-Dependent Perturbation Theory . 99 4.2.2 First-Order Susceptibility . 102 4.2.3 Nonlinear Susceptibilities and Permutation Symmetry 104 4.3 Using Feynman-Like Diagrams . 105 4.3.1 Introduction . 105 4.3.2 Elements of Feynman Diagram . 108 4.3.3 Rules of Feynman Diagram . 109 4.3.4 Example: Using Feynman Diagrams to Evaluate Sum Frequency Generation . 111 4.4 Broadening Mechanisms . 114 4.5 Introduction to Density Matrices . 117 4.5.1 Phenomenological Model of Damping . 121 4.6 Symmetry . 126 4.7 Sum Rules . 129 4.8 Local Field Model . 130 Downloaded from NLOsource.com vii 5 Using the OKE to Determine Mechanisms 135 5.1 Intensity Dependent Refractive Index . 135 5.1.1 Mechanism of χ(3) .................... 137 (3) 5.2 Tensor Nature of χijkl ...................... 140 5.3 Molecular Reorientation . 143 5.3.1 Zero Electric field . 147 5.3.2 Non-Zero Electric field . 148 5.3.3 General Case . 149 5.4 Measurements of the Intensity-Dependent Refractive Index . 150 5.5 General Polarization . 153 5.5.1 Plane Wave . 153 5.6 Special Cases . 154 5.6.1 Linear Polarization . 154 5.6.2 Circular Polarization . 154 5.6.3 Elliptical Polarization . 155 5.7 Mechanisms . 156 5.7.1 Electronic Response . 157 5.7.2 Divergence Issue . 158 5.7.3 One- and Two-Photon States . 158 6 Applications 163 6.1 Optical Phase Conjugation . 163 6.2 Phase Conjugate Mirror . 164 7 Appendix - Homework Solutions 169 7.1 Problem 1.1-1(a to c) . 169 7.2 Problem: nlosource.com/Polarizability.html . 172 7.3 Problem 4.1-1 . 173 7.4 Problem 4.2-1 . 173 7.5 Problem 4.3-1 . 174 viii Chapter 1 Introduction to Nonlinear Optics The traditional core physics classes include electrodynamics, classical me- chanics, quantum mechanics, and thermal physics. Add nonlinearity to these subjects, and the richness and complexity of all phenomena grows exponentially. Nonlinear optics is concerned with understanding the behavior of light- matter interactions when the material's response is a nonlinear function of the applied electromagnetic field. In this book, we focus on building a fun- damental understating of wave propagation in a nonlinear medium, and the phenomena that result. Such un understanding requires both an under- standing of the Nonlinear Maxwell Equations as well as the mechanisms of the nonlinear response of the material at the quantum level. We begin this chapter with a brief history of nonlinear optics, examples of some of the more common phenomena, and a non-rigorous but physically intuitive treatment of the nonlinear response. 1.1 History 1.1.1 Kerr Effect The birth of nonlinear optics is often associated with J. Kerr, who observed the change in the refractive index of organic liquids and glasses in the pres- ence of an electric field. [1, 2, 3] Figure 1.1 shows a diagram of the experi- ment. Kerr collimated sunlight and passed it through a prism, essentially mak- 1 2 Spectrometer Prism Sun CollimatingLenses E Holein Aperture V theWall Polarizer1 Polarizer2 SampleBetweenElectrodes E Screen Applied Field Polarizer2 Polarizer1 Figure 1.1: Kerr observed the change in transmittance through a sample between crossed polarizers due to an applied voltage. Inset at the bottom left shows the orientations of the polarizers and the applied electric field due to the static voltage. ing a spectrometer that could be used to vary the color of the light incident on the sample. He placed an isotropic sample between crossed polarizers (i.e. the polarizers' axes are perpendicular to each other), so that no light makes it to the screen. A static electric field is applied 45o to the axis of each polarizer as shown in the bottom left portion of Figure 1.1. He found that the transmitted intensity is a quadratic function of the applied voltage. This phenomena is called the Kerr Effect or the quadratic electrooptic effect; and, originates in a birefringence that is induced by the electric field. The invention of the laser in 1960[4] provided light sources with high-enough electric field strengths to induce the Kerr effect with a second laser in that replaces the applied voltage. This latter phenomena is called the Optical Kerr Effect (OKE). Since the intensity is proportional to the square of the electric field, the OKE is sometimes called the intensity dependent refractive index, n = n0 + n2I; (1.1) where n0 is the linear refractive index, I is the intensity and n2 the material- dependent Kerr coefficient. Downloaded from NLOsource.com 3 E2 E2 Blue Red Photon Photons E1 E1 Figure 1.2: (left)A system is excited by a photon if its energy matches the difference in energies between two states. (right) Two-photon absorption results when two photons, each of energy (E2 − E1) =2, are sequentially ab- sorbed. 1.1.2 Two-Photon Absorption Two Photon Absorption (TPA) was first predicted by Maria Goeppert- Meyer in 1931,[5] and is characterized by an intensity dependent absorption, of the form, α = α0 + α2I; (1.2) where α0 is the linear absorption coefficient and α2 the two-photon ab- sorption coefficient. Since the absorption coefficient is proportional to the imaginary part of the refractive index, a comparison of Equations 1.1 and 1.2 implies that α2 is proportional to imaginary part of n2. Thus, both the OKE and TPA are the real and imaginary parts of the same phenomena. Figure 1.2 can be used as an aid to understand the quantum origin of TPA. A transition can be excited by a photon when the photon energy matches the energy difference between two states (sometimes called the transition energy),h! ¯ = E2 − E1, where ! is the frequency of the light. Two-photon absorption results whenh! ¯ = (E2 − E1) =2, and two photons are sequentially absorbed even when there are no two states that meet the conditionh! ¯ = E2 − E1. We can understand TPA as follows. Imagine that the first photon per- turbs the electron cloud of a molecule (or any quantum system) so that the molecule is in a superposition of states j i such that the expectation value of the energy is ¯h! = h j H j i. This is called a virtual state, which will be more clearly understood when treated rigourously using perturbation the- ory. If a second photon interacts with this perturbed state, it can lead to a transition into an excited energy eigenstate. Since the process depends on one photon perturbing the system, and the second photon interacting with the perturbed system, the probability of a double absorption is pro- 4 Quartz Prism Glass Ruby Laser CollimatingLenses Fundamental Second Harmonic Photographic Plate Exposed Photographic Plate Figure 1.3: The experiment used by Franken and coworkers to demonstrate second harmonic generation. The inset shows an artistic rendition of the photograph that recorded the two beams. portional to the square of the number of photons (the probability of the first absorption times the probability of the second absorption), and thus the intensity.
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