Fundamentals of Nonlinear Optics 2nd Edition By Peter Powers, Joseph Haus (Solution Manual)

SOLUTIONS MANUAL Fundamentals of Nonlinear Optics Second Edition Peter E. Powers and Joseph W. Haus Chapter 1 Problem Solutions 1.1. This problem is an opportunity to enrich your knowledge and appreciation about historical milestones in nonlinear optics. Searching the list of Nobel prizes in physics and chemistry choose one that use lasers and/or nonlinear optics. Read the laureate’s biography and listen or read the lecture on the Nobel website and write a brief description of the laureate and a summary of the technical discovery credited by the Nobel committee. The problem allows students to explore the motivation of a distinguished scientist and learn from that person by reading their description of the discovery. This is a list of laureates with the prize directly identif ied with an optical phenomenon. Nobel Laureates in Physics: 2014 Isamu Akasaki, Hiroshi Amano and Shuji Nakamura 2012 Serge Haroche and David J. Wineland 2009 Charles Kuen Kao, Willard S. Boyle and George E. Smith 2005 Roy J. Glauber, John L. Hall and Theodor W. Hänsch 2001 Eric A. Cornell, Wolfgang Ketterle and Carl E. Wieman 2000 Zhores I. Alferov and Herbert Kroemer 1997 Steven Chu, Claude Cohen-Tannoudji and William D. Phillips 1989 Norman F. Ramsey 1981 Nicolaas Bloembergen and Arthur Leonard Schawlow 1971 Dennis Gabor 1966 Alfred Kastler 1965 Sin-Itiro Tomonaga, Julian Schwinger and Richard P. Feynman 1964 Charles Hard Townes, Nicolay Gennadiyevich Basov and Aleksandr Mikhailovich Prokhorov 1958 Pavel Alekseyevich Cherenkov , Il´ja Mikhailovich Frank and Igor Yevgenyevich Tamm 1955 Willis Eugene Lamb 1954 Max Born 1953 Frits Zernike 1933 Erwin Schrödinger and Paul Adrien Maurice Dirac 1932 Werner Karl Heisenberg 1930 Sir Chandrasekhara Venkata Raman 1924 Karl Manne Georg Siegbahn 1923 Robert Andrews Millikan 1922 Niels Henrik David Bohr 1921 Albert Einstein 1919 Johannes Stark 1918 Max Karl Ernst Ludwig Planck 1908 Gabriel Lippmann 1907 Albert Abraham Michelson 1902 Hendrik Antoon Lorentz and Pieter Zeeman Nobel Laureates in Chemist ry 2014 Eric Betzig , Stefan W. Hell and William E. Moerner 1999 Ahmed H. Zewail 1971 Gerhard Herzberg 1.2 The hydrogen atom is a prototype model to compare the size of the nonlinear coefficients with an electrostatic field related to electron binding. Use classical concepts to calculate the electric field strength due to the proton’s Coulomb field at the Bohr radius, the most probable value of the electron’s orbital radius in the ground state. Use the electric field strength to estimate the second - and third-order nonlinear coefficients based on a unit analysis. Compare the results with second -order coefficients for inorganic compounds found in Appendix B and with third -order coefficients for several inorganic compounds you find in the literature. The calculation is simply to infer an electric field produced by a point charge e at a distance a from the center. The elementary charge and Bohr’s radius are used for the two parameters. 𝑒=1.6×10−19 𝐶𝑜𝑢𝑙 . and 𝑎=52.9 𝑝𝑚. (𝜀0=8.85×10−12 𝐹𝑎𝑟𝑎𝑑 /𝑚). The result is 𝐸=𝑒
4𝜋𝜀0𝑎2=5.1×1011𝑉/𝑚 . Second-order nonlinearity estimation 𝜒(2)∝1
𝐸=1.19𝑝𝑚/𝑉 ; And the third -order nonlinearity estimate is 𝜒(3)∝1
𝐸2=3.8 [𝑝𝑚
𝑉]2
. From the Appendix here are two examples : KTP: 𝜒31(2)=2𝑑31=3.9 𝑝𝑚/𝑉; 𝜒32(2)=2𝑑32=7.8 𝑝𝑚/𝑉; 𝜒33(2)=2𝑑32=30.6 𝑝𝑚/𝑉. BBO: 𝜒22(2)=2𝑑22=−4.4 𝑝𝑚/𝑉; 𝜒31(2)=2𝑑31=0.16 𝑝𝑚/𝑉. Most cases in the appendix are within one order of magnitude of the estimate. The magnitude is also affected by atomic and molecular resonances that can further skew the estimate. A couple of values are as follows: fused silica: 𝜒𝑥𝑥𝑥𝑥(3)=258 [𝑝𝑚
𝑉]2
; carbon disulfide (liquid) cw: 𝜒𝑥𝑥𝑥𝑥(3)=
19,200 [𝑝𝑚
𝑉]2
, chalcogenide glass As 2S3: 𝜒𝑥𝑥𝑥𝑥(3)=120 ,000 [𝑝𝑚
𝑉]2
. The values are generally several orders of magnitude larger than predicted by the atomic field estimate alone. Clearly other mechanisms are responsible for third -order nonlinearities. Chapter 2 Problem Solutions 2.1. A plane wave has an electric field given by  0 ˆ E E sin kz t x   is incident on a material with a susceptibility given by   01 3i /2   . a) What is the complex amplitude of the electric field? i2
0ˆ A E e x (S2.1) b) What is the phase shift between a polarization induced by the field and the incident field? The relationship between the polarization’s complex amplitude and the field’s complex amplitude is, 0PA  (S2.2) Therefore the phase difference is the phase of the complex susceptibility, which is 3 . c) What is the real polarization in this medium induced by the field? 0 0 0 ˆ P E sin kz t x3      (S2.3) or equivalently, 0 0 0 ˆ P E cos kz t x6      (S2.4) 2.2. A plane wave in a vacuum has an electric field given by, 0 ˆ E E cos(kz t )x   . a) What i s B
? We use Maxwell’s equation, 2.2, to relate the curl of the electric field to the time derivative of the magnetic field. For a field of the form given here, ˆikz and it . Therefore,  0Eˆ B cos kz t yc  
. (S2.5) b) What is the complex amplitude of B ? i0Eˆ B e yc
. (S2.6) c) What are Sand S ?  2
20
00EEBˆ S E H cos kz t zc     
. (S2.7) 2
0
0Eˆ Sz2c
. (S2.8)