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# Electromagnetic Waves OPTI 6104

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This 8 page Class Notes was uploaded by Brandy Koch on Sunday October 25, 2015. The Class Notes belongs to OPTI 6104 at University of North Carolina - Charlotte taught by Tsing-Hua Her in Fall. Since its upload, it has received 24 views. For similar materials see /class/228885/opti-6104-university-of-north-carolina-charlotte in Optical Studies at University of North Carolina - Charlotte.

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Date Created: 10/25/15

Origin of Optical Nonlinearitx Tuesday March 17 2009 l IJSAM Reading a Butcher amp Cotter quotthe elements of nonlinear opticsquotch1 b Boyd quotNonlinear Opticsquotch1 As alluded in quotoptical properties of materialquot the optical properties of material can be understood in terms of Lorentz model in which the electrons are bounded to ions with electrostatic Coulomb force and are driven by the external Efield to undergo forced harmonic oscillation The electron displacement gives rise to induced electric polarization that oscillates and therefore radiates at the same frequency as the driven field The addition of this new field affects the behavior of the existing field ie superposition and constitutes the optical properties of material Please review section quotoptical properties of materialquot for detailed analysis In its mechanical analogy of Lorentz model a spring that attaches electrons to ions could have nonlinear response ie f kx if the displacement is large enough Similarly in the atoms the electrons could displace not totally linearly if the incident field is high enough to pull electrons too far from its equilibrium position At small displacement around equilibrium position electrons are primarily subject to the interaction from its parent ions and hence the force is linear with the distance At large displacement the interaction with parents becomes weaker and interaction with other neighboring atoms become increasingly important This is especially true for solids where atoms are closely packed a couple angstroms Considering a case where the neighboring atoms around an atom of interest are not symmetrically distributed When the electron is pulled far from the ion it sees different environment at the two ends ofthe swing which gives rise to different induced electrical polarization As a result the induced dipole is not symmetric and becomes nonlinear Fourier decomposition of the distorted polarization reveals three components 0 Adominating component at the fundamental frequency 0 A component at twice the fundamental frequency it is called second harmonic generation 0 A DC term named optical rectification 51 EU a b Fi 13 Th effect of a nonlinear dependence of the polarisation P on the electric field E is shown For small input fields a P does not depart significantly from a linear dependence dashed line At larger fields b the polarisation has a distorted waveform which contains significant cnmponenls at harmonic Irequcncics Nonlinear Waves Page 1 Oprlcal Wigwamquot Fundamental Wiamanan u 1 l39rzuueuc r 39 2 w Frequency Frmupmzy gt 0 Strength consideration of nonlinear optics Estimation of the light energy for nonlinear optics What is the electric field strength that bounds an electron to a proton inside a hydrogen atom At what is the intensity of light that delivers an intensity equal to that field strength M 4quot What is the energy required to reach such intensity if the pulse is one nanosecond long What is the energy required to reach such intensity if the pulse is one picosecond long What is the energy required to reach such intensity if the pulse is one hundred femtosecond long Nonlinear Waves Page 2 Nanincur 39 39 39 9their Tuaday Maw 17 lung 11 HAM 0 Assuming the medium at the wavelength ofinterest responses to incident light instantaneously this implies the material must be lossless and dispersionless and for simpliciw assuming the quantities are all scalar The polarization can be generalized as the sum of linear secondorder nonlinear thirdorder nonlinear etc polarizations 1mm e 1 quotEm x g z Wain w rluT 39f rank TI 0 Each components of electrical polarizations is a dipole that radiates at their respective frequency constituting a wave at that particular frequency For nonlinear polarization this results in a host of new waves and rich physics Some nonlinear processes are illustrated below m n 0 Second HarmonIc Generation SHG T f 1 EN EN 153quot 4 cm mel 61 E equot 7 A s 3 C E equotquot fc zexmge39V39f ufme I 0 1 39f p 12 145 a 4 l m 1 Zn H HGURE L 1 my Geomeuv or secumllmlmnmu gclicmliun in tinrgylmr l dw gmm Llcwrilmlg uuva um w mun 0 Sum Frequency Generation SFG Nonlinear Waves Page 3 Frau 12 Sumrfrcqu v gaucminu m ammuer quotI III Inmmcrimx 1m Ens K Jew descnpuun 0 Difference Frequency Generation DFG nu 41 m I g Frank n2 Dim m EnergyIncl I url39rcqucncy gLuvmliun m mommy uf Ihc inrrnwliun wlion 0 Thirdharmonic Generation m w 9 3r 1 3w FIGURP IFS Tlurnl gem mu m lm39clnlcx 0 SelfFocusing Nonlinear Waves Page4 FIGURE I lb Sclfrfo hingMlighl 0 Fourwave Mixing lb 0 uJ 01 101 a 012 gt Z 1 0 Stimulated Raman Nonlinear Waves Page 5 Phase Matching Tuada March l7 ZEIUB ll 12AM Nonlinear susceptibility is generally very small thus requires very large field to drive electrons far enough from the equilibrium position Usually high intensity from modealock lasers are required Otherwise such effect is extremely weak to observe For example the first exp that demonstrated second harmonic generation that opened the door of nonlinear optics is done by using a Ruby laser the first ever laser in the world in the following schematics sans 6943 A Prism and culllmuing lensas V Photographic plate FIGURE 164 Arralm m m r 39 a nation Rclcmm 1 Ruby laser beam at A 0th lm is lucused on 1 quanz crystal callslng uulcmlon at a weak beam all m 0347 m The W0 beams are then separated by a pnsm Because it is a continuous wave with very low intensity the SHG intensity is so weak that after recorded onto a photography plate producing nearly NO imagell In fact this paper was first rejected by the reviewer bc it appeared to have no signal and the reviewer must think these authors were kiddingll VOLUME 7 NUMBER lt3 PHYSICAL REVIEW LETTERS AUGUST 15 I961 GENERATION OF OPTICAL HARMONICS and G Wellll elcl Hill C Wt Peters l l tyufMlchlgan Ann Arbor Miuhlgan P A Frankel A TIIC I Irll l iaml M Randall Lnbur tol nI Pine 0 1 nl39 Received July 21 1961 l O 5455337355940 5 w 55 so lolsao lllnlmximillnlllmlulllunhmllmluulmllmllml i l i i l l i i r I ll llllllllli will lllilllilllllllllllllllllllllliliillllilll ii F l gt Illact roproducllon of the MM plate In lhich lhurv was an lnlhl llun ul sccmlrl hul39lllnlllu l h wavelenth scale is in units of 100 A The arrow al 2Hquot indicates llw l1 llIL39llSI imagl prmluctrl lH lllP second harmonic The Image of lhu pl lmary beam at 69 ls H mm to lllllzlllull Nonlinear Waves Page 6 0 However such intensity are turned out not necessarily to observe nonlinear effect Provided the assembly ofthe induced dipoles oscillate coherently the individual radiated elds can in certain circumstances add constructively to produce a much larger total intensity This is known as phase matching It can be understood in the following Nonlinear Waves Page 7 Nonlinear Wave Progagation Tuesday March 17 2009 1107 AM Coupled wave equation between w amp 2w Nonlinear Waves Page 8

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