Tutorial On Nonlinear Optics - Robert Boyd
Proceedings of the International School of Physics “Enrico Fermi”Course 190 “Frontiers in Modern Optics”, edited by D. Faccio, J. Dudley and M. Clerici(IOS, Amsterdam; SIF, Bologna) 2016DOI 10.3254/978-1-61499-647-7-31Tutorial on nonlinear opticsS. ChoudharySchool of Electrical Engineering and Computer Science, University of OttawaOttawa, Ontario, K1N 6N5 CanadaR. W. BoydSchool of Electrical Engineering and Computer Science, University of OttawaOttawa, Ontario, K1N 6N5 CanadaThe Institute of Optics, University of RochesterRochester, New York, 14627 USADepartment of Physics, University of OttawaOttawa, Ontario, K1N 6N5 CanadaSummary. — Nonlinear optics deals with phenomena that occur when a veryintense light interacts with a material medium, modifying its optical properties.Shortly after the demonstration of first working laser in 1960 by Maiman (Nature,187 (1960) 493), the field of nonlinear optics began with the observation of secondharmonic by Franken et al. in 1961 (Phys. Rev. Lett., 7 (1961) 118). Since then, theinterest in this field has grown and various nonlinear optical effects are utilized forpurposes such as nonlinear microscopy, switching, harmonic generation, parametricdownconversion, filamentation, etc. We present here a brief overview of the variousaspects on nonlinear optics and some of the recent advances in the field.c Società Italiana di Fisica 31
32S. Choudhary and R. W. Boyd1. – Introduction to nonlinear opticsAccordingˇ¡proofsAuthor please note that we have written in full the reference quotations in the abstract and reordered those in the text accordingly, following the numericalsequence. Please check, thanks to ref. [1], nonlinear optics is the study of phenomena thatoccur due to the modification of material properties in the presence of light of high intensity. The nonlinearity is associated with the fact that material response varies in a nonlinear manner with the applied optical field. To study this effect, we consider the dependenceof the dipole moment per unit volume, or polarization P̃ (t) on the applied optical fieldstrength Ẽ(t). On application of the optical field, there is displacement of both electronsand the nuclei with respect to the centre of mass of the molecule. Considering dipole approximation, an electric dipole is formed due to charge separation between the negativelycharged electron cloud and the positively charged nucleus. At optical frequencies, due toits much larger mass, the oscillations in the nucleus are much weaker than the electronicoscillations. Hence the nuclear contributions are far weaker than the electronic contributions, at least for linear polarizability. The nonlinear susceptibility on the other hand(manifested in terms of Raman scattering), might be comparable or even larger dependingon whether we are on or off resonance [2]. But for all practical purposes, we neglect thenuclear contributions for simplicity in our present discussion. The bulk polarization ofthe entire material is thus a vector sum of the dipole moments of all the molecules [3, 4].In a linear regime, the induced dipole also oscillates with the same frequency as thedriving field and each molecule of the material can be viewed as a harmonic oscillator.Due to larger mass of the nucleus, these oscillations are very weak and about the meanposition of the molecules. The induced polarization in this case can be expressed as(1)P̃ (t) 0 χ(1) Ẽ(t),where 0 is the permittivity of free space and the χ(1) is the linear susceptibility.But for larger applied fields (comparable to inter-atomic fields) and proportionatelystronger oscillations, this approximation breaks down and the behaviour deviates fromthat of a harmonic oscillator. In this anharmonic case, nonlinear terms come into playwhich give rise to different frequency components in the oscillations. To account for this,we expand the polarization P̃ (t) as a generalized power series in Ẽ(t) and include all thenonlinear contributions as P̃ (t) 0 χ(1) Ẽ 1 (t) χ(2) Ẽ 2 (t) χ(3) Ẽ 3 (t) . . . ,(2)P̃ (t) P̃ 1 (t) P̃ 2 (t) P̃ 3 (t) . . . .The constants χ(2) and χ(3) are the second- and third-order nonlinear optical susceptibilities, respectively. This is a very simplified notation and does not take into accountdispersion or losses because of the instantaneous nature of the response. Under generalcircumstances when losses and dispersion are present, the susceptibilities depend on frequency. If the vector nature of fields is also taken into account, then χ(1) is a tensor of
33Tutorial on nonlinear opticsrank 2, χ(2) a tensor of rank 3 and so on. P̃ 1 (t) is called the linear polarization whileP̃ 2 (t) and P̃ 3 (t) are called the second- and third-order nonlinear polarizations respectively. Thus, the polarization is composed of linear and nonlinear components. A timevarying nonlinear polarization is a source of newer electromagnetic field components andhence is key to the description of nonlinear optical phenomena. This is evident in thewave equation for nonlinear media:(3) 2 Ẽ n2 2 Ẽ1 2 P̃ NL .22c t 0 c2 t2Here, the nonlinear polarization P̃ NL drives the electric field Ẽ and the term 2 P̃ NL / t2represents the acceleration of charges in the medium. This is consistent with Larmor’stheorem that accelerating charges generate electromagnetic waves. It should be notedthat under certain circumstances such as resonant excitation of atomic systems or undervery high applied laser field strength, the power series representation of (2) may notconverge. Such cases are dealt with a formalism that includes the possibility of saturationeffects.Susceptibilities may be complex or real depending on whether the nonlinear processinvolves exchange of energy with the medium or not, respectively. When there is noenergy exchange between the interacting waves and the medium and the quantum stateof the medium remains unchanged in the end (there may be population transfers between real and virtual levels but they have a very short lifetime), the process is called a“parametric process”. Examples include SHG, SFG, DFG, OPA, THG, Kerr nonlinearity, SPM, XPM, FWM, etc, using standard notation that will be developed within thischapter. When the quantum state of the medium is changed in the end, the process iscalled a non-parametric process. Examples include SRS, SBS, multi-photon absorption,saturable absorption, etc. A brief description of all these processes are provided in thesections that follow.2. – Second-order nonlinear optical processesThe discovery of second-harmonic generation (SHG) in 1961 by Franken et al. [5]marked the beginning of the field of nonlinear optics. In 1965, ref. [6] reported the nonlinear light scattering in a quartz crystal generating light with frequency twice that ofthe incident beam. Difference-frequency generation by a KDP crystal using non-collinearlight beams was also reported in 1965 in ref. [7]. Apart from second-harmonic generation, the effects that result from second-order nonlinearity or a non-zero χ(2) includesum- and difference-frequency generation, optical parametric oscillation and spontaneousparametric downconversion. Material symmetry plays a significant role in determiningthe second-order response as only non-centrosymmetric materials, or materials lackinginversion symmetry show a second-order response. This will be elaborated later. A briefdescription of each of the second-order processes mentioned above is as follows.
34S. Choudhary and R. W. BoydFig. 1. – (a) Schematic showing SHG process. (b) Energy level diagram for SHG process.2 1. Second-harmonic generation (SHG). – When a monochromatic laser beam ofelectric field strength represented by(4)Ẽ(t) Ee iωt c.c.is incident on a material with non-zero value of χ(2), it induces a second-order polarization given by(5) 2P̃ (2) (t) 0 χ(2) Ee iωt c.c. , P̃ (2) (t) 0 χ(2) 2EE E 2 e 2iωt E 2 e2iωt , P̃ (2) (t) 2 0 χ(2) EE 0 χ(2) E 2 e 2iωt c.c. .The second term oscillates with a frequency 2ω and is the second-harmonic contributionto the polarization, while the constant first term represents a static electric polarizationdeveloped in the material (as 2 P̃ NL / t2 vanishes) and is called the optical rectificationterm. So we see that the second-harmonic term scales quadratically with the incidentelectric field. It is to be noted though that χ(2) has an order of magnitude value ofapproximately 10 12 m/V, and one might thus think that this contribution is not significant. But with proper experimental conditions, very high efficiencies can be obtainedsuch that nearly all the incident power is converted into the second harmonic.Figure 1b shows an energy level diagram of the SHG process. The solid line indicatesthe ground state while the dotted lines indicate virtual levels. This diagram illustratesthat two photons of frequency ω are annihilated and one photon of frequency 2ω iscreated. Some results of a laboratory demonstration of SHG are shown in fig. 2.2 1.1. Mathematical description. The mathematical treatment provided here followsthose discussed in refs. [4, 1] and [9]. To develop a mathematical description of SHG, weneed to derive the coupled wave equations for the incident pump field and the generatedsecond-harmonic field within the material. We assume that the medium is lossless at thefundamental frequency ω1 as well as the second-harmonic frequency ω2 2ω1 and thatthe input beams are collimated, monochromatic and continuous-wave. The total electric
35Tutorial on nonlinear opticsFig. 2. – SHG from lithium niobate crystal. (a) Setup, (b) Screen output. (c) Trajectories ofthe pump and the SHG [8].field within the nonlinear medium is given by(6)Ẽ(z, t) Ẽ1 (z, t) Ẽ2 (z, t),where(7)Ẽj (z, t) Ej (z)e iωj t c.c.,Ej (z) Aj (z)eikj zwith kj nj ωj /c and nj [ (1) (ωj )]1/2 .The amplitude of the second-harmonic wave A2 (z) is taken to be a slowly varyingfunction of z when the nonlinear source term is not too large, in the absence of whichA2 is constant (as it should be for a plane-wave solution). The nonlinear polarization is(8)P̃ NL (z, t) P̃1 (z, t) P̃2 (z, t),where(9)P̃j (z, t) Pj (z)e iωj t c.c.,j 1, 2and(10)P2 (z) 0 χ(2) E1 (z)2 0 χ(2) A21 e2ik1 z .As each frequency component obeys the inhomogeneous wave equation (3), we can writethe wave equation for the second harmonic as(11) 2 Ẽ2 n2 2 2 Ẽ21 2 P̃ 2 .22c t 0 c2 t2
36S. Choudhary and R. W. BoydOn expanding the first term and rewriting the equation, we get (12) 2 A2 A2n2 2 ω2 2 2 A2 i(k2 z ω2 t) k2 2 A2 2ik2 e2 z zc2 t2ω2 2 2 χ(2) A1 2 e2ik1 z ω2 t .cWe take the slowly varying amplitude approximation which allows us to neglect the firstterm as it is much smaller than the second. Also, using k2 2 n2 2 ω2 2 /c2 , we get(13)2ik2ω2 A2 22 χ(2) A1 2 eiΔkz , zcwhere Δk 2k1 k2 is known as the phase or wave vector mismatch factor and is crucialin determining the efficiency of the conversion process. It accounts for the conservationof momentum for the SHG process when we consider the quantum mechanical picture.For simplicity, we make the undepleted pump approximation which means that A1 (z)is taken to be constant. It is a valid approximation in most cases as at most a negligiblefraction of the pump power is transferred to the generated fields. This simplifies theexpression even further and we obtain(14)2ik2ω2 2dA24ω1 2 2 χ(2) A1 2 eiΔkz 2 χ(2) A1 2 eiΔkz .dzccOn integrating both sides over the length L of the medium, we obtain(15)A2 (L) 2ω1 (2) eiΔkL 1χ.n2 cΔkFor the case of perfect phase-matching or Δk 0, on taking the limit Δk 0 in theabove equation, we find(16)A2 (L) 2iω1 (2) 2χ A1 L.n2 cThe intensity is given by I2 2n2 0 c A2 (L) 2 , where(17)2 A2 (L) 4ω1 2 (2) 2χ A1 4 L2 .n2 2 c2So the SHG intensity scales quadratically with the length of the medium or crystal. Forthe more general case of a nonzero Δk, we find(18)2 A2 (L) 4ω1 2 (2) 2χ A1 4 L2 sinc2n2 2 c2ΔkL2.
37Tutorial on nonlinear opticsFig. 3. – Intensity of the second-harmonic wave versus wave vector mismatch.In this case, the intensity of the second-harmonic wave varies with the phase mismatchΔkL as [sinc2 (ΔkL/2)] as shown in fig. 3.The coherence length is defined as the distance at which the output goes out of phasewith the pump wave and is given by(19)Lcoh 2.Δk.2 2. Sum frequency generation (SFG). – Sum frequency generation is a more generalsituation than SHG in that the two input pump beams have different frequencies ω1 andω2 , leading to the generation of the sum frequency ω3 ω1 ω2 . The total electric fieldassociated with the input waves is given by(20)Ẽ(t) E1 e iω1 t E2 e iω2 t c.c.The second-order nonlinear polarization in this case is given by(21)P̃ (2) (t) 0 χ(2) Ẽ(t)2which on substitution of the expression for electric field gives(22) P̃ (2) (t) 0 χ(2) E1 2 e 2iω1 t E2 2 e 2iω2 t 2E1 E2 e i(ω1 ω2 )t 2E1 E2 e i(ω1 ω2 )t c.c. 2 0 χ(2) [E1 E1 E2 E2 ] .The polarization P̃ (2) (t) can be expanded in its Fourier series and the corresponding frequency components on both sides are equated to get the complex amplitudes of different
38S. Choudhary and R. W. Boydfrequency components of the nonlinear polarization(23)P (2ω1 ) 0 χ(2) E1 2 ;P (2ω2 ) 0 χ(2)2E2 ;(SHG),(SHG),P (ω1 ω2 ) 2 0 χ(2)E1 E2 ;P (ω1 ω2 ) 2 0 χ(2) E1 E2 ;P (0) 2 0 χ(2) (SFG),(DFG),[E1 E1 E2 E2 ] ;(OR).As we can see from the above equations, in the most general case of mixing between twopump beams, we get second harmonic (SHG), sum frequency (SFG), difference frequency(DFG) and optical rectification (OR). But all these components are not present at thesame time and it is mostly one component that is the dominant one which is determinedby the phase-matching condition (to be discussed later).2 2.1. Mathematical description. The derivation of the coupled wave equations is similarto that of second-harmonic generation except for the nonlinear source term which in thecase of two pump beams becomes(24)P̃3 (z, t) P3 (z)e iω3 t ,where P3 (z) 2 0 χ(2) A1 A2 e i(k1 k2 )z .Also,(25)Ẽ3 (z, t) A3 (z)ei(k3 z ω3 t) c.c.,ω3 ω 1 ω 2 ,where(26)k3 n3 ω3,cn3 2 (1) (ω3 ).Note that the complex envelope A3 (z) is again a slowly varying function of z in thepresence of a small nonlinear source term which would have otherwise been a constantleading to a uniform plane-wave solution. Also, we make the undepleted pump approximation for both A1 and A2 and take them as constants in the analysis. As each frequencycomponent of the electric field satisfies the inhomogeneous wave equation, we write thewave equation for the sum frequency term 2 A3 A3n3 2 ω3 2 2 A3 i(k3 z ω3 t)2 k(27) 2ikA e333 z 2 zc2 t2ω3 2 2 2 χ(2) A1 A2 ei(k1 k2 )z ω3 tcAgain, making the slowly varying envelope approximation and substituting the value ofk3 n3 ω3 /c, we obtain(28)dA3iχ(2) ω3 2 A1 A2 eiΔkz ,dzk3 c2
39Tutorial on nonlinear opticsFig. 4. – Schematic showing the process of difference frequency generation.where Δk k1 k2 k3 is the phase or wave vector mismatch factor. Integrating theabove equation along the length L of the crystal, we obtain(29)A3 (L) iχ(2) ω3 A1 A2 eiΔkL 1.n3 ciΔkThe intensity of the sum frequency wave at the output of the crystal is given by I3 (L) 2n3 0 c A3 (L) 2 where2(30)2χ(2) ω3 2 I1 I2 2L sinc2 A3 (L) n1 n2 n3 0 c22ΔkL2.So the sum-frequency intensity also shows a sinc2 dependence, as was observed for thesecond-harmonic case. Figure 3 thus also shows the variation of sum frequency intensityas a function of the phase mismatch factor.2 3. Difference Frequency Generation (DFG). – In the previous section, we saw thata difference frequency component was one of the outcomes when two beams interact in amedium with non-zero value of χ(2) . Let us now consider in detail such a situation, whichas shown in fig. 4, where two waves ω3 and ω1 interact in a lossless optical medium.We use the undepleted pump approximation for the higher-frequency input wave ω3 .The coupled wave equations for the difference frequency wave ω2 and the lower-frequencyinput wave ω1 are obtained by a method analogous to that for SFG and are as follows:(31)iω1 2 χ(2)dA1 A3 A2 eiΔkz ,dzk1 c2and(32)dA2iω2 2 χ(2) A3 A2 eiΔkz ,dzk2 c2where(33)Δk k3 k1 k2 .
40S. Choudhary and R. W. BoydFig. 5. – Spatial evolution of A1 and A2 for the case of perfect phase-matching in the undepletedpump approximation.On solving the above set of differential equations for the case of perfect phase-matching,Δk 0, we obtain(34)A1 (z) A1 (0) cosh κz,(35)A2 (z) i1/2n1 ω2n2 ω1A3A1 (0) sinh κz, A3 where the coupling constant is given by2(36)κ2 χ(2) ω1 2 ω2 2 A3 2 .k1 k2 c4Figure 5 shows the spatial evolution of A1 and A2 for the case of perfect phase-matchingin the undepleted pump approximation.It is observed that both A1 and A2 show monotonically increasing growth and thateach field asymptotically experiences an exponential growth. The input field A1 retainsits initial phase and the DFG wave A2 possesses a phase that depends on both that ofthe pump and of the ω1 waves. An intuitive explanation for this behavior is that thepresence of the ω2 wave stimulates the generation of the ω1 wave and vice versa. Thisprocess of amplification of the signal wave ω1 due to nonlinear mixing resulting in theproduction of an idler is known as “parametric amplification” as DFG is a parametricprocess (due to the initial and final quantum-mechanical states being identical).2 4. Optical parametric oscillation (OPO). – The previous section described the process of parametric amplification by DFG. This gain can be used to produce oscillationwhen it is supplied with the appropriate positive feedback. This can be done by placingmirrors that are highly reflective at one or both of the signal and idler frequencies oneither side of the nonlinear medium as shown in fig. 6. If the end mirrors are reflectingat both the signal and idler frequencies, the device is called a doubly resonant oscillator,and if it is reflecting at either the signal or the idler frequency, then it is called singlyresonant oscillator. The OPO can be used as a source of frequency-tunable radiation for
41Tutorial on nonlinear optics(a)(b)ωs ω1ω p ω3ω p ω3Lω s ω1χ(2)ω i ω2ω i ω2R1, R2R1, R2Fig. 6. – (a) Energy-level diagram for a parametric amplification process. (b) Schematic for anOPO.infrared, visible and ultraviolet spectral regions and can produce either continuous wave,nanosecond, picosecond or femtosecond pulsed outputs.2 5. Parametric downconversion. – The production of simultaneous photon pairs wasdescribed as early as 1970 [10]. Also known as parametric fluorescence [11], parametricscattering or SPDC, it is the spontaneous splitting of the pump photon ωp into signal,ωs and idler, ωp photons such that ωp ωs ωi (energy conservation) and is stimulatedby random vacuum fluctuations. The emitted photons must satisfy the phase-matchingconditions due to momentum conservation, or k p k s k i .The emitted photon pairs are simultaneously entangled in several sets of complementary degrees of freedom. Specifically, the photon pairs can be entangled in time andenergy, in position and momentum, in orbital angular. The fact that the emitted photonsdisplay entanglement has enormous implications for quantum information technologies.For example, entanglement allows one to test some of the fundamental properties inquantum mechanics such reality and non-locality. SPDC is also used to build single photon sources. Entanglement between successive pairs does not occur [12]. Figure 7a showsthe energy level diagram for this process and fig. 7b shows a typical experimental setup.There are two different configurations for SPDC depending on whether the signaland idler waves have the same or orthogonal polarizations; these are called type-I andtype-II configurations, respectively. For type I, the emission is in the form of concentriccones of s
Tutorial on nonlinear optics 33 rank 2, χ(2) a tensor of rank 3 and so on. P 1(t) is called the linear polarization while P 2(t)andP 3(t) are called the second- and third-order nonlinear polarizations respec- tively. Thus, the polarization is composed of linear and nonlinear components. A time varying nonlinear polarization
Recommended reading -lasers and nonlinear optics: Lasers, by A. Siegman (University Science Books, 1986) Fundamentals of Photonics, by Saleh and Teich (Wiley, 1991) The Principles of Nonlinear Optics, by Y. R. Shen (Wiley, 1984) Nonlinear Optics, by R. Boyd (Academic Press, 1992) Optics, by Eugene Hecht (Addison-Wesley, 1987)
Welcome to the CREOL OSE6334 course: Nonlinear Optics. II. University Course Catalog Description: Maxwell's equations in nonlinear media, frequency conversion techniques (SHG, SFG, OPO), stimulated scattering, phase conjugation, wave-guided optics, nonlinear crystals. III. Course Descr
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Nonlinear Optics Fachbereich Physik Contents JJ J I II Page 8 of 199 Go Back Full Screen Close Quit 08. Mai 2005 Klaus Betzler derstand these nonlinearities - and to use them for new effects - will be of basic importance for the further development of photonic applications. These lecture notes cover some basic topics in nonlinear optics .
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