光学二次谐波产生及光混频前三节PPT
光学混频PPT演示课件
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由边界条件，并对 E3 z 积分，可以得到倍频光在 z=L
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介质除辐射直流、基频和倍频成分，还将辐射频率为 和频与差频的光波，称为光学混频。
下面将分别具体介绍光学倍频、和频、差频以及四波混频。
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对于二阶非线性介质，两光波场 E1 , E2 作用于介质， 引起二阶极化，产生新波场 E3 。这是一个和频过程，三
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二次谐波理论相关
Engineering the second harmonic generation pattern from coupled gold nanowiresA.Benedetti,M.Centini,*C.Sibilia,and M.BertolottiDipartimento di Energetica,Universitàdi Roma,La Sapienza,Via A.Scarpa16,00161Roma,Italy*Corresponding author:marco.centini@uniroma1.itReceived November2,2009;accepted December16,2009;posted January5,2010(Doc.ID119439);published February4,2010We numerically investigated second harmonic generation from systems composed of two coupled gold nano-wires.The developed method allows one to arbitrarily change the shape of the wire cross section in order to explore the generated and scatteredfield patterns.Our results suggest that the overall second harmonic gen-eration is related to the electromagnetic energy per unit length stored in the gap area between the wires.These geometrical considerations make further optimization possible.As an example we discuss the possibility to select dipolar emission and/or quadrupolar emission patterns.The selection mechanism responsible for this kind of emission can be traced back to the interaction between nonlinear sources with the surface plasmon resonance of the metallic wires.©2010Optical Society of AmericaOCIS codes:160.4330,190.3970,190.4350,290.5850.1.INTRODUCTIONThe second order nonlinear response fromflat metal screens has been widely investigated,both theoretically and experimentally,from the late1960s–1980s[1–11]. Some of this work was devoted to the study of the second harmonic generation(SHG)in reflection by thin metal layers and/or metallic gratings,with the subsequent re-porting of the enhancement of nonlinear conversion effi-ciencies due to the surface plasmon polariton(SPP)exci-tation at the metal/dielectric interface[12–15].Regarding the SHG in systems composed of many metallic particles and/or percolated systems,several models have been pro-posed that utilize the effective medium approximation and experimental techniques have been developed to characterize the nonlinear optical responses[16–20]. Also,a large number of works have been devoted to the study of materials and single molecule behavior to en-hance and characterize the nonlinear response[21–27].The last few years witnessed renewed interest in the study of nonlinear second order properties of metal/ dielectric composites thanks to the development of nano-technologies and nanoscience.The SHG has been de-tected in metal nanoparticles and the effects of scatterer size on the dipole or multipole contributions have been ex-perimentally investigated[28,29].A strong second har-monic(SH)signal has been found emanating from sharp metal tips[30],thus opening the way to interesting appli-cations for nearfield nonlinear microscopy and spectros-copy.The SHG enhancement from nanoantennas and nanodimers has been observed both in near and farfields [31–34].Other recent works[35,36]deal with the SHG in gold nanowires.In these studies the effect offield polar-ization direction with respect to the direction of the wires is emphasized.Moreover,the SHG from metamaterials composed of split-ring resonators was experimentally re-ported by Klein et al.[37,38].Another experimental pa-per,recently published,reports the SHG from a gold plate having a periodic pattern of rectangular apertures on top [39].In[39]the effect reported is connected to the shape of the apertures and to the propagation constants of the guided electromagnetic(e.m.)modes inside the apertures. This feature has been employed to enhance the nonlinear response of GaAs using it tofill patterned metallic screens with holes,taking advantage of the high trans-mission regime and strongfield localization inside the holes[40].All these aspects,combined with the state of art technology capabilities,make these materials very in-teresting for basic and applied research.From a theoretical point of view,the farfield pattern of the generated SH has been studied in detail for a single metallic cylinder or other two-dimensional(2D)single ob-jects as well as for nanospheres and deformed nano-spheres[41–45].Also,the scattered SH generated by a SPP at a surface nanodefect was theoretically investi-gated[46].Recent works report on numerical results ob-tained by the implementation of afinite difference time domain(FDTD)algorithm[47–50].Reference[48]deals with the SHG from resonant nanoslits on a silver sub-strate,by taking into account the nonlinearity arising in the free electron response from the magnetic Lorentz force.In[47]the quantum jellium model[10]for the non-linear response of a metal was implemented in a FDTD algorithm;numerical results are shown for periodically patterned metallic screens.Limitations of the validity of the approach used in[47]for nanostructured objects are related to the parameterization procedure used for the nonlinear coefficient of the metal,performed assuming an infiniteflat surface.In[49]a semiclassical approach, based on the free electron gas model for the nonlinear op-tical response of a metal,has been adopted to successfully analyze experimental results of[50].Also,in[51]the au-thors showed that for patterned metallic membranes hav-0740-3224/10/030408-9/$15.00©2010Optical Society of Americaing thicknesses of a few tens of nanometers the farfield SH is dominated by dipole contributions of the nonlinear response.In this work we study the enhancement of the SHG due to the interaction between two2D metallic wires with sec-tions of arbitrary shape,focusing on the possibility of tai-loring the emission pattern by properly designing the shape and the position of the wires.Numerical calcula-tions on two different kinds of double-wire systems,rect-angular section(RS)and trapezoidal section(TS)of the wires,are performed.The SHG as a function of the dis-tance(air gap)between the wires is investigated in both the near and farfield regimes.The enhancement of the SH signal is then explained in terms of the pump energy stored inside the gap.Our results reveal a high sensitivity of the emission pattern and the overall power with re-spect to displacements of just a few nanometers.In par-ticular we show that the SHfield with dipole and quad-rupole symmetries can be achieved by changing the geometrical parameters.Although we only consider sys-tems composed of two wires,our approach can be applied for an arbitrary number of objects with different shapes and sizes.Wefinally investigate localization effects of the SHfield at the surface,revealing the existence of hot spots that can be used for nanoillumination and single molecule spectroscopy[52].The paper is divided into four main sections:Following this introduction,in Section2,we describe the used the-oretical model and integration method.In Section3we show the results of our numerical analysis focusing our attention on the different behaviors of emittedfields as functions of the nano-object shape and position.Finally, in Section4the main conclusions and perspectives for fu-ture work are drawn.2.THEORETICAL MODELAt optical frequencies the linear response of a metal is strongly affected by the bound valence electrons.We chose to model the electric permittivity of gold from the far infrared to the ultraviolet range with a Drude–Lorentz model by consideringfive Lorentz-like[53–55] resonators,r͑͒=D͑͒+͚k=15G kP20,k2−2−i␥k,D͑͒=1−G0P2͑+i␥0͒=1−02͑+i␥0͒,͑1͒whereD is the electric permittivity obtained by consider-ing the Drude model only.p is the plasma frequency as-sociated with the total number of electrons per unit vol-ume.G k and0,k are,respectively,the oscillator strengths and resonant frequencies and␥k are damping constants related to each oscillator.0=ͱG0p is the plasma fre-quency associated with intraband transitions(relatedonly to the free electron density)with damping constant ␥0.Indeed we assume that,in the absence of external ex-citation,the free electron density n is constant and equal to its bulk value n0,being02=͑e2n0/m0͒,where e is the absolute value of the electron charge and m is the effec-tive electron mass.The values of the parameters of the Drude–Lorentz model are reported in[53].We model the optical response of the metal by assum-ing an effective current density,Jជ=JជD+ץPជb.e.ץt,͑2͒where JជD is the current density induced by the e.m.field on the free electrons and Pជb.e.is the polarization vector due to the presence of bound electrons.Each term on the right hand side of Eq.(2)contains both linear and nonlin-ear contributions.At this stage we focus our attention on the nonlinear contributions arising from conduction elec-trons only.We study the SHG process in the undepleted pump ap-proximation from a metallic2D structure that reacts in the presence of a transverse magnetic͑p-͒polarized e.m.field;the system is completely2D.The nonlinear second order response of the conduction electrons is then evalu-ated as in[49]and the dynamic equations for JជD and n areץJជDץt=JជDneٌជ·JជD+͑JជD·ٌជ͒JជDne−␥0JជD+e2mnEជ−e0mJជD∧Hជ,͑3aٌ͒ជ·JជD=eץnץt,͑3b͒where␥0is defined in Eqs.(1)and it represents losses due to Coulomb scattering of conduction electrons with the lattice ions.By adopting a perturbative approach as in[8]we move into the frequency domain and write the current density at the SH frequency as a function of the fundamental fre-quency(FF)field by collecting terms up to the second or-der.Thus,the expressions of thefields,the current,and free electron densities areJជ͑rជ,t͒=͚m=1,2Jជm͑rជ͒e−imt+c.c.,Eជ͑rជ,t͒=͚m=1,2Eជn͑rជ͒e−imt+c.c.,Hជ͑rជ,t͒=͚m=1,2Hជm͑rជ͒e−imt+c.c.,n͑rជ,t͒=n0+͚m=1,2n m͑rជ͒e−imt+c.c.,͑4͒where the subscript m=1,2refers to the FF͑͒and SH ͑2͒angular frequencies.Due to the weak nature of the nonlinear interaction,we consider that the dynamics for the FF is completely described by the linear response,Jជ1=−i͑r,1−1͒0Eជ1,͑5͒wherer,1is calculated by using Eqs.(1).For the SHfield we haveJ 2ជ=−i 20r ,2E 2ជ−i 20ͫabE 1ជٌជ·͓D ,1E 1ជ͔+b ͑a −1͒D ,12͑E 1ជ·ٌជ͒E 1ជ+bD ,14ٌជ͓E 1ជ·E 1ជ͔ͬ,͑6͒wherea =+i ␥0,b =22+i ␥0,=−e 2m 2,͑7͒and r ͑r ជ͒=r ͑r ជ͒−1,D ͑r ជ͒=D ͑r ជ͒−1are the expressionsfor the spatially dependent electric susceptibility accord-ing to the definitions given in Eqs.(1).We note that the coefficients a and b are exactly equal to unity in the case of no losses.The first term on the right hand side of Eq.(6)represents the linear contribution,which depends on both bound and free electrons;the remaining terms are the nonlinear sources.The evaluation of the nonlinear terms of Eq.(6)requires a detailed analysis to separate bulk and surface contributions.In the bulk,the first non-linear term of Eq.(6)vanishes and the other terms,evalu-ated only for the points inside the metallic objects,are re-sponsible for the bulk contribution to the nonlinear current density,J ជNL Bulk =−i 20D ,12b ͫ͑a −1͒͑E 1ជ·ٌជ͒E 1ជ+12ٌជ͓E 1ជ·E 1ជ͔ͬ.͑8͒In order to evaluate surface contributions we assume thatthe transition from metal to surrounding air is achieved by varying the value of the parameter P 2from the value it assumes in the metal to zero in the vacuum.Thus the other parameters,i.e.,damping coefficients and electron effective mass,are considered as constant coefficients.Following the procedure outlined in [56],modified and adapted to our case (losses and linear contribution of bound electrons),we evaluate the nonlinear surface cur-rents acting as sources for the SH field,X ˆ·J ជNL Surface =−i 20abE 1,X ͑−͒E 1,Y ͑−͒͑1−D ,1͒␦͑Y ͒,Y ˆ·J ជNL Surface =−i 20ab ͓E 1,Y͑−͔͒24͓͑1−D ,1͒͑r ,1+3͔͒␦͑Y ͒.͑9͒Calling r ជЈthe vectors defining the coordinates of theboundaries,X ˆ͑r Јជ͒and Yˆ͑r Јជ͒are the local unit vectors tan-gential and normal (outgoing)to the surface,respectively.In order to provide a comparison with previous results wechoose to evaluate the continuous function r ,1͑r Јជ͒E 1,Y ͑r Јជ͒below the surface as r ,1E 1,Y ͑−͒͑r Јជ͒.Equations (9)are equivalent to the results of [56]with the exceptions of themultiplicative coefficients a and b which account forlosses;the presence of D ,1represents the response of con-ducting electrons only and r ,1accounts for the total lin-ear response of both bound and free electrons.Even though we do not use it in the calculations pre-sented in this work,we add the expression for the surface nonlinear terms evaluated at the interface metal/dielectric,when a dielectric B is considered,X ˆ·J ជNL Surface =−i 20abE 1,X ͑−͒E 1,Y͑−͒B͓r ,1͑B −1͒+B ͑1−D ,1͔͒␦͑Y ͒,Y ˆ·J ជNL Surface =−i 20ab ͓E 1,Y ͑−͔͒24B2͓3r ,12͑B −1͒+B ͓r ,1͑B −D ,1͒+3B ͑1−D ,1͔͔͒␦͑Y ͒.͑10͒This feature will be used in the future to analyze realstructures composed of metal particles on top of a dielec-tric substrate,for example.Adding the expressions of Eqs.(8)and (9)we finally write the current density for the field at the SH frequency,J 2ជ=−i ͑r ,2−1͒20E 2ជ+J ជNL Bulk +J ជNLSurface .͑11͒The generated field pattern can be calculated by insertingEq.(11)into the equation for the generated SH electric field,ٌͩជ∧ٌជ∧−͑2͒2c2I ញͪE ជ2͑r ជ͒=i ͑2͒0J ជ2͑r ជ͒,͑12͒where Iញis the identity matrix.The formal solution is ob-tained by substituting Eq.(11)into Eq.(12),E ជ2͑r ជ͒=͵⌺ЈG ញ2͑r ជ,r Јជ͒ͩ2cͪ2͓r ,2͑r Јជ͒−1͔E 2ជ͑r Јជ͒d ⌺Ј+i 20͵⌺ЈG ញ2͑r ជ,r Јជ͒J ជNL Bulk ͑r Јជ͒d ⌺Ј+i 20͵⌺ЈG ញ2͑r ជ,r Јជ͒J ជNLSurface ͑r Јជ͒d ⌺Ј,͑13͒where the Green’s tensor G is defined asG ញ2͑r ជ,r Јជ͒=ͫI ញ+ٌជٌជk 0,22ͬg 2͑r ជ,rЈជ͒,͑14͒with g 2being the 2D Green’s scalar function [57]in thehomogeneous background medium,corresponding to the Bessel function of the first order for the SH frequency.⌺Јis the section area of the 2D metallic scatterers.Effects due to the substrate can be included by calculating the Green’s tensor for a stratified medium as outlined in [58].The numerical integration of Eq.(13)is performed us-ing a procedure detailed in [58,59]and adapted to the case of the SHG as shown in [55].Once the field inside the scatterers is calculated,it is straightforward to calculate the SH field at every other point in space.In particular,itis possible to depict the near field as well as the far field pattern.Then we numerically evaluate the nonlinear scattering cross section (NSCS)Q ͑͒as defined in [41],Q ͑2͒=P ͑2͒sc ͓P ͑͒inc ͔2=͵2q ͉͑2͒d ,͑15͒where q ͉͑2͒is the differential scattering cross section.Here we consider P ͑͒inc as the power flow per unit length (watt/meter)of the FF field across the segment as shown in Fig.1.P ͑2͒sc is the generated SH power flow per unit length calculated across a circumference of ra-dius R ӷ.As already mentioned,Eq.(3a)is formally identical to Eq.(7)of [49],as we began from the same model for the nonlinear response of the metal.However,there are some differences between the two models that can be summa-rized as follows:(i)we study the phenomena in the fre-quency domain,describing the SHG from a monochro-matic pump excitation.(ii)We analytically evaluate the surface contribution by considering a hard interface(abrupt change in the dielectric constant).Using this ap-proach improves the convergence of our numerical algo-rithm.Moreover,the integration method can handle both surface and bulk nonlinear sources thus making it pos-sible to directly compare our results with other models of metal response [4,8,10].(iii)We calculate both far and near field emissions by single or multiple 2D scatterers of arbitrary shapes.The periodicity of the system is not re-quired.Our approach is aimed to the studies of nonlinear microscopy and of nonlinear generation from single nano-sized objects.It may be used,for example,to design and optimize the nonlinear response from single molecules at-tached to metallic nanoantennas for detection and sens-ing applications.In our opinion,the method of [49]ap-pears to be more suitable and efficient for the study of the SHG in periodic metamaterials and sub-wavelength pat-terned screens in the time domain.(iv)We consider the linear response of bound electrons in order to obtain a more realistic value of the linear dielectric constant of the metal both at the FF and SH frequencies.Fig.1.Sketch of the scattering geometry for the FF field.Inset:(up)dimensions of the trapezoid section wires;(down)dimen-sions of the RS wires.102030401020304050012345Gap (nm)r e l a t i v e e n e r g y p e r u n i t l e n g t h i n s i d e t h e g a p r e g i o nN o n l i n e a r s c a t t e r i n g c r o s s s e c t i o n (c m 2/W )Fig.2.Relative energy per unit length inside the gap region incases of TS (solid line)and RS (dashed line)wires.NSCS for the TS (triangles)and RS (squares)wires.Fig.3.(Color online)Modulus of the normalized electric field (a),(c)y -and (b),(d)x -components normalized with respect to the incident field amplitude for gap values of (a),(b)18and (c),(d)38nm.3.NUMERICAL ANALYSISWe consider two different configurations composed of coupled gold wires with (a)RS 300nm ϫ145nm and (b)TS “bowtie antenna type”with B =420nm,d =145nm,and h =300nm (inset of Fig.1).We chose these geom-etries because they both exhibit a resonant behavior when a monochromatic p -polarized (electric field direction along the y axis)FF field with a wavelength of 800nm im-pinges along the horizontal ͑x ͒axis.We point out that the sizes of the objects that we consider have wires dimen-sions ͑h ,B ,d ͒that are not negligible with respect to the in-cident wavelength.As a result we are exploring a different regime with respect to [28,29,51].Our analysis is per-formed by comparing the generated SH fields from the two geometries,relating the enhancement of the emission to the energy confinement of the FF field in the gap re-gion.We will also show that SH patterns are extremely sensitive to the shape of the emitters and the distance be-tween wires.As a first step we study the localization prop-erties for the FF field.We considered a p -polarized plane wave,monochromatic FF electric field of amplitude equal to 100MV/m,traveling from left to right along the x axis.Calculations of the linear FF fields were performed under the Comsol Multiphysics simulation environment.We cal-culated the average energy per unit length in the gap re-gion of area=gd (see Fig.1)as a function of the gap.In order to quantify the localization effect we normalize it with respect to the energy per unit length stored intheFig.4.Differential scattering cross section for the TS wires when different values of the distance between wires ͑g ͒areconsidered.Fig.5.Differential scattering cross section for the RS wires when different values of the distance between wires ͑g ͒areconsidered.Fig. 6.(Color online)z -component of the SH magnetic field (ampere/meter)(snapshots in time)for the (a)TS and (b)RS wires.same area for the same incident field propagating in free space by applying the formulaW =͐gap ͓0͉Eជ1͑x ,y ͉͒2+0͉H ជ1͑x ,y ͉͒2͔d x d y 20͉H 1,0͉2gd,͑16͒where H 1,0is the amplitude of the incident plane wavemagnetic field,Hជ1͑inc ͒͑x ,y ͒=͑H 1,0e i ͑/c ͒x +c.c.͒z ˆ.͑17͒The results are shown in Fig.2.In both cases the maxi-mum energy confinement is observed for a value of thegap of 18nm.We note that the TS wires exhibit a higher value of the relative energy inside the gap region.As an example,in Figs.3(a)–3(d)we depict the behavior of themodulus of the FF electric field y -and x -components nor-malized with respect to the input electric field modulus for the TS and gap values of 18and 38nm.We note the strong confinement inside the gap region.We obtain simi-lar behavior if RS wires are considered.Then we use the linear solutions for the FF fields to calculate the nonlin-ear source terms as described in the previous section.Fi-nally we calculate the generated SH field by the numeri-cal integration of Eq.(13)and the NSCS as defined in Eq.(15)for different values of the air gap between wires.Re-sults are shown in Fig.2.We note that the NSCS follows the FF energy inside the gap behavior for both cases of TS and RS wires.This means that,as expected,most of the SHG is due to FF localization effects in the gap region.TS wires provide better performances with respect to the RS[x10-20cm 2/W][x10-20cm 2/W][x10-20cm 2/W]x (µm)x (µm)x (µm)y (µm )y (µm )y (µm )(a)(b)(c)Fig.7.(Color online)Differential scattering cross section (left column)and magnetic field (ampere/meter)(snapshots in time,right column)calculated for different configurations obtained by varying the value of B (see inset of Fig.1),from (a)45to (b)245and (c)345nm.wires according to the higher values of energy stored in-side the gap.Nevertheless,the contrast between resonant ͑g =18nm ͒and non-resonant emissions is higher for the RS wires.This suggests that further optimizations of the TS scheme are possible (for example,by changing the ra-tio between the major and the minor bases B /d ),in order to increase the SHG efficiency.A deeper look at the SH far field emission pattern re-veals interesting features:Figs.4and 5present the polar plots of the differential nonlinear cross section as a func-tion of the angle of emission.We note that emission pat-terns for the TS and RS wires are very different.Also,sig-nificant differences emerge by varying the size of the gap between wires.In particular,we note that for a gap of 18nm the TS wires exhibit the typical quadrupolar emission related to the quadrupole contribution of the nonlinear re-sponse of the metal.On the other hand,for RS wires the emission pattern is reminiscent of an electric dipole oscil-lating along the x axis.Further details can be obtained by analyzing the magnetic field:In Figs.6(a)and 6(b)we plot the generated SH magnetic field,z -component,for both TS and RS wires with an air gap of 18nm.We note that TS and RS wire emissions correspond to quadrupole and dipole emitters,respectively.Moreover,the RS show aresonant behavior due to the fact that the total length is close to 3/22,with 2being the SH wavelength.The modification of the characteristics of emitted radia-tion by multipolar sources close to nanostructured metal surfaces was investigated in [60]where interesting appli-cations to single molecule spectroscopy and imaging are envisioned.Here we stress the point that the source of the SH field inside the gap is coupled to the far field through the wires and,depending on their shapes,the surface plasmon resonances can affect the field emission.For this purpose we performed a set of calculations keeping the values of h =300nm,d =145nm,and g =18nm constants and by varying the value of B .This way we can explore the transition regime from the dipolar emission to the quadrupolar regime.Results are shown in Fig.7.Figures 7(a)–7(c)describe the differential scattering cross section (left side)and the real part of the magnetic field (right side)corresponding to values of B equal to 45,245,and 345nm,respectively.We note that,by increasing the value of B ,the symmetry of the emission pattern gradu-ally shifts from dipolar [Figs.7(a)and 6(b)]to quadrupole-like in intensity but still of dipolar symmetry if we con-sider the field wave fronts [Figs.7(b)and 7(c)]to be purely quadrupolar [Fig.6(a)].Looking at the differential NSCS,we note that the efficiency of the SH generation re-mains of the same order of magnitude.These results il-lustrate that the generation process is mainly driven by the field confinement in the space between the two wires.We note that starting from Fig.7(b),the emission at the four corners of the gap area has a quadrupolar pattern.Nevertheless,the resonant behavior of the coupled wires is responsible for the dipolar emission symmetry.This can be ascertained by analyzing the case of Fig.7(c):The gen-erated field is out of resonance so that the building up of the dipolar mode is inhibited.Finally,the emission pat-tern has the typical quadrupolar symmetry for higher val-ues of B as shown in Fig.6(b).We also plot the modulus of the Poynting vector in the space near the wires for gap values of 18and 38nm for TS [Figs.8(a)and 8(b)]and RS [Figs.9(a)and 9(b)]wires corresponding to the cases discussed in Figs.4and 5.The near field emission patterns show very interesting fea-tures.In particular,for the resonant case ͑gap=18nm ͒,a sub-wavelength hot spot forms.These features could be used for nanoillumination of single dots or for molecule detection and analysis.4.CONCLUSIONSWe numerically studied the second harmonic generation (SHG)from coupled gold nanowires and highlighted the relevance of the distance between wires and the sections of the SH intensity and emission patterns.Our results suggest that the overall SHG is related to the electromag-netic (e.m.)energy per unit length stored in the gap area between the wires,although finer optimizations of the shape and size of the wires need a more accurate investi-gation and goes beyond this work.We also emphasized the effect of the shape of the wire sections and the possi-bility to obtain dipolar and/or quadrupolar emission pat-terns.Different emission symmetry properties are not re-lated to the dipolar and quadrupolar nonlinearresponsesFig.8.(Color online)Modulus of the Poynting vector ͑W/m 2͒outside the TS wires:air gaps of (a)18and (b)38nm.of metals but are determined by the interaction of the nonlinear sources with the plasmon resonance of the me-tallic wires.The numerical method we developed provides an accurate,efficient,and reliable tool for tailoring the emission of the SHG from nanoparticles and nanowires and can be applied to the study of the nonlinear response of a single molecule trapped between the two metallic wires.By introducing a periodic Green’s function formal-ism it is possible to consider periodic arrangements of wires to study nonlinear properties of metamaterials and periodic metallo-dielectric structures.This feature,as well as the extension to a full three-dimensional code,will be the subject for future work.ACKNOWLEDGMENTSWe thank M.Scalora for discussions and comparisons of different numerical tools for the evaluation of the SHG.We also thank rciprete,A.Belardini,and M.Braccini for stimulating and fruitful discussions.REFERENCES1. E.Adler,“Nonlinear optical frequency polarization in a dielectric,”Phys.Rev.134,A728–A733(1964).2.S.Jha,“Theory of optical harmonic generation at a metal surface,”Phys.Rev.140,A2020–A2030(1965).3. F.Brown,R. E.Parks,and A.M.Sleeper,“Nonlinear optical reflection from a metallic boundary,”Phys.Rev.Lett.14,1029–1031(1965).4.N.Bloembergen,R.K.Chang,S.S.Jha,and C.H.Lee,“Optical second-harmonic generation in reflection from media with inversion symmetry,”Phys.Rev.174,813–822(1968).5.N.Bloembergen,R.K.Chang,and C.H.Lee,“Second-harmonic generation of light in reflection from media with inversion symmetry,”Phys.Rev.Lett.16,986–989(1966).6. C.K.Chen,A.R.B.de Castro,and Y.R.Shen,“Coherent second-harmonic generation by counterpropagating surface plasmons,”Opt.Lett.4,393–394(1979).7.G.I.Stegeman,J.J.Burke,and D.G.Hall,“Nonlinear optics of long range surface plasmons,”Appl.Phys.Lett.41,906–908(1982).8.J. E.Sipe and G.I.Stegeman,in Nonlinear Optical Response of Metal Surfaces ,Surface Polaritons,V .M.Agranovich and ls,eds.(North-Holland,1982),pp.661–701.9.M.Corvi and L.W.Schaich,“Hydrodynamic-model calculation of second-harmonic generation at a metal surface,”Phys.Rev.B 33,3688–3695(1986).10. A.Liebsch,Electronic Excitations at Metal Surfaces (Plenum,1997),Chap 5.11.T.F.Heinz,in Second-Order Nonlinear Optical Effects at Surfaces and Interfaces ,Review Chapter in Nonlinear Surface Electromagnetic Phenomena,H.Ponath and G.Stegeman,eds.(Elsevier,1991),p.353.12.J.C.Quail and H.J.Simon,“Second harmonic generation from silver and aluminium films in total internal reflection,”Phys.Rev.B.31,4900–4905(1985).13.H.J.Simon,C.Huang,J.C.Quail,and Z.Chen,“Second-harmonic generation with surface plasmons from a silvered quartz grating,”Phys.Rev.B 38,7408–7414(1988).14.G. A.Farias and A. A.Maradudin,“Second harmonic generation in reflection from a metallic grating,”Phys.Rev.B 30,3002–3015(1984).15.R.Reinisch and M.Nevière,“Electromagnetic theory of diffraction in nonlinear optics and surface-enhanced nonlinear optical effects,”Phys.Rev.B 28,1870–1885(1983).16.K.Li,M.I.Stockman,and D.J.Bergman,“Enhanced second harmonic generation in a self-similar chain of metal nanospheres,”Phys.Rev.B 72,153401(2005).17.S.Ducourtieux,S.Grésillon,A.C.Boccara,J.C.Rivoal,X.Quelin,C.Desmarest,P .Gadenne,V .P .Drachev,W.D.Bragg,V .P .Safonov,V . A.Podolskiy,Z. C.Ying,R.L.Armstrong,and V .M.Shalaev,“Percolation and fractal composites:optical studies,”J.Nonlinear Opt.Phys.Mater.9,105–116(2000).18.V .M.Shalaev,ed.,Optical Properties of Nanostructured Random Media (Springer,2002).19.B.K.Canfield,S.Kujala,K.Jefimovs,J.Turunen,and M.Kauranen,“Linear and nonlinear optical responses influenced by broken symmetry in an array of gold nanoparticles,”Opt.Express 12,5418–5423(2004).20.J.I.Dadap,H.B.de Aguiar,and S.Roke,“Nonlinear light scattering from clusters and single particles,”J.Chem.Phys.130,214710(2009).21.H.E.Katz,G.Scheller,T.M.Putvinski,M.L.Schilling,W.Fig.9.(Color online)Modulus of the Poynting vector ͑W/m 2͒outside the RS wires:air gaps of (a)18and (b)38nm.。
(推荐)二次谐波的产生及其解
§2.3 二次谐波的产生及其解二次谐波或倍频是一种很重要二阶非线性光学效应，在实践中有广泛的应用，如Nd:YAG 激光器的基频光(1.064μm)倍频成0.532m 绿光，或继续将0.532μm 激光倍频到0.266μm 紫外区域。
本节从二阶非线性耦合波方程出发，求解出产生的二次谐波光强小信号解，并解释相位匹配对二次谐波产生的影响。
2.3.1 二次谐波的产生设基频波的频率为1ω，复振幅为1E u r；二次谐波的频率为()2212ωωω=，复振幅2E u r 。
由基频波在介质中极化产生的二阶极化强度()2P u r ，辐射出的二次谐波场()3E z u r所满足的非线性极化耦合波方程()()()222202222ik z d E z i P z e dz k μω-= u ru r (2.3.1-1) ()()()()()1222110211;,ik z P z z E z e εχωωω=-:E u r u r u r t (2.3.1-2)注意简并度1D =，212ωω=()()()()()()()()()22202110211221112112;,2;,i kzi kzd E z i E z E ze dz k i E z E z e n cμωεχωωωωχωωω∆∆=-:=-:u ru r u r t u r u r t (2.3.1-3)波矢失配量，122k k k ∆=-(2.3.1-4)写成单位矢量（光波的偏振方向或电场的振动方向）和标量的乘积形式333E a E =u r r，基频光场可能有两种偏振方向，即'1111,a E a E r r ，两种偏振方向可以是相互平行也可以是相互垂直，并有331a a ⋅=r r()()()()'222121121112;,i kz dE z i a a a E z e dz n c ωχωωω∆⎡⎤=⋅-::⎢⎥⎣⎦r r r t (2.3.1-5)基频波与产生的二次谐波耦合产生的极化场强度()21P u r ，辐射出基频光场满足的非线性极化耦合波方程。
第4讲 二次谐波产生、相位匹配考虑
dE2 z dz
i
w1
n2c
deff E
2 1
z e
ikz
(2) （ 1） eff 2deff
有效非线性光学系数
辐射基频波的极化强度： 2 2 * P w1 , z 1 0 w1;w2 , w1 2 z E 1 z 产生基频波的耦合波方程: d E1 z w1 2 ikz i P w1 , z e dz 2 0 n2c dE1 z w1 (2) * i eff E2 z E1 z e ikz dz 2n1c
等于零。 晶体长度为L；边界条 件：在入射端，基频光 电场强度幅度为E1(0)， 二次谐波的电场强度幅 度E2(0)=0，。
二次谐波的耦合波方程组为：
k k2 2k1
小信号近似：指的是基频波不发生损耗，因此方程（1）右边
L
w2
w1
E 1 (0) E 2 (0)=0
产生二次谐波的示意图和边界条件
二次谐波产 生过程的 曼利-罗关系
* * * E E E E E E 1 1 2 1 1 2 * * * E E E E E E 1 1 2 1 1 2
能量守恒（产生二次谐波的曼利——罗关系）：
I1 z I2 z cons
E1 z E2 z E1 0
第4讲 二次谐波产生和相位匹配考虑
(Second-Harmonic Generation
& Phase-Matching Consideration)
通过求解产生二次谐波的耦合方程，得到小信号和大信号二次谐波光强 表达式，讨论获得强相干信号的相位条件，分析在双折射晶体材料中实 现完全相位匹配条件的角度和温度相位匹配两种技术途径,讨论准相位匹 配技术。
二次谐波的产生及其解
§2.3 二次谐波的产生及其解二次谐波或倍频是一种很重要二阶非线性光学效应，在实践中有广泛的应用，如Nd:YAG 激光器的基频光(1.064μm)倍频成0.532μm 绿光，或继续将0.532μm 激光倍频到0.266μm 紫外区域。
本节从二阶非线性耦合波方程出发，求解出产生的二次谐波光强小信号解，并解释相位匹配对二次谐波产生的影响。
2.3.1 二次谐波的产生设基频波的频率为1ω，复振幅为1E ；二次谐波的频率为()2212ωωω=，复振幅2E 。
由基频波在介质中极化产生的二阶极化强度()2P ，辐射出的二次谐波场()3E z 所满足的非线性极化耦合波方程()()()222202222ik z d E z i P z e dz k μω-= (2.3.1-1) ()()()()()1222110211;,ik z P z z E z e εχωωω=-:E (2.3.1-2)注意简并度1D =，212ωω=()()()()()()()()()22202110211221112112;,2;,i kzi kzd E z i E z E ze dz k iE z E z e n cμωεχωωωωχωωω∆∆=-:=-: (2.3.1-3)波矢失配量， 122k k k ∆=- (2.3.1-4) 写成单位矢量（光波的偏振方向或电场的振动方向）和标量的乘积形式333E a E =，基频光场可能有两种偏振方向，即'1111,a E a E ，两种偏振方向可以是相互平行也可以是相互垂直，并有331a a ⋅=()()()()'222121121112;,i kz dE z i a a a E z e dz n c ωχωωω∆⎡⎤=⋅-::⎢⎥⎣⎦ (2.3.1-5)基频波与产生的二次谐波耦合产生的极化场强度()21P ，辐射出基频光场满足的非线性极化耦合波方程。
()()()122101112ik z d E z i P z e dz k μω-= (2.3.1-6)()()()()()21*2()12101212;,i k k z P z z E z e εχωωω-=--:E (2.3.1-7)()()()()()'21*1121121211;,::i kz dE z i a a a z E z e dz n c ωχωωω-∆⎡⎤=⋅--E ⎢⎥⎣⎦ (2.3.1-8)如果介质对频率为13,ωω的光波都是无耗的，即13,ωω远离共振区，则()()()()22311131;,,;,χωωωχωωω---都是实数。
非线性光学第三章（3.1-3.2节新）光学二次谐波的产生
第三章 光学二次谐波的产生(Franken,1961首次观察到)第3.1节 倍频原理*11123()()22i kzdA z D A z ikA A e dz α-∆=-+ *22213()()22i kzdA z D A z ikA A e dz α-∆=-+ （2—138） 33312()()22i kzdA z D A z ikA A e dz α∆=-+ 321k k k k -+=∆ 321ωωω=+1. 基频光低损耗（低倍频效率） 假设：（1）无吸收0321===ααα（2）假定只有一个偏振的基频光入射，则A 1=A 2 （3）基频光无损耗0/1=dzdA , A 1=常数, 这样方程组（2—138）只需考虑最后一个方程。
313213212102121213212,2121212k k k k k k D e ikA e ikA e A ikA e A ikA D dz dA z k i z k i zk i z k i -=-+=∆========∆∆∆∆ωωωωω（3－1）利用边界条件)1(2121)(0,02102103033-∆====∆∆==⎰kl i lkz i z z e k kA dz e ikA z A A E （3—2）因为光强ϕεcos 220E nc I = （3—3）（因1cos ,1cos ＝故以后就都取ϕϕ≈）又 202)3,2,1(mm mmmm A c I m A n E ωεω=∴==（3—4）所以22202230233031))(()(412)(2)(-∆⋅==∆kLi e l A k k c l A c l I ωεωε (3—5)321321222321321n n cd k n n n cd k eff eff ωωωωω==（3—6） 又[][])2(sin 4)cos(1214)cos(12)cos(22122kl kl kl kl ekLi ∆=∆-⋅=∆-=∆-=-∆22222103210321222410321032122120222241032132122303)2()2(sin 2)2()2(sin 42)2()2(sin 42)(kl kl d l I n n c l kl kl A n n c d c l kl kl A n n c d c l I eff eff eff ∆∆=∆∆=∆∆=∴εωεωωεωωωε （3—7） 若改用功率来表示，则有S P I /=于是22212232103213)2()2(sin 2)(kl kl S P d l n n c l P eff∆∆=∴εω (3—8)定义倍频效率2210223210321113)2()2(sin 2)()2(kl klI d l n n c P P eff ∆∆==εωωωη (3—9)讨论：（1）2103)(I l I ∝ （3—10）（2）⎪⎩⎪⎨⎧∝2210ld I eff η （3—11）22/)(sin )(x x x f =（3）22)2()2(sin kl kl ∆∆∝η，（当1,0这个因子＝=∆k ）（3—11） (4)如果,0=∆k （相位匹配），则有2l ∝η，但实际上还有许多因素，需要考虑。
二次谐波产生过程
二次谐波产生过程嘿，朋友们！今天咱来聊聊二次谐波产生过程这个神奇的玩意儿。
你说这二次谐波产生过程啊，就好像一场奇妙的音乐演奏会！咱先把激光束想象成一群有活力的小乐手，它们雄赳赳气昂昂地往前冲。
当这些小乐手们碰到特定的材料时，就像是来到了一个特别的舞台。
在这个舞台上，小乐手们开始展现它们独特的魅力。
它们相互作用，就像乐手们之间的默契配合一样。
然后呢，神奇的事情发生啦！原本单一频率的激光束，突然就产生了新的频率，这可不就是像原本演奏一首曲子的乐手们突然奏出了全新的旋律嘛！你想想看，这多有意思啊！就好比你本来只期待听到一种声音，结果却给你来个惊喜，多了一种更美妙的声音。
而且啊，这二次谐波产生过程还特别挑呢！不是随便什么材料都能让它完美展现的，这就跟好的乐手也得挑个好舞台是一个道理。
它就像是一个隐藏在科学世界里的小魔法，等着我们去发现和探索。
要是你不去了解它，那可就太可惜啦！你难道不想知道这神奇的小魔法是怎么施展的吗？咱再深入一点说，这二次谐波产生过程对于很多领域那可都是大功臣啊！在医学上，它能帮助医生更清楚地看到我们身体里的情况，就像给医生配上了一副超级厉害的眼镜。
在通信领域呢，它能让信号传输得更稳定、更高效，就像给通信搭起了一条顺畅的高速公路。
哎呀呀，这二次谐波产生过程真的是太重要啦！它就像一个默默无闻却超级厉害的幕后英雄，为我们的生活带来了那么多的便利和进步。
所以啊，朋友们，可别小瞧了这看似深奥的二次谐波产生过程。
它虽然藏在科学的海洋里，但只要我们用心去了解，就会发现它的魅力和价值简直无穷无尽！我们得好好感谢那些研究它的科学家们，是他们让我们有机会领略到这神奇的世界。
怎么样，现在是不是对二次谐波产生过程更感兴趣啦？那就赶紧去探索吧！原创不易，请尊重原创，谢谢!。
(非线性光学课件)第三章 二阶非线性光学效应
E3(z) E1(0) tanh K E1(0) z
☆
现在倍频效应的应用已经比较成熟，
如常把Nd:YAG激光器发出的波长1.06mm的红外激光
变换为波长532nm的绿色倍频激光。
14
☆
假设晶体对这两种光都没有吸收， 讨论晶体出射面的倍频光强度和倍频转换效率，
即倍频光功率与入射光功率之比。
分析两种情况研究光学倍频效应： 一种是不消耗基频光的小信号近似情况； 另一种是消耗基频光的高转换效率情况。
☆
P3 ( L) P1(0)
I3 ( L) I1(0)
8 2d 2L2 0c3n2n 2
P1(0) sin c2 k L
S
2
（5）倍频效率正比于基频光的功率密度， 可以通过聚焦基频光的办法来提高倍频效率。
26
实验图
远离相位匹配条件
Input beam
SHG crystal
Output beam
30
d
E3 (z) dz
i
2d
cn
E12 (z)
E1(z) 2 E3(z) 2 E1(0) 2
☆
d
E3(z) / E1(0) dz
K
E1(0) 1
E3(z) / E1(0) 2
两边分离变量，再积分求解，得到（附录3-6）
E3(z) E1(0) tanh K E1(0) z
E1(z) E1(0) 1 tanh2 K E1(0) z E1(0) sech K E1(0) z
☆
可以得到（附录3-3）
E3 (L) 2 E3 (L)E3*(L)
4 2d 2L2
c2n2 2
E1 (0)
4
sin2 k L / 2 k L / 22
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二、光倍频及光混频的稳态小讯号解
（1）讨论混频过程
三个电磁波表示为（设电磁波传播方向是Z轴）：
则推导出耦合波方程为：
所谓小讯号解，顾名思义，就是认为在低转换效率极限情 况下，谐波讯号极小，那么可认为A1 ，A2 不变， • 从而直接积分可得
• 考虑到功率密度公式： • 我们就得到了小讯号解下的和频波功率密度：
三、光倍频及光混频高转换效率时的稳态解
（1）倍频时的耦合波方程的解耦合波写成：
• 由方程组前两式可推得 • ，那么就可得到
• 这就表明在无损非线性介质中基波与谐波的功率密度之和 守恒。 设归一化函数为：
代入耦合波方程得：
• 又由归一化函数得 • 这就是倍频过程中的门雷——罗威关系。 • 1）相位匹配的情况 • 对前页最后一式进行形式变换，代入各条件，积分得 • 这里的倍频效率为 • 特征作用长度：
（2）讨论倍频过程
• 既然是倍频过程，那么 • 分别设基波A1与二次谐波A2的电场强度为
• 推导出耦合波方程为
• 对于小讯号解，可把 看做常数，因此直接积分得，
• 则相应的功率密度为
倍频效率为
由上式可见，光混频所产生的新波功率及倍频时所产生二 次谐波功率，在小讯号近似下与 成正比，且与 有密切的 关系。
同样令 ，代入耦合波方程，可得
• 对上面左边三式进行处理， • 可得 ，表示能量守恒
• 进而得到归一化函数
• 将上面的归一化函数代入耦合波方程，可得
• 由于snX小于1，故从上式看出， 。这就说明当 输入光波的光子数不等时（ 与光子数成正比），和频光 波的光子数不会超过两个输入光波中光子数较少的那个波。 这一结论也是门雷——罗威关系的必然结果。
• 最后考虑一个特殊情况，两个输入光波的光子数相等，此 时量子效率变为：
• 由
，可以定出特性作用长度
• 2）非完全相位匹配的情况 • 再对前页最后一式进行形式变换，代入非完全相位匹配的 条件并且积分，则得到
• 相位匹配条件下，
• 反过来，就可以得到 • 画出 的关系曲线，如图三，我们可以看到当 较 大时，v 始终保持较小的值，因此这时作小讯号处理更为 方便。
（2）混频时的耦合波方程的解
（这里只讨论相位匹配情况）
一、引言
• 标志非线性光学诞生的第一个实验是弗兰肯 （Franken）等人在1961年做的光学二次谐波产生 （即光倍频）实验。 • 1962年，乔麦特（Giordmino）以及马克尔 （Maker）等人分别提出了相位匹配技术，这才使 得光倍频有可能达到较高的转换效率。值得一提 的是，光倍频及光混频技术的发展是与激光器的 发展密切相关的。实验证明，转换效率已经达到 70～80%。 • 此外，由于非线性光学混频可以实现频率上的转 换，可使红外波长的讯号转换到可见波长。