Conversion of short optical pulses to terahertz radiation in a nonlinear medium: Experiment and theory

Transcription

1 PHYSICAL REVIEW B 76, Conversion of short optial pulses to terahertz radiation in a nonlinear medium: Experiment and theory N. N. Zinov ev* Department of Physis, University of Durham, Durham DH1 3LE, United Kingdom and A. F. Ioffe Physial Tehnial Institute, St. Petersburg, Russia A. S. Nikoghosyan Department of Mirowave Engineering, Yerevan State University, Yerevan 37505, Armenia R. A. Dudley National Physial Laboratory, Teddington, Middlesex TW11 0LW, United Kingdom J. M. Chamberlain Department of Physis, University of Durham, Durham DH1 3LE, United Kingdom Reeived 0 April 007; revised manusript reeived 19 July 007; published 13 Deember 007 We report on the frequeny and time domain analysis of eletromagneti terahertz radiation generated by nonlinear onversion of short optial pulses rossing boundaries of nonlinear material. Analysis and omparison with experiment have unequivoally established that the nature of ollinear radiation at terahertz frequenies relates to the phenomenon of transition radiation produed by the instantaneous reation at the input interfae and extintion at the output interfae of the moving polarization harge formed by the nonlinear oupling of the pump eletromagneti fields. The mehanism is analogous to the phenomena of transition radiation of moving free harges, in partiular, to the radiation mehanism disussed in the Tamm problem. DOI: /PhysRevB PACS numbers: 4.65.Ky, m, p I. INTRODUCTION In spite of remarkable ahievements in nonlinear optis, there remains a fundamental issue, related to the mehanism of nonlinear wave onversion NLWC of a short optial pulse into differene and sum frequeny radiation, that does not yet seem to have a lear interpretation. Aording to the fundamentals of nonlinear optis, 1, differene frequeny generation DFG, being a phenomenon of a oherent nonlinear three-wave interation, enables the nonlinear onversion of pump eletromagneti EM waves into eletromagneti radiation EMR at a frequeny that an be the differene ombinations of the pump wave frequenies inluding zero frequeny, optial retifiation OR. 1 3 The spetrum of a short pulse ontains a broad band of frequenies around a arrier frequeny 0. 4,5 Nonlinear onversion of suh pulse in a medium generates EMR at various frequeny ombinations: DFG/OR, sum frequeny generation SFG, and seond harmoni generation SHG. Aording to NLWC theory,,6 originally developed for interations of ontinuous plane harmoni waves, oherent onversion on the mirosopi sale is interpreted as the result of onstrutive interferene of two partial waves, the inhomogeneous wave IHW and the homogeneous wave HW. These waves are related to the mathematial onstrutions of the same name whih provide the solution of the nonlinear wave equation. When extrapolated to the ase of the interation of wave pakets, the NLWC model should deal with the interation of two wave pakets, HW and IHW. For DFG, HW should be the bunh of waves in a nonlinear medium around the onversion frequenies with wave vetors k =/. IHW is the wave paket with the same frequenies = p, but with the wave vetors k p k p + p /v g p. Here, and v g p are the phase veloities of the onverted terahertz wave and the group veloity of the IHW equal to the pump wave paket and p is the bandwidth of the pump. It is believed that a mirosopi effet of NLWC is due to the onstrutive interferene of IHW and HW,6 if the phase mathing onditions PMCs are met. PMCs ome from the onservation laws of wave vetor and energy and redue to a simple equality between the veloities of interating waves, namely, =v g p. In pratie, however, beause of veloity dispersion, these PMCs are rarely met in full, at least for a broad band of frequenies. Then, the onstrutive interferene of IHW and HW ours on the sale of the oherene length L oh. This ommonly aepted approah 1 10 assumes that both the HW and IHW begin onstrutively to interfere and build up the nonlinear effet mirosopially within L oh from the input interfae. They further walk off and, therefore, do not ontribute to the nonlinear onversion on the sale zl oh. The interferene of HW and IHW explains the Maker fringes, 8,9 one of the experiments forming the ornerstone of nonlinear optis. So far, a onsiderable number of papers published on nonlinear interations of wave pakets have used the NLWC model outlined above to interpret their results, e.g., Refs. 5 and Some of these works onsidered the problem using a CW plane wave approah;,1,17,18 the others used a method based on the d Alembert symboli approah to the wave equation, 13 or onsidered the partiular ase of exat phase mathing. 11,15,16 The NLWC model was not alled into question until the publiation of Ref. 19, where a new property of DFG was reported: twinning of pulses of terahertz EMR generated /007/763/ The Amerian Physial Soiety

2 ZINOV EV et al. with short optial pump pulses in an eletro-opti rystal. The time delay between the twins was equal to the differene of propagation times for terahertz and pump pulses. From this, evidene 19 tentatively attributed the twin pulses of terahertz EMR to EM radiation emanating from boundaries. However, the twin pulse struture ould alternatively be explained within the NLWC model as the signature of a threewave interation in the time domain by assoiating the pair of pulses with the HW and IHW, respetively. Indeed, aording to Refs. 1 and, the HW and IHW pulses are generated simultaneously in the same volume of sample adjaent to the input interfae boundary. Beause of dispersion, the HW and IHW walk off with different veloities, and v g, respetively. Thus, the differene in the arrival time of the HW and IHW pulses at the end of sample of length L would be given by t=l/ L/v g. This value of t is exatly equal to the time split measured in Ref. 19. A brief outline of NLWC model applied to DFG is given in the Appendix. An alternative to the NLWC model approah to the generation of EMR with short optial pulses has been proposed phenomenologially in 196 Ref. 0 by drawing analogies between the generation of EMR from EMR and the radiation phenomena of moving harges. 1 8 An intense optial pulse in a nonlinear material is aompanied by a loud of spae polarization harge with the density d = P NL, where P NL =ˆ :E p E p, ˆ is the nonlinear suseptibility tensor of the seond rank and E p is the eletri omponent of the inident pump eletri field. The propagation of this polarization harge, aording to the proposition of Ref. 0, ould be the soure of the EMR. In Ref. 0, two mehanisms for the generation of suh EMR were suggested. One was the nonollinear mehanism of EMR analogous to the Vavilov- Cherenkov effet. 7,8 For DFG, the low frequeny dieletri onstant normally exeeds the value of the dieletri onstant at pump arrier frequenies 0 beause of the ontribution to from phonons: 0. A similar kind of inequality holds for SHG in a semiondutor, if the pump frequeny 0 is at the mid gap position 0 ± E g / and 0 ±E g, where E g is the forbidden gap. Under suh onditions, the resonane enhanement fator 0 E g 1 inreases the value of 0. In both ases, the veloity v g of the pump pulse bearing the polarization harge beomes greater than the phase veloity of the onverted EMR, reating the onditions for the Vavilov- Cherenkov generation of EMR. For a single pulse and DFG type of nonlinear wave onversion, the onservation laws for wave vetor and energy, k 0 =k and 0 k 0 =k, lead to the PMC of a nonollinear type: k 0 k = v g k os, where v g =/k 0 is the group veloity of the pump at frequeny 0 and k is the wave vetor of the onverted wave radiated at angle. The angle obeys the relationship os =/v g, where /k= is the phase veloity of the onverted wave. For a broadband DFG at terahertz frequenies, the maximum value of normally reahes for typial rystals suh as ZnTe, GaAs, LiNbO 3, et. Although Eq. 1 PHYSICAL REVIEW B 76, seems to establish PMC, several ompliations remain. Firstly, Eq. 1 is subjet to dispersion,, 0, and, therefore, the angle varies over a large sale. Seondly, Eq. 1 is the relationship between the longitudinal projetions of wave vetors; it is not a omplete PMC. The defiit of the transverse wave vetor for a single pump pulse is derived from the unertainties of pump pulse loalization, e.g., if the pump beam is tightly foused. Thirdly, even in the most favorable ase of weak dispersion, ondition 1 leads to nonollinear EMR at the angle that normally exeeds the angle of total internal refletion. This bloks effiient oupling of this type of EMR into free spae. The Vavilov-Cherenkov type of terahertz DFG so far has been identified 0,9,30 and deteted in the same rystal by mapping the Cherenkov one formed by onverted field, but it enounters serious problems of oupling into free spae. 34,35 Aording to another proposition of Ref. 0, a high frequeny ollinear EMR ould begin with a pulse rossing the boundary between a linear and nonlinear medium or two media, one of whih has a greater nonlinearity than the other. When an optial pulse traverses the interfae between linear and nonlinear media, a net spae polarization harge is reated in the nonlinear medium, and instantaneously aelerated to the optial pulse veloity. When the optial pulse leaves the nonlinear medium, the polarization harge instantaneously disappears, whih is equivalent to the harge stopping. Between these two radiation events, the polarization harge moves uniformly with the veloity of the optial pulse. At this stage, i.e., in the bulk, the harge an radiate only if the onditions for the Vavilov-Cherenkov effet are met. This model senario is very similar to the radiation events of the start-stop motion of a harged partile in homogeneous media, whih has been onsidered earlier by Tamm. 1 This radiation problem is named as the Tamm problem and has reeived attention in many subsequent papers see, e.g., Ref. and referenes ited therein. Ifthe mehanism of EMR during nonlinear wave onversion resembling the Tamm problem ase does indeed exist, it an be related to the wide lass of transition radiation TR phenomena obtained in high energy partile physis when moving harges ross boundaries between two different media 3 5 and whih have been onfirmed experimentally in many papers e.g., see Refs However, unlike radiation phenomena with moving harges, 5 nonlinear oupling between the pump fields is the essential ondition for reation of polarization harge. Therefore, suh a radiation mehanism with optial pulses an only be observed in a nonlinear medium. TR EMR ould be generated in a ollinear geometry and, unlike Vavilov-Cherenkov EMR, TR does not require any speifi relation between the veloities. The strength of TR with moving harges is proportional to the formation length, 6 i.e., the length over whih radiation from a moving soure is established. It orresponds to a definition of oherene length. Then, TR with short optial pulses should be emitted from the regions on the sale of the formation length whih are adjaent to the input and output boundaries. It is lear that within this model, the time split between the pulses emitted from input and output surfaes of a nonlinear medium is t=l/ L/v g, where and v g are the averaged veloities over the emission bandwidth and L is the thikness of nonlinear medium

3 CONVERSION OF SHORT OPTICAL PULSES TO Thus, the brief review given above shows that understanding of NLWC phenomena is still far from omplete. The existing ambiguity primarily assoiated with the mehanism of nonlinear wave generation, loalization of the generation regions, and the time- and frequeny-domain kinetis of nonlinear wave generation demands a reappraisal of the theoretial foundations of EMR with short optial pulses in nonlinear media, together with omprehensive time domain experimental researh into the oherent generation of EMR pulses. A brief report on our researh, proving both theoretially and experimentally the mehanism of terahertz TR generation, has been published reently in Ref. 39. In this paper, we extend the theory onsidering two model ases of infinite and semi-infinite nonlinear media, extending them into the pratially important ase of terahertz generation under arbitrary wave vetor mismath WVM onditions in a nonlinear slab plaed in a linear medium. To ompare with experimentally measured signals, we derive the analytial dependenies for DFG terahertz wave forms emitted into the remote free spae zone in both forward and bakward geometries. We present experimental results for two nonlinear samples, one with WVM k0 LiNbO 3 and the other with WVM k 0 ZnTe. We also ompare the obtained data with the preditions of existing NLWC model. This paper is organized as follows. Setion II onsiders nonlinear onversion of optial pulses into terahertz EMR at arbitrary WVM in infinite and semi-infinite nonlinear media and a nonlinear slab. It losely examines the ollinear onversion proess, terahertz pulse struture, and the dependene of their temporal and spetral kinetis on slab thikness, WVM onditions, dispersion, and absorption in the slab. Setion III is devoted to experimental proedures and disussion. The Appendix briefly outlines the theoretial sheme of NLWC model and ompares the preditions of existing NLWC model with the experimental time domain data and results of our theory. Setion IV ontains a summary and onlusion. II. THEORY We start from the time-frequeny domain alulations of terahertz EMR resulting from the onversion of short optial pump pulse traversing a nonlinear material. We alulate ollinearly and antiollinearly propagating EM fields in infinite and semi-infinite nonlinear media and the wave forms of THz emission radiated in ollinear, forward and antiollinear, bakward diretions from a nonlinear slab held in air under the ondition of arbitrary value of WVM k0. The solutions for terahertz wave forms, thus obtained, are ompared with the experimental data in Se. III. A. Coherent generation of eletromagneti radiation in a nonlinear medium To investigate time domain and spetral kinetis of terahertz EMR generated with short optial pulses theoretially, we onsider the proess of nonlinear wave paket onversion, a dynami analog of OR/DFG, inside a nonlinear dieletri medium. The solution of the problem begins from the set of two equations, 40,41 the wave equation, E i r,t 1 + dtt t 0 t E ir,t + = 0 dtt te i r,t + 0 P NL r,t t + E i r,t, and the Poisson s equation, + 0 dtt t Er,t = P NL r,t. For a single pulse, nonlinear polarization is given by + PHYSICAL REVIEW B 76, P NL r,t = dt t ter,ter,t For the sake of generality, we inlude in Eqs. and 3 the term desribing the loss for onverted field E i r,t due to finite ondutivity by free arriers. In nonlinear dieletris and wide band gap materials no two-photon absorption, free arrier ondutivity is insignifiant. Therefore, in the further treatment, we omit this term. In Eqs. and 3, we assume a ase of weak nonlinearity, i.e., there is no depletion of energy from the pump wave, and also negleting the pump pulse distortion. The form of expressions 4 assumes that nonloal effets and spatial dispersion are negleted leaving only the effets of time and/or frequeny dispersion. We note that Eqs. and 3 desribe radiation phenomena of a moving polarization harge with density d = P NL. The seond term in the right-hand side rhs of Eq. is known as the time derivative of the nonlinear term of displaement urrent J NL d. Then, the expression for J NL d an be expressed in the hydrodynami form as the flux of polarization harge, 3 J NL d r,t = P NL r,t/t = P NL r,t r/t = d r,tv. 5 This provides formal justifiation of the orretness of the preeding statement. Here, v is the veloity of polarization harge. It is known that, unlike ondution harge, the polarization harge is a bound harge. In fat, this distintion is stritly limited to the stati ase and beomes meaningless for high frequeny time dependent fields. With a propagating wave paket, the envelope moves with the group veloity, so also does the polarization harge indued due to nonlinear oupling of the pump fields. When formulated in this way, Eqs. and 3 resemble the familiar set of equations from the theory of radiative phenomena of moving harges. 5 The only differene is that, instead of an external moving harge, in this ase, the polarization harge is formed in the medium due to the nonlinear relationship between fields; it subsequently moves uniformly further through the medium. The wave equations Eq. and the Poisson s equation Eq. 3 desribe both ollinear-antiollinear and nonollinear radia

4 ZINOV EV et al. tion phenomena aompanying the propagation of this polarization harge. As stated in Se. I, the output radiation oupling of nonollinear EMR is greatly restrited into free spae in the forward diretion. For this reason, we restrit ourselves to the ollinear nonlinear proess of nonlinear onversion. This takes into aount only the terms from the soure s expansion in plane waves with zero wave vetor angular omponents. Therefore, for ollinear and antiollinear interations, we may retain only the dependene of the onverted field on the z oordinate beause of axial symmetry along the z axis. This approximation is relevant to the ases of onedimensional geometry or with moderate fousing of the pump beam when the onfoal parameter of the fousing optis is muh bigger than the medium thikness. Under this ondition, the wave front of pump field within the medium volume an be onsidered as nearly plane but is restrited to the ross setion square of the pump beam. Taking into aount the tensor relationship between the terahertz field and the linear polarization, both the field Er,tEz,t and polarizations P NL r,tp NL z,t an have, in general, both longitudinal and transverse omponents. Aordingly, we separate eletri field and polarization vetors into longitudinal and transverse omponents using the following expressions: Ez,t = z 0 E z,t + E z,t, P NL z,t = z 0 P NL z,t + P NL z,t, where z 0 is the unit vetor in the diretion of the z axis. The index labels the polarization state in the plane X-Y, i.e., refers to the polarization along either the X or Y axis, or to a ombination of X and Y. After substitution of Eqs. 6 and 7 into Eqs. and 3, and grouping similar polarization omponents, we obtain two equations for E z,t and E z,t: 1 E z,t z dtt t 0 t E z,t 6 7 = 0 P NL z,t t, 8 + dtt t t E z,t = 0 t P NL z,t. The seond equation of this set, Eq. 9, is simply the operator relationship between E z,t and P NL z,t. Therefore, without loss of generality, we an hoose the geometry with zero omponents of the nonlinear suseptibility tensor ˆ responsible for the P NL z,t and neglet the transverse beam onfinement effet. Finally, we obtain the single wave equation for the terahertz eletri field E z,t in the form of Eq. 8. We represent a pump optial pulse in the rhs of Eq. 8 as the produt of the Gaussian pulse envelope and the plane arrier wave at the frequeny p : 9 E p z,t = 1 e E 0 exp t z v g 4 p expi p t k p z +.., 10 where v g is the group veloity of the pump wave paket, the pump pulse, v g = 0 n p + p dn p /d p 1, and p is a parameter related to full width at half maximum. 5,4 The salar omponent E z, of the terahertz field to be used in the alulations below is E z,=e E z,, where e is the polarization vetor in the plane perpendiular to the beam axis. To ontrat the formulas, we use the relationship ˆ = i,j=k e i ˆ ijk e j e k. For definiteness, we hoose the polarization of E z,=e x z,. Then, with the use of Eq. 10, the salar form of the nonlinear wave equation desribing the ollinear proess of terahertz generation is transformed into the form E x z,t z = 1 0 t + t dtt te x z,t + dtˆ t te 0 exp t z/v g. p 11 In Eq. 11, we leave the soure term in the form desribing DFG/OR proesses. Consideration of SHG/SFG proesses an be onduted in similar way. The solution of Eq. 11 desribes the field generated in a nonlinear medium from moving soure represented by rhs. For the sake of generality, we onsider below the generation of terahertz field in infinite and bounded media. B. Infinite nonlinear medium We first turn to a model problem of an infinite nonlinear medium lossless for the radiation of the pump pulse. This ase allows us to draw up the major onlusion on the loation of EMR generation from nonlinearly oupled pump EM fields. To obtain the solution of the wave equation Eq. 11, we apply the Fourier transform to Eq. 11 for the pair of onjugated variables t. With the notation S = / 0 ˆ E 0 p exp p / and = 0 / = 0 /n, Eq. 11 has the form of the inhomogeneous Helmholtz equation: d E z, dz PHYSICAL REVIEW B 76, E z, = Sexpi v g z. 1 The solution of Eq. 1 is given as the onvolution integral of the Green s funtion with the rhs: + E x z, = dgz,expi. S v g The Green s funtion Gz, is found from the solution of

5 CONVERSION OF SHORT OPTICAL PULSES TO R d Gz, dz + Gz, = z. 14 Equation 14 is solved using Fourier transform z k on Eq. 14. Then, the solution Gz, is given as Gz, = 1 k y + dk k e ikz. 15 The integration of Eq. 15 is performed applying the residue theorem to the ontour C R shown in Fig. 1, where the poles / were added with an infinitesimal imaginary part 0. As onsequene, the Green s funtion Gz, is obtained in the form i Gz, = exp i z. 16 It obeys the boundary onditions at infinity, whih are Sommerfeld radiation onditions Then, using Eq. 13, terahertz field generated in the bulk of infinite nonlinear medium in ollinear diretion is given by E x z, = i S z + exp i 1 expi 1 v g d. 17 The integral in rhs of Eq. 17 represents funtion of the argument 1/ 1/v g, that is, the WVM quantity. This leads to the onlusion that there is no nonvanishing radiation ontribution everywhere in the bulk of nonlinear medium in the ollinear diretion, kz, if and only if there is a WVM, namely, 1/1/v g, no matter what the strength of this inequality is. In fat, this is an expeted result that finds a lose analogy in the theory of transition radiation phenomena of uniformly moving harges. 3,5,6 It is simply the mathematial form of the well known physial priniple adopted to the ase of interating EM waves that uniformly moving harge in a homogeneous medium does not radiate, unless ondition 1 for the Vavilov-Cherenkov VC EMR is fulfilled. For ollinear radiation, =0 and Eq. 17 requires the exat math of and v g. The marked behavior of Eq. 17 means that the physis of EMR generation at nonlinear wave C R FIG. 1. Contour of integration C R used to alulate Eq. 15. R k x PHYSICAL REVIEW B 76, interation is more omplex than a trivial nonlinear mixing of EM waves. In pratie, it is quite impossible to observe the exat ondition 1/=1/v g due to dispersion. In the strit sense, this ondition an only be met for a single frequeny rather than for a frequeny band. Therefore, the EMR field generated in the ollinear diretion at nonlinear oupling of EM fields should be treated as the radiation phenomenon entirely related to the interfae boundaries. The field is emitted from the boundary of two media with different nonlinear dieletri onstants linear-nonlinear media, two different nonlinear media, rossed by optial pulse. In the bulk of the medium, however, ollinear radiation an only our if and only if v g =, that is, the exat equivalent of vanishing WVM, k 0. To prove this assertion, we further onsider the ases of a semi-infinite medium and a nonlinear slab related to nonlinear opti experiments. C. Bounded nonlinear media In a bounded medium, the solution of the wave equation with the form of the soure term Eq. 11 sums up multiple refletions of the terahertz EMR that have been onverted from the first transit flight of the pump pulse; these provide the meaningful part of the solution and ontain information on the physial proesses of nonlinear generation of terahertz EMR. The form of Eq. 11 does not inlude the ontributions to terahertz EMR from the multiple refletions of the pump pulse inside the slab, whih is an equivalent to the assumption of a sample being antirefletion oated to avoid multiple refletion of the pump pulse. Bearing in mind the quadrati dependene of the soure term on the pump field in the rhs of Eq. 11, we assume that the ontribution to the terahertz wave form from the multiple refletions of the pump pulse inside the slab is negligibly small. The general solution of Eq. 1 is the sum of the solution of the homogeneous equation and the partiular solution of the inhomogeneous equation Eq. 1. The physial meaning of the rhs of Eq. 1 is the Fourier transform of the time derivative of the nonlinear term of displaement urrent that has been reated by the moving polarization harge, i.e., a moving soure, with veloity v g Eq. 5. The differene between Eq. 1 and the equations from the theory of TR of moving harges 5 is in the rhs soure term. In the theory of radiation phenomena of moving harges, the polarization of the soure urrent and urrent diretion are parallel to eah other. Unlike this, in Eq. 1, the diretion of soure eletri field lies in the transverse plane, but the soure moves along the z axis. Another differene is in the nature of moving harge. Instead of uniformly moving external harges, in this ase, a polarization harge is formed due to nonlinear oupling of EM pump fields. The important spatial parameter, the formation length, defines the amplitude of the emission proess 6 during wave onversion. The formation length is defined in several equivalent ways, namely, as the length where the radiation from a soure is established, as the length where the radiation field is separated from the field of harge, or as the length from whih all waves are radiated with phases less than. Assuming a soure radiating EMR Er,t=E 0 expikr it at an angle and propagating with

6 ZINOV EV et al. PHYSICAL REVIEW B 76, the veloity of v g, the phase differene of EMR,, between two arbitrary points of the soure trajetory separated by distane R is given by =r/v g k r. Then, using the definition of the formation or oherene length given above, it takes the form L BW/FW oh, = v, 18 g os v g1± where the signs and in the denominator orrespond to the EMR emitted into the forward FW and bakward BW hemispheres /,/, respetively. For ollinear and antiollinear geometries, we let =0 and L BW/FW oh BW/FW =0,. =L oh 1. Semi-infinite nonlinear medium The existene of an interfae boundary in this ase introdues an important differene in the treatment of the nonlinear radiation problem. The boundary ondition BC for E and H fields requires the ontinuity of the tangential omponents of the eletri and magneti fields at the interfaes z =0 input boundary of semi-infinite medium. The BC mathes the eletri omponent in free spae with the total eletri field inside the medium at z=0. Then, the ollinear field inside a semi-infinite z nonlinear medium is E x z, = C 1 expi + L oh FW L BW oh S expi z. v g 19 The expression for eletri field outgoing into free spae from the boundary at z=0 is the solution of homogeneous wave equation d E x z, dz + 0 E xz, =0. 0 This desribes the terahertz field E z, bakward irradiated into the left-hand half of free spae at z0, E x z, = C BW exp i 0 z. 1 For the hosen geometry normal inidene, polarization of E vetor along the x axis, polarization of H vetor along the y axis, and the z axis is the propagation axis, the relationship between the eletri and magneti omponents takes the form i de x z, = H y z,. 0 dz Then, the BCs for E x z, and H y z, at z=0 interfae are given by E x z = 0, = E x z =+0,, 3 H y z = 0, = H y z =+0,. 4 The integration onstants C BW and C 1 are obtained from (a) (b) FIG.. Color online Top: Calulated wave form for terahertz far-field radiation emitting in antiollinear diretion from semiinfinite medium. Bottom: Calulated wave forms for the EM field inside a semi-infinite medium urves 1,, 3, and 4 from the ontinuous set of Ez,t. The wave forms are shifted against eah other in the z diretion orresponding to the speified propagation lengths in the medium: urve 1 orresponds z=1 mm, urve orresponds z= mm, urve 3 orresponds z=4 mm, and urve 4 orresponds z=8 mm. The lines à and B orrespond to the alulated pulse positions. The medium parameters are taken for LiNbO 3 C BW = C 1, 5 C BW = C 1 0, 6 v g where = L FW oh L BW oh S/. The propagating sum of radiation and radiationless fields inside a semi-infinite medium in the diretion of the positive z axis is Ez, = v g v g expi z exp v g z, 7 and the radiation field emitting from the sample in the bakward diretion is Ez, = v g + 0 1expi z. 8 v g Comparing with the ase of infinite nonlinear medium, the appearane of the interfae boundary linear-medium nonlinear-medium has resulted in the generation of a strong single yle EM pulse Fig. of terahertz radiation emitted bakward. The nature of this radiation phenomena is regarded to the reation of nonlinear medium interfae boundary to the reation of polarization harge d = P NL

7 CONVERSION OF SHORT OPTICAL PULSES TO. Nonlinear slab In the ase of a slab, the ontinuity at the input interfae, z=0, is added by the ontinuity of E and H fields at the interfae, z=l output boundary of nonlinear slab, where L is the thikness of the nonlinear slab. These BCs determine the radiation propagating forward and bakward from the slab. To fulfill the BCs, we math the general solution of Eq. 11 in a slab: E x z, = C 1 expi z + C exp i z + L oh FW L BW oh S expi z, 9 v g where C 1 and C are integration onstants, with the solutions of the wave equation in free spae on both sides of the slab. 46 These equations are simply the homogeneous form Eq. 1. Solving the boundary value problem BVP, we math the full solutions for the field inside the slab Eq. 9 with the radiation terahertz field propagating in the negative diretion of the z axis Eq. 1 and the forward-irradiated THz field E x z, into the right-hand half of free spae at zl, E x z, = C FW expi 0 z L. 30 The values of the integration onstants, C 1 C 1, C C, C BW C BW, and C FW C FW, are determined from the BCs for the tangential omponents: E x z = z 0 0, = E x z = z 0 +0,, 31 H y z = z 0 0, = H y z = z 0 +0,, 3 where z 0 defines the interfae boundaries, z 0 =0 or z 0 =L, respetively. The onservation of the tangential omponents of the eletri and magneti fields Eqs. 3 3 leads to the following set of BVP linear equations presented in matrix form: L 0 expi C L exp i 0 1 C BW 1 C FW 0C1 1 expi L 1 exp i L v g = expi v g L 1 expi L. v g v g 33 The solution of this matrix equation gives the values of integration onstants: C 1 = C 1, C = C, C BW = A, C FW = B. 34 In Eq. 34, is the determinant of the square 44 matrix in the left-hand side lhs of Eq. 33, and C1, C1, CBW, and CFW are the determinants obtained from by the replaement of orresponding olumns by the rhs vetor, respetively. The determinant is L = exp i 0 0 expi L. 35 The other determinants are CBW = v g 0 0 v g C1 = v g v g exp i L v g v g expi v g L, 36 C = v g v g expi v g L v g expi L, 37 0 v g expi v g L L v g exp i 0 v g + v g 0 expi L, 38 0 v g CFW = v g 0 v g PHYSICAL REVIEW B 76, exp il 1 1 v g + 0 v g 0 v g expil 1 v g v g + 0 v g The omplex amplitudes for the forward and bakward emitted terahertz fields outside the slab are given by the preexponential fators of Eqs. 1 and 30, respetively,

8 ZINOV EV et al. E x L +, = 0 v g E x 0, = 0 v g PHYSICAL REVIEW B 76, L FW oh L BW oh S 0 expil/ v g exp il exp il/ 1 1 v g 0 v g expil 1 v g + 1 v g + 0, L FW oh L BW oh S 0 expil/ + 0 exp il/v g 0 exp il v g + v g 0 exp il, v g exp il where the notations z=l+ and z=0 show the oordinates outside the slab, respetively. Solutions 40 and 41 onverge to zero if the thikness of the nonlinear material L 0. They desribe the field distributions at the output aperture that is used to find the field in a remote zone far from the slab. The inverse Fourier transform on Eqs. 40 and 41 returns the time domain wave forms of terahertz field emitted from the slab in ollinear and antiollinear diretions. D. Terahertz radiation emitted into free spae in the far field The spatial distribution of the light emitted into free spae from the input and output surfaes of a nonlinear slab an be treated within the framework of the standard diffration theory 47 using the Huygens wavelet onstrution. The ative volume inside a nonlinear rystal, where the onversion into terahertz ours, forms a small ylinder with the apertures on both input and output surfaes of the order of S 0, where S 0 is the pump beam ross setion. The salar term E x r, of terahertz field at an observation point r in free spae behind the slab is found after performing the following integration: E x r, = i E x L +, expikr osn,rds 0, 0 S 0 r 4 where E x L+, is the field at the output aperture alulated from Eq. 40. The integral in Eq. 4 is taken over the pump and/or terahertz beam aperture ross setion S 0. The time domain wave form E x r,t is obtained from Eq. 4 using the Fourier transform. The field wave form at the remote point distaned by r D is given by E THz t = E x r,t = 1 osn,r d 0 S 0 r dt E xl +,t r 0. 0dS 43 Similarly, the terahertz field emitted by the slab in the bakward diretion is found from Eqs. 4 and 43, using E x 0, alulated from Eq. 41 instead of E x L+,. III. NUMERICAL CALCULATIONS, EXPERIMENTAL PROCEDURES, AND DISCUSSION In this setion, we ompare the theoretial and experimental time domain and spetral properties of terahertz wave forms, whih unambiguously reveal the physis of nonlinear onversion and establish the spatial loalization of the soure regions emitting terahertz waves. We begin from the analysis of the evolution of wave forms of the terahertz field obtained within a gedanken experiment using a semi-infinite medium. Figure, top, depits the wave form of the terahertz field radiated into free spae in a far-field region in antiollinear diretion, obtained after inverse Fourier transform of Eq. 8. It is seen that the alulated terahertz pulse emitted in the diretion of negative z presents a single pulse of very few yles whose position orresponds to t=t 0 z/ 0, where z is the absolute value of the far-field distane and t 0 =0 is the moment of pump pulse entry into semi-infinite medium. Figure, bottom, also shows the wave form of the EM terahertz field distribution inside the sample alulated from the inverse Fourier transformed expression Eq. 7. There are only four wave forms shown from the ontinuously alulated set. The wave form pattern inside semi-infinite medium onsists of two pulses à and B of opposite polarities and with fundamentally different behaviors on time. It is seen that the first pulse, named as Ã, propagates aordingly with z=t à 0 /n opt, whereas the seond pulse, B, propagates along z as z =t B 0 /n THz. Another striking distintion is that pulse à experienes almost no dispersion and absorption. These properties of pulse à are determined by the respetive quantities alulated for the pump pulse frequeny 0. These two marked major features unequivoally demonstrate the nature of these two pulses. Pulse à is related to the EM field of polarization harge propagating with the group veloity of pump pulse v g. This field, pulse Ã, has the nature of a Coulomb field assoiated with moving polarization harge, rather than a radiation propagating field. Pulse B demonstrates that its nature is related to an EM field at terahertz frequenies that is dispersed and absorbed during its propagation inside the medium. The properties of pulse B prove that it is related to terahertz EMR generated at the entry interfae boundary

9 CONVERSION OF SHORT OPTICAL PULSES TO PHYSICAL REVIEW B 76, Ti:sapphire 00 fs, 8 MHz (a) ODL Sample Balaned photodiodes I I PM ZnTe EOS PM WP /4-plate foused to a spot whose size ould be varied from 100 m to 1.5 mm, and the average power was 300 mw. The LiNbO 3 rystal was mounted suh that the axis was parallel to the diretion of polarization of the pump optial pulse. In the hosen geometry of LiNbO 3, the nonlinear polarization is determined by the largest omponent of the dieletri suseptibility tensor ˆ ˆ 33. Terahertz EMR radiating from the sample was olleted and direted to the detetor by a pair of paraboli mirrors PM in Fig., top. THz detetion was aomplished using a free spae eletroopti sampling ell 5 54 EOSC equipped with ZnTe rystal of the 110 faet orientation and 1 mm thikness Fig., top. To ompare adequately the results of numerial alulations with experimental data, we ompute theoretial wave forms and spetra taking into aount the instrumental fators introdued by remote guiding of terahertz radiation by paraboli optis Se. II D and detetion in the EOSC. The deteted terahertz wave form S THz t is the onvolution of the signal emitted from the sample, E THz t, with the instrumental fator ft, (b) FIG. 3. Top: The sketh of experimental setup. Bottom: Comparison of experimental urve 1 and theoretial urves and 3 terahertz wave forms emitted in a forward diretion from the slab of LiNbO 3 of 1 mm thikness. Pulses labeled TA-TB and TA-TB orrespond to forward emitted terahertz EMR and their first roundtrip eho replias, respetively. Theoretial urves and 3 plot forward and bakward emitted terahertz wave forms, respetively. The mark t 0 shows the time orresponding to the entry of the pump pulse into a nonlinear slab z=0. of pump pulse into semi-infinite medium. Vis-à-vis the above assertion, the ourrene of two fields inside a nonlinear medium has been misinterpreted within the NLWC model in Refs. 1,, and 18 and also widely aepted in textbooks, e.g., see Refs The ruial and unique evidene that unambiguously establishes the nature of this nonlinear phenomenon an only be obtained from studies of the refletion replia properties of à and B pulses. If the nature of pulse à does indeed relate to the Coulomb field of the polarization harge, then after pulse à reahes the output interfae, it must hange and generate a normal radiation terahertz pulse propagating with the veloity, rather than v g and also have other normal attributes. Notably, it must be established for both propagation diretions, outward, into free spae, and inward, bak refletion from the interfae into nonequilibrium medium. Below, we shall demonstrate this onversion in both ways, in experiments and theoretial alulations. Our experimental investigations utilized LiNbO 3 and ZnTe rystals mounted in free spae. Terahertz EMR was deteted with a onventional setup shown in Fig. 3, top. LiNbO 3 and ZnTe samples had end fae dimensions of 6 10 mm with a variety of lengths up to 4 mm. Optial exitation of the rystals was performed with 00 femtoseond pulses of a Ti:sapphire laser =850 nm S THz = 0 = 0 E THz t ftdt E THz fexpid. 44 The instrumental fator ft results from the WVM between terahertz and probe pulses in the eletro-opti material. It is desribed by the following relationship: 54 f = k 0 exp 0Lt 1 C opt expik, 0L ik, 0 L In Eq. 45, k 0 is the real part of the omplex wave vetor, 0 is the attenuation oeffiient, =k+i, t 1 =/n+1 is the Fresnel oeffiient for terahertz EMR at frequeny rossing the boundary air eletro-opti detetor rystal, and C opt E probe 0 E probe 0 d. = 46 Equation 46 is the autoorrelation funtion of the probe pulse eletri field E probe 0 ; 0 is the detuning relative to the entral probe pulse frequeny 0. To ompute Eqs. 40 and 41 numerially, we use the Sellmeier equations for the evaluation of v g : in LiNbO 3, 55 1/ n = , and for ZnTe,

10 ZINOV EV et al. n = / PHYSICAL REVIEW B 76, The value of is alulated using the expression for the dieletri onstant at terahertz frequenies,, in the approximation of the single TO-osillator model related to the lowest frequeny of the infrared ative phonon modes of A 1 symmetry: 57,58 = + S TO TO i, 49 where =4.03, S=3.9, TO =7.68 THz, and =0.93 THz are the best fit parameters to the experimental data for in LiNbO For ZnTe, we use Eq. 49 with the following parameters: =7.44, S=.58, TO =5.3 THz, and =0.05 THz. 54 For numerial alulations of terahertz wave forms, we use Eqs orreted for the instrumental fator of EOSC Eqs. 44 and 45 using LiNbO 3 parameters inluded in Eqs. 47 and 49 and ZnTe parameters inluded in Eqs. 48 and 49. The wave form of terahertz radiation alulated for, and measured from, a 1 mm thik sample of LiNbO 3 plaed in free spae is shown in Fig. 3, bottom. First, we desribe the struture of the terahertz wave form and its variation with experimental onditions, suh as fousing and sample positioning. The terahertz wave form onsists of several separated pulses. Two leading pulses, TA and TB urves 1 and, Fig. 3, bottom, represent terahertz EMR emitted during a single flight of the pump pulse and are onsistent with the observation made in Ref. 19. Prefix T here indiates the transmission geometry, i.e., terahertz EMR emitted from the slab in the forward diretion. Eah pulse starts with intensive bipolar osillations defining the overall shape of the spetral envelope. In the experimental wave form, it is followed by weak prolonged osillations related to various speifi spetrosopi details, suh as the residual water absorption of the free spae atmosphere, et. The pulses TA and TB have largely opposite signs. Curve of Fig. 3, bottom, presents the alulated wave form for the LiNbO 3 sample of the same thikness. Numerial alulations are made for the sample plaed in vauo and, therefore, the theoretial wave forms do not inlude the osillations in the tail of the pulse introdued by atmospheri residual gases and water vapor absorption. From a omparison of wave forms 1 and of Fig. 3, bottom, a good fit of the theoretial wave form to the experimental graphs for all of the major features is evident. To measure the thikness dependene of terahertz DFG wave forms on thikness L, the samples were plaed on a fixed base plate that provided a ommon origin in time sale, with an auray better than ±10 fs. The series of terahertz wave forms was reorded suessively using the sample set of different thiknesses ranging from 0. up to 4 mm. For eah wave form, we find that the positions of the forward propagating pulses after a single round-trip, TA and TB, are referable to TA and TB, whih are the first round-trip eho replias. For eah thikness of measured samples, we alulate the wave form and then determine the positions of peaks TA, TB, TA, and TB. Figure 4 plots the fan diagram of pulse positions as a FIG. 4. Bottom: Fan diagram showing positions of peaks 1 TA, TB, 1 TA, and TB as funtions of slab thikness L. Top: Dependenies of the terahertz field peak amplitude on slab thikness for pulse TA 1 and pulse TB. Experimental points are shown by squares and triangles. Solid urves 1,, 1, and show theoretial dependenies. funtion of slab thikness. The position of pulse TA follows the time of flight of the pump pulse, t A =Ln opt 0 / 0. The temporal position of the pulse TB orresponds to t B =Ln THz / 0, that is, the time of flight of the terahertz EMR pulse aross the slab. The relative time of flight between pulses TA and TB is t A-B =Ln THz n opt / 0. As stated in Se. I, these data themselves do not deliver onlusive evidene on the nature of the TA and TB pulses. However, sine both models had in their theoretial footing the same form of the Maxwell s equations, their onlusions should have been onsistent regardless of whether they was built on a harmoni waves approximation,6,8 or used the time domain treatment based on the formation and propagation of wave pakets Se. II. Therefore, to identify the orrelations between these two interpretations, it is important to investigate the properties of DFG wave paket onversion in as detailed a manner as possible. In order to searh for suh evidene, we turn to the analysis of other properties of DFG EMR that, so far, have not been addressed. The amplitude dependene on thikness for forward emitted EMR, shown in Fig. 4, demonstrates linear growth of the main terahertz pulse amplitude at thiknesses LL FW oh.ata ertain thikness, orresponding to the onset of WVM, the main pulse splits into two pulses labeled as TA and TB. The amplitude of pulse TA saturates with L and remains largely onstant, irrespetive of further inrease of slab thikness. Unlike pulse TA, the amplitude of pulse TB strongly falls with L. The results of spetral analysis of pulse TA and TB obtained by Fourier transform of the time-domain wave form sampled within 10 ps of eah pulse separately are shown in

11 CONVERSION OF SHORT OPTICAL PULSES TO (a) (b) FIG. 5. Top: Experimental terahertz spetra of TA urve TA and TB urve TB pulses obtained for LiNbO 3 sample of L =1 mm. The urve marked absorptivity shows the dependene of the absorptivity in LiNbO 3 defined as the ratio of transmitted to inident eletri field amplitudes. Bottom: Theoretial terahertz spetra of TA and TB pulses alulated for the LiNbO 3 sample with L=1 mm. Fig. 5. The spetral envelope of pulse TB is shifted by 00 GHz to lower frequenies relative to spetrum of TA. It is known that the absorption oeffiient in the terahertz band normally rises with frequeny. 58 Theoretial spetra alulated with the omplex refrative index taken from Ref. 59 show the shift of the TB spetrum lose to 00 GHz and have approximately the same amplitude ratio TB/ TA as that measured experimentally Fig. 5. Therefore, the shift of pulse TB spetrum is explained as the result of reabsorption during propagation of the pulse TB aross the sample. Figures 4 and 5 emphasize that the spetrum of pulse TA remains independent of thikness within the measurement auray. The alulated spetra of pulses TA and TB Fig. 5 orrespond with a good auray to the results of experiment. The strong redution of amplitude of pulse TB with thikness may be regarded as a signature of strong attenuation and broadening of the pulse TB inside the slab. The amplitude dependenies of pulses TA and TB ould still be treated within the framework of a qualitative NLWC model,6 if the pulse TB was attributed to HW and the TA pulse was related to IHW. In this ase, the observed behavior would result from the differenes in the omplex propagation onstants, absorption oeffiients, and refrative indies for HW and IHW, respetively. PHYSICAL REVIEW B 76, To identify a onvining proof, we address the properties of the eho replias of pulses TA and TB, with the pulses TA and TB following the leading pair TA-TB. The key signifiane of the data extrated from the properties of eho replias is motivated by the idea that, for spatially separated soures TR model, the eho replias should show harateristi modifiations of properties in omparison with the properties of pulses from diret flight, whereas for the pulses generated within the same volume adjaent to the input surfae NLWC model, the eho replias should keep, at least, the same struture and amplitude ratio. Moreover, if, as suggested in Ref., the so-alled IHW does indeed represent a real propagating wave reated by nonlinear onversion along with HW, it must keep its extraordinary properties after the refletion from boundaries. These are that the frequeny is equal to the onversion frequeny and the omplex wave veloity is equal to the group veloity of the pump pulse. From Fig. 4, showing the fan diagram of arrival times for the main pulses and their refletion replias, we an see that time positions for the TB replia behave onsistently with the properties of the TB pulse. Being refleted sequentially from the output and input interfaes, the pulse TB aquires a round-trip delay Ln THz / 0 in addition to the arrival time t B =3Ln THz / 0. The arrival time of flight for the TA pulse, t A, inludes the time of flight of TA, Ln opt / 0, aomplished with a round-trip time of the terahertz pulse, Ln THz / 0, leading to the total arrival time of TA pulse t A =Ln opt / 0 +Ln THz / 0. This is onfirmed by both the experiments and theory Fig. 3, bottom. To our knowledge, this striking result has not yet been reported elsewhere. It asts a deisive vote in favor of the TR model, providing the ruial evidene for onversion of the polarization harge field into a pulse of terahertz EMR at the exit from the slab. This unequivoally establishes the physial meaning of the inhomogeneous solution of the wave equation the third term in rhs of Eq. 9 as the field of moving harge, the polarization harge, rather than an EM wave omponent related to the so-alled IHW all the way through see, e.g., Refs. and 6. This diretly shows that onversion of the harge field into a bunh of propagating EM waves ours at the entry and exit boundaries due to the radiation mehanism at harge start-stop motion. 1,5 Another argument in favor of the TR mehanism is inferred from Fig. 3, bottom, where a striking inversion of the amplitude ratio from TA/TB1 tota/tb1 Fig. 3, bottom is also evident. It shows that after a seemingly simple round-trip, the terahertz EMR pulses demonstrate a striking hange of amplitude ratio: Compare the ratio of TA to TB and the ratio TA to TB in Fig. 3, bottom. Had the soures of pulses TA and TB been loalized within the same spatial region of the nonlinear slab from the front surfae, as interpreted in the NLWC model,,6 then both TA and TB pulses would boune bak and forth keeping at least the same amplitude ratio. From the data of Fig. 3, bottom, we an estimate the ratio of formation lengths for forward to bakward emitted terahertz EMRs. Classifiation of the bakward emitted pulses is shown by using R to mark their affiliation to the bakward refletion diretion. Sine the field amplitude is proportional to the value of the formation-oherene length, we obtain the ratio =TB/RBL FW oh /L BW oh. It is lear that the

12 ZINOV EV et al. PHYSICAL REVIEW B 76, FIG. 6. Terahertz spetra alulated for LiNbO 3 samples for 1 LL FW oh and LL FW oh. The inset shows the time domain wave form related to spetrum. The terahertz spetrum 3 shows the experimental spetrum of TA and TB pulses obtained for LL FW oh. ratio TA/TB is a measure of absorption in the slab. Assuming that the absolute value of peak amplitude remains the same on both interfaes, we reover the amplitude of the TB pulse FW BW /L oh at z=0+tb z=0+ TA. Then, from Fig. 4, wefindl oh.5; this is in a good agreement with the alulated value of.46. Maker et al. 9 have demonstrated strong osillations of SHG intensity as the angle of sample position with respet to the diretion of pump propagation is hanged. The theory for these osillations has been developed in Ref. 8, and it is now widely aepted see, e.g., Refs. 49 and 50 that it is the interferene of HW and IHW at the onditions of stationary exitation, aording to Ref. 8, that leads to the osillatory behavior of the intensity of the onverted wave as funtion of sample thikness. 9 These osillation patterns have formed the basis of a widely used method to measure the absolute values of seond-order nonlinear optial oeffiients. The osillatory fator sin, where =L/L FW oh. 8,9 We note that this form of osillatory fator an be transformed to the form = 1 THzt A-B using Eq. 18. To verify this assertion, we ompare the spetra of the omplete struture of the twin pulses, A and B. Figure 6 plots experimental and theoretial spetra of the wave form inluding the TA and TB twin pulses. Both the theory and experiments demonstrate very similar, well pronouned periodi osillations that follow exatly the proportionality to sin 1 THzt A-B. As stated above, this result demonstrates the identity of these osillations with Maker fringes. 9 This learly shows that the nature of Maker osillations is due to interferene of TR omponents emanating from the entry and exit interfae boundaries. If the thikness of the sample beomes less than L FW oh, the twin wave forms merge into a single pulse wave form inset of Fig. 6. Under these onditions, the spetrum osillations disappear, and the urve beomes smooth urve in Fig. 6. The final part of this setion briefly disusses the results obtained from the ZnTe sample where the onditions L oh L are met for a substantial fration of terahertz frequenies F THz with sample length L1 mm. Figure 7 shows the FW FIG. 7. Calulated 1 and experimental and 3 THz wave forms for the ZnTe rystal 110, L=1 mm. Wave forms and 3 were reorded at environment purged with dry nitrogen gas and ambient air environment, respetively. experimental and theoretial wave forms measured and alulated for this sample. This ase learly demonstrates a hybrid mehanism of terahertz EMR where the onditions L L FW oh are met for the low frequeny part of the terahertz spetrum, F THz. For the upper frequenies, F THz, L FW oh quikly falls. The mehanism of terahertz generation hanges over this region to the ase of TR, with the onditions LL FW oh. The overall pulse shape appears more omplex, if it is ompared with the ase of L FW oh L the alulated wave form shown in the inset of Fig. 6. Part of the theoretial urve orresponding to the refletion replia the seond pulse of urve in Fig. 7 shows a fine substruture of the replia pulse, whih apparently relates to a hirp of the terahertz pulse during the round-trip propagation inside the ZnTe sample. The physis of terahertz wave generation during nonlinear oherent three-wave interation is fundamentally different from the framework of NLWC model whih is usually treated as the periodi series of suessive onstrutive and destrutive oherent nonlinear wave transformations transformations from the pump wave to onverted wave and vie versa, eah of whih ours on the sale of L oh in the bulk of medium see, e.g., Refs. 1, 49, and 50. It is quite lear that, due to the oherent nature of the nonlinear wave transformation, eah onstrutive oupling event in the bulk should generate a pulse of terahertz radiation with firmly fixed phase and/or time shift with respet to the preeding event. Then, the total terahertz wave form generated by a sample of thikness L in the ollinear diretion would show a series i.e., a omb of pulses with quantized delays of number L/L oh +1. The reason why this tentative speulation fails the test in the time domain is that nonlinear wave oupling omes into play through nonlinear polarization that, of ourse, leads to formation of a polarization harge Eqs. and 3, whih moves with the veloity of the pump light in the medium. It is the radiation of this harge that determines the mehanism of wave onversion at the onversion frequeny, rather than the series of mutual bak and forth nonlinear wave transformations. What really matters is the harater of the polarization harge motion that is, of ourse, uniform in the bulk. Thus, in the bulk, the harge an radiate if and only if its veloity exeeds the phase veloity of the

13 CONVERSION OF SHORT OPTICAL PULSES TO onverted wave. This is the VC radiation mehanism, whih is the fundamentally nonollinear oherent radiation from the entire path of pump pulse onsidered as a single radiator. Otherwise, and, in partiular, in the ollinear diretion, the radiation from the bulk is stritly forbidden Se. II B, Eq. 17. In the ollinear diretion as shown above in Ses. II C and III, the polarization harge an only radiate at the moment of its formation and of its extintion within the sale of L oh at the entry and exit boundaries, respetively. Under the ondition L oh L, these radiation events simply overlap: This an be interpreted as the ollinear phase mathing radiation. However, the properties of the radiation at suh overlap differ in some respets from the radiation at perfet PMCs that require the degeneration of veloities of moving soure, v g, and onversion wave,, at the frequenies of the pump and onversion, v g p =. IV. CONCLUSIONS To summarize, the reported experimental and theoretial results have unambiguously established the mehanism of terahertz DFG with short optial pulses. With a nonlinear slab, the burst of terahertz EMR is fired twie: at the moment of formation of the polarization harge, due to the nonlinear oupling of the fields of the short pump pulse traversing the boundary between the linear and nonlinear media, and at the moment of extintion of the polarization harge at the exit of the pump pulse from the slab. This mehanism of TR with optial pulses is very similar to that disussed within the framework of the so-alled Tamm problem for radiation aused by the start-stop motion of harged partiles in a uniform medium. It has been shown that, if the oherene length beomes omparable with the thikness of a nonlinear medium, TRs from both interfaes overlap and they merge into a single bunh of ollinear PM radiation. Effiient broadband TR terahertz generation an be obtained at the interfaes of free spae and a nonlinear sample with a high value of ˆ, even if the oherene length is small. TR auses Maker osillatory behavior that is due to interferene between the oherent terahertz EMR generated at the input and output boundaries. These interferene fringes are different in nature from the interferene arising from the round-trips multiple refletions of the terahertz pulse itself. The results of theoretial alulations fully agree with the experimental data. Although the major part of this paper is devoted to differene frequeny generation DFG, related to the radiation at terahertz frequenies, it is attributable to a muh broader family of nonlinear phenomena. ACKNOWLEDGMENTS The authors would like to aknowledge finanial support from the Department of Trade and Industry National Measurement System Poliy Unit in the UK, The Royal Soiety, UK, the Ministry of Eduation and Siene of Armenia Contrats No. 840 and No. INTAS , the Russian Foundation for Basi Researh Grant No and the grant for innovations support from the Russian Aademy of Sienes. APPENDIX: NONLINEAR WAVE CONVERSION MODEL: LINEARIZED APPROACH Obtaining analytial solutions of the wave equation, for various reasons, is not always ahievable. A linearized approah to the wave equation 43,44,59 has been known for a long time, often termed as the paraxial approximation. It has been applied to the wave equation with nonlinear soure term as early as 196, 1, and this has been followed by numerous further publiations, e.g., see Refs Below, we briefly outline the sheme of linearization aiming to draw up the relevant omparisons and onlusions to the subjet of this paper. The treatment of the set and 3 usually proeeded within the framework of the slow varying envelope amplitude approximation SVEA: E i r,t = e i A i r,texpik 0i z 0i t +.., A1 where e is the polarization vetor, A i r,t is the field envelope assumed to be slow varying on the sale of 0i and 0i, 0i is the arrier frequeny, and k 0i is the arrier wave vetor. Index i denotes the pump waves as i= pj, pk and the onverted wave as i=. Then, instead of Eqs. 8 and 9, the following equation is obtained from Eq. using Eq. A1 and the Fourier transform of Eq. : A r, 0 + A r, 0 z + k k 0 A r, 0 +ik 0 A r, 0 z = i 0 A r, 0 0 P NL r,e ik 0 z. A The last term in lhs of Eq. A desribes smearing of the pulse envelope Ar,t due to the group veloity dispersion GVD. Using the assumption of slow varying amplitude on the sale of, the following approximation is used: A r, 0 z PHYSICAL REVIEW B 76, A r, 0 k 0. A3 z The following linearized wave equation is normally used in all pratial treatments of NLWC: A r, 0 A r, 0 +ik 0i z + k k 0 A r, 0 = i 0 A i r, 0 0 P NL r,e ik 0 z. A4 Equation A4 is the paraxial nonlinear Helmholtz equation with inlusion of losses due to free arrier absorption the first term in the rhs of Eq. A4. After putting the GVD term in power series form, Eq. A4 is transformed into the formal equivalent to the Shrödinger equation and is used for the treatment of many phenomena in optis and quantum optis, inluding short pulse propagation in Kerr medium, 4,5,1 14,59 61 The first term of Eq. A4 desribes the variation of the amplitude in the transverse plane to the propagation diretion. Negleting GVD term and applying Eq. A4 to the one-dimensional problem of ollinear radia

14 ZINOV EV et al. tion lead to the linearized wave equation. If we assume that the region of interest, where nonlinear wave onversion ours, is restrited within the size of the onfoal parameter around the origin of oordinate z=±z 0 this assumption is equivalent to the ase when the thikness of the nonlinear slab Lz 0, then Eq. A4 is redued to A z, 0 z = A z, 0 0 i 1 0 P NL z,e ik 0 z, A5 0 0 where = 0 / 0 is the absorption oeffiient of the generated field. With obvious assumption 0, Eq. A5 is transformed to the equation used in the original papers 1,,18,48 followed by a huge number of papers treating seond harmoni generation SHG, sum frequeny generation SFG, and differene frequeny generation DFG or optial retifiation OR, if a single pump is used. This approah has been repliated in many textbooks e.g., see Refs Using the expression for nonlinear polarization for OR/DFG is onsidered using the linearized form of the wave equation: da,z dz = A,z i 0 d * ijka pj pj,za pk z A6 pk,zexp ikzexp j + k where d ijk = ijk, k=k pj k pk k 0, and j and k are the absorption oeffiients for the omponents of pump fields, respetively. Equation A6 is a simple linear differential equation of the first order that is easily integrated. The solution of Eq. A6 requires only single BC at the entry interfae of nonlinear medium. Most of the papers and textbooks use the trivial zero BC at z=0: A,z z=0 =0. Suh a hoie of BC is usually justified by the argument that stritly at the interfae z=0, beause of zero length of nonlinear medium, no generation of onverted field is able to build up. With suh hoie of BC, the solution of Eq. A6 for the field amplitude is given by i 0 d ijka pj A pk A z, = k + j + k ik The quantity j + k exp z 1 exp ik + j + k z. A7 0d * ijk A pj A pk A z,a z, = k + j + k exp z1 oskzexp j + k z + exp j + k z A8 determines the power of onverted wave. It is easy to see that the power, proportional to the quantity expressed by Eq. A8, osillates with z as oskz, ifk0. The power osillations deay with z with the omposite absorption oeffiient = j + k and the power of generated wave deays with the absorption oeffiient i. Using the notations for the power densities, P S = 1 0 * A A P pj S = 1 0 * A pj A P pi pj j S = 1 0 * A pk A pk, k A9 we obtain the widely used formula see, e.g., Ref. 6 for the power onversion effiieny for nonlinear generation and/or onversion at three-wave interation proess at phase mathing ondition, k=0, on the length of nonlinear medium z: P = 8 S 0 PHYSICAL REVIEW B 76, / P pj P pk n i n j n k d ijk z 1 exp z exp z z. A10 The apparent simpliity of the linearized wave equations Eqs. A6 A10 is bought at a ertain prie. First of all, it must be stated that Eqs. A7 A10 seem to establish the generation of EMR due to nonlinear oupling of EM fields everywhere in the medium. This is strikingly inonsistent with the exat result Eq. 17 of Se. II B. Another major drawbak of the linearized approah is demonstrated in Fig. 8. Figure 8 plots the wave forms alulated for a semiinfinite medium using inverse Fourier transform of Eq. A7. Although it produes a twin pulse struture of EM field pattern inside the medium, in marked omparison with Fig., this reveals physially unreasonable properties of these twins. Two more striking misfits are in predited position for pulse B and pulse à whose propagation through medium is predited with infinitely high veloity. The major diffiulties stem from the redution of the order of the differential equation that leads to the loss of orret solution. The approah based on the sheme of SVEA Eq. A1 and linearization Eq. A3 of Maxwell equations leads to Eqs. A6 A10. The linearized equation of redued order, the first-order differential equation Eq. A6, does not require the BC at the output interfae. Without going into the details of legitimay of the widely used assumption

15 CONVERSION OF SHORT OPTICAL PULSES TO FIG. 8. THz wave forms alulated for a semi-infinite medium within the sheme of a linearized SVEA approximation the medium parameters are taken for LiNbO 3. Similarly to Fig. 1, the wave forms are shifted against eah other in the z diretion orresponding to the speified propagation lengths in the medium: urve 1 orresponds z=1 mm, urve orresponds z= mm, urve 3 orresponds z=4 mm, and urve 4 orresponds z=8 mm. The lines à and B follow z=t à 0 /n opt and z=t B 0 /n THz, respetively. The line C is the guideline for the seond pulse positions. PHYSICAL REVIEW B 76, Az, z=0 =0 e.g., used in Refs. 11, 15, 16, and 63, we state that the absene of the BC at the output interfae ultimately leads to the wrong physis: The solution simply does not feel the output boundary. It is also true, even if the orret hoie of the BC at the entry interfae, the ontinuity of tangential omponents for E and H vetors, applies. As a onsequene of this, there is a misleading behavior of the power of the generated wave osillations with propagation distane in nonlinear medium z as oskz at arbitrary WVM k0. Furthermore, being applied to alulate the time domain wave form pattern, Eq. A7 generates two pulses with physially unreasonable properties. The field wave form emitted out of a slab in ollinear diretion is Eq. A7 multiplied by the Fresnel transmission oeffiient T =4n/n+1. In Ref. 63, Eq. A7 has been applied to the analysis of a struture of terahertz wave forms in ZnTe measured with exitation wavelengths m. As it is shown above, partiularly in Fig. 8, widely used formulas A6 A10, when applied to the desription of nonlinear optis phenomena, give learly wrong answers on the details of time-frequeny domains kinetis. It reates even more onfusions to the understanding of optial phenomena when one deals with short pulses, e.g., single or few yles pulses. Therefore, for the orret desription of the physis related to optial nonlinearities with single or few yles EM pulses, the full set of Maxwell, equations must be deployed without the use of the linearization proedure and the SVEA approximation. *nik.zinovev@durham.a.uk 1 J. A. Armstrong, N. Bloembergen, J. Duuing, and P. S. Pershan, Phys. Rev. 17, N. Bloembergen and P. S. Pershan, Phys. Rev. 18, M. Bass, P. A. Franken, J. F. Ward, and G. Weinreih, Phys. Rev. Lett. 9, ; M. Bass, P. A. Franken, and J. F. Ward, Phys. Rev. 138, A T. Brabe and F. Krausz, Rev. Mod. Phys. 7, T. Brabe and F. Krausz, Phys. Rev. Lett. 78, D. A. Kleinman, Phys. Rev. 18, R. C. Miller, D. A. Kleinman, and A. S. Savage, Phys. Rev. Lett. 11, J. Jerphagnon and S. K. Kurtz, J. Appl. Phys. 41, P. D. Maker, R. W. Terhune, M. Nisenoff, and M. Savage, Phys. Rev. Lett. 8, W. N. Herman and L. M. Hayden, J. Opt. So. Am. B 1, K. Wynne and J. J. Carey, Opt. Commun. 56, J. E. Bjorkholm, Phys. Rev. 14, T. K. Gustafson, J.-P. E. Taran, P. L. Kelly, and R. Y. Chiao, Opt. Commun., T. K. Gustafson, J.-P. E. Taran, H. A. Haus, J. R. Lifsitz, and P. L. Kelly, Phys. Rev. 177, Y. J. Ding, Opt. Lett. 9, A. Shneider, M. Neis, M. Stillhart, B. Ruiz, R. U. A. Khan, and P. Günter, J. Opt. So. Am. B 3, F. Zernike, Jr. and P. R. Berman, Phys. Rev. Lett. 15, J. Morris and Y. R. Shen, Opt. Commun. 3, ; K.H. Yang, P. L. Rihards, and Y. R. Shen, Appl. Phys. Lett. 19, L. Xu, X.-C. Zhang, and D. H. Auston, Appl. Phys. Lett. 61, G. A. Askaryan, Sov. Phys. JETP 15, I. E. Tamm, Sobranie Nauhnyh Trudov Nauka, Mosow, 1975, Vol. 1, p. 77; J. Phys. USSR 1, G. M. Afanasiev, V. G. Kartavenko, and J. Ruzika, J. Phys. A 33, I. M. Frank and V. L. Ginzburg, J. Phys. Mosow 9, V. L. Ginzburg, Usp. Fiz. Nauk 17, V. L. Ginzburg and V. N. Tsytovih, Transition Radiation and Transition Sattering, The Adam Hilger Series on Plasma Physis Hilger, Bristol, B. M. Bolotovskii, Pro. Tr. P.N. Lebedev Phys. Inst. 140, P. A. Cherenkov, Dokl. Akad. Nauk SSSR, ; S. Vavilov, ibid., I. M. Frank and I. E. Tamm, Dokl. Akad. Nauk SSSR 14, U. A. Abdullin, G. A. Lyakhov, O. V. Rudenko, and A. S. Chirkin, Zh. Eksp. Teor. Fiz. 66, Sov. Phys. JETP 39, D. H. Auston, Appl. Phys. Lett. 43, ; D. A. Kleinman and D. H. Auston, IEEE J. Quantum Eletron. QE-0, D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, Phys. Rev. Lett. 53, D. H. Auston and M. C. Nuss, IEEE J. Quantum Eletron. 4,

16 ZINOV EV et al. 33 J. K. Wahlstrand and R. Merlin, Phys. Rev. B 68, ; S. K. Wahlstrand, T. E. Stevens, J. Kuhl, and R. Merlin, Physia B , 5500; T. E. Stevens, J. K. Wahlstrand, J. Kuhl, and R. Merlin, Siene 91, M. Theuer, G. Torosyan, C. Rau, R. Beigang, K. Maki, C. Otani, and K. Kawase, Appl. Phys. Lett. 88, M. Wang, O. Wada, and R. Koga, Appl. Opt. 35, F. G. Bass and V. M. Yakovenko, Usp. Fiz. Nauk 86, Sov. Phys. Usp. 8, U. Happek, A. J. Sievers, and E. B. Blum, Phys. Rev. Lett. 67, W. P. Leemans, C. G. R. Geddes, J. Faure, C. Toth, J. van Tilborg, C. B. Shroeder, E. Esarey, G. Fubiani, D. Auerbah, B. Marelis, M. A. Carnahan, R. A. Kaindl, J. Byrd, and M. C. Martin, Phys. Rev. Lett. 91, ; C. B. Shroeder, E. Esarey, J. van Tilborg, and W. P. Leemans, Phys. Rev. E 69, N. N. Zinov ev, A. S. Nikoghosyan, and J. M. Chamberlain, Phys. Rev. Lett. 98, L. D. Landau and E. M. Lifshitz, The Classial Theory of Fields Nauka, Mosow, J. D. Jakson, Classial Eletrodynamis Wiley, New York, Equation 10 and the following theory do not take into aount hirping and broadening of the pump and terahertz pulses during their propagation in the nonlinear medium. 43 A. Sommerfeld, Partial Differential Equations in Physis Aademi, New York, P. M. Morse and H. Feshbahm, Methods of Theoretial Physis MGraw-Hill, New York, A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Sientists Chapman and Hall, Landon/ CRC, Boa Raton, FL, 00. PHYSICAL REVIEW B 76, We point out that very often, e.g., Refs. 11, 15, and 16 another set of BC, E z=0 =0 and de/dz z=0 =0, is used to solve the radiation problem in a slab. It is worthwhile to note that suh a hoie of BC is physially unreasonable as it auses serious artifats suh as zero bakward emission from the slab, et. 47 M. Born and E. Wolf, Priniples of Optis Pergamon, New York, N. Bloembergen, Nonlinear Optis: A Leture Note Benjamin, New York, R. W. Boyd, Nonlinear Optis Aademi, New York, Y. R. Shen, The Priniples of Nonlinear Optis Wiley, New York, A. Yariv, Quantum Eletronis, 3rd ed. Wiley, New York, Q. Wu and X.-C. Zhang, IEEE J. Sel. Top. Quantum Eletron., ; Appl. Phys. Lett. 70, H. J. Bakker, G. C. Cho, H. Kurz, Q. Wu, and X.-C. Zhang, J. Opt. So. Am. B 15, G. Gallot and D. Grishkowsky, J. Opt. So. Am. B 16, V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optial Crystals Springer, Berlin, D. T. F. Marple, J. Appl. Phys. 35, A. S. Barker, Jr. and R. Loudon, Phys. Rev. 158, S. Kojima, H. Kitahara, S. Nishizawa, and M. Wada Takeda, Phys. Status Solidi C 1, B. E. A. Saleh and M. C. Teih, Fundamentals of Photonis Wiley, New York, P. L. Kelly, IEEE J. Sel. Top. Quantum Eletron. 6, S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optis of Femtoseond Laser Pulses AIP, Melville, NY, W. Shi, Y. J. Ding, N. Fernelius, and K. Vodopyanov, Opt. Lett. 7, N. van der Valk, P. C. M. Planken, A. Buijserd, and H. Bakker, J. Opt. So. Am. B,