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ON SECOND-HARMONIC GENERATION WITH STRONGLY-FOCUSSEDULTRAFAST PULSESCISCO GOODINGAbstract. An extension of an existing theory of focussed spherical-gaussian second-harmonicgeneration from the frequency domain to the temporal domain is presented, to model pulsedsystems. The approximate second-harmonic electric field inside a nonlinear negative-uniaxialcrystal is obtained for pulsed, strongly-focussed spherical-gaussian lasers in the T EM 00 mode. Apossible application in systems of plasma THz generation is considered, and the approximate SHfield is explored for specific laboratory parameters and compared with numerical calculations onMATLAB.IntroductionThough nonlinear optics is saturated with systems that cannot be described exactly by analyticalexpressions, attempts are constantly made to find approximate solutions that are valid in certainlimiting situations. Exact field expressions for the general case of second-harmonic generation(SHG) in a nonlinear crystal are not found in the literature, though many limiting cases have beensolved for using approximate methods. The classic works of Boyd and Kleinman [1] and Kleinman,Ashkin, and Boyd [2] contain thorough treatments of SHG in the limit of monochromaticity and nopump-depletion, though few analytical expressions resulted. Here, instead of extending the Green’sfunction approaches BK and KAB have used, we will be working solely with Fourier methods. Thisshould be of no concern, since the two methods are formally equivalent, as was shown in KAB forthe case of a single temporal-frequency component.In ultrafast optics, the condition of monochromaticity is badly violated, and the pulsed nature oflasers must be taken into account. Many treatments of pulsed systems are done numerically, such asin [3]. Only in sufficiently idealized cases are analytical field expressions obtainable, and an attemptis made here to present and discuss a model of second-harmonic generation within thin nonlinearcrystals in the constant-pump limit using pulsed, strongly focussed spherical-gaussian lasers. Wewill be working in the T EM 00 mode, and standard spherical-gaussian propagation methods willbe used to propagate Fourier components of the fundamental and second-harmonic pulses underconsideration. Spherical-gaussian propagation methods are described in the treatment [4], and willplay a significant role in propagating the harmonic pulse outside of the crystal.For the case of elliptical-gaussian second-harmonic generation, [5] treats the monochromatic casewell, and [6] attempts to extend the theory to include pulses, though this is accomplished by simplymultiplying the monochromatic result with a gaussian temporal envelope. This method will beimproved upon here by taking the more realistic assumption of a gaussian frequency distributionDate: August 2006.Key words and phrases. second-harmonic generation, SHG, gaussian lasers, pulsed lasers, nonlinear optics.This work was completed with the support of Dr. Roberto Morandotti, at INRS, and NSERC in the summer of2006. Special thanks to Francois Blanchard, Professeur Tsuneyuki Ozaki, and Luca Razzari for all of their help.1

2 CISCO GOODINGmodified by the frequency-dependent monochromatic result, which results in a temporal envelopethat has a richer structure than a pure gaussian.However, since we will be focussing on thin crystals, we will ignore spatial walk-off and groupvelocity dispersion (GVD), which are negligible in this limit. A sample system will be discussed,very similar to those described in [9] and [10], which is a possible application of the field expressionsobtained. This system has a four-wave mixing process known as THz generation occurring in plasmacreated by strongly-focussed gaussian laser pulses (a fundamental and a second-harmonic pulse).1. Second-Harmonic Generation in the Frequency DomainWe begin with an established theory of Type I second-harmonic generation applicable to continuouswavegaussian lasers, including only one frequency value. This treatment closely follows the onefound in the celebrated text “Nonlinear Optics” by Robert W. Boyd [7]. The second-harmonic fieldoscillates at a frequency 2ω 1 , with ω 1 being the frequency of the fundamental field, and can bedescribed by the inhomogeneous paraxial wave equation(1) 2ik 2∂A 2 (r, z, ω 1 )∂z+ ∇ 2 t A 2 (r, z, ω 1 ) = −4πω 2c 2 χ (2) A 1 (r, z, ω 1 ) 2 e i∆kz ,where A 1 (r, z, ω 1 ) and A 2 (r, z, ω 1 ) are the slowly-varying amplitudes of the fundamental and secondharmonicfrequency fields, respectively, and ∆k is the wave-vector mismatch between the input(fundamental) and polarization (harmonic) fields. Since we have a fixed geometry, χ (2) is theeffective susceptibility tensor. All expressions use the Gaussian system of units. It is evident thatpropagation is along the z-axis, and in the constant-pump limit we can use an uncoupled solution ofthe homogeneous paraxial wave equation to formulate a trial solution for the second-harmonic fieldwith a similar structure. The uncoupled homogeneous solution, which will serve as our fundamentalfield within the nonlinear material, is given by(Â 1 (ω 1 )(2) A 1 (r, z, ω 1 ) =1 + iζ(z, ω 1 ) exp −r 2 )w0 2[1 + iζ(z, ω ,1)]where ζ(z, ω 1 ) ≡ 2z/b(ω 1 ), and the confocal parameter b(ω 1 ) ≡ k(ω 1 )w0. 2 The radial coordinate rrepresents the distance from the z-axis, k(ω 1 ) is the spatial frequency, and w 0 is the 1/e radius atthe beam waist. Equations (1) and (2) suggest to the harmonic field expressions(3) A 2 (r, z, ω 1 ) = Â2(z, (ω 1 )1 + iζ(z, ω 1 ) exp −2r 2 )w0 2[1 + iζ(z, ω 1)]and(4) Â 2 (r, z, ω 1 ) = 4πiω ∫ z1n(ω 1 )c χ(2) Â 1 (ω 1 ) 2 e i∆kz′ dz ′z 01 + iζ(z ′ , ω 1 ) ,The z ′ integral begins at the location of the closest face of the crystal, z 0 , and runs until a z valuewithin the crystal.This integral cannot be solved analytically, but it is interesting to consider the case of perfectphase-matching, in which ∆k vanishes and the second-harmonic generated within the nonlinearmaterial travels with the same phase-velocity as the fundamental. This condition is often achievedin optical systems that use negative uniaxial crystals. In this case the integral in equation (4) is asimple logarithm and we have a completely analytical expression for the second-harmonic electricfield - if, that is, we have only one frequency in the spectrum of the fundamental pulse. For ultrafast

ON SECOND-HARMONIC GENERATION WITH STRONGLY-FOCUSSED ULTRAFAST PULSES 3optical systems of shorter and shorter temporal extent, there are broader and broader distributionsof frequencies, and we must take into account field contributions from each of these frequencies.2. The Complete Second-Harmonic Field2.1. Gaussian Frequency Distributions and Dispersion. Assuming a gaussian spread of frequenciesin the spectrum of the fundamental pulse, which laser systems often display, we can obtainthe complete spatiotemporal harmonic field. From the definition of the slowly-varying amplitude,(5) E 2,tot (r, z, t) =∫ ∞ω=0A 2 (r, z, ω)e ik′ (2ω)z−2ωt dω + c.c.,where ω is the continuous frequency variable of the fundamental pulse, which has a spectrumcentered at ω 1 . This amounts to a Fourier transform, with the extra condition A ∗ 2(r, z, −ω) =A 2 (r, z, ω), to ensure that the electric field will be real. The gaussian frequency distribution entersequation (4) in the fundamental field amplitude:(6) Â 1 (ω) = Be −σ·(ω−ω1)2 .The quantity B is related to the energy of the fundamental pulse, and σ is related to the frequencybandwidth.To take into account dispersion (though only to first order in ω), we can write the spatialfrequencies of the fundamental and harmonic fields as(7) k(ω) ≃ k 0 + k 1 · (ω − ω 1 )and(8) k ′ (2ω) ≃ k ′ 0 + k ′ 1 · (2ω − 2ω 1 ),respectively. The quantities k 1 and k ′ 1 are the inverse group velocities of the pulses. Note also thatthe fundamental dispersion power series is expanded around ω 1 , whereas the harmonic is expandedaround 2ω 1 .The logarithmic solution to equation (4) describes the growing amplitude of the harmonic pulseas it is generated by the fundamental pulse within the crystal. However, perfect phase-matchingcannot occur at every frequency. If there is a group velocity mismatch between the fields, then thelogarithmic solution of the integral in equation (4) is not exact, though it is still a fine approximationif the peak frequencies are phase-matched and the group velocity mismatch is not too extreme.The approximations thus far work best in the limit of very thin crystals, which are often usedto avoid spatial and temporal walk-off effects, as well as pulse broadening. This case suites ourpurposes since we have explicitly neglected group velocity dispersion, which is responsible for pulsebroadening, by truncating the dispersion power series’ before the quadratic terms.2.2. Integrand Approximations. Putting the pieces together and noting that the index of refractionn(ω) = k(ω)c/ω, we obtain an integral expression for the (complex) harmonic field:∫ ∞ω 2 e −2σ(ω−ω ( )1) 21 + iζ(z, ω)Ẽ 2,tot = ξln0 1 + iζ(z, ω) 1 + iζ(z 0 , ω)(9)(−2r 2)· expw0 2[1 + iζ(z, ω)] + i[k′ (2ω)z − 2ωt] dω.Here, the field arguments were suppressed and a constant ξ ≡ 2πB2 w 2 0cχ (2) was defined for brevity.2Our coordinate system has the focus of the beam at the origin, and to simplify matters we will

4 CISCO GOODINGassume that all frequency components focus at approximately the same position along the z-axis(z = 0). It should be mentioned that since we are working in the constant-pump limit, in whichlosses are negligible, we are necessarily well below resonant frequencies of the nonlinear materialand thus the effective susceptibility tensor is independent of frequency and can be safely pulledoutside the integral.Again, however, this integral cannot be solved analytically, but there are limiting cases wherewe can apply approximations to obtain more manageable expressions. If we restrict our attentionto pulses that are strongly focussed within the material, then the confocal parameter b(ω) satisfies(10) b(ω) ≪ |z|, ∀z ∈ [z 0 , z 0 + L],where L is the length of the crystal. It is clear that the strong-focussing condition cannot be appliedat the focal point of the pulse, where z = 0, regardless of arbitrarily small confocal parameters.Since we are working with very thin crystals, we need only to have an offset between the positionof the crystal and the focal point of the pulse (which is ideal for the sample system to be discussed)to safely apply this condition.With this in mind, we can use the condition in equation (10) to rewrite the logarithm in equation(9) from the frequency integral as(11) ln( 1 + iζ(z, ω)1 + iζ(z 0 , ω))( ) ( )1 + 2iz/b(ω) z= ln≈ ln .1 + 2iz 0 /b(ω) z 0Since this is independent of frequency, we are free to pull the logarithm out of the frequency integraland absorb it into the constant ξ, which will now be denoted by ξ ′ (z).We can now write the integral in equation (9) in a more transparent form by introducing twodummy variables, α and β, and differentiating them under the integral sign such that(12) Ẽ 2,tot = ξ ′′ ∂ 2 ∫ ∞((r, z)∂α∂β | α,β=1 e −2σ·(αω2 −2ωω 1)+2iω·(k ′ 1 z−t) −2βr 2 )expw0 2[1 + iζ(z, ω)] dω,where again ξ has been redefined as(13) ξ ′′ (r, z) ≡ ξ ′ (z)0( ) w20e −2σω2 1 +iz·(k′ 0 −2k′ 1 ω1) ,4σrto take care of constants and terms brought down by differentiation.This integral would be solvable, if it weren’t for the ζ term. Using the definition of ζ, we canwrite the offending term as(14)11 + iζ(z, ω) ≡ 11 + 2iz/b(ω) = b(ω)/2iz1 + b(ω)/2iz .The condition in (10) also allows us to expand the denominator of equation (14) in a powerseries, and truncate the result to first order, leaving us with1(15)1 + iζ(z, ω) ≈ b(ω) (2iz · 1 − b(ω) ).2izMaking use of the approximation in equation (15) and the fact that b(ω) ≡ k(ω)w 2 0, equation(12) becomes(16) Ẽ 2,tot ≈ ξ ′′ (r, z)∂α∂β | α,β=1I ω ,∂ 2

ON SECOND-HARMONIC GENERATION WITH STRONGLY-FOCUSSED ULTRAFAST PULSES 5with the definitions∫ ∞[ iβr(17) I ω ≡ e −2σαω2 +2ωφ(z,t) 2 [k 0 + k 1 · (ω − ω 1 )]expz0and (to describe axial wave propagation and take care of a constant)(18) φ(z, t) ≡ i · (k ′ 1z − t) + 2σω 1 .(· 1 + i[k 0 + k 1 · (ω − ω 1 )]w02 )]dω2zThe integral we have arrived at in equation (17) is a simple gaussian, but in order to solve it withoutthe aid of an error function we can simply extend the lower-bound on the integration range to −∞,which is valid even for large frequency bandwidths provided ∆ω/ω 1 ≪ 1.2.3. An Analytical Solution. When all the dust settles, the (complex) second-harmonic field weare left with is((19) Ẽ 2,tot ≈ ˆξ(r, r 2ˆk )· (2iz − w 2z)P (r, z, t) exp0ˆk)2z 2 + (2z2 φ + k 1 r 2 [iz − w0ˆk]) 2 22k1 2r2 w0 2z2 + 8z 4 ,σwhere(20) ˆk ≡ k0 − k 1 ω 1 ,z(21) ˆξ(r, 3 (2π) 3/2 B 2 w 4 ( )z) ≡0z4c 2 (k1 2r2 w0 2 + 4z2 σ) 9/2 χ(2) ln e −2σω2 1 +iz·(k′ 0 −2k′ 1 ω1) ,z 0and(22)P (r, z, t) ≡ 5(k 2 1r 2 w 2 0 + 4σz 2 ){k 4 1r 4 w 2 0 + k 2 1r 2 (k 2 1w 4 0 + 8z 2 σ)+ 4z 2 [4ik 1 σ(ik 0 w 2 0 + z)(2σω 1 − φ) + 4k 0 σ 2 (k 0 w 2 0 − 2iz) + k 2 1w 2 0(σ + (2σω 1 − φ) 2 )]}+ 1 z 2 {[k4 1r 2 w 2 0(r 2 + w 2 0) + 16ik 1 z 2 σ(z + ik 0 w 2 0)(2σω 1 − φ) + 16k 0 z 2 σ 2 (k 0 w 2 0 − 2iz)+ 4k 2 1z 2 (w 2 0(2σω 1 − φ) 2 + σ(2r 2 + w 2 0))] · [r 2 k 1 (iz + k 1 ω 1 w 2 0 − k 0 w 2 0) + 2z 2 φ] 2 }− 2(k 2 1r 2 w 2 0 + 4z 2 σ) 2 [8k 0 σ(k 0 w 2 0 − 2iz) + 4ik 1 (ik 0 w 2 0 + z)(4σω 1 − φ)+ k 2 1{2r 2 + w 2 0(1 − 4ω 1 (φ − 2σω 1 ))}].To obtain the real electric field, we merely add the complex conjugate of this, as dictated by equation(5).There appears to be significant mixing of space and time throughout the harmonic field expression,which has quite a rich structure that dictates the amplitude, phase, curvature, dimensions andpropagation of the growing harmonic pulse. Some aspects of this structure will be explored in oursample system. It is clear that all time dependence is contained in the propagation term, φ(z, t),which primarily describes movement of the pulse at the group velocity (along the z-axis).A similar, though simpler, procedure can be applied to the monochromatic spherical-gaussianof equation (2) to obtain the approximate electric field of the fundamental pulse. This pulse hasa different dispersion relation, reflected by the differences of equations (7) and (8), and so willpropagate at a different group velocity. Depending on the extent of group velocity mismatch, thefields obtained using the above methods can only safely be applied for crystals of certain lengths.As the pulses propagate through the crystal, their peaks begin to separate, and it becomes less

6 CISCO GOODINGaccurate to assume that the harmonic continues to be generated with the same structure as itwould if the peaks remained together. Specifically, we may restrict crystal lengths according to∆t(23) L ≤M|k 1 − k 1 ′ |,for some positive integer M, if we want our pulses to separate at most ∆t/M while within thecrystal, where ∆t is the fundamental pulse (temporal) width. Then, satisfying the inequality forhigher values of M corresponds to a better approximation.One may now wish to find the fields outside the crystal, but the above methods only describethe fields on the inside. Outside, both the fundamental field and the harmonic field obey thehomogeneous paraxial wave equation, and we could in principle use fields at the boundary togetherwith the wave equation to propagate the fields numerically. This works well in most situations,but a more efficient method can be used based on the approximations used and standard sphericalgaussianlaser propagation. The fundamental beam is comprised of frequency components that havestructure that can be propagated through optical systems according to the ABCD-rule, describedin [4], which alters the complex radius of curvature ˜q(z, ω) in a manner given by(24) ˜q ′ (z ′ , ω ′ ) =A˜q(z, ω) + BC ˜q(z, ω) + D .This enables us to know ˜q ′ (z ′ , ω ′ ) if we know ˜q(z, ω) and the optical system matrix with elementsA, B, C, and D that connects the two complex radii. For instance, if we just want to connect thecomplex radius just before the boundary to the outside of the crystal with the complex radius justafter, we can (provided we have planar surfaces) use the planar refraction system matrix(25)( ) A B=C D( 1 00 n(ω)for some frequency component ω with index of refraction n(ω).expressions used above by acknowledging that),We can relate this to the fieldb(ω)(1 + iζ(z, ω))(26) ˜q(z, ω) = .2iThe final step in calculating the fundamental field outside the crystal is to integrate all the propagatedfrequency components, which may not lead to an analytical result. Still, it is more efficientto evaluate the integral one is left numerically than to go through the propagation method basedon the boundary field and the paraxial wave equation.For the harmonic field, the structure is not identical to the a spherical-gaussian, and we canthus expect a different propagation procedure. However, if one examines the integrand in theexpression for the harmonic field given by equation (9), one finds that after being subjected to theapproximation in equation (11) it has the exact same relevant structure as a spherical-gaussian beamgiven by equation (2). One needs only to propagate the integrand, and then evaluate the result.Again, this may not lead to an analytical expression, but it is still more efficient computationallyand simpler conceptually than the alternative.3. ApplicationsThe field information obtained in the limiting cases discussed above is well suited to describethe initial nonlinear interaction that supplies input pulses for the four-wave mixing process of THzgeneration. In this system a spherical-gaussian laser is focussed strongly, and a thin nonlinearcrystal is placed some distance before the focus of the beam. Inside the crystal, a second-harmonic

ON SECOND-HARMONIC GENERATION WITH STRONGLY-FOCUSSED ULTRAFAST PULSES 7pulse is generated, as discussed above. Because of the strong focussing, the field strength at thefocus is strong enough to break down the air and create plasma. This plasma serves as the nonlinearmaterial in which the fundamental and second-harmonic pulses interact in an optical rectificationprocess to produce a THz pulse. Here, however, we are only interested in the structure of thefundamental and harmonic pulses subsequent to their interaction at the focus.The nonlinear crystal most frequently used in such systems as [9] and [10] is Barium Borate(BBO). BBO properties have been taken from [8], and the specific parameters we will be using arefrom [10]. We will be using a 860µJ fundamental pulse with a temporal extent of 140fs and apeak frequency of 800nm. Reflections will be neglected, since in practice anti-reflective coatings areoften used. The lens has a focal distance of 5cm, and the BBO crystal, of 100µm length, will beplaced 5mm before the focus. Current models of this system have been based on the “continuouswave” approximation, as found in the works of [9] and [10], though work is in progress to extendthe model discussed here to include the plasma interactions.Figure 1 shows the harmonic pulse in the first 10µm of the crystal, in its initial stage of growth.The vertical coordinate is the electric field, in statvolts, which is normalized to have a final amplitudeequal to that of the fundamental pulse. The radial coordinate r (labeled only as ‘Coordinate’ onthe plots) is reflected to include negative values, though this is only for graphical clarity, since nonegative values of r were actually used in field expressions. Due to the strong curvature of thefundamental pulse, and the fact that the crystal is offset to be before the focus, the tails of thepulse enter the crystal first. This causes an interesting harmonic pulse formation, since there aretwo off-center peaks that initially carry most of the energy. As more of the fundamental pulseenters the crystal, the two peaks of the harmonic propagate towards the center along the structuredefined by the fundamental pulse, as illustrated in figure 1.Figure 1. The initial harmonic pulse, as the two peaks move towards the center.Eventually a stable envelope structure is formed that continues to grow in a logarithmic manner,as described by equation (11). Figure 2 shows the final structure of the harmonic pulse, at abouthalf-way through the crystal. Since there is a difference between the group velocity and peakfrequencyphase velocity of the harmonic pulse, there is a constant cycling of individual wave-frontsthrough the envelope. Similar to the monochromatic case, the harmonic pulse is generated with a∼ √ 2 times smaller 1/e beam radius, as mentioned in [7].The approximate harmonic field given by equations (19)-(22) holds up very well when comparedwith numerical calculations of equation (9), using the typical experimental parameters of [10]. The

8 CISCO GOODINGFigure 2. The final structure of the harmonic pulse.envelope structure is identical, though if one looks up close, as in figure 3, one can see that there isa slight phase shift that was likely introduced by one of the integrand approximations. The scalesin figure 3 are identical. Since amplitude is much less sensitive to the approximations used thanphase, it is no surprise that there is a slight phase difference between the analytical result obtainedand the numerical calculations. If one is only interested in the pulse envelope, or the intensity,then equations (19)-(22) represent a very accurate approximation to the exact form described byequation (9).Figure 3. The approximate (left) and numerical (right) harmonic pulses, up close.

ON SECOND-HARMONIC GENERATION WITH STRONGLY-FOCUSSED ULTRAFAST PULSES 9It should also be mentioned that since we are working in the limit of no pump-depletion, conversionefficiencies predicted by the analytical result obtained here may not be extremely accurate,and a numerical solution to the original coupled partial differential equations should be used if onewishes to obtain more realistic conversion estimates.ConclusionsWe have arrived at analytical expressions for the pulsed second-harmonic electric field generatedwithin a negative-uniaxial nonlinear crystal such as BBO, in the limit of no pump-depletion, perfectphase-matching, and strong-focussing. Though highly idealized, the conditions necessary forsuch limits to hold are found in certain experimental situations such as those studied by [9] and[10]. Other than this possible application, the field represented by equations (19)-(22) is both ofinstructive and theoretical interest, though its structure requires more than a superficial glance tobe revealed.References[1] Boyd, G.D. and Kleinman, D.A. J. of App. Phys. 39, 8 (1968).[2] Kleinman, D.A., Ashkin, A. and Boyd, G.D. Phys. Rev. 145, 1 (1966).[3] Eckardt, R.C., Masuda, H., Fan, Y.X. and Byer, R.L. J. Quantum Electron. 26, 5 (1990).[4] Pedrotti, F.L. and Pedrotti, L.S. (1993). Introduction to Optics, Prentice-Hall, Inc.[5] Freegarde et al. J. Opt. Soc. Am. 14, 8 (1972).[6] Wang, H. and Weiner, A.M. J. Quantum Electron. 39, 12 (2003).[7] Boyd, Robert W. (1992). Nonlinear Optics, Academic Press, Inc.[8] Nikogosyan, D.N. Appl. Phys. A. 52, 359-368 (1991).[9] Bartel, T., Gaal, P., Reimann, K., Woerner, M., and Elsaesser, T. Opt. Let. 30, 20 (2005).[10] Xie, X., Dai, J., and Zhang, X.C. Phys. Rev. Lett. 96, 075005 (2006).INRS-MT , 1650 blvd. Lionel-Boulet, Varennes, Q.C., J3X 1S2, Canada (c/o Dr. Roberto Morandotti)E-mail address: dwg2@sfu.ca