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PRL 96, 163904 (2006) PHYSICAL REVIEW LETTERS week ending 28 APRIL 2006 Localized Polaritons and Second-Harmonic Generation in a Resonant Medium with Quadratic Nonlinearity D. V. Skryabin,1 A. V. Yulin,1 and A. I. Maimistov2 1 Centre for Photonics and Photonic Materials, Department of Physics, University of Bath, Bath BA2 7AY, United Kingdom 2 Moscow Engineering Physics Institute, Kashirskoe shosse 31, Moscow 115409, Russia (Received 3 February 2006; published 27 April 2006) We derive model equations for the propagation of ultrashort pulses in materials with resonant linear and quadratic nonlinear responses and find approximate soliton solutions describing all-bright and dark-bright polaritons. We report the specific phase matching condition for efficient 2nd harmonic generation, which involves detuning from the resonance. We also demonstrate that the 2nd harmonic emission by the polaritonic pulses can lead to reduction of their group velocity, having zero as a theoretical limit. Our analytical results are supported by numerical simulations. DOI: 10.1103/PhysRevLett.96.163904 PACS numbers: 42.65.Tg, 42.65.Ky Studies of materials engineered by inclusion of metal nanostructures into dielectric hosts is one of the most active areas of the current fundamental research in optics underpinning ongoing technological revolution in optoelectronics and all-optical processing [1]. Applications of such materials include subwavelength photonic circuits [2], negative refractive index optics [3], nonlinear optics [4,5], and more. The linear and nonlinear responses of these materials are dominated by localized surface plasmons, exhibiting pronounced resonances in the extinction spectra. Basic nonlinear processes such as secondharmonic generation and the Raman effect have all been observed in plasmonic materials and continue to attract a significant research effort [1,4,5]. In this Letter we present a theoretical study of short pulse propagation in resonant materials with quadratic nonlinearity. The conceptual novelty of our approach, which goes beyond applications in plasmonics, is that we are considering the cases when either the fundamental wave or the 2nd harmonic is in resonance with the intrinsic material oscillations. This enhances the second order susceptibility beyond the validity of the standard expansion of the material polarization into the power series of the interacting harmonics [6] and requires the development of a new approach. Methods of slowing the pulses of light and related fundamental problems have been another area of sustained recent interest spanning various approaches ranging from classical to quantum optics [7]. Analytical and numerical results presented below demonstrate intrinsically the nonlinear method of slowing light down and bringing it to a stop in plasmonic materials, where the soliton effects and 2nd harmonic emission play paramount roles. The essence of the phenomenological model used below is that the nanoinclusions are considered as the artificial classically behaving ‘‘atoms.’’ Providing that the size of the atoms is much less then the wavelength, the response of the material to the electromagnetic waves can be described using the average resonant polarization, Pr , which obeys 0031-9007=06=96(16)=163904(4)$23.00 the Lorentz equation, see, e.g., [6], 2 @2t Pr 20 @t Pr !20 fPr 0 1 1 !0 E: (1) Here !0 is the resonant frequency of the surface plasmon and 0 is the decay rate. fPr Pr 2 P2r , where the 2 term characterizes quadratic nonlinearity of the oscillators. Linear susceptibility at the frequency ! created by 2 2 the resonant polarization is given by 1 1 1 !0 !0 1 2i0 !0 !2 1 , where 1 1 lim!0 !1 . The electric field E obeys the wave equation @2z E 1 2 @t E 0 @2t Pr Ph : c2 (2) Here Ph is the nonresonant linear polarization of the host material and 0 0 1=c2 . The idealization tacitly used here is that all nanoinclusions are identical, so that their resonant frequencies are the same. Below we will assume that the spectral width of the pulse is greater than the width of plasmonic resonance and that the propagation distances are such that the effects of losses can be disregarded, i.e., 0 0. Inclusion of losses into our model under these conditions does not qualitatively alter the soliton propagation regimes and 2nd harmonic emission discussed below. Solutions of the system (1) and (2) represent an intrinsically coupled material and field excitations, which are often referred to as polaritons. We focus our study on finding the localized solitary wave solutions of the system, which are likely to serve as natural attractors for pump pulses and play a profound role in 2nd harmonic generation. Interest in solitary solutions to models with quadratic nonlinearities has been sustained during the last decade, see [8] for a review. In particular, the experimental observations of two-frequency temporal solitons were reported in [9]. Reference [10] reported single-frequency bright solitons in the systems similar to Eqs. (1) and (2), but with purely cubic nonlinearity in the polarization. Reference [11] studied the case when the equation for the polarization is linear, while the nonresonant cubic nonlinearity is included into the wave equation. Soliton solutions 163904-1 © 2006 The American Physical Society week ending 28 APRIL 2006 PHYSICAL REVIEW LETTERS PRL 96, 163904 (2006) of Eqs. (1) and (2) having zero carrier frequency have also been studied [12]. Because of the zero frequency assumption these solutions have no or little physical meaning in the optical spectral range, where the experimental efforts are directed. In order to develop an analytical approach relevant in the optical range of frequencies, it is natural to use the slowly varying approximation. We use substitutions Pr P1 eik1 zi!t P2 eik2 zi2!t c:c: and E A1 eik1 zi!t A2 eik2 zi2!t c:c:, where kj kj!, j 1; 2 and retain only the first derivatives of Pj , Aj and slowly oscillating nonlinear terms. An effect of Ph 0 is that the dispersion k! of the host material is different from the free space one, k !=c. The resulting set of equations is 2 2 2ikj @z v1 j @t Aj 0 j ! Pj ; (3) 2i!@t P1 !20 !2 P1 2!20 2 P2 P1 eiz 0 1 !20 A1 ; (4) 4i!@t P2 !20 4!2 P2 !20 2 P21 eiz 0 1 !20 A2 ; (5) where j 1; 2, 2k1 k2 is the wave number mismatch due to the host dispersion, vj vj! and v! @! k1 is the frequency dependence of the group velocity. The simplest and most studied situation is when !0 is detuned far from the both frequencies ! and 2!. In this case, the time derivatives in Eqs. (4) and (5) can be neglected and material equations solved perturbatively, which leads to the second order susceptibility defined as 2 2 2 2 2 1 2 ! 2 where 1 1 ! =!0 1 4! =!0 , 2 1 2 2 1 lim!0 !1 22 0 0 1 c . If either ! or 2! are close to !0 the nonlinearity gets resonantly enhanced and the above expression for 2 losses its validity. We switched to a dimensionless formulation of the problem through the substitutions A1 z; t A0 A~1 ~ z; ~t, z; ~teiz , P1 z; t P0 P~1 ~ z; ~t, P2 z; t A2 z; t A0 A~2 ~ P0 P~2 ~ z; ~teiz , where all the tilde variables are dimension1 2 less. Here z~ z1 t t 1 =20 v1 n1 , 0 1 A0 P0 , ~ 2 z=v1 =0 , and 0 !=!0 , so that the time is scaled to an optical cycle. Lastly, n1;2 are the host refractive indices at ! and 2!, respectively. First we consider a resonance with the fundamental wave, i.e., ! is close to !0 . This situation closely corresponds to the recent experiments on the second-harmonic generation from arrays of gold nanoparticles embedded inside glass and irradiated with femtosecond pulses [5]. In this case, the time derivative in Eq. (5) can be neglected, which gives P~2 16 P~21 13 A~2 , where 2 1 A0 = 1 21 characterizes the strength of the nonlinearity. Detuning from the second resonance !20 4!2 was approximated with 3!20 . The resulting dimensionless system is i@z A1 P1 ; (6) i@z w1 @t A2 n1 2 P 11 A2 ; 3n2 1 1 1 i@t P1 1 P1 2 jP1 j2 P1 A2 P1 A1 : 3 3 2 (7) (8) In Eqs. (6)–(8) we dropped the tildes for brevity. 1 1 !=!0 is the dimensionless detuning from the first resonance and 1=w 2v1 v2 n21 =v2 1 1 characterizes the group velocity mismatch of the two fields. 11 1 1 2n 3n2 is the wave number mismatch which includes contributions from the dispersion of the host material (1st term), 1 2n21 0 v1 =1 1 , and from the response of the resonant inclusions at 2! only (2nd term). Equations (6)–(8) are well suited for studies of secondharmonic generation by the ultrashort pulses with a spectrum in the proximity of the resonance. There are two distinct nonlinear mechanisms in Eqs. (6)–(8). One is the effective cubic nonlinearity in the P1 equation, which is induced by 2 1 0 and is primarily responsible for the nonlinear phase rotation of P1 and A1 . The second nonlinear mechanism in Eqs. (6)–(8) is related to the parametric terms A2 P1 and P21 . The parametric effect couples the second-harmonic photons with the material excitations at the fundamental frequency. The jP1 j2 P1 term is formally analogous to, but physically different from, the effective Kerr nonlinearity of the fundamental field in the offresonant and phase mismatched 2nd harmonic generation [8]. In our case, the cubic nonlinearity appears in the material excitations instead of the optical field. Therefore the 2nd harmonic of the optical field A2 can be assumed to be exactly zero and the system remains nonlinear. On the other hand, in the off-resonant case, the effective Kerr nonlinearity of A1 is provided by a small (1=jj) but nonzero value of A2 [8]. The 2nd harmonic is small if the wave number mismatch j11 j is large. Putting A2 0, we find that in the leading order, A1 and P1 obey i@z A1 ’ P1 ; 1 1 i@t P1 1 P1 ’ 2 jP1 j2 P1 A1 : (9) 3 2 Equation (9) is exactly the system considered in [10] to describe optical resonance in nanostructured materials with intrinsic Kerr nonlinearity. Adopting the approach of [10] we find the approximate localized polariton (LP) solutions: A1 rei i1 t O2 11 ; t z=v; (10) r P1 p e3i i1 t O2 11 : 2v p p Here r2 12v 2v2 sech 2v, p arctanth v=2 . The parameter v > 0 characterizes the field amplitude and, simultaneously, the velocity shift of the nonlinear polariton relative to the linear group velocity v1 . Larger values of v result in larger soliton 163904-2 PHYSICAL REVIEW LETTERS PRL 96, 163904 (2006) amplitudes. By analyzing dispersive properties of the linearized version of Eq. (9) one can show existence of the expected polaritonic band gap [6,11]. Therefore the above solutions can be understood as gap solitons. The simple expression for the second harmonic is found by neglecting the derivatives in the left-hand side of Eq. (7) (valid for j11 j 1) giving A2 n1 2 P O3 11 : 3n2 11 1 (11) Direct numerical modeling of Eqs. (6)–(8) initialized with the approximate soliton solutions (10) and (11), confirms the validity of the latter to a very good accuracy. The same solution is also excited with the pump only at the fundamental frequency, see Figs. 1(a)–1(c). Note that the width of the pump pulses in our modeling is taken to be of the order of 10 dimensionless units. This means that the pulse covers around ten optical periods, which puts us into a regime where the slowly varying approximation is applicable. In the off-resonant system the temporal solitary waves found for the large wave number mismatch can be varied to that of the perfectly matched situation [8]. Simulation of Eqs. (6)–(8) reveals a different scenario. Figures 1(d)–1(f) show the evolution of the fundamental frequency pump pulse for 11 10, 1 0. One can see that the localized component of the second harmonic is still strong, but it acquires a noticeable tail of radiation emerging in the direction of the nonsolitonic remnant of the 2nd harmonic signal. As we will demonstrate below, the amplitude of this tail is exponentially small in 11 and therefore cannot be described using a perturbation expansion in the inverse powers of the latter. Our method of finding the tail relies on the fact that Eq. (7) for the 2nd harmonic can be integrated exactly. 15 5 0 -1000 -500 0 500 1000 time, t 10 5 0 -1000 -500 15 5 0 -1000 -500 0 500 1000 time, t (c) Localized polariton =100 z=10 5 0 500 1000 time, t 15 distance, z 10 10 0 -1000 -500 0 500 1000 time, t 15 distance, z distance, z 15 distance, z distance, z distance, z 15 10 10 5 0 -1000 -500 =10 z=10 5 0 -1000 -500 0 500 1000 time, t (f) Localized polariton 10 0 500 1000 time, t (i) (j) =3 z=10 =3 z=5 FIG. 1 (color online). Results of the numerical simulation of Eqs. (6)–(8). 1st and 2nd rows show evolution of jA1 j and jA2 j, respectively, in the t; z plane. The scale used is logarithmic. 3rd row shows jA1 tj (full red line) and jA2 tj (dashed blue line) for the propagation distances indicated by the white horizontal lines in the t; z plots. Initial conditions used in modeling are A1 p 0:13 secht=7:1 and A2 0. Parameters are (a) –(c) 11 100, (d)–(f) 11 10, (g)–(j) 11 3, and w 0:1, 1 0, 1, n1 =n2 1. week ending 28 APRIL 2006 Imposing condition A2 0 at z 0 we find A2 in terms of variables z and : n1 u i11 u Z 0 A2 e P21 0 ; zei11 u d0 : (12) i3n2 z=u Here 1=u 1=w 1=v characterizes the mismatch of the two velocity shifts. Before proceeding with further analysis, let us note, that whatever the properties of the function under the integral are, the energy in the second-harmonic field has little opportunity to grow if v is close to w. Simply because the integration interval in this case is small. If this interval is much greater than that in which the sech function is substantially different from zero, then the properties of the phase factor in the integrated function are crucial. Using P1 from Eq. (10) we find that the overall phase factor is expi6 21 11 =u0 ei . The p straight line approximation ’ v=2 for the soliton phase is satisfactory over the interval making the dominant contribution to the integral, which gives ’ 0 , where p (13) 3 2v 21 11 u: The simplest limiting case allowing explicit calculation of A2 , is obtained by assuming that all of the contributions into the phase cancel out, i.e., 0. Assuming, for 0 and z ! 1, we are left pwith u > R example,0 pthat 0 sech f 2 arctanexp 2vg= 2v d p1 2v. This function is a kink which represents the tail of the second harmonic left behind by the passing pump pulse. If is made large, then the amplitude of the tail drops down and it becomes oscillatory. Figs. 1(f) and 1(j) show the tail structure of the second harmonic calculated numerically. The amplitude of the tail, A1 2 , far from the center of the pump pulse can be estimated by assuming that ! 1, z= ! 1 and u > 0. Then the integration limits in Eq. (12) can be replaced by 1, giving for an arbitrary , 2n1 juj p sech j jA1 : (14) 2 n2 2 2v Equation (14) explicitly demonstrates that jA1 2 j decays exponentially for increasing j1 j and j11 j. An important result of this calculation is that it shows that even if j11 j is large, one still can ensure exponential increase in the efficiency of second-harmonic generation by tuning 1 to minimize . The inverse is also true, i.e., if 1 is fixed then by tuning 11 , e.g., with quasiphase matching, we can achieve 0. In fact, the 0 condition is the phase matching condition, which should be used in the resonant material to achieve efficient energy transfer into the 2nd harmonic from the short pump pulses. This condition also explicitly includes nonlinear phase shifts, because v is directly related to the pulse amplitude. It can be straightforwardly demonstrated, either form Eq. (13) or directly from Eqs. (6)–(8) that without the nonlinear corrections the phase matching condition is 11 21 =w 0. If 163904-3 PRL 96, 163904 (2006) PHYSICAL REVIEW LETTERS 1=w 0, i.e., the group velocities of the 1st and 2nd harmonics are matched (which is hard, but possible to achieve [9]), then the phase matching condition is 11 0. The latter is essentially the same condition as the one derived in the off-resonant limit [6]. It is clear from the numerical modeling that for small values of , the tail of the second harmonic starts to be a dominant feature, see Figs. 1(g)–1(i). In this regime important observations can be made about the dynamics of the optical and material excitations at the fundamental frequency. In the modeling we observed that despite the relatively small ’s used, i.e., when the 2nd harmonic is fully delocalized, the sufficiently strong initial pulse still evolves into the quasisolitonic structure in A1 and P1 fields. The structure in Fig. 1(g) is clearly solitonic because the linear nonsolitonic pulse would have a propagation velocity equal to v1 and propagate along a vertical trajectory in our (z; t) plots, see Fig. 1, while the excited pulse moves under the angle, see Fig. 1(g), as prescribed by Eq. (10) for finite values of 1=v. This quasisoliton losses its energy and momentum into the tail of the second harmonic until it completely disperses by emitting vertically propagating linear waves, see Figs. 1(g)–1(j). Note, that the drop in the pulse amplitude is accompanied by a change in its velocity [v ! 0 leads to the almost horizontal trajectory in Fig. 1(g)], as predicted by Eq. (10). The physical group 2 1 velocity of the LP is given by v1 1 1 1 =v2n1 . Therefore, for v ! 0, the true group velocity also tends to zero. Therefore, emission of the 2nd harmonic by the LPs can be considered as a method for the efficient reduction of light’s group velocity. P21 decays slower with v ! 0 than A1 , see Eq. (10). Therefore, the relative energy balance within the quasisolitonic part of the field is changing in favor of the 2nd harmonic with z increasing. Finally, we briefly describe the case, when the resonance happens at the second harmonic, i.e., 2! is close to !0 . This case closely matches the conditions of the experiment [4], where the 2nd harmonic generation in the femtosecond regime has been observed from a quasiphase matched glass sample with embedded ellipsoidal silver nanoparticles. Analytical advancement here is possible in the weak excitation limit, 1. Neglecting the time derivative in the equation for P1 and using the perturbation series in we find P~1 13 A~1 29 P~2 A~1 O2 . Here the detuning from the first resonance !20 !2 was approximated by 3!20 =4. The resulting system of dimensionless equations with dropped tildes is 1 A 1 P2 A1 ; 12 1 n i@z w@t A2 1 A2 1 P2 ; 2n2 1 i@t P2 2 P2 1 A21 A2 : 4 i@z A1 (15) Here 2 12 !!0 and 1 =18. The only nonlinearity week ending 28 APRIL 2006 in Eq. (15) is the parametric one, corresponding to the conversion of the ! photons into the material excitations at 2!, which are subsequently converted to 2! photons. To find soliton solutions we again consider the limit j1 j 1. Then, A2 can be disregarded in the leading order and the equations for A1 and P2 can be integrated analytically, which gives p A1 C v sechv1 Ceiz=12i2 t O2 1 ; P2 iC tanhv1 Ceiz=6i22 t O2 1 ; A2 n1 P2 =2n2 1 O3 1 ; (16) t z=v: Here C and v > 0 are the soliton parameters. Thus in the soliton regime, the optical field and the material response at 2! are the dark pulses, while the pump is the bright pulse. Because of the sign difference in the definitions of and , the physical group velocity of the solutions (16) is greater than the velocity v1 . Thus, in this case, the nonlinearity serves as the light accelerator. In summary, we derived model equations for the 2nd harmonic generation in the ultrashort pulse regime for the quadratically nonlinear materials exhibiting resonance response close to either the pump or second-harmonic frequencies. We found approximate quasisolitonic solutions and described their role in generation of the second harmonic. We also derived conditions for efficient 2nd harmonic generation and demonstrated that the energy emission from the quasisolitons into the 2nd harmonic is able to significantly slow down their group velocity. [1] S. A. Maier and H. A. Atwater, J. Appl. Phys. 98, 011101 (2005); A. V. Zayats and I. I. Smolyaninov, J. Opt. A Pure Appl. Opt. 5, S16 (2003); Optical Properties of Nanostructured Random Media, edited by V. M. Shalaev (Springer, New York, 2002). [2] S. A. Maier et al., Nat. Mater. 2, 229 (2003). [3] A. N. Grigorenko et al., Nature (London) 438, 335 (2005). [4] A. Podlipensky et al., Opt. Lett. 28, 716 (2003). [5] B. K. Canfield et al., Opt. Express 14, 950 (2006). [6] D. L. Mills, Nonlinear Optics (Springer, New York, 1991). [7] R. W. Boyd and D. J. Gauthier, Prog. Opt. 43, 497 (2002); J. B. Khurgin, J. Opt. Soc. Am. B 22, 1062 (2005). [8] A. V. Buryak et al., Phys. Rep. 370, 63 (2002). [9] P. Di Trapani et al., Phys. Rev. Lett. 81, 570 (1998). [10] I. Gabitov et al., J. Opt. Soc. Am. B 23, 535 (2006). [11] U. Peschel, T. Peschel, and F. Lederer, J. Opt. Soc. Am. B 14, 2994 (1997); M. P. Sorenson et al., Physica D (Amsterdam) 170, 287 (2002). [12] E. V. Kazantseva, A. I. Maimistov, and B. A. Malomed, Opt. Commun. 188, 195 (2001). 163904-4