Phase independent nonlinear amplification regime in one-dimensional photonic bandgaps

INSTITUT OF PHYSICS PUBLISHING JOURNAL OF OPTICS A: PUR AND APPLID OPTICS J. Opt. A: Pure Appl. Opt. 3 (1) S19 S PII: S1-5(1)5515-9 Phase independent nonlinear amplification regime in one-dimensional photonic bandgaps V Rugolo, G D Aguanno, C Sibilia, M Centini, M Scalora and M Bertolotti INFM at Dipartimento di nergetica, Università di Roma La Sapienza, Via A Scarpa, 1, 11 Rome, Italy -mail: concita.sibilia@uniroma1.it Received June 1, in final form 1 September 1 Published October 1 Online at stacks.iop.org/jopta/3/s19 Abstract A simple way to study parametric amplification in finite layered structures with an optical nonlinear quadratic response is theoretically analysed. We show that it is possible to write equations that are formally equivalent to those of a bulk medium and that we are able to analyse the effective behaviour of the structure. Then we give the solutions of these effective equations in the undepleted pump regime and use semiconducting materials to theoretically study a structure that gives a good amplification of the signal. Keywords: Nonlinear optics, nonlinear frequency conversion 1. Introduction In recent years photonic bandgap (PBG) structures have enjoyed a great deal of attention in the optics community because of their potentiality for significant technical advances in the field [1]. In particular, the simultaneous availability of high field localization and exact phase matching conditions near the band edge make PBG structures the best candidates for the design of highly efficient frequency-converter devices based on quadratic χ () nonlinearities [ ]. Recently, an analytical way to study second-harmonic (SH) generation in a PBG structure with deep gratings has been developed []. This method allows us to write equations that analyse the effective behaviour of the structure in the case of SH generation. Here we use the same procedure to study parametric amplification. Obviously this interaction is phase independent and it can be very useful in many practical applications. The paper is organized as follows: in section we summarize the approach that we use to study χ () interactions in layered media and in section 3 parametric interaction in the infrared region (1.55 µm) is discussed, driven by a strong pump at.775 µm.. Parametric interactions in finite layered structures: the effective-medium approach We consider a one-dimensional layered structure as described in the caption of figure 1. We start with the coupled wave equations for the three (linearly polarized) electric fields involved in the parametric interaction, 1, and 3, at frequencies ω 1, ω and ω 3 = ω 1 + ω, respectively. In the rotating wave approximation we have d 1 (z) dz d (z) dz d 3 (z) dz + ω 1 ε 1(z) c 1 (z) = λ c ω 1 χ () (z) 3, + ω ε (z) c (z) = λ c ω χ () (z)1 3, + ω 3 ε 3(z) c 3 (z) = λ c ω 3 χ () (z) 1 (1a) (1b) (1c) where ε 1,,3 is the dielectric constant for the three frequencies involved and χ () is the effective nonlinearity. In the equations a dimensionless parameter, λ, has been introduced that identifies the contributions of a given perturbative order. The physical idea that will guide us in looking for solutions of equations (1) is as follows: assuming the nonlinearity in equations (1) is weak, its effect should be to modulate the solution of the linear problem on a length scale much 1-5/1/19+5$3. 1 IOP Publishing Ltd Printed in the UK S19

Phase-independent nonlinear amplification regime in one-dimensional photonic bandgaps y d 3 d d 1 d d 1 d d 3 N periods Figure 1. Schematic representation of the one-dimensional, finite N-period, symmetric structure. The layers have thickness d 1 =.15 µm of AlAs, d =.13 µm of Al(.3)GaAs and d 3 =.1 µm of GaAs. (This figure is in colour only in the electronic version) longer than (that is the single-period dimension of the structure). Now we apply a multiple scale expansion [, 5] by introducing a new set of independent variables z α = λ α z with α =, 1,,... Consequently, the derivative operator is expanded in this way: d dz = + λ + λ + () z z where z represents the fast variable that takes into account the spatial variation on a scale of length ; consequently, the dielectric permittivity functions ε 1,, the field distribution as a solution of the linear problem and the nonlinearity d () (z) will be considered to be functions of the fast variable z, while the field envelope functions that we introduce below will be functions of the slowly varying variables z 1, z etc, where z 1 is the slow variable that takes into account the spatial variation on a scale of length L (that is the total length of the structure). Further scale lengths could be possible (z,z 3,...), but will not be considered in the following. In a first approximation, we assume that the contribution of the nonlinearity will be to modulate the solution of the linear problem on the scale of the slow variables, so we can write the solutions of equations (1) as j (z,z 1,...)= A j (z 1,z,...) j (z ), j = 1,, 3 (3) where the slowly varying amplitude A changes on long lengths (z 1,z...)only and (z ) belongs to the solution of the linear wave equations. Substituting equations () and (3) into (1), and collecting the terms to the zeroth order in λ, wefind j (z ) z + ω j ε j (z ) c j (z ) =, j = 1,, 3. () 1 (z ), (z ) and 3 (z ) are the normalized solutions of the linear problem []. Λ z To first order in λ, we obtain ( A1 (z 1 ) 1 (z ) ) z = c ω 1 χ () (z )A A 3 (z ) 3 (z ) (5a) ( A (z 1 ) (z ) ) z = c ω χ () (z )A 1 A 3 1 (z ) 3 (z ) (5b) ( A3 (z 1 ) 3 (z ) ) z = c ω 3 χ () (z )A 1 A 1 (z ) (z ) (5c) which shows that A j are instead the envelope functions that take into account the presence of a nonlinear term in the equations. If we want to take into account only the effective behaviour of the structure we can average on the fast variable z ; this procedure is performed by multiplying each equation (5) by the complex conjugate of the linear solutions j (z ) (j = 1,, 3 for equations (5a) (5c) respectively), and integrating over the coordinate z. The resulting equations are where γ 1 = 1 L γ = 1 L G 1 A 1 = i ω 1 c γ 1A A 3 G A = i ω c γ A 1 A 3 G 3 A 3 = i ω 3 c γ 3A 1 A γ 3 = 1 L χ () 1 3 dz, χ () 1 3 dz, χ () 1 3 dz (a) (b) (c) (7a) G j = ic d j j dz j = 1,, 3. (7b) ω j L dz Coupled mode equations () cannot give detailed information about the actual nonlinear dynamics of the fields inside the structure. This is evident if we recall that our approach to the problem consists in averaging over the fast variable z. As a result, we cannot expect that this model will yield both a reflected and a transmitted component. However, the solutions of equations () yield remarkably accurate energy conversion efficiencies when compared with the conversion efficiencies calculated by numerically integrating the nonlinear coupled Maxwell equations [] and also they show a good agreement with the experimental results in the case of SHG []. In particular, when the fields are tuned at the transmission resonances, as we assume here, it can be demonstrated [7] that G j = 1. The validity of the applied approximation has been numerically verified by using the same numerical method as presented in [] and []. S197

V Rugolo et al 1. 75 (a).. 5 T. B 1 3. 15.75 1. 1.5 ω /ω Figure. Transmission versus normalized frequency (ω/ω, where ω = (π/λ rif )c and λ rif = 1 µm) for the 39-period structure depicted in figure 1, with the field incident at an angle of 5. with respect to the z axis and TM polarization. The indices of refraction are those of the dispersion relations of the materials. In particular the values for the FF and the SH are n 1 (FF) =.93, n 1 (SH) = 3., n (FF) = 3.31 and n (SH) = 3., where the normalized FF is ω/ω =.1 and the normalized SH is ω/ω = 1.1. The arrow points at the first peak near the first bandgap, where the FF is tuned. Φ 1 1 z[µm ] (b) The solutions of equations () are well known, because they are formally the same as for bulk media [9]. The superposition integrals (7a) take into account the fields coupling and the nonlinearity. quations () are in general valid for a χ () interaction of three nondegenerate fields. We consider now the case of two incident fields: 3 is the pump at frequency ω 3 = ω, 1 is at the frequency (FF) ω 1 = ω and has one of the linear polarizations T or TM and is the generated field at the same frequency ω = ω but with a different polarization with respect to the field 1,TMorT. In the undepleted pump regime A 3 /, we obtain the following solutions: A 1 (z 1 ) = A 1 () cosh(kz 1 ) (a) ω A 3 A (z 1 ) = i ω 1 γ1 γ A 3 A 1 () sinh(kz 1) (b) where k is a real quantity defined as k = ω 1ω γ c γ1 A 3. (9) The role of the phase matching condition and the field localization is contained in the parameter k via the superposition integrals γ 1 and γ. All previous equations (equations (1) and ()) are scalar equations assuming T polarization of fields involved in the interaction (collinear interaction). This scalar approximation is still valid for interactions of non-collinear fields with different polarizations when the input angle is small enough that the angle of refraction inside the medium can be considered small and therefore be neglected as a first step. This situation can develop if there is a large discrepancy between the indices of refraction of the layered structure and input and output substrates. In the following we adopt this approximation. 3. Applications of parametric interactions in PBG structures in the infrared region As an application of the previously developed formalism, we consider the theoretical study of a symmetric structure with B 3 5 3 1 1 z [µm ] 1 z [µm ] Figure 3. Normalized intensity of the field distribution (a) for the FF magnetic field B 1 (TM polarization and θ i = 5. ) as a function of position z inside the device, (b) for the FF electric field (T polarization and θ i = 7 ) and (c) for the FF magnetic field B 3 (TM polarization and θ i = 5.75 ). N = 39 periods made of alternating layers of Al(.3)GaAs and AlAs, with a cladding and a substrate of GaAs, of length d 3 =.1 µm (see figure 1). The layers of AlAs are of a length d 1 =.15 µm and those of Al(.3)GaAs are of a length d =.13 µm. The total length of the structure is only L = 9.735 µm. The AlGaAs is assumed to have a nonlinear coefficient approximately equal to 1 pm V 1. The FF is taken to be λ = 1.55 µm, the FF beam with TM polarization is considered to be incident at an angle of 5. with respect to the normal to the surface of the structure; we call its field amplitude A 1.Infigure, we show the linear transmittance of the structure as a function of the frequency. The FF, indicated with an arrow, is tuned at the first transmission resonance near the first bandgap. The second field A 3, at the ω frequency, is assumed to be incident with polarization TM and at an angle (c) S19

Phase-independent nonlinear amplification regime in one-dimensional photonic bandgaps.5 x 1 1 x 1 3.5 1 I 1 [W/m ] 3.5 1.5 I 1 [W/m ] 1.5.5 1 1.5.5 3 3.5 I 3 [W/m ] x 1 1.5 1 1.5.5 3 3.5 I 3 [W/m ] x 1 1 Figure. Growth of the FF field intensity I 1, versus the pump SH field intensity I 3, for the situation described in figure. Figure. Growth of the FF field intensity I 1, versus the pump SH field intensity I 3, for the situation described in figure 3. x 1.5 1 x 1 1 3.5 I [W/m ] 3.5 I [W/m ] 1.5 1.5.5 1 1.5.5 3 3.5 I 3 [W/m ] x 1 1.5 1 1.5.5 3 3.5 I 3 [W/m ] x 1 1 Figure 5. Variation of the FF field intensity I, versus the pump intensity I 3, in the same case as figure. of 5.75 (this is a strong pump field). The third field, A, will be generated at the FF, polarization T and an angle of 7, according to the minimization of the phase mismatch. All the fields are tuned near the bandgap: the fields at the FF are tuned near the first-order gap, and the field at the SH is tuned near the second-order gap. In figures 3(a) (c), we show the square modulus of the linear field distribution of the three fields inside the PBG structure. The effective phase mismatch [3] calculated for this geometry is n eff = n eff3 n eff1 n eff =.11. With an input FF field of A in 1 = 1 Vm 1, we show in figures and 5 the growth of the FF field intensities I 1 and I respectively, versus I 3 (the pump field intensity). From the figures the amplification factor can be deduced. For example, for an input A 3 = 3 1 Vm 1 (I 3 1 GW cm ) we obtain an amplification of the FF field intensity I 1 that is a = I1 out /I1 in 5. Figure 7. Variation of the FF field intensity I, versus the pump intensity I 3, in the same case as figure. We point out that an amplification of five is obtained using a structure of only 9 µm in length, which is also not perfectly phase matched. This is possible because of the high field localization inside the PBG structure. Let us consider another situation. An FF beam (A 1 ) with ω = 1.11 1 15 s 1 and TM polarization is assumed to be incident at an angle of 33.5 with respect to the normal to the surface of the structure. As before, the FF field is tuned at the first transmission resonance near the first bandgap. The pump field A 3 at SH frequency is tuned near the second-order gap, with polarization T and at an incidence angle of 35.39. A will be generated at the FF (tuned near the first gap), with polarization T at an angle of 35.39. In this case the effective mismatch is n eff =.177. As before we use an input FF field of A in 1 = 1 Vm 1 and in figures and 7 we show the variation of the FF field intensities I 1 and I, versus the pump SH field intensity I 3. S199

V Rugolo et al The parametric interaction gives better results in this case. In particular, when at the input A 3 = 3 1 Vm 1, at the output we have a = I1 out /I1 in.3. These two examples show the very important role that phase mismatch plays in the interaction. A very small change in k can lead to significant changes in the efficiency. This can be understood considering the role of the superposition integrals that define γ 1 and γ.. Conclusions In summary, we have used a multiple scale expansion which allows us to derive equations of motion for parametric interactions in PBG structures that are formally equivalent to those of a bulk medium. These equations can be solved analytically when nonlinear interactions occur in a geometry where the fields are almost collinear. The equations describe the device as an effective structure, and accurately predict its conversion efficiency. We note that, in the case of two incident fields, the amplification is independent of the initial phase relations, because of the intrinsic phase independence of the interaction. The numerical examples that are discussed show the strong influence that phase matching conditions and field localization effects have on the sensitivity of the parametric amplification process. We believe that the model we have presented could be used to design efficient, compact parametric amplification devices based on PBG structures with quadratic nonlinearities, where the field localization inside the structures introduces a great advantage in the nonlinear performance of such devices as well as the great miniaturization. Acknowledgments We thank M J Bloemer, C M Bowden and J W Haus for interesting discussion on the subject. This study has been partially funded by an spirit-open project of the C. The authors also thank the uropean Research Office for partial support. References [1] Yeh P 19 Optical Waves in Layered Media (New York: Wiley) Yablonovitch 197 Phys. Rev. Lett. 5 59 [] Scalora M, Bloemer M J, Manka A S, Dowling J P, Bowden C M, Viswanathan R and Haus J W 1997 Phys. Rev. A 5 31 [3] Centini M, Scalora M, Sibilia C, D Aguanno G, Bertolotti M, Bloemer M, Bowden C and Nefedov I 1999 Phys. Rev. 91 [] D Aguanno G, Centini M, Sibilia C, Bertolotti M, Scalora M, Bloemer M and Bowden C M 1999 Opt. Lett. 13 [5] Nayfeh 1993 Introduction to Perturbation Techniques (New York: Wiley) [] D Aguanno G et al 1 Photonic band edge effects in finite structures and applications to χ () interactions Phys. Rev. 19 Dumeige Y et al 1 nhancement of the second harmonic generation in a semiconductor 1D PBG Appl. Phys. Lett. 7 31 [7] D Aguanno G, Centini M, Scalora M, Sibilia C, Bloemer M J, Bowden C M, Haus J W and Bertolotti M 1 Phys. Rev. 3 3 1 [] Centini M, Scalora M, D Aguanno G, Sibilia C, Bloemer M J, Bowden C M, Haus J W and Bertolotti M 1 Opt. Commun. 19 135 [9] Armstrong J A, Bloembergen N, Ducuin J and Pershan P S 19 Phys. Rev. 17 191 S