Self-induced transparency quadratic solitons

doi:10.1364/OE.20.013988

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Self-induced transparency quadratic solitons

Soodeh Haghgoo and Sergey A. Ponomarenko »View Author Affiliations

Optics Express, Vol. 20, Issue 13, pp. 13988-13995 (2012)
http://dx.doi.org/10.1364/OE.20.013988

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– Abstract
1. Introduction
2. Quantitative analysis and physical model
3. Numerical simulations
4. Conclusion
– References and links

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Abstract

We discover and theoretically explore self-induced transparency quadratic solitons (SIT-QS) supported by the media with quadratic optical nonlinearities, doped with resonant impurities. The fundamental frequency of input pulses is assumed to be close to the impurity resonance. We envision an ensemble of inhomogeneously broadened semiconductor quantum dots (QD) in the strong confinement regime grown on a substrate with a quadratic nonlinearity to be a promising candidate for the laboratory realization of SIT-QS. We also examine the influence of inhomogeneous broadening as well as wave number and group-velocity mismatches on the salient properties of the introduced solitons.

1. Introduction

The solitons have been at the heart of nonlinear physics for more than three decades. The reason being they emerge as ultimate survivors of virtually any wave evolution scenario in conservative nonlinear systems of unprecedented physical diversity, from hydrodynamic [1

G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).

, 2

G. B. Lamb, Elements of Soliton Theory (Wiley, New York, 1980).

] and plasma waves [2

G. B. Lamb, Elements of Soliton Theory (Wiley, New York, 1980).

] to pulses in optical fibers [3

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, Boston, 2003).

] and matter waves in Bose-Einstein condensates [3

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, Boston, 2003).

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Clarendon, Oxford, 2003).

]. In the optical context in particular, the interest in the quadratic solitons (QS), which are coupled self-trapped entities in the media with quadratic nonlinearities, has been recently reinvigorated thanks to the advances in fabrication of phase matching structures, see Ref. [5

A. V. Buryak, P. D. Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002). [CrossRef]

] for a review.

Although the history behind QS can be traced back to 1967 when Ostrovskii discovered self-induced phase changes in nonlinearly coupled fundamental wave (FW) and second harmonic (SH) in a noncentrosymmetric crystal [6

L. A. Ostrovskii, “Propagation of wave packets and space-time self-focusing in a nonlinear medium,” Sov. Phys. JETP 24, 797–800 (1967).

, 7

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ⁽²⁾ cascading phenomenaand their applications to all-optical signal processing, mode-locking, pulse, compression and solitions,” Opt. Quantum Electron. 28, 1691–1740 (1996). [CrossRef]

], the QS were not experimentally observed for a while. As soon as high-damage optical materials with long enough interaction lengths became available, however, spatial QS were observed in the bulk χ⁽²⁾ media [8

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, and G. I. Stegeman, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995). [CrossRef] [PubMed]

] and in planar χ⁽²⁾ waveguides [9

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138–1141 (1996). [CrossRef]

On the other hand, most common quadratic materials have very low group-velocity dispersion (GVD) and fairly large group-velocity mismatch (GVM). Thus, extremely long samples and/or short pulses are required to explore the quadratic soliton regime [5

]. These obstacles hampered temporal QS generation until the application of an ingenious technique for tailoring GVD and GVM culminated in the experimental observation of their key signatures [10

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570–573 (1998). [CrossRef]

]. Yet, in the pioneering experiment [10

] the nonlinear crystal was too short and pulses too long to unequivocally observe a dispersion-free propagation regime over several dispersion lengths. Thus, the existence of alternative routes to temporal QS formation–not relying on balancing χ⁽²⁾ nonlinearities with the GVD–is a highly relevant question, which, to our knowledge, has not yet been adequately addressed. We note, however, that in the three-wave interaction situation in χ⁽²⁾-media, neutrally stable QS were discovered [11

A. Degasperis, M. Conforti, F. Baronio, and S. Wabnitz, “Stable Control of Pulse Speed in Parametric Three-Wave Solitons,” Phys. Rev. Lett. 97, 093901 (2006) [CrossRef] [PubMed]

] which do not rely on the GVD for their existence, but require three waves with different carrier frequencies to mix.

In this paper, we present a new type of QS, self-induced transparency quadratic solitons (SIT-QS), formed in resonant χ⁽²⁾ media. The SIT-QS generation does not at all require any bulk medium GVD, thereby circumventing the major obstacle for temporal χ⁽²⁾ soliton generation. The proposed SIT-QS have a hybrid nature: the mutual self-trapping of FW and SH is achieved by the joint action of both resonant nonlinearity, induced by exciting one-exciton transitions in QDs, and the quadratic nonlinearity of the bulk medium. Specifically, we assume that an input FW pulse is nearly resonant with one-exciton transitions in QDs, grown and randomly spread over a substrate exhibiting a quadratic nonlinear optical response. As the FW pulse enters the medium, it coherently excites the QD ensemble and– provided the pulse has a right area– it can open a “transparency window” in the medium leading to the fundamental soliton formation via the SIT phenomenon [12

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover Publications Inc., New York, 1975).

]. At the same time, the SH is generated via the χ⁽²⁾ nonlinearity, provided a characteristic resonant absorption length is of the same order as a typical length associated with the bulk χ⁽²⁾ nonlinearity. The SH pulse is “dragged” by the FW into the transparency window forming a mutually self-trapped SIT-QS pair. We stress that our results reveal a hitherto unbeknown extension of quantum optical SIT solitons to the realm of quadratic solitons in χ⁽²⁾ media.

Next, we emphasize that the SIT-QS formation demands neither large GVD nor extremely high input pulse intensity–and hence ultrashort duration–which sets them apart from the conventional temporal QS in quadratic media. Further, the discovered SIT-QS are fundamentally different from the previously reported small-amplitude quasisolitons supported by resonant media with quadratic nonlinearities [13

D. V. Skryabin, A. V. Yulin, and A. I. Maimistov, “Localized polaritons and second-harmonic generation in a resonant medium with quadratic nonlinearity,” Phys. Rev. Lett. 96, 163904–163907 (2006). [CrossRef] [PubMed]

]. First, while the latter exist for large wave number mismatch, the SIT-QS require rather accurate phase matching which can be realized in quasi-phase-matched structures. More importantly, though, the QS of Ref. [13

] are supported by media with collective plasmonic resonances which, being far from saturation, can be modeled by the classical anharmonic oscillator model. In contrast in the SIT-QS case, the FW interaction with an ensemble of individual QD resonances involves a swift time evolution of the QD level populations which can only be adequately described quantum mechanically.

2. Quantitative analysis and physical model

We now present our quantitative model. We begin by examining short pulse propagation in a χ⁽²⁾ medium, randomly doped with QDs. The dopants can be modeled as two-level systems. The fundamental pulse carrier frequency ω is assumed to be very close to the resonance frequency ω₀ of a one-exciton transition in a QD; bi-exciton transitions in the QD ensemble are forbidden by restricting ourselves to the case of circularly polarized FW and SH pulses [14

We assume that a photon of a given circular polarization promotes an electron-hole pair (exciton) creation with a well-defined spin orientation such that only linearly polarized light can generate biexcitons, cf., L. Jacak, P. Hawrylak, and A. Wojs, Optical Properties of Semiconductor Quantum Dots (Springer, Berlin, 1997).

]. Thus, only is the FW resonantly coupled with the exciton transition, while the SH is only coupled with the FW via the χ⁽²⁾ nonlinearity. Provided the cubic nonlinear effects are negligible [15

The contributions of cubic and quadratic nonlinearities to the polarization can be roughly estimated as $𝒫^{(3)} / 𝒫^{(2)} ~ χ_{e ff}^{(3)} ℰ_{m} / χ_{e ff}^{(2)} ~ 5 \times 10^{- 2}$ , using $χ_{e ff}^{(3)} ≃ 10^{- 18} m^{2} / V^{2}$ for GaAs. Here the peak field amplitude ℰ_m of a picosecond 2π input pulse was estimated using the known pulse area as ℰ_m ∼ πh̄/d_egτ_p. Thus quadratic nonlinearities indeed dominate in our case.

], the slowly varying field envelopes ℰ₁ and ℰ₂ of the fundamental and second harmonic pulses, respectively, obey the coupled wave equations

(\frac{\partial}{\partial z} + \frac{1}{v_{g 1}} \frac{\partial}{\partial t}) ℰ_{1} = \frac{i N d_{e g} ω^{2}}{2 k_{1} ɛ_{0} c^{2}} < σ > - \frac{i ω^{2} χ_{e f f}^{(2) *}}{2 k_{1} c^{2}} ℰ_{1}^{*} ℰ_{2} e^{i Δ k z},

(1)

(\frac{\partial}{\partial z} + \frac{1}{v_{g 2}} \frac{\partial}{\partial t}) ℰ_{2} = - \frac{i ω^{2} χ_{e f f}^{(2)}}{k_{2} c^{2}} ℰ_{1}^{2} e^{- i Δ k z} .

(2)

Here v_g1(v_g2) is a group velocity of the fundamental (second harmonic) pulse,

χ_{e f f}^{(2)}

is an effective second-order susceptibility of the bulk medium, Δk = 2k₁ − k₂ is a wave number mismatch factor, N is a density of dopants, d_eg is a dipole matrix element of the ground-to-excited state QD transition – the states are appropriately labeled by the indices g and e – and σ is a quantum dipole envelope function describing the temporal evolution of the QD dipole moment. Further, the quantum dipole moment σ and inversion w envelope functions obey the Bloch equations which can be written in the circular polarization basis as [12

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover Publications Inc., New York, 1975).

, 16

In the circular polarization basis, the Bloch equations are exact such that no rotating wave approximation is needed.

]

\partial_{τ} σ = - (γ_{⊥} + i Δ) σ - i Ω_{1} w,

(3)

\partial_{τ} w = - γ_{‖} (w - w_{eq}) - \frac{i}{2} (Ω_{1}^{*} σ - Ω_{1} σ^{*}) .

(4)

Here we introduced the Rabi frequencies, Ω_1,2 = 2d_egℰ_1,2/h̄ of the FW and SH as well as the transverse (dipole) and longitudinal (energy) relaxation rates as γ_⊥ and γ_||, respectively.

Hereafter, it will prove convenient to transform to the reference frame moving with the fundamental pulse by changing variables viz., τ = t − z/v_g1 and ζ = z. In the dimensionless variables, Z = ζ/L_A, T = τ/τ_p and Ω̄_1,2 = Ω_1,2τ_p, the coupled wave equations take the form

\frac{\partial {\bar{Ω}}_{1}}{\partial Z} = i < σ > - \frac{i L_{A}}{4 L_{NL}} {\bar{Ω}}_{1}^{*} {\bar{Ω}}_{2} e^{i (δ Z + ϕ_{0})},

(5)

and

(\frac{\partial}{\partial Z} + \frac{s_{\pm} L_{A}}{L_{W}} \frac{\partial}{\partial T}) {\bar{Ω}}_{2} = - \frac{i k_{1} L_{A}}{2 k_{2} L_{NL}} {\bar{Ω}}_{1}^{2} e^{- i (δ Z + ϕ_{0})} .

(6)

Here τ_p is a characteristic duration of the input FW pulse, δ = ΔkL_A is the dimensionless wave number mismatch, ϕ₀ is a constant relative phase of the effective susceptibility and the dipole matrix element, and s_± = +1(−1) for ν > 0(< 0) specifies the sign of the group-velocity mismatch (GVM),

ν = v_{g 2}^{- 1} - v_{g 1}^{- 1}

. Further, we introduced three key physical length scales to this problem: the linear absorption length L_A, the walk-off length L_W associated with the group velocity mismatch (GVM) of the fundamental and second-harmonic pulses, and the characteristic nonlinear length L_NL in the χ⁽²⁾ medium defined as

L_{A} = \frac{1}{α}; L_{W} = \frac{τ_{p}}{| ν |}; L_{NL} = \frac{k_{1} c^{2} | d_{e g} | τ_{p}}{\bar{h} ω^{2} | χ_{e f f} |},

(7)

where

α = N {| d_{e g} |}^{2} ω^{2} / \sqrt{2 π} \bar{h} b k_{1} ɛ_{0} c^{2}

is a linear absorption coefficient, and b is a spectral width of inhomogeneous broadening.

Next, the average over a distribution of dimensionless frequency detunings from one-exciton resonances in QDs, Δ̄ = (ω − ω₀)τ_p, is defined as

< σ > \equiv \int d \bar{Δ} g (\bar{Δ}) σ (\bar{Δ}) .

(8)

In this work, we assume the inhomogeneous broadening distribution to be a generic Gaussian function in the form

g (\bar{Δ}) = \frac{1}{\sqrt{2 π} b τ_{p}} exp (- \frac{{\bar{Δ}}^{2}}{2 b^{2} τ_{p}^{2}}) .

(9)

In the coherent transient regime we explore, the pulses must be short enough: τ_p ≪ min(T_⊥,T_||), where

T_{⊥} = γ_{⊥}^{- 1}

and

T_{‖} = γ_{‖}^{- 1}

are the transverse and longitudinal relaxation times. Thus, the Bloch Eqs. (3) and (4) reduce to

\partial_{T} σ = - i \bar{Δ} σ - i {\bar{Ω}}_{1} w,

(10)

\partial_{T} w = - \frac{i}{2} ({\bar{Ω}}_{1}^{*} σ - {\bar{Ω}}_{1} σ^{*}) .

(11)

In our numerical simulations, we consider In(Ga)As QDs grown on a GaAs substrate as a prototype for our system. GaAs is known to have a sizable second-order nonlinearity,

χ_{e ff}^{(2)} ≃ 2 \times 10^{- 10} m / V

[17

T. Brunhes, P. Boucaud, and S. Sauvage, “Infrared second-order optical susceptibility in InAs/GaAs self-assembled quantum dots,” Phys. Rev. B 61, 5562–5570 (2000). [CrossRef]

]. Moreover, high efficiency SHG was experimentally demonstrated in the medium thanks to efficient quasi-phase matching in orientation-patterned GaAs samples [18

S. J. B. Yoo, C. Caneau, R. Bhat, M. A. Koza, A. Rajhel, and N. Antoniades, “Wavelength conversion by difference frequency generation in AlGaAs waveguides with periodic domain inversion achieved by wafer bonding,” Appl. Phys. Lett. 68, 2609–2611 (1996). [CrossRef]

, 19

T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett. 27, 628–630 (2002). [CrossRef]

]. We take N ≃ 10²¹ m⁻³, the dipole moment matrix element to be d_e_g ≃ 5 × 10⁻²⁹ Cm, all magnitudes consistent with recent simulations [20

G. Panzarini, U. Hohenester, and E. Molinari, “Self-induced transparency in semiconductor quantum dots,” Phys. Rev. B 65, 165322–165327 (2002). [CrossRef]

, 21

M. Jütte, H. Stolz, and W. von der Osten, “Linear and nonlinear pulse propagation at bound excitons in CdS,” J. Opt. Soc. Am. B 13, 1205–1210 (1996). [CrossRef]

] and experimental work [21

M. Jütte, H. Stolz, and W. von der Osten, “Linear and nonlinear pulse propagation at bound excitons in CdS,” J. Opt. Soc. Am. B 13, 1205–1210 (1996). [CrossRef]

, 22

J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81, 1110–1113 (1998). [CrossRef]

Further, we work with the input 2π-pulses in the picosecond range, say, τ_p ≃ 1 ps which is also consistent with the previous experimental and numerical work on SIT on bound [21

M. Jütte, H. Stolz, and W. von der Osten, “Linear and nonlinear pulse propagation at bound excitons in CdS,” J. Opt. Soc. Am. B 13, 1205–1210 (1996). [CrossRef]

] and free [23

H. Giessen, A. Knorr, S. Haas, S. W. Koch, S. Linden, J. Kuhl, M. Hetterich, M. Grün, and C. Klingshirn, “Self-induced transmission on a free exciton resonance in a semiconductor,” Phys. Rev. Lett. 81, 4260–4263 (1998). [CrossRef]

] exciton resonances. As the characteristic relaxation times for typical QDs are T_⊥ ≃ 2T_|| ∼ 100ps at room temperature– and are several times longer if the sample is cooled down to cryogenic temperatures [24

P. Borri, W. Langbein, S. Schneider, and U. Woggon, “Ultralong dephasing time in InGaAs quantum dots,” Phys. Rev. Lett. 87, 157401–157404 (2001). [CrossRef] [PubMed]

]– the pulses are well within the coherent transient regime. The inhomogeneous broadening width depends on the ensemble preparation; a typical estimate runs as b ≃ 10⁷ MHz [20

G. Panzarini, U. Hohenester, and E. Molinari, “Self-induced transparency in semiconductor quantum dots,” Phys. Rev. B 65, 165322–165327 (2002). [CrossRef]

]. Using the above numerical values, we estimate L_A ≃ 1.5 mm and L_N_L ≃ 0.8 mm, implying that L_A ∼ L_NL which is favorable for the SIT-QS pair formation. Taking into account that ν is typically in the range 0.1–1 ps/mm for most χ⁽²⁾ crystals [25

J. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, USA, 2006).

], we estimate L_W to fall in the range from 1 mm to 1 cm.

3. Numerical simulations

A qualitative analysis–borne out by subsequent numerical simulations–reveals that the SIT-QS formation and stability crucially depend on the magnitudes of the four factors: inhomogeneous broadening width, wave number mismatch, temporal walk-off length, and the area of the incident fundamental pulse. In our numerical simulations using Eqs. (5) and (6) as well as Eqs. (8)–(11), we consider a 2π-Gaussian input FW pulse entering the medium; we assume there is no power in the SH at Z = 0. To illustrate the SIT-QS formation in the presence of inhomogeneous broadening, we exhibit in Fig. 1 the evolution of the FW and SH pulse intensities as functions of the propagation distance Z. For simplicity, we assume perfect phase matching and no temporal walk-off. It is seen in the figure that a stable SIT-QS pair emerges over two characteristic propagation distances for an inhomogeneously broadened sample with b = 10⁷MHz (top row of Fig. 1). We stress that we have numerically propagated the solitons over several dozens of characteristic absorption lengths and found no trace of instability. For comparison, the SIT-QS pair formation is also displayed for an ideal homogeneously broadened sample in the bottom row of Fig. 1. We observe that as the resonant light-QD interaction efficiency decreases due to the inhomogeneous broadening–which is always present for realistic QD samples–the second-harmonic soliton pulse develops a tail at the trailing edge.

Fig. 1 Top row: intensity profiles of the FW and SH soliton pair. The parameters are b = 10⁷MHz, δ = 0, and L_W = ∞. Bottom row: same as the top row except there is no inhomogeneous broadening.

Next, we examine the role of inhomogeneous broadening in the SIT-QS formation with a finite phase mismatch. In Fig. 2 we compare the SIT-QS pair formation in the presence of the wave number mismatch, δ = 3 in homogeneously and inhomogeneously broadened samples. It is seen in the figure that in a perfectly clean sample even a moderate phase mismatch causes breathing of the FW and splitting of the SH pulses, respectively. In contrast, the inhomogeneous broadening inhibits the instability and promotes stable SIT-QS pair generation albeit with the SH component developing a tail at the trailing edge.

Fig. 2 Top row: intensity profiles of the FW and SH soliton pair. The parameters are b = 10⁷MHz, δ = 3, and L_W = ∞. Bottom row: same as above except b = 0.

The tail, however, grows rather slowly (adiabatically) on propagation, implying that the SH component can be interpreted as a quasi-stable soliton even in the moderate phase mismatch situation. As the phase mismatch increases, though, the inhomogeneous broadening can no longer prevent the SH component break-up into two small-amplitude pulses. This situation is clearly illustrated in Fig. 3 where the SH component intensity profile is displayed for four values of the dimensionless wave number mismatch. Thus, we conclude that while the inhomogeneous broadening plays a stabilizing role, the lack of the perfect phase matching can result in the internal dynamics of the SIT-QS leading to their eventual disintegration.

Fig. 3 Intensity profile of the SH wave for (a) δ = 2, (b) δ = 3, (c) δ = 5 and (d) δ = 7. The other parameters are b = 10⁷MHz and L_W = ∞.

Further, we study the temporal walk-off influence on the SIT-QS dynamics. In Fig. 4 we plot the SIT-QS pair formation for the case of an inhomogeneously broadened sample with a modest wave number mismatch, δ = 1 and with the walk-off length of 1 mm. As is seen in the figure, a stable SIT-QS pair is generated, although the peak amplitude of the SH component decreases on the soliton propagation as the overlap between the FW and SH is reduced due to walk-off. Thus the SIT-QS formation is possible even in the presence of a rather substantial temporal walk-off.

Fig. 4 Intensity profiles of the FW and SH soliton components. The parameters are b = 10⁷MHz, δ = 1, and L_W = 1mm.

Next, we performed a number of numerical simulations with the input FWs of various temporal profiles, including Gaussian and secant hyperbolic. The latter corresponds to a fundamental SIT soliton in the FW component. Our simulations indicate that salient features of the SIT-QS formation are largely insensitive to the initial temporal profile of the fundamental pulse. To illustrate this point, we present in Fig. 5 a comparison of the SIT-QS evolution for the 2π-secant hyperbolic (top row) and Gaussian (bottom row) input pulses for the perfectly phase-matched situation with zero GVM, for simplicity. One can see in the figure that although the efficiency of SHG with secant hyperbolic pulses is reduced, the key qualitative features of the SIT-QS pair persist. In physical terms, the different SHG efficiency is explained by the fact that powers of the input 2π Gaussian and secant hyperbolic pulses are quite different, of course.

Fig. 5 Intensity profiles of the FW and SH soliton pair for secant hyperbolic (top row) and Gaussian (bottom row) input 2π fundamental pulses. The other parameters are b = 10⁷MHz, δ = 0 and L_W = ∞.

On the other hand, the SIT-QS evolution scenario strongly depends on the input FW area,

𝒜 \equiv \int_{- \infty}^{\infty} Ω_{1} d τ

. In fact, we discovered that no SIT-QS formation is possible whenever the magnitude of 𝒜 is below π. For any input FW pulse with the area, π ≤ 𝒜 ≤ 2π, though, the area magnitude goes through several oscillatory cycles, asymptotically attaining the value of 2π; the same holds true for pulses with the area greater than 2π. The oscillatory transient dynamics of the area distinguishes the SIT-QS case from the pure SIT case in which the area monotonously approaches 2π [12

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover Publications Inc., New York, 1975).

4. Conclusion

In conclusion, we have discovered and numerically explored a new type of solitons, self-induced transparency quadratic solitons supported by noncentrosymmetric nonlinear media, doped with resonant impurities. We have shown that the SIT-QS evolution and stability does not depend on the shape of the input FW, but are strongly influenced by the interplay of inhomogeneous broadening and phase mismatch in the system. Whereas the former is a stabilizing factor, the latter causes the instability that can result in the QS disintegration. To experimentally realize the envisioned SIT-QS, we recommend that relatively long–a few centimeter long, say–quasi-phase-matched samples be used. We have also demonstrated that the GVM does not substantially affect the SIT-QS formation as long as the samples are sufficiently short. To mitigate–to the point of eliminating–the GVM effects in long samples, one can use tilted input pulses, following the technique of Ref. [10

]. Finally, we note that although we considered only quadratic bulk nonlinearities in the manuscript [15

], the third-order nonlinearities can become important for femtosecond input pulses and/or sufficiently large phase mismatch. The resulting competition between χ⁽²⁾- and χ⁽³⁾-effects, similar to those discussed in Ref. [26

A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Optical solitons supported by competing nonlinearities,” Opt. Lett. 20, 1961–1963 (1995). [CrossRef] [PubMed]

], can lead to new interesting soliton regimes in the system. We plan to address this issue in the new context of SIT-QSs in a forthcoming publication.

References and links

1.	G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).
2.	G. B. Lamb, Elements of Soliton Theory (Wiley, New York, 1980).
3.	Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, Boston, 2003).
4.	L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Clarendon, Oxford, 2003).
5.	A. V. Buryak, P. D. Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002). [CrossRef]
6.	L. A. Ostrovskii, “Propagation of wave packets and space-time self-focusing in a nonlinear medium,” Sov. Phys. JETP 24, 797–800 (1967).
7.	G. I. Stegeman, D. J. Hagan, and L. Torner, “χ⁽²⁾ cascading phenomenaand their applications to all-optical signal processing, mode-locking, pulse, compression and solitions,” Opt. Quantum Electron. 28, 1691–1740 (1996). [CrossRef]
8.	W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, and G. I. Stegeman, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995). [CrossRef] [PubMed]
9.	R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138–1141 (1996). [CrossRef]
10.	P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570–573 (1998). [CrossRef]
11.	A. Degasperis, M. Conforti, F. Baronio, and S. Wabnitz, “Stable Control of Pulse Speed in Parametric Three-Wave Solitons,” Phys. Rev. Lett. 97, 093901 (2006) [CrossRef] [PubMed]
12.	L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover Publications Inc., New York, 1975).
13.	D. V. Skryabin, A. V. Yulin, and A. I. Maimistov, “Localized polaritons and second-harmonic generation in a resonant medium with quadratic nonlinearity,” Phys. Rev. Lett. 96, 163904–163907 (2006). [CrossRef] [PubMed]
14.	We assume that a photon of a given circular polarization promotes an electron-hole pair (exciton) creation with a well-defined spin orientation such that only linearly polarized light can generate biexcitons, cf., L. Jacak, P. Hawrylak, and A. Wojs, Optical Properties of Semiconductor Quantum Dots (Springer, Berlin, 1997).
15.	The contributions of cubic and quadratic nonlinearities to the polarization can be roughly estimated as $𝒫^{(3)} / 𝒫^{(2)} ~ χ_{e ff}^{(3)} ℰ_{m} / χ_{e ff}^{(2)} ~ 5 \times 10^{- 2}$ , using $χ_{e ff}^{(3)} ≃ 10^{- 18} m^{2} / V^{2}$ for GaAs. Here the peak field amplitude ℰ_m of a picosecond 2π input pulse was estimated using the known pulse area as ℰ_m ∼ πh̄/d_egτ_p. Thus quadratic nonlinearities indeed dominate in our case.
16.	In the circular polarization basis, the Bloch equations are exact such that no rotating wave approximation is needed.
17.	T. Brunhes, P. Boucaud, and S. Sauvage, “Infrared second-order optical susceptibility in InAs/GaAs self-assembled quantum dots,” Phys. Rev. B 61, 5562–5570 (2000). [CrossRef]
18.	S. J. B. Yoo, C. Caneau, R. Bhat, M. A. Koza, A. Rajhel, and N. Antoniades, “Wavelength conversion by difference frequency generation in AlGaAs waveguides with periodic domain inversion achieved by wafer bonding,” Appl. Phys. Lett. 68, 2609–2611 (1996). [CrossRef]
19.	T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett. 27, 628–630 (2002). [CrossRef]
20.	G. Panzarini, U. Hohenester, and E. Molinari, “Self-induced transparency in semiconductor quantum dots,” Phys. Rev. B 65, 165322–165327 (2002). [CrossRef]
21.	M. Jütte, H. Stolz, and W. von der Osten, “Linear and nonlinear pulse propagation at bound excitons in CdS,” J. Opt. Soc. Am. B 13, 1205–1210 (1996). [CrossRef]
22.	J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81, 1110–1113 (1998). [CrossRef]
23.	H. Giessen, A. Knorr, S. Haas, S. W. Koch, S. Linden, J. Kuhl, M. Hetterich, M. Grün, and C. Klingshirn, “Self-induced transmission on a free exciton resonance in a semiconductor,” Phys. Rev. Lett. 81, 4260–4263 (1998). [CrossRef]
24.	P. Borri, W. Langbein, S. Schneider, and U. Woggon, “Ultralong dephasing time in InGaAs quantum dots,” Phys. Rev. Lett. 87, 157401–157404 (2001). [CrossRef] [PubMed]
25.	J. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, USA, 2006).
26.	A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Optical solitons supported by competing nonlinearities,” Opt. Lett. 20, 1961–1963 (1995). [CrossRef] [PubMed]

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.2620) Nonlinear optics : Harmonic generation and mixing
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(260.0260) Physical optics : Physical optics

ToC Category:
Nonlinear Optics

History
Original Manuscript: April 26, 2012
Revised Manuscript: May 22, 2012
Manuscript Accepted: May 23, 2012
Published: June 8, 2012

Citation
Soodeh Haghgoo and Sergey A. Ponomarenko, "Self-induced transparency quadratic solitons," Opt. Express 20, 13988-13995 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-13-13988

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References

G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).
G. B. Lamb, Elements of Soliton Theory (Wiley, New York, 1980).
Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, Boston, 2003).
L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Clarendon, Oxford, 2003).
A. V. Buryak, P. D. Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep.370, 63–235 (2002). [CrossRef]
L. A. Ostrovskii, “Propagation of wave packets and space-time self-focusing in a nonlinear medium,” Sov. Phys. JETP24, 797–800 (1967).
G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomenaand their applications to all-optical signal processing, mode-locking, pulse, compression and solitions,” Opt. Quantum Electron.28, 1691–1740 (1996). [CrossRef]
W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, and G. I. Stegeman, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett.74, 5036–5039 (1995). [CrossRef] [PubMed]
R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E53, 1138–1141 (1996). [CrossRef]
P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett.81, 570–573 (1998). [CrossRef]
A. Degasperis, M. Conforti, F. Baronio, and S. Wabnitz, “Stable Control of Pulse Speed in Parametric Three-Wave Solitons,” Phys. Rev. Lett.97, 093901 (2006) [CrossRef] [PubMed]
L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover Publications Inc., New York, 1975).
D. V. Skryabin, A. V. Yulin, and A. I. Maimistov, “Localized polaritons and second-harmonic generation in a resonant medium with quadratic nonlinearity,” Phys. Rev. Lett.96, 163904–163907 (2006). [CrossRef] [PubMed]
We assume that a photon of a given circular polarization promotes an electron-hole pair (exciton) creation with a well-defined spin orientation such that only linearly polarized light can generate biexcitons, cf., L. Jacak, P. Hawrylak, and A. Wojs, Optical Properties of Semiconductor Quantum Dots (Springer, Berlin, 1997).
The contributions of cubic and quadratic nonlinearities to the polarization can be roughly estimated as P⁽³⁾/P²~χ_eff⁽³⁾ℰ_m/χ_eff⁽²⁾~5×10^-2, using χ_eff⁽³⁾≃10^-18m²V² for GaAs. Here the peak field amplitude ℰm of a picosecond 2π input pulse was estimated using the known pulse area as ℰ_m ∼ πh̄/d_egτ_p. Thus quadratic nonlinearities indeed dominate in our case.
In the circular polarization basis, the Bloch equations are exact such that no rotating wave approximation is needed.
T. Brunhes, P. Boucaud, and S. Sauvage, “Infrared second-order optical susceptibility in InAs/GaAs self-assembled quantum dots,” Phys. Rev. B61, 5562–5570 (2000). [CrossRef]
S. J. B. Yoo, C. Caneau, R. Bhat, M. A. Koza, A. Rajhel, and N. Antoniades, “Wavelength conversion by difference frequency generation in AlGaAs waveguides with periodic domain inversion achieved by wafer bonding,” Appl. Phys. Lett.68, 2609–2611 (1996). [CrossRef]
T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett.27, 628–630 (2002). [CrossRef]
G. Panzarini, U. Hohenester, and E. Molinari, “Self-induced transparency in semiconductor quantum dots,” Phys. Rev. B65, 165322–165327 (2002). [CrossRef]
M. Jütte, H. Stolz, and W. von der Osten, “Linear and nonlinear pulse propagation at bound excitons in CdS,” J. Opt. Soc. Am. B13, 1205–1210 (1996). [CrossRef]
J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett.81, 1110–1113 (1998). [CrossRef]
H. Giessen, A. Knorr, S. Haas, S. W. Koch, S. Linden, J. Kuhl, M. Hetterich, M. Grün, and C. Klingshirn, “Self-induced transmission on a free exciton resonance in a semiconductor,” Phys. Rev. Lett.81, 4260–4263 (1998). [CrossRef]
P. Borri, W. Langbein, S. Schneider, and U. Woggon, “Ultralong dephasing time in InGaAs quantum dots,” Phys. Rev. Lett.87, 157401–157404 (2001). [CrossRef] [PubMed]
J. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, USA, 2006).
A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Optical solitons supported by competing nonlinearities,” Opt. Lett.20, 1961–1963 (1995). [CrossRef] [PubMed]

Appl. Phys. Lett.

S. J. B. Yoo, C. Caneau, R. Bhat, M. A. Koza, A. Rajhel, and N. Antoniades, “Wavelength conversion by difference frequency generation in AlGaAs waveguides with periodic domain inversion achieved by wafer bonding,” Appl. Phys. Lett.68, 2609–2611 (1996). [CrossRef]

J. Opt. Soc. Am. B

M. Jütte, H. Stolz, and W. von der Osten, “Linear and nonlinear pulse propagation at bound excitons in CdS,” J. Opt. Soc. Am. B13, 1205–1210 (1996). [CrossRef]

Opt. Lett.

Opt. Quantum Electron.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomenaand their applications to all-optical signal processing, mode-locking, pulse, compression and solitions,” Opt. Quantum Electron.28, 1691–1740 (1996). [CrossRef]

Phys. Rep.

A. V. Buryak, P. D. Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep.370, 63–235 (2002). [CrossRef]

Phys. Rev. B

T. Brunhes, P. Boucaud, and S. Sauvage, “Infrared second-order optical susceptibility in InAs/GaAs self-assembled quantum dots,” Phys. Rev. B61, 5562–5570 (2000). [CrossRef]
G. Panzarini, U. Hohenester, and E. Molinari, “Self-induced transparency in semiconductor quantum dots,” Phys. Rev. B65, 165322–165327 (2002). [CrossRef]

Phys. Rev. E

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E53, 1138–1141 (1996). [CrossRef]

Phys. Rev. Lett.

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett.81, 570–573 (1998). [CrossRef]
A. Degasperis, M. Conforti, F. Baronio, and S. Wabnitz, “Stable Control of Pulse Speed in Parametric Three-Wave Solitons,” Phys. Rev. Lett.97, 093901 (2006) [CrossRef] [PubMed]
D. V. Skryabin, A. V. Yulin, and A. I. Maimistov, “Localized polaritons and second-harmonic generation in a resonant medium with quadratic nonlinearity,” Phys. Rev. Lett.96, 163904–163907 (2006). [CrossRef] [PubMed]
W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, and G. I. Stegeman, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett.74, 5036–5039 (1995). [CrossRef] [PubMed]
J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett.81, 1110–1113 (1998). [CrossRef]
H. Giessen, A. Knorr, S. Haas, S. W. Koch, S. Linden, J. Kuhl, M. Hetterich, M. Grün, and C. Klingshirn, “Self-induced transmission on a free exciton resonance in a semiconductor,” Phys. Rev. Lett.81, 4260–4263 (1998). [CrossRef]
P. Borri, W. Langbein, S. Schneider, and U. Woggon, “Ultralong dephasing time in InGaAs quantum dots,” Phys. Rev. Lett.87, 157401–157404 (2001). [CrossRef] [PubMed]

Sov. Phys. JETP

L. A. Ostrovskii, “Propagation of wave packets and space-time self-focusing in a nonlinear medium,” Sov. Phys. JETP24, 797–800 (1967).

Other

G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).
G. B. Lamb, Elements of Soliton Theory (Wiley, New York, 1980).
Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, Boston, 2003).
L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Clarendon, Oxford, 2003).
We assume that a photon of a given circular polarization promotes an electron-hole pair (exciton) creation with a well-defined spin orientation such that only linearly polarized light can generate biexcitons, cf., L. Jacak, P. Hawrylak, and A. Wojs, Optical Properties of Semiconductor Quantum Dots (Springer, Berlin, 1997).
The contributions of cubic and quadratic nonlinearities to the polarization can be roughly estimated as P⁽³⁾/P²~χ_eff⁽³⁾ℰ_m/χ_eff⁽²⁾~5×10^-2, using χ_eff⁽³⁾≃10^-18m²V² for GaAs. Here the peak field amplitude ℰm of a picosecond 2π input pulse was estimated using the known pulse area as ℰ_m ∼ πh̄/d_egτ_p. Thus quadratic nonlinearities indeed dominate in our case.
In the circular polarization basis, the Bloch equations are exact such that no rotating wave approximation is needed.
L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover Publications Inc., New York, 1975).
J. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, USA, 2006).

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