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Resonant three-wave interactions appear in many fields of physics e.g. nonlinear optics, plasma physics, acoustics and hydrodynamics. A general theory of autoresonant three-wave mixing in a nonuniform media is derived analytically and demonstrated numerically. It is shown that due to the medium nonuniformity, a stable phase-locked evolution is automatically established. For a weak nonuniformity, the efficiency of the energy conversion between the interacting waves can reach almost 100%. One of the potential applications of our theory is the design of highly-efficient optical parametric amplifiers.
© 2013 Optical Society of America
1. Introduction
Autoresonance is a unique property of many driven or mutually interacting oscillatory or wave systems stemming from their ability to stay in resonance (phase-locking) if the parameters of the system vary in space and/or time adiabatically [1]. One of the most important advantages of autoresonant schemes is their ability to transfer energy between the interacting waves with very high efficiency. In addition to being of fundamental interest, the autoresonance effect enables numerous exciting applications in many fields of physics and engineering. The autoresonance mechanism was first employed for relativistic particle acceleration 70 years ago [2, 3], and in the last 20 years it has found widespread applications to the excitation and control of nonlinear structures in many other fields (see for instance [1] and references therein). One of the most significant recent scientific achievements accomplished by implementing autoresonant schemes was the trapping of cold anti-hydrogen atoms at CERN [4].
In the field of nonlinear optics, adiabatic conversion processes in nonuniform media were previously studied in various systems involving two-wave interaction [5–13] (in some of these references the considered interaction is between three waves in the undepleted pump regime, which is effectively a two-wave mixing process). Other related studies of three-wave interactions in one dimensional nonuniform medium were conducted in co-propagating [14–16] and counter-propagating waves configurations [17–19] as well as in a multidimensional spatio-temporal varying medium [20], addressing specific applications in plasma physics and nonlinear optics. A complete theory of autoresonant Four-Wave Mixing (FWM) in a nonuniform medium was later developed by one of the authors and co-workers [21], suggesting a new application of Optical Parametric Amplification (OPA) using tapered fibers.
The mathematical description of the aforementioned plasma three-wave interactions included at least one nonlinear term (named as plasma nonlinear frequency shift) in the evolution equations of the wave amplitudes in addition to the nonlinear coupling terms among the interacting waves. The existence of nonlinear frequency shift terms with sufficiently large magnitude (satisfying some threshold criteria) appeared to be a requirement for the creation of stable autoresonant states.
In this article, we develop a general theory of autoresonant Three-Wave Mixing (TWM) in nonuniform media. We show that due to mechanisms different from those previously reported, it is possible to reach an autoresonant state also in the absence of frequency shift effects. The theory that is derived in this article is suitable for the description of a large variety of three-wave mixing processes such as stimulated Raman scattering in inhomogeneous plasmas [22], various acoustic wave interactions in solids [23], hydrodynamic three-wave interactions [24] etc. Probably, the field in which this study is most beneficial is nonlinear optics. In particular, new conversion schemes that allow complete pump depletion could be designed based on our theory, e.g. highly efficient OPAs using a nonuniform χ(2) medium. In order to simplify the interpretation of our theory in this context, we will henceforth adopt the terminology of nonlinear optics.
2. Mathematical model
We begin our analysis with the set of equations for slow wave envelopes Aj describing the interaction between three waves at frequencies that satisfy the matching condition ω1 + ω2 = ω3[25]:
In the context of TWM in an optical medium with χ(2) nonlinearity, Aj (z) is the envelope of the electric field of the j-th wave such that Ej (z,t) = Aj (z) exp (i[kjz − ωjt]), where kj is the j-th wave wavevector. The coupling coefficients ηj are defined by where c is the speed of light in vacuum and deff is the effective nonlinear susceptibility. We assume slowly varying wavevectors kj (z) for each of the interacting waves and define the wavevectors mismatch Δk(z) = k1 + k2 − k3. For simplicity, in our theory we take into account the medium nonuniformity only from the presence of the z dependence of Δk(z) in the wavevectors mismatch, i.e. kj in the coupling coefficients in Eqs. (1)–(2) are replaced by an average value kj,avg along the medium for each j. We now define the dimensionless coordinate ζ = z/l and dimensionless complex amplitudes where l = 1/[(η1η2)1/2 |A3,0|] and the subscript ”0” denotes initial condition values (note that |a3,0| ≡ 1). Using these definitions, Eqs. (1)–(2) are transformed to the dimensionless form:
Next, we represent each one of the complex amplitudes aj by its absolute value Bj and real phase ϕj using the definitions a1,2 = B1,2 exp (iϕ1,2) and , and substitute these expressions into Eqs. (3)–(4). Decomposing each one of the resulting equations into its real and imaginary parts gives the following set of equations: where we defined the phase mismatch Φ =ϕ1 +ϕ2 − ϕ3, the sign coefficients ε1,2 = +1, ε3 = −1 and . The set of three real amplitude equations (5) is equivalent to the following set of two algebraic equations, known as Manley-Rowe relations, and one differential equation for B1:For simplicity, we assume in the following that the wavevector mismatch is a linear function of position and define α as the nonuniformity rate coefficient such that Δk = α(z − z*)/l2 = α(ζ − ζ*)/l where the subscript * denotes the point of perfect wavevector matching Δk = 0. Then, we define a shifted normalized coordinate ξ = ζ − z*/l = ζ − ζ* such that the wavevecotors are matched at ξ = 0, and rewrite our system (6)–(7) as:
Our goal in the next sections is to study the set of two differential Eqs. (9)–(10) for B1 and Φ in which the variables B2 and B3 are expressed by Eqs. (8).In Figs. 1 and 2 we present the numerical solution of Eqs. (9)–(10) as a function of the normalized distance along propagation |α|ξ. The equations were solved in the interval −6 ≤ |α|ξ ≤ 6 for the initial conditions B1,0 = 0, B2,0 = 0.01, B3,0 = 1 and Φ0 = 0. For the case α = 0.01 which is shown in Fig. 1, we observe that the dynamics can be divided into two stages. The first stage (|α|ξ < −2) is characterized by an initial phase-locking (Φ ≈ 0), in which the energy of the pump wave could be considered as constant (undepleted pump regime). Phase-locking is preserved continuously, and in the second stage (|α|ξ> −2), the pump energy is depleted with a monotonic increase in the average energy of the signal and idler fields, until almost complete pump depletion is achieved. In the second case, which is presented in Fig. 2, the nonuniformity parameter is large (α = 1). In this condition the initial phase-locking is observed again but it is then lost as |α|ξ → −2. Consequently, the conversion efficiency with respect to the pump energy is low, and it strongly depends on the position along the propagation. Next, we derive an analytical theory that explains the features observed in the numerical examples.
Fig. 1 Phase-locked evolution of (a) wave envelopes and (b) phase mismatch Φ versus normalized distance |α|ξ for a spatial nonuniformity rate α = 0.01. In (a), the idler, signal and pump are denoted by (1) green, (2) blue and (3) red respectively. The black curves represent the value of the analytical expressions Eqs. (16)–(17). Note that the curves of the signal and idler are almost overlapping.
Fig. 2 Evolution of (a) wave envelopes and (b) phase mismatch Φ for large value of nonuniformity rate α = 1. Due to the large value of α, the initial phase-locking is lost prior to reaching the point |α|ξ = −2. Note that the curves of the signal and idler (blue and green, respectively) are almost overlapping.
3. Initial phase-locking and autoresonant state
3.1. Closed-form solutions
We start by showing that the initial dynamics of a TWM process in a nonuniform medium can be described as a two-wave mixing process. This dynamics is similar to the initial dynamics of a FWM process in a similar system, as studied in Ref. [21], and will be briefly summarized in this paragraph for the sake of completeness. In spite of the similarity between TWM and FWM in the initial stage, there are essential differences between the dynamics of these processes in the autoresonant stage (|α|ξ > −2). Such differences will be analysed in the rest of this article. Our study begins at a sufficiently large negative wavevector mismatch |α|ξin. Further, we assume that the initial signal is small (B2,0 << B3,0), and that B1 (z) << B2 in the initial stage, since the idler is initially zero. Consequently, we neglect the variation of B2 and the depletion of the pump wave at this stage and focus on the evolution of B1. Then, Q ≈ B2,0B3,0/B1 = B2,0/B1, since according to our definition of variables B3,0 ≡ 1. Therefore, Eqs. (9)–(10) can be approximated by
Note that Eq. (11) has the same form as Eq. (11) in Ref. [21]. It was shown there that starting at a sufficiently large negative value of |α|ξin guarantees phase-locking in which Φ ≈ π for α < 0 and Φ ≈ 0 for α > 0 as seen in Figs. 1 and 2. As the idler wave is excited, the signal amplitude cannot be assumed as a constant, and one has to consider the following two-wave system: The two-wave system of Eqs. (12)–(13) was studied in Ref. [21] (see there Eqs. (13)–(14)) and the analytical solutions of B1 and B2 were found. It was shown that the aforementioned phase-locking continues, and while approaching the point |α|ξ = −2, the amplitudes of the idler and the signal become comparable to each other, i.e. B1 ≈ B2. Beyond this point, one must take into account the pump depletion in the TWM process, by analysing Eqs. (9)–(10), which differ from the corresponding FWM equations.In the region |α|ξ > −2 we assume |cosΦ| ≈ 1, and set B1/B2 ≈ 1, as discussed above. We define the autoresonant solution (denoted by the hat symbol) consistently to the requirement that the left hand side of the Eq. (10) vanishes:
where s ≡ sign(α) = cos Φ̂ and . can be expressed in terms of using the Manley-Rowe relations in Eqs. (8), from which we see that . Substituting the expression for Q̂ in Eq. (14) results in a quadratic equation for B̂3. The autoresonant solution for the pump is chosen as the root of the quadratic equation for which B̂3 > 0. This solution decreases monotonically with |α|ξ, and is the same for both α > 0 and α < 0, The corresponding expressions for the signal and idler evolution in autoresonance are obtained using the Manley-Rowe relations Eqs. (8) and are characterized by a monotonic increase as a function of |α|ξ: There are interesting differences between the dynamics of the autoresonant FWM process that was studied in [21] and the TWM process that is considered in this article. Although in both processes, autoresonant dynamics can be obtained for α with either a positive or a negative sign, the spatial form of the solutions is the same for either sign in TWM (see Eqs. (16)–(17)), whereas in FWM it depends on the sign of α. Furthermore, in the FWM process, the spatial range of the autoresonant stage was found to be limited to a specific range of |α|ξ depending on the sign of α, after which phase-locking was lost. In contrast, we will show that in the TWM process, once established, stable autoresonant dynamics continues without any bounds.This difference between the autoresonant states in TWM and FWM can be understood by examining the corresponding Q terms in the limit of vanishing pump power (and ). In this limit, for TWM Q → −1/B3 while for FWM Q → const. Recalling that phase-locking is characterized by the balance between Q and the detuning term |α|ξ, we see that in TWM, as |α|ξ increases, this balance may be mainatined by a continous monotonic decrease in B3 (and a suitable increase in B1,2). However, in FWM, the constant limit value of Q indicates that at some point, the necessary balance for preserving phase-locking could not be satisfied, and the system will depart from the autoresonant state.
3.2. Stability analysis
We have already discussed the quasi-steady state solution in our system and showed that the system enters this state automatically in the initial phase locking stage (|α|ξ < −2). We will show now that for small enough |α| this is followed by a continuous stable autoresonance in the system. In studying the stability, we assume a solution close to the steady state, i.e. we write B1 = B̂1 + δB1 and Φ = Φ̂ + δΦ where δB1, δΦ << 1. In the following we will use the approximate trigonometric relations sinΦ = ssinδΦ, cosΦ ≈ s (derived from sinΦ̂ = 0 and cosδΦ ≈ 1). Next, we replace cosΦ by s in Eq. (10) and expand the equation to first order in δB1, yielding
where Q̂′ denotes the derivative of Q with respect to B1 evaluated at B̂1. Phase-locking is characterized by dΦ̂/dξ = s(|α|ξ + Q̂) ≈ 0, so we define B̂1 in our system at all stages of evolution (including the initial phase-locking stage |α|ξ < −2, the autoresonant stage |α|ξ > −2 and the transition region |α|ξ ≈ −2) by Then, On the other hand, d(δB1)/dξ = dB1/dξ − dB̂1/dξ, which, by using Eq. (9) to the lowest order in δB1, δΦ and dB̂1/dξ = −|α|/Q̂′ obtained by differentiation of Eq. (19), yields where D = B̂2B̂3 (we use here the relation sin Φ = ssin δΦ). It can be observed that Eqs. (20) and (21) are Hamilton equations for the canonical variables δB1 and δΦ governed by the Hamiltonian where Veff = DcosδΦ − s|α|δΦ/Q̂′ and ξ plays the role of ”time”. Note that Veff is a tilted ”washboard” potential with slow ”time”-dependent parameters D and Q̂. Thus, our quasi-steady states are stable, as long as the effective potential has well defined minima (since cos δΦ ≈ 1 − (δΦ)2/2), i.e. where Ω2 ≡ −DQ̂′. Eq. (23) expresses the required stability criterion for phase-locking in our system. We show the dependence of Ω2 on |α|ξ in Fig. 3 for several values of the initial signal amplitude B2,0. Note that the most limiting condition on |α| occurs in the transition between the initial phase locking and the autoresonant stages, i.e. at |α|ξ ≈ −2. The Figure shows that the phase-locked states are stable for the value of α = 0.01 that was used in Fig. 1, but for the case α = 1 illustrated in Fig. 2, they become unstable for |α|ξ → −2. The spatial frequency Ω is manifested by the small spatial modulations around the slowly evolving quasi-steady state that are seen in the examples in Fig. 1. A simple expression for the spatially-dependent frequency Ω in the autoresonant stage |α|ξ> −2 can be obtained by substituting Eqs. (16)–(17) into the definition of Ω2 i.e., The minimal oscillation frequency is at |α|ξ = −2 and after this point there is a monotonic increase in the spatial frequency, as seen in Fig. 1.Fig. 3 Stability function Ω2 versus |α|ξ for different values of the initial signal amplitude B2,0 (blue: 0.01, red: 0.05, green: 0.1). The black curve represents the value of the analytical expression Eq. (24). Satisfying the criterion |α| < Ω2 continuously is required for the stability of phase-locking. The inset (b) is a zoom of (a) at the location of the minimum of Ω2 near |α|ξ = −2. Note that all the curves are almost overlapping.
4. Conclusions
In summary, we have derived in this article a theory of three-wave mixing in a nonuniform medium. We have shown that for a medium with weak nonuniformity, a spatially-continuous phase-locked evolution is established automatically. This evolution is accompanied by a monotonic conversion of the energy between the interacting waves and consequently, it is possible to achieve complete pump depletion. Our theory could be applied in various fields of physics and engineering, e.g. Optical Parametric Amplification (OPA) in nonlinear χ(2) media. An estimation for the required values of the design parameters of an autoresonant OPA is provided in the Appendix.
5. Appendix - optical parametric amplification
In order to simplify the design of autoresonant OPAs, we provide in this appendix explicit formulas in which we specify the required properties of the χ(2) medium in which the TWM process takes place as well as the experimental setup conditions. We begin by expressing the normalized nonuniformity rate α and the position |α|ξ in terms of the original physical parameters. From our definitions after Eq. (2) we find that the normalization parameter for length is
where we used the relations ωj/kj = c/nj and c/ωj = λj/2π. Here nj and λj denote the material refractive index and the vacuum wavelength at frequency j, I3,0 = 2n3,0cε0 |A3,0|2 is the initial pump intensity, ε0 is the vacuum permittivity and c is the speed of light in vacuum [25]. By substituting Eq. (25) into the definitions after Eq. (8) and using some algebraic transformations, we see that and therefore Here Δkini and Δkfin are the wavevector mismatch at the initial and the final boundaries of the χ(2) medium, separated by a distance L.In the previous sections, we have seen that the most significant part of the autoresonant dynamics occurs between |α|ξ = −2 (which is required to guarantee initial phase-locking) and |α|ξ = 2 (where according to Eq. (17) i.e. about 90% of the initial pump energy is depleted and transferred to the signal and idler waves). Hence, designing the variation of the parameters of the system to include the region −2 < |α|ξ < 2 (i.e. |α|ξini < −2 and |α|ξfin> 2) will allow efficient autoresonant amplification. Therefore, we see from Eq. (28) that it is required that sign (Δkfin) ≠ sign(Δkini) and
In order to obtain stable phase-locked dynamics in the system everywhere along the propagation, it is required that where is the local minimum value of Ω2 that depends on the initial signal amplitude B2,0. We see from Eq. (26) that this criterion can be written as
For estimating explicitly the required nonuniformity rate in a typical case of interest, we assume that the dimensionless initial signal amplitude is B2,0 = 0.01 (equivalent to an initial signal to pump intensity ratio of I2,0/I3,0 = (n2,0/n3,0)2 (ω2/ω3)(B2,0/B3,0)2 ≈ 5 · 10−5 for ω2/ω3 ≈ 1/2, n2,0/n3,0 ≈ 1, recalling that B3,0 ≡ 1). In this case, we see from Fig. 3 that the minimal value of Ω2 is that could be substituted into Eq. (30) to obtain an explicit criterion for the nonuniformity rate.Considering the fact that the expression Δk(z) ≡ k1 + k2 − k3 = (n1ω1 + n2ω2 + n3ω3) / c depends on the material dispersion relation nj (λj, z), which is a function of two independent variables λj and z, it is not easy to determine analytically what is the required functional form of the dispersion relation that satisfy the criteria expressed by Eqs. (29) and (30) (note that if the propagation is within a physically bounded domain, e.g. a waveguide, the necessary conditions are expressed in terms of a mismatch between the propagation factors that depends also on the geometry of the system). However, considering a specific experimental setup for which nj (λj, z) is known, testing the fulfilment of these criteria numerically is straightforward.
In practice, the desired nonuniformity of the wavevector mismatch may be created by various methods e.g. by nonuniform doping of the nonlinear system. A detailed analysis of feasible technologies for inducing the required nonuniformity is beyond the scope of the current paper and is left to future studies. Current investigations are ongoing to establish experimental conditions where autoresonance may strongly boost the efficiency of optical parametric amplification. Preliminary analysis suggests that the proposed autoresonant scheme may allow complete pump depletion over more than 100 nm bandwidth by properly tuning the refractive index in few centimetres long standard nonlinear crystals.
Acknowledgments
This work was supported by the FQRNT (Le Fonds Québécois de la Recherche sur la Nature et les Technologies) and NSERC (Natural Sciences and Engineering Research Council of Canada). O. Y. and L. C. wish to acknowledge the FQRNT MELS fellowship program (Files 168900 and 168739, respectively). M. C. acknowledges the support of the IOF People Programme (Marie Curie Actions) of the European Union’s FP7-2012, KOHERENT, GA 299522.
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