Complete energy conversion by autoresonant three-wave mixing in nonuniform media

doi:10.1364/OE.21.001623

« Show journal navigation

Complete energy conversion by autoresonant three-wave mixing in nonuniform media

O. Yaakobi, L. Caspani, M. Clerici, F. Vidal, and R. Morandotti »View Author Affiliations

Optics Express, Vol. 21, Issue 2, pp. 1623-1632 (2013)
http://dx.doi.org/10.1364/OE.21.001623

View Full Text Article

Acrobat PDF (883 KB)

Journal Search
Article Lookup

Article Tools

Citations

Alert me when this article is cited
Export Citation/Save

Article Navigation

▸ Article Outline
– Abstract
1. Introduction
2. Mathematical model
3. Initial phase-locking and autoresonant state
4. Conclusions
– References and links

Article Tools
Full Text
Article Info
References
Cited By
Figures
Metrics
Download PDF

Viewing Equations in the
HTML Article

Optimal Viewing Recommendations

Full Text
Article Info
References (25)
Cited By
Figures (3)
Metrics

Abstract

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.

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

L. Friedland, “Autoresonance in nonlinear systems,” Scholarpedia 4, 5473 (2009). [CrossRef]

]. 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

V. I. Veksler, “A new method of acceleration of relativistic particles,” J. Phys. USSR 9, 153–158 (1945).

, 3

E. M. McMillan, “The synchrotron - a proposed high energy particle accelerator,” Phys. Rev. 68, 143–144 (1945). [CrossRef]

], 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

L. Friedland, “Autoresonance in nonlinear systems,” Scholarpedia 4, 5473 (2009). [CrossRef]

] 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

G. B. Andresen, M. D. Ashkezari, M. Baquero-Ruiz, W. Bertsche, P. D. Bowe, E. Butler, C. L. Cesar, S. Chapman, M. Charlton, A. Deller, S. Eriksson, J. Fajans, T. Friesen, M. C. Fujiwara, D. R. Gill, A. Gutierrez, J. S. Hangst, W. N. Hardy, M. E. Hayden, A. J. Humphries, R. Hydomako, M. J. Jenkins, S. Jonsell, L. V. Jorgensen, L. Kurchaninov, N. Madsen, S. Menary, P. Nolan, K. Olchanski, A. Olin, A. Povilus, P. Pusa, F. Robicheaux, E. Sarid, S. Seif el Nasr, D. M. Silveira, C. So, J. W. Storey, R. I. Thompson, D. P. van der Werf, J. S. Wurtele, and Y. Yamazaki, “Trapped antihydrogen,” Nature 468, 673–676 (2010). [CrossRef] [PubMed]

In the field of nonlinear optics, adiabatic conversion processes in nonuniform media were previously studied in various systems involving two-wave interaction [5

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990). [CrossRef]

–13

S. Richard, “Second-harmonic generation in tapered optical fibers,” J. Opt. Soc. Am. B 27, 1504–1512 (2010). [CrossRef]

] (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

L. Friedland, “Autoresonant three-wave interactions,” Phys. Rev. Lett. 69, 1749–1752 (1992). [CrossRef] [PubMed]

–16

M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Competing collinear and noncollinear interactions in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B, 25, 1402–1413 (2008). [CrossRef]

] and counter-propagating waves configurations [17

I. Y. Dodin, G. M. Fraiman, V. M. Malkin, and N. J. Fisch, “Amplification of short laser pulses by Raman backscattering in capillary plasmas,” JETP 95, 625–638 (2002). [CrossRef]

–19

T. Chapman, S. Huller, P. E. Masson-Laborde, W. Rozmus, and D. Pesme, “Spatially autoresonant stimulated Raman scattering in inhomogeneous plasmas in the kinetic regime,” Phys. Plasmas 17, 122317 (2010). [CrossRef]

] as well as in a multidimensional spatio-temporal varying medium [20

O. Yaakobi and L. Friedland, “Multidimensional, autoresonant three-wave interactions,” Phys. Plasmas 15, 102104 (2008). [CrossRef]

], 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

O. Yaakobi and L. Friedland, “Autoresonant four-wave mixing in optical fibers,” Phys. Rev. A 82, 023820 (2010). [CrossRef]

], 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

W. L. Kruer, The Physics of Laser Plasma Interactions (reprint ed. Westview Press, Boulder, CO, 2001).

], various acoustic wave interactions in solids [23

A. P. Mayer, “Surface acoustic waves in nonlinear elastic media,” Phys. Rep. 256, 237–366 (1995). [CrossRef]

], hydrodynamic three-wave interactions [24

K. Trulsen and C. C. Mei, “Modulation of three resonating gravity-capillary waves by a long gravity wave,” J. Fluid Mech. 290, 345–376 (1995). [CrossRef]

] 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 A_j describing the interaction between three waves at frequencies that satisfy the matching condition ω₁ + ω₂ = ω₃[25

R. B. Boyd, Nonlinear Optics (Third Edition, AP 2007).

\frac{d A_{j}}{d z} = i η_{j} A_{m}^{*} A_{3} e^{- i \int_{0}^{z} Δ k (z^{'}) d z^{'}}; j, m = 1, 2; j \neq m,

(1)

\frac{d A_{3}}{d z} = i η_{3} A_{1} A_{2} e^{+ i \int_{0}^{z} Δ k (z^{'}) d z^{'}} .

(2)

In the context of TWM in an optical medium with χ⁽²⁾ nonlinearity, A_j (z) is the envelope of the electric field of the j-th wave such that E_j (z,t) = A_j (z) exp (i[k_jz − ω_jt]), where k_j is the j-th wave wavevector. The coupling coefficients η_j are defined by

η_{j} = 2 d_{eff} ω_{j}^{2} / (k_{j} c^{2})

where c is the speed of light in vacuum and d_eff is the effective nonlinear susceptibility. We assume slowly varying wavevectors k_j (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. k_j in the coupling coefficients in Eqs. (1)–(2) are replaced by an average value k_j,avg along the medium for each j. We now define the dimensionless coordinate ζ = z/l and dimensionless complex amplitudes

a_{j} = η_{j}^{- 1 / 2} A_{j} / (η_{3}^{- 1 / 2} | A_{3, 0} |)

where l = 1/[(η₁η₂)^1/2 |A_3,0|] and the subscript ”0” denotes initial condition values (note that |a_3,0| ≡ 1). Using these definitions, Eqs. (1)–(2) are transformed to the dimensionless form:

\frac{d a_{j}}{d ζ} = i a_{m}^{*} a_{3} e^{- i l \int_{0}^{ζ} Δ k (ζ^{'}) d ζ^{'}}; j, m = 1, 2; j \neq m,

(3)

\frac{d a_{3}}{d ζ} = i a_{1} a_{2} e^{+ i l \int_{0}^{ζ} Δ k (ζ^{'}) d ζ^{'}} .

(4)

Next, we represent each one of the complex amplitudes a_j by its absolute value B_j and real phase ϕ_j using the definitions a_1,2 = B_1,2 exp (iϕ_1,2) and

a_{3} = B_{3} exp (i [ϕ_{3} + l \int_{0}^{ζ} Δ k (ζ^{'}) d ζ^{'}])

, 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:

\frac{d B_{j}}{d ζ} = ε_{j} B_{m} B_{n} sin Φ; j, m, n = 1 - 3; j \neq m \neq n,

(5)

\frac{d Φ}{d ζ} = l Δ k + Q cos Φ,

(6)

where we defined the phase mismatch Φ =ϕ₁ +ϕ₂ − ϕ₃, the sign coefficients ε_1,2 = +1, ε₃ = −1 and

Q = B_{1} B_{2} B_{3} (B_{1}^{- 2} + B_{2}^{- 2} - B_{3}^{- 2})

. 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:

\frac{d B_{1}}{d ζ} = B_{2} B_{3} sin Φ,

(7)

B_{1}^{2} - B_{2}^{2} = B_{1, 0}^{2} - B_{2, 0}^{2}, B_{1}^{2} + B_{3}^{2} = B_{1, 0}^{2} + B_{3, 0}^{2} .

(8)

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:

\frac{d B_{1}}{d ξ} = B_{2} B_{3} sin Φ,

(9)

\frac{d Φ}{d ξ} = α ξ + Q cos Φ .

(10)

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 B_1,0 = 0, B_2,0 = 0.01, B_3,0 = 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

B_{j}^{2}

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

B_{j}^{2}

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

O. Yaakobi and L. Friedland, “Autoresonant four-wave mixing in optical fibers,” Phys. Rev. A 82, 023820 (2010). [CrossRef]

], 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|ξ_in. Further, we assume that the initial signal is small (B_2,0 << B_3,0), 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 ≈ B_2,0B_3,0/B₁ = B_2,0/B₁, since according to our definition of variables B_3,0 ≡ 1. Therefore, Eqs. (9)–(10) can be approximated by

\frac{d B_{1}}{d ξ} = B_{2, 0} sin Φ, \frac{d Φ}{d ξ} = α ξ + \frac{B_{2, 0}}{B_{1}} cos Φ .

(11)

Note that Eq. (11) has the same form as Eq. (11) in Ref. [21

O. Yaakobi and L. Friedland, “Autoresonant four-wave mixing in optical fibers,” Phys. Rev. A 82, 023820 (2010). [CrossRef]

]. It was shown there that starting at a sufficiently large negative value of |α|ξ_in 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:

\frac{d B_{1}}{d ξ} = B_{2} sin Φ, B_{2}^{2} = B_{1}^{2}, + B_{2, 0}^{2}

(12)

\frac{d Φ}{d ξ} = α ξ + (\frac{B_{2}}{B_{1}} + \frac{B_{1}}{B_{2}}) cos Φ .

(13)

The two-wave system of Eqs. (12)–(13) was studied in Ref. [21

O. Yaakobi and L. Friedland, “Autoresonant four-wave mixing in optical fibers,” Phys. Rev. A 82, 023820 (2010). [CrossRef]

] (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:

α ξ + s \hat{Q} = 0,

(14)

sin \hat{Φ} = \frac{1}{{\hat{B}}_{1} {\hat{B}}_{3}} \frac{d {\hat{B}}_{1}}{d ξ},

(15)

where s ≡ sign(α) = cos Φ̂ and

\hat{Q} = {\hat{B}}_{1} {\hat{B}}_{2} {\hat{B}}_{3} ({\hat{B}}_{1}^{- 2} + {\hat{B}}_{2}^{- 2} - {\hat{B}}_{3}^{- 2}) \approx {\hat{B}}_{1}^{2} {\hat{B}}_{3} (2 {\hat{B}}_{1}^{- 2} - {\hat{B}}_{3}^{- 2}) = 2 {\hat{B}}_{3} - {\hat{B}}_{1}^{2} / {\hat{B}}_{3}

{\hat{B}}_{1}^{2}

can be expressed in terms of

{\hat{B}}_{3}^{2}

using the Manley-Rowe relations in Eqs. (8), from which we see that

\hat{Q} \approx 2 {\hat{B}}_{3} - (B_{1, 0}^{2} + B_{3, 0}^{2} - {\hat{B}}_{3}^{2}) / {\hat{B}}_{3} \approx 3 {\hat{B}}_{3} - 1 / {\hat{B}}_{3}

. 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,

{\hat{B}}_{3} = \frac{- | α | ξ + \sqrt{{(| α | ξ)}^{2} + 12}}{6}, | α | ξ > - 2 .

(16)

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:

{\hat{B}}_{2}^{2} \approx {\hat{B}}_{1}^{2} = 1 - {[\frac{- | α | ξ + \sqrt{{(| α | ξ)}^{2} + 12}}{6}]}^{2}, | α | ξ > - 2 .

(17)

There are interesting differences between the dynamics of the autoresonant FWM process that was studied in [21

O. Yaakobi and L. Friedland, “Autoresonant four-wave mixing in optical fibers,” Phys. Rev. A 82, 023820 (2010). [CrossRef]

] 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

B_{3}^{2} \to 0

(and

B_{1, 2}^{2} \to 1

). 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 B_1,2). 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

\frac{d Φ}{d ξ} = s (| α | ξ + \hat{Q}) + s {\hat{Q}}^{'} δ B_{1},

(18)

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

\hat{Q} \equiv - | α | ξ .

(19)

Then,

\frac{d (δ Φ)}{d ξ} = s {\hat{Q}}^{'} δ B_{1} .

(20)

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

\frac{d (δ B_{1})}{d ξ} = s D sin δ Φ + \frac{| α |}{{\hat{Q}}^{'}},

(21)

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

H (δ B_{1}, δ Φ, ξ) = s [\frac{{\hat{Q}}^{'}}{2} {(δ B_{1})}^{2} + V_{eff} (δ Φ, ξ)],

(22)

where V_eff = DcosδΦ − s|α|δΦ/Q̂′ and ξ plays the role of ”time”. Note that V_eff 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.

| α | < Ω^{2},

(23)

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 B_2,0. 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.,

Ω^{2} \approx \frac{2 | α | ξ \sqrt{{(| α | ξ)}^{2} + 12}}{3} + \frac{{(| α | ξ)}^{2}}{3} + 4 .

(24)

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 B_2,0 (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.

Appendices

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

l = {[(\frac{c ε_{0}}{8 π^{2}}) \cdot \frac{n_{1, avg} n_{2, avg} n_{3, 0} λ_{1} λ_{2}}{d_{eff}^{2} I_{3, 0}}]}^{1 / 2},

(25)

where we used the relations ω_j/k_j = c/n_j and c/ω_j = λ_j/2π. Here n_j and λ_j denote the material refractive index and the vacuum wavelength at frequency j, I_3,0 = 2n_3,0cε₀ |A_3,0|² is the initial pump intensity, ε₀ is the vacuum permittivity and c is the speed of light in vacuum [25

R. B. Boyd, Nonlinear Optics (Third Edition, AP 2007).

]. By substituting Eq. (25) into the definitions after Eq. (8) and using some algebraic transformations, we see that

α = \frac{l^{2}}{L} (Δ k_{fin} - Δ k_{ini}),

(26)

and

ξ = \frac{l Δ k}{α} = \frac{L}{l} \cdot \frac{Δ k}{(Δ k_{fin} - Δ k_{ini})},

(27)

therefore

| α | ξ = l \cdot Δ k \cdot sign (Δ k_{fin} - Δ k_{ini}) .

(28)

Here Δk_ini and Δk_fin 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)

B_{1, 2}^{2} = 8 / 9

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|ξ_ini < −2 and |α|ξ_fin> 2) will allow efficient autoresonant amplification. Therefore, we see from Eq. (28) that it is required that sign (Δk_fin) ≠ sign(Δk_ini) and

| Δ k_{ini} |, | Δ k_{fin} | > \frac{2}{l} .

(29)

In order to obtain stable phase-locked dynamics in the system everywhere along the propagation, it is required that

| α | < Ω_{min}^{2}

where

Ω_{min}^{2}

is the local minimum value of Ω² that depends on the initial signal amplitude B_2,0. We see from Eq. (26) that this criterion can be written as

\frac{| Δ k_{fin} - Δ k_{ini} |}{L} < \frac{Ω_{min}^{2}}{l^{2}} .

(30)

For estimating explicitly the required nonuniformity rate in a typical case of interest, we assume that the dimensionless initial signal amplitude is B_2,0 = 0.01 (equivalent to an initial signal to pump intensity ratio of I_2,0/I_3,0 = (n_2,0/n_3,0)² (ω₂/ω₃)(B_2,0/B_3,0)² ≈ 5 · 10⁻⁵ for ω₂/ω₃ ≈ 1/2, n_2,0/n_3,0 ≈ 1, recalling that B_3,0 ≡ 1). In this case, we see from Fig. 3 that the minimal value of Ω² is

Ω_{min}^{2} = 0.01

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 n_j (λ_j, z), which is a function of two independent variables λ_j 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 n_j (λ_j, 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.

References and links

1.	L. Friedland, “Autoresonance in nonlinear systems,” Scholarpedia 4, 5473 (2009). [CrossRef]
2.	V. I. Veksler, “A new method of acceleration of relativistic particles,” J. Phys. USSR 9, 153–158 (1945).
3.	E. M. McMillan, “The synchrotron - a proposed high energy particle accelerator,” Phys. Rev. 68, 143–144 (1945). [CrossRef]
4.	G. B. Andresen, M. D. Ashkezari, M. Baquero-Ruiz, W. Bertsche, P. D. Bowe, E. Butler, C. L. Cesar, S. Chapman, M. Charlton, A. Deller, S. Eriksson, J. Fajans, T. Friesen, M. C. Fujiwara, D. R. Gill, A. Gutierrez, J. S. Hangst, W. N. Hardy, M. E. Hayden, A. J. Humphries, R. Hydomako, M. J. Jenkins, S. Jonsell, L. V. Jorgensen, L. Kurchaninov, N. Madsen, S. Menary, P. Nolan, K. Olchanski, A. Olin, A. Povilus, P. Pusa, F. Robicheaux, E. Sarid, S. Seif el Nasr, D. M. Silveira, C. So, J. W. Storey, R. I. Thompson, D. P. van der Werf, J. S. Wurtele, and Y. Yamazaki, “Trapped antihydrogen,” Nature 468, 673–676 (2010). [CrossRef] [PubMed]
5.	T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990). [CrossRef]
6.	K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994). [CrossRef]
7.	G. Imeshev, M. M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B, 18, 534–539 (2001). [CrossRef]
8.	H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78, 063821 (2008). [CrossRef]
9.	H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Optics Express 17, 12731–12740 (2009). [CrossRef] [PubMed]
10.	A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103, 123901 (2009). [CrossRef] [PubMed]
11.	A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant propagation of incoherent light-waves,” Optics Express , 18, 17709–17718 (2010). [CrossRef] [PubMed]
12.	S. Trendafilov, V. Khudik, M. Tokman, and G. Shvets, “Hamiltonian description of non-reciprocal light propagation in nonlinear chiral fibers,” Physica B 405, 3003–3006 (2010). [CrossRef]
13.	S. Richard, “Second-harmonic generation in tapered optical fibers,” J. Opt. Soc. Am. B 27, 1504–1512 (2010). [CrossRef]
14.	L. Friedland, “Autoresonant three-wave interactions,” Phys. Rev. Lett. 69, 1749–1752 (1992). [CrossRef] [PubMed]
15.	S. Longhi, “Third-harmonic generation in quasi-phase-matched χ⁽²⁾ media with missing second harmonic,” Optics Letters 32, 1791–1793 (2007). [CrossRef] [PubMed]
16.	M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Competing collinear and noncollinear interactions in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B, 25, 1402–1413 (2008). [CrossRef]
17.	I. Y. Dodin, G. M. Fraiman, V. M. Malkin, and N. J. Fisch, “Amplification of short laser pulses by Raman backscattering in capillary plasmas,” JETP 95, 625–638 (2002). [CrossRef]
18.	O. Yaakobi, L. Friedland, R. R. Lindberg, A. E. Charman, G. Penn, and J. S. Wurtele, “Spatially autoresonant stimulated Raman scattering in nonuniform plasmas,” Phys. Plasmas 15, 032105 (2008). [CrossRef]
19.	T. Chapman, S. Huller, P. E. Masson-Laborde, W. Rozmus, and D. Pesme, “Spatially autoresonant stimulated Raman scattering in inhomogeneous plasmas in the kinetic regime,” Phys. Plasmas 17, 122317 (2010). [CrossRef]
20.	O. Yaakobi and L. Friedland, “Multidimensional, autoresonant three-wave interactions,” Phys. Plasmas 15, 102104 (2008). [CrossRef]
21.	O. Yaakobi and L. Friedland, “Autoresonant four-wave mixing in optical fibers,” Phys. Rev. A 82, 023820 (2010). [CrossRef]
22.	W. L. Kruer, The Physics of Laser Plasma Interactions (reprint ed. Westview Press, Boulder, CO, 2001).
23.	A. P. Mayer, “Surface acoustic waves in nonlinear elastic media,” Phys. Rep. 256, 237–366 (1995). [CrossRef]
24.	K. Trulsen and C. C. Mei, “Modulation of three resonating gravity-capillary waves by a long gravity wave,” J. Fluid Mech. 290, 345–376 (1995). [CrossRef]
25.	R. B. Boyd, Nonlinear Optics (Third Edition, AP 2007).

OCIS Codes
(190.4410) Nonlinear optics : Nonlinear optics, parametric processes
(190.4970) Nonlinear optics : Parametric oscillators and amplifiers
(350.5500) Other areas of optics : Propagation
(350.7420) Other areas of optics : Waves
(190.4223) Nonlinear optics : Nonlinear wave mixing
(190.4975) Nonlinear optics : Parametric processes

ToC Category:
Nonlinear Optics

History
Original Manuscript: November 12, 2012
Revised Manuscript: December 20, 2012
Manuscript Accepted: December 21, 2012
Published: January 15, 2013

Citation
O. Yaakobi, L. Caspani, M. Clerici, F. Vidal, and R. Morandotti, "Complete energy conversion by autoresonant three-wave mixing in nonuniform media," Opt. Express 21, 1623-1632 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-2-1623

Sort: Author | Year | Journal | Reset

References

L. Friedland, “Autoresonance in nonlinear systems,” Scholarpedia4, 5473 (2009). [CrossRef]
V. I. Veksler, “A new method of acceleration of relativistic particles,” J. Phys. USSR9, 153–158 (1945).
E. M. McMillan, “The synchrotron - a proposed high energy particle accelerator,” Phys. Rev.68, 143–144 (1945). [CrossRef]
G. B. Andresen, M. D. Ashkezari, M. Baquero-Ruiz, W. Bertsche, P. D. Bowe, E. Butler, C. L. Cesar, S. Chapman, M. Charlton, A. Deller, S. Eriksson, J. Fajans, T. Friesen, M. C. Fujiwara, D. R. Gill, A. Gutierrez, J. S. Hangst, W. N. Hardy, M. E. Hayden, A. J. Humphries, R. Hydomako, M. J. Jenkins, S. Jonsell, L. V. Jorgensen, L. Kurchaninov, N. Madsen, S. Menary, P. Nolan, K. Olchanski, A. Olin, A. Povilus, P. Pusa, F. Robicheaux, E. Sarid, S. Seif el Nasr, D. M. Silveira, C. So, J. W. Storey, R. I. Thompson, D. P. van der Werf, J. S. Wurtele, and Y. Yamazaki, “Trapped antihydrogen,” Nature468, 673–676 (2010). [CrossRef] [PubMed]
T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron.26, 1265–1276 (1990). [CrossRef]
K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron.30, 1596–1604 (1994). [CrossRef]
G. Imeshev, M. M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B,18, 534–539 (2001). [CrossRef]
H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A78, 063821 (2008). [CrossRef]
H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Optics Express17, 12731–12740 (2009). [CrossRef] [PubMed]
A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett.103, 123901 (2009). [CrossRef] [PubMed]
A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant propagation of incoherent light-waves,” Optics Express, 18, 17709–17718 (2010). [CrossRef] [PubMed]
S. Trendafilov, V. Khudik, M. Tokman, and G. Shvets, “Hamiltonian description of non-reciprocal light propagation in nonlinear chiral fibers,” Physica B405, 3003–3006 (2010). [CrossRef]
S. Richard, “Second-harmonic generation in tapered optical fibers,” J. Opt. Soc. Am. B27, 1504–1512 (2010). [CrossRef]
L. Friedland, “Autoresonant three-wave interactions,” Phys. Rev. Lett.69, 1749–1752 (1992). [CrossRef] [PubMed]
S. Longhi, “Third-harmonic generation in quasi-phase-matched χ(2) media with missing second harmonic,” Optics Letters32, 1791–1793 (2007). [CrossRef] [PubMed]
M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Competing collinear and noncollinear interactions in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B,25, 1402–1413 (2008). [CrossRef]
I. Y. Dodin, G. M. Fraiman, V. M. Malkin, and N. J. Fisch, “Amplification of short laser pulses by Raman backscattering in capillary plasmas,” JETP95, 625–638 (2002). [CrossRef]
O. Yaakobi, L. Friedland, R. R. Lindberg, A. E. Charman, G. Penn, and J. S. Wurtele, “Spatially autoresonant stimulated Raman scattering in nonuniform plasmas,” Phys. Plasmas15, 032105 (2008). [CrossRef]
T. Chapman, S. Huller, P. E. Masson-Laborde, W. Rozmus, and D. Pesme, “Spatially autoresonant stimulated Raman scattering in inhomogeneous plasmas in the kinetic regime,” Phys. Plasmas17, 122317 (2010). [CrossRef]
O. Yaakobi and L. Friedland, “Multidimensional, autoresonant three-wave interactions,” Phys. Plasmas15, 102104 (2008). [CrossRef]
O. Yaakobi and L. Friedland, “Autoresonant four-wave mixing in optical fibers,” Phys. Rev. A82, 023820 (2010). [CrossRef]
W. L. Kruer, The Physics of Laser Plasma Interactions (reprint ed.Westview Press, Boulder, CO, 2001).
A. P. Mayer, “Surface acoustic waves in nonlinear elastic media,” Phys. Rep.256, 237–366 (1995). [CrossRef]
K. Trulsen and C. C. Mei, “Modulation of three resonating gravity-capillary waves by a long gravity wave,” J. Fluid Mech.290, 345–376 (1995). [CrossRef]
R. B. Boyd, Nonlinear Optics (Third Edition, AP2007).

Afeyan, B.
Andresen, G. B.
Arie, A.
Ashkezari, M. D.
Baquero-Ruiz, M.
Barak, A.
Bertsche, W.
Bowe, P. D.
Boyd, R. B.
Butler, E.
Cesar, C. L.
Chapman, S.
Chapman, T.
Charbonneau-Lefort, M.
Charlton, M.
Charman, A. E.
Deller, A.
Dodin, I. Y.
Eriksson, S.
Fajans, J.
Fejer, M. M.
Fisch, N. J.
Fraiman, G. M.
Friedland, L.
Friesen, T.
Fujiwara, M. C.
Galvanauskas, A.
Gill, D. R.
Gutierrez, A.
Hangst, J. S.
Hardy, W. N.
Harter, D.
Hayden, M. E.
Huller, S.
Humphries, A. J.
Hydomako, R.
Imeshev, G.
Jenkins, M. J.
Jonsell, S.
Jorgensen, L. V.
Kato, M.
Khudik, V.
Kruer, W. L.
Kurchaninov, L.
Lamhot, Y.
Lindberg, R. R.
Longhi, S.
Madsen, N.
Malkin, V. M.
Masson-Laborde, P. E.
Mayer, A. P.
McMillan, E. M.
Mei, C. C.
Menary, S.
Mizuuchi, K.
Nishihara, H.
Nolan, P.
Olchanski, K.
Olin, A.
Oron, D.
Penn, G.
Pesme, D.
Povilus, A.
Prabhudesai, V.
Pusa, P.
Richard, S.
Robicheaux, F.
Rozmus, W.
Sarid, E.
Sato, H.
Segev, M.
Seif el Nasr, S.
Shvets, G.
Silberberg, Y.
Silveira, D. M.
So, C.
Storey, J. W.
Suchowski, H.
Suhara, T.
Thompson, R. I.
Tokman, M.
Trendafilov, S.
Trulsen, K.
van der Werf, D. P.
Veksler, V. I.
Wurtele, J. S.
Yaakobi, O.
Yamamoto, K.
Yamazaki, Y.

IEEE J. Quantum Electron.

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron.26, 1265–1276 (1990). [CrossRef]
K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron.30, 1596–1604 (1994). [CrossRef]

J. Fluid Mech.

K. Trulsen and C. C. Mei, “Modulation of three resonating gravity-capillary waves by a long gravity wave,” J. Fluid Mech.290, 345–376 (1995). [CrossRef]

J. Opt. Soc. Am. B

S. Richard, “Second-harmonic generation in tapered optical fibers,” J. Opt. Soc. Am. B27, 1504–1512 (2010). [CrossRef]

J. Opt. Soc. Am. B,

G. Imeshev, M. M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B,18, 534–539 (2001). [CrossRef]
M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Competing collinear and noncollinear interactions in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B,25, 1402–1413 (2008). [CrossRef]

J. Phys. USSR

V. I. Veksler, “A new method of acceleration of relativistic particles,” J. Phys. USSR9, 153–158 (1945).

JETP

I. Y. Dodin, G. M. Fraiman, V. M. Malkin, and N. J. Fisch, “Amplification of short laser pulses by Raman backscattering in capillary plasmas,” JETP95, 625–638 (2002). [CrossRef]

Nature

G. B. Andresen, M. D. Ashkezari, M. Baquero-Ruiz, W. Bertsche, P. D. Bowe, E. Butler, C. L. Cesar, S. Chapman, M. Charlton, A. Deller, S. Eriksson, J. Fajans, T. Friesen, M. C. Fujiwara, D. R. Gill, A. Gutierrez, J. S. Hangst, W. N. Hardy, M. E. Hayden, A. J. Humphries, R. Hydomako, M. J. Jenkins, S. Jonsell, L. V. Jorgensen, L. Kurchaninov, N. Madsen, S. Menary, P. Nolan, K. Olchanski, A. Olin, A. Povilus, P. Pusa, F. Robicheaux, E. Sarid, S. Seif el Nasr, D. M. Silveira, C. So, J. W. Storey, R. I. Thompson, D. P. van der Werf, J. S. Wurtele, and Y. Yamazaki, “Trapped antihydrogen,” Nature468, 673–676 (2010). [CrossRef] [PubMed]

Optics Express

H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Optics Express17, 12731–12740 (2009). [CrossRef] [PubMed]
A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant propagation of incoherent light-waves,” Optics Express, 18, 17709–17718 (2010). [CrossRef] [PubMed]

Optics Letters

S. Longhi, “Third-harmonic generation in quasi-phase-matched χ(2) media with missing second harmonic,” Optics Letters32, 1791–1793 (2007). [CrossRef] [PubMed]

Phys. Plasmas

O. Yaakobi, L. Friedland, R. R. Lindberg, A. E. Charman, G. Penn, and J. S. Wurtele, “Spatially autoresonant stimulated Raman scattering in nonuniform plasmas,” Phys. Plasmas15, 032105 (2008). [CrossRef]
T. Chapman, S. Huller, P. E. Masson-Laborde, W. Rozmus, and D. Pesme, “Spatially autoresonant stimulated Raman scattering in inhomogeneous plasmas in the kinetic regime,” Phys. Plasmas17, 122317 (2010). [CrossRef]
O. Yaakobi and L. Friedland, “Multidimensional, autoresonant three-wave interactions,” Phys. Plasmas15, 102104 (2008). [CrossRef]

Phys. Rep.

A. P. Mayer, “Surface acoustic waves in nonlinear elastic media,” Phys. Rep.256, 237–366 (1995). [CrossRef]

Phys. Rev.

E. M. McMillan, “The synchrotron - a proposed high energy particle accelerator,” Phys. Rev.68, 143–144 (1945). [CrossRef]

Phys. Rev. A

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A78, 063821 (2008). [CrossRef]
O. Yaakobi and L. Friedland, “Autoresonant four-wave mixing in optical fibers,” Phys. Rev. A82, 023820 (2010). [CrossRef]

Phys. Rev. Lett.

A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett.103, 123901 (2009). [CrossRef] [PubMed]
L. Friedland, “Autoresonant three-wave interactions,” Phys. Rev. Lett.69, 1749–1752 (1992). [CrossRef] [PubMed]

Physica B

S. Trendafilov, V. Khudik, M. Tokman, and G. Shvets, “Hamiltonian description of non-reciprocal light propagation in nonlinear chiral fibers,” Physica B405, 3003–3006 (2010). [CrossRef]

Scholarpedia

L. Friedland, “Autoresonance in nonlinear systems,” Scholarpedia4, 5473 (2009). [CrossRef]

Other

W. L. Kruer, The Physics of Laser Plasma Interactions (reprint ed.Westview Press, Boulder, CO, 2001).
R. B. Boyd, Nonlinear Optics (Third Edition, AP2007).

2010, Andresen, Nature
2010, Barak, Optics Express
2010, Trendafilov, Physica B
2010, Richard, J. Opt. Soc. Am. B
2010, Chapman, Phys. Plasmas
2010, Yaakobi, Phys. Rev. A
2009, Friedland, Scholarpedia
2009, Suchowski, Optics Express
2009, Barak, Phys. Rev. Lett.
2008, Suchowski, Phys. Rev. A
2008, Charbonneau-Lefort, J. Opt. Soc. Am. B,
2008, Yaakobi, Phys. Plasmas
2008, Yaakobi, Phys. Plasmas
2007, Longhi, Optics Letters
2002, Dodin, JETP
2001, Imeshev, J. Opt. Soc. Am. B,
1995, Mayer, Phys. Rep.
1995, Trulsen, J. Fluid Mech.
1994, Mizuuchi, IEEE J. Quantum Electron.
1992, Friedland, Phys. Rev. Lett.
1990, Suhara, IEEE J. Quantum Electron.
1945, Veksler, J. Phys. USSR
1945, McMillan, Phys. Rev.

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Click here to see a list of articles that cite this paper

Figures


Fig. 1	Fig. 2	Fig. 3

« Previous Article | Next Article »

OSA is a member of CrossRef.

Select a Conference
Session or Paper #
Year

OpticsInfoBase

Optics Express

Complete energy conversion by autoresonant three-wave mixing in nonuniform media

Article Tools

Article Navigation

Abstract

1. Introduction

2. Mathematical model

3. Initial phase-locking and autoresonant state

3.1. Closed-form solutions

3.2. Stability analysis

4. Conclusions

Appendices

5. Appendix - optical parametric amplification

Acknowledgments

References and links

References

IEEE J. Quantum Electron.

J. Fluid Mech.

J. Opt. Soc. Am. B

J. Opt. Soc. Am. B,

J. Phys. USSR

JETP

Nature

Optics Express

Optics Letters

Phys. Plasmas

Phys. Rep.

Phys. Rev.

Phys. Rev. A

Phys. Rev. Lett.

Physica B

Scholarpedia

Other

Cited By

Figures