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

Optics Express, Vol. 21, Issue 2, pp. 1623-1632 (2013)

http://dx.doi.org/10.1364/OE.21.001623

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### 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.

© 2013 OSA

## 1. Introduction

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

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

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

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]

*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

*A*describing the interaction between three waves at frequencies that satisfy the matching condition

_{j}*ω*

_{1}+

*ω*

_{2}=

*ω*

_{3}[25]:

*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*−

_{j}z*ω*]), where

_{j}t*k*is the

_{j}*j*-th wave wavevector. The coupling coefficients

*η*are defined by

_{j}*c*is the speed of light in vacuum and

*d*is the effective nonlinear susceptibility. We assume slowly varying wavevectors

_{eff}*k*(

_{j}*z*) for each of the interacting waves and define the wavevectors mismatch Δ

*k*(

*z*) =

*k*

_{1}+

*k*

_{2}−

*k*

_{3}. For simplicity, in our theory we take into account the medium nonuniformity only from the presence of the

*z*dependence of Δ

*k*(

*z*) in the wavevectors mismatch, i.e.

*k*in the coupling coefficients in Eqs. (1)–(2) are replaced by an average value

_{j}*k*along the medium for each

_{j,avg}*j*. We now define the dimensionless coordinate

*ζ*=

*z/l*and dimensionless complex amplitudes

*l*= 1/[(

*η*

_{1}

*η*

_{2})

^{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: Next, we represent each one of the complex amplitudes

*a*by its absolute value

_{j}*B*and real phase

_{j}*ϕ*using the definitions

_{j}*a*

_{1,2}=

*B*

_{1,2}exp (

*iϕ*

_{1,2}) and

*ϕ*

_{1}+

*ϕ*

_{2}−

*ϕ*

_{3}, the sign coefficients

*ε*

_{1,2}= +1,

*ε*

_{3}= −1 and

*B*

_{1}:

*α*as the nonuniformity rate coefficient such that Δ

*k*=

*α*(

*z*−

*z*)/

_{*}*l*

^{2}=

*α*(

*ζ*−

*ζ*)/

_{*}*l*where the subscript

***denotes the point of perfect wavevector matching Δ

*k*= 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

*B*

_{1}and Φ in which the variables

*B*

_{2}and

*B*

_{3}are expressed by Eqs. (8).

*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 Φ

_{0}= 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.

## 3. Initial phase-locking and autoresonant state

### 3.1. Closed-form solutions

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

*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*. Further, we assume that the initial signal is small (

_{in}*B*

_{2,0}<<

*B*

_{3,0}), and that

*B*

_{1}(

*z*) <<

*B*

_{2}in the initial stage, since the idler is initially zero. Consequently, we neglect the variation of

*B*

_{2}and the depletion of the pump wave at this stage and focus on the evolution of

*B*

_{1}. Then,

*Q*≈

*B*

_{2,0}

*B*

_{3,0}/

*B*

_{1}=

*B*

_{2,0}/

*B*

_{1}, since according to our definition of variables

*B*

_{3,0}≡ 1. Therefore, Eqs. (9)–(10) can be approximated by Note that Eq. (11) has the same form as Eq. (11) in Ref. [21

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

*α*|

*ξ*guarantees phase-locking in which Φ ≈

_{in}*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

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

*B*

_{1}and

*B*

_{2}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.

*B*

_{1}≈

*B*

_{2}. 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.

*α*|

*ξ*> −2 we assume |cosΦ| ≈ 1, and set

*B*

_{1}/

*B*

_{2}≈ 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

*Q*̂ in Eq. (14) results in a quadratic equation for

*B*̂

_{3}. The autoresonant solution for the pump is chosen as the root of the quadratic equation for which

*B*̂

_{3}> 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

**82**, 023820 (2010). [CrossRef]

*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.

*Q*terms in the limit of vanishing pump power

*Q*→ −1/

*B*

_{3}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

*B*

_{3}(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

*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

*B*

_{1}=

*B*̂

_{1}+

*δB*

_{1}and Φ = Φ̂ +

*δ*Φ where

*δB*

_{1},

*δ*Φ << 1. In the following we will use the approximate trigonometric relations sinΦ =

*s*sin

*δ*Φ, cosΦ ≈

*s*(derived from sinΦ̂ = 0 and cos

*δ*Φ ≈ 1). Next, we replace cosΦ by

*s*in Eq. (10) and expand the equation to first order in

*δB*

_{1}, yielding where

*Q*̂′ denotes the derivative of

*Q*with respect to

*B*

_{1}evaluated at

*B*̂

_{1}. Phase-locking is characterized by

*d*Φ̂/

*dξ*=

*s*(|

*α*|

*ξ*+

*Q*̂) ≈ 0, so we define

*B*̂

_{1}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*(

*δB*

_{1})/

*dξ*=

*dB*

_{1}/

*dξ*−

*dB*̂

_{1}/

*dξ*, which, by using Eq. (9) to the lowest order in

*δB*

_{1},

*δ*Φ and

*dB*̂

_{1}/

*dξ*= −|

*α*|/

*Q̂*′ obtained by differentiation of Eq. (19), yields where

*D*=

*B*̂

_{2}

*B*̂

_{3}(we use here the relation sin Φ =

*s*sin

*δ*Φ). It can be observed that Eqs. (20) and (21) are Hamilton equations for the canonical variables

*δB*

_{1}and

*δ*Φ governed by the Hamiltonian where

*V*=

_{eff}*D*cos

*δ*Φ −

*s*|

*α*|

*δ*Φ/

*Q*̂′ and

*ξ*plays the role of ”time”. Note that

*V*is a tilted ”washboard” potential with slow ”time”-dependent parameters

_{eff}*D*and

*Q*̂. Thus, our quasi-steady states are stable, as long as the effective potential has well defined minima (since cos

*δ*Φ ≈ 1 − (

*δ*Φ)

^{2}/2), i.e. where Ω

^{2}≡ −

*DQ*̂′. Eq. (23) expresses the required stability criterion for phase-locking in our system. We show the dependence of Ω

^{2}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 Ω

^{2}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.

## 4. Conclusions

*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

*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

*ω*/

_{j}*k*=

_{j}*c/n*and

_{j}*c*/

*ω*=

_{j}*λ*/2

_{j}*π*. Here

*n*and

_{j}*λ*denote the material refractive index and the vacuum wavelength at frequency

_{j}*j*,

*I*

_{3,0}= 2

*n*

_{3,0}

*cε*

_{0}|

*A*

_{3,0}|

^{2}is the initial pump intensity,

*ε*

_{0}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 Δ

*k*and Δ

_{ini}*k*are the wavevector mismatch at the initial and the final boundaries of the

_{fin}*χ*

^{(2)}medium, separated by a distance

*L*.

*α*|

*ξ*= −2 (which is required to guarantee initial phase-locking) and |

*α*|

*ξ*= 2 (where according to Eq. (17)

*α*|

*ξ*< 2 (i.e. |

*α*|

*ξ*< −2 and |

_{ini}*α*|

*ξ*> 2) will allow efficient autoresonant amplification. Therefore, we see from Eq. (28) that it is required that

_{fin}*sign*(Δ

*k*) ≠

_{fin}*sign*(Δ

*k*) and

_{ini}^{2}that depends on the initial signal amplitude

*B*

_{2,0}. 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

*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}(

*ω*

_{2}/

*ω*

_{3})(

*B*

_{2,0}/

*B*

_{3,0})

^{2}≈ 5 · 10

^{−5}for

*ω*

_{2}/

*ω*

_{3}≈ 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 Ω

^{2}is

*k*(

*z*) ≡

*k*

_{1}+

*k*

_{2}−

*k*

_{3}= (

*n*

_{1}

*ω*

_{1}+

*n*

_{2}

*ω*

_{2}+

*n*

_{3}

*ω*

_{3}) /

*c*depends on the material dispersion relation

*n*(

_{j}*λ*,

_{j}*z*), which is a function of two independent variables

*λ*and

_{j}*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.

## Acknowledgments

## References and links

1. | L. Friedland, “Autoresonance in nonlinear systems,” Scholarpedia |

2. | V. I. Veksler, “A new method of acceleration of relativistic particles,” J. Phys. USSR |

3. | E. M. McMillan, “The synchrotron - a proposed high energy particle accelerator,” Phys. Rev. |

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 |

5. | T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. |

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. |

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, |

8. | H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A |

9. | H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Optics Express |

10. | A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. |

11. | A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant propagation of incoherent light-waves,” Optics Express , |

12. | S. Trendafilov, V. Khudik, M. Tokman, and G. Shvets, “Hamiltonian description of non-reciprocal light propagation in nonlinear chiral fibers,” Physica B |

13. | S. Richard, “Second-harmonic generation in tapered optical fibers,” J. Opt. Soc. Am. B |

14. | L. Friedland, “Autoresonant three-wave interactions,” Phys. Rev. Lett. |

15. | S. Longhi, “Third-harmonic generation in quasi-phase-matched |

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, |

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 |

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 |

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 |

20. | O. Yaakobi and L. Friedland, “Multidimensional, autoresonant three-wave interactions,” Phys. Plasmas |

21. | O. Yaakobi and L. Friedland, “Autoresonant four-wave mixing in optical fibers,” Phys. Rev. A |

22. | W. L. Kruer, |

23. | A. P. Mayer, “Surface acoustic waves in nonlinear elastic media,” Phys. Rep. |

24. | K. Trulsen and C. C. Mei, “Modulation of three resonating gravity-capillary waves by a long gravity wave,” J. Fluid Mech. |

25. | R. B. Boyd, |

**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

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### 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]
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