## Limits to compression with cascaded quadratic soliton compressors

Optics Express, Vol. 16, Issue 5, pp. 3273-3287 (2008)

http://dx.doi.org/10.1364/OE.16.003273

Acrobat PDF (1226 KB)

### Abstract

We study cascaded quadratic soliton compressors and address the physical mechanisms that limit the compression. A nonlocal model is derived, and the nonlocal response is shown to have an additional oscillatory component in the nonstationary regime when the group-velocity mismatch (GVM) is strong. This inhibits efficient compression. Raman-like perturbations from the cascaded nonlinearity, competing cubic nonlinearities, higher-order dispersion, and soliton energy may also limit compression, and through realistic numerical simulations we point out when each factor becomes important. We find that it is theoretically possible to reach the single-cycle regime by compressing high-energy fs pulses for wavelengths λ=1.0-1.3 *µ*m in a *β*-barium-borate crystal, and it requires that the system is in the stationary regime, where the phase mismatch is large enough to overcome the detrimental GVM effects. However, the simulations show that reaching single-cycle duration is ultimately inhibited by competing cubic nonlinearities as well as dispersive waves, that only show up when taking higher-order dispersion into account.

© 2008 Optical Society of America

## 1. Introduction

2. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. **45**, 1095–1098 (1980). [CrossRef]

3. S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B **19**, 2505–2510 (2002). [CrossRef]

12. R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. **17**, 28–30 (1992). [CrossRef] [PubMed]

3. S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B **19**, 2505–2510 (2002). [CrossRef]

6. J. Moses and F. W. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett. **31**, 1881–1883 (2006). [CrossRef] [PubMed]

11. M. Bache, J. Moses, and F. W. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B **24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

14. X. Liu, L. Qian, and F. W. Wise, “High-energy pulse compression by use of negative phase shifts produced by the cascaded χ^{(2)} : χ^{(2)} nonlinearity,” Opt. Lett. **24**, 1777–1779 (1999). [CrossRef]

16. L. Bergé, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, “Self-focusing and solitonlike structures in materials with competing quadratic and cubic nonlinearities,” Phys. Rev. E **55**, 3555–3570 (1997). [CrossRef]

14. X. Liu, L. Qian, and F. W. Wise, “High-energy pulse compression by use of negative phase shifts produced by the cascaded χ^{(2)} : χ^{(2)} nonlinearity,” Opt. Lett. **24**, 1777–1779 (1999). [CrossRef]

14. X. Liu, L. Qian, and F. W. Wise, “High-energy pulse compression by use of negative phase shifts produced by the cascaded χ^{(2)} : χ^{(2)} nonlinearity,” Opt. Lett. **24**, 1777–1779 (1999). [CrossRef]

3. S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B **19**, 2505–2510 (2002). [CrossRef]

7. J. Moses, E. Alhammali, J. M. Eichenholz, and F. W. Wise, “Efficient high-energy femtosecond pulse compression in quadratic media with flattop beams,” Opt. Lett. **32**, 2469–2471 (2007). [CrossRef] [PubMed]

6. J. Moses and F. W. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett. **31**, 1881–1883 (2006). [CrossRef] [PubMed]

4. S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett. **84**, 1055–1057 (2004). [CrossRef]

8. X. Zeng, S. Ashihara, N. Fujioka, T. Shimura, and K. Kuroda, “Adiabatic compression of quadratic temporal solitons in aperiodic quasi-phase-matching gratings,” Opt. Express **14**, 9358–9370 (2006). [CrossRef] [PubMed]

6. J. Moses and F. W. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett. **31**, 1881–1883 (2006). [CrossRef] [PubMed]

^{(2)} : χ^{(2)} nonlinearity,” Opt. Lett. **24**, 1777–1779 (1999). [CrossRef]

*stationary regime*clean compression is possible, while in the

*nonstationary regime*GVM distorts the compressed pulse too much to be of any practical use, and severe reductions in compression capabilities is observed. As an example of this, the numerical simulations in Fig. 1 compare the pulse compression performance under equal conditions in the two regimes. In the stationary regime a 6 fs compressed pulse is observed while the nonstationary regime the GVM effects are much stronger, resulting in a 17 fs compressed pulse with trailing oscillations.

**31**, 1881–1883 (2006). [CrossRef] [PubMed]

17. F. Ö. Ilday, K. Beckwitt, Y.-F. Chen, H. Lim, and F. W. Wise, “Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes,” J. Opt. Soc. Am. B **21**, 376–383 (2004). [CrossRef]

10. M. Bache, O. Bang, J. Moses, and F.W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. **32**, 2490–2492 (2007), arXiv:0706.1933. [CrossRef] [PubMed]

18. N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E **68**, 036614 (2003). [CrossRef]

19. W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B: Quantum Semiclass. Opt . **6**, s288 (2004). [CrossRef]

*N*

^{2}

_{eff}=

*N*

^{2}

_{SHG}-

*N*

^{2}

_{Kerr}, appearing as the difference between the SHG and the Kerr soliton numbers [11

11. M. Bache, J. Moses, and F. W. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B **24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

*N*

_{eff}depends only on input parameters the compressed pulse properties can be predicted using these scaling laws. Appropriate input parameters can then be found giving compression to single-cycle duration. However, neither the experiments nor the numerical simulations have ever observed single-cycle compression. Moreover, optimal compression seemed to occur at certain phase-mismatch values, which the analysis of [11

11. M. Bache, J. Moses, and F. W. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B **24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

## 2. Propagation equations

*ω*

_{1}) and SH (

*ω*

_{2}=2

*ω*

_{1}) fields

*U*

_{1,2}(

*ξ*, τ) are [11

**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

20. J. Moses and F. W. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. **97**, 073903 (2006), see also arXiv:physics/0604170. [CrossRef] [PubMed]

*δ*

^{(m)}

_{j}≡

*k*

^{(m)}

_{j}(

*T*

^{m-2}_{in}|

*k*

^{(2)}

_{1}|

*m*!)

^{-1}and

*k*

^{(m)}

_{j}≡∂

*/∂*

^{m}k_{j}*ω*|

^{m}*ω*=

*ω*. Since

_{j}*k*=

_{j}*n*/

_{j}ω_{j}*c*is known analytically through the Sellmeier equations of [21], the exact dispersion

*D̂*=

_{j}*k*(

_{j}*ω*)-(

*ω*-

*ω*)

_{j}*k*

^{(1)}

_{1}-

*k*(

_{j}*ω*) is used in the numerics [11

_{j}**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

*m*=∞.

_{d}*n*is the refractive index,

_{j}*n̅*=

*n*

_{1}/

*n*

_{2}, and the phase mismatch of the SHG process is Δ

*k*=

*k*

_{2}-2

*k*

_{1}. The Kerr cross-phase modulation (XPM) term

*B*=2 for type 0 SHG while for type I SHG

*B*=2/3 [11

**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

*ν*

_{g,1}=1/

*k*

^{(1)}

_{1}, giving the GVM term

*d*

_{12}=

*ν*

^{-1}

_{g,1}-

*ν*

^{-1}

_{g,2}. Equations (1) are reported in dimensionless form, τ=

*t*/

*T*

_{in}, where

*T*

_{in}is the FW input pulse duration,

*ξ*=

*z*/

*L*

_{D,1}, where

*L*

_{D,1}=

*T*

^{2}

_{in}/|

*k*

^{(2)}

_{1}| is the FW dispersion length, and finally

*U*

_{1}=

*E*

_{1}/ℰ

_{in}and

*U*

_{2}=

*E*

_{2}/

*√n̅ℰ*

_{in}. Here

*ℰ*

_{in}is the amplitude of the peak electric input field,

*d*′

_{12}=

*d*

_{12}

*T*

_{in}/|

*k*

^{(2)}

_{1}|, and Δ

*k*′=Δ

*kL*

_{D,1}. This scaling gives the quadratic (SHG) and cubic (Kerr) soliton numbers [6

**31**, 1881–1883 (2006). [CrossRef] [PubMed]

**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

*d*

_{eff}is the effective quadratic nonlinearity, and

*n*

_{Kerr, j}=3Re(χ

^{(3)})/8

*n*is the cubic (Kerr) nonlinear refractive index.

_{j}*N*

_{SHG}might seem poorly defined in Eqs. (1) because of the factor |Δ

*k*′|

^{1/2}in front of it, but the choice will become clear later. Self-steepening is included through the operators

*cd*

^{2}

_{12}/

*ω*

_{2}

*n*

_{2}|

*k*

^{(2)}

_{1}| [11

**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

20. J. Moses and F. W. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. **97**, 073903 (2006), see also arXiv:physics/0604170. [CrossRef] [PubMed]

*effective*SH dispersion operator, whose existence is a consequence of self-steepening and GVM. This is the price to pay in the SEWA framework to reach single-cycle resolution when diffraction is neglected. We stress that all primed symbols in our notation are the dimensionless form of the corresponding unprimed symbol.

## 3. Nonlocal model: reduced equation in the cascading limit

10. M. Bache, O. Bang, J. Moses, and F.W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. **32**, 2490–2492 (2007), arXiv:0706.1933. [CrossRef] [PubMed]

*k*′|≫1 Eqs. (1) can be reduced to a single equation for the FW

*Ŝ*=1,

_{j}*m*=2), but as shown later this can straightforwardly be relaxed. The phase-mismatch parameter must be positive, Δ

_{d}*k*′>0, as to have a self-defocusing cascaded nonlinearity. Solitons are then supported when the group-velocity dispersion (GVD) is normal, so

*k*

^{(2)}

_{j}>0 was assumed. Additionally the Kerr XPM terms were neglected, otherwise the nonlocal approach fails. This dimensionless generalized nonlinear Schrödinger equation (NLSE) shows that the cascaded quadratic nonlinearity imposes a temporal nonlocal response on the FW, governed by the nonlocal response functions

*R*

_{±}[which will be derived below, see also Fig. 2(a,c)]. This model quantified the previous qualitative definitions [14

^{(2)} : χ^{(2)} nonlinearity,” Opt. Lett. **24**, 1777–1779 (1999). [CrossRef]

17. F. Ö. Ilday, K. Beckwitt, Y.-F. Chen, H. Lim, and F. W. Wise, “Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes,” J. Opt. Soc. Am. B **21**, 376–383 (2004). [CrossRef]

*d*

^{2}

_{12}>2Δ

*kk*

^{(2)}

_{2}the system is in the nonstationary regime and the oscillatory response function

*R*

_{-}must be used. Using characteristic length scales this demand reads

*L*

^{2}

_{GVM}<

*L*

_{coh}

*L*

_{D,2}/2

*π*, which can be interpreted as follows. In the nonstationary regime the GVM dominates, and its length scale

*L*

_{GVM}=

*T*

_{in}/|

*d*

_{12}| becomes shorter than one controlled by the product of the coherence length

*L*

_{coh}=

*π*/|Δ

*k*| and the SH GVD length scale

*L*

_{D,2}=

*T*

^{2}

_{in}/|

*k*

^{(2)}

_{2}|. When

*d*

^{2}

_{12}<2Δ

*kk*

^{(2)}

_{2}the system is in the stationary regime and the localized response function

*R*

_{+}must be used. We will now derive Eq. (3) in details.

*k*′≫1 the nonlocal approach takes the ansatz

*L*

_{coh}=

*π*/Δ

*k*is much shorter than any other characteristic length scale. This is true in the cascading limit except when the FW is extremely short, in which case the GVM length

*L*

_{GVM}=

*T*

_{in}/|

*d*

_{12}| can become on the order of

*L*

_{coh}. Assuming

*N*

^{2}

_{Kerr}

*U*

_{2}Δ

*k*′

^{1/2}

*N*

_{SHG}we may discard the Kerr terms in Eq. (1b), and get an ordinary differential equation (ODE)

_{2}(Ω)=

*ℱ*[ϕ

_{2}](Ω)≡(2

*π*)

^{-1/2}

*∫*

^{∞}

_{-∞}dτ

*e*

^{iΩτ}ϕ

_{2}(τ) and ϕ

_{2}(τ)=ℱ

^{-1}[

_{2}](τ)≡(2

*π*)

^{-1/2}∫

^{∞}

_{-∞}dΩ

*e*

^{-iΩτ}

_{2}(Ω) we may solve the ODE (5) in the frequency domain

*R*(τ)=

*ℱ*

^{-1}[

*R̃*(Ω)]. Now, using Eq. (7) with the ansatz (4) in Eq. (1a) we arrive at Eq. (3) under the aforementioned approximations.

*̃R*as

*s*=+1 corresponds to the stationary regime. In this case Eq. (8) becomes a Lorentzian centered in Ω′

_{b}_{a}and with the FWHM 2Ω′

_{b}, see Fig. 2(b). The roots in the denominator of Eq. (8) are complex Ω=Ω′

_{a}±

*i*Ω′

_{b}. The dimensionless [22] temporal response function,

*R*

_{+}(τ), can readily be calculated by taking the inverse Fourier transform

*controls the width of |*

_{b}*R*

_{+}| while τ

*is the period of the phase oscillations. Note that Eq. (6) is defined so ∫*

_{a}^{∞}

_{-∞}dτ

*R*

_{+}(τ)=1.

*s*=-1, and

_{b}*R̃*(Ω) has two simple poles at Ω=Ω′

_{a}±Ω′

_{b}≡Ω′

_{±}, making

*R̃*(Ω) diverge [see Fig. 2(d)].

*R*

_{-}(τ)=

*ℱ*

^{-1}[

*R̃*

_{-}(Ω)] exists as a Cauchy principal value

*R*

_{+}, this response function is not localized, and the oscillations are a consequence of the two poles in

*R̃*

_{-}(Ω) [see the example shown in Fig. 2(c)].

17. F. Ö. Ilday, K. Beckwitt, Y.-F. Chen, H. Lim, and F. W. Wise, “Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes,” J. Opt. Soc. Am. B **21**, 376–383 (2004). [CrossRef]

**31**, 1881–1883 (2006). [CrossRef] [PubMed]

*s*changes sign. On dimensional form this happens when Δ

_{b}*k*=Δ

*k*

_{sr}, with

*k*>Δ

*k*

_{sr}. When GVM is weak compared to the phase mismatch then

*s*=+1, and the response function (

_{b}*R*

_{+}) is monotonously decaying in magnitude: the convolution in the Kerr-like SPM term in Eq. (3) provides a finite temporal response. Therefore this must correspond to the stationary regime. Instead when GVM is strong compared to the phase mismatch then

*s*=-1, and the response function (

_{b}*R*

_{-}) is oscillating and non-decaying: the temporal response from the convolution is no longer finite. Thus, this must correspond to the nonstationary regime.

## 4. The weakly nonlocal limit

*U*

^{2}

_{1}. The resulting simplified equation gives a better physical insight [23

23. W. Krolikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E **63**, 016610 (2000). [CrossRef]

*U*

_{1}so short that the nonlocal response is no longer weak). We evaluate the convolution in the frequency domain ∫

^{∞}

_{-∞}ds

*R*(

*s*)

*U*

^{2}

_{1}(

*ξ*, τ-

*s*)=∫

^{∞}

_{-∞}dΩ

*e*

^{-iΩτ}

*R̃*(Ω)

*ℱ*[

*U*

^{2}

_{1}](Ω) for convenience. In the weakly nonlocal limit

*R̃*(Ω) is approximated by a 1st order expansion around Ω=0, where

*ℱ*[

*U*

^{2}

_{1}](Ω) is non-vanishing. This holds when

*R̃*(Ω) varies slowly compared to

*ℱ*[

*U*

^{2}

_{1}](Ω). In this case

*R*(

*s*)

*U*

^{2}

_{1}(

*ξ*, τ-

*s*)≃

*R*(

*s*)[

*U*

^{2}

_{1}(

*ξ,*τ)-

*s*∂

*U*

^{2}

_{1}(

*ξ,*τ)/∂τ. However, in the nonstationary regime the frequency integral ∫

^{∞}

_{-∞}dΩ

*e*

^{-iΩτ}

*R̃*(Ω)

*ℱ*[

*U*

^{2}

_{1}](Ω) is done over two simple poles located on the Ω-axis. Using the residue theorem the integral can be evaluated as a contour integral, which has a contribution from the Cauchy principal value of the integral, and a contribution from deforming the integration contour around the poles on the real Ω-axis. The residual contributions from the poles to the frequency integral are [24]

_{±}each weighted by the spectral strength of

*U*

^{2}

_{1}at that frequency. Thus, the influence of this contribution becomes important when the FW is short enough for its spectrum to cover the range where Ω′

_{±}are located, cf. Fig. 2(d). Using Eq. (15) and (2

*π*)

^{1/2}

*R̃*(0)=1 and (2

*π*)

^{1/2}d

*R̃*/dΩ|

_{Ω=0}=2Ω′

_{a}/(Ω′

^{2}

_{a}+

*s*Ω′

_{b}^{2}

_{b}), the nonlocal convolution is

*N*

^{2}

_{eff}=

*N*

^{2}

_{SHG}-

*N*

^{2}

_{Kerr}[11

**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

*R*

_{±}[10

10. M. Bache, O. Bang, J. Moses, and F.W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. **32**, 2490–2492 (2007), arXiv:0706.1933. [CrossRef] [PubMed]

*is*τ/τ

_{a}*) in Eqs. (10) and (12). It has the characteristic dimensionless time τ*

_{a}_{R,SHG}≡4|Ω′

_{a}|/(Ω′

^{2}

_{a}+

*s*Ω′

_{b}^{2}

_{b})=2|

*d*′

_{12}|/Δ

*k*′ [6

**31**, 1881–1883 (2006). [CrossRef] [PubMed]

**32**, 2490–2492 (2007), arXiv:0706.1933. [CrossRef] [PubMed]

**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

**21**, 376–383 (2004). [CrossRef]

*d*

_{12}implies that the Raman-like perturbation disappears in absence of GVM. Ref. [10

**32**, 2490–2492 (2007), arXiv:0706.1933. [CrossRef] [PubMed]

*s*=+1) [25], but the contribution

_{b}*U*

^{*}

_{1}

*ρ*(τ,

*U*

_{1}) in the nonstationary regime (

*s*=-1) is a new result.

_{b}*N*

_{eff}can be used in the scaling laws of [11

**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

**32**, 2490–2492 (2007), arXiv:0706.1933. [CrossRef] [PubMed]

**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

**31**, 1881–1883 (2006). [CrossRef] [PubMed]

*k*, the Raman-like effect of the first term of the RHS of Eq. (17) becomes increasingly significant with increasing

*N*

^{2}

_{SHG}, thus limiting the possible compression ratio. However, it is now clear that in the nonstationary regime, the Raman-like distortion is accompanied by an oscillatory perturbation term

*U*

^{*}

_{1}

*ρ*(

*t*,

*U*

_{1}) which also increases with

*N*

^{2}

_{SHG}. In both the stationary and nonstationary regimes the Raman-like distortions place a limitation on the maximum soliton order, but in the nonstationary regime both terms on the RHS of Eq. (17) distort the compression, and the combined effect is more severe (see also Sec. 5). On the other hand, when NSHG is small these detrimental effects are weak: thus, as previously observed by both numerical simulations and experiments [3

**19**, 2505–2510 (2002). [CrossRef]

*t*≪Δ

_{b}*t*, where Δ

*t*is the FW pulse duration and

*t*=τ

_{b}

_{b}*T*

_{in}. But when does it apply in the nonstationary regime? We know that the width ΔΩ of

*ℱ*[

*U*

^{2}

_{1}] is ΔΩ∝Δ

*t*

^{-1}. Referring to Fig. 2(d) we must require that the positions of the two poles Ω

_{±}be sufficiently far away from the frequency range, where

*ℱ*[

*U*

^{2}

_{1}] is nonvanishing, i.e., |Ω

_{±}|≫ΔΩ. In physical units this implies that the weakly nonlocal limit in the nonstationary regime can be expressed by the requirement Δ

*t*≫

*t*/|

_{a}t_{b}*t*-

_{a}*t*|.

_{b}*t*diverges at the transition Δ

_{b}*k*

_{sr}, see also Fig. 3(a). Thus, in the nonstationary regime the requirement Δ

*t*≫

*t*/|

_{a}t_{b}*t*-

_{a}*t*| holds even for quite short pulses as long as

_{b}*t*and

_{a}*t*are not too similar. This is generally true close to the transition Δ

_{b}*k*

_{sr}, while away from the transition

*t*≃

_{b}*t*because Δ

_{a}*k*gets small, see Eq. (11) and Fig. 3(a). In this case we can no longer be sure to be in the weakly nonlocal limit. In the stationary regime the system will initially be in the weakly nonlocal limit Δ

*t*≫

*t*except close to the transition Δ

_{b}*k*

_{sr}, where

*t*diverges.We finally remark that

_{b}*t*also may diverge when GVM is negligible. This implies that the factor

_{a}*R*(τ) becomes real and symmetric. Thus, the 1st order correction on the RHS of Eq. (17

**21**, 376–383 (2004). [CrossRef]

*T*

_{R,SHG}=0) and a 2nd order correction must be made.

*Ŝ*′

_{1}would act on all nonlinear terms and Eq. (5) would have

*Ŝ*′

_{2}acting on the RHS. In frequency domain this would imply a self-steepening–corrected response

*R̃*

^{ss}(Ω)≡

*R̃*(Ω)[1+(

*ω*

_{2}

*T*

_{in})

^{-1}Ω]. This does not change Δ

*k*

_{sr}, and would only affect the nonlocal behavior for extremely short input pulses. Lastly, the NLS-like nonlocal Eq. (17) will have the operator

*Ŝ*′

_{1}acting on all nonlinear terms. It should also be stressed that self-steepening can affect the Raman-like term in Eq. (17) [20

20. J. Moses and F. W. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. **97**, 073903 (2006), see also arXiv:physics/0604170. [CrossRef] [PubMed]

*i.e.*, making a more elaborate ansatz than Eq. (4)].

## 5. Numerical results and discussion

*initially*be well described by the nonlocal theory, but as the pulse is compressed the GVM (and other length scales) can become so short that this is no longer true. Therefore the nonlocal model will not always quantitatively be able to predict the outcome of the numerical simulations and the experiments. However, since the nonlocal model often will be an adequate approximation for a large part of the propagation through the nonlinear medium, we can still use it to understand the physics behind the temporal dynamics until that happens.

- Fig. 3.Data from numerical simulations of the full SEWA Eqs. (1) using the same parameters as in Fig. 1 and varying Δk. (a) The FW duration Δt opt=Δt FWHM opt/1.76 at z=z opt is shown both for the full SEWA model (1), and when neglecting the Kerr XPM terms. The lines show the nonlocal time scales ta,b=T inτa,b, the characteristic Raman-like time T R,SHG=2|d 12|/Δk, and the predicted Δt opt from the scaling laws [11] as well as the predicted Δt corr opt when correcting for XPM effects on N eff. tb as calculated using only up to second-order dispersion (md=2, gray curve) is also shown. The right ordinate shows time normalized to the single-cycle pulse duration t sc=2.0 fs. Note that below Δk=10 mm-1 the cascading limit breaks down [12]. (b) The SHG and Kerr soliton numbers required to have N eff=8 fixed, achieved by adjusting I in. The corrected effective soliton number due to XPM effects N corr eff is also shown.
- In the nonstationary regime Δ
*k*<Δ*k*_{sr}the oscillatory nonlocal response function implies that compression is inefficient unless the soliton order is very low. - Competing cubic nonlinearities pose an upper limit Δ
*k*[10_{c}**32**, 2490–2492 (2007), arXiv:0706.1933. [CrossRef] [PubMed]**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]*N*_{eff}<1 always. Close to this limit detrimental cubic XPM effects are observed.

- Effects limiting the compression for a given phase-mismatch value.
- The effective soliton order
*N*_{eff}=(*N*^{2}_{SHG}-*N*^{2}_{Kerr})^{1/2}controls in the weakly nonlocal limit the compression factor*fC*=*T*_{in}/Δ*t*_{opt}=4.7(*N*_{eff}-0.86) [11**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef] - Nonlocal effects. In the stationary regime Δ
*t*_{opt}is limited by the strength of the nonlocal response function*t*. In the nonstationary regime Δ_{b}*t*_{opt}is limited by the characteristic time*T*_{R,SHG}of the GVM-induced Raman-like perturbation. - Propagation effects pertaining solely to the FW, such as higher-order dispersion, the Raman effect (negligible in nonlinear crystals) and cubic self-steepening.
- Competing cubic nonlinearities necessitate large quadratic soliton orders
*N*_{SHG}, which increases detrimental nonlocal effects such as the Raman-like perturbation.

*β*-barium-borate crystal (BBO) as the quadratic nonlinear medium. The phase mismatch was changed through angle-tuning of the crystal in a type I SHG configuration (implying

*B*=2/3, see [11

**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

*k*>0, for which GVD is normal and

*d*

_{12}<0 (so

*s*<0).

_{a}_{1}=1064 nm: the FW pulse duration at the point of optimal compression Δ

*t*

_{opt}(dark circles) is plotted as the phase mismatch Δ

*k*is sweeped. The strength of the cascaded quadratic nonlinearity

*N*

^{2}

_{SHG}∝Δ

*k*

^{-1}, while the Kerr nonlinearity remains unchanged. However, in the plot we keep

*N*

_{eff}=8 fixed by adjusting the input intensity

**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

*k*=50 mm

^{-1}[this example is shown in Fig. 1(a)-(b), and also in Fig. 4], away from this point the compression becomes sub-optimal.

*m*=2), while in the numerical simulations the dispersion is calculated exactly from the Sellmeier equations [11

_{d}**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

*R̃*in Eq. (6) with

*D̂*

_{2,eff}(evaluated in frequency domain). The transition to the nonstationary regime is when a root-pair switches from being each others complex conjugate to being purely real and nondegenerate. Then Ω

*a,b*and

*t*can be extracted from these roots. The transition to the stationary regime (13) is now simply found when

_{a,b}*t*diverges. Fig. 3 shows for comparison also

_{b}*t*calculated with

_{b}*m*=2,

_{d}*i.e.*, using Eq. (11).

*k*in Fig. 3(a) is caused by the onset of XPM-effects. As Δ

*k*is increased the cascaded quadratic nonlinearity is reduced, so in order to keep

*N*=8 the input intensity

_{eff}*I*

_{in}must be increased, see Fig. 3(b). Eventually the required intensity diverges because Δ

*k*approaches the so-called upper limit of the compression window Δ

*k*, beyond which

_{c}*N*

_{eff}<1 always [10

**32**, 2490–2492 (2007), arXiv:0706.1933. [CrossRef] [PubMed]

**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

*I*

_{in}becomes large so does the input beam fluence Φ

_{in}=2

*T*

_{in}

*I*

_{in}, and this makes XPM effects more pronounced; as shown in [11

**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

_{c}=33 mJ/cm

^{2}the onset of compression in a BBO is not

*N*

_{eff,c}=1 - as one would expect from the NLS-like Eq. (3) – but can approximately be described by the scaling law

*N*

_{eff,c}=1+Δ

*N*

_{eff}, with Δ

*N*

_{eff}=Φ

_{in}/[Φ

_{c}(1+Φ

_{c}/Φ

_{in})] as the delay in onset. The delay is caused by the XPM term creating an intensity dependent self-focusing phase-shift in addition to the one already created by the Kerr SPM term; both are counteracting the self-defocusing phase shift from the cascaded quadratic interaction. The immediate consequence is that we can no longer expect the compression factor of

*f*=33.5 predicted from

_{C}*N*

_{eff}=8; instead we must use a corrected effective soliton number

*N*

^{corr}

_{eff}=

*N*

_{eff}-Δ

*N*

_{eff}[see Fig. 3(b)], which for high fluences then will give a reduced compression performance. Figure 3(a) shows Δ

*t*

_{opt}as predicted from the scaling laws for

*N*

_{eff}=8 (dotted orange line), together with the corrected Δ

*t*

^{corr}

_{opt}as calculated using

*N*

^{corr}

_{eff}(solid orange line). As expected for high Δ

_{k}values Δ

*t*

^{corr}

_{opt}starts to deviate from Δ

*t*

_{opt}. Importantly, it seems to describe very accurately the compression performance observed numerically. An example of how the pulse looks like in this regime is shown in Fig. 4 (Δ

*k*=125 mm

^{-1}, blue curve). The compressed pulse is longer (18 fs FWHM) and we also checked that it compresses later than what one would expect with

*N*

_{eff}=8. These values correspond very well to what the reduced soliton number

*N*

^{corr}

_{eff}≃3 predicts through the scaling laws. Thus, XPM strongly degrades compression when the beam fluence becomes large. This is also confirmed by the simulations shown with open circles in Fig. 3(a), for which XPM effects were turned off: only a weak degradation in compression is seen for high Δ

*k*.

*k*=43-50mm

^{-1}the limit to compression is determined by the strength of the nonlocal response function. Close to the transition to the nonstationary regime

*t*becomes large, so the nonlocal response

_{b}*R*

_{+}is very broad. Initially, however, the 200 fs FWHM input pulse sees only a weakly nonlocal response. As the pulse compresses the nonlocal response becomes strongly nonlocal, whereby the NLS-like model (3) reduces to a linear Schrödinger equation having a potential defined by the response function [18

18. N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E **68**, 036614 (2003). [CrossRef]

23. W. Krolikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E **63**, 016610 (2000). [CrossRef]

28. A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science **276**, 1538–1541 (1997). [CrossRef]

*t*, which explains the behaviour observed for phase-mismatch values just above Δ

_{b}*k*

_{sr}. This has also been observed for spatial nonlocal solitons [18

18. N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E **68**, 036614 (2003). [CrossRef]

23. W. Krolikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E **63**, 016610 (2000). [CrossRef]

28. A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science **276**, 1538–1541 (1997). [CrossRef]

*k*<42 mm

^{-1}the system is in the nonstationary regime and as Δ

*k*is reduced Δ

*t*

_{opt}increases as Δ

*k*

^{-1}. The compression limit does therefore not follow

*t*, but instead follows the characteristic Raman-like time

_{b}*T*

_{R,SHG}=2|

*d*

_{12}|/Δ

*k*quite closely. Indeed some physical explanation can be extracted from this parameter, since it namely represents the pulse duration, where the GVM length

*L*

_{GVM}=Δ

*t*/|

*d*

_{12}| becomes shorter than the coherence length

*L*

_{coh}=π/Δ

*k*. Intuitively it seems logical that the compressed pulse duration hit a limit when the GVM length is equal to the coherence length: the cascaded nonlinear interaction can no longer build up the phase shift because the GVM will remove the FW and SH from each other before even one cascaded cycle is complete. Interestingly, similar arguments to these were initially used to define the nonstationary regime [14

^{(2)} : χ^{(2)} nonlinearity,” Opt. Lett. **24**, 1777–1779 (1999). [CrossRef]

**21**, 376–383 (2004). [CrossRef]

*k*=50 mm

^{-1}a 6.3 fs FWHMcompressed pulse is observed as expected from the scaling laws. Approaching the transition to the nonstationary regime (Δ

*k*=43 mm

^{-1}) the pulse compression degrades to 10.6 fs FWHM, which (roughly) corresponds to the nonlocal time scale

*t*; thus, the potential-barrier effect of the nonlocal response function is apparent here. Once inside the nonstationary regime, the pulse not only becomes compressed poorly, but trailing oscillations are evident. The corresponding FW and SH wavelength-spectra are shown in Fig. 4(b) and (c). For Δ

_{b}*k*=50 mm

^{-1}both the FW and SH spectra are very flat, except for a spectral FW peak and corresponding spectral SH hole. As we explain below these peaks are actually dispersive waves. Closer to the transition (Δ

*k*=43 mm

^{-1}) the SH spectrum develops a peak because

*R̃*

_{+}(Ω) here is a very narrow Lorentzian. Inside the nonstationary regime a distinct red-shifted peak grows up in the SH spectrum, which can be explained by the nonlocal theory since the spectral peak sits at the frequency Ω

_{+}. In turn, close to the transition (Δ

*k*=41 mm

^{-1}) the FW has a corresponding spectral hole at

*ω*

_{1}+

*Ω*

_{+}, while further from the transition (Δ

*k*=30 mm

^{-1}) it becomes a spectral peak. To confirm this, we show in Fig. 4(d) the red-shifted holes/peaks found numerically versus Δ

*k*, with an impressive agreement with the nonlocal theory. This FW spectral hole/peak is the main limitation to the pulse compression in the nonstationary regime.

**31**, 1881–1883 (2006). [CrossRef] [PubMed]

*k*=16

*π*/mm). These results are confirmed in Fig. 5(a), showing the peak intensity of the compressed pulse versus

*N*

_{eff}for Δ

*k*=60 mm

^{-1}. Beyond

*N*

_{eff}=8 the pulse compression deviates from the prediction of the scaling law, even for the simulations including only up to third order dispersion (TOD,

*i.e.*, dispersion order

*m*=3), or neglecting the competing Kerr nonlinearities. However, the compression, shown in Fig. 5(b) as the compressed pulse duration Δ

_{d}*t*

_{opt}, improves even beyond this point of maximum intensity. The explanation is that Δ

*t*

_{opt}=

*T*

_{in}/

*f*in (b) is determined by the compression factor

_{c}*f*alone, while the intensity in (a) is

_{c}*I*

_{1,opt}/

*I*

_{in}=

*f*, where

_{c}Q_{c}*Q*is the compressed pulse quality (the energy of the central spike relative to the input pulse energy). Thus, the drop in intensity in Fig. 5(a) is caused by a drop in

_{c}*Q*. Around

_{c}*N*

_{eff}=17 the compressed pulse is quite close to the single-cycle regime. The time profiles for this case are shown In Fig. 5(c). The simulations without Kerr nonlinearities (so

*N*

_{SHG}=

*N*

_{eff}=17) actually predict single-cycle compressed pulses, while turning on the Kerr nonlinearities the compressed pulses increase to 1.5 optical cycles. Notice also that the pulses with Kerr nonlinearities are more asymmetric. This asymmetry is caused by the GVM-induced Raman-like perturbations [1st term on the RHS of Eq. (17)] as pointed out in Ref. [6

**31**, 1881–1883 (2006). [CrossRef] [PubMed]

*N*

_{SHG}=17 while with Kerr nonlinearities

*N*

_{SHG}=22.9 must be chosen to have

*N*

_{eff}=17. Therefore the strength of the Raman-like perturbation, which scales as N

^{2}

_{SHG}, is much stronger when including the competing Kerr nonlinearities leading to a more asymmetric pulse.We also note that the simulations with exact dispersion (no polynomial expansion,

*m*=∞) have fast trailing oscillations, which are absent for the TOD simulations. The FW spectra in Fig. 5(d) offer an explanation: only with exact dispersion is there a dispersive wave appearing as a spectral peak around 3

_{d}*µ*m. With TOD the spectrum is instead smooth because the phase-matching condition for the dispersive wave is pushed far into the infrared.

34. K. C. Chan and M. S. F. Liu, “Short-pulse generation by higher-order soliton-effect compression: Effects of fiber characteristics,” IEEE J. Quantum Electron. **31**, 2226–2235 (1995). [CrossRef]

*k*

^{(2)}

_{1}≃40 fs

^{2}/mm and

*k*

^{(2)}

_{2}≃100 fs

^{2}/mm).

**31**, 1881–1883 (2006). [CrossRef] [PubMed]

36. M. Bache, H. Nielsen, J. Lægsgaard, and O. Bang, “Tuning quadratic nonlinear photonic crystal fibers for zero group-velocity mismatch,” Opt. Lett. **31**, 1612–1614 (2006), arXiv:physics/0511244. [CrossRef] [PubMed]

## 6. Conclusions

*T*

_{R,SHG}, roughly the pulse duration for which the GVM length and the coherence length become identical.

**24**, 2752–2762 (2007), arXiv:0706.1507. [CrossRef]

*λ*

_{1}=800 nm, because GVM is much stronger than what was presented here. Thus we expect the nonlocal analysis to provide more insight into this case, in particular concerning the nonstationary regime, which is the dominating one at 800 nm.

1. | G. P. Agrawal, |

2. | L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. |

3. | S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B |

4. | S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett. |

5. | J. A. Moses, J. Nees, B. Hou, K.-H. Hong, G. Mourou, and F. W. Wise, “Chirped-pulse cascaded quadratic compression of 1-mJ, 35-fs pulses with low wavefront distortions,” |

6. | J. Moses and F. W. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett. |

7. | J. Moses, E. Alhammali, J. M. Eichenholz, and F. W. Wise, “Efficient high-energy femtosecond pulse compression in quadratic media with flattop beams,” Opt. Lett. |

8. | X. Zeng, S. Ashihara, N. Fujioka, T. Shimura, and K. Kuroda, “Adiabatic compression of quadratic temporal solitons in aperiodic quasi-phase-matching gratings,” Opt. Express |

9. | G. Xie, D. Zhang, L. Qian, H. Zhu, and D. Tang, “Multi-stage pulse compression by use of cascaded quadratic nonlinearity,” Opt. Commun. |

10. | M. Bache, O. Bang, J. Moses, and F.W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. |

11. | M. Bache, J. Moses, and F. W. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B |

12. | R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. |

13. | C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and Induced Kerr effects in quasi-phase-matched Quadratic media,” Phys. Rev. Lett. |

14. | X. Liu, L. Qian, and F. W. Wise, “High-energy pulse compression by use of negative phase shifts produced by the cascaded χ |

15. | P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. |

16. | L. Bergé, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, “Self-focusing and solitonlike structures in materials with competing quadratic and cubic nonlinearities,” Phys. Rev. E |

17. | F. Ö. Ilday, K. Beckwitt, Y.-F. Chen, H. Lim, and F. W. Wise, “Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes,” J. Opt. Soc. Am. B |

18. | N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E |

19. | W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B: Quantum Semiclass. Opt . |

20. | J. Moses and F. W. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. |

21. | V. Dmitriev, G. Gurzadyan, and D. Nikogosyan, |

22. |
On dimensional form t=τ_{a,b}_{a,b}T_{in}. In the frequency domain both R̃ and R̃ are dimensionless. |

23. | W. Krolikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E |

24. | M. Bache, O. Bang, and W. Krolikowski, (2008), in preparation. |

25. | The factor |

26. | These experiments were actually done in the nonstationary regime according to the nonlocal theory. |

27. | This is a typical experimental situation: the optimal compression point z |

28. | A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science |

29. | W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E |

30. | I. V. Shadrivov and A. A. Zharov, “Dynamics of optical spatial solitons near the interface between two quadratically nonlinear media,” J. Opt. Soc. Am. B |

31. | N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A |

32. | D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science |

33. | I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Opt. Express |

34. | K. C. Chan and M. S. F. Liu, “Short-pulse generation by higher-order soliton-effect compression: Effects of fiber characteristics,” IEEE J. Quantum Electron. |

35. | K.-T. Chan and W.-H. Cao, “Improved soliton-effect pulse compression by combined action of negative thirdorder dispersion and Raman self-scattering in optical fibers,” J. Opt. Soc. Am. B |

36. | M. Bache, H. Nielsen, J. Lægsgaard, and O. Bang, “Tuning quadratic nonlinear photonic crystal fibers for zero group-velocity mismatch,” Opt. Lett. |

**OCIS Codes**

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(320.2250) Ultrafast optics : Femtosecond phenomena

(320.5520) Ultrafast optics : Pulse compression

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: January 4, 2008

Revised Manuscript: February 18, 2008

Manuscript Accepted: February 22, 2008

Published: February 25, 2008

**Citation**

M. Bache, O. Bang, W. Krolikowski, J. Moses, and F. W. Wise, "Limits to compression with cascaded quadratic soliton compressors," Opt. Express **16**, 3273-3287 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-3273

Sort: Year | Journal | Reset

### References

- G. P. Agrawal, Applications of nonlinear fiber optics (Academic Press, London, 2001).
- L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980). [CrossRef]
- S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, "Soliton compression of femtosecond pulses in quadratic media," J. Opt. Soc. Am. B 19, 2505-2510 (2002). [CrossRef]
- S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, "Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate," Appl. Phys. Lett. 84, 1055-1057 (2004). [CrossRef]
- J. A. Moses, J. Nees, B. Hou, K.-H. Hong, G. Mourou, and F. W. Wise, "Chirped-pulse cascaded quadratic compression of 1-mJ, 35-fs pulses with low wavefront distortions," In Conference on Lasers and Electro-Optics, p. CTuS5 (Optical Society of America, 2005).
- J. Moses and F. W. Wise, "Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal," Opt. Lett. 31, 1881-1883 (2006). [CrossRef] [PubMed]
- J. Moses, E. Alhammali, J. M. Eichenholz, and F. W. Wise, "Efficient high-energy femtosecond pulse compression in quadratic media with flattop beams," Opt. Lett. 32, 2469-2471 (2007). [CrossRef] [PubMed]
- X. Zeng, S. Ashihara, N. Fujioka, T. Shimura, and K. Kuroda, "Adiabatic compression of quadratic temporal solitons in aperiodic quasi-phase-matching gratings," Opt. Express 14, 9358-9370 (2006). [CrossRef] [PubMed]
- G. Xie, D. Zhang, L. Qian, H. Zhu, and D. Tang, "Multi-stage pulse compression by use of cascaded quadratic nonlinearity," Opt. Commun. 273, 207-213 (2007). [CrossRef]
- M. Bache, O. Bang, J. Moses, and F.W. Wise, "Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression," Opt. Lett. 32, 2490-2492 (2007), arXiv:0706.1933. [CrossRef] [PubMed]
- M. Bache, J. Moses, and F. W. Wise, "Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities," J. Opt. Soc. Am. B 24, 2752-2762 (2007), arXiv:0706.1507. [CrossRef]
- R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, "Self-focusing and self-defocusing by cascaded second-order effects in KTP," Opt. Lett. 17, 28-30 (1992). [CrossRef] [PubMed]
- C. B. Clausen, O. Bang, and Y. S. Kivshar, "Spatial solitons and Induced Kerr effects in quasi-phase-matched Quadratic media," Phys. Rev. Lett. 78, 4749-4752 (1997). [CrossRef]
- X. Liu, L. Qian, and F. W. Wise, "High-energy pulse compression by use of negative phase shifts produced by the cascaded |(2) : |(2) nonlinearity," Opt. Lett. 24, 1777-1779 (1999). [CrossRef]
- P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, "Focusing versus defocusing nonlinearities due to parametric wave mixing," Phys. Rev. Lett. 87, 183902 (2001). [CrossRef]
- L. Berge, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, "Self-focusing and solitonlike structures in materials with competing quadratic and cubic nonlinearities," Phys. Rev. E 55, 3555-3570 (1997). [CrossRef]
- F. O. Ilday, K. Beckwitt, Y.-F. Chen, H. Lim, and F. W. Wise, "Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes," J. Opt. Soc. Am. B 21, 376-383 (2004). [CrossRef]
- N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, "Quadratic solitons as nonlocal solitons," Phys. Rev. E 68, 036614 (2003). [CrossRef]
- W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclass. Opt. 6, s288 (2004). [CrossRef]
- J. Moses and F. W. Wise, "Controllable self-steepening of ultrashort pulses in quadratic nonlinear media," Phys. Rev. Lett. 97, 073903 (2006), see also arXiv:physics/0604170. [CrossRef] [PubMed]
- V. Dmitriev, G. Gurzadyan, and D. Nikogosyan, Handbook of Nonlinear Optical Crystals, Vol. 64 of Springer Series in Optical Sciences (Springer, Berlin, 1999).
- On dimensional formR(t)=R(t/Tin)/Tin, which is independent on Tin since |a,b in Eqs. (10,12) must be replaced by the dimensional form ta,b =|a,bTin. In the frequency domain both ˜R and ˜R are dimensionless.
- W. Krolikowski and O. Bang, "Solitons in nonlocal nonlinear media: Exact solutions," Phys. Rev. E 63, 016610 (2000). [CrossRef]
- M. Bache, O. Bang, and W. Krolikowski, (2008), in preparation.
- The factor sa on the RHS of Eq. (17) was unfortunately lost during the proofs in Eq. (12) of Ref. [10].
- These experiments were actually done in the nonstationary regime according to the nonlocal theory.
- This is a typical experimental situation: the optimal compression point zopt scales with Neff [11], and since the nonlinear crystal length is a constant parameter one adjusts the intensity so zopt coincides with the crystal length.
- A. W. Snyder and D. J. Mitchell, "Accessible solitons," Science 276, 1538-1541 (1997). [CrossRef]
- W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001). [CrossRef]
- I. V. Shadrivov and A. A. Zharov, "Dynamics of optical spatial solitons near the interface between two quadratically nonlinear media," J. Opt. Soc. Am. B 19, 596-602 (2002). [CrossRef]
- N. Akhmediev and M. Karlsson, "Cherenkov radiation emitted by solitons in optical fibers," Phys. Rev. A 51, 2602-2607 (1995). [CrossRef] [PubMed]
- D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russell, "Soliton self-frequency shift cancellation in photonic crystal fibers," Science 301, 1705-1708 (2003). [CrossRef] [PubMed]
- I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, "Dispersive wave generation by solitons in microstructured optical fibers," Opt. Express 12, 124-135 (2003). [CrossRef]
- K. C. Chan and M. S. F. Liu, "Short-pulse generation by higher-order soliton-effect compression: Effects of fiber characteristics," IEEE J. Quantum Electron. 31, 2226-2235 (1995). [CrossRef]
- K.-T. Chan and W.-H. Cao, "Improved soliton-effect pulse compression by combined action of negative thirdorder dispersion and Raman self-scattering in optical fibers," J. Opt. Soc. Am. B 15, 2371-2376.
- M. Bache, H. Nielsen, J. Lægsgaard, and O. Bang, "Tuning quadratic nonlinear photonic crystal fibers for zero group-velocity mismatch," Opt. Lett. 31, 1612-1614 (2006), arXiv:physics/0511244. [CrossRef] [PubMed]

## 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.

« Previous Article | Next Article »

OSA is a member of CrossRef.