We develop a rapid and efficient numeric technique for the design of arbitrary quasi-phase matched lattices for parametric generation of single and multiple pulses with any prescribed amplitude and phase profiles from fundamental frequency excitation in the regime of pump depletion. We examine the case of simultaneous of 2nd and 3rd harmonic generation in arbitrary quasi-phase matched gratings taking into account the group velocity mismatch and dispersion.
© 2009 Optical Society of America
In recent years, various parametric frequency conversion processes have been extensively investigated in conjunction with domain inversion in ferroelectric crystals for the tailoring of χ (2) nonlinearities in quasi-phase matched (QPM) structures [1–3]. It is widely recognized that these kind of gratings can be successfully used to achieve several goals, including pulse compression , high conversion efficiencies in the fs time-scale [5–6] and in the presence of self- and cross-action effects , quadratic solitons , simultaneous generation of various wavelengths , wavelength conversion , non reciprocal isolators , and so on . Moreover, non-periodic or aperiodic QPM domains can be designed to generate light with desired parameters, e. g. spectral bandwidth [3, 13–14], higher harmonics [15–17], amplitude and phase profiles of fs and ps pulses [18–21].
Up to date a number of theoretical approaches have been developed for the design of aperiodic QPM gratings with various objectives. Among them are the simulated annealing algorithm, [13–14, 18–19, 22], optimal control based on Lagrange multipliers, [20, 23] genetic algorithms  or a combination of the latter . The method of Lagrange multipliers, faster and relatively more efficient than previous approaches, was recently employed to designing the generation of second harmonic (SH) pulses with arbitrary profiles in quadratic crystals with spatially modulated χ (2) nonlinearities  and aperiodic QPM waveguides . Conforti and coworkers applied the method to nonlinear pulse shaping with first order QPM gratings  whereas in Ref.  Buffa assumed that the effective second-order nonlinearity could take any real value.
In this Paper we illustrate and demonstrate a numerical approach based on Lagrange multipliers for the design of engineered QPM gratings capable of generating second and third harmonic (TH) pulses of given shape from a pulsed fundamental frequency (FF) input. The approach can be applied to quasi phase matching of any order and duty-cycle, accounting for group velocity mismatch and dispersion. At variance with previous studies, we employ the optimal control technique and allow for unitary sign change in real nonlinearity.
To demonstrate the approach we choose a cascaded multistep parametric frequency conversion processes such as ω+ω=2ω and 2ω+ω=3ω. In this case, the evolution of the three interacting pulses in the moving system with the 3rd harmonic and in the slowly varying envelope approximation can be derived using coupled-mode-theory and is described by:
with boundary conditions:
with γi = 2π deff/niλ, Ai and ni being the complex field amplitudes and the refractive indices of the ith harmonic (i = 1,2,3); deff the effective nonlinearity; λ the wavelength at the FF, ρ(z) the unitary amplitude sign-changing function defining arbitrarily sized domains of the QPM grating, p 1 = exp(iΔk 1 z), p 2 = exp(iΔk 2 z),Δk 1 = k 1 ? 2k 2, Δk 2 = k 3 ? k 2 ? k 1 with ki the wavenumber of the ith harmonic; Ao the peak amplitude of the FF excitation; v 1=1/V1-1/V 3 and v 2=1/V2-1/V3 with Vi the group velocity of the ith harmonic; τo the input pulse duration (at 1/e half-width of FF intensity); βi the group velocity dispersion (GVD) of the ith harmonic; τo the input pulse duration (we assume all pulse durations to be at 1/e half-width of the intensity).
being ATi(t) the target profile and fi the complex function playing the role of Lagrange multiplier for the ith harmonic.
Setting equal to zero the functional derivatives of J with respect to Ai, we obtain the evolution equations for the Lagrange multipliers:
with boundary conditions:
Finally, from the functional derivative J with respect to ρ(z) we get (at variance with Ref.  we assume ρ(z) to be a real function):
We initially used dimensionless parameters and values γ=γ1=γ2=γ3=1, τo=1, v 1=1, v 2=0.5, β1= β2= β3=0, Ao=1, Δk1=0.5 Δk2 and qo=0.01. qo is the “ideal” domain size for effective QPM generation of 2nd harmonic light., i.e. qo=π/Δk1. Note that the relative weights of group velocities, mismatches and nonlinear strengths are arbitrarily chosen just to demonstrate the approach. We solved Eq. (1) and Eq. (4) with Eqs. (2) and (5) using the fast Fourier transform and a fourth-order Runge-Kutta integration scheme for linear and nonlinear portions, respectively .
We summarize below the main steps of the algorithm:
- Choose target profiles and an initial guess for ρ(z) sin(Δk 1 z), using dz = qo/5;
- Calculate the output pulses at the selected harmonic by integrating Eqs. (1) with ρ (z) = sign(ρ (z));
- Use the results of the previous step to solve the evolution equations for the Lagrange multipliers Eqs. (4), again with ρ(z) = sign(ρ(z)) from z=L to z=0;
- Update ρ(z) = ρ(z) + α δJ/δ ρ(z) with 0 < α ≤ 1;
- Calculate the domain sizes on sign(ρ(z)) and set the boundary for minimum domain size (in our case q(N) ≥ 0.01∗qo, q(N) is the length of the Nth domain size). Recalculate ρ(z) on variable integration steps dz(N) for positive and negative areas of ρ(z), where dz(N)=q(N)/m (in our case m=5); add/remove “additional steps” for ρ(z) if the energies of the desired pulses are smaller/larger then the targets.
- If the results are close enough to the targets (at each iteration of the algorithm, targets were moved to their desired profiles’ positions) on their root-meansquare error, the iterative procedure stops; otherwise it continues from the second step.
As a first example, we considered a Gaussian target pulse at the 2nd harmonic with duration τ 1=1 and FF → SH conversion efficiency > 45%. Figure 1 shows the results obtained numerically with the procedure outlined above. Some of the domains change size through the grating as compared to q0. The output pulse at SH perfectly matches the target profile and amplitude, with a negligible amount of third harmonic generated by cascading.
In a second case, we aimed at obtaining a Gaussian 3rd harmonic pulse with duration τ 2=1 and FF → TH conversion efficiency > 40%. Figure 2 illustrates the results, with domain sizes changing substantially as compared to q 0. These “large” changes of domain sizes as compared with q 0 relate to Δk 1 ≠ Δk 2 and the 3rd harmonic target profile.
We also tested the method aiming at the simultaneous generation of 2nd and 3rd harmonic Gaussian pulses, with duration τ 1 = 1 and τ 2=1 and conversion efficiencies > 25% at each. Figure 3 illustrates the results. Here we obtained conversions of 30% and 40% for FF → TH and FF → SH, respectively. We should emphasize that the obtained QPM design with the Lagrangian approach needs not be unique in any given case. Alternative lattices could be adopted for the same target(s) in order to modify the conversion slope at the output (Figs. 2(b) and 3(b)) and improve the fabrication tolerance on overall crystal length.
Noteworthy, the developed algorithm typically runs for times ranging from 5 to 10 minutes on a standard personal computer, depending on targets. The method can also be used to design phase-modulated (chirped) pulses - as fi boundaries can take complex values - as well as for shaping from short to longer pulses (e.g. fs to ps) in QPM lattices with a large number of domains. In such cases, in order to reduce the computation time, one can employ larger dz depending on q0. Finally, GVD can be included with reference to specified crystals and wavelengths.
We tested our approach in the realistic case of Lithium Niobate LiNbO3, using λ=1.55μm, τ0 = 150fs, I0=1GW/cm2, ee-e quadratic interactions (deff=d 33=30pm/V); the desired 2nd and 3rd harmonic pulses were Gaussian profiles of durations τ1=150fs and τ2=150fs and efficiencies > 30% each. From Sellmeier equations we calculated q0=9.48 μm, Δk1 ≈ 0.3μm-1, Δk2 ≈ 0.9 μm-1,
v 1 ≈ -1 fs/μm, v 2 ≈ -0.7 fs/μm, β1 ≈0.1 fs2/μm, β2 ≈0.4 fs2/μm, β3 ≈0.8 fs2/μm, γ1 A0=0.0034μm-1, γ2 A0=0.0033μm-1, γ3 A0=0.0032μm-1, A0=6.105V/cm; Figure 4 shows the corresponding results. Even in the case of an actual material we were able to design the lattice and obtain the desired pulses with high accuracy in a rather short computing time.
In conclusion, we developed and tested a fast and efficient numerical technique for designing one-dimensional QPM gratings capable of producing harmonic laser pulses of the desired amplitude and phase profile. Focusing on optical parametric amplification with cascaded frequencies, we were able to design not only individual second and third harmonic pulses, but also their simultaneous generation with any target profile. We believe this method greatly facilitates the design of arbitrary QPM crystals and waveguides towards any desired parametric processes and their combination.
This work was funded in part by the Italian Ministry for Scientific Research (MiUR) Project PRIN no. 2007CT355C.
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