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 [4], high conversion efficiencies in the fs time-scale [5–6] and in the presence of self- and cross-action effects [7], quadratic solitons [8], simultaneous generation of various wavelengths [9], wavelength conversion [10], non reciprocal isolators [11], and so on [12]. 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 [24] or a combination of the latter [25]. 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 [23] and aperiodic QPM waveguides [20]. Conforti and coworkers applied the method to nonlinear pulse shaping with first order QPM gratings [20] whereas in Ref. [23] 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 3^rd harmonic and in the slowly varying envelope approximation can be derived using coupled-mode-theory and is described by:

with boundary conditions:

with γ_i = 2π d_eff/n_iλ, A_i and n_i being the complex field amplitudes and the refractive indices of the i^th harmonic (i = 1,2,3); d_eff 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 k_i the wavenumber of the i^th harmonic; A_o the peak amplitude of the FF excitation; v ₁=1/V₁-1/V ₃ and v ₂=1/V₂-1/V₃ with V_i the group velocity of the i^th harmonic; τ_o the input pulse duration (at 1/e half-width of FF intensity); β_i the group velocity dispersion (GVD) of the i^th harmonic; τ_o the input pulse duration (we assume all pulse durations to be at 1/e half-width of the intensity).

Following the rules of the optimal control technique based on Lagrange multipliers, the cost function to be minimized is [20, 23]:

with

being A^T_i(t) the target profile and f_i the complex function playing the role of Lagrange multiplier for the i^th harmonic.

Setting equal to zero the functional derivatives of J with respect to A_i, 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. [20] we assume ρ(z) to be a real function):

where

We initially used dimensionless parameters and values γ=γ₁=γ₂=γ₃=1, τ_o=1, v ₁=1, v ₂=0.5, β₁= β₂= β₃=0, A_o=1, Δk₁=0.5 Δk₂ and q_o=0.01. q_o is the “ideal” domain size for effective QPM generation of 2^nd harmonic light., i.e. q_o=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 [19].

We summarize below the main steps of the algorithm:

As a first example, we considered a Gaussian target pulse at the 2^nd 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.

 

Fig. 1. (Left) time evolutions of the intensity distributions of FF (red) at z=L (solid line) and at z=0 (dashed line), SH (blue line), TH (green line) pulses and target SH pulse (blue “o” symbols). (Center) Energy evolutions of the interacting harmonics versus designed QPM grating length (red, blue, green and black lines correspond to fundamental, 2^nd, 3^rd harmonics and the sum of their energies). (Right) QPM domain size distribution (relative to the unperturbed q_o) versus domain number N.

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In a second case, we aimed at obtaining a Gaussian 3^rd 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 3^rd harmonic target profile.

 

Fig. 2. Generation of optimized 3^rd harmonic profile: lines are defined as in Fig. 1. The green line with symbols “o” is the 3^rd harmonic target pulse.

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We also tested the method aiming at the simultaneous generation of 2^nd and 3^rd 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 f_i 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.

 

Fig. 3. Generation of optimized of 2^nd and 3^rd harmonic pulses: lines are defined as in Figs. 1-2.

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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 (d_eff=d ₃₃=30pm/V); the desired 2^nd and 3^rd 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.

 

Fig. 4. Generation of optimized of 2^nd and 3^rd harmonic pulses for a real experimental case in Lithium Niobate (see text): lines and colors are as in Figs. 1–3.

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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.