1. Introduction

Nonlinear optical crystals with a χ⁽²⁾ nonlinearity can be used to generate 2^nd and higher-order harmonics from direct laser excitation. In practice, one often needs to resort to cascaded arrangements under phase-synchronism for each desired harmonic and in each individual crystal. Therefore, this approach tends to be inefficient for harmonics of order higher than the 3^rd [1], owing to a number of difficulties related to optimum propagation lengths and phase-matching conditions in each crystal. High harmonics to any desired orders have been generated from laser-plasma interactions [2–5], reaching wavelengths in the extreme ultraviolet and soft X-ray spectral regions, but with rather low energy conversion.

Nowadays, quasi-phase-matching (QPM) in periodically-poled non-centrosymmetric crystals offers a viable solution to this problem. Engineered quasi-periodic QPM gratings have been employed for the efficient generation of e. g. 3^rd [6–8] and 4^th [9] harmonics. Thanks to the unique dispersion properties of ferroelectric crystals, QPM has been engineered to generate 2^nd and 3^rd harmonics using a continuous-wave (cw) laser and a single grating in lithium niobate [10], as well as generation of red, yellow, green and blue simultaneously in aperiodically-poled lithium tantalate.[11]

In this paper we present and test an effective design routine to engineer QPM gratings for the efficient generation of harmonics of arbitrary order from a fundamental frequency (FF) input, focusing on the case of one-dimensional phase-matching in the regime of substantial depletion of a cw pump. We numerically demonstrate the method with reference to an aperiodically-poled lithium niobate crystal for the generation of up to the 7^th harmonic from a 5µm fundamental input. This approach, which could find application to every nonlinear materials where QPM can be engineered [12] and to any wavelengths of interest, could also be implemented in alternative to difference-frequency processes [13–14] for the efficient generation of near- and mid-infrared wavelengths from single sources, e. g., CO or CO₂ lasers.

The paper is organized as follows: in Section 2 we describe the theoretical approach; in Section 3 we detail the material parameters and the set conditions for the numerical experiments, as well as the algorithm employed for the solution of the problem; in Section 4 we present and discuss the results.

2. Theoretical model

The coupled-mode equations, derived in the slowly varying envelope approximation and describing the evolution of the first seven harmonics for collinear parametric interactions in the presence of quasi phase matching, read:

with A_i and n_m being the complex amplitude and the refractive index of the m ^th harmonic (m=1, 2,…, 7), d^eff the effective nonlinearity assuming Kleinman symmetry; λ the FF wavelength; δ(z) a unitary sign-changing function defining the arbitrarily-sized domains of the QPM grating, Δ ₁=exp(i(k ₂-2k ₁)z); Δ₂=exp(i(k ₃-k ₂-k ₁)z); Δ ₃=exp(i(k ₄-k ₃-k ₁)z); Δ ₄=exp(i(k ₅-k ₄-k ₁)z); Δ ₅=exp(i(k ₆-k ₅-k ₁)z); Δ ₆=exp(i(k ₇-k ₆-k ₁)z); Δ ₇=exp(i(k ₄-2k ₂)z); Δ ₈=exp(i(k ₅-k ₃-k ₂)z); Δ ₉=exp(i(k ₆-k ₄-k ₂)z); Δ ₁₀=exp(i(k ₇-k ₅-k ₂)z); Δ ₁₁=exp(i(k ₆-2k ₃)z); Δ ₁₂=exp(i(k ₇-k ₄-k ₃)z); k_m=2πn_m/λ_m and λ_m the wave-number and wavelength of the m ^th harmonic, respectively.

Since it models quadratic interactions, the set of Eq. (1) above does only depend onthe “effective” coupling strength d_effL|A ₁(0)|;[15–16] hence, using normalized amplitudes a_i(ζ)=A_i(ζ)/|A₁(0)| and propagation distance ζ=z/L they could be recast in a more generalized set. Nevertheless, in order to illustrate the method in a realistic setting, we chose to integrate and illustrate the solutions of the “physical” set (1).

For the numerical solution we adopted a simple fourth-order Runge–Kutta method; however, as also noted in Ref. [17], because of the aperiodic character of δ(z), uniform integration steps dz on the propagation coordinate z cannot yield correct solutions and a step dependence dz(z) had to be employed from domain to domain. This allowed us to account for arbitrary changes of δ(z) along the non-uniform QPM distribution.

3. Sample parameters and numerical experiment

We intend to demonstrate hereby that QPM gratings with arbitrary domain sizes can be designed to generate any high-order harmonic from a given laser source. In order to illustrate the applicability of our approach, we consider a z-cut LiNbO₃ crystal (d^eff=d₃₃≈30 pm/V) and a (FF) laser beam of wavelength λ=5µm and an intensity of 0.5GW/cm². The chosen wavelength allows us to exploit the whole transparency range of lithium niobate (between 5.2 and 0.42 µm [18]) with 2^nd, 3^rd, 4^th, 5^th, 6^th, and 7^th harmonics corresponding to 2.5, 1.666, 1.25, 1.00, 0.833 and 0.71 µm, respectively.

For the calculation of k_m and n_m we used the Sellmeier equation for LiNbO₃, as reported in Ref. [19] at room temperature. The conversion efficiencies to each harmonic were calculated as η_m(z)=100*n_m|A_m(z)|²/n ₁|A ₁(z=0)|², with |A ₁(z=0)|²=2Z_oI ₁/n ₁ and Z_o the vacuum impedance.

Starting with a uniform QPM grating of length L designed for frequency doubling and consisting of N=500 domains, the main steps of the algorithm can be summarized as follows:

The resolution in size was set to 10nm and we typically used ten steps within each domain. In all cases, we checked that energy was conserved throughout the integration by forcing ${\Delta }_m\left(z\right)=\sum _{m=1}^{m=7}{\eta }_m\left(z\right)-100\%<0.01\%$ .

4. Results and discussion

Figure 1 illustrates the results yielded by the algorithm described above when maximizing the conversion efficiency to 3^rd (Fig. 1(2)) and 4^th (Fig. 1(3)) harmonics of the fundamental frequency input. We also plot in Fig. 1(1) the conversion versus z when frequency doubling in the presence of a uniform QPM modulation (Λ=23.001µm). All the harmonics up to the 7^th (as from Eq. (1)) were taken into account in the calculations.

The graphs show that we were able to obtain remarkable conversion efficiencies as high as η ₃(L)≈90% (Fig. 1(2)) and η ₄(L)≈78.6% (Fig. 1(3))) for 3^rd and 4^th harmonics, respectively. Noteworthy, even higher efficiencies could be pursued by allowing the algorithm to refine the QPM modulation with a higher resolution (<10nm) and a larger number of domains.

 

Fig. 1. Conversion efficiencies versus propagation for an FF input at λ=5.0µm, by adopting the model Eq. (1) and (1) maintaining a uniform QPM grating optimized for SHG; (2) modulating the QPM grating in order to maximize the third harmonic; (3) modulating the QPM grating in order to maximize the fourth harmonic. Black, red, blue, green, magenta, cyan and yellow lines graph the calculated intensity evolutions of FF, 2^nd, 3^rd, 4^th, 5^th, 6^th and 7^th harmonics, respectively.

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As expected, in the process of generating a harmonic higher than the second, the cascaded interaction requires the intermediate harmonics to acquire a portion of the available (input) intensity. For example, in Fig. 1(3) 2^nd (red line) and 3^rd (blue line) harmonic waves are generated inside the sample before they supply their photons to the desired fourth harmonic. In the latter case (Fig. 1(3)) the minimum (maximum) QPM domain size was 4.02 (27.39) µm, whereas it was 2.25 (38.91) µm for the generation of the 3^rd harmonic (Fig. 1(2)).

Figure 2 displays the results of the numerical experiments for the efficient generation of 5^th, 6^th and 7^th harmonics, with conversions > 76% for each of them. We obtained η ₅(L)≈80% (Fig. 2(1)), η ₆(L)≈78.5% (Fig. 2(2)) and η ₇(L)≈70% (Fig. 2(3)). Here the minimum (maximum) required QPM domain sizes were 3.16 (32.15), 3.18 (28.92) and 3.33 (30.31) µm for the 5^th, 6^th and 7^th harmonics, respectively.

 

Fig. 2. Conversion efficiencies versus propagation for an FF input at λ=5.0µm, by adopting the model Eq. (1) and modultating the QM domain sizes in order to maximize the throughput at (1) 5^th, (2) 6^th and (3) 7^th harmonic frequencies. Black, red, blue, green, magenta, cyan and yellow lines correspond to FF, 2^nd, 3^rd, 4^th, 5^th, 6^th and 7^th harmonics, respectively.

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Since the domain size and its accuracy are crucial to the proper design of optimized QPM gratings, we investigated the role of domain resolution, e. g. tolerance, for a couple of specific harmonics, namely the third and the seventh. To this extent, we re-calculated the conversion efficiencies versus propagation increasing the best available resolution, e. g. simulating a limited accuracy in grating fabrication. Figure 3(1–2) illustrates the results: noticeably, the final conversions were not substantially lower than in Fig. 1(2) and Fig. 2(3), respectively, when adopting a resolution of 100nm (one order of magnitude worse than the previously used 10nm). Resolutions limited to 0.5 µm or worse, however, led to a significant reduction in conversion, particularly at the highest order (Fig. 3(2)) where shorter domains are required for quasi phase-matching.

 

Fig. 3. Conversion efficiencies versus z for (1) 3^rd and (2) 7^th harmonics and various domain resolutions: red-0.1 µm, green-0.5 µm and magenta-1.0 µm. (3) Conversion efficiencies versus z for the 6^th harmonic and domain resolution 0.1µm, calculated for various FF input intensities: magenta (0.1 GW/cm²), cyan (0.5 GW/cm²), black (1.0 GW/cm²), red (1.5 GW/cm²) and blue (2.0 GW/cm²).

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Finally, Fig. 3(3) illustrates the role of the input FF excitation. Since quadratic cascading depends on the available number of photons,[20–21] we studied the role of the input intensity on a given (optimally designed with a resolution of 100nm) grating for the generation of a high harmonic, namely the sixth. Lines of different colors refer for different FF intensities, I=0.1, 0.5, 1.0, 1.5 and 2.0 GW/cm², respectively. As expected, the calculated conversion efficiency critically depends on the excitation: proper QPM engineering needs therefore be optimized for a certain input level.

5. Conclusions

In conclusion, we implemented a numerical procedure for designing engineered QPM gratings for the efficient generation of high-order harmonics in a quadratically nonlinear crystal. We demonstrated the approach for upconversion of a cw fundamental frequency input ω (λ=5 µm) to 3ω, 4ω, 5ω, 6ω and 7ω in lithium niobate. The results show that engineered QPM crystals can be designed to operate at any harmonic order, with given excitation and despite limited fabrication resolutions.