# Efficient Third-Harmonic Generation by Inhomogeneous Quasi-Phase-Matching in Quadratic Crystals

^{1}

^{2}

^{*}

## Abstract

**:**

_{1}or d

_{2}, with d

_{1}and d

_{2}being the coherence lengths for the cascaded parametric interactions $2\mathsf{\omega}=\mathsf{\omega}+\mathsf{\omega}$ and $3\mathsf{\omega}=2\mathsf{\omega}+\mathsf{\omega}$, respectively. We focus on the cases with single segments formed by equal and/or different domains, showing that frequency tripling can be achieved with high conversion efficiency from an arbitrary input wavelength. The presented approach allows one to accurately determine the optimized random alternation of domain thicknesses d

_{1}and d

_{2}along the propagation length.

## 1. Introduction

_{1}and d

_{2}, with d

_{1}= l

_{c1}being the coherence length for the SHG process, and d

_{2}= l

_{c2}being the coherence length for the subsequent (cascaded) SFG process [11,12]. While genetic algorithms, simulated annealing, and optimal control based on Lagrange multipliers have been exploited earlier to design nonlinear optical lattices (see, e.g., [23] and references therein), the Monte Carlo based design presented hereby with reference to third-harmonic generation is an effective and convenient numerical approach to optimizing cascaded parametric interactions in quadratic crystals.

## 2. Materials and Methods

_{j}and k

_{j}(j = 1, 2, 3) are the electric-field amplitudes and wave vectors at ω, 2ω, and 3ω, respectively; $\Delta {k}_{SHG}={k}_{2}-2{k}_{1}$ is the SHG wavevector mismatch; $\Delta {k}_{SFG}={k}_{3}-{k}_{2}-{k}_{1}$ is the SFG wavevector mismatch; γ

_{SFG}and γ

_{SHG}are the corresponding nonlinear strengths ${\gamma}_{SHG}$ ≈ ${\gamma}_{SFG}$ ≈ $4{\pi}^{2}{d}_{eff}/\left(\mathrm{n}\right(3\omega \left)\lambda \right)$ ≈ $4{\pi}^{2}{d}_{eff}/\left(\mathrm{n}\right(2\omega \left)\lambda \right)$ ≈ $4{\pi}^{2}{d}_{eff}/\left(\mathrm{n}\right(\omega )\lambda ),$${\mathrm{d}}_{eff}={\mathsf{\chi}}^{(2)}/2$ (e.g., ${\mathrm{d}}_{eff}={\mathrm{d}}_{33}$ in Lithium Niobate); and δ(z) the sign-varying unity function with random size of either d

_{1}or d

_{2}along the propagation distance z.

_{1}or d

_{2}, respectively, with zeroes (ones) corresponding to d

_{1}(d

_{2}) from the first up to the Nth domain over the total length. Then, we numerically integrated the system of ordinary differential Equation (1) and computed the resulting conversion efficiency η

_{3}from the fundamental to the third-harmonic frequency.

_{3}was initially set to zero and updated to the value obtained after each randomization of the sequence, provided the latter η

_{3}was higher than the former. Otherwise, when the last η

_{3}result was lower than the previous one, the algorithm randomized the sequence by changing each subsequent (second, third, fourth etc.) domain type and repeating the procedure. By iterating the whole process, we increased the conversion efficiency until convergence was reached to a higher value than prescribed at the sample output or until a set number of iterations was completed.

## 3. Results

_{1}= 0.01, d

_{2}= 0.5 d

_{1}, γ

_{SHG}= γ

_{SFG}= γ = 1, Δk

_{SHG}= π/d

_{1}, Δk

_{SFG}= π/d

_{2}, A

_{1}(z = 0) = 1, A

_{2}(z = 0) = 0, and A

_{3}(z = 0) = 0. The total number N of domains was set to 2000 (N/2 for d

_{1}and N/2 for d

_{2}). We normally halted the algorithm within 5000 iterations. The typical results are displayed in Figure 2, showing the normalized intensities η

_{i}(i = 1, 2, 3) at each frequency along the crystal after the 5000th iteration, for unequal domain types in each segment. Figure 3 displays the analog results for equal domain types in each segment. At the last iteration, the THG conversion efficiency reaches the value η

_{3}≈ 65% in the first case (Figure 2), whereas it is close to 100% in the second case (Figure 3). It is noteworthy that, although the numerical algorithm ran for several loops, the above conversion efficiencies from the fundamental to the 3rd harmonic were reached well before the iteration limit. For instance, 65% and 99% were obtained after ~2500 and ~300 iterations in the first and second cases, respectively.

^{2}, a total number of domains N = 600, d

_{33}= 26 pm/V, and d

_{1}≈ 9.07 µm, d

_{2}≈ 3.23 µm [26]. The iteration limit for the Monte Carlo procedure was set to 1000.

_{3}~85% (Figure 4) and η

_{3}~90% (Figure 5) in the two cases with segments of different or equal domain types, respectively. The efficiencies are initially as low as ~3% at the first iteration in both Figure 4 and Figure 5.

_{1}-d

_{2}to d

_{2}-d

_{1}, the resulting conversion efficiency η

_{3}dropped dramatically, confirming the crucial role of each domain in the specific randomly generated sequence.

_{1}/d

_{2}= 2, while in the realistic case $\Delta {k}_{SFG}/\Delta {k}_{SHG}\approx 2.8$ since d

_{1}/d

_{2}≈ 2.8.

_{3}on the input intensity at the fundamental frequency, which is of relevance in view of actual experimental measurements. To this extent, we considered the realistic case of a QPM dual lattice in the APP−LN crystal with equal domain sizes in each segment, as discussed above (see Figure 5). Figure 10 shows some typical results for three values of input power densities, as specified on each line. Clearly, high conversion efficiencies to the third harmonic can be obtained despite intensity variations as large as 100% with respect to the nominal excitation (Figure 5).

## 4. Conclusions

_{p}= ω

_{s}+ ω

_{i}(DFG) and ω

_{i}= ω

_{s}− ω

_{i′}(SFG), with ω

_{i′}as a secondary idler wave could lead to parametric amplification, with the idler ω

_{i′}energy being fed back into the signal. This would take place with the simultaneous quasi-phase-matching of both DFG and SFG processes. For example, assuming λ

_{s}≈ 3.34 µm, λ

_{p}≈ 1.064 µm, λ

_{i}≈ 1.5486 µm, and λ

_{i′}≈ 2.844 µm in a 5% MgO-doped APP-LN, [27], double QPM would require randomized optimum sequences with d

_{1}≈ 15.03 µm and d

_{2}≈ 0.33 µm.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Armstrong, J.A.; Bloembergen, N.; Ducuing, J.; Pershan, P.S. Interactions between Light Waves in a Nonlinear Dielectric. Phys. Rev.
**1962**, 127, 1918–1939. [Google Scholar] [CrossRef] - Franken, P.A.; Hill, A.E.; Peters, C.W.; Weinreich, G. Generation of Optical Harmonics. Phys. Rev. Lett.
**1961**, 7, 118–120. [Google Scholar] [CrossRef] [Green Version] - Akhmanov, S.A.; Dmitriev, V.G.; Modenov, V.P. On the theory of frequency multiplication in nonlinear dispersive lines. Radio Eng. Electron.
**1964**, 9, 814–821. [Google Scholar] - Stegeman, G.I.; Sheik-Bahae, M.; Van Stryland, E.W.; Assanto, G. Large nonlinear phase shifts in second-order nonlinear optical processes. Opt. Lett.
**1993**, 18, 13–15. [Google Scholar] [CrossRef] - Assanto, G.; Stegeman, G.I.; Sheik-Bahae, M.; Van Stryland, E.W. All Optical Switching Devices Based on Large Nonlinear Phase Shifts from Second Harmonic Generation. Appl. Phys. Lett.
**1993**, 62, 1323–1325. [Google Scholar] [CrossRef] - Hagan, D.J.; Sheik-Bahae, M.; Wang, Z.; Stegeman, G.I.; Van Stryland, E.W.; Assanto, G. Phase Controlled Transistor Action by Cascading of Second-Order Nonlinearities in KTP. Opt. Lett.
**1994**, 19, 1305–1307. [Google Scholar] [CrossRef] [PubMed] - Assanto, G.; Stegeman, G.I.; Sheik-Bahae, M.; Van Stryland, E.W. Coherent interactions for all-optical signal processing via quadratic nonlinearities. IEEE J. Quantum Electron.
**1995**, 31, 673–681. [Google Scholar] [CrossRef] - Stegeman, G.I.; Hagan, D.J.; Torner, L. χ
^{(2)}cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons. Opt. Quantum Electron.**1996**, 28, 1691–1740. [Google Scholar] [CrossRef] - Assanto, G.; Stegeman, G.I. Nonlinear Optics Basics: Cascading. In Encyclopedia of Modern Optics; Guenther, R.D., Steel, D.G., Bayvel, L.D., Eds.; Elsevier: Oxford, UK, 2005; Volume 3, pp. 207–212. [Google Scholar]
- Chirkin, A.S.; Volkov, V.V.; Laptev, G.D.; Morozov, E.Y. Consecutive three-wave interactions in nonlinear optics of periodically inhomogeneous media. Quantum Electron.
**2000**, 30, 847. [Google Scholar] [CrossRef] - Boyd, R. Nonlinear Optics, 3rd ed.; Academic Press: San Diego, MA, USA, 2008. [Google Scholar]
- Stegeman, G.I.; Stegeman, R. Nonlinear Optics, Phenomena, Materials and Devices, 1st ed.; J. Wiley & Sons: Ney York, NY, USA, 2012. [Google Scholar]
- Fejer, M.M.; Magel, G.A.; Jundt, D.H.; Byer, R.L. Quasi-phase-matched second harmonic generation: Tuning and tolerances. IEEE J. Quantum Electron.
**1992**, 28, 2631–2654. [Google Scholar] [CrossRef] [Green Version] - Hum, D.S.; Fejer, M.M. Quasi-phase matching. Comptes Rendus Phys.
**2007**, 8, 180–198. [Google Scholar] [CrossRef] - Wang, T.; Chen, P.; Xu, C.; Zhang, Y.; Wei, D.; Hu, X.; Zhu, S. Periodically poled LiNbO
_{3}crystals from 1D and 2D to 3D. Sci. China Technol. Sci.**2020**, 63, 1110–1126. [Google Scholar] [CrossRef] - Sapaev, U.K.; Kulagin, I.A.; Satlikov, N.K.; Usmanov, T. Optimization of third harmonic generation for two coupled three-frequency interactions of waves with multiple frequencies in periodic crystals. Opt. Spectr.
**2006**, 101, 983–985. [Google Scholar] [CrossRef] - Alexandrovsky, A.L.; Chirkin, A.S.; Volkov, V.V. Realization of quasi-phase-matched parametric interactions of waves of multiple frequencies with simultaneous frequency doubling. J. Russ. Laser Res.
**1997**, 18, 101–106. [Google Scholar] [CrossRef] - Volkov, V.V.; Chirkin, A.S. Quasi-synchronous parametric amplification of waves at low-frequency pumping. Quantum Electron.
**1998**, 25, 101–102. [Google Scholar] - Zhang, C.; Zhu, Y.-Y.; Yang, S.-X.; Qin, Y.-Q.; Zhu, S.-N.; Chen, Y.-B.; Liu, H.; Ming, N.-B. Crucial effects of coupling coefficients on quasi-phase-matched harmonic generation in an optical superlattice. Opt. Lett.
**2000**, 25, 436–438. [Google Scholar] [CrossRef] [PubMed] - Komissarova, M.V.; Sukhorukov, A.P. On the properties of a parametric light amplifier with a multiple frequency ratio. Quantum Electron.
**1993**, 20, 1025–1027. [Google Scholar] - Norton, A.H.; de Sterke, C.M. Aperiodic 1-dimensional structures for quasi-phase matching. Opt. Express
**2004**, 12, 841–846. [Google Scholar] [CrossRef] - Yusupov, D.B.; Sapaev, U.K. Multistep third-harmonic generation of femtosecond laser pulses in periodically-poled and chirped-periodically-poled lithium niobate. J. Russ. Laser Res.
**2009**, 30, 321–326. [Google Scholar] [CrossRef] - Sapaev, U.K.; Assanto, G. Engineered quasi-phase matching for multiple parametric generation. Opt. Express
**2009**, 17, 3765–3770. [Google Scholar] [CrossRef] - Zhu, S.; Zhu, Y.; Ming, N. Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice. Science
**1997**, 278, 843–846. [Google Scholar] [CrossRef] - Longhi, S. Third-harmonic generation in quasi-phase-matched media with missing second harmonic. Opt. Lett.
**2007**, 32, 1791–1793. [Google Scholar] [CrossRef] [PubMed] - Dmitriev, V.G.; Gurzadyan, G.G.; Nikogosyan, D.N. Handbook of Nonlinear Optical Crystals, 3rd ed.; Springer: Berlin, Germany, 2013. [Google Scholar]
- Sabirov, O.I.; Yusupov, D.B.; Akbarova, N.A.; Sapaev, U.K. On the theoretical analysis of parametric amplification of femtosecond laser pulses in crystals with a regular domain structure. Phys. Wave Phenom.
**2022**, 30, 277–282. [Google Scholar] [CrossRef]

**Figure 1.**Sketch of the QPM lattice with domains of thicknesses d

_{1}and d

_{2}and alternating sign of the nonlinear response (${\mathsf{\chi}}^{(2)}$). Here, d

_{1}and d

_{2}are the coherence lengths of the interactions 2ω = ω + ω and 3ω = 2ω + ω, respectively. (

**a**) QPM with unequal domain thicknesses in each segment, and (

**b**) QPM with equal domains in each segment.

**Figure 2.**Results of the numerical experiments after 5000 iterations in a dimensionless quadratic sample. Normalized intensities at the three frequencies along the propagation length when the QPM lattice has (see, e.g., Figure 1a) differently sized domains in each segment.

**Figure 4.**Results of the numerical experiments after 1000 iterations with reference to an aperiodically poled (z-cut) Lithium Niobate crystal with a total length of 4 mm. Normalized intensities η

_{i}(i = 1, 2, 3) at the three frequencies along the propagation distance z for different domain sizes in each segment.

**Figure 5.**Same as in Figure 4 but for equal domain sizes in each segment.

**Figure 10.**Intensity conversion efficiency from the fundamental frequency to its 3rd harmonic, calculated for different input intensities (as marked) and the designed APP−LN lattice (Figure 5) with equal domain sizes in each segment.

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**MDPI and ACS Style**

Sabirov, O.I.; Assanto, G.; Sapaev, U.K.
Efficient Third-Harmonic Generation by Inhomogeneous Quasi-Phase-Matching in Quadratic Crystals. *Photonics* **2023**, *10*, 76.
https://doi.org/10.3390/photonics10010076

**AMA Style**

Sabirov OI, Assanto G, Sapaev UK.
Efficient Third-Harmonic Generation by Inhomogeneous Quasi-Phase-Matching in Quadratic Crystals. *Photonics*. 2023; 10(1):76.
https://doi.org/10.3390/photonics10010076

**Chicago/Turabian Style**

Sabirov, Obid I., Gaetano Assanto, and Usman K. Sapaev.
2023. "Efficient Third-Harmonic Generation by Inhomogeneous Quasi-Phase-Matching in Quadratic Crystals" *Photonics* 10, no. 1: 76.
https://doi.org/10.3390/photonics10010076