# Study of Optical Rectification in Polycrystalline Materials Based on Random Quasi-Phase Matching

^{*}

## Abstract

**:**

## 1. Introduction

_{3}, and the tilted-pulse-front method has been proposed and proved to be effective in strong-field broadband terahertz generation [12], with the expense of much higher cost and complexity. If the crystal has cubic structure, e.g., ZnTe, the most famous material for OR pumped by the Ti:Sapphire laser [11], natural anomalous dispersion from optical to terahertz is necessary, otherwise phase matching can never be achieved. It is similar for GaP pumped by Yb-doped 1 μm lasers [13], and GaAs pumped by Er-doped 1.56 μm fiber lasers [14]. An inevitable problem is that it is almost impossible to change the pump wavelength which is strictly determined by the material dispersion of such crystals. An additional electric field induced Pockels effect would be helpful to modulate the refractive indices, but it is effective only in difference frequency mixing requiring two incident beams rather than OR, as demonstrated in our previous study [15].

^{2}) of ultrashort laser pulses in the femtosecond range is much higher than that of nanosecond laser pulses, so the nonlinear gain can be obviously increased by tight focusing. Meanwhile, the most remarkable advantage of RQPM in bandwidth can be fully exploited to support frequency conversion of ultrashort laser pulses. Effective wideband generation from visible to infrared based on RQPM three-wave interactions have been reported [18,19,20], but its application in terahertz sources via OR has not been touched yet.

## 2. Modeling

#### 2.1. Polycrystalline ZnSe Sample and Grain Morphology

_{eff}of every grain. The data of the grain morphology can be directly obtained from the software output. For example, Figure 2c,d gives the grain size distribution of two different samples, which generally followed lognormal functions.

#### 2.2. Theoretical Model of RQPM OR in Polycrystalline ZnSe

_{0}is the time-domain electric field of the pump laser, z is the interaction length, χ

^{(2)}is the nonlinear susceptibility of the material, τ is the duration of the pump pulse, c is the light speed in vacuum, n

_{Ω}is the refractive index of the terahertz wave, and Δk is the wave vector mismatch. Δk can be expressed as Δk = k

_{ω}+ k

_{Ω}− k

_{ω}

_{+Ω}from the view of three-wave mixing, where k

_{ω}, k

_{Ω}and k

_{ω}

_{+Ω}are the wave vectors of two optical frequencies ω, ω + Ω and a terahertz frequency Ω. Considering Ω << ω,

_{g,ω}is the group refractive index of the pump laser, expressed as

_{ω}and λ

_{ω}are the phase refractive index and wavelength at the laser angular frequency ω. Consequently, the coherent length L

_{c}of OR is given by

_{c}is independent to crystalline orientation for cubic ZnSe material. Under undepleted pump approximation, the terahertz electric field is obtained by direct integration of Equation (1):

_{eff}is the effective nonlinear coefficient and it is assumed that χ

^{(2)}= 2d

_{eff}. d

_{eff}is dependent on z due to the variation of crystalline orientation in different grains of a polycrystalline material. If there are m grains along the beam path, the integration in Equation (6) is divided into m parts, written as

_{eff,n}is the effective nonlinear coefficient of each grain (n is an integer and 1 ≤ n ≤ m), and the integration range from z

_{n−1}to z

_{n}is the interaction length in the nth grain. The major difference of a RQPM process compared with BPM and QPM is that d

_{eff}is totally random in polycrystalline materials. The mathematical model, called spherical random pointing [29,30], can be used to describe the random nonlinearity. Once the distribution of d

_{eff}is determined, the terahertz field can be calculated either from step-to-step integration with Equation (7) or based on the Fourier transform method [31].

_{pump}and terahertz fluence F

_{THz}, which are expressed as follows [28]:

## 3. Results and Discussion

#### 3.1. Dependence of Terahertz Generation on Polycrystalline Parameters

#### 3.1.1. Grain Size Distribution

^{2}, which indicates the generated terahertz intensity in the frequency domain, is shown in Figure 3, including the simulation results of seven different polycrystalline ZnSe samples. The mean value of grain size ranged from tens to hundreds of microns while the standard deviation should allow the construction of a realistic polycrystalline model, both directly obtained from Neper output. The frequency dependence is also given in Figure 3 to depict the frequency-domain profile and the bandwidth. The central wavelength of the pump laser was λ

_{0}= 1064 nm which can be realized by Yb

^{3+}-doped fiber lasers, with the time-domain pump intensity of I

_{0}= 30 GW/cm

^{2}and the pump pulse duration of τ = 100 fs. The samples had an identical thickness of 1 mm, which is a common size for various single crystals and is also acceptable in simulation time while ensuring sufficient grain number, but different cross sections to guarantee adequate grains for statistical analysis.

_{c}, and it has been proved that the highest efficiency occurs when the average grain size is close to the coherent length. As shown in Figure 4b, the higher terahertz frequency, the smaller L

_{c}. Therefore, smaller average grain size benefits high-frequency terahertz generation, and the location of the spectral peak moves towards higher frequency monotonically.

#### 3.1.2. Sample Thickness

^{2}. Such samples would contain thousands of grains to guarantee the accuracy of statistical analysis. The simulation results are shown in Figure 5.

_{c}: at low frequencies, ∆k is smaller; thus, the signal accumulation is faster with the increase in interaction length.

#### 3.2. Dependence of Terahertz Generation on Pump Laser Parameters

#### 3.2.1. Central Wavelength of the Pump Laser

^{3}ZnSe sample with the grain size distribution of μ = 82 μm and σ = 35 μm, the terahertz generation was investigated under pump wavelengths of 800, 1064, 1550, 1900 and 2350 nm, respectively, all of which had the same pulse duration of 100 fs and the pump intensity of 30 GW/cm

^{2}. All such lasers are readily available in commercial solid-state or fiber mode-locked laser products. The simulation results are shown in Figure 6.

_{c}as well. Figure 7 gives the distribution of ∆k and L

_{c}related to both the pump wavelength from 800 to 2400 nm and terahertz frequency from 0.1 to 5 THz. At the low-frequency region of the terahertz spectrum (e.g., 0.1–1 THz), the coherent length is much larger than the grain size whatever pump wavelength is used, so that the spectral intensity is almost independent to pump wavelength. At the terahertz frequency range of 2–4 THz, L

_{c}is close to the average grain size. In this case, ∆k becomes larger and L

_{c}gets smaller with the increase in pump wavelength, causing the decline of terahertz spectral intensity and the move of the spectral peak to the high-frequency part. All the results are consistent with that in Figure 6.

#### 3.2.2. Pump Pulse Duration

^{2}at 1064 nm, its spectra with Lorentz type is shown by Figure 8a when the pump pulse duration is increased from 50 to 500 fs. Since the OR process can be regarded as intra-pulse difference frequency mixing, the shorter pump pulse, the more frequency components, so the lower peak spectral intensity and broader terahertz spectrum generation. The simulation results of RQPM OR using a 1 × 1 × 1 mm

^{3}ZnSe sample with the grain size distribution of μ = 82 μm and σ = 35 μm is shown in Figure 8b. The spectrum peak moves from 2.5 THz towards the low-frequency part to 0.6 THz when the pump pulse duration is increased from 50 to 500 fs. At the same time, the high-frequency components of terahertz filed become weaker, resulting in degraded spectrum bandwidth and enhanced peak spectral intensity when the pump pulse duration is increased.

#### 3.3. Conversion Efficiency of RQPM OR

^{2}, respectively. The conversion efficiency reaches 10

^{−4}, which is comparable to OR with single crystals [33].

^{2}. The pump fluence is given in Figure 10b, demonstrating an increasing trend as the pulse durations increases, so that the conversion efficiency can be reflected by comparing Figure 8b and Figure 10b. There is an optimized average grain size for different pulse durations, and the optimal size value is larger when the pump pulse duration gets longer. The OR efficiency varies a lot with pulse durations, where a shorter pulse duration benefits higher terahertz pulse energy. Generally, shorter pump wavelength and pulse duration is good for enhancing the conversion efficiency, but high-order nonlinear effects such as the two-photon absorption and cross-phase modulation would be major problems affecting the RQPM OR efficiency.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- O’Hara, J.F.; Ekin, S.; Choi, W.; Song, I. A perspective on terahertz next-generation wireless communications. Technologies
**2019**, 7, 43. [Google Scholar] [CrossRef] [Green Version] - Wang, H.Q.; Deng, B.; Qin, Y.L. Review of terahertz radar technology. J. Radars
**2018**, 7, 1–21. [Google Scholar] - D’Arco, A.; Di Fabrizio, M.; Dolci, V.; Petrarca, M.; Lupi, S. THz pulsed imaging in biomedical applications. Condens. Matter
**2020**, 5, 25. [Google Scholar] [CrossRef] [Green Version] - Vaks, V.L.; Anfertev, V.A.; Balakirev, V.Y.; Basov, S.A.; Domracheva, E.G.; Illyuk, A.V.; Kupriyanov, P.V.; Pripolzin, S.I.; Chernyaeva, M.B. High resolution terahertz spectroscopy for analytical applications. Physics-Uspekhi
**2020**, 63, 708–720. [Google Scholar] [CrossRef] - Bogue, R. Sensing with terahertz radiation: A review of recent progress. Sens. Rev.
**2018**, 38, 216–222. [Google Scholar] [CrossRef] - Neu, J.; Schmuttenmaer, C.A. Tutorial: An introduction to terahertz time domain spectroscopy (THz-TDS). J. Appl. Phys.
**2018**, 124, 231101. [Google Scholar] [CrossRef] [Green Version] - Zhong, K.; Liu, C.; Wang, M.R.; Shi, J.; Kang, B.; Yuan, Z.R.; Li, J.N.; Xu, D.G.; Shi, W.; Yao, J.Q. Linear optical properties of ZnGeP
_{2}in the terahertz range. Opt. Mater. Express**2017**, 7, 3571–3579. [Google Scholar] [CrossRef] - Fischer, B.; Hoffmann, M.; Helm, H.; Modjesch, G.; Uhd Jepsen, P. Chemical recognition in terahertz time-domain spectroscopy and imaging. Semicond. Sci. Technol.
**2005**, 20, S246–S253. [Google Scholar] [CrossRef] [Green Version] - Naftaly, M.; Miles, R.E. Terahertz time-domain spectroscopy for material characterization. Proc. IEEE
**2007**, 95, 1658–1665. [Google Scholar] [CrossRef] - Zhong, K.; Shi, W.; Xu, D.G.; Liu, P.X.; Wang, Y.Y.; Mei, J.L.; Yan, C.; Fu, S.J.; Yao, J.Q. Optically pumped terahertz sources. Sci. China Technol. Sci.
**2017**, 60, 1801–1818. [Google Scholar] [CrossRef] - Vidal, S.; Degert, J.; Tondusson, M.; Freysz, E.; Oberlé, J. Optimized terahertz generation via optical rectification in ZnTe crystals. J. Opt. Soc. Am. B
**2014**, 31, 149–153. [Google Scholar] [CrossRef] - Helling, J.; Stepanov, A.G.; Almási, G.; Bartal, B.; Kuhl, J. Tunable THz Pulse generation by optical rectification of ultrashort laser pulses with tilted pulse fronts. Appl. Phys. B
**2004**, 78, 593–599. [Google Scholar] [CrossRef] - Aoki, K.; Savolainen, J.; Havenith, M. Broadband terahertz pulse generation by optical rectification in GaP crystals. Appl. Phys. Lett.
**2017**, 110, 201103. [Google Scholar] [CrossRef] - Nagai, M.; Tanaka, K.; Ohtake, H.; Bessho, T.; Sugiura, T.; Hirosumi, T.; Yoshida, M. Generation and detection of terahertz radiation by electro-optical process in GaAs using 1.56 μm fiber laser pulses. Appl. Phys. Lett.
**2004**, 85, 3974–3976. [Google Scholar] [CrossRef] - Zhong, K.; Li, F.J.; Qiao, H.Z.; Zhang, X.Z.; Xu, D.G.; Yao, J.Q. Wideband collinear phase matching in cubic semiconductors via the linear electro-optic effect: A theoretical study. Crystals
**2022**, 12, 764. [Google Scholar] [CrossRef] - Baudrier-Raybaut, M.; Haïdar, R.; Kupecek, P.; Lemasson, P.; Rosencher, E. Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials. Nature
**2004**, 432, 374–376. [Google Scholar] [CrossRef] - Videl, X.; Martorell, J. Generation of light in media with a random distribution of nonlinear domains. Phys. Rev. Lett.
**2006**, 97, 013902. [Google Scholar] [CrossRef] [Green Version] - Vasilyev, S.; Moskalev, I.; Mirov, M.; Smolski, V.; Mirov, S.; Gapontsev, V. Ultrafast middle-IR lasers and amplifiers based on polycrystalline Cr:ZnS and Cr:ZnSe. Opt. Mater. Express
**2017**, 7, 2636–2650. [Google Scholar] [CrossRef] - Ru, Q.T.; Lee, N.; Chen, X.; Zhong, K.; Tsoy, G.; Mirov, M.; Vasilyev, S.; Mirov, S.B.; Vodopyanov, K.L. Optical parametric oscillation in a random polycrystalline medium. Optica
**2017**, 4, 617–618. [Google Scholar] [CrossRef] - Zhang, J.W.; Fritsch, K.; Wang, Q.; Krausz, F.; Mak, K.F.; Pronin, O. Intra-pulse difference-frequency generation of mid-infrared (2.7–20μm) by random quasi-phase-matching. Opt. Lett.
**2019**, 44, 2986–2989. [Google Scholar] [CrossRef] [Green Version] - Ru, Q.T.; Kawamori, T.; Lee, N.; Chen, X.; Zhong, K.; Mirov, M.; Vasilyev, S.; Mirov, S.B.; Vodopyanov, K.L. Optical paramet-ric oscillation in a random poly-crystalline medium: ZnSe ceramic. Proc. SPIE
**2018**, 10516, 166–174. [Google Scholar] - Kawamori, T.; Ru, Q.T.; Vodopyanov, K.L. Comprehensive model for randomly phase-matched frequency conversion in zinc-blende polycrystals and experimental results for ZnSe. Phys. Rev. Appl.
**2019**, 11, 054015. [Google Scholar] [CrossRef] - Liu, K.F.; Zhong, K.; Lu, Z.T.; Xu, D.G.; Yao, J.Q. Effects of grain morphology on nonlinear conversion efficiency of random quasi-phase matching in polycrystalline materials. IEEE Photonics J.
**2020**, 12, 2200910. [Google Scholar] [CrossRef] - Müller, J.S.; Morandi, A.; Grange, R.; Savo, R. Modeling of random quasi-phase-matching in birefringent disordered media. Phys. Rev. Appl.
**2021**, 15, 064070. [Google Scholar] [CrossRef] - Li, H.H. Refractive index of ZnS, ZnSe, and ZnTe and its wavelength and temperature derivatives. J. Phys. Chem. Ref. Data
**1984**, 13, 103–150. [Google Scholar] [CrossRef] - Quey, R.; Renversade, L. Optimal polyhedral description of 3D polycrystals: Method and application to statistical and synchrotron X-ray diffraction data. Comput. Methods Appl. Mech. Eng.
**2018**, 330, 308–333. [Google Scholar] [CrossRef] [Green Version] - Neper: Polycrystal Generation and Meshing. Available online: http://neper.info/ (accessed on 11 August 2022).
- Vodopyanov, K.L. Optical generation of narrow-band terahertz packets in periodically-inverted electro-optic crystals: Conversion efficiency and optimal laser pulse format. Opt. Express
**2006**, 14, 2263–2276. [Google Scholar] [CrossRef] - Marsaglia, G. Choosing a point from the surface of a sphere. Ann. Math. Stat.
**1972**, 43, 645–646. [Google Scholar] [CrossRef] - Muller, M.E. A note on a method for generating points uniformly on n-dimensional spheres. Commun. ACM
**1959**, 2, 19–20. [Google Scholar] [CrossRef] - Zhong, K.; Wang, S.J.; Liu, K.F.; Xu, D.G.; Yao, J.Q. Fourier transform analysis on random quasi-phase-matched nonlinear optical interactions. IEEE Photonics J.
**2022**, 14, 3005505. [Google Scholar] [CrossRef] - Li, D.; Ma, G. Pump-wavelength dependence of terahertz radiation via optical rectification in (110)-oriented ZnTe crystal. J. Appl. Phys.
**2008**, 103, 123101. [Google Scholar] [CrossRef] - Meng, Q.L.; Ye, R.; Zhong, Z.Q.; Yu, J.L.; Zhang, B. Analysis on THz radiation generation efficiency in optical rectification by tilted-pulse-front pumping. J. Infrared Millim. Terahertz Waves
**2015**, 36, 866–875. [Google Scholar] [CrossRef]

**Figure 1.**Polycrystalline ZnSe and its optical property in the terahertz range: (

**a**) photo of the sample; (

**b**) refractive index; (

**c**) absorption coefficient.

**Figure 2.**Polycrystalline ZnSe, its meshed morphology and the grain size distribution: (

**a**) original sample; (

**b**) meshed sample; (

**c**,

**d**) histograms of grain size distributions of a 3 × 3 × 1 mm

^{3}sample and a 1 × 1 × 1 mm

^{3}sample with an average grain size of approximately 800 and 40 μm, respectively.

**Figure 3.**Terahertz intensity generated by RQPM OR with different 1 mm-thick polycrystalline ZnSe samples. The grain size distribution is written as (μ, σ), where μ and σ are the mean value and standard deviation in micron, respectively. The cross section for the (780, 238) sample was 3 × 3 mm

^{2}, it was 2 × 2 mm

^{2}for the (497, 116) and (295, 61) samples, and it was 1 × 1 mm

^{2}for all the left samples.

**Figure 4.**Wave vector mismatch ∆k (

**a**) and coherent length L

_{c}(

**b**) of OR in ZnSe pumped at 1064 nm.

**Figure 5.**Terahertz intensity generated by RQPM OR in polycrystalline ZnSe samples with different thickness: (

**a**) terahertz spectrum; (

**b**) nonlinear terahertz signal growth at certain frequencies. The grain size distribution (μ, σ) was (82, 35) in micron.

**Figure 6.**Terahertz intensity generated by RQPM OR in a polycrystalline ZnSe sample pumped at different wavelengths. The grain size distribution (μ, σ) was (82, 35) in micron, the pump pulse duration was 100 fs and the pump intensity was 30 GW/cm

^{2}.

**Figure 7.**Pump-wavelength-dependent wave vector mismatch ∆k (

**a**) and coherent length L

_{c}(

**b**) of OR in ZnSe.

**Figure 8.**The spectra of the pump pulse at different pulse durations (

**a**) and the corresponding generated terahertz spectra (

**b**). The pump intensity was 30 GW/cm

^{2}at 1064 nm.

**Figure 9.**The conversion efficiency of RQPM OR versus ZnSe grain size at different central pump wavelengths of 800, 1064, 1550, 1900 and 2350 nm. Note the curves of 1900 and 2350 nm are overlapped.

**Figure 10.**The conversion efficiency of RQPM OR versus ZnSe grain size (

**a**) pumped at different pulse durations, with the pump fluence at different pulse durations (

**b**) given as a reference.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, S.; Zhong, K.; Qiao, H.; Li, F.; Li, J.; Xu, D.; Yao, J.
Study of Optical Rectification in Polycrystalline Materials Based on Random Quasi-Phase Matching. *Crystals* **2022**, *12*, 1188.
https://doi.org/10.3390/cryst12091188

**AMA Style**

Wang S, Zhong K, Qiao H, Li F, Li J, Xu D, Yao J.
Study of Optical Rectification in Polycrystalline Materials Based on Random Quasi-Phase Matching. *Crystals*. 2022; 12(9):1188.
https://doi.org/10.3390/cryst12091188

**Chicago/Turabian Style**

Wang, Sijia, Kai Zhong, Hongzhan Qiao, Fangjie Li, Jining Li, Degang Xu, and Jianquan Yao.
2022. "Study of Optical Rectification in Polycrystalline Materials Based on Random Quasi-Phase Matching" *Crystals* 12, no. 9: 1188.
https://doi.org/10.3390/cryst12091188