# Prism Couplers with Convex Output Surfaces for Nonlinear Cherenkov Terahertz Generation

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## Abstract

**:**

_{3}crystal is studied considering Si-prism-lens couplers with different output surface curvatures. A theoretical approach is developed for modeling the angular distributions of THz radiation power inside the crystal, inside the Si coupler and outside in free space. Our calculations show how the imposition of a plano-convex lens on the standard flat prism can substantially improve the THz generation efficiency. The ratio between the lens curvature radius and the distance from the curvature center to the point of generation on the lens axis is found to be one of the most important parameters. The developed general approach can be used for the further design of the optimal THz extraction elements of a different configuration.

## 1. Introduction

_{3}), due to its high nonlinear susceptibility in the THz range and the possibility of manufacturing rather long crystal samples [12,13]. However, the condition of exact phase-matching can be satisfied only for the non-collinear interaction of the optical and THz waves in this crystal. The collinear process is possible under a quasi-phase matching condition within a narrow frequency band, but with the decreased working nonlinear coefficient of the periodically poled crystal [14]. Two different approaches are currently applied to arrange the broadband non-collinear OR processes in Mg:LiNbO

_{3}. The most famous one is based on the use of the pump beam with the tilted pulse front and the prism-shaped Mg:LiNbO

_{3}[15]. The second approach uses the sideways Cherenkov generation scheme with a pump focused into a transversely limited volume in a crystal. To obtain the most radiation from the crystal, the sideways extraction schemes are commonly supplied by the special coupling prism adaptors [12,13,16,17,18]. High-resistant silicon (Si) still remains as the best material choice for such prism adaptors due to its high transparency in the THz range [19]. THz generation efficiencies achieved now by this method are slightly lower than the records of the tilted pulse front approach [6], but the possible potential of the Cherenkov scheme is far from being exhausted.

_{3}crystal, inside the Si-coupler (Section 3) and after passing the spherically shaped Si-coupler surface of the arbitrary convex parameter (Section 4). Finally, we present results calculated for the total THz power, emitted within a detectable angle cone (Section 5). They show that the efficiency of THz generation into the outer space through a convex Si-prism surface can be much more or less than in case of the flat output surface. It was found that the ratio between the lens curvature radius and the distance from the curvature center to the point of generation on the lens axis I s one of the main parameters responsible for the efficiency of THz generation from this point.

## 2. Modeling of THz Generation Inside a Nonlinear Crystal

_{3}crystal at a fixed THz frequency, we used a general theoretical expression for one of the elements of the so-called crystal nonlinear scattering matrix [25] which described the field transformation in a three-wave parametric process. Explicit forms for all the elements were obtained recently in [26] within the low-gain approximation, but taking into account the multimode type of the three-frequency parametric process. The matrix element that is necessary for our further calculations describes the linear conversion of the field amplitude at ${\omega}_{2},{k}_{2}$ to the field amplitude at ${\omega}_{THz},{k}_{THz}$ in the presence of the pump wave at ${\omega}_{1}$ with the spatially limited Gaussian profile:

_{3}, all the three waves should be extraordinarily polarized along Z axis and propagate in the XOY plane (${\phi}_{1,2}=0$, ${\phi}_{THz}={180}^{\xb0}$) to obtain the majority of the crystal nonlinear susceptibility tensor components. However, when the azimuthal angle changes significantly, one has to take into account a decrease in the working Z-projection of the corresponding field. Equation (3) accounts for a convolution of the crystal second-order susceptibility tensor ${\chi}^{(2)}$ with the polarization vectors of interacting waves in the form of ${\chi}^{(2)}\approx \chi \sqrt{1+3{\mathrm{cos}}^{2}{\mathsf{\phi}}_{THz}}$ [27]. ${\mathsf{\mu}}_{THz}$ accounts for the crystal intensity absorption coefficient ${\mathsf{\alpha}}_{THz}$ at THz frequency: ${\mathsf{\mu}}_{THz}\equiv \frac{{\mathsf{\alpha}}_{THz}}{2\mathrm{cos}{\mathsf{\vartheta}}_{THz}}$.

_{p}= 100 μm and did not take into account the crystal absorption, ${\mathsf{\alpha}}_{THz}$ = 0. Figure 1c shows how this distribution changes if we take into account the intensity absorption coefficient ${\mathsf{\alpha}}_{THz}$ = 13 cm

^{−1}. The width of the distribution along the polar angles increased, while the generation efficiency decreased substantially at all angles. The broadening of polar angles also took place in the case of the tighter focusing of the pump beam; see an example in Figure 1d calculated for w

_{p}= 5 μm and ${\mathsf{\alpha}}_{THz}$ = 13 cm

^{−1}. However, the generation efficiency (the power density) was significantly increased due to the different physical nature of this effect. The first exponential multiplier in the integrand in Equation (4) was responsible for the degree of accomplishment of the transverse phase-matching; it was maximal at zero polar angles and became broader when w

_{p}decreased. Other terms determined the longitudinal phase matching and described the observed generation maximum. Its center shifted to a non-zero angle ${\vartheta}_{THz}={\vartheta}_{ph.m.}$, and its width was mostly dependent on the crystal absorption and length, but the power density depended on how wide the wings of the first exponential multiplier were. Increasing w

_{p}inevitably led to the decreasing of this multiplier at the generation angles. According to our calculations made for 1 THz and ${\mathsf{\alpha}}_{THz}$ = 13 cm

^{−1}, when w

_{p}= 150 μm the longitudinal phase matching maximum at ${\vartheta}_{THz}={\vartheta}_{ph.m.}$ decreased to such extent that it became comparable with the weak transverse phase-matching maximum at ${\vartheta}_{THz}=0$; the further increasing of w

_{p}above 200 μm made the sideways generation maximum practically negligible.

_{p}= 100 μm). The crystal absorption decreased at this frequency to ${\mathsf{\alpha}}_{THz}$ = 4 cm

^{−1}. This led to narrowing of the generation line within the polar angle axis and to the increasing of the generation power at the phase-matching polar angle. However, the last effect was not as pronounced since the general coefficient ${\mathsf{\omega}}_{THz}{}^{2}$ in Equation (4) was less by four times. Other parameters of the angular distribution, such as ${\vartheta}_{ph.m.}$ and the shape of azimuthal distribution, practically did not change.

## 3. Angular THz Power Density Distribution Inside the Si-Prism

_{3}) and high-resistive silicon, respectively. Figure 2b shows how the initial angular divergence of the beam generated at 1 THz reshapes in the Si-prism. In our calculations, we used the data on ${n}_{Si}$ from [30]. The angular dependence of the transmittance ${T}_{LN-Si}$ at the crystal-Si interface was accounted by taking Fresnel coefficients for s-polarized and p-polarized THz waves, which were initially generated as extraordinary waves in the crystal. The angular density of THz radiation power in the prism ${S}_{\mathit{Si}}({\vartheta}_{THz}^{Si}{,\phi}_{THz}^{Si},{\omega}_{THz})$ was calculated taking into account a change in the angular parameters upon transition to a new medium, according to

## 4. THz Power Density Distribution Outside the Si-Prism with a Flat or Spherical Output Surface

## 5. Comparison of Si Couplers with Different Convex Output Surfaces

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Directions of laser and THz wave components in the crystal; angular distributions of the power density generated inside the crystal, calculated for (

**b**) 1 THz, w

_{p}= 100 μm, ${\mathsf{\alpha}}_{THz}$ = 0, (

**c**) 1 THz, w

_{p}= 100 μm, ${\mathsf{\alpha}}_{THz}$ = 13 cm

^{−1}, (

**d**) 1 THz, w

_{p}= 5 μm, ${\mathsf{\alpha}}_{THz}$ = 13 cm

^{−1}, and (

**e**) 0.5 THz, w

_{p}= 100 μm, ${\mathsf{\alpha}}_{THz}$ = 4 cm

^{−1}.

**Figure 2.**(

**a**) Directions of the THz wave components in the crystal and inside the Si coupler; (

**b**) angular distribution of the power density at 1 THz inside the Si coupler.

**Figure 3.**(

**a**) Directions of THz wave components in the crystal, inside and outside the Si-prism-lens coupler; (

**b**) angular distribution of the power density at 1 THz outside the flat Si-prism (lensless) coupler.

**Figure 4.**(

**a**) Sketch for the calculation of the lens parameter $\zeta \equiv a/R$; (

**b**–

**d**) calculated angular distributions of 1 THz radiation power in the air after passing the prism-lens coupler with $\zeta $ = 1.2 (

**b**), $\zeta $ = 1 (

**c**), and $\zeta $ = 0.68 (

**d**).

**Figure 5.**(

**a**) The total power ratio ${P}_{THz}$/${P}_{THz}^{lensless}$ at 1 THz depending on the Si lens parameter $\zeta $; (

**b**) angular distribution of 1THz power outside the prism-lens coupler calculated for $\zeta $ = 0.15.

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**MDPI and ACS Style**

Kitaeva, G.K.; Markov, D.A.; Safronenkov, D.A.; Starkova, N.V.
Prism Couplers with Convex Output Surfaces for Nonlinear Cherenkov Terahertz Generation. *Photonics* **2023**, *10*, 450.
https://doi.org/10.3390/photonics10040450

**AMA Style**

Kitaeva GK, Markov DA, Safronenkov DA, Starkova NV.
Prism Couplers with Convex Output Surfaces for Nonlinear Cherenkov Terahertz Generation. *Photonics*. 2023; 10(4):450.
https://doi.org/10.3390/photonics10040450

**Chicago/Turabian Style**

Kitaeva, Galiya Kh., Dmitrii A. Markov, Daniil A. Safronenkov, and Natalia V. Starkova.
2023. "Prism Couplers with Convex Output Surfaces for Nonlinear Cherenkov Terahertz Generation" *Photonics* 10, no. 4: 450.
https://doi.org/10.3390/photonics10040450