Investigation of the Efficiency of Terahertz Radiation Generation in a Nonlinear Crystal Depending on the Parameters of Exciting Laser Beams

Abstract

The generation of the THz radiation in a nonlinear single-domain lithium niobate crystal is considered. The lasing efficiency is investigated as a function of the angle between the directions of propagation of the exciting laser beams at different beam radii and the THz radiation wavelength. The angular distribution of the generated radiation is investigated. It is shown that it is possible to obtain the highly directional THz radiation with the possibility of adjusting the directional pattern and spectral composition with an appropriate choice of the radii of the exciting beams, the angle between them, as well as the size and shape of the nonlinear crystal.

1 INTRODUCTION

Terahertz (THz) waves (∼0.1–10 THz) are of considerable interest for spectroscopy, medical diagnostics, visualization of various objects, security systems, etc. [1, 2]. Despite this, the region remains one of the technically poorly equipped parts of the spectrum because of the lack of highly efficient and affordable terahertz sources. In recent decades, the generation of a difference frequency and optical rectification of femtosecond laser pulses in a nonlinear crystal has been widely used to obtain terahertz radiation.

For an efficient generation of THz radiation, it is necessary to ensure the wave matching of the interaction. As shown in many theoretical (see, e.g., [3, 4]) and experimental works (see, eg, [5, 6]), the THz radiation power increases in proportion to the square of the intensity of the exciting laser radiation. Consequently, a higher THz radiation power can be achieved by increasing the power and focusing on the exciting laser beams. At focusing, when the transverse dimensions of the beam region become much less than the THz wavelength r $\ll$ λ, the Cherenkov mechanism of generation comes into force [3]. In this case, sufficiently effective THz radiation is obtained at the Cherenkov angle. For example, the broadband THz radiation was obtained by focusing laser beams up to 70 u and 40 u in [6, 7]. However, when using the modern femtosecond lasers of high peak power (≳1 GW), the strong focusing of laser beams is unacceptable owing to the limited electric strength of the nonlinear crystal. Consequently, an increase in the THz radiation power can be achieved only by increasing the apertures of the exciting beams. In this case, certain methods should be applied to ensure the condition of wave synchronism to obtain an effective conversion of laser radiation into terahertz radiation.

It was shown in [8–10] that by using wide-aperture beams in a lithium niobate crystal periodically poled in the transverse direction, the generation of quasi-monochromatic terahertz radiation with a central frequency determined by the spatial period Λ of a periodically poled lithium niobate crystal (PPNL) can be achieved.

However, in this case, the generation frequency is predetermined by the spatial period of the domain structure of the PPNL and therefore it cannot be changed after the preparation of the samples. Besides, the very fabrication of PPNL crystals with a large cross-section is associated with certain difficulties.

To overcome these difficulties, single-domain lithium niobate crystals equipped with shadow or binary phase masks (PM) were used in [11, 12], and for efficient generation of broadband terahertz pulses, in [13] it was proposed to use a step phase mask.

Below, we consider the generation of THz radiation in a nonlinear single-domain lithium niobate crystal. The lasing efficiency is investigated as a function of the angle between the directions of propagation of the exciting beams at different ratios r/λ. The angular distribution of the generated radiation is investigated, too.

2 WAVE EQUATION FOR THE ANGULAR SPECTRUM OF THE THZ FIELD

Suppose that a doublet of Gaussian exciting laser beams propagate in the хz plane and the direction of propagation of one of the waves (with a frequency ω₁) coincides with the z-axis (see Fig. 1), i.e.,

$${{E}_{1}}(x,y,z,t) = {{E}_{{10}}}\exp \left[ { - \frac{{{{x}^{2}} + {{y}^{2}}}}{{{{r}^{2}}}}} \right]\exp [i({{\omega }_{1}}t - {{k}_{1}}z)],$$

(1)

$${{E}_{2}}(x,y,z,t) = {{E}_{{20}}}\exp \left[ { - \frac{{{{{x'}}^{2}} + {{{y'}}^{2}}}}{{{{r}_{2}}^{2}}}} \right]\exp [i({{\omega }_{2}}t - {{k}_{2}}z')],$$

(2)

where $x' = x{\kern 1pt} \cos {\kern 1pt} \psi + z{\kern 1pt} \sin {\kern 1pt} \psi$, $y' = y$, $z' = z{\kern 1pt} \cos {\kern 1pt} \psi + x{\kern 1pt} \sin {\kern 1pt} \psi$, $\psi$ is the angle between the directions of propagation of the exciting laser beams.

Fig. 1.

(a) the mutual arrangement of laser beams and a nonlinear crystal, (b) the coordinate system and direction of THz wave emission, and (c) the wave vectors diagram at vector synchronism.

To simplify the mathematical computations, we assume that the effective radii of both exciting beams are the same (r₁ = r₂ = r).

The wave equation for the angular spectral components of the THz field has the following form [3]:

$$\frac{{{{d}^{2}}E(\alpha ,\beta ,z,{{\Omega }_{0}})}}{{d{{x}^{2}}}} + {{g}^{2}}E(\alpha ,\beta ,z,{{\Omega }_{0}}) = - \gamma F(\alpha ,\beta ,z),$$

(3)

where: $\gamma = 4\pi {{\chi }_{{{\text{ef}}}}}{{\left( {{\Omega \mathord{\left/ {\vphantom {\Omega c}} \right. \kern-0em} c}} \right)}^{2}}$, $k = {{\Omega {{n}_{\Omega }}} \mathord{\left/ {\vphantom {{\Omega {{n}_{\Omega }}} c}} \right. \kern-0em} c}$, $\Omega = {{\omega }_{2}} - {{\omega }_{1}}$ is the frequency of THz radiation, $\alpha = k{\kern 1pt} \sin {\kern 1pt} \theta \cos {\kern 1pt} \varphi$, $\beta = k\sin \theta \sin \varphi$, $g = \sqrt {{{k}^{2}} - {{\alpha }^{2}} - {{\beta }^{2}}} = k{\kern 1pt} \cos {\kern 1pt} \theta$, ${{\chi }_{{{\text{ef}}}}}$ is the effective nonlinear susceptibility of the medium, ${{n}_{\Omega }}$ is the refractive index of the medium at terahertz frequency Ω; the function F(α, β, z) is determined by the convolution of the angular spectra of the exciting beams:

$$F(\alpha ,\beta ,z) = \int {\int\limits_{ - \infty }^\infty {{{E}_{1}}({{\xi }_{1}},z)} } E_{2}^{*}({{\xi }_{2}},z)\delta (\xi + {{\xi }_{1}} - {{\xi }_{2}})d{{\xi }_{1}}{{\xi }_{2}},$$

(4)

where $d\xi = d{{\alpha }_{j}}d{{\beta }_{j}}$.

Let us solve the problem in the approximation of given fields, which usually corresponds to a real experimental situation.

For the convolution of the angular spectra of the exciting waves, we obtain:

$$F(\alpha ,\beta ,z) = {{E}_{{10}}}{{E}_{{20}}}D(\alpha ,\beta )\exp ( - a{{z}^{2}} - ibz),$$

(5)

where

$$D(\alpha ,\beta ) = \frac{{2{{\pi }^{3}}{{r}^{2}}}}{{\sqrt {1 + {{{\cos }}^{2}}{\kern 1pt} \psi } }}\exp \left[ { - \frac{{{{r}^{2}}{{\beta }^{2}}}}{8} - {{{\frac{{{{r}^{2}}(\alpha - {{k}_{2}}{\kern 1pt} \sin {\kern 1pt} \psi )}}{{4(1 + {{{\cos }}^{2}}{\kern 1pt} \psi )}}}}^{2}}} \right],$$

$$a = \frac{{{{{\sin }}^{2}}{\kern 1pt} \psi }}{{{{r}^{2}}(1 + {{{\cos }}^{2}}{\kern 1pt} \psi )}},\,\,\,\,b = {{k}_{2}}{\kern 1pt} \cos {\kern 1pt} \psi - {{k}_{1}} + \frac{{\sin {\kern 1pt} \psi \cos {\kern 1pt} \psi ({{k}_{2}}{\kern 1pt} \sin {\kern 1pt} \psi - \alpha )}}{{1 + {{{\cos }}^{2}}{\kern 1pt} \psi }}.$$

3 ANGULAR DISTRIBUTION OF THZ RADIATION

When solving equation (3), the factor exp(–az²) in formula (5) can be replaced by unity. This follows from the following considerations: it is natural to expect that the most intense THz radiation will be obtained if the vector matching is ensured when the exciting laser beams propagate at an angle $\psi = {{\psi }_{0}}$ (see Fig. 1b).

Thus, for example, for the LINbO₃ crystal upon generation of a THz wave with a length of 0.3 mm (the wavelength of the exciting waves is ∼1 μm), this angle is only 0.05°. Consequently, even at sufficiently large crystal lengths (z ~ 20 mm), the relation $a{{z}^{2}} \ll 1$ holds to be true, that is exp(–az²) ≈ 1.

In this approximation, solving equation (1) for the angular power spectrum of the THz radiation, we obtain [4]

$$\begin{gathered} P(\alpha ,\beta ,z) = \frac{{{{\gamma }^{2}}}}{{4{{\pi }^{2}}}}{{r}^{4}}E_{{10}}^{2}E_{{20}}^{2}{{z}^{2}}\frac{{({{{\sin }}^{2}}{\kern 1pt} \varphi {{{\sin }}^{2}}{\kern 1pt} \theta + {{{\cos }}^{2}}\theta )}}{{{{{(b + g)}}^{2}}}}\frac{{{\text{sin}}{{{\text{c}}}^{2}}\left[ {(b - g){z \mathord{\left/ {\vphantom {z 2}} \right. \kern-0em} 2}} \right]}}{{1 + {{{\cos }}^{2}}{\kern 1pt} \psi }} \hfill \\ \quad \quad \quad \quad \times \exp \left[ { - \frac{{{{r}^{2}}{{\beta }^{2}}}}{4} - \frac{{{{r}^{2}}{{{(\alpha - {{k}_{2}}{\kern 1pt} \sin (\psi ))}}^{2}}}}{{2(1 + {{{\cos }}^{2}}{\kern 1pt} \psi )}}} \right]. \hfill \\ \end{gathered} $$

(6)

In formula (6), the factor $\sin{{c}^{2}}\left[ {(b - g){z \mathord{\left/ {\vphantom {z 2}} \right. \kern-0em} 2}} \right]$ determines the synchronism of the interaction of the angular components of the spectrum of exciting waves, and the exponent is caused by the angular spectrum of the nonlinear polarization wave. In solving it, it was assumed that the polarization of the exciting beams and the optical axis of the nonlinear crystal was chosen so that the nonlinear polarization vector at the THz frequency was directed along y. This circumstance is taken into account by introducing the factor $({{\sin }^{2}}{\kern 1pt} \varphi {{\sin }^{2}}{\kern 1pt} \theta + {{\cos }^{2}}{\kern 1pt} \theta )$ in (6).

Figure 2 shows the angular distributions of THz radiation generated in a nonlinear crystal LiNbO₃ (n_ω ≈ 2.23, n_Ω ≈ 5.1) by parallel exciting beams (ψ = 0), at different ratios r/λ. The laser beams are polarized in y, and the optical axis of the crystal is also directed in y. In this case, the largest nonlinear coefficient of the crystal χ₃₃ acts. Crystal length z = 5 mm, the THz radiation frequency is f = 1 THz (λ = 0.3 mm).

Fig. 2.

Angular distributions of the radiation power with λ = 0.3 mm generated in a nonlinear LiNbO₃ crystal at ψ = 0. Crystal length z = 5 (a) r = 0.05, (b) r = 0.067, (c) r = 1 mm.

As noted above, the Cherenkov radiation mechanism comes into force at ${r \mathord{\left/ {\vphantom {r {\lambda \ll 1}}} \right. \kern-0em} {\lambda \ll 1}}$, and the main power is emitted at an angle θ₀ to the z-axis, where the Cherenkov radiation mechanism takes effect and the main power is emitted at an angle to the z-axis, where

$${{\theta }_{0}} = \arccos \frac{g}{k}.$$

For a lithium niobate crystal, this angle is ∼64°.

With an increase in the radii of the exciting beams, the conversion efficiency decreases (taking into account the fact that the laser powers increase with increasing radii), and the radiation is gradually concentrated around the z-axis.

The picture changes dramatically when the laser beams propagate at the vector phase-matching angle ψ₀. In Fig. 3 the angular distributions of THz radiation at ψ = ψ₀ are represented.

Fig. 3.

Angular distributions of the radiation power with λ = 0.3 mm, generated in a nonlinear LiNbO₃ crystal at ψ = ψ₀. Crystal length is z = 5 (a) r = 0.0167, (b) r = 0.05, (c) r = 3.34 mm.

For small values of r, when the transverse dimensions of the crystal region occupied by nonlinear polarization are less than the THz wavelength (see Fig. 3a), non-directional radiation is obtained, however, because the wave of exciting dipoles propagates with light speed (C/n_ω), the radiation is mainly concentrated near the Cherenkov angle. With an increase in the apertures of laser beams, the entire volume occupied by the nonlinear polarization gradually turns into a phased antenna array, all of the dipoles of which radiate in phase at the Cherenkov angle. Thus, with an appropriate choice of the angle (ψ = ψ₀) it is possible to increase the conversion efficiency relative to the case of collinear interaction (ψ = 0) by several orders. Besides, at $r \gg \lambda$, the radiation is highly acute directional. For example, r = 3.34 mm, λ = 0.3 mm, z = 5 mm, ψ = ψ₀ = 0.393°, the uncovering of the diagram at 3 dB level is Δθ ≈ 0.458° and Δφ ≈ 0.423°. Note that the acute directivity of the radiation makes it easy to remove all the power from the nonlinear crystal with minimal losses to internal reflection, at a corresponding cut of the output end of the crystal.

When terahertz radiation is generated by the method of optical rectification of ultrashort laser pulses, individual dipoles of a nonlinear crystal emit broadband radiation. Therefore, when choosing the optimal angle ψ for a certain THz frequency, other spectral components will be generated inefficiently. In Fig. 4. the curves of the radiation efficiency are presented at the optimal angle for λ = 0.3 mm, and for different values r.

Fig. 4.

The radiation efficiency at the optimal angle for λ = 0.3 mm, (1) r = 5, (2) r = 1, (3) r = 0.2 mm.

As can be seen from the graphs, it is necessary to reduce the size of the beams to increase the spectral bandwidth.

4 CONCLUSION

An analysis of the results obtained shows that with an appropriate choice of the radii r of the exciting beams, the angle ψ between them, as well as the size and shape of the nonlinear crystal, it is possible to obtain the pencil-beam THz radiation with the possibility of adjusting the directional pattern and spectral composition. In particular, if it is necessary to generate quasi-monochromatic THz radiation, the radii of the exciting beams should be increased as much as possible.

With an optimal choice of laser beam parameters, the conversion efficiency can be increased by several orders of magnitude as compared to the collinear conversion.