1 INRODUCTION

Much attention has been paid in the last decades to problems of efficient generation of terahertz radiation. This radiation finds nontrivial applications concerning safety, image reconstruction, medicine, spectroscopy of various media, etc. [1–3].

Studies of the interaction of terahertz radiation with matter are also of interest for fundamental physics [4–7]. The intensities of terahertz signals currently generated in laboratories are so high [8, 9] that the development of terahertz nonlinear optics becomes imperative necessity [4–7].

One of the most efficient methods for the generation of terahertz radiation is based on the effect of optical rectification of subpicosecond and femtosecond laser pulses in second-order nonlinear media [10–12]. terahertz pulses generated by this method have about a single-cycle duration. Thus, they are broadband ultrashort pulses. Consequently, their dynamics cannot be described theoretically in the slowly varying envelope approximation [13]. At the same time, input optical pulses with a duration of tens and hundreds of femtoseconds can be treated with a high accuracy as quasimonochromatic with a certain carrier frequency ω. Therefore, the slowly varying envelope approximation is applicable to them.

The Cherenkov matching condition \({{v}_{{\text{g}}}}\)cosθ = c/nT is important for the efficient generation of terahertz signals described above [10–12]. Here, \({{v}_{{\text{g}}}}\) is the group velocity of an optical pulse in the considered medium, θ is the angle between the propagation directions of the given optical pulse and the generated terahertz signal, c is the speed of light in vacuum, and nT is the refractive index of the medium at terahertz frequencies.

If the angle θ is nonzero, the optical and terahertz signals propagate noncollinearly. In this case, the generated terahertz signal does not continuously acquire energy because it is separated from the optical. As a result, the energy generation efficiency is as low as about 10−6 [11, 12]. The generation efficiency can be increased by ensuring the collinear propagation of the given and generated signals, i.e., under the Cherenkov condition with θ = 0. The corresponding matching condition has the form \({{v}_{{\text{g}}}}\) = c/nT and is referred to in the theory of nonlinear waves as Zakharov–Benney resonance [14]. Under the condition of Zakharov–Benney resonance, the nonlinear generation of long-wavelength pulses can be induced by short-wavelength signals. This generation occurs in the plasma physics [15], hydrodynamics [16], and physics of magnetic media [17] and ling molecules [18].

Zakharov–Benney resonance in the optical method for the generation of terahertz radiation can be ensured in semiconductor crystals [19]. However, damping of terahertz electromagnetic waves in these crystals is very high. Consequently, a high generation efficiency cannot be achieved. The optical group velocity \({{v}_{{\text{g}}}}\) in dielectric crystals with high second-order nonlinearity is usually much higher than the terahertz phase velocity c/nT [11, 12]. Therefore, Zakharov–Benney resonance is impossible.

The generation efficiency under experimental conditions was significantly increased using optical pulses with tilted wave fronts [20–24]. In this case, the angle θ in the Cherenkov condition is the angle between the phase and group fronts/velocities of a given optical pulse and the energies of the optical and generated broadband terahertz signals are transferred collinearly. As a result, the energy generation efficiency increases by several orders of magnitude [25].

As shown in [26], the generation process is described by the nonlinear integrable Yajima–Oikawa system of equations, which has a soliton solution [27]. In the case of optical generation of terahertz radiation, this means the formation of an optical–terahertz soliton. A similar study with a generalized Yajima–Oikawa system including the dispersion of the terahertz component was performed in [28] under the assumption of Zakharov–Benney resonance, which is hardly possible in practice, as mentioned above.

The soliton regime of generation induced by optical pulses with tilted wave fronts was considered in [29]. It was shown that the process of generation in this case is described by the Yajima–Oikawa–Kadomtsev–Petviashvili system. Since the unidirectional propagation approximation was used to derive the wave equation for terahertz signals [30–32], the tilt angle θ of wave fronts was assumed to be small. This is not necessarily the case in experiments. In particular, to ensure the Cherenkov condition in a lithium niobate crystal, the tilt angle should be θ ≈ 67° [11, 12]. Hence, it is necessary to examine the soliton regime of generation of broadband terahertz signals with quasimonochromatic optical pulses with an arbitrary (not necessarily small) angle between the phase and group velocities. The corresponding system of equations called the Zakharov–Boussinesq system was derived beyond the unidirectional propagation approximation for the terahertz component in [33], where the process of generation was studied using one of the soliton-like solutions.

This work is aimed at the physical analysis of a new soliton-like solution of this system describing the generation of terahertz radiation.

According to a pronounced tendency in the physical literature in the last decades, we will use the terms soliton and soliton-like solution for the same objects regardless differences concerning the mathematical property integrability.

2 OPTICAL–TERAHERTZ SOLITONS OF THE ZAKHAROV–BOUSSINESQ SYSTEM

Let the phase velocity of an optical pulse with a tilted wave front be directed along the z axis perpendicular to the x optical axis of the uniaxial crystal. Then, the Zakharov–Boussinesq system of equations for the complex envelope \(\psi \) of the electric field of the optical pulse and the field E of the generated terahertz component has the form [33]

$$i\left( {\frac{{\partial \psi }}{{\partial z}} + \frac{1}{{{{v}_{{\text{g}}}}}}\frac{{\partial \psi }}{{\partial t}}} \right) = - \frac{\beta }{2}\frac{{{{\partial }^{2}}\psi }}{{\partial {{t}^{2}}}} + \alpha E\psi + \frac{c}{{2{{n}_{\omega }}\omega }}\frac{{{{\partial }^{2}}\psi }}{{\partial {{x}^{2}}}},$$
(1)
$$\frac{{{{\partial }^{2}}E}}{{\partial {{z}^{2}}}} - \frac{{n_{{\text{T}}}^{2}}}{{{{c}^{2}}}}\frac{{{{\partial }^{2}}E}}{{\partial {{t}^{2}}}} = \frac{{{{\partial }^{2}}}}{{\partial {{t}^{2}}}}(\mu {{E}^{2}} + \sigma {\text{|}}\psi {{{\text{|}}}^{2}}) - \gamma \frac{{{{\partial }^{4}}E}}{{\partial {{t}^{4}}}} - \frac{{{{\partial }^{2}}E}}{{\partial {{x}^{2}}}}.$$
(2)

Here, \({{n}_{\omega }}\) and \({{n}_{{\text{T}}}} = \sqrt {1 + 4\pi {{\chi }_{{\text{T}}}}} \) are the optical and terahertz refractive indices, respectively; χT is the terahertz linear susceptibility of the medium; \(\beta = \partial v_{{\text{g}}}^{{ - 1}}{\text{/}}\partial \omega \) is the group velocity dispersion of the optical component; \(\gamma = 2\pi {{({{\partial }^{2}}{{\chi }_{{\text{T}}}}{\text{/}}\partial {{\omega }^{2}})}_{{\omega = 0}}}{\text{/}}{{c}^{2}}\) > 0 is the dispersion of the terahertz component; \(\alpha = 4\pi \omega {{\chi }^{{(2)}}}(\omega ;0){{v}_{{\text{g}}}}{\text{/}}{{c}^{2}}\); μ = \(4\pi {{\chi }^{{(2)}}}(0;0){\text{/}}{{c}^{2}}\); \(\sigma = 8\pi {{\chi }^{{(2)}}}(\omega ; - \omega ){\text{/}}{{c}^{2}}\); and \({{\chi }^{{(2)}}}({{\omega }_{1}};{{\omega }_{2}})\) is the second-order nonlinear optical susceptibility depending on the frequencies \({{\omega }_{1}}\) and \({{\omega }_{2}}\).

Here, we assume that both the optical pulse and the generated terahertz signal are planar. For this reason, the second derivatives with respect to the transverse \(y\) coordinate are absent in the system of Eqs. (1) and (2).

The system of Eqs. (1) and (2) with \(\mu = \gamma = 0\) disregarding the diffraction of both components is transformed to the one-dimensional Zakharov equations [15, 17]. At \(\psi = 0\), Eq. (1) becomes the identity 0 ≡ 0, and Eq. (2) is modified to the two-dimensional Boussinesq equation [34]. For this reason, the system of Eqs. (1) and (2) was called in [33] the Zakharov–Boussinesq system; this name is also used in this work.

In [33], we obtained and analyzed in detail the solution of the system of Eqs. (1) and (2) in the form of an optical–terahertz soliton with a tilted wave front of the optical component.

In this work, we obtained another soliton-like solution of this system in the form

$$\begin{array}{*{20}{c}} {\psi = {{\psi }_{m}}{{e}^{{iqz}}}{\text{sech}}\left( {\frac{{t - z{\kern 1pt} '{\text{/}}v}}{\tau }} \right),} \\ {E = - {{E}_{m}}{\text{sec}}{{{\text{h}}}^{2}}\left( {\frac{{t - z{\kern 1pt} '{\text{/}}v}}{\tau }} \right),} \end{array}$$
(3)

where the \(z{\kern 1pt} '\) axis along which the energy of both components is transferred makes the angle θ with the z axis and is related to the latter axis and the optical x axis by the rotation transformation:

$$z{\kern 1pt} ' = z\cos \theta + x\sin \theta .$$
(4)

The cosine of the angle θ and the amplitudes of the soliton components are given by the formulas

$$\cos \theta = \frac{1}{{\sqrt {1 + \eta } }},$$
(5)
$$\eta = \frac{{{{n}_{\omega }}\omega v_{{\text{g}}}^{2}}}{c}(\beta - {{\beta }_{{\text{c}}}}),\quad {{\beta }_{{\text{c}}}} = \frac{{6\alpha \gamma }}{\mu },$$
(6)
$${{\psi }_{m}} = \frac{1}{\tau }\sqrt {\frac{{6\gamma }}{{\sigma \mu }}\left( {\frac{1}{{v_{{\text{B}}}^{2}}} - \frac{1}{{{{v}^{2}}}}} \right)} ,\quad {{E}_{m}} = \frac{{6\gamma }}{{\mu {{\tau }^{2}}}}.$$
(7)

The velocities \(v\) and \({{v}_{{\text{B}}}}\) of the considered soliton and soliton of the Boussinesq equation, respectively, are determined by the expressions

$$v = {{v}_{{\text{g}}}}\cos \theta ,\quad \frac{1}{{{{v}_{{\text{B}}}}}} = \sqrt {\frac{{n_{{\text{T}}}^{2}}}{{{{c}^{2}}}} - \frac{{4\gamma }}{{{{\tau }^{2}}}}} ,$$
(8)

and the nonlinear addition to the wavenumber of the optical pulse is given by the formula

$$q = \frac{{3\alpha \gamma }}{{\mu {{\tau }^{2}}}}.$$
(9)

The duration τ of the optical component is a free parameter in the solution given by Eqs. (3)–(9). According to Eq. (3), the duration of the terahertz component is \( \sim {\kern 1pt} \tau {\text{/}}2\).

It is important that the diffraction of the optical pulse described by the last term on the right-hand side of Eq. (1) plays a fundamental role in the formation of the optical–terahertz soliton specified by Eqs. (3)–(9). Indeed, the neglect of this diffraction is equivalent to the formal condition \({{n}_{\omega }} \to \infty \). In this case, according to Eqs. (5) and (6), \(\eta \to \infty \) and \(\cos \theta = 0\). Then, the first of Eqs. (8) gives \(v = 0\) and the soliton disappears according to Eqs. (3) and (7).

The expression for the amplitude of the terahertz component of the soliton-like solution in Eqs. (7) coincides with the expression for the amplitude of the soliton of the Boussinesq equation. At the same time, the velocity of both jointly propagating components is fixed and is independent of the free parameter τ (see Eqs. (5), (6) and the first of Eqs. (8)). The situation for solitons with tilted wave fronts of the modified nonlinear Schrödinger equation is similar [35].

It is important that the tilt angle \(\theta \) of the wave front of the optical component is also fixed in the considered solution (see Eqs. (5) and (6)).

The terahertz component (3) of the soliton is a unipolar (half-wave) electromagnetic pulse. The nonlinear optics of unipolar pulses is currently under quite fast development (see, e.g., [36]).

The second (dispersion) term in the radicand in the expression determining the velocity \({{v}_{{\text{B}}}}\) in Eqs. (8) should be considered as a small correction to the first term. Consequently, this radicand is always positive. This is particularly obvious since the frequency dependence of the susceptibility in the terahertz range has the Lorentzian form \({{\chi }_{{\text{T}}}}(\omega ) = \frac{{\omega _{{\text{T}}}^{2}{{\chi }_{{\text{T}}}}}}{{\omega _{{\text{T}}}^{2} - {{\omega }^{2}}}}\), where ωT is the characteristic resonance frequency of terahertz absorption. Then, \({{({{\partial }^{2}}{{\chi }_{{\text{T}}}}(\omega ){\text{/}}\partial {{\omega }^{2}})}_{{\omega = 0}}} = 2{{\chi }_{{\text{T}}}}{\text{/}}\omega _{{\text{T}}}^{2}\). Therefore,

$$\gamma = \frac{{n_{{\text{T}}}^{2} - 1}}{{{{c}^{2}}\omega _{{\text{T}}}^{2}}}.$$
(10)

The substitution of Eq. (10) into the second of Eqs. (8) gives

$$\frac{1}{{{{v}_{{\text{B}}}}}} = \frac{{{{n}_{{\text{T}}}}}}{c}\sqrt {1 - 4\frac{{n_{{\text{T}}}^{2} - 1}}{{{{{({{n}_{{\text{T}}}}{{\omega }_{{\text{T}}}}\tau )}}^{2}}}}} .$$
(11)

The expansion of the dispersion in the time derivatives of the electric field is valid under the condition (ωTτ)2 ≫ 1 [31, 32, 37]. It is seen that the radicand in Eq. (11) is positive.

The substitution of Eq. (10) into the second of Eqs. (6) yields

$${{\beta }_{{\text{c}}}} = 6(n_{{\text{T}}}^{2} - 1)\frac{{\omega {{v}_{{\text{g}}}}}}{{{{c}^{2}}\omega _{{\text{T}}}^{2}}}\frac{{{{\chi }^{2}}(\omega ;0)}}{{{{\chi }^{{(2)}}}(0;0)}}.$$
(12)

Expressions (5) and (6) the obligatory condition

$$\beta \;-\;{{\beta }_{{\text{c}}}} > 0.$$
(13)

Similarly, taking into account Eqs. (8), (10), and (11), the condition of the positive radicand in the first of Eqs. (7) can be represented in the form

$$\frac{{{{\chi }^{{(2)}}}(\omega ; - \omega )}}{{{{\chi }^{{(2)}}}(0;0)}}\left[ {1 - {{{\left( {\frac{c}{{{{n}_{{\text{T}}}}{{v}_{{\text{g}}}}\cos \theta }}} \right)}}^{2}} - \frac{{n_{{\text{T}}}^{2} - 1}}{{{{{({{n}_{{\text{T}}}}{{\omega }_{{\text{T}}}}\tau )}}^{2}}}}} \right] > 0.$$
(14)

The general conservation law of the electric area of a unidirectionally propagating pulse, which was established in [38] and was demonstrated on examples in [39], should be satisfied in the considered process of generation. Since the optical component is a quasimonochromatic envelope soliton, its electric area is always zero. The terahertz component is absent at the input of the crystal. Hence, its electric area is also zero. According to the conservation law of the electric area, this quantity should be zero inside the crystal as well. Thus,

$${{S}_{E}} \equiv \int\limits_{ - \infty }^{ + \infty } Edt = 0.$$
(15)

The area of the terahertz component of the soli-ton (3) is \(S_{E}^{{(s)}} = - 12\gamma {\text{/}}\mu \tau \) and does not satisfy the condition (15). Consequently, a nonsoliton terahertz component should appear in the process of generation. According to Eq. (15), the area of this nonsoliton component should be equal to \( - S_{E}^{{(s)}} = + 12\gamma {\text{/}}\mu \tau \). The form of this purely terahertz component can be determined by setting \(\psi = 0\) in Eq. (2). Takin into account that \({{\partial }^{2}}{\text{/}}\partial {{z}^{2}} + {{\partial }^{2}}{\text{/}}\partial {{x}^{2}} = {{\partial }^{2}}{\text{/}}\partial z{\kern 1pt} {{'}^{2}} + {{\partial }^{2}}{\text{/}}\partial x{\kern 1pt} {{'}^{2}}\) and considering propagation only along the \(z{\kern 1pt} '\) axis, we represent Eq. (2) in the form

$$\frac{{{{\partial }^{2}}E}}{{\partial z{\kern 1pt} {{'}^{2}}}} - \frac{{n_{{\text{T}}}^{2}}}{{{{c}^{2}}}}\frac{{{{\partial }^{2}}E}}{{\partial {{t}^{2}}}} = \mu \frac{{{{\partial }^{2}}}}{{\partial {{t}^{2}}}}({{E}^{2}}) - \gamma \frac{{{{\partial }^{4}}E}}{{\partial {{t}^{4}}}}.$$

Using the unidirectional propagation approximation applicable here [32, 37], we arrive at the Korteweg–de Vries equation

$$\frac{{\partial E}}{{\partial z{\kern 1pt} '}} + \frac{{{{n}_{{\text{T}}}}}}{c}\frac{{\partial E}}{{\partial t}} + \frac{c}{{{{n}_{{\text{T}}}}}}\mu E\frac{{\partial E}}{{\partial t}} - \frac{c}{{2{{n}_{{\text{T}}}}}}\gamma \frac{{{{\partial }^{3}}E}}{{\partial {{t}^{3}}}} = 0.$$
(16)

A self-similar (nonsoliton) sign-alternating solution of the Korteweg–de Vries equation in the form of a frequency-modulated pulse is well known [40, 41]. This solution can be represented in the form [41] \(E = z{\kern 1pt} {{'}^{{2/3}}}f(\xi {\text{/}}z{\kern 1pt} {{'}^{{1/3}}})\), where \(\xi = t - {{n}_{{\text{T}}}}z{\kern 1pt} '{\text{/}}c\) and f is the function satisfying the ordinary differential equation

$$\frac{c}{{2{{n}_{{\text{T}}}}}}\gamma {\kern 1pt} \ddot {f} + \frac{\varphi }{3}\dot {f} - \frac{c}{{{{n}_{{\text{T}}}}}}f\dot {f} - \frac{2}{3}f = 0,$$

where an overdot means the derivative with respect to the self-similar variable \(\varphi = \xi {\text{/}}z{\kern 1pt} {{'}^{{1/3}}}\).

The self-similar solution has a very wide frequency spectrum with properties of a supercontinuum [28, 29].

The electric area of this self-similar solution is nonzero. In our case, it is equal to the area of the soliton (3) with the opposite sign.

It is noteworthy that the velocity of this nonsoliton component of terahertz radiation is equal to the linear velocity c/nT.

Below, we consider particular examples.

All three second-order nonlinear susceptibilities \({{\chi }^{{(2)}}}(0;0)\), \({{\chi }^{{(2)}}}(\omega ;0)\), and \({{\chi }^{{(2)}}}(\omega ; - \omega )\) in a potassium dihydrogen phosphate (KDP) crystal are positive [42]. Therefore, inequality (14) for this case can be re-presented in the form of the “super-Cherenkov” condition

$${{v}_{{\text{g}}}}{\text{cos}}\theta {\text{ }} > {{v}_{{\text{B}}}} \approx c{\text{/}}{{n}_{{\text{T}}}}.$$
(17)

Since the velocity \(v = {{v}_{{\text{g}}}}{\text{cos}}\theta \) of the optical–terahertz soliton is higher than the linear velocity c/nT, the nonsoliton component of the generated terahertz radiation mentioned above is behind the soliton component propagating in the state coupled to the optical pulse.

According to Eq. (12), βc > 0 in this case. Then, the group velocity dispersion of the optical pulse should necessarily be positive according to inequality (13). Furthermore, the positive parameter β should satisfy the condition β > βc. Taking χ(2)(ω; 0)/χ(2)(0; 0) ~ 0.1 [42], nT ≈ 2, ω ~ 1015 s−1, ωT ~ 1014 s–1, and \({{v}_{{\text{g}}}}\) ~ c, we obtain βc ~ 10–24 s2/cm. Setting also \({{n}_{\omega }} \sim 1\) and \(\beta - {{\beta }_{{\text{c}}}} \sim 0.1\), we obtain \(\eta \sim 1\) from Eqs. (6) and (12), which, being substituted into Eq. (5), can give reasonable values for the tilt angle \(\theta \) satisfying the super-Cherenkov condition (17). For more clear conclusions, it is necessary to use the exact values of above parameters rather than their estimates.

Another example is a uniaxial lithium niobate c-rystal, where χ(2)(0; 0) ~ 10–6 CGSE unit > 0, and χ(2)(ω; 0) ~ χ(2)(ω; –ω) ~ –10–7 CGSE unit < 0 [42]. Then, the “anti-Cherenkov” condition follows from Eqs. (8), (11), and (14) in the form

$${{v}_{{\text{g}}}}{\text{cos}}\theta {\text{ }} < {{v}_{{\text{B}}}} \approx c{\text{/}}{{n}_{{\text{T}}}}.$$
(18)

Here, it is easily seen that the nonsoliton component of the terahertz radiation propagates ahead of the optical–terahertz soliton, being its precursor.

As seen in Eq. (12), βc < 0 in this case. therefore, condition (13) is valid at any positive group velocity dispersion β. The absolute value of the parameter β at a negative group velocity dispersion should satisfy the condition |β| < |βc|. In particular, this inequality is satisfied at β = 0. Then, we arrive at “dispersionless” solitons, which also exist in the solution of the system of Eqs. (1) and (2) of another type found in [33]. According to Eq. (5), η = tan2θ. Using also the first of Eqs. (6), we obtain

$${{\beta }_{{\text{c}}}} = \beta - \frac{c}{{n\omega v_{{\text{g}}}^{2}}}\mathop {\tan }\nolimits^2 \theta .$$
(19)

Thus, the quantity βc serves as the group velocity dispersion of the optical pulse. the second term on the right-hand side of Eq. (19) is due to the diffraction of the optical component (see the last term on the right-hand side of Eq. (1)). Consequently, the diffraction contributes to the effective group velocity dispersion due to the tilt of the wave front of the optical pulse. The diffraction bending of phase wave fronts propagating along the z axis results in the broadening of the optical wave packet in the projection on the z' axis, which is equivalent to the effective dispersion [35]. This is confirmed by Eq. (19).

The numerical estimates of the parameters |βc| and η for the lithium niobate crystal coincide in the order of magnitude with the respective values obtained above for the KDP crystal.

The optical–terahertz soliton is obviously formed at distances of about the dispersion length ld ~ τ2/2|βc| of the optical pulse. The substitution of the estimates presented above gives ld ~ 10–2 cm. Therefore, the scenarios described above for the generation of the soliton and nonsoliton components of terahertz radiation can be observed in crystals with a characteristic size of several millimeters.

Let the transverse sizes of the optical and terahertz pulses be D ~ 1 mm. Then, the diffraction length of the terahertz signal is ld ~ D2/(cτ) ~ 10 cm. It is much larger than the expected sizes of the crystal sample. Consequently, the formation of the optical–terahertz soliton seems to be possible here under real conditions.

The characteristic wavelength of the generated broadband terahertz signal can be estimated as λT ~ cτ, which yields λT ~ 10–3–10–2 cm for τ ~ 10–13 s. Hence, λT ≪ D. This condition is an important reason for the paraxial approximation used in the initial system of Eqs. (1) and (2).

It is easily seen that the intensities of the optical, Iω, and terahertz, IT, components of the soliton considered here can be estimated by the expressions

$${{I}_{\omega }} \sim \frac{c}{{4\pi {{{({{\omega }_{{\text{T}}}}\tau )}}^{2}}{\text{|}}{{\chi }^{{(2)}}}(0;0){{\chi }^{{(2)}}}(\omega ; - \omega ){\text{|}}}},$$
$${{I}_{{\text{T}}}} \sim \frac{c}{{4\pi {{{({{\omega }_{{\text{T}}}}\tau )}}^{4}}{\text{|}}{{\chi }^{{(2)}}}(0;0){{{\text{|}}}^{2}}}}.$$

Consequently, the efficiency of generation is

$$\frac{{{{I}_{{\text{T}}}}}}{{{{I}_{\omega }}}} \sim \frac{1}{{{{{({{\omega }_{{\text{T}}}}\tau )}}^{2}}}}\left| {\frac{{{{\chi }^{{(2)}}}(0;0)}}{{{{\chi }^{{(2)}}}(\omega ; - \omega )}}} \right|.$$

Substituting the above parameters and τ ~ 10–13 s into these expressions, we obtain Iω ~ 1012 W/cm2 and IT ~ 1011 W/cm2; i.e., IT/Iω ~ 0.1.

Thus, the efficiency of generation in the analyzed soliton-like regime can reach about 10%, which is much higher than the efficiency reached in the soliton regime analyzed in [33]. A similar conclusion also concerns the intensities of the optical and terahertz components.

An important difference of the solution considered in this work from the soliton-like solution obtained in [33] is that the duration of the generated terahertz signal is half the duration of the optical pulse (see Eq. (3)). Thus, the terahertz signal is completely captured by the optical pulse in the synchronous propagation regime. This ensures a good supply of the generated signal from the energy of the optical pulse. The both components in the soliton-like solution found in [33] have the identical durations. Therefore, the capture of the terahertz signal by the optical pulse is not such efficient as that in the case considered here and, correspondingly, the efficiency of generation is not such high.

3 CONCLUSIONS

To summarize, a new soliton-like solution of the system of Eqs. (1) and (2) of the Zakharov–Boussinesq type has been obtained. It is important that diffraction plays a fundamental role in the formation of this soliton. The soliton-like solution has been used to analyze the generation of terahertz radiation by an optical pulse with a tilted wave front in a crystal with second-order optical nonlinearity. Using the conservation law of the electric area, it has been shown that the nonsoliton (self-similar) component of terahertz radiation is generated in addition to the soliton component. The tilt angle of the wave front of the optical component of the soliton is unambiguously determined by the parameters of the crystal and cannot be arbitrary.

Depending on the signs of various second-order nonlinear susceptibilities, which are functions of the carrier frequency of the optical pulse, regimes can be implemented under the super-Cherenkov, Eq. (17), and super-Cherenkov, Eq. (18), matching conditions. In the former and latter cases, the optical terahertz soliton propagates ahead and behind the nonsoliton component of the generated terahertz radiation, respectively.

We note that the soliton-like generation regime analyzed in [33] is most efficiently implemented under the strict Cherenkov condition.

The stability of the found soliton-like solution is still unclear. In addition, it is necessary to determine the initial conditions for the implementation of the soliton-like solutions found in this work and in [33]. The fact that the intensities of the optical and terahertz components of the found soliton are two to three orders of magnitude higher than the respective intensities of the soliton analyzed in [33] can possibly point to the solution of this problem. A more strict and clear answer can be expected from numerical experiments with the system of Eqs. (1) and (2), which we are going to perform elsewhere.