# Triple to quadruple frequency radiation shift caused by the interaction of terahertz pulses with a nonlinear Kerr medium

### Experimental setup

The experimental setup diagram is shown in Fig. 5. Radiation from a femtosecond laser system based on a regenerative amplifier (pulse duration is 30 fs, center wavelength is 790 nm, pulse energy is 2 mJ, rate repetition rate is 1 kHz) is split into two beams with a beam splitter with a ratio of 98:2. The radiation from the pump passes through an optical attenuator in the TERA-AX THz generator. Pump radiation after the diffraction grating focused on MgO:(hbox {LiNbO}_{3}) crystal by the spherical mirror. The generated THz radiation is collimated by the parabolic mirror and exits the system. After that, the THz radiation focused on a 1mm thick ZnTe crystal by another parabolic mirror for detection. After the beam splitter, the probe beam goes to the delay line and intersects the THz pulse at the ZnTe crystal and is measured by the electro-optical configuration.

The generation of THz radiation by the optical rectification method occurs throughout the interaction volume of the tilted wavefront with the crystal. Thus, THz radiation is generated throughout the volume of the crystal as well as during the propagation of pump radiation through the medium with both quadratic nonlinearity, due to which the process of generating THz radiation occurs, and a cubic nonlinearity, due to which the radiation should be generated at triple frequencies compared to THz radiation. A schematic diagram of THz radiation generation is shown in Fig.1a. As a result, a diverging THz radiation beam with a Gaussian profile is formed, directed perpendicular to the crystal cut. Then, the THz radiation is collimated using a parabolic mirror with a focal length of 25 mm. The THz field amplitude is measured using a standard electro-optical detection scheme in a 1 mm thick ZnTe crystal.

### Analytical model

The field approach makes it possible to describe the dynamics of the electric field of THz radiation in a transparent dielectric medium. For example, the equation that considers quadratic and cubic nonlinearities without inertia as well as absorption and amplification can be represented by^{28}:

$$begin{aligned} frac{partial E}{partial z}+frac{N_0}{c}frac{partial E}{partial t}-afrac{partial ^3 E }{partial t^3}-(Gamma -gamma )E+ g_1 Efrac{partial E}{partial t} +g_2 E^2frac{partial E}{partial t} =0 , end{aligned}$$

(1)

where E is the electric field, z is the direction of propagation, t is the time, *vs* is the speed of light in vacuum, (hbox {N}_0) and *a* are the empirical constants characterizing the dependence of the linear refractive index of the medium on the frequency (naked) of shape (n(nu )=N_0+4pi ^2 canu ^2), (Gamma) is the amplification coefficient, (gamma) is the absorption coefficient, (g_1=chi _2/c) and (g_2=2n_2/c)are the parameters characterizing respectively the quadratic and cubic non-linearity of the mean response, (chi _2) is the quadratic susceptibility and (n_2) is the nonlinear refractive index coefficient of the medium (CGS units).

The field at the entrance of a nonlinear medium (at (z = 0)) is considered as a single-cycle pulse (see Fig.2a) of the form:

$$begin{aligned} E(t,0)=E_0frac{t}{tau }e^{-(frac{t}{tau })^2} end{aligned}$$

(2)

where (E_0) is the amplitude of the impulse electric field, (tau) is its duration. Figure 1a shows that the THz pulse observed in the experiment in the far diffraction zone has one and a half cycles. However, by writing the boundary condition in the form of a monocyclic pulse (Eq. 2), it is considered that during propagation in the crystal, as well as at its exit, the THz pulse is monocyclic and only in the area distant, due to diffraction, it acquires a half-wave^{29}. This effect, which is practically not observed for pulses with a large number of oscillations, is markedly pronounced for a single-cycle pulse, which becomes one and a half cycles in the far diffraction zone^{30}.

For further analysis, it is useful to rewrite Eq. (1) in normalized variables ({widetilde{E}}=E/E_0, {widetilde{t}}=t/tau , {widetilde{z}}=z/L_{pulse})where (L_{pulse}=ctau /N_0) is the longitudinal size of the THz pulse in the medium:

$$begin{aligned} frac{partial {tilde{E}}}{partial {tilde{z}}}+frac{partial {tilde{E}}}{partial { tilde{t}}}-mu _0frac{partial ^3{tilde{E}}}{partial {tilde{t}}^3}-mu _1 {tilde{E}}+ mu _2{tilde{E}}frac{partial {tilde{E}}}{partial {tilde{t}}}+mu _3{tilde{E}}^2frac{ partial {tilde{E}}}{partial {tilde{t}}}=0 end{aligned}$$

(3)

Here, (mu _0=1/2 cdot Delta n_{disp}/N_0). (Delta n_{disp}=4pi ^2 ac nu ^{2}_{max}) characterizes the change in refractive index due to dispersion, where (nu _{max}=sqrt{2}/tau) is the maximum spectral density frequency at the entrance of the nonlinear medium, (mu _1=frac{c tau }{N_0}(Gamma -gamma )) characterizes the amplification and absorption of the medium; (mu _2=frac{chi _{2} E_{0}}{N_0}) represents the quadratic nonlinearity contribution, where (chi _{2}=frac{m omega _{0}^2 a_{l} alpha _{T} }{32 pi ^2 q N k_{B}}(n_{0, nu }^2-1)^2), (mu _3=4Delta n_{nl}/N_0), (Delta n_{nl}=1/2 n_2 E^{2}_{0}=n^{‘}_{2}I) characterizes the variation of the refractive index of the medium due to the cubic non-linearity, (n^{‘}_{2}) is the coefficient of the nonlinear refractive index of the medium (in SI units), *I* is the intensity of the THz pulse, (n_{0,nu }^2 = sqrt{1+N_0^2 – n_{el}^2}) is the vibrational contribution to the low frequency refractive index, (n_{el}) is the refractive index in the range with non-resonant electron contribution (800 nm), (omega _0) is the fundamental vibrational frequency, (a_1) is the lattice constant, *m* is the reduced mass of the vibrational mode, *q* is the effective charge of the chemical bond, (alpha _T) is the coefficient of thermal expansion, *NOT* is the numerical density of the vibrational units and (k_B) is Boltzmann’s constant.

The normalized boundary condition (Eq. 2) takes the form

$$begin{aligned} {widetilde{E}}({widetilde{t}},0)={widetilde{t}}e^{-{widetilde{t}}^2} end{ aligned}$$

(4)

Dispersion supply, as is known, leads to a change in the temporal structure of the pulse and does not affect its spectrum. Moreover, it emerges from the experimental data that no dispersive spreading of the momentum is observed; therefore, the dispersive contribution can be neglected.

The dependence of the absorption spectrum in the studied spectral range has no characteristic features^{31}. The result of taking the absorption coefficient into account leads to a monotonous decrease in the power spectrum of the output THz radiation. In this respect, the absorption can be neglected in the calculations.

To facilitate the taking into account of the amplification of the THz pulse, the case is considered in the calculations where the energy of the pulse in interaction with the medium corresponds to the energy of the pulse at the output of the medium. Therefore, in the propagation dynamics equation, the amplification term can also be omitted.

Thus, for an analytical calculation of the dynamics of a THz pulse during propagation, one can focus on the last two terms of Eq. (3). Assuming the values (mu _2) and (mu _3) are small, the solution of Eq. (3) is found in the form of a series:

$$begin{aligned} {tilde{E}}left( {tilde{t}},{tilde{z}}right) ,=,{tilde{E}}^0 left( {tilde{t}},{tilde{z}}right) +mu _2{tilde{E}}^{(1,text {nl2})}left( {tilde{ t}},{tilde{z}}right) +mu _3{tilde{E}}^{(1,text {nl3})}left( {tilde{t}},{ tilde{z}}right) end{aligned}$$

(5)

It is easy to show that with boundary conditions (Eq. 2), the terms of such a solution take the form

$$begin{aligned} {widetilde{E}}^{(0)}({widetilde{t}},{widetilde{z}}),=, & {} ({widetilde{ t}}-{widetilde{z}})e^{-({widetilde{t}}-{widetilde{z}})^2} end{aligned}$$

(6)

$$begin{aligned} {tilde{E}}^{(1,text {nl2})}left( {tilde{t}},{tilde{z}}right),= , & {} {widetilde{z}} ({widetilde{t}} – {widetilde{z}}) (2({widetilde{t}} – {widetilde{z}})^2 -1) e^{-2({widetilde{t}} – {widetilde{z}})^2} end{aligned}$$

(seven)

$$begin{aligned} {widetilde{E}}^{(1,nl3)}({widetilde{t}},{widetilde{z}}),=, & {} {widetilde {z}} ({widetilde{t}} – {widetilde{z}})^2 (2({widetilde{t}} – {widetilde{z}})^2-1) e^{ -3({widetilde{t}} – {widetilde{z}})^2} end{aligned}$$

(8)

Here, ({widetilde{E}}^{(0)}) is the solution of the equation. (3) for the “zero” approximation, i.e. without taking into account the non-linearity of the medium, ({widetilde{E}}^{(1,nl2)}) and and ({widetilde{E}}^{(1,nl3)}) are the shape change of the pulse, associated with quadratic and cubic nonlinearity respectively.

As a result, the field spectrum (Eq. 5) (G({widetilde{nu }}, {widetilde{z}})=1/sqrt{2pi }int limits _{-infty }^{+infty } {widetilde {E}}({widetilde{t}},{widetilde{z}}) cdot e^{i{widetilde{nu }}{widetilde{t}}} d{widetilde{t} }) where ({widetilde{nu }}=nu tau) can be represented as

$$begin{aligned} {widetilde{G}}({widetilde{nu }},{widetilde{z}}),=, & {} {widetilde{G}}^{( 0)}({widetilde{nu }},{widetilde{z}})+mu _2 {widetilde{G}}^{(1,nl2)}({widetilde{nu }}, {widetilde{z}})+mu _3 {widetilde{G}}^{(1,nl3)}({widetilde{nu }},{widetilde{z}}) end{aligned} $$

(9)

$$begin{aligned} {widetilde{G}}^{(0)}({widetilde{nu }},{widetilde{z}}),=, & {} frac{i {widetilde{nu }}}{2sqrt{2}}e ^ {i {tilde{z}} {tilde{nu }} – {widetilde{nu }}^2/4} end{aligned}$$

(ten)

$$begin{aligned} G^{(1,text {nl2})}left( {tilde{nu }},{tilde{z}}right),=, & {} -frac{i {tilde{nu }} {tilde{z}} ({widetilde{nu }}^2 – 4)}{64}e^{-{tilde{nu }} ^2/8 + i {tilde{nu }} {tilde{z}}} end{aligned}$$

(11)

$$begin{aligned} {widetilde{G}}^{(1,nl3)}({widetilde{nu }},{widetilde{z}}),=, & {} frac {{tilde{z}} {tilde{nu }}^2 ({widetilde{nu }}^2 – 18)}{648 sqrt{6}}e^{-{widetilde{ nu }} ^ 2 / 12 + i {tilde{nu }} {tilde{z}}} end{aligned}$$

(12)