It is theoretically investigated the generation of higher harmonics of two-dimensional and three-dimensional terahertz electromagnetic beams in nonlinear crystals. The attention is paid to crystalline paraelectrics like SrTiO
3 under the temperatures 60 - 200 K, these crystals possess the cubic nonlinearity. The bias electric field is applied to provide the dominating quadratic nonlinearity. The initial focusing of the beams not only increases the efficiency of generation of higher harmonics, but alto makes possible to select maxima of different higher harmonics at some distances from the input. At lower temperatures the nonlinearity behaves at smaller input amplitudes, whereas at higher temperatures the harmonic generation can be observed at higher frequencies up to 1.5 THz. In three-dimensional beams the peak amplitudes of higher harmonics can be bigger than in two-dimensional beams, but the ratios of these peak values to the maximum values of the focused first harmonic are smaller than in two-dimensional beams.
Terahertz Wave Beams Nonlinear Crystalline Paraelectrics Different Temperatures Generation of Harmonics Initial Focusing1. Introduction
Now the assimilation of the terahertz (THz) range 0.1 - 30 THz takes place [1] - [17] . As the nonlinear crystals, the ferroelectrics in the non-polar phase are utilized as the nonlinear dielectrics in the lower part of THz range, so-called paraelectrics like SrTiO3, KTaO3 [18] - [40] . The crystalline SrTiO3 possesses high cubic electrodynamic nonlinearity and low losses in the lower part of THz range 0.1 - 2.5 THz at moderately low temperatures T = 60 - 200 K. There exists the frequency dispersion in THz range when the frequency is near the soft mode frequency i.e. the lowest frequency of oscillations of the optical type of the crystalline lattice [19] [21] [22] [23] [24] [27] [28] [40] [41] [42] . In the crystalline SrTiO3 the soft mode frequency decreases with the decrease of the temperature whereas the dielectric nonlinearity increases there [19] . In the absence of the frequency dispersion in the microwave range, the nonlinearity results in the generation of higher harmonics and forming the shock electromagnetic (EM) waves [29] [30] . The waveguide dispersion can be used there [31] .
Because there is a problem of excitation of powerful THz radiation, the interesting and important phenomenon is generation of higher harmonics [33] [39] of input EM waves at relatively low frequencies. The transverse bias electric field should be applied to provide the induced quadratic nonlinearity [29] [33] . Earlier the generation of higher THz harmonics was investigated in 2D geometry, or for the beams that depend on a single transverse coordinate, at the temperature T ≈ 80 K [33] . Only the quadratic nonlinearity was taken into account there. Note that the frequencies of higher harmonics are limited by the soft mode frequency. At higher temperatures the soft mode frequency increases, whereas the nonlinearity decreases. Therefore, for the practical needs it is rather better to investigate the generation of higher harmonics in the wide temperature interval. In THz range the dissipation in the paraelectrics increases compared with the microwave range. A possible way to compensate losses is using the initial focusing of the EM wave beams [33] [34] [36] [37] [39] . Thus, also 3D THz beams should be considered to produce better initial focusing. Under maximum concentration of EM energy in 3D case, the role of the cubic nonlinearity may be essential compared with 2D case, as well as the self-action due the generation of the zeroth harmonic. The cubic nonlinearity results in both the influence on generation of higher harmonics and in the self-action.
The present paper is devoted to the theoretical investigations of generation of higher harmonics of THz EM waves in paraelectric crystals like SrTiO3 in the wide temperature range T = 60 - 200 K. The bias electric field is applied to provide the dominating quadratic nonlinearity. The terms due to the cubic nonlinearity are taken into account, too. Both the focusing of 2D, or plane, beams and 3D, or cylindrical, ones are considered. At lower temperatures the dielectric nonlinearity is higher, but the possible frequency range is lower due to decreasing the soft mode frequency. At higher temperatures, it is possible to increase the frequency range. It is demonstrated that the cubic nonlinearity is not important under focusing of 2D beams, but it is important under extreme focusing of 3D beams.
2. Basic Equations
The nonlinear propagation of EM wave beams with the dominating electric field component Ey = E is considered along OZ axis within the crystalline paraelectric SrTiO3. The dielectric nonlinearity in SrTiO3 is due to the nonlinear properties of the lattice polarization ε0P, where ε0 is the electric constant, SI units. Both the cases of plane, or 2D, beams Ey(t, x, z) and symmetrical cylindrical, or 3D, ones Ey(t, ρ, z) are investigated. The basic equations that describe the nonlinear EM wave propagation in paraelectric crystalline SrTiO3 are [18] [19] [26] [32] [33] :
∂ 2 P ∂ t 2 + γ ∂ P ∂ t + ω T 2 ( 1 + P 2 P n 2 ) P = ε ( 0 ) ω T 2 E ; ∂ 2 E ∂ z 2 + Δ ⊥ E = 1 c 2 ∂ 2 P ∂ t 2 ; D ≡ ε 0 ( E + P ) ≈ ε 0 P ; Δ ⊥ E ≡ { ∂ 2 E ∂ x 2 in 2D case , 1 ρ ∂ ∂ ρ ( ρ ∂ E ∂ ρ ) in 3D case ; ρ ≡ ( x 2 + y 2 ) 1 / 2 . (1)
Below both 3D beams and 2D ones are considered. Here ω T is the soft mode frequency, which is in THz range for SrTiO3, γ is the lattice dissipation. At the temperature T ≈ 80 K it is ω T ≈ 6 × 10 12 s − 1 , and the static linear dielectric permittivity is ε ( 0 ) ≡ ε ( ω = 0 ) = 1.8 × 10 3 [19] [26] [33] . In paraelectric crystalline SrTiO3 the permittivity increases with the decrease of temperature. In Equations (1) the parameter Pn determines the value of the cubic nonlinearity of the polarization. It is connected with the characteristic magnitude of the electric field E0, where the nonlinearity is essential: P n = ε ( 0 ) E n . At the temperature T ≈ 80 K it is En = 60 kV/cm [19] [26] . The cubic nonlinearity is proportional to ε(0)3, i.e. En ~ ε(0)−3/2. Because of small dissipation in SrTiO3, the nonlinearity manifests at the amplitudes of EM waves at least one order smaller than En. The parameters of SrTiO3 used in simulations are presented in Figure 1.
For 3D, or cylindrical, EM beams the validity of the paraxial approximation is assumed and checked [37] , namely the possible focusing is moderate and the longitudinal component of the electric field is small | E z | ≪ | E y | .
In the linear case Equations (1) result in the well-known expressions of the complex dielectric permittivity ε(ω) º ε'(ω) + iε''(ω) and to the dispersion relation for the plane EM wave k = k(ω) [19] [26] [33] :
ε ( ω ) ≈ ε ( ω = 0 ) ⋅ ω T 2 ω T 2 − ω 2 + i γ ω , ω > 0 ; k ( ω ) ≡ k ′ + i k ″ = ω c ε ( ω ) 1 / 2 ; E ~ exp ( i ( ω t − k z ) ) . (2)
In the nonlinear stationary case Equations (1) result in the known formula for the nonlinear permittivity [19] [26] :
ε ( E ) = ε ( ω = 0 ) 1 + P 2 P n 2 ≈ ε ( ω = 0 ) 1 + E 2 E n 2 ≈ ε ( ω = 0 ) ( 1 − E 2 E n 2 ) . (3)
Equations (3) for the static nonlinear permittivity are equivalent in the case of moderate electric fields |E| < En, that is considered below.
3. Equations for Slowly Varying Amplitudes
Here the nonlinear EM wave propagation is investigated in the crystalline SrTiO3, when the bias electric field Es is applied [33] :
E = E s + E ˜ , P = P s + P ˜ ; P s ⋅ ( 1 + P s 2 P n 2 ) = ε ( 0 ) ω T 2 E s . (4)
In the absence of the bias field only the odd harmonics are excited, and the efficiency is low, as our simulations have been demonstrated. In the presence of the bias field the induced quadratic nonlinearity becomes dominating, but the cubic nonlinearity should be preserved generally in the equations. Generally all harmonics, even and odd ones, are essential. The equations for the variable components of the polarization P ˜ and the electric field E ˜ are:
∂ 2 P ˜ ∂ t 2 + γ ∂ P ˜ ∂ t + ω ˜ T 2 P ˜ + ω T 2 ( 3 P s P ˜ 2 P n 2 + P ˜ 3 P n 2 ) = ε ( 0 ) ω T 2 E ˜ ; Δ E ˜ ≡ ∂ 2 E ˜ ∂ z 2 + Δ ⊥ E ˜ = 1 c 2 ∂ 2 P ˜ ∂ t 2 ; ω ˜ T 2 ≡ ω T 2 ( 1 + 3 P s 2 P n 2 ) . (5)
From Equations (5) it is possible to obtain a single equation for the polarization [33] :
Δ ( ∂ 2 P ˜ ∂ t 2 + γ ∂ P ˜ ∂ t + ω ˜ T 2 P ˜ ) − ω T 2 c 2 ε ( 0 ) ∂ 2 P ˜ ∂ t 2 = − ω T 2 ( 3 P s P n 2 Δ ( P ˜ 2 ) + 1 P n 2 Δ ( P ˜ 3 ) ) . (6)
The nonlinearity is considered as moderate here, and the method of slowly varying amplitudes is applied [33] [43] [44] [45] .
Below the propagation of THz EM beams along OZ axis is considered. The solution of Equation (6) is searched as the set of harmonics including the zeroth harmonic:
P ˜ = 1 2 ∑ j = 1 , 2 , 3 , ⋯ B j ( z , r → ⊥ , t ) e i φ j + c . c . + B 0 ( z , r → ⊥ , t ) ; φ j ≡ ω j t − k ′ ( ω j ) z ; ω j = j ⋅ ω 1 ; k ( ω ) ≡ k ′ + i k ″ = ω c ε ( ω ) 1 / 2 , ε ( ω ) ≈ ε ( ω = 0 ) ⋅ ω T 2 ω ˜ T 2 − ω 2 + i γ ω . (7)
Here Bj(z, r^, t) are the slowly varying amplitudes of harmonics for the polarization; ω, k are the carrier frequencies and the corresponding linear wave numbers of EM wave [33] . It is seen that the bias electric field Es and thus the polarization Ps change the linear permittivity ε(ω)) and the dispersion relation for EM waves k = k(ω) due to the presence of ω ˜ T > ω T there.
The frequency dependencies of the real parts of the permittivity, the real and imaginary parts of wave numbers of EM waves, and the ratios of the imaginary part to the real one of the permittivity are given in Figure 2. The bias field Es = 0.3En is applied; the proper values of the characteristic nonlinear field are taken for each temperature, see Figure 1.
In Equations (7) the zeroth harmonic term is taken into account that is due to the induced quadratic nonlinearity. From Equation (6) it is possible to obtain the following expression for B0:
ω ˜ T 2 B 0 + 3 P s 2 P n 2 ω T 2 ∑ l = 1 , 2 , 3 , ⋯ | B l | 2 = 0. (8)
In another words, the term with B0 < 0 results in the effective decreasing of the bias polarization Ps, i.e.
P s → P s ⋅ ( 1 − 3 ω T 2 2 P n 2 ω ˜ T 2 ∑ l = 1 , 2 , 3 , ⋯ | B l | 2 ) . (9)
With using the standard procedure described in details in [33] , the following nonlinear parabolic equations for harmonics have been derived:
∂ B j ∂ z + i 2 k ′ j Δ ⊥ B j + Γ j B j = i k ′ j ω T 2 ( ω ˜ T 2 − ω j 2 + i γ ω j ) P n 2 { 3 P s 4 [ ∑ m B m B j − m exp ( i ( φ m + φ j − m − φ j ) ) + 2 ∑ m B m ∗ B j + m exp ( i ( − φ m + φ j + m − φ j ) ) + 4 B 0 B j ] + 1 8 [ ∑ m , p B j − m − p B m B p exp ( i ( φ m + φ p + φ j − m − p − φ j ) ) + 3 ∑ m , p B m + p − j ∗ B m B p exp ( i ( φ m + φ p − φ m + p − j − φ j ) ) + 3 ∑ m , p B m + p + j B m ∗ B p ∗ exp ( i ( φ m + φ p − φ m + p − j − φ j ) ) ] } . (10)
For simulations it is better to rewrite Equations (10) with using the slowly varying amplitudes for harmonics Aj of the electric field Ĕ. From the second Equation (5) there is the relation
( k ′ j ) 2 A j exp ( − i j k ′ 1 z ) = ω j 2 c 2 B j exp ( − i k ′ j z ) ; or B j = ε ( 0 ) α j A j exp ( i ( k ′ j − j k ′ 1 ) z ) ; α j ≡ 1 ε ( 0 ) ( k ′ j c ω j ) 2 ≈ 1. (11)
From Equation (11) the following set of equation is written down:
∂ A j ∂ z + i 2 k ′ j Δ ⊥ A j + i ( k j − j k ′ 1 ) A j = i k ′ j ω T 2 ε 2 ( 0 ) ( ω ˜ T 2 − ω j 2 + i γ ω j ) P n 2 α j { 3 P s 4 ε ( 0 ) [ ∑ m α j − m α m A j − m A m + 2 ∑ m α m − j α m A m − j ∗ A m + 4 A 0 α j A j ] + 1 8 [ ∑ m , p α j − m − p α m α p A j − m − p A m A p + 3 ∑ v , p α m + p − j α m α p A m + p − j ∗ A m A p + 3 ∑ v , p α j + m − p α m α p A j + m + p A m ∗ A p ∗ ] } ; P n / ε ( 0 ) ≡ E n , P s / ε ( 0 ) ≈ E s . (12)
In Equations (12) the expression for the zeroth harmonic A0 is
A 0 ≡ B 0 ε ( 0 ) = − 3 P s ε ( 0 ) ω T 2 2 P n 2 ω ˜ T 2 ∑ l = 1 , 2 , 3 , ⋯ α l 2 | A l | 2 . (13)
It is assumed that the first harmonic only is at the input of the system, ω1 < ωT/10. All higher harmonics and the zeroth one are excited by the nonlinearity. It is possible to excite the higher harmonics at the frequencies ωj ≥ 0.5ωT, as our simulations have demonstrated.
To increase the efficiency of generation of higher harmonics, the initial focusing by the circular antenna is applied, see Figure 3. The boundary condition for the first harmonic is in 2D case
A 1 ( z = 0 , x , t ) = A 10 F x ( x ) M ( x ) . (14)
In 3D case there is the radial distance ρ instead x. Here A10 is the maximum value of the amplitude at the input z = 0, the transverse profile is Fx(x). This transverse profile is assumed as almost rectangular with the width x0 or ρ0. At the output the matching load is assumed, i.e. there are no reflections there.
The factor M(x) is due to a possible initial focusing of the pulse due to the excitation of the circular antenna, see Figure 3. It is considered the symmetrical case with respect to x = 0.
At the input of the nonlinear crystal z = 0 the EM wave gets the phase shift − k Δ ( x ) = − k ( R − ( z c 2 + x 2 ) 1 / 2 ) , where z c is the distance from the input to the focus point z c = ( R 2 − ( L x / 2 ) 2 ) 1 / 2 , k is the wave number; thus the phase multiplier is M ( x ) = exp ( − i k Δ ( x ) ) . In 3D case x is replaced by the radial coordinate ρ.
4. Simulations of Generation of Harmonics
The simulations of generation of higher harmonics in the nonlinear crystal SrTiO3 have been realized under different temperatures, both in 2D and 3D geometries. It has been obtained that the pure cubic nonlinearity without the bias electric field is not effective for this generation. There is the optimum value of the bias electric field Es ≈ (0.2 - 0.4)En to realize the generation of higher harmonics. At smaller values the induced quadratic nonlinearity is small whereas at higher values Es ≥ 0.5En the saturation of nonlinearity occurs and the efficiency of the generation does not increase, as seen from Equation (4). For all the cases considered below it is chosen Es = 0.3En where En is the characteristic nonlinear electric field under each temperature. The length of the nonlinear crystal is Lz = 0.5 cm. The parameters of the nonlinear crystal at different temperatures are given in Table 1, see also Figure 1.
The numerical simulations have been provided by the implicit finite difference schemes [46] . Several iterations have been applied that demonstrate good convergence. The simulations have demonstrated a possibility of generation of higher harmonics with the numbers n ≥ 5, i.e. the frequency multiplication, or frequency up-conversion, in the whole temperature interval T = 60 - 200 K.
The typical results of simulations are presented in Figures 4-8. It is seen that in the focused beams the maxima of different higher harmonics are realized at different distances from the input of the crystal.
At lower temperatures the nonlinearity is higher, so the input amplitudes of the first harmonics can quite lower, of about 1 kV/cm. But the highest frequencies that can be obtained under the up-conversion are of about 0.5 THz. In turn, at higher temperatures it is possible to excite the frequencies up to f º ω/2π = 1.5 THz.
From our simulations in is seen that the concentration of the energy near the focus is naturally higher in 3D beams. For this reason, in 2D beams the influence
The used parameters of crystalline SrTiO<sub>3</sub>
T, K
ε (0)
ωT, 1012 s−1
γ, 1011 s−1
En, 104 V/cm
60
4000
5
2.5
2.1
77
2000
6
2
6
150
700
12
3
29
200
500
15
4
48
of the cubic nonlinearity and the zeroth harmonic is not essential. In 3D beams this influence should be taken into account generally, because near the maximum of the focusing the value of the amplitude of the first harmonic becomes comparable with the value of the bias electric field. The using of 3D beams makes possible to increase of the maxima of higher harmonics compared with 2D beams, but the relative values of these maxima are smaller in 3D beams than in 2D ones.
Because the value of the temperature of the crystal is the important parameter to realize the frequency multiplication, it is better to use the input pulses of durations < 1 μs to avoid the heating of crystals.
5. Conclusions
In the lower part of the terahertz range in the nonlinear crystalline paraelectrics like SrTiO3, it is possible to observe the generation of higher harmonics and thus to realize the frequency up-conversion. The crystalline paralectric SrTiO3 possesses the cubic dielectric nonlinearity, so to increase the efficiency of harmonic generation, it is rather better to apply the bias electric field and to get the induced quadratic nonlinearity. In THz range, the frequency dispersion takes place, so the number of harmonics is large but finite, of about 10 - 20. The initial focusing of the beams by the circular antenna results not only in higher efficiency of generation, but also makes possible to select the maxima of different harmonics at specified distances from the input.
The generation of higher harmonics can be realized in the wide temperature range 60 - 200 K. At lower temperatures, the possible frequencies are smaller, of about 0.5 THz, but the nonlinearity is higher. At higher temperatures, the frequency up-conversion can be observed up till the frequencies of about 1.5 THz.
Both plane and cylindrical focused beams can be used for generation of harmonics. In cylindrical beams, the absolute maxima of harmonics can be higher than in plane ones, but the ratios of these maxima to ones of the first harmonic are lower in the cylindrical beams. The influence of the cubic nonlinearity and the zeroth harmonic is essential in the cylindrical beams due to high values of the focused first harmonic.
Acknowledgements
The authors are grateful to SEP-CONAHCyT (Mexico) for a partial support of our work.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Cite this paper
Grimalsky, V., Escobedo-Alatorre, J., Castrejon-Martinez, C. and Gomez-Badillo, Y. (2023) Generation of Higher Terahertz Harmonics in Nonlinear Paraelectrics under Focusing in a Wide Temperature Range. Journal of Electromagnetic Analysis and Applications, 15, 43-58. https://doi.org/10.4236/jemaa.2023.154004
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