Modeling for extracavity-pumped terahertz parametric oscillators

Zecheng Wang; Shuzhen Fan; Xiaohan Chen; Xingyu Zhang; Zhenhua Cong; Zhaojun Liu; Zengguang Qin; Na Ming; Quanxin Guo; Liyuan Guo

doi:10.1364/OE.465429

1. Introduction

Terahertz parametric sources based on stimulated polariton scattering (SPS) are efficient THz radiation sources, for their advantages of high temporal and spatial coherence, room temperature operation and wide frequency tunability, etc. [1–7]. In SPS process, a strong fundamental laser beam enters into a nonlinear crystal which is infrared and Raman active, and a Stokes beam that is slightly red-shifted from the fundamental beam and a THz wave are parametrically generated. The three waves follow the energy conservation law and the momentum conservation law. Because of the very large refractive index of the crystal in THz region, only noncollinear phase matching condition can be achieved [8–14].

Depending on the oscillating situation of the Stokes beam, terahertz parametric sources can be sorted as terahertz parametric generators (TPGs) and terahertz parametric oscillators (TPOs). In TPGs, the Stokes beam travels through the nonlinear crystal only one time [15–18], while in TPOs, the Stokes beam oscillates in an independent oscillation cavity [19–21]. For TPGs and injection-seeded TPGs (is-TPGs), the theoretical analysis can be performed by numerically solving the coupled wave equations that describing SPS process [10,22–25]. For intracavity-pumped TPOs, where the fundamental beam and the Stokes beam oscillate in their respective cavities, a group of rate equations for the two beams can be established based on the coupled wave equations of SPS process and the rate equations of the Q-switched laser [26]. As for extracavity-pumped TPOs, both methods mentioned above are not suitable since the Stokes beam oscillates in an independent cavity and forms a stable transverse mode, while the fundamental beam propagates through the crystal only once. The models in Refs. [22–25] could not reflect the oscillating situation of the Stokes wave, so they cannot be used to analyze the extracavity-pumped TPOs. Also, in the model for intracavity-pumped TPOs in Ref. [26], the nonlinear crystal is placed inside the pump laser cavity, and the pump pulse generation is influenced by SPS process, so it is not suitable for extracavity-pumped TPOs either. In this paper we present a modelling method for extracavity-pumped TPOs. In the first step, the THz wave amplitude is expressed as a function of the fundamental and Stokes wave amplitudes. Second, the rate equation for the Stokes wave is obtained. In the third step, using the solution of the Stokes wave rate equation, the temporal evolutions of the THz and residual fundamental waves are calculated by numerically solving the coupled wave equations. This model is used to analyze an extracavity-pumped TPO based on MgO:LiNbO₃ crystal, the simulation results are basically consistent with the experimental results.

2. Modeling method

2.1 Coupled wave equations of SPS process

The noncolinear phase matching condition in SPS process is shown in Fig. 1, where the wave vectors of the fundamental, Stokes and terahertz waves are denoted as k_F, k_S, and k_T, respectively. The angle between k_F and k_S is denoted as θ, and the angle between k_F and k_T is denoted as β. Generally, β is between 50°-70°. For convenience, we assume that the fundamental wave propagates along the x axis, and the transmitting direction of the THz wave is denoted as y^T. Because θ is very small (a few tenths to a few degrees), the Stokes beam is also assumed to propagates along the x axis.

Fig. 1. Noncolinear phase matching condition in SPS process.

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The coupled wave equations of SPS process are [10,22]

(1)$$\displaystyle\frac{{\partial {I_F}}}{{\partial x}} = \frac{{2\pi }}{{{n_F}{\lambda _F}}}\left[ { - \frac{1}{{\sqrt {2c{\varepsilon_0}{n_S}{n_T}/{n_F}} }}\left( {4{d_{33}} + \sum\limits_j {{d_{Qj}}\textrm Re ({{\chi_{Qj}}} )} } \right){{({{I_F}{I_S}{I_T}} )}^{1/2}} - \frac{1}{{2c{\varepsilon_0}{n_S}}}\left( {\sum\limits_j {d_{Qj}^2{\mathop{\rm Im}\nolimits} ({{\chi_{Qj}}} )} } \right){I_F}{I_S}} \right] - {\alpha _F}{I_F},$$

(2)$$\displaystyle\frac{{\partial {I_S}}}{{\partial x}} = \frac{{2\pi }}{{{n_S}{\lambda _S}}}\left[ {\frac{1}{{\sqrt {2c{\varepsilon_0}{n_F}{n_T}/{n_S}} }}\left( {4{d_{33}} + \sum\limits_j {{d_{Qj}}\textrm Re ({{\chi_{Qj}}} )} } \right){{({{I_F}{I_S}{I_T}} )}^{1/2}} + \frac{1}{{2c{\varepsilon_0}{n_F}}}\left( {\sum\limits_j {d_{Qj}^2{\mathop{\rm Im}\nolimits} ({{\chi_{Qj}}} )} } \right){I_F}{I_S}} \right] - {\alpha _S}{I_S},$$

(3)$$\frac{{\partial {I_T}}}{{\partial {y^T}}} = \frac{{2\pi }}{{{n_T}{\lambda _T}}}\left[ {\frac{1}{{\sqrt {2c{\varepsilon_0}{n_F}{n_S}/{n_T}} }}\left( {4{d_{33}} + \sum\limits_j {{d_{Qj}}\textrm Re ({{\chi_{Qj}}} )} } \right){{({{I_F}{I_S}{I_T}} )}^{1/2}}} \right] - {\alpha _T}{I_T},$$

where I_F, I_S, I_T are the intensities of the fundamental, Stokes and terahertz waves, respectively, n_m (m = F, S, T) denote the refractive indices of the crystal for the three waves, respectively, λ_m (m = F, S, T) represent the wavelengths of the three waves, respectively, ε₀ and c are the permittivity and light speed in vacuum, α_F, α_S and α_T are the absorption coefficients of the crystal for the fundamental, Stokes and THz waves, respectively, d₃₃ is the second-order nonlinear coefficient of the nonlinear crystal. 4d₃₃ corresponds to the electronic polarization and d_Qjχ_Qj corresponds to the ionic polarization. The definitions of d_Qj, χ_Qj, α_F, α_S, and α_T are [27,28]

(4)$${d_{Qj}} = {\left[ {\frac{{32{\pi^2}{\varepsilon_0}{c^4}{n_F}{{({S_{33}}/L\Delta \Omega )}_j}}}{{{S_j}\hbar {\omega_{jTO}}{{({\omega_F} - {\omega_{jTO}})}^4}{n_S}({{\overline n }_{0j}} + 1)}}} \right]^{1/2}},$$

(5)$${\chi _{Qj}} = \frac{{{S_j}\omega _{jTO}^2}}{{\omega _{jTO}^2 - \omega _T^2 - i{\Gamma _{jTO}}{\omega _T}}},$$

(6a)$${\alpha _m} = \frac{{2\pi }}{{{n_m}{\lambda _m}}}{\mathop{\rm Im}\nolimits} \left( {{\varepsilon _m}} \right),(m = F,\,S),$$

(6b)$${\alpha _T} = \frac{{2\pi }}{{{n_T}{\lambda _T}}}{\mathop{\rm Im}\nolimits} \left( {{\varepsilon_\infty } + \sum\limits_j {{\chi_{Qj}}} } \right),$$

where ω_jTO, S_j, and Γ_jTO are the eigenfrequency, oscillating strength and damping coefficient of the jth A₁ transverse optical (TO) mode, respectively, ${\bar{n}_{0j}}$ is the Bose distribution function of the jth TO mode, (S₃₃/LΔΩ)_j represents the spontaneous Raman scattering cross section of the jth TO mode [1,29], ω_F and ω_T are the angular frequencies of the fundamental and terahertz waves, ħ is the reduced Planck constant. ε_∞ is the high frequency permittivity of the nonlinear crystal, ε_m (m = F, S) are permittivities for the fundamental and Stokes waves.

For convenience, we define

(7)$$g_m^{(2)} = \frac{\pi }{{{n_m}{\lambda _m}}}\left[ {4{d_{33}} + \sum\limits_j {{d_{Qj}}\textrm {Re} ({\chi _{Qj}})} } \right],(m = F,\,S,\,T),$$

(8)$$g_m^{(3)} = \frac{\pi }{{{n_m}{\lambda _m}}}\left[ {\sum\limits_j {d_{Qj}^2{\mathop{\rm Im}\nolimits} ({\chi _{Qj}})} } \right],(m = F,\,S,\,T),$$

(9)$${A_m} = \sqrt {\frac{{{I_m}}}{{2c{\varepsilon _0}{n_m}}}} ,(m = F,\,S,\,T),$$

where $g_m^{(2)}$ and $g_m^{(3)}$ (m = F, S, T) correspond to the second order nonlinear effect and the Raman scattering in SPS process, respectively. They are dependent on the THz wave frequency. Generally, the THz wave frequency is far away from the eigenfrequencies of the TO modes, where the Raman scattering gain and the absorption of the crystal are small while the second order nonlinear coefficient is relatively large. Therefore, the Raman scattering terms are much smaller than the second order nonlinear terms in Eqs. (1) and (2) and can be neglected. Besides, the nonlinear crystals for SPS usually have identical transparency in the near infrared region, so the absorption coefficients for the fundamental and Stokes waves, α_F and α_S can be regarded as zero. In addition, the nonlinear absorption can also be neglected. The band gap of LiNbO₃ nonlinear crystal is larger than twice the pumping photon energy, so there is no two-photon absorption. The nonlinear crystals for SPS do not have saturable absorption characteristic. For example, LiNbO₃ have saturable absorption property only when it is doped with specific ions such as Au⁺ and Ag⁺ [30].

With the conditions above, the coupled wave equations of SPS process can be rewritten as

(10)$$\frac{{{\textrm d}{A_F}}}{{{d}x}} ={-} g_F^{(2)}{A_S}{A_T},$$

(11)$$\frac{{{\textrm d}{A_S}}}{{{d}x}} = g_S^{(2)}{A_F}{A_T},$$

(12)$$\frac{{{\textrm d}{A_T}}}{{{d}{y^T}}} = g_T^{(2)}{A_F}{A_S} - \frac{1}{2}{\alpha _T}{A_T}.$$

2.2 Some approximations and assumptions

Before solving the coupled wave equations, there are some approximations and assumptions that have to be employed and claimed. Firstly, the fundamental beam is considered as a plane wave, and its cross section is a circle. This is a general approximation when analyzing some laser characteristics [31–34]. Also, in nonlinear processes with low conversion efficiencies such as SPS, the consumption of the fundamental wave is usually very small, and the residual fundamental beam basically maintains its original spatial distribution at the exit of the nonlinear crystal. Secondly, A_S has a uniform distribution in the entire interaction region and its value varies only with time. The reason for such assumption is that the Stokes beam oscillates in its cavity, and the spatial distribution of the Stokes beam quickly forms a stable, axisymmetric mode after SPS conversion process begins. Despite the fact that the Stokes beam spatial distribution is more likely to be a Gaussian distribution or higher order transverse modes, it is reasonable to consider A_S uniformly distributed. This approximation is quite common and widely used when dealing with laser rate equations with nonlinear processes [35–37]. Thirdly, the transit time for the THz wave to cross the interaction region (denoted as t_T) is much smaller than the Stokes beam roundtrip period (denoted as t_rS) in the Stokes cavity. Typically, the Stokes cavity length is around 20-30 cm, and the diameter of the fundamental beam (denoted as D) is smaller than 5 mm, as illustrated in Fig. 2, so the propagation time for the THz wave to cross the interaction region is around 10⁻¹¹ s, while the Stokes beam roundtrip period is around 10⁻⁹ s. Fourthly, the fundamental beam pulse width (denoted as t_F) is much larger than the Stokes beam roundtrip period (t_F∼10⁻⁸ s). The relation among t_T, t_rS and t_F is shown in Fig. 3. Finally, the amplitudes of the three waves hardly vary within a distance smaller than their wavelengths. This is a quite common approximation when solving coupled wave equations for nonlinear processes [38,39].

Fig. 2. The interaction region of SPS process.

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Fig. 3. Relation among t_F, t_rS, and t_T.

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For convenience, the center axis of the fundamental beam is defined as the x axis (y, z = 0). Also, the entrance of the crystal is defined as x = 0, as illustrated in Fig. 4.

Fig. 4. The THz wave traveling distance through the nonlinear interaction region when it arrives at position P(x, y, z).

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2.3 Modeling step 1

The first step of the extracavity-pumped TPO modeling is to express A_T as a function of A_F and A_S. As shown in Fig. 3, t_T is much smaller than t_rS, so it can be considered that A_F and A_S keep constant within the timescale of t_T. A_T(x, y, z) can be obtained by integrating Eq. (12).

(13)$${A_T}(x,y,z) = \frac{2}{{{\alpha _T}}}g_T^{(2)}{A_F}(x){A_S}\left\{ {1 - \exp \left[ { - \frac{1}{2}{\alpha_T}(y + \sqrt {{r^2} - {z^2}} )/\sin \beta } \right]} \right\},$$

where r is the radius of the fundamental beam cross section, $(y + \sqrt {{r^2} - {z^2}} )/\sin \beta$ corresponds to the distance the terahertz wave has travelled through the nonlinear interaction region when it arrives at the position P(x, y, z), as illustrated in Fig. 4.

Considering that the Stokes beam has the assumption of uniform distribution in yOz plane and the fundamental beam has the approximation of uniform distribution, we need to calculate the average value of A_T(x, y, z) in yOz plane.

(14)$${A_T}(x) = g_T^{(2)}{A_F}(x){A_S}K,$$

where

(15)$$K = \frac{2}{{{\alpha _T}\pi {r^2}}}\int_{ - r}^r {{\textrm d}\textrm z\int_{ - C(z)}^{C(z)} {\left\{ {1 - \exp [ - \frac{1}{2}{\alpha_T}(y + C(z))/\sin \beta ]} \right\}{\textrm d}\textrm y} } ,$$

(16)$$C(z) = \sqrt {{r^2} - {z^2}} .$$

By substituting Eq. (14) into Eq. (10), the coupled wave equation of the fundamental wave can be written as

(17)$$\frac{{{\textrm d}{A_F}}}{{{\textrm d}x}} ={-} {G_F}{A_F}(x)A_S^2,$$

where

(18)$${G_F} = g_T^{(2)}g_F^{(2)}K.$$

2.4 Modeling step 2

The second step of the modeling is to establish the Stokes wave rate equation. It is valid for the entire Stokes wave pulse duration, which is much larger than the Stokes beam roundtrip period t_rS. We begin with considering the evolutions of the residual fundamental and Stokes waves in the timescale t_rS. Equation (17) can be solved as

(19)$$\int_{{A_F}(0)}^{{A_F}(x)} {\frac{1}{{{A_F}}}{\textrm d}{A_F}} = \int_0^x { - {G_F}A_S^2{d}x} ,$$

where A_F(0) is the amplitude of the fundamental wave at the entrance of the crystal. If the length of the crystal is L,

(20)$${A_F}(L) = {A_F}(0)\exp ( - {G_F}A_S^2L).$$

Using Eq. (9), Eq. (20) can be written as

(21)$${I_F}(L) = {I_F}(0)\exp \left( { - \frac{{{G_F}L}}{{{n_S}{\varepsilon_0}c}}{I_S}} \right).$$

The consumption of the fundamental wave in one passing through the crystal is

(22)$$\Delta {I_F} = {I_F}(L) - {I_F}(0) = {I_F}(0)\left[ {\exp \left( { - \frac{{{G_F}L}}{{{n_S}{\varepsilon_0}c}}{I_S}} \right)} \right].$$

According to the Manley-Rowe relation, the Stokes wave increment from SPS process is

(23)$$\Delta {I_{S - pos}} ={-} \frac{{{\lambda _F}}}{{{\lambda _S}}}{I_F}(0)\exp \left( { - \frac{{{G_F}L}}{{{n_S}{\varepsilon_0}c}}{I_s}} \right),$$

with the relation between intensity and photon density

(24)$$I = \hbar \omega c\phi ,$$

where ϕ denotes the photon density. The increment of the Stokes wave photon density in one roundtrip period in the cavity is

(25)$$\Delta {\phi _{S - pos}} = {\phi _F}(0,t)\left[ {\exp \left( { - \frac{{{G_F}L\hbar {\omega_S}}}{{{n_S}{\varepsilon_0}}}{\phi_S}} \right) - 1} \right].$$

Now we consider a larger timescale, in which the Stokes beam oscillates in the Stokes cavity for multiple times. The variation of the Stokes wave photon density in one oscillating period is

(26)$${\phi _S}[{(N + 1){t_{rS}}} ]= \left[ {1 + \frac{{\Delta {\phi_{S - pos}}}}{{{\phi_S}(n{t_{rS}})}}} \right]{T_S}{R_S}{\phi _S}(N{t_{rS}}),$$

(27)$$\Delta \ln {\phi _S} = \ln \left\{ {\left[ {1 + \frac{{{\phi_F}(0,t)}}{{{\phi_S}(N{t_{rS}})}}\left( {\exp \left( { - \frac{{{G_F}L\hbar {\omega_S}}}{{{n_S}{\varepsilon_0}}}{\phi_S}} \right) - 1} \right)} \right]{T_S}{R_S}} \right\},$$

where N denotes the Stokes wave roundtrip ordinal number in the cavity, T_S is the product of the transmittances of all the surfaces (except the output coupler) in the Stokes cavity, R_S is the output coupler reflectivity. Following the method of Ref. [34], the Stokes wave rate equation in the extracavity-pumped TPO is

(28)$$\begin{aligned} \frac{{\textrm{d}{\phi _S}}}{{{\textrm d}t}} &= {\phi _S}\frac{{{\textrm d}\ln {\phi _S}}}{{{\textrm d}t}} \approx \frac{{{\phi _S}}}{{{t_{rS}}}}{\Delta }\ln {\phi _S}\\& = \frac{{{\phi _S}}}{{{t_{rS}}}}\ln \left\{ {\left[ {1 + \frac{{{\phi _F}(0,t)}}{{{\phi _S}(t)}}\left( {\exp \left( { - \frac{{{G_F}L\hbar {\omega _S}}}{{{n_S}{\varepsilon _0}}}{\phi _S}(t)} \right) - 1} \right)} \right]{T_S}{R_S}} \right\}, \end{aligned}$$

where ϕ_F(0, t) denotes the fundamental photon density that is injected into the cavity, which is determined by the pump pulse. By numerically solving Eq. (28), the time evolution of ϕ_S(t), as well as A_S(t) can be obtained.

2.5 Modeling step 3

The third step of the extracavity-pumped TPO modeling is to obtain the properties of the residual fundamental and terahertz waves by solving the coupled wave equations. The approach of solving the coupled wave equations is similar to the methods in Refs. [10,22–24]. The main difference is that, in Refs. [10,22–24], three coupled wave equations corresponding the fundamental, Stokes, and terahertz waves in TPGs were solved to obtain the properties of the three waves, while in this paper for extracavity-pumped TPOs, A_S(t) is obtained by solving the Stokes wave rate equation Eq. (27), and by using A_S(t) and Eqs. (10) and (12), the properties of the residual fundamental and terahertz waves are obtained.

3. Experiment and simulation

Figure 5 illustrates the optical scheme of the extracavity-pumped TPO. The pump source was a Q-switched Nd:YAG laser, which generated 1064 nm laser pulses with a flattop beam profile. The pulse repetition rate was 1 Hz. The nonlinear crystal was a trapezoid MgO:LiNbO₃ crystal and the Stokes wave cavity was composed of two flat mirrors, M1 and M2, and the nonlinear crystal reflecting surface AB. The THz wave emitted vertically from this surface (surface-emitted configuration [40,41]). M1 was HR coated for 1060-1090 nm spectral range, while M2 was partial-reflection coated for the Stokes beam (T = 30% at 1070 nm). M1 and M2 were mounted on a rotating stages so that the angle between the fundamental beam and the Stokes beam could be accurately tuned. The combination of a λ/2 plate (HWP) and a polarized beam splitter (PBS) was used to adjust the fundamental pulse energy while maintaining its polarization at z direction. A Golay cell (TYDEX, GC-1D) was used to measure the output THz wave energy.

Fig. 5. Optical scheme of the extracavity-pumped MgO:LiNbO₃ TPO.

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The specific parameters of the experimental device are listed in Table 1, along with some other parameters that will be used in the simulation [18].

Table 1. Parameters for the simulation

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Figure 6(a) demonstrates the measured pulse evolutions at the pump energy of 90 mJ. It should be clarified that the pulse widths are the real ones while the intensities do not indicate the actual intensities because the residual fundamental and Stokes pulses were measured by two detectors. At different pump energies from 40 mJ to 90 mJ, the measured Stokes pulses are illustrated in Fig. 6(b), where E_F denotes the pump energy.

Fig. 6. (a) Measured pulse evolutions of the original and residual fundamental and Stokes pulses when the pump energy is 90 mJ. (b) Stokes pulses at different pump energies.

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At the pump energy of 90 mJ, the simulated output pulses for the residual fundamental, Stokes and terahertz waves are illustrated in Fig. 7(a). It can be seen that the original and residual fundamental pulses have nearly the same shape at the beginning because the Stokes wave is weak, and the consumption of the fundamental wave is not serious. The Stokes pulse gets higher and higher with accumulation time, at the cost of the fundamental pulse consumption. After the pump pulse disappears, the Stokes pulse lives a bit longer because it oscillates in its cavity. The THz wave exits the nonlinear crystal almost immediately after its generation by interacting between the pump and Stokes waves. It can be seen from Eqs. (3), (12) that the THz growth rate depends on the product of the pump and Stokes wave intensities. Before the Stokes wave accumulates to a certain value, the THz wave gain is very small and the THz intensity remains in low level compared with its peak value. The THz wave can only get high intensity when the Stokes wave intensity is relatively large and the pump beam is significantly consumed but still has relatively large intensity. So, in Fig. 7(a), the THz output energy mainly concentrates in a small range of time when the pump and Stokes wave are both relatively large. Similar analysis was given in Ref. [42]. Figure 7(b) demonstrates the Stokes pulses at different pump energies from 40 mJ to 90 mJ.

Fig. 7. (a) Calculated pulse evolutions of the residual fundamental, Stokes and terahertz beams when the pump energy is 90 mJ. (b) Calculated Stokes pulses at different pump energies.

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Figures 8–10 demonstrate the THz pulse energy, the residual fundamental pulse width, and the Stokes pulse width as functions of the pumping energy. The dots are the experimental results and the solid lines are the simulated results. It is easy to understand that the THz pulse energy increases from 0.10 µJ to 0.74 µJ with increasing pumping energy from 40 mJ to 90 mJ. The generation efficiency from pump wave to THz wave is calculated to be 8.2×10⁻⁶ at maximum. The simulated residual fundamental pulse width slightly decreases with increasing pumping energy. This can be attributed to the stronger nonlinear interaction and the relatively rapid pumping pulse consumption for larger pumping pulse energy. The pulse width of the Stokes wave has little change with increasing pump energy. For larger pumping pulse energy, the nonlinear interaction is strong. The appearance the Stokes pulse is earlier and the pump wave consumption is earlier. So the residual pump pulse disappears earlier, which causes that the Stokes pulse ends earlier. For small pumping pulse energy, the nonlinear interaction is relatively weak. The appearance and the ending of the Stokes pulse are both relatively late. So, the Stokes pulse width hardly changes as the pumping energy increases.

Fig. 8. The residual fundamental pulse width as a function of the pumping energy.

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Fig. 9. The Stokes pulse width as a function of the pumping energy.

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Fig. 10. The THz pulse energy as a function of the pumping energy.

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Figure 11 demonstrates the measured spectra of the fundamental and Stokes waves. The central wavelengths of the fundamental and Stokes waves are 1064.2 nm and 1070.9 nm, respectively. The THz frequency can be calculated as 1.76 THz using the energy conservation law, ω_F=ω_S+ω_T. The linewidths of the fundamental and Stokes waves are 0.1 nm and 0.3 nm respectively, so the linewidth of the THz wave can be inferred as 0.05 THz.

Fig. 11. The spectra of the fundamental and Stokes waves.

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It can be seen from Figs. 6–10 that the simulated results are basically consistent with the experimental results. Some deviations of the simulated results from the experiment data can also be found. The simulated results for the residual fundamental pulse widths are a little smaller than the experiment data (around 7.2 ns) while the simulated results for the Stokes pulse width are a little larger than the experiment data (around 5.6 ns). The main deviation is that the simulated THz pulse energy is larger than the measured one when the pumping energy is large. The possible causes are as follows. First, the Stokes beam spatial distribution is assumed to be uniform in the modeling while the real spatial distribution is closer to Gaussian distribution. At high pumping energy, the efficient interaction distance of the THz wave with a Gaussian Stokes beam is shorter than that with a uniform Stokes beam, since the energy concentrates in the center part and the absorption is dominant at peripheral part, so the calculated THz energy is larger than experiment data at high pump energy. Second, the distribution of A_T in y^TOz plane is not uniform, so the method of replacing A_T(x, y^T, z) with A_T(x) by averaging A_T(x, y^T, z) in y^TOz plane will cause some errors during establishing the rate equation. Third, the nonlinear conversion efficiency and the consumption of the pump wave is assumed to be small in the modeling. When the pumping energy is small, this assumption is fairly satisfied. When the pumping energy is relatively large, this assumption is not well satisfied. This means that, in the experiment, as the pump energy increases, the consumption of the pump wave becomes lager, and less THz wave is generated compared with the simulation result. This leads to the situation that the calculated pulse energy is larger than the measured pulse energy. Fourthly, in the modelling, the fundamental beam pulse width is supposed to be much larger than the Stokes beam roundtrip period. In the experiment, the Stokes roundtrip period is about 1.4 ns. The pumping pulse width at half maximum is 7.5 ns and that at tenth maximum is about 17 ns. The assumption is not well satisfied and some error may be introduced.

So far the modeling method for extracavity-pumped TPOs, and the corresponding experimental and simulational results for a MgO:LiNbO₃ TPO extracavity-pumped by a 1064 nm Nd:YAG Q-switched laser have been demonstrated. Additionally, this modeling method for extracavity-pumped TPOs can also help the theoretical analysis of the TPOs based on the third-order nonlinear processes such as four-wave mixing [43–46]. Although the coupled wave equations and the phase matching conditions of the third-order TPOs are different from those of the TPOs based on stimulated polariton scattering, and the proposed model cannot be directly used for analyzing these third-order TPOs, some critical procedures of the proposed model are possible inspirations for the modeling of the third-order TPOs. An important procedure is to express the THz wave amplitude as a function of the pump and signal amplitudes according to the concrete third-order TPO conditions, and derive the rate equation for the signal beam.

4. Summary

We have performed the modeling for extracavity-pumped Q-switched terahertz parametric oscillators. Considering that the transit time for the THz wave to cross the interaction area t_T is much smaller than the Stokes wave roundtrip time in the cavity t_rS, the fundamental and Stokes wave amplitudes, A_F and A_S, are considered as constants within the timescale of t_T and the coupled wave equations are solved to get the THz wave amplitude A_T, whose average value in the xOy plane is expressed as a function of A_F and A_S. Then, considering that t_rS is much smaller than the pumping pulse width t_F, the coupled wave equations within the timescale of t_rS are solved to get the Stokes wave amplitude increment. The rate equation for the Stokes wave is obtained by using this increment within t_rS. By solving the rate equation, the Stokes wave temporal evolution during and after the existence of the pumping pulse is obtained. Last, by considering the Stokes wave amplitude as a known parameter, the coupled wave equations for every half t_rS timescale are solved to get the properties of the residual fundamental and terahertz waves.

An extracavity-pumped TPO based on MgO:LiNbO₃ crystal is constructed. The pumping pulse width is 7.5 ns. For pumping pulse energies from 40 to 90 mJ, the pulse widths of the residual fundamental and Stokes waves are around 7.2 ns and 5.4 ns, respectively, the THz pulse energies increase from 0.10 µJ to 0.74 µJ. This model is used to analyze this TPO, the simulation results are consistent with the experimental results on the whole. We hope the presented modeling method can play an important role in the design and optimization of extracavity-pumped Q-switched TPOs.

Funding

National Natural Science Foundation of China (12074222, 61775122); Key Technology Research and Development Program of Shandong (2019JMRH0111); Natural Science Foundation of Shandong Province (ZR2017MF038).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Symbol	Parameter	Value
λ_F	Wavelength of the fundamental wave	1064.2 nm
λ_S	Wavelength of the Stokes wave	1070.9 nm
λ_T	Wavelength of the THz wave	170.1 µm
4d₃₃+∑d_QjRe(χ_Qj)	Second order nonlinear coefficient	9.48×10⁻¹⁰ m/V
α_T	Absorption for THz wave	30.8 cm^-1
τ	Pulse width of the fundamental wave	7.5 ns
D₀	Diameter of the fundamental beam	3.3 mm
L	Length of the nonlinear crystal	42.5 mm
L_cS	Length of the Stokes cavity	200 mm
T_s	Product of transmittances of all surfaces in the cavity	0.85
R_S	Reflectivity of the output coupler	0.70
n_S, n_F	Refractive index for the fundamental and Stokes waves	2.2
n_T	Refractive index for the THz wave	5.2

Modeling for extracavity-pumped terahertz parametric oscillators

Abstract

1. Introduction

2. Modeling method

2.1 Coupled wave equations of SPS process

2.2 Some approximations and assumptions

2.3 Modeling step 1

2.4 Modeling step 2

2.5 Modeling step 3

3. Experiment and simulation

4. Summary

Funding

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

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Figures (11)

Tables (1)

Equations (29)

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