## Abstract

In this paper, the rate equations describing the operation of intracavity-pumped Q-switched terahertz parametric oscillators based on stimulated polariton scattering are given for the first time. The rate equations are obtained under the plane-wave approximation, the oscillating fundamental and Stokes waves are supposed to be round uniform beam spots. Considering the fact that the terahertz wave nearly traverses the pump and Stokes beams and using the coupled wave equations, the terahertz wave intensity is expressed as the function of the fundamental and Stokes intensities. Thus, the rate equations describing the evolution processes of the fundamental and Stokes waves are obtained in the first step. The THz wave properties are then obtained. Several curves based on the rate equations are generated to illustrate the effects of the nonlinear coefficient, the THz wave absorption coefficient, and pulse repetition rate on the THz laser characteristics. Taking the intracavity-pumped Mg:LiNbO_{3} TPO as an example, the THz frequency tuning characteristic and the dependences of the fundamental, Stokes, and THz wave powers on the incident diode pump power are calculated. The theoretical results are in agreement with the experimental results on the whole.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Terahertz (THz) technology has wide applications through THz spectroscopy [1,2] and THz spectral imaging [3–6] techniques in the fields of medicine [3,4], homeland security [5,6], and nondestructive testing [6,7], and so on. Terahertz radiation source is a key part for terahertz technology development. During the past thirty years, Terahertz-wave Parametric Oscillators (TPOs) have attracted much interest in generating compact, reliable, room temperature operation and cost-effective THz wave radiation [8–12]. The physical basis of the TPOs is Stimulated Polariton Scattering (SPS), which occurs in nonlinear crystals with both infrared and Raman active transverse optical (TO) modes [8]. So far, the SPS has been demonstrated in LiNbO_{3} [9–11], KTiOPO_{4} (KTP) [12–15], KTiOAsO_{4} (KTA) [16,17] and RbTiOPO_{4} (RTP) [18–20].

TPOs include extracavity-pumped systems and intracavity-pumped systems. In order to enhance the nonlinear effect and conversion efficiencies, Q-switched laser pulses are usually used as the pumping sources. In an intracavity-pumped Q-switched system, the TPO is placed inside the fundamental Q-switched laser cavity. It takes the advantage of high intracavity intensity to improve the SPS conversion. An additional advantage is that the coupling optics between the fundamental laser and the TPO is eliminated, further contributing to the compactness of the device. Therefore, using intracavity-pumped Q-switched TPO is an efficient and economical way to obtain high power THz waves. In 2006, Edwards [21] et al. reported the first intracavity Q-switched THz laser radiation source based on SPS. Since then, there have been many papers reporting the experimental progresses on the intracavity Q-switched THz lasers [18,20–27], and the obtained average output THz power has increased to 124.7 µW [20] from the beginning of 0.5 µW [21].

While the experimental work on the intracavity-pumped Q-switched TPO has made great progress, there have been few reports on its theoretical research. Rate equations are the effective theoretical tools to describe the laser operations. The rate equations for many lasers such as Q-switched lasers [28–32], intracavity-pumped frequency doubled lasers [33], intracavity-pumped optical parametric oscillators [34], and intracavity-pumped Raman lasers [35–37] have been developed successively and played important roles in the characteristic description and optimization of these lasers. So far the rate equations for the intracavity-pumped Q-switched TPO have not developed. The most obvious difference between the intracavity-pumped Q-switched TPO and the abovementioned lasers [28–37] is that the TPO has a non-collinear phase matching while the abovementioned lasers are in collinear phase matching. The phase matching angle between the Stokes wave and the terahertz wave can be as large as 70° in the THz parametric oscillators [12,16,38]. For guiding the design and optimization of the intracavity-pumped Q-switched TPO, a suitable modeling is urgently needed.

In this paper, we perform the modeling of the intracavity-pumped Q-switched TPOs by using the rate equations. Like the existed rate equations for other Q-switched lasers, we treat them under the plane-wave approximation, i.e., the oscillating fundamental and Stokes waves are supposed to be round uniform beam spots. The large non-collinear phase matching angle between the Stokes and terahertz beams is taken into account. In the first step, we obtain the approximate expression of the THz wave intensity as a function of the fundamental and Stokes wave intensities by using the coupled wave equations. Then, we obtain the rate equations describing the evolution processes of the fundamental and Stokes waves. The characteristics of the THz wave are calculated based on the fundamental and Stokes wave properties. Several curves based on the rate equations are generated to analyze the influences of the nonlinear coefficient, the THz wave absorption coefficient, and pulse repetition rate on the THz laser performance. The intracavity-pumped TPO with Mg:LiNbO_{3} as the nonlinear crystal is taken as an example, the THz frequency tuning characteristic and the dependences of the output fundamental, Stokes, and THz wave powers on the incident diode pump power are calculated. The theoretical results are in agreement with the experimental results on the whole.

## 2. The coupled wave equations for SPS process

The coupled wave equations are a reasonable way to simulate the SPS process. In the SPS process, a fundamental photon (wave vector ${\boldsymbol{k}_{\boldsymbol{F}}}$) is consumed inside the nonlinear crystal to generate a Stokes photon (wave vector ${\boldsymbol{k}_{\boldsymbol{S}}}$) and a polariton photon (wave vector ${\boldsymbol{k}_{\boldsymbol{T}}}$) under a non-collinear phase matching condition, and their phase matching relationship is ${\boldsymbol{k}_{\boldsymbol{F}}} = {\boldsymbol{k}_{\boldsymbol{S}}} + {\boldsymbol{k}_{\boldsymbol{T}}}$, as shown in Fig. 1. It should be pointed out that, while the non-collinear phase matching angle *φ* between the fundamental and Stokes beams is very small (usually less than several degrees), the non-collinear phase matching angle *β* between the Stokes and THz beams is rather large, it can be as large as 70°. The large angle *β* causes the spatial separation of the THz beam from the fundamental and Stokes beams.

Using the slowly varying amplitude approximation, the evolution of the coupled wave equations for the amplitudes of the fundamental, Stokes, and THz waves considering the non-collinear phase matching condition in SPS process can be expressed as [39,40]

*E*,

_{F}*E*, and

_{S}*E*are the amplitudes of the fundamental, Stokes, and THz wave amplitudes, respectively. ${\xi _m}\,(m = F,S,T)$, ${\varpi _m}\,(m = F,S,T)$, ${\vartheta _m}\,(m = F,S)$ are as follows [40]

_{T}*k*,

_{m}*ω*, and

_{m}*ɛ*denote the wave vectors, the frequencies, and the permittivities of the fundamental, Stokes, and THz waves in nonlinear crystal, respectively.

_{m}*c*is the velocity of light in vacuum. In SPS process,

*d*is originated from the electronic polarization,

_{E}*d*is originated from the ionic polarization, (

_{Qj}χ_{Qj}*d*) denotes the second-order nonlinear coefficient for parametric process, $d_{Qj}^2\chi _{Qj}^{}$ denotes the third-order nonlinear coefficient for Raman process.

_{E}+d_{Qj}χ_{Qj}*d*in cgs units and

_{Qj}*χ*can be written as [41,42]

_{Qj}*n*and

_{F}*n*are the refractive indexes of the nonlinear crystal for the fundamental and Stokes waves, $\hbar $ is the Planck constant, ${\bar{n}_0}$ is the Bose distribution function. The quantity (

_{S}*S*

_{33}/

*L*ΔΩ)

*denotes the spontaneous Stokes scattering efficiency,*

_{j}*S*

_{33}is the fraction of incident power that is scattered into a solid angle ΔΩ near a normal to the optical path length

*L*, and it is proportional to the scattering cross section [8]. Ω

*,*

_{jTO}*S*, and Γ

_{j}*are the eigenfrequency, oscillator strength, and damping coefficient of the*

_{jTO}*j*th TO mode, respectively [8,43].

*ɛ*is the high-frequency dielectric constant.

_{∞}Based on the relationship between intensity and complex amplitude,

*α*

_{T}_{ }is the absorption coefficient for the THz wave by nonlinear crystal,

*α*and

_{F}*α*(absorption coefficients for the fundamental and Stokes waves) are much smaller and can be neglected. $g_m^{(2)}$ (

_{S}*m*=

*F*,

*S*,

*T*) are gain coefficients related to the second-order coefficients for parametric process, $g_m^{(3)}$ (

*m*=

*F*,

*S*) are gain coefficients related to the third-order nonlinear coefficients for Raman process. These absorption and gain coefficients can be expressed as

All the coefficients in Eqs. (10) to (12) are frequency dependent. Their values are going to change as the THz wave frequency changes by controlling the phase matching angle in SPS process. These coefficients for Mg:LiNbO_{3} will be calculated later (see Fig. 10).

## 3. The theoretical processing for intracavity TPO

The schematic description of the intracavity Q-switched TPO is shown in Fig. 2. The fundamental laser cavity is constituted by R1 and R2 in which there are the fundamental laser gain medium, the Q-switch, and the nonlinear crystal. The TPO is constituted by R3 and R4 and shares the nonlinear crystal. There is a certain small angle between the fundamental laser cavity axis and the Stokes cavity axis (usually less than several degrees).

In the intracavity Q-switched TPO, the fundamental and Stokes beams oscillate inside the respective resonant cavities. Then the SPS process occur in both the positive direction and the reverse direction of the *x*-axis inside the nonlinear crystal and there will be two propagating directions (*y’* and -*y’*) for the THz beam according to the phase matching condition in the interaction region, as shown in Fig. 3.

Due to the fundamental and Stokes beams oscillate inside the respective resonant cavities, the fundamental and Stokes beams intensities are both axisymmetric, rather than changing monotonically with *y* like the situation in an injection-seeded terahertz parametric generator in which the pumping and Stokes beams pass through the nonlinear crystal only one time [44]. Following the assumptions in the rate equations of other lasers [28–31,35,36], the fundamental and Stokes beams are assumed to be uniform circular spots (i.e., plane-wave approximation) and the increments of the fundamental intensity *I _{F}* and the Stokes intensity

*I*are small in a round-trip transit time of light in the resonator. The assumption that the fundamental and Stokes beams are uniform round spots is reasonable, it makes the equation derivation and calculation not too complicated, but the results can reflect the basic characteristics of the described lasers [28–37].

_{S}In the intracavity-pumped Q-switched TPO, considering that the transit time Δ*t _{T}* for the THz beam passing through the fundamental beam cross section along

*y’*is much smaller than the transit time Δ

*t*for the fundamental and Stokes beams passing through the nonlinear crystal, and Δ

_{FS}*t*is much smaller than the light roundtrip transit time

_{FS}*t*in the cavity, we can consider

_{r}*I*and

_{F}*I*as constants in the time range of Δ

_{S}*t*.

_{T}Considering the THz beam propagating along the positive direction of *y'*-axis in the SPS interaction region, using *I _{Tpos}*(

*y’*) to express the THz beam intensity and ordering $f(y^{\prime}) = I_{Tpos}^{1/2}(y^{\prime})$ in Eq. (9-c), we get

*I*(

_{Fpos}*x*) and

*I*(

_{Spos}*x*) are the fundamental and Stokes beam intensities propagating in the positive direction of the

*x*-axis. Integrating Eq. (13) from the bottom of the SPS interaction region in the positive direction of

*y'*-axis, we get

*Y’*represents the bottom of the SPS interaction region in

*y'*-axis and $f( - Y^{\prime}) = 0$,

*Y’*represents the top of the SPS interaction region in

*y'*-axis,

*Y*corresponds to the position in

*y-*axis and $Y = Y^{\prime}\sin \beta $.

In the same way, we can get the intensity *I _{Tneg}*(

*y’*) of the THz beam propagating along the negative direction of

*y'*-axis from the top of the SPS interaction region

*I*(

_{Fneg}*x*) and

*I*(

_{Sneg}*x*) are the fundamental and Stokes beam intensities propagating in the negative direction of the

*x*-axis.

If *I _{F}*(

*x*) and

*I*(

_{S}*x*) represent the total fundamental and Stokes beam intensities in two directions, $I_{Fpos}^{}(x)\textrm{ = }I_{Fneg}^{}(x) = I_F^{}(x)/2$ and $I_{Spos}^{}(x)\textrm{ = }I_{Sneg}^{}(x) = I_S^{}(x)/2$.

The dependences of *I _{Tpos}*(

*y*),

*I*(

_{Tneg}*y*) and ${I_{Tsum}}(y\textrm{) = }I_{Tpos}^{}(y)\textrm{ + }I_{Tneg}^{}(y)$ on

*y*for different

*α*in units of $g_T^2{I_F}{I_S}$ when the length of SPS interaction region is 0.5 mm are shown in Fig. 4. It can be seen that

_{T}*I*(

_{Tpos}*y*) increases with

*y*and

*I*(

_{Tneg}*y*) increases with

*-y*, while

*I*(

_{Tsum}*y*) keeps approximately constant with

*y*. That is to say, the total THz wave intensity

*I*can be approximately considered to be proportional to $g_T^2{I_F}{I_S}$ during the evolution process of the Q-switched pulses. Let us imagine that, if the fundamental and Stokes waves were supposed to have square beam shapes,

_{Tsum}*I*would be nearly uniform in the entire

_{Tsum}*yz*cross section.

For round beam shapes, the *I _{Tsum}* will present relatively uniform distribution in most of the spot area. For satisfying the assumption of uniform round fundamental and Stokes beam spots, we need the average THz wave intensity in the

*yz*cross section in the SPS interaction region.

*D*is the diameter of the SPS interaction region and 2

*Y*(

*z*) is the beam size of round spot in

*y*direction in different position of

*z*.

Substituting Eq. (16) into Eqs. (9-a) and (9-b), we can obtain the coupled wave equations for the fundamental and Stokes waves oscillating in cavity

whereIt can be seen that the derivative of the fundamental or Stokes intensity with respect to *x* is proportional to *I _{F}*(

*x*)

*I*(

_{S}*x*). From this point, the coupled wave equations for the intracavity-pumped TPO are basically consistent with the coupled wave equations describing the Raman process [45]. However, the differences are obvious. Firstly, in the Raman process, the gain coefficient is the third order coefficient of the nonlinear crystal, while in the intracavity-pumped TPO, the gain coefficient is much complicated. For example, the Stokes wave gain coefficient

*G*is $({{\kappa^{1/2}}g_S^{(2)}g_T^{(2)} + g_S^{(3)}} )$, $\kappa $ is related to THz absorption coefficient and the beam sizes of the fundamental and Stokes waves [see Eq. (19-b)], $g_S^{(2)}$ and $g_T^{(2)}$ are related to the second-order coefficients for parametric process of the nonlinear crystal in the SPS progress, $g_S^{(3)}$ denotes the third-order nonlinear coefficient for Raman process in the SPS progress. [see Eqs. (9)–(12)]. Secondly, in the Raman process, there is no need for phase matching, while in the intracavity-pumped TPO, the phase matching is needed, and the phase matching is non-collinear. There is a small phase matching angle between the fundamental and Stokes beams. This angle can be neglected in deriving Eqs. (18-a) and (18-b) because it is so small (usually less than several degrees) that the overlapping between the fundamental and Stokes beams is hardly affected. But the frequency tuning can be realized by adjusting this phase matching angle and the gain coefficients are changing with this angle. Thirdly, it should be pointed out that, only in the situation of intracavity-pumped TPOs in which the oscillating fundamental and Stokes beams are both axisymmetric, can Eq. (18) be approximately obtained.

_{S}## 4. The rate equations to describe the intracavity-pumped Q-switched TPO

Following J. J. Degnan’s method in [30], using Eq. (18) and the relation $I = \hbar \omega c\phi$ between intensity *I* and photon density $\phi (t)$. The rate equations to describe the performance of the intracavity-pumped Q-switched TPO can be expressed as

*l*, and ${t_{rS}} = 2{l_{cS}}/c$ is the round-trip transit time of the Stokes wave in the Stokes cavity of the optical length

_{cF}*l*,

_{cS}*c*is light speed in vacuum.

*σ*is the stimulated emission cross section of the laser gain medium,

*γ*is the inversion reduction factor of the laser gain medium.

*ω*is the angular frequency of the Stokes wave,

_{S}*g*is the gain coefficient in SPS for the fundamental wave,

_{F}*g*is the gain coefficient in SPS for the Stokes wave.

_{S}*l*and

*l*are the lengths of the laser gain medium and the nonlinear crystal, respectively.

_{R}*R*and

_{F}*R*are the reflectivities of the fundamental wave output coupler and the Stokes wave output coupler, respectively.

_{S}*L*and

_{F}*L*are the round-trip dissipative optical losses of the fundamental and Stokes waves, respectively. Equation (20-c) represents the change of population inversion density with time in the laser gain medium.

_{S}For the population inversion density in the laser gain medium [46],

*n*is the initial population inversion density,

_{in}*n*is the threshold population inversion density.

_{th}*P*is the input pumping power,

_{in}*ω*is the angular frequency of the fundamental wave,

_{P}*f*is the pulse repetition rate,

*α*is the absorption coefficient of the laser gain medium to the pump light,

_{P}*f*is Boltzmann occupation factor and

_{a}*τ*is the upper laser level lifetime,

*w*is the spot size. We define a parameter $N = {{{n_{in}}} \mathord{\left/ {\vphantom {{{n_{in}}} {{n_{th}}}}} \right.} {{n_{th}}}}$ called normalized initial population inversion density. It can represent the input pumping level.

_{F}By solving the rate equations expressed in (20), we can obtain the time-dependent fundamental and Stokes photons densities, ${\phi _F}(t)$ and ${\phi _S}(t)$. The THz photons density can be obtained using ${\phi _F}(t)$ and ${\phi _S}(t)$ according to Eq. (16)

The peak powers *P _{m-}*

_{max}, pulse energies

*E*and average powers

_{m}*P*of the output fundamental, Stokes, and THz waves can be obtained by [30]

_{m}For the fundamental and Stokes beams whose outputs are from the output couplers, ${A_{F,S}} = \pi {({{D \mathord{\left/ {\vphantom {D \textrm{2}}} \right.} \textrm{2}}} )^2}$ and ${\eta _{F,S}} = 1/2\ln ({{{1} \mathord{\left/ {\vphantom {{1} {{R_{F,S}}}}} \right.} {{R_{F,S}}}}} )$. For the THz beam whose output is from the surface of the nonlinear crystal, *A _{T}* and

*η*are determined by the output coupling mode (such as using prism coupler), the length of the nonlinear crystal and the absorption distance from the SPS interaction region to the nonlinear crystal surface.

_{T}## 5. Numerical calculation

Equations (20-a) to (20-c) are the rate equations to describe the performance of the intracavity-pumped Q-switched TPO. By solving these rate equations numerically, the dependences of the fundamental and Stokes photons densities on time can be obtained. The time-dependent THz photons density can be obtained using Eq. (24).

We simulate an intracavity-pumped Q-switched TPO by using the parameters in Table 1. The data in Table 1 are similar to but not exact the same with the actual experimental conditions because we cannot get all the needed and accurate parameters.

The pulse evolutions of the fundamental, Stokes and THz photon densities are shown in Fig. 5. It can be seen that only the fundamental wave appears at the beginning. The Stokes wave comes to be noticeable after it accumulates for a long time. Next, the fundamental wave is consumed very soon and the Stokes wave pulse is generated. The THz wave appears in the overlapping period of the fundamental and Stokes pulses.

The basic parameters related to the SPS process are the THz absorption coefficient *α _{T}* and gain coefficient

*G*or

_{S}*G*.

_{F}*G*is related to the SPS second- and third-order nonlinear coefficients $g_S^{(2)}$, $g_S^{(3)}$ and $g_T^{(2)}$, while

_{S}*G*is related to $g_F^{(2)}$, $g_F^{(3)}$ and $g_T^{(2)}$ (see Eq. (19)). The researchers are interested in the influences of the basic parameters

_{F}*α*and

_{T}*G*on the intracavity-pumped Q-switched TPO performances.

_{S}The dependences of the peak value ${\phi _{T - \max }}$ and integration ${\phi _{T - {\mathop{\rm int}} }}$ of the THz wave on *α _{T}* for a given

*G*and different pumping levels are shown in Fig. 6. The dependences of the peak value ${\phi _{T - \max }}$ and integration ${\phi _{T - {\mathop{\rm int}} }}$ of the THz wave on

_{S}*G*for a given

_{S}*α*and different pumping levels are shown in Fig. 7. It can be seen from Figs. 6 and 7 that ${\phi _{T - \max }}$ and ${\phi _{T - {\mathop{\rm int}} }}$ decrease monotonously with

_{T}*α*while there is an optimal value of

_{T}*G*for given

_{S}*α*and

_{T}*N*. The situation is easy to understand in Fig. 6. The situation in Fig. 7 needs more explanation.

We take the situation for *N* = 15 and *α _{T}* = 40 cm

^{-1}as an example. The pulse evolutions of the fundamental and Stokes waves under different

*G*are shown in Fig. 8. In Fig. 8(a), the gain coefficient

_{S}*G*= 1.5×10

_{S}^{−9}cm/W, it is relatively small. When the Stokes wave comes to be noticeable after it accumulates for a long time, the fundamental wave intensity is relatively small because it has passed its peak value for a long time. The intensity of the generated Stokes wave by consuming the fundamental wave is also small. The THz wave intensity is then small because is it proportional to the product of the fundamental wave intensity and the Stokes wave intensity. In Fig. 8(b),

*G*= 2.5×10

_{S}^{−9}cm/W, it is larger than that in Fig. 8(a). It needs less time for the Stokes wave to accumulate to be noticeable, the fundamental wave intensity is larger than that in Fig. 8(a) when the Stokes wave appears. The generated Stokes pulse by consuming the fundamental wave is stronger and the THz wave intensity is larger than that in Fig. 8(a). Summarizing the situations in Fig. 8(a) and Fig. 8(b), when

*G*is relatively small, the Stokes wave will appear after the peak value of the fundamental wave. The smaller

_{S}*G*is, the latter Stokes wave appears, and the weaker Stokes intensity will be. The THz wave intensity increases with increasing

_{S}*G*. This corresponds to the rising period in Fig. 7.

_{S}In Fig. 8(c), *G _{S}* = 4×10

^{−9}cm/W, it is large enough so that the Stokes wave begin to appear just when the fundamental wave reaches its peak value. The intensity of the generated Stokes wave by consuming the fundamental wave reaches its peak value too. The THz wave intensity also reaches its peak value. In Fig. 8(d),

*G*= 5×10

_{S}^{−9}cm/W, it is larger than that in Fig. 8(c). It is too large to make the Stokes wave appear too early before the fundamental wave reaches its peak value. The intensity of the generated Stokes wave by consuming the fundamental wave has no chance to reach its peak value as in Fig. 8(c), neither the THz wave intensity is. Summarizing the situations in Fig. 8(c) and Fig. 8(d), when

*G*is too large, the Stokes wave will appear before the peak value of the fundamental wave. The larger

_{S}*G*is, the sooner the Stokes wave appears, the weaker the Stokes intensity will be. The THz wave intensity decreases with increasing

_{S}*G*. This corresponds to the falling period in Fig. 7. The optimal

_{S}*G*corresponds to the situation in which the Stokes wave begin to appear just when the fundamental wave reaches its peak value.

_{S}In actively Q-switched lasers [28–37], the laser performance is sensitive to the pulse repetition rate *f*. Therefore, the effect of the pulse repetition rate on the SPS process in the intracavity-pumped TPO need to investigate. Using the data in Table 1 and Eqs. (21) and (22), the initial population inversion density *n _{in}* and the corresponding

*N*for a given pumping power and a given

*f*can be calculated. Solving Eqs. (20-a) to (20-c) and using Eqs. (23) to (25), ${\phi _{T - \max }}$, ${\phi _{T - {\mathop{\rm int}} }}$, and the product of ${\phi _{T - {\mathop{\rm int}} }}$ and

*f*can be obtained. While ${\phi _{T - \max }}$ is proportional to the THz pulse peak power and ${\phi _{T - {\mathop{\rm int}} }}$ is proportional to the pulse energy, $f \cdot {\phi _{T - {\mathop{\rm int}} }}$ is proportional to the THz average power. Figure 9 shows the dependences of ${\phi _{T - \max }}$, ${\phi _{T - {\mathop{\rm int}} }}$, and $f \cdot {\phi _{T - {\mathop{\rm int}} }}$ on

*f*for different diode powers. It can be seen that the pulse peak power and energy decrease monotonously with increasing

*f*, while the average power has a maximum value corresponding the optimal

*f*for a given pump power. The larger the pump power, the larger the optimal

*f*.

In [23], A. J. Lee et al. realized a frequency tuning THz source based on stimulated polariton scattering using an intracavity-pumped Q-switched Mg:LiNbO_{3} TPO. In their experiment, the lengths of the nonlinear crystal and Nd:YAG gain medium are 25 mm and 5 mm, respectively. In this kind of Nd:YAG gain medium, the stimulated emission cross section *σ* is 6.5×10^{−19} cm^{2}, the upper level lifetime *τ* is 230 µs, and the absorption *α _{P}* of the laser gain medium to the 808 nm pump light is assumed to be 2.85 cm

^{-1}[31–37,46]. The lengths of the fundamental resonator and Stokes cavity are 124 mm and 64 mm, respectively. The round-trip dissipative optical losses of the fundamental and Stokes waves are assumed to be 0.03 and 0.05, respectively. The reflectivity of the fundamental and Stokes output coupler is 99.9%. The pulse repetition rate is 3 kHz. The fundamental wavelength is 1.064 µm. The spot diameters of the fundamental and Stokes waves are all 0.5 mm. We simulate this laser to compare the theoretical results with the experimental results. The $g_S^{(2)}$, $g_S^{(3)}$, ${\alpha _T}$ and $g_T^{(2)}$ parameters for Mg:LiNbO

_{3}crystal are obtained by using the data in [47] and [48], they are frequency dependent and shown in Fig. 10. The coefficient

*η*about Si-prisms coupling output at different THz frequency are calculated using the method in [49]. Figure 11 shows the simulated results (solid lines) and the experimental results (dots) in [23], Fig. 11(a) is the THz frequency tuning characteristic and Fig. 11(b) give the dependences of the output fundamental, Stokes, and THz wave powers on the incident diode pump power. Comparing the theoretical simulation curves with the experimental data, the stimulated results about the THz frequency tuning characteristic and output powers are basically consistent with the experimental results.

_{T}Comparing the theoretical simulation curves with the experimental data, the stimulated results about the THz frequency tuning characteristic and output powers are basically consistent with the experimental results. The small differences between the theoretical and experimental results are probably caused by the following factors. First, quite many parameters about the nonlinear crystal and cavities are needed in the calculation, maybe these parameters are not accurate enough. Second, several assumptions are made in deriving the rate equations, the conditions for an actual experimental system may not satisfy all the assumptions exactly.

## 6. Conclusion

We have established a theoretical tool to simulate the operation of intracavity-pumped Q-switched terahertz parametric oscillators based on stimulated polariton scattering using the rate equations for the first time. The SPS process is described by the coupled wave equations considering the non-collinear phase matching. The approximate expression for the THz wave intensity is obtained as a function of the fundamental and Stokes wave intensities by using the coupled wave equations under the plane-wave approximation. The rate equations to describe the evolution processes of the fundamental and Stokes waves are obtained in the first step. The characteristics of the THz wave can be calculated based on the fundamental and Stokes wave properties.

The pulse evolutions have been obtained by solving the rate equations, and they are used to demonstrate the consumption of the fundamental wave and the generations of the Stokes and THz waves visually. Several curves based on the rate equations are generated to show the influences of the SPS nonlinear coefficient, the THz wave absorption coefficient, and the pulse repetition rate on the THz laser performance. An actual intracavity-pumped Q-switched Mg:LiNbO_{3} TPO in [23] is simulated. The theoretical results about the THz frequency tuning characteristic and the dependences of the output fundamental, Stokes, and THz wave powers on the incident diode pump power are in agreement with the experimental results. We hope the rate equations can play an important role in the design and optimization of the intracavity-pumped Q-switched TPOs. We also hope that more modification and improvement will be given to enhance the accuracy of the rate equations.

## Funding

National Natural Science Foundation of China (61475087, 61775122); Key Technology Research and Development Program of Shandong (2017CXGC0809, 2017GGX10103); Natural Science Foundation of Shandong Province (ZR2014FM024).

## Disclosures

The authors declare no conflicts of interest.

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