## Abstract

Terahertz (THz) generation in a periodically poled lithium niobate crystal via cascaded difference-frequency generation based on Cherenkov-type quasi-phase matching (QPM) is proposed. Photon conversion efficiency is evaluated based on a promising structure that combines QPM and Cherenkov phase-matching with reduced wave-vector mismatch. Cascading processes contribute to photon conversion efficiency, and THz radiation with maximum photon conversion efficiency of 1154.2% in a 14-order cascaded Stokes process was obtained. Comparing the processes with and without Cherenkov-type radiation, with a 50-MW pump, power was boosted nearly 1.9 times for the former case. These results provide an experimental approach to high-energy THz-wave generation.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Difference-frequency generation (DFG) from the rapid increase in laser pulse energies with two similar wavelengths produced by solid-state laser sources is of great interest for terahertz (THz) generation in the frequency range between 0.1 and 10 THz [1,2]. Laser-driven THz generation methods based on DFG are important for their advantages of high peak power outputs, wide tuning line-widths, and room-temperature working environments [3–5]. According to previous results, a periodically poled lithium niobate (PPLN) crystal with a high nonlinearity, high damage threshold, and wide bandgap is well-suited for THz generation [6,7].

Quasi-phase matching (QPM) has been demonstrated to be an effective method of DFG [8,9]. The QPM condition was first proposed by Armstrong et al. and can address the phase-matching problem of the DFG process in periodically poled crystals. However, nonlinear material properties, high THz absorption, and quantum defects limit the optical-to-THz conversion efficiency, which is an issue that must be addressed. These factors restrict the maximum efficiency to below 1%, according to the Manley–Rowe relations.

The cascading process with QPM is a promising approach; it involves using a strong high-frequency pump pulse and a low-frequency pump pulse [10–12]. In conventional non-cascaded DFG, a single THz photon is generated from each pump photon. A cascading process in which more than one THz photon is generated from the depletion of a single pump photon can enhance the photon conversion efficiency. Cronin-Golomb analyzed cascaded nonlinear optical interactions, and a potential improvement by a factor of five was demonstrated in ZnTe at a wavelength of 824 nm by using a 25-MW/mm^{2} pump [10]. Ravi et al. described a scheme for highly efficient THz generation based on spectrally cascaded optical parametric amplification (THz-COPA) [11]. An energy conversion efficiency > 8% for 100-ps pulses in cryogenically cooled PPLN was predicted. Hemmer et al. experimentally observed a sub-THz-frequency idler wave generated by pump and signal waves spectrally separated by a frequency of a few hundreds of gigahertz [12]. Optical parametric amplification (OPA) offers an approach for the generation of sparse and high-energy frequency combs, exceeding the Manley–Rowe limit. The degree of enhancement is determined by the accepted number of cascading orders and the effective interaction length. In addition, during the cascading process, the phase-matching condition is an important factor in determining conversion efficiency.

Recently, a promising configuration for Cherenkov phase matching (ChPM) was proposed [13,14], in which the phase-matching condition is automatically satisfied at a certain angle to the pump laser path. THz generation via ChPM was first discovered by Cherenkov when irradiating a uranium salt solution with γ-rays in 1934 [13]. Suizu et al. demonstrated prism-coupled Cherenkov-type THz-wave generation using the organic crystal 4-dimethylamino-N-methyl-4-stilbazolium-tosylate (DAST) [14]. This method indicates the potential of THz-wave generation using a wide variety of nonlinear crystals. This kind of phase-matching stems from vector phase matching. Moreover, Cherenkov-type radiation provides efficient THz radiation coupling. Recently, Liu et al. proposed a scheme for efficient THz generation using a cascaded DFG process with Cherenkov-type guided waves [15]. Compared with non-cascaded DFG, the cascaded output THz power could be enhanced nearly eight-fold with a 400-MW/cm^{2} pump in a 40-mm-long Si-LiNbO_{3}-Si crystal.

In this study, a practical approach to high-energy THz generation by combining cascaded DFG, QPM, and Cherenkov-type phase-matching with laser pulses of two similar frequencies is introduced. Starting with one high- and one low-frequency pump, a cascading process with a high pump power and much smaller wave-vector mismatch results in high photon-conversion-efficiency THz generation, overcoming the quantum-defect limitation. Accordingly, we present a theoretical analysis of THz generation by PPLN based on cascaded DFG via a Cherenkov-type QPM scheme. A coupled model structure for both non-cascaded and cascaded processes is theoretically analyzed in detail. This is a promising structure owing to its potential for decreasing the wave-vector mismatch, providing an effective method for improving the THz coupling, and overcoming the quantum defects. The quantum defect is defined as the difference in photon energies: *q = hv _{pump}-hv_{laser}*.

## 2. Theoretical model

Theoretical analysis and experimental research on an enhanced THz output via the cascaded DFG process with Cherenkov-type radiation has been reported recently [15]. In cascaded DFG, two mid-infrared laser beams, a high-frequency pump *ω _{j}* and low-frequency pump

*ω*

_{j}_{+1}, are allowed to beat together in a nonlinear optical medium to generate light at the difference frequency. The initial DFG between these pump pulses generates THz radiation; the cascaded DFG process consumes the high-frequency pump pulses and amplifies the low-frequency pump, and it continues to act as a high-frequency pump, amplifying the THz radiation and producing a lower frequency

*ω*

_{j}_{+2}because of their similar phase velocities. In addition, further conditions can be satisfied that will continue the process to generate

*ω*

_{j}_{+}

*. Our processes generate a series of THz waves at the same interval*

_{n}*ω*. The interaction of Stokes light in the cascaded DFG process will generate THz radiation. At the same time, anti-Stokes light will also consume THz radiation. The conversion efficiency from left to right involves Stokes interactions (THz generation), and from right to left involves anti-Stokes interactions (THz depletion). The interaction of the Stokes light is stronger than that of anti-Stokes light, which finally enhances the THz radiation [10–12,15–18]. ChPM is a radiated type of emission [19,20]: the THz source is generated in the structure as a radiated (leaky) mode. This phase-matching condition is satisfied because of anomalous dispersion.

_{T}As illustrated in Fig. 1, a THz source is generated based on Cherenkov-type QPM for the interaction between two pump waves of different frequencies. The high-frequency pump, low-frequency pump, and THz waves can be simultaneously generated in the same PPLN, which implies that the configuration shown in Fig. 1 is compact. In addition, the effect of cascaded DFG is compared to that of non-cascaded DFG situations, and an enhanced THz wave can be generated by utilizing the high nonlinear-optical coefficients in the vicinity of polarization resonances.

Figure 2 shows a schematic of THz generation by the combination of cascaded DFG, QPM, and Cherenkov-type processes; *k _{j}*,

*k*

_{j}_{+1}, and

*k*are the wave numbers of the high-frequency pump light, low-frequency pump light, and THz wave, respectively; Δ

_{T}*k*, and Δ

_{c,1}*k*are the wave-vector mismatch of

_{c,n}*ω*,

_{j}*ω*

_{j}_{+}

*, respectively, in the cascaded DFG via ChPM without QPM, and Δ*

_{n}*k*= ΣΔ

_{c}*k*is the sum of the entire process;

_{c,j}*k*is the reciprocal vector of the periodically polarized crystals. QPM refers to the elimination of the phase mismatch by periodically changing the coefficient of the nonlinear optical crystal in a frequency conversion experiment. The “inversion period”

_{Λ}*Λ*refers to the spatial modulation period, as shown in Fig. 1 [17], and may be defined by the expression

The smaller wave-vector mismatch and high pump energy facilitate cascaded frequency down-conversion, which will improve the final overall THz-generation efficiency. A theoretical model is developed, and a parameter Δ*k _{j}*, which is derived analytically and solved numerically from the coupled wave equations of Fig. 1, indicates the phase mismatch in our model:

The velocity of the polarization wave inside the nonlinear optical crystal is greater than the phase velocity of the leaky wave outside. In Fig. 1, the “Cherenkov angle” *θ*_{C} is determined by the refractive indices of the THz and pump radiation, satisfying the ChPM condition [13,14,19].

*n*is the refractive index in the THz range and

_{T}*λ*is the wavelength of the THz wave in the DFG process. THz radiation in the PPLN is generated through QPM and ChPM when the electric fields of the high-frequency pump, low-frequency pump, and THz waves are considered to be involved in the first-order DFG process. The THz power is determined by the balance between the Stokes and anti-Stokes processes.

_{T}The cascaded Stokes and anti-Stokes processes can continue to any high order under the QPM and ChPM conditions. However, conversion efficiency is limited by the phase-matching condition and the wave-vector mismatch, which no longer exist in Cherenkov-type radiation. We consider it necessary to establish a theory to combine the cascaded DFG, QPM, and Cherenkov-type processes.

## 3. Dynamics of photon conversion efficiency

In this section, we will first evaluate the non-cascaded model. Considering the basic equations of the three-wave interaction in the second-order nonlinear process, the three-wave coupling equation for the interaction between light-waves and matter for Cherenkov-type QPM is derived [15].

In the DFG process, we adopt a Cherenkov-type THz wave with a steady transverse distribution. The theory of a non-cascaded model for THz radiation generation is now analyzed, and we denote the amplitudes of the two pump fields as *A*_{1} and *A*_{2}, respectively, and that of the THz wave as *A _{T}*. We arrive at the following three-wave coupling equation:

The result of our model is similar to the previous result [21]. *α _{1}*,

*α*, and

_{2}*α*denote the absorption coefficients of the two pump light waves and the THz wave in the nonlinear optical crystal, respectively;

_{T}*d*is the nonlinear optical coefficient;

_{33}*c*= 3 × 10

^{8}m/s is the speed of light in a vacuum; and

*n*

_{1}and

*n*

_{2}are refractive indices of the high- and low-frequency pump light, respectively.

When utilizing QPM, the output power is strongly affected. The THz energy output power is proportional to (*G _{j}*)

^{2}, which can be defined as [22]

*m*is the order under the QPM condition. In first-order QPM, where

*m*is taken as unity, we obtain

*G*equal to 2/π. Based on QPM with Cherenkov-type non-cascaded DFG (nonlinear optical three-wave coupled interaction equation:

_{j}*ω*,

_{j}*ω*

_{j}_{+1}, and

*ω*), we can take the cascading process into account.

_{T}We now consider a theoretical analysis of THz generation in PPLN based on the Cherenkov-type QPM cascaded DFG process. The coupled wave equations can be derived from nonlinear optical three-wave interaction equations:

The first and second terms on the right-hand side of Eq. (7) correspond to THz absorption and THz generation by the sum of all DFG processes, respectively. The first and second terms on the right-hand side of Eq. (8) include both the Stokes and anti-Stokes interactions, respectively.

Compared to the previous results [10–12,15–17], the main difference lies in the wave-vector mismatch and total photon conversion efficiency. Under the first-order QPM condition, THz radiation is the output and the THz power and photon conversion efficiency are proportional to *4/*π* ^{2}* [22]. The THz power and photon conversion efficiency have been calculated and the photon conversion efficiency has the following form [17,22]:

*P*and (

_{T,n}*P*

_{1},

*P*

_{2,}

*) are the output power of the THz wave and the two pumps, respectively, in the cascading process via Cherenkov-type QPM radiation.*

_{n}*λ*

_{1}is the wavelength of the first-order pump power,

*ε*

_{0}= 8.854 187 817 × 10

^{12}F/m is the permittivity of free space, the vacuum permeability is

*μ*

_{0}= 4π × 10

^{−7}N·A, and Δ

*α*=

*α*+

_{j}*α*–

_{j + 1}*α*.

_{T}The photon conversion efficiency of the cascaded DFG process via Cherenkov-type QPM in PPLN is primarily determined by the propagation distance and the absorption in the nonlinear optical crystal at THz frequencies. In Eq. (9), the output power and photon conversion efficiency are both proportional to the square of crystal length *L ^{2}* (

*L*=

*z*). Usually, QPM and ChPM are the preferred schemes to allow the use of large crystal lengths. To increase the photon conversion efficiency of THz radiation generation, the cascading process via Cherenkov-type QPM is an effective method, according to Eq. (9). Furthermore, it is necessary to select a nonlinear crystal (PPLN) with a high nonlinear coefficient and high light-loss threshold.

## 4. Numerical simulation and discussion

The theoretical values of the refractive index are calculated using the wavelength-independent Sellmeier equation for PPLN in the IR and THz ranges [23,24]:

*T*is the temperature in K, and the units of the wavelengths

*λ*and

_{j}*λ*are nm and μm, respectively.

_{T}The wave-vector mismatch and inversion periods in the PPLN crystal at the pump frequency for THz waves would be generated at 1 THz and are calculated according to Eqs. (1) and (2) by considering the three-wave interactions, as shown in Fig. 3. In the calculation, the two original pump frequencies are generated by a near-degenerate parametric pump at 1 μm, and are supposed to be about 300 THz, hence *ω _{T}* = 1 × 2π THz is taken as an example to illustrate the process. Clearly, when the inversion period is 67.61 μm, zero-mismatch can be obtained at a pump wavelength of 1 μm. Under the zero-mismatch condition, Cherenkov process can be enhanced in the cascaded DFG via QPM using a PPLN crystal. According to Eqs. (1) – (3), the Cherenkov angle is 63.4° in view of the fact that the inversion period is 67.61 μm. In the cascaded Stokes process, the wave-vector mismatch is less than 1 cm

^{–1}in the course of a seven-order cascading process with a pumping frequency of about 300 THz; it is the same in the case of the cascaded anti-Stokes process.

The total output energy is related to the absorption factor *α* and a parameter Δ*k _{j}* [17]. The amplitude change of the THz power and photon conversion efficiency is not clear for the strong absorption in the nonlinear optical crystals,

*α*. As a result, Fig. 3 represents the characteristics of our model, with Δ

*k*exhibiting a minimum and approaching zero. The part of the phase-matching condition coupled-out is treated as a type of “wave-vector mismatch,” described by the leakage coefficient Δ

_{j}*k*. Compared to previous models [10–12,15–17], ours not only decreases the numerical value of Δ

_{j}*k*but also enhances the THz output energy.

_{j}By solving Eqs. (7) and (8), we can numerically analyze the energy conversion dynamics of the three waves. 1-THz generation via a Cherenkov-type scheme based on the QPM cascaded DFG process in a PPLN crystal has been theoretically analyzed above. Here, a simulation based on Cherenkov-type QPM cascaded DFG dynamics, the THz power *P _{T,n}* in PPLN with cascading orders 2, 4, and 7 versus propagation distance is presented in Fig. 4. In the following calculations, the seven-order cascaded Stokes process is taken into account as the inversion period is larger than 8.45 μm. Assuming the two pump frequencies are

*ω*= 300 × 2π THz (1 μm) and

_{j}*ω*

_{j+}_{1}= 299 × 2π THz (1.003 μm), the pump power is

*P*

_{1}

*= P*

_{2,}

*= 50 MW, and the beam cross-sectional area*

_{n}*A*= 1 mm

^{2}. Damage threshold of PPLN material is 7.11 GW/cm

^{2}(71.1 MW/mm

^{2}) [25].The nonlinear coefficient is

*d*= 27.4 pm/V [26] and the absorption coefficients are

_{33}*α*

_{j}= α_{j}_{+1}= 0 and

*α*= 0.35 cm

_{T}^{–1}[27], according to the literature. Overall, the Stokes process dominates, similar to the previous results [10,17], and it is clear that the Stokes light power decreases as the cascading order increases. Therefore, THz radiation with a maximum power of 121.1 kW can be obtained. In the high-frequency area where high-order Stokes processes interact, optimal polarization periods are longer when considering cascading.

Using numerical simulations, THz maximum photon conversion efficiencies in the PPLN crystal are calculated when the original pump powers are changed from 0 to 50 MW, as shown in Fig. 5. THz radiation generated from high-frequency pump waves and low-frequency pump waves is supposed to be 0.5 THz (red line), 1 THz (black line), and 1.5 THz (blue line), and the high-frequency pump wave is 300 THz. According to the vector-phase matching, the effective interaction length of the structure is not affected by the absorption of the PPLN crystal. Figure 5 demonstrates that the maximum photon conversion efficiency significantly increases with the pump power. Pump power is directly related to the photon conversion efficiency in the process. THz radiation with a maximum power equal to 961.9, 554.4, and 372 kW can be generated as the pump power equals 50 MW, corresponding to a photon conversion efficiency greater than 100%. At THz-wave frequencies of 0.5, 1, and 1.5 THz, the maximum value in the 14-order, 7-order, and 6-order cascaded Stokes process becomes 1154.2%, 332.6%, and 148.8%, respectively. For longer propagation distances, the photon conversion efficiency will decrease while the pump depletion cannot compensate for the coupling output. As for THz radiation with frequencies lower than 1.5 THz, THz generation can be effectively enhanced by more than 100% in the cascading process if the collinear model via the Cherenkov-type QPM scheme is satisfied, implying that the photon conversion efficiency exceeds the Manley–Rowe limit.

Pump power and THz power are directly related to photon conversion efficiency in PPLN, according to Eq. (9). Because the wave-vector mismatch varies at different pump frequencies, it leads to a change in the photon conversion efficiency. Equations (7) and (8) include the phase-matching condition and dispersion in the THz absorption and optical region, which is the main result of our theoretical analysis. By choosing two appropriate pump frequencies (300, 299 THz) and an initial pump power (50 MW), we obtain and present in Fig. 6 the dynamics of the THz generation by the PPLN crystal for the cascaded process via QPM with Cherenkov-type radiation (solid line), without Cherenkov-type radiation (dashed line), and the non-cascaded process (dotted line). In the calculations, the THz-wave frequency is 1 THz. The peak of the photon conversion efficiency for the non-cascaded process (dotted line) is 13.4%, and when ChPM is not included (dashed line), the peak photon conversion efficiency reaches a maximum of 176.5%. It is apparent that the peak of the curve for the cascaded DFG via Cherenkov-type QPM is higher than that of both the dashed and dotted curves. Comparing the processes with and without Cherenkov-type radiation, in the cascaded process, with Cherenkov-type radiation (solid line), a peak arises on the curve at about *L* = 3.59 mm, and the enhancement to the cascading process brought about by Cherenkov-type radiation is clear: the peak is 1.9 times that of the process without Cherenkov-type radiation. The better result obtained via Cherenkov-type QPM can be attributed to its long optimal interaction lengths and much lower wave-vector mismatch. Comparing the process with and without Cherenkov radiation, cascaded process with Cherenkov-type QPM decrease the wave-vector mismatch, which provides an effective method for improving the THz output coupling and overcoming quantum defects. There is no doubt that these results also agree well with the principle of nonlinear optical frequency conversion.

## 5. Conclusions

To conclude, we have theoretically analyzed a scheme for efficient THz generation in a PPLN crystal by combining cascaded DFG, QPM, and Cherenkov-type phase-matching. Furthermore, we have developed a coupled model structure, and both non-cascading and cascading processes were theoretically analyzed in detail. Assuming the two pump frequencies are 300 and 299 THz, zero-mismatch can be achieved when the inversion period is 67.61 μm, and for THz frequencies lower than 1.5 THz, the peak THz power and photon conversion efficiency are 121.1 kW and 332.6%, respectively. At a THz-wave frequency of 0.5 THz, it was possible to obtain THz waves with a maximum photon conversion efficiency of 1154.2% in a 14-order cascaded Stokes process, which exceeds the Manley–Rowe limit. A comparison was made between the effects of THz generation with and without Cherenkov-type radiation, and the THz-output peak-photon-conversion efficiency was boosted nearly 1.9 times at a pump power of 50 MW. It can be concluded that our model is a promising structure owing to its potential for decreasing the wave-vector mismatch, which provides an effective method for improving the THz output coupling and overcoming quantum defects. When the pump beam is converted to radiation at lower frequencies, it becomes more difficult to maintain phase-matching. Multiperiodic crystal is a variable periodic crystal with the QPM structure, which can compensate for the phase mismatch. Further improvements can be realized by optimizing the non-coherent optical path structure, temperature, and crystal parameters.

## Funding

National Natural Science Foundation of China (NSFC) (11664017); Technology Project funded by Provincial Department of Education (GJJ160305).

## Acknowledgment

The authors thank P. Liu for recommending the split-step method.

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