1. Introduction

During the past several years, many significant efforts have been dedicated to the development of terahertz wave generation as well as its detection, since the distinctive characteristics of THz wave have been attracting considerable attention from a variety of applications in fundamental and applied research field [1]. Compared with the current mainstream research focusing on the ultrashort-pulse terahertz radiation through photoconduction and optical rectification [2, 3], the tunable narrowband THz-wave sources with high temporal and spatial coherence are gradually playing an key role in the various applications including chemical identification, biomedical diagnostics, and THz spectroscopy. Terahertz-wave parametric oscillators (TPOs) has been proved to be one of the most useful and promising techniques for the efficient generation of widely tunable, monochromatic, high-power, coherent THz waves. This efficient THz-wave source has a number of advantages, such as compactness, broad tunability, ease of handling, and operation at room temperature. Since the mid-1990s, many research groups have carried out several pioneering and innovative work about TPO/TPG using LiNbO₃ and the doped one, and made fruitful research achievements for various practical applications based on this potential THz-wave source [4–11].

 

Fig. 1. Schematic of a typical angle-tuning TPO.

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Conventional frequency tuning techniques in TPOs is normally involved in the angle-tuning method, through which the generation of widely tunable THz-wave radiations (typically 1–3 THz) can be achieved. The angle-tuning method has the advantages of simplicity and high speed, and can be realized through rotating the TPO cavity to vary the phase-matching angle θ between the pump and oscillated Stokes inside the crystal, as shown in Fig. 1. Nevertheless, some obvious limitations exist in this tuning method. The high-frequency THz wave can be obtained through increasing the angle θ_ext between the incident pump beam and the resonator axis, but the nonlinear interaction region of three waves is inevitably decreased, which will affect the oscillation threshold of TPO and THz-wave output power. On the other hand, in order to obtain the low-frequency THz wave, the angle θ_ext should be reduced. However, the small θ_ext leads to extending the cavity length unavoidably to completely separate the pump and Stokes beams at the output mirror, which will also undoubtedly influence the operation performance of TPO. In addition, the accuracy of the frequency tuning still requires the further improvement due to the mechanical movement for the angle tuning.

In this paper, we investigate a novel TPO tuning method through changing the pump wavelength while the pump beam is at a fixed angle of incidence relative to the resonator axis. The proposed pump-wavelength tuning method not only removes the above-mentioned limitations of the angle-tuning method, but also provides a potential tuning technique for extending the THz output frequencies further to the low THz-frequency region, beyond the tuning range obtainable with angle tuning alone under the limited cavity length. We also study the characteristics of the THz-wave parametric gain and the output direction of THz waves from an arrayed Si-prism coupler for the LiNbO₃-TPO under this tuning technique. To the best of our knowledge, it is the first time that the pump-wavelength tuning method has been introduced to the TPO frequency tuning technique.

2. Tuning principle

As we all know, the operation principle of the tunable LiNbO₃-TPO is based on the process of the small-angle tunable parametric scattering (the stimulated polariton scattering) from the long-wavelength side of the 248 cm^-1 A1-symmetry polariton mode in LiNbO₃, which is relevant to the near-forward stimulated Raman scattering process of LiNbO₃ [12]. It is the characteristics of the THz-wave parametric gain and the relationship between the dispersion characteristics of the polariton mode in both infrared and Raman-active materials and the phase-matching conditions of three interactive waves involved in this stimulated scattering process that are simultaneously responsible for the frequency tuning characteristics of TPOs. Since polariton is a quasi-particle resulting from strong coupling of electromagnetic waves with the transverse optical vibration modes (TO modes), its dispersion curve is partly phonon-like in the high-frequency region, and partly photon-like in the low-frequency region, where the predominant scattering process can be thought of as a parametric process and most of the energy in the lattice mode is electromagnetic rather than mechanical, as shown in Fig. 2. Figure 2 indicates the theoretical dispersion characteristics of the 248 cm^-1 polariton mode, determined by the infrared reflectivity data of LiNbO₃ by the expression [12]

Here, k_Polariton is the polariton mode wavevector, c is the velocity of light in vacuum and ε(ν) is the complex dielectric constant given by

where ν_0j, S_j and Γ_j are the jth eigenfrequency, oscillator strength and damping coefficient of the lowest A₁-symmetry phonon mode, respectively, and ε _∞ is the high-frequency dielectric constant.

 

Fig. 2. Dispersion curve of the 248 cm^-1 A₁-symmetry polariton mode in LiNbO₃ (red one) with the phase-matching curves for the 1064 nm laser pump (blue ones), designated in terms of wave number.

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In the stimulated scattering process, the generated far-infrared radiation, known as the THz-wave radiation (ω_T) of interest, together with the Stokes radiation (ω_s) is created parametrically from the pump (ω_p) according to the energy conservation law ω_p=ω_s+ω_T. At the same time, the strong parametric interaction of three waves also satisfies the noncollinear phase-matching condition k⃑_p=k⃑_s+k⃑_T (see the inset of Fig. 1), or can be rewritten as

where k_j is the wave vector with j=p, s, and T representing the pump, Stokes, and THz waves, respectively, and θ is the phase-matching angle between the pump and Stokes beams. If varying one of the parameters mentioned above, such as the angle θ, we can obtain a family of the phase-matching curves representing the locus of all points satisfying both the energy and wave vector matching conditions using Eq. (3), as shown in Fig. 2. When these phase-matching curves are superimposed on the dispersion curve of the polariton mode, it is under the very small angles condition that the points of the intersection of these curves are expected to determine the allowed frequencies and wave vectors of THz wave and the simulated Stokes radiation participating in the scattering process. As the angle θ is changed continuously in the nonresonant region, the frequency tuning of the THz wave and the Stokes light is realized simultaneously, which is the basic principles of the so-called angle-tuning method of TPOs.

 

Fig. 3. Relation between the dispersion curve of the 248 cm^-1 polariton mode in LiNbO₃ and the phase-matching curves for the different pump wavelengths at a fixed phase-matching angle of 0.8°, designated in terms of wave number.

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On the other hand, the noncollinear phase-matching condition of three waves involved in the process of stimulated polariton scattering will also vary with the pump wavelength according to the Eq. (3), because of the incident pump wavelength dependence of the optical refractive-index dispersion of LiNbO₃. Hence, for a certain fixed value of phase-matching angle θ, the intersection of the phase-matching curves with dispersion curve of the ploariton mode will change undoubtedly when the pump wavelength is tuned continuously. Figure 3 shows the relation between the dispersion curve of the 248 cm^-1 polariton mode in LiNbO₃ and phase-matching curves at a fixed phase-matching angle of θ=0.8° when the pump wavelengths are 532 nm, 694.3 nm, and 1064 nm, respectively. The corresponding theoretical Stokes shifts are 120.17 cm^-1, 93.67 cm^-1 and 61.98 cm^-1, respectively. In other words, the frequencies of the THz wave available under this condition are about 3.61 THz, 2.81 THz and 1.86 THz, respectively. Therefore, it is found that the shorter the pump wavelength is, the higher frequency region the intersection of phase-matching curve and the dispersion curve of polariton mode will shift to. That is to say, the larger Stokes shift or higher frequency THz wave will be achieved. On the other hand, as the longer wavelength pump source is used, the corresponding smaller Stokes shift or lower frequency THz wave will be generated. Hence, it is possible for the TPO to generate the coherent, tunable THz wave only through continuously varying the pump wavelength, defined as the pump-wavelength tuning technique, under the condition that the angle of incidence of the pump relative to the resonator axis of TPO is hold fixed at a certain value.

The tuning capability of the TPO is also decided by the characteristics of the THz-wave parametric gain as well as its absorption loss inside the crystal. According to the Refs. 5 and 13, the analytical expressions of the exponential gain for the THz waves are given by

where α_T is the absorption coefficient in the THz region; φ denotes the phase-matching angle between the pump and THz wave and g₀ is the parametric gain in the low-loss limit. In cgs units, they are written as

where I_p is the pump intensity, n_β(β=p, S, T) denotes the index of refraction at the wavelengths of pump beam, Stokes wave and THz wave. The nonlinear coefficients d′_E and d′_Qj d represent second- and third-order nonlinear optical processes, respectively, and M(ω) is the frequency-dependent damping constant of the low-frequency modes below the 248 cm^-1 A₁-symmetry mode of LiNbO₃.

3. Calculations and results

Suppose a tunable laser system with a tuning range of 0.71–1.5 µm and a high peak power (such as the optical parametric oscillator) is used as the tunable pump source for the LiNbO₃-TPO. We calculate the dependence of frequency tuning characteristics on the pump wavelength of the TPO at three fixed angle of 0.4°, 0.8° and 1.2°, as shown in Fig. 4. It can be seen that the frequencies of the THz wave available decrease gradually and smoothly with the increase of pump wavelength, and fine-frequency tuning of THz wave may be attainable. More noteworthy is the fact that the TPO tuning range can be extended to a low frequency region under a certain smaller angle condition, where the angle-tuning technique alone is difficult to approach due in part to the limitation on the separation of the pump and resonant Stokes beams for the limited cavity length. In addition, the relatively rapid frequency tuning can occur in the short-wavelength region while the frequency tuning is slightly insensitive to the variation of pump wavelength in the long-wavelength region, which means the relatively narrower THz-wave tuning range. Figure 5 illustrates that the tuning range Δv increases with the fixed phase-matching angle when the pump is tuned in the range of 0.71–1.5 µm. If a pump source with the wider wavelength tuning range is employed, we can conclude that the TPO tuning range could be further widened under the same phase-matching angle condition. It is noted, however, that as a practical matter, the efficient generation of high-frequency THz wave (as shown in Fig. 4) might not be obtained since it will suffer the large absorption losses inside LiNbO3 when the resonant frequency region is approached, where most of the energy in the lattice mode is mechanical rather than electromagnetic.

 

Fig. 4. Pump wavelength dependence of THz output frequency at the different phase-matching angles.

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Fig. 5. THz-wave tuning range vs phase-matching angle under the pump wavelength of 0.71–1.5 µm.

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The characteristics of the THz-wave parametric gain are investigated based on the frequency-tuning characteristics of the pump-wavelength tuning method. Figure 6 shows the THz-wave gain coefficient as a function of the pump wavelength and fixed phase-matching angle at a pump intensity of 200 MW/cm², calculated using Eq. (4). As can be seen, it is interesting that the THz-wave parametric gain varies irregularly when the pump wavelength is tuned under the different fixed phase-matching angle ranging from 0.1° to 1.2°. When the fixed phase-matching angle is smaller, that is, the THz-wave tuning range is mainly in the low-frequency region, the gain decreases monotonically with the increase of the pump wavelength in the tuning process. Under this condition, it is clear that the tuning of pump wavelength in the short-wavelength region is more effective for the improvement of the conversion efficiency and reduction of the oscillation threshold of TPO, which is similar to that under the angle-tuning technique of TPOs [7]. Conversely, when the fixed phase-matching angle is larger, that is, the THz-wave tuning range mainly lies in the high-frequency region where the generated THz wave suffers large absorption losses, the variation of the THz-wave gain with the pump wavelength is relatively complicated, and on the whole, however, the THz-wave gain basically increases with the tuning of pump wavelength from the short-wavelength region to the long one. The characteristics of the low gain and large absorption losses for TPO at the large fixed phase-matching angle can negatively influence the operation of the TPO based on this frequency-tuning technique. However, the conversion efficiency of TPO under this condition could be improved through appropriately increasing the pump intensity, as in the case of conventional optical parametric and stimulated Raman scattering process.

 

Fig. 6. Three-dimentional plot of the THz-wave gain coefficient as a function of the pump wavelength and fixed phase-matching angle at a pump intensity of 200 MW/cm² based on the pump-wavelength tuning method.

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In order to efficiently extract the THz wave generated in the LiNbO₃ and to substantially improve the output characteristics of a TPO, an arrayed silicon prism is usually employed as the THz wave coupler to avoid the total internal reflection of THz wave inside the LiNbO₃. By choosing the proper base angle of the Si prism, the THz wave with the stable output direction can be obtained emerging approximately normal to the exit surface of the Si prism, as shown in the inset of Fig. 7. For the pump-wavelength tuning technique, the radiation angle θ^air_out of the THz wave for the different fixed phase-matching angle are plotted in Fig. 7 (a), according to the noncollinear momentum and energy conservation conditions as well as Snell’s law. Here, for comparison the base angle of Si prism is set to be often-used 40°. We can see that the THz-wave radiation angle varies as much as several degrees with the pump wavelength, that is to say, the output direction of the THz wave is closely dependent on the pump wavelength. For the pump wavelength tuning range of 0.71–1.5 µm, the calculated changes Δθ^air_out of the radiation angle are 8.8°, 8.6°, and 8.3° when the phase-matching angle θ is 0.4°, 0.8° and 1.2°, respectively, while the angle changes for the conventional angle-tuning method of TPO is only about 3.78° in a THz-wave tuning range of 95–450 µm, as shown in Fig. 7 (b). In Fig. 7 (b), the solid curve represents the radiation angle change for the conventional angle-tuning method without considering the rotation of TPO cavity in the direction counter to the radiation angle θ^air_out under the fixed pump wavelength of 1064 nm [14]. Based on the aforementioned theoretical analyses, the stability of the output direction of the THz wave generated through the pump-wavelength tuning technique is slightly unsatisfactory, and some effective measures for improving the performance of THz-wave output direction should be taken.

 

Fig. 7. Calculated radiation angle changes for THz-wave Si-prism coupler based on pumpwavelength tuning method: (a) pump wavelength dependence of the THz-wave radiation angle, (b) comparison with the angle-tuning method for the 1.064 µm laser pump.

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According to the analyses mentioned above, if one needs to achieve the THz wave at a specific frequency, the angle of the incident pump with respect to the resonator axis will vary with the pump wavelength. Figure 8 shows the pump wavelength dependence of phase-matching angle θ under the generation of the THz wave of 1.5 THz. It can be seen that the phase-matching angle increases linearly with the pump wavelength approximately. As the pump wavelength changes from 0.71 to 1.5 µm, the corresponding phase-matching angle will vary from about 0.41° to 0.91°. It is self-evident that when the longer wavelength laser is used as pump source, it is entirely possible for a shorter resonator to completely separate the pump and resonant Stokes beams at the resonator mirror under the larger angle, which will significantly improve the TPO performance. On the other hand, the use of the shorter pump wavelengths results in the smaller phase-matching angle which can increase the parametric interaction length of three waves. However, the high-power, shorter-wavelength pump can bring about the remarkable photorefractive damage easily inside the LiNbO₃. The occurrence of this undesirable phenomenon will not only severely destroy the phase-matching condition, but also have the obviously negative effect on the spatial coherence and spot quality of THz wave in its direction of propagation. Therefore, for the practical operation, the effective measure to relieve this problem to a certain extent is to choose the proper candidate, such as MgO: LiNbO₃, with the excellent resistance to photorefractive damage.

 

Fig. 8. Phase-matching angle as a function of incident pump wavelength at a fixed THz-wave output frequency of 1.5 THz.

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It should be noted that since some parameters, such as the lattice vibration parameters, refractive indices of the LiNbO₃ crystal in the different regions, used in this paper are mainly cited from the Refs. 12 and 15 and are probably different from those adopted in other publications [4–8, 14], there may be slightly but negligible discrepancy between our results and others. But we think that the final conclusions in our paper still apply.

4. Conclusion

In summary, we have theoretically demonstrated a novel frequency tuning method for TPO through varying the incident pump wavelength continuously without the need to rotate the TPO cavity to vary the phase-matching angle between the pump and oscillated Stokes inside the crystal. The mechanism of the proposed pump-tuning method is based on the variation of the noncollinear phase-matching conditions of three interactive waves due to the pump wavelength dependence of the simulated polariton scattering process relating to the low-frequency lattice vibrational modes. This new tuning method will be a useful adjunct to the angle tuning and would provide a potential technique for widening the tuning range of TPO and extending THz output frequencies of TPOs further into low-frequency THz region, where the angle tuning method alone is difficult to reach. Through the investigation on the characteristics of the THz-wave parametric gain involved in this frequency tuning scheme, it is found that the gain coefficient decreases with the increase of pump wavelength at the smaller fixed phase-matching angle, while the opposite conclusion is drawn at the larger one. The stability of the output direction of THz waves from the Si prism is somewhat undesirable under this frequency tuning scheme, and therefore some methods should be explored to reduce this negative effect for the practical applications.