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Abstract
In recent years, high-power, tunable terahertz (THZ) radiation sources have become the key areas of research in the world. The method of THZ waves by nonlinear optical difference frequency generation (DFG) has the advantages of wide tuning, high power, room temperature operation, and compact structure. However, the conversion efficiency of the current difference frequency method is low, which needs a trade-off between conversion efficiency and tuning range. We apply the nonlinear optical cascade difference frequency conversion theory based on stimulated Raman adiabatic passage (STIRAP) and propose a theoretical scheme to generate THZ waves. Numerical simulation investigates the cascaded difference frequency process of generating THZ waves with the help of the nonlinear medium lithium niobate (LN) crystal. The theoretical analysis shows that the maximum quantum conversion efficiency from signal laser to THZ waves is 43.2 % when the wavelength of the tuned signal laser varies between 1.044 − 1.065 µm with the fixed two pump laser wavelengths constant. The tunable THZ waves of 0.48 − 5.0 THz can be obtained and the maximum output intensity of THZ waves is 2.17 MW/cm2, and the method is robust to temperature variations. It also provides a novel idea for the cascaded difference frequency generation of THZ waves.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
THZ waves refer to the electromagnetic wave with a frequency between $0.1\textrm{ - }10\textrm{ THz}$, and the wavelength range is $0.03\textrm{ - 3 mm}$, which are in the transition region between macroscopic electronics and microscopic photonics. Because of its unique position, THZ waves combine the excellent performance of electronics and photonics, and have important application prospects in medical imaging, nondestructive detection, security inspection system, wireless communication and other fields [1–8]. In recent years, more and more people hope to apply THZ source to the study of material properties, the detection of distant galaxies and quantum interactions. THZ radiation has further potential in revolutionary new applications, such as manipulating bound atoms, which has potential for future quantum computers. One of the most important is to obtain high power tunable THZ source [9–13].
Zhong et al. used the OPO system to generate around $2.128\,\mathrm{\mu}\textrm{m}$ signal laser and idler laser, and obtained $0.186\textrm{ - }3.7\,\textrm{THz}$ continuous tunable THZ waves by difference frequency in GaSe crystal. The peak power is $11\,\textrm{W}$ and the corresponding quantum conversion efficiency is $0.09\mathrm{\%}$[14]. To improve the conversion efficiency of THZ waves generated by difference frequency, Liu et al. proposed a theoretical method to generate single frequency THZ waves by the cascaded difference frequency. The cascaded amplification process of THZ wave power could be realized through the quasi-phase matching waveguide structure of Cherenkov type, and the THZ wave output power was enhanced nearly $8$ times compared to the direct difference frequency process under the action of pump laser with the intensity of $400\,\mathrm{MW}/c{\mathrm{m}^2}$[15]. The cascaded difference frequency method can significantly improve the quantum conversion efficiency when generating single frequency THZ waves, but it is difficult to obtain THZ waves with wide tuning and high quantum conversion efficiency. Kalugin et al. proposed a method to generate THZ radiation based on STIRAP. The method drove coherent scattering of infrared laser in a coherently prepared molecular gas to generate ultra-short THZ pulses. The pulse duration was not restricted by relaxation rates and could be as fast as femtoseconds, with quantum conversion efficiency as high as $\mathrm{100 \%}$ when the pulse duration exceeded picoseconds [16]. The reported methods of generating THZ waves based on STIRAP belong to the category of quantum optics, and it has not reported the nonlinear optical methods of generating THZ waves based on adiabatic particle population techniques.
U. Simon et al. explored the suitability of visible III-V single-mode cw diodes for difference-frequency generation of tunable infrared radiation by mixing a red single-mode cw III-V diode laser with a tunable single-frequency cw Ti: sapphire laser in AgGaS2[17]. In addition, the nonlinear optical frequency conversion scheme based on STIRAP is robust to the variations of the incident laser wavelength, angle, phase mismatch, temperature, and other parameters [18,19]. And it makes a wide range of applications in broadband wavelength conversion, generation of mid-infrared laser sources, white laser synthesis and other fields [20–23].
Recently, analogies were shown between the dynamics of simultaneous TWM processes and those induced in three level atoms by electromagnetic fields [24–26]. We apply the nonlinear optical cascade difference frequency conversion theory based on STIRAP to propose a theoretical scheme to generate THZ waves. The numerical simulation method is Quartic Runge-kutta method and we use Matlab to simulate. The signal laser with the wavelength of $1.064\,\mathrm{\mu}\textrm{m}$ and the strong pump laser with wavelengths of $2.7\,\mathrm{\mu}\textrm{m}$, $1.764\,\mathrm{\mu}\textrm{m}$ are used to the cascaded difference frequency in the coupling modulated LN crystal. Numerical simulation investigates the effects of the coupling delay parameters, pump laser intensity, temperature, and signal laser wavelength on the intensity of the generated THZ waves.
2. Theoretical model and numerical simulation analysis of nonlinear optical cascade difference frequency generation of THZ waves based on STIRAP
2.1 Theoretical model
Figure 1(a) shows the population transfer process in the $\varLambda $-type three-state atomic system based on STIRAP [27–29]. The pump pulse with frequency ${\omega _{pump}}$ drives the population from state $|1 \rangle$ to state $|2 \rangle$. And the Stokes pulse with frequency ${\omega _{stokes}}$ drives the population from state $|2 \rangle$ to state $|3 \rangle$. The sequence of pump pulse and Stokes pulse satisfies the counterintuitive order. The coupling between states $|2 \rangle$ and $|3 \rangle$ comes first, followed by the coupling between state $|1 \rangle$ and $|2 \rangle$. The Hamiltonian [18,22,30] of the three-state system with STIRAP is expressed
Fig. 1. (a) The $\Lambda $-type three-state atomic system. (b) Schematic illustration of cascaded difference frequency generation process in the nonlinear crystal.
The generation of THZ waves by the nonlinear optical cascade difference frequency based on STIRAP includes two simultaneous DFG processes. In the first DFG process, the signal laser ${\omega _1}$ and the pump laser ${\omega _{p1}}$ generate the intermediate laser ${\omega _2}$. Then, the intermediate laser interacts with the pump laser ${\omega _{p2}}$ to generate the THZ waves ${\omega _T}$ through the second DFG process, and the corresponding frequency conversion relationship is
Under the undepleted pump approximation, the coupling wave equation corresponding to the generation of THZ waves by the nonlinear optical cascade difference frequency based on STIRAP is [22,24,31]
In the nonlinear optical cascade difference frequency generation process based on STIRAP, in order to realize the adiabatic frequency conversion process, it is necessary to modulate the coupling coefficient of the crystal to meet the adiabatic condition while satisfying the phase matching condition. The crystal structure that satisfies both phase matching and adiabatic conditions is also called the adiabatic crystal. The commonly used Gaussian modulation is adopted here, and the corresponding coupling modulation function is
2.2 Numerical simulation analysis
LN crystal is used as the nonlinear medium of the cascaded difference frequency generation process and the nonlinear coefficient of LN crystal is $168\textrm{ pm}/\textrm{V}$ [32]. Numerical simulation studies the generation of the THZ wave process by the nonlinear optical cascade difference frequency based on STIRAP. Considering the transmission range of the LN crystal, the specific values of the incident signal laser wavelength $\lambda _1^{}$, the first pump laser wavelength $\lambda _{p1}^{}$ and the second pump laser wavelength $\lambda _{p2}^{}$ are shown in (8). The signal laser can generate the THZ waves with the wavelength ${\lambda _3}$ (frequency of $0.77\,\textrm{THz}$) after two DFG processes, and the corresponding wavelength conversion relationships are
In order to realize the adiabatic frequency conversion process, the coupling coefficient of the LN crystal is modulated under the condition of phase matching. Taking at $10\,\textrm{mm}$ of the crystal as the center, the delay parameters in the coupling modulation function are set as ${s_1} ={-} 10\,\textrm{mm}$, ${s_2} = 6\,\textrm{mm}$. Figure 2(a) shows the variation of the modulated coupling coefficient along the propagation direction. It can be seen that the peak of ${\kappa _{23}}(z )/{\kappa _{23}}$ is before ${\kappa _{12}}(z )/{\kappa _{12}}$, and the corresponding coupling sequence in a counterintuitive sequence, showing that the coupling of the intermediate laser to the second pump laser precedes that of the signal laser to the first pump laser. After coupling modulation of the LN crystal, the signal laser intensity incident into the LN crystal is chosen to be $400\,\mathrm{MW}/\mathrm{c}{\mathrm{m}^2}$, and two pump laser intensities ${I_{p1}} = 5\,\textrm{GW/c}\textrm{m}^\textrm{2}$, ${I_{p2}} = 250\,\mathrm{GW}/\mathrm{c}{\textrm{m}^\textrm{2}}$. Figure 2(b) shows the variation of the signal laser, intermediate laser and THZ wave intensities with propagation distance in the cascaded difference frequency generation process, and the upper right inset shows the enlarged graph of the THZ wave intensity variation. As seen from the figure, the signal laser converts to THZ waves at $\textrm{ - 6\,mm}$ of the crystal, and the signal laser intensity gradually decreases and the THZ wave intensity gradually increases, reaching the maximum value at $8\,\textrm{mm}$ of the crystal. Finally, the THZ wave output intensity at the end of the crystal is $0.47\,\mathrm{MW}/\mathrm{c}{\mathrm{m}^2}$, and the corresponding quantum conversion efficiency is $\textrm{43}\mathrm{.2 \%}$. The cascaded difference frequency generation process also generates intermediate laser with a peak intensity of $33\,\mathrm{MW}/\mathrm{c}{\mathrm{m}^2}$, but the intensity of the intermediate laser eventually becomes zero as the propagation distance increases.
Fig. 2. (a)Variation of the coupling modulation function with the propagation distance. (b) Variation of the signal laser, intermediate laser and THZ wave intensities with the propagation distance.
Only when the coupling modulation function satisfies the adiabatic condition, the signal laser in the cascaded difference frequency generation process can be directly converted to THZ waves. The effect of the delay parameters in the coupling modulation function on the generation process of THZ waves will be discussed below. Figure 3(a) shows the variation of the THZ wave output intensity at the end of the crystal with the delay parameters when the width parameters are $d_1^2 = 50\,\textrm{mm}_{}^{\textrm{2}}$, $d_2^2 = 100\,\textrm{mm}_{}^{\textrm{2}}$, and the color depth represents the obtained THz wave intensity. It can be seen that when changing ${s_1}$, ${s_2}$, namely in different coupling orders, the maximum intensity of THz waves is $0.47\,\mathrm{MW}/\mathrm{c}{\mathrm{m}^2}$. This value is equal to the THz wave intensity obtained by adiabatic frequency conversion, indicating that the THz wave intensity obtained by the adiabatic frequency conversion from signal laser to THz waves is the largest when the cascaded difference frequency generation process is realized. In addition, the delay parameters satisfying the adiabatic condition are not a specific value. When the delay parameters change within a certain range, they can all meet the adiabatic condition. At this time, the output THz wave intensity is $0.47\,\mathrm{MW}/\mathrm{c}{\mathrm{m}^2}$. The delay parameters satisfying adiabatic conditions are all concentrated in the counterintuitive sequence, which is consistent with the previously reported characteristics of nonlinear optical frequency conversion based on STIRAP [31]. Therefore, in order to obtain the maximum output intensity of THZ waves, the delay parameter ${s_1} < {s_2}$ needs to be set in the design of crystal, that is, the peak of modulation coupling coefficient ${\kappa _{23}}$ appears before the peak of ${\kappa _{12}}$. The high intensity of THZ waves can also be obtained in a small part of the intuitive sequence, and the maximum intensity of THZ waves is $0.45\,\mathrm{MW}/\mathrm{c}{\mathrm{m}^2}$.
Fig. 3. (a) Variation of the THZ wave intensity at the end of the crystal along the delay parameters ${s_1}$, ${s_2}$. (b) Variation of the intensities of signal laser, intermediate laser and THZ waves along the propagation distance when the delay parameters ${s_1} ={-} 13\,\textrm{mm}$, ${s_2} = 8\,\textrm{mm}$.
In order to further study the effect of delay parameters on the THZ wave generation process, we set the delay parameters ${s_1} ={-} 13\,\textrm{mm}$, ${s_2} = 8\,\textrm{mm}$. Under this condition, the signal laser, intermediate laser, and THZ wave intensities varied with propagation distance are shown in Fig. 3(b). It can be seen that the signal laser, intermediate laser, and THZ waves coexist at the end of the crystal, which corresponds to an incomplete conversion process. The output intensity of the THZ waves at the end of the crystal is only $0.34\,\mathrm{MW}/\mathrm{c}{\mathrm{m}^2}$, which is significantly lower than the intensity of THZ waves generated under satisfying adiabatic conditions. Because of the severe oscillation of the signal laser in the process of conversion to THZ waves, resulting in the signal laser is not completely converted, while generating intermediate laser of non-negligible intensity, and the intermediate laser photon energy is not completely converted to THZ waves.
Figure 4(a) shows the variation of the THZ wave output intensity at the end of the crystal with two pump laser intensities, and it can be seen that the output intensity of THZ waves at the end of the crystal increases with the increase of two pump laser intensities. When the intensity of the second pump laser ${I_{p2}}$ is greater than $180\,\mathrm{GW/c}{\textrm{m}^{\textrm{2}}}$, the output THZ wave intensity reaches its maximum value. If two pump laser intensities continue to be increased, the THZ wave output intensity remains unchanged. The variation of THZ wave output intensity with ${I_{p1}}$ is shown in Fig. 4(b) when ${I_{p2}} = 180\,\mathrm{GW/c}{\textrm{m}^{\textrm{2}}}$ is fixed, and it can be seen that when the intensity of the first pump laser ${I_{p1}}$ is greater than $0.15\,\mathrm{GW/c}{\textrm{m}^{\textrm{2}}}$, the THZ wave output intensity reaches the maximum value and then remains unchanged. Through the above analysis, it can be obtained that the minimum pump laser intensities required for the THZ wave output intensity to reach the maximum are ${I_{p1}} = 0.15\,\mathrm{GW/c}{\textrm{m}^{\textrm{2}}}$, ${I_{p2}} = 180\,\mathrm{GW/c}{\textrm{m}^{\textrm{2}}}$. Since in the simulation, we need to select the two smallest pump lasers to obtain the THz waves with maximum intensity, which also results in a non-smooth curve in Fig. 4(b).
Fig. 4. (a) Variation of the THZ wave output intensity at the end of the crystal with two pump laser intensities. (b) When ${I_{p2}} = 180\,\mathrm{GW/c}{\textrm{m}^{\textrm{2}}}$ is fixed, the variation of the THZ wave output intensity at the end of the crystal with ${I_{p1}}$.
When the THZ wave intensity reaches the maximum, we will study the effect of continuing to enhance two pump laser intensities. Figure 5 shows the variation of intermediate laser and THZ wave intensities along propagation distance under three groups of different pump laser intensities with the pump laser intensity ${I_{p1}}$ is greater than $0.15\,\mathrm{GW/c}{\textrm{m}^{\textrm{2}}}$ and ${I_{p2}}$ is greater than $180\,\mathrm{GW/c}{\textrm{m}^{\textrm{2}}}$. From Fig. 5(a), when two pump laser intensities increase, the intermediate laser intensity and the propagation distance of the peak intensity of the intermediate laser both decrease. Figure 5(b) shows that the THZ wave output intensity at the end of the crystal remains unchanged, but the propagation distance required for the THZ wave intensity to reach its maximum value is shortened when two pump laser intensities increase. This is because when two pump laser intensities increase, the coupling strengths of the signal laser to the first pump laser and the intermediate laser to the second pump laser also both increase, which can shorten the propagation distance required for the adiabatic frequency conversion process.
Fig. 5. When two pump laser intensities take different values, variation of (a) intermediate laser intensity and (b) THZ wave intensity along the propagation distance.
The change of crystal temperature or incident laser wavelength will cause the change of refractive index of crystal, thus indirectly affecting the phase mismatch of the whole cascade difference frequency generation process. And the variation of the THZ wave output intensity with the crystal temperature at the end of the crystal is given in Fig. 6. It can be seen that when the crystal temperature increases from $20\,^\circ C$ to $200\,^\circ C$, the output THz wave intensity decreases, but the decrease of the output THz wave intensity is less than $0.02\,\mathrm{MW}/\mathrm{c}{\mathrm{m}^2}$, and the output THz wave intensity at high temperature and room temperature is similar. It can be seen that the output THz wave intensity is less affected by crystal temperature, and the nonlinear optical cascade frequency difference generation process based on STIRAP is robust to phase mismatch caused by crystal temperature change.
When the wavelength of two pump lasers is fixed, and the wavelength of the tuning signal laser varies between $1.044\textrm{ - }1.065\; \,\mathrm{\mu}\textrm{m}$, the quantum conversion efficiency and output intensity of the THZ waves varying with the output frequency are shown in Fig. 7. We can observe that the maximum quantum conversion efficiency from signal laser to THZ waves is $\textrm{43}\textrm{.2\%}$. When the quantum conversion efficiency is above 30 %, the corresponding THZ waves tuning range is $0.48\textrm{ - }5.00\,\textrm{THz}$, and the maximum output THZ wave intensity is $2.17\,\mathrm{MW}/\mathrm{c}{\mathrm{m}^2}$. When the signal laser wavelength is around $1.064\,\mathrm{\mu}\textrm{m}$, and the output frequency is around $0.77\,\textrm{THz}$, the phase mismatch is smaller, and the quantum conversion efficiency is higher. The phase mismatch increases when the wavelength of the signal laser changes too much, and the quantum conversion efficiency and the output intensity of THZ waves decrease sharply when the output frequency is around $6\,\textrm{THz}$ because of the large phase mismatch. The results show that the generation of the THZ wave process by the nonlinear optical cascade difference frequency based on STIRAP is robust to phase mismatch and can get a certain range of tunable THZ waves.
2.3 Adiabatic crystal design
We call the crystal an adiabatic crystal. By modulating the coupling coefficients ${\kappa _{12}}$ and ${\kappa _{23}}$, we use the Phase-Reversal Quasi-Phase Matching (PRQPM) technique [24,35,36] to design the coupled modulated adiabatic crystal structure. In the process of modulating the coupling coefficient of the LN crystal, we first construct two multiplicative binary functions
Fourier expansion of $g(z)$, similar to QPM technology, we only retain the first term, and the corresponding modulation process is
At this time, the second-order nonlinear coefficient $\chi _{}^2$ of the crystal becomes ${g_{QPM}}\chi _{}^2$. The first term of ${g_{QPM}}(z)$ provides phase matching conditions for the first DFG process, and the second term provides phase matching conditions for the second DFG process. After modulation, the coupling matrix can be rewritten as
The coupling modulation function can then be expressed as
It is not difficult to see that the amplitude of the coupling coefficient and duty cycle are related to ${D_1}$, ${D_2}$. Figure 9 shows the schematic diagram of the adiabatic crystal structure after coupling modulation. The amplitude of the coupling coefficient can be modulated by continuously changing the duty cycles ${D_1}$, ${D_2}$ of each polarization period, thereby realizing the modulation function of the constructed coupling modulation functions ${\kappa _{12}}(z)$ and ${\kappa _{23}}(z)$.
3. Conclusion
We apply the nonlinear optical cascade difference frequency conversion theory based on STIRAP applied to propose a theoretical scheme to generate THZ waves, and numerical simulation investigates the cascade difference frequency process of generating THZ waves using LN crystal as a nonlinear medium. The results show that (1) When the delay parameter in the coupling modulation function satisfies the adiabatic condition, the nonlinear optical adiabatic frequency conversion process from signal laser to THZ waves can be realized, and the intensity of the generated THZ waves is strongest. When the adiabatic condition is not satisfied, severe oscillations will occur during the conversion of the signal laser to THZ waves, resulting in a decrease in the output intensity of the THZ wave at the end of the crystal. The delay parameters that can realize the adiabatic frequency conversion process are concentrated under the counterintuitive sequence; (2) The intensity of the output THZ waves at the end of the crystal increases with the increase of the pump laser intensity. When the intensity of the first pump laser ${I_{p1}}$ is greater than $0.15\,\mathrm{GW/c}{\textrm{m}^{\textrm{2}}}$ and the intensity of the second pump laser ${I_{p2}}$ is greater than $180\,\mathrm{GW/c}{\textrm{m}^{\textrm{2}}}$, the THZ wave output intensity reaches the maximum value and then remains unchanged. If two pump laser intensities continue to increase, the peak intensity of the intermediate laser and the propagation distance required for the adiabatic frequency conversion of signal laser to THZ waves are both decreasing; (3) The generation of the THZ wave process by the nonlinear optical cascade difference frequency based on STIRAP is robust to temperature variation; (4) When two pump laser wavelengths are fixed in $\lambda _{p1}^{} = 2.7\,\mathrm{\mu m}$ and $\lambda _{p2}^{} = 1.764\,\mathrm{\mu m}$, the wavelength of the tuning signal laser changes between $1.044\textrm{ - }1.065\,\mathrm{\mu m}$, and the maximum quantum conversion efficiency from signal laser to THZ waves is $\textrm{43}\textrm{.2\%}$. When the quantum conversion efficiency is above 30 %, tunable THZ waves of $0.48\textrm{ - }5.00\,\textrm{THz}$ can be obtained, and the maximum output intensity of THZ waves is $2.17\,\mathrm{MW}/\mathrm{c}{\mathrm{m}^2}$. This method provides a novel idea for the cascaded difference frequency generation of THZ waves.
Acknowledgments
H.J. thanks Zhonghao Zhang for insightful discussions. We also thank the anonymous reviewers for their suggestions that improve this paper substantially.
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.
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