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Numerical simulation and analysis about the influence of the time modulation of the pump laser caused by mode beating on optical parametric process are presented with OPA and OPG as examples. It is shown that the output power of the generated beams from optical parametric process is modulated in the time domain and exhibits large power fluctuations, when a Q-switched laser oscillating on several random longitudinal modes is used as the pump laser. Irregular spike sequences of the generated beams are observed. We also find that the output power of the light from optical parametric process becomes more stable and exhibits a less fluctuation, when the number of the longitudinal modes (n) increases.
©2010 Optical Society of America
1. Introduction
Optical parametric process is an efficient and reliable way to obtain coherent radiation with broadly tunable spectral range from ultraviolet to far infrared that cannot be generated directly by lasers. Nowadays optical parametric process is widely used in many fields because of its compactness, high efficiency, broad tuning, excellent beam quality and stable output [1–6]. Especially in recent years, OPA and OPG have been obtaining increasing attention for a variety of applications in science, remote sensing, military, environment, industry, medicine, spectral measurement, laser radar and so on [7–14]. Typically OPG consists of a separate pump laser and a nonlinear crystal, while OPA still needs an injection seeded signal wave. The pump laser is a key element of optical parametric process. The single longitudinal mode laser is an ideal pump laser due to its high frequency stability, high wavelength stability and very smooth time domain [15–19]. However, usually a stable and low cost Q-switched Nd3+ -doped laser with a mature technology has been used as the pump laser, because it can work reliably and stably with high power in relatively tough environments and it is easy to operate and maintain [20–24]. However, Q-switched lasers oscillating on multiple random longitudinal modes lead to mode beating, which can cause time modulation of the output power of the laser. Most of the previous research has concentrated on extending tuning range and increasing efficiency, while little attention has been paid to the influence of a multiple random longitudinal mode laser on optical parametric process which has always been one of the major concerns for practical applications, especially under the circumstances where high stability and temporally smooth pulses are needed.
In this paper, the influence of the time modulation of the pump laser caused by the mode beating on optical parametric process is numerically analyzed and simulated according to a practical method which is proposed in this paper. The time modulation of the output power of the generated waves from optical parametric process appears and irregular spike sequences of the generated beams are observed, because of the time modulation of the pump laser caused by the mode beating. The output power of the light from optical parametric process exhibits a large fluctuation with the time, when a Q-switched laser oscillating on several random longitudinal modes is used as the pump laser. However, when the number of the longitudinal modes is more than or about 100 (especially more than 500), a less fluctuation of the output power of the generated beams is obtained. To our knowledge, this is the first time to investigate the influence of the time modulation of the pump laser caused by mode beating on optical parametric process.
In Section 2, we discuss the equations of the time modulation of the amplitude. Then the time-independent three-wave coupling equations for second order optical parametric process are presented. In Section 3, we present simulations to explore the time modulation of generated beams from optical parametric process induced by the time modulation of the pump laser, which is caused by mode beating.
2. Theoretical model
2.1 Time modulation of amplitude
The amplitude and phase of various longitudinal modes of a Q-switched laser which oscillates on multiple random longitudinal modes are independent of each other and their transient distribution is random. The number of the longitudinal modes of a multi-longitudinal mode laser is determined by the gain bandwidth of the laser medium and the frequency spacing of two adjacent modes. The distribution of the amplitude and phase of various longitudinal modes is unstable and a kind of random fluctuation varying with the time rapidly. Therefore the instantaneous power, which is the superposition of each longitudinal mode, is also a random fluctuation varying with the time rapidly.
If the number of the longitudinal modes in the laser is N, the total amplitude of the radiation field is given by [25]
where νn is the angular frequency of the nth longitudinal mode and φn is the initial phase of the nth longitudinal mode, respectively. The instantaneous light intensity is presented as [25]where the first term on the right hand side is the intensity sum of each longitudinal mode, the second term (the time modulation term) is the random fluctuation varying with the time, which is induced by the uncertain relationship of the angular frequency and phase between each longitudinal mode caused by the mode beating. When two frequencies are close and the initial phase difference is temporarily stable or slowly variable, the mode beating will be generated.The peak power density is given by [26]
where c is the velocity of light in the free space, ε0 is the dielectric constant and nr is the refractive index, respectively.2.2 Three-wave coupling equations for optical parametric process
The schematic configuration of second order optical parametric process is shown in Fig. 1 . Conventionally, ωp, ωs and ωi are known as the pump frequency, the signal frequency, and the idler frequency, respectively. The field of the signal wave (ωs) is amplified by the process of difference-frequency generation and the field of the idler wave (ωi) is generated simultaneously, which is known as parametric amplification. The gain associated with the process of optical parametric amplification can produce oscillation when feedback of the signal wave and/or the idler wave exists, which is known as an OPO. Optical parametric process can produce either a continuous-wave output or pulses of nanosecond, picosecond, or femtosecond duration.
To better understand the performance of three wave mixing (second order nonlinear optical effects), we implement a model calculation based on the well-known time-independent three-wave coupling equations, which shows the essential physics without the complications of pulse propagation. The results can be directly applied to situations with long pulses (with negligible effects of temporal walk-off) and remain qualitatively correct even for rather short pulses. The time-independent propagation equations for second order optical parametric process, without absorption and walk-off, are expressed by [27]
where Ej (j = s, i, p) are scalars of the electric fields for the three waves respectively and Ej* (j = s, i, p) are their complex conjugates, nj (j = s, i, p) and kj (j = s, i, p) are the indexes of refraction and the wave numbers of Ej (j = s, i, p) respectively, deff is the effective nonlinear coefficient of the nonlinear crystal, and z is the integration parameter which is taken along the propagation of the pump beam. The phase matching condition (k p = k s + k i) is achieved commonly by using the birefringence property of the nonlinear crystal. The equations are then numerically solved by RUNGE-KUTTA method. The model is assumed that the temporal pulse shape and the spatial profile of the pump light are flat-topped.3. Numerical simulation results
In this section, we first present some experimental results to introduce the problem caused by the mode beating. The time modulation of a Q-switched laser oscillating on multiple random longitudinal modes in our experiment is shown in Fig. 2 . Both waveforms are measured with a 5 GHz oscilloscope and a 40 GHz photodetector. Figure 2(a) shows the time domain of a laser, which oscillates on several random longitudinal modes. The time modulation of the laser caused by mode beating has been observed. The time modulation of a laser oscillating on hundreds of random longitudinal modes is illustrated in Fig. 2(b). The pulse is temporally smooth, because of the overlap of the mode beating between longitudinal modes.
Fig. 2 Time domain of a laser oscillating on (a) several longitudinal modes and (b) hundreds of longitudinal modes
To bring out the influence of the time modulation of the pump laser caused by mode beating on optical parametric process, we choose an OPA/OPG based on a biaxial crystal KTA, which has been most widely used in mid-infrared and optical communication, as our example. For the sake of simplicity, we adopt the noncritical phase matching KTA in all the following simulation. Table 1 shows the physical parameters in the simulation.
Table 1. Physical parameters in the simulation
The best fit equations of refractive index for KTA are given by [28,29]
The amplitudes and the angular frequencies of various longitudinal modes are a random distribution around the central value, and the initial phases are a random distribution between –π and π. The average power, pulse width and repetition rate of the pump laser are taken as 20 W, 20 ns and 10 kHz, considering that the damage threshold of the KTA crystal and the coating is about 400 MW/cm2. If the pump laser is a single longitudinal laser, the relationship between the crystal length and the power density of the pump laser, signal laser and idler laser is obtained and shown in Fig. 3(a) according to Eq. (4), (5). The power of the generated waves increases to a maximum value along the crystal length until the power of the pump laser is full depleted. After that, the power begins to decrease because the signal wave and the idler wave flow back to the pump laser. The conversion efficiency of the generated beams is also acquired and shown in Fig. 3(b). There is a maximum value of the conversion efficiency. Therefore, the optimal length of the crystal is about 44.5 mm. The output power of the generated beams will reach a maximum at the length. It is assumed that the length of the crystal is 44.5 mm in the following calculation.
Fig. 3 The relationship between the crystal length and (a) the power density and (b) the conversion efficiency
Here we assume that the time varies from −10 ns to 10 ns with the step size of 20 ps. According to Eq. (2) and (3), the relationship of the peak power density of the pump laser (Pp) and the time (t) is obtained and shown in Fig. 4 with respect to different numbers of longitudinal modes. From Fig. 4(a) and 4(b), we find that Pp fluctuates irregularly from 180 to 580 MW/cm2 approximately, and irregular spike sequences of Pp are observed (n = 2 and n = 4). The maximal change of Pp is almost 400 MW/cm2 which is about 69% of the maximum value. Pp varies significantly with the time when the pump laser oscillates on several random longitudinal modes, and therefore even the crystal may be damaged by the higher peak power density caused by the time modulation of Pp. However, when n is greater than or about 100, the maximal change of Pp is only about several tens of MW/cm2, as shown in Fig. 4(c) and 4(d). The maximal change of Pp reaches 36 MW/cm2 (n = 500) which is only about 10% of the maximum value of Pp. Pp becomes more stable when the number of longitudinal modes increases. The reason is that the mode beating has been generated between the longitudinal modes, and therefore the time modulation of Pp appears. However, when n is very large, Pp has a less fluctuation on account of the stacking of the mode beating between longitudinal modes.
Fig. 4 The relationship between Pp and t for different numbers of longitudinal modes: (a) n = 2, (b) n = 4, (c) n = 100 and (d) n = 500; the insets are the enlarged parts of the time domain
If we use a Q-switched laser oscillating on multiple random longitudinal modes as the pump laser of optical parametric process, the relationship between the peak power density of the generated beams (Ps and Pi) and t is acquired and shown in Fig. 5 according to Eq. (4), (5). Figure 5(a) and 5(b) show that Ps and Pi vary irregularly from 10 to 222 MW/cm2 and 5 to 100 MW/cm2 respectively, and irregular spike sequences of Ps and Pi are observed (n = 2 and n = 4). The maximal changes of Ps and Pi are as high as 212 MW/cm2 and 93 MW/cm2, respectively, which is almost equal to the maximum value. Ps and Pi vary significantly with t, because of the time modulation of Pp caused by the mode beating between longitudinal modes. Thus the generated beams become extremely unstable when the pump laser oscillates on several longitudinal modes. However, the maximal changes of Ps and Pi is only about several tens of MW/cm2 when n is greater than or about 100, as shown in Fig. 5(c) and 5(d). The maximal change can reach 22 MW/cm2 and 9.8 MW/cm2 respectively (n = 500), which is only about 10% of the maximum value. Ps and Pi exhibit more stability with the increase of the number of the longitudinal modes. Ps and Pi is modulated in the time domain because of the time modulation of Pp, which is induced by the mode beating. Whereas, because the mode beating between longitudinal modes overlaps each other, Ps and Pi have a less fluctuation when n is more or about 100.
Fig. 5 The relationship between Ps, Pi and t for different numbers of longitudinal modes: (a) n = 2, (b) n = 4, (c) n = 100 and (d) n = 500; the insets are the enlarged parts of the time domain
The simulation results of the conversion efficiencies of the signal wave (ηs) and the idler wave (ηi) are shown in Fig. 6 , respectively. From Fig. 6(a) and 6(b), we can find that ηs and ηi fluctuate irregularly from 2% to 70% and 1% to 31% approximately, and irregular spike sequences of ηs and ηi appear (n = 2 and n = 4). When the pump laser oscillates on several longitudinal modes, the conversion efficiencies of the generated beams fluctuate greatly with t because of the time modulation of Pp. However, when n is greater than or about 100, the maximum changes of ηs and ηi are less than 0.28 and 0.12, respectively, as shown in Fig. 6(c) and 6(d). The maximal changes of ηs and ηi can reach 0.13 and 0.05 respectively (n = 500), which is only about 15% of the maximum value. The maximal changes decrease as the number of the longitudinal modes increases. The time modulation of ηs and ηi is generated because of the time modulation of Pp, which is caused by the mode beating between longitudinal modes. However, due to the overlap of the mode beating between longitudinal modes, ηs and ηi vary very slowly when n is more than or about 100 (especially more than 500).
Fig. 6 The relationship between ηs, ηi and t for different numbers of longitudinal modes: (a) n = 2, (b) n = 4, (c) n = 100 and (d) n = 500; the insets are the enlarged parts of the time domain
4. Conclusion
The time modulation of the pump laser on optical parametric process which is caused by mode beating is analyzed and numerically simulated. The simulation results show that the time modulation of the signal wave and the idler wave is observed, and irregular spike sequences of Ps and Pi appear. When a Q-switched laser oscillating on several random longitudinal modes is used as the pump laser, the generated beams become very unstable. Ps and Pi exhibit a large variation between the maximum value and the minimum value. However, the generated beams become more stable and fluctuate more slowly when the number of the longitudinal modes increases. The maximal changes of Ps and Pi are only about 10% of the maximum value (n = 500). This can be a very useful and effective way for studying the stability and time domain characteristics of optical parametric process and selecting the pump laser.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant 60878060 and the National High Technology Research and Development Program of China.
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