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Abstract
Considering the spatial distribution of laser beams and phonon waves, the SRS coupling wave equations in the transient regime are derived and normalized for the first time. The synchronously pumped solid-state Raman laser is simulated numerically to investigate the influences of the cavity length detuning, output coupling rate, dispersion, Raman gain and dephasing time of Raman mode on laser performances. It is found that the intensive pulse compression of first Stokes laser in synchronously pumped solid-state Raman laser stems from pulse width gain narrowing and intensity oscillation effects. The cavity length detuning, dispersion, Raman gain and dephasing time considerably affect the pulse width gain narrowing and intensity oscillation processes. The theoretical results can help the design and optimization of synchronously pumped solid-state Raman laser to generate ultrafast Raman laser output efficiently.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Picosecond pulses have been widely used in industrial, medical, military and information fields [1–4]. Solid-state Raman lasers in different configurations can efficiently generate IR, visible and UV output [5–10]. For the transient regime where the pump laser pulse duration is similar or shorter than the Raman transition dephasing time T2 (typically T2 is 1∼10 ps for crystalline Raman materials), stimulated Raman scattering (SRS) process has smaller Raman gain and higher threshold compared to the steady state regime [10]. To overcome those withdraws, synchronously pumped Raman lasers have been demonstrated to realize efficient Raman conversion with picosecond pumping laser [11–13]. For synchronously pumped Raman lasers, the resonator length of external Raman laser matches the repetition rate of the pump laser, and the pumping pulse and intracavity Stokes pulse propagate in step inside the resonator.
Synchronously pumped solid-state Raman laser is a kind of frequency convertor with good features, such as simple structure, high conversion efficiency, good stability, no need for phase matching, etc. The most attracting feature is that pulse width of Stokes lasers can be greatly compressed resulting in much enhanced peak intensity compared to pumping laser pulses. Recent years, many experimental works on synchronously pumped solid-state Raman laser in picosecond regime have been reported in IR, visible and UV spectrum span [13–18]. The research group from Macquarie University used 532nm 28ps mode-locked laser to synchronously pump KGW crystal. It was found that the pulse width of first Stokes laser can be greatly compressed by cavity length detuning (CLD) [13,17], and the shortest pulse width of first Stokes laser obtained was 6.5ps [13]. The same group adopted the 532nm 26 ps pulsed laser as pumping source to simultaneously pump 6.7mm long diamond crystal, and the pulse width was reduced to 9ps with CLD of 200μm. M. Frank et al. of the Czech Republic reported GdVO4, BaWO4 and SrMoO4 as Raman crystals to conduct synchronously pumped Raman laser experiments, and obtained the extremely compressed pulses of Stokes laser by shifting the pumping laser with two different Raman modes in sequence [19–22]. In 2019, Frank M. synchronously pumped a-cut SrMoO4 external Raman laser with ring type cavity, 1173nm first Stokes laser generated by the Raman mode of 887cm−1 was successively shifted by the Raman mode of 327cm−1 to generate 1220nm Raman laser. By cavity length detuning, the minimum pulse duration of 1220nm was 1.4ps, which was compressed by 26 times compared to the pumping pulse duration of 36ps [19].
Nonlinear coupling equations of transient SRS were adopted to investigate Raman processes with gas Raman media [11,12,23,24]. In 2010, Granados E. et al. numerically simulated synchronously pumped solid-state Raman lasers with the transient coupled wave equations [14,25], and it was found that the pulse compression effect was produced by a combination of group velocity walk-off and strong pump pulse depletion [14,25]. However, full understanding of synchronously pumped solid-state Raman laser is still lacked, and there is no research to investigate the influences of main parameters on the laser performance, i.e. output coupling rate, dispersion, Raman gain, dephasing time of Raman mode and etc. Moreover, as a third nonlinear process, SRS conversion is sensitive to the beams intensities, and plane wave theories bring about considerable discrepancy [11,12,24,25]. In [26], Boyd G. et al. investigated the threshold condition for steady-state stimulated Raman oscillation by considering the paraxial Gaussian beam distributions, and got results of better precision. Our group deduced the space-dependent rate equations for intracavity Q-switched Raman lasers in the steady-state regime, and also found the new equations have better approximation than the plane-wave ones [9]. In this paper, assuming the laser beam transverse distributions to be of the Gaussian profile, considering laser beam longitudinal spacial distribution, the space-dependent transient SRS coupling wave equations are derived and normalized for the first time. By numerically solving the normalized coupling wave equations, synchronously pumped Raman laser is simulated for the full insight of its operation. The influences of the cavity length detuning, output coupling rate of the first Stokes laser, dispersion, dephasing time of Raman mode and Raman gain on Raman laser performances are investigated systematically. The new understandings of the pulse compression mechanism are revealed. The research results can help the design and optimization of synchronously pumped solid-state Raman laser to generate ultrafast Raman laser output efficiently.
2. Theory
The optical fields are described classically in terms of Maxwell’s equations, and the molecular system is treated quantum mechanically. The SRS coupling equations in the transient regime are deduced quantum-mechanically in the Gaussian unit system [27].
In the Raman medium, the upper state population changes induced by the resonant interaction can be neglected. ωv,L,S and kv,L,S respectively denote the angular frequency and wave number of the material excitation, pumping laser and first Stokes laser, and ω0 and k0 is the frequency and wave number of the material excitation at resonance. With relations of ωv =ωL -ωS and kv = kL - kS, assuming pulse durations and dephasing time T2 of phonon is in the picosecond range or longer, the transient Raman coupling equations at the resonant condition (ωv=ω0) are deduced. For the new coordinates (t, z), which are in a frame moving with the velocity of first Stokes laser vS, the following equations are obtained in [25].
The quantum statistical Langevin operator $\hat{F}({z,t} )$ is added in (1) corresponding to random fluctuations, which accounts for the spontaneous Raman scattering [28]. The Langevin force is taken to be delta correlated, and the relative deduction obeys
The SRS is third nonlinear process, and sensitive to the field intensity. To precisely model the SRS process in the transient regime, the longitudinal and transverse spatial distributions of laser fields should be taken into consideration. Fields of fundamental and first Stokes beams are assumed to be of Gaussian transverse distribution for the TEM00 mode, and the electric field amplitudes of pumping and first Stokes lasers inside the cavity are expressed as:
The electric field amplitude of input pumping laser is expressed as:
The phonon excitation is driven by multiplication of ES* and EL according to Eq. (1). When the wavelength of phonon wave is much less than beam radius of pumping and first Stokes lasers, the phonon excitation can be reasonably assumed to be
The transvers Laplace operator in cylindrical coordinate system is expressed as:
For the generality and simplicity, we introduce the normalized time τ, normalized coordinate ζ, normalized electric field amplitudes EL and ES, normalized phonon amplitude Θ, normalized Raman gain coefficient G, normalized light velocities VL and VS, normalized dephasing time Τ of Raman mode as:
In the following simulation, the normalized intensity of first Stokes pulse IS is defined as:
ISmax is the normalized peak intensity of first Stokes pulse. The normalized intensity integration of first Stokes pulse ISinteg is defined as: which is proportional to the pulse energy and conversion efficiency of first Stokes laser. The normalized pulse duration of the first Stokes laser τS=τSr+τSf is obtained from (εS·εS*), here τSr is the normalized rising time from ISmax/2 to ISmax, and τSf is the falling time from ISmax to ISmax/2.3. Setup for simulation
Simulations are carried out for the synchronously pumped z-fold external ring SRS laser as shown in Fig. 1. The cavity consists of four mirrors, which are pumping mirror (PM), M1, M2 and flat output coupler (OC). PM is coated HT for pumping laser and HR for first Stokes laser, M1 and M2 have coating HR for first Stokes laser, and OC has partial reflectivity for first Stokes laser. All mirrors are coated HT for the second Stokes laser to prohibit its generation through cascaded SRS. The pumping source adopted is usually CW mode-locked lasers, for example, a 532nm laser with an average power of 2W, pulse duration of 10ns and repetition rate of 80 MHz in [25]. The Raman crystal is assumed to be KGW, and the wavelength of the pumping and first Stokes lasers are 532nm and 559nm, respectively. For the cavity length detuning (CLD), Δx=0 corresponds to the perfect synchronization. For positive Δx, the cavity length of Raman laser becomes longer than that of perfect synchronization, and the first Stokes pulse lags behind the pumping pulse. For negative Δx, the cavity length becomes shorter, and the first Stokes pulse goes before the pumping pulse. CLD is realized by adjusting M2.
The normalized space-dependent coupling Eqs. (14)-(16) are solved numerically to model the Raman interaction between the incident infinite train of pumping pulses and first Stokes intracavity pulse. The first Stokes laser is generated through the spontaneous Raman scattering brought about by quantum fluctuations of the Raman medium at the initial stage, then is amplified through SRS process at the following stage. The interactions of circulating first Stokes pulse with a sequence of pumping pulses are modeled iteratively, and the simulation is terminated when the Stokes pulse has reached its steady state profile after about 300 pumping pulses. The dispersive broadening of the Stokes pulse is approximately modelled by applying a discrete broadening after each pass through the Raman crystal, and is sufficiently accurate in the picosecond regime [25]. The values of normalized variables are estimated with the experiments parameters as given in Table 1.
Table 1. Estimated values of the normalized parametersa
4. Results and discussion
4.1 Effects of normalized CLD Δζ on output characteristics
The effects of normalized CLD Δζ on output characteristics of synchronously pumped solid-state Raman laser are investigated numerically as shown in Fig. 2. The normalized intensity integration ISinteg, normalized maximum intensity ISmax, and normalized pulse width τS of first Stokes pulse are shown in Fig. 2(a) and (b), with RS=80% or 90%, G=4.0×10−10 or 8.0×10−10, and other parameters given in Table 1.
Fig. 2. Effects of normalized CLD Δζ on output characteristics of first Stokes laser with RS=80% or 90%, G=4.0×10−10 or 8.0×10−10, and other parameters given in Table 1.
CLD has considerable effects on the pulse energy, pulse width and peak power of first Stokes laser outputs for the synchronously pumped Raman laser [13,17]. As shown in Fig. 2(a), normalized CLD Δζ ranges from -0.04∼0.006, and there is a maximum ISinteg for each line, which corresponds the maximum SRS conversion efficiency. From the maximum value, ISinteg declines rapidly as Δζ increases to the positive direction, and decreases gradually as Δζ varies to the negative regime. For G=4.0×10−10, ISinteg of RS=80% is greater than that of RS=90% for negative Δζ, however, for the positive Δζ, ISinteg of RS=80% declines more rapidly, and is smaller than the result of RS=90%. ISinteg of G=8.0×10−10 is much higher than that of G=4.0×10−10. As shown in Fig. 2(b), lines of normalized pulse duration τS have the similar trend. τS declines gradually as Δζ varying from -0.04 to -0.003, and has an abruptly decrease within Δζ =-0.003∼0.0009 to reach the minimum value, where ISmax rises sharply to reach the maximum value. Compared to G=4.0×10−10, the maximum value of ISmax is much higher for G=8.0×10−10. As Δζ further increase from 0.0009 to 0.006, τS increases obviously, and ISmax declines sharply. Lines of ISmax all have an abrupt peak corresponding to the minimum value of τS.
There are two interesting working points adjacent to Δζ = 0 as shown in Fig. 2(a). Under working point 1, which usually has a small negative value of Δζ, the maximum Raman conversion efficiency can be achieved, when the pulse width and peak power of first Stokes laser are close to those of pumping pulses. Under working point 2, which commonly has a small positive value of Δζ, the first Stokes laser pulse is compressed considerably, where it has the maximum peak power, minimum pulse width and moderate conversion efficiency. The relative parameters of working point 1 and 2 are listed in Table 2. The experimental parameters, such as output coupling rate, Raman gain and etc., have little impact on the value of Δζ for working point 2, which ranges from 0.0005 to 0.001. As the absolute value of Δζ increases, the performance of synchronously pumped Raman lasers deteriorates. However, synchronously pumped Raman lasers have higher tolerance for positive Δζ than for negative one.
4.2 Effects of normalized CLD Δζ on pulse temporal distributions
In Fig. 3, normalized intensity temporal distributions of first Stokes laser output and residual pumping laser are given for different Δζ with G=8.0×10−10, RS=80% and other parameters given in Table 1. Δζ is equal to 0.006, 0.0009, 0, -0.008, -0.02 and -0.004 in (a), (b), (c), (d), (e) and (f), respectively. In Fig. 3, pulses profiles are in a frame moving with the velocity of first Stokes laser VS to the left, and left edges of pulses are the leading ones as shown in Fig. 3(a).
Fig. 3. Normalized intensity temporal distributions of first Stokes laser output and residual pumping laser for different Δζ with G=8.0×10−10, RS=80% and other parameters given in Table 1. Δζ is equal to 0.006, 0.0009, 0, -0.008, -0.02 and -0.04 in (a), (b), (c), (d), (e) and (f), respectively.
It is found that pulse compression of first Stokes laser can be realized for working point 2 as depicted in Fig. 2(a) with appropriate CLD. In the transient SRS, the gain at a point on the first Stokes envelope depends on the integral of the pumping power up to that point, which leads to the delayed appearance of the transient phonon wave and first Stokes laser pulse with respect to the pumping pulse. SRS process is a third nonlinear process, and has threshold effect. The Stokes pulse usually has a leading edge which rises sharply to a maximum with some delay with respect to the maximum of the pumping pulse. For positive CLDs as given in Fig. 3(a) and (b), the first Stokes pulse temporally lags behind the pumping laser pulse by an additional detuning delay, and the first Stokes pulse will gradually catch up with the pumping pulse due to dispersion. Thus, the leading edge of the first Stokes pulse sweeps through the pumping pulse temporally in the Raman medium, and always meets the undepleted part of pumping pulse to obtain much higher Raman gain than the falling edge. The unbalanced gain leads to pulse width gain narrowing effect, which results in intensive pulse compression, and forms giant Stokes pulse of high peak intensity and small pulse width. As shown in Fig. 3(b), the first Stokes pulse has the smallest pulse width and maximum peak intensity with Δζ = 0.0009, which is unexpectedly smaller because the Stokes pulse usually lag behind the pumping pulse as mentioned above. When CLD further increases in the positive direction, the peak intensity and conversion efficiency of first Stokes laser drop dramatically, because the first Stokes pulse misses the pumping pulse peak, and large portion of pumping energy cannot be converted as shown in Fig. 3(a). Thus, the pulse compression of first Stokes laser comes from the pulse width gain narrowing, which utilizes the group velocity difference of pumping and first Stokes laser due to linear dispersion under appropriate CLD.
As shown in Figs. 3(a), (b) and (c), the pulse compression is usually accompanied by intensity modulation. As intense ultrafast laser pulses go through Raman-active media, the first Stokes and phonon fields grow so large that the pumping field is driven toward zero. Since the phonon responds slowly, the transfer of power from the pumping laser to first Stokes reverses, resulting in a flow of energy back into the pumping [23]. Thus, the depleted pumping pulse and first Stokes pulse show a series of decreasing oscillations, and more oscillations follow if the depletion level is larger as shown in Fig. 3(b). The intensity oscillation can further narrow the pulse width of first Stokes laser, however, clamp its peak intensity.
As CLD goes into negative regime with relatively small value, the pumping and first Stokes laser pulses have a good temporal overlap in the Raman crystal, resulting in high conversion efficiency as shown in Figs. 3(d) and (e). It is noted that the best temporal overlap does not occur at the Δζ = 0 due to the velocity dispersion and the behind-lagging of first Stokes pulses, and the maximum ISinteg is obtained with Δζ =-0.008 as shown in Fig. 3(d). As the absolute value of CLD further increases in the negative regime, the temporal shape of the first Stokes pulse is reshaped by the high Raman gain for its trailing edge. With negative CLD, the first Stokes pulse has a large pulse width with a long leading edge and relatively short trailing edge as shown in Fig. 3(f), and the peak intensity decreases sharply.
4.3 Effects of dispersion on pulse compression
The influence of dispersion on the transient nonlinear process of synchronously pumped solid-state Raman laser was investigated in [25], and need further study. Based on the numerical simulation, it is found that the linear dispersion has important effects on pulse compression. Figure 4 shows the effects of dispersion on pulse compression of the first Stokes laser with RS = 90%, G=1.0×10−9, Τ=0.5 and other parameters given in Table 1. Solid lines are the results with dispersion, and dashed lines are those without dispersion. With dispersion, the pulse width of first Stokes laser is compressed to the minimum value of 0.0052 for Δζ = 0.0003 reaching the maximum peak intensity of 5.51 as depicted in Fig. 4(a). Without dispersion, there is no pulse compression effect. The line of ISmax is flat, and τS decrease gradually as Δζ increasing in Fig. 4(b).
Fig. 4. Effects of dispersion on pulse compression of the first Stokes laser with RS = 90%, G=1.0×10−9, Τ=0.5 and other parameters given in Table 1. Solid lines are the results with dispersion, and dashed lines are those without dispersion.
To illustrate the effect of dispersion on pulse compression more clearly, Fig. 5 depicts the effects of dispersion (VS-VL) on pulse compression with RS = 90%, Δζ = 0.0005 and other parameters given in Table 1. VS-VL is equal to 0, 0.0013, 0.0026 and 0.0052 in (a), (b), (c) and (d), respectively. It is found that SRS amplification is favorably affected by a velocity mismatch between the pumping and first Stokes pulses due to linear dispersion. As shown in Fig. 5, as the dispersion is increasing, the greater displacement for the pumping laser pulse occurs relative to the current reference frame, and the first Stokes pulse goes through the pumping pulse profile to extract more energy from the pumping pulse. Consequently, as the dispersion increases, the intensity of the residing pumping pulse decreases, and the conversion efficiency of first Stokes laser is improved.
Fig. 5. Effects of dispersion (VS-VL) on pulse compression of the first Stokes laser with RS=90%, Δζ = 0.0005 and other parameters given in Table 1. VS-VL is equal to 0, 0.0013, 0.0026 and 0.0052 in (a), (b), (c) and (d), respectively.
The group velocity dispersion is the critical factor for the pulse compression [23,25]. Having velocity dispersion, the Stokes pulse can propagate forward through the pump pulse, and the leading edge of the Stokes field sweep through the pumping pulse profile which have not been depleted. The Stokes pulse extracts more energy from the pumping pulse, and develops into a giant pulse with a greatly compressed pulse width and a high peak power. Without dispersion, there is no obvious pulse compression of the first Stokes laser pulse (τS=0.808) as shown in Fig. 5(a). With dispersion, the pulse width compression of the first Stokes pulse occurs as shown in Fig. 5(b), (c) and (d). As dispersion (VS-VL) increases from 0.0013 to 0.0026, τS decreases from 0.296 to 0.068 as depicted in Figs. 5(b) and (c). On the other hand, the dispersion brings about broadening of pulse width, and limits the narrowest pulse width of first Stokes pulses. As dispersion (VS-VL) increases to 0.0052, τS increases to 0.096 as shown in Fig. 5(d). In a nondispersive medium, the Stokes pulse profile has no intensity oscillation as shown in Fig. 5(a). As the dispersion increases, intensity oscillation of the pumping and first Stokes laser pulses get more intensive as shown in Figs. 5(b), (c) and (d).
With dispersion, the pumping pulse moves to right relative to the current reference system to generate the time delay, and the first Stokes laser pulse actually moves to right along with the pumping pulse through the SRS amplification. The higher the dispersion is, the larger the time delay is. Passing through the Raman crystal with parameters given in Table 1, the time delay of the pumping pulse is calculated to be 0.23 relative to the current reference system for VS-VL = 0.0013 (half dispersion). For VS-VL = 0.0052 (double dispersion), the time delay is estimated to be 0.92. The first Stokes pulse with double dispersion in Fig. 5(d) lags behind that of half dispersion in Fig. 5(a) obviously. Therefore, the first Stokes pulse and the pumping pulse can propagate basically synchronously under different dispersion conditions. The experimental conditions, such as output coupling rate, Raman gain, dephasing time and dispersion, have little effect on Δζ of the working point 1 and 2 as given in Table 2. It is also noted that there are different definitions of CLD. In this paper, Δζ = 0 corresponds to the perfect synchronization, for which the round trip time of first Stokes laser is equal to the time interval between two adjacent pumping pulses. For the experimental reports, CLD was defined so that the resonator had the lowest laser threshold for Δx=0 [14–17].
Pulse width gain narrowing and intensity oscillation lead to the pulse compression of synchronously pumped solid-state Raman lasers. The dispersion can bring about pulse width gain narrowing and intensity modulation, and higher dispersion is beneficial for the pulse compression. The evidence can be found in the experiment reports. The Australia group used the 7.5 W 532nm 26 ps coherent pulsed laser [14] and 4.8 W 1064 nm 15 ps CW mode-locked Nd:YVO4 laser [16] pumped synchronously diamond Raman lasers, and minimum pulse duration of first Stokes obtained experimentally was 9 ps (pulse compression factor=2.9) and 9 ps (pulse compression factor=1.7) for appropriate CLD, respectively. Relative to infrared laser, frequency-doubled laser has higher dispersion, and can achieve higher pulse compression factor of first Stokes laser output [14].
4.4. Effects of normalized Raman gain G on output characteristics
Figure 6 shows effects of normalized Raman gain G on output characteristics of the first Stokes laser with RS=80% or 90%, Δζ = 0.0005, and other parameters given in Table 1.
Fig. 6. Effects of normalized Raman gain G on output characteristics of the first Stokes laser with RS=80% or 90%, Δζ = 0.0005, and other parameters given in Table 1.
Normalized Raman gain G varies from 1×10−10 to10×10−10. At the beginning, ISinteg increases rapidly with the increase of G, and then slows down tending to a stable value as shown in Fig. 6(a). ISmax also goes up rapidly with the increase of G at the initial stage. However, further increasingly G brings about more intensive intensity modulation, and impedes the Raman conversion. Thus, ISmax decreases with the further increase of G. τS decreases rapidly with the increase of G initially, then decreases slowly tending to a stable value as shown in Fig. 6(b). It can be seen that the increased Raman gain enhances pulse width compression, and brings about more intensive intensity oscillation of the first Stokes laser, leading to the decrease of pulse width τS.
4.5 Effects of normalized dephasing time T on pulse compression
In the numerical calculation, the normalized dephasing time defined as Τ=T2/tp is varied according to the pulse duration of pumping laser tp with T2 of Raman mode to be constant. To investigate the influence of Τ on the pulse compression effect without being interfered by transient SRS gain reduction, the simulation is carried out for the same pumping pulse energy, i.e. G/Τ=2×10−9. Figure 7 shows the output characteristics of first Stokes laser for Τ=0.2 (G=4.0×10−10) and 0.5 (G=1.0×10−10), respectively. The normalized intensity integration ISinteg, normalized maximum intensity ISmax, and normalized pulse width τS of first Stokes pulses are shown in (a) and (b). ISinteg of Τ=0.5 is slightly higher than that of Τ=0.2. Around Δζ = 0.0005, the smallest pulse width of first Stokes is obtained, and is similar for two dephasing time conditions. However, for Τ=0.5, the maximum value of ISmax is 5.51, and is much larger than that of 3.09 for Τ=0.2. The parameters of two working point1 and 2 are also listed in Table 2 for Τ=0.5 (G=1.0×10−10, RS=90%).
Fig. 7. Effects of normalized dephasing time Τ on pulse compression of the first Stokes laser with RS=90%, Τ=0.2 (G=4.0×10−10) or 0.5 (G=1.0×10−9), and other parameters given in Table 1.
Table 2. Output parameters of first Stokes laser under working point 1 and 2
Figure 8 shows normalized intensity temporal distributions of first Stokes laser output and residual pumping laser for different Τ with RS=90% and Δζ = 0.0005 for the same pumping pulse energy, i.e. G/Τ=2×10−9. Τ is equal to 0.1, 0.2, 0.5 and 1 in (a), (b), (c) and (d), respectively. Intensities of first Stokes laser pulse are modulated, and the first peak of the oscillations usually has the highest intensity. As Τ increases from 0.1 to 1, the transient effect gets more serious, and the first peak possess more energy and higher intensity. The relation between pulse width of first Stokes laser and dephasing time is somewhat complicated. As Τ increases from 0.1 to 0.2, the pulse width decreases. For Τ=0.5 and 1, however, the pulse width increases due to serious dispersion pulse broadening in the highly transient regime.
Fig. 8. For the same pumping pulse energy G/Τ=2×10−9, normalized intensity temporal distributions of first Stokes laser output and residual pumping laser for different Τ with RS=90% and Δζ = 0.0005. Τ is equal to 0.1, 0.2, 0.5 and 1 in (a), (b), (c) and (d), respectively.
With the same pumping pulse energy, smaller pulse width of pumping laser favor the generation of first Stokes laser pulse of high intensity. On the other hand, in the highly transient regime, pulse broadening due to group velocity dispersion will get serious, and limits the smallest pulse width of the first Stokes laser. Dispersion compensation is needed to realize shorter pulse width.
5. Conclusions
In this paper, the synchronously pumped solid-state Raman laser is analyzed systematically by numerically solving the normalized nonlinear coupling wave equations in the transient regime. There are two working points interesting to the actual applications. Under working point 1 with relative small negative CLD, the maximum Raman conversion efficiency can be achieved. Under working point 2 with relative small positive CLD, Stokes laser pulse is compressed to have the maximum peak power and minimum pulse. Actually, the first Stokes and the pumping pulses can propagate basically synchronously under different dispersion conditions. The experimental conditions, such as output coupling rate, Raman gain, dephasing time and dispersion, have little effect on the CLD of the working point 1 and 2.
The pulse compression of first Stokes laser stems from the pulse width gain narrowing and intensity oscillation effects. Thus, the early result in [25] is confirmed, which is pulse compression comes from a combination of group velocity walk-off and strong pump pulse depletion. Pulse width gain narrowing utilizes the group velocity difference of pumping and first Stokes laser due to linear dispersion under appropriate CLD. The intensity oscillation further narrows the pulse width of first Stokes laser, and clamp its peak intensity.
As the dispersion increases, the SRS conversion efficiency is improved. The higher dispersion or Raman gain leads to more intensive pulse width gain narrowing and intensity modulation, and favors the pulse compression. However, the dispersion brings about pulse width broadening, and limits the narrowest pulse width of first Stokes laser. The increased Raman gain enhances pulse width gain narrowing and intensity oscillation of the first Stokes laser leading to the decrease of pulse width τS, but too high Raman gain leads to the decrease of maximum peak pulse intensity. As Τ increases, the transient effect gets more serious, and the first peak of modulated first Stokes pulse possess more energy and higher intensity. Smaller pulse width of pumping laser favor the generation of first Stokes laser pulse of high intensity and small pulse width. On the other hand, in the highly transient regime, pulse broadening due to group velocity dispersion will get serious, and dispersion compensation is needed to realize shorter pulse width.
Funding
Natural Science Foundation of Shandong Province (ZR2018LF014).
Disclosures
The authors declare that there are no conflicts of interest.
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