Theoretical investigation on the pumping effect of stimulated Brillouin

Appl Phys B (2012) 106:445–451 DOI 10.1007/s00340-011-4797-4

Theoretical investigation on the pumping effect of stimulated Brillouin scattering on stimulated Raman scattering in water J. Shi · X. Chen · M. Ouyang · W. Gong · Y. Su · D. Liu

Received: 10 February 2011 / Revised version: 9 July 2011 / Published online: 17 November 2011 © Springer-Verlag 2011

Abstract The pumping effect of stimulated Brillouin scattering on stimulated Raman scattering is investigated theoretically through the coupled wave equations of stimulated Brillouin scattering and stimulated Raman scattering. The numerical simulations are in agreement with the experimental results. They indicate that the backward stimulated Raman scattering is excited and amplified collectively by both pump laser and stimulated Brillouin scattering.

1 Introduction High power laser beam can excite several stimulated types of scattering in materials [1–4]. Among of them, Stimulated Raman scattering (SRS) [1, 2] and stimulated Brillouin scattering (SBS) [4] are two processes which can be easily observed. The relationship between the two kinds of process has been widely investigated [5–10]. Generally, it is thought that SBS and SRS are in competition. However, some works show that the pumping effect exists between SBS and SRS in liquid. J.Z. Zhang et al. realized pumping effect of SBS on J. Shi () · X. Chen · M. Ouyang · W. Gong · Y. Su D. Liu () Applied Optics Beijing Area Major Laboratory, Department of Physics, Beijing Normal University, Beijing 100875, China e-mail: [email protected] D. Liu () e-mail: [email protected] D. Liu Key Laboratory of Nondestructive Test (Ministry of Education), Nanchang Hang Kong University, Nanchang 330063, China Present address: M. Ouyang School of Information Photoelectric Science & Technology, South China Normal University, Guangzhou 510631, China

SRS by focusing single mode laser into alcohol droplet [11] in which the interaction length between SBS and SRS can be increased because of the particular shape of the droplet. Barille et Al. discussed the “backward SRS” (BSRS) in water and the relation of the SRS and the OH stretching vibration of water molecules [2]. Recently, the present authors demonstrated an experimental phenomenon in which SRS can be amplified by SBS [12]. We thought that there may be two possible mechanisms: (1) the BSRS is amplified by the pump laser synchronously; (2) SBS excites its own “forward SRS” (FSRS). Because the SRS excited by both mechanism are not distinguished in temporal and frequency domain, we did not give an explicit conclusion in [12]. In recent months, we analyzed theoretically the process in detail, combining the new experimental observations in the temporal domain, and made numerical simulations. The results show that the above process of BSRS is actually the result of acting cooperatively by the SBS and the pump laser beam. The theoretical investigation is presented below.

2 Theoretical consideration In this work, the coupled wave equations for SBS, BSRS, and FSRS [13–16] are considered synthetically. Here, the direction characters “B” in BSRS and “F” in FSRS are with respect to the pump laser, that is to say, the BSRS is actually the “FSRS” with respect to SBS. In our experiments, the measurements were done with the minimum focused depth of 30 cm in water, which corresponds a time duration of 1.33 ns. The life time of the phonon for SRS is in the ps order of magnitude [17], so the transient effect of SRS can be totally neglected, and the coupled wave equations for each

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Fig. 1 The relationship between pulse energy of SBS and BSRS and focused depth in water. The solid line corresponds to SBS, the dashed line corresponds to BSRS

SRS process contain two equations. However, the life time of the phonon for SBS is about 0.3 ns [18], hence the transient effect of SBS cannot be neglected, and the coupled wave equations for SBS process contain three equations. Therefore, there are five processes that need to be considered: the SBS, BSRS, and FSRS processes pumped by the laser, the BSRS and FSRS processes pumped by SBS. The final coupled wave equations should include eleven equations: three coupled wave equations for SBS, four coupled wave equations for BSRS (pumped by laser beam and SBS, respectively), and four coupled wave equations for FSRS (pumped by laser beam and SBS, respectively). Because all the processes take place simultaneously, the above eleven equations can be simplified to the following five equations after deductions: dAp = −gp Asbs eicQ − gfsrsp Ap eicA2fsrs dz αp − gsrsp2 Ap eicA2fsrs − Ap , 2 dAsbs = gsbs Ap eicQ − gsrsp Asbs eicA2fsrs dz αsbs − gfsrsp2 Asbs eicA2fsrs − Asbs , 2

(1)

dAfsrs αfsrs = gfsrs eicAfsrs A2sbs + gsrs2 eicAfsrs A2p − Afsrs , dz 2

dAfsrs = gfsrs2 eicAfsrs A2sbs dz + gfsrs2 eicAfsrs A2p −

αfsrs Afsrs , 2

dQ Γb np = (Ap Asbs − Q) . dz 2 c Here, Ap , Asbs , Afsrs , Afsrs represent the amplitude of the electric fields of pump laser, SBS, BSRS, and FSRS. Q represents amplitude of sound field, z represents the propagating distance in z direction. gsbs , gfsrs , gfsrs , gsrs2 , gfsrs2 are the gain coefficients of SBS, forward pumping BSRS and FSRS, backward pumping BSRS and FSRS. gp , gsrsp , gfsrsp , gsrsp2 , gfsrsp2 are the depletion coefficients of the pump laser on SBS process, the pump laser and SBS on corresponding forward pumping BSRS and FSRS processes, the pump laser and SBS on corresponding backward pumping BSRS, and FSRS processes, respectively. In SRS process, gfsrsp = gfsrs λfsrs /λp , gfsrsp2 = gfsrs2 λfsrs /λp [14]. α is the attenuation coefficient, n is the refractive index of the material, c is the light speed in the vacuum, eic = 2e0 np c is the transfer coefficient between the strength of electric field and optical intensity, e0 = 8.854 × 10−12 is the value of the permittivity of the vacuum, and Γb is the line width of spontaneous Brillouin scattering expressed by angular frequency. In (1), only the spatial differential part (z) has appeared, the temporal differential part (t) does not appear, however, it does not mean that the temporal differential part

Theoretical investigation on the pumping effect of stimulated Brillouin scattering on stimulated Raman

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Fig. 2 Changes of SBS and BSRS energies with the change of laser energy. The solid line corresponds to SBS, the dashed line corresponds to BSRS

(t) is neglected. Actually, (1) are the coupled equations at a certain moment of each iteration. The temporal differential part (t) is considered when the loop is running during numerical calculation, which will be explained later.

3 Numerical calculations and discussions The calculations were done using the method similar to the split-step method introduced in [14]. The program runs in loop, and the differential calculations of t and z are split and carried out alternately. In each iteration of the loop, the waves are at a new time/moment, and they interact with each other according to (1) in the whole interaction region z (similar to the water cell in the experiment). Then the waves propagate to the next time/moment (consequently the new position) in their own directions, and begin the next iteration. Therefore, although the temporal part (t) does not appear in (1) it has been considered in actual calculations. Also, the difference between the frequencies of the two

kinds of SRS pumped by the laser beam and SBS is ignored because it is very small [12]. In our calculations, the step distance (z) was chosen to be 10∼100 µm, corresponding to a temporal duration (t) of less than 0.33 ps. It is accurate enough for calculations compared to the life time for the two kinds of phonon. The chosen parameter values are: the gain of SBS in water excited by 532 nm laser beam is gsbs = 2.94 × 10−11 m/W [18], the gain of SRS is gsrs2 = gfsrs = 0.3 × 10−11 m/W, the attenuation coefficient of light in water is αsbs = 0.1 m−1 and αfsrs = 0.4 m−1 [19], the refractive index is np = nfsrs = 1.33, the length of the water is 2 m, the line width of spontaneous Brillouin scattering is 600 MHz, the Brillouin shift in water is 7.5 GHz, the sound speed in water is 1500 m/s, the spontaneous scattering coefficient is R = 3.14 × 10−8 m−1 [20]. gsbs = gp is assumed because their frequency difference is very small. For the pump laser beam, the expanded diameter is 20 mm, the wavelength is 532 nm, the pulse width is 8 ns, the pulse energy is 10.7∼214 mJ, and the focused depth in the water

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Fig. 3 3-D plot of SBS energy vs. pump energy and focused depth in water

is 0.2 m∼1.9 m. These values are chosen according to the values in the experiments. Figure 1(a) shows the calculated results of scattering intensity with different focused depth at the pump energy of 214 mJ. For comparison, Figs. 1(b) and (c) show the experimental layout and result again given in [12]. The vertical polarized laser beam of 532 nm changed into horizontal polarized after passing through λ/2 plate, so that most of the laser energy passes through BS, and only a small fraction was reflected. The reflected intensity I1 was detected by DP0. The transmission intensity IP can be monitored in real time by the transmission/reflection ratio of BS, which is already known. The transmission beam turned into circularly polarized after passing through a λ/4 plate, then was reflected by M0 and M1 and focused at a certain point in the water cell. Since

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SBS propagates only backward, it becomes vertically polarized after passing through the λ/4 plate. Most of the energy of the SBS is reflected by M2, and then it passes through a prism P to split into a first order SBS and a BSRS, which are detected, respectively, by DP2 and DP1. The residual intensity of the laser will be transmitted through the other end of the water cell and detected by DP4. In Fig. 1(c), it can be seen that the calculated variation tendencies of SBS and BSRS are in agreement with experiments qualitatively. It should be realized that, in our experiments [12], the pulse energy of the laser was 620 mJ, eliminating the loss by optical elements, the pulse energy actually entering the water was about 400 mJ. Although the simulated results are not exactly equal to the measured results, the difference with two or three orders of magnitude of the scattered energy of SBS and BSRS is in consistency with the experimental results. Figure 2(a) shows the calculated relationship between the energy of SBS and BSRS and the pulse energy of the pump beam at the focused depth of 1.5 m. For comparison Fig. 2(b) shows the experimental result given in [12]. It can be seen that the calculated results are basically in agreement with the experimental results in the comparable range (corresponds to low pump energy), i.e. the SBS appears to linearly increase beyond the saturation effect, and the energy of BSRS is three orders lower than that of SBS and begins the tendency of nonlinear increase. We did not simulate the case of higher energy. The reason is that in our calculations only the processes of SBS, BSRS, and FSRS were considered. When the pump energy is high enough, other nonlinear effects, such as self-focusing, selfphase modulation, optical breakdown, two-photon absorption and so on, cannot be ignored. Here we give a detailed discussion. It is known that the energy of light is nearly conserved in stimulated scattering, if the phonon energy can be neglected and material attenuation is considered. However, as far as self-focusing or optical breakdown is considered, much more energy is converted to heat and the energy of light would not be conserved in principle. This phenomenon can be used as the criterion of the existence of other nonlinear effects. In our experiments, we made two accurate measurements about the energy conservation of light. In the first case, the pump energy into the water tank is 203 mJ, and the focal length is 1.5 m. The back scattered energy and the transmitted energy measured are 124.5 mJ and 9.26 mJ. Considering that the transmissivity of the front panel of the water tank is 90% and the attenuation coefficient of the water is 0.1 m−1 , the calibrated total energy scattered and transmitted is 200.7 mJ, nearly equal to the pump energy. In the second case, when the pump energy is raised to 553 mJ, the calibrated total energy scattered and transmitted is 443.1 mJ, which means that about 110 mJ light


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Fig. 4 3-D plot of calculated results of BSRS energy and FSRS energy

energy is lost. This result definitely shows that other nonlinear processes occurred under high pump energy. Those processes have strong fluctuations, their interactions with stimulated scattering is very complicated. It has been impossible to express these interactions completely and perfectly up to now [1]. Therefore, the simulation for even higher energy will not reflect the actual physical process and be beyond our possibilities. An experimental 3-D plot about SBS energy was given in [12] (see Fig. 3(a)). Figure 3(b) shows the calculated result. It can be seen that the SBS energy also decreases with the increase of focused depth in water, while it increases with the increase of the pump energy. This property is in agreement with the experimental measurements shown in Fig. 3(a). But for high pump energy the calculated result is not entirely consistent with the experimental measurements although the variation tendency is similar. The reason is still that for high pump energy it is not sufficient to consider the interaction between SBS and SRS, as mentioned above. Figure 4 shows the 3-D plot of the BSRS energy and the FSRS energy. For better understanding, Figs. 4(b) and (e) show the relationship of the BSRS and FSRS vs. laser en-

Fig. 5 The calculated relationship between the temporal orders of SBS (solid line) and BSRS (dashed line). In calculations, the focused depth was 1.6 m, pump energy was 214 mJ/pulse. The pulse durations (FWHM) for SBS and BSRS are 1.41 ns and 1.23 ns, respectively

ergy at fixed focused depth of 100 cm, Figs. 4(c) and (f) show the relationship of the BSRS and FSRS vs. focused depth at a fixed laser energy of 100.7 mJ. One can find that

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Fig. 6 The temporal orders of SBS and BSRS. In experiments, the focused depth was 1.57 m, pump energy was 182 mJ/pulse. The unit of the horizontal axis is 2 ns per division. The pulse durations (FWHM) for SBS and BSRS are 1.04 ns and 0.75 ns, respectively

both the energies, of BSRS and FSRS, increase with the increase of the pump energy, but only the BSRS increases with the increase of focused depth in water; it is just reversed trend for the FSRS. This is because the gain length of BSRS increases with the focused depth, while the gain length of FSRS decreases with the focused depth. In the following, we will discuss the time profile of SBS and BSRS. It is known that both SBS and BSRS processes have the character of pulse compression, so we cannot use this character to distinguish the relation of SBS and BSRS. The temporal order may be used to achieve this task. If these two processes are in competition, enhancing of the SBS should lead to weakening of the BSRS [1], and vice versa. However, the calculated temporal relationship between SBS and BSRS shown in Fig. 5 gives a different result. It can be seen that there exists a strong time correlation between the intensities of SBS and BSRS: the temporal sequence of SBS and BSRS has little difference, meanwhile the pulse duration (FWHM) of BSRS (1.23 ns) is a bit narrower than that of SBS (1.41 ns). This phenomenon is obviously different from the temporal relationship for the case of SBS and BSRS in competition [1]. If this simulated phenomenon is in agreement with the experiment result, we can affirm that the BSRS observed in our experiments could not be in competition with SBS. It should be pointed out that temporal profiles of SBS and BSRS were given in [12]; it seems to have a small difference in their temporal orders. This is a bit different from the calculated result. A possible reason may be that in [12] the time profile was obtained using two individual detectors connected by separate BNC cables, it definitely induced difference in the time delay of the two signals. According to our experiments, the time delay caused by electronic circuit was about 25 ns. Assuming that the measured relative uncertainty is 5%, the measured error of 25 ns will be compara-

ble to 1.5 ns induced by optical layout. Therefore, the result given in [12] might not show the accurate temporal order of SBS and BSRS. In order to compare with the calculated results shown in Fig. 5 more accurately, new experiments were designed. The basic optical layout is the same as that shown in Fig. 1(b), but a single detector connected by single BNC cable was put just after M2 , and the signals of SBS and BSRS were measured by inserting band-pass filters for 532 nm and 650 nm, respectively, between the detector and M2 . Also, the output signal of a laser pulse was used as the trigger signal of the oscilloscope to set the time origin. Therefore, the difference of optical or electrical delay mentioned above for different signal could be eliminated. The comparison of the signal of SBS and BSRS was achieved through the infinite persistence function of the oscilloscope (Agilent 54832 B model). Figure 6 shows the new experimental results of the temporal profiles. It is basically in agreement with the calculated results shown in Fig. 5. Therefore, we can further confirm that the SBS has participated in the pumping of the BSRS process. Also, the pulse duration of BSRS (0.75 ns) is a bit narrower than that of SBS (1.04 ns), following the same regulation as the computational result. It can be understood that the center of the SBS peak has a higher gain on BSRS, it makes the BSRS narrower than SBS. Two possible physical mechanisms of BSRS were suggested in [12], but the functions of pump laser and SBS cannot be distinguished. Now, we can answer this question. The above theoretical calculations in this paper are basically in agreement with the experimental results. It indicates that the two mechanisms of the BSRS pumped by pump laser and by SBS could not be distinguished; only that both the mechanisms working together, as shown by (1), can lead to the result consistent with the experimental results.


4 Summary The phenomenon of SRS enhanced by SBS is investigated theoretically based on coupled wave equations. The relationship between the energy of BSRS and SBS and the pump energy, the relationship between the energy of BSRS and SBS and the focusing depth in water, and also the temporal profile of BSRS and SBS are obtained numerically. These results are in agreement with the experimental measurements. The theoretical analysis clarifies more clearly the physical mechanism of the phenomenon of BSRS enhanced by SBS. Acknowledgements The authors would like to thank the National Advanced Technology Development program (grant No. 2009AA09Z101) and the National Natural Science Foundation of China (grant Nos. 10574016, 60778049 and 10904003) for financial support.

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