We present an experimental and theoretical study of the transition from linear to nonlinear amplification of classical pump noise in a fiber Raman generator. In particular, we focus on the conversion of fluctuations in the fine temporal structure of Q-switched pump pulses into Stokes pulse energy fluctuations. We show that there is a distinct pump power domain where large scale fluctuations in the Stokes pulse energy result from the amplification of fluctuations in the temporal structure of pump pulses with stable energies. Dramatic changes in the shape of the Stokes pulse energy probability distribution also occur as the pump power is swept through the domain of large scale energy fluctuations.
© 2005 Optical Society of America
Raman scattering has enjoyed a long and intense history of research for both applied and fundamental reasons. In recent years, Raman amplifiers and lasers have become a viable component of optical communication systems as compact, high-power pump lasers have become readily available. The enthusiasm in developing Raman amplifiers has occurred in large part because the non-resonant nature of the Raman process in silica glass creates the potential for producing a completely wavelength-agile optical amplifier [1–8]. On a more fundamental level, Raman generation in gases has provided a physical system in which to observe macroscopic quantum fluctuations when a spontaneously generated signal is linearly amplified from a few photons to macroscopic proportions [9–11]. In this paper, we focus on the fundamental physics of how stimulated Raman scattering (SRS) in optical fiber can shape an initial noise distribution.
Fluctuations in the energy of output Stokes pulses in SRS can result from both quantum initiation noise and classical noise, also referred to as technical noise, which enters a nonlinear optical system because of its coupling to a fluctuating macroscopic environment. Unlike quantum fluctuations, classical noise can, in principle, be eliminated by employing techniques that isolate the nonlinear optical system from environmental perturbations such as mechanical vibrations, thermal drift, and electrical power fluctuations. However, in practice the various couplings to the macroscopic environment are never completely severed; consequently, classical noise will still affect the stability of the system.
Q-switched lasers, which produce pulses with high peak powers that can easily drive nonlinear processes in optical fiber, are one potential source of classical noise. These pulses quite often contain multiple longitudinal modes that lead to fine temporal sub-structure under a relatively long envelope. This mode structure typically fluctuates from pulse to pulse, generating large fluctuations in the pulse-to-pulse temporal sub-structure. These fluctuations can lead to wide variations in the peak intensities contained within the pulse envelope even when the total pulse energy varies little from shot to shot. Because SRS is a nonlinear process, these variations in intensity across the pump pulse can lead to large variations in the amount of generated Stokes light, as will be verified by the experiments and simulations to follow. In other words, the Raman process can convert fluctuations in the temporal sub-structure of a pump pulse with a stable energy into large-scale energy fluctuations in the first Stokes pulse. We show that there is a distinct scattering regime where the probability distribution of the first Stokes pulse energy experiences dramatic changes in shape as a result of amplified classical pump noise.
There has been a substantial amount of work on long-pulse Raman generation in optical fiber, but this work has generally represented the pump and Stokes pulses by CW waves or by simple pulse shapes that neglect any substructure [12–17]. There have also been studies of the propagation of pump noise in fiber Raman amplifiers as used in optical communication systems. In this work both co-propagating and counter-propagating pump waves have been used, and the results indicate that a counter-propagating pump produces a more stable amplified bit stream because the pump fluctuations are essentially averaged out as each bit sweeps through the pump wave [6, 7]. Other studies have investigated noise suppression in a high bit rate communication signal using saturated Raman wavelength conversion . In this work a CW probe with a shorter wavelength was depleted by the bit 1’s in a higher power, longer wavelength signal. The result was a complementary bit stream (0’s become 1’s and 1’s become 0’s) with lower noise. This work on Raman amplifiers and wavelength converters has generally involved the interaction of CW beams with bit streams, while our work focuses on pulsed Raman generation. We are aware of no other groups who have investigated the shaping of pump pulse temporal structure fluctuations by long-pulse SRS in optical fiber.
Noise shaping by Raman generation in high-pressure gas cells has also been extensively investigated. Much of this work has focused on the shaping of quantum noise, giving special attention to macroscopic quantum fluctuations that are observed in the linear growth regime. There has also been some work on the transition from the linear to nonlinear regime of growth. In this work the full scale fluctuations observed for weak Raman scattering give way to stable Stokes pulses once the scattering becomes strong [9–11, 19, 20]. Additionally, there has been more limited theoretical and experimental work on SRS in gases that studies the impact of pump fluctuations on the generated Stokes pulses [21–23]. However, glass optical fiber presents a significantly different physical system in which to study the noise shaping properties of SRS for several reasons. The Raman gain of glass is extremely broad, extending over many terahertz. The interactions between pump and generated Stokes waves are potentially much stronger because of the guiding structure of the fiber and the very low linear scattering and absorption losses in the near infrared.
In this article we extend our previous work  on the impact of classical pump noise (in the form of fluctuations in the temporal structure of Q-switched pulses) on the Stokes pulse energy statistics in a fiber Raman generator in several ways. First, we present experiments and simulations for very weak Raman scattering that clearly define the domain over which amplified pump fluctuations have the greatest impact on the Stokes pulse energy statistics. Second, we have used our previously developed model to match not only the lower order statistical parameters but also reproduce the evolution of the underlying probability distribution of the Stokes pulse energy. Finally, we also show that the quantum initiation noise does not significantly affect the statistics of the Stokes pulses in our experiments by use of the model and experiments in which very weak SRS was driven by a single-mode pump laser.
We find that the temporal pump noise results in large energy fluctuations (greater than 200% relative noise) in an intermediate scattering strength that lies between the regime of linear amplification of the first Stokes pulse in the weak scattering limit and highly nonlinear amplification of the saturated first Stokes pulse. In this regime, the output energy fluctuations are dominated by the amplification of the classical pump noise as amplified by stimulated scattering, not the quantum initiation noise. Furthermore, this nonlinear amplification can lead to energy distributions of Stokes light that are dramatically different from the distribution of the classical noise, resulting in Stokes energy fluctuations that are much larger than pump pulse energy fluctuations, the fluctuations in the temporal structure of the pump pulses, or the fluctuations due to quantum initiation of the scattering. This transition from linear to nonlinear amplification of classical pump noise in fiber-based Raman generation and the associated evolution of the Stokes pulse energy probability distribution is the main focus of this article.
2. Background and results
The experimental apparatus is described in detail in reference ; we will give a short synopsis of the setup here, focusing our discussion on points of particular relevance to the present work. The source of the pump pulses was a Q-switched diode-pumped Nd:YAG laser with a center wavelength of 1064 nm. The pulse width was typically around 25 ns full-width at half-maximum, and the repetition rate was 1 kHz. The pump pulses were coupled into a 71 m long single-mode polarization-maintaining optical fiber, parallel to one of the principal axes of the elliptical core fiber. Dispersing prisms at the fiber output were used to separate the pump and generated Stokes orders, and apertures were used to select the first Stokes order for measurement by a large area Ge photodiode with an internal electronic amplifier (New Focus 2033 Photoreceiver).
The Ge photodiode used in these experiments has a rise time that is much longer than the Stokes pulse duration so that the temporal shapes of the detected Stokes pulses were the impulse response of the detector. The primary motivation for using this detector was that it could detect much weaker Stokes pulses than the small area fast InGaAs photodiode used in our earlier work, and this sensitivity was critical for clearly resolving the Stokes pulse energy statistics reported in this article. A long-pass filter (LPF) was mounted on the input of the Ge photodiode to reject stray pump light that was Rayleigh scattered into the first Stokes beam. The photodiode was also contained in a black cloth tunnel to prevent stray light from other parts of the setup from corrupting the Stokes signal.
The pulse energy noise was collected using a gated integrator with computer interface to integrate and buffer the electronic signal from the Ge photodiode. The gated integrator system could collect pulse areas (energies) at the repetition rate of the laser. The buffer in the computer interface had a maximum capacity of 3700 pulse energies. For each input pump power we collected a total of 10,800 first Stokes pulse energies (three consecutive sets of 3600 energies). We also collected data sets with the photodiode blocked; sets of 3600 electronic background energies were collected before and after the 10,800 first Stokes pulse energies. One reason for collecting electronic background data was to ensure that the integrator input offset, which was set so that the integrated electronic noise signal had an average value of zero, did not drift during the measurements. A second reason for collecting background data was to obtain information about the statistics of the electronic background noise and how the amplitude of the background fluctuations compared with the amplitude of the first Stokes signal.
The raw first Stokes pulse energies include contributions from the Rayleigh-scattered pump light that makes it through the long-pass filter and from the integrated electronic background noise, and these contributions can have a significant effect on the measured statistics for sufficiently weak scattering into the first Stokes order. Hence, as described below, we have taken great care to determine the relative importance of extraneous contributions to the Stokes pulse energy fluctuations. For most of the noise data presented in this article, the effects of pump leakage and electronic background noise were found to be negligible.
Rayleigh scattering in the dispersing prisms directed a small fraction of the pump light along the same path as the dispersed Stokes beam, which resulted in substantial corruption of the first Stokes signal at the lowest scattering levels. Thus, the presence of the LPF at the input of the photodiode was crucial. This filter blocked most of the pump light that leaked into the first Stokes beam and transmitted most of the Stokes light that we wanted to measure. The measured LPF power transmission factors were 1.4% for the pump and 64% for the first Stokes. The amount of pump leakage into the first Stokes beam can be found by making pulse energy measurements with and without the long-pass filter in place. Without the LPF, the average detected power can be written as
where 〈A〉u is the detected power without the filter in place, 〈S〉 is the average first Stokes power dispersed by the prism, 〈P〉 is the pump leakage power, and r is the detector responsivity at the pump wavelength relative to the responsivity at the first Stokes wavelength. With the LPF in place the average detected power can be written as
where 〈A〉f is the average detected power with the filter in place, TS is the LPF transmission coefficient at the Stokes wavelength, and TP is the LPF transmission coefficient at the pump wavelength. We can solve these two equations for the unknown optical powers 〈P〉 and 〈S〉 and use the results to obtain the fraction of the average detected power that is due to the Stokes beam:
This correction is negligible (the ratio TS〈S〉/〈A〉f is nearly unity) when the peak pump power is greater than 10 watts, but becomes more significant for very low input pump powers.
The electronic background noise added significantly to the fluctuations of the first Stokes pulse energy for low input pump powers. The average value of the integrated electronic background noise was nearly zero, but the fluctuations about this very small mean were comparable in magnitude to the fluctuations of the first Stokes pulse energy about its much larger mean. In all cases but one the background mean and the raw Stokes pulse mean were separated by at least three standard deviations. The relative importance of integrated electronic background noise was assessed by comparing ensembles of energies with the Stokes signal blocked and unblocked. The actual first Stokes pulse energy noise was estimated from the raw data by assuming that the shot-to-shot fluctuations in the electronic background, the pump leakage, and the first Stokes were uncorrelated. Our measurements indicated that the correlation between fluctuations in the pump pulse and first Stokes pulse energy is small for weak scattering, and there is no reason to expect the fluctuations in the electronic background of the detection system to be correlated with the optical pulse energies. (The instantaneous intensities of pump and first Stokes pulses are highly correlated, as the results of our work so clearly indicate, but the integrated intensities will not be highly correlated if the fluctuations in the pump pulse temporal structure and total energy are independent.)
Based on this assumption, we can relate the raw Stokes pulse energy standard deviation to the standard deviations of the electronic background, the pump leakage, and the actual Stokes pulse energy:
As discussed above, the mean value of the raw Stokes energy can be expressed as the sum of the means of the actual Stokes pulse energy and the pump leakage energy
where we have used the fact that the electronic background noise has an average of approximately zero. Using these two relationships we can back out estimates of the actual relative first Stokes pulse energy noise from the raw data:
The largest part of the correction was due to the electronic background noise.
Because the pump pulse in the experiments is polarized along a principal axis of the fiber, we use a scalar model for the pulses. At each temporal point across the pulse, the model included the pump power, P 0, at wavelength λ0, and N Raman modes, denoted by Pi, each at wavelength λi. (For the results discussed here, the model included 256 Stokes modes.) These modes were spread over a frequency shift of approximately 1800 cm-1 relative to the pump frequency in order to include several Stokes orders. The pump and each Stokes mode at each temporal point were coupled through the equations 
Here gij is the effective gain between modes i and j, which is assumed to vary inversely with wavelength [15, 16], G0=0.013 W-1m-1 is the peak value of the Raman gain at the pump wavelength, ΔΩmax=454 cm-1 is the location of the peak gain, ΔΩFW=243 cm-1 is the full-width at half-maximum of the gain curve, and ΔΩij is the frequency shift between modes i and j.
All simulations were performed for a fiber length of 71 m using an adaptive step-size integration technique to assure sufficient accuracy.
In the experiments, individual pump pulses generally contained many longitudinal modes so that the pulse envelope was modulated by mode beating. The measured frequency of the modulations was approximately 1.25 GHz, which corresponds to the spacing of adjacent modes obtained from spectrum analyzer measurements. The mode structure of the Q-switched pulses fluctuated from pulse to pulse so that the structure and depth of the ripples due to mode beating varied widely from pulse to pulse even though the total pulse energy was quite stable. For input pump pulses with the same total energy, the peak intensity will be much higher for a deeply modulated pulse than for a pulse with a relatively smooth envelope, and the amount of Raman scattering from the two pump pulses will differ markedly; this source of classical pump noise is the dominant contributor to the observed fluctuations in the generated first Stokes pulse energy.
In the simulations, we chose a temporal shape for the pulses based upon a discretization of an experimentally-measured pump pulse envelope; typically we used 2000 points over a 100 ns temporal range. Each temporal point was then independently propagated down the fiber. In this way, the model was used to produce a distinct output pulse at each Stokes order for each modulated pump pulse.
To incorporate the fluctuations in the pump pulse shape due to the multi-mode operation of the laser, we modulated the pump pulse envelope through multiplication by the factor (1-Acos2πνt), where ν=1.25 GHz, and the amplitude of modulation A varied between zero and unity. The distribution of modulation amplitudes was chosen to be Gaussian in shape with an average modulation depth of A=0.45 (45% of the peak intensity of the pump pulse) and a normalized standard deviation of 0.15; this distribution is shown in Fig. 1. The Gaussian distribution was chosen for convenience, and was based on an experimentally measured distribution of the modulations of 1500 pump pulses, which was nearly Gaussian. The values of the mean modulation depth and standard deviation used in the simulations are somewhat greater than the measured values, because we expect that the actual modulation depth is significantly larger than the measured value because of bandwidth limitations of the measurement system. Furthermore, the values used in the simulations presented here provide a slightly better “global fit” to measured quantities such as the growth curves, relative noise versus pump power curves, and the histograms of the energy distributions for the first Stokes pulse.
In our simulations we modeled the quantum initiation noise in two different ways. For some of the simulation results presented here, we chose f=1 so that ηi represented the average value of the quantum initiation noise, and did not vary from mode-to-mode, or from temporal point-to- temporal point. For these runs, the model did not include any intrinsic fluctuations arising from quantum effects, and each mode and temporal point was seeded with a value of ηi that represents the average spontaneous scattering into each mode. Thus, for these simulations the only fluctuations input into in the model were those resulting from the classical pump noise due to the shot-to-shot variation in the beating of the modes of the laser cavity.
For other simulations, we chose f to be a exponentially-distributed random variable with a mean value of unity. For these runs, the “seed” used to model the quantum initiation noise in the simulations varied from temporal point-to-temporal point within each pulse, and from pulse-to-pulse. This second set of simulations therefore includes two sources of fluctuations: temporal variations of the pump pulse envelope, and variations in the seed used to initiate the scattering for each temporal point.
This model neglects the effects of dispersion because we are modeling Q-switched pulses. Walk-off does play some role in the dynamics since the pump and Stokes pulses have sub-nanosecond structures due to mode beating. Previous measurements indicate that the walk-off between the pump and first Stokes order is between 100 and 200 ps over the full fiber length. The measured period of the beats between adjacent modes is around 800 ps. Including dispersion would make the model considerably more computationally intensive, but we have obtained excellent agreement between theory and experiment using a simpler, less computationally intensive model. This agreement includes the measured growth curves and relative pulse energy noise from the point of weak scattering to strong multi-order scattering . In the work presented in this paper, we will show that model and experiment are also in good agreement for the pulse energy histograms. The fact that we can match not only the lower order statistical moments but also the underlying probability distribution indicates that the basic conclusions drawn from our comparison are well founded and would not be changed by including dispersion.
The experiment and simulation results for the relative pulse energy noise for weak scattering into the first Stokes order are shown in Fig. 2. (For this range of peak pump powers, generation of the second Stokes order is not significant; previous measurements  indicate that second Stokes order accounts for only 2% of the output energy when the peak pump power is 30 W.) There is a clearly defined noise maximum for input pump powers between 12 and 13 watts where the effects of amplified fluctuations in the pump pulse temporal structure are most prominent. In this narrow range of input pump powers, the standard deviation of the first Stokes pulse energy fluctuations is over twice the mean pulse energy at this maximum. On either side of this range, the first Stokes noise decreases rather rapidly. The agreement between the measured noises and the simulations is remarkably good over the entire range surrounding this low power noise peak, both when the model includes fluctuations in the quantum seed and when it does not. From this we conclude that the large fluctuations seen in the first Stokes output energy can be entirely accounted for by the temporal fluctuations in the input pulse envelope, despite the fact that the variation in the input pump energy is quite small.
We can understand the origin of this noise peak qualitatively as follows. On the low pump power side of this noise maximum, the noise decreases as pump power decreases because the amplitude of the Stokes pulse never becomes large enough for stimulated scattering to become significant. In this regime the scattering process is essentially linear, and the first Stokes energy varies little from pulse to pulse because the pump pulse energy variation is small. For intermediate pump powers the noise is large because stimulated scattering has become important and this scattering depends on the product of the pump and Stokes intensities. The scattering now depends strongly on the pump pulse intensity at each time slice, and the total energy of the first Stokes pulse will depend sensitively on the modulation amplitude. Pump pulses with large modulation amplitudes will contain many intensity “spikes” that cause strong stimulated scattering into the first Stokes order, resulting in significant energy in the first Stokes pulse. Pump pulses with small modulation amplitudes will not have sufficient intensity in any temporal slice to initiate stimulated scattering, and will result in a low-energy first Stokes pulse. To the right of the peak, for the highest input pump powers shown, the intensity of the pump pulse is large enough that all pulses undergo significant stimulated scattering, regardless of the level of modulation, and the noise is relatively low.
The relative noise is the ratio of two statistical parameters of the first Stokes pulse energy, the standard deviation and the mean. A more stringent test of the agreement between experiment and simulation can be obtained by comparing the pulse energy histograms for experiment and simulations since these are proportional to the underlying probability distribution from which all of the statistical parameters are drawn. Figure 3 compares experimental and simulation histograms of the output Stokes energy for a peak pump power near the noise maximum. (The accompanying multimedia file shows an animation of the histograms for a range of pump powers around the noise maximum.) The coloring of the simulation histogram indicates the magnitude of the pump pulse temporal modulations: green for pulses with modulation depths greater than 45% and red for modulation depths less than 45%. For both the experimental and simulation histograms, the horizontal axis has been normalized so that the mean of the distribution is unity.
The overall agreement between simulation and experiment is quite good for the evolution of the overall shape of the histograms. Experiment and simulation show the evolution from a highly skewed distribution spread over a wide range of pulse energies to a distribution that is tightly concentrated around the mean pulse energy. Near the relative noise peak, both the simulated and experimental distributions show a tall peak at very low energy, and both exhibit a long tail. (In Fig. 3, the tail of both distributions has been truncated. Each of the experimental and simulated distributions contains approximately 25 pulses that have energies between ten and fifty times the mean of the distributions.) These large tails, which result from the nonlinear effects of the stimulated scattering, are generated by the pump pulses with the largest modulation amplitudes, and are responsible for the large relative noise values for peak pump powers near 11.5 watts. Conversely, for this input power, the large majority of the pump pulses do not possess a sufficient modulation amplitude to initiate stimulated scattering, resulting in the tall peak at low scattering energies. It should be noted that the only source of noise in the simulations is the variation in the pump pulse amplitude modulations; there are no variations in the pump pulse energies in the simulations, and quantum fluctuations are not included. Thus, the long tail in the simulation distribution in Fig. 3, and the large peak in relative noise seen in Fig. 2 can be directly attributed to the nonlinear amplification of the pump pulse amplitude modulations. Furthermore, the nonlinear nature of the stimulated scattering is evidenced by the very different shapes of distributions of the temporal amplitude modulations (Fig. 1) and the resulting Stokes energy distribution (Fig. 3).
For comparison, Fig. 4 shows the effect of including variations in the quantum initiation seed in the simulations. As noted previously, the variations in the quantum initiation seed were exponentially distributed and had a mean equal to the constant quantum seed used in simulations shown in Fig. 3. As can be seen from Fig. 4 and the accompanying animation, the agreement between simulation and experiment is again quite good. We therefore feel quite confident of our conclusion that the large fluctuations in the Stokes output energy are due solely to the temporal variation of input pump pulse.
The results of these simulations and experiments indicate that classical pump noise, in the form of shot-to-shot fluctuations in the fine temporal structure of the pump pulse, are the dominant source of the observed fluctuations in the first Stokes pulse energy. The quantum noise, which initiates the scattering, is apparently relatively unimportant for the scattering strengths investigated in this work. This is likely the case because the pump and Stokes pulse widths are much larger than the Raman response time of the medium, which is hundreds of femtoseconds. If we were to integrate and analyze the energy fluctuations in a very narrow slice of the first Stokes pulse rather than the whole pulse, we would almost certainly find that amplified quantum noise was important [9–11,27]. We have performed such measurements for stimulated Brillouin scattering, where the characteristic response time is comparable to the pump pulse width and within reach of the temporal resolution of our apparatus, and nearly full-scale fluctuations gave way to more stable energies as the integration time was increased .
The central role of fluctuations in the pump pulse temporal structure can also be illustrated by SRS experiments using single-mode pump pulses for which the intensity spikes due to mode beating are eliminated. Stimulated Brillouin scattering (SBS) becomes prominent for such pump pulses, and the gain for SBS is significantly higher than the gain for SRS for single-mode pump pulses of the width used in our experiments; consequently, SBS generates back-scattered intensity spikes in the early sections of the fiber before the forward propagating SRS pulses have fully developed. However, for sufficiently low pump powers SBS is weak enough that the pump pulses are not strongly depleted by the SBS process, but the SRS pulses are still measurable with our detection system. This range of powers happens to overlap with the low power end of the SRS measurements for multi-mode pump pulses that have just been presented. In the following paragraphs, we describe measurements of SRS for single-mode pump pulses in this input power regime in some detail.
The pump laser was made to oscillate on a single longitudinal mode by combining two mode selection techniques. First, we inserted an uncoated 2 mm optical flat in the cavity to reduce the number of lasing modes in the Q-switched pulses to a few adjacent modes, as confirmed by a pulsed laser spectrum analyzer. In addition to the insertion of an optical flat, we reduced the losses of the AOM Q-switch so that a weak CW background was present between the pulses. For a small range of Q-switch losses the CW background was single-mode and this background locked the forming pulses onto a single mode. The temporal envelopes of the resulting pulses were quite smooth and the standard deviation of the total pump pulse energy was typically less than 1% of the mean pulse energy. When these pulses are coupled into the fiber, SBS spikes are initiated by scattering off thermally generated acoustic vibrations of the optical fiber. These spikes propagate in the opposite direction to the pump pulse and, as a consequence, tend to deplete the trailing edge of the pump pulse. For strong SBS the trailing edge of the pump pulse at the fiber output is substantially steeper than the trailing edge at the fiber input. Our goal was to measure the statistics of the SRS generated Stokes pulses for input pump powers where the depletion of the pump by SBS was relatively small.
Figure 5 shows the relative noise for the single-mode SRS experiments along with the multimode data for the lowest input pump peak powers. The relative noises for the single-mode measurements have been corrected for pump leakage into the Stokes beam and for fluctuations due to electronic background noise using data analysis methods described earlier in the paper. The noises for the single-mode measurements are nearly an order of magnitude lower than the noises for the multimode measurements at corresponding input pump peak powers. For single-mode pump pulses, the primary contributions to the Stokes output noise are from the amplification of the very small pump fluctuations, from the competing SBS process, and from any residual amplified quantum noise. It is clear that all of these noise sources are relatively unimportant compared with the effects of the fluctuations in temporal structure associated with multimode pump pulses.
The nonlinear nature of stimulated Raman scattering has a significant effect on the Stokes pulse energy output statistics. Our measurements and simulations indicate that the energy of the output Stokes pulses depends very sensitively on the classical noise, in the form of the pump pulse temporal modulations. In the particular case studied here, multi-mode Q-switched pump pulses with stable energies produced Stokes pulses whose energy distributions displayed a relative noise as large as 220%. For multi-mode pump pulses, we identified an intermediate scattering regime, near the onset of stimulated scattering for the first Stokes order, for which the output energy was extremely sensitive to variations in the pump pulse temporal structure. It is precisely in this regime that stimulated scattering interacts with the classical pump noise to convert amplitude variation into energy fluctuations. This extreme sensitivity to classical pump noise was also reflected in the evolution of the Stokes pulse energy probability distribution. It should be noted that this same sensitivity to classical pump noise will also occur for the higher order Stokes pulses near the onset of stimulated scattering into each order. In our earlier work, we observed a prominent local maximum in the first Stokes relative noise near the onset of second order scattering, which is probably a direct consequence of this observation . In fact, because of the sequential nature of multi-order SRS, the higher orders should display even more dramatic pulse energy fluctuations and changes in their probability distributions. For example, the second Stokes order will be pumped by first Stokes pulses that fluctuate in temporal structure and in total energy even if the pump pulses have completely stable energies.
The excellent agreement between a model that includes only the classical pump noise and the experimental results indicates that amplified quantum initiation noise is not a significant contributor to Stokes pulse energy variations for the pump power ranges considered in this work. This conclusion is reinforced by the fact that SRS using single-mode pump pulses with smooth temporal envelopes results in relative Stokes pulse energy noises that are roughly an order of magnitude smaller than the noises observed using multi-mode pump pulses. While quantum noise is essential to initiate the scattering, it is not very important in driving fluctuations in the total Stokes pulse energy because the scattered pulse widths are much longer than the Raman response time of the glass which sets the characteristic time scale for quantum fluctuations. This conclusion is consistent with earlier work in Raman generation in gases, and it is explicitly supported by recent experiments where variable fractions of waveforms generated by pulsed Brillouin scattering were integrated and statistically analyzed. Large scale energy fluctuations gave way to relatively stable pulse energies as the integration time grew substantially larger than the phonon damping time . We have clearly identified and analyzed the impact of fluctuations in the fine temporal structure of Q-switched pulses on SRS; this knowledge is potentially beneficial in a wide range of nonlinear optics applications were Q-switched lasers are still used as pump sources.
This work was supported by National Science Foundation grant PHY-0140305, a summer research grant from the College of Liberal Arts and Sciences at DePaul University, and paid research leave from the University Research Council of DePaul University.
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