Competition between forward and backward scatterings
Procedure for optimizing forward Raman efficiency
Our technique to optimize forward Raman scattering is relatively simple. Our starting point is to take into account the fact that both the pump and the forward Raman scattering propagate in the same direction. We are interested in the temporal regime of pulses of a few 10 ps up to a few ns, and with fiber lengths up to a few meters. The temporal propagation delay between the pump and Stokes beams (at 532 nm and 630 nm for instance) due to dispersion is thus much lower than the pulse duration. Temporal walkoffs can thus be neglected. Therefore the interaction length for copropagating effects is equal to the fiber length L. On the contrary the backward scattered beams are initiated by the leading edge of the pump beam and interact until its falling edge. Therefore, for these counterpropagating beams, the interaction length is limited by the pump pulse length to:
$$ {L}_{eff}\approx \operatorname{Min}\left({L}_{\varDelta t}\approx \frac{c\ \varDelta t}{2\ n},\kern1.12em L\right) $$
(2)
c being the celerity of light in vacuum, Δt the temporal fullwidth at halfmaximum of the pump pulse.
Therefore, increasing the fiber length above L
_{
Δt
} should preferentially favor the desired forward Raman scattering over the backward Raman scattering and Brillouin scattering.
Our goal in the following is to estimate the efficiency of this technique and to determine how long the fiber should be to get a large Raman conversion efficiency and a negligible Brillouin scattering. We thus first study the evolution of the Raman conversion efficiency using a well known analytic approach [29]. The idea is to estimate the evolution of the Raman efficiency with the fiber length and pump power. We then include the backscattering effects using numerical simulations.
Forward Raman scattering neglecting backward scattering
Presently, we neglect Raman and Brillouin backward scatterings. We also neglect all transient effects for pulses longer than the relaxation times of the involved optical Raman transitions, that is for pulses that are typically longer than about 10 ps. We thus consider that the pump pulse can be spliced in temporal slices during which the Raman gain coefficient is equal to its value for a continuous beam. We can thus exploit the extensive analyzes of forward Raman scattering presented in the literature and developed for quasicontinuous beams [5].
A convenient parameter to scale a Raman converter is the critical parameter γ
_{
C
} that corresponds to the value γ
_{
C
} of γ = g
_{
R
} I
_{
P
} L
_{
eff
} for which the output Raman Stokes intensity I
_{
R
} equals the transmitted pump beam intensity [29]. In this expression g
_{
R
}is the Raman gain coefficient, I
_{
P
} the input pump beam intensity and L
_{
eff
} the effective interaction length. In case of the forward Raman scattering we are interested in this study, L
_{
eff
} = L, L being the fiber length.
Calculation of γ
_{
C
} is detailed in the literature [29]. We have plotted in Fig. 2 the value of γ
_{
C
} versus the gainlength product in case the Raman medium is ethanol at the pump wavelength of 532 nm. It is noticeable that for a \( {g}_R^f\ L/{A}_{eff}^R \)product spanning over 5 orders of magnitude, the critical parameter changes by a factor of two only. A thumb rule to start designing a Raman converter is to state that γ
_{
C
} ≈ 20. We can also express this feature in terms of pump power required to reach the critical parameter, \( P={\gamma}_C\ {A}_{eff}^R/\left({g}_R^f\ L\right) \): increasing the fiber length by a factor 10^{5} and decreasing the pump power by a factor 2x10^{5} allows keeping the same Raman conversion efficiency.
In conclusion, for smaller fiber length changes, we infer a very simple rule: a decrease of the pump beam power is compensated by an increase of the fiber length by the same amount in order to keep the forward Raman conversion efficiency constant.
We expect the influence of the fiber length to be different for backscattering effects, because the interaction length of counterpropagating beams is limited to L
_{
eff
} by the pulse length (eq. 2). We thus anticipate that the same decrease of the pump beam power cannot be compensated by an increase of the fiber length by the same amount. To assess this prediction, we rely on the following numerical analysis taking into account forward and backward effects.
Coupledwave equations
In the temporal regime we are interested in, with pulses larger than a few tens of picoseconds, we treat the Raman effect in the quasicw regime. We write the couplewave equations for Raman and Brillouin scatterings assuming that: the pump beam is Fourier transformed limited; all beams propagate under a single spatial mode. We are not interested in the noise statistics and thus neglect beam phase fluctuation. Our simulations will thus not be valid before stimulated scattering superseded spontaneous scattering that is at very low pump powers. Furthermore, because of the absence of optical resonant cavities, all scattering processes occur at the maximum of the gain curves so that we can write all equations with real values [30]. A
_{
P
}, A
_{
B
}, \( {A}_R^b \) and \( {A}_R^f \) respectively represent the modules of the slowly varying pump beam amplitude, Brillouin amplitude, backward and forward Raman beam amplitudes. We use normalized values so that ‖A‖^{2} directly represent the powers. All these amplitudes depend on time t and on space coordinate z along the fiber. They do not have any dependence over the transverse coordinates because we assume the beams to be single spatial modes. The transverse dependences are thus taken into account by introducing effective areas. In these equations, we consider that the effective area for the Raman process, \( {A}_{eff}^R \), may differ from the effective area for the Brillouin process, \( {A}_{eff}^B \) [18].
In this set of equations, we do not consider Brillouin scattering of the Raman Stokes waves. We indeed never observed it in our experiments. That is consistent with the fact that the spectral bandwidth of the Stokes waves is much larger than the Brillouin bandwidth. We rely on references [5, 30] to write down these equations:
$$ \left\{\begin{array}{c}\frac{\partial {A}_P}{\partial z}\kern0.5em +\frac{1}{v_g}\frac{\partial {A}_P}{\partial t}=\frac{g_B}{2\ {A}_{eff}^B}\kern0.5em {A}_B\kern0.5em Q\frac{1}{2\;{A}_{eff}^R}\frac{\lambda_S}{\lambda_P}\left({g}_R^f{\left\Vert {A}_R^F\right\Vert}^2+{g}_R^b{\left\Vert {A}_R^b\right\Vert}^2\right){A}_P\\ {}\frac{\partial {A}_B}{\partial z}\kern0.5em \frac{1}{v_g}\frac{\partial {A}_B}{\partial t}=\frac{g_B}{2\ {A}_{eff}^B}\kern0.5em {A}_P\kern0.5em Q\\ {}\frac{2}{\varGamma_B}\frac{\partial Q}{\partial t}=\left(Q{Q}_0\right)+\kern0.5em {A}_P\kern0.5em {A}_B\kern0.5em \\ {}\frac{\partial {A}_R^f}{\partial z}\kern0.5em +\frac{1}{v_g}\frac{\partial {A}_R^f}{\partial t}=\kern0.5em \frac{g_R^f}{2\kern0.5em {A}_{eff}^R}\left({A}_R^fR\right)\kern0.5em {\left\Vert {A}_P\right\Vert}^2\\ {}\frac{\partial {A}_R^b}{\partial z}\kern0.5em \frac{1}{v_g}\frac{\partial {A}_R^b}{\partial t}=\kern0.5em \frac{g_R^b}{2\kern0.5em {A}_{eff}^R}\left({A}_R^bR\right)\kern0.5em {\left\Vert {A}_P\right\Vert}^2\end{array}\right. $$
(3)
λ
_{
S
} represents the wavelength of the Stokes wave. We neglect the difference in wavelengths between the pump and Brillouin waves, both being equal to λ
_{
P
}. Γ
_{
B
} is the acoustic damping rate. Variable Q is proportional to the amplitude of the acoustic wave. We normalized it in order to express the equations versus the steady state Brillouin gain g
_{
B
} [30].
In the equation for Q, we have neglected the space derivative, ∂Q/∂z, as the acoustic velocity v
_{
a
} is much lower than the light group velocity v
_{
g
} [4]. For simplicity we assume v
_{
g
} to be the same for all wavelengths, v
_{
g
} = c/n with n the effective refractive group index and c the speed of light in vacuum.
g
_{
R
}
^{b} and g
_{
R
}
^{f} are the backward and forward Raman gain coefficients that may differ, especially in gases. Indeed, g
_{
R
}
^{b} can be significantly lower than g
_{
R
}
^{f} due to the Doppler shift [31]. They are about the same in liquids.
Raman and Brillouin effects start from noises uniformly distributed along the fiber. These noises are formally introduced here as coefficients R and Q
_{0}. The expression for coefficient R is given in the Additional file 1. Q
_{0}, \( {A}_{eff}^R \) and \( {A}_{eff}^B \)are the only fitting parameters in this set of equations. They are adjusted to best fit the measurements reported later on.
All simulations conducted in this article are done assuming an ethanol filled fiber. For this simulation, we selected parameters that are consistent with the experimental demonstration reported below. We give all details on the derivation of these coefficients in the Additional file 1. They are:
λ
_{
P
} = 532 nm; λ
_{
S
} = 630 nm; ρ = 789 kg. m^{−3}; v
_{
a
} = 1205 m. s^{−1}; n = 1.36; \( {A}_{eff}^R=200\ {\upmu \mathrm{m}}^2 \); \( {A}_{eff}^B=1000\ {\upmu \mathrm{m}}^2 \); \( {g}_R^f={g}_R^b=2.9\ {10}^{12}\ \mathrm{m}.{\mathrm{W}}^{1} \); g
_{
B
} = 80 10^{−12} m. W^{−1}; Γ
_{
B
} = 4.34 10^{9} s^{−1}; R = 1.8 10^{−4} W^{1/2}; Q
_{0} = 1.3 W.
It is worth noting that our normalized parameter Q is related to the normalized parameter S of reference [32] by \( S=Q\;{A}_{eff}^B/{g}_B \) so that Q
_{0} = 1.3 W corresponds to S
_{0} = 0.1 m^{−1}. This value is higher by a factor of about 100 than those cited by [32] for solid. Such a higher value is expected because the noise term is expected to scale as Γ
_{
B
}/v
_{
a
}
^{2} which is indeed about 100 times larger in liquids compared to solids [33].
Numerical simulations
Evolution of the backward Raman scattering
In the undepleted pump beam approximation, Raman forward and backward scatterings evolve exponentially [29] with the exponential factor being respectively \( {g_R}^f\;L\;P/{A}_{eff}^R \) and \( {g_R}^b\;{L}_{eff}\;P/{A}_{eff}^R \). Because significant forward amplification is obtained around the critical parameter \( {\gamma}_c={g}_R^f\ L\kern0.5em P/{A}_{eff}^R\approx 20 \), for which the amplification factor is exp(20) ≈ 10^{8}, only a small relative difference between g
_{
R
}
^{b}
L
_{
eff
} and \( {g}_R^f\kern0.24em L \) may lead to a nearly total vanishing of the backward Raman scattering compared to the forward scattering. In gases for which, due to the Doppler shift, g
_{
R
}
^{b} can be significantly lower than \( {g}_R^f \), backward scattering may be automatically suppressed considering that L
_{
eff
} is bounded by L. In liquids, the two forward and backward Raman gains are about the same. To minimize backward scattering, one should lower the effective length L
_{
eff
} compared to the actual length L.
This is fully confirmed by the simulations and by the experiments. For L ≥ L
_{
eff
}, we never observed significant Raman backscattering, for both experiments and simulations. Therefore we do not discussed anymore the Raman backscattering in this paper.
Analysis of competition between Raman forward and Brillouin backward scattering
For Brillouin scattering, as the gain is much higher, elimination of the backward signal is less evident and we will rely on numerical simulations.
In Fig. 3, we plotted an example of the numerical resolution of the set of eqs. (3). The temporal shape of the pump beam power is Gaussian with a fullwidth at halfmaximum of the power, Δt, equal to Δt ≈ 0.9 ns. The peak power is P
_{
peak
} ≈ 3 000 W and the fiber length is L = 1 m. Because the temporal shapes of the output beam change as a function of the interaction length, we rely on values integrated over the pulse durations to define the conversion efficiencies. The Raman conversion efficiency and the Brillouin conversion efficiency are thus expressed as:
$$ {\eta}_R=\frac{\mathrm{Transmitted}\ \mathrm{Raman}\ \mathrm{energy}}{\mathrm{Incident}\ \mathrm{pump}\ \mathrm{energy}},\kern0.5em \mathrm{and}\kern0.5em {\eta}_B=\frac{\mathrm{Reflected}\ \mathrm{Brillouin}\ \mathrm{energy}}{\mathrm{Incident}\ \mathrm{pump}\ \mathrm{energy}} $$
(4)
In Fig. 3a we represented the pump beam during its propagation through the fiber from z = 0 m to z = L = 1 m. It is strongly depleted by the forward Raman beam shown in Fig. 3b). This strong depletion is in accordance with the value of γ ≈ 21.6 > γ
_{
C
} at the peak of the pump beam. Brillouin backscattered beam is represented in Fig. 3c. For these parameters, forward Raman scattering prevails over backward Brillouin scattering: η
_{
R
} = 56% while η
_{
B
} = 10%. We observe that the Brillouin beam is significant only when the Raman beam is negligible. In order to confirm this competition between Raman and Brillouin scatterings for the same pump beam, we also plotted in Fig. 3d the backward Brillouin beam in case the Raman gain is set to zero. Although the peak power of the Brillouin scattered beam is not slightly modified, around 100 W, the Brillouin pulse lasts longer. Consequently, the Brillouin efficiency is enlarged to η
_{
B
} = 13%.
Evolution of the Raman forward and Brillouin backward scatterings versus fiber length
In order to evidence the evolution of the beam power as a function of the fiber length, we conducted a series of simulations for a constant \( \gamma ={g}_R^f\;{P}_{Peak}\;L/{A}_{eff}\approx 21.6 \) product and by varying the fiber length L. The results are summarized in Fig. 4. If the Brillouin gain is set to 0, Raman efficiency η
_{
R
} is represented by the dotted black line. As anticipated by Fig. 2, the Raman efficiency is nearly independent of the fiber length because γ is kept constant. When Brillouin scattering is present, the Raman efficiency is depressed (red dashed line) for the benefit of the Brillouin scattered energy. Similarly, when the Raman effect is absent, Brillouin efficiency η
_{
B
} is represented by the brown dotdashed line. This Brillouin efficiency is slightly depressed in presence of the Raman effect as shown by the full blue line.
This figure thus illustrates the principle of Brillouin suppression in fiber Raman converters:

for short lengths, Brillouin scattering predominates because the effective length of the Brillouin and Raman effects are comparable and because of the large gain for Brillouin scattering;

for long lengths, the saturation of the effective length for Brillouin scattering favors the competition to the benefit of the Raman conversion process.