Derivation of Raman threshold formulas for CW double-clad fiber amplifiers

Cesar Jauregui; Jens Limpert; Andreas Tünnermann

doi:10.1364/OE.17.008476

1. Introduction

In recent years the output power of fiber laser and amplifier systems has risen exponentially [1]. This allows having nowadays commercially available systems which offer output powers that were unimaginable just a few years ago. Thus, systems with 6kWatt output power out of a single mode fiber have already been demonstrated [2]. This extremely rapid scaling of the output power in conjunction with the unique characteristics of fiber systems (i.e. excellent thermal management capabilities, excellent beam quality, virtually maintenance-free, and small-size) is little by little making this technology the preferred choice for some industrial applications, such as non-contact cutting and welding. At the same time the drive of the industrial applications is encouraging further scaling up of the output powers in fiber systems.

There are some limits to the maximum achievable output power in fiber systems, though. The ultimate limit is given by the damage threshold of the optical medium [3] that, if surpassed, will physically destroy the fiber. However, fiber systems are still operating far away from that limit. Another serious limitation that threatens to cap the current growth rate of output power is the onset of non-linear effects. The same geometry that gives optical fibers many of their advantages (namely long lengths and small diameters) is responsible for the existence of long interaction lengths that favour the appearance of non-linear effects [4]. The most limiting of these effects is the stimulated Raman scattering (SRS) because it normally presents the lowest onset threshold. Once that this threshold is reached the Raman scattering process steadily transfers energy from the signal wavelength to a longer wavelength region. From a practical point of view this imposes a maximum limit to the achievable output power in fiber laser and amplifier systems.

Since the Raman threshold is dependent on the fiber characteristics, several improved fiber designs have been recently proposed. Many of these designs are based either on modifying certain guiding properties of the fiber [5] or on scaling up the core area [6]. However, these speciality fibers are starting to become difficult to work with and are normally very expensive. Thus, even though for laboratory demonstrations a fiber can be oversized to make sure that non-linear effects have no impact, for real applications it is becoming more and more important to extract the maximum possible power from each fiber design for practical and economical reasons. Unfortunately, up to now this maximum possible power was very difficult to determine beforehand and its estimation was a matter of experience and/or computer simulations. This is because, contrary to the case of passive fibers, no analytic formula was available to calculate the Raman threshold in active fibers.

In this paper we present, for the first time to our knowledge, the derivation of an analytical Raman threshold formula for CW double-clad fiber amplifiers. The paper is divided as follows: in section 2 we discuss the unsuitability of the classic Raman threshold formula [7] when used for active fibers, in section 3 we present the theoretical derivation of the new Raman threshold equations, in section 4 some approximate equations are presented, and in section 5 the formulas are used to study the dependence of the Raman threshold on several parameters, finally some conclusions are drawn.

2. Classic Raman Threshold formula applied to high-power CW fiber amplifiers

The classic SRS threshold formula as obtained by R.G. Smith [7] is:

P_{th} = 16 \cdot A_{eff} / g_{R} L_{eff}

where A_eff is the effective mode area of the fiber, g_R is the Raman gain coefficient, and L_eff = (1- e ^-α_pL)/α_p is the effective fiber length, with α_p standing for propagation losses.

To date, Eq. (1) is the only available formula to easily estimate the Raman threshold in an optical fiber. Therefore, even though this expression was originally obtained for passive fibers (and no attempt was made to extent its applicability to active fibers), it might be tempting using it for fiber amplifiers simply by substituting α_p with the fiber gain rate in the effective length term. However, there are many reasons that suggest that this simple approach doesn’t work. On the one hand, the formula above, when applied to fiber amplifiers, assumes a perfect exponential signal growth. This would restrict the usability of Eq. (1) to unsaturated contra-directionally pumped amplifiers. On the other hand, the classic formula assumes that the Raman scattering undergoes the same loss/gain (due to the active fiber) as the signal to be amplified. This is clearly not true in a conventional fiber amplifier since the wavelength shift of the Raman scattering ensures that the fiber cross-sections are very different at the signal and Raman wavelengths (see [8], for example). Furthermore, the application of the classic equation to fiber amplifiers overlooks the effect of the amplified spontaneous emission (ASE) on the initiation of the Raman process. This is a very gross approximation since the combination of the ASE and the fiber gain at the Raman Stokes wavelength are two of the factors that influence the Raman threshold the most [9]. All these facts make the simple classic formula too inaccurate to be of any practical use for fiber amplifiers.

There are also several practical constraints that further limit the usability of Eq. (1) in the context of fiber amplifiers. One of them is the fact that the threshold is defined as the input signal power at which the output powers of both signal and Raman scattering are equal. This definition is clearly unsuitable for fiber amplifiers since it means that at the Raman threshold the signal has already been strongly depleted. This implies that there is a need to redefine the Raman threshold level and to find a formula that is able to predict it accordingly. Another practical limitation is the fact that in Eq. (1) the threshold is calculated for the input signal power. In the context of fiber amplifiers it would be much more natural to define the Raman threshold as a certain level of pump power for any given input signal. Moreover, the classic formula does not use the pump power but employs the fiber gain/loss instead. The latter can be difficult to know beforehand and would possibly require the use of simulation tools which would ultimately render the formula superfluous (since the very purpose of its use is to avoid simulations).

3. Derivation of the Raman threshold formulas for high-power CW fiber amplifiers

Typically a fiber amplifier can have three different configurations according to the injection of the pump power: co-directional or counter-directional with the signal power, or bidirectional. In each of these configurations the evolution of the signal power inside of the active fiber is very different, which ultimately leads to very different Raman thresholds [9]. This implies that the Raman threshold formula has to be necessarily different in each of these cases. Thus, in the following the co-directional and counter-directional configurations will be treated independently. The bi-directional pumping configuration will not be explicitly treated in the paper. However, one approach that seems to cast good results to roughly estimate the value of the Raman threshold in this situation is to use a total pump power equal to the average value of the Raman threshold in the co- and counter-propagating directions, distributed in such a way that the pump in each direction is not higher than its corresponding Raman threshold.

The simplest form of the Raman interaction can be described by [4]:

\frac{d P_{signal}}{dz} = \frac{- ω_{signal}}{ω_{R}} g_{A} \frac{P_{R}}{A_{eff}} P_{signal} - α_{signal} (z) P_{signal}

where P_signal and P_R are the signal and Raman scattering powers respectively, ω_signal and ω_R are the angular frequencies of the signal and Raman Stokes, A_eff is the effective modal area of the fiber core (considered always as single-mode), g_R is the Raman gain coefficient (that should take into account if the signal and Raman Stokes polarizations are well aligned over the fiber, i.e. PM fiber, or not [10]), α_signal represents the z-dependent gain (if negative) or losses (if positive) for the signal and α_R is related to the gain/losses at the Raman Stokes wavelength.

Taking into account that in an amplifier the Raman threshold should be set at a relatively low power of the Raman scattering to avoid significant signal depletion, then Eq. (2) can be solved as:

P_{signal} (z) = P_{{signal}_{o}} f (z)

with P_{Signal o} being the input signal power to the amplifier, P_{R o} representing an equivalent initial power for the Raman scattering process (related with the ASE and the level of spontaneous Raman scattering) and f(z) describing the power evolution of the signal along the fiber. Additionally, it can be said that the integral on f(z) acts as an effective length (L_eff) for the Raman scattering process (which is actually valid for any non-linear process taking place in the amplifier), and the integral on α_R acts as an effective gain/loss coefficient at the Raman Stokes wavelength.

From Eq. (3) a general expression for the Raman threshold can be obtained simply by considering that the Raman power at the output of the fiber should equal a certain fraction of the output signal:

P_{R_{o}} e^{g_{R} \frac{P_{signal}}{A_{eff}} L_{eff} - γ_{R}} = \frac{P_{signal} (L)}{β}

where 1/β is the factor that selects the level of the Raman threshold with respect to the output signal power (e.g. 0.1 for a Raman output power that is 10% of the signal output power), and γ_R stands for the integral on α_R.

Thus, in the following, the task consists on determining the expressions for the unknown parameters of Eq. (4). These parameters are, namely, P_signal(L), L_eff, γ_R and P_{R o}. As commented before, the expressions for these parameters are dependent on the pump configuration of the amplifier. Therefore, from this point on, the co- and counter-propagating pump cases will be studied separately.

3.1. Co-directional pumping configuration

The easiest model for a CW fiber amplifier (ignoring Raman scattering and ASE) is [11]:

N = N_{1} + N_{2}

In these equations I_p and I_Signal are the pump and signal intensities respectively where the + or − superscripts represent the propagation direction. N is the doping-ion concentration, N₂ is the population density of the excited state and N₁ is the population density of the ground state. σ_aj and σ_ej are the absorption and emission cross-sections of the doping-ions respectively at the wavelength λ_j (with j=p or signal, to denote pump or laser signal). τ is the spontaneous emission lifetime, A is the doped area of the fibre, A_eff is the effective modal area of the laser signal and A_{eff p} is the effective area of the pump signal. Γ_p and Γ_signal are the overlapping factors between the doped area and the pump or the laser radiation respectively. σ_p and σ_signal are the attenuation factors of the fibre at the pump and laser signal wavelengths. h is the Planck constant, and c the speed of light in vacuum.

Equation (5) is valid for both the co- and counter-directional pump configurations. In order to select one or the other it is necessary to choose either the pump intensities with + (co-propagating) or with – superscripts (counter-propagating). Moreover, Eq. (5) can be solved analytically considering that I_signal ≫ I_pump all along the fiber. This approximation is reasonable for double-clad fibers since, in them, the pump core area is typically tens of times larger than the signal core area. Thus, in this case, neglecting the propagation losses at the pump wavelength and taking into account that P_i = I_i * A_{eff i} , the solutions for the co-propagating case are:

P_{p} (z) \approx {P_{p}}_{o} e^{- ζ (L - z)}

where the constants C and ζ are given by the following expressions:

ζ = - Γ_{p} N \frac{({σ_{e}}_{signal} {σ_{a}}_{p} - {σ_{a}}_{signal} {σ_{e}}_{p})}{{σ_{e}}_{signal} + {σ_{a}}_{signal}}

With Eq. (6) and Eq. (7) it is possible to immediately calculate the value of the signal at the fiber output (as required by Eq. (4)). Thus:

P_{signal} (L) = \frac{C \cdot A_{eff}}{α_{signal} + ζ} (e^{- ζ \cdot L} - e^{- α_{signal} L}) + {P_{signal}}_{o} e^{- α_{signal} L}

But the use of Eq. (6) and Eq. (7) also allows obtaining an expression for the effective length of the fiber amplifier:

L_{eff} = \frac{\int_{0}^{L} I_{signal} (z) dz}{{I_{signal}}_{o}} = \frac{C \cdot (ζ \cdot e^{- α_{signal} L} - α_{signal} e^{- ζ \cdot L} + α_{signal} - ζ)}{{I_{signal}}_{o} (α_{signal} - ζ) α_{signal} ζ} + \frac{1 - e^{- α_{signal} L}}{α_{laser}}

Note that in this formula the second term corresponds with the classic expression of the effective length for passive fibers [4]. The reader should also notice that the effective length given in Eq. (9) is not only restricted to the context of Raman scattering, but it is valid for any kind of non-linear process that might take place in a CW fiber amplifier (like Brillouin scattering, for example).

Finally, from Eq. (6) and Eq. (7) we can also obtain the average value of the signal and pump powers along the fiber. This value will be very useful later. Thus, considering that the passive effective length of the fiber is approximately equal to its physical length (which always happens in a fiber amplifier since the length of the fiber is of the order of tens of meters at most), and taking into account that the propagation losses at the signal wavelength are normally very small (α_signal≪ζ), it can be shown that the average value of the signal and pump powers along the fiber are given by:

P_{p_{ave}} = \frac{{P_{p}}_{o} (1 - e^{- (ζ + α_{p}) L})}{(ζ + α_{p}) L}

Now, to complete Eq. (4), there are only two unknown parameters left: γ_R and P_{R o}. In order to calculate the value of γ_R, that represents the fiber gain/loss at the Raman Stokes wavelength, the complete differential equation that governs the growth of the Raman Stokes signal should be used (i.e. an expanded version of Eq. (2)):

\frac{d P_{R}}{dz} = g_{R} \frac{P_{signal}}{A_{eff}} P_{R} - Γ_{R} [{σ_{a}}_{R} N_{1} - {σ_{e}}_{R} N_{2}] P_{R} - {α'}_{R} P_{R}

where α′_R represents the propagation loss coefficient of the fiber at the Raman Stokes wavelength, Γ_R is the overlapping factor between the doped area and the mode at the Raman Stokes wavelength, and σ_aR and σ_eR are the absorption and emission cross-sections at the Raman Stokes wavelength respectively. This last equation can be analytically solved assuming that the Raman Stokes is small enough as not to have a big influence on the inversion profile of the fiber. Thus, the output power of the Raman scattering is given by:

P_{R} (L) = {P_{R}}_{o} . e^{g_{R} \frac{P_{signal}}{A_{eff}} L_{eff}} . e^{Γ_{R} N \frac{{σ_{e}}_{R} {σ_{a}}_{p}}{{σ_{e}}_{signal} + {σ_{a}}_{signal}} \int_{0}^{L} \frac{I_{p} (z)}{I_{signal} (z)} dz - Γ_{R} N \frac{{σ_{e}}_{signal} {σ_{a}}_{R} - {σ_{e}}_{R} {σ_{a}}_{signal}}{{σ_{e}}_{signal} + {σ_{a}}_{signal}} L - {α'}_{R} L}

where:

\int_{0}^{L} \frac{I_{p} (z)}{I_{signal} (z)} dz = \frac{A_{eff}}{ζ A_{{eff}_{p}}} Ln (1 + \frac{{P_{p}}_{o}}{{P_{signal}}_{o}} - \frac{({P_{signal}}_{o} + {P_{p}}_{o}) e^{- ζL}}{{P_{signal}}_{o} (1 + \frac{{P_{signal}}_{o}}{{P_{p}}_{o}})})

Comparing Eq. (12) with Eq. (3), it can be seen that γ_R corresponds to the exponent of the second exponential term.

Finally, the only parameter left to calculate is P_{R o}, which is related to the spontaneous photon emissions at the Raman Stokes wavelength due to both ASE and spontaneous Raman scattering. In order to obtain an expression for this parameter it is necessary to write and solve the differential equation that governs the evolution of the power of the Raman scattering along the fiber including the spontaneous sources. Thus, using Eq. (11), adding the appropriate spontaneous emission terms, and considering that at the usual Raman Stokes wavelengths σ_aR ≈ 0, we obtain:

\frac{d P_{R}}{dz} = g_{R} \frac{P_{signal}}{A_{eff}} (P_{R} + {P_{R}}_{spon}) + Γ_{R} {σ_{e}}_{R} N_{2} P_{R} - {α'}_{R} P_{R} + \frac{2 {σ_{e}}_{R} N_{2} h c^{2} Δ λ_{ASE}}{λ_{R}^{3}}

In this equation γ_R represents the Raman Stokes wavelength and ∆λ_ASE is the ASE bandwidth at the Raman Stokes wavelength (normally coinciding with the Raman gain bandwidth, i.e. ~5nm after [11]). P_{R spon} is given by [7]:

{P_{R}}_{spon} = \frac{hc}{λ_{R}} B_{eff}

where B_eff is the effective bandwidth of the Raman gain (~1 THz). The problem at this point is that Eq. (14) has no analytical solution because both N ₂ and P_signal vary along the length of the fiber. Therefore, in order to obtain the desired analytical solution an approximation is required. In this case N ₂ and P_signal will be approximated by their average values and they will be considered constant along the fiber. Thus, the solution of Eq. (14) becomes:

P_{R} (z) \approx \frac{\frac{2 {σ_{e}}_{R} {N_{2}}_{ave} h c^{2} Δ λ_{ASE}}{λ_{R}^{3}} + g_{R} \frac{{P_{signal}}_{ave}}{A_{eff}} {P_{R}}_{spon}}{g_{R} \frac{P_{{signal}_{ave}}}{A_{eff}} + Γ_{R} {σ_{e}}_{R} {N_{2}}_{ave} - {α'}_{R}} e^{(g_{R} \frac{{P_{signal}}_{ave}}{A_{eff}} + Γ_{R} {{σ_{e}}_{R} N}_{2}_{ave} - {α'}_{R}) z}

with P_{signal ave} given by Eq. (10) and N ₂ ave being calculated from Eq. (5) by substituting P_signal and P_p by P_{signal ave} and P_{p ave} respectively. In Eq. (16) the term multiplying the exponential represents P_{R o}. Therefore, at this point all the terms required by Eq. (4) to calculate the Raman threshold have already been calculated. Thus, putting all the terms together we arrive at the following expression for the Raman threshold in the co-propagating pump configuration:

\frac{\frac{2 {σ_{e}}_{R} {N_{2}}_{ave} h c^{2} Δ λ_{ASE}}{λ_{R}^{3}} + g_{R} \frac{{P_{signal}}_{ave}}{A_{eff}} {P_{R}}_{spon}}{g_{R} \frac{P_{{signal}_{ave}}}{A_{eff}} + Γ_{R} {σ_{e}}_{R} {N_{2}}_{ave} - {α'}_{R}} e^{g_{R} \frac{{P_{p}}_{o} [\frac{α_{signal}}{ζ} (e^{- ζ L} - 1) - (e^{- α_{signal} L} - 1)]}{α_{signal} . A_{eff}} + g_{R} \frac{{P_{signal}}_{o} L}{A_{eff}} - γ_{R}} \approx \frac{{P_{p}}_{o} (e^{- α_{signal} L} - e^{- ζL}) + P_{{signal}_{o}} e^{- α_{signal} L}}{β}

This new proposed formula for the Raman threshold has to be solved iteratively much in the same fashion as the original exact formula of the Raman threshold for passive fibers given in [7]. However, one particularity of the equation above is that it can be solved either for the pump or for the signal power. This means that the formula allows obtaining the Raman threshold as a certain pump power for a fixed input signal, or as a signal power for a fixed pump power.

3.2. Counter-directional pumping configuration

In the following, the final formulas for the different parameters in the counter-directional pumping configuration will be directly presented since their derivation is identical to that presented in the previous section. Thus, the signal power at the output of the amplifier is given by:

P_{signal} (L) = \frac{C \cdot A_{eff} \cdot e^{- ζ \cdot L}}{α_{signal} + ζ} (e^{ζ \cdot L} - e^{- α_{signal} L}) + P_{{signal}_{o}} e^{- α_{signal} L}

And the effective length of the fiber amplifier in the counter-propagating pump configuration is:

L_{eff} = \frac{C \cdot e^{- ζ \cdot L} (ζ \cdot e^{- α_{signal} L} + α_{signal} e^{ζ \cdot L} - α_{signal} - ζ)}{{I_{signal}}_{o} (α_{signal} + ζ) α_{signal} ζ} + \frac{1 - e^{- α_{signal} L}}{α_{laser}}

When comparing Eq. (19) with Eq. (9) it can be seen that L_eff is smaller in the counter-propagating pump configuration, which means that the Raman scattering and any other nonlinear effect will have a higher threshold (as also reported in [9]). This is because, in this configuration, the amount of average signal energy in the fiber is smaller, as revealed by the following equations:

{P_{p}}_{ave} = \frac{{P_{p}}_{o} (1 - e^{- (ζ + α_{p}) L})}{(ζ + α_{p}) L}

Additionally, the expressions for γ_R and P_{R o} are still given by Eq. (12) and Eq. (16) respectively, with the only exception that now:

\int_{0}^{L} \frac{I_{p} (z)}{I_{signal} (z)} dz = \frac{A_{eff}}{ζ {A_{eff}}_{p} (1 + \frac{{P_{signal}}_{o}}{{P_{p}}_{o}} e^{ζ . L})} Ln (\frac{(\frac{{P_{signal}}_{o}}{{P_{p}}_{o}} - 1) e^{- ζ . L} + (\frac{{P_{signal}}_{o}}{{P_{p}}_{o}} e^{ζ . L} + 1) . (1 - \frac{{P_{p}}_{o}}{{P_{signal}}_{o}})}{(\frac{{P_{signal}}_{o}}{{P_{p}}_{o}} - 1) . (1 + e^{- ζ . L})})

Then, combining all these elements into Eq. (4), the following expression for the Raman threshold in the counter-propagating pump configuration can be obtained:

\frac{\frac{2 {σ_{e}}_{R} {N_{2}}_{ave} h c^{2} Δ λ_{ASE}}{λ_{R}^{3}} + g_{R} \frac{{P_{signal}}_{ave}}{A_{eff}} {P_{R}}_{spon}}{g_{R} \frac{{P_{signal}}_{ave}}{A_{eff}} + Γ_{R} {σ_{e}}_{R} {N_{2}}_{ave} - {α'}_{R}} e^{g_{R} \frac{{P_{p}}_{o} e^{- ζ . L} [\frac{α_{signal}}{ζ} (e^{ζ . L} - 1) + e^{- α_{signal} L} - 1]}{α_{signal} . A_{eff}}} + g_{R} \frac{{P_{signal}}_{o} L}{A_{eff}} - γ_{R} \approx \frac{{P_{p}}_{o} (1 - e^{- ζL}) + {P_{signal}}_{o} e^{- α_{signal} L}}{β}

4. Approximations of the Raman threshold formulas

Even though Eq. (17) and Eq. (22) predict the Raman threshold quite accurately, the authors acknowledge that their complexity can reduce their applicability. Sometimes it is enough to have a simple formula that gives a quick estimate of the maximum pump power that can be used in the system even if it has a lower degree of accuracy. Conscious of this fact, in the following two different approximations for the Raman threshold formulas with varying degree of accuracy and complexity will be presented. These approximations do not require iterative calculation to obtain the result. One restriction is, though, that the formulas have been solved for the pump power, which means that the Raman threshold is exclusively obtained as a certain pump power level for a given signal input power (losing in this way the flexibility given by Eq. (17) and Eq. (22)).

4.1. Approximations for the co-directional pumping configuration

It can be shown that if P_{p o}≫P_{signal o}, Eq. (17) becomes:

2 β {σ_{e}}_{R} {\overset{̅}{N}}_{2}_{ave} h c^{2} \frac{Δ λ_{ASE}}{λ_{R}^{3}} \approx {P_{p}}_{o} (e^{- α_{signal} L} - e^{- ζL}) e^{- g_{R} \frac{P_{p} [\frac{α_{signal}}{ζ} (e^{- ζ L} - 1) - e^{- α_{signal} L} - 1]}{α_{signal} . A_{eff}}} - g_{R} \frac{{P_{signal}}_{o} L}{A_{eff}} + γ_{R}

with

{\bar{N}}_{2}_{ave} = \frac{({σ_{a}}_{p} α_{signal} A_{eff} + {σ_{a}}_{signal} {A_{eff}}_{pump} (ζ + α_{p}) [\frac{α_{signal}}{ζ} (e^{- ζ . L} - 1) - (e^{- α_{signal} . L} - 1)]) N}{({σ_{a}}_{p} + {σ_{e}}_{p}) α_{signal} A_{eff} + ({σ_{a}}_{signal} + {σ_{e}}_{signal}) {A_{eff}}_{pump} (ζ + α_{p}) [\frac{α_{signal}}{ζ} (e^{- ζ . L} - 1) - (e^{- α_{signal} . L} - 1)}

However, Eq. (23) still depends on γ_R which makes it difficult to approximate. If a non-iterative formula is required, then γ_R has to be approximated by a constant value $\bar{γ}$ _R. This can be done by considering that γ_R is a function which starts at a certain value for low pump powers and then tends asymptotically to another one for larger pumps. We have found that the average of these two extreme values of γ_R works well when trying to approximate this function by a constant. Thus,

{\overset{̅}{γ}}_{R} = \frac{γ_{R} ({P_{p}}_{o} = {P_{siganl}}_{o}) + γ_{R} ({P_{p}}_{o} = 1000 * {P_{signal}}_{o})}{2}

On the other hand, it can be observed that the right-hand side of Eq. (23) follows the form f(x)=xBe ^-Ax+C . It can be shown that, in the range of parameters of interest, this function can be well approximated by a Boltzmann function with the following parameters:

f (x) = xB e^{- Ax + D} \approx \frac{A_{1}}{1 + e^{(x - x_{o}) / dx}} with {\begin{array}{c} A_{1} = \frac{0.465 e^{D} B}{A} \\ x_{o} = \frac{2.21 B}{A} \\ dx = \frac{1.07}{A} \end{array}

Thus, using Eq. (24), Eq. (25) and Eq. (26) in Eq. (23), and solving for P_p, the following approximation for the Raman threshold in the co-propagating pump case is obtained:

P_{p} \approx \frac{A_{eff} α_{signal}}{g_{R}} \frac{2.21 (e^{{- α}_{signal} L} - e^{- ζL}) + 1.07 Ln (\frac{e^{- g_{R} \frac{{P_{signal}}_{o} L}{A_{eff}} + {\bar{γ}}_{R}} (e^{- α_{signal} L} - e^{- ζL}) A_{eff} α_{signal}}{β {σ_{e}}_{R} {\overset{̅}{N}}_{2}_{ave} g_{R} [\frac{α_{signal}}{ζ} (e^{- ζ . L} - 1) - (e^{- α_{signal} L} - 1)]}) + 15}{\frac{α_{signal}}{ζ} (e^{- ζ . L} - 1) - (e^{- α_{signal} L} - 1)}

This last formula can be directly solved without the need of iterations. However, it might still feel quite complicated (even though it can be easily programmed in any scientific calculator). For those cases where only a rough estimate of the Raman threshold is required, another much simpler formula can be obtained. This formula comes from the intuitive idea that the Raman threshold would be approximately reached when the average signal power in the fiber (given by Eq. (10) in this case) equals the classical Raman threshold for passive fibers obtained by R.G. Smith in [7]. The accuracy of this approach is very low since it does not take into account either the gain at the Raman Stokes wavelength or the contribution of the ASE to the initiation of the Raman scattering. Hence, this approximation will only work on those situations where γ_R is not too high, i.e. the formula below should not be used when operating a system near the lower wavelength range of the gain bandwidth and with high doping concentrations. Additionally, as will be seen in the following sections, this approximation provides a very conservative value of the Raman threshold. This is because, given the nonlinear dependence of the Raman scattering on the signal intensity and fiber length, the propagation of a high constant power along the whole fiber length represents a worse case than a small initial signal that grows along the fiber (and which average power equals that of the constant signal) because the latter will only generate significant Raman in a much shorter section of the fiber. On the other hand, in this approach the parameter β that controls the percentage of output signal power transferred to the Raman Stokes at the threshold is lost. In this sense, the formula is only able to provide a pump power value at which it is still safe to operate the system without having any detrimental effect created by the Raman scattering. In spite of the limitations listed above, we feel that this formula, given its simplicity, can still be interesting to a group of people that only want to assess whether their system will be operating free of Raman. Therefore, using Eq. (1) and Eq. (10) this simple approximation takes the form:

{P_{p}}_{o} \approx \frac{(\frac{16 A_{eff}}{g_{R} L} - {P_{signal}}_{o}) L . α_{signal}}{[\frac{α_{signal}}{ζ} (e^{- ζ . L} - 1) - (e^{- α_{signal} L} - 1)]}

4.2. Approximations for the counter-directional pumping configuration

For the counter-propagating pump configuration it is possible to obtain the equivalent approximated formulas by following a similar procedure as that described above. Therefore, in this section we will simply present the final approximate equations equivalent to Eq. (27) and Eq. (28). Thus, the first and most accurate approximation of Eq. (22), considering that P_{p o}≫P_{signal o}, is:

Fig. 1. CW fiber amplifier operating at the Raman thresholds as calculated with the new proposed formulas for a) co-propagating and b) counter-propagating pump configurations.

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P_{p_{o}} \approx \frac{A_{eff} α_{signal}}{g_{R}} \frac{2.21 (1 - e^{- ζL}) + 1.07 Ln (\frac{e^{- g_{R} \frac{{P_{signal}}_{o} L}{A_{eff}} + {\overset{}{\bar{γ}}}_{R}} (1 - e^{- ζL}) A_{eff} α_{signal}}{β {σ_{e}}_{R} {\bar{N}}_{2}_{ave} g_{R} [\frac{α_{signal}}{ζ} (e^{ζ . L} - 1) + (e^{- α_{signal} L} - 1)] e^{- ζ . L}}) + 15}{\frac{α_{signal}}{ζ} (e^{ζ . L} - 1) + (e^{- α_{signal} L} - 1) e^{- ζ . L}}

And the simpler approximation, based on the classical Raman threshold formula for passive fibers, is:

{P_{p}}_{o} \approx \frac{(\frac{16 A_{eff}}{g_{R} L} - {P_{signal}}_{o}) L . α_{signal}}{[\frac{α_{signal}}{ζ} (e^{ζ . L} - 1) + (e^{- α_{signal} L} - 1)] e^{- ζ . L}}

Note that Eq. (30) is subject to the same limitations of Eq. (28) and therefore, its use is mainly restricted to the longer wavelength region of the gain bandwidth.

5. Simulations and discussion

In this section the accuracy of the Raman threshold formulas given above will be discussed using a practical example. Later on, once their accuracy has been investigated, the approximate formulas will be used to study the dependence of the Raman threshold in double-clad fiber amplifiers on the fiber length and the doping concentration.

5.1. Accuracy of the Raman threshold formulas

In order to have a working example to evaluate the accuracy of the Raman threshold formulas derived in the preceding section, we have chosen to simulate a 5m long Yb-doped fiber amplifier. The characteristics of the fiber are the following: it is a polarization maintaining fiber with 6μm signal core, 125μm pump core, 1*10²⁶ ions/m³ doping concentration and 1.1*10^-13 m/W Raman gain coefficient. The pump wavelength is 976nm and the signal wavelength 1030nm. The input signal power to the amplifier is set to 50 Watt. In order to carry out the simulations a model based on that described in [11] was developed.

To allow having a visual impression of the accuracy of Eq. (17) and Eq. (22), the Raman threshold of the fiber amplifier above was calculated with 1/β =0.1 (i.e ten percent of the output power contained in the Raman Stokes component). Under these circumstances the formulas predict a Raman threshold for the pump of 756 Watt in the co-directional pumping case, and of 1843 Watt for counter-directional pump. The results of the simulations using exactly the pump powers predicted by the equations are given in Fig. 1. As can be seen at first sight, the results are visually accurate with the Raman power growing up to the point of being ~10% of the signal power (with the consequent depletion on the signal power). The numerical accuracy of the Raman threshold obtained with Eq. (17) and Eq. (22) can be evaluated by taking into account that the simulations presented in Fig. 1 predict an output Raman level of 56.2 Watt and 194.1 Watt respectively. These output Raman powers represent a ~8.7% and a ~13.5% of their respective output signal powers.

Table 1 presents a more quantitative evaluation of the accuracy of the different Raman threshold formulas obtained in the previous sections. In this case the Raman threshold (also calculated as a pump power level) was obtained for an output Raman Stokes power that was 1% of the signal output power (i.e. 1/β =0.01). Besides, in this case the signal wavelength was set to 1064 nm. The last column of the table represents the Raman output power normalized to the signal output power and expressed as a percentage.

Table 1. Raman threshold predictions with the different formulas for a signal wavelength of 1064nm

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It can be seen that, as in the previous example, the accuracy of the iterative Raman threshold formulas (Eq. (17) and Eq. (22)) is very good, with their results being relatively close to the expected 1%. On the other hand, it is also clear that the approximate formulas give very conservative values for the Raman threshold. However, at 1064 nm they can still be used as a quick way to get an estimate of the Raman threshold. Things are different, though, when the signal wavelength is 1030 nm. Here the impact of ASE on the Raman signal is much higher, and, as said before, under these circumstances the simplest approximation for the Raman threshold (Eq. (28) and Eq. (30)) becomes unsuitable. This can clearly be seen in Table 2.

Table 2 shows that the accuracy of the iterative and first level approximation formulas remains unchanged, whereas, as expected, the simplest approximation cannot be employed anymore (the Raman threshold prediction is far higher than the actual value).

Studying Table 1 and Table 2 it can be seen that, in agreement with the results published in [9], the formulas presented in this paper are able to predict higher Raman thresholds for the counter-propagating pump configuration and for the longer signal wavelengths.

Table 2. Raman threshold predictions with the different formulas for a signal wavelength of 1030nm

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In spite of the good results generally obtained with the iterative formulas, it should not be forgotten that they are also based on an approximation (i.e. I_signal ≫ I_pump) and, therefore, they are also subject to limitations. Thus, these formulas loose their accuracy when predicting the Raman threshold for fibers with relatively large core areas and relatively small cladding diameters (for example a fiber with 20 μm core and 125 μm cladding diameters). However, amplifiers based on these fibers normally exhibit Raman thresholds that are well beyond the fiber damage threshold (because of the large core area and their short lengths) and, therefore, they can usually be considered Raman-free.

Another important point to take into account is that the Raman threshold formulas obtained in this paper are only valid when the fiber amplifier is prepared in such a way that its end facets present null reflectivity. The presence of spurious reflexions in the amplifier (even well below 1%) can dramatically reduce the Raman threshold [9].

5.2. Influence of fiber length and doping concentration on the Raman threshold

The formulas of the Raman threshold presented in this paper allow obtaining results that up to now demanded computationally intensive simulations. Thus, as an example, now it is possible to study the interplay between fiber length (L) and doping concentration (N) regarding the Raman threshold. These parameters are normally related to each other in a fiber amplifier since, for any given doping concentration, there is an optimum length at which the amplification efficiency is maximized. Even though this optimum length depends on many factors and its determination can be far from straightforward [12], in many cases a good approximation is to consider it roughly similar to the fiber absorption length (defined as the fiber length that presents 13dB small signal pump absorption). However, even though from the amplification point of view, similar results and efficiencies can be obtained by interchanging L and N (i.e. using a longer fiber with lower doping concentration or vice versa), this doesn’t have to be the case from the Raman threshold point of view. So the question remains open: what is in general better: using shorter fibers with higher doping concentrations or the other way around?. In order to find the answer to that question Eq. (27) and Eq. (29) can be used. Even though the absolute accuracy of these equations is not as high as that from the iterative formulas, they are able to faithfully predict the evolution of the Raman threshold (as demonstrated by the fact that under different conditions and configurations, as seen in tables 1 and 2, they always produce results with the same amount of error). Therefore, even though the absolute values obtained with these formulas might be too conservative, the trends of the values will be correct.

Thus, Fig. 2 shows the results obtained by using Eq. (27) and Eq. (29) with the same fiber as before but for a changing fiber length and doping concentration. The signal wavelength is 1064nm. As expected, the maps displayed in the figure above show that the higher the doping concentration or the longer the fiber, the lower the Raman threshold. However, the maps clearly show that from a certain level of doping concentration (~5*10²⁵ ions/m³) on, the dependence of the Raman threshold on this parameter is greatly reduced (for both co- and counter-propagating pumping configurations). This suggests that, in general it will be more advantageous to work with higher doping concentrations and shorter fiber lengths.

Fig. 2. Map showing the dependence of the Raman threshold on the fiber length and doping concentration for a signal wavelength of 1064nm for a) counter-propagating and b) counter-propagating pump configuration. The white line represents the absorption length for each doping concentration (13dB total small signal pump absorption).

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Having a closer look at Fig. 2(b) (counter-propagating pump configuration) reveals an interesting behaviour of the Raman threshold. There it can be seen that increasing the doping concentration for a fixed fiber length can raise the Raman threshold (for example the Raman threshold for 5m long fiber amplifier with N=1*10²⁶ is 3257 Watt, and for the same fiber length but N=1.6*10²⁶ the threshold is ~3900Watt). Moreover, this increase in the Raman threshold comes with no penalty of the amplification efficiency, thus allowing the extraction of higher output signal powers from the fiber amplifier. This behaviour might seem counterintuitive at first, but it can easily be explained by the evolution of the signal power along the amplifier. In the example given above, with the higher doping concentration the fiber is much longer than the absorption length (white line). This implies that the pump power remaining at the signal input end is so small that it is not able to generate the necessary inversion required to amplify the signal. Therefore, in the first few meters of the amplifier the signal propagates almost unamplified and it is only in the last half of the fiber that it undergoes strong amplification. On the contrary, with the lower doping concentration the length of the fiber is similar to the absorption length, which implies that the signal will be amplifier from the very beginning of the fiber. The result is that the effective length of the amplifier with the lower doping concentration is longer than that of the amplifier with the higher doping level. Of course this behaviour can only be seen on the counter-propagating configuration, since with a co-propagating pump the signal always undergoes the strongest amplification in the first meters of the fiber. Furthermore, if the signal wavelength changes to, say 1030nm, this behaviour is also not observed (due to the stronger influence of the ASE).

The behaviour of the Raman threshold discussed above, though interesting, has little practical application. This is because, due to practical and economical reasons, a fiber amplifier with a higher doping concentration will almost always be shorter than another one with a lower N, which automatically gives a higher Raman threshold to the fiber amplifier (see white lines in Fig. 2). However, what this behaviour does highlight is the fact that the Raman threshold in fiber amplifiers is an extremely complex parameter that depends on the interplay of many different factors.

On the practical side, from Fig. 2 it can be inferred that the use of amplifiers operating at 1064nm with highly doped fibers tend to reduce the dependence of the Raman threshold on the fiber length. On the other hand, in both the co- and counter-propagating configurations, the relative change of the threshold along the white lines (those representing the absorption length of the fiber) is similar and amounting to ~2.3. This, once more stresses the importance of using shorter highly doped fibers to optimize the Raman threshold of fiber amplifiers.

6. Conclusions

In this paper we have presented the theoretical derivation of a set of analytical formulas able to predict the Raman threshold in CW double-clad fiber amplifiers. A different set of formulas has been obtained depending on the pump configuration of the amplifier (co- other counter-propagating). For each amplifier configuration the set of formulas comprise three equations: one that has to be solved iteratively and two other approximate solutions with a varying degree of accuracy and complexity. It has been shown that the iterative formulas have a high degree of accuracy. Besides, they are very flexible since they allow defining the Raman threshold as a desired power level relative to the output signal power. Furthermore, with these equations the Raman threshold can be obtained as a pump power level for a fixed input signal power or as an input signal power for a fixed pump power. Moreover, these formulas are valid for any pump or signal wavelength. However, the iterative equations present a high degree of complexity. Acknowledging the importance of having simple formulas to quickly estimate the Raman threshold (even at the price of a lower accuracy), we have obtained and presented two different approximate formulas. The first one is an approximation of the iterative equation that works well in all circumstances, but tends to provide a conservative value of the Raman threshold. The second and simplest approximate formula is based on the used of the classic passive Raman threshold equation. This formula can only be used for amplifiers operating in the longer wavelength region of the amplification band. The accuracy and limits of the different Raman threshold formulas have been discussed in the text. Finally, using the new formulas, the dependence of the Raman threshold on the fiber length and doping concentration has been studied. The results show that, to minimize the impact of the Raman scattering, it is in general better to build amplifiers with shorter but highly doped fibers.

Acknowledgments

The authors would like to thank the German Federal Ministry of Education and Research (BMBF) for its financial support of this work through the project 13 N 9100 “FaBri”.

References and links

1. J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron . 13, 537–545 (2007). [CrossRef]

2. D. Gapontsev, IPG Photonics, “6kW CW Single Mode Ytterbium Fiber Laser in All-Fiber Format,” in “Solid State and Diode Laser Technology Review” (Albuquerque, 2008)

3. S. W. Allison, G. T. Gillies, D. W. Magnuson, and T. S. Pagano, “Pulsed laser damage to optical fibers,” Appl. Opt . 24, pp. 3140–3145, (1985). [CrossRef] [PubMed]

4. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, NY, 1995).

5. J. Kim, P. Dupriez, C. Codemard, J. Nilsson, and J. K. Sahu, “Suppression of stimulated Raman scattering in a high power Yb-doped fiber amplifier using a W-type core with fundamental mode cut-off,” Opt. Express 14, 5103–5113 (2006). [CrossRef] [PubMed]

6. J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber lasers,” Opt. Express 14, 2715–2720 (2006) [CrossRef] [PubMed]

7. R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt . 11, 2489–2494 (1972). [CrossRef] [PubMed]

8. R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-Doped Fiber Amplifiers,” IEEE J. Quantum Electron . 33, pp. 1049–1056 (1997). [CrossRef]

9. Y. Wang, “Stimulated Raman scattering in high-power double-clad fiber lasers and power amplifiers,” Opt. Eng . 44, pp. 114202-1–114202-12 (2005). [CrossRef]

10. R. H. Stolen, “Polarization effects in fiber Raman and Brillouin lasers,” IEEE J. Quantum Electron . QE-15, pp. 1157–1160 (1979). [CrossRef]

11. Y. Wang, C. Xu, and H. Po, “Analysis of Raman and thermal effects in kilowatt fiber lasers,” Opt. Commun . 242, 487–502 (2004). [CrossRef]

12. F. Röser, D. N. Schimpf, J. Rothhardt, T. Eidam, J. Limpert, A. Tünnermann, and F. Salin, “Gain limitations and consequences for short length fiber amplifiers,” in OSA Topical Meeting on Advanced Solid-State Photonics (ASSP, 2008), paper WB22.

Signal wavelength=1064 nm, co-propagating pump
Eq.	Raman threshold (Watt)	Signal output (Watt)	Raman output (Watt)	%
(17)	1284	1136	12	1.05
(27)	1128	1014	2	0.2
(28)	1153	1034	2.6	0.25
Signal wavelength=1064 nm, counter-propagating pump
Eq.	Raman threshold (Watt)	Signal output (Watt)	Raman output (Watt)	%
(22)	3257	2777	52	1.87
(29)	2871	2495	8	0.32
(30)	2837	2467	6.6	0.27

Signal wavelength=1030 nm, co-propagating pump
Eq.	Raman threshold (Watt)	Signal output (Watt)	Raman output (Watt)	%
(17)	546	518	5	0.96
(27)	383	381	0.7	0.18
(28)	1214	38.33	1015	2648
Signal wavelength=1030 nm, counter-propagating pump
Eq.	Raman threshold (Watt)	Signal output (Watt)	Raman output (Watt)	%
(22)	1382	1228	18	1.46
(29)	992	907	2	0.22
(30)	2721	182	2120	1165

Signal wavelength=1064 nm, co-propagating pump
Eq.	Raman threshold (Watt)	Signal output (Watt)	Raman output (Watt)	%
(17)	1284	1136	12	1.05
(27)	1128	1014	2	0.2
(28)	1153	1034	2.6	0.25
Signal wavelength=1064 nm, counter-propagating pump
Eq.	Raman threshold (Watt)	Signal output (Watt)	Raman output (Watt)	%
(22)	3257	2777	52	1.87
(29)	2871	2495	8	0.32
(30)	2837	2467	6.6	0.27

Signal wavelength=1030 nm, co-propagating pump
Eq.	Raman threshold (Watt)	Signal output (Watt)	Raman output (Watt)	%
(17)	546	518	5	0.96
(27)	383	381	0.7	0.18
(28)	1214	38.33	1015	2648
Signal wavelength=1030 nm, counter-propagating pump
Eq.	Raman threshold (Watt)	Signal output (Watt)	Raman output (Watt)	%
(22)	1382	1228	18	1.46
(29)	992	907	2	0.22
(30)	2721	182	2120	1165