Theoretical and numerical treatment of modal instability in high-power core and cladding-pumped Raman fiber amplifiers

Shadi Naderi; Iyad Dajani; Jacob Grosek; Timothy Madden

doi:10.1364/OE.24.016550

1. Introduction

Since the invention of the laser, there has been sustained interest in developing high-power sources with near diffraction-limited beam quality. Ideally, these lasers would have high efficiencies with small footprints. In recent years, rare-earth-doped fiber lasers have emerged as viable candidates for such sources. Despite rapid progress, there remain significant challenges. Dawson et al. have projected an upper limit of 36 kW for fiber lasers limited by combination of stimulated Raman scattering (SRS) and thermal lensing [1]. In ensuing work, Dawson et al. considered fibers based on non-silica materials such as Yb-doped single crystal or ceramic YAG fibers [2]. Despite the promise of further power scaling, significant material development is required in order for lasers based on these fibers to exceed the performance of Yb-doped silica fiber lasers. As it stands today, the highest power obtained from a continuous wave (cw) fiber laser with a reportedly single-mode output stands at 10 kW [3], and it is hard to envision that power scaling to 36 kW will be achieved anytime soon. This is especially true when considering the limitations imposed by the modal instability (MI) [4,5]. This phenomenon, which is rooted in the interplay between thermal effects and modal interference, was not considered in the work of Dawson et al. [1].

Alternatively, Raman fiber lasers have been suggested as a way to overcome some of the limitations of their rare-earth doped counterparts [6–8]. An important consideration with Raman amplification in silica fibers is that the gain is both broad and tunable, providing Raman fiber lasers access to wavelengths that are traditionally inaccessible to rare-earth-doped fiber lasers. Furthermore, since the core of a Raman fiber does not contain rare earth elements, photodarkening is not present and fabrication is relatively easy offering the potential to utilize large area cores while maintaining good beam quality due to the better control of the index of refraction. It is also worth noting that recent work has provided experimental evidence of photodarkening influence on the MI threshold of Yb-doped fiber amplifiers [9]; this is not expected to occur in Raman fiber lasers.

In recent years, there have been several impressive demonstrations of high-power Raman fiber lasers in an amplifier configuration. Raman fiber amplifiers (RFAs) can provide narrow linewidth output [10–12], allowing for further power scaling through beam combining. Most of these demonstrations were performed in core-pumped RFAs. Notably Supradeepa and Nicholson reported on a cascaded RFA whereby an output power of ~300 W (albeit spectrally broadband) was obtained at the 5th order Stokes wavelength with a conversion efficiency of 64% [13]. Zhang et al. demonstrated ~80 W of single-frequency at 1178 nm using a two-stage RFA system [12]. Furthermore, spectrally broad, kilowatt-level, integrated Yb-Raman fiber amplifiers have been reported [14]. In this case, the final stage amplifier was comprised of an Yb-doped large mode area (LMA) fiber with a core diameter of 20 µm and length of 12 m, that was spliced onto a LMA passive fiber with a core diameter of 20 µm and length of 70 m. The Yb-doped fiber was cladding pumped with diodes operating at 976 nm. It was seeded at two different wavelengths sufficiently separated such that one could act as a Raman pump source for the other as the two waves, being confined to the core of the fiber, traverse the passive fiber. More recently, Xiao et al. reported on a fiber amplifier operating at 1123 nm with an output power of 3.89 kW [15]. Here, a 30 m long Yb-doped fiber with a core diameter of 20 µm was used. Amplification was achieved through both the lasing of the Yb ions and the Raman process.

Although core-pumped RFAs can lase at wavelengths inaccessible to rare-earth-doped fibers, they offer no increase in brightness. On the other hand, cladding-pumped Raman amplifiers and oscillators can enhance brightness [16,17] and, in principle, can be pumped with either diodes or other fiber lasers. Despite the promise, the highest power obtained to date from a cladding pumped Raman fiber laser stands at ~100 W [17], limited by a combination of available pump power and the generation of higher-order Raman Stokes. Since amplification in cladding-pumped Raman lasers depends on the cladding size and not the core size, it would appear that there is little benefit in utilizing LMA fibers. However, because the higher-order Stokes light being initiated from noise depends on the intensity of the signal, LMA fibers can be utilized to suppress this conversion. While specialty fibers such as W and photonic bandgap (PBG) fiber can also be utilized, they are relatively expensive and difficult to fabricate. LMA fibers are also beneficial in suppressing stimulated Brillouin scattering (SBS), should a narrow linewidth output be desired for beam combination or other applications.

It is well-known that active LMA fibers are susceptible to MI. Since MI is initiated through the thermo-optics effect originating from the quantum defect heating and/or photodarkening, this phenomenon is also expected (though have not been reported on yet) to occur in LMA RFAs. In the absence of photodarkening, the heat generated derives from the quantum defect due to the Raman conversion process. The resulting spatio-temporal oscillations in the temperature would ultimately lead to an energy transfer from the fundamental mode into the higher-order modes (HOMs), most notably the LP₁₁ mode. The physical mechanism is similar to the energy transfer in two-beam coupling where no transfer occurs if the material responds instantaneously to the optical field. A non-zero temporal response, on the other hand, results in a complex valued perturbed index of refraction which is not in phase with the optical interference pattern [18]. In MI, this temporal response is due to thermal diffusion.

It is also well-known that a fully time-dependent approach to MI is required to completely and accurately capture the temporal dynamics of the process [5,19]. However, due to the long interaction lengths in RFAs, this approach would be computationally exhaustive. Alternatively, one can investigate this problem in the frequency domain, which is the approach we follow here. While this approach is not as powerful as a fully time-dependent one, it nevertheless provides in the case of RFAs accurate predictions of the (relative changes of the) MI thresholds in various configurations. It is a viable approach understanding both the limitations imposed by MI and the effectiveness of mitigation strategies.

In this work we investigate theoretically the MI process in both cladding-pumped and core pumped RFAs. The former is discussed in Section 2 where the MI phenomenon is formulated by solving the heat equation with a Green’s function approach and by propagating the light in the frequency domain with a single frequency offset. Unlike the work of Hansen et al. in Yb-doped fiber amplifiers [20], the method used here yields exact solutions; accounting for pump depletion due to the Raman process and depletion of the fundamental mode signal due to the MI process. This is accomplished by invoking conservation of number of photons and also of energy. The effect of Raman gain tailoring on the MI threshold in cladding-pumped RFAs is also considered in section 2. In section 3, the MI process is considered for core-pumped RFAs where both the pump and signal participate in the writing of index of refraction grating. Here, the effect of dispersion on the MI process is important and is considered in our analysis. Notably, this work on core-pumped RFAs can help in laying the foundation for understanding the MI process in integrated Yb-Raman amplifiers described in [14,15].

2. Cladding-pumped RFAs

2.1 Governing equations

The equations capturing the MI phenomenon in a step-index fiber are derived here for cladding-pumped RFAs. It is assumed that the highly multimode pump light propagates as a plane wave because of nearly instantaneous mode mixing. Moreover, any effects derived from inhomogeneous pump absorption are neglected [21]. These equations are further modified in subsection 2.2 to investigate gain-tailored designs. Thermal lensing effects are being neglected because we are considering cases whereby the mode profiles are not affected by the perturbations in the refractive index. In the case of uniform gain cladding-pumped RFAs, exact solutions for the signal power in each mode and pump power along the fiber are obtained by decoupling the pump and total signal using the conservation of number of photons, and by decoupling the signal modes using conservation of energy in the absence of loss. In order to account for thermal effects, the wave equation is coupled to the temperature equation via the change in refractive index induced by quantum defect heating.

We begin by considering a co-pumped RFA. The fiber core is assumed to be polarization-maintaining and the light is launched along one of its axis. The background loss in neglected with no considerations for differential mode loss due to fiber bending. We take the normalized transverse profiles for a given mode $k$ to be $φ_{s k}$ with the propagation constant being $β_{s k}$ . Since the Raman gain and the variation in the index of refraction are small perturbations, we can construct the electric field of the signal as a superposition of the modes in the “ideal” fiber:

E_{s} (x, y, z) = \frac{1}{2} \sum_{k} A_{s k} (z) φ_{s k} (x, y) e^{i (β_{s k} z - ω_{s k} t)} + c . c .,

where

A_{s k}

is the slowly varying complex amplitude of the signal. By considering the wave equation in the frequency domain and using the approximation

n^{2} = n_{0}^{2} + 2 n_{0} Δ n

, where

n_{0}

is the background refractive index, one can derive a system of equations describing the amplitudes of the modes. Note that the variation in refractive index

Δ n = n - n_{0}

is directly dependent on the temperature difference,

Δ T

, as described by the relation

Δ n = Δ T (d n / d T)

where

d n / d T \approx 1.29 \times 10^{- 5} / ° C

is the thermo-optic coefficient for silica. The change in temperature

Δ T

can be calculated using a Green function.

Since the polarization of the pump light is not maintained, we will use a Raman gain coefficient of $g_{R} = g_{R, 0} / 2$ , where $g_{R, 0}$ is the Raman gain coefficient for co-polarized signal and pump [22]. The heat deposition from the quantum defect in a cladding-pumped RFA with a uniform Raman gain coefficient is given by:

\tilde{Q} = \frac{n_{0} (α - 1)}{2 μ_{0} c} (\sum_{k} {| A_{s k} |}^{2} {| φ_{s k} |}^{2} + \sum_{k \neq l} A_{s k}^{*} A_{s l} φ_{s k}^{*} φ_{s l} \exp [i (Δ β_{s k l} z - Δ ω_{k l} t)] + c . c .) \frac{g_{R} P_{P}}{A_{c l a d}},

where

α = λ_{s} / λ_{p}

. Here

λ_{s}

and

λ_{p}

are the signal and the pump wavelengths, respectively,

A_{c l a d}

is the cladding area,

P_{p}

is the pump power,

Δ β_{s k l} = β_{s k} - β_{s l}

is the difference between the propagation constants of any given two modes, and where the frequency offset is

Δ ω_{k l} = ω_{s k} - ω_{s l} .

Note that the heat in an RFA is deposited over both core and cladding, as opposed to an Yb-doped amplifier where heat is generated inside the core only.

The heat equation in frequency domain is given as:

\nabla_{⊥}^{2} Δ T - q Δ T = - Q / κ_{t h},

where

Q

is the Fourier transform of

\tilde{Q}

,

κ_{t h}

is the thermal conductivity of silica, and

Δ T (r_{⊥}, Ω)

is the temperature gradient between a given point inside the fiber and its outer boundary, which is assumed to be held at a constant temperature. Here

q = i ρ C Ω / κ_{t h}

where

Ω

is the frequency of concern,

ρ

is the density of silica, and

C

is the specific heat of silica. The solution is given by:

Δ T (r_{⊥}, Ω) = \frac{1}{κ_{t h}} \iint G (r_{⊥}, r_{⊥}^{'}, q (Ω)) \cdot Q (r^{'}, Ω) d^{2} r_{⊥}^{'},

where the Green’s function satisfies the following differential equation:

\nabla_{⊥}^{2} G (r_{⊥}, r_{⊥}^{'}, q (Ω)) - q (Ω) G (r_{⊥}, r_{⊥}^{'}, q (Ω)) = - δ (r_{⊥} - r_{⊥}^{'}) .

A Dirichlet boundary condition for

Δ T

is used here (

Δ T = 0

). We proceed by assuming the field consists of two dominant modes: the fundamental mode, denoted here as mode 1, and a HOM (typically the LP₁₁ mode) denoted as mode 2. For

m \neq j

, the heat equation provides the following:

Δ T (r_{⊥}, - Δ ω_{s m j}) + Δ T (r_{⊥}, Δ ω_{s m j}) = \frac{g_{R} P_{p}}{A_{c l a d}} \sum_{m \neq j} A_{s m}^{*} A_{s j} Γ_{m j} (r_{⊥}, Δ ω_{s m j}) e^{i (Δ β_{s m j} z - Δ ω_{s m j} t)},

where

j = 1, 2

,

m = 1, 2

, and where

\begin{matrix} Γ_{m j} (r_{⊥}, Δ ω_{s m j}) = \frac{n_{0} (α - 1)}{2 μ_{0} c κ_{t h}} 〈 φ_{s m} (r_{⊥}^{'}), G (r_{⊥}, r_{⊥}^{'}, q (Δ ω_{s m j})) φ_{s j} (r_{⊥}^{'}) 〉, \\ 〈 φ_{s m} (r_{⊥}^{'}), G (r_{⊥}, r_{⊥}^{'}, q (Δ ω_{s m j})) φ_{s j} (r_{⊥}^{'}) 〉 = \iint^{​} φ_{s m}^{*} (r_{⊥}^{'}) G (r_{⊥}, r_{⊥}^{'}, q (Δ ω_{s m j})) φ_{s j} (r_{⊥}^{'}) d^{2} r_{⊥}^{'} . \end{matrix}

We follow the standard derivations of coupled mode theory. Self- and cross-phase modulation terms of the form

{| A_{s m} |}^{2}

as well as terms contained in Eq. (6) will appear in the equations describing the evolution of the field amplitudes of the two modes. The following system of 1st order nonlinear ordinary differential equations is obtained:

\frac{d A_{s j}}{d z} = \frac{g_{R} P_{p}}{A_{c l a d}} [\frac{1}{2} + i k_{0} (\sum_{m} {| A_{s m} |}^{2} κ_{j j m m} (0) + \sum_{m \neq j} {| A_{s m} |}^{2} κ_{j m j m} ({(- 1)}^{j - 1} Δ ω))] A_{s j},

where

k_{0}

is the free space wave number,

Δ ω = ω_{2} - ω_{1}

, and

κ_{j m k l} (Δ ω)

is given by:

κ_{j m k l} (Δ ω) = \frac{d n}{d T} \iint φ_{s j}^{*} (r_{⊥}) Γ_{k l} (r_{⊥}, Δ ω) φ_{s m} (r_{⊥}) d^{2} r_{⊥} .

Here each of the indices

j, k, l, m

can have a value of 1 or 2. Note that the second summation on the RHS of Eq. (8) is actually comprised of one term, but is expressed in this form for convenience. Since

κ_{j m k l} (0)

is always real and by using

P_{s j} = n_{0} {| A_{s m} |}^{2} / 2 μ_{0} c

, one can reformulate the governing amplitude equations described in Eq. (8) into the following system of power equations:

\frac{d P_{s j}}{d z} = g_{R} P_{s j} [1 + {(- 1)}^{j} \sum_{j \neq m} γ_{j m} (Δ ω) P_{s m}] \frac{P_{P}}{A_{c l a d}},

Where

P_{s j}

is the optical power contained in the

j^{t h}

mode. The coupling coefficient

γ_{j m}

is given as:

γ_{j m} (Δ ω) = 4 k_{0} \frac{μ_{0} c}{n_{0}} Im (κ_{j m j m} (Δ ω)),

Note that the following relationship is used to derive Eq. (10):

γ_{j m} (Δ ω) = - γ_{j m} (- Δ ω) .

The pump, which is approximated as being uniform in the transverse direction throughout the core and inner cladding region, is propagated using the following standard equation for a co-pumped RFA:

\frac{d P_{p}}{d z} = - α g_{R} (\sum_{j} P_{s j}) \frac{P_{p}}{A_{c l a d}} .

Next, by utilizing the conservation of the number of photons and the conservation of energy in the absence of loss, one can decouple the nonlinear equations for the spatial evolution of the `the evolution of the total signal power,

P_{t o t} = \sum_{j} P_{s j}

, along the fiber length as:

\frac{d P_{t o t}}{d z} = g_{R} P_{t o t} \frac{P_{P}}{A_{c l a d}} .

Solving Eqs. (13) and (14), one obtains the following equations:

P_{p} (z) = - α P_{t o t} (z) + P_{p} (0) + α P_{t o t} (0) .

P_{t o t} (z) = \frac{e x p (\frac{C_{1}}{A_{c l a d}} g_{R} z) P_{t o t} (0) C_{1}}{[P_{p} (0) + α e x p (\frac{C_{1}}{A_{c l a d}} g_{R} z) P_{t o t} (0)]},

where

C_{1} = P_{p} (0) + α P_{t o t} (0)

, and where

P_{t o t} (0)

and

P_{p} (0)

are the total signal and pump powers, respectively, at

z = 0.

By using the substitutions of $P_{s 2} = P_{t o t} - P_{s 1}$ and $P_{p} (0) = - α P_{t o t} (0) + C_{1}$ into Eq. (10), one can derive the following equation that captures the power in each mode:

P_{s j} (z) = \frac{P_{t o t} (z)}{1 + \frac{P_{s k \neq j} (0)}{P_{s j} (0)} [\exp [{(- 1)}^{j - 1} γ_{12} (P_{t o t} (z) - P_{t o t} (0))]]},

where

j = 1, 2 k = 1, 2

. It should be noted that

γ_{12} = γ_{21}

. The onset of MI threshold is defined here by the total output signal power where

P_{s 2} (z) = 0.05 P_{t o t} (z)

. Further exploration of this MI threshold condition yields the following inequality that defines the minimum amount of injected pump power required to reach the onset of the MI threshold for a very long co-pumped fiber (i.e. the total output signal approaches the quantum efficiency limit):

P_{p} (0) \geq \frac{α}{γ_{12, \max}} \ln (\frac{P_{s 1} (0)}{19 P_{s 2} (0)}),

where

γ_{12, \max}

is the maximum MI gain,

P_{s 1} (0)

is the initial power in the fundamental mode, and

P_{s 2} (0)

is the initial power in the HOM. As shown in [20], the MI gain is zero for

Δ ω = 0

, and increases beyond this point with

Δ ω

to a maximum value before dropping off. Also, as discussed by Smith and Smith [23], MI can be initiated from thermal noise; on the order of femtowatts injected into the HOM for a seed power on the order of watts. Interestingly, Eq. (18) indicates the MI threshold in a cladding-pumped RFA is independent of the Raman gain coefficient,

g_{R}

.

The counter-pumped RFA case follows along similar lines. Equation (10) remains the same while the RHS of Eq. (13) now has $+$ sign (as opposed to $-$ sign). Proceeding, one obtains the following equation:

P_{t o t} (z) = \frac{e x p (\frac{C_{2}}{A_{c l a d}} g_{R} z) P_{t o t} (0) C_{2}}{[C_{2} + α P_{t o t} (0) (1 - e x p (\frac{C_{2}}{A_{c l a d}} g_{R} z))]},

where

P_{p} (L) - α P_{t o t} (L) = C_{2}

.

For the counter-pumped RFA case, the equation for the modal content will then have the same form as Eq. (17) with $P_{t o t} (z)$ given by Eq. (19). For a given pump power, the length of fiber required to attain a certain conversion efficiency is larger for the counter-pumped RFA than that required for the co-pumped case. Interestingly, however, the MI threshold will be the same for the two cases.

2.2 Gain-tailored cladding-pumped RFA

In this subsection, the equations are modified to investigate gain-tailored RFAs. We consider a passive fiber that has been fabricated to create two different Raman gain regions. As shown in Fig. 1, the inner core region (shown in blue) has a Raman gain coefficient of $g_{1}$ , and the outer core region has a Raman gain coefficient of $g_{2}$ (shown in red). These regions are designed to minimize the higher HOM overlap with gain; thus, usually, $g_{1} > g_{2}$ . Also, we have taken the Raman gain coefficient in the cladding to be $g_{2}$ .

Fig. 1 Gain-Tailored design: Region with a Raman gain of $g_{1}$ in blue and region with a Raman gain of $g_{2}$ in red.

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Along the lines leading to Eq. (10), the system of power equations describing gain-tailored RFAs is the following:

\frac{d P_{s j}}{d z} = P_{s j} [g_{s j}^{(e f f)} + {(- 1)}^{j} \sum_{j \neq m} {\tilde{γ}}_{j m} (Δ ω) P_{s m}] \frac{P_{p}}{A_{c l a d}} .

Note here an effective Raman gain,

g_{s j}^{(e f f)}

, is defined based on the two gain regions, as illustrated in Fig. 1, and is given by:

{g_{s j}^{(e f f)} = \iint}^{​} [g_{1} H (R_{1} - r) + g_{2} H (r - R_{1})] {| φ_{s j} |}^{2} d^{2} r_{⊥},

where

H

is the Heaviside function and

R_{1}

is the radius of the core inner region. Since

g_{R}

is not uniform across the transverse cross-section of the fiber in this case, the coupling coefficient is now given as follows,

{\tilde{γ}}_{j m} (Δ ω) = 4 k_{0} \frac{μ_{0} c}{n_{0}} I m ({\tilde{κ}}_{j m j m} (Δ ω)),

where

{\tilde{κ}}_{j m j m} (Δ ω)

is given by:

{\tilde{κ}}_{j m j m} (Δ ω) = \frac{d n}{d T} \iint φ_{s j}^{*} (r_{⊥}) {\tilde{Γ}}_{j m} (r_{⊥}, Δ ω) φ_{s m} (r_{⊥}) d^{2} r_{⊥},

and where,

{\tilde{Γ}}_{j m} (r_{⊥}, Δ ω) = \frac{n_{0} (α - 1)}{2 μ_{0} c κ_{t h}} 〈 φ_{s m} (r_{⊥}^{'}), [g_{1} H (R_{1} - r^{'}) + g_{2} H (r^{'} - R_{1})] G (r_{⊥}, r_{⊥}^{'}, q (Δ ω)) φ_{s j} (r_{⊥}^{'}) 〉 .

The bracket notation used in Eq. (24) follows that used in Eq. (7). The evolution of the pump power along the fiber length is determined by:

\frac{d P_{p}}{d z} ≅ - α (\sum_{j} g_{e f f}^{s j} P_{s j}) \frac{P_{p}}{A_{c l a d}} .

For the case of gain-tailored RFAs, the governing system of equations consisting of Eqs. (20) and (25) is solved numerically. The fourth-order Runge-Kutta numerical scheme is sufficient to accurately solve this system.

2.3 Simulations of cladding-pumped RFAs

We consider a fiber with core radius of 15 µm and numerical aperture of 0.05. The inner and outer cladding radii (actual outer fiber radius) are fixed at 35 µm and 100 µm, respectively. The signal and pump wavelengths are 1100 nm and 1040 nm, respectively. The Raman gain coefficient is taken to be $1.5 \times 10^{- 13}$ m/W. The Green’s functions are similar to those derived for the scalar Helmholtz equation in cylindrical coordinates and are given by [24]:

G (r; r^{'}, θ^{'}) = - \frac{1}{4} {\begin{cases} \sum_{m = - \infty}^{\infty} [J_{m} (\sqrt{- q} r^{'}) Y_{m} (\sqrt{- q} r_{c l a d}) - J_{m} (\sqrt{- q} r_{c l a d}) Y_{m} (\sqrt{- q} r^{'})] \frac{J_{m} (\sqrt{- q} r)}{J_{m} (\sqrt{- q} r_{c l a d})} e^{i m (θ - θ^{'})} for r < r^{'} \\ \sum_{m = - \infty}^{\infty} [J_{m} (\sqrt{- q} r) Y_{m} (\sqrt{- q} r_{c l a d}) - J_{m} (\sqrt{- q} r_{c l a d}) Y_{m} (\sqrt{- q} r)] \frac{J_{m} (\sqrt{- q} r^{'})}{J_{m} (\sqrt{- q} r_{c l a d})} e^{i m (θ - θ^{'})} for r > r^{'} \end{cases},

where

J_{m}

and

Y_{m}

represent the Bessel functions of the first and second kind of order

m

, respectively. Here the frequency offset,

Δ ω

, is chosen to provide for maximum MI gain (typically on the order of kHz). As defined in subsection 2.1, the MI threshold occurs at the point where 5% of the total power is in the HOM. Using Eqs. (20), for a co-pumped RFA seeded with 50 W in the fundamental mode and 1 fW injected in the HOM, the MI threshold for a fiber length of 106 m occurs at 556 W. This is actually well-below the threshold for the onset of second-order Stokes for this fiber size. The amplifier efficiency (defined here as signal power at output divided by input pump power) for this configuration is 70%. To maintain the efficiency in a counter-pumped RFA, a longer fiber (160 m) is needed for the same seed and pump powers. The MI threshold occurs at the same signal power of 556 W, which is expected based on the theoretical analysis shown in subsection 2. Figure 2 shows the evolution of the total signal and the HOM content in both pumping configurations.

Fig. 2 Evolution of total signal and HOM powers as a function of position along the fiber for a co-pumped and a counter-pumped RFA. In both cases the fiber length was chosen to provide an amplifier efficiency of 70%. The dashed horizontal line (here and in some of the figures below) represents the 5% level that defines the MI threshold.

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The thermal load per unit length, ${\tilde{Q}}_{t} (z)$ , is plotted for the two pumping configurations in Fig. 3. From Eq. (2), we obtain:

\tilde{Q} {}_{t}{(z)} = \iint \tilde{Q} d x d y = \frac{(α - 1) g_{R}}{A_{c l a d}} P_{t o t} P_{p} .

Due to differences in the spatial evolution of the signal,

{\tilde{Q}}_{t} (z)

(and accordingly the temperature profile) is vastly different in the two pumping configurations. For the co-pumped RFA, the ratio of the maximum to minimum thermal load per unit length is within a factor of ~4, with the minimum load occurring at the input end of the seed and the maximum at a point located at

~ 0.75 L

. For the counter-pumped RFA, the ratio of maximum to minimum thermal load is very high with the maximum load occurring at the output end of the signal. However,

\int {\tilde{Q}}_{t} (z) d z

(i.e. the area underneath the plots in Fig. 4) is the same for both pumping configurations. This is expected as Eq. (17) is applicable to both pumping configurations and the coupling coefficient

γ_{12}

is the same. Furthermore, the fiber lengths in the two configurations are chosen to provide the same efficiency; i.e. the amount of pump light absorbed is the same, leading to the same amount of Raman amplification, which is the other determining factor appearing in Eq. (17). For the same amount of Raman amplification, the total heat load will be the same.

Fig. 3 Heat load vs. longitudinal position along the fiber for the co-pumped and counter-pumped configurations. The fiber lengths were chosen to provide in both cases 70% efficiency.

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Fig. 4 MI signal threshold as a function of seed power showing a linear dependence.

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In Yb-doped fiber amplifiers, it was shown both computationally [19] and experimentally [25] that the MI signal threshold scales linearly with seed power over the seed power investigated. The invariance of $\int {\tilde{Q}}_{t} (z) d z$ can be readily used to show that for a RFA this relation holds true and that the ratio of the change in threshold signal power to the change in seed power is equal to 1. Figure 4 shows that the threshold increases linearly as seed power increases with the predicted slope.

Along the same physical argument, the MI signal threshold for RFAs is independent of fiber length. Figure 5 shows the power in the HOM along the fiber for various injected pump powers in a co-pumped configuration. The seed power was kept the same in all cases. The dashed line shows how much power is in the HOM at the defined MI threshold. In all of these cases, the MI threshold is the same (signal output power is 556W) while the fiber lengths and optical to optical efficiencies are different. To be precise, while the pump power at MI threshold was different due to different fiber lengths, the Raman signal MI threshold remained the same.

Fig. 5 Higher order mode content as a function of longitudinal position for co-pumped RFAs with different lengths. For these cases, the pump powers were set at approximately 800, 900, 1000, 1100, 1200 W, but the MI threshold occurred at the same signal power level (556W).

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We have shown that the MI phenomenon is a limitation to the power scaling of RFAs, as it is to Yb-doped fiber amplifiers. We now investigate whether or not a gain-tailored design can effectively suppress MI in RFAs. Gain tailoring was shown experimentally to be effective in suppressing MI in Yb-doped amplifiers [26]. Using Eqs. (20)-(25) described in section 2.2, we can numerically investigate gain tailoring in RFAs. We choose Raman gain coefficients of $g_{1} = 1.5 \times 10^{- 13}$ m/W and $g_{2} = 10^{- 13}$ m/W for the inner and outer regions (shown in Fig. 1), respectively. We varied the radius of the inner region, denoted by $R_{1}$ , in order to find the optimal annular gain tailored configuration that produces the maximum amount of MI suppression. Figure 6 shows the HOM content as a function of signal power for various $R_{1}$ . As we decrease the radius of the inner region, which has a higher Raman gain, the MI threshold increases up to a certain optimal radius, denoted by $R_{O}$ . Afterwards, further reduction below $R_{0}$ actually decreases the MI threshold.

Fig. 6 Higher-order mode content in a cladding-pumped RFA as a function of signal power for various inner core radii. The Raman gain coefficients are $5 \times 10^{- 13}$ m/W and $10^{- 13}$ m/W for the inner and outer regions, respectively.

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In contrast to Yb-doped fiber amplifiers, gain-tailoring in RFAs results in lower MI suppression. For the former, the gain dopant is completely removed from the outer region. However, for RFAs, it is probably impractical to envision a material with a Raman gain of zero, especially when considering the need to maintain optical uniformity throughout the entire core. These findings indicate that gain-tailoring has limited utility in suppressing MI in RFAs. More generally, our work indicates that the MI phenomenon is a major limiting factor for power scaling with Raman amplifiers. On the positive side, photodarkening, which has been reported to play a significant role in inducing MI in Yb-doped fiber amplifiers [9], is not expected to occur in RFAs.

3. Core-pumped RFAs

The approach used in the previous section to derive the equations for cladding-pump RFAs can be altered to describe the MI phenomenon in core-pumped RFAs. One of the main differences in the governing equations originates from the treatment of pump power in each case. In a cladding-pumped RFA, the pump power is approximated as being uniform in the transverse direction throughout the core and inner cladding region. This assumption obviously will not be valid in a core-pumped RFA. The other difference is that for a core-pumped RFA, the pump can potentially participate in writing an index of refraction grating. The work in [23] proposes that MI is initiated due to thermally excited fluctuations in the optical index of refraction. These fluctuations can, in theory, lead to scattering into the HOMs for both the pump and signal light. For a core-pumped RFA, where the pump and signal are co-propagating, the heat deposition is given by:

\begin{matrix} {\tilde{Q}}_{C o r e - P u m p e d} = g_{R, 0} (α - 1) {(\frac{n_{0}}{2 c μ_{0}})}^{2} (\sum_{k} {| A_{s k} |}^{2} {| φ_{k} |}^{2} + \sum_{k \neq l} A_{s k}^{*} A_{s l} φ_{k}^{*} φ_{l} \exp [i (Δ β_{s} z - Δ ω t)] + c . c .) \\ \times (\sum_{k} {| A_{p k} |}^{2} {| φ_{k} |}^{2} + \sum_{k \neq l} A_{p k}^{*} A_{p l} φ_{k}^{*} φ_{l} \exp (i (Δ β_{p} z - Δ ω t)) + c . c .), \end{matrix}

where

k = 1, 2

and

l = 1, 2

,

Δ β_{s}

is the difference between the propagation constants of the LP₀₁ and LP₁₁ modes for the signal, and

Δ β_{p}

is the corresponding difference for the pump. The pump and signal have approximately the same transverse mode profiles as well as approximately the same offset frequencies. Note that in Eq. (28),

g_{R, 0}

is used as opposed to

g_{R} = g_{R, 0} / 2

used in Eq. (2) for a cladding-pumped RFA as the signal and pump are now taken to be co-polarized. The thermal grating, written by the interference of the pump modes and by the interference of the signal modes, is investigated below; though, in reality, only one thermal grating is written, one can consider the pump and signal interferences as separate contributions that may then interact with one another to alter the MI threshold.

It is straightforward (although somewhat tedious) to show that the system of amplitude equations in a core-pumped RFA is as follows:

\begin{array}{l} \frac{d A_{s j}}{d z} = i k_{0 s} A_{s j} [\begin{array}{l} \sum_{m} \sum_{l} {| A_{s m} |}^{2} {| A_{p l} |}^{2} ν_{j j m l m l} (r_{⊥}, 0) + 2 Re (A_{s 1}^{*} A_{p 1} A_{s 2} A_{p 2}^{*} ν_{j j 1212} (r_{⊥}, 0) e^{i (Δ β_{s} - Δ β_{p}) z}) \\ + \sum_{m} \sum_{l \neq j} {| A_{s l} |}^{2} {| A_{p m} |}^{2} ν_{j l m l m j} (r_{⊥}, {(- 1)}^{j - 1} Δ ω) - \frac{i n_{0}}{4 c k_{0 s} μ_{0}} \sum_{m} {| A_{p m} |}^{2} Λ_{j m j m} \end{array}] \\ + i k_{0 s} (A_{p j} e^{{(- 1)}^{j - 1} i (Δ β_{s} - Δ β_{p}) z} \sum_{m} \sum_{l \neq j} A_{p l}^{*} A_{s l} {| A_{s m} |}^{2} ν_{j l m l m j} (r_{⊥}, {(- 1)}^{j - 1} Δ ω)), \end{array}

\begin{array}{l} \frac{d A_{p j}}{d z} = i k_{0 p} A_{p j} [\begin{array}{l} \sum_{m} \sum_{l} {| A_{p m} |}^{2} {| A_{s l} |}^{2} ν_{j j m l m l} (r_{⊥}, 0) + 2 Re (A_{p 1}^{*} A_{s 1} A_{p 2} A_{s 2}^{*} ν_{j j 1212} (r_{⊥}, 0) e^{i (Δ β_{s} - Δ β_{p}) z}) \\ + \sum_{m} \sum_{l \neq j} {| A_{p l} |}^{2} {| A_{s m} |}^{2} ν_{j l m l m j} (r_{⊥}, {(- 1)}^{j - 1} Δ ω) + \frac{i α n_{0}}{4 c k_{0 p} μ_{0}} \sum_{m} {| A_{s m} |}^{2} Λ_{j m j m} \end{array}] \\ + i k_{0 p} (A_{s j} e^{{(- 1)}^{j} i (Δ β_{s} - Δ β_{p}) z} \sum_{m} \sum_{l \neq j} A_{s l}^{*} A_{p l} {| A_{p m} |}^{2} ν_{j l m l m j} (r_{⊥}, {(- 1)}^{j - 1} Δ ω)), \end{array}

where:

\begin{matrix} ν_{i j k l m n} (r_{⊥}, Δ ω) = ξ \iint φ_{i}^{*} (r_{⊥}) 〈 φ_{k} ({r^{'}}_{⊥}) φ_{l} ({r^{'}}_{⊥}) G (r_{⊥}, r_{⊥}^{'}, q (Δ ω)) φ_{m} ({r^{'}}_{⊥}) φ_{n} ({r^{'}}_{⊥}) 〉 φ_{j} (r_{⊥}) d^{2} r_{⊥} . \\ Λ_{i j k l} (r_{⊥}) = g_{R, 0} \iint φ_{i}^{*} (r_{⊥}) φ_{j}^{*} (r_{⊥}) φ_{k} (r_{⊥}) φ_{l} (r_{⊥}) d^{2} r_{⊥} . \\ ξ = {(\frac{n_{0}}{2 μ_{0} c})}^{2} \frac{d n}{d T} \frac{g_{R, 0}}{κ_{t h}} (α - 1) . \end{matrix}

The indices shown in the equations above take on the values 1 or 2.

The first term appearing on the RHS of Eq. (29) contributes to phase modulation, while the third term is similar to the term capturing mode coupling due to MI in cladding-pumped RFAs. The fourth term represents Raman gain. The second and fifth terms do not have equivalents in cladding-pumped RFAs. Specifically, the former represents weak coupling due to the product of the pump and signal modal interferences. This term is of no consequence as far as the MI threshold is concerned. The latter term captures the coupling of a signal mode to the other mode due to the interference of the pump modes; albeit this coupling is not perfectly phase-matched. Obviously, as the dispersion between signal and pump increases (i.e. as the wavelength separation increases), one would expect the contribution from this term to diminish. For Eq. (30) which describes the evolution of the pump modes, coupling between these modes occurs due to their own modal interference, as well as the modal interference of the signal modes.

We perform numerical simulations using a fourth-order Runge-Kutta scheme to integrate the 4 × 4 nonlinear coupled system of equations described by Eqs. (29) and (30). We consider a fiber with similar transverse dimensions and numerical aperture to the cladding-pumped RFA simulated in Section 2.3. Likewise, the signal and pump wavelengths are taken to be 1100 nm and 1040 nm, respectively. For these wavelengths, $Δ β_{s} - Δ β_{p} \approx$ 40/m. The Raman gain coefficient is taken to be $1.5 \times 10^{- 13}$ m/W. The amplifier is seeded with 50 W. Figure 7 shows plots of the evolution along the direction of propagation of both the total signal and HOM powers, respectively. At 50 m of length, the output power approaches the quantum efficiency. The MI threshold is defined as the total signal power at the point where the LP₁₁ signal mode contributes 5% to this power. As can be deduced form the figure, the threshold occurs at ~690 W. This is higher than the threshold for the cladding-pumped RFA simulated in section 2.3 and can be attributed to better overlap between the fundamental mode of the signal with the pump as well as minimal cross interaction between the two interference patterns and the optical waves due to the effect of dispersion. To investigate the latter, we artificially varied $Δ β_{s} - Δ β_{p}$ over the range 0 −320 /m. As shown in Fig. 8, at high values of $Δ β_{s} - Δ β_{p}$ (which is shown as $Δ (Δ β)$ on the horizontal axis of the figure), the threshold approaches 715 W. This is ~4% greater than the actual value for the threshold; thus implying that there is little cross interaction. Obviously as the dispersion is made to zero, one would expect the threshold to drop to 1/2 of the maximum value as conferred by the numerical results. In this case, two index of refraction gratings of approximately equal magnitude are read by the signal.

Fig. 7 Evolution of total signal and HOM powers as a function of position along the direction of propagation for a co-propagating, core-pumped, 50 m long RFA.

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Fig. 8 MI threshold in a co-propagating, core-pumped RFA as a function of an artificially induced $Δ (Δ β)$ .

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As discussed in section 2.3, the MI threshold for cladding-pumped RFAs is independent of length. Generally, this is not expected to be the case for core-pumped RFAs. In the limit of a very long fiber, the coupling between the signal and the grating written by the pump becomes very weak due to large phase mismatch. But as the fiber length is shortened, the cross interactions become more significant, leading to a drop in the MI threshold. To illustrate this point, we conducted simulations for a shorter RFA. Figure 9 portrays the evolution along the direction of propagation for both the total signal and HOM powers for a 10 m long fiber. Notably, the MI threshold is now ~640 W, a drop of ~7% from the case of the 50 m long RFA. Also, as expected, the amplifier efficiency is now lower, with the pump power at threshold being ~1140 W. Finally, it can be inferred, based on this analysis, that in core-pumped RFAs, the MI threshold can have some dependence on the Raman gain, which is in contrast to cladding-pumped RFAs.

Fig. 9 Evolution of total signal and HOM powers as a function of position along a fiber for a co-propagating, core-pumped 10 m long RFA. Here, the MI threshold is lower than what was computed for a 50 m long RFA (see Fig. 7).

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4. Summary

Raman fiber lasers have been attracting considerable attention recently as potential candidates to scaling beyond the power limitations imposed on rare-earth doped fiber lasers. This work investigated one of these limitations, namely MI. We started first by formulating the physics of MI in cladding-pumped RFAs. By comparing the extracted signal at MI threshold for the same step index-fiber, it was found that the MI threshold is independent of the length of the amplifier, or whether the amplifier is co-pumped or counter-pumped; dictated by the integrated heat load along the length of fiber. We extended our treatment to gain-tailored RFAs and we showed that this approach, which has been effective in suppressing MI in Yb-doped amplifiers, would be of limited utility. Finally, we formulated the physics of MI in core-pumped RFAs, where both pump and signal participate in writing the time-dependent index of refraction grating. Here, if cross-interactions are small, the MI threshold would be higher than it would be in a comparable cladding-pumped RFA. Furthermore, in contrast to the cladding-pumped RFA, there is a length dependence to the threshold. It would be interesting in future work to extend the core-pumped RFA formalism to integrated Yb-Raman amplifiers, which have been demonstrated recently at the multi-kW level. It may also be fruitful to investigate the differential mode loss induced by fiber bending on the MI threshold for RFAs.

Acknowledgments

The authors would like to thank the Air Force Office of Scientific Research (AFOSR) and High Energy Laser Joint Technology Office (HEL-JTO) for partial funding of this work.

References and links

1. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008). [CrossRef] [PubMed]

2. J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on non-silica materials,” Proc. SPIE 7686, 768611 (2010). [CrossRef]

3. A. Yusim, “Recent progress in scaling of high power fiber lasers at IPG Photonics,” presented at Solid State and Diode Laser Technology Review,(2009).

4. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19(14), 13218–13224 (2011). [CrossRef] [PubMed]

5. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20(10), 11407–11422 (2012). [CrossRef] [PubMed]

6. C. A. Codemard, P. Dupriez, Y. Jeong, J. K. Sahu, M. Ibsen, and J. Nilsson, “High-power continuous-wave cladding-pumped Raman fiber laser,” Opt. Lett. 31(15), 2290–2292 (2006). [CrossRef] [PubMed]

7. J. E. Heebner, A. K. Sridharan, J. W. Dawson, M. J. Messerly, P. H. Pax, M. Y. Shverdin, R. J. Beach, and C. P. J. Barty, “High brightness, quantum-defect-limited conversion efficiency in cladding-pumped Raman fiber amplifiers and oscillators,” Opt. Express 18(14), 14705–14716 (2010). [CrossRef] [PubMed]

8. B. Ward, “Solid-core photonic bandgap fibers for cladding-pumped Raman amplification,” Opt. Express 19(12), 11852–11866 (2011). [CrossRef] [PubMed]

9. H. J. Otto, N. Modsching, C. Jauregui, J. Limpert, and A. Tünnermann, “Impact of photodarkening on the mode instability threshold,” Opt. Express 23(12), 15265–15277 (2015). [CrossRef] [PubMed]

10. Y. Feng, L. R. Taylor, and D. B. Calia, “25 W Raman-fiber-amplifier-based 589 nm laser for laser guide star,” Opt. Express 17(21), 19021–19026 (2009). [CrossRef] [PubMed]

11. C. Vergien, I. Dajani, and C. Robin, “18 W single-stage single-frequency acoustically tailored Raman fiber amplifier,” Opt. Lett. 37(10), 1766–1768 (2012). [CrossRef] [PubMed]

12. L. Zhang, H. Jiang, S. Cui, J. Hu, and Y. Feng, “Versatile Raman fiber laser for sodium laser guide star,” Laser Photonics Rev. 8(6), 889–895 (2014). [CrossRef]

13. V. R. Supradeepa and J. W. Nicholson, “Power scaling of high-efficiency 1.5 μm cascaded Raman fiber lasers,” Opt. Lett. 38(14), 2538–2541 (2013). [CrossRef] [PubMed]

14. L. Zhang, C. Liu, H. Jiang, Y. Qi, B. He, J. Zhou, X. Gu, and Y. Feng, “Kilowatt Ytterbium-Raman fiber laser,” Opt. Express 22(15), 18483–18489 (2014). [CrossRef] [PubMed]

15. Q. Xiao, P. Yan, D. Li, J. Sun, X. Wang, Y. Huang, and M. Gong, “Bidirectional pumped high power Raman fiber laser,” Opt. Express 24(6), 6758–6768 (2016). [CrossRef] [PubMed]

16. J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009). [CrossRef]

17. C. A. Codemard, J. Ji, J. K. Sahu, and J. Nilsson, “100-W CW cladding-pumped Raman fiber laser at 1120 nm,” Proc. SPIE 7580, 75801N (2010). [CrossRef]

18. R. W. Boyd, Nonlinear Optics, 3rd Ed. (Academic, 2008).

19. S. Naderi, I. Dajani, T. Madden, and C. Robin, “Investigations of modal instabilities in fiber amplifiers through detailed numerical simulations,” Opt. Express 21(13), 16111–16129 (2013). [CrossRef] [PubMed]

20. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37(12), 2382–2384 (2012). [CrossRef] [PubMed]

21. C. Jauregui, H. J. Otto, S. Breitkopf, J. Limpert, and A. Tünnermann, “Optimizing high-power Yb-doped fiber amplifier systems in the presence of transverse mode instabilities,” Opt. Express 24(8), 7879–7892 (2016). [CrossRef] [PubMed]

22. Raman Amplification in Fiber Optical Communication Systems, edited by C. Headley and G. P. Agrawal (Elsevier, 2005).

23. A. V. Smith and J. J. Smith, “Spontaneous Rayleigh seed for stimulated Rayleigh scattering in high power fiber amplifiers,” IEEE Photonics J. 5(5), 7100807 (2013). [CrossRef]

24. C. A. Balanis, Advanced Engineering Electromagnetics, (John Wiley & Sons, 1989).

25. H. Otto, C. Jauregui, F. Stutzki, J. Limpert, and A. Tünnermann, “Dependence of mode instabilities on the extracted power of fiber laser systems,” in Advanced Solid-State Lasers Congress, G. Huber and P. Moulton, eds., OSA Technical Digest (online) (Optical Society of America, 2013), paper ATu3A.02. [CrossRef]

26. C. Robin, I. Dajani, and B. Pulford, “Modal instability-suppressing, single-frequency photonic crystal fiber amplifier with 811 W output power,” Opt. Lett. 39(3), 666–669 (2014). [CrossRef] [PubMed]

Theoretical and numerical treatment of modal instability in high-power core and cladding-pumped Raman fiber amplifiers

Abstract

1. Introduction

2. Cladding-pumped RFAs

2.1 Governing equations

2.2 Gain-tailored cladding-pumped RFA

2.3 Simulations of cladding-pumped RFAs

3. Core-pumped RFAs

4. Summary

Acknowledgments

References and links

Cited By

Figures (9)

Equations (31)

Optics Express