Theoretical analysis of single-frequency Raman fiber amplifier system operating at 1178nm

Christopher Vergien; Iyad Dajani; Clint Zeringue

doi:10.1364/OE.18.026214

1. Introduction

It is well-known that vastly improved sky coverage for ground-based telescopes can be achieved through the deployment of a sodium guide star beacon operating at 589 nm. Illumination of the mesosphere with such laser results in photon returns obtained through fluorescence of the D_2a line of sodium atoms. Further enhancement of the photon return can be achieved by illuminating both the D_2a and D_2b lines [1]. The return light is then utilized in an adaptive optics system to correct for atmospheric turbulence. Over the past two decades, several sodium guide star lasers have been constructed and tested beginning with systems based on dye lasers. Until recently, the most powerful system was constructed at the Air Force Research Laboratory (AFRL) and provided a maximum output power of 50 W [1]. This system is known by the acronym FASOR (Frequency Addition Source of Coherent Optical Radiation) and relies on generating 1064 nm light and 1319 nm light through the utilization of Nd:YAG gain rods. The two beams are then combined in a doubly resonant nonlinear cavity containing a non-critically phase-matched lithium triborate (LBO) crystal.

More recently, a team of researchers at the European Southern Observatory (ESO) was successful in constructing a narrow linewidth sodium guide star laser based on frequency doubling the output of a Raman fiber amplifier operating at 1178 nm [2,3]. Consequently, only a single wavelength was required to pump the nonlinear crystal cavity; thus reducing the complexity of the system. Furthermore, Raman fiber lasers are generally more compact than their bulk solid state laser counterparts with the potential of eliminating much of the free space optics while delivering diffraction-limited beams. An output power approaching 30 W at 589 nm has been reported through the use of a single core-pumped Raman amplifier [3], while > 50 W was obtained through coherent beam combination (CBC) of the outputs of three Raman fiber amplifiers [4].

In silica-based optical fibers, Raman gain is fairly broad (~40 THz) due to the amorphous nature of the material and has a peak gain at approximately 13.2 THz [5]. The Raman gain coefficient, which can be related to the imaginary part of the third-order nonlinear susceptibility, depends on the fiber dopants and is inversely proportional to the pump wavelength. The value of the gain coefficient typically lies in the range of 4 × 10⁻¹⁴ ‒ 8 × 10⁻¹⁴ m/W for pump light at 1-1.5 µm. An important general point to consider with Raman fiber amplification is that Raman gain can occur at any signal wavelength and thus Raman fiber amplifiers can offer access to a broad tunable range of wavelengths that are not traditionally accessible through laser host materials.

In this paper we analyze the scalability of Raman amplifiers operating at 1178 nm. In the mesosphere, the full width at half maximum (FWHM) of either of the Doppler broadened sodium D₂ lines is approximately 1 GHz [6]. However, since linewidths < 30 MHz are required for efficient frequency conversion in the second-harmonic generation (SHG) cavity or are typically desired in a robust CBC system, we consider here “single-frequency” amplifiers wherein the linewidths are appreciably smaller than the Brillouin gain bandwidth. In fact, the Raman amplifiers built by the ESO research team were reported to have linewidths < 2 MHz [4]. As such, any concise analysis of single-frequency Raman amplifiers would invoke consideration of stimulated Brillouin scattering (SBS). The SBS process is an intrinsically phase-matched third-order nonlinear effect that couples acoustic phonons to the optical field and its associated backscattered Stokes field.

A simple analysis of the ESO system indicates that SBS suppression is required in order to achieve the reported power levels. Details of the SBS suppression technique(s) have so far not been revealed. However, it is well known that SBS can be overcome by various methods such as temperature [7], stress gradients [8], and acoustically tailored fibers [9]. We use here optimized multi-step temperature profiles in our simulations to push the power limits of a polarization maintaining (PM) fiber. In section 2 of this paper, we discuss the two-point boundary coupled nonlinear system of differential equations used in our analysis. In section 3, we solve these equations numerically in order to analyze power scaling in relation to the fiber mode field diameter, length, and available pump power in a core pumped Raman amplifier. Section 4 covers the application of a multi-temperature profile where further analysis of power scaling is considered. Finally, in section 5 we investigate the feasibility of generating the D_2a and D_2b lines in a sodium guide star beacon from a single Raman amplifier by considering the four-wave mixing (FWM) process.

2. Theory of single-frequency Raman amplifiers

2.1 Raman amplifier power equations

For a single-frequency Raman amplifier one needs to consider the evolution of the pump, signal, and the Stokes light generated through the interaction of the Raman signal with the acoustic phonons [5]. The Raman signal interacts with the pump wave through Raman gain and also with the Stokes light through the SBS process. The counter propagating Stokes light experiences both Raman and Brillouin gain while the pump wave interacts with both the Raman signal and Stokes light through SRS. It is assumed here that all waves propagate in the lowest-order transverse mode. Furthermore, we consider only the case in which the pump and signal are co-polarized which provides maximum gain. The Raman gain for orthogonally polarized pump and signal is much smaller.

The Stokes light is considered by dividing the Brillouin gain bandwidth into several channels. The coupled set of equations describing the evolution of the Raman signal power, $P_{R}$ , the Stokes light power contained in a particular channel, $P_{S, i}$ , and the pump light power, $P_{P}$ , along the longitudinal axis of the fiber, z, is expressed by the following equations [5]:

\frac{d P_{R}}{d z} = (g_{R} P_{P} - \sum_{i} g_{B, i} P_{S, i} - α_{R}) P_{R},

\frac{d P_{S, i}}{d z} = - (g_{R} P_{P} + g_{B, i} P_{R} - α_{R}) P_{S, i},

\frac{d P_{P}}{d z} = \mp γ g_{R} (P_{R} + \sum_{i} P_{S, i}) P_{P} \mp α_{P} P_{P},

where

ω_{R}

and

ω_{p}

are the angular frequencies of the signal and pump,

γ = ω_{P} / ω_{R}

and

α_{R}

and

α_{p}

are the fiber loss for the signal and pump, respectively. The ∓signs represent co and counter propagating pump waves, respectively. The normalized Raman gain coefficient is given by

g_{R} = g_{r 0} / A_{e f f, R}

where

g_{r 0}

is the intrinsic Raman gain coefficient and the effective area is given by:

A_{e f f, R} = \frac{\iint {| φ_{R} (x, y) |}^{2} d x d y \iint {| φ_{P} (x, y) |}^{2} d x d y}{\iint {| φ_{R} (x, y) |}^{2} {| φ_{P} (x, y) |}^{2} d x d y},

where

φ_{R} (x, y)

and

φ_{P} (x, y)

are the lowest-order transverse profiles for the signal and pump, respectively. Similarly the normalized Brillouin gain coefficient is

g_{B} = g_{b o} / A_{e f f, B}

, where

g_{b o}

is the intrinsic Brillouin gain coefficient. Since the Brillouin frequency is much smaller than the optical frequencies, the effective area here is given by:

A_{e f f, B} = \frac{{(\iint {| φ_{R} (x, y) |}^{2} d x d y)}^{2}}{\iint {| φ_{R} (x, y) |}^{4} d x d y} .

It is well-known that the SBS process can be mitigated by applying a temperature gradient which leads to a shift in the center Brillouin frequency along the fiber. Experimental data suggest that this frequency shift is approximately 2 MHz/°C [10]. In order to investigate the external application of a multi-step temperature profile, Eqs. (1) ‒ (3) need to be generalized to account for multiple center Stokes frequencies. Provided that there is minimal overlap among the Brillouin gain bandwidths corresponding to each temperature segment, the system of equations becomes:

\frac{d P_{R}}{d z} = (g_{R} P_{P} - \sum_{i, j} g_{B, i, j} P_{S, i, j} - α_{R}) P_{R},

\frac{d P_{S, i, j}}{d z} = - (g_{R} P_{P} + g_{B, i, j} P_{R} - α_{R}) P_{S, i, j},

\frac{d P_{P}}{d z} = \mp γ g_{R} (P_{R} + \sum_{i, j} P_{S, i, j}) P_{P} \mp α_{P} P_{P},

where the subscript j corresponds to the fiber section, and

g_{B, i, j}

is zero everywhere except for the pertinent fiber segment.

2.2 Initiation of SBS from Brillouin and Raman noise

The SBS process is initiated predominantly by the spontaneous Brillouin scattering that occurs throughout the fiber. A precise formalism entails the solution of the time-dependent field amplitude equations of the optical and phonon fields wherein a distributed noise described by a Markovian stochastic process subject to Gaussian statistics is used [11]. An approximate treatment of this problem was presented by Smith using a time-independent treatment of the intensity (power) equations [12]. It was shown that SBS initiation from a distributed source can be approximated by injecting, near the output end of the fiber, one Stokes photon per mode multiplied by the thermal average of phonons in the orbital as described by the Bose-Einstein distribution function. Since we are considering Raman gain as well as Brillouin gain with multiple peak frequencies along the length of the fiber, we use here a distributed source term to account for the initiation of the SBS process. As such, there is no need to determine the location of the injected Stokes photons a priori for each Stokes frequency under consideration. This noise term can be incorporated by adding a term of the form $- g_{B, i, j} P_{R} δ_{S, i, j}$ on the right hand side (RHS) of Eq. (7), with $δ_{S, i, j}$ given by:

δ_{S, i, j} = \frac{ℏ ω_{S, i, j} Δ ω}{2 π (\exp [ℏ (ω_{R} - ω_{S, i, j}) / k T_{j}] - 1)},

where

Δ ω

is the frequency bin size, and

T_{j}

is the temperature of the fiber segment.

The SBS process can also be initiated through Raman noise characterized by the interaction of the pump light with background optical phonons. It is well-known that $g_{R} ≪ g_{B}$ and the relevant number of background optical phonons is smaller than that of the acoustic phonons. However, unlike the spontaneous Brillouin process, the spontaneous Raman process for a given Stokes frequency occurs throughout a temperature segmented fiber. Therefore, for the sake of accuracy or to be more precise in our formalism, we include the spontaneous Raman noise of the form $- g_{R} P_{P} δ_{S, R}$ on the RHS of Eq. (7). Here $δ_{S, R}$ is given by:

δ_{S, R} = \frac{ℏ ω_{S, i, j} Δ ω}{2 π} .

3. Analysis of uniform temperature profile

3.1 Comparison of co and counter pumped configurations

Equations (6) – (8) represent a nonlinear two point boundary problem. In a co-pumped configuration, the signal and pump are known at the input end of the fiber while the noise-initiated Stokes fields are set to zero at the output end. The counter-pumped configuration has similar boundary conditions except that the pump is known at the output end of the fiber. A numerical model was developed wherein the coupled differential equations were solved using a shooting and root finding algorithm, and verified with the boundary conditions for accuracy. The numerical results were also verified using simplified configurations where analytical solutions for Raman signal or Stokes evolution can be obtained. Additionally, we checked for numerical accuracy by accounting for the number of photons in the system. Since loss in a passive fiber is typically < 0.3 dB/km and we are considering relatively short fibers, we neglect this effect in our simulations. It can be readily shown that in the absence of loss, the conservation equation for the number of photons is given by:

\frac{d}{d z} (\frac{P_{R}}{ω_{R}} - \sum_{i} \frac{P_{S, i}}{ω_{R}} \pm \frac{P_{P}}{ω_{P}}) = 0,

where ± refers to co-pumped and counter-pumped, respectively. In all our simulations the conservation relation described by Eq. (11) was satisfied to better than 0.001%

An important consideration is to compare the efficiency and the SBS process in a single-frequency Raman amplifier for co-pumped and counter-pumped configurations. We note the effective Raman gain is higher in the co-pumped configuration than the counter-pumped configuration as can be inferred from Eqs. (12) and (13) presented in section 3.2 of this paper. A related manifestation of this effect is that the threshold for noise-initiated forward propagating SRS is lower than that of backward propagating SRS [12]. However, since the SBS process depends on the spatial evolution of the Raman signal along the length of the fiber and also the interaction of the Stokes light with the Raman pump, it is not clear which pumping configuration will lead to higher SBS thresholds for a given fiber geometry. Regardless, due to the exponential rise in Raman signal at the output end of the fiber in either pumping configuration, it can generally be argued that a lower seed power would lead to a higher SBS threshold (amplifier noise notwithstanding). We initially conducted a simulation using a fiber mode field diameter (MFD) of $7.5 μm$ which is roughly equal to that used in the experiments described in Ref. [2]. We used a Raman gain coefficient of $8 \times 10^{- 14}$ m/W and an SBS gain coefficient of $1.5 \times 10^{- 11}$ m/W. The value for the Raman gain is typical of silica fibers. Based on the experimental results reported by the ESO, a Nufern 1060XP has approximately twice the Brillouin threshold of a Corning HI1060 fiber [2]. The difference in gain coefficients between the two fibers can be attributed to the effect of the dopant types and concentrations. Recent measurements by Mermelstein on aluminum-doped fibers indicate a Brillouin gain coefficient of approximately $1.0 \times 10^{- 11}$ m/W [13]. The concentration of aluminum in the Nufern 1060XP was reported to be high [14]. It thus appears that a value of $1.5 \times 10^{- 11}$ m/W was a reasonable value for use in our simulations.

A pump power of 200 W was considered. This value exceeds by 50 W the state-of-the-art 1120 nm Raman pump laser (or for that matter Raman pump power at any wavelength) reported in Ref. [15] and thus allows for investigation of future systems. For each pumping configuration, the fiber length was optimized to allow for maximum signal output at SBS threshold. We used the standard definition of SBS threshold i.e. the Raman output power at the point where the reflectivity is approximately 1% [11]. The minimum fiber length was determined to be 16.7 m for the core co-pumped and 17.5 m for the core counter-pumped configurations. Therefore, fibers with lengths below the aforementioned values would be pump limited. Figure 1 shows a plot of the spatial evolution of the signals at SBS threshold. Note there is little difference in output power as both configurations exhibit a steep rise in signal power at the output end of the fiber. Alternatively, one can fix the fiber length and allow for sufficient pump to get to the SBS threshold. In this case, the counter-pump configuration will have slightly higher output due to the slightly steeper rise in the signal at the output end. Regardless, the results of the SBS process in a Raman amplifier are in stark contrast to the process in a rare-earth doped gain fiber. For the latter, counter-pumping provides an SBS threshold that is typically twice as high as that of co-pumping even without the inclusion of the thermal gradient obtained through quantum defect heating.

Fig. 1 Raman signal evolution at SBS threshold for co and counter-pumped cases with optimized length and a pump power of 200 W.

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3.2 Raman signal in relation to fiber length and seed power

In order to understand the output power as a function of fiber length and seed power, we conducted a series of simulations. Six fiber lengths were chosen: 25 m, 50 m, 75 m, 100 m, 125 m, and 150 m. Also, a set of seed powers were selected starting at 1 mW and ending as high as 1 W. In these simulations, we allowed for sufficient pump power to be available for the amplifier to reach SBS threshold. Figure 2 shows the results of the simulations for a co-pumped configuration. Not shown here, is the counter-pumped case. But again, since the length is fixed, the counter-pumped configuration will have slightly higher output. As shown in Fig. 2, seed power can have a significant impact on signal power at SBS threshold. For a 25 meter fiber, an increase of approximately 25% can be obtained by reducing the seed power from 91 mW to 16 mW while at the 150 meter length, approximately 1.6 times the Raman power is obtained at the lower seed power. Not shown in the figure is a simulation we conducted for the 25 m case whereby the Raman power dropped by a factor approaching three at a seed power of 1 W over a seed power of 16 mW.

Fig. 2 (a) Raman power and (b) efficiency as a function of seed power and fiber length for co-pumping. Inset in figure shows linear dependence of Raman output with pump power at SBS threshold for one of the seed cases.

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The enhancement in Raman power output due to lower seed power comes at a price of reduced amplifier efficiency as shown in Fig. 2. Note that the amplifier efficiency is approximately the same for all fiber lengths for a given seed power. That is, when optimizing fiber length, the Raman output at SBS threshold scales linearly with pump power (the inset in the Fig. 2(b) illustrates this for the case of a 16 mW seed power). Consequently, substantial power scaling can be achieved by building more powerful pumps. For example, with a 16 mW seed, approximately 8 W of output power at a length of 25 m (corresponding pump power at SBS threshold of 141 W) is obtained, while approximately 0.9 W was obtained at a length of 150 m (corresponding pump power at SBS threshold of 15.4 W). If fiber lengths of 10 m and 5 m are considered, the pump power required to reach SBS threshold for a seed power of 16 mW would be approximately 418 W and 923 W, respectively. The efficiencies in these two cases are in agreement with the results presented in Fig. 2. We note that the required pump values exceed the highest reported output power from a Raman laser at any wavelength [15] by approximately 3 and 6 times.

To explain the significant improvement in output Raman power with pump power and the linear dependence, we start first by examining the Raman amplification only. Using an undepleted pump treatment for the 1120 nm light, the small signal gain is $g_{R} P_{P} L$ and thus it would appear that a reduction of length from, for example, 150 m to 25 m will require 6 times more pump power to achieve the same Raman output. That is based on this analysis, as long as $P_{P} L$ is held constant, the same power is obtained. However, accounting for pump depletion through the Raman process, the solution for the co-pumped configuration can be readily obtained from Eqs. (1) and (11):

P_{R} (z) = \frac{C_{1} P_{R} (0) \exp [g_{R} C_{1} z]}{P_{P} (0) + γ P_{R} (0) \exp [g_{R} C_{1} z]},

where

P_{R} (0)

is the seed power,

P_{P} (0)

is the input pump power, and

C_{1} = P_{P} (0) + γ P_{R} (0)

. For the counter-pumped configuration the following equation is obtained:

P_{R} (z) = \frac{C_{2} P_{R} (0) \exp [g_{R} C_{2} z]}{C_{2} + γ P_{R} (0) (1 - \exp [g_{R} C_{2} z])},

where

C_{2} = P_{P} (L) - γ P_{R} (L)

. Note that

P_{R} (L)

is determined by solving Eq. (13) with

z = L

which results in a transcendental equation. Irrespective of the pumping configuration, these equations indicate that in the regime where pump depletion starts to become significant, an increase in pump power will lead to an increase in the Raman signal as

P_{P} L

is held constant. This effect is rather small for the range of simulations we ran. Much more significantly, when the SBS process is considered, a significant reduction in the reflectivity is obtained at higher pump powers (shorter fiber lengths) due to the relatively more rapid rise of the signal; thus allowing for further pumping and consequently even higher Raman signal.

To illustrate this, we consider the case of the 150 m and 25 m fibers in a co-pumped configuration seeded with 16 mW. For the former, the pump power at SBS threshold is 15.4 W, while for the latter the pump power at SBS threshold is 141 W. Figure 3 , shows plots of the Stokes gain per unit length for each case. The total Stokes gain is the sum of the Brillouin gain, $g_{B} P_{R}$ , and Raman gain, $g_{R} P_{P}$ . A good measure of the reflectivity at SBS threshold is the exponential of the area underneath the total gain curve normalized to the Raman output power. This is approximately the same for both gain plots.

Fig. 3 Stokes gain per unit length at SBS threshold for (a) 25 m fiber and (b) 150 m fiber. The total gain is the sum of the Brillouin and Raman gain.

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To mathematically validate these observations, Eqs. (12) or (13) can be used to obtain an approximate solution for the Stokes power at reflectivities of the order of 1% [11] or less in co-pumped or counter-pumped configurations. For simplicity, we treat here the co-pumped case and also assume one Stokes channel and negligible loss. The counter-pumped case does not have a closed form solution, but the conclusions drawn for the co-pumped case hold true. Equation (12) can be used in Eq. (2) along with the distributed noise terms to obtain a solution for the evolution of the Stokes light. While this solution is more exact and was used by the authors to validate the numerical simulations, a more instructive but less exact solution can be obtained by considering a localized noise source injected at $z = L$ . It can then be shown that the reflectivity, R, is given by:

R = \frac{P_{S} (0)}{P_{R} (L)} \propto P_{S} (L) {(\frac{P_{P} (0) + γ P_{R} (0) \exp [g_{R} C_{1} L]}{P_{P} (0) + γ P_{R} (0)})}^{\frac{g_{B}}{γ g_{R}} - 1},

where

P_{S} (L)

is the injected Stokes power. Since

P_{R} (0) < < P_{P} (0)

and the Raman output scales approximately as

\exp [g_{R} C_{1} L]

, Eq. (14) indicates a linear dependence in Raman power with pump power and in agreement with the results shown in Fig. 2.

3.3 Consideration of mode-field diameter

As the SBS threshold scales linearly with the effective area [12], it would appear that using a larger MFD fiber will lead to an increase in Raman signal. This is true if the fiber length was preset and sufficient pump power was available such that the amplifier is not pump limited. However, considering the 200 W pump limit set previously and optimizing for fiber length we conducted simulations for the MFD values of $7.5 μm$ , $12.5 μm$ and $17.5 μm$ . The corresponding fiber lengths are 16.7 m, 46.4 m, and 91 m respectively. As shown in Fig. 4 , the Raman signal output is almost equal for the three MFD values under consideration. This can be explained by noting that both the SBS and SRS processes scale similarly in terms of fiber length and effective area. For a given pump power, a larger MFD requires an increase in fiber length to offset for the reduction in Raman gain. Also shown in the figure is the spatial evolution of the Stokes light.

Fig. 4 Investigation of mode field diameter effect using a pump power of 200W while the fiber length varied until SBS threshold is reached. SBS reflectivity is shown in green and corresponds to each fiber length.

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4. Multi-step temperature profile

4.1 Determination of optimized fiber segments

In order to examine further power scaling, we investigated the application of multiple temperature steps along the length of the fiber. We note that a similar analysis would apply for SBS mitigation using a multiple stress profile. If the temperature difference is such that little overlap occurs between the Brillouin gain bandwidths corresponding to the various segments of the fiber, considerable SBS suppression can be obtained with the proper selection of the length of these segments. A typical value for the Brillouin gain bandwidth is 50 MHz and thus based on the empirical estimate of a shift in the peak Brillouin frequency of 2 MHz/°C [10], approximately 30 °C difference between the temperatures of adjacent fiber sections is required. One consideration is the maximum operating temperature of the fiber polymer which is typically in the range of 100-150 °C. Therefore, from a practical viewpoint up to 3 temperature steps can be applied.

To determine the optimal length of each segment, we require that the Stokes power generated in each segment is approximately the same. Consequently, no peak Stokes frequency is allowed to run away with the reflected power. Optimization of the segment lengths can be determined numerically. However, this can be tedious or computationally exhaustive. Instead, we developed a procedure to get an accurate estimate of the length of each segment based on calculating the integrated Brillouin gain for each segment. Consider N segments with one end of each segment located at $L_{1}$ , $L_{2}$ , $L_{3}$ ,… $L_{N}$ , where $L_{N} = L$ . The optimal lengths are determined by the following set of equations:

\int_{0}^{L_{1}} e^{g_{R} P_{p} z} d z = \frac{1}{N - 1} \int_{L_{1}}^{L} e^{g_{R} P_{p} z} d z,

\int_{L_{1}}^{L_{2}} e^{g_{R} P_{p} z} d z = \frac{1}{N - 2} \int_{L_{2}}^{L} e^{g_{R} P_{p} z} d z,

\int_{L_{2}}^{L_{3}} e^{g_{R} P_{p} z} d z = \frac{1}{N - 3} \int_{L_{3}}^{L} e^{g_{R} P_{p} z} d z,

.

\int_{L_{N - 2}}^{L_{N - 1}} e^{g_{R} P_{p} z} d z = \int_{L_{N - 1}}^{L_{N}} e^{g_{R} P_{p} z} d z .

While the equations above worked very well for the co-pumped configuration, we found that for the counter-pumped case better accuracy is obtained when the effective Raman gain per unit length of the system is used.

4.2 Numerical simulations of three-step profile

A set of simulations similar to those presented in Fig. 2 was carried out for a three-step temperature profile (i.e. four temperature regions) along the fiber. Again, the fiber lengths chosen were 25 m, 50 m, 75 m, 100 m, 125 m, and 150 m. The results for both co-pumped and counter-pumped configurations are shown in Fig. 5 . Note that counter-pumping provides higher Raman signal in accordance with our findings for the uniform temperature profile.

Fig. 5 (a) Raman power and (b) efficiency achieved for both co-pumping and counter-pumping as a function of seed power and length of fiber using a three-step temperature profile (i.e. four temperature regions).

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Furthermore, for both pumping configurations, considerably higher Raman power is obtained. For example, at a pump power of 214.5 W, fiber length of 25 m and seed power of 16 mW, the output signal is 41.5 W. Additionally, due to the increase in pump power at SBS threshold, the efficiency of the system is considerably higher for the multi-step temperature simulations.

Figure 6 shows the evolution of the four Stokes signals for a co-pumped 150 m fiber, seeded with 16 mW and pumped to provide at SBS threshold 4.1 W of Raman power. These Stokes signals correspond to the peak Brillouin frequencies in each segment. The length of each fiber segment was chosen using the optimization routine described above. Note that all four Stokes signals provide almost equal reflectivities.

Fig. 6 (a)The evolution of each Stokes signal in a 150 m fiber until SBS threshold was reached. (b) The characteristic evolution of each Stokes channel at the respective calculated length.

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The fiber lengths corresponding to the four segments were 114.8 m, 17.6 m, 10.3 m, and 7.3 m; as expected decreasing in length in the direction of the signal propagation. Figure 6(b) shows a “zoom in” for the region of the fiber identified in Fig. 6(a). The change of slope in three of the Stokes signals can be traced to the traverse of the corresponding Stokes light into a region where it encounters Raman gain but no Brillouin gain. Figure 7 is a drawing of said fiber showing a relative representation of the optimized segment lengths.

Fig. 7 Three-step temperature profile (i.e. four different temperature regions) applied to a 150 m fiber seeded with 16 mW showing the relative lengths of the fiber segments.

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5. Two-signal Raman amplifier

5.1 Field equations with FWM

Based on simulations of the photon returns from the mesosphere as well as experimental results, improved performance of the guide star system can be obtained by simultaneously illuminating the two sodium lines. We investigated the possibility of generating the D_2a and D_2b lines by utilizing the same Raman amplifier. Since the separation of the two lines is approximately 1.7 GHz, there is no overlap between the corresponding Brillouin gain bandwidths even when a multi-step temperature profile is considered. Consequently, the two signals should reach their respective SBS thresholds independently. However, since linear dispersion is small for such a separation, FWM must be considered. Theoretical analysis suggests that the ratio of the power at the D_2a line should be approximately 10 times that at D_2b. The second harmonic power is proportional to the square of the pump wavelength power, which indicates that the output power of the two signals in the 1178 nm Raman amplifier should be slightly higher than 3:1.

Without consideration of FWM, SBS and fiber loss, the equations describing the “two-color” Raman amplifier take the following form:

\frac{d P_{1}}{d z} = g_{R} P_{1} P_{P},

\frac{d P_{2}}{d z} = g_{R} P_{2} P_{P},

\frac{d P_{p}}{d z} = \mp γ g_{R} (P_{1} + P_{2}) P_{P},

where

P_{1}

, and

P_{2}

are the powers of signal 1 and signal 2, respectively. The ∓signs represent co and counter propagating pump waves, respectively. Based on these equations, it can be readily deduced that regardless of the pumping configuration:

P_{2} (z) = \frac{P_{2} (0)}{P_{1} (0)} P_{1} (z) .

Using a conservation equation similar to Eq. (11) and Eqs. (19) and (20), one can readily show that for the co-pumping configuration:

P_{1} (z) = \frac{C_{3} P_{1} (0) \exp [C_{3} g_{R} z]}{P_{P} (0) + γ (P_{1} (0) + P_{2} (0)) \exp [C_{3} g_{R} z]},

where

C_{3} = P_{P} (0) + γ (P_{1} (0) + P_{2} (0))

. Similarly, it can be shown that for the counter-pumped configuration an equation of the following form is obtained:

P_{1} (z) = \frac{C_{4} P_{1} (0) \exp [C_{4} g_{R} z]}{C_{4} + γ P_{1} (0) (1 + θ) (1 - \exp [C_{4} g_{R} z])},

where

θ = P_{2} (0) / P_{1} (0)

, and

C_{4} = P_{P} (L) - γ (1 + θ) P_{1} (L)

. Note that

P_{1} (L)

is determined by solving Eq. (24) with

z = L

which results in a transcendental equation.

These solutions were utilized to check the numerical accuracy of our full system of coupled nonlinear differential equations describing the evolution of the field amplitudes of the pump, the two Raman signals, and two FWM sidebands. We are interested here in amplifier operation below the SBS threshold, and hence our investigation of the FWM process would be accurate without including the Brillouin process.

The field amplitude equations are used here to capture the full FWM interactions including the effects of self- and cross-phase modulations. The derivation of the coupled system follows that provided in Ref. [16] except that we have here Raman gain instead of laser gain. Since there is little variation among the frequencies of the Raman signals and FWM sidebands, the spatial evolution of the pump field amplitude, $A_{P}$ , is given by:

\frac{d A_{P}}{d z} = - \frac{γ g_{r o} ε_{o} c n_{R} κ_{1}}{4} ({| A_{1} |}^{2} + {| A_{2} |}^{2} + {| A_{3} |}^{2} + {| A_{4} |}^{2}) A_{P},

where

A_{1}

,

A_{2}

,

A_{3}

and

A_{4}

are the field amplitudes for the two Raman signals and two sidebands, respectively.

n_{R}

is the linear index of refraction at the Raman wavelength. The overlap integral,

κ_{1}

is given by:

κ_{1} = \frac{\iint {| φ_{R} (x, y) |}^{2} {| φ_{P} (x, y) |}^{2} d x d y}{\iint {| φ_{P} (x, y) |}^{2} d x d y} .

For the sake of brevity, we write the spatial evolution of either of the Raman signals as:

\frac{d A_{i}}{d z} = \frac{g_{r o} ε_{o} c n_{P} κ_{2}}{4} {| A_{P} |}^{2} A_{i} + \frac{i ω_{R} n^{(2)} κ_{3}}{c} (\begin{array}{l} ({| A_{i} |}^{2} + 2 \sum_{j \neq i} {| A_{j} |}^{2}) A_{i} \\ + 2 A_{i}^{*} A_{k} A_{i + 2} \exp [i β^{(2)} {(Δ ω^{'})}^{2} z] \\ + 2 A_{k}^{*} A_{3} A_{4} \exp [2 i β^{(2)} {(Δ ω^{'})}^{2} z] \\ + A_{k}^{2} A_{k + 2}^{*} \exp [- i β^{(2)} {(Δ ω^{'})}^{2} z] \end{array}),

where here

i = 1, 2

, and

j = 1, 2, 3, 4

. The value of the index kis 2for

i = 1

, and 1for

i = 2

.

n_{P}

is the linear index of refraction at the pump wavelength,

n^{(2)}

is the nonlinear index of refraction,

Δ ω^{'}

is the frequency separation between the two signals, and

β^{(2)}

is the group velocity dispersion parameter. The second and third terms on the right hand side of the equation above are FWM terms and represent self (SPM) and cross-phase modulation (XPM), respectively, and terms (4-6) are FWM terms representing energy transfer to the sidebands. The overlap integrals,

κ_{2}

and

κ_{3}

are given by:

κ_{2} = \frac{\iint {| φ_{R} (x, y) |}^{2} {| φ_{P} (x, y) |}^{2} d x d y}{\iint {| φ_{R} (x, y) |}^{2} d x d y},

κ_{3} = \frac{\iint {| φ_{R} (x, y) |}^{4} d x d y}{\iint {| φ_{R} (x, y) |}^{2} d x d y} .

The spatial evolution of the FWM sidebands is given by:

\frac{d A_{i + 2}}{d z} = \frac{g_{r o} ε_{o} c n_{P} κ_{2}}{4} {| A_{P} |}^{2} A_{i + 2} + \frac{i ω_{R} n^{(2)} κ_{3}}{c} (\begin{array}{l} ({| A_{i + 2} |}^{2} + 2 \sum_{j \neq i + 2} {| A_{j} |}^{2}) A_{i + 2} \\ + A_{i}^{2} A_{k}^{*} \exp [- i β^{(2)} {(Δ ω^{'})}^{2} z] \\ + 2 A_{1} A_{2} A_{k + 2}^{*} \exp [- 2 i β^{(2)} {(Δ ω^{'})}^{2} z] \end{array}) .

5.2 Numerical simulations and analysis of FWM

We examined FWM for the case of a three-step temperature applied along the length of fiber. Signal 1 was seeded at 45 mW while signal 2 was seeded at 15 mW. This ensured a Raman output power ratio of approximately 3:1. The pump power for the co-pumped case was chosen to be 15% below the pump level at SBS threshold; thus allowing the two-color Raman amplifier to operate comfortably below the SBS threshold. For a fair comparison, the pump power in the counter-pumping configuration was chosen such that the same Raman power was obtained as in the co-pumped configuration. Figure 8 shows the evolution of the two Raman signals along the direction of the signal for a fiber of length 150 m as well as the evolution of the two sidebands for both the co-pumped and counter-pumped configurations. For the former, approximately 90 mW of FWM power is obtained. FWM power is defined as the total optical power in the sidebands. This represents 3.2% of the total output power. For the counter-pumping configuration less FWM power is obtained. In this case, the FWM power is approximately 70 mW corresponding to 2.5% of the total output power. Had the SPM and XPM terms not been included in our simulations, the difference would be less than a 10% increase in sideband power; thus indicating the FWM process is effectively phase-matched.

Fig. 8 (a) The spatial evolution of the two Raman signals in a 150 m fiber for co-pumped and counter-pumped configurations. (b) The spatial evolution of the corresponding FWM sidebands.

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Similar simulations were conducted for a 25 m fiber. Again, the pump power was chosen to be 15% below the SBS threshold. The results of the simulations are shown in Fig. 9 . For the co-pumped configuration, approximately 325 mW of FWM power was obtained which represents 1.76% of the total output power and is 3.6 times the 150 m simulation. For the counter-pumped configuration, 267 mW of FWM power was obtained corresponding to 1.46% of the total output power and is 3.8 times the 150 m counter-pumped simulation. It is interesting to note that the Raman output power in the case of the 25 m simulations is much higher, but that the FWM percentage is significantly lower.

Fig. 9 (a) The spatial evolution of the two Raman signals in a 25 m fiber for co-pumped and counter-pumped configurations. (b) The spatial evolution of the corresponding FWM sidebands.

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This FWM process is degenerate and should scale as $P_{1}^{2} P_{2}$ where $P_{1}$ here represents the Raman signal with the higher output power. However, for low conversion, the FWM process scales quadratically with the square of the fiber length. A “back of the envelope” calculation would then indicate the FWM power for the 25 m case would be approximately 8 times that of the 150 m case. We attribute the difference between this calculation and the numerical simulations, to the spatial profiles of the Raman signals which exhibit more of a rapid rise in the case of the 25 m fiber. Regardless, while FWM power exceeding 1% of the output power is rather high, these simulations indicate that a two-line guide star system based on a single Raman amplifier is worth further exploration.

6. Summary

In summary, we have developed a detailed core-pumped single-frequency Raman amplifier model and used it to study the scalability of generating 1178 nm for sodium guide star applications. We have shown that when the fiber length is optimized, the amplifier output scales linearly with available pump power. In order to mitigate the SBS process for further power scaling, an optimized multi-step temperature distribution was utilized. Finally, we considered the feasibility of generating a two-line Raman amplifier system for use in a sodium guide star beacon from a single Raman amplifier by examining four-wave mixing (FWM).

Acknowledgments

This work was funded by AFRL/RD Technical Council. We would like to thank Harold Miller, Gerry Moore and Allen Paxton for fruitful discussions.

References and links

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Theoretical analysis of single-frequency Raman fiber amplifier system operating at 1178nm

Abstract

1. Introduction

2. Theory of single-frequency Raman amplifiers

2.1 Raman amplifier power equations

2.2 Initiation of SBS from Brillouin and Raman noise

3. Analysis of uniform temperature profile

3.1 Comparison of co and counter pumped configurations

3.2 Raman signal in relation to fiber length and seed power

3.3 Consideration of mode-field diameter

4. Multi-step temperature profile

4.1 Determination of optimized fiber segments

4.2 Numerical simulations of three-step profile

5. Two-signal Raman amplifier

5.1 Field equations with FWM

5.2 Numerical simulations and analysis of FWM

6. Summary

Acknowledgments

References and links

Cited By

Figures (9)

Equations (30)

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