Numerical Investigation of Raman-Assisted Four-Wave Mixing in Tapered Fiber Raman Fiber Amplifier

Abstract

The generation of unwanted higher-order Raman effects is the main factor restricting the power scaling of Raman fiber amplifiers (RFAs). This phenomenon arises from an interplay of physical processes, including stimulated Raman scattering (SRS), four-wave mixing (FWM), and the intricate temporal and spectral dynamics. Tapered fibers have demonstrated excellent nonlinear effects suppression characteristics due to the varying core diameter along the fiber, which is widely used in ytterbium-doped fiber lasers. In this paper, a comprehensive numerical investigation is conducted on the core-pumping tapered fiber RFAs considering Raman-assisted FWM. The higher-order Raman power in the tapered fiber is always kept at a low level, showing a weak Raman-assisted FWM effect. A numerical investigation is conducted to study the impact of the tapering ratio, the lengths of the thin part, tapered region, and thick part on the higher-order Raman threshold of RFAs. Furthermore, the impact of phase mismatch variations caused by changes in the seed wavelength, on the output signal power and nonlinear effects is analyzed. This paper presents, for the first time, a study on core-pumped RFAs using tapered fibers, providing a novel perspective on enhancing the power of RFAs.

1. Introduction

Benefiting from the properties of broad gain bandwidth, flexible wavelength selection and cascade Raman generation, Raman fiber amplifiers (RFAs) with the absence of rare-earth ion doping have attracted research attention [1,2,3,4]. The primary advantages of RFAs are manifested in two key aspects: wavelength extension and power scaling. On one hand, the cascade Raman effect is capable of producing lasers from visible to mid-infrared wavelengths [5,6,7,8,9,10,11,12,13,14]. On the other hand, RFAs utilizing Raman gain have been able to achieve higher output power levels [15,16,17,18]. Currently, the maximum power has reached the 10-kW level [19]. However, further power enhancement is constrained by the emergence of higher-order Raman scattering. Typically, in RFAs, the generation of higher-order Stokes light leads to a swift conversion of signal power into this unwanted byproduct [15,17].

Additionally, a recent study demonstrates that the suppression of the four-wave mixing (FWM) effect in fibers can increase in the higher-order Raman threshold [20]. The FWM effect is another third-order nonlinear effect that requires specific phase-matching conditions to manifest prominently. In the absence of gain, the FWM effect is typically negligible due to the difficulty in achieving phase-matching within refractive index perturbations. However, in the presence of laser gain, strong FWM effect can occur even if there is a phase mismatch among pump, signal, and higher-order Raman [21,22,23,24]. The higher-order Raman generated by the FWM effect can be amplified by Raman gain, resulting in what is known as Raman-assisted FWM. This reduces the threshold for the generation of higher-order Raman, thereby imposing limitations on the further scaling of laser power.

Typically, two main approaches are used to suppress unwanted nonlinear effects like higher-order Raman in fiber lasers: increasing the core diameter and reducing the fiber length. However, while these methods can reduce higher-order Raman generation, they also decrease pump conversion efficiency. Moreover, an enlarged core with more mode numbers will lead to the degradation of beam quality. In the field of ytterbium-doped fiber lasers (YDFLs), special large-mode-area (LMA) fibers such as W-type fibers [25,26] and all-solid photonic bandgap fiber [27,28] have been explored for suppress stimulated Raman scattering (SRS). However, the fabrication of these special fibers is inherently more complex with stringent manufacturing tolerances. Their inflexibility in integrated applications is a widely acknowledged limitation that hinders their broad adoption. Tapered fibers offer an effective alternative for mitigating SRS and higher-order modes by varying longitudinal cladding and core dimensions. These fibers are characterized by controllable loss and manageable fabrication processes and have been successfully integrated into a variety of YDFLs with commendable results [29,30,31,32,33,34,35,36,37]. As a kind of the LMA fiber, tapered fibers significantly enhance the threshold for SRS while maintaining high beam quality [31], making them ideal for advanced fiber optic applications.

The strong nonlinear effect of tapered fiber’s thin part renders it highly suitable for Raman amplification in RFAs. The adiabatic conditions in the tapered region effectively preserve the modal characteristics of the thin part and reduce the number of modes at the thick part. Additionally, the thick part can be strategically utilized to suppress higher-order Raman conversion and convert residual pump. Currently, there are limited theoretical and experimental studies on RFAs using tapered fibers. While several models have been developed for YDFLs that are based on tapered fibers [35,38], the interplay between Raman gain and FWM effect mechanisms in RFAs is fundamentally different from that of active gain. This essential difference necessitates the development of a model specifically tapered fiber Raman amplifiers. Furthermore, the phase mismatch of the FWM effect between the thin part and the thick part of the tapered fibers is different, which may have an inhibiting influence on the FWM effect.

In this work, a model of the core-pump RFA using tapered fiber is established, taking into consideration the Raman-assisted FWM effect. The variation of the effective mode interaction area and phase mismatch with fiber position in tapered fiber has been analyzed. Compared with conventional fibers, tapered fibers exhibit reduced Raman-assisted FWM effects in power evolution. Additionally, the influence of different structural parameters of tapered fiber on higher-order Raman threshold is investigated, and the influence of phase mismatch variations caused by different seed wave-lengths on RFA is studied.

2. Numerical Model

In this section, we establish a RFA model that takes the Raman-assisted FWM effect into account and incorporates tapered fiber into the model. The changes in the propagation constant of the fundamental mode and the corresponding phase mismatch in the tapered fiber are studied.

2.1. RFA Model Considering the FWM Effect

Ignoring the influences of self-phase modulation and cross-phase modulation, the RFA model considering the Raman-assisted FWM effect can be rewritten as follows when only Raman gain is taken into account [39,40]:

$${\displaystyle \frac{d{E}_1}{dz}}=-{\displaystyle \frac{g_R}{2A_{eff}}}{\displaystyle \frac{{\lambda }_2}{{\lambda }_1}}{\left|E_2\right|}^2{E}_1+j{\gamma }_1{E}_2{E}_2{E}_3^{*}{e}^{-j\Delta kz}-{\displaystyle \frac{{\alpha }_1}{2}}{E}_1$$

(1)

$${\displaystyle \frac{d{E}_2}{dz}}=-{\displaystyle \frac{g_R}{2A_{eff}}}{\displaystyle \frac{{\lambda }_3}{{\lambda }_2}}{|E_3|}^2{E}_2+{\displaystyle \frac{g_R}{2A_{eff}}}{\left|E_1\right|}^2{E}_2+2j{\gamma }_2{E}_1{E}_3{E}_2^{\ast }{e}^{j\Delta kz}-{\displaystyle \frac{{\alpha }_2}{2}}{E}_2$$

(2)

$${\displaystyle \frac{d{E}_3}{dz}}={\displaystyle \frac{g_R}{2A_{eff}}}{\left|E_2\right|}^2{E}_3+j{\gamma }_3{E}_2{E}_2{E}_1^{*}{e}^{-j\Delta kz}-{\displaystyle \frac{{\alpha }_3}{2}}{E}_3$$

(3)

where the last and penultimate terms on the right side of the equation represent fiber loss and FWM process, respectively. Subscript k = 1, 2, 3 refers to pump, signal and higher-order Raman, E_k is the complex electric field, and the |E|² represents the power P, λ is the wavelength, g_R is the Raman gain coefficient and its unit is m/W, ${\gamma }_k=2\pi n_2/{\lambda }_k{A}_{eff}$ is nonlinear gain coefficient, where n₂ is a nonlinear refractive index, usually 2.3 × 10⁻²⁰ m²/W in silicon-based fibers. Δk is phase mismatch, defined as (2β₂ − β₁ − β₃), without considering contribution of nonlinear effects. β is the propagation constant of each wavelength, and the calculation method is ${\beta }_i=n_{eff,i}\cdot 2\pi /{\lambda }_i$, n_eff is effective refractive index of the fundamental mode. A_eff is the effective area of model interaction between the fundamental modes of pump and signal, and the calculation method is as follows [41]:

$$A_{eff}={\displaystyle \frac{{\displaystyle {\int }_0^{Rcladding}{\displaystyle {\int }_0^{2\pi }{\left|{\phi }_1\left(r,\theta \right)\right|}^{2}rdrd\theta {\displaystyle {\int }_0^{Rcladding}{\displaystyle {\int }_0^{2\pi }{\left|{\phi }_2\left(r,\theta \right)\right|}^{2}rdrd\theta }}}}}{{\displaystyle {\int }_0^{Rcladding}{\displaystyle {\int }_0^{2\pi }{\left|{\phi }_1\left(r,\theta \right)\right|}^2{\left|{\phi }_2\left(r,\theta \right)\right|}^{2}rdrd\theta }}}}$$

(4)

where φ is transverse mode field distribution, r and θ are the radial and angular coordinates of the fiber cross section, respectively, R_cladding is the radius of the cladding. Since only the fundamental mode is considered, it is assumed that the A_eff of the pump and the signal and that of the signal and the higher-order Raman are equal [40].

2.2. Consider Tapered Fiber Parameters in the Model

The RFA model using tapered fiber is obtained by changing the fiber structure. Figure 1a illustrates the schematic diagram of the tapered fiber, which consists of three parts: the thin part, the tapered region, and the thick part. This paper focuses on studying a linear tapered fiber without considering any changes in bump degree within the tapered region. So the change in the core diameter along the length of the tapered region can be expressed by the following equation [42]:

$$D_{core}\left(z\right)={\displaystyle \frac{D_2-D_1}{L}}z+D_1$$

(5)

where D₁ and D₂ are the diameters of the thin end and the thick end, respectively. The tapering ratio is defined as D₂/D₁. L is the length tapered region; z starts at the beginning of the tapered region. The simulation studies a fiber with a fixed cladding-to-core ratio, allowing for the calculation of the corresponding cladding diameter.

Considering that the core and cladding diameters of the tapered fiber change along the longitudinal direction, Δk and A_eff should be modified to Δk(z) and A_eff(z), respectively. Taking a tapered fiber with specifications ranging from 10/125 to 20/250 μm as an example, the lengths of the thin part, tapered region and thick part are 10, 40 and 10 m, respectively. The NA of the fiber is 0.08 and the core refractive index is 1.465. Figure 1b shows the changes in A_eff, and Δk at different positions in the fiber when the pump, signal, and Raman-assisted FWM wavelengths are 1070 nm, 1120 nm, and 1175 nm respectively. The three parts of the tapered fiber can be clearly identified. The A_eff(z) increases with the increase in fiber length, which can inhibit Raman conversion.

The n_eff increases as the core diameter increases, leading to a variation in the propagation constant (β).

Table 1. The β of different wavelengths at both parts of the tapered fiber.

Wavelength, nm	β in the Thin Part, m−1	β in the Thick Part, m−1
1070	8,596,483.4	8,600,420.3
1120	8,212,370.1	8,216,308.3
1175	7,828,244.0	7,832,173.1

presents the β of different wavelengths at both parts of the fiber depicted in Figure 1b. Although the β of each wavelength changes a little, the effect on the Δk cannot be ignored. As can be seen from Figure 1b, Δk increases by nearly two times (from 12.7 m⁻¹ to 23.2 m⁻¹). According to Equations (1)–(3), phase mismatch plays a significant role in the Raman-assisted FWM process. Therefore, the phase mismatch changes after passing through the tapered region may have an impact on the power evolution.

3. Results and Discussion

Before conducting the simulation analysis, it is necessary to properly describe the simulation structure. Figure 2 illustrates the schematic diagram of core-pumped RFA. The pump and signal light are injected into the thin end of the tapered fiber by wavelength division multiplexing (WDM). The pump source is a single-mode YDFL at 1070 nm with a maximum pump power limit of 300 W to prevent the WDM from overheating. The seed source is a single-mode YDFL at 1120 nm, and the power is adjustable from 0 to 30 W. The corresponding higher-order Raman wavelength is 1175 nm, and the initial noise power is set to 10⁻⁸ W. To avoid Fresnel reflection, an angle cutter is applied to the fiber’s output end. Other parameters in the simulation are shown in

Table 2. Parameters used in the simulations.

Parameter	Value	Parameter	Value
ncore	1.465	α	3 dB/km
NA	0.08	gR	6.3 × 10−14 m/W
Thin end core diameter	10 μm	n2	2.3 × 10−20 m2/W
Thin end cladding diameter	125 μm

.

3.1. Power Evolution in RFAs

The power evolution in RFA is simulated in this section, to compare performances of conventional fiber and tapered fiber on seed amplification and higher-order Raman suppression respectively. The seed power is define to be 20 W here, while the fiber length is fixed to 60 m for both fiber types. In the case of RFA based on tapered fiber, the fiber longitudinal structure is depicted in Figure 1b, whose parameters are the same as Section 2.2. For comparison, the core/cladding diameter of the conventional fiber is 10/125 μm. Thus, the A_eff and Δk of the conventional fiber are same as those of the thin end of tapered fiber. Other parameters consistent with Table 2.

The power evolution of the pump and signal in the RFAs is shown in Figure 3a, and the power evolution of the higher-order Raman is shown in Figure 3b. The Raman conversion is observed in the tapered region due to the gradually increased core diameter. The maximum power of RFA appears at 42 m and 52.4 m with 297.7 W and 294.14 W in case of conventional fiber and tapered fiber respectively, indicating the similar signal amplification ability. However, the power of higher-order Raman in the conventional fiber is always higher than that in the tapered fiber, as shown in Figure 3b. The higher-order Raman suppression ratio, defined as the ratio of the output power of the higher−order Raman to that of the signal, is calculated to be −31.7 dB at the maximum power point in conventional fiber, while it is −34.6 dB in tapered fibers, indicating that tapered fibers exhibit a lower level of Raman-assisted FWM effects.

3.2. Structural Optimization of Tapered Fiber

The tapered fiber has numerous parameters, necessitating a thorough investigation into the impact of each parameter on RFA. The higher-order Raman threshold is defined as the output power of the signal light when the higher-order Raman suppression ratio is −20 dB. In this section, the influence of various structural parameters on the higher-order Raman threshold is studied, and the design criteria of tapered fiber are summarized. The main variable parameters are the tapering ratio, the thin part length, tapered region length and thick part length. The simulation results are shown in Figure 4.

Figure 4a shows the change in higher-order Raman threshold when the length of the thin part length is fixed at 10 m, while varying the tapering ratio, tapered region length and thick part length. As the length of the tapered region increases, the total fiber length needs to decrease, resulting in a slow increase in output power. This may be attributed to the fact that when the tapered ratio is fixed, an increase in tapered region length implies a slower rate of core diameter change. Consequently, there is an increased portion of fiber with a small core diameter which enhances the nonlinear effect in the fiber. In this case, if the total fiber length remains unchanged, the power of the Raman-assisted FWM wavelength will rise rapidly.

Figure 4b shows the change in higher-order Raman threshold output power when the length of tapered region is fixed at 10 m, while the thin part length, tapering ratio and thick end length vary together. As the length of the thin part increases, corresponding reduction in the length of thick part becomes necessary to prevent power amplification of Raman-assisted FWM wavelength. The reduced length of the thick part is much longer than the increased length of the thin part, resulting in a rapid decrease of the total fiber length. This leads to higher output power when the loss is fixed. Additionally, changing the length of the thin part has a greater impact on higher-order Raman threshold output power compared to changing the length of the tapered region.

The output power decreases as the tapering ratio increases, as shown in Figure 4. This is because the increase in the tapering ratio reduces the Raman gain, necessitating a longer, thicker part for further pump conversion. However, this leads to increased total fiber loss and a decrease in signal output power.

The simulation results of the above two cases show that the higher-order Raman threshold can be as high as 285 W with a reasonable design of the tapered fiber structure. Several tapered fibers with optimized designs have been selected for reference, and their parameters are shown in

Table 3. The optimized structure of tapered fiber.

Fiber	Tapering Ratio	Thin Part Length, m	Tapered Region Length, m	Thick Part Length, m	Fiber Length, m
1	2	30	10	50	90
2	2	40	10	25	75
3	2.5	40	10	40	90
4	3	50	10	20	70

.

3.3. Influence of Seed Wavelengths on RFA

This section focuses on investigating nonlinear effects in tapered fibers through adjustments in seed parameters. The seed has a significant influence on output performance of RFA because it facilitates the pump to convert into the signal, thereby accelerating the process of stimulated amplification [8,43]. Seed parameters such as wavelength, power, and temporal stability have varying effects on RFA. In this section, the impact of different seed wavelengths on signal output power and nonlinear effects are studied. When considering Raman-assisted FWM effect in an RFA, variations in the wavelength of the seed result in frequency separation between the pump and signal, leading to two distinct frequency separation: one caused by Raman-assisted FWM effect and another by the higher-order SRS effect.

Figure 5 illustrates the corresponding Raman gain for different frequency separations. It is widely accepted that the frequency separation between the wavelength of higher-order Raman and the seed is commonly assumed to be constant at 13.2 THz, irrespective of variations in the seed wavelength. Therefore, the Raman gain coefficient for higher-order Raman is set to g_R as listed in Table 2. Meanwhile, the Raman-assisted FWM effect produces a frequency separation consistent with that of both signal and pump, so the gain coefficient for Raman-assisted FWM effect remains identical to that of the corresponding seed. The frequency separation corresponding to different seed wavelengths and the wavelengths for higher-order Raman and Raman-assisted FWM are calculated, while keeping the pump wavelength at 1070 nm. These calculations are presented in

Table 4. Parameters at different seed wavelengths.

Seed wavelength, nm	1110	1120	1130
Frequency separation with pump, THz	10.0	12.5	14.8
Higher-order Raman wavelength, nm	1167	1178	1189
Raman-assisted FWM wavelength, nm	1153	1175	1197
Raman gain coefficient of signal	0.73 gR	gR	0.97 gR

.

Fiber 1 from Table 3 is selected for simulation, and the A_eff and Δk at different wavelengths are shown in Figure 6. It can be observed that A_eff is not sensitive to seed wavelengths, while Δk exhibits strong wavelength sensitivity. The longer the seed wavelength, the greater the Δk difference between the two ends of the tapered fiber.

The simulation results are presented in Figure 7, the fraction of nonlinear effect is defined as the ratio (dB) of the power of the Raman-assisted FWM or higher-order SRS effects to the power of the signal.

Figure 7 shows the variation of signal output power and the fraction of nonlinear effect with pump power at different seed wavelengths and the seed power is fixed at 20 W. It is noted from Figure 7a that the 1110 nm signal has the highest power at the 300 W pump power, which is because the smaller quantum defect at the short wavelength. It is also found from Figure 7b that the fraction of nonlinear effect is greatly affected by the seed wavelength.

When the seed wavelength is adjusted from 1120 nm to 1130 nm, the changes of output power can be ignored. However, there is a notable reduction in the Raman-assisted FWM effect, which decreases by 4.52 dB, and a corresponding decline in the higher-order SRS effect, which drops by 1.18 dB. Similarly, adjusting the seed wavelength from 1120 nm to 1110 nm results in a significant power increase of 4.3 W. This adjustment also leads to a more substantial decrease in the FWM effect, by 14.16 dB, and a reduction in the higher-order SRS effect, by 3.33 dB. These observations demonstrate that the seed wavelength can change the gain of Raman-assisted FWM effect. Meanwhile, under conditions in which the pump is fully converted, reducing the seed wavelength may lead to an increase in output power, attributable to the decreased quantum defect.

4. Conclusions

In conclusion, the Raman-assisted FWM effect in RFA is studied by numerical simulation. The impact of tapered fibers on power evolution is investigated, and the introduction of the tapered region is found to reduce Raman-assisted FWM effects. Additionally, the influence of tapered fiber structural parameters on the performance of the amplifier is discussed. The higher-order Raman threshold is primarily influenced by the length of the thin part due to the strong nonlinear effect. By optimizing the structural parameters, similar output characteristics can be achieved across different tapered fiber structures. Furthermore, it is observed that seed wavelength significantly affect RFAs, changing the seed wavelength can reduce the gain of the Raman-assisted FWM effect. This study introduces tapered fibers into core-pumped RFA for the first time and compares their performance with that of conventional fibers, providing valuable insights for the design and optimization of RFAs.