Six-mode extended L-band EDFA with a low differential modal gain

Q. Qiu; Z. M. Gu; C. J. Shi; Y. Chen; Y. Lou; L. He; J. G. Peng; H. Q. Li; Y. B. Xing; Y. G. Chu; N. L. Dai; J. Y. Li

doi:10.1364/OSAC.418512

1. Introduction

The orthogonality of the modes in the optical fiber allows each mode to transmit information separately, creating a new dimension for optical fiber communication [1]. Compared with single-mode fiber (SMF), few-mode fiber (FMF) has a larger core which leads to a higher nonlinear threshold [2] and it integrates the coherent optical communication technologies, which can increase communication capacity significantly [3]. The long-haul transmissions of mode division multiplexing (MDM) communication system are largely founded on the outstanding characteristics of the few-mode erbium-doped fiber amplifier (FM-EDFA). Compared with the traditional EDFA, the FM-EDFA has another important parameter-differential modal gain (DMG), not just the gain and noise figure, which affects the average and outage capacities of the MDM system [4]. So far, Three main ways have been used to reduce DMG: a. managing the spatial distribution of the erbium concentration, such as double-ring doping [5–8], multi-ring doping [9], large-scale doping [10,11], and other complex doping profiles [12,13]; b. adjusting the mode field distribution by controlling the refractive index profile, for example, multi-element fiber amplifier [14,15], ring core erbium-doped fiber [12,16–18], center depressed core index profile [19] and trench-assisted fiber [17,20]; c. tailoring the pump field intensity distribution, including changing the mode content of pump light [21–23] and cladding pumping [20,24]. Apart from the strategies mentioned above, many efforts have been made to minimize the DMG such as a laser-inscribed void introducing extra mode-dependent loss [25], oversizing a core diameter or increasing numerical aperture (NA) to expand the number of modes but only use LP_n1 modes [26,27], gradient descent optimization algorithm [28] and genetic algorithm [29] as guidance of doping. These technologies have obtained DMG values of less than 1 dB across the C-band.

Generally speaking, one technique is not enough to reduce DMG, while multiple techniques used in combination are preferred, especially in higher mode count amplifiers. A trench-assisted EDF combination with the cladding-pumped schema was proposed [20] to reduce DMG for C-band, which doesn’t require the complicated fiber manufacturing process. Nevertheless, few studies on L-band have been reported, which are the best choice to address the growing need for increased data network when C-band is exhausted in the future [30]. The gain at L-band is relatively inefficient because it suffers from the excited-state absorption (ESA) after the 1580 nm wavelength [31] and has a much smaller emission cross-section than C-band. Well-chosen C-band wavelengths as auxiliary pumps [32,33] or sole pumps [34,35] were used to enhance the gain of L-band EDFA. References [36–38] reported that the utilization of higher power C-band light pumping to extend spectral bandwidth and improve power conversion efficiency (PCE) of L-band EDFAs. For L-band FM-EDFA using cladding pumped-schema, the signal cannot be sufficiently amplified if the fiber length is too short. However, cladding pumping may compromise the optimization effect of DMG because the fiber length used in L-band is usually longer, the pump light intensity will obey the eigenmode mode field distribution rather than uniform distribution at the end of the fiber. Therefore, the cladding pumping is not suitable to reduce DMG for L-band FM-EDFA especially with a low core-cladding ratio.

In this paper, we present a thorough numerical investigation, taking advantage of the mode hybrid core pumped scheme to reduce the DMG of the trench-assisted 6-mode erbium-doped fiber amplifier. The gain efficiency is improved by tightly confining the pump light in the fiber core with a deep refractive index trench and using a C-band pump instead of 1480 nm or 980 nm. Our simulations confirm that the trench and the mode hybrid core pumped scheme play a significant role in DMG optimization and in improving gain efficiency.

2. Fiber design

Figure 1 (a) shows the schematic diagram of our proposed trench-assisted 6-signal-mode EDF with a uniform erbium doping profile. Table 1 shows the definition of the fiber structure parameters where several values are left to specified. At 1600 nm wavelength, for the FM-EDF with $N{A_1} = \textrm{0}\textrm{.12}$, $a = 11\mu m$, a map of allowable 6-signal mode area for the combination of d and $N{A_2}$ is shown in Fig. 1 (b). The white region is the 6-signal-mode working area. The two gray areas from left to right indicate the LP₃₁ mode is far from the cut-off and the LP₀₂ mode is cut-off.

Fig. 1. (a) Refractive index profile of proposed trench-assisted 6-mode EDF (colored region represents the doping region). (b) Map of the allowable 6-signal mode area for the trench width ($d$) and $N{A_2}$ of trench-assisted 6-mode EDF

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Table 1. Fiber Parameters

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To demonstrate the confinement degree of the core to energy, the mode-related CF (confinement factor) is introduced [20] as the ratio between the power in the core and that of the whole fiber cross-section for a certain mode. Increasing CF by adding a refractive index trench can bring increased gain efficiency with the augment of the proportion of core power. And the energy of all modes is tightly bounded in the core combined with high-order mode pumping, which can dramatically reduce the DMG. Figure 2(a) illustrates the relationship between the CF and $N{A_2}$ for $d = 4\mu m$. The CF of all modes at 1600 nm increases with the growth of $N{A_2}$ and gradually approaches to 1. When $N{A_2}$ is greater than 0.12, the CF of all modes exceeds 90%. In the case of $d = 4\mu m$ and $N{A_2}\textrm{ = 0}\textrm{.19}$, the normalized intensity of each mode is shown in Fig. 2(b). The correlation between the pump mode and signal mode depends on their overlapping degree. The greater an overlap is, the greater the gain of signal mode is. For convenience, the overlap factor of the pump and signal intensity profile is defined as:

(1)$$\Gamma _{p,i}^{s,j} = \int\limits_0^{2\pi } {\int\limits_{ - a}^a {{I_{s,j}}(r,\varphi )} } {I_{p,i}}(r,\varphi )rdrd\varphi ,$$

where i and $j$ are the mode order of pump and signal, respectively. As listed in Table 2, the relationships of the overlap factor between four mode groups with pump modes including LP₀₁, LP₁₁, LP₂₁, and LP₀₂ are as follows:

(2)$$\begin{array}{l} \Gamma _{p,01}^{s,01} > \Gamma _{p,01}^{s,02} > \Gamma _{p,01}^{s,11} > \Gamma _{p,01}^{s,21}\\ \Gamma _{p,11}^{s,01} \approx \Gamma _{p,11}^{s,11} \approx \Gamma _{p,11}^{s,21} > \Gamma _{p,11}^{s,02}\\ \Gamma _{p,21}^{s,11} \approx \Gamma _{p,21}^{s,21} > \Gamma _{p,21}^{s,01} > \Gamma _{p,21}^{s,02},\\ \Gamma _{p,02}^{s,02} > \Gamma _{p,02}^{s,01} > \Gamma _{p,02}^{s,11} \approx \Gamma _{p,02}^{s,21} \end{array}$$

Fig. 2. (a) The CF as a function of the $N{A_2}$ for each mode at 1600 nm wavelength. (b) Normalized intensity of each mode for 980 nm pump (solid line) and 1600 nm signal (dashed line) in trench-assisted 6-mode EDF along with the fiber radius position.

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Table 2. The normalized overlap factors of the 980 nm pump (columns) and 1600 nm signal (rows) intensity

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It is due to the difference in the degree of overlap between the pump mode and the signal mode that we can equalize the DMG by changing the mode content of the pump light.

3. Gain modeling

3.1 Giles model

The evolution of the beam in the erbium-doped fiber can be described by the rate equation [39].

(3)$$\begin{array}{l} \frac{{d{P_k}}}{{dz}} = {u_k}({\sigma _{ek}} - {\sigma _{esa,k}})\int\limits_0^{2\pi } {\int\limits_0^\infty {{i_k}(r,\varphi )} } {n_2}(r,\varphi ,z)rdrd\varphi ({P_k}(z) + mh{v_k}\Delta v)\\ - {u_k}{\sigma _{ak}}\int\limits_0^{2\pi } {\int\limits_0^\infty {{i_k}(r,\varphi )} } {n_1}(r,\varphi ,z)rdrd\varphi {P_k}(z), \end{array}$$

where the ${u_k} ={\pm} 1$ means the direction of propagation and is either forward (1) or backward (-1); $mh{v_k}\Delta v$ is the contribution of the spontaneous emission due to the m degenerate states; ${\sigma _{ek}}$ and ${\sigma _{ak}}$ are the emission and absorption cross-section at wavelength position k; ${\sigma _{esa,k}}$ is the ESA cross-section; ${P_k}$ and ${i_k}$ are the total power and the normalized intensity distribution for beams; ${n_1}(r,\varphi ,z)$ and ${n_2}(r,\varphi ,z)$ are the erbium ions density in the lower level and upper level. The two integrals are respectively the overlap of the optical intensity profile. In addition, the distribution of the ions at the upper and lower levels on the cross-section of the fiber indicates that only the part overlapping with erbium ions will experience attenuation and amplification. For a two-level system, the variations on ion populations of the ground and metastable states with the k beams are shown below:

(4)$$\begin{aligned}\frac{{d{n_2}(r,\varphi ,z)}}{{dz}} &= {n_1}(r,\varphi ,z)\sum\nolimits_k {\frac{{{P_k}(z){i_k}(r,\varphi )({\sigma _{ak}})}}{{h{v_k}}}} \\ & - {n_2}(r,\varphi ,z)\sum\nolimits_k {\frac{{{P_k}(z){i_k}(r,\varphi )({\sigma _{ek}} - {\sigma _{esa,k}})}}{{h{v_k}}} - } \frac{{{n_2}(r,\varphi ,z)}}{\tau },\end{aligned}$$

(5)$${n_t}(r,\varphi ,z) = {n_1}(r,\varphi ,z) + {n_2}(r,\varphi ,z),$$

where the particle conservation for the two-level system is illustrated, and ${n_t}(r,\varphi ,z)$ represents the total erbium ion density.

3.2 Multimode rate equations for Er³⁺ doped amplifier

When the Er³⁺ dopant distribution is uniform in the longitudinal direction and the intra-mode coupling in the fiber is ignored, the simultaneous amplification of different signal modes will induce a competitive effect [40] affected by the pump field, signal field and transverse doping distribution. This can make ions redistribute on the upper and lower levels, as illustrated in Fig. 3 (a).

Fig. 3. (a) Three-level diagram of Er³⁺ in FM-EDF. (b) Schematic diagram of fiber amplifiers.

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The complicated integrals in Eq. (3) can be solved by the multilayered method [40], and the discrete model of transverse mode competition [41] in Er³⁺ doped multimode fiber amplifier after taking into consideration the influence of spontaneous emission can be expressed as follows:

(6)$$\begin{array}{l} \pm \frac{{d{P_{P,i,j}}}}{{dz}} = {\sigma _{ej}}\sum\limits_{k = 1}^M {{N_{2k}}} (z){\Gamma _{i,j,k}}{P_{P,i,j}}(z) - {\sigma _{aj}}\sum\limits_{k = 1}^M {{N_{1k}}} (z){\Gamma _{i,j,k}}{P_{P,i,j}}(z),\\ \frac{{d{P_{S,i,j}}}}{{dz}} = ({\sigma _{ej}} - {\sigma _{esa,j}})\sum\limits_{k = 1}^M {{N_{2k}}} (z){\Gamma _{i,j,k}}{P_{S,i,j}}(z) - {\sigma _{aj}}\sum\limits_{k = 1}^M {{N_{1k}}} (z){\Gamma _{i,j,k}}{P_{S,i,j}}(z),\\ \pm \frac{{d{P_{ASE,i,j}}}}{{dz}} = ({\sigma _{ej}} - {\sigma _{esa,j}})\sum\limits_{k = 1}^M {{N_{2k}}} (z){\Gamma _{i,j,k}}{P_{ASE,i,j}}(z) - {\sigma _{aj}}\sum\limits_{k = 1}^M {{N_{1k}}} (z){\Gamma _{i,j,k}}{P_{ASE,i,j}}(z)\\ + mh{v_j}\Delta v({\sigma _{ej}} - {\sigma _{esa,j}})\sum\limits_{k = 1}^M {{N_{2k}}} (z){\Gamma _{i,j,k}}, \end{array}$$

(7)$${\Gamma _{i,j,k}} = \frac{{\int\limits_0^{2\pi } {\int\limits_{{r_{k - 1}}}^{{r_k}} {{i_{i,j}}(r,\varphi )rdrd\varphi } } }}{{\int\limits_0^{2\pi } {\int\limits_0^\infty {{i_{i,j}}(r,\varphi )rdrd\varphi } } }},$$

(8)$$\frac{{{N_{2k}}(z)}}{{{N_{1k}}(z)}} = \frac{{\sum\limits_i {\sum\limits_j {\frac{{{P_{i,j}}(z){\Gamma _{i,j,k}}{\sigma _{aj}}}}{{h{v_j}}}} } }}{{\sum\limits_i {\sum\limits_j {\frac{{{P_{i,j}}(z){\Gamma _{i,j,k}}({\sigma _{\textrm{e}j}} - {\sigma _{esa,j}})}}{{h{v_j}}}} } + \frac{{{A_k}}}{\tau }}},$$

where ${P_{P,i,j}}$ and ${P_{S,i,j}}$ are the pump and signal powers of mode i and frequency j respectively, and ${P_{ASE,i,j}}$ is the i mode ASE power of the j frequency within $\Delta v$ bandwidth centered on ${v_j}$; ${\pm}$ refers to the direction of transmission, as shown in Fig. 3 (b). M is the number of divided layers. ${N_{2k}}$ and ${N_{1k}}$ are the erbium ion densities of the lower and upper levels at the ${k^{th}}$ layer, and ${A_k} = \pi (r_k^2 - r_{k - 1}^2)(k = 1,2,\ldots ,M)$ is the area of the ${k^{th}}$ layer. The power filling factor ${\Gamma _{i,j,k}}$ is defined as the ratio of the layer k to total power, which can be calculated by COMSOL when the refractive index profile of fiber is atypical. The signal gain and NF are defined as:

(9)$$Gain(dB) = 10\log 10(\frac{{{P_s}(z = L)}}{{{P_s}(z = 0)}}),$$

(10)$$NF(dB) = 10\log 10(\frac{{2{n_{sp}}(G - 1)}}{G} + \frac{1}{G}) = 10\log 10(\frac{{{P_{ASE}}}}{{hv\Delta vG}} + \frac{1}{G}),$$

(11)$${n_{sp}} = {1 / {(1 - \frac{{{\sigma _{ep}}{\sigma _{as}}}}{{{\sigma _{ap}}{\sigma _{es}}}})}},$$

where the noise factor ${n_{sp}}$ at high pump powers is simplified to Eq. (11) [39].

4. Numerical simulation results and discussion

Figure 4 (a) shows the absorption cross-section, emission cross-section and ESA cross-section of a measured extended home-made Er/Al/La/Ge/P co-doped L-band fiber from the 1500 to 1630 nm. The simulation parameters are consistent with those in Table 3 unless otherwise specified below.

Fig. 4. (a) Absorption and emission cross-sections as a function of wavelength from 1500 to 1630 nm for L-band Er³⁺ doped fiber (green line is ESA cross-section after 1580 nm). (b) Power spectrum evolutions of four mode groups with fiber length with the equal proportion of pump mode components (LP₀₁, LP₁₁, LP₂₁, LP₀₂) and a pump wavelength of 980 nm.

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Table 3. Simulation Parameters

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Figure 4 (b) shows the reabsorbing process that the pump light simultaneously excites the C-band multi-mode amplified spontaneous emission (ASE) light, and then works with the generated forward-propagating C-band ASE to provide gain for the L-band signal. References [36–38] present thorough simulations and experimental results, taking advantage of the C-band light source in the wavelength range from 1530 nm to 1550 nm to achieve higher gain than conventional laser diode pumps at 1480 nm for L-band EDFAs. Here, three wavelengths are used for L-band amplification, one is located at the absorption peak (1536 nm) while the other two are located to the left (1530 nm) and right (1550 nm) of the absorption peak. First of all, to control the DMG of four mode groups, the optimal fiber length should be determined for different pump wavelengths. The results plotted in Fig. 5 (a)-(d) show that with 980, 1530, 1536 and 1550 nm pump wavelength, the optimum gain lengths are 4.7, 10.8, 11.8, 16.2 m, and the DMG at this fiber length are 1.33, 1.19, 0.95, and 1.06 dB, respectively.

Fig. 5. Variations of gain of different signal mode groups and total gain of all modes with fiber length at (a) 980 nm, (b) 1530 nm, (c) 1536 nm, and 1550 nm pump wavelength.

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4.1 Reducing DMG by regulating pump mode content

Considering that the overlap factors of LP₀₁ pump mode and the four signal mode groups are too different from each other to be controlled, the LP₁₁, LP₂₁ and LP₀₂ modes hybrid pumped architecture is adopted. When the ratio of each pump mode is adjusted from 0 to 1, the calculated maximum DMG values depend on the pump mode contents as illustrated in Fig. 6. Three phenomena can be seen from the results in Fig. 6 (a)-(d): I) Whenever whether 980 nm or C-band wavelength pumping is used, the DMG can be lower than 0.5 dB by adjusting the pump mode contents. II) Compared to other pump wavelengths, the wavelength of 1536 nm acquires a larger region probably covering more than 2/3 of the effective area when DMG is less than 1.5 dB. DMG is less sensitive to changes in pump mode composition with 1530 nm pumping for our trench-assisted 6-mode EDF. III) No matter which pump wavelength is selected, the proportion of LP₁₁ pump mode in the region where the DMG is lower than 0.5 dB is extremely small. As the optimization effect of DMG with LP₁₁ mode pumping is not evident, only LP₂₁ and LP₀₂ pump modes are used for combined pumping. The DMG changing trend curves are plotted in Fig. 7. For all pump wavelengths, DMG is minimized when the ratio of pump power between LP₂₁ and LP₀₂ modes is around 0.7:0.3. Specifically, DMG is as low as 0.40, 0.33, 0.42 and 0.28 dB respectively when pump wavelengths are 980, 1530, 1536, and 1555 nm.

Fig. 6. Calculated maximum DMG values as a function of the ratio of LP₀₁, LP₁₁ and LP₂₁ pump mode power to total power with a pump wavelength of (a) 980 nm, (b) 1530 nm, (c) 1536 nm, and (d) 1550 nm at signal wavelength ranging from 1570 to 1618 nm.

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Fig. 7. Calculated maximum DMG values as a function of the ratio of LP₁₁ and LP₂₁ pump mode power to total power.

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Figure 8 (a)-(d) show gain and NF across the L-band for four pump wavelengths when the DMG is the smallest. The gain obtained by the 980 nm pumping is much lower than that of the C-band pumping. However, the noise performance of the 980 nm pump exhibits the best because the emission cross-section at this wavelength is 0, where the number of Er³⁺ ions can be fully reversed. Since the emission cross-section at 1550 nm has exceeded the absorption cross-section, the noise performance manifests the worst. The noise factor is closely related to the ratio of the emission cross-section and absorption cross section of pump wavelength expressed as ${{{\sigma _{ep}}} / {{\sigma _{ap}}}}$ at a fixed signal wavelength. The larger the value, the greater the noise factor, as illustrated in Eq. (10) and Eq. (11). The C-band was used because of its ability to enhance the gain of the L-band. It’s important to select the appropriate pump wavelength to balance the gain and noise figure. As shown in Fig. 8, of the use of the pump wavelengths of absorption cross-section greater than emission-section like 1536 nm or 1530 nm, the noise figure <6 dB and the gain >12 dB from the wavelength range of 1570-1616 nm was obtained.

Fig. 8. Spectral variation of gains (solid line) and noise figures (dashed line) at pump wavelength (a) 980 nm, (b) 1530 nm, (c) 1536 nm, and (d) 1550 nm at signal wavelength ranging from 1570 to 1618 nm.

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4.2 Pump power tolerance discussion

Additionally, we studied the DMG and average gain fluctuations caused by pump power changes. Figure 9 shows the effect of pump power on average gains and DMG under different pump wavelengths. The average gain initially increases linearly with the increase of pump power, and then enters the saturation state. With sufficient pumping, it has the potential to achieve higher gain as the pump wavelength increases, while DMG is maintained below 0.6 dB when the pump power varies from 500 to 1300 mW. Particularly, when using a 1530 nm pump, the average gain is nearly 20 dB, and DMG is less than 0.4 dB with a total power of 1 W.

Fig. 9. (a) Effect of pump power on average gains and DMG under different pump wavelengths and pump forms. (b) Gain coefficient versus pump power for cladding pumping and core pumping at a pump wavelength of 1530 nm.

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4.3 Comparison of core pumping and cladding pumping

Finally, we simulated and compared the influences of core pumping and cladding pumping on gain and DMG for our 6-LP EDFA. The fiber that uses cladding pumping adopts a double-clad structure. The inner cladding radius is 43.6 μm [20], and the core-cladding ratio is 6.37%. Compared with core pumping, cladding pumping has the possibility of obtaining higher gains under high power pumping condition (>1.3 W). But the core pumping is more likely to achieve higher gain when the pump power is lower. As shown in Fig. 9 (a), at a pump wavelength of 1530 nm, when the total gain reaches approximately 20 dB, the pump power required for cladding pumping and core pumping is 1.3 W and 1 W, respectively, and DMG is 0.59 dB and 0.39 dB, respectively. The gain efficiency was increased by 23.1%, and DMG was decreased by 0.2 dB through using core pumping. Owing to the existence of the threshold pump power [39], the actual part of the fiber core acting on amplification is not enough to provide gain at low power due to the limitation of the core-cladding ratio on the cladding pumping, and higher gain efficiency can be obtained only at high power. However, more than 95% of the pump energy in our FM-EDF is tied to the core by adding a refractive index trench. The maximum gain coefficient of core pumping is approximately double that of the cladding pumping, as described in Fig. 9 (b).

5. Fabrication discussion

The trench-assisted 6-mode EDF designed in this paper can adopt Er/Al/La/Ge/P co-doped methods to realize L-band amplification, and it can be fabricated based on the MCVD (Modified Chemical Vapor Deposition) technology combined with the liquid-phase doping technology. The low refractive index trench can be formed by deposition of SF6 during the cladding deposition process. The depth and width of the trench are determined by the flow rate of SF6 and the number of claddings deposited, respectively. The Erbium, aluminum and lanthanum ions were doped into the fiber core by liquid phase method while the phosphorus and germanium ions are deposited by vapor phase deposition method. The NA can be adjusted by regulating the flow of P and germanium during the process of the gas phase doping.

6. Conclusion

We report a numerical investigation on trench-assisted few-mode erbium-doped fiber amplifier supporting 6 spatial modes with experimentally measured parameters of extended L-band Er³⁺ doped fiber. Based on mode hybrid core pumped architecture, the DMG reaches its minimum when the power ratio of the LP₂₁ mode and the LP₀₂ mode is about 0.7:0.3, regardless of which pump wavelength was selected. The fluctuation of DMG caused by the change of pump power was also studied which was always maintained at a low value under the optimal pump mode contents. For four signal mode groups at a pump wavelength of 1550 nm, we achieved a gain greater than 12 dB with a DMG of 0.28 dB across the range of 1570-1616 nm. It is suggested that our investigation shows great potential for decreasing the DMG effectively. Compared with cladding pumping, our method achieved a gain efficiency increase of 23.1% and a smaller DMG when both average gains reached 20 dB. The amplifier performance can be further improved with different C-band wavelengths hybrid pumping.

Funding

National Key Research and Development Program of China (2017YFB1104400).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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Parameters	Value	Unit	Definition
a	11	$μ m$	Core radius
d	TBD	$μ m$	Trench width
$n_{0}$			Refractive index of the core
$n_{1}$			Refractive index of the cladding
$n_{2}$			Refractive index of the trench
$Δ n_{1}$			Relative refractive index difference between the core and cladding
$Δ n_{2}$			Relative refractive index difference between the trench and cladding
$N A_{1}$	0.12		$n_{0} \sqrt{Δ n_{1}}$
$N A_{2}$	TBD		$n_{1} \sqrt{Δ n_{2}}$

	LP₀₁	LP₁₁	LP₂₁	LP₀₂
LP₀₁	33.6%	22.1%	15.9%	28.4%
LP₁₁	27.9%	29.0%	26.1%	17.0%
LP₂₁	22.9%	29.7%	30.5%	16.9%
LP₀₂	28.3%	15.2%	14.2%	42.3%

Parameter	Value	Parameter	Value	Parameter	Value
$σ_{a, 980 n m}$	1.11×10^-25 $m^{2}$	$σ_{e, 1530 n m}$	2.23×10^-25 $m^{2}$	$τ$	10 ms
$σ_{a, 1530 n m}$	2.66×10^-25 $m^{2}$	$σ_{e, 1536 n m}$	4.75×10^-25 $m^{2}$	$N_{t}$	2.33×10^25 $m^{- 3}$
$σ_{a, 1536 n m}$	5×10^-25 $m^{2}$	$P_{s, 0}$	-30 dBm	$N A 2$	0.19
$σ_{a, 1550 n m}$	1.9×10^-25 $m^{2}$	$P_{p, 0}$	800 mW	$d$	4 $μ m$
$σ_{a, 1550 n m}$	2.41×10^-25 $m^{2}$	$λ_{s}$	1570∼1618 nm

Parameters	Value	Unit	Definition
a	11	$μ m$	Core radius
d	TBD	$μ m$	Trench width
$n_{0}$			Refractive index of the core
$n_{1}$			Refractive index of the cladding
$n_{2}$			Refractive index of the trench
$Δ n_{1}$			Relative refractive index difference between the core and cladding
$Δ n_{2}$			Relative refractive index difference between the trench and cladding
$N A_{1}$	0.12		$n_{0} \sqrt{Δ n_{1}}$
$N A_{2}$	TBD		$n_{1} \sqrt{Δ n_{2}}$

	LP₀₁	LP₁₁	LP₂₁	LP₀₂
LP₀₁	33.6%	22.1%	15.9%	28.4%
LP₁₁	27.9%	29.0%	26.1%	17.0%
LP₂₁	22.9%	29.7%	30.5%	16.9%
LP₀₂	28.3%	15.2%	14.2%	42.3%

Six-mode extended L-band EDFA with a low differential modal gain

Abstract

1. Introduction

2. Fiber design

3. Gain modeling

3.1 Giles model

3.2 Multimode rate equations for Er³⁺ doped amplifier

4. Numerical simulation results and discussion

4.1 Reducing DMG by regulating pump mode content

4.2 Pump power tolerance discussion

4.3 Comparison of core pumping and cladding pumping

5. Fabrication discussion

6. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Tables (3)

Equations (11)

OSA Continuum

Abstract

1. Introduction

2. Fiber design

3. Gain modeling

3.1 Giles model

3.2 Multimode rate equations for Er3+ doped amplifier

4. Numerical simulation results and discussion

4.1 Reducing DMG by regulating pump mode content

4.2 Pump power tolerance discussion

4.3 Comparison of core pumping and cladding pumping

5. Fabrication discussion

6. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Tables (3)

Equations (11)

OSA Continuum

3.2 Multimode rate equations for Er³⁺ doped amplifier