1. Introduction
Single-mode fiber (SMF) communication systems are facing a capacity limit resulting from fiber’s nonlinear effects [
1]. The core or mode division multiplexing technology has become an effective way to increase fiber transmission capacity [
2]. Just like the commercially available erbium-doped fiber amplifiers (EDFAs) used in SMF communication systems, few-mode EDFAs (FM-EDFAs) can also play an important role in mode division multiplexing (MDM) systems [
3]. As a unique performance parameter for FM-EDFAs, the differential mode gain (DMG) not only affects the transmission performance of long-haul MDM systems [
4], but also increases the complexity of the multiple-input multiple-output digital signal processing algorithm [
5].
In 2011, Y. Jung et al. first realized the simultaneous amplification of two mode groups in FM-EDFAs and investigated the influence of the refractive index and erbium-ion distribution of the erbium-doped fiber (EDF) on the gain performance [
6]. The same year, N. Bai et al. studied the gain equalization of LP
01- and LP
11-mode signals under different pumping modes according to FM-EDFA theory [
7]. In 2012, R. Ryf et al. proposed a modal gain control scheme for optimizing the thickness of the ring-doping layer in the active fiber [
8], which was also used for two-mode gain equalization [
9]. In 2017, D. Vigneswaran et al. proposed the mode-selective bidirectional pumping scheme for the uniform gain pattern for different LP modes with the minimal DMG close to 0 dB [
10]. In 2021, Qayoom Taban et al. used the center-depressed erbium-ion doping profile to improve the pumping efficiency of the FM-EDFA supporting six mode groups, with the LP
21a-mode pump power of 55.567 mW and the minimum DMG of approximately 4.41 dB [
11]. It is well-known that the DMG can be reduced by optimizing pump modes, refractive index profile, and erbium-doping concentration, or combining the above methods, in which the erbium-ion concentration distribution is the basis of optimizing the pumping scheme and the FM-EDF’s length in the simulation or design of FM-EDFAs. Unfortunately, the erbium-ion concentration distribution as a commercially confidential parameter is usually unavailable for the FM-EDFA’s designers. Therefore, it is necessary to characterize or measure the erbium-ion doping concentration by an effective method with application to the FM-EDFA’s theoretical model.
In the paper, we propose a measurement method for the effective erbium doping concentration in terms of modal absorption coefficients of FM-EDFs. The absorption coefficient of the fundamental mode was firstly measured by extrapolating the mode–gain curve dependent on the average population inversion. Then, the effective erbium-ion doping concentration of the EDF under test was estimated and used to calculate the absorption coefficients of high-order modes. The measured data were also used to optimally design the FM-EDFAs with high gain and low DMG under forward pumping.
The rest of the paper is organized as follows.
Section 2 presents the analytical expression for modal gain of FM-EDFAs in terms of the average population inversion concentration in FM-EDFs. In
Section 3, we build an experimental platform to measure the gain curve relative to the population inversion concentration by analyzing the modal splicing loss and crosstalk between the single- and few-mode fibers. Then, the absorption coefficient of LP
01 mode and the effective erbium-ion concentration is obtained by calculation. The comparison between our measurement method and the traditional transmission method is given in
Section 4. The effective erbium-ion concentration obtained above is also used to optimally design the FM-EDFAs with low DMG, and the detailed results are presented in
Section 5.
Section 6 discusses the repeatability of the experiment and the practicability of the extrapolation method for the absorption coefficients.
Section 7 gives the conclusion.
2. Analytical Expression for FM-EDFA’s Gain
The optical amplification process in FM-EDFAs associated with population inversion can be explained by the two-level model of erbium ions as follows:
where
and
are, respectively, the erbium-ion concentration at the ground and metastable states, with the total concentration
;
is the relaxation time of erbium ions at the metastable energy level;
W13,
W12, and
W21 represent the simulated transition rates of erbium ions between the corresponding energy levels.
Under the steady state, the erbium ion concentration on the metastable state can be given from Equation (1) as follows [
12]:
where
and
are respectively the normalized mode field profiles of the signal and pump light,
and
are, respectively, the frequencies of signal and pump light,
h is Plank’s constant,
and
are, respectively, the absorption and emission cross-sections of signals, and
is the absorption cross-section of pump light.
By integrating Equation (2) over the fiber cross-section, we can obtain the population inversion
averaged over the length
as follows [
12]:
Thus, the optical power of the signal mode
i output from the EDF section can be expressed by:
where
and
are, respectively, the population inversion coefficient and the absorption coefficient related to the erbium-ion concentration, that is,
Similarly, we can also give the ASE noise power output from the length
as follows:
where,
is the power of the generated ASE noise,
is the signal wavelength,
is the equivalent amplifying bandwidth, and
is the input ASE noise. In the paper, ASE is omitted in the calculation of population inversion, and the approximation is thought to be reasonable for the small ASE case [
13]. Actually, the contribution of ASE power,
, can be added to Equations (2) or (3) by substituting
with
for a large ASE case.
The output optical power of the pump mode is also of the form similar to that of the signal mode, by changing the superscript
and subscript
in Equation (4) and Equation (5). Obviously, the background loss in EDFs is easily added into Equation (4) and is negligible for the short active fiber. According to Equation (4), the absorption coefficients
can be obtained from the gain value at
,
(in dB), that is,
Furthermore, according to the overlap approximation method for FM-EDFAs [
14]. Equation (5) may also be rewritten as:
where
is the overlap factor of the signal mode
i, and
is so-called effective erbium-ion concentration. In other words,
can be calculated from the measured
for a given modal field. However, the case with
is not measurable in experiments due to the existence of the signal or the pump light in the EDFAs. In this paper, we employed the extrapolation method to obtain the gain,
or
, according to the gain curve versus
by adjusting the pump power.
4. Comparison of Experimental and Simulated Results
According to the effective erbium-ion concentration obtained previously, along with the erbium-doped fiber refractive index distribution and other experimental parameters, we simulated the gain characteristic of the setup shown in
Figure 1 by VPI software. From
Figure 3, the simulated results are basically identical to the experimental data. It is shown that the extrapolation method for the modal absorption coefficients is feasible, and then the effective erbium-ion concentration can also be estimated.
We also made use of the transmission method to measure the absorption coefficient of the LP
01 mode by the single-mode experiment platform in which only 1550 nm signal light was injected into the SMF and FM-EDF. The transmittance, defined by the ratio of output to input powers of signal modes, was obtained by adjusting the optical power input to the FM-EDF. Under the case with no pump light, both optical amplification and ASE noise did not occur. In the experimental test, the 3.2 dB coupling loss and 0.4 dB excess loss between the SMF and the FM-EDF were introduced. The measured transmission of the LP
01 mode is shown in
Figure 4. For comparison, the simulated transmission curves of the LP
01 and LP
11 modes were also plotted in
Figure 4, and the VPI simulation parameters were identical with those used in
Figure 3. From
Figure 4, the measured data were basically consistent with the simulated results for the LP
01 mode. It was reconfirmed that the effective erbium-ion concentration calculated from the absorption coefficient was credible. From
Figure 4, the transmission depends on the input optical powers and applied to the case with low input power. The absorption coefficient of the LP
01 mode at the input power of −10 dBm could be calculated to be
, which was close to the value of 2.38 m
−1 obtained by our extrapolation method. Similarly, the simulated transmission coefficient was
for the LP
11-mode signal. In contrast, our extrapolation method was especially applied to the erbium-doped fiber under the operating states of pumping and amplification.
5. Optimal Design of the 3M-EDFAs with forward Pumping LP21-Mode
The parameters of modal gain and DMG were often used to characterize the amplification performance of the FM-EDFAs. The FM-EDFAs with high gain and low DMG are especially desirable for MDM transmission systems. In what follows, according to the effective erbium-ion concentration calculated from the measured absorption coefficient, we built a VPI simulation system to optimally design 3M-EDFAs under forward pumping as shown in
Figure 5. The pump and three mode signals (i.e., LP
01, LP
11a and LP
11b) were multiplexed into the FM-EDF and demultiplexed after amplification. The output powers were measured by the optical power meters to further calculate the modal gain and the DMG. Three signal modes had the same input power of −10 dBm. The LP
21a-mode pumping was employed for a large overlap integral with LP
01 and LP
11 modes, which is also helpful for low DMG [
7]. In the VPI simulation, the pump power and the length of FM-EDF were simultaneously swept by the step sizes of 100 mW and 0.1 m, respectively. On the 2D plane of the pump power and FM-EDF’s length, we plotted the contour lines of 20 dB modal gain, and the color values represent the corresponding DMG as shown in
Figure 6. The region above the contour lines of
and
= 20 dB had a gain of more than 20 dB. Similarly, the DMG lower than 2 dB was also bounded by the contour line of DMG = 2 dB. It should be pointed out that, the mode conversion between LP
11a and LP
11b modes in the short FM-EDFs was neglected in our simulation for simplicity, that is, the transverse profile of the signal or pump modal field was kept fixed or with no azimuthal rotation [
16].
From
Figure 6, if the modal gain
G ≥ 20 dB for all three mode signals was required for the 3M-EDFAs, it should be satisfied that the FM-EDF’s length
L ≥ 1.8 m and the pump power
Pp 240 mW. For example, at the lowest pump power of 240 mW, the optimal length of the FM-EDF was 3.8 m, with a DMG of 1.8 dB. As a reference, two contour lines of DMG = 0.1 dB were also plotted in
Figure 6 for further optimizing 3M-EDFAs. Clearly, to design the 3M-EDFAs with a
G ≥ 20 dB and DMG ≤ 0.1 dB, the FM-EDF’s length and pump power should be no less than 2 m and 650 mW, respectively. This optimization process was also applied to the case with backward or bidirectional pumping.
6. Discussion
Here, we discuss the repeatability of the experiment and the practicability of the extrapolation method for the absorption coefficients of few-mode fibers.
The repeatability of the experiment mainly relies on the acquisition for the signal and pump powers’ input and output from the active erbium-doping fiber. However, the measurement of these parameters is usually taken in an indirect way due to the influence of insert loss, coupling loss, mode crosstalk, and other factors. In this paper, we calculated the coupling loss between the different fibers by the VPI software, and the insert loss was measured by experiment. The resulting coupling loss and excess loss introduced in our experiment were used to estimate the net powers for the experimental repeatability.
Before taking analysis on the practicability of the extrapolation method, we described several techniques for deriving the absorption coefficients as follows: (1) the indirect method, in which the absorption coefficients are indirectly deduced from the gain coefficients according to their relationship in terms of the known absorption and emission cross sections [
17]; (2) the direct method, in which the absorption coefficients are directly calculated from the absorption cross-section and the uniformly distributed erbium-doping concentration, which can, respectively, be determined by the gain saturation measurement and the chemistry methods [
18]; (3) the transmission method, which is equivalent to the case with no pump power. In
Section 4, it was shown that the absorption coefficients given by the extrapolation method proposed in our paper were basically identical with the transmission method [
19]. In other words, we also presented an alternative method, that is, the absorption coefficients can be estimated by the extrapolation of the gain curve dependent on the pump power in which the value of
may approximately be replaced by the case with no pump power. By comparison, our method is especially applied to the erbium-doped fiber under the operating states of pumping and amplification. Most of the relevant references reported the gain curves dependent on pump power, but the data are not applied to the extrapolation method for the accurate absorption coefficient due to the inadequate fitting points [
20,
21]. In other words, it is difficult to compare our simulation results with the data presented in the literature.