## Abstract

This paper investigates the applicability of a model of a cladding-pumped multicore erbium-doped fiber amplifier (CP-MC-EDFA) that employs the average inversion level of an erbium ion. The model is modified to include the effects of multiple cores for modifications of the background loss coefficient of the pump light. The model is validated experimentally by measurements of the pump power, gain transient, and intercore cross-gain modulation of a cladding-pumped 12-core EDFA in which the pump light is uniformly distributed in the inner cladding. The calculated and measured pump power, gain transient response, and gain change caused by the intercore cross-gain modulation agree well, suggesting the modified model is useful for characterizing a CP-MC-EDFA.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Space-division-multiplexing (SDM) transmission through a multicore fiber (MCF) is attractive for overcoming the capacity crunch in conventional single-mode fiber (SMF) transmission systems [1]. MCF transmission experiments with multicore erbium-doped fiber amplifiers (MC-EDFAs) used as optical repeaters have demonstrated high capacity [2], ultra-long-haul capability [3], and large spatial channels [4]. Among MC-EDFAs, those that are cladding-pumped (CP) have the potential to reduce power consumption [5]. In this context, the power consumption of an SDM network that employs CP-MC-EDFAs has been analyzed by using experimentally obtained constant values of the electrical power consumption of the pump laser diodes (LDs) in CP-MC-EDFAs to calculate the network power consumption [6]. However, the power consumption of an EDFA depends on the pump power, which depends on the EDFA operation condition, such as the input signal power, gain, output signal power, and the EDF parameters. In addition, the EDFA gain dynamic response may affect the performance of an optically repeated network. Furthermore, intercore cross-gain modulation caused by the depletion of the pump light may be a potential issue in SDM network design [7]. Therefore, a model for calculating the characteristics of a CP-MC-EDFA operated in an adequate condition, such as the required pump power, gain transient, and intercore cross-gain modulation, is needed to calculate the power consumption and dynamic performance of an SDM network employing CP-MC-EDFAs as optical repeaters.

A rigorous EDFA model is described by a set of coupled partial differential equations which must be solved numerically as a dual boundary value problem [8,9]. The numerical calculation needs iterations not only to include both the forward and backward pump light, signal light, and amplified spontaneous emission (ASE), but also to find the input power conditions for the EDFA operation of interest, such as a flattened-gain operation suitable for amplifying a wavelength-division-multiplexing (WDM) signal. Thus, the numerical calculation of a set of coupled partial differential equations sometimes makes the calculation time-consuming. On the other hand, an EDFA model that reduces the set of coupled partial differential equations to a single transcendental equation is available [10–12]. This model is an approximate but simple description and is widely used in the research and development of EDFAs. Furthermore, a model with an average inversion level has been derived following a method similar to the latter model [13,14]. This model is a semi-analytical solution and sometimes provides more insights into EDFA behavior than a set of coupled partial differential equations. In particular, it easily describes the EDFA operation of interest and finds the input power condition for the EDFA operation with simplified steps. This model describes the dynamic as well as steady-state behavior of an EDFA, possibly suitable for investigating the dynamic behavior of an optically amplified network. An EDFA model with an average inversion level has been applied to calculate the pump power of a core-pumped EDFA, and its accuracy was proved by comparing calculated and experimentally obtained pump powers [15]. Although the EDFA model with a set of coupled partial differential equations has been applied to a CP-MC-EDFA [7,16], no attempt has been made to apply the EDFA model with an average inversion level to a CP-MC-EDFA.

In this paper, we investigate the applicability of the EDFA model with an average inversion level to a CP-MC-EDFA. We describe the model in Section 2. Then, we validate the model experimentally in Section 3 by comparing the calculated pump power, gain transient response, and intercore cross-gain modulation with those obtained experimentally. The paper is briefly summarized in Section 4.

## 2. MODELING OF CLADDING-PUMPED MULTICORE EDFA

First, we describe the model employing the average inversion level applied to a CP single-core EDFA (CP-SC-EDFA) and then modify it so that the model can be applied to a CP-MC-EDFA. Here, we assume a noncoupled CP-MC-EDFA in which signal crosstalk between cores is negligible. The rate equation for the fractional population of the normalized upper state and the photon propagation equation for the $j$th optical channel traveling in the $z$ direction are [13,14]

By introducing the average inversion level, using EDF length $L$,

integrating Eq. (2) yields the output photon flux of the $j$th channel, described as [14]The overlap integral ${\mathrm{\Gamma}}_{j}$ is defined for the signal and ASE lights as

For the steady state, the photon flux of the pump light required to obtain a gain spectrum with any average population inversion is derived from Eq. (6) and can be expressed by using the average inversion level and the gain coefficient at the steady state, ${\overline{N}}_{2}^{\mathrm{ss}}$ and ${\overline{g}}_{j}^{\mathrm{ss}}$, as [19]

Here, a modification is needed to apply the model to a CP-MC-EDFA. Considering that the stimulated emission cross-section of a 980-nm band pump light equals zero, ${\sigma}_{p}^{e}=0$, the propagation equation for a pump light of a CP-MC-EDFA is written as

## 3. EXPERIMENTAL VALIDATION

This section describes the experimental validation of using Eqs. (9a) and (9b) with Eq. (11) to calculate the required pump power, and using Eq. (6) with Eq. (11) to calculate the gain transient and the intercore cross-gain modulation.

#### A. Cladding-Pumped 12-Core EDFA

The EDF used for the calculation and experiment was a double-clad 12-core EDF whose core arrangement was the same as our previously reported double-clad fiber [20]. The double-clad 12-core EDF had an average core pitch of 36.6 μm, an inner cladding diameter of 214 μm, an outer cladding diameter of 284 μm, and a coating diameter of 362 μm. The refractive index structure between the inner and outer cladding was equivalent to a step-index multimode fiber with numerical aperture of 0.46. Of the 12 cores, we measured the pump power and gain transient of a single core, core #8. The mode field diameter (MFD) at 1550 nm, cutoff wavelength (${\lambda}_{c}$), and erbium ion absorption at 1530 nm of each core of the double-clad 12-core EDF are listed in Table 1 and are measured values. The radius and erbium concentration of each core were estimated from the MFD, cutoff wavelength, and erbium ion absorption. The absorption and stimulated emission cross-section at the signal wavelength presented in Ref. [21] and the measured absorption cross-section of $2.22\times {10}^{-25}\text{\hspace{0.17em}}{\mathrm{m}}^{2}$ at the pump wavelength for the core #8 were used for calculation. The background loss of the inner cladding for the pump light was 74.4 dB/km, which was estimated from the measured background loss at 1.2 and 1.31 μm by using the inverse proportion with the fourth power of the wavelength. The EDF lengths used for C-band and L-band experiments were 6 and 24 m, respectively.

Figure 1 shows the CP-MC-EDFA configuration used for the experiment. Free-space coupled WDM modules containing an optical isolator were used to multiplex signal and pump lights. The WDM modules had a single-clad 12-core fiber, a 105/125 multimode fiber, and a double-clad 12-core fiber as a signal input/output, pump, and common ports, respectively. Both the single-clad and double-clad 12-core fibers had the same core arrangement as the double-clad 12-core EDF. The double-clad 12-core fibers and EDF were fusion-spliced. The pump source was a 975-nm multimode LD, and the pump light was input into with the forward pumping scheme. Pitch-reducing optical fiber arrays (PROFAs) were used as fan-in/fan-out devices.

#### B. Pump Intensity Distribution in Inner Cladding

The overlap integral for the pump light described in Eq. (8) assumes that the pump intensity is uniform in an inner cladding. To validate this overlap integral, near-field pattern (NFP) at several points along the EDF were observed to measure the intensity distribution of the pump light in the inner cladding. Figure 2 shows the intensity distribution images in the inner cladding obtained from the NFP measurement, where (a), (b), and (c) are the intensity distributions at $0.3-$, $6-$ and 24-m-long EDF outputs, which were just after the EDF input, C-band EDF output, and L-band EDF output. Each image is normalized so that the maximum peak intensity is unity. All the images had speckle patterns spread uniformly within the inner cladding, and the intensity deviation became small with increasing EDF length. The probability density distributions of the pump intensity are shown in Fig. 2, where (d), (e), and (f) are the probability densities at the $0.3-$, $6-$ and 24-m-long EDF outputs, respectively. The horizontal axis of these figures is the normalized pump intensity, so the average intensity is unity for these experimental probability densities. The red and blue lines are the experimentally obtained density and the gamma distribution fitting, respectively. All the experimental probability density distributions agree well with the gamma distribution, which is the speckle pattern distribution for a nonpolarized light guided in a step-index multimode fiber [22,23]. The expectations of those gamma distributions were 0.96, 0.99, and 1.01 for (d), (e), and (f), meaning that they coincided with the average pump intensities. Because the speckle patterns appear randomly and change rapidly, the images shown in Fig. 2 are temporal patterns. Therefore, the fact that the mean values of the speckle pattern distributions are almost the same as the average intensity means that the ratio of the intensity in a region with the area ${S}_{p}$ to the total intensity in the inner cladding with the area ${S}_{\mathrm{clad}}$ is approximately equal to ${S}_{p}/{S}_{\mathrm{clad}}$ whatever the location of the considered region. This suggests that the pump intensity is uniformly distributed within the inner cladding in our CP-MC-EDFA.

#### C. Required Pump Power

The pump power of our CP-MC-EDFA was calculated by using Eqs. (9a) and (9b) with Eq. (11) and compared to the experimentally obtained pump power. Figure 3 shows the experimental setup for measuring the pump power. An eight-channel WDM signal whose optical sources were a distributed-feedback LD (DFB-LD) was used as an input signal, and the eight channels were multiplexed with an arrayed-waveguide grating (AWG). Signal channel wavelengths were 1531.0–1561.4 nm for the C-band and 1572.0–1600.4 nm for the L-band. Each channel power was adjusted by using a variable optical attenuator. The WDM signal was input into only core #8, whose gain was measured. The other cores had no signal input, and only ASE lights were output from them. The output of the CP-MC-EDFA was measured with an optical spectrum analyzer (OSA). The pump power was determined from the forward current, which was related to the pump power at the output of the common port of the WDM module. The maximum available pump power was 6 W. The EDF gain was obtained by subtracting the insertion losses of the input and output ends from the amplifier gain. The loss of the input and output ends of the amplifier in Fig. 1, which arose from the fan-in and fan-out devices and the WDM modules, was measured with 1.55-μm light by using a single-mode fiber butt-jointed to the core of interest (core #8). Our fusion splice was optimized to avoid core misalignment and to match MFDs between EDF core #8 and the corresponding MCF cores of the WDM modules. The loss of the optimized fusion splice is less than 0.1 dB, which is negligible.

Figures 4(a) and 4(b), respectively, show the input signal power dependence of the experimentally obtained and calculated pump powers for the C-band amplification and the experimental and calculated gain spectra for the input signal power. The forward current of the pump LD was adjusted so that the same gain spectra with the same average inversion level were obtained for all the input signal powers. Because the maximum available pump power was too small to obtain a flat gain spectrum, the gain spectra have tilts along which the gain at the longer wavelengths was larger than that at the shorter wavelengths. The required pump power was calculated by using Eqs. (9a) and (9b) taking Eq. (11) into account. The details of the calculation procedure are as follows. (i) Calculate the pump power for ${\overline{N}}_{2}$ of the core of interest (core #8) using Eqs. (9a) and (9b). In this calculation, Eq. (11) is not taken into account. (ii) Calculate average inversion levels of cores other than interest ${\overline{N}}_{2,n}$ using Eq. (6) with the pump power calculated in the previous procedure at steady-state condition $d{N}_{2,n}/dt=0$, not including Eq. (11). (iii) Calculate the pump power of the core of interest using Eqs. (9a) and (9b) taking Eq. (11) into account with ${\overline{N}}_{2,n}$ calculated in step (ii). (iv) Repeat steps (ii) and (iii) until changes in the pump power and ${\overline{N}}_{2,n}$ from the previous step become sufficiently small. It should be noted that iteration is required to calculate the pump power when the impact of the other cores is included. The reason is that Eqs. (9a)–(9b) are related to Eq. (6), with which the average inversion level of the other cores (core #1–7 and #9–12) is calculated at a steady state; however, only a few iterations are needed. The experimental and calculated pump powers agree well, with a small error of about 7%. There was a slight mismatch between the experimental and calculated gain spectra. This was caused by the discrepancy between the used and real absorption and the simulated emission cross-sections of an erbium ion; therefore, we believe that the discrepancy between the calculated and experimental pump powers was caused for the same reason.

Figures 5(a) and 5(b), respectively, show the input signal power dependence of the experimentally obtained and calculated pump powers for the L-band amplification and the experimental gain spectra for the input signal power in the same way as Fig. 4. Flat gain spectra with the same average inversion level were obtained for the L-band amplification. As in the case of the C-band amplification, the experimental and calculated pump powers agree well, with a small error of about 8%.

#### D. Gain Transient Response

The gain transient response is calculated by using Eq. (6), and calculated data are compared to experimental data. Figure 6 shows the experimental setup for measuring the gain transient response. The same eight-channel WDM signals as in the pump power measurement were employed in this experiment. One of the eight channels was used as a surviving channel, while the other channels were used as drop channels. The power of the drop channels was increased by $10\mathrm{log}(39/7)=7.46\text{\hspace{0.17em}}\mathrm{dB}$ compared to the power of the surviving channel so that the channel dropping corresponded to a $40\to 1$ channel change. The measured surviving channels were 1531.0, 1548.5, and 1561.4 nm for the C-band amplification and 1572.0, 1587.9, and 1600.4 nm for the L-band amplification. The drop channels were modulated with a 20-Hz rectangular wave by using an acousto-optic modulator (AOM). The signal was input into only core #8 in the same way as in the pump power measurement. The output of the surviving channel of the CP-MC-EDFA was selected with a bandpass filter (BPF) and detected with an optical-to-electrical (O/E) converter. The waveform of the surviving channel was observed with a digital oscilloscope. Figures 7 and 8, respectively, show the gain transient response for the C-band and L-band amplification. The gain change is defined here as the gain excursion after signal channels except for the surviving channel is dropped from the gain obtained when all the signal channels are input. The red and blue lines are the calculated and experimentally obtained gain transient responses, respectively. The details of the gain transient response calculation are as follows. (a) Calculate the pump power and average inversion levels of cores other than the core of interest ${\overline{N}}_{2,n}$ for an initial average inversion level of the core of interest (core #8). (b) Calculate the transient response of the average inversion level of the core of interest ${\overline{N}}_{2}$ using Eq. (6) with the Runge–Kutta method, including Eq. (11) with ${\overline{N}}_{2,n}$ calculated in step (a) and a change in the input signal at time 0. The input signal change is the equivalent channel drop from 40 to 1. (c) Convert calculated ${\overline{N}}_{2}$ to the gain using Eq. (5) and subtract the steady-state gain of the core of interest, calculated by using the initial ${\overline{N}}_{2}$ yielding the gain transient response. In each transient response, the input signal power was changed from $+5$ to $-11\text{\hspace{0.17em}}\mathrm{dBm}$, corresponding to the change in the number of channels from 40 to 1. To emulate this input signal power change of 16 dB using the eight-channel WDM signal, we set the surviving channel power to $-11\text{\hspace{0.17em}}\mathrm{dBm}$, and the dropped channel power was $-3.54\text{\hspace{0.17em}}\mathrm{dBm}/\mathrm{ch}$. The gain transient responses exhibited surviving channel wavelength dependence. The major reason for the wavelength dependence is that the gain change in the shorter wavelength channel is larger than that in the longer wavelength channel for the same ${\overline{N}}_{2}$ change. Although a small discrepancy was observed between the calculated and experimental gain transient responses because of the error in the absorption and simulated emission cross-sections used for the calculation, the calculated gain transient responses agree well with the experimental response.

#### E. Intercore Cross-Gain Modulation

As described in Eq. (6), the average inversion level changes in both statically and dynamically through the change in the background loss coefficient of the pump light. This section compares the calculation and experimental results for the intercore cross-gain modulation of the CP 12-core EDFA.

First, we investigate the static intercore cross-gain modulation by comparing the gain spectrum of the core of interest (core #8) obtained when the signal was input into only the core of interest to that obtained when the signal was input into all the cores. Figures 9 and 10 show (a) the calculated and (b) measured gains of the core of interest for C-band and L-band, respectively. The input signal power was $-18.3\text{\hspace{0.17em}}\mathrm{dBm}/\mathrm{ch}$, $-9.3\text{\hspace{0.17em}}\mathrm{dBm}$ in total. The calculation procedure was the same as described in Section 3.D except that in the case of the intercore cross-gain modulation, the signal was input into only the core of interest on the initial condition and signal channels for cores other than the core of interest are added at 0 ms. The gain reduced slightly when the signal was input into all the cores compared to when the signal was input into the core of interest, and the gain change was much smaller than 0.1 dB for the C-band amplifier. The gain change for the L-band amplifier was larger than that for the C-band amplifier. However, the observed gain change was at most 0.1 dB.

The dynamic characteristics of the intercore cross-gain modulation were calculated and measured with the same amplification condition as that used for the static characteristics. Figure 11 shows the calculated transient response of the intercore cross-gain modulation, where the red line is for the C-band, 1548.5 nm, and the green line is for L-band, 1587.8 nm. At 0 ms, the signal input into all the cores other than the core of interest was added. The gain change is defined here as the gain excursion after signal channels for cores other than the core of interest were added from the gain obtained when only the core of interest had signal input. The transient response of the intercore cross-gain modulation for the C-band was almost negligible while that for L-band resulted in about a 0.1-dB gain change.

The measured transient response of the intercore cross-gain modulation for C-band and L-band are shown in Figs. 12(a) and 12(b), respectively. For both C-band and L-band, almost the same transient responses as the calculated ones were obtained except for a slight optical surge at the time of zero for the L-band. Both the static and dynamic results suggest that the CP-MC-EDFA model employing the average inversion level is also useful for characterization of the intercore cross-gain modulation.

## 4. CONCLUSION

We investigated the applicability of a modified CP-MC-EDFA model employing the average inversion level. The differences between the modified and conventional models are in the background loss coefficient of the pump light: the background loss coefficient includes the influence of the other cores. We validated the model experimentally by conducting pump power and gain transient response measurements with a CP 12-core EDFA in which the pump light was uniformly distributed in the inner cladding. The calculated and measured pump power and the gain transient response agree well, with a slight difference caused by the error in the absorption and simulated emission cross-sections used for the calculation. Furthermore, the CP-MC-EDFA model successfully predicted the intercore cross-gain modulation, for which the change in the gain of the core of interest was larger in the L-band than in the C-band. The results suggest that the EDFA model with the average inversion level is useful for characterizing a CP-MC-EDFA and helpful for simulating SDM network performance. Although the computational power of modern computers may make it possible to obtain a full solution of a set of the propagation equations with a rate equation more rapidly than in the era when the model was developed, this semi-analytical solution can provide both good insights into EDFA behavior and a simple calculation of the EDFA operation condition of interest.

## Acknowledgment

The authors thank K. Takenaga, K. Ichii and S. Matsuo with Fujikura Ltd. for fabricating the double-clad 12-core EDF.

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