Modified differential overlap factor and modal gain equalization criteria-based comparative analysis of 4M-EDFA 980;1480 nm towards identification of a unique erbium doping profile for the 4M-EDFA1480 nm system

A four-mode erbium-doped fiber amplifier (4M-EDFA) system with LP01;980nm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{LP}}_{01;980 \mathrm{nm}}$$\end{document} and LP01;1480nm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{LP}}_{01;1480 \mathrm{nm}}$$\end{document} pump wavelengths is explored analytically using a coupled mode equation and subsequently through simulations by implementing modified erbium-doped profile-based systems. We aim to reduce the inherent differential modal gain (DMG) and differential spectral gain (DSG) between signal modes LP01,LP11,LP21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{LP}}_{01}, {\mathrm{LP}}_{11}, {\mathrm{LP}}_{21}$$\end{document} and LP02\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{LP}}_{02}$$\end{document} while maintaining high modal gain. For in-depth performance evaluation and comparison of 4M-EDFA980nm;1480nm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{EDFA}}_{980\, \mathrm{nm};1480\,\mathrm{nm}}$$\end{document} systems, novel differential performance parameters are introduced and explored. Differential modal noise figure (DMNF) and differential spectral noise figure (DSNF) parameters quantify the impact of system amplified spontaneous emission. The conventional erbium ion inclusive transverse overlap factor is modified (ηsp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\eta }_{\mathrm{sp}}$$\end{document}) using a unit-less erbium ion profile over a scale of 0–1. Differential modal overlap factor (DMOF) and differential spectral overlap factor (DSOF) are shown to be strongly correlated with DMG and DSG, respectively, and prove to be more decisive performance evaluation parameters than ηsp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\eta }_{\mathrm{sp}}$$\end{document}. Obtaining low DMOF and DSOF values with the 1480 nm pump prompts the investigation of 4M-EDFA1480nm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{EDFA}}_{1480\,\mathrm{nm}}$$\end{document} with erbium ion profile variants: uniform Nr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\left(r\right)$$\end{document}= 1, Ninvr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N}_{\mathrm{inv}}\left(r\right)$$\end{document} and Nopt(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N}_{\mathrm{opt} }(r)$$\end{document}. Ninvr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N}_{\mathrm{inv}}\left(r\right)$$\end{document} is extracted from the inverse sum of the normalized signal intensity function. Subsequently, differential modal gain equalization criteria are used which aid in unique linearizing Nopt(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N}_{\mathrm{opt} }(r)$$\end{document} profile identification. In the 4M-EDFA system, the highest values (in dB) of DMG, DSG, DMNF and DSNF are 13.295, 3.9717, 5.9996, 11.0649, respectively, for the 980 nm uniform erbium ion profile system. The proposed Nopt(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N}_{\mathrm{opt}}(r)$$\end{document} profile significantly reduces these parameters (in dB) to 2.0558, 2.4997, 2.77 and 3.879, respectively, for the 1480 nm system.


Introduction
The data-carrying capacity of a conventional single-mode fiber (SMF) has a fundamental upper limit of around 100 Tbit/s. This corresponds to a single-mode erbiumdoped fiber amplifier (SM-EDFA) spectral bandwidth of about 11 THz covering C and L amplification bands [1]. To overcome the capacity bottleneck of SMF, space division multiplexing (SDM) has become a popular research topic in recent years as a method of improving efficiency and reducing costs [2][3][4][5]. SDM can be realized by employing multiple parallel systems based on existing single-mode single-core fiber technology or alternatively by techniques such as multimode fibers (MMFs) [6], multi-core fibers (MCFs) [2,4] and hybrid SDM fibers (e.g., few-mode MCFs) [7].
With the advent of SDM based on few-mode fibers (FMFs), mode division multiplexing (MDM) has attracted the attention of the research community as a method of increasing fiber capacities [8][9][10]. To enable cost reduction we require high-performance inline few-mode EDFA (FM-EDFA) which ensures amplification of individual guided modes. FM-EDFA has attracted tremendous attention owing to low energy consumption and low cost compared to multiple single-mode EDFA units [5,[11][12][13]. In such MDM systems, N orthogonal modes are used to support N parallel signaling channels, offering an N-fold increase in capacity over conventional single-mode single-core fiber technology. This is because it enables simultaneous amplification of all signal modes, resulting in improved efficiency and lower cost per bit. Long-haul MDM systems require few-mode optical amplifiers that should provide identical gain across all signal modes. Mode number scaling with low differential modal gain (DMG) is a paramount issue in SDM [14]. In single-core fiber transmission MDM systems, the first four linearly polarized ( LP lm ) modes (i.e., LP 01 , LP 11 , LP 21 and LP 02 ) exhibited a record spectral efficiency. However, due to the scarcity of dedicated LP lm mode amplifiers, six single-mode amplifiers and 3D waveguide spatial demultiplexers were employed to amplify the signal modes [15]. It is required to concurrently enhance all spatial modes in one device for reduction in the inherent DMG and differential spectral gain (DSG).
It is reported that gain equalization in few-mode amplifiers supporting more than two LP lm modes is becoming increasingly challenging due to their unique modal profiles [16]. So far, the two most common methods for equalizing modal gain are managing pump modal contents [11,12,17] and optimizing erbium spatial distributions [12,18,19]. Other gain equalization solutions have been offered as well, such as cladding pumped arrangement [20,21], cladding and core hybrid pumped arrangement [22] and designing an EDF with a specific refractive index profile [10,23]. Numerical modeling and testing have shown that these strategies are reliable for amplifying various LP lm modes. Instead of employing EDFs with ring doping, a four-LP lm mode amplifier is used by combining a ring-doped and a uniformdoped EDF [9]. Because of the overwhelming gain in LP 01 and LP 11 modes encountered in the second uniform EDF, the DMG of this combined amplifier is considerably high, about 4 dB. Two separate systems were identified for the extraction of overlap factors by the using FM-EDFA specific extension of the Giles model at 980 nm pump power [12]. There, a modified erbium-doped profile (MEDP) technique was implemented that used an optimum multi-well ion profile. This technique provided a large mean gain of about 24.24 dB spread over four signal modes with a satisfactory reduction in DMG to about 1.324 dB. It also implemented a mode-specific pump combination (MSPC) technique for the four-mode EDFA (4M-EDFA) system with three pump modes, where the DMG was reduced to about 1.55 dB and a large mean gain value of about 28.72 dB was achieved. The main aim of [12][13][14][15][16][17][18][19][20][21][22][23] is the reduction in the DMG of different signal modes. However, some of them use multi-pump techniques, which can pose a disadvantage by adding to the cost and complexity. Further, [24] reports high gain and low noise figure performance with the 1480 nm pump. The DMG reduction as well as high overlap of signal modes with the 1480 nm pump compared with the 980 nm pump is also reported in [25]. This paper is based on an MEDP 4M-EDFA system developed to explore and investigate the superior modal overlap properties of the 1480 nm pump over the 980 nm pump with signal modes. The primary objective is a reduction in DMG, which is a prerequisite for long-haul communication systems [8,25]. In this work, for further in-depth performance assessment, evaluation and comparison of MEDP 4M-EDFA 980 nm;1480 nm systems, additional differential performance parameters are introduced and utilized. These include differential spectral gain (DSG) to quantify the difference in the unique modal profiles propagating through FM-EDFA. Differential modal noise figure (DMNF) and differential spectral noise figure (DSNF) are introduced to quantify amplified spontaneous emission (ASE) impact in 4M-EDFA 980 nm;1480 nm systems. Modal gain mechanisms realized in the EDFA is assessed by overlap factor are described as an integral function of product of pump and signal intensities over a uniform erbium ion profile [12,21]. In this work, a modified erbium ion inclusive transverse overlap factor designated as sp is introduced where the normalized erbium ion profile is calibrated over a scale of 0-1. The resulting normalized overlap factor facilitates the mathematical operation involving differential modal gain equalization criteria. This criteria is used to propose an inverse optimum erbium ion profile where the differential performance parameters are significantly reduced. The extracted parameters related to sp introduced in this work are differential modal overlap factor (DMOF) and differential spectral overlap factor (DSOF) designated as Δη sp,λ and Δη sp,λ , respectively. DMOF and DSOF prove to be decisive parameters and show a strong correlation with DMG and DSG, respectively. According to our investigation, implementation of MEDP in 4M-EDFA with 1480 nm pump power is more beneficial than the system with 980 nm pump power. It is shown that 980 nm pump power gives unfavorably higher DMOF (Δη sp,λ ) and DSOF, (Δη sp,lm ) values than 1480 nm pump power. Reported research [12][13][14][15][16][17][18][19][20][21][22][23][24]  is focused on 4M-EDFA 1480 nm with the aim of reducing the aforementioned differential performance parameters. For the 4M-EDFA 1480 nm system, the inverse normalized sum of signal intensities of the N inv (r) profile outperforms the uniform profile of the differential performance parameters. The N opt (r) profile can be obtained by optimizing the N inv (r) profile using coupled mode equations and differential modal gain equalization criteria. It will be shown that all the differential performance parameters are significantly reduced in a 4M-EDFA 1480 nm system using theN opt (r) profile.
The organization of this work is as follows: In Sect. 2, differential modal and spectral overlap factors calculated in 4M-EDFA 980nm;1480nm systems are analysed and compared. In Sect. 3, simulation-based investigations are carried out on 4M-EDFA systems with LP 01;980nm and LP 01;1480nm pump modes, respectively. In Sect. 3.1, N(r) = 1 i.e. the uniform erbium ion profile is used. In Sect. 3.2, theN inv (r) profile is extracted from the inverse sum of the normalized signal intensity function for 4M-EDFA 1480nm . In Sect. 3.3, theN opt (r) profile is identified that satisfies differential modal gain equalization criteria. Based on the analysis and results obtained, a brief conclusion is presented in Sect. 4.

;1480
systems In this section, the EDFA-based three-level energy state diagram and associated coupled mode equations are described. The coupled mode equations are used to evaluate the differential performance parameters, namely DMG, DSG, DMNF and DSNF, for assessing and investigating 4M-EDFA 980 nm;1480 nm systems. The transverse overlap integral equation is normalized over a scale of 0-1 to include the unit-less erbium ion profile parameter N(r) . This erbium ion inclusive transverse overlap integral sp will be evaluated and identified for achieving higher gain per mode with the lowest possible DMG between them. Two new related parameters introduced in this work, namely DMOF and DSOF, have been efficiently used to analyze and compare the performance of 4M-EDFA 980nm with 4M-EDFA 1480nm with a uniform profile, N(r) = 1. In this section, study has been carried out involving four signal modes, LP 01 ;LP 11 ;LP 21 and LP 02 pumped with LP 01;980nm and LP 01;1480nm , respectively. The investigations carried out in this section show that the overlap factors of all respective signal modes at all operating wavelengths with LP 01;980nm is higher than with LP 01;1480nm . However, the main drawback observed is that the DMOF and DSOF are also higher for LP 01;980nm than for LP 01;1480nm 4M-EDFA systems. FM-EDFA considers achieving low DMOF and DSOF as more important modal and spectral gain equalization criteria than achieving high individual modal overlap factors. The low DMOF is an important factor that ensures low DMG, and hence, low DMNF and, similarly, the low DSOF are significant for achieving low DSG and low DSNF.

FM-EDFA-based coupled mode equations
A simplified representation of a three-level energy system for erbium ion-doped optical fiber is shown in Fig. 1. For 1480 nm pump power, the erbium ions are directly excited from a ground level E 1 ( 4 I 15/2 ) to the upper metastable energy level E 2 ( 4 I 13/2 ). By using 980 nm pump power, the erbium ions are excited from ground level E 1 to upper energy level E 3 ( 4 I 11/2 ) first, before quickly relaxing down non-radiatively to the metastable energy level E 2 , due to the short lifetime (5-10 µs) of theE 3 energy level. This rapid spontaneous decay renders the number of ions in energy level E 3 negligible with respect to the number of ions in metastable energy level E 2 . Thus, the ion population in energy level E 3 is neglected for calculating the gain and noise figure of an EDFA that uses 980 nm pump power. The use of 980 nm and 1480 nm pump power is beneficial and widely preferred because these avoid the excited state absorption (ESA) of the pump power. ESA reduces the effective population inversion between energy level E 2 and E 1 , thus reducing the amplification efficiency of the EDFA [26]. At steady state, the normalized ion population in the ground state and metastable state given by the following Eqs. (1) and (2) are obtained by equating the rate equation to zero [11].
(1) Here, lm defines the signal or pump modes, where l and m are the number of zeros in the azimuthal and the radial directions, respectively. For a particular mode, the number of zeros in the azimuthal direction is 2l and the number of zeros in the radial direction is m − 1 [27]. The parameters v s and v p are optical frequencies at signal and pump wavelengths, respectively, is the spontaneous emission lifetime for the excited state, h is the Planck constant and N 0 is the total erbium ion concentration ( Er 3+ ) per unit volume. The parameters as and ap are the absorption cross-section areas, whereas es and ep are emission crosssection areas at signal and pump wavelengths, respectively.
Normalized intensities of signal and pump wavelengths are designated as i s,lm (r, ) and i p,lm (r, ) , respectively, where these parameters give the distribution of power in radial and azimuthal directions. For a particular mode (LP lm ) , the normalized intensity parameter is a function of operating wavelength ( ) and waveguide-related parameters, namely core radius (a), refractive indices of core (n 1 ) and cladding (n 2 ).
The gain and noise figure along the length of FM-EDFA can be calculated in terms of the following wavelengthdependent coupled mode equations. These coupled mode equations describe the propagation of signal, pump and ASE for each transverse mode as given by Eqs. (3), (4) and (5), respectively. In these equations, the Er 3+ doping region extends from a 1 ≤ r ≤ a 2 [18,21,28]. (2) where u k =1(u k = −1) denotes a mode traveling in the forward (backward) direction, and the term 2hv s Δv denotes the spontaneous emission. The performance parameters evaluated in this work for a 4M-EDFA system are DMG, DSG, DMNF, DMOF and DSOF, whereas sp -related parameters DMOF and DSOF will be calculated and discussed in Sect. 2.3.
The signal gain of any mode is given by Thereby, DMG is expressed as where i and j are the different signal modes at the same frequency amplified by FM-EDFA. DSG is defined as the maximum deviation of gain of a mode over given range of frequency and is expressed as where i and j are different frequency positions of a signal mode.
The NF of any signal mode can be obtained by Thereby, DMNF can be calculated as (6) Gain (dB) = 10 log 10 P s,lm (z = L) P s,lm (z = 0) Similarly, DSNF is also expressed as An important factor that determines the gain per mode is the overlap factor ( sp ) , which is the product of normalized signal intensity and normalized pump intensity along with doping profile [12,29]. The transverse overlap factor integral has been modified to include normalized Er 3+ ion profile over a scale of 0 to 1. Here, the normalized Er 3+ ion profile inclusive transverse integral has been used to evaluate and compare the DMOF and DSOF performance of 4M-EDFA 980nm and 4M-EDFA 1480nm systems. It also facilitates the use of modal gain equalization criteria and thereby extraction of N opt (r) profile identification where all differential performance evaluation parameters are significantly reduced.

Erbium ion profile inclusive transverse overlap integral
In this section, Eqs. (1) to (3) will be used to describe theEr 3+ ion profile inclusive transverse overlap integral which is designated by sp . By calculating the number of Er 3+ ions at the energy levels E 1 and E 2 of FM-EDFA and substituting these values into coupled mode equations, we can determine the relation between modal signal gain and sp . Since the 4M-EDFA system in this work uses only one pump power mode ( LP 01 ) , the pump related term ∑ lm P p,lm (z)( ap + ep) i p,lm (r, ) and N 2 (r, , z) from accordingly altered Eqs. (1) and (2) into Eqs. (3), (4) and (5), we get where N (r) is erbium ion concentration (ions/m −3 ) along the core of FM-EDFA, expressed as Here, N 0 is the erbium ion concentration (ion/m −3 ) value. N(r) is normalized Er 3+ ion concentration, whose values are bounded between 0 and 1 as will be explained in Sect. 3.2.
F(r, , z) is given by The necessary and sufficient condition for modal gain equalization, and hence low DMG value, is that the signal power variation should be equal between all the modes, i.e., dP s,lm (z) dz must be the same for all signal modes at all points along the length [29]. This differential modal gain equalization criteria is described by Eq. (17) as The absorption and emission cross-sectional areas are dependent only on wavelength in FM-EDFA and are equal for all modes. Hence, from Eqs. (12) and (17) it is clear that for achieving modal gain equalization, the transverse overlap factor, sp of normalized signal intensity i s,lm (r, ), normalized pump intensity i s,lm (r, ) and erbium-doped profile in case of various modes must approach the same values. In this paper, a unique erbium-doped profile is proposed, designated by N opt (r) that enables achieving high modal gain with low DMG between the modes. The Er 3+ ion inclusive transverse overlap integral is given by Eq. (18).
The transverse modal intensity is given by is a Bessel function, K l w lm r a is the modified Bessel function and a is the core radius. U lm and W lm are eigenvalues of the lm mode in the core and cladding, respectively. For normalization of the intensity profile of any mode, the value of A 2 (A is the normalization constant) is calculated by [27].
Hence, the normalized intensity of any mode in the core of the multi-mode optical fiber is expressed as U lm and W lm are the eigenvalues of the lm mode in the core and cladding, respectively, defined as where k is the propagation constant for light in vacuum. The propagation constants of the guided modes lm lie in the range The normalized frequency is given as The normalized propagation constant b for a fiber in terms of Eq. (25) is Equations (22) to (26) will be used to calculate the eigenvalues, V number and the propagation constant for the four signal modes ( LP 01 , LP 11 , LP 21 and LP 02 ) and the single pump mode ( LP 01 ) for both 980 nm and 1480 nm. These values enable us to investigate and compare the DMOF and DSOF performance of 4M-EDFA systems using 980 nm pump power and 1480 nm pump power, respectively. ≪ 1) satisfies the field matching conditions at the core-cladding interface. This requires boundary continuity of the transverse and tangential field components at the core and cladding interface (at r = a) [27]. The eigenvalue equation for the LP lm modes may be written by Eq. (27) as Using Eqs. (19) to (26) and thereby solving Eq. (27) allows us to calculate the value of b of various modes and study the impact of the optical wavelengths and fiber parameters on them.

DMOF and DSOF
Consider the seven operating signal wavelengths from 1530 to 1560 nm equispaced at 5 nm from each other. The erbium-doped fiber which has 10 µm core radius guides four LP lm modes ( LP 01 , LP 11 , LP 21 and LP 02 ) over the frequency range 1530-1560 nm.
Since our system uses only one LP 01 pump power, Table 1 shows calculated eigenvalues for this pump mode at 980 nm and 1480 nm pump power. Table 2 shows the calculated eigenvalues of various signal wavelengths for the four signal modes.

Radial pump and signal overlap function
The propagation of signal, pump, and ASE power depends on how closely their normalized intensities overlap along the EDFA. A two-dimensional pump and signal product can be used to represent this mechanism. In this way, a two-dimensional radial and azimuthal overlap function can be obtained. It is more feasible to use a one-dimensional radial-dependent pump and signal overlap function for analysis as given in Eq. (28) and subsequently drawn in Fig. 2a   For four possible LP lm signal modes ( s =1550 nm), Eq. (28) has been determined for each pump mode at two different wavelengths ( p =980 nm and p =1480 nm), respectively. Hence, for LP 01 pump mode, four radial overlap functions are generated, each for 980 nm and 1480 nm pump wavelengths. The resulting radial overlap functions are designated as i(r, p , s ) 01p;lms , where lm = 01, 11, 21 and 02 corresponding to the four signal modes, respectively. These functions are shown plotted in Fig. 2a-d. In all the overlap functions, the 980 nm LP 01 pump mode yields a high-intensity peak, while the 1480 nm LP 01 pump mode yields a low-intensity peak.
The transverse overlap integral as given by Eq. (18) is implemented on all the above radial signal and pump overlap functions for 980 nm and 1480 nm wavelengths, respectively. The values of η sp so obtained are shown in Fig. 2a-d, with maximum in the case of 01s,01p = 1.7802 × 10 10 rad −1 m −2 and minimum in the case of 21s,01p = 2.7967 × 10 9 rad −1 m −2 for 980 nm wavelength. Similarly, for 1480 nm wavelength, the maximum is in the case of 01s,01p = 1.5986 × 10 10 rad −1 m −2 and minimum is 21s,01p = 2.8221 × 10 9 rad −1 m −2 . It can also be observed from Fig. 2a and d that the respective peaks of transverse overlap functions ( i r, p , s 01p;01s ) and ( i r, p , s 01p;02s ) are positioned at the core centre for both 980 nm and 1480 nm pumps since LP 01 pump and LP 01, LP 02 signal modes are all centered. Also, if the doping profile is concentrated at the center of the core, the gain of centered LP 01, LP 02 signal modes is high as compared to non-centered LP lm modes. Similarly, it can also be observed from Fig. 2b and c that the peaks of transverse overlap functions ( i r, p , s 01p;11s ) and ( i r, p , s 01p;21s ) are positioned at 4.4 µm and 6 µm, respectively, for both 980 nm and 1480 nm pumps. If the doping profile is concentrated at these respective non-centered points of the core, the gain of non-centered LP 11, LP 21 signal modes will be high as compared to centered LP lm modes.
However, the performance parameter DMOF ( Δ sp, ) is a strong indicator of modal gain equalization characteristics of a particular pump wavelength than the individual overlap function. Here, the DMOF is calculated where for 980 nm it is Δ sp, =15.0053 × 10 9 rad −1 m −2 which is greater than Δ sp, =13.1639 × 10 9 rad −1 m −2 for 1480 nm. This clearly indicates that 1480 nm is a better choice than 980 nm to be used as pump wavelength.
The performance parameter DMOF introduced in this work is expressed as where sp, (i) and sp, (j) are overlap factors of different signal modes (at the same frequency) with a pump mode of the FM-EDFA.
The performance parameter DSOF also introduced in this work is expressed as where i and j are different frequencies of a signal mode.
The calculated values of DMOF Δ sp, and DSOF Δ sp,lm are given in Table 3 and Table 4 for 4M-EDFA 980 nm and 4M-EDFA 1480 nm , respectively. These values have been calculated for the uniform erbium doping profile, N(r) = 1. From Tables 3 and 4, it is observed that each value of the overlap factors at each operating wavelength and for each signal mode LP 01;980 nm is greater than that for LP 01;1480 nm . However, even the DMOF and DSOF values associated with 980 nm pump power are undesirably high as compared to 1480 nm pump power.
The values of Δ sp, are plotted in Fig. 3 for the uniform doping profile at the seven operating wavelengths for the two pump power values. From Fig. 3, it is observed that Δ sp, is undesirably higher for LP 01;980nm than LP 01;1480nm pump mode over all the operating wavelengths. The tabular values and Fig. 3 show that Δ sp, does not change significantly with the operating wavelengths in the two systems. On the other hand, Fig. 4 is a bar diagram which gives the measure of Δ sp,lm for uniform doping profile at the different modes for the two pump power levels. The parameter value Δ sp,lm is higher for all signal modes with LP 01;980nm than LP 01;1480nm pump mode. Figure 4 shows the signal mode LP 02 results in higher values of Δ sp,lm than the other three modes.
In the following section, 4M-EDFA simulation setup has been used with three erbium ion profile variants: N(r) = 1, N inv (r) and N opt (r) . This system is used for the comparison of DMG, DSG, DMNF, and DSNF performance evaluation parameters of 4M-EDFA systems.

Simulation investigation: 4M-EDFA with 01−980 ;1480 pump modes
In Sect. 2.3, mathematical analysis showed that DMOF (Δ sp, ) and DSOF (Δ sp,lm ) are higher for LP 01;980nm than LP 01;1480nm pump mode. Therefore, 4M-EDFA 1480nm is further explored for the possibility of DMOF and DSOF reduction rather than aiming at increasing sp values. In this section, we will study the effect of Δ sp, on DMG and DMNF and the effect of Δ sp,lm on DSG and DSNF. For that, a 4M-EDFA simulation setup is used with alternative arrangements of LP 01;980nm and LP 01;1480nm pump modes. Initially, the abovementioned performance parameters are evaluated for uniform erbium profile N(r) = 1, with the selected pump power values. Because of the resulting low values of DMG, DSG, DMNF and DSNF performance parameters, LP 01;1480nm offers a better choice compared to theLP 01;980nm pump power system. This is followed by investigation carried out on 4M-EDFA with theLP 01;1480nm system with the objective to identify unique erbium ion profile N inv (r) that can further reduce these parameters. This section is finally concluded by successfully optimizing theN inv (r) profile and hence replacing it by a unique optimum erbium ion profile N opt (r) , which will ensure further reduction of DMG, DSG, DMNF and DSNF.

system performance with the N(r) = 1 profile
The 4M-EDFA simulation setup shown in Fig. 5 is used to compare the performance parameters for LP 01;980nm and LP 01;1480nm pump modes. The simulation is carried out in OptiSystem 16 software [30]. At the input, four mode-specific wavelength tunable lasers are used as optical signal sources. Two pump lasers have been used, one at 980 nm wavelength and the other at 1480 nm wavelength, both having same modal field (i.e., LP 01 ). The four signal modes are multiplexed with the LP 01;980nm pump power when K1 is on and with LP 01;1480nm pump power when K2 is on, respectively. The four-mode multiplexed output along with pump mode is fed into the 4M-EDFA unit, and after amplification the modal demultiplexer is used to separate the amplified signal modes.
The resulting modal gain observations made from dualport WDM analyzers utilize Eq. (7) to calculate values of DMG and, similarly, the resulting modal spatial observation made from spatial visualizers use Eq. (8) to calculate   Table 5.
These DMG and DSG values are calculated at all operating wavelengths with respect to LP 01;980nm and LP 01;1480nm pump modes. These values are shown in Tables 6 and 7.
The respective values of P + ASE,lm obtained from 4M-EDFA unit is used to calculate the NF using Eq. (9). Thereby, DMNF and DSNF are calculated form Eqs. (10) and (11).   In Tables 8 and 9, respectively, we present the calculated DMNF and DSNF values for all four signal modes at all operating wavelengths with LP 01;980nm and LP 01;1480nm pump modes.
Using Tables 6 and 7, DMG values are plotted for LP 01;980nm and LP 01;1480nm pump modes in Fig. 6. The DMG values of LP 01;980nm are greater than for the LP 01;1480nm pump mode at all wavelengths. This is because of low DMOF at all operating wavelengths in a 4M-EDFA 1480 nm system. Tables 3 and 4 show that for both the 980 nm and 1480 nm wavelengths, DMOF reduces as the wavelength increases, and as a consequence, the DMG is decreasing for both pump wavelengths as the signal wavelength increases. Hence, the simulation results prove the direct correlation of DMG and DMOF.
DSG values are measured for different signal modes in the bar diagram as shown in Fig. 7 for the given system. DSG values are higher at all signal modes for 4M-EDFA 980 nm than the 4M-EDFA 1480 nm system. For LP 01 signal mode, DSG is nearly the same for 980 nm and 1480 nm systems. This is expected because, as shown in Fig. 2a, the signal LP 01 and pump LP 01 have similar modal profiles resulting in maximum overlap factor sp at 980 nm as well as at 1480 nm pump modes. For all operating signal wavelengths, a minimum DSOF results in minimum DSG. The LP 21 signal modal profile differs from the LP 01 pump mode for both pump wavelengths as seen in Fig. 2c. Hence, it has the largest DSG among all signal modal profiles as seen in Fig. 7. In terms of preferred low DMG and DSG values, the 4M-EDFA 1480 nm system is a better choice than 4M-EDFA 980 nm when N(r) = 1, i.e., the uniform erbium-doped profile. Figure 8 gives the DMNF values at 4M-EDFA 980 nm and 4M-EDFA 1480 nm pump mode. It is observed that all signal modes have low DMNF for 4M-EDFA 1480 nm than 4M-EDFA 980 nm pump mode at all operating wavelengths except the last two wavelengths. According to Fig. 6, DMG reduces as wavelength increases, which also causes DMNF to decrease as wavelength increases. This is due to an inverse logarithm relation between gain and NF as given in Eq. (9). Also, Fig. 8 shows that the highest value of DMNF occurs for 4M-EDFA 980 nm at 1530 nm wavelength. Around 1550 nm wavelength, both systems result in the same value of DMNF. DSNF for four signal modes is plotted in Fig. 9. All signal modes except LP 01 have higher values of DSNF for the 4M-EDFA 980 nm system than the 4M-EDFA 1480 nm system. This trend can also be explained by inverse logarithm relation between gain and NF given in Eq. (9). A higher gain yields a lower NF, and a high DSG yields a high DSNF. With reference to Fig. 7, DSG is higher for LP 21 signal mode. Similarly, DSNF is also high for this signal mode as shown in Fig. 9. In terms of DSNF performance, 4M-EDFA 1480 nm outperforms the 4M-EDFA 980 nm system.
These simulations were carried out for the uniform erbium ion profile. Simulation results exhibit a strong correlation of DMG and DSG parameters to DMOF and DSOF, respectively. Since the 1480 nm pump yields smaller values of DMOF, DSOF, DMNF and DSNF, 4M-EDFA 1480 nm is a better choice than the 4M-EDFA 980 nm system. All further investigations will be carried based on the 4M-EDFA 1480 nm simulation system. In the following section, a new unique erbium ion profile N inv (r) function will be identified that results in the reduction of all the differential performance evaluation parameters for the 4M-EDFA 1480 nm system.

system performance with N (r) profile
To ensure reduction in the differential performance evaluation parameters, the proposed N inv (r) profile should be capable of linearizing the non-uniform plot of the total signal intensity curve over the EDFA core. The total signal intensity plot designated by Y(r) is obtained by adding all four normalized signal intensities and integrating the sum along the azimuthal direction over the limits 0 to 2 . The Y(r) function is plotted in Fig. 10  For this figure, we see that the maximum value of Y(r) is observed at the core center. This is expected because the centered normalized signal intensities i s,01 (r, )andi s,02 (r, ) have high intensity values as compared to non-centered normalized signal intensities i s,11 (r, ) and i s,21 (r, ) . At the core and cladding interface, the value of Y(r) is minimum, because all the signal intensities have low value at this point.
For reduction in all the differential performance parameters in the 4M-EDFA system, the proposed linearizing erbium ion profile N inv (r) should be related to the inverse of Y(r) function. Based on Y −1 (r) erbium profile, nearly equal distribution of power in all signal modes occurs, which gives low value of DMG. Corresponding to each value of Y(r) over the core radius, we calculate Y −1 (r) , shown in Fig. 10. The maximum value of Y −1 (r) is 3.199 × 10 -11 m 2 and occurs at r = 10 μm. In order to determine the most appropriate erbium ion concentration over each core segment that can ensure reduction of differential performance parameters, a normalized function is determined, given by This normalized function is plotted in Fig. 11 as a function of core radius with minimum value of 0.2672 at r = 0 μm and 1.0 at r = 10 μm. This plotted function shows 0.7801 at r = 9.5 μm as the maximum value that can occur for any mid-point position out of the ten core segments. The next step is to divide each value of the normalized function by 0.7801. This is to ensure step change in normalized erbium ion concentration values over a scale of 0 to 1, where 1 corresponding to maximum erbium ion profile value of 15 ×10 24 ions/m 3 . The resulting traced out function is also shown in Fig. 11 with a minimum value equal to 0.3426 at r = 0 μm and maximum value equal to 1.2819 at r = 10 μm. This step has shifted the value to 1.0000 at 9.5μm core radius segment position. The resulting linearizing erbium ion profile represented by N inv (r) is the rounded-off step function approximation of the normalized function. As observed in Fig. 11, the minimum value of N inv (r) is equal to 0.35 in the first core segment and the step function builds up to a maximum value of 1.00 over the last core segment. In comparison to N(r) = 1, the product of N inv (r) with each of the four signal modes yields the smallest deviation of values among them. DMG is reduced among the four signal modes for this reason. Since N o is a constant and has a value of 15 ×10 24 ions/ m 3 as already mentioned in Table 5, the actual erbium ion concentration over each core segment is given by The lowest value of N inv (r) occurs for the segment 0-1 μm and is 5.25 × 10 24 ions/m 3 . On the other hand, N inv (r) has the highest value for the segment 9-10 μm and is 15 × 10 24 ions/m 3 .
A comparison of DMG between the uniform erbium doping profile N(r) = 1 and inverse erbium doping profile N inv (r) can be seen in Fig. 12. For the 4M-EDFA 1480 nm

Fig. 11
Plot of normalized function Z(r) , Z(r)/0.7801 and erbium ion profile, N inv (r) as a function of core radius system with N inv (r) profile, DMG is lower for all operating wavelengths than with the uniform doping profile. Moreover, it can be observed that the minimum value of DMG is 2.5541 dB at an operating wavelength of 1550 nm. Figure 13 compares DSG for four signal modes over a given band of wavelength. It is observed that there is significant reduction in highest DSG for LP 21 signal mode when N inv (r) profile is used than with N(r) = 1 profile. N inv (r) profile is suitable for WDM systems wherein it is desirable to have minimum DSG amongst various WDM grid channels.
Similarly, DMNF and DSNF values have been obtained for the uniform erbium profile and compared with theN inv (r) profile of the 4M-EDFA 1480 nm system. The resulting values have been calculated using Eqs. (9) to (11). DMNF plot diagram and DSNF bar diagram are shown in Figs. 14 and 15, respectively. Here, the observation shows that DMNF is lower at all operating wavelengths of theN inv (r) profile in comparison to the uniform erbium profile of the 4M-EDFA 1480 1.72 dB). Similarly, DSNF is lower for all signal modes of theN inv (r) profile than the uniform profile of the 4M-EDFA 1480 nm system. Reduction in DSNF is maximum for the LP 21 mode on the order of 5.49 dB.
Although linearizing theN inv (r) profile of Fig. 11 has resulted in improvement of differential performance evaluation parameters, there is a scope of identifying its linearizing optimum variant designated as N opt (r) . The next objective is to identify an optimum N opt (r) profile that causes further reduction of differential performance evaluation parameters as well as improves the individual gain values at each operating wavelength.

Unique N (r) profile identification
for the 4M-

system
In case of the earlier determined N inv (r) profile DMG at 1550 nm wavelength is 2.5541 dB as shown in Fig. 13. Further reduction of DMG and DMNF is the main aim in this section by proposing N opt (r) profile which is the linearizing optimum variant of N inv (r) . For this, the modal gain equalization procedure required is described in Sect. 3.3.1. Therefore, in Sect. 3.3.2, the identified N opt (r) profile will be used to compare the system performance with respect to theN inv (r) profile for the 4M-EDFA 1480 nm system.

Modal gain equalization routine
There are four steps in using the modal gain equation routine, which are as follows: (i) Calculation of sp(lm) values for N(r) = 1: For reduction of DMG and DMNF, we refer back to Eqs. (12) to (14). From these equations, the gain and noise are dependent on the overlap factor and different fiber parameters. Hence, it is necessary to involve the coupled mode equations for further improvement of DMG. Over ten equal segments of core radius, a total of 40 combinations of overlap factors are calculated between the pump mode LP 01 . With each of the signal modes LP 01 ;LP 11 ; LP 21 and LP 02 using Eq. (18) for N(r) = 1 along with Eq. (21). The values are calculated for sp(lm) ( sp(01) ; sp (11) ; sp (21) and sp(02) ) which are shown in Fig. 16 over the ten core segments. The region 0-1 µm has highest value for sp(02) for LP 02 signal mode equal to 5.1787 × 10 9 rad −1 m −2 . Hence, the change in doping in this region gives the highest change in gain for LP 02 signal mode than other modes.
(ii) Calculating dz ]. These shortlisted extremes have to be modified until they approach the nearest common value so that the modal gain equalization criteria as given by Eq. (17) is satisfied. According to modal gain equalization criteria, the value of dP s,lm (z) dz for all modes must be equal at all points along the length of EDFA. This criterion, therefore, demands the modification of the shortlisted extremes until they approach the nearest common value, so that DMG is reduced.
(iv) Neutralizing the extreme dP s,lm (z) dz values: The overlap factor depends on dP s,lm (z) dz as given in Eq. (12). This relation is exploited for neutralizing these extreme values to the nearest common value. Locating the core segment with the maximum value of overlap factor given in Fig. 16 for both these extremes, respectively, thereby changes theN inv (r) erbium ion profile shown in Fig. 11 in that particular segment accordingly. This procedure is continued until Eq. (17) is reasonably satisfied. respective gains of LP lm signal modes as shown in Fig. 18. The result shows that each modal signal has more than 22 dB amplification over the input signal power of −10 dBm. The lowest deviation in gain for all four modes is observed at an operating wavelength equal to 1545 nm. Figure 19 shows that the N opt (r) profile has lower values of DMG for all operating wavelengths than theN inv (r) profile. This offers better results at all operating wavelengths in terms of DMG parameter. The DMG values at 1550 nm wavelength are 2.4388 dB for the N inv (r) profile and 0.3624 dB for the N opt (r) profile. Figure 19 shows a significant reduction in DMG for all operating wavelengths in theN opt (r) profile compared with theN inv (r) profile. DSG is also compared for theN opt (r) profile with theN inv (r) profile and is shown in Fig. 20. The values of DSG for the signal modes LP 21 and LP 01 is higher for N opt (r) profile than theN inv (r) profile. Also, the DSG for the signal modes LP 11 and LP 02 is higher for theN inv (r) profile than theN opt (r) profile.
The maximum value of DSG occurs at LP 11 signal mode for N inv profile is 2.7293 dB. Similarly, the maximum DSG for N opt (r) profile is 2.4997 occurs at LP 11 signal mode. The 4M-EDFA system operates in all modes, and high values of DSG prove to be detrimental for system performance; hence, the N opt (r) profile outperforms the N inv (r) profile. As a result, the erbium profile with the highest DSG value is not preferred.
As shown in Fig. 21, the DMNF of the N inv (r) profile is significantly higher than the DMNF of the N opt (r) profile at all operating points. At 1550 nm, the DMNF is 0.3279 dB, which is the lowest value and is found for the N opt (r) profile.
DSNF values are shown in Fig. 22 for four various signal modes, and the N inv (r) profile has higher DSNF values than theN opt (r) profile at LP 11 signal mode. The 4M-EDFA system operates in all four signal modes and the preferred profile is N opt (r) which results in lowest DSNF value. Table 10 presents the comparison of differential performance parameters namely DMG, DSG, DMNF, and DSNF with uniform profile (N(r)=1) when 980 nm and 1480 nm pumps are used, respectively, with the linearizing erbium ion profile (N inv (r)) and linearizing optimum erbium ion profile (N opt (r)) when the 1480 nm pump is used. Table 10 presents  Table 10.
Amongst all the systems, the 4M-EDFA system with a uniform profile and 980 nm pump exhibits the highest values of DMG, DMNF, DSG, and DSNF. All these parameter values reduce when initially in the same system (N(r) =1) the pump wavelength is changed from 980 to 1480 nm. Subsequently, for the 4M-EDFA 1480 nm system, theN inv (r) erbium ion profile is extracted from the inverse sum of normalized signal intensities which further reduces these parameters. Subsequently, differential modal gain equalization criteria are utilized that identifies a unique linearizing N opt (r) profile for the 1480 nm system. From Table 10, it is observed the highest value of DMG, DSG, DMNF and DSNF are 13.295 dB, 3.9717 dB, 5.9996 dB, 11.0649 dB, respectively, for the 980 nm uniform erbium ion profile system. The highly improved values are shown in the case of the 1480 nm pump using the linearizing N opt (r) profile with reduced values of DMG, DSG, DMNF and DSNF equal to 2.0558 dB, 2.4997 dB, 2.77 dB and 3.879 dB, respectively. Use of the 1480 nm pump, whether for uniform, inverse or optimum profile, proves to be beneficial over the use of the 980 nm pump wavelength. This justifies the already reported superior modal overlap properties of the 1480 nm pump wavelength with signal modes in comparison to the 980 nm pump wavelength. This is the main reason for selecting the 1480 nm wavelength for the 4M-EDFA system over the 980 nm wavelength. The DMG, DMNF values obtained are not the same for all operating wavelengths. In our proposed 4M-EDFA, DMG is equal to 2.0558 dB and 0.0882 dB at 1530 and 1545 nm, respectively. Similarly, DMNF is equal to 2.77 dB and 0.3279 dB at 1530 and 1550 nm, respectively. Also, DSG and DSNF are not the same at all LP lm signal modes. For example, DSG is equal to 2.4997 dB and 1.5247 dB for LP 01 and LP 11 modes, respectively. DSNF is equal to 3.879 dB and 2.8691 dB for LP 01 , LP 21 modes, respectively. In addition to reduced differential performance parameters, this system also offers an increase in average modal gain value of 24.17 dB making it suitable for 4M-EDFA SDM applications.

Conclusion and future scope
The MEDP 4M-EDFA system is investigated in this paper with the objective to achieve high gain per mode with minimum possible DMG between them. To quantify and ascertain the influence of erbium ion profile used in EDFA, a modified erbium ion profile inclusive overlap factor sp is introduced. The more decisive parameters extracted in this work related to sp are the DMOF (Δ sp, ) and DSOF (Δ sp,lm ). It has been shown that these two parameters are higher for the 980 nm pump wavelength than the 1480 nm pump wavelength. Simulation results show that differential performance parameters of the 4M-EDFA 1480nm system are favorably lower as compared to the 4M-EDFA 980nm system. The investigations based on coupled mode equations and simulation results strongly support the correlation of DMG and DSG with DMOF and DSOF, respectively.  The linearizing N inv (r) profile is developed from the normalized inverse sum of signal intensity function (Y(r)) specifically for a 4M-EDFA 1480nm system. This is in order to take advantage of the superior modal overlap properties of the 1480 nm pump as compared to the 980 nm pump wavelength. With the objective for further reduction in differential performance parameters, a linearizing optimum erbium ion profile, N opt (r) is extracted by applying modal gain equalization criteria. With theN inv (r) profile for 4M-EDFA 1480nm system at 1550 nm wavelength, the average modal gain is 24.05, DMG is 2.4388 dB, DMNF is 2.6862 dB, the highest DSG is 2.7293 dB and the highest DSNF is 4.6415 dB. Similarly, for theN opt (r) profile, improved average modal gain is 24.17, DMG is 0.3624 dB, DMNF is 0.3278 dB, the highest DSG is 2.4997 dB and the highest DSNF is 3.879 dB. Therefore, this N opt (r) profile for the 4M-EDFA 1480nm system with significantly low DMG, DMNF and higher gain per mode could be an appropriate choice for SDM application.
The future scope of this work is to identify the MEDP that reduces differential performance parameters for a system having more than for modes. MEDP and multi-pump technique MSPC can be simultaneously employed in higher order modes to increase the overall system capacity. In this system, MEDP has been employed with an LP 01 pump at 1480 nm showing better performance than theLP 01 pump at 980 nm. MSPC is another solution to be investigated at 1480 nm on the same signal modes (centered as well as offcentered) with multiple pump modes. Alternatively, implementing MSPC on all centered signal modes with multiple pump modes at 1480 nm is expected to considerably ease off utilization of mode gain equalization criteria.
Even though the differential parameters are significantly reduced by using the proposed N opt (r) profile, there is still a scope for further reduction. Future work can focus on using multiple pump power levels, high cladding power, or a combination of both, but at the expense of added pump sources and power costs.
Funding No funds, grants or other support was received.

Availability of data and material Not applicable.
Code availability Not applicable.