Azimuthal dependence of modal gain and its impact on gain equalization in multimode erbium-doped fiber amplifiers

Optik 126 (2015) 3538–3543

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Optik journal homepage: www.elsevier.de/ijleo

Azimuthal dependence of modal gain and its impact on gain equalization in multimode erbium-doped fiber amplifiers Zujun Qin a,b,∗ , Song Ye a , Wentao Zhang a , Xianming Xiong a a b

School of Electronic Engineering & Automation, Guilin University of Electronic Technology, Guilin 541004, China Guangxi Key Laboratory of Automatic Detecting Technology and Instruments (Guilin University of Electronic Technology), Guilin 541004, China

a r t i c l e

i n f o

Article history: Received 23 September 2014 Accepted 31 August 2015 Keywords: Multimode erbium-doped fiber amplifier Spatially degenerate modes Differential modal gain Azimuthal angles

a b s t r a c t Azimuthal-dependent modal gain for spatially degenerate signal modes in multimode erbium-doped fiber amplifiers (MEDFAs) has been numerically investigated by integrating the differential power rate equations dealing with multiple spatial modes. Results show that if the azimuthal mode numbers in signal modes are different from that in the pump modal, gain is nearly independent of the azimuthal orientations of the pump intensity pattern. High differential modal gain (DMG) between spatially degenerate signal modes is produced if signal and pump shares the same azimuthal mode number when modal gain varies periodically with the pump azimuthal angle. Such azimuthal dependence of modal gain is caused by the transverse azimuthal dependence of population inversion of erbium ions created by pump modes. Adjusting the power and azimuth of the pump mode has also been examined to reduce azimuthalinduced DMG and several optimal parameters for achieving gain equalization have been obtained for an MEDFA supporting the two lowest LP signal mode groups. Our results can be employed to equalize mode-dependent gain in MEDFAs. © 2015 Elsevier GmbH. All rights reserved.

1. Introduction Mode-division multiplexing (MDM) based on multimode fibers (MMFs) has been proposed as one of the promising approaches to further increase system capacity and meet future demands [1–3]. Large numbers of MDM experiments have demonstrated its excellent performance for large capacity transmission [4–7], in which MMF eigenmodes act as parallel orthogonal channels. By applying the weakly guiding approximation, the modes in MMF are well approximated by linearly polarized (LP) modes. Every LP mode with nonzero azimuthal mode number (or non-LP0v mode) has two spatially degenerate versions, both of which serve as mode channels in MDM. A data rate of 73.7 Tb/s has been reported by using the fiber’s LP01 and two spatially degenerate LP11 modes (i.e., LP11a and LP11b ) [5]. According to information theory, the system capacity scales linearly with the channel numbers [1]. In the past year, an MDM setup based on six spatial modes (i.e., LP01 , LP11a , LP11b , LP21a , LP21b , and LP02 ) has also been demonstrated, leading to a record spectral efficiency of 32 bit/s/Hz [6]. In this instance, amplification on the six spatial modes is achieved by six single-mode EDFAs, resulting in a

∗ Corresponding author at: Guilin University of Electronic Technology, School of Electronic Engineering, No. 1 Jinji Road, Guilin, China. Tel.: +86 13977381793. E-mail address: [email protected] (Z. Qin). http://dx.doi.org/10.1016/j.ijleo.2015.08.276 0030-4026/© 2015 Elsevier GmbH. All rights reserved.

restricted transmission distance of 177 km due to distinct field profiles. Therefore, the deployment of multimode amplifiers capable of amplifying the multiplexed modes simultaneously is of significance for long-haul MDM. Extensive investigations focused on developing practical multimode amplifiers have been carried out in both theoretical and experimental means, including three-mode EDFA (3MEDFA) [8], 5MEDFA [9], and 6MEDFA [10]. A distributed 3 M Raman amplifier was studied in Ref. [11] as well. In multimode amplifiers, it has been suggested that differential modal gain (DMG) significantly affects the system capacity and its potential for outage [12]. Hence, decreasing DMG to achieve mode equalization of optical gain is an important task in the design of MEDFAs. To do so, several approaches can be combined or used independently: controlling the pump modal contents [8]; tailoring the erbium spatial distribution in the erbium-doped fiber (EDF) [13–15]; constructing dual-stage MEDFAs [16]. Recently, some new methods, such as adaptive techniques [17] and a mode-selective bidirectional pumping configuration [18], have been presented to equalize modal gain in MEDFAs. In the above efforts for DMG control, it often refers to gain equalization between different signal LP mode groups. However, DMG between any pair of degenerate signal modes receives little attention, e.g., DMG between LP11a and LP11b , DMG between LP21a and LP21b . The reason may be that by using coherent receivers with multiple-input and multiple-output digital signal

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Fig. 1. Schematic diagram of a 3MEDFA supporting LP01,s , LP11a,s , and LP11b,s modes. TMF means two-mode transmission fiber.

processing degenerate modes can be distinguished, irrespective of the gain difference between them [4]. Actually, azimuthaldependent non-LP0v pump modes are usually used to obtain accurate modal gain control, which create transverse azimuthal dependence of Er3+ population inversion and hence high DMG between spatially degenerate signal modes. After multiple amplification stages by MEDFAs the accumulated effect caused by DMG may be so severe that the receiver fails to discriminate between degenerate signal modes. Meanwhile, such high DMG may lead to the generation of nonlinear optical effects for the mode with high optical gain during propagating in transmission MMFs. Consequently, as independent channels in MDM the spatially degenerate signal modes experience azimuthal-dependent modal gain, which should be taken into account to design a practical MEDFA. The organization of the manuscript is arranged as follows. First and foremost, a theoretical model governing the multimode amplification is given. Then, simulations about azimuthal-dependent DMG and its impact on gain equalization are presented and results are discussed. In the last section, we provide our conclusions. 2. Theoretical model The MEDFA configuration for our exploration is shown in Fig. 1. To facilitate the discussion of a wide range of parameters, signal modes are restricted to the first two lowest spatial mode groups, namely, LP01,s and LP11,s . Pump modes of LP11,p and LP21,p are applied to pump the 3MEDFA, both of which have azimuthaldependent transverse intensity profiles. Here, we use the notation of LPuv,s(p) to represent LPuv mode at signal (pump) wavelength, where subscript u is the azimuthal mode number and v the radial mode number. It is worth pointing out that for u ≥ 1 LPuv is not rotationally invariant and has twofold spatially degenerate, namely, LPuva and LPuvb . Fig. 2 shows the transverse intensity patterns of LP11a , LP11b , LP21a , and LP21b . Mode LPuva has a maximum intensity along the x-axis, corresponding to an azimuthal angle of zero; while LPuvb is spatially rotated by ␲/2u with respect to LPuva . It is observed that the intensity profiles of the two degenerate LPuv (u ≥ 1) modes are shown to be strongly depended on their azimuthal orientations. The multiplexed signal and pump modes are spatially combined by a dichoric mirror (DM) and coupled into an EDF. The EDF is considered with a step index and uniform erbium doping profile. In the following, we make use of an intensity model to characterize the 3MEDFA. By assuming reasonably a two-level model and neglecting the weak intermodal coupling, the power in signal mode k (pks ), the power in pump mode l (plp ), and the amplified spontaneous noise (ASE) power in signal mode k (pkase ) evolution along the EDF can be written by a set of coupled differential equations [14,15]:

uk

dpks (z) dz



2



= pks (z) 0

0

a



2

[es n2 (r, ϕ, z) − as n1 (r, ϕ, z)] sk (r, ϕ) rdrdϕ

(1)

Fig. 2. Azimuthal dependence of the transverse intensity distributions of LP11 and LP21 modes. The last column is the intensity profiles of LP11 and LP21 ( is an arbitrary azimuthal angle).

ul

uk

dplp (z) dz dpkase (z) dz



2



p

a

= −a plp (z) 0

 =

2



pkase (z)



0

2





2

n1 (r, ϕ, z)pl (r, ϕ) rdrdϕ

(2)

0 a



2

[es n2 (r, ϕ, z) − as n1 (r, ϕ, z)] sk (r, ϕ) rdrdϕ

0

 k  s (r, ϕ)2 rdrdϕ

a

,

(3)

2hs es n2 (r, ϕ, z)

+ 0

0

 2  2 where sk (r, ϕ) (pl (r, ϕ) ) denote the normalized inten-

sity distribution of the kth (lth) mode at signal wavelength s s(p) s(p) (pump wavelength p ); e and a are the emission and absorption cross sections of erbium ions at signal wavelength (pump wavelength), respectively. Parameter uk denotes the propagation direction of beam k (uk = 1 for forward propagation, while uk = 1 indicates backward propagation) and a forward copropagating case (uk = 1) is considered here to avoid solving two-point boundary value problems. The second term in equation (3) describes the ASE contribution in the kth signal mode. In the steady state, the local concentrations of erbium ions at lower and upper level n1 (r, ϕ, z), n2 (r, ϕ, z) are functions of local signal and pump intensity distribution, and can be expressed as:

  a

h ,

n2 (r, ϕ, z) = nt (r, ϕ) 1+







2

p (z) (r, ϕ)





a +e h







  (r, ϕ)2

(4)

p (z)

,

n1 (r, ϕ, z) = nt (r, ϕ) − n2 (r, ϕ, z),

(5)

where nt (r,ϕ) denotes the total ions density and has been assumed to be position independent; = {p, s, ase} and is transverse mode order;  means the optical frequency of beam ; and  is the lifetime for the excited state and 10 ms is assumed. Equations (1)-(3) are initial value problems, the initial conditions of which are given by the power in the signal and pump modes at position z = 0. For exploring the azimuthal-dependent modal gain, the differential power rate equations (1)-(5) are numerically integrated by the standard fourth-order runge-kutta method. 3. Simulations and discussions The EDF used in our calculation has a core diameter of 18 m and numerical aperture of 0.101. We assume a uniform concentration of erbium ions of 15 × 1024 m−3 inside the core over an EDF length of 3.5 m. Such short amplifier length ensures us to neglect the weak

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Table 1 Emission and absorption cross-section areas [14]. Parameters

Value

es p e as p a

3.667 × 10−25 m2 0 2.46 × 10−25 m2 1.879 × 10−25 m2

Table 2 Overlap integrals between pump and signal modes. ijs,uvp × 109/ m2

Pump modes

Signal modes

LP01,p LP11a,p LP11b,p LP21a,p LP21b,p

LP01,s

LP11a,s

LP11b,s

5.2588 3.7500 3.7500 2.8295 2.8295

2.9700 5.0891 1.6964 3.2242 3.2242

2.9700 1.6964 5.0891 3.2242 3.2242

mode coupling and background loss. The injected signal is set to be continuous wave at 1550 nm, which is amplified by a pump beam at 980 nm. It is assumed that the input signal has −10 dBm power in each mode (or a total power of −5.2 dBm for LP01,s , LP11a,s , and LP11b,s ). According to the above parameters, the resulting normalized frequency V at 1550 nm is 3.68, indicating the EDF supports two mode groups at signal wavelength (i.e., LP01,s and LP11,s ). The pump modes in the EDF are estimated as well with the assistance of Sellmeier equation and about six mode groups are found by solving the fiber eigenvalue equation [19]. The emission and absorption cross-section areas of the EDF are listed in Table 1.

Fig. 3. Modal gain versus LP11,p power. LP21,p power is fixed to be 20.8 dBm.

in LP11a,p (LP11b,p ) will give a higher gain for LP11a,s (LP11b,s ), leading to a large DMG between LP11a,s and LP11b,s . If LP11,s is pumped by LP11 ,p (see the last column of Fig. 2), 11s,11 p lies between 11as,11ap ( 11bs,11bp ) and 11as,11bp ( 11bs,11ap ), corresponding to an intermediate DMG. From Table 2 we also notice that 11as,21ap = 11as,21bp = 11bs,21ap = 11bs,21bp , despite the fact that the intensity profiles of LP11,s and LP21,p are both azimuthal dependent. In this case, DMG in the two degenerated LP11,s modes can be fully ignored. To account for this azimuthal-independent overlap inte-



uv (r)cos(uϕ)|2 . with the method of separation of variables as |Fs(p) Thus, Eq. (6) can be rearranged as:



3.1. Intensity overlap integrals between pump and signal modes In many previously reported MEDFAs, LPuv,p (u ≥ 1) was assumed to be formed by LPuva,p and LPuvb,p with equal power so that its resulting intensity profile was azimuthal independent, and hence no DMG generation between spatially degenerate signal modes [8,20]. In fact, pump modes are often excited with a certain azimuth by mode generation devices, such as phase plates [4]. Different values generate different spatial overlaps between pump and signal modes. Since modal gain is highly dependent upon the overlaps high DMG between degenerate signal modes is produced. We have calculated the degree of intensity overlap (defined by Eq. (6)) for different pump and signal mode pairs and the results are given in Table 2. A larger overlap integral implies higher optical gain. Therefore, by the overlap data in the table we are able to firstly estimate the azimuthal-dependent modal gain.



ijs,uvp =

 

2  

2

rdrds (r, ϕ) puv (r, ϕ) , ij

(6)

where ij (uv) represents mode order at signal (pump) mode of LPij,s (LPuv,p ). For the fundamental LP01,s mode, it can be seen that 01s,uvap is equal to 01s,uvbp since its transverse pattern has circular symmetry. Hence, the LP01,s mode can be considered to be independent on the pump azimuthal angles. The result applies also to the higher order LP0v,s (v > 1) modes. Likewise, if LPuv,s (u ≥ 1) is pumped by the fundamental LP01,p mode uvas,01p is the same as uvbs,01p , implying no obvious DMG between the two degenerate LPuv,s modes. However, we observe evident azimuthal-dependent overlap integrals for LP11a(b),s mode as shown in Table 2 by the bolded entries. As expected, the overlap integral is maximized when signal and pump modes have the same azimuthal angle, while its minimum is reached when their azimuthal angles are orthogonal. In our 3MEDFA, 11as,11ap ( 11bs,11bp ) is about three times as large as 11as,11bp ( 11bs,11ap ). Hence pumping

2

uv grals, we express the normalized intensity patterns of s(p) (r, ϕ)

ij(s),uv(p) = =

 

ij

2  

rdrFs (r)Fpuv (r)

  2 + ı(i − u). cos(2u ) 4



dcos(i) cos[u( +



 

2  

2

rdrFs (r) Fpuv (r) ij

2

)]

, (7)

ij

where Fs(p) (r) is the radial distribution of the electric field, and ı(i-u) represents the Kronecker delta function. If the azimuthal mode numbers of the signal and pump modes are unequal (i.e., i = / u) ı(i - u) = 0, indicating ij(s),uv(p) is independent on . Thereby, the azimuthal angle of LP21,p has a negligible effect on modal gain of LP01,s , LP11a,s , and LP11b,s , which is validated by our additional numerical computations (not show here). For convenience, LP21a,p mode is used in our calculation. It should be noticed that if signal modes of LP2j,s are involved the LP21,p -related DMG must be considered. As a result, only the azimuths of LP11,p have influences on the overlap integrals and will be varied to examine the azimuthal dependence modal gain in the 3MEDFA. 3.2. Azimuthal-dependent modal gain and gain equalization We assume both the cross-sectional intensity patterns of pump and signal modes will remain unchanged during propagating along the EDF. The assumption deals with the worst case, in which the slight difference of the propagation constants between the vector modes that form the corresponding LP mode is neglected [21]. Hence modal gain is not averaged out due to the transverse modal intensity evolution in the EDF. Fig. 3 shows modal gain as a function of LP11,p power. The gain at LP01,s , LP11a,s , and LP11b,s with LP11a,p (LP11b,p ) pumping is represented by the black, blue, and red solid (dashed) lines, respectively. The LP21,p power is set to be 20.8 dBm, which enables us to adjust the LP11,p power to obtain a desirable modal gain. As discussed above, modal gain exhibits a significant dependence on the azimuth of LP11,p . For the two degenerate LP11,s modes, pumping in LP11a,p (LP11b,p ) gives higher gain for LP11a,s

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Fig. 4. Modal gain (a) and noise figures (b) vs. azimuths of LP11,p . Power values of 11

pp

= 18.6 dBm and p21 p = 20.8 dBm are used.

(LP11b,s ) due to a better matched intensity profiles between them. The results are consistent with the estimations by the overlap integrals in Table 2. The maximum gain difference between LP11a,s , and LP11b,s amounts to 2.7 dB (5.2 dB) when the 3MEDFA is pumped by LP11a,p (LP11b,p ) within the range of investigated pump power. Gain variation of less than 0.4 dB at LP01,s mode is observed for the two pumping schemes, which is somewhat different from the finding predicted by the overlap integrals in Table 2. It is believed that the spatial mode competition accounts for this small gain change. It is also seen that pumping in LP11a,p with a power level of 72 mW (18.6 dBm) a modal gain of around 18 dB is obtained for the three signal modes (while the gain equalization is not reached with LP11b,p pumping). To further insight into the azimuthal dependence of modal (i.e., gain we consider LP11,p is excited with distinct azimuths LP11 ,p ). Fig. 4(a) shows the dependence of modal gain. The power in LP11 ,p and LP21,p is set to be 18.6 dBm and 20.8 dBm, respectively. Such power levels ensure us to obtain a modal gain of about 18 dB and a negligible DMG when = 0 (see Fig. 3). It is seen that with increasing from zero, the transverse intensity distribution of LP11 ,p gradually deviates from that of LP11a,p and approaches to that of LP11b,p . Therefore, 11as,11 p ( 11bs,11 p ) decreases (increases), resulting in a gain reduction (enhancement) at LP11a,s (LP11b,s ). When is equal to /2, LP11 ,p coincides with LP11b,p . The gain at LP11a,s reaches its minimum of 15.1 dB while the maximum gain of 20.1 dB at LP11b,s is achieved, corresponding to a maximum DMG of 5 dB between the two degenerate LP11,s modes. It is notable that if the LP11 ,p and LP21,p power are given by other values DMG between LP11a,s and LP11b,s can be as high as 10 dB, or even more. In practical MEDFA application, such high DMG values are undesirable. An inverse change occurs when the azimuth increases from /2 to , in which the gain at LP11a,s and LP11b,s for =  is identical to that for = 0. Consequently, modal gain for LP11a,s and LP11b,s oscillates with an azimuth period of . For the

Fig. 5. Contour maps for the gain of fundamental LP01,s mode (black solid lines), DMG11a-01 (blue dashed lines), and DMG11b-01 (red dashed lines) by pumping in (a) LP11a,p , (b) LP11b,p , and (c) LP11␲/4,p . The x-axis is the LP11,p power and the y-axis is the LP21 ,p power.

signal fundamental mode we also notice a small gain fluctuation since the competition in the Er3+ population inversion between signal modes. Fig. 4(b) shows the dependence of the noise figures. It is observed that the noise figures oscillate between 3.7 and 4.3 dB with the same period of . In the above discussions, the pump power of LP21,p (p21 p ) is 11 fixed (Fig. 3) or both the power of LP21,p (p21 p ) and LP11,p (pp ) are given by certain values (Fig. 4). It is essential to adjust p11 p and p21 p simultaneously to achieve desirable modal gain and DMG. 21 We have calculated the dependence of modal gain on p11 p and pp as well as on the azimuths of LP11,p and the results are shown in Fig. 5. The black solid lines denote the gain at the fundamental LP01,s mode. It can be seen that at arbitrary values the gain at LP01,s changes little in the three pumping schemes (pumping in LP11a,p , LP11b,p , and LP11␲/4,p , respectively), which is in accordance with the above results. We know that the gain at the two

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degenerate LP11,s modes is azimuthal dependent since the intensity overlap integrals between different LP11,p and LP11,s mode pairs have azimuthal dependence, the impacts of which on gain equalization are also examined. Fig. 5 illustrates DMG between LP11a,s and LP01,s (DMG11a-01 , blue dashed lines) and DMG between LP11b,s and LP01,s (DMG11b-01 , red dashed lines). It is observed that by pumping in LP11a,p , Fig. 5(a) shows a higher gain for LP11a,s com21 pared to LP01,s in most regions of parameter space (p11a p , pp ). By increasing the LP21,p power to about 130 mW a line indicating DMG11a-01 = 0 is observed (the bolded blue line), over which LP01,s and LP11a,s modes have the same optical gain. Also the line for DMG11b-01 = 0 is given (the bolded red line), the right (left) region of which indicates higher gain for LP01,s (LP11b,s ). For both bolded lines, there exists only one point of intersection (displayed by a yel21 low circle) over the given space of (p11a p , pp ), where three signal 21 modes share the same optical gain of around 19 dB with (p11a p , pp ) being equal to (75 and 126 mW). Similarly, Fig. 5b and c) shows DMG11a-01 (DMG11a-01 and DMG11b-01 ) when the pump mode of LP11b,p (LP11␲/4,p ) is utilized. Pumping in LP11b,p leads to higher gain for LP11b,s compared to LP01,s as 11bs,11bp is much larger than 01s,11bp . DMG11b-01 keeps positive, implying the zero-differential modal gain (ZDMG) line of DMG11b-01 is absent. Hence, under the assumption of no evolution about the cross-sectional intensity pattern of the LP modes, strict gain equalization between LP01,s and 21 LP11b,s modes over the calculated spaces of (p11a p , pp ) is unable to achieve. Since the overlap value 01s,11bp is larger than 11as,11bp while 01s,21p is less than 11as,21p , ZDMG line of DMG11a-01 is present. Therefore, we can equalize the gain at LP01,s and LP11a,s modes through properly choosing the pump power in LP11b,p and LP21,p modes. As shown in Fig. 5(c), if LP11,p mode is excited with an azimuthal angle of /4 ZDMG line for DMG11a-01 and DMG11b-01 are both observed, while no intersection points between them are found. The result reveals that gain equalization between LP01,s and each of the spatially degenerate modes of LP11,s can be reached, whereas it is unable to equalize modal gain across all signal modes. By the same way, we can also determine whether the conditions for equalizing modal gain are satisfied or not when the 3MEDFA is pumped by LP11,p mode excited with other azimuthal angles. Apart from adjusting the power in pump modes, azimuthal angle is another variable for achieving modal gain control. We further explore gain equalization by controlling the excited azimuthal 11 angles ( ) of LP11,p as well as its power (pp ). Parametric simula21 tions have been performed by using Pp = 20.8 dBm and the results are shown in Fig. 6. The value is set to be varied from 0 to  due to the -periodic dependence of modal gain as previously shown in Fig. 4. In Fig. 6a, the black lines represent the gain at LP01,s mode. As expected, for a given LP11 ,p power, the gain experienced by the signal fundamental mode shows little change over different values. When the LP11,p power increases to 70 mW, the gain at LP01,s mode is greater than 18 dB. To determine the values of ( , 11

pp ) for equalizing modal gain, Fig. 6(a) also gives DMG11a-01 and DMG11b-01 , which are represented by the blue and red dashed lines, respectively. Both ZDMG lines for DMG11a-01 and DMG11b-01 are present, generating two points of intersection (marked by red circles), at which signal modes have the same modal gain of 18.26 dB. The locations for the intersection points are (12◦ , 76 mW) and (168◦ , 76 mW), respectively. Analogously, if other LP21,p power is used to pump the 3MEDFA we can also find all the zero DMG points. Fig. 6(b) shows the noise figures of LP01,s and LP11b,s modes, which are denoted by the black and red lines, respectively. The noise figure of LP11a,s is similar to that of LP01,s and LP11b,s and is not shown in the figure. It is observed that noise figures decrease asymptotically with the increasing gain. All noise figures for LP01,s , LP11a,s , and LP11b,s at the points with zero DMG are less than 4 dB.

Fig. 6. Contour maps for (a) gain at LP01,s (black solid lines), DMG11a-01 (blue dashed lines), and DMG11b-01 (red dashed lines), and for (b) noise figures of LP01,s (black solid lines) and LP11b,s (red solid lines). The power in LP21 ,p is fixed to be 20.8 dBm. Both red circles in (a) indicate the positions where DMG between signal modes is zero.

4. Conclusions We have investigated the azimuthal-dependent modal gain and its impact on gain equalization in a short-length 3MEDFA. Since the intensity profiles are azimuthal dependent, non-LP0v pump modes create azimuthal dependence of Er3+ population inversion and hence azimuthal dependence of modal gain. Results show that if the azimuthal mode numbers in signal modes are different from that in the pump modal gain is nearly independent of the azimuthal orientations of the pump intensity pattern. The gain at LP0v signal modes shows a negligible dependence upon the azimuth of pump modes due to the circular symmetry in their intensity profiles. Pumping in LP11 mode with different excited azimuthal angles leads to a periodic oscillating modal gain with DMG amounting to 5 dB. The impact of azimuthal-dependent modal gain on gain equalization is also examined by adjusting the power in pump modes of LP11 and LP21 and by controlling the excited azimuthal angles of pump mode LP11 as well as its power. Optimal parameters capable of realizing gain equalization have been achieved by searching for the intersecting points between two ZDMG lines of DMG11a-01 and DMG11b-01 . For MEDFA supporting optical amplification beyond two propagating mode groups, similar azimuthal-dependent modal gain in the two degenerate signal modes of LPml (m≥2) can be caused by pump mode of LPmn . Acknowledgments This work has been supported by the Natural Science Foundation of Guangxi province, China (under grant no. 2013GXNSFBA019269, 2014GXNSFAA118389, 2013GXNSFDA019002 and 2014GXNSFGA118003), Guangxi Experiment Center of Information Science (Guilin University of Electronic Technology), Guangxi Key

Z. Qin et al. / Optik 126 (2015) 3538–3543

Laboratory of Automatic Detecting Technology and Instruments, and the program for innovative research team of Guilin University of Electronic Technology (IRTGUET). [10]

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