Journal of Modern Optics
ISSN: 0950-0340 (Print) 1362-3044 (Online) Journal homepage: http://www.tandfonline.com/loi/tmop20
Fiber Raman gain amplifier performance study using simple coupled-mode analysis Sanchita Pramanik & Somenath Sarkar To cite this article: Sanchita Pramanik & Somenath Sarkar (2015) Fiber Raman gain amplifier performance study using simple coupled-mode analysis, Journal of Modern Optics, 62:13, 1110-1113, DOI: 10.1080/09500340.2015.1021721 To link to this article: http://dx.doi.org/10.1080/09500340.2015.1021721
Published online: 17 Mar 2015.
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Date: 04 December 2015, At: 21:43
Journal of Modern Optics, 2015 Vol. 62, No. 13, 1110–1113, http://dx.doi.org/10.1080/09500340.2015.1021721
Fiber Raman gain amplifier performance study using simple coupled-mode analysis Sanchita Pramanika,b and Somenath Sarkara,c* a
Department of Electronic Science, University of Calcutta, Kolkata, India; bDepartment of Electronics, Vidyasagar University, Midnapore, India; cCentre for Research in Nanoscience and Nanotechnology, University of Calcutta, Kolkata, India
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(Received 6 September 2014; accepted 18 February 2015) The propagation characteristics of doped fiber amplifier have, already, been analyzed using simple coupled-mode analysis (CMA), which steers the interests to apply this simple theory to study the amplification performance of fiber Raman gain amplifier (FRGA). But such study is yet to be done for FRGA. In this paper, a premier report based on CMA is presented to analyze and predict the performance of FRGA for first time, as per our knowledge, using coaxial structure consisting of step-index profiles in both inner and outer cores in terms of effective area, Raman gain coefficient, and effective Raman gain coefficient. In our investigation, we use single pump to compute and compare our results with the results obtained by involved matrix method and a fairly excellent match over a wide region of frequency shift is reported. Keywords: Fiber Raman gain amplifier (FRGA); coupled-mode analysis (CMA); phase matching wavelength (PMW)
1. Introduction Various numerical techniques are being proposed to investigate the performance characteristics of optical fibers and associated devices as straightforward and analytical solutions are not possible for intricate refractive index distributions [1–5]. The optical amplifiers, like Erbium-doped fiber amplifier (EDFA), fiber Raman gain amplifier (FRGA) etc., are indispensable for huge internet traffic and promotion of existing wavelength division multiplexing system in modern optical communication. Wide use of EDFA has attracted worldwide attention. However, since the FRGA provides larger bandwidth than that of EDFA, it is well known today as an alternative potential candidate [6–8]. Also, there is no need of doping in FRGA, whereas Erbium is doped in EDFA. Also, the gain spectrum only depends on pump wavelength, which is a very important advantage of FRGA. Again, broad gain spectrum can be achieved by tailoring the fiber structure. It is reported that the coaxial FRGA with various refractive index profiles in the inner core is designed with keen interest to obtain wide gain spectrum [7,9]. In coaxial structure, FRGA has an inner core acting as the rod and an outer core, with less refractive index, acting as tube. The two cores are separated by a gap of suitable distance and the outer cladding, w.r.t Figure 1, supports two modes forming supermodes. The effect of refractive index in presence or absence of imperfections which may arise due to central or axial burning during fiber drawing process in the inner core of coaxial FRGA has, already, been reported in various investigative reports [9–11] in terms of gain performance and dispersion. Again, it is reported that using various *Corresponding author. Email:
[email protected] © 2015 Taylor & Francis
pumping schemes, one can increase the signal amplification capacity of such FRGA [6,12,13]. Further, the accurate modal field description in FRGA together with the modal distribution in rod and tube regions of the structure is very much necessary for the computation of the effective area and the effective Raman gain coefficient of FRGA. In case of EDFA, it is reported that the entire analysis has been carried out using simple coupled-mode analysis (CMA), instead of deeply involved matrix method [14]. The CMA is a simple and powerful technique to understand the effect of one mode in one waveguide on the other mode in adjacent waveguide and vice versa in terms of suitable coupling and predicts the final mode propagating as a stable mode, which is the solution of the practical system. But according to the best of our knowledge, to predict the entire FRGA performance, no such investigative report is available involving simple CMA. One can explore whether it is possible to obtain FRGA characteristics using such simple method within acceptable tolerance for the prediction of propagation dynamics as a simple and alternative development of the other available methods, like matrix method etc., which are not too simple like CMA. In this work, to obtain the modal fields in coaxial FRGA, we use CMA for first time and its effectiveness to describe FRGA performance is also presented. The results of our investigation using CMA to solve the scalar wave equations at the boundaries of the chosen FRGA structure is compared with the results obtained from matrix method and a fairly excellent match with those results is also reported in this work.
Journal of Modern Optics
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ws ðrÞ ¼
8 1 ffi < pffiffiffiffiffiffiffi ½w1 ðrÞ þ bs w2 ðrÞ 2 1þbs
1 ffi ½as w1 ðrÞ þ w2 ðrÞ : pffiffiffiffiffiffiffi 2
(4)
1þas
wa ðrÞ ¼
8 1 ffi ½w1 ðrÞ þ ba w2 ðrÞ < pffiffiffiffiffiffiffi 2 1þba
1 ffi ½aa w1 ðrÞ þ w2 ðrÞ : pffiffiffiffiffiffiffi 2
(5)
1þaa
Figure 1.
The bs , as , ba , and aa in the above equations are defined as: 9 bs ¼ bs b1 > j12 > > > 2 = as ¼ bsjb 21 (6) ba ¼ ba b1 > > j12 > > a ¼ ba b2 ;
Refractive index profile of our chosen fiber.
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a
2. Modeling and analysis In coaxial fiber structure, the rod and the tube regions individually support single mode of light propagation. In the entire composite structure, the supermodes are formed due to evanescent coupling of two individual modes of these two regions [8,9]. The field distributions of these supermodes are calculated using simple CMA. For the analysis, we excite only LP01 supermode of the complete structure and use single pump only. Our chosen coaxial FRGA structure has inner and outer cores with step-index profile. The refractive indices of the inner core, the outer gap and the outer core, and cladding are n1, n2, n3, and n2, respectively, with inner core radius a, the outer gap radius b, and the outer core radius c as shown in Figure 1. For simplicity, in the entire analysis, we concentrate our attention to the single mode regime considering the same wavelength dependence of the refractive indices. Also, in the small signal regime, the pump depletion is ignored for stimulated Raman scattering (SRS) in the fiber [8–10]. In the coaxial FRGA structure, the total field is obtained due to the superposition of individual inner core field, w1 ðrÞ, and outer core field, w2 ðrÞ, determined by matrix method and expressed as [14]: wðr; zÞ ¼ aðzÞw1 ðrÞ þ bðzÞw2 ðrÞ
(1)
In Equation (1), the amplitudes a(z) and b(z) are the functions of z. Using the slowly varying CMA, the supermodes of the composite structure are calculated and the supermodes ws ðr; zÞ and wa ðr; zÞ are written as: ws ðr; zÞ ¼ ws ðrÞejbs z
(2)
wa ðr; zÞ ¼ wa ðrÞejba z
(3)
where βs and βa are the propagation constants of these two supermodes, respectively. The corresponding field profiles ws ðrÞ and wa ðrÞ are expressed as:
j21
In Equation (6), β1 and β2 are the propagation constants of the inner and the outer cores, respectively. Here, βs and βa are also calculated using the following relation: 1=2 1 1 bs;a ¼ ðb1 þ b2 Þ ðb1 b2 Þ2 þj2 (7) 2 4 pffiffiffiffiffiffiffiffiffiffiffiffiffi where j ¼ j12 j21 . κ12 and κ21 are written as
j12
j21
k2 ¼ 0 n21 n22 2b1 k2 ¼ 0 n23 n22 2b2
Za w1 w2 rdr
(8)
w1 w2 rdr
(9)
0
Zc b
In writing the above Equations (8) and (9), the overlap integral of the modes is neglected assuming [15] Z Z1 1
w1 w2 dxxy\\
Z Z1
w1 w1 dxxy
(10)
1
which is valid for weak coupling between the waveguides. In Equation (10), w1 ðrÞis the complex conjugate of w1 ðrÞ. The following expression is used to calculate the effective area of the FRGA [8–10]: R R 2 wp ðrÞrdr w2sig ðrÞrdr Aeff ¼ 2p R 2 (11) wp ðrÞw2sig ðrÞrdr where wp and wsig are the modal fields for pump and signal wavelengths kp and ks , respectively. Further, in terms of the effective area Aeff and the Raman gain coefficient gR, the effective Raman gain coefficient γR is defined as [8–10]: gR cR ¼ (12) Aeff
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S. Pramanik and S. Sarkar
We calculate the above γR from the following wellknown equation [8–10,16] given as: CSiSi ðDmÞ cR ¼ 2p
Z1 ½1 2xðrÞIp ðrÞISig ðrÞrdr 0
CGeSi ðDmÞ þ 2p
Z1 2xðrÞIp ðrÞISig ðrÞrdr
(13)
0
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where in Equation (13), Δν represents the frequency shift from the pump frequency, whereas x(r) represents the Germania concentration. The pump and signal intensities are expressed in terms of pump and signal powers Pp and Ps, respectively, as: Ip ðrÞ ¼
ISig ðrÞ ¼
w2p ðrÞ Pp w2Sig ðrÞ PSig
(14)
(15)
where Z1 Pp ¼
w2p ðrÞrdr
(16)
w2sig ðrÞrdr
(17)
0
and Z1 PSig ¼ 0
Also, CSiSi(Δν) and CGeSi(Δν) are the Raman spectra coefficients of pure silica bonds Si–O–Si and binary Germania and Silica bonds Ge–O–Si, respectively. The peak values of CSiSi(Δν) and CGeSi(Δν) for unpolarized pump and signals, used in this entire analysis, are 3.71 × 10−14 m/W at 440 cm−1and 1.839 × 10−13 m/W at 430 cm−1, respectively.
3. Results and discussion In this paper, we choose the parameter values n1 = 1.47299 as inner core refractive index, n2 = 1.444388 as inner gap refractive index as well as the outer cladding refractive index, and n3 = 1.44871 as outer core refractive index. We also choose the values of a, b, and c as 1.0, 9.0, and 16.32 μm as inner core radius, inner gap radius, and outer core radius, respectively, w.r.t Figure 1. The pump wavelength is taken as 1.458 μm. We keep the phase matching wavelength (PMW) fixed at 1.52 μm to measure the refractive indices according to pure silica for Δ1 = 0.02 and Δ2 = 0.003 [8–10].
It may be noted that the individual inner and outer cores, in our analysis, phase matched at PMW of 1.52 μm and those cores also supported propagation of single mode only. For wavelength much smaller than PMW, the effective index of the fundamental supermode is very close to that of the individual mode of the inner core [7,8,10]. Hence, the modal fields at kp = 1.458 μm and ks PMW are tightly confined to inner core. It leads to high pump-signal overlap and then a small Aeff. However, for ks approaching PMW, the fractional power of the fundamental mode increases steadily in the outer core. It may also be attributed to the fact that normalized frequencies Vp and Vs corresponding to kp and ks , respectively, are different and give rise to higher Aeff and lower γR. We calculate the effective area for all frequency shifts using CMA for our chosen FRGA structure and then use these results to determine gR and γR. Now, Figure 2 shows the variations of the Raman gain coefficient with frequency shifts using CMA and matrix method [8,10]. The Figure 2 shows excellent matching of results for our chosen FRGA using both methods. Then, we draw the Figure 3 showing the variations of γR with Δν using both methods. It is seen from the Figure 3 that the results using matrix method [8,10] underestimate the values of γR obtained by CMA below ~300 cm−1 though our results overshoot that results within the frequency shift region from ~300 to ~480 cm−1 and an excellent match in higher Δν region is presented. However, for Δν < 300 cm−1 one can obtain the flatness in both cases. Therefore, the proposal of use of CMA in FRGA performance analysis, no doubt, deserves attention although it is yet to be refined, if necessary, for more rigorous calculation of Aeff leading to estimation of γR value and excellent match over the entire operating region of frequency shift. In this connection, it may be relevant to recall that we have considered
Figure 2. shift.
Variations of Raman gain coefficient with frequency
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Nanoscience and Nanotechnology of the University of Calcutta is acknowledged. The authors also acknowledge helpful discussions with Prof. K. Thyagarajan of Physics Department, Indian Institute of Technology, Delhi. The authors acknowledge constructive suggestions of the anonymous honourable reviewers.
Disclosure statement No potential conflict of interest was reported by the authors.
References
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Figure 3. Variations of effective Raman gain coefficient with frequency shift.
weak coupling of the two fiber waveguides following condition in Equation (10). For refinement, one may have to consider strong coupling of the overlap integral of the modes, which requires deeply involved calculation. Therefore, this approach needs further attention both by theoreticians and experimentalists. 4. Conclusion In performance analysis of FRGA, it is concluded that the simple CMA can be used in place of matrix method for its simplicity. It is seen that the results obtained by CMA match fairly excellently with the other method. So, one can adopt this simple method reliably as a potential alternative of matrix method or any other available method and refine it to study the FRGA performances. Acknowledgements The first author acknowledges financial support of Senior Research Fellowship of the Human Resource and Development Group of the Council of Scientific and Industrial Research, New Delhi. Partial support of the Centre for Research in
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