- Email: [email protected]
Modeling the optical nonlinear effects on DFB-RF laser based on the transfer matrix method
Applied Mathematical Modelling 74 (2019) 85–93
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Modeling the optical nonlinear effects on DFB-RF laser based on the transfer matrix method Maryam Aliannezhadi a,∗, Fatemeh Shahshahani b, Vahid Ahmadi c a
Faculty of Physics, Semnan University P.O. Box: 35195-363, Semnan, Iran Department of Physics, Alzahra University, Tehran, Iran c Department of Electrical Engineering, Tarbiat Modares University, Tehran, Iran b
a r t i c l e
i n f o
Article history: Received 6 January 2018 Revised 8 April 2019 Accepted 16 April 2019 Available online 23 April 2019 Keywords: Distributed feedback Fiber laser Raman laser Cross phase modulation (XPM) Self phase modulation (SPM) Transfer matrix method
a b s t r a c t In this paper, the operation of π -phase shifted distributed feedback Raman fiber (DFB-RF) laser above threshold condition is analyzed theoretically. The nonlinear optical phenomena such as self phase modulation (SPM) and cross phase modulation (XPM) have significant effect on the performance of DFB-RF laser. The numerical results show that the nonlinear effects cause to the saturation of output power and the value of saturated power is dependent on the fiber length. It is found that, the operation wavelength of stokes modes of DFB-RF laser varies in above threshold condition as a result of nonlinear optical properties of the fiber. Simulation is performed by using transfer matrix method to solve three coupled nonlinear wave equations which describe the propagation of pump, forward and backward Stokes waves. The nonlinear SPM and XPM effects are considered in the presented theoretical model. © 2019 Elsevier Inc. All rights reserved.
1. Introduction Raman fiber (RF) lasers are attractive coherent sources due to their exclusive properties. They can be used as a high power and tunable light source in a variety of applications. The operating wavelength of RF lasers is dependent on the pump wavelength and Raman gain spectrum of the fiber material [1]. Owing to this property, the RF lasers could be designed for special wavelengths in the range of visible to infrared region. Raman gain is small in the fiber lasers, so long fiber lengths (typically ten meters to several kilometers) are necessary to obtain low threshold pump power. On the other hand, Raman fiber lasers operate on multi-longitudinal modes and become more sensitive to environmental fluctuations because of long cavity length. Also, single mode lasers are required in more applications like optical communications, high resolution spectroscopy, defense, and network systems, So RF lasers with shorter fiber length are attractive. Various RF structures were proposed by researchers to resolve this conflict. Using Distributed Bragg Reflector (DBR) was the first candidate to solve the problem. DBRs are used on the input and output side of fiber to enhance the interaction between the pump and the Raman-shifted signal. They classified into two general categories, one Raman-Stokes shift and multiple Raman-Stokes shifts called cascaded Raman fiber laser. The result of enhancing the output coupler reflectivity is the lower threshold pump power [2]. A length of approximately 5 m is reported for these types of lasers [3], and recently, a high power laser with output power>25 W at 1570 nm is built by using sixthorder cascaded Raman amplification [4]. This type of lasers suffers from a number of drawbacks. A partly long length fiber ∗
Corresponding author. E-mail address: [email protected] (M. Aliannezhadi).
https://doi.org/10.1016/j.apm.2019.04.048 0307-904X/© 2019 Elsevier Inc. All rights reserved.
86
M. Aliannezhadi, F. Shahshahani and V. Ahmadi / Applied Mathematical Modelling 74 (2019) 85–93
is used in it and high reflector (HR) DBRs are linked in the system weakly, so their operation may be disrupted when high power propagates through it. Also, it is possible for the signal Raman to propagate back into the pump laser or diode, causing it to fail or hit it [5]. Distributed feedback Raman fiber (DFB-RF) lasers with short fiber length (typically < 1 m) are the most promising and practical candidates for a single mode light source for coherent communications, due to their extremely low noise and very narrow linewidth, λ = 0.018 nm [6]. The idea of DFB-RF laser was first proposed by Perlin and Winful in 2001 [7]. According Their theoretical works, a uniform 1m-long DFB-RF laser would have a threshold pump power below 1 W then, a DFB-RF laser with 20 cm cavity length, coupling coefficient of 60 m−1 and a π phase shift inside the fiber Bragg grating has been modeled. Their model has predicted low-threshold and practical laser with high sensitivity to the change of position and width of the phase shift. The threshold pump power of this structure was close to 1 W [8]. Westbrook and his colleagues demonstrated the first experimental DFB Raman fiber laser in 2011. The fiber Bragg gratings (FBG) with π phase shift was fabricated on the base of an OFS fiber. The laser had a power threshold of 39 W with a linewidth of 7.5 MHz .They improved the maximum output power, linewidth, and threshold of the structure by using highly nonlinear fiber (HNLF) to 350 mW, 4 MHz, and 4.3 W, respectively [9,10]. In 2012, Shi and their colleagues reported two DFB-RF lasers fabricated in fiber PS980 and UHNA4 (Ultra-High NA single mode fiber) [6,11]. The coupling coefficient of the structures were κ = 37 m−1 and length L = 0.3 m. The threshold pump power at a pump wavelength of 1069 nm reduced to 2 W and 1 W for PS980 and UHNA and oscillation wavelengths were around 1117.73 nm and 1109.21 nm, respectively. Also, highly efficient Raman-gain-based distributed-feedback fiber lasers with a cavity length of 30 cm and 2 W CW output-power with linewidth smaller than 0.01 nm around wavelength 1.11 μm have been demonstrated by them. Recently, single frequency DFB Raman fiber lasers based on tellurite, and chalcogenide fibers were designed and their operation at any arbitrary wavelength 2.5 −9.5 μm in the spectral region were considered in the threshold condition theoretically [12,13]. To the best of our knowledge, most of the reported studies on the DFB-RF lasers are experimental studies [6,9–11] or theoretical studies in the threshold conditions [14,15]. Due to the importance of DFB-RF lasers, it is worthwhile to investigate the performance of these structures further theoretically, in particular above threshold condition. The main goal of this paper is numerically analysis the behavior of DFB-RF laser above threshold condition. The physical model is based on a system of three nonlinear coupled wave equations that describe the propagation of pump, forward and backward Stokes waves along the fiber length. Raman active media generally exhibit various nonlinearities which can strongly affect the operation of Raman fiber laser. In the present paper we take into account two main nonlinear effects, cross phase modulation (XPM) and self phase modulation (SPM), in the three nonlinear wave equations. We have used transfer matrix method (TMM) to solve the three nonlinear wave equations. Nonlinear optical effects in Raman fiber lasers cause longitudinal variations along the fiber length. Moreover, in DFB-RF laser structures there is some non-uniformity along the fiber length, for example: phase shift in the grating or chirped grating. Therefore TMM is a suitable technique to analyze the periodic structure with non-uniformity. An analytical closed form approximation model based on the TMM has been presented in Ref. [15] for studying the DFB-RF laser structure at threshold condition. Also, the nonlinear optical effects have not been considered in Ref. [15]. The paper is organized as follow: The theoretical model and TMM are described in section II, Then the numerical results and discussion are presented in section III and finally section IV is dedicated to the conclusion 2. Theoretical model The evaluation of slow varying amplitudes of the pump, forward Stokes and backward Stokes waves along the cavity of DFB-RF laser are illustrated by the following set of three nonlinear coupled wave equations [7]:
2 ∂ AP αP gP 2 =− A f + |Ab |2 AP + iγP |AP |2 + 2A f + 2|Ab |2 AP − AP ∂z 2 2 2 ∂ A f gS 2 α = |AP | A f + iκ Ab + iδβ A f + iγS 2|AP |2 + A f + 2|Ab |2 A f − S A f ∂z 2 2 2 ∂A gS α − b = |AP |2 Ab + iκ A f + iδβ Ab + iγS 2|AP |2 + 2A f + |Ab |2 Ab − S Ab ∂z 2 2
(1-a)
(1-b)
(1-c)
where Ap , Af and Ab are the slow-varying amplitudes of the pump, forward, and backward Stokes waves, respectively. The Raman gain coefficient for Stokes waves is represented by gs , and the gain depletion coefficient for pump wave is given by gp = gs(λs/λp). The nonlinear coefficients are indicated by γ s = 2π n2 /λs and γ p = 2π n2 /λp where n2 is the nonlinear refractive index, κ is the coupling coefficient which depends on the amplitude of refractive index modulation of the grating and shows the coupling between forward and backward propagating Stokes waves. The wavelength of pump and Stokes waves are indicated by λp and λs respectively. Detuning the Stokes frequency from the Bragg frequency is defined by δβ = ωc -ωB , where ωc and ωB are the Stokes and Bragg frequency, respectively. The parameters α p and α s refer to the linear fiber loss for pump and Stokes waves, respectively.
M. Aliannezhadi, F. Shahshahani and V. Ahmadi / Applied Mathematical Modelling 74 (2019) 85–93
87
Fig. 1. A simple schematic of a DFB-RF laser with π phase shift at the center of grating.
Fig. 2. In transfer matrix method the fiber length is divided into several subsections.
The second set of terms in (1-a) –(1-c) equations, describe the fiber refractive index nonlinearity. Within the brackets, the terms with the factors of two describe the effect of XPM, whereas the last term describes the effect of SPM. A simple schematic of DFB-RF laser with a π -phase shift at the center of grating is shown in Fig. 1. The pump wave with power P0 is pumped to the fiber from the left at z = 0. The Stokes waves at λs are generated by Raman phenomena in the laser cavity when the pump wave propagates through the fiber. It is notable that, pump wavelength is far from Bragg wavelength and it does not interact with the grating. We used TMM algorithm to solve Eqs. (1-a)–(1-c). By applying the idea of TMM, which has been used for evaluation of semiconductor DFB diode lasers, we introduce and apply the TMM for DFB-RF laser structures for the first time. In TMM the fiber length is divided into several subsections, as shown in Fig. 2. In each subsection the parameters such as δβ , α s and κ are assumed to be constant but they vary from one segment to another segment. The amplitude of forward and backward waves at each segment, zk +1 , can be obtained as a function of the amplitude of forward and backward waves at the previous segment, zk .
A f (zk+1 ) = A f (zk ) Ab (zk+1 ) = A f (zk )
1 r
eiqz −
1 r rA
rB rB −iqz rB e + A f (zk ) − eiqz + e−iqz r rA r r
eiqz +
rB iqz 1 −iqz 1 −iqz e + Ab ( z ) − e + e r rA r rA r
(2)
(3)
where z = zk +1 - zk is the length of each segment, and r = 1+rB /rA , where the parameters rA and rB can be interpreted as the effective reflection coefficients of grating for the forward and backward waves, respectively. The parameter q is defined as:
q= where:
α=
−(δb − δ f ) ∓
(δb − δ f )2 − 4 (α + iδ f )(α + iδb ) + κ 2 2
gs α |AP |2 − s 2 2
(4)
(5)
2 δ f = δβ + γs 2|AP |2 + A f + 2|Ab |2
(6)
2 δb = δβ + γs 2|AP |2 + 2A f + |Ab |2
(7)
88
M. Aliannezhadi, F. Shahshahani and V. Ahmadi / Applied Mathematical Modelling 74 (2019) 85–93
The set of Eqs. (2) and (3) can be written in a matrix form as:
A f (zK+1 ) A (z ) = F k ( zK+1 |zk ) f K Ab (zK+1 ) Ab ( zK )
(8)
where F(Zk+1 |zk ) is a 2 × 2 matrix with the following complex elements:
F11 =
1 iqz rB −iqz e − e r rA
F12 =
rB −iqz e − eiqz r
(9-b)
F21 =
1 iqz e − e−iqz r rA
(9-c)
F22 =
1 −iqz rB iqz e − e r rA
(9-a)
(9-d)
Therefore, we can obtain the amplitude of forward and backward waves at the end of cavity length (z = L) as:
N N A f (L ) A (0 ) = F N F N−1 ...F 2 +1 S(θ )F 2 ...F 2 F 1 f Ab ( L ) Ab ( 0 )
(10)
where N is the total number of subsections and S(θ ) is the phase shift matrix at the center of the grating which is given by:
S (θ ) =
e iθ / 2 0
0
(11)
e−iθ /2
In this paper we have considered, θ = π . The following boundary conditions must be satisfied at the left and right facets of fiber:
A f ( 0 ) = r1 Ab ( 0 ) Ab ( L ) = r2 A f ( L )
(12)
where r1 and r2 are amplitude reflectivity of left and right facets for Stokes wave, respectively. The coupled parameters α and δβ for every mode can be obtained by using TMM and self-consistent numerical solving the boundary conditions (12). The smallest value of α denotes the threshold loss of the main mode and the next value of α is related to the first side mode. Threshold pump power for each mode can be calculated by the following equation:
Pth = 2Ae f f
α+
αs 2
/gS
(13)
where Aeff is called the effective mode area of the fiber and it is assumed constant during calculations. Threshold pump power for the main mode and the first side mode will be denoted by Pth0 , Pth1 , respectively. The intensity of pump, forward and backward Stokes waves are obtained by:
Im = |Am |2 =
Pm , Ae f f
m = p, f, b
(14)
Total intensity of Stokes waves in each z component along the fiber length are given by:
2
I (zk ) = A f (zk ) + |Ab (zk )|
2
(15)
3-Numerical results and discussion The DFB-RF structure with a π -phase shift at the center of grating is shown in Fig 1. The values of parameters that we used in our simulation are related to the commercial high germanium doped silica fiber (UHNA4 from Nufern), with high numerical aperture (NA=0.35), medium propagation loss of α s = 0.05 dB/m and coupling coefficient κ = 37 m−1 [15]. It is notable that, to achieve higher values of coupling coefficient, the host medium could be exposed with hydrogen or deuterium vapor, which increases the background loss of fiber, at IR wavelength. This in turn increases the threshold pump power [9]. For simulation, Eqs. (1-a)–(1-c) should be solve self-consistently with considering the boundary conditions. In order to obtain accurate results by TMM, we examined the numerical results with 200 segments along the fibre length (N = 200) for threshold condition and above threshold condition without considering the nonlinear effects. The errors of our simulations are less than 10−6 . Also, in the case of above threshold condition with considering nonlinear effects we have to increase N with increasing the FBG lengths to obtain the errors less than 10−6 . It is due to the fact that nonlinear effects are more
M. Aliannezhadi, F. Shahshahani and V. Ahmadi / Applied Mathematical Modelling 74 (2019) 85–93
89
Table 1 The parameters used in simulation [15]. Parameters Core refractive index Raman gain Pump wavelength Bragg wavelength Nonlinear refractive index Fiber loss Effective mode area Reflection of fiber ends
Values (nc ) (gs) ( λp ) ( λB ) (n2 ) (α s ) (Aeff ) (r1 , r2 )
1.4 1.55 × 10−13 (m/W) 1.064 (μm) 1.11(μm) 3.2 × 10−20 (m2 /W) 0.05(dB/m) 5.4 (μm2 ) 0
Fig. 3. Threshold pump power for main mode and first side mode as a function of fiber length.
important in the case of the longer fibers and above threshold condition. The two ends of the fiber are considered to be antireflective. The other parameters which are used in calculations are listed in Table 1. The variation of threshold pump power for main mode and first side mode, Pth0 and Pth1 as a function of cavity fiber length are shown in Fig. 3. According to the result, threshold pump power for κ = 37 cm−1 and L = 30 cm is 0.5 W. It is in agreement with the experimental data in Ref. [15] and the difference is due to connector loss and splice loss in experimental set up. Also, Fig. 3 shows that both Pth0 and Pth1 decrease with increasing L due to the increasing of effective volume of Raman gain medium. It is notable that for L < 20 cm, both Pth0 and Pth1 have very high values. As we see in Fig. 3, the pump power for main mode and first side mode at L < 20 cm are larger than 4 W and 150 W, respectively. So the structures with L < 20 cm is not desirable. It is also evident from Fig. 3 the difference between Pth0 and Pth1 decreases with increasing the fiber length and for very large values of L, it approaches to a constant value about 20 W. The difference between Pth0 and Pth1 (Pth = Pth0 –Pth1 ) could be a criterion factor for assessment the single mode operation of lasers in above threshold condition. If Pth is small, the main mode and side modes will compete together in above threshold condition. For example, if we choose L = 50 cm and launch pump power greater than 30 W, the side mode will appear at the output of the laser and the laser will not operate in single mode at this situation. It is notable that for L > 35 cm the slope variation of Pth0 with respect to L is very small. As a result, it is not required to apply the structures with fiber length much greater than 35 cm with κ = 37 m−1 . It seems that a DFB-RF laser with 20 < L < 40 cm to be an appropriate structure for single mode and low threshold operation. Oscillation wavelength is one of the most important laser parameters that describes above threshold stability of the laser. Variation of main mode wavelenght, λs , as a function of P0 , pump power launched at Z = 0, is plotted in Fig. 4 for four different values of L. According to this figure, small red shift is observed with increasing P0 and L. We predict that the red shift is due to the refractive index change of Raman gain material because of XPM and SPM nonlinear effects. To confirm this forecast, we calculate the oscillation laser wavelength for the main mode as a function of P0 , in the case of n2 = 0 in Eqs. (1-a)–(1-c). It is notable that the graph of λ versus P0 coincides with the horizental axis and this is marked by hollow
90
M. Aliannezhadi, F. Shahshahani and V. Ahmadi / Applied Mathematical Modelling 74 (2019) 85–93
Fig. 4. Variation of main mode wavelenght, λs , of DFB-RF structures, as a function of launched pump power, P0 .
Fig. 5. Refractive index change, n, along the fiber length, for κ = 37 m−1 , L = 30 cm and P0 = 1, 5, and 10 W.
circles in Fig. 4. It is found that when no nonlinear effect is considered and n2 = 0, lasing wavelength doesn’t change versus variation of pump power. In order to understand the physical origin of this behavior, the longitudinal variation of refractive index, n, for the structure with κ = 37 m−1 , L = 30 cm, and three values of pump power P0 = 1, 5, 10 W is investigated. The obtained numerical results are shown in Fig. 5. In this figure z = Z/L denotes the normalized position along the fiber length. So it is confirmed that the variation of oscillation wavelength is an outcome of the nonlinear effects. It should be mentioned that the refractive index at nonlinear fibers is dependent on the optical field intensity. Fig. 6 shows the optical field intensity along the fiber for the same data as used in Fig. 5. It can be seen that longitudinal variation of refractive index and optical intensity have similar profiles.
M. Aliannezhadi, F. Shahshahani and V. Ahmadi / Applied Mathematical Modelling 74 (2019) 85–93
91
Fig. 6. Longitudinal distribution of optical intensity of Stokes main mode for the structure with κ = 37 m−1 , L = 30 cm and three values of P0 = 1, 5, 10 W.
Fig. 7. The output power as a function of pump power for DFB-RF structures with κ = 37 m−1 and for different values of L with and without nonlinear effects.
Because of the π phase shift at the center of the grating, there is a peak at z = 0.5 and therefore we see a maximum value of n at the center of the fiber length. Longitudinal distribution of intensity shows that optical energy tends to accumulate at the center of the fiber with increased P0 . Finally, we study the output power, Pout , for both linear and nonlinear situations of DFB-RF structure with κ = 37 m−1 and for different values of L, as a function of input pump power. Output power of laser, Pout , is the sum of the powers of forward and backward Stokes waves at the left and right fiber ends. The numerical results for Pout are shown in Fig. 7. In this figure the results for two cases, with and without nonlinear effects, are compared. It is found from this graph that in linear situation, Pout rises linearly as P0 increases. However, Pout reaches a saturation level when P0 increases in the presence of nonlinear effects. For L = 0.25 m the saturation status takes place for P0 > 15 W, which is not shown in Fig. 7. The value of
92
M. Aliannezhadi, F. Shahshahani and V. Ahmadi / Applied Mathematical Modelling 74 (2019) 85–93
Fig. 8. Pump power variation along the fiber length, for structure with κ = 37 m−1 , L = 30 cm, and three values of P0 = 1, 5, 10 W.
the saturated output power is dependent on the fiber length. For example saturation output power for the structure with L = 30 cm is 6.59 W and saturation region starts from P0 = 12 W. Also, saturation output power is 3.2 W for L = 35 cm and the saturation condition is achieved at P0 = 6 W. According Figs. 3 and 7 in the threshold and above threshold conditions, a DFB fiber laser based on UHNA4 with κ = 37 m−1 and L = 30 cm is an optimum structure. In order to understand the saturation performance of DFB-RF structures, the pump power distribution along the fiber length for the structure with κ = 37 m−1 and L = 30 cm is shown in Fig. 8. The denoted numerical results are related to both linear (n2 = 0) and nonlinear (n2 = 0) conditions. The values of pump power are normalized to their input values at z = 0, P0 . As expected, the pump power is depleted along the fiber. The pump is completely depleted along the fiber in the absence of nonlinearity and for high pump powers. The pump power depletion along the fiber length is the direct result of energy conservation rule. In the steady state and ignoring the fiber loss, the input pump power must be equal to the sum of the output forward Stokes, backward Stokes, and pump powers. Therefore, in the presence of nonlinearity where the pump power does not completely deplete at the end of the fiber, the conversion energy between the pump wave and Stokes waves comes to an end and saturation state for output power occurs. We think that it is a significant result that could be investigated with more details. 4. Conclusion In the present work, we numerically studied the effects of XPM and SPM on the behavior of a π -phase-shifted DFB-RF laser by numerical solution of three coupled nonlinear wave equations using TMM. It was shown that the SPM and XPM effects cause a small red shift (<0.01 nm) at the lasing wavelength with increasing launched pump power. We predict that the red shift is due to the refractive index change of Raman gain material because of XPM and SPM nonlinear effects. We studied the variation of output power for both linear and nonlinear situations as a function of input pump power. It is found that in linear case Pout rises linearly as input pump power increases. However, in the presence of nonlinear effects, Pout reaches a saturation region when P0 increases. The value of the saturated output power is dependent on the fiber length. For example for the structure with L = 30 cm, saturation output power is 6.59 W and saturation region starts from P0 = 12 W. Also, saturation output power is 3.2 W for L = 35 cm and the saturation condition is achieved at P0 = 6 W. The presented TMM in this article, enable us to analyze the performance of DFB-RF lasers with different structures and above threshold condition. References [1] J. Liu, et al., High-power and highly efficient operation of wavelength-tunable Raman fiber lasers based on volume Bragg gratings, Opt. Express 22 (6) (2014) 6605–6612. [2] V. Supradeepa, Y. Feng, J.W. Nicholson, Raman fiber lasers, J. Opt. 19 (2) (2017) 023001. [3] V.R. Supradeepa, et al., A high efficiency architecture for cascaded Raman fiber lasers, Opt Express 21 (6) (2013) 7148–7155. [4] S. Arun, V. Supradeepa, High power fiber lasers in the SWIR band using Raman lasers, CSI Trans. ICT 5 (2) (2017) 143–148. [5] D.J. DiGiovanni, et al., Cascaded Raman Fiber Laser System Based on Filter Fiber, Google Patents, 2013. [6] J. Shi, S.-u. Alam, M. Ibsen, Sub-watt threshold, kilohertz-linewidth Raman distributed-feedback fiber laser, Opt. Lett. 37 (9) (2012) 1544–1546.
M. Aliannezhadi, F. Shahshahani and V. Ahmadi / Applied Mathematical Modelling 74 (2019) 85–93 [7] [8] [9] [10] [11]
[12] [13] [14] [15]
93
V.E. Perlin, H.G. Winful, Distributed feedback fiber Raman laser, IEEE J. Quantum Electron. 37 (1) (2001) 38–47. Y. Hu, N.G. Broderick, Improved design of a DFB Raman fibre laser, Opt. Commun. 282 (16) (2009) 3356–3359. P.S. Westbrook, et al., Demonstration of a Raman fiber distributed feedback laser, CLEO: Science and Innovations, Optical Society of America, 2011. P.S. Westbrook, et al., Raman fiber distributed feedback lasers, Opt. Lett. 36 (15) (2011) 2895–2897. J. Shi, S.-u. Alam, M. Ibsen, High power, low threshold, Raman DFB fibre lasers, in: Proceedings of the International Quantum Electronics Conference (IQEC) and Conference on Lasers and Electro-Optics (CLEO), IEEE, 2011 Pacific Rim incorporating the Australasian Conference on Optics, Lasers and Spectroscopy and the Australian Conference on Optical Fibre Technology. B. Behzadi, et al., Design of a new family of narrow-linewidth mid-infrared lasers, JOSA B 34 (12) (2017) 2501–2513. B. Behzadi, M. Hossein-Zadeh, R.K. Jain, Narrow-linewidth mid-infrared raman fiber lasers, Nonlinear Optics, Optical Society of America, 2017. J. Shi, S.-u. Alam, M. Ibsen, Highly efficient Raman distributed feedback fibre lasers, Opt. Express 20 (5) (2012) 5082–5091. T. Kremp, K.S. Abedin, P.S. Westbrook, Closed-form approximations to the threshold quantities of distributed-feedback lasers with varying phase shifts and positions, IEEE J. Quantum Electron. 49 (3) (2013) 281–292.
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Modeling the optical nonlinear effects on DFB-RF laser based on the transfer matrix method Maryam Aliannezhadi a,∗, Fatemeh Shahshahani b, Vahid Ahmadi c a
Faculty of Physics, Semnan University P.O. Box: 35195-363, Semnan, Iran Department of Physics, Alzahra University, Tehran, Iran c Department of Electrical Engineering, Tarbiat Modares University, Tehran, Iran b
a r t i c l e
i n f o
Article history: Received 6 January 2018 Revised 8 April 2019 Accepted 16 April 2019 Available online 23 April 2019 Keywords: Distributed feedback Fiber laser Raman laser Cross phase modulation (XPM) Self phase modulation (SPM) Transfer matrix method
a b s t r a c t In this paper, the operation of π -phase shifted distributed feedback Raman fiber (DFB-RF) laser above threshold condition is analyzed theoretically. The nonlinear optical phenomena such as self phase modulation (SPM) and cross phase modulation (XPM) have significant effect on the performance of DFB-RF laser. The numerical results show that the nonlinear effects cause to the saturation of output power and the value of saturated power is dependent on the fiber length. It is found that, the operation wavelength of stokes modes of DFB-RF laser varies in above threshold condition as a result of nonlinear optical properties of the fiber. Simulation is performed by using transfer matrix method to solve three coupled nonlinear wave equations which describe the propagation of pump, forward and backward Stokes waves. The nonlinear SPM and XPM effects are considered in the presented theoretical model. © 2019 Elsevier Inc. All rights reserved.
1. Introduction Raman fiber (RF) lasers are attractive coherent sources due to their exclusive properties. They can be used as a high power and tunable light source in a variety of applications. The operating wavelength of RF lasers is dependent on the pump wavelength and Raman gain spectrum of the fiber material [1]. Owing to this property, the RF lasers could be designed for special wavelengths in the range of visible to infrared region. Raman gain is small in the fiber lasers, so long fiber lengths (typically ten meters to several kilometers) are necessary to obtain low threshold pump power. On the other hand, Raman fiber lasers operate on multi-longitudinal modes and become more sensitive to environmental fluctuations because of long cavity length. Also, single mode lasers are required in more applications like optical communications, high resolution spectroscopy, defense, and network systems, So RF lasers with shorter fiber length are attractive. Various RF structures were proposed by researchers to resolve this conflict. Using Distributed Bragg Reflector (DBR) was the first candidate to solve the problem. DBRs are used on the input and output side of fiber to enhance the interaction between the pump and the Raman-shifted signal. They classified into two general categories, one Raman-Stokes shift and multiple Raman-Stokes shifts called cascaded Raman fiber laser. The result of enhancing the output coupler reflectivity is the lower threshold pump power [2]. A length of approximately 5 m is reported for these types of lasers [3], and recently, a high power laser with output power>25 W at 1570 nm is built by using sixthorder cascaded Raman amplification [4]. This type of lasers suffers from a number of drawbacks. A partly long length fiber ∗
Corresponding author. E-mail address: [email protected] (M. Aliannezhadi).
https://doi.org/10.1016/j.apm.2019.04.048 0307-904X/© 2019 Elsevier Inc. All rights reserved.
86
M. Aliannezhadi, F. Shahshahani and V. Ahmadi / Applied Mathematical Modelling 74 (2019) 85–93
is used in it and high reflector (HR) DBRs are linked in the system weakly, so their operation may be disrupted when high power propagates through it. Also, it is possible for the signal Raman to propagate back into the pump laser or diode, causing it to fail or hit it [5]. Distributed feedback Raman fiber (DFB-RF) lasers with short fiber length (typically < 1 m) are the most promising and practical candidates for a single mode light source for coherent communications, due to their extremely low noise and very narrow linewidth, λ = 0.018 nm [6]. The idea of DFB-RF laser was first proposed by Perlin and Winful in 2001 [7]. According Their theoretical works, a uniform 1m-long DFB-RF laser would have a threshold pump power below 1 W then, a DFB-RF laser with 20 cm cavity length, coupling coefficient of 60 m−1 and a π phase shift inside the fiber Bragg grating has been modeled. Their model has predicted low-threshold and practical laser with high sensitivity to the change of position and width of the phase shift. The threshold pump power of this structure was close to 1 W [8]. Westbrook and his colleagues demonstrated the first experimental DFB Raman fiber laser in 2011. The fiber Bragg gratings (FBG) with π phase shift was fabricated on the base of an OFS fiber. The laser had a power threshold of 39 W with a linewidth of 7.5 MHz .They improved the maximum output power, linewidth, and threshold of the structure by using highly nonlinear fiber (HNLF) to 350 mW, 4 MHz, and 4.3 W, respectively [9,10]. In 2012, Shi and their colleagues reported two DFB-RF lasers fabricated in fiber PS980 and UHNA4 (Ultra-High NA single mode fiber) [6,11]. The coupling coefficient of the structures were κ = 37 m−1 and length L = 0.3 m. The threshold pump power at a pump wavelength of 1069 nm reduced to 2 W and 1 W for PS980 and UHNA and oscillation wavelengths were around 1117.73 nm and 1109.21 nm, respectively. Also, highly efficient Raman-gain-based distributed-feedback fiber lasers with a cavity length of 30 cm and 2 W CW output-power with linewidth smaller than 0.01 nm around wavelength 1.11 μm have been demonstrated by them. Recently, single frequency DFB Raman fiber lasers based on tellurite, and chalcogenide fibers were designed and their operation at any arbitrary wavelength 2.5 −9.5 μm in the spectral region were considered in the threshold condition theoretically [12,13]. To the best of our knowledge, most of the reported studies on the DFB-RF lasers are experimental studies [6,9–11] or theoretical studies in the threshold conditions [14,15]. Due to the importance of DFB-RF lasers, it is worthwhile to investigate the performance of these structures further theoretically, in particular above threshold condition. The main goal of this paper is numerically analysis the behavior of DFB-RF laser above threshold condition. The physical model is based on a system of three nonlinear coupled wave equations that describe the propagation of pump, forward and backward Stokes waves along the fiber length. Raman active media generally exhibit various nonlinearities which can strongly affect the operation of Raman fiber laser. In the present paper we take into account two main nonlinear effects, cross phase modulation (XPM) and self phase modulation (SPM), in the three nonlinear wave equations. We have used transfer matrix method (TMM) to solve the three nonlinear wave equations. Nonlinear optical effects in Raman fiber lasers cause longitudinal variations along the fiber length. Moreover, in DFB-RF laser structures there is some non-uniformity along the fiber length, for example: phase shift in the grating or chirped grating. Therefore TMM is a suitable technique to analyze the periodic structure with non-uniformity. An analytical closed form approximation model based on the TMM has been presented in Ref. [15] for studying the DFB-RF laser structure at threshold condition. Also, the nonlinear optical effects have not been considered in Ref. [15]. The paper is organized as follow: The theoretical model and TMM are described in section II, Then the numerical results and discussion are presented in section III and finally section IV is dedicated to the conclusion 2. Theoretical model The evaluation of slow varying amplitudes of the pump, forward Stokes and backward Stokes waves along the cavity of DFB-RF laser are illustrated by the following set of three nonlinear coupled wave equations [7]:
2 ∂ AP αP gP 2 =− A f + |Ab |2 AP + iγP |AP |2 + 2A f + 2|Ab |2 AP − AP ∂z 2 2 2 ∂ A f gS 2 α = |AP | A f + iκ Ab + iδβ A f + iγS 2|AP |2 + A f + 2|Ab |2 A f − S A f ∂z 2 2 2 ∂A gS α − b = |AP |2 Ab + iκ A f + iδβ Ab + iγS 2|AP |2 + 2A f + |Ab |2 Ab − S Ab ∂z 2 2
(1-a)
(1-b)
(1-c)
where Ap , Af and Ab are the slow-varying amplitudes of the pump, forward, and backward Stokes waves, respectively. The Raman gain coefficient for Stokes waves is represented by gs , and the gain depletion coefficient for pump wave is given by gp = gs(λs/λp). The nonlinear coefficients are indicated by γ s = 2π n2 /λs and γ p = 2π n2 /λp where n2 is the nonlinear refractive index, κ is the coupling coefficient which depends on the amplitude of refractive index modulation of the grating and shows the coupling between forward and backward propagating Stokes waves. The wavelength of pump and Stokes waves are indicated by λp and λs respectively. Detuning the Stokes frequency from the Bragg frequency is defined by δβ = ωc -ωB , where ωc and ωB are the Stokes and Bragg frequency, respectively. The parameters α p and α s refer to the linear fiber loss for pump and Stokes waves, respectively.
M. Aliannezhadi, F. Shahshahani and V. Ahmadi / Applied Mathematical Modelling 74 (2019) 85–93
87
Fig. 1. A simple schematic of a DFB-RF laser with π phase shift at the center of grating.
Fig. 2. In transfer matrix method the fiber length is divided into several subsections.
The second set of terms in (1-a) –(1-c) equations, describe the fiber refractive index nonlinearity. Within the brackets, the terms with the factors of two describe the effect of XPM, whereas the last term describes the effect of SPM. A simple schematic of DFB-RF laser with a π -phase shift at the center of grating is shown in Fig. 1. The pump wave with power P0 is pumped to the fiber from the left at z = 0. The Stokes waves at λs are generated by Raman phenomena in the laser cavity when the pump wave propagates through the fiber. It is notable that, pump wavelength is far from Bragg wavelength and it does not interact with the grating. We used TMM algorithm to solve Eqs. (1-a)–(1-c). By applying the idea of TMM, which has been used for evaluation of semiconductor DFB diode lasers, we introduce and apply the TMM for DFB-RF laser structures for the first time. In TMM the fiber length is divided into several subsections, as shown in Fig. 2. In each subsection the parameters such as δβ , α s and κ are assumed to be constant but they vary from one segment to another segment. The amplitude of forward and backward waves at each segment, zk +1 , can be obtained as a function of the amplitude of forward and backward waves at the previous segment, zk .
A f (zk+1 ) = A f (zk ) Ab (zk+1 ) = A f (zk )
1 r
eiqz −
1 r rA
rB rB −iqz rB e + A f (zk ) − eiqz + e−iqz r rA r r
eiqz +
rB iqz 1 −iqz 1 −iqz e + Ab ( z ) − e + e r rA r rA r
(2)
(3)
where z = zk +1 - zk is the length of each segment, and r = 1+rB /rA , where the parameters rA and rB can be interpreted as the effective reflection coefficients of grating for the forward and backward waves, respectively. The parameter q is defined as:
q= where:
α=
−(δb − δ f ) ∓
(δb − δ f )2 − 4 (α + iδ f )(α + iδb ) + κ 2 2
gs α |AP |2 − s 2 2
(4)
(5)
2 δ f = δβ + γs 2|AP |2 + A f + 2|Ab |2
(6)
2 δb = δβ + γs 2|AP |2 + 2A f + |Ab |2
(7)
88
M. Aliannezhadi, F. Shahshahani and V. Ahmadi / Applied Mathematical Modelling 74 (2019) 85–93
The set of Eqs. (2) and (3) can be written in a matrix form as:
A f (zK+1 ) A (z ) = F k ( zK+1 |zk ) f K Ab (zK+1 ) Ab ( zK )
(8)
where F(Zk+1 |zk ) is a 2 × 2 matrix with the following complex elements:
F11 =
1 iqz rB −iqz e − e r rA
F12 =
rB −iqz e − eiqz r
(9-b)
F21 =
1 iqz e − e−iqz r rA
(9-c)
F22 =
1 −iqz rB iqz e − e r rA
(9-a)
(9-d)
Therefore, we can obtain the amplitude of forward and backward waves at the end of cavity length (z = L) as:
N N A f (L ) A (0 ) = F N F N−1 ...F 2 +1 S(θ )F 2 ...F 2 F 1 f Ab ( L ) Ab ( 0 )
(10)
where N is the total number of subsections and S(θ ) is the phase shift matrix at the center of the grating which is given by:
S (θ ) =
e iθ / 2 0
0
(11)
e−iθ /2
In this paper we have considered, θ = π . The following boundary conditions must be satisfied at the left and right facets of fiber:
A f ( 0 ) = r1 Ab ( 0 ) Ab ( L ) = r2 A f ( L )
(12)
where r1 and r2 are amplitude reflectivity of left and right facets for Stokes wave, respectively. The coupled parameters α and δβ for every mode can be obtained by using TMM and self-consistent numerical solving the boundary conditions (12). The smallest value of α denotes the threshold loss of the main mode and the next value of α is related to the first side mode. Threshold pump power for each mode can be calculated by the following equation:
Pth = 2Ae f f
α+
αs 2
/gS
(13)
where Aeff is called the effective mode area of the fiber and it is assumed constant during calculations. Threshold pump power for the main mode and the first side mode will be denoted by Pth0 , Pth1 , respectively. The intensity of pump, forward and backward Stokes waves are obtained by:
Im = |Am |2 =
Pm , Ae f f
m = p, f, b
(14)
Total intensity of Stokes waves in each z component along the fiber length are given by:
2
I (zk ) = A f (zk ) + |Ab (zk )|
2
(15)
3-Numerical results and discussion The DFB-RF structure with a π -phase shift at the center of grating is shown in Fig 1. The values of parameters that we used in our simulation are related to the commercial high germanium doped silica fiber (UHNA4 from Nufern), with high numerical aperture (NA=0.35), medium propagation loss of α s = 0.05 dB/m and coupling coefficient κ = 37 m−1 [15]. It is notable that, to achieve higher values of coupling coefficient, the host medium could be exposed with hydrogen or deuterium vapor, which increases the background loss of fiber, at IR wavelength. This in turn increases the threshold pump power [9]. For simulation, Eqs. (1-a)–(1-c) should be solve self-consistently with considering the boundary conditions. In order to obtain accurate results by TMM, we examined the numerical results with 200 segments along the fibre length (N = 200) for threshold condition and above threshold condition without considering the nonlinear effects. The errors of our simulations are less than 10−6 . Also, in the case of above threshold condition with considering nonlinear effects we have to increase N with increasing the FBG lengths to obtain the errors less than 10−6 . It is due to the fact that nonlinear effects are more
M. Aliannezhadi, F. Shahshahani and V. Ahmadi / Applied Mathematical Modelling 74 (2019) 85–93
89
Table 1 The parameters used in simulation [15]. Parameters Core refractive index Raman gain Pump wavelength Bragg wavelength Nonlinear refractive index Fiber loss Effective mode area Reflection of fiber ends
Values (nc ) (gs) ( λp ) ( λB ) (n2 ) (α s ) (Aeff ) (r1 , r2 )
1.4 1.55 × 10−13 (m/W) 1.064 (μm) 1.11(μm) 3.2 × 10−20 (m2 /W) 0.05(dB/m) 5.4 (μm2 ) 0
Fig. 3. Threshold pump power for main mode and first side mode as a function of fiber length.
important in the case of the longer fibers and above threshold condition. The two ends of the fiber are considered to be antireflective. The other parameters which are used in calculations are listed in Table 1. The variation of threshold pump power for main mode and first side mode, Pth0 and Pth1 as a function of cavity fiber length are shown in Fig. 3. According to the result, threshold pump power for κ = 37 cm−1 and L = 30 cm is 0.5 W. It is in agreement with the experimental data in Ref. [15] and the difference is due to connector loss and splice loss in experimental set up. Also, Fig. 3 shows that both Pth0 and Pth1 decrease with increasing L due to the increasing of effective volume of Raman gain medium. It is notable that for L < 20 cm, both Pth0 and Pth1 have very high values. As we see in Fig. 3, the pump power for main mode and first side mode at L < 20 cm are larger than 4 W and 150 W, respectively. So the structures with L < 20 cm is not desirable. It is also evident from Fig. 3 the difference between Pth0 and Pth1 decreases with increasing the fiber length and for very large values of L, it approaches to a constant value about 20 W. The difference between Pth0 and Pth1 (Pth = Pth0 –Pth1 ) could be a criterion factor for assessment the single mode operation of lasers in above threshold condition. If Pth is small, the main mode and side modes will compete together in above threshold condition. For example, if we choose L = 50 cm and launch pump power greater than 30 W, the side mode will appear at the output of the laser and the laser will not operate in single mode at this situation. It is notable that for L > 35 cm the slope variation of Pth0 with respect to L is very small. As a result, it is not required to apply the structures with fiber length much greater than 35 cm with κ = 37 m−1 . It seems that a DFB-RF laser with 20 < L < 40 cm to be an appropriate structure for single mode and low threshold operation. Oscillation wavelength is one of the most important laser parameters that describes above threshold stability of the laser. Variation of main mode wavelenght, λs , as a function of P0 , pump power launched at Z = 0, is plotted in Fig. 4 for four different values of L. According to this figure, small red shift is observed with increasing P0 and L. We predict that the red shift is due to the refractive index change of Raman gain material because of XPM and SPM nonlinear effects. To confirm this forecast, we calculate the oscillation laser wavelength for the main mode as a function of P0 , in the case of n2 = 0 in Eqs. (1-a)–(1-c). It is notable that the graph of λ versus P0 coincides with the horizental axis and this is marked by hollow
90
M. Aliannezhadi, F. Shahshahani and V. Ahmadi / Applied Mathematical Modelling 74 (2019) 85–93
Fig. 4. Variation of main mode wavelenght, λs , of DFB-RF structures, as a function of launched pump power, P0 .
Fig. 5. Refractive index change, n, along the fiber length, for κ = 37 m−1 , L = 30 cm and P0 = 1, 5, and 10 W.
circles in Fig. 4. It is found that when no nonlinear effect is considered and n2 = 0, lasing wavelength doesn’t change versus variation of pump power. In order to understand the physical origin of this behavior, the longitudinal variation of refractive index, n, for the structure with κ = 37 m−1 , L = 30 cm, and three values of pump power P0 = 1, 5, 10 W is investigated. The obtained numerical results are shown in Fig. 5. In this figure z = Z/L denotes the normalized position along the fiber length. So it is confirmed that the variation of oscillation wavelength is an outcome of the nonlinear effects. It should be mentioned that the refractive index at nonlinear fibers is dependent on the optical field intensity. Fig. 6 shows the optical field intensity along the fiber for the same data as used in Fig. 5. It can be seen that longitudinal variation of refractive index and optical intensity have similar profiles.
M. Aliannezhadi, F. Shahshahani and V. Ahmadi / Applied Mathematical Modelling 74 (2019) 85–93
91
Fig. 6. Longitudinal distribution of optical intensity of Stokes main mode for the structure with κ = 37 m−1 , L = 30 cm and three values of P0 = 1, 5, 10 W.
Fig. 7. The output power as a function of pump power for DFB-RF structures with κ = 37 m−1 and for different values of L with and without nonlinear effects.
Because of the π phase shift at the center of the grating, there is a peak at z = 0.5 and therefore we see a maximum value of n at the center of the fiber length. Longitudinal distribution of intensity shows that optical energy tends to accumulate at the center of the fiber with increased P0 . Finally, we study the output power, Pout , for both linear and nonlinear situations of DFB-RF structure with κ = 37 m−1 and for different values of L, as a function of input pump power. Output power of laser, Pout , is the sum of the powers of forward and backward Stokes waves at the left and right fiber ends. The numerical results for Pout are shown in Fig. 7. In this figure the results for two cases, with and without nonlinear effects, are compared. It is found from this graph that in linear situation, Pout rises linearly as P0 increases. However, Pout reaches a saturation level when P0 increases in the presence of nonlinear effects. For L = 0.25 m the saturation status takes place for P0 > 15 W, which is not shown in Fig. 7. The value of
92
M. Aliannezhadi, F. Shahshahani and V. Ahmadi / Applied Mathematical Modelling 74 (2019) 85–93
Fig. 8. Pump power variation along the fiber length, for structure with κ = 37 m−1 , L = 30 cm, and three values of P0 = 1, 5, 10 W.
the saturated output power is dependent on the fiber length. For example saturation output power for the structure with L = 30 cm is 6.59 W and saturation region starts from P0 = 12 W. Also, saturation output power is 3.2 W for L = 35 cm and the saturation condition is achieved at P0 = 6 W. According Figs. 3 and 7 in the threshold and above threshold conditions, a DFB fiber laser based on UHNA4 with κ = 37 m−1 and L = 30 cm is an optimum structure. In order to understand the saturation performance of DFB-RF structures, the pump power distribution along the fiber length for the structure with κ = 37 m−1 and L = 30 cm is shown in Fig. 8. The denoted numerical results are related to both linear (n2 = 0) and nonlinear (n2 = 0) conditions. The values of pump power are normalized to their input values at z = 0, P0 . As expected, the pump power is depleted along the fiber. The pump is completely depleted along the fiber in the absence of nonlinearity and for high pump powers. The pump power depletion along the fiber length is the direct result of energy conservation rule. In the steady state and ignoring the fiber loss, the input pump power must be equal to the sum of the output forward Stokes, backward Stokes, and pump powers. Therefore, in the presence of nonlinearity where the pump power does not completely deplete at the end of the fiber, the conversion energy between the pump wave and Stokes waves comes to an end and saturation state for output power occurs. We think that it is a significant result that could be investigated with more details. 4. Conclusion In the present work, we numerically studied the effects of XPM and SPM on the behavior of a π -phase-shifted DFB-RF laser by numerical solution of three coupled nonlinear wave equations using TMM. It was shown that the SPM and XPM effects cause a small red shift (<0.01 nm) at the lasing wavelength with increasing launched pump power. We predict that the red shift is due to the refractive index change of Raman gain material because of XPM and SPM nonlinear effects. We studied the variation of output power for both linear and nonlinear situations as a function of input pump power. It is found that in linear case Pout rises linearly as input pump power increases. However, in the presence of nonlinear effects, Pout reaches a saturation region when P0 increases. The value of the saturated output power is dependent on the fiber length. For example for the structure with L = 30 cm, saturation output power is 6.59 W and saturation region starts from P0 = 12 W. Also, saturation output power is 3.2 W for L = 35 cm and the saturation condition is achieved at P0 = 6 W. The presented TMM in this article, enable us to analyze the performance of DFB-RF lasers with different structures and above threshold condition. References [1] J. Liu, et al., High-power and highly efficient operation of wavelength-tunable Raman fiber lasers based on volume Bragg gratings, Opt. Express 22 (6) (2014) 6605–6612. [2] V. Supradeepa, Y. Feng, J.W. Nicholson, Raman fiber lasers, J. Opt. 19 (2) (2017) 023001. [3] V.R. Supradeepa, et al., A high efficiency architecture for cascaded Raman fiber lasers, Opt Express 21 (6) (2013) 7148–7155. [4] S. Arun, V. Supradeepa, High power fiber lasers in the SWIR band using Raman lasers, CSI Trans. ICT 5 (2) (2017) 143–148. [5] D.J. DiGiovanni, et al., Cascaded Raman Fiber Laser System Based on Filter Fiber, Google Patents, 2013. [6] J. Shi, S.-u. Alam, M. Ibsen, Sub-watt threshold, kilohertz-linewidth Raman distributed-feedback fiber laser, Opt. Lett. 37 (9) (2012) 1544–1546.
M. Aliannezhadi, F. Shahshahani and V. Ahmadi / Applied Mathematical Modelling 74 (2019) 85–93 [7] [8] [9] [10] [11]
[12] [13] [14] [15]
93
V.E. Perlin, H.G. Winful, Distributed feedback fiber Raman laser, IEEE J. Quantum Electron. 37 (1) (2001) 38–47. Y. Hu, N.G. Broderick, Improved design of a DFB Raman fibre laser, Opt. Commun. 282 (16) (2009) 3356–3359. P.S. Westbrook, et al., Demonstration of a Raman fiber distributed feedback laser, CLEO: Science and Innovations, Optical Society of America, 2011. P.S. Westbrook, et al., Raman fiber distributed feedback lasers, Opt. Lett. 36 (15) (2011) 2895–2897. J. Shi, S.-u. Alam, M. Ibsen, High power, low threshold, Raman DFB fibre lasers, in: Proceedings of the International Quantum Electronics Conference (IQEC) and Conference on Lasers and Electro-Optics (CLEO), IEEE, 2011 Pacific Rim incorporating the Australasian Conference on Optics, Lasers and Spectroscopy and the Australian Conference on Optical Fibre Technology. B. Behzadi, et al., Design of a new family of narrow-linewidth mid-infrared lasers, JOSA B 34 (12) (2017) 2501–2513. B. Behzadi, M. Hossein-Zadeh, R.K. Jain, Narrow-linewidth mid-infrared raman fiber lasers, Nonlinear Optics, Optical Society of America, 2017. J. Shi, S.-u. Alam, M. Ibsen, Highly efficient Raman distributed feedback fibre lasers, Opt. Express 20 (5) (2012) 5082–5091. T. Kremp, K.S. Abedin, P.S. Westbrook, Closed-form approximations to the threshold quantities of distributed-feedback lasers with varying phase shifts and positions, IEEE J. Quantum Electron. 49 (3) (2013) 281–292.