Particle swarm optimization on threshold exponential gain of stimulated Brillouin scattering in single mode fibers

doi:10.1364/OE.19.001842

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Particle swarm optimization on threshold exponential gain of stimulated Brillouin scattering in single mode fibers

H. A. Al-Asadi, M. H. Al-Mansoori, S. Hitam, M. I. Saripan, and M. A. Mahdi »View Author Affiliations

Optics Express, Vol. 19, Issue 3, pp. 1842-1853 (2011)
http://dx.doi.org/10.1364/OE.19.001842

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▸ Article Outline
– Abstract
1. Introduction
2. Implementation of PSO algorithm
3. Initiation of SBS source models and optimization
4. Experimental and simulation results
5. Conclusions
– References and links

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Abstract

We implement a particle swarm optimization (PSO) algorithm to characterize stimulated Brillouin scattering phenomena in optical fibers. The explicit and strong dependence of the threshold exponential gain on the numerical aperture, the pump laser wavelength and the optical loss coefficient are presented. The proposed PSO model is also evaluated with the localized, nonfluctuating source model and the distributed (non-localized) fluctuating source model. Using our model, for fiber lengths from 1 km to 29 km, the calculated threshold exponential gain of stimulated Brillouin scattering is gradually decreased from 17.4 to 14.6 respectively. The theoretical results of Brillouin threshold power predicted by the proposed PSO model show a good agreement with the experimental results for different fiber lengths from 1 km to 12 km.

1. Introduction

Among all optical non-linear effects observed in single-mode optical fiber, stimulated Brillouin scattering (SBS) particu1ar is of importance since it has numerous practical implications. When the intensity of the pump beam becomes important, the change in index of the environment induced by the wave and the scattering process becomes stimulated Brillouin [1

R. W. Boyd, Nonlinear Optics , 2nd ed., (Academic Press; 2 edition, 2002).

]. Basically, the environment will react to beating of optical waves through electrostriction [2

E. L. Buckland, “Mode-profile dependence of the electrostrictive response in fibers,” Opt. Lett. 24(13), 872–874 (1999). [CrossRef]

]. When an electromagnetic field is applied to an environment, material migrates through electrostriction to the regions where the electromagnetic field is most intense. Thus, the interaction between pump and scattered waves through hypersonic acoustic waves in the medium, leads to a beat frequency. The backscattered Stokes wave is down shifted in frequency with respect to the incident lightwave frequency. This effect is called as Brillouin frequency shift which depends on fiber parameters and wavelength of the incident light [3

E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fiber,” Appl. Phys. Lett. 21(11), 539–541 (1972). [CrossRef]

]. The frequency shift is directly proportional to the acoustic velocity and ranges from 12 to 13 GHz for a silica fiber at wavelength of 1.5 µm [4

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed., (Academic Press, New York, 2006).

]. When the environmental quantities, such as temperature and strain changing the acoustic velocity, the Brillouin frequency shift is also changed [5

Z. Weiwen, H. Zuyuan, K. Masato, and H. Kazuo, “Stimulated Brillouin scattering and its dependences on strain and temperature in a high-delta optical fiber with F-doped depressed inner cladding,” Opt. Lett. 3, 600–602 (2007).

]. For SBS, this feature is very useful for temperature and strain monitoring in optical fibers and it has been used widely in the design of fiber optic sensors [6

L. Zou, X. Bao, F. Ravet, and L. Chen, “Distributed Brillouin fiber sensor for detecting pipeline buckling in an energy pipe under internal pressure,” Appl. Opt. 45(14), 3372–3377 (2006). [CrossRef] [PubMed]

In spontaneous scattering regime, where wave propagates through an optical fiber, only a very small part of its intensity is backscattered due to Brillouin effect. However, when the intensity of incident pump then the wave becomes very strong, this will result the regime of SBS is achieved and the conversion efficiency of pump wave into Stokes wave can reach several tens of percent, which is called SBS efficiency [7

X. P. Mao, R. W. Tkach, A. R. Chraplyvy, R. M. Jopson, and R. M. Derosier, “Stimulated Brillouin threshold dependence on fiber type and uniformity,” IEEE Photon. Technol. Lett. 4(1), 66–69 (1992). [CrossRef]

Over the years, numerous evolutionary computational search algorithms, called global optimization (GO) methods have been proposed to solve applied electromagnetic problems with different degrees of success that depend upon practical tools to search global minimum and maximum [8

A. Yeniay, J. Delavaux, and J. Toulouse, “Spontaneous and stimulated Brillouin scattering gain spectra in optical fibers,” J. Lightwave Technol. 20(8), 1425–1432 (2002). [CrossRef]

–11

W. Wang, Y. Lu, J. S. Fu, and Y. Z. Xiong, “Particle swarm optimization and finite-element based approach for microwave filter design,” IEEE Trans. Magn. 41(5), 1800–1803 (2005). [CrossRef]

]. A well-known evolutionary computation technique that has been used in diverse fields recently, is genetic algorithm [12

J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antenn. Propag. 52(2), 397–407 (2004). [CrossRef]

]. One of the latest GO algorithms is particle swarm optimization (PSO), that was developed by Kennedy and Eberhart [13

D. Boeringer and D. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antenn. Propag. 52(3), 771–779 (2004). [CrossRef]

]. Three main operators are used in GO: recombination, mutation and selection operators [14

M. Jiang, Y. P. Luo, and S. Y. Yang, “Stochastic convergence analysis and parameter selection of the standard particle swarm optimization algorithm,” Inf. Process. Lett. 102(1), 8–16 (2007). [CrossRef]

]. The main advantage features of the PSO include the ease of implementation, no gradient information is required and it is computationally inexpensive. Moreover, PSO algorithm uses low memory and low processing speed [15

S. Mikki and A. A. Kishk, “Improved particle swarm optimization technique using Hard boundary conditions,” Microw. Opt. Technol. Lett. 46(5), 422–426 (2005). [CrossRef]

The aim of this paper is to implement the PSO algorithm in order to optimize and characterize the dependence of threshold exponential gain G_th , on the numerical aperture, pump laser wavelength and the optical loss coefficient in single mode optical fibers. These parameters are usually chosen to describe the fiber sensitivity to SBS which is initiated by spontaneous Brillouin scattering interaction along optical fibers.

2. Implementation of PSO algorithm

Nonlinear and non-continual optimization problems with continuous variables can be solved by PSO. It has been developed through simulation of simplified social models and is based on the symbol of social interaction and communication, such as bird flocking and fish schooling. In PSO, each single solution is a “bird” in the multi-dimensional search space that is called a “particle” with a specific velocity, which is constantly updated by the particle’s own experience and the experience of the particle’s neighbors or the experience of the whole swarm [15

S. Mikki and A. A. Kishk, “Improved particle swarm optimization technique using Hard boundary conditions,” Microw. Opt. Technol. Lett. 46(5), 422–426 (2005). [CrossRef]

]. All of particles have their own fitness values which are evaluated by the fitness function (known as cost function or objective function) to be optimized. They also have certain velocities which direct the flight of the particles. Then, PSO is initialized with a population of random solutions [16

X. Shenheng and Y. Rahmat-Samii, “Boundary conditions in particle swarm optimization revisited,” IEEE Trans. Antenn. Propag. 55(3), 760–765 (2007). [CrossRef]

In theory, let D be the particle swarm population, as in D-dimensional. Each the i-th particle can be represented as an object with several characteristics bounded by a D-dimensional vector,

X_{i} = {(x_{i 1}, x_{i 2}, ..., x_{i D})}^{T}

. The velocity of this particle can be bounded by another D-dimensional vector

V_{i} = {(v_{i 1}, v_{i 2}, ..., v_{i D})}^{T}

P_{i} = {(p_{i 1}, p_{i 2}, ..., p_{i D})}^{T}

is denoted as the best previously visited position of the i-th particle and also the best position explored so far known as

g_{i} = {(g_{i 1}, g_{i 2}, ..., g_{i D})}^{T}

. In all cases, T is the transpose operator. The g-th particle is the best as the index of the best particle in the swarm. Then the PSO algorithm can be described as below [17

M. Clerc and J. Kennedy, “The particle swarm: explosion, stability, and convergence in a multi-dimensional complex space,” IEEE Trans. Evol. Comput. 6(1), 58–73 (2002). [CrossRef]

];

v_{i j}^{k} = w v_{i j}^{k - 1} + c_{1} r_{1} (p b_{i j}^{k - 1} - x_{i j}^{k - 1}) + c_{2} r_{2} (g b_{j}^{k - 1} - x_{i j}^{k - 1}),

(1)

x_{i j}^{k} = x_{i j}^{k - 1} + v_{i j}^{k},

(2)

where

c_{1}

and

c_{2}

are positive constants, called acceleration constants (

c_{1}

is the self-confidence (cognitive) factor and

c_{2}

is the swarm confidence (social) factor), usually

c_{1}

and

c_{2}

are in the range from 1.5 to 2.5, and

r_{1}

and

r_{2}

are two random functions uniformly distributed in the range [0, 1]. wis the inertia weight factor that takes linearly decreasing values downward from 1 to 0 [18

M. Donelli and A. Massa, “Computational approach based on a particle swarm optimizer for microwave imaging of two-dimensional dielectric scatterers,” IEEE Trans. Microw. Theory Tech. 53(5), 1761–1776 (2005). [CrossRef]

]. The size of swarm population is ascertained by i = 1, 2, …, N and j = 1,2, …, D, where N is the size of swarm population and

k = 1, 2, ...,

determines the iteration number.

Equations (1) and (2) describe the flight trajectory of a population of particles. The velocity is dynamically updated as described by Eq. (1) and the position update of the flying particles is determined by Eq. (2). The effect of the particle inertia, the particle memory influence, and the swarm (society) influence represent the 1st term, 2nd term, and 3rd term in Eq. (1), respectively. The flowchart of the procedure is shown in Fig. 1 .

Fig. 1 Particle swarm optimization algorithm flowchart used in this work.

PSO with a local neighborhood and PSO with a global neighborhood are developed as two variants of the PSO algorithm. Each particle is updated by following two (best) values during each iteration process. Each particle (solution) moves towards its best previous position and towards the best particle in the whole swarm according to the global neighborhood, This best position is set as the current global best and is called g_best. On the other hand, the position vector of the best solution (fitness) this particle can be achieved (called l_best) when each particle (solution) moves towards its previous best position and towards the best particle in its restricted local neighborhood. The detailed process for implementing PSO was described in [9

S. Cui and D. S. Weile, “Application of a parallel particle swarm optimization scheme to the design of electromagnetic absorbers,” IEEE Trans. Antenn. Propag. 53(11), 3616–3624 (2005). [CrossRef]

–13

D. Boeringer and D. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antenn. Propag. 52(3), 771–779 (2004). [CrossRef]

3. Initiation of SBS source models and optimization

The scattering of light is functionally linked to the presence of inhomogeneities in the optical characteristic of the medium itself. Spontaneous Brillouin scattering follows from adiabatic density fluctuations, i.e. periodic perturbations of the refractive index generated by acoustic waves (acoustic phonons) of thermal origin. The amplitude of the Brillouin scattering is relatively low in the spontaneous regime, approximately 100 times less than intensity of the Rayleigh scattering [19

A. A. Fotiadi, R. Kiyan, O. Deparis, P. Mégret, and M. Blondel, “Statistical properties of stimulated Brillouin scattering in single-mode optical fibers above threshold,” Opt. Lett. 27(2), 83–85 (2002). [CrossRef]

]. It relates directly to the number of acoustic phonons in the fiber, which itself are simply determined by the thermal excitation. However, under stimulated environments, the population of phonons participating in the interaction lies in the strongly non-equilibrium conditions and therefore it grows very rapidly. As a result, the efficiency of the scattering process is thus significantly increased. So that at a certain level of intensity, optical fiber acts as a mirror and all the additional power that is injected automatically reflect. The phenomenon that is causing the stimulation of Brillouin scattering and leads to the creation of acoustics phonons in the presence of light in the fiber is electrostriction [1

R. W. Boyd, Nonlinear Optics , 2nd ed., (Academic Press; 2 edition, 2002).

In reality, the configurations can be categorized depending on whether the Stokes wave is derived from the Brillouin spontaneous circulation or whether the wave has been artificially introduced into the environment. We can classify it as the Brillouin generator, if the Stokes wave grows from the spontaneous scattering or Brillouin amplifier when an external signal is injected into the medium. The external signal must have an adequate frequency, i.e. close to that of Stokes waves released spontaneously by the community, to undergo amplification by SBS effect.

The operation of a Brillouin generator is generally characterized by SBS efficiency (r), the SBS threshold occurrence and noise properties such as the Brillouin amplification gain factor G and linewidth variations. Here, r is defined as the ratio between the average Stokes intensity and the intensity of the laser pump power at the near end of optical fiber. The transition between the spontaneous and stimulated processes is not really steep, in the sense that when the Stokes and laser pump waves are simultaneously present in the fiber, there may be a stimulation of the acoustic wave. Two different models can be distinguished, corresponding to the origin of SBS: 1. Localized, nonfluctuating source model. 2. Distributed (nonlocalized), fluctuating source model.

The origin of SBS is ascribed to spontaneous Brillouin scattering in the first model, but the actual source of the SBS phenomena by a spatially distributed fluctuation noise source in the second model. In both cases of SBS generator or amplifier the optical laser and Stokes fields describe the SBS including the initiation of noise are given by [20

R. Boyd, K. Rza̧ewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42(9), 5514–5521 (1990). [CrossRef] [PubMed]

];

\frac{\partial E_{L}}{\partial z} + \frac{n_{1}}{c} \frac{\partial E_{L}}{\partial t} + α E_{L} = \frac{i γ_{e} ω_{s}}{4 ρ_{o} n_{1} c} ρ E_{s},

(3)

\frac{\partial E_{s}}{\partial z} - \frac{n_{1}}{c} \frac{\partial E_{s}}{\partial t} - α E_{s} = - \frac{i γ_{e} ω_{L}}{4 ρ_{o} n_{1} c} ρ^{*} E_{L},

(4)

v_{A} \frac{\partial ρ}{\partial z} + \frac{\partial ρ}{\partial t} + \frac{1}{2 τ_{B}} ρ = \frac{i γ_{e} ε_{o} q^{2}}{4 Ω} E_{s} E_{L}^{*} + f,

(5)

where

E_{L}

E_{s}

are the forward laser and backward Stokes wave amplitudes in Brillouin medium,

ω_{L}

ω_{s}

are the laser and backward Stokes wave frequencies,

n_{1}

is the core refractive index, c is the velocity of light, α is the optical loss coefficient in the fiber, ρ is the acoustic density disturbance with a velocity

v_{A}

caused by the Langevin noise source f, which is responsible for the thermal excitation of acoustic waves (spontaneous Brillouin scattering) and which lead to the initiation of the SBS process,

λ_{e} = ρ \frac{\partial ε}{\partial ρ}

is the electrostrictive coupling coefficient of the medium,

ε_{o}

is the medium dielectric constant,

ρ_{o}

is the material density,

τ_{B}

is the acoustic phonon lifetime (inverse of the phonon decay rate

Γ_{B} = 2 π Δ v_{B}

Δ v_{B}

is the Brillouin linewidth), the acoustic frequency is defined by

Ω = ω_{L} - ω_{s}

, and the acoustic wave number is determined by

q = k_{L} - k_{s}

Floch et. al. solved Eqs. (3)–(5) in a low-loss medium, using Fourier Transform technique, to give the backward Stokes wave intensity

I_{s} (z)

, at the fiber input as [21

S. Le Floch and P. Cambon, “Theoretical evaluation of the Brillouin threshold and the steady-state Brillouin equations in standard single-mode optical fibers,” J. Opt. Soc. Am. A 20(6), 1132–1137 (2003). [CrossRef]

];

I_{s} (0) = \frac{K T Γ_{B}}{4 A_{e f f}} \frac{v_{L}}{v_{B}} \exp (- α L) . G . \exp (\frac{G}{2}) [I_{o} (\frac{G}{2}) - I_{1} (\frac{G}{2})]

(6)

where K is the Boltzman constant, T is the temperature, the effective mode cross-sectional area is

A_{e f f}

v_{B}

the Brillouin frequency shift, and L is the fiber length. The Brillouin amplification gain factor G is defined as

G = g_{B} I_{L} (0) L_{e f f}

. The interaction (effective) length is

L_{e f f} = \frac{1}{α} [1 - \exp (- α L)]

and

g_{B}

is a Brillouin gain coefficient.

I_{o}

I_{1}

are the modified Bessel functions of zeroth and first orders, respectively. The optical intensities of the fields can be defined as

I_{L, s} = \frac{1}{2} n_{1} ε_{o} {| E_{L, s} |}^{2}

The steady state behavior of the forward laser pump power intensity

I_{L} (z)

and backward Stokes wave intensity

I_{s} (z)

are usually reduce to two equations depending on Eqs. (3)–(5). These can be rewritten in simple form of;

\frac{\partial I_{L}}{\partial z} + α I_{L} = - g_{B} I_{L} I_{S},

(7)

\frac{\partial I_{s}}{\partial z} - α I_{s} = - g_{B} I_{L} I_{S} .

(8)

This model is called as the localized, nonfluctuating source model. Depending on this model, the backward Stokes wave intensity at near end of optical fiber is given by

I_{s} (0) = I_{s} (L) \exp (I_{L} (0) g_{B} L_{e f f} - α L)

, where

I_{L} (0)

is the forward laser pump power intensity at

z = 0

. In the case of an SBS amplifier

I_{s} (L)

(backward Stokes wave intensity at the far end of optical fiber) is generated externally and so its magnitude is known, but for the generator this arises from spontaneous scattering and is therefore generated internally and so its size is unknown. However, it is possible to estimate the size of spontaneous scattering from the threshold condition for SBS. Threshold parameters that are commonly used to characterize the SBS phenomenon are the threshold power and threshold exponential gain.

In low-loss optical fibers, the localized, nonfluctuating source model has typically used to find the SBS threshold power, when SBS is seed with noise. The fit parameters in this model are the Brillouin gain coefficient g_B and the threshold exponential gain G_th and it can be defined as [22

R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and brillouin scattering,” Appl. Opt. 11(11), 2489–2494 (1972). [CrossRef] [PubMed]

];

P_{t h} = \frac{G_{t h} A_{e f f}}{g_{B} L_{e f f}} .

(9)

G_{t h}

can be defined physically as the amplification factor of the seeded Stokes wave to the output Stokes wave power level of

r P_{t h}

at the input of optical fiber. The term SBS efficiency that is used to describe the ratio between the backward Stokes wave intensity

I_{s} (z)

and the laser pump wave intensity

I_{L} (z)

z = 0

r = I_{s} (0) / I_{L} (0)

The localized, nonfluctuating source model was apparently the first to use the term SBS efficiency equal to one (

r = 1

) to find the threshold exponential gain as a constant value (

G_{t h} = 21

), independent of the laser pump characteristic and the optical fiber. As a result of Eqs. (6) and (9), we define threshold exponential gain with the mean value of Brillouin amplification gain factor [21

];

G_{t h} = \ln [\frac{4 A_{e f f} v_{B} c G^{3 / 2} π^{1 / 2}}{g_{B} K T Γ_{B} λ_{L} L_{e f f}}] .

(10)

Equation (10) shows that the threshold exponential gain is directly and inversely proportional to the effective mode cross-sectional area and the effective length of optical fiber, respectively.

For cylindrical waveguide fibers, the properties of fiber and the pump laser wavelength lead to estimate the maximum effective area. For single mode fiber the light mode can be approximated by a Gaussian and the area is simply

(π w_{o}^{2})

, where

w_{0}

is the 1/e field radius.

w_{0}

can be calculated from the fiber core radius α and the normalized frequency parameter V, is [23

M. Artiglia, G. Coppa, P. Di Vita, M. Potenza, and A. Sharma, “Mode field diameter measurements in single-mode optical fibers,” J. Lightwave Technol. 7(8), 1139–1152 (1989). [CrossRef]

];

w_{0} = α [0.632 + \frac{1.478}{V^{3 / 2}} + \frac{4.76}{V^{6}}],

(11)

where

V = \frac{2 π a}{λ_{L}} \sqrt{n_{1}^{2} - n_{2}^{2}}

, and

n_{2}

is the cladding refraction index.

The optical fiber is also characterized by another coefficient which is the maximum half-angle accepted by the fiber. This coefficient is also known as the numerical aperture, NA that can be calculated from the refractive indices of the core (n₁ ) and cladding (n₂ ) as;

\sin θ_{\frac{1}{2}} = N A = \sqrt{n_{1}^{2} - n_{2}^{2}} .

(12)

Our objective is to find optimum parameters usually chosen to describe the sensitivity of the fiber to SBS which is initiated by spontaneous Brillouin scattering, numerical aperture, pump laser wavelength and optical loss coefficient, i.e. low additional propagation losses. In this work, different lengths of single mode fiber are utilized for a specific fitness (cost) function. The convergence behavior and stability of PSO algorithm depends on the parameter inertia weight, because it is used to control the impact of the previous velocities on the current one. Global exploration by using a large value of inertia weight leads to search new areas, while local exploration can be done by a small value. Equations (10), (11) and (12) are taken into account as the fitness function, defined as the minimization function. These parameters are randomly chosen over the ranges as shown in Table 1 .

Table 1 Ranges of parameter values used in PSO algorithm to determine SMF sensitivity on SBS effect

Parameter	Value
Numerical aperture	0.04 – 2
Pump laser wavelength (nm)	1540 – 1560
Optical loss coefficient (dB/km)	0.19 – 0.3

4. Experimental and simulation results

The experimental setup for the SBS threshold is shown in Fig. 2 . The pump wave (P_p ) is launched into a single mode fiber (SMF) through an erbium doped fiber amplifier (EDFA) and circulator (Cir). The pump power (P_p ) and its backward scattering Stokes peak power (P_s ) are measured by an optical spectrum analyzer (OSA) at port 1 and 3 respectively.

Fig. 2 Experimental setup to measure SBS threshold using single mode optical fiber.

The PSO algorithm, implemented in MATLAB, is capable of minimizing threshold exponential gain. Parameter values required in numerical calculations are listed in Table 2 .

Table 2 Simulation parameters used for PSO algorithm

Parameter	Value
P_p (mW)	30
$g_{B}$ (m/W)	2 x 10⁻¹¹
L (km)	1-29 step 2
$v_{B}$ (GHz)	11
$Δ v_{B}$ (MHz)	30
α (µm)	4.5

We have applied the implementation of PSO as mentioned in previous section to characterize SBS phenomenon in single mode fiber. PSO population size is often taken between 10 and 100 [22

R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and brillouin scattering,” Appl. Opt. 11(11), 2489–2494 (1972). [CrossRef] [PubMed]

] and in this paper, a population size of 100 is used. Due to that, initialization may take a bit longer time with more convergence and stability as reported in [24

V. Kadirkamanathan, K. Selvarajah, and P. J. Fleming, “Stability analysis of the particle dynamics in particle swarm optimizer,” IEEE Trans. Evol. Comput. 10(3), 245–255 (2006). [CrossRef]

]. However, the large population preferred for some nonlinear and complex problems. The PSO parameters that are required to run the optimization program are presented in Table 3 .

Table 3 Parameters of PSO used for the purpose of optimizations

Parameter	Value
Particle swarm size (population size)	100
Iteration number range	100–2000
Acceleration constant	2
Initial velocity	0
Initial local best	0
Initial global best	0
Initial inertia weight	0.7

The goal of the optimization algorithm is to locate all the global best values of the optical fiber parameters that used to characterize SBS, numerical aperture, pump laser wavelength and optical loss coefficient. The PSO algorithm can actually achieve that goal for fiber length from 1 to 29 km with 2 km step. In this case, the fiber length is not optimized and is just used for the purpose of comparison. The optimum values of three parameters with different optical fiber lengths are then calculated as shown in Table 4 .

Table 4 PSO results of SMF parameters sensitivity to SBS for different fiber lengths

Fiber Length (km)	Optimal NA	Optimal $λ_{L}$ (nm)	Optimal α (dB/km)
1	1.6131	1558.30	0.195988
3	1.6725	1557.60	0.193435
5	1.5983	1554.80	0.190034
7	1.7054	1554.50	0.195211
9	1.6871	1554.30	0.195881
11	1.6632	1558.20	0.195406
13	1.4874	1555.60	0.192705
15	1.5659	1555.30	0.197013
17	1.6539	1559.00	0.190662
19	1.6713	1551.30	0.190677
21	1.6733	1554.10	0.195563
23	1.6473	1558.60	0.199454
25	1.3352	1558.40	0.195389
27	1.5624	1553.60	0.194855
29	1.7445	1553.80	0.194182

The optimum threshold exponential gain

G_{t h}

, depends on the optimal value of these three parameters as presented in Table 4. In addition, the effective length of optical fiber is also influenced by these optimal parameters. Figure 3 reports the threshold exponential gain as a function of optimal NA and

λ_{L}

with different lengths of optical fiber from 1 km to 29 km with step of 2 km. When NA increases to the value over 1.4 and decreases

λ_{L}

to the value less than 1555 nm, they reduce the efficiency of the SBS interaction and the optimal

G_{t h}

decreases dramatically and it is closer ≈14.6.

Fig. 3 Characteristics of threshold exponential gain with respect to (a) optimal numerical aperture, and (b) optimal pump laser wavelength.

The dependence of the exponential gain on the optical fiber lengths using PSO model is shown in Fig. 4(a) . The constant value of G_th of 21 is commonly used in the localized, nonfluctuating source model, independent of the optical fiber lengths, which was originally calculated at (

r = 1

) [21

]. On the other hand, based on Eqs. (9) and (10), the numerical results of power and exponential gain at threshold by using the distributed (nonlocalized), fluctuating source model are plotted as in [21

]. For short fibers (L < 5 km with low optical attenuation), the threshold exponential gain is increased from 17.0 to 18.3. The similar pattern is also observed for the PSO model, the threshold exponential gain increases from 15.9 to 17.4 in the range of 1-5 km fiber lengths. For long fibers (L > 5 km with high optical attenuation), the threshold exponential gain is gradually decreased to about 16.6 and 14.6 for the distributed (nonlocalized), fluctuating source model and PSO model respectively.

Fig. 4 Characteristics of (a) threshold exponential gain and (b) threshold power of SBS with respect to fiber length.

For the experimental configuration shown in Fig. 2, the measured SBS threshold power is in the range of 37.7 mW to 8.5 mW from 1 km to 12 km of fiber lengths, respectively as depicted in Fig. 4(b). For the purpose of verification, the calculated SBS threshold powers for all models are also depicted in Fig. 4(b). For long fibers, all the theoretical values converge to 13.7 mW, 9.7 mW and 7.2 mW for the localized, nonfluctuating source model, the distributed (nonlocalized), fluctuating source model and PSO model respectively. Based on the findings, our proposed PSO model and the distributed (nonlocalized), fluctuating source model predict the SBS threshold power close to the experimental value with error of less than 20%. In addition, for short fibers, the theoretical values of SBS threshold power calculated from the PSO model show good agreement with the ones obtained experimentally. In this case, the error between these two values is less than 20%. The other two models cannot predict the threshold power characteristics close enough to the experimental values with error of larger than 20%. Overall, in comparison with all the models, the theoretical values calculated from the PSO model are in good agreement with the experimental values for the whole range of fiber lengths. The discrepancies between these two values are in the range of 9-20% which are contributed by some uncertainties of measurement errors. This might be due to the omission of polarization effect in our proposed PSO model which has been reported to be affecting Brillouin gain efficiency [25

M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” J. Lightwave Technol. 12(4), 585–590 (1994). [CrossRef]

]. The exponential gain and SBS power at threshold using the PSO algorithm is interdependent; which depends primarily on the optical fiber length (the optimal optical loss coefficient), the optimal numerical aperture and the optimal pump laser wavelength.

5. Conclusions

The threshold exponential gain is an important parameter for describing the characteristics of SBS initiated by the spontaneous Brillouin scattering in single mode fiber. We have investigated theoretically the influence of the exponential gain with various models; the localized, nonfluctuating source model, the distributed (nonlocalized), fluctuating source model and our proposed PSO algorithm. In this work, PSO algorithm which is a stochastic search technique, has exhibited good performance capabilities. For different values of optical fiber lengths, we show that the exponential gain is strongly dependent on the optimal numerical aperture, the optimal pump laser wavelength and the optimal optical loss coefficient. We found that the theoretical threshold exponential gain of SBS is in the range of 17.4 to 14.6 for fiber lengths from 1 km to 12 km respectively. Overall, these values are lower than those obtained from other two models. We also show that our calculated results of SBS threshold power from the PSO model are in good agreement with experimental results with an error of less than 20%.

Acknowledgments

This work was partly supported by the Ministry of Science, Technology and Innovation, Malaysia and the Brain Gain Malaysia Program, R&D Collaboration under research grant # MOSTI/BGM/R&D/19(3).

References and links

1.	R. W. Boyd, Nonlinear Optics , 2nd ed., (Academic Press; 2 edition, 2002).
2.	E. L. Buckland, “Mode-profile dependence of the electrostrictive response in fibers,” Opt. Lett. 24(13), 872–874 (1999). [CrossRef]
3.	E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fiber,” Appl. Phys. Lett. 21(11), 539–541 (1972). [CrossRef]
4.	G. P. Agrawal, Nonlinear Fiber Optics, 4th ed., (Academic Press, New York, 2006).
5.	Z. Weiwen, H. Zuyuan, K. Masato, and H. Kazuo, “Stimulated Brillouin scattering and its dependences on strain and temperature in a high-delta optical fiber with F-doped depressed inner cladding,” Opt. Lett. 3, 600–602 (2007).
6.	L. Zou, X. Bao, F. Ravet, and L. Chen, “Distributed Brillouin fiber sensor for detecting pipeline buckling in an energy pipe under internal pressure,” Appl. Opt. 45(14), 3372–3377 (2006). [CrossRef] [PubMed]
7.	X. P. Mao, R. W. Tkach, A. R. Chraplyvy, R. M. Jopson, and R. M. Derosier, “Stimulated Brillouin threshold dependence on fiber type and uniformity,” IEEE Photon. Technol. Lett. 4(1), 66–69 (1992). [CrossRef]
8.	A. Yeniay, J. Delavaux, and J. Toulouse, “Spontaneous and stimulated Brillouin scattering gain spectra in optical fibers,” J. Lightwave Technol. 20(8), 1425–1432 (2002). [CrossRef]
9.	S. Cui and D. S. Weile, “Application of a parallel particle swarm optimization scheme to the design of electromagnetic absorbers,” IEEE Trans. Antenn. Propag. 53(11), 3616–3624 (2005). [CrossRef]
10.	J. Perez and J. Basterrechea, “Particle swarm optimization and its application to antenna far-field pattern prediction from planner scanning,” Microw. Opt. Technol. Lett. 44(5), 398–403 (2005). [CrossRef]
11.	W. Wang, Y. Lu, J. S. Fu, and Y. Z. Xiong, “Particle swarm optimization and finite-element based approach for microwave filter design,” IEEE Trans. Magn. 41(5), 1800–1803 (2005). [CrossRef]
12.	J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antenn. Propag. 52(2), 397–407 (2004). [CrossRef]
13.	D. Boeringer and D. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antenn. Propag. 52(3), 771–779 (2004). [CrossRef]
14.	M. Jiang, Y. P. Luo, and S. Y. Yang, “Stochastic convergence analysis and parameter selection of the standard particle swarm optimization algorithm,” Inf. Process. Lett. 102(1), 8–16 (2007). [CrossRef]
15.	S. Mikki and A. A. Kishk, “Improved particle swarm optimization technique using Hard boundary conditions,” Microw. Opt. Technol. Lett. 46(5), 422–426 (2005). [CrossRef]
16.	X. Shenheng and Y. Rahmat-Samii, “Boundary conditions in particle swarm optimization revisited,” IEEE Trans. Antenn. Propag. 55(3), 760–765 (2007). [CrossRef]
17.	M. Clerc and J. Kennedy, “The particle swarm: explosion, stability, and convergence in a multi-dimensional complex space,” IEEE Trans. Evol. Comput. 6(1), 58–73 (2002). [CrossRef]
18.	M. Donelli and A. Massa, “Computational approach based on a particle swarm optimizer for microwave imaging of two-dimensional dielectric scatterers,” IEEE Trans. Microw. Theory Tech. 53(5), 1761–1776 (2005). [CrossRef]
19.	A. A. Fotiadi, R. Kiyan, O. Deparis, P. Mégret, and M. Blondel, “Statistical properties of stimulated Brillouin scattering in single-mode optical fibers above threshold,” Opt. Lett. 27(2), 83–85 (2002). [CrossRef]
20.	R. Boyd, K. Rza̧ewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42(9), 5514–5521 (1990). [CrossRef] [PubMed]
21.	S. Le Floch and P. Cambon, “Theoretical evaluation of the Brillouin threshold and the steady-state Brillouin equations in standard single-mode optical fibers,” J. Opt. Soc. Am. A 20(6), 1132–1137 (2003). [CrossRef]
22.	R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and brillouin scattering,” Appl. Opt. 11(11), 2489–2494 (1972). [CrossRef] [PubMed]
23.	M. Artiglia, G. Coppa, P. Di Vita, M. Potenza, and A. Sharma, “Mode field diameter measurements in single-mode optical fibers,” J. Lightwave Technol. 7(8), 1139–1152 (1989). [CrossRef]
24.	V. Kadirkamanathan, K. Selvarajah, and P. J. Fleming, “Stability analysis of the particle dynamics in particle swarm optimizer,” IEEE Trans. Evol. Comput. 10(3), 245–255 (2006). [CrossRef]
25.	M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” J. Lightwave Technol. 12(4), 585–590 (1994). [CrossRef]

OCIS Codes
(190.4370) Nonlinear optics : Nonlinear optics, fibers
(190.5890) Nonlinear optics : Scattering, stimulated
(290.5900) Scattering : Scattering, stimulated Brillouin

ToC Category:
Nonlinear Optics

History
Original Manuscript: September 7, 2010
Revised Manuscript: October 15, 2010
Manuscript Accepted: October 15, 2010
Published: January 18, 2011

Citation
H. A. Al-Asadi, M. H. Al-Mansoori, S. Hitam, M. I. Saripan, and M. A. Mahdi, "Particle swarm optimization on threshold exponential gain of stimulated Brillouin scattering in single mode fibers," Opt. Express 19, 1842-1853 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-1842

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References

R. W. Boyd, Nonlinear Optics, 2nd ed., (Academic Press; 2 edition, 2002).
E. L. Buckland, “Mode-profile dependence of the electrostrictive response in fibers,” Opt. Lett. 24(13), 872–874 (1999). [CrossRef]
E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fiber,” Appl. Phys. Lett. 21(11), 539–541 (1972). [CrossRef]
G. P. Agrawal, Nonlinear Fiber Optics, 4th ed., (Academic Press, New York, 2006).
Z. Weiwen, H. Zuyuan, K. Masato, and H. Kazuo, “Stimulated Brillouin scattering and its dependences on strain and temperature in a high-delta optical fiber with F-doped depressed inner cladding,” Opt. Lett. 3, 600–602 (2007).
L. Zou, X. Bao, F. Ravet, and L. Chen, “Distributed Brillouin fiber sensor for detecting pipeline buckling in an energy pipe under internal pressure,” Appl. Opt. 45(14), 3372–3377 (2006). [CrossRef] [PubMed]
X. P. Mao, R. W. Tkach, A. R. Chraplyvy, R. M. Jopson, and R. M. Derosier, “Stimulated Brillouin threshold dependence on fiber type and uniformity,” IEEE Photon. Technol. Lett. 4(1), 66–69 (1992). [CrossRef]
A. Yeniay, J. Delavaux, and J. Toulouse, “Spontaneous and stimulated Brillouin scattering gain spectra in optical fibers,” J. Lightwave Technol. 20(8), 1425–1432 (2002). [CrossRef]
S. Cui and D. S. Weile, “Application of a parallel particle swarm optimization scheme to the design of electromagnetic absorbers,” IEEE Trans. Antenn. Propag. 53(11), 3616–3624 (2005). [CrossRef]
J. Perez and J. Basterrechea, “Particle swarm optimization and its application to antenna far-field pattern prediction from planner scanning,” Microw. Opt. Technol. Lett. 44(5), 398–403 (2005). [CrossRef]
W. Wang, Y. Lu, J. S. Fu, and Y. Z. Xiong, “Particle swarm optimization and finite-element based approach for microwave filter design,” IEEE Trans. Magn. 41(5), 1800–1803 (2005). [CrossRef]
J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antenn. Propag. 52(2), 397–407 (2004). [CrossRef]
D. Boeringer and D. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antenn. Propag. 52(3), 771–779 (2004). [CrossRef]
M. Jiang, Y. P. Luo, and S. Y. Yang, “Stochastic convergence analysis and parameter selection of the standard particle swarm optimization algorithm,” Inf. Process. Lett. 102(1), 8–16 (2007). [CrossRef]
S. Mikki and A. A. Kishk, “Improved particle swarm optimization technique using Hard boundary conditions,” Microw. Opt. Technol. Lett. 46(5), 422–426 (2005). [CrossRef]
X. Shenheng and Y. Rahmat-Samii, “Boundary conditions in particle swarm optimization revisited,” IEEE Trans. Antenn. Propag. 55(3), 760–765 (2007). [CrossRef]
M. Clerc and J. Kennedy, “The particle swarm: explosion, stability, and convergence in a multi-dimensional complex space,” IEEE Trans. Evol. Comput. 6(1), 58–73 (2002). [CrossRef]
M. Donelli and A. Massa, “Computational approach based on a particle swarm optimizer for microwave imaging of two-dimensional dielectric scatterers,” IEEE Trans. Microw. Theory Tech. 53(5), 1761–1776 (2005). [CrossRef]
A. A. Fotiadi, R. Kiyan, O. Deparis, P. Mégret, and M. Blondel, “Statistical properties of stimulated Brillouin scattering in single-mode optical fibers above threshold,” Opt. Lett. 27(2), 83–85 (2002). [CrossRef]
R. Boyd, K. Rza̧ewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42(9), 5514–5521 (1990). [CrossRef] [PubMed]
S. Le Floch and P. Cambon, “Theoretical evaluation of the Brillouin threshold and the steady-state Brillouin equations in standard single-mode optical fibers,” J. Opt. Soc. Am. A 20(6), 1132–1137 (2003). [CrossRef]
R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and brillouin scattering,” Appl. Opt. 11(11), 2489–2494 (1972). [CrossRef] [PubMed]
M. Artiglia, G. Coppa, P. Di Vita, M. Potenza, and A. Sharma, “Mode field diameter measurements in single-mode optical fibers,” J. Lightwave Technol. 7(8), 1139–1152 (1989). [CrossRef]
V. Kadirkamanathan, K. Selvarajah, and P. J. Fleming, “Stability analysis of the particle dynamics in particle swarm optimizer,” IEEE Trans. Evol. Comput. 10(3), 245–255 (2006). [CrossRef]
M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” J. Lightwave Technol. 12(4), 585–590 (1994). [CrossRef]

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