1Laboratoire de Physique de la Matière Condensée, Unité de Recherche Associée Centre National de la Recherche Scientifique, Université de Nice–Sophia Antipolis, Parc Valrose, F-06108 Nice Cedex, France
2Institute of Automation and Electrometry, Siberian Branch, Academy of Sciences, 630090 Novosibirsk, Russia
Carlos Montes and Alexander M. Rubenchik, "Stimulated Brillouin scattering from trains of solitons in optical fibers: information degradation," J. Opt. Soc. Am. B 9, 1857-1875 (1992)
The limitation on information flux in soliton-based optical-fiber communication that is due to stimulated Brillouin scattering (SBS) is analyzed by means of the space–time three-wave resonant model for SBS. We show that a continuous train of well-separated solitons excites SBS above a bit-rate threshold fbitthd and that SBS degrades the information after some critical transmission length Lc. General simple formulas give fbitthd, the SBS gain, and Lc as functions of the fiber characteristics. A large range of threshold values is obtained, depending principally on the soliton phase distribution and on the fiber dispersion. For trains of solitons in phase, fbitthd is ∼0.7 Gbit/s for normally dispersive (n.d.) fibers and ∼3.3 Gbits/s for dispersion-shifted (d.-s.) fibers, whereas for trains of solitons uncorrelated in phase the bit-rate threshold goes from ∼21 Gbits/s for n.d. fibers to ∼440 Gbits/s for d.-s. fibers.
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Computation Parameters for a Soliton-Based n.d. Optical-Fiber Communication Systema
P (mW)
5
20
80
1000
Ip (MW/cm2)
0.02
0.08
0.32
4.0
Ep (MV/m)
0.324
0.648
1.29
4.56
〈Ep〉 = a/(a + b)Ep (MV/m)
0.050
0.101
0.202
0.716
τB (ns)
148
74
37
10.4
ΛB = cτB/n0 (m)
30.8
15.4
7.70
2.17
τp (ps)
31.5
15.7
7.88
2.22
τFWHM (ps)
55.5
27.7
13.8
3.90
Δtp = 20τp (ns)
0.630
0.315
0.157
4.4 × 10−2
fbit = 1/Δtp (Gbits/s)
1.58
3.17
6.36
22.5
μa = γaτB
7.43
3.71
1.85
0.523
μS = γSτB
6.97 × 10−4
3.48 × 10−4
1.74 × 10−4
4.92 × 10−5
3.34 × 10−3
6.65 × 10−3
1.33 × 10−2
4.70 × 10−2
G/Lc = 2γn0/c (km−1)
0.214
0.858
3.43
42.9
Lc(G = 20) = 10c/γn0 (km)
93.3
23.3
5.83
0.466
Lcfbit [(km Gbits)/s]
147
73.8
37.0
10.4
Nbit = n0Lc/cΔtp (kbit)
710
355
177
50.2
5.25 × 10−5
0.05
400
9000
44.6
The system uses a monomode laser at λ = 1.55 μm and a normal dispersive [D = 21(ps/nm)km] single-mode silica fiber of effective cross section S = 25 μm2. Adjacent solitons of the form sech(t/τp) have FWHM value (for the intensity) τFWHM = 1.76 τp = 3.75 × 10−4τB, where τB = (KEp)−1 is the coherent SBS characteristic time and K = 20.8 m s−1 V−1 is the SBS coupling constant. The dimensionless width of the hyperbolic secant is τp/τB − a/π = 2.13 × 10−4 [cf. Eq. (24)]. In order to minimize the soliton interaction,28 we separated the pulses by Δtp/τp = π (a + b)/a = 20 [cf. expression (26)], i.e., Δtp = 11.36τFWHM (or Δtp/τB ≃ 1/235 = 4.26 × 10− 1. The soliton bit rate is fbit 1/Δtp= 1/(20τp) = 0.088/τFWHM [expression (25)]. For these parameters, the peak power P required to support the N = 1 soliton is related to its width by expression (3b) with D = 21 ps/(nm/km), and the values of P correspond to the low-gain regime of low input intensities [cf. Eqs. (39) and (40)]. P, peak pump power coupled into the fiber [expressions (3a), (3b), (23a), and (23b)].
Table 2
SBS Limitation for Transmission Capacity in Soliton-Based Optical-Fiber Communicationa
Solitons Correlated, In Phase (δωp ≪ γa)
Case (c), Solitons with Distributed Random Phases δωp ∼ fbit ≫ γa
Case (a), In-Phase Low-Gain γ0a/(a + b) < γa
Case (b), Alternate-Phase High-Gain γ0a/(a + b) > γa
General formulas (valid for different fiber parameters, peak pump powers, and bit rates) of the bit-rate threshold fbitthd, time amplitude growth rate γ, gain G, information damage length Lc (or effective length), length–bit-rate product Lcfbit and number of solitons Nbit transmitted without SBS damage for the three different phase cases (a), (b), and (c):
δωp, spectral width for the soliton train (cf. Section 2);
τB(s) = 1/KEp, characteristic SBS time [Eq. (12)];
a = π(τp/τB), dimensionless soliton width [Eqs. (24)];
fbit(bit/s) = [(a + b)τB]−1 bit rate [expression (25)];
G ≃ 20, SBS power gain.
Tables (2)
Table 1
Computation Parameters for a Soliton-Based n.d. Optical-Fiber Communication Systema
P (mW)
5
20
80
1000
Ip (MW/cm2)
0.02
0.08
0.32
4.0
Ep (MV/m)
0.324
0.648
1.29
4.56
〈Ep〉 = a/(a + b)Ep (MV/m)
0.050
0.101
0.202
0.716
τB (ns)
148
74
37
10.4
ΛB = cτB/n0 (m)
30.8
15.4
7.70
2.17
τp (ps)
31.5
15.7
7.88
2.22
τFWHM (ps)
55.5
27.7
13.8
3.90
Δtp = 20τp (ns)
0.630
0.315
0.157
4.4 × 10−2
fbit = 1/Δtp (Gbits/s)
1.58
3.17
6.36
22.5
μa = γaτB
7.43
3.71
1.85
0.523
μS = γSτB
6.97 × 10−4
3.48 × 10−4
1.74 × 10−4
4.92 × 10−5
3.34 × 10−3
6.65 × 10−3
1.33 × 10−2
4.70 × 10−2
G/Lc = 2γn0/c (km−1)
0.214
0.858
3.43
42.9
Lc(G = 20) = 10c/γn0 (km)
93.3
23.3
5.83
0.466
Lcfbit [(km Gbits)/s]
147
73.8
37.0
10.4
Nbit = n0Lc/cΔtp (kbit)
710
355
177
50.2
5.25 × 10−5
0.05
400
9000
44.6
The system uses a monomode laser at λ = 1.55 μm and a normal dispersive [D = 21(ps/nm)km] single-mode silica fiber of effective cross section S = 25 μm2. Adjacent solitons of the form sech(t/τp) have FWHM value (for the intensity) τFWHM = 1.76 τp = 3.75 × 10−4τB, where τB = (KEp)−1 is the coherent SBS characteristic time and K = 20.8 m s−1 V−1 is the SBS coupling constant. The dimensionless width of the hyperbolic secant is τp/τB − a/π = 2.13 × 10−4 [cf. Eq. (24)]. In order to minimize the soliton interaction,28 we separated the pulses by Δtp/τp = π (a + b)/a = 20 [cf. expression (26)], i.e., Δtp = 11.36τFWHM (or Δtp/τB ≃ 1/235 = 4.26 × 10− 1. The soliton bit rate is fbit 1/Δtp= 1/(20τp) = 0.088/τFWHM [expression (25)]. For these parameters, the peak power P required to support the N = 1 soliton is related to its width by expression (3b) with D = 21 ps/(nm/km), and the values of P correspond to the low-gain regime of low input intensities [cf. Eqs. (39) and (40)]. P, peak pump power coupled into the fiber [expressions (3a), (3b), (23a), and (23b)].
Table 2
SBS Limitation for Transmission Capacity in Soliton-Based Optical-Fiber Communicationa
Solitons Correlated, In Phase (δωp ≪ γa)
Case (c), Solitons with Distributed Random Phases δωp ∼ fbit ≫ γa
Case (a), In-Phase Low-Gain γ0a/(a + b) < γa
Case (b), Alternate-Phase High-Gain γ0a/(a + b) > γa
General formulas (valid for different fiber parameters, peak pump powers, and bit rates) of the bit-rate threshold fbitthd, time amplitude growth rate γ, gain G, information damage length Lc (or effective length), length–bit-rate product Lcfbit and number of solitons Nbit transmitted without SBS damage for the three different phase cases (a), (b), and (c):
δωp, spectral width for the soliton train (cf. Section 2);
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