- Email: [email protected]
Theoretical investigation of tunable pulse broadening cancellation via doublet Brillouin gain lines in an optical fiber
Optics Communications 351 (2015) 35–39
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Theoretical investigation of tunable pulse broadening cancellation via doublet Brillouin gain lines in an optical fiber Kenshiro Nagasaka a,n, Qijin Tian b, Lai Liu b, Dan Zhao b, Guanshi Qin b, Weiping Qin b, Takenobu Suzuki a, Yasutake Ohishi a a b
Research Center for Advanced Photon Technology, Toyota Technological Institute, 2-12-1 Hisakata, Tenpaku, Nagoya 468-8511, Japan State Key Laboratory on Integrated Optoelectronics, College of Electronic Science and Engineering, Jilin University, Changchun 130012, China
art ic l e i nf o
a b s t r a c t
Article history: Received 5 November 2014 Received in revised form 1 April 2015 Accepted 4 April 2015 Available online 7 April 2015
Tunable pulse broadening cancellation via doublet Brillouin gain lines in a silica fiber was theoretically investigated. Minimum broadening factor of 0.96 with time delay of 134.5 ps is achieved by optimizing doublet Brillouin lines for 200 ps Gaussian pulse with pulse distortion. Frequency separation between two gain lines and spectrum bandwidth of single gain line and peak gain values are optimized and considered in the present paper. & 2015 Elsevier B.V. All rights reserved.
Keywords: Pulse compression Stimulated Brillouin scattering Signal precessing Slow light
1. Introduction Stimulated Brillouin scattering (SBS) provides a method of slowing down the group velocity of optical pulses [1–4] by producing the large normal dispersion associated with a laser-induced amplifying resonance of a material system [1]. Fiber-based SBS slow light techniques have many applications in the fields of all-optical telecommunications and optical signal processing, such as controllable delay lines [3], data synchronization, optical buffering [5], optical data storage [6], and all-optical switches [7]. However in SBS slow light system, pulse delay is always accompanied by a pulse broadening [1,8]. In a telecommunication system, the maximum broadening factor has to be less than 2 in order to reduce the bit error rate [9]. Recent experiment has demonstrated the possibility to achieve a zero-broadening pulse delay and even pulse compression up to 0.8 with a large pulse distortion by using a superposition of broad gain lines [9]. Zero-broadening effect has also been achieved by use of a single Brillouin gain line [8] and a cascaded SBS slow light system [10]. Recently, Qin et al. demonstrated a preliminary experiment of pulse compression via doublet Brillouin gain lines in an optical fiber [11]. This work achieved a pulse compression ratio of 0.43 for 40 ns square pulse.
However a limit of tunable pulse broadening cancellation via doublet gain lines, a way to reduce the pulse distortion and so on remain unclear. In this paper, the broadening cancellation limit of 200 ps pulse and pulse distortion based on doublet Brillouin gain lines in a silica fiber are theoretically investigated. Low broadening factor with minimum pulse distortion and large time delay was achieved by optimizing the frequency separation between two gain lines and an appropriate pump power.
2. Numerical method Pulse compression via doublet Brillouin gain lines is considered by solving the coupled pulse-propagation equations in optical fibers for the pump light, the signal light and the acoustic waves under the slowly varying envelope approximation [12]:
−
− n
corresponding author. E-mail address: [email protected] (K. Nagasaka).
http://dx.doi.org/10.1016/j.optcom.2015.04.011 0030-4018/& 2015 Elsevier B.V. All rights reserved.
∂Ap1 ∂z ∂Ap2 ∂z
+
+
⎛ α 1 ∂Ap1 = − Ap1 + iγ ⎜⎜ Ap1 vg ∂t 2 ⎝ ⎛ α 1 ∂Ap2 = − Ap2 + iγ ⎜⎜ Ap2 vg ∂t 2 ⎝
2
2
⎞ + 2 As 2 ⎟⎟ Ap1 + ig2 As Q 1 ⎠
(1)
⎞ + 2 As 2 ⎟⎟ Ap2 + ig2 As Q 2 ⎠
(2)
36
K. Nagasaka et al. / Optics Communications 351 (2015) 35–39
⎛ ∂As 1 ∂As α + = − As + iγ ⎜⎜ As vg ∂t 2 ∂z ⎝
2
+ 2 Ap1
⎞ ⎛ + ig2 ⎜⎜Ap1 Q 1⁎ + Ap2 Q 2⁎ ⎟⎟ ⎠ ⎝
2
of light; nf is the modal refractive index of the fiber mode; and is the material density. Eqs. (1)–(5) were solved numerically.
⎞ 2 + 2 Ap1 ⎟⎟ As ⎠ (3)
⎞ g1 ∂Q 1 ⎛ Γ1 + ⎜ − iΔω1⎟ Q 1 = − i Ap1 A s⁎ ∂t η ⎝2 ⎠
(4)
⎞ g1 ∂Q 2 ⎛ Γ2 + ⎜ − iΔω2 ⎟ Q 2 = − i Ap2 A s⁎ , ∂t η ⎝2 ⎠
(5)
where Ap1, Ap2, AS, and Q 1,2 are the field amplitudes of the pump wave, the signal wave, and the acoustic wave, respectively; vg is the group velocity of the signal inside the fiber; α is the loss coefficient of the fiber; γ is the nonlinear coefficient; Γ1,2 is the acoustic damping rate; Δω1,2 is the detuning from the center angular frequency of the SBS loss or gain; g1 = γe ε0 ωp1, p2 /4va ;
g2 = γe ωp1, p2 /(4 cnf ρ0 ); η = (1/2) c 2ε0 Aeff ; γe is the electrostriction coefficient of the fiber; ΩB is the Brillouin frequency shift of the stokes wave; ε0 is the vacuum permittivity; va is the speed of the acoustic wave; ωp1, p2 is the frequency of the pumps; c is the speed
ρ0
3. Numerical results and discussion We assume that experimental setup was like Ref. [11]. Numerical simulations were performed on the assumption that two same power widely tunable single frequency laser and dithering pump by Brillouin comb. Creating doublet gain lines with a frequency separation of 2δω . Input signal pulse was a Gaussian pulse with a temporal width of 200 ps and 0.1 μW peak power. The center of SBS doublet gain lines was 1550 nm. Optical fiber we used has a length of L ¼4 km SMF-28 (group velocity dispersion (GVD) parameter β2 = − 20 ps2 /km , third order dispersion (TOD) parameter β3 = 0.1 ps3/km , and γ = 2.43 W −1 km−1 at 1550 nm). The dispersion and nonlinear effects of pulse propagation in fibers are often scaled by the GVD length LD ≡ T02/|β2 |, the TOD length
LD ≡ T03/|β3 | and nonlinear length LNL ≡ 1/(γP0 ) [12], where T0 and P0 are the initial signal pulse duration and peak power. The GVD length LD ( ≅ 2000 km) , the TOD length LD ( ≅ 8 × 107 km), and the
nonlinear
length
LNL ( ≅ 4.1 × 106 km)
are
very
large
Fig. 1. (a) Dependence of the normalized temporal output pulse shape (solid line) and normalized input pulse shape (dashed line), (b) variation of normalized gain (solid line) and input pulse spectra (dashed line) with normalized frequency, and (c) variation of normalized time delay (solid line) and input pulse spectra (dashed line) with normalized frequency on factor d (related to the frequency separation between two gain lines) in the conditions of Γ equaling to 3 GHz and peak gain equaling to 50 dB.
K. Nagasaka et al. / Optics Communications 351 (2015) 35–39
37
Fig. 2. Variation of (a) time delay, (b) broadening factor, and (c) normalized sub-peak power with factor d in the conditions of Γ equaling to 3 GHz and peak gain equaling to 50 dB.
compared with the fiber length L¼ 4 km. Thus the signal pulse employed, effects of the dispersion and nonlinearity on pulse evolution arising from the fiber could be neglected. Therefore, the pulse broadening effect can be related to the gain induced dispersion associated with SBS resonances. For the doublet gain lines, a complex SBS gain function is given by
g (ω) =
ig0 Γ ω − (ω 0 − δω) + iΓ
+
ig0 Γ ω − (ω 0 + δω) + iΓ
,
(6)
where ω0 is the center angular frequency of Brillouin gain, g0 is the peak gain coefficient, 2δω is the frequency separation between two gain lines, and Γ is the bandwidth of the Brillouin resonance. According to Refs. [10,13], the time delay is given by
δt =
2g 0 ⎛ 1 − (Ω + d) 2 1 − (Ω − d) 2 ⎞ ⎜ ⎟, + Γ ⎝ 1 + (Ω + d) 2 1 + (Ω − d) 2 ⎠
(7)
Fig. 4. Power dependence of output pulse with increasing peak gain from 10 dB to 50 dB with a step of 5 dB when Γ = 3 GHz , and d = 1.0 .
where Ω ≡ 2(ω − ω0 )/Γ is the normalized frequency and d ≡ 2δω/Γ is the relative frequency separation factor. Fig. 1 shows (a) the temporal profiles of output pulses (output pulses is normalized for comparison), (b) variation of normalized gain with normalized frequency, and (c) variation of normalized time delay with normalized frequency at different relative frequency separations (factor d) in the condition that Γ equals to 3 GHz. Fig. 2 shows (a) variation of time delay, (b) broadening factor (FWHM ratio of output pulse to initial incident pulse) and (c) normalized sub-peak power (power ratio of sub-peak to main peak of output pulse) with factor d. The maximum peak gain of the signal pulse was set as 50 dB in the simulations, in order to avoid gain saturation effect [1].
When d = 0, a broadened output pulse with a FWHM was 505 ps and broadening factor was 2.5 due to narrow Brillouin gain bandwidth acts like a filter and the time delay in the line center of the pulse is higher than at the FWHM-bandwidth [10]. By increasing the factor d from 0.0 to 1.2, the broadening factor of output pulse gradually decreases due to the center gain decreases and the gain bandwidth extended with the fixed peak gain. When the center gain decreases, the time delay in the center of the pulse is smaller than that at the FWHM-bandwidth and it cancels the pulse broadening effect. With further increasing the factor d larger than 1.3, there will be strong pulse distortion effects (when d ¼1.5)
Fig. 3. Eye diagrams with (a) d ¼ 0.8 and (b) d ¼1.0. Dashed and solid lines show the input and output pulses, respectively. (a) d ¼0.8. (b) d ¼1.0.
38
K. Nagasaka et al. / Optics Communications 351 (2015) 35–39
Fig. 5. Power dependence of (a) time delay, (b) broadening factor, and (c) sub-peak power with increasing peak gain from 10 dB to 50 dB with a step of 5 dB when Γ = 3 GHz , and d = 1.0 .
Fig. 6. Dependence of the normalized temporal output pulse shape on different bandwidths of single gain line when factor d ¼1.0 and peak gain equaling to 50 dB.
due to strong spectral modulation of the probe signal by the doublet gain lines and strong phase mismatching of the probe signal. According to Fig. 2(b) and (c), it can be seen that a large pulse broadening cancellation accompanies a reduced time delay. Reduced time delay with increasing d can be attributed to the increase of the gain bandwidth and decrease of the center of the time delay. There were sub-peaks near the trailing edge at the factor d = 1.0. When the factor d = 1.0, the normalized sub-peak power equals to 0.2. The normalized sub-peak power should be less than 0.2 for telecommunication system [12].
The pulse-distortion effects can be checked with eye diagram. Signal quality is qualified in terms of eye opening (EO). The EO is defined by the maximum difference between the minimum value of high level and the maximum value of the low level in the eye diagram [14–16]. Fig. 3 shows the calculated output eye diagram at different relative frequency separations (parameter d) in the condition that Γ equals to 3 GHz, where the power of amplified output pulses was normalized for comparison. When the parameter d was less than 0.8, the output pulses were overlapped and the EO was small for real applications due to the existence of pulse broadening effects. However, when the parameter d = 1.0, the power ratio of sub-peaks was relatively small and EO was larger than red lines (the input pulses). This case was suitable for telecommunication. As stated above, an optimal pulse broadening cancellation was obtained by properly choosing the frequency separation between two gain lines. Therefore minimum broadening factor of 0.96 with time delay of 134.5 ps and large EO is achieved at factor d ¼1.0. Fig. 4 shows the power dependence of output temporal pulse at different peak gains from 10 dB to 50 dB. According to Fig. 5 with the increase of the peak gain, time delay, broadening factor and normalized sub-peak power were increasing. The reason for the larger normalized sub-peak power is that the spectral modulation of the probe signal by the doublet gain lines is enlarged. Fig. 6 shows output pulses at Γ equal to 2.5, 3.0, and 3.5 GHz under a 50 dB peak gain when the factor d ¼ 1.0. There was the lowest distortion near the trailing edge of the pulse at the bandwidth of 3.5 GHz. This can be attributed to the match between overall gain profile of 3.5 GHz single-gain bandwidth and the spectrum shape of 200 ps input pulse. Furthermore, the time delay also reduces with the increase of single-gain bandwidth due to the small time delay region near the center of the pulse which became
Fig. 7. Eye diagrams with varied Γ . Dashed and solid lines show the input and output pulses, respectively. (a) Γ = 2.5 GHz . (b) Γ = 3.5 GHz .
K. Nagasaka et al. / Optics Communications 351 (2015) 35–39
39
Fig. 8. Normalized time delay, when Γ equal to (a) 2.5 GHz, (b) 3.0 GHz, (c) 3.5 GHz when the factor d ¼1.0. The dashed lines show the normalized input signal pulse spectra.
larger when Γ is larger (Fig. 8). When Γ was equal to 3.5 GHz, the EO is lager than the others (Fig. 7) due to small broadening factor of 0.86 and relative low sub-peak power.
4. Conclusion Minimum broadening factor of 0.96 with time delay of 134.5 ps is achieved by optimizing doublet Brillouin gain lines for 200 ps incident Gaussian pulse with pulse distortion in silica fiber. This can be explained by considering the variation of time delay with frequency. When the frequency separation of doublet Brillouin gain lines is large, the pulse will be distorted due to strong spectral modulation of the probe signal. When we enlarge Γ to avoid the pulse distortion effect under doublet gain lines, output pulse time delay will be smaller due to time delay near the center frequency of the probe pulse which becomes smaller under the same relative frequency separation.
Acknowledgment The authors gratefully acknowledge support from MEXT, the Support Program for Forming Strategic Research Infrastructure (2011–2015), the NSFC (grants 51072065, 61178073, 60908031, 60908001, 61077033 and 61378004), the Program for NCET in University (No: NCET-08-0243), the Opened Fund of the State Key Laboratory on Integrated Optoelectronics, and Tsinghua National Laboratory for Information Science and TechnologyTNListCrossdiscipline Foundation.
References [1] Z. Zhu, D.J. Gauthier, Y. Okawachi, J.E. Sharping, A.L. Gaeta, R.W. Boyd, A. E. Willner, Numerical study of all-optical slow-light delays via stimulated Brillouin scattering in an optical fiber, J. Opt. Soc. Am. B 22 (11) (2005) 2378–2384. http://dx.doi.org/10.1364/JOSAB.22.002378, URL 〈http://josab.osa. org/abstract.cfm?URI ¼josab-22-11-2378〉. [2] K.Y. Song, M.G. Herráez, L. Thévenaz, Observation of pulse delaying and advancement in optical fibers using stimulated Brillouin scattering, Opt. Express 13 (1) (2005) 82–88. http://dx.doi.org/10.1364/OPEX.13.000082, URL 〈http:// www.opticsexpress.org/abstract.cfm?URI ¼ oe-13-1-82〉. [3] Y. Okawachi, M.S. Bigelow, J.E. Sharping, Z. Zhu, A. Schweinsberg, D.J. Gauthier, R.W. Boyd, A.L. Gaeta, Tunable all-optical delays via brillouin slow light in an optical fiber, Phys. Rev. Lett. 94 (15) (2005) 153902. http://dx.doi.org/10.1103/ PhysRevLett.94.153902, URL 〈http://link.aps.org/doi/10.1103/PhysRevLett.94.
153902〉. [4] M. Gonzalez-Herraez, K.Y. Song, L. Thevenaz, Optically controlled slow and fast light in optical fibers using stimulated brillouin scattering, Appl. Phys. Lett. 87 (8) (2005) 081113, doi: http://dx.doi.org/10.1063/1.2033147. URL 〈http://scita tion.aip.org/content/aip/journal/apl/87/8/10.1063/1.2033147〉. [5] R.W. Boyd, D.J. Gauthier, Controlling the velocity of light pulses, Science 326 (5956) (2009) 1074–1077, arXiv: http://www.sciencemag.org/content/326/ 5956/1074.full.pdf. URL 〈http://www.sciencemag.org/content/326/5956/1074. abstract〉. [6] Z. Zhu, D.J. Gauthier, R.W. Boyd, Stored light in an optical fiber via stimulated brillouin scattering, Science 318 (5857) (2007) 1748–1750, arXiv: http://www. sciencemag.org/content/318/5857/1748.full.pdf. URL 〈http://www.sciencemag. org/content/318/5857/1748.abstract〉. [7] H. Yang, S.B. Yoo, All-optical variable buffering strategies and switch fabric architectures for future all-optical data routers, J. Lightwave Technol. 23 (10) (2005) 3321–3330, URL 〈http://jlt.osa.org/abstract.cfm?URI ¼ jlt-23-10-3321〉. [8] A. Wiatrek, R. Henker, S. Preußler, M.J. Ammann, A.T. Schwarzbacher, T. Schneider, Zero-broadening measurement in brillouin based slow-light delays, Opt. Express 17 (2) (2009) 797–802. http://dx.doi.org/10.1364/ OE.17.000797, URL 〈http://www.opticsexpress.org/abstract.cfm?URI ¼ oe-17-2797〉. [9] T. Schneider, A. Wiatrek, R. Henker, Zero-broadening and pulse compression slow light in an optical fiber at high pulse delays, Opt. Express 16 (20) (2008) 15617–15622. http://dx.doi.org/10.1364/OE.16.015617, URL 〈http://www.optic sexpress.org/abstract.cfm?URI ¼ oe-16-20-15617〉. [10] A. Wiatrek, R. Henker, S. Preußler, T. Schneider, Pulse broadening cancellation in cascaded slow-light delays, Opt. Express 17 (9) (2009) 7586–7591. http: //dx.doi.org/10.1364/OE.17.007586, URL 〈http://www.opticsexpress.org/ab stract.cfm?URI¼ oe-17-9-7586〉. [11] G. Qin, T. Sakamoto, N. Yamamoto, T. Kawanishi, H. Sotobayashi, T. Suzuki, Y. Ohishi, Tunable all-optical pulse compression and stretching via doublet brillouin gain lines in an optical fiber, Opt. Lett. 34 (8) (2009) 1192–1194. http: //dx.doi.org/ 10.1364/OL.34.001192, URL 〈http://ol.osa.org/abstract.cfm?URI ¼ol-34-8-1192〉. [12] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego, California, USA, 2007. [13] Y. Zhu, J.A. Greenberg, N.A. Husein, D.J. Gauthier, Giant all-optical tunable group velocity dispersion in an optical fiber, Opt. Express 22 (12) (2014) 14382–14391. http://dx.doi.org/10.1364/OE.22.014382, URL 〈http://www.op ticsexpress.org/abstract.cfm?URI ¼ oe-22-12-14382〉. [14] M. Lee, Y. Zhu, D.J. Gauthier, M.E. Gehm, M.A. Neifeld, Information-theoretic analysis of a stimulated-brillouin-scattering-based slow-light system, Appl. Opt. 50 (32) (2011) 6063–6072. http://dx.doi.org/10.1364/AO.50.006063, URL 〈http://ao.osa.org/abstract.cfm?URI ¼ ao-50-32-6063〉. [15] R. Pant, M.D. Stenner, M.A. Neifeld, D.J. Gauthier, Optimal pump profile designs for broadband SBS slow-light systems, Opt. Express 16 (4) (2008) 2764–2777. http://dx.doi.org/10.1364/OE.16.002764, URL 〈http://www.optic sexpress.org/abstract.cfm?URI ¼oe-16-4-2764〉. [16] R. Pant, M.D. Stenner, M.A. Neifeld, Z. Shi, R.W. Boyd, D.J. Gauthier, Maximizing the opening of eye diagrams for slow-light systems, Appl. Opt. 46 (26) (2007) 6513–6519. http://dx.doi.org/10.1364/AO.46.006513, URL 〈http://ao.osa.org/ab stract.cfm?URI¼ ao-46-26-6513〉.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Theoretical investigation of tunable pulse broadening cancellation via doublet Brillouin gain lines in an optical fiber Kenshiro Nagasaka a,n, Qijin Tian b, Lai Liu b, Dan Zhao b, Guanshi Qin b, Weiping Qin b, Takenobu Suzuki a, Yasutake Ohishi a a b
Research Center for Advanced Photon Technology, Toyota Technological Institute, 2-12-1 Hisakata, Tenpaku, Nagoya 468-8511, Japan State Key Laboratory on Integrated Optoelectronics, College of Electronic Science and Engineering, Jilin University, Changchun 130012, China
art ic l e i nf o
a b s t r a c t
Article history: Received 5 November 2014 Received in revised form 1 April 2015 Accepted 4 April 2015 Available online 7 April 2015
Tunable pulse broadening cancellation via doublet Brillouin gain lines in a silica fiber was theoretically investigated. Minimum broadening factor of 0.96 with time delay of 134.5 ps is achieved by optimizing doublet Brillouin lines for 200 ps Gaussian pulse with pulse distortion. Frequency separation between two gain lines and spectrum bandwidth of single gain line and peak gain values are optimized and considered in the present paper. & 2015 Elsevier B.V. All rights reserved.
Keywords: Pulse compression Stimulated Brillouin scattering Signal precessing Slow light
1. Introduction Stimulated Brillouin scattering (SBS) provides a method of slowing down the group velocity of optical pulses [1–4] by producing the large normal dispersion associated with a laser-induced amplifying resonance of a material system [1]. Fiber-based SBS slow light techniques have many applications in the fields of all-optical telecommunications and optical signal processing, such as controllable delay lines [3], data synchronization, optical buffering [5], optical data storage [6], and all-optical switches [7]. However in SBS slow light system, pulse delay is always accompanied by a pulse broadening [1,8]. In a telecommunication system, the maximum broadening factor has to be less than 2 in order to reduce the bit error rate [9]. Recent experiment has demonstrated the possibility to achieve a zero-broadening pulse delay and even pulse compression up to 0.8 with a large pulse distortion by using a superposition of broad gain lines [9]. Zero-broadening effect has also been achieved by use of a single Brillouin gain line [8] and a cascaded SBS slow light system [10]. Recently, Qin et al. demonstrated a preliminary experiment of pulse compression via doublet Brillouin gain lines in an optical fiber [11]. This work achieved a pulse compression ratio of 0.43 for 40 ns square pulse.
However a limit of tunable pulse broadening cancellation via doublet gain lines, a way to reduce the pulse distortion and so on remain unclear. In this paper, the broadening cancellation limit of 200 ps pulse and pulse distortion based on doublet Brillouin gain lines in a silica fiber are theoretically investigated. Low broadening factor with minimum pulse distortion and large time delay was achieved by optimizing the frequency separation between two gain lines and an appropriate pump power.
2. Numerical method Pulse compression via doublet Brillouin gain lines is considered by solving the coupled pulse-propagation equations in optical fibers for the pump light, the signal light and the acoustic waves under the slowly varying envelope approximation [12]:
−
− n
corresponding author. E-mail address: [email protected] (K. Nagasaka).
http://dx.doi.org/10.1016/j.optcom.2015.04.011 0030-4018/& 2015 Elsevier B.V. All rights reserved.
∂Ap1 ∂z ∂Ap2 ∂z
+
+
⎛ α 1 ∂Ap1 = − Ap1 + iγ ⎜⎜ Ap1 vg ∂t 2 ⎝ ⎛ α 1 ∂Ap2 = − Ap2 + iγ ⎜⎜ Ap2 vg ∂t 2 ⎝
2
2
⎞ + 2 As 2 ⎟⎟ Ap1 + ig2 As Q 1 ⎠
(1)
⎞ + 2 As 2 ⎟⎟ Ap2 + ig2 As Q 2 ⎠
(2)
36
K. Nagasaka et al. / Optics Communications 351 (2015) 35–39
⎛ ∂As 1 ∂As α + = − As + iγ ⎜⎜ As vg ∂t 2 ∂z ⎝
2
+ 2 Ap1
⎞ ⎛ + ig2 ⎜⎜Ap1 Q 1⁎ + Ap2 Q 2⁎ ⎟⎟ ⎠ ⎝
2
of light; nf is the modal refractive index of the fiber mode; and is the material density. Eqs. (1)–(5) were solved numerically.
⎞ 2 + 2 Ap1 ⎟⎟ As ⎠ (3)
⎞ g1 ∂Q 1 ⎛ Γ1 + ⎜ − iΔω1⎟ Q 1 = − i Ap1 A s⁎ ∂t η ⎝2 ⎠
(4)
⎞ g1 ∂Q 2 ⎛ Γ2 + ⎜ − iΔω2 ⎟ Q 2 = − i Ap2 A s⁎ , ∂t η ⎝2 ⎠
(5)
where Ap1, Ap2, AS, and Q 1,2 are the field amplitudes of the pump wave, the signal wave, and the acoustic wave, respectively; vg is the group velocity of the signal inside the fiber; α is the loss coefficient of the fiber; γ is the nonlinear coefficient; Γ1,2 is the acoustic damping rate; Δω1,2 is the detuning from the center angular frequency of the SBS loss or gain; g1 = γe ε0 ωp1, p2 /4va ;
g2 = γe ωp1, p2 /(4 cnf ρ0 ); η = (1/2) c 2ε0 Aeff ; γe is the electrostriction coefficient of the fiber; ΩB is the Brillouin frequency shift of the stokes wave; ε0 is the vacuum permittivity; va is the speed of the acoustic wave; ωp1, p2 is the frequency of the pumps; c is the speed
ρ0
3. Numerical results and discussion We assume that experimental setup was like Ref. [11]. Numerical simulations were performed on the assumption that two same power widely tunable single frequency laser and dithering pump by Brillouin comb. Creating doublet gain lines with a frequency separation of 2δω . Input signal pulse was a Gaussian pulse with a temporal width of 200 ps and 0.1 μW peak power. The center of SBS doublet gain lines was 1550 nm. Optical fiber we used has a length of L ¼4 km SMF-28 (group velocity dispersion (GVD) parameter β2 = − 20 ps2 /km , third order dispersion (TOD) parameter β3 = 0.1 ps3/km , and γ = 2.43 W −1 km−1 at 1550 nm). The dispersion and nonlinear effects of pulse propagation in fibers are often scaled by the GVD length LD ≡ T02/|β2 |, the TOD length
LD ≡ T03/|β3 | and nonlinear length LNL ≡ 1/(γP0 ) [12], where T0 and P0 are the initial signal pulse duration and peak power. The GVD length LD ( ≅ 2000 km) , the TOD length LD ( ≅ 8 × 107 km), and the
nonlinear
length
LNL ( ≅ 4.1 × 106 km)
are
very
large
Fig. 1. (a) Dependence of the normalized temporal output pulse shape (solid line) and normalized input pulse shape (dashed line), (b) variation of normalized gain (solid line) and input pulse spectra (dashed line) with normalized frequency, and (c) variation of normalized time delay (solid line) and input pulse spectra (dashed line) with normalized frequency on factor d (related to the frequency separation between two gain lines) in the conditions of Γ equaling to 3 GHz and peak gain equaling to 50 dB.
K. Nagasaka et al. / Optics Communications 351 (2015) 35–39
37
Fig. 2. Variation of (a) time delay, (b) broadening factor, and (c) normalized sub-peak power with factor d in the conditions of Γ equaling to 3 GHz and peak gain equaling to 50 dB.
compared with the fiber length L¼ 4 km. Thus the signal pulse employed, effects of the dispersion and nonlinearity on pulse evolution arising from the fiber could be neglected. Therefore, the pulse broadening effect can be related to the gain induced dispersion associated with SBS resonances. For the doublet gain lines, a complex SBS gain function is given by
g (ω) =
ig0 Γ ω − (ω 0 − δω) + iΓ
+
ig0 Γ ω − (ω 0 + δω) + iΓ
,
(6)
where ω0 is the center angular frequency of Brillouin gain, g0 is the peak gain coefficient, 2δω is the frequency separation between two gain lines, and Γ is the bandwidth of the Brillouin resonance. According to Refs. [10,13], the time delay is given by
δt =
2g 0 ⎛ 1 − (Ω + d) 2 1 − (Ω − d) 2 ⎞ ⎜ ⎟, + Γ ⎝ 1 + (Ω + d) 2 1 + (Ω − d) 2 ⎠
(7)
Fig. 4. Power dependence of output pulse with increasing peak gain from 10 dB to 50 dB with a step of 5 dB when Γ = 3 GHz , and d = 1.0 .
where Ω ≡ 2(ω − ω0 )/Γ is the normalized frequency and d ≡ 2δω/Γ is the relative frequency separation factor. Fig. 1 shows (a) the temporal profiles of output pulses (output pulses is normalized for comparison), (b) variation of normalized gain with normalized frequency, and (c) variation of normalized time delay with normalized frequency at different relative frequency separations (factor d) in the condition that Γ equals to 3 GHz. Fig. 2 shows (a) variation of time delay, (b) broadening factor (FWHM ratio of output pulse to initial incident pulse) and (c) normalized sub-peak power (power ratio of sub-peak to main peak of output pulse) with factor d. The maximum peak gain of the signal pulse was set as 50 dB in the simulations, in order to avoid gain saturation effect [1].
When d = 0, a broadened output pulse with a FWHM was 505 ps and broadening factor was 2.5 due to narrow Brillouin gain bandwidth acts like a filter and the time delay in the line center of the pulse is higher than at the FWHM-bandwidth [10]. By increasing the factor d from 0.0 to 1.2, the broadening factor of output pulse gradually decreases due to the center gain decreases and the gain bandwidth extended with the fixed peak gain. When the center gain decreases, the time delay in the center of the pulse is smaller than that at the FWHM-bandwidth and it cancels the pulse broadening effect. With further increasing the factor d larger than 1.3, there will be strong pulse distortion effects (when d ¼1.5)
Fig. 3. Eye diagrams with (a) d ¼ 0.8 and (b) d ¼1.0. Dashed and solid lines show the input and output pulses, respectively. (a) d ¼0.8. (b) d ¼1.0.
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Fig. 5. Power dependence of (a) time delay, (b) broadening factor, and (c) sub-peak power with increasing peak gain from 10 dB to 50 dB with a step of 5 dB when Γ = 3 GHz , and d = 1.0 .
Fig. 6. Dependence of the normalized temporal output pulse shape on different bandwidths of single gain line when factor d ¼1.0 and peak gain equaling to 50 dB.
due to strong spectral modulation of the probe signal by the doublet gain lines and strong phase mismatching of the probe signal. According to Fig. 2(b) and (c), it can be seen that a large pulse broadening cancellation accompanies a reduced time delay. Reduced time delay with increasing d can be attributed to the increase of the gain bandwidth and decrease of the center of the time delay. There were sub-peaks near the trailing edge at the factor d = 1.0. When the factor d = 1.0, the normalized sub-peak power equals to 0.2. The normalized sub-peak power should be less than 0.2 for telecommunication system [12].
The pulse-distortion effects can be checked with eye diagram. Signal quality is qualified in terms of eye opening (EO). The EO is defined by the maximum difference between the minimum value of high level and the maximum value of the low level in the eye diagram [14–16]. Fig. 3 shows the calculated output eye diagram at different relative frequency separations (parameter d) in the condition that Γ equals to 3 GHz, where the power of amplified output pulses was normalized for comparison. When the parameter d was less than 0.8, the output pulses were overlapped and the EO was small for real applications due to the existence of pulse broadening effects. However, when the parameter d = 1.0, the power ratio of sub-peaks was relatively small and EO was larger than red lines (the input pulses). This case was suitable for telecommunication. As stated above, an optimal pulse broadening cancellation was obtained by properly choosing the frequency separation between two gain lines. Therefore minimum broadening factor of 0.96 with time delay of 134.5 ps and large EO is achieved at factor d ¼1.0. Fig. 4 shows the power dependence of output temporal pulse at different peak gains from 10 dB to 50 dB. According to Fig. 5 with the increase of the peak gain, time delay, broadening factor and normalized sub-peak power were increasing. The reason for the larger normalized sub-peak power is that the spectral modulation of the probe signal by the doublet gain lines is enlarged. Fig. 6 shows output pulses at Γ equal to 2.5, 3.0, and 3.5 GHz under a 50 dB peak gain when the factor d ¼ 1.0. There was the lowest distortion near the trailing edge of the pulse at the bandwidth of 3.5 GHz. This can be attributed to the match between overall gain profile of 3.5 GHz single-gain bandwidth and the spectrum shape of 200 ps input pulse. Furthermore, the time delay also reduces with the increase of single-gain bandwidth due to the small time delay region near the center of the pulse which became
Fig. 7. Eye diagrams with varied Γ . Dashed and solid lines show the input and output pulses, respectively. (a) Γ = 2.5 GHz . (b) Γ = 3.5 GHz .
K. Nagasaka et al. / Optics Communications 351 (2015) 35–39
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Fig. 8. Normalized time delay, when Γ equal to (a) 2.5 GHz, (b) 3.0 GHz, (c) 3.5 GHz when the factor d ¼1.0. The dashed lines show the normalized input signal pulse spectra.
larger when Γ is larger (Fig. 8). When Γ was equal to 3.5 GHz, the EO is lager than the others (Fig. 7) due to small broadening factor of 0.86 and relative low sub-peak power.
4. Conclusion Minimum broadening factor of 0.96 with time delay of 134.5 ps is achieved by optimizing doublet Brillouin gain lines for 200 ps incident Gaussian pulse with pulse distortion in silica fiber. This can be explained by considering the variation of time delay with frequency. When the frequency separation of doublet Brillouin gain lines is large, the pulse will be distorted due to strong spectral modulation of the probe signal. When we enlarge Γ to avoid the pulse distortion effect under doublet gain lines, output pulse time delay will be smaller due to time delay near the center frequency of the probe pulse which becomes smaller under the same relative frequency separation.
Acknowledgment The authors gratefully acknowledge support from MEXT, the Support Program for Forming Strategic Research Infrastructure (2011–2015), the NSFC (grants 51072065, 61178073, 60908031, 60908001, 61077033 and 61378004), the Program for NCET in University (No: NCET-08-0243), the Opened Fund of the State Key Laboratory on Integrated Optoelectronics, and Tsinghua National Laboratory for Information Science and TechnologyTNListCrossdiscipline Foundation.
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