Self-pulsing and dynamic bistability in cw-pumped Brillouin fiber ring lasers

Carlos Montes; Derradji Bahloul; Isabelle Bongrand; Jean Botineau; Gérard Cheval; Abdellatif Mamhoud; Eric Picholle; Antonio Picozzi

doi:10.1364/JOSAB.16.000932

Journal of the Optical Society of America B
Vol. 16,
Issue 6,
pp. 932-951
(1999)
•https://doi.org/10.1364/JOSAB.16.000932

Self-pulsing and dynamic bistability in cw-pumped Brillouin fiber ring lasers

Carlos Montes, Derradji Bahloul, Isabelle Bongrand, Jean Botineau, Gérard Cheval, Abdellatif Mamhoud, Eric Picholle, and Antonio Picozzi

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Carlos Montes, Derradji Bahloul, Isabelle Bongrand, Jean Botineau, Gérard Cheval, Abdellatif Mamhoud, Eric Picholle, and Antonio Picozzi, "Self-pulsing and dynamic bistability in cw-pumped Brillouin fiber ring lasers," J. Opt. Soc. Am. B 16, 932-951 (1999)

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Abstract

The nonlinear dynamics of cw-pumped Brillouin long-fiber ring lasers that contain a large number of longitudinal modes N beneath the Brillouin gain curve is controlled by a single parameter, namely, the Stokes feedback R. Below $R_{crit},$ a stable train of dissipative solitonic pulses is spontaneously structured at the round-trip frequency $f_{r}$ without any additional intracavity mode locking. Experimental observations in cw-pumped fiber ring cavities, supported by numerical simulation in a coherent space–time three-wave model that includes the optical Kerr effect, prove the universality of the self-pulsing mechanism. Stability analysis shows that below $R_{crit}$ the steady Brillouin mirror regime is destabilized through a Hopf bifurcation. For $R < R_{crit} < R_{0}$ the bifurcation is supercritical and exhibits an asymptotically monostable oscillatory regime at twice $f_{r}$ for high enough N or at $f_{r}$ for lower N, in a finite transition region. For $R_{0} < R < R_{crit},$ the bifurcation is subcritical and exhibits dynamic bistability between the steady and the pulsed regimes in a finite hysteresis region whose width is proportional to the Kerr parameter. For R small enough, the cavity longitudinal modes merge into a dissipative solitonic Brillouin pulse: the dynamic three-wave model yields self-structured asymptotically stable trains of pulses for any initial conditions, in fair quantitative agreement (for pulse width, intensity, shape, and period) with the experiments in the entire self-pulsing domain. Amplification of spontaneous emission breaks down the stable-pulse regime in long devices (i.e., high N), so the fiber noise amplitude is higher than the coherent amplitude that separates two consecutive pulses.

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$G = {gLI}_{p}$	4	8	12	16
P (mW)	$8.25 (\times σ^{2})$	$16.5 (\times σ^{2})$	$24.7 (\times σ^{2})$	$33.0 (\times σ^{2})$
$I_{p}$ (MW/cm²)	$0.115 (\times σ^{2})$	$0.230 (\times σ^{2})$	$0.343 (\times σ^{2})$	$0.460 (\times σ^{2})$
$F_{p}$ (MV/m)	$0.778 (\times σ)$	$1.106 (\times σ)$	$1.34 (\times σ)$	$1.55 (\times σ)$
$τ = ({KE}_{p})^{- 1}$ (ns)	19.5	13.8	11.2	9.75
$Ln / (c τ)$	19.7	28.0	34.2	39.5
$μ_{a} = γ_{a} τ$	9.89	7.00	5.71	4.94
$μ_{e} = γ_{e} τ$	$9.89 \times 10^{- 3}$	$7.00 \times 10^{- 3}$	$5.71 \times 10^{- 3}$	$4.94 \times 10^{- 3}$
$R_{crit}$	0.106	$2.31 \times 10^{- 2}$	$6.10 \times 10^{- 3}$	$1.60 \times 10^{- 4}$
$R_{thres}$	0.0183	$3.35 \times 10^{- 4}$	$6.14 \times 10^{- 6}$	$1.12 \times 10^{- 7}$
$K_{r} E_{p}$ $(m s^{- 1} V^{- 1})$	$0.123 (\times σ)$	$0.174 (\times σ)$	$0.213 (\times σ)$	$0.246 (\times σ)$
$κ = K_{r} E_{p} / K$	$1.86 \times 10^{- 3} (\times σ^{2})$	$2.64 \times 10^{- 3} (\times σ^{2})$	$3.23 \times 10^{- 3} (\times σ^{2})$	$3.73 \times 10^{- 3} (\times σ^{2})$
$δ t_{0} / t_{r}$	0.0340	0.0480	0.0589	0.0682

$G = {gLI}_{p}$	2	4	6	8
P (mW)	26.6	53.3	79.9	106
$I_{p}$ (MW/cm²)	0.0266	0.0533	0.0799	0.106
$E_{p}$ (MV/m)	0.374	0.529	0.648	0.746
$τ = ({KE}_{p})^{- 1}$ (ns)	131	93.1	76.0	66.0
$Ln / (c τ)$	9.12	12.8	15.8	18.2
$μ_{a} = γ_{a} τ$	9.09	6.45	5.27	4.55
$μ_{e} = γ_{e} τ$	$1.2 \times 10^{- 3}$	$8.7 \times 10^{- 4}$	$7.1 \times 10^{- 4}$	$6.1 \times 10^{- 4}$
$R_{crit}$	0.26	0.106	0.049	0.024
$R_{thres}$	0.13	0.018	0.0024	$3.3 \times 10^{- 4}$
$K_{r} E_{p}$ $(m s^{- 1} V^{- 1})$	0.0234	0.0331	0.0405	0.0466
$κ = K_{r} E_{p} / K$	$1.15 \times 10^{- 3}$	$1.63 \times 10^{- 3}$	$1.99 \times 10^{- 3}$	$2.29 \times 10^{- 3}$
$δ t_{0} / t_{r}$	0.0569	0.0800	0.0979	0.113

Mode	$G = {gI}_{p} L$	$N = 60$		$N = 30$
Mode	$G = {gI}_{p} L$	$ω / ω_{r}$	$R_{crit}$	$ω / ω_{r}$	$R_{crit}$
$M_{1}$	1	1.00387703	0.50483	1.00112969	0.48323
	2	1.01026114	0.31628	1.00729636	0.31383
	3	1.01652477	0.21242	1.01290068	0.21242
	4	1.02217886	0.14770	1.01790351	0.14951
	5	1.02700989	0.10454	1.02210368	0.10647
	6	1.03089994	0.07453	1.02537812	0.07630
	7	1.03379414	0.05314	1.02765962	0.05467
	8	1.03568776	0.03770	1.02892809	0.03895
	10	1.03666613	0.01842	1.02859032	0.01916
	12	1.03499657	0.00871	1.02538941	0.00902
	14	1.03295370	0.00426	1.02180343	0.00431
	16	1.03226963	0.00236	1.02008112	0.00233
$M_{2}$	3	2.00033952	0.17319
	4	2.00378791	0.13280
	5	2.00644718	0.09886
	6	2.00880769	0.07360
	7	2.01092437	0.05495
	8	2.01279984	0.04110
	10	2.01580128	0.02299
	11			1.95506352	0.00078
	12	2.01779406	0.01277	1.96510855	0.00138
	13			1.97144213	0.00704
	14	2.01889529	0.00704	1.97460167	0.00150
	15			1.97669027	0.00129
	16	2.01939901	0.00389	1.97836569	0.00107

Optical Kerr Constant	$ρ = \sqrt{R}$	$I_{B} / I_{p}^{0}$	$〈 I_{B} 〉 / I_{p}^{0}$	$Δ t / t_{r}$	$(δ t / t_{r})_{dyn}$
$K_{r} = 1.6 \times 10^{- 7} m s^{- 1} V^{- 2}$	0.110	6.156	0.3116	1.00461	0.04468
	0.120	6.965	0.3422	1.00357	0.04297
	0.130	7.653	0.3702	1.00257	0.04224
	0.140	8.173	0.3971	1.00152	0.04236
	0.150	8.503	0.4219	1.00033	0.04309
	0.160	8.571	0.4449	0.99909	0.04504
	0.170	8.272	0.4672	0.99733	0.04883
	0.175	7.941	0.4831	0.99614	0.05273
$K_{r} = 0$	0.120	7.380	0.3431	1.00385	0.04064
	0.140	8.782	0.3986	1.00195	0.03943
	0.150	9.281	0.4229	1.00085	0.04004
	0.160	9.319	0.4459	0.99971	0.04163
	0.170	9.113	0.4663	0.99814	0.04504

Optical Kerr Constant	$ρ = \sqrt{R}$	$I_{B} / I_{p}^{0}$	$〈 I_{B} 〉 / I_{p}^{0}$	$Δ t / t_{r}$	$(δ t / t_{r})_{dyn}$
$K_{r} = 6.2 \times 10^{- 8} m s^{- 1} V^{- 2}$	0.200	1.3082	0.2468	1.01188	0.16394
	0.220	2.3127	0.3166	1.01078	0.12036
	0.240	3.1754	0.3868	1.00864	0.10681
	0.260	4.1025	0.4540	1.00683	0.09655
	0.280	5.0448	0.51720	1.00520	0.08923
	0.300	5.9561	0.5794	1.00367	0.08422
	0.310	6.3821	0.6096	1.00292	0.08252
	0.320	6.7774	0.6382	1.00213	0.08142
	0.330	7.1315	0.6674	1.00139	0.08081
	0.340	7.4313	0.6975	1.00060	0.08093
	0.350	7.6590	0.7257	0.99976	0.08154
	0.360	7.7876	0.7532	0.99879	0.08325
	0.370	7.7704	0.7828	0.99772	0.08655
	0.380	7.5040	0.8104	0.99627	0.09253
	0.384	7.2683	0.8200	0.99548	0.09680
	0.386	7.1028	0.8259	0.99502	0.09961
	0.380	6.8852	0.8310	0.99446	0.10303
	0.390	6.5844	0.8359	0.99372	0.10815
	0.392	6.1102	0.8398	0.99257	0.11719
	0.394	5.1008	0.8439	0.99014	0.13818
$K_{r} = 1.5 \times 10^{- 7} m s^{- 1} V^{- 2}$	0.260	4.3290	0.4557	1.00711	0.09228
	0.270	4.6704	0.4877	1.00609	0.09057
	0.280	5.1497	0.5196	1.00530	0.08740
	0.290	5.7386	0.5507	1.00474	0.08388
	0.300	6.1797	0.5811	1.00399	0.08154
	0.310	6.5980	0.6109	1.00320	0.08056
	0.320	6.9878	0.6402	1.00246	0.07910
	0.330	7.3431	0.6700	1.00167	0.07861
	0.340	7.6562	0.6985	1.00088	0.07959
	0.350	7.9171	0.7277	1.00009	0.07910
	0.360	8.1116	0.7555	0.99925	0.08007
	0.370	8.2180	0.7852	0.99827	0.08203
	0.380	8.2123	0.8108	0.99725	0.08496
	0.390	7.9665	0.8394	0.99581	0.09033
	0.400	7.2834	0.8652	0.99376	0.10205
	0.402	6.9435	0.8702	0.99298	0.10645
	0.404	6.1256	0.8740	0.99121	0.12060

G	P (mW)	$I_{B} / I_{p}^{0}$	$〈 I_{B} 〉 / I_{p}^{0}$	$Δ t / t_{r}$	$(δ t / t_{r})_{dyn}$
2.8000	35.89	0.11238	0.1113	1.01190	0.50390
2.9689	38.05	0.61200	0.1928	1.00990	0.25280
3.1689	40.62	1.59681	0.2726	1.00967	0.14758
3.3689	43.18	2.43894	0.3461	1.00827	0.12377
3.5689	45.75	3.32165	0.4113	1.00715	0.10791
3.7689	48.31	4.20044	0.4675	1.00613	0.09704
3.9689	50.87	5.04545	0.5175	1.00520	0.08923
4.1689	53.44	5.83156	0.5621	1.00427	0.08361
4.3689	56.00	6.53568	0.6017	1.00330	0.07971
4.5689	58.56	7.13440	0.6368	1.00232	0.07714
4.7689	61.13	7.60208	0.6700	1.00120	0.07592
4.9689	63.70	7.90675	0.6981	0.99999	0.07604
5.1689	66.25	8.00177	0.7254	0.99846	0.07775
5.3689	68.82	7.79948	0.7474	0.99660	0.08227
5.5689	71.39	7.04869	0.7681	0.99353	0.09313
5.6000	71.78	6.83023	0.7715	0.99279	0.09619
5.7000	73.07	5.47826	0.7787	0.98842	0.12011
5.7689	73.95	3.40404	0.7781	0.98028	0.18383
5.8000	74.35	1.19745	0.7659	0.97098	0.38867

Abstract

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Journal of the Optical Society of America B