Carlos Montes,
Derradji Bahloul,
Isabelle Bongrand,
Jean Botineau,
Gérard Cheval,
Abdellatif Mamhoud,
Eric Picholle,
and Antonio Picozzi
Laboratoire de Physique de la Matière Condensée, Centre National de la Recherche Scientifique, Université de Nice—Sophia Antipolis, Parc Valrose, F-06108 Nice Cedex 2, France
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)
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 a stable train of dissipative solitonic pulses is spontaneously structured at the round-trip frequency 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 the steady Brillouin mirror regime is destabilized through a Hopf bifurcation. For the bifurcation is supercritical and exhibits an asymptotically monostable oscillatory regime at twice for high enough N or at for lower N, in a finite transition region. For 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.
Isabelle Bongrand, Carlos Montes, Eric Picholle, Jean Botineau, Antonio Picozzi, Gérard Cheval, and Derradji Bahloul Opt. Lett. 26(19) 1475-1477 (2001)
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Computation Parameters for the cw-Pumped Ar-Ion Brillouin Fiber Ring Lasera
4
8
12
16
P (mW)
(MW/cm2)
(MV/m)
(ns)
19.5
13.8
11.2
9.75
19.7
28.0
34.2
39.5
9.89
7.00
5.71
4.94
0.106
0.0183
0.0340
0.0480
0.0589
0.0682
The active medium is a polarization-maintaining single-mode fiber of length with a 3-µm-diameter core and an effective optical cross section The pump wavelength is (acoustic wavelength, ). The coherent SBS coupling constant of Eq. (1b) and the SBS gain coefficient of Eq. (5) are given by , and where stands for an acoustical antiguiding factor, reducing the SBS gain coefficient for polarized light through P is the launched cw-pump power; is the pump flux intensity; is the dimensionless SBS intensity gain–length; is the pump amplitude is the coherent SBS characteristic time; is the dimensionless acoustic damping rate is the dimensionless optical damping rate (); is the critical Stokes intensity feedback; is the laser threshold feedback; is the Kerr coefficient entering Eqs. (1); is the round-trip time; and is the dimensionless pulse width given by Eq. (21). The number of longitudinal modes beneath the gain curve is
Table 2
Computation Parameters for the cw-Pumped Nd:YAG Brillouin Fiber Ring Lasera
2
4
6
8
P (mW)
26.6
53.3
79.9
106
(MW/cm2)
0.0266
0.0533
0.0799
0.106
(MV/m)
0.374
0.529
0.648
0.746
(ns)
131
93.1
76.0
66.0
9.12
12.8
15.8
18.2
9.09
6.45
5.27
4.55
0.26
0.106
0.049
0.024
0.13
0.018
0.0024
0.0234
0.0331
0.0405
0.0466
0.0569
0.0800
0.0979
0.113
Same as Table 1, but with a single-mode non-polarization-maintaining optical fiber of length with an 11.3-µm-diameter Ge-doped core and an effective optical cross section The pump wavelength is The SBS coupling constants are given by where reduces the SBS coupling by accounting for a small measured depolarization (10%) and for a gain reduction that is due to the transverse material gradient, which experimentally measured yields and
Table 3
Solution of Dispersion Equation (10): Critical Feedback and Oscillation Frequency for Modes and Corresponding to and a
Mode
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
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
The Ar-ion experiment (cf. Table 1) corresponds to and the Nd:YAG experiment (cf. Table 2) corresponds to The feedback values determine the marginal stability shown in Fig. 3, either for mode of the period close to the round-trip time or for mode of the period close to As is shown in Fig. 3, for and for any G, decreasing R causes the steady mirror to bifurcate at through whereas for it bifurcates through for and through for The values for the marginal stability of are almost indistinguishable for but such is not the case for (dashed curves in Fig. 3).
Table 4
Pulse Characteristics for the Ar-Ion cw-Pumped Brillouin Fiber Ring Lasers from Eqs. (1)a
Optical Kerr Constant
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
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
Normalized pulse intensity mean reflected intensity period and width versus control parameter (Stokes feedback). The parameters for correspond to the second column of Table 1. The approximate expression (21) for the pulse width yields Bold face numbers indicate maximum peak intensity and minimum temporal width.
Table 5
Pulse Characteristics for the Nd:YAG cw-Pumped Brillouin Fiber Ring Laser from Eqs. (1)a
Optical Kerr Constant
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
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
Same as Table 4, but now the parameters for , correspond to the second column of Table 2.
Table 6
Pulse Characteristics for the Nd:YAG cw-Pumped Brillouin Fiber Ring Laser from Eqs. (1)a
G
P (mW)
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
Same as Table 5, but versus pump power P, i.e., versus gain as the control parameter for a fixed feedback .
Tables (6)
Table 1
Computation Parameters for the cw-Pumped Ar-Ion Brillouin Fiber Ring Lasera
4
8
12
16
P (mW)
(MW/cm2)
(MV/m)
(ns)
19.5
13.8
11.2
9.75
19.7
28.0
34.2
39.5
9.89
7.00
5.71
4.94
0.106
0.0183
0.0340
0.0480
0.0589
0.0682
The active medium is a polarization-maintaining single-mode fiber of length with a 3-µm-diameter core and an effective optical cross section The pump wavelength is (acoustic wavelength, ). The coherent SBS coupling constant of Eq. (1b) and the SBS gain coefficient of Eq. (5) are given by , and where stands for an acoustical antiguiding factor, reducing the SBS gain coefficient for polarized light through P is the launched cw-pump power; is the pump flux intensity; is the dimensionless SBS intensity gain–length; is the pump amplitude is the coherent SBS characteristic time; is the dimensionless acoustic damping rate is the dimensionless optical damping rate (); is the critical Stokes intensity feedback; is the laser threshold feedback; is the Kerr coefficient entering Eqs. (1); is the round-trip time; and is the dimensionless pulse width given by Eq. (21). The number of longitudinal modes beneath the gain curve is
Table 2
Computation Parameters for the cw-Pumped Nd:YAG Brillouin Fiber Ring Lasera
2
4
6
8
P (mW)
26.6
53.3
79.9
106
(MW/cm2)
0.0266
0.0533
0.0799
0.106
(MV/m)
0.374
0.529
0.648
0.746
(ns)
131
93.1
76.0
66.0
9.12
12.8
15.8
18.2
9.09
6.45
5.27
4.55
0.26
0.106
0.049
0.024
0.13
0.018
0.0024
0.0234
0.0331
0.0405
0.0466
0.0569
0.0800
0.0979
0.113
Same as Table 1, but with a single-mode non-polarization-maintaining optical fiber of length with an 11.3-µm-diameter Ge-doped core and an effective optical cross section The pump wavelength is The SBS coupling constants are given by where reduces the SBS coupling by accounting for a small measured depolarization (10%) and for a gain reduction that is due to the transverse material gradient, which experimentally measured yields and
Table 3
Solution of Dispersion Equation (10): Critical Feedback and Oscillation Frequency for Modes and Corresponding to and a
Mode
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
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
The Ar-ion experiment (cf. Table 1) corresponds to and the Nd:YAG experiment (cf. Table 2) corresponds to The feedback values determine the marginal stability shown in Fig. 3, either for mode of the period close to the round-trip time or for mode of the period close to As is shown in Fig. 3, for and for any G, decreasing R causes the steady mirror to bifurcate at through whereas for it bifurcates through for and through for The values for the marginal stability of are almost indistinguishable for but such is not the case for (dashed curves in Fig. 3).
Table 4
Pulse Characteristics for the Ar-Ion cw-Pumped Brillouin Fiber Ring Lasers from Eqs. (1)a
Optical Kerr Constant
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
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
Normalized pulse intensity mean reflected intensity period and width versus control parameter (Stokes feedback). The parameters for correspond to the second column of Table 1. The approximate expression (21) for the pulse width yields Bold face numbers indicate maximum peak intensity and minimum temporal width.
Table 5
Pulse Characteristics for the Nd:YAG cw-Pumped Brillouin Fiber Ring Laser from Eqs. (1)a
Optical Kerr Constant
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
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
Same as Table 4, but now the parameters for , correspond to the second column of Table 2.
Table 6
Pulse Characteristics for the Nd:YAG cw-Pumped Brillouin Fiber Ring Laser from Eqs. (1)a
G
P (mW)
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
Same as Table 5, but versus pump power P, i.e., versus gain as the control parameter for a fixed feedback .