Abstract

Passively mode-locked lasers with semiconductor saturable absorption mirrors are attractive comb sources due to their simplicity, excellent self-starting properties, and their environmental robustness. These lasers, however, can have an increased noise level and wake mode instabilities. Here, we investigate the wake mode dynamics in detail using a combination of evolutionary and dynamical methods. We describe the mode-locked pulse generation from noise when a stable pulse exists and the evolution of the wake mode instability when no stable pulse exists. We then calculate the dynamical spectrum of the mode-locked pulse, and we show that it has six discrete eigenmodes, two of which correspond to wake modes. The wake modes are unstable when the wake mode eigenvalues have a positive real part. We also show that even when the laser is stable, the wake modes lead to experimentally observed sidebands.

© 2017 Optical Society of America

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References

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2016 (3)

S. Droste, G. Ycas, B. R. Washburn, I. Coddington, and N. R. Newbury, Nanophotonics 5, 19 (2016).

C. R. Menyuk and S. Wang, Nanophotonics 5, 332 (2016).

Y. Shen, J. Zweck, S. Wang, and C. R. Menyuk, Stud. Appl. Math. 137, 238 (2016).
[Crossref]

2014 (2)

2013 (1)

2012 (1)

2011 (4)

2010 (3)

2009 (2)

O. Shtyrina, M. Fedoruk, S. Turitsyn, R. Herda, and O. Okhotnikov, J. Opt. Soc. Am. B 26, 346 (2009).
[Crossref]

M. E. Fermann and I. Hartl, IEEE J. Sel. Top. Quantum Electron. 15, 191 (2009).
[Crossref]

2008 (1)

N. N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, Phys. Lett. A 372, 3124 (2008).
[Crossref]

2007 (1)

2005 (1)

2004 (2)

1997 (3)

L. Nelson, D. Jones, K. Tamura, H. Haus, and E. Ippen, Appl. Phys. B 65, 277 (1997).
[Crossref]

J. N. Kutz, B. C. Collings, K. Bergman, S. Tsuda, S. T. Cundiff, W. H. Knox, P. Holmes, and M. Weinstein, J. Opt. Soc. Am. B 14, 2681 (1997).
[Crossref]

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, IEEE J. Sel. Top. Quantum Electron. 3, 1065 (1997).
[Crossref]

1996 (2)

F. Kärtner, I. Jung, and U. Keller, IEEE J. Sel. Top. Quantum Electron. 2, 540 (1996).
[Crossref]

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. A. der Au, IEEE J. Sel. Top. Quantum Electron. 2, 435 (1996).
[Crossref]

1994 (1)

1993 (1)

H. A. Haus and A. Mecozzi, IEEE J. Quantum Electron. 29, 983 (1993).
[Crossref]

1990 (1)

D. J. Kaup, Phys. Rev. A 42, 5689 (1990).
[Crossref]

1982 (1)

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982).
[Crossref]

Akhmediev, N. N.

N. N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, Phys. Lett. A 372, 3124 (2008).
[Crossref]

Bergman, K.

J. N. Kutz, B. C. Collings, K. Bergman, S. Tsuda, S. T. Cundiff, W. H. Knox, P. Holmes, and M. Weinstein, J. Opt. Soc. Am. B 14, 2681 (1997).
[Crossref]

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, IEEE J. Sel. Top. Quantum Electron. 3, 1065 (1997).
[Crossref]

Braun, B.

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. A. der Au, IEEE J. Sel. Top. Quantum Electron. 2, 435 (1996).
[Crossref]

Byun, H.

Carruthers, T. F.

B. S. Marks, T. F. Carruthers, and C. R. Menyuk, in Conference on Lasers and Electro-Optics (Optical Society of America, 2012), paper CTu1I.3.

Chen, C.-J.

Coddington, I.

S. Droste, G. Ycas, B. R. Washburn, I. Coddington, and N. R. Newbury, Nanophotonics 5, 19 (2016).

L. C. Sinclair, I. Coddington, W. C. Swann, G. B. Rieker, A. Hati, K. Iwakuni, and N. R. Newbury, Opt. Express 22, 6996 (2014).
[Crossref]

S. Wang, C. R. Menyuk, L. Sinclair, I. Coddington, and N. R. Newbury, in Conference on Lasers and Electro-Optics (Optical Society of America, 2014), paper SW3E.4.

S. Wang, C. R. Menyuk, S. Droste, L. Sinclair, I. Coddington, and N. R. Newbury, in Conference on Lasers and Electro-Optics (Optical Society of America, 2016), paper SM3H.5.

Collings, B. C.

J. N. Kutz, B. C. Collings, K. Bergman, S. Tsuda, S. T. Cundiff, W. H. Knox, P. Holmes, and M. Weinstein, J. Opt. Soc. Am. B 14, 2681 (1997).
[Crossref]

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, IEEE J. Sel. Top. Quantum Electron. 3, 1065 (1997).
[Crossref]

Cundiff, S.

J. Ye and S. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation and Applications (Springer, 2006).

Cundiff, S. T.

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, IEEE J. Sel. Top. Quantum Electron. 3, 1065 (1997).
[Crossref]

J. N. Kutz, B. C. Collings, K. Bergman, S. Tsuda, S. T. Cundiff, W. H. Knox, P. Holmes, and M. Weinstein, J. Opt. Soc. Am. B 14, 2681 (1997).
[Crossref]

Cunningham, J. E.

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, IEEE J. Sel. Top. Quantum Electron. 3, 1065 (1997).
[Crossref]

der Au, J. A.

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. A. der Au, IEEE J. Sel. Top. Quantum Electron. 2, 435 (1996).
[Crossref]

Diddams, S. A.

Docherty, A.

Droste, S.

S. Droste, G. Ycas, B. R. Washburn, I. Coddington, and N. R. Newbury, Nanophotonics 5, 19 (2016).

S. Wang, C. R. Menyuk, S. Droste, L. Sinclair, I. Coddington, and N. R. Newbury, in Conference on Lasers and Electro-Optics (Optical Society of America, 2016), paper SM3H.5.

Dudley, J. M.

M. C. Stumpf, S. Pekarek, A. E. H. Oehler, T. Südmeyer, J. M. Dudley, and U. Keller, Appl. Phys. B 99, 401 (2010).
[Crossref]

Feder, K.

Fedoruk, M.

Fejer, M. M.

Fermann, M. E.

Fluck, R.

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. A. der Au, IEEE J. Sel. Top. Quantum Electron. 2, 435 (1996).
[Crossref]

Fork, R. L.

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982).
[Crossref]

Fox, R. W.

Grelu, P.

N. N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, Phys. Lett. A 372, 3124 (2008).
[Crossref]

Hartl, I.

Hati, A.

Haus, H.

L. Nelson, D. Jones, K. Tamura, H. Haus, and E. Ippen, Appl. Phys. B 65, 277 (1997).
[Crossref]

Haus, H. A.

H. A. Haus and A. Mecozzi, IEEE J. Quantum Electron. 29, 983 (1993).
[Crossref]

Herda, R.

Holmes, P.

Hong, F.-L.

Honninger, C.

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. A. der Au, IEEE J. Sel. Top. Quantum Electron. 2, 435 (1996).
[Crossref]

Imeshev, G.

Inaba, H.

Ippen, E.

L. Nelson, D. Jones, K. Tamura, H. Haus, and E. Ippen, Appl. Phys. B 65, 277 (1997).
[Crossref]

Ippen, E. P.

Iwakuni, K.

Jan, W. Y.

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, IEEE J. Sel. Top. Quantum Electron. 3, 1065 (1997).
[Crossref]

Johnson, T. A.

Jones, D.

L. Nelson, D. Jones, K. Tamura, H. Haus, and E. Ippen, Appl. Phys. B 65, 277 (1997).
[Crossref]

Jørgensen, C. G.

Jung, I.

F. Kärtner, I. Jung, and U. Keller, IEEE J. Sel. Top. Quantum Electron. 2, 540 (1996).
[Crossref]

Jung, I. D.

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. A. der Au, IEEE J. Sel. Top. Quantum Electron. 2, 435 (1996).
[Crossref]

Jung, K.

Kartner, F. X.

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. A. der Au, IEEE J. Sel. Top. Quantum Electron. 2, 435 (1996).
[Crossref]

Kärtner, F.

F. Kärtner, I. Jung, and U. Keller, IEEE J. Sel. Top. Quantum Electron. 2, 540 (1996).
[Crossref]

Kärtner, F. X.

Kaup, D. J.

D. J. Kaup, Phys. Rev. A 42, 5689 (1990).
[Crossref]

Keller, U.

M. C. Stumpf, S. Pekarek, A. E. H. Oehler, T. Südmeyer, J. M. Dudley, and U. Keller, Appl. Phys. B 99, 401 (2010).
[Crossref]

F. Kärtner, I. Jung, and U. Keller, IEEE J. Sel. Top. Quantum Electron. 2, 540 (1996).
[Crossref]

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. A. der Au, IEEE J. Sel. Top. Quantum Electron. 2, 435 (1996).
[Crossref]

Kieu, K.

Kim, C.

Kim, H.

Kim, J.

Knox, W. H.

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, IEEE J. Sel. Top. Quantum Electron. 3, 1065 (1997).
[Crossref]

J. N. Kutz, B. C. Collings, K. Bergman, S. Tsuda, S. T. Cundiff, W. H. Knox, P. Holmes, and M. Weinstein, J. Opt. Soc. Am. B 14, 2681 (1997).
[Crossref]

Kobayashi, Y.

Koch, M.

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, IEEE J. Sel. Top. Quantum Electron. 3, 1065 (1997).
[Crossref]

Kolodziejski, L. A.

Kopf, D.

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. A. der Au, IEEE J. Sel. Top. Quantum Electron. 2, 435 (1996).
[Crossref]

Kutz, J. N.

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, IEEE J. Sel. Top. Quantum Electron. 3, 1065 (1997).
[Crossref]

J. N. Kutz, B. C. Collings, K. Bergman, S. Tsuda, S. T. Cundiff, W. H. Knox, P. Holmes, and M. Weinstein, J. Opt. Soc. Am. B 14, 2681 (1997).
[Crossref]

Langrock, C.

Marks, B. S.

S. Wang, A. Docherty, B. S. Marks, and C. R. Menyuk, J. Opt. Soc. Am. B 31, 2914 (2014).
[Crossref]

S. Wang, A. Docherty, B. S. Marks, and C. R. Menyuk, J. Opt. Soc. Am. B 30, 3064 (2013).
[Crossref]

B. S. Marks, T. F. Carruthers, and C. R. Menyuk, in Conference on Lasers and Electro-Optics (Optical Society of America, 2012), paper CTu1I.3.

Matsumoto, H.

Matuschek, N.

U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. A. der Au, IEEE J. Sel. Top. Quantum Electron. 2, 435 (1996).
[Crossref]

Mecozzi, A.

H. A. Haus and A. Mecozzi, IEEE J. Quantum Electron. 29, 983 (1993).
[Crossref]

Menyuk, C. R.

C. R. Menyuk and S. Wang, Nanophotonics 5, 332 (2016).

Y. Shen, J. Zweck, S. Wang, and C. R. Menyuk, Stud. Appl. Math. 137, 238 (2016).
[Crossref]

S. Wang, A. Docherty, B. S. Marks, and C. R. Menyuk, J. Opt. Soc. Am. B 31, 2914 (2014).
[Crossref]

S. Wang, A. Docherty, B. S. Marks, and C. R. Menyuk, J. Opt. Soc. Am. B 30, 3064 (2013).
[Crossref]

C.-J. Chen, P. K. A. Wai, and C. R. Menyuk, Opt. Lett. 19, 198 (1994).
[Crossref]

B. S. Marks, T. F. Carruthers, and C. R. Menyuk, in Conference on Lasers and Electro-Optics (Optical Society of America, 2012), paper CTu1I.3.

S. Wang, C. R. Menyuk, L. Sinclair, I. Coddington, and N. R. Newbury, in Conference on Lasers and Electro-Optics (Optical Society of America, 2014), paper SW3E.4.

S. Wang, C. R. Menyuk, S. Droste, L. Sinclair, I. Coddington, and N. R. Newbury, in Conference on Lasers and Electro-Optics (Optical Society of America, 2016), paper SM3H.5.

Minoshima, K.

Motamedi, A.

Nelson, L.

L. Nelson, D. Jones, K. Tamura, H. Haus, and E. Ippen, Appl. Phys. B 65, 277 (1997).
[Crossref]

Newbury, N. R.

S. Droste, G. Ycas, B. R. Washburn, I. Coddington, and N. R. Newbury, Nanophotonics 5, 19 (2016).

L. C. Sinclair, I. Coddington, W. C. Swann, G. B. Rieker, A. Hati, K. Iwakuni, and N. R. Newbury, Opt. Express 22, 6996 (2014).
[Crossref]

N. R. Newbury, Nat. Photonics 5, 186 (2011).
[Crossref]

N. R. Newbury and W. C. Swann, J. Opt. Soc. Am. B 24, 1756 (2007).
[Crossref]

B. R. Washburn, R. W. Fox, N. R. Newbury, J. W. Nicholson, K. Feder, P. S. Westbrook, and C. G. Jørgensen, Opt. Express 12, 4999 (2004).
[Crossref]

S. Wang, C. R. Menyuk, S. Droste, L. Sinclair, I. Coddington, and N. R. Newbury, in Conference on Lasers and Electro-Optics (Optical Society of America, 2016), paper SM3H.5.

S. Wang, C. R. Menyuk, L. Sinclair, I. Coddington, and N. R. Newbury, in Conference on Lasers and Electro-Optics (Optical Society of America, 2014), paper SW3E.4.

Nicholson, J. W.

Nugent-Glandorf, L.

Oehler, A. E. H.

M. C. Stumpf, S. Pekarek, A. E. H. Oehler, T. Südmeyer, J. M. Dudley, and U. Keller, Appl. Phys. B 99, 401 (2010).
[Crossref]

Okhotnikov, O.

Okhotnikov, O. G.

O. G. Okhotnikov and R. Herda, Quantum Electron. 41, 610 (2011).
[Crossref]

Onae, A.

Pekarek, S.

M. C. Stumpf, S. Pekarek, A. E. H. Oehler, T. Südmeyer, J. M. Dudley, and U. Keller, Appl. Phys. B 99, 401 (2010).
[Crossref]

Petrich, G. S.

Rieker, G. B.

Sander, M. Y.

Schibli, T. R.

Shank, C. V.

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982).
[Crossref]

Shen, H.

Shen, Y.

Y. Shen, J. Zweck, S. Wang, and C. R. Menyuk, Stud. Appl. Math. 137, 238 (2016).
[Crossref]

Shtyrina, O.

Sinclair, L.

S. Wang, C. R. Menyuk, L. Sinclair, I. Coddington, and N. R. Newbury, in Conference on Lasers and Electro-Optics (Optical Society of America, 2014), paper SW3E.4.

S. Wang, C. R. Menyuk, S. Droste, L. Sinclair, I. Coddington, and N. R. Newbury, in Conference on Lasers and Electro-Optics (Optical Society of America, 2016), paper SM3H.5.

Sinclair, L. C.

Song, Y.

Soto-Crespo, J. M.

N. N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, Phys. Lett. A 372, 3124 (2008).
[Crossref]

Stolen, R. H.

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982).
[Crossref]

Stumpf, M. C.

M. C. Stumpf, S. Pekarek, A. E. H. Oehler, T. Südmeyer, J. M. Dudley, and U. Keller, Appl. Phys. B 99, 401 (2010).
[Crossref]

Südmeyer, T.

M. C. Stumpf, S. Pekarek, A. E. H. Oehler, T. Südmeyer, J. M. Dudley, and U. Keller, Appl. Phys. B 99, 401 (2010).
[Crossref]

Swann, W. C.

Tamura, K.

L. Nelson, D. Jones, K. Tamura, H. Haus, and E. Ippen, Appl. Phys. B 65, 277 (1997).
[Crossref]

Tomlinson, W. J.

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982).
[Crossref]

Tsuda, S.

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Supplementary Material (1)

NameDescription
» Visualization 1: MP4 (20065 KB)      Visualization 1

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Figures (6)

Fig. 1.
Fig. 1.

Starting from noise, the system of Eq. (1) evolves to a stable profile in the (a) time domain, (b) frequency domain, where u ˜ ( f ) = 0 T R u ( t ) exp ( i ω t ) d t .

Fig. 2.
Fig. 2.

Stationary pulse u 0 and the net gain, G n = exp [ g ( | u 0 | ) l ρ n ( t ) ] , which exhibits an asymmetric profile due to the SESAM. A net gain window exists when G n > 1 .

Fig. 3.
Fig. 3.

Evolution profile of the wake mode instability (a) in the time domain, and (b) in the frequency domain (see Visualization 1), and the pulse amplitude profile at (c)  T = 823 , (d)  T = 948 , (e)  T = 1073 , and (f)  T = 1198 . The locations of profiles (c), (d), (e), and (f) are marked by white lines in (a).

Fig. 4.
Fig. 4.

Variation of the dynamical spectrum when the unsaturated gain g 0 increases from 1.90 to 2.70. We find that λ w ± = 7.75 × 10 4 ± 0.352 i when g 0 = 7.74 and λ w ± = 9.09 × 10 3 ± 1.19 i when g 0 = 13.5 . The dashed arrows indicate how the spectrum shifts as g 0 increases from 7.74 to 13.5. The eigenvalue λ a < 0.90 is not shown here.

Fig. 5.
Fig. 5.

Stability boundaries in the parameter space of the unsaturated gain g 0 and the group velocity dispersion β . The points (i) and (ii) indicate the cases g 0 = 7.74 and g 0 = 13.5 , respectively, with β = 0.0144 ps 2 .

Fig. 6.
Fig. 6.

Variation of the output power P out and the sidebands’ frequency offset f sb as the unsaturated gain g 0 increases. The stationary pulse becomes unstable when g 0 = 8.5 .

Equations (4)

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T R u T = i ϕ u + t s u t + g 2 ( 1 + 1 2 ω g 2 2 t 2 ) u l 2 u i β 2 2 u t 2 + i γ | u | 2 u ρ 2 n u ,
g ( | u | ) = g 0 [ 1 + w 0 / ( P sat T R ) ] 1 ,
n ( t ) t = 1 n T A | u ( t ) | 2 w A n , n ( T R / 2 ) = 0 ,
f sb = Im ( | λ w ± | ) / ( 2 π T R ) .