Abstract

There is an increasing demand for all-fiber passively mode-locked lasers with pulse repetition rates in the order of gigahertz for their potential applications in fields such as telecommunications and metrology. However, conventional mode-locked fiber lasers typically operate at fundamental repetition rates of only a few megahertz. In this paper, we report all-fiber laser operation with fundamental repetition rates of 4.24 GHz, 9.63GHz and 19.45GHz. This is, to date and to the best of our knowledge, the highest fundamental repetition rate reported for an all-fiber laser. The laser operation is based on the passive modelocking of a miniature all-fiber Fabry-Pérot laser (FFPL) by a carbon nanotube (CNT) saturable absorber. The key components for such device are a very high-gain Er:Yb phosphosilicate fiber and a fiber compatible saturable absorber with very small foot print and very low losses. The laser output of the three lasers was close to transform-limited with a pulsewidth of approximately 1ps and low noise. As a demonstration of potential future applications for this laser, we also demonstrated supercontinuum generation with a longitudinal mode-spacing of 0.08nm by launching the laser operating at 9.63GHz into 30m of a highly nonlinear dispersion shifted fiber.

© 2011 OSA

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References

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  1. M. E. Fermann and I. Hartl, “Ultrafast fiber laser technology,” IEEE J. Sel. Top. Quantum Electron. 15(1), 191–206 (2009).
    [CrossRef]
  2. E. P. Ippen, “Principle of Passive Mode Locking,” Appl. Phys. B 58(3), 159–170 (1994).
    [CrossRef]
  3. A. B. Grudinin and S. Gray, “Passive harmonic mode locking in soliton fiber lasers,” J. Opt. Soc. Am. B 14(1), 144 (1997).
    [CrossRef]
  4. M. Nakazawa, K. Tamura, and E. Yoshida, “Supermode noise suppression in a harmonically modelocked fibre laser by selfphase modulation and spectral filtering,” Electron. Lett. 32(5), 461 (1996).
    [CrossRef]
  5. J. W. Nicholson and D. J. DiGiovannni, “High-repetition-frequency low-noise fiber ring lasers mode-locked with carbon nanotubes,” IEEE Photon. Technol. Lett. 20(24), 2123–2125 (2008).
    [CrossRef]
  6. S. Yamashita, Y. Inoue, K. Hsu, T. Kotake, H. Yaguchi, D. Tanaka, M. Jablonski, and S. Y. Set, “5-GHz pulsed fiber Fabry–Pérot laser mode-locked using carbon nanotubes,” IEEE Photon. Technol. Lett. 17(4), 750–752 (2005).
    [CrossRef]
  7. S. Y. Set, H. Yaguchi, Y. Tanaka, and M. Jablonski, “Ultrafast Fiber Pulsed Lasers Incorporating Carbon Nanotubes,” IEEE J. Sel. Top. Quantum Electron. 10(1), 137–146 (2004).
    [CrossRef]
  8. J. J. McFerran, L. Nenadovic, W. C. Swann, J. B. Schlager, and N. R. Newbury, “A passively mode-locked fiber laser at 1.54 mum with a fundamental repetition frequency reaching 2 GHz,” Opt. Express 15(20), 13155–13166 (2007).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  19. P. Polynkin, A. Polynkin, D. Panasenko, N. Peyghambarian, M. Mansuripur, and J. Moloney, “All-fiber passively mode-locked laser oscillator at 1. 5 µm with watts-level average output power and high repetition rate,” Opt. Lett. 31(5), 592–594 (2006).
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    [CrossRef]

2010 (3)

2009 (3)

2008 (3)

S. Y. Set, C. S. Goh, D. Wang, H. Yaguchi, and S. Yamashita, “Non-synchronous Optical Sampling and Data-Pattern Recovery Using a Repetition-Rate-Tunable Carbon-Nanotube Pulsed Laser,” Jpn. J. Appl. Phys. 47(8), 6809–6811 (2008).
[CrossRef]

J. W. Nicholson and D. J. DiGiovannni, “High-repetition-frequency low-noise fiber ring lasers mode-locked with carbon nanotubes,” IEEE Photon. Technol. Lett. 20(24), 2123–2125 (2008).
[CrossRef]

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[CrossRef] [PubMed]

2007 (2)

2006 (1)

2005 (1)

S. Yamashita, Y. Inoue, K. Hsu, T. Kotake, H. Yaguchi, D. Tanaka, M. Jablonski, and S. Y. Set, “5-GHz pulsed fiber Fabry–Pérot laser mode-locked using carbon nanotubes,” IEEE Photon. Technol. Lett. 17(4), 750–752 (2005).
[CrossRef]

2004 (2)

S. Y. Set, H. Yaguchi, Y. Tanaka, and M. Jablonski, “Ultrafast Fiber Pulsed Lasers Incorporating Carbon Nanotubes,” IEEE J. Sel. Top. Quantum Electron. 10(1), 137–146 (2004).
[CrossRef]

T. R. Schibli, K. Minoshima, F.-L. Hong, H. Inaba, A. Onae, H. Matsumoto, I. Hartl, and M. E. Fermann, “Frequency metrology with a turnkey all-fiber system,” Opt. Lett. 29(21), 2467–2469 (2004).
[CrossRef] [PubMed]

1997 (1)

1996 (1)

M. Nakazawa, K. Tamura, and E. Yoshida, “Supermode noise suppression in a harmonically modelocked fibre laser by selfphase modulation and spectral filtering,” Electron. Lett. 32(5), 461 (1996).
[CrossRef]

1994 (1)

E. P. Ippen, “Principle of Passive Mode Locking,” Appl. Phys. B 58(3), 159–170 (1994).
[CrossRef]

1991 (1)

J. E. Townsend, W. L. Barnes, K. P. Jedrzejewski, and S. G. Grubb, “Yb3+ sensitised Er3+ doped silica optical fibre with ultrahigh transfer efficiency and gain,” Electron. Lett. 27(21), 1958–1959 (1991).
[CrossRef]

1986 (1)

D. von der Linde, “Characterization of the noise in continuously operating mode-locked lasers,” Appl. Phys. B 39(4), 201–217 (1986).
[CrossRef]

Aitchison, B.

Amezcua-Correa, R.

Barnes, W. L.

J. E. Townsend, W. L. Barnes, K. P. Jedrzejewski, and S. G. Grubb, “Yb3+ sensitised Er3+ doped silica optical fibre with ultrahigh transfer efficiency and gain,” Electron. Lett. 27(21), 1958–1959 (1991).
[CrossRef]

Benabid, F.

Brown, D. P.

Byun, H.

Coen, S.

Corwin, K. L.

Couny, F.

DiGiovannni, D. J.

J. W. Nicholson and D. J. DiGiovannni, “High-repetition-frequency low-noise fiber ring lasers mode-locked with carbon nanotubes,” IEEE Photon. Technol. Lett. 20(24), 2123–2125 (2008).
[CrossRef]

Dudley, J. M.

Fermann, M. E.

Ferrari, A. C.

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[CrossRef] [PubMed]

Genty, G.

Goh, C. S.

S. Y. Set, C. S. Goh, D. Wang, H. Yaguchi, and S. Yamashita, “Non-synchronous Optical Sampling and Data-Pattern Recovery Using a Repetition-Rate-Tunable Carbon-Nanotube Pulsed Laser,” Jpn. J. Appl. Phys. 47(8), 6809–6811 (2008).
[CrossRef]

Gray, S.

Grubb, S. G.

J. E. Townsend, W. L. Barnes, K. P. Jedrzejewski, and S. G. Grubb, “Yb3+ sensitised Er3+ doped silica optical fibre with ultrahigh transfer efficiency and gain,” Electron. Lett. 27(21), 1958–1959 (1991).
[CrossRef]

Grudinin, A. B.

Hakulinen, T.

Härkönen, A.

Hartl, I.

Hennrich, F.

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[CrossRef] [PubMed]

Hong, F.-L.

Hsu, K.

S. Yamashita, Y. Inoue, K. Hsu, T. Kotake, H. Yaguchi, D. Tanaka, M. Jablonski, and S. Y. Set, “5-GHz pulsed fiber Fabry–Pérot laser mode-locked using carbon nanotubes,” IEEE Photon. Technol. Lett. 17(4), 750–752 (2005).
[CrossRef]

Inaba, H.

Inoue, Y.

S. Yamashita, Y. Inoue, K. Hsu, T. Kotake, H. Yaguchi, D. Tanaka, M. Jablonski, and S. Y. Set, “5-GHz pulsed fiber Fabry–Pérot laser mode-locked using carbon nanotubes,” IEEE Photon. Technol. Lett. 17(4), 750–752 (2005).
[CrossRef]

Ippen, E. P.

Jablonski, M.

S. Yamashita, Y. Inoue, K. Hsu, T. Kotake, H. Yaguchi, D. Tanaka, M. Jablonski, and S. Y. Set, “5-GHz pulsed fiber Fabry–Pérot laser mode-locked using carbon nanotubes,” IEEE Photon. Technol. Lett. 17(4), 750–752 (2005).
[CrossRef]

S. Y. Set, H. Yaguchi, Y. Tanaka, and M. Jablonski, “Ultrafast Fiber Pulsed Lasers Incorporating Carbon Nanotubes,” IEEE J. Sel. Top. Quantum Electron. 10(1), 137–146 (2004).
[CrossRef]

Jedrzejewski, K. P.

J. E. Townsend, W. L. Barnes, K. P. Jedrzejewski, and S. G. Grubb, “Yb3+ sensitised Er3+ doped silica optical fibre with ultrahigh transfer efficiency and gain,” Electron. Lett. 27(21), 1958–1959 (1991).
[CrossRef]

Jones, R. J.

K. Kieu, R. J. Jones, and N. Peyghambarian, “High power femtosecond source near 1 micron based on an all-fiber Er-doped mode-locked laser,” Opt. Express 18(20), 21350–21355 (2010).
[CrossRef] [PubMed]

K. Kieu, R. J. Jones, and N. Peyghambarian, “Generation of Few-Cycle Pulses From an Amplified Carbon Nanotube Mode-Locked Fiber Laser System,” IEEE Photon. Technol. Lett. 22(20), 1521–1523 (2010).
[CrossRef]

Kärtner, F. X.

Kaskela, A.

Kauppinen, E. I.

Kieu, K.

K. Kieu, R. J. Jones, and N. Peyghambarian, “High power femtosecond source near 1 micron based on an all-fiber Er-doped mode-locked laser,” Opt. Express 18(20), 21350–21355 (2010).
[CrossRef] [PubMed]

K. Kieu, R. J. Jones, and N. Peyghambarian, “Generation of Few-Cycle Pulses From an Amplified Carbon Nanotube Mode-Locked Fiber Laser System,” IEEE Photon. Technol. Lett. 22(20), 1521–1523 (2010).
[CrossRef]

Kivistö, S.

Knabe, K.

Knight, J. C.

Kolodziejski, L. A.

Kotake, T.

S. Yamashita, Y. Inoue, K. Hsu, T. Kotake, H. Yaguchi, D. Tanaka, M. Jablonski, and S. Y. Set, “5-GHz pulsed fiber Fabry–Pérot laser mode-locked using carbon nanotubes,” IEEE Photon. Technol. Lett. 17(4), 750–752 (2005).
[CrossRef]

Light, P. S.

Lim, J.

Mansuripur, M.

Matsumoto, H.

McFerran, J. J.

Milne, W. I.

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[CrossRef] [PubMed]

Minoshima, K.

Moloney, J.

Motamedi, A.

Nakazawa, M.

M. Nakazawa, K. Tamura, and E. Yoshida, “Supermode noise suppression in a harmonically modelocked fibre laser by selfphase modulation and spectral filtering,” Electron. Lett. 32(5), 461 (1996).
[CrossRef]

Nasibulin, A. G.

Neely, W.

Nenadovic, L.

Newbury, N. R.

Nicholson, J. W.

Okhotnikov, O. G.

Onae, A.

Panasenko, D.

Petrich, G. S.

Peyghambarian, N.

Polynkin, A.

Polynkin, P.

Rozhin, A. G.

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[CrossRef] [PubMed]

Sander, M. Y.

Scardaci, V.

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[CrossRef] [PubMed]

Schibli, T. R.

Schlager, J. B.

Set, S. Y.

S. Y. Set, C. S. Goh, D. Wang, H. Yaguchi, and S. Yamashita, “Non-synchronous Optical Sampling and Data-Pattern Recovery Using a Repetition-Rate-Tunable Carbon-Nanotube Pulsed Laser,” Jpn. J. Appl. Phys. 47(8), 6809–6811 (2008).
[CrossRef]

S. Yamashita, Y. Inoue, K. Hsu, T. Kotake, H. Yaguchi, D. Tanaka, M. Jablonski, and S. Y. Set, “5-GHz pulsed fiber Fabry–Pérot laser mode-locked using carbon nanotubes,” IEEE Photon. Technol. Lett. 17(4), 750–752 (2005).
[CrossRef]

S. Y. Set, H. Yaguchi, Y. Tanaka, and M. Jablonski, “Ultrafast Fiber Pulsed Lasers Incorporating Carbon Nanotubes,” IEEE J. Sel. Top. Quantum Electron. 10(1), 137–146 (2004).
[CrossRef]

Shen, H.

Sun, Z.

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[CrossRef] [PubMed]

Swann, W. C.

Tamura, K.

M. Nakazawa, K. Tamura, and E. Yoshida, “Supermode noise suppression in a harmonically modelocked fibre laser by selfphase modulation and spectral filtering,” Electron. Lett. 32(5), 461 (1996).
[CrossRef]

Tanaka, D.

S. Yamashita, Y. Inoue, K. Hsu, T. Kotake, H. Yaguchi, D. Tanaka, M. Jablonski, and S. Y. Set, “5-GHz pulsed fiber Fabry–Pérot laser mode-locked using carbon nanotubes,” IEEE Photon. Technol. Lett. 17(4), 750–752 (2005).
[CrossRef]

Tanaka, Y.

S. Y. Set, H. Yaguchi, Y. Tanaka, and M. Jablonski, “Ultrafast Fiber Pulsed Lasers Incorporating Carbon Nanotubes,” IEEE J. Sel. Top. Quantum Electron. 10(1), 137–146 (2004).
[CrossRef]

Tillman, K. A.

Townsend, J. E.

J. E. Townsend, W. L. Barnes, K. P. Jedrzejewski, and S. G. Grubb, “Yb3+ sensitised Er3+ doped silica optical fibre with ultrahigh transfer efficiency and gain,” Electron. Lett. 27(21), 1958–1959 (1991).
[CrossRef]

von der Linde, D.

D. von der Linde, “Characterization of the noise in continuously operating mode-locked lasers,” Appl. Phys. B 39(4), 201–217 (1986).
[CrossRef]

Wang, D.

S. Y. Set, C. S. Goh, D. Wang, H. Yaguchi, and S. Yamashita, “Non-synchronous Optical Sampling and Data-Pattern Recovery Using a Repetition-Rate-Tunable Carbon-Nanotube Pulsed Laser,” Jpn. J. Appl. Phys. 47(8), 6809–6811 (2008).
[CrossRef]

Wang, F.

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[CrossRef] [PubMed]

Wang, Y.

Washburn, B. R.

White, I. H.

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[CrossRef] [PubMed]

Yaguchi, H.

S. Y. Set, C. S. Goh, D. Wang, H. Yaguchi, and S. Yamashita, “Non-synchronous Optical Sampling and Data-Pattern Recovery Using a Repetition-Rate-Tunable Carbon-Nanotube Pulsed Laser,” Jpn. J. Appl. Phys. 47(8), 6809–6811 (2008).
[CrossRef]

S. Yamashita, Y. Inoue, K. Hsu, T. Kotake, H. Yaguchi, D. Tanaka, M. Jablonski, and S. Y. Set, “5-GHz pulsed fiber Fabry–Pérot laser mode-locked using carbon nanotubes,” IEEE Photon. Technol. Lett. 17(4), 750–752 (2005).
[CrossRef]

S. Y. Set, H. Yaguchi, Y. Tanaka, and M. Jablonski, “Ultrafast Fiber Pulsed Lasers Incorporating Carbon Nanotubes,” IEEE J. Sel. Top. Quantum Electron. 10(1), 137–146 (2004).
[CrossRef]

Yamashita, S.

S. Y. Set, C. S. Goh, D. Wang, H. Yaguchi, and S. Yamashita, “Non-synchronous Optical Sampling and Data-Pattern Recovery Using a Repetition-Rate-Tunable Carbon-Nanotube Pulsed Laser,” Jpn. J. Appl. Phys. 47(8), 6809–6811 (2008).
[CrossRef]

S. Yamashita, Y. Inoue, K. Hsu, T. Kotake, H. Yaguchi, D. Tanaka, M. Jablonski, and S. Y. Set, “5-GHz pulsed fiber Fabry–Pérot laser mode-locked using carbon nanotubes,” IEEE Photon. Technol. Lett. 17(4), 750–752 (2005).
[CrossRef]

Yoshida, E.

M. Nakazawa, K. Tamura, and E. Yoshida, “Supermode noise suppression in a harmonically modelocked fibre laser by selfphase modulation and spectral filtering,” Electron. Lett. 32(5), 461 (1996).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (2)

E. P. Ippen, “Principle of Passive Mode Locking,” Appl. Phys. B 58(3), 159–170 (1994).
[CrossRef]

D. von der Linde, “Characterization of the noise in continuously operating mode-locked lasers,” Appl. Phys. B 39(4), 201–217 (1986).
[CrossRef]

Electron. Lett. (2)

M. Nakazawa, K. Tamura, and E. Yoshida, “Supermode noise suppression in a harmonically modelocked fibre laser by selfphase modulation and spectral filtering,” Electron. Lett. 32(5), 461 (1996).
[CrossRef]

J. E. Townsend, W. L. Barnes, K. P. Jedrzejewski, and S. G. Grubb, “Yb3+ sensitised Er3+ doped silica optical fibre with ultrahigh transfer efficiency and gain,” Electron. Lett. 27(21), 1958–1959 (1991).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

S. Y. Set, H. Yaguchi, Y. Tanaka, and M. Jablonski, “Ultrafast Fiber Pulsed Lasers Incorporating Carbon Nanotubes,” IEEE J. Sel. Top. Quantum Electron. 10(1), 137–146 (2004).
[CrossRef]

M. E. Fermann and I. Hartl, “Ultrafast fiber laser technology,” IEEE J. Sel. Top. Quantum Electron. 15(1), 191–206 (2009).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

J. W. Nicholson and D. J. DiGiovannni, “High-repetition-frequency low-noise fiber ring lasers mode-locked with carbon nanotubes,” IEEE Photon. Technol. Lett. 20(24), 2123–2125 (2008).
[CrossRef]

S. Yamashita, Y. Inoue, K. Hsu, T. Kotake, H. Yaguchi, D. Tanaka, M. Jablonski, and S. Y. Set, “5-GHz pulsed fiber Fabry–Pérot laser mode-locked using carbon nanotubes,” IEEE Photon. Technol. Lett. 17(4), 750–752 (2005).
[CrossRef]

K. Kieu, R. J. Jones, and N. Peyghambarian, “Generation of Few-Cycle Pulses From an Amplified Carbon Nanotube Mode-Locked Fiber Laser System,” IEEE Photon. Technol. Lett. 22(20), 1521–1523 (2010).
[CrossRef]

J. Opt. Soc. Am. B (2)

Jpn. J. Appl. Phys. (1)

S. Y. Set, C. S. Goh, D. Wang, H. Yaguchi, and S. Yamashita, “Non-synchronous Optical Sampling and Data-Pattern Recovery Using a Repetition-Rate-Tunable Carbon-Nanotube Pulsed Laser,” Jpn. J. Appl. Phys. 47(8), 6809–6811 (2008).
[CrossRef]

Nat. Nanotechnol. (1)

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[CrossRef] [PubMed]

Opt. Express (4)

Opt. Lett. (2)

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Figures (8)

Fig. 1
Fig. 1

Scheme of a fiber Fabry- Pérot laser (FFPL). WDM - wavelength division multiplexer, HR - highly reflective mirror CNT-SA - carbon nanotube saturable absorber.

Fig. 2
Fig. 2

Optical spectrum of the (a) 25mm-long laser, (b) 10mm-long laser and (c) 5mm-long laser. (d) pulsed laser scheme. Optical spectrum showing the longitudinal modes and mode-spacing for the (e) 25mm-long laser, (f) 10mm-long laser and (g) 5mm-long laser.

Fig. 3
Fig. 3

(a) Autocorrelator trace for the three lasers (offset for clarity).

Fig. 4
Fig. 4

RF signal for the laser operating at a fundamental repetition rate of 4.24 GHz, (a)fundamental repetition rate, (b) 2nd, (c) 3rd, (d) 4th, (e) 5th and (f) 6th order.

Fig. 5
Fig. 5

RF signal for the laser operating at a fundamental repetition rate of 9.63 GHz fundamental repetition rate laser, (a) fundamental frequency, (b) 2nd order frequency.

Fig. 6
Fig. 6

RF signal for the laser operating at a fundamental repetition rate of 19.45 GHz laser.

Fig. 7
Fig. 7

Time-jitter analysis for the three lasers. (a) Schematic of the measured parameters corresponding to the signal (PA), and the narrow-band (PB) and wide-band components (PC) measured at the multiples of the fundamental frequency. (b)Measured values for the three lasers, fitted n2-function for the PB/PA and PC/PA, for the 25mm-long laser.

Fig. 8
Fig. 8

Supercontinuum generation. (a) Set-up used for supercontinuum generation, HNLF-Highly Nonlinear fiber and EDFA (Erbium doped fiber amplifier. (b) seed laser source (f1-9.63GHz) and generated supercontinuum.

Equations (2)

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f r e p =     c / 2 n L ,
Δ λ =     λ 2 / 2 n L ,

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