Abstract

Using the time domain master equation for a complex electric-field pulse envelope, we find analytical results for the optical spectra of passively mode-locked semiconductor lasers. The analysis includes the effect of optical nonlinearity of semiconductor lasers, which is characterized by a slow saturable amplifier and absorber. Group velocity dispersion, bandwidth limiting, and self-phase modulation were considered as well. The FWHM of the spectrum profile was found to have a strong dependence on group velocity dispersion and self-phase modulation. For large absolute values of the chirp parameter, the optical spectra result in equispaced continuous wave frequencies, a large fraction of which have equal power.

© 1997 Optical Society of America

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  1. T. Morioka, K. Mori, M. Saruwatari, “More than 100-wavelength-channel picosecond optical pulse generation from single-laser source using supercontinuum in optical fibers,” Electron. Lett. 29, 862–864 (1993).
    [CrossRef]
  2. H. W. Messenger, “Developers unveil modelocked erbium fiber laser,” Laser Focus World 32, 15–16 (May1996).
  3. B. Y. Zhu, I. H. White, “Multiwavelength picosecond optical pulse generation using an actively mode-locked multichannel grating cavity laser,” J. Lightwave Technol. 13, 2327–2335 (1995).
    [CrossRef]
  4. H. Hasaka, Y. Yoshikuni, K. Sato, H. Ishii, H. Sanjoh, “Multiwavelength light source with precise frequency spacing using mode-locked semiconductor laser and arrayed waveguide grating filter,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 299–300.
  5. D. J. Derickson, R. J. Helkey, A. Mar, J. R. Karin, J. G. Wasserbauer, J. E. Bowers, “Short pulse generation using multisegment mode-locked semiconductor-lasers,” IEEE J. Quantum Electron. 28, 2186–2202 (1992).
    [CrossRef]
  6. H. A. Haus, “Theory of mode-locking with slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–746 (1975).
    [CrossRef]
  7. O. E. Martinez, R. L. Fork, J. P. Gordon, “Theory of passively mode-locked lasers for the case of a nonlinear complex propagation coefficient,” J. Opt. Soc. Am. B 2, 753–760 (1985).
    [CrossRef]
  8. R. G. M. P. Koumans, R. van Roijen, “Theory for passive mode-locking in semiconductor laser structures including the effects of self-phase modulation, dispersion, and pulse collisions,” IEEE J. Quantum Electron. 32, 478–492 (1996); R. G. M. P. Koumans, R. A. Salvatore, D. Eliyahu, R. van Roijen, “Correction to: Theory for passive mode-locking in semiconductor laser structures including the effects of self-phase modulation, dispersion, and pulse collisions,” IEEE J. Quantum Electron. 32, 2017 (1996).
  9. G. H. C. New, “Pulse evolution in mode-locked quasi-continuous lasers,” IEEE J. Quantum Electron. QE-10, 115–124 (1974).
    [CrossRef]
  10. E. P. Ippen, “Principles of passive mode locking,” Appl. Phys. B 58, 159–170 (1994).
    [CrossRef]
  11. R. A. Salvatore, A. Yariv, “Demonstration of down-chirped and chirp-free pulses from high-repetition-rate passively mode-locked lasers,” IEEE Photon. Technol. Lett. 7, 1151–1153 (1995).
    [CrossRef]
  12. M. Schell, J. Yu, M. Tsuchiya, T. Kamiya, “Chirp of passively and actively mode-locked semiconductor-lasers,” Appl. Phys. Lett. 67, 1797–1799 (1995).
    [CrossRef]
  13. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic, New York, 1965), p. 933.

1996

H. W. Messenger, “Developers unveil modelocked erbium fiber laser,” Laser Focus World 32, 15–16 (May1996).

R. G. M. P. Koumans, R. van Roijen, “Theory for passive mode-locking in semiconductor laser structures including the effects of self-phase modulation, dispersion, and pulse collisions,” IEEE J. Quantum Electron. 32, 478–492 (1996); R. G. M. P. Koumans, R. A. Salvatore, D. Eliyahu, R. van Roijen, “Correction to: Theory for passive mode-locking in semiconductor laser structures including the effects of self-phase modulation, dispersion, and pulse collisions,” IEEE J. Quantum Electron. 32, 2017 (1996).

1995

B. Y. Zhu, I. H. White, “Multiwavelength picosecond optical pulse generation using an actively mode-locked multichannel grating cavity laser,” J. Lightwave Technol. 13, 2327–2335 (1995).
[CrossRef]

R. A. Salvatore, A. Yariv, “Demonstration of down-chirped and chirp-free pulses from high-repetition-rate passively mode-locked lasers,” IEEE Photon. Technol. Lett. 7, 1151–1153 (1995).
[CrossRef]

M. Schell, J. Yu, M. Tsuchiya, T. Kamiya, “Chirp of passively and actively mode-locked semiconductor-lasers,” Appl. Phys. Lett. 67, 1797–1799 (1995).
[CrossRef]

1994

E. P. Ippen, “Principles of passive mode locking,” Appl. Phys. B 58, 159–170 (1994).
[CrossRef]

1993

T. Morioka, K. Mori, M. Saruwatari, “More than 100-wavelength-channel picosecond optical pulse generation from single-laser source using supercontinuum in optical fibers,” Electron. Lett. 29, 862–864 (1993).
[CrossRef]

1992

D. J. Derickson, R. J. Helkey, A. Mar, J. R. Karin, J. G. Wasserbauer, J. E. Bowers, “Short pulse generation using multisegment mode-locked semiconductor-lasers,” IEEE J. Quantum Electron. 28, 2186–2202 (1992).
[CrossRef]

1985

1975

H. A. Haus, “Theory of mode-locking with slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–746 (1975).
[CrossRef]

1974

G. H. C. New, “Pulse evolution in mode-locked quasi-continuous lasers,” IEEE J. Quantum Electron. QE-10, 115–124 (1974).
[CrossRef]

Bowers, J. E.

D. J. Derickson, R. J. Helkey, A. Mar, J. R. Karin, J. G. Wasserbauer, J. E. Bowers, “Short pulse generation using multisegment mode-locked semiconductor-lasers,” IEEE J. Quantum Electron. 28, 2186–2202 (1992).
[CrossRef]

Derickson, D. J.

D. J. Derickson, R. J. Helkey, A. Mar, J. R. Karin, J. G. Wasserbauer, J. E. Bowers, “Short pulse generation using multisegment mode-locked semiconductor-lasers,” IEEE J. Quantum Electron. 28, 2186–2202 (1992).
[CrossRef]

Fork, R. L.

Gordon, J. P.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic, New York, 1965), p. 933.

Hasaka, H.

H. Hasaka, Y. Yoshikuni, K. Sato, H. Ishii, H. Sanjoh, “Multiwavelength light source with precise frequency spacing using mode-locked semiconductor laser and arrayed waveguide grating filter,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 299–300.

Haus, H. A.

H. A. Haus, “Theory of mode-locking with slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–746 (1975).
[CrossRef]

Helkey, R. J.

D. J. Derickson, R. J. Helkey, A. Mar, J. R. Karin, J. G. Wasserbauer, J. E. Bowers, “Short pulse generation using multisegment mode-locked semiconductor-lasers,” IEEE J. Quantum Electron. 28, 2186–2202 (1992).
[CrossRef]

Ippen, E. P.

E. P. Ippen, “Principles of passive mode locking,” Appl. Phys. B 58, 159–170 (1994).
[CrossRef]

Ishii, H.

H. Hasaka, Y. Yoshikuni, K. Sato, H. Ishii, H. Sanjoh, “Multiwavelength light source with precise frequency spacing using mode-locked semiconductor laser and arrayed waveguide grating filter,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 299–300.

Kamiya, T.

M. Schell, J. Yu, M. Tsuchiya, T. Kamiya, “Chirp of passively and actively mode-locked semiconductor-lasers,” Appl. Phys. Lett. 67, 1797–1799 (1995).
[CrossRef]

Karin, J. R.

D. J. Derickson, R. J. Helkey, A. Mar, J. R. Karin, J. G. Wasserbauer, J. E. Bowers, “Short pulse generation using multisegment mode-locked semiconductor-lasers,” IEEE J. Quantum Electron. 28, 2186–2202 (1992).
[CrossRef]

Koumans, R. G. M. P.

R. G. M. P. Koumans, R. van Roijen, “Theory for passive mode-locking in semiconductor laser structures including the effects of self-phase modulation, dispersion, and pulse collisions,” IEEE J. Quantum Electron. 32, 478–492 (1996); R. G. M. P. Koumans, R. A. Salvatore, D. Eliyahu, R. van Roijen, “Correction to: Theory for passive mode-locking in semiconductor laser structures including the effects of self-phase modulation, dispersion, and pulse collisions,” IEEE J. Quantum Electron. 32, 2017 (1996).

Mar, A.

D. J. Derickson, R. J. Helkey, A. Mar, J. R. Karin, J. G. Wasserbauer, J. E. Bowers, “Short pulse generation using multisegment mode-locked semiconductor-lasers,” IEEE J. Quantum Electron. 28, 2186–2202 (1992).
[CrossRef]

Martinez, O. E.

Messenger, H. W.

H. W. Messenger, “Developers unveil modelocked erbium fiber laser,” Laser Focus World 32, 15–16 (May1996).

Mori, K.

T. Morioka, K. Mori, M. Saruwatari, “More than 100-wavelength-channel picosecond optical pulse generation from single-laser source using supercontinuum in optical fibers,” Electron. Lett. 29, 862–864 (1993).
[CrossRef]

Morioka, T.

T. Morioka, K. Mori, M. Saruwatari, “More than 100-wavelength-channel picosecond optical pulse generation from single-laser source using supercontinuum in optical fibers,” Electron. Lett. 29, 862–864 (1993).
[CrossRef]

New, G. H. C.

G. H. C. New, “Pulse evolution in mode-locked quasi-continuous lasers,” IEEE J. Quantum Electron. QE-10, 115–124 (1974).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic, New York, 1965), p. 933.

Salvatore, R. A.

R. A. Salvatore, A. Yariv, “Demonstration of down-chirped and chirp-free pulses from high-repetition-rate passively mode-locked lasers,” IEEE Photon. Technol. Lett. 7, 1151–1153 (1995).
[CrossRef]

Sanjoh, H.

H. Hasaka, Y. Yoshikuni, K. Sato, H. Ishii, H. Sanjoh, “Multiwavelength light source with precise frequency spacing using mode-locked semiconductor laser and arrayed waveguide grating filter,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 299–300.

Saruwatari, M.

T. Morioka, K. Mori, M. Saruwatari, “More than 100-wavelength-channel picosecond optical pulse generation from single-laser source using supercontinuum in optical fibers,” Electron. Lett. 29, 862–864 (1993).
[CrossRef]

Sato, K.

H. Hasaka, Y. Yoshikuni, K. Sato, H. Ishii, H. Sanjoh, “Multiwavelength light source with precise frequency spacing using mode-locked semiconductor laser and arrayed waveguide grating filter,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 299–300.

Schell, M.

M. Schell, J. Yu, M. Tsuchiya, T. Kamiya, “Chirp of passively and actively mode-locked semiconductor-lasers,” Appl. Phys. Lett. 67, 1797–1799 (1995).
[CrossRef]

Tsuchiya, M.

M. Schell, J. Yu, M. Tsuchiya, T. Kamiya, “Chirp of passively and actively mode-locked semiconductor-lasers,” Appl. Phys. Lett. 67, 1797–1799 (1995).
[CrossRef]

van Roijen, R.

R. G. M. P. Koumans, R. van Roijen, “Theory for passive mode-locking in semiconductor laser structures including the effects of self-phase modulation, dispersion, and pulse collisions,” IEEE J. Quantum Electron. 32, 478–492 (1996); R. G. M. P. Koumans, R. A. Salvatore, D. Eliyahu, R. van Roijen, “Correction to: Theory for passive mode-locking in semiconductor laser structures including the effects of self-phase modulation, dispersion, and pulse collisions,” IEEE J. Quantum Electron. 32, 2017 (1996).

Wasserbauer, J. G.

D. J. Derickson, R. J. Helkey, A. Mar, J. R. Karin, J. G. Wasserbauer, J. E. Bowers, “Short pulse generation using multisegment mode-locked semiconductor-lasers,” IEEE J. Quantum Electron. 28, 2186–2202 (1992).
[CrossRef]

White, I. H.

B. Y. Zhu, I. H. White, “Multiwavelength picosecond optical pulse generation using an actively mode-locked multichannel grating cavity laser,” J. Lightwave Technol. 13, 2327–2335 (1995).
[CrossRef]

Yariv, A.

R. A. Salvatore, A. Yariv, “Demonstration of down-chirped and chirp-free pulses from high-repetition-rate passively mode-locked lasers,” IEEE Photon. Technol. Lett. 7, 1151–1153 (1995).
[CrossRef]

Yoshikuni, Y.

H. Hasaka, Y. Yoshikuni, K. Sato, H. Ishii, H. Sanjoh, “Multiwavelength light source with precise frequency spacing using mode-locked semiconductor laser and arrayed waveguide grating filter,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 299–300.

Yu, J.

M. Schell, J. Yu, M. Tsuchiya, T. Kamiya, “Chirp of passively and actively mode-locked semiconductor-lasers,” Appl. Phys. Lett. 67, 1797–1799 (1995).
[CrossRef]

Zhu, B. Y.

B. Y. Zhu, I. H. White, “Multiwavelength picosecond optical pulse generation using an actively mode-locked multichannel grating cavity laser,” J. Lightwave Technol. 13, 2327–2335 (1995).
[CrossRef]

Appl. Phys. B

E. P. Ippen, “Principles of passive mode locking,” Appl. Phys. B 58, 159–170 (1994).
[CrossRef]

Appl. Phys. Lett.

M. Schell, J. Yu, M. Tsuchiya, T. Kamiya, “Chirp of passively and actively mode-locked semiconductor-lasers,” Appl. Phys. Lett. 67, 1797–1799 (1995).
[CrossRef]

Electron. Lett.

T. Morioka, K. Mori, M. Saruwatari, “More than 100-wavelength-channel picosecond optical pulse generation from single-laser source using supercontinuum in optical fibers,” Electron. Lett. 29, 862–864 (1993).
[CrossRef]

IEEE J. Quantum Electron.

D. J. Derickson, R. J. Helkey, A. Mar, J. R. Karin, J. G. Wasserbauer, J. E. Bowers, “Short pulse generation using multisegment mode-locked semiconductor-lasers,” IEEE J. Quantum Electron. 28, 2186–2202 (1992).
[CrossRef]

H. A. Haus, “Theory of mode-locking with slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–746 (1975).
[CrossRef]

R. G. M. P. Koumans, R. van Roijen, “Theory for passive mode-locking in semiconductor laser structures including the effects of self-phase modulation, dispersion, and pulse collisions,” IEEE J. Quantum Electron. 32, 478–492 (1996); R. G. M. P. Koumans, R. A. Salvatore, D. Eliyahu, R. van Roijen, “Correction to: Theory for passive mode-locking in semiconductor laser structures including the effects of self-phase modulation, dispersion, and pulse collisions,” IEEE J. Quantum Electron. 32, 2017 (1996).

G. H. C. New, “Pulse evolution in mode-locked quasi-continuous lasers,” IEEE J. Quantum Electron. QE-10, 115–124 (1974).
[CrossRef]

IEEE Photon. Technol. Lett.

R. A. Salvatore, A. Yariv, “Demonstration of down-chirped and chirp-free pulses from high-repetition-rate passively mode-locked lasers,” IEEE Photon. Technol. Lett. 7, 1151–1153 (1995).
[CrossRef]

J. Lightwave Technol.

B. Y. Zhu, I. H. White, “Multiwavelength picosecond optical pulse generation using an actively mode-locked multichannel grating cavity laser,” J. Lightwave Technol. 13, 2327–2335 (1995).
[CrossRef]

J. Opt. Soc. Am. B

Laser Focus World

H. W. Messenger, “Developers unveil modelocked erbium fiber laser,” Laser Focus World 32, 15–16 (May1996).

Other

H. Hasaka, Y. Yoshikuni, K. Sato, H. Ishii, H. Sanjoh, “Multiwavelength light source with precise frequency spacing using mode-locked semiconductor laser and arrayed waveguide grating filter,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 299–300.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic, New York, 1965), p. 933.

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Figures (3)

Fig. 1
Fig. 1

Pulse width τ, chirp parameter β (dashed curve), and FWHM of the optical spectra profile Δf as a function of normalized GVD. (a), (b), l/l t = 1.4, g/l t = 2.13, s = 8, α g = 4, and α l = 2; (c), (d), l/l t = 3.16, g/l t = 3.38, s = 12, α g = 6, and α l = 2. The gray areas represent unstable regions for which no stable solution was found.

Fig. 2
Fig. 2

Normalized FWHM of the optical spectra as a function of saturable absorber SPM parameter (α l) for various values of saturable gain self-phase modulation parameter (α g) and GVD. The laser parameters are l = 1.4l t, g = 2.13l t, and s = 8. The solid curve represents the result for α g = 4 and DΩ2/4l t = -5; the dot–dash curve, α g = 6 and DΩ2/4l t = -5; the dashed curve, α g = 4 and DΩ2/4l t = 5; the dot–dash curve represents α g = 6 and DΩ2/4l t = 5. The inset illustrates the dependence of Δf on the stability parameter s for α g = 4, α l = 2, and DΩ2/4l t = -5.

Fig. 3
Fig. 3

(a) Normalized spectral amplitude (Ω/E sl) 1/2 |a(ω)| and (b) spectral phase arg [a(ω)] versus the normalized angular frequency. The laser parameters are l/l t = 1.4, g/l t = 2.13, and s = 8. The solid curves (β = 4.32) illustrate the negative normalized GVD (DΩ2/4l t = -5), whereas the dashed curves represent DΩ2/4l t = 5 (β = -0.179).

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

lt1+2ΔωΩ2+i2ΔωΩ+21-4iΔωΩ1Ωddt-4Ω2d2dt2+1+iαllexp-EtEsl-1+iαgg×exp-EtEsg+iφ-Dd2dt2+ΔTddtat=0,
g=g0expT/τg-1expT/τg-exp-E/Esg,
Et=-tat2dt.
lt+4ltΔωΩ2+1+iαll-1+iαgg+i2ltΔωΩ+φ+-1+iαll+1+iαggsEtEsl+121+iαl×lEtEsl2+2lt+ΩΔT2-4iltΔωΩ1Ωddt-4lt+iΩ2D1Ω2d2dt2at=0.
at=Asecht/τ1+iβ,
β=12Ω2D-4ltαl34lt+Ω2Dαl-94lt+Ω2Dαl2+8Ω2D-4ltαl21/2,
EEsl=12γ1-γ2+γ22-4γ1γ31/2,
Ωτ=44lt-2β2lt-3βD/2Ω2l1/2EslE,
ΔωΩ=-Ωτ8lt1+2β2lβ-αl1+l2β-αlEEsl2-gsβ-αgEEsl,
ΩΔT=-2lt-8ltβΔωΩ-12Ωτl-gsEEsl+14lΩτEEsl2,
φ=-2ltΔωΩ+8ltβΩτ2+D1-β2τ2-lαl+gαg+12lαl-gαgsEEsl-14lαlEEsl2,
γ1=l8lt3-βDΩ2/lt-β22-β2-3βDΩ2/4lt+β-αl22-β2-3βDΩ2/4lt1+2β22,
γ2=-12ltl-gs+-lαl+gαg/s+βl-g/sβ-αl2-β2-3βDΩ2/4lt1+2β22,
γ3=1+l-glt+12llt1+2β22β-αll-β-αggs22-β2-3βDΩ2/4lt.
lt+l-g>0
-lt+l-g+l-gsEEsl-12lEEsl2<0.
l-g/s>0.
aω-atexp-iωtdt=A2iβτΓ1+iβ×Γ12+i2β+ωτΓ12+i2β-ωτ,
aω2=2A2τ2π3βsinhπβcoshπωτ+coshπβ,
coshΔfΔτπ22cosh-12=coshπβ+2,

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