Abstract

An analysis of the dynamical features in the output of a Fourier Domain Mode Locked laser is presented. An experimental study of the wavelength sweep-direction asymmetry in the output of such devices is undertaken. A mathematical model based on a set of delay differential equations is developed and shown to agree well with experiment.

© 2013 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Huber, M. Wojtkowski, K. Taira, J. Fujimoto, and K. Hsu, “Amplified, frequency swept lasers for frequency domain reflectometry and OCT imaging: design and scaling principles,” Opt. Express13, 3513–3528 (2005).
    [CrossRef] [PubMed]
  2. R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier domain mmode locking (FDML): a new laser operating regime and applications for optical coherence tomography,” Opt. Express14, 3225–3237 (2006).
    [CrossRef] [PubMed]
  3. A.F. Fercher, W. Drexler, C.K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys.66, 239–303 (2003).
    [CrossRef]
  4. B. R. Biedermann, W. Wieser, C. M. Eigenwillig, T. Klein, and R. Huber, “Direct measurement of the instantaneous linewidth of rapidly wavelength-swept lasers,” Opt. Lett.35, 3733–3735 (2010).
    [CrossRef] [PubMed]
  5. S. Todor, B. Biedermann, W. Wieser, R. Huber, and C. Jirauschek, “Instantaneous lineshape analysis of Fourier domain mode-locked lasers,” Opt. Express19, 8802–8807 (2011).
    [CrossRef] [PubMed]
  6. S. Todor, B. Biedermann, R. Huber, and C. Jirauschek, “Balance of physical effects causing stationary operation of fourier domain mode-locked lasers,”J. Opt. Soc. Am B29, 656–664 (2012).
    [CrossRef]
  7. R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: unidirectional swept laser sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett.31, 2975–2977 (2006).
    [CrossRef] [PubMed]
  8. M. Y. Jeon, J. Zhang, and Z. Chen, “Characterization of Fourier domain modelocked wavelength swept laser for optical coherence tomography imaging,” Opt. Express16, 3727–3737 (2008).
    [CrossRef] [PubMed]
  9. A. Bilenca, S. H. Yun, G. J. Tearney, and B. Bouma, “Numerical study of wavelength-swept semiconductor ring lasers: the role of refractive-index nonlinearities in semiconductor optical amplifiers and implications for biomedical imaging applications,” Opt. Lett.31, 760–762 (2006).
    [CrossRef] [PubMed]
  10. C. Jirauschek, B. Biedermann, and R. Huber, “A theoretical description of Fourier domain mode locked lasers,” Opt. Express17, 24013–24019 (2009).
    [CrossRef]
  11. S. Todor, B. Biedermann, R. Huber, and C. Jirauschek, “Analysis of the optical dynamics in fourier domain mode-locked lasers,” in Advanced Photonics & Renewable Energy, OSA Technical Digest (CD) (Optical Society of America, 2010), p. SWC4.
    [CrossRef]
  12. D. C. Adler, W. Wieser, F. Trepanier, J. M. Schmitt, and R. A. Huber, “Extended coherence length Fourier domain mode locked lasers at 1310 nm,” Opt. Express19, 20930–20939 (2011).
    [CrossRef] [PubMed]
  13. W. Wieser, T. Klein, D. C. Adler, F. Trépanier, C. M. Eigenwillig, S. Karpf, J. M. Schmitt, and R. Huber, “Extended coherence length megahertz FDML and its application for anterior segment imaging,” Biomed. Opt. Express3, 2647–2657 (2012).
  14. A. G. Vladimirov and D. Turaev, “Model for passive mode-locking in semiconductor lasers,” Phys. Rev. A72, 033808 (2005).
    [CrossRef]
  15. A. Vladimirov, D. Turaev, and G. Kozyreff, “Delay differential equations for mode-locked semiconductor lasers,” Opt. Lett.29, 1221–1223 (2004).
    [CrossRef] [PubMed]
  16. A. Vladimirov and D. Turaev, “A new model for a mode-locked semiconductor laser,” Radiophys. and Quantum Electronics47, 769–776 (2004).
    [CrossRef]
  17. S. Kashchenko, “Normalization techniques as applied to the investigation of dynamics of difference-differential equations with a small parameter multiplying the derivative,” Differ. Uravn.25, 1448–1451 (1989).
  18. G. Giacomelli and A. Politi, “Multiple scale analysis of delayed dynamical systems,” Physica D117, 26–42 (1998).
    [CrossRef]
  19. S. Yanchuk and M. Wolfrum, “A multiple timescale approach to the stability of external cavity modes in the lang-kobayashi system using the limit of large delay,” SIAM J. Appl. Dyn. Syst.9, 519–535 (2010).
    [CrossRef]
  20. M. Lichtner, M. Wolfrum, and S. Yanchuk, “The spectrum of delay differential equations with large delay,” SIAM J. Math. Anal.43, 788–802 (2011).
    [CrossRef]

2012 (2)

S. Todor, B. Biedermann, R. Huber, and C. Jirauschek, “Balance of physical effects causing stationary operation of fourier domain mode-locked lasers,”J. Opt. Soc. Am B29, 656–664 (2012).
[CrossRef]

W. Wieser, T. Klein, D. C. Adler, F. Trépanier, C. M. Eigenwillig, S. Karpf, J. M. Schmitt, and R. Huber, “Extended coherence length megahertz FDML and its application for anterior segment imaging,” Biomed. Opt. Express3, 2647–2657 (2012).

2011 (3)

2010 (2)

S. Yanchuk and M. Wolfrum, “A multiple timescale approach to the stability of external cavity modes in the lang-kobayashi system using the limit of large delay,” SIAM J. Appl. Dyn. Syst.9, 519–535 (2010).
[CrossRef]

B. R. Biedermann, W. Wieser, C. M. Eigenwillig, T. Klein, and R. Huber, “Direct measurement of the instantaneous linewidth of rapidly wavelength-swept lasers,” Opt. Lett.35, 3733–3735 (2010).
[CrossRef] [PubMed]

2009 (1)

2008 (1)

2006 (3)

2005 (2)

2004 (2)

A. Vladimirov and D. Turaev, “A new model for a mode-locked semiconductor laser,” Radiophys. and Quantum Electronics47, 769–776 (2004).
[CrossRef]

A. Vladimirov, D. Turaev, and G. Kozyreff, “Delay differential equations for mode-locked semiconductor lasers,” Opt. Lett.29, 1221–1223 (2004).
[CrossRef] [PubMed]

2003 (1)

A.F. Fercher, W. Drexler, C.K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys.66, 239–303 (2003).
[CrossRef]

1998 (1)

G. Giacomelli and A. Politi, “Multiple scale analysis of delayed dynamical systems,” Physica D117, 26–42 (1998).
[CrossRef]

1989 (1)

S. Kashchenko, “Normalization techniques as applied to the investigation of dynamics of difference-differential equations with a small parameter multiplying the derivative,” Differ. Uravn.25, 1448–1451 (1989).

Adler, D. C.

Biedermann, B.

S. Todor, B. Biedermann, R. Huber, and C. Jirauschek, “Balance of physical effects causing stationary operation of fourier domain mode-locked lasers,”J. Opt. Soc. Am B29, 656–664 (2012).
[CrossRef]

S. Todor, B. Biedermann, W. Wieser, R. Huber, and C. Jirauschek, “Instantaneous lineshape analysis of Fourier domain mode-locked lasers,” Opt. Express19, 8802–8807 (2011).
[CrossRef] [PubMed]

C. Jirauschek, B. Biedermann, and R. Huber, “A theoretical description of Fourier domain mode locked lasers,” Opt. Express17, 24013–24019 (2009).
[CrossRef]

S. Todor, B. Biedermann, R. Huber, and C. Jirauschek, “Analysis of the optical dynamics in fourier domain mode-locked lasers,” in Advanced Photonics & Renewable Energy, OSA Technical Digest (CD) (Optical Society of America, 2010), p. SWC4.
[CrossRef]

Biedermann, B. R.

Bilenca, A.

Bouma, B.

Chen, Z.

Drexler, W.

A.F. Fercher, W. Drexler, C.K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys.66, 239–303 (2003).
[CrossRef]

Eigenwillig, C. M.

Fercher, A.F.

A.F. Fercher, W. Drexler, C.K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys.66, 239–303 (2003).
[CrossRef]

Fujimoto, J.

Fujimoto, J. G.

Giacomelli, G.

G. Giacomelli and A. Politi, “Multiple scale analysis of delayed dynamical systems,” Physica D117, 26–42 (1998).
[CrossRef]

Hitzenberger, C.K.

A.F. Fercher, W. Drexler, C.K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys.66, 239–303 (2003).
[CrossRef]

Hsu, K.

Huber, R.

W. Wieser, T. Klein, D. C. Adler, F. Trépanier, C. M. Eigenwillig, S. Karpf, J. M. Schmitt, and R. Huber, “Extended coherence length megahertz FDML and its application for anterior segment imaging,” Biomed. Opt. Express3, 2647–2657 (2012).

S. Todor, B. Biedermann, R. Huber, and C. Jirauschek, “Balance of physical effects causing stationary operation of fourier domain mode-locked lasers,”J. Opt. Soc. Am B29, 656–664 (2012).
[CrossRef]

S. Todor, B. Biedermann, W. Wieser, R. Huber, and C. Jirauschek, “Instantaneous lineshape analysis of Fourier domain mode-locked lasers,” Opt. Express19, 8802–8807 (2011).
[CrossRef] [PubMed]

B. R. Biedermann, W. Wieser, C. M. Eigenwillig, T. Klein, and R. Huber, “Direct measurement of the instantaneous linewidth of rapidly wavelength-swept lasers,” Opt. Lett.35, 3733–3735 (2010).
[CrossRef] [PubMed]

C. Jirauschek, B. Biedermann, and R. Huber, “A theoretical description of Fourier domain mode locked lasers,” Opt. Express17, 24013–24019 (2009).
[CrossRef]

R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: unidirectional swept laser sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett.31, 2975–2977 (2006).
[CrossRef] [PubMed]

R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier domain mmode locking (FDML): a new laser operating regime and applications for optical coherence tomography,” Opt. Express14, 3225–3237 (2006).
[CrossRef] [PubMed]

R. Huber, M. Wojtkowski, K. Taira, J. Fujimoto, and K. Hsu, “Amplified, frequency swept lasers for frequency domain reflectometry and OCT imaging: design and scaling principles,” Opt. Express13, 3513–3528 (2005).
[CrossRef] [PubMed]

S. Todor, B. Biedermann, R. Huber, and C. Jirauschek, “Analysis of the optical dynamics in fourier domain mode-locked lasers,” in Advanced Photonics & Renewable Energy, OSA Technical Digest (CD) (Optical Society of America, 2010), p. SWC4.
[CrossRef]

Huber, R. A.

Jeon, M. Y.

Jirauschek, C.

S. Todor, B. Biedermann, R. Huber, and C. Jirauschek, “Balance of physical effects causing stationary operation of fourier domain mode-locked lasers,”J. Opt. Soc. Am B29, 656–664 (2012).
[CrossRef]

S. Todor, B. Biedermann, W. Wieser, R. Huber, and C. Jirauschek, “Instantaneous lineshape analysis of Fourier domain mode-locked lasers,” Opt. Express19, 8802–8807 (2011).
[CrossRef] [PubMed]

C. Jirauschek, B. Biedermann, and R. Huber, “A theoretical description of Fourier domain mode locked lasers,” Opt. Express17, 24013–24019 (2009).
[CrossRef]

S. Todor, B. Biedermann, R. Huber, and C. Jirauschek, “Analysis of the optical dynamics in fourier domain mode-locked lasers,” in Advanced Photonics & Renewable Energy, OSA Technical Digest (CD) (Optical Society of America, 2010), p. SWC4.
[CrossRef]

Karpf, S.

Kashchenko, S.

S. Kashchenko, “Normalization techniques as applied to the investigation of dynamics of difference-differential equations with a small parameter multiplying the derivative,” Differ. Uravn.25, 1448–1451 (1989).

Klein, T.

Kozyreff, G.

Lasser, T.

A.F. Fercher, W. Drexler, C.K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys.66, 239–303 (2003).
[CrossRef]

Lichtner, M.

M. Lichtner, M. Wolfrum, and S. Yanchuk, “The spectrum of delay differential equations with large delay,” SIAM J. Math. Anal.43, 788–802 (2011).
[CrossRef]

Politi, A.

G. Giacomelli and A. Politi, “Multiple scale analysis of delayed dynamical systems,” Physica D117, 26–42 (1998).
[CrossRef]

Schmitt, J. M.

Taira, K.

Tearney, G. J.

Todor, S.

S. Todor, B. Biedermann, R. Huber, and C. Jirauschek, “Balance of physical effects causing stationary operation of fourier domain mode-locked lasers,”J. Opt. Soc. Am B29, 656–664 (2012).
[CrossRef]

S. Todor, B. Biedermann, W. Wieser, R. Huber, and C. Jirauschek, “Instantaneous lineshape analysis of Fourier domain mode-locked lasers,” Opt. Express19, 8802–8807 (2011).
[CrossRef] [PubMed]

S. Todor, B. Biedermann, R. Huber, and C. Jirauschek, “Analysis of the optical dynamics in fourier domain mode-locked lasers,” in Advanced Photonics & Renewable Energy, OSA Technical Digest (CD) (Optical Society of America, 2010), p. SWC4.
[CrossRef]

Trepanier, F.

Trépanier, F.

Turaev, D.

A. G. Vladimirov and D. Turaev, “Model for passive mode-locking in semiconductor lasers,” Phys. Rev. A72, 033808 (2005).
[CrossRef]

A. Vladimirov and D. Turaev, “A new model for a mode-locked semiconductor laser,” Radiophys. and Quantum Electronics47, 769–776 (2004).
[CrossRef]

A. Vladimirov, D. Turaev, and G. Kozyreff, “Delay differential equations for mode-locked semiconductor lasers,” Opt. Lett.29, 1221–1223 (2004).
[CrossRef] [PubMed]

Vladimirov, A.

A. Vladimirov, D. Turaev, and G. Kozyreff, “Delay differential equations for mode-locked semiconductor lasers,” Opt. Lett.29, 1221–1223 (2004).
[CrossRef] [PubMed]

A. Vladimirov and D. Turaev, “A new model for a mode-locked semiconductor laser,” Radiophys. and Quantum Electronics47, 769–776 (2004).
[CrossRef]

Vladimirov, A. G.

A. G. Vladimirov and D. Turaev, “Model for passive mode-locking in semiconductor lasers,” Phys. Rev. A72, 033808 (2005).
[CrossRef]

Wieser, W.

Wojtkowski, M.

Wolfrum, M.

M. Lichtner, M. Wolfrum, and S. Yanchuk, “The spectrum of delay differential equations with large delay,” SIAM J. Math. Anal.43, 788–802 (2011).
[CrossRef]

S. Yanchuk and M. Wolfrum, “A multiple timescale approach to the stability of external cavity modes in the lang-kobayashi system using the limit of large delay,” SIAM J. Appl. Dyn. Syst.9, 519–535 (2010).
[CrossRef]

Yanchuk, S.

M. Lichtner, M. Wolfrum, and S. Yanchuk, “The spectrum of delay differential equations with large delay,” SIAM J. Math. Anal.43, 788–802 (2011).
[CrossRef]

S. Yanchuk and M. Wolfrum, “A multiple timescale approach to the stability of external cavity modes in the lang-kobayashi system using the limit of large delay,” SIAM J. Appl. Dyn. Syst.9, 519–535 (2010).
[CrossRef]

Yun, S. H.

Zhang, J.

Biomed. Opt. Express (1)

Differ. Uravn. (1)

S. Kashchenko, “Normalization techniques as applied to the investigation of dynamics of difference-differential equations with a small parameter multiplying the derivative,” Differ. Uravn.25, 1448–1451 (1989).

J. Opt. Soc. Am B (1)

S. Todor, B. Biedermann, R. Huber, and C. Jirauschek, “Balance of physical effects causing stationary operation of fourier domain mode-locked lasers,”J. Opt. Soc. Am B29, 656–664 (2012).
[CrossRef]

Opt. Express (6)

Opt. Lett. (4)

Phys. Rev. A (1)

A. G. Vladimirov and D. Turaev, “Model for passive mode-locking in semiconductor lasers,” Phys. Rev. A72, 033808 (2005).
[CrossRef]

Physica D (1)

G. Giacomelli and A. Politi, “Multiple scale analysis of delayed dynamical systems,” Physica D117, 26–42 (1998).
[CrossRef]

Radiophys. and Quantum Electronics (1)

A. Vladimirov and D. Turaev, “A new model for a mode-locked semiconductor laser,” Radiophys. and Quantum Electronics47, 769–776 (2004).
[CrossRef]

Rep. Prog. Phys. (1)

A.F. Fercher, W. Drexler, C.K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys.66, 239–303 (2003).
[CrossRef]

SIAM J. Appl. Dyn. Syst. (1)

S. Yanchuk and M. Wolfrum, “A multiple timescale approach to the stability of external cavity modes in the lang-kobayashi system using the limit of large delay,” SIAM J. Appl. Dyn. Syst.9, 519–535 (2010).
[CrossRef]

SIAM J. Math. Anal. (1)

M. Lichtner, M. Wolfrum, and S. Yanchuk, “The spectrum of delay differential equations with large delay,” SIAM J. Math. Anal.43, 788–802 (2011).
[CrossRef]

Other (1)

S. Todor, B. Biedermann, R. Huber, and C. Jirauschek, “Analysis of the optical dynamics in fourier domain mode-locked lasers,” in Advanced Photonics & Renewable Energy, OSA Technical Digest (CD) (Optical Society of America, 2010), p. SWC4.
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Experimental set-up of the ring laser. Iso: Isolator. FFP-TF: Fiber Fabry-Pérot Tunable Filter. AWG: Arbitrary Waveform Generator. SOA: Semiconductor Optical Amplifier. OSC: Oscilloscope. SMF: Single Mode Fiber. PC: Polarization Controller.

Fig. 2
Fig. 2

Example of optical spectrum in FDML regime.

Fig. 3
Fig. 3

Intensity as measured on the oscilloscope for (a) a detuning of +15mHz and (b) a detuning of −15mHz. The sweeping voltage of the filter is shown (in red) above the filter. The turning point of the sweep is approximately at 0 μs and the central wavelength of the filter is decreasing on the left and increasing on the right. The asymmetry has clearly reversed with the change of sign of the detuning.

Fig. 4
Fig. 4

Intensity as measured on the oscilloscope in a quasistatic regime. The filter sweep frequency was 100 mHz. The sweeping voltage of the filter is shown (in red) above the intensity. The turning point of the sweep is at 0 s and the central wavelength of the filter is decreasing on the left and increasing on the right.

Fig. 5
Fig. 5

Intensity of a CW solution of the model Eqs. (1) and (2) as a function of ω − Δ. In the limit T → ∞ (or α → 0) it is symmetric with respect to ω = Δ. The parameter values are γ = 0.1, κ = 0.2, α = 2.0 and T ≫ 1.

Fig. 6
Fig. 6

Real parts of the eigenvalues of the CW solution vs their imaginary parts. Panels (a) and (b) illustrate modulational and Turing instabilities, respectively. The parameter values are the same as in Fig. 5.

Fig. 7
Fig. 7

Instability boundaries for the CW solutions. Solid and dashed lines show modulational and Turing instabilities, respectively. The parameter values are given in Tab. 1.

Fig. 8
Fig. 8

Direct simulation of the model equations displaying the experimentally relevant asymmetry of the output intensity with respect to sweep direction. The parameter values are given in Tab. 1. The turning point of the modulation is at the 0 point of the x-axis and the sweeping voltage of the filter is shown (in red) above the intensity.

Tables (1)

Tables Icon

Table 1 Parameter values for simulations

Equations (22)

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

t A + A i Δ ( t ) A = κ e ( 1 i α ) G ( t T ) / 2 A ( t T ) ,
t G = γ [ g 0 G ( e G 1 ) | A | 2 ] ,
1 = κ e G 0 / 2 cos θ , Δ ω = κ e G 0 / 2 sin θ , g 0 G 0 = ( e G 0 1 ) R ,
1 + ( Δ ω ) 2 = κ e G 0 , Δ ω = tan ( α G 0 2 + T ω ) .
R n = κ [ g 0 ln ( 1 + ( Δ ω n ) 2 κ ) ] 1 κ + ( Δ ω n ) 2
ω n = Δ tan [ ω n T + α 2 ln ( 1 + ( Δ ω n ) 2 κ ) ] .
A ( t ) = a ( t ) e i 0 t Δ ( τ ) d τ .
t a + a = κ e ( 1 i α ) G ( t T ) / 2 i t T t Δ ( τ ) d τ a ( t T ) .
t a + a = κ e ( 1 i α ) G ( t T ) / 2 a ( t T ) , t G = γ [ g 0 G ( e G 1 ) | a | 2 ] .
A ( t ) = a n e i ω n t + i 0 t Δ ( τ ) d τ ,
t a + a = κ e ( 1 i α ) G ( t T ) / 2 + i ψ ( t ) a ( t T ) ,
ψ ( t ) = r T ε Ω sin ( Ω t ) .
t x = A x + x ( t T ) ,
x = ( ReA ImA G ) ,
A = ( 1 ω Δ 0 Δ ω 1 2 γ ( e G 0 1 ) R 0 γ ( 1 + e G 0 R ) ) ,
= ( 1 Δ ω R 2 ( 1 + α ω α Δ ) ω Δ 1 R 2 ( α + Δ ω ) 0 0 0 ) .
det ( λ I + A + e λ T ) = 0 ,
λ = i ν 0 + μ 1 + i ν 1 T
det ( A i ν 0 I + Y ) = 0 ,
[ γ ( 1 + e G 0 R ) + i ν 0 ] [ ( Y 1 ) 2 ( Δ ω ) 2 + ( Y 1 i ν 0 ) 2 ] = γ R Y ( e G 0 1 ) [ ( Y 1 ) ( 1 + ( Δ ω ) 2 ) + i ν 0 ( α ( Δ ω ) 1 ) ] ,
μ 1 = ν 1 = 0 , and μ 1 = Re [ ln ( 1 + e G 0 R 1 + R ) ] < 0 .
[ α ( Δ ω ) 1 ] 2 < 2 ( 1 + α 2 ) ( 1 + e G 0 R ) ( Δ ω ) 2 ( e G 0 1 ) R ,

Metrics