## Dynamics of Fourier domain mode-locked lasers |

Optics Express, Vol. 21, Issue 16, pp. 19240-19251 (2013)

http://dx.doi.org/10.1364/OE.21.019240

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### 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

## 1. Introduction

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. Express **13**, 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. Express **14**, 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. Express **19**, 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 B **29**, 656–664 (2012) [CrossRef] .

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. Express **13**, 3513–3528 (2005) [CrossRef] [PubMed] .

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. Express **16**, 3727–3737 (2008) [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 B **29**, 656–664 (2012) [CrossRef] .

8. M. Y. Jeon, J. Zhang, and Z. Chen, “Characterization of Fourier domain modelocked wavelength swept laser for optical coherence tomography imaging,” Opt. Express **16**, 3727–3737 (2008) [CrossRef] [PubMed] .

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] .

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] .

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. Express **19**, 20930–20939 (2011) [CrossRef] [PubMed] .

*f*

_{filter}is the sweep rate of the filter and

*T*is the roundtrip time. The detuning affects some aspects of the FDML laser performance such as the number of roundtrip times required for the FDML to operate in a stationary regime after self starting [10

10. C. Jirauschek, B. Biedermann, and R. Huber, “A theoretical description of Fourier domain mode locked lasers,” Opt. Express **17**, 24013–24019 (2009) [CrossRef] .

10. C. Jirauschek, B. Biedermann, and R. Huber, “A theoretical description of Fourier domain mode locked lasers,” Opt. Express **17**, 24013–24019 (2009) [CrossRef] .

## 2. Experimental setup

## 3. Experimental results

### 3.1. FDML swept source regime

*μ*s indicating the moment when the sweep changes from decreasing wavelength to increasing wavelength. An obvious qualitative change in the laser output is observed close to the turning point for both cases. For the positive detuning and forward wavelength sweep case, the output resembled a series of jumps from one stable output to another while the backward wavelength sweep resulted in complex GHz oscillations, as shown in the inset. For the negative detuning case the situation was reversed.

### 3.2. Quasistatic regimes

## 4. Theory

### 4.1. Model equations and CW solutions

14. A. G. Vladimirov and D. Turaev, “Model for passive mode-locking in semiconductor lasers,” Phys. Rev. A **72**, 033808 (2005) [CrossRef] .

16. A. Vladimirov and D. Turaev, “A new model for a mode-locked semiconductor laser,” Radiophys. and Quantum Electronics **47**, 769–776 (2004) [CrossRef] .

*A*(

*t*) is the electric field envelope at the entrance of the SOA and the carrier density is modeled via a saturable gain

*G*(

*t*). The position of the central frequency of the spectral filter is defined by the time dependent quantity Δ(

*t*) =

*r*sin(Ω

*t*), where the amplitude

*r*and frequency Ω of the sweep are normalized to the filter bandwidth Γ. The time variable

*t*is normalized to the inverse filter bandwidth Γ

^{−1}. For simplicity the gain bandwidth is assumed to be infinitely wide. The parameters

*κ*and

*α*are the linear attenuation factor per cavity round trip and the linewidth enhancement factor respectively and

*g*

_{0}and

*γ*are the linear unsaturated gain parameter and normalized gain relaxation rate in the SOA, respectively. The normalized delay time is

*T*≫ 1. Since, as mentioned above, the qualitative properties of the laser intensity time traces remain unchanged for different central wavelengths of the spectrum, we ignore chromatic dispersion together with some other phenomena such as the Kerr nonlinearity in the fiber delay line, and show that this minimal model is sufficient to understand the dynamics of the system.

*A*=

*A*

_{0}

*e*and

^{iωt}*G*=

*G*

_{0}. Substituting these into Eqs. (1) and (2) we get the following three equations for the modal amplitude

*A*

_{0}, frequency

*ω*, and saturated gain

*G*

_{0}: where

*R*= |

*A*

_{0}|

^{2}and

*n*. The intensities of these modes are given by with the frequencies

*ω*obeying the transcendental equation Figure 5 shows a typical example of a CW solution.

_{n}*t*). First let us change the reference frame to one comoving with the filter. This is achieved with the transformation Substituting this into Eq. (1) yields: Now let us consider the FDML regime where the filter is swept with a period exactly equal to the cavity round-trip time. In this case Δ(

*t*) =

*r*sin(Ω

*t*) with Ω = 2

*π/T*. With this choice for Δ(

*t*) and performing the integration in the exponential the system becomes autonomous: This is identical in form to the system with a static filter centered at Δ = 0. The CW solutions have the same form as those of Eqs. (1) and (2),

*a*(

*t*) =

*a*

_{n}*e*

^{iωnt}, in the comoving frame. However, unlike the CW solutions that appear in the genuine absence of a frequency sweep corresponding to the usual cavity longitudinal laser modes, these solutions are “FDML modes” since in the lab frame they correspond to the chirped frequency swept solutions where |

*a*|

_{n}^{2}=

*R*and

_{n}*ω*are defined by Eqs. (4) and (5). This is the formal demonstration of the earlier intuitive idea of the equivalence of the static and exactly synchronous FDML operations.

_{n}*t*) =

*r*sin(

*εt*) with

*ε*≪ 2

*π/T*. Following the above steps we find with

*ψ*(

*t*) =

*rT*sin(

*εt*) and

*dψ*(

*t*)/

*dt*=

*O*(

*ε*), where we have used sin(

*εT*) ≈

*εT*. One might expect that this should be formally equivalent to slightly detuned FDML operation. To test this we consider a period of modulation close to the cavity round trip time, Δ(

*t*) =

*r*sin(Ω

*t*) with Ω = 2

*π/T*+

*ε*and

*ε*≪ Ω. Following the same steps and using the same relation sin(

*εT*) ≈

*εT*as above we get the same Eq. (8) but with Here, as before the phase

*ψ*is a slowly varying function of time,

*dψ*(

*t*)/

*dt*=

*O*(

*ε*). The difference however is that while in the quasistatic case the results of our stability analysis will concern the usual CW longitudinal modes

*A*

_{n}e^{iωnt}, in the case of slightly detuned FDML operation these results will be applied to the chirped modes.

### 4.2. Stability of CW solutions and numerical results

*π/T*.

*ν*≈ 2

_{k}*πk/T*as

*T*→ ∞ with

*ν*

_{k}_{+1}−

*ν*= 2

_{k}*π/T*equal to the spacing of the longitudinal cavity modes. In Fig. 6 (a) a modulational-like instability is shown where eigenvalues with small nonzero

*k*-numbers, 0 < |

*k*| <

*k*, where

_{c}*k*is come critical number, have crossed the imaginary axis. The instability of a given mode is associated with the growth of perturbations of the closest longitudinal modes. In Fig. 6 (b) a Turing-like instability is shown where eigenvalues with sufficiently large

_{c}*k*-numbers centered around some |

*k*| =

*k*

_{0}acquire positive real parts.

*μ*

_{1}(

*ν*

_{0}) at

*ν*

_{0}= 0. This condition can be rewritten in the form where

*R*and

*ω*are defined by (4) and (5), respectively. It follows from this inequality that the mode at the filter centre frequency

*ω*= Δ, is always stable with respect to modulational instability. With an increase of |

*ω*− Δ| a modulation instability of the the CW solution sets in (see Fig. 6(a)).

*ω*− Δ the destabilization of the CW solution can take place either via a modulational instability or via a Turing instability. For zero

*α*due to the symmetry of the model equations, the CW solution and its stability properties depend only on the absolute value |

*ω*− Δ|. Thus, at

*α*= 0 the instability experienced by the CW regime is the same for both directions of the sweep of the filter central frequency Δ. This symmetry is lost for sufficiently large values of

*α*explaining the asymmetry of the experimentally observed time-traces with respect to the frequency sweep direction since typically SOAs such as that used in the experiment display an

*α*greater than or approximately equal to 2. When the CW state is destabilized via a (long wavelength) modulational instability a transition to a complex and possibly chaotic, oscillating solution takes place, see Fig. 8. This suggests that the modulational instability is supercritical resulting in the creation of a stable oscillating solution from the destabilization of the CW regime. On the other hand, destabilization of the CW solution via the Turing instability results in a frequency jump to another CW solution with a frequency jump of the order of 2

*πk*

_{0}/

*T*. This suggests that the Turing instability is subcritical in the sense that any solution created from the destabilization of the CW mode is also unstable. A numerical simulation of the output is shown in Fig. 8. The simulation parameters are presented in Tab. 1.

*ε*between the filter sweep rate and the inverse cavity round trip time (

*ε*= Ω − 2

*π/T*). The form is precisely that of the quasistatic case and so, for sufficient detunings, the asymmetry must be preserved with abrupt frequency jumps arising from a Turing-like instability in one sweep direction and complex oscillations from a modulational instability in the other. However, because of the very different rates in the sine arguments, the asymmetry arises within one cavity round trip rather than over the long

*ε*

^{−1}time scale of the quasistatic case. Furthermore, it is seen from Eqs. (8) and (9) that in the FDML case a change of the sign of the frequency sweep detuning

*ε*→ −

*ε*is equivalent to a reversal of the frequency sweep direction. This explains the experimental result that the sweep direction asymmetry in the intensity traces is reversed when the detuning is changed from positive to negative.

## 5. Conclusions

## Acknowledgments

## References and links

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. Express |

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. Express |

3. | A.F. Fercher, W. Drexler, C.K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. |

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. |

5. | S. Todor, B. Biedermann, W. Wieser, R. Huber, and C. Jirauschek, “Instantaneous lineshape analysis of Fourier domain mode-locked lasers,” Opt. Express |

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 B |

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. |

8. | M. Y. Jeon, J. Zhang, and Z. Chen, “Characterization of Fourier domain modelocked wavelength swept laser for optical coherence tomography imaging,” Opt. Express |

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. |

10. | C. Jirauschek, B. Biedermann, and R. Huber, “A theoretical description of Fourier domain mode locked lasers,” Opt. Express |

11. | S. Todor, B. Biedermann, R. Huber, and C. Jirauschek, “Analysis of the optical dynamics in fourier domain mode-locked lasers,” in |

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. Express |

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. Express |

14. | A. G. Vladimirov and D. Turaev, “Model for passive mode-locking in semiconductor lasers,” Phys. Rev. A |

15. | A. Vladimirov, D. Turaev, and G. Kozyreff, “Delay differential equations for mode-locked semiconductor lasers,” Opt. Lett. |

16. | A. Vladimirov and D. Turaev, “A new model for a mode-locked semiconductor laser,” Radiophys. and Quantum Electronics |

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. |

18. | G. Giacomelli and A. Politi, “Multiple scale analysis of delayed dynamical systems,” Physica D |

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. |

20. | M. Lichtner, M. Wolfrum, and S. Yanchuk, “The spectrum of delay differential equations with large delay,” SIAM J. Math. Anal. |

**OCIS Codes**

(110.4500) Imaging systems : Optical coherence tomography

(140.3430) Lasers and laser optics : Laser theory

(140.3600) Lasers and laser optics : Lasers, tunable

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: May 7, 2013

Revised Manuscript: June 27, 2013

Manuscript Accepted: June 27, 2013

Published: August 6, 2013

**Virtual Issues**

Vol. 8, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

S. Slepneva, B. Kelleher, B. O’Shaughnessy, S.P. Hegarty, A.G. Vladimirov, and G. Huyet, "Dynamics of Fourier domain mode-locked lasers," Opt. Express **21**, 19240-19251 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-16-19240

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### References

- 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]
- 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]
- 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]
- 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]
- 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, “Balance of physical effects causing stationary operation of fourier domain mode-locked lasers,”J. Opt. Soc. Am B29, 656–664 (2012). [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]
- 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]
- 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]
- 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]
- 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]
- 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).
- A. G. Vladimirov and D. Turaev, “Model for passive mode-locking in semiconductor lasers,” Phys. Rev. A72, 033808 (2005). [CrossRef]
- 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]
- 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).
- G. Giacomelli and A. Politi, “Multiple scale analysis of delayed dynamical systems,” Physica D117, 26–42 (1998). [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]
- 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]

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