## A theoretical description of Fourier domain mode locked lasers

Optics Express, Vol. 17, Issue 26, pp. 24013-24019 (2009)

http://dx.doi.org/10.1364/OE.17.024013

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### Abstract

The first theoretical model of Fourier domain mode locking operation is presented. A specially tailored dynamic equation in a moving spectral reference frame is derived, enabling efficient numerical treatment, despite the broad laser spectrum and the extremely long cavity. The excellent agreement of the presented theory with experiment over a wide range of operation parameters enables a quantitative assessment of the relevant physical effects, such as the spectral loss modulation and gain saturation dynamics, amplified spontaneous emission, linewidth enhancement, and self-phase modulation.

© 2009 OSA

## 1. Introduction

1. R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography,” Opt. Express **14**(8), 3225–3237 (2006). [CrossRef] [PubMed]

2. D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics **1**(12), 709–716 (2007). [CrossRef]

1. R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography,” Opt. Express **14**(8), 3225–3237 (2006). [CrossRef] [PubMed]

3. 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**(20), 2975–2977 (2006). [CrossRef] [PubMed]

1. R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography,” Opt. Express **14**(8), 3225–3237 (2006). [CrossRef] [PubMed]

4. L. A. Kranendonk, X. An, A. W. Caswell, R. E. Herold, S. T. Sanders, R. Huber, J. G. Fujimoto, Y. Okura, and Y. Urata, “High speed engine gas thermometry by Fourier-domain mode-locked laser absorption spectroscopy,” Opt. Express **15**(23), 15115–15128 (2007). [CrossRef] [PubMed]

5. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science **254**(5035), 1178–1181 (1991). [CrossRef] [PubMed]

*in vivo*experiments with FDML based OCT systems have already demonstrated superior performance in ophthalmology [6

6. V. J. Srinivasan, D. C. Adler, Y. L. Chen, I. Gorczynska, R. Huber, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Ultrahigh-speed optical coherence tomography for three-dimensional and en face imaging of the retina and optic nerve head,” Invest. Ophthalmol. Vis. Sci. **49**(11), 5103–5110 (2008). [CrossRef] [PubMed]

2. D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics **1**(12), 709–716 (2007). [CrossRef]

**14**(8), 3225–3237 (2006). [CrossRef] [PubMed]

## 2. Experimental setup

## 3. Theoretical model

*t*is defined with respect to a frame moving along with the optical field, and

*z*denotes the position in the laser system along the propagation direction, suppressed in the following for a more compact notation. The coefficients

*γ*describe the second and third order dispersion and self-phase modulation, assumed to be constant in the delay fiber and zero for the other optical components. The spectral gain of the optical amplifier and loss in the fiber system are characterized by

*α*, and saturation effects are also included as discussed further below.

*T*. This sweeping action, reflected by a time dependent bandpass center frequency

*instantaneous*frequency of the optical carrier wave to the sweep filter position, leading to the transformed envelopeThis approach accounts for the complex FDML dynamics, and is quite different from conventional swept laser simulations, where the filter detuning occurs over many roundtrips and it is sufficient to shift the frequency window once per roundtrip [9

9. A. Bilenca, S. H. Yun, G. J. Tearney, and B. E. 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**(6), 760–762 (2006). [CrossRef] [PubMed]

## 4. Results

9. A. Bilenca, S. H. Yun, G. J. Tearney, and B. E. 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**(6), 760–762 (2006). [CrossRef] [PubMed]

*t*not directly from the instantaneous optical power

9. A. Bilenca, S. H. Yun, G. J. Tearney, and B. E. 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**(6), 760–762 (2006). [CrossRef] [PubMed]

10. D. Cassioli, S. Scotti, and A. Mecozzi, “A time-domain computer simulator of the nonlinear response of semiconductor optical amplifiers,” IEEE J. Quantum Electron. **36**(9), 1072–1080 (2000). [CrossRef]

*T*of about

^{23}(≈8 million) grid points are used. The simulation is self-starting from ASE noise, and typically converges after 10-1000 roundtrips, depending on the laser parameters. We have ensured convergence of the solution in the time and frequency domain, only limited by the small random fluctuations due to the ASE noise floor. We emphasize that all simulations are performed self-consistently without fitting parameters, i.e., all parameters are obtained from literature or, where not available, from experimental characterization as discussed above. The results enable us to address two crucial questions of FDML operation: (1) What is the role of ASE for self-starting and for the stationary operation of FDML? (2) What physical mechanisms are important for FDML at the various operation points?

*δ*between the sweep filter and the roundtrip time, i.e.,

*A*before the sweep filter at each roundtrip. The dynamics become more complex if the sweep rate is slightly detuned with respect to the roundtrip frequency, i.e.,

## 5. Conclusions

## Acknowledgments

## References and links

1. | R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography,” Opt. Express |

2. | D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics |

3. | 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. |

4. | L. A. Kranendonk, X. An, A. W. Caswell, R. E. Herold, S. T. Sanders, R. Huber, J. G. Fujimoto, Y. Okura, and Y. Urata, “High speed engine gas thermometry by Fourier-domain mode-locked laser absorption spectroscopy,” Opt. Express |

5. | D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science |

6. | V. J. Srinivasan, D. C. Adler, Y. L. Chen, I. Gorczynska, R. Huber, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Ultrahigh-speed optical coherence tomography for three-dimensional and en face imaging of the retina and optic nerve head,” Invest. Ophthalmol. Vis. Sci. |

7. | J. M. Schmitt, R. Huber, and J. G. Fujimoto, “Limiting ischemia by fast Fourier-domain imaging,” in |

8. | G. P. Agrawal, |

9. | A. Bilenca, S. H. Yun, G. J. Tearney, and B. E. 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. | D. Cassioli, S. Scotti, and A. Mecozzi, “A time-domain computer simulator of the nonlinear response of semiconductor optical amplifiers,” IEEE J. Quantum Electron. |

**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

(140.3600) Lasers and laser optics : Lasers, tunable

(170.4500) Medical optics and biotechnology : Optical coherence tomography

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: November 6, 2009

Revised Manuscript: December 11, 2009

Manuscript Accepted: December 11, 2009

Published: December 16, 2009

**Citation**

Christian Jirauschek, Benjamin Biedermann, and Robert Huber, "A theoretical description of Fourier domain mode locked lasers," Opt. Express **17**, 24013-24019 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-24013

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### References

- R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography,” Opt. Express 14(8), 3225–3237 (2006). [CrossRef] [PubMed]
- D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1(12), 709–716 (2007). [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(20), 2975–2977 (2006). [CrossRef] [PubMed]
- L. A. Kranendonk, X. An, A. W. Caswell, R. E. Herold, S. T. Sanders, R. Huber, J. G. Fujimoto, Y. Okura, and Y. Urata, “High speed engine gas thermometry by Fourier-domain mode-locked laser absorption spectroscopy,” Opt. Express 15(23), 15115–15128 (2007). [CrossRef] [PubMed]
- D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991). [CrossRef] [PubMed]
- V. J. Srinivasan, D. C. Adler, Y. L. Chen, I. Gorczynska, R. Huber, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Ultrahigh-speed optical coherence tomography for three-dimensional and en face imaging of the retina and optic nerve head,” Invest. Ophthalmol. Vis. Sci. 49(11), 5103–5110 (2008). [CrossRef] [PubMed]
- J. M. Schmitt, R. Huber, and J. G. Fujimoto, “Limiting ischemia by fast Fourier-domain imaging,” in Optical Coherence Tomography in Cardiovascular Research, E. Regar, P. W. Serruys, and T. G. van Leeuwen, eds. (Informa HealthCare, London, 2007), p. 257.
- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).
- A. Bilenca, S. H. Yun, G. J. Tearney, and B. E. 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(6), 760–762 (2006). [CrossRef] [PubMed]
- D. Cassioli, S. Scotti, and A. Mecozzi, “A time-domain computer simulator of the nonlinear response of semiconductor optical amplifiers,” IEEE J. Quantum Electron. 36(9), 1072–1080 (2000). [CrossRef]

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