## Dynamics of ultimate spectral narrowing in a semiconductor fiber-grating laser with an intra-cavity saturable absorber

Optics Express, Vol. 14, Issue 7, pp. 2706-2714 (2006)

http://dx.doi.org/10.1364/OE.14.002706

Acrobat PDF (613 KB)

### Abstract

The ultimate spectrum-narrowing and side-mode suppression due to the presence of a saturable absorber in an external cavity of a fiber Bragg grating semiconductor laser is numerically simulated. The proposed algorithm describes an effect of absorption bleaching in a saturable absorber using earlier measurements and shows the evolution of a dynamic grating in the laser cavity. The simulations confirm for the first time an empirical theory of spectral line narrowing in a laser with an intra-cavity saturable absorber.

© 2006 Optical Society of America

## 1. Introduction

1. W. H. Loh, R. I. Laming, M. N. Zervas, M. C Farries, and U. Koren, “Single frequency erbium fiber external cavity semiconductor laser,” Appl. Phys. Lett. **66**, 3422–3424 (1995). [CrossRef]

4. R. N. Liu, I. A. Kostko, R. Kashyap, K. Wu, and P. Kiiveri, “Inband-pumped, broadband bleaching of absorption and refractive index changes in erbium doped fiber,” Opt. Commun. **255**, 65–71 (2005). [CrossRef]

1. W. H. Loh, R. I. Laming, M. N. Zervas, M. C Farries, and U. Koren, “Single frequency erbium fiber external cavity semiconductor laser,” Appl. Phys. Lett. **66**, 3422–3424 (1995). [CrossRef]

4. R. N. Liu, I. A. Kostko, R. Kashyap, K. Wu, and P. Kiiveri, “Inband-pumped, broadband bleaching of absorption and refractive index changes in erbium doped fiber,” Opt. Commun. **255**, 65–71 (2005). [CrossRef]

5. I. A. Kostko and R. Kashyap, “Modeling of self-organized coherence-collapsed and enhanced regime semiconductor fibre grating reflector lasers,” in *Photonic Applications in Telecommunications, Sensors, Software, and Lasers*,
J. C. Armitage, R. A. Lessard, and G. A. Lampropoulos, eds, Proc. SPIE **5579**, 367–374 (2004). [CrossRef]

6. C.R. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” J. Lightwave Technol. **9**, 271–283 (1991). [CrossRef]

4. R. N. Liu, I. A. Kostko, R. Kashyap, K. Wu, and P. Kiiveri, “Inband-pumped, broadband bleaching of absorption and refractive index changes in erbium doped fiber,” Opt. Commun. **255**, 65–71 (2005). [CrossRef]

^{-6}and a high coupling coefficient κL ~1 [5

5. I. A. Kostko and R. Kashyap, “Modeling of self-organized coherence-collapsed and enhanced regime semiconductor fibre grating reflector lasers,” in *Photonic Applications in Telecommunications, Sensors, Software, and Lasers*,
J. C. Armitage, R. A. Lessard, and G. A. Lampropoulos, eds, Proc. SPIE **5579**, 367–374 (2004). [CrossRef]

^{-6}) and lower coupling coefficient of the dynamic grating (κL < 0.5) [4

**255**, 65–71 (2005). [CrossRef]

_{max}L ~ 0.04) dynamic grating, and show that, even with such weaker gratings, line-narrowing is indeed possible.

## 2. Evolution of the dynamic grating

10. A. J. Lowery, “Dynamic modelling of distributed-feedback lasers using scattering matrices,” Electron. Lett. **25**, 1307–1308 (1989). [CrossRef]

5. I. A. Kostko and R. Kashyap, “Modeling of self-organized coherence-collapsed and enhanced regime semiconductor fibre grating reflector lasers,” in *Photonic Applications in Telecommunications, Sensors, Software, and Lasers*,
J. C. Armitage, R. A. Lessard, and G. A. Lampropoulos, eds, Proc. SPIE **5579**, 367–374 (2004). [CrossRef]

10. A. J. Lowery, “Dynamic modelling of distributed-feedback lasers using scattering matrices,” Electron. Lett. **25**, 1307–1308 (1989). [CrossRef]

10. A. J. Lowery, “Dynamic modelling of distributed-feedback lasers using scattering matrices,” Electron. Lett. **25**, 1307–1308 (1989). [CrossRef]

*Photonic Applications in Telecommunications, Sensors, Software, and Lasers*,
J. C. Armitage, R. A. Lessard, and G. A. Lampropoulos, eds, Proc. SPIE **5579**, 367–374 (2004). [CrossRef]

*α(P,t)*as well as refractive index modulation

*∆n(P,t)*of the dynamic grating.

**255**, 65–71 (2005). [CrossRef]

**255**, 65–71 (2005). [CrossRef]

*α*

_{min}and

*α*

_{max}are the minimum (fully bleached) and the maximum (non-bleached) absorption.

*P*

_{min}is the minimum optical power, which causes the absorption bleaching.

*P*

_{max}is the minimum optical power, which causes

*full*absorption bleaching.

*P*the bleaching of absorption begins. This power-dependent absorption bleaching and spatial hole burning caused by the standing wave in the cavity, result in the absorption modulation along the doped fiber leading to an absorption (loss) grating. In this grating, the absorption varies between bleached (

_{min}*α*

_{n+1}=

*α*(

*P*,

*t*)) according to the Eq. (1) and not bleached (

*α*=

_{n}*α*

_{max}) absorption, where

*n*is a subsection number in the grating calculation. The absorption bleaching increases the refractive index modulation

*∆n*in the SA and thus forms a dynamic refractive index grating. The absorption and the refractive index gratings form a unified “dynamic grating”.

*∆α(ω’,ω)*due to the narrow-linewidth pumping at the frequency

*ω*gives the change in refractive index,

*∆n(ω)*[4

**255**, 65–71 (2005). [CrossRef]

*c*is the velocity of light in vacuum, and

*P. V.*is the principal value of the integral calculated over the frequency range

*ω*<

_{1}*ω’*<

*ω*, where the absorption changes are significant. Using the refractive index change we obtain a coupling factor of an induced grating:

_{2}*n*is a refractive index of the doped and

*λ*is the central wavelength of the dynamic grating. Although Eqs. (2)–(3) for the dynamic grating have been presented and discussed earlier in Ref [4

**255**, 65–71 (2005). [CrossRef]

*∆n*along the SA also depends on the optical power

*P(t)*. Therefore, the increase of the power

*P(t)*inside the cavity alters the coupling factor

*κ*of the formed dynamic grating. The growth of the coupling factor with power is calculated as:

*κ*

_{min}and

*κ*

_{max}are the minimum (non-bleached) and the maximum (fully bleached) coupling constant of the dynamic grating, which may be derived from Eqs. (2)–(3) assuming that the absorption is fully bleached: ∆

*α*(

*ω*') = ∆

*α*

_{max}(

*ω*'). Only the mode that has enough optical power to bleach the absorption forms the dynamic grating. In the model we assume that the central wavelength in the reflectivity of the dynamic refractive index grating coincides with the maximum in the reflected spectrum of the FBG.

*A*and

*B*are the forward and backward propagating waves, superscripts

*i*and

*r*denotes initial and reflected parts,

*n*is the subsection number,

*η*is integer,

*k*- the time-step,

*∆L*is the subsection length. In Eq. (5) integer

*η*= 1 for a low-high step in impedance (

*n*is odd) and

*η*= -1 for the high-low impedance transition (

*n*is even). The forward and backward traveling waves in the doped fiber are multiplied by a loss factor exp(-

*α∆L*/ 2) as they traverse a subsection. To take into account the loss grating formed by the standing wave, we have assumed that

*α*=

_{n}*α*

_{max}and

*α*

_{n+1}=

*α*(

*P*,

*t*) for two adjacent subsections.

- The DFECL is simulated with the initial coupling of the dynamic grating inside the SA κL~0.001 and absorption
*α*_{max}. - The optical power, reflected from the external FBG into the SA during the “iteration” of ∆t ~ 17 nsec, is calculated. The Fast-Fourier Transform (FFTW) [11] over the time ∆t gives the internal optical spectrum
*P*(∆_{in}*t*, ω) injected into the SA (Fig. 1). Due to the long iteration time ∆t, the calculated spectrum has a resolution of 0.2 pm. - The spectrum
*P*(∆_{in}*t*,*ω*) is analyzed: the frequency*ω*_{max}and the optical power*P*(_{in}*t*,*ω*_{max}) of the dominant longitudinal mode of the internal spectrum are derived. The frequency of the dominant mode*ω*_{max}defines the central frequency of the dynamic grating. - The optical power
*P*(*t*) =*P*(_{in}*t*,*ω*_{max}) is used to calculate the new SA parameters, absorption*α*and the dynamic grating coupling*κL*, through Eq. (1) and Eq. (4). If*P*_{min}<*P*(_{in}*t*,*ω*_{max}) <*P*_{max}, the bleached absorption*α*in the absorption grating decreases following Eq. (1). At the same time, the refractive index modulation and the coupling factor*κ*of the dynamic grating increases through the Eq. (4). Hence, the reflectivity of the dynamic grating grows with increasing internal optical power, as*R*= tanh^{2}[*κ*(*P*)_{in}*L*]. - The adjusted absorption
*α*and the coupling factor*κ*of the dynamic grating are used for the next iteration. - Steps 2-5 of the algorithm are repeated for > 600 times (10 microseconds).

## 3. Spectral line-narrowing in the DFECL

1. W. H. Loh, R. I. Laming, M. N. Zervas, M. C Farries, and U. Koren, “Single frequency erbium fiber external cavity semiconductor laser,” Appl. Phys. Lett. **66**, 3422–3424 (1995). [CrossRef]

2. F. N. Timofeev and R. Kashyap, “High-power, ultra-stable, single-frequency operation of a long, doped-fiber external-cavity, grating-semiconductor laser,” Opt. Express **11**, 515–520 (2003),
http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-515 [CrossRef] [PubMed]

3. R. Liu, I. Kostko, K. Wu, and R. Kashyap, “Optical generation of microwave signal by doped fiber external cavity semiconductor laser for radio-over-fiber transmission,” in *Photonic Applications in Nonlinear Optics, Nanophotonics, and Microwave Photonics*,
R. A. Morandotti, H. E. Ruda, and J. Yao, eds, Proc. SPIE **5971**, 59711W (2005). [CrossRef]

_{min}= 0.1 mW, P

_{max}= 4 mW; α

_{min}= 0.5 dB/m, α

_{max}= 10 dB/m. Using these absorption bleaching parameters in the Kramers-Kronig relations we have derived the refractive index modulation, ∆n~0.3 × 10

^{-6}, in the SA and the coupling of the dynamic grating

*κ*

_{max}L ~ 0.04 (L=10 cm), in keeping with our earlier calculations [4

**255**, 65–71 (2005). [CrossRef]

_{th}~ 20 mA and total length of the external cavity was 24 cm. The external FBG had the maximum of the reflectivity spectrum, R

_{FBG}~ 0.6, centre wavelength of 976.4 nm, and a bandwidth ∆λ

_{FBG}= 0.15 nm. Using the model we have simulated a laser with similar parameters without the saturable absorber in the cavity. As expected, the output spectrum had over a 100 longitudinal modes and did not show any evidence of line narrowing.

*κ*L = 0.04 (R ~ 0.16%, ∆

*λ*~ 6.6 pm) is formed during the first ~ 20 iterations (340 nsec) the side-mode suppression is very low [Fig. 5(a)]. We believe that further behavior of the laser spectrum is defined by the relaxation oscillations of power in the long external cavity. An increase in the optical power leads to the oscillation of high number of external-cavity modes and increase of mode competition, reducing the power of a dominant mode and making the wavelength unstable in phases II and III in Fig. 4. The resonance in the phase III increases the output power by up to 6 mW (calculated but not shown here) and significantly suppresses the power of the dominant mode, leading to weakening of the dynamic grating (Fig. 3). Following the relaxation oscillation, the increased power in a dominant mode forms the grating for the second time and although neither coupling nor absorption of the SA changes, the spectrum of the laser continues to remain multi-mode [Fig. 5(b)]. During this time the reflection from the dynamic grating gradually increases the power in a dominant mode and, due to its narrow bandwidth, suppresses the side modes. After the dominating external-cavity mode has been stable for ~ 200 nsec (80 round-trips) the laser becomes single-mode with ~ 27 dB side-mode suppression ratio (phase IV in Fig. 4, spectrum on Fig. 6). The laser remains in a single-mode regime for the rest of the simulation time, up-to 10 microseconds.

**66**, 3422–3424 (1995). [CrossRef]

*τ*

_{sim}, depends on the drive current of the laser diode and on the initial absorption

*α*

_{max}of the saturable absorber. The time of stabilization in our simulations,

*τ*

_{sim}~2.3 microseconds, is much longer than simulated

*τ*

_{sim}of a conventional external cavity laser (~ 5 nsec) and longer than

*τ*

_{sim}in our earlier model of a DFECL with

*fixed*parameters of the “dynamic” grating [5

*Photonic Applications in Telecommunications, Sensors, Software, and Lasers*,
J. C. Armitage, R. A. Lessard, and G. A. Lampropoulos, eds, Proc. SPIE **5579**, 367–374 (2004). [CrossRef]

3. R. Liu, I. Kostko, K. Wu, and R. Kashyap, “Optical generation of microwave signal by doped fiber external cavity semiconductor laser for radio-over-fiber transmission,” in *Photonic Applications in Nonlinear Optics, Nanophotonics, and Microwave Photonics*,
R. A. Morandotti, H. E. Ruda, and J. Yao, eds, Proc. SPIE **5971**, 59711W (2005). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | W. H. Loh, R. I. Laming, M. N. Zervas, M. C Farries, and U. Koren, “Single frequency erbium fiber external cavity semiconductor laser,” Appl. Phys. Lett. |

2. | F. N. Timofeev and R. Kashyap, “High-power, ultra-stable, single-frequency operation of a long, doped-fiber external-cavity, grating-semiconductor laser,” Opt. Express |

3. | R. Liu, I. Kostko, K. Wu, and R. Kashyap, “Optical generation of microwave signal by doped fiber external cavity semiconductor laser for radio-over-fiber transmission,” in |

4. | R. N. Liu, I. A. Kostko, R. Kashyap, K. Wu, and P. Kiiveri, “Inband-pumped, broadband bleaching of absorption and refractive index changes in erbium doped fiber,” Opt. Commun. |

5. | I. A. Kostko and R. Kashyap, “Modeling of self-organized coherence-collapsed and enhanced regime semiconductor fibre grating reflector lasers,” in |

6. | C.R. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” J. Lightwave Technol. |

7. | R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. |

8. | C. Barnard, P. Myslinski, Chrostowski J., and M. Kavehrad, “Analytical model for rare-earth-doped fiber amplifiers and lasers,” IEEE J. Quantum Electron. |

9. | A. W. T. Wu and A. J. Lowery, “Efficient multiwavelength dynamic model for erbium-doped fiber amplifier,” IEEE J. Quantum Electron. |

10. | A. J. Lowery, “Dynamic modelling of distributed-feedback lasers using scattering matrices,” Electron. Lett. |

11. | M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” in |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(050.2770) Diffraction and gratings : Gratings

(060.2380) Fiber optics and optical communications : Fiber optics sources and detectors

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: December 14, 2005

Revised Manuscript: March 14, 2006

Manuscript Accepted: March 16, 2006

Published: April 3, 2006

**Citation**

Irina A. Kostko and Raman Kashyap, "Dynamics of ultimate spectral narrowing in a semiconductor fiber-grating laser with an intra-cavity saturable absorber," Opt. Express **14**, 2706-2714 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2706

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### References

- W. H. Loh, R. I. Laming, M. N. Zervas, M. C Farries, and U. Koren, "Single frequency erbium fiber external cavity semiconductor laser," Appl. Phys. Lett. 66,3422-3424 (1995). [CrossRef]
- F. N. Timofeev and R. Kashyap, "High-power, ultra-stable, single-frequency operation of a long, doped-fiber external-cavity, grating-semiconductor laser," Opt. Express 11, 515-520 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-515 [CrossRef] [PubMed]
- R. Liu, I. Kostko, K. Wu, and R. Kashyap, "Optical generation of microwave signal by doped fiber external cavity semiconductor laser for radio-over-fiber transmission," in Photonic Applications in Nonlinear Optics, Nanophotonics, and Microwave Photonics, R. A. Morandotti, H. E. Ruda, J. Yao, eds, Proc. SPIE 5971, 59711W (2005). [CrossRef]
- R. N. Liu, I. A. Kostko, R. Kashyap, K. Wu, and P. Kiiveri, "Inband-pumped, broadband bleaching of absorption and refractive index changes in erbium doped fiber," Opt. Commun. 255, 65-71 (2005). [CrossRef]
- I. A. Kostko and R. Kashyap, "Modeling of self-organized coherence-collapsed and enhanced regime semiconductor fibre grating reflector lasers," in Photonic Applications in Telecommunications, Sensors, Software, and Lasers, J. C. Armitage, R. A. Lessard, G. A. Lampropoulos, eds, Proc. SPIE 5579, 367-374 (2004). [CrossRef]
- C. R. Giles and E. Desurvire, "Modeling erbium-doped fiber amplifiers," J. Lightwave Technol. 9, 271-283 (1991). [CrossRef]
- R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, "Ytterbium-doped fiber amplifiers," IEEE J. Quantum Electron. 33,1049-1056 (1997) [CrossRef]
- C. Barnard, P. Myslinski, J. Chrostowski, M. Kavehrad, "Analytical model for rare-earth-doped fiber amplifiers and lasers," IEEE J. Quantum Electron. 30, 1817-1830 (1994). [CrossRef]
- A. W. T. Wu and A. J. Lowery, "Efficient multiwavelength dynamic model for erbium-doped fiber amplifier," IEEE J. Quantum Electron. 34, 1325-1331 (1998). [CrossRef]
- A. J. Lowery, "Dynamic modelling of distributed-feedback lasers using scattering matrices," Electron. Lett. 25, 1307-1308 (1989). [CrossRef]
- M. Frigo and S. G. Johnson, "The design and implementation of FFTW3," inProceedings of IEEE, Special Issue on Program Generation, Optimization, and Platform Adaptation, 93, 216-231 (2005).

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