## Theory of the ultrafast mode-locked GaN lasers in a large-signal regime |

Optics Express, Vol. 19, Issue 18, pp. 17114-17120 (2011)

http://dx.doi.org/10.1364/OE.19.017114

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### Abstract

Analytical theory of the high-power passively mode-locked laser with a slow absorber is developed. In distinguishing from previous treatment, our model is valid at pulse energies well exceeding the saturation energy of absorber. This is achieved by solving the mode-locking master equation in the pulse energy-domain representation. The performances of monolithic sub-picosecond blue-violet GaN mode-locked diode laser in the high-power operation regime are analyzed using the developed approach.

© 2011 OSA

## 1. Introduction

1. S. Nakamura, M. Senoh, S.- Nagahama, N. Iwasa, T. Matsushita, and T. Mukai, “Blue InGaN-based laser diodes with an emission wavelength of 450 nm,” Appl. Phys. Lett. **76**(1), 22–24 (2000). [CrossRef]

2. K. Saito, H. Watanabe, T. Miyajima, M. Ikeda, and H. Yokoyama, “Mode locking of an external-cavity bisection GaInN blue-violet laser diode producing 3 ps duration optical pulses,” Appl. Phys. Lett. **96**(3), 031112 (2010). [CrossRef]

4. G. H. C. New, “Pulse evolution in mode-locked quasi-continuous lasers,” IEEE J. Quantum Electron. **10**(2), 115–124 (1974). [CrossRef]

5. H. A. Haus, “Theory of mode-locking with a slow saturable absorber,” IEEE J. Quantum Electron. **11**, 736-746 (1975); “Mode-locking of lasers,” IEEE J. Sel,” Topics in QE 6(6), 1173–1185 (2000). [CrossRef] [CrossRef]

6. R. G. M. P. Koumans and 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. QE. **32**(3), 478–492 (1996). [CrossRef]

7. J. Leegwater, “Theory of Mode-Locked Semiconductor Lasers,” IEEE J. Quantum Electron. **32**(10), 1782–1790 (1996). [CrossRef]

*E*=

_{sA}*ħωv*/Γ

_{g}dh*s*(∂g*/∂n) of 2.5 pJ. The expected energy

*W*of the pulse travelling in the cavity is in the range 1-10pJ, which is comparable or higher then

_{p}*E*. This renders previous time-domain developments not suitable for GaN LDs.

_{sA}## 2. Haus-New’s theoretical model in time-domain representation

4. G. H. C. New, “Pulse evolution in mode-locked quasi-continuous lasers,” IEEE J. Quantum Electron. **10**(2), 115–124 (1974). [CrossRef]

5. H. A. Haus, “Theory of mode-locking with a slow saturable absorber,” IEEE J. Quantum Electron. **11**, 736-746 (1975); “Mode-locking of lasers,” IEEE J. Sel,” Topics in QE 6(6), 1173–1185 (2000). [CrossRef] [CrossRef]

6. R. G. M. P. Koumans and 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. QE. **32**(3), 478–492 (1996). [CrossRef]

7. J. Leegwater, “Theory of Mode-Locked Semiconductor Lasers,” IEEE J. Quantum Electron. **32**(10), 1782–1790 (1996). [CrossRef]

*T*in the gain-section quantum wells (QWs) is significantly longer than generated picosecond pulses so as the relaxation

_{L}*during*the pulse can be neglected (see Table 1). Even though the recovery time

*T*of absorber is only few times longer the pulse, the absorber relaxation

_{A}*during*the pulse is neglected as well. (The effect of partial relaxation within the pulse has been accounted for in the numerical model of Ref. [8

8. J. L. A. Dubbeldam, J. A. Leegwater, and D. Lenstra, “Theory of mode-locked semiconductor lasers with finite absorber relaxation times,” Appl. Phys. Lett. **70**(15), 1938 (1997). [CrossRef]

6. R. G. M. P. Koumans and 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. QE. **32**(3), 478–492 (1996). [CrossRef]

7. J. Leegwater, “Theory of Mode-Locked Semiconductor Lasers,” IEEE J. Quantum Electron. **32**(10), 1782–1790 (1996). [CrossRef]

*q*and

_{i}*g*are the saturated absorption and gain in the cavity at the beginning of a pulse,

_{i}*q*exp(-

_{i}*W*/

_{p}*E*) and

_{sA}*g*exp(-

_{i}*W*/

_{p}*E*) represent the same large-signal parameters at the end of the pulse. Solving the rate equations during the cavity roundtrip

_{sA}*T*time, when these relax back to the initial absorption

_{rep}*q*. and gain

_{i}*g*, one obtains the relationships between the steady-state saturated and unsaturated small-signal values of the absorption and gain in the cavity [6

_{i}**32**(3), 478–492 (1996). [CrossRef]

**32**(10), 1782–1790 (1996). [CrossRef]

*between*the subsequent pulses. For the model system in Fig. 1, the normalized small-signal roundtrip absorption

*V*and pump current density

_{a}*J*:where

*n*is the transparency carrier density, and

_{t}**32**(3), 478–492 (1996). [CrossRef]

**32**(10), 1782–1790 (1996). [CrossRef]

## 3. Analytic solution in the pulse-energy domain

*E*(

*t*),

*W*<<

_{p}*E*,

_{sA}*, yielding a series expansion of the gain and absorption terms in the time-domain [6*

_{sL}**32**(3), 478–492 (1996). [CrossRef]

**32**(10), 1782–1790 (1996). [CrossRef]

*x*≤1. It allows us to apply the following anzatz for the slowly varying pulse amplitudewhere

*A*is the peak pulse amplitude,

*β*is the pulse chirp parameter, and the function

*G(x)*is the pulse intensity envelope. It reaches maximum value

*G*(0) =

*G*(1) = 0. Substituting (4) in (1) and separating the real and imaginary parts, we transform it to the master equation for steady-state mode-locking regime in the energy domain representation:

*x*= 1/2. Note that its inverse transform

*x*, we obtain the following steady-state solution [Eqs. (6)-(10)], which is valid at any energy of the pulse, including the special case of our interest

*µ*and duration

*τ*

_{p}of the mode-locked pulse in a simple formwhere the indexes “+” and “-”distinguish two solutions referenced here as the high-energy and low-energy branches, respectively; the FWHM of the hyperbolic secant pulse is 1.76

*τ*

_{p}. Finally, the normalized frequency detuning from the line center and the pulse delay read

## 4. Results and discussion

10. W. G. Scheibenzuber, U. T. Schwarz, L. Sulmoni, J. Dorsaz, J.-F. Carlin, and N. Grandjean, “Recombination coefficients of GaN-based laser diodes,” J. Appl. Phys. **109**(9), 093106 (2011). [CrossRef]

^{−1}(

*τ*) are plotted in the Fig. 2(a) as a function of the pump current. The output peak power per cavity facet is calculated as a fraction

_{p, ±}*µ*= W

_{±}*/*

_{p, ±}*E*and absorber bias parameter are displayed.

_{sA}*µ*

_{±}of the mode-locked pulses traveling in the cavity is comparable to or greater than 1. Therefore our approach to solution of the Haus-New’s master equation suitable for mode-locked pulses of arbitrary energy W

*is not just an interesting feature of the model. As indicated in Fig. 2(b), in InGaN/GaN monolithic mode-locked lasers, the pulse energy systematically exceeds the absorber saturation energy*

_{p}*E*, and the absorber operates at high saturation level. Previously known solutions [6

_{sA}**32**(3), 478–492 (1996). [CrossRef]

**32**(10), 1782–1790 (1996). [CrossRef]

4. G. H. C. New, “Pulse evolution in mode-locked quasi-continuous lasers,” IEEE J. Quantum Electron. **10**(2), 115–124 (1974). [CrossRef]

**32**(10), 1782–1790 (1996). [CrossRef]

*f*(

*x*):

*x*<1 in the energy domain representation. Therefore the maximum of the roundtrip net gain

*f*(

*x*) located at

*x*=

*x*

_{max}[see the discussion after Eq. (5)] assumes that

11. A. G. Vladimirov and D. Turaev, “Model for passive mode locking in semiconductor lasers,” Phys. Rev. A **72**(3), 033808 (2005). [CrossRef]

## 5. Conclusion

**32**(3), 478–492 (1996). [CrossRef]

**32**(10), 1782–1790 (1996). [CrossRef]

*T*/2 in all equations so as to account for the reduced time elapsed between propagation of pulses. The second modification consists in substituting the double pulse energy 2

_{rep}*W*in the equation for steady state saturated absorption

_{p}*q*in Eq. (2) so as to account for the saturation effect of the two pulses overlapping in the absorber.

_{i}## References and links

1. | S. Nakamura, M. Senoh, S.- Nagahama, N. Iwasa, T. Matsushita, and T. Mukai, “Blue InGaN-based laser diodes with an emission wavelength of 450 nm,” Appl. Phys. Lett. |

2. | K. Saito, H. Watanabe, T. Miyajima, M. Ikeda, and H. Yokoyama, “Mode locking of an external-cavity bisection GaInN blue-violet laser diode producing 3 ps duration optical pulses,” Appl. Phys. Lett. |

3. | P. P. Vasil'ev, |

4. | G. H. C. New, “Pulse evolution in mode-locked quasi-continuous lasers,” IEEE J. Quantum Electron. |

5. | H. A. Haus, “Theory of mode-locking with a slow saturable absorber,” IEEE J. Quantum Electron. |

6. | R. G. M. P. Koumans and 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. QE. |

7. | J. Leegwater, “Theory of Mode-Locked Semiconductor Lasers,” IEEE J. Quantum Electron. |

8. | J. L. A. Dubbeldam, J. A. Leegwater, and D. Lenstra, “Theory of mode-locked semiconductor lasers with finite absorber relaxation times,” Appl. Phys. Lett. |

9. | J. Piprek, |

10. | W. G. Scheibenzuber, U. T. Schwarz, L. Sulmoni, J. Dorsaz, J.-F. Carlin, and N. Grandjean, “Recombination coefficients of GaN-based laser diodes,” J. Appl. Phys. |

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

12. | J. Dorsaz, D. L. Boïko, L. Sulmoni, J.-F. Carlin, W. G. Scheibenzuber, U. T. Schwarz, and N. Grandjean, “Optical bistability in InGaN-based multi-section laser diodes,” Appl. Phys. Lett. |

**OCIS Codes**

(140.2020) Lasers and laser optics : Diode lasers

(140.4050) Lasers and laser optics : Mode-locked lasers

(140.5960) Lasers and laser optics : Semiconductor lasers

(320.7090) Ultrafast optics : Ultrafast lasers

(250.5960) Optoelectronics : Semiconductor lasers

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: April 18, 2011

Revised Manuscript: June 20, 2011

Manuscript Accepted: June 22, 2011

Published: August 17, 2011

**Citation**

Igor V. Smetanin, Peter P. Vasil’ev, and Dmitri L. Boiko, "Theory of the ultrafast mode-locked GaN lasers in a large-signal regime," Opt. Express **19**, 17114-17120 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-17114

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### References

- S. Nakamura, M. Senoh, S.- Nagahama, N. Iwasa, T. Matsushita, and T. Mukai, “Blue InGaN-based laser diodes with an emission wavelength of 450 nm,” Appl. Phys. Lett. 76(1), 22–24 (2000). [CrossRef]
- K. Saito, H. Watanabe, T. Miyajima, M. Ikeda, and H. Yokoyama, “Mode locking of an external-cavity bisection GaInN blue-violet laser diode producing 3 ps duration optical pulses,” Appl. Phys. Lett. 96(3), 031112 (2010). [CrossRef]
- P. P. Vasil'ev, Ultrafast diode lasers: fundamentals and applications (Artech House, Boston, 1995).
- G. H. C. New, “Pulse evolution in mode-locked quasi-continuous lasers,” IEEE J. Quantum Electron. 10(2), 115–124 (1974). [CrossRef]
- H. A. Haus, “Theory of mode-locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11, 736-746 (1975); “Mode-locking of lasers,” IEEE J. Sel,” Topics in QE 6(6), 1173–1185 (2000). [CrossRef]
- R. G. M. P. Koumans and 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. QE. 32(3), 478–492 (1996). [CrossRef]
- J. Leegwater, “Theory of Mode-Locked Semiconductor Lasers,” IEEE J. Quantum Electron. 32(10), 1782–1790 (1996). [CrossRef]
- J. L. A. Dubbeldam, J. A. Leegwater, and D. Lenstra, “Theory of mode-locked semiconductor lasers with finite absorber relaxation times,” Appl. Phys. Lett. 70(15), 1938 (1997). [CrossRef]
- J. Piprek, Semiconductor optoelectronic devices (Academic, 2002)
- W. G. Scheibenzuber, U. T. Schwarz, L. Sulmoni, J. Dorsaz, J.-F. Carlin, and N. Grandjean, “Recombination coefficients of GaN-based laser diodes,” J. Appl. Phys. 109(9), 093106 (2011). [CrossRef]
- A. G. Vladimirov and D. Turaev, “Model for passive mode locking in semiconductor lasers,” Phys. Rev. A 72(3), 033808 (2005). [CrossRef]
- J. Dorsaz, D. L. Boïko, L. Sulmoni, J.-F. Carlin, W. G. Scheibenzuber, U. T. Schwarz, and N. Grandjean, “Optical bistability in InGaN-based multi-section laser diodes,” Appl. Phys. Lett. 98(19), 191115 (2011). [CrossRef]

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