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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 18 — Aug. 29, 2011
  • pp: 17114–17120
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Theory of the ultrafast mode-locked GaN lasers in a large-signal regime

Igor V. Smetanin, Peter P. Vasil’ev, and Dmitri L. Boiko  »View Author Affiliations


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

Compact laser sources generating sub-picosecond pulses in the blue/violet wavelength range [1

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]

] can meet the requirements for various applications in different fields of science and technology. They can be used for high-density optical data storage systems, biomedical diagnostic and fluorescence imaging, optical combs etc. For such an application, understanding the dynamics of GaN-based semiconductor lasers is particularly essential. Mode locking is often considered as the most promising technique to generate picosecond and femtosecond optical pulses with high repetition rates and low timing jitter [2

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]

,3

3. P. P. Vasil'ev, Ultrafast diode lasers: fundamentals and applications (Artech House, Boston, 1995).

].

In this paper, we develop a new analytical approach for passively mode-locked diode lasers with slow absorber and amplifier. Our approach utilizes pulse-energy domain analysis and is suitable at high pulse energies, much exceeding the saturation energy of absorber section. We find that, two analytic solutions are possible, which are referenced herewith as a high- and low-energy branches. Ultrafast pulse operation in mode-locking regime and its stability are discussed in application to InGaN/GaN monolithic multi-section laser diode.

As a model system, we utilize monolithic multisection cavity, in which absorber section is located in the middle of the laser cavity (see Fig. 1
Fig. 1 Schematic of illustration of InGaN/GaN ridge-waveguide multi-section laser diode (a). Sellmeier model predictions for the group velocity dispersion D*=ωL22k/ω2 (b) and group velocity (c).
.(a)). The gain and absorption section are treated in localized dynamic model approximation and a single pulse traveling in the cavity is assumed.

2. Haus-New’s theoretical model in time-domain representation

As a starting point, we use the steady-state master equation of the time-domain Haus-New’s theory [4

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

,5

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]

] which in the case of semiconductor laser diode can be written as [6

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

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

]
{1+(1iαA)qiexp(E(t)/EsA)(1iαL)giexp(E(t)/EsL)+ξ2+i(ψ+ξ)++(1+δ2iξ)1ωLddt(1+iD)1ωL2d2dt2}a(t)=0.
(1)
Here a(t)is the complex slow varying amplitude of the electric field component of the laser pulse, E(t)=(dh/Γ)(e2/8πc)t|a(t)|dt and Wp=(dh/Γ)(e2/8πc)|a(t)|dt are the instantaneous (cumulative) and total pulse energies, ξ=Δω/ωL is the frequency detuning of the carrier from the center of the gain line reported to the gain spectral half-width, EsA and EsL=sEsA are the saturation energies of absorber and amplifier sections, lA and lLare their lengths. All material and geometrical parameters of the structure (Table 1) are normalized with respect to the cold cavity loss αClC=αilC(1/2)ln(R1R2) [see the first term in Eq. (1)].

The carrier relaxation time TL in the gain-section quantum wells (QWs) is significantly longer than generated picosecond pulses so as the relaxation during the pulse can be neglected (see Table 1). Even though the recovery time TA of absorber is only few times longer the pulse, the absorber relaxation 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]

] and appears to be insignificant.) As such, large-signal absorption (second term in (1)) and gain (third term) are represented by instantaneous solutions of the corresponding rate equations during the build-up of an ultrafast pulse [6

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

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

]. In particular, qi and gi are the saturated absorption and gain in the cavity at the beginning of a pulse, qiexp(-Wp/EsA) and giexp(-Wp/EsA) represent the same large-signal parameters at the end of the pulse. Solving the rate equations during the cavity roundtrip Trep time, when these relax back to the initial absorption qi. and gain gi, one obtains the relationships between the steady-state saturated and unsaturated small-signal values of the absorption and gain in the cavity [6

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

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

]:

qi=q01exp(Trep/TA)1exp(Wp/EsATrep/TA),gi=g01exp(Trep/TL)1exp(Wp/EsLTrep/TL)
(2)

These expressions account for the carrier relaxation between the subsequent pulses. For the model system in Fig. 1, the normalized small-signal roundtrip absorption q0 and gain g0 in the cavity are controlled by the external bias Va and pump current density J:
q0=lAΓsvgαClC(gn)nt(1Va),g0=lLΓvgαClC(gn)[JTLehnt],
(3)
where nt is the transparency carrier density, and Va is the normalized bias parameter. The last four terms in Eq. (1) accounts for the dispersion coefficient D*=ωL22k/ω2 (Fig. 1(b)), the round trip phase shift ψ0 and time delay ΔT of the pulse. In Eq. (1), these are represented by normalized parameters D=2D*/αC, ψ=2ψ0/αClC, δ=2ωLΔT/αClC [6

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

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

].

3. Analytic solution in the pulse-energy domain

So far the analytic solutions of the master Eq. (1) have been utilizing an assumption of small pulse energy E(t),Wp<<EsA,sL, yielding a series expansion of the gain and absorption terms in the time-domain [6

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

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

]. We find an original approach to solve Eq. (1) for high pulse energies,developing our analysis in the pulse energy domain.

We introduce new variablex=E(t)/Wp, measuring the progress of instantaneous (cumulative) pulse energy towards the total energy in the pulse, so as 0≤x≤1. It allows us to apply the following anzatz for the slowly varying pulse amplitude
a=[AG1/2(x)]1+iβ,
(4)
where A is the peak pulse amplitude, A=(4πcWp/e2τp)1/2, βis the pulse chirp parameter, and the function G(x) is the pulse intensity envelope. It reaches maximum value max(G)=1 at the peak of the pulse, and vanishes at the beginning and the end of the pulse, 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:
2(1βD)B2GG''+(1β22βD)B2(G')2(1+δ+2ξβ)BG'ξ2==1+qiexp(μx)giexp(μx/s),2(β+D)B2GG''+(2β+Dβ2D)B2(G')2(β+δβ2ξ)BG'==ψ+ξαAqiexp(μx)+αLgiexp(μx/s)
(5)
where μ=Wp/EsA and B=1/4ωLτp are the normalized pulse energy and the inverse pulse width.

We obtain the analytic solution of Eq. (5) in the steady-state mode-locking regime by introducing a series expansion for the pulse envelop G(x)14(x1/2)2 in the vicinity of the peak at x = 1/2. Note that its inverse transform t/τp=1/2xdx/G(x) to the time-domain representation yields the hyperbolic secant pulse shape |a(t)|2=A2cosh2(t/τp). The term in the right-hand side of the first Eq. (5) is the net cavity gain f(x)=1qiexp(μx)+giexp(μx/s). It reaches maximum at the pulse energy xmax=sln[sqi/gi]/μ(s1). Therefore, we substitute in Eq. (5) the series expansion f(x)sΔ1112μ2Δ1(xxmax)2. In a similar way we introduce a series expansion in the right-hand side of the second Eq. (5). Finally, considering terms at each power of 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 μWp/EsA1:

The chirp β of the pulse (4) is
β=23γ+D1+D2Y.
(6)
where
γ=[αLsαA]1/(s1),Y=32(1+D2)(γ+D2){2+94(1γD)2(γ+D)232(1γD)(γ+D)}.
(7)
Using the notation
T1=Y1+4Y(β+γ1+β2)2+234(γ+D)3(γD1)1+D2,   T2=4b1+8Y(β+γ1+β2)2βb1+γb2β+γT3=16(s1Δ1)8b1216Y((β+γ)(βb1+γb2)(1+β2)(β+γ))2,,Δ1=s1s[(gi/s)sqi]1/(s1),b1=ss1ln[sqigi],b2=ss1ln[αAαLsqigi]
(8)
we can represent the normalized energy µ and duration τ p of the mode-locked pulse in a simple form
μ±=T2T1±T22T12+T3T1,τp,±=4ωLY1/2μ±Δ11/2,
(9)
where 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 1.76τ p. Finally, the normalized frequency detuning from the line center and the pulse delay read

ξ=12β+γ1+β2Y1/2Δ11/2[μ±2βb1+γb2β+γ],1+δ=121βγ1+β2Y1/2Δ11/2[μ±2b1b2βγ1βγ].
(10)

The Lyapunov’ analysis of dynamic stability of the two solutions (9) is a complicated problem and will be reported elsewhere. Instead of it, we utilize below a set of conditions, which define domains of stable mode-locking regime for each solution.

4. Results and discussion

We apply our approach to analyze the stability and performances of a monolithic mode-locked InGaN/GaN diode laser. All major laser parameters are indicated in Table 1 and the ratio of the absorber length to the overall cavity length is lA/lC=0.1. In Table 1, the short recovery time of absorber is settled by the tunneling of carriers from the QWs to the tilted QW barriers and by the carrier drift time through the intrinsic region of the multiple QW heterostructure barriers of about 30-40 nm thick. Thus, due to a large reverse bias, the absorber recovers much faster than the carrier relaxation takes place in the gain-section QWs. The group refractive index in InGaN/GaN lasers is close to 3.5 [9

9. J. Piprek, Semiconductor optoelectronic devices (Academic, 2002)

,10

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]

]. To estimate the group velocity dispersion we use Sellmaier model for the refraction index dispersion in GaN [9

9. J. Piprek, Semiconductor optoelectronic devices (Academic, 2002)

]. The predicted group velocity value (Fig. 1(c)) is in good agreement with the value from Table 1, therefore we are confident in the estimated group velocity dispersion of D*60cm−1 (D3) in Fig. 1(b).

The model predictions for the output peak power P± and FWHM pulse width (1.76τp, ± ) are plotted in the Fig. 2(a)
Fig. 2 Solution Eq. (9) of the mode-locking master Eq. (1): (a) FWHM pulse width 1.76τp, ± (left axis) and output peak power per facet P± (righ axis) v.s. pumping current for high-energy (black curves) and low energy (red curves) solution branches; (b) Bias voltage at which the self-starting modelocking regime is stable (left axis) and the relative pulse energy µ ± = Wp, ± /EsA (righ axis)
as a function of the pump current. The output peak power per cavity facet is calculated as a fraction ln(R)/2(αLlnR) of the pulse peak power in the cavity Wp/2τp. In Fig. 2(b), the corresponding normalized pulse energy µ ± = Wp, ± /EsA and absorber bias parameter are displayed.

As expected, the higher the pulse energy, the shorter the pulse width is. This applies equally to the comparison of the pulse parameters between the high- (index “+”) and low-energy (index “-”) branches of Eq. (9). The relative energy 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 Wp 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 EsA, and the absorber operates at high saturation level. Previously known solutions [6

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

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

] are not valid in this operation regime of absorber. In the mode-locking regime corresponding to the high–energy branch of Eq. (9) the pulses of ~500fs width can be produced at the output peak pulse power of ~1.5W per laser facet.

The necessary conditions for stable regime of mode-locking has been formulated in Refs [4

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

7

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

]: The roundtrip net gain in the cavity must be negative all the time but the lasing pulse so as all spurious field fluctuations are dumped. In terms of the energy-domain solution (6)-(10), this requirement leads to the boundary conditions for f(x):

1+qigi>0(x=0),1+qiexp(μ±)giexp(μ±/s)>0(x=1).
(11)

However during the pulse, the roundtrip net gain should become positive somewhere on the interval 0<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 sΔ11>0,yielding us the third condition for the steady-state mode-locking regime:

gisqi1/s/(s1)(s1)/s
(12)

It follows that there exists the minimum value of the absorption coefficient qimin=1/(s1) so as at qi<qimin both the high-energy and low pulse energy mode-locking regimes are unstable. Finally we have to add also the necessary start-up condition
1+q0<g0,
(13)
which means the net gain should be positive to launch the amplification from zero amplitude level.

In the case of GaN laser diode, the domain (11)-(13) of self-starting stable mode-locking regime for each solution (9) is very narrow, as indicated in the absorber bias map in Fig. 2(b). The domains of stability of the two solutions do not overlap, indicating that the two solutions may not coexist simultaneously. Interestingly, in the limit of very small pulse energies, the difference between two solutions vanishes, in agreement with results of Ref. [11

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

In conclusion, we have developed a new analytical approach to the theory of passively mode-locked diode lasers at moderate pulse energies μ1, which is based on the energy-domain representation of the pulse envelope evolution. In this approach, we calculate the parameters of the mode-locked pulse in the vicinity of the peak intensity at the arbitrary pulse energies, while usually [6

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

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

] the small-energy approximation μ<<1 is used which drops at pulse energies exceeding the absorber’s saturation energy.

Our approach, in fact applies well to any cavity arrangement, e.g. with absorber section located at the facet or in the external cavity configuration. Furthermore, it can be easily tailored to the case of colliding pulse mode locking (CPM) regime, which may occur in a cavity with absorber situated in the middle (as in Fig. 1 (a)). The overlap of two CPM pulses at the absorber is taken in to account by replacing the roundtrip time by Trep/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 2Wp in the equation for steady state saturated absorption qi in Eq. (2) so as to account for the saturation effect of the two pulses overlapping in the absorber.

The developed approach allows us to determine the mode-locked operation parameters for the high-power sub-picosecond blue-violet GaN diode laser.

This research is supported by the EC Seventh Framework Programme under grant agreement #238556 (FEMTOBLUE).

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. 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]

3.

P. P. Vasil'ev, Ultrafast diode lasers: fundamentals and applications (Artech House, Boston, 1995).

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]

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]

9.

J. Piprek, Semiconductor optoelectronic devices (Academic, 2002)

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]

11.

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

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. 98(19), 191115 (2011). [CrossRef]

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

  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]
  3. P. P. Vasil'ev, Ultrafast diode lasers: fundamentals and applications (Artech House, Boston, 1995).
  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]
  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]
  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]
  9. J. Piprek, Semiconductor optoelectronic devices (Academic, 2002)
  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]
  11. A. G. Vladimirov and D. Turaev, “Model for passive mode locking in semiconductor lasers,” Phys. Rev. A 72(3), 033808 (2005). [CrossRef]
  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. 98(19), 191115 (2011). [CrossRef]

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