## Self-injected semiconductor distributed feedback lasers for frequency chirp stabilization |

Optics Express, Vol. 20, Issue 23, pp. 26062-26074 (2012)

http://dx.doi.org/10.1364/OE.20.026062

Acrobat PDF (1823 KB)

### Abstract

It is well known that semiconductor distributed feedback lasers (DFB) are key devices for optical communications. However direct modulation applications are limited by the frequency chirp induced by current modulation. We demonstrate that a proper external control laser operation leads to chirp-to-power ratio (CPR) stabilization over a wide range of modulation frequencies as compared to the free-running case. Under experimentally selected optical feedback conditions, the CPR decreases significantly in the adiabatic regime from about 650 MHz/mW in the solitary case down to 65 MHz/mW. Experimental results are also confirmed by numerical investigations based on the transfer matrix method. Simulations point out the possible optimization of the CPR in the adiabatic regime by considering a judicious cavity design in conjunction with a proper external control. These results demonstrate important routes for improving the transmission performance in optical telecommunication systems.

© 2012 OSA

## 1. Introduction

*μ*m signal. Several methods and devices have been developed with the aim of extending the transmission reach. These include Electro-absorption Modulated Lasers (EMLs) and more recently Dual modulation [2

2. K. Kechaou, T. Anfray, K. Merghem, C. Aupetit-Berthelemot, G. Aubin, C. Kazmierski, C. Jany, P. Chanclou, and D. Erasme,
“Improved NRZ transmission distance at 20 Gbit/s using dual electroabsorption
modulated laser,” Electron. Lett. **48**(6), 335–336
(2012). [CrossRef]

3. D. Mahgerefteh, Y. Matsui, X. Zheng, and K. McCallion,
“Chirp managed laser and applications,” IEEE J. Sel.
Top. Quantum Electron. **16**(5), 1126–1139
(2010). [CrossRef]

4. J. Binder and U. Kohn,
“10 Gbits/s-dispersion optimized transmission at 1,55 μm wavelength on standard
single mode fiber,” IEEE Photon. Technol. Lett. **6**(4), 558–560
(1994). [CrossRef]

*α*-factor) through injection-locking under strong optical injection [5]. Therefore, a new approach to enhance the transmission distance may consist in monitoring the DFB laser’s adiabatic chirp through external control techniques. The scope of this paper is to theoretically and experimentally demonstrate a stabilization of the frequency chirp over a wide range of modulation frequencies based on a suitably controlled external optical feedback. The investigation is carried out using a conventional quantum well (QW) based DFB laser through an analysis of the chirp to power ratio (CPR).

_{H}## 2. Experimental results

*µ*m long with an active layer consisting of six InGaAsP QWs separated by 10 nm wide barriers. The QWs are 8 nm wide and have a 1.1% compressive strain. The grating is defined in a passive quaternary layer localized over the active region and was measured to be about 30 cm

^{−1}(

*κ*L ~0.8). A conventional holographic process is used to fabricate a single pitch grating over the full wafer [15

15. F. Grillot, B. Thedrez, F. Mallecot, C. Chaumont, S. Hubert, M. F. Martineau, A. Pinquier, and L. Roux,
“Analysis, fabrication and characterization of 1.5μm selection-free tapered
stripe DFB lasers,” IEEE Photon. Technol. Lett. **14**(8), 1040–1042
(2002). [CrossRef]

*I*= 8 mA with an external efficiency

_{th}*η*= 0.26 W/A at 25°C. The objective of the experiments is to determine all the characteristics of the frequency modulation (FM) contribution induced by current modulation and under optical feedback conditions.

*Γ*=

*P*of the power returned to the facet and launched into the cavity

_{1}/P_{0}*P*over

_{1}*P*the emitted one. The feedback is studied for a long external cavity [7] given by ω

_{0}_{r}τ >> 1 where

*τ*and ω

_{r}are the external round trip time (in the order of several hundred nanoseconds) and the laser relaxation frequency (a few GHz). Measurements are made for a bias current equal to 2.4 ×

*I*, which corresponds to an emitting power of P

_{th}_{0}≈3 mW. The coupling loss coefficient between the laser output and the optical fiber was kept constant at ~3 dB during the entire experiment. Since the external cavity is long, all feedback regimes could not be observed. The optical spectrum was monitored using a 10 pm high-resolution optical spectrum analyzer.

*ω*/2

*π*< 10 MHz), thermal effects are predominant. For instance, at 30 kHz, the amplitude and the phase of the CPR are ~1.0 GHz/mW and ~140° respectively. Within the range 10 MHz <

*ω*/2

*π*< 1 GHz, amplitude (AM) and FM modulations are in-phase (adiabatic regime) and the amplitude of the CPR reaches 650 MHz/mW at

*ω*/2

*π*= 500 MHz. In that case, thermal effects are no longer significant compared to the refractive index effects induced by the modulation of the carrier density. When ω/2π>1GHz, relaxation oscillations between the carrier and photon numbers lead to a transient chirp with larger CPR values of about 1.5 GHz/mW at 10 GHz and a phase difference approaching 90°.

*Γ*= 1.4 × 10

^{−6}, (b)

*Γ*= 1.5 × 10

^{−5}, (c)

*Γ*= 1.6 × 10

^{−4}, and (d)

*Γ*= 5.5 × 10

^{−3}. The results demonstrate that the use of a controlled optical feedback can modify both the thermal and adiabatic chirps. Firstly, the averaged CPR measured in the thermal regime is significantly decreased from about 565 MHz/mW in the solitary case down to 33 MHz/mW at the highest feedback level. Secondly, the CPR measured in the adiabatic regime at

*ω*/2

*π*= 500 MHz ranges from about 650 MHz/mW in the solitary case down to 65 MHz/mW at the highest feedback level. Under the selected optical feedback condition (

*Γ*= 5.5 × 10

^{−3}) the CPR does not exceed ~100 MHz/mW on average from 10 kHz to 10 GHz, which corresponds to a reduction by a factor of 6.3 when compared to the solitary case. Figure 4 illustrates the stabilization of the measured adiabatic CPR at

*ω*/2

*π*= 500 MHz as a function of the optical feedback strength.

*ω*>>

*ω*, a condition which is easily reached in our experiments (the maximum modulation frequency is around 20 GHz), 2

_{c}*β*/

*m*directly equals to the laser’s linewidth enhancement factor. For low modulation frequencies, the ratio 2

*β*/

*m*becomes inversely proportional to the modulation frequency.

*β*/

*m*ratio is plotted starting from 50 MHz (beyond the thermal effects) for the solitary case (red plot) and for a feedback level of about

*Γ*= 1.5 × 10

^{−5}(blue plot). As predicted by (1), 2

*β*/

*m*tends asymptotically to the linewidth enhancement factor value, which is estimated to be about 3.2 in the absence of external perturbation. The modification of the linewidth enhancement factor under self-injection can be explained by the change of the threshold carrier density induced by optical feedback. The measured linewidth enhancement factor decreases down to about 1.8, which can indeed explain the chirp reduction magnitude. Independently of optical feedback, the roll-off frequency was graphically determined to be about 4.5 GHz from Fig. 5. The variations of 2

*β*/

*m*with optical output power can also be used to evaluate the gain compression factor

*ε*

_{.}Based on reference [20

20. G.
P. Agrawal, “Effect of gain nonlinearities on the
dynamic response of single-mode semiconductor lasers,” IEEE Photon.
Technol. Lett. **1**(12), 419–421
(1989). [CrossRef]

^{−1}, which is in good agreement with previous published values on similar laser structures [21].

## 3. Model

*S*(

*t*), the phase

*φ*(

*t*) and the free-running laser frequency

*ω*/2

_{0}*π*. In Eq. (4),

*ω*/2

*π*is the laser frequency in the presence of external optical feedback,

*G*the modal gain,

*α*the linewidth enhancement factor,

_{H}*τ*the photon lifetime and τ the external roundtrip time. The strength of the delayed field is denoted by the parameter

_{p}*K*, which can be expressed as [1]:with

*τ*being the internal roundtrip time within the laser’s cavity,

_{i}*γ*the amplitude reflectivity of the delayed field (Γ = γ

^{2}) originating from a distant reflecting point and being assumed to be such that

*γ*<<1 with

*C*being the coupling strength coefficient of the AR-facet [24

_{AR}24. F. Grillot, B. Thedrez, and G.-H. Duan,
“Feedback sensitivity and coherence collapse threshold of semiconductor DFB
lasers with complex structures,” IEEE J. Quantum Electron. **40**(3), 231–240
(2004). [CrossRef]

*r*can also be defined as follows [25

_{AR,eq}25. F. Grillot, B. Thedrez, O. Gauthier-Lafaye, M. F. Martineau, V. Voiriot, J.
L. Lafragette, J.
L. Gentner, and L. Silvestre,
“Coherence collapse threshold of 1.3 μm semiconductor DFB
lasers,” IEEE Photon. Technol. Lett. **15**(1), 9–11
(2003). [CrossRef]

*ϕ*is the facet phase term describing the position of the facet with respect to the Bragg reflector. The feedback does not affect the description of the HR-facet whose reflectivity can be written:where

_{AR}*ϕ*is the phase term describing the position of the facet with respect to the Bragg reflector. The dynamic evolution of the carrier density is governed by the usual relation:where

_{HR}*N*(

*t*),

*τ*, and

_{e}*I*(

*t*) respectively are the carrier density within the active zone, the carrier density lifetime and the pump current respectively. In what follows Eq. (6) will be used to evaluate the feedback sensitivity of DFB lasers through the determination of the CPR in the adiabatic regime for several effective front facet reflectivities. Although the numerical simulations do not explicitly incorporate the delayed field occurring in Eq. (4), the approach based on the complex effective reflectivity has already been used to analyze both the static and dynamic DFB lasers properties operating under external control [24

24. F. Grillot, B. Thedrez, and G.-H. Duan,
“Feedback sensitivity and coherence collapse threshold of semiconductor DFB
lasers with complex structures,” IEEE J. Quantum Electron. **40**(3), 231–240
(2004). [CrossRef]

26. B. Tromborg, H. Olesen, X. Pan, and S. Saito,
“Transmission line description of optical feedback and injection-locking for
Fabry-Perot and DFB lasers,” IEEE J. Quantum Electron. **23**(11), 1875–1889
(1987). [CrossRef]

27. A. Lestra and P. Brosson,
“Design rules for a low-chirp integrated DFB laser with electroabsorption
modulator,” IEEE Photon. Technol. Lett. **8**(8), 998–1000
(1996). [CrossRef]

28. K. Bjork and O. Nilsson,
“A new exact and efficient numerical matrix theory of complicated laser
structures: properties of asymmetric phase-shifted DFB lasers,” J.
Lightwave Technol. **5**(1), 140–146
(1987). [CrossRef]

*N*sections consisting of many grating periods in which all physical parameters, like the injection current, the material gain, the photon density, the carrier density and the refractive index are assumed to be homogeneous. The laser is modeled by assuming the Bragg grating has a rectangular shape. The transfer matrix for one corrugation period is defined by:

*n*and

_{1}*n*are the refractive indices, and

_{2}*k*and

_{1}*k*are the complex propagation constants in the two refractive index regions. The real part of the propagation constant is determined by the net gain. The imaginary part depends on refractive index, which affects the frequency shift of the laser. The laser is divided into m sections where the carrier density is kept uniform. The carrier density may however vary from section to section. Section

_{2}*i*contains

*m*corrugation periods. The complete transfer matrix describing the DFB laser with coated facets is written as:where

_{i}*m*is the number of period in the ith section,

_{i}*g*,

_{0}*e*,

*d*,

_{qw}*B*,

_{rad}*J*,

_{0}*ε*,

*N*and

*P*are respectively the empirical gain coefficient, the electron charge, the thickness of one quantum well, the radiative recombination coefficient, the transparency current density, the gain compression coefficient, the carrier density and the photon density, respectively. For the TMM, the oscillation condition at threshold is computed by using:

*α*,

_{th}*λ*). The initial guesses (

_{th}*α*) of solutions correspond to Fabry-Perot modes. After calculating the value of

_{i}, λ_{i}*α*), (

_{i}, λ_{i}*α*) and, (

_{i}+ dα, λ_{i}*α*), we compute the relative variations allowing the correction of the values of

_{i}, λ_{i}+ dλ*α*and

*λ*. This procedure is iteratively repeated until a relative variation less than 10

^{−9}is reached.

29. I. Orfanos, T. Sphicopoulos, A. Tsigopoulos, and C. Caroubalos,
“A tractable above-threshold model for the design of DFB and phase-shifted DFB
lasers,” IEEE J. Quantum Electron. **27**(4), 946–956
(1991). [CrossRef]

*n*is the refractive index at threshold,

^{th}*Γ*is the confinement factor,

_{c}*N*is the carrier density at threshold and

_{th}*dn/dN*is the slope of the refractive index with respect to the carrier density. The newly obtained solution (

*α, λ*) leads to another distribution of photons and carriers. Therefore, we solve again (16). All steps are repeated until reaching an unchanged (

*α, λ*). All parameters used in simulation are given in Table 1 below.

## 4. Numerical results

*ϕ*<π), the lasing wavelength shifts towards the red. Because of the large grating coefficient, the introduction of the feedback does not affect the typical CPR power dependence but it can, however, affect its magnitude, at a fixed output power. The distribution of the internal optical power is sensitive to the effective reflection coefficient leading to a significant dependence of the CPR on the feedback strength. Hence, for a 1mW output power, the absolute value of the CPR increases from 0 to 2 GHz/mW when

_{HR}*η*is the internal quantum efficiency and

_{i}*V = N*the cavity volume. Equation (19) describes the wavelength shift induced by the gain compression effect. At high output power, the index profile gets stabilized, and the SHB induced wavelength shift gradually disappears. The remaining wavelength shift is then only due to gain compression, which happens not to be feedback dependent.

_{QW}d_{QW}Lw*Q*-factor of the cavity (lower

*κL*). Figure 8 shows the calculated adiabatic CPR for a grating coupling coefficient of

*κL*= 0.5 with

*L*= 350 μm. In this case, the front facet reflectivity is 0.1% (green plot), 0.5% (black plot), 1% (red plot) and 2% (blue plot) respectively. Compared to Fig. 6, decreasing (

*κL*) leads to an increase of the laser’s sensitivity to the optical feedback.

## 5. Conclusion

30. L. V. Asryan and R.
A. Suris, “Longitudinal spatial hole burning in a
quantum-dot laser,” IEEE J. Quantum Electron. **36**(10), 1151–1160
(2000). [CrossRef]

## Acknowledgments

## References and Links

1. | K. Petermann, |

2. | K. Kechaou, T. Anfray, K. Merghem, C. Aupetit-Berthelemot, G. Aubin, C. Kazmierski, C. Jany, P. Chanclou, and D. Erasme,
“Improved NRZ transmission distance at 20 Gbit/s using dual electroabsorption
modulated laser,” Electron. Lett. |

3. | D. Mahgerefteh, Y. Matsui, X. Zheng, and K. McCallion,
“Chirp managed laser and applications,” IEEE J. Sel.
Top. Quantum Electron. |

4. | J. Binder and U. Kohn,
“10 Gbits/s-dispersion optimized transmission at 1,55 μm wavelength on standard
single mode fiber,” IEEE Photon. Technol. Lett. |

5. | N. A. Naderi, F. Grillot, V. Kovanis, and L. F. Lester, “Simultaneous low linewidth enhancement factor and high bandwidth quantum-dash injection-locked laser,” International Photon. Conf. Arlington, USA (2011). |

6. | F. Grillot, B. Thedrez, J. Py, O. Gauthier-Lafaye, V. Voiriot, and J.
L. Lafragette, “2.5-Gb/s transmission characteristics
of 1.3-μm DFB lasers with external optical feedback,” IEEE Photon.
Technol. Lett. |

7. | D. M. Kane and K. A. Shore, |

8. | N. Schunk and K. Petermann,
“Numerical analysis of the feedback regimes for a single-mode semiconductor
laser with external feedback,” IEEE J. Quantum Electron. |

9. | R. W. Tkach and A.
R. Chraplyvy, “Regimes of feedback effects in 1.5-μm
distributed feedback lasers,” J. Lightwave Technol. |

10. | D. Lenstra, B. H. Verbeek, and A.
J. Den Boef, “Coherence collapse in single-mode
semiconductor lasers due to optical feedback,” IEEE J. Quantum
Electron. |

11. | C. Henry and R.
F. Kazarinov, “Instabilities of semiconductor lasers
due to optical feedback from distant reflectors,” IEEE J. Quantum
Electron. |

12. | J. Mork, B. Tromborg, and J. Mark,
“Chaos in semiconductor lasers with optical feedback: theory and
experiment,” IEEE J. Quantum Electron. |

13. | J. Mork, B. Tromborg, and P.
L. Christiansen, “Bistability and low-frequency
fluctuations in semiconductor lasers with optical feedback: a theoretical
analysis,” IEEE J. Quantum Electron. |

14. | G. Duan, P. Gallion, and G. Debarge,
“Analysis of frequency chirping of semiconductor lasers in the presence of
optical feedback,” Opt. Lett. |

15. | F. Grillot, B. Thedrez, F. Mallecot, C. Chaumont, S. Hubert, M. F. Martineau, A. Pinquier, and L. Roux,
“Analysis, fabrication and characterization of 1.5μm selection-free tapered
stripe DFB lasers,” IEEE Photon. Technol. Lett. |

16. | J. G. Provost and F. Grillot,
“Measuring the chirp and the linewidth enhancement factor of optoelectronic
devices with a mach-zehnder interferometer,” IEEE Photon. J. |

17. | R. Schimpe, J. E. Bowers, and T.
L. Koch, “Characterization of frequency response
of 1.5-µm InGaAsP DFB laser diode and InGaAs PIN photodiode by heterodyne measurement
technique,” Electron. Lett. |

18. | L. Olofsson and T.
G. Brown, “Frequency dependence of the chirp
factor in 1.55 |

19. | L. A. Coldren and S. W. Corzine, |

20. | G.
P. Agrawal, “Effect of gain nonlinearities on the
dynamic response of single-mode semiconductor lasers,” IEEE Photon.
Technol. Lett. |

21. | G. P. Agrawal, |

22. | P. Vankwikelberge, F. Buytaert, A. Franchois, R. Baets, P. Kuindersma, and C.
W. Fredriksz, “Analysis of the carrier-induced FM
response of DFB lasers: Theoretical and Experimental case studies,”
IEEE J. Quantum Electron. |

23. | B. Thedrez, J. M. Rainsant, N. Aberkane, B. Andre, H. Bissessur, J. G. Provost, and B. Fernier, “ Power and facet phase dependence of chirp for index and gain-coupled DFB lasers,” paper TuE41, Semiconductor Laser Conference, 175–176 (1998). |

24. | F. Grillot, B. Thedrez, and G.-H. Duan,
“Feedback sensitivity and coherence collapse threshold of semiconductor DFB
lasers with complex structures,” IEEE J. Quantum Electron. |

25. | F. Grillot, B. Thedrez, O. Gauthier-Lafaye, M. F. Martineau, V. Voiriot, J.
L. Lafragette, J.
L. Gentner, and L. Silvestre,
“Coherence collapse threshold of 1.3 μm semiconductor DFB
lasers,” IEEE Photon. Technol. Lett. |

26. | B. Tromborg, H. Olesen, X. Pan, and S. Saito,
“Transmission line description of optical feedback and injection-locking for
Fabry-Perot and DFB lasers,” IEEE J. Quantum Electron. |

27. | A. Lestra and P. Brosson,
“Design rules for a low-chirp integrated DFB laser with electroabsorption
modulator,” IEEE Photon. Technol. Lett. |

28. | K. Bjork and O. Nilsson,
“A new exact and efficient numerical matrix theory of complicated laser
structures: properties of asymmetric phase-shifted DFB lasers,” J.
Lightwave Technol. |

29. | I. Orfanos, T. Sphicopoulos, A. Tsigopoulos, and C. Caroubalos,
“A tractable above-threshold model for the design of DFB and phase-shifted DFB
lasers,” IEEE J. Quantum Electron. |

30. | L. V. Asryan and R.
A. Suris, “Longitudinal spatial hole burning in a
quantum-dot laser,” IEEE J. Quantum Electron. |

**OCIS Codes**

(140.3490) Lasers and laser optics : Lasers, distributed-feedback

(140.5960) Lasers and laser optics : Semiconductor lasers

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: May 9, 2012

Revised Manuscript: July 22, 2012

Manuscript Accepted: September 12, 2012

Published: November 2, 2012

**Citation**

Khalil Kechaou, Frédéric Grillot, Jean-Guy Provost, Bruno Thedrez, and Didier Erasme, "Self-injected semiconductor distributed feedback lasers for frequency chirp stabilization," Opt. Express **20**, 26062-26074 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-23-26062

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### References

- K. Petermann, Laser Diode Modulation and Noise (Kuwer Academic Publisher, 1991).
- K. Kechaou, T. Anfray, K. Merghem, C. Aupetit-Berthelemot, G. Aubin, C. Kazmierski, C. Jany, P. Chanclou, and D. Erasme, “Improved NRZ transmission distance at 20 Gbit/s using dual electroabsorption modulated laser,” Electron. Lett.48(6), 335–336 (2012). [CrossRef]
- D. Mahgerefteh, Y. Matsui, X. Zheng, and K. McCallion, “Chirp managed laser and applications,” IEEE J. Sel. Top. Quantum Electron.16(5), 1126–1139 (2010). [CrossRef]
- J. Binder and U. Kohn, “10 Gbits/s-dispersion optimized transmission at 1,55 μm wavelength on standard single mode fiber,” IEEE Photon. Technol. Lett.6(4), 558–560 (1994). [CrossRef]
- N. A. Naderi, F. Grillot, V. Kovanis, and L. F. Lester, “Simultaneous low linewidth enhancement factor and high bandwidth quantum-dash injection-locked laser,” International Photon. Conf. Arlington, USA (2011).
- F. Grillot, B. Thedrez, J. Py, O. Gauthier-Lafaye, V. Voiriot, and J. L. Lafragette, “2.5-Gb/s transmission characteristics of 1.3-μm DFB lasers with external optical feedback,” IEEE Photon. Technol. Lett.14(1), 101–103 (2002). [CrossRef]
- D. M. Kane and K. A. Shore, Unlocking Dynamical Diversity (Wiley, 23–54, 2005).
- N. Schunk and K. Petermann, “Numerical analysis of the feedback regimes for a single-mode semiconductor laser with external feedback,” IEEE J. Quantum Electron.24(7), 1242–1247 (1988). [CrossRef]
- R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5-μm distributed feedback lasers,” J. Lightwave Technol.4(11), 1655–1661 (1986).
- D. Lenstra, B. H. Verbeek, and A. J. Den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron.21(6), 674–679 (1985). [CrossRef]
- C. Henry and R. F. Kazarinov, “Instabilities of semiconductor lasers due to optical feedback from distant reflectors,” IEEE J. Quantum Electron.22(2), 294–301 (1986). [CrossRef]
- J. Mork, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron.28(1), 93–108 (1992). [CrossRef]
- J. Mork, B. Tromborg, and P. L. Christiansen, “Bistability and low-frequency fluctuations in semiconductor lasers with optical feedback: a theoretical analysis,” IEEE J. Quantum Electron.24(2), 123–133 (1988). [CrossRef]
- G. Duan, P. Gallion, and G. Debarge, “Analysis of frequency chirping of semiconductor lasers in the presence of optical feedback,” Opt. Lett.12(10), 800–802 (1987). [CrossRef] [PubMed]
- F. Grillot, B. Thedrez, F. Mallecot, C. Chaumont, S. Hubert, M. F. Martineau, A. Pinquier, and L. Roux, “Analysis, fabrication and characterization of 1.5μm selection-free tapered stripe DFB lasers,” IEEE Photon. Technol. Lett.14(8), 1040–1042 (2002). [CrossRef]
- J. G. Provost and F. Grillot, “Measuring the chirp and the linewidth enhancement factor of optoelectronic devices with a mach-zehnder interferometer,” IEEE Photon. J.3(3), 476–488 (2011). [CrossRef]
- R. Schimpe, J. E. Bowers, and T. L. Koch, “Characterization of frequency response of 1.5-µm InGaAsP DFB laser diode and InGaAs PIN photodiode by heterodyne measurement technique,” Electron. Lett.22(9), 453–454 (1986). [CrossRef]
- L. Olofsson and T. G. Brown, “Frequency dependence of the chirp factor in 1.55 μm distributed feedback semiconductor lasers,” IEEE Photon. Technol. Lett.4(7), 688–691 (1992). [CrossRef]
- L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).
- G. P. Agrawal, “Effect of gain nonlinearities on the dynamic response of single-mode semiconductor lasers,” IEEE Photon. Technol. Lett.1(12), 419–421 (1989). [CrossRef]
- G. P. Agrawal, Semiconductor Lasers (Van Nostrand Reinhold, 1993)
- P. Vankwikelberge, F. Buytaert, A. Franchois, R. Baets, P. Kuindersma, and C. W. Fredriksz, “Analysis of the carrier-induced FM response of DFB lasers: Theoretical and Experimental case studies,” IEEE J. Quantum Electron.25(11), 2239–2254 (1989). [CrossRef]
- B. Thedrez, J. M. Rainsant, N. Aberkane, B. Andre, H. Bissessur, J. G. Provost, and B. Fernier, “ Power and facet phase dependence of chirp for index and gain-coupled DFB lasers,” paper TuE41, Semiconductor Laser Conference, 175–176 (1998).
- F. Grillot, B. Thedrez, and G.-H. Duan, “Feedback sensitivity and coherence collapse threshold of semiconductor DFB lasers with complex structures,” IEEE J. Quantum Electron.40(3), 231–240 (2004). [CrossRef]
- F. Grillot, B. Thedrez, O. Gauthier-Lafaye, M. F. Martineau, V. Voiriot, J. L. Lafragette, J. L. Gentner, and L. Silvestre, “Coherence collapse threshold of 1.3 μm semiconductor DFB lasers,” IEEE Photon. Technol. Lett.15(1), 9–11 (2003). [CrossRef]
- B. Tromborg, H. Olesen, X. Pan, and S. Saito, “Transmission line description of optical feedback and injection-locking for Fabry-Perot and DFB lasers,” IEEE J. Quantum Electron.23(11), 1875–1889 (1987). [CrossRef]
- A. Lestra and P. Brosson, “Design rules for a low-chirp integrated DFB laser with electroabsorption modulator,” IEEE Photon. Technol. Lett.8(8), 998–1000 (1996). [CrossRef]
- K. Bjork and O. Nilsson, “A new exact and efficient numerical matrix theory of complicated laser structures: properties of asymmetric phase-shifted DFB lasers,” J. Lightwave Technol.5(1), 140–146 (1987). [CrossRef]
- I. Orfanos, T. Sphicopoulos, A. Tsigopoulos, and C. Caroubalos, “A tractable above-threshold model for the design of DFB and phase-shifted DFB lasers,” IEEE J. Quantum Electron.27(4), 946–956 (1991). [CrossRef]
- L. V. Asryan and R. A. Suris, “Longitudinal spatial hole burning in a quantum-dot laser,” IEEE J. Quantum Electron.36(10), 1151–1160 (2000). [CrossRef]

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