## Enhancement of optical Kerr effect in quantum-cascade lasers with multiple resonance levels

Optics Express, Vol. 16, Issue 17, pp. 12599-12606 (2008)

http://dx.doi.org/10.1364/OE.16.012599

Acrobat PDF (291 KB)

### Abstract

In this paper, we investigated the optical Kerr lensing effect in quantum-cascade lasers with multiple resonance levels. The Kerr refractive index *n*_{2} is obtained through the third-order susceptibility at the fundamental frequency *χ*^{(3)}(*ω*; *ω*, *ω*,-*ω*). Resonant two-photon processes are found to have almost equal contributions to *χ*^{(3)}(*ω*; *ω*, *ω*,-*ω*) as the single-photon processes, which result in the predicted enhancement of the positive nonlinear (Kerr) refractive index, and thus may enhance mode-locking of quantum-cascade lasers. Moreover, we also demonstrate an isospectral optimization strategy for further improving *n*_{2} through the band-structure design, in order to boost the multimode performance of quantum-cascade lasers. Simulation results show that the optimized stepwise multiple-quantum-well structure has *n*_{2}≈10^{-8} cm^{2}/W, a twofold enhancement over the original flat quantum-well structure. This leads to a refractive-index change Δ*n* of about 0.01, which is at the upper bound of those reported for typical Kerr medium. This stronger Kerr refractive index may be important for quantum-cascade lasers ultimately to demonstrate self-mode-locking.

© 2008 Optical Society of America

## 1. Introduction

*χ*

^{(2)}and

*χ*

^{(3)}[1–5

1. C. Gmachl, A. Belyanin, D. L. Sivco, M. L. Peabody, N. Owschimikow, A. M. Sergent, F. Capasso, and A. Y. Cho, “Optimized second-harmonic generation in quantum cascade lasers,” IEEE J. Quantum Electron. **39**, 1345–1355 (2003). [CrossRef]

6. F. Capasso, C. Gmachl, D. L. Sivco, and A. Y. Cho, “Quantum cascade lasers,” Phys. Today **55**, 34 (2002). [CrossRef]

*n*

_{2}, experiences greater refractive index in central part of the beam, where the intensity is highest, in both transverse and longitudinal directions. In the transverse direction, the higher refractive index increases the beam confinement in the central part and thus decreases the waveguide loss. Combined with the ultrafast carrier relaxations in QCLs, the net result is a fast saturable-absorber mechanism [7

7. G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller, “Frontiers in ultrashort pulse generation: pushing the limits in linear and optics,” Science **286**, 1507–1512 (1999). [CrossRef] [PubMed]

8. H. A. Haus, “Mode-locking of lasers,” IEEE J. Quantum Electron. **6**, 1173–1185 (2000). [CrossRef]

*n*

_{2}[8

8. H. A. Haus, “Mode-locking of lasers,” IEEE J. Quantum Electron. **6**, 1173–1185 (2000). [CrossRef]

9. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. **28**, 2086–2096 (1992). [CrossRef]

*n*

_{2}. [10]. While the saturable absorber can stabilize the mode-locked pulses, the SPM can shape the pulse as chirped and compress the pulse even more. Thus, analysis of both transverse and longitudinal Kerr effects suggest that pulse shortening through the Kerr medium will be enhanced by increasing

*n*

_{2}. Self-pulsation of mid-infrared QCLs had been reported in [11

11. R. Paiella, F. Capasso, C. Gmachl, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, A. Y. Cho, and H. C. Liu, “Self-mode-locking of quantum cascade lasers with giant ultrafast optical nonlinearities,” Science **290**, 1739–1742 (2000). [CrossRef] [PubMed]

12. C. Y. Yang, L. Diehl, A. Gordon, C. Jirauschek, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, M. Troccoli, J. Faist, and F. Capasso, “Coherent instabilities in a semiconductor laser with fast gain recovery,” Phys. Rev. A **75**, 031802(R) (2007). [CrossRef]

13. H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. **39**, 4662–4672 (1968). [CrossRef]

15. R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, A. Y Cho, E. A. Whittaker, and H. C. Liu, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. **79**, 2526–2528 (2001). [CrossRef]

16. A. Gordon, C.Y. Yang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to special hole burning,” Phys. Rev. A , **77**, 053804 (2008). [CrossRef]

12. C. Y. Yang, L. Diehl, A. Gordon, C. Jirauschek, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, M. Troccoli, J. Faist, and F. Capasso, “Coherent instabilities in a semiconductor laser with fast gain recovery,” Phys. Rev. A **75**, 031802(R) (2007). [CrossRef]

16. A. Gordon, C.Y. Yang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to special hole burning,” Phys. Rev. A , **77**, 053804 (2008). [CrossRef]

7. G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller, “Frontiers in ultrashort pulse generation: pushing the limits in linear and optics,” Science **286**, 1507–1512 (1999). [CrossRef] [PubMed]

1. C. Gmachl, A. Belyanin, D. L. Sivco, M. L. Peabody, N. Owschimikow, A. M. Sergent, F. Capasso, and A. Y. Cho, “Optimized second-harmonic generation in quantum cascade lasers,” IEEE J. Quantum Electron. **39**, 1345–1355 (2003). [CrossRef]

*n*up to 0.005. To further explore the mechanisms of that may underlie short-pulse generation and perhaps self-mode-locking (SML) in QCLs, it is desirable to design a QCL structure with a higher Kerr nonlinearity, an issue that has not been addressed in the literature. In the later part of this paper, we demonstrate a systematic optimization method to further increase

*n*

_{2}through tailoring the quantum-well structure composing the active region of the QCL. Simulation results show that the optimization brings additional twofold increase on the Kerr refractive index, which makes the nonlinear refractive index change higher than other QCLs studied for SML. The optimized structure is a candidate for future study on resolving the interaction between the two kinds of single mode instabilities resulting in short-pulse generation in QCLs.

## 2. Enhancement of the Kerr nonlinearity due to multi-resonance

1. C. Gmachl, A. Belyanin, D. L. Sivco, M. L. Peabody, N. Owschimikow, A. M. Sergent, F. Capasso, and A. Y. Cho, “Optimized second-harmonic generation in quantum cascade lasers,” IEEE J. Quantum Electron. **39**, 1345–1355 (2003). [CrossRef]

5. J. Bai and D. S. Citrin, “Optical and transport characteristics of quantum-cascade lasers with optimized second-harmonic generation,” IEEE J. Quantum Electron. **43**, 391–398 (2007). [CrossRef]

*E*

_{1},

*E*

_{2},

*E*

_{3},

*E*

_{4}, and

*E*

_{5}. The lasing transition takes place between levels

*E*-

_{3}*E*

_{2}. The band-structure is originally designed for enhanced resonant SHG, thus the energy separations of

*E*

_{3}-

*E*

_{4}and

*E*

_{4}-

*E*

_{5}are in resonance with the lasing energy. This band structure is solved by the finite difference method with band nonparabolicity included. The subband energies and dipole matrix elements (DMEs) obtained in our model agree well with those listed in [1

**39**, 1345–1355 (2003). [CrossRef]

*n*=

*n*+

_{o}*n*[17], with

_{2}I*n*the linear refractive index,

_{o}*n*

_{2}the nonlinear refractive index,

*I*=

*n*

_{o}*cε*

_{o}E^{2}/2 the light intensity.

*ω*, i.e. states

*E*

_{2}-

*E*

_{3},

*E*

_{3}-

*E*

_{4}, and

*E*

_{4}-

*E*

_{5}. In any lasing structure there are sequential single-photon processes contributing to

*χ*

^{(3)}(

*ω*;

*ω*,

*ω*,-

*ω*), which lead to absorption saturation. The explicit expression of

*χ*

^{(3)}

_{1p}(

*ω*;

*ω*,

*ω*, -

*ω*) obtained by summing up all the contributions from the single-photon processes is

*χ*

^{(3)}

_{1p}(

*ω*;

*ω*,

*ω*, -

*ω*) depends on the population inversion (

*N*-

_{i}*N*) between resonant levels, the dipole matrix elements (DMEs)

_{j}*M*, and energy broadening terms |

_{ij}*E*-

_{ij}*ħω*-

*iγ*|. Due to the multiple-resonance nature of the subbands, additional contributions to

_{ij}*χ*

^{(3)}(

*ω*;

*ω*,

*ω*,-

*ω*) from two-photon processes also exist for resonant cascades

*E*

_{2}-

*E*

_{3}-

*E*

_{4}and

*E*

_{3}-

*E*

_{4}-

*E*

_{5}, where two photons with frequency

*ω*are absorbed simultaneously or sequentially and stimulate the upward electronic transitions across two consecutive resonant levels

*E*

_{2}-

*E*

_{4}or

*E*

_{3}-

*E*

_{5}. The expression for resonant contributions due to these two-photon processes is [17]

*N*(

_{i}*i*=1,…, 5) at the subbands in the active region are solved from the rate-equation model [5

5. J. Bai and D. S. Citrin, “Optical and transport characteristics of quantum-cascade lasers with optimized second-harmonic generation,” IEEE J. Quantum Electron. **43**, 391–398 (2007). [CrossRef]

*n*

_{2}from the sequential single-photon processes is ~2.6×10

^{-9}cm

^{2}/W and that from two-photon processes is ~2.4×10

^{-9}cm

^{2}/W. The two kinds of contributions are both positive and of comparable magnitude, which means that the additional harmonic resonant levels in the lasing active region significantly enhance (actually double)

*n*

_{2}. The maximum optical intensity in the fundamental frequency is about 1 MW/cm

^{2}. This will result in a total refractive index change Δ

*n*up to 0.005, which is within the range of 1×10

^{-2}to 1×10

^{-4}for a typical Kerr medium [18

18. L. Ding, R. Blackwell, J. F. Künzler, and W. H. Knox, “Large refractive index change in silicone-based and non-silicone-based hydrogel polymers induced by femtosecond laser micro-machining,” Opt. Express **14**, 11901–11909 (2006). [CrossRef] [PubMed]

*n*

_{2}may exhibit some tendency toward conventional SML due to the Kerr lensing mechanism and SPM. Therefore, a QCL structure with a significantly enhanced Kerr nonlinearity merits further study of the interactions between the RNGH instability and Kerr nonlinearities. In the next section, we demonstrate an optimization strategy to further increase the nonlinear refractive index

*n*

_{2}of the structure shown in Fig. 1.

## 3. Supersymmetric optimization of the Kerr refractive index

*n*

_{2}can be determined by first observing the expressions of

*χ*

^{(3)}(

*ω*;

*ω*,

*ω*,-

*ω*). As seen from Eqs. (3) and (4), both the single- and two-photon contributions to

*χ*

^{(3)}(

*ω*;

*ω*,

*ω*,-

*ω*) are proportional to the population distribution

*N*at each energy state and the DMEs

_{i}*M*, while

_{ij}*N*depends on both the band-structure and on the doping, but

_{i}*M*relates only to the band-structure. Via band-gap engineering, the band diagram of the QCLs can be flexibly tailored by a judicious choice of the geometries and compositions of the constituent quantum wells. Overall, optimization of

_{ij}*χ*

^{(3)}(

*ω*;

*ω*,

*ω*,-

*ω*) through optimizing the DMEs is preferred. To carry out an effective optimization procedure, the DME product chosen for optimization should be associated to the most dominant terms in Eqs. (3) and (4), respectively. In Eq. (3), the broadening between subbands 2 and 3 has the smallest magnitude |

*E*

_{23}-

*ħω*-

*iγ*

_{23}| and there is

*N*

_{3}-

*N*

_{2}≈

*N*

_{3}-

*N*

_{4}≈

*N*

_{3}-

*N*

_{5}for steady-state populations, so the first term associated with

*M*

^{4}

_{23}is dominant. In Eq. (4), the much larger population

*N*

_{3}(

*i.e.*,

*N*

_{3}≫

*N*

_{2},

*N*

_{4},

*N*

_{5}) in the second term suppresses the smaller broadening magnitude |

*E*

_{23}-

*ħω*-

*iγ*

_{23}| in the first term, so the second term associated with

*M*

^{2}

_{34}

*M*

^{2}

_{45}dominates. In order to increase these DME products and at the same time to maintain the energy-level positions to preserve the important resonances and, further, to facilitate carrier relaxation and period-to-period tunneling, the optimization method we adopted here is the supersymmetric quantum mechanics (SUSYQM) technique. In this technique, a family of isospectral potentials is generated depending on a single parameter

*λ*; the

*λ*value corresponds to the maximum DME which then determines the optimized potential shape in the active region. We used this technique in the optimization of second-harmonic generation in mid-infrared QCLs [4

4. J. Bai and D. S. Citrin, “Supersymmetric optimization of second-harmonic generation in mid-infrared quantum cascade lasers,” Opt. Express **14**, 4043–4048 (2006). [CrossRef] [PubMed]

4. J. Bai and D. S. Citrin, “Supersymmetric optimization of second-harmonic generation in mid-infrared quantum cascade lasers,” Opt. Express **14**, 4043–4048 (2006). [CrossRef] [PubMed]

*M*

^{2}

_{23}and

*M*

_{34}

*M*

_{45}with dependence on

*λ*are shown in Fig. 3. In Fig. 3(a), the optimization base-function is the wavefunction of the second subband level, i.e.,

*θ*(

*z*)=

*ψ*

^{(2)}

_{0}(

*z*), while

*θ*(

*z*)=

*ψ*

^{(4)}

_{0}(

*z*) in Fig. 3(b). As seen in Fig. 3(a)

*M*

^{2}

_{23}can be enhanced by up to 1.7 times while

*M*

_{34}

*M*

_{45}remains unchanged; in Fig. 3(b),

*M*

_{34}

*M*

_{45}increased 1.35 times and

*M*

^{2}

_{23}is unchanged. Due to the fact that

*N*

_{3}≫

*N*

_{2}at steady-state for lasing QCLs, the enhancement in Fig. 3(a) is stronger than that shown in Fig. 3(b). With

*λ*=0.22, the optimized potential can then be evaluated through

_{opt}*V*

_{o}(

*z*) is the original potential shape of the active region.

*E*=324+700

_{g}*x*+400

*x*

^{2}(meV) for Ga

_{x}In

_{1-x}As and

*Eg*=357+2290

*x*(meV) for Al

_{x}In

_{1-x}As [19

19. O. Madelung, *Semiconductors—Basic Data*, 3rd Ed. (Springer, New York, 1996), Chap. 2. [CrossRef]

*n*

_{2}due to the short dephasing times. In the original structure, the Ga

_{0.47}In

_{0.53}As/Al

_{0.48}In

_{0.52}As material system is lattice-matched to the InP substrate. In the optimized structure, due to the change of mole-fractions, there will be strain generated between layers, while our simulation results show that the total strain for one module of the QCL structure is compensated. The strain-induced deformation potential is also included in the model of the optimized structure. The enhancement of

*z*

^{2}

_{23}in the digitalized optimal structure is about 1.74, while that for the ideally smooth structure is 1.67. According to Eq. (3)

*χ*

^{(3)}

_{1p}(

*ω*;

*ω*,

*ω*, -

*ω*) can increase 3.3 times. Since

*χ*

^{(3)}

_{1p}(

*ω*;

*ω*,

*ω*, -

*ω*) and

*χ*

^{(3)}

_{2p}(

*ω*;

*ω*,

*ω*-

*ω*) have almost equal contributions to

*n*

_{2}in the original structure, the overall enhancement to

*n*

_{2}due to the SUSYQM optimization is a factor of ~2, with a value of 10

^{-8}cm

^{2}/W. This will result in a refractive index change of 0.01, which is much higher than those QCLs studied for spectrum broadening or multimode dynamics [11

11. R. Paiella, F. Capasso, C. Gmachl, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, A. Y. Cho, and H. C. Liu, “Self-mode-locking of quantum cascade lasers with giant ultrafast optical nonlinearities,” Science **290**, 1739–1742 (2000). [CrossRef] [PubMed]

12. C. Y. Yang, L. Diehl, A. Gordon, C. Jirauschek, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, M. Troccoli, J. Faist, and F. Capasso, “Coherent instabilities in a semiconductor laser with fast gain recovery,” Phys. Rev. A **75**, 031802(R) (2007). [CrossRef]

## 4. Conclusion

*n*

_{2}is necessary. We therefore illustrate an optimization strategy based on SUSYQM. The resultant stepwise QW structure is expected to have two-fold enhancement of

*n*

_{2}compared with the original flat quantum-well structure, which will bring a refractive index change of 0.01. This is much higher than that of the typical QCL structures available to date. The optimized structure may be used for continuing study on the relationship between Kerr nonlinearity and RNGH instability, while both lead to multimode performance of QCLs. Higher values of

*n*

_{2}may be possible by removing some constraints inherent in SUSYQM by using other optimization procedures. The optimized structure, while incorporating strain, is strain compensated over an entire period of the QCL. Finally, the optimized structure can be fabricated by the digital alloy technique with one-monolayer DP while substantially preserving the predicted enhancement. It has to be pointed out that the optimized structure has the drawback of fabrication complexity. The SUYQM optimization will work better for longer wavelength QCLs due to the easier digital alloy growth for wider QW layers. However, the optimized structure provides a lasing medium with unique combination of ultrafast carrier dynamics and high Kerr nonlinearity, which is desired for analyzing the interplay between different SML mechanisms in QCLs as well as their dominance in pulse initiation and stabilization. Experimental demonstration of SML in some lasers has been disappointing due to the fact that one mechanism is strong enough to sustain mode-locking but is typically unable to initiate it. Based on the optimized structure presented in the current work, we will further investigate the pulse formation and generation through a QCL medium with high Kerr nonlinearity by including the SPM and group velocity dispersion (GVD) effects.

## Acknowledgment

## References and links

1. | C. Gmachl, A. Belyanin, D. L. Sivco, M. L. Peabody, N. Owschimikow, A. M. Sergent, F. Capasso, and A. Y. Cho, “Optimized second-harmonic generation in quantum cascade lasers,” IEEE J. Quantum Electron. |

2. | T. S. Mosely, A. Belyanin, C. Gmachl, D. L. Sivco, M. L. Peabody, and A. Y. Cho, “Third harmonic generation in quantum cascade laser with monolithically integrated resonant optical nonlinearity,” Opt. Express |

3. | D. Qu, F. Xie, G. Shu, S. Momen, E. Narimanov, and C. F. Gmachl, “Second-harmonic generation in quantum cascade lasers with electric field and current dependent nonlinear susceptibility,” Appl. Phys. Lett , |

4. | J. Bai and D. S. Citrin, “Supersymmetric optimization of second-harmonic generation in mid-infrared quantum cascade lasers,” Opt. Express |

5. | J. Bai and D. S. Citrin, “Optical and transport characteristics of quantum-cascade lasers with optimized second-harmonic generation,” IEEE J. Quantum Electron. |

6. | F. Capasso, C. Gmachl, D. L. Sivco, and A. Y. Cho, “Quantum cascade lasers,” Phys. Today |

7. | G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller, “Frontiers in ultrashort pulse generation: pushing the limits in linear and optics,” Science |

8. | H. A. Haus, “Mode-locking of lasers,” IEEE J. Quantum Electron. |

9. | H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. |

10. | G. P. Agrawal, |

11. | R. Paiella, F. Capasso, C. Gmachl, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, A. Y. Cho, and H. C. Liu, “Self-mode-locking of quantum cascade lasers with giant ultrafast optical nonlinearities,” Science |

12. | C. Y. Yang, L. Diehl, A. Gordon, C. Jirauschek, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, M. Troccoli, J. Faist, and F. Capasso, “Coherent instabilities in a semiconductor laser with fast gain recovery,” Phys. Rev. A |

13. | H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. |

14. | Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” |

15. | R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, A. Y Cho, E. A. Whittaker, and H. C. Liu, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. |

16. | A. Gordon, C.Y. Yang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to special hole burning,” Phys. Rev. A , |

17. | R. W. Boyd, |

18. | L. Ding, R. Blackwell, J. F. Künzler, and W. H. Knox, “Large refractive index change in silicone-based and non-silicone-based hydrogel polymers induced by femtosecond laser micro-machining,” Opt. Express |

19. | O. Madelung, |

**OCIS Codes**

(190.3270) Nonlinear optics : Kerr effect

(190.5970) Nonlinear optics : Semiconductor nonlinear optics including MQW

(140.5965) Lasers and laser optics : Semiconductor lasers, quantum cascade

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: May 5, 2008

Revised Manuscript: July 25, 2008

Manuscript Accepted: July 31, 2008

Published: August 6, 2008

**Citation**

Jing Bai and D. S. Citrin, "Enhancement of optical Kerr effect in quantum-cascade lasers with multiple resonance levels," Opt. Express **16**, 12599-12606 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-12599

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### References

- C. Gmachl, A. Belyanin, D. L. Sivco, M. L. Peabody, N. Owschimikow, A. M. Sergent, F. Capasso, and A. Y. Cho, "Optimized second-harmonic generation in quantum cascade lasers," IEEE J. Quantum Electron. 39, 1345-1355 (2003). [CrossRef]
- T. S. Mosely, A. Belyanin, C. Gmachl, D. L. Sivco, M. L. Peabody, and A. Y. Cho, "Third harmonic generation in quantum cascade laser with monolithically integrated resonant optical nonlinearity," Opt. Express 12, 2972-2976 (2004). [CrossRef] [PubMed]
- D. Qu, F. Xie, G. Shu, S. Momen, E. Narimanov, and C. F. Gmachl, "Second-harmonic generation in quantum cascade lasers with electric field and current dependent nonlinear susceptibility, " Appl. Phys. Lett, 90, 031105 (2007). [CrossRef]
- J. Bai and D. S. Citrin, "Supersymmetric optimization of second-harmonic generation in mid-infrared quantum cascade lasers," Opt. Express 14, 4043-4048 (2006). [CrossRef] [PubMed]
- J. Bai and D. S. Citrin, "Optical and transport characteristics of quantum-cascade lasers with optimized second-harmonic generation," IEEE J. Quantum Electron. 43, 391-398 (2007). [CrossRef]
- F. Capasso, C. Gmachl, D. L. Sivco, and A. Y. Cho, "Quantum cascade lasers," Phys. Today 55, 34 (2002). [CrossRef]
- G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek and U. Keller, "Frontiers in ultrashort pulse generation: pushing the limits in linear and optics," Science 286, 1507-1512 (1999). [CrossRef] [PubMed]
- H. A. Haus, "Mode-locking of lasers," IEEE J. Quantum Electron. 6,1173-1185 (2000). [CrossRef]
- H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Analytic theory of additive pulse and Kerr lens mode locking," IEEE J. Quantum Electron. 28, 2086-2096 (1992). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics, 3rd Ed. (Academic Press, San Diego, 2001), Chap. 4.
- R. Paiella, F. Capasso, C. Gmachl, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, A. Y. Cho, H. C. Liu, "Self-mode-locking of quantum cascade lasers with giant ultrafast optical nonlinearities," Science 290, 1739-1742 (2000). [CrossRef] [PubMed]
- C. Y. Yang, L. Diehl, A. Gordon, C. Jirauschek, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, M. Troccoli, J. Faist, and F. Capasso, "Coherent instabilities in a semiconductor laser with fast gain recovery," Phys. Rev. A 75, 031802(R) (2007). [CrossRef]
- H. Risken and K. Nummedal, "Self-pulsing in lasers," J. Appl. Phys. 39, 4662-4672 (1968). [CrossRef]
- Graham and H. Haken, "Quantum theory of light propagation in a fluctuating laser-active medium," Zeitschrift für Physik. A 213, 420-450 (1968).
- R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, A. Y, Cho, E. A. Whittaker, and H. C. Liu, "High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers," Appl. Phys. Lett. 79, 2526-2528 (2001). [CrossRef]
- A. Gordon, C.Y. Yang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, "Multimode regimes in quantum cascade lasers: from coherent instabilities to special hole burning," Phys. Rev. A, 77, 053804 (2008). [CrossRef]
- R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, San Diego, 2003), 528-531.
- L. Ding, R. Blackwell, J. F. Künzler, and W. H. Knox, "Large refractive index change in silicone-based and non-silicone-based hydrogel polymers induced by femtosecond laser micro-machining," Opt. Express 14, 11901-11909 (2006). [CrossRef] [PubMed]
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