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 cm2/W, a twofold enhancement over the original flat quantum-well structure. This leads to a refractive-index change Δn 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

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  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]
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    [CrossRef] [PubMed]
  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,  90, 031105 (2007).
    [CrossRef]
  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]
  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]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2008 (1)

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]

2007 (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,  90, 031105 (2007).
[CrossRef]

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]

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]

2006 (2)

2004 (1)

2003 (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]

2002 (1)

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

2001 (1)

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]

2000 (2)

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]

H. A. Haus, "Mode-locking of lasers," IEEE J. Quantum Electron. 6,1173-1185 (2000).
[CrossRef]

1999 (1)

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]

1992 (1)

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]

1968 (2)

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).

Bai, J.

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]

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]

Baillargeon, J. N.

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]

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]

Belyanin, A.

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]

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]

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]

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]

Blackwell, R.

Bour, D.

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]

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]

Capasso, F.

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]

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]

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]

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

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]

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]

Cho, A. Y.

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]

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]

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

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]

Citrin, D. S.

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]

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]

Corzine, S.

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]

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]

Diehl, L.

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]

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]

Ding, L.

Faist, J.

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]

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]

Fujimoto, J. G.

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]

Gallmann, L.

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]

Gmachl, C.

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]

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]

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

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]

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]

Gmachl, C. F.

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]

Gordon, A.

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]

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]

Haus, H. A.

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]

Höfler, G.

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]

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]

Hutchinson, A. L.

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]

Hwang, H. Y.

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]

Ippen, E. P.

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]

Jirauschek, C.

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]

Kärtner, F. X.

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]

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]

Keller, U.

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]

Knox, W. H.

Künzler, J. F.

Liu, H. C.

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. 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]

Maier, T.

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]

Martini, R.

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]

Matuschek, N.

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]

Momen, S.

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]

Mosely, T. S.

Narimanov, E.

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]

Nummedal, K.

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

Owschimikow, N.

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]

Paiella, R.

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]

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]

Peabody, M. L.

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]

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]

Qu, D.

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]

Risken, H.

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

Schneider, H.

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]

Sergent, A. M.

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]

Shu, G.

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]

Sivco, D. L.

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]

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]

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

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]

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]

Steinmeyer, G.

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]

Sutter, D. H.

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]

Troccoli, M.

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]

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]

Xie, F.

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]

Yang, C. Y.

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]

Yang, C.Y.

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]

Appl. Phys. Lett (1)

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]

Appl. Phys. Lett. (1)

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]

IEEE J. Quantum Electron. (4)

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]

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]

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]

J. Appl. Phys. (1)

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

Opt. Express (3)

Phys. Rev. A (2)

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]

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]

Phys. Today (1)

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

Science (2)

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]

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]

Zeitschrift für Physik. A (1)

Graham and H. Haken, "Quantum theory of light propagation in a fluctuating laser-active medium," Zeitschrift für Physik. A 213, 420-450 (1968).

Other (3)

G. P. Agrawal, Nonlinear Fiber Optics, 3rd Ed. (Academic Press, San Diego, 2001), Chap. 4.

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, San Diego, 2003), 528-531.

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

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Figures (4)

Fig. 1.
Fig. 1.

Band structure of the injector, active region and collector of a QCL with resonant harmonic levels.

Fig. 2.
Fig. 2.

Illustration of different resonant transitions: (a) third-harmonic generation; (b) single-photon process; (c) two-photon process.

Fig. 3.
Fig. 3.

Dependence of DME products on λ with different optimization base function θ(z) : (a) θ(z)=ψ (2) 0(z); (b) θ(z)=ψ (4) 0(z).

Fig. 4.
Fig. 4.

SUSYQM optimized band structure: (a) Optimized potential shape and its one-monolayer stepwise approximation; (b) Comparison between band-structures of the original potential and optimized potential with monolayer DP.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

n 2 = 3 4 n 0 2 ε 0 c Re [ χ ( 3 ) ( ω ; ω , ω , ω ) ] ,
χ ( 3 ) ( ω ; ω , ω , ω ) = χ 1 p ( 3 ) ( ω ; ω , ω , ω ) + χ 2 p ( 3 ) ( ω ; ω , ω , ω ) .
χ 1 p ( 3 ) ( ω ; ω , ω , ω ) = 8 π q 4 3 ε 0 [ ( N 3 N 2 ) M 23 4 ( E 32 ћ ω i γ 32 ) 2 ( E 32 ћ ω + i γ 32 )
+ ( N 4 N 3 ) M 43 4 ( E 43 ћ ω i γ 43 ) 2 ( E 43 ћ ω + i γ 43 )
+ ( N 5 N 4 ) M 54 4 ( E 54 ћ ω i γ 54 ) 2 ( E 54 ћ ω + i γ 54 ) ] ,
χ 2 p ( 3 ) ( ω ; ω , ω , ω ) = 8 π q 4 3 ε 0 [ N 2 M 23 2 M 34 2 ( E 32 ћ ω i γ 32 ) 2 ( E 42 2 ћ ω + i γ 42 )
+ N 3 M 34 2 M 45 2 ( E 43 ћ ω i γ 43 ) 2 ( E 53 2 ћ ω + i γ 53 ) ] .

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