## Laterally periodic resonator for large-area gain lasers

Optics Express, Vol. 11, Issue 6, pp. 632-638 (2003)

http://dx.doi.org/10.1364/OE.11.000632

Acrobat PDF (166 KB)

### Abstract

Laterally periodic resonators, which can be constructed by use of transversely periodic phase- or amplitude-modulating elements in a cavity, are proposed for stabilization and generation of transversely coherent output from large-area gain. Lasers with periodic resonators have the combined features of conventional cavities and laser arrays. Significant low-order transverse modes and mode discrimination of a sample resonator with intracavity periodic phase elements are investigated numerically by the iteration method. Wave-propagation calculations are carried out by use of a fast Fourier transform, and a modified Prony method is used to evaluate wave functions and losses of transverse modes. Results of numerical calculations are consistent with expectations.

© 2002 Optical Society of America

## 1. Introduction

2. R. J. Pierre, G. W. Holleman, M. Valley, H. Injeyan, J. G. Berg, G. M. Harpole, R. C. Hilyard, M. Mitchell, M. E. Weber, J. Zamel, T. Engler, D. Hall, R. Tinti, and J. Machan, “Active tracked laser (ATLAS),” IEEE. J. Sel. Top. Quantum Electron. **3**, 64–70 (1997). [CrossRef]

3. R. J. Pierre, D.W. Mordaunt, H. Injeyan, J. G. Berg, R. C. Hilyard, M. E. Weber, M. G. Wickham, G. M. Harpole, and R. Senn, “Diode array pumped kilowatt laser,” IEEE. J. Sel. Top. Quantum Electron. **3**, 53–58 (1997). [CrossRef]

4. R. J. Lang, K. Dzurko, A. A. Hardy, S. Demars, A. Schoenfelder, and D. F. Welch, “Theory of grating-confined broad-area lasers,” IEEE J. Quantum Electron. **34**, 2196–2210 (1998). [CrossRef]

8. D. Stryckman, G. Rousseau, M. D’Auteuil, and N. McCarthy, “Improvement of the lateral-mode discrimination of broad-area diode lasers with a profiled reflectivity output facet,” Appl. Opt. **35**, 5955–5959 (1996). [CrossRef] [PubMed]

5. M. Szymanski, J. M. Kubica, and P. Szczepanski, “Theoretical analysis of lateral modes in broad-area semiconductor lasers with profiled reflectivity output facets,” IEEE J. Quantum Electron. **37**,430–438 (2001). [CrossRef]

8. D. Stryckman, G. Rousseau, M. D’Auteuil, and N. McCarthy, “Improvement of the lateral-mode discrimination of broad-area diode lasers with a profiled reflectivity output facet,” Appl. Opt. **35**, 5955–5959 (1996). [CrossRef] [PubMed]

9. B. Mroziewicz, “Broad-area semiconductor lasers with spatially modulated reflectivity of mirrors,” Electron. Lett. **32**, 329–330 (1996). [CrossRef]

12. J. R. Leger, D. Chen, and K. Dai, “High modal discrimination in a Nd:YAG laser resonator with internal phase gratings,” Opt. Lett. **19**, 1976–1978 (1994). [CrossRef] [PubMed]

13. D. Botez and D. R. Scifres, *Diode Laser Arrays* (Cambridge U. Press, Cambridge, UK, 1994), Chap. 1. [CrossRef]

17. D. Mehuys, W. Streifer, R. G. Waarts, and D. F. Welch, “Modal analysis of linear Talbot-cavity semiconductor lasers,” Opt. Lett. **16**, 823–825 (1991). [CrossRef] [PubMed]

18. R. J. Beach, M. D. Feit, R. H. Page, L. D. Brasure, R. Wilcox, and S. A. Payne, “Scalable antiguided ribbon laser,” J. Opt. Soc. Am. B **19**, 1521–1533 (2002). [CrossRef]

20. M. Wrage, P. Glas, D. Fisher, M. Leitner, D. V. Vysotsky, and A. P. Napartovich, “Phase locking in a multicore fiber laser by means of a Talbot resonator,” Opt. Lett. **25**, 1436–1438 (2000). [CrossRef]

21. Y. Kono, M. Takeoka, K. Uto, A. Uchida, and F. Kannari, “A coherent all-solid-state laser array using the Talbot effect in a three-mirror cavity,” IEEE J. Quantum Electron. **36**, 607–614 (2000). [CrossRef]

24. A. Desfarges-Berthelemot, B. Colombeau, M. Vampouille, P. J. Devilder, C. Froehly, and S. Monneret, “Adjustable phase-locking of two Nd:Glass ring laser beams,” Opt. Commun. **141**, 123–126 (1997). [CrossRef]

_{2}lasers [25

25. K. M. Abramski, A. D. Colley, H. J. Baker, and D. R. Hall, “Phase-locked CO_{2} laser array using diagonal coupling of waveguide channels,” Appl. Phys. Lett. **60**, 530–532 (1992). [CrossRef]

## 2. Concept

26. V. I. Yukalov, “Optical turbulent structures,” in *High-Power Laser Ablation III*, C. R. Phipps, ed., Proc. SPIE4065, 237–244 (2001). [CrossRef]

*N*

_{f}, and the period number,

*m;*then the Fresnel number of the subresonators is ~

*N*

_{f}/

*m*. If

*N*

_{f}/

*m*is small enough (sufficiently small

*N*

_{f}/

*m*can always be obtained by means of choosing a sufficiently high

*m*value), subresonators alone are likely to generate stable and high-beam-quality outputs. Because of coupling between them, oscillations in these subresonators can be locked together. Hence coherent output from all subresonators, which means coherent output from large-area gain, is possible. This is the key point of the current concept. Thus techniques and insights from phase locking of an array can be applied to this situation.

18. R. J. Beach, M. D. Feit, R. H. Page, L. D. Brasure, R. Wilcox, and S. A. Payne, “Scalable antiguided ribbon laser,” J. Opt. Soc. Am. B **19**, 1521–1533 (2002). [CrossRef]

20. M. Wrage, P. Glas, D. Fisher, M. Leitner, D. V. Vysotsky, and A. P. Napartovich, “Phase locking in a multicore fiber laser by means of a Talbot resonator,” Opt. Lett. **25**, 1436–1438 (2000). [CrossRef]

*N*

_{f}is inevitable. However, since the current concept is based on periodic structure, no limitation on lateral dimension is expected

*a priori*, at least theoretically.

27. J. K. Butler, D. E. Ackley, and D. Botez, “Coupled-mode analysis of phase-locked injection laser arrays,” Appl. Phys. Lett. **44**, 293–295 (1984); Appl. Phys. Lett. 44, 935 (erratum) (1984). [CrossRef]

28. E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked semiconductor laser arrays,” Opt. Lett. **10**, 125–127 (1984); Opt. Lett. 10, 318 (erratum) (1984). [CrossRef]

## 3. Modal properties of an example resonator

### 3.1 Numerical model

30. E. A. Sziklas and A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE **62**, 410–412 (1974). [CrossRef]

*P*is placed close to the plane mirror

*M*

_{l}, passing twice through the grating amounts to multiplying the field distribution

*u*(

*x,y*) by a laterally periodic phase delay of twice the modulation depth

*d*:

*d*is the phase-modulation depth and λ

*m*is the modulation spatial period.

31. A. E. Siegman and H. Y. Miller, “Unstable optical resonator loss calculations using the Prony method,” Appl. Opt. **9**, 2729–2736 (1970). [CrossRef] [PubMed]

_{i},

*u*

_{i}(

*x,y*)} can be obtained, where γ

_{i}is the resonator eigenvalue, which relates to the round-trip loss of the ith mode of the bare cavity by 1-|γ

_{i}|

^{2}and

*u*

_{i}(

*x,y*) is the field distribution of the

*i*th mode. The far-field intensity pattern can be calculated by means of squaring the Fourier transform of the corresponding mode field. When applying the FFT method in wave-propagation calculation, we must pay careful attention to questions of aliasing, sampling, and windowing.

### 3.2 Results and discussion

_{m}was set to 1 mm, a phase plate with 10 periods of modulation was inserted into a plane–plane cavity, and the mirror diameter was 10 mm. Cavity length

*L*and modulation depth

*d*were two varying parameters for investigation.

*d*=π/8 and the corresponding far-field patterns are displayed in Fig. 2. The longer the resonator, the larger the coupling between subresonators. One can see that modal structures are quite diverse. At the vicinity of

*L*=0.125 m and 1.25 m, fundamental modes are in-phase beam arrays; at

*L*=0.25 m and 0.75 m, fundamental modes are out-of-phase beam arrays; at the neighborhood of

*L*=0.5 m, which is one quarter of a Talbot length, the beam profiles of fundamental modes are similar to fundamental Gaussian modes of corresponding empty resonators but with extra modulations. This is because of the Talbot effect [32

32. M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. **43**, 2139–2164 (1996). [CrossRef]

*L*=1.0 and 1.125 m, envelopes of fundamental modes correspond to the third mode of the corresponding empty cavity.

*d*=π/8,

*L*=0.625 m and corresponding far-field patterns are shown in Fig. 3. First, second, and fifth modes have profiles similar to those of the first, second, and third modes of the corresponding empty cavity. Third and fourth modes are two out-of-phase array modes with beams operating mainly at the position of maximum and minimum of cosine phase modulation, respectively.

## 4. Summary

## Acknowledgments

## References and links

1. | A. E. Siegman, |

2. | R. J. Pierre, G. W. Holleman, M. Valley, H. Injeyan, J. G. Berg, G. M. Harpole, R. C. Hilyard, M. Mitchell, M. E. Weber, J. Zamel, T. Engler, D. Hall, R. Tinti, and J. Machan, “Active tracked laser (ATLAS),” IEEE. J. Sel. Top. Quantum Electron. |

3. | R. J. Pierre, D.W. Mordaunt, H. Injeyan, J. G. Berg, R. C. Hilyard, M. E. Weber, M. G. Wickham, G. M. Harpole, and R. Senn, “Diode array pumped kilowatt laser,” IEEE. J. Sel. Top. Quantum Electron. |

4. | R. J. Lang, K. Dzurko, A. A. Hardy, S. Demars, A. Schoenfelder, and D. F. Welch, “Theory of grating-confined broad-area lasers,” IEEE J. Quantum Electron. |

5. | M. Szymanski, J. M. Kubica, and P. Szczepanski, “Theoretical analysis of lateral modes in broad-area semiconductor lasers with profiled reflectivity output facets,” IEEE J. Quantum Electron. |

6. | J. R. Marciante and G. P. Agrawal, “Lateral spatial effects of feedback in gain-guided and broad-area semiconductor lasers,” IEEE J. Quantum Electron. |

7. | C. Simmendinger, D. Preisher, and O. Hess, “Stabilization of chaotic spatiotemporal filamentation in large broad area lasers by spatially structured optical feedback,” Opt. Express |

8. | D. Stryckman, G. Rousseau, M. D’Auteuil, and N. McCarthy, “Improvement of the lateral-mode discrimination of broad-area diode lasers with a profiled reflectivity output facet,” Appl. Opt. |

9. | B. Mroziewicz, “Broad-area semiconductor lasers with spatially modulated reflectivity of mirrors,” Electron. Lett. |

10. | R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Transverse mode shaping and selection in laser resonators,” in |

11. | J. R. Leger, D. Chen, and Z. Wang, “Diffractive optical element for mode shaping of a Nd:YAG laser,” Opt. Lett. |

12. | J. R. Leger, D. Chen, and K. Dai, “High modal discrimination in a Nd:YAG laser resonator with internal phase gratings,” Opt. Lett. |

13. | D. Botez and D. R. Scifres, |

14. | D. Auerbach and J. A. Yorke, “Controlling chaotic fluctuations in semiconductor laser arrays,” J. Opt. Soc. Am. B |

15. | J. R. Leger, G. Mowry, and X. Li, “Modal properties of an external diode-laser-array cavity with diffractive mode-selecting mirrors,” Appl. Opt. |

16. | M. Cronin-Golomb, A. Yariv, and I. Ury, “Coherent coupling of diode lasers by phase conjugation,” Appl. Phys. Lett. |

17. | D. Mehuys, W. Streifer, R. G. Waarts, and D. F. Welch, “Modal analysis of linear Talbot-cavity semiconductor lasers,” Opt. Lett. |

18. | R. J. Beach, M. D. Feit, R. H. Page, L. D. Brasure, R. Wilcox, and S. A. Payne, “Scalable antiguided ribbon laser,” J. Opt. Soc. Am. B |

19. | M. Wrage, P. Glas, and M. Leitner, “Combined phase locking and beam shaping of a multicore fiber laser by structured mirrors,” Opt. Lett. |

20. | M. Wrage, P. Glas, D. Fisher, M. Leitner, D. V. Vysotsky, and A. P. Napartovich, “Phase locking in a multicore fiber laser by means of a Talbot resonator,” Opt. Lett. |

21. | Y. Kono, M. Takeoka, K. Uto, A. Uchida, and F. Kannari, “A coherent all-solid-state laser array using the Talbot effect in a three-mirror cavity,” IEEE J. Quantum Electron. |

22. | M. Oka, H. Masuda, Y. Kaneda, and S. Kubota, “Laser-diode-pumped phase-locked Nd:YAG laser arrays,” IEEE J. Quantum Electron. |

23. | S. Menard, M. Vampouille, B. Colombeau, and C. Froehly, “Highly efficient phase locking and extracavity coherent combination of two diode-pumped Nd:YAG laser beams,” Opt. Lett. |

24. | A. Desfarges-Berthelemot, B. Colombeau, M. Vampouille, P. J. Devilder, C. Froehly, and S. Monneret, “Adjustable phase-locking of two Nd:Glass ring laser beams,” Opt. Commun. |

25. | K. M. Abramski, A. D. Colley, H. J. Baker, and D. R. Hall, “Phase-locked CO |

26. | V. I. Yukalov, “Optical turbulent structures,” in |

27. | J. K. Butler, D. E. Ackley, and D. Botez, “Coupled-mode analysis of phase-locked injection laser arrays,” Appl. Phys. Lett. |

28. | E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked semiconductor laser arrays,” Opt. Lett. |

29. | A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. |

30. | E. A. Sziklas and A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE |

31. | A. E. Siegman and H. Y. Miller, “Unstable optical resonator loss calculations using the Prony method,” Appl. Opt. |

32. | M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. |

**OCIS Codes**

(140.3290) Lasers and laser optics : Laser arrays

(140.3410) Lasers and laser optics : Laser resonators

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 15, 2002

Revised Manuscript: March 11, 2003

Published: March 24, 2003

**Citation**

Yan Feng and Ken-ichi Ueda, "Laterally periodic resonator for large-area gain lasers," Opt. Express **11**, 632-638 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-6-632

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### References

- A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 22 and 23.
- R. J. Pierre, G.W. Holleman, M. Valley, H. Injeyan, J. G. Berg, G. M. Harpole, R. C. Hilyard, M. Mitchell, M. E. Weber, J. Zamel, T. Engler, D. Hall, R. Tinti, and J. Machan, �??Active tracked laser (ATLAS),�?? IEEE. J. Sel. Top. Quantum Electron. 3, 64-70 (1997). [CrossRef]
- R. J. Pierre, D.W. Mordaunt, H. Injeyan, J. G. Berg, R. C. Hilyard, M. E.Weber, M. G.Wickham, G. M. Harpole, and R. Senn, �??Diode array pumped kilowatt laser,�?? IEEE. J. Sel. Top. Quantum Electron. 3, 53-58 (1997). [CrossRef]
- R. J. Lang, K. Dzurko, A. A. Hardy, S. Demars, A. Schoenfelder, and D. F. Welch, �??Theory of grating-confined broad-area lasers,�?? IEEE J. Quantum Electron. 34, 2196-2210 (1998). [CrossRef]
- M. Szymanski, J. M. Kubica, P. Szczepanski, �??Theoretical analysis of lateral modes in broad-area semiconductor lasers with profiled reflectivity output facets,�?? IEEE J. Quantum Electron. 37, 430-438 (2001 [CrossRef]
- J. R. Marciante and G. P. Agrawal, �??Lateral spatial effects of feedback in gain-guided and broad-area semiconductor lasers,�?? IEEE J. Quantum Electron. 32, 1630-1635 (1996). [CrossRef]
- C. Simmendinger, D. Preisher, and O. Hess, �??Stabilization of chaotic spatiotemporal filamentation in large broad area lasers by spatially structured optical feedback,�?? Opt. Express 5, 48-54 (1999), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-5-3-48">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-5-3-48</a> [CrossRef] [PubMed]
- D. Stryckman, G. Rousseau, M. D�??Auteuil, and N. McCarthy, �??Improvement of the lateral-mode discrimination of broad-area diode lasers with a profiled reflectivity output facet,�?? Appl. Opt. 35, 5955-5959 (1996). [CrossRef] [PubMed]
- B. Mroziewicz, �??Broad-area semiconductor lasers with spatially modulated reflectivity of mirrors,�?? Electron. Lett. 32, 329-330 (1996). [CrossRef]
- R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, �??Transverse mode shaping and selection in laser resonators,�?? in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 2001), Vol. II.
- J. R. Leger, D. Chen, and Z. Wang, �??Diffractive optical element for mode shaping of a Nd:YAG laser,�?? Opt. Lett. 19, 108-110 (1994) [CrossRef] [PubMed]
- J. R. Leger, D. Chen, and K. Dai, �??High modal discrimination in a Nd:YAG laser resonator with internal phase gratings,�?? Opt. Lett. 19, 1976-1978 (1994). [CrossRef] [PubMed]
- D. Botez and D. R. Scifres, Diode Laser Arrays (Cambridge U. Press, Cambridge, UK, 1994), Chap. 1 [CrossRef]
- D. Auerbach and J. A. Yorke, �??Controlling chaotic fluctuations in semiconductor laser arrays,�?? J. Opt. Soc. Am. B 13, 2178-2186 (1996). [CrossRef]
- J. R. Leger, G. Mowry, and X. Li, �??Modal properties of an external diode-laser-array cavity with diffractive mode-selecting mirrors,�?? c. 34, 4302-4311 (1995). [CrossRef] [PubMed]
- M. Cronin-Golomb, A. Yariv, and I. Ury, �??Coherent coupling of diode lasers by phase conjugation,�?? Appl. Phys. Lett. 48, 1240-1242 (1986). [CrossRef]
- D. Mehuys, W. Streifer, R. G. Waarts, and D. F. Welch, �??Modal analysis of linear Talbot-cavity semiconductor lasers,�?? Opt. Lett. 16, 823-825 (1991). [CrossRef] [PubMed]
- R. J. Beach, M. D. Feit, R. H. Page, L. D. Brasure, R. Wilcox, and S. A. Payne, �??Scalable antiguided ribbon laser,�?? J. Opt. Soc. Am. B 19, 1521-1533 (2002). [CrossRef]
- M. Wrage, P. Glas, and M. Leitner, �??Combined phase locking and beam shaping of a multicore fiber laser by structured mirrors,�?? Opt. Lett. 26, 980-982 (2001). [CrossRef]
- M. Wrage, P. Glas, D. Fisher, M. Leitner, D. V. Vysotsky, and A. P. Napartovich, �??Phase locking in a multicore fiber laser by means of a Talbot resonator,�?? Opt. Lett. 25, 1436-1438 (2000). [CrossRef]
- Y. Kono, M. Takeoka, K. Uto, A. Uchida, and F. Kannari, �??A coherent all-solid-state laser array using the Talbot effect in a three-mirror cavity,�?? IEEE J. Quantum Electron. 36, 607-614 (2000). [CrossRef]
- M. Oka, H. Masuda, Y. Kaneda, and S. Kubota, �??Laser-diode-pumped phase-locked Nd:YAG laser arrays,�?? IEEE J. Quantum Electron. 28, 1142-1147 (1992). [CrossRef]
- S. Menard, M. Vampouille, B. Colombeau, and C. Froehly, �??Highly efficient phase locking and extracavity coherent combination of two diode-pumped Nd:YAG laser beams,�?? Opt. Lett. 21, 1996-1998 (1996). [CrossRef] [PubMed]
- A. Desfarges-Berthelemot, B. Colombeau, M. Vampouille, P. J. Devilder, C. Froehly, and S. Monneret, �??Adjustable phase-locking of two Nd:Glass ring laser beams,�?? Opt. Commun. 141, 123-126 (1997). [CrossRef]
- K. M. Abramski, A. D. Colley, H. J. Baker, and D. R. Hall, �??Phase-locked CO2 laser array using diagonal coupling of waveguide channels,�?? Appl. Phys. Lett. 60, 530-532 (1992). [CrossRef]
- V. I. Yukalov, �??Optical turbulent structures,�?? in High-Power Laser Ablation III, C. R. Phipps, ed., Proc. SPIE 4065, 237-244 (2001). [CrossRef]
- J. K. Butler, D. E. Ackley, and D. Botez, �??Coupled-mode analysis of phase-locked injection laser arrays,�?? Appl. Phys. Lett. 44, 293-295 (1984); Appl. Phys. Lett. 44, 935 (erratum) (1984). [CrossRef]
- E. Kapon, J. Katz, and A. Yariv, �??Supermode analysis of phase-locked semiconductor laser arrays,�?? Opt. Lett. 10, 125-127 (1984); Opt. Lett. 10, 318 (erratum) (1984). [CrossRef]
- A. G. Fox and T. Li, �??Resonant modes in a maser interferometer,�?? Bell Syst. Tech. J. 40, 453-488 (1961).
- E. A. Sziklas and A. E. Siegman, �??Diffraction calculations using fast Fourier transform methods,�?? Proc. IEEE 62, 410-412 (1974). [CrossRef]
- A. E. Siegman and H. Y. Miller, �??Unstable optical resonator loss calculations using the Prony method,�?? Appl. Opt. 9, 2729-2736 (1970). [CrossRef] [PubMed]
- M. V. Berry, and S. Klein, �??Integer, fractional and fractal Talbot effects,�?? J. Mod. Opt. 43, 2139-2164 (1996). [CrossRef]

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