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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 6 — Mar. 24, 2003
  • pp: 632–638
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Laterally periodic resonator for large-area gain lasers

Yan Feng and Ken-ichi Ueda  »View Author Affiliations


Optics Express, Vol. 11, Issue 6, pp. 632-638 (2003)
http://dx.doi.org/10.1364/OE.11.000632


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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

Although energy storage can be enhanced by use of large-volume gain media, generation of high laser-output power from conventional stable resonators, while maintaining good beam quality, is limited by fundamental Gaussian-mode volume and by the problem of damage from cavity elements. To extract available power from large-area gain, unstable resonators with graded reflective mirrors have been successful [1

1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 22 and 23.

] but are not suitable for low- and medium-gain lasers. Another approach is use of the master oscillator power amplifier (MOPA) [2

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

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]

], which has a low-power but high-beam-quality laser as a master oscillator and large-area gain media as amplification stages. Hence MOPA combines the advantages of high beam quality with MO and the high energy storage of amplifiers. However, this system is complicated, and its efficiency is generally low.

The broad-area laser (BAL) is a type of large-area gain laser in the field of semiconductor lasers, which has received both theoretical and practical attention [4

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

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]

]. To suppress the chaotic spatiotemporal filamentation and to generate stable and coherent output, the above-mentioned unstable resonators and MOPA have been used. In addition, BALs with single or several profiled reflecting stripes have been investigated [5

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

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

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

]. It has been shown that the BAL could be an effective tool for lateral mode control; moreover, stabilization of the filament location was obtained. Another interesting approach is the so-called “custom resonator,” which enables the custom fundamental mode profile by use of a diffractive mode-selecting mirror and intracavity phase plates [10

10. 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.

12

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]

] but gives good results only for cavities with quite small Fresnel numbers. However, phase locking of multiple laser outputs, as an alternative approach for scaling output power while maintaining beam quality, has been intensively investigated for semiconductor lasers [13

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

17

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]

], fiber and planar waveguide lasers [18

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

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]

], solid-state lasers [21

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

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]

], CO2 lasers [25

25. 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]

], and so on. Three coupling mechanisms exist: evanescent coupling, in which gain cores couple evanescently to their nearest neighbors; diffractive coupling, such as Talbot plane methods; and radiative coupling, in which all gain cores communicate with all others, such as antiguided laser arrays. Because of phase locking, optical fields would be confined mostly to one or two lobes for in-phase and out-of-phase locking, respectively.

2. Concept

The Fresnel number plays a similar role in optics as the Reynolds number does in hydrodynamics [26

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

]. Multimode operation and even spatiotemporal filamentation will occur in lasers with large Fresnel numbers. Our concept is based on combining features of broad-area lasers and laser arrays. The laterally periodic resonator, which can be constructed by use of transversely periodic phase- or amplitude-modulating elements in a cavity, could be considered physically as an array of equally separated identical subresonators, if diffractive coupling between oscillations of subresonators is not too strong (as with the tight binding approximation in solid-state physics). Let the Fresnel number of the whole resonator be Nf , and the period number, m; then the Fresnel number of the subresonators is ~Nf /m. If Nf /m is small enough (sufficiently small Nf /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.

The difference between this concept and that of common laser arrays is that, here, gain is uniform over the cross section instead of being spatially distributed. So obviously, this concept has the advantage of ease in fabrication of gain medium compared with multicore waveguide laser designs [18

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

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]

].

It is noteworthy that in this case the current concept is similar to previously reported approaches, such as custom resonators and resonators with intracavity binary phase elements [10

10. 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.

]. But since those approaches all aim at generating a custom mode from a conventional cavity with a certain axial symmetry, limited application on small Nf is inevitable. However, since the current concept is based on periodic structure, no limitation on lateral dimension is expected a priori, at least theoretically.

3. Modal properties of an example resonator

3.1 Numerical model

Figure 1 illustrates the sample resonator configuration. The modulated feedback could be of either phase or loss. In practice, loss modulation can be realized by modulated reflection coating or insertion of a lossy filter. By inserting a phase plate, we can realize phase modulation. For the sake of mathematical simplicity, cosine phase modulation is considered here.

Fig. 1. Schematic of the introductory resonator configuration used in numerical investigation. Ml , Mr , and P are left end mirror, right end mirror, and periodic phase plate, respectively. L is cavity length; d and λm are modulation depth and spatial period of P, respectively. u(x,y) is the optical field oscillating in the cavity.

To evaluate modes and their losses, the iteration method first introduced to mode calculation by Fox and Li is applied [29

29. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

]. A fast Fourier transform (FFT) is used for wave-propagation calculations, which was first introduced by Siegman and Miller [30

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

] for its significant increase in computational efficiency. The difference and key element here is the periodic phase plate. Assuming that the phase plate 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:

u(x,y)=u(x,y)exp[j2dcos(2πx2+y2λm)],
(1)

where d is the phase-modulation depth and λm is the modulation spatial period.

The Prony method is used to find low-order transverse modes and their associated losses [31

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]

]. Calculations using this method give higher precision for the modes that have lower loss. So only modes with the lowest loss are picked up; then another iteration is started with a new initial field distribution, which is orthogonal to all mode functions that have already been selected. Our calculations show that such a modified Prony method results in much better precision.

Thus data pairs {γi, u i(x,y)} can be obtained, where γi is the resonator eigenvalue, which relates to the round-trip loss of the ith mode of the bare cavity by 1-|γi|2 and ui (x,y) is the field distribution of the ith 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

Numerical investigation was performed on a one-dimensional model. λ was set to 1 µm, λm 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.

Fig. 2. Left, fundamental mode patterns for resonators with modulation d=π/8 and cavity length L=0.125, 0.250, …, 1.25 from top to bottom, respectively. Right, corresponding far-field patterns.

By starting with a random initial optical-field distribution, the optical field always becomes symmetric and coherent across the lateral section after a sufficiently large number of iterations, which means that the diffractive coupling among different portions of the cavity would always be strong enough to give rise to spatial coherence. This phenomenon is a numerical artifact and should not be occur for real cavities that have losses, gains, and perturbations. There must be a requirement on coupling strength for the optical field across the lateral section to be correlated.

Fig. 3. Left, near-field amplitude profiles of the five lowest-order modes at d=π/8, L=0.625 m. Right, corresponding far-field intensity patterns.

The near-field amplitude profiles of the five lowest-order modes at 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.

It is worth noting that those results are consistent with what one can expect from the combined features of laser arrays and conventional cavities. It seems that generation of array-like coherent output is possible.

4. Summary

Fig. 4. Round-trip losses of three lowest-order modes for different cavity lengths L and phasemodulation depths d.

Acknowledgments

This research has been supported by a Grant-in-Aid for Science Research from the Ministry of Education, Science and Culture of Japan. We thank J. F. Bisson for his reading of the manuscript.

References and links

1.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 22 and 23.

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]

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]

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. 32, 1630–1635 (1996). [CrossRef]

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 5, 48–54 (1999).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-5-3-48. [CrossRef] [PubMed]

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]

10.

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.

11.

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]

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]

14.

D. Auerbach and J. A. Yorke, “Controlling chaotic fluctuations in semiconductor laser arrays,” J. Opt. Soc. Am. B 13, 2178–2186 (1996). [CrossRef]

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. 34, 4302–4311 (1995). [CrossRef] [PubMed]

16.

M. Cronin-Golomb, A. Yariv, and I. Ury, “Coherent coupling of diode lasers by phase conjugation,” Appl. Phys. Lett. 48, 1240–1242 (1986). [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]

19.

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]

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]

22.

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]

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. 21, 1996–1998 (1996). [CrossRef] [PubMed]

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]

25.

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]

26.

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

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]

29.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

30.

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

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]

32.

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

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

  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 22 and 23.
  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]
  5. 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]
  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. 32, 1630-1635 (1996). [CrossRef]
  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 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]
  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]
  10. 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.
  11. 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]
  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]
  14. D. Auerbach and J. A. Yorke, �??Controlling chaotic fluctuations in semiconductor laser arrays,�?? J. Opt. Soc. Am. B 13, 2178-2186 (1996). [CrossRef]
  15. 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]
  16. M. Cronin-Golomb, A. Yariv, and I. Ury, �??Coherent coupling of diode lasers by phase conjugation,�?? Appl. Phys. Lett. 48, 1240-1242 (1986). [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]
  19. 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]
  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]
  22. 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]
  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. 21, 1996-1998 (1996). [CrossRef] [PubMed]
  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]
  25. 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]
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