Abstract

We demonstrate phase locking of a linear array of seven fiber lasers via diffractive coupling. Coupling between the lasers is achieved by a common output coupler positioned at a quarter Talbot distance from the lasers. The output beams are anti-phase locked with a measured far-field fringe contrast of 82%, and their total output power is higher than that obtained when the lasers operate individually. We measure an exponential phase decorrelation between distant lasers in the array, and discuss its fundamental limitation on scalability of this and similar local coupling methods.

© 2011 Optical Society of America

1. Introduction

A single fiber laser is limited both in peak power and average power. Currently, the peak power of pulsed fiber lasers and the average power of (single mode) CW fiber lasers are limited to several megawatts and several kilowatts, respectively. To overcome these limits different approaches for beam combining of fiber lasers were investigated over the past decade. These include active coherent beam combing of fiber amplifiers, intra-cavity passive coherent beam combining, and incoherent spectral beam combining [14]. Yet, scalability of the various approaches, both in terms of the number of combined lasers and the absolute total output power (brightness), has proved to be very challenging and so far with limited success.

Efficient coherent combining of lasers requires that their beams be phase locked. Passive phase locking techniques, generally considered to be simple compared to active techniques, have been extensively studied and developed over the years. They include self-Fourier resonators [5], intra-cavity interferometric elements [6], intra-cavity polarization elements [7], etc. Specifically, the simple Talbot coupling technique, which relies on local diffractive coupling between neighboring lasers, was only demonstrated in diode arrays [8, 9], and in multicore fiber lasers [10] where the fill factor was low, and single supermode operation was not achieved.

In this paper, we investigate phase locking of a fiber laser array via the diffractive coupling. To the best of our knowledge, we present the first experimental demonstration of phase locking a linear fiber laser array exploiting the Talbot effect. This phase locking technique is relatively simple, robust, does not require any additional intra-cavity optical elements, and in principle can be easily implemented in large 2D fiber laser arrays. In this work we focus on a small laser array (7 fiber lasers) in order to avoid the issue of common longitudinal modes, which currently limits the scalability of phased locked fiber laser arrays [11]. We show experimentally that with the Talbot coupling method there is phase decorrelation between the lasers in the array, which poses an additional fundamental limitation on scalability of this and similar local coupling methods. We believe the decorrelation is due to environmental phase noise and to the local nature of the coupling between lasers. In two dimensional arrays the number of nearest neighbors is increased so we expect the decorrelation limitation to be less stringent.

2. Basic configuration

The basic fiber laser array configuration is shown in Fig. 1. It consists of four modules: rear mirrors of the laser array, the active fiber array (1D or 2D), a micro lens array for collimating the light from each fiber in the array, and a common output coupler. The collimated light beams coming out of the micro lens array are partially reflected back from the output coupler, and due to diffraction some light from each beam is coupled into neighboring lasers. Since the fibers are positioned in a periodic manner, if the beams are phased locked with the same phase at the fiber outputs their field distribution will reconstruct itself after a Talbot distance. If the beams are phase locked in an anti-phase mode (0−π−0−π ..) their field distribution will reconstruct after half a Talbot distance. In our experiments we positioned the output coupler at a quarter of a Talbot distance from the micro lens array, and thus the anti-phase mode has relatively low losses and is expected to be dominant.

 

Fig. 1 Basic configuration for phase locking a fiber laser array via diffractive coupling.

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3. Experimental setup and results

The experimental setup included 10.0 ± 0.6 m long Yb doped polarization maintaining double clad fibers, with 125 μm pump cladding diameter and 6 μm core diameter, which were pumped via fiber couplers by 915 nm laser diodes. The rear end of each fiber was spliced to a fiber collimating lens which was aligned to the high reflecting rear mirror. The front end of each of the active fibers was spliced to a passive fiber, and all passive fibers were held in a tightly packed linear v-groove array. The pitch size of the fiber v-groove array was 250 μm, the fibers numerical aperture (NA) was 0.14, and they were angle cleaved (8 degrees). The pitch size of the micro lens array was 250 μm, the radius of curvature of each micro lens was 487 μm, and its NA 0.14. The reflectivity of the common output coupler was 30%. The resulting fill factor of the laser array was ≈0.4, and the lasers operated around 1080 nm. The Talbot distance for the beams coming out of the micro lens array was Zt = 11.57 cm and the output coupler was positioned at a distance of 2.89 cm from the micro lens array. During the experiments variations at the scale of 1 cm of the resonator length did not affect the phase locking between lasers, in good correspondence with eigenmode calculations [9]. According to the free space cavity analysis presented in [9] we calculated that at a quarter Talbot distance the losses to the anti-phase mode are 16% lower than the losses to that mode when the output coupler is at a half Talbot distance, and therefore we chose to work at a quarter Talbot distance.

The measured near field intensity distribution is shown in Fig. 2(a). As evident, the fill factor is relatively high. The far field intensity distribution is presented in Fig. 2(b) and its cross section in Fig. 2(c). The measured fringe contrast of the anti-phase mode is 82% (the fringe contrast of pure theoretical anti-phase mode is 85.4%). Using the Gerschberg Saxton algorithm [12] we retrieved the phase distribution of the near-field and calculated the far-field intensity distribution for the case where a suitable π-phase element is placed at the near-field. Figure 2(c) shows both the calculated (in-phase) far-field intensity distribution in this case, and the ideal far-field intensity distribution of an ideal in-phase mode. The beam quality (M2) of the expected in-phase far field intensity distribution beam is deteriorated by a factor of 1.61. The mean power of a single fiber laser when operated alone was measured to be 4.64 mW, while that of a single laser when all lasers were operated in an anti-phase locked mode was measured to be 5.28 mW. This can be easily explained by increased diffraction losses when a single laser operates alone.

 

Fig. 2 (a) Measured near field intensity distribution of an array of seven fiber lasers and (b) their far field intensity distribution. (c) The cross section of the far field intensity distribution of the supermode. Blue curve: measured far field intensity, fringe contrast of 82.3%. Green curve: retrieved far field intensity by Gerschberg Saxton algorithm. Red curve: calculated intensity distribution of the pure in phase supermode. Black curve: calculated (by Gerschberg Saxton algorithm) far field intensity distribution for the anti phase supermode after a π phase mask that changes it into an in phase mode.

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In our experiments, in order to avoid the problem of common longitudinal modes that arise when phase locking large laser arrays we limited our experiments to only 7 lasers where high efficiency phase locking can be obtained (i.e. phase locking lasers with different cavity lengths, where each has its own set of eigen frequencies, is becoming more difficult as the number of lasers increase, and therefore the mean detuning between all lasers in the system increases; [11, 13, 14]). To verify that the issue of common longitudinal mode does not play a dominant role in our configuration we measured the spectrum of the output laser beams when operated individually and when operated together in the anti-phased locked mode. The output spectrum of each fiber laser (when operated alone), their mean curve, and that of the whole phase locked array are shown in Fig. 3(a). As evident, the spectrum of the phase locked array is similar to the mean spectrum, indicating that it does not expand or shrink in order to find a common longitudinal mode, although the laser bandwidth is not limited and theoretically can be 70 nm wide (Yb:Silica). In Fig. 3(b) we plot the far-field intensity fringe contrast as a function of the number of lasers operating in our array. As evident, the far-field intensity fringe contrast shows a weak dependence on the number of lasers in the phase-locked array. The maximum fringe contrast is obtained with 4 lasers. A possible explanation for this is that with fewer lasers the Talbot effect is relatively weak while with more lasers the common longitudinal mode issue starts to affect the phase locking.

 

Fig. 3 (a) The spectrum of the individual lasers in the array, their mean spectrum, and the spectrum of the phase-locked array; (b) far field intensity fringe contrast as a function of the number of lasers in the array.

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When the optical lengths of the lasers in the array are not well controlled the common longitudinal mode issue severely limits the scalability to large arrays. However, in the case of systems with local coupling their exist another inherent limitation for achieving global coherence in large arrays. This phenomena is well known in the field of magnetization of solids, where magnetization of large ensembles is achieved when the temperature is larger than zero, only for a 3D array. However, the correlation function of the array, decrease exponentially for a 1D array, and can decay both exponentially and with a power law for a 2D array. In analogy, when phase locking large laser arrays via local coupling the system can start suffering from similar correlation decrease within the array [15, 16].

In order to investigate possible decorrelation in our phase locked array we experimentally measured the far field intensity fringe contrast of only two lasers within the array as a function of the distance between the two lasers (when the whole array is operating in an anti phase locked supermode). In Fig. 4 we plot the measured fringe contrast between two lasers as a function of the number of lasers between them in a semi log scale. As expected for 1D systems with nearest neighbor coupling the correlation function indeed decreases exponentially (see the red linear fit in Fig. 4). The fitted curve for the exponential decrease in the fringe contrast is fci,j = e−0.61(ij), where ij is the distance between lasers.

 

Fig. 4 Measured far field intensity fringe contrast of two lasers in the phase locked array as a function of the distance between the two lasers. As evident, although all lasers operate with the same central wavelength, the fringe contrast between two distant lasers is deteriorated. Blue dots: measured fringe contrast. Green line: mean. Red line: linear fit.

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To translate the decrease in phase correlation to more meaningful terms such as Strehl ratio of the array, we followed the analysis presented in [17]. The Strehl ratio can be written as:

S=<|ΣiEi|2>E02N2=N+2Σi,ji<cos(ϕiϕj)>N2
Where E0 is the amplitude value for all laser fields, Ei is the complex field of laser i, N is the number of lasers in the array, and ϕi is the phase of laser i. The measured far-field intensity fringe contrast of laser i and j is equal to their phase correlation:
fci,j<|Ei+Ej|2><|EiEj|2><|Ei+Ej|2>+<|EiEj|2>=<cos(ϕiϕj)>

Assuming an exponential decay eα(ij) in fringe contrast (as measured in our experiment), the dependence of the Strehl ratio on the number of lasers and the exponential decay parameter is:

S=N+2(Σi,jieα(ij))N2=N+2eα(eNαeα(eα1)2N1eα1)N2

In Fig. 5 we plot the Strehl ratio of an array for different correlation lengths and array sizes. As expected, the Strehl ratio decreases for large arrays and for large exponential decay parameters [17]. For short correlation lengths (eα =0.1), the Strehl ratio decreases like 1N, similar to an incoherent laser array. For long correlation lengths the Strehl ratio decreases linearly. For mid-sized correlation lengths and large N the Strehl ratio approaches, asymptotically, SxN, where x is a parameter depending only on the correlation length. It is interesting to point out that the scalability limitation due to common longitudinal modes also predicts for large N the same dependency of the Strehl ratio on the number of lasers [11].

 

Fig. 5 The Strehl ratio as a function of the size of the array for different correlation lengthes. eα, blue curve: eα=0.99; green curve eα=0.9; red curve eα=0.54 (similar to our experimental results); turquoise curve: eα=0.1.

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From the fringe contrast measurements between two lasers in the array we can estimate that eα ≃ 0.54 (see Fig. 4), and the corresponding Strehl ratio according to our analysis should be ≃ 0.375 (red curve, N=7 in Fig. 5). The Strehl ratio calculated from the intensity distributions in Fig. 2(c) is ∼ 0.48, which is in good agreement.

In order to overcome the strong dephasing between distant lasers in an array, one can use semi-local coupling (next nearest neighbors interaction) or, a better option, is to phase lock a 2D laser array, since every laser can correlate to every other laser through different correlation pathes, and the over all global correlation of the entire array will increase [15, 16].

4. Conclusion

We investigated phase locking of a linear fiber laser array exploiting the Talbot effect. We obtained phase locking of 7 fiber lasers in an anti-phase mode, with high fringe contrast. This configuration is relatively simple, does not include any special intra-cavity elements, and thus could be incorporated in practical applications. However, we show that locally coupled laser arrays will suffer from exponential correlation decay that will decrease the Strehl ratio of the entire array. We believe that with a 2D array, the dephasing will be less significant and high Strehl ratio could be achieved for locally coupled laser arrays with a larger number of lasers.

Acknowledgments

This research was supported by the Israel Science Foundation (grant No. 1205/08 and 1626/08).

References and links

1. S. J. Augst, T. Y. Fan, and A. Sanchez, “Coherent beam combining and phase noise measurements of ytterbium fiber amplifiers,” Opt. Lett. 29, 474–476 (2004). [CrossRef]   [PubMed]  

2. S. J. Augst, A. K. Goyal, R. L. Aggarwal, T. Y. Fan, and A. Sanchez, “Wavelength beam combining of ytterbium fiber lasers,” Opt. Lett. 28, 331–333 (2003). [CrossRef]   [PubMed]  

3. A. Ishaaya, N. Davidson, and A. Friesem, “Passive laser beam combining with intracavity interferometric combiners,” IEEE J. Sel. Top. Quantum Electron. 15, 301–311 (2009). [CrossRef]  

4. J. R. Leger, G. J. Swanson, and W. B. Veldkamp, “Coherent laser addition using binary phase gratings,” Appl. Opt. 26, 4391–4399 (1987). [CrossRef]   [PubMed]  

5. C. J. Corcoran and F. Durville, “Experimental demonstration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86, 201118 (2005).

6. L. Shimshi, A. A. Ishaaya, V. Ekhouse, N. Davidson, and A. A. Friesem, “Passive intra-cavity phase locking of laser channels,” Opt. Commun. 263, 60–64 (2006). [CrossRef]  

7. E. Ronen, M. Fridman, M. Nixon, A. A. Friesem, and N. Davidson, “Phase locking of lasers with intracavity polarization elements,” Opt. Lett. 33, 2305–2307 (2008). [CrossRef]   [PubMed]  

8. J. R. Leger, “Lateral mode control of an AlGaAs laser array in a Talbot cavity,” Appl. Phys. Lett. 55, 334–336 (1989). [CrossRef]  

9. 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]  

10. M. Wrage, P. Glas, D. Fischer, 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]  

11. J. E. Rothenberg, “Passive coherent phasing of fiber laser arrays,” Proc. SPIE 6873, 687315 (2008). [CrossRef]  

12. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

13. M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, “Passive phase locking of 25 fiber lasers,” Opt. Lett. 9, 1434–1436 (2010). [CrossRef]  

14. W. Chang, T. Wu, H. G. Winful, and A. Galvanauskas, “Array size scalability of passively coherently phased fiber laser arrays,” Opt. Express 18, 9634–9642 (2010), (all other references are cited here). [CrossRef]   [PubMed]  

15. E. du Tremolet de Lacheisserie, D. Gignoux, and M. Schlenker, Magnetism: Fundamentals (Springer, 2005), 2nd ed.

16. R. K. Pathria, Statistical Mechanics (Butterworth-Heinemann, Oxford, 1996), 2nd ed.

17. C. D. Nabros, “Effects of phase errors on coherent emitting arrays,” Appl. Opt. 33, 2284–2289 (1994). [CrossRef]  

References

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  1. S. J. Augst, T. Y. Fan, and A. Sanchez, "Coherent beam combining and phase noise measurements of ytterbium fiber amplifiers," Opt. Lett. 29, 474-476 (2004).
    [CrossRef] [PubMed]
  2. S. J. Augst, A. K. Goyal, R. L. Aggarwal, T. Y. Fan, and A. Sanchez, "Wavelength beam combining of ytterbium fiber lasers," Opt. Lett. 28, 331-333 (2003).
    [CrossRef] [PubMed]
  3. A. Ishaaya, N. Davidson, and A. Friesem, "Passive laser beam combining with intracavity interferometric combiners," IEEE J. Sel. Top. Quantum Electron. 15, 301-311 (2009).
    [CrossRef]
  4. J. R. Leger, G. J. Swanson, and W. B. Veldkamp, "Coherent laser addition using binary phase gratings," Appl. Opt. 26, 4391-4399 (1987).
    [CrossRef] [PubMed]
  5. C. J. Corcoran, and F. Durville, "Experimental demonstration of a phase-locked laser array using a self-Fourier cavity," Appl. Phys. Lett. 86, 201118 (2005).
  6. L. Shimshi, A. A. Ishaaya, V. Ekhouse, N. Davidson, and A. A. Friesem, "Passive intra-cavity phase locking of laser channels," Opt. Commun. 263, 60-64 (2006).
    [CrossRef]
  7. E. Ronen, M. Fridman, M. Nixon, A. A. Friesem, and N. Davidson, "Phase locking of lasers with intracavity polarization elements," Opt. Lett. 33, 2305-2307 (2008).
    [CrossRef] [PubMed]
  8. J. R. Leger, "Lateral mode control of an AlGaAs laser array in a Talbot cavity," Appl. Phys. Lett. 55, 334-336 (1989).
    [CrossRef]
  9. 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]
  10. M. Wrage, P. Glas, D. Fischer, 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]
  11. J. E. Rothenberg, "Passive coherent phasing of fiber laser arrays," Proc. SPIE 6873, 687315 (2008).
    [CrossRef]
  12. R. W. Gerchberg, and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttg.) 35, 237-246 (1972).
  13. M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, "Passive phase locking of 25 fiber lasers," Opt. Lett. 9, 1434-1436 (2010).
    [CrossRef]
  14. W. Chang, T. Wu, H. G. Winful, and A. Galvanauskas, "Array size scalability of passively coherently phased fiber laser arrays," Opt. Express 18, 9634-9642 (2010) (all other references are cited here.).
    [CrossRef] [PubMed]
  15. E. du Tremolet de Lacheisserie, D. Gignoux, and M. Schlenker, Magnetism: Fundamentals (Springer, 2005), 2nd ed.
  16. R. K. Pathria, Statistical Mechanics (Butterworth-Heinemann, Oxford, 1996), 2nd ed.
  17. C. D. Nabros, "Effects of phase errors on coherent emitting arrays," Appl. Opt. 33, 2284-2289 (1994).
    [CrossRef]

2010 (2)

M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, "Passive phase locking of 25 fiber lasers," Opt. Lett. 9, 1434-1436 (2010).
[CrossRef]

W. Chang, T. Wu, H. G. Winful, and A. Galvanauskas, "Array size scalability of passively coherently phased fiber laser arrays," Opt. Express 18, 9634-9642 (2010) (all other references are cited here.).
[CrossRef] [PubMed]

2009 (1)

A. Ishaaya, N. Davidson, and A. Friesem, "Passive laser beam combining with intracavity interferometric combiners," IEEE J. Sel. Top. Quantum Electron. 15, 301-311 (2009).
[CrossRef]

2008 (2)

E. Ronen, M. Fridman, M. Nixon, A. A. Friesem, and N. Davidson, "Phase locking of lasers with intracavity polarization elements," Opt. Lett. 33, 2305-2307 (2008).
[CrossRef] [PubMed]

J. E. Rothenberg, "Passive coherent phasing of fiber laser arrays," Proc. SPIE 6873, 687315 (2008).
[CrossRef]

2006 (1)

L. Shimshi, A. A. Ishaaya, V. Ekhouse, N. Davidson, and A. A. Friesem, "Passive intra-cavity phase locking of laser channels," Opt. Commun. 263, 60-64 (2006).
[CrossRef]

2005 (1)

C. J. Corcoran, and F. Durville, "Experimental demonstration of a phase-locked laser array using a self-Fourier cavity," Appl. Phys. Lett. 86, 201118 (2005).

2004 (1)

S. J. Augst, T. Y. Fan, and A. Sanchez, "Coherent beam combining and phase noise measurements of ytterbium fiber amplifiers," Opt. Lett. 29, 474-476 (2004).
[CrossRef] [PubMed]

2003 (1)

S. J. Augst, A. K. Goyal, R. L. Aggarwal, T. Y. Fan, and A. Sanchez, "Wavelength beam combining of ytterbium fiber lasers," Opt. Lett. 28, 331-333 (2003).
[CrossRef] [PubMed]

2000 (1)

M. Wrage, P. Glas, D. Fischer, 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]

1994 (1)

C. D. Nabros, "Effects of phase errors on coherent emitting arrays," Appl. Opt. 33, 2284-2289 (1994).
[CrossRef]

1991 (1)

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]

1989 (1)

J. R. Leger, "Lateral mode control of an AlGaAs laser array in a Talbot cavity," Appl. Phys. Lett. 55, 334-336 (1989).
[CrossRef]

1987 (1)

J. R. Leger, G. J. Swanson, and W. B. Veldkamp, "Coherent laser addition using binary phase gratings," Appl. Opt. 26, 4391-4399 (1987).
[CrossRef] [PubMed]

1972 (1)

R. W. Gerchberg, and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttg.) 35, 237-246 (1972).

Aggarwal, R. L.

S. J. Augst, A. K. Goyal, R. L. Aggarwal, T. Y. Fan, and A. Sanchez, "Wavelength beam combining of ytterbium fiber lasers," Opt. Lett. 28, 331-333 (2003).
[CrossRef] [PubMed]

Augst, S. J.

S. J. Augst, T. Y. Fan, and A. Sanchez, "Coherent beam combining and phase noise measurements of ytterbium fiber amplifiers," Opt. Lett. 29, 474-476 (2004).
[CrossRef] [PubMed]

S. J. Augst, A. K. Goyal, R. L. Aggarwal, T. Y. Fan, and A. Sanchez, "Wavelength beam combining of ytterbium fiber lasers," Opt. Lett. 28, 331-333 (2003).
[CrossRef] [PubMed]

Chang, W.

W. Chang, T. Wu, H. G. Winful, and A. Galvanauskas, "Array size scalability of passively coherently phased fiber laser arrays," Opt. Express 18, 9634-9642 (2010) (all other references are cited here.).
[CrossRef] [PubMed]

Corcoran, C. J.

C. J. Corcoran, and F. Durville, "Experimental demonstration of a phase-locked laser array using a self-Fourier cavity," Appl. Phys. Lett. 86, 201118 (2005).

Davidson, N.

M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, "Passive phase locking of 25 fiber lasers," Opt. Lett. 9, 1434-1436 (2010).
[CrossRef]

A. Ishaaya, N. Davidson, and A. Friesem, "Passive laser beam combining with intracavity interferometric combiners," IEEE J. Sel. Top. Quantum Electron. 15, 301-311 (2009).
[CrossRef]

E. Ronen, M. Fridman, M. Nixon, A. A. Friesem, and N. Davidson, "Phase locking of lasers with intracavity polarization elements," Opt. Lett. 33, 2305-2307 (2008).
[CrossRef] [PubMed]

L. Shimshi, A. A. Ishaaya, V. Ekhouse, N. Davidson, and A. A. Friesem, "Passive intra-cavity phase locking of laser channels," Opt. Commun. 263, 60-64 (2006).
[CrossRef]

Durville, F.

C. J. Corcoran, and F. Durville, "Experimental demonstration of a phase-locked laser array using a self-Fourier cavity," Appl. Phys. Lett. 86, 201118 (2005).

Ekhouse, V.

L. Shimshi, A. A. Ishaaya, V. Ekhouse, N. Davidson, and A. A. Friesem, "Passive intra-cavity phase locking of laser channels," Opt. Commun. 263, 60-64 (2006).
[CrossRef]

Fan, T. Y.

S. J. Augst, T. Y. Fan, and A. Sanchez, "Coherent beam combining and phase noise measurements of ytterbium fiber amplifiers," Opt. Lett. 29, 474-476 (2004).
[CrossRef] [PubMed]

S. J. Augst, A. K. Goyal, R. L. Aggarwal, T. Y. Fan, and A. Sanchez, "Wavelength beam combining of ytterbium fiber lasers," Opt. Lett. 28, 331-333 (2003).
[CrossRef] [PubMed]

Fischer, D.

M. Wrage, P. Glas, D. Fischer, 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]

Fridman, M.

M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, "Passive phase locking of 25 fiber lasers," Opt. Lett. 9, 1434-1436 (2010).
[CrossRef]

E. Ronen, M. Fridman, M. Nixon, A. A. Friesem, and N. Davidson, "Phase locking of lasers with intracavity polarization elements," Opt. Lett. 33, 2305-2307 (2008).
[CrossRef] [PubMed]

Friesem, A.

A. Ishaaya, N. Davidson, and A. Friesem, "Passive laser beam combining with intracavity interferometric combiners," IEEE J. Sel. Top. Quantum Electron. 15, 301-311 (2009).
[CrossRef]

Friesem, A. A.

M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, "Passive phase locking of 25 fiber lasers," Opt. Lett. 9, 1434-1436 (2010).
[CrossRef]

E. Ronen, M. Fridman, M. Nixon, A. A. Friesem, and N. Davidson, "Phase locking of lasers with intracavity polarization elements," Opt. Lett. 33, 2305-2307 (2008).
[CrossRef] [PubMed]

L. Shimshi, A. A. Ishaaya, V. Ekhouse, N. Davidson, and A. A. Friesem, "Passive intra-cavity phase locking of laser channels," Opt. Commun. 263, 60-64 (2006).
[CrossRef]

Galvanauskas, A.

W. Chang, T. Wu, H. G. Winful, and A. Galvanauskas, "Array size scalability of passively coherently phased fiber laser arrays," Opt. Express 18, 9634-9642 (2010) (all other references are cited here.).
[CrossRef] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg, and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttg.) 35, 237-246 (1972).

Glas, P.

M. Wrage, P. Glas, D. Fischer, 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]

Goyal, A. K.

S. J. Augst, A. K. Goyal, R. L. Aggarwal, T. Y. Fan, and A. Sanchez, "Wavelength beam combining of ytterbium fiber lasers," Opt. Lett. 28, 331-333 (2003).
[CrossRef] [PubMed]

Ishaaya, A.

A. Ishaaya, N. Davidson, and A. Friesem, "Passive laser beam combining with intracavity interferometric combiners," IEEE J. Sel. Top. Quantum Electron. 15, 301-311 (2009).
[CrossRef]

Ishaaya, A. A.

L. Shimshi, A. A. Ishaaya, V. Ekhouse, N. Davidson, and A. A. Friesem, "Passive intra-cavity phase locking of laser channels," Opt. Commun. 263, 60-64 (2006).
[CrossRef]

Leger, J. R.

J. R. Leger, "Lateral mode control of an AlGaAs laser array in a Talbot cavity," Appl. Phys. Lett. 55, 334-336 (1989).
[CrossRef]

J. R. Leger, G. J. Swanson, and W. B. Veldkamp, "Coherent laser addition using binary phase gratings," Appl. Opt. 26, 4391-4399 (1987).
[CrossRef] [PubMed]

Leitner, M.

M. Wrage, P. Glas, D. Fischer, 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]

Mehuys, D.

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]

Nabros, C. D.

C. D. Nabros, "Effects of phase errors on coherent emitting arrays," Appl. Opt. 33, 2284-2289 (1994).
[CrossRef]

Napartovich, A. P.

M. Wrage, P. Glas, D. Fischer, 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]

Nixon, M.

M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, "Passive phase locking of 25 fiber lasers," Opt. Lett. 9, 1434-1436 (2010).
[CrossRef]

E. Ronen, M. Fridman, M. Nixon, A. A. Friesem, and N. Davidson, "Phase locking of lasers with intracavity polarization elements," Opt. Lett. 33, 2305-2307 (2008).
[CrossRef] [PubMed]

Ronen, E.

E. Ronen, M. Fridman, M. Nixon, A. A. Friesem, and N. Davidson, "Phase locking of lasers with intracavity polarization elements," Opt. Lett. 33, 2305-2307 (2008).
[CrossRef] [PubMed]

Rothenberg, J. E.

J. E. Rothenberg, "Passive coherent phasing of fiber laser arrays," Proc. SPIE 6873, 687315 (2008).
[CrossRef]

Sanchez, A.

S. J. Augst, T. Y. Fan, and A. Sanchez, "Coherent beam combining and phase noise measurements of ytterbium fiber amplifiers," Opt. Lett. 29, 474-476 (2004).
[CrossRef] [PubMed]

S. J. Augst, A. K. Goyal, R. L. Aggarwal, T. Y. Fan, and A. Sanchez, "Wavelength beam combining of ytterbium fiber lasers," Opt. Lett. 28, 331-333 (2003).
[CrossRef] [PubMed]

Saxton, W. O.

R. W. Gerchberg, and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttg.) 35, 237-246 (1972).

Shimshi, L.

L. Shimshi, A. A. Ishaaya, V. Ekhouse, N. Davidson, and A. A. Friesem, "Passive intra-cavity phase locking of laser channels," Opt. Commun. 263, 60-64 (2006).
[CrossRef]

Streifer, W.

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]

Swanson, G. J.

J. R. Leger, G. J. Swanson, and W. B. Veldkamp, "Coherent laser addition using binary phase gratings," Appl. Opt. 26, 4391-4399 (1987).
[CrossRef] [PubMed]

Veldkamp, W. B.

J. R. Leger, G. J. Swanson, and W. B. Veldkamp, "Coherent laser addition using binary phase gratings," Appl. Opt. 26, 4391-4399 (1987).
[CrossRef] [PubMed]

Vysotsky, D. V.

M. Wrage, P. Glas, D. Fischer, 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]

Waarts, R. G.

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]

Welch, D. F.

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]

Winful, H. G.

W. Chang, T. Wu, H. G. Winful, and A. Galvanauskas, "Array size scalability of passively coherently phased fiber laser arrays," Opt. Express 18, 9634-9642 (2010) (all other references are cited here.).
[CrossRef] [PubMed]

Wrage, M.

M. Wrage, P. Glas, D. Fischer, 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]

Wu, T.

W. Chang, T. Wu, H. G. Winful, and A. Galvanauskas, "Array size scalability of passively coherently phased fiber laser arrays," Opt. Express 18, 9634-9642 (2010) (all other references are cited here.).
[CrossRef] [PubMed]

Appl. Opt. (2)

J. R. Leger, G. J. Swanson, and W. B. Veldkamp, "Coherent laser addition using binary phase gratings," Appl. Opt. 26, 4391-4399 (1987).
[CrossRef] [PubMed]

C. D. Nabros, "Effects of phase errors on coherent emitting arrays," Appl. Opt. 33, 2284-2289 (1994).
[CrossRef]

Appl. Phys. Lett. (2)

C. J. Corcoran, and F. Durville, "Experimental demonstration of a phase-locked laser array using a self-Fourier cavity," Appl. Phys. Lett. 86, 201118 (2005).

J. R. Leger, "Lateral mode control of an AlGaAs laser array in a Talbot cavity," Appl. Phys. Lett. 55, 334-336 (1989).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

A. Ishaaya, N. Davidson, and A. Friesem, "Passive laser beam combining with intracavity interferometric combiners," IEEE J. Sel. Top. Quantum Electron. 15, 301-311 (2009).
[CrossRef]

Opt. Commun. (1)

L. Shimshi, A. A. Ishaaya, V. Ekhouse, N. Davidson, and A. A. Friesem, "Passive intra-cavity phase locking of laser channels," Opt. Commun. 263, 60-64 (2006).
[CrossRef]

Opt. Express (1)

W. Chang, T. Wu, H. G. Winful, and A. Galvanauskas, "Array size scalability of passively coherently phased fiber laser arrays," Opt. Express 18, 9634-9642 (2010) (all other references are cited here.).
[CrossRef] [PubMed]

Opt. Lett. (6)

M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, "Passive phase locking of 25 fiber lasers," Opt. Lett. 9, 1434-1436 (2010).
[CrossRef]

S. J. Augst, T. Y. Fan, and A. Sanchez, "Coherent beam combining and phase noise measurements of ytterbium fiber amplifiers," Opt. Lett. 29, 474-476 (2004).
[CrossRef] [PubMed]

S. J. Augst, A. K. Goyal, R. L. Aggarwal, T. Y. Fan, and A. Sanchez, "Wavelength beam combining of ytterbium fiber lasers," Opt. Lett. 28, 331-333 (2003).
[CrossRef] [PubMed]

E. Ronen, M. Fridman, M. Nixon, A. A. Friesem, and N. Davidson, "Phase locking of lasers with intracavity polarization elements," Opt. Lett. 33, 2305-2307 (2008).
[CrossRef] [PubMed]

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]

M. Wrage, P. Glas, D. Fischer, 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]

Optik (Stuttg.) (1)

R. W. Gerchberg, and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttg.) 35, 237-246 (1972).

Proc. SPIE (1)

J. E. Rothenberg, "Passive coherent phasing of fiber laser arrays," Proc. SPIE 6873, 687315 (2008).
[CrossRef]

Other (2)

E. du Tremolet de Lacheisserie, D. Gignoux, and M. Schlenker, Magnetism: Fundamentals (Springer, 2005), 2nd ed.

R. K. Pathria, Statistical Mechanics (Butterworth-Heinemann, Oxford, 1996), 2nd ed.

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Figures (5)

Fig. 1
Fig. 1

Basic configuration for phase locking a fiber laser array via diffractive coupling.

Fig. 2
Fig. 2

(a) Measured near field intensity distribution of an array of seven fiber lasers and (b) their far field intensity distribution. (c) The cross section of the far field intensity distribution of the supermode. Blue curve: measured far field intensity, fringe contrast of 82.3%. Green curve: retrieved far field intensity by Gerschberg Saxton algorithm. Red curve: calculated intensity distribution of the pure in phase supermode. Black curve: calculated (by Gerschberg Saxton algorithm) far field intensity distribution for the anti phase supermode after a π phase mask that changes it into an in phase mode.

Fig. 3
Fig. 3

(a) The spectrum of the individual lasers in the array, their mean spectrum, and the spectrum of the phase-locked array; (b) far field intensity fringe contrast as a function of the number of lasers in the array.

Fig. 4
Fig. 4

Measured far field intensity fringe contrast of two lasers in the phase locked array as a function of the distance between the two lasers. As evident, although all lasers operate with the same central wavelength, the fringe contrast between two distant lasers is deteriorated. Blue dots: measured fringe contrast. Green line: mean. Red line: linear fit.

Fig. 5
Fig. 5

The Strehl ratio as a function of the size of the array for different correlation lengthes. eα, blue curve: eα=0.99; green curve eα=0.9; red curve eα=0.54 (similar to our experimental results); turquoise curve: eα=0.1.

Equations (3)

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S = < | Σ i E i | 2 > E 0 2 N 2 = N + 2 Σ i , j i < cos ( ϕ i ϕ j ) > N 2
f c i , j < | E i + E j | 2 > < | E i E j | 2 > < | E i + E j | 2 > + < | E i E j | 2 > = < cos ( ϕ i ϕ j ) >
S = N + 2 ( Σ i , j i e α ( i j ) ) N 2 = N + 2 e α ( e N α e α ( e α 1 ) 2 N 1 e α 1 ) N 2

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