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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 27 — Dec. 17, 2012
  • pp: 28163–28170
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Phase locking of lasers with self-stabilized minimal coupling

E. Ronen, A. A. Ishaaya, M. Nixon, A. Godel, A. A. Friesem, and N. Davidson  »View Author Affiliations


Optics Express, Vol. 20, Issue 27, pp. 28163-28170 (2012)
http://dx.doi.org/10.1364/OE.20.028163


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Abstract

A novel configuration for phase locking two ring lasers with self-stabilized minimal exchange of power between them is presented. We show experimentally that losses introduced between the lasers are self compensated in order to maintain minimal power exchange between them. The experimental results are in good agreement with numerical results.

© 2012 OSA

1. Introduction

Two coupled oscillators will phase lock only if the coupling strength between them exceeds a critical threshold which is proportional to the detuning between their resonance frequencies and to their noise level [1

1. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer Verlag, 1984). [CrossRef]

].Typically, in systems of coupled oscillators, and in particular in systems of coupled lasers, the coupling strength (i.e. fractional amount of light exchanged between the lasers) is an externally imposed parameter which remains essentially constant and does not depend on the ”internal” dynamics of the systems [2

2. B. Wang, E. Mies, M. Minden, and A. Sanchez, “all-fiber 50 w coherently combined passive laser array,” Opt. Lett. 34, 863–865 (2009). [CrossRef] [PubMed]

9

9. 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]

]. In order to maintain stable phase locking in the presence of fluctuating system parameters, the coupling strength between the oscillators must either be constantly stabilized externally or at least be set above a certain threshold value.

In this paper we introduce a novel system of two coupled ring laser oscillators where the coupling strength between them can be self-stabilized by the system dynamics, and is near the critical value required for phase locking. Experimentally, we demonstrate phase locking of two ring lasers with self-stabilized power exchange of only 13000 of their combined power. When loss is added in the optical path between the coupled lasers, there is a corresponding internal change that compensates for that loss and maintains a constant power exchange between them. We exploited a relatively simple model that uses nonlinear mode competition between internal degrees of freedom of each laser to explain our experimental results. It predicts that as the frequency detuning between any one pair of resonant frequencies of two multi-frequency ring lasers approaches zero, phase locking can occur with essentially no light exchanged.

2. Basic configurations and principle of operation

The basic configurations for a single and two coupled ring lasers are presented in Fig. 1. Figure 1(A) shows a single ring laser with external feedback that serves to explain the principle of self-stabilized coupling. The ring laser can support lasing modes in both the clockwise (CW) and the counter clockwise (CCW) directions, having the same frequencies and powers. These counter propagating modes interfere, resulting in spatial hole burning and many lasing frequencies, even for a homogenously broadened gain medium [10

10. A. E. Siegman, Lasers (University Science Books, 1986).

, 11

11. F. R. Faxvog, “Modes of a unidirectional ring laser,” Opt. Lett. 5, 285–287 (1980). [CrossRef] [PubMed]

]. The spatial hole burning can be eliminated, in order to obtain single frequency operation, by introducing an auxiliary mirror (AM) that reflects the light from the CCW mode back into the cavity where it is added to the CW mode. Thus, the CW ”acceptor” mode has effectively lower losses than the CCW ”donor” mode, and due to nonlinear mode competition, is the only one to survive [10

10. A. E. Siegman, Lasers (University Science Books, 1986).

, 11

11. F. R. Faxvog, “Modes of a unidirectional ring laser,” Opt. Lett. 5, 285–287 (1980). [CrossRef] [PubMed]

].

Fig. 1 Basic configurations for coupled ring lasers. A: Single laser self-coupled by an auxiliary external mirror. B: Two coupled ring lasers. Blue solid line [green dashed line] indicates the lasing direction of the donor modes [acceptor modes]. Inset: experimental interference fringe pattern of light from the two acceptor modes with 80% fringe visibility indicating high level of phase locking. AM- auxiliary mirror; M- mirror; GM- gain medium; OC- output coupler;

Figure 1(B) shows two ring lasers whose outputs face each other so that the CCW donor mode of laser 1 is added the CCW acceptor mode of laser 2. Conversely, the CW donor mode of laser 2 is added to the CW acceptor mode of laser 1. When the resonant frequencies of the two lasers are identical, Fig. 1(B) can be viewed as an ”unfolded” version of Fig. 1(A). We thus expect the amplitude of the light from the two donor modes to reduce to zero, again because of mode competition with the acceptor modes, so the two ring lasers do not exchange any light. Yet, assuming no noise, there is a constant phase relation between the lasers because of symmetry. We note that in Fig. 1(A) the donor mode is suppressed because it strengthens its own “acceptor” while in Fig. 1(B) each donor mode strengthens the acceptor mode of the other laser.

For a finite frequency detuning between the two ring lasers, the enhancement of the two acceptor modes by the donor modes (and at their expense) will occur only as long as they are phase locked to them. When the initial conditions of the two lasers are identical, the process for suppressing the donor modes should terminate when both donor modes reach the minimal power required for phase locking the acceptor mode, i.e. when the coupling strength between the lasers reaches a critical value. However, as the deviation from identical initial conditions increases, one of the donor modes will fully vanish before the other, and consequently the remaining donor mode will cease to decrease. We will demonstrate this intriguing phenomenon experimentally and numerically.

3. Experimental arrangement and results

Our experimental arrangement for coupling two ring lasers is presented in Fig. 2. It included a ring laser in a rectangular configuration with three flat high reflection mirrors, a 50% reflecting output coupler and two consecutive intra-cavity telescopes for obtaining a self-imaging (degenerate) cavity [12

12. J. A. Arnaud, “Degenerate optical cavities,” Appl. Opt. 8, 189–195 (1969). [CrossRef] [PubMed]

, 13

13. M. Nixon, M. Fridman, E. Ronen, I. Kanter, A.A Friesem, and N. Davidson, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106, 223–227 (2011). [CrossRef]

], where any field distribution is exactly repeated after a round trip propagation. The gain medium was 10 mm diameter Nd:Yag rod with 1.1% doping so as to obtain a wavelength of 1.06μm, pumped for 100μs at a frequency of 2 Hz to suppress thermal effects. Two apertures of 0.4 mm diameters and separated by 0.65 mm defined two independent ring lasers each operating at a single Gaussian transverse mode (but with many longitudinal modes). The separation between the two apertures and their perfect imaging to the gain medium was sufficient to ensure no overlap of the laser beams in the gain medium. We verified that without the auxiliary mirror there are no interference fringes between the two lasers indicating that they are not phase locked. Finally, an auxiliary flat mirror with reflectivity of 94% (located at the focal plane of lens F1) reflected the CCW donor mode of each ring laser into the CW acceptor mode of the other laser.

Fig. 2 Experimental arrangement for coupling two ring lasers, experimentally implementing the configuration of Fig. 1(B). Ap: two apertures defining laser 1 and 2; F1: lens, 30 cm focal length; F2: lens, 15 cm focal length; M: mirror; OC: 50% output coupler; GM: gain medium; At: calibrated optical attenuators; AM: auxiliary mirror, 94% reflectivity; solid (blue) line: propagation direction of the CCW donor mode; dashed (green) line: propagation direction of the CW acceptor mode. Insets: Detected intensity distributions of different modes; A: one donor mode of zero intensity, and the other of extremely low intensity (surrounded by circles). B: the two acceptor modes have nearly equal high intensities (surrounded by circles).

We began our experiments with one of the intra-cavity apertures blocked, implementing the single ring laser configuration of Fig. 1(A). When operating close to the lasing threshold we observed essentially full suppression of the donor mode ( IaId>4*106), which is 4 to 5 orders of magnitude higher than previously reported [10

10. A. E. Siegman, Lasers (University Science Books, 1986).

, 14

14. J. Hohimer, G. Vawter, and D. Craft, “Unidirectional lasing in a ring semiconductor diode laser,” Appl. Phy. Lett. 62, 1185–1187 (1993). [CrossRef]

]. Such a high suppression is probably due to the near-perfect overlap of the counter propagating modes in our degenerate cavity. Note, that without the auxiliary mirror the gain and loss of the two counter-propagating modes are identical due to symmetry considerations. We also verified that the coherence length of the laser increased significantly, due to suppression of spatial hole burning.

Next, with the two intra cavity apertures unblocked, implementing the two coupled ring laser configuration of Fig. 1(B), we measured the power of each of the four lasing modes (the two donor modes and two acceptor modes). Working close to threshold we found that the powers of the two acceptor lasers were nearly equal with Ia1Ia21.2, whereas the power of one donor laser was almost completely suppressed ( Ia2Id2>4*106) and that of the other donor laser was significantly but not completely suppressed ( Ia1Id1>1400) (see insets in Fig. 2). Accordingly, the relative power of exchanged light between the two ring lasers Id1+Id2Ia1+Ia2 was about 13000. Yet, when we interfered the light from the two acceptor modes we detected interference fringes with 80% contrast (see inset in Fig. 1(B)), indicating high level of phase coherence. We observed that the level of suppression of the donor modes and the fringe contrast of the interference between the two acceptor modes, slightly decreased when the pumping power was increased. Note that there is always some finite coupling between the surviving donor mode and the acceptor mode in the same laser cavity (zero detuning) and this coupling has a constant phase which results in the phase locking of the two acceptors modes.

To qualitatively demonstrate how the coupling self-stabilizes, we inserted calibrated optical attenuators before the auxiliary mirror (see Fig. 2) in order to introduce controlled loss between the two lasers. We measured the power of the surviving donor mode as a function of the round trip external loss. The results, presented in Fig. 3, reveal that the power of the surviving donor laser increases as the attenuation (loss) between the lasers increases (blue dots). As a result, the power of the light reflected back into the laser (after passing twice through the attenuator) has relatively small dependence on the attenuation (red squares). For example, with 80% round trip attenuation, the power of the light emerging from the surviving donor mode increased by 8.4, so the power of the light injected back, increased only by 1.68 [15

15. The attenuators are not optically flat so in addition to the absortion loss they add phase abberations to the donor mode which further reduces its overlap with the acceptor mode causing an additional loss that is not included in Fig. 3.

]. Intriguingly, there is self-stabilization of the surviving donor mode so as to compensate for the attenuation and maintain a relatively constant power of exchanged light. Also shown, are the corresponding calculated results from our theoretical model (described below) which follow the trend of the experimental data.

Fig. 3 Power of the surviving donor mode as a function of the round trip external loss introduced by the optical attenuators. Blue dots: measured power emerging from the cavity. Red squares: measured power injected back into the cavity (after passing twice through the optical attenuators). Blue solid line and red dashed line: corresponding calculated results (when the lasers have nearly identical initial conditions).

4. Theoretical model

We exploited a relatively simple theoretical model to explain our experimental results.We started by considering two coupled single frequency ring lasers with frequencies ω1 and ω2, and identical pumping rate p, round trip loss γ, cavity time tc and fluorescence time tf (without rotation, the CCW and CW modes of each ring laser have exactly the same frequency). Accordingly, the rate equation for the fields of the two donor modes Ed1/2, the two acceptor modes Ea1/2, and the two gains g1/2 are [16

16. L. Fabiny, P. Colet, R. Roy, and D. Lenstra, “Coherence and phase dynamics of spatially coupled solid-state lasers,” Phys. Rev. A 47, 4287–4296 (1993). [CrossRef] [PubMed]

, 17

17. M. Fridman, V. Eckhouse, N. Davidson, and A. A. Friesem, “effect of quantum noise on coupled laser oscillators,” Phys. Rev. A 77061803 (2008). [CrossRef]

]:
E˙d1/2=1tc((g1/2γ)Ed1/2)+iω1/2Ed1/2,
(1)
E˙a1/2=1tc((g1/2γ)Ea1/2+κEd2/1)+iω1/2Ea1/2,
(2)
and
g˙1/2=p1tf(g1/2(Id1/2+Ia1/2+1),
(3)
where κ is the geometrical coupling strength, defined as the ratio (in field amplitudes) of the self feedback light of the acceptor mode to the injected light from the donor mode, when both modes have the identical amplitudes. Note that the effective coupling between the lasers is κAdAp and depends on the internal system dynamics, whereas κ itself is externally defined. The phase of κ depends on the optical path to the auxiliary mirror and back. Nevertheless, because of the unidirectionality of the system the phase difference will only change the constant phase difference between the master and slaved modes [11

11. F. R. Faxvog, “Modes of a unidirectional ring laser,” Opt. Lett. 5, 285–287 (1980). [CrossRef] [PubMed]

]. The coupling delay time in our system was 0.5 ns, much shorter than any other time scale in our system, so instabilities that arise from delayed feedback need not to be considered [18

18. I. Kanter, Y. Aviad, I. Igor, E. Cohen, and M. Rosenblu, “An optical ultrafast random bit generator,” Nat. Phys. 4, 58–61 (2010). [CrossRef]

]. Since the donor modes are extremely weak in the steady state, the nonlinear coupling terms between the counter propagating modes inside each laser, caused by spatial hole burning [19

19. S. Schwartz, G. Feugnet, P. Bouyer, E. Lariontsev, A. Aspect, and J.-P. Pocholle, “Mode-coupling control in resonant devices: application to solid-state ring lasers,” Phys. Rev. Lett. 97, 093902 (2006). [CrossRef] [PubMed]

, 20

20. S. Schwartz, G. Feugnet, E. Lariontsev, and J.-P. Pocholle, “Oscillation regimes of a solid-state ring laser with active beat-note stabilization: From a chaotic device to a ring-laser gyroscope,” Phys. Rev. A 76, 023807 (2007). [CrossRef]

], are not included.

Denoting the fields by E = Ae, we calculated the field amplitudes and phases of the four lasing modes, Aa/d1/2 and ϕa/d1/2, by numerical integration of Eqs. (1)(3). The results are presented in Fig. 4. Figure 4(A) shows the calculated field amplitudes as a function of ttc for two ring lasers with Δωtc = 0.015, κ = 0 15 and near identical initial conditions for the two lasers (1% difference between donor modes). As evident, initially (0 < t < 4000tc) both donor modes lase and each one increases the acceptor mode of the other laser, and hence decreases the amplitude of the other donor mode. Then (at t ≃ 4000tc), Ad2 decreases below some critical value and cannot slave Aa1, and consequently cannot increase it. However, Ad1 is still strong enough to slave Aa2 and increase its amplitude, and hence Ad2 continues to decrease to zero at t ≃ 8000tc (as in the experimental results, where one donor mode vanishes). Only then, the system reaches a steady state. Figure 4(B) shows the phase difference between the two ring lasers. Here again, a constant value is reached, indicating stable phase locking around t ≃ 8000tc.

Fig. 4 Calculated field amplitudes and phase difference of relevant donor and acceptor modes as a function of time for a system of two ring lasers.A: Amplitudes of donor and acceptor modes.B: The phase difference between the surviving donor mode and its slaved acceptor mode. The detuning is 0.015tc, the coupling κ is 0.15, τc = 10ns, τf = 230μs, γ = 1 and p = 2.

We repeated the numerical simulations for many different system parameters. For example, with nearly identical initial conditions for the two lasers, we found that after a period of oscillations a steady state solution with the following features is reached:
  1. The two acceptor modes survive with nearly identical amplitude Aa1Aa2 and with a constant phase difference (i.e. phase locking) between them.
  2. Only one donor mode vanishes, say Ad20.
  3. The steady state amplitude of the other surviving donor mode, Ad1, reaches very close (within few percents) to Ad1ηAa2, with η=Δωτcκ. This amplitude is exactly the critical amplitude required to phase lock the other acceptor mode. Specifically, the real effective coupling κeffκAdAa is equal to the critical coupling κc = Δω · τc.

When the initial conditions are not the same, our simulations indicate that one donor mode vanishes much earlier than the other, so the steady state amplitude of the surviving donor mode is somewhat above the critical amplitude. Averaging over a wide range of initial conditions, where we allowed an order of magnitude variation in the initial amplitudes ratio of the two donor modes and 20% difference in their pump power p, we calculated the average ratio of the surviving donor mode power to that of either acceptor modes for η = 0.01. We found that the calculated average ratio is 11300, comparable to that obtained experimentally 11400. The calculated maximum ratio is 110000 when the initial condition for both lasers are nearly identical.

To determine the relation between the steady state amplitude of the surviving donor mode and the steady state phase difference between slaved acceptor mode and the the donor mode, we solved the rate equations analytically after assuming that one of the donor modes completely vanishes. We found that the steady state amplitude of the surviving donor mode is [21

21. It should be noted that the usual solution for detuned coupled oscillators, where both oscillate with the mean frequency and the same intensity [16], does not yield a steady state solution for our coupled ring lasers. Experimentally there is always some finite breaking of symmetry between the two donor lasers that causes one of them to vansih.

]:
Ad1=ηAa2sin(Δϕss).
(4)
where Δϕss=ϕa2ϕd1 is the steady state phase difference between the slaved acceptor mode and the surviving donor mode. Equation (4) provides a family of solutions, where for each steady state phase difference there is a corresponding surviving donor mode amplitude. This correspondence was supported in our experiments where we detected stable high fringe contrast interference between the two acceptor modes (inset in Fig. 1), indicating steady state phase differences, when the power of the donor mode was low and stable.

When the two lasers have nearly identical initial conditions, the surviving donor mode has minimal amplitude Ad1=ηAa2, and the phase difference is π2 as seen in Fig. 4 Recall that Ad1 is inversely proportional to the geometrical coupling κ, so κAd1 and consequently the effective coupling κeff=κAdAa remains essentially constant, as equivalently shown by the dashed red line in Fig. 3.

Finally, we extended our numerical calculations to include two ring lasers, each of which can support two different frequencies, with one common to both lasers. Specifically, ω0 and ω1 are the frequencies of ring laser a, and ω0 and ω2 are the frequencies of ring laser b. The calculated amplitudes for the four acceptor modes as a function of time are presented in Fig. 5. These results indicate that, as a consequence of mode competition, only the two acceptor modes with the common frequency ω0 survive, while the two acceptor modes with different frequencies ω1 and ω2 vanish. The amplitudes of the four donor modes (not shown in the figure) all vanish. These results can be extended to coupled ring lasers each having many different oscillation frequencies, whereby only the modes with identical frequencies survive and phase lock with near vanishing power of exchanged light (as in our experiment).

Fig. 5 Calculated amplitudes of the four acceptor modes as function of time for two ring lasers each with two different frequencies. A0a and A0b: amplitudes of lasers a and b with a common frequency; A1a and A2b: amplitudes of lasers a and b each with different frequency. Only the two acceptor modes with common frequency survive. The amplitudes of all four donor modes reach zero (not shown).

5. Concluding remarks

To conclude, we presented a configuration for phase locking two ring lasers with self-stabilized minimal power exchanged between them. We also showed both theoretically and experimentally that two multi frequency ring lasers with even one common frequency can phase lock with negligible power exchanged between them. We showed that losses introduced between the lasers are self compensated by increased power in the surviving donor mode in order to maintain a constant and critical power exchange between the lasers.

We believe that the concept of self-stabilized critical coupling in coupled lasers can be extended to a broad range of coupled oscillators. Such an extension requires internal degrees of freedom within each oscillator (analogous to the CW and CCW modes in ring lasers) with nonlinear mode competition that distributes the energy between them so as to affect the coupling strength.

Acknowledgments

The authors wish to acknowledge the ISF Bikura foundation and BSF foundation for their support in this research.

References and links

1.

Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer Verlag, 1984). [CrossRef]

2.

B. Wang, E. Mies, M. Minden, and A. Sanchez, “all-fiber 50 w coherently combined passive laser array,” Opt. Lett. 34, 863–865 (2009). [CrossRef] [PubMed]

3.

F. Jeux, A. Desfarges-Berthelemot, V. Kermne, J. Guillot, and A. Barthelemy, “Passive coherent combining of lasers with phase-contrast filtering for enhanced efficiency,” App. Phys. B , 108, 1–7 (2012).

4.

M. Khajavikhan and J. Leger, “Modal analysis of path length sensitivity in superposition architectures for coherent laser beam combining,” IEEE Sel. Top. Quantum Electron. 15, 281–290 (2009). [CrossRef]

5.

F. X. D’Amato, E. T. Siebert, and C. Roychoudhuri, “Coherent operation of an array of diode lasers using a spatial filter in a Talbot cavity,” App. Phys. Lett. 55, 816–818 (1989). [CrossRef]

6.

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

7.

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).

8.

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]

9.

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]

10.

A. E. Siegman, Lasers (University Science Books, 1986).

11.

F. R. Faxvog, “Modes of a unidirectional ring laser,” Opt. Lett. 5, 285–287 (1980). [CrossRef] [PubMed]

12.

J. A. Arnaud, “Degenerate optical cavities,” Appl. Opt. 8, 189–195 (1969). [CrossRef] [PubMed]

13.

M. Nixon, M. Fridman, E. Ronen, I. Kanter, A.A Friesem, and N. Davidson, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106, 223–227 (2011). [CrossRef]

14.

J. Hohimer, G. Vawter, and D. Craft, “Unidirectional lasing in a ring semiconductor diode laser,” Appl. Phy. Lett. 62, 1185–1187 (1993). [CrossRef]

15.

The attenuators are not optically flat so in addition to the absortion loss they add phase abberations to the donor mode which further reduces its overlap with the acceptor mode causing an additional loss that is not included in Fig. 3.

16.

L. Fabiny, P. Colet, R. Roy, and D. Lenstra, “Coherence and phase dynamics of spatially coupled solid-state lasers,” Phys. Rev. A 47, 4287–4296 (1993). [CrossRef] [PubMed]

17.

M. Fridman, V. Eckhouse, N. Davidson, and A. A. Friesem, “effect of quantum noise on coupled laser oscillators,” Phys. Rev. A 77061803 (2008). [CrossRef]

18.

I. Kanter, Y. Aviad, I. Igor, E. Cohen, and M. Rosenblu, “An optical ultrafast random bit generator,” Nat. Phys. 4, 58–61 (2010). [CrossRef]

19.

S. Schwartz, G. Feugnet, P. Bouyer, E. Lariontsev, A. Aspect, and J.-P. Pocholle, “Mode-coupling control in resonant devices: application to solid-state ring lasers,” Phys. Rev. Lett. 97, 093902 (2006). [CrossRef] [PubMed]

20.

S. Schwartz, G. Feugnet, E. Lariontsev, and J.-P. Pocholle, “Oscillation regimes of a solid-state ring laser with active beat-note stabilization: From a chaotic device to a ring-laser gyroscope,” Phys. Rev. A 76, 023807 (2007). [CrossRef]

21.

It should be noted that the usual solution for detuned coupled oscillators, where both oscillate with the mean frequency and the same intensity [16], does not yield a steady state solution for our coupled ring lasers. Experimentally there is always some finite breaking of symmetry between the two donor lasers that causes one of them to vansih.

OCIS Codes
(140.3560) Lasers and laser optics : Lasers, ring
(140.3298) Lasers and laser optics : Laser beam combining

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: June 26, 2012
Revised Manuscript: August 23, 2012
Manuscript Accepted: August 25, 2012
Published: December 5, 2012

Citation
E. Ronen, A. A. Ishaaya, M. Nixon, A. Godel, A. A. Friesem, and N. Davidson, "Phase locking of lasers with self-stabilized minimal coupling," Opt. Express 20, 28163-28170 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-27-28163


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References

  1. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer Verlag, 1984). [CrossRef]
  2. B. Wang, E. Mies, M. Minden, and A. Sanchez, “all-fiber 50 w coherently combined passive laser array,” Opt. Lett.34, 863–865 (2009). [CrossRef] [PubMed]
  3. F. Jeux, A. Desfarges-Berthelemot, V. Kermne, J. Guillot, and A. Barthelemy, “Passive coherent combining of lasers with phase-contrast filtering for enhanced efficiency,” App. Phys. B, 108, 1–7 (2012).
  4. M. Khajavikhan and J. Leger, “Modal analysis of path length sensitivity in superposition architectures for coherent laser beam combining,” IEEE Sel. Top. Quantum Electron.15, 281–290 (2009). [CrossRef]
  5. F. X. D’Amato, E. T. Siebert, and C. Roychoudhuri, “Coherent operation of an array of diode lasers using a spatial filter in a Talbot cavity,” App. Phys. Lett.55, 816–818 (1989). [CrossRef]
  6. A. Ishaaya, N. Davidson, and A. Friesem, “Passive laser beam combining with intracavity interferometric combiners,” Sel. Top. Quantum Electron.15, 301–311 (2009). [CrossRef]
  7. 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).
  8. 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]
  9. 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]
  10. A. E. Siegman, Lasers (University Science Books, 1986).
  11. F. R. Faxvog, “Modes of a unidirectional ring laser,” Opt. Lett.5, 285–287 (1980). [CrossRef] [PubMed]
  12. J. A. Arnaud, “Degenerate optical cavities,” Appl. Opt.8, 189–195 (1969). [CrossRef] [PubMed]
  13. M. Nixon, M. Fridman, E. Ronen, I. Kanter, A.A Friesem, and N. Davidson, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett.106, 223–227 (2011). [CrossRef]
  14. J. Hohimer, G. Vawter, and D. Craft, “Unidirectional lasing in a ring semiconductor diode laser,” Appl. Phy. Lett.62, 1185–1187 (1993). [CrossRef]
  15. The attenuators are not optically flat so in addition to the absortion loss they add phase abberations to the donor mode which further reduces its overlap with the acceptor mode causing an additional loss that is not included in Fig. 3.
  16. L. Fabiny, P. Colet, R. Roy, and D. Lenstra, “Coherence and phase dynamics of spatially coupled solid-state lasers,” Phys. Rev. A47, 4287–4296 (1993). [CrossRef] [PubMed]
  17. M. Fridman, V. Eckhouse, N. Davidson, and A. A. Friesem, “effect of quantum noise on coupled laser oscillators,” Phys. Rev. A77061803 (2008). [CrossRef]
  18. I. Kanter, Y. Aviad, I. Igor, E. Cohen, and M. Rosenblu, “An optical ultrafast random bit generator,” Nat. Phys.4, 58–61 (2010). [CrossRef]
  19. S. Schwartz, G. Feugnet, P. Bouyer, E. Lariontsev, A. Aspect, and J.-P. Pocholle, “Mode-coupling control in resonant devices: application to solid-state ring lasers,” Phys. Rev. Lett.97, 093902 (2006). [CrossRef] [PubMed]
  20. S. Schwartz, G. Feugnet, E. Lariontsev, and J.-P. Pocholle, “Oscillation regimes of a solid-state ring laser with active beat-note stabilization: From a chaotic device to a ring-laser gyroscope,” Phys. Rev. A76, 023807 (2007). [CrossRef]
  21. It should be noted that the usual solution for detuned coupled oscillators, where both oscillate with the mean frequency and the same intensity [16], does not yield a steady state solution for our coupled ring lasers. Experimentally there is always some finite breaking of symmetry between the two donor lasers that causes one of them to vansih.

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