Abstract

Two coupled microlasers display instabilities in the intermediate regime between locked and independent states of oscillations. The frequencies and amplitudes of these oscillations exhibit strongly asymmetric behaviors for positive and negative detunings. Our experimental findings cannot be accounted for by coupled-equation models with purely real or imaginary coupling coefficients but are well described if complex coupling coefficients are introduced. We describe a method to determine experimentally the coupling coefficient and study its dependence as a function of the separation between lasers.

© 2000 Optical Society of America

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References

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  1. H. G. Winful and L. Rahman, “Synchronized chaos and spatiotemporal chaos in arrays of coupled lasers,” Phys. Rev. Lett. 52, 1575–1578 (1990); S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52, 1774–1776 (1988); H. G. Winful and S. S. Wang, “Stability of phase locking in coupled semiconductor laser arrays,” Appl. Phys. Lett. APPLAB 53, 1894–1896 (1988).
    [CrossRef]
  2. M. B. Spencer and W. E. Lamb, Jr., “Theory of two coupled lasers,” Phys. Rev. A 5, 893–898 (1972).
    [CrossRef]
  3. T. Sugawara, M. Tachikawa, T. Tsukamoto, and T. Shimizu, “Observation of synchronization in laser chaos,” Phys. Rev. Lett. 72, 3502–3505 (1994); Y. Liu, P. C. de Oliveira, M. B. Danailov, and J. R. Rios Leite, “Chaotic and periodic passive Q switching in coupled CO2 lasers with a saturable absorber,” Phys. Rev. A 50, 3464–3470 (1994); J. R. Terry, K. S. Thornburg, Jr., D. J. DeShazer, G. D. VanWiggeren, S. Zhu, P. Ashwin, and R. Roy, “Synchronization of chaos in an array of three lasers,” Phys. Rev. E PLEEE8 59, 4036–4043 (1999).
    [CrossRef] [PubMed]
  4. K. S. Thornburg, Jr., M. Möller, R. Roy, T. W. Carr, R.-D. Li, and T. Erneux, “Chaos and coherence in coupled lasers,” Phys. Rev. E 55, 3865–3869 (1997).
    [CrossRef]
  5. 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]
  6. P. Mandel, L. Ruo-Ding, and T. Erneux, “Pulsating self-coupled lasers,” Phys. Rev. A 39, 2502–2508 (1989); L. Ruo-Ding, P. Mandel, and T. Erneux, “Periodic and quasiperiodic regimes in self-coupled lasers,” Phys. Rev. A 41, 5117–5126 (1990); T. Erneux, T. W. Carr, and R.-D. Li, “Coupled lasers asymptotics,” Proc. SPIE PSISDG 2039, 80–90 (1993).
    [CrossRef] [PubMed]
  7. K. Otsuka, “Self-induced phase turbulence and chaotic itinerancy in coupled laser systems,” Phys. Rev. Lett. 65, 329–332 (1990); J. Xu, S. Li, K. K. Lee, and Y. C. Chen, “Phase locking in a two element laser array: a test of the coupled-oscillator model,” Opt. Lett. 18, 513–515 (1993); F. Prati, D. Vecchione, and G. Vendramin, “Frequency locking of supermodes and stability of out-of-phase-locked state in one-dimensional and two-dimensional arrays of vertical-cavity surface-emitting lasers,” Opt. Lett. OPLEDP 22, 1633–1635 (1997).
    [CrossRef] [PubMed]
  8. J. Katz, E. Kapon, C. Lindsey, S. Margalit, and A. Yariv, “Coupling coefficient of gain-guided lasers,” Appl. Opt. 23, 2231–2235 (1984); J. K. Butler, D. E. Ackley, and M. Ettenberg, “Coupled-mode analysis of gain and wavelength oscillation characteristics of diode laser phased arrays,” IEEE J. Quantum Electron. QE-21, 458–463 (1985); E. Kapon, C. Lindsey, J. Katz, S. Margalit, and A. Yariv, “Coupling mechanism of gain-guided integrated semiconductor laser arrays,” Appl. Phys. Lett. APPLAB 44, 389–391 (1984).
    [CrossRef] [PubMed]
  9. P. Ru, P. K. Jakobsen, J. V. Moloney, and R. A. Indik, “Generalized coupled-mode model for the multistripe index-guided laser arrays,” J. Opt. Soc. Am. B 10, 507–515 (1993).
    [CrossRef]
  10. J. Xu, K. K. Lee, and Y. C. Chen, “Phase locking in a two-element laser array with detuning,” Opt. Commun. 117, 198–206 (1995).
    [CrossRef]
  11. R. Kuske and T. Erneux, “Localized synchronization of two coupled solid state lasers,” Opt. Commun. 139, 125–131 (1997).
    [CrossRef]
  12. H. Laabs and B. Ozygus, “The influence of transverse structures on the coupling of solid-state lasers,” Opt. Laser Technol. 29, 401–406 (1997).
    [CrossRef]
  13. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1988), p. 628.

1997 (3)

K. S. Thornburg, Jr., M. Möller, R. Roy, T. W. Carr, R.-D. Li, and T. Erneux, “Chaos and coherence in coupled lasers,” Phys. Rev. E 55, 3865–3869 (1997).
[CrossRef]

R. Kuske and T. Erneux, “Localized synchronization of two coupled solid state lasers,” Opt. Commun. 139, 125–131 (1997).
[CrossRef]

H. Laabs and B. Ozygus, “The influence of transverse structures on the coupling of solid-state lasers,” Opt. Laser Technol. 29, 401–406 (1997).
[CrossRef]

1995 (1)

J. Xu, K. K. Lee, and Y. C. Chen, “Phase locking in a two-element laser array with detuning,” Opt. Commun. 117, 198–206 (1995).
[CrossRef]

1993 (2)

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]

P. Ru, P. K. Jakobsen, J. V. Moloney, and R. A. Indik, “Generalized coupled-mode model for the multistripe index-guided laser arrays,” J. Opt. Soc. Am. B 10, 507–515 (1993).
[CrossRef]

1972 (1)

M. B. Spencer and W. E. Lamb, Jr., “Theory of two coupled lasers,” Phys. Rev. A 5, 893–898 (1972).
[CrossRef]

Carr, T. W.

K. S. Thornburg, Jr., M. Möller, R. Roy, T. W. Carr, R.-D. Li, and T. Erneux, “Chaos and coherence in coupled lasers,” Phys. Rev. E 55, 3865–3869 (1997).
[CrossRef]

Chen, Y. C.

J. Xu, K. K. Lee, and Y. C. Chen, “Phase locking in a two-element laser array with detuning,” Opt. Commun. 117, 198–206 (1995).
[CrossRef]

Colet, P.

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]

Erneux, T.

K. S. Thornburg, Jr., M. Möller, R. Roy, T. W. Carr, R.-D. Li, and T. Erneux, “Chaos and coherence in coupled lasers,” Phys. Rev. E 55, 3865–3869 (1997).
[CrossRef]

R. Kuske and T. Erneux, “Localized synchronization of two coupled solid state lasers,” Opt. Commun. 139, 125–131 (1997).
[CrossRef]

Fabiny, L.

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]

Indik, R. A.

Jakobsen, P. K.

Kuske, R.

R. Kuske and T. Erneux, “Localized synchronization of two coupled solid state lasers,” Opt. Commun. 139, 125–131 (1997).
[CrossRef]

Laabs, H.

H. Laabs and B. Ozygus, “The influence of transverse structures on the coupling of solid-state lasers,” Opt. Laser Technol. 29, 401–406 (1997).
[CrossRef]

Lamb Jr., W. E.

M. B. Spencer and W. E. Lamb, Jr., “Theory of two coupled lasers,” Phys. Rev. A 5, 893–898 (1972).
[CrossRef]

Lee, K. K.

J. Xu, K. K. Lee, and Y. C. Chen, “Phase locking in a two-element laser array with detuning,” Opt. Commun. 117, 198–206 (1995).
[CrossRef]

Lenstra, D.

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]

Li, R.-D.

K. S. Thornburg, Jr., M. Möller, R. Roy, T. W. Carr, R.-D. Li, and T. Erneux, “Chaos and coherence in coupled lasers,” Phys. Rev. E 55, 3865–3869 (1997).
[CrossRef]

Möller, M.

K. S. Thornburg, Jr., M. Möller, R. Roy, T. W. Carr, R.-D. Li, and T. Erneux, “Chaos and coherence in coupled lasers,” Phys. Rev. E 55, 3865–3869 (1997).
[CrossRef]

Moloney, J. V.

Ozygus, B.

H. Laabs and B. Ozygus, “The influence of transverse structures on the coupling of solid-state lasers,” Opt. Laser Technol. 29, 401–406 (1997).
[CrossRef]

Roy, R.

K. S. Thornburg, Jr., M. Möller, R. Roy, T. W. Carr, R.-D. Li, and T. Erneux, “Chaos and coherence in coupled lasers,” Phys. Rev. E 55, 3865–3869 (1997).
[CrossRef]

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]

Ru, P.

Spencer, M. B.

M. B. Spencer and W. E. Lamb, Jr., “Theory of two coupled lasers,” Phys. Rev. A 5, 893–898 (1972).
[CrossRef]

Thornburg Jr., K. S.

K. S. Thornburg, Jr., M. Möller, R. Roy, T. W. Carr, R.-D. Li, and T. Erneux, “Chaos and coherence in coupled lasers,” Phys. Rev. E 55, 3865–3869 (1997).
[CrossRef]

Xu, J.

J. Xu, K. K. Lee, and Y. C. Chen, “Phase locking in a two-element laser array with detuning,” Opt. Commun. 117, 198–206 (1995).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

J. Xu, K. K. Lee, and Y. C. Chen, “Phase locking in a two-element laser array with detuning,” Opt. Commun. 117, 198–206 (1995).
[CrossRef]

R. Kuske and T. Erneux, “Localized synchronization of two coupled solid state lasers,” Opt. Commun. 139, 125–131 (1997).
[CrossRef]

Opt. Laser Technol. (1)

H. Laabs and B. Ozygus, “The influence of transverse structures on the coupling of solid-state lasers,” Opt. Laser Technol. 29, 401–406 (1997).
[CrossRef]

Phys. Rev. A (2)

M. B. Spencer and W. E. Lamb, Jr., “Theory of two coupled lasers,” Phys. Rev. A 5, 893–898 (1972).
[CrossRef]

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]

Phys. Rev. E (1)

K. S. Thornburg, Jr., M. Möller, R. Roy, T. W. Carr, R.-D. Li, and T. Erneux, “Chaos and coherence in coupled lasers,” Phys. Rev. E 55, 3865–3869 (1997).
[CrossRef]

Other (6)

H. G. Winful and L. Rahman, “Synchronized chaos and spatiotemporal chaos in arrays of coupled lasers,” Phys. Rev. Lett. 52, 1575–1578 (1990); S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52, 1774–1776 (1988); H. G. Winful and S. S. Wang, “Stability of phase locking in coupled semiconductor laser arrays,” Appl. Phys. Lett. APPLAB 53, 1894–1896 (1988).
[CrossRef]

T. Sugawara, M. Tachikawa, T. Tsukamoto, and T. Shimizu, “Observation of synchronization in laser chaos,” Phys. Rev. Lett. 72, 3502–3505 (1994); Y. Liu, P. C. de Oliveira, M. B. Danailov, and J. R. Rios Leite, “Chaotic and periodic passive Q switching in coupled CO2 lasers with a saturable absorber,” Phys. Rev. A 50, 3464–3470 (1994); J. R. Terry, K. S. Thornburg, Jr., D. J. DeShazer, G. D. VanWiggeren, S. Zhu, P. Ashwin, and R. Roy, “Synchronization of chaos in an array of three lasers,” Phys. Rev. E PLEEE8 59, 4036–4043 (1999).
[CrossRef] [PubMed]

P. Mandel, L. Ruo-Ding, and T. Erneux, “Pulsating self-coupled lasers,” Phys. Rev. A 39, 2502–2508 (1989); L. Ruo-Ding, P. Mandel, and T. Erneux, “Periodic and quasiperiodic regimes in self-coupled lasers,” Phys. Rev. A 41, 5117–5126 (1990); T. Erneux, T. W. Carr, and R.-D. Li, “Coupled lasers asymptotics,” Proc. SPIE PSISDG 2039, 80–90 (1993).
[CrossRef] [PubMed]

K. Otsuka, “Self-induced phase turbulence and chaotic itinerancy in coupled laser systems,” Phys. Rev. Lett. 65, 329–332 (1990); J. Xu, S. Li, K. K. Lee, and Y. C. Chen, “Phase locking in a two element laser array: a test of the coupled-oscillator model,” Opt. Lett. 18, 513–515 (1993); F. Prati, D. Vecchione, and G. Vendramin, “Frequency locking of supermodes and stability of out-of-phase-locked state in one-dimensional and two-dimensional arrays of vertical-cavity surface-emitting lasers,” Opt. Lett. OPLEDP 22, 1633–1635 (1997).
[CrossRef] [PubMed]

J. Katz, E. Kapon, C. Lindsey, S. Margalit, and A. Yariv, “Coupling coefficient of gain-guided lasers,” Appl. Opt. 23, 2231–2235 (1984); J. K. Butler, D. E. Ackley, and M. Ettenberg, “Coupled-mode analysis of gain and wavelength oscillation characteristics of diode laser phased arrays,” IEEE J. Quantum Electron. QE-21, 458–463 (1985); E. Kapon, C. Lindsey, J. Katz, S. Margalit, and A. Yariv, “Coupling mechanism of gain-guided integrated semiconductor laser arrays,” Appl. Phys. Lett. APPLAB 44, 389–391 (1984).
[CrossRef] [PubMed]

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1988), p. 628.

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Figures (7)

Fig. 1
Fig. 1

Experimental setup. The lasers are separately imaged on two detectors; Ii refers to the time series of the laser i intensity.

Fig. 2
Fig. 2

Experimental observations on the different locking regimes of two coupled microlasers. (a) The relative peak-to-peak amplitudes and (b) the fundamental components F1 and F2 of the fast Fourier transform spectrum of the intensities are plotted versus the frequency detuning of the two lasers. Symbols refer to laser 1 (●), to laser 2 (▲), and to behaviors common to both lasers (■). The insert gives an example of the fast Fourier transform spectrum obtained for a frequency detuning equal to 18.5 MHz (FB). The distance between lasers is d=194 µm.

Fig. 3
Fig. 3

Far-field patterns for different detunings: Figures (a)–(c) are obtained for a laser separation d=150 µm; the lasers are phase locked, and the cavity detunings are estimated to (a) -60 MHz, (b) 0 MHz, and (c) 50 MHz. The arrow indicates the position of the central fringe for a detuning equal to zero. Figures (d)–(f) are obtained for d=194 µm: (d) synchronized-pulsed regime [arrow 1 of Fig. 2(a)]; (e)–(f) in the almost-independent regime [arrows 2 and 3 of Fig. 2(a)]. Reverse contrast.

Fig. 4
Fig. 4

(a) Relative amplitude and (b) phase of the modulation at the beat frequency of the two lasers for large detuning versus beat frequency. Symbols refer to laser 1 (●) and to laser 2 (▲), and the solid curve refers to the best fit to the hyperbolic law given by Eq. (4.3).

Fig. 5
Fig. 5

Evolution of (a) the modulus and (b) the argument of the coupling coefficient versus laser separation d.

Fig. 6
Fig. 6

(a) Real part of the coupling coefficient versus the laser separation d (♦) and its fit (solid curve) obtained with a pump waist wp=50 µm and a laser waist w0=54.5 µm; (b) imaginary part versus d.

Fig. 7
Fig. 7

Numerical predictions of the dynamical behavior of the two coupled microlasers’ (a) amplitudes and (b) resonance frequencies with a complex coupling coefficient. Symbols refer to laser 1 (●), to laser 2 (▲), and to behaviors common to both lasers (■). The coupling coefficient (κr=-0.0013, κi=-0.0029) is measured in the conditions of Fig. 2.

Equations (13)

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

E¯it=(Di-1-iδi)E¯i+κE¯j,
Dit=γ[Ai-(1+|E¯i|2)Di],
Eit=(Di-1)Ei+KEj cos[φ+(-1)jθ],
φt=Δ-KE2E1sin(φ+θ)+E1E2sin(φ-θ),
Δ+C2+D2 sin(φS+Φ)=0,
C=-κrE1SE2S+E2SE1S,D=-κiE1SE2S-E2SE1S,
Φ=arctanDC.
|Δ|C2+D2.
φS=-arcsinΔC2+D2-Φ.
Ei=Ei0+ei+O(2),
Di=Di0+di+O(2),
Ii=Ei2=Ii01+2KνRjΔνRisin[Δt+(-1)jθ]+O(2),
κr=(2η(2+α)/(1+α)-η-η(5+α)/(1+α)),

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