Abstract

We apply the modal coherence theory to evaluate the spatial mode structure of a 2×2 phase-coupled array of vertical cavity surface emitting lasers (VCSELs). The eigenmode structure is extracted for different pump currents by measuring the degree of spatial coherence of all VCSEL pairs in the array. The results reveal the impact of optical disorder and spatial hole burning on the modal discrimination. The approach is useful more generally for the evaluation of spatial mode content of other laser array.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, Appl. Phys. Lett. 60, 1535 (1992).
    [CrossRef]
  2. H. Pier, E. Kapon, and M. Moser, Nature 407, 880 (2000).
    [CrossRef] [PubMed]
  3. A. C. Lehman, J. J. Raftery, A. J. Danner, P. O. Leisher, and K. D. Choquette, Appl. Phys. Lett. 88, 021102 (2006).
    [CrossRef]
  4. L. D. A. Lundeberg, G. P. Lousberg, D. L. Boiko, and E. Kapon, Appl. Phys. Lett. 90, 021103 (2007).
    [CrossRef]
  5. E. Kapon, J. Katz, and A. Yariv, Opt. Lett. 9, 125 (1984).
    [CrossRef] [PubMed]
  6. J. K. Butler, D. E. Ackley, and D. Botez, Appl. Phys. Lett. 44, 293 (1984).
    [CrossRef]
  7. P. Debernardi and G. P. Bava, IEEE J. Sel. Top. Quantum Electron. 9, 905 (2003).
    [CrossRef]
  8. A. Cutolo, T. Isernia, I. Izzo, R. Pierri, and L. Zeni, Appl. Opt. 34, 7974 (1995).
    [CrossRef] [PubMed]
  9. E. Tervonen, J. Turunen, and A. T. Friberg, Appl. Phys. B 49, 409 (1989).
    [CrossRef]
  10. C. M. Warnky, B. L. Anderson, and C. A. Klein, Appl. Opt. 39, 6109 (2000).
    [CrossRef]
  11. G. P. Lousberg, L. D. A. Lundeberg, D. L. Boiko, and E. Kapon, Opt. Lett. 31, 990 (2006).
    [CrossRef] [PubMed]
  12. E. Wolf and G. S. Agarwal, J. Opt. Soc. Am. A 1, 541 (1984).
    [CrossRef]
  13. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  14. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, J. Opt. Soc. Am. A 20, 78 (2003).
    [CrossRef]
  15. F. Gori and M. Santarsiero, Opt. Lett. 31, 858 (2006).
    [CrossRef] [PubMed]
  16. M. Santarsiero, F. Gori, R. Borghi, and G. Guattari, J. Opt. A 9, 593 (2007).
    [CrossRef]
  17. H. Pier and E. Kapon, Opt. Lett. 22, 546 (1997).
    [CrossRef] [PubMed]
  18. A. Golshani, H. Pier, M. Moser, and E. Kapon, J. Appl. Phys. 85, 2454 (1999).
    [CrossRef]

2007

L. D. A. Lundeberg, G. P. Lousberg, D. L. Boiko, and E. Kapon, Appl. Phys. Lett. 90, 021103 (2007).
[CrossRef]

M. Santarsiero, F. Gori, R. Borghi, and G. Guattari, J. Opt. A 9, 593 (2007).
[CrossRef]

2006

2003

2000

1999

A. Golshani, H. Pier, M. Moser, and E. Kapon, J. Appl. Phys. 85, 2454 (1999).
[CrossRef]

1997

1995

1992

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

1989

E. Tervonen, J. Turunen, and A. T. Friberg, Appl. Phys. B 49, 409 (1989).
[CrossRef]

1984

Ackley, D. E.

J. K. Butler, D. E. Ackley, and D. Botez, Appl. Phys. Lett. 44, 293 (1984).
[CrossRef]

Agarwal, G. S.

Anderson, B. L.

Bava, G. P.

P. Debernardi and G. P. Bava, IEEE J. Sel. Top. Quantum Electron. 9, 905 (2003).
[CrossRef]

Boiko, D. L.

L. D. A. Lundeberg, G. P. Lousberg, D. L. Boiko, and E. Kapon, Appl. Phys. Lett. 90, 021103 (2007).
[CrossRef]

G. P. Lousberg, L. D. A. Lundeberg, D. L. Boiko, and E. Kapon, Opt. Lett. 31, 990 (2006).
[CrossRef] [PubMed]

Borghi, R.

Botez, D.

J. K. Butler, D. E. Ackley, and D. Botez, Appl. Phys. Lett. 44, 293 (1984).
[CrossRef]

Butler, J. K.

J. K. Butler, D. E. Ackley, and D. Botez, Appl. Phys. Lett. 44, 293 (1984).
[CrossRef]

Choquette, K. D.

A. C. Lehman, J. J. Raftery, A. J. Danner, P. O. Leisher, and K. D. Choquette, Appl. Phys. Lett. 88, 021102 (2006).
[CrossRef]

Cutolo, A.

Danner, A. J.

A. C. Lehman, J. J. Raftery, A. J. Danner, P. O. Leisher, and K. D. Choquette, Appl. Phys. Lett. 88, 021102 (2006).
[CrossRef]

Debernardi, P.

P. Debernardi and G. P. Bava, IEEE J. Sel. Top. Quantum Electron. 9, 905 (2003).
[CrossRef]

Florez, L. T.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

Friberg, A. T.

E. Tervonen, J. Turunen, and A. T. Friberg, Appl. Phys. B 49, 409 (1989).
[CrossRef]

Golshani, A.

A. Golshani, H. Pier, M. Moser, and E. Kapon, J. Appl. Phys. 85, 2454 (1999).
[CrossRef]

Gori, F.

Guattari, G.

Harbison, J. P.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

Isernia, T.

Izzo, I.

Kapon, E.

L. D. A. Lundeberg, G. P. Lousberg, D. L. Boiko, and E. Kapon, Appl. Phys. Lett. 90, 021103 (2007).
[CrossRef]

G. P. Lousberg, L. D. A. Lundeberg, D. L. Boiko, and E. Kapon, Opt. Lett. 31, 990 (2006).
[CrossRef] [PubMed]

H. Pier, E. Kapon, and M. Moser, Nature 407, 880 (2000).
[CrossRef] [PubMed]

A. Golshani, H. Pier, M. Moser, and E. Kapon, J. Appl. Phys. 85, 2454 (1999).
[CrossRef]

H. Pier and E. Kapon, Opt. Lett. 22, 546 (1997).
[CrossRef] [PubMed]

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

E. Kapon, J. Katz, and A. Yariv, Opt. Lett. 9, 125 (1984).
[CrossRef] [PubMed]

Katz, J.

Klein, C. A.

Lehman, A. C.

A. C. Lehman, J. J. Raftery, A. J. Danner, P. O. Leisher, and K. D. Choquette, Appl. Phys. Lett. 88, 021102 (2006).
[CrossRef]

Leisher, P. O.

A. C. Lehman, J. J. Raftery, A. J. Danner, P. O. Leisher, and K. D. Choquette, Appl. Phys. Lett. 88, 021102 (2006).
[CrossRef]

Lousberg, G. P.

L. D. A. Lundeberg, G. P. Lousberg, D. L. Boiko, and E. Kapon, Appl. Phys. Lett. 90, 021103 (2007).
[CrossRef]

G. P. Lousberg, L. D. A. Lundeberg, D. L. Boiko, and E. Kapon, Opt. Lett. 31, 990 (2006).
[CrossRef] [PubMed]

Lundeberg, L. D. A.

L. D. A. Lundeberg, G. P. Lousberg, D. L. Boiko, and E. Kapon, Appl. Phys. Lett. 90, 021103 (2007).
[CrossRef]

G. P. Lousberg, L. D. A. Lundeberg, D. L. Boiko, and E. Kapon, Opt. Lett. 31, 990 (2006).
[CrossRef] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Moser, M.

H. Pier, E. Kapon, and M. Moser, Nature 407, 880 (2000).
[CrossRef] [PubMed]

A. Golshani, H. Pier, M. Moser, and E. Kapon, J. Appl. Phys. 85, 2454 (1999).
[CrossRef]

Orenstein, M.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

Pier, H.

H. Pier, E. Kapon, and M. Moser, Nature 407, 880 (2000).
[CrossRef] [PubMed]

A. Golshani, H. Pier, M. Moser, and E. Kapon, J. Appl. Phys. 85, 2454 (1999).
[CrossRef]

H. Pier and E. Kapon, Opt. Lett. 22, 546 (1997).
[CrossRef] [PubMed]

Pierri, R.

Piquero, G.

Raftery, J. J.

A. C. Lehman, J. J. Raftery, A. J. Danner, P. O. Leisher, and K. D. Choquette, Appl. Phys. Lett. 88, 021102 (2006).
[CrossRef]

Santarsiero, M.

Simon, R.

Stoffel, N. G.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

Tervonen, E.

E. Tervonen, J. Turunen, and A. T. Friberg, Appl. Phys. B 49, 409 (1989).
[CrossRef]

Turunen, J.

E. Tervonen, J. Turunen, and A. T. Friberg, Appl. Phys. B 49, 409 (1989).
[CrossRef]

Warnky, C. M.

Wolf, E.

E. Wolf and G. S. Agarwal, J. Opt. Soc. Am. A 1, 541 (1984).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Yariv, A.

Zeni, L.

Appl. Opt.

Appl. Phys. B

E. Tervonen, J. Turunen, and A. T. Friberg, Appl. Phys. B 49, 409 (1989).
[CrossRef]

Appl. Phys. Lett.

J. K. Butler, D. E. Ackley, and D. Botez, Appl. Phys. Lett. 44, 293 (1984).
[CrossRef]

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

A. C. Lehman, J. J. Raftery, A. J. Danner, P. O. Leisher, and K. D. Choquette, Appl. Phys. Lett. 88, 021102 (2006).
[CrossRef]

L. D. A. Lundeberg, G. P. Lousberg, D. L. Boiko, and E. Kapon, Appl. Phys. Lett. 90, 021103 (2007).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

P. Debernardi and G. P. Bava, IEEE J. Sel. Top. Quantum Electron. 9, 905 (2003).
[CrossRef]

J. Appl. Phys.

A. Golshani, H. Pier, M. Moser, and E. Kapon, J. Appl. Phys. 85, 2454 (1999).
[CrossRef]

J. Opt. A

M. Santarsiero, F. Gori, R. Borghi, and G. Guattari, J. Opt. A 9, 593 (2007).
[CrossRef]

J. Opt. Soc. Am. A

Nature

H. Pier, E. Kapon, and M. Moser, Nature 407, 880 (2000).
[CrossRef] [PubMed]

Opt. Lett.

Other

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Measurement of the degree of coherence for I = 46 mA . (a) Far-field pattern of a lasing 2 × 2 VCSEL array; (b) interference patterns measured between the lower left VCSEL and all other VCSELs; (c) far-field line scan of the two solitary VCSELs (dashed red and blue curves) and their corresponding interference pattern (solid black curve); and (d) corresponding degrees of coherence for the interference pattern represented in (b).

Fig. 2
Fig. 2

Histograms of the eigenvalues (modal power) α i , { i = 1 , 2 , 3 , 4 } including representation of each eigenmode amplitude distribution (wave function) ψ i for three different injection currents above threshold current I th = 40 mA . The corresponding measured far-field patterns are displayed on the right panel.

Fig. 3
Fig. 3

Calculated eigenmodes using coupled-mode theory for an ideal, uniform 2 × 2 VCSEL array with Gaussian modes supported by each individual VCSEL. (a) Normalized near-field amplitude, (b) corresponding normalized far-field intensity.

Fig. 4
Fig. 4

Modal power (eigenvalues α i ; i = 1 , 2 ,3, 4) versus the injection current. Above a current value of I = 46 mA , the eigenvalue α 4 (full blue curve) corresponding to the out-of-phase mode increases.

Equations (6)

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

Γ ( r 1 , r 2 , τ ) = 0 W ( r 1 , r 2 , v ) e i 2 π v τ d v .
W ( r 1 , r 2 , v ) = n λ n ( v ) ψ n * ( r 1 , v ) ψ n ( r 2 , v ) ,
D W ( r 1 , r 2 , v ) ψ n ( r 1 , v ) d 3 r 1 = λ n ( v ) ψ n ( r 2 , v ) .
V ( r , t ) = n α n ( v o ) ψ n ( r , v o ) e i 2 π v o t .
D Γ ( r 1 , r 2 , τ ) ψ n ( r 1 , v o ) d 3 r 1 = λ n ( v o ) ψ n ( r 2 , v o ) .
Γ 11 Γ 12 * Γ 13 * Γ 14 * Γ 21 Γ 22 Γ 23 * Γ 24 * Γ 31 Γ 32 Γ 33 Γ 34 * Γ 41 Γ 42 Γ 43 Γ 44 ψ n ( 1 ) ( v o ) ψ n ( 2 ) ( v o ) ψ n ( 3 ) ( v o ) ψ n ( 4 ) ( v o ) = α n ( v o ) ψ n ( 1 ) ( v o ) ψ n ( 2 ) ( v o ) ψ n ( 3 ) ( v o ) ψ n ( 4 ) ( v o ) ,

Metrics