Abstract

We explain a technique that recovers the structure and the modal weights of spatial modes of lasers from a limited number of spatial coherence measurements. Our approach interpolates the unobserved spatial coherence measurements via the low-rank matrix completion algorithm based on nuclear norm minimization and then extracts the set of modes via singular value decomposition. Numerical examples are provided on a variety of lasers to demonstrate the effectiveness of the method, and it is shown that the proposed method can further reduce the number of measurements by a factor of 2 for a moderate data size.

© 2013 Optical Society of America

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References

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  1. R.-J. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, IEEE Photon. J. 5, 0701307 (2013).
    [CrossRef]
  2. F. Gori, M. Satarsiero, R. Borghi, and G. Guattari, Opt. Lett. 23, 989 (1998).
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    [CrossRef]
  5. R. Borghi and M. Santasiero, Opt. Lett. 23, 313 (1998).
    [CrossRef]
  6. P. Spano, Opt. Commun. 33, 265 (1980).
    [CrossRef]
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    [CrossRef]
  8. L. J. Pelz and B. L. Anderson, Opt. Eng. 34, 3323 (1995).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. C. Iaconis and I. A. Walmsley, Opt. Lett. 21, 1783 (1996).
    [CrossRef]
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    [CrossRef]
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  15. L. J. Pelz and B. L. Anderson, Proceedings of IEEE Lasers and Electro-Optics Society 6th Annual Meeting (IEEE, 1993), pp. 163–164.
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    [CrossRef]
  17. E. J. Candès and B. Recht, Found. Comput. Math. 9, 717 (2009).
  18. E. J. Candès and T. Tao, IEEE Trans. Inf. Theory 56, 2053 (2010).
    [CrossRef]
  19. M. Grant, S. Boyd, and Y. Ye, “CVX: Matlab software for disciplined convex programming,” 2008, http://cvxr.com/cvx/ .
  20. L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, Opt. Express 20, 8296 (2012).

2013 (1)

R.-J. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, IEEE Photon. J. 5, 0701307 (2013).
[CrossRef]

2012 (1)

2010 (1)

E. J. Candès and T. Tao, IEEE Trans. Inf. Theory 56, 2053 (2010).
[CrossRef]

2009 (1)

E. J. Candès and B. Recht, Found. Comput. Math. 9, 717 (2009).

2007 (1)

Y. Meja and A. I. Gonzlez, Opt. Commun. 273, 428 (2007).
[CrossRef]

2003 (1)

2000 (1)

1998 (3)

1996 (1)

1995 (2)

1993 (1)

B. L. Anderson and P. L. Fuhr, Opt. Eng. 32, 926 (1993).
[CrossRef]

1989 (1)

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

1980 (2)

P. Spano, Opt. Commun. 33, 265 (1980).
[CrossRef]

P. Spano, J.-M. Dumant, and Y. Guillausseau, Ann. Télécommun. 35, 231 (1980).

Anderson, B. L.

C. M. Warnky, B. L. Anderson, and C. A. Klein, Appl. Opt. 39, 6109 (2000).
[CrossRef]

L. J. Pelz and B. L. Anderson, Opt. Eng. 34, 3323 (1995).
[CrossRef]

B. L. Anderson and P. L. Fuhr, Opt. Eng. 32, 926 (1993).
[CrossRef]

L. J. Pelz and B. L. Anderson, Proceedings of IEEE Lasers and Electro-Optics Society 6th Annual Meeting (IEEE, 1993), pp. 163–164.

Barbastathis, G.

Borghi, R.

Candès, E. J.

E. J. Candès and T. Tao, IEEE Trans. Inf. Theory 56, 2053 (2010).
[CrossRef]

E. J. Candès and B. Recht, Found. Comput. Math. 9, 717 (2009).

Cutolo, A.

de la Torre, L.

Dumant, J.-M.

P. Spano, J.-M. Dumant, and Y. Guillausseau, Ann. Télécommun. 35, 231 (1980).

Essiambre, R.-J.

R.-J. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, IEEE Photon. J. 5, 0701307 (2013).
[CrossRef]

Fontaine, N. K.

R.-J. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, IEEE Photon. J. 5, 0701307 (2013).
[CrossRef]

Friberg, A. T.

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

Fuhr, P. L.

B. L. Anderson and P. L. Fuhr, Opt. Eng. 32, 926 (1993).
[CrossRef]

Gonzlez, A. I.

Y. Meja and A. I. Gonzlez, Opt. Commun. 273, 428 (2007).
[CrossRef]

Gori, F.

Guattari, G.

Guillausseau, Y.

P. Spano, J.-M. Dumant, and Y. Guillausseau, Ann. Télécommun. 35, 231 (1980).

Iaconis, C.

Isernia, T.

Izzo, I.

Klein, C. A.

Konforti, N.

Lee, J.

Lohmann, A. W.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), p. 215.

Meja, Y.

Y. Meja and A. I. Gonzlez, Opt. Commun. 273, 428 (2007).
[CrossRef]

Mendlovic, D.

Oh, S. B.

Pelz, L. J.

L. J. Pelz and B. L. Anderson, Opt. Eng. 34, 3323 (1995).
[CrossRef]

L. J. Pelz and B. L. Anderson, Proceedings of IEEE Lasers and Electro-Optics Society 6th Annual Meeting (IEEE, 1993), pp. 163–164.

Pierri, R.

Randel, S.

R.-J. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, IEEE Photon. J. 5, 0701307 (2013).
[CrossRef]

Recht, B.

E. J. Candès and B. Recht, Found. Comput. Math. 9, 717 (2009).

Ryf, R.

R.-J. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, IEEE Photon. J. 5, 0701307 (2013).
[CrossRef]

Santasiero, M.

Satarsiero, M.

Shabtay, G.

Spano, P.

P. Spano, Opt. Commun. 33, 265 (1980).
[CrossRef]

P. Spano, J.-M. Dumant, and Y. Guillausseau, Ann. Télécommun. 35, 231 (1980).

Tao, T.

E. J. Candès and T. Tao, IEEE Trans. Inf. Theory 56, 2053 (2010).
[CrossRef]

Tervonen, E.

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

Tian, L.

Turunen, J.

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

Walmsley, I. A.

Warnky, C. M.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), p. 215.

Zeni, L.

Ann. Télécommun. (1)

P. Spano, J.-M. Dumant, and Y. Guillausseau, Ann. Télécommun. 35, 231 (1980).

Appl. Opt. (2)

Appl. Phys. B (1)

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

Found. Comput. Math. (1)

E. J. Candès and B. Recht, Found. Comput. Math. 9, 717 (2009).

IEEE Photon. J. (1)

R.-J. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, IEEE Photon. J. 5, 0701307 (2013).
[CrossRef]

IEEE Trans. Inf. Theory (1)

E. J. Candès and T. Tao, IEEE Trans. Inf. Theory 56, 2053 (2010).
[CrossRef]

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

Opt. Commun. (2)

Y. Meja and A. I. Gonzlez, Opt. Commun. 273, 428 (2007).
[CrossRef]

P. Spano, Opt. Commun. 33, 265 (1980).
[CrossRef]

Opt. Eng. (2)

B. L. Anderson and P. L. Fuhr, Opt. Eng. 32, 926 (1993).
[CrossRef]

L. J. Pelz and B. L. Anderson, Opt. Eng. 34, 3323 (1995).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Other (3)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), p. 215.

L. J. Pelz and B. L. Anderson, Proceedings of IEEE Lasers and Electro-Optics Society 6th Annual Meeting (IEEE, 1993), pp. 163–164.

M. Grant, S. Boyd, and Y. Ye, “CVX: Matlab software for disciplined convex programming,” 2008, http://cvxr.com/cvx/ .

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Figures (4)

Fig. 1.
Fig. 1.

Twin-fiber apparatus for measuring spatial coherence.

Fig. 2.
Fig. 2.

Reconstruction of spatial modes for four lasers. The result for each laser is in a separate column. First row: the measured mutual intensity matrix with missing main diagonals. Second row: reconstructed mutual intensity matrix. Third row: reconstructed spatial modes via MC.

Fig. 3.
Fig. 3.

LD20: (a) recovered modes by the MC method and the Jacobian method and (b) recovered beam from the MC method and the measured intensity profile.

Fig. 4.
Fig. 4.

Normalized mean-squared error of the reconstructed matrix using a subset of the measured data and that using all measured data.

Equations (8)

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

J12=mλmψm*(x1)ψm(x2),
J12r=mλmϕm(x1)ϕm(x2),
|J12|=(ImaxImin)/4|K1||K2|,
R=ΦΛΦT.
minVrank(V)s.t.PΩ(R)=PΩ(V),
minVV*s.t.PΩ(R)=PΩ(V),
minVTr(V)s.t.PΩ(R)=PΩ(V),V0,
minVTr(V)s.t.PΩ(RV)Fϵ,V0,

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