Abstract

We present experimental investigations on spatially resolved Stokes parameters of vertical-cavity surface-emitting lasers (VCSELs) with a small aperture diameter of 3 μm and a monolithically integrated surface grating on top of the structure to technologically control the polarization. As expected, the grating fixes the state of polarization, but still shows both a spatially nonuniform linear polarization distribution of the fundamental transverse mode as well as an interesting eight-lobe pattern of circular polarization in terms of change of sign. These experimental findings are reproduced by numerical simulations using a fully vectorial three-dimensional model.

© 2013 Optical Society of America

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  1. W. E. J. Neal and R. W. Fane, J. Phys. E 6, 409 (1973).
    [CrossRef]
  2. S. Inoué, J. Cell Biol. 89, 346 (1981).
    [CrossRef]
  3. T. Nomura, B. Javidi, S. Murata, E. Nitanai, and T. Numata, Opt. Lett. 32, 481 (2007).
    [CrossRef]
  4. R. Dom, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef]
  5. T. A. Nieminen, N. R. Hechenburg, and H. Rubinsztein-Dunlop, Opt. Lett. 33, 122 (2008).
    [CrossRef]
  6. P. Debernardi, J. M. Ostermann, M. Feneberg, C. Jalics, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 11, 107 (2005).
    [CrossRef]
  7. M. S. Torre, A. Valle, and L. Pesquera, Opt. Quantum Electron. 38, 445 (2006).
    [CrossRef]
  8. J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
    [CrossRef]
  9. M. B. Willemsen, M. P. van Exter, and J. P. Woerdman, Phys. Rev. Lett. 84, 4337 (2000).
    [CrossRef]
  10. M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
    [CrossRef]
  11. R. Michalzik, VCSELs—Fundamentals, Technology and Applications of VCSELs (Springer, 2013).
  12. L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsässer, Phys. Rev. A 64, 031803(R) (2001).
    [CrossRef]
  13. A. Molitor, S. Hartmann, and W. Elsässer, Opt. Lett. 37, 4799 (2012).
    [CrossRef]
  14. P. Debernardi, IEEE J. Quantum Electron. 45, 979 (2009).
    [CrossRef]
  15. P. Debernardi, G. P. Bava, F. Monti di Sopra, and M. Willemsen, IEEE J. Quantum Electron. 39, 109 (2003).
    [CrossRef]
  16. H. G. Berry, G. Gabrielse, and A. E. Livingston, Appl. Opt. 16, 3200 (1977).
    [CrossRef]
  17. B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, Am. J. Phys. 75, 163 (2007).
    [CrossRef]
  18. P. Debernardi, R. Orta, T. Gründl, and M. C. Amann, IEEE J. Quantum Electron. 49, 137 (2013).
    [CrossRef]
  19. P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
    [CrossRef]

2013 (1)

P. Debernardi, R. Orta, T. Gründl, and M. C. Amann, IEEE J. Quantum Electron. 49, 137 (2013).
[CrossRef]

2012 (1)

2009 (1)

P. Debernardi, IEEE J. Quantum Electron. 45, 979 (2009).
[CrossRef]

2008 (1)

2007 (3)

T. Nomura, B. Javidi, S. Murata, E. Nitanai, and T. Numata, Opt. Lett. 32, 481 (2007).
[CrossRef]

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, Am. J. Phys. 75, 163 (2007).
[CrossRef]

P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
[CrossRef]

2006 (1)

M. S. Torre, A. Valle, and L. Pesquera, Opt. Quantum Electron. 38, 445 (2006).
[CrossRef]

2005 (1)

P. Debernardi, J. M. Ostermann, M. Feneberg, C. Jalics, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 11, 107 (2005).
[CrossRef]

2003 (2)

R. Dom, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

P. Debernardi, G. P. Bava, F. Monti di Sopra, and M. Willemsen, IEEE J. Quantum Electron. 39, 109 (2003).
[CrossRef]

2001 (1)

L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsässer, Phys. Rev. A 64, 031803(R) (2001).
[CrossRef]

2000 (1)

M. B. Willemsen, M. P. van Exter, and J. P. Woerdman, Phys. Rev. Lett. 84, 4337 (2000).
[CrossRef]

1997 (1)

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

1995 (1)

M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
[CrossRef]

1981 (1)

S. Inoué, J. Cell Biol. 89, 346 (1981).
[CrossRef]

1977 (1)

1973 (1)

W. E. J. Neal and R. W. Fane, J. Phys. E 6, 409 (1973).
[CrossRef]

Abraham, N. B.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

Ackemann, T.

P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
[CrossRef]

Amann, M. C.

P. Debernardi, R. Orta, T. Gründl, and M. C. Amann, IEEE J. Quantum Electron. 49, 137 (2013).
[CrossRef]

Barrett, D.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, Am. J. Phys. 75, 163 (2007).
[CrossRef]

Bava, G. P.

P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
[CrossRef]

P. Debernardi, G. P. Bava, F. Monti di Sopra, and M. Willemsen, IEEE J. Quantum Electron. 39, 109 (2003).
[CrossRef]

L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsässer, Phys. Rev. A 64, 031803(R) (2001).
[CrossRef]

Berry, H. G.

Collett, E.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, Am. J. Phys. 75, 163 (2007).
[CrossRef]

Debernardi, P.

P. Debernardi, R. Orta, T. Gründl, and M. C. Amann, IEEE J. Quantum Electron. 49, 137 (2013).
[CrossRef]

P. Debernardi, IEEE J. Quantum Electron. 45, 979 (2009).
[CrossRef]

P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
[CrossRef]

P. Debernardi, J. M. Ostermann, M. Feneberg, C. Jalics, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 11, 107 (2005).
[CrossRef]

P. Debernardi, G. P. Bava, F. Monti di Sopra, and M. Willemsen, IEEE J. Quantum Electron. 39, 109 (2003).
[CrossRef]

L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsässer, Phys. Rev. A 64, 031803(R) (2001).
[CrossRef]

Degen, C.

L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsässer, Phys. Rev. A 64, 031803(R) (2001).
[CrossRef]

Dom, R.

R. Dom, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Elsässer, W.

A. Molitor, S. Hartmann, and W. Elsässer, Opt. Lett. 37, 4799 (2012).
[CrossRef]

L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsässer, Phys. Rev. A 64, 031803(R) (2001).
[CrossRef]

Fane, R. W.

W. E. J. Neal and R. W. Fane, J. Phys. E 6, 409 (1973).
[CrossRef]

Feneberg, M.

P. Debernardi, J. M. Ostermann, M. Feneberg, C. Jalics, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 11, 107 (2005).
[CrossRef]

Feng, Q.

M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
[CrossRef]

Fischer, I.

L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsässer, Phys. Rev. A 64, 031803(R) (2001).
[CrossRef]

Fraher, B.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, Am. J. Phys. 75, 163 (2007).
[CrossRef]

Fratta, L.

L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsässer, Phys. Rev. A 64, 031803(R) (2001).
[CrossRef]

Gabrielse, G.

Gründl, T.

P. Debernardi, R. Orta, T. Gründl, and M. C. Amann, IEEE J. Quantum Electron. 49, 137 (2013).
[CrossRef]

Hartmann, S.

Hechenburg, N. R.

Inoué, S.

S. Inoué, J. Cell Biol. 89, 346 (1981).
[CrossRef]

Jalics, C.

P. Debernardi, J. M. Ostermann, M. Feneberg, C. Jalics, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 11, 107 (2005).
[CrossRef]

Javidi, B.

Kaiser, J.

L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsässer, Phys. Rev. A 64, 031803(R) (2001).
[CrossRef]

Leuchs, G.

R. Dom, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Livingston, A. E.

Martin-Regalado, J.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

Michalzik, R.

P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
[CrossRef]

P. Debernardi, J. M. Ostermann, M. Feneberg, C. Jalics, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 11, 107 (2005).
[CrossRef]

R. Michalzik, VCSELs—Fundamentals, Technology and Applications of VCSELs (Springer, 2013).

Molitor, A.

Moloney, J. V.

M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
[CrossRef]

Monti di Sopra, F.

P. Debernardi, G. P. Bava, F. Monti di Sopra, and M. Willemsen, IEEE J. Quantum Electron. 39, 109 (2003).
[CrossRef]

Murata, S.

Neal, W. E. J.

W. E. J. Neal and R. W. Fane, J. Phys. E 6, 409 (1973).
[CrossRef]

Nieminen, T. A.

Nitanai, E.

Nomura, T.

Numata, T.

Orta, R.

P. Debernardi, R. Orta, T. Gründl, and M. C. Amann, IEEE J. Quantum Electron. 49, 137 (2013).
[CrossRef]

Ostermann, J. M.

P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
[CrossRef]

P. Debernardi, J. M. Ostermann, M. Feneberg, C. Jalics, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 11, 107 (2005).
[CrossRef]

Pesquera, L.

M. S. Torre, A. Valle, and L. Pesquera, Opt. Quantum Electron. 38, 445 (2006).
[CrossRef]

Prati, F.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

Quabis, S.

R. Dom, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Rubinsztein-Dunlop, H.

San Miguel, M.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
[CrossRef]

Schaefer, B.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, Am. J. Phys. 75, 163 (2007).
[CrossRef]

Smyth, R.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, Am. J. Phys. 75, 163 (2007).
[CrossRef]

Sondermann, M.

P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
[CrossRef]

Torre, M. S.

M. S. Torre, A. Valle, and L. Pesquera, Opt. Quantum Electron. 38, 445 (2006).
[CrossRef]

Valle, A.

M. S. Torre, A. Valle, and L. Pesquera, Opt. Quantum Electron. 38, 445 (2006).
[CrossRef]

van Exter, M. P.

M. B. Willemsen, M. P. van Exter, and J. P. Woerdman, Phys. Rev. Lett. 84, 4337 (2000).
[CrossRef]

Willemsen, M.

P. Debernardi, G. P. Bava, F. Monti di Sopra, and M. Willemsen, IEEE J. Quantum Electron. 39, 109 (2003).
[CrossRef]

Willemsen, M. B.

M. B. Willemsen, M. P. van Exter, and J. P. Woerdman, Phys. Rev. Lett. 84, 4337 (2000).
[CrossRef]

Woerdman, J. P.

M. B. Willemsen, M. P. van Exter, and J. P. Woerdman, Phys. Rev. Lett. 84, 4337 (2000).
[CrossRef]

Am. J. Phys. (1)

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, Am. J. Phys. 75, 163 (2007).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (4)

P. Debernardi, R. Orta, T. Gründl, and M. C. Amann, IEEE J. Quantum Electron. 49, 137 (2013).
[CrossRef]

P. Debernardi, IEEE J. Quantum Electron. 45, 979 (2009).
[CrossRef]

P. Debernardi, G. P. Bava, F. Monti di Sopra, and M. Willemsen, IEEE J. Quantum Electron. 39, 109 (2003).
[CrossRef]

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

P. Debernardi, J. M. Ostermann, M. Feneberg, C. Jalics, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 11, 107 (2005).
[CrossRef]

P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
[CrossRef]

J. Cell Biol. (1)

S. Inoué, J. Cell Biol. 89, 346 (1981).
[CrossRef]

J. Phys. E (1)

W. E. J. Neal and R. W. Fane, J. Phys. E 6, 409 (1973).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

M. S. Torre, A. Valle, and L. Pesquera, Opt. Quantum Electron. 38, 445 (2006).
[CrossRef]

Phys. Rev. A (2)

L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsässer, Phys. Rev. A 64, 031803(R) (2001).
[CrossRef]

M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
[CrossRef]

Phys. Rev. Lett. (2)

M. B. Willemsen, M. P. van Exter, and J. P. Woerdman, Phys. Rev. Lett. 84, 4337 (2000).
[CrossRef]

R. Dom, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Other (1)

R. Michalzik, VCSELs—Fundamentals, Technology and Applications of VCSELs (Springer, 2013).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Schematic drawing of the VCSEL structure showing the active layer (red), the oxide aperture (black), the high reflecting DBR mirrors framed by the electrical contacts (yellow) and the surface grating with its parameters etching depth (i) and grating period (ii) on top. (b) A photographical picture of the top view of the VCSEL, showing the orientation of the grating grooves with respect to the crystal axis [110] and [110] here depicted schematically with a red and a blue arrow, respectively. (c) Stokes parameters as a function of pump current measured using a photo detector. The polarization control due to the surface grating is clearly visible regarding the constant state of polarization represented by a dominant contribution of the negative value of S1 (red plus) and small but nonzero contributions of S3 (green circle) and S2 (blue cross). Due to the single-mode operation the VCSEL’s total emission is fully polarized reflected in the DOP value close to one above laser threshold at 2 mA.

Fig. 2.
Fig. 2.

Schematic of the experimental setup showing the VCSEL with the crystal axes [110] and [110]. The emitted light is collimated by a lens, passes a combination of a quarter-wave plate with revolvable fast axis and a polarizer with fixed transmission axis and finally the spatial intensity distribution is detected by a CCD camera. In the lower part, a representative set of the intensity of three selected pixels (1—green, 2—blue, 3—red) of the near-field image (top left) is shown as a function of the of the angle β together with a fit to the measured data (lines) to extract the Stokes parameters. The 2D Stokes parameter distribution is calculated for each pixel using this procedure.

Fig. 3.
Fig. 3.

Spatially resolved Stokes parameters: experimental (left) and numerically simulated (right) results.

Fig. 4.
Fig. 4.

Spatially resolved Ix(x,y) and Iy(x,y) calculated from the experimentally obtained Stokes parameters (left) in comparison with their numerically simulated counterparts (right).

Equations (9)

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

DOP=S12+S22+S32S02,
S̲(x,y)=LP*WP(β)*S̲(x,y).
S0(β,x,y)=12(S0(x,y)+S1(x,y)cos2(2β)+S2(x,y)cos(2β)sin(2β)+S3(x,y)sin(2β)).
S0=ExEx*+EyEy*,
S1=ExEx*EyEy*,
S2=ExEy*+ExEy*,
S3=i(ExEy*ExEy*).
Ix(x,y)=12(S0(x,y)+S1(x,y))
Iy(x,y)=12(S0(x,y)S1(x,y)).

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