Abstract

A general model is proposed for a Vertical Cavity Surface Emitting Laser (VCSEL) with medium aspect ratio whose field profile can be described by a limited set of Gauss-Laguerre modes. The model is adapted to self-mixing schemes by supposing that the output beam is reinjected into the laser cavity by an external target mirror. We show that the self-mixing interferometric signal exhibits features peculiar of the spatial distribution of the emitted field and the target-reflected field and we suggest an applicative scheme that could be exploited for experimental displacement measurements. In particular, regimes of transverse mode-locking are found, where we propose an operational scheme for a sensor that can be used to simultaneously measure independent components of the target displacement like target translations along the optical axis (longitudinal axis) and target rotations in a plane orthogonal to the optical axis (transverse plane).

© 2012 OSA

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    [CrossRef]
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2011

2010

2009

2008

2007

2005

2004

C-M. Wu and Y-T. Chuang, “Roll angular displacement measurement system with microradian accuracy,” Sens. Actuators, A116, 145–149 (2004).
[CrossRef]

A. C. Tropper, H. D. Foreman, A. Garnache, K. G. Wilcox, and S. H. Hoogland, “Vertical-external-cavity semiconductor lasers,” J. Phys. D: Appl. Phys.37, R75–R85 (2004).
[CrossRef]

2003

“Z. Liu, D. Lin, H. Jiang, and C. Yin, “Roll angle interferometer by means of wave plates,” Sens. Actuators, A104, 127–131 (2003).
[CrossRef]

2002

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Kndl, M. Miller, and R. Jger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

M. S. Torre, C. Masoller, and P. Mandel, “Transverse mode dynamics in vertical-cavity surface-emitting lasers with optical feedback,” Phys. Rev. A66, 053817 (2002).
[CrossRef]

W. S. Park and H. S. Cho, “Measurement of fine 6-degrees-of-freedom displacement of rigid bodies through splitting a laser beam: experimental investigation,” Opt. Eng.41, 860–871 (2002).
[CrossRef]

S.-H. Park, Y. Park, H. Kim, H. Jeon, S. M. Hwang, J. K. Lee, S. H. Nam, B. C. Koh, J. Y. Sohn, and D. S. Kim, “Microlensed vertical-cavity surface-emitting laser for stable single fundamental mode operation,” Appl. Phys. Lett.80, 183–185 (2002).
[CrossRef]

2001

F. A. Chollet, G. M. Hegde, A. K. Asundi, and A. Q. Liu, “Simple extra-short external cavity laser self-mixing interferometer for acceleration sensing,” Proc. SPIE4596, 272–279 (2001).
[CrossRef]

G. Giuliani, S. Donati, M. Passerini, and T. Bosch, “Angle measurement by injection detection in a laser diode,” Opt. Eng.40, 95–99 (2001).
[CrossRef]

2000

S. Wolff and H. Fouckhardt, “Intracavity stabilization of broad area lasers by structured delayed optical feedback,” Opt. Express7, 222–227 (2000).
[CrossRef] [PubMed]

J. U. Nöckel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, “Mode structure and ray dynamics of a parabolic dome microcavity,” Phys. Rev. E62, 8677–8699 (2000).
[CrossRef]

G Slekys, I Ganne, I Sagnes, and R Kuszelewicz, “Optical pattern formation in passive semiconductor microresonators,” J. Opt. B: Quantum Semiclassical Opt.2, 443–446 (2000).
[CrossRef]

1997

J. Y. Law and G. P. Agrawal, “Effects of optical feedback on static and dynamic characteristics of vertical-cavity surface-emitting lasers,” IEEE J. Sel. Top. Quantum Electron.3, 353-3-58 (1997).

F. Prati, G. Tissoni, M. San Miguel, and N. B Abraham, “Vector vortices and polarization state of low-order transverse modes in a VCSEL,” Opt. Commun.143, 133–146 (1997).
[CrossRef]

J Martń-Regalado, S. Balle, M. San Miguel, A. Valle, and L. Pesquera, “Polarization and transverse-mode selection in quantum-well vertical-cavity surface-emitting lasers: index- and gain-guided devices,” Quantum Semi-classic. Opt.9, 713–736 (1997).
[CrossRef]

F. Prati, M. Travagnin, and L. A. Lugiato, “Logic gates and optical switching with vertical-cavity surface-emitting lasers,” Phys. Rev. A55, 690–700 (1997).
[CrossRef]

1995

M. San Miguel, Q. Feng, and J. V. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A52, 1728–1739 (1995).
[CrossRef] [PubMed]

A. Valle, J. Sarma, and K. A. Shore, “Dynamics of transverse mode competition in vertical cavity surface emitting laser diodes,” Opt. Commun.115, 297–302 (1995).
[CrossRef]

S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback inteferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron.31, 113–119 (1995).
[CrossRef]

1994

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A49, 1427–1451 (1994).
[CrossRef] [PubMed]

A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, J. R. Tredicce, M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, and G.-L. Oppo, “Dynamical transverse laser patterns. II. Experiments,” Phys. Rev. A49, 1452–1466(1994).
[CrossRef] [PubMed]

F. Prati, M. Brambilla, and L. A. Lugiato, “Pattern formation in lasers,” Riv. Nuovo Cimento17, 1–85 (1994).
[CrossRef]

H. Lia, T. L. Lucas, J. G. McInerney, and R. A. Morgan, “Transverse modes and patterns of electrically pumped vertical-cavity surface-emitting semiconductor lasers,” Chaos, Solitons Fractals4, 1619–1636 (1994).
[CrossRef]

1993

C. O. Weiss, H. R. Telle, K. Staliunas, and M. Brambilla, “Restless optical vortex,” Phys. Rev. A47, R1616–R1619 (1993).
[CrossRef] [PubMed]

1992

G. Oppo and G. Dalessandro, “Gauss–Laguerre modes - a sensible basis for laser dynamics,” Opt. Commun.88, 130–136 (1992).
[CrossRef]

L. A. Lugiato, “Spatio-temporal structures. Part I,” Phys. Rep.219, 293–310 (1992).
[CrossRef]

1990

C. J. Chang-Hasnain, M. Orenstein, A. Von Lehmen, l. T. Florez, J. P. Harbison, and N. G. Stoffel, “Transverse mode characteristics of vertical cavity surface-emitting lasers,” Appl. Phys. Lett.57, 218–220 (1990).
[CrossRef]

1988

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, “Role of transverse effects in laser instabilities,” Phys. Rev. A37, 3847–3866 (1988).
[CrossRef] [PubMed]

1980

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser proprieties,” IEEE J. Quantum Electron.16, 347–355 (1980).
[CrossRef]

Abraham, N. B

F. Prati, G. Tissoni, M. San Miguel, and N. B Abraham, “Vector vortices and polarization state of low-order transverse modes in a VCSEL,” Opt. Commun.143, 133–146 (1997).
[CrossRef]

Abram, I.

J. U. Nöckel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, “Mode structure and ray dynamics of a parabolic dome microcavity,” Phys. Rev. E62, 8677–8699 (2000).
[CrossRef]

Adams, R.

J. U. Nöckel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, “Mode structure and ray dynamics of a parabolic dome microcavity,” Phys. Rev. E62, 8677–8699 (2000).
[CrossRef]

Agrawal, G. P.

J. Y. Law and G. P. Agrawal, “Effects of optical feedback on static and dynamic characteristics of vertical-cavity surface-emitting lasers,” IEEE J. Sel. Top. Quantum Electron.3, 353-3-58 (1997).

Ancona, A.

Asundi, A. K.

F. A. Chollet, G. M. Hegde, A. K. Asundi, and A. Q. Liu, “Simple extra-short external cavity laser self-mixing interferometer for acceleration sensing,” Proc. SPIE4596, 272–279 (2001).
[CrossRef]

Balle, S.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Kndl, M. Miller, and R. Jger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

J Martń-Regalado, S. Balle, M. San Miguel, A. Valle, and L. Pesquera, “Polarization and transverse-mode selection in quantum-well vertical-cavity surface-emitting lasers: index- and gain-guided devices,” Quantum Semi-classic. Opt.9, 713–736 (1997).
[CrossRef]

Bandy, D. K.

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, “Role of transverse effects in laser instabilities,” Phys. Rev. A37, 3847–3866 (1988).
[CrossRef] [PubMed]

Baque, J. L.

Barland, S.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Kndl, M. Miller, and R. Jger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

Bertling, K.

Bosch, T.

G. Giuliani, S. Donati, M. Passerini, and T. Bosch, “Angle measurement by injection detection in a laser diode,” Opt. Eng.40, 95–99 (2001).
[CrossRef]

Bourdon, G.

J. U. Nöckel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, “Mode structure and ray dynamics of a parabolic dome microcavity,” Phys. Rev. E62, 8677–8699 (2000).
[CrossRef]

Brambilla, M.

F. P. Mezzapesa, L. Columbo, M. Brambilla, M. Dabbicco, A. Ancona, T. Sibillano, F. De Lucia, P. M. Lugará, and G. Scamarcio, “Simultaneous measurement of multiple target displacements by self-mixing interferometry in a single laser diode,” Opt. Express19, 16160–16173 (2011).
[CrossRef] [PubMed]

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Kndl, M. Miller, and R. Jger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
[CrossRef] [PubMed]

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A49, 1427–1451 (1994).
[CrossRef] [PubMed]

A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, J. R. Tredicce, M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, and G.-L. Oppo, “Dynamical transverse laser patterns. II. Experiments,” Phys. Rev. A49, 1452–1466(1994).
[CrossRef] [PubMed]

F. Prati, M. Brambilla, and L. A. Lugiato, “Pattern formation in lasers,” Riv. Nuovo Cimento17, 1–85 (1994).
[CrossRef]

C. O. Weiss, H. R. Telle, K. Staliunas, and M. Brambilla, “Restless optical vortex,” Phys. Rev. A47, R1616–R1619 (1993).
[CrossRef] [PubMed]

Cattaneo, M.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A49, 1427–1451 (1994).
[CrossRef] [PubMed]

A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, J. R. Tredicce, M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, and G.-L. Oppo, “Dynamical transverse laser patterns. II. Experiments,” Phys. Rev. A49, 1452–1466(1994).
[CrossRef] [PubMed]

Cha, M. T.

Chang-Hasnain, C.

Chang-Hasnain, C. J.

C. J. Chang-Hasnain, M. Orenstein, A. Von Lehmen, l. T. Florez, J. P. Harbison, and N. G. Stoffel, “Transverse mode characteristics of vertical cavity surface-emitting lasers,” Appl. Phys. Lett.57, 218–220 (1990).
[CrossRef]

Chen, C. J.

Cho, H. S.

W. S. Park and H. S. Cho, “Measurement of fine 6-degrees-of-freedom displacement of rigid bodies through splitting a laser beam: experimental investigation,” Opt. Eng.41, 860–871 (2002).
[CrossRef]

Chollet, F. A.

F. A. Chollet, G. M. Hegde, A. K. Asundi, and A. Q. Liu, “Simple extra-short external cavity laser self-mixing interferometer for acceleration sensing,” Proc. SPIE4596, 272–279 (2001).
[CrossRef]

Chuang, Y-T.

C-M. Wu and Y-T. Chuang, “Roll angular displacement measurement system with microradian accuracy,” Sens. Actuators, A116, 145–149 (2004).
[CrossRef]

Coates, A. B.

A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, J. R. Tredicce, M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, and G.-L. Oppo, “Dynamical transverse laser patterns. II. Experiments,” Phys. Rev. A49, 1452–1466(1994).
[CrossRef] [PubMed]

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A49, 1427–1451 (1994).
[CrossRef] [PubMed]

Columbo, L.

Dabbicco, M.

Dalessandro, G.

G. Oppo and G. Dalessandro, “Gauss–Laguerre modes - a sensible basis for laser dynamics,” Opt. Commun.88, 130–136 (1992).
[CrossRef]

DAngelo, E. J.

A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, J. R. Tredicce, M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, and G.-L. Oppo, “Dynamical transverse laser patterns. II. Experiments,” Phys. Rev. A49, 1452–1466(1994).
[CrossRef] [PubMed]

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A49, 1427–1451 (1994).
[CrossRef] [PubMed]

De Lucia, F.

di Vietro, M.

S. Ottonelli, M. Dabbicco, F. De Lucia, M. di Vietro, and G. Scamarcio, “Laser-self-mixing interferometry for mechatronics applications,” Sensors9, 3527–3548 (2009).
[CrossRef]

Donati, S.

G. Giuliani, S. Donati, M. Passerini, and T. Bosch, “Angle measurement by injection detection in a laser diode,” Opt. Eng.40, 95–99 (2001).
[CrossRef]

S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback inteferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron.31, 113–119 (1995).
[CrossRef]

Feng, Q.

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G. Giuliani, S. Donati, M. Passerini, and T. Bosch, “Angle measurement by injection detection in a laser diode,” Opt. Eng.40, 95–99 (2001).
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M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A49, 1427–1451 (1994).
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K. Green, B. Krauskopf, and D. Lenstra, “External cavity mode structure of a two-mode VCSEL subject to optical feedback,” Opt. Commun.277, 359–371 (2007).
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C. J. Chang-Hasnain, M. Orenstein, A. Von Lehmen, l. T. Florez, J. P. Harbison, and N. G. Stoffel, “Transverse mode characteristics of vertical cavity surface-emitting lasers,” Appl. Phys. Lett.57, 218–220 (1990).
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F. A. Chollet, G. M. Hegde, A. K. Asundi, and A. Q. Liu, “Simple extra-short external cavity laser self-mixing interferometer for acceleration sensing,” Proc. SPIE4596, 272–279 (2001).
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A. C. Tropper, H. D. Foreman, A. Garnache, K. G. Wilcox, and S. H. Hoogland, “Vertical-external-cavity semiconductor lasers,” J. Phys. D: Appl. Phys.37, R75–R85 (2004).
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F. Prati, A. Tesei, L. A. Lugiato, and R.J. Horowicz, “Stable states in surface-emitting semiconductor lasers,” Chaos, Solitons Fractals4, 1637–1654 (1994).
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A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, J. R. Tredicce, M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, and G.-L. Oppo, “Dynamical transverse laser patterns. II. Experiments,” Phys. Rev. A49, 1452–1466(1994).
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S.-H. Park, Y. Park, H. Kim, H. Jeon, S. M. Hwang, J. K. Lee, S. H. Nam, B. C. Koh, J. Y. Sohn, and D. S. Kim, “Microlensed vertical-cavity surface-emitting laser for stable single fundamental mode operation,” Appl. Phys. Lett.80, 183–185 (2002).
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K. Green, B. Krauskopf, and D. Lenstra, “External cavity mode structure of a two-mode VCSEL subject to optical feedback,” Opt. Commun.277, 359–371 (2007).
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S.-H. Park, Y. Park, H. Kim, H. Jeon, S. M. Hwang, J. K. Lee, S. H. Nam, B. C. Koh, J. Y. Sohn, and D. S. Kim, “Microlensed vertical-cavity surface-emitting laser for stable single fundamental mode operation,” Appl. Phys. Lett.80, 183–185 (2002).
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K. Green, B. Krauskopf, and D. Lenstra, “External cavity mode structure of a two-mode VCSEL subject to optical feedback,” Opt. Commun.277, 359–371 (2007).
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“Z. Liu, D. Lin, H. Jiang, and C. Yin, “Roll angle interferometer by means of wave plates,” Sens. Actuators, A104, 127–131 (2003).
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Liu, A. Q.

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“Z. Liu, D. Lin, H. Jiang, and C. Yin, “Roll angle interferometer by means of wave plates,” Sens. Actuators, A104, 127–131 (2003).
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F. Prati, A. Tesei, L. A. Lugiato, and R.J. Horowicz, “Stable states in surface-emitting semiconductor lasers,” Chaos, Solitons Fractals4, 1637–1654 (1994).
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S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Kndl, M. Miller, and R. Jger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
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S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback inteferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron.31, 113–119 (1995).
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Miller, M.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Kndl, M. Miller, and R. Jger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
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J. U. Nöckel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, “Mode structure and ray dynamics of a parabolic dome microcavity,” Phys. Rev. E62, 8677–8699 (2000).
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M. San Miguel, Q. Feng, and J. V. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A52, 1728–1739 (1995).
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H. Lia, T. L. Lucas, J. G. McInerney, and R. A. Morgan, “Transverse modes and patterns of electrically pumped vertical-cavity surface-emitting semiconductor lasers,” Chaos, Solitons Fractals4, 1619–1636 (1994).
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S.-H. Park, Y. Park, H. Kim, H. Jeon, S. M. Hwang, J. K. Lee, S. H. Nam, B. C. Koh, J. Y. Sohn, and D. S. Kim, “Microlensed vertical-cavity surface-emitting laser for stable single fundamental mode operation,” Appl. Phys. Lett.80, 183–185 (2002).
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L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, “Role of transverse effects in laser instabilities,” Phys. Rev. A37, 3847–3866 (1988).
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M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A49, 1427–1451 (1994).
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C. J. Chang-Hasnain, M. Orenstein, A. Von Lehmen, l. T. Florez, J. P. Harbison, and N. G. Stoffel, “Transverse mode characteristics of vertical cavity surface-emitting lasers,” Appl. Phys. Lett.57, 218–220 (1990).
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S.-H. Park, Y. Park, H. Kim, H. Jeon, S. M. Hwang, J. K. Lee, S. H. Nam, B. C. Koh, J. Y. Sohn, and D. S. Kim, “Microlensed vertical-cavity surface-emitting laser for stable single fundamental mode operation,” Appl. Phys. Lett.80, 183–185 (2002).
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G. Giuliani, S. Donati, M. Passerini, and T. Bosch, “Angle measurement by injection detection in a laser diode,” Opt. Eng.40, 95–99 (2001).
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J Martń-Regalado, S. Balle, M. San Miguel, A. Valle, and L. Pesquera, “Polarization and transverse-mode selection in quantum-well vertical-cavity surface-emitting lasers: index- and gain-guided devices,” Quantum Semi-classic. Opt.9, 713–736 (1997).
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M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A49, 1427–1451 (1994).
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A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, J. R. Tredicce, M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, and G.-L. Oppo, “Dynamical transverse laser patterns. II. Experiments,” Phys. Rev. A49, 1452–1466(1994).
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F. Prati, G. Tissoni, M. San Miguel, and N. B Abraham, “Vector vortices and polarization state of low-order transverse modes in a VCSEL,” Opt. Commun.143, 133–146 (1997).
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F. Prati, M. Travagnin, and L. A. Lugiato, “Logic gates and optical switching with vertical-cavity surface-emitting lasers,” Phys. Rev. A55, 690–700 (1997).
[CrossRef]

F. Prati, M. Brambilla, and L. A. Lugiato, “Pattern formation in lasers,” Riv. Nuovo Cimento17, 1–85 (1994).
[CrossRef]

A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, J. R. Tredicce, M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, and G.-L. Oppo, “Dynamical transverse laser patterns. II. Experiments,” Phys. Rev. A49, 1452–1466(1994).
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L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, “Role of transverse effects in laser instabilities,” Phys. Rev. A37, 3847–3866 (1988).
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G Slekys, I Ganne, I Sagnes, and R Kuszelewicz, “Optical pattern formation in passive semiconductor microresonators,” J. Opt. B: Quantum Semiclassical Opt.2, 443–446 (2000).
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F. Prati, G. Tissoni, M. San Miguel, and N. B Abraham, “Vector vortices and polarization state of low-order transverse modes in a VCSEL,” Opt. Commun.143, 133–146 (1997).
[CrossRef]

J Martń-Regalado, S. Balle, M. San Miguel, A. Valle, and L. Pesquera, “Polarization and transverse-mode selection in quantum-well vertical-cavity surface-emitting lasers: index- and gain-guided devices,” Quantum Semi-classic. Opt.9, 713–736 (1997).
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M. San Miguel, Q. Feng, and J. V. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A52, 1728–1739 (1995).
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A. Valle, J. Sarma, and K. A. Shore, “Dynamics of transverse mode competition in vertical cavity surface emitting laser diodes,” Opt. Commun.115, 297–302 (1995).
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A. Valle, J. Sarma, and K. A. Shore, “Dynamics of transverse mode competition in vertical cavity surface emitting laser diodes,” Opt. Commun.115, 297–302 (1995).
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S.-H. Park, Y. Park, H. Kim, H. Jeon, S. M. Hwang, J. K. Lee, S. H. Nam, B. C. Koh, J. Y. Sohn, and D. S. Kim, “Microlensed vertical-cavity surface-emitting laser for stable single fundamental mode operation,” Appl. Phys. Lett.80, 183–185 (2002).
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S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Kndl, M. Miller, and R. Jger, “Cavity solitons as pixels in semiconductor microcavities,” Nature419, 699–702 (2002).
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C. J. Chang-Hasnain, M. Orenstein, A. Von Lehmen, l. T. Florez, J. P. Harbison, and N. G. Stoffel, “Transverse mode characteristics of vertical cavity surface-emitting lasers,” Appl. Phys. Lett.57, 218–220 (1990).
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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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Supplementary Material (1)

» Media 1: MPEG (194 KB)     

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Figures (7)

Fig. 1.
Fig. 1.

Sketch of a semiconductor laser subject to optical feedback provided by an external target in a self-imaged configuration.

Fig. 3.
Fig. 3.

Parametric regime: α = 5, α0 = 3.0, Ψ = 50.0, γ = 0.001, τ = 0.16ns. (a) Region C (Ip = 2.4, k = 0.4). Left: Temporal evolution of the modal intensities of the TEM10 (B2) and TEM01 (B3) modes. Right: Fast Fourier trasform (FFT) of the TEM10 modal intensity. We plot the FFT of a single transverse mode, the FFT of the other mode is analogous. The dashed lines denote the external cavities resonances ωi. (b) Region A (Ip = 1.05, k = 0.0008) and (c) Region D (Ip = 0.9, k = 0.4). Left: Temporal evolution of the modal intensities of the TEM10 and TEM01 modes. The blue trace represents the sum of the modal intensities. The intensity profile of the two modes is reported in the inset together with the total intensity distribution averaged over ∼ 50ns. The dimension of the transverse plane is 5W0 × 5W0 in the simulation. Right: FFT of the TEM10 modal intensity. The dashed line in part (b) denotes the relaxation oscillation frequency ωr while in part (c) it denotes the first external cavity resonance.

Fig. 2.
Fig. 2.

Parametric regime: α = 5, α0 = 3.0, Ψ = 50.0, γ = 0.001. (a) τ = 0.16ns. Summary of the numerical results in the (k,Ip) plane for the family (q = 1). The red crosses represent the values of k and Ip considered in the simulations. The vertical and horizontal gray lines denote the k = 0 and Ip = 0 axis. The different dynamical domains A, B, C, D and E separated by the black continuous lines are described in the text. The red crosses corresponding to region A are not shown in the picture for sake of clarity. (b) Ip = 1.05. Each symbol represents a numerical simulation. The red circles correspond to regular oscillations of the field intensity while the black squares are associated to an irregular dynamical behavior.

Fig. 4.
Fig. 4.

Parametric regime as in Fig. 3(c). (a) Top: Time plot of the modal intensities during a feedback mask rotation from Θ = 0 (left inset mask) to Θ = π (right inset mask) (rotation steps of Θ = 0.17rad each 32ns). Bottom: Single frame extracted from the numerical simulation of the field intensity variation ( Media 1. (b) Time plot of the modal intensities during a feedback mask rotation from Θ = 0 to Θ = 3π (rotation steps of Θ = 0.05rad each 0.32ns).

Fig. 5
Fig. 5

Parametric regime: α = 5, α0 = 3.0, Ψ = 50.0, γ = 0.001, Ip = 0.9, k = 0.4. (a) Primary (low frequency) peak variation in the FFT of the TEM10 modal intensity (that of TEM01 mode is very similar) with the displacement dL. (b) Variation of the average total intensity with the displacement dL. The circles highlight the frequency discontinuity and the corresponding values of the average total intensity.

Fig. 6.
Fig. 6.

Parametric regime: α = 5, γ = 0.001, Ip = 0.9, k = 0.4. Modal intensities sum (Isum) and normalized difference (Idiff) during: (a) a target rotation of π/2 in the transverse plane for a fixed external cavity length L = 2.4cm (We rotate the feedback mask of Θ = 0.17rad each 48ns); (b) a target longitudinal displacement of 1μm (dL = 0 corresponds to L = 2.4cm, λ = 830nm) for a fixed angle Θ = 0rad.

Fig. 7.
Fig. 7.

Parametric regime: α = 5, γ = 0.001, Ip = 1.05, k = 0.002. Plot of Isum and Idiff during: (a) a target rotation of π/2 in the transverse plane for a fixed external cavity length L = 2.4cm (We rotate the feedback mask of Θ = 0.17rad each 160ns); (d) a target longitudinal displacement of 1μm (dL = 0 corresponds to L = 2.4cm, λ = 830nm) for a fixed angle Θ = π/4. In part (a) for sake of clarity not all the data points in the Idiff trace are plotted. The plot in part (b) represents a zoom of the traces in part (a). Figure 7(c) has been obtained by low-pass filtering the oscillations shown in part (a) with a cut-off at 1MHz, corresponding to a detector integration time of about 1μs.

Equations (32)

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n ( r ) = n ( 0 ) 1 r 2 / h 2
E ( x , y , t ) = E F ( x , y , z , t ) exp ( i ( k 0 z + ω 0 t ) ) + E B ( x , y , z , t ) exp ( i ( k 0 z + ω 0 t ) ) + c . c . Y ( x , yz , t ) = Y ¯ 1 ( x , y , z , t ) exp ( i ( k 0 z + ω 0 t ) ) + Y ¯ 2 ( x , y , z , t ) exp ( i ( k 0 z + ω 0 t ) ) + c . c .
1 2 i k 0 ( 2 k 0 r 2 h 2 ) E F , B ± E F , B z + 1 v E F , B t = 1 2 g n ( 1 + i α ) ( N N 0 ) E F , B
𝒜 p , m ( ρ , ϕ , z ) = A p , m ( ρ , ϕ ) exp ( i ( 2 p + | m | + 1 ) z / h )
A p , m ( ρ , ϕ ) = 2 π ( 2 ρ 2 ) | m | / 2 [ p ! ( p + | m | ) ! ] 1 / 2 L p | m | ( 2 ρ 2 ) e ρ 2 e im ϕ
ω s , q = ω s + ω q = v l s π + v h ( 2 p + | m | + 1 )
E F , B ( ρ , ϕ , z , t ) = p , m ε F , B ; p , m ( z , t ) 𝒜 p , m ( ρ , ϕ , z ) , ε F , B ; p , m
± ε F , B ; p , m z + 1 v ε F , B ; p , m t = 1 2 g n ( 1 + i α ) 0 2 π d ϕ 0 ρ d ρ 𝒜 p , m * ( ± z ) ( N N 0 ) E F , B
E F ( x , y , z = 0 , t ) = R E B ( x , y , z = 0 , t ) E B ( x , y , z = l , t ) = R exp ( 2 i k 0 l ) E F ( x , y , z = l , t ) + T Y 2 ( x , y , z = l , t )
ε F ; p , m ( 0 , t ) = R ε B ; p , m ( 0 , t ) ε B ; p , m ( l , t ) = R exp ( i δ 0 ) ε F ; p , m ( l , t ) + T Y 2 , p , m ( l , t )
Y 2 ( ρ , ϕ , z , t ) = p , m Y 2 , p , m ( z , t ) 𝒜 p , m ( ρ , ϕ , z ) , Y 2 , p , m 𝒞
dE p , m dt = ( iq Δ ω T 1 2 τ p ( 1 + i α ) ) E p , m + 1 2 G n ( 1 + i α ) 0 2 π d ϕ 0 ρ d ρ A p , m * ( N N 0 ) E + 1 τ c 0 2 π d ϕ 0 ρ d ρ A p , m * k ( ρ , ϕ ) E ( ρ , ϕ , t τ ) exp ( i ω 0 τ )
k ( ρ , ϕ ) = ε ( 1 R ) R ext ( ρ , ϕ ) R
dN ( ρ , ϕ , t ) dt = I eV N ( ρ , ϕ , t ) τ e n ( 0 ) 2 ε 0 2 ω 0 G n ( N ( ρ , ϕ , t ) N 0 ) | E ( ρ , ϕ , t ) | 2
G n τ e n ( 0 ) 2 ε 0 2 ω 0 E E , ( N N 0 ) G n τ p N , t / τ p t χ ( ρ ) = I p e 2 ρ 2 / Ψ 2 , Ψ with I p = G n τ p N 0 ( I τ e e V N 0 1 ) , γ = τ p τ e
dE p , m ( t ) dt = ( iq Δ ω T τ p 1 2 ( 1 + i α ) ) E p , m ( t ) + 1 2 ( 1 + i α ) 0 2 π d ϕ 0 ρ d ρ A p , m * N ( ρ ϕ , t ) E ( ρ ϕ , t ) + τ p τ c 0 2 π d ϕ 0 ρ d ρ A p , m * k ( ρ , ϕ ) E ( ρ , ϕ , t τ ) exp ( i ω 0 τ )
dN ( ρ ϕ , t ) dt = γ ( χ ( ρ ) N ( ρ ϕ , t ) ( 1 + | E ( ρ ϕ , t ) | 2 ) )
B p , m , 0 ( ρ , ϕ ) = A p , 0 ( ρ , ϕ ) B p , m , 1 ( ρ , ϕ ) = 1 2 [ A p , m ( ρ , ϕ ) + A p , m ( ρ , ϕ ) ] B p , m , 2 ( ρ , ϕ ) = 1 2 i [ A p , m ( ρ , ϕ ) A p , m ( ρ , ϕ ) ]
E ( ρ , ϕ , t ) = p , m , o g p , m , o ( t ) B p , m , o ( ρ , ϕ ) , g p , m , o 𝒞
ε F ; p , m = exp ( i δ 0 + ln ( R ) ( z l ) / l ) ε F ; p , m + z l T R Y 2 , p , m ε B ; p , m = exp ( ln ( R ) z / l ) ε B ; p , m
ε F ; p , m ( 0 , t ) = ε B ; p , m ( 0 , t )
ε F ; p , m ( l , t ) = ε B ; p , m ( l , t )
ε F ; p , m z + 1 v ε F ; p , m t z l T R Y 2 , p , m z z vl T R Y 2 , p , m t = T l ( ( ε F ; p , m z l T R Y 2 , p , m ) ( i δ 0 T + ln ( R ) T ) + ( l 2 T g n ( 1 + i α ) p , m ( ε F ; p , m z l T R Y 2 , p , m ) exp ( 2 i ( q q ) z h ) 0 2 π d ϕ 0 ρ d ρ 𝒜 p , m * ( z ) 𝒜 p , m ( z ) ( N N 0 ) ) + Y 2 , p , m R )
ε B ; p , m z + 1 v ε B ; p , m t = T l ( ε B ; p , m ln ( R ) T + ( l 2 T g n ( 1 + i α ) p , m ε B ; p , m 0 2 π d ϕ 0 ρ d ρ 𝒜 p , m * ( z ) 𝒜 p , m ( z ) ( N N 0 ) ) )
dE p , m dt = v T 2 l ( E p , m ( i θ T ) + l T g n ( 1 + i α ) 0 2 π d ϕ 0 ρ d ρ A p , m * ( N N 0 ) E ¯ + Y 2 , p , m R )
dE p , m dt = ( iq Δ ω T 1 2 τ p ( 1 + i α ) ) E p , m + 1 2 G n ( 1 + i α ) 0 2 π d ϕ 0 ρ d ρ A p , m * ( N N 0 ) E + T R τ c Y 2 , p , m
Y 2 ( l , t ) = exp ( i ( k 0 k 0 ) l ) n ( 0 ) Y ¯ 2 ( l + L , t τ / 2 ) exp ( 2 i k 0 L ) = R ext 1 R exp ( 2 i k 0 l ) exp ( i ω 0 τ ) E F ( l , t τ ) R ext R exp ( i ω 0 τ ) Y 2 ( l , t τ )
Y 2 ( l , t ) = ε R ext 1 R exp ( i ω 0 τ ) E ( t τ ) ε R ext R exp ( i ω 0 τ ) Y 2 ( l , t τ )
Y 2 ( l , t ) = ε R ext 1 R exp ( i ω 0 τ ) p , m E p , m ( t τ ) 𝒟 A p , m ε R ext R exp ( i ω 0 τ ) p , m Y 2 , p , m ( l , t τ ) 𝒟 A p , m
dE p , m dt = ( iq Δ ω T 1 2 τ p ( 1 + i α ) ) E p , m + 1 2 G n ( 1 + i α ) 0 2 π d ϕ 0 ρ d ρ A p , m * ( N N 0 ) E + 1 τ c ( ε R ext ( 1 R ) R exp ( i ω 0 τ ) E p , m ( t τ ) ε R ext 1 R exp ( i ω 0 τ ) Y p , m ( l , t τ ) )
dE p , m ( t ) dt = ( iq Δ ω T 1 2 τ p ( 1 + i α ) ) E p , m ( t ) + 1 2 G n ( 1 + i α ) 0 2 π d ϕ 0 ρ d ρ A p , m * ( N ( ρ , ϕ , t ) N 0 ) E ( ρ , ϕ , t ) + 1 τ c ( s = 1 k s E p , m ( t s τ ) exp ( is ω 0 τ ) )
k s = ε s ( 1 R ) R ext R ( R R ext ) s 1 , s 𝒩

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