Abstract

We consider polarization changes of randomly fluctuating electromagnetic pulsed light in temporal imaging. The polarization properties of pulses formed by the time lens are formulated in terms of the Stokes parameters. For Gaussian Schell-model pulses we show that the degree and state of polarization of the time-imaged pulse can be tailored in versatile ways, depending on the temporal polarization and coherence of the input pulse and the system parameters. In particular, weakly polarized central region of the pulse may become fully polarized without energy absorption. The results have potential applications in optical communication, micromachining, and light–matter interactions.

© 2013 OSA

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  1. J. W. Goodman, Statistical Optics (Wiley, 1985).
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  3. E. Collett, Polarized Light in Fiber Optics (SPIE, 2003).
  4. R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).
    [CrossRef]
  5. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento13, 1165–1181 (1959).
    [CrossRef]
  6. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2001).
  7. R. Barakat, “n-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Optica Acta30, 1171–1182 (1983).
    [CrossRef]
  8. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun.204, 53–58 (2002).
    [CrossRef]
  9. H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A21, 2117–2123 (2004).
    [CrossRef]
  10. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A22, 1536–1545 (2005).
    [CrossRef]
  11. K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A80, 053804 (2009).
    [CrossRef]
  12. T. Voipio, T. Setälä, and A. T. Friberg, “Partial polarization theory of optical pulses,” J. Opt. Soc. Am. A30, 71–81 (2013).
    [CrossRef]
  13. B. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett.14, 630–632 (1989).
    [CrossRef] [PubMed]
  14. I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon.1, 308–437 (2009).
    [CrossRef]
  15. V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” in Progress in Optics, ed. E. Wolf, vol. 56 (Elsevier, 2011), pp. 1–80.
    [CrossRef]
  16. C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett.24, 783–785 (1999).
    [CrossRef]
  17. Ch. Dorrer and I. Kang, “Complete temporal characterization of short optical pulses by simplified chronocyclic tomography,” Opt. Lett.28, 1481–1483 (2003).
    [CrossRef] [PubMed]
  18. M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456, 81–85 (2008).
    [CrossRef] [PubMed]
  19. A. A. Godil, B. A. Auld, and D. M. Bloom, “Time-lens producing 1.9 ps optical pulses,” Appl. Phys. Lett.62, 1047–1049 (1992).
    [CrossRef]
  20. L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron.36, 795–801 (2000).
    [CrossRef]
  21. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).
  22. C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up conversion time microscope,” Appl. Phys. Lett.65, 2513–2515 (1994).
    [CrossRef]
  23. A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys.11, 073004 (2009), http://iopscience.iop.org/1367-2630/11/7/073004 .
    [CrossRef]
  24. T. Voipio, T. Setälä, A. Shevchenko, and A. T. Friberg, “Polarization dynamics and polarization time of random three-dimensional electromagnetic fields,” Phys. Rev. A82, 063807 (2010).
    [CrossRef]
  25. K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nature Phot.1, 228–231 (2007).
    [CrossRef]
  26. Ph. Réfrégier, T. Setälä, and A. T. Friberg, “Maximal polarization order of random optical beams: reversible and irreversible polarization variations,” Opt. Lett.37, 3750–3752 (2012).
    [CrossRef] [PubMed]
  27. Ph. Réfrégier, T. Setälä, and A. T. Friberg, “Temporal and spectral degrees of polarization of light,” Proc. SPIE8171, 817102 (2011).
    [CrossRef]
  28. A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford University, 2007).

2013 (1)

2012 (1)

2011 (1)

Ph. Réfrégier, T. Setälä, and A. T. Friberg, “Temporal and spectral degrees of polarization of light,” Proc. SPIE8171, 817102 (2011).
[CrossRef]

2010 (1)

T. Voipio, T. Setälä, A. Shevchenko, and A. T. Friberg, “Polarization dynamics and polarization time of random three-dimensional electromagnetic fields,” Phys. Rev. A82, 063807 (2010).
[CrossRef]

2009 (3)

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A80, 053804 (2009).
[CrossRef]

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys.11, 073004 (2009), http://iopscience.iop.org/1367-2630/11/7/073004 .
[CrossRef]

I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon.1, 308–437 (2009).
[CrossRef]

2008 (1)

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456, 81–85 (2008).
[CrossRef] [PubMed]

2007 (1)

K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nature Phot.1, 228–231 (2007).
[CrossRef]

2005 (1)

2004 (1)

2003 (1)

2002 (1)

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun.204, 53–58 (2002).
[CrossRef]

2000 (1)

L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron.36, 795–801 (2000).
[CrossRef]

1999 (1)

1994 (1)

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up conversion time microscope,” Appl. Phys. Lett.65, 2513–2515 (1994).
[CrossRef]

1992 (1)

A. A. Godil, B. A. Auld, and D. M. Bloom, “Time-lens producing 1.9 ps optical pulses,” Appl. Phys. Lett.62, 1047–1049 (1992).
[CrossRef]

1989 (1)

1983 (1)

R. Barakat, “n-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Optica Acta30, 1171–1182 (1983).
[CrossRef]

1959 (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento13, 1165–1181 (1959).
[CrossRef]

Andrés, P.

V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” in Progress in Optics, ed. E. Wolf, vol. 56 (Elsevier, 2011), pp. 1–80.
[CrossRef]

Auld, B. A.

A. A. Godil, B. A. Auld, and D. M. Bloom, “Time-lens producing 1.9 ps optical pulses,” Appl. Phys. Lett.62, 1047–1049 (1992).
[CrossRef]

Barakat, R.

R. Barakat, “n-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Optica Acta30, 1171–1182 (1983).
[CrossRef]

Barthélémy, A.

L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron.36, 795–801 (2000).
[CrossRef]

Bennett, C. V.

C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett.24, 783–785 (1999).
[CrossRef]

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up conversion time microscope,” Appl. Phys. Lett.65, 2513–2515 (1994).
[CrossRef]

Bloom, D. M.

A. A. Godil, B. A. Auld, and D. M. Bloom, “Time-lens producing 1.9 ps optical pulses,” Appl. Phys. Lett.62, 1047–1049 (1992).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2001).

Collett, E.

E. Collett, Polarized Light in Fiber Optics (SPIE, 2003).

Dorrer, C.

Dorrer, Ch.

Foster, M. A.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456, 81–85 (2008).
[CrossRef] [PubMed]

Friberg, A. T.

T. Voipio, T. Setälä, and A. T. Friberg, “Partial polarization theory of optical pulses,” J. Opt. Soc. Am. A30, 71–81 (2013).
[CrossRef]

Ph. Réfrégier, T. Setälä, and A. T. Friberg, “Maximal polarization order of random optical beams: reversible and irreversible polarization variations,” Opt. Lett.37, 3750–3752 (2012).
[CrossRef] [PubMed]

Ph. Réfrégier, T. Setälä, and A. T. Friberg, “Temporal and spectral degrees of polarization of light,” Proc. SPIE8171, 817102 (2011).
[CrossRef]

T. Voipio, T. Setälä, A. Shevchenko, and A. T. Friberg, “Polarization dynamics and polarization time of random three-dimensional electromagnetic fields,” Phys. Rev. A82, 063807 (2010).
[CrossRef]

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A80, 053804 (2009).
[CrossRef]

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys.11, 073004 (2009), http://iopscience.iop.org/1367-2630/11/7/073004 .
[CrossRef]

K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nature Phot.1, 228–231 (2007).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun.204, 53–58 (2002).
[CrossRef]

Froehly, C.

L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron.36, 795–801 (2000).
[CrossRef]

Gaeta, A. L.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456, 81–85 (2008).
[CrossRef] [PubMed]

Geraghty, D. F.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456, 81–85 (2008).
[CrossRef] [PubMed]

Godil, A. A.

A. A. Godil, B. A. Auld, and D. M. Bloom, “Time-lens producing 1.9 ps optical pulses,” Appl. Phys. Lett.62, 1047–1049 (1992).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Kaivola, M.

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys.11, 073004 (2009), http://iopscience.iop.org/1367-2630/11/7/073004 .
[CrossRef]

K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nature Phot.1, 228–231 (2007).
[CrossRef]

Kang, I.

Kh. Mouradian, L.

L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron.36, 795–801 (2000).
[CrossRef]

Kolner, B.

Kolner, B. H.

C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett.24, 783–785 (1999).
[CrossRef]

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up conversion time microscope,” Appl. Phys. Lett.65, 2513–2515 (1994).
[CrossRef]

Lajunen, H.

Lancis, J.

V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” in Progress in Optics, ed. E. Wolf, vol. 56 (Elsevier, 2011), pp. 1–80.
[CrossRef]

Lindfors, K.

K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nature Phot.1, 228–231 (2007).
[CrossRef]

Lipson, M.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456, 81–85 (2008).
[CrossRef] [PubMed]

Louradour, F.

L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron.36, 795–801 (2000).
[CrossRef]

Mandel, L.

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

Martínez-Herrero, R.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).
[CrossRef]

Mejías, P. M.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).
[CrossRef]

Messager, V.

L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron.36, 795–801 (2000).
[CrossRef]

Nazarathy, M.

Pääkkönen, P.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun.204, 53–58 (2002).
[CrossRef]

Piquero, G.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).
[CrossRef]

Priimägi, A.

K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nature Phot.1, 228–231 (2007).
[CrossRef]

Réfrégier, Ph.

Saastamoinen, K.

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A80, 053804 (2009).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Salem, R.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456, 81–85 (2008).
[CrossRef] [PubMed]

Scott, R. P.

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up conversion time microscope,” Appl. Phys. Lett.65, 2513–2515 (1994).
[CrossRef]

Setälä, T.

T. Voipio, T. Setälä, and A. T. Friberg, “Partial polarization theory of optical pulses,” J. Opt. Soc. Am. A30, 71–81 (2013).
[CrossRef]

Ph. Réfrégier, T. Setälä, and A. T. Friberg, “Maximal polarization order of random optical beams: reversible and irreversible polarization variations,” Opt. Lett.37, 3750–3752 (2012).
[CrossRef] [PubMed]

Ph. Réfrégier, T. Setälä, and A. T. Friberg, “Temporal and spectral degrees of polarization of light,” Proc. SPIE8171, 817102 (2011).
[CrossRef]

T. Voipio, T. Setälä, A. Shevchenko, and A. T. Friberg, “Polarization dynamics and polarization time of random three-dimensional electromagnetic fields,” Phys. Rev. A82, 063807 (2010).
[CrossRef]

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys.11, 073004 (2009), http://iopscience.iop.org/1367-2630/11/7/073004 .
[CrossRef]

K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nature Phot.1, 228–231 (2007).
[CrossRef]

Shevchenko, A.

T. Voipio, T. Setälä, A. Shevchenko, and A. T. Friberg, “Polarization dynamics and polarization time of random three-dimensional electromagnetic fields,” Phys. Rev. A82, 063807 (2010).
[CrossRef]

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys.11, 073004 (2009), http://iopscience.iop.org/1367-2630/11/7/073004 .
[CrossRef]

K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nature Phot.1, 228–231 (2007).
[CrossRef]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Tervo, J.

Torres-Company, V.

V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” in Progress in Optics, ed. E. Wolf, vol. 56 (Elsevier, 2011), pp. 1–80.
[CrossRef]

Turner-Foster, A. C.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456, 81–85 (2008).
[CrossRef] [PubMed]

Turunen, J.

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A80, 053804 (2009).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun.204, 53–58 (2002).
[CrossRef]

Vahimaa, P.

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A80, 053804 (2009).
[CrossRef]

H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A22, 1536–1545 (2005).
[CrossRef]

H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A21, 2117–2123 (2004).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun.204, 53–58 (2002).
[CrossRef]

Voipio, T.

T. Voipio, T. Setälä, and A. T. Friberg, “Partial polarization theory of optical pulses,” J. Opt. Soc. Am. A30, 71–81 (2013).
[CrossRef]

T. Voipio, T. Setälä, A. Shevchenko, and A. T. Friberg, “Polarization dynamics and polarization time of random three-dimensional electromagnetic fields,” Phys. Rev. A82, 063807 (2010).
[CrossRef]

Walmsley, I. A.

Wolf, E.

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento13, 1165–1181 (1959).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2001).

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

Wyrowski, F.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun.204, 53–58 (2002).
[CrossRef]

Yariv, A.

A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford University, 2007).

Yeh, P.

A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford University, 2007).

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (2)

A. A. Godil, B. A. Auld, and D. M. Bloom, “Time-lens producing 1.9 ps optical pulses,” Appl. Phys. Lett.62, 1047–1049 (1992).
[CrossRef]

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up conversion time microscope,” Appl. Phys. Lett.65, 2513–2515 (1994).
[CrossRef]

IEEE J. Quantum Electron. (1)

L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron.36, 795–801 (2000).
[CrossRef]

J. Opt. Soc. Am. A (3)

Nature (1)

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456, 81–85 (2008).
[CrossRef] [PubMed]

Nature Phot. (1)

K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nature Phot.1, 228–231 (2007).
[CrossRef]

New J. Phys. (1)

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys.11, 073004 (2009), http://iopscience.iop.org/1367-2630/11/7/073004 .
[CrossRef]

Nuovo Cimento (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento13, 1165–1181 (1959).
[CrossRef]

Opt. Commun. (1)

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun.204, 53–58 (2002).
[CrossRef]

Opt. Lett. (4)

Optica Acta (1)

R. Barakat, “n-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Optica Acta30, 1171–1182 (1983).
[CrossRef]

Phys. Rev. A (2)

T. Voipio, T. Setälä, A. Shevchenko, and A. T. Friberg, “Polarization dynamics and polarization time of random three-dimensional electromagnetic fields,” Phys. Rev. A82, 063807 (2010).
[CrossRef]

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A80, 053804 (2009).
[CrossRef]

Proc. SPIE (1)

Ph. Réfrégier, T. Setälä, and A. T. Friberg, “Temporal and spectral degrees of polarization of light,” Proc. SPIE8171, 817102 (2011).
[CrossRef]

Other (8)

A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford University, 2007).

V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” in Progress in Optics, ed. E. Wolf, vol. 56 (Elsevier, 2011), pp. 1–80.
[CrossRef]

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2001).

J. W. Goodman, Statistical Optics (Wiley, 1985).

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

E. Collett, Polarized Light in Fiber Optics (SPIE, 2003).

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).
[CrossRef]

Supplementary Material (4)

» Media 1: MOV (1853 KB)     
» Media 2: MOV (2351 KB)     
» Media 3: MOV (2356 KB)     
» Media 4: MOV (2356 KB)     

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Figures (6)

Fig. 1
Fig. 1

Illustration of the Poincaré sphere. The dashed circle shows the intersection of the s3(t) = 0 plane with the sphere. Dots indicate intersections of the sphere with the coordinate axes, with the respective fully polarized states indicated (beam propagates towards the viewer).

Fig. 2
Fig. 2

Schematic of a temporal imaging system. The middle row illustrates the arrangement with two single-mode fibers (SMF) whose GDD parameters are Φa and Φb, and the quadratic phase modulator (QPM). The phase ϕ(t) imparted by the QPM is determined by the parameter γ. The three figures in the top row illustrate the behavior of the electric field E(t) of an originally transform-limited pulse with center frequency ω0. The first plot shows the electric field at the input of the system. The second plot shows that propagation through the first SMF has broadened the pulse and introduced chirp. Finally, propagation through the second SMF compresses the pulse. The bottom row shows the spectral densities S(ω) of the pulse before and after propagation through the QPM. In the case of an originally transform-limited pulse, the spectrum is broadened by the phase modulation.

Fig. 3
Fig. 3

(a) Intensities of x (blue dash-dotted) and y (green dashed) components and the degree of polarization (red solid) before and after temporal imaging, shown with thin and thick curves, respectively. The vertical axis represents intensity normalized such that the peak intensity of the components at the input of the temporal imaging system is 1.0. (b) Behavior of the normalized Stokes parameters as a function of time. Solid blue line, dashed green line, and dash-dotted red line indicate values of s1(t), s2(t), and s3(t), respectively. (c) Evolution of the polarization state (solid curve) represented on the Poincaré sphere. The dashed circles show the intersection of the s1(t) = 0, s2(t) = 0, and s3(t) = 0 planes with the surface of the sphere. Correspondence between the intensities, the Stokes parameters, and the respective locations on the Poincaré sphere is further illustrated in Media 1.

Fig. 4
Fig. 4

Polarization state of a pulse after propagation through a temporal imaging system which imparts a quadratic phase difference between the orthogonal components ( Media 2). (a) Behavior of the normalized Stokes parameters as a function of time. Solid blue line, dashed green line, and dash-dotted red line indicate values of s1(t), s2(t), and s3(t), respectively. (b) Evolution of the polarization state (solid curve) represented on the Poincaré sphere. The dashed circles show the intersection of the s1(t) = 0, s2(t) = 0, and s3(t) = 0 planes with the surface of the sphere.

Fig. 5
Fig. 5

(a) Component intensities and the degree of polarization before (thin lines) and after (thick lines) temporal imaging with opposite magnifications for x and y components. Intensities of the x and y components and the degree of polarization are shown with blue dash-dotted, green dashed, and red solid curves, respectively. Intensities before and after imaging are represented on the same scale to show the temporal compression-induced change in peak intensity. The main difference to the case Mx = My in Fig. 3 is the short period of full polarization midway between the intensity peaks. (b) Normalized Stokes si(t) parameters of a temporally imaged pulse with opposite magnifications for the orthogonally polarized components. (c) Polarization state represented using a Poincaré sphere, solid curve, and the intersections of the planes s1(t) = 0, s2(t) = 0, and s3(t) = 0 with the surface of the Poincaré sphere, dashed circles. The locus of the points representing the polarization states of the beam at different instants of time forms a closed curve. The connection between the Poincaré sphere, the component intensities, and the Stokes parameters is illustrated in Media 3.

Fig. 6
Fig. 6

(a) Intensities of the x and y components (blue dash-dotted and green dashed lines, respectively), and the degree of polarization (red solid lines) before and after imaging (thin and thick lines, respectively). (b) Normalized Stokes parameters s1(t), s2(t), and s3(t) drawn with blue solid, green dashed, and red dash-dotted lines, respectively. (c) Polarization state (red solid curve) in terms of the Poincaré sphere. The dashed circles depict the intersections of the planes s1(t) = 0, s2(t) = 0, and s3(t) = 0 with the surface of the Poincaré sphere. The time evolution of the polarization state is demonstrated in Media 4.

Equations (75)

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Γ i j ( r 1 , r 2 , t 1 , t 2 ) = E i * ( r 1 , t 1 ) E j ( r 2 , t 2 ) ,
P ( r , t ) = 1 4 det J ( r , t ) tr 2 J ( r , t ) .
S 0 ( r , t ) = J x x ( r , t ) + J x x ( r , t ) = I x ( r , t ) + I y ( r , t ) ,
S 1 ( r , t ) = J x x ( r , t ) J x x ( r , t ) = I x ( r , t ) I y ( r , t ) ,
S 2 ( r , t ) = J x y ( r , t ) + J y x ( r , t ) = I α ( r , t ) I β ( r , t ) ,
S 3 ( r , t ) = i [ J y x ( r , t ) J x y ( r , t ) ] = I r ( r , t ) I l ( r , t ) ,
s i ( r , t ) = S i ( r , t ) / S 0 ( r , t ) ,
P 2 ( r , t ) = s 1 2 ( r , t ) + s 2 2 ( r , t ) + s 3 2 ( r , t ) .
A i ( out ) ( z , t ) = A i ( in ) ( z , t ) exp [ i t 2 / ( 2 γ i ) ] , i = x , y ,
A i ( z , t ) = A i ( 0 , t ) K i ( f ) ( t t ; z ) d t , i = x , y ,
K i ( f ) ( t t ; z ) = e i β 0 z i 2 π β 2 i z exp { i [ ( t t ) + β 1 z ] 2 2 β 2 i z } , i = x , y .
A i ( z , t ˜ ) = A i ( 0 , t ) K i ( f ) ( t ˜ t ; z ) d t , i = x , y ,
K i ( f ) ( t ˜ t ; z ) = e i β 0 z i 2 π Φ i exp [ i ( t ˜ t ) 2 2 Φ i ] , i = x , y ,
A i ( out ) ( t ) = A i ( in ) ( t ) K i ( t , t ) d t , i = x , y .
K i ( t , t ) = e i ϕ i ( t , t ) i 2 π Φ a i i 2 π Φ b i exp [ i 2 L i t 2 + i ( t Φ a i + t Φ b i ) t ] d t ,
1 γ i = 1 Φ a i + 1 Φ b i , i = x , y ,
K i ( t , t ) = e i ϕ i ( t , t ) i Φ a i i Φ b i | Φ a i | δ ( t + Φ a i Φ b i t ) , i = x , y .
A i ( out ) ( t ) = exp { i [ β 0 a z a + β 0 b z b + π 4 ( sgn Φ a i + sgn Φ b i ) + M i 1 1 2 Φ b i t 2 ] } 1 | M i | 1 / 2 A i ( in ) ( t M i ) ,
M i = Φ b i / Φ a i , i = x , y .
Γ i j ( e ) ( t 1 , t 2 ) = A i * ( t 1 ) A j ( t 2 ) = Γ i j ( t 1 , t 2 ) e i ω 0 ( t 2 t 1 ) , i , j = x , y ,
Γ i j ( e ) ( t 1 , t 2 ) = Γ 0 i j ( e ) ( t 1 , t 2 ) K i j ( t 1 , t 2 ; t 1 , t 2 ) d t 1 d t 2 , i , j = x , y ,
K i j ( t 1 , t 2 ; t 1 , t 2 ) = K i * ( t 1 , t 1 ) K j ( t 2 , t 2 ) , i , j = x , y ,
Γ i j ( e ) ( t 1 , t 2 ) = exp [ i ϕ i j ( t 1 , t 2 ) ] | M i M j | 1 / 2 Γ 0 i j ( e ) ( t 1 M i , t 2 M j ) , i , j = x , y .
J i j ( t ) = exp [ i ϕ i j ( t ) ] | M i M j | 1 / 2 Γ 0 i j ( t M i , t M j ) exp [ i ω 0 ( M i 1 M j 1 ) t ] , i , j = x , y ,
ϕ i j ( t ) = π 4 [ sgn ( Φ a j ) + sgn ( Φ b j ) sgn ( Φ a i ) sgn ( Φ b i ) ] + ( M j 1 1 2 Φ b j M i 1 1 2 Φ b i ) t 2 ,
I i ( t ) = 1 | M i | I 0 i ( t M i ) , i = x , y ,
P ( t ) = P 0 ( t M ) ,
S 0 ( t ) = 1 | M | S 00 ( t M ) ,
S 1 ( t ) = 1 | M | S 01 ( t M ) ,
S 2 ( t ) = 1 | M | [ S 02 ( t M ) cos ϕ x y ( t ) S 03 ( t M ) sin ϕ x y ( t ) ] ,
S 3 ( t ) = 1 | M | [ S 03 ( t M ) cos ϕ x y ( t ) + S 02 ( t M ) sin ϕ x y ( t ) ] ,
s 1 ( t ) = s 01 ( t M ) ,
s 2 ( t ) = s 02 ( t M ) cos ϕ x y ( t ) s 03 ( t M ) sin ϕ x y ( t ) ,
s 3 ( t ) = s 03 ( t M ) cos ϕ x y ( t ) + s 02 ( t M ) sin ϕ x y ( t ) ,
E ( t ) = A 0 a ( t ) exp [ t 2 / ( 2 T 0 2 ) ] exp ( i ω 0 t )
a * ( t 1 ) a ( t 2 ) = exp [ ( t 2 t 1 ) 2 / T c 2 ] ,
I ( t ) = A 0 2 exp ( t 2 / T 0 2 ) .
Γ 0 ( t 1 , t 2 ) = [ Γ ( t 1 , t 2 ) Γ ( t 1 , t 2 τ d ) Γ ( t 1 τ d , t 2 ) Γ ( t 1 τ d , t 2 τ d ) ] ,
Γ ( t 1 , t 2 ) = [ I ( t 1 ) I ( t 2 ) ] 1 / 2 exp [ ( t 2 t 1 ) 2 / T c 2 ] exp [ i ω 0 ( t 2 t 1 ) ]
J x x ( t ) = I x ( t ) = 1 | M x | I ( t M x ) ,
J y y ( t ) = I y ( t ) = 1 | M y | I ( t M y τ d ) ,
J x y ( t ) = J y x * ( t ) = [ I x ( t ) I y ( t ) ] 1 / 2 exp { i [ ϕ x y ( t ) + ω 0 τ d ] Δ 2 ( t ) / T c 2 } ,
Δ ( t ) = ( M y 1 M x 1 ) t τ d .
S 0 ( t ) = 1 | M x | I ( t M x ) + 1 | M y | I ( t M y τ d ) ,
S 1 ( t ) = 1 | M x | I ( t M x ) 1 | M y | I ( t M y τ d ) ,
S 2 ( t ) = 2 | M x M y | 1 / 2 [ I ( t M x ) I ( t M y τ d ) ] 1 / 2 exp [ Δ 2 ( t ) / T c 2 ] cos [ ϕ x y ( t ) + ω 0 τ d ] ,
S 3 ( t ) = 2 | M x M y | 1 / 2 [ I ( t M x ) I ( t M y τ d ) ] 1 / 2 exp [ Δ 2 ( t ) / T c 2 ] sin [ ϕ x y ( t ) + ω 0 τ d ] .
J x x ( t ) = 1 | M | I ( t M ) ,
J y y ( t ) = 1 | M | I ( t M τ d ) ,
J x y ( t ) = 1 | M | [ I ( t M ) I ( t M τ d ) ] 1 / 2 exp ( i ω 0 τ d τ d 2 / T c 2 ) ,
S 0 ( t ) = 1 | M | [ I ( t M ) + I ( t M τ d ) ] ,
S 1 ( t ) = 1 | M | [ I ( t M ) I ( t M τ d ) ] ,
S 2 ( t ) = 2 | M | [ I ( t M ) I ( t M τ d ) ] 1 / 2 exp ( τ d 2 / T c 2 ) cos ( ω 0 τ d ) ,
S 3 ( t ) = 2 | M | [ I ( t M ) I ( t M τ d ) ] 1 / 2 exp ( τ d 2 / T c 2 ) sin ( ω 0 τ d ) ,
s 1 ( t ) = tanh [ ( t / M τ d / 2 ) τ d / T 0 2 ] ,
s 2 ( t ) = exp ( τ d 2 / T c 2 ) cosh [ ( t / M τ d / 2 ) τ d / T 0 2 ] cos ( ω 0 τ d ) ,
s 3 ( t ) = exp ( τ d 2 / T c 2 ) cosh [ ( t / M τ d / 2 ) τ d / T 0 2 ] sin ( ω 0 τ d ) .
P ( t ) = { tanh 2 [ ( τ d 2 t / M ) τ d 2 T 0 2 ] + sech 2 [ ( τ d 2 t / M ) τ d 2 T 0 2 ] exp ( 2 τ d 2 / T c 2 ) } 1 / 2 ,
ϕ x y ( t ) = ϕ q t 2 ,
ϕ q = M 1 1 2 ( 1 Φ b y 1 Φ b x )
J x y ( t ) = 1 | M | [ I ( t M ) I ( t M τ d ) ] 1 / 2 exp [ i ( ω 0 τ d + ϕ q t 2 ) τ d 2 / T c 2 ] .
S 2 ( t ) = 2 | M | [ I ( t M ) I ( t M τ d ) ] 1 / 2 exp ( τ d 2 / T c 2 ) cos ( ω 0 τ d + ϕ q t 2 ) ,
S 3 ( t ) = 2 | M | [ I ( t M ) I ( t M τ d ) ] 1 / 2 exp ( τ d 2 / T c 2 ) sin ( ω 0 τ d + ϕ q t 2 ) ,
s 2 ( t ) = exp ( τ d 2 / T c 2 ) cosh [ ( t / M τ d / 2 ) τ d / T 0 2 ] cos ( ω 0 τ d + ϕ q t 2 ) ,
s 3 ( t ) = exp ( τ d 2 / T c 2 ) cosh [ ( t / M τ d / 2 ) τ d / T 0 2 ] sin ( ω 0 τ d + ϕ q t 2 ) ,
J x x ( t ) = 1 | M | I ( t M ) ,
J y y ( t ) = 1 | M | I ( t M + τ d ) ,
J x y ( t ) = 1 | M | [ I ( t M ) I ( t M + τ d ) ] 1 / 2 exp { i [ ϕ x y ( t ) + ω 0 τ d ] ( 2 t / M + τ d ) 2 / T c 2 } .
S 0 ( t ) = 1 | M | [ I ( t M ) + I ( t M + τ d ) ] ,
S 1 ( t ) = 1 | M | [ I ( t M ) I ( t M + τ d ) ] ,
S 2 ( t ) = 2 | M | [ I ( t M ) I ( t M + τ d ) ] 1 / 2 exp [ ( 2 t / M + τ d ) 2 / T c 2 ] cos ( ϕ x y + ω 0 τ d ) ,
S 3 ( t ) = 2 | M | [ I ( t M ) I ( t M + τ d ) ] 1 / 2 exp [ ( 2 t / M + τ d ) 2 / T c 2 ] sin ( ϕ x y + ω 0 τ d ) ,
s 1 ( t ) = tanh [ ( t / M + τ d / 2 ) τ d / T 0 2 ] ,
s 2 ( t ) = exp [ ( 2 t / M + τ d ) 2 / T c 2 ] cosh [ ( t / M + τ d / 2 ) τ d / T 0 2 ] cos ( ϕ x y + ω 0 τ d ) ,
s 3 ( t ) = exp [ ( 2 t / M + τ d ) 2 / T c 2 ] cosh [ ( t / M + τ d / 2 ) τ d / T 0 2 ] sin ( ϕ x y + ω 0 τ d ) .

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