Abstract

Recently the optical transmission matrix (TM) has been shown to be useful in controlling the propagation of light in highly scattering media. In this paper, we present the vector transmission matrix (VTM) which, unlike the TM, captures both the intensity and polarization transmission property of the scattering medium. We present an experimental technique for measuring the absolute values of the VTM elements which is in contrast to existing techniques whereby the TM elements are measured to within a scaling factor. The usefulness of the VTM is illustrated by showing that it can be used to both predict and control the magnitude of the complex polarization ratio of the light focused through the scattering medium. To the best of our knowledge, this is the first study to show the possibility of controlling the polarization of the light transmitted through highly scattering media.

© 2012 OSA

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References

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  1. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
    [CrossRef] [PubMed]
  2. S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
    [CrossRef] [PubMed]
  3. S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
    [CrossRef] [PubMed]
  4. I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
    [CrossRef]
  5. S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2012).
    [CrossRef] [PubMed]
  6. D. B. Conkey, A. M. Caravaca-Aguirre, and R. Piestun, “High-speed scattering medium characterization with application to focusing light through turbid media,” Opt. Express 20, 1733–1740 (2012).
    [CrossRef] [PubMed]
  7. Y. Choi, T. Yang, C. Fang-Yen, P. Kang, K. Lee, R. Dasari, M. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett. 107, 023902 (2011).
    [CrossRef] [PubMed]
  8. A. Dolginov, Y. Gnedin, and N. Silant’ev, “Photon polarization and frequency change in multiple scattering,” J. Quantum Spectrosc. Ra. 10, 707–754 (1970).
    [CrossRef]
  9. T. Kohlgraf-Owens and A. Dogariu, “Finding the field transfer matrix of scattering media,” Opt. Express 16, 13225–13232 (2008).
    [CrossRef] [PubMed]
  10. T. Kohlgraf-Owens and A. Dogariu, “Transmission matrices of random media: Means for spectral polarimetric measurements,” Opt. Lett. 35, 2236–2238 (2010).
    [CrossRef] [PubMed]
  11. C. Prada and M. Fink, “Eigenmodes of the time reversal operator: A solution to selective focusing in multiple-target media,” Wave Motion 20, 151–163 (1994).
    [CrossRef]
  12. C. Prada and J.-L. Thomas, “Experimental subwavelength localization of scatterers by decomposition of the time reversal operator interpreted as a covariance matrix,” J. Acoust. Soc. Am. 114, 235–243 (2003).
    [CrossRef] [PubMed]
  13. J. Pasquesi, S. Schlachter, M. Boppart, E. Chaney, S. Kaufman, and S. Boppart, “In vivo detection of exercise-induced ultrastructural changes in genetically-altered murine skeletal muscle using polarization-sensitive optical coherence tomography,” Opt. Express 14, 1547–1556 (2006).
    [CrossRef] [PubMed]
  14. R. Rao, M. Mehta, and K. Toussaint, “Fourier transform-second-harmonic generation imaging of biological tissues,” Opt. Express 17, 14534–14542 (2009).
    [CrossRef] [PubMed]
  15. C. Krafft, B. Dietzek, and J. Popp, “Raman and cars microspectroscopy of cells and tissues,” Analyst 134, 1046–1057 (2009).
    [CrossRef] [PubMed]
  16. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley and Sons, 2007).
  17. R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2007).
  18. B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
    [CrossRef]
  19. R. H. Byrd, J. Nocedal, and R. A. Waltz, “Knitro: An integrated package for nonlinear optimization,” in Large Scale Nonlinear Optimization, G. D. Pillo and F. Giannessi, eds. (Springer Science+Business Media, 2006).
    [CrossRef]

2012 (2)

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2012).
[CrossRef] [PubMed]

D. B. Conkey, A. M. Caravaca-Aguirre, and R. Piestun, “High-speed scattering medium characterization with application to focusing light through turbid media,” Opt. Express 20, 1733–1740 (2012).
[CrossRef] [PubMed]

2011 (1)

Y. Choi, T. Yang, C. Fang-Yen, P. Kang, K. Lee, R. Dasari, M. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef] [PubMed]

2010 (4)

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[CrossRef] [PubMed]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

T. Kohlgraf-Owens and A. Dogariu, “Transmission matrices of random media: Means for spectral polarimetric measurements,” Opt. Lett. 35, 2236–2238 (2010).
[CrossRef] [PubMed]

2009 (2)

R. Rao, M. Mehta, and K. Toussaint, “Fourier transform-second-harmonic generation imaging of biological tissues,” Opt. Express 17, 14534–14542 (2009).
[CrossRef] [PubMed]

C. Krafft, B. Dietzek, and J. Popp, “Raman and cars microspectroscopy of cells and tissues,” Analyst 134, 1046–1057 (2009).
[CrossRef] [PubMed]

2008 (1)

2007 (2)

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
[CrossRef] [PubMed]

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

2006 (1)

2003 (1)

C. Prada and J.-L. Thomas, “Experimental subwavelength localization of scatterers by decomposition of the time reversal operator interpreted as a covariance matrix,” J. Acoust. Soc. Am. 114, 235–243 (2003).
[CrossRef] [PubMed]

1994 (1)

C. Prada and M. Fink, “Eigenmodes of the time reversal operator: A solution to selective focusing in multiple-target media,” Wave Motion 20, 151–163 (1994).
[CrossRef]

1970 (1)

A. Dolginov, Y. Gnedin, and N. Silant’ev, “Photon polarization and frequency change in multiple scattering,” J. Quantum Spectrosc. Ra. 10, 707–754 (1970).
[CrossRef]

Aubry, A.

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2012).
[CrossRef] [PubMed]

Barrett, D.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

Boccara, A. C.

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2012).
[CrossRef] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef] [PubMed]

Boppart, M.

Boppart, S.

Byrd, R. H.

R. H. Byrd, J. Nocedal, and R. A. Waltz, “Knitro: An integrated package for nonlinear optimization,” in Large Scale Nonlinear Optimization, G. D. Pillo and F. Giannessi, eds. (Springer Science+Business Media, 2006).
[CrossRef]

Caravaca-Aguirre, A. M.

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef] [PubMed]

Chaney, E.

Choi, W.

Y. Choi, T. Yang, C. Fang-Yen, P. Kang, K. Lee, R. Dasari, M. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef] [PubMed]

Choi, Y.

Y. Choi, T. Yang, C. Fang-Yen, P. Kang, K. Lee, R. Dasari, M. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef] [PubMed]

Collett, E.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

Conkey, D. B.

Dasari, R.

Y. Choi, T. Yang, C. Fang-Yen, P. Kang, K. Lee, R. Dasari, M. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef] [PubMed]

Dietzek, B.

C. Krafft, B. Dietzek, and J. Popp, “Raman and cars microspectroscopy of cells and tissues,” Analyst 134, 1046–1057 (2009).
[CrossRef] [PubMed]

Dogariu, A.

Dolginov, A.

A. Dolginov, Y. Gnedin, and N. Silant’ev, “Photon polarization and frequency change in multiple scattering,” J. Quantum Spectrosc. Ra. 10, 707–754 (1970).
[CrossRef]

Fang-Yen, C.

Y. Choi, T. Yang, C. Fang-Yen, P. Kang, K. Lee, R. Dasari, M. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef] [PubMed]

Feld, M.

Y. Choi, T. Yang, C. Fang-Yen, P. Kang, K. Lee, R. Dasari, M. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef] [PubMed]

Fink, M.

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2012).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[CrossRef] [PubMed]

C. Prada and M. Fink, “Eigenmodes of the time reversal operator: A solution to selective focusing in multiple-target media,” Wave Motion 20, 151–163 (1994).
[CrossRef]

Fraher, B.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

Gigan, S.

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2012).
[CrossRef] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef] [PubMed]

Gnedin, Y.

A. Dolginov, Y. Gnedin, and N. Silant’ev, “Photon polarization and frequency change in multiple scattering,” J. Quantum Spectrosc. Ra. 10, 707–754 (1970).
[CrossRef]

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2007).

Kang, P.

Y. Choi, T. Yang, C. Fang-Yen, P. Kang, K. Lee, R. Dasari, M. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef] [PubMed]

Kaufman, S.

Kohlgraf-Owens, T.

Krafft, C.

C. Krafft, B. Dietzek, and J. Popp, “Raman and cars microspectroscopy of cells and tissues,” Analyst 134, 1046–1057 (2009).
[CrossRef] [PubMed]

Lagendijk, A.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

Lee, K.

Y. Choi, T. Yang, C. Fang-Yen, P. Kang, K. Lee, R. Dasari, M. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef] [PubMed]

Lerosey, G.

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2012).
[CrossRef] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef] [PubMed]

Mehta, M.

Mosk, A. P.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
[CrossRef] [PubMed]

Nocedal, J.

R. H. Byrd, J. Nocedal, and R. A. Waltz, “Knitro: An integrated package for nonlinear optimization,” in Large Scale Nonlinear Optimization, G. D. Pillo and F. Giannessi, eds. (Springer Science+Business Media, 2006).
[CrossRef]

Pasquesi, J.

Piestun, R.

Popoff, S.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[CrossRef] [PubMed]

Popoff, S. M.

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2012).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef] [PubMed]

Popp, J.

C. Krafft, B. Dietzek, and J. Popp, “Raman and cars microspectroscopy of cells and tissues,” Analyst 134, 1046–1057 (2009).
[CrossRef] [PubMed]

Prada, C.

C. Prada and J.-L. Thomas, “Experimental subwavelength localization of scatterers by decomposition of the time reversal operator interpreted as a covariance matrix,” J. Acoust. Soc. Am. 114, 235–243 (2003).
[CrossRef] [PubMed]

C. Prada and M. Fink, “Eigenmodes of the time reversal operator: A solution to selective focusing in multiple-target media,” Wave Motion 20, 151–163 (1994).
[CrossRef]

Rao, R.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley and Sons, 2007).

Schaefer, B.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

Schlachter, S.

Silant’ev, N.

A. Dolginov, Y. Gnedin, and N. Silant’ev, “Photon polarization and frequency change in multiple scattering,” J. Quantum Spectrosc. Ra. 10, 707–754 (1970).
[CrossRef]

Smyth, R.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley and Sons, 2007).

Thomas, J.-L.

C. Prada and J.-L. Thomas, “Experimental subwavelength localization of scatterers by decomposition of the time reversal operator interpreted as a covariance matrix,” J. Acoust. Soc. Am. 114, 235–243 (2003).
[CrossRef] [PubMed]

Toussaint, K.

Vellekoop, I. M.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
[CrossRef] [PubMed]

Waltz, R. A.

R. H. Byrd, J. Nocedal, and R. A. Waltz, “Knitro: An integrated package for nonlinear optimization,” in Large Scale Nonlinear Optimization, G. D. Pillo and F. Giannessi, eds. (Springer Science+Business Media, 2006).
[CrossRef]

Woods, R. E.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2007).

Yang, T.

Y. Choi, T. Yang, C. Fang-Yen, P. Kang, K. Lee, R. Dasari, M. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef] [PubMed]

Am. J. Phys. (1)

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

Analyst (1)

C. Krafft, B. Dietzek, and J. Popp, “Raman and cars microspectroscopy of cells and tissues,” Analyst 134, 1046–1057 (2009).
[CrossRef] [PubMed]

J. Acoust. Soc. Am. (1)

C. Prada and J.-L. Thomas, “Experimental subwavelength localization of scatterers by decomposition of the time reversal operator interpreted as a covariance matrix,” J. Acoust. Soc. Am. 114, 235–243 (2003).
[CrossRef] [PubMed]

J. Quantum Spectrosc. Ra. (1)

A. Dolginov, Y. Gnedin, and N. Silant’ev, “Photon polarization and frequency change in multiple scattering,” J. Quantum Spectrosc. Ra. 10, 707–754 (1970).
[CrossRef]

Nat. Commun. (1)

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[CrossRef] [PubMed]

Nat. Photonics (1)

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. Lett. (3)

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2012).
[CrossRef] [PubMed]

Y. Choi, T. Yang, C. Fang-Yen, P. Kang, K. Lee, R. Dasari, M. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett. 107, 023902 (2011).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef] [PubMed]

Wave Motion (1)

C. Prada and M. Fink, “Eigenmodes of the time reversal operator: A solution to selective focusing in multiple-target media,” Wave Motion 20, 151–163 (1994).
[CrossRef]

Other (3)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley and Sons, 2007).

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2007).

R. H. Byrd, J. Nocedal, and R. A. Waltz, “Knitro: An integrated package for nonlinear optimization,” in Large Scale Nonlinear Optimization, G. D. Pillo and F. Giannessi, eds. (Springer Science+Business Media, 2006).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Experimental setup to measure the vector transmission matrix. Refer to text for details.

Fig. 2
Fig. 2

A simplified block diagram outlining the procedure followed in calculating the phase image for polarization optimization. A given phase profile is changed iteratively until the polarization predicted by the VTM corresponding to the phase profile is similar to the desired polarization. At each iteration, corrections to the phase image are calculated based on both the nature of the change in polarization error and the phase image from previous iterations.

Fig. 3
Fig. 3

Speckle field before and after optimization. An enhancement of 41× was observed with 256 control segments.

Fig. 4
Fig. 4

Plot of the observed versus predicted polarization state to illustrate the value of the vector transmission matrix in predicting the output polarization state.

Fig. 5
Fig. 5

Plot comparing the experimentally obtained (closed markers) and targeted (open markers) ratios of the intensities of the polarization components y and x. The dashed curve shows the trajectory of all possible ratios. Each ratio can represent any point on a unique circle on the Poincaré sphere; the inset shows the circle corresponding to the targeted polarization ratio of 1 outlined on the sphere.

Equations (8)

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( t 1 , 1 x x t 1 , M x x t 1 , 1 x y t 1 , M x y t N , 1 x x t N , M x x t N , 1 x y t N , M x y t 1 , 1 y x t 1 , M y x t 1 , 1 y y t 1 , M y y t N , 1 y x t N , M y x t N , 1 y y t N , M y y ) ( E 1 x ( I ) E M x ( I ) E 1 y ( I ) E M y ( I ) ) = ( E 1 x ( O ) E N x ( O ) E 1 y ( O ) E N y ( O ) ) ,
T n , m i j = I n , m R i I n , m S i e i Δ ϕ ,
I n , m R x = 1 2 ( S n , m R , 0 + S n , m R , 1 )
I n , m R y = 1 2 ( S n , m R , 0 S n , m R , 1 ) ,
φ = 1 2 ( S n , m R , 2 + i S n , m R , 3 ) .
t n , m x j = T n , m x j I n , m R x ,
t n , m y j = T n , m y j I n , m R y e i φ ,
Maximize E m i ( I ) I ( E m i ( I ) ) subject to { | R 2 | E n y ( O ) | 2 | E n x ( O ) | 2 | < ε r ( ϕ E n y ( O ) + E n x ( O ) ) mod 2 π < ε p .

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