Abstract

Laser speckle is generated by the multiple interference of light through a disordered medium. Here we study the premise that the speckle pattern retains information about the polarisation state of the incident field. We analytically verify that a linear relation exists between the Stokes vector of the light and the resulting speckle pattern. As a result, the polarisation state of a beam can be measured from the speckle pattern using a transmission matrix approach. We perform a quantitative analysis of the accuracy of the transmission matrix method to measure randomly time-varying polarisation states. In experiment, we find that the Stokes parameters of light from a diode laser can be retrieved with an uncertainty of 0.05 using speckle images of 150×150 pixels and 17 training states. We show both analytically and in experiment that this approach may be extended to the case of more than one laser field, demonstrating the measurement of the Stokes parameters of two laser beams simultaneously from a single speckle pattern and achieving the same uncertainty of 0.05.

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  9. M. Mazilu, T. Vettenburg, A. Di Falco, and K. Dholakia, “Random super-prism wavelength meter,” Opt. Lett. 39(1), 96–99 (2014).
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    [Crossref]
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    [Crossref]
  13. G. D. Bruce, L. O’Donnell, M. Chen, and K. Dholakia, “Overcoming the speckle correlation limit to achieve a fiber wavemeter with attometer resolution,” Opt. Lett. 44(6), 1367–1370 (2019).
    [Crossref]
  14. G. D. Bruce, L. O’Donnell, M. Chen, M. Facchin, and K. Dholakia, “Femtometer-resolved simultaneous measurement of multiple laser wavelengths in a speckle wavemeter,” Opt. Lett. 45(7), 1926–1929 (2020).
    [Crossref]
  15. L. O’Donnell, K. Dholakia, and G. D. Bruce, “High speed determination of laser wavelength using Poincaré descriptors of speckle,” Opt. Commun. 459, 124906 (2020).
    [Crossref]
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    [Crossref]
  18. B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013).
    [Crossref]
  19. B. Redding, S. M. Popoff, and H. Cao, “All-fiber spectrometer based on speckle pattern reconstruction,” Opt. Express 21(5), 6584–6600 (2013).
    [Crossref]
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    [Crossref]
  21. A. Mourka, M. Mazilu, E. Wright, and K. Dholakia, “Modal characterization using principal component analysis: application to bessel, higher-order gaussian beams and their superposition,” Sci. Rep. 3(1), 1422 (2013).
    [Crossref]
  22. M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100(23), 231115 (2012).
    [Crossref]
  23. I. Freund, “Stokes-vector reconstruction,” Opt. Lett. 15(24), 1425–1427 (1990).
    [Crossref]
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    [Crossref]
  29. R. Azzam, E. Masetti, I. Elminyawi, and F. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59(1), 84–88 (1988).
    [Crossref]
  30. Y. Arita, E. M. Wright, and K. Dholakia, “Optical binding of two cooled micro-gyroscopes levitated in vacuum,” Optica 5(8), 910–917 (2018).
    [Crossref]

2020 (2)

G. D. Bruce, L. O’Donnell, M. Chen, M. Facchin, and K. Dholakia, “Femtometer-resolved simultaneous measurement of multiple laser wavelengths in a speckle wavemeter,” Opt. Lett. 45(7), 1926–1929 (2020).
[Crossref]

L. O’Donnell, K. Dholakia, and G. D. Bruce, “High speed determination of laser wavelength using Poincaré descriptors of speckle,” Opt. Commun. 459, 124906 (2020).
[Crossref]

2019 (1)

2018 (1)

2017 (2)

N. K. Metzger, R. Spesyvtsev, G. D. Bruce, B. Miller, G. T. Maker, G. Malcolm, M. Mazilu, and K. Dholakia, “Harnessing speckle for a sub-femtometre resolved broadband wavemeter and laser stabilization,” Nat. Commun. 8(1), 15610 (2017).
[Crossref]

H. Cao, “Perspective on speckle spectrometers,” J. Opt. 19(6), 060402 (2017).
[Crossref]

2015 (1)

2014 (2)

2013 (4)

D. Briers, D. D. Duncan, E. R. Hirst, S. J. Kirkpatrick, M. Larsson, W. Steenbergen, T. Stromberg, and O. B. Thompson, “Laser speckle contrast imaging: theoretical and practical limitations,” J. Biomed. Opt. 18(6), 066018 (2013).
[Crossref]

A. Mourka, M. Mazilu, E. Wright, and K. Dholakia, “Modal characterization using principal component analysis: application to bessel, higher-order gaussian beams and their superposition,” Sci. Rep. 3(1), 1422 (2013).
[Crossref]

B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013).
[Crossref]

B. Redding, S. M. Popoff, and H. Cao, “All-fiber spectrometer based on speckle pattern reconstruction,” Opt. Express 21(5), 6584–6600 (2013).
[Crossref]

2012 (2)

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100(23), 231115 (2012).
[Crossref]

B. Redding and H. Cao, “Using a multimode fiber as a high-resolution, low-loss spectrometer,” Opt. Lett. 37(16), 3384–3386 (2012).
[Crossref]

2010 (2)

T. W. Kohlgraf-Owens and A. Dogariu, “Transmission matrices of random media: means for spectral polarimetric measurements,” Opt. Lett. 35(13), 2236–2238 (2010).
[Crossref]

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. 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(10), 100601 (2010).
[Crossref]

2009 (2)

2008 (1)

2006 (1)

1990 (2)

I. Freund and R. Berkovits, “Surface reflections and optical transport through random media: coherent backscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B 41(1), 496–503 (1990).
[Crossref]

I. Freund, “Stokes-vector reconstruction,” Opt. Lett. 15(24), 1425–1427 (1990).
[Crossref]

1988 (1)

R. Azzam, E. Masetti, I. Elminyawi, and F. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59(1), 84–88 (1988).
[Crossref]

1987 (1)

1970 (1)

E. Archbold, J. Burch, and A. Ennos, “Recording of in-plane surface displacement by double-exposure speckle photography,” Opt. Acta 17(12), 883–898 (1970).
[Crossref]

1968 (1)

J. Burch and J. Tokarski, “Production of multiple beam fringes from photographic scatterers,” Opt. Acta 15(2), 101–111 (1968).
[Crossref]

Archbold, E.

E. Archbold, J. Burch, and A. Ennos, “Recording of in-plane surface displacement by double-exposure speckle photography,” Opt. Acta 17(12), 883–898 (1970).
[Crossref]

Arita, Y.

Azzam, R.

R. Azzam, E. Masetti, I. Elminyawi, and F. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59(1), 84–88 (1988).
[Crossref]

Beiderman, Y.

Berkovits, R.

I. Freund and R. Berkovits, “Surface reflections and optical transport through random media: coherent backscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B 41(1), 496–503 (1990).
[Crossref]

Bianchi, S.

Boccara, A.

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. 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(10), 100601 (2010).
[Crossref]

Briers, D.

D. Briers, D. D. Duncan, E. R. Hirst, S. J. Kirkpatrick, M. Larsson, W. Steenbergen, T. Stromberg, and O. B. Thompson, “Laser speckle contrast imaging: theoretical and practical limitations,” J. Biomed. Opt. 18(6), 066018 (2013).
[Crossref]

Bruce, G. D.

G. D. Bruce, L. O’Donnell, M. Chen, M. Facchin, and K. Dholakia, “Femtometer-resolved simultaneous measurement of multiple laser wavelengths in a speckle wavemeter,” Opt. Lett. 45(7), 1926–1929 (2020).
[Crossref]

L. O’Donnell, K. Dholakia, and G. D. Bruce, “High speed determination of laser wavelength using Poincaré descriptors of speckle,” Opt. Commun. 459, 124906 (2020).
[Crossref]

G. D. Bruce, L. O’Donnell, M. Chen, and K. Dholakia, “Overcoming the speckle correlation limit to achieve a fiber wavemeter with attometer resolution,” Opt. Lett. 44(6), 1367–1370 (2019).
[Crossref]

N. K. Metzger, R. Spesyvtsev, G. D. Bruce, B. Miller, G. T. Maker, G. Malcolm, M. Mazilu, and K. Dholakia, “Harnessing speckle for a sub-femtometre resolved broadband wavemeter and laser stabilization,” Nat. Commun. 8(1), 15610 (2017).
[Crossref]

R. K. Gupta, G. D. Bruce, S. J. Powis, and K. Dholakia, “Deep learning enabled laser speckle wavemeter with a high dynamic range,” arXiv: 1910.10702 (2019).

Burch, J.

E. Archbold, J. Burch, and A. Ennos, “Recording of in-plane surface displacement by double-exposure speckle photography,” Opt. Acta 17(12), 883–898 (1970).
[Crossref]

J. Burch and J. Tokarski, “Production of multiple beam fringes from photographic scatterers,” Opt. Acta 15(2), 101–111 (1968).
[Crossref]

Cao, H.

H. Cao, “Perspective on speckle spectrometers,” J. Opt. 19(6), 060402 (2017).
[Crossref]

B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013).
[Crossref]

B. Redding, S. M. Popoff, and H. Cao, “All-fiber spectrometer based on speckle pattern reconstruction,” Opt. Express 21(5), 6584–6600 (2013).
[Crossref]

B. Redding and H. Cao, “Using a multimode fiber as a high-resolution, low-loss spectrometer,” Opt. Lett. 37(16), 3384–3386 (2012).
[Crossref]

Carminati, R.

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. 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(10), 100601 (2010).
[Crossref]

Chakrabarti, M.

M. Chakrabarti, M. L. Jakobsen, and S. G. Hanson, “Speckle-based spectrometer,” Opt. Lett. 40(14), 3264–3267 (2015).
[Crossref]

S. G. Hanson, M. L. Jakobsen, and M. Chakrabarti, “Speckle-based wavemeter,” in SPECKLE 2015: VI International Conference on Speckle Metrology, vol. 9660 (International Society for Optics and Photonics, 2015), p. 96600U.

Chen, M.

Dholakia, K.

G. D. Bruce, L. O’Donnell, M. Chen, M. Facchin, and K. Dholakia, “Femtometer-resolved simultaneous measurement of multiple laser wavelengths in a speckle wavemeter,” Opt. Lett. 45(7), 1926–1929 (2020).
[Crossref]

L. O’Donnell, K. Dholakia, and G. D. Bruce, “High speed determination of laser wavelength using Poincaré descriptors of speckle,” Opt. Commun. 459, 124906 (2020).
[Crossref]

G. D. Bruce, L. O’Donnell, M. Chen, and K. Dholakia, “Overcoming the speckle correlation limit to achieve a fiber wavemeter with attometer resolution,” Opt. Lett. 44(6), 1367–1370 (2019).
[Crossref]

Y. Arita, E. M. Wright, and K. Dholakia, “Optical binding of two cooled micro-gyroscopes levitated in vacuum,” Optica 5(8), 910–917 (2018).
[Crossref]

N. K. Metzger, R. Spesyvtsev, G. D. Bruce, B. Miller, G. T. Maker, G. Malcolm, M. Mazilu, and K. Dholakia, “Harnessing speckle for a sub-femtometre resolved broadband wavemeter and laser stabilization,” Nat. Commun. 8(1), 15610 (2017).
[Crossref]

M. Mazilu, T. Vettenburg, A. Di Falco, and K. Dholakia, “Random super-prism wavelength meter,” Opt. Lett. 39(1), 96–99 (2014).
[Crossref]

A. Mourka, M. Mazilu, E. Wright, and K. Dholakia, “Modal characterization using principal component analysis: application to bessel, higher-order gaussian beams and their superposition,” Sci. Rep. 3(1), 1422 (2013).
[Crossref]

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100(23), 231115 (2012).
[Crossref]

R. K. Gupta, G. D. Bruce, S. J. Powis, and K. Dholakia, “Deep learning enabled laser speckle wavemeter with a high dynamic range,” arXiv: 1910.10702 (2019).

Di Falco, A.

Dogariu, A.

Duncan, D. D.

D. Briers, D. D. Duncan, E. R. Hirst, S. J. Kirkpatrick, M. Larsson, W. Steenbergen, T. Stromberg, and O. B. Thompson, “Laser speckle contrast imaging: theoretical and practical limitations,” J. Biomed. Opt. 18(6), 066018 (2013).
[Crossref]

Elminyawi, I.

R. Azzam, E. Masetti, I. Elminyawi, and F. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59(1), 84–88 (1988).
[Crossref]

Ennos, A.

E. Archbold, J. Burch, and A. Ennos, “Recording of in-plane surface displacement by double-exposure speckle photography,” Opt. Acta 17(12), 883–898 (1970).
[Crossref]

Facchin, M.

Fink, M.

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. 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(10), 100601 (2010).
[Crossref]

Freund, I.

I. Freund, “Stokes-vector reconstruction,” Opt. Lett. 15(24), 1425–1427 (1990).
[Crossref]

I. Freund and R. Berkovits, “Surface reflections and optical transport through random media: coherent backscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B 41(1), 496–503 (1990).
[Crossref]

Garcia, J.

Gigan, S.

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. 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(10), 100601 (2010).
[Crossref]

Gingold, S.

Grosz, F.

R. Azzam, E. Masetti, I. Elminyawi, and F. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59(1), 84–88 (1988).
[Crossref]

Gupta, R. K.

R. K. Gupta, G. D. Bruce, S. J. Powis, and K. Dholakia, “Deep learning enabled laser speckle wavemeter with a high dynamic range,” arXiv: 1910.10702 (2019).

Hanson, S. G.

Hirst, E. R.

D. Briers, D. D. Duncan, E. R. Hirst, S. J. Kirkpatrick, M. Larsson, W. Steenbergen, T. Stromberg, and O. B. Thompson, “Laser speckle contrast imaging: theoretical and practical limitations,” J. Biomed. Opt. 18(6), 066018 (2013).
[Crossref]

Ishijima, R.

Jakobsen, M. L.

M. Chakrabarti, M. L. Jakobsen, and S. G. Hanson, “Speckle-based spectrometer,” Opt. Lett. 40(14), 3264–3267 (2015).
[Crossref]

S. G. Hanson, M. L. Jakobsen, and M. Chakrabarti, “Speckle-based wavemeter,” in SPECKLE 2015: VI International Conference on Speckle Metrology, vol. 9660 (International Society for Optics and Photonics, 2015), p. 96600U.

Joseph W., G.

G. Joseph W., Speckle Phenomena in Optics (Roberts and Company Publishers, 2007).

Kim, K.

Kirkpatrick, S. J.

D. Briers, D. D. Duncan, E. R. Hirst, S. J. Kirkpatrick, M. Larsson, W. Steenbergen, T. Stromberg, and O. B. Thompson, “Laser speckle contrast imaging: theoretical and practical limitations,” J. Biomed. Opt. 18(6), 066018 (2013).
[Crossref]

Kohlgraf-Owens, T.

Kohlgraf-Owens, T. W.

Larsson, M.

D. Briers, D. D. Duncan, E. R. Hirst, S. J. Kirkpatrick, M. Larsson, W. Steenbergen, T. Stromberg, and O. B. Thompson, “Laser speckle contrast imaging: theoretical and practical limitations,” J. Biomed. Opt. 18(6), 066018 (2013).
[Crossref]

Lerosey, G.

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. 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(10), 100601 (2010).
[Crossref]

Liew, S. F.

B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013).
[Crossref]

Maker, G. T.

N. K. Metzger, R. Spesyvtsev, G. D. Bruce, B. Miller, G. T. Maker, G. Malcolm, M. Mazilu, and K. Dholakia, “Harnessing speckle for a sub-femtometre resolved broadband wavemeter and laser stabilization,” Nat. Commun. 8(1), 15610 (2017).
[Crossref]

Malcolm, G.

N. K. Metzger, R. Spesyvtsev, G. D. Bruce, B. Miller, G. T. Maker, G. Malcolm, M. Mazilu, and K. Dholakia, “Harnessing speckle for a sub-femtometre resolved broadband wavemeter and laser stabilization,” Nat. Commun. 8(1), 15610 (2017).
[Crossref]

Mandel, L.

Margalit, I.

Masetti, E.

R. Azzam, E. Masetti, I. Elminyawi, and F. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59(1), 84–88 (1988).
[Crossref]

Mazilu, M.

N. K. Metzger, R. Spesyvtsev, G. D. Bruce, B. Miller, G. T. Maker, G. Malcolm, M. Mazilu, and K. Dholakia, “Harnessing speckle for a sub-femtometre resolved broadband wavemeter and laser stabilization,” Nat. Commun. 8(1), 15610 (2017).
[Crossref]

M. Mazilu, T. Vettenburg, A. Di Falco, and K. Dholakia, “Random super-prism wavelength meter,” Opt. Lett. 39(1), 96–99 (2014).
[Crossref]

A. Mourka, M. Mazilu, E. Wright, and K. Dholakia, “Modal characterization using principal component analysis: application to bessel, higher-order gaussian beams and their superposition,” Sci. Rep. 3(1), 1422 (2013).
[Crossref]

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100(23), 231115 (2012).
[Crossref]

Metzger, N. K.

N. K. Metzger, R. Spesyvtsev, G. D. Bruce, B. Miller, G. T. Maker, G. Malcolm, M. Mazilu, and K. Dholakia, “Harnessing speckle for a sub-femtometre resolved broadband wavemeter and laser stabilization,” Nat. Commun. 8(1), 15610 (2017).
[Crossref]

Mico, V.

Miller, B.

N. K. Metzger, R. Spesyvtsev, G. D. Bruce, B. Miller, G. T. Maker, G. Malcolm, M. Mazilu, and K. Dholakia, “Harnessing speckle for a sub-femtometre resolved broadband wavemeter and laser stabilization,” Nat. Commun. 8(1), 15610 (2017).
[Crossref]

Mourka, A.

A. Mourka, M. Mazilu, E. Wright, and K. Dholakia, “Modal characterization using principal component analysis: application to bessel, higher-order gaussian beams and their superposition,” Sci. Rep. 3(1), 1422 (2013).
[Crossref]

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100(23), 231115 (2012).
[Crossref]

O’Donnell, L.

Popoff, S.

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. 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(10), 100601 (2010).
[Crossref]

Popoff, S. M.

Powis, S. J.

R. K. Gupta, G. D. Bruce, S. J. Powis, and K. Dholakia, “Deep learning enabled laser speckle wavemeter with a high dynamic range,” arXiv: 1910.10702 (2019).

Redding, B.

Sarma, R.

B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013).
[Crossref]

Spesyvtsev, R.

N. K. Metzger, R. Spesyvtsev, G. D. Bruce, B. Miller, G. T. Maker, G. Malcolm, M. Mazilu, and K. Dholakia, “Harnessing speckle for a sub-femtometre resolved broadband wavemeter and laser stabilization,” Nat. Commun. 8(1), 15610 (2017).
[Crossref]

Steenbergen, W.

D. Briers, D. D. Duncan, E. R. Hirst, S. J. Kirkpatrick, M. Larsson, W. Steenbergen, T. Stromberg, and O. B. Thompson, “Laser speckle contrast imaging: theoretical and practical limitations,” J. Biomed. Opt. 18(6), 066018 (2013).
[Crossref]

Stromberg, T.

D. Briers, D. D. Duncan, E. R. Hirst, S. J. Kirkpatrick, M. Larsson, W. Steenbergen, T. Stromberg, and O. B. Thompson, “Laser speckle contrast imaging: theoretical and practical limitations,” J. Biomed. Opt. 18(6), 066018 (2013).
[Crossref]

Takeda, M.

Teicher, M.

Thompson, O. B.

D. Briers, D. D. Duncan, E. R. Hirst, S. J. Kirkpatrick, M. Larsson, W. Steenbergen, T. Stromberg, and O. B. Thompson, “Laser speckle contrast imaging: theoretical and practical limitations,” J. Biomed. Opt. 18(6), 066018 (2013).
[Crossref]

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J. Burch and J. Tokarski, “Production of multiple beam fringes from photographic scatterers,” Opt. Acta 15(2), 101–111 (1968).
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Vettenburg, T.

M. Mazilu, T. Vettenburg, A. Di Falco, and K. Dholakia, “Random super-prism wavelength meter,” Opt. Lett. 39(1), 96–99 (2014).
[Crossref]

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100(23), 231115 (2012).
[Crossref]

Wang, W.

Wolf, E.

Wright, E.

A. Mourka, M. Mazilu, E. Wright, and K. Dholakia, “Modal characterization using principal component analysis: application to bessel, higher-order gaussian beams and their superposition,” Sci. Rep. 3(1), 1422 (2013).
[Crossref]

Wright, E. M.

Y. Arita, E. M. Wright, and K. Dholakia, “Optical binding of two cooled micro-gyroscopes levitated in vacuum,” Optica 5(8), 910–917 (2018).
[Crossref]

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100(23), 231115 (2012).
[Crossref]

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Appl. Opt. (1)

Appl. Phys. Lett. (1)

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100(23), 231115 (2012).
[Crossref]

J. Biomed. Opt. (1)

D. Briers, D. D. Duncan, E. R. Hirst, S. J. Kirkpatrick, M. Larsson, W. Steenbergen, T. Stromberg, and O. B. Thompson, “Laser speckle contrast imaging: theoretical and practical limitations,” J. Biomed. Opt. 18(6), 066018 (2013).
[Crossref]

J. Opt. (1)

H. Cao, “Perspective on speckle spectrometers,” J. Opt. 19(6), 060402 (2017).
[Crossref]

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

Nat. Commun. (1)

N. K. Metzger, R. Spesyvtsev, G. D. Bruce, B. Miller, G. T. Maker, G. Malcolm, M. Mazilu, and K. Dholakia, “Harnessing speckle for a sub-femtometre resolved broadband wavemeter and laser stabilization,” Nat. Commun. 8(1), 15610 (2017).
[Crossref]

Nat. Photonics (1)

B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013).
[Crossref]

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[Crossref]

J. Burch and J. Tokarski, “Production of multiple beam fringes from photographic scatterers,” Opt. Acta 15(2), 101–111 (1968).
[Crossref]

Opt. Commun. (1)

L. O’Donnell, K. Dholakia, and G. D. Bruce, “High speed determination of laser wavelength using Poincaré descriptors of speckle,” Opt. Commun. 459, 124906 (2020).
[Crossref]

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Opt. Lett. (8)

Optica (1)

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Sci. Rep. (1)

A. Mourka, M. Mazilu, E. Wright, and K. Dholakia, “Modal characterization using principal component analysis: application to bessel, higher-order gaussian beams and their superposition,” Sci. Rep. 3(1), 1422 (2013).
[Crossref]

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Figures (7)

Fig. 1.
Fig. 1. Diffusion geometry. An input laser beam is incident on a single rough surface, which diffuses the light in the same half-space. The diffused light is collected on a surface denoted as observation plane.
Fig. 2.
Fig. 2. Polarisation measurement setup. A laser beam passes through three waveplates, rotating with incommensurate angular speeds, enabling a randomly time-varying state of polarisation. The light is split into two paths using a non-polarising beam splitter (BS): on one path the state of polarisation is measured using a commercial polarimeter, on the other path the laser is diffused on a single highly-reflective rough surface, and the produced speckle pattern is recorded on a CMOS camera. A $150\times 150$ -pixel image of a speckle pattern is shown, with each pixel being 8 $\mu$ m $\times$ 8 $\mu$ m. The scale bar denotes 50 pixels (0.4 mm), and the colour bar shows the intensity normalised to maximum. For the two-beam version of the experiment (see section 5), a second laser joins the optical path via a pellicle beam splitter (grey), after passing through a waveplate so that the state of polarisation of laser 2 is different to that of laser 1 before passing through the three waveplates.
Fig. 3.
Fig. 3. Single-beam polarisation measurement. (a) Poincaré sphere representation of the 17 training states. (b) Trajectory of the polarisation state across the Poincaré sphere from t=160 s to t=190 s, measured by the commercial polarimeter (black) and retrieved from the speckle patterns (red). (c-e) The Stokes parameters $S_{1}$ to $S_{3}$ as a function of time, measured by the commercial polarimeter (black) and retrieved from the speckle patterns (red). (f) Measurement error. For convenience, we display the error as the absolute residual, averaged over the Stokes parameters, so that it can be plotted in a single graph. The estimation was performed using 150 $\times$ 150-pixel images and 17 training states.
Fig. 4.
Fig. 4. Measurement uncertainty for a single beam. The uncertainty is given by the standard deviation of the residuals. It is shown as a function of the number of training images (four being the minimum required), for different image sizes ranging from $20\times 20$ to $150\times 150$ pixels. We see that the uncertainty reaches a minimum of 0.05 after about 15 training images and an image size of 100 $\times$ 100 pixels.
Fig. 5.
Fig. 5. Two-beam polarisation measurement. The Stokes parameters $S_{1}$ to $S_{3}$ are given as a function of time, measured by the commercial polarimeter (black) and retrieved from the speckle patterns (red and blue, one for each beam). The estimation was performed using 150 $\times$ 150-pixel images and 17 training states.
Fig. 6.
Fig. 6. Measurement uncertainty for two beams. The uncertainty is given by the standard deviation of the residuals. It is shown as a function of the number of training images (eight being the minimum required), for different image sizes ranging from $20\times 20$ to $150\times 150$ pixels. Here the uncertainty also reaches a minimum of 0.05 after about 15 training images.
Fig. 7.
Fig. 7. Speed and sampling regularity. Left column: The Stokes parameters $S_{1}$ to $S_{3}$ as a function of time, measured by the commercial polarimeter (black) and retrieved from the speckle patterns (red), when a 10 Hz modulation is applied. The acquisition rates are respectively 150 Hz (on average) and 1000 Hz. Right column: The Stokes parameters as a function of time retrieved from the speckle patterns, when a 500 Hz modulation is applied. At this point the modulation is no longer visible on the commercial polarimeter.

Equations (13)

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E i j = E i α i j ,
E j = i E i α i j .
E j = i e ρ i e i ϕ i + i φ ( t ) α i j = e ( i ρ i e i ϕ i α i j ) e i φ ( t ) = e α j e i φ ( t ) ,
C j = E j E j = E j E j = ( e α j e i φ ( t ) ) ( e α j e i φ ( t ) ) = ( α j e ) ( e α j ) = α j ( e e ) α j = α j ( e e ) α j = α j C 0 α j ,
S j m = T r ( C j σ m ) = T r ( α j C 0 α j σ m ) = T r ( α j n 1 2 S n σ n α j σ m ) = n S n 1 2 T r ( α j σ n α j σ m ) S j = S M j .
I = S M ,
I 0 = S 0 M ,
M = S 0 + I 0 ,
S = I M + .
C j = E j E j = ( e 1 α 1 , j e i φ 1 ( t ) + e 2 α 2 , j e i φ 2 ( t ) ) ( e 1 α 1 , j e i φ 1 ( t ) + e 2 α 2 , j e i φ 2 ( t ) ) = α 1 , j ( e 1 e 1 ) α 1 , j + α 2 , j ( e 2 e 2 ) α 2 , j = α 1 , j C 1 α 1 , j + α 2 , j C 2 α 2 , j ,
S j m = T r ( C j σ m ) = T r ( α j C 1 α j σ m + α j C 2 α j σ m ) = T r ( α j C 1 α j σ m ) + T r ( α j C 2 α j σ m ) S j = S 1 M 1 , j + S 2 M 2 , j ,
I j = S ¯ M ¯ j ,
I = S ¯ M ¯ ,

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