Abstract

In this paper we present an experimental setup based on laser optical feedback imaging (LOFI) and on synthetic aperture with translational scanning by galvanometric mirrors for the purpose of making deep and resolved images through scattering media. We provide real two-dimensional optical synthetic aperture image of a fixed scattering target with a moving aperture and an isotropic resolution. We demonstrate theoretically and experimentally that we can keep microscope resolution beyond the working distance. A photometric balance is made, and we show that the number of photons participating in the final image decreases with the square of the reconstruction distance. This degradation is partially compensated by the high sensitivity of LOFI.

© 2012 Optical Society of America

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    [CrossRef]
  2. M. Pernot, J.-F. Aubry, M. Tanter, A.-L. Boch, F. Marquet, M. Kujas, D. Seilhean, and M. Fink, “In vivo transcranial brain surgery with an ultrasonic time reversal mirror,” J. Neurosurg. 106, 1061–1066 (2007).
    [CrossRef]
  3. I. M. Vellekoop and C. M. Aegerter, “Scattered light fluorescence microscopy: imaging through turbid layers,” Opt. Lett. 35, 1245–1247 (2010).
    [CrossRef]
  4. A. Dubois and C. Boccara, “OCT plein champ,” Med. Sci. 22, 859–864 (2006).
  5. P. Pantazis, J. Maloney, D. Wu, and S. E. Fraser, “Second harmonic generating (SHG) nanoprobes for in vivo imaging,” Proc. Natl. Acad. Sci. U.S.A. 107, 14535–14540 (2010).
    [CrossRef]
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  7. S. Vertu, J. Flugge, J. J. Delaunay, and O. Haeberle, “Improved and isotropic resolution in tomographic driffractive microscopy combining sample and illumination rotation,” Central Eur. J. Phys. 44, 969–974 (2011).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2012 (1)

2011 (4)

O. Hugon, F. Joud, E. Lacot, O. Jacquin, and H. Guillet de Chatellus, “Coherent microscopy by laser optical feedback imaging (LOFI) technique,” Ultramicroscopy 111, 1557–1563 (2011).
[CrossRef]

S. Vertu, J. Flugge, J. J. Delaunay, and O. Haeberle, “Improved and isotropic resolution in tomographic driffractive microscopy combining sample and illumination rotation,” Central Eur. J. Phys. 44, 969–974 (2011).
[CrossRef]

K. Otsuka, “Self-mixing thin-slice solid-state laser metrology,” Sensors 11, 2195–2245 (2011).
[CrossRef]

O. Jacquin, E. Lacot, W. Glastre, O. Hugon, and H. Guillet de Chatellus, “Experimental comparison of autodyne and heterodyne laser interferometry using Nd:YVO4 microchip laser,” J. Opt. Soc. Am. A 28, 1741–1746 (2011).
[CrossRef]

2010 (4)

E. Lacot, O. Jacquin, G. Roussely, O. Hugon, and H. Guillet de Chatellus, “Comparative study of autodyne and heterodyne laser interferometry for imaging,” J. Opt. Soc. Am. A 27, 2450–2458 (2010).
[CrossRef]

P. Pantazis, J. Maloney, D. Wu, and S. E. Fraser, “Second harmonic generating (SHG) nanoprobes for in vivo imaging,” Proc. Natl. Acad. Sci. U.S.A. 107, 14535–14540 (2010).
[CrossRef]

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]

I. M. Vellekoop and C. M. Aegerter, “Scattered light fluorescence microscopy: imaging through turbid layers,” Opt. Lett. 35, 1245–1247 (2010).
[CrossRef]

2008 (1)

2007 (1)

M. Pernot, J.-F. Aubry, M. Tanter, A.-L. Boch, F. Marquet, M. Kujas, D. Seilhean, and M. Fink, “In vivo transcranial brain surgery with an ultrasonic time reversal mirror,” J. Neurosurg. 106, 1061–1066 (2007).
[CrossRef]

2006 (3)

2005 (1)

2003 (1)

O. Hugon, E. Lacot, and F. Stoeckel, “Submicrometric displacement and vibration measurement using optical feedback in a fiber laser,” Fib. Integr. Opt. 22, 283–288 (2003).
[CrossRef]

2002 (1)

2001 (1)

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815 (2001).
[CrossRef]

1999 (1)

1995 (1)

1994 (1)

1988 (1)

M. Minsky, “Memoir on inventing the confocal scanning microscope,” Scanning 10, 128–138 (1988).
[CrossRef]

1987 (1)

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser,” Proc. SPIE 783, 29–40 (1987). http://adsabs.harvard.edu/abs/1987SPIE..783...29A

Abshier, J. O.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser,” Proc. SPIE 783, 29–40 (1987). http://adsabs.harvard.edu/abs/1987SPIE..783...29A

Accetta, J. S.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser,” Proc. SPIE 783, 29–40 (1987). http://adsabs.harvard.edu/abs/1987SPIE..783...29A

Aegerter, C. M.

Aleksoff, C. C.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser,” Proc. SPIE 783, 29–40 (1987). http://adsabs.harvard.edu/abs/1987SPIE..783...29A

Aubry, J.-F.

M. Pernot, J.-F. Aubry, M. Tanter, A.-L. Boch, F. Marquet, M. Kujas, D. Seilhean, and M. Fink, “In vivo transcranial brain surgery with an ultrasonic time reversal mirror,” J. Neurosurg. 106, 1061–1066 (2007).
[CrossRef]

Bashkansky, M.

Beck, S. M.

Bleistein, N.

N. Bleistein and R. Handelsman, “Fourier integrals and the method of stationary phase,” in Asymptotic Expansions of Integrals (Dover, 1975), pp. 219–223.

Boccara, A. C.

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]

Boccara, C.

A. Dubois and C. Boccara, “OCT plein champ,” Med. Sci. 22, 859–864 (2006).

Boch, A.-L.

M. Pernot, J.-F. Aubry, M. Tanter, A.-L. Boch, F. Marquet, M. Kujas, D. Seilhean, and M. Fink, “In vivo transcranial brain surgery with an ultrasonic time reversal mirror,” J. Neurosurg. 106, 1061–1066 (2007).
[CrossRef]

Buck, J. R.

Buell, W. F.

Callens, N.

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]

Colella, B. D.

Curlander, J. C.

J. C. Curlander and R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, 1991).

Day, R.

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815 (2001).
[CrossRef]

E. Lacot, R. Day, and F. Stoeckel, “Laser optical feedback tomography,” Opt. Lett. 24, 744–746 (1999).
[CrossRef]

Delaunay, J. J.

S. Vertu, J. Flugge, J. J. Delaunay, and O. Haeberle, “Improved and isotropic resolution in tomographic driffractive microscopy combining sample and illumination rotation,” Central Eur. J. Phys. 44, 969–974 (2011).
[CrossRef]

Dickinson, R. P.

Dubois, A.

A. Dubois and C. Boccara, “OCT plein champ,” Med. Sci. 22, 859–864 (2006).

Dubois, F.

Fee, M.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser,” Proc. SPIE 783, 29–40 (1987). http://adsabs.harvard.edu/abs/1987SPIE..783...29A

Fink, M.

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]

M. Pernot, J.-F. Aubry, M. Tanter, A.-L. Boch, F. Marquet, M. Kujas, D. Seilhean, and M. Fink, “In vivo transcranial brain surgery with an ultrasonic time reversal mirror,” J. Neurosurg. 106, 1061–1066 (2007).
[CrossRef]

Flugge, J.

S. Vertu, J. Flugge, J. J. Delaunay, and O. Haeberle, “Improved and isotropic resolution in tomographic driffractive microscopy combining sample and illumination rotation,” Central Eur. J. Phys. 44, 969–974 (2011).
[CrossRef]

Fraser, S. E.

P. Pantazis, J. Maloney, D. Wu, and S. E. Fraser, “Second harmonic generating (SHG) nanoprobes for in vivo imaging,” Proc. Natl. Acad. Sci. U.S.A. 107, 14535–14540 (2010).
[CrossRef]

Funk, E.

Gigan, S.

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]

Glastre, W.

Green, T. J.

Guillet de Chatellus, H.

Haeberle, O.

S. Vertu, J. Flugge, J. J. Delaunay, and O. Haeberle, “Improved and isotropic resolution in tomographic driffractive microscopy combining sample and illumination rotation,” Central Eur. J. Phys. 44, 969–974 (2011).
[CrossRef]

Handelsman, R.

N. Bleistein and R. Handelsman, “Fourier integrals and the method of stationary phase,” in Asymptotic Expansions of Integrals (Dover, 1975), pp. 219–223.

Hugon, O.

Jacquin, O.

Joud, F.

O. Hugon, F. Joud, E. Lacot, O. Jacquin, and H. Guillet de Chatellus, “Coherent microscopy by laser optical feedback imaging (LOFI) technique,” Ultramicroscopy 111, 1557–1563 (2011).
[CrossRef]

Klooster, A.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser,” Proc. SPIE 783, 29–40 (1987). http://adsabs.harvard.edu/abs/1987SPIE..783...29A

Kozlowski, D. A.

Kujas, M.

M. Pernot, J.-F. Aubry, M. Tanter, A.-L. Boch, F. Marquet, M. Kujas, D. Seilhean, and M. Fink, “In vivo transcranial brain surgery with an ultrasonic time reversal mirror,” J. Neurosurg. 106, 1061–1066 (2007).
[CrossRef]

Lacot, E.

W. Glastre, E. Lacot, O. Jacquin, and H. Guillet de Chatellus, “Sensitivity of synthetic aperture laser optical feedback imaging,” J. Opt. Soc. Am. A 29, 476–485 (2012).
[CrossRef]

O. Hugon, F. Joud, E. Lacot, O. Jacquin, and H. Guillet de Chatellus, “Coherent microscopy by laser optical feedback imaging (LOFI) technique,” Ultramicroscopy 111, 1557–1563 (2011).
[CrossRef]

O. Jacquin, E. Lacot, W. Glastre, O. Hugon, and H. Guillet de Chatellus, “Experimental comparison of autodyne and heterodyne laser interferometry using Nd:YVO4 microchip laser,” J. Opt. Soc. Am. A 28, 1741–1746 (2011).
[CrossRef]

E. Lacot, O. Jacquin, G. Roussely, O. Hugon, and H. Guillet de Chatellus, “Comparative study of autodyne and heterodyne laser interferometry for imaging,” J. Opt. Soc. Am. A 27, 2450–2458 (2010).
[CrossRef]

A. Witomski, E. Lacot, O. Hugon, and O. Jacquin, “Two dimensional synthetic aperture laser optical feedback imaging using galvanometric scanning,” Appl. Opt. 47, 860–869(2008).
[CrossRef]

A. Witomski, E. Lacot, O. Hugon, and O. Jacquin, “Synthetic aperture laser optical feedback imaging using galvanometric scanning,” Opt. Lett. 31, 3031–3033 (2006).
[CrossRef]

O. Hugon, E. Lacot, and F. Stoeckel, “Submicrometric displacement and vibration measurement using optical feedback in a fiber laser,” Fib. Integr. Opt. 22, 283–288 (2003).
[CrossRef]

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815 (2001).
[CrossRef]

E. Lacot, R. Day, and F. Stoeckel, “Laser optical feedback tomography,” Opt. Lett. 24, 744–746 (1999).
[CrossRef]

Lerosey, G.

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]

Lucke, R. L.

Majwski, R. M.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser,” Proc. SPIE 783, 29–40 (1987). http://adsabs.harvard.edu/abs/1987SPIE..783...29A

Maloney, J.

P. Pantazis, J. Maloney, D. Wu, and S. E. Fraser, “Second harmonic generating (SHG) nanoprobes for in vivo imaging,” Proc. Natl. Acad. Sci. U.S.A. 107, 14535–14540 (2010).
[CrossRef]

Marechal, N. J.

Markus, S.

Marquet, F.

M. Pernot, J.-F. Aubry, M. Tanter, A.-L. Boch, F. Marquet, M. Kujas, D. Seilhean, and M. Fink, “In vivo transcranial brain surgery with an ultrasonic time reversal mirror,” J. Neurosurg. 106, 1061–1066 (2007).
[CrossRef]

McDonough, R. N.

J. C. Curlander and R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, 1991).

Minsky, M.

M. Minsky, “Memoir on inventing the confocal scanning microscope,” Scanning 10, 128–138 (1988).
[CrossRef]

Otsuka, K.

K. Otsuka, “Self-mixing thin-slice solid-state laser metrology,” Sensors 11, 2195–2245 (2011).
[CrossRef]

Pantazis, P.

P. Pantazis, J. Maloney, D. Wu, and S. E. Fraser, “Second harmonic generating (SHG) nanoprobes for in vivo imaging,” Proc. Natl. Acad. Sci. U.S.A. 107, 14535–14540 (2010).
[CrossRef]

Pasmurov, A. Ja

A. Ja Pasmurov and J. S. Zimoview, Radar Imaging and Holography (Institution of Electrical Engineers, 2005).

Pernot, M.

M. Pernot, J.-F. Aubry, M. Tanter, A.-L. Boch, F. Marquet, M. Kujas, D. Seilhean, and M. Fink, “In vivo transcranial brain surgery with an ultrasonic time reversal mirror,” J. Neurosurg. 106, 1061–1066 (2007).
[CrossRef]

Peterson, L. M.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser,” Proc. SPIE 783, 29–40 (1987). http://adsabs.harvard.edu/abs/1987SPIE..783...29A

Popoff, S. M.

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]

Reintjes, J.

Rickard, L. J.

Roussely, G.

Schockaert, C.

Schroeder, K. S.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser,” Proc. SPIE 783, 29–40 (1987). http://adsabs.harvard.edu/abs/1987SPIE..783...29A

Seilhean, D.

M. Pernot, J.-F. Aubry, M. Tanter, A.-L. Boch, F. Marquet, M. Kujas, D. Seilhean, and M. Fink, “In vivo transcranial brain surgery with an ultrasonic time reversal mirror,” J. Neurosurg. 106, 1061–1066 (2007).
[CrossRef]

Stoeckel, F.

O. Hugon, E. Lacot, and F. Stoeckel, “Submicrometric displacement and vibration measurement using optical feedback in a fiber laser,” Fib. Integr. Opt. 22, 283–288 (2003).
[CrossRef]

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815 (2001).
[CrossRef]

E. Lacot, R. Day, and F. Stoeckel, “Laser optical feedback tomography,” Opt. Lett. 24, 744–746 (1999).
[CrossRef]

Tai, A. M.

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser,” Proc. SPIE 783, 29–40 (1987). http://adsabs.harvard.edu/abs/1987SPIE..783...29A

Tanter, M.

M. Pernot, J.-F. Aubry, M. Tanter, A.-L. Boch, F. Marquet, M. Kujas, D. Seilhean, and M. Fink, “In vivo transcranial brain surgery with an ultrasonic time reversal mirror,” J. Neurosurg. 106, 1061–1066 (2007).
[CrossRef]

Vellekoop, I. M.

Vertu, S.

S. Vertu, J. Flugge, J. J. Delaunay, and O. Haeberle, “Improved and isotropic resolution in tomographic driffractive microscopy combining sample and illumination rotation,” Central Eur. J. Phys. 44, 969–974 (2011).
[CrossRef]

Witomski, A.

Wright, T. J.

Wu, D.

P. Pantazis, J. Maloney, D. Wu, and S. E. Fraser, “Second harmonic generating (SHG) nanoprobes for in vivo imaging,” Proc. Natl. Acad. Sci. U.S.A. 107, 14535–14540 (2010).
[CrossRef]

Yourassowsky, C.

Zimoview, J. S.

A. Ja Pasmurov and J. S. Zimoview, Radar Imaging and Holography (Institution of Electrical Engineers, 2005).

Appl. Opt. (4)

Central Eur. J. Phys. (1)

S. Vertu, J. Flugge, J. J. Delaunay, and O. Haeberle, “Improved and isotropic resolution in tomographic driffractive microscopy combining sample and illumination rotation,” Central Eur. J. Phys. 44, 969–974 (2011).
[CrossRef]

Fib. Integr. Opt. (1)

O. Hugon, E. Lacot, and F. Stoeckel, “Submicrometric displacement and vibration measurement using optical feedback in a fiber laser,” Fib. Integr. Opt. 22, 283–288 (2003).
[CrossRef]

J. Neurosurg. (1)

M. Pernot, J.-F. Aubry, M. Tanter, A.-L. Boch, F. Marquet, M. Kujas, D. Seilhean, and M. Fink, “In vivo transcranial brain surgery with an ultrasonic time reversal mirror,” J. Neurosurg. 106, 1061–1066 (2007).
[CrossRef]

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

Med. Sci. (1)

A. Dubois and C. Boccara, “OCT plein champ,” Med. Sci. 22, 859–864 (2006).

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (1)

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

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]

Proc. Natl. Acad. Sci. U.S.A. (1)

P. Pantazis, J. Maloney, D. Wu, and S. E. Fraser, “Second harmonic generating (SHG) nanoprobes for in vivo imaging,” Proc. Natl. Acad. Sci. U.S.A. 107, 14535–14540 (2010).
[CrossRef]

Proc. SPIE (1)

C. C. Aleksoff, J. S. Accetta, L. M. Peterson, A. M. Tai, A. Klooster, K. S. Schroeder, R. M. Majwski, J. O. Abshier, and M. Fee, “Synthetic aperture imaging with a pulsed CO2 TEA laser,” Proc. SPIE 783, 29–40 (1987). http://adsabs.harvard.edu/abs/1987SPIE..783...29A

Scanning (1)

M. Minsky, “Memoir on inventing the confocal scanning microscope,” Scanning 10, 128–138 (1988).
[CrossRef]

Sensors (1)

K. Otsuka, “Self-mixing thin-slice solid-state laser metrology,” Sensors 11, 2195–2245 (2011).
[CrossRef]

Ultramicroscopy (1)

O. Hugon, F. Joud, E. Lacot, O. Jacquin, and H. Guillet de Chatellus, “Coherent microscopy by laser optical feedback imaging (LOFI) technique,” Ultramicroscopy 111, 1557–1563 (2011).
[CrossRef]

Other (3)

N. Bleistein and R. Handelsman, “Fourier integrals and the method of stationary phase,” in Asymptotic Expansions of Integrals (Dover, 1975), pp. 219–223.

J. C. Curlander and R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, 1991).

A. Ja Pasmurov and J. S. Zimoview, Radar Imaging and Holography (Institution of Electrical Engineers, 2005).

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Figures (6)

Fig. 1.
Fig. 1.

Experimental setup of the synthetic aperture LOFI-based imaging system. The laser is a cw Nd:YVO4 microchip collimated by lens L1. A beam splitter sends 10% of the beam on a photodiode connected to a lock-in amplifier, which gives access to the amplitude and phase of the signal. The frequency shifter is made of two acousto-optic modulators which diffract respectively in orders 1 and 1 and give a net frequency shift of Fe/2=1.5MHz. XY plane is scanned (see Fig. 2) by galvanometric mirrors MX (scan in the X direction) and MY (scan in the Y direction) conjugated by a telescope made by lenses L3 and L4. f3, f4, and f5 are the focal lengths of lenses L3, L4, and L5. αX and αY are the angular positions of galvanometric mirrors MX and MY.

Fig. 2.
Fig. 2.

Scanning mechanism of the setup of Fig. 1 after lens L4. The collimated laser beam is focused by the final imaging lens L5 in its focal image plane with a waist radius r. Mirror MX (image of MX through telescope, see Fig. 1) and MY are in the focal object plane of imaging lens L5, which implies a displacement of the beam waist in the image focal plane of L5. The laser beam presents a slight misalignment with L5 optical axis and rotation axis of galvanometric mirror of dX in the X direction and dY in the Y direction. αX and αY are the angular position of galvanometric mirrors MX and MY. When αX=αY=0, the center of the beam is passing through the target (convention). x and y are the position of the waist in the focal image plane of L5. θX and θY are the angles of the center of the beam relative to the direction normal to the focal image plane of L5 (0 due to dX and dY). The target at a position (LθX, LθY) is scanned by a defocused beam at a distance L from the beam waist.

Fig. 3.
Fig. 3.

Comparison in terms of resolution of the (a) raw acquisition and (b) after numerical refocusing in the plane of object 1. δ is the distance between two objects of interest. SR(L)=π RESR(L)2 is the surface of the beam at a distance L from the beam waist and SSA(δ)=π RESSA(δ)2 is the surface of the beam at a distance δ from the plane of refocusing (here plane of object 1).

Fig. 4.
Fig. 4.

Example of synthetic aperture LOFI. (a) Object under microscope. It is made of reflective silica beads of 40 μm diameter behind a circular aperture of 1 mm diameter. The bright field transmission image is made through a Zeiss microscope objective with a magnification of 10 and a 0.25 numerical aperture (focal length of 20 mm); (b) Raw image of the object with LOFI setup r=13.5μm, L=1cm, 512*512 pixels. The beam size on the target plane is equal to 180 μm: beads are not resolved; (c) Recorded image in the space frequency domain; (d) Image after numerical refocusing, a resolution RESR(0)=RESSA(0)=r/2=9.5μm is expected and verified experimentally.

Fig. 5.
Fig. 5.

Evolution of the resolution in the X direction with the defocus δ, fitted by the theoretical expression of RESR(δ). The resolution is calculated by fitting a section along X of the image of one bead in the image of Fig. 4 (which is the PSF since the bead can be considered as a punctual scatterer); (a) raw case, the defocus corresponds to the L=δ in Figs. 1 and 2; (b) Synthetic case, the defocus corresponds to the difference between the parameter L during the acquisition and the numerical retro propagation distance L: δ=LL [see Fig. 3(b)].

Fig. 6.
Fig. 6.

Evolution of the signal power reflected by the object of Fig. 4(a) versus the defocus L (see Fig. 3(a)]. Positive defocus corresponds to an increase of the distance between the object and the lens L5. The vertical axis has an arbitrary normalized unit.

Equations (25)

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x=2f5αx,y=2f5αy.
θX=dXf5,θY=dYf5.
hR(L,x,y)(exp(x2+y2r2(1+(LZR)2))2exp(j2π(xLθx)2+(yLθy)22R(L)λ))2exp(x2+y2RESR(L)2)exp(j2π(xLθx)2+(yLθy)22R(L)2λ),
RESR(L)=r21+(LZR)2ZR=πr2λR(L)=L(1+(ZRL)2).
HR(L,ν,μ)exp((νFX)2Δν2+(μFY)2Δμ2)exp(jπLλ((νFX)2+(μFY)2)2),
FX=2θXλ,FY=2θYλ,Δν=Δμ=2πr.
Hfilter(L,ν,μ)=exp(jπLλ(ν2+μ2)2).
HSA(L,ν,μ)=HR(L,ν,μ)Hfilter(L,ν,μ)exp((νFx)2+(μFy)2Δν2)exp(jπλL(νFX+μFY)).
|hSA(L,x,y)|=|TF1(HSA)(x,y)||exp(π2Δν2((x+Lθx)2+(y+Lθy)2))exp(j2π(xFx+yFy))|exp(((x+Lθx)2+(y+Lθy)2)(r2)2).
HSA(Lδ,L,ν,μ)=HR(Lδ,ν,μ)Hfilter(L,ν,μ).
|hSA(Lδ,L,x,y)|=|TF1(HSA(Lδ,L,ν,μ))(x,y)|exp((x+Lθx)2+(y+Lθy)2RESSA(δ)),
RESSA(δ)=r1+(δ2ZR)2,r=r2,ZR=πr2λ.
Tacq,R(L=0)LXLYΔνΔμTLXLYTr2.
Tacq(L)(LX+2RESR(L))(LY+2RESR(L))ΔνΔμT(LX+2RESR(L))(LY+2RESR(L))Tr2.
PR=ρLRGR=ρERπGR=ρP0πSR(0)Sπ(NA)2=ρP0S(NA)2SR(0).
SR(0)=πRESR(0)2=πr22,NA=RESR(δ)δ|δZR=λπr2=λ2πSR(0).
PR=ρP0S(NA)2SR(0)=ρP0S(λ2πSR(0))2SR(0)=ρP0Sλ24πSR(0)2.
PSA(L)=PRawSA(L)Npixels(L),
PRawSA(L)=ρLSA(L)GSA(L)=ρESA(L)πGSA(L)=ρP0πSR(L)SSR(0)L2.
Npixels(L)=SR(L)SR(0).
PSA(L)=PRawSA(L)Npixels(L)=ρP0πSR(L)SSR(0)L2SR(L)SR(0)=ρP0SπL2.
SR(L)=πRESR(L)2=π(λL2πr)2=λ2L24π(r2)2=λ2L24SR(0)L=2SR(0)SR(L)λ.
PSA(L)PR=ρP0SπL2ρP0Sλ24πSR(0)2=4SR(0)2L2λ2=SR(0)2SR(0)SR(L)=SR(0)SR(L).
PSA(L,L)PSA(Lδ,L)=PSA(L,L)PRSR(0)SR(δ)PSA(Lδ,Lδ)PR=SR(δ)SR(0)SR(0)SR(L)SR(0)SR(Lδ)=SR(δ)SR(0)SR(Lδ)SR(L).
PSA(Lδ,L)=SR(0)SR(δ)PSA(Lδ,Lδ).

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