Abstract

We demonstrate a wavefront sensor that unites weak measurement and the compressive-sensing, single-pixel camera. Using a high-resolution spatial light modulator (SLM) as a variable waveplate, we weakly couple an optical field’s transverse-position and polarization degrees of freedom. By placing random, binary patterns on the SLM, polarization serves as a meter for directly measuring random projections of the wavefront’s real and imaginary components. Compressive-sensing optimization techniques can then recover the wavefront. We acquire high quality, 256 × 256 pixel images of the wavefront from only 10,000 projections. Photon-counting detectors give sub-picowatt sensitivity.

© 2014 Optical Society of America

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  1. F. Roddier, Adaptive Optics in Astronomy (Cambridge university press, 1999).
    [CrossRef]
  2. L. N. Thibos and X. Hong, “Clinical applications of the Shack-Hartmann aberrometer,” Optometry Vision Sci. 76, 817–825 (1999).
    [CrossRef]
  3. M. J. Booth, “Adaptive optics in microscopy,” Philos. T. R. Soc. A 365, 2829–2843 (2007).
    [CrossRef]
  4. M. Levoy, “Light fields and computational imaging,” IEEE Comput. 39, 46–55 (2006).
    [CrossRef]
  5. R. Tyson, Principles of Adaptive Optics (CRC Press, 2010).
    [CrossRef]
  6. J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wave-function,” Nature (London) 474, 188–191 (2011).
    [CrossRef]
  7. B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17, S573–S577 (2001).
    [PubMed]
  8. R. Lane and M. Tallon, “Wave-front reconstruction using a Shack-Hartmann sensor,” Appl. Opt. 31, 6902–6908 (1992).
    [CrossRef] [PubMed]
  9. S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
    [CrossRef] [PubMed]
  10. J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics 7, 316–321 (2013).
    [CrossRef]
  11. D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in “Electronic Imaging 2006,” (International Society for Optics and Photonics, 2006), pp. 606509.
  12. R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Proc. Mag. 83, 914730 (2008).
  13. Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351 (1988).
    [CrossRef] [PubMed]
  14. O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 787–790 (2008).
    [CrossRef] [PubMed]
  15. P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
    [CrossRef] [PubMed]
  16. A. N. Jordan, J. Martínez-Rincón, and J. C. Howell, “Technical advantages for weak-value amplification: When less is more,” Phys. Rev. X 4, 011031 (2014).
  17. J. Lundeen and A. Steinberg, “Experimental joint weak measurement on a photon pair as a probe of hardys paradox,” Phys. Rev. Lett. 102, 020404 (2009).
    [CrossRef]
  18. J. Dressel and A. Jordan, “Significance of the imaginary part of the weak value,” Phys. Rev. A 85, 012107 (2012).
    [CrossRef]
  19. J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307 (2014).
    [CrossRef]
  20. J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: Quantum and classical treatments,” Phys. Rev. A 81, 033813 (2010).
    [CrossRef]
  21. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory 52, 1289–1306 (2006).
    [CrossRef]
  22. E. J. Candes, “The restricted isometry property and its implications for compressed sensing,” C. R. Math. 346, 589–592 (2008).
    [CrossRef]
  23. A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math. 76, 167–188 (1997).
    [CrossRef]
  24. E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inform. Theory 52, 5406–5425 (2006).
    [CrossRef]
  25. M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
    [CrossRef] [PubMed]
  26. J. Bobin, J.-L. Starck, and R. Ottensamer, “Compressed sensing in astronomy,” IEEE J. Sel. Top. Signa. 2, 718–726 (2008).
    [CrossRef]
  27. D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
    [CrossRef]
  28. G. A. Howland and J. C. Howell, “Efficient high-dimensional entanglement imaging with a compressive-sensing double-pixel camera,” Phys. Rev. X 3, 011013 (2013).
  29. E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Proc. Mag. 25, 21–30 (2008).
    [CrossRef]
  30. J. Romberg, “Imaging via compressive sampling [introduction to compressive sampling and recovery via convex programming],” IEEE Signal Proc. Mag. 25, 14–20 (2008).
    [CrossRef]
  31. C. Li, “Compressive sensing for 3D data processing tasks: applications, models and algorithms,” Ph.D. thesis, Rice University (2011).
  32. C. Li, W. Yin, and Y. Zhang, “Users guide for TVAL3: TV minimization by augmented lagrangian and alternating direction algorithms,” CAAM Report (2009).
  33. P. S. Considine, “Effects of coherence on imaging systems,” J. Opt. Soc. Am. 56, 1001–1007 (1966).
    [CrossRef]
  34. J. Goodman, Introduction to Fourier Optics (Roberts and Company, 2008).
  35. M. A. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Sig. 1, 586–597 (2007).
    [CrossRef]
  36. G. A. Howland, D. J. Lum, M. R. Ware, and J. C. Howell, “Photon counting compressive depth mapping,” Opt. Express 21, 23822–23837 (2013).
    [CrossRef] [PubMed]
  37. J. D. Hunter, “Matplotlib: A 2D graphics environment,” IEEE Comput. Sci. Eng. 9, 90–95 (2007).
    [CrossRef]

2014 (2)

A. N. Jordan, J. Martínez-Rincón, and J. C. Howell, “Technical advantages for weak-value amplification: When less is more,” Phys. Rev. X 4, 011031 (2014).

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307 (2014).
[CrossRef]

2013 (3)

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics 7, 316–321 (2013).
[CrossRef]

G. A. Howland and J. C. Howell, “Efficient high-dimensional entanglement imaging with a compressive-sensing double-pixel camera,” Phys. Rev. X 3, 011013 (2013).

G. A. Howland, D. J. Lum, M. R. Ware, and J. C. Howell, “Photon counting compressive depth mapping,” Opt. Express 21, 23822–23837 (2013).
[CrossRef] [PubMed]

2012 (1)

J. Dressel and A. Jordan, “Significance of the imaginary part of the weak value,” Phys. Rev. A 85, 012107 (2012).
[CrossRef]

2011 (2)

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef] [PubMed]

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wave-function,” Nature (London) 474, 188–191 (2011).
[CrossRef]

2010 (2)

J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: Quantum and classical treatments,” Phys. Rev. A 81, 033813 (2010).
[CrossRef]

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
[CrossRef]

2009 (2)

J. Lundeen and A. Steinberg, “Experimental joint weak measurement on a photon pair as a probe of hardys paradox,” Phys. Rev. Lett. 102, 020404 (2009).
[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef] [PubMed]

2008 (6)

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef] [PubMed]

E. J. Candes, “The restricted isometry property and its implications for compressed sensing,” C. R. Math. 346, 589–592 (2008).
[CrossRef]

R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Proc. Mag. 83, 914730 (2008).

J. Bobin, J.-L. Starck, and R. Ottensamer, “Compressed sensing in astronomy,” IEEE J. Sel. Top. Signa. 2, 718–726 (2008).
[CrossRef]

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Proc. Mag. 25, 21–30 (2008).
[CrossRef]

J. Romberg, “Imaging via compressive sampling [introduction to compressive sampling and recovery via convex programming],” IEEE Signal Proc. Mag. 25, 14–20 (2008).
[CrossRef]

2007 (4)

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef] [PubMed]

M. A. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Sig. 1, 586–597 (2007).
[CrossRef]

J. D. Hunter, “Matplotlib: A 2D graphics environment,” IEEE Comput. Sci. Eng. 9, 90–95 (2007).
[CrossRef]

M. J. Booth, “Adaptive optics in microscopy,” Philos. T. R. Soc. A 365, 2829–2843 (2007).
[CrossRef]

2006 (3)

M. Levoy, “Light fields and computational imaging,” IEEE Comput. 39, 46–55 (2006).
[CrossRef]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory 52, 1289–1306 (2006).
[CrossRef]

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inform. Theory 52, 5406–5425 (2006).
[CrossRef]

2001 (1)

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17, S573–S577 (2001).
[PubMed]

1999 (1)

L. N. Thibos and X. Hong, “Clinical applications of the Shack-Hartmann aberrometer,” Optometry Vision Sci. 76, 817–825 (1999).
[CrossRef]

1997 (1)

A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math. 76, 167–188 (1997).
[CrossRef]

1992 (1)

1988 (1)

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351 (1988).
[CrossRef] [PubMed]

1966 (1)

Agnew, M.

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics 7, 316–321 (2013).
[CrossRef]

Aharonov, Y.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351 (1988).
[CrossRef] [PubMed]

Albert, D. Z.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351 (1988).
[CrossRef] [PubMed]

Bamber, C.

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wave-function,” Nature (London) 474, 188–191 (2011).
[CrossRef]

Baraniuk, R. G.

R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Proc. Mag. 83, 914730 (2008).

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in “Electronic Imaging 2006,” (International Society for Optics and Photonics, 2006), pp. 606509.

Baron, D.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in “Electronic Imaging 2006,” (International Society for Optics and Photonics, 2006), pp. 606509.

Becker, S.

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
[CrossRef]

Bobin, J.

J. Bobin, J.-L. Starck, and R. Ottensamer, “Compressed sensing in astronomy,” IEEE J. Sel. Top. Signa. 2, 718–726 (2008).
[CrossRef]

Bolduc, E.

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics 7, 316–321 (2013).
[CrossRef]

Booth, M. J.

M. J. Booth, “Adaptive optics in microscopy,” Philos. T. R. Soc. A 365, 2829–2843 (2007).
[CrossRef]

Boyd, R. W.

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307 (2014).
[CrossRef]

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics 7, 316–321 (2013).
[CrossRef]

Braverman, B.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef] [PubMed]

Candes, E. J.

E. J. Candes, “The restricted isometry property and its implications for compressed sensing,” C. R. Math. 346, 589–592 (2008).
[CrossRef]

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inform. Theory 52, 5406–5425 (2006).
[CrossRef]

Candès, E. J.

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Proc. Mag. 25, 21–30 (2008).
[CrossRef]

Chambolle, A.

A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math. 76, 167–188 (1997).
[CrossRef]

Considine, P. S.

Dixon, P. B.

J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: Quantum and classical treatments,” Phys. Rev. A 81, 033813 (2010).
[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef] [PubMed]

Donoho, D.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef] [PubMed]

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory 52, 1289–1306 (2006).
[CrossRef]

Dressel, J.

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307 (2014).
[CrossRef]

J. Dressel and A. Jordan, “Significance of the imaginary part of the weak value,” Phys. Rev. A 85, 012107 (2012).
[CrossRef]

Duarte, M. F.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in “Electronic Imaging 2006,” (International Society for Optics and Photonics, 2006), pp. 606509.

Eisert, J.

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
[CrossRef]

Figueiredo, M. A.

M. A. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Sig. 1, 586–597 (2007).
[CrossRef]

Flammia, S. T.

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
[CrossRef]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (Roberts and Company, 2008).

Gross, D.

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
[CrossRef]

Hong, X.

L. N. Thibos and X. Hong, “Clinical applications of the Shack-Hartmann aberrometer,” Optometry Vision Sci. 76, 817–825 (1999).
[CrossRef]

Hosten, O.

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef] [PubMed]

Howell, J. C.

A. N. Jordan, J. Martínez-Rincón, and J. C. Howell, “Technical advantages for weak-value amplification: When less is more,” Phys. Rev. X 4, 011031 (2014).

G. A. Howland and J. C. Howell, “Efficient high-dimensional entanglement imaging with a compressive-sensing double-pixel camera,” Phys. Rev. X 3, 011013 (2013).

G. A. Howland, D. J. Lum, M. R. Ware, and J. C. Howell, “Photon counting compressive depth mapping,” Opt. Express 21, 23822–23837 (2013).
[CrossRef] [PubMed]

J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: Quantum and classical treatments,” Phys. Rev. A 81, 033813 (2010).
[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef] [PubMed]

Howland, G. A.

G. A. Howland, D. J. Lum, M. R. Ware, and J. C. Howell, “Photon counting compressive depth mapping,” Opt. Express 21, 23822–23837 (2013).
[CrossRef] [PubMed]

G. A. Howland and J. C. Howell, “Efficient high-dimensional entanglement imaging with a compressive-sensing double-pixel camera,” Phys. Rev. X 3, 011013 (2013).

Hunter, J. D.

J. D. Hunter, “Matplotlib: A 2D graphics environment,” IEEE Comput. Sci. Eng. 9, 90–95 (2007).
[CrossRef]

Johnson, A. S.

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics 7, 316–321 (2013).
[CrossRef]

Jordan, A.

J. Dressel and A. Jordan, “Significance of the imaginary part of the weak value,” Phys. Rev. A 85, 012107 (2012).
[CrossRef]

Jordan, A. N.

A. N. Jordan, J. Martínez-Rincón, and J. C. Howell, “Technical advantages for weak-value amplification: When less is more,” Phys. Rev. X 4, 011031 (2014).

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307 (2014).
[CrossRef]

J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: Quantum and classical treatments,” Phys. Rev. A 81, 033813 (2010).
[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef] [PubMed]

Kelly, K. F.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in “Electronic Imaging 2006,” (International Society for Optics and Photonics, 2006), pp. 606509.

Kocsis, S.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef] [PubMed]

Kwiat, P.

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef] [PubMed]

Lane, R.

Laska, J. N.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in “Electronic Imaging 2006,” (International Society for Optics and Photonics, 2006), pp. 606509.

Leach, J.

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics 7, 316–321 (2013).
[CrossRef]

Levoy, M.

M. Levoy, “Light fields and computational imaging,” IEEE Comput. 39, 46–55 (2006).
[CrossRef]

Li, C.

C. Li, “Compressive sensing for 3D data processing tasks: applications, models and algorithms,” Ph.D. thesis, Rice University (2011).

C. Li, W. Yin, and Y. Zhang, “Users guide for TVAL3: TV minimization by augmented lagrangian and alternating direction algorithms,” CAAM Report (2009).

Lions, P.-L.

A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math. 76, 167–188 (1997).
[CrossRef]

Liu, Y.-K.

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
[CrossRef]

Lum, D. J.

Lundeen, J.

J. Lundeen and A. Steinberg, “Experimental joint weak measurement on a photon pair as a probe of hardys paradox,” Phys. Rev. Lett. 102, 020404 (2009).
[CrossRef]

Lundeen, J. S.

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wave-function,” Nature (London) 474, 188–191 (2011).
[CrossRef]

Lustig, M.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef] [PubMed]

Malik, M.

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307 (2014).
[CrossRef]

Martínez-Rincón, J.

A. N. Jordan, J. Martínez-Rincón, and J. C. Howell, “Technical advantages for weak-value amplification: When less is more,” Phys. Rev. X 4, 011031 (2014).

Miatto, F. M.

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307 (2014).
[CrossRef]

Mirin, R. P.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef] [PubMed]

Nowak, R. D.

M. A. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Sig. 1, 586–597 (2007).
[CrossRef]

Ottensamer, R.

J. Bobin, J.-L. Starck, and R. Ottensamer, “Compressed sensing in astronomy,” IEEE J. Sel. Top. Signa. 2, 718–726 (2008).
[CrossRef]

Patel, A.

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wave-function,” Nature (London) 474, 188–191 (2011).
[CrossRef]

Pauly, J. M.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef] [PubMed]

Platt, B. C.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17, S573–S577 (2001).
[PubMed]

Ravets, S.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef] [PubMed]

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge university press, 1999).
[CrossRef]

Romberg, J.

J. Romberg, “Imaging via compressive sampling [introduction to compressive sampling and recovery via convex programming],” IEEE Signal Proc. Mag. 25, 14–20 (2008).
[CrossRef]

Salvail, J. Z.

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics 7, 316–321 (2013).
[CrossRef]

Sarvotham, S.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in “Electronic Imaging 2006,” (International Society for Optics and Photonics, 2006), pp. 606509.

Shack, R.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17, S573–S577 (2001).
[PubMed]

Shalm, L. K.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef] [PubMed]

Starck, J.-L.

J. Bobin, J.-L. Starck, and R. Ottensamer, “Compressed sensing in astronomy,” IEEE J. Sel. Top. Signa. 2, 718–726 (2008).
[CrossRef]

Starling, D. J.

J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: Quantum and classical treatments,” Phys. Rev. A 81, 033813 (2010).
[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef] [PubMed]

Steinberg, A.

J. Lundeen and A. Steinberg, “Experimental joint weak measurement on a photon pair as a probe of hardys paradox,” Phys. Rev. Lett. 102, 020404 (2009).
[CrossRef]

Steinberg, A. M.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef] [PubMed]

Stevens, M. J.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef] [PubMed]

Stewart, C.

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wave-function,” Nature (London) 474, 188–191 (2011).
[CrossRef]

Sutherland, B.

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wave-function,” Nature (London) 474, 188–191 (2011).
[CrossRef]

Takhar, D.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in “Electronic Imaging 2006,” (International Society for Optics and Photonics, 2006), pp. 606509.

Tallon, M.

Tao, T.

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inform. Theory 52, 5406–5425 (2006).
[CrossRef]

Thibos, L. N.

L. N. Thibos and X. Hong, “Clinical applications of the Shack-Hartmann aberrometer,” Optometry Vision Sci. 76, 817–825 (1999).
[CrossRef]

Tyson, R.

R. Tyson, Principles of Adaptive Optics (CRC Press, 2010).
[CrossRef]

Vaidman, L.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351 (1988).
[CrossRef] [PubMed]

Vudyasetu, P. K.

J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: Quantum and classical treatments,” Phys. Rev. A 81, 033813 (2010).
[CrossRef]

Wakin, M. B.

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Proc. Mag. 25, 21–30 (2008).
[CrossRef]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in “Electronic Imaging 2006,” (International Society for Optics and Photonics, 2006), pp. 606509.

Ware, M. R.

Wright, S. J.

M. A. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Sig. 1, 586–597 (2007).
[CrossRef]

Yin, W.

C. Li, W. Yin, and Y. Zhang, “Users guide for TVAL3: TV minimization by augmented lagrangian and alternating direction algorithms,” CAAM Report (2009).

Zhang, Y.

C. Li, W. Yin, and Y. Zhang, “Users guide for TVAL3: TV minimization by augmented lagrangian and alternating direction algorithms,” CAAM Report (2009).

Appl. Opt. (1)

C. R. Math. (1)

E. J. Candes, “The restricted isometry property and its implications for compressed sensing,” C. R. Math. 346, 589–592 (2008).
[CrossRef]

IEEE Comput. (1)

M. Levoy, “Light fields and computational imaging,” IEEE Comput. 39, 46–55 (2006).
[CrossRef]

IEEE Comput. Sci. Eng. (1)

J. D. Hunter, “Matplotlib: A 2D graphics environment,” IEEE Comput. Sci. Eng. 9, 90–95 (2007).
[CrossRef]

IEEE J. Sel. Top. Sig. (1)

M. A. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Sig. 1, 586–597 (2007).
[CrossRef]

IEEE J. Sel. Top. Signa. (1)

J. Bobin, J.-L. Starck, and R. Ottensamer, “Compressed sensing in astronomy,” IEEE J. Sel. Top. Signa. 2, 718–726 (2008).
[CrossRef]

IEEE Signal Proc. Mag. (3)

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Proc. Mag. 25, 21–30 (2008).
[CrossRef]

J. Romberg, “Imaging via compressive sampling [introduction to compressive sampling and recovery via convex programming],” IEEE Signal Proc. Mag. 25, 14–20 (2008).
[CrossRef]

R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Proc. Mag. 83, 914730 (2008).

IEEE Trans. Inform. Theory (2)

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inform. Theory 52, 5406–5425 (2006).
[CrossRef]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory 52, 1289–1306 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Refract. Surg. (1)

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17, S573–S577 (2001).
[PubMed]

Magn. Reson. Med. (1)

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef] [PubMed]

Nat. Photonics (1)

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics 7, 316–321 (2013).
[CrossRef]

Nature (London) (1)

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wave-function,” Nature (London) 474, 188–191 (2011).
[CrossRef]

Numer. Math. (1)

A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math. 76, 167–188 (1997).
[CrossRef]

Opt. Express (1)

Optometry Vision Sci. (1)

L. N. Thibos and X. Hong, “Clinical applications of the Shack-Hartmann aberrometer,” Optometry Vision Sci. 76, 817–825 (1999).
[CrossRef]

Philos. T. R. Soc. A (1)

M. J. Booth, “Adaptive optics in microscopy,” Philos. T. R. Soc. A 365, 2829–2843 (2007).
[CrossRef]

Phys. Rev. A (2)

J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: Quantum and classical treatments,” Phys. Rev. A 81, 033813 (2010).
[CrossRef]

J. Dressel and A. Jordan, “Significance of the imaginary part of the weak value,” Phys. Rev. A 85, 012107 (2012).
[CrossRef]

Phys. Rev. Lett. (4)

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
[CrossRef]

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351 (1988).
[CrossRef] [PubMed]

J. Lundeen and A. Steinberg, “Experimental joint weak measurement on a photon pair as a probe of hardys paradox,” Phys. Rev. Lett. 102, 020404 (2009).
[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef] [PubMed]

Phys. Rev. X (2)

A. N. Jordan, J. Martínez-Rincón, and J. C. Howell, “Technical advantages for weak-value amplification: When less is more,” Phys. Rev. X 4, 011031 (2014).

G. A. Howland and J. C. Howell, “Efficient high-dimensional entanglement imaging with a compressive-sensing double-pixel camera,” Phys. Rev. X 3, 011013 (2013).

Rev. Mod. Phys. (1)

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86, 307 (2014).
[CrossRef]

Science (2)

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef] [PubMed]

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef] [PubMed]

Other (6)

F. Roddier, Adaptive Optics in Astronomy (Cambridge university press, 1999).
[CrossRef]

R. Tyson, Principles of Adaptive Optics (CRC Press, 2010).
[CrossRef]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in “Electronic Imaging 2006,” (International Society for Optics and Photonics, 2006), pp. 606509.

J. Goodman, Introduction to Fourier Optics (Roberts and Company, 2008).

C. Li, “Compressive sensing for 3D data processing tasks: applications, models and algorithms,” Ph.D. thesis, Rice University (2011).

C. Li, W. Yin, and Y. Zhang, “Users guide for TVAL3: TV minimization by augmented lagrangian and alternating direction algorithms,” CAAM Report (2009).

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Figures (5)

Fig. 1
Fig. 1

Weak Value Wavefront Sensing on the Bloch Sphere: A state with transverse field ψ(x⃗) is initially in polarization |pi〉. At x⃗ = x⃗0, the polarization is rotated by small angle θ in the (î, ĵ) plane to state |pf〉; this weakly measures ψ(x⃗0). Following a post selection on transverse momentum k⃗ = k⃗0, the real and imaginary parts of ψ(x⃗0) are mapped to a polarization rotation of the post-selected state |ψps〉. The real part generates a rotation αR in the (î, ĵ) plane (b) and the imaginary part generates a rotation αI in the (, ĵ) plane (c). The weak measurement of ψ(x⃗0) is read-out by measuring 〈σ̂j〉 and 〈σ̂k〉 respectively.

Fig. 2
Fig. 2

Experimental Setup for Wavefront Sensing: An input field is prepared by illuminating an object SLM with a collimated, attenuated HeNe laser. The input beam is polarized in |a〉 to produce a nearly pure phase object with some intensity coupling. A 4F imaging system reproduces the field on a pattern SLM, with a polarizer setting the initial polarization to horizontal (|pi〉 = |h〉). A sequence of M random, binary patterns are placed on the SLM; pattern pixels with value 1 have their polarization rotated a small amount to |pf〉 This constitutes a weak measurement of the projection of the input field onto the random pattern. A spatial filter performs the k⃗ = 0 momentum post-selection. Polarization analyzers take expected values of σ̂j and σ̂k, which are proportional to the real and imaginary parts of the projection.

Fig. 3
Fig. 3

SLM Polarization Calibration: The polarization rotation performed by the SLM on a horizontally polarized input state on the Bloch sphere is given in (a) and (b). (c) gives the angle θ/2 between the initial horizontal state |pi〉 and output state |pf〉. Point p0 corresponds to the smallest rotation away from |pi〉. Point p1 is the SLM intensity and corresponding angle used for the rotated state |pf〉, approximately 25 degrees.

Fig. 4
Fig. 4

256 × 256 pixel ħ Character Wavefront: An interferogram of the object field (a) taken with an 8-bit CCD camera confirms a near-phase-only input wavefront. The CS-reconstructed real and imaginary parts are shown in (b,c), where the wavefront is normalized to unit total intensity. From the real and imaginary parts, we find intensity (d) and phase (e) images. A masked phase image (f) removes all intensities below 5 × 10−10, where it is not meaningful to asign a phase. Only M = 0.15N = 10,000 random projections are needed for a high quality reconstruction.

Fig. 5
Fig. 5

Phase Grid Test Pattern: A 16 square grid of increasing SLM intensity (a) was placed on the object SLM, which converts it to a phase image. A CCD image of the object SLM (b) shows nearly uniform intensity, while a dark-port interferogram (c) shows the increasing phase of each square. The phase angle for a 256 × 256 reconstruction is given in (d–e) for M = 10000. The increasing phase of each square can be seen riding the illuminating beam’s Gaussian phase profile.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

A w = f | A ^ | ψ f | ψ .
A w ( x 0 ) = k 0 | x 0 x 0 | ψ k 0 | ψ = e i k 0 x 0 ψ ( x 0 ) ψ ˜ ( k 0 ) .
| ψ = d x ψ ( x ) | x | p i .
ψ ( x 0 ) | x 0 | p f = ψ ( x 0 ) e i σ ^ k θ / 2 | x 0 | p i ,
ψ ( x 0 ) | x 0 | p f = ψ ( x 0 ) ( 1 i σ ^ k θ / 2 ) | x 0 | p i .
| ψ = d x ψ ( x ) | x | p i ψ ( x 0 ) i σ ^ k θ / 2 | x 0 | p i .
| ψ ps = k 0 | ψ = ψ ˜ ( k 0 ) | p i e i k 0 x 0 ψ ( x 0 ) i σ ^ k θ / 2 | p i .
| ψ ps = ψ ˜ ( k 0 ) e i e i k 0 x 0 ψ ( x 0 ) ψ ˜ ( k 0 ) σ ^ k θ / 2 | p i = ψ ˜ ( k 0 ) e i A w ( x 0 ) σ ^ k θ / 2 | p i
ψ ps | σ ^ j | ψ ps Re { ψ ( x 0 ) }
ψ ps | σ ^ k | ψ ps Im { ψ ( x 0 ) } .
| f i = d x f i ( x ) | x i .
A i = k 0 | f i f i | ψ k 0 | ψ = k 0 | f i Y i ψ ˜ ( k 0 ) ,
Y i = d x f i ( x ) ψ ( x ) .
| ψ i ps = ψ ˜ ( k 0 ) e i Y i 2 ψ ( k 0 ) σ ^ k θ | p i .
ψ i ps | σ ^ j | ψ i ps Y i Re
ψ i ps | σ ^ k | ψ i ps Y i Im .
Y = FX + Γ ,
min X μ 2 Y FX 2 2 + g ( X ) .
TV ( X ) = adj . i , j X i X j ,
T ^ ( x 0 ) = | d d | + exp [ i Φ ( I SLM ( x 0 ) ) ] | a a | ,
Y = Y Re + i Y Im = FX .

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