Abstract

The phasor field has been shown to be a valuable tool for non-line-of-sight imaging. We present a formal analysis of phasor-field imaging using paraxial wave optics. Then, we derive a set of propagation primitives—using the two-frequency, spatial Wigner distribution—that extend the purview of phasor-field imaging. We use these primitives to analyze a set of simple imaging scenarios involving occluded and unoccluded geometries with modulated and unmodulated light. These scenarios demonstrate how to apply the primitives in practice and reveal what kind of insights can be expected from them.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Phasor field waves: a mathematical treatment

Jeremy A. Teichman
Opt. Express 27(20) 27500-27506 (2019)

Phasor field waves: experimental demonstrations of wave-like properties

Syed Azer Reza, Marco La Manna, Sebastian Bauer, and Andreas Velten
Opt. Express 27(22) 32587-32608 (2019)

Phasor field waves: A Huygens-like light transport model for non-line-of-sight imaging applications

Syed Azer Reza, Marco La Manna, Sebastian Bauer, and Andreas Velten
Opt. Express 27(20) 29380-29400 (2019)

References

  • View by:
  • |
  • |
  • |

  1. A. Kirmani, T. Hutchison, J. Davis, and R. Raskar, “Looking around the corner using ultrafast transient imaging,” Int. J. Comput. Vision 95, 13–28 (2011).
    [Crossref]
  2. A. Velten, T. Willwacher, O. Gupta, A. Veeraraghavan, M. G. Bawendi, and R. Raskar, “Recovering three-dimensional shape around a corner using ultrafast time-of-flight imaging,” Nat. Commun. 3, 745 (2012).
    [Crossref] [PubMed]
  3. F. Heide, L. Xiao, W. Heidrich, and M. B. Hullin, “Diffuse mirrors: 3D reconstruction from diffuse indirect illumination using inexpensive time-of-flight sensors,” in Proc. IEEE Conf. Comput. Vis. Pattern Recog., pp. 3222–3229 (2014).
  4. M. Buttafava, J. Zeman, A. Tosi, K. Eliceiri, and A. Velten, “Non-line-of-sight imaging using a time-gated single photon avalanche diode,” Opt. Express 23, 20997–21011 (2015).
    [Crossref] [PubMed]
  5. G. Gariepy, F. Tonolini, R. Henderson, J. Leach, and D. Faccio, “Detection and tracking of moving objects hidden from view,” Nat. Photonics 10, 23–27 (2015).
    [Crossref]
  6. A. Kadambi, H. Zhao, B. Shi, and R. Raskar, “Occluded imaging with time-of-flight sensors,” ACM Trans. Graph. 35, 1–12 (2016).
    [Crossref]
  7. J. Klein, M. Laurenzis, and M. Hullin, “Transient imaging for real-time tracking around a corner,” Proc. SPIE 9988, 998802 (2016).
    [Crossref]
  8. M. O’Toole, D. B. Lindell, and G. Wetzstein, “Confocal non-line-of-sight imaging based on the light-cone transform,” Nature 555, 338–341 (2018).
    [Crossref]
  9. S. A Reza, M. La Manna, and A. Velten, “A physical light transport model for non-line-of-sight imaging applications,” arXiv:1802.1823 [physics.optics].
  10. X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, D. Gutierrez, A. Jarabo, and A. Velten, “Virtual wave optics for non-line-of-sight imaging,” arXiv:1810.07535 [cs.CV].
  11. S. A. Reza, M. La Manna, S. Bauer, and A. Velten, “Wave-like properties of phasor fields: experimental demonstrations,” arXiv:190401565 [physics.optics].
  12. F. Xu, G. Shulkind, C. Thrampoulidis, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Revealing hidden scenes by photon-efficient occlusion-based opportunistic active imaging,” Opt. Express 26, 9945 (2018).
    [Crossref] [PubMed]
  13. C. Thrampoulidis, G. Shulkind, F. Xu, W. T. Freeman, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Exploiting occlusion in non-line-of-sight active imaging,” IEEE Trans. Comput. Imag. 4, 419 (2018).
    [Crossref]
  14. The short-time average z-plane irradiance is the instantaneous irradiance averaged over a time Ta satisfying ω0Ta ≫ 1 and ΔωTa ≪ 1.
  15. In what follows, integrals without explicit limits are over the integration variable’s entire domain.
  16. Because h0(ρ) is a zero-mean Gaussian process, its samples at ρ0 and ρ′0 are zero-mean jointly Gaussian random variables whose joint characteristic function is as given in Eq. (12).
  17. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1: Single Scattering and Transport Theory (Academic, New York, 1978).
  18. A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).
    [Crossref]
  19. E. H. Adelson and J. R. Bergen, “The plenoptic function and the elements of early vision,” in M. S. Landy and J. A. Movshon, eds., Computational Models of Visual Processing, (MIT Press, 1991), pp. 3–20.
  20. M. Levoy and P. Hanrahan, “Light field rendering,” in Proc. SIGGRAPH (ACM, New York, NY, USA, 1996), pp. 31–42.
  21. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256 (1968).
    [Crossref]
  22. M. J. Bastiaans, “Wigner distribution and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1980).
    [Crossref]
  23. M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particles,” Adv. Opt. Photon. 3, 272–365 (2011).
    [Crossref]
  24. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2: Multiple Scattering, Turbulence, Rough Surfaces, and Remote Sensing (Academic, New York, 1978).
  25. For notational convenience, we have assumed that the diffraction takes place between the z = 0 and z = L planes, but the result we obtain will apply for +z-going Fresnel diffraction over a distance L starting from an arbitrary z plane.

2018 (3)

M. O’Toole, D. B. Lindell, and G. Wetzstein, “Confocal non-line-of-sight imaging based on the light-cone transform,” Nature 555, 338–341 (2018).
[Crossref]

F. Xu, G. Shulkind, C. Thrampoulidis, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Revealing hidden scenes by photon-efficient occlusion-based opportunistic active imaging,” Opt. Express 26, 9945 (2018).
[Crossref] [PubMed]

C. Thrampoulidis, G. Shulkind, F. Xu, W. T. Freeman, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Exploiting occlusion in non-line-of-sight active imaging,” IEEE Trans. Comput. Imag. 4, 419 (2018).
[Crossref]

2016 (2)

A. Kadambi, H. Zhao, B. Shi, and R. Raskar, “Occluded imaging with time-of-flight sensors,” ACM Trans. Graph. 35, 1–12 (2016).
[Crossref]

J. Klein, M. Laurenzis, and M. Hullin, “Transient imaging for real-time tracking around a corner,” Proc. SPIE 9988, 998802 (2016).
[Crossref]

2015 (2)

M. Buttafava, J. Zeman, A. Tosi, K. Eliceiri, and A. Velten, “Non-line-of-sight imaging using a time-gated single photon avalanche diode,” Opt. Express 23, 20997–21011 (2015).
[Crossref] [PubMed]

G. Gariepy, F. Tonolini, R. Henderson, J. Leach, and D. Faccio, “Detection and tracking of moving objects hidden from view,” Nat. Photonics 10, 23–27 (2015).
[Crossref]

2012 (1)

A. Velten, T. Willwacher, O. Gupta, A. Veeraraghavan, M. G. Bawendi, and R. Raskar, “Recovering three-dimensional shape around a corner using ultrafast time-of-flight imaging,” Nat. Commun. 3, 745 (2012).
[Crossref] [PubMed]

2011 (2)

A. Kirmani, T. Hutchison, J. Davis, and R. Raskar, “Looking around the corner using ultrafast transient imaging,” Int. J. Comput. Vision 95, 13–28 (2011).
[Crossref]

M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particles,” Adv. Opt. Photon. 3, 272–365 (2011).
[Crossref]

1980 (1)

1968 (1)

1939 (1)

A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).
[Crossref]

Adelson, E. H.

E. H. Adelson and J. R. Bergen, “The plenoptic function and the elements of early vision,” in M. S. Landy and J. A. Movshon, eds., Computational Models of Visual Processing, (MIT Press, 1991), pp. 3–20.

Alonso, M. A.

Bastiaans, M. J.

Bauer, S.

S. A. Reza, M. La Manna, S. Bauer, and A. Velten, “Wave-like properties of phasor fields: experimental demonstrations,” arXiv:190401565 [physics.optics].

Bawendi, M. G.

A. Velten, T. Willwacher, O. Gupta, A. Veeraraghavan, M. G. Bawendi, and R. Raskar, “Recovering three-dimensional shape around a corner using ultrafast time-of-flight imaging,” Nat. Commun. 3, 745 (2012).
[Crossref] [PubMed]

Bergen, J. R.

E. H. Adelson and J. R. Bergen, “The plenoptic function and the elements of early vision,” in M. S. Landy and J. A. Movshon, eds., Computational Models of Visual Processing, (MIT Press, 1991), pp. 3–20.

Buttafava, M.

Davis, J.

A. Kirmani, T. Hutchison, J. Davis, and R. Raskar, “Looking around the corner using ultrafast transient imaging,” Int. J. Comput. Vision 95, 13–28 (2011).
[Crossref]

Eliceiri, K.

Faccio, D.

G. Gariepy, F. Tonolini, R. Henderson, J. Leach, and D. Faccio, “Detection and tracking of moving objects hidden from view,” Nat. Photonics 10, 23–27 (2015).
[Crossref]

Freeman, W. T.

C. Thrampoulidis, G. Shulkind, F. Xu, W. T. Freeman, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Exploiting occlusion in non-line-of-sight active imaging,” IEEE Trans. Comput. Imag. 4, 419 (2018).
[Crossref]

Gariepy, G.

G. Gariepy, F. Tonolini, R. Henderson, J. Leach, and D. Faccio, “Detection and tracking of moving objects hidden from view,” Nat. Photonics 10, 23–27 (2015).
[Crossref]

Gershun, A.

A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).
[Crossref]

Guillén, I.

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, D. Gutierrez, A. Jarabo, and A. Velten, “Virtual wave optics for non-line-of-sight imaging,” arXiv:1810.07535 [cs.CV].

Gupta, O.

A. Velten, T. Willwacher, O. Gupta, A. Veeraraghavan, M. G. Bawendi, and R. Raskar, “Recovering three-dimensional shape around a corner using ultrafast time-of-flight imaging,” Nat. Commun. 3, 745 (2012).
[Crossref] [PubMed]

Gutierrez, D.

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, D. Gutierrez, A. Jarabo, and A. Velten, “Virtual wave optics for non-line-of-sight imaging,” arXiv:1810.07535 [cs.CV].

Hanrahan, P.

M. Levoy and P. Hanrahan, “Light field rendering,” in Proc. SIGGRAPH (ACM, New York, NY, USA, 1996), pp. 31–42.

Heide, F.

F. Heide, L. Xiao, W. Heidrich, and M. B. Hullin, “Diffuse mirrors: 3D reconstruction from diffuse indirect illumination using inexpensive time-of-flight sensors,” in Proc. IEEE Conf. Comput. Vis. Pattern Recog., pp. 3222–3229 (2014).

Heidrich, W.

F. Heide, L. Xiao, W. Heidrich, and M. B. Hullin, “Diffuse mirrors: 3D reconstruction from diffuse indirect illumination using inexpensive time-of-flight sensors,” in Proc. IEEE Conf. Comput. Vis. Pattern Recog., pp. 3222–3229 (2014).

Henderson, R.

G. Gariepy, F. Tonolini, R. Henderson, J. Leach, and D. Faccio, “Detection and tracking of moving objects hidden from view,” Nat. Photonics 10, 23–27 (2015).
[Crossref]

Hullin, M.

J. Klein, M. Laurenzis, and M. Hullin, “Transient imaging for real-time tracking around a corner,” Proc. SPIE 9988, 998802 (2016).
[Crossref]

Hullin, M. B.

F. Heide, L. Xiao, W. Heidrich, and M. B. Hullin, “Diffuse mirrors: 3D reconstruction from diffuse indirect illumination using inexpensive time-of-flight sensors,” in Proc. IEEE Conf. Comput. Vis. Pattern Recog., pp. 3222–3229 (2014).

Hutchison, T.

A. Kirmani, T. Hutchison, J. Davis, and R. Raskar, “Looking around the corner using ultrafast transient imaging,” Int. J. Comput. Vision 95, 13–28 (2011).
[Crossref]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1: Single Scattering and Transport Theory (Academic, New York, 1978).

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2: Multiple Scattering, Turbulence, Rough Surfaces, and Remote Sensing (Academic, New York, 1978).

Jarabo, A.

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, D. Gutierrez, A. Jarabo, and A. Velten, “Virtual wave optics for non-line-of-sight imaging,” arXiv:1810.07535 [cs.CV].

Kadambi, A.

A. Kadambi, H. Zhao, B. Shi, and R. Raskar, “Occluded imaging with time-of-flight sensors,” ACM Trans. Graph. 35, 1–12 (2016).
[Crossref]

Kirmani, A.

A. Kirmani, T. Hutchison, J. Davis, and R. Raskar, “Looking around the corner using ultrafast transient imaging,” Int. J. Comput. Vision 95, 13–28 (2011).
[Crossref]

Klein, J.

J. Klein, M. Laurenzis, and M. Hullin, “Transient imaging for real-time tracking around a corner,” Proc. SPIE 9988, 998802 (2016).
[Crossref]

La Manna, M.

S. A Reza, M. La Manna, and A. Velten, “A physical light transport model for non-line-of-sight imaging applications,” arXiv:1802.1823 [physics.optics].

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, D. Gutierrez, A. Jarabo, and A. Velten, “Virtual wave optics for non-line-of-sight imaging,” arXiv:1810.07535 [cs.CV].

S. A. Reza, M. La Manna, S. Bauer, and A. Velten, “Wave-like properties of phasor fields: experimental demonstrations,” arXiv:190401565 [physics.optics].

Laurenzis, M.

J. Klein, M. Laurenzis, and M. Hullin, “Transient imaging for real-time tracking around a corner,” Proc. SPIE 9988, 998802 (2016).
[Crossref]

Le, T. H.

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, D. Gutierrez, A. Jarabo, and A. Velten, “Virtual wave optics for non-line-of-sight imaging,” arXiv:1810.07535 [cs.CV].

Leach, J.

G. Gariepy, F. Tonolini, R. Henderson, J. Leach, and D. Faccio, “Detection and tracking of moving objects hidden from view,” Nat. Photonics 10, 23–27 (2015).
[Crossref]

Levoy, M.

M. Levoy and P. Hanrahan, “Light field rendering,” in Proc. SIGGRAPH (ACM, New York, NY, USA, 1996), pp. 31–42.

Lindell, D. B.

M. O’Toole, D. B. Lindell, and G. Wetzstein, “Confocal non-line-of-sight imaging based on the light-cone transform,” Nature 555, 338–341 (2018).
[Crossref]

Liu, X.

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, D. Gutierrez, A. Jarabo, and A. Velten, “Virtual wave optics for non-line-of-sight imaging,” arXiv:1810.07535 [cs.CV].

Nam, J. H.

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, D. Gutierrez, A. Jarabo, and A. Velten, “Virtual wave optics for non-line-of-sight imaging,” arXiv:1810.07535 [cs.CV].

O’Toole, M.

M. O’Toole, D. B. Lindell, and G. Wetzstein, “Confocal non-line-of-sight imaging based on the light-cone transform,” Nature 555, 338–341 (2018).
[Crossref]

Raskar, R.

A. Kadambi, H. Zhao, B. Shi, and R. Raskar, “Occluded imaging with time-of-flight sensors,” ACM Trans. Graph. 35, 1–12 (2016).
[Crossref]

A. Velten, T. Willwacher, O. Gupta, A. Veeraraghavan, M. G. Bawendi, and R. Raskar, “Recovering three-dimensional shape around a corner using ultrafast time-of-flight imaging,” Nat. Commun. 3, 745 (2012).
[Crossref] [PubMed]

A. Kirmani, T. Hutchison, J. Davis, and R. Raskar, “Looking around the corner using ultrafast transient imaging,” Int. J. Comput. Vision 95, 13–28 (2011).
[Crossref]

Reza, S. A

S. A Reza, M. La Manna, and A. Velten, “A physical light transport model for non-line-of-sight imaging applications,” arXiv:1802.1823 [physics.optics].

Reza, S. A.

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, D. Gutierrez, A. Jarabo, and A. Velten, “Virtual wave optics for non-line-of-sight imaging,” arXiv:1810.07535 [cs.CV].

S. A. Reza, M. La Manna, S. Bauer, and A. Velten, “Wave-like properties of phasor fields: experimental demonstrations,” arXiv:190401565 [physics.optics].

Shapiro, J. H.

F. Xu, G. Shulkind, C. Thrampoulidis, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Revealing hidden scenes by photon-efficient occlusion-based opportunistic active imaging,” Opt. Express 26, 9945 (2018).
[Crossref] [PubMed]

C. Thrampoulidis, G. Shulkind, F. Xu, W. T. Freeman, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Exploiting occlusion in non-line-of-sight active imaging,” IEEE Trans. Comput. Imag. 4, 419 (2018).
[Crossref]

Shi, B.

A. Kadambi, H. Zhao, B. Shi, and R. Raskar, “Occluded imaging with time-of-flight sensors,” ACM Trans. Graph. 35, 1–12 (2016).
[Crossref]

Shulkind, G.

C. Thrampoulidis, G. Shulkind, F. Xu, W. T. Freeman, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Exploiting occlusion in non-line-of-sight active imaging,” IEEE Trans. Comput. Imag. 4, 419 (2018).
[Crossref]

F. Xu, G. Shulkind, C. Thrampoulidis, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Revealing hidden scenes by photon-efficient occlusion-based opportunistic active imaging,” Opt. Express 26, 9945 (2018).
[Crossref] [PubMed]

Thrampoulidis, C.

F. Xu, G. Shulkind, C. Thrampoulidis, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Revealing hidden scenes by photon-efficient occlusion-based opportunistic active imaging,” Opt. Express 26, 9945 (2018).
[Crossref] [PubMed]

C. Thrampoulidis, G. Shulkind, F. Xu, W. T. Freeman, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Exploiting occlusion in non-line-of-sight active imaging,” IEEE Trans. Comput. Imag. 4, 419 (2018).
[Crossref]

Tonolini, F.

G. Gariepy, F. Tonolini, R. Henderson, J. Leach, and D. Faccio, “Detection and tracking of moving objects hidden from view,” Nat. Photonics 10, 23–27 (2015).
[Crossref]

Torralba, A.

C. Thrampoulidis, G. Shulkind, F. Xu, W. T. Freeman, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Exploiting occlusion in non-line-of-sight active imaging,” IEEE Trans. Comput. Imag. 4, 419 (2018).
[Crossref]

F. Xu, G. Shulkind, C. Thrampoulidis, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Revealing hidden scenes by photon-efficient occlusion-based opportunistic active imaging,” Opt. Express 26, 9945 (2018).
[Crossref] [PubMed]

Tosi, A.

Veeraraghavan, A.

A. Velten, T. Willwacher, O. Gupta, A. Veeraraghavan, M. G. Bawendi, and R. Raskar, “Recovering three-dimensional shape around a corner using ultrafast time-of-flight imaging,” Nat. Commun. 3, 745 (2012).
[Crossref] [PubMed]

Velten, A.

M. Buttafava, J. Zeman, A. Tosi, K. Eliceiri, and A. Velten, “Non-line-of-sight imaging using a time-gated single photon avalanche diode,” Opt. Express 23, 20997–21011 (2015).
[Crossref] [PubMed]

A. Velten, T. Willwacher, O. Gupta, A. Veeraraghavan, M. G. Bawendi, and R. Raskar, “Recovering three-dimensional shape around a corner using ultrafast time-of-flight imaging,” Nat. Commun. 3, 745 (2012).
[Crossref] [PubMed]

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, D. Gutierrez, A. Jarabo, and A. Velten, “Virtual wave optics for non-line-of-sight imaging,” arXiv:1810.07535 [cs.CV].

S. A Reza, M. La Manna, and A. Velten, “A physical light transport model for non-line-of-sight imaging applications,” arXiv:1802.1823 [physics.optics].

S. A. Reza, M. La Manna, S. Bauer, and A. Velten, “Wave-like properties of phasor fields: experimental demonstrations,” arXiv:190401565 [physics.optics].

Walther, A.

Wetzstein, G.

M. O’Toole, D. B. Lindell, and G. Wetzstein, “Confocal non-line-of-sight imaging based on the light-cone transform,” Nature 555, 338–341 (2018).
[Crossref]

Willwacher, T.

A. Velten, T. Willwacher, O. Gupta, A. Veeraraghavan, M. G. Bawendi, and R. Raskar, “Recovering three-dimensional shape around a corner using ultrafast time-of-flight imaging,” Nat. Commun. 3, 745 (2012).
[Crossref] [PubMed]

Wong, F. N. C.

F. Xu, G. Shulkind, C. Thrampoulidis, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Revealing hidden scenes by photon-efficient occlusion-based opportunistic active imaging,” Opt. Express 26, 9945 (2018).
[Crossref] [PubMed]

C. Thrampoulidis, G. Shulkind, F. Xu, W. T. Freeman, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Exploiting occlusion in non-line-of-sight active imaging,” IEEE Trans. Comput. Imag. 4, 419 (2018).
[Crossref]

Wornell, G. W.

C. Thrampoulidis, G. Shulkind, F. Xu, W. T. Freeman, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Exploiting occlusion in non-line-of-sight active imaging,” IEEE Trans. Comput. Imag. 4, 419 (2018).
[Crossref]

F. Xu, G. Shulkind, C. Thrampoulidis, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Revealing hidden scenes by photon-efficient occlusion-based opportunistic active imaging,” Opt. Express 26, 9945 (2018).
[Crossref] [PubMed]

Xiao, L.

F. Heide, L. Xiao, W. Heidrich, and M. B. Hullin, “Diffuse mirrors: 3D reconstruction from diffuse indirect illumination using inexpensive time-of-flight sensors,” in Proc. IEEE Conf. Comput. Vis. Pattern Recog., pp. 3222–3229 (2014).

Xu, F.

F. Xu, G. Shulkind, C. Thrampoulidis, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Revealing hidden scenes by photon-efficient occlusion-based opportunistic active imaging,” Opt. Express 26, 9945 (2018).
[Crossref] [PubMed]

C. Thrampoulidis, G. Shulkind, F. Xu, W. T. Freeman, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Exploiting occlusion in non-line-of-sight active imaging,” IEEE Trans. Comput. Imag. 4, 419 (2018).
[Crossref]

Zeman, J.

Zhao, H.

A. Kadambi, H. Zhao, B. Shi, and R. Raskar, “Occluded imaging with time-of-flight sensors,” ACM Trans. Graph. 35, 1–12 (2016).
[Crossref]

ACM Trans. Graph. (1)

A. Kadambi, H. Zhao, B. Shi, and R. Raskar, “Occluded imaging with time-of-flight sensors,” ACM Trans. Graph. 35, 1–12 (2016).
[Crossref]

Adv. Opt. Photon. (1)

IEEE Trans. Comput. Imag. (1)

C. Thrampoulidis, G. Shulkind, F. Xu, W. T. Freeman, J. H. Shapiro, A. Torralba, F. N. C. Wong, and G. W. Wornell, “Exploiting occlusion in non-line-of-sight active imaging,” IEEE Trans. Comput. Imag. 4, 419 (2018).
[Crossref]

Int. J. Comput. Vision (1)

A. Kirmani, T. Hutchison, J. Davis, and R. Raskar, “Looking around the corner using ultrafast transient imaging,” Int. J. Comput. Vision 95, 13–28 (2011).
[Crossref]

J. Math. Phys. (1)

A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).
[Crossref]

J. Opt. Soc. Am. (2)

Nat. Commun. (1)

A. Velten, T. Willwacher, O. Gupta, A. Veeraraghavan, M. G. Bawendi, and R. Raskar, “Recovering three-dimensional shape around a corner using ultrafast time-of-flight imaging,” Nat. Commun. 3, 745 (2012).
[Crossref] [PubMed]

Nat. Photonics (1)

G. Gariepy, F. Tonolini, R. Henderson, J. Leach, and D. Faccio, “Detection and tracking of moving objects hidden from view,” Nat. Photonics 10, 23–27 (2015).
[Crossref]

Nature (1)

M. O’Toole, D. B. Lindell, and G. Wetzstein, “Confocal non-line-of-sight imaging based on the light-cone transform,” Nature 555, 338–341 (2018).
[Crossref]

Opt. Express (2)

Proc. SPIE (1)

J. Klein, M. Laurenzis, and M. Hullin, “Transient imaging for real-time tracking around a corner,” Proc. SPIE 9988, 998802 (2016).
[Crossref]

Other (12)

E. H. Adelson and J. R. Bergen, “The plenoptic function and the elements of early vision,” in M. S. Landy and J. A. Movshon, eds., Computational Models of Visual Processing, (MIT Press, 1991), pp. 3–20.

M. Levoy and P. Hanrahan, “Light field rendering,” in Proc. SIGGRAPH (ACM, New York, NY, USA, 1996), pp. 31–42.

F. Heide, L. Xiao, W. Heidrich, and M. B. Hullin, “Diffuse mirrors: 3D reconstruction from diffuse indirect illumination using inexpensive time-of-flight sensors,” in Proc. IEEE Conf. Comput. Vis. Pattern Recog., pp. 3222–3229 (2014).

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2: Multiple Scattering, Turbulence, Rough Surfaces, and Remote Sensing (Academic, New York, 1978).

For notational convenience, we have assumed that the diffraction takes place between the z = 0 and z = L planes, but the result we obtain will apply for +z-going Fresnel diffraction over a distance L starting from an arbitrary z plane.

S. A Reza, M. La Manna, and A. Velten, “A physical light transport model for non-line-of-sight imaging applications,” arXiv:1802.1823 [physics.optics].

X. Liu, I. Guillén, M. La Manna, J. H. Nam, S. A. Reza, T. H. Le, D. Gutierrez, A. Jarabo, and A. Velten, “Virtual wave optics for non-line-of-sight imaging,” arXiv:1810.07535 [cs.CV].

S. A. Reza, M. La Manna, S. Bauer, and A. Velten, “Wave-like properties of phasor fields: experimental demonstrations,” arXiv:190401565 [physics.optics].

The short-time average z-plane irradiance is the instantaneous irradiance averaged over a time Ta satisfying ω0Ta ≫ 1 and ΔωTa ≪ 1.

In what follows, integrals without explicit limits are over the integration variable’s entire domain.

Because h0(ρ) is a zero-mean Gaussian process, its samples at ρ0 and ρ′0 are zero-mean jointly Gaussian random variables whose joint characteristic function is as given in Eq. (12).

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1: Single Scattering and Transport Theory (Academic, New York, 1978).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1 Unfolded geometry for three-bounce NLoS active imaging. Scalar, paraxial diffraction theory is assumed, with {Ek(ρk, t) : 0 ≤ k ≤ 2} being the baseband complex-field envelopes illuminating the z = 0, z = L1, and z = L1 + L2 planes, respectively, written as functions of the transverse spatial coordinates, {ρk = (xk, yk) : 0 ≤ k ≤ 2}, in those planes and time, t. The blue rectangles represent thin transmissive diffusers, and the black line represents a thin transmission screen whose intensity transmission pattern, T(ρ1), is to be imaged using the light that emerges from the z = L1 + L2 plane.
Fig. 2
Fig. 2 Thin-lens imaging setup. A focal-length f thin lens casts an inverted image of the intensity pattern that emerges from the diffuser at z = L1 + L2. The image is located in the plane—shown as a black dashed line—a distance Lim behind the lens, where 1/f = 1/L3 + 1/Lim.
Fig. 3
Fig. 3 Unfolded geometry for three-bounce, occlusion-aided NLoS active imaging. Scalar, paraxial diffraction theory is assumed, with E0(ρ0, t) being the baseband complex-field envelope illuminating the z = 0 plane and E′2(ρ2, t) being the baseband complex-field envelope emerging from the z = L1 + L2 plane. These fields are written as functions of their transverse spatial coordinates, {ρk = (xk, yk) : k = 0, 2}, in their respective planes and time, t. The blue rectangles represent thin transmissive diffusers, and the black line at z = L1 represents a thin specular-plus-diffuser transmission mask with field-transmission function F(ρ1), whose associated intensity-transmission pattern is to be imaged using the light that emerges from the z = L1 + L2 plane. That imaging process is aided by the presence of occluders in the z = L1Ld and z = L1 + L′d planes, whose field-transmission functions are P(ρd) and P′(ρ′d), respectively.
Fig. 4
Fig. 4 Plots of Gps(ρ)/Gps(∞) for the Gaussian pinspeck versus ρ/ρres(Ω) for ρ = (x, 0) and Ω = 0.1, 1, and 10.

Equations (113)

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

0 ( ρ 0 , ω ) d t E 0 ( ρ 0 , t ) e i ω t ,
0 ( ρ 0 , ω ) = 0 ( ρ 0 , ω ) exp [ i ( ω 0 + ω ) h 0 ( ρ 0 ) / c ] ,
1 ( ρ 1 , ω ) = d 2 ρ 0 0 ( ρ 0 , ω ) exp [ i ( ω 0 + ω ) L 1 / c + i ( ω 0 + ω ) | ( ρ 1 ρ 0 ) | 2 / 2 c L 1 ] ( ω 0 + ω ) i 2 π c L 1 ,
1 ( ρ 1 , ω ) = 1 ( ρ 1 , ω ) T ( ρ 1 ) exp [ i ( ω 0 + ω ) h 1 ( ρ 1 ) / c ] ,
2 ( ρ 2 , ω ) = d 2 ρ 1 1 ( ρ 1 , ω ) exp [ i ( ω 0 + ω ) L 2 / c + i ( ω 0 + ω ) | ( ρ 2 ρ 1 ) | 2 / 2 c L 2 ] ( ω 0 + ω ) i 2 π c L 2 ,
2 ( ρ 2 , ω ) = 2 ( ρ 2 , ω ) exp [ i ( ω 0 + ω ) h 2 ( ρ 2 ) / c ] ,
I 1 ( ρ 1 , t ) = d ω 2 π d ω 2 π 1 ( ρ 1 , ω ) 1 * ( ρ 1 , ω ) e i ( ω ω ) t
= d ω 2 π [ d ω + 2 π 1 ( ρ 1 , ω + + ω / 2 ) 1 * ( ρ 1 , ω + ω / 2 ) ] e i ω t
= d ω 2 π 𝒫 1 ( ρ 1 , ω ) e i ω t ,
𝒫 1 ( ρ 1 , ω ) = d 2 ρ 0 d 2 ρ 0 d ω + 2 π 0 ( ρ 0 , ω ) 0 * ( ρ 0 , ω ) e i [ ( ω 0 + ω ) h 0 ( ρ 0 ) ( ω 0 + ω ) h 0 ( ρ 0 ) ] / c × ( ω 0 + ω ) ( ω 0 + ω ) e i ( ω ω ) L 1 / c + i [ ( ω 0 + ω ) | ρ 1 ρ 0 | 2 ( ω 0 + ω ) | ρ 1 ρ 0 | 2 ] / 2 c L 1 / ( 2 π c L 1 ) 2 ,
𝒫 1 ( ρ 1 , ω ) = d 2 ρ 0 d 2 ρ 0 d ω + 2 π 0 ( ρ 0 , ω ) 0 * ( ρ 0 , ω ) e i ω 0 [ h 0 ( ρ 0 ) h 0 ( ρ 0 ) ] / c ω 0 2 / ( 2 π c L 1 ) 2 × e i ( ω ω ) L 1 / c + i [ ( ω 0 + ω ) | ρ 1 ρ 0 | 2 ( ω 0 + ω ) | ρ 1 ρ 0 | 2 ] / 2 c L 1 .
e i ω 0 [ h 0 ( ρ 0 ) h 0 ( ρ 0 ) ] / c = exp { ω 0 2 [ σ h 2 K h ( | ρ 0 ρ 0 | ) ] / c 2 } .
e i ω 0 [ h 0 ( ρ 0 ) h 0 ( ρ 0 ) ] / c λ 0 2 δ ( ρ 0 ρ 0 ) ,
𝒫 1 ( ρ 1 , ω ) = d 2 ρ 0 d ω + 2 π 0 ( ρ 0 , ω ) 0 * ( ρ 0 , ω ) e i ( ω ω ) L 1 / c + i ( ω ω ) | ρ 1 ρ 0 | 2 / 2 c L 1 / L 1 2 .
= d 2 ρ 0 𝒫 ( ρ 0 , ω ) e i ω L 1 / c + i ω | ρ 1 ρ 0 | 2 / 2 c L 1 / L 1 2 .
𝒫 0 ( ρ 0 , ω ) = d ω + 2 π 0 ( ρ 0 , ω + + ω / 2 ) 0 * ( ρ 0 , ω + ω / 2 ) ,
I 1 ( ρ 1 , t ) = d 2 ρ 0 I 0 ( ρ 0 , t L 1 / c | ρ 1 ρ 0 | 2 / 2 c L 1 ) / L 1 2 ,
exp [ i ω L 1 2 + | ρ 1 ρ 0 | 2 / c ] L 1 2 + | ρ 1 ρ 0 | 2 exp ( i ω L 1 / c + i ω | ρ 1 ρ 0 | 2 / 2 c L 1 ) L 1 , for | ω | Δ ω
𝒫 2 ( ρ 2 , ω ) d ω + 2 π 2 ( ρ 2 , ω + + ω / 2 ) 2 * ( ρ 2 , ω + ω / 2 )
= d 2 ρ 1 𝒫 1 ( ρ 1 , ω ) T ( ρ 1 ) exp ( i ω L 2 / c + i ω | ρ 2 ρ 1 | 2 / 2 c L 2 ) / L 2 2 ,
𝒫 2 ( ρ 2 , ω ) = d 2 ρ 1 ( d 2 ρ 0 𝒫 0 ( ρ 0 , ω ) exp ( i ω L 1 / c + i ω | ρ 1 ρ 0 | 2 / 2 c L 1 ) / L 1 2 ) × T ( ρ 1 ) exp ( i ω L 2 / c + i ω | ρ 2 ρ 1 | 2 / 2 c L 2 ) / L 2 2 .
𝒫 2 ( ρ 2 , 0 ) = d 2 ρ 1 T ( ρ 1 ) d 2 ρ 0 𝒫 0 ( ρ 0 , 0 ) / ( L 1 L 2 ) 2 ,
im ( ρ im , ω ) = | ρ 3 | D / 2 d 2 ρ 3 e i ( ω 0 + ω ) L im / c + i ( ω 0 + ω ) | ρ im ρ 3 | 2 / 2 c L im i ( ω 0 + ω ) | ρ 3 | 2 / 2 c f i λ 0 L im × d 2 ρ 2 2 ( ρ 2 , ω ) e i ( ω 0 + ω ) L 3 / c + i ( ω 0 + ω ) | ρ 3 ρ 2 | 2 / 2 c L 3 i λ 0 L 3
= e i ( ω 0 + ω ) | ρ im | 2 / 2 c L im d 2 ρ 2 2 ( ρ 2 , ω ) e i ( ω 0 + ω ) ( L 3 + L im ) / c + i ( ω 0 + ω ) | ρ 2 | 2 / 2 c L 3 i λ 0 L 3 × | ρ 3 | D / 2 d 2 ρ 3 e i ( ω + ω 0 ) ρ 3 ( ρ 2 / L 3 + ρ im / L im ) / c i λ 0 L im .
im ( ρ im , ω ) = e i ( ω 0 + ω ) | ρ im | 2 / 2 c L im × d 2 ρ 2 2 ( ρ 2 , ω ) e i ( ω 0 + ω ) ( L 3 + L im ) / c + i ( ω 0 + ω ) | ρ 2 | 2 / 2 c L 3 λ 0 2 L 3 L im π D 2 4 J 1 ( π D λ 0 | ρ 2 L 3 + ρ im L im | ) π D 2 λ 0 | ρ 2 L 3 + ρ im L im | ,
2 ( ρ 2 , ω ) 2 * ( ρ 2 , ω ) λ 0 2 2 ( ρ 2 , ω ) 2 * ( ρ 2 , ω ) δ ( ρ 2 ρ 2 ) ,
𝒫 im ( ρ im , ω ) = d 2 ρ 2 𝒫 2 ( ρ 2 , ω ) × e i ω ( L 3 + L im ) / c + i ω | ρ 2 | 2 / 2 c L 3 + i ω | ρ im | 2 / 2 c L im [ π D 2 4 λ 0 L 3 L im J 1 ( π D λ 0 | ρ 2 L 3 + ρ im L im | ) π D 2 λ 0 | ρ 2 L 3 + ρ im L im | ] 2 .
I im ( ρ im , t ) = d 2 ρ 2 I 2 ( ρ 2 , t ( L 3 + L im ) / c | ρ 2 | 2 / 2 c L 3 | ρ im | 2 / 2 c L im ) × [ π D 2 4 λ 0 L 3 L im J 1 ( π D λ 0 | ρ 2 L 3 + ρ im L im | ) π D 2 λ 0 | ρ 2 L 3 + ρ im L im | ] 2 .
E 0 ( ρ 0 , t ) = { 8 P 0 π d 2 e 4 | ρ 0 | 2 / d 2 cos ( Δ ω t / 2 ) , for | t | t 0 / 2 , 0 , otherwise ,
I 0 ( ρ 0 , t ) = { 8 P 0 π d 2 e 8 | ρ 0 2 / d 2 cos 2 ( Δ ω t / 2 ) = 4 P 0 π d 2 e 8 | ρ 0 2 / d 2 [ 1 + cos ( Δ ω t ) ] , for | t | t 0 / 2 , 0 , otherwise ,
𝒫 0 ( ρ 0 , ω ) = 8 P 0 t 0 π d 2 e 8 | ρ 0 | 2 / d 2 [ sin ( ω t 0 / 2 ) ω t 0 / 2 + sin [ ( ω + Δ ω ) t 0 / 2 ] ( ω + Δ ω ) t 0 + sin [ ( ω Δ ω ) t 0 / 2 ] ( ω Δ ω ) t 0 ] ,
𝒫 1 ( ρ 1 , Δ ω ) d 2 ρ 0 4 P 0 t 0 π d 2 e 8 | ρ 0 | 2 / d 2 exp ( i Δ ω L 1 / c + i Δ ω | ρ 1 ρ 0 | 2 / 2 c L 1 ) L 1 2 ,
𝒫 ˜ 2 ( ρ 2 , Δ ω ) ( L im / L 3 ) 2 𝒫 im ( ρ 2 L im / L 3 , Δ ω ) e i Δ ω ( L 3 + L im ) / c i Δ ω | ρ 2 | 2 / 2 c L 3 i ω | ρ im | 2 / 2 c L im ,
T ˜ ( ρ ˜ 1 ) | 𝒫 1 ( ρ ˜ 1 , Δ ω ) | = | | ρ 2 | D / 2 d 2 ρ 2 𝒫 ˜ 2 ( ρ 2 , Δ ω ) e i Δ ω | ρ 2 | 2 / 2 c L 2 + i Δ ω ρ 2 ρ ˜ 1 / c L 2 Δ λ 2 | ,
𝒫 ˜ 2 ( ρ 2 , Δ ω ) 𝒫 2 ( ρ 2 , Δ ω ) ,
T ˜ ( ρ ˜ 1 ) | 𝒫 1 ( ρ ˜ 1 , Δ ω ) | = | d 2 ρ 1 𝒫 1 ( ρ 1 , Δ ω ) T ( ρ 1 ) e i Δ ω | ρ 1 | 2 / 2 c L 2 × π 4 ( D Δ λ L 2 ) 2 J 1 ( π D | ρ ˜ 1 ρ 1 | / Δ λ L 2 ) π D | ρ ˜ 1 ρ 1 | / 2 Δ λ L 2 | .
I z ( ρ + , s , t ) d 2 ρ λ 0 2 E z ( ρ + + ρ / 2 , t ) E z * ( ρ + ρ / 2 , t ) e i 2 π s ρ / λ 0 .
W ( ρ + , k ) d 2 ρ E z ( ρ + + ρ / 2 ) E z * ( ρ + ρ / 2 ) e i k ρ ,
I z ( ρ + , t ) = d 2 s I z ( ρ + , s , t ) ,
I z ( ρ + , s , t ) = d ω 2 π d ω 2 π d 2 ρ λ 0 2 z ( ρ + + ρ / 2 , ω ) z * ( ρ + ρ / 2 , ω ) e i 2 π s ρ / λ 0 e i ( ω ω ) t
= d ω 2 π [ d ω + 2 π ( d 2 ρ λ 0 2 z ( ρ + + ρ / 2 , ω ) z * ( ρ + ρ / 2 , ω ) e 2 π s ρ / λ 0 ) ] e i ω t ,
W z ( ρ + , k , ω + , ω ) d 2 ρ z ( ρ + + ρ / 2 , ω + + ω / 2 ) z * ( ρ + ρ / 2 , ω + ω / 2 ) e i k ρ ,
I z ( ρ + , s , t ) = 1 λ 0 2 d ω 2 π d ω + 2 π W z ( ρ + , 2 π s / λ 0 , ω + , ω ) e i ω t .
Γ z ( ρ 1 , ρ 2 , t 1 , t 2 ) E z ( ρ 1 , t 1 ) E z * ( ρ 2 , t 2 ) ,
I z ( ρ + , s , t ) = d 2 ρ λ 0 2 Γ z ( ρ + + ρ / 2 , ρ + ρ / 2 , t , t ) e i 2 π s ρ / λ 0 ,
W z ( ρ + , k , ω + , ω ) = d 2 ρ d t 1 d t 2 Γ z ( ρ + + ρ / 2 , ρ + ρ / 2 , t + + t / 2 , t + t / 2 ) × e i ( ω + t + ω t + k ρ ) ,
Γ z ( ρ + + ρ / 2 , ρ + ρ / 2 , t + + t / 2 , t + t / 2 ) = d 2 k ( 2 π ) 2 d ω + 2 π d ω 2 π W z ( ρ + , k , ω + , ω ) × e i ( ω + t + ω t + k ρ ) .
E z ( ρ , t ) = d τ d 2 ρ E z ( ρ , τ ) h ( ρ , ρ ; t , τ ) ,
I z ( ρ + , s , ω + , t ) 1 λ 0 2 d ω 2 π W ( ρ + , 2 π s / λ 0 , ω + , ω ) e i ω t ,
𝒫 z ( ρ + , ω ) = d ω + 2 π d 2 k ( 2 π ) 2 W z ( ρ + , k , ω + , ω ) .
W 0 ( ρ + , k , ω + , ω ) = λ 0 2 d 2 k ( 2 π ) 2 W 0 ( ρ + , k , ω + , ω ) .
W L 1 L d ( ρ + , k , ω + , ω ) = d 2 k ( 2 π ) 2 W L 1 L d ( ρ + , k , ω + , ω ) W P ( ρ + , k k ) .
W L 1 ( ρ + , k , ω + , ω ) = d 2 k ( 2 π ) 2 W L 1 ( ρ + , k , ω + , ω ) W F ( ρ + , k k ) + λ 0 2 ( ρ + ) d 2 k ( 2 π ) 2 W L 1 ( ρ + , k , ω + , ω ) .
W L 1 L d ( ρ + , k , ω + , ω ) = W 0 ( ρ + c ( L 1 L d ) k / ω 0 , k , ω + , ω ) e i [ ω ( L 1 L d ) / c ] ( 1 + c 2 | k | 2 / 2 ω 0 2 ) .
W 0 ( ρ + , k , ω + , ω ) = W in ( ω + , ω ) ( 2 π / λ 0 ) 2 δ ( k ) ,
W in ( ω + , ω ) = λ 0 2 d t I 0 ( t ) e i ( ω + + ω / 2 ) t d u I 0 ( u ) e i ( ω + ω / 2 ) u .
W 0 ( ρ + , k , ω + , ω ) = W in ( ω + , ω ) ,
W L ( ρ + , k , ω + , ω ) = W in ( ω + , ω ) e i ( ω L / c ) ( 1 + c 2 | k | 2 / 2 ω 0 2 ) .
W L ( ρ + , k , ω + , ω ) = ( ρ + ) W in ( ω + , ω ) e i ω L / c 2 π i c / ω L .
W 3 L / 2 ( ρ + , k , ω + , ω ) = ( ρ + c L k / 2 ω 0 ) W in ( ω + , ω ) e i ω 3 L / 2 c e i ω c L | k | 2 / 4 ω 0 2 2 π i c / ω L ,
W 3 L / 2 ( ρ + , k , ω + , ω ) = W in ( ω + , ω ) d 2 k ( 2 π ) 2 ( ρ + c L k / 2 ω 0 ) e i ω 3 L / 2 c e i ω c L | k | 2 / 4 ω 0 2 × W P ( ρ + , k k ) 2 π i c / ω L .
W 2 L ( ρ + k , ω + , ω ) = W in ( ω + , ω ) d 2 k ( 2 π ) 2 ( ρ + c L ( k + k ) / 2 ω 0 ) e i ω 2 L / c × e i ω c L ( | k | 2 + | k | 2 ) / 4 ω 0 2 W P ( ρ + c L k / 2 ω 0 , k k ) 2 π i c / ω L ,
𝒫 2 L ( ρ + , ω ) = d ω + 2 π W in ( ω + , ω ) d 2 k ( 2 π ) 2 d 2 k ( 2 π ) 2 ( ρ + c L ( k + k ) / 2 ω 0 ) e i ω 2 L / c × e i ω c L ( | k | 2 + | k | 2 ) / 4 ω 0 2 W P ( ρ + c L k / 2 ω 0 , k k ) 2 π i c / ω L .
𝒫 0 ( ρ + , ω ) = d ω + 2 π d 2 k ( 2 π ) 2 W 0 ( ρ + , k , ω + , ω ) = d t I 0 ( t ) e i ω t ,
𝒫 2 L ( ρ + , ω ) = λ 0 2 𝒫 0 ( ω ) e i ω 2 L / c d 2 k + ( 2 π ) 2 d 2 k ( 2 π ) 2 ( ρ + c L k + / ω 0 ) × e i ω c L ( 2 | k + | 2 + | k | 2 / 2 ) / 4 ω 0 2 W P ( ρ + c L ( k + / 2 + k / 4 ) / ω 0 , k ) 2 π i c / ω L ,
G ( ρ , ω ) = d 2 k ( 2 π ) 2 e i ω c L | k | 2 / 8 ω 0 2 W P ( ρ / 2 c L k / 4 ω 0 , k ) 2 π i c / ω L .
𝒫 2 L ( ρ + , ω ) = λ 0 2 𝒫 0 ( ω ) e i ω 2 L / c d 2 k + ( 2 π ) 2 ( ρ + c L k + / ω 0 ) × G ( 2 ρ + + c L k + / ω 0 , ω ) e i ω c L | k + | 2 / 2 ω 0 2 .
𝒫 2 L ( ρ + , ω ) = 𝒫 0 ( ω ) e i ω 2 L / c d 2 ρ ˜ ( ρ ˜ ) G ( ρ + ρ ˜ , ω ) e i ω | ρ + ρ ˜ | 2 / 2 c L L 2 .
W E 0 ( ρ + , k ) d 2 ρ E 0 ( ρ + + ρ / 2 ) E 0 * ( ρ + ρ / 2 ) e i k ρ ,
I 2 L ( ρ + ) | E 2 L ( ρ + ) | 2 = I 0 d 2 ρ ˜ ( ρ ˜ ) G ( ρ + ρ ˜ ) ,
G ( ρ ) π L 2 d 2 k ( 2 π ) 2 W P ( ρ / 2 c L k / 4 ω 0 , k ) ,
P ph ( ρ ) = e | ρ | 2 / 2 ρ 0 2 ,
P ps ( ρ ) = 1 e | ρ | 2 / 2 ρ 0 2 ,
G ph ( ρ ) = π Ω 2 L 2 ( 1 + Ω 2 ) exp [ Ω 2 1 + Ω 2 | ρ | 2 4 ρ 0 2 ] ,
G ph opt ( ρ ) = π exp ( π | ρ | 2 / λ 0 L ) 2 L 2 ,
G ph ( ρ ) / G ph ( 0 ) = exp [ π | ρ 2 | ρ res 2 ( Ω ) ] ,
G ps ( ρ ) = π L 2 | 1 Ω 1 + Ω 2 exp [ Ω 1 + Ω 2 | ρ | 2 8 ρ 0 2 ( Ω i ) i tan 1 ( 1 / Ω ) ] | 2 .
G ps opt ( ρ ) = π L 2 | 1 exp ( π | ρ | 2 ( 1 i ) / 2 λ 0 L i π / 4 ) 2 | 2 .
G ps ( ρ ) / G ps ( ) = [ 1 exp ( | ρ | 2 / 8 ρ 0 2 ) ] 2 ,
P ( ρ ) = circ ( 2 ρ / d ) { 1 , for | ρ | d / 2 0 , otherwise .
z ( ρ , ω ) = z ( ρ , ω ) e i ( ω 0 + ω ) h z ( ρ ) / c z ( ρ , ω ) e i ω 0 h z ( ρ ) / c ,
e i ω 0 [ h z ( ρ ) h k ( ρ ) ] / c λ 0 2 δ ( ρ ρ ) .
W z ( ρ + , k , ω + , ω ) = d 2 ρ z ( ρ + + ρ / 2 , ω + + ω / 2 ) z * ( ρ + ρ / 2 , ω + ω / 2 ) e i k · ρ
= d 2 ρ z ( ρ + + ρ / 2 , ω + + ω / 2 ) z * ( ρ + ρ / 2 , ω + ω / 2 ) × e i ω 0 [ h z ( ρ + + ρ / 2 ) h k ( ρ + ρ / 2 ) ] / c e i k ρ
= λ 0 2 z ( ρ + , ω + + ω / 2 ) z * ( ρ + , ω + ω / 2 )
= λ 0 2 d 2 k ( 2 π ) 2 W z ( ρ + , k , ω + , ω ) .
z ( ρ , ω ) = z ( ρ , ω ) P ( ρ ) .
W z ( ρ + , k , ω + , ω ) = d 2 ρ z ( ρ + + ρ / 2 , ω + + ω / 2 ) z * ( ρ + ρ / 2 , ω + ω / 2 ) e i k · ρ
= d 2 ρ z ( ρ + + ρ / 2 , ω + + ω / 2 ) z * ( ρ + ρ / 2 , ω + ω / 2 ) × P ( ρ + + ρ / 2 ) P * ( ρ + ρ / 2 ) e i k ρ
= d 2 k ( 2 π ) 2 W z ( ρ + , k , ω + , ω ) d 2 ρ P ( ρ + + ρ / 2 ) P * ( ρ + ρ / 2 ) e i ( k k ) ρ
= d 2 k ( 2 π ) 2 W z ( ρ + , k , ω + , ω ) W P ( ρ + , k k ) ,
W P ( ρ + , k ) d 2 ρ P ( ρ + + ρ / 2 ) P * ( ρ + ρ / 2 ) e i k ρ
𝒫 z ( ρ + , ω ) = d ω + 2 π d 2 k ( 2 π ) 2 W z ( ρ + , k , ω + , ω )
= d ω + 2 π d 2 k ( 2 π ) 2 d 2 k ( 2 π ) 2 W z ( ρ + , k , ω + , ω ) W P ( ρ + , k k )
= d ω + 2 π d 2 k ( 2 π ) 2 W z ( ρ + , k , ω + , ω ) | P ( ρ + ) | 2 = 𝒫 z ( ρ + , ω ) | P ( ρ + ) | 2 ,
W z ( ρ + , k , ω + , ω ) = d 2 ρ z ( ρ + + ρ / 2 , ω + + ω / 2 ) z * ( ρ + ρ / 2 , ω + ω / 2 ) e i k ρ
= d 2 ρ z ( ρ + + ρ / 2 , ω + + ω / 2 ) z * ( ρ + ρ / 2 , ω + , ω / 2 ) × F ( ρ + + ρ / 2 ) F * ( ρ + ρ / 2 ) e i k ρ .
W z ( ρ + , k , ω + , ω ) = d 2 ρ z ( ρ + + ρ / 2 , ω + + ω / 2 ) z * ( ρ + ρ / 2 , ω + ω / 2 ) × ( F ( ρ + + ρ / 2 ) F * ( ρ + ρ / 2 ) + Δ F ( ρ + + ρ / 2 ) Δ F * ( ρ + ρ / 2 ) ) e i k ρ
= d 2 k ( 2 π ) 2 W L 1 ( ρ + , k , ω + , ω ) W F ( ρ + , k k ) + λ 0 2 ( ρ + ) d 2 k ( 2 π ) 2 W L 1 ( ρ + , k , ω + , ω ) .
L ( ρ L , ω ) = d 2 ρ 0 0 ( ρ 0 , ω ) ( ω 0 + ω ) e i ( ω 0 + ω ) ( L / c + | ρ L ρ 0 | 2 / 2 c L ) i 2 π c L ,
W L ( ρ + , k , ω + , ω ) = d 2 ρ d 2 ρ 0 d 2 ρ 0 0 ( ρ 0 , ω + + ω / 2 ) 0 * ( ρ 0 , ω + ω / 2 ) × e i ω L / c e i k ρ ( ω 0 + ω + + ω / 2 ) e i ( ω 0 + ω + + ω / 2 ) | ρ + + ρ / 2 ρ 0 | 2 / 2 c L i 2 π c L × ( ω 0 + ω + ω / 2 ) e i ( ω 0 + ω + ω / 2 ) | ρ + ρ / 2 ρ 0 | 2 / 2 c L i 2 π c L .
W L ( ρ + , k , ω + , ω ) = d 2 ρ d 2 ρ 0 + d 2 ρ 0 ( ρ 0 + + ρ 0 / 2 , ω + + ω / 2 ) 0 * ( ρ 0 + ρ 0 / 2 , ω + ω / 2 ) × e i ω L / c ( λ 0 L ) 2 e i ( ω 0 + ω + ) + ( ρ + ρ 0 + ) ( ρ ρ 0 ) / c L e i ω ( | ρ + ρ 0 + | 2 + | ρ ρ 0 | 2 / 4 ) / 2 c L e i k ρ .
W L ( ρ + , k , ω + , ω ) = d 2 ρ 0 + d 2 ρ 0 0 ( ρ 0 + + ρ 0 / 2 , ω + + ω / 2 ) 0 * ( ρ 0 + ρ 0 / 2 , ω + ω / 2 ) e i ω L / c ( λ 0 L ) 2 × e i ( ω 0 + ω + ) ( ρ + ρ 0 + ) ρ 0 / c L e i ω ( | ρ + ρ 0 + | 2 / 2 c L + | ρ 0 | 2 / 8 c L ) × d 2 ρ e i ω | ρ | 2 / 8 c L e i [ k ( ω 0 + ω + ) ( ρ + ρ 0 + ) / c L + ω ρ 0 / 4 c L ] ρ .
W L ( ρ + , k , ω + , ω ) = d 2 ρ 0 + d 2 ρ 0 0 ( ρ 0 + + ρ 0 / 2 , ω + + ω / 2 ) 0 * ( ρ 0 + ρ 0 / 2 , ω + ω / 2 ) e i ω L / c ( λ 0 L ) 2 × e i ( ω 0 + ω + ) ( ρ + ρ 0 + ) ρ 0 / c L e i ω | ρ + ρ 0 + | 2 / 2 c L e i ω | ρ 0 | 2 / 8 c L ( i 8 π c L / ω ) × e 2 i c L | k ( ω 0 + ω + ) ( ρ + ρ 0 + ) / c L + ω ρ 0 / 4 c L | 2 / ω ,
W L ( ρ + , k , ω + , ω ) = d 2 ρ 0 + d 2 ρ 0 0 ( ρ 0 + + ρ 0 / 2 , ω + + ω / 2 ) 0 * ( ρ 0 + ρ 0 / 2 , ω + ω / 2 ) × e i ω L / c ( λ 0 L ) 2 e i ω | ρ + ρ 0 + | 2 / 2 c L e 2 i c L | k ( ω 0 + ω + ) ( ρ + ρ 0 + ) / c L | 2 / ω e i k ρ 0 ( i 8 π c L / ω )
= d 2 ρ 0 + W 0 ( ρ 0 + , k , ω + , ω ) e i ω L / c ( λ 0 L ) 2 e i ω | ρ + ρ 0 + | 2 / 2 c L × e 2 i c L | k ( ω 0 + ω + ) ( ρ + ρ 0 + ) / c L | 2 / ω ( i 8 π c L / ω ) .
e 2 i c L | k ( ω 0 + ω + ) ( ρ + ρ 0 + ) / c L | 2 / ω i 8 π c L / ω ( λ 0 L ) 2
W L ( ρ + , k , ω + , ω ) = W 0 ( ρ + c L k / ( ω 0 + ω + ) , k , ω + , ω ) e i ( ω L / c ) ( 1 + c 2 | k | 2 / 2 ( ω 0 + ω + ) 2 ) .
W L ( ρ + , k , ω + , ω ) = W 0 ( ρ + c L k / ω 0 , k , ω + , ω ) e i ( ω L / c ) ( 1 + c 2 | k | 2 / 2 ω 0 2 ) .
𝒫 L ( ρ + , ω ) = d ω + 2 π d 2 k ( 2 π ) 2 W 0 ( ρ + c L k / ω 0 , k , ω + , ω ) e i ( ω L / c ) ( 1 + c 2 | k | 2 / 2 ω 0 2 ) .
𝒫 L ( ρ + , ω ) = λ 0 2 d ω + 2 π d 2 k ( 2 π ) 2 d 2 k ( 2 π ) 2 W 0 ( ρ + c L k / ω 0 , k , ω + , ω ) e i ( ω L / c ) ( 1 + c 2 | k | 2 / 2 ω 0 2 ) .
𝒫 L ( ρ + , ω ) = d ω + 2 π d 2 ρ 0 d 2 k ( 2 π ) 2 W 0 ( ρ 0 , k , ω + , ω ) e i ( ω L / c ) ( 1 + | ρ + ρ 0 | 2 / 2 L 2 ) L 2 .
𝒫 L ( ρ + , ω ) = d 2 ρ 0 𝒫 0 ( ρ 0 , ω ) e i ω L / c e i ω | ρ + ρ 0 | 2 / 2 c L L 2 ,

Metrics