Abstract

Due to multiple light scattering inside biological tissues, deep non-invasive optical medical imaging is very challenging. Acousto-optic imaging is a technique coupling ultrasound and light that allows recovering optical contrast at depths of few centimeters with a millimeter resolution. Recent advances in acousto-optic imaging are using short focused ultrasound pulses often averaged over several hundred or thousand pulses. As the pulsing rate of commercial probes is limited to about few ultrasound cycles every 100 μs, acquiring an acousto-optic image usually takes several tens of seconds due to the high number of acoustic pulses excitation. We propose here a new acousto-optic imaging technique based on the use of ultrasound plane waves instead of focused ones that allows increasing drastically the imaging rate.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  22. S. Dunne, S. Napel, and B. Rutt, “Fast reprojection of volume data,” in Proceedings of the First Conference on Visualization in Biomedical Computing (IEEE, 1990), pp 11–18.
    [Crossref]
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2014 (2)

J.-B. Laudereau, E. Benoità, La Guillaume, V. Servois, P. Mariani, A. A. Grabar, M. Tanter, J.-L. Gennisson, and F. Ramaz, “Multi-modal acousto-optic/ultrasound imaging of ex vivo liver tumors at 790 nm using a Sn2P2S6 wavefront adaptive holographic setup,” J. Biophotonics 8, 429–436 (2014).
[Crossref]

S. Resink, E. Hondebrink, and W. Steenbergen, “Solving the speckle decorrelation challenge in acousto-optic sensing using tandem nanosecond pulses within the ultrasound period,” Opt. Lett. 39(122), 6486–6489 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (2)

J. E. P. Honeysett, E. Stride, J. Deng, and T. S. Leung, “An algorithm for sensing venous oxygenation using ultrasound-modulated light enhanced by microbubbles,” Proc. SPIE 8223, 82232Z (2012).
[Crossref]

P. Lai, X. Xu, and L. V. Wang, “Ultrasound-modulated optical tomography at new depth,” J. Biomed. Opt. 17(6), 066006 (2012).
[Crossref] [PubMed]

2010 (3)

P. Kuchment and L. Kunyansky, “Synthetic focusing in ultrasound modulated tomography,” Inverse Probl. Imaging 4(14), 665–673 (2010).
[Crossref]

S. Farahi, G. Montemezzani, A. A. Grabar, J.-P. Huignard, and F. Ramaz, “Photorefractive acousto-optic imaging in thick scattering media at 790 nm with a Sn2P2S6:Te crystal,” Opt. Lett. 35(111), 1798–1800 (2010).
[Crossref] [PubMed]

M. Fink and M. Tanter, “Multiwave imaging and super resolution,” Phys. Today 63(12), 28–33 (2010).
[Crossref]

2009 (1)

G. Montaldo, M. Tanter, J. Bercoff, N. Benech, and M. Fink, “Coherent plane-wave compounding for very high frame rate ultrasonography and transient elastography,” IEEE Trans. Ultrason. Ferr. 56(13), 489–506 (2009).
[Crossref]

2008 (1)

2007 (1)

2005 (3)

2004 (1)

J. Li and L. V. Wang, “Ultrasound-modulated optical computed tomography of biological tissues,” Appl. Phys. Lett. 84(9), 1597 (2004).
[Crossref]

2001 (1)

L. V. Wang, “Mechanisms of ultrasonic modulation of multiply scattered coherent light: an analytic model,” Phys. Rev. Lett. 87(14), 043903 (2001).
[Crossref] [PubMed]

2000 (1)

1999 (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(12), R41 (1999).
[Crossref]

1997 (1)

1995 (1)

W. Leutz and G. Maret, “Ultrasonic modulation of multiply scattered light,” Physica B 204(1–4), 14–19 (1995).
[Crossref]

1976 (1)

D. C. Solmon, “The X-ray transform,” J. Math. Anal. Appl. 56(11), 61–83 (1976).
[Crossref]

1971 (1)

N. G. Ramachandran and V. A. Lakshminarayanan, “Three-dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms,” PNAS 68(19), 2236–2240 (1971).
[Crossref] [PubMed]

Arridge, S. R.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(12), R41 (1999).
[Crossref]

Atlan, M.

Bach, T.

Benech, N.

G. Montaldo, M. Tanter, J. Bercoff, N. Benech, and M. Fink, “Coherent plane-wave compounding for very high frame rate ultrasonography and transient elastography,” IEEE Trans. Ultrason. Ferr. 56(13), 489–506 (2009).
[Crossref]

Benoità, E.

J.-B. Laudereau, E. Benoità, La Guillaume, V. Servois, P. Mariani, A. A. Grabar, M. Tanter, J.-L. Gennisson, and F. Ramaz, “Multi-modal acousto-optic/ultrasound imaging of ex vivo liver tumors at 790 nm using a Sn2P2S6 wavefront adaptive holographic setup,” J. Biophotonics 8, 429–436 (2014).
[Crossref]

Bercoff, J.

G. Montaldo, M. Tanter, J. Bercoff, N. Benech, and M. Fink, “Coherent plane-wave compounding for very high frame rate ultrasonography and transient elastography,” IEEE Trans. Ultrason. Ferr. 56(13), 489–506 (2009).
[Crossref]

Blonigen, F. J.

Boccara, A. C.

Bossy, E.

Delaye, P.

Deng, J.

J. E. P. Honeysett, E. Stride, J. Deng, and T. S. Leung, “An algorithm for sensing venous oxygenation using ultrasound-modulated light enhanced by microbubbles,” Proc. SPIE 8223, 82232Z (2012).
[Crossref]

DiMarzio, C. A.

Dunne, S.

S. Dunne, S. Napel, and B. Rutt, “Fast reprojection of volume data,” in Proceedings of the First Conference on Visualization in Biomedical Computing (IEEE, 1990), pp 11–18.
[Crossref]

Farahi, S.

Fink, M.

M. Fink and M. Tanter, “Multiwave imaging and super resolution,” Phys. Today 63(12), 28–33 (2010).
[Crossref]

G. Montaldo, M. Tanter, J. Bercoff, N. Benech, and M. Fink, “Coherent plane-wave compounding for very high frame rate ultrasonography and transient elastography,” IEEE Trans. Ultrason. Ferr. 56(13), 489–506 (2009).
[Crossref]

J. Provost, W. Kwiecinski, M. Fink, M. Tanter, and M. Pernot, “Ultrafast acoustoelectric imaging,” in Proceedings of IEEE 11th International Symposium on Biomedical Imaging (IEEE, 2014), pp. 702–705.

Forget, B. C.

Genack, A. Z.

Gennisson, J.-L.

J.-B. Laudereau, E. Benoità, La Guillaume, V. Servois, P. Mariani, A. A. Grabar, M. Tanter, J.-L. Gennisson, and F. Ramaz, “Multi-modal acousto-optic/ultrasound imaging of ex vivo liver tumors at 790 nm using a Sn2P2S6 wavefront adaptive holographic setup,” J. Biophotonics 8, 429–436 (2014).
[Crossref]

Gnter, P.

Grabar, A. A.

Gross, M.

Guillaume, La

J.-B. Laudereau, E. Benoità, La Guillaume, V. Servois, P. Mariani, A. A. Grabar, M. Tanter, J.-L. Gennisson, and F. Ramaz, “Multi-modal acousto-optic/ultrasound imaging of ex vivo liver tumors at 790 nm using a Sn2P2S6 wavefront adaptive holographic setup,” J. Biophotonics 8, 429–436 (2014).
[Crossref]

Hemmer, P.

Hondebrink, E.

Honeysett, J. E. P.

J. E. P. Honeysett, E. Stride, J. Deng, and T. S. Leung, “An algorithm for sensing venous oxygenation using ultrasound-modulated light enhanced by microbubbles,” Proc. SPIE 8223, 82232Z (2012).
[Crossref]

Huignard, J.-P.

Jazbinsek, M.

Kempe, M.

Kim, C.

Kuchment, P.

P. Kuchment and L. Kunyansky, “Synthetic focusing in ultrasound modulated tomography,” Inverse Probl. Imaging 4(14), 665–673 (2010).
[Crossref]

Kunyansky, L.

P. Kuchment and L. Kunyansky, “Synthetic focusing in ultrasound modulated tomography,” Inverse Probl. Imaging 4(14), 665–673 (2010).
[Crossref]

Kwiecinski, W.

J. Provost, W. Kwiecinski, M. Fink, M. Tanter, and M. Pernot, “Ultrafast acoustoelectric imaging,” in Proceedings of IEEE 11th International Symposium on Biomedical Imaging (IEEE, 2014), pp. 702–705.

Lai, P.

P. Lai, X. Xu, and L. V. Wang, “Ultrasound-modulated optical tomography at new depth,” J. Biomed. Opt. 17(6), 066006 (2012).
[Crossref] [PubMed]

Lakshminarayanan, V. A.

N. G. Ramachandran and V. A. Lakshminarayanan, “Three-dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms,” PNAS 68(19), 2236–2240 (1971).
[Crossref] [PubMed]

Larionov, M.

Laudereau, J.-B.

J.-B. Laudereau, E. Benoità, La Guillaume, V. Servois, P. Mariani, A. A. Grabar, M. Tanter, J.-L. Gennisson, and F. Ramaz, “Multi-modal acousto-optic/ultrasound imaging of ex vivo liver tumors at 790 nm using a Sn2P2S6 wavefront adaptive holographic setup,” J. Biophotonics 8, 429–436 (2014).
[Crossref]

Lesaffre, M.

Leung, T. S.

J. E. P. Honeysett, E. Stride, J. Deng, and T. S. Leung, “An algorithm for sensing venous oxygenation using ultrasound-modulated light enhanced by microbubbles,” Proc. SPIE 8223, 82232Z (2012).
[Crossref]

Leutz, W.

W. Leutz and G. Maret, “Ultrasonic modulation of multiply scattered light,” Physica B 204(1–4), 14–19 (1995).
[Crossref]

Li, J.

J. Li and L. V. Wang, “Ultrasound-modulated optical computed tomography of biological tissues,” Appl. Phys. Lett. 84(9), 1597 (2004).
[Crossref]

Li, Y.

Maguluri, G.

Manneville, S.

Maret, G.

W. Leutz and G. Maret, “Ultrasonic modulation of multiply scattered light,” Physica B 204(1–4), 14–19 (1995).
[Crossref]

Mariani, P.

J.-B. Laudereau, E. Benoità, La Guillaume, V. Servois, P. Mariani, A. A. Grabar, M. Tanter, J.-L. Gennisson, and F. Ramaz, “Multi-modal acousto-optic/ultrasound imaging of ex vivo liver tumors at 790 nm using a Sn2P2S6 wavefront adaptive holographic setup,” J. Biophotonics 8, 429–436 (2014).
[Crossref]

Montaldo, G.

G. Montaldo, M. Tanter, J. Bercoff, N. Benech, and M. Fink, “Coherent plane-wave compounding for very high frame rate ultrasonography and transient elastography,” IEEE Trans. Ultrason. Ferr. 56(13), 489–506 (2009).
[Crossref]

Montemezzani, G.

Murray, T. W.

Napel, S.

S. Dunne, S. Napel, and B. Rutt, “Fast reprojection of volume data,” in Proceedings of the First Conference on Visualization in Biomedical Computing (IEEE, 1990), pp 11–18.
[Crossref]

Nieva, A.

Pernot, M.

J. Provost, W. Kwiecinski, M. Fink, M. Tanter, and M. Pernot, “Ultrafast acoustoelectric imaging,” in Proceedings of IEEE 11th International Symposium on Biomedical Imaging (IEEE, 2014), pp. 702–705.

Provost, J.

J. Provost, W. Kwiecinski, M. Fink, M. Tanter, and M. Pernot, “Ultrafast acoustoelectric imaging,” in Proceedings of IEEE 11th International Symposium on Biomedical Imaging (IEEE, 2014), pp. 702–705.

Ramachandran, N. G.

N. G. Ramachandran and V. A. Lakshminarayanan, “Three-dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms,” PNAS 68(19), 2236–2240 (1971).
[Crossref] [PubMed]

Ramaz, F.

Resink, S.

Roosen, G.

Roy, R. A.

Rutt, B.

S. Dunne, S. Napel, and B. Rutt, “Fast reprojection of volume data,” in Proceedings of the First Conference on Visualization in Biomedical Computing (IEEE, 1990), pp 11–18.
[Crossref]

Servois, V.

J.-B. Laudereau, E. Benoità, La Guillaume, V. Servois, P. Mariani, A. A. Grabar, M. Tanter, J.-L. Gennisson, and F. Ramaz, “Multi-modal acousto-optic/ultrasound imaging of ex vivo liver tumors at 790 nm using a Sn2P2S6 wavefront adaptive holographic setup,” J. Biophotonics 8, 429–436 (2014).
[Crossref]

Solmon, D. C.

D. C. Solmon, “The X-ray transform,” J. Math. Anal. Appl. 56(11), 61–83 (1976).
[Crossref]

Steenbergen, W.

Stride, E.

J. E. P. Honeysett, E. Stride, J. Deng, and T. S. Leung, “An algorithm for sensing venous oxygenation using ultrasound-modulated light enhanced by microbubbles,” Proc. SPIE 8223, 82232Z (2012).
[Crossref]

Sui, L.

Tanter, M.

J.-B. Laudereau, E. Benoità, La Guillaume, V. Servois, P. Mariani, A. A. Grabar, M. Tanter, J.-L. Gennisson, and F. Ramaz, “Multi-modal acousto-optic/ultrasound imaging of ex vivo liver tumors at 790 nm using a Sn2P2S6 wavefront adaptive holographic setup,” J. Biophotonics 8, 429–436 (2014).
[Crossref]

M. Fink and M. Tanter, “Multiwave imaging and super resolution,” Phys. Today 63(12), 28–33 (2010).
[Crossref]

G. Montaldo, M. Tanter, J. Bercoff, N. Benech, and M. Fink, “Coherent plane-wave compounding for very high frame rate ultrasonography and transient elastography,” IEEE Trans. Ultrason. Ferr. 56(13), 489–506 (2009).
[Crossref]

J. Provost, W. Kwiecinski, M. Fink, M. Tanter, and M. Pernot, “Ultrafast acoustoelectric imaging,” in Proceedings of IEEE 11th International Symposium on Biomedical Imaging (IEEE, 2014), pp. 702–705.

Vysochanskii, Y. M.

Wang, L. V.

P. Lai, X. Xu, and L. V. Wang, “Ultrasound-modulated optical tomography at new depth,” J. Biomed. Opt. 17(6), 066006 (2012).
[Crossref] [PubMed]

Y. Li, P. Hemmer, C. Kim, H. Zhang, and L. V. Wang, “Detection of ultrasound-modulated diffuse photons using spectral-hole burning,” Opt. Express 16(119), 14862–14874 (2008).
[Crossref] [PubMed]

J. Li and L. V. Wang, “Ultrasound-modulated optical computed tomography of biological tissues,” Appl. Phys. Lett. 84(9), 1597 (2004).
[Crossref]

L. V. Wang, “Mechanisms of ultrasonic modulation of multiply scattered coherent light: an analytic model,” Phys. Rev. Lett. 87(14), 043903 (2001).
[Crossref] [PubMed]

G. Yao and L. V. Wang, “Theoretical and experimental studies of ultrasound-modulated optical tomography in biological tissue,” Appl. Opt. 39(14), 659–664 (2000).
[Crossref]

Xu, X.

P. Lai, X. Xu, and L. V. Wang, “Ultrasound-modulated optical tomography at new depth,” J. Biomed. Opt. 17(6), 066006 (2012).
[Crossref] [PubMed]

Yao, G.

Zaslavsky, D.

Zhang, H.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

J. Li and L. V. Wang, “Ultrasound-modulated optical computed tomography of biological tissues,” Appl. Phys. Lett. 84(9), 1597 (2004).
[Crossref]

IEEE Trans. Ultrason. Ferr. (1)

G. Montaldo, M. Tanter, J. Bercoff, N. Benech, and M. Fink, “Coherent plane-wave compounding for very high frame rate ultrasonography and transient elastography,” IEEE Trans. Ultrason. Ferr. 56(13), 489–506 (2009).
[Crossref]

Inverse Probl. (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(12), R41 (1999).
[Crossref]

Inverse Probl. Imaging (1)

P. Kuchment and L. Kunyansky, “Synthetic focusing in ultrasound modulated tomography,” Inverse Probl. Imaging 4(14), 665–673 (2010).
[Crossref]

J. Biomed. Opt. (1)

P. Lai, X. Xu, and L. V. Wang, “Ultrasound-modulated optical tomography at new depth,” J. Biomed. Opt. 17(6), 066006 (2012).
[Crossref] [PubMed]

J. Biophotonics (1)

J.-B. Laudereau, E. Benoità, La Guillaume, V. Servois, P. Mariani, A. A. Grabar, M. Tanter, J.-L. Gennisson, and F. Ramaz, “Multi-modal acousto-optic/ultrasound imaging of ex vivo liver tumors at 790 nm using a Sn2P2S6 wavefront adaptive holographic setup,” J. Biophotonics 8, 429–436 (2014).
[Crossref]

J. Math. Anal. Appl. (1)

D. C. Solmon, “The X-ray transform,” J. Math. Anal. Appl. 56(11), 61–83 (1976).
[Crossref]

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

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

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

L. V. Wang, “Mechanisms of ultrasonic modulation of multiply scattered coherent light: an analytic model,” Phys. Rev. Lett. 87(14), 043903 (2001).
[Crossref] [PubMed]

Phys. Today (1)

M. Fink and M. Tanter, “Multiwave imaging and super resolution,” Phys. Today 63(12), 28–33 (2010).
[Crossref]

Physica B (1)

W. Leutz and G. Maret, “Ultrasonic modulation of multiply scattered light,” Physica B 204(1–4), 14–19 (1995).
[Crossref]

PNAS (1)

N. G. Ramachandran and V. A. Lakshminarayanan, “Three-dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms,” PNAS 68(19), 2236–2240 (1971).
[Crossref] [PubMed]

Proc. SPIE (1)

J. E. P. Honeysett, E. Stride, J. Deng, and T. S. Leung, “An algorithm for sensing venous oxygenation using ultrasound-modulated light enhanced by microbubbles,” Proc. SPIE 8223, 82232Z (2012).
[Crossref]

Other (2)

J. Provost, W. Kwiecinski, M. Fink, M. Tanter, and M. Pernot, “Ultrafast acoustoelectric imaging,” in Proceedings of IEEE 11th International Symposium on Biomedical Imaging (IEEE, 2014), pp. 702–705.

S. Dunne, S. Napel, and B. Rutt, “Fast reprojection of volume data,” in Proceedings of the First Conference on Visualization in Biomedical Computing (IEEE, 1990), pp 11–18.
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Schematic of the US plane wave propagating inside an illuminated sample with an angle θ. The US are focused along the z-direction so that they propagate in a plane perpendicular to light propagation direction. The figure represents this section defined by the US probe position. The three plain blue lines represent the ultrasonic pulse propagating at VUS. At time t, the wave is at the position of the dashed blue line. A voxel (black square) is located through its coordinates (x, y). Photons are indistinctly tagged along the entire wavefront so that the signal recorded on a single detector is the integrated light intensity along the dashed line.
Fig. 2
Fig. 2 Illustration of the projection-slice theorem and the limitation of the Fourier space (a) The temporal FT of the signal recorded on the photodiode is taken and represents a slice at θ of spatial 2D ℱ�� of the light irradiance. (b) The slice is placed back in its rightful place in the spatial 2D Fourier space and it is done for all angles. The inverse 2D ℱ�� of the reconstructed Fourier space gives the expected image. (c) The receiving and emitting bandwidths limit the highest accessible frequency modulus and the probe limits the angular range so that the accessible region is limited to a segment of the Fourier plane (darker gray portion).
Fig. 3
Fig. 3 (a) PSF for an angular range of ±20° which is the typical range for a commercial linear ultrasound probe. One can see here that the limited angular range degrades the lateral resolution. (b) Slice along the ϕ = 0° direction (x-axis). (c) Slice along the ϕ = 90° direction (y-axis). (d) Plot of the resolution (FWHM) along both directions as a function of the angular range. Crosses stand for the computed values. Plain lines correspond to the FWHM of the main lobe assuming the 1D PSF is a section of an Airy disk.
Fig. 4
Fig. 4 Schematic of the experimental setup. The laser source is a laser diode added to a 2 W tapered amplifier in order to have a MOPA system (Sacher Lasertechnik GmBH). The wavelength is 790 nm. The light beam is split into two beams thanks to a beam splitter (BS). The signal beam is collected in a Thorlabs multimode optical fiber (MMOF) through a commercial collimator and guided to the scattering sample. The scattered light is collected through a liquid core optical fiber (LCOF) with a collection area of 1 cm2 and 0.4 NA and guided to the SPS photorefractive crystal (PRC). The reference beam (few dozens of mW) is used to perform two wave mixing (TWM) and filter the untagged photons [6]. The AO signal is measured on a photodiode (PD) and processed on a computer.
Fig. 5
Fig. 5 (a) AO image using focused pulses. The white arrow points to the inclusion. (b) 1D vertical profile along the black dashed line. (c) Picture of the inclusions taken before it was embedded. (d) 1D horizontal profile along the white dashed line.
Fig. 6
Fig. 6 (a) AO image using plane waves ranging from −20° to +20° with 1° steps. (b) vertical profile along the black dashed line. (c) horizontal profile along the white dotted line. (d) Evolution of the CNR as a function of the averaging. The CNR level for focused US averaged 2000 times is indicated by the blue line for comparison purpose.
Fig. 7
Fig. 7 (a) Schematic of the sample with two inclusions and the positions of the two probes. Numbers 1 and 192 on each probe represent the locations of the first and last elements. The two inclusions are close enough so that they can’t be distinguished in the plane wave image using only probe 1. (b), (c), (d) and (e) show the results on this gel. The colorbar is the same for each AO image and is showed on the left side of (c). The black dashed line on each image is the line along which the profiles of Fig. 8 were plotted. (b) Picture of the gel with the two black inclusions. (c) AO image using focused US. (d) AO image using plane waves from one probe. (e) AO image using plane waves from two perpendicular probes.
Fig. 8
Fig. 8 Profiles along the black dashed lines of Fig. 7. The black arrows point the positions of the two inclusions. As expected, with one probe, it is impossible to distinguish the two inclusions as they clearly appear with 2 probes.

Equations (34)

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P US ( r , t ) = P 0 ( t K US r ω US ) sin ( ω US t )
Δ φ US ( t , l ) = s l , 0 + δ s l sin ( ω US t )
δ s l = α P 0 ( t K US r ω US )
G ( r , t , τ ) = A 0 β Ω ( r , r ) Ψ ( r ) P 0 2 ( t K US r ω US ) ( 1 cos ( ω US τ ) ) d x d y
I T = β I ( r , r ) P 0 2 ( t K US r ω US ) d x d y
s ( t , θ ) I ( x , y ) P 0 2 ( t x cos θ + y sin θ V US ) d x d y
s ( t , θ ) I ( x , y ) [ P 0 2 ( t ) * δ ( t x cos θ + y sin θ V US ) ] d x d y
s ˜ ( ν , θ ) = FT { s ( t , θ ) } = P ˜ ( ν ) I ( x , y ) exp ( 2 i π ν x cos θ + y sin θ V US ) d x d y
s ˜ ( k , θ ) = R ˜ ( k ) P ˜ ( k ) I ( k x , θ , k y , θ )
( x , y ) = 𝒯 1 { θ s ˜ ( k , θ ) } = 𝒯 1 { R ˜ ( k ) P ˜ ( k ) } * 𝒯 1 { θ I ˜ ( k x , θ , k y , θ ) }
( x , y ) = 𝒯 1 { θ [ θ m , θ m ] s ˜ ( ν , θ ) }
PSF ( x , y ) = 𝒯 1 { θ [ θ m , θ m ] δ ˜ ( k x , θ , k y , θ ) }
PSF ( r , φ ) = k m k m θ m θ m δ ˜ ( k , θ ) exp [ 2 i π k r cos ( θ φ ) ] | k | d k d θ
PSF ( r , φ ) = n = + e in ( π 2 φ ) θ m θ m e in θ d θ k m k m J n ( 2 π k r ) | k | d k
PSF ( r , φ ) = n = + 2 n sin ( n θ m ) e in ( π 2 φ ) k m k m J n ( 2 π k r ) | k | d k
PSF ( r , φ ) = 2 θ m k m 2 1 1 J 0 ( 2 π k m r u ) | u | d u + k m n = 1 + 2 n sin ( n θ m ) ( e in ( π 2 φ ) + ( 1 ) n e in ( π 2 φ ) ) 1 1 J n ( 2 π k m r u ) | u | d u
PSF ( r , φ ) = 4 θ m k m 2 0 1 J 0 ( 2 π k m r u ) | u | d u + k m 2 p = 1 + 4 p ( 1 ) p sin ( 2 p θ m ) cos ( 2 p φ ) 0 1 J 2 p ( 2 π k m r u ) | u | d u
PSF ( r , φ ) 4 θ m k m 2 = J 1 ( 2 π k m r ) 2 π k m r
PSF ( r , φ ) 4 θ m k m 2 = A ( r ) + p = 1 B p ( r , φ ) + p = 1 C p ( r , φ )
{ A ( r ) = J 1 ( 2 π k m r ) 2 π k m r B p ( r , φ ) = 2 ( 1 ) p + 1 sinc ( 2 p θ m ) cos ( 2 p φ ) J 2 p 1 ( 2 π k m r ) 2 π k m r C p ( r , φ ) = 2 p ( 1 ) p π k m r sinc ( 2 p θ m ) cos ( 2 p φ ) c p ( r )
{ c 0 ( r ) = J 0 ( 2 π k m r ) 1 2 π k m r c p + 1 ( r ) = c p ( r ) 1 π k m r ( J 2 p ( 2 π k m r ) J 2 p ( 0 ) )
{ Δ x = 0.7 k m Δ y = 0.7 k m sin θ m
f ( k ) = | k | cos ( π k 2 k m ) rect k m ( k )
Σ = Σ diff Σ incl + C
PSF ( r , φ ) 4 θ m k m 2 = J 1 ( 2 π k m r ) 2 π k m r + p = 1 + 2 ( 1 ) p sinc ( 2 p θ m ) cos ( 2 p φ ) 0 1 J 2 p ( 2 π k m r u ) u d u
J n + 1 ( x ) = n J n ( x ) x J n ( x )
0 1 J 2 p ( 2 π k m r u ) u d u = 2 p 1 2 π k m r 0 1 J 2 p 1 ( 2 π k m r u ) d u 1 2 π k m r 0 1 d d u { J 2 p 1 ( 2 π k m r u ) } u d u
0 1 d d u { J 2 p 1 ( 2 π k m r u ) } u d u = J 2 p 1 ( 2 π k m r ) 0 1 J 2 p 1 ( 2 π k m r u ) d u
0 1 J 2 p ( 2 π k m r u ) u d u = 2 p 2 π k m r 0 1 J 2 p 1 ( 2 π k m r u ) d u J 2 p 1 ( 2 π k m r ) 2 π k m r
J n 1 ( x ) = J n + 1 ( x ) + 2 J n ( x )
c p ( r ) = 0 1 J 2 p 1 ( 2 π k m r u ) d u
c p ( r ) = 0 1 J 2 p + 1 ( 2 π k m r u ) d u + 1 π k m r 0 1 d d u { J 2 p ( 2 π k m r u ) } d u
c p + 1 ( r ) = c p ( r ) 1 π k m r ( J 2 p ( 2 π k m r ) J 2 p ( 0 ) )
c 0 ( r ) = 0 1 J 1 ( 2 π k m r u ) d u = J 0 ( 2 π k m r ) 1 2 π k m r

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