Abstract

The time-reversed ultrasonically encoded (TRUE) optical focusing technique is a method that is capable of focusing light deep within a scattering medium. This theoretical study aims to explore the depth limits of the TRUE technique for biological tissues in the context of two primary constraints – the safety limit of the incident light fluence and a limited TRUE’s recording time (assumed to be 1 ms), as dynamic scatterer movements in a living sample can break the time-reversal scattering symmetry. Our numerical simulation indicates that TRUE has the potential to render an optical focus with a peak-to-background ratio of ~2 at a depth of ~103 mm at wavelength of 800 nm in a phantom with tissue scattering characteristics. This study sheds light on the allocation of photon budget in each step of the TRUE technique, the impact of low signal on the phase measurement error, and the eventual impact of the phase measurement error on the strength of the TRUE optical focus.

© 2014 Optical Society of America

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2013 (1)

B. Judkewitz, Y. M. Wang, R. Horstmeyer, A. Mathy, C. Yang, “Speckle-scale focusing in the diffusive regime with time-reversal of variance-encoded light (TROVE),” Nat. Photonics 7(4), 300–305 (2013).
[CrossRef] [PubMed]

2012 (5)

D. B. Conkey, A. M. Caravaca-Aguirre, R. Piestun, “High-speed scattering medium characterization with application to focusing light through turbid media,” Opt. Express 20(2), 1733–1740 (2012).
[CrossRef] [PubMed]

Y. M. Wang, B. Judkewitz, C. A. Dimarzio, C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat Commun 3, 928 (2012).
[CrossRef] [PubMed]

K. Si, R. Fiolka, M. Cui, “Fluorescence imaging beyond the ballistic regime by ultrasound pulse guided digital phase conjugation,” Nat. Photonics 6(10), 657–661 (2012).
[CrossRef] [PubMed]

K. Si, R. Fiolka, M. Cui, “Breaking the spatial resolution barrier via iterative sound-light interaction in deep tissue microscopy,” Sci. Rep. 2, 748 (2012).
[CrossRef] [PubMed]

A. Mosk, A. Lagendijk, G. Lerosey, M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6(5), 283–292 (2012).
[CrossRef]

2011 (1)

X. Xu, H. Liu, L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics 5(3), 154–157 (2011).
[CrossRef] [PubMed]

2010 (7)

V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods 7(8), 603–614 (2010).
[CrossRef] [PubMed]

M. Cui, E. J. McDowell, C. Yang, “An in vivo study of turbidity suppression by optical phase conjugation (TSOPC) on rabbit ear,” Opt. Express 18(1), 25–30 (2010).
[CrossRef] [PubMed]

M. Cui, C. Yang, “Implementation of a digital optical phase conjugation system and its application to study the robustness of turbidity suppression by phase conjugation,” Opt. Express 18(4), 3444–3455 (2010).
[CrossRef] [PubMed]

I. Vellekoop, C. Aegerter, “Focusing light through living tissue,” Proc. SPIE 7554, 755430 (2010).

M. Cui, E. J. McDowell, C. Yang, “An in vivo study of turbidity suppression by optical phase conjugation (TSOPC) on rabbit ear,” Opt. Express 18(1), 25–30 (2010).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[CrossRef] [PubMed]

M. O. Culjat, D. Goldenberg, P. Tewari, R. S. Singh, “A review of tissue substitutes for ultrasound imaging,” Ultrasound Med. Biol. 36(6), 861–873 (2010).
[CrossRef] [PubMed]

2008 (1)

Z. Yaqoob, D. Psaltis, M. S. Feld, C. Yang, “Optical Phase Conjugation for Turbidity Suppression in Biological Samples,” Nat. Photonics 2(2), 110–115 (2008).
[CrossRef] [PubMed]

2007 (1)

2004 (1)

D. Dalecki, “Mechanical bioeffects of ultrasound,” Annu. Rev. Biomed. Eng. 6(1), 229–248 (2004).
[CrossRef] [PubMed]

2001 (1)

L. V. Wang, “Mechanisms of Ultrasonic Modulation of Multiply Scattered Coherent Light: An Analytic Model,” Phys. Rev. Lett. 87(4), 043903 (2001).
[CrossRef] [PubMed]

1998 (1)

1997 (1)

1994 (1)

1992 (1)

J. Huang, J. Nissen, E. Bodegom, “Diffraction of light by a focused ultrasonic wave,” J. Appl. Phys. 71(1), 70–75 (1992).
[CrossRef]

1981 (1)

1980 (1)

M. Moharam, T. Gaylord, R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980).
[CrossRef]

1967 (1)

W. Riley, W. Klein, “Piezo Optic Coefficients of Liquids,” J. Acoust. Soc. Am. 42(6), 1258–1261 (1967).
[CrossRef]

Aegerter, C.

I. Vellekoop, C. Aegerter, “Focusing light through living tissue,” Proc. SPIE 7554, 755430 (2010).

Boccara, A. C.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[CrossRef] [PubMed]

Bodegom, E.

J. Huang, J. Nissen, E. Bodegom, “Diffraction of light by a focused ultrasonic wave,” J. Appl. Phys. 71(1), 70–75 (1992).
[CrossRef]

Caravaca-Aguirre, A. M.

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[CrossRef] [PubMed]

Conkey, D. B.

Cui, M.

Culjat, M. O.

M. O. Culjat, D. Goldenberg, P. Tewari, R. S. Singh, “A review of tissue substitutes for ultrasound imaging,” Ultrasound Med. Biol. 36(6), 861–873 (2010).
[CrossRef] [PubMed]

Dalecki, D.

D. Dalecki, “Mechanical bioeffects of ultrasound,” Annu. Rev. Biomed. Eng. 6(1), 229–248 (2004).
[CrossRef] [PubMed]

Dimarzio, C. A.

Y. M. Wang, B. Judkewitz, C. A. Dimarzio, C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat Commun 3, 928 (2012).
[CrossRef] [PubMed]

Feld, M. S.

Z. Yaqoob, D. Psaltis, M. S. Feld, C. Yang, “Optical Phase Conjugation for Turbidity Suppression in Biological Samples,” Nat. Photonics 2(2), 110–115 (2008).
[CrossRef] [PubMed]

Feng, T. C.

Fink, M.

A. Mosk, A. Lagendijk, G. Lerosey, M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6(5), 283–292 (2012).
[CrossRef]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[CrossRef] [PubMed]

Fiolka, R.

K. Si, R. Fiolka, M. Cui, “Fluorescence imaging beyond the ballistic regime by ultrasound pulse guided digital phase conjugation,” Nat. Photonics 6(10), 657–661 (2012).
[CrossRef] [PubMed]

K. Si, R. Fiolka, M. Cui, “Breaking the spatial resolution barrier via iterative sound-light interaction in deep tissue microscopy,” Sci. Rep. 2, 748 (2012).
[CrossRef] [PubMed]

Gaylord, T.

M. Moharam, T. Gaylord, R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980).
[CrossRef]

Gaylord, T. K.

Gigan, S.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[CrossRef] [PubMed]

Goldenberg, D.

M. O. Culjat, D. Goldenberg, P. Tewari, R. S. Singh, “A review of tissue substitutes for ultrasound imaging,” Ultrasound Med. Biol. 36(6), 861–873 (2010).
[CrossRef] [PubMed]

Haskell, R. C.

Horstmeyer, R.

B. Judkewitz, Y. M. Wang, R. Horstmeyer, A. Mathy, C. Yang, “Speckle-scale focusing in the diffusive regime with time-reversal of variance-encoded light (TROVE),” Nat. Photonics 7(4), 300–305 (2013).
[CrossRef] [PubMed]

Huang, J.

J. Huang, J. Nissen, E. Bodegom, “Diffraction of light by a focused ultrasonic wave,” J. Appl. Phys. 71(1), 70–75 (1992).
[CrossRef]

Judkewitz, B.

B. Judkewitz, Y. M. Wang, R. Horstmeyer, A. Mathy, C. Yang, “Speckle-scale focusing in the diffusive regime with time-reversal of variance-encoded light (TROVE),” Nat. Photonics 7(4), 300–305 (2013).
[CrossRef] [PubMed]

Y. M. Wang, B. Judkewitz, C. A. Dimarzio, C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat Commun 3, 928 (2012).
[CrossRef] [PubMed]

Klein, W.

W. Riley, W. Klein, “Piezo Optic Coefficients of Liquids,” J. Acoust. Soc. Am. 42(6), 1258–1261 (1967).
[CrossRef]

Lagendijk, A.

A. Mosk, A. Lagendijk, G. Lerosey, M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6(5), 283–292 (2012).
[CrossRef]

Lerosey, G.

A. Mosk, A. Lagendijk, G. Lerosey, M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6(5), 283–292 (2012).
[CrossRef]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[CrossRef] [PubMed]

Lin, S. P.

Liu, H.

X. Xu, H. Liu, L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics 5(3), 154–157 (2011).
[CrossRef] [PubMed]

Magnusson, R.

M. Moharam, T. Gaylord, R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980).
[CrossRef]

Marquez, G.

Mathy, A.

B. Judkewitz, Y. M. Wang, R. Horstmeyer, A. Mathy, C. Yang, “Speckle-scale focusing in the diffusive regime with time-reversal of variance-encoded light (TROVE),” Nat. Photonics 7(4), 300–305 (2013).
[CrossRef] [PubMed]

McAdams, M. S.

McDowell, E. J.

Moharam, M.

M. Moharam, T. Gaylord, R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980).
[CrossRef]

Moharam, M. G.

Mosk, A.

A. Mosk, A. Lagendijk, G. Lerosey, M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6(5), 283–292 (2012).
[CrossRef]

Mosk, A. P.

Nissen, J.

J. Huang, J. Nissen, E. Bodegom, “Diffraction of light by a focused ultrasonic wave,” J. Appl. Phys. 71(1), 70–75 (1992).
[CrossRef]

Ntziachristos, V.

V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods 7(8), 603–614 (2010).
[CrossRef] [PubMed]

Piestun, R.

Popoff, S. M.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[CrossRef] [PubMed]

Psaltis, D.

Z. Yaqoob, D. Psaltis, M. S. Feld, C. Yang, “Optical Phase Conjugation for Turbidity Suppression in Biological Samples,” Nat. Photonics 2(2), 110–115 (2008).
[CrossRef] [PubMed]

Riley, W.

W. Riley, W. Klein, “Piezo Optic Coefficients of Liquids,” J. Acoust. Soc. Am. 42(6), 1258–1261 (1967).
[CrossRef]

Schwartz, J. A.

Si, K.

K. Si, R. Fiolka, M. Cui, “Fluorescence imaging beyond the ballistic regime by ultrasound pulse guided digital phase conjugation,” Nat. Photonics 6(10), 657–661 (2012).
[CrossRef] [PubMed]

K. Si, R. Fiolka, M. Cui, “Breaking the spatial resolution barrier via iterative sound-light interaction in deep tissue microscopy,” Sci. Rep. 2, 748 (2012).
[CrossRef] [PubMed]

Singh, R. S.

M. O. Culjat, D. Goldenberg, P. Tewari, R. S. Singh, “A review of tissue substitutes for ultrasound imaging,” Ultrasound Med. Biol. 36(6), 861–873 (2010).
[CrossRef] [PubMed]

Svaasand, L. O.

Tewari, P.

M. O. Culjat, D. Goldenberg, P. Tewari, R. S. Singh, “A review of tissue substitutes for ultrasound imaging,” Ultrasound Med. Biol. 36(6), 861–873 (2010).
[CrossRef] [PubMed]

Thomsen, S. L.

Tromberg, B. J.

Tsay, T. T.

Vellekoop, I.

I. Vellekoop, C. Aegerter, “Focusing light through living tissue,” Proc. SPIE 7554, 755430 (2010).

Vellekoop, I. M.

Wang, L. V.

X. Xu, H. Liu, L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics 5(3), 154–157 (2011).
[CrossRef] [PubMed]

L. V. Wang, “Mechanisms of Ultrasonic Modulation of Multiply Scattered Coherent Light: An Analytic Model,” Phys. Rev. Lett. 87(4), 043903 (2001).
[CrossRef] [PubMed]

G. Marquez, L. V. Wang, S. P. Lin, J. A. Schwartz, S. L. Thomsen, “Anisotropy in the absorption and scattering spectra of chicken breast tissue,” Appl. Opt. 37(4), 798–804 (1998).
[CrossRef] [PubMed]

Wang, Y. M.

B. Judkewitz, Y. M. Wang, R. Horstmeyer, A. Mathy, C. Yang, “Speckle-scale focusing in the diffusive regime with time-reversal of variance-encoded light (TROVE),” Nat. Photonics 7(4), 300–305 (2013).
[CrossRef] [PubMed]

Y. M. Wang, B. Judkewitz, C. A. Dimarzio, C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat Commun 3, 928 (2012).
[CrossRef] [PubMed]

Xu, X.

X. Xu, H. Liu, L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics 5(3), 154–157 (2011).
[CrossRef] [PubMed]

Yamaguchi, I.

Yang, C.

B. Judkewitz, Y. M. Wang, R. Horstmeyer, A. Mathy, C. Yang, “Speckle-scale focusing in the diffusive regime with time-reversal of variance-encoded light (TROVE),” Nat. Photonics 7(4), 300–305 (2013).
[CrossRef] [PubMed]

Y. M. Wang, B. Judkewitz, C. A. Dimarzio, C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat Commun 3, 928 (2012).
[CrossRef] [PubMed]

M. Cui, E. J. McDowell, C. Yang, “An in vivo study of turbidity suppression by optical phase conjugation (TSOPC) on rabbit ear,” Opt. Express 18(1), 25–30 (2010).
[CrossRef] [PubMed]

M. Cui, E. J. McDowell, C. Yang, “An in vivo study of turbidity suppression by optical phase conjugation (TSOPC) on rabbit ear,” Opt. Express 18(1), 25–30 (2010).
[CrossRef] [PubMed]

M. Cui, C. Yang, “Implementation of a digital optical phase conjugation system and its application to study the robustness of turbidity suppression by phase conjugation,” Opt. Express 18(4), 3444–3455 (2010).
[CrossRef] [PubMed]

Z. Yaqoob, D. Psaltis, M. S. Feld, C. Yang, “Optical Phase Conjugation for Turbidity Suppression in Biological Samples,” Nat. Photonics 2(2), 110–115 (2008).
[CrossRef] [PubMed]

Yaqoob, Z.

Z. Yaqoob, D. Psaltis, M. S. Feld, C. Yang, “Optical Phase Conjugation for Turbidity Suppression in Biological Samples,” Nat. Photonics 2(2), 110–115 (2008).
[CrossRef] [PubMed]

Zhang, T.

Annu. Rev. Biomed. Eng. (1)

D. Dalecki, “Mechanical bioeffects of ultrasound,” Annu. Rev. Biomed. Eng. 6(1), 229–248 (2004).
[CrossRef] [PubMed]

Appl. Opt. (2)

J. Acoust. Soc. Am. (1)

W. Riley, W. Klein, “Piezo Optic Coefficients of Liquids,” J. Acoust. Soc. Am. 42(6), 1258–1261 (1967).
[CrossRef]

J. Appl. Phys. (1)

J. Huang, J. Nissen, E. Bodegom, “Diffraction of light by a focused ultrasonic wave,” J. Appl. Phys. 71(1), 70–75 (1992).
[CrossRef]

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

Nat Commun (1)

Y. M. Wang, B. Judkewitz, C. A. Dimarzio, C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat Commun 3, 928 (2012).
[CrossRef] [PubMed]

Nat. Methods (1)

V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods 7(8), 603–614 (2010).
[CrossRef] [PubMed]

Nat. Photonics (5)

X. Xu, H. Liu, L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics 5(3), 154–157 (2011).
[CrossRef] [PubMed]

K. Si, R. Fiolka, M. Cui, “Fluorescence imaging beyond the ballistic regime by ultrasound pulse guided digital phase conjugation,” Nat. Photonics 6(10), 657–661 (2012).
[CrossRef] [PubMed]

Z. Yaqoob, D. Psaltis, M. S. Feld, C. Yang, “Optical Phase Conjugation for Turbidity Suppression in Biological Samples,” Nat. Photonics 2(2), 110–115 (2008).
[CrossRef] [PubMed]

B. Judkewitz, Y. M. Wang, R. Horstmeyer, A. Mathy, C. Yang, “Speckle-scale focusing in the diffusive regime with time-reversal of variance-encoded light (TROVE),” Nat. Photonics 7(4), 300–305 (2013).
[CrossRef] [PubMed]

A. Mosk, A. Lagendijk, G. Lerosey, M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6(5), 283–292 (2012).
[CrossRef]

Opt. Commun. (1)

M. Moharam, T. Gaylord, R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. Lett. (2)

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[CrossRef] [PubMed]

L. V. Wang, “Mechanisms of Ultrasonic Modulation of Multiply Scattered Coherent Light: An Analytic Model,” Phys. Rev. Lett. 87(4), 043903 (2001).
[CrossRef] [PubMed]

Proc. SPIE (1)

I. Vellekoop, C. Aegerter, “Focusing light through living tissue,” Proc. SPIE 7554, 755430 (2010).

Sci. Rep. (1)

K. Si, R. Fiolka, M. Cui, “Breaking the spatial resolution barrier via iterative sound-light interaction in deep tissue microscopy,” Sci. Rep. 2, 748 (2012).
[CrossRef] [PubMed]

Ultrasound Med. Biol. (1)

M. O. Culjat, D. Goldenberg, P. Tewari, R. S. Singh, “A review of tissue substitutes for ultrasound imaging,” Ultrasound Med. Biol. 36(6), 861–873 (2010).
[CrossRef] [PubMed]

Other (4)

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE Press, 2010).

Laser Institute of America, American National Standard for Safe Use of Lasers ANSI Z136.1–2000 (American National Standards Institute Inc., 2000).

J. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company Publishers, 2007).

U.S. Department of Health and Human Services, “Guidance for Industry and FDA Staff—Information for Manufacturers Seeking Marketing Clearance of Diagnostic Ultrasound Systems and Transducers,” (2008).

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Figures (8)

Fig. 1
Fig. 1

Schematic of the TRUE focusing principle with digital optical phase conjugation system (DOPC). (a) Collimated incident beam propagates through scattering medium. Light component passing through the ultrasound focus is encoded with ultrasound frequency. (b) Ultrasound-modulated light propagates back to the tissue surface. The distorted wave front is measured by a sensor in the DOPC system. (c) Spatial light modulator (SLM) reproduces a phase-conjugated copy of the measured wave front. The OPC beam with time-reversal characteristic is focused back into the US spot.

Fig. 2
Fig. 2

Flow chart of simulation procedure. 2D photon flux map emerging from the tissue surface is calculated from the first three steps regarding light propagation and ultrasonic light modulation. Then, wavefront measurement error resulting from shot-noise is determined to calculate the field contribution from M g r i d number of modes to the TRUE focal spot ( A O P C _ L U T e i ϕ O P C _ L U T ). In the last step, PBR is calculated by summing up the field contributions from each simulation grid cell. Here, we use the lookup table approach (field contribution is interpolated from) that we will describe in detail at Section 2.3.2 and 2.3.3.

Fig. 3
Fig. 3

(a) Longitudinal sectional photon fluence map of the light beam propagating through the biological tissue corresponding to a single light pulse at the safety limit. The map is plotted in log scale. The wavelength is 800 n m . We calculated the number of photons passing through the US spot ( Φ ( r U S ) × p d u r / ( h c / λ ) ) by multiplying the photon flux at target depth with the longitudinal cross-sectional area of US spot. (b) Number of photons passing through the US spot is plotted along depth. The scale on the right axis represents the corresponding photon numbers normalized by the number of incident photons. The plot is in log scale.

Fig. 4
Fig. 4

Dependence of ultrasonic modulation efficiency ( η ( θ ) ) on the incident light angle as calculated by the Raman-Nath theory for an 800 n m light source and an ultrasonic pressure of 2.35 M P a . As θ max ( 88 ) is around 90 and the light irradiance is nearly isotropic in the diffusing regime, we averaged out the modulation efficiency for incidence angles of [ 0 , θ max ] . This results in a value of ~ 0.0067 (for 532 n m and 633 n m , ~ 0.013 and ~ 0.010 , respectively). The number of modulated photon numbers ( N U S M ) is calculated by multiplying the averaged modulation efficiency with the number of photons passing through the US spot ( N U S P ). The plot is in log scale.

Fig. 5
Fig. 5

(a) 2D photon flux map of ultrasonically modulated light emerging from the tissue surface from a single incident light pulse. The map is plotted in log scale. The target depth is 5 c m and the wavelength is 800 n m . We calculated the average number of photons per mode ( N mode ( x l , m , y l , m ) ) by dividing the photon flux at each grid cell ( J ( x l , m , y l , m ) ) by the number of modes inside each grid cell ( M grid ). (b) The total number of emerging photons at the surface is plotted for various depths. The scale on the right axis represents corresponding photon numbers normalized by the number of incident photons. The plot is in log scale.

Fig. 6
Fig. 6

Normalized probability density function of the phase measurement error at different average signal level ( I s i g / ( h c / λ ) = A s i g 2 / ( h c / λ ) ) – 100, 10, 1, 0.1, 0.01 photon(s). The PDF is built with the Monte-Carlo simulation of 4 step phase shifting method for 10 5 number of modes. The error is increased as the signal is decreased. Though it is wide, peak around 0 is observed even at signal photon number smaller than 1.

Fig. 7
Fig. 7

(a) Normalized intensity contribution from the single grid cell ( 0.1 m m × 0.1 m m ) to a single input mode ( A O P C _ L U T 2 )) when OPC is performed for M g r i d ( = 6.25 × 10 4 ) number of modes. The plot is normalized with the ideal intensity contribution in the case without wavefront measurement error which has a linear dependence on the average signal photon number. As the average signal photon number decreases, the wavefront measurement error across M g r i d number of modes increases, resulting in a dramatic reduction in intensity contribution. (b) Dependence of PBR on the average signal photon number when OPC is performed on single grid cell for the single input mode. By assuming plane wave front, background intensity contribution is calculated. Then, PBR is calculated by dividing peak intensity contribution with the background intensity contribution. PBR drops to 1 at low signal photon limit and saturates to π M g r i d / 4 ( = 4.9 × 10 4 ) when there is enough signal photon for an accurate phase measurement. The insets show the resultant phase distribution (PDF) of the phase-conjugated field ( ϕ O P C _ L U T ) at different signal photon level. The plots are in log-log scale.

Fig. 8
Fig. 8

(a) Dependence of PBR of TRUE focal spot on the target depth. Penetration depth limit is ~ 103 m m ( d e p t h l o c a l where P B R T R U E = 2 ) with 800 n m (red line, circle marker). (b) Dependence of fluence at TRUE focal spot normalized by the incident playback light intensity at surface. Light power on the TRUE focal spot becomes weaker than the incident light power from ~ 85 m m ( d e p t h f l u e n c e ) for 800 n m . Penetration depth limits for both standards are reduced to ~ 34 m m and ~ 75 m m ( d e p t h l o c a l ), ~ 25 m m and ~ 62 m m ( d e p t h f l u e n c e ), for 532 n m (green line, triangle marker) and 633 n m (blue line, square marker) light, respectively. The plots are in log scale.

Tables (1)

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Table 1 Coordinate System and Parameters Used During the Calculation

Equations (24)

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( D 2 + μ a ) Φ ( r ) = q 0 ( r )
D = 1 3 ( μ a + μ s ) , a n d q 0 ( r ) = 4 π ε ( r , s ^ ) d Ω = 4 π ε ( r ) .
Φ point ( r ) = 1 4 π r D e μ e f f r
Φ ( r ) = Φ ( x , y , z ) = i p o int s o u r c e s P point 4 π | r ( x i , y i , 1 / μ s ) | D e μ e f f | r ( x i , y i , 1 / μ s ) | P point 4 π | r ( x i , y i , 1 / μ s ) | D e μ e f f | r ( x i , y i , 1 / μ s ) |
N U S P = Φ ( r U S ) × A U S × p d u r / ( h c / λ )
n = n p P
Q ( θ ) τ ( θ ) 2 1
η ( θ ) = J 1 2 ( τ ( θ ) )
N U S M = ( N U S P 0 θ max η ( θ ) d θ ) / θ max
J ( r ) = D Φ ( r ) = Φ ( r U S ) A U S η 4 π | r r U S | 2 ( 1 | r r U S | + μ e f f ) e μ e f f | r r U S | ( r r U S ) Φ ( r U S ) A U S η 4 π | r + r U S | 2 ( 1 | r + r U S | + μ e f f ) e μ e f f | r + r U S | ( r + r U S )
N mode ( x l , m , y l , m ) = J z ( x l , m , y l , m ) × ( λ / 2 ) 2 × p d u r / ( h c / λ ) .
I i = | E r e f + E s i g + E unmod | 2 = A r e f 2 + A s i g 2 + A unmod 2 + 2 A r e f A s i g cos ( ϕ r e f i ϕ s i g ) + 2 A r e f A unmod cos ( ϕ r e f i ϕ unmod ) + 2 A s i g A unmod cos ( ϕ s i g ϕ unmod )
S N R = 2 A r e f A s i g A r e f 2 + A unmod 2 + A s i g 2 + 2 A r e f A unmod + 2 A s i g A unmod 1 h c / λ
A 0 (input mode) A k e i ϕ k (k th mode measured at OPC plane) E k (k th mode played back at OPC plane) A k A 0 e i ϕ k E k (time-reversed input mode)
E O P C = k M m o d e A k e i ϕ k A 0 E k = k M m o d e A k A 0 A R e f
E B a c k = k M m o d e A k e i ϕ k A 0 A R e f
P B R s i n g l e = k M m o d e E O P C E O P C * k M m o d e E B a c k E B a c k * π 4 M m o d e
E O P C E O P C * = | k M m o d e A k e i ϕ k A 0 A R e f e i ψ k | 2 with phase measurement error
E O P C = k M m o d e A k A 0 A R e f e i ( ϕ k ψ k ) = l , m k c e l l l , m M g r i d A k A 0 A R e f e i ( ϕ k ψ k ) = l , m A O P C _ L U T ( N mode ( x l , m , y l , m ) ) e i ϕ O P C _ L U T ( N mode ( x l , m , y l , m ) )
M U S = 2 A U S ( λ / 2 ) 2
P B R T R U E = π 4 k M m o d e E O P C E O P C * / k M m o d e E B a c k E B a c k * M U S with phase measurement error .
1st standard : P B R T R U E ( d e p t h l o c a l ) = 2
Φ T R U E = Φ B a c k ( r U S ) × P B R T R U E
2nd standard : Φ T R U E ( d e p t h f l u e n c e ) I p l a y b a c k = 1.

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