Abstract

The focal field distribution of tightly focused laser beams in turbid media is sensitive to optical scattering and therefore of direct relevance to image quality in confocal and nonlinear microscopy. A model that considers both the influence of scattering and diffraction on the amplitude and phase of the electric field in focused beam geometries is required to describe these distorted focal fields. We combine an electric field Monte Carlo approach that simulates the electric field propagation in turbid media with an angular-spectrum representation of diffraction theory to analyze the effect of tissue scattering properties on the focal field. In particular, we examine the impact of variations in the scattering coefficient (µs), single-scattering anisotropy (g), of the turbid medium and the numerical aperture of the focusing lens on the focal volume at various depths. The model predicts a scattering-induced broadening, amplitude loss, and depolarization of the focal field that corroborates experimental results. We find that both the width and the amplitude of the focal field are dictated primarily by µs with little influence from g. In addition, our model confirms that the depolarization rate is small compared to the amplitude loss of the tightly focused field.

© 2011 OSA

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  1. V. Tuchin, Tissue Optics (SPIE Press, 2007).
  2. P. Theer, M. T. Hasan, and W. Denk, “Two-photon imaging to a depth of 1000 µm in living brains by use of a Ti:Al2O3 regenerative amplifier,” Opt. Lett. 28(12), 1022–1024 (2003).
    [CrossRef] [PubMed]
  3. M. Balu, T. Baldacchini, J. Carter, T. B. Krasieva, R. Zadoyan, and B. J. Tromberg, “Effect of excitation wavelength on penetration depth in nonlinear optical microscopy of turbid media,” J. Biomed. Opt. 14(1), 010508 (2009).
    [CrossRef] [PubMed]
  4. M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
    [CrossRef] [PubMed]
  5. L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206(1), 65–71 (2002).
    [CrossRef] [PubMed]
  6. T. M. Nieuwenhuizen, A. Lagendijk, and B. A. van Tiggelen, “Resonant point scatterers in multiple scattering of classical waves,” Phys. Lett. A 169(3), 191–194 (1992).
    [CrossRef]
  7. A. K. Dunn and R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Sel. Top. Quantum Electron. 2(4), 898–905 (1996).
    [CrossRef]
  8. R. Drezek, A. Dunn, and R. Richards-Kortum, “Light scattering from cells: finite-difference time-domain simulations and goniometric measurements,” Appl. Opt. 38(16), 3651–3661 (1999).
    [CrossRef] [PubMed]
  9. C. Liu, C. Capjack, and W. Rozmus, “3-D simulation of light scattering from biological cells and cell differentiation,” J. Biomed. Opt. 10(1), 014007 (2005).
    [CrossRef] [PubMed]
  10. I. R. Çapoglu, A. Taflove, and V. Backman, “Generation of an incident focused light pulse in FDTD,” Opt. Express 16(23), 19208–19220 (2008).
    [CrossRef] [PubMed]
  11. M. S. Starosta and A. K. Dunn, “Three-dimensional computation of focused beam propagation through multiple biological cells,” Opt. Express 17(15), 12455–12469 (2009).
    [CrossRef] [PubMed]
  12. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vols. I and II (Academic Press, 1978).
  13. L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
  14. A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20(1), 92–98 (2003).
    [CrossRef] [PubMed]
  15. G. W. Kattawar and G. N. Plass, “Radiance and polarization of multiple scattered light from haze and clouds,” Appl. Opt. 7(8), 1519–1527 (1968).
    [CrossRef] [PubMed]
  16. B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10(6), 824–830 (1983).
    [CrossRef] [PubMed]
  17. I. Lux, and L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, 1991).
  18. X. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: a Monte Carlo study,” J. Biomed. Opt. 7(3), 279–290 (2002).
    [CrossRef] [PubMed]
  19. J. S. You, C. K. Hayakawa, and V. Venugopalan, “Frequency domain photon migration in the δ- P1 approximation: analysis of ballistic, transport, and diffuse regimes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2), 021903 (2005).
    [CrossRef] [PubMed]
  20. J. M. Schmitt and K. Ben-Letaief, “Efficient Monte Carlo simulation of confocal microscopy in biological tissue,” J. Opt. Soc. Am. A 13(5), 952–961 (1996).
    [CrossRef]
  21. C. M. Blanca and C. Saloma, “Monte carlo analysis of two-photon fluorescence imaging through a scattering medium,” Appl. Opt. 37(34), 8092–8102 (1998).
    [CrossRef] [PubMed]
  22. Z. Song, K. Dong, X. H. Hu, and J. Q. Lu, “Monte carlo simulation of converging laser beams propagating in biological materials,” Appl. Opt. 38(13), 2944–2949 (1999).
    [CrossRef] [PubMed]
  23. L. V. Wang and G. Liang, “Absorption distribution of an optical beam focused into a turbid medium,” Appl. Opt. 38(22), 4951–4958 (1999).
    [CrossRef] [PubMed]
  24. X. S. Gan and M. Gu, “Effective point-spread function for fast image modeling and processing in microscopic imaging through turbid media,” Opt. Lett. 24(11), 741–743 (1999).
    [CrossRef] [PubMed]
  25. A. K. Dunn, V. P. Wallace, M. Coleno, M. W. Berns, and B. J. Tromberg, “Influence of optical properties on two-photon fluorescence imaging in turbid samples,” Appl. Opt. 39(7), 1194–1201 (2000).
    [CrossRef] [PubMed]
  26. X. Deng, X. Gan, and M. Gu, “Monte Carlo simulation of multiphoton fluorescence microscopic imaging through inhomogeneous tissuelike turbid media,” J. Biomed. Opt. 8(3), 440–449 (2003).
    [CrossRef] [PubMed]
  27. X. Deng and M. Gu, “Penetration depth of single-, two-, and three-photon fluorescence microscopic imaging through human cortex structures: Monte Carlo simulation,” Appl. Opt. 42(16), 3321–3329 (2003).
    [CrossRef] [PubMed]
  28. X. Deng, X. Wang, H. Liu, Z. Zhuang, and Z. Guo, “Simulation study of second-harmonic microscopic imaging signals through tissue-like turbid media,” J. Biomed. Opt. 11(2), 024013 (2006).
    [CrossRef] [PubMed]
  29. A. Leray, C. Odin, E. Huguet, F. Amblard, and Y. Le Grand, “Spatially distributed two-photon excitation fluorescence in scattering media: Experiments and time-resolved Monte Carlo simulations,” Opt. Commun. 272(1), 269–278 (2007).
    [CrossRef]
  30. J. M. Schmitt and A. Knuttel, “Model of optical coherence tomography of heterogeneous tissue,” J. Opt. Soc. Am. A 14(6), 1231–1242 (1997).
    [CrossRef]
  31. D. J. Smithies, T. Lindmo, Z. Chen, J. S. Nelson, and T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43(10), 3025–3044 (1998).
    [CrossRef] [PubMed]
  32. A. Tycho, T. M. Jørgensen, H. T. Yura, and P. E. Andersen, “Derivation of a Monte Carlo method for modeling heterodyne detection in optical coherence tomography systems,” Appl. Opt. 41(31), 6676–6691 (2002).
    [CrossRef] [PubMed]
  33. G. Xiong, P. Xue, J. Wu, Q. Miao, R. Wang, and L. Ji, “Particle-fixed Monte Carlo model for optical coherence tomography,” Opt. Express 13(6), 2182–2195 (2005).
    [CrossRef] [PubMed]
  34. D. G. Fischer, S. A. Prahl, and D. D. Duncan, “Monte Carlo modeling of spatial coherence: free space diffraction,” J. Opt. Soc. Am. A 25(10), 2571–2581 (2008).
    [CrossRef]
  35. V. R. Daria, C. Saloma, and S. Kawata, “Excitation with a focused, pulsed optical beam in scattering media: diffraction effects,” Appl. Opt. 39(28), 5244–5255 (2000).
    [CrossRef] [PubMed]
  36. M. Xu, “Electric field Monte Carlo simulation of polarized light propagation in turbid media,” Opt. Express 12(26), 6530–6539 (2004).
    [CrossRef] [PubMed]
  37. K. Phillips, M. Xu, S. K. Gayen, and R. R. Alfano, “Time-resolved ring structure of circularly polarized beams backscattered from forward scattering media,” Opt. Express 13(20), 7954–7969 (2005).
    [CrossRef] [PubMed]
  38. J. Sawicki, N. Kastor, and M. Xu, “Electric field Monte Carlo simulation of coherent backscattering of polarized light by a turbid medium containing Mie scatterers,” Opt. Express 16(8), 5728–5738 (2008).
    [CrossRef] [PubMed]
  39. C. K. Hayakawa, V. Venugopalan, V. V. Krishnamachari, and E. O. Potma, “Amplitude and phase of tightly focused laser beams in turbid media,” Phys. Rev. Lett. 103(4), 043903 (2009).
    [CrossRef] [PubMed]
  40. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems 2: structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A 253(1274), 358–379 (1959).
    [CrossRef]
  41. L. Novotny, and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).
  42. C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26(17), 1335–1337 (2001).
    [CrossRef] [PubMed]
  43. C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, 1983).
  44. T. L. Troy and S. N. Thennadil, “Optical properties of human skin in the near infrared wavelength range of 1000 to 2200 nm,” J. Biomed. Opt. 6(2), 167–176 (2001).
    [CrossRef] [PubMed]
  45. S. L. Jacques, “Skin Optics,” http://omlc.ogi.edu/news/jan98/skinoptics.html .
  46. B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part I: General discussion of the geometrical aberrations,” Physica 10(8), 679–692 (1943).
    [CrossRef]
  47. T. Wilson and A. R. Carlini, “Aberrations in confocal imaging systems,” J. Microsc. 154, 243–256 (1998).
  48. C. K. Tung, Y. Sun, W. Lo, S. J. Lin, S. H. Jee, and C. Y. Dong, “Effects of objective numerical apertures on achievable imaging depths in multiphoton microscopy,” Microsc. Res. Tech. 65(6), 308–314 (2004).
    [CrossRef] [PubMed]
  49. C. Y. Dong, K. Koenig, and P. So, “Characterizing point spread functions of two-photon fluorescence microscopy in turbid medium,” J. Biomed. Opt. 8(3), 450–459 (2003).
    [CrossRef] [PubMed]
  50. N. Ghosh, H. S. Patel, and P. K. Gupta, “Depolarization of light in tissue phantoms - effect of a distribution in the size of scatterers,” Opt. Express 11(18), 2198–2205 (2003).
    [CrossRef] [PubMed]

2009 (3)

M. Balu, T. Baldacchini, J. Carter, T. B. Krasieva, R. Zadoyan, and B. J. Tromberg, “Effect of excitation wavelength on penetration depth in nonlinear optical microscopy of turbid media,” J. Biomed. Opt. 14(1), 010508 (2009).
[CrossRef] [PubMed]

C. K. Hayakawa, V. Venugopalan, V. V. Krishnamachari, and E. O. Potma, “Amplitude and phase of tightly focused laser beams in turbid media,” Phys. Rev. Lett. 103(4), 043903 (2009).
[CrossRef] [PubMed]

M. S. Starosta and A. K. Dunn, “Three-dimensional computation of focused beam propagation through multiple biological cells,” Opt. Express 17(15), 12455–12469 (2009).
[CrossRef] [PubMed]

2008 (3)

2007 (1)

A. Leray, C. Odin, E. Huguet, F. Amblard, and Y. Le Grand, “Spatially distributed two-photon excitation fluorescence in scattering media: Experiments and time-resolved Monte Carlo simulations,” Opt. Commun. 272(1), 269–278 (2007).
[CrossRef]

2006 (1)

X. Deng, X. Wang, H. Liu, Z. Zhuang, and Z. Guo, “Simulation study of second-harmonic microscopic imaging signals through tissue-like turbid media,” J. Biomed. Opt. 11(2), 024013 (2006).
[CrossRef] [PubMed]

2005 (4)

J. S. You, C. K. Hayakawa, and V. Venugopalan, “Frequency domain photon migration in the δ- P1 approximation: analysis of ballistic, transport, and diffuse regimes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2), 021903 (2005).
[CrossRef] [PubMed]

C. Liu, C. Capjack, and W. Rozmus, “3-D simulation of light scattering from biological cells and cell differentiation,” J. Biomed. Opt. 10(1), 014007 (2005).
[CrossRef] [PubMed]

K. Phillips, M. Xu, S. K. Gayen, and R. R. Alfano, “Time-resolved ring structure of circularly polarized beams backscattered from forward scattering media,” Opt. Express 13(20), 7954–7969 (2005).
[CrossRef] [PubMed]

G. Xiong, P. Xue, J. Wu, Q. Miao, R. Wang, and L. Ji, “Particle-fixed Monte Carlo model for optical coherence tomography,” Opt. Express 13(6), 2182–2195 (2005).
[CrossRef] [PubMed]

2004 (2)

M. Xu, “Electric field Monte Carlo simulation of polarized light propagation in turbid media,” Opt. Express 12(26), 6530–6539 (2004).
[CrossRef] [PubMed]

C. K. Tung, Y. Sun, W. Lo, S. J. Lin, S. H. Jee, and C. Y. Dong, “Effects of objective numerical apertures on achievable imaging depths in multiphoton microscopy,” Microsc. Res. Tech. 65(6), 308–314 (2004).
[CrossRef] [PubMed]

2003 (6)

2002 (4)

X. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: a Monte Carlo study,” J. Biomed. Opt. 7(3), 279–290 (2002).
[CrossRef] [PubMed]

M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[CrossRef] [PubMed]

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206(1), 65–71 (2002).
[CrossRef] [PubMed]

A. Tycho, T. M. Jørgensen, H. T. Yura, and P. E. Andersen, “Derivation of a Monte Carlo method for modeling heterodyne detection in optical coherence tomography systems,” Appl. Opt. 41(31), 6676–6691 (2002).
[CrossRef] [PubMed]

2001 (2)

2000 (2)

1999 (4)

1998 (3)

C. M. Blanca and C. Saloma, “Monte carlo analysis of two-photon fluorescence imaging through a scattering medium,” Appl. Opt. 37(34), 8092–8102 (1998).
[CrossRef] [PubMed]

T. Wilson and A. R. Carlini, “Aberrations in confocal imaging systems,” J. Microsc. 154, 243–256 (1998).

D. J. Smithies, T. Lindmo, Z. Chen, J. S. Nelson, and T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43(10), 3025–3044 (1998).
[CrossRef] [PubMed]

1997 (1)

1996 (2)

J. M. Schmitt and K. Ben-Letaief, “Efficient Monte Carlo simulation of confocal microscopy in biological tissue,” J. Opt. Soc. Am. A 13(5), 952–961 (1996).
[CrossRef]

A. K. Dunn and R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Sel. Top. Quantum Electron. 2(4), 898–905 (1996).
[CrossRef]

1992 (1)

T. M. Nieuwenhuizen, A. Lagendijk, and B. A. van Tiggelen, “Resonant point scatterers in multiple scattering of classical waves,” Phys. Lett. A 169(3), 191–194 (1992).
[CrossRef]

1983 (1)

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10(6), 824–830 (1983).
[CrossRef] [PubMed]

1968 (1)

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems 2: structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A 253(1274), 358–379 (1959).
[CrossRef]

1943 (1)

B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part I: General discussion of the geometrical aberrations,” Physica 10(8), 679–692 (1943).
[CrossRef]

Adam, G.

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10(6), 824–830 (1983).
[CrossRef] [PubMed]

Albert, O.

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206(1), 65–71 (2002).
[CrossRef] [PubMed]

Alfano, R. R.

Amblard, F.

A. Leray, C. Odin, E. Huguet, F. Amblard, and Y. Le Grand, “Spatially distributed two-photon excitation fluorescence in scattering media: Experiments and time-resolved Monte Carlo simulations,” Opt. Commun. 272(1), 269–278 (2007).
[CrossRef]

Andersen, P. E.

Backman, V.

Baldacchini, T.

M. Balu, T. Baldacchini, J. Carter, T. B. Krasieva, R. Zadoyan, and B. J. Tromberg, “Effect of excitation wavelength on penetration depth in nonlinear optical microscopy of turbid media,” J. Biomed. Opt. 14(1), 010508 (2009).
[CrossRef] [PubMed]

Balu, M.

M. Balu, T. Baldacchini, J. Carter, T. B. Krasieva, R. Zadoyan, and B. J. Tromberg, “Effect of excitation wavelength on penetration depth in nonlinear optical microscopy of turbid media,” J. Biomed. Opt. 14(1), 010508 (2009).
[CrossRef] [PubMed]

Ben-Letaief, K.

Berns, M. W.

Bevilacqua, F.

Blanca, C. M.

Booth, M. J.

M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[CrossRef] [PubMed]

Capjack, C.

C. Liu, C. Capjack, and W. Rozmus, “3-D simulation of light scattering from biological cells and cell differentiation,” J. Biomed. Opt. 10(1), 014007 (2005).
[CrossRef] [PubMed]

Çapoglu, I. R.

Carlini, A. R.

T. Wilson and A. R. Carlini, “Aberrations in confocal imaging systems,” J. Microsc. 154, 243–256 (1998).

Carter, J.

M. Balu, T. Baldacchini, J. Carter, T. B. Krasieva, R. Zadoyan, and B. J. Tromberg, “Effect of excitation wavelength on penetration depth in nonlinear optical microscopy of turbid media,” J. Biomed. Opt. 14(1), 010508 (2009).
[CrossRef] [PubMed]

Chen, Z.

D. J. Smithies, T. Lindmo, Z. Chen, J. S. Nelson, and T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43(10), 3025–3044 (1998).
[CrossRef] [PubMed]

Coleno, M.

Daria, V. R.

Deng, X.

X. Deng, X. Wang, H. Liu, Z. Zhuang, and Z. Guo, “Simulation study of second-harmonic microscopic imaging signals through tissue-like turbid media,” J. Biomed. Opt. 11(2), 024013 (2006).
[CrossRef] [PubMed]

X. Deng, X. Gan, and M. Gu, “Monte Carlo simulation of multiphoton fluorescence microscopic imaging through inhomogeneous tissuelike turbid media,” J. Biomed. Opt. 8(3), 440–449 (2003).
[CrossRef] [PubMed]

X. Deng and M. Gu, “Penetration depth of single-, two-, and three-photon fluorescence microscopic imaging through human cortex structures: Monte Carlo simulation,” Appl. Opt. 42(16), 3321–3329 (2003).
[CrossRef] [PubMed]

Denk, W.

Dong, C. Y.

C. K. Tung, Y. Sun, W. Lo, S. J. Lin, S. H. Jee, and C. Y. Dong, “Effects of objective numerical apertures on achievable imaging depths in multiphoton microscopy,” Microsc. Res. Tech. 65(6), 308–314 (2004).
[CrossRef] [PubMed]

C. Y. Dong, K. Koenig, and P. So, “Characterizing point spread functions of two-photon fluorescence microscopy in turbid medium,” J. Biomed. Opt. 8(3), 450–459 (2003).
[CrossRef] [PubMed]

Dong, K.

Drezek, R.

Duncan, D. D.

Dunn, A.

Dunn, A. K.

Fischer, D. G.

Gan, X.

X. Deng, X. Gan, and M. Gu, “Monte Carlo simulation of multiphoton fluorescence microscopic imaging through inhomogeneous tissuelike turbid media,” J. Biomed. Opt. 8(3), 440–449 (2003).
[CrossRef] [PubMed]

Gan, X. S.

Gayen, S. K.

Ghosh, N.

Gu, M.

Guo, Z.

X. Deng, X. Wang, H. Liu, Z. Zhuang, and Z. Guo, “Simulation study of second-harmonic microscopic imaging signals through tissue-like turbid media,” J. Biomed. Opt. 11(2), 024013 (2006).
[CrossRef] [PubMed]

Gupta, P. K.

Hasan, M. T.

Hayakawa, C. K.

C. K. Hayakawa, V. Venugopalan, V. V. Krishnamachari, and E. O. Potma, “Amplitude and phase of tightly focused laser beams in turbid media,” Phys. Rev. Lett. 103(4), 043903 (2009).
[CrossRef] [PubMed]

J. S. You, C. K. Hayakawa, and V. Venugopalan, “Frequency domain photon migration in the δ- P1 approximation: analysis of ballistic, transport, and diffuse regimes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2), 021903 (2005).
[CrossRef] [PubMed]

C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26(17), 1335–1337 (2001).
[CrossRef] [PubMed]

Hu, X. H.

Huguet, E.

A. Leray, C. Odin, E. Huguet, F. Amblard, and Y. Le Grand, “Spatially distributed two-photon excitation fluorescence in scattering media: Experiments and time-resolved Monte Carlo simulations,” Opt. Commun. 272(1), 269–278 (2007).
[CrossRef]

Jee, S. H.

C. K. Tung, Y. Sun, W. Lo, S. J. Lin, S. H. Jee, and C. Y. Dong, “Effects of objective numerical apertures on achievable imaging depths in multiphoton microscopy,” Microsc. Res. Tech. 65(6), 308–314 (2004).
[CrossRef] [PubMed]

Ji, L.

Jørgensen, T. M.

Juskaitis, R.

M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[CrossRef] [PubMed]

Kastor, N.

Kattawar, G. W.

Kawata, S.

Keller, J. B.

Kim, A. D.

Knuttel, A.

Koenig, K.

C. Y. Dong, K. Koenig, and P. So, “Characterizing point spread functions of two-photon fluorescence microscopy in turbid medium,” J. Biomed. Opt. 8(3), 450–459 (2003).
[CrossRef] [PubMed]

Krasieva, T. B.

M. Balu, T. Baldacchini, J. Carter, T. B. Krasieva, R. Zadoyan, and B. J. Tromberg, “Effect of excitation wavelength on penetration depth in nonlinear optical microscopy of turbid media,” J. Biomed. Opt. 14(1), 010508 (2009).
[CrossRef] [PubMed]

Krishnamachari, V. V.

C. K. Hayakawa, V. Venugopalan, V. V. Krishnamachari, and E. O. Potma, “Amplitude and phase of tightly focused laser beams in turbid media,” Phys. Rev. Lett. 103(4), 043903 (2009).
[CrossRef] [PubMed]

Lagendijk, A.

T. M. Nieuwenhuizen, A. Lagendijk, and B. A. van Tiggelen, “Resonant point scatterers in multiple scattering of classical waves,” Phys. Lett. A 169(3), 191–194 (1992).
[CrossRef]

Le Grand, Y.

A. Leray, C. Odin, E. Huguet, F. Amblard, and Y. Le Grand, “Spatially distributed two-photon excitation fluorescence in scattering media: Experiments and time-resolved Monte Carlo simulations,” Opt. Commun. 272(1), 269–278 (2007).
[CrossRef]

Leray, A.

A. Leray, C. Odin, E. Huguet, F. Amblard, and Y. Le Grand, “Spatially distributed two-photon excitation fluorescence in scattering media: Experiments and time-resolved Monte Carlo simulations,” Opt. Commun. 272(1), 269–278 (2007).
[CrossRef]

Liang, G.

Lin, S. J.

C. K. Tung, Y. Sun, W. Lo, S. J. Lin, S. H. Jee, and C. Y. Dong, “Effects of objective numerical apertures on achievable imaging depths in multiphoton microscopy,” Microsc. Res. Tech. 65(6), 308–314 (2004).
[CrossRef] [PubMed]

Lindmo, T.

D. J. Smithies, T. Lindmo, Z. Chen, J. S. Nelson, and T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43(10), 3025–3044 (1998).
[CrossRef] [PubMed]

Liu, C.

C. Liu, C. Capjack, and W. Rozmus, “3-D simulation of light scattering from biological cells and cell differentiation,” J. Biomed. Opt. 10(1), 014007 (2005).
[CrossRef] [PubMed]

Liu, H.

X. Deng, X. Wang, H. Liu, Z. Zhuang, and Z. Guo, “Simulation study of second-harmonic microscopic imaging signals through tissue-like turbid media,” J. Biomed. Opt. 11(2), 024013 (2006).
[CrossRef] [PubMed]

Lo, W.

C. K. Tung, Y. Sun, W. Lo, S. J. Lin, S. H. Jee, and C. Y. Dong, “Effects of objective numerical apertures on achievable imaging depths in multiphoton microscopy,” Microsc. Res. Tech. 65(6), 308–314 (2004).
[CrossRef] [PubMed]

Lu, J. Q.

Miao, Q.

Milner, T. E.

D. J. Smithies, T. Lindmo, Z. Chen, J. S. Nelson, and T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43(10), 3025–3044 (1998).
[CrossRef] [PubMed]

Neil, M. A. A.

M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[CrossRef] [PubMed]

Nelson, J. S.

D. J. Smithies, T. Lindmo, Z. Chen, J. S. Nelson, and T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43(10), 3025–3044 (1998).
[CrossRef] [PubMed]

Nieuwenhuizen, T. M.

T. M. Nieuwenhuizen, A. Lagendijk, and B. A. van Tiggelen, “Resonant point scatterers in multiple scattering of classical waves,” Phys. Lett. A 169(3), 191–194 (1992).
[CrossRef]

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part I: General discussion of the geometrical aberrations,” Physica 10(8), 679–692 (1943).
[CrossRef]

Norris, T. B.

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206(1), 65–71 (2002).
[CrossRef] [PubMed]

Odin, C.

A. Leray, C. Odin, E. Huguet, F. Amblard, and Y. Le Grand, “Spatially distributed two-photon excitation fluorescence in scattering media: Experiments and time-resolved Monte Carlo simulations,” Opt. Commun. 272(1), 269–278 (2007).
[CrossRef]

Patel, H. S.

Phillips, K.

Plass, G. N.

Potma, E. O.

C. K. Hayakawa, V. Venugopalan, V. V. Krishnamachari, and E. O. Potma, “Amplitude and phase of tightly focused laser beams in turbid media,” Phys. Rev. Lett. 103(4), 043903 (2009).
[CrossRef] [PubMed]

Prahl, S. A.

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems 2: structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A 253(1274), 358–379 (1959).
[CrossRef]

Richards-Kortum, R.

R. Drezek, A. Dunn, and R. Richards-Kortum, “Light scattering from cells: finite-difference time-domain simulations and goniometric measurements,” Appl. Opt. 38(16), 3651–3661 (1999).
[CrossRef] [PubMed]

A. K. Dunn and R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Sel. Top. Quantum Electron. 2(4), 898–905 (1996).
[CrossRef]

Rozmus, W.

C. Liu, C. Capjack, and W. Rozmus, “3-D simulation of light scattering from biological cells and cell differentiation,” J. Biomed. Opt. 10(1), 014007 (2005).
[CrossRef] [PubMed]

Saloma, C.

Sawicki, J.

Schmitt, J. M.

Sherman, L.

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206(1), 65–71 (2002).
[CrossRef] [PubMed]

Smithies, D. J.

D. J. Smithies, T. Lindmo, Z. Chen, J. S. Nelson, and T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43(10), 3025–3044 (1998).
[CrossRef] [PubMed]

So, P.

C. Y. Dong, K. Koenig, and P. So, “Characterizing point spread functions of two-photon fluorescence microscopy in turbid medium,” J. Biomed. Opt. 8(3), 450–459 (2003).
[CrossRef] [PubMed]

Song, Z.

Spanier, J.

Starosta, M. S.

Sun, Y.

C. K. Tung, Y. Sun, W. Lo, S. J. Lin, S. H. Jee, and C. Y. Dong, “Effects of objective numerical apertures on achievable imaging depths in multiphoton microscopy,” Microsc. Res. Tech. 65(6), 308–314 (2004).
[CrossRef] [PubMed]

Taflove, A.

Theer, P.

Thennadil, S. N.

T. L. Troy and S. N. Thennadil, “Optical properties of human skin in the near infrared wavelength range of 1000 to 2200 nm,” J. Biomed. Opt. 6(2), 167–176 (2001).
[CrossRef] [PubMed]

Tromberg, B. J.

Troy, T. L.

T. L. Troy and S. N. Thennadil, “Optical properties of human skin in the near infrared wavelength range of 1000 to 2200 nm,” J. Biomed. Opt. 6(2), 167–176 (2001).
[CrossRef] [PubMed]

Tung, C. K.

C. K. Tung, Y. Sun, W. Lo, S. J. Lin, S. H. Jee, and C. Y. Dong, “Effects of objective numerical apertures on achievable imaging depths in multiphoton microscopy,” Microsc. Res. Tech. 65(6), 308–314 (2004).
[CrossRef] [PubMed]

Tycho, A.

van Tiggelen, B. A.

T. M. Nieuwenhuizen, A. Lagendijk, and B. A. van Tiggelen, “Resonant point scatterers in multiple scattering of classical waves,” Phys. Lett. A 169(3), 191–194 (1992).
[CrossRef]

Venugopalan, V.

C. K. Hayakawa, V. Venugopalan, V. V. Krishnamachari, and E. O. Potma, “Amplitude and phase of tightly focused laser beams in turbid media,” Phys. Rev. Lett. 103(4), 043903 (2009).
[CrossRef] [PubMed]

J. S. You, C. K. Hayakawa, and V. Venugopalan, “Frequency domain photon migration in the δ- P1 approximation: analysis of ballistic, transport, and diffuse regimes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2), 021903 (2005).
[CrossRef] [PubMed]

C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26(17), 1335–1337 (2001).
[CrossRef] [PubMed]

Wallace, V. P.

Wang, L. V.

X. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: a Monte Carlo study,” J. Biomed. Opt. 7(3), 279–290 (2002).
[CrossRef] [PubMed]

L. V. Wang and G. Liang, “Absorption distribution of an optical beam focused into a turbid medium,” Appl. Opt. 38(22), 4951–4958 (1999).
[CrossRef] [PubMed]

Wang, R.

Wang, X.

X. Deng, X. Wang, H. Liu, Z. Zhuang, and Z. Guo, “Simulation study of second-harmonic microscopic imaging signals through tissue-like turbid media,” J. Biomed. Opt. 11(2), 024013 (2006).
[CrossRef] [PubMed]

X. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: a Monte Carlo study,” J. Biomed. Opt. 7(3), 279–290 (2002).
[CrossRef] [PubMed]

Wilson, B. C.

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10(6), 824–830 (1983).
[CrossRef] [PubMed]

Wilson, T.

M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[CrossRef] [PubMed]

T. Wilson and A. R. Carlini, “Aberrations in confocal imaging systems,” J. Microsc. 154, 243–256 (1998).

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems 2: structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A 253(1274), 358–379 (1959).
[CrossRef]

Wu, J.

Xiong, G.

Xu, M.

Xue, P.

Ye, J. Y.

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206(1), 65–71 (2002).
[CrossRef] [PubMed]

You, J. S.

J. S. You, C. K. Hayakawa, and V. Venugopalan, “Frequency domain photon migration in the δ- P1 approximation: analysis of ballistic, transport, and diffuse regimes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2), 021903 (2005).
[CrossRef] [PubMed]

C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26(17), 1335–1337 (2001).
[CrossRef] [PubMed]

Yura, H. T.

Zadoyan, R.

M. Balu, T. Baldacchini, J. Carter, T. B. Krasieva, R. Zadoyan, and B. J. Tromberg, “Effect of excitation wavelength on penetration depth in nonlinear optical microscopy of turbid media,” J. Biomed. Opt. 14(1), 010508 (2009).
[CrossRef] [PubMed]

Zhuang, Z.

X. Deng, X. Wang, H. Liu, Z. Zhuang, and Z. Guo, “Simulation study of second-harmonic microscopic imaging signals through tissue-like turbid media,” J. Biomed. Opt. 11(2), 024013 (2006).
[CrossRef] [PubMed]

Appl. Opt. (9)

G. W. Kattawar and G. N. Plass, “Radiance and polarization of multiple scattered light from haze and clouds,” Appl. Opt. 7(8), 1519–1527 (1968).
[CrossRef] [PubMed]

C. M. Blanca and C. Saloma, “Monte carlo analysis of two-photon fluorescence imaging through a scattering medium,” Appl. Opt. 37(34), 8092–8102 (1998).
[CrossRef] [PubMed]

Z. Song, K. Dong, X. H. Hu, and J. Q. Lu, “Monte carlo simulation of converging laser beams propagating in biological materials,” Appl. Opt. 38(13), 2944–2949 (1999).
[CrossRef] [PubMed]

R. Drezek, A. Dunn, and R. Richards-Kortum, “Light scattering from cells: finite-difference time-domain simulations and goniometric measurements,” Appl. Opt. 38(16), 3651–3661 (1999).
[CrossRef] [PubMed]

L. V. Wang and G. Liang, “Absorption distribution of an optical beam focused into a turbid medium,” Appl. Opt. 38(22), 4951–4958 (1999).
[CrossRef] [PubMed]

A. K. Dunn, V. P. Wallace, M. Coleno, M. W. Berns, and B. J. Tromberg, “Influence of optical properties on two-photon fluorescence imaging in turbid samples,” Appl. Opt. 39(7), 1194–1201 (2000).
[CrossRef] [PubMed]

V. R. Daria, C. Saloma, and S. Kawata, “Excitation with a focused, pulsed optical beam in scattering media: diffraction effects,” Appl. Opt. 39(28), 5244–5255 (2000).
[CrossRef] [PubMed]

A. Tycho, T. M. Jørgensen, H. T. Yura, and P. E. Andersen, “Derivation of a Monte Carlo method for modeling heterodyne detection in optical coherence tomography systems,” Appl. Opt. 41(31), 6676–6691 (2002).
[CrossRef] [PubMed]

X. Deng and M. Gu, “Penetration depth of single-, two-, and three-photon fluorescence microscopic imaging through human cortex structures: Monte Carlo simulation,” Appl. Opt. 42(16), 3321–3329 (2003).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

A. K. Dunn and R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Sel. Top. Quantum Electron. 2(4), 898–905 (1996).
[CrossRef]

J. Biomed. Opt. (7)

C. Liu, C. Capjack, and W. Rozmus, “3-D simulation of light scattering from biological cells and cell differentiation,” J. Biomed. Opt. 10(1), 014007 (2005).
[CrossRef] [PubMed]

M. Balu, T. Baldacchini, J. Carter, T. B. Krasieva, R. Zadoyan, and B. J. Tromberg, “Effect of excitation wavelength on penetration depth in nonlinear optical microscopy of turbid media,” J. Biomed. Opt. 14(1), 010508 (2009).
[CrossRef] [PubMed]

X. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: a Monte Carlo study,” J. Biomed. Opt. 7(3), 279–290 (2002).
[CrossRef] [PubMed]

X. Deng, X. Gan, and M. Gu, “Monte Carlo simulation of multiphoton fluorescence microscopic imaging through inhomogeneous tissuelike turbid media,” J. Biomed. Opt. 8(3), 440–449 (2003).
[CrossRef] [PubMed]

X. Deng, X. Wang, H. Liu, Z. Zhuang, and Z. Guo, “Simulation study of second-harmonic microscopic imaging signals through tissue-like turbid media,” J. Biomed. Opt. 11(2), 024013 (2006).
[CrossRef] [PubMed]

T. L. Troy and S. N. Thennadil, “Optical properties of human skin in the near infrared wavelength range of 1000 to 2200 nm,” J. Biomed. Opt. 6(2), 167–176 (2001).
[CrossRef] [PubMed]

C. Y. Dong, K. Koenig, and P. So, “Characterizing point spread functions of two-photon fluorescence microscopy in turbid medium,” J. Biomed. Opt. 8(3), 450–459 (2003).
[CrossRef] [PubMed]

J. Microsc. (2)

T. Wilson and A. R. Carlini, “Aberrations in confocal imaging systems,” J. Microsc. 154, 243–256 (1998).

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206(1), 65–71 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (4)

Med. Phys. (1)

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10(6), 824–830 (1983).
[CrossRef] [PubMed]

Microsc. Res. Tech. (1)

C. K. Tung, Y. Sun, W. Lo, S. J. Lin, S. H. Jee, and C. Y. Dong, “Effects of objective numerical apertures on achievable imaging depths in multiphoton microscopy,” Microsc. Res. Tech. 65(6), 308–314 (2004).
[CrossRef] [PubMed]

Opt. Commun. (1)

A. Leray, C. Odin, E. Huguet, F. Amblard, and Y. Le Grand, “Spatially distributed two-photon excitation fluorescence in scattering media: Experiments and time-resolved Monte Carlo simulations,” Opt. Commun. 272(1), 269–278 (2007).
[CrossRef]

Opt. Express (7)

Opt. Lett. (3)

Phys. Lett. A (1)

T. M. Nieuwenhuizen, A. Lagendijk, and B. A. van Tiggelen, “Resonant point scatterers in multiple scattering of classical waves,” Phys. Lett. A 169(3), 191–194 (1992).
[CrossRef]

Phys. Med. Biol. (1)

D. J. Smithies, T. Lindmo, Z. Chen, J. S. Nelson, and T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43(10), 3025–3044 (1998).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

J. S. You, C. K. Hayakawa, and V. Venugopalan, “Frequency domain photon migration in the δ- P1 approximation: analysis of ballistic, transport, and diffuse regimes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2), 021903 (2005).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

C. K. Hayakawa, V. Venugopalan, V. V. Krishnamachari, and E. O. Potma, “Amplitude and phase of tightly focused laser beams in turbid media,” Phys. Rev. Lett. 103(4), 043903 (2009).
[CrossRef] [PubMed]

Physica (1)

B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part I: General discussion of the geometrical aberrations,” Physica 10(8), 679–692 (1943).
[CrossRef]

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

M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems 2: structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A 253(1274), 358–379 (1959).
[CrossRef]

Other (7)

L. Novotny, and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).

S. L. Jacques, “Skin Optics,” http://omlc.ogi.edu/news/jan98/skinoptics.html .

I. Lux, and L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, 1991).

V. Tuchin, Tissue Optics (SPIE Press, 2007).

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vols. I and II (Academic Press, 1978).

L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).

C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, 1983).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic of the diffraction geometry. The wavefront of the initial field E far is modified to Ed far, which captures the effects of a given medium. Waves launched from a Lambertian source (symbolized by semi-circle) with a wave vector k are allowed to scatter in a medium of thickness T, and the amplitude and phase at each exit wave vector k′ is determined.

Fig. 2.
Fig. 2.

(a) Lateral and (b) axial dimension of the focal field (FWHM) as a function of slab thickness T in units of l* for anisotropy coefficients g = 0, 0.6, 0.8 for a numerical aperture NA = 1.16. l* = 495µm in all samples.

Fig. 3.
Fig. 3.

(a) Lateral and (b) Axial FWHM as a function of slab thickness in terms of ls for anisotropy coefficients g = 0,0.6,0.8 and numerical apertures NA = 0.81 (○), 1.16 (△), 1.31 (□).

Fig. 4.
Fig. 4.

Normalized maximum amplitude as a function of slab thickness in terms of (a) l* and (b) ls for numerical apertures NA = 0.81, 1.16, 1.31 and anisotropy coefficients g = 0(○), 0.6(△), 0.8(□).

Fig. 5.
Fig. 5.

CADF for slab thicknesses of (a) 1ls , (b) 3ls , and (c) 5ls for a single scattering anisotropy of g = 0.8. θ and θ′ represent the incident and exiting angle of the wavefront, respectively. The color bar represents a logarithmic scale.

Fig. 6.
Fig. 6.

Normalized E amplitude versus exiting angle θ′ for slab thicknesses of T = 1ls (○),3ls (△),5ls (□) for a single scattering anisotropy g = 0.8.

Fig. 7.
Fig. 7.

(a) E normalized by maximum value, and (b) (1−E /E ), as a function of the wave’s exiting angle (θ′) for anisotropy coefficients g = 0,0.6,0.8 and slab thicknesses of T = 1ls (○),3ls (△),5ls (□).

Equations (17)

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E f ( x , y , z ) = i f e i k f 2 π ( k x 2 + k y 2 ) k 2 E far d ( k x , k y ) e i ( k x x + k y y + k z z ) 1 k z d k x d k y
E f ( ρ , φ , z ) = i k f e i k f 2 π ϕ = 0 2 π θ = 0 θ max E far d ( θ , ϕ ) e i k z cos θ e i k ρ sin θ cos ( ϕ φ ) sin θ d θ d ϕ .
E far d ( θ , ϕ ) = ϕ = 0 2 π θ = 0 π 2 G ( θ , ϕ θ , ϕ ) E far ( θ , ϕ ) sin θ d θ d ϕ
E far ( θ , ϕ ) = ( n 1 n 2 ) 1 2 cos θ ( cos ϕ cos θ cos ( ϕ γ ) + sin ϕ sin ( ϕ γ ) sin ϕ cos θ cos ( ϕ γ ) cos ϕ sin ( ϕ γ ) sin θ cos ( ϕ γ ) ) E inc ( θ , ϕ ) .
( m ̂ n ̂ s ̂ ) = M ( θ , ϕ ) ( m ̂ n ̂ s ̂ )
M ( θ , ϕ ) = ( cos θ cos ϕ cos θ sin ϕ sin ϕ sin ϕ cos ϕ 0 sin θ cos ϕ sin θ sin ϕ cos θ ) .
p ( θ , ϕ ) = F ( θ , ϕ ) π x 2 Q sca
F ( θ , ϕ ) = ( S 2 2 cos 2 ϕ + S 1 2 sin 2 ϕ ) E 2 + ( S 2 2 sin 2 ϕ + S 1 2 cos 2 ϕ ) E 2 +
2 ( S 2 2 S 1 2 ) cos ϕ sin ϕ Re [ E ( E ) * ] ,
p ( θ ) = 0 2 π p ( θ , ϕ ) d ϕ = S 1 ( θ ) 2 + S 2 ( θ ) 2 x 2 Q sca .
( E E ) = L ( θ , ϕ ) ( E E )
L ( θ , ϕ ) = 1 F ( θ , ϕ ) ( S 2 ( θ ) cos ϕ S 2 ( θ ) sin ϕ S 1 ( θ ) sin ϕ S 1 ( θ ) cos ϕ ) .
ϕ j = [ ( t j / t cycle ) floor ( t j / t cycle ) ] * 2 π .
( m ̂ f n ̂ f s ̂ f ) = M ( θ p , ϕ p ) M f ( m ̂ n ̂ s ̂ )
( E f E f ) = ( cos ϕ p sin ϕ p sin ϕ p cos ϕ p ) L f ( E E )
Re ( E p ) = 1 N p S p Σ j = 1 N p Re ( E p , j )
Im ( E p ) = 1 N p S p Σ j = 1 N p Im ( E p , j )

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