Abstract

We introduce a high performance parallelization to the PSTD solution of Maxwell equations by employing the fast Fourier transform on local Fourier basis. Meanwhile a reformatted derivative operator allows the adoption of a staggered-grid such as the Yee lattice in PSTD, which can overcome the numerical errors in a collocated-grid when spatial discontinuities are present. The accuracy and capability of our method are confirmed by two analytical models. In two applications to surface tissue optics, an ultra wide coherent backscattering cone from the surface layer is found, and the penetration depth of polarization gating identified. Our development prepares a tool for investigating the optical properties of surface tissue structures.

© 2010 OSA

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References

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  1. S. H. Tseng and B. Huang, “Comparing Monte Carlo simulation and pseudospectral time-domain numerical solutions of Maxwell’s equations of light scattering by a macroscopic random medium,” Appl. Phys. Lett. 91(5), 051114 (2007).
    [CrossRef]
  2. F. Voit, J. Schäfer, and A. Kienle, “Light scattering by multiple spheres: comparison between Maxwell theory and radiative-transfer-theory calculations,” Opt. Lett. 34(17), 2593–2595 (2009).
    [CrossRef] [PubMed]
  3. H. Subramanian, P. Pradhan, Y. L. Kim, Y. Liu, X. Li, and V. Backman, “Modeling low-coherence enhanced backscattering using Monte Carlo simulation,” Appl. Opt. 45(24), 6292–6300 (2006).
    [CrossRef] [PubMed]
  4. V. Backman, R. Gurjar, K. Badizadegan, L. Itzkan, R. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Sel. Top. Quantum Electron. 5(4), 1019–1026 (1999).
    [CrossRef]
  5. Y. Liu, Y. L. Kim, X. Li, and V. Backman, “Investigation of depth selectivity of polarization gating for tissue characterization,” Opt. Express 13(2), 601–611 (2005).
    [CrossRef] [PubMed]
  6. Y. L. Kim, P. Pradhan, H. Subramanian, Y. Liu, M. H. Kim, and V. Backman, “Origin of low-coherence enhanced backscattering,” Opt. Lett. 31(10), 1459–1461 (2006).
    [CrossRef] [PubMed]
  7. A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Second Edition (Artech House, 2000).
  8. Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett. 15(3), 158–165 (1997).
    [CrossRef]
  9. S. H. Tseng, Y. L. Kim, A. Taflove, D. Maitland, V. Backman, and J. T. Walsh., “Simulation of enhanced backscattering of light by numerically solving Maxwell’s equations without heuristic approximations,” Opt. Express 13(10), 3666–3672 (2005).
    [CrossRef] [PubMed]
  10. G. J. P. Correa, M. Spiegelman, S. Carbotte, and J. C. Mutter, “Centered and Staggered Fourier derivatives and Hilbert transforms,” Geophysics 67, 1558–1563 (2002).
    [CrossRef]
  11. K. Reuter, F. Jenko, C. B. Forest, and R. A. Bayliss, “A parallel implementation of an MHD code for the simulation of mechanically driven, turbulent dynamos in spherical geometry,” Comput. Phys. Commun. 179(4), 245–249 (2008).
    [CrossRef]
  12. M. Israeli, L. Vozovoi, and A. Averbuch, “Spectral multidomain technique with local Fourier basis,” J. Sci. Comput. 8(2), 135–149 (1993).
    [CrossRef]
  13. Q. B. Liao and G. A. McMechan, “2-D pseudo-spectral viscoacoustic modeling in a distributed-memory multi-processor computer,” Bull. Seismol. Soc. Am. 83, 1345–1354 (1993).
  14. T. W. Lee and S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Antennas Wirel. Propag. Lett. 3(14), 253–256 (2004).
    [CrossRef]
  15. Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospetral time-domain(PSTD) algorithm,” IEEE Trans. Geosci. Rem. Sens. 37(2), 917–926 (1999).
    [CrossRef]
  16. Y. F. Leung and C. H. Chan, “Combining the FDTD and PSTD methods,” Microw. Opt. Technol. Lett. 23(4), 249–254 (1999).
    [CrossRef]
  17. E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
    [CrossRef] [PubMed]
  18. K. M. Koo, Y. Takiguchi, and R. R. Alfano, “Weak localization of photons: contributions from the different scattering pathlengths,” IEEE Photon. Technol. Lett. 58, 94–96 (1989).

2009

2008

K. Reuter, F. Jenko, C. B. Forest, and R. A. Bayliss, “A parallel implementation of an MHD code for the simulation of mechanically driven, turbulent dynamos in spherical geometry,” Comput. Phys. Commun. 179(4), 245–249 (2008).
[CrossRef]

2007

S. H. Tseng and B. Huang, “Comparing Monte Carlo simulation and pseudospectral time-domain numerical solutions of Maxwell’s equations of light scattering by a macroscopic random medium,” Appl. Phys. Lett. 91(5), 051114 (2007).
[CrossRef]

2006

2005

2004

T. W. Lee and S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Antennas Wirel. Propag. Lett. 3(14), 253–256 (2004).
[CrossRef]

2002

G. J. P. Correa, M. Spiegelman, S. Carbotte, and J. C. Mutter, “Centered and Staggered Fourier derivatives and Hilbert transforms,” Geophysics 67, 1558–1563 (2002).
[CrossRef]

1999

Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospetral time-domain(PSTD) algorithm,” IEEE Trans. Geosci. Rem. Sens. 37(2), 917–926 (1999).
[CrossRef]

Y. F. Leung and C. H. Chan, “Combining the FDTD and PSTD methods,” Microw. Opt. Technol. Lett. 23(4), 249–254 (1999).
[CrossRef]

V. Backman, R. Gurjar, K. Badizadegan, L. Itzkan, R. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Sel. Top. Quantum Electron. 5(4), 1019–1026 (1999).
[CrossRef]

1997

Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett. 15(3), 158–165 (1997).
[CrossRef]

1993

M. Israeli, L. Vozovoi, and A. Averbuch, “Spectral multidomain technique with local Fourier basis,” J. Sci. Comput. 8(2), 135–149 (1993).
[CrossRef]

Q. B. Liao and G. A. McMechan, “2-D pseudo-spectral viscoacoustic modeling in a distributed-memory multi-processor computer,” Bull. Seismol. Soc. Am. 83, 1345–1354 (1993).

1989

K. M. Koo, Y. Takiguchi, and R. R. Alfano, “Weak localization of photons: contributions from the different scattering pathlengths,” IEEE Photon. Technol. Lett. 58, 94–96 (1989).

1986

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[CrossRef] [PubMed]

Akkermans, E.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[CrossRef] [PubMed]

Alfano, R. R.

K. M. Koo, Y. Takiguchi, and R. R. Alfano, “Weak localization of photons: contributions from the different scattering pathlengths,” IEEE Photon. Technol. Lett. 58, 94–96 (1989).

Averbuch, A.

M. Israeli, L. Vozovoi, and A. Averbuch, “Spectral multidomain technique with local Fourier basis,” J. Sci. Comput. 8(2), 135–149 (1993).
[CrossRef]

Backman, V.

Badizadegan, K.

V. Backman, R. Gurjar, K. Badizadegan, L. Itzkan, R. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Sel. Top. Quantum Electron. 5(4), 1019–1026 (1999).
[CrossRef]

Bayliss, R. A.

K. Reuter, F. Jenko, C. B. Forest, and R. A. Bayliss, “A parallel implementation of an MHD code for the simulation of mechanically driven, turbulent dynamos in spherical geometry,” Comput. Phys. Commun. 179(4), 245–249 (2008).
[CrossRef]

Carbotte, S.

G. J. P. Correa, M. Spiegelman, S. Carbotte, and J. C. Mutter, “Centered and Staggered Fourier derivatives and Hilbert transforms,” Geophysics 67, 1558–1563 (2002).
[CrossRef]

Chan, C. H.

Y. F. Leung and C. H. Chan, “Combining the FDTD and PSTD methods,” Microw. Opt. Technol. Lett. 23(4), 249–254 (1999).
[CrossRef]

Correa, G. J. P.

G. J. P. Correa, M. Spiegelman, S. Carbotte, and J. C. Mutter, “Centered and Staggered Fourier derivatives and Hilbert transforms,” Geophysics 67, 1558–1563 (2002).
[CrossRef]

Dasari, R. R.

V. Backman, R. Gurjar, K. Badizadegan, L. Itzkan, R. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Sel. Top. Quantum Electron. 5(4), 1019–1026 (1999).
[CrossRef]

Feld, M. S.

V. Backman, R. Gurjar, K. Badizadegan, L. Itzkan, R. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Sel. Top. Quantum Electron. 5(4), 1019–1026 (1999).
[CrossRef]

Forest, C. B.

K. Reuter, F. Jenko, C. B. Forest, and R. A. Bayliss, “A parallel implementation of an MHD code for the simulation of mechanically driven, turbulent dynamos in spherical geometry,” Comput. Phys. Commun. 179(4), 245–249 (2008).
[CrossRef]

Gurjar, R.

V. Backman, R. Gurjar, K. Badizadegan, L. Itzkan, R. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Sel. Top. Quantum Electron. 5(4), 1019–1026 (1999).
[CrossRef]

Hagness, S. C.

T. W. Lee and S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Antennas Wirel. Propag. Lett. 3(14), 253–256 (2004).
[CrossRef]

Huang, B.

S. H. Tseng and B. Huang, “Comparing Monte Carlo simulation and pseudospectral time-domain numerical solutions of Maxwell’s equations of light scattering by a macroscopic random medium,” Appl. Phys. Lett. 91(5), 051114 (2007).
[CrossRef]

Israeli, M.

M. Israeli, L. Vozovoi, and A. Averbuch, “Spectral multidomain technique with local Fourier basis,” J. Sci. Comput. 8(2), 135–149 (1993).
[CrossRef]

Itzkan, L.

V. Backman, R. Gurjar, K. Badizadegan, L. Itzkan, R. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Sel. Top. Quantum Electron. 5(4), 1019–1026 (1999).
[CrossRef]

Jenko, F.

K. Reuter, F. Jenko, C. B. Forest, and R. A. Bayliss, “A parallel implementation of an MHD code for the simulation of mechanically driven, turbulent dynamos in spherical geometry,” Comput. Phys. Commun. 179(4), 245–249 (2008).
[CrossRef]

Kienle, A.

Kim, M. H.

Kim, Y. L.

Koo, K. M.

K. M. Koo, Y. Takiguchi, and R. R. Alfano, “Weak localization of photons: contributions from the different scattering pathlengths,” IEEE Photon. Technol. Lett. 58, 94–96 (1989).

Lee, T. W.

T. W. Lee and S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Antennas Wirel. Propag. Lett. 3(14), 253–256 (2004).
[CrossRef]

Leung, Y. F.

Y. F. Leung and C. H. Chan, “Combining the FDTD and PSTD methods,” Microw. Opt. Technol. Lett. 23(4), 249–254 (1999).
[CrossRef]

Li, X.

Liao, Q. B.

Q. B. Liao and G. A. McMechan, “2-D pseudo-spectral viscoacoustic modeling in a distributed-memory multi-processor computer,” Bull. Seismol. Soc. Am. 83, 1345–1354 (1993).

Liu, Q. H.

Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospetral time-domain(PSTD) algorithm,” IEEE Trans. Geosci. Rem. Sens. 37(2), 917–926 (1999).
[CrossRef]

Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett. 15(3), 158–165 (1997).
[CrossRef]

Liu, Y.

Maitland, D.

Maynard, R.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[CrossRef] [PubMed]

McMechan, G. A.

Q. B. Liao and G. A. McMechan, “2-D pseudo-spectral viscoacoustic modeling in a distributed-memory multi-processor computer,” Bull. Seismol. Soc. Am. 83, 1345–1354 (1993).

Mutter, J. C.

G. J. P. Correa, M. Spiegelman, S. Carbotte, and J. C. Mutter, “Centered and Staggered Fourier derivatives and Hilbert transforms,” Geophysics 67, 1558–1563 (2002).
[CrossRef]

Perelman, L. T.

V. Backman, R. Gurjar, K. Badizadegan, L. Itzkan, R. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Sel. Top. Quantum Electron. 5(4), 1019–1026 (1999).
[CrossRef]

Pradhan, P.

Reuter, K.

K. Reuter, F. Jenko, C. B. Forest, and R. A. Bayliss, “A parallel implementation of an MHD code for the simulation of mechanically driven, turbulent dynamos in spherical geometry,” Comput. Phys. Commun. 179(4), 245–249 (2008).
[CrossRef]

Schäfer, J.

Spiegelman, M.

G. J. P. Correa, M. Spiegelman, S. Carbotte, and J. C. Mutter, “Centered and Staggered Fourier derivatives and Hilbert transforms,” Geophysics 67, 1558–1563 (2002).
[CrossRef]

Subramanian, H.

Taflove, A.

Takiguchi, Y.

K. M. Koo, Y. Takiguchi, and R. R. Alfano, “Weak localization of photons: contributions from the different scattering pathlengths,” IEEE Photon. Technol. Lett. 58, 94–96 (1989).

Tseng, S. H.

S. H. Tseng and B. Huang, “Comparing Monte Carlo simulation and pseudospectral time-domain numerical solutions of Maxwell’s equations of light scattering by a macroscopic random medium,” Appl. Phys. Lett. 91(5), 051114 (2007).
[CrossRef]

S. H. Tseng, Y. L. Kim, A. Taflove, D. Maitland, V. Backman, and J. T. Walsh., “Simulation of enhanced backscattering of light by numerically solving Maxwell’s equations without heuristic approximations,” Opt. Express 13(10), 3666–3672 (2005).
[CrossRef] [PubMed]

Voit, F.

Vozovoi, L.

M. Israeli, L. Vozovoi, and A. Averbuch, “Spectral multidomain technique with local Fourier basis,” J. Sci. Comput. 8(2), 135–149 (1993).
[CrossRef]

Walsh, J. T.

Wolf, P. E.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

S. H. Tseng and B. Huang, “Comparing Monte Carlo simulation and pseudospectral time-domain numerical solutions of Maxwell’s equations of light scattering by a macroscopic random medium,” Appl. Phys. Lett. 91(5), 051114 (2007).
[CrossRef]

Bull. Seismol. Soc. Am.

Q. B. Liao and G. A. McMechan, “2-D pseudo-spectral viscoacoustic modeling in a distributed-memory multi-processor computer,” Bull. Seismol. Soc. Am. 83, 1345–1354 (1993).

Comput. Phys. Commun.

K. Reuter, F. Jenko, C. B. Forest, and R. A. Bayliss, “A parallel implementation of an MHD code for the simulation of mechanically driven, turbulent dynamos in spherical geometry,” Comput. Phys. Commun. 179(4), 245–249 (2008).
[CrossRef]

Geophysics

G. J. P. Correa, M. Spiegelman, S. Carbotte, and J. C. Mutter, “Centered and Staggered Fourier derivatives and Hilbert transforms,” Geophysics 67, 1558–1563 (2002).
[CrossRef]

IEEE Antennas Wirel. Propag. Lett.

T. W. Lee and S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Antennas Wirel. Propag. Lett. 3(14), 253–256 (2004).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

V. Backman, R. Gurjar, K. Badizadegan, L. Itzkan, R. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Sel. Top. Quantum Electron. 5(4), 1019–1026 (1999).
[CrossRef]

IEEE Photon. Technol. Lett.

K. M. Koo, Y. Takiguchi, and R. R. Alfano, “Weak localization of photons: contributions from the different scattering pathlengths,” IEEE Photon. Technol. Lett. 58, 94–96 (1989).

IEEE Trans. Geosci. Rem. Sens.

Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospetral time-domain(PSTD) algorithm,” IEEE Trans. Geosci. Rem. Sens. 37(2), 917–926 (1999).
[CrossRef]

J. Sci. Comput.

M. Israeli, L. Vozovoi, and A. Averbuch, “Spectral multidomain technique with local Fourier basis,” J. Sci. Comput. 8(2), 135–149 (1993).
[CrossRef]

Microw. Opt. Technol. Lett.

Y. F. Leung and C. H. Chan, “Combining the FDTD and PSTD methods,” Microw. Opt. Technol. Lett. 23(4), 249–254 (1999).
[CrossRef]

Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett. 15(3), 158–165 (1997).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[CrossRef] [PubMed]

Other

A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Second Edition (Artech House, 2000).

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Figures (10)

Fig. 1
Fig. 1

(a). The curve of function f ( x ) stacked on top of a schematic diagram illustrating overlapping domain decomposition and the bell function; and (b) the first derivative of f ( x ) given by Eq. (4) vs. analytical results. In (a), the x axis is divided by the vertical dashed lines into multiple subdomains, which are in turn mapped into different computation nodes. Each node obtains from its neighbors a copy of data in the thin neighborhood just outside its boundaries, referred to as the overlapping region; meanwhile the subdomain itself forms the non-overlapping region. The data in the overlapping and non-overlapping regions are then multiplied by the bell function to produce the local data for the local FFT. The bell function, as depicted by the blue curve, has a flat top 1 and gradually decreases to 0 at both the left and the right ends, and imposes the periodic condition on the local data as required by FFT. The derivatives concatenated from all the non-overlapping regions reproduce the derivatives in the whole computation region. Here for illustrative purposes, the width of the subdomains in (a) is not proportional to what we employed in actual calculations. In (b), note there are actually two curves in the plot, but they agree so well that they essentially coincide. Because Eq. (1) has an analytical expression, its exact first derivative can be strictly calculated. So the discrete results in (b) demonstrate the remarkable accuracy of Eq. (4) and FFT with local Fourier basis. Here for clarity, only region [-50,50] is shown in (b). Situations in other regions are similar.

Fig. 2
Fig. 2

Differential scattering cross section . d σ / d θ . as predicted by SLPSTD vs. the exact Mie solution, in linear scale (left axis) and logarithmic scale (right axis). Gird resolution is set to 0.098 μm, corresponding to 5 samplings per wavelength in polystyrene. The two curves coincide when plotted on linear scale. Their discrepancy is only visible on the logarithmic scale at the order of 10 6 .

Fig. 3
Fig. 3

Radiation intensities from an electric dipole embedded in a concentric dielectric sphere, as predicted by SLPSTD and the analytical solution. As shown in the inset, the concentric sphere simulates a biological cell and is composed of two layers. The inner core and the outer shell have the diameters and refractive indices of cell nucleus and cytoplasm, respectively. The surrounding medium is water. A harmonic electric dipole off-centered along the z axis simulates a Raman emitter or fluorophore. Origin of the coordinates is at the sphere center and θ is the polar angle.

Fig. 4
Fig. 4

Snapshots of the electric field E x on the plane z = 1.4  μm , (a) parallel solution by SLPSTD and (b) sequential solution by collocated-grid PSTD with global Fourier basis. In (b), the actual signal is completely overwhelmed by spurious artifacts.

Fig. 5
Fig. 5

Schematic diagram of the setup for EBS simulation. The tissue-like medium is a thin rectangular suspension of polystyrene beads, with volume equal to 100 × 100 × 50  μm 3 . Note the medium itself is immerged in water, not air. Microspheres of 2-μm diameter are uniformly and randomly positioned in the rectangle at 2.9% vol. concentration. A plane wave linearly polarized in the incidence plane is delivered at 15° from normal to avoid the specular reflection. Backscattering angle θ is defined as the angular deviation from the reverse of incidence. Scattered outgoing wave is absorbed by setting perfectly matched layers in the PSTD program. To suppress speckles, results are averaged over 21 different frequencies centered at f 0 = 3.82 × 10 14 H z .

Fig. 6
Fig. 6

EBS cone from a top layer of 4 optical depths described in Fig. 5. The FWHM of the enhanced cone is about 1.4°.

Fig. 7
Fig. 7

The calculated EBS cone using the same parameters in Kim’s experiment [6], averaged over 4 sets of scattering medium realizations. The origin of the slow descending slop in the tail of the curve (0.9°<θ<3°) is still unknown.

Fig. 8
Fig. 8

Schematic diagram of the setup for polarization gating modeling. The medium parameters are the same as Fig. 5, except the normal incidence and a varying z .

Fig. 9
Fig. 9

Simulation results for a series of τ: (a) the total signal I t o t ( θ ) and (b) the difference signal I d i f ( θ ) . I 0 is the intensity of the incident light. Peaks at θ 0.17 0 contain specular reflection components.

Fig. 10
Fig. 10

The difference signal collected in the 5° cone in the backward direction vs. the penetration depth τ. The saturation curve reaches a plateau at τ c = 4 .

Tables (1)

Tables Icon

Table 1 Performance comparison between SLPSTD and gPSTD on two model calculations.

Equations (10)

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

f ( x ) = 1 2 π σ [ sin x + sin ( 1.2 x ) + 1 ] exp [ x 2 / 2 σ 2 ] .
b ( ) = { sin [ π / n ovl ] , 0 < n ovl / 2 ; 1 , n ovl / 2 2 + n ovl / 2 ; sin [ ( 2 ) π / n ovl ] , 2 + n ovl / 2 < 2 + n ovl ,
f ˜ ( x + Δ / 2 ) = F 1 [ j k e j k Δ / 2 F ( f ˜ ) ] ( x ) ,
ε E x t + σ E x = ( H y z H z y ) ,
{ E x t | i + 1 / 2 , j , k n + 1 / 2 = E x | i + 1 / 2 , j , k n + 1 E x | i + 1 / 2 , j , k n Δ t E x | i + 1 / 2 , j , k n + 1 / 2 = 1 2 ( E x | i + 1 / 2 , j , k n + 1 + E x | i + 1 / 2 , j , k n ) .
( 2 c 0 Δ t sin ( ω Δ t 2 ) ) 2 = 1 ε μ ( k x 2 + k y 2 + k z 2 )
x 1 x 2 y 1 y 2 J ( x , y , z 0 ) e j ( k x x + k y y + k z z 0 ) d x d y = 1 N x N y e j k z z 0 m , n J m n ( z 0 ) e j ( k x , m + k x ) x 2 e j ( k x , m + k x ) x 1 j ( k x , m + k x ) e j ( k y , n + k y ) y 2 e j ( k y , n + k y ) y 1 j ( k y , n + k y ) .
I ( θ ) 0 2 π S ( θ , ϕ ) d ϕ , I ( θ ) 0 2 π S ( θ , ϕ ) d ϕ   .
I t o t ( θ ) I ( θ ) + I ( θ ) , I d i f ( θ ) I ( θ ) I ( θ ) .
I ( τ ) = 0.2 o 5 o I d i f τ ( θ ) sin θ d θ .

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