## Staggered-grid PSTD on local Fourier basis and its applications to surface tissue modeling

Optics Express, Vol. 18, Issue 9, pp. 9236-9250 (2010)

http://dx.doi.org/10.1364/OE.18.009236

Acrobat PDF (1521 KB)

### 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

## 1. Introduction

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]

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 Jr., “Simulation of enhanced backscattering of light by numerically solving Maxwell’s equations without heuristic approximations,” Opt. Express **13**(10), 3666–3672 (2005). [CrossRef] [PubMed]

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]

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]

*x*direction) [11

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]

*x*,

*y*and

*z*directions, data exchanges occur only between adjacent nodes concurrently, and local FFT are conducted simultaneously within each node without degradation of accuracy.

## 2. Methods

### 2.1 FFT with local Fourier basis

*conceived*global nature of the derivative operator. Therefore in traditional implementations of PSTD, FFT is performed on the global data set. The spectral multi-domain technique with local Fourier basis is an established method to break the one-single global computation into multiple independent concurrent local ones after a carefully designed “patching” procedure on the data [12

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

*N*grids and

*m*subdomains. Each subdomain obtains

*ℓ*denotes the local grid index. The overlapping region is

*N*number of grids into

*m*subdomains results in

*m*, and

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]

*Δ*denotes the grid size,

*F*and

*within the local subdomain*, respectively. Unlike existing PSTD implementations, in Eq. (4) we offset the coordinate of the derivative by half a grid size so that a Yee lattice can be employed directly (see below). To be more specific about the meaning of Eq. (4), the

*x*gives the first derivate at coordinate

### 2.2 PSTD on staggered-grid with local Fourier basis and overlapping domain decomposition

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]

*δ*-functions must be replaced with extended ones to avoid the Gibbs phenomenon [14

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]

**67**, 1558–1563 (2002). [CrossRef]

*Δ*be the grid spacing, and the maximum frequency of FFT is the Nyquist frequency

*ψ*, i.e.

*N*) and the scale of supercomputer (

*m*) go up.

*ε*and

*σ*the permittivity and conductivity, respectively. Same as in the conventional PSTD, the left hand side of Eq. (5) is digitized at time step

### 2.3 Elimination of numerical dispersions

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

*ε*and

*μ*the permittivity and permeability of the medium respectively. To speed up code execution, a large

*t*. However, this would increase the numerical dispersion according to Eq. (8) and cause a large numerical error. Fortunately, if we define

*ε*with

*μ*with

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

### 2.4 Reduction of numerical error in the near-to-far-field transformation

## 3. Applications to surface-tissue optics modeling

### 3.1 Verification of the SLPSTD algorithm

### 3.2 Verification on a point source: harmonic dipole emission in a simple cell model

*a single grid in 3D*, without resorting to the double-grid or point-smoothing tricks in literature [14

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]

### 3.3 Analysis on the angular profile of the intensity cone of EBS from surface tissue layer

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]

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

*λ*is wavelength of the incident light and

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]

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]

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

**31**(10), 1459–1461 (2006). [CrossRef] [PubMed]

### 3.4 Analysis on the penetration depth of polarization gating

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]

*τ*is given by Mie theory. The size of the rectangle is kept constant in the horizontal directions with

*τ*ranging from 1 to 8.

*τ*to obtain the final results.

*θ*resolved collection geometry, the co- and cross-polarized Poynting components

*ϕ*, i.e.The overall factor comes from the detector response. In literature, two more quantities are defined [4

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]

*τ*. The irregular oscillations are due to residual speckle effects. One obvious trend in Fig. 9b is the gap between adjacent curves decreases as

*τ*increases, indicating a saturation as the increased depth does not contribute to

*τ*on the right hand side of Eq. (12) indexes the curve associated with

*τ*. The 0.2° lower limit of the integral excludes the specular reflection. The

*saturation curve*, first introduced in Ref [5

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]

**13**(2), 601–611 (2005). [CrossRef] [PubMed]

^{3}and 100 × 100 × 100 μm

^{3}, respectively. The polystyrene sphere diameter, volume concentration and illumination configuration were the same as in Fig. 8. On a per time step and per node base, the time spent on data exchange and inner-node computation as well as their summation after averaged over 200 leapfrogs are listed in Table 1 . To facilitate FFT computation, the grid numbers were slightly different for gPSTD and SLPSTD, resulting in slight different grid resolutions. The advantage of SLPSTD becomes more prominent as the model size scales up. Also can be seen is that the size of the second model is 8 times that of the first one. The inner-node computation time of SLPSTD scales at a factor of 8, while the communication time scales at 4, proportional to the area of subdomain surfaces. Due to limitations to access 8 times as many CPUs, we were unable to test the scalability of SLPSTD with respect to an increased number of computation nodes. In Table 1, when the grid number increases from 512

^{3}to 1024

^{3}, the computation time of gPSTD scales disproportionally at a factor of

## 4. Summary

## Acknowledgement

## References and links

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. |

2. | F. Voit, J. Schäfer, and A. Kienle, “Light scattering by multiple spheres: comparison between Maxwell theory and radiative-transfer-theory calculations,” Opt. Lett. |

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. |

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. | Y. Liu, Y. L. Kim, X. Li, and V. Backman, “Investigation of depth selectivity of polarization gating for tissue characterization,” Opt. Express |

6. | Y. L. Kim, P. Pradhan, H. Subramanian, Y. Liu, M. H. Kim, and V. Backman, “Origin of low-coherence enhanced backscattering,” Opt. Lett. |

7. | A. Taflove, and S. C. Hagness, |

8. | Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett. |

9. | S. H. Tseng, Y. L. Kim, A. Taflove, D. Maitland, V. Backman, and J. T. Walsh Jr., “Simulation of enhanced backscattering of light by numerically solving Maxwell’s equations without heuristic approximations,” Opt. Express |

10. | G. J. P. Correa, M. Spiegelman, S. Carbotte, and J. C. Mutter, “Centered and Staggered Fourier derivatives and Hilbert transforms,” Geophysics |

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. |

12. | M. Israeli, L. Vozovoi, and A. Averbuch, “Spectral multidomain technique with local Fourier basis,” J. Sci. Comput. |

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. |

14. | T. W. Lee and S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Antennas Wirel. Propag. Lett. |

15. | Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospetral time-domain(PSTD) algorithm,” IEEE Trans. Geosci. Rem. Sens. |

16. | Y. F. Leung and C. H. Chan, “Combining the FDTD and PSTD methods,” Microw. Opt. Technol. Lett. |

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. |

18. | K. M. Koo, Y. Takiguchi, and R. R. Alfano, “Weak localization of photons: contributions from the different scattering pathlengths,” IEEE Photon. Technol. Lett. |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(260.5430) Physical optics : Polarization

(290.1350) Scattering : Backscattering

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: March 19, 2010

Revised Manuscript: April 12, 2010

Manuscript Accepted: April 13, 2010

Published: April 19, 2010

**Virtual Issues**

Vol. 5, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Ming Ding and Kun Chen, "Staggered-grid PSTD on local Fourier basis and its applications to surface tissue modeling," Opt. Express **18**, 9236-9250 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-9-9236

Sort: Year | Journal | Reset

### References

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

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.