## Three-Dimensional Computation of Focused Beam Propagation through Multiple Biological Cells

Optics Express, Vol. 17, Issue 15, pp. 12455-12469 (2009)

http://dx.doi.org/10.1364/OE.17.012455

Acrobat PDF (1491 KB)

### Abstract

The FDTD method was used to simulate focused Gaussian beam propagation through multiple inhomogeneous biological cells. To our knowledge this is the first three dimensional computational investigation of a focused beam interacting with multiple biological cells using FDTD. A parametric study was performed whereby three simulated cells were varied by organelle density, nuclear type and arrangement of internal cellular structure and the beam focus depth was varied within the cluster of cells. Of the organelle types investigated, it appears that the cell nuclei are responsible for the greatest scattering of the focused beam in the configurations studied. Additional simulations to determine the optical scattering from 27 cells were also run and compared to the three cell case. No significant degradation of two-photon lateral imaging resolution was predicted to occur within the first 40 *µ*m of imaging depth.

© 2009 Optical Society of America

## 1. Introduction

1. W. Denk, J. Strickler, and W. Webb, “2-Photon Laser Scanning Fluorescence Microscopy,” Science **248(4951)**, 73–76 (1990). [CrossRef]

2. W. Denk and K. Svoboda, “Photon upmanship: Why multiphoton imaging is more than a gimmick,” Neuron **18(3)**, 351–357 (1997). [CrossRef]

3. 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]

8. P. Theer and W. Denk, “On the fundamental imaging-depth limit in two-photon microscopy,” J. Opt. Soc. Am. A **23(12)**, 3139–3149 (2006). [CrossRef]

*µ*m [4

4. 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]

5. 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]

7. M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods **112(2)**, 205–205 (2001). [CrossRef]

5. 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]

7. M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods **112(2)**, 205–205 (2001). [CrossRef]

8. P. Theer and W. Denk, “On the fundamental imaging-depth limit in two-photon microscopy,” J. Opt. Soc. Am. A **23(12)**, 3139–3149 (2006). [CrossRef]

3. 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]

5. 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]

8. P. Theer and W. Denk, “On the fundamental imaging-depth limit in two-photon microscopy,” J. Opt. Soc. Am. A **23(12)**, 3139–3149 (2006). [CrossRef]

13. J. Q. Lu, P. Yang, and X. H. Hu, “Simulations of light scattering from a biconcave red blood cell using the finite-difference time-domain method,” J. Biomed. Opt. **10**, 024,022 (2005). [CrossRef]

*µ*m to 1

*µ*m play a significant role in the far-field scattering from a single cell, especially at angles greater than 90 degrees. Liu and Capjack [12

12. C. Liu and C. E. Capjack, “Effects of cellular fine structure on scattered light pattern,” IEEE Trans. Nanobiosci. **5(2)**, 76–82 (2006). [CrossRef]

15. H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, X. Li, J. D. Rogers, A. Heifetz, D. Kunte, H. K. Roy, A. Taflove, and V. Backman, “Optical methodology for detecting histologically unapparent nanoscale conse-quences of genetic alterations in biological cells,” Proc. National Acad. Sci. **105(51)**, 20,118–20,123 (2008).

16. R. Gauthier, “Computation of the optical trapping force using an FDTD based technique,” Opt. Express **13(10)**, 3707–3718 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-10-3707. [CrossRef]

18. W. Sun, S. Pan, and Y. Jiang, “Computation of the optical trapping force on small particles illuminated with a focused light beam using a FDTD method,” J. Mod. Opt. **53(18)**, 2691–2700 (2006). [CrossRef]

19. K. Choi, H. Kim, Y. Lim, S. Kim, and B. Lee, “Analytic design and visualization of multiple surface plasmon resonance excitation using angular spectrum decomposition for a Gaussian input beam,” Opt. Express **13(22)**, 8866–8874 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-22-8866. [CrossRef]

## 2. Focused Beam FDTD Model

### 2.1. Scattered-Field Only Formulation

*ε*≠1. Because the source term in Eq. (1) is nonzero only when the grid cell contains a scattering medium, the incident field need not be computed for any grid cells containing the background medium. Because the incident field is computed analytically and does not propagate via the FDTD update equations, it is immune from any phase errors that may occur due to numerical dispersion often associated with FDTD [10].

_{r}21. J.-P. Berenger, “Three-Dimensional Perfectly Matched Layer for the Absorption of Electromagnetic Waves,” J. Comput. Phys. **127(2)**, 363–379 (1996). [CrossRef]

22. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. **23(4)**, 377–382 (1981). [CrossRef]

### 2.2. Focused Beam Formulation

*z*direction at a free space wavelength of 800 nm was simulated. The beam waist radius was set to be one wavelength, corresponding to a numerical aperture of 0.5. We note that the focused beam was treated as a continuous wave source. We did not model the temporal properties of the pulsed source, as described in Eq. 4.

23. I. R. Capoglu, A. Taflove, and V. Backman, “Generation of an incident focused light pulse in FDTD,” Opt. Express **16(23)**, 19,208–19,220 (2008). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19208.

25. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. **66**, 2800 (1989). [CrossRef]

25. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. **66**, 2800 (1989). [CrossRef]

25. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. **66**, 2800 (1989). [CrossRef]

*y*and

*z*directions. For a beam waist size of one wavelength (

*λ*), the average percent error over all grid locations as calculated in Ref. [25

**66**, 2800 (1989). [CrossRef]

### 2.3. Cell Configuration and Optical Properties

*n*=1.36 [26

26. A. Brunsting and P. F. Mullaney, “Differential Light Scattering from Spherical Mammalian Cells,” Biophys. J. **14(6)**, 439 (1974). [CrossRef]

*n*=1.40 [26

26. A. Brunsting and P. F. Mullaney, “Differential Light Scattering from Spherical Mammalian Cells,” Biophys. J. **14(6)**, 439 (1974). [CrossRef]

*n*=1.38 [27

27. H. Liu, B. Beauvoit, M. Kimura, and B. Chance, “Dependence of tissue optical properties on solute-induced changes in refractive index and osmolarity,” J. Biomed. Opt. **1(2)**, 200–211 (1996). [CrossRef]

*µ*m and minor diameters of 13

*µ*m. The major and minor diameters of the ellipsoidal nuclei are 6

*µ*m and 5

*µ*m respectively. The organelles are divided evenly by volume into two groups: one ellipsoidal group with major and minor diameters of 1.5

*µ*m and 0.5

*µ*m and another spherical group with diameters of 0.5

*µ*m. The location and orientation of cellular components are chosen via random number generation. The seed for the random number generator can be fixed to allow a particular geometry to be repeated, or it may be changed to allow for original configurations and to test for sensitivity of results to particular cellular configurations.

*µ*m diameter. The refractive indices of each sphere are randomly assigned and vary from

*n*-Δ

_{n}*n*to

*n*+Δ

_{n}*n*where

*n*=1.4 [26

_{n}26. A. Brunsting and P. F. Mullaney, “Differential Light Scattering from Spherical Mammalian Cells,” Biophys. J. **14(6)**, 439 (1974). [CrossRef]

*n*is the maximum variation of the refractive index of the inhomogeneities. In all cases Δ

*n*was chosen to be 0.03.

### 2.4. Simulation Parameters

28. A. Taflove and M. Brodwin, “Numerical Solution of Steady-State Electromagnetic Scattering Problems Using the Time-Dependent Maxwell’s Equations,” IEEE Trans. Microwave Theory Tech. **23(8)**, 623–630 (1975). [CrossRef]

*N*

^{4}, with

*N*as the one dimensional size of the problem grid. Once steady state is reached, the code calculates and records the steady-state optical intensity values along the

*x*=0 and

*y*=0 planar slices of the 3D space. Additionally, the 3D optical intensity within the focal volume is recorded for accurate calculation of relative two-photon signal strengths.

## 3. Simulations and Results

### 3.1. Focused Beam Analysis

*µ*m. Fig. 3(b) shows the total optical intensity for the same incident field. The scattering effect of the cells is evident in the shape of the beam focus and in the increased off-axis signal level in the focal plane. Fig. 3(c) shows the total optical intensity for the case of an incident beam with a focal depth of 22

*µ*m.

*µ*m. In Fig. 4(b), it is seen that the radial half-power beamwidth does not not significantly change with focal depth. Increased scattering with depth is seen, however, in the increased optical intensity in the lateral sidelobes of the focused beam. If this trend is extrapolated to greater imaging depths, it is apparent that there will be some maximum imaging depth beyond which this scattered off-axis light will degrade the PSF beyond recovery.

**23(12)**, 3139–3149 (2006). [CrossRef]

*I*

^{2}) over the entire imaged volume, the majority of the two-photon fluorescence will arise from within the beam focal volume. As seen in Eq. (3), the two-photon excitation can therefore be approximated as that excitation that occurs within the focal volume. In this case, the boundaries of the focal volume were defined by the FWHM, as determined in Eq. (3). As seen in the rightmost part of Eq. (4), the intensity within the focal volume was squared, integrated via rectangular quadrature in three dimensions and compared to that of the incident beam to give an indication of the degradation of the induced two-photon fluorescence signal due to scattering.

### 3.2. Parametric Study

*µ*m respectively, moving from left to right. For each focal depth, the density of the smaller organelles in each cell was varied between 0% and 10% by volume in 2% increments. The organelle types were distributed evenly by volume between the ellipsoidal scatterers representing mitochondria and the spherical scatterers that represent other organelles such as lysosomes. In addition, the nuclei of the three cells were changed to be either a homogeneous ellipsoid of refractive index of 1.40 or that of the cytoplasm background. This latter refractive index of 1.36 effectively represents the no nucleus case. Placement of the subcellular components in each cell is done via random number generation, but they are constrained to be entirely within each cuboudal cell volume. To investigate the sensitivity of the scattering results to cellular internal configuration, the parametric study was conducted on two different 3-cell geometries, each of which was generated using a different starting random number seed. The total number of simulations initially run was 144. Each three cell simulation required approximately 500 processor-hours.

### 3.3. Effects of Increasing Focal Depth

*µ*m and 1000

*µ*m, depending on a number of factors including tissue type [29

29. M. Oheim, D. J. Michael, M. Geisbauer, D. Madsen, and R. H. Chow, “Principles of two-photon excitation fluorescence microscopy and other nonlinear imaging approaches,” Adv. Drug Del. Rev. **58(7)**, 788–808 (2006). [CrossRef]

*Ae*

^{-2µsz}yields scattering coefficients of 3.41

*mm*

^{-1}, 6.56

*mm*

^{-1}, 9.99

*mm*

^{-1}, 13.31

*mm*

^{-1}and 16.39

*mm*

^{-1}for organelle volume densities of 2%, 4%, 6%, 8% and 10% respectively. The values were consistent when the placement of the organelles within the cells was varied. These scattering coefficients are also consistent with those reported for dermis or epidermis [30].

### 3.4. Effects of Volume Organelle Density

12. C. Liu and C. E. Capjack, “Effects of cellular fine structure on scattered light pattern,” IEEE Trans. Nanobiosci. **5(2)**, 76–82 (2006). [CrossRef]

### 3.5. Effects of Cell Configuration

*µ*m. This trend may be due to the placement of the nuclei with respect to the axis of the incident beam. When the simulation was re-run using different random number seeds for nuclear placement, it was found that the effect was sensitive to the location of internal cellular structure, particularly as the focal depth was increased. At a focal depth of 22

*µ*m, the fluorescence excitation signal could vary as much as 6% depending on cellular structure. At a depth of 42

*µ*m, the maximum variation increased to 23%. From Fig. 1, it may be inferred that the greater variation at this depth could be due to the nearness of that focal depth to the center of the deepest cell. In certain configurations, this could put the focal plane immediately behind the nucleus of the deepest cell, causing a sharp focusing in these cases and leading to a larger fluctuation in the fluorescence excitation signals from this depth. This lensing effect persisted in the case of cells containing heterogeneous nuclei as well as the homogeneous nuclei case.

### 3.6. Point Spread Function Size

### 3.7. Increasing Number of Cells

*N*=1750, resulting in an approximate memory requirement of 130 GB and a computational requirement of about 5300 processor-hours. The cube of cells is centered on the z-axis and the three cells along that axis are in the exact configuration as those using one of the random seeds in the parametric study. This allows a more direct comparison to be made between the three cell and 27 cell cases. A cell organelle density of 6% by volume was chosen and the beam focus was placed at depths of both 12

*µ*m and 42

*µ*m, corresponding to the shallowest and deepest of the focal planes shown in Fig. 1.

*µ*m the axial and radial beam profiles are nearly identical, suggesting that off-axis scatterers do not play a significant role in determining focal characteristics for shallow imaging depths. As may also be expected, at deeper focal depths the off-axis cells have a larger role in scattering the incident light. The three cell case resulted in radial PSF beamwidths of 107.1% and 72.6% of the incident width, respectively, for the 12

*µ*m and 42

*µ*m focal depths. The 27 cell case resulted in radial beamwidths of 107.1% and 66.0% of the incident width, showing good agreement with the three cell case and suggesting that additional scattering by off-axis cells does not cause an appreciable degradation in the size of the two-photon imaging PSF as seen in Fig. 12(b). This is consistant with reported measurements of two photon PSF trends with depth which state that radial PSF width is not noticably increased by scattering within the imaging medium within the first 100

*µ*m of imaging depth [4

4. 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]

**39(7)**, 1194–1201 (2000). [CrossRef]

*µ*m and 42

*µ*m were 71.6% and 18.0% of the incident signal, respectively. Those same signals in the 27 cell case were 71.6% and 9.8%., suggesting that off-axis scatterers contribute to the reduction of two-photon excitation signal in the focal volume. This is again consistant with reported measurements of two-photon fluorescence signal versus depth [5

**39(7)**, 1194–1201 (2000). [CrossRef]

**39(7)**, 1194–1201 (2000). [CrossRef]

## 4. Conclusion

*µ*m of imaging depth. The results also demonstrate the need for an accurate method to measure the refractive indices of subcellular components to facilitate more accurate modeling results.

## Acknowledgments

## References and links

1. | W. Denk, J. Strickler, and W. Webb, “2-Photon Laser Scanning Fluorescence Microscopy,” Science |

2. | W. Denk and K. Svoboda, “Photon upmanship: Why multiphoton imaging is more than a gimmick,” Neuron |

3. | 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. |

4. | C.-Y. Dong, K. Koenig, and P. So, “Characterizing point spread functions of two-photon fluorescence microscopy in turbid medium,” J. Biomed. Opt. |

5. | 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. |

6. | X. Gan and M. Gu, “Effective point-spread function for fast image modeling and processing in microscopic imaging through turbid media,” Opt. Lett. |

7. | M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods |

8. | P. Theer and W. Denk, “On the fundamental imaging-depth limit in two-photon microscopy,” J. Opt. Soc. Am. A |

9. | K. Yee, “Numerical solution of inital boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. |

10. | A. Taflove, |

11. | A. Dunn and R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Sel. Top. Quantum Electron. |

12. | C. Liu and C. E. Capjack, “Effects of cellular fine structure on scattered light pattern,” IEEE Trans. Nanobiosci. |

13. | J. Q. Lu, P. Yang, and X. H. Hu, “Simulations of light scattering from a biconcave red blood cell using the finite-difference time-domain method,” J. Biomed. Opt. |

14. | X. Li, A. Taflove, and V. Backman, “Recent progress in exact and reduced-order modeling of light-scattering properties of complex structures,” IEEE J. Sel. Top. Quantum Electron. |

15. | H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, X. Li, J. D. Rogers, A. Heifetz, D. Kunte, H. K. Roy, A. Taflove, and V. Backman, “Optical methodology for detecting histologically unapparent nanoscale conse-quences of genetic alterations in biological cells,” Proc. National Acad. Sci. |

16. | R. Gauthier, “Computation of the optical trapping force using an FDTD based technique,” Opt. Express |

17. | T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical Modelling of Optical Trapping,” Comp. Phys. Comm. |

18. | W. Sun, S. Pan, and Y. Jiang, “Computation of the optical trapping force on small particles illuminated with a focused light beam using a FDTD method,” J. Mod. Opt. |

19. | K. Choi, H. Kim, Y. Lim, S. Kim, and B. Lee, “Analytic design and visualization of multiple surface plasmon resonance excitation using angular spectrum decomposition for a Gaussian input beam,” Opt. Express |

20. | R.W. Ziolkowski, “FDTD modeling of Gaussian beam interactions with metallic and dielectric nano-structures,” in |

21. | J.-P. Berenger, “Three-Dimensional Perfectly Matched Layer for the Absorption of Electromagnetic Waves,” J. Comput. Phys. |

22. | G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. |

23. | I. R. Capoglu, A. Taflove, and V. Backman, “Generation of an incident focused light pulse in FDTD,” Opt. Express |

24. | W. Challener, I. Sendur, and C. Peng, “Scattered field formulation of finite difference time domain for a focused light beam in dense media with lossy materials,” Opt. Express |

25. | J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. |

26. | A. Brunsting and P. F. Mullaney, “Differential Light Scattering from Spherical Mammalian Cells,” Biophys. J. |

27. | H. Liu, B. Beauvoit, M. Kimura, and B. Chance, “Dependence of tissue optical properties on solute-induced changes in refractive index and osmolarity,” J. Biomed. Opt. |

28. | A. Taflove and M. Brodwin, “Numerical Solution of Steady-State Electromagnetic Scattering Problems Using the Time-Dependent Maxwell’s Equations,” IEEE Trans. Microwave Theory Tech. |

29. | M. Oheim, D. J. Michael, M. Geisbauer, D. Madsen, and R. H. Chow, “Principles of two-photon excitation fluorescence microscopy and other nonlinear imaging approaches,” Adv. Drug Del. Rev. |

30. | A. J. Welch and M. J. C. van Gemert, eds., |

31. | R. Drezek, A. Dunn, and R. Richards-Kortum, “Light scattering from cells: finite-difference time-domain simulations and goniometric measurements,” Appl. Opt. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

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

(290.5850) Scattering : Scattering, particles

(300.6410) Spectroscopy : Spectroscopy, multiphoton

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: March 19, 2009

Revised Manuscript: June 26, 2009

Manuscript Accepted: June 28, 2009

Published: July 8, 2009

**Virtual Issues**

Vol. 4, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Matthew S. Starosta and Andrew K. Dunn, "Three-Dimensional Computation of Focused Beam Propagation through Multiple Biological Cells," Opt. Express **17**, 12455-12469 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12455

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### References

- W. Denk, J. Strickler, and W. Webb, "2-Photon Laser Scanning Fluorescence Microscopy," Science 248(4951), 73-76 (1990). [CrossRef]
- W. Denk and K. Svoboda, "Photon upmanship: Why multiphoton imaging is more than a gimmick," Neuron 18(3), 351-357 (1997). [CrossRef]
- 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]
- 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]
- 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]
- X. 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]
- M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, "Two-photon microscopy in brain tissue: parameters influencing the imaging depth," J. Neurosci. Methods 112(2), 205 (2001). [CrossRef]
- P. Theer and W. Denk, "On the fundamental imaging-depth limit in two-photon microscopy," J. Opt. Soc. Am. A 23(12), 3139-3149 (2006). [CrossRef]
- K. Yee, "Numerical solution of inital boundary value problems involving maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. 14(3), 302-307 (1966).
- A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House Inc., Norwood MA, 2005).
- A. Dunn and R. Richards-Kortum, "Three-dimensional computation of light scattering from cells," IEEE J. Sel. Top. Quantum Electron. 2(4), 898-905 (1996).
- C. Liu and C. E. Capjack, "Effects of cellular fine structure on scattered light pattern," IEEE Trans. Nanobiosci. 5(2), 76-82 (2006). [CrossRef]
- J. Q. Lu, P. Yang, and X. H. Hu, "Simulations of light scattering from a biconcave red blood cell using the finite-difference time-domain method," J. Biomed. Opt. 10, 024,022 (2005). [CrossRef]
- X. Li, A. Taflove, and V. Backman, "Recent progress in exact and reduced-order modeling of light-scattering properties of complex structures," IEEE J. Sel. Top. Quantum Electron. 11(4), 759-765 (2005).
- H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, X. Li, J. D. Rogers, A. Heifetz, D. Kunte, H. K. Roy, A. Taflove, and V. Backman, "Optical methodology for detecting histologically unapparent nanoscale consequences of genetic alterations in biological cells," Proc. National Acad. Sci. 105(51), 20,118-20,123 (2008).
- R. Gauthier, "Computation of the optical trapping force using an FDTD based technique," Opt. Express 13(10), 3707-3718 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-10-3707. [CrossRef]
- T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, "Numerical Modelling of Optical Trapping," Comp. Phys. Comm. 142, 468-471 (2001). [CrossRef]
- W. Sun, S. Pan, and Y. Jiang, "Computation of the optical trapping force on small particles illuminated with a focused light beam using a FDTD method," J. Mod. Opt. 53(18), 2691-2700 (2006). [CrossRef]
- K. Choi, H. Kim, Y. Lim, S. Kim, and B. Lee, "Analytic design and visualization of multiple surface plasmon resonance excitation using angular spectrum decomposition for a Gaussian input beam," Opt. Express 13(22), 8866-8874 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-22-8866. [CrossRef]
- R.W. Ziolkowski, "FDTD modeling of Gaussian beam interactions with metallic and dielectric nano-structures," in Proc. 2004 URSI International Symposium on Electromagnetic Theory, pp. 27-29 (2004).
- J.-P. Berenger, "Three-Dimensional Perfectly Matched Layer for the Absorption of Electromagnetic Waves," J. Comput. Phys. 127(2), 363-379 (1996). [CrossRef]
- G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23(4), 377-382 (1981). [CrossRef]
- I. R. Capoglu, A. Taflove, and V. Backman, "Generation of an incident focused light pulse in FDTD," Opt. Express 16(23), 19,208-19,220 (2008). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19208.
- W. Challener, I. Sendur, and C. Peng, "Scattered field formulation of finite difference time domain for a focused light beam in dense media with lossy materials," Opt. Express 11(23), 3160-3170 (2003). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-23-3160 [CrossRef]
- J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800 (1989). [CrossRef]
- A. Brunsting and P. F. Mullaney, "Differential Light Scattering from Spherical Mammalian Cells," Biophys. J. 14(6), 439 (1974). [CrossRef]
- H. Liu, B. Beauvoit, M. Kimura, and B. Chance, "Dependence of tissue optical properties on solute-induced changes in refractive index and osmolarity," J. Biomed. Opt. 1(2), 200-211 (1996). [CrossRef]
- A. Taflove and M. Brodwin, "Numerical Solution of Steady-State Electromagnetic Scattering Problems Using the Time-Dependent Maxwell’s Equations," IEEE Trans. Microwave Theory Tech. 23(8), 623-630 (1975). [CrossRef]
- M. Oheim, D. J. Michael, M. Geisbauer, D. Madsen, and R. H. Chow, "Principles of two-photon excitation fluorescence microscopy and other nonlinear imaging approaches," Adv. Drug Del. Rev. 58(7), 788-808 (2006). [CrossRef]
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