Abstract

We demonstrate how a tightly-focused coherent TEMmn laser beam can be computed in the finite-difference time-domain (FDTD) method. The electromagnetic field around the focus is decomposed into a plane-wave spectrum, and approximated by a finite number of plane waves injected into the FDTD grid using the total-field/scattered-field (TF/SF) method. We provide an error analysis, and guidelines for the discrete approximation. We analyze the scattering of the beam from layered spaces and individual scatterers. The described method should be useful for the simulation of confocal microscopy and optical data storage. An implementation of the method can be found in our free and open source FDTD software (“Angora”).

© 2013 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 2005), 3rd ed.
  2. 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.2691–2700 (2006).
    [CrossRef]
  3. I. R. Capoglu, A. Taflove, and V. Backman, “Generation of an incident focused light pulse in FDTD,” Opt. Express16, 19208–19220 (2008).
    [CrossRef]
  4. L. Jia and E. L. Thomas, “Radiation forces on dielectric and absorbing particles studied via the finite-difference time-domain method,” J. Opt. Soc. Am. B Opt. Phys.26, 1882–1891 (2009).
    [CrossRef]
  5. J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett.95 (2009).
    [PubMed]
  6. C. Guiffaut and K. Mahdjoubi, “A perfect wideband plane wave injector for FDTD method,” in “Antennas and Propagation Society International Symposium, 2000. IEEE,”, (Salt Lake City, UT, USA, 2000), 1, 236–239.
  7. T. Tan and M. Potter, “FDTD discrete planewave (FDTD-DPW) formulation for a perfectly matched source in TFSF simulations,” IEEE Trans. Antennas Propag.58, 2641–2648 (2010).
    [CrossRef]
  8. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  9. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Govt. Print. Off., 1964).
  10. J. W. Goodman, Statistical Optics (Wiley, New York, NY, 2000).
  11. M. Born and E. Wolf, Principles of Optics : Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, Cambridge, 1999), 7th ed.
  12. L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2006).
    [CrossRef]
  13. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci.253, 358–379 (1959).
    [CrossRef]
  14. E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci.253, 349–357 (1959).
    [CrossRef]
  15. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999), 7th ed.
  16. R. Cools and K. Kim, “A survey of known and new cubature formulas for the unit disk,” J. Appl. Math. Comput.7, 477–485 (2000).
  17. R. Cools and P. Rabinowitz, “Monomial cubature rules since Stroud: A compilation,” J. Comput. Appl. Math.48, 309–326 (1993).
    [CrossRef]
  18. R. Cools, “Monomial cubature rules since Stroud: A compilation - part 2,” J. Comput. Appl. Math.112, 21–27 (1999).
    [CrossRef]
  19. R. Cools, “An encyclopaedia of cubature formulas,” J. Complexity19, 445–453 (2003).
    [CrossRef]
  20. L. E. R. Petersson and G. S. Smith, “On the use of a Gaussian beam to isolate the edge scattering from a plate of finite size,” IEEE Trans. Antennas Propag.52, 505–512 (2004).
    [CrossRef]
  21. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1986).
  22. R. Cools, “Encyclopaedia of Cubature Formulas,” (2012). http://nines.cs.kuleuven.be/ecf .
  23. L. Novotny, Private communication.
  24. I. R. Capoglu and G. S. Smith, “A total-field/scattered-field plane-wave source for the FDTD analysis of layered media,” IEEE Trans. Antennas Propag.56, 158–169 (2008).
    [CrossRef]
  25. I. R. Capoglu, A. Taflove, and V. Backman, “A frequency-domain near-field-to-far-field transform for planar layered media,” IEEE Trans. Antennas Propag.60, 1878–1885 (2012).
    [CrossRef]
  26. J. A. Roden and S. D. Gedney, “Convolution PML (CPML): an efficient FDTD implementation of the CFD-PML for arbitrary media,” Microw. Opt. Technol. Lett.27, 334–9 (2000).
    [CrossRef]
  27. I. R. Capoglu, J. D. Rogers, A. Taflove, and V. Backman, “Chapter 1 - The Microscope in a Computer: Image Synthesis from Three-Dimensional Full-Vector Solutions of Maxwell’s Equations at the Nanometer Scale,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2012), 57, 1–91.
    [CrossRef]
  28. I. R. Capoglu, “Angora: A free software package for finite-difference time-domain (FDTD) electromagnetic simulation,” (2012). http://www.angorafdtd.org .
  29. I. R. Capoglu, A. Taflove, and V. Backman, “Angora: A free software package for finite-difference time-domain electromagnetic simulation,” accepted for publication in the IEEE Antennas and Propagation Magazine.
  30. I. R. Capoglu, “Binaries and configuration files used for the manuscript “Computation of tightly-focused laser beams in the FDTD method”,” (2012). http://www.angorafdtd.org/ext/tflb/ .

2012 (1)

I. R. Capoglu, A. Taflove, and V. Backman, “A frequency-domain near-field-to-far-field transform for planar layered media,” IEEE Trans. Antennas Propag.60, 1878–1885 (2012).
[CrossRef]

2010 (1)

T. Tan and M. Potter, “FDTD discrete planewave (FDTD-DPW) formulation for a perfectly matched source in TFSF simulations,” IEEE Trans. Antennas Propag.58, 2641–2648 (2010).
[CrossRef]

2009 (2)

L. Jia and E. L. Thomas, “Radiation forces on dielectric and absorbing particles studied via the finite-difference time-domain method,” J. Opt. Soc. Am. B Opt. Phys.26, 1882–1891 (2009).
[CrossRef]

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett.95 (2009).
[PubMed]

2008 (2)

I. R. Capoglu and G. S. Smith, “A total-field/scattered-field plane-wave source for the FDTD analysis of layered media,” IEEE Trans. Antennas Propag.56, 158–169 (2008).
[CrossRef]

I. R. Capoglu, A. Taflove, and V. Backman, “Generation of an incident focused light pulse in FDTD,” Opt. Express16, 19208–19220 (2008).
[CrossRef]

2006 (1)

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.2691–2700 (2006).
[CrossRef]

2004 (1)

L. E. R. Petersson and G. S. Smith, “On the use of a Gaussian beam to isolate the edge scattering from a plate of finite size,” IEEE Trans. Antennas Propag.52, 505–512 (2004).
[CrossRef]

2003 (1)

R. Cools, “An encyclopaedia of cubature formulas,” J. Complexity19, 445–453 (2003).
[CrossRef]

2000 (2)

R. Cools and K. Kim, “A survey of known and new cubature formulas for the unit disk,” J. Appl. Math. Comput.7, 477–485 (2000).

J. A. Roden and S. D. Gedney, “Convolution PML (CPML): an efficient FDTD implementation of the CFD-PML for arbitrary media,” Microw. Opt. Technol. Lett.27, 334–9 (2000).
[CrossRef]

1999 (1)

R. Cools, “Monomial cubature rules since Stroud: A compilation - part 2,” J. Comput. Appl. Math.112, 21–27 (1999).
[CrossRef]

1993 (1)

R. Cools and P. Rabinowitz, “Monomial cubature rules since Stroud: A compilation,” J. Comput. Appl. Math.48, 309–326 (1993).
[CrossRef]

1959 (2)

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

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci.253, 349–357 (1959).
[CrossRef]

Backman, V.

I. R. Capoglu, A. Taflove, and V. Backman, “A frequency-domain near-field-to-far-field transform for planar layered media,” IEEE Trans. Antennas Propag.60, 1878–1885 (2012).
[CrossRef]

I. R. Capoglu, A. Taflove, and V. Backman, “Generation of an incident focused light pulse in FDTD,” Opt. Express16, 19208–19220 (2008).
[CrossRef]

I. R. Capoglu, A. Taflove, and V. Backman, “Angora: A free software package for finite-difference time-domain electromagnetic simulation,” accepted for publication in the IEEE Antennas and Propagation Magazine.

I. R. Capoglu, J. D. Rogers, A. Taflove, and V. Backman, “Chapter 1 - The Microscope in a Computer: Image Synthesis from Three-Dimensional Full-Vector Solutions of Maxwell’s Equations at the Nanometer Scale,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2012), 57, 1–91.
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999), 7th ed.

M. Born and E. Wolf, Principles of Optics : Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, Cambridge, 1999), 7th ed.

Capoglu, I. R.

I. R. Capoglu, A. Taflove, and V. Backman, “A frequency-domain near-field-to-far-field transform for planar layered media,” IEEE Trans. Antennas Propag.60, 1878–1885 (2012).
[CrossRef]

I. R. Capoglu, A. Taflove, and V. Backman, “Generation of an incident focused light pulse in FDTD,” Opt. Express16, 19208–19220 (2008).
[CrossRef]

I. R. Capoglu and G. S. Smith, “A total-field/scattered-field plane-wave source for the FDTD analysis of layered media,” IEEE Trans. Antennas Propag.56, 158–169 (2008).
[CrossRef]

I. R. Capoglu, A. Taflove, and V. Backman, “Angora: A free software package for finite-difference time-domain electromagnetic simulation,” accepted for publication in the IEEE Antennas and Propagation Magazine.

I. R. Capoglu, J. D. Rogers, A. Taflove, and V. Backman, “Chapter 1 - The Microscope in a Computer: Image Synthesis from Three-Dimensional Full-Vector Solutions of Maxwell’s Equations at the Nanometer Scale,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2012), 57, 1–91.
[CrossRef]

Cools, R.

R. Cools, “An encyclopaedia of cubature formulas,” J. Complexity19, 445–453 (2003).
[CrossRef]

R. Cools and K. Kim, “A survey of known and new cubature formulas for the unit disk,” J. Appl. Math. Comput.7, 477–485 (2000).

R. Cools, “Monomial cubature rules since Stroud: A compilation - part 2,” J. Comput. Appl. Math.112, 21–27 (1999).
[CrossRef]

R. Cools and P. Rabinowitz, “Monomial cubature rules since Stroud: A compilation,” J. Comput. Appl. Math.48, 309–326 (1993).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1986).

Gedney, S. D.

J. A. Roden and S. D. Gedney, “Convolution PML (CPML): an efficient FDTD implementation of the CFD-PML for arbitrary media,” Microw. Opt. Technol. Lett.27, 334–9 (2000).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, NY, 2000).

Guiffaut, C.

C. Guiffaut and K. Mahdjoubi, “A perfect wideband plane wave injector for FDTD method,” in “Antennas and Propagation Society International Symposium, 2000. IEEE,”, (Salt Lake City, UT, USA, 2000), 1, 236–239.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 2005), 3rd ed.

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2006).
[CrossRef]

Huang, Z.

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett.95 (2009).
[PubMed]

Jia, L.

L. Jia and E. L. Thomas, “Radiation forces on dielectric and absorbing particles studied via the finite-difference time-domain method,” J. Opt. Soc. Am. B Opt. Phys.26, 1882–1891 (2009).
[CrossRef]

Jiang, Y.

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.2691–2700 (2006).
[CrossRef]

Kim, K.

R. Cools and K. Kim, “A survey of known and new cubature formulas for the unit disk,” J. Appl. Math. Comput.7, 477–485 (2000).

Lin, J.

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett.95 (2009).
[PubMed]

Lu, F.

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett.95 (2009).
[PubMed]

Mahdjoubi, K.

C. Guiffaut and K. Mahdjoubi, “A perfect wideband plane wave injector for FDTD method,” in “Antennas and Propagation Society International Symposium, 2000. IEEE,”, (Salt Lake City, UT, USA, 2000), 1, 236–239.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

Novotny, L.

L. Novotny, Private communication.

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2006).
[CrossRef]

Pan, S.

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.2691–2700 (2006).
[CrossRef]

Petersson, L. E. R.

L. E. R. Petersson and G. S. Smith, “On the use of a Gaussian beam to isolate the edge scattering from a plate of finite size,” IEEE Trans. Antennas Propag.52, 505–512 (2004).
[CrossRef]

Potter, M.

T. Tan and M. Potter, “FDTD discrete planewave (FDTD-DPW) formulation for a perfectly matched source in TFSF simulations,” IEEE Trans. Antennas Propag.58, 2641–2648 (2010).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1986).

Rabinowitz, P.

R. Cools and P. Rabinowitz, “Monomial cubature rules since Stroud: A compilation,” J. Comput. Appl. Math.48, 309–326 (1993).
[CrossRef]

Richards, B.

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

Roden, J. A.

J. A. Roden and S. D. Gedney, “Convolution PML (CPML): an efficient FDTD implementation of the CFD-PML for arbitrary media,” Microw. Opt. Technol. Lett.27, 334–9 (2000).
[CrossRef]

Rogers, J. D.

I. R. Capoglu, J. D. Rogers, A. Taflove, and V. Backman, “Chapter 1 - The Microscope in a Computer: Image Synthesis from Three-Dimensional Full-Vector Solutions of Maxwell’s Equations at the Nanometer Scale,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2012), 57, 1–91.
[CrossRef]

Sheppard, C. J. R.

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett.95 (2009).
[PubMed]

Smith, G. S.

I. R. Capoglu and G. S. Smith, “A total-field/scattered-field plane-wave source for the FDTD analysis of layered media,” IEEE Trans. Antennas Propag.56, 158–169 (2008).
[CrossRef]

L. E. R. Petersson and G. S. Smith, “On the use of a Gaussian beam to isolate the edge scattering from a plate of finite size,” IEEE Trans. Antennas Propag.52, 505–512 (2004).
[CrossRef]

Sun, W.

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.2691–2700 (2006).
[CrossRef]

Taflove, A.

I. R. Capoglu, A. Taflove, and V. Backman, “A frequency-domain near-field-to-far-field transform for planar layered media,” IEEE Trans. Antennas Propag.60, 1878–1885 (2012).
[CrossRef]

I. R. Capoglu, A. Taflove, and V. Backman, “Generation of an incident focused light pulse in FDTD,” Opt. Express16, 19208–19220 (2008).
[CrossRef]

I. R. Capoglu, J. D. Rogers, A. Taflove, and V. Backman, “Chapter 1 - The Microscope in a Computer: Image Synthesis from Three-Dimensional Full-Vector Solutions of Maxwell’s Equations at the Nanometer Scale,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2012), 57, 1–91.
[CrossRef]

I. R. Capoglu, A. Taflove, and V. Backman, “Angora: A free software package for finite-difference time-domain electromagnetic simulation,” accepted for publication in the IEEE Antennas and Propagation Magazine.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 2005), 3rd ed.

Tan, T.

T. Tan and M. Potter, “FDTD discrete planewave (FDTD-DPW) formulation for a perfectly matched source in TFSF simulations,” IEEE Trans. Antennas Propag.58, 2641–2648 (2010).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1986).

Thomas, E. L.

L. Jia and E. L. Thomas, “Radiation forces on dielectric and absorbing particles studied via the finite-difference time-domain method,” J. Opt. Soc. Am. B Opt. Phys.26, 1882–1891 (2009).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1986).

Wang, H.

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett.95 (2009).
[PubMed]

Wolf, E.

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

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci.253, 349–357 (1959).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

M. Born and E. Wolf, Principles of Optics : Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, Cambridge, 1999), 7th ed.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999), 7th ed.

Zheng, W.

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett.95 (2009).
[PubMed]

Appl. Phys. Lett. (1)

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett.95 (2009).
[PubMed]

IEEE Trans. Antennas Propag. (4)

T. Tan and M. Potter, “FDTD discrete planewave (FDTD-DPW) formulation for a perfectly matched source in TFSF simulations,” IEEE Trans. Antennas Propag.58, 2641–2648 (2010).
[CrossRef]

L. E. R. Petersson and G. S. Smith, “On the use of a Gaussian beam to isolate the edge scattering from a plate of finite size,” IEEE Trans. Antennas Propag.52, 505–512 (2004).
[CrossRef]

I. R. Capoglu and G. S. Smith, “A total-field/scattered-field plane-wave source for the FDTD analysis of layered media,” IEEE Trans. Antennas Propag.56, 158–169 (2008).
[CrossRef]

I. R. Capoglu, A. Taflove, and V. Backman, “A frequency-domain near-field-to-far-field transform for planar layered media,” IEEE Trans. Antennas Propag.60, 1878–1885 (2012).
[CrossRef]

J. Appl. Math. Comput. (1)

R. Cools and K. Kim, “A survey of known and new cubature formulas for the unit disk,” J. Appl. Math. Comput.7, 477–485 (2000).

J. Complexity (1)

R. Cools, “An encyclopaedia of cubature formulas,” J. Complexity19, 445–453 (2003).
[CrossRef]

J. Comput. Appl. Math. (2)

R. Cools and P. Rabinowitz, “Monomial cubature rules since Stroud: A compilation,” J. Comput. Appl. Math.48, 309–326 (1993).
[CrossRef]

R. Cools, “Monomial cubature rules since Stroud: A compilation - part 2,” J. Comput. Appl. Math.112, 21–27 (1999).
[CrossRef]

J. Mod. Opt. (1)

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.2691–2700 (2006).
[CrossRef]

J. Opt. Soc. Am. B Opt. Phys. (1)

L. Jia and E. L. Thomas, “Radiation forces on dielectric and absorbing particles studied via the finite-difference time-domain method,” J. Opt. Soc. Am. B Opt. Phys.26, 1882–1891 (2009).
[CrossRef]

Microw. Opt. Technol. Lett. (1)

J. A. Roden and S. D. Gedney, “Convolution PML (CPML): an efficient FDTD implementation of the CFD-PML for arbitrary media,” Microw. Opt. Technol. Lett.27, 334–9 (2000).
[CrossRef]

Opt. Express (1)

Proc. R. Soc. Lond. A Math. Phys. Sci. (2)

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

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci.253, 349–357 (1959).
[CrossRef]

Other (15)

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999), 7th ed.

C. Guiffaut and K. Mahdjoubi, “A perfect wideband plane wave injector for FDTD method,” in “Antennas and Propagation Society International Symposium, 2000. IEEE,”, (Salt Lake City, UT, USA, 2000), 1, 236–239.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Govt. Print. Off., 1964).

J. W. Goodman, Statistical Optics (Wiley, New York, NY, 2000).

M. Born and E. Wolf, Principles of Optics : Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, Cambridge, 1999), 7th ed.

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2006).
[CrossRef]

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1986).

R. Cools, “Encyclopaedia of Cubature Formulas,” (2012). http://nines.cs.kuleuven.be/ecf .

L. Novotny, Private communication.

I. R. Capoglu, J. D. Rogers, A. Taflove, and V. Backman, “Chapter 1 - The Microscope in a Computer: Image Synthesis from Three-Dimensional Full-Vector Solutions of Maxwell’s Equations at the Nanometer Scale,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2012), 57, 1–91.
[CrossRef]

I. R. Capoglu, “Angora: A free software package for finite-difference time-domain (FDTD) electromagnetic simulation,” (2012). http://www.angorafdtd.org .

I. R. Capoglu, A. Taflove, and V. Backman, “Angora: A free software package for finite-difference time-domain electromagnetic simulation,” accepted for publication in the IEEE Antennas and Propagation Magazine.

I. R. Capoglu, “Binaries and configuration files used for the manuscript “Computation of tightly-focused laser beams in the FDTD method”,” (2012). http://www.angorafdtd.org/ext/tflb/ .

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 2005), 3rd ed.

Supplementary Material (5)

» Media 1: MOV (2203 KB)     
» Media 2: MOV (2226 KB)     
» Media 3: MOV (2246 KB)     
» Media 4: MOV (2153 KB)     
» Media 5: MOV (2186 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

(a) Intensity maps of several TEMmn modes on the beam waist. (b) The geometry of the incidence and focusing of the beam.

Fig. 2
Fig. 2

Cubature rules for the approximation in Eq. (8) for the 2D integral in Eq. (3). The top graphs show the placement of the quadrature points in the unit disk. The bottom graphs show the weights along sy=0. (a) 188 points on an equally-spaced Cartesian grid of (sx, sy) positions inside the illumination cone. (b) Separation of the 2D integral on the (sx, sy) plane into two 1D integrals over the radial coordinates (s, ϕ). The s integral is evaluated using Gauss-Legendre quadrature, and the ϕ integral is evaluated using the midpoint rule. A total of 20×8=160 quadrature points are used. (c) A custom 127-point quadrature rule for the unit disk [16], exact for polynomials s x i s y j where i + j < 25.

Fig. 3
Fig. 3

An example surface A over which the computed and theoretical beams are compared. The error ε in Eq. (22) is calculated over this area.

Fig. 4
Fig. 4

Snapshots of the electric field amplitude from FDTD simulations for focused laser beams traveling in the +z direction. The x component of the electric field on the xz plane is plotted linearly in grayscale at 4.975 fs intervals from left to right. The maximum brightness corresponds to 1.059 × 105 V/m. (a) (0, 0) mode. [ Media1] (b) (1, 0) mode. [ Media2] (c) (2, 0) mode. [ Media3]

Fig. 5
Fig. 5

The distribution of the error on the measurement (xz) plane for the FDTD parameters in the rightmost column of Table 2. (a) The ∞-norm of the theoretical incident field. (b) The ∞-norm of the error. The grayscale upper limit is 1/10th of that in (a) for accentuation. The error is seen to be concentrated at the corners of the plane.

Fig. 6
Fig. 6

Snapshots of the electric field amplitude for a focused laser beam traveling in the +z direction in a two-layered space. The x component of the electric field on the xz plane is plotted linearly in grayscale at 4.975 fs intervals from left to right. The maximum brightness corresponds to 5 × 104 V/m. [ Media4]

Fig. 7
Fig. 7

Snapshots of the electric field amplitude for a focused laser beam traveling in the +z direction in a two-layered space containing rectangular scatterers. The x component of the electric field on the xz plane is plotted linearly in grayscale at 4.975 fs intervals from left to right. The maximum brightness corresponds to 5 × 104 V/m. [ Media5]

Fig. 8
Fig. 8

Numerical microscope images of two rectangular scatterers buried inside the upper half space, under focused-beam illumination. (a) Refractive index map of the xy cross section at z = 300 nm. (b) The bright-field image of the structure, dominated by the light reflected from the interface. (c) The image with the reflection from the interface removed. This resembles the procedure followed in dark-field microscopy.

Tables (3)

Tables Icon

Table 1 Normalized Euclidean-norm error ε2 [given by Eq. (22)] over the xz plane for the approximation in Eq. (8).

Tables Icon

Table 2 Normalized ∞-norm error εinf [given by Eq. (23)] over the xz plane for the approximation in Eq. (8).

Tables Icon

Table 3 The normalized Euclidean-norm error (Eq. (22)) for different grid spacings. The middle and right columns show the error with and without dispersion correction, respectively.

Equations (23)

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

E inc ( x , y ; t ) = e ^ ψ ( t ) γ m n ( x , y ) = e ^ ψ ( t ) H m ( 2 x / w 0 ) H n ( 2 y / w 0 ) e ( x 2 + y 2 ) / w 0 2 ,
E ( r , θ , ϕ , t ) = a ^ ( θ , ϕ ) E ( θ , ϕ , t + n 2 r / c ) / r ,
E ( r , t ) = n 2 2 π c Ω ill a ^ ( θ , ϕ ) E ˙ ( θ , ϕ , t n 2 s ^ r / c ) d Ω ,
s ^ = ( s x , s y , s z ) = ( sin θ cos ϕ sin θ sin ϕ , cos θ )
r = ( x , y , z )
E ( θ , ϕ , t ) = ( n 1 / n 2 ) 1 / 2 f cos 1 / 2 ( θ ) ψ ( t t c ) γ m n ( x , y ) ,
f 0 = w 0 / a = w 0 / ( f sin θ ill )
E ˜ ( r , t ) = n 2 2 π c n α n a ^ ( θ n , ϕ n ) E ˙ ( θ n , ϕ n , t n 2 s ^ n r / c ) ,
s ^ n = ( s x n , s y n , s z n ) = ( sin θ n cos ϕ n , sin θ n sin ϕ n , cos θ n ) .
E ( r ) = i k 2 π P ( s x , s y ) a ^ ( s ^ ) E ( s ^ ) e i k s ^ r d s x d s y / cos θ ,
D = 2 π k Δ
Δ < 2 π k max { W 0 , T 0 } .
E ( r , t ) = 1 2 π c ϕ = 0 π d ϕ s = sin θ ill sin θ ill s d s 1 s 2 a ^ ( s , ϕ ) E ˙ ( s , ϕ , t n 2 s ^ r / c )
E x th ( ρ , ϕ , z ) = i k f 2 n 1 n 2 ( I 00 + I 02 cos 2 ϕ ) [ ( 0 , 0 ) mode ]
E x th ( ρ , ϕ , z ) = i k f 2 2 w 0 n 1 n 2 ( i I 11 cos ϕ + i I 14 cos 3 ϕ ) [ ( 1 , 0 ) mode ]
E x th ( ρ , ϕ , z ) = i k f 2 2 w 0 n 1 n 2 ( i ( I 11 + 2 I 12 ) sin ϕ + i I 14 sin 3 ϕ ) [ ( 0 , 1 ) mode ]
I 00 ( ρ , z ) = 0 θ ill e 1 f 0 2 sin 2 θ sin 2 θ ill cos 1 2 θ sin θ ( 1 + cos θ ) J 0 ( k ρ sin θ ) e i k z cos θ d θ
I 02 ( ρ , z ) = 0 θ ill e 1 f 0 2 sin 2 θ sin 2 θ ill cos 1 2 θ sin θ ( 1 cos θ ) J 2 ( k ρ sin θ ) e i k z cos θ d θ
I 11 ( ρ , z ) = 0 θ ill e 1 f 0 2 sin 2 θ sin 2 θ ill cos 1 2 θ sin 2 θ ( 1 + 3 cos θ ) J 1 ( k ρ sin θ ) e i k z cos θ d θ
I 12 ( ρ , z ) = 0 θ ill e 1 f 0 2 sin 2 θ sin 2 θ ill cos 1 2 θ sin 2 θ ( 1 cos θ ) J 1 ( k ρ sin θ ) e i k z cos θ d θ
I 14 ( ρ , z ) = 0 θ ill e 1 f 0 2 sin 2 θ sin 2 θ ill cos 1 2 θ sin 2 θ ( 1 cos θ ) J 3 ( k ρ sin θ ) e i k z cos θ d θ .
ε 2 = ( d t A d r | E ˜ x ( r , t ) E x th ( r , t ) | 2 ) 1 / 2 ( d t A d r | E x th ( r , t ) | 2 ) 1 / 2
ε inf = max A , t | E ˜ x ( r , t ) E x th ( r , t ) | max A , t | E x th ( r , t ) |

Metrics