## Computation of tightly-focused laser beams in the FDTD method |

Optics Express, Vol. 21, Issue 1, pp. 87-101 (2013)

http://dx.doi.org/10.1364/OE.21.000087

Acrobat PDF (1693 KB)

### Abstract

We demonstrate how a tightly-focused coherent TEM* _{mn}* 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

## 1. Introduction

3. I. R. Capoglu, A. Taflove, and V. Backman, “Generation of an incident focused light pulse in FDTD,” Opt. Express **16**, 19208–19220 (2008). [CrossRef]

*i*) The range of beams that can be introduced into the FDTD grid is significantly expanded. The results in [3

3. I. R. Capoglu, A. Taflove, and V. Backman, “Generation of an incident focused light pulse in FDTD,” Opt. Express **16**, 19208–19220 (2008). [CrossRef]

*f*

_{0}= ∞ within the context of the present paper. (

*ii*) The focused beam is computed in spaces with planar layers and inhomogeneities. A microscopy example is given to demonstrate the possibility for simulating a confocal imaging scenario. (

*iii*) The error analysis performed in the present paper is much more comprehensive and general. (

*iv*) The results described here have been implemented in an open-source FDTD software (Angora), which can be freely downloaded under the GNU Public License.

*Angora*) is briefly introduced. The paper is concluded by final remarks in Section 8.

## 2. Focusing of laser beams

*m*,

*n*) (also called a Hermite-Gaussian mode, or a TEM

*mode) is a solution of the paraxial wave equation [8], which assumes that the energy in the beam propagates mainly in a single direction along parallel rays. The electric field of a TEM*

_{mn}*mode at the beam waist, which is assumed to lie in the (*

_{mn}*x*,

*y*) plane, is given by the following expression: where

**ê**is the constant transverse unit vector in the (

*x*,

*y*) plane that determines the uniform polarization,

*ψ*(

*t*) is the time waveform of the beam,

*w*

_{0}is the beam width, and

*H*(

_{n}*x*) are the

*n*

^{th}-order Hermite polynomials [9]. The first two Hermite polynomials are

*H*

_{0}(

*x*) = 1,

*H*

_{1}(

*x*) = 2

*x*. The intensity maps of the the time-independent parts of the (0, 0), (0, 1), (1, 0), and (1, 1) modes on the beam waist are shown in Fig. 1(a). In real situations, the time dependence

*ψ*(

*t*) is a randomly-fluctuating waveform; which can be assumed statistically stationary in time [8]. This random process will have a wavelength spectrum, which might consist merely of a very narrow wavelength band for a traditional laser, or span a wide range of wavelengths for a su-percontinuum laser. If the entire optical system (including the illuminated object) is linear and time invariant, all second-order coherence properties at the output (e.g., power-spectral density at a point, mutual coherence function between two points, etc.) are completely determined by the second-order coherence properties of the input waveform and the deterministic spectral response of the system [8, 10, 11]. The latter can be obtained by sending a deterministic time pulse with a finite duration and a predefined spectral content through the system. The parameters of a modulated Gaussian waveform, for example, can easily be adjusted to manipulate its spectral content; since the cutoff wavelengths of this waveform are expressible in closed form. This is a suitable approach for a deterministic numerical method such as FDTD that operates directly in time domain.

*w*

_{0}is sufficiently smaller than the pupil radius

*a*so that the beam is contained within the pupil. Following [12

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

*filling factor*as the following ratio: In the remainder of this analysis, we assume that the filling factor

*f*

_{0}is less than 0.6. Increasing

*f*

_{0}beyond this number causes the focal fields to have a more oscillatory behavior, which makes the approximation methods introduced here less accurate. A more uniform method of approximation that is valid for all values of

*f*

_{0}is the subject of future study.

## 3. FDTD implementation using the TF/SF method

**ŝ**. Let’s write the integral in Eq. (3) as a Riemann sum over a finite collection of plane waves: in which the index

*n*is used to enumerate the individual plane waves. The spherical incidence angles are

*θ*and

_{n}*ϕ*, and the incidence directions

_{n}**ŝ**

*are The weight*

_{n}*α*replaces the differential

_{n}*d*Ω in Eq. (3). The above form is not necessarily the optimal solution for the approximation of the focal fields, since the choice of

**ŝ**

*and*

_{n}*α*are independent of the image-space position

_{n}**r**′ and the time

*t*. However, this arrangement has the advantage that the beam is expressed as a sum of plane waves, each of which can be introduced into the FDTD grid using well-documented approaches such as the scattered-field (SF) or the total-field/scattered-field (TF/SF) methods [1]. We have chosen the TF/SF method for our implementation, mainly because its computational cost is proportional to the surface area of the TF/SF boundary. The cost of the SF method is usually much higher, since it is proportional to the volume of the region over which the beam is calculated.

*s*,

_{x}*s*inside the unit disk

_{y}**ŝ**

*and the weights*

_{n}*α*in Eq. (8) for the optimal approximation of Debye-Wolf diffraction integral in Eq. (3) is the subject of

_{n}*two-dimensional cubature*[17

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

19. R. Cools, “An encyclopaedia of cubature formulas,” J. Complexity **19**, 445–453 (2003). [CrossRef]

### 3.1. Equally-spaced s_{x}, s_{y}

**ŝ**

*inside the unit disk*

_{n}*s*,

_{x}*s*) points are spaced Δ

_{y}*s*and Δ

_{x}*s*apart in the

_{y}*s*and

_{x}*s*directions, respectively; with equal cubature weight

_{y}*α*= Δ

_{n}*s*Δ

_{x}*s*for each point. In addition to the simplicity of this arrangement, useful guidelines can be derived for the spacings Δ

_{y}*s*and Δ

_{x}*s*. These guidelines rely on the Whittaker-Shannon sampling theorem [20

_{y}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]

*iωt*) is assumed in the diffraction integral in Eq. (3): in which

*k*′ =

*n*

_{2}

*ω*/

*c*is the wavenumber in the object space, and

*P*(

*s*,

_{x}*s*) is equal to 1 for

_{y}*z*′, the observation coordinate

**r**′ depends only on

*x*′ and

*y*′. The integral in Eq. (10) is then in the form of a two-dimensional

*Fourier transform*from the (

*s*,

_{x}*s*) domain to the (

_{y}*x*′,

*y*′) domain. Since the center of the beam is usually the region of interest, we can proceed by taking

*z*′ = 0. The Whittaker-Shannon sampling theorem says that, if the integral in Eq. (10) is approximated by a finite sum at a Cartesian grid of (

*s*,

_{x}*s*) points as shown in Fig. 2(a), the result is an

_{y}*infinitely replicated*(or

*aliased*) version of

**E**(

**r**′) [20

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]

*s*=Δ

_{x}*s*=Δ, the period of this replication is given by If the fields on the focal plane (

_{y}*z*′ = 0) can be contained in a square region of dimensions

*W*

_{0}×

*W*

_{0}, an overlap can be avoided with

*D*>

*W*

_{0}. From vectorial diffraction theory, we know that the focal fields decay in the lateral direction at a distance scale of

*d*

_{0}=

*λ*/(

*n*

_{2}

*f*

_{0}sin

*θ*

_{ill}) around the focus [12

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

*f*

_{0}is inside our range of interest 0 <

*f*

_{0}< 0.6. Our extensive numerical experiments suggest that the beam is always well contained within 5

*d*

_{0}– 6

*d*

_{0}of the focus. In our implementation, we choose

*W*

_{0}to be 5.2

*d*

_{0}. Another length scale to be taken into account is the lateral size of the TF/SF boundary. If the TF/SF boundary is too wide, the beam may be replicated inside the boundary. If the lateral diagonal length of the TF/SF boundary is

*T*

_{0},

*D*should be larger than

*T*

_{0}to avoid this replication. In summary, the condition on the spacing Δ is In the following, we will denote this quadrature scheme by the acronym

**EQ**.

### 3.2. Gauss-Legendre quadrature

*s*,

_{x}*s*) into the radial coordinates (

_{y}*s*,

*ϕ*), where

*ϕ*is the angle between the

*s*axis and the vector (

_{x}*s*,

_{x}*s*): Note that the limits for the usual radial coordinates are modified such that

_{y}*s*ranges from −sin

*θ*

_{ill}to sin

*θ*

_{ill}, and

*ϕ*ranges from 0 to

*π*. In this way, the integral over the unit disk is transformed into an integral over the rectangular region {|

*s*| < sin

*θ*

_{ill}, 0 <

*ϕ*<

*π*} [16]. The integral over

*s*can be approximated using Gauss-Legendre quadrature [21]. The

*ϕ*integral can be approximated trivially by the midpoint rule, since the integrand becomes periodic with period

*π*once the

*s*dependence is integrated out. For periodic functions, the midpoint rule is similar to the Gauss-Legendre quadrature in terms of accuracy [21]. In the following, we will denote this two-dimensional cubature rule by the acronym

**GL**. An example

**GL**quadrature rule with 20×8=160 total points is shown in Fig. 2(b).

### 3.3. Custom cubature rules

**EQ**and

**GL**rules for the same level of accuracy; thus reducing the number of plane waves in Eq. (8). The downside of these rules is that they have limited availability, with no readily available software for computing them. One has to to resort to tabulated values for the positions and weights for the cubature points, such as R. Cools’ online encyclopedia of cubature formulas [22

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

*d*of a quadrature rule is the maximum total degree of the two-dimensional polynomials for which the cubature formula is exact. In the following, we will label the results obtained using this cubature rule by the acronym

**CC**. The positions and weights for the 127-point quadrature are shown in Fig. 2(c).

*θ*

_{ill}, and the weights by sin

^{2}

*θ*

_{ill}. In the following, we will present an error analysis for the cubature rules shown in Fig. 2.

## 4. Error analysis

*E*

_{xth}(

**r**′,

*t*) around the focus should be computed to a high degree of accuracy. For this purpose, we use previous results from Novotny & Hecht [12

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

*iωt*),

*E*

_{xth}(

*ρ*,

*ϕ*,

*z*,

*t*) is given by in which

*E*

_{xth}(

**r**′) is expressed in cylindrical coordinates (

*ρ*,

*ϕ*,

*z*). The

*ρ*and

*z*dependencies are contained entirely in the functions

*I*

_{00},

*I*

_{02},

*I*

_{11},

*I*

_{12}, and

*I*

_{14}; which are given by [12

*Principles of Nano-optics* (Cambridge University Press, 2006). [CrossRef]

*J*(·) is the n

_{n}*order Bessel function. For the error analysis, these integrals are evaluated with very high accuracy using an adaptive Gauss-Kronrod quadrature rule. A general time dependence in*

^{th}*ψ*(

*t*) is handled by multiplying Eqs. (14)–(16) by the temporal spectrum of

*ψ*(

*t*), and taking the inverse temporal Fourier transform.

*A*, such as the one shown in Fig. 3: where

**r**′ is the position variable on the surface

*A*. The normalized Euclidean-norm error

*ε*

_{2}is a measure of the root-mean-square (rms) average of the error on

*A*compared to the rms average of the theoretical field

*E*

_{xth}(

**r**′,

*t*) on the same surface. The normalized ∞-norm error

*ε*

_{inf}is a measure of the maximum error on

*A*compared to the maximum amplitude of the theoretical field

*E*

_{xth}(

**r**

*′*,

*t*) on the same surface. We assume in the examples to follow that the incident beam is

*x*-polarized [i.e.,

**ê**=

**x**̂ in Eq. (1)], and we only compare the dominant (

**x̂**) components

*Ẽ*(

_{x}**r**′,

*t*) and

*E*

_{xth}(

**r**′,

*t*) of the computed and theoretical electric fields. Our numerical experiments have shown that the comparison of the dominant components provides a very reliable estimate of the accuracy of the entire beam. Furthermore, we have found that the error calculated over any vertical plane of the beam (as long as the beam does not vanish on the plane, e.g., the

*yz*plane for the (1,0) mode) is highly representative of the total error in the beam.

*x*=Δ

*y*=Δ

*z*=Δ=6.59 nm,

*μ*m×1.713

*μ*m×3.295

*μ*m, no absorbing boundary. The grid was filled with a lossless, non-dispersive, non-magnetic dielectric material representing immersion oil (

*n*

_{2}= 1.518). The focused laser beam was assumed to be

*x*-polarized, and propagating in the +

*z*direction. The aperture half angle

*θ*

_{ill}was 68.96°, resulting in a numerical aperture of 1.4. The total-field/scattered-field (TF/SF) surface was located 5 cells away from the grid boundaries. The waveform

*ψ*(

*t*) of the paraxial beam incident on the entrance pupil of the focusing system [see Eq. (1)] was a modulated Gaussian function

*ψ*(

*t*) = sin(2

*πf*

_{0}

*t*) exp(−

*t*

^{2}/(2

*τ*

^{2})) with

*τ*=3 fs and

*f*

_{0}=5.889 × 10

^{14}Hz. This waveform has a Gaussian temporal spectrum that falls to 1% of its maximum amplitude (0.01% of its maximum power) at free-space wavelengths 400 nm and 700 nm. In order to reduce the errors caused by the inherent grid anisotropy and grid dispersion, the grid spacing was chosen to be 1/40

^{th}of the wavelength in the immersion oil at 400 nm. This is much stricter than the usual

*λ*/20-

*λ*/15 rule-of-thumb for the grid spacing. From Eqs. (14)–(21), it follows that the back focal length

*f*of the lens is merely a constant scaling factor in all resulting field values, as long as the filling factor

*f*

_{0}is kept constant. We have used the somewhat arbitrary value of 0.1 m for the back focal length. The

*x*component of the electric field is shown in Fig. 4 at several time instants for a filling factor of

*f*

_{0}= 0.4. Figs. 4(a)–4(c) are for the (0, 0), (1, 0), and (2, 0) beams, respectively.

*A*in Fig. 3 over which the error is calculated was the

*xz*plane. We recorded the

*x*component of the electric field in the FDTD grid over a rectangular grid on the

*xz*plane, with a spacing of 12 cells in the

*z*dimension and 8 cells in the

*x*dimension. This amounts to a total of 1271 recording points. The normalized errors in Eqs. (22)–(23) were then approximated as a sum over these recording points. The results for a range of filling factors [see Eq. (7)] and for

**EQ**,

**GL**, and

**CC**cubature rules are tabulated in Table 1 and Table 2. Table 1 is for the normalized Euclidean-norm error

*ε*

_{2}, while Table 2 is for the normalized ∞-norm error

*ε*

_{inf}.

**EQ**rule has the best performance for small

*f*

_{0}values, while this performance deteriorates much faster than

**GL**and

**CC**as

*f*

_{0}increases. The normalized errors are generally higher for the (1, 0) mode, with the exception of the Euclidean error

*ε*

_{2}for the

**GL**rule. The

**GL**rule is also seen to have superior performance for high

*f*

_{0}. The overall performance of the

**CC**rule is particularly noteworthy. Although it has 30%-40% less number of points than the

**EQ**and

**GL**rules, it results in comparable error for the (0, 0) mode. On the negative side, the performance of the

**CC**rule deteriorates much faster than the others as the focal fields are evaluated farther away from the focus. Controlling the lateral dimensions of the TF/SF boundary is therefore much more important for the

**CC**rule. The normalized ∞-norm error

*ε*

_{inf}resulting from the

**CC**rule is also significantly higher for the (1, 0) mode. A quick comparison of Table 1 and Table 2 shows that the two measures of error in Eqs. (22) and (23) are not drastically different from each other. One notable exception is the significantly increased error in the rightmost column in Table 2. The contribution to this error comes mainly from the corners of the measurement plane, as seen in Fig. 5(b). Although not shown here, it was observed that the error is distributed in a similar way regardless of the cubature rule employed. This reaffirms the importance of the limits of the TF/SF boundary in the approximation in Eq. (8).

^{th}of the wavelength in the immersion oil at 400 nm, which is much stricter than usual. Normally, a grid spacing of ≈

*λ*/20 would be enough for most purposes [1]. As the grid spacing is made larger, inherent FDTD errors caused by grid anisotropy and grid dispersion become more prominent. These effects are demonstrated in Table 3, in which the normalized Euclidean-norm error (Eq. (22)) is shown for grid spacings ranging from

*λ*/40 to

*λ*/10. The wavelength

*λ*is taken to be 400 nm, which is the lower −40-dB wavelength of the excitation waveform in the immersion oil. Each of the plane waves in Eq. (8) suffer from grid anisotropy and dispersion while propagating from the TF/SF surface toward the center of the grid. If no correction is applied to these plane waves, the errors increase significantly, as seen in the right column of Table 3. In our FDTD implementation (see Section 7), we have used a dispersion-correction algorithm called the matched-numerical-dispersion method [6]. The middle column in Table 3 shows that the error is drastically reduced by this dispersion correction algorithm.

*Quest*system (see Acknowledgments). The TF/SF focused beam calculations accounted for 77%, 75%, and 70% of the total simulation times for the

**EQ**,

**GL**, and

**CC**rules, respectively. The additional memory requirements for the focused beams in our FDTD simulations were not significant, thanks to the low storage requirements of TF/SF sources. Because of the low spatial step (

*λ*/40) used, the simulations took much longer than necessary (7–10 minutes). At

*λ*/20, the simulation took about 3 minutes on the same number of processors.

## 5. Inhomogeneous spaces

### 5.1. Two-layered space

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]

### 5.2. Numerical microscope image of a scatterer

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]

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]

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]

*z*= 300 nm. In Figs. 8(b)–8(c), two microscope images of the scatterers are shown under different microscope modalities. Both images are synthesized at wavelength

*λ*= 509 nm using the light scattered into the lower half space, which amounts to epi (or re-flectance) microscopy. The bright-field microscope image, shown in Fig. 8(b), is obtained by integrating the reflectance spectrum at each pixel over the excitation spectrum. The image is saturated because the grayscale limits are chosen to be the same as in Fig. 8(c), which shows the image obtained when the reflection from the planar interface is removed, leaving only the reflection from the scatterers. This is closely analogous to the procedure followed in dark-field microscopy. As a result of the weak scattering from the two blocks, the image in Fig. 8(c) is much dimmer than the total bright-field image in Fig. 8(b). The overlap of the images of the two blocks is a consequence of the diffraction limit at

*λ*= 509 nm.

## 6. Future work

**GL**and

**CC**rules for an arbitrary FDTD setting would be very useful. One could also seek alternatives to expressing the Debye-Wolf diffraction integral in Eq. (3) as a fixed sum of plane waves as in Eq. (8). Any method of computing Eq. (3) efficiently and accurately on the TF/SF boundary for an arbitrary

*ψ*(

*t*) in Eq. (1) could be a good alternative to the method described in this paper. The computational cost of such a method would still be inherently proportional to the surface area of the TF/SF boundary, rather than its volume.

## 7. FDTD implementation: *Angora*

*Angora*[28

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

*Angora*is currently available for the GNU/Linux operating system. It supports full parallelization in all three dimensions, allowing it to be run easily on high-performance computing systems.

*Angora*operates by reading a text-based con-figuration file that specifies all details of the simulation. The

*Angora*binaries and configuration files used to generate the results in this paper can be found on the Angora website [30

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/.

## 8. Summary

*Angora*), which features the method described in this paper. The binaries and configuration files used for the examples in this paper have been made available on the

*Angora*website [30

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/.

## Acknowledgments

*Quest*high-performance computing resource.

## References and links

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

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. Express |

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. |

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. |

6. | C. Guiffaut and K. Mahdjoubi, “A perfect wideband plane wave injector for FDTD method,” in “ |

7. | T. Tan and M. Potter, “FDTD discrete planewave (FDTD-DPW) formulation for a perfectly matched source in TFSF simulations,” IEEE Trans. Antennas Propag. |

8. | L. Mandel and E. Wolf, |

9. | M. Abramowitz and I. A. Stegun, eds., |

10. | J. W. Goodman, |

11. | M. Born and E. Wolf, |

12. | L. Novotny and B. Hecht, |

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. |

14. | E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

15. | M. Born and E. Wolf, |

16. | R. Cools and K. Kim, “A survey of known and new cubature formulas for the unit disk,” J. Appl. Math. Comput. |

17. | R. Cools and P. Rabinowitz, “Monomial cubature rules since Stroud: A compilation,” J. Comput. Appl. Math. |

18. | R. Cools, “Monomial cubature rules since Stroud: A compilation - part 2,” J. Comput. Appl. Math. |

19. | R. Cools, “An encyclopaedia of cubature formulas,” J. Complexity |

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. |

21. | W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, |

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. |

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. |

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. | 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 |

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 |

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/. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(180.1790) Microscopy : Confocal microscopy

(210.0210) Optical data storage : Optical data storage

(050.1755) Diffraction and gratings : Computational electromagnetic methods

**ToC Category:**

Physical Optics

**History**

Original Manuscript: October 8, 2012

Revised Manuscript: December 2, 2012

Manuscript Accepted: December 3, 2012

Published: January 2, 2013

**Virtual Issues**

Vol. 8, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

İlker R. Çapoğlu, Allen Taflove, and Vadim Backman, "Computation of tightly-focused laser beams in the FDTD method," Opt. Express **21**, 87-101 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-1-87

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