## FDTD method for laser absorption in metals for large scale problems |

Optics Express, Vol. 21, Issue 21, pp. 25467-25479 (2013)

http://dx.doi.org/10.1364/OE.21.025467

Acrobat PDF (2729 KB)

### Abstract

The FDTD method has been successfully used for many electromagnetic problems, but its application to laser material processing has been limited because even a several-millimeter domain requires a prohibitively large number of grids. In this article, we present a novel FDTD method for simulating large-scale laser beam absorption problems, especially for metals, by enlarging laser wavelength while maintaining the material’s reflection characteristics. For validation purposes, the proposed method has been tested with in-house FDTD codes to simulate *p*-, *s*-, and circularly polarized 1.06 μm irradiation on Fe and Sn targets, and the simulation results are in good agreement with theoretical predictions.

© 2013 Optical Society of America

## 1. Introduction

1. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media,” IEEE Trans. Antenn. Propag. **14**(3), 302–307 (1966). [CrossRef]

2. C. M. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express **18**(20), 21427–21448 (2010). [CrossRef] [PubMed]

7. H. Ki and J. Mazumder, “Numerical simulation of femtosecond laser interaction with silicon,” J. Laser Appl. **17**(2), 110–117 (2005). [CrossRef]

8. H. Li and H. Ki, “Effect of ionization on femtosecond laser pulse interaction with silicon,” J. Appl. Phys. **100**(10), 104907 (2006). [CrossRef]

*et al*. [7

7. H. Ki and J. Mazumder, “Numerical simulation of femtosecond laser interaction with silicon,” J. Laser Appl. **17**(2), 110–117 (2005). [CrossRef]

8. H. Li and H. Ki, “Effect of ionization on femtosecond laser pulse interaction with silicon,” J. Appl. Phys. **100**(10), 104907 (2006). [CrossRef]

_{2}laser is generally irradiated on a metal plate that is at least several millimeters thick. In the case of a Nd:YAG laser, the typical wavelength is 1.06 μm, and assuming that 10 grids per wavelength is required and the plate thickness is 1 mm, the grid number in one dimension is roughly 10,000, which leads to ~10

^{12}grids in three dimensions. If a CO

_{2}laser beam is used, which has a wavelength of 10.6 μm, the total grid number can be decreased to ~10

^{9}, but even this grid number can be handled only by the most powerful supercomputing systems.

*standard FDTD*, hereafter) and an FDTD algorithm for dispersive media with the Drude model [6] (or

*dispersive FDTD*, hereafter). We have also proposed a scheme that enables the use of the standard FDTD method for dispersive media by obtaining a new set of refractive index and extinction coefficient. Numerical tests have been performed for 1.06 μm laser beam interaction with iron (Fe) and tin (Sn) targets. The obtained simulation results are in good agreement with the theoretical predictions.

## 2. Changing wavelength for standard FDTD algorithm

*R*) formulas for

*s*- and

*p*- polarized lights where subscripts

*s*and

*p*denote

*s*- and

*p*-polarizations,

*n*and

*κ*are the real and imaginary parts of the complex refractive index

*n*and

*κ*are known (

*n*and

*κ*can be expressed in terms of primitive variables as where

*n*and

*κ*if

*n*and

*κ*values fixed, there are two equations in two unknowns,

*λ*,

*ε*,

*μ*, and

*σ*values by using Eq. (6). Note that in this study we let

## 3. Changing wavelength for dispersive FDTD algorithm

*κ*is larger than refractive index

*n*, and therefore, Eqs. (4) and (5) become inappropriate and the strategy presented in Section 2 cannot be employed. (Comparing Eqs. (4) and (5), everything is exactly the same except for the sign between the two terms in parentheses, so the latter cannot be larger than the former.) In this case, the standard FDTD method cannot be used to simulate metals, and the dispersive FDTD method needs to be used [6]. In this section, we will present a scheme to use an enlarged wavelength, which can be used with the dispersive FDTD method.

*λ*(i.e., decrease

*ω*) while maintaining the same

*n*and

*κ*values. From Eqs. (7)-(10), if

*λ*as long as

*ε*and

_{1}*ε*are unchanged. In other words, from Eq. (9), when

_{2}*ω*is changed, we can select proper values of

*ε*and

_{1}*ε*values, which will in turn lead to the same

_{2}*n*and

*κ*values.

## 4. Using standard FDTD Scheme for n ≤ *κ*

*n*and

*κ*values. In this section, we will present a scheme that enables the use of the standard FDTD scheme for

*n*and

*κ*. Here, for the sake of simplicity, the incident angle will be fixed at

*p*- and

*s*-polarizations become identical. Figure 1 shows the reflectance value contour lines plotted using Eqs. (1) and (2). In this figure, for a reflectance value, there exist infinitely many sets of

*n*and

*κ*can be used to maintain the exact same reflectance according to Eqs. (1) and (2). In most problems, however, this is not the case and the incident angle cannot be assumed constant because the structure geometry could be very complicated: the incident angle can assume any values between 0° and 90°, and thus, different sets of

*n*

**=**3.81 and

*κ*= 4.44 [9]. In this case, apparently

*n*and

*κ*, which reproduces the reflection characteristic of Fe at 1.06 μm over the entire range of incident angle. Figure 2 shows angle-dependent reflectance values of iron (Fe) under 1.06 μm irradiation. Here, the red curves represent the actual reflectance patterns of Fe constructed for

*s*- and

*p*-polarizations by using Eqs. (1)-(2) and

*n*

**=**3.81 and

*κ*= 4.44. In this case, the reflectance value at normal incidence is found to be 0.644.

*n*and

*κ*, several of which are listed in Table 1. Using these new

*n*and

*κ*values, we can re-evaluate Eqs. (1) and (2) as a function of incident angle

*s*-polarized lights, regardless of

*n*and

*κ*values, the calculated angle-dependent reflectance curves are very close to the actual reflectance curve of Fe at 1.06 μm irradiation (red curve) over the entire incident angle range from 0° to 90°. Secondly, for

*p*-polarized lights, as the

*n*value increases from the smallest to the largest, the reflectance curve approaches the actual reflectance curve from above and passes it eventually. Therefore, in between, there exist some

*n*and

*κ*values that very closely approximate the actual reflectance pattern of the material. Here, our strategy is to find a set of

*n*and

*κ*, which gives the most accurate approximation to the actual and also satisfies

*n*

**=**4.62 and

*κ*= 4.51, the approximated reflectance curve is reasonably close to the actual especially if the incident angle is away from the minimum reflectance point.

*s*- and

*p*-polarizations. The original

*n*and

*κ*values of Fe are shown at the intersection of two curves as a green circle. In finding a new set of

*n*and

*κ*values that satisfies

*n*and

*κ*values need to be used. If a light polarization is well defined and fixed in a given problem, it is not a problem at all. However, in most problems, light polarization is arbitrary and/or changes from one place to another, so that having to select different

*n*and

*κ*values for different polarizations is impractical. Furthermore, as shown in Fig. 4, the calculated reflectance curves, especially for

*p*-polarization, is much worse except when

*p*- or

*s*-polarizations calculated by Eqs. (1) and (2), and

*n*and

*κ*values. Figure 5 presents the relative errors for

*s*- and

*p*-polarizations as a function of the newly chosen

*n*value. Here, we considered two

*n*the error is small and it increases as it moves away from this point. The minimum errors occur at around

*s*-polarized lights the error is very small for the almost entire range of

*n*values while for

*p*-polarization errors are much larger but still reasonably small if

*n*is chosen near the original

*n*value. For

*n*

**=**4.62 and

*κ*= 4.51, the relative errors for

*s*- and

*p*-polarizations are 0.04% and 3.4%, respectively. For

*p*-polarization, the relative errors will be much smaller if the incident angle is not very large. Note that, although the scheme has been explained for a Fe target under 1.06 μm irradiation it can be used for other metal/wavelength combinations.

## 5. Results and discussion

*n*= 4.7 and

*κ*= 1.6 [9]) and Fe targets under 1.06 μm irradiation (

*n*= 3.81 and

*κ*= 4.44). For both materials, four types of simulations were performed: (a) standard FDTD simulation with the given wavelength, (b) dispersive FDTD simulation with the given wavelength, (c) standard FDTD simulation with an increased wavelength, and (d) dispersive FDTD simulation with an increased wavelength. In the case of the Sn target,

*n*is already larger than

*κ*, so a new set of

*n*and

*κ*does not have to be obtained for standard FDTD simulations ((a) and (c)). On the other hand, for the Fe target, a new set of

*n*and

*κ*needs to be used because

*z*-direction, and the surface tilting angle

*θ*is increased from 0° to 70°. For all cases,

*p*-,

*s*-, and circularly polarized Gaussian beams were considered.

*n*and

*κ*) with λ = 1.06 μm, (b) dispersive FDTD simulation with λ = 1.06 μm, (c) standard FDTD simulation (new

*n*and

*κ*) with an increased wavelength of 21.2 μm, and (d) dispersive FDTD simulation with an increased wavelength of 21.2 μm. For each case, three different polarizations were considered:

*p*-,

*s*-, and circular polarizations. Apparently, if the electric field of the beam is aligned in the

*y*-direction, even though the cylinder surface is curved, the laser beam is 100%

*s*-polarized. Also, if the electric field is in the

*x*-direction, the laser beam is entirely

*p*-polarized. All simulation parameters are listed in Table 4 and Table 5.

*s*- and

*p*- polarizations as follows:where

*n*and

*κ*values were used for the standard FDTD simulations of Fe. For dispersive and standard FDTD simulations, the maximum errors are 3.27% and 4.59%, respectively, both of which occurred for

*p*-polarization. Also, from the results, we can see that the reflectance for circular polarization is exactly the average of the

*p*- and

*s*- polarizations.

## 6. Conclusions

## Acknowledgments

## References and links

1. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media,” IEEE Trans. Antenn. Propag. |

2. | C. M. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express |

3. | K. Kitamura, K. Sakai, and S. Noda, “Finite-difference time-domain (FDTD) analysis on the interaction between a metal block and a radially polarized focused beam,” Opt. Express |

4. | S. Buil, J. Laverdant, B. Berini, P. Maso, J. P. Hermier, and X. Quélin, “FDTD simulations of localization and enhancements on fractal plasmonics nanostructures,” Opt. Express |

5. | C. Lundgren, R. Lopez, J. Redwing, and K. Melde, “FDTD modeling of solar energy absorption in silicon branched nanowires,” Opt. Express |

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

7. | H. Ki and J. Mazumder, “Numerical simulation of femtosecond laser interaction with silicon,” J. Laser Appl. |

8. | H. Li and H. Ki, “Effect of ionization on femtosecond laser pulse interaction with silicon,” J. Appl. Phys. |

9. | W. M. Steen and J. Mazumder, |

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

**OCIS Codes**

(260.3910) Physical optics : Metal optics

(350.3390) Other areas of optics : Laser materials processing

(050.1755) Diffraction and gratings : Computational electromagnetic methods

**ToC Category:**

Physical Optics

**History**

Original Manuscript: August 26, 2013

Revised Manuscript: October 8, 2013

Manuscript Accepted: October 8, 2013

Published: October 17, 2013

**Citation**

Chun Deng and Hyungson Ki, "FDTD method for laser absorption in metals for large scale problems," Opt. Express **21**, 25467-25479 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-25467

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### References

- K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media,” IEEE Trans. Antenn. Propag.14(3), 302–307 (1966). [CrossRef]
- C. M. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express18(20), 21427–21448 (2010). [CrossRef] [PubMed]
- K. Kitamura, K. Sakai, and S. Noda, “Finite-difference time-domain (FDTD) analysis on the interaction between a metal block and a radially polarized focused beam,” Opt. Express19(15), 13750–13756 (2011). [CrossRef] [PubMed]
- S. Buil, J. Laverdant, B. Berini, P. Maso, J. P. Hermier, and X. Quélin, “FDTD simulations of localization and enhancements on fractal plasmonics nanostructures,” Opt. Express20(11), 11968–11975 (2012). [CrossRef] [PubMed]
- C. Lundgren, R. Lopez, J. Redwing, and K. Melde, “FDTD modeling of solar energy absorption in silicon branched nanowires,” Opt. Express21(S3), A392–A400 (2013). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
- H. Ki and J. Mazumder, “Numerical simulation of femtosecond laser interaction with silicon,” J. Laser Appl.17(2), 110–117 (2005). [CrossRef]
- H. Li and H. Ki, “Effect of ionization on femtosecond laser pulse interaction with silicon,” J. Appl. Phys.100(10), 104907 (2006). [CrossRef]
- W. M. Steen and J. Mazumder, Laser Material Processing, 4th ed. (Springer-Verlag, 2010).
- M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

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