## Efficient curvilinear coordinate method for grating diffraction simulation |

Optics Express, Vol. 21, Issue 21, pp. 25236-25247 (2013)

http://dx.doi.org/10.1364/OE.21.025236

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### Abstract

The article presents a new method for rigorous simulation of the light diffraction on one-dimensional gratings. The method is capable to solve metal-dielectric structures in linear time and consumed memory with respect to structure complexity. Exceptional performance and convergence for metal gratings are achieved by implementing a curvilinear coordinate transformation into the generalized source method previously developed for dielectric gratings.

© 2013 OSA

## 1. Introduction

1. A. V. Tishchenko, “Generalized source method: new possibilities for waveguide and grating problems,” Opt. Quantum Electron. **32**(6/8), 971–980 (2000). [CrossRef]

2. A. A. Shcherbakov and A. V. Tishchenko, “Fast numerical method for modeling one-dimensional diffraction gratings,” Quantum Electron. **40**(6), 538–544 (2010). [CrossRef]

3. A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2d periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transf. **113**(2), 158–171 (2012). [CrossRef]

2. A. A. Shcherbakov and A. V. Tishchenko, “Fast numerical method for modeling one-dimensional diffraction gratings,” Quantum Electron. **40**(6), 538–544 (2010). [CrossRef]

3. A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2d periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transf. **113**(2), 158–171 (2012). [CrossRef]

## 2. Diffraction problem

## 3. Curvilinear coordinates

2. A. A. Shcherbakov and A. V. Tishchenko, “Fast numerical method for modeling one-dimensional diffraction gratings,” Quantum Electron. **40**(6), 538–544 (2010). [CrossRef]

3. A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2d periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transf. **113**(2), 158–171 (2012). [CrossRef]

*g*denotes determinant

## 4. GSM implementation in curvilinear coordinates

**J**(

**z**) and magnetic

**M**(

**z**) sources:Note that these equations do not describe any real electromagnetic field as they do not follow from the Maxwell’s equations in the Cartesian coordinates. However, these equations are mathematically equivalent to the Maxwell’s equations in the Cartesian coordinates. This means that Eq. (3) can be formally obtained by replacing Cartesian coordinates and vector components in (1) by curvilinear ones leaving the differential operators unchanged. Thus, solutions to Eq. (3) are readily known. Presence of the magnetic current in Eq. (3) is new with respect to the former GSM implementations [2

**40**(6), 538–544 (2010). [CrossRef]

**113**(2), 158–171 (2012). [CrossRef]

**M**is very similar to that of electric sources

**J**, and the fields emitted by magnetic currents are well known [8

8. J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. **56**(1), 99–107 (1939). [CrossRef]

8. J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. **56**(1), 99–107 (1939). [CrossRef]

9. A. V. Tishchenko, “A generalized source method for wave propagation,” Pure Appl. Opt. **7**(6), 1425–1449 (1998). [CrossRef]

**40**(6), 538–544 (2010). [CrossRef]

**113**(2), 158–171 (2012). [CrossRef]

9. A. V. Tishchenko, “A generalized source method for wave propagation,” Pure Appl. Opt. **7**(6), 1425–1449 (1998). [CrossRef]

*f*denotes a decomposed function,

**40**(6), 538–544 (2010). [CrossRef]

**113**(2), 158–171 (2012). [CrossRef]

## 5. Numerical method

10. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A **13**(5), 1019–1023 (1996). [CrossRef]

12. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A **13**(9), 1870–1876 (1996). [CrossRef]

**40**(6), 538–544 (2010). [CrossRef]

**113**(2), 158–171 (2012). [CrossRef]

**113**(2), 158–171 (2012). [CrossRef]

*n*) in each slice (index

*p*):Four matrices R, P, V, and Q are defined by (17), (16), (8), and (14), respectively. The size of vectors

**40**(6), 538–544 (2010). [CrossRef]

**113**(2), 158–171 (2012). [CrossRef]

**113**(2), 158–171 (2012). [CrossRef]

**40**(6), 538–544 (2010). [CrossRef]

**113**(2), 158–171 (2012). [CrossRef]

**40**(6), 538–544 (2010). [CrossRef]

**113**(2), 158–171 (2012). [CrossRef]

## 6. Numerical examples

14. A. V. Tishchenko, “Numerical demonstration of the validity of the Rayleigh hypothesis,” Opt. Express **17**(19), 17102–17117 (2009). [CrossRef] [PubMed]

5. J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical applications,” J. Optics (Paris) **11**(4), 235–241 (1980). [CrossRef]

*n*= 0.25 + 6.25

_{g}*i*placed on a substrate of index

*n*= 1.5, and covered by air (

_{s}*n*= 1) [Fig. 3(a)]. Profile parameters are

_{c}*c*= 0.1 μm,

*b*= 0.11 μm, and Λ = 1 μm. We calculate the diffraction of a plane wave of wavelength λ = 0.6328 μm incident under 10° from the substrate side. Figure 4(a) shows the convergence of the calculated diffraction efficiency, and comparison with the Rayleigh method versus increasing number of slices

*N*with fixed number of diffraction orders

_{S}*N*= 64 (this number of harmonics guarantees the accuracy of the Rayleigh method solution to be better than the single floating point precision

_{O}**40**(6), 538–544 (2010). [CrossRef]

**113**(2), 158–171 (2012). [CrossRef]

*N*= 2048. It is seen that the method quite rapidly and almost monotonically converges to better than a single floating point precision. Note also that for the both TE and TM polarizations the convergence is near the same.

_{S}*N*= 63 and

_{O}*N*= 1024. It is seen that solutions coincide up to the 5th digit.

_{S}*n*= 0.25 + 6.25

_{g}*i*) [15

15. I. Avrutsky, Y. Zhao, and V. Kochergin, “Surface-plasmon-assisted resonant tunneling of light through a periodically corrugated thin metal film,” Opt. Lett. **25**(9), 595–597 (2000). [CrossRef] [PubMed]

^{st}grating orders. A detailed description of the phenomenon can be found, for example, in [16

16. Y. Jourlin, S. Tonchev, A. V. Tishchenko, C. Pedri, C. Veillas, O. Parriaux, A. Last, and Y. Lacroute, “Spatially and polarization resolved plasmon mediated transmission through continuous metal films,” Opt. Express **17**(14), 12155–12166 (2009). [CrossRef] [PubMed]

## 7. Conclusion

## Acknowledgment

## References and links

1. | A. V. Tishchenko, “Generalized source method: new possibilities for waveguide and grating problems,” Opt. Quantum Electron. |

2. | A. A. Shcherbakov and A. V. Tishchenko, “Fast numerical method for modeling one-dimensional diffraction gratings,” Quantum Electron. |

3. | A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2d periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transf. |

4. | E. Popov, |

5. | J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical applications,” J. Optics (Paris) |

6. | K. Edee, J. P. Plumey, and G. Granet, “On the Rayleigh-Fourier method and the Chandezon method: comparative study,” Opt. Commun. |

7. | J. A. Schouten, |

8. | J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. |

9. | A. V. Tishchenko, “A generalized source method for wave propagation,” Pure Appl. Opt. |

10. | G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A |

11. | P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A |

12. | L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A |

13. | Y. Saad, |

14. | A. V. Tishchenko, “Numerical demonstration of the validity of the Rayleigh hypothesis,” Opt. Express |

15. | I. Avrutsky, Y. Zhao, and V. Kochergin, “Surface-plasmon-assisted resonant tunneling of light through a periodically corrugated thin metal film,” Opt. Lett. |

16. | Y. Jourlin, S. Tonchev, A. V. Tishchenko, C. Pedri, C. Veillas, O. Parriaux, A. Last, and Y. Lacroute, “Spatially and polarization resolved plasmon mediated transmission through continuous metal films,” Opt. Express |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(050.2770) Diffraction and gratings : Gratings

(050.1755) Diffraction and gratings : Computational electromagnetic methods

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: July 9, 2013

Revised Manuscript: August 21, 2013

Manuscript Accepted: September 7, 2013

Published: October 15, 2013

**Citation**

Alexey A. Shcherbakov and Alexandre V. Tishchenko, "Efficient curvilinear coordinate method for grating diffraction simulation," Opt. Express **21**, 25236-25247 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-25236

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### References

- A. V. Tishchenko, “Generalized source method: new possibilities for waveguide and grating problems,” Opt. Quantum Electron.32(6/8), 971–980 (2000). [CrossRef]
- A. A. Shcherbakov and A. V. Tishchenko, “Fast numerical method for modeling one-dimensional diffraction gratings,” Quantum Electron.40(6), 538–544 (2010). [CrossRef]
- A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2d periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transf.113(2), 158–171 (2012). [CrossRef]
- E. Popov, Gratings: Theory and Numerical Applications (Presses Universitaires de Provence, 2012).
- J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical applications,” J. Optics (Paris)11(4), 235–241 (1980). [CrossRef]
- K. Edee, J. P. Plumey, and G. Granet, “On the Rayleigh-Fourier method and the Chandezon method: comparative study,” Opt. Commun.286, 34–41 (2013). [CrossRef]
- J. A. Schouten, Tensor Analysis for Physicists (Dover Publications, 1954).
- J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev.56(1), 99–107 (1939). [CrossRef]
- A. V. Tishchenko, “A generalized source method for wave propagation,” Pure Appl. Opt.7(6), 1425–1449 (1998). [CrossRef]
- G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A13(5), 1019–1023 (1996). [CrossRef]
- P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A13(4), 779–784 (1996). [CrossRef]
- L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A13(9), 1870–1876 (1996). [CrossRef]
- Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, 2003).
- A. V. Tishchenko, “Numerical demonstration of the validity of the Rayleigh hypothesis,” Opt. Express17(19), 17102–17117 (2009). [CrossRef] [PubMed]
- I. Avrutsky, Y. Zhao, and V. Kochergin, “Surface-plasmon-assisted resonant tunneling of light through a periodically corrugated thin metal film,” Opt. Lett.25(9), 595–597 (2000). [CrossRef] [PubMed]
- Y. Jourlin, S. Tonchev, A. V. Tishchenko, C. Pedri, C. Veillas, O. Parriaux, A. Last, and Y. Lacroute, “Spatially and polarization resolved plasmon mediated transmission through continuous metal films,” Opt. Express17(14), 12155–12166 (2009). [CrossRef] [PubMed]

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