## Stable algorithm for the computation of the electromagnetic field distribution of eigenmodes of periodic diffraction structures |

JOSA A, Vol. 29, Issue 11, pp. 2307-2313 (2012)

http://dx.doi.org/10.1364/JOSAA.29.002307

Acrobat PDF (363 KB)

### Abstract

In the present work, a stable algorithm for the calculation of the electromagnetic field distributions of the eigenmodes of one-dimensional diffraction gratings is presented. The proposed approach is based on the method for the computation of the propagation constants of Bloch waves of such structures previously presented by Cao *et al.* [J. Opt. Soc. Am. A 19, 335 (2002)] and uses a modified

© 2012 Optical Society of America

## 1. INTRODUCTION

1. M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A **12**, 1068–1076 (1995). [CrossRef]

3. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A **13**, 1024–1035 (1996). [CrossRef]

4. P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. **25**, 1092–1094 (2000). [CrossRef]

7. M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Modified **28**, 1364–1371 (2011). [CrossRef]

4. P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. **25**, 1092–1094 (2000). [CrossRef]

8. Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A **19**, 335–338 (2002). [CrossRef]

9. G. Lecamp, J. P. Hugonin, and P. Lalanne, “Theoretical and computational concepts for periodic optical waveguides,” Opt. Express **15**, 11042–11060 (2007). [CrossRef]

10. T. Vallius, J. Tervo, P. Vahimaa, and J. Turunen, “Electromagnetic field computation in semiconductor laser resonators,” J. Opt. Soc. Am. A **23**, 906–911 (2006). [CrossRef]

8. Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A **19**, 335–338 (2002). [CrossRef]

11. E. L. Tan, “Hybrid-matrix algorithm for rigorous coupled-wave analysis of multilayered diffraction gratings,” J. Mod. Opt. **53**, 417–428 (2006). [CrossRef]

12. J. Ning and E. L. Tan, “Generalized eigenproblem of hybrid matrix for Bloch–Floquet waves in one-dimensional photonic crystals,” J. Opt. Soc. Am. B **26**, 676–683 (2009). [CrossRef]

## 2. RIGOROUS COUPLED-WAVE ANALYSIS AND MODE COMPUTATION

### A. Geometry of the Structure and Field Representation

4. P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. **25**, 1092–1094 (2000). [CrossRef]

5. E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A **18**, 2865–2875 (2001). [CrossRef]

8. Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A **19**, 335–338 (2002). [CrossRef]

13. Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-Fa Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. **43**, 1460–1463 (1995). [CrossRef]

5. E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A **18**, 2865–2875 (2001). [CrossRef]

5. E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A **18**, 2865–2875 (2001). [CrossRef]

13. Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-Fa Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. **43**, 1460–1463 (1995). [CrossRef]

1. M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A **12**, 1068–1076 (1995). [CrossRef]

**18**, 2865–2875 (2001). [CrossRef]

### B. Scattering Matrix Formalism

1. M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A **12**, 1068–1076 (1995). [CrossRef]

3. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A **13**, 1024–1035 (1996). [CrossRef]

## 3. COMPUTATION OF THE PROPAGATION CONSTANTS AND THE FIELD DISTRIBUTIONS OF THE EIGENMODES

### A. Computation of the Propagation Constants

**19**, 335–338 (2002). [CrossRef]

### B. Computation of the Field Distribution and Numerical Instability

**18**, 2865–2875 (2001). [CrossRef]

**18**, 2865–2875 (2001). [CrossRef]

### C. Modification of the Algorithm for the Stable Field Computation

14. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A **12**, 1077–1086 (1995). [CrossRef]

15. E. L. Tan, “Note on formulation of the enhanced scattering- (transmittance-) matrix approach,” J. Opt. Soc. Am. A **19**, 1157–1161 (2002). [CrossRef]

15. E. L. Tan, “Note on formulation of the enhanced scattering- (transmittance-) matrix approach,” J. Opt. Soc. Am. A **19**, 1157–1161 (2002). [CrossRef]

## 4. NUMERICAL EXAMPLES

**25**, 1092–1094 (2000). [CrossRef]

**19**, 335–338 (2002). [CrossRef]

16. K. C. Chang, V. Shah, and T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. **70**, 804–813 (1980). [CrossRef]

**19**, 335–338 (2002). [CrossRef]

**19**, 335–338 (2002). [CrossRef]

18. P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. **45**, 1357–1374 (1998). [CrossRef]

17. E. A. Bezus, D. A. Bykov, L. L. Doskolovich, and I. I. Kadomin, “Diffraction gratings for generating varying-period interference patterns of surface plasmons,” J. Opt. A Pure Appl. Opt. **10**, 095204 (2008). [CrossRef]

19. E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings,” Microelectron. Eng. **88**, 170–174 (2011). [CrossRef]

## 5. CONCLUSION

**19**, 335–338 (2002). [CrossRef]

## ACKNOWLEDGMENTS

## REFERENCES

1. | M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A |

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

3. | L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A |

4. | P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. |

5. | E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A |

6. | M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Aperiodic Fourier modal method in contrast-field from finite structures,” J. Opt. Soc. Am. A |

7. | M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Modified |

8. | Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A |

9. | G. Lecamp, J. P. Hugonin, and P. Lalanne, “Theoretical and computational concepts for periodic optical waveguides,” Opt. Express |

10. | T. Vallius, J. Tervo, P. Vahimaa, and J. Turunen, “Electromagnetic field computation in semiconductor laser resonators,” J. Opt. Soc. Am. A |

11. | E. L. Tan, “Hybrid-matrix algorithm for rigorous coupled-wave analysis of multilayered diffraction gratings,” J. Mod. Opt. |

12. | J. Ning and E. L. Tan, “Generalized eigenproblem of hybrid matrix for Bloch–Floquet waves in one-dimensional photonic crystals,” J. Opt. Soc. Am. B |

13. | Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-Fa Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. |

14. | M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A |

15. | E. L. Tan, “Note on formulation of the enhanced scattering- (transmittance-) matrix approach,” J. Opt. Soc. Am. A |

16. | K. C. Chang, V. Shah, and T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. |

17. | E. A. Bezus, D. A. Bykov, L. L. Doskolovich, and I. I. Kadomin, “Diffraction gratings for generating varying-period interference patterns of surface plasmons,” J. Opt. A Pure Appl. Opt. |

18. | P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. |

19. | E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings,” Microelectron. Eng. |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(050.1970) Diffraction and gratings : Diffractive optics

(130.2790) Integrated optics : Guided waves

(230.7390) Optical devices : Waveguides, planar

(050.1755) Diffraction and gratings : Computational electromagnetic methods

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: July 3, 2012

Revised Manuscript: September 13, 2012

Manuscript Accepted: September 17, 2012

Published: October 11, 2012

**Citation**

Evgeni A. Bezus and Leonid L. Doskolovich, "Stable algorithm for the computation of the electromagnetic field distribution of eigenmodes of periodic diffraction structures," J. Opt. Soc. Am. A **29**, 2307-2313 (2012)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-29-11-2307

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### References

- M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995). [CrossRef]
- L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996). [CrossRef]
- L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996). [CrossRef]
- P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. 25, 1092–1094 (2000). [CrossRef]
- E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001). [CrossRef]
- M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Aperiodic Fourier modal method in contrast-field from finite structures,” J. Opt. Soc. Am. A 27, 2423–2431 (2010). [CrossRef]
- M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Modified S-matrix algorithm for the aperiodic Fourier modal method in contrast-field formulation,” J. Opt. Soc. Am. A 28, 1364–1371 (2011). [CrossRef]
- Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002). [CrossRef]
- G. Lecamp, J. P. Hugonin, and P. Lalanne, “Theoretical and computational concepts for periodic optical waveguides,” Opt. Express 15, 11042–11060 (2007). [CrossRef]
- T. Vallius, J. Tervo, P. Vahimaa, and J. Turunen, “Electromagnetic field computation in semiconductor laser resonators,” J. Opt. Soc. Am. A 23, 906–911 (2006). [CrossRef]
- E. L. Tan, “Hybrid-matrix algorithm for rigorous coupled-wave analysis of multilayered diffraction gratings,” J. Mod. Opt. 53, 417–428 (2006). [CrossRef]
- J. Ning and E. L. Tan, “Generalized eigenproblem of hybrid matrix for Bloch–Floquet waves in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 26, 676–683 (2009). [CrossRef]
- Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-Fa Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995). [CrossRef]
- M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995). [CrossRef]
- E. L. Tan, “Note on formulation of the enhanced scattering- (transmittance-) matrix approach,” J. Opt. Soc. Am. A 19, 1157–1161 (2002). [CrossRef]
- K. C. Chang, V. Shah, and T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–813 (1980). [CrossRef]
- E. A. Bezus, D. A. Bykov, L. L. Doskolovich, and I. I. Kadomin, “Diffraction gratings for generating varying-period interference patterns of surface plasmons,” J. Opt. A Pure Appl. Opt. 10, 095204 (2008). [CrossRef]
- P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45, 1357–1374 (1998). [CrossRef]
- E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings,” Microelectron. Eng. 88, 170–174 (2011). [CrossRef]

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