## Comparative study of total absorption of light by two-dimensional channel and hole array gratings |

Optics Express, Vol. 20, Issue 19, pp. 21702-21714 (2012)

http://dx.doi.org/10.1364/OE.20.021702

Acrobat PDF (1586 KB)

### Abstract

A detailed study of light absorption by silver gratings having two-dimensional periodicity is presented for structures constructed either of channels or of holes with subwavelength dimensions. Rigorous numerical modelling shows a systematic difference between the two structures: hole (cavity) gratings can strongly absorb light provided the cavity is sufficiently deep, when compared to the wavelength, whereas very thin channel gratings can induce total absorption. A detailed analysis is given in the limit when the period tends towards zero, and an explanation of the differences in behavior is presented using the properties of effective optical index of the metamaterial layer that substitutes the periodical structure in the limit when the period tend to zero.

© 2012 OSA

## 1. Introduction

2. U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),” J. Opt. Soc. Am. **31**(3), 213–222 (1941). [CrossRef]

3. M. C. Hutley and D. Maystre, “Total absorption of light by a diffraction grating,” Opt. Commun. **19**(3), 431–436 (1976). [CrossRef]

4. D. A. Weitz, T. J. Gramila, A. Z. Genack, and J. I. Gersten, “Anomalous low-frequency Raman scattering from rough metal surfaces and the origin of the surface-enhanced Raman scattering,” Phys. Rev. Lett. **45**(5), 355–358 (1980). [CrossRef]

5. R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,” Phys. Rev. B **28**(4), 1870–1885 (1983). [CrossRef]

6. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature **391**(6668), 667–669 (1998). [CrossRef]

*plasmonics’*of the domain.

7. J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett. **100**(6), 066408 (2008). [CrossRef] [PubMed]

9. R.-L. Chern, Y.-T. Chen, and H.-Y. Lin, “Anomalous optical absorption in metallic gratings with subwavelength slits,” Opt. Express **18**(19), 19510–19521 (2010). [CrossRef] [PubMed]

7. J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett. **100**(6), 066408 (2008). [CrossRef] [PubMed]

8. E. Popov, S. Enoch, and N. Bonod, “Absorption of light by extremely shallow metallic gratings: metamaterial behavior,” Opt. Express **17**(8), 6770–6781 (2009). [CrossRef] [PubMed]

## 2. Comparative study of channel and hole array gratings

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

12. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A **14**(10), 2758–2767 (1997). [CrossRef]

7. J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett. **100**(6), 066408 (2008). [CrossRef] [PubMed]

8. E. Popov, S. Enoch, and N. Bonod, “Absorption of light by extremely shallow metallic gratings: metamaterial behavior,” Opt. Express **17**(8), 6770–6781 (2009). [CrossRef] [PubMed]

**100**(6), 066408 (2008). [CrossRef] [PubMed]

8. E. Popov, S. Enoch, and N. Bonod, “Absorption of light by extremely shallow metallic gratings: metamaterial behavior,” Opt. Express **17**(8), 6770–6781 (2009). [CrossRef] [PubMed]

_{0}is the wavenumber in the superstrate, assumed to be air. If the eigenvalues of the transmission matrix are denoted by

_{p}are the eigenvalues of the diffraction matrix [11

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

16. 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]

_{x}and d

_{y}, the periods in x- and y-direction, respectively.

_{h}determined by the real part of the mode propagation constant:For example, when f = 0.6,

6. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature **391**(6668), 667–669 (1998). [CrossRef]

## 3. Metamaterial analysis of channel and hole arrays

17. J. C. Maxwell-Garnett, “Colors in metal glasses and in. metallic films,” Philos. Trans. R. Soc. London Ser. A **203**(359-371), 385–420 (1904). [CrossRef]

### 3.1. Numerical results for short periods

### 3.2. Equivalent effective-index analysis

17. J. C. Maxwell-Garnett, “Colors in metal glasses and in. metallic films,” Philos. Trans. R. Soc. London Ser. A **203**(359-371), 385–420 (1904). [CrossRef]

18. D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE **68**(2), 248–263 (1980). [CrossRef]

_{2}embedded in a medium with permittivity ε

_{1}. The incident field

**G**. This tensor has a singular

**L**and a principal value

**P**

_{v}parts:where

18. D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE **68**(2), 248–263 (1980). [CrossRef]

17. J. C. Maxwell-Garnett, “Colors in metal glasses and in. metallic films,” Philos. Trans. R. Soc. London Ser. A **203**(359-371), 385–420 (1904). [CrossRef]

**Q**. In what follows, numerical study will distinguish the choice between Eqs. (13) and (15).

**203**(359-371), 385–420 (1904). [CrossRef]

_{eff}of the inhomogeneous medium, based on a simple hypothesis that it gives the link between the mean values of electric and displacement fields:where angular brackets stand for mean value and, in general, ε

_{eff}is a tensor. On the other hand, in the first-order approximation, inside the inclusions the electric field is equal to the total field

_{eff}takes the formIn the case of electrically isotropic media, ε

_{1}and ε

_{2}are scalars. In addition, in the case of highly symmetrical inclusions, the

**Q**tensor is diagonal (see above), Eq. (21) is simplified into:A well-known conclusion implies that a mixture of isotropic substances lead to anisotropic effective permittivity. Less obvious is the observation that the exchange of both ε

_{1}with ε

_{2}and f with (1 – f) does not keep the result for ε

_{eff}the same. In particular, considering long cylindrical inclusions, as described by Eqs. (11) and (13), the effective permittivity represent anisotropic uniaxial medium with axis along the cylinder axis. The ordinary part of the permittivity (in the x-y plane) is given by:While the extraordinary part along the z-axis represents the mean arithmetic value of the two permittivities:If, instead of long object, we consider flat ones, Eqs. (14) and (15), the resulting tensor of effective permittivity is also uniaxial, but it contains the mean arithmethic value of the two permittivities in the x-y plane:While along the z-axis we obtain the mean harmonic value:

20. G. W. Milton and K. Golden, “Representations for the Conductivity Functions of Multicomponent Composites,” Commun. Pure Appl. Math. **43**(5), 647–671 (1990). [CrossRef]

## 4. Single-mode model and effective index behavior

_{eff,xx}are given in Fig. 9 . For the channel gratings, as f increases, the real part of the effective index (and thus the real part of the normalized propagation constant of the vertical mode) increases, which explains why the distance in h of the resonances in Fig. 8(a) decreases with h (see Eq. (4)). In fact, for f < 0.7, the metamaterial behaves like a lossy dielectric with large optical index, having real part that varies from 1 to 6. The same behavior was observed for d = 250 nm in sec.1. As f approaches 0.7, both real and imaginary part grow significantly, thus a very thin optical layer (several nanometers thick) can totally absorb the incident light.

## Conclusions

**100**(6), 066408 (2008). [CrossRef] [PubMed]

9. R.-L. Chern, Y.-T. Chen, and H.-Y. Lin, “Anomalous optical absorption in metallic gratings with subwavelength slits,” Opt. Express **18**(19), 19510–19521 (2010). [CrossRef] [PubMed]

## References and links

1. | R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phylos. Mag. |

2. | U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),” J. Opt. Soc. Am. |

3. | M. C. Hutley and D. Maystre, “Total absorption of light by a diffraction grating,” Opt. Commun. |

4. | D. A. Weitz, T. J. Gramila, A. Z. Genack, and J. I. Gersten, “Anomalous low-frequency Raman scattering from rough metal surfaces and the origin of the surface-enhanced Raman scattering,” Phys. Rev. Lett. |

5. | R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,” Phys. Rev. B |

6. | T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature |

7. | J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett. |

8. | E. Popov, S. Enoch, and N. Bonod, “Absorption of light by extremely shallow metallic gratings: metamaterial behavior,” Opt. Express |

9. | R.-L. Chern, Y.-T. Chen, and H.-Y. Lin, “Anomalous optical absorption in metallic gratings with subwavelength slits,” Opt. Express |

10. | R. C. McPhedran, G. H. Derrick, and L. C. Botten, “Theory of crossed gratings,” in |

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

12. | L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A |

13. | M. Nevière and E. Popov, “Crossed gratings,” in |

14. | M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. |

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

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

17. | J. C. Maxwell-Garnett, “Colors in metal glasses and in. metallic films,” Philos. Trans. R. Soc. London Ser. A |

18. | D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE |

19. | G. W. Milton, |

20. | G. W. Milton and K. Golden, “Representations for the Conductivity Functions of Multicomponent Composites,” Commun. Pure Appl. Math. |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(240.6680) Optics at surfaces : Surface plasmons

(260.3910) Physical optics : Metal optics

(050.5745) Diffraction and gratings : Resonance domain

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: July 11, 2012

Revised Manuscript: August 21, 2012

Manuscript Accepted: August 21, 2012

Published: September 6, 2012

**Citation**

Anne-Laure Fehrembach and Evgeny Popov, "Comparative study of total absorption of light by two-dimensional channel and hole array gratings," Opt. Express **20**, 21702-21714 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-19-21702

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### References

- R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phylos. Mag.4, 396–402 (1902).
- U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),” J. Opt. Soc. Am.31(3), 213–222 (1941). [CrossRef]
- M. C. Hutley and D. Maystre, “Total absorption of light by a diffraction grating,” Opt. Commun.19(3), 431–436 (1976). [CrossRef]
- D. A. Weitz, T. J. Gramila, A. Z. Genack, and J. I. Gersten, “Anomalous low-frequency Raman scattering from rough metal surfaces and the origin of the surface-enhanced Raman scattering,” Phys. Rev. Lett.45(5), 355–358 (1980). [CrossRef]
- R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,” Phys. Rev. B28(4), 1870–1885 (1983). [CrossRef]
- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature391(6668), 667–669 (1998). [CrossRef]
- J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett.100(6), 066408 (2008). [CrossRef] [PubMed]
- E. Popov, S. Enoch, and N. Bonod, “Absorption of light by extremely shallow metallic gratings: metamaterial behavior,” Opt. Express17(8), 6770–6781 (2009). [CrossRef] [PubMed]
- R.-L. Chern, Y.-T. Chen, and H.-Y. Lin, “Anomalous optical absorption in metallic gratings with subwavelength slits,” Opt. Express18(19), 19510–19521 (2010). [CrossRef] [PubMed]
- R. C. McPhedran, G. H. Derrick, and L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980).
- L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A13, 1870–1876 (1996). [CrossRef]
- L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A14(10), 2758–2767 (1997). [CrossRef]
- M. Nevière and E. Popov, “Crossed gratings,” in Light Propagation in Periodic Media, Differential Theory and Design (Marcel Dekker, New York, 2003) Chap. 9.
- M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am.72, 1780–1787 (1986). [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]
- 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]
- J. C. Maxwell-Garnett, “Colors in metal glasses and in. metallic films,” Philos. Trans. R. Soc. London Ser. A203(359-371), 385–420 (1904). [CrossRef]
- D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE68(2), 248–263 (1980). [CrossRef]
- G. W. Milton, The Theory of Composites (Cambridge Univ. Press, 2002).
- G. W. Milton and K. Golden, “Representations for the Conductivity Functions of Multicomponent Composites,” Commun. Pure Appl. Math.43(5), 647–671 (1990). [CrossRef]

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