## Total absorption of light by lamellar metallic gratings

Optics Express, Vol. 16, Issue 20, pp. 15431-15438 (2008)

http://dx.doi.org/10.1364/OE.16.015431

Acrobat PDF (1247 KB)

### Abstract

Lamellar gratings illuminated in conical (off-plane) mounting can achieve with suitable optogeometrical parameters (grating profile, angle of incidence and wavelength) a total absorption of light for any polarization provided there is only the zeroth propagating order. A detailed analysis shows that electromagnetic resonances are involved and their nature strongly depends on the polarization. When the incident electric field is parallel to the cross-section of the grating, the resonance is provoked by the excitation of surface plasmons. For the orthogonal polarization, total absorption occurs for deep gratings only, when the grooves behave like resonant optical cavities. It is possible to reduce the optimal grating height by filling the grooves with a high refractive index material.

© 2008 Optical Society of America

## 1. Introduction

2. A. Hessel and A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. **4**, 1275–1297 (1965). [CrossRef]

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

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

6. A. Wirgin and T. López-Rios, “Can surface-enhanced Raman scattering be caused by waveguide resonance?,” Opt. Commun. **48**, 416–420 (1984). [CrossRef]

11. S. Collin, F. Pardo, R. Teissier, and J. L. Pelouard, “Efficient light absorption in metal-semiconductor-metal nanostructures,” Appl. Phys. Lett. **85**, 194–196 (2004). [CrossRef]

12. E. Popov, D. Maystre, R. C. McPhedran, M. Nevière, M. C. Huthley, and G. H. Derrick, “Total absorption of unpolarized light by crossed gratings,” Opt. Express **16**, 6146–6155 (2008). [CrossRef] [PubMed]

13. T. V. Teperik, F. J. García De Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surface,” Nat. Photonics **2**, 299–301 (2008). [CrossRef]

13. T. V. Teperik, F. J. García De Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surface,” Nat. Photonics **2**, 299–301 (2008). [CrossRef]

14. E. Popov, L. Tsonev, and D. Maystre, “Lamellar diffraction grating anomalies,” Appl. Opt. **33**, 5214–5219 (1994). [CrossRef] [PubMed]

## 2. Notations

*z*axis. The groove height, groove width and period are denoted by

*h*,

*c*and

*d*respectively. The filling factor

*c*/

*d*and the wavelength of light λ=2π/

*k*are taken equal to 0.5 and 650 nm respectively. Figure 1 shows the notations used to describe the incidence parameters. We denote by φ the angle between the

*xy*plane and the plane of incidence. Except in the last part of the paper, we will deal with pure conical mountings,

*i.e*. φ=90°. The incident wavevector

**k**

^{i}makes an angle θ with the normal (

*y*-axis). The polarization angle δ is the angle between the electric field and the normal

**n**to the plane of incidence. It is equal to zero or 90° according to whether the electric or magnetic field is perpendicular to the plane of incidence.

## 3. Total absorption of light by deep lamellar gratings in conical mounting: numerical optimization

15. S. E. Sandstrom, G. Tayeb, and R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings - an exact eigenfunction solution,” J. Electromagn. Waves Appl. **7**, 631–649 (1993). [CrossRef]

*h*, the period of the grating

*d*and the angle of incidence θ. We restrict ourselves to the case of a single reflected propagating order, which imposes limits on the wavelength-to-period ratio as a function of the incident angle. Numerical optimization consists in searching a set of parameters for which the reflectivity is minimal. For the best result, it turns out that an absorption rate larger than 99.9% for both fundamental polarizations is reached for

*d*=644.8 nm (=0.992λ),

*h*=1360 nm, and θ=30°. The results have been checked using two independent methods for modeling of diffraction by diffraction gratings. These are the integral and the differential methods. The differential theory [17] is based on a Fourier decomposition of the electromagnetic field, which transfers Maxwell equations into a system of ordinary differential equations, which is solved by the shooting method. In the case of lamellar grating, it reduces to the Fourier modal method. The integral theory [17] describes the electromagnetic field as unknown values taken on the grating surface and uses the Green-functions method to establish the field everywhere in space, resulting in a system of coupled integral equations, which require special analysis of singularities.

## 4. Physical origin of total absorption

*yz*plane, i.e. to the groove sides, and even though polarization states are coupled, it can be conjectured that the total electric field remains nearly parallel to the same plane (it is rigorously satisfied for the 0th order, due to the symmetry with respect to the

*yz*plane). The groove can be considered as an air slab truncated at the top and filled with metal at the bottom. The cut-off of the fundamental mode is reached at a wavelength λ=2

*c*=

*d*=644.8 nm for infinitely conducting metal but the cut-off wavelength is longer for real metals [19

19. E. Popov, M. Nevière, P. Boyer, and N. Bonod, “Light transmission through single apertures,” Opt. Commun. **255**, 338–348 (2005). [CrossRef]

*x*and

*z*components of the electric field vanish at the bottom of the groove (for perfectly conducting metals) and form interference pattern between the modes propagating down- and upwards which gives the maximum of light intensity in Fig. 2(a). With a perfectly conducting grating, the distance between two successive nodes of the standing wave is given by λ/(2 cosθ). It must be stressed that in the case of real metal gratings, this distance is slightly shorter [20

20. E. Popov, N. Bonod, and S. Enoch, “Non-Bloch plasmonic stop-band in real-metal gratings,” Opt. Express **15**, 6241–6250(2007) [CrossRef] [PubMed]

*h*. Four other absorption peaks appear when the height of the grating is decreased, i.e. the number of nodes in Fig. 2(b), a fact which confirms our conjecture. Another confirmation can be found in Fig. 3(b) which presents the map of the electric intensity for

*h*=1032 nm, at the second minimum of the reflected efficiency. The nodes have been shifted upwards, the upper node has been cancelled from the groove and the number of nodes is reduced to 3 (plus the one present below the bottom of the groove).

*t*

_{1}=65 nm. The necessary groove depth to achieve this with more nodes can easily be evaluated in the following way. At first, it is necessary to take into account that the bottom of the groove does not correspond to a node. Elementary calculations lead to a formula in closed form giving the shift between the metal surface and the node when a plane wave with electric field perpendicular to the plane of incidence is reflected by a plane metallic surface. It turns out from this formula that the node is located below the surface (here it is necessary to consider the continuation of the incident and reflected waves inside the metal). For a wavelength of 650 nm and an incidence of 30° on a gold plane, the shift

*t*

_{0}reaches a value of 29 nm. Second, let us note with

*t*

_{1}to

*t*

_{5}the distances in

*y*direction between the top of the groove and the corresponding node numbered in Fig. 2(b). Using these values, the groove depth for which the upper node lies at the same distance from the top of the groove is simply given by:

*h*

*=*

_{n}*t*

*-*

_{n}*t*

_{0}, where

*n*=1,…,5 corresponds to the node number.

*h*

*) and the exact groove depth corresponding to absorption peaks in Fig. 3(a). It can be concluded from Table 1 that the heights deduced from absorption peaks and from the distances*

_{n}*h*

*are very close to each other, except for the first resonance, which corresponds to a shallow grating. This shallow grating has a groove height/period ratio of the same order as the sinusoidal gold grating described in [4*

_{n}4. M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. **19**, 431–436 (1976). [CrossRef]

21. E. Popov, N. Bonod, and S. Enoch, “Comparison of plasmon surface waves on shallow and deep metallic 1D and 2D gratings,” Opt. Express **15**, 4224–4237 (2007). [CrossRef] [PubMed]

21. E. Popov, N. Bonod, and S. Enoch, “Comparison of plasmon surface waves on shallow and deep metallic 1D and 2D gratings,” Opt. Express **15**, 4224–4237 (2007). [CrossRef] [PubMed]

## 5. Total absorption of unpolarized light by lamellar gratings with moderate heights

*yz*plane (δ=90°, φ=90°) a resonance inside the groove must be excited and since the electric field must be small at the four sides of the groove, large groove heights and widths are needed. If one wants to reduce the groove height, two solutions can be envisaged. First, one can enlarge the groove width. However, since it is necessary not to increase significantly the period in order to maintain a single possible reflected order, we are led to use

*c*/

*d*ratios greater than 0.5 and obviously this condition entails for this first solution a strong limitation. On the other hand, it is well known that it is possible to reduce the geometrical size of a resonant cavity by increasing its optical index and this remark leads to a second solution.

*n*=3.5+i 0.02), and, in normal incidence, the groove depth has been increased until a cavity resonance for δ=90° is reached. This resonance has been obtained for a groove height close to λ/5. Then, the grating geometry has been optimized to achieve a good absorption for both polarizations. For the set of parameters

*d*=542 nm,

*c*=287 nm,

*h*=120 nm, the residual reflection in TE polarization (electric field parallel to the grooves) remains less than 5% and in TM polarization (magnetic field parallel to the grooves) the reflection is about 1.2%. These values can further be decreased if we use silver (refractive index n=0.07+i4.2) instead of gold for the grating materials. Grating parameters that give maximum absorption are quite similar,

*d*=542 nm,

*c*=287 nm,

*h*=120 nm and they lead to an absorption of 98.58% averaged on both fundamental polarizations in normal incidence.

^{st}order represented by a black line. When the electric field is parallel to the grooves (Fig. 4(a)), the absorption poorly depends on the grating period, which confirms that the resonance is linked to the groove dimensions. On the contrary, for the other polarization the resonance position strongly depends on both parameters

*d*and θ. The plasmon surface wave that propagates along a flat silver surface has a normalized propagation constant α

*,*

_{p}_{0}(=k

_{x}/k

^{i}) approximately equal to 1.032, and the position of its optimal excitation through order −1 is given by the equation θ=asin(λ/

*d*-α

_{p},

_{0}), presented by a solid violet curve in Fig. 4(b). When the grating depth is varied, the plasmon propagation constant changes. In order to determine it, the pole of the scattering matrix S has been calculated. Let us remind that S is defined as: S(α) I=D, where I and D represent column vectors made of the Fourier components of the x components of the incident (I) and diffracted (D) electric and magnetic fields. The so-called pole is the complex solution of the homogeneous problem [22]:

_{p}. It has been calculated has a function of the period

*d*for

_{h}=120 nm and

_{c}=287 nm, and the its values almost coincide with those determined for a flat surface, the curve θ=asin(λ/

*d*-

*α*

_{p}) has been superposed in Fig. 4(b) (dark circles). It does not match exactly the maximum of absorption because the latter is determined by the zero of the reflected order, which differs slightly from the pole [22], but the behaviour of both of them stays the same as a function of

*d*, as observed in Fig. 4(b).

## 6. Conclusion

## References and links

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

2. | A. Hessel and A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. |

3. | D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in |

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

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

6. | A. Wirgin and T. López-Rios, “Can surface-enhanced Raman scattering be caused by waveguide resonance?,” Opt. Commun. |

7. | E. Popov, L. Tsonev, and D. Maystre, “Losses of plasmon surface wave on metallic grating,” J. Mod. Opt. |

8. | T. López-Rios, D. Mendoza, F. J. Garcia-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in lamellar metallic gratings,” Phys. Rev. Lett. |

9. | F. J. Garcia-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Rios, T. Fournier, and B. Pannetier, “Localized surface plasmons in lamellar metallic gratings,” J. Lightwave Technol. |

10. | R. Hooper and J. R. Sambles, “Surface plasmon polaritons on narrow-ridged short-pitch metal gratings in the conical mount,” J. Opt. Soc. Am. |

11. | S. Collin, F. Pardo, R. Teissier, and J. L. Pelouard, “Efficient light absorption in metal-semiconductor-metal nanostructures,” Appl. Phys. Lett. |

12. | E. Popov, D. Maystre, R. C. McPhedran, M. Nevière, M. C. Huthley, and G. H. Derrick, “Total absorption of unpolarized light by crossed gratings,” Opt. Express |

13. | T. V. Teperik, F. J. García De Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surface,” Nat. Photonics |

14. | E. Popov, L. Tsonev, and D. Maystre, “Lamellar diffraction grating anomalies,” Appl. Opt. |

15. | S. E. Sandstrom, G. Tayeb, and R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings - an exact eigenfunction solution,” J. Electromagn. Waves Appl. |

16. | B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. |

17. | M. Nevière and E. Popov, |

18. | D. Maystre, “Integral methods,” in |

19. | E. Popov, M. Nevière, P. Boyer, and N. Bonod, “Light transmission through single apertures,” Opt. Commun. |

20. | E. Popov, N. Bonod, and S. Enoch, “Non-Bloch plasmonic stop-band in real-metal gratings,” Opt. Express |

21. | E. Popov, N. Bonod, and S. Enoch, “Comparison of plasmon surface waves on shallow and deep metallic 1D and 2D gratings,” Opt. Express |

22. | M. Neviere, “The homogeneous problem,” in |

**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: August 12, 2008

Revised Manuscript: September 10, 2008

Manuscript Accepted: September 10, 2008

Published: September 15, 2008

**Citation**

Nicolas Bonod, Gérard Tayeb, Daniel Maystre, Stefan Enoch, and Evgeny Popov, "Total absorption of light by lamellar metallic gratings," Opt. Express **16**, 15431-15438 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15431

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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).
- A. Hessel and A. A. Oliner, "A new theory of Wood's anomalies on optical gratings," Appl. Opt. 4, 1275-1297 (1965). [CrossRef]
- D. Maystre, "General study of grating anomalies from electromagnetic surface modes," in Electromagnetic Surface Modes, A. D. Boardman, ed. (John Wiley, 1982), Chap. 17.
- M. C. Hutley and D. Maystre, "The total absorption of light by a diffraction grating," Opt. Commun. 19, 431-436 (1976). [CrossRef]
- J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Rios, "Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light," Phys. Rev. Lett. 100, 066408 (2008). [CrossRef] [PubMed]
- A. Wirgin, T. López-Rios, "Can surface-enhanced Raman scattering be caused by waveguide resonance?," Opt. Commun. 48, 416-420 (1984). [CrossRef]
- E. Popov, L. Tsonev, and D. Maystre, "Losses of plasmon surface wave on metallic grating," J. Mod. Opt. 37, 379-387 (1990). [CrossRef]
- T. López-Rios, D. Mendoza, F. J. Garcia-Vidal, J. Sánchez-Dehesa, and B. Pannetier, "Surface shape resonances in lamellar metallic gratings," Phys. Rev. Lett. 81, 665-668 (1998). [CrossRef]
- F. J. Garcia-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Rios, T. Fournier, and B. Pannetier, "Localized surface plasmons in lamellar metallic gratings," J. Lightwave Technol. 17, 2191-2195 (1999). [CrossRef]
- R. Hooper, J. R. Sambles, "Surface plasmon polaritons on narrow-ridged short-pitch metal gratings in the conical mount," J. Opt. Soc. Am. 20, 836-843 (2003). [CrossRef]
- S. Collin, F. Pardo, R. Teissier, J. L. Pelouard, "Efficient light absorption in metal-semiconductor-metal nanostructures," Appl. Phys. Lett. 85, 194-196 (2004). [CrossRef]
- E. Popov, D. Maystre, R. C. McPhedran, M. Nevière, M. C. Huthley, G. H. Derrick, "Total absorption of unpolarized light by crossed gratings," Opt. Express 16, 6146-6155 (2008). [CrossRef] [PubMed]
- T. V. Teperik, F. J. García De Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, J. J. Baumberg, "Omnidirectional absorption in nanostructured metal surface," Nat. Photonics 2, 299-301 (2008). [CrossRef]
- E. Popov, L. Tsonev, and D. Maystre, "Lamellar diffraction grating anomalies," Appl. Opt. 33, 5214-5219 (1994). [CrossRef] [PubMed]
- S. E. Sandstrom, G. Tayeb, and R. Petit, "Lossy multistep lamellar gratings in conical diffraction mountings - an exact eigenfunction solution," J. Electromagn. Waves Appl. 7, 631-649 (1993). [CrossRef]
- B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, "Theoretical study of photonic band gaps in woodpile crystals," Phys. Rev. E 67, 66601 (2003).
- M. Nevière and E. Popov, Light Propagation in Periodic Media: Diffraction Theory and Design (Marcel Dekker, New York, 2003).
- D. Maystre, "Integral methods," in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3.
- E. Popov, M. Nevière, P. Boyer, and N. Bonod, "Light transmission through single apertures," Opt. Commun. 255, 338-348 (2005). [CrossRef]
- E. Popov, N. Bonod, and S. Enoch, "Non-Bloch plasmonic stop-band in real-metal gratings," Opt. Express 15, 6241-6250 (2007) [CrossRef] [PubMed]
- E. Popov, N. Bonod, and S. Enoch, "Comparison of plasmon surface waves on shallow and deep metallic 1D and 2D gratings," Opt. Express 15, 4224-4237 (2007). [CrossRef] [PubMed]
- M. Neviere, "The homogeneous problem," in Electromagnetic theory of gratings, R. Petit ed. (Springer-Verlag, 1980), Chap. 5.

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