## Analytical description of subwavelength plasmonic MIM resonators and of their combination |

Optics Express, Vol. 21, Issue 6, pp. 7025-7032 (2013)

http://dx.doi.org/10.1364/OE.21.007025

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### Abstract

We show that a periodic array of metal-insulator-metal resonators can be described as a high refractive index metamaterial. This approach permits to obtain analytically the optical properties of the structure and thus to establish conception rules on the quality factor or on total absorption. Furthermore, we extend this formalism to the combination of two independent resonators.

© 2013 OSA

## 1. Introduction

1. J. Le Perchec, Y. Desieres, and R. E. de Lamaestre, “Plasmon-based photosensors comprising a very thin semiconducting region,” Appl. Phys. Lett. **94**, 181104 (2009). [CrossRef]

9. J. Yang, S. Sauvan, A. Jouanin, S. Collin, J.-L. Pelouard, and P. Lalanne, “Quality factor of metal-insulator-metal nanoresonators,” Opt. Express **20**, 16880–16891 (2012). [CrossRef]

1. J. Le Perchec, Y. Desieres, and R. E. de Lamaestre, “Plasmon-based photosensors comprising a very thin semiconducting region,” Appl. Phys. Lett. **94**, 181104 (2009). [CrossRef]

10. Y. Wang, T. Sun, T. Paudel, Y. Zhang, Z. Ren, and K. Kempa, “Metamaterial-plasmonic absorber structure for high efficiency amorphous silicon solar cells,” Nano Lett. **12**, 440–445 (2012). [CrossRef]

11. A. Cattoni, P. Ghenuche, A. Haghiri-Gosnet, D. Decanini, J. Chen, J. Pelouard, and S. Collin, “*λ*^{3}/1000 plasmonic nanocavities for biosensing fabricated by soft uv nanoimprint lithography,” Nano Lett. **11**, 3557–3563 (2011). [CrossRef] [PubMed]

12. Z. Jiang, S. Yun, F. Toor, D. Werner, and T. Mayer, “Conformal dual band near-perfectly absorbing mid-infrared metamaterial coating,” ACS Nano **5**, 4641–4647 (2011). [CrossRef] [PubMed]

6. C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J. Pelouard, and R. Haidar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. **99**, 241104–241104 (2011). [CrossRef]

13. J. Le Perchec, Y. Desieres, N. Rochat, and R. Espiau de Lamaestre, “Subwavelength optical absorber with an integrated photon sorter,” Appl. Phys. Lett. **100**, 113305–113305 (2012). [CrossRef]

7. P. Bouchon, C. Koechlin, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Wideband omnidirectional infrared absorber with a patchwork of plasmonic nanoantennas,” Opt. Lett. **37**, 1038–1040 (2012). [CrossRef] [PubMed]

8. M. Nielsen, A. Pors, O. Albrektsen, and S. Bozhevolnyi, “Efficient absorption of visible radiation by gap plasmon resonators,” Opt. Express **20**, 13311–13319 (2012). [CrossRef] [PubMed]

12. Z. Jiang, S. Yun, F. Toor, D. Werner, and T. Mayer, “Conformal dual band near-perfectly absorbing mid-infrared metamaterial coating,” ACS Nano **5**, 4641–4647 (2011). [CrossRef] [PubMed]

14. J. Shen, P. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. lett. **94**, 197401 (2005). [CrossRef] [PubMed]

## 2. Metamaterial equivalence

14. J. Shen, P. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. lett. **94**, 197401 (2005). [CrossRef] [PubMed]

*t*̄ and of effective index

*n*̄. In this paper we extend Shen’s approach to subwavelength periodic arrays of real (i.e. lossy) MIM absorbers, which is an important step towards the conception and applicability of such integrated structures.

*n*, width

_{I}*w*and thickness

*h*, sandwiched between a gold substrate and an upper gold ribbon of the same width. The thickness of this latter has almost no influence on the optical properties of the structure as soon as it is larger than the skin depth

_{I}*δ*; its value is set at 50 nm in the following of this paper (

*δ*≃ 25 nm for gold at

*λ*= 10

*μ*m). The ribbons are periodically arranged with a subwavelength period

*d*. The computed spectral response of such a structure for a normally incident transverse magnetic (TM) polarized wave, with parameters

*n*= 4 (corresponding to germanium in the infrared range [15]),

_{I}*w*= 1.087

*μ*m,

*h*= 200 nm,

_{I}*d*= 3.8

*μ*m, is given in Fig. 1(c) in blue. It was computed with a B-spline modal method on a non uniform discretization that allows fast and accurate computations [16

16. P. Bouchon, F. Pardo, R. Haïdar, and J. Pelouard, “Fast modal method for subwavelength gratings based on b-spline formulation,” J. Opt. Soc. Am. A **27**, 696–702 (2010). [CrossRef]

*ε*= 1 − [(

_{Au}*λ*/

_{p}*λ*+

*iγ*)

*λ*/

_{p}*λ*]

^{−1}, with

*λ*= 159 nm and

_{p}*γ*= 0.0048 [6

6. C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J. Pelouard, and R. Haidar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. **99**, 241104–241104 (2011). [CrossRef]

*λ*= 10 μm. It is due to a Fabry-Perot like resonance of the guided mode in the insulator layer. Its resonance wavelength is roughly given by

_{r}*λ*= 2

_{r}*n*+

_{eff}w*λ*, where

_{ϕ}*λ*accounts for the phase shifts and

_{ϕ}*n*is the effective index of the guided mode in the MIM cavity approximated by [17

_{eff}17. S. Collin, F. Pardo, and J. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express **15**, 4310–4320 (2007). [CrossRef] [PubMed]

*i.e.*a Fabry-Perot resonance) is similar to the one which occurs in slits. This indicates that the formalism developed by Shen et al. [14

14. J. Shen, P. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. lett. **94**, 197401 (2005). [CrossRef] [PubMed]

*α*and

*β*represented in dashed line in Fig. 1(a). We propose to limit the problem to an unit cell of width

*d*/2 with a half resonator. Since in MIM structure the in-plane electric field component is zero on

*α*, a perfect conductor can be introduced here without changing the optical response. Thus in a period

*d*, we have two Fabry-Perot resonators, which appear following Shen’s formalism, similar to a grating of grooves having a period

*d*/2 deposited on a perfect conductor, see Fig. 1(b). The electric and magnetic fields plots (data not shown) for this geometry of slits are very similar to those obtained for the initial MIM resonator geometry, which underpins the analogy between these two structures. This slits grating can be described by an equivalent layer (see Fig. 1(c)), and we propose to write its index as: The period of the new unit cell is

*d*/2. Besides, metal losses have to be taken into account in order to describe nearly perfect absorbers. They are included in our model through the complex effective index of the MIM cavity given by Eq. (1). The metal skin depth

*δ*was added to

*h*to take into account the penetration of the electromagnetic wave inside the non-perfect metal. Besides, so that the structure has a resonance at the same wavelength

_{I}*λ*as the MIM cavity, the thickness of the equivalent layer is:

_{r}*n*̄+

*ik*̄ = 33.76+0.508

*i*and a thickness

*t*̄ = 74 nm. It must be highlighted that

*n*̄ is much higher than natural refractive index in the mid-infrared range. The reflexion coefficient amplitude

*r*of the equivalent structure for a TM-polarized, normally incident plane wave is written as: where

*r*

_{12}= (1 −

*n*̄ −

*ik*̄)/(1 +

*n*̄ +

*ik*̄) is the classical Fresnel coefficient. The absorption spectrum

*A*of the effective layer computed thanks to Eq. (4) (i.e.

*A*= 1 − |

*r*|

^{2}) is plotted in red dashed line in Fig. 1(d) and compared to the numerical calculations made for the grating of MIM ribbons. The perfect agreement between the two curves validates our extension of Shen’s formalism for horizontal MIM resonators with lossy metals.

*n*is positive), which are interesting for optoelectronic devices [1

_{I}1. J. Le Perchec, Y. Desieres, and R. E. de Lamaestre, “Plasmon-based photosensors comprising a very thin semiconducting region,” Appl. Phys. Lett. **94**, 181104 (2009). [CrossRef]

*n*= 4 + 0.2

_{I}*i*and (d) MIM slits.

## 3. Total absorption condition

*n*,

_{I}*h*,

_{I}*d*,

*w*) on the spectral response of the MIM resonators and in particular, the conditions necessary to perform nearly total absorption,

*r*≃ 0. By neglecting the low imaginary part of

*r*

_{12}in Eq. (4), there is nearly total absorption when

*λ*= 4

_{r}*n*̄

*t*̄, and

*r*

_{12}= −

*e*

^{−πk̄/n̄}. The first condition sets the Fabry-Perot resonance wavelength, while the second condition corresponds to the impedance matching between the MIM resonators and the free space. Using the approximate expression of the Fresnel coefficient

*r*

_{12}and Eq. (2), this gives a condition on the metamaterial index: As a consequence, for a given choice of wavelength, metal and cavity parameters (i.e.

*h*and

_{I}*n*), there exists only one period that permits to obtain a nearly total absorption. To illustrate this conception law, Fig. 3 represents the period values computed both analytically (plain lines) and numerically (stars,

_{I}*A*> 99.9%) that are needed to obtain a nearly total absorption at 10

*μ*m, as a function of the dielectric thickness

*h*, for various

_{I}*n*. The agreement is rather fair except when the period is greater than 8 μm. This is explained by the fact that its value goes near the resonance wavelength

_{I}*λ*. Indeed surface plasmons excited by the grating are known to couple strongly with cavity resonance [18

_{r}18. P. Jouy, A. Vasanelli, Y. Todorov, A. Delteil, G. Biasiol, L. Sorba, and C. Sirtori, “Transition from strong to ultrastrong coupling regime in mid-infrared metal-dielectric-metal cavities,” Appl. Phys. Lett. **98**, 231114–231114 (2011). [CrossRef]

## 4. Quality factor of MIM resonators

*Q*=

*λ*/

_{r}*FWHM*(where

*FWHM*is the full width at half maximum) is an essential characteristic of a resonator, which determines how losses are managed in the system. Thanks to the previous analysis, we show how this quality factor can be engineered in MIM ribbons. The quality factors of MIM structures exhibiting a nearly total absorption at 10

*μ*m for insulator index of

*n*= 4 (red) and

_{I}*n*= 2 (blue) calculated numerically (stars) and analytically (circles) are represented as a function of

_{I}*h*on Fig. 4(a). It must be emphasized that the quality factor appears to depend linearly on the insulator thickness and to be independent of

_{I}*n*. As an illustration, Fig. 4(b) shows the spectra of three of such resonators for various values of

_{I}*h*, and

_{I}*n*. This is consistent with the expansion at first order of Eq. (4), which gives the absorption spectra. Indeed, from this formula, the quality factor can be fairly approximated by

_{I}*Q*≃

*n*/4

_{eff}*k*. By an expansion at first order of Eq. (1), it comes that

_{eff}*h*, the independence on

_{I}*n*and

_{I}*d*and the range of the obtained values. To conclude, the quality factor of these MIM resonators, limited by metal losses [19

19. J. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett. **99**, 211106–211106 (2011). [CrossRef]

*h*in the range from 10 to 25.

_{I}## 5. Combination of resonators

*d*was proposed [6

6. C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J. Pelouard, and R. Haidar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. **99**, 241104–241104 (2011). [CrossRef]

7. P. Bouchon, C. Koechlin, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Wideband omnidirectional infrared absorber with a patchwork of plasmonic nanoantennas,” Opt. Lett. **37**, 1038–1040 (2012). [CrossRef] [PubMed]

*w*

_{1}and

*w*

_{2}within the same subwavelength period

*d*, and separated by a distance

*l*= (

*d*−

*w*

_{1}−

*w*

_{2})/2. To avoid a slight coupling between the two MIM, the distance

*l*between them must be greater than 300 nm. It is not possible to describe such biMIM structure by a single equivalent layer as previously done.

*A*of such biMIM structures exhibit two peaks at the resonance wavelengths of the single MIM resonators taken alone in the same period

*d*(we call their individual spectra

*A*

_{1}and

*A*

_{2}). Indeed thanks to the localized nature of Fabry-Perot resonance, the two resonators behave independently in the biMIM structure. First, we propose to treat them separately thanks to our model (i.e. to compute analytically

*A*

_{1}and

*A*

_{2}). Then, to express the probability (1 −

*A*) for a photon to escape from absorption by the biMIM as the product of the independent probabilities (1 −

*A*

_{1}) and (1 −

*A*

_{2}) to be non absorbed by the single MIM resonators taken separately. This leads to the following law: The analytically computed spectra of three biMIM structures in which the relative differences of the peak wavelengths Δ

*λ*/

_{r}*λ*are respectively 0.2, 0.1 and 0.05, are represented in Figs. 5(b–d) for insulator thickness, index, and period of respectively

_{r}*h*= 250 nm,

_{I}*n*= 4, and

_{I}*d*= 5.8

*μ*m. We add, on the same figures, the spectra of the biMIM calculated numerically. When the relative difference in resonance wavelength is large (see Figs. 5(b–c), where Δ

*λ*/

_{r}*λ*is respectively 0.2, and 0.1), our fully analytical model fits perfectly the biMIM behavior and confirms the validity of Eq. (6). Slight discrepancies appear when Δ

_{r}*λ*/

_{r}*λ*decreases (typically Δ

_{r}*λ*/

_{r}*λ*≤ 0.05, see near the resonance peaks on Fig. 5(d)), which evidences a weak coupling between the two resonators. The analytical model for one MIM resonator used together with Eq. (6), which is based on their independence when they are inserted in the same period, allow the fast conception of dual band absorbers.

_{r}## 6. Conclusion

## Acknowledgments

## References and links

1. | J. Le Perchec, Y. Desieres, and R. E. de Lamaestre, “Plasmon-based photosensors comprising a very thin semiconducting region,” Appl. Phys. Lett. |

2. | J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett. |

3. | J. Hao, L. Zhou, and M. Qiu, “Nearly total absorption of light and heat generation by plasmonic metamaterials,” Phys. Rev. B |

4. | P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. |

5. | F. Pardo, P. Bouchon, R. Haïdar, and J. Pelouard, “Light funneling mechanism explained by magnetoelectric interference,” Phys. Rev. Lett. |

6. | C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J. Pelouard, and R. Haidar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett. |

7. | P. Bouchon, C. Koechlin, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Wideband omnidirectional infrared absorber with a patchwork of plasmonic nanoantennas,” Opt. Lett. |

8. | M. Nielsen, A. Pors, O. Albrektsen, and S. Bozhevolnyi, “Efficient absorption of visible radiation by gap plasmon resonators,” Opt. Express |

9. | J. Yang, S. Sauvan, A. Jouanin, S. Collin, J.-L. Pelouard, and P. Lalanne, “Quality factor of metal-insulator-metal nanoresonators,” Opt. Express |

10. | Y. Wang, T. Sun, T. Paudel, Y. Zhang, Z. Ren, and K. Kempa, “Metamaterial-plasmonic absorber structure for high efficiency amorphous silicon solar cells,” Nano Lett. |

11. | A. Cattoni, P. Ghenuche, A. Haghiri-Gosnet, D. Decanini, J. Chen, J. Pelouard, and S. Collin, “ |

12. | Z. Jiang, S. Yun, F. Toor, D. Werner, and T. Mayer, “Conformal dual band near-perfectly absorbing mid-infrared metamaterial coating,” ACS Nano |

13. | J. Le Perchec, Y. Desieres, N. Rochat, and R. Espiau de Lamaestre, “Subwavelength optical absorber with an integrated photon sorter,” Appl. Phys. Lett. |

14. | J. Shen, P. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. lett. |

15. | E. Palik and G. Ghosh, |

16. | P. Bouchon, F. Pardo, R. Haïdar, and J. Pelouard, “Fast modal method for subwavelength gratings based on b-spline formulation,” J. Opt. Soc. Am. A |

17. | S. Collin, F. Pardo, and J. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express |

18. | P. Jouy, A. Vasanelli, Y. Todorov, A. Delteil, G. Biasiol, L. Sorba, and C. Sirtori, “Transition from strong to ultrastrong coupling regime in mid-infrared metal-dielectric-metal cavities,” Appl. Phys. Lett. |

19. | J. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett. |

**OCIS Codes**

(130.3060) Integrated optics : Infrared

(230.1950) Optical devices : Diffraction gratings

(160.3918) Materials : Metamaterials

(050.6624) Diffraction and gratings : Subwavelength structures

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Metamaterials

**History**

Original Manuscript: October 16, 2012

Revised Manuscript: December 22, 2012

Manuscript Accepted: December 29, 2012

Published: March 13, 2013

**Citation**

Charlie Koechlin, Patrick Bouchon, Fabrice Pardo, Jean-Luc Pelouard, and Riad Haïdar, "Analytical description of subwavelength plasmonic MIM resonators and of their
combination," Opt. Express **21**, 7025-7032 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-6-7025

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### References

- J. Le Perchec, Y. Desieres, and R. E. de Lamaestre, “Plasmon-based photosensors comprising a very thin semiconducting region,” Appl. Phys. Lett.94, 181104 (2009). [CrossRef]
- J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, “High performance optical absorber based on a plasmonic metamaterial,” Appl. Phys. Lett.96, 251104 (2010). [CrossRef]
- J. Hao, L. Zhou, and M. Qiu, “Nearly total absorption of light and heat generation by plasmonic metamaterials,” Phys. Rev. B83, 165107 (2011). [CrossRef]
- P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett.98, 191109 (2011). [CrossRef]
- F. Pardo, P. Bouchon, R. Haïdar, and J. Pelouard, “Light funneling mechanism explained by magnetoelectric interference,” Phys. Rev. Lett.107, 93902 (2011). [CrossRef]
- C. Koechlin, P. Bouchon, F. Pardo, J. Jaeck, X. Lafosse, J. Pelouard, and R. Haidar, “Total routing and absorption of photons in dual color plasmonic antennas,” Appl. Phys. Lett.99, 241104–241104 (2011). [CrossRef]
- P. Bouchon, C. Koechlin, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Wideband omnidirectional infrared absorber with a patchwork of plasmonic nanoantennas,” Opt. Lett.37, 1038–1040 (2012). [CrossRef] [PubMed]
- M. Nielsen, A. Pors, O. Albrektsen, and S. Bozhevolnyi, “Efficient absorption of visible radiation by gap plasmon resonators,” Opt. Express20, 13311–13319 (2012). [CrossRef] [PubMed]
- J. Yang, S. Sauvan, A. Jouanin, S. Collin, J.-L. Pelouard, and P. Lalanne, “Quality factor of metal-insulator-metal nanoresonators,” Opt. Express20, 16880–16891 (2012). [CrossRef]
- Y. Wang, T. Sun, T. Paudel, Y. Zhang, Z. Ren, and K. Kempa, “Metamaterial-plasmonic absorber structure for high efficiency amorphous silicon solar cells,” Nano Lett.12, 440–445 (2012). [CrossRef]
- A. Cattoni, P. Ghenuche, A. Haghiri-Gosnet, D. Decanini, J. Chen, J. Pelouard, and S. Collin, “λ3/1000 plasmonic nanocavities for biosensing fabricated by soft uv nanoimprint lithography,” Nano Lett.11, 3557–3563 (2011). [CrossRef] [PubMed]
- Z. Jiang, S. Yun, F. Toor, D. Werner, and T. Mayer, “Conformal dual band near-perfectly absorbing mid-infrared metamaterial coating,” ACS Nano5, 4641–4647 (2011). [CrossRef] [PubMed]
- J. Le Perchec, Y. Desieres, N. Rochat, and R. Espiau de Lamaestre, “Subwavelength optical absorber with an integrated photon sorter,” Appl. Phys. Lett.100, 113305–113305 (2012). [CrossRef]
- J. Shen, P. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. lett.94, 197401 (2005). [CrossRef] [PubMed]
- E. Palik and G. Ghosh, Handbook of optical constants of solids (Academic press, 1985).
- P. Bouchon, F. Pardo, R. Haïdar, and J. Pelouard, “Fast modal method for subwavelength gratings based on b-spline formulation,” J. Opt. Soc. Am. A27, 696–702 (2010). [CrossRef]
- S. Collin, F. Pardo, and J. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express15, 4310–4320 (2007). [CrossRef] [PubMed]
- P. Jouy, A. Vasanelli, Y. Todorov, A. Delteil, G. Biasiol, L. Sorba, and C. Sirtori, “Transition from strong to ultrastrong coupling regime in mid-infrared metal-dielectric-metal cavities,” Appl. Phys. Lett.98, 231114–231114 (2011). [CrossRef]
- J. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett.99, 211106–211106 (2011). [CrossRef]

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