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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 15 — Jul. 16, 2012
  • pp: 16690–16703
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Complex Bloch-modes calculation of plasmonic crystal slabs by means of finite elements method

Giuseppe Parisi, Pierfrancesco Zilio, and Filippo Romanato  »View Author Affiliations


Optics Express, Vol. 20, Issue 15, pp. 16690-16703 (2012)
http://dx.doi.org/10.1364/OE.20.016690


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Abstract

We present a Finite Element Method (FEM) to calculate the complex valued k(ω) dispersion curves of a photonic crystal slab in presence of both dispersive and lossy materials. In particular the method can be exploited to study plasmonic crystal slabs. We adopt Perfectly Matched Layers (PMLs) in order to truncate the open boundaries of the model, including their related anisotropic permittivity and permeability tensors in the weak form of Helmholtz's eigenvalue equation. Results of the model are presented in the interesting case of a holey metal film enabling to study the observed extraordinary optical transmission properties in term of the plasmonic Bloch modes of the structure.

© 2012 OSA

1. Introduction

Periodic metallic nanostructures have received great attention in recent years due to their surprising optical properties. The interaction of electromagnetic fields with the free electron gas of metals gives rise to particular electromagnetic field waves which are known as Surface Plasmon Polaritons (SPPs) [1

1. H. Rather, Surface Plasmons (Springer-Verlag, 1988).

]. Due to their properties of extreme light confinement and sensitivity to the dielectric environment, SPPs have found lots of interesting application in the fields of sensing [2

2. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008). [CrossRef] [PubMed]

], thin film photovoltaics [3

3. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). [CrossRef] [PubMed]

], near field imaging [4

4. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005). [CrossRef]

], subwavelength waveguides [5

5. M. L. Brongersma and P. G. Kik, Surface Plasmon Nanophotonics, (Springer, 2007).

] etc.

In particular, periodically nanostructured metal-dielectric interfaces have been subject of many researches [1

1. H. Rather, Surface Plasmons (Springer-Verlag, 1988).

,4

4. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005). [CrossRef]

] due to their unexpected optical properties such as Extraordinary Optical Transmission [6

6. F. J. Garcia de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007). [CrossRef]

10

10. J. Bravo-Abad, L. Martín-Moreno, F. J. García-Vidal, E. Hendry, and J. Gómez Rivas, “Transmission of light through periodic arrays of square holes: From a metallic wire mesh to an array of tiny holes,” Phys. Rev. B 76(24), 241102 (2007). [CrossRef]

] and negative refraction [11

11. A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garcia-Vidal, “Plasmonic metamaterials based on holey metallic films,” J. Phys. Condens. Matter 20(30), 304215 (2008).

]. Surface Plasmons propagation across such structures exhibits analogous properties of light propagation in dielectric photonic crystals [4

4. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005). [CrossRef]

]. A complicated system of band gaps has indeed been calculated and measured using different metallic gratings [4

4. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005). [CrossRef]

]. Shallow surface features can be assigned an effective refractive index describing their interaction with SPPs, thus enabling a complete analogy with 2D photonic crystals. The description of plasmonic Bloch waves propagating in a plasmonic crystal slab, however, is a much more challenging task with respect to the description of photonic Bloch waves in fully periodic dielectric systems.

The analysis of Photonic Crystal Slabs (PCS), however, requires proper handling not only of truly bound modes but also of leaky modes. This term addresses eigenmodes whose crystal momentum lies inside the light cone. In these cases the modes can couple to waves propagating in the semi-infinite half spaces surrounding the slab, resulting in radiative losses [12

12. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and D. Robert, Meade, Photonic Crystals - Molding the Flow of Light, 2nd ed. (Princeton University Press, 2007).

,13

13. K. Sakoda, Optical Properties of Photonic Crystal, ser. Optical Sciences. (Springer, New York, 2001).

]. Simulation of open space boundaries is a tricky task both for methods based on discretization of real space (like Finite Elements and Finite Differences) as well as for methods which discretize the wave vectors space (like Plane Wave Expansion method). A common approach introduces a fictitious periodicity in the direction normal to the slab (Super-cell approach [13

13. K. Sakoda, Optical Properties of Photonic Crystal, ser. Optical Sciences. (Springer, New York, 2001).

]). The period is chosen sufficiently large in order to decouple the bound modes of the slabs. However, the artificial periodicity produces many spurious modes and moreover it perturbs the physical leaky eigenmodes profile. Plane Wave Expansion Method has been extended by Shi et al. [15

15. S. Shi, C. Chen, and D. Prather, “Revised plane wave method for dispersive material and its application to band structure calculation of photonic crystal slabs,” Appl. Phys. Lett. 86(4), 043104 (2005). [CrossRef]

] including Perfectly Matched Layers (PMLs) which correctly absorb leakage radiation. These layers are characterized by slowly space varying relative permittivity and permeability, properly designed in order to minimize reflection of plane waves impinging on them at arbitrary incidence. The method can deal with photonic crystal slabs with arbitrarily dispersive materials but only in case of absence of losses, excluding therefore the important class of plasmonic crystal slabs. Other methods based on FDTD can handle the lossy dispersive problem but are computationally very expensive [15

15. S. Shi, C. Chen, and D. Prather, “Revised plane wave method for dispersive material and its application to band structure calculation of photonic crystal slabs,” Appl. Phys. Lett. 86(4), 043104 (2005). [CrossRef]

]. To our knowledge, a complete and efficient numerical treatment of metallic planar crystals still lacks.

2. The FEM simulation

The aim of the present section is to briefly revisit the FEM weak formulation [17

17. M. Davanco, Y. Urzhumov, and G. Shvets, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express 15(15), 9681–9691 (2007). [CrossRef] [PubMed]

,18

18. C. Fietz, Y. Urzhumov, and G. Shvets, “Complex k band diagrams of 3D metamaterial/photonic crystals,” Opt. Express 19(20), 19027–19041 (2011). [CrossRef] [PubMed]

] of the following Helmholtz’s eigenvalue equation
×(p^×V)ω2q^V=0,
(1)
whereV=E,p^=1/μ^(r,ω),q^=ε^(r,ω)for the electric field formulation and V=H,p^=1/ε^(r,ω),q^=μ^(r,ω) for the magnetic field formulation. Hereε^=ε^(r,ω)andμ^=μ^(r,ω)are respectively the frequency-dependent permittivity and permeability tensors of the plasmonic crystal. They are supposed to be periodic in x, y and z direction. The problem reduces to find Bloch-waves solutions of the form [12

12. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and D. Robert, Meade, Photonic Crystals - Molding the Flow of Light, 2nd ed. (Princeton University Press, 2007).

,13

13. K. Sakoda, Optical Properties of Photonic Crystal, ser. Optical Sciences. (Springer, New York, 2001).

]
V(r;k)=eikru(r)=ei(kxx+kyy+kzz)u(r),
(2)
where k is the Bloch-vector. The Finite Elements method relies on the so-called weak formulation of Eq. (1), which consists in annulling the following residues
Rk(v,u)=Ωv[×(p^×V(r;k))ω2q^V(r;k)]d3r,
(3)
where v is a weight function and the integration is over the unitary cell volume [20

20. J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (John Wiley & Sons, Inc., 2002).

]. After inserting expression (2) in Eq. (3) and some straightforward algebra, Eq. (3) yields to

Ωd3r(v[k×p^(k×u)]iv[k×p^(×u)]i(×v)p^(k×u))++Ωd3r((×v)p^(×u)ω2c2vq^u)+ΩdAv[n^×p^(ik×u+×u)]=0,
(4)

In order to turn Eq. (4) into an eigenvalue problem, the three degrees of freedom that comprise the Bloch wave-vector k must be reduced to one; i.e by fixing a particular k-vector direction,λ^. In this case Eq. (4) turns out to be a quadratic eigenvalue equation in the k-vector projection along the chosen direction, i.e. λ=kλ^. Following the usual FEM discretization procedure [20

20. J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (John Wiley & Sons, Inc., 2002).

], Eq. (4) is then turned into a matrix equation [17

17. M. Davanco, Y. Urzhumov, and G. Shvets, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express 15(15), 9681–9691 (2007). [CrossRef] [PubMed]

,18

18. C. Fietz, Y. Urzhumov, and G. Shvets, “Complex k band diagrams of 3D metamaterial/photonic crystals,” Opt. Express 19(20), 19027–19041 (2011). [CrossRef] [PubMed]

,20

20. J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (John Wiley & Sons, Inc., 2002).

] and opportunely linearized.

The presented weak formulation of the eigenvalue problem is inherently handled by several Finite Elements software packages. All the models and examples in this paper have been performed and calculated with COMSOL Multiphysics, which allows the user to specify a custom field equation to be solved. In particular the software automatically turns the input Eq. (4) into an algebraic linear equation system. For more details about FEM we remand to Ref [20

20. J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (John Wiley & Sons, Inc., 2002).

,21

21. F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev. 43(2), 235–286 (2001). [CrossRef]

].

3. The perfect matched layer (PML) implementation

We notice that PMLs are not the only option in order to properly truncate open boundaries. Absorbing Boundary Conditions (ABC) [20

20. J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (John Wiley & Sons, Inc., 2002).

] can also be used to absorb plane wave leaky radiation. A detail comparison between effectiveness of PMLs and ABCs in the present method is out of the scope of the present work. However, we notice that if the leakage radiation is not in the form of a single plane wave, ABCs are expected to be not suitable.

4. 2D sinusoidal grating and PML test

We first start with a 2D example (Fig. 2(a)
Fig. 2 (a) Scheme of unit cell for the sinusoidal plasmonic grating; (b) Variation of a sample eigenvalue (λL = d = 0.0905356 at ω = 1.7∙1015Hz) as a function of PML thickness L and σ, ∆λ is defined as ∆λm = λL = (m + 1)d - λL = md (m>0), scale is in decibels.
) which serves as a guideline to set the PMLs parameters, optimizing them in order to minimize reflections. The unit cell consists of a sinusoidal metal-dielectric interface in the x-z plane, infinitely extended in y-direction (out of plane). In the z-direction the unitary cell is limited above by a PML layer and below by the bulk metal. The period (d) in the x-direction is set to 600nm while the peak-valley amplitude of the sinusoid (a) is increased from 0 to 150nm. We adopt silver as metal, its permittivity being taken from Palik [24

24. D. Edward, Palik, Handbook of optical constants of solids, (Academic Press, 1985).

]. Periodic boundary conditions are set in both x and z directions. In z direction this condition is not mandatory because of the presence of the PML. This choice is made in order to reduce to zero the last term in Eq. (4). Since the PMLs are expected to absorb all radiation impinging on them, the particular boundary condition set at the end of the PML layer actually does not influence the resulting field distribution.

In the case of mono-periodic gratings, as in the considered example, one is typically interested in studying surface plasmon modes whose k-vector is parallel to the grating vector, G=(2π/d)x^. In this case the weak form of Helmholtz’s equation, Eq. (4), reduces to the following simplified form
Ωd2r[λ2uvε33+λiε33(uxvuvx)+uzvz1ε11+uxvx1ε33(ωc)2uvμ22]=0.
(8)
where λ = kx, εii and μii are the diagonal components of the relative permittivity and permeability tensors of the medium involved (both physical domains and PMLs).

The need of terminating the computational domain with proper boundary condition, in any case, results in the generation of undesired spurious modes, even in presence of PMLs. Most of these nonphysical modes look like guided modes within the PML domains or between slab and absorbing boundary. A common way to discriminate physical leaky or bound modes from PML modes is to employ an average field intensity based filter [15

15. S. Shi, C. Chen, and D. Prather, “Revised plane wave method for dispersive material and its application to band structure calculation of photonic crystal slabs,” Appl. Phys. Lett. 86(4), 043104 (2005). [CrossRef]

]. The physical modes, even leaky modes, are expected to be those ones mostly confined in proximity of the plasmonic slab and rapidly decaying within the PML domain. We calculated the following quantity for all eigenmodes
κ=|H|2(1)|H|2(2)
(9)
where <|H|2>(1), <|H|2>(2) are the averages of the squared H field norm in a region far from the grating in which all evanescent fields are vanished and within a thin air layer (tens of nm) close to the metal surface respectively, i.e. regions marked with (1) and (2) in Fig. 2(a) respectively. We found out that κ values ranging from 0 to 20% are typically obtained in case of physical bound or leaky eigenmodes, whereas higher values are found for nonphysical PML–related modes. We used therefore the quantity κ to filter out non-physical modes.

With regard to the n parameter, it was shown for example in [23

23. S. D. Gedney, “An Anisotropic Perfectly Matched Layer-Absorbing Medium for the truncation of FDTD Lattices,” IEEE Trans. Antenn. Propag. 44(12), 1630–1639 (1996). [CrossRef]

,27

27. M. Imran and G. Andrew, Kirk, “Accurate determination of quality factor of axisymmetric resonators by use of perfectly mathed layer,” arXiv:1101.1984v2 [physics.optics], 1–7(2011).

], that a simple quadratic or cubic turn-on of the PML absorption usually produces negligible reflections for a PML only few wavelengths thick.

For frequencies above the upper gap edge (ω > 1.7∙1015Hz) the imaginary part of the Bloch-wave-vector has values much greater than its corresponding values at frequencies below the gap. This is because the folded plasmonic band enters the light cone and the radiative coupling to propagating waves in the upper air half space becomes possible.

5. 2-D periodic array

Holey thin metal films exhibit very interesting plasmonic properties of extraordinary optical transmission which have been extensively studied in a vast literature [6

6. F. J. Garcia de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007). [CrossRef]

11

11. A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garcia-Vidal, “Plasmonic metamaterials based on holey metallic films,” J. Phys. Condens. Matter 20(30), 304215 (2008).

]. However, what is usually omitted in literature is the detailed visualization of both the real and imaginary dispersions of the SPP Bloch modes of the considered structures. This is due the strong leakage radiation damping that affects the modes for large holes sizes, which is hardly handled by standard numerical techniques.

In case of large holes, we note that the (1,0)- mode strongly deviates from the flat SPP dispersion curves, whereas the (1,0)+ is almost unperturbed. It is also evident how the (1,0)- one is correlated to a much higher transmittance peak than the (1,0)+ (Fig. 6(c)).

In Fig. 7(a)
Fig. 7 (a) Hy and (b) |Ex| fields of the TM modes found at frequency of 1.6∙1015Hz for a = 250nm (I,II) and a = 500nm (III,IV). The calculated eigenvalues in the four cases are respectively kx/(π/d) = 0.3828 + 0.0019i, 0.3878 + 0.0011i, 0.2792 + 0.1251i, 0.3829 + 0.0025i. Modes are classified as antisymmetric (I,III) and symmetric (II,IV) according to the symmetry of Hy field with respect to the z = 0 plane. PML size is fixed at 2d for convenience.
and 7(b) we report respectively the fields Hy and |Ex| in the x-z plane, for the two TM modes observed at frequency ω = 1.6∙1015Hz with a = 250nm (I-II) and a = 500nm (III-IV). As can be expected, for small hole sizes, the modes are well confined close to the metal slab. The mode confinement decreases in case of a = 500nm and a stronger leakage radiation is observed in the cladding regions. The presence of the PMLs, however, makes it possible to properly absorb this radiation and allows reconstructing the correct mode field profiles. This is clearly seen in Fig. 7(b). In fact, the almost uniform |Ex| in the region far from the slab indicates that no waves reflected from the PMLs are present. Comparing Fig. 7(b) I and III with 7(b) II and IV, we observe that the electric field intensity is mainly concentrated within the hole in the anti-symmetric modes, while the field has a node at the plane z = 0 in the symmetric case.

As just pointed out in [7

7. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010). [CrossRef]

11

11. A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garcia-Vidal, “Plasmonic metamaterials based on holey metallic films,” J. Phys. Condens. Matter 20(30), 304215 (2008).

], peaks in optical transmission through a holey metal film can be described in terms of symmetric and antisymmetric coupling of plasmonic modes between the two horizontal metal-dielectric interfaces of the slab. In case of symmetric coupling, SPPs at the two horizontal metal-dielectric interfaces are weakly coupled via evanescent fields inside the hole. The mechanism depends mainly on the periodicity of the structure and is the main EOT channel for arrays of small holes. In the antisymmetric coupling, instead, single-interface SPPs are strongly coupled via a Fabry-Pérot resonance inside the hole. This resonance depends more on single hole characteristics (size and depth) rather than on the periodicity of the structure, and turns out to be the dominant EOT mechanism for arrays of large holes [10

10. J. Bravo-Abad, L. Martín-Moreno, F. J. García-Vidal, E. Hendry, and J. Gómez Rivas, “Transmission of light through periodic arrays of square holes: From a metallic wire mesh to an array of tiny holes,” Phys. Rev. B 76(24), 241102 (2007). [CrossRef]

]. We refer to [7

7. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010). [CrossRef]

11

11. A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garcia-Vidal, “Plasmonic metamaterials based on holey metallic films,” J. Phys. Condens. Matter 20(30), 304215 (2008).

] for a detailed description of these EOT mechanisms.

Black dots in Fig. 8(a) mark the TE and TM (0,1) ± modes. Their imaginary parts present strong increases in correspondence of the edge of the first Brillouin zone around the frequency of 2.2∙1015 Hz (Fig. 8(b) inset). This behavior indicates the presence of gaps similar to the ones observed for the TM modes. In the insets of Fig. 8(a) we report the dominant magnetic field component profiles of the (|m|,|n|)- modes for a = 250 nm in a x-y plane laying 10 nm above the slab at frequency of 2.05∙1015 Hz. We see that the TM (1,0)- mode is a transversal mode with the magnetic field polarized in the y-direction, while both the TE and TM (0,1)- modes are longitudinal modes with the magnetic field mostly polarized in the x-direction of propagation.

Looking at the real dispersions of the modes it can be noticed a band bending at frequencies around ω = 2∙1015 Hz near kx = 0. Correspondingly, a divergence in the imaginary parts is observed. Similar deviations are typical of real-frequency (and complex propagation constant) eigenvalue methods. They are not physical and were already reported elsewhere [18

18. C. Fietz, Y. Urzhumov, and G. Shvets, “Complex k band diagrams of 3D metamaterial/photonic crystals,” Opt. Express 19(20), 19027–19041 (2011). [CrossRef] [PubMed]

,31

31. K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, “Nature of lossy Bloch states in polaritonic photonic crystals,” Phys. Rev. B 69(19), 195111 (2004). [CrossRef]

,33

33. Y. Ding and R. Magnusson, “Band gaps and leaky-wave effects in resonant photonic-crystal waveguides,” Opt. Express 15(2), 680–694 (2007). [CrossRef] [PubMed]

].

6. Conclusions

Acknowledgments

This work has been supported by SPLENDID project (Surface Plasmonics for Enhanced Nano-Detectors and Innovative Devices, CARIPARO foundation) and PLATFORMS project of Padova University.

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H. Rather, Surface Plasmons (Springer-Verlag, 1988).

2.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008). [CrossRef] [PubMed]

3.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). [CrossRef] [PubMed]

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F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010). [CrossRef]

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11.

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Y. Ding and R. Magnusson, “Band gaps and leaky-wave effects in resonant photonic-crystal waveguides,” Opt. Express 15(2), 680–694 (2007). [CrossRef] [PubMed]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(240.6680) Optics at surfaces : Surface plasmons
(260.2030) Physical optics : Dispersion
(050.5298) Diffraction and gratings : Photonic crystals
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Photonic Crystals

History
Original Manuscript: May 9, 2012
Revised Manuscript: May 29, 2012
Manuscript Accepted: June 13, 2012
Published: July 9, 2012

Citation
Giuseppe Parisi, Pierfrancesco Zilio, and Filippo Romanato, "Complex Bloch-modes calculation of plasmonic crystal slabs by means of finite elements method," Opt. Express 20, 16690-16703 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16690


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